General Equilibrium and Welfare Economics
James C. Moore
General Equilibrium and Welfare Economics An Introduction
With 40 Figures and 11 Tables
12
Professor James C. Moore Purdue University Department of Economics 100 S. Grant Street West Lafayette, IN 479072076 USA
[email protected] ISBN10 3540314075 Springer Berlin Heidelberg New York ISBN13 9783540314073 Springer Berlin Heidelberg New York CataloginginPublication Data Library of Congress Control Number: 2006932262 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com ° Springer Berlin ´ Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. HardcoverDesign: WMXDesign GmbH, Heidelberg SPIN 11660033
42/31535 4 3 2 1 0 ± Printed on acidfree paper
To Donna, Donovan, Brian, Jerry, Linda, Ted, Julie, and Bradley, . . . and to the University of Minnesota Faculty who taught me General Equilibrium and Welfare Economics: John Chipman, Leo Hurwicz, Ket Richter, and Hugo Sonnenschein, this book is most aﬀectionately, and respectfully, dedicated.
Preface This book is intended as a graduate (or perhaps, advanced undergraduate) level textbook in general equilibrium and welfare economics. General equilibrium theory is, of course, at the very heart of our ﬂedgling science of economics, and welfare economics provides the normative basis for all professional policy recommendations, as well as most applied work. In developing this text, I hope that I have not slighted the needs of the aspiring economic theorist, but at the same time, I have tried to take account of the fact that most of the students who have studied or will study this text will not go on to specialize in advanced theory. Consequently, I have attempted to include and concentrate upon that material which I believe would be most useful to students who will go on to specialize in, for example, international economics, public economics, or economic development. How well I have succeeded in this endeavor only time will tell. This book has been developed from lecture notes and handouts which I have used over the past several years in the course, ‘General Equilibrium and Welfare Economics,’ (Economics 609) which I have taught at Purdue University. Before going further, however, let me quickly confess that I have never covered all of the material in this book in one semester. On the other hand, I have taught all of it at one time or another, so the whole book has been classroomtested to some extent. The course for which the book was written is the second semester of microtheory required of students in the ﬁrst year of our PhD program. Consequently, I have written the book assuming that the reader is familiar with, say, the partial equilibrium portion of MasColell, Whinston, and Green [1985], which is used as the text in the ﬁrst semester of our microtheory sequence. I also assume that the reader has the usual mathematical background required of a ﬁrstyear graduate student in economics: competence in calculus, and some background in Linear Algebra, as well as familiarity with the elementary concepts of set theory: membership, union, intersection, and settheoretic diﬀerence. I do not often use game theory in any very essential way in this work, but the reader should be familiar with the deﬁnitions of Nash equilibrium and the core. I have included a glossary of the basic mathematical notation which is used in this book at the end of this preface. I have included a number of exercises at the end of each chapter, and I would strongly recommend that a student who is encountering this material for the ﬁrst time work through as many of these problems as her or his schedule permits. In Chapter 19 I have also included solutions for a number of these problems, but I hope that it goes without saying that a student should make every eﬀort to work through a problem on her or his own before consulting Chapter 19 for its solution!
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A number of people have contributed to this project in various ways, and I very much want to express my gratitude for their help. In particular, Dan Kovenock, John Ledyard and Bill Novshek have read various parts of the manuscript, and have made a number of helpful comments thereon. Several research assistants have done yeoman work in trying to rid this manuscript of all the ‘typo’s’ and other errors which I always manage to accumulate. I particularly want to thank Dan Nguyen, Jennifer Pate Oﬀenberg, Daniela Puzzello, and Brian Roberson, who have gone ‘above and beyond’ the usual requirements of a research assistant in helping to clean up this manuscript. Thanks are also due Paola Boel and Curtis Price for their help in this regard, as well as to my secretary, Karen Angstadt, who has handled the various organizational chores which I have inﬂicted upon her with her usual eﬃciency and dispatch. In addition, of course, several ‘generations’ of graduate students in our economics program have endured assignments in, and lectures oriented toward this material with no (or little) complaint. I would also like to thank my colleagues in the economics group of the Krannert School here at Purdue, who have been remarkably tolerant of the death grip I have maintained on Economics 609 over the past several years. I would also like to thank Deans Rick Cosier and Bob Plante, who maintained an atmosphere which encourages scholarly work in a variety of dimensions and directions. Finally, of course, I must thank my wife, Donna, without whose tolerance and encouragement this book could not possibly have been written.
Mathematical Notation I will use ‘Rn ’ to denote ndimensional Euclidean space, and I will use bold letters to denote elements therein (vectors). Thus, if x ∈ Rn , x is of the form: x = (x1 , . . . , xj , . . . , xn ), j th
with ‘xj ’ denoting its coordinate. It will only very rarely make any diﬀerence whether we consider elements of Rn to be row or column vectors, but on those few occasions in which it does, I will take them to be column vectors, despite the fact that I will almost always write them as in the above equation (it does, after all, save a lot of space). I use what seems to be the standard notation for vector inequalities on Rn : x ≥ y ⇐⇒ xi ≥ yi , for i = 1, . . . , n, x > y ⇐⇒ x ≥ y & y x, and x y ⇐⇒ xi > yi , for i = 1, . . . , n. Making use of these inequalities, we deﬁne the: nonnegative orthant : Rn+ = {x ∈ Rn  x ≥ 0} semipositive orthant : Rn+ \ {0} = {x ∈ Rn  x > 0}, and strictly positive orthant : Rn++ = {x ∈ Rn  x 0},
Preface
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where ‘0’ denotes the origin in Rn , and we use the symbol ‘\’ to denote settheoretic diﬀerence; that is: A \ B = {x ∈ A  x ∈ / B}. Since we will often be considering ordered pairs, for example, (p, w) ∈ Rn++ ×R+ , where p ∈ Rn++ and w ∈ R+ , and in general need to distinguish between the ordered pair (x, y) ∈ R2 and the open interval in R bounded by x and y; we will use a somewhat unorthodox notation for intervals of real numbers, thus: [x, y] ={z ∈ R  x ≤ z ≤ y}, [x, y[ ={z ∈ R  x ≤ z < y} ]x, y] ={z ∈ R  x < z ≤ y}, and ]x, y[ ={z ∈ R  x < z < y}. Incidentally, in the above material I have made use of the notation ‘x y,’ to indicate that it is not the case that x ≥ y, and whenever possible I will use a similar notation, a diagonal line through a symbol, to denote the negation of the relation indicated. Unfortunately, the limitations on the symbols available to me in the typesetting program will mean that I can’t always do this. Thus, for example, we will often use the notation ‘xGy’ to mean that a consumer considers the commodity bundle x to be at least as good as y. However, we will have to use the notation ‘¬xGy’ to indicate the opposite situation (the negation); that is, to indicate that the consumer does not consider x to be at least as good as y. I will make fairly extensive use of universal and existential quantiﬁers. Thus we might write, assuming that A and B are sets of real numbers: (∀x ∈ A)(∃y ∈ B) : y ≥ x; which is read verbally as, “for every x in the set A, there exists an element, y, in the set B such that y is at least as great as x.” In general, the end of a string of quantiﬁers will be indicated by a colon (:), and you should be careful to take note of the order in which the quantiﬁers occur. Thus, for example, the statement: (∀x ∈ R)(∃y ∈ R) : y > x, is true, whereas the statement: (∃y ∈ R)(∀x ∈ R) : y > x, is not! If you have not been introduced to this notation previously, it may be quite intimidating at ﬁrst; but I think that you will quickly ﬁnd that its use is very advantageous in stating complicated conditions. In fact, you might begin to convince yourself of this by comparing the equation in which I introduced this notation with the verbal interpretation which follows it. W. Lafayette, IN June, 2006
J. C. M.
Contents Preface
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1 An 1.1 1.2 1.3
1 1 2 11
Introduction to Preference Theory Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Relations and Orderings . . . . . . . . . . . . . . . . . . . . . Preference Relations and Utility Functions . . . . . . . . . . . . . . .
2 Algebraic Choice Theory 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The General Algebraic Theory of Choice . . . . . . . . . . . . . . 2.3 Some Criticisms of the Model . . . . . . . . . . . . . . . . . . . . 2.4 Stated Preferences versus Actual Choices . . . . . . . . . . . . . 2.5 The Speciﬁcation of the Primitive Terms . . . . . . . . . . . . . . 2.6 Weak Separability of Preferences . . . . . . . . . . . . . . . . . . 2.7 Additive Separability . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Sequential Consumption Plans . . . . . . . . . . . . . . . . . . . 2.9 The BPL Experiment Reconsidered . . . . . . . . . . . . . . . . . 2.10 Probabilistic Theories of Choice . . . . . . . . . . . . . . . . . . . 2.11 Are Preferences Total? . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Are Preferences Transitive? . . . . . . . . . . . . . . . . . . . . . 2.12.1 ‘Just Noticeable Diﬀerence,’ or ‘Threshold Eﬀects’ . . . . 2.12.2 Decision Rules Based On Qualitative Information . . . . . 2.12.3 Priorities and Measurement Errors . . . . . . . . . . . . . 2.12.4 Group Decisions: The Dr. Jekyll and Ms. Jekyll Problem 2.13 Asymmetric Orders . . . . . . . . . . . . . . . . . . . . . . . . . .
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21 21 22 25 28 30 33 39 41 44 45 47 50 50 51 52 53 54
3 Revealed Preference Theory 3.1 Introduction . . . . . . . . . . . . . . . . . . . 3.2 Choice Correspondences and Binary Relations 3.3 RegularRational Choice Correspondences . . 3.4 Representable Choice Correspondences . . . . 3.5 Preferences and Observed Demand Behavior . 3.6 The Implications of Asymmetric Orders* . . .
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59 59 59 64 67 70 77
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4 Consumer Demand Theory 4.1 Introduction . . . . . . . . . . . . . . . . 4.2 The Consumption Set . . . . . . . . . . 4.3 Demand Correspondences . . . . . . . . 4.4 The Budget Balance Condition . . . . . 4.5 Some Convexity Conditions . . . . . . . 4.6 Wold’s Theorem . . . . . . . . . . . . . 4.7 Indirect Preferences and Indirect Utility 4.8 Homothetic Preferences . . . . . . . . . 4.9 CostofLiving Indices . . . . . . . . . . 4.10 Consumer’s Surplus . . . . . . . . . . . 4.11 Appendix . . . . . . . . . . . . . . . . .
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85 85 85 88 90 94 96 97 104 108 111 125
5 Pure Exchange Economies 5.1 Introduction . . . . . . . . . . . . . . . . . . . . 5.2 The Basic Framework . . . . . . . . . . . . . . 5.3 The Edgeworth Box Diagram . . . . . . . . . . 5.4 Demand and Excess Demand Correspondences 5.5 Pareto Eﬃciency . . . . . . . . . . . . . . . . . 5.6 Pareto Eﬃciency and ’NonWastefulness’ . . .
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131 131 131 133 138 142 150
6 Production Theory 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.2 Basic Concepts of Production Theory . . . . . . . . 6.3 Linear Production Sets . . . . . . . . . . . . . . . . . 6.4 InputOutput Analysis . . . . . . . . . . . . . . . . . 6.5 Proﬁt Maximization . . . . . . . . . . . . . . . . . . 6.6 Proﬁt Maximizing with Constant Returns to Scale* . 6.7 Production in General Equilibrium Theory . . . . . 6.8 Activity Analysis* . . . . . . . . . . . . . . . . . . .
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155 155 155 161 166 171 176 178 182
7 Fundamental Welfare Theorems 7.1 Introduction . . . . . . . . . . . . . . . . . . . 7.2 Competitive Equilibrium with Production . . 7.3 Some Diagrammatic Techniques . . . . . . . . 7.4 Walras’ Law with Production . . . . . . . . . 7.5 The ‘First Fundamental Theorem’ . . . . . . 7.6 ‘Unbiasedness’ of the Competitive Mechanism 7.7 A Stronger Version of ‘The Second Theorem’
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191 191 191 196 201 205 211 219
8 The 8.1 8.2 8.3 8.4 8.5
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227 227 229 235 239 241
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Existence of Competitive Equilibrium Introduction . . . . . . . . . . . . . . . . . . Examples, Part 1 . . . . . . . . . . . . . . . Assumption (c) and the Attainable Set . . . The Gale and MasColell Theorem . . . . . An (Especially) Simple Existence Theorem
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CONTENTS 8.6
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Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9 Examples of General Equilibrium Analyses 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Optimal Commodity Taxation: Initial Formulation . . 9.3 A Reconsideration of the Problem . . . . . . . . . . . 9.4 The Simplest Model of Optimal Commodity Taxation 9.5 Some Results . . . . . . . . . . . . . . . . . . . . . . . 9.6 Optimal Income Taxation . . . . . . . . . . . . . . . . 9.7 Monopoly in a General Equilibrium Model . . . . . . . 9.8 Money in a General Equilibrium Model . . . . . . . . 9.9 Indivisible Commodities . . . . . . . . . . . . . . . . .
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249 249 249 252 255 256 259 269 272 276
10 Comparative Statics and Stability 10.1 Introduction . . . . . . . . . . . . . . . . 10.2 Aggregate Excess Demand . . . . . . . . 10.3 The ‘Law of Demand’ . . . . . . . . . . 10.4 Gross Substitutes . . . . . . . . . . . . . 10.5 Qualitative Economics . . . . . . . . . . 10.6 Stability in a Single Market . . . . . . . 10.7 MultiMarket Stability . . . . . . . . . . 10.8 A Note on NonTˆatonnement Processes 11 The 11.1 11.2 11.3 11.4 11.5 11.6
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281 281 282 286 291 294 298 302 307
Core of an Economy Introduction . . . . . . . . . . . . . . . . . . . . Convexity and the Attainable Consumption Set The Core of a Production Economy . . . . . . The Core in Replicated Economies . . . . . . . Equal Treatment . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . .
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311 311 314 317 320 329 330
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12 General Equilibrium with Uncertainty 12.1 Introduction . . . . . . . . . . . . . . . . 12.2 ArrowDebreu Contingent Commodities 12.3 Radner Equilibrium . . . . . . . . . . . 12.4 Complete Markets . . . . . . . . . . . . 12.5 Complete Markets and Eﬃciency . . . . 12.6 Concluding Notes . . . . . . . . . . . . .
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333 333 333 339 345 350 355
13 Further Topics 13.1 Introduction . . . . . . . . . . . . 13.2 Time in the Basic Model . . . . . 13.3 An Inﬁnite Time Horizon . . . . 13.4 Overlapping Generations . . . . . 13.5 A Continuum of Traders . . . . . 13.6 Suggestions for Further Reading
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359 359 359 366 369 372 379
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14 Social Choice and Voting Rules 14.1 Introduction . . . . . . . . . . . . . . . . . . . . 14.2 The Basic Setting . . . . . . . . . . . . . . . . . 14.3 Voting Rules . . . . . . . . . . . . . . . . . . . 14.4 Arrow’s General Possibility Theorem . . . . . . 14.5 Appendix. A More Sophisticated Borda Count
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383 383 384 387 392 402
15 Some Tools of Applied Welfare Analysis 15.1 Introduction . . . . . . . . . . . . . . . . . . 15.2 The Framework . . . . . . . . . . . . . . . . 15.3 Measurement Functions . . . . . . . . . . . 15.4 Social Preference Functions . . . . . . . . . 15.5 The Compensation Principle . . . . . . . . 15.6 Indirect Preferences: Individual and Social . 15.7 Measures of Real National Income . . . . . 15.8 Consumers’ Surplus . . . . . . . . . . . . .
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407 407 408 409 411 416 420 422 428
16 Public Goods 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 16.2 A Simple Model . . . . . . . . . . . . . . . . . . . . . 16.3 Public Goods . . . . . . . . . . . . . . . . . . . . . . 16.4 A Simple Public Goods Model . . . . . . . . . . . . 16.5 Lindahl and Ratio Equilibria . . . . . . . . . . . . . 16.6 The ‘Fundamental Theorems’ for Lindahl Equilibria 16.6.1 The ‘First Fundamental Theorem’ . . . . . . 16.6.2 The ‘Second Fundamental Theorem’ . . . . . 16.6.3 The ‘Metatheorem’ . . . . . . . . . . . . . . .
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437 437 437 441 442 446 455 456 458 462
17 Externalities 17.1 Introduction . . . . . . . . . 17.2 Externalities: A First Look 17.3 Extending the Model . . . . 17.4 The ‘Coase Theorem’ . . . . 17.5 Lindahl and Externalities . 17.6 Postscript . . . . . . . . . .
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467 467 468 475 480 484 487
18 Incentives and Implementation Theory 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 18.2 Game Forms and Mechanisms . . . . . . . . . . . . 18.3 The GibbardSatterthwaite Theorem . . . . . . . . 18.4 Implementation Theory . . . . . . . . . . . . . . . 18.5 SinglePeaked Preferences and Dominant Strategies 18.5.1 SinglePeaked Preferences . . . . . . . . . . 18.5.2 The Bowen Model . . . . . . . . . . . . . . 18.6 QuasiLinearity and Dominant Strategies . . . . . 18.7 Implementation in Nash Equilibria . . . . . . . . .
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489 489 490 495 502 504 504 509 511 520
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18.8 Nash Implementation with Public Goods . . . . . . . . . . . . . . . . 522 18.9 The Revelation Principle Reconsidered . . . . . . . . . . . . . . . . . 525 18.10Notes and Suggestions for Further Reading . . . . . . . . . . . . . . 527 19 Appendix. Solutions for Selected Exercises 19.1 Chapter 1 . . . . . . . . . . . . . . . . . . . 19.2 Chapter 2 . . . . . . . . . . . . . . . . . . . 19.3 Chapter 3 . . . . . . . . . . . . . . . . . . . 19.4 Chapter 4 . . . . . . . . . . . . . . . . . . . 19.5 Chapter 5 . . . . . . . . . . . . . . . . . . . 19.6 Chapter 6 . . . . . . . . . . . . . . . . . . . 19.7 Chapter 7 . . . . . . . . . . . . . . . . . . . 19.8 Chapter 8 . . . . . . . . . . . . . . . . . . . 19.9 Chapter 9 . . . . . . . . . . . . . . . . . . . 19.10Chapter 10 . . . . . . . . . . . . . . . . . . 19.11Chapter 11 . . . . . . . . . . . . . . . . . . 19.12Chapter 12 . . . . . . . . . . . . . . . . . . 19.13Chapter 13 . . . . . . . . . . . . . . . . . . 19.14Chapter 14 . . . . . . . . . . . . . . . . . . 19.15Chapter 15 . . . . . . . . . . . . . . . . . . 19.16Chapter 16 . . . . . . . . . . . . . . . . . . 19.17Chapter 17 . . . . . . . . . . . . . . . . . . 19.18Chapter 18 . . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . .
531 531 533 533 536 537 540 542 543 548 548 548 550 551 551 551 551 552 553
References
555
Author Index
569
Subject Index
573
Chapter 1
An Introduction to Preference Theory 1.1
Introduction
Choice, or more precisely, choice under constraint, is central to economic theory. Choice theory is the foundation of the economic theory of demand, of welfare and public economics, and is crucially important in decision and game theory. Since so many of these topics are critical parts of this course, it is only appropriate that we begin our study by investigating the foundations of choice theory itself; namely abstract preference theory. For us, of course, the most important single application of choice theory, is to consumer demand theory. In its most basic general equilibrium form, this theory postulates that we can think of consumers as making choices of ‘commodity bundles,’ which for us will be vectors x = (x1 , . . . , xj , . . . , xn ) ∈ Rn , where the j th coordinate of x, xj , denotes the quantity of the j th commodity available for consumption. We suppose further that, irrespective of prices and income or wealth, the consumer’s choice is constrained (presumably by physiological and/or technological requirements) to some subset, X of Rn . It is also usual to suppose that the consumer’s choice of commodity bundles in X is consistent with the consumer’s preferences over the set X; which preferences are modeled as a binary relation over X. This, of course, leads very naturally into the next section, which is concerned with beginning our investigation of binary relations in the abstract. You are probably already familiar with the fact that binary relations are used as an abstract representation of consumers’ preference relations in economic theory. What you may not be aware of is that the theory of binary relations is also central to welfare economics, and to index number theory, as well as to a number of other applications in economic theory. Consequently, we will devote a considerable amount of time to the study of binary relations in the abstract. This will represent a bit of ‘overkill,’ insofar as consumer demand theory is concerned, but we will be developing the theoretical foundations for much of our work in welfare economics as well as for the theory of consumer behavior.
2
Chapter 1. An Introduction to Preference Theory
1.2
Binary Relations and Orderings
Whether or not you have encountered a formal deﬁnition of a ‘binary relation,’ you certainly have encountered examples of such before this. The weak and strong inequalities for the real number system, for instance, are both examples of binary relations. Informally, a binary relation, R, on a set X, is simply a rule such that for each x and y in X, we can determine whether xRy, yRx, or neither, or both. Thus, for example, for any nonempty set, X, we can deﬁne the relation E (for equality) by: xEy ⇐⇒ x = y. Another example is the relation G deﬁned on R by: yGx ⇐⇒ y ≥ x2 . Notice that this last example is a special case of the following. Suppose f : R → R, and deﬁne the relation G on R by: yGx ⇐⇒ y ≥ f (x). In this section, we will consider the following properties of binary relations. In the deﬁnition to follow, and throughout the remainder of this chapter, we shall suppose that the set on which the binary relation is deﬁned is nonempty. 1.1 Deﬁnition. Let G be a binary relation on a nonempty set X. We shall say that G is: 1. total iﬀ: (∀x, y ∈ X) : xGy or yGx or x = y. 2. reﬂexive iﬀ: (∀x ∈ X) : xGx. 3. irreﬂexive iﬀ: (∀x ∈ X) : ¬xGx. 4. symmetric iﬀ: (∀x, y ∈ X) : xGy ⇒ yGx. 5. asymmetric iﬀ: (∀x, y ∈ X) : xGy ⇒ ¬yGx. 6. antisymmetric iﬀ: (∀x, y ∈ X) : [xGy & yGx] ⇒ x = y. 7. transitive iﬀ: (∀x, y, z ∈ X) : [xGy & yGz] ⇒ xGz. Notice that a number of the relations which appear to be negations of one another actually are not. For example, irreﬂexivity is not the negation of reﬂexivity; that is, if a relation is not reﬂexive, it is nonetheless not necessarily irreﬂexive, and conversely. Similarly, a relation which is not symmetric is not necessarily asymmetric; conversely, a relation may fail to satisfy asymmetry, yet not be symmetric.
1.2. Binary Relations and Orderings
3
1.2 Examples/Exercises. 1. Let X be the set of all persons alive on earth at the present date, and deﬁne the relation R on X by: xRy ⇐⇒ x is the brother of y; that is, xRy if, and only if: (a) x is a male person, and (b) x and y have the same (pair of) parents. Insofar as the normal English deﬁnition of the phrase ‘is the brother of’ is concerned, the relation is irreﬂexive; whereas, in the way we have deﬁned it here, the relation is not irreﬂexive. How could you modify the deﬁnition in order to make it correspond more closely to normal English usage? 2. Let X be the set of all physical objects on the earth at the present time, and deﬁne the relation R on X by: xRy ⇐⇒ x has at least as much mass as y. Show that R is total, reﬂexive, and transitive. (This is something of a trick question, since it is really an empirical, and not a mathematical issue. In order to arrive at something which you can prove, assume that mass can be measured to any degree of accuracy that we choose.) 3. Consider the usual weak inequality relation, ≥, on the real numbers. Show that ≥ is total, reﬂexive, antisymmetric, and transitive. Incidentally, here is an example of a binary relation which is neither symmetric nor asymmetric. 4. Show that the usual strict inequality relation, >, on the real numbers is total, irreﬂexive, asymmetric (and thus antisymmetric, since asymmetry implies antisymmetry), and transitive. 5. Let f : X → R, where X is any nonempty set, and deﬁne E on X by: xEy ⇐⇒ f (x) = f (y). Show that E is reﬂexive, symmetric, and transitive.
Incidentally, before proceeding further with our discussion of binary relations, I should mention that my insistence on having the set X be nonempty in Deﬁnition 1.1 is, essentially, for one reason; namely, a binary relation on the empty set satisﬁes all of the conditions, 1–7, in Deﬁnition 1.1. Consequently, if we include binary relations on the empty set in our deﬁnitions, the relationships among the conditions deﬁned in 1.1 become somewhat confused! Most of the binary relations which we encounter in economic theory are orderings of one type or another, where we use the term ordering to mean any transitive binary relation. Before considering the types of orderings which we will study in connection with consumer preference relations, however, let’s take a look at some orderings from mathematics which we will ﬁnd particularly useful. 1.3 Deﬁnitions. For x, y ∈ Rn , we deﬁne: 1. x ≥ y [read ‘x is greater than or equal to y’] iﬀ: xi ≥ yi
for i = 1, . . . , n.
4
Chapter 1. An Introduction to Preference Theory 2. x > y [read ‘x is semigreater than y’] iﬀ x ≥ y, but y x. 3. x y [read ‘x is strictly greater than y’] iﬀ: xi > yi
for i = 1, . . . , n.
Notice that if n = 1, the distinction between > and disappears. On the other hand, for n ≥ 2, there is a real diﬀerence between the two; for example, in the case of R3 , if we take: x = (1, 1, 1), y = (1, 2, 0), and z = (0, 0, 0), we have: x z, y > z, but ¬(y z). The weak inequality relation for Rn is an example of a partial order; that is, it is reﬂexive, antisymmetric, and transitive. This is stated formally in Theorem 1.4, which follows. The proof of 1.4 is fairly easy, and will be left as an exercise (those of you who have not been through a proof of this result before, however, should be sure to try to work out a proof now). 1.4 Theorem. The weak inequality (≥) for Rn is a partial order (that is, it is reﬂexive, antisymmetric, and transitive). However, ≥ is not total for n ≥ 2. 1.5 Deﬁnitions. We shall say that x ∈ Rn is: 1. nonnegative iﬀ x ≥ 0, 2. semipositive iﬀ x > 0, and 3. (strictly) positive iﬀ x 0, where ‘0’ denotes the origin in Rn in each of the above statements. 1.6 Deﬁnitions. We deﬁne Rn+ , the nonnegative orthant in Rn , as the set of all nonnegative vectors in Rn ; that is: Rn+ = {x ∈ Rn  x ≥ 0}; and Rn++ , the strictly positive orthant in Rn , by: Rn++ = {x ∈ Rn  x 0}. Be careful to note the distinction between the strictly positive orthant and the ‘semipositive orthant:’ Rn+ \ {0} = {x ∈ Rn  x > 0}; although in R (that is, in the case where n = 1), the two sets coincide. In economic theory, it is quite usual to base consumer demand theory on the assumption that an individual consumer’s (weak) preference relation over the set of commodity bundles, X, is a weak order; which we formally deﬁne as follows. 1.7 Deﬁnition. Let G be a binary relation on a set X. We shall say that G is a weak order (or that G is a weak ordering of X) iﬀ G is total, reﬂexive, and transitive.
1.2. Binary Relations and Orderings
5
1.8 Examples/Exercises. 1. It follows immediately from Example 1.2.3–4 that the usual weak inequality, ≥, on the real numbers is a weak order, but that the strict inequality for the real numbers, >, is not. Since ≥ is antisymmetric, it is an example of a more restrictive type of order than a weak order, called a linear order (which, by deﬁnition, is a relation which is total, reﬂexive, antisymmetric, and transitive). 2. Let X be any nonempty set, and let f be a realvalued function deﬁned on X. If we deﬁne the relation G on X by: xGy ⇐⇒ f (x) ≥ f (y), show that G is a weak order on X. 3. Is the weak inequality, ≥, a weak order of Rn , for n ≥ 2?
1.9 Proposition. Let G be a binary relation on a set X, and deﬁne P and I on X by: xP y ⇐⇒ [xGy & ¬yGx], and xIy ⇐⇒ [xGy & yGx], respectively. Then P is asymmetric and irreﬂexive, and I is symmetric. Proof. We will only prove that P is asymmetric; the proof that P is irreﬂexive is immediate, and the proof that I is symmetric will be left as an exercise. In all three cases the argument is almost so simple as to not need doing, but when one ﬁrst encounters this sort of material, it is diﬃcult to know exactly where to begin in constructing a proof of these facts. Consequently, we will illustrate. Let x and y be arbitrary elements of X, and suppose that xP y. Then by deﬁnition of P , we have: xGy and ¬yGx. (1.1) But then we see that we cannot have yP x; because by deﬁnition of P , this would require that yGx and ¬xGy; and by (1.1), neither of these conditions holds. Thus, if xP y, we cannot have yP x as well, and it follows that P is asymmetric. From the proposition just established, we see that the terminology in the following is indeed justiﬁed. 1.10 Deﬁnitions. If G is a binary relation on X, we deﬁne: 1. the asymmetric part of G, P , by: xP y ⇐⇒ [xGy and ¬yGx]. 2. the symmetric part of G, I, by: xIy ⇐⇒ [xGy & yGx]. If ‘G’ denotes a consumer’s weak preference relation over the set of commodity bundles, X, then the asymmetric part of G, P , would clearly be interpretable as the consumer’s strict preference relation, and the symmetric part, I, is the consumer’s indiﬀerence relation. Under these assumptions, the indiﬀerence relation is an example of an equivalence relation, deﬁned as follows.
6
Chapter 1. An Introduction to Preference Theory
1.11 Deﬁnition. If X is a nonempty set, and R is a binary relation on X, we shall say that R is an equivalence relation on X iﬀ R is reﬂexive, symmetric, and transitive. 1.12 Examples/Exercises. 1. In the terminology just introduced, you were asked in Example 1.2.5 to show that the relation, E, deﬁned there is an equivalence relation. 2. Let X = R+ , and deﬁne the relation R on X by: xRy ⇐⇒ x − y  < 1. Is R an equivalence relation? Explain. Is R an equivalence relation if, instead of X = R+ , we take X to be the set of nonnegative integers; that is, X = {0, 1, 2, . . . }? 3. Suppose we take X to be the set of people in this room, and deﬁne f : X → R by: f (x) = the height of x to the nearest inch (rounding up to n + 1 if the exact height is n.500 . . . 0 . . . .). If we now deﬁne E on X by: xEy ⇐⇒ f (x) − f (y) < 1, is E an equivalence relation?
1.13 Theorem. If G is a transitive binary relation, then: 1. the asymmetric part of G, P, is irreﬂexive, asymmetric, and transitive. 2. the symmetric part of G, I, is symmetric and transitive; 3. for any w, x, y and z in X: [wGx, xP y, & yGz] ⇒ wP z. and, if G is reﬂexive (as well as being transitive), then: 4. I is an equivalence relation. Proof. We will only prove part 1 of the conclusion; leaving parts 2–4 as exercises. It follows at once from Proposition 1.9 that P is irreﬂexive and asymmetric. To prove that P is transitive, let x, y and z be elements of X such that: xP y & yP z. Then, from the deﬁnition of P , we have: xGy & ¬yGx,
(1.2)
yGz & ¬zGy.
(1.3)
and: From the ﬁrst parts of (1.2) and (1.3), and the transitivity of G, we then have: xGz.
(1.4)
1.2. Binary Relations and Orderings
7
Suppose that we were to have zGx as well. Then from the ﬁrst part of (1.3) and the transitivity of G, we would have yGx; which contradicts the second part of (1.2). Therefore we must have ¬zGx, and combining this with (1.4), we see that xP z. Notice that the assumptions of the above result do not require G to be a weak order; if G is a weak order, the asymmetric part satisﬁes somewhat stronger properties. In particular, in this case, P will satisfy the following, as we will prove shortly. 1.14 Deﬁnition. We shall say that a relation, P , on a set X is negatively transitive iﬀ, for all x, y, z ∈ X, we have: if xP z, then either xP y or yP z. While the condition deﬁning negative transitivity undoubtedly appears odd at ﬁrst reading, notice that if P is a strict preference relation, what it says is the following. If x is preferred to z, and y is any other alternative, then if x is not preferred to y (so that, in the usual interpretation of preference, it must be true that y is at least as good as x), it must be the case that y is preferred to z. 1.15 Theorem. If G is a weak order on X, and P and I are the asymmetric and symmetric parts of G, respectively, then for all x, y ∈ X: 1. we have: ¬xP y ⇐⇒ yGx, (1.5) or, equivalently: ¬yGx ⇐⇒ xP y.
(1.6)
2. exactly one of the following conditions holds: xP y, yP x, or xIy. 3. P is negatively transitive. Proof. I will leave the proof of parts 1 and 2 as exercises. To prove part 3, suppose that xP z, but that ¬xP y. Then by (1.5), yGx; and, since xP z, it then follows from part 3 of Theorem 1.13 that yP z In our work thus far, we have generally been considering the properties which will be satisﬁed by the asymmetric part of a (usually reﬂexive) binary relation. Suppose we turn things around, and begin with an asymmetric binary relation which we use to deﬁne a reﬂexive relation, as follows. 1.16 Deﬁnition. Suppose P is a binary relation on X. We deﬁne the negation of P , which we will denote by ‘G,’ by: xGy ⇐⇒ ¬yP x.
(1.7)
Why are we interested in the negation of a binary relation? Well, suppose we begin with the idea of a strict preference relation for a consumer; instead of ﬁrst introducing the idea of a weak preference (or ‘at least as good as’) relation, and using it to deﬁne the strict preference relation. If we once again denote this strict preference relation by ‘P ,’ then we can deﬁne the weak preference relation as the
8
Chapter 1. An Introduction to Preference Theory
negation of P . What is the point of this? The basic reason is that many scholars have expressed doubts concerning the transitivity of the weak preference relation; and in any event, most economists and psychologists are much more comfortable in assuming that strict preference relations are transitive than they are assuming that the weak preference relation is transitive. We will discuss some of the reasons for this in the next chapter; in the meantime, let’s investigate some of the properties of the negation of an asymmetric binary relation.1 1.17 Proposition. If P is an asymmetric binary relation on X, then its negation, G, is total and reﬂexive. Moreover, P is the asymmetric part of G, and G is the only total and reﬂexive binary relation having P as its asymmetric part. Proof. To prove that G is total, suppose x and y are elements of X such that ¬xGy. Then it follows from (1.7) that we must have yP x; and, since P is asymmetric, it then follows that ¬xP y. Therefore yGx. That G is reﬂexive follows immediately from the fact that an asymmetric relation is also irreﬂexive.2 To prove that P is the asymmetric part of G, let x, x ∈ X. If we then have xGx and ¬x Gx, it is obvious from the deﬁnition of G that we have xP x . Conversely, suppose xP x . Then by deﬁnition of G, we must have ¬x Gx. Moreover, since P is asymmetric, we also have ¬x P x. Therefore, xGx , and we see that we have: xGx & ¬x Gx. It now follows that P is the asymmetric part of G. In order to establish uniqueness, suppose now that G∗ is a total and reﬂexive relation having P as its asymmetric part, and let x, x ∈ X be arbitrary. If xG∗ x , then, since P is the asymmetric part of G∗ , it follows that ¬x P x, and thus by deﬁnition of the negation that xGx . Conversely, suppose we have ¬xG∗ x . Then, since G∗ is total and reﬂexive, we must also have x G∗ x. But then, since P is the asymmetric part of G∗ , it follows that x P x. Therefore, using the deﬁnition of the negation, we see that ¬xGx ; and we conclude that G ≡ G∗ . While this last proposition establishes that there can be only one total and reﬂexive binary relation of which P is the asymmetric part, there may be other reﬂexive and possibly transitive binary relations of which P is the asymmetric part, even if P is transitive, as well as asymmetric. This is shown by the following example. 1.18 Example. Consider the ‘semigreaterthan’ relation on R2 , deﬁned by: x > x ⇐⇒ [x ≥ x & x x]. Obviously > is the asymmetric part of ≥. On the other hand, we will show that G, the negation of >, is given by: xGx ⇐⇒ x > x ⇐⇒ max{x1 − x1 , x2 − x2 } > 0 or x = x . 1
Strict preferences are asymmetric, by the very deﬁnition of the word ‘prefer.’ Therefore it also follows that G will be reﬂexive if P is simply irreﬂexive, and not necessarily asymmetric. 2
1.2. Binary Relations and Orderings
9
To prove the above statement, suppose ﬁrst that: ¬ max{x1 − x1 , x2 − x2 } > 0 or x = x . Then: max{x1 − x1 , x2 − x2 } ≤ 0 and x = x , from which it follows that x > x; and we conclude that: x > x ⇒ max{x1 − x1 , x2 − x2 } > 0 or x = x . Conversely, if: max{x1 − x1 , x2 − x2 } > 0 or x = x , then it is apparent that x > x. Consequently, since G is diﬀerent from the weak inequality on R2 , and is obviously reﬂexive, we see that there is in this case more than one reﬂexive binary relation of which ≥ is the asymmetric part. Our next result shows the reason that the property deﬁned in 1.14 is called ‘negative transitivity.’ 1.19 Proposition. If P is a binary relation on X, and G is its negation, then G is transitive if, and only if, P is negatively transitive. Proof. To prove that the negation of P is transitive, let x, y, and z be elements of X such that: xGy & yGz, (1.8) and suppose, by way of obtaining a contradiction, that ¬xGz. Then it follows from the deﬁnition of G that zP x. But this is impossible; for it would then follow from negative transitivity that either zP y or yP x, and either of these conditions contradicts (1.8). Consequently, we see that G is transitive. To prove the converse, suppose that G, the negation of P , is transitive; let x and y be elements of X such that xP y, and let z ∈ X. If ¬xP z, then we have, by deﬁnition, zGx. If we also have ¬zP y, then it would necessarily be the case that yGz, and the transitivity of G would imply yGx; which contradicts the assumption that xP y. The last couple of results we have established have several interesting implications, which we will state as corollaries; the proof of which I will leave as exercises. 1.20 Corollary. If P is an asymmetric and negatively transitive binary relation on a nonempty set X, then its negation, G, is a weak order on X, and P is its asymmetric part. Our next corollary is an immediate consequence of 1.20 and 1.13. 1.21 Corollary. If P is a binary relation which is asymmetric and negatively transitive, then P is also transitive.
10
Chapter 1. An Introduction to Preference Theory
While Corollary 1.21 may suggest that negatively transitive binary relations are also transitive, there exist binary relations which are negatively transitive and irreﬂexive, but which are not transitive. A very simple example of such a relation is the usual inequality relation, =. It is easy to show that this relation is irreﬂexive and negatively transitive, but it is not transitive. It is important to notice that a binary relation may be asymmetric and transitive without being negatively transitive, as is shown by the following (generic) example. 1.22 Examples. 1. Let X be any nonempty set, let f : X → R be any realvalued function deﬁned on X, and let δ be a strictly positive real number. If we then deﬁne the relation P on X by: xP y ⇐⇒ f (x) > f (y) + δ, (1.9) you should have no diﬃculty in proving that P is irreﬂexive, asymmetric, and transitive. On the other hand, P will generally not be negatively transitive. For example, let X = R+ , let f be the identity function, and let δ = 1. Then, letting x = 3/2, y = 3/4, and z = 0, we have xP z, but neither xP y nor yP z. 2. Let X = R2+ , let z = (a, b) be a ﬁxed vector, where a > 0 and b > 0, and deﬁne: Z = {x ∈ R2+  (∃λ ≥ 0) : x = λz}, Imagine now an extremely cautious consumer who has been maximizing preferences at the bundle z, and who is only willing to compare bundles whose proportions are the same as those at z; in other words, his eﬀective strict preference relation is deﬁned by: xP x ⇐⇒ x, x ∈ Z & x x . In this case, is P asymmetric? Is it transitive? 3. Suppose our cautious consumer of the previous example now decides that, given a bundle, x ∈ Z, any bundle x∗ which is not on the ray Z should be preferred to x if some of one or both commodities could be taken away from x∗ to yield a point x on Z which is such that x x. Suppose further that our consumer now notices that, for a given x∗ not on Z, the best (largest) bundle he can obtain on Z by giving up one of the commodities is the bundle x deﬁned by: x∗ x∗ 1 x = min , 2 z. a b Because of this, our consumer now decides that for x ∈ Z, any bundle x∗ ∈ R2+ satisfying: x∗ x∗ 1 min , 2 z x, a b should be preferred to x. Is the relation P so deﬁned asymmetric? Is it transitive? 4. Continuing with our cautious friend of the previous two examples, suppose he now realizes that any bundle in R2+ can be converted to one having the right proportions via the formula in the previous example. Because of this, our consumer
1.3. Preference Relations and Utility Functions
11
now decides that a bundle x should be preferred to a bundle x∗ ∈ R2+ if, and only if: x∗ x∗ x x 1 2 1 , z min min , 2 z. a b a b Is the relation P so deﬁned asymmetric? Is it transitive? Can you think of a simpler way of representing it?
1.3
Preference Relations and Utility Functions
In general equilibrium theory it is usual to suppose that a consumer’s choice of a ‘commodity bundle’ is limited to some nonempty subset, X, of Rn ; which subset we will refer to as the consumption set. If x = (x1 , . . . , xn ) is an element of X (a ‘commodity bundle’), then ‘xj ’ will denote the quantity of the j th commodity available to the consumer per unit of time, if xj ≥ 0. If, on the other hand, xj < 0, then we will take this to mean that the consumer is oﬀering to supply the j th commodity in the amount −xj = xj  per unit of time. In this context, the consumer is supposed to choose according to his or her3 ‘preference relation,’ G, deﬁned over the consumption set. Typically one assumes that this preference relation is a weak order (although we will introduce a weaker assumption in the next chapter). It is important to note that, while we refer to G as a ‘preference relation,’ it would be more appropriate to call it something like the ‘atleastasgoodas relation;’ since if x and y are elements of X, we would say that the consumer considers x at least as good as y (or that y is no better than x) if, and only if xGy (notice that this is consistent with the assumption that G is reﬂexive). Where it is important to make this distinction, we will refer to G as the consumer’s weak preference relation. In any event, the asymmetric part of G, P , is called the consumer’s strict preference relation, and the symmetric part of G, I, is called the consumer’s indiﬀerence relation. It follows from 1.13 and 1.20 that if G is a weak order, then P is irreﬂexive, asymmetric and negatively transitive (and transitive as well); while I is an equivalence relation. 1.23 Deﬁnition. If G is a binary relation on X, we deﬁne the upper contour set for x, Gx, and the lower contour set for x, xG, by: Gx = {y ∈ X  yGx} and xG = {z ∈ X  xGz}, respectively. In the case where G is a consumer’s (weak) preference relation, we will often wish to consider the sets P x and xP , where P is the asymmetric part of G. We will refer to these two sets as the strictly preferred and strictly inferior to x sets, respectively. 3 Hereafter we will use the word ‘its’ in place of this awkward circumlocution. The word is more appropriate in any case, since the term ‘consumer,’ as generally used in economics, should not necessarily be interpreted to be an individual. A safer, more correct, general identiﬁcation is to interpret ‘consumer’ to mean ‘household.’
12
Chapter 1. An Introduction to Preference Theory
Notice that if G is a binary relation on a set X, that the upper and lower contour set ideas deﬁne very natural correspondences from X into itself; speciﬁcally, we can deﬁne the correspondences Γ : X → X and Φ : X → X by: Γ(x) = xG and Φ(x) = Gx, respectively. I have used symbols diﬀerent from G to denote these correspondences, because (a) there are two such correspondences, and (b) in principle, the correspondences should be distinguished, to some extent, from the binary relation. However, notice that either of these two correspondences completely deﬁnes the binary relation. Conversely, if Γ is any correspondence such that Γ : X → X, then Γ deﬁnes a binary relation, G, on X by:
or, by:
xGy ⇐⇒ y ∈ Γ(x);
(1.10)
xG y ⇐⇒ x ∈ Γ(y),
(1.11)
for that matter. In the ﬁrst instance, we are identifying Γ(x) with the lower contour set for x (xG), while in the second deﬁnition, we are identifying Γ(y) with the upper contour set for y (that is, with G y). In practice, we will often ﬁnd it very convenient to use a correspondence to deﬁne a binary relation; although when we do, we will generally identify the values of the correspondence with the upper contour sets, rather than the lower; that is, we will generally deﬁne the binary relation as in (1.11), above, rather than as in (1.10). xG y ⇐⇒ y ∈ Γ(x). However, things are greatly simpliﬁed, and confusion minimized, by using the upperor lowercontour set notation in the ﬁrst place; and this is what we will do hereafter. As an example of this method of deﬁning a binary relation, notice that the preference relation of Example 1.22.3 of the previous section can be deﬁned by the upper contour set correspondence, given by: ⎧ ⎪ ⎨ x∗ ∈ R2  min x∗1 , x∗2 > x1 , for x ∈ Z, + a b a Gx = ⎪ ⎩ ∅ for x ∈ R2+ \ Z. Hopefully, you will agree that this represents a much simpler method of deﬁning the consumer’s (strict) preference relation than was used in our original development of the example. (For other examples of deﬁning preference relations by this method, see Examples 1.31, below.) 1.24 Deﬁnitions. If G is a reﬂexive binary relation on X, we shall say that a function f : X → R represents G on X iﬀ, for all x and y in X, we have: xGy ⇐⇒ f (x) ≥ f (y).
(1.12)
If a function exists which represents G on X, we shall say that there exists a representation for G, or that G admits of a realvalued representation. In the special case in which G is a consumer’s (weak) preference relation, we shall say that a function f which represents G on X is an (ordinal) utility function for G.
1.3. Preference Relations and Utility Functions
13
In Example 1.8, we showed that if a binary relation, G admits of a realvalued representation, then G is a weak order. Thus, a necessary condition for a binary relation to be representable by a realvalued function is that G be a weak order. Shortly, we will consider suﬃcient conditions for a binary relation to be representable, but before we do, let’s consider a further aspect of the deﬁnition of representability itself. Notice that (1.12) can be written as the compound statement: xGy ⇒ f (x) ≥ f (y),
(1.13)
f (x) ≥ f (y) ⇒ xGy.
(1.14)
and: Once again letting ‘P denote the asymmetric part of G, we see that the contrapositive of (1.13) is: f (y) > f (x) ⇒ yP x (recall Theorem 1.15), while the contrapositive of (1.14) is: yP x ⇒ f (y) > f (x). Therefore condition (1.12) is equivalent to: xP y ⇐⇒ f (x) > f (y);
(1.15)
which leads us to the following. 1.25 Deﬁnition. We shall say that an asymmetric relation, P , on X is representable iﬀ there exists a function f : X → R satisfying (1.15), above. Thus, if G is a weak order, then G is representable by Deﬁnition 1.24 if, and only if, its asymmetric part, P , is representable according to Deﬁnition 1.25. Accordingly, the two deﬁnitions are equivalent in the sense just stated, and where we ﬁnd it more convenient to use (1.15) rather than (1.12) as our deﬁnition of representability, we shall not hesitate to do so. 1.26 Proposition. Suppose that X is a ﬁnite set, and that G is a reﬂexive binary relation on X. Then there exists a function, f : X → R which represents G on X if, and only if, G is a weak order. Proof. It follows at once from 1.8.2 that if there exists a function representing G, then G must be a weak order. To prove the converse, suppose G is a weak order, and deﬁne the realvalued function f on X by: f (x) = #{y ∈ X  xGy} = #xG;
(1.16)
that is, f (x) is the number of elements, y, of X such that xGy. To prove that f represents G, suppose ﬁrst that xGy. Then if z ∈ X is such that yGz, it follows from the transitivity of G that xGz as well. Therefore: yG = {z ∈ X  yGz } ⊆ {z ∈ X  xGz} = xG,
14
Chapter 1. An Introduction to Preference Theory
and it follows that: f (x) = #xG ≥ #yG = f (y). Conversely, suppose that x and y are such that x P y . Then:
and:
x Gy ,
(1.17)
¬y Gx
(1.18)
From (1.17) and the transitivity of G, it is easy to see that: y G ⊆ x G;
(1.19)
while from the reﬂexivity of G and (1.18), we see that: / y G. x ∈ x G and x ∈
(1.20)
From (1.19) and (1.20) it follows that: f (x ) = #x G > #y G = f (y ). Thus we have shown that; x P y ⇒ f (x ) > f (y ), which, since G is total, is equivalent to: f (y ) ≥ f (x ) ⇒ y Gx .
1.27 Corollary. Suppose that X is a ﬁnite set, and that P is an asymmetric binary relation on X. Then there exists a function, f : X → R satisfying: xP y ⇐⇒ f (x) > f (y), for all x, y ∈ X if, and only if, P is negatively transitive. In other words, an asymmetric binary relation on a ﬁnite set, X, is representable if, and only if, it is negatively transitive. While the results just presented provide very simple and straightforward necessary and suﬃcient conditions for a binary relation to be representable for the case in which X is a ﬁnite set, things get more complicated if X is an inﬁnite set, as is demonstrated by the following example. 1.28 Example. (The lexicographic order.) Let X = R2+ , and deﬁne >L , the lexicographic order, on X, by: x1 > y1 or: (1.21) (x1 , x2 ) >L (y1 , y2 ) ⇐⇒ x1 = y1 and x2 > y2 . It is easy to show that >L is total and asymmetric (and thus is antisymmetric). We will prove that it is negatively transitive, from which it will follow that it is also transitive.
1.3. Preference Relations and Utility Functions
15
To prove negative transitivity, suppose x >L z. Then either: x1 > z1 ,
(1.22)
x1 = z1 and x2 > z2 .
(1.23)
or: Now let y ∈
R2+ ,
and suppose y ≯L z. Then either: y1 < z1 ,
(1.24)
y1 = z1 and y2 ≤ z2 .
(1.25)
or: However, if (1.22) holds, then either (1.24) or (1.25) implies x1 > y1 , and thus x >L y. Similarly, if (1.23) and (1.24) hold, then x >L y. On the other hand, if (1.23) and (1.25) hold, then we have x1 = y1 and x2 > y2 ; so that x >L y in this case as well. One can show, however, that >L does not admit of a realvalued representation. We will present only an outline of a proof of this here. For details, see Debreu [1959, pp. 72–3]. Hopefully, the basic idea of the argument will be clear enough, despite the fact that it formally depends upon some cardinal number concepts which you may not have previously encountered. Suppose, by way of obtaining a contradiction, that >L admits of a realvalued representation, so that there exists a function f : R2+ → R satisfying: (∀x, y ∈ R2+ ) : x >L y ⇐⇒ f (x) > f (y);
(1.26)
and for the sake of convenience in the remainder of our argument, let us use the generic notation ‘(x, y)’ to denote elements of R2+ . Then, for each x ∈ R+ , we can deﬁne real numbers ax and bx by: ax = f (x, 0) and bx = sup f (x, y).
(1.27)
y∈R+
Moreover, from (1.26) we see that, since for each x ∈ R+ , (x, 1) >L (x, 0), we must have: ax = f (x, 0) < f (x, 1) < bx ; while for x, x∗ ∈ R+ such that x∗ > x, similar considerations establish that bx ≤ ax∗ . Thus we see that the family, I given by: I = {[ax , bx [ x ∈ R+ }, is a family of disjoint, nondegenerate intervals of real numbers; a distinct such interval for each nonnegative real number, x. But this is impossible, because there are only a countable number of such intervals; whereas there are an uncountable number of nonnegative real numbers. More crudely put, there are simply too many nonnegative real numbers to obtain a nondegenerate interval for each, such that no two (distinct) intervals have any points in common!
16
Chapter 1. An Introduction to Preference Theory
In order to pursue the question of when a weak order on an inﬁnite set will be representable by a realvalued function, we will need to begin by considering the following generalization of the notion of a closed set in Rn . 1.29 Deﬁnition. If X is a nonempty subset of Rn , and A is a nonempty subset of X, we shall say that A is closed relative to X, or that A is closed in X, iﬀ, whenever xq is a sequence of points from A which converges to a point x which is an element of X, then we must have x ∈ A. 1.30 Deﬁnitions. Let X be a nonempty subset of Rn , and let G be a weak order on X. We shall say that G is: 1. upper semicontinuous on X iﬀ, for each x ∈ X, the set Gx is closed in X. 2. lower semicontinuous on X iﬀ, for each x ∈ X, the set xG is closed in X. 3. continuous on X iﬀ G is both upper and lower semicontinuous on X. 1.31 Examples. 1. Let X = R2+ , and deﬁne the correspondence Γ on X by: if x = 0, R2+ Γ(x) = R2+ \ {0} if x = 0. We then let G be the binary relation deﬁned by Γ; that is: xGx∗ ⇐⇒ x ∈ Γ(x∗ ). Then G is lower semicontinuous on X, but is not upper semicontinuous on X. Notice, however, that G is representable; for example by the function f : X → R deﬁned by: 0 for x = 0, and: f (x) = 1 for x > 0, is a function which represents G on X. Notice that in deﬁning the preceding example, the correspondence Γ is simply the upper contour correspondence. Hereafter, where there appears to be no danger of confusion, we will use the same symbol to refer to both the correspondence and the relation; as is done in the next example. 2. Let X = R2+ , and deﬁne G on X by: {x} for x = (1, 1), and G(x) = R2+ for x = (1, 1). In this case, it is easy to see that G is upper semicontinuous on X, but is not lower semicontinuous on X. Once again G is representable on X, however; for example, by the function f deﬁned on X by: 0 for x = (1, 1), and: f (x) = 1 for x = (1, 1).
1.3. Preference Relations and Utility Functions
17
3. The lexicographic ordering, >L , deﬁned in 1.28, above, is neither upper, nor lower semicontinuous on X. 4. Let X = R2+ , and deﬁne the subset, D, by: D = {x ∈ R2+  x1 = x2 }. We then deﬁne G by: G(x) =
X {x ∈ D  x ≥ x}
if x ∈ / D, if x ∈ D.
Is G representable on X? by a continuous realvalued function?
One can make use of the above deﬁnitions to prove the following result; although here we will simply state the result without providing a proof. 1.32 Proposition. If G is a binary relation on a nonempty set, X, and if there exists a continuous function, f : X → R, which represents G on X, then G is a continuous weak order on X. The next deﬁnition is not one which I will expect you to remember, and is stated only for completeness. 1.33 Deﬁnition. Let X be a subset of Rn . A pair of subsets of X, A and B, will be said to be a separation for X iﬀ A and B satisfy: 1. A and B are both closed in X, 2. A and B are both nonempty, and 3. A ∩ B = ∅ and X = A ∪ B. The set X will be said to be connected iﬀ there exists no separation for X. Intuitively, a subset of Rn is connected if it is ‘of one piece.’ The space Rn itself is connected (even if n = 1), and Rn+ is connected; in fact, any convex subset of Rn is connected. Debreu has proved the following theorem, which we will state without proof.4 1.34 Theorem. If X is a connected subset of Rn , and G is a continuous weak order on X, then G is representable on X. In fact, there exists a continuous realvalued function which represents G on X. Be sure to note that the above result establishes the fact that, if X is a connected subset of Rn , then the continuity of G is a suﬃcient condition for G to be representable. The fact that it is not necessary is shown by the following example. 1.35 Example. Let X = R2+ , and deﬁne the upper contour correspondence G : X → X by: ⎧ 2 ⎪ if x = 0, ⎨R+ G(x) = R2+ \ {0} if x ∈ R2+ \ {0, (1, 1)}, ⎪ ⎩ {(1, 1)} if x = (1, 1). 4
For a proof, see Debreu [1959, pp. 56–9].
18
Chapter 1. An Introduction to Preference Theory
Here it can be shown that the binary relation, G is neither upper, nor lower semicontinuous. However, it is clear that the function f : R2+ → R+ deﬁned by: ⎧ ⎪ ⎨0 f (x) = 1 ⎪ ⎩ 2 represents G on X.
for x = 0, for x > 0 & x = (1, 1), for x = (1, 1).
Exercises. 1. Prove Theorem 1.4. 2. Show that the inequality, >, on R2 is not negatively transitive. 3. Prove directly (that is, without using Proposition 1.17) that the ‘semigreaterthan’ relation on R2 , >, is the asymmetric part of the relation G deﬁned in Example 1.18. 4. Let X be any nonempty set, let f : X → R, and deﬁne the relation G on X by: xGy ⇐⇒ f (x) ≥ f (y). Show that G is a weak order. 5. Show that ≥, the usual weak inequality on Rn is reﬂexive, antisymmetric, and transitive. 6. Show that if f : X → R represents the weak order G, and F : f (X) → R is any strictly increasing function, then the composition of F and f , the function g deﬁned by: g(x) = F [f (x)] for x ∈ X, also represents G. It is because of this consideration that such representations are called ordinal utility functions in consumer demand theory. 7. Suppose X is a nonempty set, that P is a binary relation on X, and that f : X → R is a function satisfying: (∀x, x ∈ X) : xP x ⇐⇒ f (x) > f (x ). Show that P is asymmetric and negatively transitive. 8. Suppose X is a nonempty set, that P is a binary relation on X, and that f : X → R is a function satisfying: (∀x, y ∈ X) : xP y ⇒ f (x) > f (y). Prove the following statements, or provide a counterexample: a. P is irreﬂexive. b. P is asymmetric. c. P is transitive. d. P is negatively transitive. 9. Assume the same conditions as in Exercise 8, above, except this time assume: (∀x, y ∈ X) : f (x) > f (y) ⇒ xP y.
1.3. Preference Relations and Utility Functions
19
Answer the same questions as in Exercise 8. 10. Let G be the relation deﬁned on R by: xGy ⇐⇒ x ≥ f (y), where f : R → R. Can you provide suﬃcient conditions for G to be: a. reﬂexive? b. total? c. transitive? d. asymmetric? e. antisymmetric? (If you have found suﬃcient conditions, don’t worry at this point about whether they’re necessary as well.) 11. Show that if G is a weak order on a ﬁnite set, X, then the following function represents G on X: u(x) = #X − #P x, where P is the asymmetric part of G. How does this function compare with that used in the proof of Proposition 1.26? 12. Show that if G is a weak order on a ﬁnite set, X, then the following function represents G on X:
#P x , u∗ (x) = 1 − #X where P is the asymmetric part of G. 13. Show that if G is a weak order on a ﬁnite set, X, then the following function represents G on X: 1 , u ¯(x) = (#P x) + 1 where P is the asymmetric part of G. 14. Deﬁne the set E ⊆ R2+ by: E = {x ∈ R2+  x1 = x2 }, and deﬁne the relation P on R2+ by: x ∈ E  min{x1 , x2 } > min{x1 , x2 } Px = x ∈ R2+  min{x1 , x2 } > min{x1 , x2 }
if x ∈ R2+ \ E, if x ∈ E.
Is the relation P asymmetric? Is the relation P transitive? In each case, provide a justiﬁcation for your answer, either a brief proof or a counterexample. 15. Suppose is a linear order on a nonempty set, X, that {Y, Z} is a partition of X,5 and that Z is a binary relation on Y . Deﬁne the relation R on X by: ⎧ ⎪ if x, y ∈ Y, ⎨xQy xRy ⇐⇒ x y if x, y ∈ Z, or ⎪ ⎩ x ∈ Y & y ∈ Z. . a. Show that if Q is a weak order on Y , then R is a weak order on X. b. Show that if Q is a linear order on Y , then R is a linear order on X. 5
So that Y and Z are both nonempty, Y ∩ Z = ∅, and X = Y ∪ Z.
Chapter 2
Algebraic Choice Theory 2.1
Introduction
In this chapter we will examine the foundations of the economic theory of consumer demand. In particular, we will begin a critical examination of the appropriate interpretation of two of the primitives (undeﬁned, basic terms) in general equilibrium theory; namely ‘consumer’ and ‘commodity.’ One’s initial tendency is to identify ‘consumers’ in the theory with individual ‘consumers,’ as we deﬁne the term in the popular press. That is, a ‘consumer’ would be an individual adult human being who is not in the care of others. As it turns out, however, in economic application, we are on ﬁrmer ground (for reasons to be explained shortly) if an individual consumer in the theory is identiﬁed with a household in the ‘real world.’ This creates some potential problems in our theoretical development, and is one of the things which we will discuss at some length in this chapter. Insofar as the primitive ‘commodity’ is concerned, in most of abstract general equilibrium theory, commodities are diﬀerentiated by four characteristics: • physical characteristics, • time of availability, • location at which the commodity is available, and • state of the world in which the commodity is available. Thus, suppose that in a given economy, we have only two physically distinct commodities; say No. 1 paper clips, and sheets of 8 1/2 by 11 one hundred per cent rag content bond paper. However, suppose that we also are considering two distinct locations, two time periods (today and tomorrow), and two possible states of the world (with each a possibility both today and tomorrow). Then in our general equilibrium model we would distinguish 24 commodities; so that n, the number of commodities in our analysis (and the dimension of the commodity space), is equal to 16. Thus, in most of our basic theory we can be considered to be taking into account location, time, and undertainty. There is, however, a problem with this; if, for example, we are going to analyze the eﬀects of location, we need to put more
22
Chapter 2. Algebraic Choice Theory
structure in the model than we will be doing. In particular, for example, we need to take into account the fact that No. 1 paper clips available today at location one and given state of the world one, as opposed to paper clips available today at location two, given state of the world one, are diﬀerentiated from one another in a diﬀerent way than are paper clips available today at location one and given state of the world one versus paper clips available tomorrow at location one, given state of the world one. In other words, we need to add more structure to the model than we will generally be doing in this book in order to analyze the eﬀect of location, time, or uncertainty. In later chapters, we will devote some attention to the analysis of the eﬀects of time and of undertainty, but we will do very little in the way of analyzing the eﬀects of location. In this omission I am not alone, basic economic theory and application is rather remiss in analyzing the eﬀect of location, and I will have to leave a more thorough analysis of this topic to a specialized course in location theory. In any case, the notions of ‘commodity’ and quantities thereof can be interpreted in many diﬀerent ways in the context of the bulk of the general equilibrium theory which we will be studying, and in this chapter we will be taking what can reasonably be termed the ‘applied microeconomics’ interpretation of these notions. Speciﬁcally, in this chapter we will generally suppose that commodities are diﬀerentiated solely by physical characteristics, that there is no uncertainty as to availability; and, correspondingly, that the j th coordinate of a commodity bundle represents the quantity of the j th commodity available for consumption ‘now,’ per unit of time. In particular, then, we will suppose that quantities of commodities represent ‘ﬂows’ per unit of time. In the next section we will introduce what we will call the general algebraic choice model, which can be regarded as setting forth the ‘bare bones’ of the theory. The remaining sections of the chapter can essentially be regarded as a critique of this basic model within the context of the applied microeconomics interpretation of the commodity space; we will be concerned with a critical appraisal of some standard interpretations of the basic model, and with a number of criticisms which have been leveled at this type of choice theory.
2.2
The General Algebraic Theory of Choice
The term ‘algebraic’ is used to distinguish the theory to be studied here from probabilistic choice theories, which will be considered brieﬂy later in the chapter. Roughly speaking, an algebraic theory is deterministic in nature; in the sense that the basic assumption of the theory is that, if a decisionmaker were repeatedly oﬀered a choice between a given pair of alternatives, he or she would make the same choice from the pair each time it was oﬀered. In a probabilistic theory, the basic assumption is that there is a probability that one of the alternatives, call it ‘x,’ would be chosen over the other (denoted by ‘y’); and that if the choice set {x, y} were oﬀered a large number of times, the proportion of times that x would be chosen from this set would be approximately equal to this probability. We will discuss the distinction between these two types of choice theory in more detail in Section 10 of this chapter. In the basic algebraic theory of choice, it is assumed that the decisionmaker has
2.2. The General Algebraic Theory of Choice
23
welldeﬁned preferences, , over a nonempty set of alternatives, X; and that, if her choice is constrained to a nonempty subset, B, of X, the alternative chosen will be at least as good (in terms of her preferences, ) as any other alternative in B. More formally, the theory deals with: X a nonempty set, the set of alternatives, the ‘preference relation,’ assumed to be a weak order on X, and B a nonempty family of nonempty subsets of X. Sets contained in (elements of) B will be called budget sets; and the pair X, B will be called a budget space. The fundamental assumption of the theory is that, if the decisionmaker’s choice is conﬁned to the set B ∈ B, the element, or alternative actually chosen from B will be an element of the set h(B) deﬁned as: h(B) = {x ∈ B  (∀y ∈ B) : x y}.
(2.1)
The elements of X should be considered to be distinct and mutually exclusive alternatives. In the economic theory of consumer demand, X is usually taken to be a subset of Rn ; with the ordered ntuples: x = (x1 , x2 , . . . , xn ) ∈ X, taken to be commodity bundles, a list of quantities (per unit of time) of the n commodities. In this context, the budget sets, B, are usually taken to be of the form: B = b(p, w) = {x ∈ X  p · x ≤ w}, where p is an element of Rn , and represents a vector of prices, and w represents the consumer’s wealth (or income, per time period, depending upon the context). However, the framework being presented here is, in principle, applicable to many other situations; for example the elements of X might be interpreted as cash ﬂows, military strategies, inventory policies, legislative programs, potential marriage partners, sound energies (at ﬁxed frequency, but varied decibel levels; or at a ﬁxed decibel level and varying frequencies), and so on. More detailed examples (speciﬁc interpretations) representing typical applications of the general algebraic choice model, are presented in the following. 2.1 Examples. 1. Let ‘Z = {z1 , z2 , z3 , z4 }’ denote a set of objects. We might formulate the problem of analyzing a consumer’s choice of the objects in Z in one of two diﬀerent ways, depending upon the choice context. a. Deﬁne the entities ai (i = 0, 1, . . . , 15) in the following way: a0 = ∅, ai = {zi } for i = 1, . . . , 4, a5 = {z1 , z2 }, a6 = {z1 , z3 }, . . . , a9 = {z2 , z4 }, a10 = {z3 , z4 }, a11 = {z1 , z2 , z3 }, a12 = {z1 , z2 , z4 }, a13 = {z1 , z3 , z4 }, a14 = {z2 , z3 , z4 }, a15 = {z1 , z2 , z3 , z4 }; and deﬁne the set of alternatives, X, by: X = {a0 , a1 , . . . , a15 }.
(2.2)
24
Chapter 2. Algebraic Choice Theory
If the consumer has welldeﬁned preferences over this set of alternatives (in particular, if the consumer’s preference relation is a weak order), and if she chooses an element from X in accordance with these preferences, then the general algebraic choice model is applicable here. We would expect that in a typical case, the actual choice would be constrained to some proper subset of X; for example: B = {a0 , a1 , . . . , a10 } (‘you can have at most any two of the four elements. . . ’). The prediction of the theory would then be that the consumer’s actual choice would be an element of the set h(B) given by: h(B) = {ai ∈ B  ai aj , for j = 0, 1, . . . , 10}. Moreover, if the objects are all desirable, then in this case we would expect to ﬁnd that: h(B) ⊆ {a5 , . . . , a10 }. b. Suppose now that the four objects are four diﬀerent kinds of (new) refrigerators. In this case, it would appear that the general algebraic choice model is applicable to the consumer’s choice of a refrigerator. However, the structure of the problem is greatly simpliﬁed by taking what we might call the ‘marketing approach’ to the problem. The general idea here is to restrict the claimed applicability of the theory to the case wherein the consumer has already decided to buy (choose) exactly one refrigerator. The set of alternatives then becomes: Z = {z1 , . . . , z4 }. The relationship between the sort of preference relation considered in part (a) of this example and that on Z is of particular interest at this point. Notice ﬁrst that a weak order, , on the set X deﬁned in (2.2) induces a weak order, G, on the set Z by the deﬁnition: zi Gzj ⇐⇒ ai = {zi } aj = {zj }; however, the converse is not true. More speciﬁcally, if G is the consumer’s preference relation on Z, then G will generally provide very little information about the consumer’s preference relation, , on X. In fact, while it might at ﬁrst glance appear that if: z 1 P z 2 P z3 P z 4 (where P is the asymmetric part of G, the consumer’s ‘strict preference relation’), then we would surely have: a5 = {z1 , z2 } a8 = {z2 , z3 }, a little thought should convince you that even this is not the case. Even in the special case of diﬀerent kinds of refrigerators, we may well have a8 a5 in this case; for suppose z1 and z2 are fullsize refrigerators, and z3 is a smaller (‘apartmentsize’ (?)) refrigerator. Abstracting from questions of price, the consumer might very well
2.3. Some Criticisms of the Model
25
prefer to have the combination of a fullsize and smaller refrigerator (which could, perhaps, ﬁt in a rec room in the basement) to having two fullsize refrigerators. Returning to the ‘marketing approach,’ however, notice that the consumer may have welldeﬁned preferences over the set Z (over the four refrigerators), and make a choice consistent with these preferences, without having given a thought to the question of whether, for example, the combination {z1 , z2 } is preferred to {z2 , z3 }. 2. Let X be a set of lottery tickets of the form (more correctly, which can be denoted by): x = (π; a, b), where ‘π’ denotes the probability of winning the prize a; the probability of winning the alternative prize, b, being equal to 1 − π. If a decisionmaker can be regarded as having a preference relation over these alternatives which is total, reﬂexive, and transitive (and thus is a weak order), then the general algebraic choice model is applicable to the analysis of this situation. 3. Let X be the collection of all legislation of a particular type which has been proposed on the ﬂoor of the U. S. House of Representatives as of a certain date. Would you expect to be able to apply the general algebraic choice model to the actions of the House subcommittee having jurisdiction over this type of legislation? How about to the President’s choice of legislative policy in the area? 4. Let ‘x1 ’ denote the quantity of food available for a speciﬁc period of time (say a month), and let ‘x2 ’ denote the quantity of clothing available during the same period of time. What kinds of diﬃculties might we encounter in trying to analyze a particular consumer’s choice of food and clothing within the context of the general algebraic choice model, taking X = R2+ ? Before jumping to any conclusions here, carefully consider the problem of comparing a pair x = (x1 , x2 ) with a second pair, x∗ = (x∗1 , x∗2 ). 5. Suppose there are n types of soft drinks (excluding coﬀee and tea) available in a particular locality as of the beginning of a given month, and that we label these soft drinks with the numbers 1, . . . , n (for example, ‘1’ might be CocaCola, ‘2’ PepsiCola, ‘3’ Royal Crown Cola, ‘4’ Diet coke, etc.). Letting X = Rn+ , the general algebraic choice model might be applicable to a particular consumer’s choice of soft drinks for the month, if said consumer has welldeﬁned preferences over X. This simple example can be used to illustrate a number of diﬃculties in the applicability of the model, however, and we will return to a consideration of various aspects of this example in the following sections.
2.3
Some Criticisms of the Model
Somewhat paradoxically, the general algebraic choice model suﬀers from two seemingly contradictory ﬂaws: it is so general as to have very little predictive power, yet at the same time, there is a very real question as to whether the assumptions of the model are satisﬁed in very common individual choice situations. We will return to the issue of the predictive power of the model in the next chapter; in this section and the remainder of this chapter, we will brieﬂy consider a number of criticisms which have been levied at the model regarding its applicability.
26
Chapter 2. Algebraic Choice Theory
Among many objections which have been raised concerning the realism or applicability of the assumptions of the general algebraic choice model, there are seven types of criticisms which we will consider in this chapter. 1. The model may be inapplicable to certain kinds of choices under uncertainty; speciﬁcally, to situations in which the decisionmaker’s choice does not uniquely determine the outcome. 2. The model, as set forth here, does not admit of a genuinely dynamic analysis. 3. Individuals’ preferences over alternatives may depend upon the way the alternatives are presented; that is, a given alternative may be describable or representable in two diﬀerent ways, and an individual’s preference for it as compared to a second alternative may depend upon which of these representations is chosen. This is the issue of ‘framing,’ and will be discussed in the next section. 4. Individuals’ stated preferences may be inconsistent with their actual choices. 5. In actual choice situations (particularly in experiments), individuals often exhibit inconsistencies. Thus a probabilistic (as opposed to an algebraic) theory of choice may be needed. 6. It may be unreasonable to suppose that preferences are total. 7. It may be unreasonable to suppose that preferences are transitive. We will provide only an extremely cursory consideration of the ﬁrst three of these objections here. The remaining four diﬃculties will be given a more extensive consideration in the remaining sections of this chapter. As an illustration of the kind of diﬃculties presented by choice under uncertainty, consider the situation in which a business manager has a choice among three inventory policies, a1 , a2 and a3 ; with policy ai best if event Ei occurs, for i = 1, 2, 3. Without further assumptions about (or better, knowledge of) the policies and the probabilities of the events Ei , we could hardly assert with any conﬁdence that our manager has welldeﬁned deterministic preferences over the three policies. The difﬁculty here, notice, is that we would suppose that the decisionmaker’s preferences are not deﬁned directly over the objects of choice (a1 , a2 and a3 ), but rather over outcomes, which are determined jointly by the policy variables ai and the random events, Ej . It is usual in economics to analyze choice under uncertainty (more correctly, under risk) via the expected utility (EU) model, which we can develop here as follows. Denote the outcome associated with action ai , given state j, Ej by ‘xij ,’ and let pj denote the probability of the occurence of state Ej , where: p1 + p2 + p3 = 1. If the von NeumannMorgenstern utility of the the outcomes is given by a function u : X → R, then action ai is preferred to action ak if, and only if: p1 u(xi1 ) + p2 u(xi2 ) + p3 u(xi3 ) > p1 u(xk1 ) + p2 u(xk2 ) + p3 u(xk3 ).
2.3. Some Criticisms of the Model
27
While this model creates an elegant theory of decisionmaking under risk, and has been widelyused in economics, there is a great deal of empirical evidence which casts doubt on some of the assumptions of this model. We will discuss the model, and some of this empirical evidence later in the course; in the meantime, let me mention that Loomes, Starmer, and Sugden [1991], Machina [1987], Starmer [1996, 2000], and Tversky and Thaler [1990] all provide quite readable and useful discussions of the empirical ﬁndings which have been at odds with this theory, as well as some alternative theories which have been proposed in response to the empirical ﬁndings.
Insofar as the diﬃculties with a dynamic analysis are concerned, suppose we consider the typical formulation in economics, where X is taken to be a nonempty subset of Rn , and ‘xi ’ denotes the quantity of the ith commodity (per unit of time) for i = 1, . . . , n. In this context, we can distinguish between the physically identical commodity available now, as opposed to T periods from now.1 Thus ‘x1 ’ might denote the quantity of #1 paper clips available ‘now’ (at the beginning of period 1), ‘x2 ’ the quantity of #1 paper clips available at the beginning of the next period (t = 2), and so on. The trouble with this is that in order to apply this interpretation, we need to assume that the decisionmaker knows his or her budget set now (at t = 1) even though it involves commodities available only at later dates. This would appear to be reasonable only (if at all) in the presence of much more pervasive and eﬃcient futures markets than appear to exist currently.2 Furthermore, actual preferences may change over time (if a learning process takes place as commodities are consumed, for example), and there is no adequate allowance for this eﬀect in the present formulation. We will return to a more complete discussion of some diﬃculties connected with the dynamic analysis of consumer choice in Section 8 of this chapter. As to the ‘framing’ problem, consider an example/experiment which was reported by Tversky and Kahneman [1988]. A group of subjects was presented with the following material (their Problem 3). Problem 3 [N = 150] Imagine that you face the following pair of concurrent decisions. First examine both decisions, then indicate the options you prefer. Decision (i) Choose between: A. a sure gain of $240 B. 25% chance to gain $1000, and a 75% chance to gain nothing. Decision (ii) Choose between: C. a sure loss of $750, D. 75% chance to lose $1000, and 25% chance to lose nothing.
When this problem was presented to 150 subjects, 84% chose (A) as the ﬁrst decision and 87% chose (D) as Decision (ii). However, consider the following, which is, notice, exactly the same as the concurrent choice in the previous problem. 1 We might equally well want to distinguish on the basis of location and on the basis of the state of the world in which the commodity will be available, but we will postpone a consideration of these complications for the moment. 2 For excellent discussions of this kind of interpretation of the model, and some of the pitfalls involved therein, see Debreu [1959, pp. 28–36 & pp. 50–5], or Chapter 20 of MasColell, Whinston, and Green [1995].
28
Chapter 2. Algebraic Choice Theory Problem 4. Choose between: A & D. 25% chance to win $ 240, and 75% chance to lose $ 760. B & C. 25% chance to win $ 250, and 75% chance to lose $750.
When the problem was presented in this way to 86 subjects, all of them chose B & C; but notice that the problem is exactly the same as Problem 3, it is simply presented (‘framed’) in a diﬀerent way. This is only one type of ‘framing’ diﬃculty which has been investigated in the literature. A much more complete discussion of the problem, as well as some other anomalies, is provided in the references mentioned earlier: Loomes, Starmer, and Sugden [1991], Machina [1987], Starmer [1996, 2000], and Tversky and Thaler [1990], as well as Tversky and Kahneman [1988] and Tversky et al [1990].
2.4
Stated Preferences versus Actual Choices
The general algebraic choice model is applicable to a speciﬁc choice situation only if the decisionmaker has welldeﬁned preferences (a weak order) over the underlying set of alternatives, X; and if, when presented with a subset of X from which a choice must be made, said decisionmaker always makes a choice consistent with these preferences. In terms of the notation of Section 2, if B represents the available or feasible set, then the actual choice must be an element of h(B), where: h(B) = {x ∈ B  (∀y ∈ B) : x y}.
(2.3)
The diﬃculty to be considered in this section is the claim that, even in cases where the weak order condition is satisﬁed, the actual choice from some subset, B, of X may not be an element of h(B). We are all familiar with the fact that stated preferences may be very diﬀerent from actual choices in situations where there are ‘facesaving’ motives, or group pressure present. Thus, for example, we have all heard of situations in which an individual who was actually watching a particular (lowbrow) television show would claim, when asked by a pollster, that he was watching some other show which he perceived to be more socially respectable. Similarly, we have probably all, at one time or another when out with a group, yielded to group preferences, and attended a movie that was our second choice or lower. Neither of these phenomena really represents a fundamental theoretical diﬃculty with the general algebraic choice model, however. The ﬁrst situation is simply a matter of not stating true preferences, while the second situation may be reconciled with the model by noting that there is nothing inconsistent about the fact that an individual may prefer the alternative of attending movie x in the company of friends to the alternative of attending movie y alone, even though he prefers attending movie y alone to attending movie x alone.3 In contrast, the objection to be considered in this section is that, even when these facesaving, or group pressure diﬃculties are absent, there may nonetheless 3 However, to dismiss this sort of diﬃculty this glibly is to ignore the possibility that the model may be extremely sensitive to the correct speciﬁcation of the alternative set, X. We will return to this question in the next section.
2.4. Stated Preferences versus Actual Choices
29
be inconsistencies between stated preferences and actual choices. For example, in an experiment conducted at Purdue University by Professors F. M. Bass, E. A Pessemier, and D. R. Lehmann [1972], 280 subjects were required to select a 12ounce can of soft drink four days a week for three weeks from the set of alternatives shown in Table 1. Cola
NonDiet CocaCola PepsiCola
Diet Tab Diet Pepsi
7Up Sprite
Like Fresca
LemonLime
Table 2.1: Soft Drink Choices. For participating in the experiment, the subjects each received $3 in addition to 12 cans of soft drink. To quote from the study (p. 533): In order to keep the selection as natural as possible and to control the eﬀect of speciﬁc purchase and use contexts, subjects were allowed to make their choice any time between 9:00 a.m. and 12:30 p.m. in a room which adjoined the student lounge where soft drinks, candy, etc., are available in vending machines. All the subjects had a reason to be in the building daily between those times.4
In addition to making the choices, participants were required to ﬁll out three diﬀerent questionnaires at various times; in one of which (Questionnaire 2), they were asked to rankorder the eight brands in terms of their own preferences. This questionnaire was ﬁlled out at the beginning of the experiment, and at the end of each of the three weeks. The accuracy of the choice predictions based on the preference rankings, as well as on the basis of the lastperiod choice, are summarized in Table 2, below (Bass, Pessemier, and Lehmann [1972], Table 4, p. 537).5
Model Stated ﬁrst choice (Post) Stated ﬁrst choice (Pre) Last period choice Random
Percentage of Correct Choice Predictions 52.1 50.8 37.1 12.5
Table 2.2: Choice Probabilities. There are a number of facets of this experiment which deserve further consideration, and in fact in the next four sections we will be discussing various aspects of this 4
The subjects were all Purdue students and/or secretaries. In Table 2, ‘Post’ means that the ﬁrst choice is based upon Questionnaire 2 the ﬁrst time it was asked after the choice, while ‘Pre’ means the Questionnaire 2 response obtained most recently before the actual choices. 5
30
Chapter 2. Algebraic Choice Theory
experiment and their relationship to the conventional economic theory of demand. For the sake of convenience, and with apologies to Professors Bass, Pessemier, and Lehmann, we will hereafter refer to their experiment as the ‘BPL experiment.’
2.5
The Speciﬁcation of the Primitive Terms
Suppose we begin by considering the basic framework of the economic theory of demand, as developed in a typical textbook. From a formal point of view, we can say that this theory is the special case of the general algebraic choice model which is obtained by taking X to be a nonempty subset of Rn (for the sake of convenience at this point, we will take X = Rn+ ), and B to be the family of all subsets of X having the form: B = b(p, w) = {x ∈ Rn+  p · x ≤ w}
for p ∈ Rn++ , w ∈ R+ .
If x = (x1 , x2 , . . . , xn ) is an element of Rn+ , the ith coordinate of x, xi , denotes the quantity of the ith commodity available per unit of time. Similarly, ‘pi ’ denotes the price per unit of the ith commodity. In terms of the formal development of the theory, however, ‘commodity,’ ‘price,’ ‘wealth,’ ‘unit of time,’ and ‘consumer’ are all primitives of the theory; that is, they are undeﬁned terms, just as ‘point’ and ‘line’ are undeﬁned terms, or primitives, in Euclidean geometry. To be sure, since economics is (or at least is partially) an empirical science, rather than a branch of mathematics, there is an implicit claim that there are empirical counterparts, or speciﬁc interpretations, of these primitives such that the assumptions of the theory are satisﬁed in actual choice situations (given these interpretations).6 However, most textbooks are conveniently vague as to the claimed applicability of the theory; that is, most textbooks never state explicitly under which interpretations of the primitives it is being claimed that the assumptions of the model will hold in actual choice situations. It would appear, however, that most members of the economics profession would feel safest with something like the following speciﬁcations of these primitive terms. S.1 ‘Individual consumer’ is (for economists in the U. S.) taken to be an individual household, as deﬁned by the U. S. Bureau of the Census.7 S.2 The list of consumer commodities should be exhaustive in terms of the immediate locality involved (including everything available within, say, a twohour drive, or by mail order, or the web, in the locality in question). S.3 The ‘commodities’ should be speciﬁed precisely enough so that diﬀerent units of what we are calling the ‘same’ commodity should be essentially indistinguishable, with regard to physical characteristics, location, time of availability, 6 Hopefully it is obvious that this distinction between a primitive term in a theory and an empirical counterpart, or possible interpretation of the term, has nothing whatsoever to do with whether we are undertaking a ‘mathematical’ or a ‘nonmathematical’ development of the theory. 7 Because of this, to remind ourselves that the term ‘consumer’ is a primitive, and to avoid sexist connotations, in theoretical discussions we will generally use the pronoun ‘it’ in referring to an individual consumer.
2.5. The Speciﬁcation of the Primitive Terms
31
and state of the world in which the commodity is available. This last aspect of the deﬁnition of a commodity brings uncertainty into consideration, and will be ignored for the remainder of this chapter; although we will return to this problem in a later chapter. S.4 The time involved will be taken to be one month; thus we would interpret xi to be the quantity of the ith commodity available per month. S.5 The ‘unit’ in which a given commodity is measured can be any convenient unit in terms of which the commodity is actually sold, implicitly or explicitly (thus we might use ‘ﬂuid ounces’ as the unit of measurement for milk or soft drinks, weight in ounces [or grams] for bread, fruits, vegetables, etc.). S.6 ‘Price’ will be interpreted as price per unit at the beginning of the month. S.7 ‘Wealth’ will be interpreted as ‘planned total consumption expenditure’ for the month. The above list of interpretations (or speciﬁcations) is not claimed to be deﬁnitive, and in fact, the deﬁnitions set out are not really suﬃciently precise for econometric work; although they should be suﬃciently precise and detailed for the purposes of our present discussion. Furthermore, while (for what it’s worth) I feel most conﬁdent about the empirical validity of the theory under the above list of speciﬁcations, I am certainly not claiming that the economic theory of demand is only valid under the above list of speciﬁcations of the primitives. I have set out this list here primarily to make one basic point: even if we are conﬁdent that the economic theory of consumer demand is empirically valid under one set of speciﬁcations of the primitives of the theory, there is no reason to suppose, on a priori grounds, that it is empirically valid under some alternative speciﬁcation of the primitives, unless we can demonstrate (deductively) that its validity under the ﬁrst speciﬁcation implies its validity under the second speciﬁcation as well. We will refer to the theory with the above list of speciﬁcations as the standard economic theory of demand. This terminology is simply a convenient label, and should not be taken to mean that all economists feel most conﬁdent about the validity of the model with these speciﬁcations; however, as the reader will probably agree, this speciﬁcation has some claim to a consensus status. Since the reader has no doubt already encountered many discussions presenting a priori and/or introspective reasons for believing the theory to be empirically valid under something like the above speciﬁcations, we will conﬁne our discussion here to one or two remarks about these speciﬁcations. First of all, insofar as item S.1 is concerned, the basic reason for this speciﬁcation is that most economists’ conﬁdence as to the validity of the theory is inversely proportional to the size of the decisionmaking unit; in fact, the theory seems to have been developed with the idea that ‘individual consumer’ should be speciﬁed to mean an ‘individual human being.’ On the other hand, many individuals (most notably, dependent children) do not make consumption decisions for themselves, and adult members of a multipleperson household presumably make joint decisions on
32
Chapter 2. Algebraic Choice Theory
many consumption items. Consequently, it would appear that the household is the smallest entity that can be treated as an independent consumption unit. The reason for specifying the unit of time to be a month hinges around two basic considerations. First, the time period should be long enough to allow for expected varietyseeking behavior (most people would make a diﬀerent selection from a given restaurant menu at breakfast than they would at lunch even if the expenditure involved was exactly the same), but not so long as to allow for expected changes in taste (a given individual’s preferences over commodities are likely to be very diﬀerent at 60 than they were at 20 years of age). Secondly, it appears that most households actually do some formal budget planning for each month (or so I hear). Insofar as the speciﬁcation of price is concerned, pi would probably be best interpreted as the ‘expected average (per unit) price of the ith commodity for the forthcoming month.’ However, because of the diﬃculty in predicting a household’s expected average price, most economists would probably generally settle for interpreting pi to be the price per unit at the beginning of the month (since this is when most formal budget planning seems to be done) and hope for the best. Finally, it should be mentioned that the ‘wealth’ speciﬁcation is a rather tautological deﬁnition that I have used here only for the sake of convenience. A more meaningful speciﬁcation of the wealth variable can only be made, however, after we have gone into some aspects of the speciﬁcation of the consumption set, X, which we will not take up until a later chapter. Suppose we now reconsider the BPL experiment in light of the above discussion. As we shall see, this single experiment could be regarded as a test of many diﬀerent theories; and what is more important, from the standpoint of our present discussion, it can be regarded as a test of the empirical validity of the algebraic choice model under a number of diﬀerent speciﬁcations of the primitives. However, it is probable that the simplest way of viewing the experiment is as a test of the theory obtained when items S.1 and S.4 in the speciﬁcation of the ‘standard theory’ are changed to: S.1 ‘Individual consumer’ is taken to be an ‘individual subject’ of the experiment. S.4 The time unit involved is taken to be one day: thus we will interpret xi to be the quantity of the ith soft drink (in number of 12ounce cans) available per day. We will also simplify things drastically by assuming that there are only two brands of soft drinks available. While this is obviously unrealistic, the presence of two distinct brands in our theoretical model will suﬃce to illustrate the points to be made in our discussion. We will also suppose that the consumers’ consumption set, X, can be taken to be Rn+ , and that the ﬁrst two coordinates measure the quantities available for consumption of these two diﬀerent brands of soft drink; with xj = the quantity of brand j in number of 12ounce cans, for j = 1, 2. We will hereafter refer to the special case of the economic theory of demand in which S.1 and S.4 are substituted for S.1 and S.4, respectively, and with the convention indicated for the ﬁrst two coordinates as the soft drink model.
2.6. Weak Separability of Preferences
33
Before proceeding further with our discussion, we should take note of the fact that a consumer could, in principle, satisfy all of the assumptions of the standard economic theory of demand, yet not have welldeﬁned preferences over daily consumption. We will ignore this possibility for the moment (more correctly, we will simply assume that assumption S.4 holds), but we will return to this issue in Section 8 of this chapter. There remains a further bit of diﬃculty concerning the nature of the speciﬁcation tested in the BPL experiment, however, stemming from the fact that, in the context of the model, the notion of brand preference is not necessarily welldeﬁned. In the next several sections of this chapter, we will consider several diﬀerent ways of deﬁning what is meant by the statement that a consumer prefers one brand of soft drink over all others. The issue which is our initial concern is this: if we ask a given subject to rankorder Brands 1 and 2 in order of preference, and he or she responds that Brand 1 is preferred to Brand 2, what should we take this to mean? The ﬁrst problem we face in trying to deﬁne a straightforward interpretation of brand preference is that a given subject’s preference for soft drinks might depend upon his or her other consumption for the day. If this possibility sounds slightly farfetched to you, consider the preferences commonly expressed by wine aﬃciandos: white wine, rather than red, with ﬁsh or fowl; red wine with red meat. In particular, in the BPL experiment, any given subject’s preferences over brands of soft drinks might depend upon what he or she was having for lunch. In any event, in the next section we will tackle a formal consideration of this problem.
2.6
Weak Separability of Preferences
As noted in the previous section, in the ‘usual’ case, preferences over, for example, soft drinks will depend upon the consumption of other items. The case in which such dependence does not occur is, by deﬁnition, that in which preferences are separable. Our discussion of separability will be conﬁned to a very simple situation, as compared to other such discussions in the literature; in that we will limit our consideration to the case in which the (exhaustive) list of commodities available can be divided into two groups in such a way that preferences over one of the commodity groups is weakly separable. Other authors deal with many commodity groups, and with other forms of separability.8 Throughout the remainder of this section, we will suppose that X is a subset of Rn , where n ≥ 2, and that X can be written in the form: X = Y × Z, where: Y ⊆ Rk1 and Z ⊆ Rk2 ,
ki ≥ 1, for i = 1, 2, and k1 + k2 = n;
8 For more comprehensive treatments, see Blackorby and Davidson [1991], Blackorby and Russell [1994], and Mak [1986]. For earlier surveys, see Katzner [1970], pp. 27–32 and 78–90; and, for the deﬁnitive treatment of the mathematics of separability, see Blackorby, Primont, and Russell [1978].
34
Chapter 2. Algebraic Choice Theory
and that is a weak order on X. We will then follow the convention of denoting points (commodity bundles) in X by: x = (y, z)
where y ∈ Y and z ∈ Z.
z∗
∈ Z, we deﬁne the conditional preference relation, 2.2 Deﬁnition. For each z∗ , on Y by: y z∗ y ⇐⇒ (y, z ∗ ) (y , z ∗ ). Similarly, given any y ∗ ∈ Y , we deﬁne y∗ on Z by: z y∗ z ⇐⇒ (y ∗ , z) (y ∗ , z ). The proof of the following result is more or less immediate, and will be left as an exercise. 2.3 Proposition. Given any (y ∗ , z ∗ ) ∈ Y × Z, the conditional preference relations, z∗ and y∗ , are weak orders on Y and Z, respectively. 2.4 Example. Let n = 3, X = R3+ , and consider the weak order, deﬁned on X by: x x∗ ⇐⇒ u(x) ≥ u(x∗ ), where the utility function u(·) is deﬁned on X by: u(x) = (x1 x2 )1/2 + (x2 x3 )1/2
for x ∈ X.
Notice that X can be written in the form: X = Y × Z, where: Y = R2+ and Z = R+ (and thus k1 = 2 and k2 = 1). Clearly, if z ∗ ∈ Z = R+ (since Z = R+ , we will use simply ‘z,’ rather than ‘z,’ to denote the second subvector of x), the conditional preference relation, z ∗ , is given by: √ √ √ y z ∗ y ⇐⇒ (y1 y2 )1/2 + ( z ∗ ) y2 ≥ (y1 y2 )1/2 + ( z ∗ ) y2 . Thus, for example, if z ∗ = 1, z ∗ is representable by the conditional utility function: √ u(y; z = 1) = (y1 y2 )1/2 + y2 ; while if z ∗ = 9, z ∗ is representable by the conditional utility function: √ u(y; z = 9) = (y1 y2 )1/2 + 3 y2 . Since u(·; z = 1) is not an increasing transformation of u(·; z = 9), it is clear that the corresponding conditional preference relations are not the same. We can verify this by considering the points: y = (16, 16) and y ∗ = (1, 64).
2.6. Weak Separability of Preferences
35
We have: u1 (y; z = 1) = 16 + 4 = 20 > u(y ∗ ; z = 1) = 8 + 8 = 16, so that for z = 1, we have y z y ∗ . On the other hand, for z = 9: u(y; z = 9) = 28 < u(y ∗ ; z = 9) = 32; and thus, with z ∗ = 9: y ∗ z ∗ y. We see then, that if is the preference relation of a consumer, then said consumer prefers having y = (16, 16) to having y ∗ = (1, 64) if there is only one unit of the third commodity available; but the consumer prefers having y ∗ to having y if it has 9 units of the third commodity. The above example is illustrative of the usual case; we will usually ﬁnd that if, say z = z ∗ , then the conditional preference relations, z and z∗ , will not be the same. Thus, if ‘hours of automotive use’ and ‘shoes’ are included in the ﬁrst group of commodities, and ‘gallons of gasoline per month’ in the second group, we would likely ﬁnd that the marginal rate of substitution between the former two commodities would depend upon the quantity of gasoline available, so that z would be diﬀerent from z∗ , for z = z ∗ . On the other hand, if, for example, the ﬁrst commmodity group contained all foodstuﬀs, while the second group contained all other commodities, then we might ﬁnd the assumption that all marginal rates of substitution between commodities in the ﬁrst group are independent of the quantities in the second group to be a little more plausible. More formally, we might in this latter case expect the following condition to hold. 2.5 Deﬁnition. If X = Y × Z, and Z ∗ is a nonempty subset of Z, we shall say that is weakly separable in y over Z ∗ iﬀ, for each z and z ∗ in Z ∗ , we have: z ≡ z∗ . If is weakly separable in y over Z, we will simply say that is weakly separable on Y . Similarly, if Y ∗ is a nonempty subset of Y , we shall say that is weakly separable in z over Y ∗ iﬀ, for each y and y ∗ in Y ∗ , we have: y ≡ y∗ ; and if Y ∗ = Y , we will say that is weakly separable on Z. In other words, for example, is weakly separable in y if the consumer’s preferences over subbundles from Y are independent of how much z is available to it. We have already looked at a case in which preferences were not weakly separable; the concept will probably be a great deal more clear, however, if we also take a look at a case in which preferences are weakly separable.
36
Chapter 2. Algebraic Choice Theory
2.6 Example. Let X = Rn+ , where n ≥ 2, let k1 and k2 be positive integers such that k1 + k2 = n, and let be representable on X by the CobbDouglas function: n a u(x) = xj j , (2.4) j=1
where: def
aj ≥ 0, for j = 1, . . . , n; and α =
n j=1
aj = 1.
(2.5)
We also deﬁne two subsets of Rk+2 , Z1 and Z2 , by: n Z1 = z ∈ Rk+2  zj > 0 , j=k1 +1
n Z2 = z ∈ Rk+2 
and:
j=k1 +1
zj = 0 ,
respectively. If z ∗ ∈ Z1 , we have: y z∗ y ⇐⇒
k1 j=1
n (yj )aj ·
(zj∗ )aj n
k1 ≥ (yj )aj ·
j=k1 +1
j=k1 +1
j=1
or equivalently, since
n
∗ aj j=k1 +1 (zj )
y z∗ y ⇐⇒
(zj∗ )aj ;
> 0: k1 j=1
(yj )aj ≥
k1 j=1
(yj )aj
Thus, deﬁning the function u1 : Rk+1 → R+ by: u1 (y) =
k1 j=1
(yj )aj ,
(2.6)
we see that if we let ‘1 ’ denote the weak order induced on Y by u1 (·), we have, for each z from Z1 : z ≡ 1 . I will leave as an exercise the task of showing that is also weakly separable in y over Z2 , but that, for each z ∈ Z2 , z is the trivial ordering of Y ≡ Rk+1 . It is obvious that any weak order will be weakly separable in y over Z ∗ if Z ∗ is a singleton; and, for less obvious reasons, the concept of weak separability in y is also not very interesting9 in the case where k1 = 1. Consequently, our interest in this deﬁnition centers around the situation in which k1 > 1, and Z ∗ contains more than one element. However, the known and interesting results concerning weak separability do not generally require these restrictions (except that they may 9 ‘Most’ weak orders of interest in connection with demand theory are weakly separable in a single variable. In particular, you should have no diﬃculty in showing that if is strictly increasing in the ﬁrst variable, or ﬁrst commodity, then it is weakly separable in that variable.
2.6. Weak Separability of Preferences
37
require that Z ∗ = Z). Consequently, there is no reason to complicate our deﬁnition of weak separability by excluding these two cases. Another fact of which we should take notice in connection with weak separability is that the condition is not symmetric. That is, for example, we may ﬁnd that is weakly separable in y over Z, but that it is not weakly separable in z over Y . Our next example illustrates such a case. 2.7 Example. Let X = R4+ , k1 = k2 = 2, Y = Z = R2+ , deﬁne the function u : X → R+ by: u(x) = min{x1 , x2 } · x3 + 1 + x4 ; and let be the weak order induced on X by u(·). It is then very easy to show that, while is weakly separable in y over Z, is not weakly separable in z over Y. Now, one might very well argue that there is no a priori reason to suppose that a given individual’s preferences would be weakly separable in any commodity subgroup; but notice that the theory would be much more useful in application if weak separability were the rule, rather than the exception! In fact, the data requirements in dealing with an exhaustive list of ﬁnelydiﬀerentiated commoties are so enormous that I know of no empirical study which has been undertaken in such a context. Moreover, the notion of separability of preferences has a number of interesting implications; for example to provide suﬃcient conditions for ‘twostage budgeting’ (Blackorby and Russell [1997]), and to some issues of Social Choice (LeBreton and Sen [1999]). Now, the question of immediate concern is, what does this notion of weak separability have to do with deﬁning brand preference? Well, even before attempting a formal deﬁnition of brand preference, we can already note that it is apparent that unless consumer preferences are weakly separable in the soft drink component, the notion of brand preference is going to be more than a bit ambiguous. In other words, if the consumer’s conditional preferences over soft drinks is not independent of other consumption, then it is not immediately apparent how one could unambiguously deﬁne what is meant by the statement that one brand of soft drink is preferred to the others. In order to formally deﬁne a connection between stated brand preference and preferences over commodity bundles, let’s begin by supposing that X = Rn+ . Given that this is the case, we can write: X = R2+ × Rm +, where m = n − 2. We will then use the generic notation: x = (y, z), to denote commodity bundles, x ∈ X, where y ∈ R2+ and z ∈ Rm + . As already suggested, we will suppose throughout the remainder of this section, that each subject’s preference ordering is weakly separable in y over Rm +.
38
Chapter 2. Algebraic Choice Theory
Now, suppose a subject tells us that Brand 1 (of soft drink) is preferred to Brand 2.10 The question is, what can we interpret this to mean? In developing a formal deﬁnition, let’s introduce the notation: 1, 2 as shorthand for the statement (by the subject), “Brand 1 is preferred to Brand 2;” with 2, 1 indicating that the subject’s response is: “Brand 2 is preferred to Brand 1.” Furthermore, making use of the assumption of weak separability, we deﬁne the relation G on R2+ by: yGy ⇐⇒ (y, z ∗ ) (y , z ∗ ),
(2.7)
11 using ‘P ’ and ‘I’ to denote the asymmetric for some ‘reference bundle,’ z ∗ ∈ Rm +; and symmetric parts of G, respectively. Using this notation, we can make the following assumption.
A1. The response 1, 2 by a given subject implies that, in terms of this subject’s preference ordering, we must have: (1, 0), z ∗ (0, 1), z ∗ , for some ‘reference bundle,’ z ∗ ∈ Rm + ; or, more compactly: (1, 0)P (0, 1). Similarly, the response 2, 1 implies that:12 (0, 1)P (1, 0). Now the question is, if the other assumptions of the soft drink model are satisﬁed, and if A1 is correct, will the subjects’ statements about brand preferences correlate perfectly with the choices actually made? A moment’s thought will undoubtedly suﬃce to raise some doubt in your mind about this. The basic diﬃculty (given separability of preferences) is that the can of soft drink chosen as part of the experiment may or may not be the only can of soft drink consumed by the subject in a given day; and it may well be, for example, that for a given subject: (1, 1)P (2, 0), even though: (1, 0)P (0, 1). It would appear that what is happening in the experiment is that something (namely, a can of soft drink) is being added to the subject’s budget for the day. Consequently, the diﬃculty alluded to in the above paragraph may well cause a discrepancy between stated preferences and actual choice unless preferences are additive in the soft drink component; a condition which we will consider in the next section. 10 Remember that for the sake of simplicity we are now supposing that there are only two brands of soft drink available. 11 Notice that, given weak separability of preferences, it makes no diﬀerence what reference bundle is chosen. 12 Recall that each subject was allowed to choose exactly one can of soft drink per day, as a part of the experiment.
2.7. Additive Separability
2.7
39
Additive Separability
The formal diﬃculty with the assumption concerning brand preferences which was discussed in the previous section is that, in the notation of that section, we may ﬁnd that, for some y, y ∈ R2+ , we have yP y , yet, for some x∗ ∈ Rn+ :
(y , 0) + x∗ (y, 0) + x∗ .
In other words, even if a consumer’s preferences are weakly separable in y on all of Z, and we ﬁnd that commodity bundle one is preferred to commodity bundle two, it does not follow that if we add bundle one to her other consumption that she will consider herself better oﬀ than if bundle two had been added to her other consumption. This assertion is veriﬁed by the following. 2.8 Example. Let X = R3+ , and let be the weak order induced on X by the utility function: u(x) = (x1 + 1)2 · (x2 + 1) · (x3 + 1). In this case, if we deﬁne Y = R2+ and Z = R+ , it is easily veriﬁed that is weakly separable in y over all of Z = R+ , and that the conditional preference ordering on Y is representable by the function v : R2+ → R+ given by: v(y) = (y1 + 1)2 · (y2 + 1). We also have: v(1, 0) = 4 > v(0, 1) = 2; so that, if the ﬁrst two coordinates of each commodity bundle represent quantities of Brands 1 and 2 of soft drinks, respectively, then according to our assumption A1, this consumer prefers Brand 1 to Brand 2. However, letting: x∗ = (2, 0, 1), we have:
(1, 0), 0 + x∗ = (3, 0, 1),
and thus u(3, 0, 1) = 32. On the other hand:
(0, 1), 0 + x∗ = (2, 1, 1),
and u(2, 1, 1) = 36, so that: (0, 1), 0 + x∗ (1, 0), 0 + x∗ .
So, the above example indicates a problem with our deﬁnition/assumption A1 even if preferences are weakly separable in soft drinks. However, consider the following deﬁnition.
40
Chapter 2. Algebraic Choice Theory
2.9 Deﬁnition. Suppose is a weak order on a nonempty subset of Rn , X, which is of the form: X = Y × Z, where Y ⊆ Rk1 , Z ⊆ Rk2 , with ki ≥ 1, for i = 1, 2, k1 + k2 = n; and suppose Y is a convex cone. We shall say that is additively separable in Y iﬀ: 1. is weakly separable in y over Z, and: 2. for every y 1 , y 2 ∈ Y , and every x∗ ∈ X, we have:13 y 1 Gy 2 ⇐⇒ (y 1 , 0) + x∗ (y 2 , 0) + x∗ . where G is the conditional weak preference order on Y derived from . 2.10 Example. Let X = R3+ = R2+ × R+ , and let be the weak order induced on X by the function: √ u(x) = x1 + x2 + 2 x3 , and deﬁne Y = R2+ and Z = R+ . It is then easy to see that, in terms of the notation introduced in the above deﬁnition, if y i ∈ Y , for i = 1, 2, and x∗ is any element of X = R3+ , then: y 1 Gy 2 ⇐⇒ (y 1 , 0) + x∗ (y 2 , 0) + x∗ . Consequently, is additively separable in Y in this case.
Our assumption about the meaning of stated brand preference says that if a subject states that brand 1 is preferred to brand 2 (retaining the assumption that there are only two brands to be concerned with), then this means that, in terms of conditional strict preference: (1, 0)P (0, 1). (2.8) This brings us back to the problem mentioned earlier, namely, if (8) holds, does this also mean that: (2, 0)P (1, 1)? Moreover, up to this point we have ignored a possible stronger deﬁnition of preference: we might interpret the statement, “Brand 1 is preferred to Brand 2,” to mean that if y and y are elements of R2+ for which: y1 + y2 = y1 + y2 then: 13
y1 > y1 ⇒ yP y
Notice that, since x∗ is of the form:
x∗ = (y ∗ , z ∗ ), the sum (y i , 0 + x∗ is of the form: (y i , 0 + x∗ = y i + y ∗ , z ∗ ). Therefore, since Y is a convex cone, (y i , 0 + x∗ is an element of X. See Proposition 6.6, in Chapter 6.
2.8. Sequential Consumption Plans
41
So, how does this stronger deﬁnition of brand preference compare with the deﬁnition originally set out? Well, as it turns out, there is no conﬂict between these two deﬁnitions of brand preference if preferences are additively separable in Y , and are representable; for, given some technical qualiﬁcations which needn’t concern us here, under these conditions, there must be a utility function, u, which takes the form: u(y, z) = φ(a · y) + ψ(z),
(2.9)
R k1 ,
where a is a (ﬁxed) semipositive vector in φ : R+ → R+ , and ψ : Z → R+ . I will leave it to you to show that, given that preferences can be represented by a utility function of the form (2.9), if we ﬁnd that (maintaining our assumption that there are only two brands of soft drinks, and thus that k1 = 2): (1, 0)P (0, 1),
(2.10)
then, for any two vectors, y, y ∈ Y , we have that: [y1 + y2 = y1 + y2 & y1 > y1 ] ⇒ yP y .
(2.11)
Thus, if preferences over daily consumption are additively separable in the soft drink component, and are constant from day to day, it would appear that either interpretation of the meaning of brand preference which we have considered would imply perfect agreement between stated brand preference, and actual choice in the soft drink experiment. However, this brings us back to the issue of whether it is reasonable to suppose that preferences over daily consumption remain constant from day to day. We will consider this question in the next section.
2.8
Sequential Consumption Plans
It was noted in Section 5 that a consumer might satisfy all the assumptions of the standard economic theory of demand, and yet not have welldeﬁned preferences over daily consumption. Thus, a given consumer may satisfy all the assumptions of the standard economic theory of demand, and yet not satisfy the Soft Drink Model; at least not in the sense of having invariant preferences from one day to the next. In order to establish this fact, and to explore the reasons for it, we will develop a model in this section which explains the standard economic theory of demand. Because this model explains the standard theory (that is, its assumptions imply those of the standard theory, with the appropriate speciﬁcations of the primitives) it is, in eﬀect, a special case of the standard theory; and, it should be emphasized, there are other special cases of the standard economic theory of demand in which the consumer’s daily preferences would be welldeﬁned. On the other hand, the model to be developed here [which we will call the sequential consumption plan (SCP) model] seems suﬃciently plausible and interesting as to merit the time which we will spend on its development. In order to motivate our discussion, let’s begin by considering a consumer that satisﬁes all of the assumptions of the standard theory. Let us further assume, for
42
Chapter 2. Algebraic Choice Theory
the sake of simplicity, that when said consumer chooses the bundle x∗ from the budget set, B, this means that it purchases the bundle x∗ at the beginning of the month. In terms of the context in which the standard theory was formulated, this means that the consumer makes x∗ available for its consumption14 over the month to come. The next question which arises, however, is how, more speciﬁcally, at what rate, will the consumer choose to consume the bundle x∗ ? We would probably be quite surprised if the consumer proceeded to consume x∗ at the rate: z ∗ ≡ (1/30)x∗ , per day (assuming, for the sake of convenience, that there are 30 days in a month); but isn’t this exactly what would happen if the consumer’s preferences over daily consumption were exactly the same from day to day? Let’s take a look at this question from a bit diﬀerent point of view. Suppose that at the beginning of each month, our consumer considers alternative sequences of consumption of the form: z = (z 1 , z 2 , . . . , z 30 ), where z t ∈ Rn+ denotes planned consumption of the tth day (t = 1, . . . , 30). Suppose further that our consumer’s preferences over such sequences of consumption plans constitutes a weak order, G, on the set Z of admissible sequences of this type, where Z is a subset of R30n + . In fact, will suppose that Z is of the form: Z=
30
Zt
(2.12)
for t = 1, . . . , 30.
(2.13)
t=1
where: Zt = Rn+
In particular, we will suppose that Zt , the feasible consumption set for day t, is invariant from day to day. For the sake of convenience, we will also suppose that G is representable by a continuous utility function, U (·); that is, we will suppose that U : Z → R, and satisﬁes, for all z, z ∈ Z: zGz ⇐⇒ U (z) ≥ U (z ). We can then relate this situation to the standard theory in the following way. Deﬁne the function u : Rn+ → R by: 30 u(x) = max U (z)  z ∈ Z & zt ≤ x . t=1
(2.14)
In other words, u(x) is the maximum utility which could be obtained from an admissible sequence, z, whose daily components add up to a bundle less than or 14 There is a bit of confusion between stock and ﬂow going on here, which, it is hoped, will not cause you undue confusion. Strictly speaking, we should distinguish between x∗ , which is a ﬂow, and the consumer’s inventory of commodities at the beginning of the month, which is a stock. However, since xj is measured in terms of quantity per month, the real number xj will here also be the quantity of the j th commodity on hand at the beginning of the month.
2.8. Sequential Consumption Plans
43
equal to x.15 The function, u(·) deﬁned in (2.14) obviously induces a weak order, , on Rn+ . Consequently, this model is actually a special case of the standard economic theory of demand. The next question, however is this: under what conditions will the corresponding daily preferences be invariant from day to day? After our discussion in Section 6, it should be clear to you that it is not apparent that we can speak of ‘daily preferences’ in an unambiguous fashion here unless the weak order, G, is separable. To be more speciﬁc, consider an arbitrary t ∈ {1, . . . , 30}, and denote values of z = (z 1 , . . . , z 30 ) by: z = (z −t , z t ), where: z −t = (z 1 , . . . , z t−1 , z t+1 , . . . , z 30 ). Notice that, for a ﬁxed value of z −t , G induces a weak order on Zt = Rn+ , Gt (z −t ), deﬁned by: (2.15) z t Gt (z −t )z t ⇐⇒ (z −t , z t )G(z −t , z t ). However, if z ∗−t = z −t , there is no reason to suppose that we will necessarily have: Gt (z −t ) ≡ Gt (z ∗−t ). To see the point of this statement, simply ask yourself whether or not your preferences for pizza versus meatloaf for dinner tonight might not be diﬀerent if you had had pizza each night for the preceding twentynine days than would be the case if you had had meatloaf for each of the preceding twentynine dinners. In the special case in which Gt (z −t ) is independent of the value of z −t , for t = 1, . . . , 30, we shall say that G is weakly separable in daily consumption. However, notice that, even if G is weakly separable in daily consumption, it will not necessarily be the case that: Gt = G1
for t = 2, . . . , 30.
(2.16)
In other words, daily preferences may be diﬀerent even if G is weakly separable in daily consumption. If, on the other hand, G is weakly separable in daily consumption, and, in addition satisﬁes (2.16), we shall say that G is stationary. It is no doubt abundantly clear to you that there are a number of subtle and diﬃcult problems connected with the analysis and interpretation of this sort of model, and that we have no more than begun to analyze these problems here. However, our goal was simply to introduce the general idea of the SCP model, and to point out some of the reasons why the appropriate speciﬁcation of the ‘unit of time’ is so important in the empirical testing of the general algebraic choice model. Rabin [1998] provides a very interesting little example which is of particular interest in connection with the SCP model. I quote his introduction as follows. Say you eat at one of two restaurants every night, either Blondie’s or Fat Slice. You enjoy Fat Slice more, but because you also enjoy variety, your utility each evening is as follows. 15 It can also be shown that, under the assumptions being employed here, the function u(·) will be (well deﬁned and) continuous on Rn +.
44
Chapter 2. Algebraic Choice Theory Utility Utility Utility Utility
from from from from
Fat Slice = 7 if you ate at Blondie’s last night, Fat Slice = 5 if you ate at Fat Slice last night, Blondie’s = 4 if you ate at Fat Slice last night, Blondie’s = 3 if you ate at Blondie’s last night.
Suppose now that you have eaten at Blondie’s last night, and let’s just consider consumption decisions over a ﬁveday horizon. Obviously, your utilitymaximizing choice for tonight is to dine at Fat Slice; however, what about tomorrow? If, in fact, your preferences satisfy the assumptions of the SCP model, you will dine at Blondie’s tomorrow, at Fat Slice the next night, and so on; since this provides a total utility of 29 for the ﬁve nights together. On the other hand, if you only consider tomorrow’s preferences tomorrow, and so on, you will dine at Fat Slice each of the following four nights, since this provides a marginal utility of 5 for each night. However, your total utility for the ﬁve nights is then only 27! In general, if you alternate consumption between the two restaurants from night to night, you obtain an average utility of 5.5; whereas if you always eat at Fat Slice, your average utility will be only 5.16 The case discussed here is, I’m afraid, a situation similar to what a lot of us face in real life; a similar sort of anomaly arises in connection with procrastination, for example. For the classic development of this idea, see Phelps and Pollak [1968]. In general this sort of example highlights the possibility of an inconsistency between planning for the future and carrying out those plans; a theoretical possibility which seems to have ﬁrst been pointed out by Strotz [1955].17 We will not discuss the theory of intertemporal choice further here, but let me recommend Koopmans [1972a, 1972b], Goldman [1979, 1980], and Loewenstein and Prelec [1992].
2.9
The BPL Experiment Reconsidered
In Section 5, we noted that a subject might satisfy all of the assumptions of the standard economic theory of demand, and Assumption A1 as well, yet nonetheless display inconsistencies between stated brand preference and actual choice in the BPL experiment. In the last three sections, we have discussed a number of factors which could lead to such inconsistencies in this context; moreover, in the process, we have implicitly developed a special case of the standard economic theory of demand which would predict perfect agreement between stated brand preference and actual choice. This special theory occurs when the subject satisﬁes the assumptions of the standard economic theory of demand18 (call this Assumption A0), Assumption A1 from Section 6, and Assumptions A2 and A3, deﬁned as follows. A2. The subject satisﬁes the stationary SCP model;, and, denoting the common value of the daily preferences, Gt , by ‘G,’ the weak order, G: A3. is additively separable in the soft drink component. 16
This is an example of ‘melioration.’ See Herrnstein and Prelec [1992] and Rabin [2002]. For more recent discussions of this sort of inconsistency, see Goldman [1979, 1980]; and for an excellent general discussion of anomalies connected with intertemporal choice, see Loewenstein and Prelec [1992]. 18 Assuming that each of the subjects is a 1person household. 17
2.10. Probabilistic Theories of Choice
45
If the subject satisﬁes Assumptions A0–A3, then, as you can readily verify for yourself, there should be perfect agreement between stated brand preferences and actual choice. Since the results of the BPL experiment do not substantiate this agreement, we can regard the BPL experiment as having rejected the joint hypothesis: H0 ≡ A0 & A1 & A2 & A3. Thus we can conclude that at least one of the Assumptions A0–A3 is not satisﬁed, at least for the population sampled in the experiment. The experiment does not, of course, tell us which of the Assumptions A0–A3 is false; although from other experiments and/or statistical studies, we might be inclined to believe that A0 is true, and therefore that the culprit must be one or more of Assumptions A1–A3. Further experiments may yet shed some light on which of these latter assumptions, if any, is empirically tenable. However, just now the broader lesson to be gained from our study of the BPL experiment is that, even if we consider the standard economic theory of demand to be empirically correct, we must guard against the presumption that this implies that the theory is empirically correct under alternative speciﬁcations of the primitives of the theory.
2.10
Probabilistic Theories of Choice
Returning to the general algebraic choice model of Section 2, and recalling our convention of denoting the asymmetric and symmetric parts of by ‘’ and ‘∼,’ respectively; we can give an operational deﬁnition of (strict preference) as follows: ‘x y’ means that if the decisionmaker were presented with repeated choices between x and y, he/she would always choose x. (2.17) If we introduce the notation ‘p(x, y)’ to denote the probability that x would be chosen over y, given that only x and y are available (so that the budget set is {x, y}), (2.17) is equivalent to the statement: x y ⇐⇒ p(x, y) = 1.
(2.18)
While it is a bit harder to come up with a satisfactory operational deﬁnition of indiﬀerence along these lines, the following is certainly a possibility: x ∼ y ⇐⇒ p(x, y) = 1/2.
(2.19)
If we accept (2.18) and (2.19) as operational deﬁnitions of and ∼, then the assumption made implicitly in the algebraic choice model is that for each x, y ∈ X, we have: p(x, y) ∈ {0, 1/2, 1}. In contrast, a probabilistic theory of choice assumes only that p(x, y) ∈ {0, 1}. 2.11 Deﬁnition. Let X be a nonempty set. We shall say that a function, p : X × X → [0, 1] is a binary preference probability (on X) iﬀ p(·) satisﬁes: ∀(x, y) ∈ X × X : p(x, y) + p(y, x) = 1.
46
Chapter 2. Algebraic Choice Theory
Probabilistic, as opposed to algebraic theories of choice are the rule, rather than the exception, in psychology. As to the reasons for this, we can probably do no better than to quote from a classic text in the ﬁeld (Coombs, Dawes, and Tversky [1970, p. 148]): Inconsistency is one of the basic characteristics of individual choice behavior. When faced with the same alternatives, under seemingly identical conditions, people do not always make the same choice. Although the lack of consistent preferences may be attributable to factors such as learning, saturation, or changes in taste over time, inconsistencies exist even when the eﬀects of such factors appear negligible. One is led, therefore, to the hypothesis that the obeserved inconsistency is a consequence of an underlying random process. The randomness may reﬂect uncontrolled momentary ﬂuctuations such as attention shifts, or it may correspond to a choice mechanism that is inherently probabilistic. Be that as it may, the most natural way of coping with inconsistent preferences is by replacing the deterministic notion of preference by a probabilistic one.. . .
Certainly a great many choice experiments have revealed an inconsistency in stated binary preferences. To quote Starmer [2000, p. 374]: . . . a common ﬁnding is that individuals confronted with the same pairwise choice problem twice within a given experiment frequently give diﬀerent responses on the two occasions. Stochastic choice is more convincing than indiﬀerence as an account for such intrinsic variability. . .
Starmer goes on to cite a number of such recent experimental studies, among them Hey and Orme [1994] and Ballinger and Wilcox [1997], in which between onequarter and onethird of subjects ‘switch’ preferences on repeated questions. Despite all of this experimental evidence, it cannot fairly be said that the algebraic choice model has been obviously refuted, for several reasons. For us, the most important of these reasons is that the objects over which choice has been made in the experiments cited have not been those appearing in the economic theory of demand; namely commodity bundles. It is entirely possible that preferences over gambles, for example, are much less consistent than are preferences over commodity bundles; and all of the studies cited here have involved choices over uncertain prospects.19 In any event, in the remainder of the course, our focus will be upon algebraic choice theory, since the concensus of the profession seems to be that this theory is appropriate for the issues to be examined in our remaining discussion. Things are a bit diﬀerent when it comes to applied work and/or forecasting, however. As noted by McFadden [2001], before the 1960’s empirical applications of demand theory generally proceeded by positing the existence of a ‘. . . representative agent [consumer], with marketlevel behavior given by the representative agent’s behavior writ large.’ (McFadden [2001, p. 351]). Deviations from preferencemaximizing behavior in the market as a whole were then attributed to an error term, generally assumed to be additive, with zero mean. All of this is consistent, to a certain extent, with the fact that the focus of interest in economics is upon market behavior, rather than individual choice behavior; at least 19 It should be noted, however, that choice over commodity bundles often involves risk; for example, in assessing the desirability of a new product.
2.11. Are Preferences Total?
47
in the theory of demand.20 However, the conditions under which aggregate demand will exhibit the same qualitative properties as are implied for individual consumer behavior by the economic theory of demand are extremely restrictive; as we will demonstrate in a later chapter. In contrast, McFadden has pioneered the theoretical and statistical development of methods for extending estimates of individual behavior to arrive at estimates of market demand. An integral part of such methods is the assumption that consumer preferences can be represented by a wellbehaved function of the (p. 357) ‘. . . characteristics of the consumer, and consumption levels and attributes of goods.’ Diﬀerences in choice (randomness) are then attributed to unobserved characteristics. This technique has been particularly eﬀective in predicting choice over discrete alternatives. For details, see McFadden [2001]
2.11
Are Preferences Total?
Suppose there are exactly n commodities available, and that the consumers’ budget sets are subsets of Rn , but that consumers’ rankings of these commodity bundles may depend upon a vector of ‘environmental variables,’ y; which might include such variables as season of the year, the expected mean temperature over the planning period, the number of interesting concerts scheduled in the area during the planning period, and so on. More precisely, suppose the environmental variables, y, can take on any value in some set Y . It then seems natural to suppose that consumers have preference orderings, , over the set: def
Z = X × Y.
(2.20)
We then have a situation analogous to that discussed in Section 6. In particular, if a consumer’s preference relation, , is a weak order, then for each y ∈ Y , induces a weak order, y , on X, deﬁned by: x y x ⇐⇒ (x, y) (x , y);
(2.21)
y∗,
we may have y =y∗ . however, for y = Thus in such a context we may ﬁnd that in one period, the consumer chose x1 ∈ X at prices p1 , while in the next period it chose x2 ∈ X at prices p2 , where: p1 · x1 > p1 · x2
(2.22)
p2 · x2 > p2 · x1 .
(2.23)
while: This is, of course, apparently inconsistent behavior, in that (2.22) indicates that (since x1 was chosen when x2 would have cost less); x1 x2 ; while (2.23) would appear to indicate that x2 x1 . In the context of the present discussion, however, there is nothing inconsistent about such a situation; for the 20 As we shall ﬁnd later, however, the theory of individual choice which we have been developing here, and will continue to develop in the next chapter, will be of considerable value in our analysis of social, or group choice.
48
Chapter 2. Algebraic Choice Theory
environmental variable prevailing in period one (which we will denote by ‘y 1 ’) may be diﬀerent from that in period two (call in ‘y 2 ’), and it may be that: (x1 , y 1 ) (x2 , y 1 ), and yet: (x2 , y 2 ) (x1 , y 2 ). Of course, this diﬃculty disappears if is weakly separable in x on Y , but this is an assumption that is very diﬃcult to justify as a universal rule; most consumers’ preferences for ‘cutoﬀs’ versus coats, or for swim suits versus ski pants, are likely to be quite diﬀerent in June than in January. However, if the consumer’s preference relation varies from month to month, the standard economic theory of demand loses virtually all of its predictive power; since, as was pointed out by Samuelson some time ago (Samuelson [1938], [1947], [1948]), the entire operational content of the standard economic theory of demand is bound up in revealedpreference conditions like: p1 · x1 ≥ p1 · x2 ⇒ p2 · x1 ≥ p2 · x2 (2.24) If x1 and x2 are the commodity bundles chosen by the consumer in two successive periods, however, such a statement will hold21 (that is, the implication will be logically correct) only if the consumer’s preference relation is the same in the two periods.22 There are several ways in which we might attempt to circumvent the diﬃculty under discussion here. The simplest way out of it, which constitutes the reason for the title of this section, is to drop the assumption that the consumer’s preference relation is total. Why is this? Well, if is a weak order on X × Y , we can use it to deﬁne a relation, G, on X by: xGx∗ ⇐⇒ [(∀y ∈ Y ) : (x, y) (x∗ , y)].
(2.25)
It is easy to prove that if is a weak order on X × Y , then G will be reﬂexive and transitive, but, in general, will not be total on X. If fact, G will be total if, and only if, is weakly separable in x on Y . On the other hand, there is a problem with this approach; speciﬁcally, in relating this relation, G, to demand behavior. If we return to the framework set out in Section 2, with a family of budget sets, B, we see that we cannot characterize the consumer’s demand correspondence, h, by: h(B) = {x ∈ B  (∀x ∈ B) : xGx }
for B ∈ B.
(2.26)
21
Given certain additional assumptions (for example, strict quasiconcavity) on the preference relation, we can strengthen (2.24) to: [p1 · x1 ≥ p1 · x2 & x1 = x2 ] ⇒ p2 · x1 > p2 · x2 . 22 Even if we assume that the consumer’s preferences change from month to month, however, it might be well worth testing the hypothesis that these changes cycle with the seasons of the year. Thus, for example, we might test the hypothesis that a consumer’s preferences were the same in June, 2000, as in June, 2001; compare July, 2000, with July, 2001, and so on.
2.11. Are Preferences Total?
49
In fact, the set deﬁned in (2.26) will be empty, for most budgets B ∈ B; and thus the consumer’s actual choice will not generally be an element of the set. Suppose we instead deﬁne a correspondence, h, by / B} for B ∈ B; h(B) = {x ∈ B  (∀x ∈ X) : x P x ⇒ x ∈
(2.27)
where P is the asymmetric part of G. With this deﬁnition we have almost the same problem; in general, the set deﬁned in (2.27) will not be empty,23 but for a given budget, B, the consumer may make a choice which is not an element of the set h(B) deﬁned in (2.27). I will leave the task of explaining why this may happen under the present assumptions as an exercise. Suppose, however, that instead of deﬁning the relation P as the asymmetric part of G, we deﬁne a new relation P by: xP x ⇐⇒ (∀y ∈ Y ) : (x, y) (x , y);
(2.28)
and then take (2.27) to be our deﬁnition of the correspondence h, using this new deﬁnition of P . A moment’s reﬂection should then suﬃce to convince you that the consumer’s choice will now always be an element of h(B), for each B ∈ B. The natural question to ask about this, however, is whether a theory based on these assumptions and this deﬁnition has any real predictive power. We will consider this question further in the next chapter. In the meantime, let’s take a look at some alternative approaches to the solution of this environmental variable problem which are also of interest. The ﬁrst alternative, and perhaps that most consistent with the literature on demand theory, is to reinterpret the notion of a commodity. Thus, we might think of ‘food,’ ‘clothing,’ ‘housing,’ ‘recreational goods,’ and so on, as individual commodities; rather than using the speciﬁcation S.3 (of Section 5) of what we have been calling the ‘standard economic theory of demand.’ The point of this change is that it is a very intuitively appealing notion that preferences over broadlydeﬁned commodity groups like ‘food’ versus ‘clothing,’ might exhibit much less variability (over seasons of the year, in particular) than preferences for more narrowlydeﬁned items, like ‘cutoﬀs’ versus ‘chinos,’ ‘coldcuts’ versus clam chowder, or beer versus (hotbuttered) rum. Unfortunately, this leaves us with the very messy scientiﬁc problem of determining exactly which such speciﬁcation of the primitive ‘commodity’ will work; and with the concomitant problem of how to measure such conglomerates. In all fairness, it would appear that we would have to admit that the economics profession has not succeeded in fully solving this problem. Another interesting way of handling the diﬃculty under consideration here is to treat the variable y ∈ Y as random. If the weak order on X × Y is representable by a utility function, U (x, y), one is then led to the notion of random utility (over X). This is one of two alternative assumptions underlying probabilistic theories of choice. We will not be able to pursue this topic further in this course, but let me recommend the book by Train [1986] as a very readable introduction to both the theory and estimation techniques which have been developed in this area.24 23
We will provide a justiﬁcation for this statement in Chapter 4. See also McFadden [2001]. It is he who developed most of the theory and many of the estimation techniques which are used in this area. 24
50
Chapter 2. Algebraic Choice Theory
Yet another way of reacting to this diﬃculty is to attempt to systematically exploit this variability in preferences. After all, as economists we are fundamentally more concerned with market demand than with individual consumer demand, and to the extent that individual preferences change from month to month (or from quarter to quarter) in a systematic manner which is much the same for diﬀerent individuals (and casual observation suggests that there is at least some basis for believing this to be the case), we may be able to develop a much more useful and powerful theory of market demand because of this variability than would be possible without it.
2.12
Are Preferences Transitive?
Over the years, a number of writers have questioned the correctness of assuming that preferences are transitive. In this section, we will consider four kinds of objections, or diﬃculties, which have been raised regarding the transitivity assumption.
2.12.1
‘Just Noticeable Diﬀerence,’ or ‘Threshold Eﬀects’
More than sixty years ago, W. E. Armstrong [1939] objected to the transitivity assumption on the grounds that an alternative, x, may be indiﬀerent to y, y may be indiﬀerent to z; and yet x may be preferred to z. The reasoning behind this sort of contention is that the diﬀerence between x and y may be too small to notice, and such may also be the case as regards y and z; yet the diﬀerence between x and z may nonetheless be suﬃciently great as to result in the preference of x over z. As a theoretical device to deal with this and similar phenomena in the area of psychophysics, R. Duncan Luce [1956] developed the notion of a semiorder. An example of this sort of binary relation was presented in Chapter 1, but we will brieﬂy review the example here.25 Let f : X → R, and suppose that X = Rn+ , and that a consumer’s preferences on X satisfy: (2.29) x x ⇐⇒ f (x) ≥ f (x ) − δ, where δ is a positive constant. I will leave it as an exercise to show that the asymmetric part of (the strict preference relation) is given by: x y ⇐⇒ f (x) > f (y) + δ;
(2.30)
while the symmetric part (the indiﬀerence relation) is given by: x ∼ y ⇐⇒ f (y) − f (x) ≤ δ.
(2.31)
The constant, δ, is therefore identiﬁed with the threshold level of perception, or ‘just noticeable diﬀerence.’ Thus x and y are indiﬀerent if the absolute value of the diﬀerence between the value of f at x and its value at y is not greater than δ; whereas, if, say, f (x) > f (y) + δ, then x is preferred to y. We showed in Chapter 1 that, while strict preference is transitive in this case, indiﬀerence is not. 25 The reader interested in learning more about semiorders should be warned that later authors have used a set of axioms diﬀerent (logically equivalent, but somewhat more transparent) from those originally formulated by Luce. Luce’s and Suppes’ [1965] survey article, or Fishburn’s [1970] book both contain good, and extensive introductions to the concept.
2.12. Are Preferences Transitive?
2.12.2
51
Decision Rules Based On Qualitative Information
Suppose a prospective home buyer has a choice of three houses, labeled ‘A,’ ‘B,’ and ‘C;’ all of which are selling at the same price. Suppose further that our home buyer is interested in three attributes of a home, other than price: square feet of ﬂoor space, location (convenience with respect to school, shopping centers, and so on), and appearance; and ranks these three houses with respect to these characteristics as follows. A B C
Floor Space Best Middle Poorest
Location Middle Poorest Best
Appearance Poorest Best Middle
Table 2.3: Housing Attributes.
Consider the following decision rule: x y ⇐⇒ x is better than y in at least two characteristics; and show, on the basis of this decision rule, that we have:26
A B, B C, and C A. In an experiment involving 62 college students, K. O. May [1954] examined a similar problem. We quote from his description of the experiment (p. 6). . . . The alternatives were three hypothetical marriage partners, x, y and z. In intelligence they ranked xyz, in looks yzx, in wealth zxy. The structure of the experiment was not explained, but subjects were confronted at diﬀerent times with pairs labeled with randomly chosen letters. On each occasion, x was described as very intelligent, plain looking, and well oﬀ; y as intelligent, very good looking, and poor; z as fairly intelligent, good looking, and rich. All prospects were described as acceptable in every way, none being so poor, plain, or stupid as to be automatically eliminated. . . . Parts of the experiment were repeated to test for consistency and possible capriciousness. The results, as well as the behavior of the subjects, indicated practically no random element in the choices. In terms of the probability deﬁnition of preference given in the ﬁrst section, it was evident that 0 and 1 were the only possible probabilities, and that repeated trials were not necessary. Since indiﬀerence is ruled out, there are six possible orderings and two circular patterns designated by xyzx and xzyx. If group preferences be deﬁned by majority vote, the results indicate a circular pattern, since x beat y by 39 to 23, y beat z by 57 to 5, and z beat x by 33 to 29. The number of individuals having each of the possible patterns was xyz: 21; xyzx: 17; yzx: 12; yxz: 7; 26 Before leaving this discussion, you may wish to consider the following question. Suppose that, in a given local housing market, all prospective buyers are interested in the attributes, or characteristics, discussed here, and only those (other than price). Would you expect to ﬁnd any conﬁguration very diﬀerent from that shown in Table 3 for any three houses which were all selling at the same price?
52
Chapter 2. Algebraic Choice Theory zyx: 4; xzy: 1; zxy: 0; xzyx: 0. The intransitive pattern is easily explained as the result of choosing the alternative that is superior in two out of three criteria. The orders xyz and yzx seem to have resulted from giving heavier weight to intelligence and looks, respectively. The four who chose inversely with respect to intelligence (zyx) were men, and may indicate the extent of male fear of intelligent women.. . . What is the signiﬁcance of this experiment? Of course it does not prove that individual patterns are always intransitive. It does, however, suggest that where choice depends on conﬂicting criteria, preference patterns may be intransitive unless one criteria dominates.. . .
2.12.3
Priorities and Measurement Errors
The following is an example of what Tversky [1969] has dubbed a ‘lexicographic semiorder.’ Suppose a decisionmaker is attempting to rankorder a group of college applicants, having available only three pieces of information: their examination scores on tests of intelligence, emotional stability, and social facility. Our decisionmaker decides that for him the order of importance of these scores is the order in which they are listed above. On the other hand, he also recognizes the fact that all of these tests are subject to a great deal of measurement error; thus he arrives at the following rule (perhaps based upon his perception of the standard errors of the testing techniques). If individual 1’s intelligence score is more than 3 points higher than individual 2’s score, he will rank 1 above 2, whatever their remaining two scores. If their intelligence scores diﬀer by no more than 3 points, he will look at their emotional stability examination scores. If this diﬀerence is more than 6 points (this examination being somewhat less reliable than the ﬁrst), he will rank the individual having the higher score above the other, and ignore the third score. Finally, if the emotional stability scores for the two individuals diﬀer by no more than 6 points, he will look at the third score; ranking 1 above 2 if 1’s social facility score is 9 or more points higher than 2’s. Formally, if we denote the intelligence, emotional stability, and social facility scores of applicant x by ‘Ix ,’ ‘Ex ,’ and ‘Sx ,’ respectively, the decision rule we have just described verbally supposes that there exist positive constants, δ1 , δ2 , and δ3 such that the decisionmaker will order pairs of candidates, x and y, in the following way: ⎧ ⎪ or ⎨Ix > Iy + δ1 x y ⇐⇒ Ix − Iy  ≤ δ1 and Ex > Ey + δ2 , (2.32) or ⎪ ⎩ Ix − Iy  ≤ δ1 , Ex − Ey  ≤ δ2 and Sx > Sy + δ3 . Of course, in the special case which we have described verbally, the decision rule is of the form (2.32), with: δ1 = 3, δ2 = 6, and δ3 = 9. Now suppose that the candidates have the following test score proﬁles set out on the next page. If our decisionmaker only makes adjacent pair comparisons of the candidates (that is, compares a with b, b with c, and so on) he will end up ranking the candidates inversely with respect to their intelligence scores; despite the fact that the
2.12. Are Preferences Transitive?
53 I 69 72 75 78 81
Applicant a b c d e
E 84 78 72 66 60
S 75 65 55 45 35
Table 2.4: Test Scores.
decision rule ostensibly gives ﬁrst priority to intelligence! Furthermore, as in the case of the semiorder considered earlier, the decisionmaker will exhibit intransitive indiﬀerence. However, in contrast to the simple semiorder considered earlier, this decisionmaker will also display intransitive strict preference.27
2.12.4
Group Decisions: The Dr. Jekyll and Ms. Jekyll Problem
Yet a further problem in our economic theory of choice arises from the fact that we usually specify our individual decisionmaking unit (‘individual consumer’) as being, or corresponding to ‘individual household’ in the census data. The diﬃculty with this is that most households contain more than one individual, and the collective choices of a group may not be transitive even though all the individuals in the group have a transitive ordering over the alternatives available, as the following example demonstrates.28 Suppose three individuals, A, B, and C, rank three alternatives (for example, political candidates, proposed budgets, etc.) in the following way, and that majority voting is to be used to rankorder the alternatives. If we denote group preference A x y z
B y z x
C z x y
Table 2.5: A Preference Proﬁle. by ‘P ,’ it is easy to show that in this case we have: xP y, yP z, and zP x. An interesting aspect of this sort of situation (although it has no necessary connection with the question of whether a household’s preferences can reasonably be assumed to be transitive) is that if a group choice of one of the three alternatives is to be made by pairwise elimination: 27 A diﬀerent sort of systematic violation of transitivity has been observed in experiments involving choice over risky prospects. For an excellent summary, as well as some particularly interesting experimental results, see Loomes, Starmer, and Sugden [1991]. 28 Notice the formal similarity between this example and that used in the May [1954] experiment discussed earlier.
54
Chapter 2. Algebraic Choice Theory one pair of alternatives is compared ﬁrst, and the alternative which wins the majority vote is then voted upon visavis the third alternative,
then the alternative actually chosen depends upon which pair was compared ﬁrst. For example, given the preference proﬁle in Table 5, if x and y are compared ﬁrst, we have xP y; so that x would then be compared with z. Since zP x, the ﬁnal choice (that is, the alternative actually chosen) would be z. On the other hand, if we ﬁrst compared y and z, the ﬁnal choice would be x, and so on. Returning to the issue of whether a multipleperson household will display transitive preferences, consider, for a moment, the simplest sort of nonsingleperson household; namely one containing just two persons, whom we will suppose are husband and wife. Suppose further that each of them has a welldeﬁned preference relation, Gi (i = 1 for the wife, and i = 2 for the husband), over the household consumption set, X, and denote the asymmetric part of Gi by ‘Pi ,’ for i = 1, 2. Then if x∗ is the commodity bundle chosen from b(p, w) for some period, it seems quite unlikely that this choice will satisfy: (2.33) ∀x ∈ b(p, w) : x∗ Gi x, for either i = 1 or i = 2 [although a lot of Ms. 1’s and Mr. 2’s might claim that (2.33) is satisﬁed with i = 2 or i = 1, respectively]; rather, it seems that some sort of compromise solution is likely to be reached. However, regardless of how Ms. 1 and Mr. 2 go about reconciling their diverse preferences, it seems likely that the ﬁnal choice, x∗ , will be such that x∗ ∈ b(p, w) and will be such that there exists no ∈ b(p, w) such that: x P2 x ∗ . P1 x∗ & x (2.34) x The considerations of the above paragraph suggest that it may be worthwhile to pursue the following approach. We ﬁrst deﬁne the binary relation, P , on X by: xP x ⇐⇒ [xP1 x & xP2 x ].
(2.35)
The relation P can be thought of (and we will often refer to it as) the household’s unanimity ordering. Furthermore, it makes sense to think of P as the household’s strict preference relation, in that, if we accept the argument of the preceding paragraph, then the household will behave as if it attempted to maximize the binary relation, P ; that is, the household will, given a pricewealth pair (p, w), choose a commodity bundle x∗ ∈ b(p, w) satisfying: (∀x ∈ X) : xP x∗ ⇒ p · x > w.
2.13
Asymmetric Orders
Because of the diﬃculties discussed in the previous two sections, we will wish whenever possible to consider a more general form of ordering than a weak order as our ‘model’ of consumer preferences. Speciﬁcally, we will whenever possible (especially in developing the theory of welfare economics), suppose only that a consumer’s strict preference relation is an asymmetric order; where we deﬁne this as follows.29 29
Recall that the asymmetric part of a weak order is also negatively transitive.
2.13. Asymmetric Orders
55
2.12 Deﬁnition. Let P be a binary relation on a nonempty set, X. We shall say that P is an asymmetric order iﬀ P is asymmetric and transitive. Making use of this deﬁnition, it is easy to prove the following. 2.13 Proposition. Suppose y is an asymmetric order on the nonempty set X, for each y ∈ Y , and deﬁne the binary relation, P on X by: xP x ⇐⇒ [(∀y ∈ Y ) : x y x ]. Then P is also an asymmetric order on X. Thus it follows from this proposition that the binary relations deﬁned in equation (2.28) of Section 11, equation (2.30) of Section 12, and equation (2.35) of Section 12, are all asymmetric orders (given the assumptions of the respective sections). Consequently, we can see that there is a real gain in generality in assuming, wherever possible, that a consumer’s (strict) preference relation, P , is an asymmetric order; and, correspondingly, that, given a budget space, X, B, that the consumer’s demand correspondence takes the form: h(B) = {x ∈ B  (∀x ∈ X) : x P x ⇒ x ∈ / B} for B ∈ B.
(2.36)
In the next chapter, we will investigate the implications of these assumptions. In the meantime, let’s take a look at the way in which our continuity assumptions need to be reformulated in order to apply to asymmetric orders. 2.14 Deﬁnitions. Let X be a nonempty subset of Rn , and let P be an asymmetric binary relation on X. We shall say that P is: 1. upper semicontinuous on X iﬀ, for each x, y ∈ X, if xP y, then there exists a neighborhood, N (y), such that, for all y ∈ N (y) ∩ X, xP y . 2. lower semicontinuous on X iﬀ, for each x, y ∈ X, if xP y, then there exists a neighborhood, N (x), such that, for all x ∈ N (x) ∩ X, x P y. 3. continuous on X iﬀ it is both upper and lower semicontinuous on X. 4. strongly continuous on X, iﬀ, for each x, y ∈ X, if xP y, then there exist neighborhoods, M (x) and N (y), respectively, such that, for all x ∈ M (x) ∩ X, and for all y ∈ N (y) ∩ X: x P y . One can prove the following relationships. 2.15 Theorem. Let X be a nonempty subset of Rn , and let P be an asymmetric binary relation on X. If P is also negatively transitive, and if we let ‘G’ denote the negation of P, then: 1. P is upper semicontinuous on X if, and only if, G is upper semicontinuous on X. 2. P is lower semicontinuous on X if, and only if, G is lower semicontinuous on X. 3. P is strongly continuous on X if, and only if, G is continuous on X.
56
Chapter 2. Algebraic Choice Theory
The last part of Theorem 2.15 probably looks a bit strange, but may be cleared up by noting the following facts: (a) there exist asymmetric orders (which are not negatively transitive) which are continuous, but which are not strongly continuous, and (b) if an asymmetric order is negatively transitive and continuous, then it is strongly continuous. An example of an asymmetric order, and one of which we will make a great deal of use, is deﬁned in the following. 2.16 Deﬁnition. We will say that a relation P on a nonempty set X is a semiorder iﬀ there exists a function f : X → R amd a postive constant, δ ∈ R++ such that, for all x, y ∈ X: xP y ⇐⇒ f (x) > f (y) + δ. We will say that P is a continuous semiorder iﬀ the function f is continuous. It is fairly easy to prove that a continuous semiorder is strongly continuous. (See Exercise 2, at the end of this chapter.) A useful generic example of an asymmetric order is provided in the following example. 2.17 Example. Let X be any nonempty subset of Rn , let u : X → R be any realvalued function deﬁned on X, and let α and β be any nonnegative constants. If we then deﬁne P on X by: xP y ⇐⇒ u(x) > u(y) + αx − y + β, then P is an asymmetric order.
Given that we have an interest in asymmetric orders, it is obvious that the following type of binary relation is of interest. 2.18 Deﬁnition. We will say that a binary relation, G, on a nonempty set, X, is a quasi order iﬀ G is total, reﬂexive, and its asymmetric part, P , is transitive.30 Notice that it follows from Proposition 1.17 that a binary relation, P , is an asymmetric order if, and only if, its negation is a quasi order. Obviously a weak order is a special case of a quasi order. Exercises. 1. Show that the relation > is an asymmetric order on Rn . 2. Show that a continuous semiorder is strongly continuous. (Where we say that a semiorder is continuous iﬀ the function f by which it is deﬁned is a continuous function.) 3. Show that the binary relation, , deﬁned in equation (2.30) of Section 12 is irreﬂexive, asymmetric, and transitive; and that ∼ [the symmetric part of the 30 There does not seem to be an established term to denote a binary relation satisfying these properties. However, Sen [1986] refers to binary relations whose asymmetric part is transitive as being ‘quasitransitive.’ Consequently, it seems reasonable to apply the term ‘quasi order’ in the present case.
2.13. Asymmetric Orders
57
relation deﬁned in equation (2.29) of Section 12] is reﬂexive and symmetric, but not transitive. 4. Prove that the binary relation, P , deﬁned in Example 2.17 is an asymmetric order. 5. Show that if a consumer’s preferences are representable by a utility function of the form (2.9) of Section 7 (with a 0), then the condition of equation (2.10) [(1, 0)P (0, 1)] insures the implication given in equation (2.11) of Section 7. 6. As in Section 12.4, consider a twoperson household, with Gi being the (weak) preference relation of the ith person (i = 1, 2). Deﬁne the relation G by: xGy ⇐⇒ [xG1 y & xG2 y], and let P be the asymmetric part of G. Do you think that it will be the case that the household consumption choice will always be an element of: h(p, w) = {x ∈ b(p, w)  (∀y ∈ X) : yP x ⇒ p · y > w}? Why or why not? How does the correspondence deﬁned here compare with that deﬁned in Section 12.4?
Chapter 3
Revealed Preference Theory 3.1
Introduction
We noted earlier that if a decisionmaker has (a) a welldeﬁned preference relation which is, in mathematical terms, a weak order, and (b) always makes a choice consistent with said preferences, then the general algebraic choice model is applicable as a description of the choice situation. In eﬀect, then, these two conditions together constitute suﬃcient conditions for the application of the general algebraic choice model. They are not necessary conditions, but nonetheless the model can be applied only if the decisionmaker behaves as if (a) and (b) hold. In this chapter, we will investigate the implications of this last statement; that is, the implications of the assumption that a decisionmaker behaves as if (a) and (b) hold. In Sections 2 through 4 we will look at the implications of conventional demand theory; that is, the implications of the assumption that consumer preferences can be modeled as a weak order. In Section 5 we will consider the testable implications of the model when only a ﬁnite number of observations of quantities demanded can be made. Finally, in Section 6, we will take a brief look at the implications of the assumption that consumer strict preferences are assumed only to be an asymmetric order. The approach to be followed in the ﬁrst four sections, as well as most of the deﬁnitions and results, are due to Richter [1966, 1971].1
3.2
Choice Correspondences and Binary Relations
In traditional demand theory, we suppose that a consumer makes a unique choice from a budget set of the form: b(p, w) = {x ∈ Rn+  p · x ≤ w}, where ‘w’ denotes the consumer’s wealth, or income. These choices result in a demand function, h : Rn++ × R+ → Rn+ . The following deﬁnition extracts the key elements of these concepts, and generalizes the idea to a very broad concept of choice. 1
Although I have taken the liberty of changing Richter’s terminology slightly.
60
Chapter 3. Revealed Preference Theory
3.1 Deﬁnitions. A budget space, X, B, is a nonempty set, X, and a family, B, of nonempty subsets, B, of X. A choice correspondence on a budget space X, B is a correspondence, h, which to each B ∈ B, assigns a nonempty subset, h(B), satisfying: (∀B ∈ B) : h(B) ⊆ B. We will sometimes, particularly in Sections 4 and 6 of this chapter (and in later chapters), be interested in a special kind of choice correspondence; those satisfying the condition: (∀B ∈ B)(∃x ∈ B) : h(B) = {x}. (3.1) In this case, we will refer to h as a choice function, and (at the expense of strictly proper mathematics useage), for B ∈ B we will think of h(B) as being an element of B rather than a subset of B. That is, in the case where h is a choice function, if B ∈ B and x ∈ B satisfy (3.1), above, we will write: h(B) = x, rather than ‘h(B) = {x}.’ 3.2 Deﬁnition. Let h be a choice correspondence on a budget space, X, B. We shall say that a binary relation, G, on X, rationalizes h on X, B iﬀ: (∀B ∈ B) : h(B) = {x ∈ B  (∀y ∈ B) : xGy}.
(3.2)
In the economic theory of consumer choice, it is generally assumed that a decisionmaker has a preference relation, R, which is a weak order, and that given a budget set, B ∈ B, chooses an element from the set h(B) deﬁned by: h(B) = {x ∈ B  (∀y ∈ B) : xRy}. Obviously in this case h will be a choice correspondence. Furthermore, the weak order R will in this case rationalize h on X, B. We will now begin examining the converse question of whether a given choice correspondence can be rationalized by some binary relation. 3.3 Deﬁnition. Let h be a choice correspondence on a budget space X, B. We shall say that h is: 1. rational iﬀ there exists a binary relation, G, which rationalizes h on X, B. 2. reﬂexiverational iﬀ there exists a reﬂexive binary relation, G, on X which rationalizes h on X, B. 3. transitiverational iﬀ there exists a transitive binary relation, G, on X which rationalizes h on X, B. 4. regularrational iﬀ there exists a weak order, G, on X which rationalizes h on X, B. 5. irrational iﬀ it is not rational; that is, iﬀ there exists no binary relation, G, which rationalizes h on X, B. 3.4 Proposition. There exist irrational choice correspondences; that is, there exist choice correspondences which cannot be rationalized by any binary relation.
3.2. Choice Correspondences and Binary Relations
61
In order to prove this proposition, it obviously suﬃces to exhibit an irrational choice correspondence. This is done in the following example. 3.5 Example. Let X = {a, b, c}, and B = {B1 , B2 }, where: B1 = X = {a, b, c}
h(B1 ) = {b},
B2 = {a, b}
h(B2 ) = {a}.
Suppose, by way of obtaining a contradiction, that there exists a binary relation, G, which rationalizes h on X, B. Then by (3.2) and the deﬁnition of h(B1 ), we see that: bGa, bGb, and bGc. However, it then follows that: (∀x ∈ B2 ) : bGx, which implies, if G rationalizes h, that b ∈ h(B2 ); contrary to the deﬁnition of h(B2 ). 3.6 Proposition. There exist choice correspondences which can be rationalized by a reﬂexive binary relation, but not by any total binary relation. In order to prove this, it again suﬃces to produce an example, as in the following. 3.7 Example. Let X = {a, b, c}, B = {B1 , B2 , B3 }, and: B1 = {a, c}
h(B1 ) = {a, c},
B2 = {b, c}
h(B2 ) = {b, c},
B3 = {a, b, c}
h(B3 ) = {c}.
Suppose that G is a binary relation which rationalizes h on X, B. Then, from the deﬁnition of h, it follows that G must satisfy: a b c
a b c aGa . . . aGc . . . bGb bGc cGa cGb cGc.
(3.3)
The entries in the ﬁrst row of the above matrix follow from the deﬁnition of h(B1 ), those in the second row from h(B2 ), and so on. Notice that the particular binary relation deﬁned in (3.3) [that is, if we take (3.3) to be the deﬁnition of G] rationalizes h on X, B; which establishes the fact that h is rational. On the other hand, if G is deﬁned as in (3.3), then it is not total, since we have neither bGa, nor aGb. Now suppose we try to extend G in such a way as to make it total. If we have aGb, then it follows that (∀x ∈ B3 ) : aGx, so that G no longer rationalizes h [since a ∈ / h(B3 )]. On the other hand, if we let bGa, then we have: (∀x ∈ B3 ) : bGx;
62
Chapter 3. Revealed Preference Theory
and, since b ∈ / h(B3 ), G no longer rationalizes h. Since any binary relation which rationalizes h must satisfy (3.3), it then follows that there exists no binary relation which is both total and rationalizes h in this case. The logic of the argument developed in the last paragraph of the preceding example may not be all that clear at this point. In any case, consideration of the following material may make said logic clearer, as well as improving our understanding of the theory of choice correspondences in general. 3.8 Deﬁnitions. Let h be a choice correspondence on a budget space X, B. We then deﬁne the relations V and W on X by: xV y ⇐⇒ (∃B ∈ B) : x ∈ h(B) & y ∈ B,
(3.4)
[read ‘x is directly revealed preferred to y’], and xW y iﬀ there is a ﬁnite sequence, ui m i=1 , satisfying: (3.5) xV u1 V . . . V um V y. [read ‘x is revealed preferred to y’]. Our immediate concern at the moment is with the V relation (we will return to a discussion of the W relation later on). The ﬁrst, and most important, thing to notice about the V relation is that if h is a choice correspondence, and G rationalizes h on X, B, then G must extend V on X, deﬁned as follows. 3.9 Deﬁnition. If R and S are binary relations of a nonempty set X, we shall say that S extends R on X iﬀ we have: (∀x, y ∈ X) : xRy ⇒ xSy. In the case at hand, then, if G rationalizes h on X, B, we must have: (∀x, y ∈ X) : xV y ⇒ xGy.
(3.6)
[I will leave the veriﬁcation of (3.6) as an exercise; it follows at once from the deﬁnitions.] Returning now to Example 3.7, notice that the relation deﬁned in (3.3) is actually the V relation corresponding to the given h; and what we established in the last paragraph of the example is that any binary relation which extends V and is also total cannot rationalize h on X, B. If we think about the Examples 3.5 and 3.7 in connection with the V relation, it quickly becomes apparent that if h can be rationalized by the V relation, then h is a rational choice correspondence [let G = V in Deﬁnition 3.2]. We might also suspect that h is rational only if h can be rationalized by V , and it turns out that this is indeed the case, as we shall now establish. 3.10 Deﬁnition. Let h be a choice correspondence on the budget space X, B. We shall say that h satisﬁes the Vaxiom (Richter [1971, p. 33]) iﬀ: (∀x ∈ X)(∀B ∈ B) : [x ∈ B & (∀y ∈ B) : xV y] ⇒ x ∈ h(B).
(3.7)
3.2. Choice Correspondences and Binary Relations
63
3.11 Theorem. A choice correspondence, h, on a budget space X, B, satisﬁes the Vaxiom if, and only if, it is rational. Proof. 1. Suppose h satisﬁes the Vaxiom, and let B ∈ B be arbitrary. If x ∈ h(B), then we obviously have (by deﬁnition of the V relation): (∀y ∈ B) : xV y. Conversely, if x ∈ B satisﬁes: (∀y ∈ B) : x V y, then it follows from the assumption that h satisﬁes the Vaxiom that x ∈ h(B). Consequently, since B ∈ B was arbitrary, it follows that: (∀B ∈ B) : h(B) = {x ∈ B  (∀y ∈ B) : xV y}; and thus the relation V rationalizes h. Therefore h is rational. 2. Suppose h is rational, and that G is a binary relation on X which rationalizes h (so that G satisﬁes 3.2 [equation (3.2)]). If B ∈ B and x ∈ B satisfy: (∀y ∈ B) : xV y. then, since G must extend V , we have: (∀y ∈ B) : xGy. Since G rationalizes h, it then follows that x ∈ h(B). Therefore, h satisﬁes the Vaxiom. Notice that in the ﬁrst part of the above proof we have established that a choice correspondence, h, is rational if, and only if, it can be rationalized by the V relation which it deﬁnes. However, let me hasten to add that a rational choice correspondence can generally be rationalized by many diﬀerent binary relations, as is illustrated by the following example.2 3.12 Example. Let X = {a, b, c}, and B = {B1 , B2 }, where: B1 = {a, b}
h(B1 ) = {a},
B2 = {a, c}
h(B2 ) = {a}.
In this case the V relation determined by h, which does rationalize h, is given by: a b c
a b c aV a aV b aV c ... ... ... ... ... ....
2 For conditions implying that the binary relation rationalizing a given choice correspondence is unique, see Arrow [1959] (for the case in which X is ﬁnite) and Chipman and Moore [1977] (for the case in which X is inﬁnite).
64
Chapter 3. Revealed Preference Theory
However, each of the following two relations also rationalize h (and there are many other relations which rationalize h as well): a b c
a b c aGa aGb aGc . . . bGb . . . ... . . . cGc
a b c aG a aG b aG c ... bG b bG c ... . . . cG c.
Probably a few moments’ thought will suﬃce to convince you that if a choice correspondence is rational, then it is reﬂexiverational. However, we will nonetheless take the time to prove this. 3.13 Proposition. If a choice correspondence, h is rational, then it is reﬂexiverational. Proof. Suppose h is rational. Then by (the proof of) Theorem 3.11, h can be rationalized by the direct preference relation, V . Deﬁne the binary relation G on X by: Gx = V x ∪ {x} for each x ∈ X. Then G is reﬂexive, and we can show that it rationalizes h, as follows. First, let B ∈ B be arbitrary, and let x ∈ h(B). Then, by deﬁnition we have: (∀y ∈ B) : xV y; and thus, since G extends V , we also have (∀y ∈ B) : xGy. / h(B). Then it follows from the Conversely, suppose x ∈ B is such that x ∈ VAxiom that there exists x∗ ∈ B such that x∗ = x and: ¬x V x∗ . But then, since: we see that
Gx∗ = V x∗ ∪ {x∗ },
¬x Gx∗
as well. Consequently, we conclude that: / h(B) ⇒ (∃x∗ ∈ B) : ¬x Gx∗ ; x ∈
or, equivalently: if x ∈ B satisﬁes: (∀x ∈ B) : x Gx, then x ∈ h(B).
3.3
RegularRational Choice Correspondences
In order to study regular rational choice correspondences, we begin by establishing the following. 3.14 Lemma. Suppose h is a choice correspondence on X, B. If G is a transitive binary relation on X which rationalizes h, then G must extend W.
3.3. RegularRational Choice Correspondences
65
Proof. Suppose G is a transitive binary relation which rationalizes h, and let x, y ∈ X be such that xW y. Then, by deﬁnition of the W relation, there exist u1 , . . . , us ∈ X such that: xV u1 V . . . V us V y. Thus, since G must extend V, we have: xGu1 G . . . Gus Gy; and, since G is transitive, it then follows that xGy.
3.15 Proposition. There exist choice correspondences which can be rationalized by a total and reﬂexive binary relation, but not by any transitive binary relation. 3.16 Example. Let X = {a, b, c}, B = {B1 , B2 , B3 }, and: B1 = {a, b}
h(B1 ) = {a},
B2 = {b, c}
h(B2 ) = {b},
B3 = {a, c}
h(B3 ) = {c}.
It is easy to show that the following relation, which is the V relation deﬁned from h, is total and reﬂexive, and rationalizes h on X, B: a b c
a b c aV a aV b . . . . . . bV b bV c cV a . . . cV c.
(3.8)
(notice that G is identical to the V relation in this case). The fact that h cannot be rationalized by any transitive binary relation follows easily from 3.18, below. 3.17 Deﬁnition. (Richter [1966]). We shall say that a choice correspondence, h, on X, B, satisﬁes the Congruence Axiom iﬀ we have: (∀x, y ∈ X)(∀B ∈ B) : [x ∈ h(B) & y ∈ B & yW x] ⇒ y ∈ h(B). 3.18 Theorem. (Richter) Let h be a choice correspondence on a budget space, X, B. Then there exists a transitive binary relation rationalizing h on X, B if, and only if, h satisﬁes the Congruence Axiom. Proof. 1. Suppose h can be rationalized by a transitive binary relation, G, and suppose B ∈ B and x, y ∈ B satisfy: x ∈ h(B) & yW x. (3.9) Then, by Lemma 3.13, we have: yGx. Furthermore, since x ∈ h(B), it follows that: (∀u ∈ B) : xW u;
(3.10)
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Chapter 3. Revealed Preference Theory
and, again using Lemma 3.13, we then have: (∀u ∈ B) : xGu.
(3.11)
Combining (3.10) and (3.11) with the fact that G is transitive, we then have: (∀u ∈ B) : yGu; and, since G rationalizes h, it then follows that y ∈ h(B). Therefore, h satisﬁes the Congruence Axiom. 2. Suppose h satisﬁes the Congruence Axiom, and let B ∈ B. From the deﬁnition of the W relation it is obvious that: ∀x ∈ h(B) ∀u ∈ B : xW u. If, on the other hand, y ∈ B satisﬁes: (∀u ∈ B) : yW u, then it follows at once from the Congruence Axiom, and the fact that h(B) = ∅, that y ∈ h(B). Since B ∈ B was arbitrary, we have shown that h satisﬁes: (∀B ∈ B) : h(B) = {x ∈ B  (∀u ∈ B) : xW u}; that is, W rationalizes h. Since W is obviously transitive, our result follows.
(3.12)
The relation W is the transitive closure of V , for a given choice correspondence. For our purposes, the transitive closure of a relation, R, is deﬁned as follows. 3.19 Deﬁnition. Let R be a binary relation on a nonempty set, X. We will say that a binary relation, G, on X is the transitive closure of R iﬀ: 1. G is transitive, 2. G extends R on X, and: 3. given any transitive binary relation, , which extends R on X, must also extend G on X. Suppose now that R is a binary relation on a nonempty set, X, and deﬁne the relation G on X by xGy iﬀ there exists a ﬁnite sequence, ui m i=1 ⊆ X satisfying: xRu1 & u1 Ru2 & . . . & um Ry, for x, y ∈ X. It is easy to show that G is then transitive, and obviously G extends R on X. Furthermore, one can establish, by an argument similar to the proof of Lemma 3.13 (details are left as an exercise), that if is a transitive binary relation which extends R on X, then must also extend G. Consequently, it follows that G is the transitive closure of R,3 and, as a special case of this result, it follows that for a given choice correspondence, h, the revealed preference relation, W , determined by h is the transitive closure of the relation V determined by h. From these considerations and a careful study of the proof of Theorem 3.18, one can easily prove the following (again the details will be left as an exercise). 3
Notice also that R is its own transitive closure if it is itself transitive.
3.4. Representable Choice Correspondences
67
3.20 Proposition. If h is a choice correspondence on a budget space X, B, then h is transitiverational if, and only if, it can be rationalized by W. In light of the above proposition, and the discussion which preceded it, let’s consider another example of a choice correspondence which can be rationalized by a total and reﬂexive binary relation, but not by any transitive binary relation. This example will also be particularly useful to us in our consideration of social choice functions in Chapter 14. 3.21 Example. Let X = {a, b, c}, B = {B1 , B2 , B3 , B4 }, and: B1 = {a, b}
h(B1 ) = {a, b},
B2 = {b, c}
h(B2 ) = {b, c},
B3 = {a, c}
h(B3 ) = {a}.
B4 = X
h(B4 ) = {a, b}.
(3.13)
In this case, the V relation determined by h is given by the following table: a b c
a b c aV a aV b aV c bV a bV b bV c . . . cV b cV c.
(3.14)
It is then easily seen that V is total, reﬂexive, and rationalizes h. However, it is also more or less immediate that the W relation is in this case the trivial relation deﬁned by: xW y ⇐⇒ x, y ∈ X. in particular, we have cW a, and from this fact you can easily show that (a) W does not rationalize h, or (b) h does not satisfy the Congruence Axiom (take your pick). In any case it follows that h cannot be rationalized by any transitive binary relation.
Theorem 3.18 has been extended (in one direction) by Richter [1966] to the form presented in Theorem 3.21, below. Since the proof of this extended result involves a considerably more sophisticated argument than that used in the proof of 3.18, however, we will not provide a proof here. On the other hand, notice that Theorem 3.18 is not a special case of 3.21. In fact, while the suﬃciency part of 3.21 generalizes the suﬃciency part of 3.18, the necessity part of 3.21 is a special case of the necessity part of 3.18. 3.22 Theorem. (Richter [1966, p. 639]). Let h be a choice correspondence on a budget space X, B. Then h is regular rational if, and only if, h satisﬁes the Congruence Axiom.
3.4
Representable Choice Correspondences
In modern discussions of demand theory, authors often make the statement that the economic theory of consumer behavior assumes that consumers behave as if they
68
Chapter 3. Revealed Preference Theory
were maximizing a realvalued utility function. The following deﬁnition provides a precise deﬁnition of this statement. 3.23 Deﬁnition. A choice correspondence, h, on a budget space X, B will be said to be representable iﬀ there exists a function, g : X → R satisfying: (∀B ∈ B) : h(B) = {x ∈ B  (∀y ∈ B) : g(x) ≥ g(y)}.
(3.15)
It follows at once from Theorem 3.17 that if h is representable, then h must satisfy the Congruence Axiom. On the other hand, it is possible for a choice function to satisfy the Congruence Axiom, and yet not be representable (see Richter [1971, pp. 467]). In order to state suﬃcient conditions for representability, we consider a special class of choice functions, deﬁned as follows. 3.24 Deﬁnition. A choice correspondence, h, will be said to be competitive iﬀ h is a choice correspondence on the budget space Rn+ , B∗ , where: B∗ = B ⊆ Rn+  ∃(p, w) ∈ Ω : B = b(p, w) ; where we deﬁne: Ω = {(p, w) ∈ Rn+1  p ∈ Rn++ & w ∈ R+ }, and where, for (p, w) ∈ Ω, we deﬁne: b(p, w) = {x ∈ Rn+  p · x ≤ w}. [Notation: For competitive choice correspondences, we will write B = b(p, w) and h(B) = h(p, w).] For a competitive choice correspondence, h, deﬁne: h(Ω) = h(p, w). (p,w)∈Ω
3.25 Examples/Exercises. 1. Consider the CobbDouglas utility function, g : Rn+ → R+ , given by: g(x) =
n i=1
(xi )ai ,
where: ai > 0
for i = 1, . . . , n; and
n i=1
ai = 1.
(3.16)
In this case, as is well known, the corresponding demand functions are given by: hi (p, w) =
ai w pi
for i = 1, . . . , n.
(3.17)
It should then be clear that here we have: h(Ω) ⊆ Rn++ ∪ {0};
(3.18)
3.4. Representable Choice Correspondences that is:
69
∀(p, w) ∈ Ω : h(p, w) ∈ Rn++ ∪ {0}.
Conversely, suppose x∗ is a arbitrary element of Rn++ ∪ {0}. If x∗ = 0, then obviously: x∗ = h(p, 0). On the other hand, if x∗ ∈ Rn++ , and we deﬁne: p∗i = ai /xi then it is easy to show that:
for i = 1, . . . , n;
x∗ = h(p∗ , 1).
Thus it follows from the arguments of this paragraph that: Rn++ ∪ {0} ⊆ h(Ω); and combining this with (18), we then have that: h(Ω) = Rn++ ∪ {0}. 2. Suppose we change the speciﬁcation in (3.25) to: n ai ≥ 0 for i = 1, . . . , n; and ai = 1. i=1
(3.19)
What is the form of h(Ω) in this case? 3. Suppose we consider a case in which a consumer has a continuously diﬀerentiable and strictly quasiconcave utility function, having the property that: (∀x ∈ Rn++ ) : ∇u(x) 0. Can you then prove that we will have Rn++ ∪ {0} ⊆ h(Ω)?
Richter has established the following result. 3.26 Theorem. (Richter [1966]). Let h be a competitive choice correspondence, suppose that D(h) is a convex set, and that: ∀(p, w) ∈ Ω : h(p, w) is a closed set. If h also satisﬁes the Congruence Axiom, then h is representable. 3.27 Examples/Exercises. Suppose h is a competitive choice function4 having the property that the proportion of income spent on the ith commodity is equal to some constant ai ≥ 0, for i = 1, . . . , n, where: n ai = 1. i=1
Show that h is representable. 4 Recall the terminology introduced in Section 2. A choice function is a singlevalued choice correspondence, but we also think of h(B) as being an element, rather than a subset of B in this case.
70
Chapter 3. Revealed Preference Theory
3.5
Preferences and Observed Demand Behavior
Let’s return to the issue of determining what, exactly, are the implications of preferencemaximizing behavor. We’ll start by considering the following issue. Suppose we are given a function, h : Ω → Rn+ . How can we tell if it is consistent with preference maximization; that is, how can we tell whether it might be the demand function of a preferencemaximizing consumer? Obviously, if it is a demand function, it needs to be positively homogeneous of degree zero in (p, w) and satisfy the condition: (∀(p, w) ∈ Ω) : p · h(p, w) ≤ w. (3.20) We can take this one step further: consider the following deﬁnition. 3.28 Deﬁnition. Let X be a nonempty subset of Rn , and let P be a binary relation on X. We shall say that P is locally nonsaturating iﬀ, given any x ∈ X, and any > 0, there exists y ∈ N (x, ) ∩ X such that yP x. It is then easy to prove that if h : Ω → X is the demand function generated by a locally nonsaturating preference relation, P , then h must satisfy the following condition. 3.29 Deﬁnition. Let h : Ω → Rn+ be a competitive demand function. We shall say that h satisﬁes the budget balance condition iﬀ we have, for all (p, w) ∈ Ω: (∀(p, w) ∈ Ω) : p · h(p, w) = w.
(3.21)
Thus, to return to our earlier discussion, if there is a (p, w) pair for which: p · h(p, w) < w,
(3.22)
then this function is not consistent with the maximization of a locally nonsaturating preference relation. So, let’s specialize our question a bit, to consider a function, h : Ω → Rn+ , which is homogeneous of degree zero and satisﬁes the bundget balance condition. How can we then tell whether or not h is consistent with locally nonsaturatingpreferencemaximizing behavior? To avoid repeating this rather awkward phrase innumerable times in our discussion, let’s begin by deﬁning the following. 3.30 Deﬁnition. We shall say that a function h : Ω → Rn+ is Srational iﬀ it can be rationalized (Deﬁnition 3.2) by a locally nonsaturating weak order on Rn+ . Almost 70 years ago, Paul Samuelson provided a partial answer to this question (Samuelson [1938]). To be Srational, h must satisfy what is now called the Weak Axiom of Revealed Preference (WARP). In order to state this, we begin by deﬁning the relation S on Rn+ by: xSy ⇐⇒ x = y and [(∃(p, w) ∈ Ω) : x = h(p, w) & p · y ≤ w]. We can then deﬁne the axiom as follows.
(3.23)
3.5. Preferences and Observed Demand Behavior
71
3.31 Deﬁnition. We say that the function h : Ω → Rn+ satisﬁes the Weak Axiom of Revealed Preference (WARP) iﬀ the relation S deﬁned in (3.23) is asymmetric. Thus, if the function h is Srational, it must be postively homogeneous of degree zero and satisfy equation 3.21 and WARP. However, this leaves open the question of whether or not these three conditions exhaust the implications of the assumption that h is Srational. Writing some time after Samuelson, Houthakker [1950] noted, in eﬀect,5 that an Srational function must also satisfy what is now known as the Strong Axiom of Revealed Preference. To state this, we begin by deﬁning the relation H as the transitive closure of S; that is, we deﬁne H on Rn+ by: (3.24) xHy ⇐⇒ xSy or (∃u1 , . . . , us ∈ Rn+ ) : xSu1 Su2 S . . . Sus Sy [read: ‘x is revealed preferred to y’], 3.32 Deﬁnition. We say that the function h : Ω → Rn+ satisﬁes the Strong Axiom of Revealed Preference (SARP) iﬀ the relation H deﬁned in (3.24) is asymmetric. A question which the Houthakker paper left unresolved was whether SARP and WARP were independent conditions. This question was answered in the aﬃrmative by David Gale [1960], who exhibited a function satisfying WARP, but not SARP.6 Moreover, the question of whether homogeneity, (3.21), and SARP fully exhausted the implications of the assumption that h is Srational was not deﬁnitively answered until the publication of Richter’s [1966] paper. The issue here is this: Suppose h : Ω → Rn+ . Let’s agree to call h a dfunction if it (a) is positively homogeneous of degree zero, (b) satisﬁes budget balance [equation (3.21)], and (c) satisﬁes SARP. If we are given a function h : Ω → Rn+ which fails any one of conditions (a)– (c), we can be sure that it is not Srational. However, this leaves unanswered the question of whether every dfunction is Srational (that is, whether it might be the demand function of a preferencemaximizing consumer whose preferences are a locally nonsaturating weak order on Rn+ ). Richter’s article answers this aﬃrmatively and deﬁnitively. In order to demonstrate this, we need ﬁrst to show that, in the present context, SARP and Richter’s Congruence Axiom are equivalent. We can do this as follows. Proof of equivalence, for h : Ω → Rn+ satisfying (3.21) We begin by noting that, under the present conditions, if x, y ∈ Rn+ are such that xW y, and x = y, then xHy. For, if xW y, then there exist u1 , . . . , ur ∈ Rn+ such that, deﬁning u0 = x and ur+1 = y, we have: ui V ui+1 5
for i = 0, 1, . . . , r.
(3.25)
Both Samuelson and Houthakker framed their investigations in terms of utilitymaximization. This investigation was extended and expanded by Kihlstrom, MasColell, and Sonnenschein [[1976]. They developed a whole class of functions satisfying WARP, but not SARP; but, more importantly developed necessary, and suﬃcient conditions for the matrix of substitution terms to be symmetric and negative semideﬁnite. 6
72
Chapter 3. Revealed Preference Theory
Furthermore, if x = y, then we must have uj = uj+1 for at least one j ∈ {0, 1, . . . , r}, and thus: uj Suj+1 . Moreover, for each remaining index, i, for which ui = ui+1 , we can eliminate ui+1 , to obtain a set {v 0 , . . . , v t+1 } ⊆ {u0 , u1 , . . . , ur+1 }, with: v 0 = x & v t+1 = y, and satisfying: v k Sv k+1
for k = 0, . . . , t.
Now suppose that h satisﬁes SARP, and that (p, w) ∈ Ω and x, y ∈ Rn+ satisfy: x = h(p, w) & p · y ≤ w. If x = y, it then follows that xHy, and thus by SARP and the argument of the above paragraph we cannot have yW x as well. Therefore, h satisﬁes the Congruence Axiom. Conversely, suppose h satisﬁes the Congruence Axiom, and that x, y ∈ Rn+ are such that: xHy. Then we can distinguish two cases. First, suppose xSy. Then there exists (p, w) ∈ Ω such that: x = h(p, w) & p · y ≤ w.
(3.26)
If we were then also to have yHx, we would obviously also have yW x, and it would follow from the Congruence Axiom that y = h(p, w), which, since xSy implies x = y, contradicts (3.26). Otherwise (that is, if ¬xSy), there will exist u1 , . . . , ur ∈ Rn+ such that: xSu1 , u1 Su2 , . . . , ur Sy. If we also were to have yHx, then there would exist
v1, . . . , vs
ySv 1 , v 1 Sv 2 , . . . , v s Sx.
(3.27) ∈
Rn+
such that:7 (3.28)
Then, combining (3.27) and (3.28), we see that u1 Hx. However, this implies u1 W x, which is impossible; for by the fact that xSu1 , there exists (p, w) ∈ Ω such that: x = h(p, w) & p · u1 ≤ w; which, by the Congruence Axiom would
imply u1
(3.29)
= h(p, w), contradicting (3.27).
Given the equivalence just established, the following is easily established, using Richter’s Theorem 3.22. The formal proof will be left as an exercise. 7 Strictly speaking, we should allow for the case in which ySx. However, this leaves the basic argument unaﬀected.
3.5. Preferences and Observed Demand Behavior
73
3.33 Proposition. Suppose h : Ω → Rn+ . If h is Srational, then it is positively homogeneous of degree zero, and satisﬁes equation (3.21) (budget balance) and SARP. Conversely, if h is positively homogeneous of degree zero, and satisﬁes equation (3.21) and SARP, then there exists a weak order on Rn+ , which is locally nonsaturating on: h(Ω) = {x ∈ Rn+  (∃(p, w) ∈ Ω) : x = h(p, w)}, and which rationalizes h. The converse statement in the above proposition tells us that the conditions listed exhaust the implications of the assumption that h is Srational, To put this another way, if we have a function h : Ω → Rn+ which is positively homogeneous of degree one and satisﬁes budget balance and SARP, we can be certain that it could be the demand function of a preferencemaximizing consumer, whose preference relation is a weak order on Rn+ , and locally nonsaturating on the portion of Rn+ relevant to her or his demand behavior. However, the proposition just established leaves some issues still unaddressed. First of all, we have been looking at demand functions; presuming, in eﬀect, that given the same (p, w) pair in repeated choices, the consumer would always pick the same bundle from the budget set. However, in Chapter 2 we saw that in experimental situations, subjects often varied their choices when given the same budget set in repeated situations. In fact, notice that SARP actually implies that the choice correspondence is singlevalued. In the context of consumer demand theory, this means that if, for example a consumer faced the same prices in two successive periods (two successive months in the standard interpretation of the theory), and if the consumer’s wealth (money income) were the same in the two preiods, then she/he/it will choose exactly the same commodity bundle in the two periods. It is doubtful whether anyone really believes that this would happen, however. Casual observation suggests that the choice actually made in the two time periods would be inﬂuenced by a myriad of factors not taken into account in the standard theory; for example, whether the consumer ‘owes’ or ‘is owed’ dinner invitations the season of the year, and so on and so on. We can, of course, treat all such aberrations as ‘changes in taste,’ but to do so is to imply that the currently received theory of demand has no empirical (predictive) content whatever. As Richter has pointed out [1966, p. 3a], however, such complications can be allowed for in the following way. Suppose we view the consumer’s choice as a twostep process. A ‘viable set’ of alternatives is chosen from the budget set, and then a ﬁnal (and unexplained, from the point of view of the standard formal theory) choice is made from this viable set. If the initial choice of a ‘viable set’ can be regarded as being guided by a weak order, and this viable set is the set of maximal elements of the budget set, then we may still have a theory with empirical content, but one which allows for variations in the ﬁnal choice. Richter’s Congruence Axiom, then provides a complete characterization of the viable set in the context of consumer demand theory. However, there may be a problem with this characterization. Speciﬁcally, the diﬃculty is, that while one can characterize situations in which an investigator may be able to observe values of a consumer’s demand function, it is diﬃcult to
74
Chapter 3. Revealed Preference Theory
imagine scenarios in which one can observe the values of the viable sets. In fact, in experimental or statistical studies of actual individual (competitive) choice behavior, all that one can generally hope to observe is a data set, D = (pt , wt ), xt Tt=1 , where T is a positive integer, and xt is the bundle chosen at (pt , wt ), for t = 1, . . . , T . How can we determine whether such a data set is consistent with (one or twostep) preference maximization? To see the problem that the diﬃculty in observing the viable set creates, let’s return to Example 3.7; which, as you may recall, exhibited a choice correspondence which could not be rationalized by any total binary relation. This time, however, suppose that the individual makes a choice of a viable set (which we will identify with the correspondence h deﬁned in the example initially), and then makes a ﬁnal choice according to some unknown criterion. We will indicate this ﬁnal choice by ‘d(B),’ and will suppose that the investigator observes the pairs Bt , d(Bt ), for t = 1, 2, 3. Then we may have the situation exhibited in the following example. 3.34 Example. Let X = {a, b, c}, B = {B1 , B2 , B3 }, and: B1 = {a, c}
h(B1 ) = {a, c}
d(B1 ) = a,
B2 = {b, c}
h(B2 ) = {b, c}
d(B2 ) = b,
B3 = {a, b, c}
h(B3 ) = {c}
d(Bt ) = c.
Given what we are supposing can be observed in this case, we cannot distinguish between the consumer whose choices we have just been describing, and the consumer who maximizes in one step, and whose preference relation is such that a is preferred to b and b is preferred to c. In particular, we could not refute the hypothesis that the consumer’s choice of a viable set involves maximization of a weak order. This last example shows that problems are created by the fact that we may not observe the entirety of a consumer’s viable sets, if our description of the twostep maximization process is a fair description of reality, and probably makes you wonder whether this sort of hypothesis would have any observable implications at all! However. the fact is that in the context of demand theory, the hypothesis does have observable implications, and the Congruence Axiom will help us determine what they are. To see this, suppose that the criterion used to determine the viable set (we will hereafter refer to this as the ﬁrststep criterion and we will denote the ‘viable set correspondence’ by ‘h(·)’) is a locally nonsaturating weak order. Then when faced with a budget pair (p, w), the consumer’s ﬁnal choice, which we will denote by ‘d(p, w),’ must satisfy: p · d(p, w) = w.
(3.30)
We will refer to this property of a data set as budget balance. Now suppose that we observe a data set D = (pt , wt ), xt Tt=1 , satisfying budget balance, and suppose that for some subset, D∗ = (ps , ws ), y s Ss=1 ⊆ D = (pt , wt ), xt Tt=1 , we have: ps · y s+1 ≤ ws
for s = 1, . . . , S − 1.
(3.31)
3.5. Preferences and Observed Demand Behavior
75
Then, in terms of the (direct) revealed preference (V ) relation corresponding to the viable sets (that is, corresponding to h), we have: y s V y s+1
for s = 1, . . . , S − 1;
and therefore: y1W yS .
(3.32)
y1
is in the budget set b(pS , wS ), and if we suppose that the ﬁrstConsequently, if step criterion is a locally nonsaturating weak order, then y 1 must be in the viable set, h(pS , wS ); which implies: pS · y 1 = w S . Thus, whether or not y 1 is in the ﬁnal budget set, we must have: pS · y 1 ≥ w S . This implication is called the Generalized Axiom of Revealed Preference. More properly, we state the following deﬁnition. 3.35 Deﬁnition. We will say the data set, D = (pt , wt ), xt Tt=1 satisﬁes the Generalized Axiom of Revealed Preference (GARP) if, given any subset, D∗ = (ps , ws ), y s Ss=1 ⊆ D = (pt , wt ), xt Tt=1 satisfying: ps · y s+1 ≤ ws
for s = 1, . . . , S − 1,
(3.33)
we have: pS · y 1 ≥ w S .
(3.34)
Afriat [1967, 1973] developed the very subtle and insightful theorem which we state as follows. 3.36 Theorem. Afriat If the data set D = (pt , wt ), xt Tt=1 satisﬁes budget balance and GARP , then there exist real numbers u1 , . . . , uT and positive real numbers λ1 , . . . , λT satisfying: uj ≤ ui + λi (pi · xj − wi )
for i, j = 1, . . . , T.
(3.35)
While we will not provide a proof of Afriat’s theorem here, let me recommend that those of you with a particular interest in theory consult the excellent article by Fostel, Scarf, and Todd [2004] in which they provide an elegant, and much shorter and simpler proof than Afriat’s original argument.8 We can also (and again following Afriat, although not so literally this time) state something which is a sort of converse of the above result. However, we need to begin with some considerations involving the meaning of a preference relation (or utility function) rationalizing demand in the situation under consideration. Since we are supposing that we would not generally observe all of h(p, w), but rather only an element thereof, the following deﬁnitions become more important in our current discussion than the deﬁnitions of a preference relation (or a utility function) rationalizing h. 8 Such readers should also consult the articles by Diewert [1973] and Varian [1982], who provide alternative arguments and tests for GARP.
76
Chapter 3. Revealed Preference Theory
3.37 Deﬁnition. Let D = (pt , wt ), xt Tt=1 be a data set. We will say that a binary relation, G, on Rn+ (respectively, a function, u : Rn+ → R) is consistent with D iﬀ we have: (3.36) (∀x ∈ Rn+ ) : pt · x ≤ wt ⇒ xt Gx for t = 1, . . . , T. [respectively: (∀x ∈ Rn+ ) : pt · x ≤ wt ⇒ u(xt ) ≥ u(x)
for t = 1, . . . , T.]
(3.37)
We can then state a second theorem due to Afriat (albeit our statement is a bit diﬀerent from Afriat’s) as follows. 3.38 Theorem. Let the data set D = (pt , wt ), xt Tt=1 satisfy budget balance, and suppose the real numbers u1 , . . . , uT and the positive real numbers λ1 , . . . , λT satisfy (3.35) of Theorem 3.36. Then the function u : Rn+ → R deﬁned by: u(x) = min[ut + λt (pt · x − wt )]
(3.38)
t
is consistent with D. Proof. Notice, ﬁrst of all, that it follows from from budget balance that: ut + λt (pt · xt − wt ) = ut ; and thus from (3.35) and the defnition of u, we see that: u(xt ) = ut
for t = 1, . . . , T.
Next we note that if pt · x ≤ wt , then: u(x) ≤ ut + λt (pt · x − wt ) ≤ ut = u(xt ).
Notice that the functions: ut + λt (pt · x − wt ), are continuous, strictly increasing, and concave in x. Consequently, the minimum function, u, deﬁned in (3.38) is concave, strictly increasing, and concave as well. Therefore, Theorem 3.36 tells us that GARP and the budget balance condition completely exhaust the observable implications of the assumption that the viable correspondence can be rationalized by a strictly increasing, continuous, and concave utility function. Interestingly, Matzkin and Richter [1991] have shown that if D = (pt , wt ), xt Tt=1 satisﬁes budget balance and the Strong Axiom of Revealed Preference, then (the consumer’s choice correspondence is a function, and) a strengthened version of Afriat’s Theorem can be deduced in that all of the inequalities in (3.35) can be taken to be strict, for xi = xj . These inequalities are then used to construct a function, u(·), which is strictly concave, strictly increasing, continuous, and rationalizes the data set.
3.6. The Implications of Asymmetric Orders*
3.6
77
The Implications of Asymmetric Orders*
In this section we will take a brief look at some of the implications of the assumption that consumer (strict) preferences can only be assumed to be an asymmetric order. In our treatment here, we will follow Kim and Richter [1986]. 3.39 Deﬁnitions. If h is a choice correspondence on a budget space X, B, and is a binary relation on X, then is said to motivate h iﬀ, for every B ∈ B: h(B) = {x ∈ B  (∀y ∈ B) : y x}.
(3.39)
Equivalently, we can say that motivates h iﬀ, for every B ∈ B: h(B) = {x ∈ B  (∀y ∈ X) : y x ⇒ y ∈ / B}.
(3.40)
In either case we will say that h is motivated by , and if there exists a binary relation, which motivates h, we will say that h is motivated. If there exists a binary relation, , which motivates h, and which is, respectively: irreﬂexive, asymmetric, transitive, or asymmetric and transitive, we will say that h is irreﬂexive, asymmetric, transitive, or asymmetric ordermotivated, respectively. Now, it is easy to show formally that if a choice correspondence, h is motivated by a binary relation, , on X, and if we deﬁne the binary relation, , on X by: x y ⇐⇒ y x,
(3.41)
then h is rationalized by (as deﬁned in Deﬁnition 3.2). Conversely, if h is rationalized by the relation , and we deﬁne by: x y ⇐⇒ ¬[y x],
(3.42)
then h is motivated by . Thus the proof of the ﬁrst of the following results is fairly immediate. Similarly, we know that if is asymmetric, and we deﬁne as in (3.41), then is total and reﬂexive (Proposition 1.17 of Chapter 1); while if is total and reﬂexive, and is deﬁned as in (3.42), then is asymmetric. These considerations provide the basis of the proof of Theorem 3.22. 3.40 Theorem. (Kim and Richter Theorem 3, p. 333) A choice correspondence, h, is motivated iﬀ h satisﬁes the VAxiom. Hence, h is motivated if, and only if, it is rational. 3.41 Theorem. (Kim and Richter Theorem 5, p. 334) A choice correspondence, h, is asymmetricmotivated iﬀ it is totalrational, and iﬀ it is totalreﬂexiverational. 3.42 Theorem. (Kim and Richter Theorem 6, p. 334) Let h be a competitive choice correspondence satisfying the budget balance condition. Then h is asymmetricmotivated if, and only if, h satisﬁes the VAxiom. Proof. See Kim & Richter [1986, pp. 334–5].
3.43 Proposition. If h satisﬁes the Congruence Axiom, then h is asymmetric ordermotivated.
78
Chapter 3. Revealed Preference Theory
Proof. Suppose h satisﬁes the Congruence Axiom, let W be the revealed preference relation deﬁned by h, and deﬁne on X by: x y ⇐⇒ [xW y & ¬yW x].
(3.43)
It follows from Theorem 1.13 of Chapter 1 that is an asymmetric order (notice that W is a reﬂexive and transitive relation), and thus our proof will be complete if we can show that motivates h. Accordingly, let B ∈ B, and suppose ﬁrst that x ∈ h(B). Then it follows at once from the deﬁnition of W that for all y ∈ B, we must have xW y; and from this it is immediate that: (∀y ∈ B) : y x ⇒ y ∈ / B. Conversely, suppose z is an element of B satisfying: (∀y ∈ X) : y z ⇒ y ∈ / B. Then in particular, for x ∈ h(B) [and recall that h(B) must be nonempty, by the deﬁnition of a choice correspondence], we must have: x z.
(3.44)
However, since x ∈ h(B), it follows from the deﬁnition of the W relation that we must have xW z. If it were also the case that ¬zW x, then it would follow that x z; which contradicts (3.44). Thus we must have zW x, and it then follows from the Congruence Axiom that z ∈ h(B). Notice that in the above result we have shown that the satisfaction of the Congruence Axiom is a suﬃcient condition for the choice function h to be asymmetric ordermotivated. Necessary and suﬃcient conditions for h to be asymmetric ordermotivated are apparently not known; however, in the remainder of this section we will investigate some aspects of this question in more detail. We begin with a useful deﬁnition which is often used in the revealed preference literature.9 3.44 Deﬁnition. Let P be an irreﬂexive binary relation on a nonempty set, X. We shall say that P is cyclic iﬀ, for some positive integer, n, there exists points x1 , x2 , . . . , xn ∈ X such that: x1 P x2 & x2 P x3 & . . . & xn−1 P xn , but xn P x1 . If no such cycle exists (that is, if P is not cyclic), we shall say that P is acyclic. It is easy to see that if P is acyclic, then it is asymmetric. Conversely, if it is asymmetric and transitive (and thus is an asymmetric order), then it is acyclic (and irreﬂexive as well). It is, however, easy to construct examples of irreﬂexive binary relations which are acyclic (and thus are also asymmetric), but which are not transitive. For example consider the following. 9 This deﬁnition is also used very frequently in the literature on social choice, as we shall discover in Chapter 14.
3.6. The Implications of Asymmetric Orders*
79
3.45 Example. Let X be the three element set, X = {a, b, c}, and let P be as indicated in the following table. a b c
a ... ... ...
b c aP b . . . . . . bP c ... ....
Then, while this relation is irreﬂexive and acyclic, it is not transitive; transitivity would require that we also have aP c. Notice also that if we were to ﬁll in the lower left cell of the table, specifying that cP a, then the relation would be cyclic. We can use the deﬁnition of acyclicity to completely characterize choice on ﬁnite sets, as follows. 3.46 Proposition. Suppose X, B is a budget space, where X is a ﬁnite set, that P is an irreﬂexive binary relation on X, and deﬁne the correspondence, h, on B by: h(B) = {x ∈ B  (∀y ∈ X) : yP x ⇒ y ∈ / B}. If P is acyclic, then h is decisive; that is, it is nonemptyvalued. Furthermore, if B includes all subsets of X containing two or more elements, and h is decisive, then P is acyclic. Proof. 1. Suppose that h is not decisive. We wish to prove that it must then be the case that P is cyclic. Accordingly if h is not decisive, then there exists B ∗ ∈ B such that h(B ∗ ) = ∅. Since X is ﬁnite, we may suppose without loss of generality that #B ∗ = k, where k is an integer greater than or equal to one. If k = 1, that is, if B ∗ is of the form: B ∗ = {x∗ }, for some x∗ ∈ X, then, since h(B ∗ ) = ∅, we must have: x∗ P x∗ ; and we see that P is not irreﬂexive, contrary to our hypothesis. Consequently, we must have k ≥ 2. Now, if we choose an arbitrary element of B ∗ to label ‘x1 ,’ then there must exist an element of B ∗ , x2 , such that: x2 P x1 . If also x1 P x2 , then we have established that P is cyclic. Otherwise, since h(B ∗ ) = ∅, there must exist x3 ∈ B ∗ , distinct from x1 , such that: x3 P x2 . However, if x2 P x3 , or x1 P x3 , we have established that P is cyclic, and we can stop. Otherwise, since P is irreﬂexive and, since we have already noted that we must have x3 = x1 , it follows that x1 , x2 and x3 are distinct elements of B ∗ satisfying: x3 P x2 & x2 P x1 .
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Chapter 3. Revealed Preference Theory
Proceeding in this way, suppose we have found m distinct elements, x1 , . . . , xm ∈ B ∗ , where m ≥ 2, satisfying: xj+1 P xj
for j = 1, . . . , m − 1.
Then, since h(B ∗ ) = ∅, there must exist xm+1 ∈ B ∗ such that xm+1 P xm . However, if xm+1 = xj , for some j ∈ {1, . . . , m}, then we have: xm P xm−1 & . . . xj+1 P xj and xj P xm , and we have shown that P is cyclic. Consequently, we see that at the mth step, we will either have shown that P is cyclic, or we will obtain an element, xm+1 ∈ B ∗ such that xm+1 , xm , . . . , x1 are all distinct elements of B ∗ and satisfy: xj+1 P xj
for j = 1, . . . , m.
(3.45)
However, since there are only k elements in B ∗ , this process can continue at most until k = m + 1. On the other hand, since h(B ∗ ) = ∅, it then follows that we must have xj P xk , for some j ∈ {1, . . . , k − 1}, and the same basic argument as was presented earlier in this paragraph establishes that P is cyclic. 2. Suppose B includes all subsets of X containing two or more elements, and suppose P is irreﬂexive but not acyclic; that is, suppose P is irreﬂexive and cyclic. Then there exists an integer, n ≥ 2, and elements x1 , . . . , xn ∈ X such that: x1 P x2 & . . . & xn−1 P xn and xn P x1 . B∗
But then, deﬁning = {x1 , . . . , xn }, we see that for each j ∈ {1, . . . , n}, there exists k ∈ {1, . . . , n} such that xk P xj .10 It follows, therefore, that for each x ∈ B ∗ , there exists x ∈ B ∗ such that x P x; and thus h(B ∗ ) = ∅. Since B ∗ ∈ B under the present assumptions, we see that h is not decisive. It therefore follows that if h is decisive (given the extra assumptions on B), then P is acyclic. As was noted earlier, full necessary conditions for a choice correspondence to be asymmetrictransitivemotivated are apparently not known.11 We can, however, make some progress toward the solution of this problem by considering the following examples. In the ﬁrst of the two, we develop a choice correspondence which can be motivated by a strict preference relation, P , which is acyclic, but such that h does not satisfy the congruence axiom;12 while in the second example, h is asymmetric ordermotivated, but nonetheless does not satisfy the congruence axiom. 3.47 Example. Let X = {a, b, c}, B = {B1 , B2 , B3 , B4 }, and: B1 = {a, b}
10
h(B1 ) = {a},
B2 = {a, c}
h(B2 ) = {a, c},
B3 = {b, c}
h(B3 ) = {b},
B4 = {a, b, c}
h(B4 ) = {a}.
If j ∈ {2, . . . , n}, let k = j − 1; while if j = 1, let k = n. 11 However, see Kim [1987]. 12 It is also true that in this example h is not motivated by the transitive closure of P ; which is actually my main reason for presenting it in addition to Example 3.39
3.6. The Implications of Asymmetric Orders*
81
Notice that B includes all subsets of X containing two or more elements. It is easy to prove that the choice correspondence, h is motivated by the following preference, P , wich is obviously irreﬂexive and acyclic:13
a b c
a ... ... ...
b c aP b . . . . . . bP c ... ....
In order to show that h does not satisfy the congruence axiom, we begin by noting that the V relation determined by h is as follows (notice that V is total).
a b c
a b c aV a aV b aV c . . . bV b bV c cV a . . . cV c.
Thus the W relation is as follows: a b c
a b c aW a aW b aW c bW a bW b bW c cW a cW b cW c.
/ h(B1 ); which But then we have, for example, b ∈ B1 , a ∈ h(B1 ), and bW a, but b ∈ shows that h does not satisfy the congruence axiom. 3.48 Example. Here we take X = {a, b, c, d}, B = {B1 , . . . , B11 }, and: B1 = {a, b}
h(B1 ) = {a},
B2 = {a, c}
h(B2 ) = {a},
B3 = {a, d}
h(B3 ) = {a, d},
B4 = {b, c}
h(B4 ) = {b},
B5 = {b, d}
h(B5 ) = {b, d},
B6 = {c, d}
h(B6 ) = {d},
B7 = {a, b, c}
h(B7 ) = {a}
B8 = {a, b, d}
h(B8 ) = {a, d}
B9 = {a, c, d}
h(B9 ) = {a, d}
B10 = {b, c, d}
h(B10 ) = {b, d}
B11 = {a, b, c, d}
h(B11 ) = {a, d}.
Notice that, as in the previous example, B includes all subsets of X which contain at least two elements. Moreover, it is a straightforward exercise to show that the 13 The transitive closure of P, P ∗ , is identical to P except that we also have aP ∗ c. It is easy to see, however, that P ∗ does not motivate h.
82
Chapter 3. Revealed Preference Theory
following preference motivates h:
a b c d
a ... ... ... ...
b c d aP b aP c . . . . . . bP c . . . ... ... ... . . . dP c . . . .
Notice that the relation P deﬁned in the above table is an asymmetric order. It is, however, not negatively transitive; since, for example, we have aP b, but neither aP d nor dP b.14 The V relation generated by h is then as follows.
a b c d
a b c d aV a aV b aV c aV d . . . bV b bV c bV d ... ... ... ... dV a dV b dV c dV d.
Consequently, the W relation for h is then given by:
a b c d
a b c d aW a aW b aW c aW d bW a bW b bW c bW d ... ... ... ... dW a dW b dW c dW d.
it is now easy to see that h does not satisfy the congruence axiom; for we have, for / h(B1 ). example, b ∈ B1 , a ∈ h(B1 ), and bW a, but b ∈ Exercises. In each of the following three problems, a choice correspondence is presented. In each case, answer the following questions, and provide a justiﬁcation for each answer. Is h (1) rational? (2) totalreﬂexiverational? (3) transitiverational? (4) regularrational? 1. Let X = {a, b, c, d}, B = {B1 , B2 , B3 , B4 }, and: B1 = {a, b}
h(B1 ) = {a, b},
B2 = {b, c}
h(B2 ) = {b},
B3 = {a, c}
h(B3 ) = {c}.
B4 = {b, d}
h(B4 ) = {d}.
14 Kim and Richter [1986] show that if h is asymmetricnegatively transitivemotivated, then it is regularrational, and thus must satisfy the congruence axiom.
3.6. The Implications of Asymmetric Orders*
83
2. Let X = {a, b, c, d}, and B and h be deﬁned by: B1 = {a, d}
h(B1 ) = {a},
B2 = {b, d}
h(B2 ) = {b},
B3 = {a, c, d}
h(B3 ) = {c}.
B4 = {a, b}
h(B4 ) = {a}.
3. Let X = {a, b, c, d}, and B and h be deﬁned by: B1 = {a, b}
h(B1 ) = {a},
B2 = {b, d}
h(B2 ) = {d},
B3 = {c, d}
h(B3 ) = {d},
B4 = {a, c, d}
h(B4 ) = {d}.
B5 = {b, c}
h(B5 ) = {b, c}
B6 = {a, c}
h(B6 ) = {a, c}.
4. Can you ﬁnd the (competitive) choice, or demand function for a consumer having the lexicographic preferences deﬁned in Example 1.28 of Chapter 1? Is the resulting choice function representable? 5. Suppose X is a nonempty set, that P is a binary relation on X, and that f : X → R is a function satisfying: (∀x, y ∈ X) : xP y ⇒ f (x) > f (y). Is P acyclic? Prove or provide a counterexample. (See also Problem 5, at the end of Chapter 1.)
Chapter 4
Consumer Demand Theory 4.1
Introduction
In this chapter we will add structure to our choice theory model by examining the additional implications which follow from some standard structural/geometric assumptions used in economics. Consequently, much of this chapter will probably be review material. We will suppose throughout the chapter that the ith consumer has a (strict) preference relation, Pi , which is asymmetric and transitive (so that Pi is an asymmetric order); but, as already suggested, we will generally assume that Pi satisﬁes other assumptions as well. We begin our study by considering the interpretations of the consumers’ consumption sets which are used in General Equilibrium Theory.
4.2
The Consumption Set
We will suppose that Pi is deﬁned over the consumer’s consumption set, Xi , where Xi ⊆ Rn . In much of this chapter, we will suppose that Xi ⊆ Rn+ ; however, we will allow for the more general case when it is convenient, and we will often need to allow for the (mathematically) more general case when we talk about equilibrium in a production economy. We will use the generic notation, ‘xi ,’ ‘x∗i ,’ etc. to denote the commodity bundle chosen by (and available for consumption by) the ith consumer. Thus we write: xi = (xi1 , . . . , xij , . . . , xin ) ∈ Xi ; where ‘xij ’ denotes the quantity of the j th commodity (j = 1, . . . , n) available to the ith consumer, if xij ≥ 0. If xij < 0, then we will take this to mean that the ith consumer is oﬀering to supply the j th commodity in the amount −xij = xij . There are two basic conventions with respect to the interpretation of the ith consumer’s consumption set which are used in general equilibrium theory. The ﬁrst, which is the one used in the above paragraph, is that the amounts xij represent the total amounts available for consumption, or to be supplied by the ith consumer. The second convention involves the idea of interpreting Xi to be a trading set or a net demand set. We can relate these two ideas in the following way. Suppose
86
Chapter 4. Consumer Demand Theory
we assume that the consumer’s consumption set is necessarily a subset of Rn+ , and let’s denote this consumption set by ‘Ci ;’ so that: Ci ⊆ Rn+ . This convention may necessitate redeﬁning some of our commodities: for example, if, under the interpretation of the above paragraph, xi1 represents the quantity of labor services to be oﬀered by the ith consumer (so that the ﬁrst commodity is ‘labor’), the convention being followed in this second approach is that if ci ∈ Ci , then ci1 will represent the amount of leisure time being enjoyed by the consumer. We then suppose that the consumer has an initial endowment of the n commodities, which we shall denote by r i ∈ Rn+ . In particular, ‘ri1 ’ would here denote the total amount of leisure available to the consumer in the time period under consideration, if no labor services were oﬀered at all. We would then suppose that any commodity bundle, ci , available to the consumer (that is, any ci ∈ Ci ) would necessarily satisfy the condition: ci1 ≤ ri1 ; and we would interpret the quantity: def
i = ri1 − ci1 , to be the quantity of labor services being oﬀered by the consumer, given the total consumption bundle ci . However, we can conveniently represent the conventions of the above paragraph in a diﬀerent way, as follows. Let’s deﬁne the set Xi as: Xi = Ci − r i . The natural interpretation of Xi is that if xi ∈ Xi , then the quantity xij represents the quantity of the j th commodity being demanded from the rest of the economy (if xij ≥ 0), or being oﬀered to the rest of the economy (if xij < 0). In particular, recalling our earlier interpretation of the ﬁrst commodity as representing leisure, notice that if xi ∈ Xi , then: xi1 = − i . For future reference, notice that with this deﬁnition of Xi , it will be the case that Xi will satisfy: for all xi ∈ Xi : xi ≥ −r i . That this is so follows from the fact that if xi ∈ Xi , then the consumer’s total consumption (or commodity bundle available for consumption), ci is given by: c i = xi + r i ; Rn+ ,
and, since Ci ⊆ we necessarily have ci ≥ 0. Continuing our discussion of the trading set, notice that if the consumer’s preferences can be represented as an asymmetric order, i , on Ci , then we can represent the consumer’s preferences on Xi by the relation Pi deﬁned as follows: xi Pi xi ⇐⇒ (xi + r i ) i (xi + r i ).
4.2. The Consumption Set
87
It is easy to see that if i is asymmetric, or transitive, or negatively transitive, then Pi will satisfy exactly the same properties.1 Now, having read all of this discussion, you may be inclined to ask this question: “If some elements of X have some negative coordinates, doesn’t this mean that we need to use the ‘trading set’ interepretation of X, and suppose that given a commodity bundle x ∈ X, the consumer’s actual commodity bundle available for consumption is given by c = x + r, where r is the consumer’s initial commodity endowment, and where c ∈ Rn+ ?” Well, the answer to this question is “not necessarily.” Suppose we wish to allow for the fact that the consumer may be able to supply two diﬀerent types of labor, and suppose these two types of labor are commodities one and two (measured in labor hours), that commodity three is, say, ‘food,’ while for convenience we suppose that there are just these three commodities in the economy. If our consumer needs at least two units of food to survive, and can supply no more than 16 hours of the two types of labor per period then a natural representation of the consumer’s consumption set is: X = {x ∈ R3  16 + x1 + x2 ≥ 0, xj ≤ 0 for j = 1, 2, and x3 ≥ 2}. The key thing here is that the consumer’s choice of leisure (say the quantity 24+x1 + x2 ) does not enable us to determine the quantities of either x1 of x2 . Consequently, the net trading set representation does not work in this context. So, the next question is, how do these distinctions aﬀect our analysis. The fact is, that in most of our analysis, we won’t need to worry very much about which interpretation of the consumption set should be used. The budget constraint for the consumer will normally be deﬁned by a pair (p, w), where p ∈ Rn+ is the vector of prices of the n commodities, and we suppose the consumer’s choice is constrainted to be in the set: def b(p, w) = {x ∈ X  p · x ≤ w}. Under the ‘trading set’ interpretation, or under the sort of deﬁnition of the consumption set indicated in the preceding paragraph, w is interpreted as ‘wealth,’ or income from sources other than the supply of labor. On the other hand, in the ‘ﬁnal consumption’ interpretation (where we take Xi to be a subset of Rn+ ), w will need to include receipts from the ‘sale of leisure,’ that is, if we return to the case in which we take Xi ⊆ Rn+ , and let the ﬁrst commodity be the consumer’s labor/leisure, with the consumer’s total endowment of leisure being given by ri1 > 0, then the consumer’s budget constraint can be expressed as: p · xi ≤ p1 ri1 + wi , where now ‘wi ’ denotes income from sources other than the sale of labor. Alternatively, in this case we can simply deﬁne: wi = p1 r11 + wi ; and express the budget constraint exactly as before. 1 We do, however, need to be careful to note that if r i should change, then so will Pi ; even if i remains the same!
88
Chapter 4. Consumer Demand Theory
4.3
Demand Correspondences
Suppose the ith consumer faces the price vector p = (p1 , . . . , pn ) ∈ Rn++ . Whether one interprets Xi as the consumption set or as a trading set, the consumer’s budget set can, in the absence of nonlabor income, be represented as the set βi (p) deﬁned as: βi (p) = {xi ∈ Xi  p · xi ≤ 0}. Under either the consumption set or the trading set interpretation, however, we will often want to allow for the fact that the consumer may have income or wealth, wi from other sources; that is, purchasing power which is derived from something (possibly the proﬁts of ﬁrms) other than the sale of the consumer’s labor services, or initial endowment of commodities. Consequently, we will handle the consumer’s budget constraint as follows. We begin by deﬁning the set Ωi , a subset of Rn+1 , by: Ωi = {(p, wi ) ∈ Rn+1  p ∈ Rn++ & (∃xi ∈ Xi ) : p · xi ≤ wi }. We then deﬁne the consumer’s budget set, bi (p, wi ), for (p, wi ) ∈ Ωi by: bi (p, wi ) = {xi ∈ Xi  p · xi ≤ wi }.
(4.1)
This last equation deﬁnes a correspondence, which we deﬁne formally in the following. 4.1 Deﬁnitions. We deﬁne the consumer’s budget correspondence, bi : Ωi → Xi , by equation (4.1), for (p, wi ) ∈ Ωi . We then deﬁne the consumer’s demand correspondence, hi , by: hi (p, wi ) = {xi ∈ bi (p, wi )  (∀xi ∈ Xi ) : xi Pi xi ⇒ p · xi > wi },
(4.2)
for (p, wi ) ∈ Ωi . Formally (and sometimes this much formality will be convenient, if not necessary), we shall refer to the correspondence hi : Ωi → Xi deﬁned in (4.2) as the demand correspondence determined by P i .2 In the remainder of this, and the next four sections, however, we will be dealing with the theory of demand for a single consumer, so that we can drop the subscript ‘i’ wherever it appears; writing simply ‘x,’ ‘b(p, w),’ ‘h(p, w),’ etc. We begin our investigation of the theory of consumer demand with the most basic consideration of all; namely, under what conditions will the consumer’s demand correspondence be welldeﬁned? More precisely, our concern in the remainder of this section is to investigate the conditions under which we will have: ∀(p, w) ∈ Ω : h(p, w) = ∅. 4.2 Deﬁnition. We shall say that a subset, X, of Rn is bounded below iﬀ there exists a point z ∈ Rn satisfying: (∀x ∈ X) : x ≥ z. 2
(4.3)
It might be objected that the demand correspondence is jointly determined by Pi and Xi as well, but a part of the deﬁnition of Pi is a speciﬁcation of its domain; that is, of Xi .
4.3. Demand Correspondences
89
4.3 Proposition. Let X be a closed, nonempty subset of Rn , which is bounded below, and let Ω and b : Ω → X be deﬁned as in Deﬁnition 4.1. Then, given any (p∗ , w∗ ) ∈ Ω, b(p∗ , w∗ ) is compact and nonempty. Proof. It is obvious from the deﬁnition of Ω that b(p∗ , w∗ ) is nonempty. To prove that it is also compact, we begin by noting that b(p∗ , w∗ ) is the intersection of the closed halfspace: H = {x ∈ Rn  p∗ · x ≤ w∗ }, def
with X. Since both of these sets are closed, it follows that b(p∗ , w∗ ) is closed as well. To prove that b(p∗ , w∗ ) is bounded, we begin by recalling that, since X is bounded below, there exists a point z satisfying (4.3), above. Next deﬁne w† by: p∗ · z = w† , and note that it follows from (4.3), the deﬁnition of Ω, and the fact that that: w† ≤ w∗ .
(4.4) p∗
≥0 (4.5)
If we now deﬁne the vector y ∈ Rn by: yj = z j +
w∗ − w† p∗j
for j = 1, . . . , n,
(4.6)
it follows from (4.5) that y ≥ z. We will prove that, deﬁning: Y = {x ∈ Rn  z ≤ x ≤ y}, we must have:
b(p∗ , w∗ ) ⊆ Y ;
(4.7)
b(p∗ , w∗ )
is bounded. from which it follows that To prove (4.7), suppose, by way of obtaining a contradiction, that there exists / Y . Then, in view of (4.3), it must be that y ≥ x∗ ; so x∗ ∈ b(p∗ , w∗ ) such that x∗ ∈ that, for some j ∈ {1, . . . , n}: (4.8) x∗j > yj . However, if x∗ satisﬁes (4.8), then we have, making use also of (4.3) and (4.6): p∗ · x∗ = p∗ · (x∗ − z + z) = p∗ · (x∗ − z) + p∗ · z ≥ p∗j (x∗j − zj ) + w† > p∗j (yj − zj ) + w† = p∗j (w∗ − w† )/p∗j + w† = w∗ ; that is:
p∗ · x ∗ > w ∗ ,
contradicting the assumption that x∗ ∈ b(p∗ , w∗ ). Thus we see that (4.7) must hold; and thus that b(p∗ , w∗ ) is bounded. Since we also showed that it was closed, it now follows that b(p∗ , w∗ ) is compact. The following is a repetition of Deﬁnition 2.14, and is repeated here for the sake of having a convenient reference.
90
Chapter 4. Consumer Demand Theory
4.4 Deﬁnitions. We shall say that an asymmetric binary relation, P , deﬁned on a nonempty subset, X, of Rn , is: 1. upper semicontinuous iﬀ, for each x ∈ X, the set xP is open relative to X; that is, for each x ∈ X such that xP x , there exists a (Euclidean) neighborhood of x , N (x ), such that: ∀y ∈ N (x ) ∩ X : xP y. 2. lower semicontinuous iﬀ, for each x ∈ X, the set P x is open relative to X. 3. continuous iﬀ it is both upper and lower semicontinuous. 4. strongly continuous iﬀ, for each x, y ∈ X, if xP y, then there exist neighborhoods of x and y, N (x) and M (y), respectively such that, for all x ∈ N (x) ∩ X and all y ∈ M (y) ∩ X, we have x P y . The following result is formally proved in the appendix to this chapter. From it we can see that the consumer’s demand correspondence is welldeﬁned under very general conditions indeed! 4.5 Theorem. If X is a nonempty, closed subset of Rn which is bounded below, and P is an asymmetric ordering on X which is upper semicontinuous, then h(·), the demand correspondence determined by P, satisﬁes: ∀(p, w) ∈ Ω : h(p, w) = ∅; that is, for each (p, w) ∈ Ω, there exists a bundle x ∈ b(p, w) satisfying: ∀x ∈ X : x P x ⇒ p · x > w. While it has seemed to me to be worthwhile to state and prove (albeit in an appendix) the above result, the fact is that in most of our work with demand correspondences we will be assuming that the consumer’s (strict) preference relation satisﬁes more stringent conditions than are assumed in Theorem 4.5. In fact, more often than not we will be assuming that P is negatively transitive, as well as being asymmetric; in which case, its negation, G, is a weak order. In any case, whether or not P is negatively transitive, the demand correspondence which it determines can equally well be deﬁned by: h(p, w) = x ∈ b(p, w)  ∀y ∈ b(p, w) : xGy , (4.9) where G is the negation of P . Since this is the more conventional way of deﬁning demand correspondences in any event, we shall hereafter generally speak of demand correspondences as being determined by a (presumably reﬂexive) binary relation, G, as per equation (4.9).
4.4
The Budget Balance Condition
A condition which is normally assumed to characterize consumer demand correspondences is budget balance, deﬁned as follows.
4.4. The Budget Balance Condition
91
4.6 Deﬁnition. Let G be a (weak) preference relation, and let h be the demand correspondence determined by G. We shall say that the demand correspondence determined by G, h(·), satisﬁes the budget balance condition (or that G satisﬁes the budget balance condition) if, and only if, for all (p, w) ∈ Ω, we have: ∀x ∈ h(p, w) : p · x = w. It is easily seen that the second of the following conditions implies that the consumer’s demand correspondence satisﬁes the budget balance condition. While it is the assumption used most often in economic theory to justify the budget balance condition, we will consider a more general condition shortly. 4.7 Deﬁnitions. Let G be a binary relation on Rn+ , and let P be its asymmetric part. We will say that G is: 1. nondecreasing iﬀ, given any x, y ∈ Rn+ : x ≥ y ⇒ xGy, or equivalently, yP x ⇒ x ≥ y; 2. increasing iﬀ G is nondecreasing and, in addition, satisﬁes the following condition: (∀x, y ∈ Rn+ ) : x y ⇒ xP y. 3. strictly increasing iﬀ, for every x, y ∈ Rn+ : x > y ⇒ xP y. 4.8 Deﬁnitions. Let X be a nonempty subset of Rn , and let G be a binary relation on X, with P its asymmetric part. We shall say that G is: 1. nonsaturating iﬀ, given any x ∈ X, there exists y ∈ X such that yP x. 2. locally nonsaturating iﬀ, given any x ∈ X, and any > 0, there exists y ∈ N (x, ) ∩ X such that yP x. Notice that a preference relation which is increasing, as deﬁned in 4.7 above, is locally nonsaturating. Moreover, any locally nonsaturating binary relation is nonsaturating, but the converse is not necessarily true. For instance, Example 1.31.4 of Chapter 1 features a nonsaturating binary relation which is not locally nonsaturating. Another such example is presented as Example 4.12.1, below. 4.9 Proposition. Suppose the preference relation, G, is a locally nonsaturating weak order, and that x∗ ∈ h(p∗ , w∗ ), for some (p∗ , w∗ ) ∈ Ω. Then for all x ∈ X: xGx∗ ⇒ p∗ · x ≥ w∗ . ¯ ∈X Proof. Suppose, by way of obtaining a contradiction, that there exists x such that: ¯ Gx∗ , (4.10) x and:
¯ < w∗ . p∗ · x
92
Chapter 4. Consumer Demand Theory
From this last inequality and the continuity of the inner product, there exists > 0 such that ∀x ∈ N (¯ x, ) : p∗ · x < w∗ . (4.11) Now, since G is locally nonsaturating, there exists x ∈ N (¯ x, ) ∩ X such that: ¯, x P x where P is the asymmetric part of G; and, making use of (4.10) and the fact that G is a weak order, it follows that x P x∗ . However, by (4.11) we also have: p∗ · x < w∗ ; which contradicts the assumption that x∗ ∈ h(p∗ , w∗ ).
The next result is easily proved by a modiﬁcation of the argument just presented. Notice, however, that the assumptions used here are much weaker than those used in 4.9. 4.10 Proposition. If G is locally nonsaturating on X, then G satisﬁes the budget balance condition. It should be apparent that G cannot be locally nonsaturating if all commodities are indivisible. However, it is only necessary that one commodity be more or less completely divisible in order that G be locally nonsaturating. In our next deﬁnition, we will present a condition which will be particularly useful to us, and which implies local nonsaturation. In order to present it, however, we need to remind ourselves of a bit of notation. In Rn we deﬁne the n unit coordinate vectors, ej (j = 1, . . . , n), by: ej = (δj1 , . . . , δjn ), where δjk is the Kronecker delta function deﬁned by: 0 for j = k, δjk = 1 for j = k. We can then deﬁne the following. 4.11 Deﬁnition. Let G be a preference relation on a consumption set, X, with P eraire good its asymmetric part. We shall say that the j th commodity is a num´ for G iﬀ, for all x ∈ X, and all θ ∈ R++ , we have: x + θej ∈ X and (x + θej )P x. We shall say that G admits a num´ eraire iﬀ, for some j ∈ {1, . . . , n}, the j th commodity is a num´eraire good for G. Notice that if G admits a num´eraire, then G is locally nonsaturating. Notice also that if G is strictly increasing (with X = Rn+ ), then all commodities are num´eraire goods for G.
4.4. The Budget Balance Condition
93
4.12 Examples. 1. Let f : Rn+ → R be any nondecreasing function, let δ be a positive constant, and deﬁne P on Rn+ by: xP y ⇐⇒ f (x) > f (y) + δ. Here P (or, more correctly, its negation, G) will be nondecreasing, but neither increasing nor locally nonsaturating. 2. Let f be deﬁned on R2+ by: f (x) = 10x1 − (x1 )2 + x2 , and deﬁne G on R2+ by:
xGy ⇐⇒ f (x) ≥ f (y).
Here G is locally nonsaturating, but is not nondecreasing. 3. Let the functions f and g be deﬁned on R2+ by: f (x) = x1 + (x2 )2 and g(x) = (x1 )2 + x2 , respectively; and deﬁne P on R2+ by: xP y ⇐⇒ [f (x) > f (y) & g(x) > g(y)]. In this case, P is a strictly increasing asymmetric order. 4. Deﬁne the functions f and g on R2+ by: f (x) = 2x1 − x2 and g(x) = x2 , respectively; and deﬁne P on R2+ by: xP y ⇐⇒ [f (x) > f (y) & g(x) > g(y)]. Here P is nondecreasing and locally nonsaturating, but not increasing. (To see that P is nondecreasing, notice that if xP y, then we must have g(x) = x2 > g(y) = y2 . Thus, obviously, we cannot have y ≥ x.) 5. Suppose G is representable by the (utility) function, u, deﬁned on Rn+ by: u(x) =
n j=1
(xj + cj )aj ,
where: cj , aj > 0 for j = 1, . . . , n; and
n j=1
aj = 1.
In this case, it is easy to show that G is strictly increasing; probably the easiest way to show this being that the partial derivatives of u are all strictly positive, at any x ∈ Rn+ . What happens, however, if one of the cj ’s is equal to zero?
94
Chapter 4. Consumer Demand Theory
4.5
Some Convexity Conditions
In this section we will explore the implications of some convexity conditions which are very often used in general equilibrium theory. A deﬁnition from mathematics which will be very useful to us, both in this section and in the remainder of this book is the following. 4.13 Deﬁnition. We deﬁne the unit simplex for Rn , denoted by ‘∆n ,’ by: n ∆n = p ∈ Rn+  pj = 1 . j=1
4.14 Deﬁnitions. Suppose X is a convex subset of Rn , that G is a weak order on X, and that P is its asymmetric part. We shall say that G is: 1. weakly convex iﬀ, for all x ∈ X, Gx is a convex set. 2. convex iﬀ it is weakly convex, and in addition, for all x, y ∈ X, we have that if xP y, then: (∀θ ∈ ]0, 1[) : [θx + (1 − θ)y]P y. 3. strictly convex iﬀ, for all x, y, z ∈ X, we have that if yGx, zGx, and y = z, then: (∀θ ∈ ]0, 1[) : [θy + (1 − θ)z]P x. Notice that if x, y ∈ X, and θ ∈ ]0, 1[, then the vector (or ‘commodity bundle’); z = θx + (1 − θ)y, can be viewed as a weighted average of the commodity bundles x and y. Thus we can see, for example, that weak convexity can be interpreted as stating that if x and y are both considered to be at least as good as some third bundle, z, then any weighted average of the two bundles will also be considered to be at least as good as z. 4.15 Deﬁnitions. Let X be a nonempty and convex subset of Rn , and suppose f : X → R. We shall say that f is: 1. concave (respectively, convex) iﬀ, for each x, y ∈ X, and each θ ∈ ]0, 1[, we have: f [θx + (1 − θ)y] ≥ θf (x) + (1 − θ)f (y) respectively, f [θx + (1 − θ)y] ≤ θf (x) + (1 − θ)f (y) . 2. strictly concave (respectively, strictly convex) iﬀ, for each x, y ∈ X, and each θ ∈ ]0, 1[, we have that if x = y, then: f [θx + (1 − θ)y] > θf (x) + (1 − θ)f (y) respectively, f [θx + (1 − θ)y] < θf (x) + (1 − θ)f (y) . 3. quasiconcave (respectively, quasiconvex) iﬀ, for each x, y ∈ X, and each θ ∈ ]0, 1[, we have: f [θx + (1 − θ)y] ≥ min{f (x), f (y)} respectively, f [θx + (1 − θ)y] ≤ max{f (x), f (y)} .
4.5. Some Convexity Conditions
95
4. strictly quasiconcave (respectively, strictly quasiconvex) iﬀ, for each x, y ∈ X, and each θ ∈ ]0, 1[, we have that if x = y, then: f [θx + (1 − θ)y] > min{f (x), f (y)} respectively, f [θx + (1 − θ)y] < max{f (x), f (y)} . 5. semiconcave3 (respectively, semiconvex) iﬀ f is quasiconcave (respectively, quasiconvex) and in addition, for each x, y ∈ X, and each θ ∈ ]0, 1[, we have that if f (x) > f (y), then:
f [θx + (1 − θ)y] > f (y) (respectively, f [θx + (1 − θ)y] < f (x)). Notice that if f is strictly concave, then f is concave and strictly quasiconcave. Similarly, if f is concave, then f is semiconcave; however, it is also true that if f is strictly quasiconcave, then f is semiconcave. It is, of course, obvious that if f is semiconcave, then it is quasiconcave; on the other hand, any strictly increasing transformation of a linear function is semiconcave, but not strictly quasiconcave. The proof of the following result will be left as an exercise. 4.16 Proposition. Suppose G is a weak order on a nonempty, convex subset, X, of Rn , and suppose f : X → R represents G on X. Then: 1. G is weakly convex if, and only if, f is quasiconcave. 2. G is strictly convex if, and only if, f is strictly quasiconcave. 3. G is convex if, and only if, f is semiconcave. 4.17 Proposition. If X is a nonempty convex subset of Rn , and G is a weak order on X which is weakly convex, then for each (p, w) ∈ Ω, h(p, w) is a convex set. Proof. Let (p∗ , w∗ ) ∈ Ω be given, let x and x be elements of h(p∗ , w∗ ), and let θ ∈ [0, 1] be given. If we then deﬁne y = θx + (1 − θ)x , we see that: p∗ · y = p∗ · [θx + (1 − θ)x ] = θp∗ · x + (1 − θ)p∗ · x ≤ θw∗ + (1 − θ)w∗ = w∗ ; where the inequality is by the fact that both x and x must be in the budget set. Furthermore, by the weak convexity of G and the deﬁnition of the consumer’s demand correspondence, we see that: yGx.
(4.12)
Thus, if z ∈ b(p∗ , w∗ ) it follows from the fact that x ∈ h(p∗ , w∗ ) that xGz. From the transitivity of G and (4.12), it then follows that yGz. Therefore, y ∈ h(p∗ , w∗ ), and it follows that h(p∗ , w∗ ) is a convex set. 4.18 Proposition. If X is a nonempty, convex subset of Rn which is closed and bounded below, and G is a weak order on X which is upper semicontinuous and strictly convex, then for each (p, w) ∈ Ω, h(p, w) is a singleton; in other words, under these conditions the consumer’s demand correspondence is actually a function. 3
This is not a standard deﬁnition, but it will be useful to us in our remaining work.
96
Chapter 4. Consumer Demand Theory
Proof. Letting (p∗ , w∗ ) ∈ Ω be arbitrary, it follows from Theorem 4.5 of this chapter that h(p∗ , w∗ ) = ∅. Suppose, by way of obtaining a contradiction, that there exist distinct points, x and y which are both elements of h(p∗ , w∗ ). Then yGx and xGx, so that by the strict convexity of G we must have: [(1/2)x + (1/2)y]P x.
(4.13)
However, since both x and y are elements of b(p∗ , w∗ ), it is easy to see that: (1/2)x + (1/2)y ∈ b(p∗ , w∗ ) as well; and thus (4.13) contradicts the assumption that x ∈ h(p∗ , w∗ ).
j th
When h(·) is a function, we will denote the coordinate function (the demand function for the j th commodity) by ‘hj (·).’ Hopefully, you will have no trouble in distinguishing between this and the ith consumer’s demand function, which will be denoted by ‘hi (·)’ [the ith consumer’s demand function for the j th commodity will be denoted by ‘hij (·)’]. The proof of the following two results will be left as exercises. 4.19 Proposition. If X is a nonempty, convex subset of Rn , and G is a nonsaturating and convex weak order on X, then G is locally nonsaturating. 4.20 Corollary. If X is a nonempty convex subset of Rn , and G is a nonsaturating and convex weak order on X, then the demand correspondence determined by G satisﬁes the budget balance condition.
4.6
Wold’s Theorem
In this section, we will state and prove the ﬁrst result to appear in the economics literature which established suﬃcient conditions for a preference relation to be representable by a realvalued utility function; and which is due to Herman Wold [1943]. It has been generalized since (in particular, by Debreu; see Theorem 1.34); but Wold’s original proof is much simpler than the later generalizations, and since we will want to make use of his result in some of our later work, it seems quite appropriate to state and prove his result here. 4.21 Theorem. (Wold [1943]) Let G be a continuous and increasing weak order on Rn+ . Then there exists a continuous function, u : Rn+ → R+ , which represents G on Rn+ . Proof. Let x∗ ∈ Rn++ be a (ﬁxed) strictly positive vector in Rn+ , and deﬁne: L = {x ∈ Rn+  (∃µ ∈ R+ ) : x = µx∗ }; in other words, let L be the halfray determined by x∗ . We will make use of L to deﬁne our utility function in the following way. Let x ∈ Rn+ be arbitrary. Then we note that there exists x ∈ L such that: x x,
4.7. Indirect Preferences and Indirect Utility
97
and thus, since G is increasing, x P x. Moreover, we also have 0 ∈ L and xG0. Thus if we deﬁne the subsets of R+ , A and B by: A = {µ ∈ R+  xGµx∗ }, and:
B = {µ ∈ R+  µx∗ Gx},
we see that both A and B are nonempty sets. Obviously the set A is bounded above (by any element of B), and thus A has a least upper bound, call in u(x). On the other hand, it is clear that u(x) is also the greatest lower bound for the set B. Consequently, since it follows easily from the continuity of G that both sets are closed, we see that: u(x) ∈ A ∩ B; and thus:
u(x)x∗ Ix.
(4.14)
The argument of the above paragraph establishes the existence of a function u : Rn+ → R+ satisfying (4.14). I will leave as an exercise the task of proving that this function represents G on Rn+ . To prove that u(·) is continuous, let a be an arbitrary real number. If a < 0, then: {x ∈ Rn+  f (x) ≤ a} = ∅ and {x ∈ Rn+  f (x) ≥ a} = Rn+ ; both of which are closed sets. On the other hand, if a ≥ 0, notice that if we deﬁne xa by: xa = ax∗ , we have: u(xa ) = a. Therefore, since u(·) represents G, it follows that: {x ∈ Rn+  f (x) ≤ a} = xa G and {x ∈ Rn+  f (x) ≥ a} = Gxa ; and since both sets are closed relative to Rn+ by our continuity assumption, it now follows that u(·) is a continuous function (see Moore [1999], Proposition 3.32, p. 137).
4.7
Indirect Preferences and Indirect Utility
In this section we will examine some aspects of indirect preferences and indirect utility. We will begin with some very general considerations, and then sharpen our results by considering the implications of some stronger assumptions. Throughout the material to follow, we deﬁne the set Z by: Z = h(Ω) = x ∈ X  ∃(p, w) ∈ Ω : x ∈ h(p, w) , and where we deﬁne: Ω = {(p, w) ∈ Rn+1  p ∈ Rn++ & (∃x ∈ X) : p · x ≤ w};
(4.15)
98
Chapter 4. Consumer Demand Theory
We will be denoting the consumer’s (weak) preference relation over the consumption set, X, by ‘G,’ but we will also denote the restriction of G to Z by ‘G.’ In our initial deﬁnition, we suppose the following assumptions hold. Assumptions I.1. We suppose that the consumer’s demand correspondence, h, is welldeﬁned on Ω. In other words, we suppose that for each (p, w) ∈ Ω, there exists x∗ ∈ b(p, w) (where b : Ω → X is the consumer’s budget correspondence), satisfying: (∀x ∈ X) : xP x∗ ⇒ p · x > w. We also suppose that the restriction of G to Z is a weak order, with asymmetric and symmetric parts P and I, respectively. In the context of these assumptions, we then deﬁne the following. 4.22 Deﬁnition. Given Assumptions I.1, we deﬁne the consumer’s indirect preference relation, G∗ , on Ω as follows: for (p, w), (p , w ) ∈ Ω: (p, w)G∗ (p , w ) ⇐⇒ ∃x ∈ h(p, w) & x ∈ h(p , w ) : xGx . We then denote the symmetric and asymmetric parts of G∗ by ‘I ∗ ’ and ‘P ∗ ,’ respectively. I will leave the proof of the following proposition as an exercise. 4.23 Proposition. Given Assumptions I.1, the indirect preference relation G∗ is a weak order. Of course, it follows from the above result and Theorem 1.15 that P ∗ is negatively transtive (as well as being asymmetric), and that I ∗ is an equivalence relation. It can readily be seen that if X is a subset of Rn+ which contains the origin, then the set Ω deﬁned in (4.15), above, is given by: Ω = Rn++ × R+ . In any case, it will be convenient for us to assume that this condition holds throughout the remainder of this section. Assumptions I.2. We suppose that Ω = Rn++ × R+ and that h satisﬁes the budget balance condition on Ω; that is, for all (p, w) ∈ Ω and all x ∈ h(p, w), we have p · x = w. 4.24 Proposition. Given Assumptions I.1 and I.2, G∗ and P ∗ satisfy the following conditions (in addition to those set out in Proposition 4.23, above): 1. given any (p, w) ∈ Ω, and any w ∈ R, we have: w > w ⇒ [(p, w ) ∈ Ω & (p, w )P ∗ (p, w)]. 2. given (p1 , w1 ), (p2 , w2 ) ∈ Ω such that (p2 , w2 )G∗ (p1 , w1 ), any θ ∈ ]0, 1[, and deﬁning (p∗ , w∗ ) by: (p∗ , w∗ ) = θ(p1 , w1 ) + (1 − θ)(p2 , w2 ),
4.7. Indirect Preferences and Indirect Utility we have: and if
99
(p2 , w2 )G∗ (p∗ , w∗ ),
(p2 , w2 )P ∗ (p1 , w1 ),
then (p2 , w2 )P ∗ (p∗ , w∗ ).
Proof. I will leave the proof of part 1 as an exercise. To prove part 2,4 we begin by showing that: (4.16) b(p∗ , w∗ ) ⊆ [b(p1 , w1 ) ∪ b(p2 , w2 )]. To prove (4.16), suppose that x ∈ X is such that: / b(p2 , w2 ). x∈ / b(p1 , w1 ) and x ∈ Then: p1 · x > w1 and p2 · x > w2 ; and therefore, since 0 < θ < 1: p∗ · x = θp1 · x + (1 − θ)p2 · x > θw1 + (1 − θ)w2 = w∗ ; and, consequently, x ∈ / b(p∗ , w∗ ), so we see that (4.16) holds. Having established (4.16), it follows that, since (p2 , w2 )G∗ (p1 , w1 ), we must have (p2 , w2 )G∗ (p∗ , w∗ ). Now suppose that (p2 , w2 )P ∗ (p1 , w1 ). Then it follows readily from the deﬁnition of indirect preferences that there exist xt ∈ h(pt , wt ), for t = 1, 2 such that: x2 P x 1 .
(4.17)
¯ ∈ X is such that Then by (4.17) and the transitivity Suppose, then, that x ¯ P x1 ; and therefore: of G, x ¯ > w1 . (4.18) p1 · x ¯ Gx2 . x
¯ ≤ w2 , it follows from the transitivity of G that x ¯ ∈ On the other hand, if p2 · x ¯ = w2 . Thus, in any case we must h(p2 , w2 ), and thus by budget balance that p2 · x have: ¯ ≥ w2 ; p2 · x and combining this with (4.18) we see that: ¯ + (1 − θ)p2 · x ¯ > θw1 + (1 − θ)w2 = w∗ . ¯ = θp1 · x p∗ · x It then follows that for all x ∈ b(p∗ , w∗ ), x2 P x, and therefore: (p2 , w2 )P ∗ (p∗ , w∗ ).
An indirect utility function is simply a representation of G∗ , as we formally note in the following. 4.25 Deﬁnition. We say that a function V : Ω → R is an indirect utility function representing G∗ (and P ∗ ) iﬀ V satisﬁes: ∀(p, w), (p , w ) ∈ Ω : V (p, w) ≥ V (p , w ) ⇐⇒ (p, w) G∗ (p , w ). (4.19) 4 A careful reading of this part of the proof will reveal that it holds under the weaker assumption that Ω is a convex cone; that is, that Ω is a cone and a convex set as well.
100
Chapter 4. Consumer Demand Theory
Having now deﬁned what we mean by an indirect utility function, we can obtain the following corollary of 4.24. I will leave the details of the proof as an exercise. 4.26 Corollary. Given Assumptions I.1 and I.2, if V : Ω → R is an indirect utility function representing P ∗ , then it must be strictly increasing in w, for each p ∈ Rn++ . Moreover, it must be positively homogeneous of degree zero and semiconvex on Ω; that is, it must satisfy the following condition: given any (pt , wt ) ∈ Ω (t = 1, 2) such that: (4.20) V (p2 , w2 ) ≥ V (p1 , w1 ), and any θ ∈ ]0, 1[: V θp1 + (1 − θ)p2 , θw1 + (1 − θ)w2 ≤ V (p2 , w2 ); and if V
(p2 , w2 )
>V
(p1 , w1 ),
(4.21)
the strict inequality holds in (4.21).
It is tempting to replace the second part of the conclusion of the corollary with the statement that V (·) is strictly quasiconvex. However, this is not necessarily the case, as is shown by the ﬁrst of the following examples. 4.27 Examples. 1. Suppose a consumer’s preferences can be represented on Rn+ by the Leontief utility function: x j u(x) = min , aj where aj > 0 for j = 1, . . . , n. Then the consumer’s demand function is given by (see Exercise 3, at the end of this chapter): aj w hj (p, w) = n , k=1 ak pk for j = 1, . . . , n. Consequently, the function: V (p, w) = n
w
j=1 aj pj
,
(4.22)
is an indirect utility function for the consumer in this case. Now suppose that (p1 , w1 ) and (p2 , w2 ) are such that: V (p1 , w1 ) = V (p2 , w2 ) = β > 0,
(4.23)
and let θ ∈] 0, 1[. Then, making use of (4.22) and (4.23), we see that: V θ(p1 , w1 ) + (1 − θ)(p2 , w2 ) = n
θw1 + (1 − θ)w2 n 1 2 a j=1 j θpj + j=1 aj (1 − θ)pj
=
θ
θw1 + (1 − θ)w2 θw1 + (1 − θ)w2 n 1 + (1 − θ) 2 = θ(1/β)w 1 + (1 − θ)(1/β)w 2 = β; a p a p j j j=1 j=1 j j
n
whether or not (p1 , w1 ) = (p2 , w2 ). Therefore, we see that V (·) is not strictly quasiconvex in this case.
4.7. Indirect Preferences and Indirect Utility
101
2. Suppose a consumer’s preferences can be represented by the CobbDouglas utility function: n a xj j , u(x) = j=1
where, as usual, we assume that all of the aj ’s are positive and sum to one. Then (see exercise 4, at the end of this chapter), the consumer’s indirect preferences can be represented by the function: V (p, w) = n
w
aj j=1 pj
.
It can be shown that in this case, the indirect utility function is strictly quasiconvex. Notice that if indirect utility is set equal to some positive value, denoted by v ∗ , then the associated indiﬀerence curve for G∗ can be represented by the equation: n a w = v∗ · pj j ; (4.24) j=1
and it is of interest to consider the contour curves of this function in the special case in which n = 2. Consider ﬁrst the representation of indirect preferences when we normalize the price of the second commodity; setting p2 = 1. If, for the sake of convenience we then denote the price of the ﬁrst commodity by p, we see that equation (4.24) reduces to: w = v ∗ pa1 = φ(p); def
and, since 0 < a1 < 1, φ(·) is strictly concave. I will leave it to you to consider the shape of the contour curves in this case, as well as what happens when v ∗ increases. It is also of interest to consider the indiﬀerence map for G∗ if we ﬁx (or normalize) w, and consider the contour (indiﬀerence) curves of G∗ in (p1 , p2 )space. However, I will also leave this as an exercise. The following result generalizes propositions established by Antonelli [1886], Allen [1933], and Roy [1942].5 4.28 Theorem. Suppose G satisﬁes Assumptions I.1 and I.2, that the demand correspondence generated by G is singlevalued (and thus is a function), and that V : Ω → R is a diﬀerentiable indirect utility function representing G∗ . Then, if (p∗ , w∗ ) ∈ Ω is such that w∗ > 0, we must have:
∂V ∂V ∗ ∗ =− ∗ ∗ hk (p∗ , w∗ ) ∂pk (p ,w ) ∂w (p ,w )
for k = 1, . . . , n.
(4.25)
Proof. Since V represents G∗ , it is clear that, for all (p, w) ∈ Ω, we must have: p · h(p∗ , w∗ ) ≤ w ⇒ V (p, w) ≥ V (p∗ , w∗ ). 5 In its present form, the result is from Chipman and Moore [1976a, Lemma 3, p. 74]; although the present, simpler, proof is from Chipman and Moore [1990, p. 71]. The assumption that the budget balance condition holds is not needed, as a careful reading of the proof will disclose. Some of the other assumptions can be weakened as well.
102
Chapter 4. Consumer Demand Theory
Thus we see that (p∗ , w∗ ) minimizes V (·) subject to [(p, w) ∈ Ω and]: p · h(p∗ , w∗ ) = w. From the classical Lagrangian method, it then follows that there exists a Lagrangian multiplier, λ ∈ R such that: ∂V − λhk (p∗ , w∗ ) = 0 for k = 1, . . . , n; (4.26) ∂pk (p∗ ,w∗ ) and:
∂V + λ = 0. ∂w (p∗ ,w∗ )
(4.27)
Solving for λ in (4.27), and substituting into (4.26), we obtain the desired result.
Notice that in the result just presented we have not assumed that the indirect utility function was deﬁned as a composite function: V (p, w) = u[h(p, w)], where u(·) is a continuously diﬀerentiable direct utility function representing G. There is some question as to whether every diﬀerentiable indirect utility function representing G∗ is obtainable in this way, even when there exists at least one continuously diﬀerentiable utility function representing G. More to the point, however, there may exist a continuously diﬀerentiable indirect utility function representing G∗ even when there exists no utility function (diﬀerentiable or not) representing G. Recall, for example, the lexicographic preference relation deﬁned on R2+ by: or x1 > x1 xP x ⇐⇒ x1 = x1 & x2 > x2 . In Chapter 1, we noted that Debreu [1959] has proved that there exists no realvalued utility function representing P in this case. However, it is easy to prove that the function V (·) deﬁned by: V (p, w) = w/p1 , is a continuously diﬀerentiable indirect utility function representing P ∗ . In fact, all of the assumptions of the theorem, that is, both of Assumptions I.1 and I.2, are satisﬁed here. The discussion of the previous paragraph not withstanding, the most obvious way of deﬁning an indirect utility function is, of course, as a composite function: V (p, w) = u[h(p, w)]; at least in the case in which the preference relation, G, is representable by a utility function, and the correspondence h is actually a function. We will, in fact, often make use of this representation in the remainder of this course, but I want to conclude this section by considering a somewhat diﬀerent way of deﬁning an indirect utility function. The following result is proved in the Appendix.
4.7. Indirect Preferences and Indirect Utility
103
4.29 Proposition. Suppose, in addition to Assumptions I.1 and I.2, that the set Z = h(Ω) is a subset of Rn+ , and is closed, convex, and contains 0. Suppose further that the restriction of the consumer’s (weak) preference relation to Z is a continuous ¯ ∈ Rn++ , and any (p , w ) ∈ Ω, there exists weak order on Z. Then, given any p w ¯ ∈ R+ such that: w ¯ = min{w ∈ R+  (¯ p, w)G∗ (p , w )}, and we have:
(¯ p, w)I ¯ ∗ (p , w ).
Because of the above proposition, we see that the following function, which was originally introduced by Hurwicz and Uzawa [1971], is welldeﬁned. 4.30 Deﬁnition. Given the assumptions of Proposition 4.29, we deﬁne the incomecompensation function, µ : Rn++ × Ω → R+ by: p, w)G∗ (p, w)}. µ(¯ p; p, w) = min{w ∈ R+  (¯ Verbally, the value of µ(¯ p; p, w) is the minimum level of income, or wealth which ¯ , as with the would leave the consumer exactly as well oﬀ, given the price vector p pricewealth pair (p, w). The following sets out the basic properties of the incomecompensation function. 4.31 Theorem. The incomecompensation function satisﬁes the following conditions, given the assumptions of Proposition 4.29,: 1. For all (¯ p; p, w) ∈ Rn++ × Ω, and all w ¯ ∈ R+ : ¯ = µ(¯ p; p, w). (¯ p, w)I ¯ ∗ (p, w) ⇐⇒ w ¯; 2. For all (p, w) ∈ Ω, µ(· ; p, w) is positively homogeneous of degree one in p that is: (∀λ ∈ R+ ) : µ(λ¯ p; p, w) = λµ(¯ p; p, w). ¯ ∈ Rn++ , µ(¯ p; · ) is an indirect utility function representing 3. For any ﬁxed p G∗ ; that is, for all (p, w), (p , w ) ∈ Ω: µ(¯ p; p, w) ≥ µ(¯ p; p , w ) ⇐⇒ (p, w)G∗ (p , w ). ¯ ∈ Rn++ , µ(¯ 4. For any ﬁxed p p; · ) is strictly increasing in w, and positively homogeneous of degree zero in (p, w). Moreover, µ(¯ p; · ) is semi convex in (p, w). Proof. 1. It follows at once from 4.29 that: ¯ , µ(¯ p p; p, w) I ∗ (p, w). The converse follows readily from the fact that the budget balance condition implies that G∗ must be strictly increasing in w, for ﬁxed p. 2. I will leave the proof that µ(·; p, w) must be positively homogeneous of degree ¯ as an exercise. one in p
104
Chapter 4. Consumer Demand Theory ¯ ∈ Rn++ be ﬁxed, and suppose (p, w), (p , w ) ∈ Ω are such that: 3. Let p µ(¯ p; p, w) > µ(¯ p; p , w ).
(4.28)
¯ ∈ Rn++ , we have: Then, since G∗ is strictly increasing in w, for ﬁxed p
¯ , µ(¯ ¯ , µ(¯ p p; p, w) P ∗ p p; p , w ) .
(4.29)
The fact that (p, w)P ∗ (p , w ) then follows from part 1 and the transitivity of G∗ . The proof that (4.29) implies (4.28) can proceed by essentially reversing the above steps. 4. Given Part 3, Part 4 of our conclusion is an immediate consequence of Corollary 4.26.
4.8
Homothetic Preferences
In this section we will study a special kind of consumer preference relation; the case in which preferences satisfy a condition called ‘homotheticity.’ Empirical studies have often cast doubt upon the realism of assuming that consumer preferences are homothetic; at least there are reasons to suppose that preferences are not generally homothetic globally. On the other hand, almost everyone who has ever written anything for consumers on how to do budget planning has lent support to the belief that there must be some way of aggregating over commmodities which results in a homothetic preference relation; for, as we will see, if a consumer’s preferences are homothetic, then the consumer’s expenditures on each commodity category (given ﬁxed prices) is a constant percentage of income. 4.32 Deﬁnition. Let H be a binary relation on a cone,6 X ⊆ Rn . We shall say that H is homothetic iﬀ for all x, y ∈ X, and every θ ∈ R++ , we have: xHy ⇒ (θx)H(θy).
(4.30)
Since we will often be working with strict preference relations for a consumer, we will often be assuming that the consumer’s strict preference relation, P , is homothetic. However, if P is homothetic, then its negation is homothetic as well (and conversely), that is, its negation will also satisfy (4.30); as is shown in the following result. 4.33 Proposition. Suppose H is a binary relation deﬁned on a cone, X ⊆ Rn , and let Q be its negation. Then H is homothetic if, and only if, Q is homothetic as well. Proof. Suppose H is homothetic, but, by way of obtaining a contradiction, that there exist x, y ∈ X and θ > 0 such that: xQy, 6
n
A set X ⊆ R is said to be a cone iﬀ, for each x ∈ X, and every θ ∈ R++ , θx ∈ X.
(4.31)
4.8. Homothetic Preferences
105
and yet: ¬[(θx)Q(θy)].
(4.32)
Then from (4.32) and the fact that Q is the negation of H, it follows that: (θy)H(θx). However, from the homotheticity of H, it then follows that: [(1/θ)(θy)]H[(1/θ)(θx)]; that is, yHx, which contradicts (4.31). By reversing the roles of Q and H in the argument of the above paragraph, it follows that if Q is homothetic, then H is as well. 4.34 Deﬁnition. Suppose X ⊆ Rn is a cone,7 and let f : X → R. We shall say that f is homothetic iﬀ there exist functions, g : X → R and F : Y → R, where g(X) ⊆ Y , satisfying the following three conditions: 1. g is positively homogeneous of degree one, 2. F is strictly increasing, and: 3. for all x ∈ X, we have: f (x) = F [g(x)]. I will leave the proof of the following proposition as an exercise. 4.35 Proposition. Suppose X ⊆ Rn is a cone, and that P is an asymmetric and negatively transitive binary relation on X. If there exists a homothetic function, f : X → R such that f represents P on X, then P is homothetic. As you probably suspect, continuous (and increasing) homothetic preferences can be represented by a utility function which is homogeneous of degree one. What may not be so apparent is that any two such representations of a given preference relation must be scalar multiples of one another. Both facts are established in the following theorem. 4.36 Theorem. Suppose G is a weak order on Rn+ , and that G is: 1. homothetic, 2. continuous, and 3. increasing. Then there exists a function u : Rn+ → R+ representing G, and such that u is contin : Rn+ → R is uous and positively homogeneous of degree one on Rn+ . Moreover, if u another function representing G which is also positively homogeneous of degree one, then there exists a positive constant a > 0 such that: (∀x ∈ Rn+ ) : u (x) = au(x).
(4.33)
7 We say that X ⊆ Rn is a cone iﬀ, for each positive real number, θ > 0, and each x ∈ X, we have θx ∈ X as well.
106
Chapter 4. Consumer Demand Theory
Proof. It follows from the proof of the Wold Representation Theorem (4.21) that if we let x∗ ∈ Rn++ be arbitrary, the function u : Rn+ → R+ deﬁned implicitly by the equation: (4.34) xIu(x)x∗ , is a continuous utility function representing G. Now suppose x ∈ Rn+ and let λ ∈ R+ . If λ = 0, then it is clear from the deﬁnition of u(·) and (4.34), above, that: u(λx) = u(0) = 0 = λu(x). On the other hand, if λ > 0, it follows from (4.34) and the homotheticity of G that: (λx)I[λu(x)x∗ ]. Thus since u(λx) is that unique nonnegative real number such that: (λx)I[u(λx)x∗ ], it follows that u(λx) = λu(x); and therefore that u(·) is positively homogeneous of degree one. Finally, suppose u is another function which is both positively homogeneous of degree one and represents G, deﬁne the positive real number a by:8 a=u (x∗ ), represents G, we must and let x ∈ Rn+ be arbitrary. Then, since xI[u(x)x∗ ], and u have: u (x) = u [u(x)x∗ ]. However, since u is positively homogeneous of degree one, we also have: u(x∗ ) = au(x); u [u(x)x∗ ] = u(x) and equation (4.33) now follows.
Rather surprisingly we can make use of the theorem just proved to establish suﬃcient conditions for the existence of a concave utility function representing given preferences. 4.37 Proposition. If, an addition to the other assumptions of Theorem 4.36, G is weakly convex, then any function which represents G and is positively homogeneous of degree one is also concave. Proof. Suppose f : Rn+ → R represents G and is positively homogeneous of degree one. For any λ > 0, we have: f (λ0) = f (0) = λf (0); 8
Notice that, since u is positively homogeneous of degree one, we must have u (0) = 0. Consequently, since x∗ P 0, it follows that u (x∗ ) > 0.
4.8. Homothetic Preferences
107
and therefore f (0) = 0. Moreover, since G is increasing and f represents G, it then follows that: (∀x ∈ Rn++ ) : f (x) > 0; while, by the weak convexity of G, we see that f must be quasiconcave (see Proposition 4.15). Consequently, it follows from Corollary 5.101, p. 334, of Moore [1999] that f is concave. Homothetic preferences yield demand correspondences having a particularly interesting and tractable form, as is established in the following two results. 4.38 Theorem. If G is a homothetic preference relation on a cone X ⊆ Rn+ , then the demand correspondence determined by G, h, satisﬁes the following condition: for all (p, w) ∈ Ω, and all λ ∈ R++ : h(p, λw) = λh(p, w). Proof. Suppose (p∗ , w∗ ) ∈ Ω, λ > 0, and that x∗ ∈ h(p∗ , w∗ ). Then: p∗ · x∗ ≤ w∗ , and thus: p∗ · (λx∗ ) = λp∗ · x∗ ≤ λw∗ . Furthermore, if x ∈ X is such that xP (λx∗ ), then, by the homotheticity of P , we have: (1/λ)xP x∗ . But then, since x∗ ∈ h(p∗ , w∗ ), we have: p∗ · (1/λ)x > w∗ ; and therefore: p∗ · x > λw∗ . It follows that λx∗ ∈ h(p∗ , λw∗ ), and therefore we conclude that: λh(p∗ , w∗ ) ⊆ h(p∗ , λw∗ ). Conversely, suppose x ∈ h(p∗ , λw∗ ). Then, by reversing the roles of λx and x in the argument of the above paragraph, we can conclude that: (1/λ)x ∈ h(p∗ , w∗ ); and thus that x ∈ λh(p∗ , w∗ ). Therefore we see that: h(p∗ , λw∗ ) ⊆ λh(p∗ , w∗ ), and our result follows.
108
Chapter 4. Consumer Demand Theory
4.39 Theorem. Suppose G is a homothetic, upper semicontinuous, and strictly convex weak order on a nondegenerate convex cone,9 X ⊆ Rn+ . Then: 1. G generates a demand function of the form: h(p, w) = g(p)w,
(4.35)
→ X, and where g : 2. if, in addition, G is increasing, then h satisﬁes the budget balance condition, and thus: (∀p ∈ Rn++ ) : p · g(p) = 1. Rn++
3. if g(·) is diﬀerentiable, then at each p ∈ Rn++ , we have: ∂gj ∂gk = ∂pj ∂pk
for j, k = 1, . . . , n.
Proof. It follows from Proposition 4.18 of this chapter that the demand correspondence generated by G is a function. Furthermore, if we deﬁne g : Rn++ → X by: g(p) = h(p, 1) for p ∈ Rn++ , we have from Theorem 4.38 that, for all (p, w) ∈ Ω, if w > 0, then: h(p, w) = wh(p, 1) = wg(p). If w = 0, then the equality in (4.35) obviously holds as well, and thus the ﬁrst part of our result follows. I will leave the proof of part 2 of the result as an exercise. Part 3 of our result follows readily from the Slutsky symmetry conditions. Details of the argument will be left as an exercise. We noted at the beginning of this section that, if a consumer’s preference relation is homothetic, then his/her/its expenditure on any given category of commodities was a constant percentage of income (constant, that is, with respect to income). In fact, suppose a consumer’s preference relation satisﬁes the assumptions of Theorem 4.39, and let J be a nonempty subset of {1, . . . , n}. Then, according to 4.39, the consumer’s expenditures on the commodities corresponding to the set J, as a fraction of income (or wealth) is given by:
1 w
j∈J
1 pj xj = pj gj (p)w = pj gj (p); w j∈J
j∈J
which is clearly independent of w.
4.9
CostofLiving Indices
In this section, we shall suppose throughout that Ω = Rn++ × R+ , and that the consumer’s indirect preference relation, G∗ , is a weak order on Ω. 9
A cone X is nondegenerate if it contains a point x = 0.
4.9. CostofLiving Indices
109
4.40 Deﬁnition. We shall say that a function γ : Rn++ → R++ is a costofliving index for G∗ iﬀ, for all p, p ∈ Rn++ , and all w, w ∈ R++ , we have: w w ≥ ⇐⇒ (p, w)G∗ (p , w ) γ(p) γ(p )
(4.36)
Notice that γ(·) is a costofliving index for G∗ if, and only if, the function v : Ω → R+ deﬁned by: v(p, w) = w/γ(p), is an indirect utility function representing G∗ . Consequently, it follows that if γ(·) is a costofliving index for G∗ , then it must be positively homogeneous of degree one on Rn++ . Given the assumptions of Theorem 4.36 and 4.39 of this chapter (in particular, that the consumer’s preference relation, G is homothetic), there exists a utility function, u : X → R+ representing G, and such that u(·) is positively homogeneous of degree one on X. By Theorem 4.39, the consumer’s demand function, h takes the form: h(p, w) = g(p)w; and, as always, the composite function: v(p, w) = u[h(p, w)], is an indirect utility function representing G∗ on Ω. However, in this case, we have, for (p, w) ∈ Ω: v(p, w) = u[h(p, w)] = u[g(p)w] = w · u[g(p)]. Consequently, if we deﬁne: γ(p) = 1/u[g(p)], we see that we can write: v(p, w) = w/γ(p); and thus γ(·) is a costofliving index for G∗ . Unfortunately for the generality of the concept of a costofliving index, it can be shown that if there exists a costofliving index for G∗ , then G must be homothetic on Z = h(Ω); in particular, G∗ must be homothetic, where this is deﬁned as follows. 4.41 Deﬁnition. We shall say that an indirect preference relation, G∗ , is homothetic iﬀ, for all (p , w ), (p, w) ∈ Ω and all λ > 0, we have: (p , w )G∗ (p, w) ⇒ (p , λw )G∗ (p, λw). Of course, if G is homothetic, then it follows readily from Theorem 4.38 that G∗ is homothetic as well. In principle, however, G∗ may be homothetic, as just deﬁned, even though G is not. There are a few more facts which are of interest regarding costofliving indices in the homothetic case, however, and we will investigate some of them in the remainder of this section.
110
Chapter 4. Consumer Demand Theory
4.42 Proposition. Suppose G∗ is homothetic, and that the incomecompensation p; p, w) ∈ Rn++ × function for G∗ satisﬁes the condition (see Theorem 4.31): for all (¯ Ω, and all w ¯ ∈ R+ : ¯ = µ(¯ p; p, w). (¯ p, w)I ¯ ∗ (p, w) ⇐⇒ w
(4.37)
Then, given any (¯ p; p, w) ∈ Rn++ × Ω, and any λ ∈ R++ , we have: µ(¯ p; p, λw) = λ · µ(¯ p; p, w). Proof. Given (¯ p; p, w) ∈ Rn++ × Ω, it follows from (4.37) that: (¯ p, µ(¯ p; p, w))I ∗ (p, w). Thus, it follows from the fact that G∗ is homothetic that, for λ ∈ R++ : ¯ , λµ(¯ p p; p, w) I ∗ (p, λw). Making use of (4.37) once again, it now follows that: µ(¯ p; p, λw) = λµ(¯ p; p, w).
The above result implies that we can deﬁne a costofliving index for G∗ along the lines of the procedure set out on the previous page, and taking µ(¯ p; ·) as the indirect utility function. It turns out, however, that we can turn things around a bit; as is set out in the following proposition. 4.43 Proposition. If G∗ satisﬁes the hypotheses of Proposition 4.42, then the incomecompensation function for G∗ satisﬁes the following condition: for any (ﬁxed) (p0 , w0 ) ∈ Ω, the function γ : Rn++ → R++ deﬁned by: γ(p) = µ(p; p0 , w0 ), is a costofliving index for G∗ . Proof. Let (p0 , w0 ) ∈ Ω be ﬁxed, and let (p, w) ∈ Ω be arbitrary. Then by (4.37), above, we have: (p0 , w0 )I ∗ p, µ(p; p0 , w0 ) . Therefore, since G∗ (and therefore I ∗ ) is homothetic, we have: w w 0 0 ∗ 0 0 p , w I p, µ(p; p , w ) ; µ(p; p0 , w0 ) µ(p; p0 , w0 ) or, equivalently:
p0 ,
w · w0 I ∗ p, w . µ(p; p0 , w0 )
Therefore, by (4.37), we have: µ(p0 ; p, w) = w0
w . 0 0 µ(p; p , w )
4.10. Consumer’s Surplus
111
Since (p, w) ∈ Ω was arbitrary, and since we know from Theorem 4.31 that µ(p0 ; ·) is an indirect utility function representing G∗ on Ω, it now follows that the function v : Ω → R+ deﬁned by: v(p, w) =
w , µ(p; p0 , w0 )
is an indirect utility function representing G∗ on Ω, and that: γ(p) = µ(p; p0 , w0 ), is a costofliving index for G∗ .
4.10
Consumer’s Surplus
If I may be allowed the rather egocentric action of quoting an article of which I was a coauthor, consumer’s surplus has been described as follows [Chipman and Moore (1976a, p.69)]:10 The concept of consumer’s surplus is one of the oldest in neoclassical economics, even predating the development of marginal utility theory; and it has proved to be one of the most durable. It has great intuitive appeal to the applied economist, for it promises to provide an objective money measure of a person’s satisfaction, in terms of the amount of money he would, as proved by his actions, pay for a thing rather than go without it. . . In this section we will begin by following the train of thought set out in the above quotation in that we will seek a measure of the beneﬁt (or loss), B, of a change from one vector of prices p1 to a second vector p2 , which is such that, supposing that the change is achievable at a monetary cost C, the change can be regarded as beneﬁcial if, and only if: B ≥ C. In terms of indirect preferences, we can express our initial goal as follows: ﬁnd a beneﬁt function, Φ(p1 , p2 ; w1 ) such that: Φ(p1 , p2 ; w1 ) ≥ w1 − w2 ⇐⇒ (p2 , w2 )G∗ (p1 , w1 );
(4.38)
as − C. In the following example, we where, of course, we are interpreting will consider the case in which this all works out most nicely. w2
w1
4.44 Example. In this example, it will be convenient to suppse that there are n + 1 commodities, and to use the generic notation ‘(x0 , x), (x∗0 , x∗ ),’ etc., to denote commodity bundles. The commodity whose quantity is denoted by ‘x0 ’ we will suppose is a num´eraire good, as was deﬁned earlier (Deﬁnition 4.11).11 We will 10 If I remember correctly, this introduction was actually written by John Chipman, so that I am not being quite as egocentric here as it appears at ﬁrst glance. 11 It could also be thought of as ‘expenditure on other commodities.’ We will return to this interpretation later.
112
Chapter 4. Consumer Demand Theory
n denote price vectors as (p0 , p) ∈ R1+n ++ , where p0 is the price of x0 , and p ∈ R++ is the vector of prices of the remaining n commodities. We will often normalize the price of x, however, and deﬁne:
q = (1/p0 )p; referring to qj as the normalized price of commodity j. Suppose a consumer’s prefby the utility function: erences can be represented on R1+n + n
u(x0 , x) = x0 +
j=1
φj (xj );
(4.39)
where for each j we have: (∀xj ∈ R+ ) : φj (xj ) > 0 & φj (xj ) ≤ 0.
(4.40)
If you do the mathematics, you can easily verify the fact that the consumer maximizes utility by choosing x∗j such that: φ (x∗j ) = pj /p0 = qj def
and: x∗0 =
for j = 1, . . . , n,
w − p · x∗ w = − q · x∗ . p0 p0
Thus, we can consider the graph of the function φ (·) to be the consumer’s inverse demand curve for the j th commodity, as a function of the normalized price. Now suppose that the consumer’s pricewealth situation changes from (p10 , p1 , w1 ) to (p20 , p2 , w2 ), and deﬁne: q t = (1/pt0 )pt
for t = 1, 2.
Then, letting: φj (xtj ) = qjt
for j = 1, . . . , n; t = 1, 2,
the consumer’s change in utility is given by: ∆u = w2 /p20 − q 2 · x2 +
n j=1
n φj (x2j ) − w1 /p10 − q 1 · x1 +
j=1
φj (x1j ) . (4.41)
Suppose we now deﬁne: ∆Sj = (qj1 − qj2 )x1j − qj2 · (x2j − x1j ) + φj (x2j ) − φj (x1j ), and: ∆S =
n j=1
∆Sj .
Then by (4.41), we have: ∆u =
w2 w1 − 1 + ∆S. p20 p0
4.10. Consumer’s Surplus However, we have: x2 j
x1j
113
φj (xj )dxj = φj (x2j ) − φj (x1j )
for j = 1, . . . , n;
and therefore: def
∆S =
n j=1
def
∆Sj =
n j=1
= q 1 · x 1 − q 2 · x2 +
1 (qj − qj2 ) · x1j − qj2 · (x2j − x1j ) +
n j=1
φj (x2j ) − φj (x1j ) .
(4.42)
x2j
x1j
φj (xj )dxj
(4.43) See Figure 4.1, below, for a graphical depiction of ∆Sj . It may nonetheless not be pj
p j1
∆Sj p j2
φ′ j
xj1
xj2
xj
Figure 4.1: Consumer’s Surplus in the Simplest Case. clear why this works out to be a correct measure of the change in the consumer’s utility. Perhaps we can clear things up a bit, however, by expressing everything in terms of the demand and indirect utility functions, rather than the inverse demand and direct utility functions. Let’s denote the demand functions for the xj ’s [which are simply the inverses of the φj (·)’s] by ‘δj ;’ so that: φj [δj (qj )] = qj
for j = 1, . . . , n.
(4.44)
We will denote the indirect utility function by ‘v(p0 , p, w);’ or, upon normalizing by the price of the num´eraire: def V (q, w/p0 ) = v 1, (1/p0 )p, w/p0 . It is obvious, of course, that what we wish to evaluate in this case is: V (q 2 , w2 /p20 ) − V (q 1 , w1 /p10 ).
(4.45)
114
Chapter 4. Consumer Demand Theory
Now, in this case, [using (4.41)] it is easy to see that V (q, w/p0 ) takes the form: V (q, w/p0 ) = w/p0 − q · δ(q) +
n
def
j=1
ψj (qpj ) = w/p0 +
n j=1
ϕj (qj ),
(4.46)
where: ψj (qj ) = φj [δj (qj )], and: ϕ(qj ) = ψj (qj ) − qj · δj (qj )
for j = 1, . . . , n.
Consequently: ∂V = ϕj (qj ) = ψ (qj ) − δj (qj ) − qj · δj (qj ). ∂qj
(4.47)
However, we have: ψj (qj ) =
d φj [δj (qj )] = φj [δj (qj )]δj (qj ); dqj
and from (4.44) and (4.47) it then follows that: ϕj (qj ) = −δj (qj ). Another examination of Figure 4.1 will then make it clear that in the situation under analysis there: ∆Sj =
qj1
qj2
δj (qj )dqj =
qj2
qj1
−δj (qj )dqj = ϕj (qj2 ) − ϕj (qj1 ).
(4.48)
Notice in particular, that if only the j th (normalized) price is changed (all other prices and income remaining the same); say in the amount ∆qj , then we can unambiguously interpret the beneﬁt (or cost) to the consumer of the price change as the diﬀerence: qj +∆qj
∆Sj =
−δj (t)dt = ϕj (qj + ∆qj ) − ϕj (qj ).
(4.49)
qj
In fact, it is worth noting that in this situation, if only a subset, J ⊆ {1, . . . , n}, of prices change, then we will have: ∆u =
j∈J
qj +∆qj
qj
−δj (t)dt .
The situation which we analyzed in the above example is, obviously, very special.12 If you look back at the analysis we have done to this point, probably the ﬁrst thing which stands out is that the fact that the demand for the j th commodity was a function of pj alone meant that we were able to make an unambiguous interpretation of the monetary value of a change in the j th price. Moreover, it allowed us to 12
However, see also Example 4.47, below.
4.10. Consumer’s Surplus
115
evaluate an integral of an (n + 1)dimensional function as the sum of n + 1 simple Riemann integrals. Let’s see if we can extend the deﬁnitions used in the example to some extent, and seek to ﬁnd a function, W (·), of (p1 , w1 ) and (p2 , w2 ) which is such that: W [(p1 , w1 ), (p2 , w2 )] ≥ 0 ⇐⇒ (p2 , w2 )G∗ (p1 , w1 );
(4.50)
in which case we will say that W (·) provides an acceptable indicator of welfare change for the consumer. We will begin by considering the possibility of obtaining such a function by integrating an observable function of some kind. In this connection, the natural notion of integration to use, since we want to allow for the fact that p1 and p2 may diﬀer in several coordinates, is that of the line integral. Without going into too many technical details,13 the concept of a line integral works by reducing the integral of a multidimensional function to a standard RiemannStieljes integral; doing so by making use of the notion of a ‘path function;’ which, for the situation we are considering here is simply a function ω : [0, 1] → Ω, where we deﬁne Ω = Rn++ × R+ . We will say that such a path is ‘polygonal’ if its graph is the union of a ﬁnite number of line segments; and that it connects (p1 , w1 ) and (p2 , w2 ) iﬀ we have: ω(0) = (p1 , w1 ) and ω(1) = (p2 , w2 ). (Incidentally, we are going back to our usual supposition that there are n commodities in the remainder of our discussion.) 4.45 Example. Two particularly simple polygonal paths connecting (p1 , w1 ) and (p2 , w2 ) are given by: ω(t) = t(p2 , w2 ) + (1 − t)(p1 , w1 ), and, in the case in which n = 2: ⎧ 1 1 ∗ 1 1 1 ⎪ ⎨(p , w ) + 3t[(p , w ) − (p , w )] ∗ 1 2 1 ω(t) = (p , w ) + (3t − 1)[(p , w ) − (p∗ , w1 )] ⎪ ⎩ 2 1 (p , w ) + (3t − 2)[(p2 , w2 ) − (p2 , w1 )] where we deﬁne p∗ by:
p∗ = (p21 , p12 ).
for 0 ≤ t ≤ 1/3, for 1/3 < t ≤ 2/3, for 2/3 < t ≤ 1;
To continue, given a function, f : Ω → Rn+1 , and a path function, ω, the integral of f (·) from (p1 , w1 ) to (p2 , w2 ), given a path function ω, satisfying: ω(0) = (p1 , w1 ) and ω(1) = (p2 , w2 ), is given by:
0
13
1
f [ ω(t)] · dω(t) =
n+1 j=1
1
f j [ω(t)]dωj (t),
0
For more detailed analysis, see Chipman and Moore [1976a, 1990].
(4.51)
116
Chapter 4. Consumer Demand Theory
whenever each of the RiemannStieltjes on the righthandside of the equation exists.14 It should also be noted that if n = 0; that is, if the integrand function is realvalued, then the line integral reduces to a standard RiemannStieljes integral. Now, the trouble with trying to make use of a line integral to obtain an acceptable indicator of welfare change is that the value of the line integral may be dependent upon the path function chosen; an obviously unacceptable attribute for a measure of consumer welfare. However, if the integrand function, f (·), is continuously diﬀerentiable, then given any two polygonal path functions, ω and ω, which connect (p1 , w1 ) and (p2 , w2 ), we have:
1
f [ ω(t)] · dω(t) =
0
1
f [ ω(t)] · dω(t).
(4.52)
0
In this case, we will say that f : Ω → Rn+1 provides an acceptable indicator of welfare change on Ω iﬀ, for all (p1 , w1 ), (p2 , w2 ) ∈ Ω and any polygonal path function connecting the two points, we have:
1
f [ ω(t)] · dω(t) ≥ 0 ⇐⇒ (p2 , w2 )G∗ (p1 , w1 );
(4.53)
0
where G∗ is the consumer’s indirect preference relation. The remarkable thing about this is that if the line integral is independent of path on Ω, then there exists a continuouslydiﬀerentiable function, V : Ω → R, called a potential function, such that, given any polygonal path function, ω connecting (p1 , w1 ) and (p2 , w2 ), we have: 1
f [ ω(t)] · dω(t) = V (p2 , w2 ) − V (p1 , w1 );
(4.54)
0
and for all (p, w) ∈ Ω: f j (p, w) = Vj (p, w)
for j = 1, . . . , n + 1;
(4.55)
that is, the j th partial derivative of the potential function must equal the j th coordinate function of f (·) at each (p, w) ∈ Ω, and for each j = 1, . . . , n + 1. In fact, the converse is also true: if there exists a continuously diﬀerentiable function V : Ω → R which satisﬁes the system (4.55), then (it is a potential function for the integrand f (·) and) equation (4.54) holds for any ‘piecewise smooth’ path function connecting (p1 , w1 ) and (p2 , w2 ). But now if we combine (4.53) and (4.54), we see that if a continuously diﬀerentiable function, f , provides an acceptable indicator of welfare change on Ω, then the corresponding potential function must be an indirect utility function representing G∗ ! It then follows from the AntonelliAllenRoy theorem (Theorem 4.28) that we must have, for all (p, w), ∈ Ω: f j (p, w) = −f n+1 (p, w)hj (p, w),
(4.56)
where ‘hj (·)’ denotes the demand function for the j th commodity. 14 If you haven’t studied RiemannStieltjes integrals, don’t worry. We aren’t going to do anything technical with such integrals.
4.10. Consumer’s Surplus
117
Now suppose we look at the functions most often used in this integrand when investigators are doing empirical estimates of consumer’s surplus. In such investigations, the integrand most often used is given by:15 f j (p, w) = −hj (p, w)
for j = 1, . . . , n,
(4.57)
and: f n+1 (p, w) = 1.
(4.58)
As was pointed out in Chipman and Moore [1976a, p. 79], there exists no preference relation for which this integrand function provides an acceptable indicator of welfare change! Why? because the (n + 1)st function must be the marginal utility of income for an indirect preference relation representing the consumer’s (indirect) preferences, and as was established more than sixty years ago by Paul Samuelson (Samuelson [1942]), there exists no indirect preference relation yielding a marginal utility of income which is independent of both prices and income.16 Incidentally, while we considered an integrand [in equations (4.57) and (4.58)] ostensibly appropriate17 for analyzing changes in both prices and income, the method discussed in the preceding paragraph is no more appropriate if only prices have changed; for, as we have seen, the function corresponding to the j th price must take the form of the product of a valid marginal utility of income times the demand function for the j th commodity. However, in Example 4.44 we considered an admittedly very special situation in which something very close to the procedure discussed in the above paragraph worked just ﬁne. There is another case in which a similar integrand deﬁnes a valid measure of welfare change; the case in which the consumer’s preferences are homothetic. To see this, recall ﬁrst that if a function f : Ω → Rn+1 is continuously diﬀerentiable, then its line integral is independent of path, and there exists a potential function, V (·) satisfying (4.54) and (4.55). Suppose, then, that a consumer’s preferences are homothetic, and yield a continuously diﬀerentiable demand function. Then, as was noted in Section 4.8, above, we can write this function in the form: h(p, w) = g(p)w. deﬁned by: Consequently, the integrand function f : Ω → Rn+1 + f j (p, w) = −g j (p)
for j = 1, . . . , n,
(4.59)
and: f n+1 (p, w) = 1/w,
(4.60)
yields line integrals independent of path. Moreover, as noted in the previous section, in the homothetic case, indirect preferences can be represented by a function of the 15 Typically only a subset of prices, and correspondingly, of demand functions are used in the investigation. This fact does nothing to invalidate the argument presented here, however. 16 A somewhat simpler argument than Samuelson’s is presented in Chipman and Moore [1976, pp. 79–80] who simply note that, since any indirect utility function must be positively homogeneous of degree zero, the marginal utility of income must necessarily be homogeneous of degree minus one. 17 Since it is n + 1dimensional.
118
Chapter 4. Consumer Demand Theory
form w/γ(p). Taking the log of this function yields an indirect utility function also representing the consumer’s indirect preferences, and is given by: V (p, w) = log w − log γ(p).
(4.61)
It is easily shown that this is a potential function for the integrand deﬁned in (4.59) and (4.60); and, consequently, it follows that said integrand function provides an acceptable indicator of welfare change on Ω. A more detailed discussion of valid possibilities for the DupuitMarshall (line integral) type of consumer’s surplus can be found in Chipman and Moore [1976, 1990], and we will consider the possibilies for extending the results we’ve considered here to the multiconsumer case (to obtain consumers’ surplus) in Chapter 15. However, before leaving this topic, I should mention the fact that there is a silver lining to these dark clouds. While conventional consumer’s surplus analysis is not theoretically correct, this fact is sometimes not as critical as it appears at ﬁrst glance. The reason is this: if one is estimating the demand function for a consumer (or for a representative consumer), the parameters which must be estimated to deﬁne the demand function (more speciﬁcally, the vector of demand functions) are often suﬃcient to deﬁne (identify) the corresponding indirect utility function, which can then be used to evaluate the desirability of the change. For example, suppose we have determined that a CobbDouglas demand function ﬁts the data well, and are attempting to derive the desirability of a change from (p1 , w1 ) to (p2 , w2 ). In this situation, the consumer’s demand functions are given by: xj = aj w/pj
for j = 1, . . . , n;
and estimating the demand functions amounts to determining the values of the aj for this consumer. But, the values of the aj determine an indirect utility function for the consumer; in fact, we know that the function: v(p, w) = n
w
aj j=1 (pj /aj )
,
is the corresponding indirect utility function. In order to evaluate the desirability of the change, therefore, we need only evaluate: v(p2 , w2 ) − v(p1 , w1 ). The bad news associated with this point is that it may be necessary to estimate the whole system of demand functions in order to obtain the values of the parameters which determine the indirect utility function. The fact that applied economists often do consumer’s surplus analysis for cases where only one or two prices change, and, correspondingly, only one or two demand functions need to be estimated in order to do conventional consumer’s surplus estimates, is surely one of the principal reasons that this sort of work appears so often in the literature (and that so many articles have been published analyzing the size of the error involved in using the area under conventional [Marshallian] demand curves as an estimate of ‘true’ consumer’s surplus). However, in some cases the evaluation of changes in indirect utility can
4.10. Consumer’s Surplus
119
be done using only the parameters associated with the demand functions for the commodities whose prices change. This point is pursued further in Example 4.48, below, and in Exercises 8–10 at the end of this chapter. Let’s now turn our attention to two alternative indicators of welfare change which were originally introduced by Hicks [1942]. We will ﬁrst consider compensating variation, which is deﬁned as the amount that would need to be added to a consumer’s wealth after a price change in order to make the consumer exactly as well oﬀ after the change as before. Having already studied the properties of the income compensation function, it should be apparent that we can deﬁne the compensating variation of a proposed change from (p1 , w1 ) to (p2 , w2 ) as: CV (p1 , w1 ), (p2 , w2 )] = µ(p2 ; p1 , w1 ) − w2 .
(4.62)
Since this quantity can equivalently be expressed as: µ(p2 ; p1 , w1 ) − µ(p2 ; p2 , w2 ); it follows from the fact that µ(p2 ; ·) is an indirect utility function representing G∗ that the change should be undertaken if, and only if: CV (p1 , w1 ), (p2 , w2 )] ≤ 0, although it is probably more natural to turn this around to deﬁne the compensating variation criterion for welfare improvement, W C (·), by: W C (p1 , w1 ), (p2 , w2 ) = −CV (p1 , w1 ), (p2 , w2 )] = w2 − µ(p2 ; p1 , w1 ).
(4.63)
Not only does this provide an acceptable indicator of welfare change, but in the case in which w2 = w1 − C, with C being the cost of the change, notice that: W C (p1 , w1 ), (p2 , w2 ) ≥ 0 ⇐⇒ w1 − µ(p2 ; p1 , w1 ) ≥ C; which means that we obtain a valid criterion of the general form of equation (4.38), with w1 − µ(p2 ; p1 , w1 ) as a measure of benﬁt. So, we have seen the good news regarding the theory of compensating variation. The question now is, what is the bad news? Well, there are problems with respect to actually estimating µ(p2 ; p1 , w1 ) in reallife practical situations; but our concern here will be with the theory, and the theoretical diﬃculty with this welfare criterion is that it cannot generally be used to rank projects. That is, if two diﬀerent policies or projects are being considered; with project/policy t yielding the pricewealth combination (pt , wt ), for t = 1, 2, and if the status quo pricewealth combination is (p0 , w0 ), it is tempting to say that project/policy 2 is the better one if: W C (p0 , w0 ), (p2 , w2 ) > W C (p0 , w0 ), (p1 , w1 ) .
(4.64)
Unfortunately, this inference is not generally valid, as is shown by the following example.
120
Chapter 4. Consumer Demand Theory
4.46 Example. Let X = R2+ , and, using the generic notation ‘(x, y)’ to denote points in R2 , suppose a consumer’s preferences can be represented by the utility function: u(x, y) = (x + 2) · y. Then (see Exercise 5, at the end of this chapter) the demand functions for x and y are given by: w − 2p1 w + 2p1 x= and y = , 2p1 2p2 respectively. Consequently, an indirect utility function for the consumer is given by: v(p, w) =
w − 2p w + 2p w2 − 4(p )2 1 1 1 = . 2p1 2p2 4p1 p2
Now suppose the status quo (current) pricewealth situation for the consumer is (p0 , w0 ), where: p0 = (1, 1) and w0 = 2; and suppose project one will result in the pricewealth pair (p1 , w1 ), where: p1 = (2, 1) and w1 = 8. Then, since v(p0 , w0 ) = (4 − 4)/4 = 0, while: v(p1 , w) =
w2 − 16 =0 8
iﬀ w = 4, it follows that µ(p1 ; p0 , w0 ) = 4, and therefore: W C (p0 , w0 ), (p1 , w1 ) = 8 − 4 = 4. Now suppose a second project results in the pricewealth pair (p2 , w2 ), where: p2 = (1, 2) and w2 = 7. Then similar considerations to those of the above paragraph establish the fact that µ(p2 ; p0 , w0 ) = 2, and therefore: W C p0 , w0 ), (p2 , w2 ) = 7 − 2 = 5 > W C (p0 , w0 ), (p1 , w1 ) = 4. However, as you can (and should) readily verify: 5 v(p1 , w1 ) = 6 > v(p2 , w2 ) = 5 . 8 Thus project one should be preferred, despite the contradictory compensating variation comparison.
4.10. Consumer’s Surplus
121
Despite the fact that compensating variation comparisons will not generally allow one to rank projects, there are two special cases in which such a ranking is valid. The ﬁrst case is that in which the consumer’s preference relation is homothetic,if we normalize appropriately. Suppose project t results in the pricewealth pair (pt , wt ), for t = 1, 2, and that normalizing by wealth, we ﬁnd that: W C (p0 , w0 ), (p1 /w1 , 1) > W C (p0 , w0 ), (p2 /w2 , 1) . Then it follows from the deﬁnitions that: 1 − µ p1 /w1 ; p0 , w0 > 1 − µ p2 /w2 ; p0 , w0 ; so that 1/µ p1 /w1 ; p0 , w0 > 1/µ p2 /w2 ; p0 , w0 , and thus, making use of the homogeneity property of the income compensation function: w2 w1 > . 1 0 0 2 µ p ;p ,w µ p ; p0 , w 0 ∗ 2 2 But then it follows from Proposition 4.43 that (p1 , w1 )P (p , w ). Thus, if indirect preferences are homothetic, the relative values of W C (p0 , w0 ), (pt /wt , 1) can be used to rank projects.18 You can also show that similar considerations allow projects to be ranked by compensating variation comparisons if they result in the same wealth (that is, if w1 = w2 ); given that indirect preferences are homothetic. A second situation in which such rankings are valid is set out in the following example.19
4.47 Example. Suppose we once again assume that the consumer’s consumption set is Rn+1 + , and use the generic notation ‘(x0 , x)’ to denote points (commodity bundles) in Rn+1 . We then suppose that the consumer’s utility function takes the form: u(x0 , x) = x0 + ϕ(x), where ϕ : Rn+ → R is concave, continuous, and positively homogeneous of degree a, where 0 < a < 1. For future reference, we note that ϕ can be considered as a utility function on Rn+ , and we will denote the demand function it generates by ‘h,’ which can be written in the form: h(p, m) = g(p)m. From our work in Section 9 of this chapter, we know that if we deﬁne the function ψ by: 1/a ψ(x) = ϕ(x) for x ∈ Rn+ , then the indirect utility function corresponding to ψ can be written in the form: v ∗ (p, m) = m/γ(p), 18 To the best of my knowledge, this, and the fact that compensating variation cannot generally be used to rank projects, was ﬁrst pointed out in Chipman and Moore [1980]. 19 I should mention, however, that the next example does not represent the only additional case in which compensating variation can be used to rank projects. If the consumer’s utility function is of the form used in Example 4.44, which is not a special case of our next example, the compensating variation criterion can be used to rank projects. See also, Chipman and Moore [1980].
122
Chapter 4. Consumer Demand Theory
where: γ(p) = 1/ψ[g(p)]. Consequently, it follows that the indirect utility function corresponding to ϕ can be written in the form: a (4.65) v(p, m) = m/γ(p) . Now, given (p∗0 , p∗ , w∗ ), it can be shown that: ap∗ 1 1−a def 0 m∗ = min w∗ , , γ(p∗ )a maximizes the function:
w ∗ − m m a + ; p∗o γ(p∗ )
and thus we see that, if:
ap 1 1−a 0 , γ(p)a then the consumer’s demand is given by: w≥
(4.66)
x0 = (w − m)/p0 and x = h(p, m), where:
ap 1 1−a 0 , (4.67) γ(p)a and h : Ω → Rn+ is the demand function for ϕ. Consequently, we see that the consumer’s indirect preferences can be represented by the function V , given by: a 1 ap0 /γ(p)a 1−a w − ap0 /γ(p)a 1−a + V (p0 , p, w) = γ(p)a p0 a
ap0 1−a w + (1 − a) = γ(p) p0 m=
Given that we have found the functional form of the indirect utility function, one can then show by straightforward substitution that: ap a ap a µ(p0 , p; p0 , p, w) w 0 1−a 0 1−a = + (1 − a) − (4.68) p0 p0 γ(p) γ(p) Now suppose the current pricewealth situation for the consumer is (p0 , p, w), and that two projects are contemplated which will result in pricewealth situations (pt0 , pt , wt ), for t = 1, 2. If we normalize prices and wealth for the new situations, to obtain: q t = (1/pt0 )pt and y t = wt /pt0 for t = 1, 2; it follows from (4.68) and the homogeneity of the incomecompensation function that: W C (1, q 1 , y 1 ), (p0 , p, w) − W C (1, q 2 , y 2 ), (p0 , p, w) a a a a 1−a 1−a 1 2 = y + (1 − a) − y + (1 − a) . γ(q 1 ) γ(q 2 )
4.10. Consumer’s Surplus
123
Thus we see that: W C (1, q 1 , y 1 ), (p0 , p, w) ≥ W C (1, q 2 , y 2 ), (p0 , p, w)
⇐⇒ V (1, q 1 , y 1 ) ≥ V (1, q 2 , y 2 ). (4.69) Since:
(pt0 , pt , wt )I ∗ (1, q t , y t )
for t = 1, 2,
it follows that project one is preferred if, and only if: W C (1, q 1 , y 1 ), (p0 , p, w) > W C (1, q 2 , y 2 ), (p0 , p, w)
In our discussion of DupuitMarshall consumer’s surplus, we noted that in situations where only a subset of prices change, it may be possible to determine whether indirect utility has increased even if the investigator knows the values of only that subset of parameters corresponding to the commodities whose price has changed. It may also be possible to determine the compensating variation associated with such a change, as the following example demonstrates. 4.48 Example. Suppose a consumer’s preferences can be represented by a CobbDouglas utility function (with initialy unknown parameter values), and that a proposed policy will change a subset, J, of prices and income/wealth from the status quo, (p0 , w0 ) to (p1 , w1 ), where: / J. p1j = p0j for all j ∈ In order to conduct traditional consumer’s surplus analysis in this case, one would need to estimate the J th demand function: aj w xj = for each j ∈ J. pj Since such estimation involved determining (estimated) values of aj , for each j ∈ J, for the remainder of our discussion we will assume that these parameter values are known. To obtain the value of the compensating variation associated with the change, we begin by noting that µ(p1 ; p0 , w0 ) is the value of w which solves the equation: w nj=1 (aj )aj w0 nj=1 (aj )aj n = n 1 0 aj . aj j=1 (pj ) j=1 (pj )
Solving, we ﬁnd: µ(p1 ; p0 , w0 ) =
w0 j∈J (p1j )aj 0 aj . j∈J (pj )
Therefore: w0 j∈J (p1j )aj W C (p0 , w0 ), (p1 , w1 ) = w0 − 0 aj j∈J (pj ) =
w0 (p0 )aj − (p1 )aj . 0 )aj j∈J j j∈J j (p j∈J j
124
Chapter 4. Consumer Demand Theory
It is also worth noting that if we were allow for a change in all prices (that is, if J = {1, . . . , n}), then the above formula becomes: n n w0 W C (p0 , w0 ), (p1 , w1 ) = n (p0 )aj − (p1 )aj ; 0 )aj j=1 j j=1 j (p j=1 j that is, the compensating variation criterion is given by the costofliving indirect utility function, evaluated at (p0 , w0 ), times the change in the costofliving index. Hicks’ second measure of ‘consumer surplus,’ equivalent variation, is deﬁned verbally as the amount which could be added to a consumer’s wealth before a price change in order to leave her or him exactly as well oﬀ without the change as with it. Thus we can deﬁne: EV (p1 , w1 ), (p2 , w2 ) = µ(p1 ; p2 , w2 ) − w1 . (4.70) It is then easy to see that EV (·) is an acceptable indicator of welfare change. Moreover, if we consider the problem of ranking projects/policies as before; with project/policy t yielding the pricewealth combination (pt , wt ), for t = 1, 2, and with the status quo pricewealth combination (p0 , w0 ), we see that EV (p0 , w0 ), (p2 , w2 ) > EV (p0 , w0 ), (p1 , w1 ) , (4.71) if, and only if: µ(p0 ; p2 , w2 ) > µ(p0 ; p1 , w1 ).
(4.72)
µ(p0 ; ·)
It then follows at once from the fact that is a valid indirect utility function (Theorem 4.31) that project two is preferred if inequality (4.71) holds. So, as we have just seen, there is a real advantage in using equivalent variation, rather than compensating variation as a welfare criterion if we are dealing with a single consumer, or if we are comfortable with the assumption of a ‘representative consumer.’ However, when dealing with multiple consumers, the advantages are somewhat reversed. For, while neither criterion can generally be used to rank projects which aﬀect multiple consumers, compensating variation provides the better indicator of welfare change in this situation. To see this, suppose a policy is being contemplated which would change the ith consumer’s pricewealth situation from (p1 , wi1 ) to (p2 , wi2 ), and suppose: m def m 2 wi − µi (p2 ; p1 , wi1 ) > 0. WiC (p1 , wi1 ), (p2 , wi2 ) = (4.73) i=1
i=1
In this case, it can be shown (and is intuitively apparent) that wealth can be redistributed (and/or the costs of the project can be allocated) in such a way as to make each consumer better oﬀ after the change than before. Unfortunately, no such inference follows from the fact that: m m EVi (p1 , wi1 ), (p2 , wi2 ) = µi (p1 ; p2 , wi2 ) − wi1 > 0. (4.74) i=1
i=1
I will leave this discussion at this point, but we will return to a reconsideration of some of the issues taken up in this last paragraph in a later chapter.
4.11. Appendix
4.11
125
Appendix
Before presenting a proof of Theorem 4.5, we must ﬁrst introduce a deﬁnition and some facts from Topology. A.1. Deﬁnition. We shall say that a family of sets, G = {Ga  a ∈ A}, satisﬁes the ﬁnite intersection property iﬀ, for each ﬁnite subset, {G1 , . . . , Gm }, of G, we have: m Gi = ∅. i=1
A.2. Fact (Theorem) from General Topology. If F is a compact set, and G = {Ga  a ∈ A} is a family of closed (and nonempty) subsets of F which satisfy the ﬁnite intersection property, then: Ga = ∅; a∈A
that is, there must exist at least one element of F which is a member of each Ga . Proof of Theorem 4.5 Let (p∗ , w∗ ) ∈ Ω; and, for each x ∈ b(p∗ , w∗ ), deﬁne: Gx = {y ∈ b(p∗ , w∗ )  ¬xP y} = [X \ xP ] ∩ b(p∗ , w∗ ). Then we note that, since P is upper semicontinuous, Gx is a closed subset of b(p∗ , w∗ ), for each x ∈ b(p∗ , w∗ ). Now suppose x1 , . . . , xm is a ﬁnite subset of b(p∗ , w∗ ), and deﬁne: i1 = 1. Then, since P is asymmetric: ¬xi1 P xi1 , and therefore xi1 ∈ Gxi1 . Consequently, if: m / Gxi , xi1 ∈ i=1
/ Gxi2 ; that is: then there exists i2 ∈ {1, . . . , m} \ {i1 } such that xi1 ∈ xi2 P xi1 .
(4.75)
Therefore, since P is asymmetric and irreﬂexive: xi2 ∈ Gxi1 ∩ Gxi2 ; and thus if: / xi2 ∈
m i=1
Gxi ,
then there exists i3 ∈ {1, . . . , m} \ {i1 , i2 } such that: xi3 P xi2 .
(4.76)
126
Chapter 4. Consumer Demand Theory
It now follows from (4.75), (4.76), and the transitivity of P that we also have: xi3 P xi1 .
(4.77)
Now, using (4.76), (4.77), and the asymmetry of P , it now follows that: xi3 ∈ and thus, if / xi3 ∈
3 j=1
m i=1
Gxij ;
Gxi ,
there exists i4 ∈ {1, . . . , m} \ {i1 , i2 , i3 } such that: xi4 P xi3 . We can now show, as before, that: xi4 P xij
for j = 1, 2, 3;
and thus, from the deﬁnition of the Gxj sets: xi4 ∈
4 j=1
Gxij .
Continuing in this fashion, we will, after the k th step (k ≥ 2), have obtained a ﬁnite sequence of distinct points: {xi1 , . . . , xik } ⊆ {x1 , . . . , xm }, such that: xik P xij
for j = 1, . . . , k − 1.
Moreover, having obtained xik , it will either be true that: xik ∈
m i=1
Gxi ,
or there exists: xik+1 ∈ / {xi1 , . . . , xik }
(4.78)
xik+1 P xik .
(4.79)
such that: However, it follows from (4.78), (4.79), and the asymmetry of P that this process must terminate after, at most, m steps. Therefore, there exists j ∈ {1, . . . , m} such that: m xj ∈ Gxi . i=1
To this point, we see that we have shown that the family of sets: G = {Gx  x ∈ b(p∗ , w∗ )},
4.11. Appendix
127
is a family of closed subsets of b(p∗ , w∗ ) which satisﬁes the ﬁnite intersection property. Since we have from Proposition 4.3 of this chapter that b(p∗ , w∗ ) is compact, it therefore follows from Theorem A.2, above, that there exists x∗ ∈ b(p∗ , w∗ ) such that: x∗ ∈ Gx; x∈b(p∗ ,w∗ )
and it is then an immediate consequence of the deﬁnition of the Gx sets that we must have, for all x ∈ b(p∗ , w∗ ): ¬xP x∗ .
The following is Proposition 4.29. We repeat its statement here for convenient reference, before providing a proof. Proposition 4.29. Suppose, in addition to Assumptions I.1 and I.2, that the set Z = h(Ω) is a subset of Rn+ , and is closed, convex, and contains 0. Suppose further that the restriction of the consumer’s (weak) preference relation to Z is a ¯ ∈ Rn++ , and any (p , w ) ∈ Ω, there continuous weak order on Z. Then, given any p exists w ¯ ∈ R+ such that: p, w)G∗ (p , w )}, w ¯ = min{w ∈ R+  (¯ and we have:
(¯ p, w)I ¯ ∗ (p , w ).
¯ ∈ Rn++ , let (p , w ) ∈ Ω be arbitrary, and consider: Proof. Let p W = {w ∈ R+  (¯ p, w)G∗ (p , w )}. Clearly, W is a nonempty subset of R+ , so that inf W exists and is nonnegative. Deﬁne: p, w)G∗ (p , w )}. w ¯ = inf W = inf{w ∈ R+  (¯ We wish ﬁrst to prove that w ¯ ∈ W. Accordingly, deﬁne the sequence wq by: ¯ + 1/q wq = w
for q = 1, 2, . . . ,
p, wq ) xq ∈ h(¯
for q = 1, 2, . . . .
and let xq be such that:
Since xq is contained in b(¯ p, w1 ) ∩ Z = b(¯ p, w ¯ + 1) ∩ Z, which is a compact set, we ¯ ∈ Z such that: may assume, without loss of generality, that there exists x ¯. lim xq = x
q→∞
Moreover, letting x ∈ h(p , w ), we see that (since wq ∈ W for each q): xq Gx
for q = 1, 2, . . . .
128
Chapter 4. Consumer Demand Theory
¯ , we must have: Therefore, since G is continuous on Z and xq → x ¯ Gx . x
(4.80)
Now let x ∈ b(¯ p, w)∩Z ¯ be arbitrary. Then x ∈ b(¯ p, wq ), for each q, and therefore: xq Gx
for q = 1, 2, . . . .
¯ Gx; and, since Once again using the fact that G is continuous on Z, it follows that x ¯·x ¯ = w, ¯ ∈ h(¯ p ¯ we now conclude that x p, w). ¯ Making use of (4.80), we can also conclude that: (¯ p, w)G ¯ ∗ (p , w ). ¯ P x . Then, since h Now suppose, by way of obtaining a contradiction, that x ¯ > 0. Moreover, since G is satisﬁes the budget balance condition, we must have x lower semicontinuous on Z, there exists a neighborhood, N (¯ x), such that: x) ∩ Z : xP x . ∀x ∈ N (¯ Furthermore, since 0 ∈ Z and Z is convex, there exists θ ∈ ]0, 1[ such that: ¯ ∈ N (¯ θx x) ∩ Z, and thus:
¯ P x . θx
However, this cannot be, for then it follows that: def
¯ = θw p·x ¯ < w, ¯ w ˆ = θ¯ is an element of W ; which contradicts the deﬁnition of w. ¯ We conclude, therefore, ¯ Gx , it must be the case that x G¯ xP x ; and thus, since x that ¬¯ x as well. Therefore, ¯ Ix , and we see that (¯ p, w)I ¯ ∗ (p , w ). x Exercises. 1. Let X = Rn+ , and let G be the (weak) preference relation representable by: n xi . u(x) = i=1
Is G nondecreasing? increasing? strictly increasing on Rn+ ? What is the demand function generated by G? Is G homothetic in this case? 2. Suppose a consumer’s (direct) preferences are representable by the function: x i u(x) = min , i ai where ai > 0 for i = 1, . . . , n. Find the consumer’s demand function and a costofliving function for the consumer. 3. Follow the instructions for problem 3, except take: n a xj j , u(x) = j=1
4.11. Appendix
129
with: aj > 0 for j = 1, . . . , n, 4. Show that if a function f : θ > 0, then it is homothetic.
Rn+
and
n j=1
aj = 1.
→ R+ is positively homogeneous of degree
5. Suppose a consumer’s (direct) preferences are representable on Rn+ by the function: n (xj + cj )aj , u(x) = j=1
where: aj > 0 for j = 1, . . . , n,
and
n j=1
aj = 1;
and cj ≥ 0, for j = 1, . . . , n. Show that, deﬁning: c = (c1 , . . . , cn ), the consumer’s demand functions are given by: hj (p, w) =
aj w + p · c − cj , pj
for j = 1, . . . , n. 6. Show that the function V (·) deﬁned in equation (4.61) is a potential function for the integrand function deﬁned in (4.59) and (4.60). 7. Complete the details of the analysis of Example 4.46. 8. Prove part 1 of Proposition 4.24. 9. Show that if a consumer’s preferences can be represented by a CobbDouglas utility function, as in Problem 4, above, and, as in Example 4.48, only a subset, J, of prices are aﬀected by a policy change, then the sign of the change in indirect utility can be determined if only the values of aj , for j ∈ J (and the original and new values for prices and income) are known. In fact, show that in this situation, the indicated information is suﬃcient to determine the ratio of V (p1 , w1 ) to V (p0 , w0 ). 10. Complete the proof of Theorem 4.39 11. Given the conditions of Example 4.48, derive the formula for equivalent variation in this case. 12. Suppose once again that the conditions of Example 4.48 hold, but that in this case the consumer’s preferences can be represented by the function: u(x) = min{xj /aj }, j
where aj > 0 for j = 1, . . . , n. What information is needed to determine equivalent and compensating variation in this case?
Chapter 5
Pure Exchange Economies 5.1
Introduction
In this chapter, we will consider general equilibrium models of ‘pure exchange.’ Such models were justiﬁed in the neoclassical literature by the rationale that what one was doing in such a model was analyzing exchange after production had taken place. It may well be that it is more properly portayed as the analysis of the aggregate eﬀects of consumer demand. In any event, this basic model is a fundamental tool in public economics, international trade models and welfare economics. The reason for this is quite simple; a surprising number of fundamental economic principles can be illustrated and analyzed in the context of a pure exchange economy. Moreover, we are able to examine these principles in a context simpler than that of a production economy, and we gain a bonus in that our study of pure exchange economies will make it easier to understand the theoretical analysis of a production economy which we undertake in Chapters 7 and 8.
5.2
The Basic Framework
In dealing with pure exchange economies, we will always suppose, unless otherwise explicitly stated, that Xi , the ith consumer’s consumption set, is equal to Rn+ . Thus we can think of a (nonprivateownership) exchange economy with m consumers (agents) as being completely speciﬁed by an mtuple of (strict) preference relations and an aggregate commodity, or resource endowment, r. Notationally we will indicate this as follows. When we say that: E = Pi , r , is an exchange economy, we will mean that Pi is the ith consumer’s (strict) preference relation, for i = 1, . . . , m, and that the total commodity bundle available to the economy, collectively, is given by the aggregate resource endowment, r ∈ Rn+ . In dealing with such an economy, we will always suppose (at least) that each Pi is an irreﬂexive binary relation on Rn+ . Occasionally (primarily in Chapter 11) we may wish to emphasize the number of consumers in the economy, and we
132
Chapter 5. Pure Exchange Economies
will do so by writing: E = (Pi m i=1 , r), or: E = (Pi i∈M , r), where we deﬁne M = {1, . . . , m}. 5.1 Deﬁnition. Let E = Pi , r be an exchange economy. We shall say that an 1 msequence, xi i∈M is: 1. an allocation for E iﬀ: xi ∈ Rn+
for i = 1, . . . , m.
2. an attainable (or feasible) allocation for E iﬀ xi i∈M is an allocation for E satisfying: m xi = r. i=1
We shall denote the set of all attainable, allocations for E by ‘A(E).’ As or feasible in our deﬁnition of an economy, E = Pi , r , however, we will generally not need to exhibit the number of consumers, and thus we will denote allocations simply by ‘xi ’, rather than ‘xi i∈M .’ Thus we deﬁne the set of attainable allocations for E by: m n A(E) = xi ∈ Rmn xi = r . +  xi ∈ R+ , for i = 1, . . . , m, & i=1
When we are considering competitive equilibria for an exchange economy, however, we will need to specify a distribution of ownership for the aggregate resource endowment, r; that is, we will deal with private ownership exchange economies, where individual resource endowments are speciﬁed for each of the m consumers. Formally, when we say ‘E = Pi , r i is a private ownership exchange economy,’ we shall mean that the associated economy: m E = Pi , ri , i=1
is an exchange economy, and we will let ‘rij ’ denote the ith consumer’s initial endowment of the j th commodity. We will refer to r i as the ith consumer’s endowment (or resource endowment). In the remainder of this chapter, we will be concerned much of the time with competitive equilibria for a pure exchange economy. It is assumed that, in a competitive exchange economy, consumers take the vector of prices, p ∈ Rn+ , as given, and choose the best available commodity bundle, given this price vector and their wealth, which will now be given by: wi = p · r i Thus we will assume that the
ith
for i = 1, . . . , m.
consumer chooses that (or a) bundle, xi satisfying: def
xi ∈ Rn+ , p · xi ≤ wi = p · r i , 1
(5.1)
The notation ‘ xi ’ is, of course, intended to suggest a ﬁnite sequence.
(5.2)
5.3. The Edgeworth Box Diagram
133
and: (∀xi ∈ Rn+ ) : xi Pi xi ⇒ p · xi > wi ;
(5.3)
so that xi ∈ hi (p, wi ), where hi (·) is the consumer’s demand correspondence. 5.2 Deﬁnition. Let E = Pi , r i be aprivate ownership exchange economy. We shall say that an (m + 1)ntuple, xi , p is a competitive ( or Walrasian) equilibrium iﬀ: 1. p ∈ Rn+ \ {0}, 2. xi ∈ hi (p, wi ), where wi = p · r i for i = 1, . . . , m,, 3. xi is an attainable allocation for E. ith
Often in the general equilibrium literature, an allocation for a pure exchange economy is said to be ‘feasible’ if: m xi ≤ r. i=1
Where this deﬁnition of feasibility is used, an additional requirement is added to the deﬁnition of a competitive, or Walrasian equilibrium; namely that: p · (r − x) = 0, where:
m i=1
def
xi = x.
We will discuss this alternative deﬁnition in Chapter 7.
5.3
The Edgeworth Box Diagram
Surprisingly enough, a great many of the important results in the theory of pure exchange economies can be illustrated quite handily in the context of a twoconsumer, two commodity economy; and, thanks to a very clever invention of the economist F. Y. Edgeworth, we can illustrate much of the analysis diagrammatically. The device in question is the socalled ‘Edgeworth Box’ diagram, and is developed as follows. In the diagram on the next page, we have supposed that the consumers, Ms. 1 and Mr. 2, have the initial endowments, r 1 and r 2 , respectively, and we have then used the parallelogram law of addition to ﬁnd the aggregate resource endowment, r. In our diagram we would like to graph the set of all possible allocations of commodity bundles between the two consumers; that is, we would like to graph the set: A(E) = {xi ∈ R4+  x1 + x2 = r}. Unfortunately, it is a bit diﬃcult to graph a fourdimensional space, especially on a twodimensional page. However, Edgeworth developed the basic trick which allows us to construct such a graph; we do this by inserting a second set of coordinate axes in our graph. More speciﬁally, we will indicate the commodity bundle available to Ms. 1 in the usual way; reading the quantitites of the two commodities available to her in the usual way on the axes in Figure 5.1. However we will read the quantities available to Mr. 2 on axes oriented to (that is, with the origin at) the aggregate resource endowment, r, and reading from right to left for the quantity of the ﬁrst commodity
134
Chapter 5. Pure Exchange Economies x2
r
r2
r1
x1
Figure 5.1: The Allocation Space.
(x21 ) and reading down for the quantity of the second commodity available to Mr. 2 (x22 ). Thus, in Figure 5.2, on the next page, the point x∗i in the diagram represents an attainable allocation, with quantities as indicated in the diagram; since the quantities of the two commodities going to the two consumers necessarily add up to the totals available in the aggregate resource endowment. Notice also that, using the axes we have constructed for Mr. 2 that his resource endowment will now coincide (reading the quantities along the axes labled ‘x21 ’ and ‘x22 ’) with r 1 . The slightly tricky thing about this sort of diagram is the representation of the consumers’ respective indiﬀerence maps. Once again there is no particular problem in connection with Ms. 1’s indiﬀerence map, we can represent it in the usual way; only remembering that Ms. 1’s consumption set extends to the ‘north’ and ‘east’ of the boundaries of the box. Mr. 2’s indiﬀerence map may look a bit strange, however, if this is the ﬁrst time you have enountered an Edgeworth Box diagram, and will, if Mr. 2 has the sort of preference relation favored in textbook diagrams, look something like the curves labled ‘I2 ,’ ‘I2 ,’ and so on, in Figure 5.2. The ﬁrst thing to keep in mind explaining this graph is that we would expect these indiﬀerence curves to be convex to the two axes along which we measure Mr. 2’s consumption quantities. Secondly, of course, Mr. 2’s prefences will generally increase as we move downward and to the left in our box diagram; and thirdly, we need to keep in mind the fact that Mr. 2’s consumption set (and indiﬀerence curves) will extend to the south and west beyond the boundaries of the box (as is indicated in Figure 5.2). Now, “for our next trick,” let’s see if we can obtain a graphical depiction of a competitive (or Walrasian) equilibrium in such a diagram. Suppose that the price vector p∗ prevails in our economy, as indicated in Figure 5.3. Then Ms. 1’s budget
5.3. The Edgeworth Box Diagram
135
x12 x*21
r
x21 x*12
x*22
I'2
〈x*i〉
I2
.
〈r i〉
x*11
x11
x22
Figure 5.2: The Edgeworth Box. line will be perpendicular to (the directed line segment) p∗ and go through r 1 . x12 r x21 x' 2
x'1 p*
〈r 1〉
x22
x1
Figure 5.3: A Nonequilibrium Price. Similarly, 2’s budget line will be perpendicular to the price vector p∗ , and pass through r 2 . The handy thing about all of this is that if we measure quantities along the new axes we’ve constructed for Mr. 2, his budget line will coincide with that for Ms. 1. To see this, note ﬁrst that the line we have constructed for Ms. 1 passes through 2’s resource endowment, r 2 (when we measure quantities along the new axes). Secondly, recall that the slope of a line is uniquely determined by its angle of incidence with the horizontal axis. Moreover, we know that a transversal between two parallel lines forms equal angles of incidence with the two parallel lines (thus the two angles marked in Figure 5.3 are equal to one another). On the other hand, keep in mind the fact that the consumption bundles chosen by the two consumers need
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Chapter 5. Pure Exchange Economies
not coincide. Thus, we may have the sort of situation depicted in Figure 5.3, with the bundles demanded by the two consumers denoted by ‘x1 ’ and ‘x2 ,’ respectively. Notice that we then have excess demand for the ﬁrst commodity and excess supply of the second. In order to have a competitive equilibrium, the bundles demanded by the two consumers must then coincide, in order that demands for the two commodities add up to the exact amounts available of the two goods. Thus, with the indiﬀerence maps indicated in Figure 5.4, it can easily be seen that both consumers are maximizing preferences, subject to their budget constraint, at x∗i ; which is, therefore, a competitive equilibrium allocation. x12
x21
.
〈x*i 〉
〈r i〉
. x11 x 22
Figure 5.4: A Competitive Equilibrium. Having examined the graphical depiction of a competitive equilibrium, let’s take a look at an algebraic analysis of an example in the twobytwo exchange case. (While we will only go through an analytic solution for this example, you should also try to depict the equilibrium in an Edgeworth Box diagram.) 5.3 Example. Here we will be considering a twoconsumer, twocommodity economy, E, in which the consumers’ preferences can be represented by the utility functions: ui (x1 ) = x11 + x12 , and u2 (x2 ) = x21 x22 , respectively. We will consider two situations, and determine whether a competitive equilibrium exists in each case, and if so, what are the competitive allocations and
5.3. The Edgeworth Box Diagram
137
prices. I will list the two speciﬁcations, and recommend that you try to determine the answers before reading the analyses which follow. a. Suppose ﬁrst that the consumers’ initial endowments are given by: r 1 = r 2 = (5, 0). b. Suppose this time that the consumers’ initial endowments are given by: r 1 = (16, 4) and r 2 = (16, 0), respectively. Analyses. a. In this situation, there will be no competitive equilibrium. To see this, notice that if the price of the ﬁrst good is zero, then consumer one (Ms. 1) will demand an inﬁnite amount of the commodity. On the other hand, if p1 is positive, then Mr. 2 will have a positive income, and will then demand a positive quantity of good 2, of which there is none available. b. If there is to be a competitive equilibrium in this situation, then it is clear that the price of the ﬁrst commodity will have to be positive. Consequently, we can normalize to set p1 = 1. If we then were to have p2 < 1, consumer one would demand more than 4 units of the commodity (consumer one’s income/wealth will be equal to 16 + 4p2 , and given p2 < 1 = p1 , Ms. 1 will demand 16/p2 + 4 units of commodity 2). Now, since comsumer 2 has a CobbDouglas utility function, Mr. 2’s demand for the second commodity is given by: x22 =
p · r2 8 = . 2p2 p2
Consequently, if p2 = 1, Mr. 2 will demand 8 units of the second commodity, of which there are only 4 units available. Therefore, we see that if an equilibrium exists, we must have p2 > 1; in which case, Ms. 1 will demand only commodity one, and equilibrium will thus require that: x22 = 8/p2 = 4; that is, p2 = 2. If we check this out, we see that with p = (1, 2), Mr. 2 demands 8 units of commodity one and 4 units of commodity two. Furthermore, Ms. 1, with linear preferences, will spend all of her income on commodiy one. So, Ms. 1’s income is: p1 · 16 + p2 · 4 = 16 + 8 = 24, and thus Ms. 1’s demand for commodity one is: x11 = 24/p1 = 24. Adding, we then see that demand equals supply for each commodity, and therefore we do have a competitive equilibrium in this case.
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Chapter 5. Pure Exchange Economies
5.4
Demand and Excess Demand Correspondences
In the context of a pure exchange economy, and given a price vector p ∈ Rn++ , a pricetaking consumer will choose a commodity bundle, xi , satisfying: def
xi ∈ hi (p, p · r i ) = di (p).
(5.4)
Making use of this, we deﬁne the following. 5.4 Deﬁnition. Given the ith consumer’s demand correspondence, as deﬁned in equation (5.4), we deﬁne the ith consumer’s excess demand correspondence, ei : Rn++ → Rn , deﬁned by: ei (p) = di (p) − r i . (5.5) 5.5 Deﬁnitions. If E = Pi , r i is a private ownership exchange economy, we deﬁne the aggregate demand correspondence, d(·), for E, by: d(p) =
m i=1
di (p),
(5.6)
and the aggregate excess demand correspondence for E, e(·), by: e(p) =
m i=1
[di (p) − r i ] =
m i=1
di (p) −
m i=1
r i = d(p) − r.
(5.7)
The proof of the following facts will be left as an exercise. Facts regarding the aggregate demand correspondence: 1. The aggregate demand correspondence, d(·), will be positively homogeneous of degree zero in p, 2. The aggregate demand correspondence, d(·), will satisfy: for all p ∈ Rn++ :
∀x ∈ d(p) : p · x ≤ p · r,
3. The price vector p ∈ Rn++ deﬁnes a competitive equilibrium for E if, and only if, there exists x ∈ d(p) satisfying: x = r. Making use of the deﬁnition of a consumer’s excess demand correspondence, you can easily prove the following, very fundamental results. 5.6 Proposition. Let E = Pi , r i be a private ownership economy. Then for n each i, each p ∈ R++ , and each z i ∈ ei (p): p · z i ≤ 0.
(5.8)
Furthermore, if Pi is locally nonsaturating, then we will have:
∀p ∈ Rn++ ∀z i ∈ ei (p) : p · z i = 0.
(5.9)
5.4. Demand and Excess Demand Correspondences
139
5.7 Proposition. [Walras’ Law (Weak Form)] Let E = Pi , r i be a private n n ownership exchange economy, and e : R++ → R be the aggregate excess demand correspondence for E. Then, given any p ∈ Rn++ , and any z ∈ e(p), we have: p · z ≤ 0.
(5.10)
5.8 Corollary. [Walras’ Law (Strong Form)] If, in addition to the other hypotheses of 5.7 we have: Pi is locally nonsaturating, for i = 1, . . . , m, then, given any p ∈ Rn++ , and any z ∈ e(p): p · z = 0. In our next result, we will say that the j th market is in equilibrium, given p ∈ Rn++ and z ∈ e(p), iﬀ zj = 0. 5.9 Corollary. [Walras’ Law (Original Form)] Let E = Pi , r i be a private ownership exchange economy in which Pi is locally nonsaturating, for i = 1, . . . , m; and suppose that p∗ ∈ Rn++ and z ∈ e(p∗ ) are such that n − 1 of the n markets are in equilibrium. Then the nth market must be in equilibrium as well. Proof. Suppose that for all j = k, the j th market is in equilibrium. Then by Corollary 5.8, above, we have that: p j z j + p k zk . (5.11) 0=p·z = j=k
However, by assumption we have:
pj zj = 0;
j=k
and thus it follows from (5.11) that pk zk = 0. Since pk > 0, it now follows that zk = 0. Walras’ Law is a very useful property of the aggregate excess demand correspondence, and we have shown that it holds under quite general conditions. Moreover, it is also true that this correspondence will be positively homogeneous of degree zero, and because of this, we will often ﬁnd it convenient to suppose that the domain of the correspondence is ∆n (as will be done in Theorem 5.9, below). We will later study conditions suﬃcient to imply that the aggregate excess demand correspondence satisﬁes suﬃciently strong continuity properties as to enable one to prove the existence of a competitive equilibrium. In the meantime, let’s consider some additional examples of competitive equilibrium in a pure exchange economy. 5.10 Examples. 1.Once again we consider a 2person, 2commodity economy; this time supposing that the ith consumer’s preferences can be represented by the utility function: ui (xi ) = (xi1 )ai · (xi2 )1−ai ,
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Chapter 5. Pure Exchange Economies
with 0 < ai < 1, for i = 1, 2; and suppose the consumers’ initial endowments are given by: r 1 = (r11 , 0) and r 2 = (0, r22 ), where: r11 > 0 and r22 > 0. Show that, if we normalize to set p2 = 1, we can ﬁnd equilibrium p1 as a function of a11 , a21 , r11 and r22 , and that: a2 r22 ∂p1 ∂p1 a2 =− and = . ∂r11 (1 − a1 )(r11 )2 ∂r22 (1 − a1 )(r11 ) (And thus ∂p1 /∂r11 < 0 and ∂p1 /∂r22 > 0.) Analysis. Recalling Walras Law (original form), we see that it suﬃces to ﬁnd equilbrium in the market for the ﬁrst commodity. With the price of the second commodity normalized to p2 = 1, and with the given initial endowments, the demands of the two consumers for the ﬁrst commodity are given by: x11 =
a1 p1 r11 a2 r22 and x21 = , p1 p1
respectively. Equilibrium in the ﬁrst market thus requires: a1 p1 r11 a2 r22 = r11 . + p1 p1 Solving, we then obtain: p1 =
a2 r22 ; r11 (1 − a1 )
which, when diﬀerentiated, yields the indicated values of the partial derivatives. 2. This time we generalize the last example to consider m consumers with initial endowments r i , along with the utility functions used in the last example: ui (xi ) = (xi1 )ai · (xi2 )1−ai
for i = 1, . . . , m.
Show that, once again normalizing to set p2 = 1, the equilibrium price for the ﬁrst commodity is given by: m ai ri2 p1 = m i=1 . i=1 (1 − ai )ri1
The facts concerning aggregate demand which we noted earlier raise an interesting question, namely: are there further qualitative conditions of aggregate excess demand correspondences which hold under the kinds of assumptions we have been making here and in the previous chapter. Unfortunately, H. Sonnenschein [1973 and 1974] established results which pretty much showed that there are no other qualitative implications for market (aggregate) demand which follow from the standard assumptions about individual preferences. Sonnenschein’s original results have been
5.4. Demand and Excess Demand Correspondences
141
extended and reﬁned in various ways; but our next result is what Shafer and Sonnenschein refer to in their 1982 survey as a ‘state of the art’ result, and is due to Debreu [1974]. For a proof, consult the original article, or the Shafer and Sonnenschein survey, where a somewhat simpler argument than Debreu’s is presented. In the Debreu result, ‘∆ ’ will be used to denote that portion of ∆n satisfying: pj > for j = 1, . . . , n. 5.11 Theorem. (Debreu). Let F : ∆n → Rn+ be a continuous function satisfying: (∀p ∈ ∆n ) : p · F (p) = 0. Then, for any ∈ ]0, 1/n [, there exists an nconsumer exchange economy, E = Pi , r i , such that each Pi is asymmetric, negatively transitive, continuous, strictly convex, and increasing, and such that F is the aggregate excess demand function for E on ∆ . While the Debreu result stresses the limitations on the qualitative properties of aggregate demand correspondences in general, the structure of a general equilibrium model (in particular, of a pure exchange model) places functional restrictions on incomes. More exactly, in a pure exchange economy, individual wealth is determined solely by prices (given resource endowments), and thus incomes cannot vary independently. Because of this, one can in some cases establish stronger properties for aggregate demand correspondences. Thus, for example, the following result follows from Theorem 4.2 and Example 5.2 of Chipman and Moore [1979]. 5.12 Theorem. Let E = Pi , r i be a private ownership exchange economy in which Pi is an asymmetric and negatively transitive binary relation which is: 1. continuous, 2. nondecreasing, and 3. homothetic for i = 1, . . . , m; and suppose δ ∈ ∆m and r ∈ Rn+ are such that: r i = δi r
for i = 1, . . . , m.
(5.12)
Then the aggregate demand correspondence for the economy is that generated by the utility function, U , given by: m m U (x) = max [ui (xi )]δi  xi ≤ x ; (5.13) i=1
i=1
where ui is any positively homogeneous of degree one utility function representing Pi , for i = 1, . . . , m. 5.13 Example. Return to the situation described in Example 5.10.2, except assume now that: r i = δi r for i = 1, . . . , m, where: δi ≥ 0, for i = 1, . . . , m,
m i=1
δi = 1,
(5.14)
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Chapter 5. Pure Exchange Economies
and r ∈ R2++ is the aggregate resource endowment. Show that in this case, aggregate demand for the ﬁrst commodity is given by: p r + r m ai (δi p1 r1 + δi r2 ) m 1 1 2 = ai δi ; i=1 i=1 p1 p1 where we have set p2 = 1. Compare this with the demand function of a single consumer having the resource endowment r, and the utility function: u(x) = (x1 )a¯ · (x2 )1−¯a , where: a ¯=
m i=1
ai δi .
Notice also that, since 0 < ai < 1 for each i, we have: m m ai δi < δi = 1, i=1
where the inequality is by (5.14)
5.5
i=1
Pareto Eﬃciency
In this section, we will be studying some orderings which are intended to be candidates for a ‘universally acceptable’ criterion for economic improvement for an economy as a whole. Since our concern here is with normative criteria for economic improvement, we will want to abstract from ownership in most of our present considerations, and deal with ‘exchange economies’ (as opposed to ‘private ownership exchange economies’). 5.14 Deﬁnitions. Let E = Pi , r be an exchange economy. We then deﬁne: ma. the unanimity ordering (the strong Pareto ordering), Q, on X = i=1 Xi by: xi Qxi ⇐⇒ [xi Pi xi for i = 1, . . . , m]. (5.15) b. the Pareto (atleastasgoodas) ordering, R, on X by: xi Rxi ⇐⇒ [xi Gi xi
for i = 1, . . . , m].
(5.16)
c. the strict Pareto ordering, P , on X by: xi P xi ⇐⇒ [(xi Rxi and ¬xi Rxi ].
(5.17)
In dealing with these three orderings, we will use the following terminology. If xi Qxi , we will say that (xi ) is unanimously preferred to (xi ). If xi Rxi , we will say that (xi ) is Pareto noninferior to xi (or that xi is no better than xi in the Pareto sense). If xi P xi , we will say that xi is Pareto superior to xi (or that xi Pareto dominates xi ).
5.5. Pareto Eﬃciency
143
5.15 Proposition. If each Pi is an asymmetric order, then Q, as deﬁned in 5.14.a, above, is also an asymmetric order; and R, as deﬁned in 5.14.b, is reﬂexive. If each Pi is negatively transitive, then R is also transitive; and its asymmetric part, P , is transitive as well (in addition to being asymmetric). [Neither Q nor P will generally be negatively transitive, however.] Proof. I will prove that if each Pi is negatively transitive, then P is transitive (in fact, I will prove a slightly stronger statement, as you will see). I will leave the remainder of the proof as an exercise (see also Exercise 7); although we will look at an example to show that neither Q nor P is necessarily negatively transitive following this proof. Suppose xi , x∗i , and xi are such that: xi Rx∗i and x∗i P xi .
(5.18)
Then from the ﬁrst relation in (5.18 ), we have: xi Gi x∗i
for i = 1, . . . , m,
(5.19)
where Gi is the negation of Pi ; while from the second relationship: x∗i Gi xi
for i = 1, . . . , m,
(5.20)
and, for some h ∈ {1, . . . , m}, we have: x∗h Ph xh .
(5.21)
But then from (5.19), (5.20), and the fact that each Pi is negatively transitive: xi Gi xi while from (5.19) and (5.21): Therefore
xi P xi .
for i =, . . . , m; xh Ph xi .
Generally economists have proceeded as if the following is a universally acceptable value judgment A.1. If xi Qxi , for two feasible allocations, xi and xi , then society should choose xi over xi . We will refer to the acceptance of A.1, above, as a criterion for economic improvement as the ‘unanimity principle;’ and to the acceptance of the corresponding statement when the strict Pareto criterion, P , is substituted for Q in A.1 as the ‘Pareto principle.’ Earlier I emphasized the fact that in dealing with welfare economics, we would make every eﬀort to develop as much of the material as possible assuming only that consumers’ strict preference relations were asymmetric orders (and not necessarily negatively transitive). It is probably already apparent that under these assumptions the strict Pareto ordering deﬁned in 5.14.c will be of somewhat dubious value. The following example illustrates the problem.
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Chapter 5. Pure Exchange Economies
5.16 Example. Consider the exchange economy with 3 consumers (m = 3), 2 commodities (n = 2), and individualistic preferences, and the allocations: x1i = (4, 1), (5, 1), (6, 1) , (5.22) x2i = (1/10, 50), (3, 2), (2, 2) , x3i = (120, 1/20), (40, 1/10), (100, 1/20) . Clearly, it may be diﬃcult for the consumers to make comparisons between these, quite widely dispersed, commodity bundles. Suppose, in fact that the consumers’ preferences are deﬁned by: xi Pi xi ⇐⇒ u(xi ) > u(xi ) + 1
for i = 1, 2, 3;
(5.23)
where: u(xi ) = xi1 · xi2 .
(5.24)
If we then denote the values of this ‘utility function’ at the three allocations by: def u(xti ) = u(xt1 ), u(xt2 ), u(xt3 ) for t = 1, 2, 3, (5.25) we have: u(x1i ) = (4, 5, 6), u(x2i ) = (5, 6, 4),
(5.26)
u(x3i ) = (6, 4, 5). Consequently, according to the strict Pareto ordering deﬁned in Deﬁnition 5.14, we have: x1i P x2i & x2i P x3i & x3i P x1i ; (5.27) in other words, the strict Pareto relation is cyclic (Deﬁnition 3.44) in this case. 5.17 Deﬁnition. Let E = Pi , r be an exchange economy. We shall say that a ∗ feasible allocation for E, xi , is Pareto eﬃcient for E (respectively, strongly Pareto eﬃcient for E) iﬀ there exists no alternative feasible allocation for E, xi , satisfying: xi Qx∗i [respectively, xi P x∗i ], or alternatively:
/ X∗ (E); xi Qx∗i ⇒ xi ∈
where the orderings Q and P are the unanimity and strict Pareto orderings, respectively, and ‘X∗ (E)’ denotes the set of feasible consumption allocations for E. The favorite textbook picture of Pareto eﬃciency in an Edgeworth Box diagram tends to look like Figure 5.5, on the next page. In the diagram, the heavy curve running from the southwest corner to the northeast corner of the box is (usually) called the contract curve,2 and is the locus of the Pareto eﬃcient allocations for the 2
We will use a somewhat more restrictive deﬁnition of the contract curve in Chapter 11.
5.5. Pareto Eﬃciency
145
x12 x 21
x 11 x 22
Figure 5.5: The Contract Curve. economy. Notice that it is the locus of the tangency points of the indiﬀerence curves of the two consumers. We can show that this is necessarily the case in two ways. First we’ll consider a mathematical development. We can characterize a Pareto eﬃcient allocation (for the twoconsumer, twocommodity exchange case) as the solution of the problem: max u1 (x11 , x12 ), (5.28) w.r.t.x11 ,x12
subject to:
u2 (x21 , x22 ) ≥ u∗ ,
(5.29)
x1 + x2 ≤ r,
(5.30)
and: for some feasible value of u∗ .3 If we assume that these utility functions are increasing, as well as being diﬀerentiable, then we can simplify things a bit because we can then replace the inequalities in the two constraints [equations (5.29) and (5.30)] by equalities. This in turn will enable us to use the classical Lagrangian multiplier method to derive the necessary conditions for an interior solution (that is, a solution in which both consumers receive positive quantities of both commodities). I will then leave it to you to verify the fact that the necessary conditions imply that at an interior Pareto eﬃcient allocation, we must have: ∂u1 ∂u1 ∂u2 ∂u2 = . (5.31) ∂x11 ∂x12 ∂x21 ∂x22
Which veriﬁes the fact that the slopes of the two individuals’ indiﬀerence curves have to be equal at a Pareto (actually strongly Pareto) eﬃcient allocation in this case. 3 Insofar as the derivation of necessary conditions are concerned, the numerical value of u∗ is not important. Consequently, for this demonstration, we need not worry about which values of u∗ are ‘feasible.’
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Chapter 5. Pure Exchange Economies
Moreover, if u1 and u2 are both quasiconcave, then any (interior) allocation at which (5.31) is satisﬁed is strongly Pareto eﬃcient, given the assumptions we have been making here. Our second development is geometric, and in a sense is much more general than the mathematical development which we have just presented; although we will be considering the solution to the same problem as before, that is, the problem set out in equations (5.28)–(5.30), above. Moreover, once again we will assume that the consumers’ utility functions are increasing, so that we can replace the inequalities in the two constraints by equalities. Now, the question is, how do we handle these constraints in our geometric development? x12 x21
〈x*i 〉
u2 = u*
x22
x 11
Figure 5.6: An Interior Pareto Eﬃcient Allocation. Considering the constraints in reverse order, we can see that the second constraint is automatically satisﬁed within the conﬁnes of the Edgeworth Box, for every point in the box satisﬁes this constraint. Therefore we can concentrate our attention upon the problem of maximizing u1 on the set of points satisfying the ﬁrst constraint; maximizing 1’s utility on 2’s indiﬀerence curve for the value u2 = u∗ . Thus, it is easy to see that the allocation x∗i in Figure 5.6 is Pareto eﬃcient. In Figure 5.6 we have shown for a second time that an interior Pareto eﬃcient allocation occurs at a point of tangency of the two individuals’ indiﬀerence curves. What happens, however, if the utility functions are not diﬀerentiable? It is in its ability to handle this contingency that the geometric method of analysis is more general (in a sense) than the mathematical development which we went through earlier. For example, in Figure 5.7 we have illustrated a case in which both individuals have Leontieftype utility functions, which are, of course, not diﬀerentiable. However, by concentrating our attention on the problem of maximizing u1 subject to being on 2’s indiﬀerence curve for u2 = u∗ , it is easy to see that any allocation on the heavilyshaded line segment between xi and x∗i is Pareto eﬃcient. In the following examples, we present two illustrative cases in which there are allocations which are Pareto eﬃcient for E, but which are not strongly Pareto eﬃcient.
5.5. Pareto Eﬃciency
147
x12
x21
〈xi 〉
〈x*i 〉
u2 = u^*
x 22
x11
Figure 5.7: Pareto Eﬃciency in the Leontief Case. 5.18 Examples. 1. Deﬁne: Z = {x ∈ R2  xj is an integer, for j = 1, 2}; and let: Xi = R2+ ∩ Z
for i = 1, 2;
R2
having each coordinate a nonnegative integer, that is, Xi is the set of all vectors in for i = 1, 2. We also suppose that the initial endowments of the two consumers are given by: r 1 = (1, 0) and r 2 = (1, 1); and that the two consumers have preference relations which can be represented on: Xi∗ = {xi ∈ Xi  0 ≤ xi ≤ r} def
by the functions f1 and f2 deﬁned in the table below. f1 f2 xi (0, 0) 1 1 (1, 0) 3 3 (0, 1) 3 2 (1, 1) 5 4 (2, 0) 5 5 (2, 1) 6 6 We can show in this case that the two allocations (1, 0), (1, 1) and (2, 0), (0, 1) are both Pareto eﬃcient, but are not strongly Pareto eﬃcient.
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Chapter 5. Pure Exchange Economies
2. Consider the same example as before, except that the preference relations Pi can be represented by the functions f1 and f2 set out in the following. f1 f2 xi (0, 0) 1 1 (1, 0) 2 3 (0, 1) 4 2 (1, 1) 5 4 (2, 0) 3 5 (2, 1) 6 6 In this example, all of the Pareto eﬃcient allocations will be strongly Pareto eﬃcient. x12 〈x*i 〉
0'
x21
x11 x22
Figure 5.8: Pareto Eﬃciency without Strong Pareto Eﬃciency. 3. Lest you get the idea that the divergence between Pareto eﬃciency and strong Pareto eﬃciency can only occur when consumers have discrete consumption sets (commodities only available in discrete quantities), consider the example presented in Figure 5.8, above. In this example, we have a classic Edgeworth Box case with both consumers having increasing utility functions. However, you should have no diﬃculty in verifying the fact that all of the allocations lying on the open line segment connecting x∗i and 0 are Pareto eﬃcient, but not strongly Pareto eﬃcient. The signiﬁcance of our next result resides in the fact that it provides suﬃent conditions for Pareto eﬃciency to imply strong Pareto eﬃciency. 5.19 Proposition. Suppose E = Pi , r i is such that Pi is lower semicontinuous, strictly increasing, asymmetric, and negatively transitive, for i = 1, . . . , m. Then ∗ mn for all xi , x∗i ∈ Rmn + such that xi P xi , there exists xi ∈ R+ such that: m m xi = xi and xi Qx∗i . i=1
i=1
5.5. Pareto Eﬃciency
149
Proof. Suppose that xi P x∗i . Then by deﬁnition of the strict Pareto ordering, we have: xi Gi x∗i for i = 1, . . . , m; (5.32) and, for some k ∈ {1, . . . , m}:
xk Pk x∗k .
(5.33)
Now, since Pk is strictly increasing, we have: (∀xk ∈ Rn+ ) : xk Gk 0, where we have denoted the origin in Rn by ‘0;’ and thus from (5.33) we see that: xk > 0.
(5.34)
Furthermore, since Pk is lower semicontinuous, it also follows that there exists θ ∈ ]0, 1[ satisfying: θxk Pk x∗k . (5.35) But now consider the allocation (xi ) ∈ Rmn + deﬁned by: xk = θxk ,
(5.36)
and: xi = xi + [(1 − θ)/(m − 1)]xk
for all i = k.
xi
> xi ; and thus, since each Pi is strictly From (5.34) we have, for each i = k, increasing: xi Pi xi for all i = k. (5.37) Therefore, using (5.35) and (5.37), we see that: xi Qx∗i . Furthermore, from the deﬁnition of xi , we have: m i=1
xi = θxk +
xi + [(1 − θ)/(m − 1)]xk
i=k
= θxk +
i=k
xi + (m − 1)[(1 − θ)/(m − 1)]xk =
m
xi .
i=1
Our next proposition is now an easy consequence of the result just proved. Details will be left as an exercise. 5.20 Proposition. If E = Pi , r i is such that Pi is lower semicontinuous, strictly increasing, asymmetric, and negatively transitive, for i = 1, . . . , m, then an allocation x∗i is Pareto eﬃcient for E if, and only if, it is strongly Pareto eﬃcient for E.
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Chapter 5. Pure Exchange Economies
5.6
Pareto Eﬃciency and ’NonWastefulness’
In the terminology introduced by Hurwicz [1960], we will demonstrate that, loosely speaking: 1. the competitive mechanism is nonwasteful, in the sense that any competitive equilibrium is Pareto eﬃcient. 2. the competitive mechanism is unbiased, in the sense that (given some additional assumptions) any Pareto eﬃcient allocation can be made a competitive equilibrium. The ﬁrst of the above two results is often called the ‘First Fundamental Theorem of Welfare Economics,’ and seems to have been originally established by Enrico Barone [1908]. The second result is the ‘Second Fundamental Theorem of Welfare Economics,’ and was originally formulated and proved by Kenneth Arrow [1951a]. We will conclude this chapter with two versions of the ‘First Fundamental Theorem,’ but we will postpone our study of the ‘Second Fundamental Theorem’ until Chapter 7. 5.21 Theorem. If x∗i , p∗ is a competitive equilibrium for a private ownership ∗ economy, E, then xi is Pareto eﬃcient for E. ∗ Proof. Suppose xi ∈ Rmn + is such that xi Qxi , so that:
xi Pi x∗i
for i = 1, . . . , m.
Then, since x∗i ∈ hi (p∗ , p∗ · r i ), for each i, we must have: p∗ · xi > p∗ · r i
for i = 1, . . . , m.
But then, by summing over i, we see that we must have:
m m 0< (p∗ · xi − p∗ · r i ) = p∗ · xi − r . i=1
i=1
However, it then follows that we cannot have: m xi = r; i=1
and thus xi is not feasible for E. Therefore, x∗i is Pareto eﬃcient for E.
Our alternative version of the ‘First Fundamental Theorem’ is as follows. 5.22 Theorem. Suppose x∗i , p∗ is a competitive equilibrium for a private ownership economy, E, and that each Pi is locally nonsaturating, asymmetric, and negatively transitive. Then x∗i is strongly Pareto eﬃcient for E ∗ Proof. Suppose xi ∈ Rmn + is such that xi P xi . Then:
xi Gi x∗i
for i = 1, . . . , m;
5.6. Pareto Eﬃciency and ’NonWastefulness’ and, for some k ∈ {1, . . . , m}:
151
xk Pk x∗k .
It then follows from Proposition 4.9 and the deﬁnition of hi (·), respectively, that: p∗ · xi ≥ p∗ · r i
for i = 1, . . . , m,
(5.38)
and: p∗ · xk > p∗ · r k .
(5.39)
Adding (5.38) and (5.39) over all i, we then see that:
m m (p∗ · xi − p∗ · r i ) = p∗ · xi − r ; 0< i=1
i=1
from which we see that we cannot have: m i=1
xi = r.
Therefore, (xi ) ∈ / A(E), and we conclude that x∗i is strongly Pareto eﬃcient for E. Exercises Each of the following four problems deals with a twoperson, twocommodity exchange economy, in which we suppose the consumers’ preferences can be represented by the utility functions given in the problem. 1. Suppose the consumers’ utility functions are given by: 3/4 1/4
u1 (x1 ) = x11 x12 , and: 1/4 3/4
u2 (x2 ) = x21 x22 , respectively; and suppose the income distribution in the economy is given by: wi = (1/2)W
for i = 1, 2,
where W = W (p) = p · r for p ∈ R2++ . a. Find the aggregate demand function in this case, if one exists. b. Does the market demand behave as if there is a single utilitymaximizing individual in the economy in this case? Explain. 2. Suppose the consumers’ utility functions are given by: x x i1 i2 ui (xi ) = min , for i = 1, 2, ai1 ai2 where aij > 0 for i, j = 1, 2. a. Supposing that the ith consumer’s initial resource endowment is given by r = (ri1 , ri2 ), ﬁnd the ith consumer’s demand function, di (p).
152
Chapter 5. Pure Exchange Economies b. If we now suppose that: a11 = a12 = a21 = 1, a22 = 4, r12 = r21 = 0, r11 = 5, and r22 = 10,
ﬁnd the (or a) competitive equilibrium for the economy, or show that no competitive equilibrium exists in this case. c. Now suppose that all the data assumed in part b still holds, except that we now have a22 = 2, and ﬁnd the (or a) competitive equilibrium for the economy, or show that no competitive equilibrium exists. 3. Suppose the consumers’ utility functions are given by: i ui (xi ) = xai1i · x1−a i2 ,
for i = 1, 2,
where: 0 < ai < 1,
for i = 1, 2.
a. Supposing that the ith consumer’s initial resource endowment is given by r = (ri1 , ri2 ), ﬁnd the ith consumer’s demand function for the ﬁrst commodity, di1 (p). b. If we now suppose that: r12 = r21 = 0, while r11 > 0 & r22 > 0, ﬁnd the (or a) competitive equilibrium for the economy. c. With the values for r i as speciﬁed in part b, can you ﬁnd ∂p1 /∂r22 ? Does its value make sense to you? Explain. 4. Suppose the consumers’ utility functions are given by: x x22 11 u1 (x1 ) = min , x12 and u2 (x2 ) = min x21 , , 2 2 while the initial resource endowments are given by: r 1 = (3, 0) and r 2 = (0, 3). On the basis of this information, answer the following questions. a. Is the ﬁrst consumer’s preference relation homothetic? b. Is the ﬁrst consumer’s preference relation weakly convex? convex? strictly convex? c. Find the ith consumer’s demand function for the ﬁrst commodity, di1 (p); normalizing the price of the second commodity to equal one, that is, setting p2 = 1. d. Find the (or a) competitive equilibrium for the economy, or show that no competitive equilibrium exists in this case. 5. Consider a pure exchange economy in which Xi = R2+ ∩ Z, for i = 1, 2; where: Z = {x ∈ R2  xj is an integer, for j = 1, 2}, that is, Xi is the set of all vectors in R2 having each coordinate a nonnegative integer. Suppose that the initial endowments are given by r 1 = (1, 0) and r 2 = (1, 1); and
5.6. Pareto Eﬃciency and ’NonWastefulness’
153
that the two consumers have preference relations which can be represented on Xi∗ by the functions f1 and f2 , respectively, given by Table 1, below. xi f1 f 2 (0, 0) 1 1 (1, 0) 3 3 (0, 1) 3 2 (0, 2) 4 3 (1, 1) 5 4 (2, 0) 5 5 (2, 1) 6 6 Table 1. Show that
(x∗i ), p∗
is a competitive equilibrium for this economy, where:
x∗1 = (1, 0), x∗2 = (1, 1), and p∗ = (3/5, 2/5). Is this allocation Pareto eﬃcient? Is it strongly Pareto eﬃcient? Explain your answers brieﬂy. 6. Complete the proof of Proposition 5.14. 7. Show that neither the unanimity relationship, nor the strict Pareto ordering is necessarily negatively transitive, even if individual preferences are negatively transitive. (Note: it suﬃces to produce an example for each [or possibly one dualpurpose example] in which negative transitivity fails. This can be done with simple Edgeworth Box diagrams.) 8. Suppose Pi is asymmetric and negatively transitive for each i, and let R and P be the Pareto atleastasgoodas, and strict Pareto dominance relation. Show that if xi , x∗i , xi . and xi are such that: xi Rx∗i , x∗i P xi , and xi Rxi , then
xi P xi .
(5.40)
Chapter 6
Production Theory 6.1
Introduction
In this chapter we will develop the ‘bare bones’ of production theory as it is utilized in the remaining chapters of this book. The next section covers the most basic topics, while in section 3 we consider the special case of linear production sets. Linear production sets play a major role in many portions of applied general equilibrium analysis; particularly in the area of public economics, where the assumption of linear production sets greatly simpliﬁes the analysis in the literature on ‘optimal commodity taxation,’ for example. InputOutput analysis, which has played a key role in development and planning models, also utilizes the assumption of an aggregate linear production set; and we will undertake a very brief study of one version of this model in Section 4. In Section 5 we will examine the issue of proﬁt maximization for a competitive ﬁrm, as well as some of the properties of the ﬁrm’s proﬁt function in this case. We then move on in Section 6 to a consideration of the speciﬁcally general equilibrium development of production theory; in particular, the relationship of the aggregate production set and the aggregate proﬁt function to the individual production sets, and individual proﬁt functions, respectively. Finally, in Section 6 we present a brief development of the theory of ‘activity analysis.’ This is, in eﬀect, a special case of a linear production model, and is a topic which can easily be skipped on one’s ﬁrst passage through this text; although some of the mathematical results in this section will be utilized in our study of general equilibrium under uncertainty.
6.2
Basic Concepts of Production Theory
In our development of the theory of the ﬁrm in a general equilibrium context, we suppose that the set of technologically feasible production vectors for the ﬁrm is a subset, Y , of Rn , which we shall call a production set. The producer chooses a production plan (or production vector, or ‘netput’ vector): y = (y1 , . . . , yn ), from Y , with the interpretation: if yj is:
156
Chapter 6. Production Theory
positive, then the vector y speciﬁes a net production of the j th commodity in the amount yj > 0; negative, then the j th commodity is being used as an input in the amount: −yj = yj  > 0; and, of course, if yj = 0, then there is neither net production nor a net useage of commodity j as an input. In this section, we will examine some of the implications of several assumptions which are commonlyused in connection with production in general equilibrium models, as well as some deﬁnitions we will use a great deal in our discussion of production. In intermediate theory courses one of the most important and signiﬁcant assumptions which one examines is the assumption of decreasing returns to scale; or, in the shortrun, diminishing returns. In general equilibrium developments, the corresponding condition is that the production set is convex. However, this is getting a bit ahead of our story; let’s begin by considering the following conditions. 6.1 Deﬁnitions. A production set, Y ⊆ Rn is said to satisfy (or to exhibit): a. nonincreasing returns to scale iﬀ, for any y ∈ Y and any θ ∈ ]0, 1], we have θy ∈ Y as well. b. nondecreasing returns to scale iﬀ, for any y ∈ Y and any θ ≥ 1, we have θy ∈ Y as well. c. constant returns to scale iﬀ Y is a cone; that is, for all y ∈ Y and all θ > 0, we have θy ∈ Y . d. increasing returns to scale iﬀ Y satisﬁes nondecreasing returns to scale, and does not satisfy nonincreasing returns to scale. e. decreasing returns to scale iﬀ Y does not satisfy nondecreasing returns to scale, and does satisfy nonincreasing returns to scale. In words, a production set satisﬁes nonincreasing returns if whenever we decrease all input and output quantities of a feasible production vector in the same proportion, we arrive at another feasible production vector. I will leave it to you to develop analagous verbal statements for the other properties set out in the above deﬁnition. In the ﬁgures on the next page, we present examples satisfying nonincreasing and nondecreasing returns to scale, respectively, for production sets in R2 . In fact, in Figure 6.1.a, Y satisﬁes decreasing returns; while in Figure 6.1.b, Y exhibits increasing returns. While the deﬁnitions presented in 6.1 are frequently seen in the literature, definitions (d) and (e) (increasing and decreasing returns to scale, respectively) are not very satisfactory. For example the production set in Figure 6.2.a satisﬁes the deﬁnition of increasing returns, while that in 6.2.b satisﬁes decreasing returns. The essential relationship between nonincreasing returns and convexity is set out in the following proposition. 6.2 Proposition. If Y ⊆ Rn is convex, and contains the origin (that is, 0 ∈ Y ), then Y satisﬁes nonincreasing returns to scale.
6.2. Basic Concepts of Production Theory
157
Y
Y
Figure 6.1.a
Figure 6.1.b
Figure 6.1: Decreasing and Increasing Returns.
Y
Figure 6.2.a
Y
Figure 6.2.b
Figure 6.2: Increasing and Decreasing Returns: Another Example.
158
Chapter 6. Production Theory
Proof. If y ∈ Y and θ ∈ ]0, 1], then making use of the fact that 0 ∈ Y , and the convexity of Y, we have: θy + (1 − θ)0 = θy ∈ Y.
While the above result shows that convexity, together with the assumption that 0 ∈ Y is suﬃcient to imply nonincreasing returns to scale, these conditions are not necessary for same. In fact, in Figure 6.3, Y contains the origin and satisﬁes decreasing returns to scale even though it is not convex. On the other hand, convexity of the production set can also be viewed as the counterpart of the assumption that the shortrun production function satisﬁes diminishing returns; as is illustrated in the examples which follow.
Y
Figure 6.3: Decreasing Returns without Convexity. 6.3 Examples. → R+ be a production function, and deﬁne Y ⊆ Rn by: 1. Let ϕ : Rn−1 + Y = {y = (v, x) ∈ R(n−1)+1  v ∈ −Rn−1 & 0 ≤ x ≤ ϕ(−v)}. + If ϕ is concave, then Y is a convex set. Moreover, if ϕ(0) = 0, then (as established by Proposition 6.2, above) Y satisﬁes nonincreasing returns to scale. → R+ be a production function, but this time suppose 2. As above, let ϕ : Rn−1 + that ϕ is positively homogeneous of degree one. If we then deﬁne Y ⊆ Rn by: & 0 ≤ x ≤ ϕ(−v)}, Y = {y = (v, x) ∈ R(n−1)+1  v ∈ −Rn−1 +
(6.1)
it is easy to show that Y satisﬁes constant returns to scale. 3. Suppose once again that ϕ is concave, but that, say v1 is ﬁxed at the level v1 = v1∗ in the short run. In this case the set Y deﬁned in (1), above, will be convex; however, the shortrun production set will be given by: Y = {y = (v, x) ∈ R(n−1)+1  v ∈ −Rn−1 & 0 ≤ x ≤ ϕ(−v) & v1 = v1∗ }. + On the other hand, we can deﬁne the set Y by: Y = Y ∩ {y ∈ Rn  y1 = v1∗ };
6.2. Basic Concepts of Production Theory
159
so that we see that Y is the intersection of two convex sets, and is, therefore, also convex. The following deﬁnition sets out some further conditions which are often used in the general equilibrium theory of production. 6.4 Deﬁnitions. 1. Impossibility of Free Production: Y ∩ Rn+ ⊆ {0}. 2. Irreversibility: Y ∩ (−Y ) ⊆ {0}; that is, if y ∈ Y , and −y ∈ Y as well, then y = 0. 3. Additivity: if y ∈ Y and y ∗ ∈ Y , then y + y ∗ ∈ Y as well. 4. Possibility of inaction: 0 ∈ Y . 5. Disposability: a. Limited: If y ∈ Y, y ∗ ∈ Rn , and for all j we have: 0 ≤ yj∗ ≤ yj or yj∗ ≤ yj < 0, then y ∗ ∈ Y . b. SemiFree: Y − Rn+ ⊆ Y ; that is, if y ∈ Y and y ∈ Rn are such that y ≤ y, then y ∈ Y . c. Free: −Rn+ ⊆ Y .1 Let’s begin our consideration of these deﬁnitions by noting some relationships among the disposability conditions. The proofs of the ﬁrst and third of the following facts we will leave as exercises. Fact 2 is demonstrated by Figures 6.4.a and 6.4.b. (In Figure 6.4.a, Y is the set Y = {y ∈ R2  y ≤ y ∗ }; while in Figure 6.4.b, Y is the union of the third quadrant and the ray indicated.)
Y Y
Figure 6.4.a
Figure 6.4.b
Figure 6.4: SemiFree and Free Disposability. Facts Regarding Disposability. 1. If Y satisﬁes ‘semifree disposability,’ then it satisﬁes limited disposability. However, the converse is not true. 1
This is an assumption which is usually applied only to the aggregate production set.
160
Chapter 6. Production Theory
2. A production set may satisfy semifree disposability, but not satisfy free disposability. Furthermore, a production set may satisfy free disposability but not limited disposability (and hence not semifree disposability). 3. If Y satisﬁes semifree disposability and 0 ∈ Y , then Y satisﬁes free disposability. 4. If Y satisﬁes free disposability and is closed and convex, then Y also satisﬁes semifree disposability. Proof of Fact 4. Let y ∗ ∈ Y and y ∈ Rn be such that y ≤ y ∗ . Then: ¯ = y − y ∗ ≤ 0; y def
¯ ∈ Y , by free disposability. But then, since −Rn+ ⊆ Y and −Rn+ is a cone, and thus y we see that for all µ > 0 (in particular, for µ > 1), we must have µ¯ y ∈ Y . Therefore, given an arbitrary µ > 1, if we let θ = 1/µ, it follows from the convexity of Y that: ¯ + y∗ − θ(µ¯ y ) + (1 − θ)y ∗ = y But: y −
1 µ
1 µ
y∗ = y − y∗ + y∗ −
1 µ
y∗ = y −
1 µ
y ∗ ∈ Y.
y ∗ → y as µ → +∞;
and, since Y is closed, it follows that y ∈ Y .
6.5 Deﬁnition. We shall say that a set Y ⊆ Rn is a convex cone iﬀ Y is a cone which is also a convex set. Within the context of production theory, an interesting property of convex cones is set out in the following. 6.6 Proposition. If a set Y ⊆ Rn is a convex cone, then Y is additive. Proof. Suppose y and y are elements of Y . Then, since Y is convex: (1/2)y + (1/2)y ∈ Y. But then, since Y is also a cone: 2 (1/2)y + (1/2)y = y + y ∈ Y.
In the next section, we will investigate the properties of a particular kind of production set in detail. In the meantime, we close this section by presenting some alternative useful methods for characterizing a production set. q 6.7 Example. Let P : Rq+ → Rm + be a correspondence; where, for v ∈ R+ we m interpret P (v) as the set of all output vectors, x ∈ R+ such that v can produce x. In fact, in this case, for v ∈ Rq+ , we call P (v) the production possibility set for v (and we will refer to such a correspondence as a production possibility correspondence). In this case, if n = q + m, then one approach to the task of
6.3. Linear Production Sets
161
deﬁning Y from P is to more or less deﬁne Y as the graph of P ; more exactly, we can deﬁne: Y = {(v, x) ∈ Rq+m  v ∈ −Rq+ & x ∈ P (−v)}. A better approach, however, from the standpoint of general equilibrium theory is to do the following. We can always assume, without loss of generality, that q = m = n. With this convention, we can then deﬁne Y by: Y = {y ∈ Rn  (∃v, x ∈ Rn+ ) : x ∈ P (v) & y = x − v}. Rm +
Rq+
6.8 Example. Let V : → be a correspondence, where for x ∈ Rm + we interpret the set V (x) as the set of input vectors, v ∈ Rq+ , such that v can produce x, and we call the set V (x) the inputrequirement set for x. In this case the correspondence, V , is called the inputrequirement correspondence, and the corresponding production set, Y , is often deﬁned by: Y = {(v, x) ∈ Rq+m  x ∈ Rm + & − v ∈ V (x)}. However, once again a better approach is to suppose, without loss of generality, that q = m = n, and to deﬁne Y by: Y = {y ∈ Rn  (∃v, x ∈ Rn+ ) : v ∈ V (x) & y = x − v}.
6.3
Linear Production Sets
In order to get a better feel for the meaning of the conditions set out in Deﬁnitions 6.4, we will examine them within the context of a particular type of production set; one which we will often be considering in the material to follow. 6.9 Deﬁnition. We shall say that a production set, Y ⊆ Rn is linear iﬀ there exists a nonzero m × n matrix, A such that: Y = {y ∈ Rn  Ay ≤ 0}.
(6.2)
Before considering the general properties of linear production sets, let’s take a look at what is apparently a quite diﬀerent sort of linear production relationship. 6.10 Example. Let B be a semipositive q × r matrix. Deﬁne a pair (v, x) ∈ Rq+r + to be technologically feasible iﬀ: v ≥ Bx. The situation here is that we suppose that B is an inputrequirement matrix, with the interpretation that the matrixvector product Bx gives the minimal amounts of the q inputs which are needed to produce x. Notice that with this speciﬁcation, the corresponding inputrequirement correspondence (Example 6.8 of the previous section) is given by: V (x) = {v ∈ Rq+  v ≥ Bx} for x ∈ Rr+ .
162
Chapter 6. Production Theory
If we now suppose that n = q + r, we can deﬁne a production set for this case by (assuming that the ﬁrst q commodities are the inputs used in the production process, and that the last n − q = r commodities are the outputs): Y = {(v, x) ∈ Rn  −v ≥ Bx & x ∈ Rr+ } = {(v, x) ∈ Rn  v + Bx ≤ 0 & x ∈ Rr+ }. Alternatively, we can express this production set in the general form of equation (6.2) above, by deﬁning the n × n matrix A as: ! Iq B , A= O −I r where Iq and Ir are the q × q and r × r identity matrices, respectively; and then deﬁning Y by: ! ! v v Y = y= ∈ Rn  A ≤0 . x x Returning to the general deﬁnition of a linear production set, we can prove the following. 6.11 Proposition. If Y is a linear production set, then 0 ∈ Y , and Y is a closed convex cone. Thus, in particular, Y is additive. Proof. Suppose Y is linear; so that there exists an m × n nonzero matrix, A, such that: Y = {y ∈ Rn  Ay ≤ 0}. Obviously we then have 0 ∈ Y . To prove that Y is convex, let y, y ∈ Y and θ ∈ [0, 1]. We then have: Ay ≤ 0 & Ay ≤ 0, and thus:
A[θy + (1 − θ)y ] = θAy + (1 − θ)Ay ≤ 0,
and it follows that:
θy + (1 − θ)y ∈ Y.
Similarly, if y ∈ Y and λ > 0, we have Ay ≤ 0, and therefore: Aλy = λAy ≤ 0. To prove that Y is closed, denote the ith row of A by ‘ai· ,’ for i = 1, . . . , m, and deﬁne Hi ⊆ Rn by: Hi = {y ∈ Rn  ai· · y ≤ 0} for i = 1, . . . , m. Then we note that: Y =
m i=1
Hi ;
and, since each Hi is a closed lower halfspace, it follows that Y is closed as well. Finally, the fact that Y is additive follows from Proposition 6.6, now that we have shown Y to be a convex cone.
6.3. Linear Production Sets
163
The generic example presented in Example 6.10, above, is obviously very special in at least three senses (in addition to the linearity of the technology). First, it takes the ﬁrst q commodities as inputs and produces the last n−q commodities as outputs. Secondly, every commodity is either an input or an output of the production process. Third, each production process uses the inputs in ﬁxed proportions; there is no substitution possible between inputs. In principle, however, all of these deﬁciencies are readily correctible while retaining the assumption of linearity, as is shown in the following examples. 6.12 Examples. 1. Suppose n = 6, and that a ﬁrm operates two production processes. Process 1, we will suppose, produces commodity two using the ﬁrst and third commodities as inputs, with inputoutput combinations feasible if, and only if, they satisfy the production constraint: y2 ≤ −a11 y1 − a13 y3 , where a11 > 0 and a13 > 0. Process 2 produces the fourth commodity using the sixth commodity as an input; with production constraint: y4 ≤ −a26 y6 , where a26 > 0. Commodity ﬁve, we will suppose, does not enter into this ﬁrm’s production processes at all. Notice that in this case the ﬁrst production process allows possible substitution between the ﬁrst and the third commodities as inputs. We can characterize this ﬁrm’s technology in the form of equation (6.2) by deﬁning the 9 × 6 matrix A by: ⎛ ⎞ a11 1 a13 0 0 0 ⎜ 0 0 0 1 0 a26 ⎟ ⎜ ⎟ ⎜ 1 0 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 0 −1 0 0 0 0 ⎟ ⎜ ⎟ 0 1 0 0 0 ⎟ A=⎜ ⎜ 0 ⎟. ⎜ 0 0 0 −1 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 0 0 1 0 ⎟ ⎜ ⎟ ⎝ 0 0 0 0 −1 0 ⎠ 0 0 0 0 0 1
(6.3)
I will leave it to you to verify that the ﬁrm’s production set, Y , can be represented as: Y = {y ∈ R6  Ay ≤ 0}, and that Y satisﬁes the properties set out at the beginning of this paragraph. In this example, we have only one commodity produced by each of the production processes, but this is not at all necessary for the type of representation being considered here; we can allow for joint production of two or more commodities in a single production process in much the same way that we allow for more than one input.
164
Chapter 6. Production Theory
2. In connection with the preceding example, notice that if we deﬁne the matrix B by: ! a11 1 a13 0 0 0 B= , 0 0 0 1 0 a26 then we can equally well deﬁne the production set by: Y = {y ∈ R6  By ≤ 0, y1 ≤ 0, y2 ≥ 0, y3 ≤ 0, y4 ≥ 0, y5 = 0 & y6 ≤ 0}. In fact, if one is actually to construct a detailed or numerical example, it is usually more convenient to represent the production set by deﬁning a q × n semipositive matrix, B (where q is the number of production processes), and nonempty sets I and J, where, deﬁning N = {1, . . . , n}, we have: I ∪ J = N, in such a way that Y can be deﬁned as: Y = y ∈ Rn  By ≤ 0 & (∀i ∈ I) : yi ≤ 0 & (∀j ∈ J) : yj ≥ 0 .
(6.4)
(6.5)
Letting K = I ∩ J, we then see that these sets have the following interpretations: I \ J = i ∈ {1, . . . , n} such that commodity i is used as an input by the technology, J \ I = j ∈ {1, . . . , n} such that commodity j is produced by the technology, and K = k ∈ {1, . . . , n} such that commodity k does not enter the technology. In the present example: I = {1, 3, 5, 6} & J = {2, 4, 5}. While the representation in (6.5) is often more convenient than the representation in (6.2), and is probably more intuitive as well, it is important to notice that if a production set Y can be deﬁned as in (6.5), then one can deﬁne an m × n matrix, A, using the example developed in (6.3) as a model, to equivalently deﬁne Y in the form of equation (6.2). Thus, in particular, if a production set can be deﬁned as in equation (6.5), then it is linear, as we have deﬁned the term. 3. Suppose this time that n = 5, and consider a somewhat more complicated example, as follows. Once again we suppose that the ﬁrm operates two production processes. Process one produces the second commodity, using the ﬁrst and third commodities as inputs, with production constraint: y2 ≤ −b11 y1 − b13 z1 ,
(6.6)
where b11 and b13 are both positive, and we are denoting the quantity of the third commodity used in the ﬁrst production process by ‘z1 .’ We will suppose that the second process produces the ﬁfth commmodity, using the third commodity as input, with the production constraint given by: y5 ≤ −b23 z2 ,
(6.7)
6.3. Linear Production Sets
165
where b23 > 0, and z2 ≤ 0 is the quantity of the third commodity used as an input in the second process. Commodity four, we then suppose, does not enter into this ﬁrm’s technology at all. On the other hand, the total amount of commodity three used as an input by this ﬁrm must satisfy the constraint: y 3 = z 1 + z2 ,
(6.8)
if the ﬁrm is paying a positive price for commodity three, and if the ﬁrm is to maximize proﬁts. Here, in order to express this production set in the form of equation (6.5), we can proceed as follows. Deﬁne b15 = b13 /b23 , the matrix B by: ! b11 1 b13 0 b15 , 0 0 b23 0 1
(6.9)
I = {1, 3, 4} & J = {2, 4, 5}.
(6.10)
B= and the sets I and J by:
(See Exercise 3, at the end of this chapter.) 4. An interesting oddity of the deﬁnitions being used here stems from the fact that, given a set Y of the form indicated in (6.5), we can deﬁne an almost equivalent production set as follows. Deﬁne the set J ∗ as the set of indices of commodities which are produced by the technology; that is, in the notation of the previous two examples, let: J ∗ = J \ I, and deﬁne I ∗ = N \ J ∗ . Now consider the set Y ∗ deﬁned by: Y ∗ = y ∈ Rn  By ≤ 0 & (∀i ∈ I ∗ ) : yi ≤ 0 . Is the production set Y ∗ the same as the set Y deﬁned in (6.5)? It is probably pretty obvious that in general, these two sets will not be the same. However, what happens if, say, n = 2, I = I ∗ = {1}, and J = J ∗ = {2}? Alternatively, consider the set Y † deﬁned by: Y † = y ∈ Rn  By ≤ 0 & (∀i ∈ J ∗ ) : yi ≥ 0 . Does the set Y † = Y ? Does Y † = Y ∗ ?
Turning now to the issue of which of the remaining conditions set out in Deﬁnition 6.4 are satisﬁed by linear technologies, I will leave it as an easy exercise to show that if Y is speciﬁed as in equations (6.4) and (6.5), above, then Y will satisfy Irreversibility: Y ∩ (−Y ) ⊆ {0}. The question of whether Y will satisfy 6.4.2, Impossibility of Free Production, in this case is a little more complicated, however, as is shown in the following example.
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Chapter 6. Production Theory
6.13 Example. In the notation of Example 6.12.2, let n = 4, I = {1, 2}, J = {3, 4}, and suppose the matrix B is given by: ! 1 0 1 0 B= 0 1 3 0 In this case, will Y satisfy Impossibility of Free Production? What if B is given by the following matrix? ! 1 0 1 2 B= 0 0 0 0 At the risk of sounding rather too English, let’s coin the following deﬁnition. 6.14 Deﬁnition. We shall say that Y is a proper linear technology iﬀ there exist a semipositive m × n matrix, B, and sets I, J ⊆ N ≡ {1, . . . , n} such that: 1. I ∪ J = N, I \ J = ∅, J \ I = ∅, 2. deﬁning K = I ∩ J, we have: (∀j ∈ N \ K)(∃i ∈ {1, . . . , m}) : bij > 0, and 3. Y = {y ∈ Rn  By ≤ 0 & (∀i ∈ I) : yi ≤ 0 & (∀j ∈ J) : yj ≥ 0}. I will leave the proof of the following result as an exercise. 6.15 Proposition. If Y ⊆ Rn is a proper linear technology, then Y satisﬁes Impossibility of Free Production, Irreversibility, and Limited Disposability, in addition to the properties set out in Proposition 6.11. Suppose a production set Y takes the form: Y = {y ∈ Rn  (∃z ∈ Rk+ ) : y = Bz}, where k is a positive integer, and B is an n × k matrix. Obviously it would make sense to call Y a linear production set in this case, but this seems to be a very diﬀerent speciﬁcation of technology than that which we have termed ‘linear’ in this section. One can show, however, that this new speciﬁcation of the production set is linear, as we have deﬁned the term in this section. While the proof of this is deferred until Section 8 of this chapter (a starred section), we will devote the next section to a discussion of a very important example of this second formulation of a linear technology.
6.4
InputOutput Analysis
InputOutput analysis has been a primary tool of applied economics and planning models for the past 60 years. It was developed by W. Leontief during the 1930’s, and is still a basic component of many computational general equilibrium models today. In applications it may be used to model the aggregate production possibilities of a nation or a region, or even a large ﬁrm; however, hereafter in this discussion
6.4. InputOutput Analysis
167
we will refer to the entity being analyzed as an ‘economy,’ and proceed as if we are attempting to characterize the aggregate production set of the economy. The basic assumption needed is that the products produced in the economy can be classiﬁed into n nonoverlapping and exhaustive categories. In some applications, this categorization might be very broad: for example Agricultural Products, Manufactured Goods, and Services; while in other applications the categorization may be much ﬁner. In any case, given a categorization of products, the production sector of the economy can be divided into n corresponding production sectors, with each producing exactly one (aggregated) commodity. It is then postulated that the technology of the j th production sector can be characterized by the (Leontief) production function: y ynj z1j zmj 1j yj = min ,... , ,..., ; (6.11) z1j anj b1j bmj where: yij is the amount of the ith sector’s output used as an input in the j th sector (i = 1, . . . , n), zkj is the amount of the k th primary (nonproduced) input used in the j th sector (k = 1, . . . , m), and:2
aij ≥ 0
for i = 1, . . . , n, and
bkj ≥ 0
for k = 1, . . . , m.
If no goods are free, then eﬃcient production in the j th sector will require that: yij = aij xj for i = 1, . . . , n, and zkj = bkj xj for k = 1, . . . , m.
(6.12)
On the other side of the coin, the ith sector’s output may be used as in input in the production of any of the the other n−1 sectors’ outputs, or it may be delivered to the consumer sector, or to government, or exported as a ﬁnal good. In our discussion we will lump these three sectors together as (exogenous) ‘ﬁnal demand,’ and we will denote the quantity of this ﬁnal demand by ‘cj .’ Now, if we ignore primary factor inputs for the moment, we can picture the structure of the aggregate production technology as in the following table.
Delivering Sector 1 (y1 ) 2 (y2 ) ... n (yn )
Receiving Sector y11 y12 . . . y1n c1 y21 y22 . . . y2n c2 ... yn1 yn2 . . . ynn cn
2 If, for some i, j, we have aij = 0, we deﬁne xy /aij = +∞, and similarly if, for some k, j, bkj = 0. Notice that in this eventuality, for example, if ahj = 0, then min{xj /aij } is never equal to xj /ahj .
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Chapter 6. Production Theory
Consequently, we see that the net output, that is, the quantity of the ith good available for ﬁnal consumption, is given by: n aij yj ; yi − j=1
and if the vector, c, of ﬁnal demands is feasible, it must be that: n ci = yi − aij yj for i = 1, . . . , n. j=1
(6.13)
Alternatively, if we deﬁne the n × n matrix A = [aij ] (the technology matrix), we can express this feasibility requirement as: y − Ay = c; or, denoting the n × n identity matrix by ‘I:’ (I − A)y = c. (The matrix I − A is called the Leontief matrix.) Since we are treating ﬁnal demand as exogenous, we can express the basic problem with which inputoutput analysis deals as: Problem A. Given c ∈ Rn+ , does there exist y ∈ Rn+ such that: (I − A)y = c,
(6.14)
By ≤ z,
(6.15)
and: where ‘z’ denotes the vector of available primary input quantities, and B is the m × n matrix, B = [bkj ] In the remainder of our discussion, however, we will concentrate our attention upon the equality (6.14), and ignore inequality (6.15), for the following reason. Suppose that for some c∗ and y ∗ , equation (6.14) is satisﬁed;, but that for some subset, K, of the primary resources, we have: (∀k ∈ K) : bk· · y ∗ > z k ; where ‘bk· ’ denotes the k th row of the matrix B. If we then deﬁne the numbers µk by: zk µk = for k ∈ K, bk· · y ∗
let: µ = min µk , k∈K
and deﬁne: we will have:
y = µy ∗ and c = µx∗ , (I − A)y = µ(I − A)y ∗ = µc∗ = c ,
6.4. InputOutput Analysis
169
while:3 By ≤ z.
(6.16)
Because of this, the key problem is to ﬁnd a solution to (6.15). If this can be done, it may then be necessary to scale back demand in order that the resource constaints are satisﬁed, but the primary problem is to determine whether the proportions involved in a vector of ﬁnal demands is or is not feasible. Consequently, instead of Problem A, we will be concentrating our attention upon the following. Problem B. Given c ∈ Rn+ , does there exist y ∈ Rn+ such that: (I − A)y = c? For anyone who has taken a course in linear algebra, Problem B may appear rather trivial, at least at ﬁrst glance. One’s inclination is probably to simply remark that, in order to solve Problem B, we can simply let: y = (I − A)−1 c, and move on to other things. However, there are two major diﬃculties here. First the Leontief matrix, (I − A) may be singular. Secondly, even if the Leontief matrix is nonsingular, it may be that the vector: (I − A)c, is not nonnegative; and thus the mathematical solution found may not be economically meaningful. In connection with these two points, consider the following conditions regarding equation (6.15) and the Leontief matrix. Condition I. There exists y ∈ Rn+ such that: (I − A)y > 0. Condition II. For every c ∈ Rn+ , there exists y ∈ Rn+ such that: (I − A) = c. Condition III. The n upper lefthand corner principal minors of the matrix (I −A) are all positive; that is: 1 − a11 . . . ... −ak1 . . . 3
−a1k >0 1 − akk
for k = 1, . . . , n.
The student should verify this inequality: see Excercise 7, at the end of this chapter.
170
Chapter 6. Production Theory
We will say that the technology matrix is productive if Condition (I) holds. Now consider what is, in eﬀect, the dual of the problem we have been examining. Suppose we consider the j th production sector as a potentially proﬁtmaximizing entity. If a vector of prices for the n outputs, p ∈ Rn+ , is given, and the j th sector’s output is yj , its input cost for intermediate goods inputs (the other yi ’s) is given by: Cj (yj ) =
n i=1
pi aij yj .
Consequently, value added in the j th sector, given the price vector p is given by: n pj − pi aij yj = p (I − A)j yj , i=1
column of the Leontief matrix by ‘(I − A)j ; and ‘p ’ where I am denoting the denotes the transpose of the vector p; that is, the row vector: j th
p = (p1 , . . . , pn ). In relation to this issue, consider the following conditions. Condition I . There exists p ∈ Rn+ such that: p (I − A) > 0.
(6.17)
Condition II . Given any v ∈ Rn+ , there exists p ∈ Rn+ such that: p (I − A) = v . We will say that the j th production sector is viable, given the price vector p ∈ Rn+ iﬀ: p (I − A)j ≥ 0; and that the production sector is sustainable, iﬀ there exists a price vector p ∈ Rn+ such that: p (I − A) > 0. In other words, if the production sector is sustainable then there exists a price vector, p ∈ Rn+ such that the j th production sector is viable, for each j = 1, . . . , n; more simply, the production sector is sustainable iﬀ Condition (I ) holds. One can prove the following, quite remarkable theorem; which establishes the aptness of the terminology we have introduced in this section. (For a proof, see Nikaido [1968, pp. 90–4].) 6.16 Theorem. Given that the technology matrix, A, is nonnegative, Conditions (I)–(III) and (I ) and (II ) are mutually equivalent. Thus if the technology matrix is productive, it is also sustainable; in fact, it will satisfy Condition (II ). Conversely, if the technology is sustainable, then it will satisfy Condition (II) as well. Because of this, it turns out that there is a very
6.5. Proﬁt Maximization
171
simple pair of conditions, either of which is suﬃcient to insure that all of Conditions (I)–(III) and (I ) and (II ) hold. Deﬁne: n ri = aij for i = 1, . . . , n, j=1
and: sj =
n i=1
aij
for j = 1, . . . , n.
6.17 Theorem. (BrauerSolow Conditions) Either of the following implies Conditions (I)–(III), as well as Conditions (I ) and (II ) ri < 1, for i = 1, . . . , n, or :
(6.18)
sj < 1, for j = 1, . . . , n.
(6.19)
Proof. Let y ∗ = (1, 1, . . . , 1) . Then if (6.18) holds; n aij = 1 − ri > 0 for i = 1, . . . , n. (I − A)y ∗ i = 1 − j=1
Thus Condition (I) holds, and it follows at once from Theorems 6.16 that Conditions (II), (III), (I ) and (II ) hold as well. A similar argument proves that (6.19) implies that Condition (I ) holds. In the proof of the above result we have shown that the technology matrix is productive if (6.18) holds. A similar argument establishes the fact that the technology matrix is sustainable if (6.19) holds. Before concluding this section, let’s return to the formulation of ‘Problem A,’ which was set out earlier in this section. In particular, recall (6.14) and (6.15) in the speciﬁcation of the problem. Let’s change notation a bit here to require that the matrices I and A in (6.14) are both r × r, where r is a positive integer; while the matrix B in (6.15) which deﬁnes the requirement of primary inputs is s × r, where s is a positive integer, and where n = r + s. Finally, we note that the aggregate production set assumed to hold in inputoutput analysis is given by: ( ) * + I −A Y = y ∈ Rn  (∃z ∈ Rr+ ) : y = z . −B
6.5
Proﬁt Maximization
In this section, we will consider some basic results concerning proﬁt maximization, and the relationship between proﬁt maximization and eﬃciency. I will leave the proof of the result following the deﬁnition as an exercise. 6.18 Deﬁnition. Given a production set, Y ⊆ Rn , we shall say that y ∗ ∈ Y is eﬃcient (in Y ) iﬀ there exists no y ∈ Y satisfying y > y ∗ . 6.19 Proposition. If y ∗ ∈ Y and p∗ ∈ Rn++ are such that: (∀y ∈ Y ) : p∗ · y ∗ ≥ p∗ · y, then y ∗ is eﬃcient in Y .
172
Chapter 6. Production Theory
6.20 Example. Suppose f : − R+ → R+ , and deﬁne Y ⊆ R2 by: Y = {y ∈ R2  y1 ≤ 0 & 0 ≤ y2 ≤ f (y1 )}. While the deﬁnition just presented yields a quite conventional production set, it follows from Proposition 6.19 that if we are analyzing the behavior of a proﬁtmaximizing ﬁrm, then we can conﬁne our attention to eﬃcient points in the production set. Obviously, a production vector y ∈ Y is eﬃcient if, and only if y2 = f (y1 ). Consequently, we can analyze this sort of situation with the simplest sort of production function; one showing a single output as a function of a simple input. The only unfamiliar aspect of this example is that in order to insure that Y is convex, it is usual to assume that for all y1 ∈ −R+ : f (y1 ) ≤ 0 & f (y1 ) ≤ 0.
The following two mathematical results will be used several times in the remainder of this book, but will be stated here without proof. For those interested, proofs are given in, for example, Moore [1999, pp. 297–300]. 6.21 Theorem. If A and B are disjoint and nonempty convex subsets of Rn , then there exists a hyperplane separating A and B; that is, there exists p∗ ∈ Rn \ {0} such that: sup p∗ · a ≤ inf p∗ · b. a∈A
b∈B
In other words, there exists a nonzero p∗ ∈ Rn and α ∈ R such that, for all a ∈ A and all b ∈ B: p∗ · a ≤ α ≤ p∗ · b. 6.22 Theorem. If A and B are nonempty, disjoint, closed convex sets, at least one of which is bounded, then there exists a hyperplane strongly separating A and B; that is, there exists a nonzero p ∈ Rn and β ∈ R such that: sup p · a < β < inf p · b.
a∈A
b∈B
6.23 Theorem. If Y is a convex (production) subset of Rn , and y ∗ ∈ Y is eﬃcient, then there exists p∗ ∈ Rn+ \ {0} such that: (∀y ∈ Y ) : p∗ · y ≤ p∗ · y ∗ . Proof. Deﬁne the set B by: B = {y ∈ Rn  y > y ∗ }. Then B is a convex set, and, since y ∗ is eﬃcient in Y , Y ∩B = ∅. It then follows from Theorem 6.21 that there exists a nonzero (price) vector in Rn and a real number α satisfying: (6.20) (∀y ∈ Y )(∀z ∈ B) : p∗ · y ≤ α ≤ p∗ · z. Our proof will therefore be complete if we can show that: p∗ > 0,
(6.21)
6.5. Proﬁt Maximization
173
and: p∗ · y ∗ = α.
(6.22)
To prove (6.21), suppose, by way of obtaining a contradiction, that for some h ∈ {1, . . . , n}, we have p∗h < 0; and deﬁne z ∗ ∈ Rn by: z ∗ = y ∗ − p∗h eh , where eh is the hth unit coordinate vector. Then, since p∗h < 0, it follows that z ∗ > y ∗ , and thus z ∗ ∈ B. However, α − p∗ · z ∗ ≥ p∗ · y ∗ − p∗ · z ∗ = p∗ · (y ∗ − z ∗ ) = p∗ · (p∗h eh ) = (p∗h )2 > 0. Therefore, α − p∗ · z ∗ > 0, that is: α > p∗ · z ∗ ; which, since z ∗ ∈ B, contradicts (6.20). Consequently, we see that (6.21) must hold. To prove (6.22), let > 0 be given, and let i be such that p∗i > 0. Deﬁning: z = y ∗ + (/p∗i )ei , we have z > y ∗ , so that z ∈ B, and thus: p∗ · z ∗ ≥ α. However, we then have: p∗ · y ∗ ≤ α ≤ p∗ · z = p∗ · [y ∗ + (/p∗i )ei ] = p∗ · y ∗ + (/p∗i )p∗i = p∗ · y ∗ + . Thus we see that, for all > 0, we have: p∗ · y ∗ ≤ α ≤ p∗ · y ∗ + , and it follows that p∗ · y ∗ = α.
Given an arbitrary price vector, p, there may or may not exist a production vector, y ∗ ∈ Y which maximizes proﬁts on Y . Obviously, however, we have a particular interest in those price vectors for which such a proﬁtmaximizing output vector exists. 6.24 Deﬁnitions. For a production set, Y , we deﬁne: 1. Π = Π(Y ) = {p ∈ Rn  (∃y ∗ ∈ Y )(∀y ∈ Y ) : p · y ∗ ≥ p · y}, 2. and, for p ∈ Π, we then deﬁne: π(p) = max p · y, y∈Y
and: σ(p) = {y ∈ Y  p · y = π(p)}.
174
Chapter 6. Production Theory
Π(Y) Y
Figure 6.5: Π(Y ), for linear Y . Figure 6.5, above, illustrates the relationship between Y and Π(Y ) for a linear production set. I will leave the proof of the ﬁrst of the following two results as an exercise. 6.25 Proposition. If Y satisﬁes free or semifree disposability, and p∗ ∈ Π(Y ), then p∗ ≥ 0. 6.26 Proposition. Whatever the form of Y, Π = Π(Y ) will be a nonempty cone; and π(·) and σ(·) will be positively homogeneous of degrees one and zero, respectively. Furthermore, if Π is convex, then the proﬁt function, π(·) is convex on Π. Proof. I will only prove the last part of this result here; leaving the remainder of the proof as an exercise. Let p , p∗ ∈ Π(Y ), and let θ ∈ [0, 1]. Then, given an arbitrary y ∈ Y , we have, since both θ and 1 − θ are nonnegative: θp + (1 − θ)p∗ · y = θp · y + (1 − θ)p∗ · y ≤ θπ(p ) + (1 − θ)π(p∗ ). Since Π(Y ) is convex:
θp + (1 − θ)p∗ ∈ Π(Y ),
and since y was an arbitrary element of Y , it follows that: π θp + (1 − θ)p∗ ≤ θπ(p ) + (1 − θ)π(p∗ ).
The following rather remarkable result ﬁrst appeared in Debreu [1959, p. 47]. Interestingly enough, however, Debreu credits the result to Samuelson [1947, Chapter 4]. (Samuelson’s Chapter 4 is concerned with the theory of revealed preference for consumer demand.) 6.27 Theorem. If p , p ∈ Π(Y ), y ∈ σ(p ), and y ∈ σ(p ), then: ∆p · ∆y ≥ 0;
6.5. Proﬁt Maximization where we deﬁne: Furthermore, if
175
∆p = p − p and ∆y = y − y .
p
takes the form: p = p + ∆pj ej ,
where ∆pj is a nonzero real number, and ej is the j th unit coordinate vector, then: ∆yj /∆pj ≥ 0.
(6.23)
Proof. By proﬁt maximization, we have: p · y ≥ p · y , and thus: Similarly,
p · (y − y ) = p · ∆y ≥ 0. p
·
y
≥
p
·
y ,
(6.24)
and thus: 0 ≥ p · y − p · y = p · ∆y.
(6.25)
Combining (6.24) and (6.25), we have: p · ∆y ≥ 0 ≥ p · ∆y; and thus:
0 ≤ p · ∆y − p · ∆y = ∆p · ∆y.
Now suppose that
p
(6.26)
takes the form: p = p + ∆pj ej ,
where ∆pj is nonzero. Then in this case, ∆p takes the form: ∆p = ∆pj ej = (0, . . . , 0, ∆pj , 0, . . . , 0); and thus, using (6.26): ∆p · ∆y = ∆pj ∆yj ≥ 0 def
(where ∆yj =
yj
−
yj ).
Dividing both sides of (6.27) by (∆pj
(6.27) )2
yields (6.23).
The following simple little result is useful surprisingly often in examples and applications of general equilibrium theory. I will leave the proof as an exercise. 6.28 Proposition. If 0 ∈ Y , and p ∈ Rn satisﬁes: (∀y ∈ Y ) : p · y ≤ 0, then p ∈ Π(Y ), and π(p) = 0. By making use of the preceding result, we can give a complete characterization of Π(Y ) and π(·) for production sets containing the origin and satisfying nondecreasing returns to scale; as follows.
176
Chapter 6. Production Theory
6.29 Proposition. If Y ⊆ Rn satisﬁes nondecreasing returns to scale and 0 ∈ Y , then: Π(Y ) = {p ∈ Rn  (∀y ∈ Y ) : p · y ≤ 0}. Furthermore, if p ∈ Π(Y ) and y ∈ σ(p), then p · y = 0. Thus we have: ∀p ∈ Π(Y ) : π(p) = 0. Proof. Suppose p ∈ Π(Y ) and y ∗ ∈ σ(p). Then, since 0 ∈ Y , we must have p · y ∗ ≥ 0 Suppose, by way of obtaining a contradiction, that p · y ∗ > 0. Since Y satisﬁes nondecreasing returns to scale, 2y ∗ ∈ Y . However, since p · y ∗ > 0: p · (2y ∗ ) = 2p · y ∗ > p · y ∗ ; contradicting the assumption that y ∗ ∈ σ(p). Thus we see that we must have p · y ∗ = π(p) ≤ 0; and thus: (∀y ∈ Y ) : p · y ≤ 0. Combining this with Proposition 6.28 completes our proof.
6.6
Proﬁt Maximizing with Constant Returns to Scale*
Recall that, in terms of the deﬁnitions in this book, a production set satisﬁes constant returns to scale if, and only if, it is a cone; that is, if and only if for all y ∈ Y , and all θ > 0, we have θy ∈ Y . The case in which Y is linear is a special case of this, and our ﬁrst result of this section deals with a special sort of cone. 6.30 Proposition. If Y ⊆ Rn is a proper linear production set; so that there exists a semipositive m × n matrix, B, and sets I and J satisfying the conditions of Deﬁnition 6.14, then deﬁning K = I ∩ J, we have the following. Given any a ∈ ∆m ∩ Rn++ and any p ∈ Rn satisfying: (∀i ∈ N \ K) : pi = 0 & (∀k ∈ K) : pk > 0, the price vector
p∗
∈ Rn deﬁned by:4 p∗ = B a + p,
is a strictly positive element of Π. Proof. Suppose p∗ has the indicated form, and let y ∈ Y be arbitrary. Then: p∗ · y = (p∗ ) y = (a B + p )y = a By + p y = a By; where the last equality is by the deﬁnition of K and the fact that pj = 0 for all j ∈ / K. However, since a 0 and By ≤ 0 by the fact that y ∈ Y , we then see that p · y ≤ 0. Thus it follows from Proposition 6.28 that p∗ ∈ Π(Y ). The fact 4
We denote the transpose of a matrix, B, by ‘B .’
6.6. Proﬁt Maximizing with Constant Returns to Scale*
177
that p∗ 0 follows from the deﬁnition of a proper linear technology, the fact that a 0 and the speciﬁcation of p. Details will be left as an exercise. A type of cone which is of particular interest for us is the cone dual to another cone, where we deﬁne this as follows.5 6.31 Deﬁnition. If K ⊆ Rn is a cone, we deﬁne the dual cone for K, K ∗ , by: K ∗ = {y ∈ K  (∀x ∈ K) : y · x ≤ 0}. A reason for our particular interest in the dual cone is that if Y is a cone which contains the origin in Rn , then Π(Y ) = Y ∗ , a fact whose proof I will leave as an exercise. Having established a reason for an interest in the dual cone, let’s take a look at some of the properties of same. The proof of the ﬁrst of the following two results will be left as an exercise. A closed cone, incidentally, is simply a cone which is also a closed set. Notice that, while a cone does not necessarily contain the origin, a closed cone necessarily does. 6.32 Proposition. If Y ⊆ Rn is a cone, its dual, Y ∗ , is a closed, convex cone.6 If K ∗ is the dual cone to a cone, K, K ∗ itself has a dual cone, which we denote by ‘K ∗∗ ;’ that is: def K ∗∗ = (K ∗ )∗ . The following result sets out the basic properties of this second dual. 6.33 Proposition. If K ⊆ Rn is a cone, then: i. we have K ⊆ K ∗∗ , and: ii. K = K ∗∗ if, and only if, K is closed and convex. Proof. I will leave the proof of part (i) of this result and the of the fact that K ∗∗ = K only if K is a closed convex coneas an exercise. To prove that K ∗∗ ⊆ K if K is a closed convex cone, suppose K is a closed convex cone, and that a ∈ Rn is a point which is not an element of K. Then by Theorem 6.22 of the previous section, there exists a nonzero p ∈ Rn , and β ∈ R such that: p · a > β = sup p · x. x∈K
(6.28)
Now, we must then have: (∀x ∈ K) : p · x ≤ 0; x∗
for, suppose that for some ∈ K we have p · x∗ > 0. Then for all θ ∈ R++ , θx∗ ∈ K; and: lim p · θx∗ = lim θ p · x∗ = +∞, θ→+∞
5
θ→+∞
In the mathematical literature, it is quite common to see the dual cone deﬁned as the set of points having a nonnegative inner product with each element of Y , rather than, as here, those having a nonpositive inner product with each element. The mathematical properties of the two types of dual are, however, precisely the same, and for us the more directly useful deﬁnition is the one given here. 6 Remember that a convex cone is simply a cone which is also a convex set. The result stated here does not, as you may have noticed, really need the assumption that Y is a cone.
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Chapter 6. Production Theory
which contradicts (6.28). Since 0 ∈ K by virtue of the fact that K is closed, it now follows that β = 0; and, therefore, that p ∈ K ∗ . However, since p · a > 0, we can now see that a ∈ / K ∗∗ . Therefore, if y ∈ K ∗∗ , we must have y ∈ K; that is, K ∗∗ ⊆ K Since, by part (i) we always have K ⊆ K ∗∗ , it now follows that the two sets are equal in this case. This last result is the basis of one type of duality result in production theory; namely, if Y is a closed convex cone, then Y = [Π(Y )]∗ . In other words, if we specify, or have empirical data providing a reason to believe that a production set Y is a closed convex cone, then Y is completely determined by Π(Y ), the set of prices for which a maximum proﬁt output exists. As one simple application: in such a situation, if we assume or specify that a maximum proﬁt level exists for each p ∈ Rn+ , then Y can only be −Rn+ ; in other words, no positive production can take place. Another useful mathematical result concerning cones and their duals is the following theorem; a proof of which can be found in Nikaido [1968, pp. 35–6]. 6.34 Theorem. Let K ⊆ Rn be a closed convex cone. If K contains no semipositive element, then K ∗ contains a positive element, and viceversa. Thus, if Y is a closed convex production set satisfying constant returns to scale, and also satisﬁes 6.4.1 (Impossibility of Free Production), then Π(Y ) contains a strictly postive price, p 0.
6.7
Production in General Equilibrium Theory
In our basic general equlibrium model of the next chapter, we suppose that the number of producers is a given positive integer, ; and producers are indexed by k (k = 1, . . . , ). The k th producer chooses a production plan (or production vector, or netput vector), y k , from some nonempty subset of Rn , Yk . We refer to the set Yk as the k th producer’s production set. 6.35 Deﬁnitions. If y k ∈ Yk for k = 1, . . . , , then the vector: y= yk , k=1
is called the aggregate (or total) production vector; and: def Y = Yk , k=1
is called the aggregate (or total) production set. The notation of the previous sections is, in this context, extended as follows. Notation/Deﬁnitions. For k = 1, . . . , , we deﬁne: 1. Πk = {p ∈ Rn \ {0}  (∃y ∗ ∈ Yk )(∀y ∈ Yk ) : p · y ∗ ≥ p · y}, 2. and for p ∈ Πk , we then deﬁne: πk (p) = max p · y and σk (p) = {y ∈ Yk  p · y = πk (p)}. y∈Yk
6.7. Production in General Equilibrium Theory
179
3. We use a similar notation, simply dropping the subscript ‘k,’ to denote the corresponding concepts for the aggregate production set, Y . If one is given production sets, Yk , it may be very diﬃcult to characterize the aggregate production set, Y . Fortunately, we will generally not need to do so; it is usually suﬃcient to use the formal deﬁnition of the summation set. There are some cases, however, in which it is easy and may be useful to characterize the set Y . One interesting example, which corresponds to a model used quite frequently in the theoretical public economics literature, is presented in the following. 6.36 Example. Suppose there are n − 1 ﬁrms (that is, = n − 1); with the k th ﬁrm’s production set given by: / {k, n} ; (6.29) Yk = y k ∈ Rn  ykk ≥ 0, ykn ≤ 0, ck ykk + ykn ≤ 0 & ykj = 0 for j ∈ where we suppose: ck > 0 for k = 1, . . . , n − 1. In other words, we suppose the k th ﬁrm produces only the k th commodity, and uses only the nth commodity (which we generally suppose to be labor) as an input. In this case, the aggregate production set is given by: −1 Y = y ∈ Rn  ck yk + yn ≤ 0, yk ≥ 0, for k = 1, . . . , n − 1, & yn ≤ 0 . (6.30) k=1
In other words, in the framework of the terminology introduced in Deﬁnition 6.14, the aggregate production set is a proper linear technology, with I = {n} and J = {1, . . . , n − 1}. We can prove this for the special case in which = 2 as follows (the basic argument for the case in which is an arbitrary positive integer is basically the same; the notation is just messier). Suppose ﬁrst that y k ∈ Yk , for k = 1, 2. Then y 1 and y 2 are of the form: y 1 = (y11 , 0, y13 ) and y 2 = (0, y22 , y23 ), with: y11 ≥ 0, y13 ≤ 0, c1 y11 + y13 ≤ 0,
(6.31)
y22 ≥ 0, y23 ≤ 0, c2 y22 + y23 ≤ 0.
(6.32)
and: It is then easy to check (I will leave it to you to verify the details) that: y 1 + y 2 = (y11 , y22 , y13 + y23 ), is an element of the set Y deﬁned in (6.30). Conversely, suppose y ∈ Y . In this case, if we deﬁne y 1 and y 2 by: y11 = y1 , y12 = 0, y13 = −c1 y1 ,
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Chapter 6. Production Theory
and: y21 = 0, y22 = y2 , y23 = y3 + c1 y1 , it is easy to show that y k ∈ Yk , for k = 1, 2, and that: y 1 + y 2 = y.
I will leave the proof of the following mathematical result as an exercise. It will often be useful in our work in general equilibrium theory. 6.37 Proposition. If Aj is a convex subset of Rn , for j = 1, . . . , m, then the set A deﬁned by: m Aj , A= j=1
is a convex set. One implication of this last result is this: if each production set, Yk , is convex, then so is the aggregate production set. This implication is particularly interesting in view of our next result. Suppose we actually knew the exact form of the aggregate production set in an economy, and also knew the sum of the individual production vectors chosen by the ﬁrms in the economy, given a price vector, p∗ . Could we then tell if the individual ﬁrms were maximizing proﬁts, given p∗ , even given that we did not know the production vectors chosen by the individual ﬁrms? The answer is, yes, we could; as is established in the following theorem. 6.38 Theorem. Suppose p∗ is a nonzero price vector, let Yk be a production set (a nonempty subset of Rn ), for k = 1, . . . , , and let: def
Y =
k=1
Yk ,
be the corresponding aggregate production set. Then: 1. if y ∗ ∈ Y is such that: (∀y ∈ Y ) : p∗ · y ≤ p∗ · y ∗ ,
(6.33)
and y k ∈ Yk (k = 1, . . . , ) are such that: y∗ =
k=1
y k ,
then for each k, we have: (∀y k ∈ Yk ) : p∗ · y k ≤ p∗ · y k . 2. Conversely, if (6.34) holds, for
y k
y∗ = then y ∗ will satisfy (6.33).
∈ Yk (k = 1, . . . , ), and we deﬁne:
k=1
y k ,
(6.34)
6.7. Production in General Equilibrium Theory
181
Proof. Suppose y ∗ ∈ Y is such that: (∀y ∈ Y ) : p∗ · y ≤ p∗ · y ∗ , and that: y∗ =
k=1
(6.35)
y k ,
with y k ∈ Yk , for k = 1, . . . , . We will prove that y 1 maximizes proﬁts over Y1 . A similar argument then establishes the general case. Accordingly, let y 1 ∈ Y1 be arbitrary. Then if we deﬁne y ∈ Y by: y = y 1 + y k , k=2
we see from (6.34) that we must have: p∗ · y ∗ ≥ p∗ · y . However: p∗ · y ∗ = p∗ · while:
k=1
p∗ · y = p∗ · y 1 +
(6.36)
y k = p∗ · y 1 +
k=2
k=2
y k = p∗ · y 1 +
p∗ · y k ,
k=2
p∗ · y k ,
(6.37) (6.38)
and thus, from (6.36)–(6.38), we see that; p∗ · y 1 ≥ p∗ · y 1 . Since y 1 was an arbitrary element of Y1 , it now follows that y 1 maximizes proﬁts on Y1 . Now suppose that: y∗ = y k , (6.39) k=1
where y k maximizes proﬁts on Yk , for k = 1, . . . , , and let y ∈ Y be arbitrary. Then there exist y k ∈ Yk , for k = 1, . . . , , such that: y= yk . k=1
Since y k maximizes proﬁts over Yk , for each k, it then follows easily from (6.39) that: p∗ · y ∗ ≥ p∗ · y. The following is a more or less immediate implication of Theorem 6.38. 6.39 Corollary. Given the production sets, Y1 , . . . , Y and corresponding aggregate production set, Y , we have: Πk ; Π= k=1
and, for all p ∈ Π: π(p) =
k=1
πk (p) and σ(p) =
k=1
σk (p).
182
6.8
Chapter 6. Production Theory
Activity Analysis*
In this section, we will discuss another method of specifying a linear production set; namely Activity Analysis. Activity Analysis no longer commands the attention of economists to the extent which it did thirtyﬁve years or so ago, but it still can be quite useful in applied work, as well as involving some interesting theoretical problems. We will, however, consider only a very abbreviated presentation of this theory here. Consider a production process, which we will refer to as a productive activity, and suppose that if the activity is operated at the level z ∈ R+ , then the input requirements are given by: u = az, while output is given by: x = bz, where a and b are semipositive q and mvectors, respectively. We can express the corresponding production set, Y , as: (6.40) Y = (v, x) ∈ Rq+m  (∃z ∈ R+ ) : (v, x) = (−a, b)z . For example, suppose that q = m = 1, a = 2, and b = 1. Then: Y = (v, x) ∈ R2  (∃z ∈ R+ ) : (v, x) = (−2, 1)z = (v, x) ∈ R2  v ≤ 0 & x = −(1/2)v . We can usefully modify and generalize this example in two directions, as follows. Suppose ﬁrst that there are such productive activities, and denote the inputrequirement and output vectors for the k th such activity by ‘ak ’ and ‘bk ,’ respectively; with the level at which the k th activity is being operated being denoted by ‘zk ,’ for k = 1, . . . , . If we assume that all activities can be operated simultaneously with no loss of productive eﬃciency (that is, if there are no externalities in production), then with the activity levels given by the vector z = (z1 , . . . , z ), output, x, is given by: x= bk zk , k=1
while the required input vector is given by: u=
k=1
ak zk .
More compactly, if we deﬁne the q × matrix A, and the m × matrix B by: A = −a1 −a2 . . . −a and B = b1 b2 . . . b , respectively; then we see that the inputoutput vector (v, x) is feasible (where we are now returning to the general equilibrium convention of denoting input quantities by nonpositive numbers) if, and only if, there exists z ∈ R+ such that: (6.41) Y = (v, x) ∈ Rq+m  (∃z ∈ R+ ) : v = Az & x = Bz .
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183
The formulation of the above paragraph is very convenient and useful in a great many contexts. In the standard general equilibrium model which we are in the process of considering, there is a better way of proceeding, however. We begin by noting that we can assume here, without loss of generality, that q = m = n, where n is the total number of commodities available. Thus, suppose that a productive activity uses two inputs to produce one output; with a and b given by: a = (2, 3) and b = 1, respectively. Suppose further that there are only four commodities available in a given economy, and that the two inputs being used here are the ﬁrst and the third commodities, while the commodity being produced is the second commodity. Let’s then deﬁne the new inputrequirement and output vectors, a∗ and b∗ by: a∗ = (2, 0, 3, 0) and b∗ = (0, 1, 0, 0), respectively. Next, deﬁne the vector c as: c = b∗ − a∗ , and notice that the feasible production set can now be deﬁned as: Y = y ∈ R4  (∃z ∈ R+ ) : y = cz . We can complete this line of generalization by allowing for several productive activities once again, leading to a production set of the form: Y = {y ∈ Rn  (∃z ∈ R+ ) : y = Cz},
(6.42)
where C is taken to be a nonzero n × matrix. By allowing for disposability, we are led to the two further generic examples: Y = {y ∈ Rn  (∃z ∈ R+ ) : y ≤ cz},
(6.43)
or, for the activity case, we can consider: Y = {y ∈ Rn  (∃z ∈ R+ ) : y ≤ Cz}.
(6.44)
Denoting the ith row of the matrix C by ‘ci· ,’ for i = 1, . . . , n; we generally assume in this context that there exists h ∈ {1, . . . , n} such that: ch· < 0.
(6.45)
Notice that in the case of a single activity, and where c is given by: c = b − a, condition (6.45) will be guaranteed by the requirement: a·b=0
(6.46)
184
Chapter 6. Production Theory
(remember that we are assuming that both a and b are semipositive vectors). Verbally interpret this condition. If (6.45) [or the special case in (6.46)] holds, which of the conditions of Deﬁnition 6.4 will Y satisfy? Clearly the sort of production set being considered here, in particular, as speciﬁed by (6.42), above, is in some sense linear. On the other hand, this seems to be quite a diﬀerent sort of set from our deﬁnition of a linear production set, as presented in Deﬁnition 6.9. Remarkably enough, however, the two speciﬁcations are, in a formal mathematical sense, equivalent. In order to demonstrate this, we will, in the remainder of this section, consider some results from linear algebra which are of interest in their own right. The ﬁrst three of these results are known collectively, as ‘theorems of the alternative,’ and our proofs of them are adapted from Nikaido [1968, pp. 36–9]. 6.40 Theorem. (Stiemke, 1915). If A is an m × n matrix, then exactly one of the following holds. The equation: Ax = 0, (6.47) has a (strictly) positive solution, or the inequality: p A > 0,
(6.48)
has a solution. ¯ ∈ Rn++ Proof. Suppose, by way of obtaining a contradiction, that there exist x ¯ ∈ Rm satisfying (6.47) and (6.48), respectively. Then we have: and p ¯ · 0 = 0. ¯ A¯ ¯ A¯ x=p x =p p ¯ 0: On the other hand, by (6.48) and the fact that x ¯A x ¯ > 0; ¯ A¯ x= p p giving a contradiction. Now suppose there exists no p ∈ Rm which satisﬁes (6.48), and deﬁne: L = {y ∈ Rn  (∃p ∈ Rm ) : A p = y}. Then, by hypothesis, L does not contain a semipositive point. But then, by Theorem 6.34, L∗ contains a strictly positive element, x∗ . Moreover, since L is a linear subspace, L∗ = L⊥ ; and, given our deﬁnition of L, it is clear that: L⊥ = {x ∈ Rn  Ax = 0}; which establishes the desired result.
The nifty thing about Stiemke’s theorem is that it often provides a much simpler way of establishing whether or not a system of linear equations has a positive solution than trying to solve it directly. Our next result is probably even more interesting, from a theoretical point of view.
6.8. Activity Analysis*
185
6.41 Theorem. (Tucker, 1956) If A is an m × n matrix, then the system of linear inequalities: A p ≥ 0, and the homogeneous linear equation: Ax = 0, always have a pair of solutions,
p∗
∈ Rm and x∗ ∈ Rn such that:
x∗ ≥ 0 & A p∗ + x∗ 0. Proof. For any p ∈ Rm , let ‘ A p j ’ denote the j th coordinate of A p, and deﬁne: N (p) = j ∈ {1, . . . , n}  A p j > 0 . Next, deﬁne the set P by: P = {p ∈ Rm  A p ≥ 0}; and note that, since for all p ∈ P , we have N (p) ⊆ {1, . . . , n}, the number of ∈ P . We distinguish elements in N (p), #N (p), is maximized at some point p several cases, depending upon the value of #N ( p). 1. #N ( p) = 0. Here it follows from Stiemke’s Theorem that there exists x∗ ∈ Rn++ such that x∗ 0, and Ax∗ = 0. Consequently, if we deﬁne p∗ = 0 ∈ Rm , we have: A p∗ ≥ 0, Ax∗ = 0, x∗ ≥ 0, and A p∗ + x∗ 0, as desired. 0, and setting x = 0 ∈ Rn , we have: 2. #N ( p) = n. Here we must have A p +x 0, A x = 0 and A p once again. def 3. 1 ≤ #N ( p) = k < n. Here we can assume, without loss of generality, that: ]j > 0 for j = 1, . . . , k, [A p
]j = 0 for j = k + 1, . . . , n. and [A p
Deﬁne B as the submatrix consisting of the ﬁrst k columns of A, and C as the submatrix consisting of columns k + 1, . . . , n. Thus we can write: A = [B C], and we have:
0 and C p = 0. Bp
Now suppose, by way of obtaining a contradiction, that there exists y ∈ Rm such that: C y > 0, and deﬁne p ∈ Rm by: p = θ p + y,
186
Chapter 6. Production Theory
where: θ = max
1≤j≤k
−[B y]j ]j [B p
+ 1.
Then, for each j ∈ {1, . . . , n}, we have: ]j ≥ −[B y]j + [B p ]j > −[B y]j ; p)]j = θ[B p [B (θ so that:
p + y) = B (θ p) + B y > 0. B (θ
Moreover:
+ C y = C y 0, p + y) = θC p C (θ
so that p ∈ P . But this is impossible, since #N (p) > #N ( p). From the argument of the preceding paragraph we conclude that there exists no y ∈ Rm satisfying: C y > 0. n−k such Consequently, it follows from Stiemke’s Theorem that there exists z ∈ R++ that: Cz = 0.
If we now deﬁne x∗ ∈ Rn+ by: x∗ =
! 0 , z
we see that: Ax∗ = [B C] = [B C] and: + x∗ = A p as desired.
! ! B 0 + = p z C
0 z
! = B0 + Cz = 0,
! ! 0 Bp + = z C p
! Bp 0, z
By making use of Tucker’s Theorem, we can prove yet another sort of ‘theorem of the alternative,’ which in this case is known as the ‘MinkowskiFarkas Lemma.’7 6.42 Theorem. (Farkas, 1902; Minkowski, 1910) If A is an m × n matrix and b ∈ Rm , then exactly one of the following holds. Either the equation: Ax = b,
(6.49)
has a nonnegative solution, or the inequalities: p A ≥ 0 & p b = p · b < 0,
(6.50)
have a solution. 7 For yet additional results of this sort, see Nikaido [1968, pp. 38–9], or Mangasarian [1969, Chapter 2].
6.8. Activity Analysis*
187
Proof. This time I will leave as an exercise the proof that there cannot exist both x∗ ∈ Rn+ satisfying (6.49) and p∗ ∈ Rm satisfying (6.50). To show that one of the two must hold, we note that by Tucker’s Theorem, there exist p∗ ∈ Rm and z ∗ ∈ Rn+1 ++ such that: ) * ) ∗* A p A ∗ = p ≥ 0, (6.51) −b −b p∗ A − b z ∗ = 0, (6.52) and:
* ) ∗* A p A ∗ ∗ p + z ∗ 0. + z = −b −b p∗
)
(6.53)
Now, if in fact there exists no p ∈ Rm such that: A p ≥ 0 and p b < 0, then it follows from (6.51) that:
p∗ · b = 0;
in which case we have from (6.53) that zn+1 > 0; so that, deﬁning x∗ ∈ Rn+ by: x∗j = it follows from (6.52) that:
zj zn+1
for j = 1, . . . , n,
Ax∗ = b.
6.43 Deﬁnition. If a vector a is an element of Rn , we deﬁne the ray generated by a, denoted by ‘(a),’ by: (a) = {x ∈ Rn  (∃θ ∈ R+ ) : x = θa}. Given aj ∈ Rm , for j = 1, . . . , n, we follow the usual set summation rule in deﬁning (a1 )+(a2 )+. . . (an ) as the set of all y ∈ Rm such that there exist y j ∈ (aj ), for j = 1, . . . , n, such that: n yj . y= j=1
The deﬁnition toward which I have been aiming can now be set forth as follows. 6.44 Deﬁnition. A set K ⊆ Rm is said to be a polyhedral cone iﬀ there exist vectors a1 , . . . , an ∈ Rm such that: K = (a1 ) + · · · + (an ). I will leave it to you to prove that a polyhedral cone is a convex cone. I will also leave the proof of the following proposition as an exercise. 6.45 Proposition. A set K ⊆ Rm is a polyhedral cone if, and only if there exists an m × n matrix, A such that: K = y ∈ Rm  (∃x ∈ Rn+ ) : y = Ax .
188
Chapter 6. Production Theory
Now, having established Proposition 6.45, let’s return to a consideration of the general activity analysis model developed at the beginning of this section. Recall that, given the assumptions of that model, the production set took the form: Y = {y ∈ Rn  (∃x ∈ Rn+ ) : y = Cx}; which we now see is a polyhedral cone. In fact, every one of the ‘theorem of the alternative’ type results which we have just established has an economically meaningful application/interpretation in terms of the activity analysis model of production; although I will leave as an exercise the task of verifying this statement. I will close this section with some results which eﬀectively deﬁne the relationship between the activity analysis model and the linear production set deﬁnitions set forth earlier in this chapter. 6.46 Theorem. If K ⊆ Rm is a polyhedral cone, then K ∗∗ = K. Proof. We have already noted the fact that, for any cone, K, we must necessarily have: K ⊆ K ∗∗ . To complete our proof, let the m × n matrix A be such that: K = {y ∈ Rm  (∃x ∈ Rn+ ) : y = Ax}.
(6.54)
Then it is easy to prove that: K ∗ = {z ∈ Rm  z A ≤ 0} = {z ∈ Rm  z (−A) ≥ 0}. Now, suppose b ∈ / K. Then it follows from (6.54) that there exists no nonnegative x such that: Ax = b, or (−A)x = −b; ¯ ∈ Rm such that: and thus by the MinkowskiFarkas Lemma, there exists z ¯ (−A) ≥ 0 and z ¯ · (−b) < 0. z ¯ ∈ K ∗ , while by the second, it By the ﬁrst of these two inequalities, we see that z then follows that b ∈ / K ∗∗ . Consequently, if b ∈ K ∗∗ , then b ∈ K. The following is now an immediate implication of Proposition 6.33 and Theorem 6.46. 6.47 Corollary. A polyhedral cone is a closed set. For a proof of our next, and ﬁnal result of this section, see, for example, Nikaido [1968, p. 42]. 6.48 Theorem. If A is an m × n matrix, the set Y ⊆ Rn deﬁned by: Y = {y ∈ Rn  Ay ≤ 0}, is a polyhedral cone.
6.8. Activity Analysis*
189
As you have probably already noticed, it follows immediately from Theorem 6.45 that any linear production set can be generated by an activity analysis model. The converse is also true. In this case, the proof of the statement is a little more tricky, but we can proceed as follows. Let Y ⊆ Rn be generated by an activity analysis model, so that there exists an n × m matrix, A, such that: Y = y ∈ Rn  (∃x ∈ Rm + ) : y = Ax . Then it is easy to prove (although I will leave this as an exercise) that: Y ∗ = Π(Y ) = {p ∈ Rn  A p ≤ 0}.
(6.55)
∗
But then it follows from Theorem 6.48 that Y is a polyhedral cone; so that there exists a positive integer, q, and an n × q matrix, B, such that: Y ∗ = p ∈ Rn  (∃z ∈ Rq+ ) : p = Bz . However, by the same reasoning as yielded (6.55), we can see that: Y ∗∗ = (Y ∗ )∗ = {y ∈ Rn  B y ≤ 0}. Since Theorem 6.46 tells us that Y ∗∗ = Y , we can now see that Y is a linear production set, as per Deﬁnition 6.9. Exercises. 1. Let A be a nonnegative m × q matrix; and deﬁne the production set, T , by: T = {(v, x) ∈ Rm+q  v + Ax ≤ 0 & x ∈ Rq+ } On the basis of this information, answer the following ﬁve questions. In each case, you should try to prove the property directly, and not by appealing to results established in this chapter. a. Does this production process satisfy nondecreasing returns to scale? Explain brieﬂy. b. Is this production set convex? Is it additive? Explain your answer. c. Suppose now that the matrix, A, satisﬁes the following condition: for each j ∈ {1, . . . , q}, there exists i ∈ {1, . . . , m} such that aij > 0. Can you characterize the eﬃcient pairs, (v, x) for this production set? d. Suppose now that w ∈ Rm ++ is the vector of prices for the inputs, v, and that p ∈ Rq++ is the vector of prices of the outputs, x. What must be the relationship between w and p if a pricetaking (and proﬁtmaximizing) ﬁrm operating this technology is to produce a nonzero (but ﬁnite) output? e. Given that A satisﬁes the condition deﬁned in part (c), above, does T satisfy Impossibility of Free Production? Does it satisfy Irreversibility? Explain your answers. 2. Prove the ﬁrst and third of the ‘Facts Regarding Disposability’ set out in Section 2.
190
Chapter 6. Production Theory
3. Show that production constraints (6.6)–(6.8) are equivalent to (6.9)–(6.10) in the example in Section 3. 4. Prove Proposition 6.15. 5. Complete the proof outlined in Example 6.36. 6. Prove Proposition 6.37 7. Verify inequality (6.16) in Section 4.
Chapter 7
Fundamental Welfare Theorems 7.1
Introduction
In this chapter, we will be extending the development of the ‘Fundamental Theorems of Welfare Economics‘ from the pure exchange economy case which we discussed in Chapter 5 to a production economy. Much of the analysis will be almost unchanged from the corresponding material set out in Chapter 5. The main diﬀerence is that we will present here a detailed proof and discussion of the ‘Second Fundamental Theorem of Welfare Economics’ (the unbaisedness result), neither of which was included in Chapter 5.
7.2
Competitive Equilibrium with Production
We will suppose in our discussion here that there are given ﬁnite (integer) numbers of commodities, consumers, and ﬁrms; and we will denote these quantities by ‘n, m,’ and ‘ ,’ respectively. The commodity space then becomes Rn , and we will employ the following system of notation. First, we will denote the set of consumers by ‘M ,’ and will use ‘K’ to denote the set of producers; in other words: M = {1, . . . , m} and K = {1, . . . , }. The remaining basic notation is as follows: Xi ⊆ Rn denotes the ith consumer’s consumption set, and we will assume throughout that Xi = ∅, for i = 1, . . . , m (that is, Xi = ∅, for all i ∈ M ); ‘Pi ’ denotes the ith consumer’s (strict) preference relation on Xi , and we will assume throughout that Pi is irreﬂexive, for i = 1, . . . , m; Yk ⊆ Rn denotes the k th ﬁrm’s production set, and we will asssume that Yk = ∅, for k = 1, . . . , ; r ∈ Rn will denote the aggregate resource endowment of the economy. 7.1 Deﬁnition. When we write ‘E is an economy,’ we will mean that E is a tuple of the form: E = (Xi , Pi , Yk , r),
192
Chapter 7. Fundamental Welfare Theorems
where Xi , Pi (i = 1, . . . , m), Yk (k = 1, . . . , ), and r ∈ Rn satisfy the above conditions. In dealing with allocations for an economy, we will denote the coordinates of the ith consumer’s commodity bundle, xi , by ‘xij ’, (j = 1, . . . , n); that is, we write: xi = (xi1 , . . . , xin ), where ‘xij ’ denotes the quantity of the j th commodity available to (or being made available by) the ith consumer. We then follow the convention that if: a. xij ≥ 0, then the j th commodity is available for i’s consumption in the amount xij , while if: b. xij < 0, then the consumer is oﬀering to supply the j th commodity (or service), in the amount xij  = −xij . With this convention, notice that if commodity prices are given by the vector p ∈ Rn+ , then the net expenditure necessary for the consumer to obtain the bundle xi ∈ Xi is given by the inner product of p and xi , that is, p · xi . A similar convention will be followed with respect to production vectors y k ∈ Yk ; which, where necessary for clarity, we will write out as: y k = (yk1 , . . . , ykn ). In this case, if: a. ykj ≥ 0, then the k th producer is producing (or planning to produce) the j th commodity in the net amount ykj ; while if b. ykj < 0, then the producer is using the j th commodity as an input1 in the amount ykj  = −ykj . Consequently, given prices p ∈ Rn+ , the proﬁt to the k th producer yielded by the choice of y k ∈ Yk is given by p · y k . We will, in fact, deﬁne the proﬁt function πk on Rn+ by: πk (p) = max{p · y k  y k ∈ Yk }, and assume that the producer attempts to maximize proﬁts, taking the price vector as given.2 In dealing with an economy, E = (Xi , Pi , Yk , r), we denote the cartesian product of the Xi ’s, the consumption allocation space, by ‘X’, or by ‘X(E),’ if it appears that a reminder might be needed as to the association of the set with the economy; that is, m Xi = Xi . X ≡ X(E) = i=1
i∈M
Similarly, we denote the product of the Yk ’s, the production allocation space by Y, or Y(E): Y ≡ Y(E) = Yk = Yk . k=1
1
k∈K
For this particular production plan; other technologically feasible production plans for the producer may have ykj = 0, or, indeed, have ykj > 0. 2 So that the producers are taken to be pure competitors, that is ‘pricetakers.’
7.2. Competitive Equilibrium with Production
193
We use the generic notation, ‘xi , xi , x∗i ,3 ’ and so on to denote elements of X; and, similarly, ‘y k , y k , y ∗k ,’ will be used to denote elements of Y. Combining these, we will use ‘(xi , y k ), (xi , y k ), (x∗i , y ∗k ),’ and so on, to denote elements of the allocation space:
m X×Y≡ Xi × Yk . i=1
k=1
7.2 Deﬁnitions. Let E be an economy. An (m + ) · ntuple, xi , y k ∈ R(m+)n will be said to be a feasible (or attainable) allocation for E iﬀ: 1. xi ∈ Xi for i = 1, . . . , m, 2. y k ∈ Yk for k = 1, . . . , , and: 3. m i=1 xi = r + k=1 y k In other words, xi , y k is feasible for E iﬀ xi , y k ∈ X × Y, and: i∈M
xi = r +
k∈K
yk .
We will denote the set of all feasible or attainable allocations for E by ‘A(E)’. We are going to want to discuss competitive, or Walrasian equilibria4 for an economy, E; but of course we cannot deﬁne such an equilibrium without specifying what the consumers have to spend in such an economy. Our next deﬁnition will provide us with a great deal of ﬂexibility in this respect. 7.3 Deﬁnition. Given an economy, E = (Xi , Pi , Yk , r), we shall say that a vector w = (w1 , . . . , wm ) is an assignment of wealth (or a wealthassignment vector) for E, given the price vector p∗ ∈ Rn , iﬀ for each i we have: ¯ i ≤ wi , (∃¯ xi ∈ Xi ) : p∗ · x and w satisﬁes:
m i=1
wi = p∗ · r +
k=1
πk (p∗ ).
(7.1)
A wealth assignment vector, given a vector of prices, p∗ , must provide each consumer with enough wealth to purchase something in its consumption set, and must also exhaust the sum of the value of resources plus aggregate proﬁts, given p∗ . We then use the concept of a wealth assignment vector to deﬁne a competitive (or Walrasian) equilibrium for an economy, E, as follows. 7.4 Deﬁnitions. An (m + + 1) · ntuple, (x∗i , y ∗k , p∗ ), is a competitive (or Walrasian) equilibrium for the economy E = (Xi , Pi , Yk , r), iﬀ there is an assignment of wealth for E, given p∗ , w = (w1 , . . . , wm ), such that: ∗ 1. p = 0, 2. x∗i , y ∗k ∈ A(E), 3. for each k ∈ K, we have: p∗ · y ∗k = πk (p∗ ), and 3
Or sometimes, for example, xi m i=1 , or xi i∈M , as was indicated in Chapter 5. We will use these two terms, competitive equilibrium and Walrasian equilibrium, as synonyms throughout the remainder of this book; except in Chapter 8, as will be explained there. 4
194
Chapter 7. Fundamental Welfare Theorems
4. for each i ∈ M , we have: a. p∗ · x∗i ≤ wi , and: b. (∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > wi . In this case we shall also say that (x∗i , y ∗k , p∗ ) is a competitive (or Walrasian) equilibrium for the economy E, given the wealth assignment, w = (w1 , . . . , wm ). I should at this point take a moment to explain the inclusion of the condition p∗ = 0 in the above deﬁnition. It should be clear that if at each attainable consumption allocation, xi ∈ X(E), at least one consumer is not satiated, then the satisfaction of condition 4 of Deﬁnition 7.4 implies p∗ = 0. Consequently, the requirement that p∗ = 0 is almost redundant in our deﬁnition of a competitive equilibrium. However, we will need this last condition as a part of our deﬁnition of a ‘quasicompetitive equilibrium,’ a concept we will be deﬁning later in this chapter; and in order to ensure that a competitive equilibrium is always a special case of a quasicompetitive equilibrium, I have included condition 1 in our deﬁnition. The most usual way of specifying a wealth assignment for an economy is based upon the following deﬁnition. 7.5 Deﬁnitions. Let E = (Xi , Pi , Yk , r) be an economy. We shall say that for E iﬀ: (r i , [sik ]) is a distribution of ownership 1. r i ∈ Rn , for i = 1, . . . , m, and m i=1 r i = r, 2. sik ≥ 0, for i = 1, . . . , m, k = 1, . . . , , and 3. m i=1 sik = 1, for k = 1, . . . , . We shall then say that E = (Xi , Pi , Yk , r i , [sik ]) is a private ownership economy iﬀ E = (Xi , Pi , Yk , r) is an economy, and (r i , [sik ]) is a distribution of ownership for E. Notice that the above deﬁnition allows for the possibility that the k th ﬁrm is a sole proprietorship (in which case, there exists some i such that sik = 1), an equalshares partnership (in which case there would exist some h, i such that shk = sik = 1/2), or a publiclytraded corporation (in which we would have sik > 0 for many of the consumers). If E = (Xi , Pi , Yk , r i , [sik ]) is a private ownership economy, and a price vector, p ∈ Rn is given, the usual deﬁnition of the ith consumer’s income (or wealth) is given by: sik πk (p). (7.2) wi (p) = p · r i + k=1
Notice that if 0 ∈ Yk (although in this chapter we will not usually be requiring this condition), then for any price vector, p, we will have wi (p) ≥ p · r i . The deﬁnition of a private ownership economy is due to Debreu, and private ownership economies, as thus deﬁned, are the principal subjects of investigation in Debreu’s Theory of Value (Debreu [1959]). As we have just indicated, Debreu treated individual resource endowments as something distinct from the production sets in the economy. From a formal mathematical point of view, however, we could easily eliminate the explicit inclusion of individual resource endowments by deﬁning m new production sets: Yk = {r k− }
for k = + 1, . . . , + m;
7.2. Competitive Equilibrium with Production
195
and correspondingly adding m shares of ownership, with: 1 for i = k − , sik = 0 otherwise, for i = 1, . . . , m; k = + 1, . . . + m. (Or alternatively, by making use of the consumption sets Xi deﬁned as Xi = Xi − r i .) Why then, you may well ask, do we complicate our notation by explicitly including the individual resource endowments? Well, the discussion in the above paragraph indicates one reason; if we add to the assumption that 0 ∈ Yk , for each K, the condition r i ∈ Xi , for each i, then given any price vector, p, each consumer will have a nonempty budget set. A second reason is that the inclusion of individual resource endowments allows us to consider private ownership pure exchange economies as the special case of a private ownership economy in which = 1, and Y = {0}. On the other hand we obviously gain no generality in our model by including individual resource endowments in the speciﬁcation of an economy, and in later chapters we will often simplify our notation by not explicitly taking such endowments into account. As we have seen, the deﬁnition of a private ownership economy provides a natural deﬁnition of a wealth assignment for the economy, given a price vector, p. However, the more abstract notion of a wealthassignment for E provides us with much more ﬂexibility in our analysis, and allows us to deal with many diﬀerent situations with more or less the same arguments. Some of those diﬀerent situations are set out in the following examples. 7.6 Examples. 1. Let E = (Xi , Pi , Yk , r) be an economy, let (r i , [sik ]) be a distribution of ownership for E, and denote the resultant private ownership economy by ‘E.’ We will say that (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E if it satisﬁes Deﬁnition 7.4 with the wealth assignment: sik πk (p∗ ), for i = 1, . . . , m. wi = wi (p∗ ) = p∗ · r i + k=1
However, we will sometimes wish to modify this arrangement, as follows. Deﬁne a vector t = (t1 , . . . , tm ) to be a system of lumpsum transfers for E iﬀ: m ti = 0. i=1
We then say that (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E, given the system of lumpsum transfers, t, iﬀ, deﬁning the assignment of wealth w = (w1 , . . . , wm ) by: wi = p∗ · r i + sik πk (p∗ ) + ti , for i = 1, . . . , m, k=1
it is true that (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E, given the assignment of wealth w, 2. Let E = (Xi , Pi , Yk , r) be an economy, and suppose (x∗i , y ∗k , p∗ ) satisﬁes conditions 1–3 of Deﬁnition 7.4, above; and, in addition, that for each i we have: (∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > p∗ · x∗i .
196
Chapter 7. Fundamental Welfare Theorems
Then if we assign the wealth levels: wi∗ = p∗ · x∗i
for i = 1, . . . , m;
it follows that (x∗i , y ∗k , p∗ ) is a competitive ∗ ). assignment vector w∗ = (w1∗ , . . . , wm
equilibrium for E with the wealth
3. Exercise. Show that in the above examples, the vectors w and w∗ are wealth assignments for E, respectively, given the price vector p∗ ; that is, that they satisfy Deﬁnition 7.3
7.3
Some Diagrammatic Techniques
In this and the next chapter, we will be looking at a great many simple examples of competitive or Walrasian equilibria. Our examples will deal primarily with two special cases. One is that in which m = n = 2, = 1, and Y = {0} or Y = −R2+ . Either of these cases can be interpreted as the classical twoperson, twocommodity pure exchange model, and the primary diagrammatic technique which we will use in the analysis of such examples will be the Edgeworth Box diagram with which you are already familiar. The other special case with which we shall be dealing is that in which: m = = 1, n = 2, and in which there is nonzero production. In this case, it will be convenient to use some diagrammatic techniques with which you may not be familiar.5 Consider the economy, E, in which m = = 1, n = 2, and: X = {x ∈ R2  (−2, 1) ≤ x & x1 ≤ 0}, Y = {y ∈ R2  y1 + y2 ≤ 0 & y1 ≤ 0}, and r = (0, 3/2). We can graph this production set and consumption set as in Figure 7.1.a, on the next page. In the righthand diagram (Figure 7.1.b) on the next page, we have indicated the attainable consumption set, which we will denote by ‘X ∗ ,’ and which is simply the set of all consumption bundles which correspond to an attainable allocation for the economy. In other words, it is the set of all consumption bundles which can be attained with the resources and production technology available in the economy. Formally, we deﬁne X ∗ as follows. s Suppose a pair (x, y) is in A(E) for this economy, so that (x, y) ∈ X × Y , and satisﬁes: x = r + y. Then x must be a member of the set: def
r + Y = {z ∈ R2  (∃y ∈ Y ) : z = r + y}. Consequently, deﬁning X ∗ , the attainable consumption set, by: X ∗ = {x ∈ X  (∃y ∈ Y ) : (x, y) ∈ A(E)}, 5
This type of diagram was introduced, and used extensively in Koopmans [1957].
7.3. Some Diagrammatic Techniques
197
x2
x2 r+ Y X
X*
r
Y
Y
x1
x1 Figure 7.1.a
Figure 7.1.b
Figure 7.1: A Production Economy. we see that we must have:
X ∗ ⊆ X ∩ [r + Y ].
(7.3)
x ∈ X ∩ [r + Y ],
(7.4)
Conversely, if: then x ∈ X, and, by deﬁnition of r + Y , there exists y ∈ Y such that x = r + y. We see, therefore, that if (7.4) holds, then (x, y) ∈ A(E), and it follows that: X ∩ [r + Y ] ⊆ X ∗ .
(7.5)
Combining (7.3) and (7.5), we see, therefore, that: X ∗ = X ∩ [r + Y ].
(7.6)
It is also easy to see that in our example, the set r + Y will be given by: r + Y = {z ∈ R2  z1 ≤ 0 & z1 + z2 ≤ 3/2}, as is indicated in Figure 7.1.b; and thus that the attainable consumption set will be as shown in that diagram. Since the attainable consumption set is given by (7.6), it is clear that the set r + Y will be quite important in the analysis of our examples. In the examples we will present, the set r + Y will also be quite easy to represent graphically. For example, it is easy to see that if the commodity space is R2 , 0 ∈ Y , and r is of the form r = (0, r2 ), where r2 > 0, then r + Y will simply be the translate of Y
198
Chapter 7. Fundamental Welfare Theorems
obtained by sliding the production set upward along the vertical axis until its vertex coincides with r, rather than the origin. In fact, it is easy to prove the following (and it would be a good exercise to do so). Suppose Y is a subset of R2 of the form: Y = {y ∈ R2  y1 ≤ 0 & a1 y1 + a2 y2 ≤ 0}, where a1 and a2 are positive constants. Then r + Y is the set whose upper boundary is given by: (7.7) y2 = (−a1 /a2 )y1 , whose vertical intercept is at the origin; and if r is of the form: r = (0, r2 ), then: r + Y = {z ∈ R2  z1 ≤ 0 & a1 z1 + a2 z2 ≤ a2 r2 }. The upper boundary of r + Y is therefore given by the line: z2 = (−a1 /a2 )z1 + r2 ,
(7.8)
which is parallel to the line deﬁned in (7.7), and which has the vertical intercept r2 . The set r + Y can, for much of our analysis of this type of example, actually be thought of as a kind of production set. In fact, if we are trying to determine the level of supply, given a price vector p, we can actually eﬀectively ignore the set Y , and search for a ‘proﬁtmaximizing’ vector, z, in the set Z = r + Y . That this is so stems from the following fact, which is actually a special case of Theorem 6.38, from Chapter 6; however, it might nonetheless be a good exercise to prove this fact directly. 7.7 Proposition. If z ∗ = r + y ∗ maximizes ‘proﬁts’ on Z, given the price vector p∗ , then y ∗ maximizes p∗ · y on Y . Conversely, if y maximizes p∗ · y on Y, then z = r + y maximizes ‘proﬁts,’ p∗ · z, on Z. Now consider a private ownership economy, E, where m = = 1. Since there is only one consumer and one ﬁrm, we have: s ≡ s11 = the share of the ﬁrst consumer in the proﬁts of the ﬁrst ﬁrm = 1, and: r 1 ≡ the ﬁrst consumer’s resource endowment = r, the aggregate resource endowment. Therefore, if z ∗ ∈ Z maximizes ‘proﬁts’ on Z, given the price vector p∗ , the consumer’s budget line (or hyperplane), given p∗ , is given by: b(p∗ ) = {x ∈ X  p∗ · x = p∗ · z ∗ }. Thus, in the case where n = 2, we have the sort of situation illustrated in Figure 7.2, below, which depicts a Walrasian equilibrium at (x∗ , y ∗ , p∗ ).
7.3. Some Diagrammatic Techniques
199
.
x*
r+Y b(p*)
r
y* X p*
Y
Figure 7.2: A Competitive Equilibrium. We can simplify our discussion of examples, and gain some further understanding of a Walrasian equilibrium, by consideration of the following material. Earlier we deﬁned the sets Πk and Π by: Πk = {p ∈ Rn  (∃y ∗k ∈ Yk )(∀y k ∈ Yk ) : p · y ∗k ≥ p · y k } for k = 1, . . . , ; and:
Π = {p ∈ Rn  (∃y ∗ ∈ Y )(∀y ∈ Y ) : p · y ∗ ≥ p · y},
(7.9)
where ‘Y ’ denotes the aggregate production set. We then deﬁned the aggregate proﬁt function, π(·), and the aggregate supply correspondence, σ(·): π(p) = max p · y y∈Y
for p ∈ Π,
(7.10)
and: σ(p) = {y ∈ Y  p · y = π(p)} for p ∈ Π. Recall also that it follows from Theorem 6.38 that: 0∈Π&Π= Πk ;
(7.11)
(7.12)
k=1
and, for p ∈ Π: π(p) =
k=1
πk (p) & σ(p) =
k=1
σk (p).
(7.13)
200
Chapter 7. Fundamental Welfare Theorems
Furthermore, it follows easily from the same theorem that if (x∗i , y ∗k , p∗ ) is a competitive equilibrium for an economy, E, then we must have p∗ ∈ Π, and, deﬁning: y∗ = y ∗k , k=1
we must have: Now deﬁne the set Π∗ by:
p∗ · y ∗ = π(p∗ ) & y ∗ ∈ σ(p∗ ). Π∗ = Π ∩ ∆n ,
where ‘∆n ’ denotes the unit simplex in Rn : n ∆n = p ∈ Rn+ 
j=1
pj = 1 .
The homogeneity of the consumers’ demand correspondences and the producers’ supply correspondences, together with Proposition 6.25 and Theorem 6.38 of Chapter 6, imply the following; the proof of which will be left as an exercise. 7.8 Proposition. Suppose the economy E satisﬁes either: Y − Rn+ ⊆ Y,
(7.14)
−Rn+ ⊆ Y,
(7.15)
or: and that (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E. If we deﬁne: 1 ¯ = n p p∗ , ∗ j=1 pj ¯ is also a competitive equilibrium for E. ¯ ∈ Π∗ and x∗i , y ∗k , p then p We will often employ a variant of the model we have been discussing here, one which also deals with the case in which there is one producer, one consumer, and two commodities as before, but makes use of the assumption that X = R2+ . Here we will also generally suppose that r takes the form: r = (r, 0), and we will use the generic notation: p = (w, p), to denote price vectors; although we will frequently normalize the price of the ﬁrst commodity; that is, set w = 1. In this case we interpret ‘x1 ’ as the quantity of leisure demanded by the consumer; while: = x1 − r, is the quantity of labor oﬀered. A typical competitive equilibrium in this case might look something like the following, in diagrammatic presentation.
7.4. Walras’ Law with Production
201
x*2
Y
Y+r π(p)/p
x*1
r ᒂ = x*1  r
Figure 7.3: Competitive Equilibrium for the Labor/Leisure Model.
7.4
Walras’ Law with Production
We begin this section by extending our deﬁnition of a wealthassignment vector to a wealthassignment function. In the deﬁnition, recall that we have deﬁned Π for an economy as the collection of all price vectors for which a maximum proﬁt exists on each production set, and, for each i ∈ M : Ωi = {(p, w) ∈ Rn+1  (∃x ∈ Xi ) : p · x ≤ w}. 7.9 Deﬁnition. Let E = (Xi , Pi , Yk , r) be an economy. We shall say that a function, w : Π → Rm is a feasible wealthassignment function for E iﬀ, for each p ∈ Π, we have: p, wi (p) ∈ Ωi for i = 1, . . . , m, and:
m i=1
wi (p) = p · r +
k=1
πk (p).
We have already considered competitive equilibria with an arbitrary assignment of wealth. The notion of a feasible wealthassignment function simply extends this basic idea by allowing for a wealth assignment which is a function of prices.6 7.10 Examples. 1. Let E = (Xi , Pi , Yk , r i , [sik ]) be a private ownership economy satisfying: r i ∈ Xi
for i = 1, . . . , m,
0 ∈ Yk
for k = 1, . . . , ;
and: 6
The very useful notion of a wealthassignment function was introduced in Gale and MasColell [1975].
202
Chapter 7. Fundamental Welfare Theorems
and, as we have done previously, deﬁne w : Π → Rm by: wi (p) = p · r i + sik πk (p) for i = 1, . . . , m. k=1
(7.16)
Then, since 0 ∈ Yk for each k, we see that for any p ∈ Π; πk (p) ≥ 0
for k = 1, . . . , .
Therefore, for each p ∈ Π and each i: wi (p) = p · r i +
k=1
sik πk (p) ≥ p · r i .
Since r i ∈ Xi , and (as I will leave you to verify), for each p ∈ Π: m wi (p) = p · r + πk (p), i=1
k=1
it follows that w(·) is a feasible wealthassignment function for E. 2. It will sometimes be useful to make use of the idea of a wealthassignment function to formally relate results for pure exchange economies to production economies with linear technologies, as follows. Let E = (Xi , Pi , Yk , r) be an economy in which Yk is linear, for k = 1, . . . , . We can then deﬁne the function w : Π → Rm by: (7.17) wi (p) = p · r i for i = 1, . . . , m. If (r i ) satisﬁes the condition: r i ∈ Xi
for i = 1, . . . , m;
then, since we showed in Chapter 6 that we must have πk (p) = 0 for each p ∈ Π and each k, it follows that w(·) is a feasible wealthassignment function for E. 3. Suppose once again that E = (Xi , Pi , Yk , r i , [sik ]) satisﬁes: r i ∈ Xi
for i = 1, . . . , m;
0 ∈ Yk
for k = 1, . . . , ;
and that: and let’s introduce government as an (m + 1)st consumer; using the subscript ‘0’ to denote government’s income and consumption. We will then suppose that X0 = Rn+ , so that: Ω0 = {(p, w) ∈ Rn+1  p ∈ Rn++ & w ≥ 0}. We will suppose that the ith consumer pays the tax:
ti = τ i · sik πk (p) , k=1
where:7 0 ≤ τi < 1 7
for i = 1, . . . , m.
In practice, there would typically be only three or four diﬀerent tax rates; with the tax rate paid by the ith consumer depending upon her or his income. However, the formulation here incorporates this situation as a special case, and is simpler to deal with in this example.
7.4. Walras’ Law with Production
203
We then deﬁne s0k by: s0k =
m i=1
τi · sik
for k = 1, . . . , ;
and w : Π → R1+m by: w0 (p) =
s0k πk (p), and: wi (p) = p · r i + sik πk (p) − ti k=1
for i = 1, . . . , m.
k=1
I will leave it to you to verify that this is a feasible wealth assignment function for E. If a feasible wealthassignment function is deﬁned for an economy, individual and aggregate demand and excess demand become functions of prices alone; as is set out formally in the following deﬁnition. 7.11 Deﬁnition. Let E = (Xi , Pi , Yk , r), and let w : Π → Rm be a feasible wealthassignment function for E. We deﬁne the excess demand correspondence for E, given w, η : Π → Rn , by: η(p) =
m i=1
δi (p) − r −
k=1
σk (p),
(7.18)
where we deﬁne δ i (·) by: δ i (p) = hi [ p, wi (p)] where hi (·) is the
ith
for i = 1, . . . , m;
consumer’s demand correspondence.
7.12 Proposition. [Walras’ Law (Weak Form)] Let E = (Xi , Pi , Yk , r) be an economy, let w : Π → Rm be a feasible wealthassignment function for E, and let η : Π → Rn be the aggregate excess demand correspondence for E, given w. Then for any p ∈ Π and any z ∈ η(p) we have p · z ≤ 0. Proof. Under the stated conditions, there exist xi , y k such that: xi ∈ hi [ p, wi (p)] y k ∈ σ k (p) and: z=
m i=1
for i = 1, . . . , m,
for k = 1, . . . , , xi − r −
k=1
(7.19) (7.20)
yk .
(7.21)
Since y k ∈ σ k (p), for each k, we also have: p · y k = πk (p)
for k = 1, . . . , ;
(7.22)
while from (7.30), we have: p · xi − wi (p) ≤ 0
for i = 1, . . . , m.
(7.23)
204
Chapter 7. Fundamental Welfare Theorems
From (7.23) we have, upon adding over i, and making use of (7.31), (7.21), and the deﬁnition of a feasible wealthassignment function:
m m m p · xi − wi (p) = p · xi − p · r + πk (p) 0≥ i=1 i=1 i=1 k=1
m xi − r − y k = p · z. =p· i=1
k=1
The proof of the next two results will be left as exercises. 7.13 Proposition. [Walras’ Law (Strong Form)] Let E = (Xi , Pi , Yk , r) be an economy, let w : Π → Rm be a feasible wealthassignment function for E, let η : Π → Rn be the aggregate excess demand correspondence for E, given w, and suppose that: Pi is locally nonsaturating, for i = 1, . . . , m. Then for any p ∈ Π and any z ∈ η(p) we have p · z = 0. 7.14 Corollary. [Walras’ Law (Original Form)] Let E = (Xi , Pi , Yk , r) be an economy, let w : Π → Rm be a feasible wealthassignment function for E, let η : Π → Rn be the aggregate excess demand correspondence for E, given w, and suppose that Pi is locally nonsaturating, for i = 1, . . . , m. Then if p∗ ∈ Π ∩ Rn++ and z ∗ ∈ η(p∗ ) are such that for some k ∈ {1, . . . , n}, we have ∀j ∈ {1, . . . , n} \ {k} : zj∗ = 0, we must have zk∗ = 0 as well. In other words if n−1 of the markets are in equilibrium, then the remaining market must be in equilibrium as well. 7.15 Example. Suppose that in the economy, E = (Xi , Pi , Yk , r i , [sik ]), the th ﬁrm represents government production. We are not excluding the possibility that we may have Y ⊆ −Rn+ ; that is, that government is simply a consumer, as was the case in Example 7.10.3; however, we can equally well suppose that some elements y ∈ Y have some positive coordinates. In any case, an allocation, xi , y k will be feasible iﬀ: m m xi = ri + yk . (7.24) i=1
i=1
We deﬁne: Π=
−1 k=1
k=1
Πk ;
Rm
and, deﬁning the function w : Π → by: −1 sik πk (p) − ti (p) wi (p) = p · r i + k=1
for i = 1, . . . , m,
(7.25)
we suppose that the tax/transfer function t : Π → Rm has been deﬁned in such a way that, for each p ∈ Π: (7.26) p, wi (p) ∈ Ωi for i = 1, . . . , m. We will suppose that the government’s choice of ‘production’ can be characterized by a function of price, σ : Π → Y . We will then say that (x∗i , y ∗k , p∗ ) is a competitive equilibrium, given the governmental policy (σ , t) iﬀ:
7.5. The ‘First Fundamental Theorem’ 1. 2. 3. 4.
205
∗ p 0, = x∗i , y ∗k is feasible, y ∗k ∈ σ k(p∗ ), for k= 1, . . . , , and: x∗i ∈ hi p∗ , wi (p∗ ) , for i = 1, . . . , m.
Now suppose that each consumer’s demand correspondence satisﬁes the budget balance condition (Deﬁnition 4.6). We can then derive a rather interesting conclusion, as follows. By the budget balance condition, we have: −1 sik πk (p∗ ) − ti (p∗ ), for i = 1, . . . , m. p∗ · x∗i = wi (p∗ ) = p∗ · r i + k=1
Thus, adding over i, and deﬁning
x∗
=
p∗ · x∗ = p∗ · r +
m
∗ k=1 xi
−1 k=1
and τ (p) =
m
i=1 ti (p),
we obtain:
p∗ · y ∗k − τ (p∗ ).
(7.27)
On the other hand, since x∗i , y ∗k is feasible, we obtain from (7.24) that:
p∗ · x∗ = p∗ · r +
−1 k=1
p∗ · y ∗k + p∗ · y ∗
(7.28)
Combining (7.27) and (7.28), we see that: p∗ · y ∗ = −τ (p∗ ); that is, the government’s budget is necessarily balanced! How is it that this conclusion, which is so very unlike our recent experience in the U. S., can be reached? Well, it is rather a variant of Walras’ Law; and comes about essentially because the value of demand must be equal to the value of supply. To put this another way, we cannot have an unbalanced governmental budget in this sort of model unless we introduce a ﬁnancial sector into the model. Notice ﬁnally that it follows from (7.27) and (7.28) that the function deﬁned in (7.24) is a feasible wealthassignment function for E.
7.5
The ‘First Fundamental Theorem’
The following deﬁnitions are essentially unchanged from those presented in Chapter 5. I repeat them here largely for the sake of providing a convenient reference. 7.16 Deﬁnitions. Let E be an economy. We then deﬁne: 1. the unanimity ordering (or the strong Pareto ordering), Q, on X by: xi Qxi ⇐⇒ [xi Pi xi
for i = 1, . . . , m].
(7.29)
2. the Pareto (atleastasgoodas) ordering, R, on X, by: xi Rxi ⇐⇒ [¬xi Pi xi , for i = 1, . . . , m].
(7.30)
3. the strict Pareto ordering, P , on X, by: xi P xi ⇐⇒ [xi Rxi & ¬xi Rxi ].
(7.31)
206
Chapter 7. Fundamental Welfare Theorems We will use the following terminology in dealing with these three orderings. If: x∗i Qxi , we shall say that x∗i is unanimously preferred to xi , x∗i Rxi , we shall say that x∗i (weakly) Pareto dominates xi , x∗i P xi , we shall say that x∗i strictly Pareto dominates xi .
7.17 Deﬁnitions. Let E = (Xi , Pi , Yk , r) be an economy. We shall say that a feasible allocation for E, x∗i , y ∗k is Pareto eﬃcient for E [respectively, strongly Pareto eﬃcient for E] iﬀ there exists no alternative feasible allocation for E, xi , y k , satisfying: xi Qx∗i [respectively, xi P x∗i ]; where the orderings Q and P are deﬁned in equations (7.29) and (7.31), above. While the above deﬁnitions are stated for an economy, the corresponding deﬁnitions for a private ownership economy, E = (Xi , Pi , Yk , r i , [sik ]), are obvious, and will be used where needed without further comment. In the terminology just introduced, a feasible allocation, x∗i , y ∗k , will be said to be Pareto eﬃcient for allocation which exists no alternative feasible E iﬀ there all consumers prefer to x∗i , y ∗k . The feasible allocation x∗i , y ∗k is strongly Pareto eﬃcient for E iﬀ there exists no alternative feasible allocation which strictly Pareto dominates x∗i , y ∗k . Since x∗i , y ∗k may be such that, while no feasible alternative allocation is unanimously preferred, there nonetheless is another feasible allocation, xi , y no consumer is worse oﬀ, and at least one consumer is k , where better oﬀ than at x∗i , y ∗k , there are in principle more Pareto eﬃcient allocations than there are strongly Pareto eﬃcient allocations for a given economy, E. This is the reason for the terminology used here. Making use of the terminology introduced by Hurwicz [1960], we will demonstrate that, loosely speaking: 1. the competitive mechanism is nonwasteful, in the sense that any competitive equilibrium is Pareto eﬃcient, and 2. the competitive mechanism is unbiased, in the sense that (given some additional assumptions) any Pareto eﬃcient allocation can be made a competitive equilibrium. Roughly speaking, these two results respectively constitute what are known as the ‘First’ and ‘Second Fundamental Theorems of Welfare Economics.’ In the material to be presented here, we will concentrate on Pareto eﬃcient allocations, as opposed to strongly Pareto eﬃcient allocations; for the reasons set out in Chapter 5. However, the following result provides suﬃcient conditions for Pareto eﬃcient allocations to be strongly Pareto eﬃcient. I will leave the proof as an exercise, since it can be done in essentially the same way as we proved Proposition 5.19. 7.18 Proposition. If E = (Xi , Pi , Yk , r) is an economy in which Xi = Rn+ and semicontinuous, and strictly increasPi is asymmetric, negatively transitive, lower ing, for i = 1, . . . , m; then an allocation x∗i , y ∗k is Pareto eﬃcient for E if, and only if, it is strongly Pareto eﬃcient for E.
7.5. The ‘First Fundamental Theorem’
207
Turning now to the ‘nonwastefulness’ property of the competitive mechanism, we begin by deﬁning a slight generalization of the idea of a competitive, or Walrasian equilibrium for an economy. The deﬁnition presented here will be critical to our initial development of the ‘unbiasedness’ result; in fact, the usual statement of the ‘Second Fundamental Theorem’ is something like: ‘given the appropriate convexity conditions it is the case that, given any Pareto eﬃcient allocation x∗i , y ∗k , there ∗ ∗ ∗ ∗ exists a price vector, p , and a wealth assignment, w , such that (xi , y k , p∗ ) is a quasicompetitive equilibrium for E, given w∗ . In the deﬁnition we make use of a bit of notation which we will continue to use throughout the remainder of this book; given a vector p∗ ∈ Rn and a set Z ⊆ Rn , we use the expression ‘min p∗ · Z’ as shorthand for: min{p∗ · z  z ∈ Z}. 7.19 Deﬁnition. If E is an economy, we shall say that (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for E, iﬀ there exists a wealthassignment for E, given p∗ , w = (w1 , . . . , wm ), such that: 1. p∗ = 0, 2. x∗i , y ∗k ∈ A(E), 3. p∗ · y ∗k = πk (p∗ ), for k = 1, . . . , , 4. for each i (i = 1, . . . , m), we have p∗ · x∗i ≤ wi , and either: wi == min p∗ · Xi ,
(7.32)
(∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > wi
(7.33)
or: (x∗i , y ∗k , p∗ )
(or both). In this case, we shall also say that equilibrium for E, given the wealthassignment, w.
is a quasicompetitive
If you compare Deﬁnition 7.19 with the deﬁnition of a competitive equilibrium for E (Deﬁnition 7.4), you will see that the two deﬁnitions diﬀer only in condition 4: condition 4 of Deﬁnition 7.4 says that any commodity bundle xi which is preferred to x∗i must cost more than wi (given the price vector, p∗ ) while condition 4 of 7.19 says that the former condition can fail only if every commodity bundle in Xi costs at least as much, given the price vector p∗ , as does x∗i . The following presents the properties upon which our proof of the ‘First Fundamental Theorem’ is based. 7.20 Proposition. If (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for an econ∗ ), then: omy, E, given the wealthassignment, w∗ = (w1∗ , . . . , wm p∗ · x∗i = wi∗
for i = 1, . . . , m;
(7.34)
and, for any feasible allocation, xi , y k , we have:
m i=1
wi∗ = p∗ ·
m
i=1
m m m
x∗i = p∗ · x∗i ≥ p∗ · xi = p∗ · xi . i=1
i=1
i=1
(7.35)
208
Chapter 7. Fundamental Welfare Theorems
Proof. Since (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for E, x∗i , y ∗k is feasible for E, and thus: 0 = p∗ ·
m i=1
ri +
y ∗k − m
m
x∗i
k=1
i=1
=
p∗ · r i +
i=1
k=1
p∗ · y ∗k −
m i=1
p∗ · x∗i . (7.36)
However, by the deﬁnition of a quasicompetitive equilibrium, we have that: wi∗ − p∗ · x∗i ≥ 0 for i = 1, . . . , m, m wi∗ = p∗ · r + πk (p∗ ),
(7.37)
p∗ · y ∗k = πk (p∗ )
(7.39)
i=1
and:
(7.38)
k=1
for k = 1, . . . , .
Adding the terms in (7.37) and making use of (7.38) and (7.39), we have:
m m m wi∗ − p∗ · x∗i wi∗ − p∗ · x∗i = i=1 i=1 i=1 m = p∗ · r + πk (p∗ ) − p∗ · x∗i k=1 i=1 m = p∗ · r + p∗ · y ∗k − p∗ · x∗i k=1 i=1
m = p∗ · r + y ∗k − x∗i = 0. k=1
i=1
Since the sum of nonnegative terms can only be zero if all of these terms are zero, we can then conclude that p∗ · x∗i = wi∗ , for i = 1, . . . , m; and furthermore:
m m p∗ · x∗i = p∗ · x∗i = p∗ · r + p∗ y ∗k . (7.40) i=1
i=1
k=1
Now suppose that xi , y k is feasible for E. Then we have: p∗ ·
m i=1
xi = p∗ ·
m i=1
r i + p∗ ·
k=1
y k = p∗ · r +
k=1
p∗ · y k .
(7.41)
However, since (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for E, we have: k=1
p∗ · y k ≤
k=1
p∗ · y ∗k ;
and combining (7.40)–(7.42), we then obtain: m m p∗ · xi ≤ p∗ · x∗i . i=1
i=1
(7.42)
The following is our ﬁrst version of the ‘First Fundamental Theorem.’ 7.21 Theorem. If (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for an economy, E, and:
m p∗ · x∗ > min p∗ · X, (7.43) i=1 i ∗ ∗ then xi , y k is Pareto eﬃcient for E.
7.5. The ‘First Fundamental Theorem’
209
Proof. Suppose (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for E, given the wealthassignment w∗ , and that xi , y k is such that xi Qx∗i . By (7.43), there exists h ∈ {1, . . . , m} such that: p∗ · x∗h > min p∗ · Xh ;
(7.44)
and, for any h ∈ {1, . . . , m} satisfying (7.44), we must have, by the deﬁnition of a quasicompetitive equilibrium and Proposition 7.20: p∗ · xh > wh∗ = p∗ · x∗h .
(7.45)
On the other hand, for i ∈ {1, . . . , m} not satisfying (7.44), we obviously have: p∗ · xi ≥ min p∗ · Xi = p∗ · x∗i .
(7.46)
Combining (7.45) and (7.46), we see that:
m
m p∗ · xi > p∗ · x∗i ; i=1
i=1
and it follows from Proposition 7.20 that xi , y k is not an attainable allocation for E In Chapter 5, we showed that there are undesirable properties of the strict Pareto ordering, P in the case where each Pi is an asymmetric order, but not necessarily negatively transitive (Example 5.16). Correspondingly, it seems that there is a case to be made for concentrating on Pareto eﬃcient allocations, as opposed to strongly Pareto eﬃcient allocations for an economy. It is only fair, therefore, that we present Theorem 7.21 as our ﬁrst version of the ‘First Fundamental Theorem;’ for clearly a quasicompetitive equilibrium can involve a very undesirable allocation, in that any consumer minimizing expenditure over Xi at x∗i can be very badly oﬀ indeed! Of course, from our discussion in Chapter 5 we already knew that some Pareto eﬃcient allocations may be quite undesirable. Since a competitive equilibrium for E is necessarily also a quasicompetitive equilibrium for E, it is immediately apparent that Theorem 7.21 remains correct if we substitute ‘competitive equilibrium for E’ for ‘quasicompetitive equilibrium for E’ in its statement. However, we can prove a somewhat stronger result for competitive equilibria, as follows. The proof will be left as an (easy) exercise, since it is an almost immediate corollary of the proof of Theorem 7.21. 7.22 Theorem. If (x∗i , y ∗k , p∗ ) is a competitive equilibrium for an economy, E, then x∗i , y ∗k is Pareto eﬃcient for E. Our third (and basic alternative) version of the ‘First Fundamental Theorem’ is a little more complicated, but still fairly simple to state and prove. 7.23 Theorem. If (x∗i , y ∗k , p∗ ) is a competitive equilibrium for an economy, E, and∗ each∗ Pi is asymmetric, negatively transitive, and locally nonsaturating, then xi , y k is strongly Pareto eﬃcient for E.
210
Chapter 7. Fundamental Welfare Theorems
Proof. Suppose (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E, given the assignment of wealth levels w = (w1 , . . . , wm ), and that xi is a consumption allocation such that xi P xalas. Then, by deﬁnition of the strict Pareto dominance relation: xi Gi x∗i for i = 1, . . . , m, (7.47) and, for some h ∈ {1, . . . , m}:
xh Ph x∗h .
(7.48)
However, by (7.47) and Proposition 4.9, we have: p∗ · xi ≥ wi
for i = 1, . . . , m;
(7.49)
while by (7.48) and the deﬁnition of a competitive equilibrium: p∗ · xh > wh .
(7.50)
Adding over equations (7.49) and (7.50), we have: m m p∗ · xi > wi ; i=1
i=1
and it then follows from Proposition 7.20 that xi cannot be feasible for E. Therefore, x∗i , y ∗k is strongly Pareto eﬃcient for E. As we showed earlier, a quasicompetitive equilibrium is Pareto eﬃcient if the aggregate nonminimum expenditure condition [equation (7.43)] is satisﬁed. However, a quasicompetitive equilibrium is not necessarily strongly Pareto eﬃcient, even if (7.43) is satisﬁed and, in addition, the individual preference relations satisfy the assumptions of the above result. This is shown by the following example. 7.24 Example. Consider the twoperson, twocommodity exchange economy in which the two consumers’ preferences can be represented by the utility functions: u1 (x1 ) = x11 + x12 , and: u2 (x2 ) = min{x21 , x22 }, respectively; and let: r 1 = (1, 0) and r 2 = (2, 1). Then, as you can easily show, if we deﬁne: x∗i = r i for i = 1, 2, and p∗ = (0, 1), then x∗i , p∗ is a quasicompetitive equilibrium for E. Moreover, the two consumers’ preferences satisfy the assumptions of 7.23, while:
p∗ · x∗1 + p∗ · x∗2 = 1 > min p∗ · X = 0. Nonetheless, x∗i is not strongly Pareto eﬃcient for E.
7.6. ‘Unbiasedness’ of the Competitive Mechanism
211
This last example also illustrates a problem involved in using the word ‘equilibrium’ as a part of the phrase ‘quasicompetitive equilibrium.’ The ‘equilibrium’ deﬁned in the example is not a situation from which there is ‘no net tendency to change;’ in fact, consumer one will demand an indeﬁnitely large quantity of the ﬁrst commodity, given its zero price, despite having zero income. Nonetheless, the concept of a quasicompetitive equilibrium will provide a useful ‘stepping stone’ in developing the results of the next section. The three versions of the ‘First Fundamental Theorem’ which were presented here, in particular, the last two, state conditions under which the competitive mechanism is ‘nonwasteful.’ While the assumptions used in the two results are quite general (particularly in the case of Theorem 7.22), it should be noted that two assumptions which were used implicitly, but not stated as explicit hypotheses are: 1. Pi is individualistic, for i = 1, . . . , m, and: 2. there are no external eﬀects in production, in the sense that if y k ∈ Yk , for k = 1, . . . , , then the aggregate production level: y=
k=1
yk ,
can be achieved. These two assumptions were not stated explicitly for the simple reason that we have been implicitly maintaining them throughout our discussion of competitive equilibrium; and in fact, we would have to redeﬁne what we mean by such an equilibrium if either of these conditions fails. It is nonetheless important to keep in mind that these two assumptions are implicitly used in the result. It is also worth noting that we can construct an example, based on Example 5.18 of Chapter 5, of a competitive equilibrium which is not strongly Pareto eﬃcient, as follows. Let: r 1 = (1, 0) and r 2 = (1, 1), and let P1 and P2 be as set out in the example, with f2 (0, 2) = 3 (that is, we deﬁne 2’s utility at the bundle (0, 2) to be 3). If we let: p∗ = (3/2, 1), x∗1 = r 1 , and x∗2 = r 2 ,
then x∗i , p∗ is a competitive equilibrium for E, but x∗i is not strongly Pareto eﬃcient for E.
7.6
‘Unbiasedness’ of the Competitive Mechanism
While our proof of the ‘First Fundamental Theorem’ was straightforward indeed, establishing the ‘Second Fundamental Theorem’ is a bit more complicated. A ﬁrst diﬃculty is that a reallocation of initial endowments and shares of ownership in ﬁrms may be necessary in order to make a competitive equilibrium of a Pareto eﬃcient allocation. One way of dealing with this diﬃculty, which is the approach we will follow here, is to seek a wealthassignment vector for E, which enables equilibrium to be achieved. A second diﬃculty arises in that standard assumptions do not imply that an arbitrary Pareto eﬃcient allocation can actually be made a
212
Chapter 7. Fundamental Welfare Theorems
competitive equilibrium. The results that people loosely interpret as establishing this implication usually actually establish the existence of a weakened form of competitive equilibrium. We will follow this pattern initially; making use of the concept of a quasicompetitive equilibrium, as deﬁned in the previous section. In doing our ﬁrst version of the ‘Second Fundamental Theorem,’ we will need a supporting result and one further deﬁnition, as follows. 7.25 Proposition. If Pi is a lower semicontinuous binary relation on a convex set, Xi , and x∗i ∈ Xi and p∗ ∈ Rn satisfy: (∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi ≥ p∗ · x∗i ,
(7.51)
p∗ · x∗i > min p∗ · Xi ,
(7.52)
and
then: (∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > p∗ · x∗i . Proof. Suppose, by way of obtaining a contradiction, that there exists xi ∈ Xi such that xi Pi x∗i , but: p∗ · xi ≤ p∗ · x∗i . Since Pi is lower semicontinuous, there exists a neighborhood, N (xi ) such that: (∀xi ∈ N (xi ) ∩ Xi ) : xi Pi x∗i .
(7.53)
Now, by (7.52) there exists xi ∈ Xi such that p∗ · xi < p∗ · x∗i , and we then have: (∀θ ∈ ]0, 1]) : p∗ · [θxi + (1 − θ)xi ] < p∗ · x∗i .
(7.54)
However, it is clear that, since Xi is convex, there exists a value of θ > 0 and small enough so that: θxi + (1 − θ)xi ∈ N (xi ) ∩ Xi ; which, given (7.53) and (7.54), contradicts (7.51).
7.26 Deﬁnition. Let Xi be a convex subset of Rn , and let Pi be an irreﬂexive binary relation on Xi . We shall say that Pi is weakly convex iﬀ, for each x∗i ∈ Xi , the set Pi x∗i deﬁned by: Pi x∗i = {xi ∈ Xi  xi Pi x∗i }, is convex. The initial version of the ‘Second Fundamental Theorem’ which we will consider is a generalization of the theorem originally developed by Arrow [1951]. Early variations of Arrow’s result were published by Debreu [1954] and Koopmans [1957].
7.6. ‘Unbiasedness’ of the Competitive Mechanism
213
7.27 Theorem. Let E = (Xi , Pi , Yk , r) be an economy such that: a. Xi is convex, b. Pi is weakly convex, locally nonsaturating, and lower semicontinuous, for each i = 1, . . . , m; and suppose that: def c. Y = k=1 Yk is a convex set. Then if x∗i , y ∗k is Pareto eﬃcient for E, there exists a price vector, p∗ ∈ Rn such that (x∗i , y ∗k , p∗ ) is a quasi competitive equilibrium for E given the assignment of wealth w∗ deﬁned by: wi∗ = p∗ · x∗i for i = 1, . . . , m. (7.55) ∗ ∗ Proof. We note ﬁrst that, since xi , y k is Pareto eﬃcient for E, we must have: m i=1
Deﬁne:
x∗ =
and the subset, P, of
Rn ,
x∗i =
m i=1
m i=1
by: P=
ri +
i=1
k=1
m
x∗i , r =
m
i=1
y ∗k .
(7.56)
ri ;
Pi x∗i ;
where we recall the notation: Pi x∗i = {xi ∈ Xi  xi Pi x∗i }
for i = 1, . . . , m.
By way of completing our preliminaries, we note also that it follows immediately from the assumption that each Pi is weakly convex and the fact that the sum of convex sets is convex, that P is a convex set. Next, we note that, by assumption (c) and using Proposition 6.37 once again, the set r +Y is convex; and it is easy to see that, since x∗i , y ∗k is Pareto eﬃcient for E, we must have: P ∩ [ r + Y ] = ∅. Thus, by Theorem 6.21 (the Separating Hyperplane Theorem), there exists a nonzero p∗ ∈ Rn such that: α = sup{p∗ · z  z ∈ r + Y } ≤ β = inf{p∗ · x  x ∈ P}. def
def
(7.57)
Now, it follows at once from (7.56), (7.57), and our deﬁnition of x∗ , that: p∗ · x∗ ≤ α.
(7.58)
We are going to prove that we also must have p∗ · x∗ ≥ β. To do this, let > 0 be given. Then, using the continuity of the inner product function and the fact that each Pi is locally nonsaturating, we see that, for each i, there exists x†i satisfying: x†i Pi x∗i & p∗ · x†i < p∗ · x∗i + /m for i = 1, . . . , m.
(7.59)
Adding the inequalities on the right in (7.59), we then obtain: m
m m x†i = p∗ · x†i < (p∗ · x∗i + /m) = p∗ · x∗ + , (7.60) β ≤ p∗ · i=1
i=1
i=1
214
Chapter 7. Fundamental Welfare Theorems
where the ﬁrst inequality in (7.60) is by the deﬁnitions of β and P and the lefthand part of (7.59). However, since (7.60) has been shown to hold for any positive real number, , it follows that β ≤ p∗ · x∗ ; and, combining this with (7.57) and (7.58), we see that: α = β = p ∗ · x∗ . (7.61) Now let j ∈ K, let y j be an arbitrary element of Yj , and consider the production allocation y †k ∈ Y deﬁned by: y †k =
yj y ∗k
for k = j, for k = j.
From (7.56), (7.57), and (7.61), we have: p∗ · r +
p∗ · y ∗k + p∗ · y j ≤ p∗ · r +
k=j
k=1
p∗ · y ∗k ;
from which we obtain: p∗ · y j ≤ p∗ · y ∗j ; and we conclude that: p∗ · y ∗k = πk (p∗ )
for k = 1, . . . , .
(7.62)
∗ ) by: Next, deﬁning w∗ = (w1∗ , . . . , wm
wi∗ = p∗ · x∗i
for i = 1, . . . , m,
we note that it follows from (7.55) and (7.56) that w∗ is a wealthassignment for E, given p∗ ; and thus we have shown that (x∗i , y ∗k , p∗ ) satisﬁes the ﬁrst three of the conditions deﬁning a quasicompetitive equilibrium for E, given the wealth assignment w∗ ; and the ﬁrst part of Condition 4 as well. Therefore, to complete our proof, we need only establish that, for each i, either (7.32) or (7.33) of Deﬁnition 7.19 must hold. Accordingly, let i ∈ M be arbitrary, suppose x†i ∈ Xi is such that x†i Pi x∗i , and let > 0 be given. Since each Ph is locally nonsaturating, and making use of the ¯ h ∈ Xh continuity of the inner product, we see that for each h = i, there exists x such that: ¯ h < p∗ · x∗h + /(m − 1). ¯ h Ph x∗h and p∗ · x x (7.63) If we then deﬁne ( xh ) by: h = x we see that
¯h x x†
for h = i, for h = i,
m
h h=1 x
∈ P , and thus by (7.57) and (7.61):
m m ¯ h + p∗ · x†i ≥ β = h = p∗ · p∗ · x p∗ · x∗h x h=1
h=i
h=1
7.6. ‘Unbiasedness’ of the Competitive Mechanism
215
However, by (7.63) we have: ∗ ∗ ¯h < p∗ · x p · xh + ; p∗ · x∗h + /(m − 1) = h=i
h=i
(7.64)
h=i
and from (7.63) and (7.64), we then obtain: m p∗ · x∗h + + p∗ · x†i ≥ p∗ · x∗h ; h=1
h=i
so that:
p∗ · x†i ≥ p∗ · x∗i − .
Since > 0 was arbitrary, we can now conclude that: p∗ · x†i ≥ p∗ · x∗i . From the argument of the above paragraph, we conclude that for all xi ∈ Xi : xi Pi x∗i ⇒ p∗ · xi ≥ p∗ · x∗i
for i = 1, . . . , m.
(7.65)
Condition 4 of Deﬁnition 7.19 now follows from Proposition 7.25. This last result can be generalized to the extent of allowing for the commodity space to be inﬁnitedimensional.8 It is also possible that it could be generalized within the context of Rn . However, the conclusion of the result does not hold if any one of the conditions of Theorem 7.27 is simply dropped. In fact, a series of examples presented in the next chapter show that none of the assumptions of Theorem 7.27 can be dispensed with: 1. Example 8.3 shows that we cannot drop the assumption that each of the consumption sets, Xi , is convex. 2. Example 8.7 shows that we cannot drop the assumption that each Pi is locally nonsaturating. 3. Example 8.8 shows that we cannot drop the assumption that each Pi is weakly convex. 4. Example 8.9 shows that we cannot drop the assumption that each Pi is lower semicontinuous. 5. Example 8.14 shows that we cannot drop the assumption that Y is convex. Under the hypotheses ofTheorem 7.27, we have shown that given any Pareto eﬃ cient allocation, x∗i , y ∗k , there exists a price vector, p∗ , such that (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium. Unfortunately, the hypotheses of that result are not suﬃciently strong to imply that we can obtain a competitive equilibrium. In fact, under the hypotheses of Theorem 7.27 an allocation may be strongly Pareto eﬃcient, yet there may nonetheless be no price vector, p∗ , such that the allocation becomes a competitive equilibrium with this price vector. That this is so is demonstrated by the following, which is based upon a famous example by Arrow [1951]. 8
Although some of the other assumptions then need to be strengthened.
216
Chapter 7. Fundamental Welfare Theorems
7.28 Example. Let E be a private ownership economy in which m = n = 2, = 1, and Y = −R2+ . If the preference relations Pi are such that either P1 or P2 is strictly increasing, then it is clear that any competitive equilibrium, x∗i , y ∗ , p∗ , if one exists, must satisfy p∗ 0. However, this in turn implies that in any such competitive equilibrium, we must have y ∗ = 0. Consequently, in this case we can analyze the possible existence of a competitive equilibrium within the context of a traditional Edgeworth Box diagram. Suppose, then, that P1 , P2 , r 1 , and r 2 are as in 2 Figure 7.4, on the next page, and that X1 = X2 = R+ , and consider the price vector p∗ = (0, 1). It is easy to see that x∗i , y ∗ , p∗ is a quasicompetitive equilibrium. However, consumer 1’s demand for the is unbounded, given a zero ﬁrst commodity price for that commodity. Therefore, x∗i , y ∗ , p∗ is not a competitive equilibrium. Furthermore, it is easily seen in this case that if a nonnegative price vector deﬁnes a line separating P1 x∗1 and P2 x∗2 , it must be a scalar multiple of p∗ = (0, 1); and thus it follows that no price vector, p∗ , exists which is such that x∗i , y ∗ , p∗ is a competitive equilibrium. x12
x21
(ri)
x 22
x 11
Figure 7.4: Arrow’s ‘Exceptional Case.’ Now let’s turn our attention to ﬁnding conditions ensuring that (x∗i , y ∗k , p∗ ), as obtained in the conclusion of Theorem 7.27, is a competitive equilibrium with the given wealth assignment; as opposed to the weaker conclusion established in the theorem. Actually, one can easily obtain the stronger conclusion by making some rather stringent additional assumptions; the simplest of which makes use of the following mathematical condition. 7.29 Deﬁnition. The interior of a set, A ⊆ Rn , denoted int(A) is deﬁned as the set of all x ∈ A such that there exists > 0 such that N (x, ) ⊆ A.
7.6. ‘Unbiasedness’ of the Competitive Mechanism
217
While the following corollary is very easy to prove, it is not a very satisfactory result, for reasons we will discuss shortly. 7.30 Corollary. Suppose, in addition to the assumptions and conditions of Theorem 7.27, that the allocation x∗i , y ∗k satisﬁes: x∗i ∈ intXi p∗
Rn
∈ Then there exists E given the assignment of
for i = 1, . . . , m.
such that (x∗i , y ∗k , p∗ ) wealth w∗ deﬁned by: wi∗ = p∗ · x∗i
(7.66)
is a competitive equilibrium for
for i = 1, . . . , m.
In order to prove the corollary, we need only note that a linear function, in this case fi (xi ) ≡ p∗ · xi , can only be minimized at an interior point of a set if it is identically zero on the space. However, since we know that p∗ = 0, we see that this cannot be the case; that is, we must have: p∗ · x∗i > min p∗ · Xi
for i = 1, . . . , m.
It then follows from the deﬁnitions that the quasicompetitive equilibrium obtained in the theorem must, with this additional assumption, actually be a competitive equilibrium (with the assignment of wealth, w∗ ). The problem with this result is, of course, that the sort of Pareto eﬃcient allocation which would satisfy this assumption is strange and rare indeed! Notice, in particular, equation (7.66) implies that, for each consumer, i, there exists a point, xi ∈ Xi satisfying: xi x∗i . Consequently, it follows that at the allocation (x∗i ), and given any commodity, j, each consumer must either possess a strictly positive quantity of the j th commodity, or be supplying less of it than he or she is capable of supplying. It is not at all clear that any such point could ever be a Pareto eﬃcient allocation;9 and in any case, it is clear that most of the eﬃcient allocations of interest will not satisfy this property. In fact, to carry this argument one step further, remember that we would like to be able to claim that the model allows for a ﬁnite number of time periods; with commodities being diﬀerentiated by time of availability as well as physical characteristics. In this context, it is worth noting that in most of our work in this and the next chapter the assumptions of the model incorporate as a special case the situation in which there are T time periods, and G physically distinguishable commodites, so that n = G × T ; and where, for each i there exist positive integers ti and ti such that: 1 ≤ ti < ti ≤ T and such that Xi takes the form: Xi = 0ni × Ci × 0ni , 9 Unless consumer preferences are identical, the satisfaction of equation (7.66) will almost certainly indicate that Paretoimproving trades among consumers are possible; and thus that ∗ ((xi ), (y ∗k )) cannot be Pareto eﬃcient.
218
Chapter 7. Fundamental Welfare Theorems
where: ni = ti · G, ni = (T − ti ) · G, and, deﬁning ni = ti − ti :
Ci ⊆ Rni ·G .
The idea here is that, for a consumer who is born in the tth i period and dies in period ti , only its consumption in periods ti + 1, . . . , ti aﬀects its survival or preferences. Under these conditions, the sets Xi do not even possess interiors! In the next section we will consider a diﬀerent sort of strengthening of Theorem 7.27, but before ending this discussion it may be of some interest to show how easily we can now obtain a version of the ‘Second Fundamental Theorem’ for private ownership economies. In doing this we will deﬁne a quasicompetitive equilibrium for E, given a system of lumpsum transfers; where we shall say that t ∈ Rm is a system of lumpsum transfers for E iﬀ: m i=1
ti = 0.
(7.67)
7.31 Deﬁnition. Let E = (Xi , Pi , Yk , r i , [sik ]) be a private ownership economy. We shall say that (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for E with lumpsum transfers t iﬀ: 1. p∗ = 0, 2. ((x∗i ), (y ∗k )) ∈ A(E), 3. p∗ · y ∗k = πk (p∗ ), for k = 1, . . . , , 4. for each i (i = 1, . . . , m), we have p∗ · x∗i ≤ wi (p∗ ), and either: wi (p∗ ) = min{p∗ · xi  xi ∈ Xi } = min p∗ · Xi ,
(7.68)
(∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > wi (p∗ )
(7.69)
def
or: (or both), where: wi (p∗ ) = p∗ · r i +
k=1
sik πk (p∗ ) + ti
for i = 1, . . . , m;
I will leave as an exercise the following corollary of Theorem 7.27. 7.32 Corollary. Let E = (Xi , Pi , Yk , r i , [sik ]) be a private ownership economy such that: a. Xi is convex, b. Pi is weakly convex, locally nonsaturating, and lower semicontinuous, for each i = 1, . . . , m; and suppose that: def c. Y = k=1 Yk is a convex set. Then if x∗i , y ∗k is Pareto eﬃcient for E, there exists a price vector, p∗ ∈ Rn and a vector t∗ ∈ Rm such that (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium for E with the lumpsum transfers t∗ .
7.7. A Stronger Version of ‘The Second Theorem’
7.7
219
A Stronger Version of ‘The Second Theorem’
In order to develop a better version of the ‘Second Fundamental Theorem,’ we will make use of two principal conditions, the ﬁrst of which is deﬁned as follows. 7.33 Deﬁnition. We shall say that the economy, E = (Xi , Pi , Yk , r) is irreducible at the allocation x∗i , y ∗k iﬀ, given any partition of the consumers, {I1 , I2 },10 there exists xi ∈ X and z ∈ Rn such that: m i=1
and:
z ∈ r + Y, xi = z,
(∀i ∈ I1 ) : xi Pi x∗i .
(7.70) (7.71)
(7.72)
The condition just deﬁned is developed from the ‘irreducibility condition’ introduced by L. McKenzie [1959, 1961], and was generalized somewhat in Moore [1970, 1975]. Eﬀectively, it implies that at the allocation ((x∗i ), (y ∗k )), there is no deﬁnable subgroup of consumers, I1 , who could not make themselves better oﬀ, collectively, if they were simply allowed to exploit the remaining consumers, and, possibly, to reorganize the means of production. In the simplest situation, a twoperson, twocommodity exchange economy, E is irreducible at an allocation in the Edgeworth box if, and only if, each consumer prefers some other attainable allocation (not, of course, necessarily the same one).11 A similar interpretation applies in the case of a pure exchange economy with an arbitrary ﬁnite number of consumers; and thus, in this case the satisfaction of the irreducibility condition at x∗i implies that there is no subgroup of consumers (I2 ) so poor that the remaining consumers could not (if allowed to do so) exploit them in such a way as to make themselves better oﬀ. In other words, and somewhat loosely interpreting, the condition guarantees that no subgroup of consumers is so poor as to have nothing which is valued by the remaining consumers. We will consider the meaning of the condition in more detail shortly, but ﬁrst let me introduce the second of the key conditions mentioned earlier. 7.34 Deﬁnition. Let E = (Xi , Pi , Yk , r) and E = (Xi , Pi , T ) be economies. We will say that E is aggregatively similar to E at an allocation xi , y k ∈ A(E) iﬀ: Xi , Pi = Xi , Pi for i = 1, . . . , m, r+ Yk ⊆ T, k=1
(7.73) (7.74)
and the allocation x∗i , z ∗ is Pareto eﬃcient for E.
Clearly = (Xi , Pi , T ) is aggregatively similar to E at x∗i , y ∗k ∈ A(E), ∗ if E then xi , y ∗k is Pareto eﬃcient for E. The following presents what are probably the simplest examples of this aggregatively similar relationship.
10 By a partition of the consumers, {I1 , I2 }, we mean Ij ⊆ I & Ij = ∅, for i = 1, 2, I1 ∩ I2 = ∅, and I1 ∪ I2 = I. 11 Notice that this condition fails at the allocation x∗i in Example 7.28.
220
Chapter 7. Fundamental Welfare Theorems
7.35 Examples. 1. Let E = (Xi , Pi , Yk , r) be an economy, x∗i , y ∗k be a Pareto eﬃcient allocation for E, and deﬁne the set T by: T =r+
k=1
Yk .
Then E = Xi , Pi , T is aggregatively similar to E at x∗i , y ∗k . ∗ 2. Let E = (Xi , Pi , Yk , r) be an economy, suppose (xi , y ∗k , p∗ ) is a competitive equilibrium for E, and deﬁne: πk (p∗ ) . T = z ∈ Rn  p∗ · z ≤ p∗ · r + k=1
Then, can easily prove, E = Xi , Pi , T ∗ as∗ you xi , y k .
is aggregatively similar to E at
7.36 Theorem. Let E = (Xi , Pi , Yk , r) be an economy such that: a. Xi is convex, and lower semicontinuous, for b. Pi is weakly convex,locally nonsaturating, for E, and suppose there each i = 1, . . . , m; suppose x∗i , y ∗k is Pareto eﬃcient exists set T ⊆ Rn such that E = Xi , Pi , T is aggregatively similar to E ∗a convex at xi , y ∗k , c. int(X) ∩ T = ∅, and d. E is irreducible at (x∗i , z ∗ ), where: z∗ = r +
k=1
y ∗k .
Then there exists a price vector, p∗ ∈ Rn such that (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E given the assignment of wealth w∗ , where: wi∗ = p∗ · x∗i
for i = 1, . . . , m;
(7.75)
and we have: wi∗ > min p∗ · Xi
for i = 1, . . . , m.
(7.76)
Proof. It follows from Theorem 7.27 that there exists a price vector, p∗ = 0 such that (x∗i , z ∗ , p∗ ) is a quasicompetitive equilibrium for E, given the wealth assignment: wi∗ = p∗ · x∗i for i = 1, . . . , m. (7.77) ∈ X ∩ T and θ ∈ R++ such Moreover, from Assumption c, we see that there exists x that: def − θp∗ ∈ X, x† = x (7.78) and thus: − θp∗ · p∗ < p∗ · x ≤ p∗ · x∗ , x − θp∗ ] = p∗ · x p∗ · x† = p∗ · [
7.7. A Stronger Version of ‘The Second Theorem’
221
where the last inequality is from Proposition 7.20. Therefore since (again by 7.20) at a quasicompetitive equilibrium, each consumer’s consumption expenditure must be equal to wealth, it must be the case that for some i ∈ M : wi∗ = p∗ · x∗i > min p∗ · Xi .
(7.79)
Now deﬁning the sets of consumers Ih ⊆ M (h = 1, 2) by: I1 = {i ∈ I  wi∗ > min p∗ · Xi }, and:
I2 = {i ∈ I  wi∗ = min p∗ · Xi },
respectively, it follows from (7.79) that I1 = ∅. Suppose by way of obtaining a contradiction, that I2 = ∅ as well. Then, since E is irreducible at (x∗i , z ∗ ), there exists (xi , z) ∈ A(E) satisfying:
z ∈ T, xi = z −
i∈I2
and:
(7.80) xi ,
(7.81)
i∈I1
(∀i ∈ I1 ) : xi Pi x∗i .
(7.82)
For future reference, we note that it follows from the deﬁnition of I1 that for each i ∈ I1 : (7.83) p∗ · xi > wi∗ = p∗ · x∗i . Now, from (7.81) and (7.83), we have p∗ · xi = p∗ · z − p∗ · xi < p∗ · z − wi∗ . i∈I2
i∈I1
(7.84)
i∈I1
Moreover, since z ∈ T , we see that we must have p∗ · z ≤ p∗ · z ∗ ; and, since w∗ is a feasible wealth assignment for E, we also have: wi∗ = p∗ · z ∗ . (7.85) i∈M
Therefore, it now follows from (7.84) that: p∗ · xi < wi∗ − wi∗ = wi∗ . i∈I2
i∈M
i∈I1
i∈I2
But this contradicts the deﬁnition of I2 . It follows, therefore, that I2 is empty, and, consequently, that (7.76) holds and that (x∗i , z ∗ , p∗ ) is a competitive equilibrium for E. Since: Yk ⊆ T, r+ k=1
you can now (making use of Theorem 6.38) easily prove that (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E as well, given the wealth distribution, w∗ .
222
Chapter 7. Fundamental Welfare Theorems
One might question whether Theorem 7.36 is of much signiﬁcance, even from a purely theoretical point of view. After all, under the assumptions of Theorem 7.27, we have shown that if x∗i is Pareto eﬃcient, then there exists p∗ such that (x∗i , y ∗k , p∗ ) is a quasicompetitive equilibrium. Moreover, (x∗i , y ∗k , p∗ ) will be a competitive equilibrium unless for one or more consumeres, we have p∗ · x∗i = min p∗ · Xi ; and, since p∗ = 0, the only way this can happen is if x∗i is on the boundary of Xi . However, as I suggested in the discussion at the end of the previous section, if we deﬁne commodities ﬁnely (as we usually specify that we are when dealing with competitive behavior), all reasonable allocations would result in each consumer’s commodity bundle being on the boundary of its consumption set. After all, does anyone consume a positive quantity of each commodity available in the U. S. each month? I think not! Consequently, even though we tend to think of boundary values as being a very special case, they are the norm in reality. Perhaps it is because we use Edgeworth Box diagrams so frequently in our analysis that we think of consumption values on the boundary of Xi as being an ‘exceptional case,’ and if it were reasonable to assume that in reality there are only two or three commodities available in an economy, this attitude would probably be correct. However, one needs to allow for a large number of commodities in order to justify our competitive assumptions; and, correspondingly, we need to treat boundary values as the norm. The assumption that there exists a convex set, T , such that E = (Xi , Pi , T ) is aggregatively similar to E = (Xi , Pi , Yk , r) at x∗i , y ∗k is obviously critical in the proof of Theorem 7.36. Figure 7.5, on the next page, in which the set P is intended to represent the set: m Pi x∗i , P = i=1
and Y represents the aggregate production set, conveys some idea of the generality of the assumption (in the diagram, we are supposing that r = 0). We can obtain another strengthened version of the Second Fundamental Theorem by making use of the following deﬁnition (which is partially repeated from Chapter 4). eraire good for 7.37 Deﬁnitions. We will say that the j th commodity is a num´ P i iﬀ for all x ∈ Xi and all θ ∈ R++ , we have: x + θej ∈ Xi and (x + θej )Pi x, j th
where ej is the unit coordinate vector. We shall say that the j th commodity is a num´ eraire good for the economy, E, at an allocation x∗i , y ∗k ∈ A(E) iﬀ it is a num´eraire good for each i ∈ M , and for each i ∈ M there exists θi > 0 such that: x∗i − θi ej ∈ Xi . Since it is easily seenthat the num´ eraire good assumption in the following implies that E is irreducible at x∗i , y ∗k , and that each preference relation is locally nonsaturating, this next result is a corollary of 7.36. Details of the proof will be left as an exercise. 7.38 Theorem. Let E = (Xi , Pi , Yk , r) be an economy such that:
7.7. A Stronger Version of ‘The Second Theorem’
223
P
T
Y
O Figure 7.5: Aggregatively Similar Economies. a. Xi is convex, b. Pi is weakly convex and lower semicontinuous, for each i = 1, . . . , m; def c. Y = k=1 Yk is a convex set, and suppose that: d. int(X) ∩ (r + Y ) = ∅. Then we have the following. ¯th If x∗i , y ∗k is Pareto eﬃcient for E and for some ¯j ∈ {1, . . . , n}, the j commodity is a num´eraire good for the economy at x∗i , y ∗k , then there exists a price vector, p∗ ∈ Rn such that (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E given the assignment of wealth w∗ , where: wi∗ = p∗ · x∗i and we have:
wi∗ > min p∗ · Xi
for i = 1, . . . , m;
for i = 1, . . . , m.
(7.86)
(7.87)
Notes on the Literature. The notion of Pareto dominance was apparently formally introduced into economics in Pareto [1894]; although it seems to have been Pareto’s friend and colleague, Enrico Barone who ﬁrst stated and proved a version of the ‘First Fundamental Theorem’ (Barone [1908]). Kenneth Arrow ﬁrst stated and proved a version of the ‘Second Fundamental Theorem’ (Arrow [1951a]): although Gerard Debreu independently published a closelyrelated result
224
Chapter 7. Fundamental Welfare Theorems
(Debreu [1951]). Early reﬁnements/generalizations of the ‘Second Fundamental Theorem’ were done by Debreu [1954], Koopmans [1957], and Koopmans and Bausch [1959]. Hurwicz [1960] was the ﬁrst to examine the issue of nonwastefulness and unbiasedness of abstract resource allocation mechanisms generally. As mentioned in the text, the irreducibility condition used in Theorem 7.36, while adapted from an earlier condition introduced by Lionel McKenzie. It was generalized in various ways in Moore[1970], and further generalized and reﬁned in Moore [1973, 1975]. However, the irreducibility condition introduced here incorporates and generalizes all of these conditions.
Exercises. 1. Consider the economy, E, in which we have one consumer, one producer, and two commodities, and where X, r and Y are given by: X ={x ∈ R2  −4 ≤ x1 ≤ 0 & x2 ≥ 3}, r =(0, 2), and Y ={y ∈ R2  y1 ≤ 0 ≤ y2 & y1 + y2 ≤ 0}, respectively; and suppose the consumer’s preferences can be represented by the utility function: u(x) = min{8 + 2x1 , x2 }. (a) Show that the allocation (x∗ , y ∗ ) is Pareto eﬃcient for E, where: x∗ = (−2, 4) and y ∗ = (−2, 2). [Note: There are various hard ways to verify this answer, as well as an easy way. Try to ﬁnd the easy way, but before concluding that you have found it, answer part (b).] (b) Is the allocation (x , y ), where: x = (−4, 6) and y = (−4, 4), Pareto eﬃcient for E? 2. Consider the pure exchange economy, E, in which m = n = 2, and in which the ith consumer’s preference relation, Pi , is representable by the utility function: ui (xi ) = xi1 · xi2
for i = 1, 2;
and where r = (1, 1). Show that an allocation, (xi ), is Pareto eﬃcient for E if, and only if, there exists some θ ∈ [0, 1] such that: x1 = (θ, θ) and x2 = (1 − θ, 1 − θ). Once again there is an easy way to do this. 3. Consider the twoconsumer, twocommodity exchange economy in which the consumer’s preferences can be represented by the utility functions: u1 (x1 ) = x11 + x12 ,
7.7. A Stronger Version of ‘The Second Theorem’
225
and: u2 (x2 ) = min{x21 , x22 }, respectively; and with r, the aggregate resource endowment, given by r = (10, 10). a. Find, either graphically or algebraically, all Pareto eﬃcient allocations for this economy. b. Let θ ∈ ]0, 1[, and consider the function w : R2++ → R2+ deﬁned by: w1 (p) = 10θ(p1 + p2 ) and w2 (p) = 10(1 − θ)(p1 + p2 ), respectively. Is w(·) a feasible wealthassignment funtion for E? c. Given the wealthassignment function deﬁned in part b, above, can you ﬁnd a Walrasian equilibrium for E, given w(·) and an arbitrary θ ∈]0, 1[? 4. Prove Proposition 7.13. 5. Prove Corollary 7.14. 6. Prove Corollary 7.32 7. In the literature on general competitive equilibrium, it is quite common to ﬁnd the condition: m xi ≤ r + yk , (7.88) i=1
k=1
used in place of Condition 3 in the deﬁnition of an attainable allocation for an economy, E. The tuple (x∗i , y ∗k , p∗ ) is then said to be a competitive equilibrium for E if it satisﬁes this modiﬁed attainability condition; and, in addition to the other conditions of Deﬁnition 7.4, it satisﬁes:
m p∗ · r + yk∗ − x∗i = 0. l=1
i=1
Show that this approach is equivalent to using the deﬁnitions in the present chapter, while maintaining the assumption that Y satisﬁes ’semifree disposability’ (Deﬁnition 6.4.5.b)
Chapter 8
The Existence of Competitive Equilibrium 8.1
Introduction
In this chapter, our main concern is to analyze the following theorem concerning the existence of Walrasian equilibrium for a private ownership economy. It is based upon (and is a special case of) Gale and MasColell [1975], and is a generalization of Theorem 5.7.1, pp. 83–4 of Debreu [1959]. 8.1 Theorem. The private ownership economy, E = (Xi , Pi , Yk , r i , [sik ]), has a Walrasian equilibrium if: for each i (i = 1, . . . , m): a. Xi is closed, convex, and bounded below, b. Pi is (irreﬂexive and): 1. nonsaturating, 2. weakly convex, and: 3. strongly continuous; ¯ i ri . c. (∃¯ xi ∈ X i ) : x d.1. 0 ∈ Yk , for k = 1, . . . , , d.2. Y ≡ k=1 Yk is closed and convex, d.3. Y ∩ (−Y ) ⊆ {0}, and: d.4. −Rn+ ⊆ Y . In the Sections 2–4 of this chapter, we will go through the assumptions of this theorem one by one; showing in each case that the assumption cannot simply be dispensed with. In Section 4 we will present at statement and brief discussion of the original Gale and MasColell theorem; and in Section 5 we will prove an especially simple version of an existence theorem. Returning to our discussion of Theorem 8.1, let me explain that by the statement, “an assumption cannot be dispensed with,” in a given theorem, I mean the following. Suppose we have a theorem of the form: A1 & A2 & A3 ⇒ C,
228
Chapter 8. The Existence of Competitive Equilibrium
where ‘Ai ’ denotes the statement of an assumption, for i = 1, 2, 3; and ‘C’ denotes the statement of the conclusion. We will say that, for example, A1 cannot be dispensed with in this result if we can ﬁnd an example satisfying A2 and A3 , but where the conclusion, C, does not hold. Notice that the existence of such an example does not preclude the possibility of their being an assumption A∗1 such that: A∗1 & A2 & A3 ⇒ C, and where A∗1 generalizes A1 (that is, A1 ⇒ A∗1 , but not conversely). Thus, to show, for example, that assumption (b.1) cannot be dispensed with in Theorem 8.1, we need to ﬁnd an example of a private ownership economy satisﬁng all of the remaining assumptions of the theorem, but for which no Walrasian equilibrium exists. Our examples will deal primarily with two special cases introduced in the previous chapter: the classical twoperson, twocommodity pure exchange model; and the oneperson, one producer, two commodity model. Recall the notation used in the previous chapter: we deﬁne Π by: Π = {p ∈ Rn  (∃¯ y ∈ Y )(∀y ∈ Y ) : p · y¯ ≥ p · y},
(8.1)
where ‘Y ’ denotes the aggregate production set. Furthermore, just as we deﬁned the proﬁt functions, πk (·) and the supply correspondences, σk (·) on Πk , we deﬁned an aggregate proﬁt function, π(·), and an aggregate supply correspondence, σ(·), on Π by: π(p) = max p · y and σ(p) = {y ∈ Y  p · y = π(p)} y∈Y
for p ∈ Π,
(8.2)
respectively. Recall also that it follows immediately from Proposition 7.8, that, under the assumptions of the present Theorem 8.1: Π ⊆ Rn+ ,
(8.3)
Π∗ = Π ∩ ∆ n ,
(8.4)
so that, if we deﬁne the set Π∗ by:
Rn :
where ‘∆n ’ denotes the unit simplex in n ∆n = p ∈ Rn+ 
j=1
pj = 1 ,
(8.5)
we can make use of the homogeneity of the producers’ supply and consumers’ demand correspondences to conﬁne our search for equilibrium prices to the set Π∗ . In this chapter, although nowhere else in this book, we will distinguish between competitive and Walrasian equilibria. A Walrasian equilibrium will be one which satisﬁes Deﬁnition 7.4; while a competitive equilibrium will be deﬁned as is set out in Exercise 7, at the end of Chapter 7. In the examples to follow, it will be shown that no Walrasian equilibrium exists; however, it can also be shown that no competitive equilibrium (as just deﬁned) exists either. We will discuss this issue further in the Appendix to this chapter.
8.2. Examples, Part 1
8.2
229
Examples, Part 1
In this section we will develop a number of examples in which we drop only one of the assumptions of Theorem 8.1, and then show that a Walrasian equilibrium does not exist. One particular thing to keep in mind as you study the examples to follow is this: we know from our work in the previous chapter that if a Walrasian equilibrium exists, then the allocation involved must be Pareto eﬃcient. Consequently, if there is only one consumer involved in the example, then the only commodity bundle which could be involved in a Walrasian equilibrium is the one which maximizes the consumer’s preferences over the attainable consumption set (or a member of the maximal set, if the preferencemaximizing bundle is not unique). Correspondingly, if there is no bundle in the attainable, or feasible consumption set at which the consumer’s preferences are maximized, then there will be no Walrasian equilbrium in the example. This fact is the basis of our ﬁrst example, which shows that the closure of each Xi is an assumption which cannot be dispensed with in Theorem 8.1). 8.2 Example. Let m = = 1, n = 2, and suppose X, Y and r are as indicated in Figure 8.1, below. While the consumer can almost achieve the level of satisfaction (or utility) corresponding to the indiﬀerence curve I1 , we are supposing that the leftmost boundary of the consumption set (the dashed vertical line) is not contained in the consumption set. Thus, in particular, the point at which I1 and the upper boundary of r + Y appear to intersect is not an element of the consumptions set, and thus no maximal point exists within the attainable consumption set.
I1 r+Y
X r
Figure 8.1: Consumption Set not Closed. The example just presented is, admittedly, highly artiﬁcial in appearance. This is an inevitable consequence of the fact that continuity and set closure are not
230
Chapter 8. The Existence of Competitive Equilibrium
generally assumptions which have any empirical content; that is, in general closure and continuity cannot be directly refuted by empirical observation. For example, in Rn a set is closed if, given any convergent sequence of points from the set, the limit of the sequence is also an element of the set. Since we cannot observe all the terms in an inﬁnite sequence, we cannot determine whether a ﬁnite sequence of empirical observations is or is not drawn from a convergent inﬁnite sequence. Consequently, we cannot generally refute the assumption that individual consumption sets are closed. The reason which I have slightly qualiﬁed my statements in the above paragraph is that there may be real choice situations in which one can see that the choice set (or the attainable consumption set in our general equilibrium examples) is not closed. One such example, which is, I believe, due to Marcel K. Richter, runs as follows. Suppose I am given a gold bar, and told that I need to cut it into two parts, giving one part to you, and keeping the other for myself—with no stipulations about relative size, and no choice of part by yourself. In this case, I, being the greedy person that I am, will try to cut oﬀ and give you as thin a slice of the bar as possible in order that I maximize the amount of gold which I get to keep for myself. Well, as you can see, there is no maximal point in this problem; no matter how thin the slice I give you, I might have been able to cut oﬀ a still thinner slice. Formally, this is an example in which the attainable consumption set is not closed, and, correspondingly, in which no preferencemaximizing choice exists for me. 8.3 Example. (Showing that the convexity of each Xi is an assumption which cannot be dispensed with in 8.1.) Let m = = 1, n = 2, and suppose X, Y and r are as indicated in Figure 8.2, below.
X x**
r+Y x*
Y P* r
Π∗
Figure 8.2: Nonconvexity of the Consumption Set. In Figure 8.2, we can see that if p ∈ Π∗ is such that p1 ≥ p∗1 , then the consumer’s preferencemaximizing commodity bundle will be on the lefthand boundary of the consumption set (at some x for which x1 = x∗∗ 1 , which is outside the attainable
8.2. Examples, Part 1
231
consumption set, X ∗ ). On the other hand, if p ∈ Π∗ is such that p1 < p∗1 , then the producer will maximize proﬁts at some point y ∈ Y where r + y ∈ / X (in particular, for any such p, we will have y1 < x∗1 , for the proﬁtmaximizing value of y). Therefore, no Walrasian equilibrium exists for E in this case. 8.4 Example. (Showing that the assumption that Xi is bounded below for each i cannot be dispensed with in Theorem 8.1) Let m = = 1, n = 2, and suppose X, Y and r are as indicated in Figure 8.3, below; where X is the set of all points in the second quadrant lying above the heavy horizontal line, and the consumer’s indiﬀerence curves are the family of parallel lines drawn with slope ﬂatter than the upper boundary of Y (which is indicated by the heavy upwardsloping line in the ﬁgure). We can see from the ﬁgure that if p ∈ Π∗ is such that p1 > p∗1 , then the producer will maximize proﬁts at the origin (that is, at y = 0); while if p = p∗ , then the producer’s proﬁts are maximized at all points y on the upper boundary of the production set. However, in either of these cases, the consumer’s demand is unbounded. On the other hand, if p ∈ ∆n is such that p1 < p∗1 , then the producer’s proﬁts are unbounded (and no proﬁtmaximizing production vector exists). Therefore, no Walrasian equilibrium exists in this case.
r
X
p*
Y
Π∗
Figure 8.3: An Unbounded Consumption Set. Notice that the attainable consumption set, X ∗ , is unbounded in Example 8.4. Intuitively, we would probably suspect that the attainable consumption set is going to have to be bounded if there is to exist a Walrasian equilibrium. As a matter of fact, however, it is possible for an equilibrium to exist even if X ∗ is unbounded. On the other hand, it is very diﬃcult to specify nontrivial, yet meaningful conditions which are suﬃcient to guarantee that an equilibrium exists in such a case; in fact, nearly all of the existence proofs with which I am acquainted make use of assumptions which insure that X ∗ is bounded. Our next example shows an even more insurmountably diﬃcult situation which can arise when X ∗ is unbounded, and notice that this time X is bounded below.
232
Chapter 8. The Existence of Competitive Equilibrium
8.5 Example. (Showing that (d.3), the aggregate irreversibility of production assumption, cannot be dispensed with in 8.1) Let m = = 1, n = 2, and suppose X, Y and r are as indicated in Figure 8.4, below; where Y is the halfspace consisting of all points on or to the left of the vertical axis. Notice that, since Y is a convex cone in this case, and r ∈ Y , we will have r + Y = Y . You should have no diﬃculty in establishing the fact that no Walrasian equilibrium exists in this case.
.
X
r
Y
∆1
.
Π∗
Figure 8.4: Reversible Production. Debreu has proved the following result, which I will simply state without proof. It is proved on p. 77 of Debreu [1959], for those of you who might be interested. 8.6 Proposition. (Debreu) Let E be an economy n which X is bounded below, Y is closed and convex, and Y ∩Rn+ = {0}. If, in addition, = 1, and/or Y ∩(−Y ) = {0}, then A(E) is bounded; as then are X ∗ and Y ∗ as well. This result, together with Examples 4 and 5, show the role played by the assumptions: Xi is bounded below, for i = 1, . . . , m, and: Y ∩ (−Y ) ⊆ {0}. In Theorem 8.1; as well as indicating one of the roles played by each of the assumptions (d.1), (d.2), and (d.4) in the result. Notice, incidentally, that if Y satisﬁes: −Rn+ ⊆ Y and Y ∩ (−Y ) ⊆ {0}, then we will also have: Rn+ ∩ Y = {0}.
8.2. Examples, Part 1
233
8.7 Example. (Showing that the assumption that each Pi is nonsaturating cannot be dispensed with in Theorem 8.1) Once again we consider a case in which there is one consumer, two commodities, and one producer; this time with the consumption and production sets as indicated in Figure 8.5, below. The concentric ovals represent the consumer’s indiﬀerence map in this case; with the consumer essentially ordering bundles in terms of their distance from his or her ‘bliss point,’ which is x∗ . If there were a Walrasian equilibrium in this situation, we would need to be able to ﬁnd a def nonnull price vector, p, such that the producer maximizes proﬁts at y ∗ = x∗ − r. Obviously, however, there is no such price vector.
r+Y
.
x*
r
.p*
Figure 8.5: Saturating Preferences. 8.8 Example. (Showing that the assumption that each Pi is weakly convex cannot be dispensed with in 8.1). Let m = = 1, n = 2, and suppose X, Y and r are as indicated in Figure 8.6, on the next page. The indiﬀerence curves for P , the consumer’s preference relation, are the kinked lines with vertices on the upwardsloping dashed line emanating from the lower left corner of X. Notice that for p ∈ Π∗ such that p1 > p∗1 , the consumer’s optimal commodity bundle will have x1 = x∗1 and x2 ≥ x∗2 . On the other hand, for p ∈ Π∗ satisfying 0 < p1 < p∗1 , the consumer’s optimal commodity bundle, x, will satisfy x1 = 0. Finally, for p = p∗ , h(p∗ , w) = {x∗ , x† }. Thus we can see that no Walrasian Equilibrium exists in this case. 8.9 Example. (Showing that the assumption that each Pi is strongly continuous cannot be dispensed with in Theorem 8.1) In Figure 8.7, on the next page, the line with the arrows represents the consumer’s ‘behavior line’ through x∗ ; that is, everything above the line is preferred to anything on or below the line, while anything
234
Chapter 8. The Existence of Competitive Equilibrium
x*
x†
r
p*
Y
Π∗
Figure 8.6: Nonconvex Upper ontour Sets. on the line is preferred to anything below the line. On the line itself, however, there is an ordering, as indicated by the direction of the arrows.
x* r+Y
r p*
Figure 8.7: Noncontinuous Preferences. In this case, there is a maximal consumption bundle (Pareto eﬃcient consumption allocation), namely x∗ . Moreover, there is a wide range of price vectors, among them p∗ , such that the producer will maximize proﬁts at x∗ − r. However, if the price vector is anything (in ∆2 ) other than p∗ , the consumer will maximize satisfaction at a point on one of the boundaries of X. On the other hand, with p = p∗ , the consumer will maximize satisfaction at the point where the ‘behavior line’ intersects the lefthand boundary of X. Therefore, no Walrasian equilibrium exists in this case.
8.3. Assumption (c) and the Attainable Set
8.3
235
Assumption (c) and the Attainable Set
A condition which we have not as yet considered, but which is obviously a necessary condition for the existence of a Walrasian equilibrium for an economy, E, is that: A(E) = ∅. Our next result shows how this property is guaranteed under the hypotheses of Theorem 8.1. 8.10 Proposition. If E satisﬁes: ¯ ≤ r, x ∈ X) : x c . (∃¯ and (d.4): −Rn+ ⊆ Y, then A(E) = ∅. ¯ i ∈ Xi , for i = 1, . . . , m, satisfying: ¯ satisfy (c ). Then there exist x Proof. Let x m ¯= ¯ i ≤ r. x (8.6) x i=1
Deﬁning: ¯=x ¯ − r, y ¯ ∈ we see by (8.6) that y k = 1, . . . , , satisfying:
−Rn+ .
¯ k ∈ Yk , for Therefore, by (d.4) there exists y
¯ . ¯ −r =y ¯= y x k=1 k y k ) ∈ A(E). It follows at once from (8.7) that (¯ xi ), (¯
(8.7)
The above proposition indicates another role which assumption (d.4) plays in Theorem 8.1, as well as one role played by assumption (c).1 The role played by assumption (c) in Theorem 8.1 is much more complicated and more subtle than this, however. Consider, for instance, the following example. 8.11 Example. Let E be the private ownership economy in which = 1, m = n = 2, and: Xi = {xi ∈ R2  −1 ≤ xi1 ≤ 0 & xi2 ≥ 2} for i = 1, 2; Y = {y ∈ R2  y1 + y2 ≤ 0 & y1 ≤ 0}, let: r 1 = (0, 5),
s1 = 1,
r 2 = (0, 0),
s2 = 0;
and consider the commodity bundles: x∗1 = x∗2 = (−1, 2) ∈ Xi 1
Obviously (c) implies (c ).
for i = 1, 2.
236
Chapter 8. The Existence of Competitive Equilibrium
We have:
x∗ = x∗1 + x∗2 = (−2, 4) r = r 1 + r 2 = (0, 5), def
def
and thus we see that E satisﬁes the condition: c.
¯ r. (∃¯ x ∈ X) : x
Moreover, it is easy to show that E satisﬁes (a), (d.1), (d.2), (d.3), and (d.4) of Theorem 8.1. We will demonstrate, however, that, if P1 is a locally nonsaturating binary relation, then whatever the form of P2 , no Walrasian equilibrium exists in this case. We begin by noting that in this case the set Π∗ will be given by: Π∗ = {p ∈ R2+  1/2 ≤ p1 ≤ 1 & p2 = 1 − p1 }. Now, suppose p ∈ Π∗ is such that 1/2 < p1 ≤ 1. Then the producer will maximize proﬁts at y = 0, and the ﬁrst consumer’s budget constraint will be of the form: p1 x11 + p2 x12 ≤ w1 (p) = p · r 1 + s1 π(p) = 5p2 . However, if P1 is any locally nonsaturating preference relation, then any which maximizes P1 , given p and w1 (p) will satisfy (8.8) with an equality: p1 x∗11 = (5 − x∗12 )p2 . x∗
x∗11
(8.8) x∗1
∈ X1 (8.9)
x∗12
Since ∈ X1 implies ≤ 0, we see that (8.9) implies that ≥ 5. However, in order that x2 be an element of X2 , we must have x22 ≥ 2, and thus it is clear that there exists no x2 ∈ X2 satisfying: x∗1 + x2 = r + y = (0, 5) + 0 = (0, 5); and, consequently, that no Walrasian equilibrium exists in which 1/2 < p1 ≤ 1.
(8.10)
Now, if p ∈ Π∗ , the only alternative to its satisfying (8.10) is that p = (1/2, 1/2). However, for this value of p, the second consumer’s budget constraint is give by: p · x2 = (1/2)(x21 + x22 ) ≤ w2 (p) = 0.
(8.11)
On the other hand, if x2 ∈ X2 , we have x21 ≥ −1 and x22 ≥ 2, so that: (∀x2 ∈ X2 ) : p · x2 = (1/2)(x21 + x22 ) ≥ (1/2)(−1 + 2) = 1/2.
(8.12)
Upon comparing (8.11) and (8.12), we see that the second consumer’s budget set is empty if p = (1/2, 1/2); and since we have now considered all values of p which are consistent with proﬁtmaximization for the producer, it follows that no Walrasian equilibrium exists for this economy. As is made clear by the last example, one of the functions of assumption (c) in Theorem 8.1 is that it [together with (d.1)] guarantees that each consumer will have suﬃcient wealth to participate in a market economy. However, consider the following result.
8.3. Assumption (c) and the Attainable Set
237
8.12 Proposition. Suppose E satisﬁes: ¯ i ≤ r i for i = 1, . . . , m; xi ∈ X i ) : x c. (∃¯ and (d.1): 0 ∈ Yk for i = 1, . . . , . Then we have: def
(∀p ∈ Π∗ ) : Bi (p) = {x ∈ Xi  p · x ≤ wi (p)} = ∅
for i = 1, . . . , m.
Proof. Let p∗ be an arbitrary element of Π∗ . Then, by (d.1) and the deﬁnition of Π∗ , it follows that for each k: πk (p∗ ) ≥ 0.
(8.13)
¯ i ∈ Xi satisﬁes (c ), it follows from (8.13) and the fact that p∗ ≥ 0 that: Thus, if x ¯ i ≤ p∗ · r i ≤ p∗ · r i + p·x
k=1
sik πk (p∗ ) = wi (p∗ ).
Since p∗ was an arbitrary element of Π∗ , and i was arbitrary, our result follows.
Propositions 8.10 and 8.12 together suggest that we might be able to generalize Theorem 8.1 by replacing hypothesis (c) with (c ). However, this in not the case; in fact consider the somewhat stronger assumption: m m def ¯i r = ¯ i < r i , for i = 1, . . . , m; and x (∃¯ xi ∈ X i ) : x ri . c∗ . i=1
That Theorem 8.1 does not remain correct if strated by the following.2
(c∗ )
i=1
is substituted for (c) is demon
8.13 Example. Consider the economy, E, in which m = n = 2, X1 = X2 = R2+ , and the two preferences can be represented by the utility functions: u1 (x1 ) = min{x11 /2, x12 }, and: u2 (x2 ) = x22 , respectively. Suppose further that: r 1 = (4, 4), r 2 = (0, 4), s1 = s2 = 1/2, & Y = −R2+ . You can easily conﬁrm the fact that E satisﬁes all of the assumptions of Theorem 8.1, except assumption (c), and that it does satisfy assumption (c*).3 We will show that no Walrasian equilibrium exists for E in this case. Accordingly, suppose, by way of obtaining a contradiction, that (x∗i ), y ∗ , p∗ is a Walrasian equilibrium for E. Then, since Y satisﬁes (d.4) and both consumers’ preferences are increasing, it must be the case that p∗ > 0; and we may therefore 2 See also Example 7.28, which provides a bit diﬀerent insight into the role played by assumption (c) in the theorem. 3 And yes, it is essentially a pure exchange economy.
238
Chapter 8. The Existence of Competitive Equilibrium
suppose that p∗ ∈ ∆2 . If we suppose that p∗ 0, then the ﬁrst consumer’s demand for the ﬁrst commodity is given by (verify this): x∗11 =
8 . 2p∗1 + p∗2
Now, given the form of Y , it then follows that we must have: 8 ≤ 4; 2p∗1 + p∗2 and, since p∗2 = 1 − p∗1 , this requires p∗1 = 1, and, correspondingly, p∗2 = 0. However, with p∗2 = 0, the second consumer has unbounded demand for the second commodity! Therefore, no Walrasian equilibrium exists in this case. What the simple analytics of this example brings out in sharp relief is this: given any ﬁnite price ratio of p1 /p2 , the ﬁrst consumer has strictly positive excess demand for the ﬁrst commodity; which means that this consumer’s demand for the ﬁrst commodity exceeds the total quantity available. If, on the other hand, p1 rises to p1 = 1 (thereby making p1 /p2 = +∞), consumer one no longer has positive excess demand for the ﬁrst commodity, but the second consumer now has unbounded demand for the second commodity. On the other hand, if we change the second consumer’s endowment to set r21 > 0,4 then, however small the quantity of the ﬁrst commodity we add to his endowment, consumer one’s excess demand for the ﬁrst commodity can be accommodated with a ﬁnite ratio of p1 /p2 , and a Walrasian equilibrium exists for E (as I will leave you to verify). Insofar as the remaining hypotheses of 8.1 are concerned, our ﬁnal example of this section shows that the convexity of Y cannot be dispensed with. We will not include an example showing that the closure of Y cannot be dispensed with, but such examples are easy to construct. Bergstrom [1976] has shown that the free disposal assumption can actually be dispensed with. (See also Shafer [1976].) 8.14 Example. Let m = = 1, n = 2, and suppose X, Y r, and P are as indicated in Figure 8.8, on the next page. Notice that if p ∈ Π∗ satisﬁes: p∗1 < p1 ≤ 1,
(8.14)
then the producer will maximize proﬁts at y = 0; whereas the consumer’s preferencemaximizing commmodity bundle will not equal r. Hence, no Walrasian equilibrium can exist for any p ∈ Π∗ satisfying (8.14). On the other hand, for p ∈ Π∗ satisfying: 0 ≤ p1 ≤ p∗1 , the producer will maximize proﬁts at some y ∈ Y such that r + y ∈ / X. Therefore, no Walrasian equilibrium exists in this case. This example shows the way in which the competitive pricing system can break down in the presence of nonconvexity. Notice that, from the consumer’s point of view, a best consumption vector exists in X ∗ ; namely at x = x∗ . However, no price vector exists such that the producer will maximize his proﬁts at y = x∗ − r. 4
Thereby satisfying assumption (c).
8.4. The Gale and MasColell Theorem
239
x* y*
X Y
r
p*
Π∗
Figure 8.8: Nonconvexity of the Production Set.
8.4
The Gale and MasColell Theorem
As indicated in the introduction, Theorem 8.1 is a special case of the main theorem in Gale and MasColell [1975]. Their theorem makes use of a wealthassignment function, as per Deﬁnition 7.9, and is stated as follows. 8.15 Theorem. The economy E has a Walrasian equilibrium if: for each i (i = 1, . . . , m): a. Xi is closed, convex, and bounded below, b. Pi is (irreﬂexive and): 1. nonsaturating, 2. weakly convex, and: 3. strongly continuous; c. the feasible wealth assignment function, w : Π ∩ ∆n → Rm is continuous [that is each wi (·) is a continuous realvalued function], and satisﬁes: (∀p ∈ Π∗ ) : wi (p) > min p · X,
(8.15)
Π∗
where = Π ∩ ∆n , and the aggregate production set, Y ≡ k=1 Yk : d.1. is closed and convex, d.2. has a bounded intersection with Rn+ , and: d.3. contains −Rn+ . While I don’t propose to go through the assumptions of this result, showing each assumption cannnot be dispensed with, in the way that we did in connection with Theorem 8.1, a few comments may be in order.
240
Chapter 8. The Existence of Competitive Equilibrium
First of all, if you compare the statements of this and the earlier result, you will see that the diﬀerences occur in assumption (c) and (d) in the respective statements. Assumptions (d.1) and (d.3) of the present theorem are used in the earlier result as well. Moreover, it is obvious that the assumption Y ∩ (−Y ) ⊆ 0, together with assumption (d.3) of the present result, implies that Y has a bounded intersection with Rn+ (in fact, that Y ∩Rn+ ⊆ 0) Consequently, the present assumptions regarding the production sector generalize those in the earlier result. On the other hand, in Example 8.5 we have already shown that the present assumption (d.2) cannot be dispensed with; indeed that it cannot be weakened to: Y ∩ Rn++ = ∅. The relationship between the two diﬀerent assumptions (c) is more intriguing, however. We saw in Section 8.3 that one of the roles played by Assumption (c) in Theorem 8.1 is to guarantee that the set of attainable allocations is nonempty. One can be forgiven a little headscratching over the puzzle of what it is that Gale and MasColell have assumed which guarantees this same condition. However, it is their assumption (c) which does the trick here as well, for notice that if X ∩ Y = ∅,5 then it follows from the ‘separating hyperplane theorem’ (Theorem 6.21) that there exists p† ∈ ∆n such that: sup p† · y ≤ inf p† · x. (8.16) y∈Y
x∈X
However, since w(·) is required to be a feasible wealth assignment function, we must have: m wi (p† ) = π(p† ) = sup p† · y. i=1
y∈Y
Combining this with (8.16), we see that we must have: m wi (p† ) ≤ inf p · x; i=1
x∈X
which clearly contradicts equation (8.15) in the statement of the Gale and MasColell theorem. Consequently, we see that if E satisﬁes the present assumption (c), then we must have X ∩ Y = ∅. We saw in Example 8.13 that we cannot weaken the present assumption (c) by replacing equation (8.15) with: (∀p ∈ Π∗ ) : wi (p) ≥ min p · X i ,
(8.17)
Our next example, with which we will close this section, shows that we cannot dispense with the assumption that w(·) is continuous. 8.16 Example. Let E be an economy in which m = n = 2, = 1, Xi = R2+ , for i =, 2, and: Y = {y ∈ R2  y ≤ (6, 6)}. 5 We will follow Gale and MasColell in dispensing with aggregate resource endowment, r, in this discussion. One can, of course, allow for such an endowment by deﬁning an ‘extra’ production set, Y0 = {r}.
8.5. An (Especially) Simple Existence Theorem
241
We suppose also that the consumers’ preferences can be represented by the utility functions: u1 (x1 ) = (x11 )1/3 · (x12 )2/3 , and u2 (x2 ) = (x21 )2/3 · (x22 )1/3 , respectively. Since we know that we can, without loss of generality, conﬁne our attention to price vectors p ∈ ∆2 , we can deﬁne the wealth function to be used as a function of p1 alone (implicitly assuming that p2 = 1 − p1 ). We then we suppose that the wealthassignment function, w : [0, 1] → R2+ , is given by: 4 for 1 ≥ p1 ≥ 1/2, w1 (p1 ) = 2 for 1/2 > p1 ≥ 0; and:
2 for 1 ≥ p1 ≥ 1/2, w2 (p1 ) = 4 for 1/2 > p1 ≥ 0.
It is easy to show that w(·) is a feasible wealthassignment function for E, and it satisﬁes: (∀p1 ∈ [0, 1]) : wi (p1 ) > min p · X i = 0, for i = 1, 2. However, for this wealthassignment function, aggregate demand for the ﬁrst commodity, given p1 ≥ 1/2, satisﬁes: 4/3 + (2/3) · 2 1 δ(p1 ) ≤ δ(p1 ) = = 16/3 = 5 ; 1/2 3 while for 0 < p1 < 1/2:
2 δ(p1 ) > lim δ(p1 ) = 6 . 3 p1 1/2
Consequently, no Walrasian equilibrium exists in this case.
8.5
An (Especially) Simple Existence Theorem
In this section we will study a very simple theorem establishing the existence of a Walrasian equilibrium. While the result assumes a very special case of a private ownership economy, it incorporates one which is often used in the Public Economics literature (although often only implicitly); and, I believe that working through the proof of the result which we are going to study may help you to attain some valuable insights into the meaning and nature of a general competitive equilibrium. The basic model which we are going to be studying here builds upon the model presented at the end of Section 7.3, in that we take the consumers’ consumption sets to be a subset of the nonnegative orthant of the commodity space, and suppose that only the initial endowments of leisure are positive. We will deal with a private ownership economy in which there are m consumers and n+1 commodities, with the 0th commodity being leisure/labor. The production
242
Chapter 8. The Existence of Competitive Equilibrium
sector will be characterized by an aggegate Leontief technology, as described in Section 6.4. Thus, the aggregate production set can be expressed as: ! −c n+1 n z , (8.18) Y = y∈R (∃z ∈ R+ ) : y = I −A where A is an n × n semipositive matrix, and c is a strictly positive nvector. We suppose that each consumer’s initial endowment takes the form: r i = (ri0 , 0), where ri0 > 0 and ‘0’ denotes the origin in Rn , and we will use the generic notation ‘(xi0 , xi )’ to denote the ith consumer’s commodity bundle. Using this notation, we suppose that the ith consumer’s consumption set is given by: Xi = {(xi0 , xi ) ∈ Rn+1  0 ≤ xi0 ≤ ri0 }; + and, given (xi0 , xi ) ∈ Xi , the
ith
(8.19)
consumer’s labor oﬀer is: i = xi0 − ri0 .
We will suppose that leisure is a num´ eraire for E, a condition we deﬁne as follows; for each i ∈ M , each (xi0 , xi ) ∈ Xi and each ∆x0 ∈ R++ , we have: 0 ≤ xi0 + ∆x0 < ri0 ⇒ (xi0 + ∆x0 , xi )Pi (xi0 , xi ).
(8.20)
The deﬁnitions of feasible allocations and competitive equilibria adapt easily to this context, although it is easier and more natural to identify the production level with the nvector z than with the (n + 1)vector y. Accordingly, we will say that an allocation, (xi0 , xi ), z is feasible for E iﬀ: (xi0 , xi ) ∈ Xi , for i = 1, . . . , m; z ∈ Rn+ , and:
m i=1
i xi
! =
! −c z; I −A
that is: m i=1
(xi0 − ri0 ) ≡
m i=1
i = −c · z and
m i=1
xi = I − A z.
Turning now to the issue of deﬁning a Walrasian equilibrium for this economy, let me begin by noting that we will always normalize to set the price of leisure (the wage rate) equal to one, so that prices are completely determined by the nvector, p ∈ Rn+ , consisting of the prices of the n produced goods. Thus, wewill say that a tuple (x∗i0 , x∗i ), z ∗ , p∗ is a Walrasian equilibrium for E iﬀ (a) (x∗i0 , x∗i ), z ∗ is feasible for E, (b) zj∗ maximizes proﬁts in the j th sector, given p∗ , and (c) for each i ∈ M , (x∗i0 , x∗i ) maximizes Pi , given: x∗i0 + p∗ · x∗i ≤ ri0
(8.21)
8.5. An (Especially) Simple Existence Theorem
243
Since the production technology is linear, there will be a proﬁtmaximizing output in sector j only if there is zero proﬁt in producing commodity j; and in order that a nonzero net output of commodity j be produced, it is necessary that its price be equal to the unit cost of production; that is, p∗ must satisfy p∗j zj − C(zj ) = 0, so that:
n n p∗j akj zj − cj zj = p∗j − p∗j akj − cj zj = 0 (8.22) p∗j zj − k=1
k=1
With these considerations in mind, we can turn to our existence theorem. 8.17 Theorem. Suppose the private ownership economy, E = Xi , Pi , r i , Y , satisﬁes the following conditions: for each i (i = 1, . . . , m);  0 ≤ xi0 ≤ ri0 }, a. Xi = {(xi0 , xi ) ∈ Rn+1 + b. Pi is: 1. asymmetric, 2. transitive, 3. locally nonsaturating, and: 4. upper semicontinuous; c. r i is of the form; r i = (ri0 , 0), where ri0 > 0, d. the aggregate production set takes the form set out in (8.18), where c 0, and the matrix A is semipositive, and satisﬁes: n akj < 1 for j = 1, . . . , n, (8.23) k=1
and: e. leisure is a num´eraire for E. Then E has a Walrasian equilibrium.. Proof. Since A satisﬁes (8.23), it follows from Theorem 6.17 that there exists a unique vector p∗ ∈ Rn+ satisfying: (p∗ ) (I − A) = c .
(8.24)
Moreover, since c 0, we must have p∗ 0 as well, for we can write the j th equation in (8.24 as: n p∗j − p∗ akj = cj ; k=1
and, since akj ≥ 0 for all k, j, it follows that we must have p∗j > 0. Since p∗ 0, it now follows from Theorem 4.5 that, for each i, there exists (x∗i0 , x∗i ) ∈ Xi satisfying: x∗i0 + p∗ · x∗i = ri0 , and, for all (xi0 , xi ) ∈ Xi : (xi0 , xi )Pi (x∗i0 , x∗i ) ⇒ xi0 + p∗ · xi > ri0 .
(8.25)
244
Chapter 8. The Existence of Competitive Equilibrium
Notice also that it follows from (8.22) and Theorem 6.17 that, deﬁning x∗ ∈ Rn+ by: x∗ =
m i=1
x∗i ,
there exists z ∗ ∈ Rn+ satisfying: x∗ = (I − A)z ∗ .
(8.26)
Now, it follows from (8.25) that: m i=1
p∗ · x∗i = p∗ · x∗ =
m i=1
(ri0 − x∗i0 ) = −
m
∗ . i=1 i
(8.27)
On the other hand, from the deﬁnition of z ∗ and (8.24), we have: p∗ · x∗ = (p∗ ) (I − A)z ∗ = c · z ∗ .
(8.28)
Combining (8.26)–(8.28) with the fact that, for each i ∈ M , (x∗i0 , x∗i ) ∈ Xi , we see that (x∗i0 , x∗i ), z ∗ is feasible for E. Finally, we note that for each j: p∗j −
n
p∗ akj k=1 k
− cj = 0,
so that proﬁts in the j th sector are zero (and thus maximized) when zj = zj∗ . There fore (x∗i0 , x∗i ), z ∗ , p∗ is a Walrasian equilibrium for E. It is worth noting that if we strengthen the hypotheses of this existence theorem by requiring that, in addition to the hypotheses of Theorem 8.17, each Pi is negatively transitive and strictly convex, then the Walrasian equilibrium established in our proof is unique. In the special case of the model used here which is often used in public economics literature, the aggregate production set takes the form: Y = {y ∈ Rn  c · y ≤ 0 & yj ≥ 0, for j = 1, . . . , n − 1}. See Exercise 7, at the end of this chapter.
8.6
Appendix
Making use of the distinction introduced in Section 1, we have been showing that no Walrasian equilibrium exists for the economies in the examples presented in this chapter. This raises the question of whether we might have been able to ﬁnd a competitive (free disposal) equilibrium in some cases. However, recall that in most of our examples, Y satisﬁed the semifree disposability condition: if y, y ∈ Rn are such that y ∈ Y and y ≤ y, then y ∈ Y ; and, in addition, in nearly all of our examples, we assumed 0 ∈ Y , so that: −Rn+ ⊆ Y.
(8.29)
8.6. Appendix
245
Under these conditions, suppose (x∗i , y ∗k , p∗ ) is a competitive equilibrium for the economy, but that: m x∗i < r + y ∗k . (8.30) i=1
k=1
From (8.29) and Proposition 7.8 of Chapter 7, it follows that we must have p∗ > 0. However, if in fact, p∗ 0, then from (8.30) we have: m y ∗k − x∗i > 0; p∗ · r + k=1
i=1
contradicting the deﬁnition of a competitive equilibrium. If, on the other hand, p∗ > 0, but one or more p∗i = 0, then it will typically be the case that the consumer(s) could not be maximizing preferences at x∗i . The example in which it may seem most likely that a competitive (free disposal) equilibrium, as opposed to a Walrasian equilibrium, may exist is Example 8.7. If you return to Figure 8.5, you can easily verify the fact that we can ﬁnd a price vector such that the producer maximizes proﬁts at a point y satisfying: r + y ≥ x∗ .
(8.31)
However, notice that the only price vector yielding this relationship (that is, the only one in ∆2 ) is p∗ ; and, since p∗ 0, any y ∈ Y and satisfying equation (8.31) is such that: p∗ · (r + y − x∗ ) > 0. Therefore, no competitive equilibrium exists in this case either.
Exercises. 1. Suppose there are two commodities, and a consumer has the consumption set: Xi = {xi ∈ R2  −2 ≤ xi1 & xi2 ≥ 2}. Answer the following questions. a. Show that the consumer’s demand correspondence can be deﬁned only for pairs (p, w) ∈ R3+ satisfying w ≥ µ(p), where: µ(p) = 2(p2 − p1 ), for p ∈ R2++ . b. Given that the consumer’s preferences can be represented by the utility function ui (xi ) = xi1 , ﬁnd the consumer’s demand correspondence. c. Given that the consumer’s preferences can be represented by the utility function ui (xi ) = xi2 , ﬁnd the consumer’s demand corresponcence. d. Consider the private ownership economy, E, in which we have one producer, two consumers and two commodities, and where Xi , r i , si , and Y are given by: Xi ={xi ∈ R2  −2 ≤ xi1 & xi2 ≥ 2} and r i = (0, 1) s1 =s2 = 1/2, and Y ={y ∈ R2  y1 ≤ 0 ≤ y2 & y1 + y2 ≤ 0},
for i = 1, 2;
246
Chapter 8. The Existence of Competitive Equilibrium
respectively; and suppose that the consumers’ preferences can be represented by the utility functions: u1 (x1 ) = x11 and u2 (x2 ) = x22 , respectively. Find the (a) competitive equilibrium for E, if one exists, or show that no competitive equilibrium exists for E in this case; and (b) answer the following question: Are all of the assumptions of Theorem 8.1 satisﬁed by the economy presented in this problem? If not, which assumptions are violated? 2. Follow the same sequence of questions as in problem 1 for the private ownership economy, E having one consumer, one producer, two commodities, and where X, r and Y are given by: X ={x ∈ R2  −2 ≤ x1 & 2 ≤ x2 }, r = (0, 1), and Y ={y ∈ R2  y1 ≤ 0 ≤ y2 & y1 + 2y2 ≤ 0}, and where the consumer’s preferences can be represented by the utility function: u(x) = min{4 + x1 , x2 }. 3. Consider the private ownership economy, E, in which we have one consumer, one producer, two commodities, and in which X, r, and Y are given by: X ={x ∈ R2  −4 ≤ x1 & x2 ≥ 4}, r =(0, 2), and Y ={y ∈ R2  y1 ≤ 0 & 3y1 + y2 ≤ 0}, respectively; and suppose the consumer’s preferences can be represented by the utility function: u(x) = min{2x1 + 12, x2 }. On the basis of this information, a. Find the consumer’s demand function (correspondence) as a function of p ∈ R2++ and w ≥ 0. b. Find the (or a) Walrasian equilibrium for this economy, or show that no Walrasian equilibrium exists in this case. 4. Consider the private ownership economy, E, in which we have one consumer, one producer, two commodities, and where X, r, and Y are given by: X =R2+ , r =(24, 0), and Y ={y ∈ R2  y1 ≤ 0 & y1 + y2 ≤ 0}, respectively; and suppose the consumer’s preferences can be represented by the utility function: u(x) = (x1 )2 · (x2 ).
8.6. Appendix
247
On the basis of this information, answer the following questions. a. Find the consumer’s demand function (correspondence) as a function of p ∈ R2++ and w ≥ 0. b. Find the (or a) Walrasian equilibrium for this economy. c. If we interpret the ﬁrst commodity as the consumer’s leisure, how much labor is being oﬀered in the Walrasian equilibrium? 5. Here we will consider a case in which we have two consumers, two commodities, and one producer. We will depart from our usual notation to denote quantities of the ﬁrst commodity by ‘x’ (interpreted as ’leisure’), and the second (produced) commodity by ‘y.’ We will suppose the two consumers’ preferences can be represented by the utility functions: u1 (x1 , y1 ) = (x1 )1/4 · (y1 )3/4 and u2 (x2 , y2 ) = (x2 )3/4 · (y2 )1/4 , respectively; and have the initial endowments: r i = (24, 0), for i = 1, 2. Finally, we suppose that the producer’s production function is given by: √ y = 2 −z, where ‘z’ denotes the aggregate labor supplied by the consumers, and we suppose that the consumers’ shares of ownership in the ﬁrm are given by: si = 1/2, for i = 1, 2. Given this information, ﬁnd the competitive equilbrium for this economy, or show that none exists. 6. In this question, we will be considering a twoconsumer, twocommodity economy, E, in which the consumers’ preferences can be represented by the utility functions: ui (xi ) = xi1 + xi2 for i = 1, 2, with the initial endowments: r 1 = (16, 4) and r 2 = (16, 0), respectively. On the basis of this information, answer the following two questions. a. Suppose Y = {0}; that is, that this is a pure exchange economy. Find the (or the set of) Walrasian equilibrium (or equilibria) in this case, or show that no such equilibrium exists. b. Now suppose there is one producer, whose production set is given by: Y = {y ∈ R2  y1 ≤ 0 & y1 + 2y2 ≤ 0}, and that the shares of ownership in the ﬁrm are given by: s1 = s2 = 1/2.
248
Chapter 8. The Existence of Competitive Equilibrium
Find the (or the set of) Walrasian equilibrium (or equilibria) in this case, or show that no such equilibrium exists. 7. Show that the production set Y deﬁned by: Y = {y ∈ Rn  c · y ≤ 0 & yj ≥ 0, for j = 1, . . . , n − 1}, where c ∈ Rn++ , is a special case of the aggregate production set speciﬁed in Theorem 8.17.
Chapter 9
Examples of General Equilibrium Analyses 9.1
Introduction
In this chapter, we are going to consider some applications of general equilibrium theory to policy analysis. While our treatment here will stop far short of the current frontiers of the related policy analyses, I hope that it will illustrate some of the ﬂavor of such analysis, and the usefulness of general equilibrium models therein. We will begin by presenting the elements of the basic theory of ‘optimal taxation.’ We will ﬁrst take up the theory of ‘optimal commodity taxation,’ and we will then take an even briefer look at the theory of ‘optimal income taxation.’ While neither discussion will take us very far toward the current frontier of research in the respective ﬁelds, the analysis to follow should provide a bit of insight into the role of general equilibrium theory in current policy analysis. Moreover, in the discussion of optimal income taxation, we will for the ﬁrst time in this book encounter the problem of ‘incentive compatibility,’ an issue which will play a key role in much of our work in Chapters 16–18. After our consideration of optimal taxation, we will examine some extensive examples incorporating monopoly, and then money in a general equilibrium model. We then conclude the chapter with an example incorporating indivisible commodities into a general equilibrium model.
9.2
Optimal Commodity Taxation: Initial Formulation
The problem which we will be examining initially is the ‘eﬃciency aspect’ of optimal commodity taxation. For this, it is customary to suppose that there is only one consumer, or that the consumption sector of the economy as a whole behaves as if it were a single consumer. In principle, this enables us to separate the eﬃciency aspect from equity considerations, which occur because there is more than one consumer in the economy. We will ﬁrst set out the standard ‘text book model’ used in this literature, we will then analyze the workings of this model in the simplest case possible, and ﬁnally, we will set out a few of the conclusions reached in this literature.
250
Chapter 9. Examples of General Equilibrium Analyses
The standard ‘text book model’ assumes one consumer, n + 1 commodities (n produced commodities plus labor), and constant returns to scale in production, with each commodity produced with the use of labor alone. In each industry, it is assumed that the inputrequirement function is given by: j = −cj yj ,
(9.1) j th
commodity, and where cj > 0, for each j. Consequently, given the price pj for the a wage w for labor, proﬁtmaximization at nonzero production will require that: pj yj + w j = pj yj − wcj yj = (pj − wcj )yj ≡ 0;
(9.2)
and thus, normalizing to set w = 1, we must have: pj = cj
for j = 1, . . . , n
(9.3)
Moreover, for future reference, notice that the aggregate production set is here given by: (9.4) Y = {(y0 , y) ∈ R1+n  y ∈ Rn+ & y0 + c · y ≤ 0} (compare Example 6.27 of Chapter 6); and notice that, if (y0 , y) ∈ Y , and (w, p) satisﬁes (9.3) (and with w = 1), we have: π(y0 , y) = (w, p) · (y0 , y) = wy0 + p · y = y0 + c · y ≤ 0. Now, the consumer can (and will) pay a positive tax on the j th commodity, and we will denote the price paid by the consumer by ‘qj ,’ where: q j = p j + tj
for j = 1, . . . , n,
and where tj is the tax levied on the j th commodity, for each j. Denoting the consumer’s consumption bundle, generically, by ‘(x0 , x),’ where 0 ≤ x0 ≤ r and x ∈ Rn+ , with r > 0 representing the consumer’s endowment of leisure, and: = x0 − r,
(9.5)
denoting the consumer’s oﬀer of labor; the tax revenue raised by the government, given the tax vector t ∈ Rn+ , is given by: n tj xj . (9.6) T =t·x= j=1
Notice that our formulation allows some coordinates of t to be negative; on the other hand, the problem becomes analytically somewhat more tractable in some ways if we require that: tj ≥ 0 for j = 1, . . . , n. However, we will ignore this complication for the moment; coming back to it at the end of the next section. Initially, we will suppose that the government needs to raise a given amount, R, of funds; and thus we require that: T = t · x ≥ R.
(9.7)
9.2. Optimal Commodity Taxation: Initial Formulation
251
However, we will also sometimes specialize this requirement to assume that the government uses the tax revenue to purchase a commodity bundle, xg ∈ Rn+ (which may be used as an input in the production of ‘governmental services’); in which case the government’s budget constraint becomes: T = t · x ≥ p · xg .
(9.8)
We suppose the (representative) consumer maximizes a utility function, u(x0 , x) subject to the budget constraint: q · x + wx0 = wr; or, equivalently: q · x = w(r − x0 ) = −w ; which, since we are normalizing throughout with w = 1, becomes: q · x = (r − x0 ) = − .
(9.9)
In our treatment, we will suppose that for each vector of commodity prices, q ∈ Rn++ ,1 there exists a unique utilitymaximizing bundle, h0 (q), h(q) .2 Finally, we denote the consumer’s indirect utility function by ‘v(q);’ and we note that in this case, we can take v(·) to be given by: v(q) = u h0 (q), h(q) . Notice that there is no income term in our expression for the consumer’s demand function and indirect utility function. This is because in our analysis of optimal commodity taxation, we must necessarily take the consumer’s nonlabor income to be zero; proﬁts from the production sector are necessarily zero because of our linearity assumption regarding production, and we are assuming a closed general equilibrium system. Given our assumptions about the consumer, however, we can certainly deﬁne a demand function which allows income to vary, and an indirect utility function which treats income as an independent variable as well. We will denote these functions by ‘ H0 (q, I), H(q, I) ’ and ‘V (q, I),’ respectively (where ‘I’ denotes nonlabor income). We then have the following relationships: for any q ∈ Rn++ : (9.10) H0 (q, 0), H(q, 0) = h0 (q), h(q) and V (q, 0) = v(q). The usual formulation of the optimal commodity taxation problem is then to maximize the consumer’s utility (with respect to t), given R. Thus we can formulate the problem as: max u h0 (p + t), h(p + t) subject to t · h(p + t) ≥ R; w.r.t. t
or, equivalently: max v(p + t) subject to t · h(p + t) ≥ R
w.r.t. t
(9.11)
While this is the normal statement of the problem which is used in the literature, the ﬁrst question which I want to investigate here is ‘what happened to the production constraint?’ We will pursue an answer to this question in the next section. 1 2
Because of our normalization, we are suppressing the variable w throughout our treatment. In other words, we suppose that the consumer’s demand correspondence is a function.
252
9.3
Chapter 9. Examples of General Equilibrium Analyses
A Reconsideration of the Problem
In the previous sections of this chapter we have been considering a oneconsumer, oneproducer economy with n + 1 commodities; in this section we will generalize this model to the extent of allowing an arbitrary ﬁnite number, m, of consumers. Thus we consider a private ownership economy: E = R1+n + , Gi , r i , Y , where Gi is a continuous weak order on Rn+1 + , and r i is of the form: r i = ri0 , 0), where ri0 > 0. We shall also suppose that each Gi is strictly convex; so that, 1+n ∗ ∗ satisfying for each (w, q) ∈ R1+n ++ , there exists a unique vector (xi0 , xi ) ∈ R+ ∗ ∗ wxi0 + q · xi ≤ wri0 and: ∀(x0 , x) ∈ R1+n : (x0 , x)Pi (x∗i0 , x∗i ) ⇒ wx0 + q · x > w∗ ri0 . + Continuing as per the discussion in the previous section, we will always normalize to set w = 1; and, given q ∈ Rn++ , we denote the values of x∗i0 and x∗i which satisfy the above conditions (with w = 1) by ‘hi0 (q)’ and ‘hi (q),’ respectively. 9.1 Deﬁnition. We shall say that (x∗i0 , x∗i ), (y0∗ , y ∗ ) is feasible for E, given g n the governmental demand x ∈ R+ , iﬀ: . . . , n, (y0∗ , y ∗ ) ∈ Y , 1. (x∗i0 , x∗i ) ∈ R1+n + , for i = 1, m ∗ − r ) = y ∗ and ∗ g ∗ (x 2. m i0 0 i=1 i0 i=1 xi + x = y . We then make use of this deﬁnition of feasibility to introduce the following equilibrium concept; where, as in the previous section, we normalize to set wages (the price of the 0th commodity) equal to one; and where we say that a pair (t, xg ) ∈ R2n is a level of governmental activity in E iﬀ xg ∈ Rn+ and t ∈ Rn . 9.2 Deﬁnition. If xg ∈ Rn+ and t ∈ Rn , we shall say that (x∗i0 , x∗i ), (y0∗ , y ∗ ), p∗ is a competitive equilibrium for E, given the governmental activity (t, xg ), iﬀ: def 1. p∗ ∈ Rn+ \ {0} and q ∗ = p∗ + t ∈ Rn++ , ∗ ∗ ∗ ∗ 2. (xi0 , xi ), (y0 , y ) is feasible for E, given xg , ∗ proﬁts on 3. (y0∗ , y ∗ ) maximizes Y , given (1, p ), ∗ ∗ ∗ ∗ for i = 1, . . . , m, and 4. (xi0 , xi ) = hi0 (q ), hi (q ) , ∗ 5. t · x∗ ≥ p∗ · xg , where x∗ = m i=1 xi . 9.3 Proposition. Suppose that Y is linear; that is, that there exists c ∈ Rn+ \ {0} such that: (9.12) Y = {(y0 , y) ∈ R1+n  y ∈ Rn+ & y0 + c · y ≤ 0}, and that (x∗i0 , x∗i ) and the level of governmental activity (t, xg ) ∈ R2n satisfy: c + t ∈ Rn++ , (x∗i0 , x∗i ) = hi0 (c + t), hi (c + t) for each i and t · x∗ = c · xg .
(9.13) (9.14)
9.3. A Reconsideration of the Problem
253
Then, deﬁning: p∗ = c, ∗
y = x∗ + xg , j = −cj yj∗
for j = 1, . . . , n, and n j ; y0∗ = j=1
(9.15) (9.16) (9.17) (9.18)
(x∗i0 , x∗i ), (y0∗ , y ∗ ), p∗ is a competitive equilibrium for E, given the level of governg mental activity (t, x ).
Proof. If (x∗i0 , x∗i ) and the level of governmental activity (t, xg ) ∈ R2n satisfy (9.13) and (9.14), and we deﬁne p∗ = c and q ∗ = p∗ + t, it follows immediately that (x∗0 , x∗ ), (y0∗ , y ∗ ), p∗ and (t, xg ) satisfy conditions 1, 4, and 5 of Deﬁnition 9.2. Furthermore, deﬁning (y0∗ , y ∗ ) as in (9.16) and (9.18), we have: n j + c · y ∗ = −c · y ∗ + c · y ∗ = 0; y0∗ + c · y ∗ = j=1
so that (y0 , y) ∈ Y . Moreover, since w = 1 and p∗ = c, we have: wy0∗ + p∗ · y ∗ = y0∗ + c · y ∗ = 0; ∗ proﬁts over and it also follows that (y0∗ , y ∗ ) maximizes Y , given (w, p ) = (1, c). Now, since, for each i, (x∗i0 , x∗i ) = hi0 (q ∗ ), hi (q ∗ ) , we have:
x∗i0 = ri0 − q ∗ · x∗i
for i = 1, . . . , m.
Therefore: m i=1
(x∗i0 − ri0 ) = −
i=1
q ∗ · x∗i = −q ∗ · x∗ = −c · x∗ − t · x∗
= −c · x∗ − c · xg = −c · y ∗ = y0∗ ; and it now follows that (x∗i0 , x∗i ), (y0∗ , y ∗ ), p∗ satisﬁes condition 2 of Deﬁnition 9.2.
Our next result is stated in a bit more generality than is needed; that is, it is somewhat more general than the context in which we have been working. In it, we will drop our assumption that Y is linear; which means in turn that we will have to allow for the possibility that the ﬁrm may make a proﬁt. As usual, we will denote the maximum proﬁt achievable in Y , given a price vector, p ∈ Rn+ by ‘π(p),’ and denote the ith consumer’s share of these proﬁts by ‘si .’ In our model, this proﬁt becomes the nonlabor income that we considered at the end of the previous section; and, in terms of the notation introduced there, the ith consumer’s preferencemaximizing commodity bundle at consumer prices q ∈ Rn++ and producer prices p ∈ Rn+ will be given by: (xi0 , xi ) = Hi0 [q, si π(p)], H i [q, si π(p)] . This is the notation utilized in our next result.
254
Chapter 9. Examples of General Equilibrium Analyses
9.4 Proposition. Suppose (x∗i0 , x∗i ), (y0∗ , y ∗ ), p∗ and the level of government acg tivity (t, x ) are such that: def
∗ n ∗ ∗ n 1. p ∈ R+ \ {0} and q = p + t ∈ R++ , 2. (x∗i0 , x∗i ), (y0∗ , y ∗ ) is feasible for E, given xg , proﬁts on Y , given p∗ (and 3. (y0∗ , y ∗ ) maximizes w = 1), 4. (x∗i0 , x∗i ) = Hi0 [q ∗ , si π(p∗ )], H i [q ∗ , si π(p∗ )] , for i = 1, . . . , m. Then t · x∗ = p∗ · xg , and (x∗i0 , x∗i ), (y0∗ , y ∗ ), p∗ is a competitive equilibrium for E, given the level of governmental activity (t, xg ).
Proof. Upon rechecking Deﬁnition 9.2, we see that it suﬃces to prove that t · x∗ = p∗ · xg . To establish this equality, we make use of the deﬁnitions and conditions 1–4 to obtain: t · x∗ = (q ∗ − p∗ ) · x∗ = q ∗ · x∗ − p∗ · (y ∗ − xg ) m = q ∗ · x∗ − p∗ · y ∗ + p∗ · xg = ri0 − x∗i0 + si π(p∗ ) − p∗ · y ∗ + p∗ · xg i=1
= −y0∗ + [y0∗ + p∗ · y ∗ ] − p∗ · y ∗ + p∗ · xg = p∗ · xg .
Returning to the one consumer case, the thrust of the above two results is that we can equivalently formulate our optimal commodity taxation problem either as (going back to the assumption that Y is linear [that is, satisﬁes (9.12)], and setting p = c): max v(p + t) subject to t · h(p + t) = c · xg (and p + t ∈ Rn++ );
w.r.t. t
(9.19)
or as: max v(c + t) subject to: h0 (q) − r, h(q) + xg ∈ σ(c) (and q ≡ c + t ∈ Rn++ ); (9.20)
w.r.t. t
which, as is easily veriﬁed, given the form of Y , is equivalent to: max v(c+t) subject to: h0 (q)−r +c·[h(q)+xg ] = 0 (& q ≡ c+t ∈ Rn++ ). (9.21)
w.r.t. t
Formally, it will sometimes be more convenient to state the problem as follows: max v(c + t)
(9.22)
x0 − r + c · [x + xg ] = 0, (x0 , x) = h0 (c + t), h(c + t) , and:
(9.23)
q ≡ c + t ∈ Rn++ .
(9.25)
w.r.t. t
subject to:
(9.24)
It should be apparent that the problem stated in equations (9.22)–(9.25) is equivalent to both (9.20) and (9.21). On the other hand, the longer formulation perhaps makes some aspects of the problem stand out in sharper relief. In particular, consider
9.4. The Simplest Model of Optimal Commodity Taxation
255
the last constraint, (9.25). Since we are assuming that c 0, it will automatically be satisﬁed if we require that: t ≥ 0. (9.26) Moreover, if we substitute (9.26) for (9.25), the constraint set becomes closed (in fact, it is also bounded, and thus is compact); and therefore a solution will necessarily exist. On the other hand, using (9.25), rather than (9.26) allows us to characterize the solution via calculus, which is done in most of the literature on optimal commodity taxation. We will ﬁnd the long statement of the problem useful in our discussion of the next section as well.
9.4
The Simplest Model of Optimal Commodity Taxation
In this section, we will examine the simplest special case of the optimal commodity taxation problem formulated in the two previous sections; one in which we have one consumer, one produced commodity and labor, and where the commodity is produced under conditions of constant returns to scale. The consumer will choose a bundle, (x0 , x) ∈ R2+ , where: ‘x0 ’ denotes the quantity of leisure, and ‘x’ denotes the quantity of the produced good, chosen by the consumer. We then denote the initial endowment of leisure by ‘r;’ and the quantity of labor supplied by the consumer is given by: = x0 − r (thus the quantity of labor supplied is given by a negative number). The production set will be given by: Y = {y ∈ R2  0 ≤ y1 ≤ 0 & y0 + cy1 ≤ 0}, where c > 0 is a constant. Thus the ‘production function’ is given by: y1 = −y0 /c, for y0 ∈ R− . Now, in fact, we will always be concerned with allocations in which: y0 = = x0 − r; and hereafter we will write production vectors as a pair ( , y), where ∈ R− , and y ∈ R+ denotes the quantity of the good being produced. Correspondingly, we can write the production set as: Y = {( , y) ∈ R2  0 ≤ y ≤ − /c} = {( , y) ∈ R2  y ≥ 0 & + cy ≤ 0}. We will denote the vector of prices faced by the producer by ‘(w, p),’ where (w, p) ∈ R2+ ; and we will normalize to set w = 1. The price vector faced by the consumer will
256
Chapter 9. Examples of General Equilibrium Analyses
x r/c  xg
(Displ aced) P roduction Function: x = − (1/c)(x0 − r) − xg Offer Curve
x*
x0 + qx = r
x*0
r
x0
x g
Figure 9.1: Basic Solution. be denoted by q = (q1 , q2 ); although in the present analysis we will be supposing that q1 = w = 1, so that we can (and will) denote the vector of prices faced by the consumer by ‘(1, q).’ As before, we suppose that the government is to purchase an amount xg of the produced good, and will be levying a tax, t ∈ R+ in such a way that: pxg = tx = (q − p)x. The basic diagrammatic solution to the problem: max v(c + t),
(9.27)
x0 − r + c · (x + xg ) = 0, (x0 , x) = h0 (c + t), h(c + t) , and:
(9.28)
c + t > 0,
(9.30)
w.r.t. t
subject to: (9.29)
is indicated in the diagram above.
9.5
Some Results
In this section we will derive two of the more famous results concerning optimal commodity taxation in the oneconsumer (eﬃciency) case (for a development of
9.5. Some Results
257
[equity] results for the multipleconsumer case, see Myles [1995, pp. 108–14[). We will suppose throughout that the consumer’s indirect utility function and demand functions are all continuously diﬀerentiable. We will say that t∗ ∈ Rn is an optimal tax vector if t∗ solves the problem stated in (9.22)–(9.25), above. 9.5 Proposition. (DiamondMirlees [1971]) If t∗ is an optimal tax vector, then there exists β ∈ R such that: n
∂hj ∗ t = −βhi (p + t∗ ) ∂qi j
j=1
for i = 1, . . . , n;
(9.31)
where the partial derivatives are evaluated at p + t∗ . Proof. Since t∗ is optimal, it solves the problem stated as equation (9.19) in the previous section; or, equivalently, recalling the discussion in Section 1 of this chapter: max V (p + t, 0) subject to t · h(p + t) = p · xg (and p + t ∈ Rn++ ).
w.r.t. t
Forming the appropriate Lagrangian: ϕ(t, λ) = V (p + t, 0) + λ p · xg − t · h(p + t) , and setting the partial derivatives equal to zero, we have:
n ∂hj ∂V − λ hi (p + t∗ ) + tj = 0 for i = 1, . . . , n, j=1 ∂qi ∂qi with all partial derivatives evaluated at (p + t∗ , 0); from which we obtain: λ·
n j=1
tj
∂hj ∂V = λhi (p + t∗ ) − ∂qi ∂qi
for i = 1, . . . , n.
(9.32)
However, by the AntonelliAllenRoy conditions (see Theorem 4.28 of Chapter 4) we have, evaluating all partial derivatives at (p + t∗ , 0): ∂V ∂V hi (q) =− ∂qi ∂I
for i = 1, . . . , n.
(9.33)
Deﬁning: α=
∂V , ∂I
and substituting into (9.32), we obtain: λ·
n j=1
tj
∂hj = λhi (p + t∗ ) + α · hi (p + t∗ ); ∂qi
or, deﬁning: β = (λ + α)/λ, we obtain:
n j=1
tj
∂hj = β · hi (p + t∗ ) ∂qi
for i = 1, . . . , n.
(9.34)
258
Chapter 9. Examples of General Equilibrium Analyses In our next result, we make use of the compensated demand functions: gj (q, u)
for j = 1, . . . , n,
and we recall that: def
Sij (q, u) =
∂Hj def ∂hj ∂hi ∂Hi + hj (q) + hi (q) = Sji (q, u) = ∂qj ∂I ∂qi ∂I
for i, j = 1, . . . , n; (9.35)
and that, if we deﬁne u∗ = V (c + t∗ , 0), we must have: gj (c + t∗ , u∗ ) = hj (c + t∗ ) 9.6 Proposition. (Ramsey [1927]) If a real number, θ, such that: n
∂gj t∗ j=1 j ∂qi
t∗
for j = 1, . . . , n.
(9.36)
is an optimal tax vector, then there exists
= −θ · gi (c + t∗ , u∗ )
for i = 1, . . . , n;
(9.37)
where all partial derivatives are evaluated at c + t∗ . Moreover, we have θT ∗ ≥ 0; where T ∗ = t∗ · h(c + t∗ ) = t∗ · g(c + t∗ , u∗ ). Proof. If we substitute from (9.35) and (9.36) into (9.34), we obtain: n
t∗ j=1 j
Sji −gi (c+t∗ , u∗ )
∂Hj = −(1+α/λ)gi (c+t∗ , u∗ ) ∂I
Rearranging, we have:
n n t∗j Sji = − 1 + α/λ +
∂Hj t∗ j=1 j ∂I
j=1
which, deﬁning: θ = 1 + α/λ +
gi (c + t∗ , u∗ )
fori = 1, . . . , n. (9.38)
for i = 1, . . . , n; (9.39)
n
∂Hj t∗ ; j=1 j ∂I
establishes the ﬁrst part of our result. To prove the ‘moreover’ portion of our result, we begin by deﬁning t∗0 = 0, to obtain from (9.39), making use of the symmetry of the Slutsky matrix, [Sij ]: n Sij t∗j = −θ · gi (c + t∗ , u∗ ) for i = 1, . . . , n; (9.40) j=0
But then, multiplying both sides of (9.40) by t∗i , and adding over i: n n n Sij t∗i t∗j = −θ gi (c + t∗ , u∗ )t∗i . i=1
j=0
However, with t∗0 ≡ 0, we have: n n i=1
so that:
n n i=0
j=0
j=0
i=1
Sij t∗i t∗j =
Sij t∗i t∗j = −θ
n n
n i=1
i=0
j=0
(9.41)
Sij t∗i t∗j ;
gi (c + t∗ , u∗ )t∗i = −θ · T ∗ .
(9.42)
9.6. Optimal Income Taxation
259
Since [Sij ] is negative semideﬁnite, our result follows.
Going back to (9.39) of the above proof, and making use of the symmetry of [Sij ], we obtain: n n ∗ ∗ j=1 Sij tj j=1 Sij tj ≡ = −θ for i = 1, . . . , n. (9.43) gi (c + t∗ , u∗ ) x∗i Roughly speaking: n j=1
Sij t∗j =
n j=1
∂gi ∗ n ∂gi t = ∆qj , j=1 ∂qj ∂qj j
is the total diﬀerential of the compensated demand function for the ith commodity; so that the basic interpretation of (9.43) is that the proportionate reduction in compensated demand which results from the imposition of the commodity tax scheme t∗ should be the same for all commodities.
9.6
Optimal Income Taxation
The literature on this topic begins with the seminal work of Mirrlees [1971], and the model to be presented here is an adaptation of the one originally developed by him. It is typical of the framework used throughout much of the recent discussion of this topic.3 The fundamental assumption is that there are a ﬁnite number of consumer types in the economy; all of whom have the same (continuously diﬀerentiable) utility function, but who have diﬀering labor productivities. We also follow the vast majority of recent articles by supposing that there are only two goods in the economy, a consumption good and labor/leisure. Thus consumer i has the utility function: Ui = u(xi ), where u : R2+ → R; and we suppose each consumer has the same initial endowment: r = (r, 0), where r > 0. As already mentioned, however, the consumers diﬀer in their labor productivity, with this productivity being indexed by si , where we assume, without loss of generality, that: 1 = s1 ≤ s2 ≤ · · · ≤ sm . The (aggregate) production set is assumed to be linear: Y = {y ∈ R2  y1 + y2 ≤ 0 & y1 ≤ 0}, where in equilibrium (that is, with labor used in production equal to the labor oﬀered by consumers): m y1 = si (xi1 − r). i=1
3
For surveys of the literature on this topic, see Stiglitz [1987], Mirrlees [1986] and Auerbach and Hines [2002].
260
Chapter 9. Examples of General Equilibrium Analyses
Thus eﬃcient (equilibrium) production requires: y2 =
m i=1
si (r − xi1 ).
If we normalize to set the price of the produced good equal to one, then it follows from our previous work with linear production sets that proﬁt maximization at positive output can only occur if wi , the wage rate paid the ith consumer, is equal to si . We suppose that government plans to purchase xg units of the second good, to be paid for with taxes: m t= ti , i=1
with ti the tax to be levied on the ith consumer. A balanced budget for the government then requires that: m xg = ti . i=1
The utility maximization problem for the ith consumer thus becomes: max u(xi1 , xi2 )
subject to : xi2 = si (r − xi1 ) − ti .
The ith consumer’s income (before taxes) is given by: zi = wi (r − xi1 ) = si (r − xi1 ). The standard assumption in this literature is that government can observe zi , for each i, but cannot observe either xi1 or si . We will not attempt a fullscale analysis of this problem here, but will be content to consider the case in which there are only two consumer types, with mh consumers of each type, where m1 + m2 = m, and mh ≥ 1, for h = 1, 2. Many, if not most of the key issues raised in the recent literature arise even in this simple context. In fact, we will begin by considering a particular example which illustrates many of the basic issues.4 In our example, we will suppose that the consumers’ utility function is additively separable; taking the form: u(xi ) = φ(xi1 ) + ψ(xi2 ),
(9.44)
φ (xi1 ) > 0, ψ (xi2 ) > 0, φ (xi1 ) < 0 and ψ (xi2 ) < 0.
(9.45)
where, for all xi ∈ R2+ :
Now, if all consumers of a given type face the same price for x2 , and are paid the same wage, they will each make the same consumption choice (and labor oﬀer) as every other consumer of the same type. Moreover, as per the general assumptions 4
This example is an adaptation of one preseented in Stiglitz [1987].
9.6. Optimal Income Taxation
261
stated earlier, we will assume that the marginal products of labor are constant, and equal to 1 and s > 1, for the two types. Thus the production function is given by:5 def
y = m1 (r − x11 ) + m2 s(r − x21 ) = m1 1 + m2 s 2 ,
(9.46)
where xi1 denotes the consumption of leisure by type i; with equality of supply and demand requiring that: (9.47) y = m1 x12 + m2 x22 + xg . However, if the government’s budget balance condition: m1 t1 + m2 t2 = xg ,
(9.48)
is satisﬁed, and if each consumer is paid the value of her/his marginal product, we will have: m1 x12 + m2 x22 = m1 (r − x11 ) − t1 + m2 s(r − x21 ) − t2 (9.49) = m1 1 + m2 s 2 − m1 t1 − m2 t2 = m1 1 + m2 s 2 − xg ; and thus: m1 x12 + m2 x22 + xg = m1 1 + m2 s 2 . We will suppose that the government (IRS, or whatever; hereafter we will refer to this entity as ‘the policymaker’) wishes to maximize the sum of utilities (which, as we will see in Chapter 15, amounts to maximizing a utilitarian social welfare function). Consequently, given pricetaking behavior by the consumers, the policymaker’s optimization problem reduces to the following: max
w.r.t. x11 ,x21 ,t1 ,t2
m1 φ(x11 ) + ψ(r − x11 − t1 ) + m2 φ(x21 ) + ψ[s(r − x21 ) − t2 ] , (9.50)
subject to: m1 t1 + m2 t2 − xg = 0.
(9.51)
If we write out the standard Lagrangian function, and take ﬁrstorder conditions, we see that the optimal values must satisfy: m1 φ (x∗11 ) − ψ (r − x∗11 − t∗1 ) = 0 ∗ m2 φ (x21 ) − sψ [(s(r − x∗21 ) − t∗2 ] = 0
(9.52) (9.53)
−m1 ψ (r − x∗11 − t∗1 ) + m1 λ = 0,
(9.54)
−m2 ψ [s(r − x∗21 ) − t∗2 ] + m2 λ = 0
(9.55)
and: From (9.54) and (9.55), we have: ψ [s(r − x∗21 ) − t∗2 ] = ψ (r − x∗11 − t∗1 ); 5 In this discussion, we will simplify our notation by denoting the production sector’s output of the produced good by ‘y,’ rather than ‘y2 .’
262
Chapter 9. Examples of General Equilibrium Analyses
which, since ψ < 0 everywhere, implies: x∗2 = s(r − x∗21 ) − t∗2 = r − x∗11 − t∗1 ; def
(9.56)
and thus all consumers have the same consumption of good 2 at the optimum. We then have from (9.52) and (9.53) that:
and:
φ (x∗11 ) = ψ (x∗2 ),
(9.57)
φ (x∗21 ) = sψ (x∗2 ),
(9.58)
respectively. Consequently, it follows that: φ (x∗21 ) = sφ (x∗11 ) > φ (x∗11 ). But, since φ < 0, this means that: x∗21 < x∗11 ; which, in turn implies that: u2 (x∗2 ) = φ(x∗21 ) + ψ(x∗2 ) < φ(x∗11 ) + ψ(x∗2 ) = u(x∗1 ); and thus that consumers of type 2 are worse oﬀ at the optimum than are consumers of type 1. Now, suppose for the moment that the policymaker is able to observe a consumer’s type. In this event, the policymaker could simply levy a tax of t∗i on each consumer of type i (i = 1, 2). Then, for example, consumers of type 2 would seek to: max φ(x21 ) + ψ[s(r − x21 ) − t∗2 ]; w.r.t. x21
which we see, upon taking derivatives, would result in a choice of x21 , call it ‘ x21 ,’ satisfying: x21 ) = sψ [s(r − x 21 ) − t∗2 ]. (9.59) φ ( Diagrammatically, the consumer’s solution would look like the situation illustrated in Figure 9.2, on the next page. However, given our assumptions, there is a unique point of intersection of the curves φ (x21 ) and sψ [s(r − x21 ) − t∗2 ]. Consequently, if we return to (9.58), we see that x 21 = x∗21 ; that is, the consumers’ choices, once their tax was announced, would coincide with the optimal quantities. Unfortunately for the policymaker, however, in our scenario we are supposing that, while the policymaker can observe consumers’ incomes, it cannot observe either consumers’ types or their choice of leisure ( xi1 ). Consequently, if, for example, the policymaker were to announce the tax rule: t1 if you are type 1 t= (9.60) t2 if you are type 2, consumers of type 2 would have every incentive to lie, and claim to be of type 1. Notice that this is true even if a large penalty is assessed for lying, assuming that
9.6. Optimal Income Taxation
263
sψ'[s(r  x2 1)  t2]
φ'(x2 1)
x*21
x 21
Figure 9.2: Type 2 Consumers’ Optimum. the policymaker can only observe income and tax paid; for, as you can easily prove, consumers of type 2 are better oﬀ setting:
s − 1 x∗ x21 = r + 11 , s s thereby accepting the income and consumption level of good 2 enjoyed by consumers of type 1, rather than admitting to be of type 2, and paying the higher tax, t∗2 . Thus, as we say, the mechanism being employed here is not incentivecompatible; consumers of type 2 will ﬁnd it preferable to deny being of type 2. This is, of course, disastrous for the policymaker’s plans, for the situation which will result will not yield a balanced budget, and will not maximize the sum of utilities. Now let’s generalize this example a bit. We will retain the assumption that there are only two types of consumers; however, we will drop the assumption that their common utility function is additively separable. We will suppose instead that u(·) is continuously diﬀerentiable, satisﬁes, for all x ∈ R2+ : def
u1 (x) =
∂u def ∂u > 0, u2 (x) = > 0, ∂x1 x ∂x2 x
and that u(·) is strictly quasiconcave. We will also suppose that u(·) satisﬁes a fourth assumption, but this last condition requires a little explanation. First of all, in our remaining discussion, it will be convenient to concentrate our attention upon the incomex2 space. However, in dealing with this space, let’s modify our previous notation slightly to use the generic notation ‘(z, x)’ to denote points in this space, where ‘x’ denotes the quantity of the produced good. In this space the utilitymaximization problem faced by a consumer having an index of labor productivity s, and facing a tax of t can be expressed as: def
max U (z, x) = u(r − z/s, x),
(9.61)
264
Chapter 9. Examples of General Equilibrium Analyses
subject to: x = z − t.
(9.62)
If we form the appropriate Lagrangian expression and take ﬁrstorder conditions, we ﬁnd that the optimizing values, (z ∗ , x∗ ) must satisfy: u1 (r − z ∗ /s, x∗ ) =1 su2 (r − z ∗ /s, x∗ )
(9.63)
Since it is easily shown that the slope of the indiﬀerence curve through an arbitrary point (z, x) is given by: −
U1 (z, x) u1 (r − z/s, x) = , U2 (z, x) su2 (r − z/s, x)
(9.64)
it follows that the slope of the indﬀerence curve must equal one at the optimal point, (z ∗ , x∗ ). This brings us to our fourth assumption, a condition which is standard in the optimal income taxation literature, and which is called ‘agent monotonicity.’ 9.7 Deﬁnition. The utility function u(·) satisﬁes agent monotonicity iﬀ the marginal rate of substitution; −
U1 (z, x) u1 (r − z/s, x) = , U2 (z, x) su2 (r − z/s, x)
is a decreasing function of s. In other words, the consumer’s indiﬀerence curves in (z, x)space will be ﬂatter, the higher is the agent’s productivity index. Suppose, for example, that: u(x0 , x) = (x0 )a (x)1−a , for some real number, a, satisfying: 0 < a < 1, and where we are using ‘x0 ’ and ‘x’ to denote the quantities of leisure and the produced good, respectively. Then, as you can easily verify: −
u1 (r − z/s, x) ax U1 (z, x) = = ; U2 (z, x) su2 (r − z/s, x) (1 − a)(sr − z)
which is obviously decreasing in s. Therefore, agent monotonicity is satisﬁed in the CobbDouglas case. Suppose, that the policymaker has calculated optimal values, x∗10 , x∗20 , x∗1 , x∗2 , t∗1 , and t∗2 . It can then calculate the optimal (pretax) income values: z1∗ = r − x∗10 and z2∗ = s(r − x∗20 ), and we will suppose that:6 6
t∗1 < t∗2 & z1∗ < z2∗ .
We will return to a discussion of this assumption later.
(9.65)
9.6. Optimal Income Taxation
265
Suppose our policymaker now chooses any number (income), z † , satisfying: z1∗ < z † < z2∗ ,
(9.66)
and then attempts to implement the following tax schedule: t∗ for 0 ≤ z ≤ z † t = 1∗ t2 for z † < z.
(9.67)
In order to analyze the consumers’ choices here, let’s begin by considering the optimization problem for a consumer of type 1, and supposing for the moment, that the tax schedule is simply t = t1 , for all z. In this case consumers of type one face the optimization problem: max
w.r.t. z1 , x1
u(r − z1 , x1 ) subject to: x1 = z1 − t1 .
This yields the ﬁrstorder necessary conditions: x 12
x*2
z*
z†
z
t1 t2
Figure 9.3: Type 1 Consumers’ Optimum. u1 (r − z1∗ , x∗1 ) = 1, u2 (r − z1∗ , x∗1 )
(9.68)
where ‘z1∗ ’ and ‘x∗1 ’ denote the optimal quantities of income and consumption of the produced good, respectively. Recalling that the lefthand fraction is the slope of the
266
Chapter 9. Examples of General Equilibrium Analyses
indiﬀerence curve through (z1∗ , x∗1 ), we see that the utilitymaximizing solution looks like that shown in Figure 9.3, on the previous page. As the diagram makes clear, this is the optimizing value for consumers of type one; that is, we do not have to consider the portion of the consumption schedule corresponding to z > z † in determining the consumptionincome choice for consumers of type one. However, what about consumers of type 2? By the assumption of agent monotonicity, consumers of type 2 will have an indiﬀerence curve through (z1∗ , x∗1 ) which looks something like that shown in Figure 9.4. Consequently, the diﬀerence between the tax levied on the productive group (t2 ) and that levied on the lower productivity group (t1 ) can be no greater than that indicated in Figure 9.4, since a higher value for t2 − t1 (which would move the t2 schedule downward parallel to itself) would result in the consumers of type 2 achieving a higher utility by mimicking the behavior of type 1 consumers, than they would by achieving the higher income associated with t2 . In fact, of course, with the tax diﬀerential and schedules shown in Figure 9.4, consumers of type 2 would maximize utility by setting z2 = z † . The policymaker can correct for this by charging a tax of t2 for all persons with an income greater than z1∗ ; however, this amounts to choosing the tax schedule: t1 for 0 ≤ z ≤ z1∗ , t= (9.69) t2 for z1∗ < z; which would leave consumers of type 2 indiﬀerent between z1∗ and z2∗ . Moreover, the policy maker can set this schedule only if she knows the exact value of z1∗ . x 22
z1 *
z†
z 2*
z
t1
t2
Figure 9.4: Limiting Tax Value for Consumer 2.
9.6. Optimal Income Taxation
267
Let’s see if we can extend this analysis a bit. In our example, we supposed that the policymaker wished to maximize the sum of the consumers’ utilities. As one would expect, however, maximizing according to a diﬀerent objective function will yield diﬀerent optimal consumption and tax rates for the two consumer types. Let’s take a look at what sorts of solutions will be selfenforcing, in the sense that, if the policymaker determines an optimal tax and income, (t∗i , zi∗ ), for consumers of type i, an announcement of an appropriate tax schedule will lead to consumers maximizing utility at the policymaker’s optimal values. If we go back to take another look at Figure 9.3, it becomes apparent that the policymaker’s tax choices must satisfy t∗2 ≥ t∗1 ; since in the opposite case, consumers of type 1 would maximize utility at some point along the type 2 schedule. Thus, whatever the objective function which the policymaker is attempting to maximize, the tax for the type 2 consumers needs to be at least as high as for type one consumers if the solution is to be consistent with utility maximization; that is, if it is to be ‘incentive compatible.’ I1
x
I'2
I2
x*1
z*1
z
 t'1 z'  t*1  t' 2
−τ
Figure 9.5: A Pareto Ineﬃcient Solution. The next question is, what must be true of z2∗ visavis z1∗ ? A moment’s study of Figure 9.3 should suﬃce to convince you that, given that the utility function satisﬁes agent monotonicity (and since we must have t∗2 ≥ t∗1 ), the policymaker’s choice of an optimal income for type 2 consumers must be at least as high as for type one
268
Chapter 9. Examples of General Equilibrium Analyses
consumers. In fact, consider the limiting case in which the policymaker’s choice of (t∗1 , z1∗ ) and (t∗2 , z2∗ ) satisfy: t∗1 = t∗2 and z1∗ = z2∗ . Then a tax schedule of the form: t∗ T = 1 τ
for 0 ≤ z ≤ z1∗ , and for z1∗ < z,
(9.70)
with τ as indicated in Figure 9.5 will induce the consumers to choose the incomeconsumption pairs which the policymaker views to be optimal. However, the resulting situation cannot be optimal for any objective function which is an increasing function of consumers’ utilities! To see this, notice that the tax schedule indicated will result in that utility for type 1 consumers corresponding to the the indiﬀerence curve I1 , while type 2 consumers will achieve the utility associated with indiﬀerence curve I2 . However, if the tax schedule: t for 0 ≤ z ≤ z , and (9.71) T = 1 t2 for z < z, is instituted instead, the same tax revenue will be raised, and both consumers will be better oﬀ! The perceptive reader may have noticed from the outset that the type of taxation we have been analyzing in this section is not a conventional sort of income tax at all, but is eﬀectively a lumpsum tax. That is, in our discussion we have, for all practical purposes, been assuming that the policymaker wishes to assess a tax of ti on consumers of type i. What happens if we instead consider a tax schedule more typical of that used in actual practice? In particular, suppose we consider the simplest sort of income tax schedule; one deﬁned by the function: T = tz, where 0 < t < 1.7 In this case, the consumers’ consumption schedule (for the produced good) takes the form: x = z − T = (1 − t)z. Consequently, as you can easily demonstrate, consumers of the two types will choose incomes z1∗ and z2∗ satisfying: − and: −
U1 u1 [r − z1∗ , (1 − t)z1∗ ] = = 1 − t, U2 u2 [r − z1∗ , (1 − t)z1∗ ]
u1 [r − z2∗ /x, (1 − t)z2∗ ] U1 = = 1 − t, U2 su2 [r − z2∗ /s, (1 − t)z1∗ s]
7 For discussion of the ’optimal’ tax rates for schedules of this type, see Hellwig[1986] and Strawczynski [1998].
9.7. Monopoly in a General Equilibrium Model
269 x=z
x
x = z  t1
x = z  t2
z*1
z*2
z
 t1
 t2
Figure 9.6: Ineﬃciency of the Conventional Tax Schedule.
respectively. Thus, utilitymaximization by the two consumers will result in the sort of situation depicted in Figure 9.6, above. However, the solution, while simple and incentivecompatible, is not eﬃcient. As indicated in the diagram, the same tax revenue would be raised if consumers of type 1 were to pay a tax of t1 , while consumers of type 2 pay a tax of t2 (I will leave the details of the reasoning to you). Unfortunately, of course, this second solution may not be implementable, and certainly will not be practicable!
9.7
Monopoly in a General Equilibrium Model
In this section we’ll look at a very simple general equilibrium model with three commodities (two produced goods and leisure), two consumers, and two ﬁrms. My goal here is to develop a bit diﬀerent perspective as regards the ’First Fundamental Theorem,’ and the secondbest optimality of the optimal commodity tax solution. Consumer (agent) one (‘the worker’) will be assumed to have preferences repre
270
Chapter 9. Examples of General Equilibrium Analyses
sentable by the utility function: u1 (x1 ) = (x11 ) · (x12 + 1)(x13 ), while consumer (agent) two (‘the capitalist’) has preferences representable by: u2 (x2 ) = x21 · x23 . Thus, consumer two cares nothing for the second commodity. We will suppose that consumer one faces the usual budget constraint: p1 x11 + p2 x12 + wx13 ≤ wr1 , and 0 ≤ x13 ≤ r1 , where we have denoted the price of the third commodity by ‘w,’ to suggest ‘wages,’ and ‘r1 ’ denotes the consumer’s initial endowment of leisure. Consumer 2 (‘the capitalist’), on the other hand, has the budget constraint: p1 x21 ≤ π2 , and 0 ≤ x23 ≤ r2 , where ‘π2 ’ denotes the proﬁts of the second ﬁrm, and r2 is the consumer’s endowment of leisure. I will leave to you the task of showing that, in the absence of government intervention, with leisure chosen as the num´eraire (w = 1), and setting: r1 = r2 = 32, the consumers’ demand functions are given by (see Exercise 5, at the end of Chapter 4): 32 + p2 32 + p2 32 + p2 x11 = , x12 = − 1, x13 = , (9.72) 3p1 3p2 3 and: x21 =
π2 . p1
(9.73)
Finally, we will suppose that both production technologies are linear, with labor requirement functions given by: j = xj
for j = 1, 2.
(9.74)
We will ﬁrst consider the case in which Firm 2 behaves as a monopolist; while Firm 1 behaves as a pricetaker (setting price equal to marginal cost). Firm 2’s proﬁt function is given by: π2 =
32 + p2 32 + p2 − p2 − + 1, 3 3p2
and we see that proﬁts (and consumer 2’s utility) are maximized when p2 = 4, resulting in π2 = 6. Since Firm 2 sets price equal to marginal cost, we must have (given that w = 1): p1 = 1;
9.7. Monopoly in a General Equilibrium Model
271
and thus: 36 + 6 = 18, 3 36 x12 = − 1 = 2, 12 36 x13 = = 12 3 1 + 2 = x1 + x2 = 20,
x1 = x11 + x21 =
so that: r1 − x13 = 32 − 12 = 20 = 1 + 2 . Thus, we have found an equilibrium for the economy, and in this equilibrium the consumers’ utilities are given by: u1 = 12 × 3 × 12 = 432,
(9.75)
u2 = 32 × 6 = 192.
(9.76)
and:
Now suppose government regulates the monopolist; requiring that p2 = marginal cost = 1, while compensating consumer 2, and paying for this compensation by taxing consumer one’s consumption of the ﬁrst commodity with a tax of t per unit. Then consumer one pays a price of 1 + t per unit of good one, while consumer 2 pays a price of 1 per unit. In this case, the demand for the ﬁrst commodity becomes: x1 = x11 + x21 =
32 + p2 11 t · x11 = + (1 + t) = 11. 3(p1 + t) p1 1+t
(9.77)
In order that agent two achieve her/his previous utility level, we must have x21 = 6, or:
11 t · x11 /p1 = t = 6; 1+t so that t = 6/5. We then have: x11 = 5, x12 = 10, and x13 = 11. I will leave it to you to verify that this is indeed an equilibrium, and that the ﬁrst consumer’s utility is u1 = 550; which is considerably higher than the utility of 432 which the ﬁrst consumer achieved in the monopoly situation. Thus the secondbest situation with a commodity tax and subsidy strictly Pareto dominates the original unregulated monopoly equilibrium. As a matter of fact, it can easily be shown (although I will leave it as an exercise) that if government were to set t = 7/4, then the consumers’ utilities at the new equilibrium are: u1 = 440 and u2 = 224; so that both consumers are better oﬀ than in the monopoly situation!
272
9.8
Chapter 9. Examples of General Equilibrium Analyses
Money in a General Equilibrium Model
It is often postulated that individual’s have a ‘transactions demand’ for money balances. Typically this would mean that, for each i, there exists αi ∈ ] 0, 1[ such that i s demand for money balances is related to commodity demands by the equation mi = αi p · xi
for i = 1, . . . , I;
(9.78)
where xi is the vector of consumer i’s commodity demands. In this section we will consider two questions. First, is (9.78) consistent with utility maximization? If so, what sort of utility function yields the relation set out in (9.78)? Secondly, what are some of the implications of (9.78) for the economy as a whole? Turning our attention to the ﬁrst question, suppose the ith consumer’s utility function takes the form: ui (xi , mi ) = φi (xi ) · (mi )αi ,
(9.79)
→ R+ is an increasing function which is continuously where αi ∈ ] 0, 1[, and diﬀerentiable and positively homogeneous of degree one, and ‘mi ’denotes agent i’s desired money balance, or the quantity of gold to be held, if you prefer. We suppose that the ith consumer maximizes utility subject to: φi :
n j=1
Rn+
pj xij + mi = p · xi + mi = wi (p) + mi0 ,
(9.80)
where, as usual ‘wi ’ demotes the ith consumer’s wealth, and ‘mi0 ’ denotes its initial money ( or gold) balance. [The assumption that φi is increasing allows us to use an equality sign, rather than inequality in (9.80).] We will also suppose that w : Rn+ → RI+ is a feasible wealthassignment function. Now, suppose that (x∗i , m∗i ) is the (n + 1)tuple which maximizes i’s utility, given (p∗ , m∗i0 ). If we form the relevant Lagrangian for i’s utility maximization problem, we see that there must exist a scalar, µ such that at (x∗i , m∗i ) we will have: φij (x∗i ) · (m∗i )αi = µp∗j
for j = 1, . . . , n,
(9.81)
and: αi φi (x∗i )(m∗i )αi −1 = µ. Multiplying each of the equations in (9.81) by the corresponding have: n (m∗i )αi x∗ij φij (x∗i ) = µp∗ · x∗i .
(9.82) x∗ij
j=1
and adding, we (9.83)
Substituting (9.82) into (9.83), we then obtain: (m∗i )αi
n
or: m∗i
j=1
x∗ij φij (x∗i ) = αi φi (x∗i )(m∗i )αi −1 p∗ · x∗i ,
n j=1
x∗ij φij (x∗i ) = αi φi (x∗i )p∗ · x∗i .
(9.84)
9.8. Money in a General Equilibrium Model
273
However, by Euler’s theorem, we have: n x∗ij φij (x∗i ) = φi (x∗i ); j=1
and thus we obtain from (9.84): m∗i = αi p∗ · x∗i .
(9.85)
Thus, transactions demand for money of the form (9.78) is consistent with maximization of a utility function of the form (9.79). However, in this case, we can express the transactions demand for money balances in a somewhat more useful way as follows. From the budget constraint and (9.85), we have: mi = αi [wi (p) + mi0 − mi ]; so that: mi = or:
α i wi (p) + mi0 , 1 + αi
mi = βi wi (p) + mi0 ,
(9.86)
where we have deﬁned βi as:
αi . (9.87) 1 + αi Notice that we necessarily have 0 < βi < 1. , Now suppose we have an equilibrium at (x∗i , m∗i ) , p∗ , m0 . Then we have: βi =
I i=1 I
I i=1
x∗i =
K k=1
σk (p∗ ),
m∗i =M K wi (p∗ ) = p∗ · σk (p∗ );
(9.88)
i=1
k=1
where σk is the k th ﬁrm’s supply correspondence, ‘M ’ denotes the aggregate money supply, and: m0 = (m10 , . . . , mi0 , . . . , mI0 ). Notice that it follows from the last relation in (9.88) that wi (·) is positively homogeneous of degree one, for each i. For future reference, we also note that, adding the budget constraints over i: I I I I p∗ · x∗i + m∗i = wi (p∗ ) + mi0 , (9.89) i=1
i=1
i=1
i=1
and using the last equation in (9.88), we have: I i=1
wi (p∗ ) +
I i=1
mi0 =
K k=1
p∗ · σ k (p∗ ) +
I i=1
mi0
(9.90)
Furthermore, from the ﬁrst and second equations in (9.88), we have: I i=1
p∗ · x∗i +
I i=1
m∗i =
K k=1
p∗ · σ k (p∗ ) + M.
(9.91)
274
Chapter 9. Examples of General Equilibrium Analyses
Combining (9.89)–(9.91), we have, therefore: I i=1
mi0 = M.
(9.92)
Now suppose the money supply is multiplied by λ > 0, and that this is done by multiplying each consumer’s initial money balance by λ. Then from (9.86), we have: m†i = βi wi (p† ) + λmi0 , where p† is the new price vector, and m†i is consumer i’s new demand for money balances. However, if we set p† = λp∗ , then: mi = βi [wi (λp∗ ) + λmi0 ] = λ[wi (p∗ ) + mi0 ]; and thus:
I i=1
mi =
I i=1
λm∗i = λM,
where the last inequality frollows from (9.88). Furthermore, since consumer commodity demand and producer’s supply functions are positively homogeneous of degree zero in (p, wi ) and p, respectively, it now follows from (9.88) that the commodity market is also in equilibrium with p† = λp∗ . Thus, in this new equilibrium, all commodity prices are multiplied by λ. Now let’s see if we can generalize this a bit. Suppose now that the ith consumer’s utility function can be written in the form (9.79), as before, but let’s now drop the assumption that φi is positively homogeneous and continuously diﬀerentable; supposing only that it is,increasing.  Using the same notation as before, suppose we have an equilibrium at (x∗i , m∗i ) , p∗ , m0 . Then (9.88) and (9.89) are satisﬁed; and, for each i, (x∗i , m∗i ) maximizes ui , given (p∗ , wi (p∗ ), m∗i0 ). Now let λ > 0. Then we note that: (λp∗ ) · x∗i + λm∗i = λ(p∗ · xi + m∗i ) = λ · [wi (p∗ ) + m∗i0 ] = wi (λp∗ ) + λm∗i0 , where the last inequality follows from the fact that a feasible wealthassignment function must be positively homogeneous of degree one in p. Thus we see that (x∗i , λm∗i ) is in the budget set deﬁned by (λp∗ , λm∗i0 ): n+1 B(λp∗ , λm∗i0 ) = (xi , mi ) ∈ R+  p∗ · xi + mi ≤ wi (λ p∗ ) + λm∗i0 . Now suppose (xi , mi0 ) is such that: (λp∗ ) · xi + mi ≤ wi (p∗ ) + λm∗i0 . Then we have: p∗ · xi + mi /λ ≤ wi (p∗ ) + m∗i0 . Therefore, since (x∗i , m∗i ) maximizes utility subject to: p∗ · xi + mi ≤ wi (p∗ ) + m∗i0 ,
9.8. Money in a General Equilibrium Model
275
it follows that: α ui (xi , mi /λ) = φi (xi ) · mi /λ i ≤ φi (x∗i ) · (m∗i )αi ; or:
φi (xi ) · (m∗i )αi ≤ φi (x∗i ) · (λm∗i )αi .
Therefore, denoting i’s demand function for money balances by ‘hi,n+1 (p, mio ),’ we have: hi,n+1 (λp, λmio ) = λhi,n+1 (p, mio ). (9.93) , ∗ ∗  ∗ I will leave it as an exercise to show that if (xi , mi ) , p , m0 is an equilibrium for the economy, and the money supply is multiplied by a positive number, λ, by means of multiplying each consumer’s initial money balance by λ, then , (x∗i , λm∗i ) , λp∗ , λm0 will be a new competitive equilibrium. One thing that may be slightly troubling about the above analysis is the assumption that each individual’s initial money balance is increased (or decreased) in exactly the same proportion. This is consistent with government’s changing the oﬃcial exchange rate, or the amount of gold backing, for example; but is not consistent with the way that the money supply is adjusted by the Federal Reserve in the U. S., for example. So, it would seem to be of interest to investigate the question of what happens to the price level if the aggregate money supply is changed, but individual initial holdings are not. (This may be a situation in which it is more realistic to suppose that we are dealing with the total supply of and demand for gold balances, rather than money balances per se.) Suppose, then, that the aggregate money (or gold) supply, M , is multiplied by a positive number λ, which for deﬁniteness we will suppose is greater than one. In order to simplify our analysis we will also suppose that all the βi ’s are equal: βi = β
for i = 1, . . . , I,
where 0 < β < 1. If, in fact, the vector of commodity prices, p is changed in the proportion µ, then equilibrium requires that: λM =
I
mi =
i=1
I I I
β wi (µp) + mi0 = βµ wi (p) + β mi0 . i=1
i=1
i=1
However, by (9.92) we have: λM =
I i=1
mi0 ;
and if the economy was initially in equilibrium, then: I M =β wi (p) + mi0 . i=1
Substituting these last two equations into (9.94), we obtain: βλ
I i=1
I I
wi (p) + β mi0 , wi (p) + mi0 = βµ i=1
i=1
(9.94)
276 or:
Chapter 9. Examples of General Equilibrium Analyses µ = λ + (λ − 1) I
M
i=1 wi (p)
;
and thus we see that prices in the new equilibrium have gone up more than proportionately to the increase in the money supply.
9.9
Indivisible Commodities
In this section I will simply present an example of general equilibrium analysis with indivisible commodities. The example itself was inspired by Ellickson [1993, pp. 111–14]. We consider an economy in which we have four commodities and one hundred consumers. The fourth commodity will be assumed to be perfectly divisible (perhaps a ‘composite commodity’); while the ﬁrst three commodities are types of rental properties (hereafter called “apartments”) of low (L), medium (M), and high (H) quality, respectively. We will suppose that each of the consumers has preferences representable by the utility function: u(xi ) = (1 + xi1 + 2xi2 + 5xi3 )xi4 . While this utility function is quite conventional, we are going to suppose that consumers rent (consume) at most one apartment. We do this by deﬁning their (common) consumption set in the following way. We ﬁrst deﬁne the set C by: C = x ∈ R4+  xj ∈ {0, 1} for j = 1, 2, 3 , and then deﬁne X, the consumption set, by: X = {x ∈ C  x1 + x2 + x3 ≤ 1}. We will label the individuals from lowest to highest wealth; supposing the individual commodity endowments, ω i , are given by: ⎧ ⎪ ⎨(0, 0, 0, 199 + 5i) for i = 1, . . . , 80, ω i = (2, 0, 0, 5i) for i = 81, . . . , 90, ⎪ ⎩ (2, 4, 2, 2i) for i = 91, . . . , 100. Obviously the fourth commodity is a num´eraire for the economy in this case, and consequently, in considering competitive allocations we can normalize to set p4 = 1. Thus consumer i’s wealth, wi will be given by: wi = p1 ωi1 + p2 ωi2 + p3 ωi3 + ωi4 ; so that, for example consumer one’s wealth is, w1 , is equal to 204 (units of commodity 4). To analyze the workings of this type of example, notice that, for example, a consumer will prefer to be homeless, as opposed to renting a lowquality apartment
9.9. Indivisible Commodities
277
if, and only if the utility obtained with no housing exceeds that obtained with one unit of good 1: wi > 2(wi − p1 ); that is: 2p1 > wi .
(9.95)
Similarly, consumer i will prefer to rent lowquality housing rather than medium iﬀ: 2(wi − p1 ) > 3(wi − p2 ), or: 3p2 − 2p1 > wi .
(9.96)
Finally, consumer i will prefer to rent a medium, as opposed to a high quality apartment, if and only if: 3(wi − p2 ) > 6(wi − p3 ), or: 2p3 − p2 > wi .
(9.97)
We will ﬁrst consider a competitive equilibrium, and then examine a disequilibrium situation. Consider the price vector p∗ = (100, 200, 400, 1). Since our poorest consumer has a wealth equal to 204, it follows from (9.95) that consumer 1 will prefer to rent a lowquality apartment to the alternative of being homeless. On the other hand: w40 = 199 + 5 · 40 = 399 < 3p2 − 2p1 = 400 < w41 = 404; so we see that the ﬁrst forty consumers will choose to rent lowquality housing, while the fortyﬁrst consumer will prefer to rent a mediumquality apartment. Since each consumer with a label larger than 41 will have a higher wealth, it follows that exactly 40 consumers will demand a low quality apartment, and this is exactly equal to the supply of same. Similarly, we ﬁnd that: w80 = 599 < 2p3 − p2 = 600 < w81 = 2 · 100 + 5 · 81 = 605; so that only consumers 41–80 will rent medium quality apartments, and since there are 40 such apartments available, we once again have demand equal to supply. Finally, it is easy to show that consumers 81–100 will demand highquality apartments, and since there are 20 such available, we have found a competitive equilibrium. However, suppose a (bad) mistake is made in pricing the highquality apartments, so that p3 is set equal to 500, instead of 400. Then it is easy to see that, with p1 and p2 set at the levels just considered, we will have: w90 = 650 < 2p3 − p2 = 800; so that consumers 81–90 will now opt for medium, rather than highquality apartments. However, since there are only 40 such units available, the price of mediumquality housing will have to rise by enough to induce consumers 41–50 to choose
278
Chapter 9. Examples of General Equilibrium Analyses
lowquality housing. But this in turn means that the price of lowquality housing will have to rise by enough to induce consumers 1–10 to prefer homelessness to renting a lowquality apartment. Are there prices which will accomplish all of this? In order to answer this question, let’s begin by considering the price of lowquality housing. In order to induce 10 consumers to opt for homelessness, we need to have p1 satisfy: 2p1 > w10 = 249. Consequently, the desired result will be achieved with p1 = 125. In order to induce consumers 41–50 to choose lowquality housing, we see from (9.96) that we need p2 to satisfy: 3p2 − 2p1 = 3p2 − 250 > w50 = 449, or: p2 > 233. However, we also need to have the ﬁftyﬁrst consumer choose mediumquality housing, so that we need: w51 = 454 > 3p2 − 2p1 = 3p2 − 250, or: 3p2 < 704. Thus, supply for mediumquality housing will be equal to demand if p2 = 234. I will leave it to you to show that consumers 81–90 will continue to opt for mediumquality housing, despite their increased income. Are there then market forces which will tend to move prices back toward the equilibrium levels which we found earlier? Before considering this question as such, let’s take a look at the utility levels of these 100 consumers when prices are at the (nonequilibrium) level p = (125, 234, 500, 1). This means that, for example, consumers 110 in the new situation attain a utility of: u(xi ) = wi = 199 + 5i; whereas with p∗ , the corresponding utility values were: u(xi ) = 2(wi − 100) = 2(99 + 5i) = 198 + 5i; so that each of these consumers was strictly better oﬀ in the equilibrium situation, as compared to that with p = p . In fact, although I will leave you to show this (Exercise 3), every consumer has a higher utility at the equilibrium prices than in the disequilibrium situation! Returning to the issue of whether there are forces tending to push this market toward equilibrium, as you have probably already noticed, with p = p , there are 10 highquality apartments standing empty. I will leave it to you to trace out the forces which will then move this market toward equilibrium.
9.9. Indivisible Commodities
279
Exercises. 1. Let E be an economy with one consumer, whose preferences can be represented by the utility function: u(x0 , x1 ) =
√
x0 · x1 ,
and whose initial endowment is given by r = (20, 0); and suppose there is one producer whose production set is given by: Y = {y ∈ R2  y0 ≤ 0 & y0 + y1 ≤ 0}. Find the competitive equilibrium for E, given the level of governmental activity (t, xg ), where: t = (0, 1) and xg = (0, 5); or show that no such equilibrium exists in this case. 2. Let E be an economy with one consumer, whose preferences can be represented by the utility function: u(x0 , x1 , x2 ) = x0 · x1 · x2 , and whose initial endowment is given by r = (24, 0, 0); and suppose there is one producer whose production set is given by: Y = {y ∈ R3  y1 , y2 ≥ 0 & y0 + 2y1 + y2 ≤ 0}. Find the optimal commodity tax vector, t = (t1 , t2 ); supposing that government demand, xg , is given by: xg = (0, 3, 2). 3. Verify the claims made in the Example of Section 9.7 regarding the situation in which government sets a tax of t = 7/4. 4. Show that, in the Example of Section 9, every consumer has a higher utility at the equilibrium prices than in the disequilibrium situation.
Chapter 10
Comparative Statics and Stability 10.1
Introduction
We are accustomed to saying that if a commodity is a normal good (positive income eﬀect), and given a normal supply curve (increasing with respect to price), an increase in the demand for the good will result in an increase in both the price of the good and the quantity of the good traded. However, this is a partial equilibrium analysis. In a general equilibrium context, the increased demand for good i must have repercussions for, or come about because of, changes in excess demand for one or more other goods. Moreover, as the price of the ith good increases, there will be changes in the quantity demanded of other commodities, which in turn will have feedback eﬀects on the market for the ith good. Consequently, can we still be sure the the new equilibrium price for the ith good will be higher? You’re probably thinking that it surely will be higher; it is, after all, only ‘common sense.’ Unfortunately, the theoretical conditions under which one can verify this simple analysis in the context of a general equilibrium model are rather more restrictive than one would like, although I am speaking here of known suﬃcient conditions for the analysis to hold; the known necessary conditions are less discouraging. In this chapter we will investigate a portion of what is known of comparative statics in the context of general equilibrium, as well as giving brief consideration to the issues of the uniqueness of equilibrium and the stability of general competitive equilibrium. It is particularly reasonable to combine these topics; for, ﬁrst of all, even in the context of partial equilibrium analysis, comparative statics analysis is likely to become meaningless unless the equilibria in a market (both before and after a demand change, say) are unique. In turn, comparative statics analysis will also break down if the equilibria in a market are not stable; after all, comparative statics analysis proceeds by comparing equilibria before and after a change has taken place. There is little point in such a comparison unless the market price and quantities approach the new equilibrium. Moreover, there is another aspect of this relationship; the suﬃcient conditions for stability of general competitive equilbria may enable us to deduce the sign of price changes following a change in underlying
282
Chapter 10. Comparative Statics and Stability
demand or supply conditions. This is the ‘correspondence principle,’ ﬁrst stated and discussed by Samuelson [1947]. Comparative statics analysis in general equilibrium proceeds along essentially two diﬀerent lines; global analysis, and local analysis. We will begin by considering some global analysis; which as we will develop it here, is based upon a variation of the weak axiom of revealed preference for aggregate excess demand. This approach has admitted weaknesses, and we will consider some of these as well. However, the analysis is, I believe, interesting, intuitive, and the conclusions seem to be in accord with the stylized facts of real economies. In any event, we will present the basic approach and the fundamental analysis based upon this approach in the next section. We will then look at two conditions which imply this form of the weak axiom; the ‘law of demand,’ and gross substitutability. We will discuss the ‘law of demand’ in Section 3, where we will show that it is implied by homothetic preferences. We will also show that it is unfortunately diﬃcult to justify the assumption that aggregate excess demand functions satisfy the law of demand, even if preferences are homothetic. On the other hand, we will show that this law will hold under plausible empirical conditions, and as such it has strong and interesting implications. In Section 4 we will discuss the assumption of ‘gross substitutability,’ and we will show that it implies the weak axiom condition of Section 2. In Section 5, we will look at an alternative approach to comparative statics analysis; the local (diﬀerential) approach constituting a portion of what is called ‘qualitative economics.’ Finally, in the last two sections of this chapter, we provide a very brief introduction to the literature on the stability of general competitive equilibrium.
10.2
Aggregate Excess Demand
We will assume throughout this and the next two sections that individual preferences are continuous, strictly convex, and that the nth commodity is a num´eraire good for the economy (and thus consumers will have demand functions satisfying the budget balance condition); in fact, we will suppose that the aggregate excess demand correspondence for the economy is singlevalued (that is, is a function). More formally, we suppose that the producers in the economy are pricetakers, and that the aggregate supply correspondence is singlevalued, and thus is a function, s(p). We denote the portion of the domain of this function which lies within Rn++ by Π, we let π : Π → R be the aggregate proﬁt function, and we will assume throughout this, and the next two sections of this chapter, that Π is a convex cone.1 We further suppose that consumer wealth is deﬁned by a feasible wealthassignment function, w : Π → Rm (see Deﬁnition 7.9). Thus, for p ∈ Π, we deﬁne the aggregate excess demand function for the economy, z : Π → Rn . by: z(p) = hi [p, wi (p)] − r − s(p), (10.1) i∈M
where r ∈ 1
Rn
is the aggregate resource endowment for the economy.
The fact that Π is a cone follows from the deﬁnitions without any special assumptions. The issue of whether or not Π is also convex is discussed in Chapter 6.
10.2. Aggregate Excess Demand
283
A particularly important special case of this occurs when E is a private ownership, pure exchange economy. In this situation, both the supply and proﬁt functions are null, ri , r= i∈M
where r i is the ith consumer’s initial resource endowment, and the feasible wealthassignment function is given by: wi (p) = p · r i
for p ∈ Π, and i = 1, . . . , m.
In this case, it is also useful to deﬁne the individual excess demand functions, z i : Π → Rn by: z i (p) = hi [ p, p · r i ] − r i
for i = 1, . . . , m.2
Returning to the general case, notice that under the assumptions being utilized here, p∗ ∈ Π is an equilibrium price vector if, and only if: z(p∗ ) = 0. It is also important to notice that, given our assumptions, the excess demand function will be positively homogeneous of degree zero in p (recall that Π will be a cone), and will satisfy the strong form of Walras Law, which can be expressed as the condition: (∀p ∈ Π) : p · z(p) = 0. A key consideration in many investigations of conditions under which an economy will have a unique competitive equilibrium revolves around the question of whether or not the aggregate excess demand function satisﬁes the same properties as are satisﬁed by individual excess demand functions; in particular, the following condition. 10.1 Deﬁnition. The aggregate excess demand function, z(·), satisﬁes the Weak Axiom of Revealed Preference (abbreviated WA) iﬀ, for any pair of price vectors, p and p , we have: z(p) = z(p ) & p · z(p ) ≤ 0 ⇒ p · z(p) > 0. In fact, MasColell, Whinston, and Green prove that the satisfaction of WA by the aggregate excess demand function is, in a sense, both necessary and suﬃcient for the uniqueness of competitive equilibrium in the situation in which the aggregate production set is a convex cone (see Proposition 17.F.2, p. 609, of MasColell, Whinston, and Green [1995]). However, in our analysis we will make use of the slightly weaker conditon deﬁned as follows. 10.2 Deﬁnition. The aggregate excess demand function, z(·), satisﬁes the Weak* Axiom of Revealed Preference (abbreviated WA*) iﬀ, given any equilibrium price vector, p∗ , and any second price vector, p ∈ Π such that z(p) = 0, we have: p∗ · z(p) > 0. 2
Notice also that in this case we can take Π to be equal to Rn ++ .
284
Chapter 10. Comparative Statics and Stability
It is an easy exercise to show that if the aggregate demand function satisﬁes the Weak Axiom (WA), then it satisﬁes WA* (see Exercise 1, at the end of this chapter). We can then prove the following variant of the suﬃciency portion of the MWG result. 10.3 Proposition. Suppose the aggregate excess demand function, z(·), satisﬁes WA*. Then the set of equilibrium price vectors is convex. Proof. Suppose p and p are both equilibrium price vectors for the economy E, let θ ∈ [0, 1], and deﬁne: p∗ = θp + (1 − θ)p . Suppose, by way of obtaining a contradiction, that p∗ is not a competitive equilibrium price. Then z(p∗ ) = 0, and it follows from WA* that: p · z(p∗ ) > 0 and p · z(p∗ ) > 0.
(10.2)
But this is impossible, for by Walras’ Law: 0 = p∗ · z(p∗ ) = [θp + (1 − θ)p ] · z(p∗ ) = θp · z(p∗ ) + (1 − θ)p · z(p∗ ); which contradicts (10.2) As noted in MasColell, Whinston, and Green [1995], if the set of normalized equilibria is ﬁnite, then the above result implies that, given WA*, the set of normalized equilibrium prices is a singleton (that is, there is a unique normalized equilibrium price). When the aggregate excess demand satisﬁes WA*, increases in the demand for a given commodity will result in an increase in the price of that commodity. In demonstrating this, we will more or less follow McKenzie [2002, pp. 143–4]. We begin with the following deﬁnition. 10.4 Deﬁnition. Let p∗ be an equilibrium price vector, given the aggregate excess demand function z(·), let z(·) be a second excess demand function, and let j ∈ {1, . . . , n − 1} be arbitrary. We will say that z exhibits increased excess demand for the j th commodity (alone), as compared with z(·) iﬀ we have: zj (p∗ ) > 0, and, for all k ∈ {1, . . . , n − 1} \ {j}: zk (p∗ ) = 0. The idea of the above deﬁnition corresponds to the situation described in the introduction to this chapter. We begin with a situation in which the economy is in equilibrium, given the excess demand function z, and then we suppose that one or more consumers’ tastes change in favor of the j th commodity (relative to the num´eraire), or perhaps there has been a wealth transfer from a given consumer to a second consumer who values the j th commodity (relative to the num´eraire) more than did the ﬁrst.3 Given such a change, we will have a new excess demand function for the economy which exhibits increased excess demand for the j th commodity. For instance, consider the following examples. 3
Given the other conditions of the deﬁnition, we will necessarily have zn (p∗ ) < 0. Why?
10.2. Aggregate Excess Demand
285
10.5 Examples. We consider a pure exchange economy with m = 2, n = 3, and suppose initially that both consumers have the utility functions: 1/3 . (10.3) ui (xi ) = xi1 · xi2 · xi3 Suppose also that initial endowments are given by: r 1 = (4, 0, 2) and r 2 = (0, 4, 2).
(10.4)
In this case, as I will leave it to you to demonstrate, if we set p = p∗ = (1, 1, 1), then z(p∗ ) = 0; so that p∗ is an equilibrium price, given z. Now suppose that the ﬁrst consumer’s utility changes to: 1/3
1/2
1/6
u1 (x1 ) = x11 · x12 · x13 ,
(10.5)
while consumer 2’s utility function remains as in (10.3). Then, as I will leave for you to demonstrate, the aggregate excess demand function for the economy changes to (normalizing to set p3 = 1):
4p + 4p + 4 6p1 + 4p2 + 5 2p1 + 4p2 + 3 1 2 z(p) = − 4, − 4, −4 ; 3p1 3p2 3 so that:
z(p∗ ) = (0, 1, −1);
and we see that z exhibits increased demand for the second commodity. I will leave it to you to construct a similar example in which a transfer of a quantity of the num´eraire good (the third commodity in this case) from one consumer to the other results in an increase in demand for the second commodity (see Exercise 4, at the end of this chapter). Making use of WA* and our (McKenzie’s) deﬁnition of an increase in excess demand for the j th commodity, we can establish the following.4 10.6 Proposition. Suppose p∗ is an equilibrium price for an economy, given the excess demand function z(·), and that excess demand changes to z(·), which exhibits increased demand for commodity j, and satisﬁes WA*. Then if p is the equilibrium price for E, given z, we will have pj > p∗j . Proof. Since z exhibits increased excess demand for commodity j, and satisﬁes WA*, we have: 0 < p · z(p∗ ) = pj zj (p∗ ) + zn (p∗ ). However, we also have, by (the strong form of) Walras’ Law: 0 = p∗ · z(p∗ ) = p∗j zj (p∗ ) + zn (p∗ ); so that:
p∗j zj (p∗ ) < pj zj (p∗ ),
and thus: Since zj 4
(p∗ )
( pj − p∗j ) · zj (p∗ ) > 0. > 0, our result follows.
The statement and proof of which are lifted almost verbatim from McKenzie [2002, Theorem 10, p. 144].
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Thus, the satisfaction of WA* by the aggregate excess demand function has quite strong and useful implications. Unfortunately, even in the case of a pure exchange economy, and with homothetic preferences, the aggregate excess demand function does not necessarily satisfy WA*, as is demonstrated by the following example. 10.7 Example. Suppose m = n = 2, that X1 = X2 = R2+ , and that the consumers’ preferences are represented by the utility functions: u1 (x1 ) = min{x11 , x12 }, and u2 (x2 = min{4x21 , x22 }
(10.6)
respectively; and suppose: r 1 = (2, 0) and r 2 = (0, 5). If p = (1, 1), then: x11 + x21 = 1 + 1 = r11 , and thus (by Walras’ Law) p is a competitive equilibrium price. Now consider the price vector p = (1, 1/4). We have:
8 5 9 −9 8 5 z(p ) = + − 2, + − 5 = . , 5 8 5 2 40 10 Therefore: p · z(p ) = 9/40 − 9/10 = −27/40 < 0;
which violates WA*.
It is quite reasonable at this point for you to be wondering just why it is that we are bothering with the condition WA* if there is such a simple example, with preferences so wellbehaved, in which the condition is violated. Well, there are two cases of some interest in which WA* is satisﬁed.5 The ﬁrst of these two cases is that in which aggregate excess demand satisﬁes the ‘Law of Demand.’ We will study this case in the next section, and while we will ﬁnd that this condition is not satisﬁed in general, we can specify intuitive empirical conditions in which it is satisﬁed. The second case in which WA* is satisﬁed is that in which the commodities are all gross substitutes. We will consider this condition in section 4 of this chapter.
10.3
The ‘Law of Demand’
One often sees the phrase ‘the law of demand’ used in the economics literature to mean that the aggregate demand function for a commodity is downwardsloping. The following condition reduces to this in the situation in which only one price has changed.6 In the second part of the following deﬁnition, we will use the notation H(·) to denote the aggregate demand function deﬁned by: m hi (p, wi ). H(p, w) = i=1
5
It should also be noted that, insofar as I am aware, no one has proved that these are the only two conditions implying WA*. 6 I believe that this label was ﬁrst attached to this condition by J. R. Hicks [1956]. The treatment here owes much to Section 4.C of MasColell, Whinston, and Green [1995], however.
10.3. The ‘Law of Demand’
287
10.8 Deﬁnition. We will say that the ith consumer’s demand function satisﬁes the law of demand iﬀ, given any p, p ∈ Rn++ , and any wi ∈ R+ such that hi (p, wi ) = hi (p , wi ), we have: p − p · hi (p, wi ) − hi (p , wi ) < 0.
(10.7)
Similarly, we will say that aggregate demand satisﬁes the law of demand iﬀ given any p, p ∈ Rn++ , and any w ∈ Rm + such that H(p, w) = H(p , w), we have: p − p · H(p, w) − H(p , w) < 0.
(10.8)
As already suggested, the above condition implies that the demand function for each commodity is downwardsloping. As you know, an individual demand function does not necessarily satisfy the law of demand; there is always the possibility that the infamous Giﬀen good case may arise. However, if an individual’s preferences are homothetic, then the consumer’s demand function will satisfy the law of demand, and thus the demand function for each commodity is necessarily downwardsloping in this case. 10.9 Theorem. Suppose G is homothetic, continuous, strictly convex, and locally nonsaturating on Rn+ . Then the demand function determined by G satisﬁes the law of demand on Ω. Proof. Recall from Theorem 4.39 that, given the present assumptions, the demand function takes the form: h(p, w) = g(p)w. Now, let p, p ∈ Rn++ , and deﬁne: µ = p · g(p). Then we note that it follows from the fact that p · g(p) ≤ µ, that: p · g(p )µ ≥ 1 [remember that h(p , µ) = g(p )µ]. Therefore, we see that: p · g(p) + p · g(p ) ≥ µ + 1/µ.
(10.9)
Now consider the function f : R++ → R++ deﬁned by: f (µ) = µ + 1/µ.
(10.10)
If you check ﬁrst and secondorder conditions for an extremum, it is easy to show that f has a unique minimum at µ = 1; and that f is strictly convex, so that for all µ ∈ R++ : µ = 1 ⇒ f (µ) > f (1) = 2. (10.11)
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Thus it follows from (10.9)–(10.11) that if p · g(p) = 1, then:7 p · g(p) + p · g(p ) > 2. p
(10.12)
g(p ),
On the other hand, if µ = · g(p) = 1, and g(p) = then it follows from WA that p · g(p ) > 1, so that (10.12) holds in this case as well. Now let w be arbitrary, and let p, p ∈ Rn++ be such that: h(p, w) = g(p)w = h(p , w) = g(p )w. Then obviously, g(p) = g(p ), so it follows from equation (10.12) that: p · g(p)w + p · g(p )w > 2w, so that:
w − p · g(p)w + w − p · g(p )w < 0.
But then we see that: 0 > p · g(p )w − p · g(p)w + p · g(p)w − p · g(p )w = p · h(p , w) − h(p, w) − p · h(p , w) − h(p, w) = (p − p) · h(p , w) − h(p, w) .
The good news about the Law of Demand is that if it holds for each individual consumer’s demand function, then it holds for the aggregate demand function as well. Thus we obtain the following as an easy corollary of Theorem 10.9. I will leave the details of the proof as an exercise. 10.10 Corollary. If each individual demand function, hi (·) saisﬁes the Law of Den mand, then the aggregate demand function H : Rn++ × Rm + → R+ also satisﬁes the Law of Demand. While the aggregate demand function, H, is welldeﬁned whenever all individual demand functions are welldeﬁned, economists often (especially in applied work) assume that aggregate demand can be expressed as a function of price and aggregate income. The following condition provides one method for justifying such an assumption. 10.11 Deﬁnition. We will say that a function ω : R+ → Rm + is an income distribution function iﬀ it is positively homogeneous of degree one, and satisﬁes: m (∀w ∈ R+ ) : ωi (w) = w. i=1
Given such an income distribution function, we can deﬁne an aggregate demand function, h, on Rn++ × R+ by: h(p, w) = 7
m i=1
Notice that if g(p) = g(p ), then p · g(p) = 1.
hi [p, ωi (w)].
10.3. The ‘Law of Demand’
289
It is easy to show that, given any such ω, the function h(p, w) will be positively homogeneous of degree zero in (p, w); and if each individual demand function satisﬁes the budget balance condition (as we are assuming is the case throughout this section), the aggregate demand function will satisfy this condition as well, that is: (∀(p, w) ∈ Rn++ × R+ ) : p · h(p, w) = w. It is also easy to show that if each individual demand function, hi satisﬁes the Law of Demand, then so will the aggregate demand function, h. Of course, the most interesting special case of this is where we have the following 10.12 Deﬁnition. Let E = (Xi , Pi , Yk , r) be an economy, suppose aggregate supply is welldeﬁned8 on the set Π ⊆ Rn++ , and let w : Π → Rm + be a feasible wealth function for E. We will say that E and w satisfy the income distribution condition iﬀ, there exists an income distribution function, ω : R+ → Rm + such that, for all p ∈ Π we have: wi (p) = ωi [w(p)]
for i = 1, . . . , n,
where w : Π → R+ is deﬁned by: w(p) = p · r + p · σ(p). 10.13 Example. Suppose E = (Xi , Pi , Yk , r i , [sik ]) is a private ownership economy, and let a ∈ ∆m be such that for each i ∈ {1, . . . , m}: r i = ai
m j=1
r j ≡ ai r and sik = ai , for k = 1, . . . , .
If we now deﬁne w(·) in the usual way: wi (p) = p · r i +
k=1
sik πk (p)
for i = 1, . . . , m,
it is easy to show that w and E satisfy the incoe distribution condition.
The following result sets forth an interesting property guaranteed by the Law of Demand. 10.14 Proposition. Let ω : R+ → Rm + be an income distribution function, let h be the corresponding aggregate demand function, and suppose h satisﬁes the Law of def
Demand.9 Then h satisﬁes the weak axiom (WA) on Ω = Rn++ × R+ . Proof. Suppose (p, w), and (p , w ) are elements of Ω such that: h(p, w) = h(p , w ) and p · h(p , w ) ≤ w. 8 9
That is, for each p ∈ Π, there exists a unique y ∈ Y which maximizes proﬁts on Y , given p. As will be the case, remember, if each hi satisﬁes the Law of Demand.
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Then, deﬁning p = (w/w )p , we have, by the homogeneity of degree zero of the aggregate demand function, that h(p , w) = h(p , w ). It then follows from the fact that aggregate demand satisﬁes the law of demand on Ω, that: 0 > [p − p] · [h(p , w) − h(p, w)] = p · h(p , w) − p · h(p, w) − p · h(p , w) + p · h(p, w) = w − p · h(p, w) + w − p · h(p , w), (10.13) where the last equality is by the fact that each hi satisﬁes the budget balance condition. Moreover, by the fact that h(p , w) = h(p , w ), and our hypothesis, we have that: w − p · h(p , w) ≥ 0. Thus it follows from (10.13) that p · h(p, w) > w; and using the deﬁnition of p , it now follows that: p · h(p, w) > w .
Returning to the notsogood news about the Law of Demand, notice that, under the assumptions which we’re employing, we can write aggregate excess demand as: m z i (p) − s(p), z(p) = i=1
where z i (·) is the ith consumer’s excess demand function, for i = 1, . . . , m.10 Thus, given p, p ∈ Π, we will have: m (p −p)· z i (p )−z i (p) −(p −p)· s(p )−s(p) (10.14) (p −p)· z(p )−z(p) = i=1
Now, it follows from Theorem 6.27 that we necessarily have: −(p − p) · s(p ) − s(p) ≤ 0; with strict inequality if s(p) = s(p ). (See Exercise 6, at the end of this chapter.) Consequently, if each individual excess demand function satisﬁes the Law of Demand, then the aggregate excess demand function will satisfy the Law as well. Unfortunately, individual excess demand functions do not necessarily satisfy the Law of Demand even if the consumer’s preferences are homothetic, as is shown by the following example. 10.15 Example. Let n = 2, Xi = R2+ , and r i = (1, 0); while the consumer’s utility function is given by: ui (xi ) = xαi1 · x1−α i2 , 10 If we are allowing for individually owned resource endowments, r i ∈ Rn , then z i (p) = h[p, wi (p)] − r i . If we are not allowing for such ownership, then we can add r to s(p).
10.4. Gross Substitutes
291
where 0 < α < 1, and let: p = (1, 1) and p = Then we have:
4 ,2 (1 − α)
z i (p ) = (α − 1, 1 − α),
while: z i (p ) =
α ×
4 (1−α) 4 1−α
− 1,
4 (1 − α) (1−α)
2
= (α − 1, 2).
Therefore: 3 + α , 1 · (0, 1 + α) = 1 + α > 0. (p − p ) · z i (p ) − z i (p ) = 1−α Thus we see that z i (·) does not satisfy the Law of Demand.
As the above example demonstrates, the reason that the ith consumer’s excess demand function may not satisfy the Law of Demand is that, in general, when p changes, wi (p) changes as well. In fact, while the above example is discouraging to be sure, notice that to achieve the violation we changed the consumer’s income from 1 to 4/(1 − α). Since 0 < α < 1, this is an extremely large change, in percentage terms. In order to consider this issue further, suppose p changes from p to p , and consider the ith consumer’s demand and excess demand functions. We have: (p − p ) · z i (p ) − z i (p ) = (p − p ) · hi p , wi (p ) − hi p , wi (p )
= (p − p ) · hi p , wi (p ) − hi p , wi (p ) + hi p , wi (p ) − hi p , wi (p )
= (p − p ) · hi p , wi (p ) − hi p , wi (p )
+ (p − p ) · hi p , wi (p ) − hi p , wi (p ) (10.15) If the consumer’s demand function satisﬁes the Law of Demand, and if: hi p , wi (p ) = hi p , wi (p ) , then the ﬁrst inner product on the righthandside of (10.15) is negative. Consequently, if the income change, wi (p )−wi (p ), is suﬃciently small, the inner product on the lefthandside of (10.15) will be negative as well. Thus, to oversimplify things just a bit, we can say that if individual consumers’ demand functions satisfy the Law of Demand, then the aggregate excess demand will also satisfy the Law of Demand for price changes which do not induce ‘large’ income changes.
10.4
Gross Substitutes
The idea behind the formal deﬁnition of gross substitutes is exactly the intuitive idea of substitute commodities; we say that commodities i and j are gross substitutes if,
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whenever the price of i goes up, other things being equal, the demand for j goes up as well. This is as opposed to the more sophisticated (Hicksian) idea of substitutes, which says that commmodities i and j are substitutes iﬀ: Sij ≡
∂hi ∂hi + hj > 0. ∂pj ∂w
The formal deﬁnition which we will use is as follows. 10.16 Deﬁnition. Given the excess demand function, z : Π → Rn , we shall say that the commodities i and j are gross substitutes (ﬁnite increment form)11 iﬀ, for any p∗ ∈ Π and any positive real numbers, ∆pi and ∆pj we have: zi (p∗ + ∆pj ej ) > zi (p∗ ) and zj (p∗ + ∆pi ei ) > zj (p∗ ), ith
(10.16)
j th
and unit coordinate vectors, respectively. We where ‘ei ’ and ‘ej ’ denote the will say that z(·) satisﬁes (S) iﬀ commodities j and k are gross substitutes, for each j, k such that k = j. We will also be interested in a diﬀerential version of Deﬁnition 10.16, stated as follows. 10.17 Deﬁnition. Given the diﬀerentiable excess demand function, z : Π → Rn , we shall say that the commodities i and j are gross substitutes iﬀ, for any p∗ ∈ Π, we have: ∂ ∂ [zi (p∗ )] > 0 and [zj (p∗ )] > 0. (10.17) ∂pj ∂pi Clearly, if z(·) is diﬀerentiable and satisﬁes Deﬁnition 10.17, it also satisﬁes 10.16; although the converse is not quite true, even for diﬀerentiable excess demand functions.12 The following example shows that our deﬁnitions are not vacuous. 10.18 Example. Suppose E is a pure exchange economy in which the ith consumer has the CobbDouglas utility function: n n a xijij , where aij > 0 for all i, j, and aij = 1, for i = 1, . . . , m. ui (xi ) = j=1
Then the
ith
j=1
consumer’s excess demand function for the j th commodity is given by: zij (p) =
aij p · r i − rij . pj
Thus we see that if k = j, then: aij rij ∂ [zij (p)] = ≥ 0. ∂pk pj 11 While we will not be dealing with the alternative deﬁntion here, gross substitutes are sometimes deﬁned by substituting weak inequalities for the strict inequalities we have used in (10.16). Usually, however, i and j are then said to be ‘weak gross substitutes.’ 12 If z(·) is diﬀerentiable and satisﬁes Deﬁnition 10.17, it may nonetheless be true that there exist price vectors at which the partial derivatives appearing in 10.17 are zero. There cannot, however, be any neighborhoods in which the partials are zero throughout the neighborhood.
10.4. Gross Substitutes
293
Therefore, if: def
r =
m i=1
r i 0,
it follows that, for k = j, we will have: 1 m ∂ m ∂ [zj (p)] = zij (p) = aij rij > 0, for j = 1, . . . , n. i=1 i=1 ∂pk ∂pk pj
It is easily shown that if each consumer’s excess demand function satisﬁes condition (S), then the aggregate excess demand function for the economy will satisfy condition (S) as well. Unfortunately, in an economy with production each individual consumer’s demand function may satisfy condition (S), while the aggregate excess demand function fails to satisfy the condition.13 It can be shown that, under the assumptions which we have been employing, if z(·) satisﬁes (S), and p∗ ∈ Π is an equilibrium for z(·) [so that z(p∗ ) = 0], then p∗ 0 (for a proof, see Arrow, Block, and Hurwicz [1959]). This fact is employed in the next result. 10.19 Proposition. If z(·) is a continuous excess demand function satisfying (S), and if p∗ and p are equilibria for z(·), then there exists θ ∈ R++ such that p = θp∗ . Proof. Let:
µ = min{p1 /p∗1 , . . . , pn /p∗n }.
By the homogeneity of z, we have: z(µp∗ ) = z(p∗ ) = 0. If we suppose, by way of obtaining a contradiction, that have, for some i, k ∈ {1, . . . , n}:
(10.18) p
=
µp∗ ,
then we must
pi = µp∗i , pk > µp∗k , and pj ≥ µp∗j for all j = i. But then it follows from (S) that: zi (p ) > zi (µp∗ ); which, together with (10.18), contradicts the assumption that p is an equilibrium for z(·). The following result is of particular interest in connection with stability analysis. (See particularly, Proposition 10.24 of Section 7.) While the conclusion of the following result holds in an economy with any ﬁnite number of commodities, we will conﬁne our argument to the case in which n = 2. (For a proof for the case of an arbitrary ﬁnite number of economies, see Arrow, Block, and Hurwicz [1959].) 10.20 Proposition. If z(·) is a continuous excess demand function satisfying (S) and the strong form of Walras’ Law, and if p∗ is an equilibrium for z(·), then for any p ∈ R++ which is not a scalar multiple of p∗ , we must have p∗ · z(p) > 0. 13 For a more detailed discussion of this diﬃculty, see MasColell, Whinston, and Green [1995], pp. 612–14.
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Chapter 10. Comparative Statics and Stability Proof. (For the case in which n = 2). Suppose: p∗2 /p∗1 > p2 /p1 ,
and deﬁne p† and p by: p† = (1, p∗2 /p∗1 ) and p = (1, p2 /p1 ), respectively. Then by homogeneity and (S), we have: 0 = z1 (p∗ ) = z1 (p† ) > z1 (p ) = z1 (p). Since z1 (p) < 0, it follows readily from (the strong form of) Walras’ Law that we also have z2 (p) > 0. Therefore: (1/p∗1 )p∗ · z(p) = p† · z(p ) = z1 (p ) + (p∗2 /p∗1 )z2 (p )
> z1 (p ) + (p2 /p1 )z2 (p ) = p · z(p ) = 0;
where the ﬁrst equality is by the homogeneity of z, and the last is by Walras’ Law. Consequently: (1/p∗1 )p∗ · z(p) > 0, and thus p∗ · z(p) > 0. On the other hand, if: then:
p∗2 /p∗1 < p2 /p1 , p∗1 /p∗2 > p1 /p2 ;
so that, if we deﬁne: p† = (p∗1 /p∗2 , 1) and p = (p1 /p2 , 1), it follows from the homogeneity of z and (S) that: 0 = z2 (p† ) > z2 (p ) = z2 (p). Proceeding as in the argument of the previous paragraph, we can then show that z1 (p ) > 0, and thus that: p∗ · z(p) > 0.
10.5
Qualitative Economics
We will conﬁne our investigation of the ‘local’ approach to the study of comparative statics in a general equilibrium context to a discussion of what is known as ‘qualitative economics.’ This is an area of investigation which was originally proposed and given its initial development by Samuelson [1947, Chapters 2 and 3], and further developed in its early stages by Gorman [1964], Lancaster [1962], and James Quirk and various collaborators (for example, Bassett, Maybee, and Quirk [1968]).
10.5. Qualitative Economics
295
In our present discussion, we will borrow heavily from Lang, Moore, and Whinston [1995].14 The basic issue with which qualitative economics is concerned is the development of comparative statics results in an essentialy general equilibrium context, while making use of only purely qualitative information. We can illustrate the basic idea which motivated this work with a very simple illustration. Suppose in a given market, we can write the demand and supply functions as: δ(p, I) and σ(p), respectively, where ‘p’ and ‘I’ denote the price of the product and consumer income, respectively; and suppose further that for all relevant values of the variables, we have: ∂δ ∂δ < 0, > 0, and σ (p) > 0. ∂p ∂I As usual, we suppose the market is in equilibrium at (p∗ , I ∗ ) if: δ(p∗ , I ∗ ) − σ(p∗ ) = 0. What I want to do now is to see whether we can deduce the eﬀect on p of an increase in I, making use only of the information given above. Of course, I know that you probably already know perfectly well how to do this, but bear with me; I want to do this in a way which fairly naturally extends to a system of equations (and a general equilbrium system). Formally, we know from the Implicit Function Theorem that if: ∂ (10.19) δ(p, I) − σ(p) ∗ ∗ = 0, ∂p (p ,I ) then we can solve for p as a function of I in a neighborhood of (p∗ , I ∗ ). Moreover, if we denote this function by ‘ρ(I),’ then the functional solution obtained will satisfy the identity: δ ρ(I), I − σ ρ(I) ≡ 0. (10.20) I will leave it to you to verify that (10.19) is necessarily satisﬁed at an equilibrium. Diﬀerentiating the identity in equation (10.20), we have: ∂δ d ∂δ δ[ρ(I), I] − σ[ρ(I)] = · ρ (I) + − σ · ρ (I) = 0; dI ∂p ∂I
from which we obtain: ρ (I) =
−∂δ/∂I . ∂δ/∂p − σ
(10.21)
(10.22)
From the assumptions we have made regarding the signs of the relevant derivatives, it now follows that ρ (I) > 0; that is, that the eﬀect of an increase in I will be to increase the equilibrium price. 14 For a more complete review of most aspects of this research, see Quirk and Saposnik [1968, Chapter 6], and for an alternative recent development of this type of material, see Fontaine, Garbely, and Gilli [1991], as well as McKenzie [2002, Chapter 2].2
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Now, the intriguing thing about the derivation which we have just gone through is that we deduced the direction of change which will take place in equilibrium price solely on the basis of qualitative information and/or assumptions about the demand and supply functions. Given the diﬃculty in obtaining reliable numerical estimates in economics, this is obviously very important. So now the issue is, how well can we do in extending this mode of analysis to a general equilibrium context. In general equilibrium analysis, we often deal with equilibrium conditions of the form: (10.23) f i (x1 , . . . , xm ; z1 , . . . , zn ) = 0 for i = 1, . . . , m; or, more compactly: f (x; z) = 0,
(10.24)
where: f : X × Z → Rm , X ⊆ Rm , and Z ⊆ Rn . In this context, the vector x would generally represent endogenous variables, while the vector z would consist of n exogenous (possibly governmental policy variables); and we will say that (x∗ , z ∗ ) ∈ X × Z is an equilibrium of the system iﬀ f (x∗ ; z ∗ ) = 0. Writing: ∂f i for i, j = 1, . . . , m, fji (x∗ ; z ∗ ) = ∂xj (x∗ ,z∗ ) and, similarly: fki (x∗ ; z ∗ ) =
∂f i ∂zk (x∗ ;z∗ )
for i = 1, . . . , m, and k = m + 1, . . . , m + n,
(for the sake of convenience, we will label the coordinates of vectors z by m + 1, . . . , m + n)15 it is common in applied general equilibrium analysis to specify the sign of each of these derivatives, or to specify that one or more of these derivatives is identically zero. Now, if at an equilibrium (x∗ ; z ∗ ) ∈ X × Z, we have: J  = 0, where:
⎡
f11 (x∗ ; z ∗ ) f21 (x∗ ; z ∗ ) ⎢ f12 (x∗ ; z ∗ ) f22 (x∗ ; z ∗ ) J =⎢ ⎣ ... ... f1m (x∗ ; z ∗ ) f2m (x∗ ; z ∗ )
... ... ... ...
⎤ 1 (x∗ ; z ∗ ) fm 2 (x∗ ; z ∗ ) ⎥ fm ⎥, ⎦ ... m (x∗ ; z ∗ ) fm
and ‘J ’ denotes the determinant of J , it follows from the Implicit Function Theorem that there exists a neighborhood, N of z ∗ (contained in Z), and a function g : N → X, such that g has continuous ﬁrst partials, and satisﬁes: (∀z ∈ N ) : f g(z); z = 0. (10.25) 15
Thus, in particular, we write: ∗ ∗ z ∗ = (zm+1 , . . . , zm+n ).
10.5. Qualitative Economics
297
From (10.25) and the diﬀerentiability of f and g, we then have, for each z ∈ N : m
f i [g(z); z]gkj (z) j=1 j
+ fki g(z), z = 0
Thus, we have, in particular: m fji (x∗ ; z ∗ )gkj (z ∗ ) = −fki (x∗ ; z ∗ ) j=1
for i = 1, . . . , m; k = m + 1, . . . , m + n.
for i = 1, . . . , m; k = m + 1; . . . , m + n;
so that, deﬁning: g k (z ∗ ) = gk1 (z ∗ ), . . . , gkn (z ∗ ) and f k (x∗ ; z ∗ ) = fk1 (x∗ ; z ∗ ), . . . , fkn (x∗ ; z ∗ ) we obtain the system: J g k (z ∗ ) = −f k (z ∗ )
for k = m + 1, . . . , m + n.
(10.26)
From (10.26), we then obtain: g k (z ∗ ) = −J −1 f k (z ∗ )
for k = m + 1, . . . , m + n.
(10.27)
Now, it is often possible to determine whether J  = 0 (and thus whether J −1 exists); in fact, to determine the sign of J , solely on the basis of qualitative assumptions regarding f ; that is, solely on the basis of a speciﬁcation of the signs of the partial derivatives, fji = ∂f i /∂xj and fki = ∂f i /∂zk . Sometimes, though less often, it is actually possible to determine the signs of gki = ∂g i /∂zk solely on the basis of this sort of qualitative information. The idea is this: let us specify only that each fi (globally) takes on one of the three values: ⎧ ⎪ ⎨+ fi = 0 ⎪ ⎩ −
;
(10.28)
and we can agree to use the following arithmetic for these symbols: (+) · (+) = (−) · (−) = +, (+) · (−) = (−) · (+) = −, (+) · (0) = (−) · (0) = (0) · (+) = (0) · (−) = 0, (+) + (+) = (+) − (−) = +, (−) + (−) = (−) − (+) = − (+) − (+) = (+) + (−) = (−) + (+) = (−) − (−) = ?, (+) + (0) = (0) + (+) = +, (0) + (0) = (0) − (0) = 0, (0) − (+) = (−) + (0) = − We can illustrate the principles involved here with an example.
(10.29)
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Chapter 10. Comparative Statics and Stability
10.21 Example. Consider the system: f i (x1 , x2 ; z1 , z2 ) = 0
for i = 1, 2;
(10.30)
and suppose:
Then:
f11 = +
f21 = −
f31 = +
f41 = −,
f12
f22
f32
=−
f42 = 0.
=+
=+
1 f f 1 + − = (+) · (+) − (+) · (−) = +. J  = 12 22 = f1 f2 + +
(10.31)
(10.32)
Consequently we know that the function g : N → X (giving x as a function of z) will exist, for any system of functions satisfying the qualitative speciﬁcation in (10.31). For this example, and for k = 1 (or k = m + 1 = 2 + 1 = 3) equation (10.27) becomes: ) ! ! * ! ! −1 f22 −f21 f31 −1 f22 f31 − f21 f32 −f31 g11 = J −1 · = = 2 2 2 1 2 2 1 1 2 g1 −f3 f3 J  −f1 f1 J  −f1 f3 + f1 f3 (10.33) ! ! ! (+) · (+) − (−) · (−) (−) · (?) ? = (−) · = = . (−) · (+) · (+) + (+) · (−) (−) · (−) + Similarly: ! ! * ! ! ) −1 f22 −f21 f41 −1 f22 f41 − f21 f42 g21 −f41 = J −1 · = = 2 2 2 1 2 2 1 1 2 g2 −f4 f4 J  −f1 f1 J  −f1 f4 + f1 f4 (10.34) ! ! ! (+) · (−) − (−) · (0) (−) · (−) + = (−) · = = . (−) · (+) · (−) + (+) · (0) (−) · (+) − Thus, in this case we see that only ∂x2 /∂z1 = g12 has an indeterminate sign.
I will not pursue this topic further here. For those interested, it is probably still true that the best survey of this material, and introduction to the correspondence principle, is contained in Quirk and Saposnik [1968, Chapter 6]; and for a diﬀerent, and quite promising approach, see Milgrom and Roberts [1994].
10.6
Stability in a Single Market
Turning our attention to the stability of competitive equilibrium, in this section we will introduce our topic by considering a markeet for a single commodity; which we suppose is characterized by demand and supply curves D(p) and S(p), respectively, where here ‘p’ denotes the (scalar) price of the good in question. We then denote the excess demand function by ‘E(p);’ that is: def
E(p) = D(p) − S(p). Stability (of competitive equilibrium) analysis is concerned with two related issues: (1) how does the market (or markets) behave out of equilibrium? and (2) does the market ( or markets) approach equilibrium over time? The two classic (continuous
10.6. Stability in a Single Market
299
time) adjustment mechanisms developed for the analysis of stability in a single market are due to Walras and Marshall. The Walrasian adjustment mechanism, in its simplest formulation, postulates: dp ≡ p˙ = k · [E(p)], dt
(10.35)
where k is a positive constant. The Marshallian adjustment mechanism is a bit more complicated to formulate, however we proceed as follows. First, we will suppose that both D(·) and S(·) are invertible; which, of course, they will be if D is everywhere downwardsloping, and S is everywhere upwardsloping. We can then interpret the values of D−1 (x), for a given quantity of the commodity, x, as being the demand price of the quantity x; that is, the (maximum) price at which the quantity x can be sold. Similarly, for a given quantity, x, S −1 (x), the supply price of x is the (minimum) price suﬃcient to bring the quantity x onto the market. The Marshallian adjustment mechanism is then, in its simplest form, given by: dx = µ · [D−1 (x) − S −1 (x)], dt
(10.36)
where µ is a positive constant. 10.22 Deﬁnitions. Suppose p∗ ∈ R++ is such that D(p∗ ) = S(p∗ ) and let x∗ = D(p∗ ) = S(p∗ ). We shall say that the equilibrium (x∗ , p∗ ) is: 1. Walrasian Stable iﬀ, for each p > p∗ , S(p) > D(p), and for each p < p∗ , S(p) < D(p). 2. Marshallian stable iﬀ, for each x > x∗ , S −1 (x) > D−1 (x), and for each x < x∗ , S −1 (x) < D−1 (x). If we compare these deﬁnitions with the adjustment mechanisms deﬁned in (10.35) and (10.36), respectively, the logic of the deﬁnitions should be clear enough: the equilibrium is said to be Walrasian stable iﬀ whenever p > p∗ , dp/dt < 0, and whenever p < p∗ , dp/dt > 0. Marshallian stability has an analogous interpretation in terms of changes in the quantity of the commodity on the market. p S
p*
D x*
x
Figure 10.1: Demand and Supply: The Textbook Case.
300
Chapter 10. Comparative Statics and Stability
In Figure 1, on the previous page, the market is both Marshallian stable and Walrasian stable. In Figure 2a, below, we have Marshallian, but not Walrasian stability; while in Figure 2b, we have Walrasian, but not Marshallian stability. p
p
p*
p*
D
S S
D x*
x
Fi gure 2a
x*
x
Figure 2b
Figure 10.2: Stability and Instability.
p
p z(p)
p*
p*
z(p) z Figure 3a
z Figure 3b
Figure 10.3: Excess Demand Functions. Notice, however, that both Figures 1 and 2b correspond to an excess demand conﬁguration like that depicted in Figure 3a; while Figure 2b corresponds to the sort of excess demand function shown in Figure 3b. (Note: in these two diagrams, we have used ‘z(·)’ to denote the excess demand function for this market; that is, for each p, we deﬁne z(p) = D(p) − S(p).] In order to study this question of stability, or lack thereof, in greater generality,
10.6. Stability in a Single Market
301
we will need to set out our relationships more formally. We will begin our analysis with a brief consideration of a more formal development of stability analysis and the Walrasian adjustment mechanism along the lines just discussed, but for an economy with only two goods. In such an economy it suﬃces to look at stability in one of the two markets, because, by Walras’ law, if one market is in equilibrium, the other must be also. In fact, it can be shown that (in an exchange economy) if we are moving toward equilibrium in one market, we have to be moving toward equilibrium in the other as well. In dealing with stability of competitive equilibrium, it is very convenient to normalize prices; in fact, many of our deﬁnitions will become simpler, and it is much easier to actually work through the mathematics in this case. Moreover, since excess demand functions are necessarily positively homogeneous of degree zero, it suﬃces to study normalized prices; which is what we will do in the remainder of our present discussion. We consider an economy, E, in which n = 2, and normalize prices by setting p2 = 1, and deﬁne: p = p1 /p2 = p1 , (10.37) where the second equality arises from our nornalization. Suppose further that the rules of change in market one are given by: p˙ = dp/dt = f [ζ(p)];
(10.38)
where f : R → R, and where ‘ζ(p)’ denotes excess demand in market one as a function of p; that is, we deﬁne: ζ(p) = z1 (p, 1).
(10.39)
Question for Discussion. If we deﬁne equilibrium in a market as a situation from which there is no net tendency to change, and if the rules of change in market one are given by (10.38), then consider the folllowing question (due to L. Hurwicz): “Under what conditions is it true that the market is in equilibrium if, and only if, supply equals demand?” 10.23 Deﬁnition. Let f : R → R. We will say that f is signpreserving iﬀ, for all z ∈ R: 1. f (z) = 0 iﬀ z = 0, and: 2. zf (z) > 0 for z = 0. A particular example of a signpreserving function is the identity function i(·), deﬁned on R by: i(z) = z for z ∈ R. (10.40) Other examples are: f (z) = arctan z, and f (z) = exp(z) − 1.
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Chapter 10. Comparative Statics and Stability
Of course, if we take any signpreserving function, and multiply it by a positive constant, we get another signpreserving function. Thus, if α > 0 is a positive constant, the function: f (z) = α[exp(z) − 1], is also a signpreserving function. Now, if the rules of change in market one are given by (10.38), where f is a signpreserving function, then: pζ(p) ˙ > 0, (10.41) for all p ∈ R+ such that ζ(p) = 0 Thus, if p∗ ∈ R++ is such that ζ(p∗ ) = 0, and if ζ(·) is downwardsloping, as in Figure 10.3.a, above (and thus the market is Walrasian Stable), then we will have: ⎧ ∗ ⎪ ⎨p˙ > 0 ⇐⇒ p < p , (10.42) p˙ = 0 ⇐⇒ p = p∗ , and ⎪ ⎩ p˙ < 0 ⇐⇒ p > p∗ ; that is, price will increase over time if, and only if, price is initially below the equilibrium level, and so on. Thus, in the case of just two markets, there isn’t much to (tˆ atonnement) stability analysis. Essentiallly, if price in market one changes according to a rule of the form of (10.38), above, where f is a signpreserving function, and if the excess demand function is downwardsloping, then an equilibrium of the economy must be (at least locally) stable. Complications arise, however, in connection with the parenthetic insertions in the statement just made, and in connection with intermarket reactions when there are more than two commodities, as we will see in the next section.16
10.7
MultiMarket Stability
The material in this section is only a sampling of results conerning multimarket stability, and is largely drawn from the work of Arrow and Hurwicz [1958] and Arrow, Block, and Hurwicz [1959]. More complete surveys are provided by Negishi [1962], Quirk and Saposnik [1968], Hahn [1982], Takayama [1985], and McKenzie [2002,Chapter 2]. In our discussion, we will use the same normalization that was introduced in the beginning of this section, except that we will suppose for notational convenience that there are n + 1 commodities. In fact, we will normalize to consider price vectors of the form: P = (p, 1), where p ∈ Rn++ . We will assume throughout the remainder of this section that the excess demand function, z(·) satisﬁes the properties which were introduced in Section 2; so that it satisﬁes the strong form of Walras’ Law (W), homogeneity of degree zero (H), and continuity (C). It follows from Walras’ Law (W) that we 16 Complications also arise in connection with the treatment of time, continuous or discrete, but we will consider only continuous time models here.
10.7. MultiMarket Stability
303
need only consider equilibrium and price adjustments for the ﬁrst n commoditities. Consequently, in considering stability issues we will consider a price adjustment mechanism of the form: dpj = fj [ζj (p)] dt
for j = 1, . . . , n;
(10.43)
where we will take fj (·) to be a continous and signpreserving function, and where we deﬁne ζ : Rn++ → Rn+ by: ζ(p) = ζi (p), . . . , ζn (p) = zi (p, 1), . . . , zn (p, 1) = z(p, 1). (10.44) The system of equations (10.43) is a system of diﬀerential equations;17 a solution of which is a function ρ : R+ → Rn++ , satisfying: dρj = fj zj [ρ(t)] dt
for j = 1, . . . , n.
(10.45)
Under the conditions we will be assuming to hold here, the system (10.43) will always possess a solution. However, the solution will not generally be unique unless we specify a starting value for p; that is, a value p0 ∈ Rn++ , which we take to be ρ(0). Thus a function ρ : R+ → Rn++ is said to be a solution of (10.43), given the initial value p0 , iﬀ: dρj = fj zj [ρ(t)] dt
for j = 1, . . . , n, and all t ∈ R+ ,
(10.46)
ρ(0) = p0 .
(10.47)
and: Under the conditions which we are assuming to hold here, and subject to some mild technical qualiﬁcations (which we will ignore in this discussion), the system (10.43) will have a unique solution, for each initial value p0 ∈ Π. For purposes of the present discussion, let us agree to call a pair ζ, f a price adjustment mechanism iﬀ ζ(·) is a continuous function, and f : Rn → Rn is continuous and signpreserving (that is, each fj is a signpreserving function). Given a pair, ζ, f , we shall say that a function ρ is a solution for the mechanism, given the initial value p0 ∈ Rn++ iﬀ ρ satisﬁes (10.46) and (10.47). We shall say that p∗ ∈ Rn++ is an equilibrim for ζ, f iﬀ ζ(p∗ ) = 0. Notice that, since f is signpreserving, this last condition is equivalent to requiring that: fj [ζj (p∗ )] = 0
for j = 1, . . . , n.
(10.48)
10.24 Deﬁnitions. If ζ, f is a price adjustment mechanism, we shall say that: 1. an equilibrium for ζ, f , p∗ , is globally stable for ζ, f iﬀ we have: lim ρ(t) = p∗ ,
t→∞
(10.49)
17 The system corresponds to the Walrasian idea of tˆ atonnement. We will brieﬂy consider the idea of a nontˆ atonnement process in Section 8, below.
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Chapter 10. Comparative Statics and Stability
given any p0 ∈ Rn++ , and any ρ(·) which is a solution for ζ, f , given the initial value p0 . 2. an equilibrium for ζ, f , p∗ , is locally stable for ζ, f iﬀ there exists a neighborhood of p∗ , N (p∗ ), such that for all p0 ∈ N (p∗ ), and any ρ(·) which is a solution for ζ, f , given the initial value p0 , we have: lim ρ(t) = p∗ .
t→∞
3. the mechanism ζ, f is stable, or that ζ, f posesses system stability, iﬀ for each p0 ∈ Rn++ , and any ρ(·) which is a solution for ζ, f , given the initial value p0 , there exists an equilibrium for ζ, f , p∗ ,, such that: lim ρ(t) = p∗ .
t→∞
In connection with Deﬁnitions 10.21.1 and 10.21.2, it is worth noting that if P ∗ is an equilibrium for z(·), then it follows from (H) that, for all λ ∈ R++ , λP ∗ is also an equilibrium for z(·). Consequently, if we were to deﬁne concepts here for nonnormalized [and (n + 1)dimensional] prices, we cannot have a unique equilibrium price vector; the most we can have is uniqueness up to a scalar multiple. Correspondingly, if we were to deﬁne concepts here for nonnormalized price vectors, we could never have global stability in the sense of Deﬁnition 10.21.1. In the analysis to follow, we will work with a special case of a price adjustment mechanism: namely those ζ, f for which f is the identity function: i(ζ) = ζ
for all ζ ∈ Rn .
(10.50)
Before proceeding further, let’s take a moment to consider the form which Walras’ Law, in the strong version, will take here. If we denote the excess demand function for the (n+1)st commodity by ‘ζn+1 (p);’ then, remembering that we are taking pn+1 to be identically equal to one, we will have, for all p ∈ Π: 0 = p, 1 · ζ(p), ζn+1 (p) = p · ζ(p) + ζn+1 (p); or: p · ζ(p) = −ζn+1 (p).
(10.51)
As a special, but particularly useful case of this, notice that if ρ(·) is a solution for a mechanism, ζ, f , given p0 ∈ Π, then we will have, for all t ∈ R+ : ρ(t) · ζ[ρ(t)] = −ζn+1 [ρ(t)].
(10.52)
Now, if p∗ is an equilibrium for z i i∈M , and ρ(·) is a solution for z i i∈M , given p0 ∈ Π, the distance of ρ(t) from p∗ will be given by: ρ(t) − p∗ = It can be shown that:
n j=1
[ρj (t) − p∗j ]2
lim ρ(t) = p∗ ,
t→∞
1/2
.
(10.53)
10.7. MultiMarket Stability
305
if, for all t, t ∈ R+ such that 0 < t < t , we have: ρ(t ) − p∗ < ρ(t) − p∗ .
(10.54)
Inequality (10.54) will hold in turn if the function V (·) deﬁned: V (t) = (1/2)ρ(t) − p∗ 2 = (1/2)
n j=1
[ρj (t) − p∗j ]2 ,
(10.55)
is strictly decreasing. Because of this relationship, the basic tool which we will use in our analysis is the following result. 10.25 Proposition. Suppose ζ(·) satisﬁes (W) and (C), and that p∗ is an equilibrium for z i i∈M which satisﬁes the following condition (WA*): for all p ∈ Π such that p = p∗ : (10.56) p∗ · ζ(p) + ζn+1 (p) > 0. Then, given any p0 ∈ Π, if ρ(·) is a solution for z i i∈M , given the initial value p0 : dV = − p∗ · ζ[ρ(t)] + ζn+1 [ρ(t)] < 0. dt
(10.57)
Proof. Since ρ(·) satisﬁes (10.46) and (10.47) for the mechanism z i i∈M , the function V (·) is diﬀerentiable in t, and we have: n
n dV [ρj (t) − p∗j ] · [dρj /dt] = [ρj (t) − p∗j ] · ζj [ρ(t)] = (1/2) 2 j=1 j=1 dt n n (10.58) ρj (t) · ζj [ρ(t)] − p∗ · ζj [ρ(t)] = j=1 j=1 j n p∗j · ζj [ρ(t)] = −ζn+1 [ρ(t)] − j=1
or:
dV = − p∗ · ζ[ρ(t)] + ζn+1 [ρ(t)] ; dt
(10.59)
where the second and last equalities in (10.58) are by (10.46) and (10.52), respectively. Our conclusion is then an immediate consequence of (10.59), given (10.56).
The last part of the following result is a more or less immediate consequence of Proposition 10.22, above. We will simply accept the ﬁrst part of the result without a formal proof; hopefully, however, our earlier discussion at least renders it intuitively plausible. 10.26 Theorem. (Arrow and Hurwicz [1958]) If z(·) satisﬁes (W) and (C), then the mechanism z i i∈M posesses a unique solution, ρ(·), for each initial value, p0 ∈ Π. Furthermore, if there exists an equilibrium for z i i∈M , p∗ , which satisﬁes (10.56) of Proposition 10.22, then p∗ is globally stable for z i i∈M . The following result is now a more or less immediate consequence of the above Theorem and Proposition 10.20.
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Chapter 10. Comparative Statics and Stability
10.27 Theorem. (Arrow, Block, and Hurwicz [1959]) If z(·) satisﬁes (H), (W), (C), and the diﬀerential version of gross substitutability, as deﬁned in Section 4, then the mechanism z i i∈M posesses a unique solution, ρ(·), for each initial value, p0 ∈ Π. Furthermore, if p∗ is an equilibrium for ζ(·), then given any p0 ∈ Π, the solution ρ(t; p∗ ) for z i i∈M , given p0 , satisﬁes: lim ρ(t; p0 ) = p∗ .
t→+∞
Unfortunately, not all aggregate excess demand functions satisfy condition (S), nor do they necessarily posess an equilibrium price satisfying (10.56). As a matter of fact, not all mechanisms are stable, as the following example demonstrates. 10.28 Example. (Scarf [1960]) Consider the pure exchange economy in which m = 3, n = 3, and the consumers have the respective utility functions: u1 (x1 ) = min{x11 , x12 }, u2 (x2 ) = min{x22 , x23 }, and
(10.60)
u3 (x3 ) = min{x31 , x33 }; and that: r 1 = (1, 0, 0), r 2 = (0, 1, 0), and r 3 = (0, 0, 1). In this example, we will let the vector p be threedimensional. Thus, for example, the ﬁrst consumer’s excess demand function will be given by:
p −p2 1 , z 1 (p) = ,0 ; p1 + p2 p1 + p2 while the aggregate excess demand equations will be given by: p3 p2 − p 1 + p 3 p1 + p 2 p1 p3 − z2 (p) = p 1 + p 2 p2 + p 3 p1 p2 − z3 (p) = p 2 + p 3 p1 + p 3 z1 (p) =
p1 (p3 − p2 ) , (p1 + p2 )(p1 + p3 ) p2 (p1 − p3 ) = , (p1 + p2 )(p2 + p3 ) p3 (p2 − p1 ) = , (p1 + p3 )(p2 + p3 ) =
(10.61)
respectively; and it is then easy to show that if p∗ is an equilibrium for z(·), we must have: (10.62) p∗1 = p∗2 = p∗3 . Now suppose we have an initial value, p0 ∈ Π satisfying: p0 2 = 3 and p01 · p02 · p03 = 1 (10.63) 4 √ [for example, p0 = (1/ 3, 5/3, 1)]. Then we note ﬁrst that if a solution, ρ(·), exists for z i i∈M , then we must have: 3
d d ρ(t)2 = ρj (t)2 = 2 ρ1 (t)z1 [ρ(t)] + ρ2 (t)z2 [ρ(t)] + ρ3 (t)z3 [ρ(t)] = 0, dt dt j=1
(10.64)
10.8. A Note on NonTˆatonnement Processes
307
for all t; where the last equality is by Walras’ Law. Thus we see that the norm of ρ must remain constant, and it then follows from (10.62) and (10.63) that if ρ(·) converges to an equilibrium, it must converge to the vector: p∗ = (1, 1, 1). def
0 However, again supposing that ρ(·) is a solution for z i i∈M , given p , consider the expression 3j=1 ρj (t; p0 ). We have:
d 3 d 3 ρj (t; p0 ) = pj j=1 j=1 dt dt dp1 dp2 dp3 = · (p2 p3 ) + · (p1 p3 ) + · (p1 p2 ) dt dt dt = z1 (p) · (p2 p3 ) + z2 (p) · (p1 p3 ) + z3 (p) · (p1 p2 );
(10.65)
where the last equation in (10.65) is by the assumption that ρ(·) is a solution for z i i∈M , given p0 . Substituting from (10.61), we then obtain: d 3 ρj (t; p0 ) j=1 dt p1 p2 p3 (p3 − p2 )(p2 + p3 ) + (p1 − p3 )(p1 + p3 ) + (p2 − p1 )(p1 + p2 ) = (p1 + p2 )(p1 + p3 )(p2 + p3 ) p 1 p2 p 3 = (p3 )2 − (p2 )2 + (p1 )2 − (p3 )2 + (p2 )2 − (p1 )2 = 0. (p1 + p2 )(p1 + p3 )(p2 + p3 ) Thus we see that if ρ(·) is a solution for z i i∈M , given p0 , we must have: 3 j=1
ρj (t; p0 ) =
3
p0 j=1 j
= 1;
where the last equality is by our choice of p0 . We see, therefore, that no such solution can converge to p∗ .
10.8
A Note on NonTˆ atonnement Processes
The idea of tˆ atonnement, as originally set forth by Walras, is this. Imagine that all individuals interested in trading in a given commodity gather in a room at an appointed time. An oﬃcial (‘auctioneer’ or ‘referee’) then announces a price for the commodity, and each individual writes on a card the amount he or she would buy (a positive number) or sell (a negative number) at that price. These cards are then passed in to the auctioneer, who adds up the totals. If the excess demand is positive, then the auctioneer announces a new price higher than the original, and if excess demand is negative, price is lowered. This process continues until excess demand is zero at some price; and then, and only then, does any trading actually take place. In our study of stability in the preceding sections, we can be said to have studied whether this tˆ atonnement process will ever come to a halt. Of course, no real market functions exactly like the process just described; and, fortunately, our analysis did
308
Chapter 10. Comparative Statics and Stability
not need explicit assumptions about the institutional structure of the markets’ functionings. However, we did assume throughout our analysis that the excess demand functions remained ﬁxed for the duration of the adjustment process; an assumption which is diﬃcult to justify unless we assume that no trading actually takes place until an equilibrium is achieved. To see the signiﬁcance of this last point, consider a pure exchange economy in which the ith individual’s excess demand function is given by: z i (p) = di (p) − r i .
(10.66)
Agent i’s initial resource endowment obviously aﬀects this excess demand function: in fact, it does so in two ways: (a) directly, as it enters equation (10.66), and (b) indirectly, since in a pure exchange economy, wi (p) = p · r i , and: di (p) = hi [p, wi (p)].
(10.67)
We can formally indicate this dependence by writing agent i’s excess demand function as: (10.68) z i = z i (p; r i ) for i = 1, . . . , m. The basic idea of a nontˆ atonnement process is simply that trade is allowed in the process of achieving equilibrium. Thus, our tˆ atonnementtype adjustment equation [(10.43) of Section 7): dpj = fj [zj (p)] dt
for j = 1, . . . , n;
(10.69)
is no longer valid, and must b e modiﬁed to reﬂect trades. Accordingly, we might replace (10.69) by: dpj for j = 1, . . . , n, = fj zj [p, (r i )] dt drij for i = 1, . . . , m; j = 1, . . . , n; = gij [p, (r i )] dt
(10.70)
where once again each fj is taken to be a signpreserving function, while gij (·) reﬂects the transaction rules governing trade out of equilibrium, and thus must satisfy: m gij [p, (r i )] = 0 for j = 1, . . . , n; and all (p, (r i )). (10.71) i=1
A solution, given the initial values p0 and (r i ), is a function (ρ, γ) mapping R+ into Rn(1+m) satisfying: dρj for j = 1, . . . , n, = fj zj [ρ(t), γ(t)] dt dγij for i = 1, . . . , m; j = 1, . . . , n; = gij [ρ(t), γ(t)] dt
(10.72)
and: ρ(0) = p0 and γ(0) = (r i ).
(10.73)
10.8. A Note on NonTˆatonnement Processes
309
While we will not pursue this topic further here, good introductions to nontˆ atonnement process are provided in Arrow and Hahn [1971, Chapter 13], Negishi [1962, Sections 8  10], and Quirk and Saposnik [1968, pp. 191–3]. Exercises. 1. Show that if the aggregate demand function satisﬁes WA, then it satisﬁes WA*. 2. Verify the details of Example 10.5. 3. Suppose a consumer’s preferences can be represented by the utility function: x x 1 2 u(x) = min . , a1 a2 where a1 , a2 > 0. a. Find the consumer’s demand functions for the two commodities. b. Are the commodities gross substitutes in this case? 4. Construct an example with two consumers and three commodities in which a transfer of a unit of the num´eraire (the third commodity) from the ﬁrst consumer to the second results in an increase in excess demand for the second commodity. 5. Prove Corollary 10.10. 6. Show that if the aggregate supply correspondence is singlevalued, so that we can consider it to be a function, s : Π → Rn , then for all p, p ∈ Π, we have: −(p − p) · s(p ) − s(p) ≤ 0. Furthermore, if s(p) = s(p ), then the above inequality is strict. 7. Verify the details of Example 10.22
Chapter 11
The Core of an Economy 11.1
Introduction
In this chapter, we will be concerned with the core of a production economy. Some of the results and concepts become somewhat more diﬃcult to deal with in this context than would be the case if we conﬁned our attention to a pure exchange economy, but the generality we will gain, and the additional insights obtained in this context more than compensate for the slight added diﬃculty. We will begin our discussion by considering a private ownership economy. In our discussion here, however, we will suppose that E takes the form: E = Xi , Pi , Yk , [sik ] . that is, we will dispense with the explicit display of the consumers’ resourse endowments.1 We will be assuming thoughout that, for each i ∈ M : (Please note that I am changing the notation slightly here; ‘I,’ rather than ‘M ’ was used to denote this set in earlier chapters.) Pi is irreﬂexive, where:
def
M = {1, . . . , m}. We will often be concerned with the allocation of consumption bundles to the consumers; denoting such an allocation by, for example, ‘xi i∈M ,’ which we can think of as vectors in Rmn , or as a sequence of m vectors from Rn . Formally, we deﬁne: 11.1 Deﬁnitions. If E = Xi , Pi , Yk , [sik ] is a private ownership economy, we mn is a consumption allocation for E iﬀ: will say that xi i∈M ∈ R xi ∈ X i
for i = 1, . . . , m;
and is an attainable consumption allocation for E iﬀ there exist y k ∈ Yk (k = 1, . . . , ) such that: m xi = yk . (11.1) i=1
1
k=1
We can account for such endowments either by supposing that the ﬁrst m production sets are of the form Yi = {r i }, with sih = 1 for i = h and 0 for i = h; or by interpreting the Xi as ‘trading sets,’ ` a la Chapter 4.
312
Chapter 11. The Core of an Economy
In other words, xi i∈M is an attainable consumption allocation for E iﬀ there exists y k k∈L such that xi i∈M , y k k∈L ∈ A(E). We will denote the set of all attainable consumption allocations for E by ‘X ∗ (E),’ or simply by ‘X ∗ ’ if the type of economy is understood. In this chapter, we will generally express equation (11.1) as: xi = yk , i∈M
(11.2)
k∈L
where we deﬁne: L = {1, . . . , }. We will be considering possible actions of coalitions of consumers, where a coalition of consumers can be identiﬁed with a subset, S, of M ; the idea here, of course, being that the coalition S ⊆ M consists of those agents (consumers), i, such that i ∈ S. We will denote the collection of all such coalitions, that is, the collection of all nonempty subsets of M , by ‘S.’ When dealing with coalitions, we will need to concern ourselves with the issue of what they could accomplish as a group if they operated as a separate subeconomy, independently of the other consumers. In doing this, we will take a bit diﬀerent approach to the idea of ownership of ﬁrms than has been our custom. Up to this point, when we have considered a private ownership economy, E = (Xi , Pi , Yk , r i , [sik ]), we have supposed that sik (i = 1, . . . , m; k = 1, . . . , ) represented the ith consumer’s (proportionate) share of the proﬁts of the k th ﬁrm. In this chapter, however, we suppose instead that the ith consumer controls the production set Zik , which, in the case of a private ownership economy would generally be deﬁned as: def
Zik = sik Yk = {z ∈ Rn  (∃y k ∈ Yk ) : z = sik y k }.
(11.3)
With this deﬁnition, it is easy to prove the following (see the exercises at the end of this chapter). 11.2 Proposition. Let E = Xi , Pi , Yk , [sik ] be an economy, and p∗ ∈ Rn \ {0}. Then: 1. if y ∗k maximizes p∗ · y on Yk , then: def
z ∗ik = sik y ∗k , maximizes p∗ · z on Zik ; and: 2. if sik > 0, and z ik ∈ Zik maximizes p∗ · z on Zik , then: def
y k = (1/sik )z ik , maximizes proﬁts on Yk . Moreover, 3. if we deﬁne, for p ∈ Π(Yk ) ≡ Πk : π ik (p) = max p · z, z∈Zik
(11.4)
11.1. Introduction
313
then (a) Πk ⊆ Π(Zik ), (b) if sik > 0, then Π(Zik ) = Πk , and (c) for any p ∈ Πk : π ik (p) = sik πk (p) ≡ sik max p · y. y∈Yk
With the ideas of the above paragraph and proposition in mind, we can deﬁne the ith consumer’s production set, Zi , for a private ownership economy, E as: Zik = sik Yk . (11.5) Zi = k=1
k=1
It must be confessed at the outset that the ideas just presented are only completely consistent with our deﬁnition of the attainable set for the economy, A(E), if each Yk is convex and contains 0. We will come back to this point shortly; in the meantime, the following result notes how neatly things do work out if each Yk is convex. 11.3 Proposition. Suppose E = Xi , Pi , Yk , [sik ] is such that Yk is convex, for k = 1, . . . , , and that Zi , (i = 1, . . . , m) is deﬁned as in equation (12.5), above. Then the following holds: If z i ∈ Zi , for i = 1, . . . , m, then: zi ∈ Y ≡ Yk ; i∈M
k∈L
and conversely, if y ∗ ∈ Y , then there exist z ∗i ∈ Zi for i = 1, . . . , m, such that: y∗ = z ∗i . i∈M
Proof. Suppose ﬁrst that z i ∈ Zi , for i = 1, . . . , m. Then, by the deﬁnitions of Zi and Zik , for each i there exist y ik , for k = 1, . . . , , such that: zi = sik y ik . (11.6) k∈L
However, since each Yk is convex, and
def
yk =
i∈M
sik = 1:
sik y ik ,
i∈M
is an element of Yk , for each k ∈ L. Moreover: zi = sik y ik = sik y ik = yk sik = yk . i∈M
i∈M k∈L
k∈L i∈M
k∈L
i∈M
k∈L
Conversely, suppose y ∗ ∈ Y . Then there exist y ∗k ∈ Yk , for k = 1, . . . , , such that: y ∗k . y∗ = k∈L
But then, if for each i, we deﬁne z ∗ik by: z ∗ik = sik y ∗k , it follows from the deﬁnition of Zik that z ∗ik ∈ Zik , for k = 1, . . . , . Furthermore, we have: z ∗ik = z ∗ik = sik y ∗k = y ∗k sik = y ∗k ≡ y ∗ . i∈M k∈L
k∈L i∈M
k∈L i∈M
k∈L
i∈M
k∈L
We will explore these relationships in more depth in the next section.
314
11.2
Chapter 11. The Core of an Economy
Convexity and the Attainable Consumption Set
If you go back over the proof of Proposition 11.3, you can readily verify the fact that the convexity of the Yk sets was not used in the second part of the proof. Consequently, the following corollary follows easily for those who are comfortable with the idea of set summation. 11.4 Corollary. If E = Xi , Pi , Yk , [sik ] is a private ownership economy, then, given the deﬁnitions of the previous section: Y ≡
k∈L
Yk ⊆
Zi .
(11.7)
i∈M
Moreover, if Yk is convex, for each k ∈ L, then we also have:
Zi ⊆ Y.
(11.8)
i∈M
Now let’s return to the issue of why it is that our deﬁnitions are a bit inconsistent unless the Yk are both convex and contain 0. First of all, without convexity of the individual production sets, Yk , the inclusion in equation (12.8) will not necessarily hold; which means that without convexity, we cannot suppose consumer i can produce whatever net output vector z i ∈ Zi is desired, for i = 1, . . . , m. That is, the combined results of such production will not necessarily be feasible, in the aggregate. There is a further diﬃculty, however. In our treatment of production, we have supposed that Yk contains all of the production vectors which can be produced by the k th ﬁrm, and only those production vectors. In other words, Yk contains all feasible production vectors, given the technology available to the k th ﬁrm, and given any ﬁxed factors embodied in the ﬁrm’s production facilities. Suppose now ¯ k ∈ Yk such that for all θ ∈ ]0, 1[, we have: that there exists a production vector y / Yk ; θ¯ yk ∈ in other words, suppose Yk does not satisfy nonincreasing returns to scale. Then, ¯ k ∈ Zik , for each i, sik y ¯k ∈ / Yk . Thus in this case there will be elements while sik y of Zik which cannot actually be produced; for, given the deﬁnition of the sets Zik which was presented in the previous section, it would seem to be logical to deﬁne: ∗ Zik = Zik ∩ Yk ;
(11.9)
as the k th production set actually feasible for i, and the ith consumer’s production set as: ∗ Zik . (11.10) Zi∗ = k=1
If this deﬁnition is reasonable, then it presents some interesting insights into the potential gains from cooperation by the consumers. Consider, for instance, the following example.
11.2. Convexity and the Attainable Consumption Set
315
11.5 Example. Let E be the private ownership economy in which m = n = 2, and in which there is one ﬁrm whose production set is given by: Y = {y ∈ R2  0 ≤ y2 ≤ −y1 & y1 ≤ −4}. Suppose further that: s1 = s2 = 1/2, and that: X1 = X2 = {xi ∈ R2  −3 ≤ xi1 ≤ 0 & xi2 ≥ 1}. Then we have: (−2, 2) ∈ Xi ∩ Zi , if we simply deﬁne Zi = si Y . However, if Zi∗ is deﬁned as in equation (11.10), then it is easy to see that: Zi∗ = Y for i = 1, 2; which means that, for each i:
Xi ∩ Zi∗ = ∅.
In other words, neither consumer can survive utilizing only her or his own resources! On the other hand, suppose the consumers combine forces to form a ﬁrm with production set Y as originally given, so that: Y = Z1 + Z2 = (1/2)Y + (1/2)Y1 = Y. Then, for example the consumption bundles xi given by: x1 = x2 = (−2, 2), are such that xi ∈ Xi for each i, and: x1 + x2 = (−4, 4) ∈ Y ; so that (xi , z i )i∈M is attainable.
If, however, 0 ∈ Yk and Yk is convex (in other words, if Yk satisﬁes nonincreasing ∗ disappears; returns, see Proposition 6.2), then the distinction between Zik and Zik in fact, we have the following, the proof of which I will leave as an exercise. ∗ are 11.6 Proposition. If Yk is convex and contains the origin, and Zik and Zik ∗. deﬁned as in (11.3) and (11.9), respectively, then Zik = Zik , In the remainder of this chapter, we consider economies E = (Xi , Pi , Zi ) , where Xi and Zi are nonempty subsets of Rn , and Pi is an irreﬂexive binary relation on Xi , for i = 1, . . . , m. We will also assume that, for each i:
Xi ∩ Zi = ∅. We will not be assuming that the Zi sets are necessarily of the form: Zik = sik Yk , Zi = k∈L
k∈L
(11.11)
316
Chapter 11. The Core of an Economy
and thus that the economy E is derived from a private ownership economy, E; however, we will not be ruling this possibility out either. Consequently, it will be worth our while to consider some further aspects of the relationship between E and E if the former is derived from the latter. We can begin by introducing the deﬁnition , we will use for attainable allocations and competitive equilibria in economies E = (Xi , Pi , Zi ) . 2mn is a feasible allocation 11.7 Deﬁnitions. We  will say that (xi , z i )i∈M ∈ R , for E = (Xi , Pi , Zi ) iﬀ:
(xi , z i ) ∈ Xi × Zi and:
for i = 1, . . . , m,
(xi − z i ) = 0.
i∈M
As we did in the case of private ownership economies, we will denote the set of all attainable consumption allocations for E by ‘X ∗ (E).’ ∗ ∗ ∗ 11.8 Deﬁnition. We will say that , a tuple, xi , z i , p is a competitive (or Walrasian) equilibrium for E = (Xi , Pi , Zi ) iﬀ 1. p∗ = 0, 2. (x∗i , z ∗i )i∈M is an attainable allocation for E, 3. for each i, we have: (∀z i ∈ Zi ) : p∗ · z i ≤ p∗ · z ∗i , p∗ · x∗i ≤ p∗ · z ∗i and:
(∀xi ∈ Xi ) : xPi x∗i ⇒ p∗ · xi > p∗ · z ∗i .
The following proposition, the proof of which I will leave as an exercise, sets forth the relationship between  equilibria for a private ownership economy, E, , competitive and an economy E = (Xi , Pi , Zi ) , assuming that the latter is derived from E as per equation (11.11). 11.9 Proposition., Suppose E = Xi , Pi , Yk , [sik ] is a private ownership economy, and that E = (Xi , Pi , Zi ) is derived from E as per equation (11.5). Then we have the following. 1. If (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E, and we deﬁne z ∗i by: z ∗i = sik yk∗ , k∈L
for i = 1,. . . , m, then x∗i , z ∗i , p∗ is a competitive equilibrium for E. ∗ ∗ ∗ 2. If xi , z i , p is a competitive equilibrium for E, then z ∗i is of the form: sik y ik , z ∗i = k∈L
for i = 1, . . . , m: and if each Yk is convex, and we deﬁne y ∗k by: sik y ik , y ∗k = i∈M
for i = 1, . . . , m, then (x∗i , y ∗k , p∗ ) is a competitive equilibrium for E.
11.3. The Core of a Production Economy
11.3
317
The Core of a Production Economy
, As mentioned earlier,in this chapter when we say E = (Xi , Pi , Zi ) is an economy, we will always suppose that the following condition holds: Xi ∩ Zi = ∅
for i = 1, . . . , m;
(11.12)
¯ i ∈ Zi such ¯ i ∈ Xi and z in other words, for each i ∈ M , we suppose that there exist x that: ¯i. ¯i = z (11.13) x The assumption expressed as equation (11.12) is fairly restrictive; in a modern industrialized society, individuals specialize in the expectation of being able to purchase (or trade for) necessities which they themselves do not produce. On the other hand, in much of the literature, it is supposed that Xi = Rn+ for each i, and that r i ∈ Rn+ as well; consequently, the condition in (11.12) generalizes the assumption commonlyused. In any event, the present assumption will be important in our development here; so much so that we will make use of the following deﬁnition. , 11.10 Deﬁnition. In an economy E = (Xi , Pi , Zi ) we deﬁne the (individually) ∗ attainable set for i, Xi , by:
that is: The deﬁnition of
Xi∗ = Xi ∩ Zi ;
(11.14)
Xi∗ = xi ∈ Xi  (∃z i ∈ Zi ) : xi = z i .
(11.15)
Xi∗
is extended to coalitions of consumers in the following.2
11.11 Deﬁnition. Let S be a nonempty subset of M (so that S ∈ S). We will say that (xi , z i )i∈S is attainable for S, or feasible for S, iﬀ: xi ∈ Xi and z i ∈ Zi for all i ∈ S, and:
i∈S
xi =
zi.
(11.16)
(11.17)
i∈S
11.12 Deﬁnition. Let x∗i i∈M be a consumption allocation for E, and let S ∈ S be a coalition. We shall say that x∗i i∈M can be improved upon by the coalition S (or is blocked by S) iﬀ there exists an allocation, (xi , z i )i∈S , which is feasible for S, and satisﬁes: (11.18) (∀i ∈ S) : xi Pi x∗i . , 11.13 Deﬁnition. The core of an economy E = (Xi , Pi , Zi ) , is deﬁned as the set of all attainable consumption allocations for E which cannot by improved upon (or blocked) by any coalition, S ∈ S. We will denote the set of all core allocations for E by ‘C(E).’ 2 Remember that a ‘coalition’ is simply a nonempty subset of M . The collection of all coalitions available in an economy is then the set of all nonempty subsets of M , which set we denote by ‘S.’
318
Chapter 11. The Core of an Economy
Notice that C(E) ⊆ X ∗ (E). In the material to follow, we will frequently be concerned with another subset of X ∗ (E), deﬁned as follows. , 11.14 Deﬁnition. Given an economy, E = (Xi , Pi , Zi ) , and a consumer h ∈ M , ∗ we shall say that a consumption allocation xi is individually rational for h iﬀ for every xh ∈ Xh∗ : x∗h Gh xh , where ‘Gh ’ denotes the negation of Ph . We denote the set of all individually rational allocations for h by I ∗h (E); and, ﬁnally, we deﬁne the set of individually rational allocations for E, denoted by ‘I(E),’ by: I ∗i (E). I(E) = i∈M
∗
Notice that C(E) ⊆ I(E) ⊆ X (E) [to see the ﬁrst inclusion, consider the coalition S = {i}, for an arbitrary i ∈ M ]. Moreover, if we denote the set of all Pareto eﬃcient consumption allocations for E by ‘P (E),’ then we also have: C(E) ⊆ P (E) ∩ I(E). Of course, if M = 2, then the only nonempty subsets of M are {1}, {2}, and {1, 2}; and thus C(E) = I(E) ∩ P (E), which should explain the Edgeworth Box diagram, Figure 11.1, below.3 x12 x 21 C(Ᏹ)
I(Ᏹ) (ri )
x 11 x 22
Figure 11.1: Individually Rational Allocations and the Core. It may be helpful at times to consider a slightly diﬀerent, but logically equivalent way of deﬁning the core of an economy. We begin by deﬁning, for each S ∈ S, the 3 In which we have an exchange economy in mind, of course. Notice that in terms of the framework , being utilized here, E = (Xi , Pi , Zi ) is an exchange economy if, and only if for each i ∈ M, Xi = n n R+ and there exists r i ∈ R+ such that Zi = {r i }.
11.3. The Core of a Production Economy
319
collection of all attainable allocations for E which can be improved upon by S, by ‘D(S), that is: xi = zi D(S) = x∗i i∈M ∈ X ∗ (E)  ∃(xi , z i )i∈S : i∈S
i∈S
and (∀i ∈ S) : xi ∈ Xi , z i ∈ Zi & xi Pi x∗i
(11.19)
We can then deﬁne the core of E by: C(E) = X ∗ (E) \
D(S).
(11.20)
S∈S
By analogy with the deﬁnition of ‘individually rational’ allocations, we might say that the collection of core allocations, C(E), is exactly the set of all allocations which are ‘coalitionrational’ for each possible coalition, S. Because of this, we will think of the core as being primarily a welfare criterion; that is, it would appear that there is at least some interest in studying C(E) on the grounds that many, if not most people might accept the criterion that a ‘good’ allocation should be in the core.4 However, many economists, for example W. Hildenbrand and A. P. Kirman, have felt that the core of an economy is of great interest as a solution concept; that is, they feel that the natural outcome of unfettered economic activity is that one attains an outcome in C(E). In fact, this is the case if the economy is competitive in the classical sense, as is shown by the following result.5 We will not prove Theorem 11.15 here, incidentally, since it is a special case of Theorem 11.17, which we will present and prove in the next section. 11.15 Theorem. If x∗i , z ∗i , p∗ is a competitive (or Walrasian) equilibrium for E, then x∗i i∈M is a core allocation for E; that is: x∗i i∈M ∈ C(E). , 11.16 Deﬁnitions. Given an economy, E = (Xi , Pi , Zi ) , we deﬁne the set of all Walrasian allocations for E, W(E), by: W(E) = (x∗i , z ∗i )i∈M ∈ A(E)  ∃ p∗ ∈ Rn : ∗ ∗ ∗ xi , z i , p is a Walrasian equilibrium for E We then deﬁne the set of Walrasian consumption allocations for E, W (E), by: W (E) = x∗i i∈M ∈ X ∗ (E)  (∃z ∗i ) : (x∗i , z ∗i )i∈M ∈ W(E) . Thus, it follows from Theorem 11.15 that W (E) ⊆ C(E). Notice also that Theorem 11.15 generalizes the ‘First Fundamental Theorem of Welfare Economics;’ or, more speciﬁcally, the version of this result presented as Theorem 7.22. In general, 4 Although many would also specify the further condition that this is true only after an appropriate redistribution of initial endowments takes place. 5 We will also consider the idea of the core as a solution, or equilibrium concept in more detail in Chapter 16.
320
Chapter 11. The Core of an Economy
of course, the set C(E) is considerably larger than W (E). We will show, however, that with a suﬃciently large number of agents, the situation changes; in some sense, as the economy ‘grows large,’ the core ‘shrinks’ to the set of competitive equilibria. In the literature on the core, there are two basic approaches to an exact statement and proof of this last assertion; the ﬁrst of which derives from a result which was ﬁrst rigorously stated and proved by Debreu and Scarf [1963], although it was suggested earlier by Edgeworth, and the second of which is derived from results by Aumann [1964] and Arrow and Hahn [1971]). These two approaches yield results which can be (and have been) rather imperfectly stated in two distinct fashions, as follows. 1. As the number of agents in the economy grows, C(E) ‘shrinks’ to the set of Walrasian allocations, W (E). 2. If an economy is ‘suﬃciently large,’ then any allocation in the core is a Walrasian (competitive equilibrium) allocation. The DebreuScarf result is a formalization of statement 1, and in which the number of agents in the economy grows large in a very speciﬁc way; while the Aumann/ArrowHahn theorem more directly relates to statement 2. We will conﬁne our discussion here to results along the lines of the DebreuScarf result; that is, to results along the lines of statement 1, above. For discussions of the second approach, let me recommend Anderson [1978, 1986], and Hildenbrand [1982].
11.4
The Core in Replicated Economies
, Given an economy, E = (Xi , Pi , Zi ) , we consider the sequence of related economies, Eq , deﬁned in the following way. E1 = E, ...... Eq = (Xhi , Phi , Zhi )(h,i)∈ Q×M , where Q = {1 . . . , q}, and : Xhi = Xi , Phi = Pi , and Zhi = Zi for h = 1, . . . , q; i = 1, . . . , m. Thus, in Eq , the agents (consumers) have a double index; agent (h, i) is the hth agent of the ith type. If h, h ∈ Q, and i ∈ M , then agents (h, i) and (h , i) are ‘economic twins,’ in the sense of having precisely the same economic characteristics. We will refer to Eq as the qfold replication of E. In dealing with Eq , we will use the notation ‘xhi (h,i)∈Q×M ’ to denote consumption allocations for Eq ; and we deﬁne X ∗ (Eq ) as the set of all consumption allocations xhi (h,i)∈Q×M such that there exist z i i∈M satisfying: xhi ∈ Xi & z hi ∈ Zi and:
h∈Q i∈M
for h = 1, . . . , q; i = 1, . . . , m,
xhi =
h∈Q i∈M
z hi
(11.21)
(11.22)
11.4. The Core in Replicated Economies
321
We will show that, in a sense to be explained shortly, as q → ∞, C(Eq ) ‘shrinks to W (E).’ Our basic approach will revolve around the study of the sets C q , deﬁned as the set of all feasible allocations, xi i∈M ∈ X ∗ (E) such that the allocation xhi (h,i)∈Q×M given by: xhi = xi
for h = 1, . . . , q; i = 1, . . . , m;
(11.23)
∗
is in C(Eq ). In other words, C q is the projection on X (E) of the allocations xhi (h,i)∈Q×M from C(Eq ) which have the property that: xhi = xh i
for h, h = 1, . . . , q; i = 1, . . . , m.
(11.24)
Notice, incidentally, that if xi i∈M is a feasible consumption allocation for E, and we deﬁne xhi (h,i)∈Q×M as in equation (11.23), then xhi (h,i)∈Q×M is a feasible consumption allocation for Eq . The following result generalizes the ‘First Fundamental Theorem of Welfare Economics,’ in the version presented as Theorem 7.22. Moreover, it establishes the fact that if W (E) = ∅, then C q = ∅, for q = 1, 2, . . . . , 11.17 Theorem. For any economy, E = (Xi , Pi , Zi ) , we have: 1. W (E) ⊆ C q , and 2. C q+1 ⊆ C q , for q = 1, 2, . . . . Proof. 1. Suppose x∗i i∈M ∈ W (E), let z ∗i i∈M and p∗ be such that x∗i , z ∗i , p∗ is a Walrasian equilibrium for E, and deﬁne the consumption allocation ¯ xhi (h,i)∈Q×M for Eq by: ¯ hi = x∗i for h = 1, . . . , q; i = 1, . . . , m. x (11.25) If we suppose, by way of obtaining a contradiction, that ¯ xhi (h,i)∈Q×M is not in C(Eq ), then there exists a coalition S ⊆ Q × M which can improve upon ¯ xhi (h,i)∈Q×M ; and thus there exists xhi , z hi (h,i)∈Q×M such that:
and:
¯ hi xhi Pi x
for all (h, i) ∈ S,
(11.26)
z hi ∈ Zi
for all (h, i) ∈ S,
(11.27)
xhi =
(h,i)∈S
z hi .
(11.28)
(h,i)∈S
However, since x∗i , z ∗i , p∗ is a competitive equilibrium for E, it follows from (11.26) and (11.27) that: p∗ · xhi > p∗ · z ∗hi
for all (h, i) ∈ S.
(11.29)
From (11.29) we then obtain, upon adding over all (h, i) ∈ S: p∗ · xhi = p∗ · xhi > p∗ · z hi = p∗ · z hi ; (h,i)∈S
(h,i)∈S
(h,i)∈S
(h,i)∈S
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Chapter 11. The Core of an Economy
which contradicts (12.28). 2. To see why condition 2 must hold, consider an allocation, x∗i i∈M , in C q+1 . Then the allocation xhi (h,i)∈(Q+1)×M deﬁned by: xhi = x∗i
for h = 1, . . . , q + 1; i = 1, . . . , m,
is such that no coalition, S, from (Q + 1) × M , can improve upon xhi (h,i)∈(Q+1)×M . But then it follows that the consumption allocation xhi (h,i)∈Q×M deﬁned by: xhi = x∗i
for h = 1, . . . , q; i = 1, . . . , m,
must be in the core for Eq as well; since any coalition from Q×M which could improve upon it could also improve upon the consumption allocation xhi (h,i)∈(Q+1)×M in Eq+1 . Therefore, x∗i i∈M is in C q , and it follows that: C q+1 ⊆ C q .
Notice that it is an immediate consequence of Theorem 11.17 that: ∞ W (E) ⊆ Cq; q=1
and thus that if W (E) = ∅, then: ∞ q=1
C q = ∅.
It also follows from 11.17 that for all q: Cq =
q s=1
C s;
and thus it is natural to write: ∞ q=1
C q = lim C q . q→∞
(11.30)
Debreu and Scarf [1963] showed that given any exchange economy, E = (Pi , r i )i∈M , satisfying certain assumptions, we will have: ∞ Cq ⊆ W (E); q=1
which, when combined with Theorem 11.17 and equation (11.30) means that under the DebreuScarf conditions, we have: lim C q = W (E).
q→∞
(11.31)
We will prove a generalization of their result; one which applies to a private ownership economy with production. However, as was the case when we studied the ‘Second Fundamental Theorem of Welfare Economics,’ we will begin by introducing the idea of a ‘quasicompetitive equilibrium;’ this time deﬁning said equilibrium for a private ownership economy. Since you can probably guess exactly how this deﬁnition will be stated, we will present it here in abbreviated form.
11.4. The Core in Replicated Economies
323
∗ ∗ ∗ 11.18 Deﬁnition. We shall say that xi , z i, p is a quasicompetitive equi , librium for the economy E = (Xi , Pi , Zi ) , iﬀ x∗i , z ∗i , p∗ satisﬁes conditions 1–3 of Deﬁnition 11.8, and: 4 . for each i ∈ M , we have: a. p∗ · x∗i ≤ p∗ · z ∗i , and: b. either: p∗ · z ∗i = min p∗ · Xi , or: (∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > p∗ · z ∗i . We will denote the set of all consumption allocations, x∗i ∈ X ∗ (E), for which there exists a production allocation z ∗i i∈M and a price vector p∗ such that x∗i , z ∗i , p∗ is a quasicompetitive equilibrium for E by ‘W † (E).’ In our initial result, we will establish conditions suﬃcient to ensure that: ∞ q=1
C q ⊆ W † (E).
In our proof, which owes a great deal to McKenzie [1988] and Nikaido [1968, Theorem 17.4, p. 291], we will need to make use of the following mathematical result, the proof of which is provided in the appendix to this chapter. 11.19 Proposition. If Ci ⊆ Rn is convex and nonempty, for i = 1, . . . , m, then def 5 the convex hull of C = m i=1 Ci , co(C), is given by: m ai xi . co(C) = x ∈ Rn  (∃a ∈ ∆m & xi ∈ Ci , for i = 1, . . . , m) : x = i=1 (11.32) The proposition just stated is one of those rather frustrating little results which appears too obvious to really need a formal proof, but for which the development of a rigorous proof is nonetheless a somewhat tricky task. Please note, however, that the conclusion no longer holds if the Ci ’s are not all convex; that is, the convex hull of C is not generally given by the formula in equation (11.32) if the sets Ci are not all convex.6 , 11.20 Theorem. If E = (Xi , Pi , Zi ) is an economy such that: 1. Zi is convex, for i = 1, . . . , m; and, for each i ∈ M : 2. Xi is convex and Pi is locally nonsaturating, lower semicontinuous, and weakly convex, and: 3. Xi ∩ Zi = ∅, then: ∞ C q ⊆ W † (E). q=1
6 We provide an example in which the formula of equation (11.32) does not hold in the appendix to this chapter.
324
Chapter 11. The Core of an Economy
Proof. Suppose x∗i i∈M ∈ C q for all q, deﬁne Pi = Pi x∗i − Zi , for each i ∈ M ,7 and:
m P = co Pi ; i=1
that is, P is the convex hull of the union of the Pi ’s. The tricky part of the proof is to establish the fact that 0 ∈ / P. Suppose, by way of establishing a contradiction, that 0 ∈ P. Then, since each Pi is a convex set (and nonempty, by the assumption that each Pi is locally nonsaturating), it follows from Proposition 11.19 that there exist a ∈ ∆m , xi ∈ Xi , and z i ∈ Zi for i = 1, . . . , m, such that: m ai (xi − z i ) = 0, (11.33) i=1
and: xi Pi x∗i
for i = 1, . . . , m.
(11.34)
We will show that these two conditions allow us to construct a coalition in Eq∗ , for some (ﬁnite) integer, q ∗ , which can improve upon x∗i i∈M ; contradicting the assumption that x∗i i∈M ∈ C q , for all q. Accordingly, we begin by noting that (11.33) implies: m i=1
ai xi =
m i=1
ai z i .
(11.35)
We then deﬁne I = {i ∈ M  ai > 0}, and, for each i ∈ M and each positive integer, q, we let bqi be the smallest integer greater than or equal to qai . Now, by assumption i ∈ Zi such that: i ∈ Xi and z 3, for each i ∈ M there exist x i = z i . x
(11.36)
i to deﬁne, for each i ∈ M and each positive integer, q: We make use of the x
qa
qa i i i; xqi = (11.37) x q xi + 1 − bi bqi and note that, since each Pi is lower semicontinuous, and since: qai → 1 as q → ∞, bqi it follows from (11.34) that for each i ∈ M , there exists a positive integer, qi such that for all q ≥ qi , xqi Pi x∗i . (11.38) But now let: q ∗ = max qi , i∈M
7
That is:
Pi = v ∈ Rn  (∃xi ∈ Xi & z i ∈ Zi ) : xi Pi x∗i & v = xi − z i .
11.4. The Core in Replicated Economies ∗
325
∗
let b∗ = max{bq1 , . . . , bqm } and consider the coalition, S, in Eb∗ consisting of bqi consumers of each type i ∈ M , and the allocation ¯ xhi (h,i)∈S deﬁned by: ¯ hi = xqi x
∗
∗
for h = 1, . . . , bqi , and each i ∈ M.
∗
(11.39)
¯ hi Phi x∗i for each h and each i ∈ I; while by using (11.37), (11.36), and We have x (11.35) in turn, we have: bqi ∗
i∈M
h=1
∗ ∗ ∗ ∗ ¯ hi = x bq xqi = xi (q ai )xi + (bqi − q ∗ ai ) i∈M i i∈M ∗ i + i = q∗ ai xi − q ∗ ai z bq z i∈M i∈M i
i∈M ∗ i + i = q∗ ai z i − q ∗ ai z bq z i∈M i∈I i∈M i
q∗a ∗ q ai i q∗ i . z =+ b ∗ q∗ z i + z i − i∈M i bi bqi
Thus, since each Zi is convex, it follows that the coalition S can improve upon x∗i i∈M ; contradicting the assumption that x∗i i∈M ∈ Cq for all positive integers, q. Therefore 0 ∈ / P. Since we have now established the fact that 0 ∈ / P, it follows from the Separating Hyperplane Theorem (Theorem 6.21) that there exists a nonzero p∗ ∈ Rn satisfying: (∀v ∈ P) : p∗ · v ≥ 0.
(11.40)
From the deﬁnition of P, it then follows immediately that for each i ∈ M , we have: (∀xi ∈ Xi & z i ∈ Zi ) : xi Pi x∗i ⇒ p∗ · xi ≥ p∗ · z i .
(11.41)
Moreover, since Pi is locally nonsaturating, it then follows easily that, for each i and each z i ∈ Zi : (11.42) p∗ · x∗i ≥ p∗ · z i . Now, since x∗i i∈M ∈ C q , for each q, it follows from the deﬁnitions that there exists z ∗i i∈M such that: x∗i = z ∗i . (11.43) i∈M
i∈M
But then, letting i ∈ M and z †i ∈ Zi be arbitrary, we see that it follows from (11.42) and (11.43) that, if we deﬁne z i i∈M ∈ i∈M Zi by: z ∗h for h = i, zh = z †i for h = i, we have:
h∈M
so that:
p∗ · z ∗h ≥
p∗ · z ∗h + p∗ · z †i ;
h=i
p∗ · z ∗i ≥ p∗ · z †i .
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Chapter 11. The Core of an Economy
Finally, we let i ∈ M be arbitrary once again. Then from (11.42), (11.43) we have: def p∗ · x∗i = wi (p∗ ) = p∗ · z ∗i .. Furthermore, it follows from (11.41) that: (∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi ≥ wi (p∗ ); and thus from Proposition 7.25, it follows that either: wi (p∗ ) = min p∗ · Xi , or:
(∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > wi (p∗ ). Therefore, x∗i , z ∗i , p∗ is a quasicompetitive equilibrium for E.
We can strengthen the conclusion of the theorem just proved to conclude that if:
x∗i ∈
∞ q=1
Cq,
and satisﬁes an irreducibility condition, then x∗i ∈ W (E). In order to introduce this condition, we ﬁrst deﬁne the following. , 11.21 Deﬁnition. If E = (Xi , Pi , Zi ) is an economy, and p∗ ∈ Rn , we will say m that w ∈ R is a feasible wealth distribution for E iﬀ for each i ∈ M , there exists xi ∈ Xi such that p∗ · xi ≤ wi , and: wi = sup p∗ · z, z∈Z
i∈M
where we deﬁne Z by: Z=
Zi .
i∈M
We can then deﬁne x∗i , z ∗i , p∗ to be a quasi competitive equilibrium for E, ∗ given the wealth distribution, w ∈ Rm in the obvious way: requiring that w∗ be a feasible wealth distribution for E, given p∗ , and requiring that Deﬁnition 11.18 be satisﬁed, with wi∗ replacing p∗ · z ∗i in condition 4 of the deﬁnition. Our irreducibility condition is then deﬁned as follows. , 11.22 Deﬁnition. We shall say that the economy, E = (Xi , Pi , Zi ) is irreducible at the consumption allocation x∗i ∈ X ∗ (E) iﬀ, given any partition of the consumers, {S1 , S2 },8 there exists (xi , z i )i∈S such that: (xi , z i ) ∈ Xi × Rn for i = 1, . . . , m, m z i ∈ Z, i=1 (xi − z i ) = (z i − xi ), i∈S1
(11.44) (11.45) (11.46)
i∈S2
8 By a partition of the consumers, {S1 , S2 }, we mean Sj ⊆ M & Sj = ∅, for i = 1, 2, S1 ∩ S2 = ∅, and S1 ∪ S2 = M .
11.4. The Core in Replicated Economies and:
(∀i ∈ S1 ) : xi Pi x∗i .
327
(11.47)
We will denote the set of consumption allocations at which E is irreducible by ‘X I (E).’ Those with good memories will undoubtedly already have noticed that the above deﬁnition is a straightforward modiﬁcation of the condition of the same name which was presented in Chapter 7 (Deﬁnition 7.33). in the present context the vectors z i have a more natural interpretation than was the case in Chapter 7, in that here the natural interpretation of the points z i is to suppose that they are elements of Zi ; in which case, equation (11.45) is necessarily satisﬁed. However, notice that our condition does not require that z i ∈ Zi . In any case, the following result is a more or less immediate application of Theorem 11.20 and the proof of Theorem 7.36. I will leave the details of the proof as an exercise. , 11.23 Theorem. If E = (Xi , Pi , Zi ) is an economy such that: 1. int(X) ∩ Z = ∅, and: for each i ∈ M : 2. Xi and Zi are convex sets, 3. Pi is locally nonsaturating, lower semicontinuous, and weakly convex, and 4. Xi ∩ Zi = ∅, then: ∞ C q ⊆ W (E). X I (E) ∩ q=1
We can also make good use of the deﬁnition of a num´eraire good in this context. For convenience, I will repeat the deﬁnition here (modiﬁed for our deﬁnition of an economy, E). 11.24 Deﬁnitions. We will say that the j th commodity is a num´ eraire good for Pi iﬀ for all x ∈ Xi and all θ ∈ R++ ,9 we have: x + θej ∈ Xi and (x + θej )Pi x,
(11.48)
th where ej is the j th unit coordinate vector.10 We , shall saythat the j commodity is a num´ eraire good for the economy, E = (Xi , Pi , Zi ) iﬀ it is a num´eraire good for each Pi , and for each i ∈ M there exists θi > 0 such that:
Xi ∩ [Zi − θi ej ] = ∅. 11.25 Theorem. If E = (Xi , Pi , Zi ) is an economy such that: 1. int(X) ∩ Z = ∅, and: for each i ∈ M : 2. Xi and Zi are convex sets, 3. Pi is weakly convex and lower semicontinuous, and:
(11.49)
,
9 10
one.
Where R++ = {x ∈ R  x > 0}. The vector having all coordinates equal to zero except for the j th coordinate, which is equal to
328
Chapter 11. The Core of an Economy
4. for some j ∈ {1, . . . , n}, the commodity j is a num´eraire good for E, then: ∞ C q = W (E). q=1
6∞Proof. Since it is an immediate implication of Theorem 11.17 that W (E) ⊆ q=1 C q , we need only prove the converse. Accordingly, let j ∈ {1, . . . , n} be the num´eraire good for E, and note that it then follows from (11.48) and (11.49) of Deﬁnition 11.24 that, for each i ∈ M : Xi ∩ Zi = ∅. Consequently, since each Pi is locally nonsaturating by virtue of the fact that commodity j is a num´eraire good for E, it follows from Theorem 11.20 that if x∗i i∈M ∈ 6∞ ∗ n ∗ ∗ ∗ ∗ is a C q , then there exists p ∈ R \ {0} and z i i∈M such that xi , z i , p q=1 quasicompetitive equilibrium for E. It also follows from assumption 2 that there def exists x ∈ X = i∈M Xi , θ ∈ R++ , and z ∈ i∈M Zi such that: x − θp∗ ∈ X and x = z. Thus, as in the proof of Theorem ?? we see that there must exist at least one h ∈ M such that: p∗ · z ∗h > min p∗ · Xi ; so that, by deﬁnition of a quasicompetitive equilibrium: (∀xh ∈ Xh ) : xh Ph x∗h ⇒ p∗ · xh > p∗ · z ∗h . However, since commodity j is a num´eraire good for Ph , we recall that for any ∆xj > 0, we have: (x∗h + ∆xj ej )Ph x∗h , where ej is the (j )th unit coordinate vector. It then follows that we must have p∗j > 0. Now let i ∈ M be arbitrary. Then, by deﬁnition of a num´eraire good for E, there ¯ i ∈ Xi , z¯i ∈ Zi , and θj > 0 such that: exists x ¯ i − θj ej = z ¯i. x and, since
p∗j
> 0, it then follows that: ¯ i < p∗ · z ¯i . p∗ · x
(11.50)
Moreover, it follows from the deﬁnition of a quasicompetitive equilibrium that: ¯i ≤ p∗ · z ∗i . p∗ · z
(11.51)
From (11.50), (11.51), and the deﬁnition of a quasicompetitive equilibrium, it now follows that: (∀xi ∈ Xi ) : xi Pi x∗i ⇒ p∗ · xi > p∗ · z ∗i , ∗ ∗ ∗ and we see that xi , z i , p is a Walrasian equilibrium for E.
11.5. Equal Treatment
329
This last result has an interesting corollary for the case of a pure exchange economy, E = (Pi , r i )i∈M ; where, remember that if E is a pure exchange economy, we will always assume that: Xi ⊆ Rn+ and r i ∈ Xi
(11.52)
(and, of course, that Pi is irreﬂexive). I will leave the details of the proof as an exercise. 11.26 Corollary. If E = (Pi , r i )i∈M is a pure exchange economy such that: 1. r = i∈M r i 0, 2. Pi is weakly convex and lower semicontinuous, for i = 1, . . . , m; and 3. for some j ∈ {1, . . . , n}, the commodity j is a num´eraire good for E, then: ∞ C q = W (E). q=1
11.5
Equal Treatment
Under somewhat stronger assumptions than those we used, the only allocations in C(Eq ) are those generated from Cq . More precisely, under familiar convexity assumptions, one can show that, if (xhi , z hi )(h,i)∈Q×M is in C(Eq ), then: xhi = xh i
for h, h = 1, . . . , q, and i = 1, . . . , m.
(11.53)
The basic fact from which this statement can be proved is the following. The proof of the result, which is an easy consequence of the deﬁnitions, I will leave as an exercise. , 11.27 Proposition. Suppose E = (Xi , Pi , Zi ) is an economy, that Eq is the qfold replication of E, and that (xhi , z hi )(h,i)∈Q×M is a feasible allocation for Eq , whiere q ≥ 2. Then, letting η be any function mapping M into Q, and letting S be the coalition formed by taking the η(i)th agent of type i, for each i; that is: S = η(1), 1 , η(2), 2 , . . . , η(m), m , the allocation (x∗i , z ∗i )i∈S deﬁned by: m (x∗η(i),i , z ∗η(i),i ) = (1/q) (xhi , z hi ) h=1
for i = 1, . . . , m,
is feasible for S. One can then use the following result, together with Proposition 11.27, to prove the ‘equal treatment’ result just quoted. 11.28 Proposition. Let P be an asymmetric, negatively transitive, and convex binary relation on the convex set X ⊆ Rn , let q be a positive integer greater than one, and let x, x1 , . . . , xq ∈ X and a ∈ ∆q satisfy: ¬xP xh and, for some
h
for h = 1, . . . , q,
∈ {1, . . . , q}: xh P x and ah > 0.
Then we have:
q h=1
ah xh P x.
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Chapter 11. The Core of an Economy
11.6
Appendix
Proof of Proposition 11.14. In our proof, we will make use of the sets Di deﬁned by: i−1 D1 = C1 , and Di = Ci \ Ch for i = 2, . . . , m. h=1
¯ i ∈ Ci . We also choose, for each i, an arbitrary element, x Since C ⊆ Rn , the convex hull of C, co(C), is given by:11 n+1 co(C) = x ∈ Rn  (∃b ∈ ∆n+1 & xj ∈ C, for each j) : x = bj xj .
(11.54)
j=1
Thus, if x is an arbitrary element of co(C), there exist b ∈ ∆n+1 , and xj ∈ C, for j = 1, . . . , n + 1 such that:12 n+1 x= bj xj . (11.55) j=1
However, each xj is an element of one of the Ci ’s. Consequently, we can represent x by a diﬀerent formula, as follows. For each i ∈ {1, . . . , m}, we deﬁne the set J(i) ⊆ {1, . . . , n + 1} by: J(i) = j ∈ {1, . . . , n + 1}  xj ∈ Di & bj = 0 .
Next deﬁne:
j∈J(i) bj
ai =
if J(i) = ∅,
and: x∗i =
(11.57)
if J(i) = ∅,
0
j∈J(i) (bj /aj )xj
if J(i) = ∅,
(11.58)
if J(i) = ∅.
x¯i
(11.56)
Since Ci is convex (and, since Di ⊆ Ci , for each i), x∗i ∈ Ci , for each i; and from our deﬁnitions, it is obvious that: m x= ai x∗i ; i=1
and that: ai ≥ 0 for i = 1, . . . , m, and
m i=1
ai = 1.
In the text of this chapter, I promised to present an example showing that the conclusion of Proposition 11.14 does not necessarily hold if the sets Ci are not all convex. Here is the promised example. 11.29 Example. Deﬁne the sets C1 and C2 in R3 by: C1 = {x ∈ R3+  0 ≤ x1 ≤ 1, & x2 = x3 = 0}∪{x ∈ R3+  x1 = x3 = 0 & 0 ≤ x2 ≤ 1}, 11 12
For a proof, see, for example, Moore [1999, Theorem 5.13, p. 268]. Some of the bj ’s may, of course, be zero.
11.6. Appendix
331
and: C2 = {x ∈ R3+  x2 = 0 ≤ x1 ≤ 1, & x3 = 1} ∪ {x ∈ R3+  x1 = 0 ≤ x2 ≤ 1 = x3 }, Since the points (1, 0, 0) and (0, 1, 0) are both elements of C1 , and (1, 0, 1) and (0, 1, 1) are both contained in C2 , and since: 3 1 3 1 (1, 0, 0) + (0, 1, 0) + (1, 0, 1) + (0, 1, 1) = (3/4, 1/4, 1/2), 8 8 8 8 it follows that the vector x∗ = (3/4, 1/4, 1/2) is contained in co(C), where: C = C1 ∪ C2 . However, suppose, by way of obtaining a contradiction, that there exist y i ∈ Ci (i = 1, 2) and a ∈ ∆2 such that: a1 y 1 + a2 y 2 = x∗ = (3/4, 1/4, 1/2). Then, since y 1 and y 2 are in C1 and C2 , respectively, we must have: y13 = 0 & y23 = 1. consequently, we see that we must have a2 = 1/2; from which it follows that we must also have a1 = 1/2. We now consider two cases. 1. y11 = 0 Here we see that we must have: (1/2)y21 = 3/4, which implies y21 = 3/2; contradicting the deﬁnition of C2 . 2. y11 > 0 In this case we see that we must have y12 = 0. But then we see that we must have: (1/2)y22 = 1/4; so that y22 = 1/2. But it must then also be true that y21 = 0; which would imply that: y11 = 3/2; yielding a contradiction once again. Exercises. 1. Prove Proposition 11.2.
, 2. Suppose in a given economy, E = (Xi , Pi , Zi ) , one of the consumers, say the ﬁrst, has continuously representable and strictly convex preferences. Show that , if the tuple x∗(h,i) , z ∗h,i (h,i)∈Q×M , p∗ is a competitive equilibrium for Eq , then we must have: x∗h1 = x∗11 for h = 1, . . . , q. 3. Consider a twoconsumer, twocommodity pure exchange economy, with: r 1 = (10, 0) and r 2 = (0, 10), and suppose the consumers’ preferences can be represented by: u1 (x1 ) = 10x11 x12 ,
332
Chapter 11. The Core of an Economy
and: u2 (x2 ) = log x21 + log x22 , respectively. Is the allocation: x1 = x2 = (5, 5), in the core for E? Why or why not? 4. Consider a pure exchange economy in which m = n = 2, and suppose that the two consumer’s preferences can be represented by the utility functions: u1 (x1 ) = min{x11 , x12 }, and: u2 (x2 ) = x21 + x22 , respectively. Suppose further that the consumers’ initial endowments are given by: r 1 = (5, 5) = r 2 . Find the set of core allocations for E, C(E). 5. Consider a pure exchange economy in which m = n = 2, and suppose that the two consumers’ preferences can be represented by the utility functions: u1 (x1 ) = min{x11 , x12 }, and: u2 (x2 ) = 2 min{x21 , x22 }, respectively. Suppose further that the consumers’ initial endowments are given by: r 1 = (10, 0) and r 2 = (0, 10), respectively. Given this information: (a) Find C(E). (b) Consider replicating this economy, and the replicative cores, Cq , for q = 1, 2, . . . . Find Cq , for an arbitrary q. 6. Prove Corollary 11.26 7. Prove the following proposition. 11.30 Proposition. If E = (Pi , r i )i∈M is an exchange economy such that, 1. for each i ∈ M, Pi is weakly convex, lowever semicontinuous, and strictly increasing, and 2. r = i∈M r i 0, then: ∞ C q = W (E). q=1
Chapter 12
General Equilibrium with Uncertainty 12.1
Introduction
In this chapter we will make a brief foray into the theory of general equilibrium with uncertainty. If you remember the discussion of Chapter 2, you will recall that in general equilibrium theory, a commodity is deﬁned by (1) its physical description, (2) its location, (3) the time at which it is available, and (4) the state of the world in which it is available. Consequently, in most of this chapter we are specializing the theory which we have been studying; putting more structure into the model in order to account for the eﬀects of uncertainty. Of course, when one delves more deeply into this theory, questions arise which did not appear to be relevant in our earlier studies. Moreover, if we were to pursue the subject to its current frontiers, we would ﬁnd that new theoretical concepts and tools are needed to provide answers for these questions. However, in the interests of practicality, we will only attempt to provide a ‘bare bones’ introduction to this theory. Fortunately, in even this cursory introduction to the topic, we will ﬁnd that some interesting issues and applications can be discussed. We will begin our discussion with what is known as the ‘ArrowDebreu Contingent Commodities Model.’
12.2
ArrowDebreu Contingent Commodities
The crux of this model is that we suppose that there are two periods, t = 0, 1. At t = 0, it is supposed that we can set forth all possibilities for the state of the world at the second date, t = 1. We assume that there is a ﬁnite set, S, of such possible states, and we will also write S = #S; denoting states by lower case ‘s, s ,’ etc. Each ‘state’ is a complete description of the world, and in this theory, we suppose that every agent will know which state, s ∈ S has occurred once we reach t = 1. We will suppose that there are G physically distinguishable commodities (which, in principle could also be distinguished by location), so that ‘n,’ the dimension of our commodity space becomes: n = S · G.
334
Chapter 12. General Equilibrium with Uncertainty
Commodity bundles then take the form: x = (x11 , . . . , x1G , x21 , . . . , xsg , . . . , xSG ), which is understood to be an entitlement to receive the commodity bundle: xs = (xs1 , . . . , xsG ), if state s occurs. Thus ‘xsg ’ denotes the amount of commodity g to be received (or supplied, if xsg < 0) if state s occurs. In further specifying the economy, we will depart from our previous notation to denote consumer i’s resource endowment, by ‘ω i ,’ which now takes the form: ω i = (ωi11 , . . . , ωi1G , ωi21 , . . . , ωisg , . . . , ωiSG );
(12.1)
that is, ‘ωisg ’ denotes consumer i’s endowment of the g th commodity if state s occurs. Fortunately, we will rarely have to write out the full vector as we’ve done in (12.1), above. Deﬁning ω is = (ωis1 , . . . ωisG ) for s = 1, . . . , S; that is, letting ‘ω is ’ denote i’s endowment if state s occurs, the ﬁnest detail we will usually write out is: ω i = (ω i1 , . . . , ω is , . . . , ω iS ). We suppose also that the consumer’s preferences describe a weak order over Xi , denoted by ‘i . Furthermore, we denote the ith consumer’s consumption bundle, contingent upon the occurrence of state s by ‘xis ;’ so that we can write: xi = (xi1 , . . . , xis , . . . , xiS ). Similarly, we will let ‘Yk ⊆ Rn ’ denote the feasible production plans for the k th ﬁrm, and we will use the generic notation: y k = (y k1 , . . . , y ks , . . . , y kS ),
(12.2)
to denote elements of Yk , where ‘y ks ’ denotes the production vector of the ﬁrm, contingent upon the occurence of state s. We then complete the model, departing from our previous notation,1 by letting ‘θik ’ denote the ith consumer’s share in the k th ﬁrm’s proﬁts. We will have to be a bit careful in dealing with individual consumption and production sets. One is tempted, for example to express the ith consumer’s consumption set as: S Xi = Xis , (12.3) s=1
where ‘Xis ’ denotes the ith consumer’s feasible consumption set if state s occurs; with similar speciﬁcations for the ﬁrms’ production sets. That this will not quite do is perhaps best illustrated by considering the following production example, which is inspired by MasColell, Whinston, and Green [1995, Example 19.B.2, p. 689]. 1 This change is made in order that the ith consumer’s shares not be confused with the ith state of the world.
12.2. ArrowDebreu Contingent Commodities
335
12.1 Example. Suppose there are two states, s1 and s2 , representing good and bad weather. There are two physical commodities: seeds (g = 1) and crops (g = 2). In this case, the elements of Yk are fourdimensional vectors. Assume that seeds must be planted before the resolution of the uncertainty about the weather and that if the weather is good, the ﬁrm’s production possibilities are given by: Yk1 = {y k1 ∈ R2  yk12 ≥ 0 & 2yk11 + yk12 ≤ 0}; whereas in bad weather, production is given by: Yk2 = {y k2 ∈ R2  yk22 ≥ 0 & yk21 + 2yk22 ≤ 0}. Recalling our assumption that the seed must be planted before the resolution of uncertainty, we see that we can represent the ﬁrm’s production set as: Yk = {y k ∈ Yk1 × Yk2  yk11 = yk21 }.
(12.4)
Thus, for example, the production vector: y k = (yk11 , yk12 , yk21 , yk22 ) = (−2, 4, −2, 1), is a feasible plan; whereas neither of the production plans: y k = (−4, 8, −2, 1) and y = (−2, 4, 0, 0), is feasible.
While the above example deals with a production set, the diﬃculty applies equally to consumption sets; after all, someone has to plant the seeds, and this labor must also be undertaken before the resolution of uncertainty. In order to allow for this fact, while yet being able to assume on some occasions that consumers’ preferences are weakly separable over states, we will assume that for each consumer there exist sets: Xis ⊆ RG , representing the consumer’s feasible consumption possibilities if state s occurs (for i (presumably a proper subset of G), such that: s = 1, . . . , S), and a set G S i ) : xi1g = xi2g = · · · = xiSg (12.5) X i = xi ∈ Xis  (∀g ∈ G s=1
Thus, with this speciﬁcation, one can make sense of the following example. 12.2 Example. Suppose that, for a given consumer, i, there exist S utility functions: uis : RG → R, 7
such that: xi i xi ⇐⇒
s∈S
πis uis (xis ) ≥
8 πis uis (xis ) .
(12.6)
s∈S
where ‘πis ’ denotes i’s subjective (or objective) probability of the occurrence of state s. Notice that, even though we have statedependent utility here, preferences are weakly separable on Xis , for each state, s.
336
Chapter 12. General Equilibrium with Uncertainty
We will not need to assume much about the form of the ﬁrms’ production sets,2 we will simply suppose that the k th ﬁrm’s technological production possibilities are given by a production set Yk ⊆ Rn . We will make use of the following deﬁnition of feasible allocations for the economy. 12.3 Deﬁnition. We will say that an allocation, x∗i , y ∗k ∈ R(m+)n is feasible for E iﬀ: x∗i ∈ Xi for i = 1, . . . , m, (12.7) y ∗k ∈ Yk for k = 1, . . . , , and:
i∈M
x∗i =
ωi +
i∈M
y ∗k .
(12.8)
k∈L
Now, at this point, you may be saying, or thinking, “Hold on! That’s exactly the deﬁnition of a feasible allocation which was presented in Chapter 7!” And in saying this you are absolutely right! All we have done here so far is to present a somewhat more detailed and specialized speciﬁcation of what the commodity space is. However, notice that equation (12.8) of the above deﬁnition implies that: i∈M
x∗is =
i∈M
ω is +
y ∗ks
for s = 1, . . . , S;
(12.9)
k∈L
so that in each state, consumption equals net supply. To continue our interpretation of the ArrowDebreu Contingent Commodities Model, the interpretation of the equilibrium which we are now going to discuss is that at time t = 0 there is a futures market for each contingent commodity. Equilibrium will require that supply equals demand for each contingent commodity. 12.4 A system of prices, p∗ = (p∗11 , . . . , p∗SG ) ∈ Rn and an allocation, ∗ Deﬁnition. xi , y ∗k will be said to be an ArrowDebreu equilibrium iﬀ: 1. x∗i , y ∗k is a feasible allocation, 2. for every k ∈ L, y ∗k satisﬁes: (∀y k ∈ Yk ) : p∗ · y ∗k ≥ p∗ · y k , and: 3. for every i ∈ M : p∗ · x∗i ≤ p∗ · ω i +
θik p∗ · y ∗k , and :
k∈L
(∀xi ∈ Xi ) : xi i x∗i ⇒ p∗ · xi > p∗ · ω i +
θik p∗ · y ∗k .
(12.10)
k∈L 2 Other than to keep in mind the fact that it is probably inappropriate to suppose that they can be written as a cartesian product of statespeciﬁc production sets.
12.2. ArrowDebreu Contingent Commodities
337
Once again, the deﬁnition is formally identical to that presented in Chapter 7; the only diﬀerence is in the interpretation. The beauty of the situation, however, is that we can immediately deduce some important results. In particular, we can see that if (x∗i , y ∗k , p∗ ) is an ArrowDebreu equilibrium then x∗i , y ∗k must be Pareto eﬃcient; at least in terms of the ex ante consumer preferences, i . The following example may help you to get a better ‘feel’ for the model and the meaning of the deﬁnition of competitive equilibrium being used here. 12.5 Example. Suppose E is an exchange economy with m = 2 = S, and G = 1; that is we have two consumers, one physically distinguishable commodity, and two states of the world to consider. We will also suppose that the ith consumer has a twicediﬀerentiable Bernoullian utility function, ui : R+ → R+ such that, for all x ∈ R+ : ui (x) > 0 and ui (x) < 0; so that ui is strictly increasing and strictly concave. If xi = (xi1 , xi2 ) and xi = (xi1 , xi2 ) are two commodity bundles in Xi = R2+ , consumer i will prefer xi to xi if, and only if: Ui (xi ) = πi1 ui (xi1 ) + πi2 ui (xi2 ) > Ui (xi ) = πi1 ui (xi1 ) + πi2 ui (xi2 ), where ‘πis ’ denotes i’s (subjective) probability that state s will occur, for s = 1, 2. Supposing that these probabilities are strictly positive, and that prices for the two goods at t = 0 are given by p = (p1 , p2 ) ∈ R2++ , the ith consumer will maximize utility by setting: πi1 ui (xi1 ) πi2 ui (xi2 ) = , (12.11) p1 p2 and: p · xi = p · ω i .
(12.12)
Assuming that the two consumers agree on the probabilities of the two states (so that π1s = π2s ≡ πs , for s = 1, 2, it is easily seen that in competitive (ArrowDebreu) equilibrium: u1 (x11 ) u (x21 ) π2 p 1 = 2 = ; (12.13) u1 (x12 ) π1 p2 u2 (x22 )
and thus it is easy to see that the allocation will be Pareto eﬃcient. Now suppose that: ω11 + ω21 = ω12 + ω22 ;
(12.14)
that is, that the total endowment in the two states is exactly the same. Suppose further that π1 = π2 ; that is, that the both consumers consider the two states to be equally probable. Then (12.13) becomes: u1 (x11 ) u (x21 ) p1 = 2 = . u1 (x12 ) p2 u2 (x22 )
(12.15)
Suppose then, by way of obtaining a contradiction, that: x11 > x12
(12.16)
338
Chapter 12. General Equilibrium with Uncertainty
Then by the assumed properties of the ui functions: u1 (x11 ) < 1. u1 (x12 ) However, it then follows from (12.15) that u2 (x21 )/u2 (x22 ) < 1 also; in which case it follows from the assumed properties of the ui that: x21 > x22 as well. But then it follows that: x11 + x21 > x12 + x22 ; which, given (12.14), contradicts the assumption that xis , p is a competitive (ArrowDebreu) equilibrium. A symmetric argument shows that we cannot have xi1 < x12 for either i = 1 or i = 2. Therefore, we must have: xi1 = xi2
for i = 1, 2;
and from (12.15) we see that this implies that we must have p1 = p2 . Maintaining the assumption that (12.14) holds, arguments similar to those of the above paragraph establish that if both individuals believe the ﬁrst state to be more probable than the second, then we must have p1 > p2 in equilibrium. Next suppose that we have: ω12 = ω21 = 0, but that (12.14) continues to hold (so that we have private risk, but we do not have social risk). Then it follows from the reasoning above that both consumers fully insure; that is, they each sell oﬀ rights to half of their endowments in order to equalize expected consumption in the two states. Finally, suppose that we have social risk; that is, suppose we have: def
def
ω1 = ω11 + ω21 = ω2 = ω12 + ω22 ,
(12.17)
but that π1 = π2 . I will leave it as an exercise to show that in this situation we must have: (12.18) (p1 − p2 )(ω1 − ω2 ) < 0. The scenario involved in the usual interpretation of the model we have been discussing is that all markets operate and are cleared in the initial period (t = 0), while all consumption takes place at t = 1. There are a couple of points which should be made with respect to this interpretation. First of all, there is a question about ex ante versus ex post eﬃciency. Suppose we have an ArrowDebreu equilibrium, (x∗i , y ∗k , p∗ ), but that markets are reopened at t = 1, after the uncertainty is resolved, but before consumption takes place.3 What would happen then? Strictly 3 The markets in question here are called spot markets, while the markets at t = 0 are called forward markets.
12.3. Radner Equilibrium
339
speaking, we cannot say without assuming that preferences are weakly separable on Xis and that all consumers’ ex post preferences are the same as their ex ante preferences over Xis . However, both of these assumptions seem to be eminently reasonable, and if both are true, then there would be no incentive for trades to take place in this situation. Why is this? Well, each consumer must be maximizing satisfaction, given the expenditure p∗s · xis at x∗is ; for if, for some consumer i there were some xis such that: p∗s · xis ≤ p∗s · xis and xis is x∗is , the consumer would have preferred to replace the bundle x∗is with xis at t = 0.4 Since the allocation in state s is therefore a competitive equilibrium, given the price vector p∗s , it follows that it is also Pareto eﬃcient. Consequently, there are no mutually beneﬁcial trades which consumers can make among themselves after the resolution of uncertainty. A serious objection to the interpretation of the model which we set out in the preceding paragraph is that it is clearly unrealistic to expect the existence of forward markets in each commodity. However, suppose we have an ArrowDebreu equilibrium, (x∗i , y ∗k , p∗ ). If prices in each state are correctly anticipated by all agents, and we have a futures market for only one commodity, with trading only in that one commodity at t = 0, then we can achieve that same consumption allocation, x∗i , if retrading is possible (at the anticipated prices) at t = 1. This remarkable fact was ﬁrst noted by Arrow [1953]. The formal extension of this idea which we will be studying in the next section was, however, developed by Professor Roy Radner [1968, 1982].
12.3
Radner Equilibrium
For the sake of simplicity, in the remainder of this chapter we will conﬁne our discussion to the context of a pure exchange economy, and we will retain the notation and basic assumptions regarding consumers which were introduced in the previous section; so that consumer i has a preference relation i on Xi , and has the initial endowment ω i , as before. Once again we will deal with a twoperiod model; with uncertainty being resolved in the second period (t = 1). This time, however, we will allow no commodity trading in the ﬁrst period (t = 0). We will, however, introduce the idea of some tradeable assets, which can be purchased (or sold short) in the ﬁrst period. There are three pivotal assumptions which we will make in this context. First, we will suppose that at t = 0 consumers have expectations of the prices which will occur (and at which trading will take place) at t = 1, for each possible state of nature (each s ∈ S). Secondly, we suppose that all consumers expect the same vector of prices to prevail at t = 1 if state s ∈ S occurs; we denote this vector by ‘ps ,’ and we denote the full vector of such prices by ‘p;’ that is: p = (p1 , . . . , ps , . . . , pS ). 4 Notice, however, that both weak separability and the identity of ex post and ex ante preferences are needed to make this argument correct.
340
Chapter 12. General Equilibrium with Uncertainty
Thirdly, we suppose that at t = 0 there are K assets available, all of which pay oﬀ a conditional return in the ﬁrst commodity; which we take to be a num´eraire. 12.6 Deﬁnition. A unit of an asset is a title to receive an amount rs of good 1 at date t = 1 if state s occurs. An asset is therefore characterized by its return vector r = (r1 , . . . , rS ) ∈ RS . Thus, a checking account (in a fullyinsured bank) might be characterized as having the return vector r = 1 , where 1 ∈ RS+ is the vector all of whose coordinates equal 1. Very useful examples are also provided by assets whose return vectors are of the form r s = es , where es ∈ RS+ is the sth unit coordinate vector. In other words, such an asset provides a return of one unit of the num´eraire if state s occurs, and nothing otherwise. We suppose there is a given set of K assets (an asset structure), which can be traded at t = 0. We denote the return vector associated with the k th asset by r k ∈ RS . We assume that there are no initial endowments of assets, and that short sales are possible. The price of the k th asset at t = 0 is denoted by qk , and a vector of quantities of the K assets, z ∈ RK is called a portfolio. Thus the expenditure required to obtain the portfolio z at t = 0, given the asset price vector q is equal to q · z. Since we assume that consumers have no initial endowments of assets, consumer i must choose a portfolio z i satisfying: q · z i ≤ 0. Notice that we have put no restriction on the sign of z i ; a negative value for zik means the consumer is selling the k th asset short; that is, the consumer will owe rks zik units of the num´eraire if state s should occur. Since we are assuming that the ﬁrst commodity is a num´eraire, we can normalize to set ps1 = 1, for s = 1, . . . , S.5 Given an asset structure, we can then deﬁne the S × K return matrix, R, by: ⎞ ⎛ r11 r12 . . . r1k . . . r1K ⎜. . . . . . . . . . . . . . . . . . ⎟ ⎟ ⎜ ⎟ R=⎜ (12.19) ⎜ rs1 rs2 . . . rsk . . . rsK ⎟ . ⎝. . . . . . . . . . . . . . . . . . ⎠ rS1 rS2 . . . rSk . . . rSK Notice that in this matrix, the rows correspond to states, while the columns correspond to assets. Thus, if r k is the vector of returns for the k th asset, r k becomes the k th column of the return matrix. We will be a bit sloppy in our terminology in dealing with asset structures and the return matrices which they determine in that we will often identify the asset structure with the return matrix which it determines. That is, we will often refer to a nonnegative S × K matrix as an asset structure. Now, if a consumer holds the portfolio z ∈ RK , the return from this portfolio if state s occurs is given by: K rsk zk ; (12.20) r s· · z = k=1
5
It may not be apparent at this point why it is that we can normalize in this manner, but it will be clear once we deﬁne consumer equilbrium for the model to be presented in this section. See Exercise 5, at the end of this chapter.
12.3. Radner Equilibrium
341
where ‘r s· ’ denotes row s of R.6 Thus, we can make use of this return matrix (and given the price normalization just mentioned), to write the ith consumer’s consumption budget constraint, given the price vector, p∗ = (p∗1 , p∗2 , . . . , p∗s , . . . , p∗S ) and the portfolio, z ∗i ∈ RK as: ⎞⎛ ∗ ⎞ ⎞ ⎛ ⎛ ∗ zi1 r11 r12 . . . r1k . . . r1K p1 · (xi1 − ω i1 ) ⎜ ⎟ ⎜. . .⎟ ⎟ ⎜ . . . . . . . . . . . . . . . . . . . . . ⎟⎜ ⎟ ⎟ ⎜ ⎜ ∗ ⎜ ps · (xis − ω is ) ⎟ ≤ ⎜ rs1 rs2 . . . rsk . . . rsK ⎟ ⎜ z ∗ ⎟ = Rz ∗i . (12.21) ⎟ ⎜ ik ⎟ ⎟ ⎜ ⎜ ⎝ ⎠ ⎝ . . . . . . . . . . . . . . . . . . ⎠ ⎝. . .⎠ ... ∗ ∗ rS1 rS2 . . . rSk . . . rSK ziK pS · (xiS − ω iS ) Thus, in each state s, the consumer’s expenditure on commodities, minus the value of its endowment, must be no greater than the value of the return on its portfolio. Thus, given the vector of asset prices, q ∗ at t = 0, the vector of commodity prices, p∗ ∈ Rn , and the return matrix, R, the ith consumer’s budget set, Bi (p∗ , q ∗ , R), can be written as: Bi (p∗ , q ∗ , R) = xi ∈ Xi  (∃z i ∈ RK ) : q ∗ · z i ≤ 0, & (xi , z i ) satisﬁes (12.21) . 12.7 Example. An important asset structure for comparisons and examples is given by the set S of assets (often called Arrow securities), where the sth asset has the return vector: r s = es for s = 1, . . . , S. (12.22) That is, the sth asset pays a return of 1 unit of the ﬁrst good if state s occurs, and nothing otherwise. In this case, the return matrix is the S × S identity matrix, I S . Thus, if we suppose that there are only two states of the world in period 1, the ith consumer’s budget set, given the commodity price vector p∗ = (p∗1 , p∗2 ), and security prices q ∗ = (q1∗ , q2∗ ), is given by: !+ ! ( z p∗1 · (xi1 − ω i1 ) Bi (p∗ , q ∗ ; R) = xi ∈ Xi (∃z ∈ R2 ) : q ∗ ·z ≤ 0 & . ≤ 1 ∗ z2 p2 · (xi2 − ω i2 ) In this context, we will be considering the following deﬁnition of equilibrium. 12.8 Deﬁnition. A tuple, x∗i , z ∗i , p∗s , q ∗ , where q ∗ ∈ RK is a vector of asset ∗ G prices at t = 0, ps ∈ R is a vector of spot prices at state s (s = 1, . . . , S), x∗i ∈ Rn is the consumption bundle (at t = 1) and z ∗i ∈ RK the asset portfolio of the ith consumer, is a Radner equilibrium given the asset structure R iﬀ: 1. for each i (i = 1, . . . , m), the pair (x∗i , z ∗i ) solves the consumer’s problem: max w.r.t.(xi ,zi )
i
subject to:
q ∗ · z i ≤ 0 and p∗s · xis ≤ p∗s · ω is +
∗ 2. m i=1 z i ≤ 0, and: m 3. i=1 x∗is ≤ m i=1 ω is for s = 1, . . . , S.
K k=1
rsk zik , for s = 1, . . . , S.
6 More exactly, of course, the sum in (12.20) is the number of units of the ﬁrst commodity which the consumer will receive if state s occurs. However, since we are setting ps1 = 1, for each s ∈ S, this is also the value of the return.
342
Chapter 12. General Equilibrium with Uncertainty
While the above deﬁnition is a bit diﬃcult to state, the ﬁrst condition simply says that each x∗i is an element of Bi (p∗ , q ∗ ; R), and that given any xi ∈ Bi (p∗ , q ∗ ; R), we must have: x∗i i xi . The second condition says that trading of assets must balance; that is, positive purchases of assets by some consumers must be balanced by short sales of the assets by others. Finally, the third condition simply says that aggregate consumption in each state of the world must be equal to the aggregate commodity endowment in that state. If you reexamine the deﬁnition of the consumers’ maximization problem in 12.8, it will be apparent that an equilibrium vector of asset prices can be multiplied by a ∗ ∗ ∗ ∗ positive scalar without changing the consumers’ choices; that is, if xi , z i , ps , q is a Radner equilibrium, and θ is a positive real number, then x∗i , z ∗i , p∗ , θq ∗ is also a Radner equilibrium. Consequently, we can normalize to set the price of one of the assets equal to one (1), and we will sometimes ﬁnd it convenient to do so. 12.9 Deﬁnition. We shall say that a vector of asset prices, q ∈ RK is arbitragefree, given the return matrix, R, iﬀ there exists no portfolio, z ∈ RK satisfying: ! −q z > 0. (12.23) R Notice that if z satisﬁes (12.23), then either: q · z < 0 and Rz ≥ 0,
(12.24)
q · z ≤ 0 and Rz > 0.
(12.25)
or We will generally be assuming that each column of the return matrix, R, is semipositive (each asset provides a positive return in some state, and nonnegative returns in all other states), and given this, it can be shown (see Exercise 4, at the end of this chapter) that if (12.24) holds, then there exists a portfolio, z satisfying (12.25). On the other hand, if (12.25) holds, consumers can obtain an unbounded return in at least one state, while earning a nonnegative return in all other states. Consequently, if an asset price vector q is not arbitragefree, and if preferences are increasing, or if the ﬁrst commodity is a num´eraire good for the economy in each state, there will exist no solution for the consumers’ maximization problems in our deﬁnition of a Radner equilibrium. We will assume throughout the remainder of this chapter that preferences are weakly separable over states, and increasing in consumption goods within each state. As a consequence of this second assumption it will follow that if x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, given the asset structure, R, then q ∗ must be arbitrage free, given R. (We will revisit this statement shortly in a more formal fashion.) If the asset price vector q is arbitrage free, given R, then, as we will prove shortly, q must be in the polyhedral cone determined by the rows of R; that is, there must exist µ ∈ RS+ such that: (12.26) µ R = q .
12.3. Radner Equilibrium
343
Following Duﬃe [2001], we will refer to such a vector as a state price vector for (R.q). 12.10 Examples. 1. Let S = 2 and K = 3, and suppose: R=
1 0 1 0 1 1
!
⎛ ⎞ 1 and q = ⎝1⎠ . 2
Now the question is, is q arbitrage free, given R? To answer this, suppose z ∈ R3 is such that q · z ≤ 0. Then from the deﬁnitions we see that we must have: z1 ≤ −z2 − 2z3 . ! ! ! −z2 − 2z3 + z3 −z2 − z3 z1 + z3 ≤ = . z2 + z 3 z 2 + z3 z2 + z 3 Thus, it is clear that if Rz 1 > 0, then Rz 2 < 0, and conversely. Therefore, q is arbitragefree, given R. If we deﬁne µ = (1, 1), then we see that: ! 1 0 1 µ R = (1 1) = (1 1 2) = q ; 0 1 1
But then:
Rz =
and thus µ is a stateprice vector for (R, q). 2, Suppose the asset structure is that deﬁned by the set of Arrow securities; that is, suppose R = I S . Then if q is a vector of security prices, and µ is a stateprice vector for (R, q), we must have: µ = µ I S = q . In other words, µs = qs , for s = 1, . . . , S.
While we earlier interpreted equation (12.26) as meaning that q must be in the polyhedral cone generated by the rows of R; the economic interpretation of (12.26) is that, given such a stateprice vector for (R, q), the price of each asset is then simply the sum over the possible states of the world of the statepriceweighted return of the asset in that state; that is (12.26) implies: S s=1
µs rsk = qk
for k = 1, . . . , K.
In the following proposition we establish the fact that there always exists a stateprice vector if q is arbitrage free, given R. 12.11 Proposition. Suppose R is an S × K return matrix and that q ∈ RK + is an asset price vector. Then q is arbitragefree, given R if, and only if, there exists a stateprice vector for (R, q), µ ∈ RS++ , such that: µ R = q .
(12.27)
344
Chapter 12. General Equilibrium with Uncertainty Proof. Deﬁne the K × (S + 1) matrix A by: A = R −q ,
and suppose ﬁrst that there exists a stateprice vector, µ, satisfying (12.27). Then we have: ! ! µ µ = R µ − q = 0; A = R −q 1 1 and it follows from Stiemke’s Theorem (Theorem 6.40) that there exists no z ∈ RK such that: z A = z R −q > 0. Thus we see that q is arbitragefree, given R. On the other hand, suppose q is arbitragefree. Then there exists no z ∈ RK such that: z A > 0. Then using Stiemke’s Theorem (Theorem 6.40) once again, it follows that there exists x∗ ∈ RS+1 such that x∗ 0, and such that, writing: ! y∗ , x∗ = x∗S+1 we have:
0 = Ax∗ = R −q
y∗ x∗S+1
!
= R y ∗ − x∗S+1 q.
(12.28)
Consequently, deﬁning µ ∈ RS by: µ = (1/x∗S+1 )y ∗ , we have µ 0, and from (12.28): µ R = q .
12.12 Example. Suppose there is an asset, say the ﬁrst, which is riskfree; that is, suppose the return vector, r 1 ∈ RS , is given by: r1 = 1 , RS
where 1 is the column vector in having all of its entries equal to 1. It seems entirely natural in this case to normalize by setting the price of this asset equal to 1; and, as we noted earlier, we can do this without loss of generality, insofar as our deﬁnition of a Radner equilibrium is concerned. Notice also that if an asset price vector, q ∗ , is arbitragefree, then so is the asset price vector q , given by: q = (1/q1∗ )q ∗ . If this normalized asset price vector is used in the proof of Proposition 12.11, then the ﬁrst of equations (12.28) becomes: S S rs1 ys∗ − x∗S+1 q1 = ys∗ − x∗S+1 = 0. (12.29) s=1
s=1
12.4. Complete Markets
345
Therefore, the stateprice vector obtained in the proof satisﬁes: s µs = 1. s=1
(12.30)
Consequently, it is easy to show that for all k, we must have: min rsk ≤ qk ≤ max rsk . s
s
Suppose the ﬁrst commodity is a num´eraire in each state of the world. If any consumer, say the ﬁrst, has preferences representable by a von NeumannMorgenstern utility function, and is able to ﬁnd a utilitymaximizing choice of (xi , z i ), given prices (p, q), then the asset price vector, q, must be arbitragefree, given R. This is the content of the following proposition, whose proof will be left as an exercise. 12.13 Proposition. Suppose an agent, i, has the von NeumannMorgenstern utility function: S πs uis (xIs ), Ui (xi ) = s=1
where πs is the probability of state s, for s = 1, . . . , S; and maximizes utility at (x∗i , z ∗i ) 0, given prices (p∗ , q ∗ ) and the return matrix, R; and that: ∂uis (x∗is ) def ∂uis = >0 ∂xis1 ∂xis1 x∗i
for s = 1, . . . , S.
Then there exists a positive scalar, νi , such that the vector µ∗i given by: µ∗is =
πs ∂uis (x∗ia ) × νi p∗s1 ∂xis1
for s = 1, . . . , S,
is a stateprice vector for (R, q ∗ ) (and thus q ∗ is arbitragefree, given R). While the above proposition is, I believe, of some interest, it can obviously be generalized considerably. In particular, if preferences are weakly separable over states, and nonsaturating within a state, then it is clear that if x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, then q ∗ must be arbitragefree.
12.4
Complete Markets
An economy of the type studied here and in the previous section has very good normative properties if the return matrix is complete, as deﬁned in the following. 12.14 Deﬁnition. An asset structure with the S × K return matrix R is said to be complete iﬀ the rank of R is S; that is, iﬀ there is some subset of the K assets, containing S elements, whose return vectors are linearly independent. Obviously there needs to be at least as many assets available as there are possible states in order for this condition to hold. Perhaps the simplest example of a complete asset structure is provided by the Arrow securities of Example 12.7. However, it is all too easy to construct examples in which there are at least as many securites as states, but in which the return matrix has rank less than S.
346
Chapter 12. General Equilibrium with Uncertainty
12.15 Example. Suppose there are three states of the world at t = 1, and consider the return matrix: ⎛ ⎞ 2 1 1 R = ⎝1 0 1⎠ . 1 1 0 Remembering that the rows in the matrix correspond to states, while the columns correspond to assets (and thus the ﬁrst asset yields a return of 2 units if state one occurs, 1 if s = 2, and so on, we see that there are as many assets here as there are states. However, R does not have full rank,7 and thus the asset structure is not complete. On the other hand, the asset structure corresponding to the return matrix: ⎛ ⎞ 2 1 0 R = ⎝0 0 1 ⎠ 1 0 1 is complete, since the matrix has full rank.
The importance of the asset structure’s being complete is brought out by the following result. 12.16 Theorem. Suppose that the asset structure, R, is complete. Then we have the following. 1. If x∗i , p∗ is an ArrowDebreu equilibrium for E, then there are asset prices ∗ K q ∈ R++ and portfolio plans z ∗ = (z ∗1 , . . . , z ∗m ) ∈ RmK such that x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, given the asset structure R 2. Conversely, if x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium given R, then there exist a stateprice vector, µ = (µ1 , . . . , µS ) ∈ RS++ , such that, deﬁning p by:
ps = µs p∗s
for s = 1, . . . , S, x∗i , p is an ArrowDebreu equilibrium for E.
Proof. Since the two deﬁnitions of equilibrium have the same feasibility requirement, we needn’t bother to prove that the allocation is feasible in either part of our proof. In the argument to follow, we will denote the budget set for the ith consumer, given the ArrowDebreu price vector, p, by ‘BiA (p);’ while, given a vector of commodity prices, p and security prices, q, we will denote the ith consumer’s budget set R in the Radner sense, and given the return matrix, R, by ‘Bi (p, q; R).’ 1. Suppose xi , p is an ArrowDebreu equilibrium, where: p = (p1 , . . . , pS ) ∈ RGS + , and deﬁne q ∈ RK by:
q = 1 R;
(12.31)
where once again ‘1 ’ denotes the column vector all of whose coordinates equal 1. Now, deﬁning: y i = p1 · (xi1 − ω i1 ), . . . , pS · (xiS − ω iS ) , 7
For example, yR = 0, where y = (1, −1, −1).
12.4. Complete Markets
347
we have by deﬁnition of an ArrowDebreu equilibrium that, for each i: 1 · y i = 0, and, by feasibility:
m
(12.32)
y i = 0.
i=1
(12.33)
From the assumption that the asset structure is complete, R has full rank (equal to S), and thus, for each i = 1, . . . , m, we can ﬁnd z i such that: Rz i = y i . Therefore, if the ith consumer has the portfolio z i , we have: ps · (xi − ω i ) =
K k=1
rsk z ik
for s = 1, . . . , S,
and, by the deﬁnition of q and (12.32): q · z i = 1 Rz i = 1 y i = 0
for i = 1, . . . , m.
Thus xi ∈ BiR (p, q; R). Furthermore, it follows from (12.33) that:
m m m zi ; 0= yi = Rz i = R i=1
i=1
i=1
and thus, since R has full rank: m i=1
zi = 0
(12.34)
Now let i be arbitrary, and let xi ∈ Xi and z i ∈ RK be such that: q · z i ≤ 0, and, for each s ∈ S: ps · xis ≤ ps · ω is +
rsk zik .
k∈K
Then we have: ps · (xis − ω is ) ≤ rsk zik = zik rsk = qk zik ≤ 0; s∈S
s∈S k∈K
k∈K
s∈S
k∈K
where the last equality is by deﬁnition of q. Consequently, if xi ∈ BiR (p, q, R), then xi ∈ BiA (p); and since xi ∈ BiR (p, q, R), and maximizes i over BiA (p) it follows that xi maximizes i on BiR (p, q, R) as well. Combining this fact with (12.34), it follows that xi , z i , p, q is a Radnerequilibrium, given R. 2. Now suppose x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, given the return matrix R. Then q ∗ must be arbitragefree, given R, and it follows from Proposition 12.11 that there exists a stateprice vector µ 0 such that:
µ R = (q ∗ ) .
(12.35)
348
Chapter 12. General Equilibrium with Uncertainty
Making use of this stateprice vector, we deﬁne p ∈ RSG by: ps = µs p∗s
for s = 1, . . . , S.
(12.36)
Now let i ∈ {1, . . . , m} be arbitrary, and suppose that xi ∈ p · (xi − ω i ) =
S s=1
ps · (xis − ω is ) =
S s=1
BiA (p);
so that:
µs p∗s · (xis − ω is ) ≤ 0;
(12.37)
and deﬁne y i ∈ RS by: ys = p∗s · (xis − ω is )
for s = 1, . . . , S.
Since R is complete, there exists z i ∈ RK such that: Rz i = y i ;
(12.38)
and we note that: q ∗ · z i = (q ∗ ) z i = µ Rz i = µ y i S S = µs p∗s · (xis − ω is ) = s=1
s=1
ps · (xis − ω is ) ≤ 0.
Therefore xi ∈ BiR (p∗ , q ∗ ; R), and it follows that: BiA (p) ⊆ BiR (p∗ , q ∗ ; R).
(12.39)
Furthermore, deﬁning y ∗i ∈ RS by: ys∗ = p∗s · (x∗is − ω is )
for s = 1, . . . , S,
we have: p · (x∗i − ω i ) =
S s=1
µs p∗s · (x∗is − ω is ) =
S s=1
µs ys∗ ≤ µ Rz ∗i = q ∗ · z ∗i ≤ 0.
Therefore x∗i ∈ BiA (p); and, since x∗i maximizes i over BiR (p∗ , q ∗ ; R), it now follows from (12.39) that x∗i maximizes i over BiA (p) as well.
Notice that it follows from the above result that, given the assumptions of the theorem, the allocation associated with a Radner equilibrium is Pareto eﬃcient; at least in the ex ante sense. As in the case of ArrowDebreu equilibrium, it also follows that if preferences are weakly separable over states, and the ex post preferences, given that a state has occurred are the same as the ex ante preferences, then the allocation will be Pareto eﬃcient in the ex post sense as well (recall the discussion at the end of Section 2). Of course, this is only half of the story in any case; with appropriate convexity assumptions, and given a complete asset structure, it also follows from Theorem 12.16 (and our work in Chapter 7) that, given a Pareto eﬃcient allocation, there exist endowments as well as asset and (spot) commodity prices such that the allocation is achieved as a Radner equilibrium.
12.4. Complete Markets
349
Clearly, the assumption that the asset structure is complete is of critical importance in establishing Theorem 12.16 and thus for the implications discussed in the above paragraph. Moreover, it is clear why this assumption is so important; it enables consumers to transfer consumption between states in any desired fashion that is consistent with their beginning wealth. We will discuss this assumption, and some of its implications further in the next section. In the meantime, let’s consider a technical aspect of Theorem 12.16. If you will recall the relationship between the stateprice vector obtained in the second part of our proof and the vector of asset prices, q ∗ [see equation (12.36)], you will note that, since the return matrix, R, is assumed to have full rank, the stateprice vector is unique. Consequently, it may appear that the vector of prices which yields an ArrowDebreu equilibrium at the allocation x∗i is also unique. It is, of course, true that equation (12.36) uniquely deﬁnes a vector of ArrowDebreu prices, but there may be other vectors of prices which deﬁne an ArrowDebreu equilibrium at the same allocation, as is shown in the following. 12.17 Example. We will suppose there are two consumers, two states of the world at t = 1 (S = 2), and two physically distinguishable commodities (G = 2). We suppose further that each consumer has the von NeumannMorgenster utility function:
where u(·) is given by:
Ui (xi ) = π1 u(xi1 ) + π2 u(xi2 ),
(12.40)
1/2 u(xis ) = min{xis1 , xis2 }
(12.41)
We will suppose that the return matrix is given by: ! 1 0 R= , 0 1 and that the initial endowments are given by: ω 1 = (ω 11 , ω 12 ) = (2, 2; 0, 0) and ω 2 = (0, 0; 2, 2);
(12.42)
Now, if ps is the vector of spot prices at t = 1, given that s is the state of the world, and the consumer has allocated income ys to state s, then the indirect utility obtained is given by: vis (ps , ys ) =
1/2 ys ps1 + ps2
for s = 1, 2.
(12.43)
Thus, to maximize expected utility, given the probabilities π1 , π2 and security prices q ∗ = (q1∗ , q2∗ ), the consumer must maximize (with respect to y, z): π1 subject to:
1/2 1/2 y1 y2 + π2 , p11 + p12 p21 + p22
ps · ω is + zs − ys = 0 for s = 1, 2, and q ∗ · z ≤ 0.
(12.44)
(12.45)
350
Chapter 12. General Equilibrium with Uncertainty
In the special case in which π1 = π2 = 1/2 and q1∗ = q2∗ = 1 [I will leave you the problem of ﬁnding the solution for arbitrary positive π1 , π2 and (q1∗ , q2∗ )], we see that y ∗ must satisfy: y1∗ y2∗ = . (12.46) p21 + p22 p11 + p12 Thus, if we set: p∗sj = 1 for s, j = 1, 2; ∗ ∗ we see that xi , z i , p∗s , q ∗ , where: x∗is = (1, 1), for i, s = 1, 2, z ∗1 = (−2, 2), and z ∗2 = (2, −2); is a Radner equilibrium. Furthermore, in this case the unique stateprice vector for (R, q) is given by µ = (1, 1); and thus the price vector for the ArrowDebreu case, as deﬁned in (12.36) of the proof of Theorem 12.16 is given by p = p∗ . However, consider the ArrowDebreu price vector p = (1, 2, 1, 2). You can easily verify the fact that (x∗i , p) is an ArrowDebreu equilibrium.
12.5
Complete Markets and Eﬃciency
In the previous section, we have seen that Radner equilibria are both eﬃcient and unbiased if the asset structure is complete. In this section we will begin by presenting an example which shows that completeness of the asset structure is not a necessary condition for the Pareto eﬃciency of a Radner equilibrium. We will then present an example (with an incomplete asset structure) in which a Radner equilibrium is not Pareto eﬃcient, before moving on to discuss a bit more of the theory. 12.18 Example. Consider an economy in which there are two consumers, two physically distinguishable commodities, and three states; which we will suppose to be equally probable. We will suppose that each of the two consumers has the von NeumannMorgenstern utility function: U (xi ) = (1/3) u(xi1 )1/2 + u(xi2 )1/2 + u(xi3 )1/2 , (12.47) where: u(xis ) = min{xis1 , xis2 ).
(12.48)
The consumers’ initial endowment vectors are given by: ω 1 = (4, 4; 0, 0; 4, 4) and ω 2 = (0, 0; 4, 4; 4, 4). We further suppose that the return matrix, R, is given by: ⎛ ⎞ 1 0 ⎝ R = 0 1⎠ ; 0 0 and that the vector of asset prices is given by: ! 1 ; q∗ = 1
(12.49)
(12.50)
(12.51)
12.5. Complete Markets and Eﬃciency
351
and we note that q ∗ is arbitragefree, given R (see Exercise 10). From the form of the utility functions and the initial endowments, it is clear that we will have supply equal to demand within each of the three possible states if we have: p∗s = (1, 1)
for s = 1, 2, 3.
Thus consumer 1 can maximize expected utility by choosing c11 , consumption in period 1, and z12 , so as to maximize (see Exercise 10): √
c11 +
√
z12 ,
subject to: c11 = 8 − z12 . c∗11
= Solving, we ﬁnd consumer 2 will set:
∗ z12
=
c∗12
∗ ; and, of course, c∗ = 8. By symmetry, = 4 = −z11 13
∗ ∗ c∗21 = z21 = 4 = −z22 , and c∗23 = 8.
If we then deﬁne x∗is by: x∗isj =
c∗is 2
for i, j = 1, 2, & s = 1, 2, 3,
it is easy to show that x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, given R. Now suppose that there are futures markets for each commodity, and consider the price vector: √ √ p = (p1 , p2 , p3 ) = (1, 1; 1, 1; 1/ 2, 1/ 2). I will leave it to you to verify the fact that x∗i , p is an ArrowDebreu equilibrium. ∗ Consequently, it follows that xi is Pareto eﬃcient.
In the above example, both individuals would be made better oﬀ if each of their endowments were reduced by taking away a unit of each commodity in state 3, while adding a unit of each commodity to the aggregate endowment in both states 1 and 2. However, the structure of the model does not allow this sort of transfer; nor, of course, does the reallife situation we are trying to model. In eﬀect, the existence of the assets in this model allows consumers to trade amounts from endowments within states, but the aggregate endowment must remain ﬁxed within each state. In our next example, we will consider a situation in which the initial Radner equilibrium is not Pareto eﬃcient, but in which the addition of some additional securities allows a Radner equilibrium to be attained which is (ex ante) Pareto eﬃcient. 12.19 Example. We will here modify the previous example by considering 4 equally probable states, keeping the consumers’ (Bernoullian) utility functions the same, while letting the endowments be given by: ω 1 = (4, 4; 0, 0; 3, 3; 1, 1) and ω 2 = (0, 0; 4, 4; 1, 1; 3, 3),
352
Chapter 12. General Equilibrium with Uncertainty
and the return matrix be given by: ⎛ 1 ⎜0 ⎜ R=⎝ 0 0
⎞ 0 1⎟ ⎟. 0⎠ 0 ∗ ∗ I will leave you to show that xi , z i , p∗s , q ∗ is a Radner equilibrium, with: x∗1 = (2, 2; 2, 2; 3, 3; 1, 1), x∗2 = (2, 2; 2, 2; 1, 1; 3, 3),
z ∗1 = (−4, 4), z ∗2 = (4, −4), p∗ = (1, 1; 1, 1; 1, 1; 1, 1) and q ∗ = (1, 1). However, suppose we introduce two new assets, having the return vectors: ⎛ ⎞ ⎛ ⎞ 0 0 ⎜0⎟ ⎜ 0⎟ ⎜ ⎟ ⎜ . r 3 = ⎝ ⎠ and r 4 = ⎝ ⎟ 1 0⎠ 0 1 You can now verify the fact that xi , z i , p∗ , q is a Radner equilibrium, with: z 1
x1 = x2 = (2, 2; 2, 2; 2, 2; 2, 2),
= (−4, 4, −2, 2), z 2 = (4, −4, 2, −2), q = (1, 1, 1, 1),
and the vector of commodity prices remains as it was, p∗ = 1 . It is easy to see that the allocation xi is unanimously preferred to x∗i . Obviously, the original Radner equilibrium did not result in a Pareto eﬃcient allocation. While the above example shows that a Radner equlibrium is not necessarily Pareto eﬃcient, it follows from Theorem 12.16 that, if the asset structure is complete, then a Radner equilibrium necessarily yields a Pareto eﬃcient allocation. In fact, in the sort of situation we have been considering in examples (von NeumannMorgenstern utility functions and a concave Bernoullian utility function), any Pareto eﬃcient allocation can (with possible redistributions within states) be attainable as an ArrowDebreu equilibrium. But this means that, if the asset structure is complete, and given the same assumptions about consumer preferences, that any Pareto eﬃcient allocation can be attained as a Radner equilibrium. This sets the stage for the ﬁnal result of this chapter, which makes use of the following deﬁnition. 12.20 Deﬁnition. If R is an S × K return matrix, we deﬁne r(R), the range of R, by: r(R) = {v ∈ RS  (∃z ∈ RK ) : v = Rz}. In the next result, we make use of the following notation. If x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, given the asset structure, R, we deﬁne the vectors y ∗i (i = 1, . . . , m) by: ∗ = p∗ · (x∗i − ω is ) for s = 1, . . . , S; (12.52) yis in other words, under our maintained assumptions: y ∗i = Rz ∗i
for i = 1, . . . , m.
(12.53)
12.5. Complete Markets and Eﬃciency
353
12.21 Proposition. Suppose x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, given an asset structure with the S ×K return matrix R, and suppose a second asset structure for E has the S × K return matrix, R . If r(R) = r(R ), then x∗i , z i , p∗ , q is a Radner equilibrium for E, given the asset structure R ; where, for each i, z i is any vector in RK satisfying: R z i = y ∗i , and where:
q = (R ) µ,
and µ is a state price vector associated with (R, q ∗ ).8 Proof. Suppose x∗i , z ∗i , p∗s , q ∗ is a Radner equilibrium, given the asset struc ture R, and that R is a second asset structure for E with r(R) = r(R ). Under the maintained assumptions of this chapter, it must be the case that q ∗ is arbitrage free, and thus there exists a state price vector associated with (R, q ∗ ), µ, satisfying: R µ = q ∗ .
We then deﬁne the asset price vector for R ,
(12.54)
q,
by:
q = (R ) µ.
(12.55)
y ∗i
Next, with deﬁned as in (12.52) and (12.53), above, we make use of the assumption that r(R ) = r(R) to deﬁne assert the existence of z i satisfying: R z i = y ∗i
for i = 1, . . . , m − 1,
and deﬁne z m by: z m = −
m−1 i=1
z i .
(12.56)
(12.57)
and note that:
m−1 m−1 m−1 z i = − R z i = − y ∗i R z m = R − i=1 i=1 i=1 ⎞ ⎞ ⎛ m−1 ∗ ⎛ − i=1 p1 · (x∗i1 − ω i1 ) p∗1 · (x∗m1 − ω m1 ) ⎟ ⎟ ⎜ ⎜ .. .. ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ m−1 . ⎟ ⎜ ∗ ∗ ∗ ∗ ∗ ⎟ ⎟ ⎜ =⎜ ⎜ − i=1 ps · (xis − ω is ) ⎟ = ⎜ ps · (xms − ω ms ) ⎟ = y m . ⎟ ⎜ ⎟ ⎜ . . . . ⎠ ⎝ ⎠ ⎝ . . ∗ ∗ p∗S · (x∗mS − ω mS ) − m−1 i=1 pS · (xiS − ω iS ) We wish now to prove that x∗i , z i , p∗ , q is a Radner equilibrium for E, given R . Accordingly, we begin by noting that, for each i: q · z i = (q ) z i = µ Rz i = µ y ∗i = µ Rz ∗i = (q ∗ ) z ∗i = 0; while from (12.57), we have:
m i=1
z i = 0.
8 Both the statement and the proof of this result are derived from Proposition 19.E.2 of MasColell, Whinston, and Green [1995].
354
Chapter 12. General Equilibrium with Uncertainty
Our proof that x∗i , z i , p∗ , q is a Radner equilibrium is completed by showing that ∗ ∗ for each i, if xi ∈ Bi (p , q , R), then it is also in Bi (p∗ , q , r ), and conversely. Since this demonstration can pretty much follow along the same lines as the argument of the preceding paragraph, I will leave this part of the proof as an exercise. Notice that as a corollary of the above proof, it follows that if R is a complete asset structure, then any allocation attainable as a Radner equilibrium, given R, is also attainable as such an equilibrium with the Arrow security asset structure, I S , and conversely. Moreover, if we look back at the initial steps of the above proof, letting R = I S , then q becomes: q = (R ) µ = I S µ = µ. Thus, given a complete asset structure, R, and arbitragefree asset price vector, q, the associated state price vector, µ, can be interpreted as the asset price vector for the (equivalent) Arrow security asset structure. Before leaving this topic, it should be mentioned that it is sometimes possible to add socalled derivative securities to an incomplete asset structure in such a way as to result in a complete asset structure. A favorite example is the (European) call option. Such a security is, essentially, a guarantee of the right to buy a basic security at a ﬁxed price after the resolution of uncertainty. If the price at which the option can be exercised is denoted by q ∗ , and it is an option to buy the k th primiary security then the new security then has the return vector: rs∗ = max{rsk − q ∗ , 0}
for s = 1, . . . , S;
since the option will only be exercised if it is proﬁtable to do so. Thus, let’s return to Example 12.15. We there considered the asset structure: ⎛ ⎞ 2 1 1 R = ⎝1 0 1⎠ , 1 1 0 which is incomplete. Suppose we now add an option to buy the ﬁrst of these assets at the price of 1. Then this derived security has the return vector: r ·0 = (1, 0, 0) . If we add this to the initial asset structure, we obtain: ⎛ ⎞ 1 2 1 1 R∗ = ⎝0 1 0 1⎠ , 0 1 1 0 You can easily show that the asset structure is now complete. (However, see Exercise 14, below.)
12.6. Concluding Notes
12.6
355
Concluding Notes
As I warned you in the introduction to this chapter, I have not attempted to provide any more than a very elementary introduction to this topic. Those interested in pursuing the topic further will ﬁnd a more complete introduction provided in Chapter 19 of MasColell, Whinston, and Green [1995], and I have used a notation very nearly identical to theirs; which should make it relatively easy for you to avail yourself of that source. For those interested in pursuing their study of this material still further, let me recommend Duﬃe [2001] for the ﬁnancial aspects of this sort of model, and Magill and Quinzii [1996] for a more advanced development of the theory of general equilibrium under uncertainty than is presented here. Let me also recommend the survey by Magill and Shafer [1991]. Exercises. The ﬁrst three of the following exercises, are set within the general context of Example 12.5. 1. Show that, in the context of Example 12.5, utility maximization implies that equations (12.11) and (12.12) must hold. 2. Show that, in the context of Example 12.5, if (12.14) holds, but π1 > π2 , then in equilibrium we must have p1 > p2 . 3. In the context of Example 12.5, show that, given (12.17), (12.18) must hold. 4. Show that if equation (12.24) holds, and each column of R is semipositive, then there exists z ∈ RK satisfying (12.25). 5. Show that if the ﬁrst commodity is a num´eraire good for each state of the economy, then we can normalize to set ps1 = 1, for each state s, in dealing with Radner equilibria. [Pay close attention to equation (12.21) of the text.) 6. Prove Proposition 12.13. (Hint: Notice that you can apply the classical Lagrangian method here.) 7. Suppose there are two states, and that a consumer has a Bernoulli utility G function, us : RG + → R, which is concave on R+ . Show that, for ﬁxed probabilities for the two states, (π1 , π2 ), the expected utility function: U (x1 , x2 ) = π1 U1 (x1 ) + π2 u2 (x2 ), is concave on RG +. Now generalize your result by supposing that there are S states. 8. In this example, we consider a contingentcommodity model of pure exchange, with two possible states of nature, two commodities, and two consumers. Denoting commodity bundles as: xi = (xi1 , xi2 ) = (xi11 , xi12 , xi21 , x122 ),
356
Chapter 12. General Equilibrium with Uncertainty
we suppose that the ith consumer has the preferences described by: xi Pi xi ⇐⇒ π1 (log xi11 + log xi12 ) + π2 (log xi21 + log xi22 ) > π1 (log xi11 + log xi12 ) + π2 (log xi21 + log xi22 )
for 1 = 1, 2.
where ‘πs ’ denotes the probability that state s will occur, for s = 1, 2 a. Denoting the ith consumer’s initial resource endowment by ‘ω i ’. for i = 1, 2, show that for a given vector of prices, the ith consumer’s demand for xsj is given by: xisj =
πs p · ω i 2psj
for i, s, j = 1, 2.
b. Suppose now that the two consumers’ endowments are given by: ω 1 = (2, 1, 1, 2) and ω 2 = (1, 2, 2, 1), respectively; and that both individuals have a subjective probability of 1/2 for the occurrence of both states (so that for both agents, π1 = π2 = 1/2). Show that if: psj = 1 for s = 1, 2, & j = 1, 2, then an ArrowDebeu equilibrium is sattained with: xisj = 3/2
for i = 1, 2, s = 1, 2, & j = 1, 2.
c. Now suppose that only the ﬁrst commodity can be traded initially, but that both consumers correstly anticipate the price vector at t = 1 for each of the two possible states. Find a Radner equilibrium corresponding to the ArrowDebreu equilibrium found in part b, above. 9. Suppose there are G physically distinguishable commodities, that there are S states of nature, and that a consumer has the von NeumannMorgenstern utility function: S a U (xi ) = u(xis ) , s=1
where: 0 < a < 1, ∗ ∗ and u : RG + → R+ is positively homogeneous of degree one. Show that if p and q are the vectors of commodity and securities prices, respectively, and R is the return matrix for the economy (and we denote row s of this matrix by ‘r s· ’, for s = 1, . . . , S, then the consumer will maximize expected utility by choosing ci ∈ RS+ and z i ∈ RK + so as to maximize; c a S is πs s=1 γ(p∗s )
subject to: and:
p∗ · ω is + r s· · z i − cis = 0 for s = 1, . . . , S, q ∗ · z i = 0,
12.6. Concluding Notes and then setting:
357
xis = cis g(p∗s )
for s = 1, . . . , S;
where γ : RG ++ → R++ is the costofliving index and g(·) is the ‘unit income demand function’ associated with u(·) (see Section 4.9). Thus, in particular, if p∗s = p∗1 and πs = 1/S, for s = 2, . . . , S, then the consumer can choose ci and z i so as to maximize: S
ca , s=1 is
subject to the above constraints. 10. Verify the details of Example 12.18. 11. In the context of Example 12.18, suppose we add a third asset, with return vector: ⎛ ⎞ 0 r 3 = ⎝0⎠ ; 1 so that the asset structure then corresponds to Arrow securities, and the return matrix becomes the 3 × 3 identity matrix. Can you ﬁnd a Radner equilibrium which yields the same consumption allocation, x∗i , as was obtained in the original example? 12. Return to Example 12.8, and, keeping all other data the same, suppose now √ that the two consumers have the Bernoullian utility function u(xis ) = xis1 · xis2 . Find the new Radner equilibrium. 13. Complete the details of the proof of Proposition 12.21 14. Consider the asset structure: ⎛
⎞ 1 0 1 ⎝ R = 1 1 0⎠ 1 0 1 Show that this asset structure is incomplete. Can you add a (one) call option in such a way as to make the resultant asset structure complete? Follow the same procedure and question for the asset structure: ⎛ ⎞ 2 0 1 0 ⎜1 1 0 0⎟ ⎟ R=⎜ ⎝3 0 1 1⎠ . 1 1 0 0
Chapter 13
Further Topics in General Equilibrium Theory 13.1
Introduction
In this chapter, we will consider the explicit introduction of time into the model; beginning with a ﬁnite time horizon, and then brieﬂy considering two extensions to an inﬁnite time horizon. In our second such extension, the ‘overlapping generations model,’ we not only consider an inﬁnite number of time periods, but also an inﬁnite number of consumers. In Section 5 we will return to the case of a ﬁnite number of time periods, but suppose that there are a continuum of consumers. We will undertake only a very brief introduction to each of these topics. This should not be construed as an implicit commentary on their relative importance; indeed, all of these topics are important, and much interesting research is currently being conducted in each of these areas. However, the introduction of time into a general equilibrium model is the primary focus of the required courses in macroeconomic theory in most graduate programs; which is my reason for not pursuing the topic at great length here. The reason for making the introduction to the ‘continuum of traders’ approach so brief is somewhat diﬀerent; the fact of the matter is that one cannot proceed very far in the development of this approach without making use of some general measure and integration theory, topics with which most graduate students in economics are unlikely to be familiar.
13.2
Time in the Basic Model
We found in the last chapter that we could introduce uncertainty into the model simply by appropriately interpreting some of the variables in the standard general equilibrium model. In fact, we can also begin to examine the role of time in the model in much the same way. We consider a ﬁnite number of distinct time periods, t = 1, . . . , T , and distinguish commodities by both physical characteristics and time of availability. If, as in the last chapter, we suppose that there are G physically distinguishable commodities, then the total number of commodities, n, is given by n = T × G. We will also use a notation very similar to that developed in the
360
Chapter 13. Further Topics
previous chapter; however, we will make a change in our notation for commodity bundles, using c, c , etc., as our generic notation for commodity bundles. Thus the commodity bundle available to the ith consumer is denoted by ci ∈ Rn , and we write: ci = (ci1 , . . . , cit , . . . , ciT ), with ‘cit ∈ RG ’ denoting the commodity bundle available to the ith consumer in the tth period. Consistently with this notation, in this chapter we will denote the ith consumer’s consumption set by ‘Ci ,’ rather than ‘Xi ’ (the reasons for this change will soon be apparent). It is also worth noting that in most of our work in this and the next section the assumptions of the model incorporate as a special case the situation in which, for each i there exist positive integers ti and ti such that: 1 ≤ ti < ti ≤ T
(13.1)
Ci = 0ni × Ci∗ × 0ni ,
(13.2)
ni = ti · G, ni = (T − ti ) · G,
(13.3)
and such that Ci takes the form:
where: and, deﬁning ni = ti − ti :
Ci∗ ⊆ Rni ·G .
(13.4) tth i
The idea here is that, for a consumer who is born in the period and dies in period ti , only its consumption in periods ti + 1, . . . , ti aﬀects its survival or preferences. In general, a set of this form can satisfy all of the assumptions which we used in Chapters 5, 7 and 8. For example, if Ci∗ is bounded below, closed, or convex, then Ci is bounded below, closed, or convex, respectively. It is also common practice in the literature to assume that the ith consumer’s preferences can be represented by a utility function of the form: Ui (ci ) =
T
δ t−1 ui (ci ), t=1 i
(13.5)
where 0 < δi < 1. This also is not inconsistent with a consumption set of the form set out in equation (13.2); although such a utility function does imply that the consumer’s preferences are weakly separable and stationary over time periods (see Section 2.8). We will look at a special case of this sort of model, the ‘overlapping generations’ model, later on in this chapter, but for now let’s examine some additional general considerations. It is clear that the model we set out in chapters 7 and 8 incorporates the situation under examination here. As was true in the previous chapter, we are simply being more speciﬁc in our interpretation of the variables. We can deﬁne competitive equilibrium for the present case, and, clearly, the First and Second Fundamental Theorems developed in Chapter 7 apply to this case with no need for any modiﬁcation of assumptions. There is, however, a bit of diﬃculty in the interpretation of equilibrium in this case. The simplest interpretation, and the one which seems to be the most often used, is that all consumption and production plans and contracts
13.2. Time in the Basic Model
361
are made in the ﬁrst period. This is not very plausible as a description of how real economies operate, however. Moreover, if we allow for the possibility of consumption sets of the form speciﬁed in (13.1)–(13.4), this interpretation loses all plausibility. In any case, the ‘ﬁrstperiod contracts with forward prices’ interpretation/treatment does not allow us to investigate the eﬀects of time at all. Consequently, economists have specialized the model in order to carry out this investigation, and we will undertake a rather cursory examination of a fairly standard model of this type. In our treatment, we will not initially concern ourselves with individual consumers; concentrating instead on aggregate consumption, and the aggregate consumption set: m Ci . (13.6) C= i=1
We will also suppose that all of the individual consumption sets can be written in the form: T Ci = Cit ; (13.7) t=1
which, of course, implies that the aggregate consumption set can also be expressed as a similar cartesian product: T m C= Ct where Ct = Cit , for t = 1, . . . , T. (13.8) t=1
i=1
Notice that consumption sets of the form speciﬁed in (13.1)–(13.4) are of this form if, and only if, the sets Ci∗ can be written as cartesian products. The simplest sort of i ⊆ RG assumption which guarantees the desired form is that for each i there exists C i , for t = 1, . . . , T ; but such that Ci takes the form speciﬁed in (13.7) with Cit = C our assumptions do not require this to be the case. Turning now to the production side of the economy, the fundamental empirical fact of which the theory attempts to take account is that production takes time. One could take account of this fact by taking the duration of each time period to be suﬃciently long that all production processes can be completed within the period, but this is rather evading the issue, and certainly does not allow us to analyze the eﬀect of time lags in production. In this section we will make a very common assumption; namely that production processes can all be completed by the end of the period in which they are initiated (and thus the output from same is available at the beginning of the following period). We will also consider only the aggregate production set for the economy; supposing that production can be characterized by T − 1 sets Yt ⊆ R2G (t = 1, . . . , T − 1); where: Yt = {(xt , y t ) ∈ R2G +  xt can produce y t }.
(13.9)
Thus, the aggregate vector of inputs for the economy is denoted by ‘xt ,’ and the output vector chosen in the tth period (which becomes available at the beginning of period t + 1) is denoted by ‘y t .’ As indicated in the deﬁnition of Yt , we will follow the convention of supposing both of these vectors to be nonnegative elements of RG . The assumption that all production processes which are initiated in a period can be completed by the end of the period is considerably less stringent than it appears to be at ﬁrst glance, for one can include partiallyﬁnished goods (goods in process) as outputs in one period and inputs in the next. Thus, consider the following example.
362
Chapter 13. Further Topics
13.1 Example. Suppose we have two ﬁrms, the ﬁrst of which has a production process which takes one period to complete, while the second process requires two periods for completion. Allowing for labor, we then suppose that there are four physically distinguishable goods in the economy. We will let the ﬁrst coordinate of production vectors measure the output of the ﬁrst ﬁrm, the second coordinate will measure the quantity of goods in process for the second ﬁrm, while the third coordinate will apply to the ﬁnished good production of this ﬁrm, and, ﬁnally, the fourth coordinate will measure quantities of labor. We suppose that the inputrequirement function of the ﬁrst ﬁrm is given by 1 = g1 (y1 ). The labor requirement for goods in process for the second ﬁrm will be supposed to be given by: 2 = g2 (y2 ), while the production function for ﬁnal goods output for the second ﬁrm is given by: y3 = min{y2 /a1 , 3 /a2 }, where a1 and a2 are both positive. The production set for the economy can then be deﬁned by: Yt = (xt , y t ) ∈ R8+  xt2 = a1 · yt3 & xt4 = g1 (yt1 ) + g2 (yt2 ) + a2 yt3 & yt4 = 0 If we denote the net production chosen in the tminusﬁrst period (and which then becomes available at the beginning of the tth period) by ‘y t−1 ,’ the vector of inputs to be applied to production in the tth period by ‘xt ,’ and the commodity bundle available for consumption during the tth period by ‘ct ,’ feasibility requires that: (13.10) xt + ct = y t−1 for t = 1, . . . , T. As suggested by this feasibility requirement, we will be ignoring initial commodity endowments. 13.2 Example. Consider an economy with two commodities, labor and a produced good which can either be consumed or used as an input for this period’s production (‘capital’).1 Suppose further that technology can be characterized by the production function f : R2+ → R+ , where we take the ﬁrst coordinate to be the quantity of labor, and the second to be the quantity of the produced good (‘capital’) which is applied to production. We can then take Yt to be the set: Yt = {(x1 , x2 ; y1 , y2 ) ∈ R4+  y1 = 0 & y2 = f (x1 , x2 )}. In this case, if (xt−1 , y t−1 ) ∈ Yt−1 and (xt , y t ) ∈ Yt , we must have: xt + ct = y t−1 , and thus xt1 + ct1 = 0. In other words, by taking xt ∈ RG + , we are correspondingly following the convention that negative coordinates of consumption vectors ct represent quantities of primary inputs (particularly labor) supplied by consumers to the production sector. 1 It may help to think of the produced good as ‘wheat,’ which can either be consumed this year or planted as seed to produce next year’s crop.
13.2. Time in the Basic Model
363
We will take y 0 , which is the initial stock of commodities, as given, and then make use of the following deﬁnition of feasibility in the present context. 13.3 Deﬁnitions. Let y 0 ∈ RG + be given. We will say that the sequence (program) , (xt , y t , ct ) ∈ R3GT is feasible iﬀ: 1. ct ∈ Ct , 2. (xt , y t ) ∈ Yt and 3. xt + ct = y t−1 , for t = 1, . . . , T . , We will say that a sequence (xt , y t )Tt=1 is a production program iﬀ it satisﬁes condition 2, above. T , In the material to follow, we will refer to any ﬁnite sequence (xt , y t , ct ) t=1 such that: xt , y t , ct ∈ RG for t = 1, . . . , T, as a program. In dealing with such programs, we will often ﬁnd it convenient to use the notation x, y and c to denote the whole vector of corresponding variables; that is, for example: c = (c1 , c2 , . . . , ct , . . . , cT ). We make use of this notation in the following deﬁnition. , , (x 13.4 ,Deﬁnition.  Let (xt , y t ,,ct ) and  t , y t , ct ) be two programs. We will say that (xt , y t , ct ) dominates (xt , y t , ct ) iﬀ c > c ; that is, iﬀ: ct ≥ ct for t = 1, . . . , T, and (∃t∗ ∈ {1, . . . , T }) : ct∗ > ct∗ . , 13.5 Deﬁnition. We will say that a feasible, program,  (xt , y t , ct ) , ,is eﬃcient, given y T iﬀ there exists no feasible program, (xt , y t , ct ) , dominating (xt , y t , ct ) and satisfying y T ≥ y T . As indicated in the above deﬁnition, we only consider the resultant stream of consumption vectors in determining whether a feasible program is or is not eﬃcient. However, notice the qualiﬁcation, “given y T ,” in the above deﬁnition. Without this qualiﬁcation, and supposing that (xT , y T ) ∈ YT and y T > 0 implies xT > 0, no , program (xt , y t , ct ) having y T > 0 could be eﬃcient! 13.6 Deﬁnition. Let p = (p1 , . . . , pt , . . . , pT , pT +1 ) be a vector (or,, if you prefer, a ﬁnite sequence) of prices. We will say that a production program, (xt , y t )Tt=1 is competitive, given p, iﬀ: y t−1 − xt ∈ Ct for t = 1, . . . , T, , and for all production programs, (xt , y t )Tt=1 satisfying: y t−1 − xt ∈ Ct we have:
for t = 1, . . . , T,
pt+1 · y t − pt · xt ≥ pt+1 · y t − pt · xt
for t = 1, . . . , T.
364
Chapter 13. Further Topics
Notice that we make use of a sequence of T + 1 (Gdimensional) price vectors in the above deﬁnition. In the deﬁnition, we are essentially requiring that a competitive program maximizes proﬁts in each period, so it may look a bit strange to require that this maximization be subject to y t−1 −xt ∈ Ct for each t. However, this requirement can be justiﬁed by the fact that full competitive equilibrium will require that this condition is satisﬁed. In connection with the deﬁnition just presented, we deﬁne: πt (pt , pt+1 ; y t−1 ) = max pt+1 · y t − pt · xt  (xt , y t ) ∈ Yt & y t−1 − xt ∈ Ct (13.11) , We will demonstrate shortly , that if (xt , y t , ct ) is a program, p is,a price vector, and the production program (xt , y t ) is competitive, given p, then (xt , y t , ct ) is 2 eﬃcient, given y T . However, it will be convenient to prove this by making use of the following result. , 13.7 Proposition. Suppose (x∗t , y ∗t ) is a production program which is competitive, given the price vector p∗ , and deﬁne c∗ by: c∗t = y ∗t−1 − x∗t for t = 1, . . . , T. , Then (x∗t , y ∗t , c∗t ) is feasible, and given any feasible program, (xt , y t , ct ) , we must have: T T p∗ · c∗t + p∗T +1 · y ∗T ≥ p∗ · ct + p∗T +1 · y T . (13.12) t=1 t t=1 t , ∗ ∗ ∗ Proof. Obviously (xt , y t , ct ) is feasible, and since it is, we have: ,
p∗1 · c∗1 + p∗1 · x∗1 = p∗1 · y 0 , and:
p∗2 · c∗2 + p∗2 · x∗2 = p∗2 · y 1 ;
Thus, deﬁning: πt∗ = πt (p∗t , p∗t+1 ; y ∗t−1 ) we have:
for t = 1, . . . , T − 1,
p∗1 · c∗1 + p∗2 · c∗2 = π1∗ − p∗2 · x∗2 + p∗1 · y 0 .
Suppose now that we have, for 2 ≤ t ≤ T − 1 : t t−1 p∗s · c∗s = πs∗ − p∗t · x∗t + p∗1 · y 0 . s=1
,
s=1

, Then by the feasibility of and the fact that (x∗t , y ∗t ) is competitive, ∗ given p , we have: t−1 t p∗ · c∗s + p∗t+1 · c∗t+1 = π ∗ − p∗t · x∗t − p∗1 · y 0 + p∗t+1 · y ∗t − p∗t+1 · x∗t+1 s=1 s s=1 s t = πs∗ − p∗t+1 · x∗t+1 + p∗1 · y 0 . (x∗t , y ∗t , c∗t )
s=1
2
In choosing the approach taken here, in the development of this and the next result, I am exhibiting my indebtedness to my study of lecture notes prepared by Professor Mukul Majumdar for his course in Intertemporal Economics at Cornell University.
13.2. Time in the Basic Model Therefore:
T t=1
p∗t · c∗t =
365 T −1 t=1
πt∗ − p∗T · x∗T + p∗1 · y 0 ,
and consequently: T t=1
p∗t · c∗t + p∗T +1 · y ∗T =
T t=1
πt∗ + p∗1 · y 0 ,
(13.13)
A similar argument establishes: T t=1
p∗t · ct + p∗T +1 · y T =
T t=1
(p∗t+1 · y t − p∗t · xt ) + p∗1 · y 0 ;
and thus, since (xt , y t ) ∈ Yt , for t = 1, . . . , T : T t=1
p∗t · ct + p∗T +1 · y T ≤
T t=1
πt∗ + p∗1 · y 0 ..
Combining (13.13) and (13.14) yields the desired result.
(13.14)
One can then prove the following; although I will leave the proof as an exercise. , 13.8 Proposition. Suppose (x∗t , y ∗t , c∗t ) is a feasible program, that p∗ is a price vector satisfying: p∗t 0 for t = 1, . . . , T + 1, , and that for all feasible (xt , y t , ct ) , we have: T t=1
p∗t · c∗t + p∗T +1 · y ∗T +1 ≥
T t=1
p∗t · ct + p∗T +1 · y T +1 .
, Then (x∗t , y ∗t , c∗t ) is eﬃcient, given y ∗T . Now suppose that at each time period there is a production manager whose job it is to choose a pair (x∗t , y ∗t ) ∈ Yt which maximizes proﬁts over Yt , given (pt , pt+1 ; y t−1 ), and subject to y t−1 − x∗t ∈ Ct , and suppose a strictly positive price vector, p∗ , is given. Then the tth production manager needs to know only her/his own production set, the production (endowment), y t−1 , available at the beginning of the period, and the pair of price vectors (p∗t , p∗t+1 ) in order to carry out her/his assignment. Moreover, and most remarkably, it almost follows from these last two +1 , the propositions that, given any sequence of strictly positive price vectors, p∗t Tt=1 independent actions of these T production managers will result in a production , ∗ ∗ program (xt , y t ) which is eﬃcient, given the T th production manager’s choice of y ∗T ! Unfortunately, the statement of the above paragraph is not quite true, much as we would like it to be! The problem is that the choice of output, y ∗t may not enable both production and consumption to take place in period t + 1. It should also be mentioned that if a particular value for y ∗T is speciﬁed in advance, that is, at t = 1, then the sequence of prices will have to be chosen in a way that is consistent with this desired outcome. We will return to this second point, but ﬁrst consider the following example.
366
Chapter 13. Further Topics
13.9 Example. We consider an economy with two commodities, labor and a produced good, and suppose consumption sets are given by: Ct = {c ∈ R2  −16t ≤ c1 ≤ 0 & c2 ≥ 2t}
for t = 1, 2, . . . , T ;
while the production sets are given by: Yt = {(x, y) ∈ R4+  y1 = 0 & y2 = 2(x1 x2 )1/4 }, and y 0 = (0, 3). Basically, the idea here is that the economy is growing over time, in terms of numbers of consumers. This growth leads to diﬃculties if production managers simply maximize proﬁts in a myopic fashion. In fact, I will leave it to you to verify (Exercise 2) that if p1 = (1/16, 1), then the production manager for the ﬁrst period maximizes proﬁts at x∗1,1 = 16, x∗1,2 = 1, and y ∗1 = (0, 4). This choice would be ﬁne if C2 = C1 , but given the actual form of C2 , the set of pairs (x, y) ∈ Y such that y ∗1 − x ∈ C2 is equal to {0}! As mentioned earlier, the second barrier to a decentralized development of an eﬃcient intertemporal program is that the calculation of the price sequence which will yield a speciﬁc, predetermined value of y T will generally require a knowledge of both Ct and Yt for each t (t = 1, . . . , T ). We will discuss this issue further in the next section.
13.3
An Inﬁnite Time Horizon
As we have suggested in the previous section, whether or not a program is eﬃcient is conditional upon the value speciﬁed for y T ; in fact, it is generally true that the values of all the variables appropriate for an eﬃcient program are dependent upon the speciﬁed value for y T . Moreover, it is not clear what value one should take for T . But of course, the two problems are very much interrelated. If one is using the model to develop, say, a ‘ﬁveyear plan,’ then the appropriate value for T is, of course, clear. In this case, however, the consumption values which can be attained in each period will generally depend upon the targeted value for y T . Moreover, given a speciﬁcation of y T , the values for consumption which are achieved by a feasible program will generally depend upon the value chosen for T (whether we are dealing with a 5year, or a 10year plan, for example). The price vector which will implement a feasible plan is also dependent upon the targeted value of y T , as well as the value chosen for T ; as is demonstrated by the following example.. 13.10 Example. We consider en economy in which there are two commodities; labor and a produced good. The produced good can be consumed, used as a current input in production, or used to create capital. The production function for the economy is given by: 1/2 yt,2 = kt · (xt,1 · xt,2 )1/4 , where: kt = k0
t−1 s=1
δs & δs = max 1, ys−1,2 − xs,2 − cs,2 , for s = 1, . . . , t − 1; t = 1, . . . , T ;
13.3. An Inﬁnite Time Horizon
367
where k0 = 4. Thus the production sets are given by: 1/2
Yt = {(xt , y t ) ∈ R4+  yt,1 = 0 & yt,2 = kt
· (xt,1 · xt,2 )1/4 } for t = 1, . . . , T ;
and we suppose Ct is given by: Ct = {c ∈ R2  −16 ≤ c1 ≤ 0 & c2 ≥ 0} for t = 1, . . . , T, while T = 4, and y 0 = (0, 3). We will concentrate most of our attention here on two programs, as set out in the following tables. Period t=1 t=2 t=3 t=4
yt,2 4 16 16 64
xt,1 1 4 4 16
xt,2 1 4 4 16
ct,1 1 4 4 16
ct,2 0 0 10 –
δt 2 1 2 –
Table 13.1: Program 1. I will leave it to you to demonstrate the fact that Program 1 is competitive, given prices pt = (1, 1), for t = 1, . . . , 5. Consequently, it follows from Propositions 13.7 and 13.8 that Program 1 is eﬃcient, given y ∗4 = (0, 64). Program 2 is set out in the following table. Period t=1 t=2 t=3 t=4
yt,2 8 8 8 8
xt,1 8 8 8 8
xt,2 2 2 2 2
ct,1 8 8 8 8
ct,2 1 6 6 –
δt 1 1 1 –
Table 13.2: Program 2. Once again I will leave it to you to demonstrate the fact that Program 2 is competitive; this time with the prices pt = (1/4, 1), for t = 1, . . . , 5. Consequently, it follows that Program 2 is eﬃcient, given y 4 = (0, 8). Thus these two very different programs are both competitive, and therefore are both eﬃcient, given their respective target terminal values. (See also Exercise 3, at the end of this chaper.) Because of the sensitivity of competitive programs to the speciﬁcation of the target terminal value of y T , and the awkwardness stemming from the fact that the T th period must be treated diﬀerently from the other periods in the analysis, many researchers have argued that to have a satisfactory model, one must take T = +∞. While this change to an inﬁnite time horizon (technically, we are here assuming a countable number of time periods) eliminates the asymmetry associated with the ﬁnite time horizon model, and eliminates some other ambiguities encountered there as well, it also creates some new problems. For example, consider the following example.
368
Chapter 13. Further Topics
13.11 Example. ( Hurwicz and Majumdar [1988, p. 237].) ) Here we consider an economy with one produced good, and suppose f : R+ → R+ is strictly concave and satisﬁes: (∀x ∈ R++ ) : f (x) > 0, f (x) > 0 & f (x) < 0. We assume that: C = R+ , and that y0 > 0. The program given by: xt = yt−1 , yt = f (xt ), and ct = 0 for t = 1, 2, . . . , is clearly not eﬃcient. However, if we deﬁne the sequence of prices, pt by: p1 = 1 and pt+1 = pt /f (xt ) for t = 2, 3, . . . ; it is easy to show that the program is competitive.
As this last example demonstrates, with an inﬁnite time horizon a competitive program is not necessarily eﬃcient. However, consider the following result, where we deﬁne a feasible program exactly ,  as in Deﬁnition 13.3, except that we now deﬁne an inﬁnite sequence (xt , y t , ct ) (t = 1, 2, . . . ); and similarly for a competitive program. , 13.12 Proposition. Let (x∗t , y ∗t , c∗t ) be a competitive program, given the price ∗ sequence pt , where: p∗t 0 for t = 1, 2, . . . , and suppose:
lim p∗t · y ∗t = 0.
Then (x∗t , y ∗t , c∗t ) is eﬃcient. ,
t→∞
(13.15)
Proof. ,Suppose, by  way of obtaining a contradiction, that there exists a feasible program, (xt , y t , ct ) , which is such that: ct ≥ c∗t for t = 1, 2, . . . , and, for some positive integer, T: cT = c∗T + d, where d > 0. Then, since p∗ 0, we have for all T ≥ T: T
T t=1
p∗t · (ct − c∗t ) ≥ p∗T · (cT − c∗T ) = p∗T · d > 0.
(13.16)
However, if we deﬁne = (1/2)(p∗T · d), it follows from (13.15) that there exists T ∗ such that for all t ≥ T ∗ : (13.17) p∗t · y ∗t < But then, if we let T ≥ max{T, T ∗ }, it follows from Proposition 13.7 that: T t=1
p∗t · (ct − c∗t ) ≤ p∗T · y ∗T − p∗T · y T ≤ p∗T · y ∗T < ;
13.4. Overlapping Generations which contradicts (13.16).
369
How reasonable is the condition expressed by equation (13.15)? Well, if we think in terms of the possibility of a continually increasing production program, it seems very unreasonable. On the other hand, our economic universe (and, apparently, our physical universe) is actually of ﬁnite duration. ,Consequently,  one can argue that, realistically, we only need to consider programs, (xt , y t , ct ) , such that there exists a ﬁnite integer, T such that: ct = xt = y t = 0 for t = T + 1, T + 2, . . . ;
(13.18)
and such programs necessarily satisfy (13.15). However, if we conﬁne our attention to just those programs satisfying (13.18) for some ﬁxed value of T , we are back to the ﬁnite model of the previous section; with the attendant diﬃculties already discussed. On the other hand, if we require only that any feasible program satisfy (13.18) for some value of T (where the critical value of T may vary from program to program), then, from a mathematical point of view, we are dealing with what is called the space of ﬁnitely nonzero sequences; which is a particularly nasty space to deal with, from a mathematical point of view. However, we can always normalize the price vectors; requiring, for example, that: pt = 1
for t = 1, 2, . . . .
Since the CauchySchwarz inequality3 implies: pt · y t ≤ pt · y t for all t = 1, 2, . . . , will hold if we simply assume that, for all feasible programs, ,condition (13.15) (xt , y t , ct ) , we must have: lim y t = 0. t→∞
However, the student should be warned that not all economists would agree with this assessment.
13.4
Overlapping Generations
The basic ‘overlapping generations’ model was introduced by Samuelson [1958],4 and has since become the ‘workhorse’ of macro economics.5 In this model, we once again consider a countably inﬁnite number of periods (t = 1, 2, . . . ), but we now allow a countably inﬁnite number of agents as well. In our treatment here, we will deal only with almost the simplest variety of such a model; an exchange economy in which each consumer is alive exactly two periods; and in which there exist only two 3
For a statement and proof of this inequality, see, for example, Moore [1999a, p. 35]. It had been developed and analyzed earlier by Allais [1947], but this work seems to have remained unknown to nonnativeFrenchspeaking economists until sometime after 1958. 5 I am here more or less quoting from Heijdra and van der Ploeg [2002, p. 590]; although, strictly speaking, they are referring to the extension of the model (with production) which was developed by Peter Diamond. 4
370
Chapter 13. Further Topics
consumers in each period. We will, however, suppose that there is a ﬁnite number, n, of commodities. We will denote the consumer ‘born’ in period t by ‘it ,’ and denote it ’s initial endowment by ‘ω it ,’ where: ω it = (ω ty , ω to ), where the subscripts y and o are intended to suggest ‘young’ and ‘old,’ respectively, and ω ty and ω to are elements of Rn+ .. Formally, the tth consumer’s consumption set, Ct , is of the form set out in equations (13.1)–(13.4), with: Cit = {0} × {0} × . . . {0} × Rn+ × Rn+ × {0} × . . . ; but, since we will be assuming that each consumer has a consumption set of the same general form, we will speak of the tth consumer’s choosing a pair xt = (xty , xto ) ∈ R2n + ; and we will suppose that each consumer has an asymmetric preference relation on R2n + . To make things work out symmetrically, we will also need to suppose that there is a consumer i0 who is old in period 1; although to avoid some really cumbersome notation, we will denote this consumer’s initial endowment by ‘w0 ,’ and the consumption bundle available to consumer i0 by ‘x0 .’ We can then deﬁne competitive equilibrium in more or less the usual way: 13.13 Deﬁnition. We will say that x∗0 , x∗t , p∗t is a competitive equilibrium for E = Pt , ω t iﬀ:: ∗ n 1. x∗0 ∈ Rn+ , x∗t ∈ R2n + and pt ∈ R+ for t = 1, 2, . . . , 2. x∗1y + x∗0 = ω 1y + w0 , 3. x∗ty + x∗t−1,o = ω ty + ω t−1,o for t = 2, 3, . . . , 4. p∗1 · x∗0 ≤ p∗1 · w0 & (∀x ∈ Rn+ ) : xP0 y ∗ ⇒ p∗1 · x > p∗1 · w0 , and 5. for each t: p∗t · x∗ty + p∗t+1 · x∗to ≤ p∗t · ω ty + p∗t+1 · ω to , while: ∗ ∗ ∗ ∗ ∗ ∗ ∀(x1 , x2 ) ∈ R2n + : (x1 , x2 )Pt (xty , xto ) ⇒ pt · x1 + pt+1 · x2 > pt · ω ty + pt+1 · ω to . Since this deﬁnition appears to be simply a very natural extension of the notion of competitive equilibrium to the present context, and in fact probably looks very familiar, it is quite distressing to discover that it does not have the same normative properties that such an equilibrium has in ﬁnite economies. In fact, such a competitive equilibrium may not be Pareto eﬃcient, as is demonstrated by the following example (which is taken from Geanakoplos’ and Polemarchakis’ survey [1991, p. 1927]). 13.14 Example. We suppose that there is only one commodity in each period, and that the consumers’ utility functions are given by: 1/3 ui0 (x0 ) = x0 and uit (xt ) = x2ty · xto for t = 1, 2, . . . ; while the initial endowments are given by: w0 = 1, ω t = (ωty , ωto ) = (5, 1), for t = 1, 2, . . . .
13.4. Overlapping Generations
371
It is easy to verify that if we set: p∗1 = 1, p∗2 = 5/2, . . . , p∗t+1 = (5/2) · p∗t , . . . , then x∗0 , x∗t , p∗t is a competitive equilibrium for E, where:
x∗0 = 1, x∗t = (5, 1) = ω t for t = 1, 2, . . . . However, the allocation (x0 , xt ) deﬁned by: x0 = 2, xt = (xty , xto ) = (4, 2) for t = 1, 2, . . . , is both feasible and unanimously preferred to (x∗0 , x∗t ).
As if this example weren’t troublesome enough, Hendricks et. al [1980] show that the core may be empty in even this sort of simple overlapping generations economy, while Kovenock [1984] presents an example of an economy in which there are two commodities and two consumers for t = 1, 2, , . . . , where there exists a Pareto eﬃcient allocation which is also Walrasian, but which is not in the core. However, there is also some good news in this context. The following result can be proved by essentially the same argument which established Theorem 7.22; the details will be left as an exercise. In the result, we deﬁne a feasible allocation as being Pareto eﬃcient if there exists no alternative eﬃcient allocation in which each consumer is better oﬀ. 13.15 Theorem. Under the assumptions of this section, if x∗0 , x∗t , p∗t is a competitive equilibrium for E which is such that: ∞ p∗ · (ω ty + ω t−1,o ), (13.19) p∗1 · (x∗0 + ω 1y ) + t=2 t ∗ 6 ∗ is ﬁnite, then x0 , xt is Pareto eﬃcient for E. If we deﬁne strong Pareto eﬃciency for E in the obvious way, then we can establish the following, although once again I will leave the proof as an exercise. 13.16 Corollary. If, in addition to the other assumptions of Theorem 13.15, we suppose that each relation, Pt is locally nonsaturating and negatively preference transitive, then x∗0 , x∗t is strongly Pareto eﬃcient for E. Once again, if we view the economic universe as being fundamentally ﬁnite, then competitive equilibria are Pareto eﬃcient (and contained in the core; see Exercise 9). However, a great many economists ﬁrmly believe that in an overlapping generations model one must assume an inﬁnite time horizon (see, for example, Geanakoplos and Polemarchakis [1991, p. 1900]). In this case, the sum in equation (13.19) is an inﬁnite series which will converge only if: lim p∗t · (ω ty + ω t−1,o ) = 0.
t→∞
Of course, this can be true even if T = ∞, but it is an awkward and unintuitive assumption if T is inﬁnite; and it is not suﬃcient to guarantee that the sum in equation (13.19) is ﬁnite in any case. However, I will leave this topic here, and proceed to the consideration of models in which we have an uncountably inﬁnite number (a continuum) of consumers. 6
In other words, if the value of the aggregate endowment is ﬁnite, given the prices p∗t .
372
Chapter 13. Further Topics
13.5
A Continuum of Traders
There are, it would seem, two main arguments that one might make to justify the study of markets wtih a continuum of traders. In order to present the ﬁrst reason, we can do no better than to quote the theorist who introduced the continuum of traders model into the literature, Robert Aumann. In his seminal article (Aumann [1964]), he argues as follows: The notion of perfect competition is fundamental in the treatment of economic equilibrium. The essential idea of this notion is that the economy under consideration has a “very large” number of participants, and that the inﬂuence of each individual participant is “negligible.” . . . Though writers on economic equilibrium have traditionally assumed perfect competition, they have, paradoxically, adopted a mathematical model that does not ﬁt this assumption. Indeed, the inﬂuence of an individual participant on the economy cannot be mathematically negligible, as long as there are only ﬁnitely many participants. Thus, a mathematical model appropriate to the intuitive notion of perfect competition must contain inﬁnitely many participants. We submit that the most natural model for this purpose contains a continuum of participants, similar to the ontinuum of points on a line or the continuum of particles in a ﬂuid. Very succintly, the reason for this is that one can integrate over a continuum, and changing the integrand at a single point does not aﬀect the value of the integral, that is, the actions of a single individual are negligible. Aumann’s argument is certainly eloquently stated, and has been very inﬂuential, but there is, of course, a counterargument; individuals may behave as if they believe they have no inﬂuence on markets when they, in fact, do. For example, an individual may feel that the expected gain from haggling over price does not justify the time and trouble involved in the negotiations. There is, however, another reason for being interested in continuum of traders models. To introduce this second reason, consider the following, which is a modiﬁcation of an example due to Scotchmer [2002, p. 2010]. 13.17 Example. Consider an exchange economy, in which agents each have a choice of living in one of two locations. There are two tradeable commodities, and agent i attains a utility of: u(xi ) = xi1 + xi2 , if she/he resides in Location One (L1), and a utility of: √ u(xi ) = ( 2)xi1 + xi2 /2, if Location Two (L2) is the chosen location.7 We will suppose that each agent has an initial endowment of one unit of each of the two commodities. 7 We might, for example, take L1 to be California, and L2 to be the Upper Peninsula of Michigan; while good 1 is brandy and good 2 is wine. Wine is okay as far as ‘Yoopers’ are concerned (and, yes, I am a Yooper), but no good for warming you up on a cold winter’s day. On the other hand cold is not a problem in sunny California, so (with an appropriate choice of units of measurement), an agent residing in California may be indiﬀerent between the two beverages.
13.5. A Continuum of Traders
373
Interestingly enough, if no √ trade is allowed, then each agent would prefer L1. We can verify this by noting that 2 + 1/2 < 2. However, such a situation is not Pareto eﬃcient. In fact, if an agent moves to L2, and trades 1/2 + units of good 2 to an agent in L1 in√exchange for 1/2 √− units of good one, both agents will gain as long as 0 < < (2 2 − 1)/2(1 + 2 2). The question is, however, can a market system work in this environment; that is, can we ﬁnd prices for the two commodities which result in a competitive equilibrium? Proceeding with this question, we note ﬁrst that we can obviously take the second commodity to be a num´eraire, and set its price equal to one. It is also apparent that with this normalization, the price of good one, which we will denote by ‘p,’ must be greater than one. This being the case, each agent in L1 will wish to sell her/his √ endowment of good 1, and consume only good 2. On the other hand, if p ≤ 2 an agent in L2 will sell her/his endowment of good 2, and consume only good 1. However, if we are to have equilibrium, then these two situations have to result in the same utility; in other words, we must have: √ p + 1 1+p= 2 , p √ that is, p = 2. On the other hand, the supply of good one on the market must equal the demand for same. Thus, denoting the number of agents choosing L1 by ‘m1 , the number choosing L2 by ‘m2 ,’ and, as usual, denoting the total number of agents by ‘m;’ we see that the number of units of good √ one oﬀered on the market will be equal to m1 .√On the other hand, with p = 2, each agent at L2 will have excess demand of 1/ 2 units of good 1. Thus, in order to have equilibrium, we must have: √ √ m1 = m2 / 2 = (m − m1 )/ 2;
so that we must have:
m1 1 √ . = m 1+ 2 But this means that no competitive equilibrium exists in this case, for if m1 and √ m are both positive integers, then m1 /m is a rational number; whereas 1/(1 + 2) is not!
As we noted in the above example, a competitive equilibrium would exist √if it were possible for the proportion of agents choosing L1 to be equal to 1/(1 + 2). While this is not possible if there are only a ﬁnite number of agents, a continuum of traders model handles this with ease, as we shall see. In fact, in a continuum of traders model this proportion could (with other choices of parameters) be any number between zero and one. Since it is very natural to think in terms of proportions of consumers or households having this or that property, or making this or that choice, this makes a continuum of traders model a very ﬂexible and convenient tool. Moreover, those of you who have had a real analysis course know that, given any real number between zero and one, there is a sequence of rational numbers converging to that number. This means that if we obtain a result in the continuum model which involves a certain proportion of consumers making some speciﬁc choice, we can be conﬁdent that there is a ﬁnite model in which approximately the same
374
Chapter 13. Further Topics
result obtains. We will present only the rudiments of such a model here, and we will conﬁne our attention to pure exchange economies; but, hopefully, this development will be useful in and of itself, as well as providing a basic idea of how such models work. In our discussion to this point, we have always eﬀectively identiﬁed the set of consumers with M = {1, . . . , m}, the ﬁrst m positive integers. In contrast, in this section we will identify the set of consumers with the set A = [0, 1], the unit interval in R+ .8 Again, in our work to this point, we have denoted consumption allocations by xi , xi , and so on; where the notation has been intended to suggest a ﬁnite sequence of commodity bundles (elements of Rn ), one for each consumer. However, a ﬁnite sequence with m terms is, from a formal point of view, a function from M = {1, . . . , m} into Rn . Thus, it should not cause great confusion if we now denote allocations by ‘xa , xa , and so on, where we think of, say, xa as being a function from A into Rn , that is, x : A → Rn . However, we will often denote the value of the function at a point a ∈ A by ‘xa ,’ rather than x(a); since this is the commodity bundle available to consumer (agent) a ∈ A. In a pure exchange economy a (competitive) consumer is fully identiﬁed by her/his characteristics (Pi , ω i ); and, in eﬀect, an mconsumer exchange economy is fully deﬁned by the ﬁnite sequence, (Pi , ω i ). Similarly, in the case at hand, an economy is fully identiﬁed by E = (Pa , ω a ); where, from a formal point of view, this is a function from A into ‘characteristics space.’ However, the meaning and signiﬁcance of the notation should be clear enough without worrying about a formal deﬁnition of ‘characteristics space.’ The next question is, how do we deﬁne feasible allocations for such an economy? It is pretty clear that we cannot compare the sum of individual commodity bundles with the sum of individual endowments. On the other hand, what criterion can we use to deﬁne a feasible allocation function? In order to consider this question with some precision, let’s suppose for the moment that there is only one commodity, and to simplify things still further, consider an allocation xa having the property that there exist m numbers, d1 , . . . , dm , and xi m i=1 such that: 0 < d1 < d2 < · · · < dm−1 < dm = 1,
(13.20)
and such that, deﬁning d0 = 0, we have: (∀a, ∈ [di−1 , di [ ) : xa = xi
for i = 1, . . . , m.
(13.21)
In other words, our allocation function is constant on each subinterval Ii = [ di−1 , di [. In this case, the proportion of consumers who will receive the quantity xi of the commodity is given by mi = di − di−1 . Consequently, it is consistent with standard useage to deﬁne per capita consumption of the commodity by: x= 8
m i=1
mi xi =
m i=1
(di − di−1 )xi .
Denoting this set by ‘A’ is intended to suggest the set of agents.
(13.22)
13.5. A Continuum of Traders
375
But in fact, given the form of the function x(·), the expression on the right in equation (13.22 ) is just:9 1 x= x(a)da (13.23) 0
Now suppose that the initial endowment of the (single) commodity has a similar distribution; so that ω : A → R+ and there exist 2m numbers, d1 , . . . , dm , and ω1 , . . . , ωm such that: 0 < d1 < · · · < . . . dm −1 < dm = 1, and, deﬁning
d0
(13.24)
= 0: (∀a ∈ [dj−1 , dj [) : ωa = ωj
for j = 1, . . . , m .
(13.25)
Then the per capita initial endowment is given by: 1 m (dj − dj−1 )ωj = ω(a)da; j=1
0
and feasibility will require that:
1
1
x(a)da =
x= 0
ω(a)da
(13.26)
0
(see Exercise 13, at the end of this chapter). Now, in terms of fully characterizing an economy of this type, we are left with two obvious problems. First, how do we deal with more than one commodity? Secondly, how can we characterize distributions of the commodities, and the endowments, in such a way as to guarantee that the integrals we need are always welldeﬁned? The ﬁrst of these problems is easy to handle. If x : A → Rn+ is a commodity distribution, it deﬁnes n coordinate functions, xj : A → R+ , for j = 1, . . . , n. We simply deﬁne the desired per capita commodity bundle by: ⎞ ⎛9 1 0 x1 (a)da ⎟ ⎜ .. ⎟ ⎜ 1 ⎟ ⎜9 1 . ⎟ x(a)da = ⎜ (13.27) ⎜ 0 xj (a)da ⎟ ; 0 ⎟ ⎜ .. ⎠ ⎝ . 91 0 xn (a)da although I will omit the limits of integration hereafter, as they will always be the same. The feasibility requirement: x(a)da = ω(a)da, (13.28) 9
Or, to make this expression look more familiar: 1 x= x(t)dt. 0
376
Chapter 13. Further Topics
then simply asserts that the per capita consumption of each commodity is equal to the per capita endowment of that commodity. The second problem is much more diﬃcult. You probably remember that if we interpret our integrals to be standard Riemann integrals, then we need the allocation and endowment functions to be continuous, except, possibly, at a ﬁnite set of points, in order for the integrals to exist.10 This is, in principle, a very troublesome and restrictive requirement. If we were worried about one, and only one distribution function, assuming it to be continuous except at a ﬁnite number of points is fairly innocuous; for, if we stretch our imaginations a bit, we can imagine that the agents have been labeled in such a way as to ensure that this continuity occurs. However, we may ﬁnd that, for example, the endowment function has to have the agents labeled in a diﬀerent order in order to ensure that it is continuous. In practice, more advanced developments of models of this type deal with distribution and endowment functions which are measurable, which ensures that the Lebesgue integrals of these functions, which we will denote by: x, A
always exist. A very important aspect of this assumption is that the set of all measurable functions from A to Rn is a real linear space; and thus one can add or consider scalar multiples of such functions.11 Moreover, and again most conveniently, it turns out that if, say, x and x∗ are both measurable functions from A to Rn , then: (x + x∗ ) = x+ x∗ , A
A
A
Since I do not expect most students to have had any previous exposure to the idea of measurable functions and Lebesgue integration, we will not pursue these ideas further here. In any case, however, most applications of continuum of traders models make use of distribution and endowment functions which are step functions; that is, functions of the type deﬁned in equations (13.20) and (13.21), above. In order to deal further with this theory, however, let’s deﬁne such functions in a bit more abstract fashion. First, we deﬁne the following. 13.18 Deﬁnition. If X is a nonempty interval of real numbers, we will say that a family, D = {D1 , . . . , Dm }, of sets is a ﬁnite interval partition of X iﬀ: 1. for all i ∈ {1, . . . , m}, Di is a nonempty subinterval of X, 2. 5 for all i, j ∈ {1, . . . , m}, such that i = j, Di ∩ Dj = ∅, 3. m i=1 Di = X. We make use of this to state our (slightly) more abstract deﬁnition of a step function. 10 More correctly, we need the set of points at which the functions are discontinuous to be a set of measure zero. 11 Which are deﬁned in the obvious way; for example, if x and x∗ are two such functions, we deﬁne x + x∗ by: (x + x∗ )(a) = x(a) + x∗ (a) for a ∈ A.
13.5. A Continuum of Traders
377
13.19 Deﬁnition. Let A be a nonempty interval of real numbers. We will say that x : A → Rn is a step function iﬀ there exist a ﬁnite interval partition of A, D = {D1 , . . . , Dm }, and a subset, {x1 , . . . , xm } such that: (∀a ∈ Di ) : x(a) = xi
for i = 1, . . . , m.
Of course, in the remainder of this section, we will always take A to be equal to the unit interval, A = [0, 1]. In this context, the following should be fairly obvious, although I will leave the proof as an exercise (Exercise 10, at the end of this chapter.) } are two 13.20 Proposition. Suppose D = {D1 , . . . , Dm } and D = {D1 , . . . , Dm ﬁnite interval partitions of A. If we then deﬁne the family of sets D∗ by:
D∗ = {D ⊆ A  (∃Di ∈ D & Dj ∈ D ) : [D = Di ∩ Dj & D = ∅]. Then
D∗
(13.29)
is a ﬁnite interval partition of A.
Given two step functions, x and x∗ on A, and a real number α ∈ R, we deﬁne x + x and αx by: (x + x∗ )(a) = x(a) + x∗ (a) for a ∈ A,
(13.30)
(αx)(a) = αx(a) for a ∈ A,
(13.31)
and: respectively. One can then make use of Proposition 13.20 to prove the following. 13.21 Proposition. Let S be the family of all step functions (into Rn ) deﬁned on A. Then S is a real linear space. The key to proving Proposition 13.21, and the primary reason for stating it here, is that if we deﬁne the sum of two step functions as in equation (13.30), above, we get another step function on A. Similarly, the scalar multiple of a step function on A is again a step function on A. The proof of the latter statement is obvious, and the proof of the ﬁrst statement follows fairly easily from Proposition 13.20. It also is an immediate application of the elementary theory of Riemann integration that for all x, x∗ ∈ S and all α ∈ R: 1 1 1 (x + x∗ )(a)da = x(a)da + x∗ (a)da, 0
and:
0
0
1
0
(αx)(a)da = α ·
1
x(a)da. 0
Consequently, we can almost deal with our simpliﬁed continuum of traders model in the same way that we have dealt with economies with a ﬁnite number of traders. The diﬃculty is, of course, that we can only deal with allocation and initial endowment functions which treat many consumers in exactly the same way. Consider the problem of deﬁning a competitive equilibrium for the simpliﬁed continuum of traders economy, as we have set it out here. We can make use of the following deﬁnitions.
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Chapter 13. Further Topics
13.22 Deﬁnitions. We shall say that E = (Pa , ω a ) is a (simpliﬁed) continuum of traders economy iﬀ Pa is an asymmetric relation on Rn+ , ω a ∈ Rn+ , for each a ∈ A, and the function ω : A → Rn+ deﬁned by: ω(a) = ω a for a ∈ A, is a step function. A function x : A → Rn+ is said to be an allocation for E iﬀ if is a step function (on A), and is said to be feasible for E iﬀ, in addition, it satisﬁes:
1
x(a)da = 0
1
ω(a)da. 0
We can then deﬁne a competitive equilibrium as follows. 13.23 Deﬁnition. We shall say that (x∗ , p∗ ) is a competitive equilibrium for a (simpliﬁed) continuum of traders economy, E = (Pa , ω a ), iﬀ: 1. p∗ ∈ Rn , 2. x∗ : A → Rn+ is a feasible allocation for E, and 3. for each a ∈ A, p∗ · x∗a ≤ p∗ · ω a , and: (∀x ∈ Rn+ ) : xPa x∗a ⇒ p∗ · x > p∗ · ω a . Everything works out just ﬁne if we apply the deﬁnitions here to the example with which we introduced this section, for consider the following. 13.24 Example. Let E = (Pa , ω a ) be as in Example 13.17, except that we take the set of agents to be A = [0, 1] insteacd of M = {1, . . . , m} (notice that the initial endowment function here is constant, and thus is a step function on A). If we deﬁne √ the price vector p∗ by p∗ = ( 2, 1), the intervals: D1 = [0,
1 1 √ and D2 = √ ,1 , 1+ 2 1+ 2
and the allocation function x∗ : A → R2+ by: x∗a
=
√ (0, 1 + 2) √
( 1+√2 2
for a ∈ D1 for a ∈ D2 .
it is easy to verify the fact that (x∗ , p∗ ) is a competitive equilibrium for E. (Formally, we also need to specify (ua , ω a ) = (u1 , ω), for a ∈ D1 , and (ua , ω a ) = (u2 , ω), for a ∈ D2 , where ω = (0, 0).) While the above example works out as desired, notice that our deﬁnitions imply that if E = (Pa , ω a ) is a (simpliﬁed) continuum of traders economy, then there can be only a ﬁnite number of diﬀerent consumer endowments. In fact, no competitive equilibrium is possible unless our continuum of consumers demand only a ﬁnite number of diﬀerent commodity bundles. The simplest way to ensure that both of these things will be so is to assume that there are only a ﬁnite number of consumer types; that is, suppose there is a ﬁnite interval partition, D = {D1 , . . . , Dm } and
13.6. Suggestions for Further Reading
379
a ﬁnite sequence of consumer characteristics, (Pi , ω i )m i=1 , such that the economy E = (Pa , ω a ) satisﬁes: (∀a ∈ Di ) : (Pa , ω a ) = (Pi , ω i )
for i = 1, . . . , m.
If we also assume that Pi is negatively transitive (as well as asymmetric), continuous, and strictly convex on Rn+ , then, given any (strictly positive) price vector, p∗ , each consumer of a given characteristic will demand the same commodity bundle. More precisely, if x∗i satisﬁes: p∗ · x∗i ≤ p∗ · ω i , and:
(∀x ∈ Rn+ ) : xPi x∗i ⇒ p∗ · x > p∗ · ω i ;
then each consumer in Di will demand x∗i (i = 1, . . . , m). Thus, the function x∗ deﬁned by: x∗ (a) = x∗i for each a ∈ Di , and for i = 1, . . . , m, is an allocation function for E, and if: 1 1 x∗ (a)da = ω(a)da, 0
(13.32)
0
then (x∗ , p∗ ) is a competitive equilibrium for E. As will no doubt have occurred to you, once we have added all of the assumptions set out in the previous paragraph, the economy will look very much like the ﬁnite exchange economy, E∗ = (Pi , ω i )m i=1 . However, there is a very important diﬀerence. Suppose we denote the length of the interval Di by ‘µi ,’ for each i. Then the integral on the lefthandside of equation (13.32) is given by: 1 m x∗ (a)da = µi x∗i . 0
In eﬀect, the economy i = 1, . . . , m.
13.6
E∗
i=1
corresponds to the special case in which µi = 1/m, for
Suggestions for Further Reading
As you were warned in the introduction to this chapter, we have barely scratched the surface of the areas of literature being introduced in this chapter. For those of you interested in pursuing the sort of analysis we developed in Sections 2 and 3 of this chapter, let me recommend the symposium in the Journal of Economic Theory; 45, 2; August, 1988. MGW oﬀers a considerably more extensive treatment of this material (Sections 2 and 3) than was presented here, and of the ‘Overlapping Generations’ model as well. The most complete recent survey of the latter topic is the Geanakoplos and Polemarchakis article [1991], which I cited earlier; however, a very readable development of the topic from the point of view of macroeconomics is presented in Heijdra and van der Ploeg [2002]. In his textbook, Ellickson [1993] presents a very enthusiastic and extensive coverage of the continuum of traders
380
Chapter 13. Further Topics
model. An interesting topic related to the material in this chapter is known as ‘temporary equilibrium,’ and is surveyed by Grandmont [1982]. A work which I cited in the last chapter will also be of interest here: Magill and Quinzii [1996] develop general equilibrium theory with both uncertainty and an explicit representation of time, and do so in a quite readable fashion. Exercises. 1. Prove Proposition 13.8. 2. Verify the details of Example 13.9 3. Given the assumptions of Example 13.10, show that the following program is competitive: given the prices p∗t = (1, 1) for t = 1, . . . , 5. Compare this program to Period t=1 t=2 t=3 t=4
yt,2 4 4 4 4
xt,1 1 1 1 1
xt,2 1 1 1 1
ct,1 1 1 1 1
ct,2 2 3 3 –
δt 1 1 1 –
Table 13.3: Program 3. Program 1 of Example 13.10. 4. Show that if T > 1, the ﬁnite program deﬁned in Example 13.11 for t = 1, . . . , T is eﬃcient, given yT = f (xT −1 ). 5. Verify the details of Example 13.14 6. Show that if we deﬁne a ﬁnite economy by letting preferences and endowments be as in Example 13.14, except that we have only, say 4 periods (T = 4, and no consumer ‘born’ in the last period), then the allocation: x∗0 = 1 and x∗t = ω t , for t = 1, . . . , 4, is Pareto eﬃcient. What happens if you add a T th consumer, who lives only one period, and has the utility function uT (xT ) = xT and ωT = 5? 7. Prove Theorem 13.15. 8. Prove Corollary 13.16. 9. Show that if the deﬁnition of the core of an economy is extended to the context of an overlapping generations economy in the natural way, then, under the assumptions of Theorem 13.15, a competitive equilibrium is in the core. 10. Prove Proposition 13.4. 11. Prove Proposiion 13.5.
13.6. Suggestions for Further Reading
381
12. Show that the pair (x∗ , p∗ ) deﬁned in Example 13.8 is a competitive equilbrium. 13. Specialize the situation set out in equations (13.20)–(13.25) of Section 5, by supposing that: di − di−1 = ai /bi for i = 1, . . . , m, and that:
dj − dj−1 = aj /bj
for j = 1, . . . , m ;
where ai , bi , aj and bj are positive integers, for all i, j, suppose (13.26 holds, and let k be any integer such that for each i there exists a positive integer, qi such that: k = q i bi
for i = 1, . . . , m;
while for each j there exists a positive integer rj such that: k = rj bj
for j = 1, . . . , m .
Suppose E = (Ph , ω h )kh=1 is any ﬁnite exchange economy such that rj a j consumers have the initial endowment ω j (j = 1, . . . , m ), and x∗h ki=1 is a distribution such that qi aj consumers receive the bundle xi , for i = 1, . . . , m [where xi m i=1 is from (13.21)]. Show that the allocation is feasible; that is: k h=1
x∗h =
k h=1
ωh.
Chapter 14
Social Choice and Voting Rules 14.1
Introduction
In this chapter we will be spending most of our time exploring the borderline between Political Science and Economics, or perhaps the intersection of the two disciplines. The origins of this study stemmed from: 1. Eﬀorts in economics to deﬁne ‘the economic good.’ 2. The study of political processes: how do they work and how should they work? and can be traced as far back as the investigations of JeanCharles de Borda [1781], the Marquis de Condorcet [1785], C. L. Dodgson (Lewis Carroll) [1876], and E. J. Nanson [1882]. In the economics literature per. se. the notion of aggregating individual preferences to obtain a social preference relation apparently originated in the writings of Jeremy Bentham [1789], who originated the idea of a utilitarian social welfare function, and thought of society’s utility as being literally the sum of individual (cardinal) utilities. In the next chapter, we will see that there were numerous problems with this approach. The introduction of the idea of a Pareto improvement was a major innovation in this development, and enabled economists to begin to put normative economic analysis on a much ﬁrmer footing than had previously been possible. However, economists were also frustrated by the fact that most allocations could not be compared via the Pareto criterion. The next major innovation in welfare (normative) economics was the Compensation Principle, which appeared momentarily to allow a much broader class of cases to be compared. However, it was eventually pointed out, as will be shown in Chapter 15, that this criterion did not really allow many more allocations to be compared than did the Pareto criterion. Consequently, the publication of Bergson’s classic article on social welfare functions [Bergson (Burk), 1938] was a very promising development (and one which we will study in Chapter 15). While the idea of a social welfare function was a major innovation, and once again helped economists clarify much of their thinking on policy issues, it fell short of providing a ‘universally acceptable’ criterion for economic improvement; as we will see. However, it led Kenneth Arrow to ask some fundamental questions regarding
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Chapter 14. Social Choice and Voting Rules
the possibility of developing such a function which would be widely acceptable, or more generally, ﬁnding a social preference relation as a function of individual preference relations. More speciﬁcially, Arrow [1950, 1951b] presented three basic properties which it would seem eminently reasonable that we would require such a function to satisfy. First, he wanted it to be deﬁned for any mtuple of weak orders (where m is the number of consumers [agents] in the economy), and to provide a weak order of the social alternatives available as a function of these individual preferences. Secondly, he wanted this function to extend the Pareto criterion. Thirdly, he asked that the social preference between two allocations depend only upon the individual preferences regarding these same two allocations. The very startling conclusion which he derived from his analysis is that any function which satisﬁes all three of these criteria must be dictatorial; that is, the function must simply pick out one of the individuals’ preference relations, and order the social alternatives according to this individual’s preferences. This is Arrow’s famous ‘general possibility theorem,’ and we will be studying this result in some detail in Section 4 of this chapter. Before turning to a formal development of Arrow’s result, however, we will ﬁrst discuss the general idea of voting rules, and majority voting over two alternatives in particular; which is the main subject of the next section.1
14.2
The Basic Setting
In the remainder of this chapter we will be considering a situation in which there is a nonempty, but ﬁnite set of alternatives, X, from which a (group) choice is being contemplated. We will denote the number of elements in X by ‘#X;’ and we will generally assume that #X ≥ 3. We suppose that there are m agents who have preferences deﬁned on X, and who are concerned with the social choice that is to be made from X. We will always assume that m is an integer greater than or equal to 2, and we will often denote the set {1, . . . , m} by ‘M ’ (we will refer to this as the ‘set of agents’). In our treatment here it will be convenient to deal with asymmetric relations; or, if you prefer, the asymmetric parts of some familiar orderings. We will let: Q = the family of all asymmetric orderings of X, P = the family of all (asymmetric parts of) weak orders on X,2 and: L = the family of all strict linear orders on X; where by a strict linear order, I mean a relation which is asymmetric, transitive, and total (I will leave it as an exercise for you to show that such a relation is also negatively transitive). We will denote the mfold cartesian products of these families by ‘Qm , Pm ,’ and ‘Lm ,’ respectively. We will generally use the generic notation, ‘Q, Q , P ,’ and ‘P ,’ etc., to denote elements of Qm , Pm , and Lm . Thus, when we write, for example, Q ∈ Qm , we will mean that Q is of the form: Q = (Q1 , . . . , Qm ), 1 For an elegant, readable, and much more complete introduction to this area than I have provided here, see Suzumura [2002]. 2 That is, P is the family of all asymmetric and negatively transitive binary relations on X.
14.2. The Basic Setting
385
where: Qi ∈ Q for i = 1, . . . , m (that is, each Qi is an asymmetric ordering of A); and similarly for Q ∈ Pm , or Q ∈ Lm . We will refer to elements of Qm , Pm , and Lm as proﬁles, or preference proﬁles. Notice that it follows from our work in Chapter 1 that: L ⊆ P ⊆ Q, and thus: Lm ⊆ Pm ⊆ Qm . We will seek a ‘good’ means by which an element of X, the set of alternatives, can be selected, given the preferences of the m agents. Formally, we can describe this search as a matter of ﬁnding a ‘voting rule’ with desirable properties, where we deﬁne a ‘voting rule’ as follows. 14.1 Deﬁnition. Let X be a nonempty set and m be a positive integer, let Q be the family of all asymmetric orders on X, and D be a nonempty subset of Q. We will say that a function, f : Dm → X is a voting rule (with admissible preferences D). In our treatment we will generally suppose that m, the number of agents, is greater than one, and that #X, the number of alternatives in X is at least three. However, we will begin our analysis by considering the most familiar example of a voting rule, namely majority voting,3 and, strictly speaking, majority voting is only applicable to the situation in which X, the set of alternatives, has two elements. Consequently, let’s begin our considerations here by supposing that we are interested in deﬁning a voting rule for a group of m individuals over a set, X, with #X = 2, and where we write X = {x, y}. We suppose that each of the m individuals has a weak order, Pi , on X,4 and we will consider the formal deﬁnition of majority voting in this case. Since there are only 3 possible weak orders over a set of two elements, X = {x, y}, we can usefully characterize the three possibilities by one of the three numbers, 1, 0, −1, as follows. Given Pi ∈ P, we replace Pi by d(Pi ) = di , where: ⎧ ⎪ if xPi y, ⎨1 (14.1) di = d(Pi ) = 0 if xIi y, ⎪ ⎩ −1 if yPi x. A preference proﬁle can then be characterized as a ﬁnite sequence, d = di m i=1 def
drawn from D = {1, 0, −1} (that is, a preference proﬁle is an element of D = Dm ); and, in this context, a voting rule can be characterized as a mapping from D to X. However, we will shift our focus a bit here initially, to consider a social preference function, which in this context can be deﬁned as a mapping δ : D → D. Thus, the 3 The treatment here, particularly in the ﬁrst part of this section, owes a great deal to Kelly [1988]. 4 Notice that any asymmetric order on a two element set is negatively transitive.
386
Chapter 14. Social Choice and Voting Rules
social preference function corresponding to simple majority rule, which we will call the simple majority social preference, is the mapping, δ s : D → D deﬁned by: ⎧ m ⎪ di > 0, if ⎨1 i=1 s m δ (d) = 0 (14.2) if di = 0, ⎪ i=1 ⎩ m −1 if i=1 di < 0, . While the simple majority rule is probably the one most frequently used in practice, a variant is sometimes adopted; namely, absolute majority voting. Working with the same representation of preference proﬁles as was just introduced, we deﬁne: M (x, y; d) = # i ∈ {1, . . . , m}  di = 1 , and (14.3) M (y, x; d) = # i ∈ {1, . . . , m}  di = −1 . The absolute majority social preference, δ a : D → D, is then deﬁned by ⎧ ⎪ if M (x, y; d) > m/2, ⎨1 a δ (d) = −1 if M (y, x; d) > m/2, (14.4) ⎪ ⎩ 0 otherwise. Simple majority rule and absolute majority rule share a common defect in many social choice situations where the choice is over pairs: they may (in fact often will) fail to pick a winner. Consequently, neither actually deﬁnes a voting rule, as we have deﬁned the term, unless D = L. In fact, the kind of situation to which they are most applicable is that in which each of the m individuals is assumed to have a linear order over the two alternatives, and in addition, where m, the number of individuals, is odd. Given these assumptions, each of the two voting methods will always pick a unique winner. Interestingly enough, however, simple and absolute majority rule produce identical social preference relations in this case (the proof of these statements I will leave as an exercise). Fiftyodd years ago, K. O. May [1952] published an interesting and important characterization of simple majority rule, and it may be useful for us to begin our more formal analysis of voting rules and social preference functions by considering May’s development. Retaining our characterization of preference proﬁles as elements of D, May considered social preference functions, δ, such that δ : D → D. Obviously both simple majority and absolute majority social preference functions are examples of such functions. May Introduces three properties which it appears one would like such a function to satisfy, the ﬁrst of which, anonymity, is deﬁned as follows. 14.2 Deﬁnition. We say that a function δ : D → D satisﬁes anonymity iﬀ whenever two proﬁiles in D, d and d , are such that d is a permutation of d, we have δ(d) = δ(d ). It is easily shown that both δ s and δ a satisfy this condition; and that both satisfy the condition of neutrality, deﬁned as follows. 14.3 Deﬁnition. We say that a function, δ : D → D satisﬁes neutrality iﬀ whenever d, d ∈ D are such that d = −d, we have δ(d ) = −δ(d).
14.3. Voting Rules
387
In order to state May’s third condition, we introduce a bit of notation which we will frequently ﬁnd useful. Given d ∈ D and di ∈ D, we denote the proﬁle d∗ ∈ D deﬁned by: dk for k = i, d∗k = di for k = i, by ‘(di , d−i ).’ We then deﬁne the following. 14.4 Deﬁnition. The social preference function δ : D → D satisﬁes positive responsiveness iﬀ for all d ∈ D, all i ∈ M , and all di ∈ D, we have: [δ(d) ≥ 0 & di > di ] ⇒ δ(di , d−i ) = 1. I will leave as an exercise the task of showing that, while δ s satisﬁes positive responsiveness, δ a does not. In fact, May established the following.5 14.5 Theorem. (May) The only social preference function, δ : D → D, satisfying anonymity, neutrality, and positive responsiveness is δ s , simple majority rule. Thus, simple majority rule fares very well as a social preference function when X, the set of alternatives, contains only two elements. The next question is, however, how do we extend this rule to cover the case in which #X, the number of disinct alternatives in X is greater than or equal to three? We will take up this issue in the next section.
14.3
Voting Rules
In this section we will be considering some voting rules which can be regarded as extensions of simple or absolute majority voting to the situation in which the number of alternatives, #X ≥ 3. 14.6 Example. The Condorcet Winner. The ﬁrst extension of simple majority voting that we’ll consider is that analyzed by one of the ﬁrst people to systematically investigate this problem; the Marquis de Condorcet [1785]. We will say that x ∈ B ⊆ X is a Condorcet winner on B, given the proﬁle P ∈ Pm iﬀ, for all y ∈ B \ {x} x wins, or at least ties in a simple majority vote against y. In other words, x is a Condorcet winnner on B iﬀ there is no alternative element in B which is a clear simple majority winner over x. It will simplify our discussion of this, and other extensions of majority voting to introduce the following notation. Let’s return to the representation of preferences which we used in discussing May’s theorem, and given a proﬁle P ∈ Dm , and x, y ∈ X, deﬁne C v ({x, y}), the majority voting choice from the pair {x, y}, by: ⎧ m ⎪ if di > 0, ⎨{x} i=1 v m (14.5) C ({x, y}) = {y} if di < 0, and ⎪ i=1 ⎩ m {x, y} if d = 0. i i=1 5
For a proof, see May [1952], or Kelly [1988, pp. 12–13].
388
Chapter 14. Social Choice and Voting Rules
An alternative, x ∈ B ⊆ X is then a Condorcet winner on B iﬀ, for every y ∈ B\{x}: x ∈ C v ({x, y}).
While it seems to be perfectly natural and appropriate to pick a Condorcet winner, as a way of extending simple majority voting from pairs to arbitrary ﬁnite sets of alternatives, this deﬁnition does not yield a welldeﬁned voting rule, as the following example demonstrates. 14.7 Example. Suppose #X ≥ 3, with x, y, and z three distinct elements of X, let m = 2q + 1, where q ≥ 1, and let Q be any preference proﬁle for which the rankings of the three distinct alternatives, x, y, and z, is as follows: Agent 1 Group A Group B x y z y z x z x y, where we suppose that both Groups A and B have exactly q members. You can easily verify that in this case, we will have: C v ({x, y}) = {x}, C v ({x, z}) = {z}, and C v ({y, z}) = {y}. Thus the set B = {x, y, z} does not contain a Condorcet winner.
The above example shows that we cannot simply deﬁne a voting rule by taking f (P ) to be the Condorcet winner in X; the problem being that a set X may not contain a Condorcet winner, if #X ≥ 3.6 One way of overcoming this diﬃculty, while retaining the spirit of majority voting is to use a staging procedure, deﬁned as follows. Let be an arbitrary strict linear order on X, which we will call the agenda ordering. We begin by using to label the alternatives in X according to ; that is, we write X = {x1 , . . . , xn }, where: x1 x2 · · · xj xj+1 · · · xn . We then deﬁne a voting rule as follows. We ﬁrst compare x1 with x2 . The simple majority vote winner is then compared with x3 , and so on. If there is a tie at any stage, we take the element with the smaller index to use in our next pairwise vote, or as the singleton element in f (P ), if the voting has progressed to a choice between xn and the winner of the preceding stage.. It is fairly easily shown that this procedure does result in a welldeﬁned voting rule, for each Q ∈ Qm . On the other hand, the staging procedure has some rather severe defects. Consider the following example. 6 Another problem, of course, is that even if a Condorcet winner does exist, it may not be unique. Consequently, one also needs to have a tiebreaking rule to obtain a voting rule from this procedure. However, this is probably a much less serious defect than the fact that there may not be a Condorcet winner at all!
14.3. Voting Rules
389
14.8 Example. Suppose X = {x1 , x2 , x3 , x4 }, that m = 3, and consider f (P ) when the three agents have the preference proﬁle set out as follows. Agent 1 Agent 2 Agent 3 x3 x2 x1 , x2 x4 x4 x4 x3 x1 x3 x2 x1 Here the staging procedure chooses x4 . But x2 is the unique Condorcet winner on B! To illustrate a second defect with the staging procedure, consider the following example. 14.9 Example. Once again we suppose X = {x1 , x2 , x3 , x4 }, and that m = 3; this time considering f (P ) when the three agents have the preference proﬁle set out as follows. Agent 1 Agent 2 Agent 3 x2 x3 x1 x1 x2 x4 x4 x1 x3 x3 x4 x2 Here the staging procedure yields C(B) = {x4 }, as you can readily verify. However, all three agents strictly prefer x1 to x4 ! The staging procedure is also very sensitive to the choice of the agenda (the strict linear ordering, , which determines the order of the pairwise votes). To see this, if you return to Example 14.7, you can readily verify that if the agenda ordering coincides with agent one’s ordering of the alternatives, then the staging procedure will choose z. On the other hand, if the agenda ordering coincides with Group A’s ordering of alternatives, then the staging procedure selects x; while if Group B’s order constitutes the agenda, then y wins. A much more complete discussion of the defects of the staging procedure, as well as a development of several other attempts to consistently extend simple majority voting to a voting rule on sets containing 3 or more elements is provided in Kelly [1988], Chapters 2 and 5, 15–22 and pp 50 –6. Let’s now consider a diﬀerent sort of variant of majority voting; namely, plurality voting. Plurality voting deﬁnes a voting rule on Pm as follows. For P ∈ Pm , F (P ) consists of that element (or those elements) which is (or which are) the most preferred choice in X for the largest number of voters.7 The following example should make the idea of plurality voting clearer, as well as illustrating a very serious problem with the procedure. 14.10 Example. Let X = {x, y, z}, and suppose m = 15, with the agents’ orderings over X as follows: Group 1, 6 agents Group 2, 5 agents Group 3, 4 agents x y z z z y y x x 7
Once again a tiebreaking procedure is needed to obtain a welldeﬁned voting rule.
390
Chapter 14. Social Choice and Voting Rules
In this case, plurality voting yields F (P ) = {x}, but x is a Condorcet loser in X! That is, a majority prefers y to x, and a majority prefers z to x. In fact, a majority of the agents consder x to be the worst of the three alternatives! Moreover, there is a Condorcet winner in this case, namely z, which is not chosen by the plurality rule. 14.11 Example. The Borda Count. You are undoubtedly already familiar with the essential idea of the Borda count, because it is very often used in situations in which one is trying to obtain some sort of aggregate ranking of some alternatives. If, for example, we have four distinct alternatives, w, x, y, and z; each agent assigns a weight of 4 to her or his ﬁrstchoice alternative, 3 to the second, 2 to the third, and 1 to the lastplace alternative. The weight assigned to each alternative is then added over individuals to obtain social weights, with the alternative receiving the largest total being the socially mostpreferred alternative, the alternative with the second largest total being society’s secondranked alternative, and so on.8 Thus suppose we have seven individuals, whose ranking of the alternatives is as in the following table.9 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Agent 7 w x y w x y w x y z x y z x y z w y z w y z w x z w x z If we follow the formulas just set out, we get the social (aggregate) utility values of the rankings as follows: W (w) = 18, W (x) = 19, W (y) = 20, and W (z) = 13; so that the social ranking is: y x w z. However, suppose z is eliminated from consideration,10 and the voting is over just the ﬁrst three alternatives, with the ﬁrstplace alternative receiving an individual weight 3, and individual’s secondplace alternative receiving a weight of 2, and so on. If you carry out the calculations, you will ﬁnd that the social utility values are now given by: W (w) = 15, W (x) = 14, and W (y) = 13. 8 It should be noted that the Borda count is often developed by assigning weights in a manner exactly opposite to that used here; that is, an agent’s ﬁrst choice is assigned a weight of one, second choice a weight of two, and so on. The social ordering is then given by: xP y iﬀ x has received a lower total than has y. It should also be noted that the social preference relation over the set of alternatives deﬁned by the Borda count is not necessarily antisymmetric; it may be that two alternatives are tied for the ﬁrstplace ranking, and so on. 9 This example was ﬁrst developed and reported by Fishburn [1974]. 10 There are a number of reasons why it might be eliminated. If the alternatives were political candidates, it might be that z recognizes before a formal vote that he or she is likely to receive the fewest votes. If the alternatives are public projects, z might be eliminated from consideration because everyone prefers y to it, and so on.
14.3. Voting Rules
391
Thus the social ordering is: w x y; which is exactly the reverse of the original social ranking!
Since it would seem desirable to choose a Condorcet winner, if one is available, the following would seem to be a very reasonable axiom to require of a ‘good’ voting rule. 14.12 Deﬁnition. A Condorcetconsistent voting rule is a voting rule which always chooses a Condorcet winner, if one exits. Example 14.10 shows that the plurality voting rule is not Condorcetconsistent; and it can also be shown that the Borda count method fails this test as well. However, both plurality voting and the Borda count are examples of positional voting, the basic idea of which is to assign weights wn to each ﬁrst place vote, wn−1 to each second place vote, and so on down to w1 for the lastplace alternative; where w1 , . . . , wn satisfy:11 0 ≤ w1 ≤ w2 ≤ · · · ≤ wn and wn > w1 , with the choice over the set being the alternative with the highest point total when we add these weights over individual agents. Thus, for a threeelement set, X, plurality voting uses a weight of 1 for ﬁrstplace votes, and 0 for second or thirdplace votes. The Borda count uses a weight of 3 for ﬁrstplace votes, 2 for secondplace votes, and 1 for thirdplace votes.12 Now, even though neither plurality voting nor the Borda count deﬁnes a Condorcetconsistent voting rule, one might hope that some other positional voting rule might be Condorcetconsistent. Unfortunately, we are once again doomed to disappointment. Fishburn [1973, Chapter 17] has proved the following proposition.13 14.13 Proposition (Fishburn). There are proﬁles for which the Condorcet winner is never elected by any positional voting rule. Proof. In order to prove this, we only need to present a proﬁle satisfying the indicated property. Consider the following proﬁle, in which we have seven agents and three alternatives, X = {x, y, z}:14 3 agents 2 agents 1 agent 1 agent z x x y x y z z y z y x 11 An apparently more general deﬁnition would not require that w1 ≥ 0. However, since there are only a ﬁnite number of alternatives in X, we can assume nonnegativity without loss of generality. 12 Or equvalently, 2 for ﬁrstplace votes, 1 for secondplace votes, and 0 for thirdplace votes. See Exercise 3, at the end of this chapter. 13 The treatment here follows Moulin [1988, pp. 231–2]. 14 This example was actually presented in Fishburn [1984].
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Chapter 14. Social Choice and Voting Rules
In this case, alternative z is the Condorcet winner. However, if the weights assigned to the alternatives satisfy: 0 ≤ w1 < w2 < w3 , the totals for x and z will satisfy: W (x) = 3w3 + 3w2 + w1 > W (z) = 3w3 + 2w2 + 2w1 . While this establishes our result for the case of strictly increasing weights, we can exhibit a proﬁle in which the same thing happens with nondecreasing weights; although in this case we need 17 agents and 3 alternatives. The proﬁle in question is: 6 agents 3 agents 4 agents 4 agents x z y y y x x z z y z x In this case, x is the Condorcet winner. However, if the weights satisfy: 0 ≤ w1 ≤ w2 ≤ w3 and w3 > w1 ≥ 0, then we have: W (y) − W (x) = 8w3 + 6w2 + 3w1 − (6w3 + 7w2 + 4w1 ) = 2w3 − w2 − w1 ≥ w3 − w1 > 0.
While the result just proved shows that no positional voting rule is Condorcetconsistent, there do exist voting rules which are. The two most wellknown are the ‘Copeland rule’ and the ‘Simpson rule.’ These are deﬁned in the exercises at the end of this chapter, where you are asked to prove that the rules are indeed Condorcetconsistent. It turns out that even though the Borda count method has the kind of defect illustrated in Example 14.11, Saari [1996] is able to make a very strong case for its being superior to other positional voting methods. The reason that the Borda count has some desirable properties which are generally not posessed by other positional voting methods in that the Borda count method posesses an internal consistency which the others generally (unless they are equivalent to the Borda count method) lack. The internal consistency which Saari has in mind relates to the fact that the Borda count (and other positional voting schemes as well) actually deﬁnes a social preference relation, and/or a social choice function; concepts which we will take up in the next section.
14.4
Arrow’s General Possibility Theorem
In the previous section we saw that the Borda count, in fact, any positional voting rule, can be used to obtain a social preference ranking. Such a social preference
14.4. Arrow’s General Possibility Theorem
393
ranking can, of course, be used to determine a social choice function, or correspondence. In principle, such a rule could be extremely useful in a situation where a society is to be faced with a succession of choices from a known alternative set. In this section we will consider social preference functions and social choice functions, beginning with social preference functions. The concept of a social preference function was introduced into the economics (and political science) literature by Kenneth Arrow [1951].15 The basic problem introduced by Arrow was that of somehow arriving at a social ordering of alternatives which took account, in a ‘reasonable fashion,’ of individual preferences over those states. Obviously this can be viewed as a problem of arriving at some sort of ordering as a function of the individual agents’ orderings of the alternative set. In our discussion, we will consider this to be the problem of deﬁning a ‘reasonable’ social preference function, or an Arrovian social preference function, deﬁned as follows, where we use the notation introduced in Section 2. 14.14 Deﬁnition. If D is a nonempty subset of Qm , we shall say that a function, f : D → Q is a social preference function. In the special case in which f : Pm → P (that is, where D = Pm , and f maps into P as well), we shall say that f is an Arrovian social preference function. It is easy to deﬁne examples of such functions; the diﬃculty, as we will see, is to deﬁne such functions which also satisfy apparently ‘reasonable’ properties. In the meantime, consider the following; the ﬁrst two of which are the simplest possible examples of social preference functions.. 14.15 Examples. 1. Let P ∗ ∈ Q be ﬁxed, and deﬁne f ∗ : Qm → Q by: f ∗ (Q) = P ∗
for each Q = (Q1 , . . . , Qm ) ∈ Qm ;
in other words, f ∗ is a constant function. Arrow [1951] called this type of social preference function an imposed social preference function. In the extreme, this is the case in which society is ruled by convention, or a ‘sacred code.’ 2. Let j ∈ {1, . . . , m}; and deﬁne fj : Qm → Q by: fj (Q1 , . . . , Qj , . . . , Qm ) = Qj ; in other words, let fj be the j th projection function. Notice that if we take the domain of this function to be Pm , rather than Qm , then this will be an Arrovian social preference function, as we have just deﬁned the term. However, this is the undesirable (unless you are agent j) situation in which agent j is a dictator. 3. The Unanimity ordering: A social preference function. Given a preference proﬁle, Q = (Q1 , . . . , Qm ) ∈ Qm , deﬁne P = F (Q) on X by: xP y ⇐⇒ [xQi y for i = 1, . . . , m]. 15
(14.6)
One should also mention, however, the contribution of Duncan Black [1958], which was almost contemporaneous with Arrow’s; and the work of Julian Blau (for example, [1972]), which helped to clarify much of this area early on.
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Chapter 14. Social Choice and Voting Rules
4. The Pareto ordering: An (almost) Arrovian social preference function. We deﬁne a function f : Pm → Q in the following way: given a proﬁle P ∈ Pm , we deﬁne Q = f (P ) by: xQy ⇐⇒ [¬yPk x for k = 1, . . . , m & (∃i ∈ {1, . . . , m}) : xPi y].
(14.7)
Notice that this second variation is not quite an Arrovian social preference function, because, as you can easily verify, Q = f (P ) is not generally negatively transitive, although it is an asymmetric order.16 Basically, Arrow’s General Possibility Theorem states that no Arrovian social preference function can satisfy all of three very reasonable (and independent) conditions, if #X ≥ 3. In this section, we will state and discuss the three conditions, then state, discuss, and prove a version of Arrow’s General Possibility Theorem. We will then go on to discuss the formal idea of a social choice function. . In our statements of the properties introduced by Arrow, we will suppose only that f is a social preference function; so that f : D → Q, where D is simply taken to be a nonempty subset of Qm . Property 1. The (Weak) Pareto Principle (WPP). For each proﬁle, Q ∈ D, and each x, y ∈ X, Q = f (Q) extends the unanimity ordering; that is: [xQi y for i = 1, . . . , m] ⇒ xQy. Property 2. Independence of Irrelevant Alternatives [IIA]. For each Q, Q ∈ D, and any x, y ∈ X, we have that if: Qi{x,y} = Qi{x,y}
for i = 1, . . . , m,
then, writing Q = f (Q) and Q = f (Q ), we must have: Q{x,y} = Q{x,y} ; where, for a binary relation, G, and {x, y} ⊆ X, we denote the restriction of G to {x, y} by ‘G{x,y} .’ In order to deﬁne our third property, we will need a deﬁnition. 14.16 Deﬁnition. If f : D → Q, we shall say that i ∈ {1, . . . , m} is a dictator for f iﬀ, for all Q ∈ D, and for all x, y ∈ X, we have, writing P = f (Q): xQi y ⇒ xP y. Property 3. Absence of a dictator. No individual, i ∈ {1, . . . , m}, is a dictator for f . You should have no trouble in proving that our Example 14.15.1 satisﬁes Properties 2 and 3 of the above list; and that Example 14.15.2 satisﬁes Properties 1 and 2. As you have probably already noticed (but I will leave the proof as an exercise), the Borda Count function (Example 14.11) satisﬁes Properties 1 and 3, but not 2. These three examples collectively demonstrate that the three conditions are independent of one another, and that none (in fact, no pair) of the three conditions is selfcontradictory. It is also an easy exercise to prove the following. 16
Both of these properties were established in Section 5 of Chapter 5.
14.4. Arrow’s General Possibility Theorem
395
14.17 Proposition. The Pareto ordering, Variation 1, as deﬁned in Example 14.15.3, is a social preference function which satisﬁes all three of the above properties (and with D = Qm ). The problem with the Pareto ordering, of course, is that for most alternatives, x and y, we will have neither xP y, nor yP x; in other words, most of the alternatives in X will be noncomparable. Because of this diﬃculty, investigators in the area spent a great deal of time trying to ﬁnd (in eﬀect) an Arrovian social preference function. Unfortunately, as we will demonstrate, if #X ≥ 3, there is no Arrovian social preference function which satisﬁes all of Properties 1–3. We will prove this by establishing several supporting results, which, I believe, are of some interest in their own right. The basic strategy of our proof of Arrow’s Theorem is adapted from Blau [1972] (see also Blau [1957], and Arrow [1963, 98–100]). In it we will often use the following notation. Suppose E is a nonempty subset of agents, that is suppose: E ⊆ {1, . . . , m} and E = ∅. and denote the complement of E by
‘E c ,’
(14.8)
that is:
E = {1, . . . , m} \ E. c
(14.9)
If x, y ∈ X, for example, and Q ∈ Qm , we will write: E Ec x y y x,
(14.10)
as shorthand for the statements: (∀i ∈ E) : xQi y and (∀j ∈ E c ) : yQj x. Secondly, if we have a social preference function f : D → Q, where Lm ⊆ D and Q ∈ D, we will always denote f (Q) by ‘P ’ (even though, remember, we are dealing with social preference functions, and not Arrovian social preference functions, in our ﬁrst several results). Given a social preference function, f : D → Q, and a nonempty subset of agents, E, we deﬁne the relation DE on X by: xDE y ⇐⇒ (∃Q ∈ D) : Q satisﬁes (14.10) and xP y.
(14.11)
(In the special case in which E c = ∅, we will say that (14.10) holds if the lefthand column is true for E. Notice that in this case, xDE y iﬀ there exists Q ∈ D such that x is Pareto superior to y, given Q. Notice also that, since E is nonempty, and, for each Q ∈ D and each i ∈ E, Qi is irreﬂexive, it follows that DE is irreﬂexive. In both of the next two results, we will suppose that we are given a social preference function, f , satisfying Property 2 (IIA). Notice that, in this case, if we have xP y for one Q ∈ D satisfying (14.10), then we will have xP y for P = f (Q ) and Q any proﬁle from D which satisﬁes (14.10). We can formalize this a bit for later reference, as follows. We begin with a deﬁnition.
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Chapter 14. Social Choice and Voting Rules
14.18 Deﬁnition. Let f be a social preference function, E be a nonempty subset of agents, and x and y be a pair of distinct alternatives in X. If, for each Q ∈ D satisfying (14.10), above, we have xP y, where P = f (Q), then we shall say that E is decisive for {x, y} (given f). The fact which we have just noted can then be stated as follows. The proof is immediate. 14.19 Proposition. Suppose f : D → Q satisﬁes IIA, let E be a nonempty subset of agents, and x and y be a pair of distinct alternatives in X. Then xDE y if, and only if, E is decisive for {x, y}. We can now make use of this result, and these last deﬁnitions, to obtain a sharper characterization of social preference functions, as follows. (From here on, however, we will suppose that D takes the form D = Dm , for some D satisfying L ⊆ D ⊆ P.) 14.20 Theorem. Suppose f : Dm → Q satisﬁes WPP and IIA, where D contains L, that #X ≥ 3, and let E be a nonempty subset of agents. Then either E is decisive for all pairs of distinct x, y ∈ X, or E is decisive for no such pair. In order to prove this result, we ﬁrst prove the following lemma. 14.21 Lemma. Suppose f : Dm → Q satisﬁes WPP and IIA, where D contains L, and that #X ≥ 3, let E be a nonempty subset of agents, and let a, b, c and d be elements of X. Then if aDE b, and c = a [respectively, d = b], then aDE c [respectively, dDE b]. Proof. Suppose ﬁrst that c = a. If b = c, then it follows at once that aDE c. On the other hand, if b = c, and using the assumption that Lm ⊆ D, consider a proﬁle Q ∈ D for which we have: E Ec a b b c c a. Then, writing P = f (Q), we have, applying aDE b and WPP in turn: aP b and bP c. Therefore, since P is transitive, it follows that: aP c; and, since we have aQi c for all i ∈ E, and cQj a for all j ∈ E c , it then follows that aDE c. Suppose now that d = b. If a = d, it follows trivially that dDE b. Otherwise, consider a proﬁle Q ∈ D in which E Ec d b a d b a;
14.4. Arrow’s General Possibility Theorem
397
and deﬁne P = f (Q ). Applying WPP and the assumption that aDE b in turn, we have: dP a and aP b. Therefore, since P is transitive, dP b; and it then follows that dDE b.
Proof of Theorem 14.20. Suppose E is decisive for some pair of elements, a, b ∈ X. Then we must have aDE b. Suppose now that x, y ∈ X are such that x = y. We wish to prove that xDE y, but to do so we consider several cases. 1. Suppose b = x. Then from the lemma (14.21), we see that xDE b. But then, since x = y, it also follows from the lemma (letting x take the place of a and y the place of c) that xDE y. 2. Suppose b = x. Since we must here have aDE x, and y = x, it then follows from the lemma that yDE x. 3. Suppose a = x. Then, since y = x, it follows from the lemma that xDE y. 4. Suppose a = x Then by the lemma, we must have aDE x, and then, since x = y, we have, by making use of the respective statement of the lemma, we have yDE x. We have now shown that if E is decisive for one pair of distinct alternatives, then for any x, y ∈ X such that x = y we must have either xDE y or yDE x. It then follows from IIA that E is decisive for the pair {x, y}. Therefore we see that if E is decisive for some pair of distinct alternatives, it is decisive for any pair of distinct alternatives in X. 14.22 Deﬁnition. Let f : D → Q be a social preference function. We shall say that f is neutral (with respect to alternatives) if, given any nonempty subset of agents, E, and any x, y ∈ X, we have that if E is decisive for {x, y}, then it is decisive for any pair of distinct points from X. It is certainly not clear that this neutrality condition is a desirable property for a social preference function to satisfy; there may be some ‘local’ issues which are part of the overall choice set. X. but which should be decided by a subset of the agents who would not be decisive for other choices.17 For example, a decision as to whether or not to build a city park should presumably be determined only by the preferences of those living in the area, and be independent of agents’ preferences who live well outside the area. However, we have shown in Theorem 14.20 that if f satisﬁes the hypotheses set out there, then f is neutral with respect to alternatives. We can state this formally as follows (the result follows immediately from Theorem 14.20). 14.23 Corollary. If #X ≥ 3, if f : Dm → Q is a social preference function satisfying WPP and IIA, and if L ⊆ D, then f is neutral. Now, deﬁne the set M by: M = {1, . . . , m}; 17
On this issue, see Sen [1970] and Sen [1986, pp. 11556].
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Chapter 14. Social Choice and Voting Rules
that is, we will think of the set M as being the set of all agents; and suppose that a social preference function, f , and alternative set, X, satisfy the hypotheses of Theorem 14.20. By that result we have that each nonempty set of agents, E, is either decisive for all distinct pairs of alternatives, x and y, or it is decisive for no such pair. [It is also worth noting, incidentally, that M is necessarily decisive for all distinct pairs, by the Weak Pareto Principle.] Consequently, we can partition the collection of all subsets of M into two subcollections: the set W (for ‘winners’), consisting of all subsets of M which are decisive for all distinct pairs, and the set N (for ‘nonwinners’) consisting of all subsets of M which are decisive for no distinct pair of alternatives. We can then establish the following result. 14.24 Proposition. Suppose that #X ≥ 3, that f : Dm → P is a social preference function satisfying IIA and WPP, and that L ⊆ D. Then if N is deﬁned as above, the union of any ﬁnite collection of pairwise disjoint sets from N is again an element of N. Proof. We will prove this for the case of two sets, E, F ∈ N; the general case follows by an easy induction argument. Denote {1, . . . , m}\(E ∪F ) by ‘H,’ let x and y be distinct alternatives from X; and suppose, by way of obtaining a contradiction, that E∪F is decisive for {x, y}. Accordingly, let (using the assumption that #X ≥ 3) z be an alternative from X which is distinct from both x and y, and let Q ∈ D be such that: E F H x z y (14.12) y x z z y x. Then, letting P = f (Q), it follows from our assumption that E ∪ F is decisive for {x, y}, that xP y. (14.13) However, since z is distinct from both x and y, it follows from (14.13) and the negative transitivity of P that either xP z,
(14.14)
zP y.
(14.15)
or But neither (14.14) nor (14.15) can hold! For example, if (14.14) holds, then notice that it follows from (14.12) that: xDE z; and it then follows from Proposition 14.20 that E is decisive for {x, z}, contradicting the assumption that E ∈ N. Similarly, if (14.15) holds, it would follow that F is decisive for {z, y}; contradicting the assumption that F ∈ N. Since either possibility [(14.14) or (14.15)] involves us in a contradiction, it follows that (14.13) cannot hold, given (14.12); and thus that E ∪ F cannot be decisive for {x, y}. In our ﬁnal two results, we will suppose that D satisﬁes the following property.
14.4. Arrow’s General Possibility Theorem
399
14.25 Deﬁnition. We shall say that a set D ⊆ Qm satisﬁes the Arrow condition iﬀ: 1. L ⊆ D, and 2. given any Q ∈ D, and any {x, y} ⊆ X, there exists Q ∈ D and z ∈ X such that: Q{x,y} = Q{x,y} , zQ x, and zQ y. While the condition just stated is admittedly a bit strangelooking, notice that the family of all linear orders on X satisﬁes the Arrow Condition. That is, if we set D = L, then D satisﬁes the Arrow Condition. At rather another extreme, if we set D = P, or if D = Q, then the Arrow Condition is satisﬁed. To give an example of a case in which the condition is not satisﬁed, suppose X = {x, y, z}, where all three points are distinct; and that D contains all 6 linear orderings of X, together with the relation in which all three elements are indiﬀerent to one another. Then D does not satisfy the Arrow Condition, since it does not satisfy condition 2 of the deﬁnition. 14.26 Proposition. Suppose that #X ≥ 3, that f : Dm → Q is a social preference function satisfying IIA and WPP, and that D satisﬁes the Arrow condition. Then if E ∈ W; x, y ∈ X, and Q ∈ D are such that: (∀i ∈ E) : xQi y,
(14.16)
then, deﬁning P = f (Q), we have xP y. Proof. Let Q ∈ D be a proﬁle satisfying (14.16). By the fact that D satisﬁes the Arrow condition, there exists Q ∈ D and z ∈ X such that: Qi{x,y} = Qi{x,y}
for i = 1, . . . , m;
(14.17)
while (using part 1 of the Arrow condition): (∀i ∈ E) : xQi z and zQi y, and (using part 2 of the Arrow condition): (∀j ∈ E c ) : zQj x and zQj y. We can indicate this in our shorthand notation as: E Ec x z z y {x, y}. Then writing P = f (Q ), and using the fact that E ∈ W, and WPP in turn,we see that: xP z and zP y. Therefore, since P is transitive:
xP y.
(14.18)
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Letting P = f (Q), and using the fact that f satisﬁes IIA, it then follows from (14.17) that xP y as well. Notice the distinction involved in the conclusion of Proposition 14.26. If E ∈ W, and x and y are two distinct alternatives from X, then E is decisive for {x, y}; meaning that whenever every agent in E prefers x to y, and every agent in E c prefers y to x, then xP y in terms of the social preference, P . In Proposition 14.26, however, we have extended this idea to show that it must be true that whenever all the agents in E prefer x to y, then society will prefer x to y whatever the preferences of the agents in E c . This is only ‘common sense,’ but in saying this what we mean is that it seems ‘right,’ in a normative sense, that if society should prefer x to y whenever the agents in E feel this way and the agents in E c have the opposite preferences, then social preferences should also be this way when the agents in E have the same ranking for x visavis y, but the agents in E c are not necessarily so unalterably opposed. Thus, this is a ‘reasonable’ property to require of a social preference function; but, since we have not directly assumed that f will satisfy it, we had to prove that the other conditions we required f to satisfy do imply this property. We can now prove the following, which is a slight generalization of Arrow’s classic ‘General Possibility Theorem.’ 14.27 Theorem. (Arrow) If #X ≥ 3, and D satisﬁes the Arrow condition, then any social preference function f : Dm → P which satisﬁes IIA and WPP must be dictatorial. Proof. Suppose, by way of obtaining a contradiction, that f satisﬁes IIA, WPP, and, in addition, that no individual is a dictator for f . Then, using the notation of Propositions 14.24 and 14.26, we see from Proposition 14.26 and the deﬁnition of N that: i ∈ N, for i = 1, . . . , m; and by Proposition 14.24 it then follows that m i=1
{i} = M ≡ {1, . . . , m} ∈ N.
But this is impossible, since by WPP we must have M ∈ W.
We then obtain as an immediate corollary, the following result; which is the original statement of the ‘General Possibility Theorem.’ 14.28 Theorem. Arrow’s ‘General Possibility Theorem.’ If #X ≥ 3, then any social preference function f : Pm → P which satisﬁes IIA and WPP must be dictatorial. Social preference functions and social choice functions are often lumped together into one category, but in principle there are signiﬁcant diﬀerences between the two. We can illustrate the diﬀerences and similarities between the two ideas by making use of much of the notation and some of the deﬁnitions developed in Chapter 3.
14.4. Arrow’s General Possibility Theorem
401
Let B be a family of nonempty subsets of X (to be held ﬁxed in this discussion), and let ‘C’ denote the family of all choice correspondences, C on X, B. That is, C is the family of all correspondences C : B → X satisfying, for all B ∈ B: C(B) = ∅ and C(B) ⊆ B.
(14.19)
14.29 Deﬁnitions. If D is a nonempty subset of Qm , we shall say that a function g : D → C is a social choice rule. Given such a function, and a proﬁle Q ∈ D, we will refer to the correspondence, C = g(Q) as the social choice correspondence determined by (g, Q). While the above deﬁnition is a bit ambiguous on this score, it would appear that there would be no good reason for trying to develop a theory of social choice functions for any context other than that in which B consists of all nonempty subsets of X. Of course, we can weaken this stipulation by requiring only that B consist of all subsets of X containing two or more distinct elements of X; however diﬀerent two social choice correspondences may be on nonsingleton sets, their values must coincide on all singleton sets. From our work in Chapter 3, we know that if f : D → Q is a social preference function, then it will always determine a social choice rule in the following way. 14.30 Deﬁnition. Let D be a nonempty subset of Qm , and f : D → Q be a social preference function. We deﬁne the social choice rule corresponding to f , C f on B by: C f (B; Q) = {x ∈ B  (∀y ∈ X) : yQx ⇒ y ∈ / B}, where Q = f (Q). On the other hand, we also know from our work in Chapter 3 that there must be social choice rules which determine social choice correspondences which cannot be derived from any social preference function. However, one would like the social choice correspondence to display some consistency.18 For example, it would certainly appear to be reasonable to require that, if an alternative, x, is chosen when a second alternative, y, is also available, then whenever we have y ∈ C(B) and x ∈ B as well, then we should also have x ∈ C(B). But, in terms of the concepts introduced in Chapter 3, what we are saying here is that the social choice correspondence, C, should satisfy Richter’s VAxiom. Furthermore, it then follows at once from Theorem 3.11 that if C satisﬁes this condition, then it can be derived from a social preference function. In fact, a very complete theory of social choice rules can be built up by straightforward applications of the results from Chapter 3. For example, if g : Pm → C is derived from an Arrovian social preference function, then, for each P ∈ Pm , C(·) must satisfy Richter’s Congruence Axiom. One can make use of the results of Chapter 3 to derive a a number of additional results involving social choice rules, but I will leave this as a ‘project for the interested reader.’ 18
In this connection, see Plott [1973].
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Chapter 14. Social Choice and Voting Rules
14.5
Appendix. A More Sophisticated Borda Count
In order to more generally deﬁne the Borda count relation, we begin by recalling the following notation. For Qi ∈ Q, we deﬁne Qi : X → X by: Qi (x) = {y ∈ X  yQi x} for x ∈ X. We then deﬁne the function ui : X → R+ by: ui (x) = N − #Qi (x)
for x ∈ X.
(14.20)
where N is the total number of distinct elements in X; that is: N = #X. Next, we deﬁne the function n : X → R by: n(x) = #{y ∈ X  ui (y) = ui (x)},
(14.21)
and we then deﬁne u ¯i : X → R++ by: u ¯i (x) = ui (x) −
n(x) − 1 . 2
(14.22)
In eﬀect, the functions u ¯i are individual utility functions quasirepresenting (I will explain this term shortly) the asymmetric orders Qi . We use these individual ‘utility functions’ to deﬁne a social welfare function, W : X → R+ by: m W (x) = u ¯i (x). (14.23) i=1
Finally, we deﬁne B = β(Q) by: x B y ⇐⇒ W (x) > W (y).
(14.24)
Interestingly enough, we have actually deﬁned a social preference function here whose domain is Qm and whose range is P; that is, β : Qm → P. Of greater interest for the matter at hand, however, is the fact that the restriction of β to Pm is an Arrovian social preference function. Unfortunately, as we will see shortly, it does not satisfy all of the remaining conditions considered by Arrow. Moreover this function is particularly sensitive to certain kinds of strategies which might be employed by the individual agents, as we saw in Section 6. It is worth noting a couple of things about the development which we used here. First of all, notice that the function ui deﬁned in the above material is a welldeﬁned function for any Qi ∈ Q; whether or not Qi satisﬁes negative transitivity. In fact, it is easy to show that ui satisﬁes the following condition: for any x, y ∈ X: xQi y ⇒ ui (x) > ui (y).
(14.25)
14.5. Appendix. A More Sophisticated Borda Count
403
This is what I meant earlier by the comment that the function ui ‘quasirepresents Qi .’ In fact, if Qi is negatively transitive, then it can be shown that ui actually represents Qi . Of course, if Qi does not satisfy negative transitivity, then ui cannot represent Qi ; and thus it cannot satisfy ui (x) > ui (y) ⇒ xQi y,
(14.26)
as well as (14.25) [why is this?] It is also worthwhile to take a moment to consider why it is that one might wish to take the extra step to go from the nice simplydeﬁned functions ui to use the more complicated functions in deﬁning the social welfare function W (·), and thus the social preference relation, B . The reason amounts to this: each individual’s preferences are used to determine a set of weights, W (x), to be assigned to the elements x ∈ X. In eﬀect, individual i gets to vote for the desirability of alternative x, and can add the amount u ¯i (x) to the social evaluation of x. The extra step (using u ¯i in place of ui ) ensures that each agent gets to cast the same total number of votes. We can show this, for the case in which each Qi satisﬁes negative transitivity (and thus is a weak order), as follows. Let: ui (x) ≡ u† , n(x) = p ≥ 1, and consider the utility assigned to the next best alternative which is not indiﬀerent to x.19 If y is such an alternative, then clearly the p alternatives tied with x in the ranking are preferred to y, but not to x, while every other alternative preferred to x is also preferred to y. Thus: ui (y) ≡ u∗ = N − [p + #{z ∈ X  zQi x}] = [N − #{z ∈ X  zQi x}] − p = u† − p. (14.27) If the alternatives tied with x in agent i’s ranking were ordered linearly, then one of these alternatives would have the utility (under the ui function) of u∗ + 1, the next u∗ + 2, and so on up to u∗ + p. Remembering the formula for the sum of the ﬁrst n positive integers,20 we see that the total utility weights assigned to the elements tied with x (that is, in the equivalence class for x, [x ]) is given by: u∗ + 1 + u∗ + 2 + · · · + u∗ + p = pu∗ + p(p + 1)/2.
(14.28)
By using the function u ¯i deﬁned in (14.22), above, we give each of these elements the weight u† − (p − 1)/2; and thus the total of the weights assigned to these elements is (p − 1) p2 − p = pu† − p u† − . 2 2 Substituting from (14.27) into (14.29), we see that this sum is: pu† − 19 20
p2 − p p2 − p p(p + 1) = p(u∗ + p) − = pu∗ + , 2 2 2
Technically, the utility assigned the next equivalence class down from [x]. Which is equal to n(n + 1)/2.
(14.29)
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Chapter 14. Social Choice and Voting Rules
which agrees with (14.28). Thus what the use of the function accomplishes, is to give everyone the same total weights to be allocated over the alternatives.21 Exercises. 1. Suppose #X = 2, that the number of agents, m is odd (m = 2q + 1, for some integer q ≥ 1) and that D = Lm . Show that in this case the simple majority voting and absolute majority voting rules produce identical results. 2. Suppose #X = 2, and that the domain over which we deﬁne a voting rule, D, is equal to Pm . Show that the simple majority voting rule satisﬁes May’s positive responsiveness condition, but that the absolute majority voting rule does not. 3. Prove that the following Borda count methods are equivalent, where #X = n, and we have m ≥ 2 agents, each of whom has a preference relation on X which is a linear order. a. Each agent assigns a weight n to her/his most preferred alternative, n − 1 to her/his secondplace alternative, and so on. b. Each agent assigns a weight of n − 1 to her/his most preferred alternative, n − 2 to her/his secondpland alternative, and so on. 4. Suppose X = {w, x, y, z}, that m = 3, and consider the set B = X when the three agents have the preference proﬁles set out as follows. Agent 1 Agent 2 Agent 3 w x x x y y y z w z w z (a) Is there a Condorcet winner in this case? If there is, what is it? (b) What is the plurality winner? (c) Find the Borda count ranking of the four alternatives. 5. Show that if #X = 2, and m is an odd number greater than or equal to 3, then there exists an Arrovian social preference function satisfying IIA and WPP which is nondictatorial. 6. Suppose Y is a nonempty subset of X, and that Q is an asymmetric order (respectively a weak order) on Y . Show that there exists an asymmetric order on X (respectively, a weak order on X), Q∗ , such that Q is the restriction of Q∗ to Y . [Hint: See Exercise 7, at the end of Chapter 1.] 7. In this exercise, we return to the notation utilized in our discussion of May’s 21 Technically, what we have shown (or what you can show with very little additional work), is that the u ¯i function assigns the same total weights as if the alternatives were ordered linearly. Consequently, it follows that the total is the same for any weak order.
14.5. Appendix. A More Sophisticated Borda Count theorem; that is, for {x, y} ⊆ X, we deﬁne ⎧ ⎪ ⎨1 di (x, y) = 0 ⎪ ⎩ −1
405
di (x, y) by: if xPi y, if xIi y, and if yi P x;
and we consider the domain Pm , that is the set of all proﬁles of weak orders on X. Given a preference proﬁle, P , and for a given x ∈ X, we deﬁne, for each y ∈ X \ x, f (y; x) by: m 1 if i=1 di (x, y) ≥ 0, f (y; x) = 0 otherwise. We then deﬁne the Copeland score, C(x), of an alternative x ∈ X by: f (y; x). C(x) = y∈X\{x}
The Copeland winner is then the alternative which has the highest Copeland score (although a tiebreaking procedure is needed here). Show that this voting method is Condorcetconsistent, and satisﬁes May’s anonymity and neutrality conditions. 8. Given the context of the previous exercise, and for a given preference proﬁle, P , we deﬁne, for each x ∈ X, and y ∈ X \ x, the number N (x, y) by: N (x, y) = #{i  xGi y}, where Gi is the negation of Pi . We then deﬁne the social utility of x, U (x), by: U (x) =
min N (x, y). y∈X\{x}
The alternative chosen is then one which has the highest ‘social utility’ (once again, however, we must have a tiebreaking procedure). Show that this voting method (the Simpson Rule) is Condorcetconsistent, and satisﬁes May’s anonymity and neutrality conditions.22 9. Returning to the context of Exercise 7, suppose we deﬁne a voting rule in the following way: for a given proﬁle, P ∈ Pm , deﬁne, for each x ∈ X: D(x) =
m
di (x, y);
y∈X\{y} i=1
and let F (P ) be the alternative having the highest value for D(x) (with an appropriate tiebreaking rule). Is this voting rule Condorcetconsisten? 10. Show that the Borda count does not satisfy IIA. Notice that an appropriate example to show this must involve a ﬁxed choice set. 22 Both the Copeland and the Simpson rules satisfy a ‘Pareto optimality’ rule as well. For an excellent, and much more thorough discussion, see Moulin [1988, pp. 233–40].
Chapter 15
Some Tools of Applied Welfare Analysis 15.1
Introduction
In this chapter, we will examine a number of tools which are, or can be used in applied welfare economics. We will begin by continuing our consideration of social preference functions with an investigation of the socalled ‘BergsonSamuelson Social Welfare Function.’ This function was introduced into the Englishlanguage economics literature by Abram Bergson [1938], while Samuelson [1947] emphasized its importance and illustrated new usages for such a function. Interestingly enough, Pareto had introduced the idea earlier, although he did not analyze and develop the idea as extensively as did Bergson (who developed the idea independently in any case); consequently, we will give credit to all three economists. We deﬁne a ‘ParetoBergsonSamuelson (PBS) Social Welfare Function’ as a realvalued function whose domain is the space of allocations in an economy and which can be written as a composition, W = F ◦ u, where u is a vector of individual utility functions, with ui deﬁned over individual i’s commodity bundle (i = 1, . . . , m), and F is an increasing realvalued function on Rm (which we will call the aggregator function); that is, a function whose domain is utility space. Perhaps no tool of theoretical normative economics has been used as much or criticized as severely as has the ‘ParetoBergsonSamuelson Social Welfare Function.’ On the one hand, such functions are commonlyused in applied normative analysis, and are frequently used in theoretical policy analyses as well. On the other hand, we have been trained to think of preferences as being only (at most) ordinally measurable, whereas the concept seems to require that individual utilities be not only ‘cardinally measurable,’ but ‘interpersonally comparable.’ The obvious question then arises, is there a way out of this dilemma? We will explore this issue in Sections 2–4. In Section 5 we continue our exploration of tools of applied welfare analysis by examining the ‘compensation principle:’ the principal tool of what was once known as ‘The New Welfare Economics.’ In Section 6, we extend the ideas of Sections 2–4 to deﬁne indirect social preferences and indirect social welfare functions. We then
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Chapter 15. Some Tools of Applied Welfare Analysis
make use of these notions to examine some ideas about the measurement of ‘real national income’ in Section 7. We then continue our exploration of the applications of indirect social preferences and welfare functions by considering consumers’ surplus in Section 8.
15.2
The Framework
In the remainder of this chapter, we will, as usual, suppose that there are m consumers and n commodities in the economy. We will also assume throughout (except where otherwise explictly stated) that each consumer has a (weak) preference relation Gi , which will always be assumed to be a continuous, strictly convex, and increasing weak order; and, as usual we will use ‘Ii ’ and ‘Pi ’ to denote the symmetric and asymmetric parts of Gi , respectively. More formally, we will always be assuming that each Gi is an element of a family, Gc , deﬁned as follows. 15.1 Deﬁnitions. We denote by ‘Gc ’ the family of all weak orders on Rn+ which are also: a. continuous, b. increasing,1 and c. strictly convex.2 We then let ‘Gc ’ denote the collection of all mtuples of elements of Gc ; that is: Gc = (Gc )m , and we will refer to mtuples, G = (G1 , . . . , Gm ) ∈ Gc as preference proﬁles. While we will always assume that each consumer’s preference relation is an element of Gc , we will often assume that each relation is homothetic as well. 15.2 Deﬁnition. We denote by ‘Gh ’ the subset of Gc consisting of all elements of Gc which are also homothetic; that is, Gh is the family of all weak orders on Rn+ which are continuous, increasing, strictly convex, and homothetic. We use ‘Gh ’ to denote the collection of all mtuples of elements of Gh . Since each consumer’s consumption set is equal to Rn+ , an allocation xi will be a ﬁnite sequence (of m terms), with: xi ∈ Rn+
for i = 1, . . . , m.
Thus we could consider allocations to be elements of Rmn + . However, to avoid possible confusion, we will denote the allocation space by ‘X;’ that is: X = xi  xi ∈ Rn+ , for i = 1, . . . , m . 1 2
That is, if x, x ∈ Rn + are such that x x , then xP x . ∗ ∗ That is: if x, x∗ ∈ Rn + are such that xGi x and x = x , and if 0 < θ < 1, then: θx + (1 − θ)x∗ Pi x∗ .
Strict convexity will not really be needed in the vast majority of our work in this chapter, but making use of it greatly simpliﬁes many of our deﬁnitions and proofs.
15.3. Measurement Functions
409
In this chapter, we will use the generic notation ‘G’ to denote the weak Pareto ordering of allocations; that is, if G = (G1 , . . . , Gm ) is an element of Gc , we deﬁne G on X by: xi Gxi ⇐⇒ xi Gi xi , for i = 1, . . . , m. Where needed, we will use the generic notation ‘P ’ to denote the asymmetric part of G (the strict Pareto ordering).
15.3
Measurement Functions
In this chapter, we will look at utility functions in a bit diﬀerent way than is usual. Let U be deﬁned by: U = {f  f : Rn+ → R+ }. We then deﬁne the following: 15.3 Deﬁnition. We will say that a function µ : Gc → U is a utility measurement function (for Gc ) iﬀ for each G ∈ Gc , f = µ(G) satisﬁes: (∀x, x ∈ Rn+ ) : f (x) ≥ f (x ) ⇐⇒ xGx .
(15.1)
While the above deﬁnition may seem a bit strange to you, the next example may help to clear things up a bit. In fact, the method of utility measurement, or class of utility measurement functions of which we will make use, is based upon the classic representation theorem of Herman Wold (Theorem 4.21, of chapter 4), and is set out in the following example. 15.4 Example. We will deﬁne a function ϕ : Gc × Rn++ → U: given G ∈ Gc , and x∗ ∈ Rn++ ,3 we deﬁne u = ϕ(G, x∗ ) as follows. For x ∈ Rn+ , there exists a unique value of θ satisfying: (15.2) x I θx∗ , where I is the indiﬀerence relation for G, and we let u(x) = θ. In other words, u(x) = ϕ(G, x∗ )(x) is that unique real number satisfying: x I [u(x)x∗ ],
(15.3)
In terms of this notation, we showed in chapter 4 (Theorem 4.21) that u = ϕ(G, x∗ ) is a continuous function satisfying (15.1), above; that is, it represents G on Rn+ . Thus, for a ﬁxed x∗ ∈ Rn++ , the function µ(·) = ϕ(·; x∗ ) is a utility measurement function for Gc . It is important to notice that, given an element, x∗ ∈ Rn++ (a unit of measure), the function µ(·) = ϕ(·, x∗ ) is a utility measurement function for Gc ; that is, for each G ∈ Gc , u = µ(G) is a continuous utility function which represents G. In fact, 3
n Where ‘Rn ++ ’ denotes the set of strictly positive elements of R ; that is: n Rn ++ = {x ∈ R  x 0}.
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Chapter 15. Some Tools of Applied Welfare Analysis
this is the only type of utility measurement function which we will consider in this chapter. When we say that µ : Gc → U is a measurement function for Gc , we will mean that µ is deﬁned as in Example 15.4; that is, there exists x∗ ∈ Rn++ such that µ(·) = ϕ(·, x∗ ), and even though this is the only type of measurement function we’ll consider, we will often refer to such a function as a Wold measurement function. Continuing with this idea, if we speak of µ∗ and µ† as being two diﬀerent measurement functions for G, we will mean that, while both are Wold measurement functions, there exist x∗ , x† ∈ Rn++ such that x∗ = x† and for each G ∈ G: µ∗ (G) = ϕ(G, x∗ ) & µ† (G) = ϕ(G, x† ), where ϕ is deﬁned in Example 15.4. That is, µ∗ and µ† will be obtained by the same process, but may use diﬀerent ‘units of measure’ (x∗ versus x† in this example). The utility measurement function just deﬁned has especially interesting properties in the homothetic case. Recall that in chapter 4, we proved the following (Theorem 4.36). 15.5 Proposition. If x∗ ∈ Rn++ then, given any G ∈ Gh , the function u∗ = ϕ(P, x∗ ) deﬁned in Example 15.4 [satisﬁes (15.1), above, and] is concave, continuous, increasing, and positively homogeneous of degree one. Our next result shows that, eﬀectively, we lose no generality in conﬁning our attention to measurement functions of the Wold type when dealing with homothetic preference relations. 15.6 Proposition. Let G ∈ Gh , and let u : Rn+ → R+ be any function representing G which is positively homogeneous of degree one. Then there exists x∗ ∈ Rn++ such that u = ϕ(G; x∗ ), where ϕ : Gc × Rn++ → U is deﬁned as in Example 15.4. Proof. Since G is increasing and u(·) is positively homogeneous of degree one, there exists x∗ ∈ Rn++ such that u(x∗ ) = 1, and we let u∗ = ϕ(G; x∗ ); where ϕ(·) is from Example 15.4. We then note that, for an arbitrary x ∈ Rn+ , we have, since xI[u∗ (x)x∗ ] and u(·) represents G: u(x) = u[u∗ (x)x∗ ]. However, since u(·) is positively homogeneous of degree one, we then have: u(x) = u∗ (x)u(x∗ ) = u∗ (x), and our result follows.
Our last result of this section restates a fact which we had already established in chapter 4. Given its importance in our endeavors of this chapter, it has seemed worthwhile to reprise both its statement and proof. 15.7 Proposition. If G ∈ Gh , and u : Rn+ → R+ and u∗ : Rn+ → R+ are any two functions representing G which are also positively homogeneous of degree one, then there exists a ∈ R++ such that for all x ∈ Rn+ , we have u(x) = au∗ (x).
15.4. Social Preference Functions
411
Proof. Let u and u∗ satisfy the stated hypotheses. Making use of Proposition 15.6, we let x∗ ∈ Rn++ be such that u∗ = ϕ(G; x∗ ), and deﬁne a = u(x∗ ). As in the proof of Proposition 15.6, we note that for arbitrary x ∈ Rn+ , we must have xI[u∗ (x)x∗ ]; and thus, since u is positively homogeneous of degree one and represents G, we then conclude that: u(x) = u[u∗ (x)x∗ ] = u∗ (x)u(x∗ ) = au∗ (x).
15.4
Social Preference Functions
As in Chapter 14, we will refer to a function which maps preference proﬁles into asymmetric orders on the allocation space as a social preference function. However, this time we will always take the domain of such a function to be a subset of Gc . Formally, by a social preference function we will mean a function ω : G → Q, where G ⊆ Gc , and ‘Q’ denotes the family of asymmetric orders on the allocation space X = Rmn . Moreover, in this chapter, our principal concern will be with a special case of such functions, deﬁned as follows. 15.8 Deﬁnition. We will say that a social preference function, ω : Gc → Q, is a ParetoBergsonSamuelson (PBS) Social Preference Function iﬀ there exists c µ : Gc → U and an increasing function F : Rm + → R such that for all G ∈ G , if we deﬁne Q = ω(G) and ui = µ(Gi ) for i = 1, . . . , m; we have: (∀xi , x∗i ∈ X) : xi Qx∗i ⇐⇒ F u xi > F u x∗i , where we deﬁne: u xi = (u1 (x1 ), . . . , um (xm )) and u x∗i = (u1 (x∗1 ), . . . , um (x∗m )). In dealing with PBS Social Preference Functions, we will refer to the function µ as the measurement function, F , as the aggregator function, and the composite function, W = F ◦u as a ParetoBergsonSamuelson (PBS) Social Welfare Function. Since each such social preference function has the property that it can be represented by the composition of a measurement function, µ, and an aggregator function, F , we shall speak of such a social preference function as being determined by a pair (µ, F ), where µ is a measurement function, and F is an aggregator function.4 Notice that if ω is a PBS social preference function, then, for any preference proﬁle, G, in the domain of ω, the allocation ordering, Q = ω(G), extends the Pareto ordering, G, determined by G. 15.9 Examples. Consider the aggregator function F : Rm + → R+ deﬁned by: F (u) =
m i=1
ui .
(15.4)
4 Of course, such a function can generally be determined by many such (eﬀectively equivalent) pairs.
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Chapter 15. Some Tools of Applied Welfare Analysis
The function F deﬁnes a BergsonSamuelson Social Welfare Function when paired with any Wold measurement function;5 as does the aggregator function F ∗ deﬁned by: m F ∗ (u) = (ui )ai , (15.5) i=1 m where ai ∈ R++ , for i = 1, . . . , m, and i=1 ai = 1. We shall refer to the ﬁrst of these two examples as the utilitarian aggregator function, and any function of the type in equation (15.5) as a CobbDouglasEisenberg (CDE) aggregator function.6 A ﬁnal example of interest is the Rawlsian aggregator function, deﬁned by: (15.6) F (u) = min{u1 , . . . , um }. In connection with the examples just presented, it should be noted that, using the method of Example 15.4, one obtains a signiﬁcantly diﬀerent measurement function for each diﬀerent value of x∗ ∈ Rn++ . This in turn means that, for a given aggregator function, F , one may obtain very diﬀerent social preference functions if one combines one measurement function, µ∗ , deﬁned from x∗ , than one does from µ† , say, deﬁned from a second unit of measure, x† ∈ Rn++ ; as is demonstrated by the following example. 15.10 Example. Consider the twocommodity, twoconsumer economy in which the preferences of the consumers can be represented by the CobbDouglas utility functions: (15.7) u1 (x1 ) = A1 (x11 )2 · x12 and u2 (x2 ) = A2 x21 · (x22 )2 , respectively, and where Ai > 0 is a positive constant, for i = 1, 2. We begin by ˜ ), where: considering the measurement function µ ˜ = ϕ(·, x ˜ = (1, 1). x For an arbitrary x ∈ R2+ , u ˜1 (x1 ) can be found by solving the equation: A1 (x11 )2 · x12 = A1 (˜ u1 (x1 ) · 1)2 · (˜ u1 (x1 ) · 1) = A1 [˜ u1 (x1 )]3 ; so that: u ˜1 (x1 ) = (x11 )2/3 (x12 )1/3 .
(15.8)
Similarly, with this measurement function, consumer 2’s utility function is given by: u ˜2 (x2 ) = (x21 )1/3 (x22 )2/3 .
(15.9)
¯ by: If we now deﬁne the allocations x and x ¯ = (8, 8), (1, 27) , x = (27, 1), (8, 8) and x 5 Of course, one could equally well pair this aggregator function with a measurement function which is not of the Wold type, but we will not be considering such a possibility in this chapter. 6 The twoperson and symmetric version of this function was introduced by John Nash in his analysis of the bargaining problem (1950). For this reason, Moulin refers to the symmetric version of this function (all ai = 1/m) as the ‘Nash CUF’ (See Moulin [1988]). As to the inclusion of Eisenberg’s name in my labeling of the function, see Eisenberg [1961].
15.4. Social Preference Functions
413
and make use of the utility functions deﬁned in (15.8) and (15.9), the corresponding vectors of utilities are given by: u ˜(x) = (9, 8) and u ˜(¯ x) = (8, 9), respectively. Thus, if a decisionmaker has the PBS social preference function deﬁned by the pair (F, µ ˜), where the aggregator function, F , is the utilitarian aggregator given by: m ui , (15.10) F (u) = i=1
said decisionmaker will be indiﬀerent between the two allocations. This same social indiﬀerence will occur if the decisionmaker has the PBS social preference function deﬁned by the pair (F R , µ ˜), with F R being the Rawlsian function: F R (u) = min ui . i
(15.11)
The situation changes, however, if we take our unit of measure to be the bundle x∗ given by: x∗ = (64, 1), while continuing to use the measurement function [ϕ(·, P )] deﬁned in Example 15.4. In this case we can obtain u1 (x1 ), for an arbitrary bundle x1 ∈ R2+ , by solving the equation: 2/3 ∗ 1/3 · u1 (x1 ) · 1 = 16u∗1 (x1 ); (x11 )2/3 (x12 )1/3 = u∗1 (x1 ) · 64 so that:7
u∗1 (x1 ) = (1/16)(x11 )2/3 (x12 )1/3 = (1/16)˜ u1 (x1 ).
Similarly, we ﬁnd
u∗2 (x2 )
(15.12)
by solving:
1/3 ∗ 2/3 (x21 )1/3 · (x22 )2/3 = u∗2 (x2 ) · 64 · u2 (x2 ) · 1 = 4u∗2 (x2 ). u2 (x2 ). Our vectors of utilities at the two allocations now Therefore, u∗2 (x2 ) = (1/4)˜ become: x) = (1/2, 9/4); u∗ (x) = (9/16, 2) and u∗ (¯ ¯ to x, while our Rawlsian decisionmaker now so that our utilitarian now prefers x ¯. prefers x to x , where: On the other hand, suppose we take our unit of measure equal to x = (1, 64), x while continuing to use the measurement function deﬁned in Example 15.4. Here ) is given by: the same basic reasoning as before establishes that u i = ϕ(Pi , x ui (x1 ) and u 2 (x2 ) = (1/16)˜ u2 (x2 ).. u i (x1 ) = (1/4)˜
(15.13)
7 We knew from Proposition 15.7 that the new function was going to be a scalar multiple of the old one.
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Chapter 15. Some Tools of Applied Welfare Analysis
Thus our vectors of utilities at the allocations of interest become: (x) = (9/4, 1/2) and u (¯ u x) = (2, 9/16), ¯ , while our respectively. Thus our utilitarian decisionmaker now prefers x to x ¯ to x; reversing their previous preferences. Rawlsian now prefers x In connection with the preceding example it is worth noting, ﬁrst of all, how and why the utilitarian ordering can be manipulated as per the above example. In the example, consumer one basically likes commodity one better than commodity two; while consumer two has the opposite preferences. Suppose, then, that you are consumer two and that, while you are bound by sacred oath, or in some other fashion, to tell the truth in response to questions about your individual preferences, but you know that the aggregator function to be used is the utilitarian aggregator function, and you are to be allowed to choose the unit of measure for the measurement function. You can then gain by choosing a unit of measure with the proportions of the two commodities more to consumer one’s liking than to your own. It will then tend to take a larger multiple of the unit of measure to yield a commodity bundle indiﬀerent to an arbitrary bundle in your case than it will for consumer one. Consequently, your utility numbers will tend to be higher than consumer one’s, and the utilitarian aggregator will tend to favor you. Of course, if you know that the aggregator to be used is the Rawlsian one, then you want to choose a bundle as the unit of measure which has proportions which you like. Your utility numbers will then tend to be smaller than consumer one’s, and the Rawlsian aggregator will then tend to favor you. In fact, I suspect that any of us who have siblings have implicitly tried to apply these principles in bargaining with parents; sometimes trying ‘I should get that rather than his getting it, because it is my favorite and he likes other things better’ (thus exploring the possibility that the family social welfare aggregator is utilitarian), at other times the alternative, ‘I should get that rather than his getting it, because he has (or has had) lots of such things compared to my paltry few’ (the Rawlsian approach). Now, while Example 15.10 and the above discussion emphasize the dependence of the social ordering upon the unit of measure used, in the utilitarian and the Rawlsian cases, this is not to say that a decisionmaker might not have and use a social ordering of one of these types. The salient point is that the social ordering in these cases is determined jointly by the aggregator and the unit of measurement used in the utilitymeasurement process; and both are critical in deﬁning the resulting social preference ordering. Interestingly, however, it follows from Propositions 15.6 and 15.7 that if we conﬁne our attention to Gh , then a PBS function of the CDE form induces a social preference ordering which is independent of the measurement function, µ, with which it is paired. The proof of this fact is quite simple, and will be left as an exercise. A more diﬃcult question to answer, however, is whether this is the only PBS function which has this property. Unfortunately, however, it can be shown that this is the only one.8 The example just considered demonstrates the fact that PBS social preference 8
This can be proved by a slight modiﬁcation of Moulin’s proof of his Theorem 2.3 [1988, p. 37]
15.4. Social Preference Functions
415
functions do not satisfy Arrow’s Independence of Irrelevant Alternatives condition.9 As we have already noted, however, any PBS function satisﬁes the Weak Pareto Principle. The nondictatorship condition is a bit tricky, for if we only require the aggregator function to be increasing, then a PBS function may be dictatorial; after all, the function F : Rm + → R+ deﬁned by: F (u) = u1 , Rm +.
is an increasing function on In order to avoid this possibility, we could require the aggregator function to be strictly increasing, but this would eliminate the CDE aggregator function deﬁned in (15.5), above, which is not strictly increasing on the boundary of Rm + . A reasonable way to eliminate this diﬃculty would be to require aggregator functions to be increasing, and to be strictly increasing on Rm ++ ; on the other hand, this requirement would eliminate the Rawlsian aggregator function from consideration. As it turns out, however, all of the formal theory of PBS functions which we will be considering requires only that the aggregator function be increasing; and consequently we make use of this condition in our deﬁnition of PBS social preference functions, rather than the stronger condition that F is strictly increasing. It is important to notice, however, that this means that some of what we are calling PBS social preference functions may have some highly undesirable properties. Before concluding this section, let’s consider the Independence of Irrelevant Alternatives issue a bit further. Suppose that you are, in fact, a benign dictator/social planner for an economy, and that you want to make economic decisions in a manner which takes into account the preferences of the individual agents comprising the economy. How can you then determine whether one allocation should be chosen over another (that is, is better, in terms of your social preference) without knowing the full preference relation of every consumer in the economy? It was the profound insight of Bergson (1938) that, whatever the form of your social preferences in other respects, if you believe that one allocation is better that a second whenever every consumer in the economy considers it to be so [that is, if your social preference relation extends the (unanimity) Pareto order), then whatever allocation you consider best will necessarily be Pareto eﬃcient, and if two allocations are such that every consumer is better oﬀ in one than in the other, then you (in fact, anyone ordering allocations by a PBS social preference function) would prefer the former allocation to the latter. As we all know, however, being able to compare allocations only when one Pareto dominates another is extremely limiting, and indeed when appeal is made to a PBS social welfare function in economic policy analyses, the motivation for its use is usually precisely in order to be able to compare allocations on a broader basis than is possible using only Pareto dominance. But, as a practical matter, wouldn’t such extended comparisons require that you know each consumer’s full preference relation? or, more stringently still, that you develop a utility function to represent each consumer’s preferences? On the face of it, it may appear that a PBS social welfare function does require the development/estimation of a utility function for each 9 While the CDE aggregator induces a social preference relation which is independent of which Wold measurement function with which it is paired, given that preferences are in Gh , other measurement functions (in particular, if they could yield non homogeneous utility functions) may yield a diﬀerent social preference relation when paired with such an aggregator.
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consumer in order to allow comparisons to be made on anything other than Pareto dominance; and it was to avoid such astronomical informational requirements that Arrow introduced the requirement that a social preference function should satisfy independence of irrelevant alternatives. However, while PBS social welfare functions do not satisfy IIA, they also do not require full knowledge of each consumer’s utility function in order to compare two allocations. In fact, suppose the social preference function is determined by the pair (µ, F ), where µ = ϕ(·, x∗ ), and that it is desired to compare the allocations x1i and x2i . In order to determine which of these allocations is to be preferred, it is only necessary to determine u(xt ), for t = 1, 2, and this can be done by comparisons of the xti with x∗ . This is a vastly weaker informational/estimation requirement than determining the form of each ui (·)! Of course, the process of ﬁnding the value of θt [ = u(xt )] is a great deal more complex an operation than I am making it sound, but it is at least theoretically possible.
15.5
The Compensation Principle
The Compensation Principle has a long history in economics, having been proposed in slightly diﬀerent form independently by Hicks [1939] and Kaldor [1939]. The basic idea was that if we were to contemplate adopting a policy which would result in a change from situation A to a second situation, B, the change could be viewed as desirable if those who gained from the change from A to B could compensate those who lost by the change, and still be better oﬀ. While this statement may appear at ﬁrst glance to be rather unambiguous, it was soon pointed out that this was not the case. In the ﬁrst place, if we were regarding the compensation as being monetary, then a large change would likely change prices, so that monetary compensation which at ﬁrst seemed adequate might fail to be so after the price changes. Consequently, it was soon decided that the compensation should be in real terms, at least in theory; although to analyze this idea, we will need to add a little notation to that set out in Section 2, as follows. Suppose the allocation which initially prevails in the economy is x1i , and that the adoption of a policy measure under consideration would result in the new consumption allocation, x2i . Deﬁne: y2 =
m i=1
x2i ,
and, for y ∈ Rn+ , deﬁne: m A(y) = xi ∈ Rn+  xi = y . i=1
Then the policy would be said to result in an improvement if there exists x3i ∈ A(y 2 ) such that x3i P x1i ; where ‘P ’ denotes the strict Pareto ordering. The basic motivation behind the introduction of this condition as a criterion for inprovement is that, while economists cannot, as economists, recommend one consumption allocation over another (this being a ‘political question’), the satisfaction of the criterion would mean that the government could, if it so desires, redistribute the gains from
15.5. The Compensation Principle
417
the change in such a way as to result in an allocation Pareto superior to the original allocation. Probably something already strikes you as being odd about such a criterion, and in any event it was soon discovered that this criterion is intransitive; in fact it is not acyclic. If you think about it carefully, one of the things you may notice is that if x∗i and xi are two allocations such that: m m x∗i = xi , i=1
i=1
then the two are equivalent, insofar as this criterion is concerned. From a formal point of view, for a given aggregate commodity bundle, y ∈ Rn+ , every allocation in A(y) is in the same equivalence class, insofar as the ordering of allocations which is induced by this criterion is concerned. Another way of putting this is that the criterion should be viewed as an ordering of aggregate bundles, not of allocations; a point which was made in a similar fashion by Samuelson [1950], and emphasized by Chipman and Moore [1971]. In fact, in the latter paper it was noted that the criterion could easily be extended and formalized as follows. First let’s extend the attainable allocations notion to sets, in the obvious way: m A(Y ) = xi ∈ Rmn xi ∈ Y +  i=1
(in general, we would interpret Y as being the aggregate production set). We then deﬁne , which we will refer to as the ‘KaldorHicksSamuelson (KHS) ordering’ on the subsets of Rn+ by: Y Y ⇐⇒ ∀xi ∈ A(Y ) ∃xi ∈ A(Y ) : xi Gxi , where ‘G’ denotes the weak Pareto ordering: xi Gxi ⇐⇒ xi Gi xi for i = 1, . . . , m. Following the terminology of Chipman and Moore [1971], we will refer to subsets, Y , of Rn+ as ‘situations.’ The idea here is that an economic policy change, other than one which is purely redistributive, will usually result in a diﬀerent potential aggregate supply set. For example, if a country adopts a free trade policy, as opposed to autarky, the set of potentially available aggregate supply vectors will now consist of those reﬂecting the net results of trading with other countries. The KHS ordering, in principle, would then allow a comparison between the situation attainable before the change and the possibilities attainable after the change. As it is formulated here, the KHS criterion does in fact correct some of the problems connected with the earlier KaldorHicks formulations. For example, you can easily prove the following. 15.11 Proposition. Suppose Gi is reﬂexive and transitive, for i = 1, . . . , m. Then the KHS ordering, , will be reﬂexive and transitive, and its asymmetric part will be (asymmetric and) transitive. Furthermore, for each Y1 , Y2 ⊆ Rn+ , we have: Y1 ⊆ Y2 ⇒ Y2 Y1 .
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While the above proposition establishes the fact that the KHS ordering has some nice properties, it is important to notice that it is not generally total. In fact, strengthening the assumptions of the proposition to require that each Gi be total, as well as reﬂexive and transitive, does not correct this shortcoming. The following is a simple example demonstrating the fact that this is the case. 15.12 Example. Suppose m = 2, and that the two consumers’ preference relations can be represented by the utility functions: u1 (x1 ) = min{x11 , x12 /2} and u2 (x21 /2, x22 }; and let Yt = y t , for t = 1, 2, where: y 1 = (2, 4) and y 2 = (4, 2). In this case it is easy to show (see Exercise 2, at the end of this chapter) that we have neither Y1 Y2 nor Y2 Y1 . While the above example rules out the possibility of the KHS ordering’s always being total, there remains the chance that one could compare situations on some basis other than the rather trivially obvious set inclusion criterion mentioned in Proposition 15.11. In fact, one can prove the following (see Chipman and Moore [1971], Theorem 3, p. 9). 15.13 Theorem. Suppose there exists an increasing, continuous, concave and homogeneous of degree one function, g : Rn+ → R+ which is such that the ith consumer’s preferences can be represented by the utility function: ui = g(xi )
for i = 1, . . . , m.
Then we have the following, for all Y1 , Y2 ⊆ Rn+ : Y1 Y2 ⇐⇒ (∀y ∈ Y2 )(∃y ∈ Y1 ) : g(y ) ≥ g(y). If we are willing to conﬁne our attention to situations which are compact and nonempty subsets of Rn+ , then we can simplify the statement of the conclusion to: Y1 Y2 ⇐⇒ max g(y) ≥ max g(y).10 y∈Y1
y∈Y2
Obviously, the assumptions of Theorem 15.13 allow a comparison of situations on a much broader basis than mere set inclusion. The price of this expansion of comparability is high, however; in the result we not only make quite strong assumptions about individual preferences, we also require all m individuals to have the same preferences! In fact, however, the diﬃculty with Theorem 15.13 goes beyond the strong assumptions regarding individual preferences; in order to make use of the criterion, one needs to know the function g(·), which is tantamount to requiring that we know 10 Notice that this means that, under the assumptions of the theorem, the relation is total on the family of compact and nonempty subsets of Rn +.
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419
the preference relation of each and every consumer in the economy! Obviously this is a hopelessly impractical requirement. In general we are at most likely to be willing to assume only that each preference relation, Gi , satisﬁes some speciﬁc qualitative properties; for example, that each Gi is a continuous weak order. This is equivalent to requiring that the preference proﬁle, G = (G1 , . . . , Gm ) be such that each Gi is an element of some welldeﬁned family of preference relations, G. Let’s formalize this idea a bit as follows. First of all, as in Section 2 of this chapter let ‘Gc ’ denote the family of continuous, increasing, and strictly convex weak orders on Rn+ , and let ‘Gc ’ denote the mfold cartesian product of Gc ; that is, let: m Gc = Gi , where Gi = Gc , for i = 1, . . . , m; i=1
For the remainder of this discussion, let’s denote the family of situations (that is, the family of nonempty subsets of Rn+ ) which are weakly disposable by ‘Y;’ that is, we let Y be the family of all nonempty subsets of Rn+ satisfying the condition: (∀y, y ∈ Rn ) : [y ∈ Y & 0 ≤ y ≤ y] ⇒ y ∈ Y.
(15.14)
The reason we will want to deal only with sets satisfying the weak disposability condition stems from the fact that we can trivially extend the last part of the conclusion of Proposition 15.11 to note that if each Gi is nondecreasing (as we will usually be assuming to be the case), and if Y1 and Y2 are such that there exist y t ∈ Rn+ such that: Yt = {y t } for t = 1, 2, where y 2 ≥ y 1 , then Y2 Y1 . We will not include such singleton sets in Y (unless y = 0), but we will include all sets of the form: Yt = {y ∈ Rn+  y ≤ y t }, for y t ∈ Rn+ . This enables us to incorporate the trivial extension of Proposition 15.11 just discussed within the statement: (∀Y1 , Y2 ∈ Y) : Y2 ⊇ Y1 ⇒ Y2 Y1 ; or, more succinctly, by the statement: ‘the KHS order extends ⊇ on Y.’ Now consider the following deﬁnition. 15.14 Deﬁnition. If G ⊆ Gc , we deﬁne the extended KHS relation for G, G, on Y by: (15.15) Y1 G Y2 ⇐⇒ (∀G ∈ G) : Y1 Y2 . It is then very easy to show that for any G ⊆ Gc , G will extend ⊇ on Y; that is, given any G ⊆ Gc , and for all Y1 , Y2 ∈ Y: Y1 ⊇ Y2 ⇒ Y1 G Y2 .
(15.16)
The question is, can we ﬁnd an admissible preference space such that G signiﬁcantly extends ⊇? Unfortunately, a result established in Chipman and Moore [1971, Theorem 4, p. 13] dashes our hopes here.
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15.15 Theorem. Let G† ⊆ Gc be the set of all preference proﬁles, G such that: G1 = G 2 = · · · = G m , and Gi is homothetic and strictly increasing (in addition to being a continuous and strictly convex weak order). Then for all Y1 , Y2 ∈ Y, we have: Y1 G† Y2 ⇒ Y 1 ⊇ Y2 ; where ‘ Y 1 ’ denotes the closure of Y1 . In particular, then, if we conﬁne our attention to the subset of situations, Yc , consisting of only those elements of Y which are closed, then the relation G† coincides with ⊇ on Yc . While the above theorem doesn’t deﬁnitively rule out the possiblity of ﬁnding a subset of Y for which G signiﬁcantly extends ⊇, it certainly suggests that a search for such a subset is extremely likely to be a fruitless endeavor. After all, we know that the assumption that preferences are homothetic generally yields much stronger aggregative conclusions than we can obtain without this hypothesis; and, in addition, Theorem 15.13 establishes the fact that if each consumer’s preference relation is the same homothetic, continuous, increasing, and strictly convex weak order that characterizes every other consumer’s preferences, then the KHS relation is a very signiﬁcant extension of ⊇. More to the point, however, the theorem states that unless we are willing to rule out the case of identical homothetic preferences, then the extended KHS relation will not signiﬁcantly extend ⊇; for notice that if G is a subset of Gc which contains G† , then for all Y1 , Y2 ∈ Y: Y1 G Y2 ⇒ Y1 G† Y2 . On that note, let’s turn our attention to indirect social preferences.
15.6
Indirect Preferences: Individual and Social
In many economic contexts, both theoretical and applied, one can fruitfully make use of the concept of an indirect social preference relation. In fact, in the next section we will be considering measures of ‘real national income,’ a topic which depends crucially on such indirect preferences; and in Section 8 of this chapter we will be making use of this idea in deriving some results concerning consumers’ surplus. In our analysis here, and generally in making use of the idea of indirect social preferences, we will be concentrating our attention upon the problem of comparing situations which are competitive equilibria from the standpoint of the consumers in the economy. We deﬁne this sort of equilibrium as follows. 15.16 Deﬁnition. Let G ∈ Gc . We will say that a tuple (x∗i , p∗ ) ∈ X × Rn++ is a consumers’ competitive equilibrium for G iﬀ, for each i (i = 1, . . . , m), the following condition holds: (∀xi ∈ Rn+ ) : p∗ · x∗i ≥ p∗ · xi ⇒ x∗i Gi xi .
15.6. Indirect Preferences: Individual and Social
421
Our development of the idea of social indirect prefererences is based upon the concept of individual indirect prefernces, which we studied extensively in Section 4.7. For convenient reference, however, we will present a summary of this material here. Given any weak order, Gi ∈ Gc , Gi induces an indirect preference relation, G∗i , on: def
Ω = Rn++ × R+ by:
(p, w)G∗i (p , w ) ⇐⇒ hi (p, w)Gi hi (p , w );
where ‘hi (·)’ denotes the ith consumer’s demand function (the demand function determined by) Gi .11 We say that a function vi : Ω → R is an indirect utility function corresponding to Gi iﬀ vi represents G∗i on Ω. In the present context, one most conveniently obtains an indirect utility function in the following way: if ui is a utility function representing Gi , and if hi (·) is the demand function determined by Gi , then the composite function vi : Ω → R+ deﬁned by: vi (p, w) = ui [hi (p, w)]
for (p, w) ∈ Ω,
is an indirect utility function representing G∗i on Ω. Recall also that, if G ∈ Gh , then there exists an indirect utility function for G which takes the particularly simple and useful form: vi∗ (p, wi ) =
wi ; γi∗ (p)
where γi∗ (·) is a costofliving function for Gi ; that is, For Gi ∈ Gc , the function γi∗ is deﬁned as: 1 γi∗ (p) = ∗ for p ∈ Rn++ , ui [hi (p, 1)] where u∗i is any positively homogeneous of degree one function representing Gi .12 Now, given any preference proﬁle G ∈ Gc , any social preference function ω deﬁned on Gc induces an indirect social preference relation, Q∗ on: def
Ω = Rn++ × Rm +. deﬁned by:
(p , w )Q∗ (p , w ) ⇐⇒ H(p , w )QH(p , w );
where Q = ω(G) is the social preference relation determined by (ω, G), and we deﬁne H : Ω → Rmn + by: H(p, w) = h1 (p, w1 ), . . . , hm (p, wm ) . We will then refer to Q∗ as the indirect social preference relation induced by ω(G). 11 12
The ith consumer’s demand function for the j th commodity will then be denoted by ‘hij (·).’ See Section 4.9 for details.
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So, what is going on here is very much the same kind of procedure by which we deﬁne indirect (individual) preferences from direct preferences. Two vectors of prices and (mtuples of) incomes determine allocations via the demand functions of the individual consumers. The ﬁrst (p, w)pair is (indirectly) preferred to the second if the ﬁrst allocation is socially preferred to the second allocation. In much the same way that we deﬁne an indirect utility function to represent (individual) indirect preferences, we can then deﬁne an indirect social welfare function. In particular, if ω is of the PBS form, we can deﬁne an indirect social welfare function to represent Q∗ , as follows. If ω is determined by (µ, F ), we deﬁne v : Ω → Rm + by: v(p, w) = u1 [h1 (p, w1 )], . . . , um [hm (p, wm )] = v1 (p, w1 ), . . . , vm (p, wm ) , where ui = µ(Gi ) and vi = ui ◦ hi , for i = 1, . . . , m. It is then easily seen that F ◦ v represents the indirect allocation ordering, Q∗ ; that is, for any G ∈ Gc , and all (p, w), (p , w ) ∈ Ω: (p, w)Q∗ (p , w ) ⇐⇒ F v(p, w) > F v(p , w ) . We shall refer to the composite function, F ◦ v, as the indirect social welfare function for ω.13
15.7
Measures of Real National Income
In this section we will investigate the problem of measuring real national income for an economy; exploring the implications for this problem of the ideas we have been presenting in this chapter.14 If G ∈ Gc , each element of Ω deﬁnes a unique (consumers’) competitive equilibrium, and conversely. In this section we exploit this fact in that, when we say that (p, w) is an element of Ω, we will always suppose that consumer i is choosing the bundle: xi = hi (p, wi ) for i = 1, . . . , m. We will make use of the following notation: for (p, w) ∈ Ω [respectively, (p , w ) ∈ Ω, etc.], we will use the notation ‘w’ and ‘w’ [respectively, ‘w ’ and ‘w ,’ etc.] to denote total and average income; that is: m m w= wi and w = (1/m) wi = (1/m)w. i=1
i=1
We will assume throughout this section that we are concerned with maximizing a social preference function, ω, of the PBS form; where ω is determined by a pair (µ, F ), and F is increasing, quasiconcave, and positively homogeneous of degree 13 There are, of course, other functions which represent Q∗ , but this function is uniquely determined by the pair (µ, F ). 14 The relationship between improvements in ‘real national income,’ as conventionally measured, and the KHS criterion is examined extensively in Chipman and Moore [1973], [1976b]. The fact that the results obtained in those studies were essentially negative provided the motivation for the rather unconventional approach to be developed in this section.
15.7. Measures of Real National Income
423
one. We will also begin with the assumption that the admissible preference space is the set G∗ ⊆ Gc given by: G∗ = {G ∈ Gh  G1 = G2 = · · · = Gm };
(15.17)
in other words, we will suppose that each preference proﬁle consists of identical homothetic preferences. Now, given G ∈ G∗ , let γ(·) be the costofliving function for G1 determined by µ. Then the indirect social welfare function for ω, denote it by ‘V ,’ can be written: w
w wm w wm 1 1 V (p, w; G) = F ,..., = F ,..., . (15.18) γ(p) γ(p) γ(p) w w
Thus, V factors into the product of a measure of ‘real national income,’ w/γ(p), and a function of the distribution of income, f : ∆m → R+ , where f is simply the restriction of F to ∆m . The vector, d ∈ ∆m deﬁned by:
w wm 1 , ,..., d = (1/w)w = w w can be thought of as the income distribution vector associated with w; and, since we are supposing that the aggregator function, F , is positively homogeneous of degree one, it is of great interest to see at what value of d ∈ ∆m the aggregator function is maximized. Deﬁne F as the set of all functions F : Rm + → R+ which are increasing, quasiconcave, and positively homogeneous of degree one; and then deﬁne δ : F → ∆m by: δ(F ) = {d ∈ ∆m  (∀a ∈ ∆m ) : F (d) ≥ F (a)}.
(15.19)
If F is strictly quasiconcave, then δ(F ) will be singlevalued, and, as we have done in similar contexts, we can think of δ as being a function. However, it will not be necessary to do this in our present discussion. Given that F is positively homogeneous of degree one it is easy toshow that if d ∈ δ(F ), then, given any value of w ∈ R+ , F is maximized, subject to m i=1 wi = w, at w = (wd1 , . . . , wdm ) = wd. We can therefore deﬁne a measure of the eﬃciency of the income distribution, which we will denote by ‘E(w; F )’ by: E(w; F ) =
F (w) F (w1 , . . . , wm ) = . F (wd) F (wd1 , . . . , wdm )
(15.20)
Since any F ∈ F is quasiconcave and positively homogeneous of degree one, it is easy to show that, for all w ∈ Rm + \ {0}, we must have: 0 ≤ E(w; F ) ≤ 1; with: E(w; F ) = 1 ⇐⇒ (1/w)w ∈ δ(F ). From the standpoint of making use of F ◦ v as a social preference function, it is clear that one wants to keep E(w; F ) as close to one as possible; in fact, the larger the value of E(w; F ), the better. We formalize our deﬁnition of this index in the following.
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15.17 Deﬁnition. Given F ∈ F, and d ∈ δ(F ), we deﬁne E(w; F ), for w ∈ Rm + \ {0}, the distribution (eﬃciency) index, by: E(w; F ) =
F (w) F (w1 /w, . . . , wm /w) = . F (wd) F (d)
(15.21)
Before proceeding further, let’s take a look at some examples. 15.18 Example. Suppose F takes the CES form: F (w) =
m i=1
(di )1−a · (wi )a
where: di > 0 for i = 1, . . . , m,
m
1/a
,
di = 1, and a ≤ 1.
(15.22)
(15.23)
i=1
It is an easy exercise toprove (see Exercise 3, at the end of this chapter) that F is maximized, subject to m i=1 wi = 1, when: w = d; that is, in this case, we can take δ(F ) = d = (d1 , . . . , dm ). The distribution index, E(w; F ) is then given by: E(w; F ) =
F (w) F (w1 /w, . . . , wm /w) = = F (wd) F (d)
m
1−a · (w )a 1/a i i=1 (di ) . 1/a m w i=1 di
(15.24)
Of course, in this particular case, F (d) = 1 for a = 0. so that the index reduces to:
wi a 1/a . i=1 w On the other hand, when a = 0 (the CDE case), the index is given by: m (wi )di . E(w : F ) = mi=1 di (d i=1 i w) E(w : F ) = F (w1 /w, . . . , wm /w) =
m
(di )1−a ·
(15.25)
Now, in the special case in which (1/m)1 ∈ δ(F ), we will say that F is egalitarian;15 for in this case, for a given aggregate income, w, F will always be maximized when individual incomes are all the same. Notice also that in this case, the distribution index becomes: F (w) E(w; F ) = ; (15.26) F (w, . . . , w) Consequently, in this egalitarian case, the inequality index generally used in the literature as a measure of income inequality (see Exercise 5, at the end of this chapter), J(w; F ) is given by: J(w; F ) = 1 − E(w; F ). 15
Recall that we use ‘1 ’ to denote the vector each of whose coordinates equals 1.
(15.27)
15.7. Measures of Real National Income
425
Under these assumptions, it is usual in this literature to argue that the lower the value of this index (which corresponds to the higher the value of E(w; F )), the better.16 At ﬁrst glance, it is diﬃcult to think of a good reason why one shouldn’t favor an egalitarian aggregator; after all, if you were to be one of the consumers, but were not sure which number you would be labeled by (what value of i would be yours), it would be safer for you to argue in favor of an equallyweighted (egalitarian) aggregator function than otherwise. In other words, if you were in favor of using an aggregator function of the form indicated in (15.22), wouldn’t you want to set the weights all equal; that is, let: di = 1/m
for i = 1, . . . , m?
Upon reﬂection, however, it isn’t so clear that this is reasonable. After all, would it really be reasonable for someone who is able, but unwilling to work to have the same income as someone who has a diﬃcult and dangerous occupation? or, perhaps more to the point, would anyone elect to pursue a diﬃcult and dangerous (but necessary) occupation if she or he could earn the same income without working at all? In fact, a move toward a more equal income distribution in an economy may mean that the labor market is not functioning eﬀectively; with a corresponding decline in economic eﬃciency. In any event, we will not conﬁne our attention to egalitarian aggregator functions in the discussion to follow; on the other hand, we will not rule out this case either. Despite the fact that the cost of living function conveniently cancels out in the derivation of the distribution index under the assumptions being utilized here, there are problems involved with the use of this index when prices have changed. To be more precise, if one observes two priceincome pairs, (p1 , w1 ) and (p2 , w2 ), social welfare may have decreased in the move from situation 1 to situation 2 under the present assumptions even if individual preferences are unchanged and: E(w2 ; F ) > E(w1 ; F ). On the other hand, we do have the following, fairly obvious, proposition; the proof of which I will leave as an exercise. 15.19 Proposition. Suppose the social preference function, ω, is determined by a pair (µ, F ), where F is positively homogeneous of degree one, and that G ∈ G∗ , V is an indirect social welfare function for ω, that γ is a cost of living index for each Gi , and let (p1 , w1 ), (p2 , w2 ) ∈ Ω be two (consumer) competitive equilibrium situations. Then we have the following. 1. If: w2 w1 ≥ , (15.28) γ(p2 ) γ(p1 ) and: E(w2 ; F ) ≥ E(w1 ; F ), 16
(15.29)
For more on the measurement of inequality of income, and the properties of the inequality index, see Dutta [2002], Foster and Sen [1997], Moulin [1988, pp. 5152], or Myles [1995, Chapter 3].
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then: V (p2 , w2 ) ≥ V (p1 , w1 ).
(15.30)
2. Conversely, if (15.30) holds, and w2 /γ(p2 ) ≤ w1 /γ(p1 ), then (15.29) holds as well. The above proposition shows that, under the assumptions of that result, the comparison of values of the distribution index, E(w; F ), in two diﬀerent equilibria has an unambiguous interpretation in terms of the deﬁning social welfare function in the case where prices are unchanged. That is, if prices are unchanged, and E(w2 ; F ) > E(w1 ; F ), we can be sure that V (p2 , w2 ; G) > V (p1 , w1 ; G) as well. In order to begin to extend this analysis, consider the product y(p, w; F ), deﬁned by: y(p, w; F ) =
w × E(w; F ). γ(p)
(15.31)
It is easily seen from (15.18) that this product is an increasing transformation (in fact, a positive scalar multiple) of the indirect social welfare function, V (p, w). Consequently, a comparison of values of y(·; F ) at diﬀerent (consumer) competitive equilibria has unambiguous welfare implications. Moreover, notice that the function y(·; F ) deﬁned in (15.31) has an interesting and natural interpretation: we can think of it as income distributionadjusted real national income. The above discussion, and Proposition 15.19 shows that, under the assumptions of the proposition, the income distributionadjusted real national income function has an unambiguous meaning in terms of the underlying social welfare function. However, suppose we take the admissible preference space to be all of Gh , rather than just G∗ . In this case, we do not get the convenient factorization which we have presented in (15.18); in fact the best we can do is something like: w
w /γ (p) wm wm /γm (p) 1 1 1 V (p, w) = F =w·F ,..., ; (15.32) ,..., γ1 (p) γm (p) w w
or, perhaps:
7
8 8 7 w w1 /γ1 (p) wm /γm (p) V (p, w) = , ,..., ·F w/γ(p) w/γ(p) γ(p)
(15.33)
where γ(p) is any sort of average of the cost of living functions, for example: m 1/m m γ(p) = (1/m) γi (p) or γ ∗ (p) = γi (p) . (15.34) i=1
i=1
However we express the equation, the basic problem remains; once we allow individual preferences to diﬀer, our distribution index, which is deﬁned as a function of nominal income, may have no dependable relationship with the value of the indirect social preference function; even when comparing equilibria with unchanged prices. Consider the following example: 15.20 Example. Consider an economy with two consumers, whose utility functions are given by: u1 (x1 ) = min{x11 /2, x12 } and u2 (x2 ) = min{x21 , x22 /4};
15.7. Measures of Real National Income
427
and suppose the social welfare function is given by: 2 W (xi ) = [u1 (x1 )]1/2 + [u2 (x2 )]1/2 ,
(15.35)
(we suppose that the given utility functions are the appropriate functions to be paired with the aggregator function in order to deﬁne W ). Now suppose the vector of prices is given by p = (1, 2), and consider the income ﬁgures w1 = (144, 36) and w2 = (36, 144). Then (see Exercise 6, at the end of this chapter): V (p, w1 ) = 64 while V (p, w2 ) = 49. However, the distribution index is given by: E(w1 ; F ) = 9/10 = E(w2 ; F ). It should also be noted that, since the aggregator function is egalitarian, the inequality index, J(w; F ), is equal to 1/10 in both equilibria. This last example shows that even if the aggregator function is egalitarian, and both prices and aggregate incomes are unchanged in two situations, a change in the distribution of income (which leaves both the distribution index and inequality index unchanged) may nonetheless result in a decline in welfare. However, there is at least one case in which things work out much better. Consider the generic CDE case with the aggregator function: m a F (u) = ui i , (15.36) i=1
where:
m i=1
ai = 1 and ai > 0,
for i = 1, . . . , m.
In this case, the corresponding indirect social welfare function becomes: 7 8 7 8 m wi ai m wi ai w V (p, w) = = m · ; ai i=1 γi (p) i=1 w i=1 γi (p) so that: y(p, w; F ) = m
w
ai i=1 γi (p)
× E(w; F );
(15.37)
(15.38)
(15.39)
is once again a scalar multiple of V (p, w). Consequently, if two consumers’ competitive equilibria, (p1 , w1 ) and (p2 , w2 ) are such that: p1 = p2 , our decisionmaker will prefer the equilibrium having the higher value of y(p, w; F ). Of course, it should also be noted that in this situation, that is, with p1 = p2 , we will have: m m V (p2 , w2 ) ≥ V (p1 , w1 ) ⇐⇒ (wi2 )ai ≥ (wi1 )ai . i=1
i=1
428
Chapter 15. Some Tools of Applied Welfare Analysis
However, it nonetheless seems worthwhile to consider the index number idea a bit further. Deﬁne γ ∗ (p) by: m γ ∗ (p) = γi (p)ai , (15.40) i=1
then we can write:
w y(p, w; F ) = ∗ × E(w; F ). (15.41) γ (p) ∗ The γ function could reasonably be estimated by taking a manageable sample of consumers, and estimating cost of living functions for the sample. Of course the function γ ∗ is determined by the parameters a1 , . . . , am as well; however, one might hope that the function values might not be too sensitive to the values of the parameters chosen, since γ ∗ is, essentially, a geometric mean. Indeed, in the special case in which γ1 (p) ≡ γ2 (p) ≡ · · · ≡ γm (p), the function is independent of the values of the parameters ai . Consequently, it might be possible to construct such an aggregate index which would be meaningful whatever one’s feelings as to what were the appropriate values of the ai parameters. However, this is probably not the place for further speculation along these lines, and it is time we turned our attention to consumers’ surplus.
15.8
Consumers’ Surplus
As we saw in Section 10 of Chapter 4, the analysis of consumer’s surplus for a single consumer gets complicated very quickly if one attempts to do the analysis carefully. Obviously even more complications arise when one is trying to develop a justiﬁable measure of (aggregate) consumers’ surplus, and in fact in most applied work it appears that investigators implicitly assume the existence of a ‘representative consumer.’17 However, it turns out that we can make some progress in the development of such a measure without making such a restrictive assumption. In our analysis we will make use of the general framework and assumptions which we have been using throughout the rest of this chapter. We now deﬁne: Ω = Rn++ × Rm +, and expand upon our deﬁnition of an acceptable (integral) measure of welfare change, as presented in Section 10 of Chapter 4, as follows.18 We will take as given a preference proﬁle G ∈ Gc and a PBS social welfare function, W , determined by a pair (µ, F ), and we will denote the indirect social preference relation determined by W , given G, by ‘Q∗ .’ In the following, we will let Ω∗ be a nonempty open subset of Ω, and let P(Ω∗ ) be the set of all polygonal paths, ω : [0, 1] → Ω∗ , connecting points of Ω∗ , and lying entirely within the set (see Section 4.10 for a discussion of polygonal paths, as well as an explanation of the idea of line integrals, which will be used in the following deﬁnition). 17 That is, the method used is typically only theoretically correct in the situation where consumers as a whole behave as if they were maximizing a single utility function. See Section 3 of Chapter 5 for a discussion of this topic. 18 In our treatment of integral measures of change in Q∗ , we are generally following the development presented in Chipman and Moore [1994]; although in a somewhat simpliﬁed form.
15.8. Consumers’ Surplus
429
15.21 Deﬁnition. We will say that a function f : Ω∗ → Rn+m furnishes an ac+ ceptable indicator of change in Q∗ on Ω∗ iﬀ: 1. for all ω, ω ∗ ∈ P(Ω∗ ) satisfying ω(0) = ω ∗ (0) and ω(1) = ω ∗ (1), we have: 1 1 f [ω(t)] · dω(t) = f [ω ∗ (t)] · dω ∗ (t), (15.42) 0
0
(independence of path) and: 2. for all (p0 , w0 ), (p1 , w1 ) ∈ Ω∗ and for all ω ∈ P(Ω∗ ) satisfying: ω(0) = (p0 , w0 ) and ω(1) = (p1 , w1 ), we have:
1
f [ω(t)] · dω(t) ≥ 0 ⇐⇒ (p1 , w1 )Q∗ (p0 , w0 ).
(15.43)
(15.44)
0
The basic rationale for the above deﬁnition is a straightforward extension of the deﬁnition for the singleconsumer case which was presented in Chapter 4. We have simply allowed for m consumers, rather than just one, in the speciﬁcation of the range of the polygonal paths; and we have substituted an indirect social preference relation for a single consumer’s indirect preference relation. I will refer you to Section 10 of Chapter 4 for a justiﬁcation of the use of line integrals in this deﬁnition, as well as for the independence of path requirement. Very much as was the case in Chapter 4, standard results on line integrals tell us that if f satisﬁes Condition 1 of the above deﬁnition, then there exists a twicediﬀerentiable potential function, V : Ω∗ → R such that, for all (p0 , w0 ), (p1 , w1 ) ∈ Ω∗ and all ω ∈ P(Ω∗ ) satisfying (15.43), we will have: 1 f [ω(t)] · dω(t) = V (p1 , w1 ) − V (p0 , w0 ); (15.45) 0
and, for all (p, w) ∈ Ω∗ , V and f will satisfy: ∂V = fj (p, w) for j = 1, . . . , n, ∂pj (p,w) ∂V = fn+i (p, w) for i = 1, . . . , m, ∂wi (p,w)
(15.46) (15.47)
and, for example (remember that V is twicediﬀerentiable): ∂fj ∂fn+i = ∂wi (p,w) ∂pj (p,w)
for j = 1, . . . , n; i = 1, . . . , m.
(15.48)
It then follows from (15.45), that if f furnishes an acceptable indicator of welfare change in Q∗ on Ω∗ , then the corresponding potential function, V , must be an indirect social welfare function representing Q∗ . Now that we know this to be the case, the following generalization of the ‘AntonelliAllenRoy’ equation (Theorem 4.28)19 shows us what form the function f must take. 19
A slightly more general result is established in Chipman and Moore [1990, Theorem 4, p. 484].
430
Chapter 15. Some Tools of Applied Welfare Analysis
15.22 Theorem. Let Ω∗ be a nonempty open subset of Ω, let Q∗ be an indirect social preference relation on Ω∗ which extends the weak Pareto ordering, and let V : Ω∗ → R be a diﬀerentiable function representing Q∗ on Ω∗ . Then, given any (p0 , w0 ) ∈ Ω∗ , V and H satisfy: m ∂V ∂V (15.49) 0 0 =− 0 0 hij (p0 , wi0 ) for j = 1, . . . , n. i=1 ∂wi (p ,w ) ∂pj (p ,w ) Proof. From the deﬁnitions of individual demand and indirect preferences, it follows that, for each i (i = 1, . . . , m); (15.50) ∀(p, wi ) ∈ Ω : p · hi (p0 , wi0 ) ≤ wi ⇒ (p, wi )G∗i (p0 , wi0 ), where G∗i is the ith consumer’s indirect preference relation. Therefore, since V represents Q∗ on Ω∗ , and Q∗ extends the weak Pareto ordering, it follows that V is minimized at (p0 , w0 ), subject to: p · hi (p0 , wi0 ) ≤ wi
for i = 1, . . . , m.
Consequently, it follows from the classical theory of constrained minimization that there exist multipliers λi ∈ R, for i = 1, . . . , m, such that: m ∂V − λi hij (p0 , wi0 ) = 0 for j = 1, . . . , n; (15.51) i=1 ∂pj (p0 ,w0 ) and:
∂V + λi = 0 for i = 1, . . . , m. ∂wi (p0 ,w0 )
We then obtain (15.49) by substituting (15.52) into (15.51).
(15.52)
From the above result and our earlier discussion we can now see a great deal about what form a function which furnishes an acceptable (integral) indicator of change in Q∗ on Ω∗ must take. We must have: m fj (p, w) = − fn+i (p, w)hij (p, wi ), (15.53) i=1
and: fj (p, w) =
∂V ∂wi (p,w)
for j = n + 1, . . . , n + m;
(15.54)
where V is the potential function associated with f . Turning things around, suppose Q∗ is the indirect social preference relation determined by a PBS social welfare function which is, in turn, determined by the aggregatormeasure function pair (µ, F ). Suppose further that F is twicediﬀerentiable, and that each utility function ui = µ(Gi ) is twicediﬀerentiable as well. Then we know that the function f : Ω → Rn+m deﬁned by: m ∂F ∂vi fj (p, w) = − hij (p, wi ) for j = 1, . . . , n; (15.55) i=1 ∂ui ∂wi (p,w) ∂F ∂vi fn+i (p, w) = for i = 1, . . . , m. (15.56) ∂ui ∂wi (p,w)
15.8. Consumers’ Surplus
431
furnishes an acceptable measure of change in Q∗ on Ω∗ . The problem with the formulas in (15.55) and (15.56), from an applied standpoint is that there are some really thorny estimation problems that must be dealt with in order to evaluate the integrals of interest. In the ﬁrst place, each individual consumer’s demand function must be estimated; a quite impractical chore in and of itself; but in fact, if each such function is estimated, then, generally speaking, one could then obtain each consumer’s indirect utility function. Given these functions, and a knowledge of the appropriate aggregator function, the issue of whether a given (p, w)pair does or does not dominate a second such pair can be determined directly and exactly without resort to any line integral! Once again, however, things are a bit better in some special cases; one of which we set out in the following example. 15.23 Example. Suppose the PBS social welfare function of interest is of the CDE form, and that each ui is positively homogeneous of degree one (as well as twicediﬀerentiable). In this case, the indirect social preference function can be expressed in the form: m wi ai V (p, w) = , (15.57) i=1 γi (p) where a ∈ Rm + is such that: m i=1
ai = 1 and ai > 0, for i = 1, . . . , m;
(15.58)
and γi (·) is the ith consumer’s costofliving function. It will be convenient in this development, however, to take the log of the function in (15.57), to obtain the equivalent indirect social preference function given by: m ai log wi − log γi (p) . (15.59) v(p, w) = i=1
In this case it follows from equations (15.55) and (15.56) that the function f : Ω → Rn+m deﬁned by: m (ai /wi )hij (p, wi ) for j = 1, . . . , m; (15.60) fj (p, w) = − i=1
fn+i (p, w) = ai /wi
for i = 1, . . . , m;
(15.61) Q∗
furnishes an acceptable (integral) measure of change in on Ω. We can actually simplify this function a bit further. Recall that, since each consumer’s preferences are homothetic, hi (·) can be expressed in the form: hi (p, wi ) = g i (p)wi
for i = 1, . . . , m.
Consequently, we can simplify (15.60) to: m m fj (p, w) = − ai · gij (p) = − hij (p, ai ) i=1
i=1
for i = 1, . . . , m.
(15.62)
Now, in order to estimate fj (·), one would presumably need to estimate the function: m def h(p, w) = hi (p, wi ). (15.63) i=1
432
Chapter 15. Some Tools of Applied Welfare Analysis
However, notice that, while to estimate this function one needs observations on individual consumer incomes, one only needs observations on aggregate commodity demands. Of course, in practice, one would need to aggregate consumers into income classes, or occupation, or in some other meaningful fashion, as well as aggregating over commodities in some standard fashion. It is worth noting, however, that if one obtains an estimate of the function deﬁned in (15.63), then one can deﬁne a function f to furnish an acceptable indicator of change in Q∗ , for any indirect social preference function of the CDE form. That is, any a ∈ Rm + satisfying (15.58) determines a PBS social welfare function of the CDE form, and one can then deﬁne a function which furnishes an acceptable indicator of change for the associated indirect social preference by: m fj (p, w) = −hj (p, a) = − gij (p)ai for j = 1, . . . , m, (15.64) i=1
fn+i (p, w) = ai /wi
for i = 1, . . . , m.
(15.65)
Let’s now turn our attention to the development of a measure of consumers’ surplus which is of a diﬀerent form; namely adding values of compensating variation. Recall that in Section 10 of Chapter 4, we deﬁned the compensating variation criterion for welfare improvement for a single consumer by: WiC (p1 , wi1 ), (p2 , wi2 ) = wi2 − µi (p2 ; p1 , wi1 ). (15.66) This deﬁnition is then easily extended to obtain the aggregate compensating variation obtained in moving from (p1 , w1 ) to (p2 , w2 ) as: m 2 W C (p1 , w1 ), (p2 , w2 ) = (15.67) wi − µi (p2 ; p1 , wi1 ) . i=1
It is intuitively appealing to say that if a project or policy would result in a change 2 2 from a ﬁrst pricewealth pair, (p1 , w1 ), to a second, (p , w ), and it is estimated (accurately, we will assume) that W C (p1 , w1 ), (p2 , w2 ) is positive, then the change should be made. However, it is doubtful whether anyone would advocate this as a welfare criterion in and of itself; in fact, a quite persuasive critique of its use as a welfare measure is contained in Blackorby and Donaldson [1990], and I will refer you to their article for a detailed critique of the use of aggregate compensating variation in applied welfare analysis. In the remainder of this section, we will turn our attention to a closely related concept; aggregate ‘willingness to pay.’ In our discussion we will make use of some new concepts and notation, as follows. It is reasonable to suppose that a consumer’s preferences depend upon not only her or his privatelypurchased consumption goods but also on publiclyprovided commodities (for example, parks, bridges, roads, and so on). Consequently, in the remainder of this section, we will suppose the ith consumer’s consumption set takes the form Ci = Rn+ × Yi , where Yi is something which we will call the ith consumer’s ‘public goods space,’ and we will use the generic notation ‘(xi , y i )’ to denote elements of Ci . We will study public goods in some detail in the next chapter, but for now we will simply suppose that elements yi ∈ Yi are things which contribute to consumer i’s wellbeing, but which are not privately purchased. The vector of private goods chosen by consumer i, given a price vector for private goods, p, and an income
15.8. Consumers’ Surplus
433
(or wealth), wi , may depend not only on these variables, but also upon the public goods available to her or him. Consequently, we should in principle consider the ith consumer’s demand function hi to be deﬁned on Ω × Yi , and write ‘hi (p, wi , yi )’ in place of ‘hi (p, wi )’ to denote values of i’s demand function. Allowing for this complication in the analysis to follow will not complicate things for us at all, for we will be making use of indirect preferences, which are now deﬁned on Ω × Yi for the ith consumer by: (p2 , wi2 , y 2i )G∗i (p1 , wi1 , y 1i ) ⇐⇒ hi (p2 , wi2 , y 2 ), y 2i Gi hi (p1 , wi1 , y 1 ), y 1i . We will make use of the notational framework just introduced in the example which follows. It is a fairly extended discussion of the notion of ‘aggregate willingness to pay,’ which is a concept closely related to aggregate compensating variation. 15.24 Example. Suppose a project or a policy change is being considered for the economy, which we will characterize as involving a change from (p1 , w1 , y 1 ) to (p2 , w2 , y 2 ); where: yit ∈ Yi
for i =, 1, . . . , m; t = 1, 2.
The change may, in fact involve the building of some public project, and be such as to cause no change (or a negligible change) in the prices of marketed commodities; in which case we would suppose y 2 = y 1 , while p1 = p2 (although we might nonetheless have w1 = w2 ). As an opposite case, we could be dealing with a policy change which would result in y 1 = y 2 , while changing some prices and income. As it turns out, in our treatment we needn’t distinguish between such cases; the change might be a public project or a policy change, or involve changes in both. However, we will refer to the change being contemplated as being the construction of some public project. We will suppose that, in order to assess the desireability of undertaking the project, a survey is taken in which each consumer is questioned about her or his willingness to pay for this project. We will denote by ‘ci ’ the ith consumer’s stated willingness to pay for the project; which may be a negative number if the consumer does not wish the project to be undertaken; and we will suppose that ci , is such that if wi2 is the income consumer i expects to earn after the change, in the absence of any tax assessed to pay for the project, that we have: (p2 , wi2 − ci , y 2i )G∗i (p1 , wi1 , y 1i ),
(15.68)
for each i. In other words, we are supposing that consumer i states an amount ci which would leave her or him at least as well oﬀ after the change as before even if she or he were to have the amount ci subtracted from her or his income after the change. Before proceeding further, we should note that if we ignore the presence of public goods, then the condition in (15.68) will hold if, and only if: wi2 − ci = µi (p2 ; p2 , wi2 − ci ) ≥ µi (p2 ; p1 , w1 ); so that:
ci ≤ WiC (p1 , wi1 ), (p2 , wi2 ) for i = 1, . . . , m.
(15.69)
434
Chapter 15. Some Tools of Applied Welfare Analysis Returning to the general case, suppose that we have a project for which: m ci > C, (15.70) i=1
where C is the cost of carrying out the project; and let δ > 0 be any number such that: m 0 wi2 − ci . Furthermore, we note that it follows from (15.71) that m γ−λ−C −δ = ci − C − δ ≥ 0, i=1
and therefore we have: 0 < a ≤ 1. Consequently, for i ∈ I2 , we see that: ti = a · ci ≤ ci , and thus: w i = wi2 + δ/m − ti > wi2 − ci ;
(15.76)
15.8. Consumers’ Surplus
435
and thus it follows readily from (15.68) that each individual i ∈ I2 is also better oﬀ than before the change. Finally, we have: m ti = ti + ti = −λ + a · ci = −λ + a · γ = −λ + C + δ + λ = C + δ i=1
i∈I1
i∈I2
i∈I2
Therefore, the tax bill covers both the cost of the project and the income subsidy.
In the above example we have shown that if aggregate willingness to pay for a project is greater than the cost of the project, then there may be a strongly Pareto superior improvement if the project is undertaken, and if there is an appropriate allocation of the cost and appropriate compensation is paid. However, it is obvious that the fact that aggregate willingness to pay for a project exceeds its cost does not guarantee that carrying it out will result in Pareto improvement in the absence of such cost allocation and compensation. It should also be noted that if consumers are aware of the plan to allocate costs and compensation after the adoption of the project or policy, then they have a very strong incentive to misrepresent their ‘willingnesstopay’ amounts, ci . In fact, there is every incentive for consumers to claim that they will suﬀer large damage if the policy is adopted; given the scheme we have set out in the above example. We will return to a discussion of this issue in Chapter 18 Exercises. 1. Prove Proposition 15.11 2. Verify the assertion of Example 15.12 3. Verify the calculations in Example 15.18. 4. Show that if we conﬁne our admissible preference proﬁles to Gh , then a PBS social welfare function deﬁned from an aggregator function of the CDE form induces the same social preference relation on X regardless of the measurement function with which it is paired. 5. Moulin [1988, pp. 51–2] deﬁnes, for a given w ∈ Rm + , the equally distributed equivalent income, e(w; F ), as that income, which given any p ∈ Rn++ , solves the equation: e(w; F ) w e(w; F ) wm 1 F =F ,..., . (15.77) ,..., γ(p) γ(p) γ(p) γ(p) Show that, given the assumptions of Section 6, this yields: e(w; F ) =
F (w) . F (1 )
(15.78)
Moulin then deﬁnes the inequality index, J(w; F ), by: J(w; F ) = 1 −
e(w; F ) . w
(15.79)
Show that, if F is egalitarian, then this agrees with the income distribution index deﬁned in Section 6.
436
Chapter 15. Some Tools of Applied Welfare Analysis 6. Verify the ﬁgures given in Example 15.20
7. Returning to the sort of situation contemplated in Example 15.24, show that if we ignore the public goods aspect of that example, then it is true that if aggregate compensating variation is greater than the cost of undertaking a policy change, there is a way of allocating the cost of the change which guarantees that the net eﬀect will be strongly Paretoimproving. [Hint: consider equation (15.69).]
Chapter 16
Public Goods 16.1
Introduction
In this chapter we will examine a bit of the economic theory of public goods. We will begin our study by conducting a brief analysis of a simple general equilibrium model which can be used in the analysis of both public goods and externalities (and thus will be used in Chapter 17, as well as the present chapter). In Section 3 we discuss the basic deﬁnition of public goods, as well as the distinction between public and private goods. Section 4 deals with the simple general equilibrium model which will be the primary tool used in our analysis of public goods allocation. This model is then used in Section 5 to present the basic theory of Lindahl and Ratio equilibrium. In Section 6 we develop a much more general model of public goods production, and show that in this context, the Lindahl equilibrium is both ‘nonwasteful’ and ‘unbiased;’ in other words, we show that Lindahl equilibria in this model are Pareto eﬃcient; and, with appropriate assumptions, given any Pareto eﬃcient allocation, there exist Lindahl prices such that the allocation is (theoretically) attainable as a Lindahl equilibrium.
16.2
A Simple Model
In this section we consider an magent, (1 + n)commodity economy, E. We will use the generic notation, ‘(wi , y i ),’ where wi ∈ R and y i ∈ Rn to denote the commodity bundle available to the ith agent, for i = 1, . . . , m. We will suppose that the ith consumer’s ‘consumption’ set, Zi , takes the form: Zi = Wi × Yi , where Yi is a nonempty subset of Rn , and Wi takes the form: Wi = {wi ∈ R  wi ≥ w i }, for some (ﬁxed) w i ∈ R. We then suppose that the ith agent’s payoﬀ, or utility function, ui , is of the ‘quasilinear’ form: ui (wi , y i ) = wi + vi (y i );
438
Chapter 16. Public Goods
and we take as given a ‘cost function;’ c : Y → R+ [in some applications we may take c(·) to be identically zero], where Y is a nonempty subset of m i=1 Yi . Finally, we suppose that the attainable set for E, A(E), takes the form: , m wi + c(y) ≤ w ¯ , A(E) = (wi , y i ) ∈ Z  y i ∈ Y & i=1
where: Z=
m i=1
Zi ,
w ¯ > 0 is the total endowment of the ‘commodity,’ w, and we assume that: m w i . w ¯> i=1
We will use the generic notation:
to denote elements of
m
y = (y 1 , . . . , y m ),
i=1 Yi ,
we deﬁne: m Wi , W = i=1
and we use the generic notation: w = (w1 , . . . , wm ), to denote elements of W . The model, and the variables therein, can be interpreted in many diﬀerent ways, and this is one of the strengths of the model. On the other hand, the absence of a speciﬁc interpretation may also make it a bit more diﬃcult to understand what is going on in the discussion to follow. Consequently, I think it is helpful as we begin our analysis to look at one speciﬁc interpretation and application for which the model can be utilized. Let’s suppose that there are n public goods and one private good in an economy, and in this application we can take: Yi = Rn+
for i = 1, . . . , m.
In a pure public goods model, which we will consider here, each consumer consumes the same amount of the public goods; and to express this in the present framework, we will take Y to be the set deﬁned by: m Yi  y 1 = y 2 = · · · = y m . Y = y∈ i=1
The production possibilities for public goods are then summarized by the cost function, c(·), which can then be viewed as expressing the quanitity of the private good which must be used as an input to produce the public goods vector y (which in this
16.2. A Simple Model
439
context can be viewed as an element of Rn+ ). Finally, in this particular interpretation, we would probably want to take w i = 0, for each i. Now, returning to the generic model, if we assume that Y is compact,1 that vi : Yi → R is continuous, for i = 1, . . . , m and that c is continuous as well, then there exists y ∗ ∈ Y satisfying:2 m m vi (y ∗i ) − c(y ∗ ) ≥ vi (y i ) − c(y). (16.1) ∀y ∈ Y : i=1
i=1
This fact provides the motivation for the following. 16.1 Proposition. If y ∗ ∈ Y satisﬁes (16.1), and if c(y ∗ ) ≤ w, ¯ then for all wi∗ ∈ W satisfying: m wi∗ = w ¯ − c(y ∗ ), (16.2) i=1
the allocation (wi∗ , y ∗i ) is strongly Pareto eﬃcient for E. Proof. Obviously such an allocation is feasible for E. To complete our proof, suppose, by way of obtaining a contradiction, that there exists (wi , y i ) ∈ A(E) such that: (16.3) ui (wi , y i ) ≥ ui (wi∗ , y ∗i ) for i = 1, . . . , m, and, for some j ∈ {1, . . . , m}: uj (wj , y j ) > uj (wj∗ , y ∗j ).
(16.4)
Adding (16.3) and (16.4) over i, we obtain: m m ui (wi , y i ) > ui (wi∗ , y ∗i ); i=1
i=1
which, making use of the quasilinear form of the utility functions and equation (16.2), implies: m m m vi (y i ) + wi > vi (y ∗i ) + w ¯ − c(y ∗ ). (16.5) i=1
i=1
i=1
However, since (wi , y i ) is feasible for E, we have: m w ¯− wi − c(y) ≥ 0, i=1
and thus from (16.5), we obtain: m i=1
so that:
vi (y i ) +
m
wi + w ¯−
i=1
m i=1
m
wi − c(y) >
i=1
vi (y i ) − c(y)