General Equilibrium and Welfare Economics James C

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 47907-2076 USA [email protected]

ISBN-10 3-540-31407-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-31407-3 Springer Berlin Heidelberg New York

Cataloging-in-Publication 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 illus- trations, 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 Springer-Verlag. 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. Hardcover-Design: WMXDesign GmbH, Heidelberg SPIN 11660033 42/3153-5 4 3 2 1 0 ± Printed on acid-free paper To Donna, Donovan, Brian, Jerry, Linda, Ted, Julie, and Bradley,

...andtotheUniversityofMinnesotaFaculty who taught me General Equilibrium and Welfare Economics: John Chipman, Leo Hurwicz, Ket Richter, and Hugo Sonnenschein,

this book is most aﬀectionately, and respect- fully, 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 hand-outs 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 classroom-tested 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 Mas-Colell, 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 first-year 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 set-theoretic 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! viii Preface

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 n-dimensional 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),

th with ‘xj’denotingitsj 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:

n n nonnegative orthant: R+ = {x ∈ R | x ≥ 0} n n semipositive orthant: R+ \{0} = {x ∈ R | x > 0}, and n n strictly positive orthant: R++ = {x ∈ R | x 0}, Preface ix where ‘0’ denotes the origin in Rn, and we use the symbol ‘\’ to denote set-theoretic diﬀerence; that is: A \ B = {x ∈ A | x/∈ B}. n Since we will often be considering ordered pairs, for example, (p,w) ∈ R++ ×R+, n where p ∈ R++ 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

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 J. C. M. June, 2006 Contents

Preface vii

1 An Introduction to Preference Theory 1 1.1Introduction...... 1 1.2BinaryRelationsandOrderings...... 2 1.3 Preference Relations and Utility Functions ...... 11

2 Algebraic Choice Theory 21 2.1Introduction...... 21 2.2TheGeneralAlgebraicTheoryofChoice...... 22 2.3SomeCriticismsoftheModel...... 25 2.4 Stated Preferences versus Actual Choices ...... 28 2.5TheSpecificationofthePrimitiveTerms...... 30 2.6 Weak Separability of Preferences ...... 33 2.7 Additive Separability ...... 39 2.8SequentialConsumptionPlans...... 41 2.9TheBPLExperimentReconsidered...... 44 2.10 Probabilistic Theories of Choice ...... 45 2.11 Are Preferences Total? ...... 47 2.12ArePreferencesTransitive?...... 50 2.12.1 ‘Just Noticeable Difference,’ or ‘Threshold Effects’ ...... 50 2.12.2 Decision Rules Based On Qualitative Information ...... 51 2.12.3PrioritiesandMeasurementErrors...... 52 2.12.4 Group Decisions: The Dr. Jekyll and Ms. Jekyll Problem . . 53 2.13AsymmetricOrders...... 54

3 Revealed Preference Theory 59 3.1Introduction...... 59 3.2ChoiceCorrespondencesandBinaryRelations...... 59 3.3 Regular-Rational Choice Correspondences ...... 64 3.4 Representable Choice Correspondences ...... 67 3.5PreferencesandObservedDemandBehavior...... 70 3.6TheImplicationsofAsymmetricOrders*...... 77

xi xii CONTENTS

4 Consumer Demand Theory 85 4.1Introduction...... 85 4.2TheConsumptionSet...... 85 4.3DemandCorrespondences...... 88 4.4TheBudgetBalanceCondition...... 90 4.5SomeConvexityConditions...... 94 4.6Wold’sTheorem...... 96 4.7 Indirect Preferences and Indirect Utility ...... 97 4.8HomotheticPreferences...... 104 4.9 Cost-of-Living Indices ...... 108 4.10Consumer’sSurplus...... 111 4.11Appendix...... 125

5 Pure Exchange Economies 131 5.1Introduction...... 131 5.2TheBasicFramework...... 131 5.3TheEdgeworthBoxDiagram...... 133 5.4DemandandExcessDemandCorrespondences...... 138 5.5ParetoEﬃciency...... 142 5.6 Pareto Eﬃciency and ’Non-Wastefulness’ ...... 150

6 Production Theory 155 6.1Introduction...... 155 6.2BasicConceptsofProductionTheory...... 155 6.3LinearProductionSets...... 161 6.4 Input-Output Analysis ...... 166 6.5ProﬁtMaximization...... 171 6.6ProﬁtMaximizingwithConstantReturnstoScale*...... 176 6.7 Production in General Equilibrium Theory ...... 178 6.8 Activity Analysis* ...... 182

7 Fundamental Welfare Theorems 191 7.1Introduction...... 191 7.2 Competitive Equilibrium with Production ...... 191 7.3SomeDiagrammaticTechniques...... 196 7.4Walras’LawwithProduction...... 201 7.5 The ‘First Fundamental Theorem’ ...... 205 7.6 ‘Unbiasedness’ of the Competitive Mechanism ...... 211 7.7AStrongerVersionof‘TheSecondTheorem’...... 219

8 The Existence of Competitive Equilibrium 227 8.1Introduction...... 227 8.2Examples,Part1...... 229 8.3 Assumption (c) and the Attainable Set ...... 235 8.4TheGaleandMas-ColellTheorem...... 239 8.5 An (Especially) Simple Existence Theorem ...... 241 CONTENTS xiii

8.6Appendix...... 244

9 Examples of General Equilibrium Analyses 249 9.1Introduction...... 249 9.2OptimalCommodityTaxation:InitialFormulation...... 249 9.3AReconsiderationoftheProblem...... 252 9.4TheSimplestModelofOptimalCommodityTaxation...... 255 9.5SomeResults...... 256 9.6OptimalIncomeTaxation...... 259 9.7 Monopoly in a General Equilibrium Model ...... 269 9.8 Money in a General Equilibrium Model ...... 272 9.9 Indivisible Commodities ...... 276

10 Comparative Statics and Stability 281 10.1Introduction...... 281 10.2 Aggregate Excess Demand ...... 282 10.3The‘LawofDemand’...... 286 10.4GrossSubstitutes...... 291 10.5 Qualitative Economics ...... 294 10.6 Stability in a Single Market ...... 298 10.7 Multi-Market Stability ...... 302 10.8 A Note on Non-Tˆatonnement Processes ...... 307

11 The Core of an Economy 311 11.1Introduction...... 311 11.2ConvexityandtheAttainableConsumptionSet...... 314 11.3TheCoreofaProductionEconomy...... 317 11.4TheCoreinReplicatedEconomies...... 320 11.5EqualTreatment...... 329 11.6Appendix...... 330

12 General Equilibrium with Uncertainty 333 12.1Introduction...... 333 12.2Arrow-DebreuContingentCommodities...... 333 12.3 Radner Equilibrium ...... 339 12.4CompleteMarkets...... 345 12.5CompleteMarketsandEﬃciency...... 350 12.6ConcludingNotes...... 355

13 Further Topics 359 13.1Introduction...... 359 13.2TimeintheBasicModel...... 359 13.3AnInﬁniteTimeHorizon...... 366 13.4OverlappingGenerations...... 369 13.5AContinuumofTraders...... 372 13.6SuggestionsforFurtherReading...... 379 xiv CONTENTS

14 Social Choice and Voting Rules 383 14.1Introduction...... 383 14.2TheBasicSetting...... 384 14.3VotingRules...... 387 14.4 Arrow’s General Possibility Theorem ...... 392 14.5Appendix.AMoreSophisticatedBordaCount...... 402

15 Some Tools of Applied Welfare Analysis 407 15.1Introduction...... 407 15.2TheFramework...... 408 15.3MeasurementFunctions...... 409 15.4SocialPreferenceFunctions...... 411 15.5TheCompensationPrinciple...... 416 15.6 Indirect Preferences: Individual and Social ...... 420 15.7 Measures of Real National Income ...... 422 15.8Consumers’Surplus...... 428

16 Public Goods 437 16.1Introduction...... 437 16.2ASimpleModel...... 437 16.3PublicGoods...... 441 16.4ASimplePublicGoodsModel...... 442 16.5 Lindahl and Ratio Equilibria ...... 446 16.6 The ‘Fundamental Theorems’ for Lindahl Equilibria ...... 455 16.6.1 The ‘First Fundamental Theorem’ ...... 456 16.6.2 The ‘Second Fundamental Theorem’ ...... 458 16.6.3 The ‘Metatheorem’ ...... 462

17 Externalities 467 17.1Introduction...... 467 17.2Externalities:AFirstLook...... 468 17.3ExtendingtheModel...... 475 17.4The‘CoaseTheorem’...... 480 17.5LindahlandExternalities...... 484 17.6Postscript...... 487

18 Incentives and Implementation Theory 489 18.1Introduction...... 489 18.2GameFormsandMechanisms...... 490 18.3TheGibbard-SatterthwaiteTheorem...... 495 18.4 Implementation Theory ...... 502 18.5 Single-Peaked Preferences and Dominant Strategies ...... 504 18.5.1Single-PeakedPreferences...... 504 18.5.2TheBowenModel...... 509 18.6Quasi-LinearityandDominantStrategies...... 511 18.7 Implementation in Nash Equilibria ...... 520 CONTENTS xv

18.8 Nash Implementation with Public Goods ...... 522 18.9TheRevelationPrincipleReconsidered...... 525 18.10NotesandSuggestionsforFurtherReading...... 527

19 Appendix. Solutions for Selected Exercises 531 19.1Chapter1...... 531 19.2Chapter2...... 533 19.3Chapter3...... 533 19.4Chapter4...... 536 19.5Chapter5...... 537 19.6Chapter6...... 540 19.7Chapter7...... 542 19.8Chapter8...... 543 19.9Chapter9...... 548 19.10Chapter10...... 548 19.11Chapter11...... 548 19.12Chapter12...... 550 19.13Chapter13...... 551 19.14Chapter14...... 551 19.15Chapter15...... 551 19.16Chapter16...... 551 19.17Chapter17...... 552 19.18Chapter18...... 553

References 555

Author Index 569

Subject Index 573