Foundations of Mathematical Economics

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FOUNDATIONS OF MATHEMATICAL ECONOMICSMichael CarterThe MIT PressCambridge, MassachusettsLondon, England(2001 Massachusetts Institute of TechnologyAll rights reserved. No part of this book may be reproduced in any form by any electronic ormechanical means (including photocopying, recording, or information storage and retrieval)without permission in writing from the publisher.This book was set in Times New Roman in `3B2' by Asco Typesetters, Hong Kong.Printed and bound in the United States of America.Library of Congress Cataloging-in-Publication DataCarter, Michael, 1950Foundations of mathematical economics / Michael Carter.p. cm.Includes bibliographical references and index.ISBN 0-262-03289-9 (hc. : alk. paper) ISBN 0-262-53192-5 (pbk. : alk. paper)1. Economics, Mathematical. I. Title.HB135 .C295 2001330/.01/51dc21 2001030482To my parents, Merle and Maurice Carter, who provided a rmfoundation for lifeContentsIntroduction xiA Note to the Reader xvii1 Sets and Spaces 11.1 Sets 11.2 Ordered Sets 91.2.1 Relations 101.2.2 Equivalence Relations and Partitions 141.2.3 Order Relations 161.2.4 Partially Ordered Sets and Lattices 231.2.5 Weakly Ordered Sets 321.2.6 Aggregation and the Pareto Order 331.3 Metric Spaces 451.3.1 Open and Closed Sets 491.3.2 Convergence: Completeness and Compactness 561.4 Linear Spaces 661.4.1 Subspaces 721.4.2 Basis and Dimension 771.4.3 Ane Sets 831.4.4 Convex Sets 881.4.5 Convex Cones 1041.4.6 Sperner's Lemma 1101.4.7 Conclusion 1141.5 Normed Linear Spaces 1141.5.1 Convexity in Normed Linear Spaces 1251.6 Preference Relations 1301.6.1 Monotonicity and Nonsatiation 1311.6.2 Continuity 1321.6.3 Convexity 1361.6.4 Interactions 1371.7 Conclusion 1411.8 Notes 1422 Functions 1452.1 Functions as Mappings 1452.1.1 The Vocabulary of Functions 1452.1.2 Examples of Functions 1562.1.3 Decomposing Functions 1712.1.4 Illustrating Functions 1742.1.5 Correspondences 1772.1.6 Classes of Functions 1862.2 Monotone Functions 1862.2.1 Monotone Correspondences 1952.2.2 Supermodular Functions 1982.2.3 The Monotone Maximum Theorem 2052.3 Continuous Functions 2102.3.1 Continuous Functionals 2132.3.2 Semicontinuity 2162.3.3 Uniform Continuity 2172.3.4 Continuity of Correspondences 2212.3.5 The Continuous Maximum Theorem 2292.4 Fixed Point Theorems 2322.4.1 Intuition 2322.4.2 Tarski Fixed Point Theorem 2332.4.3 Banach Fixed Point Theorem 2382.4.4 Brouwer Fixed Point Theorem 2452.4.5 Concluding Remarks 2592.5 Notes 2593 Linear Functions 2633.1 Properties of Linear Functions 2693.1.1 Continuity of Linear Functions 2733.2 Ane Functions 2763.3 Linear Functionals 2773.3.1 The Dual Space 2803.3.2 Hyperplanes 2843.4 Bilinear Functions 2873.4.1 Inner Products 2903.5 Linear Operators 2953.5.1 The Determinant 2963.5.2 Eigenvalues and Eigenvectors 2993.5.3 Quadratic Forms 302viii Contents3.6 Systems of Linear Equations and Inequalities 3063.6.1 Equations 3083.6.2 Inequalities 3143.6.3 InputOutput Models 3193.6.4 Markov Chains 3203.7 Convex Functions 3233.7.1 Properties of Convex Functions 3323.7.2 Quasiconcave Functions 3363.7.3 Convex Maximum Theorems 3423.7.4 Minimax Theorems 3493.8 Homogeneous Functions 3513.8.1 Homothetic Functions 3563.9 Separation Theorems 3583.9.1 Hahn-Banach Theorem 3713.9.2 Duality 3773.9.3 Theorems of the Alternative 3883.9.4 Further Applications 3983.9.5 Concluding Remarks 4153.10 Notes 4154 Smooth Functions 4174.1 Linear Approximation and the Derivative 4174.2 Partial Derivatives and the Jacobian 4294.3 Properties of Dierentiable Functions 4414.3.1 Basic Properties and the Derivatives of ElementaryFunctions 4414.3.2 Mean Value Theorem 4474.4 Polynomial Approximation 4574.4.1 Higher-Order Derivatives 4604.4.2 Second-Order Partial Derivatives and the Hessian 4614.4.3 Taylor's Theorem 4674.5 Systems of Nonlinear Equations 4764.5.1 The Inverse Function Theorem 4774.5.2 The Implicit Function Theorem 4794.6 Convex and Homogeneous Functions 4834.6.1 Convex Functions 483ix Contents4.6.2 Homogeneous Functions 4914.7 Notes 4965 Optimization 4975.1 Introduction 4975.2 Unconstrained Optimization 5035.3 Equality Constraints 5165.3.1 The Perturbation Approach 5165.3.2 The Geometric Approach 5255.3.3 The Implicit Function Theorem Approach 5295.3.4 The Lagrangean 5325.3.5 Shadow Prices and the Value Function 5425.3.6 The Net Benet Approach 5455.3.7 Summary 5485.4 Inequality Constraints 5495.4.1 Necessary Conditions 5505.4.2 Constraint Qualication 5685.4.3 Sucient Conditions 5815.4.4 Linear Programming 5875.4.5 Concave Programming 5925.5 Notes 5986 Comparative Statics 6016.1 The Envelope Theorem 6036.2 Optimization Models 6096.2.1 Revealed Preference Approach 6106.2.2 Value Function Approach 6146.2.3 The Monotonicity Approach 6206.3 Equilibrium Models 6226.4 Conclusion 6326.5 Notes 632References 635Index of Symbols 641General Index 643x ContentsIntroductionEconomics made progress without mathematics, but has made faster progress with it.Mathematics has brought transparency to many hundreds of economic arguments.Deirdre N. McCloskey (1994)Economists rely on models to obtain insight into a complex world. Eco-nomic analysis is primarily an exercise in building and analyzing models.An economic model strips away unnecessary detail and focuses attentionon the essential details of an economic problem. Economic models comein various forms. Adam Smith used a verbal description of a pin factoryto portray the principles of division of labor and specialization. IrvingFisher built a hydraulic model (comprising oating cisterns, tubes, andlevers) to illustrate general equilibrium. Bill Phillips used a dierenthydraulic model (comprising pipes and colored water) to portray the cir-cular ow of income in the national economy. Sir John Hicks developed asimple mathematical model (IS-LM) to reveal the essential dierencesbetween Keynes's General Theory and the ``classics.'' In modern eco-nomic analysis, verbal and physical models are seen to be inadequate.Today's economic models are almost exclusively mathematical.Formal mathematical modeling in economics has two key advantages.First, formal modeling makes the assumptions explicit. It claries intu-ition and makes arguments transparent. Most important, it uncovers thelimitations of our intuition, delineating the boundaries and uncoveringthe occasional counterintuitive special case. Second, the formal modelingaids communication. Once the assumptions are explicit, participantsspend less time arguing about what they really meant, leaving more timeto explore conclusions, applications, and extensions.Compare the aftermath of the publication of Keynes's General Theorywith that of von Neumann and Morgenstern's Theory of Games andEconomic Behavior. The absence of formal mathematical modeling in theGeneral Theory meant that subsequent scholars spent considerable energydebating ``what Keynes really meant.'' In contrast, the rapid developmentof game theory in recent years owes much to the advantages of formalmodeling. Game theory has attracted a predominance of practitionerswho are skilled formal modelers. As their assumptions are very explicit,practitioners have had to spend little time debating the meaning of others'writings. Their eorts have been devoted to exploring ramications andapplications. Undoubtedly, formal modeling has enhanced the pace ofinnovation in game-theoretic analysis in economics.Economic models are not like replica cars, scaled down versions of thereal thing admired for their verisimilitude. A good economic model stripsaway all the unnecessary and distracting detail and focuses attention onthe essentials of a problem or issue. This process of stripping awayunnecessary detail is called abstraction. Abstraction serves the same rolein mathematics. The aim of abstraction is not greater generality butgreater simplicity. Abstraction reveals the logical structure of the mathe-matical framework in the same way as it reveals the logical structure of aneconomic model.Chapter 1 establishes the framework by surveying the three basicsources of structure in mathematics. First, the order, geometric and alge-braic structures of sets are considered independently. Then their interac-tion is studied in subsequent sections dealing with normed linear spacesand preference relations.Building on this foundation, we study mappings between sets or func-tions in chapters 2 and 3. In particular, we study functions that preservethe structure of the sets which they relate, treating in turn monotone,continuous, and linear functions. In these chapters we meet the threefundamental theorems of mathematical economicsthe (continuous)maximum theorem, the Brouwer xed point theorem, and the separatinghyperplane theorem, and outline many of their important applications ineconomics, nance, and game theory.A key tool in the analysis of economic models is the approximation ofsmooth functions by linear and quadratic fun