04 Foundations-of-mathematical-economics

FOUNDATIONS OF MATHEMATICAL ECONOMICS

Michael Carter

The MIT Press Cambridge, Massachusetts London, England

( 2001 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical 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 Data Carter, Michael, 1950± Foundations of mathematical economics / Michael Carter. p. cm. Includes bibliographical references and index. ISBN 0-262-03289-9 (hc. : alk. paper) Ð ISBN 0-262-53192-5 (pbk. : alk. paper) 1.Economics, Mathematical. I. Title. HB135 .C295 2001 2001030482 330 H .01 H 51Ðdc21

To my parents, Merle and Maurice Carter, who provided a ®rm foundation for life

Contents

Introduction A Note to the Reader 1 Sets and Spaces 1.1 Sets 1.2 Ordered Sets 1.2.1 Relations 1.2.2 Equivalence Relations and Partitions 1.2.3 Order Relations 1.2.4 Partially Ordered Sets and Lattices 1.2.5 Weakly OrderedSets 1.2.6 Aggregation and the Pareto Order 1.3 Metric Spaces 1.3.1 Open and Closed Sets 1.3.2 Convergence: Completeness and Compactness 1.4 Linear Spaces 1.4.1 Subspaces 1.4.2 Basis and Dimension 1.4.3 A½ne Sets 1.4.4 Convex Sets 1.4.5 Convex Cones 1.4.6 Sperner's Lemma 1.4.7 Conclusion 1.5 Normed Linear Spaces 1.5.1 Convexity in Normed Linear Spaces 1.6 Preference Relations 1.6.1 Monotonicity andNonsatiation 1.6.2 Continuity 1.6.3 Convexity 1.6.4 Interactions 1.7 Conclusion 1.8 Notes Functions 2.1 Functions as Mappings 2.1.1 The Vocabulary of Functions

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2.1.2 Examples of Functions 2.1.3 Decomposing Functions 2.1.4 Illustrating Functions 2.1.5Correspondences 2.1.6 Classes of Functions 2.2 Monotone Functions 2.2.1 Monotone Correspondences 2.2.2 Supermodular Functions 2.2.3 The Monotone Maximum Theorem 2.3 Continuous Functions 2.3.1 Continuous Functionals 2.3.2 Semicontinuity 2.3.3 Uniform Continuity 2.3.4 Continuity of Correspondences 2.3.5 The Continuous Maximum Theorem 2.4 Fixed Point Theorems 2.4.1 Intuition 2.4.2 Tarski Fixed PointTheorem 2.4.3 Banach Fixed Point Theorem 2.4.4 Brouwer Fixed Point Theorem 2.4.5 Concluding Remarks 2.5 Notes 3 Linear Functions 3.1 Properties of Linear Functions 3.1.1 Continuity of Linear Functions 3.2 A½ne Functions 3.3 Linear Functionals 3.3.1 The Dual Space 3.3.2 Hyperplanes 3.4 Bilinear Functions 3.4.1 Inner Products 3.5 Linear Operators 3.5.1 The Determinant 3.5.2 Eigenvalues andEigenvectors 3.5.3 Quadratic Forms

156 171 174 177 186 186 195 198 205 210 213 216 217 221 229 232 232 233 238 245 259 259 263 269 273 276 277 280 284 287 290 295 296 299 302

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3.6 Systems of Linear Equations and Inequalities 3.6.1 Equations 3.6.2 Inequalities 3.6.3 Input±Output Models 3.6.4 Markov Chains 3.7 Convex Functions 3.7.1 Properties of Convex Functions 3.7.2 QuasiconcaveFunctions 3.7.3 Convex Maximum Theorems 3.7.4 Minimax Theorems 3.8 Homogeneous Functions 3.8.1 Homothetic Functions 3.9 Separation Theorems 3.9.1 Hahn-Banach Theorem 3.9.2 Duality 3.9.3 Theorems of the Alternative 3.9.4 Further Applications 3.9.5 Concluding Remarks 3.10 Notes 4 Smooth Functions 4.1 Linear Approximation and the Derivative 4.2 Partial Derivatives and the Jacobian 4.3 Properties ofDi¨erentiable Functions 4.3.1 Basic Properties and the Derivatives of Elementary Functions 4.3.2 Mean Value Theorem 4.4 Polynomial Approximation 4.4.1 Higher-Order Derivatives 4.4.2 Second-Order Partial Derivatives and the Hessian 4.4.3 Taylor's Theorem 4.5 Systems of Nonlinear Equations 4.5.1 The Inverse Function Theorem 4.5.2 The Implicit Function Theorem 4.6 Convex and Homogeneous Functions...