Mathematics for Economics

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1 Mathematics for Economics third edition Michael Hoy John Livernois Chris McKenna Ray Rees Thanasis Stengos The MIT Press Cambridge, Massachusetts London, England

2 c 2011 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 Helvetica and Times Roman by Westchester Book Group. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Mathematics for economics / Michael Hoy... [et al.]. 3rd ed. p. cm. Includes index. ISBN (hbk. : alk. paper) ISBN (pbk. : alk. paper) 1. Economics, Mathematical. I. Hoy, Michael, 1953 Sept. 22 HB135. M dc

3 Contents Preface Part I xiii Introduction and Fundamentals Chapter 1 Introduction What Is an Economic Model? How to Use This Book Conclusion 9 Chapter 2 Review of Fundamentals Sets and Subsets Numbers Some Properties of Point Sets in R n Functions 41 Chapter 3 Sequences, Series, and Limits Definition of a Sequence Limit of a Sequence Present-Value Calculations Properties of Sequences Series 84 Part II Univariate Calculus and Optimization Chapter 4 Continuity of Functions Continuity of a Function of One Variable Economic Applications of Continuous and Discontinuous Functions 113

4 viii CONTENTS Chapter 5 The Derivative and Differential for Functions of One Variable Definition of a Tangent Line Definition of the Derivative and the Differential Conditions for Differentiability Rules of Differentiation Higher Order Derivatives: Concavity and Convexity of a Function Taylor Series Formula and the Mean-Value Theorem 185 Chapter 6 Optimization of Functions of One Variable Necessary Conditions for Unconstrained Maxima and Minima Second-Order Conditions Optimization over an Interval 220 Part III Linear Algebra Chapter 7 Systems of Linear Equations Solving Systems of Linear Equations Linear Systems in n-variables 250 Chapter 8 Matrices General Notation Basic Matrix Operations Matrix Transposition Some Special Matrices 293 Chapter 9 Determinants and the Inverse Matrix Defining the Inverse Obtaining the Determinant and Inverse of a 3 3 Matrix The Inverse of an n n Matrix and Its Properties Cramer s Rule 329 Chapter 10 Some Advanced Topics in Linear Algebra Vector Spaces The Eigenvalue Problem Quadratic Forms 378

5 CONTENTS ix Part IV Multivariate Calculus Chapter 11 Calculus for Functions of n-variables Partial Differentiation Second-Order Partial Derivatives The First-Order Total Differential Curvature Properties: Concavity and Convexity More Properties of Functions with Economic Applications Taylor Series Expansion* 464 Chapter 12 Optimization of Functions of n-variables First-Order Conditions Second-Order Conditions Direct Restrictions on Variables 491 Chapter 13 Constrained Optimization Constrained Problems and Approaches to Solutions Second-Order Conditions for Constrained Optimization Existence, Uniqueness, and Characterization of Solutions 520 Chapter 14 Comparative Statics Introduction to Comparative Statics General Comparative-Statics Analysis The Envelope Theorem 554 Chapter 15 Concave Programming and the Kuhn-Tucker Conditions The Concave-Programming Problem Many Variables and Constraints 575 Part V Integration and Dynamic Methods Chapter 16 Integration The Indefinite Integral The Riemann (Definite) Integral Properties of Integrals 605

6 x CONTENTS 16.4 Improper Integrals Techniques of Integration 623 Chapter 17 An Introduction to Mathematics for Economic Dynamics Modeling Time 634 Chapter 18 Linear, First-Order Difference Equations Linear, First-Order, Autonomous Difference Equations The General, Linear, First-Order Difference Equation 656 Chapter 19 Nonlinear, First-Order Difference Equations The Phase Diagram and Qualitative Analysis Cycles and Chaos 673 Chapter 20 Linear, Second-Order Difference Equations The Linear, Autonomous, Second-Order Difference Equation The Linear, Second-Order Difference Equation with a Variable Term 708 Chapter 21 Linear, First-Order Differential Equations Autonomous Equations Nonautonomous Equations 731 Chapter 22 Nonlinear, First-Order Differential Equations Autonomous Equations and Qualitative Analysis Two Special Forms of Nonlinear, First-Order Differential Equations 748 Chapter 23 Linear, Second-Order Differential Equations The Linear, Autonomous, Second-Order Differential Equation The Linear, Second-Order Differential Equation with a Variable Term 772 Chapter 24 Simultaneous Systems of Differential and Difference Equations Linear Differential Equation Systems Stability Analysis and Linear Phase Diagrams Systems of Linear Difference Equations 825

7 CONTENTS xi Chapter 25 Optimal Control Theory The Maximum Principle Optimization Problems Involving Discounting Alternative Boundary Conditions on x(t ) Infinite Time Horizon Problems Constraints on the Control Variable Free-Terminal-Time Problems (T Free) 909 Answers 921 Index 953

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