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Problems Book to AccompanyMathematics for Economists Tamara Todorova JOHN WILEY & SONS, INC. ” Library of Congress Cataloging in Publication Data: ISBN-13 978-0-470-59181-9 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ii . please contact your local representative. recording. Return instructions and a free of charge return shipping label are available at www. All rights reserved.wiley. please return the evaluation copy to Wiley. These copies are licensed and may not be sold or transferred to a third party. (201)748-6011.ACQUISITIONS EDITOR MARKETING MANAGER EDITORIAL ASSISTANT SENIOR DESIGNER MEDIA EDITOR PRODUCTION MANAGER PRODUCTION EDITOR Lacey Vitetta Diane Mars Emily McGee Jim O’Shea Lauren Sapira Micheline Frederick Amy Weintraub This book was printed and bound by Courier Westford. “Evaluation copies are provided to qualified academics and professionals for review purposes only. electronic. NJ 07030-5774. scanning or otherwise. fax (201)748-6008. website http://www. Outside of the United States. photocopying. Hoboken. Inc. stored in a retrieval system or transmitted in any form or by any means. except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act.com/go/returnlabel.wiley. MA 01923.com/go/permissions.. without either the prior written permission of the Publisher. or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center. website www. 111 River Street. The cover was printed by Courier Westford. John Wiley & Sons. Requests to the Publisher for permission should be addressed to the Permissions Department. Upon completion of the review period. Inc.com. This book is printed on acid free paper. 222 Rosewood Drive. ∞ Copyright © 2011 John Wiley & Sons. mechanical. Danvers.copyright. Inc. for use in their courses during the next academic year. No part of this publication may be reproduced. Advanced Differential and Difference Equations 615 electronic website: www. Comparative Statics I: Partial Derivatives 3. Comparative Statics II: Differentials 4. Simple Difference Equations vi viii 1 67 125 202 265 311 375 464 512 579 11. Constrained Optimization 8.wiley.com/college/todorova iii . Functions of More than One Variable 7. Introduction to Dynamic Optimization: The Calculus of Variations electronic website: 701 www. Simple Differential Equations 10.Table of Contents Preface Suggested Readings 1. Matrix Algebra 2.wiley. Optimization of Functions of One Variable 5. Exponential and Logarithmic Functions 6. Integration 9.com/college/todorova 12. problems iv . IS-LM model. total derivatives of composite functions. CES production function. inequality constraints. cubic cost function. Cobb-Douglas production function. utility maximization. rules of integration. problems Chapter 6: Functions of More than One Variable – functions of two variables. definite integrals. input decisions of a firm. discounting. implicit-function rule. Gauss method. determinants. conditions for optimization without constraints. differentiation of functions of more than one variable. rate of growth of composite functions. instantaneous rate of growth. n-variable quadratic forms. multiproduct and multimarket firm. n-variable and multiconstraint case. homogeneous functions. base conversion. interest compounding. the number e . Leontief input-output model. derivatives of exponential and logarithmic functions. probability and definite integrals. logarithmic functions. partial and total differentials. Domar growth model. market equilibrium analysis. transposes and inverses. problems Chapter 4: Optimization of Functions of One Variable – first-derivative test. matrix inversion. improper integrals. problems Chapter 8: Integration – indefinite integral. profit maximization. Keynesian national-income model. partial derivatives. secondderivative test. nonsingularity of a matrix. implicit-function theorem. Jacobian determinants. logarithms. conditions for optimization with constraints. inverse-function rule. rules of differentials. problems Chapter 3: Comparative Statics II: Differentials – point elasticity. elasticity of substitution. problems Chapter 2: Comparative Statics I: Partial Derivatives – derivative. cubic profit function. investment and capital formation. functions of three and more variables. Slutsky equation. rules of logarithms. present value of a cash flow. consumer and producer surplus. point elasticity revisited. optimal timing. rules of differentiation.Detailed Table of Contents Chapter 1: Matrix Algebra – linear models. Cramer’s rule. input decisions and comparative static aspects of optimization. problems Chapter 7: Constrained Optimization – Lagrange-multiplier method. n-th derivative test for relative extremum. problems Chapter 5: Exponential and Logarithmic Functions – exponential functions. general-function models – market equilibrium model. characteristic-root test for sign definiteness. constrained dynamic optimization. higher-order difference equations. a glimpse of optimal control theory. dynamic stability of equilibrium. problems Chapter 12: Introduction to Dynamic Optimization: The Calculus of Variations – introduction to dynamic optimization. problems Chapter 10: Simple Difference Equations – first-order difference equations. integrating factors. market equilibrium with price expectations. Bernoulli equations. exact differential equations. separation of variables. Ramsey growth model. multiplier-accelerator model. problems Chapter 11: Advanced Differential and Difference Equations – second-order linear differential equations. phase diagrams. problems v .Chapter 9: Simple Differential Equations – first-order linear differential equations with constant coefficient and constant term. general method of solving. the Arrow-Pratt measure of risk aversion. inflation and unemployment in discrete time. the Cobweb model. second-order difference equations. dynamic model of market price. Euler’s equation. higher-order differential equations. the case of variable coefficient and variable term. cost-minimizing firm. the relationship between inflation and unemployment. Solow growth model. the calculus of variations. many problems can be seen as simple economic models that challenge student thinking and knowledge. short theoretical summaries of the relevant theory and descriptions of economic applications are provided in the beginning of each chapter so as to introduce students to the essentials of the subject. it is not perfectly suited for them. mathematical economics for undergraduate students. with no claim to originality on my part except in some sections. However.and macroeconomics are suitable prerequisites for this book. the problems book is a practice text illustrating a large number of mathematical tools to be applied in economics. Students from such areas as business and finance interested in applying various quantitative tools to their fields of study may also use the book. courses in intermediate micro. I regard the problems and solutions as the essence of the book and the justification for it. While the problems book may be used by beginning graduate students of economics. which I have tried to make as wideranging as possible. students might use the book as a reference guide to refresh some lost knowledge. at this advanced level. This book reduces the intimidation with math and helps make economics a more interesting and approachable subject. Rather. While these books are predominantly theoretical texts. particularly amid a global financial and economic crisis. it is not a perfect substitute for a book in applied mathematics for business or financial mathematics. the book is appropriate for the master’s level and perhaps only the beginning of a doctoral program in economics. While most undergraduate textbooks in the area stress the math theory and illustrate it with some economic applications. An essential aim of the book is to popularize the subject of mathematical economics and make it more accessible to beginning economists. herein. which will enable students to reinforce and strengthen the theory they have learnt from standard textbooks in mathematical economics. such as Mathematics for Economists by Carl Simon and Laurence Blume and Fundamental Methods of Mathematical Economics by Alpha Chiang. My students have had no substantial learning aids or tools. Students of economics lack robust homework-type problems with which to test their knowledge. and the rest from macroeconomics. The book is designed for students of economics who are taking mathematical economics as part of their education. this book is as much an economics book as it is a math book. The theory is only meant to orient the readers before they plunge into the problems. Hard-core mathematics bordering on the graduate level scares a great many students and deters them from studying economics. The book supplements any of the most widely used texts. although many of the problems in this book are drawn from such a course. About 70 percent of the economics content.Preface This supplement is intended to complement any course in quantitative methods in economics. A first course in calculus is desirable as well. Yet. and no problem is purely mathematical without relevance to economics. The key purpose is to provide a variety of problems with fully worked out solutions. All problems are economic applications. fewer are eager to pursue economics because of the math they would have to do as economics majors. In fact. Therefore. these theoretical summaries are only prefatory material. secondary to the main purpose of the book. Problems Book to Accompany Mathematics for Economists was written as a response to an insufficiency in the field of mathematical economics of appropriate study tools and exercise books. For completeness. and hence have vi . As a professor I have found that to be a serious handicap in my teaching. is drawn from microeconomics. While a large number of students around the world are intrigued by economics and finance. or introductory mathematical economics for graduate students. Thus. Neither is it a substitute for a course in intermediate microeconomics. The last part of the book discusses differential and difference equations. Emily McGee. they would be tempted to follow the solution passively and without much understanding. Students can try to solve a problem without looking at its solution – this is what I call active solving. Brian Gross (University of California. thus. I wish to thank my family. Only afterward should they look up the solution and compare it with their own results. Mina Baliamoune-Lutz (University of North Florida). as well as for some beginning graduate students who want to brush up on their knowledge of differential calculus. Yordanka Kostova and Martin Milev meticulously solved and corrected problems for me. Alex was the first to write those in Microsoft Equation. There is no stress on sets and vectors. and Wendy Wu (Wilfrid Laurier University). Berkeley).asked for a book that gives exercises at the level of exams in the course. The book covers the essential parts of theory in the simplest possible terms. and integral calculus. These last topics are discussed in bonus chapters located at the Wiley website www. Special thanks go to Amy Weintraub who was my Production Editor and who followed her deadline strictly. especially in moments of distress and disappointment. Rossitza Wooster (Portland State University). Tamara Todorova American University in Bulgaria e-mail: ttodorova@aubg. There is also introduction to basic dynamic optimization. who grew along with the book and was delivered several months before the manuscript was complete. they are included for those students taking rigorous math courses or double-majoring in mathematics and economics. Wing Suen (University of Hong Kong). who gave me the idea of writing my problems electronically. I am also truly grateful to three work-study students of mine. George Lobell. His skilful guidance and editing of the book helped me overcome numerous problems and my inexperience as an author. starting with first-order equations and moving to higher-order ones. and Amy Weintraub from WileyBlackwell for coordinating the process of book preparation and assisting my work. He was always there to advise me and encourage me. who believed in this project and gave me a helping hand. Lacey Vitetta.bg vii . for example. but there is extensive coverage of matrix algebra. my mom and Iliya. and there is rarely time to discuss them in a one-semester course. The problems are easy for students to work with as they closely follow the theory covered in the beginning of each chapter. Problems are arranged thematically. with a focus on the calculus of variations and its parallel with optimal control theory. This ultimately brought me to the idea of writing the book. In terms of content. sophisticated mathematical proofs are omitted. unconstrained and constrained optimization. I dedicate this book to my daughter Tatyana. exponential functions. These last topics are more challenging for undergrads. This resulted in the design of a large number of problems. I am also thankful to Constance Adler. starting with the easier ones and moving to the more challenging ones. I am indebted to John Drexel for proofreading a manuscript full of equations and variables. Mak Arvin (Trent University. The students from the last four years helped me the most as the book was tested on them numerous times. who had patience in the eternal writing of the book. I should also mention that all my students taking Quantitative Methods in Economics with me at the American University in Bulgaria receive my appreciation. Many problems expand on the economic models and applications provided in the theoretical part. Kareem Ismail (Johns Hopkins University). Yet.wiley. Otherwise. I thank my editor.com/college/todorova. Monica Costa Dias (University College London). the problems book covers the most essential applications of mathematics in economics that are key to undergraduate learning. Canada). one of my first work-study students. Finally. logarithmic functions. I express my thanks to Alexandru Andronic. gave me much inspiration in writing the book. the product of years of work. the first for the early version of the book and the second for the later. who is my spiritual mentor and father. Bharath Ramachandran. I also thank the following reviewers whose helpful comments contributed to improvements in the book: Steven Antler (Roosevelt University). Michael D. and Wing Suen. in Schaum's Outline Series. Fort Worth. The Structure of Economics: A Mathematical Analysis. Chenery. 13. Domar.. 2. “A Contribution to the Theory of Economic Growth. 21. Paul. 1956): 65–94. 1996. Alban H. 1986. 2000. UK: Manchester University Press.Suggested Readings 1. Phillips. NY: Prometheus Books. and Employment. “The Relationship between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom. Oxford University Press. 19. NJ: Prentice Hall. Jeffrey. 9.” Economica (November 1958) 283–299. and Ngo Van Long. Bedient. M. 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Mathematical Methods and Models for Economists. New York: McGraw-Hill. Interest and Money. MA: Harvard University Press. 1998. Input-Output Economics. 5th ed. Carl. Simon. Olivier. 1957. “Capital Expansion. Earl David. Englewood Cliffs. 2000. Jacques. “Capital-Labor Substitution and Economic Efficiency. and Richard E. 15. Malcolm. Mathematics for Economists.. Minhas. Dowling. Alpha C. New York: W. 2nd ed. Bedient. Englewood Cliffs. 1992. Arrow. Robert M. Avinash. 5. 2006. Foundations of Economic Analysis. 3rd ed. W. 2001. Whinston. Leonard. de la Fuente. 1994. Optimal Control Theory and Static Optimization in Economics. New York: McGraw-Hill/Irwin. Solow. Angel. Edward T. 137–147. Manchester. 2000. Samuelson. 3. 17. Mathematical Economics. Boston. Cambridge: Cambridge University Press. and Gerry R. 20. reprinted in Domar. 2nd ed. 2nd ed. Rainville. April 1946. 1990. 70–82. James Bradfield. Upper Saddle River. Blanchard. Essays in the Theory of Economic Growth. Silberberg. Baldani. NJ: Prentice Hall. Chiang. Cambridge: Cambridge University Press.” Quarterly Journal of Economics (February. Walter. Stiglitz. and Carl E. 39: no. 1984. Joseph E. New York: W. Norton and Company. Englewood Cliffs. Walsh. 3rd ed. W. and Peter Hammond. Hal. 2002. Microeconomic Analysis. New York: W.” The Review of Economics and Statistics. Robert M.22.. Solow. Essential Mathematics for Economic Analysis. “Technical Change and the Aggregate Production Function. Knut. 2006. 3 (August 1957): 312–320. Sydsaeter.W. ix . NJ: Prentice Hall. 25. Economics. Norton. 2nd ed. Varian. 23. 24.