Schaum's Outline Theory and Problems of Introduction to ...

SCHAUM'S OUTLINE OF

Theory and Problems of

INTRODUCTION TO

MATHEMATICAL ECONOMICS

Third Edition

EDWARD T. DOWLING, Ph.D.

Professor and Former Chairman Department of Economics Fordham University

Schaum's Outline Series

McGRAW-HILL New York San Francisco Washington, D.C. Auckland Bogota? Caracas

Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto

To the memory of my parents, Edward T. Dowling, M.D. and Mary H. Dowling

EDWARD T. DOWLING is professor of Economics at Fordham University. He was Dean of Fordham College from 1982 to 1986 and Chairman of the Economics Department from 1979 to 1982 and again from 1988 to 1994. His Ph.D. is from Cornell University and his main areas of professional interest are mathematical economics and economic development. In addition to journal articles, he is the author of Schaum's Outline of Calculus for Business, Economics, and the Social Sciences, and Schaum's Outline of Mathematical Methods for Business and Economics. A Jesuit priest, he is a member of the Jesuit Community at Fordham.

Copyright ? 2001, 1992 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.

ISBN: 978-0-07-161015-5

MHID: 0-07-161015-4

The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-135896-5, MHID: 0-07-135896-X.

All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps.

McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To contact a representative please e-mail us at bulksales@mcgraw-.

Copyright 1980 by McGraw-Hill, Inc. Under the title Schaum's Outline of Theory and Problems of Mathematics for Economists. All rights reserved.

TERMS OF USE

This is a copyrighted work and The McGraw-Hill Companies, Inc. ("McGraw-Hill") and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill's prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms.

THE WORK IS PROVIDED "AS IS." McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.

PREFACE

The mathematics needed for the study of economics and business continues to grow with each passing year, placing ever more demands on students and faculty alike. Introduction to Mathematical Economics, third edition, introduces three new chapters, one on comparative statics and concave programming, one on simultaneous differential and difference equations, and one on optimal control theory. To keep the book manageable in size, some chapters and sections of the second edition had to be excised. These include three chapters on linear programming and a number of sections dealing with basic elements such as factoring and completing the square. The deleted topics were chosen in part because they can now be found in one of my more recent Schaum books designed as an easier, more detailed introduction to the mathematics needed for economics and business, namely, Mathematical Methods for Business and Economics.

The objectives of the book have not changed over the 20 years since the introduction of the first edition, originally called Mathematics for Economists. Introduction to Mathematical Economics, third edition, is designed to present a thorough, easily understood introduction to the wide array of mathematical topics economists, social scientists, and business majors need to know today, such as linear algebra, differential and integral calculus, nonlinear programming, differential and difference equations, the calculus of variations, and optimal control theory. The book also offers a brief review of basic algebra for those who are rusty and provides direct, frequent, and practical applications to everyday economic problems and business situations.

The theory-and-solved-problem format of each chapter provides concise explanations illustrated by examples, plus numerous problems with fully worked-out solutions. The topics and related problems range in difficulty from simpler mathematical operations to sophisticated applications. No mathematical proficiency beyond the high school level is assumed at the start. The learning-by-doing pedagogy will enable students to progress at their own rates and adapt the book to their own needs.

Those in need of more time and help in getting started with some of the elementary topics may feel more comfortable beginning with or working in conjunction with my Schaum's Outline of Mathematical Methods for Business and Economics, which offers a kinder, gentler approach to the discipline. Those who prefer more rigor and theory, on the other hand, might find it enriching to work along with my Schaum's Outline of Calculus for Business, Economics, and the Social Sciences, which devotes more time to the theoretical and structural underpinnings.

Introduction to Mathematical Economics, third edition, can be used by itself or as a supplement to other texts for undergraduate and graduate students in economics, business, and the social sciences. It is largely self-contained. Starting with a basic review of high school algebra in Chapter 1, the book consistently explains all the concepts and techniques needed for the material in subsequent chapters.

Since there is no universal agreement on the order in which differential calculus and linear algebra should be presented, the book is designed so that Chapters 10 and 11 on linear algebra can be covered immediately after Chapter 2, if so desired, without loss of continuity.

This book contains over 1600 problems, all solved in considerable detail. To get the most from the book, students should strive as soon as possible to work independently of the solutions. This can be done by solving problems on individual sheets of paper with the book closed. If difficulties arise, the solution can then be checked in the book.

iii

iv

PREFACE

For best results, students should never be satisfied with passive knowledgethe capacity merely to follow or comprehend the various steps presented in the book. Mastery of the subject and doing well on exams requires active knowledgethe ability to solve any problem, in any order, without the aid of the book.

Experience has proved that students of very different backgrounds and abilities can be successful in handling the subject matter presented in this text if they apply themselves and work consistently through the problems and examples.

In closing, I would like to thank my friend and colleague at Fordham, Dr. Dominick Salvatore, for his unfailing encouragement and support over the past 25 years, and an exceptionally fine graduate student, Robert Derrell, for proofreading the manuscript and checking the accuracy of the solutions. I am also grateful to the entire staff at McGraw-Hill, especially Barbara Gilson, Tina Cameron, Maureen B. Walker, and Deborah Aaronson.

EDWARD T. DOWLING

CONTENTS

CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5

CHAPTER 6

Review

1

1.1 Exponents. 1.2 Polynomials. 1.3 Equations: Linear and Quadratic. 1.4 Simultaneous Equations. 1.5 Functions. 1.6 Graphs, Slopes, and Intercepts.

Economic Applications of Graphs and Equations

14

2.1 Isocost Lines. 2.2 Supply and Demand Analysis.

2.3 Income Determination Models. 2.4 IS-LM Analysis.

The Derivative and the Rules of Differentiation

32

3.1 Limits. 3.2 Continuity. 3.3 The Slope of a Curvilinear Function. 3.4 The Derivative. 3.5 Differentiability and Continuity. 3.6 Derivative Notation. 3.7 Rules of Differentiation. 3.8 Higher-Order Derivatives. 3.9 Implicit Differentiation.

Uses of the Derivative in Mathematics and Economics 58

4.1 Increasing and Decreasing Functions. 4.2 Concavity and Convexity. 4.3 Relative Extrema. 4.4 Inflection Points. 4.5 Optimization of Functions. 4.6 Successive-Derivative Test for Optimization. 4.7 Marginal Concepts. 4.8 Optimizing Economic Functions. 4.9 Relationship among Total, Marginal, and Average Concepts.

Calculus of Multivariable Functions

82

5.1 Functions of Several Variables and Partial Derivatives.

5.2 Rules of Partial Differentiation. 5.3 Second-Order Partial

Derivatives. 5.4 Optimization of Multivariable Functions.

5.5 Constrained Optimization with Lagrange Multipliers.

5.6 Significance of the Lagrange Multiplier. 5.7 Differentials.

5.8 Total and Partial Differentials. 5.9 Total Derivatives.

5.10 Implicit and Inverse Function Rules.

Calculus of Multivariable Functions in Economics 110

6.1 Marginal Productivity. 6.2 Income Determination Multipliers and Comparative Statics. 6.3 Income and Cross Price Elasticities of Demand. 6.4 Differentials and Incremental Changes. 6.5 Optimization of Multivariable Functions in Economics. 6.6 Constrained Optimization of Multivariable

v

vi

CONTENTS

CHAPTER 7 CHAPTER 8 CHAPTER 9 CHAPTER 10 CHAPTER 11

Functions in Economics. 6.7 Homogeneous Production Functions. 6.8 Returns to Scale. 6.9 Optimization of Cobb-Douglas Production Functions. 6.10 Optimization of Constant Elasticity of Substitution Production Functions.

Exponential and Logarithmic Functions

146

7.1 Exponential Functions. 7.2 Logarithmic Functions.

7.3 Properties of Exponents and Logarithms. 7.4 Natural

Exponential and Logarithmic Functions. 7.5 Solving Natural

Exponential and Logarithmic Functions. 7.6 Logarithmic

Transformation of Nonlinear Functions.

Exponential and Logarithmic Functions in

Economics

160

8.1 Interest Compounding. 8.2 Effective vs. Nominal Rates of

Interest. 8.3 Discounting. 8.4 Converting Exponential to

Natural Exponential Functions. 8.5 Estimating Growth Rates

from Data Points.

Differentiation of Exponential and Logarithmic

Functions

173

9.1 Rules of Differentiation. 9.2 Higher-Order Derivatives.

9.3 Partial Derivatives. 9.4 Optimization of Exponential and

Logarithmic Functions. 9.5 Logarithmic Differentiation.

9.6 Alternative Measures of Growth. 9.7 Optimal Timing.

9.8 Derivation of a Cobb-Douglas Demand Function Using a

Logarithmic Transformation.

The Fundamentals of Linear (or Matrix) Algebra

199

10.1 The Role of Linear Algebra. 10.2 Definitions and

Terms. 10.3 Addition and Subtraction of Matrices.

10.4 Scalar Multiplication. 10.5 Vector Multiplication.

10.6 Multiplication of Matrices. 10.7 Commutative, Associative,

and Distributive Laws in Matrix Algebra. 10.8 Identity and

Null Matrices. 10.9 Matrix Expression of a System of Linear

Equations.

Matrix Inversion

224

11.1 Determinants and Nonsingularity. 11.2 Third-Order

Determinants. 11.3 Minors and Cofactors. 11.4 Laplace

Expansion and Higher-Order Determinants. 11.5 Properties of

a Determinant. 11.6 Cofactor and Adjoint Matrices.

11.7 Inverse Matrices. 11.8 Solving Linear Equations with the

Inverse. 11.9 Cramer's Rule for Matrix Solutions.

CONTENTS

vii

CHAPTER 12 CHAPTER 13 CHAPTER 14 CHAPTER 15 CHAPTER 16 CHAPTER 17

Special Determinants and Matrices and Their Use in

Economics

254

12.1 The Jacobian. 12.2 The Hessian. 12.3 The

Discriminant. 12.4 Higher-Order Hessians. 12.5 The

Bordered Hessian for Constrained Optimization.

12.6 Input-Output Analysis. 12.7 Characteristic Roots and

Vectors (Eigenvalues, Eigenvectors).

Comparative Statics and Concave Programming

284

13.1 Introduction to Comparative Statics. 13.2 Comparative

Statics with One Endogenous Variable. 13.3 Comparative

Statics with More Than One Endogenous Variable.

13.4 Comparative Statics for Optimization Problems.

13.5 Comparative Statics Used in Constrained Optimization.

13.6 The Envelope Theorem. 13.7 Concave Programming and

Inequality Constraints.

Integral Calculus: The Indefinite Integral

326

14.1 Integration. 14.2 Rules of Integration. 14.3 Initial

Conditions and Boundary Conditions. 14.4 Integration by

Substitution. 14.5 Integration by Parts. 14.6 Economic

Applications.

Integral Calculus: The Definite Integral

342

15.1 Area Under a Curve. 15.2 The Definite Integral.

15.3 The Fundamental Theorem of Calculus. 15.4 Properties of

Definite Integrals. 15.5 Area Between Curves. 15.6 Improper

Integrals. 15.7 L'Ho^ pital's Rule. 15.8 Consumers' and

Producers' Surplus. 15.9 The Definite Integral and Probability.

First-Order Differential Equations

362

16.1 Definitions and Concepts. 16.2 General Formula for

First-Order Linear Differential Equations. 16.3 Exact

Differential Equations and Partial Integration. 16.4 Integrating

Factors. 16.5 Rules for the Integrating Factor.

16.6 Separation of Variables. 16.7 Economic Applications.

16.8 Phase Diagrams for Differential Equations.

First-Order Difference Equations

391

17.1 Definitions and Concepts. 17.2 General Formula for

First-Order Linear Difference Equations. 17.3 Stability

Conditions. 17.4 Lagged Income Determination Model.

17.5 The Cobweb Model. 17.6 The Harrod Model.

17.7 Phase Diagrams for Difference Equations.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download