Schaum's outline of theory and problems of introduction to mathematical economics

532 Pages·2001·2.621 MB·English

by  Dowling, Edward Thomas

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
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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.
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PREFACE
Themathematicsneededforthestudyofeconomicsandbusinesscontinuestogrowwitheachpassing
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 oneon optimal control
theory.Tokeepthebookmanageableinsize,somechaptersandsectionsofthesecondeditionhadto
beexcised.Theseincludethreechaptersonlinearprogrammingandanumberofsectionsdealingwith
basic elements such as factoring and completing the square. The deleted topics were chosen in part
because they can now be found in oneofmymorerecentSchaumbooks 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.Thosewhoprefermorerigorandtheory,ontheotherhand,mightfinditenrichingtowork
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
toothertextsforundergraduateandgraduatestudentsineconomics,business,andthesocialsciences.
It is largely self-contained. Starting with a basic review of high school algebrain Chapter1,thebook
consistently explains all the concepts and techniques needed for the material in subsequent
chapters.
Sincethereisnouniversalagreementontheorderinwhichdifferentialcalculusandlinearalgebra
shouldbepresented,thebookisdesignedsothatChapters10and11onlinearalgebracanbecovered
immediately after Chapter 2, if so desired, without loss of continuity.
Thisbookcontainsover1600problems,allsolvedinconsiderabledetail.Togetthemostfromthe
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 knowledge(cid:1)the capacity merely
tofolloworcomprehendthevariousstepspresentedinthebook.Masteryofthesubjectanddoingwell
on exams requires active knowledge(cid:1)the ability to solve any problem, in any order, without the aid
of the book.
Experiencehas provedthatstudentsofverydifferentbackgroundsandabilitiescanbesuccessful
in handling the subject matter presented in this text if they apply themselves and work consistently
through the problems and examples.
Inclosing,IwouldliketothankmyfriendandcolleagueatFordham,Dr.DominickSalvatore,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 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.
CHAPTER 2 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.
CHAPTER 3 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.
CHAPTER 4 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.
CHAPTER 5 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.
CHAPTER 6 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
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.
CHAPTER 7 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.
CHAPTER 8 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.
CHAPTER 9 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.
CHAPTER 10 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.
CHAPTER 11 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 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).
CHAPTER 13 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.
CHAPTER 14 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.
CHAPTER 15 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’Hoˆpital’s Rule. 15.8 Consumers’ and
Producers’ Surplus. 15.9 The Definite Integral and Probability.
CHAPTER 16 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.
CHAPTER 17 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.
viii CONTENTS
CHAPTER 18 Second-Order Differential Equations and
Difference Equations 408
18.1 Second-Order Differential Equations. 18.2 Second-Order
Difference Equations. 18.3 Characteristic Roots.
18.4 Conjugate Complex Numbers. 18.5 Trigonometric
Functions. 18.6 Derivatives of Trigonometric Functions.
18.7 Transformation of Imaginary and Complex Numbers.
18.8 Stability Conditions.
CHAPTER 19 Simultaneous Differential and Difference Equations 428
19.1 Matrix Solution of Simultaneous Differential Equations,
Part 1. 19.2 Matrix Solution of Simultaneous Differential
Equations, Part 2. 19.3 Matrix Solution of Simultaneous
Difference Equations, Part 1. 19.4 Matrix Solution of
Simultaneous Difference Equations, Part 2. 19.5 Stability and
Phase Diagrams for Simultaneous Differential Equations.
CHAPTER 20 The Calculus of Variations 460
20.1 Dynamic Optimization. 20.2 Distance Between Two
Points on a Plane. 20.3 Euler’s Equation and the Necessary
Condition for Dynamic Optimization. 20.4 Finding Candidates
for Extremals. 20.5 The Sufficiency Conditions for the
Calculus of Variations. 20.6 Dynamic Optimization Subject to
Functional Constraints. 20.7 Variational Notation.
20.8 Applications to Economics.
CHAPTER 21 Optimal Control Theory 493
21.1 Terminology. 21.2 The Hamiltonian and the Necessary
Conditions for Maximization in Optimal Control Theory.
21.3 Sufficiency Conditions for Maximization in Optimal
Control. 21.4 Optimal Control Theory with a Free
Endpoint. 21.5 Inequality Constraints in the Endpoints.
21.6 The Current-Valued Hamiltonian.
Index 515
CHAPTER 1
Review
1.1 EXPONENTS
Given n a positive integer, xn signifies that x is multiplied by itself n times. Here x is referred to
as the base and n is termed an exponent. By convention an exponent of 1 is not expressed: x1(cid:3)x,
81(cid:3)8. By definition, any nonzero number or variable raised to the zero power is equal to 1: x0(cid:3)1,
30(cid:3)1. And 00 is undefined. Assuming a and b arepositiveintegers andxand yarereal numbersfor
whichthefollowingexist,therulesofexponentsareoutlinedbelowandillustratedinExamples1and
2 and Problem 1.1.
1
1. xa(xb)(cid:3)xa(cid:4)b 6. (cid:3)x(cid:5)a
xa
xa
2. (cid:3)xa(cid:5)b 7. (cid:1)x(cid:3)x1/2
xb
3. (xa)b(cid:3)xab 8. (cid:1)a x(cid:3)x1/a
4. (xy)a(cid:3)xaya 9. (cid:1)b xa(cid:3)xa/b(cid:3)(x1/b)a
(cid:2)x(cid:3)a xa 1
5. (cid:3) 10. x(cid:5)(a/b)(cid:3)
y ya xa/b
EXAMPLE1. FromRule2,itcaneasilybeseenwhyanyvariableornonzeronumberraisedtothezeropower
equals1. Forexample,x3/x3(cid:3)x3(cid:5)3(cid:3)x0(cid:3)1; 85/85(cid:3)85(cid:5)5(cid:3)80(cid:3)1.
EXAMPLE2. In multiplication, exponents of the same variable are added; in division, the exponents are
subtracted; whenraisedtoapower, theexponentsaremultiplied, asindicatedbytherulesabove and shown in
theexamplesbelowfollowedbyillustrationsinbrackets.
a) x2(x3)(cid:3)x2(cid:4)3(cid:3)x5(cid:6)x6 Rule1
[x2(x3)(cid:3)(x·x)(x·x·x)(cid:3)x·x·x·x·x(cid:3)x5]
x6
b) (cid:3)x6(cid:5)3(cid:3)x3(cid:6)x2 Rule2
x3
(cid:4)x6 x·x·x·x·x·x (cid:5)
(cid:3) (cid:3)x·x·x(cid:3)x3
x3 x·x·x
c) (x4)2(cid:3)x4·2(cid:3)x8(cid:6)x16 orx6 Rule3
[(x4)2(cid:3)(x·x·x·x)(x·x·x·x)(cid:3)x8]
1