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Mathematics for Economics, third edition.pdf
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Mathematics for Economics
third edition
Michael Hoy
John Livernois
Chris McKenna
Ray Rees
Thanasis Stengos
Contents
Preface xiii
Part I Introduction and Fundamentals
Chapter 1
Introduction 3
1.1 What Is an Economic Model? 3
1.2 How to Use This Book 8
1.3 Conclusion 9
Chapter 2
Review of Fundamentals 11
2.1 Sets and Subsets 11
2.2 Numbers 23
2.3 Some Properties of Point Sets in Rn 31
2.4 Functions 41
Chapter 3
Sequences, Series, and Limits 61
3.1 Definition of a Sequence 61
3.2 Limit of a Sequence 65
3.3 Present-Value Calculations 69
3.4 Properties of Sequences 79
3.5 Series 84
Part II Univariate Calculus and Optimization
Chapter 4
Continuity of Functions 103
4.1 Continuity of a Function of One Variable 103
4.2 Economic Applications of Continuous and Discontinuous Functions 113
viii CONTENTS
Chapter 5
The Derivative and Differential for Functions of One Variable 127
5.1 Definition of a Tangent Line 127
5.2 Definition of the Derivative and the Differential 134
5.3 Conditions for Differentiability 141
5.4 Rules of Differentiation 147
5.5 Higher Order Derivatives: Concavity and Convexity of a Function 175
5.6 Taylor Series Formula and the Mean-Value Theorem 185
Chapter 6
Optimization of Functions of One Variable 195
6.1 Necessary Conditions for Unconstrained Maxima and Minima 196
6.2 Second-Order Conditions 211
6.3 Optimization over an Interval 220
Part III Linear Algebra
Chapter 7
Systems of Linear Equations 235
7.1 Solving Systems of Linear Equations 236
7.2 Linear Systems in n-Variables 250
Chapter 8
Matrices 267
8.1 General Notation 267
8.2 Basic Matrix Operations 273
8.3 Matrix Transposition 288
8.4 Some Special Matrices 293
Chapter 9
Determinants and the Inverse Matrix 301
9.1 Defining the Inverse 301
9.2 Obtaining the Determinant and Inverse of a 3×3 Matrix 318
9.3 The Inverse of an n×n Matrix and Its Properties 324
9.4 Cramer’s Rule 329
Chapter 10
Some Advanced Topics in Linear Algebra 347
10.1 Vector Spaces 347
10.2 The Eigenvalue Problem 363
10.3 Quadratic Forms 378
CONTENTS ix
Part IV Multivariate Calculus
Chapter 11
Calculus for Functions of n-Variables 393
11.1 Partial Differentiation 393
11.2 Second-Order Partial Derivatives 407
11.3 The First-Order Total Differential 415
11.4 Curvature Properties: Concavity and Convexity 436
11.5 More Properties of Functions with Economic Applications 451
11.6 Taylor Series Expansion* 464
Chapter 12
Optimization of Functions of n-Variables 473
12.1 First-Order Conditions 474
12.2 Second-Order Conditions 484
12.3 Direct Restrictions on Variables 491
Chapter 13
Constrained Optimization 503
13.1 Constrained Problems and Approaches to Solutions 504
13.2 Second-Order Conditions for Constrained Optimization 516
13.3 Existence, Uniqueness, and Characterization of Solutions 520
Chapter 14
Comparative Statics 529
14.1 Introduction to Comparative Statics 529
14.2 General Comparative-Statics Analysis 540
14.3 The Envelope Theorem 554
Chapter 15
Concave Programming and the Kuhn-Tucker Conditions 567
15.1 The Concave-Programming Problem 567
15.2 Many Variables and Constraints 575
Part V Integration and Dynamic Methods
Chapter 16
Integration 585
16.1 The Indefinite Integral 585
16.2 The Riemann (Definite) Integral 593
16.3 Properties of Integrals 605
x CONTENTS
16.4 Improper Integrals 613
16.5 Techniques of Integration 623
Chapter 17
An Introduction to Mathematics for Economic Dynamics 633
17.1 Modeling Time 634
Chapter 18
Linear, First-Order Difference Equations 643
18.1 Linear, First-Order, Autonomous Difference Equations 643
18.2 The General, Linear, First-Order Difference Equation 656
Chapter 19
Nonlinear, First-Order Difference Equations 665
19.1 The Phase Diagram and Qualitative Analysis 665
19.2 Cycles and Chaos 673
Chapter 20
Linear, Second-Order Difference Equations 681
20.1 The Linear, Autonomous, Second-Order Difference Equation 681
20.2 The Linear, Second-Order Difference Equation with aVariableTerm 708
Chapter 21
Linear, First-Order Differential Equations 715
21.1 Autonomous Equations 715
21.2 Nonautonomous Equations 731
Chapter 22
Nonlinear, First-Order Differential Equations 739
22.1 Autonomous Equations and Qualitative Analysis 739
22.2 Two Special Forms of Nonlinear, First-Order Differential Equations 748
Chapter 23
Linear, Second-Order Differential Equations 753
23.1 The Linear, Autonomous, Second-Order Differential Equation 753
23.2 The Linear, Second-Order Differential Equation with aVariableTerm 772
Chapter 24
Simultaneous Systems of Differential and Difference Equations 781
24.1 Linear Differential Equation Systems 781
24.2 Stability Analysis and Linear Phase Diagrams 803
24.3 Systems of Linear Difference Equations 825
CONTENTS xi
Chapter 25
Optimal Control Theory 845
25.1 The Maximum Principle 848
25.2 Optimization Problems Involving Discounting 860
25.3 Alternative Boundary Conditions on x(T ) 872
25.4 Infinite–Time Horizon Problems 886
25.5 Constraints on the Control Variable 899
25.6 Free-Terminal-Time Problems (T Free) 909
Answers 921
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