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Mathematics for Economics 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 33 2.4 Functions 43 2.5 Proof, Necessary and Sufficient Conditions* Chapter 3 Sequences, Series, and Limits 67 3.1 Definition of a Sequence 67 3.2 Limit of a Sequence 70 3.3 Present-Value Calculations 75 3.4 Properties of Sequences 84 3.5 Series 89 Part II Univariate Calculus and Optimization Chapter 4 Continuity of Functions 115 4.1 Continuity of a Function of One Variable 115 4.2 Economic Applications of Continuous and Discontinuous Functions 125 4.3 Intermediate-Value Theorem 143 60 Chapter 5 The Derivative and Differential for Functions of One Variable 155 5.1 Definition of a Tangent Line 155 5.2 Definition of the Derivative and the Differential 162 5.3 Conditions for Differentiability 169 5.4 Rules of Differentiation 175 5.5 Higher-Order Derivatives: Concavity and Convexity of a Function 208 5.6 Taylor Series Formula and the Mean-Value Theorem 218 Chapter 6 Optimization of Functions of One Variable 227 6.1 Necessary Conditions for Unconstrained Maxima and Minima 227 6.2 Second-Order Conditions 253 6.3 Optimization over an Interval 265 Part Ill Linear Algebra Chapter 7 Systems of Linear Equations 279 7.1 Solving Systems of Linear Equations 279 7.2 Linear Systems in n -Variables 293 Chapter 8 Matrices 317 8.1 General Notation 317 8.2 Basic Matrix Operations 324 8.3 Matrix Transposition 340 8.4 Some Special Matrices 345 Chapter 9 Determinants and the Inverse Matrix 353 9.1 Defining the Inverse 353 9.2 Obtaining the Determinant and Inverse of a 3 x 3 Matrix 370 9.3 The Inverse of an n x n Matrix and Its Properties 376 9.4 Cramer's Rule 386 Chapter 10 Some Advanced Topics in Linear Algebra* 405 10.1 Vector Spaces 405 10.2 The Eigenvalue Problem 421 10.3 Quadratic Forms 436 Part IV Multivariate Calculus Chapter 11 Calculus for Functions of n-Variables 455 11.1 Partial Differentiation 455 11.2 Second-Order Partial Derivatives 469 11.3 The First-Order Total Differential 477 11.4 Curvature Properties: Concavity and Convexity 498 11.5 More Properties of Functions with Economic Applications 513 11.6 Taylor Series Expansion* 534 Chapter 12 Optimization of Functions of n-Variables 545 12.1 First-Order Conditions 545 12.2 Second-Order Conditions 560 12.3 Direct Restrictions on Variables 569 Chapter 13 Constrained Optimization 585 13.1 Constrained Problems and Approaches to Solutions 585 13.2 Second-Order Conditions for Constrained Optimization 616 13.3 Existence, Uniqueness, and Characterization of Solutions 622 Chapter 14 Comparative Statics 631 14.1 Introduction to Comparative Statics 631 14.2 General Comparative-Statics Analysis 643 14.3 The Envelope Theorem 660 Chapter 15 Concave Programming and the Kuhn-Tucker Conditions 677 15.1 The Concave-Programming Problem 677 15.2 Many Variables and Constraints 686 Part V Integration and Dynamic Methods Chapter 16 Integration 701 16.1 The Indefinite Integral 701 16.2 The Riemann (Definite) Integral 709 16.3 Properties of Integrals 721 16.4 Improper Integrals 733 16.5 Techniques of Integration 742 Chapter 17 An Introduction to Mathematics for Economic Dynamics 753 17.1 Modeling Time 754 Chapter 18 Linear, First-Order Difference Equations 763 18.1 Linear, First-Order, Autonomous Difference Equations 763 18.2 The General, Linear, First-Order Difference Equation 780 Chapter 19 Nonlinear, First-Order Difference Equations 789 19.1 The Phase Diagram and Qualitative Analysis 789 19.2 Cycles and Chaos 799 Chapter 20 Linear, Second-Order Difference Equations 811 20.1 The Linear, Autonomous, Second-Order Difference Equation 811 20.2 The Linear, Second-Order Difference Equation with a Variable Term 838 Chapter 21 Linear, First-Order Differential Equations 21.1 Autonomous Equations 849 21.2 Nonautonomous Equations 870 849 Chapter 22 Nonlinear, First-Order Differential Equations 879 22.1 Autonomous Equations and Qualitative Analysis 879 22.2 Two Special Forms of Nonlinear, First-Order Differential Equations 888 Chapter 23 Linear, Second-Order Differential Equations 897 23.1 The Linear, Autonomous, Second-Order Differential Equation 897 23.2 The Linear, Second-Order Differential Equation with a Variable Term 919 Chapter 24 Simultaneous Systems of Differential and Difference Equations 929 24.1 Linear Differential Equation Systems 929 24.2 Stability Analysis and Linear Phase Diagrams 951 24.3 Systems of Linear Difference Equations 976 Chapter 25 Optimal Control Theory 999 25.1 The Maximum Principle 1002 25.2 Optimization Problems Involving Discounting 1014 25.3 Alternative Boundary Conditions on x (T) 1026 25.4 Infinite--Time Horizon Problems 1040 25.5 Constraints on the Control Variable 1053 25.6 Free-Terminal-Time Problems (T Free) 1063 Appendix: Complex Numbers and Circular Functions 1081 Answers 1091 Index 1123