Essential Mathematics for Economic Analysis, 6th edition

Preface

I PRELIMINARIES

1. Essentials of Logic and Set Theory
   • 1.1 Essentials of Set Theory
   • 1.2 Essentials of Logic
   • 1.3 Mathematical Proofs
   • 1.4 Mathematical Induction

   Review Exercises

2. Algebra
   • 2.1 The Real Numbers
   • 2.2 Integer Powers
   • 2.3 Rules of Algebra
   • 2.4 Fractions
   • 2.5 Fractional Powers
   • 2.6 Inequalities
   • 2.7 Intervals and Absolute Values
   • 2.8 Sign Diagrams
   • 2.9 Summation Notation
   • 2.10 Rules for Sums
   • 2.11 Newton's Binomial Formula
   • 2.12 Double Sums

   Review Exercises

3. Solving Equations
   • 3.1 Solving Equations
   • 3.2 Equations and Their Parameters
   • 3.3 Quadratic Equations
   • 3.4 Some Nonlinear Equations
   • 3.5 Using Implication Arrows
   • 3.6 Two Linear Equations in Two Unknowns

   Review Exercises

4. Functions of One Variable
   • 4.1 Introduction
   • 4.2 Definitions
   • 4.3 Graphs of Functions
   • 4.4 Linear Functions
   • 4.5 Linear Models
   • 4.6 Quadratic Functions
   • 4.7 Polynomials
   • 4.8 Power Functions
   • 4.9 Exponential Functions
   • 4.10 Logarithmic Functions

   Review Exercises

5. Properties of Functions
   • 5.1 Shifting Graphs
   • 5.2 New Functions From Old
   • 5.3 Inverse Functions
   • 5.4 Graphs of Equations
   • 5.5 Distance in The Plane
   • 5.6 General Functions

   Review Exercises

II SINGLE-VARIABLE CALCULUS

6. Differentiation
   • 6.1 Slopes of Curves
   • 6.2 Tangents and Derivatives
   • 6.3 Increasing and Decreasing Functions
   • 6.4 Economic Applications
   • 6.5 A Brief Introduction to Limits
   • 6.6 Simple Rules for Differentiation
   • 6.7 Sums, Products, and Quotients
   • 6.8 The Chain Rule
   • 6.9 Higher-Order Derivatives
   • 6.10 Exponential Functions
   • 6.11 Logarithmic Functions

   Review Exercises

7. Derivatives in Use
   • 7.1 Implicit Differentiation
   • 7.2 Economic Examples
   • 7.3 The Inverse Function Theorem
   • 7.4 Linear Approximations
   • 7.5 Polynomial Approximations
   • 7.6 Taylor's Formula
   • 7.7 Elasticities
   • 7.8 Continuity
   • 7.9 More on Limits
   • 7.10 The Intermediate Value Theorem
   • 7.11 Infinite Sequences
   • 7.12 L’Hôpital’s Rule Review Exercises

   Review Exercises

8. Concave and Convex Functions
   • 8.1 Intuition
   • 8.2 Definitions
   • 8.3 General Properties
   • 8.4 First Derivative Tests
   • 8.5 Second Derivative Tests
   • 8.6 Inflection Points

   Review Exercises

9. Optimization
   • 9.1 Extreme Points
   • 9.2 Simple Tests for Extreme Points
   • 9.3 Economic Examples
   • 9.4 The Extreme and Mean Value Theorems
   • 9.5 Further Economic Examples
   • 9.6 Local Extreme Points

   Review Exercises

10. Integration
    • 10.1 Indefinite Integrals
    • 10.2 Area and Definite Integrals
    • 10.3 Properties of Definite Integrals
    • 10.4 Economic Applications
    • 10.5 Integration by Parts
    • 10.6 Integration by Substitution
    • 10.7 Infinite Intervals of Integration

    Review Exercises

11. Topics in Finance and Dynamics
    • 11.1 Interest Periods and Effective Rates
    • 11.2 Continuous Compounding
    • 11.3 Present Value
    • 11.4 Geometric Series
    • 11.5 Total Present Value
    • 11.6 Mortgage Repayments
    • 11.7 Internal Rate of Return
    • 11.8 A Glimpse at Difference Equations
    • 11.9 Essentials of Differential Equations
    • 11.10 Separable and Linear Differential Equations

    Review Exercises

III MULTI-VARIABLE ALGEBRA

12. Matrix Algebra
    • 12.1 Matrices and Vectors
    • 12.2 Systems of Linear Equations
    • 12.3 Matrix Addition
    • 12.4 Algebra of Vectors
    • 12.5 Matrix Multiplication
    • 12.6 Rules for Matrix Multiplication
    • 12.7 The Transpose
    • 12.8 Gaussian Elimination
    • 12.9 Geometric Interpretation of Vectors
    • 12.10 Lines and Planes

    Review Exercises

13. Determinants, Inverses, and Quadratic Forms
    • 13.1 Determinants of Order 2
    • 13.2 Determinants of Order 3
    • 13.3 Determinants in General
    • 13.4 Basic Rules for Determinants
    • 13.5 Expansion by Cofactors
    • 13.6 The Inverse of a Matrix
    • 13.7 A General Formula for The Inverse
    • 13.8 Cramer's Rule
    • 13.9 The Leontief Mode
    • 13.10 Eigenvalues and Eigenvectors
    • 13.11 Diagonalization
    • 13.12 Quadratic Forms

    Review Exercises

IV MULTI-VARIABLE CALCULUS

14. Multivariable Functions
    • 14.1 Functions of Two Variables
    • 14.2 Partial Derivatives with Two Variables
    • 14.3 Geometric Representation
    • 14.4 Surfaces and Distance
    • 14.5 Functions of More Variables
    • 14.6 Partial Derivatives with More Variables
    • 14.7 Convex Sets
    • 14.8 Concave and Convex Functions
    • 14.9 Economic Applications
    • 14.10 Partial Elasticities

    Review Exercises

15. Partial Derivatives in Use
    • 15.1 A Simple Chain Rule
    • 15.2 Chain Rules for Many Variables
    • 15.3 Implicit Differentiation Along A Level Curve
    • 15.4 Level Surfaces
    • 15.5 Elasticity of Substitution
    • 15.6 Homogeneous Functions of Two Variables
    • 15.7 Homogeneous and Homothetic Functions
    • 15.8 Linear Approximations
    • 15.9 Differentials
    • 15.10 Systems of Equations
    • 15.11 Differentiating Systems of Equations

    Review Exercises

16. Multiple Integrals
    • 16.1 Double Integrals Over Finite Rectangles
    • 16.2 Infinite Rectangles of Integration
    • 16.3 Discontinuous Integrands and Other Extensions
    • 16.4 Integration Over Many Variables

    Review Exercises

V MULTI-VARIABLE OPTIMIZATION

17. Unconstrained Optimization
    • 17.1 Two Choice Variables: Necessary Conditions
    • 17.2 Two Choice Variables: Sufficient Conditions
    • 17.3 Local Extreme Points
    • 17.4 Linear Models with Quadratic Objectives
    • 17.5 The Extreme Value Theorem
    • 17.6 Functions of More Variables
    • 17.7 Comparative Statics and the Envelope Theorem

    Review Exercises

18. Equality Constraints
    • 18.1 The Lagrange Multiplier Method
    • 18.2 Interpreting the Lagrange Multiplier
    • 18.3 Multiple Solution Candidates
    • 18.4 Why Does the Lagrange Multiplier Method Work?
    • 18.5 Sufficient Conditions
    • 18.6 Additional Variables and Constraints
    • 18.7 Comparative Statics

    Review Exercises

19. Linear Programming
    • 19.1 A Graphical Approach
    • 19.2 Introduction to Duality Theory
    • 19.3 The Duality Theorem
    • 19.4 A General Economic Interpretation
    • 19.5 Complementary Slackness

    Review Exercises

20. Nonlinear Programming
    • 20.1 Two Variables and One Constraint
    • 20.2 Many Variables and Inequality Constraints
    • 20.3 Nonnegativity Constraints

    Review Exercises

Appendix

Solutions to the Exercises

Index

Publisher's Acknowledgments