Essential Mathematics for Economic Analysis

I PRELIMINARIES

1. Essentials of Logic and Set Theory
   • Essentials of Set Theory
   • Essentials of Logic
   • Mathematical Proofs
   • Mathematical Induction
   Review Exercises
2. Algebra
   • The Real Numbers
   • Integer Powers
   • Rules of Algebra
   • Fractions
   • Fractional Powers
   • Inequalities
   • Intervals and Absolute Values
   • Sign Diagrams
   • Summation Notation
   • Rules for Sums
   • Newton's Binomial Formula
   • Double Sums
   Review Exercises
3. Solving Equations
   • Solving Equations
   • Equations and Their Parameters
   • Quadratic Equations
   • Some Nonlinear Equations
   • Using Implication Arrows
   • Two Linear Equations in Two Unknowns
   Review Exercises
4. Functions of One Variable
   • Introduction
   • Definitions
   • Graphs of Functions
   • Linear Functions
   • Linear Models
   • Quadratic Functions
   • Polynomials
   • Power Functions
   • Exponential Functions
   • Logarithmic Functions
   Review Exercises
5. Properties of Functions
   • Shifting Graphs
   • New Functions From Old
   • Inverse Functions
   • Graphs of Equations
   • Distance in The Plane
   • General Functions
   Review Exercises

II SINGLE-VARIABLE CALCULUS

• Differentiation
  • Slopes of Curves
  • Tangents and Derivatives
  • Increasing and Decreasing Functions
  • Economic Applications
  • A Brief Introduction to Limits
  • Simple Rules for Differentiation
  • Sums, Products, and Quotients
  • The Chain Rule
  • Higher-Order Derivatives
  • Exponential Functions
  • Logarithmic Functions
  Review Exercises
• Derivatives in Use
  • Implicit Differentiation
  • Economic Examples
  • The Inverse Function Theorem
  • Linear Approximations
  • Polynomial Approximations
  • Taylor's Formula
  • Elasticities
  • Continuity
  • More on Limits
  • The Intermediate Value Theorem
  • Infinite Sequences
  • LH�pitals Rule Review Exercises
  Review Exercises
• Concave and Convex Functions
  • Intuition
  • Definitions
  • General Properties
  • First Derivative Tests
  • Second Derivative Tests
  • Inflection Points
  Review Exercises
• Optimization
  • Extreme Points
  • Simple Tests for Extreme Points
  • Economic Examples
  • The Extreme and Mean Value Theorems
  • Further Economic Examples
  • Local Extreme Points
  Review Exercises
• Integration
  • Indefinite Integrals
  • Area and Definite Integrals
  • Properties of Definite Integrals
  • Economic Applications
  • Integration by Parts
  • Integration by Substitution
  • Infinite Intervals of Integration
  Review Exercises
• Topics in Finance and Dynamics
  • Interest Periods and Effective Rates
  • Continuous Compounding
  • Present Value
  • Geometric Series
  • Total Present Value
  • Mortgage Repayments
  • Internal Rate of Return
  • A Glimpse at Difference Equations
  • Essentials of Differential Equations
  • Separable and Linear Differential Equations
  Review Exercises III MULTI-VARIABLE ALGEBRA
• Matrix Algebra
  • Matrices and Vectors
  • Systems of Linear Equations
  • Matrix Addition
  • Algebra of Vectors
  • Matrix Multiplication
  • Rules for Matrix Multiplication
  • The Transpose
  • Gaussian Elimination
  • Geometric Interpretation of Vectors
  • Lines and Planes
  Review Exercises
• Determinants, Inverses, and Quadratic Forms
  • Determinants of Order 2
  • Determinants of Order 3
  • Determinants in General
  • Basic Rules for Determinants
  • Expansion by Cofactors
  • The Inverse of a Matrix
  • A General Formula for The Inverse
  • Cramer's Rule
  • The Leontief Mode
  • Eigenvalues and Eigenvectors
  • Diagonalization
  • Quadratic Forms
  Review Exercises

  IV MULTI-VARIABLE CALCULUS

• Multivariable Functions
  • Functions of Two Variables
  • Partial Derivatives with Two Variables
  • Geometric Representation
  • Surfaces and Distance
  • Functions of More Variables
  • Partial Derivatives with More Variables
  • Convex Sets
  • Concave and Convex Functions
  • Economic Applications
  • Partial Elasticities
  Review Exercises
• Partial Derivatives in Use
  • A Simple Chain Rule
  • Chain Rules for Many Variables
  • Implicit Differentiation Along A Level Curve
  • Level Surfaces
  • Elasticity of Substitution
  • Homogeneous Functions of Two Variables
  • Homogeneous and Homothetic Functions
  • Linear Approximations
  • Differentials
  • Systems of Equations
  • Differentiating Systems of Equations
  Review Exercises
• Multiple Integrals
  • Double Integrals Over Finite Rectangles
  • Infinite Rectangles of Integration
  • Discontinuous Integrands and Other Extensions
  • Integration Over Many Variables
  Review Exercises

  V MULTI-VARIABLE OPTIMIZATION

• Unconstrained Optimization
  • Two Choice Variables: Necessary Conditions
  • Two Choice Variables: Sufficient Conditions
  • Local Extreme Points
  • Linear Models with Quadratic Objectives
  • The Extreme Value Theorem
  • Functions of More Variables
  • Comparative Statics and the Envelope Theorem
  Review Exercises
• Equality Constraints
  • The Lagrange Multiplier Method
  • Interpreting the Lagrange Multiplier
  • Multiple Solution Candidates
  • Why Does the Lagrange Multiplier Method Work?
  • Sufficient Conditions
  • Additional Variables and Constraints
  • Comparative Statics
  Review Exercises
• Linear Programming
  • A Graphical Approach
  • Introduction to Duality Theory
  • The Duality Theorem
  • A General Economic Interpretation
  • Complementary Slackness
  Review Exercises
• Nonlinear Programming
  • Two Variables and One Constraint
  • Many Variables and Inequality Constraints
  • Nonnegativity Constraints
  Review Exercises Appendix
• Geometry
• The Greek Alphabet
• Bibliography
Solutions to the Exercises Index Publisher's Acknowledgments