Essential Mathematics for Economic Analysis

Title Page
Copyright Page
About the Authors
Contents
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
		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 More on Limits
		7.11 More on Limits
		7.12 More on Limits
		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 Improper Integrals
		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 MULTIVARIABLE 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 Model
		13.10 Eigenvalues and Eigenvectors
		13.11 Diagonalization
		13.12 Quadratic Forms
		Review Exercises
IV MULTIVARIABLE CALCULUS
	14 Functions of Many Variables
		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 n Variables
		14.6 Partial Derivatives with Many 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
V MULTIVARIABLE 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
	Geometry
	The Greek Alphabet
	Bibliography
Solutions to the Exercises
Index
Publisher’s Acknowledgements