Essential Mathematics for Economic Analysis

Knut Sydsaeter (1937-2012) was Emeritus Professor of Mathematics in the Economics Department at the University of Oslo, where he had taught mathematics to economists for over 45 years. Peter Hammond is currently a Professor of Economics at the University of Warwick, where he moved in 2007 after becoming an Emeritus Professor at Stanford University. He has taught Mathematics for Economists at both universities, as well as the universities of Oxford and Essex. Arne Strøm is Associate Professor Emeritus at the University of Oslo and has extensive experience in teaching mathematics to economists at the University Department of Economics. Andrés Carvajal is an Associate Professor in the Department of Economics at the University of California, Davis.

Preface I PRELIMINARIES

Essentials of Logic and Set Theory

Essentials of Set Theory
Essentials of Logic
Mathematical Proofs
Mathematical Induction

Review Exercises
 
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
Solving Equations

Solving Equations
Equations and Their Parameters
Quadratic Equations
Some Nonlinear Equations
Using Implication Arrows
Two Linear Equations in Two Unknowns

Review Exercises
Functions of One Variable

 
Introduction
Definitions
Graphs of Functions
Linear Functions
Linear Models
Quadratic Functions
Polynomials
Power Functions
Exponential Functions
Logarithmic Functions
 

Review Exercises
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
L'Hôpital's 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