Essential Mathematics for Economic Analysis

Acquire the key mathematical skills you need to master and succeed in Economics.

Essential Mathematics for Economic Analysis, 6^th edition by Sydsaeter, Hammond, Strøm, and Carvajal is a global best-selling text providing an extensive introduction to all the mathematical resources you need to study economics at an intermediate level.

This book has been applauded for covering a broad range of mathematical knowledge, techniques, and tools, progressing from elementary calculus to more advanced topics.

With a plethora of practice examples, questions, and solutions integrated throughout, this latest edition provides you a wealth of opportunities to apply them in specific economic situations, helping you develop key mathematical skills as your course progresses.

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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

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

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