# Essential Mathematics for Economic Analysis, 6th edition

Published by Pearson (April 15, 2021) © 2021

**Knut Sydsaeter**University of Oslo**Peter Hammond**Stanford University**Arne Strom**University of Oslo**Andrés Carvajal**

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**Provide your students with the key mathematical skills to master and succeed in Economics.**

**Essential Mathematics for Economic Analysis, 6 ^{th} edition** offers an extensive introduction to the discipline. Applauded for the broad range of knowledge and techniques, this book will support your teaching as you provide your students with all the mathematical tools they need to study Economics.

From elementary calculus to more advanced topics, this edition includes a plethora of practice examples, questions, and solutions integrated throughout, giving your students a wealth of opportunities to apply them in specific economic situations and develop key mathematical skills as the course progresses.

**Hallmark features of this title**

**A wide range of mathematical techniques offers your students essential knowledge from theory to practice.**

- Some of the topics studied in the book include Algebra, functions, optimisation, derivatives, and linear and non-linear programming.
- The text further discusses the practical application of mathematical knowledge, introducing your students to the economist way of thinking.

**A clear pedagogical structure supports better understanding.**

- Numerous exercises and worked examples throughout each chapter allow students to practice skills and improve techniques.
- Review exercises at the end of each chapter to test your students' understanding of a topic, allowing them to progress confidently.
**Solutions**to exercises are provided in the book and online.

**New and updated features of this title**

**Updated extensive content coverage, exercises, and worked examples **

**NEW!** A new Chapter 8 **considers concave and convex functions of one variable**, including results on supergradients of concave functions and subgradients of convex functions that play a key role in the theory of optimisation.

**NEW!** A new Chapter 16 on **multiple integrals.**

**REVISED!** Chapter 13 has been extended to include **eigenvalues** and **quadratic forms.**

**REVISED! Numerous exercises and worked examples** throughout each chapter allow your students to practice skills and improve techniques.

**REVISED! Review exercises** at the end of each chapter to test your students' understanding of a topic before they progress with confidence.

**A revised structure enhances student engagement **

**NEW! Matrix algebra is introduced earlier in this edition** and now precedes the chapter on multivariate calculus. This allows new tools to be used in the treatment of multivariate calculus and in the last four chapters, now devoted exclusively to optimization.

**NEW!** The chapter on **constrained optimisation** has been divided to form two new chapters: Chapters 18 & 20.

**REVISED! Solutions** to exercises are provided in the book and online, breaking down the steps needed to arrive at the correct answer.

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**Preface**

**I PRELIMINARIES**

**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

**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

**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

**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

**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**

**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

**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

**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

**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

**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

**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**

**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

**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**

**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

**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

**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**

**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

**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

**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

**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**

**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.

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