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Mathematics for Economics

Michael Hoy, John Livernois, Chris McKenna, Ray Rees, and Thanasis Stengos
Publisher: 
MIT Press
Publication Date: 
2011
Number of Pages: 
959
Format: 
Hardcover
Edition: 
3
Price: 
90.00
ISBN: 
9780262015073
Category: 
Textbook
[Reviewed by
Fernando Q. Gouvêa
, on
08/25/2011
]

Economics students need mathematics: single-variable and multivariable calculus, linear algebra, basic optimization, difference and differential equations. Ideally, they should take mathematics courses. In fact, the economics professors at my college tell me that students who want to go to graduate school in economics should be double majors in mathematics and economics.

But not all students want to go to graduate school, and not all colleges want to have economics students take standard mathematics courses. This textbook is intended for a course that covers the basic mathematical skills required for economics. The content is basically what I listed above, plus a little bit of optimal control theory. The mathematics is presented using epsilon-delta definitions and with some proofs.

Some things are a little strange. The one-variable differential is introduced on the same page as the derivative, as a function of two variables (base point and increment). It is interpreted in terms of approximating a function by its tangent line. At that point, linear transformations haven't been introduced yet, so the linearity is not highlighted. Alas, when the differential of a function of two variables is introduced (at which point some linear algebra has been covered) it is just given as a symbol and then used to approximate errors. In particular, the authors never specify what the variables are, a point that I have found really confuses students.

The economic focus shows up mainly in the sequencing of topics and the examples. It is worth pondering the table of contents. For example, infinite series come up very early in the context of present value computations. "Univariate calculus" is entirely differential, and integration only appears much later, as a preliminary to the study of differential equations.

I find this textbook interesting in three ways. First, it shows that our colleagues in the Economics Department think that undergraduates can and should be taught real mathematics. Second, it provides a wide array of concrete examples of how mathematics is used. Finally, it is interesting to see the material presented in a different sequence. Perhaps getting to know textbooks like this one will help us mathematicians shake up the long-frozen calculus curriculum.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He is preparing to teach single-variable calculus in the Fall of 2011, and expects that many of the students in his class will eventually become economics majors.

Preface

Part I Introduction and Fundamentals

Chapter 1: Introduction
1.1 What Is an Economic Model?
1.2 How to Use This Book
1.3 Conclusion

Chapter 2: Review of Fundamentals
2.1 Sets and Subsets
2.2 Numbers
2.3 Some Properties of Point Sets in Rn
2.4 Functions

Chapter 3: Sequences, Series, and Limits
3.1 Definition of a Sequence
3.2 Limit of a Sequence
3.3 Present-Value Calculations
3.4 Properties, of Sequences
3.5 Series

Part II Univariate Calculus and Optimization

Chapter 4: Continuity of Functions
4.1 Continuity of a Function of One Variable
4.2 Economic Applications of Continuous and Discontinuous Functions

Chapter 5: The Derivative and Differential for Functions of One Variable
5.1 Definition of a Tangent Line
5.2 Definition of the Derivative and the Differential
5.3 Conditions of Differentiability
5.4 Rules of Differentiation
5.5 Higher Order Derivatives: Concavity and Convexity of a Function
5.6 Taylor Series Formula and the Mean-Value Theorem

Chapter 6: Optimization of Functions of One Variable
6.1 Necessary Conditions for Unconstrained Maxima and Minima
6.2 Second-Order Conditions
6.3 Optimization over an Interval

Part III Linear Algebra

Chapter 7: Systems of Linear Equations
7.1 Solving Systems of Linear Equations
7.2 Linear Systems in n-Variables

Chapter 8: Matrices
8.1 General Notation
8.2 Basic Matrix Operations
8.3 Matrix Transposition
8.4 Some Special Matrices

Chapter 9: Determinants and the Inverse Matrix
9.1 Defining the Inverse
9.2 Obtaining the Determinant and Inverse of a 3×3 Matrix
9.3 The Inverse of an n×n Matrix and Its Properties
9.4 Cramer’s Rule

Chapter 10: Some Advanced Topics in Linear Algebra
10.1 Vector Spaces
10.2 The Eigenvalue Problem
10.3 Quadratic Forms

Part IV Multivariate Calculus

Chapter 11: Calculus for Functions of n-Variables
11.1 Partial Differentiation
11.2 Second-Order Partial Derivatives
11.3 The First-Order Total Differential
11.4 Curvature Properties: Concavity and Convexity
11.5 More Properties of Functions with Economic Applications
11.6 Taylor Series Expansion*

Chapter 12: Optimization of Functions of n-Variables
12.1 First-Order Conditions
12.2 Second-Order Conditions
12.3 Direct Restrictions on Variables

Chapter 13: Constrained Optimization
13.1 Constrained Problems and Approaches to Solutions
13.2 Second-Order Conditions for Constrained Optimization
13.3 Existence, Uniqueness, and Characterization of Solutions

Chapter 14: Comparative Statics
14.1 Introduction to Comparative Statics
14.2 General Comparative-Statics Analysis
14.3 The Envelope Theorem

Chapter 15: Concave Programming and the Kuhn-Tucker Conditions
15.1 The Concave-Programming Problem
15.2 Many Variables and Constraints

Part V Integration and Dynamic Methods

Chapter 16: Integration
16.1 The Indefinite Integral
16.2 The Riemann (Definite) Integral
16.3 Properties of Integrals
16.4 Improper Integrals
16.5 Techniques of Integration

Chapter 17: An Introduction to Mathematics for Economic Dynamics
17.1 Modeling Time

Chapter 18: Linear, First-Order Difference Equations
18.1 Linear, First-Order, Autonomous Difference Equations
18.2 The General, Linear, First-Order Difference Equation

Chapter 19: Nonlinear, First-Order Difference Equations
19.1 The Phase Diagram and Qualitative Analysis
19.2 Cycles and Chaos

Chapter 20: Linear, Second-Order Difference Equations
20.1 The Linear, Autonomous, Second-Order Difference Equation
20.2 The Linear, Second-Order Difference Equation with a Variable Term

Chapter 21: Linear, First-Order Differential Equations
21.1 Autonomous Equations
21.2 Nonautonomous Equations

Chapter 22: Nonlinear, First-Order Differential Equations
22.1 Autonomous Equations and Qualitative Analysis
22.2 Two Special Forms of Nonlinear, First-Order Differential Equations

Chapter 23: Linear, Second-Order Differential Equations
23.1 The Linear, Autonomous, Second-Order Differential Equation
23.2 The Linear, Second-Order Differential Equation with a Variable Term

Chapter 24: Simultaneous Systems of Differential and Difference Equations
24.1 Linear Differential Equation Systems
24.2 Stability Analysis and Linear Phase Diagrams
24.3 Systems of Linear Difference Equations

Chapter 25: Optimal Control Theory
25.1 The Maximum Principle
25.2 Optmiziation Problems Involving Discounting
25.3 Alternative Boundary Conditions on x(T)
25.4 Infinite-Time Horizon Problems
25.5 Constraints on the Control Variable
25.6 Free-Terminal-Time Problems (T Free)

Answers

Index

Tags: 
Mathematics for Economists