# Mathematics for Economists. Made Simple

**subjects:**
mathematics and statistics

paperback, 366 pp., 1. edition

**published:** august 2010

**ISBN:** 978-80-246-1657-5

**recommended price:** 275 czk

### summary

As the field of economics becomes ever more specialized and complicated, so does the mathematics required of economists. With Mathematics for Economists, expert mathematician Viatcheslav V. Vinogradov offers a straightforward, practical textbook for students in economics?for whom mathematics is not a scientific or philosophical subject but a practical necessity. Focusing on the most important fields of economics, the book teaches apprentice economists to apply mathematics algorithms and methods to economic analysis, while abundant exercises and problem sets allow them to test what they?ve learned.

?For non-mathematicians who just use math in their professional activity I believe this is a very helpful source of knowledge, and also a very efficient reference.??Elena Kustova, Saint Petersburg University

Viatcheslav V. Vinogradov is a researcher at the Economics Institute of the Academy of Sciences of the Czech Republic and a consultant to the World Bank.

### table of contents

Preface

0 Preliminaries

0.1 Basic Mathematical Notation

0.2 Methods of Mathematical Proof.

0.3 Powers, Exponents, Logs and Complex Numbers

1 Linear Algebra

1.1 Matrix Algebra.

1.2 Systems of Linear Equations

1.3 Quadratic Forms

1.4 Eigenvalues and Eigenvectors

1.5 Diagonalization and Spectral Theorems

1.6 Appendix: Vector Spaces

2 Calculus

2.1 The Concept of Limit.

2.2 Differentiation - the Case of One Variable.

2.3 Rules of Differentiation

2.4 Maxima and Minima of a Function of One Variable

2.5 Integration.

2.6 Functions of More than One Variable

2.7 Multivariate Unconstrained Optimization

2.8 The Implicit Function Theorem.

2.9 (Quasi)Concavity and (Quasi)Convexity

2.10 Appendix: Matrix Derivatives

2.11 Appendix: Topological Structure and Its Implications

2.12 Appendix: Correspondences and Fixed-Point Theorems

3 Constrained Optimization

3.1 Optimization with Equality Constraints

3.2 The Case of Inequality Constraints

3.2.1 Non-Linear Programming

3.2.2 Kuhn-Tucker Conditions.

3.3 Appendix: Linear Programming.

4 Dynamics

4.1 Differential Equations.

4.1.1 Differential Equations of the First Order

4.1.2 Qualitative Theory of First-Order Differential Equations

4.1.3 Linear Differential Equations of a Higher Order with Constant Coefficients

4.1.4 Systems of First-Order Linear Differential Equations

4.1.5 Simultaneous Differential Equations and Types of Equilibria.

4.2 Difference Equations.

4.2.1 First-Order Linear Difference Equations

4.2.2 Second-Order Linear Difference Equations

4.2.3 The General Case of Order n.

4.2.4 Systems of Simultaneous First-Order Difference Equations with Constant Coefficients

4.3 Introduction to Dynamic Optimization

4.3.1 First-Order Conditions

4.3.2 Present-Value and Current-Value Hamiltonians

4.3.3 Dynamic Problems with Inequality Constraints

5 Exercises

5.1 Practice Problems

5.1.1 Problems

5.1.2 Answers.

5.2 Solved Problems

5.2.1 Linear Algebra.

5.2.2 Calculus.

5.2.3 Constrained Optimization

5.2.4 Dynamics

5.3 Economics Applications

5.4 Written Assignments

5.5 Sample Problem Sets

5.6 Unsolved Problems

5.6.1 More Problems

5.6.2 Sample Tests

### reviews

To my knowledge, in modern economics most scholars use mathematics, but for the majority of them math is rather a kind of toolbox than a rigorous science they would like to advance (I am not talking here about prominent cases such as, for instance, Arrow, Debrey, Intrilligator, Kamien, or Samuelson, though). Therefore, from my point of view the book exactly serves the purpose. It gives the reader fairly precise understanding of how to apply mathematical algorithms, tools and methods in economic analysis without getting into deep details of math background.

If I were asked whether this textbook would be helpful for Ph.D. students majoring in malhematics, I would probably be more skeptieal. but for non-mathematicians who just use math in their professional activity I believe this is very helpful source ofknowledge, and also a very efficient reference point. Furthermore, it was indeed a great idea to include a chapter solely dedicated to problems and solutions. Another value added is that the structure of this book clearly makes it a good text for distance learning students.

Elena Kustova, Professor at the Department of Mathematics and Mechanic, Saint Peterburg University

The book under review is designed as a textbook for the first-year doctoral students in economics, but at least some parts of it are equally suitable for mathematically oriented students in the MA program in economics and in other social sciences. It consists of four chapters covering virtually all areas of mathematics relevant to modern economic analysis. In addition, there is an introductory section reviewing basic mathematical prerequisites needed for entering the doctoral program, and a lengthy final section of exercises.

The introductory section,-labelled as Chapter 0-starts with outlining basic mathematical notation, continuing with explaining methods of mathematical proofs, and finishing with describing powers, logs and complex numbers. The knowledge of the contents ofthis section is indispensable for anybody claiming to be mathematically literate. A shiningjewel of it is the part on mathematical proofs which is exhaustive, remarkably clear and beautifully logical.

The first chapter covers linear algebra which is absolutely essential for most econometric work. The formulation of systems of linear equations in terms of matrix algebra is well explained and clearly shows the efficiency gained. Particularly commendable is the explanation of eigenvalues and eigenvectors whose abstract nature is frequently perplexing for students. The second chapter deals with basic calculus, starting with the concept of limit and ending with the notions of concavity and convexity. This is a crucial part of mathematics for economists as recognized by the leading US universities where the knowledge of calculus is a prerequisite for all undergraduates majoring in economics. The author does a very good job of covering this topic.

The subject ofthe third chapter ofthe book is constrained optimization, the basic and universal problem of economics. The recognition of this forces the students to approach each economic problem by identifying the economic agents, their objective functions and their constraints. The chapter deals well with both equality and inequality constraints and gives a good exposition of linear programming in the appendix.

The fourth chapter extends the realm of economic problems to dynamic situations. It starts with differential equations, goes on to difference equations, leading to dynamic optimisation which is quite a fashionable topic in modern economic research. The final section on dynamic programming is a special bonus for students and researchers interested in dynamic problems. An appendix on optimal control theory helps students comprehend its popularity initiated by Gregory Chow in the 1970s.

The last section ofthe book-labelled as Chapter 5-contains several sets ofexercises. These include practice problems (with solutions), solved problems for each of the preceding four chapters, problems involving economic applications, written assignments, sample problem sets, and unsolved problems. The whole section takes up almost one half of the whole book and is not only unique for a text of this kind but also invaluable. It is extremely well done and provides a wonderful resource for students in mastering mathematics needed for serious study of economics.

In my opinion the book is definitely commendable for the purpose for which it was written. The author has wisely decided to put emphasis on understanding over abstract proofing, which for economists would be more of an intellectual luxury than of practical use. This general orientation of the book makes it also a good reference text on the bookshelves of economic researchers.

Jan Kmenta, Professor Emeritus of Economics, University of Michigan and Visiting Professor CERGE-EI, Prague