Mathematical Economics

Kelvin Lancaster

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486145044
• Topic: Mathematics
• Subtopic: Mathematics Research & Study
• Index: Mathematics

`An excellent book which should find wide use.` — Mathematics Reviews.
In this classic volume, a noted economist and teacher has combined a modern text for graduate courses in mathematical economics with a valuable reference book of analytical economics for professional economists.
Unique in its unified and careful presentation of a variety of techniques of economic analysis, the book is divided into two parts: chapters on mathematical economics (i.e. economic models analyzed primarily from the point of view of their mathematical properties) and appropriate mathematical reviews. To keep the exposition as smooth as possible, the economic analysis has been separated from the purely mathematical material — permitting flexible use of the book as a text. Moreover, the chapters and reviews are designed as a self- contained system, wherein the reviews contain all the mathematics required for the chapters and the chapters illustrate the use of almost all the techniques set out in the reviews. An extensive mathematical background is not required; however, it is assumed the reader has some acquaintance with elementary calculus.
The economic analysis covers linear and nonlinear optimizing techniques, input-output, activity analysis, neoclassical and set- theoretic static economic models, modern general equilibrium theory, the Von Neumann and other models of balanced growth, efficient growth and turnpike theorems, and modern stability analysis.
The mathematical reviews include discussions of set theory, linear algebra, matrices, linear equations and inequalities, convex sets and functions, continuous functions and mappings (including neoclassical calculus methods), topological ideas, differential and difference equations, calculus of variations, and related topics.
Every attempt has been made to give a complete and rigorous exposition (except in topological methods where the approach is descriptive and heuristic) which omits no essential proofs or steps in the argument.

CHAPTER
1

Introduction

1.1 MATHEMATICAL ECONOMICS

Mathematical economics is not a subject but an area of study within economics, closely affiliated with economic theory. Its scope is changing constantly, since it acts as a port of entry for new analytical techniques imported from mathematics (or engineering, or even other social sciences) on their way into the main body of economic analysis. Yesterday’s advanced mathematical economics is today’s mathematical economics, and will be tomorrow’s economic analysis. We have seen this process happen in the past, as undergraduate courses come to contain what was once regarded as too technical for the ordinary professor of economics, and the process will continue.

The more rapid the rate of import, the larger will be the inventory in transit. In the last twenty years we have had a tremendous flow of new techniques, with a corresponding growth in the scope of mathematical economics. In addition (to change the metaphor), mathematical economics is developing a rate of natural increase. As in other disciplines, economists are developing their own mathematical methods which are something more than simple direct applications of techniques well-known in other fields.

The well-equipped economic theorist must now know much more of mathematics than a few chapters from an advanced calculus text. One of his problems is the variety of different mathematical tools now at his disposal. These come from several areas of mathematics and are nowhere assembled together in a suitable fashion. A course in linear algebra will contain many things of no great interest to the economist and may leave out some things that are important to the economist but peripheral to the mathematician or physical scientist. The same is true of other areas of mathematics. One of the purposes of this book is to assemble and unify as many of these fragments of mathematics as possible.

However, mathematical economics is not just pieces of mathematics, it is the application of these to economic analysis and the development of new techniques to solve new problems. Economists have become genuine innovators in their adaptation of mathematical techniques to their own needs. This book aims to show this process of application, adaptation, and innovation at work.

The history of mathematical economics is yet to be fully written, since few historians of economic thought have been mathematically inclined. Briefly, we can distinguish three chief phases. The first was a period of important individual contributions, almost entirely neglected by the economics profession in general and the Anglo-American branch in particular. This was followed in the thirties by the growth and flowering of neoclassical mathematical economics which continued into the early fifties. The period of fifteen years or so since then has been a period of tremendous absorption of new techniques, leading to what can be regarded as the “new” mathematical economics.

Neoclassical mathematical economics, whose chief tools are the derivative and the equation, can be considered to be digested into the main body of economic theory, even though there remain many technical problems to interest the mathematically inclined. For the new mathematical economics, whose tools are vectors, convexity, sets, and inequalities, digestion has barely commenced. This book is designed to speed the process.

1.2 OUTLINE OF THE BOOK

The book contains, apart from this introduction, eleven chapters on topics in mathematical economics proper (that is, on economic models analyzed primarily from the point of view of their mathematical properties), followed by eleven mathematical reviews¹ designed to cover the required mathematics. The chapters and reviews are designed as a self-contained system, wherein the reviews contain all the mathematics required for the chapters and the chapters illustrate the use of almost all the techniques set out in the reviews. The mathematical background required, with references to the appropriate reviews, is specified at the beginning of each chapter.

The main chapters are grouped into three parts. Part One (Chapters 2 through 5) discusses optimizing theory generally, including linear programming, classical calculus, and Kuhn-Tucker theory. Part Two (Chapters 6 through 9) discusses various static economic models, including input-output (Leontief) models, activity analysis, advanced neoclassical models, set theory formulations, and modern general equilibrium theory. Part Three (Chapters 10 through 12) contains discussions of multisector dynamic models, including Von Neumann and other balanced growth models, optimal growth and turnpike theorems, and stability analysis.

The mathematical reviews (R1 through R11) cover linear algebra, inequalities, convex sets and cones, matrices, functions and mappings, some topological ideas, properties of special matrices, differential and difference equations, calculus of variations, and related topics. In all cases except in the sections involving topological methods and in Review R11 the treatment is complete, with no essential proofs omitted.

Throughout the book, the most complete and rigorous analysis possible is presented. On a few topics the treatment is heuristic only because rigor would require methods beyond the scope of the book or space that is not available.

There are a variety of other topics that might have claimed a place in a book of this kind. Space required selection, but the author believes he has included all topics that are of major importance in the current and recent literature of economic theory.

As a course text, the author suggests the following order:

(a)General Background and Optimizing Theory. Reviews R1, R2, R3, R4 (omitting Section R4.7), and R8 (omitting Sections R8.7 and R8.8), followed by Chapters 2, 3, 4 (noting only results in Section 4.5), and 5 (Sections 5.1 through 5.5).

(b)Basic Economic Models. Reviews R5, R6 (perhaps omitting Section R6.3), and R7 (Sections R7.1 through R7.3), followed by Chapters 6, 7, 8, and 10 (omitting Section 10.5).

(c)More Advanced Topics (in any order), (i) Growth Theory: Reviews R8 (Sections R8.7 and R8.8), and R11 (Sections R11.1 and R11.2) followed by Chapters 10 (Section 10.5) and 11; (ii) Stability Theory: Reviews R10, and R7 (Section R7.4), followed by Chapter 12; (iii) General Equilibrium: Review R9, followed by Chapter 9.

(d)Tidying Up. Review R6 (Section R6.3), followed by serious study of Section 4.5 of Chapter 4, Chapter 5 (Sections 5.6 and 5.7), Review 7 (Section R7.5), and Review R11 (Sections R11.3 and R11.4).

From the author’s experience, the basic material and some of the advanced topics can be fitted into a two-semester course, given the availability of the present text. The material in the mathematical reviews often exceeds the minimum background required for the relevant chapters and some discretion can be exercised in treating this material in greater or less depth.

1.3 NOTES ON THE LITERATURE

There is no book with coverage similar to this, but the following books in the same general field are valuable additional reading. All are referred to in the text at appropriate places. They are classified as more elementary, at approximately the same level, or more advanced, in comparison with this book. The clas...