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By Kelvin Lancaster
Dover Publications, Inc.Copyright © 1968 Kelvin Lancaster
All rights reserved.
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 reviews 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 classification refers only to level of exposition, not to content.
Baumol, Economic Theory and Operations Research (Baumol)
Baumol gives an excellent conspectus of many of the techniques developed in recent years and of some of their application to economics. It is especially useful for obtaining a general idea of the nature of certain topics not covered in this book. Its chief use is as a survey.
Dorfman, Samuelson, and Solow, Linear Programming and Economic Analysis (Dorfman, Samuelson, and Solow)
In the earlier chapters on linear programming and its direct applications, the authors tied their hands by avoiding the use of matrices and vectors. Later chapters contain material of great importance, especially Chapters 12 (growth theory) and 13 (general equilibrium). Chapter 13 is the only attempt at an exposition of general equilibrium theory at a level more elementary than Chapter 9 of this book.
Allen, Mathematical Economics (Allen)
Allen contains an excellent coverage (Chapters 1 through 9) of single-sector economic models with complex dynamic specifications, as contrasted with the models of this book, which are multisector with simple dynamic specifications. The coverage of linear methods is broad, but the exposition is not in the most desirable form.
Approximately the Same Level
Gale, The Theory of Linear Economic Models (Gale)
The author regards this as the best exposition of the theory of linear programming among the books listed. Gale also discusses balanced growth models. Gale's notation is not standard, since he uses a row vector where column vectors are more usual, and vice versa. His matrix-vector relationships must be transposed to be comparable with the usage of this book.
Karlin, Mathematical Methods and Theory in Games, Programming, and Economics (Karlin)
This book covers much ground, in a compressed fashion. Chapters 8 and 9 are filled with miniature gems of mathematical economics, none of which is quite complete as an exposition. Karlin's notes are of interest and his references extensive. This is highly recommended as a next book to read after the present one, although only Chapters 5, 7, 8, and 9 and the three appendices are concerned with topics covered here.
Debreu, Theory of Value (Debreu)
Debreu gives the most complete account of the modern generalized treatment of production, consumption, and general equilibrium theory. The exposition is rigorous in style but often has important steps in the argument left almost unexplained. It is essential but difficult reading for those who wish to follow up the discussion on general equilibrium given in Chapter 9 of this book.
Morishima, Equilibrium Stability and Growth (Morishima)
This contains important material on growth theory. It is quite closely argued, and should not be read until after Chapters 10 and 11 of this book.
In addition to the works mentioned, more specialized references are given in footnotes throughout the book. These are not designed to be a bibliography of the subject but merely to lead the reader into further literature in the field. Sometimes the references are to original papers, sometimes to secondary sources. Any issue of Econometrica will supply the reader with other applications of the methods set out in this book.CHAPTER 2
The General Optimizing Problem
Most of this chapter requires only a general background on sets and functions, such as given in Review R1. Section 2.6 requires additional knowledge of convex and concave functions (Review R8, Section Sections R8.5) and convex sets (Review R4, Sections R4.1 and R4.2).
Optimizing, a catch-all term for maximizing, minimizing, or finding a saddle point, lies at the heart of economic analysis. In passive economic models (such as general equilibrium studies), we are interested in the optimizing behavior of decision makers in the economy. In active models (such as models of efficient growth), we are interested in finding the optimum ourselves. Much of the history of economic theory has been a process of movement from deterministic to optimizing models. In recent years, this trend has been apparent in the movement from deterministic input-output to optimizing activity analysis models, and from naive deterministic growth models to the study of optimal and efficient growth paths. Even in macroeconomic policy-oriented models, where optimizing has been mainly confined to parameter estimation, the trend is toward more sophisticated optimal policy models.
Indeed, optimization subject to constraint has been considered by many as defining the essential nature of economics.
Because of the importance of optimizing, and because the considerable development of optimizing theory in the last twenty years has led to a variety of different techniques of analysis, it seems desirable here to give a general description of the structure of the optimizing problem and of how the different techniques fit into this structure.
2.2 THE GENERAL STRUCTURE
The variables of the problem will be considered to be in the form of a vector in Rn. In addition to this vector, x, we have:
(a) a feasible set K. Only x [member of] K is to be taken into account in the problem.
(b) a single valued continuous objective function, f(x), whose value for x [member of] K is to be optimized.
Thus we can state a typical maximizing problem in formal terms as
Find x* [member of] K such that f(x*) [??] f(x), all x [member of] K.
If such an x* exists, the problem has a weak global maximum—weak because it satisfies the weak inequality, global because the inequality is satisfied for all x [member of] K. A global optimum should not be confused with an unconstrained optimum. The latter implies that K = Rn. We would have a strong maximum if we could find x* such that f(x*) >f(x), all x [member of] k.
A weak optimum is equivalent to a nonunique optimum point since any x satisfying f(x) = f(x*) is also an optimum point. A strong global optimum implies a unique optimum.
If we reverse the inequalities we obtain a minimum, weak or strong as the case may be. A minimum for f(x) implies a maximum for [—f(x)]. The value x* is often called simply the solution of the optimum problem. To avoid confusion with other claimants for the same name in many economic models, we shall usually call it the optimal solution or optimum point.
Most calculus techniques cannot solve the problem as set out above, but can only solve a problem of the following kind:
Find x* [member of] K such that f(x*) [??] f(x), all x [member of] (E [intersection] K), where E is a neighborhood of x*.
Such a point is a weak local maximum. We can have a strong local maximum and weak or strong local minima. Some authors use the terms relative and absolute rather than local and global.
It is obvious that, if f(x) has an optimum at all, it must have a global optimum and that this must also be a local optimum. On the other hand a local optimum is not necessarily global. Our interest is primarily in the global optimum which, after all, is the optimum. Thus we are interested in conditions on the structure of the problem that will guarantee that a local optimum is also global. Such conditions (applicable to many economic cases) are given later in the chapter. If they are not satisfied, we may need to adopt ad hoc procedures (such as enumerating and comparing all local optima) to locate the global optimum.
The problem itself determines whether the optimum is weak or strong—there is no question of trying to locate the strong optimum if it is essentially weak. It should also be noted that a strong local optimum does not imply a unique optimum, since f(x) may take on the optimal value at several distinct points, each being a strong local optimum.
2.3 CONSTRAINTS AND THE FEASIBLE SET
The feasible set, over which the variables are permitted to range, may be defined in any suitable way. In the case of discrete variables, the feasible set may even be described by enumeration. Typically, however, the feasible set will be defined by equalities or inequalities involving relationships between the variables. The relationships which define the feasible set are the constraints of the problem.
A single constraint defines some set of values of the variables. If there is more than one constraint, the permissible values of the variables must satisfy all constraints, so that the feasible set is the intersection of all the sets defined by the individual constraints. In Figure 2–1 (a) we have two constraints:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The first constraints the variables to the disk of radius k, and the second to a rectangular hyperbola with two branches and to the area of the plane cut off from the origin by the hyperbola.
The feasible region in this case consists of two lens-shaped areas, heavily outlined in the diagram; one is in the positive and one is in the negative quadrant. Typically, we would be interested in only one of these areas, usually because of other constraints of the form x1, x2 [??] 0, which would eliminate the area in the negative quadrant.
We can distinguish constraints of the type x1, x2 [??] 0 as direct constraints on the variables. Usually they appear as nonnegativity constraints and can always be put in this form by a simple linear transformation of the variables. We shall refer to the type of constraint represented by the circular and hyperbolic areas as functional constraints. A more typical relationship between functional and direct constraints than in Figure 2–1 (a) is given by Figure 2–1 (b), where the functional constraints give a disk and the direct constraints confine the variables to a quadrant of the disk.
Excerpted from Mathematical Economics by Kelvin Lancaster. Copyright © 1968 Kelvin Lancaster. Excerpted by permission of Dover Publications, Inc..
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