Mathematical Optimiation in Economics

Selon Schumpeter (History of Economic Analysis) le premier auteur qui a tenté une définition mathématique de l'EQUILIBRE est ISNARD sans son Traiteé des Richesses (1781) “qui attend encore dans l'histoire de la science économique la situation qui lui est due” (celle de précurseur de Walras).

II s'agit de montrer comment les valeurs se constituent à partir des quantités disponibles et des besoins. Dans l'échange entre deux agents économiques, c'est clair: si l'on échange tant de litres de vin contre tant de mesures de blé, les termes mêmes de cet échange définissent le rapport en question. Ce qui importe c'est la généralisation, pour laquelle l'expression mathématique est nécessaire.

1. Introduction. The most simple locational problem has its mathematical origin in classical geometry, where it is known as Steiner's Problem [l]. It appeared, in a slightly generalized form, in the pioneering work Über den Standort der Industrien of Alfred Weber [2]. This form qf the problem, which we shall call the Steiner-Weber Problem, asks for a point in the plane that will minimize the weighted sum of distances to n given points in the plane. In spite of the simple and explicit form of the problem, relatively little is known about its solution, either analytically or computationally. The purpose of this paper is to discuss the problem from the point of view of mathematical programming. In Section 2, certain general properties and a set of necessary and sufficient conditions for a solution are derived. In Section 3, a problem dual to the Steiner-Weber Problem is formulated. This problem has a linear objective function and quadratic constraints; it possesses all of the desirable properties of the dual in linear programming and its solution yields a solution of the Steiner-Weber Problem trivially. In Section 4, some preliminary conclusions concerning computation are presented. Economic applications are the subject of a joint paper with R.E. Kuenne [3], to be published shortly; detailed computational methods will be treated in a later paper.

I devote this lecture to presenting the classical Walrasian model in its smallest scale and to examining it for the existence of a growth equilibrium. We assume that the economy we are going to be concerned with consists of many firms that are classified into two industries: the consum-tion-good industry and the capital-good industry.

It is assumed that a finite number of manufacturing processes are available to each industry. Let ∝_ι and λ_ι be the capital- and labour- input coefficients of the ι-th process of the consumtion-good industry (ι, = 1,…μ), and a_i and l_i the corresponding coefficients of the i-th process of the capital-good industry (i=l …, m). (We are following Professor Hicks in denoting prices and quantities referring to the consumption-good sector by Greek letters and those to the capital-good sector by the corresponding Latin letters). For the sake of simplicity we assume that all ∝'s, λ's, a's and l's are positive and that the capital-good does not suffer wear and tear.

The aim of our lecture is to make known some experiments which were and are carried out by Hungarian economists and mathematicians in order to apply the mathematical programming methods to the preparation of economic plans aiming at more favourable proposals, than those based on traditional planning methods.

We do not mention in our lecture other mathematical planning methods, - e.g. the input-output analysis - which are also used in Hungary. We speak only about the application of mathematical programming methods. Furthermore we pass over the wide-spread, intrafirm results in operations research, which apply sometimes programming techniques as well. We deal only with more wide-ranging problems: planning a whole industry, more industries or the economy as a whole.

Different kinds of stochastic programming models are formulated in the present mathematical programming literature. Their solutions lead to linear or non-linear deterministic programming problems. There are, however, a number of problems, mainly probability distribution problems, which remained unsolved which are nevertheless important and necessary to solve in order to be able to handle effectively these stochastic optimization problems. The main types of stochastic programming problems are the following.

a) Chance constrained programming. The problem is to minimize the expectation of a functional z(c, x) under the condition that where g is a certain function of the elements of the matrix A and of the elements of the unknown vector x. ∝ is a prescribed probability usually near 1. In many practical cases the above problem reduces to the simpler form: subject to the condition The matrix A and the vector b are partly or entirely random. Among several papers where this problem is investigated we mention [20] and [2l].

I shall start by giving you a bird's-eye view of the whole problem, adopting a general and philosophical attitude with absolutely no mathematics. Later you will have mathematics and just as much as you want. But today I will speak in general terms.

What is rational and democratic planning ? I will give you my personal views on this matter. It will be my personal views because I think that much of what is currently called economic planning does not deserve that name.