Activity Analysis, Quadratic Programming and General Equilibrium

Abstract

THE TASK UNDERTAKEN in this paper is the presentation of a quadratic programming problem whose solution if it exists is a competitive equilibrium for the economy described by the model. In the first section, some general comments concerning the relationship between the model and accepted economic theory will be made. The second section is devoted to two theorems which constitute the necessary mathematical ground work. The first is a duality theorem for concave programming; the second deals with the basic characteristics of a special form of a quadratic programming problem. The third section consists of outlining the model and proving the main proposition. Finally, a hypothetical numerical example is given. In examining the relations between our model and the relevant body of economic theory, it will be useful to classify the decision-makers in the economy under consideration into two groups: suppliers and consumers. There is, of course, nothing new about such a classification. In fact, such classification is used in every work cited in the context of the present discussion. Starting with the suppliers, the point of origin is the famous article by Professor Koopmans [91 in which he presented the linear activity analysis model. Dealing with the producing sector of the economy, as described by a finite set of linear production activities, it is proven that under a certain set of postulates an efficient production pattern exists. Furthermore, after identifying the necessary and sufficient conditions for efficiency, Koopmans shows that every efficient point can be found by solving a linear programming problem with the appropriate set of prices and that the solution of every linear programming problem is efficient. Later on, Koopmans and Bausch [11] showed this two-way connection between efficiency and linear - programming as a special case of the relation between competitive equilibrium and Pareto Optimum. The limitation which makes it a special case results from the fact that no decision-making by consumers is present. Hence the definition of competitive equilibrium, as given for instance by Koopmans [10 (46)], is transformed into one suitable for linear activity analysis,' and the concept of Pareto Optimum is translated to efficiency. As will be seen shortly, this is the main point of departure in the following analysis, for in the proposed model the consuming sector is explicitly treated. The second relevant body of economic theory is the theory of consumer demand. The point of origin here is Debreu's [3] proof of the existence of