Proramming, Pareto Optimum and the Existence of Competitive Equilibria

Abstract

One of the important recent developments of economic theory is the modern formulation of the theory of competitive equilibrium. In parallel with this development we have also encountered a marvelous development of the theory of non-linear programming. The purpose of this paper is to relate these two developments. Specifically, we shall prove the optimality and the existence of competitive equilibria as a straightforward application of the theory of non-linear programming. Not only will this paper give a unified treatment of the problem of existence, welfare economics and the theory of programming, but also it will give an analysis which is fairly simple and straightforward, and yet it shall retain the generality of the model of competitive market fairly well.We should mention one important predecessor in such an attempt. Negishi, in his ingenious paper [IO], proved the existence of competitive equilibria by using the theory of non-linear propgramming (I). His formulation is based on the quasi-saddle point characterization of the constraint maximum problem. Hence he assumcd, among others, the existence of the right hand and left hand derivatives of each producer's production function (his F J . Moreover the use of the quasi-saddle point characterization complicated his formulation and the proof of the existence a great deal. In the present paper, we entirely avoid the use of each producer's production functionhence we do not impose any conditions on such functions I;, (conditions such as concavity and the existence of the right hand and left hand derivatives of each Fk as imposed by Negishi [IO], except for the (*) This paper is developed from the discussions between the authors during the summer of 1966. The actual writing is done by Takayama for his lecture at Purdue University in the fall of 1966, hence hc takes the entire responsibility for any errors which may exist in the paper. We are grateful to Professor Takashi Negishi for valuable comments.(1) One of the important b product of such an approach is that we can avoid the concepts of deman~-~orrespondence and supply correspondence altogether.

Negishi's contributions to the development of economic analysis: Research programs and outcomes

“…What has been called Negishi's “approach” or “method” (1960a) has been widely utilized on both theoretical and applied levels, the former in the context of dynamic models and infinite dimensional general equilibria; the latter in the context of their numerical computation, as will be seen below. Similar results to Negishi's 1960 proof of the existence of equilibria based on what can be termed mathematical programming techniques, were obtained by Takayama and El Hodiri (1968) and Diewert (1970).…”

Computing Equilibria in General Equilibrium Models via Interior-point Methods

“…for a certain δ * such that δ * 0 and transfers Takayama and El Hodiri (1968) and Diewert (1970) also provided a similar result based on these techniques. A demonstration of the existence of equilibrium when firms have a general objective to maximize can be found in Esteban-Bravo (2000).…”