Existence of Marginal Cost Pricing Equilibria in Economies with Several Nonconvex Firms

Abstract

This paper considers a general equilibrium model of an economy where some firms may exhibit increasing returns to scale or more general types of nonconvexities. The firms are instructed to follow the standard pricing rule or to fulfill the first-order necessary conditions for profit maximization. A general existence theorem of equilibria is proved in the case of an arbitrary number of firms. No assumption is made to imply the aggregate productive efficiency of equilibria, a condition that must be excluded in the nonconvex case. THE FAILURE OF THE COMPETITIVE MECHANISM in the presence of increasing returns to scale has been recognized since Marshall (1920) and the search for alternative mechanisms has resulted in the so-called theory of marginal pricing. This theory was developed by Allais (1953), Hotelling (1938), Lange (1936), Lerner (1936), Pigou (1932) (to remain in the thirties and forties), but has a long history dating back to the discussion on public utility pricing by Dupuit (1844); it states that an optimum of welfare corresponds to the sale of everything at cost (Hotelling (1938)). This statement of the second welfare theorem has since been formulated at a high level of generality, starting with the work by Guesnerie (1975) and more recently by Bonnisseau and Cornet (1988b), Cornet (1986, 1988a, b, c), Khan and Vohra (1987, 1988), Quinzii (1986). None of these papers, however, is concerned with the existence of equilibria; this has been considered in the past ten years by Beato (1979, 1982), Beato and Mas-Colell (1985), Brown and Heal (1982), Brown et al. (1986), Cornet (1982), Dierker, Guesnerie, and Neuefeind (1985), Mantel (1979), and more recently by Bonnisseau, Bonnisseau-Cornet, Kamiya, and Vohra in the special issue of the Journal of Mathematical Economics (1988) on general equilibrium theory and increasing returns. The purpose of this paper is to provide a general existence theorem for an economy with an arbitrary number of convex and nonconvex producers which may exhibit increasing returns to scale or even more general types of nonconvexities. In our model, (a) the consumers maximize their preferences subject to their budget constraints, (b) the convex producers maximize their profits while the nonconvex producers are instructed to follow the pricing rule, i.e., they