Edgeworth Bounds for Oligopoly Prices

Abstract

Edgeworth's analysis of the theory of exchange by two individuals with different initial holdings of two commodities, where he determines the set of outcomes of mutually advantageous trading from which no further mutually advantageous trading is possible (the contract curve), has long been a standard part of the literature of economic theory. His extension of this analysis to the case of an arbitrarily large number of individuals of each of the two (that is, having the same initial holding of the two commodities and the same tastes) and his demonstration that the contract curve shrinks to the set of competitive equilibria was long neglected. However, during the past decade the importance of this extension has become widely appreciated, and much highly sophisticated work has been done on its generalization. It was first generalized to cover exchange of any finite number of commodities and any finite number of types of individual, provided only that the number of individuals of each type became simultaneously arbitrarily large. Later, it was shown that, in markets containing a continuum of individuals, there was no need to restrict these individuals to a finite number of types. The possibility of production as well as pure exchange was also incorporated, and the level of mathematical rigour was increased.2 Thus, Edgeworth's simple model was so generalized that it could take its deserved place of honour in modern general equilibrium analysis. However, valuable though all this generalization was, it is not general but partial equilibrium analysis that has been, and probably still is, the main preoccupation of economists working at less than the highest level of abstraction; and the application of the Edgeworth analysis to the problems of partial equilibrium analysis remains neglected. It is the object of this paper to consider one of the more important questions in partial equilibrium analysis-the effect on the price and output of an oligopolistic industry of a modest increase in the number of firms-using simple, largely geometrical techniques, closely based on Edgeworth's own analysis.