Cournot Equilibrium with Free Entry

Abstract

Despite the fact that the assumptions underlying perfect competition never actually hold, the use of the competitive model, as an idealization, is justified if the predictions of the model approximate the outcomes of situations it is used to represent. In partial equilibrium analysis, this justification is embodied in the Folk which states that if firms are small relative to the market, then the market outcome is approximately competitive. This paper provides a precise statement and proof of the Folk for competitive markets with a single homogeneous good, and free entry and exit. It is shown that if firms are small relative to the market then there is a Cournot equilibrium with free entry; furthermore, any Cournot equilibrium with free entry is approximately competitive. More specifically, if we consider an appropriate sequence of markets in which firms become arbitrarily small relative to the market, then there is a Cournot equilibrium with free entry for all markets in the tail of the sequence, and aggregate equilibrium output converges to perfectly competitive output. If firms have strictly U-shaped average cost curves, then individual firm behaviour converges to competitive behaviour. The treatment of free entry distinguishes this paper from other papers dealing with the Folk , where either the number of firms is exogeneous, ruling out free entry, or free entry is treated as being equivalent to a zero profit condition, ignoring the integer problem that arises when the number of firms is finite but unspecified. Firms may become small relative to the market in two ways: through changes in technology, absolute firm size (the smallest output at which minimum average cost is attained) may become small, or, through shifts in demand, the absolute size of the market (the market demand at competitive price) may become large. We allow both types of changes here, though shifts in demand, especially in the form of replication of the consumer sector, may be more familiar. In his conclusion, Ruffin (1971) presents a verbal argument for the Folk which is based on replication of demand and entry. Hart (1979), though not concerned with existence, shows that in a general equilibrium model with differentiated products and free entry, equilibria are approximately competitive (Pareto optimal) when consumers have been replicated a sufficient number of times. The paper is organized as follows: Section 1 contains the perfectly competitive model and its assumptions, Section 2 contains the assumptions and definitions for the imperfectly competitive model, Section 3 contains an example contrasting the usual treatment of the Folk Theorem and the present approach, Section 4 contains the proofs of the main results, and Section 5 contains remarks on the results and indicates how some of the assumptions that are used can be weakened.