On the Existence of Cournot Equilibrium without Concave Profit Functions*

This communication is concerned with the existence of equilibrium in Cournot’s model of oligopoly [2, Chap. VII]. This question, of course, has been examined often (see, e.g., [l-3]). To our knowledge, however, all previous treatments of the problem have assumed (either directly or indirectly) that the reaction curves of the firms are single-valued continuous functions or convex-valued, upper hemicontinuous correspondences, so that the Brouwer-Kakutani fixed point theorem may be used. In the constant marginal cost case, this assumption amounts to a condition that marginal revenue always be decreasing. Given some regularity conditions, marginal revenue will be falling at any profit-maximizing output. However, to assume that this condition holds globally is extremely restrictive. For example, one can easily construct examples in which it is not met even though the demand arises from a single competitive consumer with homothetic preferences.l We will consider here the case in which the price of the single homogeneous product is given by an upper hemicontinuous correspondence of the total production. Although this assumption is classical, it is nevertheless still restrictive, since it does not allow for general equilibrium effects. Moreover, we also assume costless production (although this