The Existence of Equilibrium with Price-Setting Firms

Ever since Joseph Bertrand (1883), economists have been interested in static models of oligopoly where firms set prices. Francis Edgeworth's 1925 critique of Bertrand recognized, however, that, except in the case of constant marginal costs, there are serious equilibrium existence problems when firms produce a homogeneous good. In particular, Edgeworth proposed a modification of Bertrand's model in which firms have zero marginal cost up to some fixed capacity. He showed that, unless demand is highly elastic, price equilibrium may fail to exist. Mixed strategies provide one way of avoiding this nonexistence problem, as various authors have noted. Martin Beckmann (1965), for instance, explicitly calculated mixed strategy equilibria in a symmetric example of the Bertrand-Edgeworth model. However, a general treatment of mixed strategies has suffered from the fact that the standard equilibrium existence lemmas (see, for example, K. Fan, 1952, and I. Glicksberg, 1952), require continuous payoff functions, whereas, in price-setting oligopoly, payoffs are inherently discontinuous the firm charging the lowest price captures the whole market. Recently, Partha Dasgupta and I (1986) and Leo Simon (1984) developed several existence theorems for discontinuous games. Dasgupta and I used one of the theorems to establish the general existence of mixed strategy equilibrium in the Bertrand-Edgeworth model when market demand as a function of price is continuous, downward sloping, and equal to zero for a sufficiently high price. In their study of Bertrand-Edgeworth competition in large economies, Beth Allen and Martin Hellwig (1983) extended this result to demand curves that do not necessarily intersect the horizontal axis and need not slope downward. There have been several treatments of cost functions more general than the BertrandEdgeworth variety. R. Gertner (1985) established the existence of symmetric equilibrium in a model where firms are identical, have convex or concave costs, and choose output levels at the same time as prices. Also in a model of identical firms, H. Dixon (1984) proved existence when firms have convex costs and produce to order, that is, produce after other firms' prices are realized. In this paper, I present some existence results that do not require symmetry and permit fairly general cost functions. These findings pertain both to the simultaneous choice of price and production level and to the formulation where a firm's output is set only after it knows others' prices. Existence in the former case is proved by direct application of the Dasgupta-Maskin/Simon theorems, as are the existence results mentioned above. The latter case, however, requires some additional argument. I give a sketch of this argument below; the details and more general results can be found in Dixon and myself (1986).