The competitive outcome as the equilibrium in an Edgeworthian price-quantity model

This paper considers a model of price-setting oligopoly with perfectly informed consumers, where firms have strictly-convex cost functions. In the standard Bertrand-Edgeworth model, there exists no pure-strategy Nash equilibrium. The author allows firms to choose both price and the quantity that they are willing to sell, output being the minimum of this quantity and demand. Firms cannot offer to sell a quantity that would bankrupt them. The paper shows that if there are enough firms, then an equilibrium exists and, in all equilibria, firms set the competitive price and each produce their competitive output