A general model of price competition with soft capacity constraints

We propose a general model of oligopoly with firms relying on a two factor production function. In a first stage, firms choose a certain fixed factor level (capacity). In the second stage, firms compete on price, and adjust the variable factor to satisfy all the demand. When the factors are substitutable, the capacity constraint is "soft", implying a convex cost function in the second stage. We show that there is a unique equilibrium prediction in pure strategies, whatever the returns to scale, characterized by a price that increases with the number of firms up to a threshold. The main propositions are established under the general assumption that the production function is quasi-concave but the paper provides a general methodology allowing the model to be solved numerically for special parametrical forms.