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. 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 exists a continuum of subgame perfect equilibria in pure strategies, whatever the returns to scale. Among them, a payoff dominant one can always be selected. The equilibrium price may increase with the number of firms.