A General Model of Price Competition with Soft Capacity Constraints

Abstract

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.

Chamberlin without differentiation: Soft capacity constrained price competition with free entry

“…Because the capacity factor and the variable factor are substitutable, above‐capacity production is always possible but at an increasing marginal cost (i.e., the capacity constraint is “soft”). As demonstrated by Cabon‐Dhersin and Drouhin (2020) under very general assumptions, this framework produces highly unconventional results, two of which are particularly important to illustrate our point: (1) the noncooperative equilibrium of the nonrepeated game is collusive (and thus can be found as a solution to a profit maximization program). (2) The model has an equilibrium whatever the returns to scales.…”

“…Thus, when $$\rho $$ is strictly lower than one, there is a threshold value of $$z$$ at which the returns to scale changes from being increasing (beneath the threshold) to decreasing (above), leading to a U‐shaped average cost function, as illustrated in Beattie and Aradhyula (2015). Because our general model of price competition (Cabon‐Dhersin & Drouhin, 2020) disentangles the existence of equilibrium from the nature of the returns to scale, the Stone–Geary production function with a minimum firm size can rightfully be used to investigate certain industrial organization problems.…”

“…The results described in Proposition 1 make the equilibrium quite easy to compute because the equilibrium of the noncooperative game is a solution of the constrained optimization program. In the following, we use the general calibration procedure outlined on page 106 of Cabon‐Dhersin and Drouhin (2020).…”

“…To do this, we use a model of price competition with soft capacity constraints (Cabon‐Dhersin & Drouhin, 2014, 2020) that expand on Dastidar (1995) by assuming that firms rely on two substitutable factors of production chosen sequentially. In the first stage, firms choose a capacity factor.…”

“…To stay as close as possible to the original Chamberlinian framework, we use a Stone–Geary production function that generates a U‐shaped average cost function. Since this function is nonhomothetic, this function is particularly interesting to study free‐entry equilibria not considered in Cabon‐Dhersin and Drouhin (2020).…”