Asymmetric Bertrand-Edgeworth Oligopoly and Mergers

Abstract

This paper investigates mixed strategy equilibria in a capacity-constrained price competition among three firms. It is shown that the equilibria in an asymmetric oligopoly are substantially different from those in a duopoly and symmetric oligopoly. In an asymmetric triopoly, it is possible that (i) a continuum of equilibria exists and that (ii) the lowest price of the smallest firm is higher than that of the others and the smallest firm earns more than the max-min profit in undominated strategies. In particular, the second finding sheds light on a new pricing incentive in Bertrand competitions. As an application, the equilibrium characterizations give rise to a new class of merger paradoxes.

Horizontal Mergers in the Presence of Capacity Constraints

“…Then the demand at the price equal to marginal cost is D (0), which is assumed to be a finite number. As is common in the literature on the Bertrand‐Edgeworth model (e.g., De Francesco and Salvadori , Hirata , Vives ), we assume that D (·) is decreasing, concave, and twice continuously differentiable. These assumptions imply that pD (·) is strictly concave in p .…”

“…Surprisingly, very few authors have studied the effects of merger in the Bertrand‐Edgeworth model despite its prominence in the oligopoly theory . Our literature search has uncovered only two such studies, namely Davidson and Deneckere () and Hirata (). Davidson and Deneckere () study the impact of merger on tacit collusion in a Cournot model and a Bertrand‐Edgeworth model with a linear demand function.…”

“…Davidson and Deneckere () study the impact of merger on tacit collusion in a Cournot model and a Bertrand‐Edgeworth model with a linear demand function. Hirata () contains a brief analysis on the possibility that a merger could be unprofitable when firms face capacity constraints. Neither of them, however, directly studied the price effect of a merger when firms act independently (i.e., the unilateral effect of merger)…”

Does timing of decisions in a mixed duopoly matter?

“…Even the existence of an equilibrium of multi-period games with exogenously given ordering of moves is not known for the case in which at least pairs of firms move in different time periods. Most recent results on the mixed-strategy equilibria of the simultaneous-move Bertrand-Edgeworth oligopoly game by Hirata (2009) andDe Francesco andSalvadori (2010) point to the difficulty of the problem.…”

A Bertrand–Edgeworth oligopoly with a public firm

“…Although the equilibrium of the relatively simple symmetric oligopoly game with identical firms was solved by Vives (1986), the equilibrium of the general triopolistic case was obtained only recently, independently by Hirata (2009) andDe Francesco andSalvadori (2010). In case of n ≥ 4, the full characterization of the equilibrium is still undone but some of its important properties were derived by De Francesco and Salvadori (2010), which enable us to compare the purely private oligopoly with the mixed oligopoly in terms of social surplus.…”