Multilateral Limit Pricing in Price-Setting Games

Abstract

It is well-known that the existence and uniqueness of the Cournot equilibrium extend to environments where firms prefer not to be active. However, we show that differentiated Bertrand oligopolies with linear demand, constant unit costs, and continuous best responses do not need to satisfy supermodularity, the single-crossing property (SCP), or log-supermodularity (LS). Moreover, best responses might have negative slopes and there is a continuum of pure strategy Bertrand-Nash equilibria on a wide range of parameter values when the number of firms is more than two. These results are very different from those in the existing literature on Bertrand models with differentiated products, where uniqueness, supermodularity, the SCP, and LS usually hold under a linear market demand assumption, and best response functions slope upward. We fully characterize the set of pure strategy equilibria. We provide an iterative algorithm to find the set of players that are active in any equilibrium and show that this set is the same in all equilibria. Various applications and extensions of the findings are provided in the contexts of market entry and exit.