Optimal symmetric punishments in a Bertrand differentiated products duopoly

This paper studies how product differentiation affects collusive pricing in a duopoly when products are horizontally differentiated and firms compete in prices. In contrast to earlier work, the punishment mechanism supporting collusion is assumed to be the optimal symmetric punishment proposed by Abreu (1986, Journal of Economic Theory 39, 191-225). Such punishments have previously been characterized only for Cournot oligopolies and for Bertrand oligopolies with capacity constraints. The optimal symmetric punishment turns out to have a simple two-phase structure. After one period of an intense price war, firms return to the collusive outcome. If prices are restricted to be non-negative and the discount factor is large, the price war will be extended to more than one period. The best collusive price is shown to be lower, the more substitutable are products.