Equilibria in an infinite horizon game with an incumbent, entry and switching costs

We consider an infinite horizon game in which a consumer incurs a switching cost when changing supplier. There is an incumbent firm at the beginning of the game, and a new entrant in each period, all with a production cost of 0. We identify the set of possible profits of the incumbent when the common discount rate of all the agents is greater than 1/2: it is the interval [0, Sigma/(1 - delta)]. There exist equilibria in which the incumbent sells the good to the consumer in every period at any price in [0, Sigma]. The result also holds with several entrants and several consumers.