An incumbent firm may be a low-cost type, with constant marginal cost of production 10, or a high-cost type, with marginal cost of production 20, with probabilities t and , respectively. The incumbent's type is private information. The incumbent produces as a monopolist in the first period. An entrant who has marginal cost 15 may enter the market between periods. Entry requires at least a small fixed investment. If the entrant comes in the market, it learns what the incumbent's marginal cost is, and firms engage in Bertrand competition in homogeneous products in the second period (see Chapter 12 for a discussion of Bertrand competition). Consumer demand is the same in each period. Suppose there is no discounting between periods, so the incumbent's objective is to maximize the sum of first- plus second-period profit.
What is the Nash equilibrium of the second-stage game if the entrant enters? Solve the game for each type of incumbent.