More Limit Pricing:
An incumbent firm (player 1) is either a low-cost type θ1 = θL or a high-cost type θ1 = θH, each with equal probability. In period t = 1 the incumbent is a monopolist and sets one of two prices pL or pH , and its profits in this period depend on its type and the price it chooses, given by the following table:
After observing the period t = 1 price, a potential entrant (player 2), which does not know the incumbent's type but knows the distribution of types, can choose to enter the market (E) or stay out (O) in period t = 2. The payoffs of both players in period 2 depend on the entrant's choice and on the incumbent's type and are given by the following table:
At the beginning of the game the incumbent discounts profits for period t = 2 using a discount factor δ ≤ 1.
a. Draw the extensive-form game tree of this game and write down the corresponding matrix.
b. For δ = 1 find a pooling perfect Bayesian equilibrium of the game in which both types of player 1 choose pL in period t = 1.
c. Find the range of discount factors for which a separating perfect Bayesian equilibrium of the game exists in which type θL chooses pL and type θH chooses pH in period t = 1.