Excercise:3 Two firms, i = 1, 2, sell differentiated goods. The firms set prices PI and P2 simultaneously. We assume that prices have to be non-negative, pi, p2 > 0. The demand function facing firm i is given by
qi(pi,pj) = 2 — pi + pj
(do not worry about negative demand when doing this exercise - in equilibrium demand will be positive).
Firm 1 can have low marginal cost, cL, or high marginal cost, cH, where 0 < cL. < CH < 2. Firm 1 knows whether its marginal cost is low or high. Firm 2 only knows that the probability that Firm 1 has high marginal cost is 0, where < 8 < 1. Firm 2's marginal cost is c = 1 and this is known by both firms. (All this is common knowledge for the firms).
Formulate the situation as a static game of incomplete information, i.e., specify action spaces, type spaces, beliefs, and payoff functions for the two firms.
- Find the best response functions.
- Find the Bayesian Nash equilibrium of the game (i.e., find the equilibrium prices).
- Compare the BNE prices if Firm l's realized cost is high with the equilibrium prices in the game where it is common knowledge that Firm 1 has high marginal cost (cH).
- Suppose now that Firm 1 does not observe its own marginal cost before it sets its price. I.e., Firm 1 only knows that the probability that it has high marginal cost (cH) is 0. Everything else is as in the original set-up. Which equilibrium concept would you use to solve this game? Explain.