Question: Firms 1 and 2 compete on price. The demand is Q = 100 - p, where p is the price. Consumers buy from the firm that sells more cheaply. If both firms sell at the same price,
demand is divided equally. The marginal cost is 0. The discount factor is d. The firms compete for T periods. Strategy C is defined as follows: firm i sets its price equal to the monopoly price if it has always observed the other firm setting the monopoly price in the past; if firm i ever observes the other firm setting a different price, it sets its price equal to its marginal cost (pi=0). Observe that if both firms follow Strategy C, they achieve collusion.
(a) Suppose T = 30. Does an equilibrium exist in which the firms follow Strategy C? If so, for what values of d? Suppose for the remaining parts of this question that T = 8:
(b) Does an equilibrium exist in which the firms follow Strategy C? If so, for what values of d?
(c) Suppose that the other firm's prices are observed with delay. If firm j sets a price other than the monopoly price in period t, firm i only observes this in period t+2. Does an equilibrium exist in which the firms follow Strategy C? If so, for what values of d? Compare your answer to part (b).
(d) Suppose that prices are observed without delay, however, there are N firms competing. Does an equilibrium exist in which the firms follow Strategy C? If so, for what values of N and d?
(e) Suppose prices are observed without delay and that there are only 2 firms. But, suppose, there is a 50% chance firm 2 will leave the market in period 2. If it does not leave the market in period 2, it remains in the market with probability 1. Does an equilibrium exist in which the firms follow Strategy C? If so, for what values of d?