Exercise 3. Consider a homogeneous-product duopoly. The two firms in the mar- ket are assumed to have constant marginal costs of production equal to c. The two firms compete possibly over an infinite time horizon. In each period they simultaneously set price p;, i = 1;2. After each period the market is closed down with probability 1 4. Market demand Q (p) is decreasing, where p = min {p;; po}. Suppose, furthermore, that the monopoly problem is well defined, i.e. there is a solution p = argmax (p ) Q (p). If firms set the same price, they share total demand with weight A for firm 1 and 1 A for firm 2. Suppose that A [1/2;1]. Suppose that firms use trigger strategies and Nash punishment. 1. Suppose that 6 = 0. Derive the equilibrium of the game. 2. Suppose that > 0 and A = 1/2. Derive the condition according to which firm 1 and firm 2 do not find it profitable to deviate from the collusive price p*. 3. Suppose that > 0 and A > 1/2. Derive the condition according according to which pM is played along the equilibrium path. Show that the condition is the more stringent the higher A. 2 4. Show that previous results in (3) also hold for any collusive price pC (c;pM). 5. Suppose that > 0 and A > 1/2 and that firm 1 can only adjust its price every 7 periods. Derive the condition according to which p is played along the equilibrium path. How does the time span infuence the condition