Exercise 4: Cournot-Nash and Stackelberg Quantity-Setting Games Consider a market with aggregate demand 100 - 4p if p 25 D(p) if p > 25 There are two firms. Each firm 1's cost function is er(gi) = 1 and firm 2's cost fumetion is ca() = $42- 1. What is the inverse demand function, P(Q), for this market? 2. Suppose the firms act as " Cournot-Nash Duopolists", simultaneously and independently choosing their output quantities. If firms 1 and 2 choose output quantities q 2 0 and 92 2 0 respectively, then the market clearing price would be given by the inverse demand function evaluated at the total quantity, P(gi + 92). (a) Find the payoff functions for both firms, that is compute s(qi, ge) and a2(qi, 42)- (b) Find the best responses for the first firm, Bi(g2), and the best response for the second firm, By(q). (c) Find all Nash Equilibria (of, 4;). What are the corresponding market price, consumers' surplus, and profits for each firm? 3. Now suppose the same firms act as "Stackelberg" (leader-follower) quantity-setters. That is, they choose their output quantities sequentially. Suppose the firms move sequentially, with firm 1 choosing an output level first. Then firm 2 observes firm 1's output choice ; before choosing its own output level Q3. (a) Use Backward Induction to solve for the subgame-perfect Nash equilibrium of this sequential game. That is, first find the optimal output strategy for the second firm Q:(Q,). Then, suppose firm 1 correctly anticipates the behavior of firm 2 (Le. firm 1 knows 2's optimal strategy Q;(Q, ) above) and find firm 1's optimal output level. (b) What are the resulting market price, consumers' surplus, and profit levels for each firm