Question

1 (Bertrand Model with sequential move) Consider a Bertrand duopoly model with two firms, Fi and Fa selling two varieties of a product. The demand curve for Fi's product is 91 (pi,P2) = 10-Pl + 0.5p2: and the demand for F's product is where p is the price charged by F). Both firms have a constant marginal cost of (a) Write down the profits of F1 and F2 as a function of prices P1 and P2. You have b) Derive the reaction functions for Fi and Fo. [You have done it before in problem production equal to 0. Suppose the firms compete in prices. done it before in problem set 2 set 2 (c) Suppose Fi is the market leader and F2 is the follower. That is, Fi first sets its price pi, and observing Fi's price F2 sets its price p2. Using the backward induc tion method, find the pair of prices (pi.P) that constitutes the Nash Equilibrium of this game. d) Find the profits earned by the two firms in equilibrium. Which firm earns more, the leader or the follower? Explain the economic intuition behind your finding. PROBLEM SET e) Look at your solutions to problem set 2, and write down the profits the two firms make when they set their prices simultaneously. Compare these profits with the profits calculated in part (d) above. Does the leader firm makes more profit than what it was making when firms moved simultaneously? Why or why not?

Answer 1

Firm 1: MC = 0 q_1(p_1 p_2) = 10 - p_1 + 0.5 p_2 TR_1 = p_1.q_1 = p_1(10 - p_1 + 0.5 p_2) = 10p_1 - p^2_1 + 0.5 p_1 p_2 MR_1 = partial differential TR_1/partial differential p_1 = 10 - 2p_1 + 0.5 p__2 setting MR_1 = MC 10 - 2p_1 + 0.5 p_2 = 0 Implies 2 p_1 = 10 + 0. p_2 Implies p_1 = 5 + 0.25 p_2 Reaction function Firm 2: MC = 0 q_2 (p_1, p_2) = 10 - p_2 + 0.5 p_1 TR_2 = p_2, q_2 = p_2(10 - p_2 + 0.5 p_1) = 10 p_2 - p^2_2 + 0.5 p_1 p_2 MR_2 = partial differential TR_2/partial differential p_2 = 10 - 2p_2 + 0.5 p_2 setting MR_1 = MC 10 - 2p_2 + 0.5 p_1 = 0 10 + 0.5 p_1 = 2p_2 or p_2 = 5 + 0.25 p_1 Reaction function \r\n p_1 = 5 + 0.25 (5 + 0.25 p_1) = 5 + 1.25 + 0.0625 p_1 p_1 = 0.0625 p_1 = 6.25 Implies 0.9375 p_1 = 6.25 Implies p_1 = $ 6.67 Therefore q_1 = 6.66 & q_2 = 6.66 Therefore p_2 = 5 + 0.25 times 6.67 p_2 = $ 6.6675 Therefore pi_1 = p_1 Q_2 = 44.42 & pi_2 = TR_2 = p_2 Q_2 = 4.40

From calculation we see that firm 2 makes more profit than firm 1 in sequential game