Homework #4 from Managerial Economics 1. Let the inverse demand function and the cost function be given by P = 502QandC= 10+2Q respectively, where Q is total industry output and P is the price of a product. Suppose the total industry is regulated by a monopolist. Then, find: a) Profitmaximizin g output level of the monopolist, Q34. b) Price,P,*w, the monopolist charges to produce the output level in (a). c) The maximum prot of the monopolist, 11;". d) The Consumer Surplus, CS. e) The Total Surplus, T5 2 CS + TERI. f) Now, suppose the industry is regulated by rworms, or duopoly. So, let ql be the output of rm 1 and (12 the output of rm 2. Then, the total industry output is Q = q1 + qz. The cost of each firm is given by (IL-(qi) = 10 + Zqi for i = 1,2. Write down the profit function of each firm. (Cournot Model with identical costs I. g) Derive the reaction function of rm 1, Q1 = R1012). Show what happens to firm 1 output when firm 2 increases qz. That is, %. 3'12 h) Find the Nash equilibrium outputs of both rms, q; and (1;. i) Find the total industry output, Q*, produced by a duopoly. j) Find the price charged by duopolists, PD _ k) Find the maximum profit of each firm, It; and TIE. 1) Find the new consumer surplus, CS. m)Find the total surplus, TS = C5 + 11': + 11;. 2. Cournot Model with different costs: Let again the demand function be P = 50 2Q. But the firms have different costs such as C1011) = 10 + qu and 52012) = 12 + 3'12- 3) Find the Nash equilibrium outputs of both rms, (1; and (1;. b) Find the total industry output, Q* , produced by a duopoly. c) Find the price charged by duopolists, PD . d) Find the maximum profit of each firm, TI: and HE. 3. Stackelber LeaderFollower Model with identical costs: Let again the demand function and the cost function be P = 50 ZQ and Ci(qi) = 10 + Zqifor i = 1,2. Assume the rm 1 is a leader and the rm 2 is a follower. In other words, first firm 1 enters the market, then firm 2 does. 3) Find the Nash equilibrium outputs of both firms, q: and (1;. b) Find the total industry output, 0*, produced by a leader and a follower. c) Find the maximum profit of each firm, Tr: and TEE. 4. Stackelber Leader-Follower R'Iodel with different costs: Let again the demand function be P = 50 ZQ and But the firms have different costs such as 61(Q1) = 10 + 2th and 02%) = 12 + 8'12- Assume the firm 1 is a header and the firm 2 is a follower. In other words, first firm 1 enters the market, then firm 2 does. a) Find the Nash equilibrium outputs of both firms, q: and (1;. b) Find the total industry output, Q *, produced by a leader and a follower. c) Find the maximum profit of each firm, TI; and 11

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