Question

5. Consider a version of the Cournot duopoly game, which will be thoroughly analyzed in Chapter 10. Two firms (1 and 2) compete in a homogeneous goods market, where the firms produce exactly the same good. The firms simultaneously and independently select quantities to produce. The quantity selected by firm i is denoted q, and must be greater than or equal to zero, for i - 1,2. The market price is given by p-2 - q1 -q2. For simplicity, as sume that the cost to firm i of producing any quantity is zero. Further, assume that each firm's payoff is defined as its profit. That is, firm i's payoff is pqi, where j denotes firm i's opponent in the game. Describe the normal form of this game by expressing the strategy spaces and writing the payoffs as func- tions of the strategies.

Answer 1

Cournot Game M k t d d: P = 2 - Q, Q = = q_1 + q_2, M C_i = 0 then pi_1 = P MIDDOT q = (2 - q_1 - q_2) q_1 = (2 - q_2) q_1 - q_2 B R_1 (q_2): partial differential pi/ partial differential q_1 = > (2 - q_2) - 2 q_1 = 0, so q_1 B R = 2 - q_2/2 similarly q_2 B R = 2 - q_1/2, then solving simultaneously q_1 = 1 - 1/2 [1 - q_1/2] = > q_1 = 1 - 1/2 + q_1/4 so 3 q_1/4 = 1/2 = > q_1 C N = 2/3, q_2 C N = 2/3, thus Q = 4/3, P = 2 - 4/3 = 2/3 Now pi_1 C N = 2/3 middot 2/3 = 4/9 = pi_2 C N Now If Both Cooperates & produces half of monopoly o/p, Then at monopoly eq^n: M R = M C