Question

3. [Cournot mergers with efficiency gains] Consider an industry with three identical firms each selling a homogenous good and producing at a cost c> 0. Industry demand is given by p(Q)1-Q, where Q1 2+93 denotes aggregate output. Competition in the marketplace is in quantities (a la Cournot) (a) Find the equilibrium quantities, price and profits (b) Consider now a merger between two of the three firms, resulting in duopolistic structure of the market. The merger might give rise to efficiency gains, in the sense that the firm resulting from the merger produces at a cost e x c, with e 1 (whereas the outsider still has a cost c) 1. Find the post-merger equilibrium quantities, price and profits 2. Under which conditions does the merger reduce prices? 3. Under which conditions is the merger beneficial to the merging firms?

Answer 1

Cournot Mergers P(Q) = 1 - Q, Q = sigma Q_i M_c = C pi_1 = (P-C)q_1 = (1-c-q_1-q_2-q_3)q_1 = (1-c-q_2-q_3)q_1 - q_1^2 BR_1 (q_2, q_3): partial differential pi_1/ partial differential q_1 = > (1-c-q_2-q_3) - 2q_1 = 0 thus q_1^BR = 1-C-q_2-q_3/2 similarly q_2^BR = 1-C-q_1 - q_3 q_3^BR = 1-c-q_1-q_2/2 Now at eq^m q_1 = q_2 = q_3 = q_1 so q_1 = 1-c-2q_1/2 = > 2q_1 + 2q_1 = 1-c 4q_2 = 1-c thus q_1^CN = q_2^CN = q_3^CN = 1-C/4 P = 1-sigma Q_i