Question

1. Consider a Cournot game between two firms. The firms face an inverse demand function described by the equation P(Q) = α − Q if Q ≤ α, P(Q) = 0 if Q > α, where P is the price of output and Q is the total output produced by the two firms. Firm 1 produces its output q1 at a constant unit cost c1 (i.e, the total cost to firm 1 of producing q1 units of output is c1q1). Firm 2 produces output q2 at a constant unit cost c2.

(a) Solve for the Nash equilibrium output choices of each firm, (q ∗ 1 , q∗ 2 ), as a function of α, c1, and c2. You may assume that both firms choose a positive quantity of output in the Nash equilibrium (this will be true as long as c1 and c2 are not too big, or not too far apart). Solve also for each firm’s profit, total output, and the Nash equilibrium price, also as functions of α, c1 and c2.

(b) Suppose firm 2 has an innovation that reduces its cost (that is, makes c2 smaller). What is the directional effect (up or down) on the Nash equilibrium outputs and profits of the firms, and on the Nash equilibrium price?

Answer 1