Kate and Alice are small-town ready-mix concrete duopolists.The market demand function is Q^d = 20,000 - 200P,where P is the price of a cubic yard of concrete and Q^d is the number of cubic yards demanded per year.Marginal cost is $80 per cubic yard.The Cournot model describes the competition in this market.How much does Alice produce in the Nash equilibrium?

Question

Quizplus · Accepted Answer

In a Cournot model, each firm produces an output level that maximizes their profit given the output of the other firm. The profit function of each firm is given by: π1 = (P1 - 80)Q1 π2 = (P2 - 80)Q2 where Q1 + Q2 = Q^d, the market demand. Taking the derivative of each profit function with respect to Qi and solving for Qi in terms of Qj, we get: Q1 = (1/3) (20,000 - 200P1 - 400P2) Q2 = (1/3) (20,000 - 400P1 - 200P2) Substituting Q2 into the demand equation to eliminate Q2 and solving for P1, we get: P1 = 100 - 0.25Q1 Substituting this into the expression for Q1, we get: Q1 = 1,333.33 Similarly, we can find Q2: Q2 = 1,333.33 Therefore, Alice produces 1,333.33 cubic yards of concrete in the Nash equilibrium.

Kate and Alice Are Small-Town Ready-Mix Concrete Duopolists