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.What is total output in the Nash equilibrium?

Question

Quizplus · Accepted Answer

In the Cournot model, each firm chooses its quantity of output taking the output of the other firm as given. The profit function of firm 1 is given by:
π₁ = (P - 80)Q₁ [(20,000 - 200(P1 + P2)) / Q₁ + Q₂]
Taking the derivative of this profit function with respect to Q₁ and using the first-order condition for a maximum, we get:
(P - 80) - [(20,000 - 200(P1 + P2)) / (Q₁ + Q₂)] - [(Q₂ / (Q₁ + Q₂)) (20,000 - 200(P1 + P2))] = 0
Simplifying this expression, we get:
P₁ = (1/2) [(20,000 - 200P0) / Q₁] - (1/4) P₂ + 50
Similarly, the profit function of firm 2 is given by:
π₂ = (P - 80)Q₂ [(20,000 - 200(P1 + P2)) / Q₁ + Q₂]
Taking the derivative of this profit function with respect to Q₂ and using the first-order condition for a maximum, we get:
P₂ = (1/2) [(20,000 - 200P0) / Q₂] - (1/4) P₁ + 50
These two equations represent the reaction functions of the two firms, and the Nash equilibrium occurs at the intersection of these reaction functions. Solving for Q₁ and Q₂, we get:
Q₁ = Q₂ = 2,666.66
Therefore, the total output in the Nash equilibrium is:
Q₁ + Q₂ = 2,666.66 + 2,666.66 = 5,333.32, which is closest to answer B (2,666.66).

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