Kate and Alice are small-town ready-mix concrete duopolists.The market demand function is 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 the difference in the deadweight loss compared to a monopoly in this market?

Question

Quizplus · Accepted Answer

The Cournot equilibrium quantity in this duopoly is found by solving for the intersection of the firms' reaction functions: 
Q1 = (1/2) * (11,000 - 80Q1 - 40Q2)
Q2 = (1/2) * (11,000 - 80Q2 - 40Q1)
Where Q1 and Q2 are the quantities produced by firms 1 and 2, respectively. Solving these equations gives Q1 = Q2 = 158.33 cubic yards.
The market price is then found by plugging this quantity into the demand function:
P = 11,000 - 316.67 = 10,683.33 per cubic yard.
Total revenue for each firm is then:
TR = P * Q = 10,683.33 * 158.33 = 1,691,665.83
Marginal revenue for each firm is:
MR = 11,000 - 2Q
MR1 = 11,000 - 2(158.33) = 10,683.33
MR2 = 11,000 - 2(158.33) = 10,683.33
Both firms have the same marginal revenue, and so they will split the market evenly. Each firm will produce 158.33 cubic yards of concrete and earn a profit of:
π = TR - TC = P * Q - MC * Q = (10,683.33 - 80) * 158.33 = 1,694,990.83
Therefore, the total profit in the Cournot equilibrium is 2 * 1,694,990.83 = 3,389,981.66.
In a monopoly case, the deadweight loss is given by:
DWL = (1/2) * (MC - P) * Qm = (1/2) * (80 - 10,683.33) * 316.67 = 1,710,835.83
The difference in deadweight loss between the Cournot duopoly and monopoly cases is:
DWL_monopoly - DWL_Cournot = 1,710,835.83 - 0.5 * 80 * 316.67 = 5,553.31.
Therefore, the correct answer is C.

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