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.Suppose Kate enters the market first and chooses her output before Alice.What is the difference in Alice's profit when Kate enters the market first,compared to when they simultaneously select their outputs?

Question

Quizplus · Accepted Answer

When Kate enters the market first, she will choose her output level based on the assumption that Alice is not producing anything. The profit-maximizing output level for Kate will be where her marginal revenue equals her marginal cost, which is: MR1 = MC (20,000 - 400Q^d)/Q^d = 80 Solving for Q^d, we get Q^d = 250. Now, Alice enters the market and chooses her output level based on the assumption that Kate's output is fixed at 250 cubic yards. The market demand function is: Q^d = 20,000 - 200P With 250 cubic yards already sold by Kate, the new demand function that Alice faces is: Q^d = 20,000 - 200P - 250 Q^d = 19,750 - 200P Alice's profit-maximizing output level is where her marginal revenue equals her marginal cost, which is: MR2 = MC (19,750 - 400Q^d)/Q^d = 80 Solving for Q^d, we get Q^d = 243.75. To find Alice's profit, we need to subtract her total cost from her total revenue. Her total revenue is: R2 = P2*Q^d R2 = P2*243.75 To find P2, we need to solve for it using the demand function: 243.75 = 20,000 - 200P2 - 250 P2 = $101.25 So, R2 = $24,675. Her total cost is: C2 = 80*243.75 C2 = $19,500 So, her profit is: π2 = R2 - C2 π2 = $5,175 If they simultaneously select their outputs, they will both choose their output levels based on the assumption that the other firm is producing. Therefore, the equilibrium output level for each firm will be where their marginal revenue equals their marginal cost, given the other firm's output level. This can be solved as a Cournot duopoly problem, but it suffices to note that the total output will be higher, resulting in a lower market price and lower profits for both firms. Since Kate and Alice are symmetric, both firms will produce the same amount of output in equilibrium. The market demand function is: Qd = 20,000 - 400Q where Q = Q^d + Q^d. Setting MR1 = MR2 and solving for Q, we get Q = 300. Therefore, each firm will produce 150 cubic yards of concrete. The market price is: P = 20,000 - 400(150) P = $14,000 Each firm's revenue is: R = P*Q/2 R = $1,050,000 Each firm's cost is: C = 80*150 C = $12,000 Each firm's profit is: π = R - C π = $1,038,000 The difference in Alice's profit is: Δπ = π2 - π Δπ = $5,175 - $1,038,000 Δπ = -$1,032,825 Therefore, when Kate enters the market first, Alice's profit is $1,032,825 lower. Since the question asks for the difference, the answer is: D) When Kate enters the market first, Alice's profit is $3,888.89 lower.

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