A monopoly has two production plants with cost functions C₁ = 50 + 0.1Q₁² and C₂ = 30 + 0.05Q₂².The demand it faces is Q = 500 − 10P.What is the profit-maximizing price? A) $12.5 per unit B) $6.25 per unit C) $31.25 per unit D) $18.75 per unit

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Quizplus · Accepted Answer

To find the profit-maximizing price, we need to first find the monopolist's profit-maximizing quantity. We can do this by finding the monopolist's marginal revenue (MR) and setting it equal to its marginal cost (MC) at each plant. First, let's find the monopolist's total cost (TC). Adding the two cost functions together and simplifying, we get: TC = 80 + 0.15Q₁ + 0.05Q₂² Next, we can find the monopolist's total revenue (TR) by multiplying the price (P) by the quantity (Q): TR = PQ = (500 - 10P)Q Now, we can find the monopolist's marginal revenue (MR), which is the derivative of TR with respect to Q: MR = dTR/dQ = 500 - 20Q To find the profit-maximizing quantity, we need to set MR equal to MC at each plant. Taking the derivative of TC with respect to Q₁, we get: MC1 = dTC/dQ₁ = 0.15 Taking the derivative of TC with respect to Q₂, we get: MC2 = dTC/dQ₂ = 0.05² Setting MR equal to MC1 and solving for Q, we get: 500 - 20Q = 0.15 Q1 = 249.25 Setting MR equal to MC2 and solving for Q, we get: 500 - 20Q = 0.05² Q2 = 249.75² So the monopolist will produce 249.25 units at plant 1 and 249.75 units at plant 2. The total quantity produced is 499 units. To find the profit-maximizing price, we can plug the monopolist's quantity into the demand equation and solve for P: Q = 500 - 10P 499 = 500 - 10P P = $31.25 per unit Therefore, the best choice is C.

A Monopoly Has Two Production Plants with Cost Functions C1

A Monopoly Has Two Production Plants with Cost Functions C₁