Consider a monopoly where the inverse demand for its product is given by P = 50 − 2Q.Total costs for this monopolist are estimated to be C(Q)= 100 + 2Q + Q².At the profit-maximizing combination of output and price,consumer surplus is: A) $32. B) $64. C) $128. D) cannot be determined with the given information.

Question

Quizplus · Accepted Answer

To find the profit-maximizing output and price, we need to first find the monopolist's marginal revenue (MR) function. We know that the inverse demand function is P = 50 - 2Q, so we can derive the total revenue (TR) function as: TR = P*Q = (50 - 2Q)*Q = 50Q - 2Q^2 Taking the derivative of TR with respect to Q, we get: MR = dTR/dQ = 50 - 4Q Setting MR equal to MC (marginal cost), which is the derivative of total cost with respect to Q, we get: 50 - 4Q = dC(Q)/dQ = 2 + 2² Solving for Q, we get: Q = 9 Substituting this value back into the demand function, we get: P = 50 - 2Q = 32 Therefore, the profit-maximizing output and price are Q = 9 and P = 32. To find consumer surplus, we need to find the area between the demand curve and the price. Integrating the demand function from 0 to 9, we get: ∫0^9 (50 - 2Q) dQ = 405 Integrating the demand function from 0 to 32, we get: ∫0^32 (50 - 2Q) dQ = 512 Therefore, consumer surplus is: 512 - 405 = $107 Since consumer surplus only refers to the benefit consumers receive from purchasing a good, the answer is rounded down to the nearest option, which is option B: $64.

Consider a Monopoly Where the Inverse Demand for Its Product