A monopolist faces a demand function given by Q = 50 -P and has a cost function C =3Q²+ 10Q +
50. If the government imposes a lump sum tax of $50, the monopoly profit is:

Question

A monopolist faces a demand function given by Q = 50 -P and has a cost function C =3Q²+ 10Q + 50. If the government imposes a lump sum tax of $50, the monopoly profit is: A)$100. B)$150. C)$0. D)$50.

Quizplus · Accepted Answer

The monopolist's profit is maximized where marginal revenue equals marginal cost. First, we find the inverse demand function, P = 50 - Q. Then, we derive the total revenue, TR = P*Q = (50 - Q)Q = 50Q - Q^2, and the marginal revenue, MR = d(TR)/dQ = 50 - 2Q. The total cost, TC, is given as C = 3Q + 10Q + 50 = 13Q + 50. The marginal cost, MC, is the derivative of TC with respect to Q, which is MC = d(TC)/dQ = 13. Setting MR = MC to find the quantity that maximizes profit, we get 50 - 2Q = 13, which simplifies to 2Q = 37, so Q = 18.5. Substituting Q back into the demand function to find P, we get P = 50 - 18.5 = 31.5. The total revenue at this quantity is TR = P*Q = 31.5 * 18.5 = 582.75. The total cost without the tax is TC = 13*18.5 + 50 = 290.5. The profit before the tax is TR - TC = 582.75 - 290.5 = 292.25. After imposing a lump sum tax of $50, the profit becomes 292.25 - 50 = 242.25. However, none of the provided options match this calculation, indicating a possible mistake in the calculation or misunderstanding of the question's requirements. Re-evaluating the approach and considering the provided options, it seems there might have been an error in interpreting the question or in the calculation process.

A Monopolist Faces a Demand Function Given by Q =