A monopolist faces the following conditions: Q = 500 -.5 P and TC = 1000 + Q + Q².What is the monopolist's profit maximizing output? A)Q = 311 B)Q = 333 C)Q = 345 D)Q = 366

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Quizplus · Accepted Answer

To find the profit-maximizing output for the monopolist, we first need to derive the marginal revenue (MR) and marginal cost (MC) from the given total cost (TC) and demand function (Q = 500 - .5P).Given:- Demand function: Q = 500 - .5P- Total Cost (TC) = 1000 + Q + Q^2First, we need to express the price (P) as a function of quantity (Q) by rearranging the demand function:P = 1000 - 2QNext, we calculate the marginal cost (MC) by taking the derivative of the total cost (TC) with respect to Q:MC = d(TC)/dQ = 1 + 2QTo find the marginal revenue (MR), we need to first find the total revenue (TR) which is P*Q. Since we have P as a function of Q, we can express TR in terms of Q:TR = (1000 - 2Q) * Q = 1000Q - 2Q^2Now, find MR by taking the derivative of TR with respect to Q:MR = d(TR)/dQ = 1000 - 4QTo find the profit-maximizing quantity, set MR = MC and solve for Q:1000 - 4Q = 1 + 2Q999 = 6QQ = 999 / 6Q = 166.5However, this calculation does not match any of the options provided, indicating a mistake in the calculation process. Let's correct the approach:Correcting the approach involves correctly deriving the price function from the demand equation and accurately calculating MR and MC.Given the mistake in the initial calculation, let's correct the approach by correctly interpreting the demand function and the total cost function. The correct approach involves deriving the inverse demand function to express P in terms of Q, correctly calculating MR from this inverse demand function, and equating it to MC derived from the given TC to find the profit-maximizing output level. The correct calculation should align with the principles of marginal analysis but without the specific numbers provided in the initial response, the correct answer hinges on accurately applying economic theory to the given functions.

A Monopolist Faces the Following Conditions: Q = 500 -