A firm with market power faces the demand function q = 2,000 - 20P. The firm's total cost function is TC(q) = 20q + 0.02q² + 2,000. If the firm behaves as a single-price monopoly, it will charge a price of $____. A) 71.53 B) 52.65 C) 48.34 D) 42.72

Question

Quizplus · Accepted Answer

To find the monopoly price, we first need to derive the marginal revenue (MR) and marginal cost (MC) from the given demand and cost functions, and then set MR = MC to solve for the optimal quantity (q*), and finally use the demand function to find the price.Given demand function: $q = 2000 - 20P$To find the inverse demand function, solve for P: $P = 100 - \frac{q}{20}$Total cost function: $TC(q) = 20q + 0.02q^2 + 2000$Marginal cost (MC) is the derivative of the total cost function with respect to q: $MC = \frac{dTC}{dq} = 20 + 0.04q$To find marginal revenue, we need to first get the revenue function (R = P*q), which requires the inverse demand function:$R = (100 - \frac{q}{20})q = 100q - \frac{q^2}{20}$Marginal revenue (MR) is the derivative of the revenue function with respect to q: $MR = \frac{dR}{dq} = 100 - \frac{q}{10}$Setting MR = MC to find the optimal quantity (q*):$100 - \frac{q}{10} = 20 + 0.04q$Solving this equation for q gives us the quantity that maximizes profit. Let's rearrange and solve for q:$80 = \frac{q}{10} + 0.04q$$800 = q + 0.4q$$800 = 1.4q$$q = \frac{800}{1.4} = 571.43$Using the demand function to find the price at this quantity:$P = 100 - \frac{571.43}{20} = 100 - 28.57 = 71.43$The correct price is approximately $71.43, which due to rounding and the way the question and options are presented, matches closest to option A) 71.53.

A Firm with Market Power Faces the Demand Function Q