Suppose P = 20 − 2Q is the market demand function for a local monopoly.The marginal cost is 2Q.If fixed costs are zero and the firm engages in two-part pricing,the most profits the firm will earn is:
A) $5.
B) $10.
C) $25.
D) $50.

Question

Quizplus · Accepted Answer

To maximize profits, the local monopoly should charge a price that is equal to its marginal cost of production. In this case, the marginal cost is 2Q.

The monopoly is engaging in two-part pricing, which means it is charging a fixed fee (F) in addition to per-unit pricing (P). The formula for total revenue in this case is:

TR = (P - 2Q)Q + F

To maximize profits, the monopoly should set P = 2Q, which means that the per-unit price should be equal to the marginal cost.

Substituting P = 2Q into the formula for total revenue, we get:

TR = (2Q - 2Q)Q + F 
TR = F

Therefore, the total revenue generated by the monopoly is equal to the fixed fee (F). To maximize profits, the monopoly should set F as high as possible without discouraging consumers from participating.

Since the demand function is P = 20 - 2Q, we can solve for Q in terms of P:

P = 20 - 2Q 
2Q = 20 - P 
Q = 10 - 0.5P

We can plug this equation for Q into the formula for total revenue:

TR = (P - 2Q)Q + F 
TR = (P - 2(10 - 0.5P))(10 - 0.5P) + F

To maximize profits, we need to find the value of P that maximizes total revenue. We can do this by taking the derivative of TR with respect to P and setting it equal to zero:

dTR/dP = -3P/2 + 30 = 0 
P = 20

Therefore, the per-unit price should be $20, and the quantity demanded will be Q = 10 - 0.5P = 5.

To find the optimal fixed fee, we can plug in the values of P and Q into the formula for total revenue:

TR = (P - 2Q)Q + F 
TR = (20 - 2(5))(5) + F 
TR = 50 + F

Therefore, the total revenue is $50 + F. To maximize profits, the monopoly should set F = $25, which will result in total revenue of $75.

The monopoly's profit will be equal to total revenue minus total cost, which in this case is just total revenue since fixed costs are zero:

Profit = TR - TC 
Profit = $75 - $0 
Profit = $75

Therefore, the most profits the firm will earn is $75, which corresponds to choice C.

Suppose P = 20 − 2Q Is the Market Demand