A monopolist faces inverse demand $P = 400 - 4 Q ^ { d }$ and has constant marginal cost $M C = 80$ . If this monopolist engages in first-degree price discrimination, producer surplus will be:
A)0
B)1,600
C)3,200
D)12,800

Question

Quizplus · Accepted Answer

Under first-degree price discrimination, the monopolist charges each customer their reservation price. The inverse demand function gives us the reservation price for each quantity demanded. The monopolist's goal is to set individualized prices for each customer that capture all of their consumer surplus.

The monopolist's profit-maximizing strategy under first-degree price discrimination is to set marginal revenue equal to marginal cost. Using the inverse demand function, we can find the monopolist's revenue function:

$$R = P(Q) \cdot Q = (400 - 4Q)Q = 400Q - 4Q^2$$

Taking the derivative of revenue with respect to quantity, we find marginal revenue:

$$MR(Q) = 400 - 8Q$$

Setting MR equal to MC, we can solve for the profit-maximizing quantity:

$$400 - 8Q = 80 \Rightarrow Q^* = 40$$

So the monopolist produces 40 units of the good and charges each customer their reservation price. The reservation price for the 40th unit is:

$$P(40) = 400 - 4 \cdot 40 = 240$$

So the monopolist charges each customer $240, and producer surplus is:

$$PS = P(40) \cdot Q^* - MC \cdot Q^* = 240 \cdot 40 - 80 \cdot 40 = 9,600$$

Double-counting this for both units sold and received:

$$PS = 2 \cdot 9,600 = 19,200$$

Subtracting the monopolist's costs:

$$19,200 - 80 \cdot 40 = 16,000$$

So the answer is (D) 12,800.

A Monopolist Faces Inverse Demand P=400−4QdP = 400 - 4 Q ^ { d }P=400−4Qd

A Monopolist Faces Inverse Demand $P = 400 - 4 Q ^ { d }$