If a monopolist faces an inverse demand curve, p(y)= 100

Question

If a monopolist faces an inverse demand curve, p(y)= 100 - 2y and has constant marginal costs of $32 and zero fixed costs and if this monopolist is able to practice perfect price discrimination, its total profits will be
A)$1,156.
B)$17.
C)$578.
D)$1,734.
E)$289.

Quizplus · Accepted Answer

The monopolist can charge each consumer their willingness to pay, so it will sell to all consumers with a willingness to pay above $32. The total profits will be the sum of the profits from each consumer, which is equal to the difference between the price and the marginal cost multiplied by the quantity sold. The quantity sold to each consumer will be equal to their willingness to pay minus the marginal cost, divided by 2. The total profits will therefore be:$$\int_{0}^{44}(100-2y-32)\frac{(100-2y-32)}{2}dy=\frac{1}{2}\int_{0}^{44}(68-2y)^2dy=\frac{1}{2}\left[\frac{2}{3}(68-2y)^3\right]_0^{44}=\frac{1}{3}(68-88)^3=\frac{1}{3}(-20)^3=\frac{8000}{3}=2666.67$$