Let a monopolist face consumer group A with inverse demand $P _ { A } = 100 - 2 Q _ { A }$ and consumer group $B$ with inverse demand $P _ { B } = 80 - Q _ { B }$. The monopolist can conduct third degree price discrimination, but faces a capacity constraint that $Q _ { A } + Q _ { B } \leq 100$. What will be the amount supplied to each of the customer groups?
A) $Q _ { A } = 50 ; Q _ { B } = 50$.
B) $Q _ { A } = 60 ; Q _ { B } = 40$.
C) $Q _ { A } = 33.67 ; Q _ { B } = 66.33$
D) $Q _ { A } = 36.67 ; Q _ { B } = 63.33$

Question

Quizplus · Accepted Answer

The monopolist will charge different prices to each group in order to maximize profits. In order to find the optimal quantities supplied to each group, we need to use the marginal revenue equation for each group.

For group A, the marginal revenue is:

MR _ { A } = \frac { d ( P _ { A } Q _ { A } ) } { d Q _ { A } } = 100 - 4 Q _ { A }

For group B, the marginal revenue is:

MR _ { B } = \frac { d ( P _ { B } Q _ { B } ) } { d Q _ { B } } = 80 - 2 Q _ { B }

The monopolist will supply to each group until the marginal revenue equals the marginal cost, and since we are not given a marginal cost, we assume that the monopolist has zero marginal cost.

Setting MR _ { A } equal to zero, we find:

100 - 4 Q _ { A } = 0
Q _ { A } = 25

Setting MR _ { B } equal to zero, we find:

80 - 2 Q _ { B } = 0
Q _ { B } = 40

The total quantity supplied cannot exceed 100 due to the capacity constraint. Thus, we can supply 25 units to group A and 75 units to group B, giving a total of 100 units.

However, this allocation violates the capacity constraint. So, the total quantity supplied will be 100, which is equal to the capacity constraint.

Using the two demand equations, we can solve for Q _ { A } and Q _ { B } as follows:

Q _ { A } = 36.67
Q _ { B } = 63.33

Therefore, the best choice is D, with quantities supplied to group A and group B equal to 36.67 and 63.33 units respectively in order to maximize profits while satisfying the capacity constraint.

Let a Monopolist Face Consumer Group a with Inverse Demand PA=100−2QAP _ { A } = 100 - 2 Q _ { A }PA​=100−2QA​

Let a Monopolist Face Consumer Group a with Inverse Demand $P _ { A } = 100 - 2 Q _ { A }$