Question

An online shopping website considers offering its customers a mail-in rebate program. By this way, it aims to differentiate between different groups of customers. After the completion of the purchase of a wireless router, consumers can mail a rebate form to receive $ back. In other words, the net price after the rebate is p*-a for those who are interested in the mail-in-rebate. The shopping website is a monopoly with no fixed cost. Its marginal cost is $30.

The Market consists of two different consumer groups whose demand functions are as follows:

The demand of the two groups are as follows;

P1=150-2q1

P2=100-5q2

Assuming that the consumers of group 1 are not interested in rebate forms,

a. What is the optimal rebate amount, which differentiates between different consumer groups?

b. Assume that the firm has enough information about its consumers to implement a perfect price discrimination strategy. Calculate the equilibrium price and quantities and the profit if the firm uses perfect price discrimination. (Do not derive the kinked market demand curve. Simple assume that the market demand is Q=95-07P)

Answer 1

a) Given the demand curves for the two different types of consumers, we can get the profit maximizing monopoly price for each by finding the MR curves for each and equating with the marginal cost. Then we can get the optimal price and quantity for each type of consumer.
we have, p1= 150-2q1 or, pi*q1 = 150q1 - 2q1^2 or, MR1 = 150-4q1 (differentiating with respect to q1)
equating MR1 = MC, we get 150-4q1 = 30 . SO, q1 = 30 and p1 = 90. ( plugging the value of q1 in the demand curve)
We do the similar exercise with second type of consumer, and get MR2= MC
we get 100-10q2= 30 So, q2 = 7 and p2 = 65 .

So the price differential is 90-65 = 25. Since the monopolist cant differentiate, the price should be set at 90 and the optimal rebate or the value of 'a' should be 25. This way, the second group will but the rebate and buy the good from the monopolist.

b) If the firm can implement perfect price discrimination, then it can charge different prices from the two different groups.
then for the first group, as calculated, p1 = 90, q1= 30.
So, profit = p1*q1 - C*q1 = 30(90-30)= 1800
For the second group, as calculated, p2= 65 q2=7
so Profit = p2*q2-C*q2 = 455-210=245

So total profit = $2045