Question

A monopolist produces gizmos at constant marginal cost of 4 and no fixed cost. It recognizes that it has two types of customers. The demand curve of each type of customer is given by: Type-1 customer: ( ) p1 q1 = 18 − 2q1 Type-2 customer: ( ) p2 q2 =15 − q2 where qi (i = 1, 2) are the quantity demanded of type i consumer per period. While the monopolist knows the demand curves, it cannot identify the type of a given consumer. It is assumed that it has 10 customers in total per period, and five of them are of Type 1. a) The monopolist wishes to sell both types of consumers. Assuming it charges the same two-part tariff schedule to both types of consumers; derive the optimal two-part tariff schedule. What would be the resulting profit? b) Suppose that the monopolist engages in second-degree price discrimination. Derive the two-part tariff schedules that are incentive compatible for both types, and the resulting profit in this case

Answer 1

p1(q1) = 18 - 2q1

The demand function can be plotted as

Consumer Surplus for Type 1 Customers = 1/2 * (18-4)*7 = 49

p2(q2) = 15 - q2

The demand function can be plotted as

Consumer Surplus for Type 2 Customers = 1/2 * (15-4)*11 = 60.5

Since the consumer surplus for the type 1 customer is lower, so the monopolist has to go ahead with the demand function of type 1 customer if it has to charge the same 2 part tariff.

So Consumer Surplus for Consumer 1 = 1/2 * (18-p) * q = 1/2 * (15 - p) * (18 - p)/2 = 1/4 * (18-p)^2

Profit of the Monopolist = 2 * (Consumer Surplus for Consumer 1) + (p-MC) * (q1 + q2)

Since the same consumer surplus will be obtained for Consumer 1 and consumer 2, so total profit will have 2 * Consumer Surplus.

p is the profit maximizing price. MC is the marginal cost=4

So now Profit = 2 * 1/4 * (18-p)^2 + (p - 4) * ((18-p)/2+ 15 - p)

P = 1/2 * (18-p)^2 + (p-4) *(24 -3p/2)

To maximize profit, dP/dp = 0

or 2 * 1/2 * (18-p) * (-1) + (p-4) * (-3/2) + (24-3p/2)*(1) = 0

or p - 18 - 3p/2 + 6 + 24 - 3p/2 = 0

or 2p = 12

or p = 6

So q1 = (18 - p) / 2 = (18 - 6) / 2 = 6

q2 = 15 - p = 15 - 6 = 9

Access Fee = Consumer Surplus of consumer 1 = 1/2 * (18 - p) * q1 = 1/2 * (18-6) * 6 = 36

So the optimal 2 part tariff will be a price of 6 and an access fee of 36.

The total profit will be

P = 1/2 * (18-p)^2 + (p-4) *(24 -3p/2)

= 1/2 * (18-6)^2 + (6-4)*(24 - 3*6/2)

= 1/2 * 144 + 2 * 15

= 72 + 30

= 102

This is the profit per customer.

So Total Profit = 102 * 10 = 1020