Question

In a market of a certain good, there is a monopolist who has a constant marginal cost c = 8. There are two consumers (i = 1, 2) in this market, and their demand functions are given by p = 20 − 4q for consumer 1 and p = 25 − 5q for consumer 2. Suppose that the monopolist is trying to design an optimal two-part tariff.

A. If the monopolist maximizes its profit while it wants to attract both consumers, what fixed fee does the monopolist charge to the consumers?

(a) 121/8

(b) 116/3

(c) 95/4

(d) 17/2

(e) 191/11

B. If the monopolist maximizes its profit while it wants to attract both consumers, what price per unit of the good does the monopolist charge to the consumers?

(a) 8

(b) 11

(c) 9

(d) 15

(e) 17

C. What is the maximum profit under a two-part tariff when the monopolist attracts both consumers?

(a) 123/4

b) 71/2

(c) 242/7

(d) 38

(e) 181/5

D. What is the maximum profit under a two-part tariff when the monopolist attracts only consumer 2?

(a) 173/8

(b) 311/13

(c) 56/3

(d) 289/10

(e) 71/2

Answer 1

Demand functions are Q1 = 5 – 0.25P1 and Q2 = 5 – 0.2P2. Marginal cost is 8.

In this case the two part tariff would have a fixed fee equal to the consumer surplus of the light users at a price of P. The profit under that condition is given by:

Profit function = 2 x F + (P - MC)*(Q1 + Q2) where F = CS of consumer type 1.

= 2 x 0.5*(20 - P)*(5 – 0.25P) + (P - 8)*((10 – 0.45P)

= 100 – 5P – 5P + 0.25P^2 + 10P – 0.45P^2 – 80 + 3.6P

= 20 + 3.6P – 0.20P^2

Profit is maximum when

3.6 = 0.4P

P* = 9

Hence the price per unit is 9. Fees amount is F = 0.5*(20 - 9)*(5 – 0.25*9) = 121/8

Profit = 20 + 3.6*9 – 0.20*(9^2) = 181/5

When only consumer 2 is served, profit = CS = 0.5*(25 - 8)*(5 - 0.20*8) = 289/10

Select A) 121/8, B) 9, C) 181/5 D) 289/10