Question

PROBLEM II. 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 dennand 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 Q4. 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 Q5. 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

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

Select 4) 121/8, 5) 9