Question

1) All 20 consumers are alike and each has a demand curve for a monopolist's product of p=12-2q. The cost of production C(Q)=Q. Let the monopolist charge a price of $r per unit purchased and a subscription fee of $F that must be paid by each producer. Find the r and F that maximize profits.

a) What is r?

b) What is F?

c) What is the maximum profit the monopolist can earn in this market? (pi)

2)All 200 consumers are alike and each has a demand curve for a monopolist's product of p=20-q. The cost of production C(Q)=2Q. Let the monopolist charge a price of $PM for qM unit purchased. Find the menu prices that maximize profits. (The buyer pays menu price PM for quantity qM)

a) What is qM?

b) What is PM?

c) What is the maximum profit the monopolist can earn in this market?

d) What is the consumer surplus?

Answer 1

(1)

With two-part pricing, r = MC and F = Consumer surplus (CS) = Profit.

MC = dC(Q)/dQ = 1

(a)

r = MC = 1

(b)

12 - 2q = 1

2q = 11

q = 5.5

When q = 0, p = 12

F = (1/2) x (12 - 1) x 5.5 = 2.75 x 11 = 30.25

(c)

Profit = CS = F = 30.25

(2)

Profit is maximized when MR = MC.

MC = dC(Q)/dQ = 2

(a)

TR = p x q = 20q - q2

MR = dTR/dq = 20 - 2q

20 - 2q = 2

2q = 18

q = 9

(b)

p = 20 - 9 = 11

(c)

Profit = q x (p - MC) = 9 x (11 - 2) = 9 x 9 = 81

(d)

When q = 0, p = 20

CS = (1/2) x (20 - 11) x 9 = 4.5 x 9 = 40.5