Question

A monopolist faces different demand from two groups of consumers. Their demand functions and total cost of the monopolist are as follows:

Demand from group 1: Q_{1}=20-P_{1}

Demand from group 2: Q_{2}=22-0.5P_{2}

Total cost function: TC=100+0.5Q^{2} where
Q=Q_{1}+Q_{2}

Suppose the firm is able to price discriminate between the two groups of consumers.

(a) What is the optimal output level of the monopolist for group 1?

(b) What price should be charged for group 1 by the monopolist?

(c) What is the optimal output level of the monopolist for group 2?

(d) What price should be charged for group 2 by the monopolist?

(e) What is the total profit for the monopolist?

Answer #1

Q1 = 20 - P1, so P1 = 20 - Q1

Q2 = 22 - 0.2P2, so P2 = 44 - 2Q2

TR1 = P1 x Q1 = 20Q1 - Q1^{2}, so MR1 = dTR1/dQ1 = 20 -
2Q1

TR2 = P2 x Q2 = 44Q2 - 2Q2^{2}, so MR2 = dTR2/dQ2 = 44 -
4Q2

TC = 100 + 0.5(Q1 + Q2)^{2}

MC1 = TC/Q1 = Q1 + Q2

MC2 = TC/Q2 = Q1 + Q2

(a) - (d)

For Group 1, Setting MR1 = MC1,

20 - 2Q1 = Q1 + Q2

3Q1 + Q2 = 20.............(1)

For Group 2, Setting MR2 = MC2,

44 - 4Q2 = Q1 + Q2

Q1 + 5Q2 = 44.............(2)

Multiplying (2) by 3,

3Q1 + 15Q2 = 132.........(3)

3Q1 + Q2 = 20...........(1)

14Q2 = 112

Q2 = 8

Q1 = 44 - 5Q2 [from (2)] = 44 - (5 x 8) = 44 - 40 = 4

P1 = 20 - 4 = 16

P2 = 44 - (2 x 8) = 44 - 16 = 28

(e)

TR = TR1 + TR2 = (P1 x Q1) + (P2 x Q2) = (16 x 4) + (28 x 8) = 64 + 224 = 288

TC = 100 + 0.5 x (4 + 8)^{2} = 100 + 0.5 x
(12)^{2} = 100 + 0.5 x 144 = 100 + 72 = 172

Profit = TR - TC = 288 - 172 = 116

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