Suppose a monopoly seller of mineral water is able to segment its market into th

Question

Suppose a monopoly seller of mineral water is able to segment its market into three consumer groups: 1, 2, and 3. The (inverse) demand for mineral water on the part of each group is given by: Group 1 Demand: P1 = 1000 – (1/2)Q1 Group 2 Demand: P2 = 1000 – (1/3)Q2 Group 3 Demand: P3 = 1000 – (1/5)Q3 The total cost faced by the monopolist is: TC = 100Q, where the quantity produced (Q) is distributed across the 3 groups such that Q1 + Q2 + Q3 = Q. Having the ability to charge each group a unique price, determine the profit-maximizing price and quantity the monopolist should set for each group, as well as the firm’s profit.

Explanation / Answer

TC = 100Q = 100 x (Q1 + Q2 + Q3) = 100Q1 + 100Q2 + 100Q3

Profit is maximized by setting MR1 = MC1, MR2 = MC2 & MR3 = MC3

Group 1: P1 = 1000 - 0.5Q1

Total revenue, TR1 = P1 x Q1 = 1000Q1 - 0.5Q12

MR1 = dTR1 / dQ1 = 1000 - Q1

MC1 = dTC / dQ1 = 100

Equating MR1 with MC1:

1000 - Q1 = 100

Q1 = 900

P1 = 1000 - (900 x 0.5) = 1000 - 450 = 550

Profit, Z1 = Q1 x (P1 - MC1) = 900 x (550 - 100) = 900 x 450 = 405,000

Group 2: P2 = 1000 - (Q2 / 3)

TR2 = P2 x Q2 = 1000Q2 - (Q22 / 3)

MR2 = dTR2 / dQ2 = 1000 - (2/3)Q2

MC2 = dTC / dQ2 = 100

1000 - (2/3)Q2 = 100

(2/3)Q2 = 900

Q2 = 1350

P2 = 1000 - (1350 / 3) = 1000 - 450 = 550

Profit, Z2 = Q2 x (P2 - MC2) = 1350 x (550 - 100) = 1350 x 450 = 607,500

Group 3: P3 = 1000 - (Q3 / 5)

TR3 = P3 x Q3 = 1000Q3 - (Q32 / 5)

MR3 = dTR3 / dQ3 = 1000 - (2/5)Q3

MC3 = dTC / dQ3 = 100

1000 - (2/5)Q3 = 100

(2/5)Q3 = 900

Q3 = 2250

P3 = 1000 - (2250 / 5) = 1000 - 450 = 550

Profit, Z3 = Q3 x (P3 - MC3) = 2250 x (550 - 100) = 2250 x 450 = 1,012,500

Firm's total profit = Z1 + Z2 + Z3 = 405,000 + 607,500 + 1,012,500 = 2,025,000

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