Question

A monopolist produces a product in one central production facility using the cost structure: TC = (1/2) Q2 +300 and sells it in two different markets with the following demand functions:

Market 1: P1 = 60 – (1/4)Q1

Market 2: P2 = 80 – (1/2)Q2
where Q =Q1 + Q2

Calculate the amounts of outputs, Q1 and Q2 that the monopolist should produce and the prices that it should charge if it wants to maximize total profit. Calculate the amount of total profit.

Answer 1

C = (Q²/2) + 300

In market 1,

C = (Q1²/2) + 300

MC1 = dC/dQ1 = Q1

In market 2,

C = (Q2²/2) + 300

MC2 = dC/dQ2 = Q2

Profit is maximized when MR1 = MC1 and MR2 = MC2

In market 1,

P1 = 60 - (Q1/4) = 60 - 0.25Q1

Total revenue (TR1) = P1 x Q1 = 60Q1 - 0.25Q1²

MR1 = dTR1/dQ1 = 60 - 0.5Q1

Equating with MC1,

60 - 0.5Q1 = Q1

1.5Q1 = 60

Q1 = 40

P1 = 60 - (0.25 x 40) = 60 - 10 = 50

In market 2,

P2 = 80 - (Q2/2) = 80 - 0.5Q2

TR2 = P2 x Q2 = 80Q2 - 0.5Q2²

MR2 = dTR2/dQ2 = 80 - Q2

Equating with MC2,

80 - Q2 = Q2

2Q2 = 80

Q2 = 80

P2 = 80 - (0.5 x 80) = 80 - 40 = 40

Q = Q1 + Q2 = 40 + 80 = 120

Total revenue (TR) = (P1 x Q1) + (P2 x Q2) = (50 x 40) + (40 x 80) = 2,000 + 3,200 = 5,200

Total cost (TC) = [(120 x 120) / 2] + 300 = 7,200 + 300 = 7,500

Profit = TR - TC = 5,200 - 7,500 = -2,300 (loss)