Question

1. Multiplant monopoly problem: Assume the firm has two plants with the following marginal cost functions:

MC₁= 20 + 2Q₁

MC₂= 10 + 5Q₂

Assume that the inverse demand curve is P = 500-Q.

What is the maximum profit?  Assume total Fixed Costs for plant 1 (TFC₁) = $0 and Total Fixed Costs for plant 2 (TFC₂) = $0.

Answer 1

P = 500 - Q

Total revenue (TR) = PQ = 500Q - Q²

Marginal revenue (MR) = dTR/dQ = 500 - 2Q

MC1 = 20 + 2Q1, therefore Q1 = (MC1 - 20)/2 = 0.5MC1 - 10

MC2 = 10 + 5Q2, therefore Q2 = (MC2 - 10)/5 = 0.2MC2 - 2

Since Q = Q1 + Q2,

Q = 0.5MC1 - 10 + 0.2MC2 - 2

Q = 0.5MC1 + 0.2MC2 - 12

Setting MC1 = MC2 = MC,

Q = 0.5MC + 0.2MC - 12

Q = 0.7MC - 12

0.7MC = Q + 12

MC = (Q + 12)/0.7

Profit is maximized by equating MR and MC.

500 - 2Q = (Q + 12)/0.7

350 - 1.4Q = Q + 12

2.4Q = 338

Q = 140.83

P = 500 - 140.83 = 359.17

MC = (140.83 + 12)/0.7 = 152.83/0.7 = 218.33

Maximum profit = Q x (P - MC) = 140.83 x (359.17 - 218.33) = 140.83 x 130.84 = 19,834.50