Question

A multi-plant monopolist faces a demand P = 460 - 6Q . The first plant (call it Plant 1) has total cost, TC1 = 2.5Q2 and the second plant (call it Plant 2) has total cost, TC2 = 5Q2 . Suppose that the monopolist owns both, plants 1 and 2. Determine the optimal price this monopolist should charge, the total number of units the monopolist will supply, and the number of units the monopolist should produce at each plant. Also, calculate the profit of the multi-plant monopolist.

Answer 1

P = 460 - 6Q

TR = P x Q = 460Q - 6Q²

MR = dTR/dQ = 460 - 12Q

TC1 = 2.5Q1²

MC1 = dTC1/dQ1 = 5Q1, so Q1 = MC1 / 5 = 0.2MC1

TC2 = 5Q2²

MC2 = dTC2/dQ2 = 10Q2, so Q2 = MC2 / 10 = 0.1MC2

Since Q = Q1 + Q2,

Q = 0.2MC1 + 0.1MC2

Equating MC1 = MC2 = MC,

Q = 0.2MC + 0.1MC = 0.3MC

Q = MC / 0.3

MC = 0.3Q

Equating MR = MC,

460 - 12Q = 0.3Q

12.3Q = 460

Q = 37.4

MC = 0.3 x 37.4 = 11.22

Q1 = 0.2 x 11.22 = 2.244

Q2 = 0.1 x 11.22 = 1.122

Q = 2.244 + 1.122 = 3.366

P = 460 - 6 x 3.366 = 460 - 20.196 = 439.804

TR = P x Q = 439.804 x 3.366 = 1480.38

TC = TC1 + TC2 = 2.5 x 2.244 x 2.244 + 5 x 1.122 x 1.122 = 12.59 + 6.29 = 18.88

Profit = TR - TC = 1480.38 - 18.88 = 1,461.5