Question

# Multiplant monopoly problem: Assume the firm has two plants with the following marginal cost functions: MC1...

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 profit maximizing outputs produced in each plant? Show your work.
What is the profit maximizing price? Show your work.
What is the maximum profit?

The monopolist's demand curve is

P = 500 – Q

The costs of the two plants are

MC1 = 20 + 2Q1 and MC2 = 10 + 5Q2

The goal of the Monopolist is to maximize profit

TR = Total Revenue

TR = PQ

TR = (500 – Q)Q

TR = 500Q –Q2

Marginal revenue (MR) = ΔTR/ΔQ = 500 – 2Q = 500 – 2(Q1 + Q2)

Equating each MC to the common MR

MC1 = MR

20 + 2Q1 = 500 – 2Q1 – 2Q2

4Q1 + 2Q2 = 480

MC2 = MR

10 + 5Q2 = 500 – 2Q1 – 2Q2

2Q1 + 7Q2 = 490

Solving for Q1 and Q2 we find

Q1 = 99.17

Q2 = 41.67

So, the total Q is 140.83 units. This total output will be sold at price P defined by

P = 500 – Q

P = 359.2

The Monopolit’s Profit is

Π = TR – C1 – C2

C1 = ∫MC1 = ∫20 + 2Q1 = 20Q1 + Q21

C2 = ∫MC2 = ∫10 + 5Q2 = 10Q2 + 5/2(Q22)

Π = (359.2 × 140.83) – 1983.4 – 9834.69 – 416.7 – 4340.97

Π = 34,010.38