Question

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?

Answer 1

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