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

MC_{1} = 20 + 2Q_{1} and MC_{2} = 10 +
5Q_{2}

The goal of the Monopolist is to maximize profit

TR = Total Revenue

TR = PQ

TR = (500 – Q)Q

TR = 500Q –Q^{2}

Marginal revenue (MR) = ΔTR/ΔQ = 500 – 2Q = 500 –
2(Q_{1} + Q_{2})

Equating each MC to the common MR

MC_{1} = MR

20 + 2Q_{1} = 500 – 2Q_{1} – 2Q_{2}

4Q_{1} + 2Q_{2} = 480

MC_{2} = MR

10 + 5Q_{2} = 500 – 2Q_{1} – 2Q_{2}

2Q_{1} + 7Q_{2} = 490

Solving for Q_{1} and Q_{2} we find

**Q _{1} =
99.17**

**Q _{2} =
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 – C_{1} – C_{2}

C_{1} = ∫MC_{1} = ∫20 + 2Q_{1} =
20Q_{1} + Q^{2}_{1}

C_{2} = ∫MC_{2} = ∫10 + 5Q_{2} =
10Q_{2} + 5/2(Q^{2}_{2})

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

**Π =
34,010.38**

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