Answer to Question #279343 in Microeconomics for tom

Question

Think about a monopolist, the market (inverse) demand function is: P = 30-2Q, his cost function is: C(Q) = 5+ Q²

1. What is the monopolist’s optimal quantity and price? 
2. What is the monopolist’s highest profit? 

Answer 1

Solution:

1.). Profit maximizing quantity is where MR = MC

Derive MR:

TR = P $\times$ Q

TR = (30 – 2Q) Q = 30Q – 2Q²

MR = $\frac{\partial TR} {\partial Q}$ = 30 – 4Q

 

MC = $\frac{\partial TC} {\partial Q}$ = 2Q

Set MR = MC

30 – 4Q = 2Q

30 = 2Q + 4Q

30 = 6Q

Q = 5

Optimal quantity = 5 units

Substitute in the demand function to derive optimal price:

P = 30 – 2Q

P = 30 – 2(5) = 30 – 10 = 20

P = 20

Optimal price = 20

 

2.). Monopolist highest profit = TR – TC

TR = P $\times$ Q = 20 $\times$ 5 = 100

TC = 5+ Q² = 5 + 5² = 5 + 25 = 30

Monopolist highest profit = 100 – 30 = 70

Monopolist highest profit = 70