Question

Consider a monopoly whose total cost function is TC=Q3-30Q2+302Q, whose marginal cost function is MC= 3Q2-60Q+302, whose demand is P=329-30Q, and whose marginal revenue function MR= 329-60Q, where Q is output and P is price.

Assume that the firm maximizes profit but cannot practice price discrimination 

How much does the firm produce?        

How much does the firm charge?

How large are the firms profit?

Answer 1

Solution:

a.). A monopolist will produce an output where MR = MC, which is the profit maximizing quantity.

MR = 329 – 60Q

MC = 3Q² – 60Q + 302

Set MC = MR to derive profit maximizing output:

MC = MR

3Q² – 60Q + 302 = 329 – 60Q

3Q² – 60Q + 60Q = 329 – 302

3Q² = 27

Q² = 9

Q = 3

The firm will produce 3 units.

 

b.). Derive price from the demand function:

P = 329 – 30Q

P = 329 – 30(3)

P = 329 – 90

P = 239

The firm will charge a price of 239

 

c.). Profit = TR – TC

TR = P $\times$ Q

TR = 239 $\times$ 3 = 717

TC = Q³ - 30Q² + 302Q

TC = 3³ – 30(3²) + 302(3)

TC = 27 – 270 + 906

TC = 27 + 906 – 270

TC = 663

Profit = 717 – 663 = 54

The firm’s profit is 54