A firm can produce any quantity of good X with the following

A firm can produce any quantity of good X with the following cost structure: TC = 450,000 + 20Q, where Q measures units of output.

A) The industry demand for good X is Q = 100,000 - 500P. At the profit-maximizing output level, calculate the firm\'s ATC of production.

B) Suppose the profit-maximizing output level you calculated to answer part A is split evenly between two firms, each with the cost structure given by TC = 450,000 + 20Q. What is the ATC of production in this two-firm industry?

Solution

A) Q = 100,000 - 500P

    The inverse demand function is

      P = 200 - 0.002Q

TR = PxQ = 200Q - 0.002Q²

MR = 200 - 0.004Q

MC = 20

At the profit-maximizing output level MR = MC

200 - 0.004Q = 20

0.004Q = 180

Q = 180/0.004 = 45000 units

TC = 450,000 + 20Q

ATC = TC/Q = 450,000/Q + 20

ATC = 450,000/45000 + 20 = $30

B) If the total output is split evenly between two firms, each firm will produce 22500 units. Then,

    ATC = 450,000/22500 + 20 = $40