Question

Newsprint (the paper used for newspapers) is produced in a perfectly competitive, constant cost, market. Each identical firm has total cost equal to TC = 32+ 20q + 0.5q2, where q is the amount of output produced by each firm. What is the quantity produced by each firm in the long-run? What is the long-run price?

Answer 1

In long-run, a perfectly competitive firm produce that level of output corresponding to which average total cost is at its minimum.

The price in long-run in a perfectly competitive market is equal to minimum average total cost.

The total cost function is as follows -

TC = 32 + 20q + 0.5q2

The average total cost function is as follows -

ATC = TC/q = [32 + 20q + 0.5q2]/q = (32/q) + 20 + 0.5q

The marginal cost function is as follows -

MC = dTC/dq = d(32 + 20q + 0.5q2)/dq = 20 + q

Average total cost is at its minimum when it is equal to the marginal cost.

ATC = MC

(32/q) + 20 + 0.5q = 20 + q

(32/q) + 0.5q - q = 20 - 20

(32/q) - 0.5q = 0

32/q = 0.5q

0.5q2 = 32

q2 = 32/0.5 = 64

q = 8

The average total cost is at its minimum when 8 units of output are produced.

So,

The quantity produced by each firm in the long run is 8 units.

Calculate minimum ATC -

ATC = (32/q) + 20 + 0.5q

ATC = (32/8) + 20 + (0.5 * 8)

ATC = 4 + 20 + 4

ATC = 28

The minimum ATC is $28.

Thus,

The long-run price is $28 per unit.