Question

Firm Z, operating in a perfectly competitive market, can sell as much or as little as it wants of a good at a price of $16 per unit. Its cost function is C = 50 + 4Q + 2Q2.

(a) Determine the firm’s profit-maximizing level of output. Compute its profit.

(b) The industry demand curve is QD = 200 - 5P. What is the total market demand at the current $16 price? If all firms in the industry have cost structures identical to that of firm Z, how many firms will supply the market?

(c) The outcomes in part (a) and (b) cannot persist in the long run. Explain why. Find the market’s price, total output, number of firms, and output per firm in the long run.

(d) Comparing the short-run and long-run results, explain the changes in the price and in the number of firms.

Answer 1

C = 50 + 4Q + 2Q^2

ATC = C/Q = 50/Q + 4 + 2Q

MC = dC/dQ = 4 + 4Q

Equilibrium at P=MC

16 = 4 + 4Q

4Q = 12

Q = 3 units (at equilibrium each firm will produce 3 units)

ATC = C/Q = 50/Q + 4 + 2Q at Q = 3

ATC = 50/3 + 4 + 2*3 = 26.67

Profits can be calculated by using formula

(P-ATC)*Q = (16-26.67)*3 = -32.01 (The firm suffers a loss)

Total Market demand at P = 16, QD = 200 - 5*16 = 120 units

The number of firms will be QD/Q = 120/3 = 40 firms in the market

In the long run, the firms will exit the market considering the loss being suffered. The firms will exit and supply curve will shift to the left bringing the price up to the level equal to the ATC and all firms at that point will earn normal profits. In the long run the firms will earn normal profits. The market will keep adjusting in the short run but in the long run the firms will make ZERO economic profits.

In the long run price will equal min ATC

ATC = C/Q = 50/Q + 4 + 2Q

for min ATC we take first order derivative of the ATC function

-50/Q^2 + 2 = 0

50/Q^2 = 2

Q^2 = 25 or Q = 5 units in the long run (Output per firm) (Q=+-5, we ignore negative value of output)

P = ATC = 50/5+4+2*5 = 24

at P = 24, market Qd = 200 - 5*24 = 80 units (Market output)

No of Firms = 80/5 = 16 Firms

In the long run the no of firms are less than the no of firms when market price was lower and firms were suffering losses. Due to exit of firms from 40 in numbers to only 16, the supply fell and price rose from 16 to 24 so that the firm earns normal profits in the long run.