Question

1. Suppose a firm has the total cost function T C = 3/8Q^2 + 400 (a) Is this firm in the short run or long run?

(b) Suppose this firm is facing a perfectly competitive market where the price is P = 24. What is the firm’s marginal revenue?

(c) Write the firm’s profit function and solve for the profit-maximizing quantity of production Q∗ . (3 points) (d) How much profit does the firm make at the profit-maximizing level of output? (1 point) (e) Write an equation for the firm’s short run average variable costs (AV C). (1 point) (f) Write the condition for whether a firm should continue to operate or shut down. Should this firm shut down? If not, when should it shut down? (2 points)

2. Suppose a firm faces an inverse demand curve P = 6 − 1/2Q and has a total cost function T C = 1/4Q^2 − Q. (a) Is this firm a price-taker or does it have market power? Explain. (2 points) (b) Write an equation for the firm’s profit function. (1 point) (c) Solve for the firm’s profit-maximizing level of output, Q∗ . (2 points) (d) What price does the firm sell its product at? (1 point)

Answer 1

Given,

For a firm in a perfectly competitive industry,

Total Cost, TC = (3/8)Q2 + 400

a)

Total cost at zero output, TC(0) = (3/8)*02 + 400 = $400

Therefore, Fixed Cost, FC = $400

As there are fixed costs for the firm, the firm is in short-run

b)

Market price = $24

For a firm in perfect competition, Price = Marginal Revenue = Average Revenue

Therefore, the marginal revenue for the given firm = $24

c)

Total Revenue = Price * Quantity = 24 * Q = 24Q

Profit, P(Q) = Total Revenue - Total Cost = 24Q - ((3/8)Q2+400) = 24Q - (3/8)Q2 - 400

Profit is maximized when P'(Q) = 0

=>

=> 24 - (3/8)*2Q - 0 = 0

=> 24 - (3/4)Q = 0

=> Q = 24*4/3

=> Q = 32

P'(Q) = -3/4 < 0. Therefore, the function P(Q) attains a maximum at Q = 32

Profit maximising quantity, Q* = 32 units

d)

Profit at the profit-maximizing quantity i.e., Q = 32 is

P(32) = 24*32 - (3/8)*322 - 400 = -16(loss)

Therefore, the firm experiences a loss of $16 at the profit-maximizing(loss-minimizing) quantity.