Question

The cost function for a firm is C(q) = 8 + 1/2q². In addition, the price of the good

produced is $5 and the marginal cost is MC = 1/2q

If there is a $2 tax in place and the price of the good is still $5. The firm’s marginal cost is: MC(q) = q +2. Will the firm shutdown or continue to operate in the short run? And, in the long run, will the firm shutdown or continue to operate?

Answer 1

Cost function is C(q) = 8 + (1/2)q^2. Marginal cost = dC/dq = 2*(1/2)q = q. AVC =  (1/2)q^2/q = q/2. Minimum of AVC is when q = 0 and so the shut down price is $0.

Given that price is $5. Hence at this price MC = P = q + 2 = 5 or q = 3. AVC is now q/2 + 2. Minimum AVC is now $2. Since price at $5 is greater than minimum AVC and current AVC, it will not shut down in the short run. ATC after the tax is 8/q + q/2 + 2 and so its minimum value in the long run is -8/q^2 + 1/2 = 0 or q* = 4. At this level ATC is 8/4 + 4/2 + 2 = $6. Now Price is less than minimum ATC so the firm will leave in the long run if price is $5.