Question

The total cost function of a perfectly competitive firm is TC= 12+2q^2+4q. What is the firm's optimal quantity at a market price of $12? If over time, the firm could exit and not pay its fixed cost, what would the optimal quantity be?

Answer 1

In order to maximize profit a firm produces that quantity at which P = MC. If at that quantity P < AVC(i.e. fail to recover its variable cost) then firm will shut down in the short run.

Here, P = Price = 12, MC = Marginal Cost = d(TC)/dq = 4q + 4

Thus, Profit maximizing condition: P = MC => 12 = 4q + 4 => q = 2

Variable cost(VC) is the portion of TC which depends on Quantity(q).

So, VC = 2q² + 4q => AVC = Average Variable cost = VC/q = (2q² + 4q)/q = 2q + 4 and as calculated above q = 2 => AVC = 2*2 + 4 = 8 < 12. Thus P > AVC and thus it should not shut down and produce 2 units.

Thus, the firm's optimal quantity is 2 units.

At profit maximizing quantity q = 2 and thus Profit = TR - TC where TR = Total Revenue = P*q = 12*2 = 24

=> Profit = P*q - (12 + 2q² + 4q) = 12*2 - (12 + 2*2² + 4*2) = -4 < 0

Thus he is incurring a Loss.

If it exit the market it is given that he will not have to pay any fixed cost and and it will not producing any quantity, Variable cost will also be 0 and thus Total Cost = 0.Also as Quantity = 0, Total revenue will also 0. Hence If it exits its profit = TR - TC = 0 - 0 = 0 which is greater than if it produces. Thus over time he should exit and produce 0 units(i.e. do not produce).

Hence, over time the optimal quantity will be 0 units