Total costs for a firm are given by: TC = 3Q22 + 12

In the long-run, under the perfect competition, the firm's optimal price level is determined at a level where its price is the same as the minimum of the average total cost.

Here,

TC=3Q2+12.6TC=3Q2+12.6

Therefore, the average total cost (ATC) can be estimated as:

ATC=TC/Q=3Q+(12.6/Q)ATC=TC/Q=3Q+(12.6/Q)

Now, derivate the ATC function and equate it to zero. That is

∂ATC∂Q=3−(12.6/Q2)=0∂ATC∂Q=3−(12.6/Q2)=0

By solving this, we get, Q = 2.05

Hence, ATC=3(2.05)+(12.6/2.05)ATC=3(2.05)+(12.6/2.05)

ATC = $12.30 (approx.)

Therefore, in the long-run, if the price falls below $12.30, the firm will exit the market.

Average Total Cost (ATC):

In economics, the term ATC is associated with the average cost that a firm incurs to produce certain units of a product. Mathematically, it can be estimated as TC/Q where TC is the total cost and Q is the quantity produced.