A firm has the long-run cost function C(Q)= 4Q2 + 64.In the long run, it will supply a positive amount of output, so long as the price is greater than
A)$64.
B)$72.
C)$16.
D)$32.
E)$37.

Question

Quizplus · Accepted Answer

In the long run, a firm will supply output where marginal cost (MC) equals the minimum of the AVC and ATC curves. 
We can find the firm's long-run AVC and ATC by dividing the long-run cost function by Q: 
AVC(Q) = [F1F1F1S1 + F1F1F10/64] / Q = (1/64)F1F1F1S1/Q + F1F1F10/64
ATC(Q) = [F1F1F1S1 + F1F1F10]/Q = (1/Q)F1F1F1S1 + F1F1F10/Q
To find the minimum of AVC and ATC, we need to take the derivative of each with respect to Q: 
AVC'(Q) = - (1/64)F1F1F1S1/Q^2 
ATC'(Q) = - (1/Q^2)F1F1F1S1 + F1F1F10/Q^2
Setting these derivatives equal to each other, we get: 
-(1/64)F1F1F1S1/Q^2 = -(1/Q^2)F1F1F1S1 + F1F1F10/Q^2 
Simplifying and multiplying through by Q^2, we get: 
F1F1F1S1 = 64F1F1F10/Q 
Solving for Q, we get: 
Q = √(64F1F1F10/F1F1F1S1)
Plugging in the given values, we get Q = √(64*64/4) = 32
Therefore, the firm will supply output if P > MC = C'(Q) = (1/32)F1F1F1S1 + F1F1F10/64 = $32.

A Firm Has the Long-Run Cost Function C(Q)= 4Q2