Question

A firm has the following average cost 𝑨𝑪 = 𝟐𝟎𝟎 𝑸 + 𝟐𝑸 − 𝟑𝟔.

a. Find the stationary point and determine if it is a maximum or a minimum.

b. Find the marginal cost function.

Answer 1

$AC=200/Q+2Q-36$

A) Taking the FOC

$AC'=-200Q^{-2}+2=0$

$200=2Q^2$

$100=Q^2$

$Q=-10$ $Q=10$

Taking the SOC

$AC''=400Q^{-3}$

$AC''(10)=400(10)^{-3}$

$AC''=0.4>0.$

The function is a relative minimum.

B) $MC=\frac{dTC}{dQ}$

$TC=AC×Q=(200/Q+2Q-36)Q$

$TC=200+2Q^2-36Q$

$MC=4Q-36$