Question

A manufacturer determines that his total cost function is $c=\frac{q^3}{3}+2q+300$. Where $q$ is the number of units produced.
(i) Find the marginal cost function.
(ii) Find the average cost function.
(iii) Find the level of output at which the averages cost is minimum.

Answer 1

(a) Taking the first derivative of the total cost function wrt to quantity (q) gives the marginal cost function

$MC=\frac{dc}{dq}=q^2+2$

(b) Average cost function is given by the total cost function divided by the number of units (q).

$AC=\frac{c}{q}=\frac{q^2}{3}+2+\frac{300}{q}$

(c)To find the level output at the average cost is minimum we have to set

$\frac{d\left(AC\right)}{dq}=0\Rightarrow q^3=450\Rightarrow q=\sqrt[3]{3450}$