Question

Suppose the total cost function of a firm is:

C = 3q3 – 12q2 + 20q a. Find the marginal cost function. Show your steps. (2 marks)

b. Find the average total cost function. Show your steps. (2 marks)

c. Find the output level that correspond to the efficient scale of the firm. Show your

steps. (2 marks)

d. Show your results in (c) in a diagram. (2 marks)

Answer 1

Part (a)

$Marginal\ cost,MC=\frac{dTC}{dq}=3*3*q^{3-1}-12*2q^{2-1}+20=9q^{2}-24q+20\\ MC=9q^{2}-24q+20\\$

Part (b)

$Average\ cost,AC=\frac{TC}{q}=\frac{3q^{3}-12q^{2}+20q}{q}=3q^{2}-12q+20$

Part (c)

efficient scale of firm is level of output where average cost is minimized

$AC=3q^{2}-12q+20\\ differentiating\ with\ respect\ to\ q\ and\ equating\ to\ 0\\ \frac{dAC}{dq}=6q-12=0\\ q=\frac{12}{6}=2\\ \frac{d^{2}AC}{dq^{2}}=6>0\\ \therefore\ Average\ cost(AC)\ is\ minimized\ at\ q=2$

Therefore, at efficient scale of production average cost is minimized and equal to marginal cost

Part (D)