Question

Consider the non-linear inverse demand function, p= -q2-q+I. Given the total cost function: TC =9q2+2q+
i. Find the marginal cost (MC) and average cost (AC) as functions of q
ii. Find the output at which average cost is minimized

Answer 1

i. Find the marginal cost (MC) and average cost (AC) as functions of q

The total cost function is:

$$TC =9q^2+2q+ 81$$

The marginal cost is:

$$MC = \dfrac{TC}{\Delta Q}= 18q + 2$$

The average cost is:

$$AC = \dfrac{TC}{q} = 9q + 2 + \dfrac{81}{q}$$

ii. Find the output at which average cost is minimized

The marginal cost crosses the average cost at its minimum. Thus, at the minimum of the average cost, $MC = AC.$

$$18q + 2 = 9q + 2 + \dfrac{81}{q}$$

$$9q = \dfrac{81}{q}$$

$$q^2 = 9$$

$$q^* = \sqrt{9} = 3$$