Question

Suppose the short-run cost function of Tyrell Corporation is 𝒄 = 𝟓𝟎𝟎𝟎 + 𝟏𝟎𝟎𝒒 + 𝟖𝒒^𝟐. 

a.  Find the level/s of  that satisfies/satisfy the first-order and second-order conditions for the minimisation

of Tyrell’s average cost (𝒂𝒄). [4]

Show that 𝒂𝒄 equals marginal cost (𝒎𝒄) at the level/s of  obtained in (a). [2

Answer 1

(a)

Given cost function:

$C = 5000 + 100q + 8q^2$

Average cost (AC)

$=\frac{ C}{q}$

$=\frac{5000}{q}+100+8q$

To minimize AC, FOC :-

$=\frac{\delta(AC)}{\delta q}=0$

or, $5000(-1)q^{-2} + 8 = 0$

or, $q = 25$

(b)

SOC :-

$\frac{d^{2}(AC)}{dq^{2}}=-5000(-2)q^{-3}$

Substituting q = 25 we get 0.64, which is > 0. Hence SOC is satisfied. AC must be minimum at q = 25.

Substituting q=25 in AC we get 500.

Now, MC:-

$\frac{\delta C}{\delta q}=100+16q$

Substituting q=25 in MC we get 500.

Hence MC = AC at q=25.