Question

Given a firms demand function Q-90+2P=0 and it's average cost function AC=Q²-8Q+57+2/Q. Find the level of output which maximises marginal cost.

Answer 1

Given Information

Demand function

$Q-90+2P=0$

Average cost function

$AC=Q²-8Q+57+\frac{2}{Q}$ 

$Total\space cost = AC \times quantity$

        $=( Q²-8Q+57+\frac{2}{Q} ) \times Q$

$=Q^{3} - 8Q^{2} +57Q + 2$

      Differentiate Total Cost with respect of Quantity

$MC = \frac{\delta TC}{\delta Q}$

$MC = \frac{Q^{3} − 8Q^{2} +57Q + 2}{Q}$

$= 3Q^{2} -16Q +57$

To calculate Maximization of MC of output we equate Mc with 0

MC = 0

$3Q^{2} -16Q +57 = 0$

a = 3

b = -16

c = 57

$b^{2} - 4ac = (-16)2 - 4 \times 3\times 57$

$= 256 - 684$

$= -428.$

$\frac{−b ±√b2 − 4ac}{2a}$

 $= \frac{−(−16) ±√ −428}{2 x 3}$

$= 36.68i$

And another value will be -4.68i

 Negative output is not possible hence the 36.68 output will be maximize MC.