Question

Suppose the monopolist faces the market demand function given by Q=144/p2

The AVC of the firm is given as AVC = Q ½ and the firm has a fixed cost of $ 5

a) determine equilibrium P&Q

b) determine the maximum profit

Answer 1

To identify the profit, there is a need to set the equation: MR = MC

Given:

$Q = \frac{144}{P^2}, \space P^2 = \frac{144}{Q}$

Or $P=(\frac{144}{Q})^{\frac{1}{2}} ...........(i)$

Therefore, $P=(\frac{12}{\sqrt{Q}})$

Now TR = Price and Quantity sold

TR = PQ

$=\frac{12}{\sqrt Q}\times Q$

$=12\sqrt Q..........................(ii)$

Now, MR=

$\frac{\Delta(TR)}{\Delta Q}$ [MR derivative of $\frac{TR}{\Delta Q}$ ]

$=\frac{\Delta 12\sqrt Q}{\Delta Q}$

$=\frac{1}{2}(12Q)\\=\frac{6}{\Delta Q}.........................(iii)$

Given the total fixed cost of $5

Average variable cost (AVC)$=Q^{\frac{1}{2}}=\sqrt Q$

TC = TFC + TVC

TVC = (AVC)(Q)

Therefore, total cost  $TC=5+(\sqrt Q)(Q)$

$=5+Q^{\frac{1}{2}+1}\\=5+Q^{\frac{3}{2}}.......................(iv)$

with TC, Now we can compute MC by taking the derivation within respect to Q

Now MR = MC,

Therefore to find the profit-maximizing level of output

$\frac{6}{\sqrt Q}=\frac{3}{2}\sqrt Q\\or\space Q=\frac{12}{3}$

Q = 4units

by putting Q = 4

$P=\frac{12}{\sqrt Q}$

$P=\frac{12}{\sqrt 4}$

$P=\frac{12}{2}$

P = $6

Then profit is:

Profit = PQ - TC

$=(6)(4) - [5 + (4)^{3}{2}]$

$=24-13\\ = \$11$