Question

If the inverse demand curve of profit maximizing monopolist is given as P =1200 − 2Q , and cost function as 

 C = Q3 − 61.25Q2+1528.5Q + 2000, find equilibrium output level, monopolist price, and profit.

Answer 1

$P= 1200-2Q$

$C= Q^3-61.25Q^2+1528.5Q$ +2000

Total Revenue= Price ×Quantity ($P×Q)$

Therefore TR = $(1200-2Q)Q$ = 1200Q-2Q²

Marginal Revenue$= \frac{\delta TR}{\delta Q}$ =

$1200- 4Q$

Marginal Cost $= \frac{\delta TC}{\delta Q}$ =

$3Q^2 - 122.5Q+1528.5$

At equilibrium output $MR=MC$

$1200- 4Q =3Q^2-122.5Q+1528.5\implies$

$3Q^2-118.5Q+328.5=0$

$Q= 36.5$ or 3

Equilibrium output= $36.5$

Using Q = 36.5, the monopolist price is

$P= 1200-2(36.5) = 1127$

Profit is obtained by getting the difference between total revenue and the total cost.

Profit = $TR- TC$

$TR = 1200(36.5)-2(36.5)^2$

$= 41135.5$

$.TC = 36.5^3- 61.25(36.5^2)$ + 1528.5(36.5)+2000 = 24817.0625

Profit = $41135.5- 24817.0625$ = 16318.4375