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

$P=1200-2Q$

$TR=1200Q-2Q^2$

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

$MR=\frac {\delta TR}{\delta Q}=1200-4Q$

$MC=\frac {\delta TC}{\delta Q}= 3Q^2-122.5Q+1528.5$

(1) To find the equilibrium output

Equate MR=MC:

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

$3Q^2-118.5Q+328.5$ =0 …(equation 1)

solving equation 1 quadratically,

$Q=\frac {118.5+/- ([118.5]^2-[4\times 3\times 328.5])^\frac{1}{2}}{2\times3}$

$Q=37$

Equilibrium output level= 37 .

The profit maximizing quantity= 37

(2) To find the monopolistic price,

Substitute the value of Q in the demand curve equation:

$P=1200-2(37)$

$P=1126$

The monopolistic price is 1126.

(3) To find profit , we work out the difference between Total Revenue(TR) and Total Cost(TC)

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

substituting Q=37;

$TC=25,356.25$

$TR=1200Q-2Q^2$

substituting Q =37;

$TR=41,662$

$\therefore$ Profit = Total Revenue- Total Cost

= $41662-25356.25$

$=16,305.75$