Question

The demand and cost function for a company is estimated to be as follows: 

P = 100-8Q; TC = 50+80Q-10Q 2 + 0.6Q 3.

A.   What Price should it charge if it wants to maximize its profit in the short run?

B.    What price should it charge if it wants to maximize its revenue in the short run?

Answer 1

A) $P=100-8Q$

At maximum, $TR=PQ$

$TR=(100-8Q)Q=100Q-8Q^2$

$TC=50+80Q-10Q^2+0.6Q^3$

Now $MR=MC$

$\Delta TR/\Delta Q=\Delta TC/\Delta Q$

$100-16Q=80-20Q+1.8Q^2$

$80-20Q+1.8^2-100+16Q=0$

$1.8Q^2-4Q-20=0$

Use quadratic formula to solve

$Q=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=-\frac{(-4)\pm \sqrt{(-4)^2-4(1.8)(-20)}}{2(1.8)}=\frac{4\pm 12.64}{3.6}$

$Q=4.62$ or -2.4

Since the quantity demanded is not negative, the profit-maximizing level of output=4.62 units.

Now, $P=100-8Q=100-8(4.62)=\$63.04$

B) For revenue maximizing firm $MR=0$

$100-16Q=0$

$16Q=100$

$Q=6.25$

Now, $P=100-8Q=100-8(6.25)=\$50$