Question

A monopoly firm has the following demand and cost conditions:P = 100 -0.5qTC=1000+10q+q²MC = 10 + 2qa. Find the output level q* which maximizes profit

Answer 1

Given P, TC, MC, the profit is given by

$\pi=Pq-TC-MC$

Now, to find the q that maximizes the profit, consider the first and the second derivative of the last function.

$\frac{d}{dq}\pi=P+q\frac{d}{dq}P-\frac{d}{dq}(TC)-\frac{d}{dq}(MC)=100-0.5q+q(-0.5)-(2q+10)-2$

after solving this part we get that

$88-3q=0\text{ will give us the critical points for the profit function.}$

So,

$q=\frac{88}{3};$

Finally, the second derivative of the profit function gives us

$\frac{d^2}{dq^2}\pi=\frac{d}{dq}(88-3q)=-3<0$

It means that the profit function has a maximum local point at q=88/3.