Question

Suppose a firm's total revenues depends on the amount produced (q) according to the functoins TR=70q-q, total cost also depends on q; TC=q²+30q+100 a. What level of output should the firm produce in order to maximize profit? b. Show that the second order conditions for a maximum are satisfied at the out put level found in part (a). c. Does the solution calculated here obey the condition "marginal revenue equal marginal cost" rule? Explain.

Answer 1

a.

$\pi=TR-TC\\=(70q-q^2)-(q^2+30q+!00)\\=70q-q^2-q^2-30q-100\\=40q-2q^2-100$

necessary condition will be as dollows

$\frac{\delta \pi}{\delta q}=0\\40-4q=0\\4q=40\\q=10$

therefore the profit will be

$\pi=TR-TC\\=40q-2q^2-100\\=40(10)-2(10)^2-100\\400-200-100\\=\$100$

10 units will be produced to get a profit of $100

b.

the second order is $\frac{\delta ^2\pi}{\delta q^2}=0$

$\frac{\delta ^2\pi}{\delta q^2}=0\\ \frac{\delta ^2(40-4q)}{\delta q^2}\leq0\\-4\leq0$

c.

$MR=\frac{d(TR)}{dq}\\=70-2q\\MC=\frac{d(TC)}{dq}\\=2q+30\\MR=MC\\70-2q+30\\4q=40\\q=10$

the solution obeys the rule of mrginal revenue equa;s to marginal cost