Question

A manufacturer in a monopolistically competitive industry produces two different brands of a product for which the demand functions are P1=56-4Q1 and P2=48-2Q2. and the joint cost function is

TC=Q12+5Q1Q2+Q22. Find the profit maximizing level of output and the price that should be charged for each brand.

Answer 1

$P_1= 56-4Q_1$

$P_2= 48-2Q_2$

$TC=Q_1^2+5Q_1Q_2+Q_2^2$

$\Pi=TR-TC$

$= P_1Q_1+P_2Q_2$ - $(Q_1^2+5Q_1Q_2+Q_2^2)$

$=Q_1(56-4Q_1)+Q_2(48-2Q_2)-Q_1^2-5Q_1Q_2-Q_2^2$

=$56Q_1-4Q_1^2+48Q_2-2Q_2^2-Q_1^2-5Q_1Q_2-Q_2^2$

First Order Condition

$\frac{\Delta\Pi}{\Delta Q_1}=56-10Q_1-5Q_2=0$

$\frac{\Delta\Pi}{\Delta Q_2}=48-6Q_2-5Q_1=0$

Solve the two Equations Simultaneously by eliminating $Q_1$

$56-10Q_1-5Q_2=0$

$(48-6Q_2-5Q_1=0)2$

=$56-10Q_1-5Q_2=0$

$96-10Q_1-12Q_2=0$

40=7Q$_2$

$Q_2= 5.7$

Eliminate Q$_2 from the above equations$

=$(56-10Q_1-5Q_2=0)6$

$=(48-6Q_2-5Q_1=0)5$

$336-60Q_1-30Q_2=0$

$240-30Q_2-25Q_1= 0$

$35Q_1=96$

$Q_1= 2.7$

From the Inverse Demand functions

$P_1=56-4(2.7)=45.2$

$P_2=48-2(5.7)=36.6$