Answer to Question #219815 in Microeconomics for Prince Ahmed

Question

Suppose that , firm under perfectly competition market produce two commodities X1 and X2 with corresponding prices birr 10 and birr 15 . If cost function of the firm is

C= 2x12 +x1x2+2x22 where x1 and x2 denote the level of output, then, determine the following questions.

i. Profit maximizing level of output x1 and x2

Answer 1

Price of$\space X_1 =P_1=10$

Price of $X _ 2 ​ =P_ 2 ​ =15$

Cost function $C=2X_1^2+X_1X_2+2X_2^2$

$Total\space Revenue\space R=P_1X_1+P_2X_2\\=10X_1+15X_2$

$Profit\space \pi=R-C\\=10X_1+15X_2-2X_1^2-X_1X_2-2X_2^2$

To find the profit maximizing output of $X_1:$

$\frac{\delta \pi}{\delta X_1}=0$

$\implies\frac{\delta \pi}{\delta X_1}(10X_1+15X_2-2X_1^2-X_1X_2-2X_2^2)=0$

$\implies10+0-4X_1-X_2-0=0$

$\implies4X_1+X_2=10.........................................1$

To find the profit maximizing output of $X_2:$

$\frac{\delta \pi}{\delta X_2}=0$

$\implies\frac{\delta \pi}{\delta X_2}(10X_1+15X_2-2X_1^2-X_1X_2-2X_2^2)=0$

$\implies0+15-0-X_1-4X_2=0$

$\implies X_1+4X_2=15...........................................2$

$Eq1\times4\implies16X_1+4X_2=40\\\underline{Eq2\times1\implies X_1\space \space \space \space +4X_2=15}\\\space \space \space \space \space \space\space\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space 15X_1\space \space \space \space \space \space\space \space \space \space \space \space =25\\X_1=\frac{25}{15}\\\implies X=\frac{g5}{3}\\X_1\approx1\space unit$

In Eq

1, $4X_1+X_2=10\\\implies X_2=10-4(\frac{5}{3})\\X_2=\frac{30-20}{3}\\X_2=(\frac{10}{3})\\X_2\approx3 \space units$