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 TC= 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

ii. The amount of maximum profit

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