Answer to Question #293602 in Microeconomics for Ermi 09

Question

10. The Bok Chicken Factory is trying to figure out how to minimize the cost of producing 1200 units of chicken parts. The production function is q = 100L0.5 K0.5. The wage rate is birr 9 per hour and the rental rate on capital is birr 4 per machine hour

A.B. Find the maximum output that can be produced for a total cost of birr 720.

Answer 1

w=9

r=4

q=1200

$q=100L^\frac{1}{2}K^\frac{1}{2}$

$(\frac{\alpha}{\beta})(\frac{K}{L})= (\frac{w}{r})$

$(\frac{0.5}{0.5})(\frac{K}{L})=(\frac{9}{4})$

$\frac{K}{L}=\frac{9}{4}$

K=$\frac{9}4L$

Substitute K in the production function

$q=100(\frac{13}{4})^\frac{1}{2}(L^\frac{1}{2})$

$1200=100(\frac{13}{4}L)^\frac{1}{2}$

$L^\frac{1}{2}=(\frac{1200}{(100)(3.25^\frac{1}2)})$

$L=(\frac{1200}{(100)(3.25^\frac{1}2)})^2$

$= \frac{1440000}{32500}= 44.31$

$K= \frac{9}{4}L$

$K= \frac{9}{4}\times 44.31= 99.69$

TC= wL+rK

720=9L+4K

But L=$\frac {9}{4}K$

$720= 9L+\frac{9}{4}L$

$720=\frac{45}{4}L$

$L=\frac{(720\times 4)}{45}= 64$

$K=\frac{9}{4}\times 64= 144$

Total Output=144+64= 208