Question

1. 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 = 100L³5 Kº". The wage rate is birr 9 per hour and the rental rate on capital is birr 4 per machine hour.

A. Find the minimum cost of producing 1200 units.

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 this value 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