Question

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.

Find the minimum cost of producing 1200 units.

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

Answer 1

w=9

r=4

q=1200

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

Derive MP_L and MP_K

MP_L = $\frac{\partial Q} {\partial L}$ = 50L^-0.5K^0.5

MP_K = $\frac{\partial Q} {\partial K}$ = 50L^0.5K^-0.5

$\frac{MPL}{MPK}$ = $\frac{w}{r}$

w = 9

r = 4

50L^-0.5K^0.5 $\div$ 50L^0.5K^-0.5 = $\frac{9}{4}$

$\frac {50L^{-0.5}K^{0.5}}{50L^{0.5}K^{-0.5}}=\frac{9}{4}$

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

K=$\frac{9}4L$

Substitute this value in the production function

$q=100(\frac{9}{4}L)^{0.5}L^{0.5}$

$1200=100(\frac{13}{4}L)^{0.5}$

$L^{0.5}=\frac{1200}{(100)(3.25)^{0.5}}$

$L=(\frac{1200}{(100)(3.25^{0.5}})^2$

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

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

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

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