Answer to Question #289419 in Macroeconomics for Shalke

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