Question #289419

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.

1
Expert's answer
2022-01-24T11:02:43-0500

w=9

r=4

q=1200

q=100L12K12q=100L^\frac{1}{2}K^\frac{1}{2}

Derive MPL and MPK


MPL = QL\frac{\partial Q} {\partial L} = 50L-0.5K0.5


MPK = QK\frac{\partial Q} {\partial K} = 50L0.5K-0.5


MPLMPK\frac{MPL}{MPK} = wr\frac{w}{r}

w = 9

r = 4

50L-0.5K0.5 ÷\div 50L0.5K-0.5 = 94\frac{9}{4}


50L0.5K0.550L0.5K0.5=94\frac {50L^{-0.5}K^{0.5}}{50L^{0.5}K^{-0.5}}=\frac{9}{4}


KL=94\frac{K}{L}=\frac{9}{4}


K=94L\frac{9}4L


Substitute this value in the production function

q=100(94L)0.5L0.5q=100(\frac{9}{4}L)^{0.5}L^{0.5}


1200=100(134L)0.51200=100(\frac{13}{4}L)^{0.5}


L0.5=1200(100)(3.25)0.5L^{0.5}=\frac{1200}{(100)(3.25)^{0.5}}


L=(1200(100)(3.250.5)2L=(\frac{1200}{(100)(3.25^{0.5}})^2

=144000032500=44.31= \frac{1440000}{32500}= 44.31


K=94LK= \frac{9}{4}L

K=94×44.31=99.69K= \frac{9}{4}\times 44.31= 99.69


TC= wL+rK

720=9L+4K


But L=94K\frac {9}{4}K

720=9L+94L720= 9L+\frac{9}{4}L

720=454L720=\frac{45}{4}L


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


K=94×64=144K=\frac{9}{4}\times 64= 144


Total Output=144+64= 208

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS