Answer to Question #289771 in Microeconomics for Honey Grace

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 = 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. 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

Solution:

A.). Q = 100L^0.5K^0.5

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 = $\frac{9}{4}$

K = $\frac{9L}{4}$ = 2.25L

1200 = 100L^0.5K^0.5

1200 = 100L^0.5(2.25L^0.5)

L = 5.3

K = 2.25L = 2.25(5.3) = 11.925

C = wL + rK

C = (9 $\times$ 5.3) + (4 $\times$ 11.925) = 47.7 + 47.7 = 95.4

The minimum cost to produce 1200 units = 95.4