Question

1. A product’s production function is given as Q=50KL. Assume all factors of production are variable. What would be the minimum level of output of production if a firm wants to produce 2000 units of output. Price of labor is 20, price of capital is 80.

Answer 1

Solution:

Cost minimization:

MRTS = $\frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

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

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

$\frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

w = 20

r = 80

$\frac{50K }{50L} = \frac{20 }{80}$

$\frac{K }{L} = 0.25$

K = 0.25L

Q = 50KL

2000 = 50(0.25L) (L) = 12.5L²

2000 = 12.5L²

L² = 160

L = 12.65

K = 0.25L = 0.25(12.65) = 3.16

The minimum level of output of production (K,L) = (3.16, 12.65)

Labor = 12.65

Capital = 3.16