Question

Suppose the production function of a firm is given as X  0.5L1/ 2K1/ 2 prices of labor and
capital are given as $ 5 and $ 10 respectively, and the firm has a constant cost out lay of $
600.Find the combination of labor and capital that maximizes the firm’s out put and the
maximum out put.

Answer 1

Solution:

Production function: X = 0.5L^0.52K^0.5

Derive MPL and MPK:

MPL = $\frac{\partial X} {\partial L} = 0.25L^{0.5} 2K^{0.5}$

MPK = $\frac{\partial X} {\partial K} = 0.5L^{0.5} K^{0.5}$

Then derive MRTS:

MRTS = $\frac{MPL}{MPK} = \frac{w}{r}$

w = 5

r = 10

= $\frac{0.25L^{0.5} 2K^{0.5}}{0.5L^{0.5} K^{0.5}}$

$\frac{K}{L} = \frac{1}{2}$

K = 2L

 

Find the combination of labor and capital that maximizes the firm’s output:

C = wL + rK

We know: K = 2L

600 = 5(L) + 10(2L)

600 = 5L + 20L

600 = 25L

L = $\frac{600}{25} = 24$

L = 24

K = 2L = 2(24) = 48

K = 48

The combination of labor and capital that maximizes the firm’s output is: (w,r) = (24, 48)

Derive the maximum output:

X = 0.5L^0.52K^0.5

Where X is the maximum output

Substitute with labor and capital values:

X = 0.5(24)^0.52(48)^0.5

X = (2.45) (13.86)

X = 33.96

The maximum output = 33.96