Question

Production function of a firm is given as X=0.5 L ^0.5 K^0.5 price of labor and capital are 5$ and 10$ respectively the firm has a constant cost out lay of $600 find the combination of labor and capital that maximize the farm output and the maximum outputs?

Answer 1

$MPL=\frac{K^{0.5}}{4L^{0.5}}$

$MPK=\frac{L^{0.5}}{4K^{0.5}}$

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

$MRTS = \frac{ \frac{K^{0.5}}{4L^{0.5}}}{\frac{L^{0.5}}{4K^{0.5}}}$

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

$\frac{K}{L}= \frac{10}{5}$

10L=5K

2L=K

$600=0.5L^{0.5} K^{0.5}$

$600=0.5L^{0.5} (2L)^{0.5}$

$1200=L^{0.5}L^{0.5}2^{0.5}$

$1200=(2)^{0.5}L$

L=848.53

K=1697.06