Answer to Question #308672 in Microeconomics for Janbare

Question

Suppose the production function of a firm is given as X=0.5L ½ K½ price of labor and capital are given as $5 and $10 respectively, and the firm has cos out lay of$ 600

I. The profit maximizing level of L & K to employ.

II. The MRTSL,K at optimum

Answer 1

production function, X= $0.5l^{0.5}2k^{0.5}$

MPL=$\frac{dx}{dl}$ =$0.25l^{0.5}2k^{0.5}$

MPK=$\frac{dx}{dk}$ =$.5l^{0.5}k^{0.5}$

MRTS=MPL/MPK=w/r

w=5

r=10

MRTS = ( $0.25l^{0.5}2k^{0.5}$ / ($0.5l^{0.5}k^{0.5}$ )

$\frac{k}{l}$ = $\frac{1}{2}$

K=2L

Calculating the combination of labor that maximizes the firm's output

C = wL + rK, and K=2L

600=5L + $10\times2l$

600=25L

L=24

K=2L = 48

Combination of labor and capital that maximizes the firm's output is;

w,r)=(24,48)

Maximum output, X= $0.5l^{0.5}2k^{0.5}$

X= $0.5(24)^{0.5}2(48)^{0.5}$

^=33.96

^{Maximum output= 33.96}