Answer to Question #311308 in Microeconomics for Abas

Question

Suppose the production function of a firm is given ×=0.5lthe power of 1/2*k the power of 1/2 price of labor and capital are given $5 and$10 respectively,and the firm has constant cost out lay of$600

1; find profit maximize level of L and K to employee

2; find MRTS of L to K at optimum

Answer 1

A)

$x=0.5l^{\frac{1}{2}}k^{\frac{1}{2}}$

Given that P_l=$5 and P_k=$10 and C=$600

$X=0.5l^{\frac{1}{2}}k^{\frac{1}{2}}+\lambda(60-5l-10k)$

$X_l=0.25l^{-\frac{1}{2}}k^{\frac{1}{2}}-5\lambda=0$ .......1)

$X_k=0.25l^{\frac{1}{2}}k^{-\frac{1}{2}}-10\lambda=0$ .....2)

$X_{\lambda}=60-5l-10k=0$ .......3)

$\frac{0.25l^{-\frac{1}{2}}k^{\frac{1}{2}}}{0.25l^{\frac{1}{2}}k^{-\frac{1}{2}}}=\frac{5}{10}$

$\frac{k}{l}=\frac{5}{10}$

$l=2k$

Substituting l in (3)

$600-5(2k)-10k=0$

$600-20k=0$

$\bar{k}=30$

$\bar{l}=2(30)=60$

B)$MRTS_{lk}=\frac{-k}{l}$

At the optimum,

$MRTS_{lk}=\frac{-30}{60}=\frac{-1}{2}$