Question

Suppose a firm produces according to the production function Q = 2L^0.6K^0.2, and faces wage rate ₵10, a rental cost of capital ₵5, and sells output at a price of ₵20.

a. Obtain and expression for the factor demand functions.

b. Compute the profit-maximizing factor demands for capital and labour.

Answer 1

$Q=2L^{0.6}K^{0.2} \\ w = 10 \\ r = 5 \\ p = 20$

a. Input demand for labor L = f(w,r,p)

Input demand for capital K=f(w,r,p)

Profit

$Π = p(2L^{0.6}K^{0.2}) -wL -rK$

At maximum profit and low cost (minimum) ratio of capital to labor is

$\frac{K}{L} = \frac{(w/p)^5}{(r/p)^{5/3}} = \frac{(1/2)^5}{(1/2)^{10/3}} \\ = (\frac{1}{2})^{5/3} = 0.315$

Implies more capital required for one unit increase in wage rate.

b. Profit $Π = pQ-wL-rK$

$Π = p(2L^{0.6}K^{0.2}) -wL -rK$

For maximim profit

$\frac{dΠ}{dL} = \frac{dΠ}{dL} =0 \\ \frac{d}{dL}(p(2L^{0.6}K^{0.2}))-wL-rK = 0 \\ pK^{0.2} -w= 0 \\ K=(\frac{w}{p})^5 = (\frac{10}{20})^5 = (\frac{1}{2})^5 = 0.03125 \\ \frac{d}{dK}(p(2L^{0.6}K^{0.2}))-wL-rK =0 \\ pL^{0.6} -r=0 \\ L=(\frac{r}{p})^{5/3} = (\frac{5}{20})^{5/3} \\ L = (\frac{1}{4})^{5/3} = 0.099$