Suppose a firm produces according to the production function Q = 2L0.6K0.2, 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.
"Q=2L^{0.6}K^{0.2} \\\\\n\nw = 10 \\\\\n\nr = 5 \\\\\n\np = 20"
a. Input demand for labor L = f(w,r,p)
Input demand for capital K=f(w,r,p)
Profit
"\u03a0 = 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}} \\\\\n\n= (\\frac{1}{2})^{5\/3} = 0.315"
Implies more capital required for one unit increase in wage rate.
b. Profit "\u03a0 = pQ-wL-rK"
"\u03a0 = p(2L^{0.6}K^{0.2}) -wL -rK"
For maximim profit
"\\frac{d\u03a0}{dL} = \\frac{d\u03a0}{dL} =0 \\\\\n\n\\frac{d}{dL}(p(2L^{0.6}K^{0.2}))-wL-rK = 0 \\\\\n\npK^{0.2} -w= 0 \\\\\n\nK=(\\frac{w}{p})^5 = (\\frac{10}{20})^5 = (\\frac{1}{2})^5 = 0.03125 \\\\\n\n\\frac{d}{dK}(p(2L^{0.6}K^{0.2}))-wL-rK =0 \\\\\n\npL^{0.6} -r=0 \\\\\n\nL=(\\frac{r}{p})^{5\/3} = (\\frac{5}{20})^{5\/3} \\\\\n\nL = (\\frac{1}{4})^{5\/3} = 0.099"
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