Given a cobb – Douglas production functionQ = L0.5 K0.4And the prices of capital and labour are ksh3 and ksh 4 respectively while the firm outlay is Kshs.108, calculate the optimal combination of factors inputs.
"Q = L0.5 K0.4"
Cost minimization combination of labor and capital
"MRTS =\\frac{MP_L}{MP_K} = \\frac{w}{r}"
Marginal product of labor
"\\frac{dQ}{dL} = 0.2K"
Marginal product of capital
"\\frac{dQ}{dK} = 0.2L"
"\\therefore" the marginal rate of technical substitution
"\\frac{0.2K}{0.2L} = \\frac{K}{L}"
Optimal capital - labor ratio
"\\frac{K}{L} = \\frac{3}{4} or L = \\frac{3}{4} K"
Substitution
"Q = L0.5 K0.4"
"108 =K \\frac{3}{4}0.5 K 0.4"
"108 = 0.15K^2"
"K^2 = 720"
"K = 26.83"
"108 = L0.5 (26.83) 0.4"
"108 = 5.37L"
"L = 20.13"
"TC = wL + rK"
"TC = (3)(20.13) + (4)(26.83)"
"TC = Ksh 168"
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