Answer to Question #270237 in Microeconomics for Paul

Question #270237

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.


1
Expert's answer
2021-11-23T10:57:51-0500

"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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