Answer to Question #238821 in Macroeconomics for SPG

Question #238821
When production function of a firm is Q = 20K^0.5L^0.5 , price of capital is 5 per unit and price of labour is 4 per unit . What will be the expansion path ratio for the firm?
1
Expert's answer
2021-09-19T18:34:31-0400

Given

Q=20K0.5L0.5Q = 20 K^{0.5}L^{0.5}

MPL=dqdl=20×0.5×L0.5×K0.5=10×L0.5×K0.5MPL = \frac{dq}{dl}\\ = 20\times 0.5×L^{-0.5} ×K^{0.5}\\ =10×L^{-0.5} ×K^{0.5}


MPK=dqdl=20×0.5×L0.5×K0.5=10×L0.5×K0.5MPK = \frac{dq}{dl}\\ = 20\times 0.5×L^{0.5} ×K^{-0.5}\\ =10×L^{0.5} ×K^{-0.5}


Now expansion path = MPL/Price of labor= MPK/Price of capital

[10×L0.5×K0.5]4=[10×L0.5×K0.5]5K4=L5K=4L5\frac{[10×L^{-0.5} ×K^{0.5}]}{ 4} =\frac{ [ 10×L^{0.5}×K^{-0.5} ]}{5}\\ \frac{K}{4} =\frac{L}{5}\\ K=\frac{4L}{5}



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