Answer to Question #238821 in Macroeconomics for SPG

Question

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?

Answer 1

Given

$Q = 20 K^{0.5}L^{0.5}$

$MPL = \frac{dq}{dl}\\ = 20\times 0.5×L^{-0.5} ×K^{0.5}\\ =10×L^{-0.5} ×K^{0.5}$

$MPK = \frac{dq}{dl}\\ = 20\times 0.5×L^{0.5} ×K^{-0.5}\\ =10×L^{0.5} ×K^{-0.5}$

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

$\frac{[10×L^{-0.5} ×K^{0.5}]}{ 4} =\frac{ [ 10×L^{0.5}×K^{-0.5} ]}{5}\\ \frac{K}{4} =\frac{L}{5}\\ K=\frac{4L}{5}$