Question

Find the elasticity of substitution for the following production function:Q=√LK where Q represent quantity of output produced L and K represent labour and capital used.

Answer 1

$Q=√LK$ this is the same as $Q=L^{\frac{1}{2}}K^{\frac{1}{2}}$

$Elasticity Of Substitution=\frac{input Ratio}{MRS}$

$Inputs Ratio=\frac{K^{\frac{1}{2}}}{L^{\frac{1}{2}}}$

$MRS=\frac{-∆L}{∆K}$

$∆L=\frac{1}{2}(K^{\frac{1}{2}})$

$∆K=\frac{1}{2}(L^{\frac{1}{2}})$

$MRS=\frac{(\frac{1}{2}(K^{\frac{1}{2}})}{(\frac{1}{2}(L^\frac{1}{2})}=\frac{-K^\frac{1}{2}}{L^\frac{1}{2}}$

Therefore:

$E.S=\frac{K^{\frac{1}{2}}}{L^{\frac{1}{2}}}÷\frac{-K^{\frac{1}{2}}}{L^{\frac{1}{2}}}$

$=\frac{K^{\frac{1}{2}}}{L^{\frac{1}{2}}}×\frac{L^{\frac{1}{2}}}{-K^{\frac{1}{2}}}=-1$