Example

Consider the production function $P(L,K)=5L^{\frac{1}{5}}K^{\frac{1}{3}}+L$. We determine the partial elasticity with respect to labor at $(L,K)=(1024,27)$.

$$\begin{align} \epsilon_L & = P'_L(L,K)\cdot \frac{L}{P(L,K)}\\ & = (L^{-\frac{4}{5}}K^{\frac{1}{3}}+1)\frac{L}{5L^{\frac{1}{5}}K^{\frac{1}{3}}+L}\\ &=\frac{L^{\frac{1}{5}}K^{\frac{1}{3}}+L}{5L^{\frac{1}{5}}K^{\frac{1}{3}}+L}\\ \end{align}$$

Hence, at $(L,K)=(1024,27)$ we have $\epsilon_L=\dfrac{1024^{\frac{1}{5}}27^{\frac{1}{3}}+1024}{5\cdot1024^{\frac{1}{5}}27^{\frac{1}{3}}+1024}=\dfrac{1036}{1084}=\dfrac{259}{271}$.

This implies that if capital remains constant at $K=27$ and at $L=1024$ labor increases with $1 \%$, then production will increase by approximately $\frac{259}{271} \%$.