Price elasticity of demand of CES

Question

Anyone would like to help me show the following ( or a book/paper reference would be a great help)

" The price elasticity of demand is equal to $\sigma$ for the demand function of CES preference, $d(p_i,I,P)=\dfrac{p_i^{-\sigma} I}{P^{1-\sigma}}$, where $P=\left(\sum_{j=1}^N p_j^{1-\sigma}\right)^{\dfrac{1}{1-\sigma}}$ and $I=\sum_{j=1}^Np_jx_j$."

The formula for the price elasticity of demand is given by $\epsilon_i(p_i,I,P)=-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)}$.

I have tried many times but couldn't get the desired answer. Thank you very much in advance.

Answer 1

Assuming that $p_i \neq p_j$ you just apply the formula;

$$\epsilon_i(p_i,I,P) =-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)} \\ = -\left( \frac{-\sigma p_i^{-\sigma-1} I}{P^{1-\sigma}} \right)\frac{p_i}{\frac{p_i^{-\sigma} I}{P^{1-\sigma}}} =\sigma$$.

The way how you gave the problem neither $P$ or $I$ contain $p_i$ so you treat them as constants during differentiation.