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.

1muflon1 · Answer

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.

Price elasticity of demand of CES

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