<p>Assuming that <span class="math-container">$p_i \neq p_j$</span> you just apply the formula;</p>
<p><span class="math-container">$$\epsilon_i(p_i,I,P) =-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)} \\ = -\left( \frac{-\sigma p_i^{-\sigma-1} I}{P^{1-\sigma}} \right)\frac{p_i}{\frac{p_i^{-\sigma} I}{P^{1-\sigma}}} =\sigma$$</span>.</p>
<p>The way how you gave the problem neither <span class="math-container">$P$</span> or <span class="math-container">$I$</span> contain <span class="math-container">$p_i$</span> so you treat them as constants during differentiation.</p>

CC BY-SA 4.0
2020-11-07T01:45:51.657Z
2020-11-07T01:45:51.657Z
1muflon1
<p>Anyone would like to help me show the following ( or a book/paper reference would be a great help)</p>
<p>&quot; The price  elasticity of demand is equal to <span class="math-container">$\sigma$</span> for the demand function of CES preference, <span class="math-container">$d(p_i,I,P)=\dfrac{p_i^{-\sigma} I}{P^{1-\sigma}}$</span>, where <span class="math-container">$P=\left(\sum_{j=1}^N p_j^{1-\sigma}\right)^{\dfrac{1}{1-\sigma}}$</span> and <span class="math-container">$I=\sum_{j=1}^Np_jx_j$</span>.&quot;</p>
<p>The formula for the price elasticity of demand is given by <span class="math-container">$\epsilon_i(p_i,I,P)=-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)}$</span>.</p>
<p>I have tried many times but couldn't get the desired answer. Thank you very much in advance.</p>

<macroeconomics><mathematical-economics><elasticity><price><elasticity-of-substitution>
Price elasticity of demand of CES
stochastic learner
# Price elasticity of demand of CES

(Asked by: username_0 on 2020-11-07)

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.

**Tags:** macroeconomics, mathematical-economics, elasticity, price, elasticity-of-substitution


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## 1 answers

### Answer by username_1 on 2020-11-07. Score: 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.


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