<p>I'm currently reviewing my prof's slides in preparation for an exam. In one of them, he talks about Euler's Homogenous Function Theorem: </p>

<blockquote>
  <p>Let $f(x_1, x_2, ..., x_n)$ be a function homogenous in degree $\rho$. </p>
  
  <p>$$\rho f(x)=\sum_{i=1}^n x_if_i(x)$$
  Where $f_i(x)$ is the partial derivative with respect to $x_i$</p>
</blockquote>

<p>In the next slide, the following consequence is stated (the slides clearly state that the result is obtained by applying Euler's theorem to Marshallian demand):</p>

<blockquote>
  <p>$$\sum_{j=1}^n e_{i,p_j}+e_{i,I}=0$$</p>
</blockquote>

<p>I assume that this is a case where the function is homogenous in degree 0, as the same slide states that, if a demand function is homogenous in degree 0, then there is no monetary illusion. However, I have no idea how this result was derived using Euler's theorem.</p>

CC BY-SA 3.0
2017-02-21T16:54:53.590Z
Hint: Divide by $f(x)$.
Alecos Papadopoulos
CC BY-SA 3.0
2017-02-21T19:24:18.697Z
@AlecosPapadopoulos Just to clarify: in this case, $f(x)$ would be the utility function that leads to the marshallian demand $x_i$?
Grizzly0111
CC BY-SA 3.0
2017-02-21T20:00:25.173Z
The equation that troubles you includes the elasticities of demand, not of the utility function
Alecos Papadopoulos
<microeconomics><mathematical-economics>
Euler's Homogenous Function Theorem with elasticity
Grizzly0111
# Euler's Homogenous Function Theorem with elasticity

(Asked by: username_0 on 2017-02-21)

I'm currently reviewing my prof's slides in preparation for an exam. In one of them, he talks about Euler's Homogenous Function Theorem:

> Let $f(x\_1, x\_2, ..., x\_n)$ be a function homogenous in degree $\rho$.
>
> $$\rho f(x)=\sum\_{i=1}^n x\_if\_i(x)$$
> Where $f\_i(x)$ is the partial derivative with respect to $x\_i$

In the next slide, the following consequence is stated (the slides clearly state that the result is obtained by applying Euler's theorem to Marshallian demand):

> $$\sum\_{j=1}^n e\_{i,p\_j}+e\_{i,I}=0$$

I assume that this is a case where the function is homogenous in degree 0, as the same slide states that, if a demand function is homogenous in degree 0, then there is no monetary illusion. However, I have no idea how this result was derived using Euler's theorem.

**Tags:** microeconomics, mathematical-economics

> **Comments:**
> **username_1**: Hint: Divide by $f(x)$.  
> **username_0**: @username_1 Just to clarify: in this case, $f(x)$ would be the utility function that leads to the marshallian demand $x_i$?  
> **username_1**: The equation that troubles you includes the elasticities of demand, not of the utility function  

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## 0 answers

