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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.

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    $\begingroup$ Hint: Divide by $f(x)$. $\endgroup$ – Alecos Papadopoulos Feb 21 '17 at 16:54
  • $\begingroup$ @AlecosPapadopoulos Just to clarify: in this case, $f(x)$ would be the utility function that leads to the marshallian demand $x_i$? $\endgroup$ – Grizzly0111 Feb 21 '17 at 19:24
  • $\begingroup$ The equation that troubles you includes the elasticities of demand, not of the utility function $\endgroup$ – Alecos Papadopoulos Feb 21 '17 at 20:00

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