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From Mas-Colell, Whinston and Green (Microeconomic Theory), we have this proposition on page 27. If homogeneity holds,

$$ \Sigma_{k=1}^{L} \frac{\partial x_l (p,w)}{\partial p_k} p_k + \frac{\partial x_l (p,w)}{\partial w} w = 0 $$

That is, the price and wealth derivatives, when weighted by the price, $ p_k$ and wealth $w$, equate to 0. What is the economic logic behind this? Homogeneity implies this -- but for this to hold, we don't require that prices and wealth increase proportionally (am I right?)

From this, we can also derive that the sum of price elasticities (own and cross price elasticities) equate to 0. I understand we can derive this mathematically-- but again, why'd these elasticities equate to 0, economically?

(Also, if these results are held only when prices and wealth increase proportionally, I completely understand -- but the book doesn't make that distinction).

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As mentioned by Mas-Colell, Whinston and Green, the equality is true "for all $p$ and $w$". It is a consequence of the budget constraint, which is satisfied for any prices and income values: $$ \Sigma_{k=1}^{L} p_k x_k (p,w) = w. $$ As you mention, the equality $$ \Sigma_{k=1}^{L} \frac{\partial x_l (p,w)}{\partial p_k} p_k + \frac{\partial x_l (p,w)}{\partial w} w = 0 $$ arises when applying the Euler-theorem to the budget constrained demand functions. As demand functions are homogeneous of degree zero in $(p,w)$, they are left unchanged when prices and income are up- or downscaled by any number $\kappa>0$: $$ x_l (\kappa p,\kappa w) = x_l ( p, w) , $$ and this equality holds for any value of $p$ and $w$. This implies the former equality and also explains why it is not only true for the upscaled new prices and income $(p,w)=(\kappa p_0,\kappa w_0)$ but actually for any prices and income because both $\kappa$ and $(p_0,w_0)$ can be arbitrary.

What about the economic interpretation? Hmm... I would say that any simultaneous (marginal and arbitrary) change in $p$ and $w$ triggers changes in the values of demands such that the (own- and cross-) price reactions and the income response compensate. It is equivalent saying that all price elasticities and income elasticity add to zero for any demand function $x_l$.

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The homogeneity mentioned in the text actually refers to homogeneity of degree zero as the book is talking about comparative statics. If it's not homogeneity of degree zero then the equation actually doesn't hold.

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