# Why does the price and wealth derivatives, weighted by the price and wealth, sum to 0?

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).

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 $$x_l$$.