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

<p>What about the economic interpretation? Hmm... I would say that any simultaneous (marginal and arbitrary) change in <span class="math-container">$p$</span> and <span class="math-container">$w$</span> 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 <span class="math-container">$x_l$</span>.</p>

CC BY-SA 4.0
2019-09-15T16:02:45.290Z
2019-09-16T06:39:57.267Z
2019-09-16T06:39:57.267Z
Bertrand
<p>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.</p>

CC BY-SA 4.0
2019-09-16T02:58:02.920Z
2019-09-16T02:58:02.920Z
Arvind Chandrashekar
<p>From Mas-Colell, Whinston and Green (Microeconomic Theory), we have this proposition on page 27. If homogeneity holds,</p>

<p><span class="math-container">$$ \Sigma_{k=1}^{L} \frac{\partial x_l (p,w)}{\partial p_k} p_k + \frac{\partial x_l (p,w)}{\partial w} w = 0 $$</span></p>

<p>That is, the price and wealth derivatives, when weighted by the price, <span class="math-container">$ p_k$</span> and wealth <span class="math-container">$w$</span>, 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?)</p>

<p>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?</p>

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

<microeconomics>
Why does the price and wealth derivatives, weighted by the price and wealth, sum to 0?
WorldGov
# Why does the price and wealth derivatives, weighted by the price and wealth, sum to 0?

(Asked by: username_0 on 2019-09-14)

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).

**Tags:** microeconomics


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## 2 answers

### Answer by username_1 on 2019-09-15. Score: 1 | Accepted answer

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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### Answer by username_2 on 2019-09-16. Score: 0

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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