<p>Consider a change in the distribution of wealth <span class="math-container">$dw_i$</span> that leaves aggregate wealth <span class="math-container">$\sum_i w_i = W$</span> unchanged:
<span class="math-container">$$
\sum_i dw_i = 0,
$$</span>
Let <span class="math-container">$X(p, w_1, \ldots, w_N)$</span> be aggregate demand. Now assume this can be written in terms of prices and aggregate wealth <span class="math-container">$W$</span>. Then we get:
<span class="math-container">$
X(p,\underbrace{W}_{\sum_i w_i})$</span>.</p>
<p>As such,
<span class="math-container">$$
\sum_i x_i(p, w_i) = X(p,W).
$$</span>
Taking the differential gives:
<span class="math-container">$$
\sum_i \frac{\partial x_i(p,w_i)}{\partial w_i} dw_i = \frac{\partial X(p,W)}{\partial W} \sum_i \underbrace{\frac{\partial W}{\partial w_i}}_{=1} dw_i. = 0
$$</span></p>

CC BY-SA 4.0
2024-06-10T14:49:20.867Z
This is false. Consider the degenerate distribution dwᵢ = δ(wᵢ) (that is, all wealth held by one person). The aggregate demand goes to something that looks like C₀ + log(wᵢ).
Joshua
CC BY-SA 4.0
2024-06-11T17:11:12.570Z
In this case, $W = w_i$, so $dW = dw_i = 0$.
tdm
CC BY-SA 4.0
2024-06-10T05:24:03.687Z
2024-06-10T05:24:03.687Z
tdm
<p><span class="math-container">$x(p,\sum_{i=1}^{I}w_{i}) = \sum_{i=1}^{I}x_{i}(p,w_{i})$</span> hold in all generality if for any two distribution of same wealth <span class="math-container">$(w_{1},...,w_{I})$</span> and <span class="math-container">$(w_{1}',...,w_{I}')$</span>,</p>
<p><span class="math-container">$\sum_{i=1}^{I}x_{i}(p,w_{i}) = \sum_{i=1}^{I}x_{i}(p,w_{i}')$</span>, s.t. <span class="math-container">$\sum_{i=1}^{I}dw_{i} = 0$</span> is true</p>
<p>Now,
Consider two distribution of same wealth <span class="math-container">$(w_{1},...,w_{I})$</span> and <span class="math-container">$(w_{1}+dw_{1},...,w_{I}+dw_{I})$</span> satisfying <span class="math-container">$\sum_{i=1}^{I}dw_{i} = 0$</span> then we must have,</p>
<p><span class="math-container">$\sum_{i=1}^{I}x_{i}(p,w_{i}) = \sum_{i=1}^{I}x_{i}(p,w_{i}+dw_{i})$</span></p>
<p><span class="math-container">$\sum_{i=1}^{I}x_{i}(p,w_{i}) = \displaystyle\sum_{i=1}^{I} \left( x_{i}(p,w_{i}) + \dfrac{\partial x_{i}(p,w_{i})}{\partial w_{i}}.dw_{i} \right) $</span></p>
<p>which gives us:</p>
<p><span class="math-container">$\displaystyle\sum_{i=1}^{I} \left( \dfrac{\partial x_{i}(p,w_{i})}{\partial w_{i}}.dw_{i} \right) = 0$</span></p>

CC BY-SA 4.0
2024-11-11T11:44:59.037Z
There is not reference to the aggregate demand $x(p,\sum_{i=1}^I w_i)$ in your calculus.
Bertrand
CC BY-SA 4.0
2024-11-11T11:48:27.620Z
for that to hold the aggregate demand must be same for any two distribution of the same total wealth. So i have started from there.
hr08
CC BY-SA 4.0
2024-11-11T11:19:51.313Z
2024-11-11T11:55:17.170Z
2024-11-11T11:55:17.170Z
hr08
<p>I'm self studying MWG chapter 4, and it says the aggregate demand <span class="math-container">$x(p,w_1,...,w_I)=\Sigma_{i=1}^Ix_i(p,w_i)$</span> can be written as function of <span class="math-container">$W=\Sigma_i^I w_i$</span>, that is <span class="math-container">$\Sigma_{i=1}^Ix_i(p,w_i)=x(p,W)$</span>, we must have <span class="math-container">$$\Sigma_i\frac{\partial x_{li}(p,w_i)}{\partial w_i} dw_i=0$$</span> under <span class="math-container">$\Sigma dw_i=0$</span>.</p>
<p>I want to know why this is the case mathematically, where does it come from. Can anyone prove it?</p>

<microeconomics><aggregate-demand>
where does the condition of aggregate demand can be written as function of aggregate wealth come from
Nonenicht
# where does the condition of aggregate demand can be written as function of aggregate wealth come from

(Asked by: username_0 on 2024-06-10)

I'm self studying MWG chapter 4, and it says the aggregate demand $x(p,w\_1,...,w\_I)=\Sigma\_{i=1}^Ix\_i(p,w\_i)$ can be written as function of $W=\Sigma\_i^I w\_i$, that is $\Sigma\_{i=1}^Ix\_i(p,w\_i)=x(p,W)$, we must have $$\Sigma\_i\frac{\partial x\_{li}(p,w\_i)}{\partial w\_i} dw\_i=0$$ under $\Sigma dw\_i=0$.

I want to know why this is the case mathematically, where does it come from. Can anyone prove it?

**Tags:** microeconomics, aggregate-demand


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## 2 answers

### Answer by username_1 on 2024-06-10. Score: 3

Consider a change in the distribution of wealth $dw\_i$ that leaves aggregate wealth $\sum\_i w\_i = W$ unchanged:
$$
\sum\_i dw\_i = 0,
$$
Let $X(p, w\_1, \ldots, w\_N)$ be aggregate demand. Now assume this can be written in terms of prices and aggregate wealth $W$. Then we get:
$
X(p,\underbrace{W}\_{\sum\_i w\_i})$.

As such,
$$
\sum\_i x\_i(p, w\_i) = X(p,W).
$$
Taking the differential gives:
$$
\sum\_i \frac{\partial x\_i(p,w\_i)}{\partial w\_i} dw\_i = \frac{\partial X(p,W)}{\partial W} \sum\_i \underbrace{\frac{\partial W}{\partial w\_i}}\_{=1} dw\_i. = 0
$$

> **Comments:**
> **username_2**: This is false. Consider the degenerate distribution dwᵢ = δ(wᵢ) (that is, all wealth held by one person). The aggregate demand goes to something that looks like C₀ + log(wᵢ).  
> **username_1**: In this case, $W = w_i$, so $dW = dw_i = 0$.  

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### Answer by username_3 on 2024-11-11. Score: 1

$x(p,\sum\_{i=1}^{I}w\_{i}) = \sum\_{i=1}^{I}x\_{i}(p,w\_{i})$ hold in all generality if for any two distribution of same wealth $(w\_{1},...,w\_{I})$ and $(w\_{1}',...,w\_{I}')$,

$\sum\_{i=1}^{I}x\_{i}(p,w\_{i}) = \sum\_{i=1}^{I}x\_{i}(p,w\_{i}')$, s.t. $\sum\_{i=1}^{I}dw\_{i} = 0$ is true

Now,
Consider two distribution of same wealth $(w\_{1},...,w\_{I})$ and $(w\_{1}+dw\_{1},...,w\_{I}+dw\_{I})$ satisfying $\sum\_{i=1}^{I}dw\_{i} = 0$ then we must have,

$\sum\_{i=1}^{I}x\_{i}(p,w\_{i}) = \sum\_{i=1}^{I}x\_{i}(p,w\_{i}+dw\_{i})$

$\sum\_{i=1}^{I}x\_{i}(p,w\_{i}) = \displaystyle\sum\_{i=1}^{I} \left( x\_{i}(p,w\_{i}) + \dfrac{\partial x\_{i}(p,w\_{i})}{\partial w\_{i}}.dw\_{i} \right) $

which gives us:

$\displaystyle\sum\_{i=1}^{I} \left( \dfrac{\partial x\_{i}(p,w\_{i})}{\partial w\_{i}}.dw\_{i} \right) = 0$

> **Comments:**
> **username_4**: There is not reference to the aggregate demand $x(p,\sum_{i=1}^I w_i)$ in your calculus.  
> **username_3**: for that to hold the aggregate demand must be same for any two distribution of the same total wealth. So i have started from there.  

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