<p>The issue here is that market clearing can be defined in <em>gross</em> as well as in <em>net</em> terms.</p>
<p><strong>Gross vs. net</strong></p>
<p>Suppose I need 10 eggs for the next two weeks, and I don't intend to have any left over (I am going on vacation afterwards). I already have 3 at home, so I buy 7 at the local farmers market.</p>
<p>My <em>gross</em> demand was 10 eggs, this is how many I wanted given the current parameters. <br>
My <em>net</em> demand was 7 eggs, this is how much I purchased from the market. <br>
My net demand is my gross demand minus my endowment, thus in this case <span class="math-container">$10-3 = 7$</span>.</p>
<p><strong>Market clearing</strong></p>
<p>Market clearing can be phrased in both gross and net terms. Let <span class="math-container">$x_i$</span> denote the gross demand of consumer <span class="math-container">$i$</span> for some good <span class="math-container">$x$</span>, and let <span class="math-container">$\omega_{x,i}$</span> denote the endowment or gross supply of consumer <span class="math-container">$i$</span> of this same good <span class="math-container">$x$</span>. Then the market for <span class="math-container">$x$</span> clears if
<span class="math-container">$$
\sum_i x_i = \sum_i \omega_{x,i}. \tag{1}
$$</span>
This was phrased in gross terms: total gross demand equals total gross supply: every unit available was consumed by someone. (In the terms of the previous example you could say that I consumed 10 eggs; I bought 3 from myself and 7 from other agents.) <br>
But we could also phrase market clearing in net terms. The net demand of consumer <span class="math-container">$i$</span> for good <span class="math-container">$x$</span> is
<span class="math-container">$x_i - \omega_{x,i}$</span>. If this is a negative number, then consumer <span class="math-container">$i$</span> actually wants to sell, not buy this good, so they will have a net supply of <span class="math-container">$\omega_{x,i} - x_i$</span>. <br>
Let us denote the set of consumers with non-negative net demand by <span class="math-container">$I_D$</span>, and the set of consumers with negative net demand/positive net supply by <span class="math-container">$I_S$</span>, thus
<span class="math-container">$$
I_D = \left\{i | x_i - \omega_{x,i} \geq 0 \right\} \\
I_S = \left\{i | \omega_{x,i} - x_i &gt; 0 \right\}
$$</span>
It follows from <span class="math-container">$(1)$</span> that
<span class="math-container">$$
\sum_i (x_i - \omega_{x,i}) = 0.
$$</span>
Splitting up the summation
<span class="math-container">$$
\sum_{i\in I_D} (x_i - \omega_{x,i}) + \sum_{i\in I_S} (x_i - \omega_{x,i}) = 0
$$</span>
and rearranging the equation we have
<span class="math-container">$$
\sum_{i\in I_D} (x_i - \omega_{x,i}) = \sum_{i\in I_S} (\omega_{x,i} - x_i),
$$</span>
thus total net demand equals total net supply. The steps are reversible, you can also start from net market clearing and prove that the market is also cleared in gross terms, so these two conditions are equivalent.</p>

CC BY-SA 4.0
2022-02-03T23:23:02.503Z
Ah this makes so much sense. Thank you for your help!
Nick
CC BY-SA 4.0
2022-02-02T08:37:16.853Z
2022-02-03T17:32:57.387Z
2022-02-03T17:32:57.387Z
Giskard
<p>My question relates to an endowment economy. We assume perfect competition and markets clear, i.e. supply = demand. The way my professor defined it, he said endowment (per good) = supply (per good) = demand (per good). I am confused by this, because from my understanding, markets clearing implies that the quantity demanded <em>at some price</em> will equal the amount a seller is willing to sell <em>at this same price</em>. From the way my professor described this, he is basically saying that the entire endowment is going to be sold.</p>
<p>Can someone explain the intuition?</p>

CC BY-SA 4.0
2022-02-03T17:36:13.427Z
I removed your other question from the body of this one and [posted it separately](https://economics.stackexchange.com/questions/50305/homothetic-preferences-for-international-trade).
Giskard
<microeconomics><pure-exchange-economy>
Net and gross market clearing in endowment economy
Nick
# Net and gross market clearing in endowment economy

(Asked by: username_0 on 2022-02-02)

My question relates to an endowment economy. We assume perfect competition and markets clear, i.e. supply = demand. The way my professor defined it, he said endowment (per good) = supply (per good) = demand (per good). I am confused by this, because from my understanding, markets clearing implies that the quantity demanded *at some price* will equal the amount a seller is willing to sell *at this same price*. From the way my professor described this, he is basically saying that the entire endowment is going to be sold.

Can someone explain the intuition?

**Tags:** microeconomics, pure-exchange-economy
> **Comments:**
> **username_1**: I removed your other question from the body of this one and [posted it separately](https://economics.stackexchange.com/questions/50305/homothetic-preferences-for-international-trade).  

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## 1 answers

### Answer by username_1 on 2022-02-02. Score: 2

The issue here is that market clearing can be defined in *gross* as well as in *net* terms.

**Gross vs. net**

Suppose I need 10 eggs for the next two weeks, and I don't intend to have any left over (I am going on vacation afterwards). I already have 3 at home, so I buy 7 at the local farmers market.

My *gross* demand was 10 eggs, this is how many I wanted given the current parameters.   
My *net* demand was 7 eggs, this is how much I purchased from the market.   
My net demand is my gross demand minus my endowment, thus in this case $10-3 = 7$.

**Market clearing**

Market clearing can be phrased in both gross and net terms. Let $x\_i$ denote the gross demand of consumer $i$ for some good $x$, and let $\omega\_{x,i}$ denote the endowment or gross supply of consumer $i$ of this same good $x$. Then the market for $x$ clears if
$$
\sum\_i x\_i = \sum\_i \omega\_{x,i}. \tag{1}
$$
This was phrased in gross terms: total gross demand equals total gross supply: every unit available was consumed by someone. (In the terms of the previous example you could say that I consumed 10 eggs; I bought 3 from myself and 7 from other agents.)   
But we could also phrase market clearing in net terms. The net demand of consumer $i$ for good $x$ is
$x\_i - \omega\_{x,i}$. If this is a negative number, then consumer $i$ actually wants to sell, not buy this good, so they will have a net supply of $\omega\_{x,i} - x\_i$.   
Let us denote the set of consumers with non-negative net demand by $I\_D$, and the set of consumers with negative net demand/positive net supply by $I\_S$, thus
$$
I\_D = \left\{i | x\_i - \omega\_{x,i} \geq 0 \right\} \\
I\_S = \left\{i | \omega\_{x,i} - x\_i > 0 \right\}
$$
It follows from $(1)$ that
$$
\sum\_i (x\_i - \omega\_{x,i}) = 0.
$$
Splitting up the summation
$$
\sum\_{i\in I\_D} (x\_i - \omega\_{x,i}) + \sum\_{i\in I\_S} (x\_i - \omega\_{x,i}) = 0
$$
and rearranging the equation we have
$$
\sum\_{i\in I\_D} (x\_i - \omega\_{x,i}) = \sum\_{i\in I\_S} (\omega\_{x,i} - x\_i),
$$
thus total net demand equals total net supply. The steps are reversible, you can also start from net market clearing and prove that the market is also cleared in gross terms, so these two conditions are equivalent.

> **Comments:**
> **username_0**: Ah this makes so much sense. Thank you for your help!  

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