<ol>
<li><p>No the identity would be preserved, since <span class="math-container">$I_2=S_2$</span> when inventory is reduced by <span class="math-container">$\Delta I_2$</span> =-\$2 at the same time the second person dissaves <span class="math-container">$\\\$2$</span>, so <span class="math-container">$\Delta S_2$</span> = -\$2 as well. If then that person saves 2 dollars that they got from their customers then that will become some sort of investment (depending on how it is saved) so then we have <span class="math-container">$ \Delta S_2 = -2 + 2 =0$</span> and <span class="math-container">$ \Delta I_2 = -2 + 2 =0$</span>.</p>
<p>This is as it should be, since the sale was made from previously made inventory investment the output <span class="math-container">$Y_2$</span> must be zero this time period.</p>
</li>
<li><p>That does not need to necessarily happen based on your text. You are making extra assumption that the bread was bought from previous period savings and that baker saves what they earn. But if you make those extra assumptions then it would be correct.</p>
</li>
<li><p>In the last example if it is an intermediate good then it does not count directly. You would have to first get to final sale and then you can try to allocate it to <span class="math-container">$I=S$</span>, <span class="math-container">$C$</span> or <span class="math-container">$G$</span> depending on what happens to when we get to the final sale.</p>
</li>
</ol>

CC BY-SA 4.0
2021-09-04T23:16:53.283Z
thank you @1muflon1 much appreciated, I think you misunderstood my notation though, the subscript indicate the person who saves/dissaves and so on...And then I have a question, in 1. we have also that person 1 dissaves 2 dollars to consume them but then given that, as you wrote, $\Delta S_2 = \Delta I_2$  the identity wouldn't be preserved, what am I missing?
Tortar
CC BY-SA 4.0
2021-09-04T23:32:08.503Z
how I would explain it is this series of variations : $\Delta S_2 = 0$ because person 2 is dissaving inventory but he is also taking the money from the customer, $\Delta I_2 = -2 $ for reduction of inventory, $\Delta S_1 = -2$ and $ \Delta C_1 = 2$, the identity would be preserved. Besides at the end of the transaction $\Delta Y = \Delta S + \Delta C = \Delta I + \Delta C = 0$ , everything seems fine in the end, Is it like so?
Tortar
CC BY-SA 4.0
2021-09-04T23:42:14.413Z
@Tortar I understand the subscript indicates person. Also it would be preserved, the identity says I=S, if person dissaves 2 USD and then saves 2 USD saving should be zero and the same should hold for investment.
1muflon1
CC BY-SA 4.0
2021-09-04T23:44:15.570Z
@Tortar Taking money from customer does not automatically means S goes up, if the baker would spend money on some consummation good S could be zero or only half of that, by definition (assuming no taxes) S=Y-C, if the baker sells 2 usd bread you only know that Y increases by 2 in order to say that S increases by 2 you have to assume that C=0. In any case the identity is still preserved
1muflon1
CC BY-SA 4.0
2021-09-04T23:45:03.157Z
yes I'm just considering the state of the world after the transaction, so $\Delta C_2 = 0$ I'm not implying that in the future he won't consume
Tortar
CC BY-SA 4.0
2021-09-04T23:46:14.130Z
@Tortar that fine, I am just saying that’s something that was missing so I am notifying you about that in case you did not know
1muflon1
CC BY-SA 4.0
2021-09-04T23:47:13.507Z
just to be clear , the way I reported the variations in my second comment is appropriate to represent the situation ? Thank you for the help anyway
Tortar
CC BY-SA 4.0
2021-09-04T23:50:31.163Z
@Tortar yes the Y should be zero because the way how you set up the problem there is no production so output is 0, you only have people transacting some output produced in the past
1muflon1
CC BY-SA 4.0
2021-09-04T20:41:14.183Z
2021-09-04T20:41:14.183Z
1muflon1
<p>Say we are in a closed economy where there is no government.</p>
<p>It is claimed that</p>
<p><span class="math-container">$$I = S$$</span></p>
<p>is an accounting identity and so it's always valid.</p>
<p>I find it difficult to understand why this identity is always true even though I know how to derive it analytically.</p>
<p>Then, I created some examples to understand it better. All my examples are based on the fact that in GDP only final transactions are counted in consumption.</p>
<hr />
<p>Edit:</p>
<p>I'm sorry for the confusion, the subscript under every variable refers to the person who is performing the action and what I'm considering is the state of the system after the transaction is made.</p>
<hr />
<ul>
<li>Say person 1 goes to the bakery to buy <span class="math-container">$\\\$2$</span> of bread from person 2: there can be 2 cases</li>
</ul>
<ol>
<li>the bread person 1 buys is in inventory so what should happen is this series of variations :</li>
</ol>
<p><span class="math-container">$$\Delta S_1 = -$2 \:,\: \Delta I_2 =  -$2 \:,\: \Delta C_1 = +$2$$</span></p>
<p>and also <span class="math-container">$\Delta S_{\color{red}2} = +{$}2$</span> , but that would mean that the identity is not preserved, so why instead <span class="math-container">$\Delta S_{\color{red}2} = {$}0 $</span> ?</p>
<ol start="2">
<li>the bread person 1 buys is &quot;just baked&quot; so what should happen is this series of variations :</li>
</ol>
<p><span class="math-container">$$\Delta S_1 = -$2 \:,\: \Delta S_2 =  +$2 \:,\: \Delta C_1 = +$2$$</span></p>
<p>Am I right about this case?</p>
<p>The last example I'd like to show you is this one :</p>
<ul>
<li>Tha owner of the bakery (person 1) buys some flour  to produce bread for <span class="math-container">$\\\$100$</span> from person 2, the problem is that here we have an intermediate good so, what I thought should happen, is :</li>
</ul>
<p><span class="math-container">$$\Delta S_1 = -$100 \:,\: \Delta S_2 =  +$100 \:,\: \Delta I_1 =  +$100 \:,\: \Delta I_2 =  -$100  $$</span></p>
<p>what about this one?</p>
<p>Thanks to all of you</p>

<macroeconomics><gdp><national-accounts><accounting>
How to show the identity $I = S$ is valid in these applied cases?
Tortar
# How to show the identity $I = S$ is valid in these applied cases?

(Asked by: username_0 on 2021-09-04)

Say we are in a closed economy where there is no government.

It is claimed that

$$I = S$$

is an accounting identity and so it's always valid.

I find it difficult to understand why this identity is always true even though I know how to derive it analytically.

Then, I created some examples to understand it better. All my examples are based on the fact that in GDP only final transactions are counted in consumption.

---

Edit:

I'm sorry for the confusion, the subscript under every variable refers to the person who is performing the action and what I'm considering is the state of the system after the transaction is made.

---

* Say person 1 goes to the bakery to buy $\\\$2$ of bread from person 2: there can be 2 cases

1. the bread person 1 buys is in inventory so what should happen is this series of variations :

$$\Delta S\_1 = -$2 \:,\: \Delta I\_2 = -$2 \:,\: \Delta C\_1 = +$2$$

and also $\Delta S\_{\color{red}2} = +{$}2$ , but that would mean that the identity is not preserved, so why instead $\Delta S\_{\color{red}2} = {$}0 $ ?

2. the bread person 1 buys is "just baked" so what should happen is this series of variations :

$$\Delta S\_1 = -$2 \:,\: \Delta S\_2 = +$2 \:,\: \Delta C\_1 = +$2$$

Am I right about this case?

The last example I'd like to show you is this one :

* Tha owner of the bakery (person 1) buys some flour to produce bread for $\\\$100$ from person 2, the problem is that here we have an intermediate good so, what I thought should happen, is :

$$\Delta S\_1 = -$100 \:,\: \Delta S\_2 = +$100 \:,\: \Delta I\_1 = +$100 \:,\: \Delta I\_2 = -$100 $$

what about this one?

Thanks to all of you

**Tags:** macroeconomics, gdp, national-accounts, accounting

---

## 1 answers

### Answer by username_1 on 2021-09-04. Score: 1 | Accepted answer

1. No the identity would be preserved, since $I\_2=S\_2$ when inventory is reduced by $\Delta I\_2$ =-\$2 at the same time the second person dissaves $\\\$2$, so $\Delta S\_2$ = -\$2 as well. If then that person saves 2 dollars that they got from their customers then that will become some sort of investment (depending on how it is saved) so then we have $ \Delta S\_2 = -2 + 2 =0$ and $ \Delta I\_2 = -2 + 2 =0$.

   This is as it should be, since the sale was made from previously made inventory investment the output $Y\_2$ must be zero this time period.
2. That does not need to necessarily happen based on your text. You are making extra assumption that the bread was bought from previous period savings and that baker saves what they earn. But if you make those extra assumptions then it would be correct.
3. In the last example if it is an intermediate good then it does not count directly. You would have to first get to final sale and then you can try to allocate it to $I=S$, $C$ or $G$ depending on what happens to when we get to the final sale.

> **Comments:**
> **username_0**: thank you @1muflon1 much appreciated, I think you misunderstood my notation though, the subscript indicate the person who saves/dissaves and so on...And then I have a question, in 1. we have also that person 1 dissaves 2 dollars to consume them but then given that, as you wrote, $\Delta S_2 = \Delta I_2$  the identity wouldn't be preserved, what am I missing?  
> **username_0**: how I would explain it is this series of variations : $\Delta S_2 = 0$ because person 2 is dissaving inventory but he is also taking the money from the customer, $\Delta I_2 = -2 $ for reduction of inventory, $\Delta S_1 = -2$ and $ \Delta C_1 = 2$, the identity would be preserved. Besides at the end of the transaction $\Delta Y = \Delta S + \Delta C = \Delta I + \Delta C = 0$ , everything seems fine in the end, Is it like so?  
> **username_1**: @Tortar I understand the subscript indicates person. Also it would be preserved, the identity says I=S, if person dissaves 2 USD and then saves 2 USD saving should be zero and the same should hold for investment.  
> **username_1**: @Tortar Taking money from customer does not automatically means S goes up, if the baker would spend money on some consummation good S could be zero or only half of that, by definition (assuming no taxes) S=Y-C, if the baker sells 2 usd bread you only know that Y increases by 2 in order to say that S increases by 2 you have to assume that C=0. In any case the identity is still preserved  
> **username_0**: yes I'm just considering the state of the world after the transaction, so $\Delta C_2 = 0$ I'm not implying that in the future he won't consume  
> **username_1**: @Tortar that fine, I am just saying that’s something that was missing so I am notifying you about that in case you did not know  
> **username_0**: just to be clear , the way I reported the variations in my second comment is appropriate to represent the situation ? Thank you for the help anyway  
> **username_1**: @Tortar yes the Y should be zero because the way how you set up the problem there is no production so output is 0, you only have people transacting some output produced in the past  

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