How to show the identity $I = S$ is valid in these applied cases?

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 $$\Delta S_{\color{red}2} = {}0$$ ?

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

$$\Delta S_1 = -2 \:,\: \Delta S_2 = +2 \:,\: \Delta C_1 = +2$$

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 :

$$\Delta S_1 = -100 \:,\: \Delta S_2 = +100 \:,\: \Delta I_1 = +100 \:,\: \Delta I_2 = -100$$

Thanks to all of you

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.

• 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? Sep 4 at 23:16
• 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? Sep 4 at 23:32
• @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
Sep 4 at 23:42
• @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
Sep 4 at 23:46
• @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
Sep 4 at 23:50