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I am trying to fully understand the savings-investment identity; both the Y=C+I+G=C+S+T and inventory accumulation perspectives make sense to me, but I came up with two examples that are a bit confusing and look like are maybe related to intermediate consumption.

Let's say that companies A and B pay employees A and B \$1 respectively, who each spend \$1 on final goods from company A, which spends $1 on X from company B.

If X = "The ordinary, regular maintenance and repair of fixed assets used in production" then it is intermediate consumption according to UNSNA and I would assume Y = C = \$2.

If X = "Major renovations, reconstructions, or enlargements of existing fixed assets enhancing their efficiency or capacity, or prolonging their expected working lives" then X is not intermediate consumption according to UNSNA. Is it then investment? This would mean C = \$2, I = \$1, Y = \$3. This checks out with the intuition that there is more "value added" here than in the first example. However this would require that S = \$1 and I don't see:

  • Where does S come from? It looks to me like X is an investment that doesn't come from savings and I can't seem to fit it on any variant of the circular flow model
  • employees' total salary is \$2 and I don't see a reason here why personal income should be different from national income in this case

Did I get the national income wrong?

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Where does S come from? It looks to me like X is an investment that doesn't come from savings and I can't seem to fit it on any variant of the circular flow model

Saving on national account doesn't mean the same as saving as normally understood by layman.

Private savings is defined thusly:

$$S= Y-T-C$$

In your case you don't specify any taxes so T=0 and you specify that Y=3 and C=2, hence by definition society created S=1.

employees' total salary is $2 and I don't see a reason here why personal income should be different from national income in this case

Well the money for renovation had to came from somewhere, GDP is not just equivalent to all spending (i.e. $Y=C+I+G$) it is also equivalent to all income $Y= w+ i+\pi+r$ where $w$ would be wages, $i$ interest income, $\pi$ profits and $r$ rent.

If you claim that $Y=3$ there must have been some other income than those two wages $w=2$. Otherwise $Y$ cannot be 3 and it would be 2 and then there cannot be any spending on renovation.

By the way you should remember the fundamental truth about national income: Someone's spending is always someone else's income.

You cannot claim there were \$3 spent in an economy without allowing for \$3 of income. That \$1 of investment spending had to create corresponding \$1 income for some firm, or workers doing the reconstruction. In your case if we are talking about major renovation that firm would either get $\pi=1$ or workers in that firm would get $w=1$ or some combination of the two such that $\pi+w=1$.

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  • $\begingroup$ In my case I have chosen for company B 𝜋 = 0, w = 1 (revenue 1 labor expense 1). For company A I have 𝜋 = 0, w = 1 as well (revenue 2 labor expense 1 investment 1). Unless the investment by company A is not considered a cost, such that 𝜋 = 1, w = 1. This is the only way I can see "Someone's spending is always someone else's income" working, i.e. the \$2 spent on company A created only \$1 of personal income so it must have created \$1 in "undistributed profits" even though that \$1 "profit" was spent and is no longer with the company $\endgroup$
    – llllvvuu
    Jun 28, 2022 at 13:01
  • $\begingroup$ Ah, it seems this is indeed the case: investopedia.com/terms/c/capitalize.asp $\endgroup$
    – llllvvuu
    Jun 28, 2022 at 13:05
  • $\begingroup$ @llllvvuu 1. someone spending is always someone else’s income is based on the following $C+I+G=Y=w+i+r+\pi$ left side C+I+G is all spending and w+i+r+\pi is all income. So definitionally all spending = all income 2. That 1 investment must become income of someone else if there is no 3rd company then that investment spending done within company will have to fund wages of some other workers in the company. You can’t choose arbitrary numbers here. For example from physics we know that d=ts where d is distance t time and s speed. It’s impossible for distance to be 100km time be 10h and speed be $\endgroup$
    – 1muflon1 ♦
    Jun 28, 2022 at 13:12
  • $\begingroup$ 100km/h. In the same way it’s impossible for \Delta I=1 and not have corresponding \Delta in either w i r or pi or combination of those such that \Delta of income is 1 $\endgroup$
    – 1muflon1 ♦
    Jun 28, 2022 at 13:14
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The broken assumption was that the company A had 0 profits because it earned and spent \$2. The key concept I missed was capitalization, which implies:

  • Company A only realized an expense of \$1 (employee A's wage)
  • Thus, company A had \$1 in "undistributed profits" (explains the personal income vs national income gap)
  • This "undistributed profits" is considered savings
  • \$1 was saved and invested by company A
  • There was no savings by individuals

I'm accepting @1muflon1's answer because it helped me realize this, in particular its last paragraph.

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