S = I + (S – I) : The Most Important Equation in Economics

A while back, I had a laugh with someone I don’t know and have never met.

A very smart professional economist had a great, great post that purported to be a takedown of operational Modern Monetary Realism. But he started with I = S.

I had a laugh with the mysterious JKH over this absurdity, because it’s like saying “I’ll prove you wrong. We all know 2 + 2 =5, so therefore…”

We know S does not equal I, because it doesn’t when you look at the entire sector balances equation. We know this is definitionally true:

(S – I) + (G – T) + (M- X) = 0

Therefore, the only way S = I is if the next two parts of the equation net to zero. But we know this never ever happens. And we also know that if we want growth, we need G – T > 0 over the long run.  It’s an iron law we’re doomed to obey. It’s a theorem.

In short S = I if we want our economy to never grow, and insist on living in the Yukon off the land.

I was confident I had this all figured out.

But frankly, nothing prepared me for the fireworks over at Steve Roth’s place. Nothing could have prepared me.

JKH and Steve Waldman showed up and started shining spotlights on all that is hidden. It turns out that we know S doesn’t equal I, we know  S = I + (S-I). Well, at least we do now that JKH and Steve Waldman applied themselves to the task of figuring stuff out.

I’ll turn it over to the master himself for the coup de grace:

S = I + (S – I) is not a mistake.

And rather than being a small error, it’s a critical analytical decomposition of private sector saving, most particularly in the context of interpreting the (sometimes ambiguous) MMT intended meaning of saving. Not understanding it can be the source of much confusion around MMT. The fact that you thought it was an error might hint at some. Moreover, it’s the saving algebra that encapsulates what Steve W. is driving at in his comment. That’s partly why I started my opening comment with it.

It’s an obvious identity/truth from the algebra alone, just as a self-referencing rearrangement of terms.

But it’s much more than that in the context of deconstructing MMT’s approach to the interpretation of saving.

Before getting to that interpretation, here is the national income accounting/algebra that gives background context:

C + I + G + (X – M) = C + S + T

(S – I) = (G – T) + (X – M)

That’s one permutation of “sector financial balances”, as a derivation of national income accounting.

That particular permutation says:

Private sector net financial asset accumulation = the government deficit plus the current account surplus.

This is the SFB formulation that defines NFA (actually the flow change in NFA) as the left hand side – it’s the excess of private sector saving over the amount required to fund investment.

Private sector NFA is MORE than just the government deficit in an open economy. Only in a closed economy does it collapse to (G – T) as you suggested. The always correct definition starts with (S – I), not (G – T).

Now, in parallel, but at the core of the relationship between private sector saving, investment, and “net financial asset” accumulation:

S = I + (S – I)

Just looking at the algebra, this is an obvious identity (or truth, really) just as a result of being a mere rearrangement of entirely equal terms. But it has considerable meaning in the context of how the pieces of the saving puzzle come together.

Using the earlier definition of net financial asset accumulation from the sector balances equation above:

Private sector saving = investment plus private sector net financial asset accumulation

That equation is the foundation algebra for illustrating the point Steve W. was making, in my view, as per my explanation above. I agree with SW’s point, as per my comment. And I also covered this same aspect (among many others) in recent comments under previous posts here at Steve R.’s site.

Just check out the effortless logic, as though everyone had known this for the last  30 years and we can’t get Paul Krugman to shut up about it. And it’s 100% obvious now that JKH writes it in such clear terms, but was a murky unknown mess prior to a few days ago.

This is one of our goals over here at MMR. We want to illuminate, we want to make it easier to think about our economy. And does this formulation make it easier to think about how the world works? Well, yes it does!

Private sector saving = investment plus private sector net financial asset accumulation

So a tip of the coffee mug to JKH this Sunday morning. I hope you’re kicking back, feeling good this morning, wherever you are.


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31 Comments on "S = I + (S – I) : The Most Important Equation in Economics"

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5 years 4 months ago

Regarding nonprofits etc.:

There are certainly issues of data organization and classification within the Fed Flow of Funds reports. But understanding the logical structure of the accounts is a prerequisite to dealing effectively with these data issues. Data sub-strata can always be estimated and backed out as desired, but getting the account structure right is the first priority. As Keynes said, better to be vaguely right than precisely wrong.

5 years 4 months ago
Thanks, Mike Steve Waldman said: “There is no “to the penny” sort of accounting relationship between household-sector financial saving and government issue of NFA” This is correct as it stands. It’s about a possible relationship between two aggregate flows. The problem is that household financial saving and/or the change in household net financial assets as a flow is affected by more than just “government issue NFA”. Most importantly, it can be affected by business investment. It IS POSSIBLE for there to be a direct 1:1 (“to the penny”) theoretical association between government-issued NFA and household NFA, IF one holds “I” constant (as well as the current account), but this required restriction on I is not very often revealed as such, which is much to Waldman’s point. There are an infinite number of examples that are possible to demonstrate this, but a particularly glaring one should suffice: Suppose the government runs a balanced budget and the country runs a balanced current account. By the sector balances identity, that means the private sector must run a balanced NFA account – i.e. zero NFA accumulation for the accounting period. And it also means S = I (in amounts): ………… C + I + G + (X – M) = C + S + T (S – I) = (G – T) + (X – M) = 0 + 0 S = I ………… One way this condition (S = I) can come about is as the result of business issuing new debt and equity claims to households in the amount of S = I. (This route would also force household saving and net financial saving to S = I and business saving to zero by implication.) Those same financial claims in the amount S = I become the change in household NFA. So… Read more »
5 years 4 months ago

“There are other ways for household financial savings to increase, apart from NFA”

should read, “apart from private sector NFA”

Clonal Antibody
5 years 4 months ago

Mike, just a small quibble.

You said

(S – I) + (G -T) + (N- X) = 0

But the National Income accounting equation as you further rightly said is

C + I + G + (X – M) = C + S + T

which rearranges to

(S-I) + (T-G) + (M-X) = 0

and that is quite different from what you wrote.

If savings are greater than or equal to investment, and a country is running a fiscal budget surplus (or a small deficit), then the economy has to export itself out of trouble. Example Germany today.

If a country is a net importer, it has to attract investments, or run a fiscal deficit or both.

Investments can come from out of the country, or they can come from the government deficit. In my view investments from government deficits are preferable to foreign governments or foreign bankers owning your country

5 years 4 months ago

Not sure what blogging software here, but need a Comments Feed widget.

Actually, I don’t even see the Posts Feed link…

5 years 4 months ago

SRW said: “There is no “to the penny” sort of accounting relationship between household-sector financial saving and government issue of NFA.”

JKH responded: “Right…”

If those statements are correct, then I don’t think I properly understand “Private sector saving = investment plus private sector net financial asset accumulation.” I say this because if I conceptualize the equation as S = I + NFAinjection, then there would still seem to be a “to the penny” relationship. In other words, at the moment a deficit adds NFA to the private sector, S looks like it will still rise by the NFA injection, to the penny.

Some issues I might be having here could include:

1) I don’t fully understand what S and I mean (a la Roth’s and Rowe’s recent posts)
2) That I am confusing stocks with flows
3) Perhaps SRW is suggesting that an injection of NFA will have an impact on I, such that the final impact on S will not be the same as (S-I) to the penny.

5 years 4 months ago

BTW, I think for people to fully appreciate this, more context is needed on the following issues-

1) What S and I actually mean
2) Clarification on what SRW is claiming to be MMT’s ‘rhetorical overreach’ (which I am not sure is fair, since a quick Google of “Mosler to the penny” only pulls up statements claiming deficits add NFA to the penny, which is accurate)