Briefly Revisiting S = I + (S – I)


Some readers of Pragmatic Capitalism and Monetary Realism have been underwhelmed by the aesthetics of this equation:

S = I + (S – I)

This says that private sector saving consists of an amount required to fund investment I plus an additional amount (S – I). The structure of the equation is a tautology that cannot be falsified in purely symbolic form, of course. There has been some criticism of it for that reason. For example, one reaction (from a few people) is that a 6 year old could derive the same thing. That criticism is perhaps understandable – absent further consideration of the reason for this decomposition.

Those interested in reviewing the reasoning behind the tautological form – or perhaps seeing it explained for the first time – may wish to read further.

But before that, some earlier references:

Michael Sankowski provided an excellent, brief introduction here:

I elaborated at length here:

And Cullen Roche summarized it in part 6 of his paper here:

With that background, we briefly revisit the reasoning behind the equation. Despite its simple appearance, there is a more nuanced explanation behind it, based on an element of Keynesian style macroeconomic intuition.

Expenditure/Income Sector Decomposition

The standard expenditure/income model:

C + I + G + (X – M) = C + S + T

For purposes of this discussion, the pivotal variable is S, which corresponds to private sector saving.

S = I + (G – T) + (X – M)    *

This says that private sector saving is an amount required to fund investment I, the government budget deficit (G – T) and a current account surplus (X – M).

Subtracting investment I from both sides,

(S – I) = (G – T) + (X – M)  **

This decomposes (S – I) into its two components – funding for the government budget deficit and a current account surplus.

Substituting the left side of **  into the right side of *:

S = I + (S – I)

And that is the equation in question. It says that private sector saving is the amount required to fund investment I plus a residual amount in excess of that, equal to (S – I). The excess amount is deliberately left un-decomposed. The occasional complaint about this form is that it is a self-referencing tautology, since it can also be derived by simple rearrangement of symbols without reference to their underlying meaning.

The ‘Keynesian Skew’

Assume temporarily that the government budget and the current account are both in balance.

Then, from the Keynesian expenditure income model above:

S = I

This equation does not mean that S is the same ‘thing’ as I.

Suppose private sector saving in the current period consists entirely of household saving of S, with the result being an addition to household net worth and a corresponding bank deposit. Suppose also that a corporation has borrowed an amount equal to S to make a new investment I during the same period. Under these assumptions, S = I. But it is obvious that the substance of I (the material substance whose value is recorded on the corporate balance sheet) is different than the substance of S (a net worth increase whose value is measured on the household balance sheet and which equals an amount held in bank deposit form).

It is measured value and not substance that is being equated in S =I.

While this may seem too obvious, it is an important distinction in following the meaning of national income accounting construction and sector financial balances using such symbols.

For purposes of this post, we’ll coin the phrase “Keynesian skew”. This will refer to the idea that government deficits are a rational response to shortages in aggregate demand, as roughly prescribed by Keynes. This translates directly to corresponding saving dynamics. According to the Keynesian skew, the private sector generally desires to save in excess of what might achievable in the absence of government deficits. This means that, unless the country is running a sufficiently large current account surplus that adds to saving, there will be a tendency for S to be insufficient in quantity to satisfy private sector saving desires in full.

Our temporary assumption above was that saving S equals investment I. The Keynesian skew suggests that the quantity of investment I will be insufficient in allowing for enough private sector saving. Aggregate demand and economic output and employment will be stopped out below potential – because the private sector is starving for more saving. A temporary equilibrium has been reached where S = I, but the economy has not yet created GDP sufficient to reach potential.

Again, suppose S = I now holds in a real world situation, and that the economy is performing below potential. There are two ways in which the economy can expand from here. Either investment I increases or something else has to give. In framing the situation, we also make the upfront assumption that investment I has pretty well maxed out.

So assume the government starts to run a deficit as a deliberate policy response.


S = I + (G – T)

Recalling the distinction mentioned above between substance and the measure of substance, the above equation means:

Private sector saving S equals an amount of saving equal to the amount of investment I, plus saving in the amount of the budget deficit (G – T). Thus, private sector saving funds both private sector investment and the government budget deficit.

(The term “fund” is used here in the sense of standard flow of funds accounting and sources and uses of funds accounting. This does not contradict the dynamic of macroeconomic construction, which is that for any accounting period, it is the expenditure on investment and the act of government deficit spending (as well as foreign sector effects in the more general case) that allows the actual private sector saving result. This dual interpretation is analogous to that emphasized in the case of banking, as described in a previous post on ‘loans create deposits’, where it is also the case that ‘deposits fund loans’:

Now bring an active current account into the picture:

S = I + (G – T) + (X – M)

That says that the quantity of private sector saving funds private sector investment, the government budget deficit, and a current account surplus.

This equation might be represented as:


Where SAVEGAP = (G – T) + (X – M)


Why not just write:

S = I + (G – T) + (X – M)



Instead of:

S = I + (S – I)?

This is the question.

The Keynesian skew suggests that investment alone is not capable of delivering an adequate supply of saving to the private sector. Notwithstanding the amount of saving that must be generated by the same amount of investment, there is a residual shortage of saving relative to private sector desires in total – a shortage that can only be alleviated by finding outlets other than investment.

Thus, there need to be two components of private sector saving:

a) The amount corresponding to investment I – which is the amount of saving that at the macro level is created by investment, and which funds investment in the sense of both macro/micro flow of funds accounting and micro level competition for the form of financial intermediation

b) An additional amount, which by residual (tautological) equivalence, is (S – I)

And according to that decomposition,

S = I + (S – I)

This decomposition is logical, according to the assumption of the Keynesian skew. The equation is derived without direct reference to further detail in the national income equations. And that accounts for the simple term (S – I), instead of the explicit sector decomposition (government deficit and current account surplus) noted earlier.

As an alternative, the notation might have been something like:


But the logical association still holds:



Those whose instinct is to dismiss the relevance of such a tautology might consider that the chosen decomposition has a meaning that supersedes the mere observation that it is a tautology. The message of the decomposition is that the Keynesian skew suggests a natural inclination by the private sector to save more than the amount required to fund private sector investment alone – i.e. an excess amount which is (S – I) by residual decomposition.

More generally, the basic idea behind a residual or tautological decomposition is found in Boolean Algebra and Venn diagrams – where a given set is split in two subsets, according to the defined outer set (S), a known subset (I), and the residual gap that remains (S – I).


S = I + (S – I) delineates the idea that (S – I) is the additional saving component, when investment I alone is insufficient to deliver enough saving to achieve economic capacity. That said, a good deal of saving comes from investment I, not (S – I), and that should be a point of emphasis as well. The comparison between those two quantities is important. From there, further sector decomposition of the component (S – I) is naturally of interest. And the full expansion as derived earlier, is:

S = I + (G – T) + (X – M).

The Keynesian skew suggest that the private sector wants to save more than the amount that corresponds to investment I alone. It doesn’t exactly specify that desire as a desire for net financial assets (NFA). The fact that the demand for saving gets satisfied through net financial assets is a consequence of monetary configuration. If investment I is assumed to be maxed out, then additional private sector saving is forced into net financial asset form. But that’s not necessarily because the private sector seeks net financial assets because of their financial form alone. It’s because net financial assets is the only form in which that additional saving can be manifested, given the assumption of maxed out investment I. If investment I could be expanded, a similar aggregate demand impetus and overall private sector saving result might be achieved. In this sense, the NFA ‘solution’ is the result of the ‘failure’ of private sector investment to produce enough saving on its own. It is consistent with the judgment that government needs to act in these circumstances, absent additionally compensating export expansion.

Moreover, the circumstantial nature of NFA compositional demand can be revealed further by looking one more level down in sector decomposition terms. The private sector is composed of the household sub-sector and the business sub-sector. From the perspective of their own balance sheets, households save in the form of both real assets (e.g. residential real estate) and net financial assets (e.g. bank accounts, bonds, stocks; net of financial liabilities). And the household NFA component is present even when private sector S = I. This is because much of investment I is present on corporate balance sheets, from where it is intermediated back to household wealth through financial claims. And that puts the household sector into its own ‘NFA long position’, even when S = I. So when the private sector as a whole is in a state of seeking additional saving beyond the level of investment I  – i.e. seeking positive (S – I) – one should remember that the household sector already holds considerable NFA of its own via financial intermediation from the business sector. It is saving that is the primary quest – not so much NFA – and the NFA result at the private sector level is a function of the assumption that investment has been maxed out, with NFA expansion being the private sector saving outlet beyond that.

(MMT was the original blogosphere promoter of the NFA concept. Like most ideas, it’s been subject to scrutiny and interpretation. The equation in question has been involved in that process.)


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388 Comments on "Briefly Revisiting S = I + (S – I)"

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4 years 3 months ago

You can count me as one who was initially “underwhelmed” by the aesthetics of the equation. I might have even called it ugly. But I stuck with it and eventually came around to understand its true power.

I do like the S = ISAVE + SAVEGAP terminology in that it emphasizes that private sector Saving has two components. The first is the main one, i.e. Investment. The second is sort of a residual that fills the gap when Investment is too low.

Nice job, as usual, JKH

4 years 3 months ago

That’s a really nice summary of the way I think a lot of people feel about this. It’s a very powerful idea that gets the orders of magnitude right. We have an economy where I is the backbone of private saving. But we also have tools at our disposal that can be used when that backbone is weak.

I wish MMTers wouldn’t get so defensive about this discussion and would instead try to see the balance in it and how well it relates to the real economy. It’s perfectly consistent with the realism (wink) of the way the monetary system works. And yes, contrary to some people’s opinions, it is a “game changer” with regards to how we should approach not only our understanding, but also our policy.

4 years 3 months ago

Yeah, I saw a couple of very defensive tweets from some prominent MMTers recently. Your exchange with Dr.K was particulalry interesting. They clearly think you’re trying to attack them, which is just weird. Your gratitude and respect for them was obvious in your primer.

4 years 3 months ago

HI Geoff,

I was pretty similar to you- the first few times through reading JKH and SRW over at Steve Rs place, I was not sure why they were spending time on this.

But one major point is once you accept the possibility of a savings gap, then thinking about what to do for the economy becomes much more complex, and much more like the real world. If you don’t accept a savings gap, then the neo-classical approach is everything, and you want to get government out of the way because it just crowds out investment.

But what happens instead of crowding out, there is a savings gap? The neo-classical approach – and really the hard core libertarian approach too – is to assume it’s all crowding out. But that means the most primitive societies should out grow the most governed societies. What we see in the real world is nothing like that. That to me says the “savings gap” idea probably explains more of the S I issue we see in the real world.

Jan Hatzius has a good approach to this. I can’t find the article right now, but he says something like “The accounting identities are just the beginning. The analysis comes in with the projection of the levels of the inputs and how these changes will feed on each other.” This was Godleys approach, and this savings gap identity helps to think about how we can best structure the rules of our society so we see high, sustainable, and enjoyable growth.

4 years 3 months ago

Interesting, Michael. It sounds like the “savings gap” idea is not generally accepted by the mainstream?

4 years 3 months ago


I was actually thinking of including the phrase “ugly duckling” in the post.

At the time, we talked about ‘I’ being the ‘backbone’ of the economy and of saving – in contrast to the (S – I) piece, which at least in the case of the government deficit is sort of a filler.

I’m still not sure how to think about vimothy’s question – for one thing, housing as part of the ‘backbone’ investment got a little too strong.

4 years 4 months ago


This is a neat defense of your equation from a Keynesian point of view. But I have a (slightly tangential) question: What does your “Keynesian skew” analysis suggest should be the response when private saving is very low (perhaps even negative), and/or quantity (S – I) is negative? I’m thinking of recent US experience in particular.

4 years 3 months ago

This was more of an explanation for the framing of the equation, partly in response to at least one recent request for it.

I don’t think I have a great answer to your specific question.

The MMT people have done a close study of the Clinton era government surpluses, which in combination with current account deficits produced negative (S – I), and associate that with the recession that followed.

The subsequent housing boom added to saving, but was also associated with a current account deficit – so didn’t do much for private sector saving.

Private sector saving was too low throughout both periods, but the current account deficit was a consistent factor.

I think this was a complex pattern overall. It certainly warrants a decomposition of (S – I) in order to assess best policy responses. It’s not clear to me that (higher) government deficits would have been the best policy response. But I just haven’t considered that question enough.

4 years 3 months ago


Thanks. I realise that this is a bit off-topic for the post.


One idea that you often come away from PK blogs with is that the basic macroeconomic problem facing society is that people want to save a greater amount than is invested, or can be invested, and so this is either going to lower aggregate demand cause unemployment, or the government is going to have to step in and fill the “saving gap”. But it’s not always the case that saving exceeds investment. Just as saving can exceed investment, so investment can exceed saving.

[We could even rewrite the above expression as I = S + (I – S) to show the sources of finance for investment exceed private saving in this scenario. But maybe we shouldn’t go there 🙂 ]

In fact, the whole approach at times seems to lead some of the more naive commenters to a point of view from which saving is bad generally.


And even though, as Philip Pilkington has been keen to point out, these identities are only tautologies (which is, itself, a kind of tautology), I find that they are quite helpful when trying to understand why saving is not bad generally, what it actually does and where it goes.

4 years 3 months ago
“The MMT people… in combination with current account deficits produced negative” The MMTers have some Monetarist inclinations. For example, I have seen Mosler arguing that the government doesn’t control the money stock but “controls” the NFA! The private sector deficit cannot be said to be the result of the budget surplus or the current account deficit. The private sector deficit is the result of decision/behaviour of the private sector in the discussed scenario. More importantly the budget surplus/deficit is not under the control of the government. The governments sets the expenditure and the tax rates. If however the propensity to save is not low enough and the government’s fiscal stance is low, unemployment results. Also their tendency to argue about deficit level itself is counterproductive. One has to look at the story – as it moves forward in time. The deficits in the Clinton era was problematic because growth was the result of rising private expenditure relative to income. (As in PX/PI kept rising). And so if the private expenditure falls relative to private income, a recession is likely because the motor of the growth itself was a rise in PX/PI, rather than increase in the fiscal stance. Same thing happened before the crisis. Private expenditure again rose relative to private income in the mid 2000s and a sharp reversal led to a recession. Of course fiscal policy is important in this because if fiscal policy doesn’t take these into account, and the stance of fiscal policy is not relaxed (policy ≠ deficit), recession is likely to be long. There is however some logic to discussions around private sector deficits. Typically the private sector has a small surplus – historically speaking. There is a nice 2001 exchange between Wynne Godley and Jan Hatzius and he initially misinterpreted the sectoral balances… Read more »
Jose Guilherme
4 years 3 months ago

A great way to visualise these issues is Krugman’s cross:

By plotting the “desirable”, full employment GDP on the horizontal axis one can clearly see what kind of behavioral and policy changes by the government and/or the private sector that would be needed to end the recession.

Actual GDP is at the intersection of the private surplus and government deficit lines; the 2008 crisis, for instance, can be explained by a sharp increase in the desired private sector surplus at every level of income (the S-I line) that in the end led to a lower level of GDP, higher (G-T) and higher (S-I).

Scott Fullwiler even said that Krugman’s diagram “…would be a much better framework for understanding macroeconomics than the traditional IS-LM model”.

No mean achievement for a neoclassical economist 🙂

4 years 3 months ago

agreed – too much ‘reasoning from identities’ is dangerous – my sentence read that way (although MMT leans that way I think)

although ignoring identities as constraints may be more dangerous

4 years 3 months ago

Yes of course I understood/noted that.

Tobin’s quote comes to mind – identities should be respected (and we saw a violation recently elsewhere 🙂 and they tell nothing about causation.

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