Monetary Realism

Understanding The Modern Monetary System…

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 Responses

  1. stone says

    , I must say there is an ideological tinge to the survival of that misunderstanding – as in those bad banks who must keep on lending in order to get the interest paid, and as in Marx ensuring the collapse of capitalism because interest simply can’t get paid. ……. I also don’t recall seeing where the MMTers have written about this, but quite possibly I’ve just missed it.

    Does Michael Hudson count as a MMTer? He is a standard bearer for the “miracle of compound interest = disaster” brigade. To my mind however Michael Hudson’s stuff seems entirely compatible with all you are saying because Michael Hudson is so clear about unsustainability being a consequence of those receiving interest NOT choosing to spend it back to those paying the interest. He is repeatedly very explicit about that. After all “compound interest” by definition means compounding the interest rather than spending it on tattoos or whatever :) . To my mind all the issues regarding wealth inequality etc are actually brought into sharper focus by the clarity you are giving this.
    Before I started reading this blog I had a go at wrapping my head around the whole “miracle of compound interest” thing; it is on page 11 “Monetary expansion attempts to realise the miracle of compound interest” of this link:
    Looking back at it, I still think it is compatible with what you are saying because you are talking about servicing a steady state of debt that can be funded out of the current economy whilst “the miracle of compound interest” is all about creditors not wanting to spend and more debt being issued to compound “imaginary” wealth.

    • JKH says

      I think you’re right.

      I’ve noticed Hudson writing on compound interest but haven’t read his stuff closely.

      I think he’s pretty much associated with MMT.

      What he writes may be compatible, or perhaps more accurately, not necessarily incompatible.

      What I’m responding to I think is a “theory” that it is impossible to earn enough interest to service loans without more borrowing. That’s VERY rough, and as I understand it the idea is associated with Marx and with the Circuitist school (according to Steve Keen). I believe Keen is correct in identifying the theoretical error as one of stock/flow consistency – which is basically accounting. I can’t follow his demonstration of this but I agree with the premise.

      The Hudson issue I think is more associated with how debt and interest payments play a role in economic cycles. I can’t disagree with that – except that I don’t think the problem is with compounding of interest per se, although that doesn’t help. The problem is one of distribution of credit and debt, which is always an issue in recessions and depressions.

      • Ramanan says

        I think circuit theory doesn’t conclude it happens like that because outside sectors can add to profits for example but take issue if anyone comes with a resolution with a purely private economy because they think the attempts are not satisfactory enough – at least there is no consensus. They believe however that it can be explained – unlike Marx who I believe thought capitalism is doomed because of this.

        The idea has however helped them come up with great insights.

        • JKH says

          My interpretation of the circuitist interpretation flows through a Keenian filter.


      • stone says

        JKH, I think Michael Hudson was saying that compounding of interest was symptomatic of the problematic distribution of credit and debt you mention and that it exacerbated that problem. AFAIK Michael Hudson says that although Marx also said the same thing about distribution of credit and debt, Marx mistakenly presumed that industrialists would force the finance system to serve industrialization and so problems with finance would NOT be the problems with capitalism in the future. I don’t know anything about Marxism though (apart from it didn’t seem to work :) ).
        Perhaps (as Tom Hickey mentioned above) Ellen Brown is the one to blame for so many people being muddled about this. Unlike Michael Hudson, she doesn’t give much (if any) prominence to the issue just being one of distribution and spending patterns rather than being an intrinsic accounting property of debt based money.

        • Tom Hickey says

          I didn’t intend to single out Ellen, since Michael Hudson and others are also factors in that they are widely read and quoted. But this is also a pervasive idea I have seen all over the place without attribution. It’s an idea that seems intuitive and has caught on. In fact it has sort of become a meme that self-propagates.

    • stone says

      Further to what I just wrote, I do see that debt funded real wealth can sustainably grow in a compounding way if debt is used to finance real expansion of the means of servicing that debt through improvements in technology, infrastructure and organisation that can find paying customers. Those are crucial caveats though.

  2. Oilfield Trash says


    Some comments sometime else but: In the most optimistic scenario of the world, Steve Keen and his fans will barely manage to produce what stock-flow models already say (with simulations in excel sheets!).

    Very bold, below is a link to the developement of his model.

    If you can do all he is proposing in excel, my hat is off to you. I guess time will tell. A “beta” version of Minsky is out on the web running if you are interested.

    • Ramanan says

      Yes have seen that. A complicated analysis isn’t necessarily right if one gets basics wrong.

      • Oilfield Trash says


        Most criticisms of the new theory are often based on alternative theories, rejection of the abstract-mathematical method, misunderstandings, and alleged errors in the theory.

        As Minsky pointed out

        Can “It”—a Great Depression—happen again? And if “It” can happen, why didn’t “It” occur in the years since World War II? These are questions that naturally follow from both the historical record and the comparative success of the past 44 thirty-five years. To answer these questions it is necessary to have an economic theory which makes great depressions one of the possible states in which our type of capitalist economy can find itself. (Minsky, 1982a, p. 5)

        At a minimum Keen’s economic theory and models can product and explain a Great Depression with out the need for black swains. DSGE models or Krugman’s attempt to breathe life back into IS-LM cannot.

        I doubt he is Einstein of economics but I also doubt he is the Silberstein.

        • Fed Up says

          “Can “It”—a Great Depression—happen again? And if “It” can happen, why didn’t “It” occur in the years since World War II?”

          Yes, it can.

          … because World War II destroyed excess supply and excess labor so economies went back to being AS constrained instead of AD constrained. It took awhile to go from AS constrained back to AD constrained again. I think the 30-year mortgage also played a role.

        • Ramanan says

          Oilfield Trash,

          I have given up on him.

          He definitely needs to get his accounting right. He had this fashionable idea called anti-money but seems to have gone out of fashion.

          He now has this

          aggregate demand = GDP + change in debt

          which looks like the engine of his model.

          Aggregate demand is simple. It is C – IM + I + G + X and differs from aggregate supply by change in inventories. That is the former doesn’t include change in inventories but the latter does and one can be higher than the other. (although in neoclassical theory it is a curve which intersect the aggregate supply at a price).

          The bad thing about accounting models is that the accounting needs to be tight – else one can produce strange self-inconsistent results. Also from a behavioural point of view, he uses the same variable for the actual and expectation of the variable. He can write big models and even manage to get a supercomputer to solve models but unless he gets the simple things right, he is not going anywhere. He will keep getting Dirac’s runaway solutions.

        • Oilfield Trash says


          Yes I know, I followed JKH and your spirited debate with Matheus, who does not seem to have the same concerns with Steve’s model as you do.

          In the end all of you are smart and my hope is in collaboration ya’ll can work it out.

        • JKH says

          “aggregate demand = GDP + change in debt”

          People may have forgotten this, but that was the equation that set off the Krugman/Keen debate a year ago.

          That’s the equation that PK called Keen on very specifically in PK’s initial post – i.e. that it made no sense in terms of an internally consistent conceptual framework. And he’s right about that.

          From there it generated into a peeing contest about endogenous money. And Krugman strayed off course on that. But that stuff was all a red herring relative to the original issue, which was the equation.

        • Fed Up says

          Ramanan said: “Aggregate demand is simple. It is C – IM + I + G + X and differs from aggregate supply by change in inventories.”

          JKH said: ““aggregate demand = GDP + change in debt”

          And, “That’s the equation that PK called Keen on very specifically in PK’s initial post – i.e. that it made no sense in terms of an internally consistent conceptual framework. And he’s right about that.”

          I believe that Keen includes financial assets in aggregate demand. I think that is a poor definition, but definitions matter once again.

        • Ramanan says

          Yes right.

          He later changed it to effective demand – as if the change makes any difference.

          There was a strange defense of his accounting here – written at the time:

        • JKH says

          strange and mystical

        • wh10 says

          Ramanan, but do you think it’s useful to at least have a robust computer model to work off of, even if the accounting isn’t right? For example, some one down the road could re-engineer it to get the accounting right, but it’d maybe be useful to have what Keen’s doing as a starting point.

        • Tom Hickey says

          Getting an open source model up and running is hugely important since it will attract “crowd-wisdom.” One person working alone and not full time at that is not going to solve this.

        • Ramanan says

          Yes I think he needs to make small changes in the sense that it won’t affect his infrastructure – but if irons out some starting points, he can come to the right track. Simple tweaks can be useful to him – such as assuming a econometric relationship – variables at any time depending on variables at t – delta so that accounting constraints are not violated. My pessimism is because he doesn’t seem to realize it in spite of various people saying it.

          He knows a lot of literature/history of economics etc., so won’t be stuck by those things – it’s just that someone needs to put him in the right track but he has a track record of going off track.

  3. vimothy says

    JKH, Excellent (pair of) comments (on interest payments in the steady state, somewhere above (if I’m nesting this right)).

  4. stone says

    By inference, that means the circulation of all actual transactions in the economy has allowed all interest to be received and paid from the bank’s perspective. And because the money was available in total at the bank level, it must have been available at the non-bank user level in aggregate – specifically for the borrower to pay that interest.
    The rest of it is just a matter of how the original deposit was used in real economy transactions in order to facilitate all that. It becomes a matter of stock/flow reconciliation. Most people make the mistake of thinking that the payment of interest (a flow) requires a corresponding increase in the stock of money. That is not the case.

    Doesn’t it all depend on there being enough flow back from the creditors to the debtors? If creditors only buy stuff from other creditors (eg they sell each other stocks, mansions and art works) then things get messy. I guess the idea that interest means more and more money is needed is based on the impression that in practice creditors don’t buy enough from debtors? Is there a good historical example of a debt based financial system persisting in a fairly stable way for many decades?

    • JKH says

      not sure I can answer those bigger systemic history questions, but one way to think about the paradox is that the payment of interest in general is a flow of funds – it is the result of a transaction that is part of the overall velocity of money moving around the system – and in itself, it doesn’t require an increase in the stock of money outstanding. Other things put upward pressure on the demand for an increased stock of money more generally – but interest at the end of the day just moves money from one pocket to another – except that intermediaries like banks take their cut – but as I said to Ramanan above, all of that gets dividended back to deposit form EXCEPT for what is retained as incremental capital – and the ONLY reason to retain incremental capital is to support growth in the banking system and growth in the economy – and it is that general growth that puts upward pressure on the need for the creation of new money and new deposits – again from new lending – so that’s where the pressure for more money creation comes from – not necessarily from the current level of interest payments in the economy

      … hey, that wasn’t too bad for some mid-morning scribble

      • stone says

        JKH, you describe it beautifully but I’m still left thinking that real life tends to mess it up. I guess the issue is that creditors almost by definition have surplus money over what they choose to spend. As such, the interest they receive is also likely to add to that surplus and so I guess just sit there rather than flowing back. I suppose it is very important to describe how it could work sustainably just as you described but equally important to describe how it goes off the rails when it does.

        • JKH says

          I certainly agree that its important to consider the “off the rails” piece as well, but that the problem of paying interest as its sometimes perceived may not be quite the right analysis to do that.

  5. vimothy says

    “The dollar is both it seems to me.”

    Right, and what I am saying is that it’s this dual rule that makes “dollars” a macro-economically important variable. Because the price of every good is quoted in dollars, when the price of dollars changes, the price of *every* good changes. (This is why the naive comparison between prices at different times fails–the units are different. Hence the need for price indices to measure changes in the value of the dollar.)

    Since changes in the value of the dollar result in changes in the price of every good, changes in supply and demand for the dollar can affect supply and demand throughout the whole economy. That’s what makes money so significant. This is the channel through which you get Keynesian / monetary business cycles. But you do need both sides of money–the MOE *and* the MOA function (or whatever terms you prefer)–so that supply and demand in the money market can have system-wide effects.

    If prices were quoted in, I don’t know, some other good X rather than dollars, then supply and demand for good X would affect supply and demand in all other markets, but supply and demand for dollars would not be significant on a global level–at least, no more than any other financial asset.

    • Greg says

      Well certainly the value of the dollar changes every minute literally, but one must ask relative to what of course, and we dont see prices change by the minute, so Im not sure the connection you make between the dollars prices and goods prices is as strong as you suggest. In a given day a currency might fluctuate 10% but prices of things in that currency stay the same.

      “f prices were quoted in, I don’t know, some other good X rather than dollars, then supply and demand for good X would affect supply and demand in all other markets, but supply and demand for dollars would not be significant on a global level–at least, no more than any other financial asset.”

      There would be no need for dollars at all in this example, there would be a global currency of that good you speak of. Which is essentially what a gold standard was and why we no longer use gold standards.

      • vimothy says

        Greg: How could the value of the dollar fluctuate when no prices change?

        Dollars are priced in “goods” (i.e., in everything else available for sale). If the price of these goods goes up, then the value of the dollar comes down. The value of the dollar going up (or down) is logically equivalent to prices going down (or up). That’s money’s unit of account function.

        Because dollars are also exchanged for goods, demand for dollars equates to supply of everything else and supply of dollars equates to demand for everything else. That’s money’s medium of exchange function.

        Taking both of these functions together, the result is that if supply and demand for money changes, but prices do not (because prices adjust gradually, for example), then there can be excess supply or demand for goods across the whole economy.

  6. stone says

    This is a really stark example of how investment doesn’t increase nominal wealth if there are not customers paying enough to support the financing costs:
    I know that this is an example of dumbness on a fairly small scale but it does illustrate that there is the potential for the private sector to spend millions building real assets and to end up with not a penny more in nominal wealth. I guess it is the prospect of the same phenomenon that is stopping the private sector from doing all of the sensible things that everyone wants done. For instance I guess the two billion people without access to clean water or sanitation are largely within reach of global capitalism since they are fully able to buy coca cola and Marlboro cigarettes BUT providing them with sewers would be as financially dumb as building that Scottish climbing gym for the same reason, the end users don’t have the money.

    To me the bottom line is not whether there can be flows of saving by private investment in building real assets but whether or not those flows subsequently reverse due to lack of customers. That prospect is what often stops investment happening in the first place.

  7. stone says

    From Cullen vs Jose :) above:

    ” “No, govt’s who run a budget surplus generally run a trade surplus”.
    Yeah, I remember those magnificent U.S. trade surpluses of the Clinton budget surplus era. ”

    My recollection of the billyblog message about budget surpluses was that they were sometimes in exceptional circumstances needed to destroy money so as to reduce aggregate demand so as to prevent inflation. Bill Mitchell gave the example of Norway where he said that the massive oil revenue needed to be taken out of the economy so as to prevent inflation. I realize that that fits in with Cullen’s caveat that it is a big trade surplus. Also the Norway government buys foreign stock market shares with the budget surplus. I think Bill Mitchell would say that that is an afterthought and that the primary driver of the taxation is to reduce aggregate demand BUT Cullen could also make the case that the tax money is being used to buy stocks rather than going into a shredder.

    Bill Mitchell also used the example of Iceland in the 2000 to 2006 period. Then Icelandic banks were raking in crazy amounts from global asset bubble blowing. The Icelandic government ran big surpluses from all of the tax money coming in from the banks. I guess that is very much like the Clinton surpluses that were driven by capital gains tax receipts from the tech stock bubble. Speculative capital flows into a country can result in government surpluses that are not really driven by the government seeking to have spending money ?????