THE EQUATION: A Reconciliation


OK, “The Equation” seems to be under renewed discussion. Again. And someone seems to have been upset about comments made here some weeks ago.

by L. Randall Wray

Let’s all be nice and cuddly. We won’t name names and we won’t cast dispersions. So far as I can tell, “The Equation” is this:

S = I + (S-I).

It comes from this: S = I in a closed economy with no government. We know this is an identity, accepted by Keynesians and Neoclassicals alike. Their arguments are about causation, not the identity.

S=I tells us that at the aggregate level the total spending on investment equals the total income that is NOT spent on consumption (AKA “saving”). It has been common in the Post Keynesian and Institutionalist literature to say that “saving is the pecuniary accountancy of investment” or “saving is the accounting record of investment”. I’ve been saying that since around 1980 when I took a course from Marc Tool, who propounded the theories of his teacher, J. Fagg Foster—a link between Keynes and Veblen. Foster wrote the clearest–ever–piece on the relation between saving and investment, published in the JEI in 1981: “the Reality of the Present and the Challenge of the Future”.

While we generally think of Households doing the Saving and Firms doing the Investment, the National Accounts don’t quite treat it that way because Households also “Invest” in homes.

The MMR folks like to write the S=I equation as The Equation:

S = I + (S-I).

This is supposed to shed additional light because it “disaggregates” the private sector between Households and Firms. However, except for the caveat that Households “Invest” in Homes, the Original Keynesian Equation already disaggregated: Saving is the Accounting Record of Investment; it is the Household Income NOT Consumed. At the aggregate level, Household Saving accumulates as claims on the Business sector (in the 2 sector model, there is no other sector on which the Household Sector can accumulate claims). When we account for the Caveat that Households also Invest in Homes, then their Saving is augmented by their Investment Spending on the Homes.

(Note there are other definitions of saving; for example, we can include rising values of homes and other assets as a kind of “saving”. For the purposes of the exposition here, I’m sticking with the most conventional Keynesian explication—what is presented in the macro textbooks and which corresponds reasonably well with NIPA accounting. The difference between the two approaches to measuring saving becomes really important when it comes to valuing prices of existing assets.)

Before moving on to this, let us dispense with one very strange misunderstanding held by the MMR folks.

MMR folks claim that the MMT version of The Equation holds that

S= (S-I).

I do not know how they got that idea. This is not an MMT equation. This is not an identity. This is not even an equality (usually). If you simplify the equation it says:

I = 0.

I do not know what to make of such a claim. Investment might be zero, but that would be an exceedingly rare event.

The MMR critics go on to argue (apparently from that equation) that:

“The MMT focus on (S – I) disregards the massive accumulation of US private sector wealth that arises from private sector investment over time.”

That is false. US household financial claims on the US business sector are in the trillions; US nonfinancial wealth is also in the trillions—for example, there used to be about $10 trillion of net real housing wealth (remember, new home production represents investment and thus Household Saving). What we argue is that US private sector “inside” financial wealth nets to zero—claims by households and firms on households and firms net to zero by definition. US private sector net “real” wealth is large and positive—it does not net to zero. All of this is explained in detail in my Modern Money Theory primer. You can check for yourself (and look at the notes below).

Let’s try to unpack the confusion. If we open the economy and add a government sector, the Keynesian equation becomes this:

S = I + Def + NX. (Def is government deficit; NX is net exports)

Now the arguments about causality become more pointed. Keynesians argue that causation goes from right to left, and….SO….if the government deficit goes up, that raises saving. WOW. Controversy ensues. Mainstreamers argue that if the government deficit goes up, given a fixed supply of saving, then investment gets crowded-out (Investment falls by some proportion—or even all of—the amount of the government’s deficit.)

Let’s return to “The Equation” to compare it to the expanded Keynesian Equation. Use the Simplest Model:


True, by identity given assumptions (no government, closed economy).

Now, let’s subtract I from both sides:

0 = S-I.

Using our identity that I=S we can add an S to one side and an I to the other:

S = I + (S-I).

According to MMR, this exposes additional insight. The (S-I) tells us that if we relax the assumption of a closed economy with no investment, then S can be greater than I by the amount of Def+NX. Saving is Investment PLUS the Deficit PLUS net exports. Yes, we knew that, from above, from the original expanded Keynesian equation.

The (S-I) is a stand-in for Def+NX. To be clear, define it this way: (S-I) = Def + NX.

How did we get that? Return to the equation above:

S = I + Def + NX.

Now, subtract I from both sides:

(S-I) = Def + NX.

OK, now pay careful attention because the next part is tricky. Substitute the right hand side of this equation into our Keynesian equation above:

S = I + Def + NX


S = I + (S-I).

That clarifies everything, right? Saving is equal to Investment plus, in the open economy with a government, the excess of saving OVER investment, which must, by identity be equal to the Deficit plus Net Exports.

Can we understand that?

Yes, We Can!

Look MMT agrees. Identities are Identities. In the simple economy, saving is the “accounting record of investment”. If we add government, then its deficit adds to saving. If we add a foreign sector and a current account surplus, that adds to saving, too. “Saving is the Accounting Record of Investment plus the Deficit plus Net Exports.”

And, YES We Can! agree that much of domestic saving represents claims on the domestic business sector. (That is obvious from our simplest Keynesian equation, I=S, since saving is the “accounting record of investment”—the household claim on investment.)

It can be useful to separate Households from Firms. Normally Households run financial surpluses (with a strong countercyclical bias—saving rates rising in bad times and declining in good times), accumulating claims on Firms, Government, and Foreigners. For a decade after 1996, US Households did not—they ran up deficits most of the time while the US government (and firms) ran surpluses some of that time.

We were there with Wynne Godley, doing the sectoral balances at that time, noticing that the government surplus and the current account deficit resulted in a negative private sector balances. And we were there in the “recovery” after 2000 when the household sector’s balance went hugely negative again but the business sector’s balance did not. We knew that you can subdivide the private sector into a household sector and a business sector, and the two do not need to move in the same direction. Yes!

And we worried that the household sector deficits were dangerous and “unsustainable”.

Here is what Wynne and I wrote in 1999:

Recent economic statistics confirm that our Goldilocks economy continues to grow at a relatively swift pace, in spite of financial turmoil in Asia, Latin America, and Russia and economic recession in a third of the world. The longevity of the expansion is record- setting; it is already the longest peacetime expansion in U.S. history and is about to break the record set by the 1960s Vietnam-era expansion. The expansion’s longevity and its strength have tightened labor markets, allowing unemployment rates to remain below 4.5 percent for the past year and raising real wages at a good clip for the first time in a generation. Perhaps the most potent symbol of the strength of the expansion has been the remarkable turnabout of the federal government’s budget, from a chronically large deficit to a substantial surplus. One has to go all the way back to the demilitarization of the economy after World War II to find a comparable shift in the fiscal stance. By most accounts, the surplus will continue indefinitely. The Congressional Budget Office (CBO) is projecting a rise in the federal budget surplus through the next 10 years from 1.2 percent of GDP for 1999 to 2.8 percent for 2009. Such projections are, of course, contingent on continued economic growth and unchanged budget policies. What we wish to do here is to take the CBO’s projections (which are not substantially different from those used by the administration) at face value and determine what they mean for the private sector. As we will explain, government budget surpluses imply that the private sector must have an offsetting deficit. The financial situation of the domestic private sector is made worse because of the United States’s international payments imbalance. Indeed, it is becoming widely recognized that there are two black spots that blemish the appearance of our Goldilocks economy: low household saving (which actually has fallen to zero) and the burgeoning trade deficit. However, commentators have not yet discovered the links between public sector surpluses, domestic private sector deficits, and international current account deficits. Once these are understood, it will become clear that Goldilocks is doomed. .

It should be pretty clear WHY we focused on the budget surplus, which was REDUCING the PRIVATE SECTOR’S net financial saving. We went on to explain WHY we consolidated the Households and Firms into a Private Sector:

Let us begin with an analysis of the fiscal stance. While most discussion focuses solely on the federal government budget, we prefer to include the state and local government budgets, which show substantial surpluses. This consolidated government balance provides a more accurate assessment of the impact that the overall government budget has on the economy. We also consolidate households and firms for the purposes of our analysis. When the consolidated government balance is negative (i.e., in deficit), the public sector’s expenditures exceed its revenues. Ignoring for a moment the foreign sector, this must mean that the private sector’s income exceeds its spending, with the difference equal to its net acquisition of government debt. On the other hand, when the consolidated government balance is positive (i.e., in surplus), the public sector’s revenues exceed its expenditures, leading to the retirement of public debt. In this case, the private sector must be in deficit, with income less than spending. The private sector deficit will equal the public sector surplus, and this shows up on balance sheets as a reduction of private sector wealth by an amount equal to the retirement of outstanding public sector debt. Thus no one should be surprised that as the federal budget moved to surplus last year, American saving rates fell to zero. While it is possible for household saving to be positive when the public sector runs a surplus, this can happen only when firms run large enough deficits to offset the combined government and household surpluses. As it happens, the business sector as a whole is not in deficit, so the private sector’s deficit is entirely due to household expenditures that greatly exceed incomes. (ibid)

In 1998 I had already written:

The huge grizzly that will gobble up the last bits of our poor Goldilocks is the U.S. budget surplus. It is sheer folly to attempt to maintain a budget surplus in the face of a looming world crisis-which many commentators, including President Clinton, have already admitted is the worst since the 1930s. The obvious solution is to abandon the commitment to a surplus, which is depressing income and sucking profits out of the economy. .

(And poor little Goldilocks was never heard from again!)

In addition to adding to net financial saving of the private sector, we have always argued that budget deficits add to profits of the business sector (all else equal)—as I argued in another piece in 1998:

As economists have long understood, government spending generates private income. Government spending can contribute to productivity when it adds to the nation’s stock of physical and human capital, when it finances innovation, when it encourages research and development, when it improves health and education, and when it lowers private sector costs. Government budget deficits can be beneficial. Not only do deficits result in a net increase of private sector disposable income (since the income created by the government exceeds the taxes collected whenever there is a deficit), they directly add to corporate profits. Everyone understands that the behavior of a capitalist economy depends on profits. The fundamental profits equation shows national (gross, after-tax) profits are, by definition, equal to spending on investment plus the government’s deficit spending plus net export spending minus household saving. In other words, all of the major categories of spending except household consumption add to the flow of profit income, and household saving detracts from profits. It might appear strange that saving reduces profits, but if one remembers that saving is income received but not spent, then it follows that saving cannot add to business sector profits; it is not a revenue source, but a leakage that diminishes profits. Government spending in excess of tax revenue (that is, deficit spending) is a net revenue source to firms. Government adds to profits directly when it purchases output of the private sector and adds to profits indirectly by providing transfers to households to purchase more output.

That is another reason to focus on government budgets; budget surpluses suck profits out of the business sector.

We also of course recognized that for many purposes it is useful to separate households from firms. (See the Notes below, especially the final bolded paragraph that deals explicitly with this.) What was very unusual about Goldilocks was that it was the Household Sector whose behavior was so unusual—leveraging up its debt and running unprecedented debts. As I wrote in 2000:

How would Minsky explain the processes that brought the economy to this point, and what would he think about the prospects for continued Goldilocks growth? First, I think he would observe that consumers became able and willing to borrow to a degree not seen since the 1920s. Credit cards became readily and widely available; lenders expanded credit to subprime borrowers; publicity about redlining provided the stick and the Community Reinvestment Act provided the carrot to expand the supply of loans to lower income homeowners; deregulation of financial institutions enhanced competition. All these things made it easier for consumers to borrow. Consumers were also more willing to borrow. As memories of the Great Depression faded, people became less reluctant to commit future income flows to debt service. The last general debt deflation is beyond the experience of almost the whole population and the last recession was almost half a generation ago. With only one recession in nearly a generation, it is not hard for people to convince themselves that downside risks are small. Add to that the stock market’s irrational exuberance and the wealth effect, and you can pretty easily explain consumer willingness to borrow. I would add one more point, which is that until recently the average American family had not regained its real 1973 income. Even during the Clinton expansion, real wage growth has been low. Americans are not used to living through a quarter of a century without rising living standards. The first reaction to the slow growth was to increase the number of earners per family, but that has resulted in only a small increase in real income. Thus, it is not surprising that consumers ran out and borrowed as soon as they became reasonably confident that the expansion would last. The result has been consistently high growth of consumer credit. The debt service burden increased by about 1.8 percentage points, from about 11.7 percent of disposable income in 1992 to 13.5 percent in 1999; almost all of that growth was due to growth in servicing consumer debt and almost none to changes in servicing mortgages. Still, thanks to relatively low interest rates, the burden is not at record highs. It was above 14 percent at some points during the late 1980s. Of course, interest rates are rising, and everyone expects the Fed to continue to tighten, so we might yet break the 1980s record.

And, finally, we always knew that a private sector can have net financial claims on the “outside” sector: the government plus foreigners. Today, the net claims on foreigners is negative (but as Godley noted, somewhat puzzled, the flow of income is still positive because US assets abroad generate more income flows than foreigners earn on their holdings of US assets). But US private sector net claims on the US government remain large and positive.

We often focus on this to counter the argument that US government deficits and debts “burden” the American private sector. In fact, we see this as net financial wealth, that Americans should be—and mostly are—glad to hold.

We separate net Household claims on sovereign government from net Household claims on firms because sovereign government’s IOUs are the most liquid and safest assets one can hold. The IOUs of firms can become illiquid and firms can become insolvent. That does not happen to US Federal Government IOUs. Hence it makes sense to argue there is a desire to hold net claims on the US government. This is not at all controversial. Everyone operating in financial markets knows this to be true. Federal government debt is pledged as collateral; currency and short-term claims on government are held as “cash”. During the Clinton years, the Fed held conferences, worrying about what they’d do once all the Federal government debt was retired (Clinton predicted 15 years—which would be RIGHT ABOUT NOW, FOLKS). Markets were extremely concerned about this—what would they hold. This helped to spur the securitization of mortgages. The rest, as they say, is history (a sorry one, at that).

To Close…..

Note 1:Financial Assets and Liabilities, Net Financial Assets, Net Wealth, and NonFinancial Wealth (Explication from the Modern Money Theory primer):

One’s financial asset is another’s financial liability. It is a fundamental principle of accounting that for every financial asset there is an equal and offsetting financial liability. The checking deposit (also called a demand deposit or a sight deposit) is a household’s financial asset, offset by the bank’s liability (or IOU). A government or corporate bond is a household asset, but represents a liability of the issuer (either the government or the corporation). The household has some liabilities, too, including student loans, a home mortgage, or a car loan. These are held as assets by the creditor, which could be a bank or any of a number of types of financial institutions such as pension funds, hedge funds, or insurance companies. A household’s net financial wealth is equal to the sum of all its financial assets (equal to its financial wealth) less the sum of its financial liabilities (all of the money-denominated IOUs it issued). If that is positive, it has positive net financial wealth.

Inside wealth versus outside wealth. It is often useful to distinguish among different types of sectors in the economy. The most basic distinction is between the public sector (including all levels of government) and the private sector (including households and firms). If we were to take all of the privately issued financial assets and liabilities, it is a matter of logic that the sum of financial assets must equal the sum of financial liabilities. In other words, net financial wealth would have to be zero if we consider only private sector IOUs. This is sometimes called “inside wealth” because it is “inside” the private sector. In order for the private sector to accumulate net financial wealth, it must be in the form of “outside wealth”, that is, financial claims on another sector. Given our basic division between the public sector and the private sector, the outside financial wealth takes the form of government IOUs. The private sector holds government currency (including coins and paper currency) as well as the full range of government bonds (short-term bills, longer maturity bonds) as net financial assets, a portion of its positive net wealth.

A note on nonfinancial wealth (real assets). One’s financial asset is necessarily offset by another’s financial liability. In the aggregate, net financial wealth must equal zero. However, real assets represent one’s wealth that is not offset by another’s liability, hence at the aggregate level net wealth equals the value of real (nonfinancial) assets. To be clear, you might have purchased an automobile by going into debt. Your financial liability (your car loan) is offset by the financial asset held by the auto loan company. Since those net to zero, what remains is the value of the real asset – the car. In most of the discussion that follows we will be concerned with financial assets and liabilities, but will keep in the back of our minds that the value of real assets provides net wealth at both the individual level and at the aggregate level. Once we subtract all financial liabilities from total assets (real and financial) we are left with nonfinancial (real) assets, or aggregate net worth.

Net private financial wealth equals public debt. Flows (of income or spending) accumulate to stocks. The private sector accumulation of net financial assets over the course of a year is made possible only because its spending is less than its income over that same period. In other words, it has been saving, enabling it to accumulate a stock of wealth in the form of financial assets. In our simple example with only a public sector and a private sector, these financial assets are government liabilities – government currency and government bonds. These government IOUs, in turn, can be accumulated only when the government spends more than it receives in the form of tax revenue. This is a government deficit, which is the flow of government spending less the flow of government tax revenue measured in the money of account over a given period (usually a year). This deficit accumulates to a stock of government debt – equal to the private sector’s accumulation of financial wealth over the same period.

A complete explanation of the process of government spending and taxing will be provided later. What is necessary to understand at this point is that the net financial assets held by the private sector are exactly equal to the net financial liabilities issued by the government in our two-sector example. If the government always runs a balanced budget, with its spending always equal to its tax revenue, the private sector’s net financial wealth will be zero. If the government runs continuous budget surpluses (spending is less than tax receipts), the private sector’s net financial wealth must be negative. In other words, the private sector will be indebted to the public sector.

Deficitsin one sector create the surpluses of another. Earlier we showed that the deficits of one sector are by identity equal to the sum of the surplus balances of the other sector(s). If we divide the economy into three sectors (domestic private sector, domestic government sector, and foreign sector), then if one sector runs a deficit at least one other must run a surplus. Just as in the case of our analysis of individual balances, it “takes two to tango” in the sense that one sector cannot run a deficit if no other sector will run a surplus. Equivalently, we can say that one sector cannot issue debt if no other sector is willing to accumulate the debt instruments.

Of course, much of the debt issued within a sector will be held by others in the same sector. For example, if we look at the finances of the private domestic sector we will find that most business debt is held by domestic firms and households. In the terminology we introduced earlier, this is “inside debt” of those firms and households that run budget deficits held as “inside wealth” by those households and firms that run budget surpluses. However, if the domestic private sector taken as a whole spends more than its income, it must issue “outside debt” held as “outside wealth” by at least one of the other two sectors (domestic government sector and foreign sector). Because the initiating cause of a budget deficit is a desire to spend more than income, the causation mostly goes from deficits to surpluses and from debt to net financial wealth. While we recognize that no sector can run a deficit unless another wants to run a surplus, this is not usually a problem because there is a propensity to net save financial assets. That is to say, there is a desire to accumulate financial wealth – which by definition is somebody’s liability.

Note 2: Further Fun with Identities

Begin with the closed economy and no government, where

S = I.

Since S=I we can add S to one side and I to the other:

S + I = I + S.

Add an H and an F to each side:

F+ H + S + I = I + S + H + F

Finally, rearrange:

F+I+S+H = H + S + I + F.

Give me an F!

Give me an I!

Give me an S!

Give me an H!

What’s that spell!




OK, you won’t understand that unless you were around in the days of the Summer of Love and Woodstock Nation.


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