I’ve mentioned before, I think a useful frame to think about the secular stagnation debate through is what’s become known as Harrod’s growth model. [1] My presentation here is a bit different from his.

Start with the familiar equation:

*S – I + T – G = X – M*

Private savings minus private investment, plus taxes minus government spending, equal exports minus imports. [2] If the variables refer to the actual, realized values, then this is an accounting identity, always true by definition. Anything that is produced must be purchased by someone, for purposes of consumption, investment, export or provision of public services. (Unsold goods in a warehouse are a form of investment.) If the variables refer to expected or intended values, which is how Harrod used them, then it is not an identity but an equilibrium condition. It describes the condition under which businesses will be “satisfied that they have produced neither more nor less than the right amount.”

The next step is to rearrange the equation as *S – (G – T) – (X – M) = I*. We will combine the government and external balances into *A = (G – T) + (X – M)*. Now divide through by *Y*, writing *s = S/Y* and *a = A/Y.* This gives us:

*s – a = I/Y*

Private savings net of government and foreign borrowing, must equal private investment. Next, we decompose investment. Logically, investment must either provide the new capital goods required for a higher level of output, or replace worn-out or obsolete capital goods, or be a shift toward a more capital-intensive production technique. [3] So we write:

*s – a = gk + dk + delta-k*

where *g* is the growth rate of the economy, *k* is the current capital-output ratio, *d* is the depreciation rate (incorporating obsolescence as well as physical wearing-out) and *delta-k* is the change in the capital-output ratio.

What happens if this doesn’t hold? Realized net savings and investment are always equal. So if desired savings and desired investment are different, that means that somebody’s expectations were not fulfilled. For a situation to arise in which desired net savings are greater than desired investment, either people must have saved less than they wish they had in retrospect, or businesses must have investment more than they wish they had in retrospect. Either way, expenditure in the next period will fall.

What prevents output from falling to zero, in this case? Remember, some consumption is linked to current income, but some is not. This means that when income falls, consumption falls less than proportionately. Which is equivalent to saying that when income falls, there is also a fall in the fraction of income that is saved. In other words, if the marginal propensity to save out of income is less than one, then *s* — which, remember, is *average* saving rate — must be a positive function of the current level of output. So the fall in output resulting from a situation in which *s > I/Y* will eventually cause *s* to fall sufficiently to bring desired saving into equality with desired investment. The more sensitive is consumption to current income, the larger the fall in income required; if investment is also sensitive to current income, then a still larger fall in income will be required. (If investment is more sensitive than saving to current income, this adjustment process will not work and the decline in output will continue until investment reaches zero.) This is simply the logic of the Keynesian multiplier.

In addition to current income, saving is also a function of the profit rate. Saving is higher out of profits than out of wages, partly because profit recipients are typically richer than wage-earners, but also because are large fraction of profits remain within the business sector and are not available for consumption. [4] Finally, saving is usually assumed to be a function of the interest rate. The desired capital output ratio may also be a function of the interest rate. All the variables are of course also subject to longer term social, technological and economic influences.

So we write

*s(u, i, p) – a = gk + dk + delta-k(i, p)*

where *u* is the utilization rate (i.e. current output relative to some measure of trend or potential), *i* is some appropriate interest rate, and *p* is the profit share. *s* is a positive function of utilization, interest rates and the profit share, and *delta-k* is a negative function of the interest rate and a positive function of the profit rate. Since the profit share and interest rate are normally positive functions of the current level of output, their effects on savings are stabilizing — they reduce the degree to which output must adjust to maintain equality of desired net savings equal and investment. The effect of interest rates on investment is also stabilizing, while the effect of the profit share on investment (as well as any direct effect of utilization on investment, which we are not considering here) are destabilizing.

How does this help make sense of secular stagnation?

In modern consensus macroeconomics, it is implicitly assumed that savings and/or investment are sufficiently sensitive to interest rates that equilibrium can be normally be maintained entirely by changes in interest rates, with only short-term adjustments of output while interest rates move to the correct level. The secular stagnation idea — in both its current and original 1940s edition, as well as the precursor ideas about underconsumption going back to at least J. A. Hobson — is that at some point interest rate adjustment may no longer be able to play this role. In that case, desired investment will not equal desired saving at full employment, so there will be a persistent output gap.

There are a number of reasons that *s – a* might rise over time. As countries grow richer, the propensity to consume may fall simply because people’s people’s desires for goods and services are finite. This was what Keynes and Alvin Hansen (who coined the term “secular stagnation”) believed. Desired saving may also rise as a result of an upward redistribution of income, or a shift from wage income to profit income, or an increase in the share of profits retained by firms. [5] Unlike the progressive satiation of consumption demand, these three factors could in principle just as easily evolve in the other direction. Finally, government deficits or net exports might decline — but again, they might also increase.

On the right side of the equation, growth may fall for exogenous reasons, slowing population growth being the most obvious. This factor has been emphasized in recent discussions. Depreciation is hardly mentioned in today’s secular stagnation debate, but it is prominent in the parallel discussion of underconsumption in the Marxist tradition. The important point here is to remember that depreciation refers not only to the physical wearing-out or using-up of capital goods, but also to capital goods displaced by competition or obsolescence. In competitive capitalism, businesses invest not only to increase aggregate capacity, but to win market share from each other. Much of depreciation represents capital that goes out of use not because it has ceased to be physically productive, but because it is attached to businesses that have lost out in the competitive struggle. Under conditions of monopoly, the struggle over market share is suppressed, so effective depreciation rates, and hence desired investment, will be lower. Physical depreciation does also exist, and will change as the production technology changes. If there is a secular tendency toward longer-lived means of production, that will pull down desired investment. As for *delta-k*, it is clearly the case that the process of industrialization involves a large upward shift in the capital-output ratio. But it’s hard to imagine it continuing to rise indefinitely; there are reasons (like the shift toward services) to think it might reach a peak and then decline.

So for secular, long-term trends tending to raise desired saving relative to desired investment we have: (1) the progressive satiation of consumption demand; (2) slowing population growth; (3) increasing monopoly power; and (4) the end of the industrialization process. Factors that might either raise or lower desired savings relative to investment are: (5) changes in the profit share; (6) changes in the fraction of profits retained in the business sector; (7) changes in the distribution of income; (8) changes in net exports; (9) changes in government deficits; and (10) changes in the physical longevity of capital goods. Finally, there are factors that will tend to raise desired investment relative to desired saving. The include: (11) consumption as status competition (this may offset or even reverse the effect of greater inequality on consumption); (12) social protections (public pensions, etc.) that reduce the need for precautionary and lifecycle saving; (13) easier access to credit, for consumption and/or investment; and (14) major technological changes that render existing capital goods obsolete, increasing the effective depreciation rate. These final four factors will offset any tendency toward secular stagnation.

It’s a long list, but I think it’s close comprehensive. Different versions of the stagnation story emphasize various of these factors, and their relative importance has varied in different times and places. I don’t think there is any *a priori* basis for saying that any of them are more or less important in general.

One problem with this conversation, from my point of view, is that people have a tendency to pick out a couple items from this list as *the* story, without considering the whole question systematically. For instance, there’s a very popular story in left Keynesian circles that makes it all about (7), offset for a while by (13) and perhaps (11). I don’t doubt that greater income inequality has increased desired private saving. It may be that this is the main factor at work here. But people should not be confidently asserting it is before clearly posing the question and analyzing the full range of possible answers.

In a future post we will think about how to assess the relative importance of these factors empirically.

*g*: sometimes it is the change in output from one period to the next, while at other times it is the normal or usual change in output expected by business. Furthermore, as Joan Robinson pointed out, his famous knife-edge results depend on using the average savings rate as a parameter, which only makes sense if we are describing a long-run equilibrium. In the short period, it’s the marginal savings rate that is stable, while the average savings rate varies with output. So while it is true that Harrod thought he was writing about economic dynamics, the model he actually wrote is inconsistent. One way to resolve this inconsistency is to treat it as a model of equilibrium long-run growth, as Samuelson did; the other way, which I take here, is to treat it as a Keynesian short-run model in which the current, usual or expected growth rate appears as a parameter.

EDIT: I think I’ve been misled by reading too much of the Keynesian classics from the 1930s and 40s. The dynamic I describe in this post is correct for that period, but not quite right for the US economy today. Since 1980, the average private savings rate has moved countercyclically, rather than procyclically as it did formerly and as I suggest here. So the mechanism that prevents booms and downturns from continuing indefinitely is no longer — as Keynes said, and I unthinkingly repeated — the behavior of private savings, but rather of the government and external balances. I can’t remember seeing anything written about this fundamental change in business cycle dynamics, which is a bit surprising, but it’s unambiguous in the data.

Fortunately we are interested here in longer term changes rather than cyclical dynamics, so the main argument of this post and the sequel shouldn’t be too badly undermined.

EDIT 2: Of course this change has been written about, what was I thinking. For example, Andrew Glyn, Capitalism Unleashed:

From Marx to Keynes at least, consumption was viewed as an essentially passive component of the growth process. Capital accumulation, investment spending on machinery and buildings, was the essential driving force on the demand as well as on the supply side. It was the capitalists’ access to finance which allowed capital spending to exceed the previous period’s savings and fuelled the expansion of demand; future profits ensured that such borrowing was repaid with a real return. Deficit spending by the government could, in wartime for example, impart a similar impulse to demand, at least till capital markets took fright at the growing debt interest burden and worries about inflation. However household consumption, some two-thirds of aggregate demand, was seen as playing the role of sustaining the current output level rather than driving it up.

Savings ratios often fell during recessions, as consumers attempted to maintain spending in the face of falling incomes.Indeed, Milton Friedman criticized the Keynesians for exaggerating the dependence of consumption on current income and ignoring the extent to which savings could be used to ‘smooth’ out the path of consumption.More recently, rather than acting as a stabilizing influence, sharp falls in the savings ratio have occurred during expansions.By boosting consumption proportionately more than the rise in incomes this has intensified upswings, with the danger of sharp falls in demand if savings rebound sharply when the expansion slackens and pessimism builds up.

Great post.

"Unsold goods in a warehouse are a form of investment."

But what about goods and services that cost money to produce but cannot be warehoused such as tables at a restaurant, seats on a flight, designs for a product, hours willing to work and so on. How are they accounted for? Also, how does this deal with million dollar cans of investment grade sardines?

what about goods and services that cost money to produce but cannot be warehoused such as tables at a restaurant, seats on a flight, designs for a productIn these cases, goods that are not purchased are not produced at all.

"Start with the familiar equation:

S – I + T – G = X – M"

This seems like a safe place to start but it is NOT safe. The terms S and I include financial property which is subject to spontaneous creation and destruction. The source of the spontaneous creation and destruction is held within the terms T and G.

If the equation was simply about physical production, hours worked, and some measure of output, the equation would hold as you suggest. The problem is that the terms of the equation are all measured in money; money is created with lending or simply printing.

Here is a scenario where money increases but goods do not increase: Government gives all employees a nice wage increase, finding the money by borrowing from it's own central bank. All employees, being good savers, simply buy government bonds which are for sale from the central bank. This happens year after year.

Under this scenario, investment and savings increase each year. Unfortunately, there is no real physical improvement because the only thing that is happening is that the monetary measure is increasing, not true physical improvement.

So, we should agree, this equation should only be correct if G EQUALS T so that there is no change in the monetary measure.

@Roger

"All employees, being good savers, simply buy government bonds which are for sale from the central bank. This happens year after year."

That's an ex-post identity. Assuming (for simplicity) balanced trade, then the bond issuance (where the employees direct their "extra" wage) equals to the penny the difference G – T which equals to the penny S – I.

I don't see how you can conclude that investment rises though.Only S rises in this case.

@Crossover

Yes, only S rises in my comment scenario.

The term I is PRIVATE Investment. Government investment is already included in the G term. The S term in the base relationship (C + S + T = GDP = C + I + G + (X – M) therefore is correctly composed of both private investment and the "investment" resulting from the difference between G and T.

A commonly made error is to go on to say that (I – S) + (G – T) + (X – M) = 0 as if it were true for all values (when it is only true when G – T = 0).

@Roger

Not sure I'm following.

This identity accouts for nominal values.

You claim that if the government increases wages (and thus increase G) and the employees purchase bonds with their additional income then somehow S – I will not equal G – T, assuming a balanced trade or S – I will not equal (G – T) + (X – M) assuming non balanced trade.

I'm wondering how is that possible ? As G rises due to the increase in wages so will S rise due to the increase in private savings.

Crossover is correct.

Thank you for pursuing this discussion.

I believe that GDP is a measure of transactions, each transaction to be a trade of labor or materials for monetary property. One party gives money, receives goods or labor; the second party receives money, gives goods or labor. It is intended to NOT count purely financial transactions such as debt exchanges.

When we count I with G, we are looking at the private investment SPENT and government investment SPENT. This is the SPENT side of the equation.

Next we look at the receive side and see that term S is Savings. Hmmm. S is on the side with T so the receive side is counting the Tax received by government and the money received by the I exchange. Therefore, S less I is G – T which is the government debt.

But we just argued that GDP does not count purely financial transactions! Government Debt is a purely financial transaction!

My argument here is that we have mixed apples and oranges. We have mixed debt into the GDP tabulation! No longer is GDP only measuring exchanges between parties. It is also measuring an increase in money supply.

An alternative argument could be made (and I think this is your argument) that the I term is still exchanging property for property (not debt for property) but government has simply created new money (thereby creating new property) which is then RECEIVED identical to

the RECEIVED side of I. With this nuance added to our definitions, term S is not debt exchanged for goods or labor, but property for property, identical to all the other terms in the equation.

I can accept this. At this juncture, the equation is correct. (I – S) + (G – T) + (X – M) = 0 is correct for all values. The only addition is that government is acknowledged as having created monetary property when G exceeds T.

@Roger

You seem to mix government debt with Government Spending.G increases because government spending increases.

Let me give an example.

Government increases wages resulting in an increase in gvt spending by $100.

Employees do not buy bonds with the additional $100, instead they consume.The producers that offered their goods/services in exchange buy bonds thus S increases.

Again S increased as G increased and given a closed economy then G – T = S – I

Assuming that real output increased by $100 because the producers actually produced more output due to increased demand then nominal and real output grew together.

Turning to your own example:

Government increases wages to employees by $100 however the employees themselves do not increase their actual output and additionally they don't consume the extra $100 but use it to buy gvt. bonds.

G is up by 100. S is up by 100 and ceteris paribus real output stays constant but NOMINAL output has increased.

Again in a closed economy G – T = S – I.

@Crossover

Again, thanks for pursuing this discussion.

Here is another way of presenting my concern: Mathematically speaking, if we have two terms of a three term identity in an equation, then the final third term lays hidden in the equation.

In the base equation (C + S + T = GDP = C + I + G + (X – M), we find two terms of the government identity "G – T = change in Government Debt". Government Debt is, by mathematically necessity, incorporated into this base equation.

We can therefore simplify the base equation to C + S = GDP = C + I + Government Debt + (X – M). At this point we can further simplify the base equation to write S – I = Government Debt = G – T, (assuming that X – M equals zero) which is identical to what you wrote.

So we agree.

Now it bothers me that the units of S – I are Government Debt because financial transactions are not supposed to be part of GDP. But clearly, financial transactions ARE part of GDP — we just proved it mathematically.

The only way out of this dimensional dilemma (that I see) is to allow that government, when it issues debt, is creating financial property. This is new property identical to the property being traded by other economy participants.

@Roger

"Now it bothers me that the units of S – I are Government Debt because financial transactions are not supposed to be part of GDP."

I don't understand where in the equation you see government debt being added to the GDP.

GDP = C + G + I + NX

The increase in NOMINAL GDP happens beacuse G increases.G as you also correctly noted aobve includes Government spending and government investment.

In the case of an increase in public employee wages its the "government spending" component of G that increases.Still the wages count towards the gdp since essentially the government pays employees in exchange for their services.

The fact that a rise in employee wages without an equal rise in their output happens, means that only nominal GDP rises.

Correct me if I am wrong but I have a suspicion that you consider S as a part of GDP.

Howevere GDP is C + G + I + NX

S derives from the uses of income: S + C + T.

The whole equation stands true because Spending = Income

Thus if government increases its spending all other things equal someone's income will by definition increase.

In this case it's the employees income that increases and since they choose to save this income instead of consuming it the S component of the income equation increases.

Btw the government does create financial property.I would call it Net Financial Assets for the private sector in the form of t-bills and bonds.But the purchase of the bonds from the savers is a transaction that is not added to the gdp.

And this is pretty evident: If government increased wages by $100, all other things equal

newGDP = OldGDP + 100.

If the purchase of debt was accounted for in the gdp then newGDP = OldGDP + 100 + 100 which is obviously not true.

I also see a flaw in this:

We can therefore simplify the base equation to C + S = GDP = C + I + Government Debt + (X – M)

You say "C + S = GDP = C + I + Gov.Debt + NX

However C + S is not equal to GDP.C + S +T is because Spending equals Income

Therfore the correct way to put is : C + S + T = GDP =>

C + S = GDP – T =>

C + S = (C + G + I + NX ) – T.

If you want to convert G – T = gov debt then

C + S = Gvt. Debt + C + I + NX

What this equation says is that Income spent on consumption plus income saved equals the change in gvt debt plus spending on consumption plus spending on inveestment plus net exports

if we re-arrange one more time we end up with the usual equation:

S – I = Gvt Debt + NX or S – I = G – T + NX.

I believe this was the point of your confusion, you assumed that C +S = GDP whereas C +S = GDP – T

@Crossover

Yet again, thanks for continuing this discussion.

Yes, you are correct that I error-ed in writing "C + S = GDP = C + I + Government Debt + (X – M)". The entire paragraph should have read "We can therefore simplify the base equation to C + S = C + I + Government Debt + (X – M). At this point we can further simplify the base equation to write S – I = Government Debt = G – T, (assuming that X – M equals zero) which is identical to what you wrote."

As I re-read the entire discussion, perhaps it is only me that is mixing apples and oranges. OF COURSE GDP is reporting the results of all spending including spending enabled by financial exchanges. The fact that G – T may or may not, by itself, "create money" does not alter the fact that additional spending to the amount of G – T has occurred and thereby increased GDP.

We seem to be in agreement that government borrowing DOES increase the quantity of financial property available in the economy.

You have read my comments very carefully, and found error. Thank you for helping me to a more robust grasp of the the relation C + S + T = GDP = C + I + G + (X – M).