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.  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.  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.  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.  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.  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.
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.