The blue line is gross private savings, the red line is gross private investment. Krugman interprets this as showing an increase in people's desire to save relative to their desire to invest:
This huge move into surplus reflects the end of the housing bubble, a sharp rise in household saving, and a slump in business investment due to lack of customers.
Given this reality, it’s not hard to see why massive government borrowing hasn’t led to soaring interest rates; we’re awash in saving with no place to go. And that’s also why we’re in a liquidity trap, in which large increases in the monetary base don’t lead to inflation. ... Unless the confidence fairy arrives, causing households and businesses to suddenly ramp up their spending despite high unemployment and weak sales, deficit reduction will only intensify the problem of excessive savings relative to perceived investment opportunitiesNo, no, no! You cannot do that!
Don't get me wrong, I agree completely with his conclusion here. But this graph does not give it an iota of support. The fact that private saving has increased relative to private investment tells you nothing about what is causing what. It's rather shocking, actually, that Krugman would do this: Anyone familiar with Keynes, or who understand the national accounts, both of which he surely is, should not be making this mistake.
See here's the problem: The national income identity is an accounting identity. It's by definition always true, no matter what, that
S - I = G - T + X - M
Here's another version of Krugman's graph:
|Blue = Gross Private Saving Less Gross Private Investment|
Red = Net Government Saving Less Current Account Balance
The two lines are mirror images of each other because they are showing the same thing.  To say that the blue line went up -- that the private sector is now saving more than it is borrowing -- is exactly the same as saying the red line went down -- that the government is now borrowing more than the rest of the world is lending. But since this relationship always holds by definition, it tells us nothing at all about which of these changes is cause, and which is effect. Anything -- anything at all -- that causes the government budget deficit to increase, or the current account deficit to decrease, will result in an increase in private savings relative to private investment. That's just the way the national accounts work.
Another way of looking at it: If the federal government did balance its budget, then the gap between private savings and private investment would necessarily close. Now Krugman (and I) think it would close because the resulting fall in national income would force households to reduce their savings. A conservative might say that it would be because private investment would increase. But one way or another, in a closed economy, if the government is not borrowing then private saving and private investment must converge since there's no one else for the aggregate private sector to lend to. So when the gap closed -- as, again, it would have to -- that would tell us nothing whatsoever about which of these stories was true.
This is a long post, and not as clear as it should be. But the point is important. An increase in desired savings relative to desired investment (at a given level of income) will always reduce aggregate demand and income. But what change, if any, it leads to in the gap between actual savings and investment depend entirely on what happens to the government and external balances. In a closed economy without government deficits, private savings will always equal private investment, but people's decisions to save more or less, or invest more or less, will still produce fluctuations in income. Understanding this -- that the equality of savings and investment is maintained by adjustments in income -- was, arguably, the key advance made by Keynes in the General Theory. Suggesting that S and I can vary independently is not just wrong, it is, as they say, unacceptable dirty pedagogy. It's a way of making a valid (and important) point that leaves Krugman's readers less able to reason coherently about the economy.
UPDATE: Trying again.
In a closed economy with a balanced government budget, private saving always equals private investment. Keynesians and (neo)classical economists agree on this. We also agree, in principle, that savings and investment are both functions of income (Y) and interest rates (i), and we all think that dS/dY > dI/dY > 0, dI/di < 0 and dS/di > 0. When you speak of a change in investment or savings, you implicitly mean investment and savings at any given income and interest rate, i.e. and upward or downward shift of the whole S(Y,i) or I(Y,i) function. Given a shift in one of these functions, the condition S=I then requires a change in Y and/or i.
The difference between the macro schools is that Keynesians think the equality of I and S is maintained mainly by shifts in Y, because i can't adjust freely since it also needs to equilibrate the asset market. (And also, I and especially S don't respond much to i.) Whereas the (neo)classicals think the equality of I and S is maintained entirely by shifts in i, because Y is fixed on the supply side. (And also, S and especially I don't respond much to Y.) But both of these stories are stories of what keeps S equal to I. Whether S and I in fact diverge, depends entirely on the behavior of the government budget and the current account. In a country where the government adopted a pro-cyclical budget, we would see private investment run ahead of private savings in a recession. But that would't in any way invalidate Keynesian theory or be evidence against the claim that it was an increase in desired savings that caused the recession. On the other hand, in the actual world where government budgets behave countercyclically, that fact in itself will cause private savings to rise relative to private investment in recessions. But even a dyed in the wool Keynesian  like me cannot accept the argument -- the logical implication of Krugman's post -- that the fact that the government is running a deficit is itself sufficient proof that a government deficit is desirable.
Is that any better?
SECOND UPDATE: You know what? I think I'm wrong on this.
I mean, I'm not literally wrong. (That would be embarrassing.) Everything I said here is true in principle. But I've committed the common economist's sin of getting caught up in logic and missing what's actually at stake.
It is true, as I said, that there no reason that a fall in private expenditure MUST lead to an increase in private savings relative to private investment. In principle, it could just as easily have the opposite effect. But in practice it is true, as long as we add some other assumption: (1) Desired private savings responds more strongly to changes in income than does private investment; (2) the government balance has a strong tendency to move toward surplus when incomes rise, and towards deficit when income falls; (3) the trade balance has a strong tendency to move toward deficit when income rises, and toward surplus when income falls; and (4) there are no large exogenous shocks to net exports or the government balance. That's a lot of assumptions, and Krugman doesn't spell any of them out. Well, but it's a blog; the important thing is, they are all good assumptions for the contemporary US. (Tho 4 in particular is definitely not a good assumption for other times and places.) So as a matter of fact using the gap between private savings and private investment to support our (shared) preferred story of the recession, is perfectly reasonable. And he is certainly right to point out that ISLM tells you that such a fall in private demand should be associated with lower market interest rates unless some other source of autonomous demand (i.e. government deficits above and beyond the automatic stabilizers) rises by more.
 Well, the same thing up to the statistical discrepancy. Why the discrepancy was unusually large in the mid-2000s, I don't know.
 Post Keynesian? Structuralist? What should we call ourselves?