As I have argued before: "Does the macroeconomy self-equilibrate?" is a stupid question, because the answer depends on the monetary policy being followed. Your answer also depends on your model of the economy. And that's what I want to look at here.
"Compare and contrast Old Keynesian and New Keynesian views on whether the economy self-equilibrates to full employment" is my exam question. (The definition of "full employment"/"potential output"/"the natural rate of unemployment" may be problematic, but that doesn't matter for this question, as long as our definition isn't completely vacuous.)
What Paul Krugman says isn't wrong. But it's not quite right either. It's easy to explain the Old Keynesian models' answers to this question; but the New Keynesian model's answer is a bit of a mess, even ignoring the ZLB.
Here's my answer to the exam question:
1. Old Keynesian Income-Expenditure model. This is the first macro model I learned in school in the 1970's. It's still taught as the macro model in many first year university texts. It's the one with desired real Aggregate Expenditure AE on the vertical axis, real income Y on the horizontal, an upward-sloping AE curve (because AE depends on Y), a 45 degree line where AE=Y for the "equilibrium condition" (which says only that output=output demanded, and says nothing about the supply-side), and "equilibrium" where the AE curve cuts the 45 degree line.
This model has zero tendency to self-equilibrate at full employment. There is a unique equilibrium level of AE and Y, but there is nothing in the model that says that this equilibrium has any relation whatsoever with anything we might (non-vacuously) call full-employment. That's because the supply-side of the economy doesn't affect that equilibrium at all. If people save too much, or invest too little, equilibrium Y will be below full-employment Y, and will stay there forever. Unless the economy gets lucky, or fiscal policy gets the economy to full employment, it simply won't ever get there by itself. (In this model, monetary policy doesn't even appear.)
2. Old Keynesian ISLM model. If you start with the Old Keynesian Income Expenditure model, assume that investment (and maybe savings too) depends on the rate of interest, and add in the assumption that money supply = money demand, and that money demand depends on Y and the rate of interest, you get the Old Keynesian ISLM model. This model is the main model in most second year university texts, and has been for decades.
This model may or may not have a tendency towards full employment "in the long run". It depends. It depends on the monetary policy being followed. (With a really stupid monetary policy, like where the central bank cuts the money supply in the same proportion as any fall in the price level, there will be zero tendency towards long run full employment.) But it depends on other things too. It depends on the shapes of the IS and LM curves, because it may be that no fall in the price level or increase in the money supply can get the rate of interest low enough to get the economy to full employment. It also depends on the interaction between expected inflation/deflation and actual inflation/deflation. It may be that a fall in the price level causes people to expect further falls in the price level, which increases the wedge between real and nominal interest rates (makes real interest rates higher for any given nominal rate), which reduces demand still further.
The basic mechanism is that if Y is less than full employment the price level will eventually fall, which will increase the real money supply M/P, which will shift the LM curve right, and increase output demanded, and increase output.
The ISLM model answers the question. It's not a simple answer, but it is a clear answer. The answer is "it depends", but the model tells you what it depends on.
3. The New Keynesian model. There are lots of "New Keynesian" models, going all the way back to the late 1970's. They have changed quite a bit over the years. And different New Keynesian models would have answered this question differently. What I am talking about here is the simplest version of the "canonical" New Keynesian model, as taught for the last decade or so to upper-year undergrads or graduate students. This model has replaced ISLM as the new "workhorse" model.
It has three equations. The new "IS curve" is an equation in which the ratio of current demand to expected future demand is a negative function of the real rate of interest. There is an equation for the expectations-augmented Phillips Curve, which both defines "full-employment" and tells you the relationship between deviations of output from full employment and deviations of actual from expected inflation. And there is an equation for the monetary policy reaction function, which tells you how the central bank sets the nominal interest rate, typically as a function of output and inflation. (You can call the assumption of rational expectations a fourth equation, if you wish.)
What I will say next may be a little controversial.
This model has zero tendency towards full employment, even if you assume a sensible monetary policy. New Keynesian modellers just assume a tendency towards "long run" full employment, even though it's not there in the model.
I explained why in my old post. Here's the short version: even if monetary policy is perfect, so the central banks always sets an interest rate that's exactly right, all that tells you is that the ratio of today's demand to tomorrow's demand will be exactly right. But we will only be at full employment today if people expect full employment tomorrow; and they will only expect full employment tomorrow if they expect full employment the day after tomorrow, and so on. A cut in real interest rates does not increase today's demand; it only increases the ratio of today's demand to expected tomorrow's demand. And there are two ways a ratio can increase: a rise in the numerator; or a fall in the denominator. New Keynesian modellers just assume that the people in their model expect a long run tendency towards full employment.
Think back to the ISLM model. In that model, if monetary policy wasn't really stupid, there was normally a force that tended to push the economy eventually towards full employment. (It might be offset by forces pushing the other way, but no matter.) At less than full employment, prices would fall, so the real money supply M/P would rise, shifting the LM curve right, and increasing demand for output.
That equilibrating mechanism in the Old Keynesian ISLM model is simply not there in the canonical New Keynesian model. Because M isn't there in the model. (And P isn't really there in the model either; only the rate of change in P — the inflation rate — is there in the model. And changes in the inflation rate, as opposed to the price level, are normally a disequilibrating force, because expected deflation increases real interest rates for given nominal rates, which in the ISLM model reduces demand.)
4. My reflections on all this:
As a teenager I looked at the Old Keynesian Income-Expenditure model, and thought: "Hmmm. Something's not quite right with this model. Because for most of history, we haven't had fiscal policy being done by sensible governments that understood this model. So how come we ever spend as much time as we do anywhere near full employment?"
I was like a pre-Darwinian biologist being taught The Argument From Design, and that God had only existed since the 1940's. It didn't make sense.
Nowadays, looking at the canonical New Keynesian model, I have a similar sort of question to the one I had as a teenager. We've only had a Designer who understood the model for the last decade or so, and according to this model even a perfect Designer can't get it right, unless we all have Faith. It doesn't make sense.
Worthwhile Canadian Initiative 
Don’t overlook multiple equilibria. http://www.worldcomplex.blogspot.ca/2012/02/unemployment.html
There’s no reason to suppose that an equilibrium point should be anywhere near full employment. More likely the employment situation tailors itself to the number of jobs available. If millions of new jobs were suddenly created somewhow, I think we would find the candidates to fill them, even if that number were much greater than the number of officially unemployed people.
Even when the labour force participation rate falls back to 1950s levels, after a suitable period of time, unemployment will officially be back to historical levels, as so many will be dropped from the list.
If you look at unemployment since the dawn of time, can you really say it has typically been low? Can you say that,for example, a typical peasant in Leicestershire in 1290 was fully employed? Or in 1620? Why did so many agricultural workers leave Europe in the 19th century? Was it simply because land was available in America, Australia and Africa or perhaps because they would have been underemployed if they stayed in Europe?
Perhaps the wondrous succession of positive technology shocks from the mid 18th century forward has masked some of system instability? Or perhaps the system is so changed that it is completely different an not comparable to what previously existed?
Can you assume that we will always be on the saddle point express train to equilibrium?
The DSGE framework sets up a great benchmark — but it needs to be tested against a variety of data sets and time periods — without imposing paradigmatic assumptions (either way) in the testing process.
“Because for most of history, we haven’t had fiscal policy being done by sensible governments that understood this model”
For most of history, monetary policy was set up by emperors and kings. Emperors and kings had a tendency to spend more than they could, to start endless wars, to finance their debt through seignorage or simply through violence. It worked sometimes, when it didn’t work, it was Zimbabwe all along.
It’s been a long way, but we ended up in the other extreme: central banks are independent and debt financing is a no-go policy, governments are run by people who see Keynes as a communist and, at the same time, they have been kidnapped by financial markets. The positive thing is that, if you are not in currency area with the Germans, a financial collapse could be stopped. Things can’t go Zimbabwe, but what we see is an irreversible economic and social decline.
To Mickeyman:
Interesting observation. If you read the classics, structural unemployment was indeed impossible for a reason: starvation or, more in general, demographic change caused by economic factors. The classics thought that the economic system was in a long-run equilibrium because unemployed people would simply die and/or their sons and daughter dies. Read Chantillon for example. But why did they get unemployed before the Industrial Revolution?
Not because they could not find a job in the labour market, but because they could not sell enough of their product in order to survive. Up to the late 18th century and well into the 19th, 99% of the population was composed of small capital/land owners. Hence, the classics were thinking to the system equilibrating mechanism as affecting the number of producers, surely not wage workers.
The huge transformation came when people were convinced to accept wage jobs. Read Polanyi, for example. People were either convinced or forced by legislation to supply their labour to whom owned big capital equipment (machineries). But this move involved huge risks.
If you have your peace of land and/or your small capital equipment, you starve if you can’t sell enough of your products for a very long period of time. But, even in time of crisis, you still produce something, you have probably a small piece of land which gives you at least some food. You can eventually adapt and start to produce something else. Starvation is in theory possible, but a lot of people have the means to survive the crisis, maybe some of the children died for diseases caused by reduced food or worsened material conditions.
But once you sell your workforce in the labour market and you can’t find a job, you simply die if there is no social safety nets. Even if real wages are much higher than what people could expect to earn as small capital/land owners, wage workers choose a riskier option even if justified in terms of inter-temporal optimization under the assumption of an infinite time horizon, that is, you can’t die whatever you choose!!!!!!!
So, basically, if you want a market economy, you need social safety net and protect workers from starvation in times of crisis. But if you have a social safety net, there is no equilibrating mechanism. Then a huge recession must be fought through expansionary monetary and fiscal policy.
So, not only Keynes was right, but the evolution of society shows that he was right.