To misquote Milton Friedman: macroeconomics has only regressed one derivative since Wicksell.

The old Wicksellian indeterminacy has long been known: the equilibrium price level is indeterminate if the central bank sets an interest rate. The neo-Wicksellian/New Keynesian model has two indeterminacies. I've been worrying about the indeterminacy of the level of output; John Cochrane has been worrying about the indeterminacy of the rate of inflation. New Keynesian macroeconomists have just assumed them away, without us realising they were doing so. This is a major problem.

This is hard. I've been trying to get my own head straight on this, and trying to explain it clearly and simply. Bear with me.

Define P* as what the price level would be, Y* as what real output would be, and r* as what the real interest rate would be, if all prices were perfectly flexible and there were no nominal rigidities.

Monetarist models, and Old Keynesian models like ISLM if the central bank targets a nominal variable like the money supply, have a well-defined P* and Y* and r*. If prices are sticky, and the money supply is too low, actual P will be above P* and actual Y will be below Y*. We get a recession because actual P only falls slowly to P*. (Keynesians would say that Y<Y* means r>r*, but monetarists might disagree.)

The old Wicksellian indeterminacy was that P* was indeterminate if the central bank targets a rate of interest rather than a nominal variable like the money supply. If the central bank sets the rate of interest too low, so r<r*, P* is infinite. If the central bank sets the rate of interest too high, so r>r*, P* is zero. If the central bank sets the rate of interest just right, so r=r*, P* can be anything whatsoever, but Y=Y* and Y* is well-defined.

Now define y as the growth rate in Y, and p (the inflation rate) as the growth rate in P. Define y* and p* similarly.

In an Old Keynesian model, with an Old keynesian IS curve, Y-Y* is a negative function of r-r*. If the central bank sets the interest rate too high, the level of output will be too low. We get a recession.

In a New Keynesian model, with a New Keynesian IS curve, y-y* is a positive function of r-r*. If the central bank sets the interest rate too high, the growth rate of output will be too high. The level of output is not determined by r. Even if the central bank always sets r=r*, all we know is that output is growing at the frictionless equilibrium rate y*, but the level of output, and Y-Y*, could be anything whatever. New Keynesian modellers "solve" this problem by just assuming that Y(t) asymptotes towards Y*(t) as time goes to infinity, even though there is nothing in the model that says it should. In other words, they use the solution from monetarist or Old Keynesian models to pick one of the infinite number of solutions from the New Keynesian model. But those are different models.

That's the Neo-Wicksellian/New Keynesian indeterminacy that I have been complaining about in several recent posts.

John Cochrane has been complaining about a different Neo-Wicksellian/New Keynesian indeterminacy. I think he's right. Let me try to explain it simply.

If P* were well-defined, a sensible Phillips Curve would say that if P* made any sudden unexpected moves, P would get left behind at first, but would eventually catch up to P*, whatever P* was doing, as long as people expected it. Which would mean that Y would be less than Y* if P* were below what people had forecast, but Y would eventually asymptote to Y* "in the long run", which means when expectations are correct.

If you take that sort of Phillips Curve, and match it with some old Keynesian ISLM or monetarist model, assume there's a negative AD shock, or the central bank unexpectedly tightens monetary policy, so that P* falls unexpectedly, we get falling inflation and a recession, until P eventually catches up with P*, and so Y eventually returns to Y*. Unless monetary policy is really stupid, and moves P* away from P faster than P can catch up.

I think that's the sort of model that most of us macroeconomists have at the back of our minds.

But that's not the Phillips Curve in the New Keynesian model. Since P* is either zero, infinite, or indeterminate in the New Keynesian model, you cannot use P* in the New Keynesian Phillips Curve. You can only use Y and Y*. Instead, the New Keynesian Calvo Phillips curve says that the rate of inflation is a positive function of the expected present value of the output gap Y-Y*.

One well-known weird feature of this New Keynesian Calvo Phillips Curve is that a declining actual-and-expected rate of inflation is associated with a boom where Y>Y*.

What John Cochrane is saying is that inflation is indeterminate in the Neo-Wicksellian/New Keynesian model, even if you just assume that the output gap asymptotes to zero.

He gives an example. Suppose initially the economy has r=r* and Y=Y* and p=0%. Then suddenly and unexpectedly r* drops to minus 5%, and people know it will stay there for 5 years, then return to where it was before. And the central bank cannot set a negative nominal interest rate because of the Zero Lower Bound. When the shock hits, the central bank sets the nominal interest rate at 0% for 5 years, then sets it at r* again. What happens?

The standard answer simply assumes that Y=Y* and p=0% and r=r* when the 5 years are up, and solves backwards from there. Which means both p and Y jump down and r jumps up when the bad news hits. We have deflation and a recession for 5 years, with that deflation and recession slowly ending as we approach the end of the 5 years.

John Cochrane proposes an alternative solution to the same set of New Keynesian equations. He simply assumes that Y does not jump when the bad news hits, and solves forwards from there. If Y does not jump when the shock hits, that means that inflation must jump to 5% immediately, so that real interest rate can be minus 5% when the nominal rate is 0%. But the inflation rate can only jump to 5% if the expected present value of the output gap jumps too. Hitting the ZLB causes a boom, as Y slowly rises, then slowly falls, and only asymptotes to Y* well after the 5 years are past. And inflation slowly falls, and only asymptotes to zero well after the 5 years are past.

John Cochrane is not (as I read him) saying his solution is the right one. He is saying it is no less right than the standard solution.

Both those solutions are consistent with the IS curve and Phillips curve of the New Keynesian model. Both are mathematically correct. Neither of these solutions is "pathological" in the sense of causing inflation to go to plus or minus infinity. Both solutions would be equally stable or unstable in the sense of staying or not staying on that path if the central bank threatened to respond if they strayed from the equilibrium path. And those are just two solutions from an infinite number of solutions. And there is nothing in the model itself that tells us which of those solutions is the "right" one. There are multiple equilibria.

Now there are some models with multiple equilibria where the modeller knew there were multiple equilibria right from the start, and proudly called the readers' attention to the fact that the model had multiple equilibria, because he thought it was an important and desirable feature of the model that reflected something about the world. The Diamond-Dybvig model of bank runs (which has two equilibria, one with a bank run and one without) is like that.

But the New Keynesian model is not like that. Nobody said "Hey look! I've just built this New Keynesian model which is a really neat model because it has an infinite number of equilibria, and so it can explain why sometimes we get recessions and sometimes we get booms, and those recessions and booms just happen; they aren't caused by anything at all, except animal spirits, or sunspots! And you can get totally different responses to exactly the same shocks and exactly the same monetary policy responses to those shocks, just because!" The New Keynesian model was never taught that way. It was taught as saying that recessions and booms and inflation and deflation were caused by bad monetary policy which didn't or couldn't respond to shocks correctly and quickly enough.

How should we respond to all this? Here are some options:

1. Try to argue that the standard solution is in some sense the "right" solution, and that all other solutions are in some sense "wrong". And to my intuition, the standard solution does feel more "right". But then my intuition (and I think others' too) is coming from a very different implicit model, which does have a well-defined P*, where P will approach P* and Y will approach Y* in the long run. I think John Cochrane is right when he says that New Keynesian economists are unwittingly using their Old Keynesian intuitions to pick one of the many equilibria as being the "right" equilibrium and sweeping all the other equilibria under the rug. But there is something deeply inconsistent about using an implicit monetarist or Old Keynesian model to choose an equilibrium in an explicit New Keynesian model, where those models are totally different.

2. Maybe use learning models to show that just one equilibrium is learnable and all the others aren't? Dunno.

3. Try to argue that the standard solution is empirically better than the other solutions. We did in fact get a recession and not a boom when central banks hit the ZLB, therefore the standard solution must be the right one. Maybe, but I'm not comfortable with this. Models are supposed to explain the world; not vice versa. And maybe we just got unlucky this time around, and there are other times we got lucky. And what we thought were the macroeconomic consequences of shocks and bad monetary policy were really just the macroeconomic consequences of animal spirits and sunspots.

4. Embrace the multiplicity of equilibria as a neat feature of the model which is telling us something important about the real world. There are multiple equilibria out there, and it's part of the job of the central bank to help us all coordinate on the best of those equilibria, not just by setting an interest rate, because that alone won't work, but by using "open mouth" policies like the conductor of an orchestra or coxswain in a racing eight, so we can all converge on a good Schelling focal point in a pure coordination game. Simply announce a desired path for (say) NGDP then sit down again. Hope that if you say it clearly enough and loudly enough everyone will believe it simply because they expect everyone else will believe it. Maybe.

5. Change the New Keynesian Calvo Phillips curve to some other equation where inflation cannot jump, and get a variant on the New Keynesian model which resolves the indeterminacy of inflation rates. Maybe. But sometimes inflation does seem to jump. And even if you resolved the inflation indeterminacy that way, the indeterminacy of real output is still unresolved.

6. Throw away the model. Put money back into the model explicitly. And throw away any policy that uses interest rates. Use the central bank's control over its own monetary liabilities to target some nominal variable like P or NGDP. So P* is determinate, non-zero, and finite. Destroy the original sin of old Wicksellian indeterminacy, and you get rid of both Neo-Wicksellian indeterminacies.

(7. And no, "fiscal policy" does not resolve this problem with the New Keynesian model. Any fiscal policy will still have an infinite set of equilibria associated with it, just a different infinite set of equilibria for different fiscal policies.)