### John Maynard Keynes and the Chamber of Secrets

9/27/2013 08:00:00 AM
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 Those aren't snake heads. They're multiple equilibria.
John Cochrane is out with a new paper which I can only describe as unlocking a chamber of secrets in the New Keynesian model. As in the famed children's novel, its not really that no one knew the chamber existed, but that wasn't something you admitted in front of the grad students.

What does this chamber contain? Extra equilibria! In particular, a hideous extra equilibrium-monster intent on ridding the economics profession of all the mud-blood Keynesians.

Ok, actually, there's more equilibria hiding in the chamber than anyone ever wanted. And they don't necessarily overturn the standard New Keynesian analysis, but they can't be totally dismissed either. Nick Rowe has a better explanation of the extra equilibria here than I could possibly give you.

The monster in this chamber isn't what I expected when I first read Cochrane's blog post here. Several papers had used non-linearized versions of the New Keynesian model to examine multiple equilibria, some of which have reversed all the comparative statics relative to the "standard" liquidity trap model. I expected Cochrane's paper to be more of this--it's easy to make non-linearized models produce multiple equilibria--but that's not what Cochrane did.

When you linearize the New Keynesian model you end up with an almost linear program. It's almost linear because the zero-lower-bound is an important non-linearity in the system. But you can otherwise solve it like a linear program. There are three possibilities: either there are no solutions (something we economists don't like to talk about), exactly one solution, or a continuum of infinitely many solutions indexed by one or more indeterminate variables. As Rowe notes, what Cochrane did was point out that inflation is an indeterminate variable, and then pick an equilibrium with the opposite policy implications from the typical New Keynesian liquidity trap.

Now, this all sounds very dramatic. Many will think that Cochrane has found a counterpoint to those liberal New Keynesians. But, I don't think so. And if you think about the liquidity trap for a while (a long while, as Nick Rowe and I have found out), Cochrane's equilibrium-monster is both obvious and irrelevant.

Here's the essence of the story. In a liquidity trap, there is a negative "natural interest rate," which is to say that the real interest rate--we will call it $r_t$--has to be negative in order for the economy to be at full employment. So the natural interest rate during the liquidity trap is, say, $$-r<0$$ but there is a zero lower bound on the nominal interest rate $$i_t\geq 0$$ for all $t$. If you consider the fisher equation of interest rates, which is true in all New Keynesian models, then the nominal interest rate is the sum of the real interest rate and inflation: $$i_t=r_t+\pi_t$$ where $\pi_t$ is inflation. Plugging in our liquidity trap conditions with a zero nominal rate and negative natural rate gives $$0=-r+\pi_t.$$ This condition can only be true if $$\pi_t=r.$$ That is, we can have a negative real interest rate with a zero nominal interest rate if prices are rising--this result arises because prices and interest rates are redundant at the macro level, something I've written about before. Now, if $\pi_t<r$ then what we have is $r_t>-r$, implying that the real interest rate is higher than the one at which we would have full employment. The higher interest rate causes people to attempt to save too much and spend too little relative to what the economy is capable of producing, so we have "unemployment" of resources, including labor.

So, in all these New Keynesian liquidity trap models, we have some version of: negative natural interest rate strikes in period $t=0$ and returns to normal in period $t=T$. What Cochrane has done is simply asked "What if $\pi_o=r$?" There's no reason why that can't be the case. There's also no reason why it would be the case. Cochrane is not saying, necessarily, that the arrival of a negative natural rate caused inflation to spike to $r$, nor is he saying that the central bank or any policy makers possess the means to cause inflation to be $\pi_o=r$. As in Krugmanite/Woodfordian models, monetary policy is ineffective at the zero-lower-bound, and the Fed can only influence inflation through forward guidance. Unlike Krugman/Woodford, inflation is already high enough to ensure greater-than-potential output at the zero-lower-bound.

Let's look at Krugman/Woodford policy recommendations. Basically they say that in a liquidity trap the only way to lift output is to raise inflation, and the only way monetary policy can do that is to credibly commit to higher future inflation once the liquidity trap is ended. Now look at the transition dynamics of Cochrane's equilibrium-monster:
the $t=5$ on the horizontal axis is where the liquidity trap ends in this graph. Notice that the red line, representing the path of inflation, remains elevated for several periods following the end of the liquidity trap. Basically, that's exactly the forward guidance Krugman and Woodford are looking for--a promise to retain elevated inflation for a while after the liquidity trap ends. So while New Keynesians have been clamoring for more inflation to eliminate the output gap, Cochrane's equilibrium says "this is the inflation you're looking for" (note, Cochrane is not arguing this is the way it is, only that this could be the way it is).

So why am I confident we are in a Krugman/Woodford equilibrium and not a Cochrane equilibrium-monster? That's easy. Cochrane's equilibrium says 1) there's no output gap (actually, that we're producing too much), and 2) inflation is greater than normal. These are empirically false. We had a large contraction of output despite the fact nothing happened to our technological capacity. We also had a large decline in inflation, not an increase. These are sufficient facts to rule out Cochrane's equilibrium.

The Chamber of Secrets has been opened. But the monster was already dead.
Nick Rowe 9/27/2013 08:24:00 AM
But *why* did we not go to Cochrane's equilibrium? (Or to one of many other equilibria?)

Was it that central banks did a really bad job of open mouth policy, and told us we would go to the bad equilibria, so we all went there?

Was it that we all just got unlucky this time, and next time we might get lucky?

Or maybe, somebody up there doesn't like us?

The model doesn't say.

Or maybe the model is wrong. It's a nearly useless model anyway, since it can't say what happens.
Matthew Martin 9/27/2013 01:26:00 PM
Three thoughts:
1)I have suspected for a while now that Calvo pricing is the source of most of the New Keynesian model's predictive failures in this recession--in particular the failure to predict low inflation rather than massive deflation. Im not sure if an alternative specification of nominal rigidity would change much in Cochrane's paper though.

2) This could all be about date-zero problems. In all these liquidity trap models, the economy is "born" in the liquidity trap at date-zero, and eventually transitions out of it. A wide class of models--including most overlapping generations models--make goofy predictions as a result of the finite start date. Accordingly, Cochrane's equilibrium is "born" with inflation just high enough to mitigate the ZLB. I'd like to see whether it's really possible to transition into Cochrane's equilibrium, since I'm somewhat skeptical that at the micro level, massive price hikes are a rational response to an exogenous decrease in aggregate demand (ie, in monopolistic competition can you really convince someone to buy more today by promising a big price hike tomorrow?).

3) Maybe its all about the expected non-linearities in the Taylor Rule. Cochrane's equilibrium requires a promise to be "irresponsible," allowing output to spike well above potential and an extended period of elevated inflation. In the real world, the Fed does not actually follow a Taylor Rule, and it's possible people simply feel the Fed would deviate from the rule in a way to clamp down on inflation as soon as it appears. Empirically, this would be justified: the Fed usually practices "opportunistic disinflation," using recessions to clamp down on inflation. This would fit nicely with the Freidmanite narrative of the business cycle being a primarily monetary phenomenon.