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"But the standard New Keynesian model, with the Calvo Phillips Curve, doesn't have inflation inertia in it."

I'm gobsmacked. Suppose only 1% of firms can revise prices in each period. There's no inflation inertia?

Kevin: with the Calvo fairy, who visits (say) 1% of firms each period ***at random***, there is no inflation inertia. ZERO. The price ***level*** is very sticky. The inflation rate is not sticky.

Intuition: the smaller the percentage of firms allowed to change their prices, the bigger they will jump their prices if expected inflation increases, because they know it will probably be a long time before they can change them again.

"...the bigger they will jump their prices...."

But they are just the 1%, so inertia remains. I'm looking at Fig 3.3 of Gali's texbook and if that curve isn't showing inflation inertia then the word doesn't mean what I think it means. (He's assuming 33% of firms revise prices every quarter.)

My best guess at what you're saying is, there's no inertia in the rate of increase of newly-set prices. But that's not a statement about the price index.

Kevin: if the equilibrium price level jumps up (say it doubles) then 1% of firms double their prices every period, so it takes almost forever for the actual price level to double. But the Phillips curve equation is p(t)=BE[p(t+1)] + (1/n)aE[y(t+1)] where p(t) is inflation, and B is very close to one, so if expected inflation jumps up, then actual inflation jumps up too by (almost) the same amount.

As John Cochrane says: "As you can see, it's perfectly possible, despite the price-stickiness of the new-Keynesian Phillips curve, to see the super-neutral result, inflation rises instantly."

I'm not the first person to have noted this. It's an old problem. It's why Mankiw did his paper on inflation with sticky information.

Nick, okay, thanks. I guess there are NK models and NK models. In Gali's version the PC equation relates inflation, expected inflation and the output gap. But surely in any version, all three of these variables are endogenous. You can't describe their behaviour without solving the whole system. Of course you can get a jump in inflation in Gali's model too, if you hit it with a suitable shock.

Since I'm not sure what Cochrane is on about or why you think his post is good I'll bow out. On Twitter, Brad DeLong tells me Cochrane has simply failed to read Brock (1975) and that's all there is to it. I'm hoping he means Brock's "A Simple Perfect Foresight Monetary Model" and not "The Global Asymptotic Stability of Optimal Control...."(!)

I'm not sure BDL is right, but I don't usually bet against him.

Kevin: a lot of these young whippersnappers could benefit from something like Brock. But Brock has M in the model, and NK models don't. Plus, JC has the Calvo Phillips Curve in his model. Plus, JC seems to be the only person around (except me) who takes the indeterminacy problems in NK models seriously.

So am I right to think that Cochran is another person who has not paid sufficient attention to Brock?

'Eventually'

Brad: I don't know. But I don't think you can say that from his current post. If he had M in the model, so you could say what is happening to M when i and pi change, you could talk about Brock. But the standard New Keynesian /Neo-Wicksellian model doesn't have M in it. If we wanted to, we could add a standard money demand function, let M be demand-determined, and just assume M adjusts however it needs to along each of his equilibrium paths. For example: if i and pi both jump up together, with no jump in P, we could have M jump down, and Mdot jump up, and it would all be consistent with Brock's stable path.

Ritwik: is there a typo? I couldn't find it.

Nick, I have a question. Does the Cochrane post translate into monetarist economics in the way the Williamson post seems to? I.e. If the Fed reduces the money supply, I can see two possible counterintuitive claims:

1. It fails to produce a liquidity effect, the fed funds rate falls, so does inflation, and the Fisher effect applies in the short run.

2. The liquidity effect occurs, the fed funds rate rises, but so does inflation. In this case the QTM no longer holds. Reducing the money supply is inflationary (which seemed to be Williamson claim (in the other direction of course) about QE.)

I had trouble understanding which of these two counterintuitive scenarios intrigued Cochrane.

Scott: It doesn't really translate at all. Because the main action is with a New Keynesian model, where money doesn't appear in the model. The Fed sets i, but that doesn't tell us what is happening with M. What he is saying is that, in the NK model, there are multiple equilibrium paths following a change in i.

As wonks anonymous(?) noted in comments: this is compatible with what you have been saying: low nominal interest rates don't necessarily mean loose money.

Thanks NIck, I did a post:

http://www.themoneyillusion.com/?p=25613

"And suppose he is a sensible type, and is targeting 2% inflation. Every year, the 99% of firms set their prices for the following year 4% higher than the price level in the preceding year."

If firms are setting their prices 4% higher, how is that not 4% inflation (plus/minus any noise from the 1%)? I'm wondering if maybe you took the 4% price rise and converted it into a "real rate" of 2%, but when you're talking about inflation, that's redundant.

Is there something I'm missing here?

A Young: that's 4% over 2 years, so 2% inflation per year. (The 99% of firms know last year's price level, when they set next year's prices, but they don't know this year's price level yet, because they don't know what prices the 1% of firms with flexible prices will set this year).

Nick I love your part about the evil CBers who want as much deflation as possible. Finally realistic assumptions.

Kevin,

The apparent discrepancy between Gali getting apparent inflation inertia and Nick's claim that Calvo pricing doesn't generate it are probably in the thought experiment being considered. The difference is the same as why Williamson and Cochrane are all confused.

If the "shock" being studied is a change in the inflation target (assume it's entirely believed and assume we start with inflation at the old target and no output gap) then under Calvo pricing the inflation rate does in fact jump instantaneously to the new rate and nominal rates would need to instantaneously rise. (Cochrane even puts it this way at one point in his post, he says that one way to raise current inflation is to raise the target.)

The reason for this comes from the fact that for each firm the arrivals of the Calvo fairy are Poisson. So, if the model says that 1/3 of firms change price in a period then this also means that each firm expects the price they choose to last 3 periods. Thus, in the case of a changed inflation target and no expected output gaps they raise prices by 3*(expected inflation). In general if the number of firms changing prices is 1/theta then they raise prices by theta*(expected inflation). (I'm not saying anything different than what Nick said, I jus think the detail of how each firms sees arrivals of the fairy as important.)

I'd imagine that Gali gets apparent inertia because he's showing you a change in the nominal interest rate without a change in target. Such a shock implies firms don't change future inflation expectations to a new constant rate, they expect future inflation to be somewhere between target and where current inflation will move to, they expect current inflation to move because if it doesn't we get a higher real rate and an output gap and the output gap means lower inflation.

There is an interaction among all the equations in the model to determine what inflation you get but the result will be inflation moving around slowly, appearing inertial, but that is because of the dynamics of everything working together.

Nick,

This a good post and good explanation but I think the source of confusion is simpler. SW is implicitly talking about a change in nominal rates that coincides with a changed inflation target as is Cochrane. SW appears not to be aware of this, as usual he just doesn't know what he's talking about.

In the real world the Fed has lowered nominal rates while explicitly not changing their inflation target. Neither SW nor Cochrane has said anything that applies to that case.

Thanks indeed, that clarifies things quite a bit. Of course you're right about Gali, he is showing the impact of a stochastic shock, not a permanent change in the policy rule. I'm not even sure that changing policy rules is a permissible thought-experiment in his kind of RatEx model. One would need a "super-rule" which defines the rules which can come into force, with possible rules being drawn from a hat or something like that.

Optimal Price Setting and Inflation Inertia in a Rational Expectations Model
Michael Juillard, Ondra Kamenik, Michael Kumhof, Douglas Laxton

http://www.douglaslaxton.org/sitebuildercontent/sitebuilderfiles/JKKL2.pdf

A THEORY OF RATIONAL INFLATIONARY INERTIA
Guillermo Calvo, University of Maryland and NBER
Oya Celasun, International Monetary Fund
Michael Kumhof, Stanford University

http://www.stanford.edu/~kumhof/ratinfinert.pdf

"In our view,firms in such environments can more usefully be thought of as continuously adjusting their prices according to some pricing rule which is only updated at infrequent intervals, again because of adjustment costs or a Calvo or Taylor staggering rule.

In our model we therefore give firms one more choice variable, by letting them choose both today’s price level and the rate at which they will update prices in the future, a ’firm-specific inflation rate’. In terms of the regression analogy, it aamounts to fitting a weighted least squares regression line through future optimal prices. In an environment of non-zero steady state inflation this assumption is much less restrictive than the standard one."

Yep. It matters a lot that the fairy arrives at random. If the fairy arrives at each firm at fixed intervals, you get some inflation inertia.

Peter N: good finds. Yep, there are other ways to build inflation inertia into the model. Mine was quick and dirty, but very simple. Plus, in my model, inertia disappears after one period if firms learn the new inflation target. Most of the inertia comes from slow learning (even though it's fully rational expectations, given their information).

Nick,
Thanks a lot. That makes sense now.

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