This is in response to John Cochrane's good post.

First off, you need to understand that Keynesians don't actually believe their own New Keynesian models. When they reason informally, they always assume inertia in actual or expected inflation. But the standard New Keynesian model, with the Calvo Phillips Curve, doesn't have inflation inertia in it. So I am going to build a New Keynesian model that does have inflation inertia in it.

If there's inflation inertia, so actual and expected inflation can't jump, then an upward jump in nominal interest rates means an upward jump in real interest rates, and that means a recession. What happens next?

The second thing you need to understand is that Keynesians don't actually believe their New Keynesian models. When they reason informally, they worry about the economy going off on an equilibrium path that does not eventually converge to the natural rate. They don't believe the New Keynesian model when it just assumes eventual convergence to the natural rate. Your diagram was neat, but it only showed the equilibrium paths *that converge eventually to the natural rate*. Keynesians believe that a permanent rise in the nominal interest rate could cause the economy to have ever-accelerating deflation and an ever-worsening recession. My model shows that as possible.

The third thing you (and the Keynesians too) need to understand is that the nominal interest rate is a signal. A really stupid way for central banks to signal, but a signal nevertheless. And the effects of any signal depend on how it is interpreted. How does it change people's beliefs about the central bank's beliefs and preferences? My model illustrates that.

So, here is my model:

At the beginning of every period, 99% of firms announce their prices for the beginning of the next period. There is a one-period lag in changing prices. This is a quick and dirty way to get inflation inertia into the model, but it's no worse than the Calvo fairy as a "microfoundation". **The inflation rate is 99% pre-determined one period in advance.** The remaining 1% of firms can change prices any time they like.

Next, Nature chooses the natural rate of interest. She gives the central bank, and each individual firm, imperfect information about her choice, but she tells them all different things, so nobody knows what she told the central bank.

Next, the central bank announces the nominal interest rate.

Next, people choose how much to buy. Standard Euler equation stuff, with a stochastic natural rate of interest.

Central bankers differ by type. All the sensible types of central banker have a (different) inflation target. They don't like inflation deviating from their target, and they don't like output deviating from the natural rate. Standard assumption, except firms don't observe the inflation target and must infer it from the nominal interest rate. *But there is also an evil type of central banker who wants ever-increasing deflation and an ever-worsening recession, and so always wants lower inflation than firms think he is targeting.*

The Phillips Curve for this economy will be (roughly):

p(t) = E(t-1)[p*(t)] + 0.01y(t)

where p(t) is actual inflation, p*(t) is the central banker's inflation target, E(t-1)[p*(t)] is firms' prior expectation of the central banker's inflation target, and y(t) is the output gap. The output gap has a very small effect on current inflation, because only 1% of firms can adjust their prices any time they like. **All the action in this model comes from the remaining 99% of firms updating their priors about the central banker's inflation target.**

Suppose we start in an equilibrium where the central banker has been on the job long enough for firms to have inferred his type with certainty. 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. Actual inflation will always equal 2%, plus or minus a small amount of noise from the 1% of firms who have perfectly flexible prices. The central banker sets the nominal interest rate equal to his best estimate of the natural rate, plus target inflation. But his estimate will be imperfect, because Nature did not give him perfect information. If he sets the nominal rate too high, there is a recession, the 1% of firms cut their prices below those of the 99% of firms who can't, and actual inflation comes in slightly below target. If he sets the nominal rate too low, there is a boom, the 1% of firms raise their prices above those of the 99% of firms who can't, and actual inflation comes in slightly above target.

That's what we see in normal times, when the inflation target is known. And we will observe zero correlation between nominal interest rates and inflation. (That is an immediate implication of rational expectations on the part of the central banker, since his inflation forecast errors must be orthogonal to his information set).

Now suppose that the old central banker retires, and a new one is appointed. Firms don't know his type. They will need to infer it from how he sets nominal interest rates.

Initially, just to keep it simple, suppose that firms do know the new central banker is one of the sensible types. They just don't know his preferred inflation target.

Suppose the new central banker has a higher inflation target than the old one. But firms don't know this. So he has a higher inflation target than they expect. He observes the inflation rate implied by the 99% of firms who have already announced their prices for next period. He sets a nominal interest rate such that the real interest rate will be below what he expects the natural rate to be, so the 1% of firms will raise their prices. There is a boom, and inflation rises above the previous target, part way towards the new target.

Firms reason that he might have a higher inflation target than the old central banker, or he might just have made a mistake about the natural rate of interest. They slowly revise upwards their Bayesian priors about his inflation target, if he makes repeated "mistakes" in the same direction.

When the inflation target changes, we should observe an immediate negative correlation between nominal interest rates and the inflation rate. Then a transition period with a negative correlation between real interest rates and the rate of change of the inflation rate. Then finally a positive correlation between nominal interest rates and inflation, once firms have figured out the new inflation target.

Now let's introduce the possibility of an evil type of central banker.

Suppose firms see that the new central banker has set nominal interest rates too high, given the 99% predetermined inflation rate, so that inflation falls, and there is a recession. There are three possibilities: maybe the new central banker just over-estimated the natural rate; maybe the new central banker is sensible, but has a lower inflation target than the old one; or maybe he is evil. They will adjust their priors each period according to the evidence.

If they see the central banker set real interest rates too low in subsequent periods, so that inflation rises back to target, they figure it was just a mistake, and the inflation target has not changed.

If they see the central banker cut nominal interest rates as inflation falls, so that real interest rates return to normal as inflation stabilises at the new target, they figure the inflation target is lower.

If they see the central banker keep real interest rates permanently too high, they figure he must be evil. Inflation falls without limit. *If he wanted ever-increasing deflation and ever-worsening recession he would set the nominal interest rate too high and keep it there.*

Any competent grad student could do the math and solve this model formally.

Of course, if we changed the model, so that new central banks could simply announce their type, or else signal their type in a more sensible way, the relation between nominal interest rates and inflation could be very different.

"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?

Posted by: Kevin Donoghue | December 20, 2013 at 02:01 PM

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.

Posted by: Nick Rowe | December 20, 2013 at 02:15 PM

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.

Posted by: Nick Rowe | December 20, 2013 at 02:22 PM

"...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.)

Posted by: Kevin Donoghue | December 20, 2013 at 02:28 PM

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.

Posted by: Kevin Donoghue | December 20, 2013 at 02:33 PM

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.

Posted by: Nick Rowe | December 20, 2013 at 02:58 PM

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.

Posted by: Kevin Donoghue | December 20, 2013 at 03:22 PM

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.

Posted by: Nick Rowe | December 20, 2013 at 03:34 PM

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

Posted by: Brad DeLong | December 20, 2013 at 05:03 PM

'Eventually'

Posted by: Ritwik | December 21, 2013 at 06:03 AM

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.

Posted by: Nick Rowe | December 21, 2013 at 06:59 AM

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.

Posted by: Scott Sumner | December 21, 2013 at 11:37 AM

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.

Posted by: Nick Rowe | December 21, 2013 at 01:33 PM

Thanks NIck, I did a post:

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

Posted by: Scott Sumner | December 21, 2013 at 02:45 PM

"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?

Posted by: A Young | December 21, 2013 at 02:47 PM

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).

Posted by: Nick Rowe | December 21, 2013 at 03:08 PM

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

Posted by: Mike Sax | December 22, 2013 at 08:31 AM

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.

Posted by: Adam P | December 22, 2013 at 08:44 AM

Adam,

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.

Posted by: Kevin Donoghue | December 22, 2013 at 09:23 AM

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."

Posted by: Peter N | December 22, 2013 at 09:58 AM

AdamP: Thanks!

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).

Posted by: Nick Rowe | December 22, 2013 at 10:25 AM

Nick,

Thanks a lot. That makes sense now.

Posted by: A Young | December 23, 2013 at 12:33 AM