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Interesting post Nick.

1. In your 2nd paragraph I at first thought you were describing (confessing) your method for doing algebra. (Shoot, I know he says he's bad at math, but what the...!). Lol. Good to know I was wrong.

2. You taught me a new word: "tatonnement." Thanks. (I discovered about a year ago that MS Excel has that built in ... it's called "auto iterate" or some such).

3. Did you see Stephen Williamson's comment to Noah? He seemed a little annoyed to me.

4. Didn't Marcus Nunes do a post once about inflation NOT having any inertia? (The answer is "Yes, yes, he did".) A difference in views, or is it that I'm not understanding something (as usual)?

Tom Brown: "tâtonnement" is a french word. Date from the time french economist were the best in the world and worth listening to. So long ago, I wasn't even born.

I think everything about this Neo-Fisherian stuff breaks down if you think about monetary policy in terms of the evolution of quantity of money (which is more intuitive anyway) and not interest rates.

Jacques, thanks: I should have known with that "onne" in the middle, but the "chapeau" makes it obvious. Now I have an inkling of how to pronounce it without embarrassing myself quite as much as I would have otherwise (although I'd love to hear Nick's pronunciation).

Tom: 1. well,....sometimes I do solve by guessing, then checking my guess. And the method of undetermined coefficients is half lucky guess and half algebra. There was one economics paper that contained the immortal line: "By sheer dumb luck, we found the solution to be ..."

3. I hadn't seen Steve's comment. I have read it now. I think rational expectations vs adaptive expectations is a partly false dichotomy. (I did a post on that once). But Steve has a point in this case. The sort of tatonnement learning in this model is a little peculiar.

4. I half agree with Marcus. I think it depends. A big enough public shock can break inflation inertia. Just like a conductor leading an orchestra. There's a coordination game.

Makroint: I agree. But we have to imagine a world in which the central bank holds the nominal interest rate constant, allowing the stock of money to adjust. And M could go either way, depending on what people expect. There is no nominal anchor.

Tom: roughly: ta ton a mon, with both a's like "hat". That's how I say it, anyway. Jacques Rene would know better. Biologists (I think) sometimes talk of "gradient climbing" in the evolutionary process, which is sort of similar to tatonnement. My French is a bad mixture of English schoolboy and local Gatineau valley (which has lots of english and very old french mixed in, like "mon char a un flat"). My daughter (who can work in French) cringes when I speak.

Lol... love the "sheer dumb luck" thing.

"Tom: 1. well,....sometimes I do solve by guessing, then checking my guess"

I do something similar in my line of work that I call "solution by Matlab" (i.e. a quick simulation). Sometimes it's just easier.


P = Pe + aY where a > 0 (Phillips Curve)
Y = -c(i - Pe - r) where c > 0 (IS Curve)

P = Pe - ac(i - Pe - r)
P = Pe(1+ac) - ac(i-r)

Before we get to a tatonnement process, what do "a" and "c" really represent? Can individuals adjust "a" independently of "c" so that "(1 + ac)" is always less than one?

It appears that "a" is a price adjustment factor - individuals have more goods than they can sell, so they mark down prices below their expectations to be able to sell goods as the output gap gets more negative. Likewise they can have too few goods to sell and mark up prices above their expectations to maximize profits.

It appears that "c" is a multiplier on how large the output gap gets as the central bank increases the nominal interest rate above the natural rate + expected inflation. As i rises above the natural rate + expected inflation, more people save (focus on debt reduction?) and fewer people spend (borrow to fund new spending?). As i falls below the natural rate + expected inflation, more people spend (borrow to fund new spending?) and fewer people save (focus on debt reduction?).

And so do people make pricing decisions that are interconnected with their savings (debt reduction) / spending (debt increase) decisions? Maybe not at first, but what about learned behavior? People learn that they can adjust their "a" and can adjust their "c", but for them to collectively seek P = Pe = i - r via tatonnement, (1+ac) must always be less than one.

For instance, Joe chooses to have an "a" of -2 and a "c" of +2. He raises prices above expectation as the output gap rises, and lowers prices below expectations as the output gap falls. He increases his spending (borrowing) as the central bank lowers the nominal interest rate below expected inflation plus the natural rate and reduces his spending (borrowing) as the central bank raises the nominal interest rate.

On the other hand, Sally chooses to have an "a" of +2 and a "c" of -2. She lowers prices below expectation as the output gap rises, and raises prices above expectations as the output gap falls. She decreases her spending (borrowing) as the central bank lowers the nominal interest rate below expected inflation plus the natural rate and increases her spending (borrowing) as the central bank raises the nominal interest rate above elected inflation plus the natural rate.

I realize that neither Sally or Joe is using a profit maximizing strategy. A profit maximizer borrows when the nominal interest rate is below the inflation rate + natural rate and raises prices as the output gap goes from negative to positive. But both can be profitable strategies and allow a tatonnement process to get P = Pe = i - r.

Nick, Tom: this humble IO guy was never that strong on macro. During my graduate days, we once had a problem that I was one of the few to have solved. Prof called me to his office. Explained my method. After reflexion, he said: "Have now idea what grade to give you.The only thing right is the answer...". Dumb luck more prevalent than thought.
Other macro anecdotes on demand.

I'm stuck at Y = -c(i - Pe - r) where c > 0

i = actual interest rate = 5%
Pe = expected inflation = 0%
r = natural interest rate = 5%


(i - Pe - r) = (5 - 0 - 5) = 0


Y = -c(0)

so whatever c is, Y will always be 0.

I assume I screwed-up somewhere. Can someone spot where ?

MF: you didn't screw up. There is no uncertainty in this little model, so the rational expectations equilibrium always has Y=0.

Frank: think of a and c as technology and preference parameters. We don't choose them, they are just given. And the model says a > 0 and c > 0.

MF: and Y is not output, it's the gap between actual and "potential" output.


Okay thanks. But even with "a" being technology and "c" being preference, I still assume we chose preference and we work towards technological progress?

"And the model says a > 0 and c > 0."

But why are they both greater than 0?
It seems to me that a company can be financial wizards, terrible at technology, and still turn a profit.
Likewise a company can be technological wizards, terrible at finance, and still turn a profit.

I understand why a firm would need either a or c to be greater than 0. I don't understand why they both need to be greater than 0. Yes, I know that economists like to treat firms as profit maximizers. But that seems to ignore the real world where we are not all some perfect mix of Thomas Edison and Charles Schwab.

Frank: you are off-topic. Stop.

Awesome post, Nick. I'm really impressed at how you can make little monetary models that have useful insight. I never learned how to do that. I never even learned how to think conceptually about monetary economies or monetary policy. My grad program just went straight to New Keynesian models as if they were just a little tweak of RBC. I am jealous.

Thanks Noah! But I always had to do stuff like this, because I was never smart enough to be able to do the math. Plus I'm old.

"MF: and Y is not output, it's the gap between actual and "potential" output." , thanks - that is what I missed.

Market Fiscalist,

If the central bank screws up - they set i greater than expected inflation plus the natural rate - this will increase the output gap (Y becomes more negative). The degree to which this happens is affected by the multiplier "c".

"If there is inflation inertia in the model, then a central bank that fixed the nominal interest rate too high, even if only for a finite period of time, might see a cumulative decline in inflation from which it would be impossible to recover."

This seems important from a practical standpoint - making it difficult to effect a successful transition from current policy to a neo-Fisherian policy?

JKH: I think inflation inertia, and how it can be broken, is very important in practice. For example, if you have high inflation, and you want to stop it quickly, unless you break inflation inertia you will get a recession. And it's an awful lot easier to understand why Neo-Fisherianism is wrong, and how the current practice of central banks works, if you have inflation inertia in the model. With inflation inertia, if the central bank raises the nominal rate unexpectedly, actual inflation (and hence expected inflation) doesn't change much initially, so the increase in nominal rates must increase real rates too, which causes demand to fall, which causes inflation to slowly fall over time. In short, with inflation inertia in the model, inflation can't jump (much). Which rules out the Neo-Fisherian equilibrium, where the central bank jumps the nominal interest rate up and the inflation rate jumps up too. And that is pretty much how central bankers think.

The reason we rarely have inflation inertia in our macro models is very embarrassing. If you take the standard Calvo Phillips curve, where the fairy has a probability 1/n of visiting each period, and tweak it very slightly so the fairy visits every n periods (instead of every n periods on average), you get inflation inertia. (And it's slightly less unrealistic at the micro level too.) But it makes the math much harder, and the equations horribly messy, so we don't do it.

Is there some limitation that the tatonnement couldn't just be run "backwards" when b>1, thus converging? (I don't really understand if tatonnement itself is being modeled as something that economic actors must be doing. But even if so, the smart (rational?) ones would figure this out, no?)

In particular, where you have
P = bPe + stuff, where b > 1
convert it to
Pe = (1/b)P - (1/b)stuff, where (1/b) < 1
and tatonnement on that *will* converge, right?

(Again, if there is some arrow-of-time dependency that I'm missing that prevents this, never mind.)

"You can see the problem. If people try to solve for the rational expectations equilibrium using the tatonnement process, it won't converge, because b > 1. The rational expectations equilibrium does exist, and is unique, but there is almost zero chance the agents in the model will solve for it by tatonnement (unless they are extremely lucky and guess it exactly right first time)."

Nick, in Woodford's notes he mentions temporary equilbrium. Is the above quote an example of temporary equilibrium analysis?

Nick: Heh, there's that Canadian modesty. ;)

Maths time!

What you have described here is known formally as fixed point iteration. In this iteration, you plug a candidate solution back into your model and use the residual to correct your guess. Your result about bad things happening if b >= 1 is a very general result, and we can formally get that by saying that b>=1 does not make this a contraction mapping. That is, if we start in a neighbourhood of the solution, after one iteration our neighbourhood is bigger and not smaller.

The interesting thing is that fixed point iteration such as this also extends to nonlinear systems. Instead of a nice simple b coefficient, we get a more complicated thing. Near the solution itself, b is the partial derivative of the right-hand-side (the economic model) with respect to the variable.

One particularly cool facet of fixed point iteration is that when we get to nonlinear systems, we can get chaotic behaviour in addition to simple convergence or divergence. The logistic map is a famous one-parameter example, where the iteration changes character as that parameter changes.

Now, with regards to the economic application, you highlight a key facet of rational expectations here:

> This model has a unique rational expectations equilibrium, Xe = X = 0. But if agents try to solve for the rational expectations equilibrium by tatonnement, they will fail if b > 1.

Rational expectations are model-consistent expectations; we assume that if a model has a unique solution then the representative agent acts with respect to that solution.

This tatonnement approach is actually one of irrational expectations, or at least minimally-rational expectations -- the agents have no idea of the inner workings of the model and are only able to observe one iteration at a time.

When b < 1 -- when the economic model is a contraction map or self-stabilizing -- then this difference is marginal. Agents that are dumb as rocks will eventually converge to the rational expectations solution, so if we're talking about equilibrium behaviour over the moderate to long term then we don't need to care that the agents are dumb.

If b > 1 or the model is fundamentally unstable, then we do care about this difference. Super-rational agents will still act in accordance with the equilibrium, even if that equilibrium is unstable.

Even more interesting are cases where we may have hysteresis, where b<1 near the solution but b>1 further away, or where there are two or more b<1 solutions bridged by a b>1 region. Here, there is no guarantee that a "dumb" population will converge to the desired solution, and converging to an unexpected solution would be surprising to modellers.

> My little model, with no lags or leads, will explode instantly if the central bank sets a fixed nominal interest rate for just one "period" (unless people guess exactly right first time).

Not really. A linear model like this that "explodes instantly" usually means that nonlinear factors that were neglected aer actually important.

You've already covered a case where these nonlinear factors enter into it, namely the nominal anchor under the gold standard. This is a perfect example where over the short, small-deviation term your linear model holds, but it is constrained within bounds by the nominal anchor. In this particular case, I'd venture to say that b=1 near the equilibrium such that the economy can drift, but b<1 further away as the nominal anchor becomes more important.

> And it's an awful lot easier to understand why Neo-Fisherianism is wrong, and how the current practice of central banks works, if you have inflation inertia in the model.

You are very true here. Inflation intertia, or really any time-varying model of the economy, imposes a particular model for the tatonnement. That turns an unstable fixed-point iteration into an unstable time series, which makes things far more obvious.

We approached some of these ideas from a more practical angle last year in a discussion of a Schmitt-Grohe & Uribe paper, where aside from an overdetermined system their models would have been unstable in a dynamical sense.

Noah: not modesty in this case. All things considered, I'm pretty sure you are smarter than me. I know your brain works a lot quicker than mine (even adjusting for age). Writing posts like this is one of the few things I can do better than you (fixing cars is probably another). I don't think this is an accident. Joan Robinson once said (roughly) "I was never good at maths. In some ways this was an advantage; it forced me to think." I think there's something to that, though I'm not sure precisely what.

You are not alone in having been taught NK macro as a minor modification of RBC. Steve Williamson says this too. Being older, I see NK macro as a cumulative outgrowth of Monetarism with a Keynesian policy angle. Monopolistic competition was added, then Woodford introduced the Wicksellian angle. But above all, there is a massive difference between RBC and NK Macro. RBC makes perfect sense in a barter economy. NK macro models make no sense whatsoever outside of a monetary exchange economy. The underemployed worker-producers in a recession would instantly barter their way back to full employment even if barter had to be done at Calvo prices. I tried to convince Steve of this, but got nowhere.

Jeff: "Is there some limitation that the tatonnement couldn't just be run "backwards" when b>1, thus converging?"

Schmidt/Woodford just assume it is only run forward. But there is a sort of time arrow argument that could be made here. Imagine a one-shot game, where each of us is trying to decide what we will do. And to keep it simple, imagine a game where all players are identical.

This is how we might reason: "I don't know what you are going to do. But if I think you will play X1 ((my first guess), my best response would be X2. But then if you are thinking like me, you would also guess I will play X1 and so you would choose X2. But if I expect you to play X2 (my second guess), I will respond with X3. And so on."

Running it backwards sounds weird. "If I choose X1, that would only be a sensible choice if I expect you to choose X2..."

There is something similar to this in game theory. But I forget what they call it.

JP: "Nick, in Woodford's notes he mentions temporary equilbrium. Is the above quote an example of temporary equilibrium analysis?"

Yes and no. Schmidt/Woodford aren't quite right when they compare what they are doing to temporary equilibrium analysis. When the old guys did temporary equilibrium analysis, people had expectations about the future, and you solved for the equilibrium in the current period taking those expectations as exogenous. Then in the next period, looking back and seeing how some of their expectations were false, because other people acted differently than how they thought they would act, they would revise their expectations. (The old guys would wave their hands about precisely how they would revise their expectations).

What S&W are doing is more like a Walrasian tatonnement. They all revise their expectations in metatime before play actually begins. They are not learning from experience. And if the tatonnement converges, and if there are no subsequent exogenous shocks, they never revise their expectations in real time.

It's like the difference between Walrasian stability, where no trades take place until the auctioneer solves for the market-clearing price vector, and Leijonhufvud et al stability, where trade begins at disequilibrium prices, so people learn they are unable to buy or sell as much as they want, and adjust their prices and their expectations while play proceeds.

JKH, could you check my comments here:



Majro: I think I followed most of that!

But this bit isn't what is going on here: "This tatonnement approach is actually one of irrational expectations, or at least minimally-rational expectations -- the agents have no idea of the inner workings of the model and are only able to observe one iteration at a time."

No. In this case the agents *observe* nothing. They are not learning from watching what actually happens to inflation. These iterations are all happening in their heads, before they observe anything. They actually do know the model; it's just that they can't solve it except by guessing Pe, figuring out what the model says P would be, then repeating.

Yep, it's weird. This is not learning. It's math for dummies, that doesn't always converge on the right answer.

> These iterations are all happening in their heads, before they observe anything.

Okay, so you're talking about one way by which the rational expectation can be formed.

In this case, you've come up with a useful necessary condition for the validity of a rational-expectation equilibrium in an economic model. If this tantonnement approach does not work, then it will also not work in a real, dynamical economy where model is unobservable and actions must be based on observation.

This is a useful rather than trivial condition because rational expectations is independently useful -- "dumb" agents acting through a market over time tend to act as if they are rational.

Majro: except, in economics, the model itself changes when people learn about it. It's not like planets that rotate at the same speed whether we understand them or not. So I'm not sure if tatonnement convergence is either necessary or sufficient for real time learning convergence. Except maybe in simple little models like the one I sketched above, with no lags or leads, where it should be both necessary and sufficient.

Thanks Nick. Is the following...

"Most of us mortals have to watch what happens, and revise our expectations in the light of experience, and what the central bank tells us it's targeting. It's maybe more sensible to worry about tatonnement convergence than to assume people can somehow all coordinate their beliefs by magic, but I think worrying about real time convergence makes more sense still."

...an example of thinking in terms of temporal equilibrium?

@Nick, Regarding running tatonnement "backwards", in game theory it is called "backward induction":


I am most familiar with it from chess "endgame tablebases", by which chess has now been "solved" completely for virtually all positions with up to seven pieces on the board (counting the two kings).

But why invoke any type of induction as a rational-actor mechanism in the first place when a straight algebraic solution exists (solve for P=Pe)?

The output gap seems to be given as known, yet what if it is really not known? What might happen if we simply change the perception of the output gap?

Y = -c(i - Pe - r) where c > 0 (IS Curve)

Pe = Y/c + i - r

Hold c constant, fix i over a long time, make r very slow moving.
Then, as the perceived output gap becomes more negative, Pe falls.
What if the perceived output gap all of sudden became less negative based on an intention from the central bank to raise i? The central bank sending a message that people should see a smaller output gap? Then Pe would rise up as Y was calculated as less negative.
The smaller that c is, the more that Pe will change as the perceived output gap adjusts.
Also might changes in the perceived output gap affect c? A change in c could offset a change in the perceived output gap... holding all other domestic and international factors and consequences constant.

JP: yes.

Jeff: that's not really the same. Because you are considering a game with multiple moves, so first you solve for equilibrium in all the possible last moves, then second last, and so on. That's how you find the subgame perfect equilibrium, for example. Here we are trying just to solve for one move, when we can't do simultaneous equations.

Edward: Y is endogenous to the model, along with P and Pe. The exogenous variables are the three parameters a, c, and r, along with i, if the central bank holds it fixed. That gives us 3 unknowns, and 3 equations (Phillips Curve, IS, and the RE restriction Pe=P), so it can be solved for those 3 endogenous variables.


I think we have to assume that people are not blind automotons here; blindly substituting in. Rather they evaluate the errors and take corrective action.

In your X=2X, starting from X=1, we get 2, 4, 8, ... or deltaX = 1, 3, 7, ... Thus, someone trying such a simple method will realize that it's not converging, and beyond which as there input increases, the size of the error grows. Thus, they may then start substituing in the other direction:
Let me try X=0.5... ok back to X=1
X = 0.25... back to 0.5
But as they do so they notice that each time their starting number has lower error; thus, they will keep going until they converge near enough to 0.

Putting this back to real-world terms, if there's an easily identifiable trend, people will easily adjust (i.e., my estimate was too high, I need to lower future estimates). If there's no such trend or it's not strong enough to be easily detected (i.e., my measurement error is too high because I have very little information), then they'll just keep on doing what they're doing. It is rule of thumb, but it's maybe something like tatonnement++.

My 0.02 (and as usual, I'm likely missing the point).

Interesting. Now that I read this I think you have hit on the head what Schmidt and Woodford were saying. It's a technical point about how to sensibly interpret the behavior of certain rational expectations models, as having sort of short-circuited rather than as having given a result. A bit of rational expectationist theory inside baseball, really.

It's astonishing to me that the proposal that increasing the cost of borrowing would accelerate inflation has 1) been attached to Fisher's name 2) been carried so far by such ostensibly qualified economists and 3) provoked so much mathematics in the process of refutation. But I guess it's good to know that when challenged there are people who can bat that inside baseball.

Nick, OK I think I understand the idea better, thanks. Sort of a black-box model of how an actor may "think" without doing math. Where "solving algebraically" would be doing "math" and would sort of correspond to your "lucky first guess" improbable. Like ignoramaus, I still don't understand, if a "multiple iteration thought experiment in metatime" is the idea, why that couldn't also include a "reverse gear if diverging" rule, or even just be a "guess and check"-only iteration until the right "first guess" is hit upon...

ignormaus: if we interpret this tatonnement at least semi-literally, as a thought-process in each individual's head, in metatime, I think you are right. If an individual is trying to solve a math problem (or adjust a car's Ebrake) by trial and error, and notices it's getting worse, he would likely try turning the screw counterclockwise instead of clockwise. I do this all the time.

On the other hand, if we had a large number of people learning from experience in real time, this ain't going to happen. It would only happen if they all got together and said "Look, we seeme to be getting further and further away from the rational expectations equilibrium each year, so let's all try something different. If inflation turns out to be higher than we expected, let's all expect **lower** inflation in future. OK? That might get us to the rational expectations equilibrium."

It ain't going to happen in real time because: they don't all meet together to coordinate their learning; it would not be in any individual's interest to stick to the deal; it would likely not be in their collective interest to stick to the deal (unless the rational expectations equilibrium is Pareto Optimal, which it often won't be).

In fact, this is perhaps a rather strong criticism of Schmidt/Woodford. Understood literally, their tatonnement makes no sense. Understood as a metaphor, the process it's a metaphor of is very different.

Tom: I'm still gobsmacked by the whole affair. Noah's to blame for calling it after Fisher (who must be rolling in his grave), but it did need a name, and nobody could come up with a better one, so we must all share blame for following Noah, so we are stuck with it. But, you see, there's a grain of truth in the whole thing. If the Bank of Canada suddenly raised the nominal interest rate, the effects of that action depend totally on how people interpret that action. "WTF? Why did they do that?". And *one* answer to that question is "We at the BoC have decided to raise the inflation target from 2% to 3%, and we figure that once we announce this, people will now expect 3% rather than 2% inflation, so we need to raise the nominal interest rate to stop real interest rates falling and actual inflation rising above the new 3% target." Which gives us the Neo-Fisherian result.

Monetary policy is a socially constructed of reality. It's like a language. The effects of doing something depend on what it is understood to mean. I keep yammering on at this point, but most people don't get it. (Though to John Cochrane's credit, when I showed him my sketch of a formal model where the nominal interest rate is a signal of the central banker's unknown type, he did say he liked it, which probably means he got the point, in one sort of way.)

I should probably recycle this old post with "Neo-Fisherian" in the title.

Argh. Why don't we call it Neo-Friedmanist? It would make just as much sense. I guess Neo-Fisherian doesn't jar so much because fewer people these days have read him. I think it wasn't Noah that started it.

I have to disagree with your Bank of Canada example and your magical understanding of the interaction between reality and expectations. Effects of a rate-hike announcement don't "depend totally" on interpretation - the increased cost of borrowing is a real and important fact. The range of possible mainstream interpretations isn't unbounded and your example is outside that range. Central banks aren't regarded as super-human or possessed of deep secrets, so they have very limited ability to change market expectations of macro parameters other than through policies that objectively move parameters in that direction - eg raising rates objectively dampens inflation. When a central bank raises rates, inflation expectations are adjusted if at all downward. If rates and inflation expectations rise together, that's because something else is happening to push up inflation expectations. No central bank can talk up inflation expectations out of whole cloth.

Tom: it's called "Neo-Fisherian" because it's based on the Fisher equation: nominal interest rate = real interest rate + inflation

But instead of reading that equation causally from right to left (raising the inflation rate causes nominal interest rates to rise), it reads it from left to right (raising nominal interest rates causes inflation to rise).

Think of a ball in a bowl. It's a stable equilibrium. Now turn the bowl upside down, and put the ball on top. It's an unstable equilibrium. The slightest hint of a breath of wind will cause the ball to move in either direction.

Think of a crowd of people, where every individual wants to be at the font of the crowd. If just a few people expect the crowd to run north, the crowd does run north.

Think of a bank run. If people expect a run there is a run.

Two people play a game, where if they write down the same number they win a prize, but they can't communicate. If I say "7", they both write down 7. I have no mechanical power over them at all, but I am like the conductor of an orchestra, or the coxswain on a boat.

" eg raising rates objectively dampens inflation."

Nope. Raising *real* rates may objectively dampen inflation. But the central bank sets nominal rates.

We are talking about people, not gearboxes. What one person does depends on what he expects everyone else to do.

Thanks Nick, I understand all that, but I don't think you've said anything that contradicts me. When I say that raising (nominal) rates objectively dampens inflation, I think it should be understood that I mean with all else equal. After all, I also said in the same comment that if (nominal) rates and inflation rise together, there must be something else pushing up inflation. That fully covers your objection that a rise in nominal rates that's not more than the acceleration of inflation, and hence doesn't increase real rates, won't dampen inflation.

I understand that people are people and that expectations are very important. But mainstream expectations are after all rational, and though they can't be precisely predicted, they do actually follow rules. One of the rules they follow is that an increase in policy rates is a factor lowering expectations for inflation. Most people have learned so from others, deduced so by logic and/or remember the last cycle. It's not the only or even all-important factor, but definitely a factor lowering expectations for inflation. There is a wealth of market data demonstrating that. It is not ambiguous.

And actually these days it is a machine. The market reaction is literally pre-programmed. Computers read the rate release news and move the quotes up or down within a tiny fraction of a second.

I understand you mostly agree and you're just trying to inject a little hypothetical "yeah, but, what if somehow ..." to make a point about the power of expectations. Maybe I'm being a bit mean having none of it. But I'm sorry it just doesn't happen that way. Markets react to (unexpected) rate hikes by adjusting their inflation expectations downwards, period.

I can't make sense of the original argument. If everyone actually knows the values of a, c, r and i, what motivation do we have for assuming that they try to solve for P, Pe and Y using this dumb approach? Why wouldn't they just use the obvious solution Pe = i-r?

As a description of what might happen in real time, does it make more sense? i.e. assume that everyone agrees that the relationships between P, Pe and Y are of this form, but they don't know what the actual values of a, c and r are. Then people guess at what r is, and set their prices accordingly. Let's suppose i = 5%, we guess that r is 3% and increase prices by 2%. Suppose the guess turns out to be too high: we learn that by observing that Y is negative, and actual inflation is lower than 2%. We revise our guess of r to 2%, say. The model says that if the new guess is correct, we should increase prices by 3% (relative to the realized price level). This is counter-intuitive, but do we really have any better option? If we continue to expect that inflation will be 2% (or less), than we know that actual inflation will be lower still, so we know that expecting 2% or lower inflation is irrational.

This doesn't appear to correspond to what happens in real economies, but within this model's assumptions, it would appear that learning in real time will in fact converge, with people eventually discovering what the natural interest rate r is.

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