The collective speed limit game

Neo-Fisherite fun. Plus concrete steppes fun.

1. Suppose you don't care what speed you drive. Anything between 0 and 200 is all the same to you. But the cops do care what speed you drive. They want you to drive at exactly the speed limit S*, neither faster nor slower. (They have a symmetric target.) So the cops announce that if your speed is Si, you must pay a fine Fi, where Fi=(Si-S*)², for driving either faster or slower than the speed limit. In equilibrium you drive at exactly the speed limit, and never pay the fine.

In equilibrium, the cops do nothing, and you expect them to do nothing. It is your counterfactual conditional expectations that matter -- what you expect the cops would do if you did do something you won't in fact do.

And if the cops announce they have increased the speed limit S*, you would increase your speed too, and yet the cops actually do nothing.

1a. But if the cops got the sign wrong in the formula, and announced that the fine would be Fi=minus[(Si-S*)²], it wouldn't work. It's a negative fine. The cops pay you money if they catch you driving above or below the speed limit. There is no interior equilibrium. You would either drive at 0, or else at 200.

2. Now let's change the game slightly.

Suppose you don't care what speed you drive. But you do care about your speed relative to the average speed of everyone else. And everyone else is just like you. Except that some drivers like driving faster than average, and some drivers like driving slower than average. But the average driver likes driving at exactly the average speed, regardless of what that average speed is.

And suppose the cops don't care what speed you drive either. But they do care about the average speed of all drivers, Sbar. They want the average speed to equal the speed limit S* exactly, neither faster nor slower. If some want to drive faster than average, and some want to drive slower than average, that's fine by the cops.

So the cops think about it for a bit, then announce that each driver must pay a fine Fi=(Sbar-S*)(Si-Sbar). [Update: note that this is fiscal-neutral, both in and out of equilibrium. Total net revenue from fines is always zero.]

If the average speed is greater than the speed limit, those driving faster than average pay a positive fine, and those driving slower than average pay a negative fine. So each individual driver has an incentive to slow down relative to other drivers.

If the average speed is less than the speed limit, those driving faster than average pay a negative fine, and those driving slower than average pay a positive fine. So each individual driver has an incentive to speed up relative to other drivers.

And if the average speed is exactly equal to the speed limit, nobody pays any fines, and so the average driver chooses to drive at the average speed (by assumption), with some driving slower and some driving faster.

The only equilibrium to this game is where the average speed is exactly equal to the speed limit.

In equilibrium, the cops do nothing, and everyone expects them to do nothing. And if the number of cars is very large, so that each individual driver has no effect on the average speed, each individual driver expects the cops would do nothing even if he did do something that he won't in fact do. The counterfactual conditional expectations that create this equilibrium are expectations about what the cops would do if everyone did something that they won't in fact do.

And if the cops announce they have increased the speed limit S*, all drivers increase their speed too, and yet the cops actually do nothing.

2a. But what would happen if the cops get the sign wrong in the formula, and announce that each driver must pay a fine Fi=(S*-Sbar)(Si-Sbar)? And would all the drivers still speed up if the cops raised S*?

Neo-Fisherites say it makes no difference if the cops get the sign wrong. There's still an equilibrium where Sbar=S*, and nobody pays any fines, so nobody has any incentive to drive either faster or slower than they are actually driving. And Neo-Fisherites say that if the cops raise S*, all the drivers will speed up.

I say there are two other equilibria: a second where everyone drives as slowly as possible; and a third where everyone drives as fast as possible. And we are much more likely to observe the second or third equilibria than the first. Because there is no way that all the individual drivers, alone in their cars, could coordinate on that first equilibrium. It's unstable. (If the average driver is driving faster than the speed limit, those driving slower than average pay a positive fine, and those driving faster than average pay a negative fine, so every driver has an incentive to speed up relative to other drivers, so the average speed rises even further above the speed limit.) [Update: is it a trembling right foot equilibrium?? I think it maybe isn't.] And if by sheer fluke they were initially at that first equilibrium, and the cops raised S*, we would see all the drivers slow down, and all drive as slowly as possible.

[S* is the inflation target, Sbar is average inflation, the cops are the central bank, and the average firm wants to increase its price by more/less than average inflation if the real interest rate is less/more than the natural rate. No firm cares about its dollar price; it only cares about its price relative to other prices. And the central bank cares only about the average inflation rate. And the cops getting the sign right means the central bank follows the Howitt/Taylor principle.

And to make it even more fun for the people from the concrete steppes, we could change the game slightly, so that the average driver wants to drive at what he expects the average speed will be next period. So even if there are no cops at all on this stretch of road, the possibility that there might be cops in the very far distance will be enough to make the average driver drive now at the distant future speed limit. By backward induction. Forward guidance works.]

Comments

[Miami Vice]

What if you have public and private transportation? What if once speeders have had their cars seized and are taking the bus which used to be driving fast enough?

[Nick Rowe]

Miami: that's like asking: what if Godzilla ate all the cars? You can stretch metaphors, but only so far.

[Frank Restly]

Nick,

I realize that this is just a metaphor but

"And we are much more likely to observe the second or third equilibria than the first. Because there is no way that all the individual drivers, alone in their cars, could coordinate on that first equilibrium. Because there is no way that all the individual drivers, alone in their cars, could coordinate on that first equilibrium."

Are we presuming that drivers don't know the value of S*? They only find out they are above / below S* when they receive / pay a fine?

"And Neo-Fisherites say that if the cops raise S*, all the drivers will speed up." Aren't the Neo-Fisherites saying that if the cops raise S* and announce to drivers what that new S* value is, drivers will speed up?

[Nick Rowe]

Frank: The cops announce S*. They post visible "S* = 100" signs all along the road. But those signs don't do anything concrete. Those signs don't hit your car if you drive over or under the speed limit.

I am saying that if the cops increase S* by 10kms/hr, then all drivers will increase their speeds by 10kms/hr. But only if the cops get the sign right in their fining formula. The Neo Fisherites say it will work even if the cops get the sign wrong in the fining formula.

[Miami Vice]

OK forget the car seized bit. Why is public and private transportation stretching the metaphor too far?

[Nick Rowe]

Miami: because if we force everyone to take the bus, they all have to go at the same speed. And some want to go faster than average and some want to go slower than average. Which is why forcing everyone to take the bus is a BAD THING.

C'mon guys. Here I come up with a superb automotive metaphor for New Keynesian macro, and you want to talk about commie buses?

[Frank Restly]

Nick,

Okay. I get what you are saying.

Fi = (S* - Sbar) x (Si - Sbar)

If the average is lower than the posted limit, then I will be the recipient of a positive fine when I drive faster to try to raise the average.
If the average is higher than the posted limit, then I will be the recipient of a positive fine when I drive slower to try to lower the average.

Positive fines are assessed on people that try bring the average closer to the speed limit when you really want the opposite.

It may also depend on the the type of fine. If suspension / revocation of driving privileges is the fine, then coordination may become easier as fewer and fewer drivers are permitted on the highway.

Taking the metaphor through translation, if a central bank (cop) is able to close down firms (extract fines), then the remaining firms can more easily coordinate price changes.

[psummers]

I reckon you nailed this one, Nick! And you say you can't do math...

[Nick Rowe]

p*** ! How are you!? Hope you are well. I'm so glad you like it.

Yeah, I had some trouble with the quadratics, and decided to simplify by assuming everyone was indifferent about absolute speed. Then I realised (duh!) that the metaphor worked even better if drivers don't care about absolute speed, because firms don't care about their absolute prices, only relative prices! So it was a happy accident, caused by my bad math!

(I'm wondering though, what the reaction function of the representative driver looks like. I think it's some sort of saddlepoint? The equilibrium here is similar to the Darwinian equilibrium for the 50-50 ratio of males and females. If everyone else is having mostly boys, you want to have girls. If everyone else is having mostly girls, you want to have boys. If everyone else is 50-50, you don't care if you have girls or boys.)

[primedprimate]

Indeed, a brilliant metaphor.

[Nick Rowe]

primed: THANKS!!

[Rajiv Sethi]

Terrific post Nick. Among other things it explains very clearly that the rational expectations hypothesis is an equilibrium condition and not a behavioral assumption. Also reveals that when predictions based on rational expectations differ sharply from those based on adaptive expectations, it is the latter that are robust while the former are fragile. You can model agents as econometricians, using the most sophisticated forecasting methods available, and you still won't converge to the interior equilibrium if adaptive expectations would drive you away. This, of course, was Peter Howitt's point in his wonderful 1992 JPE paper. It seems like we keep having this debate every few months.

[Nick Rowe]

Rajiv: Yes! THANKS!

[Majromax]

@Nick:

> what if Godzilla ate all the cars?

Godzilla can't eat all the cars, it will trip on the concrete steppes first.

> I'm wondering though, what the reaction function of the representative driver looks like. I think it's some sort of saddlepoint?

Well, a single, non-representative driver who wants to drive at X above/below the average speed will actually drive at (X+ε), where ε is governed by fine-avoidance (or payment-seeking) behaviour.

We can fully capture the model dynamics with two agents, one who wishes to drive X above the average speed and one who wishes to drive X below.

For a single representative agent, we obviously have to look at one who wishes to drive at precisely the average speed. (They'll then be surprised to find out that they're right.) Now, the cops raise S*.

This is where expectations come into play. If our agent assumes that everyone immediately adjusts, then we're still on the equilibrium. However, that knowledge is equivalent to saying that the representative agent knows that they are the representative agent.

So what are the expectations? Let's say that our agent thinks that some of the other drivers will be slow to get the news, so the real average speed will be S' < S*. Our agent then would intend to drive S' (since preferences are relative to average). That's consistent with preferences and gives no fine (since they drive the average speed), which is great.

Now, go one layer deeper: our agent knows their model is imperfect, so S' is not fully known. Model that as a possible perturbation of speed: our agent expects to drive (S'+ε) and pay a fine of (S* - S')(ε). That means that the representative agent would expect to pay a fine for being on the high side.

So, the representative agent should err on the side of caution and ensure that, to the extent that they're uncertain, ε should be negative and they drive ever so slightly slower than the expected average. (Then it turns out that the average speed is precisely S'-ε, making the caution seemingly justified. They still don't pay a fine.)

[Nick Rowe]

Majro: " Let's say that our agent thinks that some of the other drivers will be slow to get the news, so the real average speed will be S' < S*. Our agent then would intend to drive S' (since preferences are relative to average)."

No. If the representative agent/driver expects the average speed will be S' < S*, then he will choose to drive a little faster than S', because then he will pay a negative fine (be paid money by the police) if he drives faster than average.

[Majromax]

> then he will pay a negative fine (be paid money by the police) if he drives faster than average.

I was separating those effects; in the absence of fines our agent would want to drive the average speed. Fine avoidance then modifies that preference.

In situation #2, with proper control, what you suggest works. To the extent that our agent thinks that the average is unknowable, and they're forced to be off-average, then they'd prefer to err on the high side (towards the target speed) rather than away.

In #2a, with backwards control, the agent would prefer to err away from the target speed.

I think the representative agent thought process illustrates the fundamental problem: even if equilibrium is possible, any imperfection of information drives the system away from that equilibrium.

[Squeeky Wheel]

Nice analogy Nick.

I was wondering how heterogenous agents might alter behaviour. Consider the scenario where the cops collect fines from those over S* and give them to those under S* minus a cut (borrowers pay lenders minus seignorage or something). Borrowers drive fast and lenders drive slow. If the cops change the S* point upwards, borrowers pay less to lenders - I could imagine some (on either side) driving faster - trying to borrow more / lend less - and some slowing down - trying to lend more / borrow less.

My first thought (and my brain can't really wrap around this) is that it seems like the composition of preferences for borrowing/lending with the heterogenous changes could result in an aggregate reduction in speed even if S* goes up - a negative sign - if the lenders try to slow down more than the try to borrowers speed up.

My second thought is that I can't imagine borrowers wanting borrowing less at low interest rates. That suggests that there would be an asymmetric (upwards) reaction function - what we expected.

I'm probably stretching this too far.

[Nick Rowe]

Majro: OK, I *think* I'm following you now.

Squeeky: Thanks. I'm not sure I'm following you. But I did assume heterogenous agents. There's a whole distribution, with some liking to drive faster than average and some liking to drive slower than average. We could even imagine they are heterogenous in how strongly they like their preferred relative speed, relative to their dislike of paying the fine. But since nobody pays any fines in equilibrium, that latter preference cannot affect the equilibrium.

Suppose the cops raise S*, but forget to post the news. They cut i, so the real rate falls, so Y rises above Y*, so all firms want to raise their prices faster than before, relative to the expected average inflation. I don't think it matters if individual firms respond by different amounts to Y being above Y*.

[JRHullls]

Interesting metaphor. However, the moment you describe the nature of the roadway, it changes the nature of the flow of traffic, e.g.. one way, one lane, with no place to pass, then the average speed will be that of the slowest driver. If it is like a multi-laned motorway, and there is light traffic, then there is not necessarily an equilibrium, just the average of the limit of the individual driver's abilities to control his car and extract its maximum performance. Of course, this assumes they want to arrive at their destination as soon as possible and not slow down to look at the scenery.
If, however, the highway is near capacity, then the familiar bunching and unbunching of traffic can occur, much studied by highway engineers. Perhaps you have uncovered a reasonable metaphor for RBC's. The point I would make is that the nature and capacity of the roadway, rather than just the drivers behavior and response to regulation, are equally important in determining if there is an equilibrium state that can be regulated, or simply a physical limit that drivers will attempt to achieve, And a crash that blocks traffic can cause massive slowdowns and tailbacks…and in that case, the speed limit signs and enforcement have zero impact on speed.

Fun to play with.

[Nick Rowe]

JR: Sure, but this metaphor, like all metaphors, can only stretch so far. But the debate between monetarists/keynesians vs Neo-Fisherians is at its clearest and starkest when we boil it down to the essentials, and abstract away from everything except the link between monetary policy and inflation. Cops and drivers.

[J.V. Dubois]

This is an excellent post. And while I have nothing else to add that was not already said. However I like how you made this analogy work without all the confusing stuff with interest rates (short term vs long term; liquidity prefference vs loanable funds) and drives the point home in a pretty straightforward setup. It made my thinking clearer, thanks for that.

[Jeremy Fox]

Nice post Nick. Though it surprises me to think that any economist would fail to distinguish between what folks in my field would call an ESS (an evolutionarily stable strategy, i.e. an equilibrium point) and a CSS (a convergence stable strategy, i.e. a *stable* equilibrium point).

[Nick Rowe]

Thanks JV!

I think the thing it clarified for my own thinking was this: we need to think about the incentives facing the individual driver/firm, given what other drivers are doing. It's the representative agent/fallacy of composition that can mislead us into thinking it is easy for all drivers to coordinate on the first equilibrium. Writing down a macro equilibrium condition from macro equations slides that one under the rug.

[Nick Rowe]

BTW: are there any serious game theorists reading this? I updated the post to add a very short bit about trembling hand (right foot) equilibria. Am I right?

[Majromax]

@Wheel:

> if the lenders try to slow down more than the try to borrowers speed up.

You've missed the section of drivers that switch between being borrowers and being lenders, by now falling on the wrong side of S*.

The fines also aren't the sole incentive. Drivers do have an intrinsic preference for how quickly they wish to drive, relative to the average speed.

@Nick:

> Cops and drivers.

I think the true question about the metaphor's relevance is "how closely does monetary policy resemble these fines?"

What you've described looks a great deal like NGDP futures. On the other hand, it doesn't much resemble paying new money exclusively as interest on old money.

[Nick Rowe]

Jeremy: thanks!

I had heard of ESS (you sometimes see the concept used by economists, I think), but CSS was a new one on me (though I think I can see the point). But what complicates it is rational expectations. The Darwinian process is "blind" gradient-climbing, while rational agents can jump to the top of a new hill, if they see the hill and expect all others to jump too. But I think my unstable hill here is more like a saddlepoint, of some sort.

If the cops get the sign right, I think my equilibrium here is like the ESS equilibrium for (roughly) 50-50 males-females. At equilibrium, the individual's sex ratio for kids doesn't matter. But out of equilibrium it does matter, and it moves the population sex ratio back towards the 50-50 equilibrium. So it's a CSS at the population level. Is that right?

Majro: "What you've described looks a great deal like NGDP futures."

Maybe.

"On the other hand, it doesn't much resemble paying new money exclusively as interest on old money."

Yes.

[Nick Rowe]

One thing that makes it easier for me to explain why the Neo-Fiserite view is wrong, in this metaphor, is that I have 3 equilibria: S*; 0; and 200. If a Neo-Wicksellian had 3 equilibria too, it would be easier too. Maybe 200=Zimbabwe, where they stop using dollars because the stock of dollars is growing so fast, and 0=barter?

[Majromax]

> If a Neo-Wicksellian had 3 equilibria too,

The Schmitt-Grohe and Uribe paper with a "Taylor rule" did have three equilibria: the (unstable) full employment+inflation state, the (stable, because the ZLB acts as an interest floor) liquidity trap state, and the rapidly-growing hyperinflationary state. The paper did not discuss hyperinflation.

The liquidity trap state does correspond to your "0=barter" condition, because their labour model made wages more flexible as the output gap grew. That makes their system more barter-like.

[Phil Koop]

"Here I come up with a superb automotive metaphor for New Keynesian macro, and you want to talk about commie buses?"

OK, that had me laughing! If I used twitter, I would certainly retweet.

Oh yes; for the record, I think this is a great post.

[Nick Rowe]

Majro: but if the central bank sets i=0%, the unemployment equilibrium is unique, IIRC. Because the natural rate is too *high*, so you need deflation to make r=the natural rate, and you can only get deflation with Y < Y*. (Which also means that anything that raised the natural rate (like fiscal loosening) would *reduce* Y even further below Y*.)

Thanks Phil!

[Jeremy Fox]

@Nick:

"But what complicates it is rational expectations. The Darwinian process is "blind" gradient-climbing, while rational agents can jump to the top of a new hill, if they see the hill and expect all others to jump too."

Aha, good point. Still, as your metaphor shows, even with rational expectations there's still an important distinction between an unstable equilibrium and a stable one. Isn't this distinction routinely made in the macroeconomics literature? Honest question. And in asking it, I don't mean to imply that economists are fools who don't know how to analyze their own models. Ecologists and evolutionary biologists make analytical mistakes and oversights too--just not this particular one of neglecting to consider stability of equilibria. I'm just an outsider who's genuinely puzzled as to how this particular mistake could be possible.

Probably, my puzzlement just indicates my lack of familiarity with macroeconomics models and the associated notions of "stability". Thanks to rational expectations the notion of "stability" here seems to be somewhat different from notions like convergence stability in evolutionary game theory. If so, don't feel any obligation to try to teach me all of intro macro in a blog comment. :-)

[Nick Rowe]

Jeremy: good question.

In the olden days, yes we would carefully distinguish between stable and unstable equilibria. And by that we meant (for example) "does the demand curve slope down, and the supply curve slope up? So that if the price is above the model's equilibria, will there be excess supply that would cause the price to fall back down again?" Sometimes the stability analysis was formal, but mostly it was informal discussion.

But young people nowadays don't think like that. They say the model is always in equilibrium, by definition. And the only question you can ask is how that equilibrium moves over time, if a shock hits it.

I did a post on this, trying to defend the old view.

The economists who do something close to what you guys do with ESS and CSS are those who do "agent-based modelling". Get a computer, fill it with agents who follow rules of thumb, and who slowly learn from experience, let them interact, and see what happens over time.

[Jeremy Fox]

@Nick:

"If the cops get the sign right, I think my equilibrium here is like the ESS equilibrium for (roughly) 50-50 males-females. At equilibrium, the individual's sex ratio for kids doesn't matter. But out of equilibrium it does matter, and it moves the population sex ratio back towards the 50-50 equilibrium. So it's a CSS at the population level. Is that right?"

Hmm. I'm not sure that's quite right. If you're imagining an evolutionary context in which the population sex ratio is 50-50, but only because there are some individuals producing mostly male offspring, some producing mostly female offspring, and some producing a 50-50 ratio, I'm not sure that's a state in which there's no selection so that everybody has equal relative fitness. Which if so may just be another sign of the mismatch between the sorts of games that rational agents can play in economics and the sorts of games that evolving species can play. I'm not really a game theory guy, though, so I'd have to go look up the answer...

[Jeremy Fox]

@Nick:

"In the olden days, we would carefully distinguish between stable and unstable equilibria...But young people nowadays don't think like that. They say the model is always in equilibrium, by definition. And the only question you can ask is how that equilibrium moves over time, if a shock hits it. I did a post on this, trying to defend the old view."

FWIW (probably not much), I'm totally with you on the merits of the old view. Thanks to the background and training I have, the new view seems very weird to me.

And it's interesting that agent-based modeling (which seems like it hasn't gained much of a foothold in mainstream macro) is actually a throwback in terms of how it treats stability. It hasn't gained much of a foothold in mainstream ecology and evolution either, for various reasons which probably overlap with but aren't identical to the reason why it hasn't gained a foothold in mainstream macro.

[PJ]

This is brilliant. All of this talk about whether we even have the sign right on monetary policy was beginning to depress me. This makes the whole thing crystal clear and I feel much better about the state of macro. Hopefully it catches on quickly. It gets right at my fundamental problem with the neo-Fisherian view which is that there's an awkward (or absent) story about how you get to the new equilibrium inflation rate. Inflation is, after all, the result of millions of price setting decisions and not the result of an arbitrage condition.

[Min]

Case 1: “each driver must pay a fine Fi=(Sbar-S*)(Si-Sbar)”

If driver i drives at the speed limit, then Si = S* and Fi = -(Sbar - S*)^2, which is non-positive with a maximum of 0.

If driver i drives at speed 2.Sbar - S*, then Fi = (Sbar - S*)^2, which is positive with a minimum of 0. Not a good idea, even if he could do so. (2.Sbar - S* could be negative or greater than the maximum speed.)

If the average speed of all the other drivers equals S*, and driver i is not driving at S*, then (Sbar - S*) and (Si - Sbar) have the same sign, and Fi is positive. So driver i should also drive at S*.

If the average speed of all the other drivers is greater than S*, then driver i can get a negative fine by going less than the speed limit, as long as Sbar > S*, which is always possible. Similarly, if the average speed of everyone else is below the speed limit, driver i can benefit by going above the speed limit. Drivers are rewarded for bucking the trend.

Case 2: “each driver must pay a fine Fi=(S*-Sbar)(Si-Sbar)”

Now if Si = S*, then Fi = (S* - Sbar)^2, which is non-negative. Not a good idea.

If Si = 2.Sbar - S*, then Fi = -(S* - Sbar)^2, which is non-positive. Not a bad idea if it is possible. However, driver i does not know Sbar, and 2.Sbar - S* may be out of range. (Besides, there are better ideas.)

If the average speed of all the other drivers is less than S*, then driver i maximizes her payoff by driving at Si = 0. Conversely, if the average speed of all the other drivers is greater than S*, then driver i maximizes her payoff by driving at the maximum speed. Drivers are rewarded for front-running the trend.
——
Main text: “S* is the inflation target, Sbar is average inflation, the cops are the central bank, and the average firm wants to increase its price by more/less than average inflation if the real interest rate is less/more than the natural rate.”

If Case 2 is the analog of the Cochranite policy, then it is a destabilizing policy. Such a policy is not a bad idea when we are stuck in a suboptimal equilibrium. However, as a destabilizing policy it has the wrong sign.

Let’s stick with the analogy for a second. As an experienced driver I know that there is a general bias for going faster than the speed limit. Therefore in case 2 I go the maximum speed. Back to the Cochranite policy. As an adult over 30 I know that there is an anti-inflation bias. Therefore I go for low inflation or for deflation. That does not get us unstuck. Since the Cochranites believe otherwise, I am not sure that the analogy is apt.

How about case 1 as the analogy for normal CB policy? If the average inflation is below target inflation, where is the payoff to firms for price hikes above the target, i. e., for bucking the trend? Apparently the firms do not see it, since the trend is continuing. In fact, the whole idea that for the CB to announce an inflation target can alter payoffs in the manner of case 1 under current circumstances is brought into question.

[Majromax]

@Min:

> As an experienced driver I know that there is a general bias for going faster than the speed limit.

Not quite, you've just made money non-neutral in the long term. Your bias must relate to the average speed, not to any fixed speed.

[JKH]

Nick,

I haven't seen you acknowledge anywhere that Cochrane gets the sign "right" by contending that the "right" sign assumes an implied fiscal backing of monetary policy - i.e. the "wrong" sign is only the case without that assumed fiscal backing

[Too Much Fed]

Nick said: "But I did assume heterogenous agents."

I believe here is what I was trying to say in an earlier post:

Whom or What Does the Representative Individual Represent?

http://econ2.econ.iastate.edu/tesfatsi/WhomOrWhatDoesRepIndRepresent.AKirman1992.pdf

Check the conclusion at the end.

[Min]

Moi: "As an experienced driver I know that there is a general bias for going faster than the speed limit."

Majromax: "Not quite, you've just made money non-neutral in the long term. Your bias must relate to the average speed, not to any fixed speed."

Well, as I indicate later, I question the aptness of the analogies. So I am talking about driving, not about money.

One thing I like to do on long trips on major highways is to estimate the median speed of cars (when there is enough traffic). Cars in northern Colorado, with a speed limit of 75 mph, I found going 85 - 90. IMX, median speeds of 10-15 mph above the limit are common. In both northern Colorado and northern California, I have found stretches where the median speed is around 85, even when the posted limit is 65. :)

[Frank Restly]

Nick,

Driving Mode A (The Good Samaritans): Each driver tries to drive so that no fines (either positive or negative) are assessed
Driving Mode B (The Gamblers): Each driver tries to drive so that he / she maximizes the amount of negative fine he / she receives over time.

Going back to your case 2.

Fi=(Sbar - S*)(Si - Sbar)

You mention that even in off equilibrium, the net fiscal position of the cops is zero. But what about the fiscal position of drivers? Could a significant portion of drivers in Mode B prevent the system from ever reaching equilibrium?

If I am in driving mode A, I will try to drive the average speed or as best I can gauge the average speed - presumably by looking at what speed other drivers are going and keeping pace.

If I am in driving mode B, I will try to do two things - drive at a speed very far from the average and drive at a speed maximum / minimum depending on whether I think the average will be higher / lower than the speed limit. If I think the average will be higher than the speed limit, I will drive at 0 (or as close to 0 to be considered driving). If I think the average will be less than the speed limit, I will drive at 200.

Do the cops care about a volatile average velocity? We know they don't care if some people drive faster than the speed limit and some people drive slower than the speed limit as long as:
1. The average is below the speed limit or
2. Everyone is driving the average

But do the cops care if everyone drives 200 one day and 0 the next? Taking this thru to monetary policy, should the central bank concern itself with a volatile price level? Could firms adjust prices not to hit a central bank target, but to maximize the amount of negative fines they receive (Mode B firms)?

[Nick Rowe]

Jeremy: "Thanks to the background and training I have, the new view seems very weird to me."

It seems very weird to me too. But if "equilibrium" means "that which the model predicts", it's hard to talk about what would happen if we were out of equilibrium. It's all about counterfactual conditionals. I think it comes down to "trembling hand" equilibria, where an equilibrium must also be robust is some players make a slightly irrational move.

PJ: Thanks!

"All of this talk about whether we even have the sign right on monetary policy was beginning to depress me."

It depresses and upsets me too. Non-economists must think we are total idiots, if we can't even agree on basics like that. Plus I really worry about what is happening to (some of the) next generation of economists.

"Inflation is, after all, the result of millions of price setting decisions and not the result of an arbitrage condition."

Aha! Yes. I must tell that to David Andolfatto. Or you should.

Min: "How about case 1 as the analogy for normal CB policy? If the average inflation is below target inflation, where is the payoff to firms for price hikes above the target, i. e., for bucking the trend?"

If inflation were below target, the central bank would cut the nominal interest rate more than the drop in inflation, which would cut the real rate, which would increase demand, which would give each firm an incentive to raise its price more than inflation. Those are all counterfactual conditionals (because there are no exogenous shocks in this simple little model).

JKH: "I haven't seen you acknowledge anywhere that Cochrane gets the sign "right" by contending that the "right" sign assumes an implied fiscal backing of monetary policy - i.e. the "wrong" sign is only the case without that assumed fiscal backing."

Sorry. You lost me there.

Frank: your questions sound random and pointless to me.

[Min]

Moi: "How about case 1 as the analogy for normal CB policy? If the average inflation is below target inflation, where is the payoff to firms for price hikes above the target, i. e., for bucking the trend?"

Nick Rowe: "If inflation were below target, the central bank would cut the nominal interest rate more than the drop in inflation, which would cut the real rate, which would increase demand, which would give each firm an incentive to raise its price more than inflation. Those are all counterfactual conditionals (because there are no exogenous shocks in this simple little model)."

As I said, for the past couple of years US firms are not seeing sufficient payoffs to make them buck the trend, and I am not, either.

[Min]

Or are you saying that fiscal policy is necessary at the Zero Lower Bound? (I don't think you are. :))

[Steve Roth]

Hey Nick not sure if you're still reading here, but this is darned interesting:

http://cdn.static-economist.com/sites/default/files/imagecache/original-size/images/print-edition/20141025_FBC963.png

http://www.economist.com/news/briefing/21627625-politicians-and-central-bankers-are-not-providing-world-inflation-it-needs-some?frsc=dg%7Ca&fsrc=scn%2Ftw_app_ipad

Why is Canada (so consistently) right-on? All those countries, the single one that's really banging the sweet spot.

Something(s) structural? Cultural? Material/geographic? Absence of same?

Could this fact contribute to your belief in the primacy of monetary?

Maybe you've written about this. If not, would be interesting.

[Nick Rowe]

Steve: Short answer -- I don't know. (And we don't always get macro right.) The bigger puzzle is banks. Why has the Canadian banking system always been so stable, with very very few bank failures, ever? (And the answer is not "Because Canadians are a bunch of lefties who always regulate everything more than the free market US!")

[Jacques René Giguère]

Whatever the critics about its view of the U.S, banking system , that paper
http://www.ledevoir.com/documents/pdf/political_foundations.pdf
by Calomiris has interesting to say about Canada.

[JKH]

NR:

"Why has the Canadian banking system always been so stable ..."

Quick and dirty thoughts:

Fundamentally different approach (compared to the US) to the pipeline/portfolio management system for residential mortgages + an inherently conservative Canadian approach to regulatory positioning relative to the global "norm" + a regulatory system that is comparatively integrated for issues facing the banking system (less so for the securities markets)

[JKH]

It's always refreshing to see somebody from the US asking genuinely interested questions about the Canadian banking system

The Royal Bank of Canada is about the size of the 3 largest US banks combined - in proportion to the size of the economies

Which says something about the US left-wing fetish around "too big to fail"

And US hedge funds have had their heads handed to them time and again as they try and short the Canadian banks

Of course, stopped clocks remain hopeful ....

[JKH]

And concentrated banking systems are easier to regulate

[Min]

Nick Rowe: "If inflation were below target, the central bank would cut the nominal interest rate more than the drop in inflation, which would cut the real rate, which would increase demand, which would give each firm an incentive to raise its price more than inflation."

In your model #1, where Fi = (Sbar - S*)(Si - Sbar), the incentive for the greedy firm is not just to raise its price above inflation, but to raise its price above the target.

Toy example: 10 firms, Sbar = 90, S* = 100. If a greedy firm raises its price to 150 while every other firm keeps its the same, Sbar will rise to 96, and the firm will make 216 through the negative fine. If all the others raise their price to 100, Sbar will rise to 105, and the firm will lose 225. A risky strategy for the greedy firm.

Now suppose that the greedy firm raises its price to 150, but the other firms raise prices so that their new average is only 95. That makes Sbar 100.5, and the greedy firm loses only 24.75. Not so risky.

Now suppose that the greedy firm guesses that the other firms will raise their prices to 95 on average, which they do. The greedy firm raises its price to 102, making Sbar equal 95.7. The greedy firm gains 27.09. In the worst case, from the greedy firm's point of view, the other firms raise their prices to average 100 8/9. Then the greedy firm loses 1. Them's pretty good odds, as Maverick's pappy used to say. ;)

Anyway, in real life it seems unlikely that an announcement by the CB of an inflation target will lead greedy firms to overshoot the target. :)

[Miami Vice]

What's the difference (for the private sector) between paying $100 in interest to the central bank vs $100 in taxes?

[Nick Rowe]

Miami: incentives. It's not how much you pay; it's what actions you would choose to take to pay less (or get paid more).

[Miami Vice]

Are you talking about the incentives to lend or borrow?

I thought you were the quantity demanded, quantity supplied guy;)

Junior miners vs government?
Are the returns, adjusted for risk, different?

[Mike Freimuth]

Didn't make it through all the comments so this might have come up but regarding this:

"[Update: is it a trembling right foot equilibrium?? I think it maybe isn't.]"

I think it is, basically, although getting technical about trembling hand with an infinite number of drivers is a little dicey since one trembling hand/foot technically still has no effect (and with a finite number, I believe it's not an equilibrium at all). Incidentally, when I came across this concept in my graduate game theory book and asked my professor (a very good economist) something like "so is _______ not trembling-hand perfect?" his response was basically "beats me, I don't really know what that means, you'll have to just look it up." And yet economics is full of such equilibria. It's a concept that we should probably take more notice of.

[Nick Rowe]

Mike: It didn't come up in comments, IIRC. I started doing a post on it, but abandoned it. Thanks for your thoughts on this. I'm still puzzling it over.

I think you are right, that with a finite number of players, Sbar=S* is not a Nash equilibrium if the cops get the sign wrong. Good point.

With an infinite number of players, if a mass of players trembled in the same direction, and if the other players observed that tremble, it is not trembling hand perfect. But I don't know if that counts for "not trembling hand perfect".

[Nick Rowe]

Mike: I don't know if there is any formal definition of "fragility/robustness" of equilibria in game theory. But there should be:

Suppose a small random fraction of the drivers make a mistake, and all the other drivers observe this mistake and respond rationally. We get a new equilibrium.

If the cops get the sign right, that new equilibrium will be close to the S* Nash equilibrium. (And will approach it in the limit, as small goes to zero.) It is robust.

If the cops get the sign wrong, that new equilibrium will be very distant from the S* Nash equilibrium. (And will not approach it in the limit, as small goes to zero.) It is fragile.

[Mike Freimuth]

Nick,

Yes I think you essentially have it right. I don't have my textbook handy but I looked it up on Wikipedia to refresh my memory about the technical definition.

"First we define a perturbed game. A perturbed game is a copy of a base game, with the restriction that only totally mixed strategies are allowed to be played. A totally mixed strategy is a mixed strategy where every pure strategy is played with non-zero probability. This is the "trembling hands" of the players; they sometimes play a different strategy than the one they intended to play. Then we define a strategy set S (in a base game) as being trembling hand perfect if there is a sequence of perturbed games that converge to the base game in which there is a series of Nash equilibria that converge to S."
The reason I don't think this technically works as "not trembling hand perfect" is actually more subtle than I originally thought. It's not that the effect of any individual driver is infinitesimal since we would be assuming that every driver had a small chance of hitting the "wrong" speed. However, I think the *average* of their speeds will still converge to the equilibrium speed as the number goes to infinity. There should be some refinement that rules that equilibrium out though. One of us should invent one haha.

[Nick Rowe]

Mike: please check out my new post, where I attempt to define a "fragile/robust" Nash equilibrium, based on the trembling hand idea. Tell me what you think.