Boy racers in Neo-Fisherian Equilibrium with upper and lower bounds

This is not just about monetary policy; this is about how we use the concept of "equilibrium". But it is also about Narayana Kocherlakota's recent note and working paper.

There is a road from Ottawa to Toronto. There are 100 boy racers in Ottawa. The race starts at midnight. Each boy racer has a utility function U(i) = minus (S(i)-1.1Sbar)² , where S(i) is his speed, and Sbar is the average speed of all boy racers. He wants to drive to Toronto 10% faster than the average boy racer, and this is common knowledge between all boy racers.

If each boy racer has a car with an unlimited top speed, there is a unique Nash equilibrium. All the boy racers drive at 0 km/hr. Because 110% of zero equals zero. It doesn't work for any other set of numbers. This proves they will all stay in Ottawa. (That last sentence was not serious).

If each boy racer has a car with a maximum top speed, there are two Nash Equilibria: all drive at 0km/hr; all drive at maximum speed to Toronto (except maybe for the driver in the very fastest car, if his car can go more than 10% faster than the average top speed of all the other cars).

We can imagine a Mexican standoff, where all the drivers are revving their engines at midnight, just in case one driver defects from the 0km/hr equilibrium, not wanting to be left behind. But if one driver's foot trembles on the clutch pedal, all the other drivers will launch too. And the driver with the fastest car figures he might as well launch now anyway, just in case some other driver's foot trembles. Or the driver whose car can accelerate the quickest to its top speed might launch a pre-emptive strike launch. If one driver launches the rest will follow.

Each boy racer has a reaction function S(i) = 1.1Sbar. In the olden days we would have said the S(i)=Sbar=0 Nash equilibrium is "unstable", because the marginal propensity to speed exceeds one, but we aren't allowed to say that any more. We have to say it's "unlearnable" by adaptive learning, or talk about trembling hand or sequential perfection. But we end up in the same place. There is something very implausible about that "unstable" equilibrium. No individual driver has an incentive to defect from that equilibrium, but if one defects all will defect. And if each fears another might defect, all will defect. But until we modify the game to put an upper bound on cars' top speeds, that unstable Nash equilibrium is the only Nash equilibrium.

The Neo-Fisherian equilibrium is like the equilibrium where all the boy racers stay in Ottawa. (Except maybe the race starts in Peterborough - halfway between Ottawa and Toronto - and racing from Peterborough to Ottawa counts as a negative speed.) The speed of an individual car is the inflation rate chosen by an individual firm, and the average speed is the economy-wide inflation rate. The higher the inflation rate, the lower the real interest rate for a given nominal interest rate, and the faster each firm wants to raise its price relative to the average. And some firms will always be able to change their prices more quickly than others, just like some cars will be faster than others.

The equilibrium inflation rate is not something that just happens; it is something that is the outcome of individuals' choices, where each individual's choice depends on what he expects others to choose. Macro really does have, or ought to have, some sort of foundation in individual choice. Hooray for microfoundations!

If a central bank pegs the nominal interest rate permanently, the Neo-Fisherian equilibrium is the only equilibrium. That's what makes it so attractive to equilibrium theorists. Until, that is, we put upper and lower bounds on the problem and create two more equilibria, that look a lot more attractive than the interior equilibrium.

In a monetary exchange economy, I think we can see what those upper and lower bounds might be. The upper bound is perhaps what happened in Zimbabwe, where the inflation rate got so high that people stopped using the Zimbabwe dollar. Or perhaps the central bank abandons the nominal interest rate peg. The lower bound is perhaps what started to happen in Greece, or in the US during the Depression, where people resort to barter, or alternative monetary systems, because nobody can sell their goods or labour for the official money.

Introducing an upper and lower bound to the monetary equilibrium creates a game with three equilibria, where only the upper and lower bound equilibria are "stable" with a nominal interest rate peg. The unstable Neo-Fisherian equilibrium in the middle, where individual firms somehow coordinate so they raise their prices on average at exactly that inflation rate to keep the economy balanced on the knife-edge, suddenly looks a lot less plausible when there are two alternative equilibria.

Narayana Kocherlakota, like me and many others, has doubts about the plausibility of the Neo-Fisherian equilibrium. His recent brief note is simpler (though less general) than his recent working paper. Here's the intuitive gist of his note:

The central bank pegs the nominal interest rate permanently.

The equilibrium real interest rate is an increasing function of the growth rate of output (standard Consumption-Euler equation).

If the time-path of output is pinned down exogenously, the Neo-Fisherian equilibrium is then the only equilibrium. The equilibrium inflation rate must rise one-for-one with the pegged nominal interest rate.

But Narayana then puts a very small twist in the model. He introduces a small non-super-neutrality of money. The level of output is an increasing function of the inflation rate (the long run Phillips curve is not quite vertical). This makes the equilibrium real interest rate an increasing function of the rate of increase in the inflation rate.

In a finite horizon model there are now multiple equilibria. We can pick one of those equilibria by pinning down (somehow) the terminal inflation rate p(T). For a given p(T), the current inflation rate p(t) is now a negative function of the nominal interest rate i, which is contrary to the Neo-Fisherian hypothesis. By raising the nominal interest rate by 1%, the terminal equilibrium real interest rate r(T) must increase by 1% too (since p(T) is pinned down by assumption), which means that output must be growing faster, which means that inflation must be growing faster, which means that current inflation must be lower than if the central bank had not raised the nominal interest rate.

Here's the kicker: in the limit, as we stretch the horizon out to infinity, this negative effect of interest rates on inflation does not go away; it gets bigger.

Here's the second kicker: in the limit, as the Long Run Phillips Curve gets steeper and becomes vertical, this effect does not go away; it gets bigger.

(Though I should add: if the Long Run Phillips curve slopes the "wrong" way, so higher inflation causes lower output (which is perhaps more plausible due to higher transactions costs), an increase in the nominal interest rate will cause increased inflation, though not one-for-one, but the two kickers about what happens in the limit would still apply.)

Narayana, quite correctly I think, finds these results troubling. Models should have relatively robust predictions. If what happens in the limit is totally different from what happens at the limit, we have a problem.

Back to my boy racers.

If each boy racer had wanted to drive at 90% of the average speed, we get exactly the same Nash equilibrium, where they all drive at 0km/hr and stay in Ottawa, only now it's a "stable" equilibrium. We do not get multiple equilibria by adding an upper (or negative lower) bound to their speed. Any plausible equilibrium should be like that; it should be robust to minor changes in the boundary conditions. But if each boy racer wants to drive at 110% of the average speed, so driving at 0km/hr becomes an unstable equilibrium, adding boundary conditions creates new equilibria that are more plausible than the original unstable equilibrium, simply because they are stable.

And I think we can see what Narayana is doing, when he considers a finite horizon version of the same game, as being like adding boundary conditions. If the game's equilibrium is very fragile when you add or subtract or change those boundary conditions, there is something wrong with that equilibrium. We ought to get the same results in the limit as at the limit. If we don't, we have a problem. Something like the Howitt/Taylor principle (or controlling a monetary aggregate or NGDP rather than a nominal interest rate) can convert an unstable equilibrium into a stable one.

Comments

[Steve Williamson]

Actually, Kocherlakota's note is more along the lines of: Change the utility function so that it includes the speed of some boy racer who goes to Toronto no matter what, with a large weight. And the only equilibrium is where everyone goes to Toronto - and the speed of the character who always goes to Toronto has huge effects on everyone else's speed.

[Nick Rowe]

Steve: let's start in Peterborough, to make it symmetric. In my original game, with no speed limits, if one boy racer has a trembling foot and drives to Toronto at 100km/hr, the Neo-Fisherian equilibrium says the other 99 boy racers immediately head off to Ottawa at approx 10km/hr, (if I did the arithmetic right).

I'm not sure you're right about the large weight in NK's model. As Toronto gets more and more distant, the exogenous speed they all drive on the last kilometer matters more and more for the speed at which they leave Ottawa.

[Mike63]

It sure takes a lot of piling on of debt by governments (all levels), businesses and households in this country to keep our way of life going.

The following numbers are taken from the Statistics Canada website:

At the end of September, 2015 the total debt outstanding in Canada (bottom line of the credit market summary data table) was $6.70 trillion.

At the end of September, 2014 the total debt outstanding was $6.17 trillion. In the one year period from the end of September, 2014 to the end of September, 2015 it increased by $530 billion. This is an increase of 8.5%.

The approximate beginning of the global financial crisis was June, 2007. At the end of June, 2007 the total debt outstanding was $3.99 trillion. In the last 8-1/4 years it has increased by $2.71 trillion. This is an increase of 67.9%.

Looking at the total debt outstanding in Canada of domestic non-financial sectors (17th line up from the bottom of the credit market summary data table):

At the end of September, 2015 the total debt outstanding of domestic non-financial sectors was $4.86 trillion.

At the end of September, 2014 the total debt outstanding of domestic non-financial sectors was $4.54 trillion. In the one year period from the end of September, 2014 to the end of September, 2015 it increased by $323 billion. This is an increase of 7.1%.

At the end of June, 2007 the total debt outstanding of domestic non-financial sectors was $2.84 trillion. In the last 8-1/4 years it has increased by $2.02 trillion. This is an increase of 70.9%.

The start date of this Statistics Canada data table can be changed by clicking on the “add/remove data” tab at the top of the page.

http://www5.statcan.gc.ca/cansim/pick-choisir?lang=eng&p2=33&id=3780122

[Nick Rowe]

Mike63: knuckledragger implied fallacy of composition, as well as totally off-topic.

[Roger Sparks]

I am sorry but this boy-racer analogy does not work for me.

As I understand it, the Neo-Fisherian idea is that an increase in interest rates will result in an increase in inflation at some future time. If this has any possibility of realism, it seems to me that we must look to the consumption side of interest payments for the logical reasons.

Why might I suggest this?

Interest payments consist of the payment side and the consuming side (the receiver of the payment).

The payment side of interest cash flows are provided by the borrower. The cash flow needed by the payer requires additional work by him or a corresponding reduction in his spendable income stream. This can never be anything but a discouragement for the borrower and will always be a disincentive for future borrowing.

On the other hand, the consumer of interest payments receives an interest payment. This interest payment will increase his cash flow position, increasing his spendable income. This benefit will always be present so long as interest payments are continuing. With more spendable income, the consumer of interest payments has incentive to buy more products thereby increasing the potential for inflation. The products purchased may include additional income producing assets.

Summing up this comment, I think if we are to use a boy-racer analogy to demonstrate the Neo-Fisherian idea, we would need to change the goals of the race. The winner of the race should get a prize dependent upon the percentage of excellence over his nearest competitors. With this goal, the race will always be run and only the potential winners will start.

But this change still has a problem. Why should the race be run at all? Why bring the players together to play this game? The organizer needs some kind of reward. But who would provide a source of rewards? There seems to be a need for an endogenous or exogenous spark of inspiration.

[Tom Brown]

Nick,

"if one boy racer has a trembling foot and drives to Toronto at 100km/hr, the Neo-Fisherian equilibrium says the other 99 boy racers immediately head off to Ottawa at approx 10km/hr"

10km/hr or 110km/hr?

[Tom Brown]

... there's a certain "boy racer" I'd like to see chime in on this post.

[Nick Rowe]

Tom: 10km/hr. Think it should be 12.36km/hr, according to my little calculator. But remember that's a negative speed, because they are driving in the opposite direction to the trembling foot guy.

Because 1 boy racer drives at 100, and 99 drive at 1.1xSbar, and Sbar is the average speed so Sbar = (99x1.1xSbar + 1x100)/100

Roger: we are talking about intertemporal substitution, not throwing cash at people to make them richer so they spend it.

[Tom Brown]

Nick, yes, your arithmetic is correct (ignoring the sign). Thanks.

[Herbert]

Nick, I believe to are totally right with your criticism of Neo-Fisherism. When reading the boy racing story, I wished such a perfect analogue had been my own idea.

However, after reading the comments, I get the impression that no one gets your point: That RE, NK, and Neo-Fisherism are simply absurd. Benhabib et al. pointed out as early as 2001 the "perils" of Taylor rules, i. e. the fact that you get a liquidity trap or hyperinflation after the smallest deviation from the equilibrium. Such equilibrium concepts make no sense.

PS: Why do you think that stability analysis is forbidden today?

PPS: There is one minor error in your analysis: If each car has an individual speed maximum, the properties of the second, non-trivial equilibrium depend in a complicated manner on the speed distribution. With a uniform distribution, for instance, one sixth of the drivers drives at maximum speed, the other five sixth drive at a speed 10 percent higher.

[Majromax]

> Benhabib et al. pointed out as early as 2001 the "perils" of Taylor rules, i. e. the fact that you get a liquidity trap or hyperinflation after the smallest deviation from the equilibrium.

Taylor rules are proportional controllers, and with an output gap they are proportional-derivative controllers with basic Phillips-curve-like assumptions.

Absent ZLB considerations, these controllers are stable with enough gain (ensured over the long run with a greater-than-one coefficient on interest rate versus current inflation less target). However, while they are stable their equilibrium value is not necessarily equal to the target. In the specific case of a Taylor Rule, achieving the target inflation rate as equilibrium requires correctly estimating both the true natural rate and the output gap (or alternately having offsetting errors here).

From control theory, adding an integral term to a Taylor rule would over time correct for misestimation of its parameters. In economics terms, this would make a Taylor rule look like:

R(t) = A*(I(t)+I*) + B*(Y(t)-Y*) + C*(P(t)-P*(t)) + R0

where I is current inflation with target inflation rate I*, Y is current output with Y* its estimated potential, P is the current price level with P* the target path, and R0 is the estimated natural rate. For consistency, Y and P would likely be expressed in log terms.

For fixed coefficients A, B, and C, a persistent undershoot of the target inflation rate would result in arbitrarily large deviation from the target price level path, which would act to decrease the interest rate set-point as if R0 were re-estimated.

If all parameters were correctly estimated, at the ZLB this functional form would cause a central bank to maintain low interest rates for a longer period than a traditional Taylor rule. The price-level term would suppress interest rates (after "liftoff") until the price level was back on its pre-ZLB track.

[Nick Rowe]

Herbert: thanks!

When I first learned of Neo-Fisherism, I was flabbergasted. Looking back on it, I thinking it came from an overemphasis on equilibrium. "Is that the equilibrium? Then that is where we must be. This is what th model says, and we cannot interrogate the model further." No thought of how individuals would coordinate on that equilibrium. But with NK now onside, this battle is now largely won, and we are beginning to explore the interesting stuff.

You are right about that minor error.

Majro: yes, but remember one thing about control theory: it is normally applied to inanimate things that do not have expectations.

"From control theory, adding an integral term to a Taylor rule would over time correct for misestimation of its parameters."

Yes. And I think this is why we see a lagged interest rate in the estimated Taylor Rule.

[jonathan]

My perspective on Neo-fisherism has changed a bit recently. I used to agree with you that the central issue was the stability of equilibrium. But now I think it's really about commitment.

Suppose we think in terms of setting the growth rate of the money supply, rather than the interest rate. Then the Neo-fisherian claim amounts to saying that if the Fed announces a higher future growth rate of money, this will raise both inflation and the nominal interest rate.

Would this occur in practice? If people believed that the Fed would raise money growth in the future, they would expect future inflation, which would raise current inflation and the nominal interest rate. If people didn't believe the Fed, there would be no effect, and if the Fed raised the nominal interest rate anyway (which involves contracting the current money supply) there would be deflation.

So I think the confusion clears up once we bring money back into the conversation, and realize that we're talking about two different policies. The answer depends on whether the Fed raises interest rates by contracting the current money supply, or by credibly promising future money growth. Both raise nominal interest rates, but the former is contractionary while the latter is expansionary.

[Frank Restly]

Nick,

"But Narayana then puts a very small twist in the model. He introduces a small non-super-neutrality of money. The level of output is an increasing function of the inflation rate (the long run Phillips curve is not quite vertical)."

Is there a time lag between changes in the inflation rate and changes in the level of output?

"In a finite horizon model there are now multiple equilibria. By raising the nominal interest rate by 1%, the terminal equilibrium real interest rate r(T) must increase by 1% too (since p(T) is pinned down by assumption)"

In a finite horizon model with a time lag between changes in the level of output and changes in the inflation rate, you never know if your output at terminal time T has fully adjusted to prior changes in the inflation rate. And so your real interest rate at T may not be an equilibrium value, it may be a transitional value.

[Nick Rowe]

jonathan: I agree it helps to bring the stock of money back into the picture. When the CB "raises the nominal interest rate", does that mean (temporarily) raising or lowering the money growth rate, and what does it signal about the expected future time-path of the money supply?

Woodford thought he could do monetary policy analysis without money. But money keeps wanting to get back into the model.

Frank: see the note I linked to. There is no lag between output and inflation in his second equation.

[Herbert]

@Majromax: Think in terms of the simplest flexible price model with exogenous endowments. From the Euler equation you get 1/beta = R/pi_t+1.

The Taylor rule is R(pi_t). Substituting yields pi_t+1 = R(pi_t)*beta, a first oder difference equation. The equilibrium is a fixed point. Under the Taylor principle R' beta > 1, the equilibrium is locally unique. That is your point.

However, the equilibrium is also unstable: Starting with pi_t somewhat above the equilibrium, you get hyperinflation. Starting below yields the ZLB. Hence, the entire model makes no sense and confirms Friedman's dictum that interest pegs do not work.

[jonathan]

Nick:

"Woodford thought he could do monetary policy analysis without money. But money keeps wanting to get back into the model."

I would say that good macroeconomics is eclectic, and we should use different models for thinking about different issues.

Abstracting from money is quite useful in many cases, but at other times it is misleading. I think that the neo-fisherian discussion is an example of the latter.

[Frank Restly]

Nick,

From his short paper:

Y(t) = E(t) * Y(t+1) - ( 1 / sigma ) * ( i%(t) - E(t) * Inf(t+1) - r% )
Inf(t) = Kappa * Y(t)
i%(t) = i%

Y(t) = E(t) * Y(t+1) - ( 1 / sigma ) * ( i% - E(t) * Inf(t+1) - r% )

Strange that while inflation expectations (E) is expressed as time varying, Kappa is treated as fixed in time (constant).

Reading a little more here:
https://en.wikipedia.org/wiki/New_Keynesian_economics#The_New_Keynesian_Phillips_curve

Kappa = [ h * ( 1 - ( 1 - h ) * Beta ) / ( 1 - h ) ] * Gamma

From the article: "The less rigid nominal prices are (the higher is h), the greater the effect of output on current inflation."

This statement seems a little bizarre - it almost sounds like it is saying that if all prices are perfectly flexible (h is very, very big) then any change in output (real growth) will "cause" an equal / even larger change in the inflation rate. Am I reading that right?

[Tel]

The upper bound is perhaps what happened in Zimbabwe, where the inflation rate got so high that people stopped using the Zimbabwe dollar.

You presume that the boy racers know the capability of their vehicles (and each other), but there's no way a central bank can really know how far it will get away with pushing the general public on price inflation, nor can it know what the other countries will get away with. When hyper-inflation does happen, it tends to be a little at first then all of a sudden without an easily measurable event horizon.

That said, in the real world, the impact of price inflation at street level hasn't been all that bad (yet) and the central banks have sworn an oath to achieve more inflation, so either there are psych games and fakery afoot, or there's a problem with your "foot to the floor" equilibrium.

[Nick Rowe]

Tel: If boys racers are uncertain about the top speeds of their own and others' cars, it won't affect my point that there is a second equilibrium. It just makes it more complicated to figure out what it is.

[Tel]

Nick, it also makes it more difficult for the participants to figure out where the equilibrium is... but wait, the utility function does depend on what other participants are doing, so every boy racer must be pretty darn cautious about evaluating that utility function. How to calculate a derivative on a function you cannot evaluate? There might be ways to do it, but we are in somewhat different territory I think.