Interest Rate Pegs with a Finite Horizon Omega Point

[I was trying to write a post on Narayana Kocherlakota's fascinating but difficult paper and post on closely-related topics, but I'm not quite ready yet, and I got distracted.]

What I used to think.

This is what I used to think about what would happen if the central bank tried to peg the nominal interest rate forever (and I'm pretty sure most macroeconomists thought the same):

If the inflation rate was not just sticky, but permanently stuck, it would be possible for the central bank to peg the nominal interest rate forever. If it pegged it too high there would be a permanent recession; if it pegged it too low there would be a permanent boom.

But if there was any price-flexibility whatsoever, so that actual and expected inflation eventually rose and kept on rising in a boom, and eventually fell and kept on falling in a recession, it would be impossible for the central bank to peg the nominal rate of interest forever. (Because higher actual and expected inflation reduces the real interest rate for any pegged nominal interest rate, which causes an even bigger boom.) The monetary system would eventually be destroyed because it would either explode into hyperinflation, or implode into hyperdeflation.

And as we changed the model to increase the degree of price-flexibility, so that actual and expected inflation adjusted more and more quickly to boom or recession, the more quickly the monetary system would explode or implode. In the limit, as we changed the model so that actual and expected inflation adjusted infinitely quickly, the monetary system would explode or implode instantly, so it would be impossible for the central bank to peg the nominal rate of interest even for a very short time. (I didn't think about what would happen at the limit of perfect price flexibility, as opposed to in the limit as price flexibility became perfect, which are not necessarily the same thing.)

Pegging the nominal interest rate creates an unstable system. The central bank can only do it for a short period of time, and only because actual and expected inflation do not adjust instantly. And so it needs to follow something like the Howitt-Taylor principle, where it adjusts the nominal interest rate more than one-for-one in response to changes in actual and expected inflation, to keep the system more or less stable.

I still think that is roughly right. But at the same time, expectations of the distant future price level matter for today's inflation rate, and not just what the central bank does today. And it is very difficult to know how those expectations of the distant future are related to what the central bank does or says today. (Which is why I keep writing daft posts about signalling, Chuck Norris, the social construction of monetary policy, and WTF!? moments.)

And the Calvo Phillips curve typically assumed by New Keynesian macroeconomists (because it makes the math easy) put a spanner in the works of modelling this. Because even though the price level is sticky in that model, the inflation rate isn't. The price level can't jump; but the inflation rate can jump, if expected inflation changes. So the formal modelling didn't fit my informal reasoning.

Finite Horizon.

Then a year ago I started thinking about finite horizon models, where the expected future price level was pinned down exogenously. [JP found an earlier old post I had forgotten about.] (Because I was wondering how central banks had managed to keep nominal interest rates pegged for many decades under the gold standard.) Here's a simple example:

Suppose that everybody know that 70 years from now (to keep the math simple) the price level will be exactly 100. Because, maybe, the government and central bank promise to retire all the paper money in exchange for real goods at that price level in 2086, and then start afresh with a completely new monetary system. So the price level at the terminal date is pinned down exogenously. That changes everything. A "permanent" (i.e. 70 year) nominal interest rate peg now becomes a stable monetary system. The monetary system cannot explode into hyperinflation or implode into hyperdeflation, because everyone knows the 2086 price level is pinned down at 100. Brad DeLong would call that 2086 price level of 100 an "Omega Point" ["page not found" when I tried to link thanks JP]. It's somewhere we know we will end up, eventually.

If the central bank pegs the nominal interest rate equal to the natural interest rate, the equilibrium 2016 price level is also 100, and people expect 0% inflation "forever" (70 years).

Finite Horizon with Flexible Prices.

If the central bank suddenly raises the nominal interest rate by 1% (one percentage point), and promises to keep it there "forever", and if prices are perfectly flexible, the price level instantly jumps down to 50 when people hear the news, and inflation is now 1% "forever".

Setting aside the initial jump up or down in the price level, the inflation rate in this model behaves exactly like Neo-Fisherians say it will. An increase in the nominal interest rate set by the central bank causes a one-for-one increase in the inflation rate (but also a one-time drop in the price level).

To get pure Neo-Fisherian results (to eliminate the one-time jump in the price level), you would need to change the model slightly. So when the central bank announces a 1% increase in the nominal interest rate, it also announces that the 2086 price level will be doubled to 200.

What happens in this model if we make the finite horizon longer and longer? The answer is that nothing much happens at all, except we get a bigger and bigger initial jump in the price level, or would need a bigger and bigger change in the promised horizon price level to offset that change. For a 140 year horizon, the same 1% "permanent" increase in the nominal interest rate would now cause the 2016 price level to drop to 25, unless the central bank promised a price level of 400 in 2156.

Another way to say the same thing is that a given change in the end-of-horizon price level matters less for the current price level, the longer is the horizon. In the limit, as the finite horizon approaches infinity, the effect of doubling the end-of-horizon price level approaches zero. But we cannot say the same about what happens at the limit, when there is nothing to pin down an Omega point. Because without any Omega point, there is nothing to pin down the current price level. Any current price level, including zero and infinity, is equally possible.

Finite Horizon with Sticky Prices.

What happens if we assume the price level is sticky, but the inflation rate is perfectly flexible (like with the Calvo Phillips curve)? So the price level can't jump, but the inflation rate can.

If the central bank doubles the 2086 price level to 200, at the same time it raises the nominal interest rate peg by 1%, we still get pure Neo-Fisherian results. There is no temporary recession, because the price level doesn't need to jump, so it doesn't matter if it can't jump.

But if the central bank holds the 2086 price level fixed at 100 when it "permanently" raises the nominal interest rate peg by 1%, there will be a temporary recession. The price level cannot jump down instantly; it can only fall slowly. So initially there will be deflation, so the real interest rate will rise, and rise by more than 1%, which is what causes the temporary recession, which lasts until the price level stops falling and starts to rise again at 1% inflation. So the short run effects of increasing the nominal interest rate are very standard; but the monetary system does not implode into ever-increasing deflation, because the 2086 price level is pinned down at 100, so it cannot.

What happens if we re-run the same experiment with double the length of the finite horizon? So the central bank now holds the 2156 price level fixed at 100 when "permanently" raises the nominal interest rate peg by 1%. The price level needs to fall by more before it starts rising again (remember it would fall to 25 rather than 50 if the price level could jump). So the deflationary recession would be longer and deeper with a 140 year horizon that with a 70 year horizon.

Now take the limit of that model as the horizon gets longer and longer. In the limit, if the central bank raises the nominal interest rate by 1% and holds it there for an arbitrarily long time, the price level approaches zero, and stays arbitrarily close to zero for an arbitrarily long time. And there is an arbitrarily long and arbitrarily deep recession.

Yep. That sounds pretty much like what I used to think would happen. If the central bank pegs the nominal interest rate too high for a very very long time, the monetary system implodes into hyperdeflation.

Someone else can do the math. Please.

Comments

[Herbert]

Very good post, Nick. Perhaps one should point out that what you consider is essentially Wicksell' celebrated cumulative process. Curiously, this process cannot be represented in a fricionless modell (because no stable equilibrium exists under an interest peg).

[Nick Rowe]

Thanks Herbert! Yep, this is Wicksell, with an extra time-derivative. I can't remember if Wicksell implicitly had a gold-standard Omega Point at the back of his mind?

[Majromax]

> Now take the limit of that model as the horizon gets longer and longer. In the limit, if the central bank raises the nominal interest rate by 1% and holds it there for an arbitrarily long time, the price level approaches zero, and stays arbitrarily close to zero for an arbitrarily long time. And there is an arbitrarily long and arbitrarily deep recession.

I think this isn't quite correct since for extreme changes you need to use logarithmic math. For an omega point in year Y and current year y, the recession ends when the price level is given by 100*1.01^(y-Y).

If sticky prices limit deflation to -d% during a recession, the price level of an initial recession will be given by 100*(1-d)^y. (1-d < 1)

The intersection of these curves determines the length of the recession, giving (y-Y)*log(1.01) = y*log(1-d), or y*(log(1.01)-log(1-d))=Y*log(1.01), or y=Y*log(1.01)/log(1.01/(1-d))

That means that given a fixed downward price flexibility, the recession will always last for a fixed portion of the time to omega-point. For example, if prices can fall by 5% per year during a recession, an arbitrary 1% increase in the nominal interest rate will cause a recession for log(1.01)/log(1.01/0.95) = 16% of the time to the omega point.

Incidentally, this omega-point thinking strongly suggests that a central bank that targets a crawling price level is superior to one that targets the inflation rate. An inflation-targeting central bank that raises interest rates by 1% but has the explanation lost on the way to the printers will send a mixed message on whether it is changing its omega-target or reacting to current conditions; one that targets a price level must be reacting to current conditions.

[Nick Rowe]

Majro: good simple math example. Don't think it really contradicts what I said though, it's just you need to interpret my "arbitrarily close to zero" relative to the initial 100, in absolute (not log) units. The bit about the deflationary recession lasting for a fixed portion of the time until the horizon is good enough for my conclusion.

"Incidentally, this omega-point thinking strongly suggests that a central bank that targets a crawling price level is superior to one that targets the inflation rate. An inflation-targeting central bank that raises interest rates by 1% but has the explanation lost on the way to the printers will send a mixed message on whether it is changing its omega-target or reacting to current conditions; one that targets a price level must be reacting to current conditions."

Exactly. See my very short old post on this.

[JP Koning]

"Then a year ago I started thinking about finite horizon models, where the expected future price level was pinned down exogenously."

You had a previous post on this, I think:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/10/sign-wars-with-price-level-targeting.html

[Market Fiscalist]

If I understand this post correctly I think you are saying that within a model with a Finite Horizon Omega Point Neo-Fisherisms eventually holds, it just might take a very long time , and a very deep recession to reach the point where it does hold.

Do you think the world we live in has a Finite Horizon Omega Point ? Do people form any expectations of the distant future price level that affects today's behavior in a meaningful way ?

[Nick Rowe]

JP: Oooops! My memory is failing. I can't even remember what *I* have written. Well-spotted!

[Tom Brown]

@JP Koning, that's hilarious!... he even used the same "70 years to keep the math simple" ... almost like he's plagiarizing himself.

@Nick, I like the post. Pretty clear, even for me to follow. And I've often wondered how you keep straight what you've written before (since you're so prolific). (but I would think the 70 year thing would ring a bell!) ;D

One thought on this: Noah Smith here in his point #2 speaks of "Scope conditions" in the natural sciences and macro. Personally I'm a huge fan of taking models to the extreme limits to see what the model says happens (I do that all the time myself). However, I'm wondering if "Scope conditions" might make models invalid outside their intended scopes. In this case, perhaps this model isn't inherently valid over all time scales. Just a thought.

[JP Koning]

"Yep. That sounds pretty much like what I used to think would happen. If the central bank pegs the nominal interest rate too high for a very very long time, the monetary system implodes into hyperdeflation."

So what do you think now? Is there a real-world counterpoint for the Omega point, and do you think this ties down the inflation rate? I believe you've been skeptical in the past about the existence of this sort of point (In your old post, you say:"Unfortunately, we don't live in a world like that. Nothing pins down the price level in 2084, or in any future year."). Has the failure of a hyperdeflation to emerge over the last five years changed your mind, or tilted it a bit?

Very well explained post, by the way. I had been reading your interaction with Kocherlakota on Twitter with much interest; this answered all my questions.

[Tom Brown]

... well, I can answer my own question: of course models are "invalid" outside their intended scope. That's not your point here: I understand that. I can't quite put my finger on what bothers me about this... I guess it's a vague hope that in a real model that actually affects policy, that somebody has been very careful about reasoning from results at infinity.

[JP Koning]

By the way, here is the DeLong link:

http://www.bradford-delong.com/2015/06/new-economic-thinking-hicks-hansen-wicksell-macro-and-blocking-the-back-propagation-induction-unraveling-from-the-long-run.html

[Nick Rowe]

MF: "Do you think the world we live in has a Finite Horizon Omega Point ? Do people form any expectations of the distant future price level that affects today's behavior in a meaningful way ?"

Under inflation targeting, there is an Omega point for the inflation rate, but not for the price level. There would be an Omega point for both under Price level path targeting.

Tom: a few weeks back I had composed a post in my head, while in the shower and driving to work, and then found by chance I had already written the almost exact same post, right down to the metaphors, a couple of years back. Old age. Things ring a bell, but I never know if it's in my head or on the blog.

Regarding "scope". The key distinction here is between what is true *in the limit* and *at the limit*, as time goes to infinity. (I daren't even think about probabilistic limits yet, but I'm planning a post where P can hit an upper or lower bound, at which point the monetary regime collapses because people resort to barter or a different money.)

JP: Thanks. And thanks again for finding the Brad DeLong link. That was a great post of his, that needs integrating into this stuff.

I don't think there is an (effective) Omega point under the current regime. Gold used to be the Omega point. NGDP level path targeting (or price level path targeting) would create one.

[Tom Brown]

"Things ring a bell, but I never know if it's in my head or on the blog." Well you can try running it past JP: he's apparently pretty good about keeping track. ;D Plus it never hurts to redo a good post. Also I think that coming up with the same metaphors and examples shows that you're at least remarkably consistent.

[Nick Edmonds]

FTPL provides an omega point for the price level, provided we see fiscal policy as a fully independent variable. If a) government freely sets real surpluses, b) the interest rate is the rate paid on government debt; and c) the relationship between inflation between one period and the next is determined by the Phillips curve relationship, then (at any point in time) there is only one price level that ensures the long-term government budget constraint is met.

The key thing here is accepting the idea that the government freely sets fiscal policy and the path of prices follows. If instead, fiscal policy is designed to try to meet the budget constraint it becomes indeterminate again.

[Tom Brown]

@Nick E: FTPL = Fiscal Theory of the Price Level?

[Nick Rowe]

Nick E: FTPL depends also on r > g, doesn't it? And how would FTPL determine the price level if government bonds were indexed, or if net government financial liabilities were zero, or even negative? Or if the central bank gave all its seigniorage profits to the United Way (a charity) instead of to the federal government? And a given Present Value of seigniorage profits is compatible with many different time-paths for inflation.

But you could perhaps say I am invoking FTPL when I assume the government-owned central bank buys back all the outstanding money at a price of 100 in 2086.

Tom: yep.

[John Handley]

Nick,

I think you'd agree with me by saying the primary difficulty with an interest rate peg (at least in a rational expectations and forward looking price setting context) is pinning down the initial level of the price level (or the initial inflation rate, depending on how you want to think about it). In a basic two (or one, without a monetary policy rule) equation friction-less model, the problem becomes clear:

You, of course, have the euler equation

(1) i_t = r + pi_t+1

and then some monetary policy rule. Say, for example, the central bank sets the path of interest rates exogenously at period zero which, given (1) completely defines the path of inflation rates except for pi_0. pi_0 can take any value in equilibrium given an interest rate peg. This is pretty uninteresting when prices are completely flexible and the path of interest rates is completely exogenous, but bear with me.

In a normal infinite horizon model, there are only two solutions to this problem. Either, the central bank can commit to an explosive path for inflation (by following the Taylor Principle) so long as inflation is not on target or, provided there is some channel through which the central bank pays seigniorage revenue to the fiscal authority, the fiscal authority can commit to blowing up the stock of government debt if inflation is not on target. Neither of these solutions are desirable, since they require either the central bank or the fiscal authority to threaten off-equilibrium paths for either inflation or debt.

From here, I can see why you might turn to a finite horizon approach. After all, it solves the indeterminacy problem, but, if anything, it brings forth even more misgivings than the previous two solutions (to me, at least). It seems strange to exogenously fix the final price level, especially since money is neutral in the long run in both flexible price and NKPC models. Furthermore, the data at which the price level is fixed is completely arbitrary.

I guess I'd really just like to ask one question: why finite horizon instead of some other way of dealing with indeterminacy at an interest rate peg?

[John Handley]

In my last comment, I accidentally wrote "data" instead of "date," hope that clears it up.

[Nick Rowe]

John: I was initially interested in the finite horizon approach because I was thinking about the fact that central banks in the gold exchange standard world could peg the interest rate for a very long time without the system exploding or imploding. Gold provided an Omega Point. Then it was a way to think about the difference between inflation targeting and price level path targeting. But the present post was inspired by Narayana Kocherlakota's paper and post, where he has a finite horizon. I simply wanted to get my own head clear on the finite horizon case, and the limit of the finite horizon case as the horizon goes to infinity, before trying to figure out what is going on in his model.

I'm not 100% sure I want to "deal with" the indeterminacy problem. Maybe the theory is trying to tell us something about real world indeterminacy. But it seems sensible to explore many possible ways of "dealing with" the problem, in order to understand it better. What would it take to deal with it? What are the implications of various different ways of dealing with it? Are any of those ways remotely plausible? What policies would plausibly deal with it?

[Nick Edmonds]

"FTPL depends also on r > g, doesn't it?"

No. Not in this context. It merely requires that r < > g (but if r < g, we have to replace surpluses with deficits). That's because I'm not taking the constraint as being any kind of limit on what the government can do, but rather as what determines the long run steady state. But if r = g, then it is indeterminate. As it is if debt is indexed or there is no debt. But, in general......

[Nick Rowe]

Nick E: If r < g, the present value of primary surpluses is undefined in an infinite horizon, so it cannot determine the price level. You are taking the sum of an infinite series whose items do not converge to zero in the limit. I think that's right.

[JKH]

very clearly explained

[Nick Edmonds]

OK - I see that.

[Tel]

Because, maybe, the government and central bank promise to retire all the paper money in exchange for real goods at that price level in 2086, and then start afresh with a completely new monetary system.

A government promise? Stack it in the corner with all the others.

Seriously, the general rule is that the further an event is into the future, the less predictable it is. Although things like a solar eclipse might be exceptions to that rule, when it comes to human affairs they were arguing up to the last day about each and every Fed meeting where interest rates could have risen. Large numbers of pundits got it wrong each and every time. Although the finite horizon makes an interesting mathematical exercise, that's all it is.

[Tel]

Yep. That sounds pretty much like what I used to think would happen. If the central bank pegs the nominal interest rate too high for a very very long time, the monetary system implodes into hyperdeflation.

I think this is kind of silly. How can the central bank "peg" the rate at which people loan money to each other? If general bank rates are high, people will obviously search around amongst relatives and friends to borrow from and those relatives and friends will be enthusiastic about lending at such high rates. That's what high rates do, they are an incentive to the lenders. The only reason banks have an advantage in this area is government support and a supply of cheap money. As soon as that dries up, a million other avenues of lending will emerge just like they have for all of history.

In order to have the type of deflation you talk about, money needs to be sucked out of the system... but the central bank can't suck the cash out of my hand, even if they could do, I could still lend my labour, or I could accept someone else's IOU note if I trust them, or go back to gold and silver coins. Government can of course tax hard and spend little and use that mechanism to suck money out of the system, but any government that tried it would get booted.

[W. Peden]

Tel,

The central bank can sell assets for base money or encourage banks to hold base money as reserves, thus reducing the total cash available in the system. So they can.

That people could then switch to gold coins etc. is exactly what I think Nick means by the monetary system "imploding". It's the deflation equivalent of how people in extreme hyperinflations start to (and sometimes complete) abandoning the currency.

[Nick Rowe]

W Peden: Yep, though I hadn't thought of people switching to gold coins when the monetary system implodes, but I think it's possible. What I had in mind was something closer to Tel's "or I could accept someone else's IOU note if I trust them," or various forms of private monetary systems, or even direct barter. That fits exactly with my monetary disequilibrium perspective, and seems to be born out empirically in e.g. Greece, or the US in the Great Depression.

JKH: Thanks.

[John Handley]

Nick,

"I was initially interested in the finite horizon approach because I was thinking about the fact that central banks in the gold exchange standard world could peg the interest rate for a very long time without the system exploding or imploding. Gold provided an Omega Point."

Now that I think more about it, indeterminacy problems shouldn't be worrying in a real world context, since we've already had a monetary policy that pins down the 'initial price level' before the regime change to an interest rate peg. Suppose the central bank, at period zero, commits to blowing up the price level in the future unless pi_0 is exactly on target. After this, the path of future interest rates is set by the central bank, regardless of the rate of inflation. This equilibrium should be stable and, as long as there are no frictions in the economy, completely Neo-Fisherian. This is effectively the exact opposite of the finite horizon case, since pi_0 is fixed instead of terminal inflation.

[Nick Rowe]

John: last period's price level is pinned down by history, but the *initial* price level, on the first day of the new policy, can be anything whatsoever (with perfect flexibility). Stock prices jump on news; the inverse of the price of money can jump on news of a new policy.

[John Handley]

Nick,

I guess this question might be better addressed to Kocherlakota than you, but how exactly is fixing the terminal price level less egregious than fixing the initial price level? Both seem useful as a way of coming up with rationalizations (e.g., I can conclude that, in a friction-less model, raising the nominal interest rate increases the ratio of the future price level to the current price level, which could mean, in a finite horizon case, that the current price level falls while the future price level remains fixed, or, if the current price level (or current inflation) is fixed, the future price level must rise), but is it really reasonable to draw any more serious inferences from either case?

[Nick Rowe]

John: with any asset, the price you would be willing to pay for it today depends on your expectation of the price you could sell it for tomorrow, and not vice versa. My current choice depend on my expectations about the future, and not vice versa. It would be a strange world in which individuals thought: "I am choosing to pay P(t) today, therefore I must rationally expect P(t+1) tomorrow".

The whole Neo-Fisherian question gets resolved if we think about central banks as setting M rather than i. Does raising i mean increasing or reducing the growth rate of M?

[Tel]

The central bank can sell assets for base money or encourage banks to hold base money as reserves, thus reducing the total cash available in the system. So they can.

If we are talking about the USA and the Federal Reserve, they purchased financial assets during a period of low interest, and you are suggesting they might sell into a period of high interest. Problem is, this entails selling at a loss. Not strictly a problem for the Fed to make a loss, other than the question of whether you can dry up the supply of money by printing new money and then handing it out. Seems like right now the Fed are going the "Reverse Repo" path which is kind of similar to selling assets, but also promising to buy the same assets back again... and that to me sounds a bit like an alternative way for the Fed to make a loss.

In terms of paying "Interest on Reserves"; once again the Fed is attempting to dry up the money supply, by printing more money and injecting it as interest payments. Dunno how that's going to work... as interest rates go up, this money injection has to get larger in order to maintain that interest on reserves. Seems a bit inflationary, although I admit the higher interest rates are themselves deflationary, could potentially balance out.

Inflation is always and everywhere a monetary phenomenon.

That certainly is the thing that bothers me.