Finite Horizon models of inflation as the horizon goes to infinity

You have an infinite horizon model of inflation. Your model tells you what happens to the time-path of the price level P(t) if the central bank changes monetary policy.

Please do something:

Convert your infinite horizon model into a finite horizon model. Suppose the price level at time T, when the world ends, is pinned down at some fixed number P(T). Figure out what happens to the time-path of the price level P(t) if the central bank changes monetary policy, holding P(T) fixed.

Now figure out the results of a change in monetary policy in your finite horizon model, in the limit as T goes to infinity.

Do you get the same results as in your original infinite horizon model? If not, I think there is something seriously wrong with your infinite horizon model.

Take an example:

Suppose your infinite horizon model says that the real interest rate is [exogenous and] constant, the Fisher equation always holds exactly (nominal rate = real rate + expected inflation), rational expectations, the central bank sets a nominal interest rate, and if the central bank unexpectedly raises the nominal interest rate by 1% above the real rate at time t₀, and holds it there forever, the price level will start rising at 1% per year. So if the price level is initially 100 before the central bank changes policy, and would have stayed at 100 forever, it instead rises to 101 next year, and so on.

A finite horizon version of your model will give different results. Suppose that P(T) is pinned down at 100, and let T=70 years. If the central bank unexpectedly raises the nominal interest rate by 1% above the real rate, as before, the price level immediately drops to 50, and only then starts rising at 1% per year, so it doubles to 100 over the next 70 years.

If T=140, the price level immediately drops to 25.

If T=210, the price level immediately drops to 12.5.

And so on.

In the limit, as T approaches infinity, the results for the time-path P(t) in the finite horizon model differ by a larger and larger amount from the results in the original infinite horizon model.

That is not good. Your infinite horizon model has a problem.

A standard monetarist model, where the central bank sets the money supply, with a standard Cagan-style money demand function (M/P a negative function of expected inflation), would not have that problem. Proof is left as an exercise for the reader.

[This is my version of what I think Narayana Kocherlakota has recently been saying about the Neo-Fisherian model. My version differs in details, but I think it's the same in spirit.]

Comments

[rsj]

This is very clear and convincing!

[Adam P]

Nick, to my reading your interpretation of Kocherlakota is exactly correct and I think, both of you, are exactly correct in your conclusion.

[Nick Rowe]

Thanks rsj and Adam!

[John Handley]

Nick:

I think you're giving too much credit to the Neo-Fisherians when it comes to the infinite horizon model. Since prices are flexible, it's perfectly possible for the price level to jump down immediately, the Neo-Fisherian result is just one of literally infinite equilibria. All we know is how fast the price level grows.

That being said, choosing whether the future price level or the present price level is fixed as a means of choosing equilibria does seem to illustrate this point, but perhaps we should be thinking about how we select for equilibria in the infinite horizon model. This means chucking out the basic New Keynesian model in place of one with a money demand function.

[Nick Rowe]

John: I think you are onto something.

Could we think of my post as maybe helping us answer the question how we select for equilibrium, and whether we can get a sensible/robust answer to that question in a particular model? And what it says about the model if we can't?

[John Handley]

Nick:

In some sense simply have multiple equilibria in a model is enough to throw up your hands and say there's no way to figure out what the correct one is (which is exactly the opposite of what Cochrane and Williamson have done). Ideally there would be some non-arbitrary way to deal with the problem. On one hand, you can add money and then note that the Neo-Fisherian result only applies if the central bank unexpectedly increases the growth rate of the money supply forever, leaving the current money supply constant, which is 1) not at all what the Federal Reserve is currently doing and 2) impossible in a liquidity trap (because money no longer affects the price level until the indeterminate end of said liquidity trap). The conventional wisdom (in which the central bank reduces the money supply now and grows it faster for x amount of time) is always possible, because central banks can reliably increase interest rates with money supply reductions even in liquidity traps. On the other hand, you can add government debt and assume the government will never pay it off, thus forcing higher inflation so that the government doesn't violate its budget constraint, but I think you and I both agree that that is a suboptimal way to select an equilibrium at best.

[Nick Rowe]

John: good comment. But a bit unsure about your last 2 sentences.

[Roger Sparks]

As I read about the interaction of the "infinite horizon model" and the "Neo-Fisherian model", I am reminded of the use of the map direction "go north".

Of course, if we followed the direction to "go north", at the infinite limit we would be at the north pole. That is seldom the intent of the direction giver.

Intellectually, we can extend the direction of "go north" by asking the question "what happens if we keep on the same course after reaching the north pole?" Of course, the answer is that the course reverses and the path continues smoothly to the south. It seems to me that the "Neo-Fisherian model" attempts to describe what happens when, following the same path, we continue past the north pole in macroeconomic theory.

[John Handley]

Last two sentences paraphrased:
Central banks can always reduce the price level, even if they can't always increase it; FTPL is weird.

[Adam P]

Guys,

Nick's finite horizon model here, as written, does not have multiple equilibria. There is one and only one equilibrium path for prices and it's the one Nick gave.

This part:

the real interest rate is [exogenous and] constant, the Fisher equation always holds exactly (nominal rate = real rate + expected inflation), rational expectations, the central bank sets a nominal interest rate, and if the central bank unexpectedly raises the nominal interest rate by 1% above the real rate at time t0, and holds it there forever

combined with a constant P(T) does the job. There are only multiple equilibria if you vary P(T), one for each possible value of P(T).

[Adam P]

actually ignore me, I see John was saying the same thing...

[Avon Barksdale]

Nick,

“Do you get the same results as in your original infinite horizon model? If not, I think there is something seriously wrong with your infinite horizon model.”

Ohhh...., be very careful here. Consider the infinite sum:

\sum_{n=1}^\infty \frac{1}{n^s} = 1/1^s + 1/2^s + 1/3^s +...

This thing is called the Riemann zeta function: \zeta(s). Now, what happens when we let s-> -1? Well the sum becomes the sum of the integers:

1 + 2 + 3 + 5 + 6 +....

which would sensibly be infinite (divergent series). But, by analytic continuation \zeta(-1) = -1/12. So, the zeta function implies that

1 + 2 + 3 + 5 + 6 +.... = -1/12

What happened? Is there something wrong with the zeta function? Of course not! Be careful when you interchange limiting operations – sometimes limiting operations commute and sometimes they don't. The problem with trying to grab infinity by the balls is that you always miss!

[FN]

I think Kocherlakota’s post is little bit different from yours. What you claim, in my opinion being right, is that the finite and infinite versions of the two models have the same implications for first differences (except for the first term in the series!) but very different predictions for levels.
What Kocherlakota is saying is that Neo-Fisherians are working on different equilibria, assuming that the transversality condition holds in every possible equilibrium path, while moving from one solution to the other changes the terminal condition as well.

[Gilles]

This is not what Kocherlakota did. Prices are not stationary; inflation is (at least in models). So you cannot do that P(T) thing. You are starting with a fallacy then running with it.

Now, do the same thing with the simplest NK model, but with a 2 percent inflation target. Do not change interest rates or shock anything. You still end-up with the same "wrong" result. In the end, your argument is that ANY model with a non-zero inflation target is wrong.

[Steve Williamson]

"Convert your infinite horizon model into a finite horizon model. Suppose the price level at time T, when the world ends, is pinned down at some fixed number P(T)."

Now you have a problem. With a finite horizon, 1/P(T) = 0, unless you're going to fix P(T) through some government intervention, e.g. government levies taxes to retire the money at period T. But now that's a different model - the limit of the finite horizon economy is not the infinite horizon economy.

[Robert Driskill]

I'm a long-time proponent of using the limit of the equilibrium of an (analogous) finite-horizon model as the horizon goes to infinity as a selection criterion to eliminate multiple equilibria in infinite-horizon models. I spelled out the argument in detail in "Multiple equilibria in dynamic rational expectations models: A critical review," Eur. Ec. Rev., (50), 2006. Steve Williamson is I think correct: getting the finite-horizon terminal condition requires an extra assumption. But I think there are good reasons to add that assumption-or something like it.

[Scott Sumner]

Unless I'm mistaken, this is related to an argument I've been making. The best way to understand NeoFisherism is in an open economy context. In the absence of shocks, we can assume that interest parity (IPT) and PPP both hold. That really simplifies these thought experiments. A change in the interest rate target does produce NeoFisherian results regarding the rate of change in the exchange rate (IPT), and, if we assume PPP, also with the inflation rate. But the central bank can actually impact exchange rates in two ways; levels and rates of change. In the real world, most unexpected increases in the policy interest rate will cause a sharp once and for all appreciation in the currency, and then expected depreciation from that point forward. The NeoFisherians are focusing on the expected depreciation over time (associated with the higher interest rates), while the NKs focus on the once and for all appreciation, due to the sudden rise in interest rates. This once and for all appreciation shows up first, and, since prices are sticky, in the real world we see a lower inflation rate for a period of time.

My theory allows for the NeoFisherian theory to hold true in some cases, even in the short run, if the central bank adjusts the spot exchange rate enough to offset any "NK effects" and so you merely end up with the NeoFisherian results. The Swiss did exactly that in January 2015. They sharply cut interest rates to negative 0.75%. Because of the IPT, the Swiss franc was now expected to appreciate strongly over time, against the euro. Normally that action would also cause the Swiss franc to sharply depreciate on the announcement, which would create the sort of inflationary pressures predicted by NK economists. But the Swiss did something weird---on the same day that they cut interest rates sharply, they did a once and for all appreciation of the SF by about 10% or 15%. That prevented the near-term inflationary effects from the "easy money" policy of lower rates, and only the NeoFisherian effects showed up. Note that the news media treated this new policy as a "tight money" announcement; for once they got it right! The currency appreciation outweighed the lower interest rates.

In contrast, when the Fed adopts a lower than expected interest rate target, the dollar is usually allowed to depreciate, and so you get NK results in the short run.

[Tel]

Krugman used the Hotelling model to get a similar result for gold prices reacting to interest rate changes. Krugman's assumption is not so much a finite time horizon, as a finite amount of gold that gradually gets consumed (these models were also applied to fossil fuels for example). At any rate, it cannot go out to infinity.

http://krugman.blogs.nytimes.com/2011/09/06/treasuries-tips-and-gold-wonkish/

Mind you, the Fed did raise rates (a tiny bit) at the end of last year and gold prices have gone UP in response to that, so it isn't as easy as Krugman thinks.

[Nick Rowe]

Avon: I'm afraid I don't understand that zeta function stuff. But I don't think I am taking any infinite sums in this post.

FN: Yes, Narayana Kocherlakota's post is about inflation rates, while mine is about price levels. I *think* that is just a "detail", but am still thinking about this.

Gilles: Take my simple monetarist model.

In the infinite horizon version, if we increase the money supply growth rate to 1%, we get an initial jump up in P(t), then 1% inflation thereafter.

In the finite horizon version, with P(T) pinned down at 100, we get a smaller initial jump in P(t), then inflation at less than 1%, then inflation slowly falls, eventually goes negative, and P slowly falls back to 100. But as T gets bigger, the *initial part* of the time-path for P(t) gets closer and closer to the infinite horizon solution, for a longer and longer time.

So I think I disagree with you.

There is an important underlying problem with NK (Neo-Wicksellian, i.e. Woodfordian) models. Monetarist models are models of the price level (from which the inflation rate can of course be derived). NK models are models of inflation. There is always that pesky constant of integration. The only thing in NK models that lets us say anything about the price level is the Calvo Phillips Curve, which tells us that the price level cannot jump (though the inflation rate can jump). And the underlying problem here is that NK (Neo-Wicksellian) models see central banks as setting an interest rate (not, e.g. a monetary aggregate, or an exchange rate, or the price of gold). And that is what opens the door to the Neo-Fisherian problem, by leaving the equilibrium price level indeterminate.

Steve: I think I see what you are getting at. If we interpret "the world ends at time T" literally, then money (and all assets) will become worthless at time T-1, because everyone will want to sell everything to have a last consumption binge.

But I don't want to interpret it that literally. It's really just a metaphor for long run expectations. If we want to make it more concrete, let's think of it as the *monetary regime* ending at time T. The central bank (or it might even be a magic fairy) redeems all the money at a price level P(T), then issues a totally new money and starts afresh with a totally new monetary policy.

Or, maybe think about it another way: how robust is a model's conclusions to assumptions about expectations about the very distant future?

Robert: Ah! So you have been here before! (I should have expected that someone would have.) I will have a look at your paper, and see if I can understand it (hoping it's not *too* technical).

Scott: I think it is indeed related.

Yes, in a similarly simple model of an open economy, if the central bank unexpectedly announced a new crawling peg 1% per year depreciation, we would observe results that an outside observer (who did not know whether the central bank was pegging an interest rate or a time-path for the exchange rate) would be unable to distinguish from the Neo-Fisherian result. Similarly, if the central bank in my simple monetarist model announced a one-time cut in M of exactly the right size, to be followed by M growing at 1% per year, we would get the same result.

Back to the Social Construction of Monetary Policy. It's all what the CB is seen to be doing.

[Market Fiscalist]

If the natural rate of interest was 5% and the monetary authorities said "we are setting an interest rate target of 2% and we are going to keep increasing taxes and burning the money raised until we hit the target", wouldn't that drive neo-Fisherian results ?

[Scott Sumner]

Thanks Nick, I'm glad I'm on the right track. And I agree about the thought experiment involving a one time cut in the money supply, combined with an increase in the growth rate. But here's why I keep bringing up exchange rates:

1. It's really hard to observe policy announcements that clearly represent changes in the expected growth rate of money. All we know is that, ex post, a rapid increase in the money supply over a sustained period is correlated with (and causes) higher inflation and higher nominal rates. We don't have futures markets for the money supply.

2. In contrast, we have many, many policy announcements that have immediate impacts on both the level of exchange rates, and the expected future changes in exchange rates--all easily observable. We have good forward markets in exchange rates. Interest parity tends to hold up quite well (ex ante.)

3. We know that in the vast majority of cases, assets markets react to unanticipated changes in the central bank interest rate target exactly as monetarists and NKs would expect. In those cases, the foreign exchange rate value of the currency moves in the same direction as the interest rate.

4. We know that in a few cases the foreign exchange rate moves in a "perverse" direction, and in these cases we seem to get NeoFisherian results.

5. We know that in order for the foreign exchange rate to move in a "perverse" direction the government must take affirmative actions to make it do so.

Thus to use your "socially constructed" concept, central banks have succeeded in convincing the public that NeoFisherism is wrong, and that unexpected rate cuts are a signal of the government's intention to take the steps needed to gradually boost inflation over time. When they want to enact a NeoFisherian policy, they must take affirmative steps to move on two fronts, as the Swiss did in January 2015, when they simultaneously sharply revalued the franc, and cut rates sharply.

So while I agree that the thought experiment involving simultaneously changes in the quantity of money and its growth rate is a good way to think about the contrast between NK and NeoFisherian approaches, in the case of the exchange rate thought experiment we actually have a mountain of evidence to back up our intuitions.

One final point. If Cochrane, et all, succeeded in convincing the public that NeoFisherianism was correct, he would also succeed in totally destroying the EMH and the rational expectations hypothesis, as almost all asset markets would be systematically responding in the "wrong" way to almost every single monetary policy announcement. Thus NeoFisherism is a profoundly anti-market "left wing" idea, far more so than MMT or post-Keynesianism.

[Nick Rowe]

Scott: I agree that in practice, the only way I can see getting the Neo-Fisherian result would be with the crawling peg exchange rate target (or something similar). Roosevelt could (sorta) do it with the price of gold, but I don't think that carries the same signal nowadays.

MF: basically, yes, I think so.

[JKH]

Nick,

“Convert your infinite horizon model into a finite horizon model. Suppose the price level at time T, when the world ends, is pinned down at some fixed number P(T). Figure out what happens to the time-path of the price level P(t) if the central bank changes monetary policy, holding P(T) fixed.”

I follow the comparison of the two models, and what happens in the limit. The math and the logic seem reasonably straight forward.

But, not being familiar with the models, can you tell me what the intuition is for assuming that the price level is “pinned down” in the finite model? I understand what happens as a result in the comparison with the infinite model, but the resulting math (seems) so obvious, it almost appears like this “pinning down” device is equivalent to assuming the conclusion.

(I fear this is very basic to the construction of the finite model, but again, I don’t sense the intuition behind it.)

[Nick Rowe]

JKH: if we interpret "P(T) is pinned down at 100" *literally*, then we have to tell some story like: "Everyone knows that at time T the central bank will withdraw all the money by exchanging it for real goods at a price P(T), and then start afresh with a whole new monetary system".

Or we could just interpret it metaphorically, as "what would happen if expectations about the price level in the very distant future did not adjust as the model assumes they will? Do we get a *totally* different result in the model?"

Or, "Are there multiple, very different equilibria, that depend a lot on people's arbitrary expectations about the very distant future?"

[Nick Rowe]

I found an open access version of Robert Driskill's paper (see his comment above), though the equations are messed up:

https://www.researchgate.net/publication/222576476_'Multiple_Equilibria_in_Dynamic_Rational_Expectations_Models_a_Critical_Review'

He's obviously way ahead of me. But I think I'm on the same page. He actually works through the Cagan (monetarist) model I mention above (and lots of other models too). What's most interesting (to me) about his paper is he shows the relation between the different ways of solving the problem of multiple solutions, and defining a "stable" equilibrium.

[Nick Rowe]

My post is simpler and shorter though :-)

[notsneaky]

"Everyone knows that at time T the central bank will withdraw all the money by exchanging it for real goods at a price P(T), and then start afresh with a whole new monetary system".

I don't understand this part. Why would this make a difference? Suppose at time T the CB withdraws the old money from circulation and introduces new money into circulation at some explicit or implicit parity (which is known ahead of time). The world goes on. Isn't this just the infinite horizon model - not the finite one - except that before T everyone agrees we call money "old dollars" and after T everyone we call money "new dollars"? So there is no contradiction here.

I don't think you can hand-wave away Steve's point that in a finite horizon model 1/P=0. I mean, that's like almost the point of a finite horizon model.

If a new money does get introduced then to make your point you have to model what happens after the introduction.

(Incidentally, I've never seen a NK/Woodford style model which allows for changes in policy regimes - like shifts in the weights on inflation vs. output gap, for example, which obviously happen all the time in the real world (Grexit?). I'm sure they're out there)

[Mike Freimuth]

Nick,

You're just anchoring it on the front end in one and the back end in the other. To me it's not clear (at least from this) that one is more appropriate than the other. Seems to me you need some sort of theory about what anchors the price level at all in order to determine which (if either) is appropriate, which is what I've been trying to say (though meekly) for a while now. This is exactly why liquidity preference isn't enough of an explanation. It only explains differences in the rate of change in one asset (money) compared to another but this can mean very different things for the overall value depending on what else you assume.