Price-level targeting as an automatic stabiliser for inflation

Targeting the price level could mean lower variation of inflation than targeting inflation. The best way to target inflation might be to target the price level instead. It's one of those paradoxes of pre-commitment. A promise to do something you don't want to do can affect others' expectations, and others' actions, and help you get what you want. Price-level targeting acts like an automatic stabiliser, which works even if there's a lag between the shock and the central bank's response to that shock. If expectations of the future price level matter, lags get offset by leads. And it could be better than letting bygones be bygones, depending on the parameters.

This is supposed to be a simple "teaching" post, to try to explain a point.

Here's a simple example, to show how it works:

Suppose the structure of the economy is P(t) = bP^e(t+1) - R(t) + S(t)

where P(t) is the price level at time t, P^e(t+1) is the expected future price level, R(t) is the nominal interest rate instrument, S(t) is an unforecastable serially uncorrelated shock, and 0 < b < 1 is some parameter. Assume the central bank only observes the shock with a one-period lag, so it won't be able to keep the inflation rate perfectly on target by responding immediately.

To keep it simple, suppose the central bank either adopts a 0% inflation target, or else adopts a constant price level target.

If the inflation target is credible, we can set P^e(t+1)=P(t) (people expect the central bank will keep the price level at its new higher level), so the shock has a multiplier of dP(t)/dS(t)= 1/(1-b).

If the price level target is credible, P^e(t+1) is unaffected by the shock (people expect the central bank will ensure the price level goes back down to its previous level), so the shock has a multiplier of dP(t)/dS(t) = 1.

For 0 < b < 1, the same shock has a bigger effect on inflation with an inflation-targeting central bank than with a price-level targeting central bank. But it's not quite over. Because a single shock will only affect inflation for one period if the bank targets inflation. It affects inflation for two periods if the bank targets the price level; the price level first rises, then falls again.

[Update: this might be underselling the case for price-level targeting. Because in the second period the inflation is anticipated, which is presumably less damaging than the unanticipated inflation in the first period.]

If the central bank has a loss function equal to the absolute value of the deviation of inflation from target, the critical value is b=0.5 . Targeting the price level gives twice as many losses, but each one only half the size, so it's a wash. For 0.5 < b < 1, price level targeting means that inflation is less "variable" (as defined by that loss function) than inflation targeting.

If the loss function is a quadratic, so the marginal loss is an increasing function of the deviation of inflation from target, the critical value of b will be lower than 0.5 . Two small deviations are better than one deviation that is twice as big. [Someone else can do the math, because I will mess it up.]

If b > 1, (as it will be for a long-enough time horizon) there's a problem with the stability of the equilibrium in this model (using a nominal interest rate instrument to target inflation or the price level is not feasible). So let's not go there. [Update: and I don't think b > 1 makes sense theoretically anyway.]

The important assumption is that b > 0. This is plausible. An increase in the expected future price level (for a given nominal interest rate) means a bigger incentive to spend now, which raises prices now. And if firms expect higher prices in future, they will want to raise their prices now.

(In this simple model, the central bank does not respond to the shock, either under inflation targeting or under price-level targeting. What makes the target credible is people's belief about how the bank would respond if the target lost credibility (expectations became "unanchored"), so the inflation rate or price level drifted way from the target even without a shock.)

Everything I say here should also apply to an NGDP level-path target. But it's simpler to explain the point with a price-level target.

Comments

[Matthew Young]

Eliminate t. Adding in time just tells us the central bank is not doing time travel.

So, like NGDP, it seems to me that when the price level is too high, the central bank dings borrows for an interest charge to rebalance. That means the Fed takes a gain, so there is no seigniorage. When the price level is loo low, the central banker takes a loss, not covered by government.

Who knows the current value of NGDP? Everyone, the loans to deposits are a known tradebook. This is a straight two sides corridor system, (we can do three sided also). Nothing called yield/time, no guarantees the government budget is made, not a central banking as we know it without a government budget.

[Majromax]

> The best way to target inflation might be to target the price level instead. It's one of those paradoxes of pre-commitment.

This should not be a surprise in general.

An inflation target that looks at just inflation is a proportional controller. Adding in the rate of change of inflation or the output gap (as a proxy for the rate of change of inflation) makes this a proportional-derivative controller.

However, P-D controllers have no particular guarantee of meeting their target, even if the underlying system is well-behaved. A proportional controller that (for example) tries to keep a pot of water at 80° will undershoot that target (or oscillate around it and miss in the mean) because it is 'blind' to evaporation and other heat losses.

In economics, a simple analogy is the Taylor rule: a central bank that overestimates the natural rate of interest will invariably undershoot its inflation target.

In engineering and control theory, the solution to this is to add an integral term to the controller, so as to correct for persistent undershoots or overshoots. In economics, of course, the integral of inflation is the price level.

> If the central bank has a loss function equal to the absolute value of the deviation of inflation from target, the critical value is b=0.5 .

This also depends on your time-frame. If the loss function is based on long-term inflation and the central bank targets short-term inflation, then price-level targeting almost automatically wins out.

[Nick Rowe]

Majro: It's always interesting to see the perspective of someone (unlike me) who understand control theory (right name?). But isn't there one important difference though? There's an expectation (and hence lead as opposed to lag) in economic systems. And that's why precommitment (promises to do something in future you won't want to do) matters.

I get what you are saying about the Taylor Rule. But another way to fix the Taylor Rule (the one that seems to be used in practice) is to add the lagged nominal interest rate on the right hand side. So if you keep on undershooting the inflation target (because the natural rate is lower than you think it is) the nominal interest rate keeps on falling.

[Nick Rowe]

Frank: No. It's a simple stripped-down New Keynesian model where the central bank sets a nominal interest rate and not the money stock or its growth rate. STOP.

[Nick Rowe]

Frank: because this post is not about New Keynesian vs Monetarist models of AD. And nominal interest rate is the instrument that central banks currently use, and I want it to be understood by people who think in those terms. Final warning: stop commenting on this post.

[Thaomas]

I agree with this, but I think there is a more basic way to get the same result. Presumably the damage from "inflation" is that it makes it more difficult to predict future relative prices (not rates of change of relative prices), so a price LEVEL target is a better match between target and the ultimate objective.

And there is a second reason for price Level targeting, as well. A rate of change target leaves too much uncertainty in how the CB will react to rates of change that are above or below target, as shown by the US Fed's inaction and even perverse action in the face of constant "missing" of its 2% rate of change target.

[Spencer]

Monetary policy objectives should be formulated in terms of desired rates-of-change, RoC's, in monetary flows, M*Vt (volume X’s velocity), relative to RoC's in R-gDp. RoC's in N-gDp (though "raw materials, intermediate goods and labor costs, which comprise the bulk of business spending are not treated in N-gDp"), can serve as a proxy figure for RoC's in all transactions, P*T, in Professor Irving Fisher's truistic: "equation of exchange".

And Alfred Marshall's cash-balances approach (viz., a schedule of the amounts of money that will be offered at given levels of "P"), viz., where at times "K" is the reciprocal of Vt, or “K” has the dimension of a “storage period” and "bridges the gaps of transition periods" in Yale Professor Irving Fisher’s model. RoC's in R-gDp have to be used, of course, as a policy standard.

Neither financial transactions not “animal spirits” are random:

American, Yale Professor Irving Fisher – 1920 2nd edition: “The Purchasing Power of Money”:

“If the principles here advocated are correct, the purchasing power of money — or its reciprocal, the level of prices — depends exclusively on five definite factors:

(1)the volume of money in circulation;
(2) its velocity of circulation;
(3) the volume of bank deposits subject to check;
(4) its velocity; and
(5) the volume of trade.

“Each of these five magnitudes is extremely definite, and their relation to the purchasing power of money is definitely expressed by an “equation of exchange.”

“In my opinion, the branch of economics which treats of these five regulators of purchasing power ought to be recognized and ultimately will be recognized as an EXACT SCIENCE, capable of precise formulation, demonstration, and statistical verification.”

-- Michel de Nostredame

[Majromax]

> It's always interesting to see the perspective of someone (unlike me) who understand control theory (right name?).

Right, it's control theory. I'm far from an expert myself (for that you'll want to talk to an engineer), but I have a passing familiarity with it from a mathematical background.

The entire field tries to do with physical systems what central banks try to do with the economy: control the path of some important indicator within constraints based on a combination of incomplete data and an imperfect model of the system's real workings.

> But isn't there one important difference though? There's an expectation (and hence lead as opposed to lag) in economic systems. And that's why precommitment (promises to do something in future you won't want to do) matters.

Being a bit more mathematical, a central bank that targets inflation, even almost perfectly, gives us a price-level series that contains a unit root. If businesses do not care about the instantaneous rate of inflation but instead care about inflation integrated over some time (say they're visited by the Calvo fairy every k periods on average), then they must change their prices going forward to incorporate some of the central bank's aggregate 'miss' so far because that 'miss' becomes part of the central expectation of price level going forward.

If the central bank instead targets the price level with equal skill, then the expected future price level will be indifferent to errors thus far and a Calvo-visited businesses would be indifferent to these random shocks when setting its going-forward prices. It may have lost out because of shocks in the interim, but those are sunk costs.

> But another way to fix the Taylor Rule (the one that seems to be used in practice) is to add the lagged nominal interest rate on the right hand side.

How is that stated? i(t) = π(t) + r + α(π(t) - π*) + β(y(t) - y*) + γ(i(t-Δt) - π(t-Δt) - r)? That might implicitly add some integral memory to the system, although I'd want to math out the real effects. Delay terms in control equations are scary because they can sometimes destabilize the system.

[Nick Rowe]

Spencer: I'm a fan of Irving Fisher. I could have written this post from his perspective (replace -R with +M, and let the shock S be a shock to V). A central bank using M to target the inflation rate would face a similar problem, since it won't be able to forecast V exactly.

[Nick Rowe]

Majro: remember that the Calvo fairy flies at random, so the subset of firms she touches with her wand are a perfectly representative sample of the population of firms. That (cooked-up for math convenience) assumption is important. Because it means the deviation of inflation from expected is a perfect signal of whether there is excess demand in the economy, which is what the central bank wants to minimise. ("Divine Coincidence"). But I noticed after writing this post (and added the update) that in the second period, when the price level falls under price-level targeting, this is not a deviation of actual from expected inflation, so it might be costless.

Simplest revised Taylor Rule: i = alpha(inflation gap) + beta(output gap) + i(t-1). So it's the change (not the level) of nominal interest rate that depends on the two gaps.

Or maybe just put a coefficient on i(t-1) that is close to but less than one.

[Majromax]

> Because it means the deviation of inflation from expected is a perfect signal of whether there is excess demand in the economy, which is what the central bank wants to minimise. ("Divine Coincidence").

I'm not entirely sure this works if we're talking about the reaction to shocks with the ongoing expectation of 0% inflation. If a business expects r% inflation and is visited by the Calvo fairy with probability p, it will raise its prices by r/p so that the overall going-forward inflation rate is p*r/p=r%.

However, we're talking about different ways of dealing with 0% inflation. If we're in the first world and the price level has increased by r%, a business visited by the Calvo fairy will not increase its prices by r/p%, but instead it will increase its prices by just r%: going forward it still expects no inflation. That means that there is an intrinsic 'inertia' of inflation to the tune of p*r% – although we have to be careful about just how the price level changed if not as the product of Calvo-fairy decisions.

In the second scenario, even having experienced a price-level increase of r%, a business visited by the Calvo fairy would expect the CB to make up for the shock and would choose to leave its prices unchanged going forward. The intrinsic inflation inertia is p*0=0%, which is more consistent with the CB's target in that it doesn't introduce autocorrelation to the inflation-generating process.

> : i = alpha(inflation gap) + beta(output gap) + i(t-1)

Okay, so Δi = α(inflation gap) + β(output gap) means that the central bank controls the derivative of the interest rate rather than the interest rate itself. That definitely introduces more lag into the system, although as you note it does eliminate the estimate of r. Intuitively, this feels right about how CBs behave, where the beatings (interest rate increases) continue until inflation improves and there are periodic overshoots/undershoots.

[Tel]

MM: "An inflation target that looks at just inflation is a proportional controller."

I'll agree with you on that one.

MM: "Adding in the rate of change of inflation or the output gap (as a proxy for the rate of change of inflation) makes this a proportional-derivative controller."

But, the original post was talking about attempting to target the price level, and I'm fairly sure that price is not the rate of change of inflation.

Indeed, I think that inflation might be the rate of change of price (expressed in terms of exponential growth of course)... making it a proportional-integral controller (and that's a fine thing, I have no beef against PI controllers, other than their lack of stability, but let us not speak of such things).

Anyhow, I believe that depending on how you process the Fedspeak, possibly central banks already have something like this in operation. Their argument being that after a period of unusually low inflation, they are entitled to some high inflation to compensate. That could perhaps be a target price under the hood (or at least a mapped increase in price which is then the moving target price as a function of time... no this is not the same as target inflation when it comes to control theory, once you consider steady state error).

[Nicolas]

You can find simple math for the loss-function case that you mention in this 2013 paper: https://ideas.repec.org/p/bdr/borrec/783.html

[Nicolas]

Sorry I forgot to mention the exact page (page 9) in my previous post:
You can find simple math for the loss-function case that you mention in this 2013 paper: https://ideas.repec.org/p/bdr/borrec/783.html p.9

[Majromax]

@Tel:

> But, the original post was talking about attempting to target the price level, and I'm fairly sure that price is not the rate of change of inflation.

I was thinking more about the mechanics of the Taylor Rule in specific and the Phillips-Curve arguments behind interest rate adjustments in general. The argument is that neither of these end up explicitly using integrated error so they may have a persistent bias.

At minimum, they leave the price level random walk with a unit root, which means that expected inflation over the next N periods will have mean zero but standard deviation proportional to sqrt(N).

> Their argument being that after a period of unusually low inflation, they are entitled to some high inflation to compensate. That could perhaps be a target price under the hood

Is it a target price, an explicit medium-term overshooting goal, or a reaffirmation that they believe their response function errs on the side of being underdamped (and thus prone to oscillation around the target)?

[Nick Rowe]

Nicolas: Good find!

Their model looks very similar to mine, but there is an important difference. They assume the central bank observes the shock immediately (and so responds immediately); I assume there is an information lag, so the central bank cannot respond immediately. (And their central bank does not fully offset the shock, despite observing it immediately, because doing so would cause output to deviate from potential.)

[Nick Rowe]

Thaomas: My apologies. I just found your comment in our (hyperactive) spam filter.

I agree there are other good reasons for price-level targeting rather than inflation targeting. (I would re-describe your second argument as saying a price level target makes it easier to hold central banks accountable, because they have to fix their past "mistakes".) But in this post I wanted to focus on this particular reason, which I think gets forgotten.

[Tel]

"Is it a target price, an explicit medium-term overshooting goal, or a reaffirmation that they believe their response function errs on the side of being underdamped (and thus prone to oscillation around the target)?"

Well, none of the above... and also all of the above... as is the tradition with Fedspeak they never really explain the details. Here is a fairly lucid explanation though, and I point out that at no time does he actually come out and say this is Fed policy, he only wafts around the general topic being entirely speculative and noncommittal. My explanation is that the Fed has the purpose of destroying information, but that's another discussion. Here's the quote:

The Makeup Principle

The academic literature on monetary policy suggests a variety of prescriptions for preventing a lower neutral rate of interest from eroding longer-run inflation expectations. The paper argues convincingly that many of these proposals present practical difficulties that would create a very high bar for their adoption. For instance, raising the inflation target sufficiently to provide meaningfully greater policy space could engender public discomfort or, at the other extreme, risk unmooring inflation expectations. The transition to a notably higher target is likely to be challenging and could heighten uncertainty.

As I have noted previously, the persistence of the shortfall in inflation from our objective is an important consideration for monetary policy. The makeup principle, in which policy would make up for past misses of the inflation target, is not reflected in most standard monetary policy frameworks, although it is an important precept in theory. Some of the proposals that have been advanced to implement this principle present some difficulties. For example, while price-level targeting would be helpful in the aftermath of a recession that puts the economy at the effective lower bound, it could require tightening into a negative supply shock, which is a very unattractive feature, as Bernanke points out.

Bernanke proposes a framework that avoids this undesirable possibility by implementing a temporary price-level targeting framework only in periods where conventional policy is constrained by the lower bound. Bernanke's proposal thus has the advantage of maintaining standard practice in normal times while proposing a makeup policy in periods when the policy rate is limited by the lower bound and inflation is below target. His proposed temporary price-level target would delay the liftoff of the policy rate from the lower bound until the average inflation over the entire lower bound episode has reached 2 percent and full employment is achieved. This type of policy, which would result in temporary overshooting of the inflation target in order to make up for the previous period of undershooting, is designed to, in Bernanke's words, "calibrate the vigor of the policy response...to the severity of the episode."

Now that idea of making up for what you previously lost, does sound a leeeeetle bit like an integrated error term... and there he is using the very phrase "price-level targeting" which sounds a log like the topic of this thread. In fact, I think it is the topic of this thread.

Oh, and reference linky, make me look like a scholar, n at: https://www.federalreserve.gov/newsevents/speech/brainard20171012a.htm

Similar concepts turn up in the Australian RBA supposed "dual mandate" where they talk about a long term average inflation. Note than an "average" is the sum total of a series of samples divided by the number of samples, so if you take the average of perhaps the previous 10 years of inflation numbers and use this as your target, that's not entirely removed from an integral term... it's not exactly an integral term but still contains the sum of error values. A true integral term would be an infinite sum, and they are talking (probably) about a finite sum, but you see what I mean.

[Matthew Young]

The shock term is not visible to the currency banker except via the member banks. The limited vision is by construction, no time travel, no secret information channels. You caused this when you introduce time.

Central banking is different because it is about the monopoly tax dollar, the monopoly license allows time travel. Government can do claw backs and bailouts, and central banks have advanced warning.