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) = bPe(t+1) – R(t) + S(t)
where P(t) is the price level at time t, Pe(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 Pe(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, Pe(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.
Worthwhile Canadian Initiative 
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.
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.
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.
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.
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.
Monetary policy objectives should be formulated in terms of desired rates-of-change, RoC’s, in monetary flows, MVt (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, PT, 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
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.
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.
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).
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
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
@Tel:
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.)
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.
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:
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.
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.