# Some Skepticism About Level Targeting

The conventional Market Monetarist view of monetary policy can be summarized by two points:

1. Target the forecast.
2. Target the level.

David Beckworth articulates the latter point as follows:

What these inflation critics miss is that the Fed could actually raise the level of aggregate nominal spending by a meaningful amount without jeopardizing long-run inflation expectations. This is possible if one uses a price level or a NGDP level target that provides a credible nominal anchor.

If this is indeed correct, then perhaps it is a conundrum as to why the FOMC doesn’t adopt a level target. However, it is not necessarily clear that this statement is true. If the Fed were to adopt a target of the price level or the level of nominal GDP, could they keep inflation expectations stable? It is likely that this depends on the aggressiveness of monetary policy. In fact, if the Fed were to adopt the type of policy described by points 1 and 2 above, it would actually be the case that more aggressive monetary policy would lead to the potential for self-fulfilling expectations. As a result, there is reason to be skeptical of level targeting.

To illustrate this point, consider a simple model. First suppose that the price level is governed by a Wicksellian process:

$P_t = \alpha E_t P_{t + 1} + \beta(r - i_t) + e_t$

where $P_t$ is the price level, $r$ is the natural rate of interest, $i_t$ is the market rate of interest, $\beta$ and $\alpha$ are parameters, $E_t$ is the expectations operator, $e_t$ is a stochastic shocks, and all variables are expressed as logarithms.

In addition, suppose that monetary policy is governed by points 1 and 2 above such that:

$i_t = \delta(E_t P_{t + 1} - P^*) + u_t$

where $P^*$ is the target for the price level, $u_t$ is a monetary policy disturbance, and the variables are expressed as logarithms. This rule captures the market monetarist objectives of a level target and a forecast target.

To simplify the analysis, it is assumed above that the price level target and the natural rate of interest are constant over time. To simplify this even further, let’s re-write the equations above without these constant terms:

$P_t = \alpha E_t P_{t + 1} - \beta i_t + e_t$

$i_t = \delta E_t P_{t + 1} + u_t$

Now, substituting the second equation into the first, we can get a rational expectations difference equation for the price level:

$P_t = (\alpha - \beta \delta)E_t P_{t + 1} + \epsilon_t$

where $\epsilon_t = -\beta u_t + e_t$. A necessary condition for a unique rational expectations solution is that $|(\alpha - \beta \delta)| < 1$. Thus, the more aggressive the response of monetary policy to deviations of the expected price level from its target, the less likely this condition is to hold. In addition, if this condition is not satisfied, then the price level will be subject to self-fulfilling expectations. In other words, if the Fed responds too aggressively to the deviation of the expected price level from target, it is possible that price level expectations could become unanchored thereby generating self-fulfilling fluctuations.

It is important to note that the conclusion above is not unique to an interest rate rule. We can re-write this as a modified Cagan model with rational expectations. Thus re-write the first equation:

$P_t = \alpha E_t P_{t + 1} + \beta m_t + e_t$

where $m_t$ is the money supply. Now describe a rule for the money supply that is consistent with points 1 and 2 above (agains, suppressing constants for simplicity):

$m_t = -\theta E_t P_{t + 1} + u_t$

Substituting the monetary policy rule into the first equation yields:

$P_t = (\alpha - \beta \theta)E_t P_{t + 1} + \epsilon_t$

where the variables are as defined above. Notice that the greater the responsiveness of monetary policy, the less likely there is a unique, rational expectations equilibrium.

So what is the source of this instability? Why is it that price level expectations can become unanchored? The reason is that an aggressive monetary policy is designed to rapidly converge to the desired price level. For example, if inflation expectations have been 2% for some time and the inflation rate rises to say 7%, the public might lose confidence in the central bank to maintain price level stability.

The idea that rapid convergence is behind the results above can be illustrated by modifying our basic framework. Suppose that rather than focusing exclusively on the price level, the Fed were to also place emphasis in the expected rate of inflation in their monetary policy rule. Thus, the rule could modified such that:

$i_t = \delta(E_t P_{t + 1} - P^*) + \lambda (E_t P_{t + 1} - P_t) + u_t$

Again, ignoring constants and substituting this rule into our equation for the price level, we arrive at:

$P_t = {{(\alpha - \beta \delta - \beta \lambda)}\over{1 + \lambda}} E_t P_{t + 1} + \epsilon_t$

Now the condition for a unique equilibrium is that $|{{(\alpha - \beta \delta - \beta \lambda)}\over{1 + \lambda}}| < 1$. Thus, for a given responsiveness of monetary policy to the expected price level, a greater responsiveness of the central bank to the expected rate of inflation can ensure a unique, rational expectations equilibrium.

What this very simple model illustrates is that a monetary policy that is consistent with points 1 and 2 above does not necessarily keep price level expectations anchored. If monetary policy is too aggressive, price level expectations can become unanchored thereby resulting in the potential for self-fulfilling price level fluctuations. The lesson then is that a policy consistent with points 1 and 2 above must also be mindful of the rate of change in the variable for which the Fed is targeting the forecast. A policy consistent with points 1 and 2 alone might not be sufficient to keep the price level anchored.

### 24 responses to “Some Skepticism About Level Targeting”

1. wehat nonsense. Next suppose that the price level is governed by the movement of venus in the heavens. Then what.

You assumr what you are trying to prove with those wixcksellian processes and rational expectsations, etc.

If one rejects the money is neutral axiom -as Keynes did, and even Milton Friedman did at least for the short run — then there is no direct relationship betwee monetary policy and the forthcoming level of prices.

So lewt us talk real world economics for a change

2. I think this model is wrongheaded.

An excessively strong reaction to errors might cause lots of problems (finanical and real output instablity) but not an unstable price level.

I don’t really understand the argument that if people are used to 2% inflation and they observe 7% inflation they will lose confidence in the central bank.

That is assuming some kind of adaptive expectations where everyone is focused on predicting the rate of change in the price level.

This might make sense if there was no clear target for the price level. That is, if there was inflation targeting or even unemployment or quantity of money targeting.

But with price level targeting, the target for Pt+1 is fixed. Assuming it is constant, then if Pt is not on target, then of course the expected inflation rate will not be zero. Confidence in the central bank would mean expecting the inflation rate will be the negative of the last one.

Basically, I think alpha will be really big and beta really small. While interest rates might impact other things, like output, prices would end up very sticky due to the level target.

I don’t see how the instability can be anything other than ossilations, and that is implausible because why would people buy when prices are high rather when they are low? Why do they sell when prices are low rather than when they are high?

You would need to create big changes in aggregate demand to get people to change prices (which is the selling part) and big changes in interest rates to get people to change spending.

Interest rates are super low this period, so I should buy, but prices are higher now and will be lower if I wait. And prices are really high, so firms want to sell now, expecially because they will be lower if they wait.

High alpha and low beta, I think. And the expected price level is the target.

But whether or not I am right, the focus on inflation is a mistake. It is the price level that is determined and inflation is always expected to just return. And there are powerful forces dampening ossilations.

3. “That is assuming some kind of adaptive expectations where everyone is focused on predicting the rate of change in the price level.”

But this isn’t at all what is happening. The individuals in the model have rational expectations.

What is happening in the model is that the price level is determined by expectations of the price level and the deviation of the market interest rate and the natural rate of interest. The central bank is targeting the forecast. Both the household and the central bank are forward-looking. The result of the model comes from this fact. Individuals expectations of the price level are dependent on the monetary policy response.

“Basically, I think alpha will be really big and beta really small.”

Now we are getting somewhere. If that is the case, then the response of the central bank has lower weight. Suppose that alpha = 0.9 and beta = 0.1, for example. Now suppose that delta = 1.5. In this case, (alpha – beta*delta) = 0.9 – .15 = 0.75. This is clearly below 1 in absolute value and therefore the level target leads to a unique equilibrium.

BUT, the problem here is that I have simply posited the price level equation. We cannot know what the values of these parameters are because we don’t know where they come from. So how can we form inferences about alpha and beta?

“But whether or not I am right, the focus on inflation is a mistake.”

Let’s consider the monetary policy rule with money (so we don’t have to talk about negative nominal interest rates). To do so, however, we need to incorporate constants again. Here is the rule:

$m_t = m - \theta(E_t P_{t + 1} - P^*) - \Lambda (E_t P_{t + 1} - P_t)$

The monetary policy rule governs the behavior of the money supply around the steady state, m. Now, suppose that the price level is 10% below target. and the expected inflation rate is 2% (i.e. the Fed has fallen behind the path — or belatedly adopted the price level target). Suppose that they place no weight on expected inflation. This implies that the money supply would be:

$m_t = m - 0.10 \theta$

Whereas, if the central bank places weight on expected inflation, the money supply is given by:

$m_t = m - 0.10 \theta + 0.02 \lambda$

This rule simply suggests that central bank should use a more tempered response in targeting the price level. I don’t know why this is a mistake. All it implies is that the Fed should move more gradually toward its target.

4. Yet another reason to directly target the SPX – TIPS spread beta at zero.

5. Josh: what happens when you replace m(t) on the LHS of your money supply rule with m(t+1) ? (Or maybe m(t)-m(t-1) )?

You see, there is something funny about your money supply rule. If you want to target Ep(t+1) your control variable should be m(t+1) too.

6. Nick,

I don’t think that is correct. You are choosing the money supply today that will produce the desired price level in the future. If you are choosing the money supply tomorrow that will produce the desired price level in the future, then you might be subject to the circularity critique. In addition, this is consistent with the literature. A forward-looking inflation target rule is often given as $i_t = \phi_{\pi} E_t \pi_{t + 1}$. What would it be different for monetary policy.

Even so, suppose that I modified the model such that:

$M_t = \rho M_{t - 1} - \theta E_t P_{t + 1} + u_t$

Substituting this into the price equation now yields,

$P_t = (\alpha - \beta \theta)E_t P_{t + 1} + \beta \rho M_{t - 1} + \epsilon_t$

The conditions for a unique equilibrium are still the same.

The problem lies in the fact that both households and the central bank are entirely forward-looking. And this is my point. Suppose that we modified the rule so that the Fed were adjusting the money supply to both expectations and the inflation rate:

$M_t = -\theta E_t P_{t + 1} - \Theta (E_t P_{t + 1} - P_t) + u_t$

Substituting this into the price level equation yields:

$P_t = {{(\alpha - \beta \theta - \beta \Theta)}\over{(1 + \Theta)}} P_{t + 1} + u_t$

7. Oops. I cut myself off. That last equation shows that when you incorporate contemporary variables, then it is more likely that you get a stable solution.

8. Josh: What bothers me is that you have a model in which monetary policy (whether defined as i or m) has an immediate impact on contemporaneous P. If we lived in such a world, with no lags, why would we target expected future P? Why not just target current P?

Normally, central banks target expected future prices (or inflation) because they think that prices are sticky (or inflation is inertial) so it takes time for m to affect have its full effect on P.

Maybe that’s consistent with what you are saying.

9. Or, if we did want to target expected future P, we would make a commitment about how future m would respond to future shocks.

10. Nick,

I think we might be talking past one another.

The view of many market monetarists is that the central bank should target the forecast and level of the target variable. The purpose of my post is to express skepticism about this view. I think that you are similarly expressing skepticism about this view, but it is a slightly different form of skepticism.

With that being said, let me respond to some of your comments. The conventional way to target the forecast is to adjust the monetary instrument today so that the forecast of the target variable is equal to the desired level. Scott Sumner makes the point in his Economic Inquiry paper that this is one way to ensure that a policy that targets the forecast is immune from the circularity critique levied by Bernanke and Woodford.

In virtually every micro-founded model we can write down, money is going to have a contemporary impact on the price level — even if prices are sticky. Thus, in the monetary policy rule above, if $\beta$ is close to zero, one interpretation could be that prices are sticky. Alternatively, if there is inertia in the monetary policy rule (as I alluded to in a comment above), then there will be a lag in the effect of money on the price level. (This is one problem with the model above. There are no micro foundations so it is hard to interpret the parameters.)

But here is the most important point. You stated that if money has a contemporaneous effect on the price level, then you should place weight on the current price level. That is precisely what my conclusion is above. Incorporating the expected inflation rate implicitly incorporates the contemporaneous price level.

11. David Beckworth

Two thoughts Josh.
(1) You initial Wicksellian process equation has a real natural interest rate and a nominal market interest rate. The nominal interest rate is i = r + (E(t)P(t+1) – P(t). If you account for these other price level terms what does this do to your solution?

“What is happening in the model is that the price level is determined by expectations of the price level and the deviation of the market interest rate and the natural rate of interest. The central bank is targeting the forecast. Both the household and the central bank are forward-looking. The result of the model comes from this fact. Individuals expectations of the price level are dependent on the monetary policy response.”

I think what Bill is saying is that if the price level target is highly credible and widely understood (i.e. the policy rate is not expected to systematically deviate from the natural rate so that the price level target is hit on average) then it should automatically be the case that E(t)P(t+1) = P*. Why not make that assumption? There would be no problem in this case.

12. 1. I never said that the natural rate of interest was in real terms. Since I’m assuming that the natural rate is constant, this point isn’t crucial to the analysis. In fact, if there is a unique equilibrium in this model, there is no inflation because I’m assuming that the central bank is targeting a constant price level.

2. Why would I make this assumption? The only reason that there is no problem in this case is because it is assumed that there is no problem. However, what is needed is an explicit account of how expectations are formed and how the central bank intends to achieve its target.

13. David Beckworth

You would make that assumption because one of the key reasons for doing level targeting in the first place is better expectation management. Yes, the solution is trivial, but that is the point. A truly credible price level target should anchor price level expectations around the targeted path. The model should reflect that.

14. Bill Woolsey

Josh:

David gets to my point.

And I think your response is worthy, what happens if people don’t believe that the system will work? What would happen then?

Further, the previous analysis of this question _did_ assume at least partly backward looking expectations and a projection of inflation rates into the future, which is really destructive.

I understood that your model was 100% rational and forward looking, my comment about the backward looking part refered to your initial explanaion in words, which I still don’t understand. If it is true that fully foward looking is problematic, it is a useful addition.

If it is really true that any model we write down…then we can’t do formal modeling of phenonemon inconsistent with that assumption. Of course, “any” is a bit strong. It is just, “none that any of us know about,” or “none that anyone has thought up.” Perhaps we will come up with a way to model it. But we can’t look for the keys under the streetlight just because that is where we can see.

Anyway, I do think there is something very wrong with your approach, but I want to encourage you to pursue this sort of work.

As a practical matter, of course, some kind of excessive response would be destructive. For example, if an expecation that the price level will be above target resulting in the quantity of money immediately falling to zero, it would be bad.

But currently, my puzzle would be that this doesn’t target the forecast.

I do think there are puzzles regarding the circularity problem. There is a whole lot of the money supply expected in t+2 determining the price level in t+1. And what the heck does that have to do with the money supply in t?

One other note. At the liberty fund conference last weekend, McCallum was criticial of the standard of unique stable equilibirum. He pitched learnability. I suspect you are already up on this debate.

15. Bill Woolsey

” For example, if inflation expectations have been 2% for some time and the inflation rate rises to say 7%, the public might lose confidence in the central bank to maintain price level stability.”

It seems to me that “inflation expectations _had been_ 2% for some time” refers to the past. And further, the plain meaning of “and the inflation rate rises to say 7%” also refers to the recent past, or maybe the present.

Lets stick with the contant price level scenario. The price level is targeted at 100. With rational expectations, any past history of inflation targeting at 2% doesn’t matter.

Further, any inflation of the recent past would also be irrelevant. So, if the inflation rate was 7% and so the price level is 107, the expectation is for the price level to fall slighly less than 7% back to 100.

Any notion that people would expect the price level to rise further is inconsisent with rational expecations.

On the other hand, if the price level is currently 93, then the rational expectation is that there would sure enough be approximately 7% inflation. (Slightly more because of the lower base.)

But, I think you are, plausibly enough, not going with the assumption that the coefficient in the policy rules (is it gamma for interest rate and Beta for money?) are consistent with getting the price level back to target.

They could be “too big” so that everyone expects overshooting. Or “too small” and everyone expects undershooting. I am not sure who it works that everyone expects this, but the central bank is off, but OK.

With undershooting, it takes more periods to get back to target, I guess.

With overshooting, we get osciliations, and these are either dampening, permanent, or explosive. Now, if you assume gamma (?) or beta (for the money one, are fixed in stone, then there is nothing to be done.

But that is one of the things market monetarists reject. Most of us reject the notion that it is the coeffiecients in the policy rule that tie things down with inflation allowed to do a random walk. It is the target for nominal GDP that ties it down, and the policy “rule” is varied as needed.

So, I would say that if people expect the price level to be below target, people expect the monetary authority to make such a large change in monetary conditions, that the will expect the price level to end up above target and set prics accordingly. Then it is above target, and they will expect the monetary authority to make such a large change in monetary conditions that it will be below target, and set prices accordingly.

Now, when you add inflation, then suppose the price level far is below target. This, by assumption, is going to result in massively expansionary monetary conditions. But, of course, the expected inflation to get back to target (or perhaps a price level that is over target due to overshooting) is high, and you propose adding to the reaction function a term that dampens the massive expansion due the price level being far below target.

But why not just set the original terms in the reaction function to that we just get back to target?

Also, does the inflation term you propose allow the price level to eventually return to 100? Is it just a dampener? Or does it mean that the price level settles at something new?

To me, “level targeting” doesn’t mean that a mechanical reaction function must be generated where interest rates or the quantity of money only depends on devlations of something from a target level. It just means that the goal is for a level (s) and adjustments are _somehow_ made until tthere is a return to the target path. If adding some term referring to growth rates helps by helping to dampen (or avoid) oscilaitions, then that is fine.

But my view is that giving some formula to people and reallhy expecting them to calculate the results and set prices accordingly is.. absurd.

In the real world, central banks don’t follow exact formulas anyway.

I think what is most important is a target rule, not some specific formula that ties a policy instrument to a goal. In other words, the modeling approach here seems almost tailored to something like a Taylor rule. In my view, when the Taylor rule broke down in 2008/2009, it should have just been dropped. The entire notion that we need to adjust the short term rate by 1.5 times the inflation rate, because otherwise the price level will be indeterminate is just an artifact of the models, not reality.

16. David,

“You would make that assumption because one of the key reasons for doing level targeting in the first place is better expectation management. Yes, the solution is trivial, but that is the point.”

Yes, but that is the point of this post. How do we know that the solution is trivial?

Bill,

“I understood that your model was 100% rational and forward looking, my comment about the backward looking part refered to your initial explanaion in words, which I still don’t understand.”

Yes, perhaps this explanation is wrong, but not the model. If we have equilibria that can result from self-fulfilling expectations, then we have to specify a belief process.

“If it is really true that any model we write down…then we can’t do formal modeling of phenonemon inconsistent with that assumption. Of course, “any” is a bit strong. It is just, “none that any of us know about,” or “none that anyone has thought up.” Perhaps we will come up with a way to model it. But we can’t look for the keys under the streetlight just because that is where we can see.”

I am aware of the sweeping generalization I made. However, I am not actually wiling to back down from this statement. I said that any model we write down suggests that a change in the money supply has a contemporaneous impact on the price level. This was in response to a comment by Nick in which he implied that with sticky prices, money cannot affect the price level contemporaneously. This is only correct if all prices are fixed. Not even the New Keynesians believe that prices are that rigid. You are correct that my statement doesn’t imply we cannot write a model down with this characteristic. But why don’t we? Because this assumption seems implausible.

“One other note. At the liberty fund conference last weekend, McCallum was criticial of the standard of unique stable equilibirum. He pitched learnability. I suspect you are already up on this debate.”

I am aware of the debate. I think that it is interesting, but I’m not completely convinced.

17. Bill,

“Any notion that people would expect the price level to rise further is inconsisent with rational expecations.”

No, its not. If the equilibrium is indeterminate, there is a potential for self-fulfilling fluctuations that are independent of the fundamentals of the model.

“It is the target for nominal GDP that ties it down, and the policy “rule” is varied as needed.”

If the rule is changing over time, then we need to talk about regime shifts (i.e. time-varying parameters). This complicates the analysis and (ironically) may likely lead to adaptive expectation formation.

“Also, does the inflation term you propose allow the price level to eventually return to 100? Is it just a dampener?”

It is just a dampener.

“But my view is that giving some formula to people and reallhy expecting them to calculate the results and set prices accordingly is.. absurd.

In the real world, central banks don’t follow exact formulas anyway.”

Agreed, but this is a model designed to capture the behavior of individuals and the central bank. The formulas keep me logically consistent.

“The entire notion that we need to adjust the short term rate by 1.5 times the inflation rate, because otherwise the price level will be indeterminate is just an artifact of the models, not reality.”

Most likely, yes.

18. David Beckworth

Josh,

Let me put it this way. If we accept your approach then the conclusion is that in order for a price level target to work the central bank (CB) must not too aggressively return the price level to its target. Say the CB does just that so that all three conditions above are met and it is therefore able to run a successful price level target. If so, the public should come to expect any deviations from price level growth path would be corrected. But if they come to expect this, then we end up with E(t)P(t+1) = P*. That is

19. Yes, if there is a unique equilibrium.

20. Josh:

I can barely imagine an interest rate rule to be simply explosive, but not a quantity of money rule.

The price level is above target, and the quantity of money is reduced (a little or alot,) and this makes the price level go up more?

With interest rates, it would be possible that the price level rises, the nominal intereset rate rises, but the inflation rate results in a lower real rate. But this explosive process is getting you further and further away from target.

The problem has to be oscilations. The price level rises above target, and so the quantity of money falls or the interest rate rises to a point where the price level ends up below target. And then, it goes the other way. So the quantity of money and or the interest rate are oscilating, which causes the price level to oscilate as well.

To me, if that really happened, you just change the rule.

The point isn’t to have a feed back rule for people to use the determine what price level is going to occur. It is rather people should just be able to expect a particular price level. If the rule is generating oscilations, then stop oscilating the quantity of money and/or interest rates.

At some point, I would really like to shift to a discussion of nominal GDP growth paths rather than the price level.

21. By the way, I think coming up with a mechanical rule for the central bank is just fine. And if the mechanical rule includes an growth term, that is OK. As long as it gets us back to the target growth path.

Further, it is important that the central bankers understand that it is the target growth path that is important, and not the rule. The rule should be revised when it looks like it is not working very well.

Another way to say that is that if people work out the rule and so expect that nominal GDP will be away from target 5 quarters from now, there should be no weight put upon this, and if a revision of the rule puts nominal GDP closer to target, then that should be done. Everyone should know that the target level of nominal GDP is primary, and some feedback mechanism is just a tentative rule of thumb, and there is no commitment to let nominal GDP be at the level based upon the change in the interest rate or quantity of money following from the rule.

So, if nominal GDP is expected to be 2% above target, the growth rate of base money will be reduced 1/3 percent, for example. That is the rule of thumb. People expect that this the implications of this rule for base money is that nominal GDP ends up below target, or above target but less than 2% above, or even more than 2% above target. They should know that the rule is subject to change at any time. There no committment to the formula that relates 2% above target to a 1/3% decrease in the growth of base money. The only committment would be to something like nominal GDP will 17 trilliion in the third quarter of 2014, and then, say, 3%/4 higher than that in the next quarter.

22. Bill,

“I can barely imagine an interest rate rule to be simply explosive, but not a quantity of money rule.”

There isn’t explosiveness in the rule. The equilibrium is indeterminate. There are multiple, stable equilibria.

23. By “stable” equilibrium do you mean a price level?

For example, the target is 100, but we could have a stable equlibrium where the price level is not 100. It could be 90 or 110?

Or do you mean that if the target is 100, but there might be an equilibirum inflation rate of 3%, with the price level going to 103, 106.09, etc.?

In either case, the “rule” would need to be adjusted. The target rule is a price level of 100. But the growth path of base money is such that the price level is 105. Or rising. Well, then you change the growth path of base money because it is a _target rule_, not base money remains at some level or on some growth path based upon the value of the price level.

24. Bill,

“By “stable” equilibrium do you mean a price level?

For example, the target is 100, but we could have a stable equlibrium where the price level is not 100. It could be 90 or 110?”

Yes and Yes.

“In either case, the “rule” would need to be adjusted.”

Correct. And the way that you would adjust the model is to reduce the parameter attached to $E_t P_{t + 1}$ in the monetary policy rule or to attach some weight to the inflation rate.