[Updated to fix arithmetic errors spotted by Min. A big thank you to Min! (I did not leave my embarrassing original mistakes in, because I wanted to keep it clear). The effects I am talking about are even bigger now.]
Sometimes I think that US monetary policy is too important to be left to the Americans. If you see your neighbour thinking of doing something daft, apparently unaware of one of the problems, you ought to speak up. Especially if it will affect you too, because you do a lot of trade with your neighbour.
[Update: there is something weird about this. I have read three recent criticisms of the US proposal for a legislated Taylor Rule: by Simon Wren-Lewis, (though Simon posted his critique of Taylor Rules presumably just before knowing about the US proposal), Tony Yates, and now Gavyn Davies. That's three Brits, plus me, a (British-)Canadian. Did I miss the American critics? Is this a Brit thing??]
A fixed Taylor Rule multiplies your mistakes in estimating a margin of safety for avoiding the Zero Lower Bound by a factor of three. It makes the danger of hitting the ZLB bigger than you think it is. And Taylor Rules don't work at the ZLB.
Suppose you thought that the natural rate of interest was r^, and you thought that potential output was y^. And you wanted to target an inflation rate p^. Then you might (or might not) tell your central bank to implement a Taylor Rule, and set the nominal interest rate i(t), as the following function of actual inflation p(t) and actual output y(t):
1. Set i(t) = r^ + p^ + 1.5(p(t)-p^) + 0.5(y(t)-y^)
Theory and evidence suggest that following such a rule might then result in a reasonable outcome, in which actual inflation will equal the inflation target on average. It might not be the best way to implement that inflation target, but it won't be the worst either.
But what happens if you are wrong about the natural rate of interest, or wrong about potential output? You think they are r^ and y^, but they are actually r* and y*. So the correct Taylor Rule would be:
2. Set i(t) = r* + p* + 1.5(p(t)-p*) + 0.5(y(t)-y*)
What happens to inflation in that case?
Subtracting the second equation from the first, we get:
3. 0 = (r^-r*) + (p^-p*) + 1.5(p*-p^) + 0.5(y*-y^)
Rearranging terms, we get:
4. (p*-p^) = 2(r*-r^) – (y*-y^)
Assuming that Taylor Rules actually work as they are supposed to work, equation 4 tells us what determines the gap (p*-p^) between the inflation rate you are actually targeting, p*, and the inflation rate you intended to target, p^.
If the actual natural rate is one percentage point higher than you think it is, you will actually be targeting an inflation rate two percentage points above what you intended to target.
If actual potential output is one percent higher than you think it is, you will actually be targeting an inflation rate one percentage points below what you intended to target.
The intuition is straightforward:
If the actual natural rate is higher than you think it is, that makes you set the nominal rate too low, and so inflation would need to be above target on average to have an offsetting effect to cancel out your mistake.
And if actual potential output is higher than you think it is, that makes you set the nominal rate too high, and so inflation would need to be below target on average to have an offsetting effect to cancel out your mistake.
We do not observe either the natural rate of interest, nor potential output. These are both theoretical constructs, and we need to estimate them. Our estimates will be wrong because, for one thing, both the natural rate and potential output will be changing over time in ways we cannot perfectly foresee. So we will in fact make mistakes about the natural rate of interest and potential output, and we will end up targeting an inflation rate that is either higher or lower than the one we want to target, until we figure out our mistakes.
For a normal central bank, that is a problem, but it is not a big problem. Because normal central banks learn from their past mistakes. If they see output persistently below potential, given what they thought was the correct real rate of interest to keep output at potential, they revise down their estimate of the natural rate. If they see inflation persistently below target, given what they thought was output at potential, they revise up their estimate of potential output.
They fix mistakes in their Taylor Rule as they go along. Normal inflation-targeting central banks do this all the time. That's probably the main reason why we always observe a lagged interest rate in the equation when we estimate a central bank's reaction function. If inflation comes in below target, they don't just cut the nominal rate of interest once. They cut once, and then cut again and again, if inflation comes in persistently below target, and keep on cutting until inflation comes back up to target. Persistently below target inflation causes not a low but a falling nominal rate of interest, as the central bank slowly revises its estimates.
But if the parameter values of the Taylor Rule are fixed by law, central banks are not allowed to learn from their mistakes. That means that inflation can be above target on average, or below target on average.
If inflation comes in below target on average, that can be a problem, because the danger of the Zero Lower Bound becoming a binding constraint gets bigger.
Set the Left Hand Side of equation 2 to be greater than zero, then substitute for p* (the de facto inflation target) from equation 4:
5. 0 < r* + p^ + 2(r*-r^) – (y*-y^) + 1.5(p(t)-p*) + 0.5(y(t)-y*)
Since (p(t)-p*) will equal zero on average, and (y(t)-y*) will also equal zero on average, (assuming Taylor Rules work as they are supposed to), this simplifies to, on average:
6. 0 < (r* + p^) + 2(r*-r^) – (y*-y^)
Equation 6 contains three terms:
The first term, (r* + p^), represents what would normally be the margin of safety for avoiding the ZLB. If the true natural rate is 2%, and the intended inflation target is 2%, then the nominal interest rate should be 4% on average, which gives you a 4% margin of safety to avoid hitting the ZLB. The higher the natural rate, and the higher the intended inflation target, the bigger the margin of safety.
The second term, + 2(r*-r^), shows the effects of mistakes about the natural rate on the margin of safety. If you think the natural rate is smaller than it really is, you get average inflation higher than you intended, and the margin of safety is bigger. But if you think the natural rate is bigger than it really is, you get average inflation lower than you intended, and the margin of safety is smaller.
The third term, – (y*-y^), shows the effects of mistakes about potential output on the margin of safety. If you think potential output is bigger than it really is, you get average inflation higher than you intended, and the margin of safety is bigger. But if you think potential output is smaller than it really is, you get average inflation lower than you intended, and the margin of safety is smaller.
It might be more useful if we rearrange equation 6 to read:
7. 0 < (r^ + p^) + 3(r*-r^) – (y*-y^)
The first term, (r^ + p^), represents the margin of safety you think you have, based on your estimate of the natural rate and your intended inflation target. But if your estimate of the natural rate is wrong, and if the true natural rate is one percentage point lower than you think it is, your actual margin of safety will be three percentage points smaller than you think it is. There's the original one percentage point mistake, plus the additional two percentage points that comes from a lower effective inflation target than intended inflation target.
A fixed Taylor Rule multiplies your mistakes in estimating a margin of safety for avoiding the ZLB by a factor of three.
This is from John Taylor's blog post:
According to the legislation “The term ‘Reference Policy Rule’ means a calculation of the nominal Federal funds rate as equal to the sum of the following: (A) The rate of inflation over the previous four quarters. (B) One-half of the percentage deviation of the real GDP from an estimate of potential GDP. (C) One-half of the difference between the rate of inflation over the previous four quarters and two. (D) Two."
That means a 2% inflation target, which many macroeconomists think is already too low to provide a big enough margin of safety for avoiding the ZLB. But that's not the biggest problem with the proposed legislation.
The big problem is it assumes the natural rate of interest is fixed at 2%. The Fed is allowed to revise its estimate of potential output, but is not allowed to revise its estimate of the natural rate of interest. The legislation implicitly estimates the natural rate of interest at 2%. That estimate is fixed by law.
r^+p^ = 2%+2% = 4% estimated margin of safety. A 4% margin of safety wasn't big enough even when central banks were allowed to revise their estimates of the natural rate. If central banks are not allowed to revise their estimates of the natural rate, that 4% margin of safety will be much too small.
If you really really want to legislate a Taylor Rule, OK. But there's a price you must pay, if you want to maintain the same margin of safety against hitting the ZLB. That price is a higher average rate of inflation built right into that legislated Taylor Rule.
Your choice: legislated Taylor Rules; hitting the ZLB more frequently; a higher rate of inflation. Pick any two. [That wasn't clear. What I meant to say is that if you choose a legislated Taylor Rule, you must also choose either hitting the ZLB more frequently or a higher inflation rate.]
And all of the above assumes that Taylor Rules actually do work the way they are supposed to work.
Can somebody please tell the Americans? (And can somebody please check my arithmetic, because I always get it wrong. And I really did try to make this as clear as possible, but I don't know if I have succeeded.)
Worthwhile Canadian Initiative 
Majromax and Min, does the Taylor rule assume real AD is unlimited?
@Nick:
@TMF:
Majromax: “Really, in an equilibrium sense the entire (y-y) term can be neglected, and the Taylor Rule has the same long-term behaviour as strict inflation targeting. Long-term errors in (y-y^) look just like a flawed assumption about the [natural] real rate [r], save that we *should get a credible [estimate of] y* from simply observing trends [in y].”
Yes.
Majromax: “Nick’s analysis works in the long-term equilibrium.”
Oh, great! Now none of the differences are observable.
Majromax said: “[Price, my add] Inflation goes up if production goes up above some long-run capacity (which may be growing), down if production is below the long-run capacity. Production goes up if interest rates go down, and down if interest rates go up.”
If real demand stays the same, won’t real production (supply) stay the same assuming there is enough real supply?
@TMF:
“Sure, but then you’re not talking about any world where the Taylor Rule works,”
Yes, but would economists admit this.
Have you seen any economic models where real AD is not assumed to be unlimited? In other words, if real AD is below potential output (real AS), then something needs to be done to get real AD back up to potential output (real AS).
Does the Taylor Rule work as advertised, in the sense that it tends to lead inflation to approach a steady state? For the sake of argument, let us say yes.
In the Taylor Rule the target inflation rate, p^, is a constant, and does not contribute any errors to the interest rate, i(t), set by the rule. However, can we solve the Taylor Rule for the target inflation rate and use that formula to predict how errors in the interest rate will translate to errors in meeting the target rate? That sounds iffy, but for the sake of argument, let us say yes.
When we do that we find that
Δp^ = 2Δi(t)
That is, errors in the interest rate are doubled in achieving the target inflation rate.
Now, suppose that there is a constant bias in setting the interest rate. It is amplified and passed on to the error in achieving the target inflation rate. And, to be redundant, it is never corrected. Furthermore, by inspection of the Taylor Rule, we see that the error in achieving the target rate is multiplied by 1.5, yielding a positive feedback loop and increasing the amplification factor from 2 to 3. See (7) in the main text.
If the Taylor Rule has a constant bias that it amplifies in a feedback loop –not how it is expected to work normally–, will it work as advertised and tend to converge to a steady state of inflation? The answer is pretty clearly no. In such a case it will not work as advertised.
Now, it may be argued that the amplification only applies to the eventual steady state. But that assumes that there is an eventual steady state. And if there is one, how does the amplification occur? All at once? How? Or is there an amplification of the constant error in every iteration of the rule? If so, how can there be an eventual steady state? It is assumed that there is an amplification of the error in the eventual steady state, regardless of other initial conditions. But suppose that we start from the steady state. Then our current error is amplified. There is no eventual steady state. We have a positive feedback loop.
It may also be argued that the constant error has other effects upon the achievement of the target rate, so that we do not get a positive feedback loop. Fine. But the the assumption that the Taylor Rule tells us how errors in setting the interest rate translate to errors in achieving the target inflation rate is false, and the whole argument fails.
Ah! I think I’ve got it!
The Taylor Rule, quoting the main text:
“1. Set i(t) = r^ + p^ + 1.5(p(t)-p^) + 0.5(y(t)-y^)
. . . .
“But what happens if you are wrong about the natural rate of interest, or wrong about potential output? You think they are r^ and y^, but they are actually r* and y*. So the correct Taylor Rule would be:
“2. Set i(t) = r* + p* + 1.5(p(t)-p) + 0.5(y(t)-y)”
Wait a second! Why the change from p^ to p? Why is i(t) the same? If we are wrong about r and y* we should have this:
2′. Set i(t) = r + p^ + 1.5(p(t)-p^) + 0.5(y(t)-y)
where i(t) is the correct setting for the interest rate.
Subtracting 2′ from 1 we get
(i(t) – i(t)) = (r^ – r) – 0.5(y^ – y)
This is the step that was skipped over. We are interested in the error in setting the interest rate, so we use the absolute values of the differences and get this:
Δi(t) = Δr* + 0.5Δy*
Now, we are also interested in the error in achieving the target inflation rate, p^, or Δp^. To do so we may assume that the interest rate is set correctly and use equation 2 in the text.
i(t) = r* + p* + 1.5(p(t)-p) + 0.5(y(t)-y)
It is now equal to equation 1, and instead of getting the difference between the interest rates, we get the difference between p^ and p*.
Quoting the main text again:
“4. (p-p^) = 2(r-r^) – (y-y^)
“Assuming that Taylor Rules actually work as they are supposed to work, equation 4 tells us what determines the gap (p-p^) between the inflation rate you are actually targeting, p*, and the inflation rate you intended to target, p^.”
I did not understand this explanation. I was assuming that at each iteration we would plug p^ into the Taylor Rule. But in that case the rule would not work as expected. For it to work as expected we would have to plug p* into the rule. That is, we would have to change our target. (Besides, we do not know what p* is, anyway.) So what (p* – p^) indicates is not our error in achieving our target rate, p^, but our error in choosing that target instead of choosing p* as our target. It seems rather strange to think of that as an error, but if we like our target, we could interpret it as our error in choosing p* as our target instead of p^. Our error in choosing the implicit target of the Taylor Rule would be particularly bad if it targeted deflation, i. e., if p* < 0.
I was going to write a bit more about these errors, but, in the immortal words of the song, “Here’s a nickel. Call someone who cares.” Why should we care about an error that A) we are not about to make, because we are going to stick with our target, p^, and B) we could not make if we wanted to, because we do not know what p* is.
But, you may protest, what about the bias in the proposed legislation in the estimate of r? Surely we care about it, and about whether it makes the Taylor target too low. Yes, we care about it, but of more practical significance is how it affects setting the interest rate, i(t), in the first place. And, having solved for Δi(t), we know that Δr is directly proportional to it. OC, we don’t know what Δr* is, either, but if we did, or we could guess, we could use that knowledge or guess to correct the interest rate.
Min, I’m not sure I’m following you.
The advocates of the Taylor Rule say (with some plausibility), that if r* is the true natural rate, and if y* is the true potential output, then for ANY number p*, if we set nominal interest rate according to:
i(t) = r* + p* + 1.5(p(t)-p) + 0.5(y(t)-y)
the average inflation rate will equal p*. So we will act as if we were targeting p* inflation, regardless of what inflation rate we intended to target.
In other words, you can’t tell the difference between: a central bank that is right about r* and y* and wants to target p* inflation; and a central bank that wrong about r* and/or y* and wants to target some other inflation rate.
Take a really simple model of the economy where p(t) = m(t) + v(t), and the central bank sets m(t). We can’t tell the difference between a central bank that has low money growth m(t) because it wants low inflation, and a second central bank that wants high inflation, but ends up with low inflation because it thinks v(t) is bigger than it really is.
@Min:
@Majromax
Thanks for responding. 🙂 Yes, the reasoning appears circular, but there are methods or processes of stepwise refinement that work that way. As I said, I did not understand what Nick meant by the Taylor Rule working as advertised. I thought that at each iteration we would use our original target, but the assumption is that we would switch to p*. In that case, there is no feedback loop. There is no error in approaching p^, since we no longer try to do so.
Looking at this case, I do not see any plausible scenario in which it can. The bias in the estimate of r is enough to prevent that, I think. Also the ignorance of p.
@Nick Rowe
Thank you so much for responding. I know that I have taken up a good bit you your time (and mine, and Majromax’s) on this.
Nick Rowe: “The advocates of the Taylor Rule say (with some plausibility), that if r* is the true natural rate, and if y* is the true potential output, then for ANY number p*, if we set nominal interest rate according to:
“i(t) = r* + p* + 1.5(p(t)-p) + 0.5(y(t)-y)
“the average inflation rate will equal p*. So we will act as if we were targeting p* inflation, regardless of what inflation rate we intended to target.”
Gotcha! But what they do not say is that if your target is p* and you set the interest rate according to this:
i(t) = r* + p” + 1.5(p(t)-p”) + 0.5(y(t)-y)
where p” != p,
that you will approach an inflation rate of p*.
Nor do they say that if you set it according to this:
i(t) = r” + p* + 1.5(p(t)-p) + 0.5(y(t)-y)
where r” != r*,
that you will approach that p*, either.
But the latter is the case we are considering, with r” set by law, regardless of what r* may be. Changing the target will not guarantee that we will converge on that target, either. In fact, there may be no inflation rate to which we will converge.
In any event, in neither case does the Taylor Rule work as advertised. 🙂
Min: “But the latter is the case we are considering, with r” set by law, regardless of what r* may be. Changing the target will not guarantee that we will converge on that target, either. In fact, there may be no inflation rate to which we will converge.”
If Taylor Rules work as advertised, there will be an inflation rate to which we converge, and that inflation rate will be p”, where p” is determined by p”-p* = -2(r”-r).
Because the maths can’t tell the difference between the Taylor Rule that targets p” using r as an estimate, and a Taylor Rule that targets p* using r” as an estimate.
Moi: “But the latter is the case we are considering, with r” set by law, regardless of what r* may be. Changing the target will not guarantee that we will converge on that target, either. In fact, there may be no inflation rate to which we will converge.”
Nick Rowe: “If Taylor Rules work as advertised, there will be an inflation rate to which we converge, and that inflation rate will be p”, where p” is determined by p”-p* = -2(r”-r).”
Let me see if this is what you are claiming.
Here is the Taylor Rule that converges to p”, assuming that it works as advertised:
(1) i(t) = r + p” + 1.5(p(t)-p”) + 0.5(y(t)-y)
Your claim is that this rule:
(2) i(t) = r” + p + 1.5(p(t)-p) + 0.5(y(t)-y)
also converges to p” if p* – p” = 2(r” – r). (I multiplied both sides by -1.) Let us rewrite the condition as
p = p” + 2r” – 2r*.
Now let us substitute for p* in (2).
(3) i(t) = r” + p” + 2r” – 2r* + 1.5p(t) – 1.5p” – 3r” + 3r* + 0.5(y(t)-y)
Yielding
(4) i(t) = r + p” + 1.5(p(t)-p”) + 0.5(y(t)-y)
which indeed equals (1). Bravo! 🙂
Applying this to your main text above, we start with
i(t) = r^ + p^ + 1.5(p(t)-p^) + 0.5(y(t)-y^)
If y^ is an unbiased estimate of y, and r^ is the legislated value, then, assuming that the Taylor Rule works as advertised, inflation should converge, not to p^, but to
p^ – 2(r^ – r).
Assuming convergence, p(t) = p^ – 2(r^ – r). Then
i(t) = r^ + p^ – 3(r^ – r*) + 0.5(y(t)-y^)
🙂
I did not know what you meant by the Taylor Rule working as advertised until your last note.
BTW, if a Taylor like rule were legislated, it might be prudent to lowball r^. For instance, if r^ were legislated to be 0, at convergence we would have this:
i(t) = p^ + 3r* + 0.5(y(t)-y^)
That looks reasonably safe, eh?
The question now is, how safe is it before convergence? 😉