[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.)