Evan Soltas
Mar 16, 2012

On Target

Generalizing some monetary policy rules

A friend of mine recently posed a very interesting question about monetary policy rules which target a certain variable, such as inflation or nominal output. In short, the question was whether the one-to-one weights of real output and inflation "inside" of an NGDP growth rate target are optimal, and if not, what are the optimal weights. This raises a bunch of very interesting theoretical (and warning: highly technical) possibilities, given that such an NGDP rule can be described as:

ΔY + π = k

What if you assign a weight other than one to either real output or inflation?

aΔY + bπ = k, which can be rewritten as:

a(ΔY + π) + (b-a)π = k or b(ΔY + π) + (a-b)ΔY = k

(Note: a and b ≥ 1. If a > 1, b = 1; if b > 1, a = 1.)

Let's explain the math: we can think of any rule which weights the real output or inflation components as an NGDP target plus either an additional sensitivity to changes in inflation or in real output respectively.

To take a concrete example, Scott Sumner has long called for a 5 percent annual NGDP growth rate target, and he expects real output potential and inflation to grow at 3 and 2 percent respectively:

ΔY + π = 5, as 3 + 2 = 5

But what if we wanted our policy rule to be especially sensitive to inflation, say twice as much, with the same trends for inflation, real output, and thus NGDP as before (I discuss rationale for different weights later):

ΔY + 2π = 7, which is equivalent to (ΔY + π) + π = 5 + 2, as 3 + 2*2 = 7

In the event that real output growth was zero in a given year, Sumner's rule would respond by raising the rate of inflation 5 percent that year, whereas our variant would only raise inflation to 3.5 percent. Naturally, we could also construct a variant in which the rate of inflation would be above 5 percent. As a reminder, neither of these variants would stabilize NGDP growth, they'd stabilize our variant measure.

Assigning the value zero to "a," of course, constructs a strict inflation target. I note here that our ability to construct monetary policy rules is limited by the fact that assigning very high values to "a," which makes policy relatively inflation-insensitive, is much more likely to result in high inflation rather than an effective "real output target."

This construction, I think, could be extraordinarily useful because it can be extended to analyze all manner of monetary policy arrangements -- in fact, assigning a relatively high weight to "b" creates a sort of explicit "flexible inflation target," which is common. With central banks which practice flexible inflation targeting, most seem to use divergences in real output as their rationale to deviate from their inflation target. For a few examples, the Reserve Bank of Australia refers to hitting its target on average "over the cycle" of real output swings, the Bank of Canada considers "mitigating volatility in other dimensions of the economy that matter for welfare, such as employment and financial stability," and the Fed rather cryptically alludes to the "magnitude" and "time horizons" of deviations in real output, employment, and inflation.

We can apply our framework to these countries and render mathematically explicit their flexible inflation targets using multiple regression to derive their "a" or "b" value:

aΔY + bπ + ε = k, where ε is an error term

What I did was I took a country's real output and inflation data from FRED and fit coefficients to either variable, such that the coefficients minimized a quantity equal to the standard deviation of ε divided by the sum of the coefficients. (I have chosen start year based on the appearance of a consistent monetary policy regime as apparent in the datasets.)

Bank of Canada, 1993-2012: ΔY + 6.22π = 14.51
U.S. Federal Reserve, 1992-2012: ΔY + 11.1π = 25.91
Reserve Bank of Australia, 1993-2011: 2.22ΔY + π = 10.81
Bank of England, 1993-2011: ΔY + 4.55π = 13.14
Bank of Japan, 1995-2011: ΔY + 5.00π = -4.58

One of the common points in the scholarly literature on monetary policy is that optimal monetary policy minimizes the sum of the variance of inflation and real output over the long term, weighting the variance of inflation and real output in accordance with the costs of such deviations from trend.

If the cost of a deviation from trend in inflation exceeds the cost for real output, then, all else equal, it makes sense to construct one of our policy rules (whether one calls it an "variant NGDP growth trend target" or an "explicit flexible inflation target") in which b > 1, i.e. not an NGDP target. The reverse is true if a deviation in real output is more costly, i.e. a > 1. The only way an NGDP target is optimal is if the costs are equal. I imagine one would have to use as "cost" the short-term cost (real output lost, for example) plus the long-term cost (lower real potential output) reduced by a discount rate.

What is amazing is the huge variation in policymakers' implied assessments of these relative costs, ranging from the Fed, which apparently sees the cost of a 1 percent deviation in real output from trend as less than a tenth of the cost of a 1 percent deviation in inflation from trend, to the RBA, whose actions imply that 1 percent deviation in real output is twice as costly as a one percent deviation in inflation.

Surely they cannot all be right, although perhaps the relative costs could differ somewhat based on dissimilarities in their economies. For example, if the Fed felt that prices in the U.S. adjusted more quickly, or that American labor markets were more efficient, it might weight its rule in the manner it has, although perhaps not so sharply.

The other place we can go with this is using these coefficients to derive a single measure of the stance of monetary policy -- a sort of "hawkishness" index, if you will:

H = log (b/a)

I can imagine this being used in a variety of applications. One might be to calculate a central bank's H-value as it evolves over time. For a quick example, the Bank of England's H-value in the time series above is 0.658, but if we consider everything after January 2008 as a new policy regime -- as I have in a prior post -- then the H-value is -0.033, showing that the Bank became sharply more worried about inflation just as real output was collapsing. This seems to be true for the other banks, to varying degrees. Another opportunity for further research would be to see if there are long-run effects on real output growth, structural unemployment, or other variables between countries which correlate to the H-values of their central banks. Perhaps this could lead us to ask if there is an optimal H-value for all economies, or for particular economies, and what are the determinants of the optimum. (Indeed, I will try to get a start on all of these questions in the coming days and weeks. If you have any others, write me a note in the comment section!)

This whole method of monetary policy analysis, however, makes several assumptions which strike me as questionable -- first, that there is no "added value" in a target that is easy for the public to comprehend; second, that the central bank only considers inflation and real output, as the addition of plausible potential variables (financial stability, unemployment, exchange rates) may significantly change the coefficients. But I think it is a pretty good "rough cut" view of monetary policy -- our relative standard deviation measures of ε are consistently low.

The argument for NGDP targeting in the U.S. is quite clear from the descriptive function of Fed policy -- although perhaps it is better said that the Fed is miles out of the mainstream in its near-singleminded focus on inflation, and that significantly more emphasis on ensuring the stability of real output is required, given more reasonable assumptions for the relative costs of their deviations.