By magic number, Nick Rowe means scale of the theory

Nick Rowe has a speculative post up about "magic numbers". It's really about scales, but economics doesn't quite know how to use scales yet.

If you start with nothing and want to talk about an inflation rate, the only rate you could sensibly discuss zero. It's "magic" in Rowe's description.

In physics, we call this, colloquially, a "what else could it be?" argument. Given the scales in your theory, what can you construct as the scale for your answer?

By fiat, we start with nothing. No fundamental scales. Therefore if the inflation rate π that we stick in the exponential factor defines the time scale t0

exp(π t) = exp(t/t0)

... then t0 could only be zero (essential singularity ... aaah!) or infinity (no fundamental scales to anchor t0). If t0 → ∞, then π → 0. There's our "magic" number: the only scale available.

Another way to think about it is that the inflation rate has units of percent per year. We don't have anything with units of years (i.e. t0), so we can't construct an inflation rate. Note that you can think of t0 as proportional to the doubling time of the economy.

As an aside, in the case of the 2-player dictator game, you get the "magic" 50% per player from 100% and the 2 players who set the scale of the division.

Conformal anomalies are a way to get around this need for an explicit scale and generate a scale in a more complex way, but let's leave that alone for now.

Nick's speculation is that since the central bank has set 1/t0 = 2% per year, inflation is now π ~ 1/t0 no matter what economic growth is, destroying the so-called divine coincidence where monetary policy targets inflation and output is magically stabilized.

Note that you can use this idea of a scale to organize more than just Nick's view. In Nick's view, the scale t0 could only come from the mouth of central bankers (as it did with Scott Sumner). In old school monetarism (quantity theory of money), it comes from the growth rate of the money supply μ = 1/tμ with μ ~ π. You could even say that in a liquidity trap, t0 comes from the growth rate of fiscal expansion since monetary policy targets no longer give us relevant scales.

Nick's presentiation assumes 1) the central bank has set the scale t0 and 2) the central bank scale is the only possible scale for t0. The question immediately arises: What sets the scale for growth? Why isn't growth zero?

One interesting resolution is that the scale for real growth rate ρ ~ 1/t0 ... i.e. nominal growth rate ν = π + ρ with ρ ~ π so that  ν = 2 π. Note that this is exactly what happens in the IT model when k = 2 (the quantity theory of money) where π = μ and ν = 2 π = 2 μ and ρ = π = μ. There's only one scale μ. This is therefore one way to realize the divine coincidence.

But in Nick's (speculative, by his own account) presentation has a scale for inflation π ~ 1/t0, but no scale for growth. Therefore growth should be either be: a) 1/t0 (per the previous paragraph, aka the divine coincidence), b) set by its own time scale T, or c) zero. He disallows b) and says a) fails, so growth should be zero. That's the only magic number allowed since the central bank controls ν and π (in his view), i.e. ν = π, ρ = 0.

Basically, failure of the divine coincidence implies that real growth should be zero unless it is set by it's own scale ρ = 1/T. But then we get into the problems ...

Let's say we start with an economy where there is some outside scale T (the scale of the concrete steppes); there is a divine coincidence so that 1/T ~ ρ ~ π. Now the central bank decides to start targeting π, setting a scale t0. Do we have

ρ ~ 1/T and π ~ 1/t0

or

ρ ~ 1/t0 and π ~ 1/t0

?

If the former, there is a scale (magic number) T besides t0 so monetary policy is not the only question about the economy. Growth is set by the real factors that operate on timescale T outside of monetary policy targets.

If the latter, did T just disappear? And in that case, why and how does the divine coincidence fail so that we end up with:

ρ ~ 0 and π ~ 1/t0

?

If the divine coincidence goes from operating (ie. ρ ~ 1/t0 and π ~ 1/t0) to failure (ie. ρ ~ 0 and π ~ 1/t0), what sets the time scale for that to happen? (My guess is T. But you see the need for an additional scale.)

Or does the divine coincidence fail and we have

ρ ~ 1/T and π ~ 1/t0

If that is the case, why didn't that happen in the first place?

What is important to understand here is that in order for Nick's view to make sense there basically must be some other scale (T, the scale of the concrete steppes) in the economy that controls economic growth that is independent of monetary policy targets (in the long run). It's the only thing that allows the divine coincidence to fail and still have non-zero economic growth. It could still be a monetary variable [1], it just has to be concrete.

Or another way: what is the "magic number" for real growth and why is it not zero in the failure of the divine coincidence?

The real magic number is of course 3:

Footnotes:

[1] In the IT model, we have three scales: 1/σ (nominal shock/labor growth time scale), 1/μ (money growth time scale) and m0 (economy size). Roughly speaking, if M < m0 (large k) then μ sets ρ and π. If M > m0 (k ~ 1), σ sets ρ, and π ~ 0.