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.

300 years of interest rates

In this post I just did a bit of educated guessing about where the monetary regime breaks should go over the past 300+ years in the UK (time series from here). Turns out all but one (in 1790) roughly correspond to points where interest rates changed from being/not being "pegged" (see here, here, and here):

I plan on doing a future post where I see if I can make this into a more scientific proposition rather than some vague empirical eyeballing ...

Principal component = information equilibrium model?

John Cochrane has an interesting paper/blog post about forecasting interest rates. I'm not sure I've absorbed it all quite yet, but I have a quick take.

The key point Cochrane is making is that the reason adding an inflation term to forecast models of interest rates improves them is really just because inflation has a trend -- a trend that roughly follows the first principal component term (PC1 at the link). Adding a trend (with principal components) allows you to get a really good fit -- and in fact it is this trend that captures most of the forecasting capability of the model. Cochrane says this means there a strong one-factor model of bond yields across all maturity. Basically, one interest rate describes them all pretty well.

This is just a cheesy overlay on the principal component graph, but that first principal component seems to be well described by the information equilibrium model:

Maximum entropy better than game theory (again)

Today Nick Rowe mentions the dictator/ultimatum game (I choose to divide a pot and if you refuse the division, we both get nothing ... or the dictator version where you get no input). It's another case where the maximum entropy guess is better than game theory. Game theory says the solution is 99.9% (or more) for the dictator and x = 0.1% (or less) for the other person if people were truly rational. Maximum entropy guesses x ≈ 50%, but allows x ≤ 50% if information transfer is non-ideal. It also would guess x = 33% for three players, x = 25% for four, etc.

More on the rest of Nick's post later, but it brings up this again. And this.

Department of Huh? Rent control edition

From Wikimedia Commons.

I got a couple paragraphs into this post at Project Syndicate by Brad DeLong and hit a stumbling block:

The reason that rent control is disliked is that it forbids transactions that would benefit both the renter and the landlord. When a government agency imposes a rent ceiling, it prohibits landlords from charging more than a set amount. This distorts the market, leaving empty apartments that landlords would be willing to rent at higher prices and preventing renters from offering what they are truly willing to pay.

Huh? DeLong is usually quite good at this sort of thing, so it's possible I'm missing something.

Why would a landlord choose zero rent and an empty apartment over renting at the rent ceiling? It's possible upkeep costs more than the rent ceiling with a renter rather than leaving the apartment empty, but I thought this was Econ 101 analysis.

And why would getting an apartment at the rent ceiling rate be worse for the renter than getting an apartment at the market rate? Sure Veblen goods come to mind (a thousand dollar a month NY loft apartment isn't as 'impressive' as a ten thousand dollar one ... to those who care about such things), as well as the ability to outbid someone else for the apartment.

I was under the impression that the reason price ceilings were bad in the Econ 101 sense is that they discouraged the building of new supply and acted as an implicit subsidy (basically producing shortages). If you can only get five hundred dollars a month for a one bedroom, it doesn't necessarily make as much sense to build a new building as it would if you could get a thousand. But additionally, more people are able to afford apartments, so fewer would be available. The result isn't empty apartments, but no apartments. And that's also what Wikipedia says.

So what does Info Econ 101 have to say about price ceilings?

In the case of a price minimum p0, there's a pretty simple solution. You basically restrict the price to p ∈ [p0, ∞) rather than p ∈ (0, ∞). However as the two spaces  (0, ∞)  and  [p0, ∞) are diffeomorphic to each other (except for the single point p0), you really end up with the same solution, just shifted. We can ignore the tiny detail of the single point at p0.

Taking the price to have a maximum value p0 is basically restricting the price to the domain (0, p0], which is more complicated. However there is a great function for mapping the space  (0, ∞) to (0, p0) where we'll again ignore the detail of including p0. We'll use an arctangent map so that the differential equation for the price with a price ceiling is the same as the differential equation without a price ceiling after the map arctan: (0, ∞) → (0, p0). This introduces an extra scale parameter (the "width" of the arctangent transition), but it can essentially be absorbed into the information transfer index. Now there are three regimes for the price ceiling. The first is where the ceiling is well above the equilibrium price:

The dashed line represents the demand curve with no price ceiling and the solid curves are the supply and demand curves with the price ceiling. For small shifts, this case is not very different than the original solution where the equilibrium price is marked with a black dot.

If the ceiling is comparable to (and below) the equilibrium price, you get a new equilibrium that is below -- but close to -- the ceiling:

And finally, if the price ceiling is well below the equilibrium, you get the standard Econ 101 result where the price is basically at the ceiling and the market is unresponsive to supply or demand shocks:

Update 11/29/2015

I just want to add that I am not advocating rent control, but rather the naive Info Econ 101 approach to understanding the effects that is more well-defined than the naive Econ 101 approach -- best would be a much more complex model.

Generally, my intuition is that a subsidy (or progressive taxation) would probably be the optimal approach. But it's also just a complex problem since space/land is genuinely limited.

Non-deflation non-surprise

Why didn’t the sustained high unemployment after 2008 push us into deflation? There are some popular stories — downward nominal wage rigidity that makes the long-run Phillips curve non-vertical at low inflation rates, “anchored” inflation expectations — and I cite those stories myself. But standard discourse on macroeconomics has not fully taken the non-deflation surprise into account.

That was Paul Krugman in his post from today.

Non-deflation isn't as much of a surprise in the information equilibrium model. Over time the response to inflation from output shocks has become approximately zero. And it's related to the liquidity trap -- the lack of a response to the price level from monetary expansion.

The minimum wage in Info Econ 101

I added an update to this post about using supply and demand diagrams for the minimum wage, but thought it would be good to do a post on its own. The traditional Econ 101 use of supply and demand diagrams looks like this:

If you set a price above the equilibrium price P* then the new equilibrium market solution becomes the intersection of the demand curve and the minimum wage and the difference between the intersection of the demand curve and the supply curve with the minimum wage value is the amount of unemployment.

Some questions arise:

Why the demand curve intersection point? Why not the supply curve intersection point? This basically assumes that employers are at their limits, rather than say people are staying out of the labor force because a minimum wage job at a lower minimum wage isn't worth it.

What do the curves below the minimum mean? These represent the "illegal" supply and demand for labor -- the desire to employ people at less than the legal minimum wage. I personally would like to drive well over the speed limit, but that piece of the solution space doesn't necessarily enter into my speed decisions that are more based on traffic and road conditions. I'd really like to go a speed that is impossible for my car to attain ... but that should have even less impact.

Anyway, the Info Econ 101 (or Info Econ 1100101, ha! there's a nerd joke for you) picture is different:

The original solution to the information equilibrium condition is no longer a solution in the case of a minimum wage. The intersection point with the minimum wage and the original demand curve (or supply curve) is no longer a meaningful point. The general information equilibrium solution just says both supply and demand go up ... at least in the simplified presentation. There could be all kinds of more complex factors going on.

However that is the point. The Econ 101 analysis is not just oversimplified but wrong according to data. The Econ 101 view above is also wrong in the Info Econ 101 view which is itself simplified, but isn't inconsistent with data.

Econ 101: simple but inconsistent with data
Info Econ 101: simple and consistent with data

Speaking of math ...

... maybe you should prefer to use the IT model for quantitative predictions of things like inflation.

The new core PCE data for October says 0.1% inflation. The error is looking pretty good for the IT model as well (the IT model is doing as well as a model that consists of a smoothed version of the data -- gray dashed line):

David Beckworth's inflation corridor

I mentioned David Beckworth's corridor in comments on an earlier post and realized (again) I hadn't updated the graph from here. This one correctly plots quarterly data (instead of monthly) along with quarterly error (using the IT model prediction for core PCE inflation listed here; see links for model details).

Here you go:

Beckworth's model does show where QE1 and QE2 take off, but might be construed to have predicted QE4 at the end of 2014 -- which did not happen.

Looks like next year or two we'll start to get some real action (in the sense of which model is better) on this one. Basically, the IT model plus Beckworth's model (they're not inconsistent with each other if you don't take QE-n to cause anything) predicts a high likelihood of QE4. Considering how the Fed seems anxious to raise rates, this could be interesting.

Math up

My view of why you should include math is best captured by Paul Krugman:

My point is that there seems to be a lot of implicit theorizing going on here — and at least at first glance, the implicit theorizing doesn’t make a lot of sense. I could be wrong, but that’s the whole point of simple models: to lay bare what you’re assuming, and make it clear what, specifically, is driving your conclusions.

Krugman says "models", and that's what I mean by math -- some sort of formal construct that organizes the various factors you are trying to explain. It doesn't have to be a precise DSGE model. Generic supply and demand diagrams are math. Working with orders of magnitude and asymptotic behavior qualify as math. Basically, math lets other people use your model.

Math is the basic universal language for talking about how things relate to each other. If you are saying X is related to Y, you are implicitly using math. If you don't want to use math, you're not allowed to say X is related to Y.

Of course, some people think math doesn't help illuminate anything. In its most dogmatic form, we can see the opposition to mathematics from Ludwig von Mises:

All of mathematical economics, with its beautiful curves and equations, is idle flirtation. The setting up of equations and the drawing of curves must be preceded by nonmathematical considerations; the setting up of equations does not broaden our understanding. Mechanical equations can be used to solve practical problems through the introduction of empirically acquired constants and data; but equations of mathematical catallactics cannot in the same way be of service to practical problems in the area of human action where constant relations do not exist.

This is of course completely disproved by the success of the theories of how auctions work. As well as the information equilibrium model ...

But in general the reason for mathematics is not that humans are mathematical, but that economists' models of the macroeconomy are mathematical -- whether they want them to be or not. When you start saying effect E has cause C, then you are assuming a functional relationship between E and C: E = f(C). From that we can say that if C is small, we have E ~ f(0) + a C ... and we're off to the races. I've just turned your statement of C causes E into a linear model with an equilibrium value E0 = f(0) and an "elasticity" a. If you say "but a and/or f(0) changes" ... then is it really C that is causing E? Note that linear function E ~ f(0) + a C has exactly the same form if E = f(a) where C is the elasticity. And f(0) changing means that something else is causing E [1].

The reason people don't use math is either a) they can't think clearly enough to use math or b) they want to pull the wool over your eyes. The stated defense is that math doesn't clarify (that's false; it does) or that human actions don't have constant relations (my question to von Mises would be "then why even write anything ... once written down as math or words, human actions will change"). If human actions don't have constant relations, then what does it matter? Even history is studied because people think certain things can cause e.g. wars in a fairly consistent way like competition for natural resources. People study WWI and WWII because they think the things that happened then could have benefit in future conflicts (preventing wars or fighting them). That is the entire point behind "history repeating itself". If history never repeated itself, then there wouldn't really be any reason to study it.

Not using mathematics for your model:

• Puts an upper bound on the complexity of your model (makes it easier to compete with smart people)
• Allows you to say what you're saying is novel even if it is equivalent to existing models (allows you to fool people into thinking you have new insight)
• Makes it impossible for anyone else to use your model (limits the use of your model to yourself)
• Allows your model to change in respose to new information without saying you are changing it in response to new information (so you can fudge any data)
• Allows you to get around stating clear predictions and precise conditions on those predictions (so you can fudge any prediction)
• Allows you to stay away from quantitative statements about the data (so you can fudge any data)
• Allows you to have effects of different magnitudes from causes of the same size (and vice versa) (so you can fudge any data)

A lack of math is not depth of argument or intuitive understanding, it is flim flam.

...

Update 7 September 2016

I should really have rephrased the part about "history repeating" in terms of "lessons of history" instead. People tend to take historical analogies too literally sometimes. An historical situation may enlighten you about the possibilities in a current situation (lessons), but there is almost never an exact analog of some historical event (repeating).

...

Footnotes:

[1] This is my opinion of those expectation operators in models. An E[u(c)] term may look like your model depends on consumption or utility, but really it just depends on E.