Sunday, December 21, 2014

A strange phrenology of money

Gah! I always seem to miss the big macro debate flare ups when I'm on vacation.

So Paul Krugman called out monetarism and Scott Sumner and Nick Rowe responded (among others). I will add some links when I have access to a real computer and am not writing on my iPhone waiting to get on a bus to go Stonehenge for the winter solstice.

Summer touts his model getting e.g. the US and Australia right. Japan and Switzerland, too.

But why can't the BoJ get inflation, whereas Australia can reach NGDP targets? No, really. Why? The central banks are just organizations of humans ... If I took the BoJ and put it in charge of Australia and vice versa, would I get different results? Why? That actually implies that the specific humans in charge matter! Literally it matters that Greenspan or whoever is in charge -- the math is irrelevant, and it becomes a question similar to whether a president or general was a 'good leader'. Macroeconomics becomes a strange phrenology of money with lots of detailed equations and models that all purport to divine that ineffable quality of a "great man" (or woman) that could get the inflation target he or she wanted at any central bank.

And Nick Rowe's word games posit a central bank that "does nothing" that is massively deflationary. I don't think anyone has said that a central bank can't achieve deflation. But what really is problematic is that the definition of doing nothing is irrelevant to Japan. M(t) and m(t) both define the same path of the monetary base, both of which are increasing, and neither of which seem to affect the price level. It really doesn't matter how you word it.

However, the information transfer model shows how a Japan or an Australia can exist simultaneously in the same framework. If they traded central banks, they'd get largely the same results as they're getting now. And it shows how 'doing nothing' -- keeping a constantly increasing monetary base -- eventually leads to impotent monetary policy. And it shows how major increases in the base can have zero impact on inflation.

And the model isn't even that complicated.

A section of something cool I saw at the Tate Modern yesterday that is mildly relevant. Details to follow when I have a real computer.

Saturday, December 13, 2014

Echoes of the financial crisis

I'm procrastinating packing for my trip.

But Scott Sumner mentioned the Big Bang today, which as a physicist, gives me license to opine about the subject at hand. He says:
Instead, the recent deflation [2014] is a distinct echo of the actual NGDP “deflation” ... that occurred in the early part of this decade [2010-2011].

At the top of this post is the (spatial) power spectrum of the fluctuations in the cosmic microwave background (CMB) radiation. This (spatial) ringing (echo) is similar to the ringing you'll sometimes see in a jpeg image that derives from e.g. "overshooting".

Sumner's story is that earlier changes in NGDP are showing up now in inflation after being obscured by commodity prices. I'd like to put forward an alternative story using the information transfer model. The current deflation is mostly part of a long run trend, but there are fluctuations that are echoes of the real big bang: the 2008 financial crisis.

First, let's separate out the contributions to the price level from NGDP terms and monetary base terms (we can do this in the information transfer model):

There is a faint hint of a ringing from the late 2008 financial crisis. I did a fit to a simple ~ exp(t) sin(t) model for these two components:

This actually works really well. However you'd only be able to see it if you can separate out these two components because the fluctuations are almost too small to pull out in the sum relative to the magnitude of the noise. 

This model puts the source of the ringing at the financial crisis -- the commodity booms of the early part of the decade likely follow from it (basically a rebound from the low) as well as the recent deflation (which is on top of a long run trend towards deflation).

It's still not a perfect model, but it's an interesting take. Here's the graph of all the pieces together:

Friday, December 12, 2014

An information transfer traffic model

David Glasner is getting into complexity theory with his two recent posts. In the post from today he talks about traffic as a non-equilibrium complex system and quotes from a paper by Brian Arthur. In an effort to win Glasner over to an information theory view of economics, I'd like to show that a traffic system can be understood in a first order analysis with the information transfer model (or information equilibrium model). The power of the framework is demonstrated by the fact that I put this entire post together in a little over an hour on my lunch break.

Let me quote from the abstract of the paper by Fielitz and Borchardt [1] that formulates the information transfer model -- non-equilibrium complex systems is exactly what the model was designed to work with:
Information theory provides shortcuts which allow one to deal with complex systems. The basic idea one uses for this purpose is the maximum entropy principle developed by Jaynes. However, an extension of this maximum entropy principle to systems far from thermodynamic equilibrium or even to non-physical systems is problematic because it requires an adequate choice of constraints. In this paper we discuss a general concept of natural information equilibrium which does not require any choice of adequate constraints. It is, therefore, directly applicable to systems far from thermodynamic equilibrium and to non-physical systems/processes (e.g. biological processes and economical processes).
[Fielitz and Borchardt added the "economical" after learning of my blog and we periodically discuss how to properly interpret the model for economics. They have been a valuable resource in my research on this subject.]

We will set up the model as a set of distance slots (X) transferring information to a set of time (T) slots -- these become our process variables (see the preceding diagram). [Another way, X is in information equilibrium with T.] The other key ingredient is the "detector", a differential relationship between the process variables that detects information transfer [or changes in equilibrium], which we identify as the velocity:

V = \frac{dX}{dT}

If we assume that the information transfer is ideal so that the information in the distribution of the occupation over the distance slots is equal to the information in the distribution over the time slots, i.e. $I(T) = I(X)$

We can use the theory [1] to state:

\text{(1) } V = \frac{dX}{dT} = \frac{1}{\kappa} \frac{X}{T}

where $\kappa$ is constant (if the slots in the diagram above don't change) that can be worked out from theory, but can also be taken from empirical observations. It's called the information transfer index. This equation represents an abstract diffusion process [1] and we have

(X)^{2} \sim (T)^{2 \kappa}

And for $\kappa = 1/2$, you recover Fick's law of diffusion. However other relationships are allowed (sub- and super-diffusion) for different values of $\kappa$. It accounts for e.g. the number of lanes or the speed limits [or number of vehicles]. This is the equilibrium model of Arthur:
A typical model would acknowledge that at close separation from cars in front, cars lower their speed, and at wide separation they raise it. A given high density of traffic of N cars per mile would imply a certain average separation, and cars would slow or accelerate to a speed that corresponds. Trivially, an equilibrium speed emerges, and if we were restricting solutions to equilibrium that is all we would see.
Additionally, "supply and demand" curves follow from the equation (1) for movements near equilibrium. The "demand curve" is the distance curve and the "supply curve" is the time curve. Some simple relationships follow: an increase in time means a fall in speed at constant distance (increase in supply means a fall in price at constant demand), and an increase in distance results in an increase in speed at constant time. These are not necessarily useful for traffic, but are far more valuable for economics. The parameter $\kappa$ effectively sets the price elasticities.

However, if we look at non-ideal information transfer we have $I(T) \leq I(X)$ and equation (1) becomes

V = \frac{dX}{dT} \leq \frac{1}{\kappa} \frac{X}{T}

In this case the velocity falls below the ideal equilibrium price. Glasner continues his quote of Arthur
But in practice at high density, a non-equilibrium phenomenon occurs. Some car may slow down — its driver may lose concentration or get distracted — and this might cause cars behind to slow down.
Our model produces something akin to Milton Friedman's plucking model where there is a level given by the theory and then the "price" (velocity) falls below that level during recessions (traffic jams):

The key to the slowdown is coordination [2]. When there is no traffic jam, drivers drive at some speed that has e.g. a normal distribution centered near the speed limit -- their speeds are uncoordinated with each other (just coordinated with the speed limit -- the drivers' speeds represent independent samples). For whatever reason (construction, an accident, too many cars on the road), drivers velocities become coordinated -- they slow down together. This coordination can be associated with a loss of entropy [2] as drivers' velocities are no longer normally distributed near the speed limit but become coordinated in a slow crawl.

This isn't a complete model -- it is more like a first order analysis. It allows you to extract trends and can be used to e.g. guide the development of a theory for how coordinations happen based on microfoundations like reaction times and following distances. In a sense, the information transfer model might be the macrofoundations necessary to study the microeconomic model.

For the information transfer model of economics, one just has to change $X$ to NGDP, $T$ to the monetary base, and $V$ to the price level. There's also an application to interest rates and employment. As a special aside, Okun's law drops out of this framework with just a few lines of algebra.

Also, the speed limit can coordinate the distribution of velocities -- much like the central bank can coordinate expectations. I'd also like to note that no matter what the speed limit is, the speed of the traffic may not reach that because there are too many cars. This may be an analogy for e.g. Japan, the US and the EU undershooting their inflation targets.

And finally, there may be two complementary interpretations of this framework for economics. One as demand transferring information to the supply via the market mechanism and another as the future allocation of goods and services transferring information to the present allocation via the market mechanism.


Thursday, December 11, 2014

On travel for fun for once

There's going to be a long pause on the blog. I'm going to be on vacation through the beginning of January 2015 -- I'm headed to the UK for a little over three weeks. Mostly in London and then up to Scotland for a bit. I've never been to the UK; if anyone has any travel tips, feel free to leave them in comments. Restaurant recommendations in London (we'll be staying in Pimlico) or Edinburgh are very welcome -- especially kid-friendly places.

When I get back, I'd like to start off the new year with how the predictions I've made with the information transfer model are coming along. I'm also considering switching over from Mathematica to iPython in order to more readily distribute source code to those that are interested.

And don't forget: 2015 is supposed to be the year information theory starts to have an influence on macroeconomics.

Many thanks for reading, and have a happy new year!

Inflation is information theory not ethical philiosophy

Scott Sumner argues that there has been both zero inflation and that 100% of NGDP growth was inflation (both since 1964). Tyler Cowen makes a good point about these arguments changing with income. However, Sumner's main conclusion is that inflation is a pointless concept -- in terms of understanding macroeconomic systems.

I think the real answer here is that economists don't really know what inflation is and they fall back on 19th century concepts like utility to ground them. Cowen and Sumner both ask if you'd rather live in 1964 or 2014 with a given nominal income. If that's what defines inflation -- hedonic adjustments and utility -- then I'd totally agree with Sumner: that's pointless.

But it is in this arena that the information transfer model may provide its most important insight (regardless of whether you picture money as transferring information from demand to supply or from the future to the present). The idea is spread over two posts, with the second being the main result:
  1. What is inflation?
  2. Expectations (rational or otherwise) and information loss
When money (M) is added to an economy, that means more information is being moved around. The difference between how much more information could theoretically be moved around with that money (proportional to M^k) and how much more information is empirically observed to be moved around (proportional to measured NGDP) is inflation.

Inflation has nothing to do with the specific goods and services being sold at a given time. It doesn't matter whether it's an iPhone or a fancy dinner. It doesn't matter whether it's toilet paper or bacon. Inflation is an information theoretical concept, not a philosophical utilitarian one.

Wednesday, December 10, 2014

How money transfers information from the future to the present

Widgets and money flow from their present allocation to their new future allocation while information in the future distribution flows to the present, transferred by the exchange. Widgets and coins (isotype) are from original work by Gerd Arntz.

Continuing along this path where we re-interpret the information transfer model as a picture of information flowing from the immediate future to the present through exchange. The diagram at the top of this post is meant to be a re-interpretation of the diagrams from this post.

Overall, this does not represent a mathematical difference, but rather a conceptual difference. Instead of organizing supply and demand "spatially" we're organizing it "temporally". This way of thinking about the process does help with the question of which way the information flows. There have been a few comments on this blog and elsewhere questioning how we know the information flows from the demand to the supply. In this way of picturing the model, information flows from the future to the present through market exchanges (the market is figuring out the future allocation of goods and money and uncovering that distribution means information is flowing from the future to the present).

This picture also gives us a new way to think about "aggregate demand": it derives from the allocation of goods and services in the future.

The information transfer model (information equilibrium) can also be stated more clearly using this picture. The ITM equates the information flowing from the future to the present though money to the information flowing from the future to the present through the goods. Of course you could always re-arrange this and say that the amount of information sent by the demand is equal to the amount of information received by the supply. However, I think the former description is much clearer.

The total amount of information that flows from the future to the present (in a given period of time) could be measured by e.g. the KL divergence between the future and present allocations. Since our knowledge of the future is imperfect, our expectations of the future allocation represent a loss of information, and this representation gives us a framework to start talking about that more explicitly.

Tuesday, December 9, 2014

Meeting expectations halfway


Proceeding in the spirit that theories that are correct have several different formulations, and in conjunction with my fever-dream post from yesterday, I thought I'd start a project where I re-interpret the information transfer model in terms of expectations. Maybe it will lead nowhere. Maybe this is how I should have started. I've tried to make contact with macro theories that contain expectations before, but my general attitude is that "expectations" terms as components in a model are unconstrained and allow you to get any result you'd like.

But maybe that is a strength? If potential expectations are unconstrained, then they could be anything -- and we can simply assume complete ignorance about the expectations that produce a given macrostate (i.e. all expectations microstates consistent with the macrostate defined by NGDP, inflation, interest rates, etc are equally probable).

Let's go back to the original model and instead of calling the information source "the demand" and the destination "the supply" let's set it up as a system where information about an expected future is received by the present via the market mechanism. Our diagram is at the top of this post. From there, everything else on this blog follows through essentially with a re-labeling of demand as "expected demand".

I will do a couple of short posts as I think about the implications of this idea in terms of previous results. If this concept triggers any flashes of insight from anyone out there, let me know in comments. My initial feeling is that these expectations are unlike anything economists currently think of as expectations. A central bank still cannot target an inflation rate over the long run. In the normal formulation, if the central bank sets a target, expectations should anchor on that target. There is no reason for these expectations in the theory in the diagram to anchor on some other number. But maybe the undershooting in inflation is a sign that some information is being lost?

I don't know the answers and maybe this will lead nowhere, but I thought this is a more coherent description of what I was going for in my previous post.