Kappa and I

I was recently going through some of my previous posts in a bit of 2013 navel gazing and thought I'd add a graph of interest rates (i) and the information transfer index (κ) in order to do a bit of thinking out loud. Here is the graph with constant interest rates as red lines and constant κ as black lines (the actual path of the economy appears as a blue line and the information trap criterion ∂P/∂MB = 0 is a gray curve):

One thing this shows is that constant κ is associated with increasing interest rates (with a slower rate increase for higher values of κ). Increasing κ can decrease the rate of rate increase, keep rates constant, or lower interest rates depending on how fast you increase κ. All else being equal, constant interest rates will be associated with a slowly increasing κ.

No major conclusions to draw at this time that are any different from here or here; just observing.

Visualizing the interest rate

Following this post on the information transfer plucking model for the interest rate, I wanted to try to show the interest rate in the context of this information transfer plucking model for RGDP and inflation. Note that the interest rate and inflation rate "interact" through the Fisher equation (I say interact because I'm a physicist and the Fisher equation reminds me of a minimal coupling in QED). Eventually, I hope to be able to describe the price level and the interest rate as a particular path defined by the nominal gross domestic product (NGDP) and the monetary base (MB) across a manifold. Here is a picture of the model interest rate (black, acting as a bound as it does here) and the empirical interest rate (blue) in (logarithmic) NGDP-MB space:

Lines of constant interest rate (gray) are shown as well. Note this maximum path is the exact same empirical path shown in this post. Note that this empirical path appears to approximate an upper bound for e.g. RGDP on its own. It is possible this path through NGDP-MB space represents an upper bound from which RGDP (hence the inflation rate) and interest rate are deviations coupled by the Fisher equation. If I use the blue empirical interest rate path above with the original fit to the price level you can see it doesn't deviate too strongly from the empirical price level (green):

PS The videos are still coming. I just haven't finished narrating them.

Top three posts of 2013

I've only had this blog running since April of 2013, but here are the top three most popular posts since then:

#1: Deriving the IS-LM model from information theory. I was actually quite proud of this post so I'm glad it takes the top spot. The post itself goes through the derivation of the IS-LM model in the information transfer framework at a high level. The IS-LM is a standard model taught in economics classes and as a model that came out of the Great Depression has come to the forefront of economic discourse during the Great Recession.

#2: The long run in the UK. Paul Krugman put up a link to three centuries of economic data from the UK; this post tests the information transfer model against that data and shows that the model does an excellent job.

#3: The information transfer model. This is one of my first posts and basically summarizes the results of this paper so I can't really take much credit for it. My contribution has been using this information transfer model to describe economics rather than the physical processes in the original paper. However, it is quite amazing that a model powerful enough to derive the IS-LM framework (#1) or empirically describe 140 years of inflation data (#2) can be described in less than 300 words (and 3 equations).

Three inequality analogies

With an interesting thought experiment, Nick Rowe makes inequality officially the issue of the week (see Noah Smith for another good take). Ever since I can remember thinking about it, I've always used evolution as an analogy -- but not as survival of the fittest!

See, that is the popular misconception about evolution. Those organisms alive today are not the "best" and they all didn't "out-compete" extinct organisms. Evolution is highly path dependent. Outside factors play a major role. Mostly, extant species are just plain lucky.

However, since this is a blog about using information theory to describe economics, I'd like to put forward a couple more analogies. First, money is basically a tool to allow the economy to move information around and solve an optimization problem. Think of money like beads on an abacus [1]. Spending money is like participating in a calculation. Now think of a person with an abacus the size of a room, but only ever uses a few of the beads to do any calculating. Now imagine she passes those beads on to her children. (Of course, no one lives with a pile of beads under a mattress; they give them to someone else to do calculations. And then they feel important for their ability to allocate beads for which they receive more beads.)

The second analogy has to do with bike/car sharing. I'm not sure if this is apocryphal, but there was a story about the free bike sharing plan in Austin. Basically, it initially fell apart (although it still exists) because the bikes were all taken from the dense downtown core and ended up in the sparse suburbs. The fact that car sharing like Car2Go has to deal with this problem by hiring people to bring the cars back to the downtown core gives some evidence that this apocryphal story might have been true. In any case, money, like a car, is a resource that chiefly enables you to do other things. Because the car sharing service lives in a world with cities and suburbs (themselves the result of government policy and regulations), the cars become mis-allocated and pile up in low density suburbs. The cars must be redistributed in order to make the car-sharing system work efficiently.

"Yes, but!" say they defenders of inequality, "the beads and the cars are not going unused. Prices for cans of soup and other things the 99% buy are fairly efficient already and don't need extra beads allocated to optimizing their allocation. And it could well be more efficient to have a Car2Go parked outside a member of the 1%'s mansion in case she would ever want to use it than parked near the 99%."

My response is that there is no way of knowing that without a complete macroeconomic theory! And in the case of this blog's raison d'être, we could potentially answer it by seeing how far from ideal information transfer we are versus e.g. Gini coefficient.

[1] I think bitcoin makes this analogy into a reality.

Plucking RGDP growth

I mentioned in this post that I had hypothesized earlier that the RGDP growth was operating like a bound; I decided to re-do some of the graphs from the second link as well as this link using a "plucking" framework. So first is the the implementation -- instead of showing S(y) as a path (where y is the time variable) through NGDP-MB space fit to a line, I instead fit to a linear bound. The result is below (the bound S is shown as a line tangent to the black empirical path):

The expected RGDP along S(y) is shown in black on this graph (the empirical RGDP is shown in green and the model calculation along the path is shown in blue):

Already you can see some hint of an upper bound (the recessions all appear to be sharp downward falls with overcompensating rises afterwards). We'll take out the trend and look at the deviations from it to make this a little clearer (the recessions are shown in red):

If we excise the recessions, it becomes even clearer (and the distribution looks more like random fluctuations around a mean):

Here is the original distribution of fluctuations (black line) and with the recessions excised (purple):

The distribution becomes noticeably more symmetric without the recessions (with a mean just below the trend, lending support to the plucking model). It is still not a normal distribution as it is much narrower than would be expected; it is not as narrow as e.g. a Cauchy distribution.

We can also look at deviations from inflation in this plucking framework:

Inflation it seems deviates systematically in both directions, in particular being unexpectedly low during the 1960s and rising during the oil shocks of the 1970s. The deviation from the expected inflation accounts for most of the deviation from NGDP growth:

In the posts linked above, I pointed to the lack of inflation in the 1960s being a mystery (which shows up as lower NGDP growth than expected). The plucking model translated into an ideal information transfer bound (IS ≤ ID, with IS being the information received by the supply and ID being the information transmitted by the demand) could give a potential explanation. The US economy increased information transfer efficiency from IS ~ say 10% of ID to IS ~ say 50% of ID in the immediate post-war period (the numbers 10% and 50% are for concreteness; I don't know what the exact values are or even if they can be determined). While this didn't affect real growth very strongly if at all, it manifested as low inflation (and hence low NGDP growth). After about 1980 we reached the bound given by S(y). (Per the fit above, is the fit a bound at IS = ID? or is it just IS ~ ID? Again, I can't answer that.) From that point on we had the "Great Moderation" where inflation and NGDP followed the expected path given by the bound S(y) until the "Great Recession", a major fall in information transfer efficiency.

One final note is that all the graphs here are only slight changes from the graphs in the posts linked above.

This plucking model

There was a recent post by Noah Smith that led me down the rabbit hole to a couple of older posts on Milton Friedman's "plucking" model (which is actually similar to the Keynesian concept of the output gap). Since the information theory model with imperfect information transfer leads to prices being systematically lower than an "ideal" price with perfect information transfer, I thought I'd see how everything works from this point of view.

I had already mentioned a few months ago that RGDP growth can be seen as noise plus negative deviations from some upper bound (i.e. the plucking model), but here is some more evidence (at least in unemployment).

If we take the unemployment model and use the "theoretical price" (aka the price level, in green below) as a bound for the model (blue), we obtain the following fit:

This already looks pretty good. Here is a version of the same information but in terms of unemployment rate:

And in terms of a "plucking" shock deviation away from the trend (recessions are shown in red):

The beginnings of the drops (i.e. rising unemployment) line up nicely with the recessions. Now what about a different model, say, interest rates? This one is a bit inconclusive. Here is the fit to the 3 month US treasury rate (gray) using the model as a theoretical bound (black):

I had previously observed that the model could act as a bound back in September. Here is the analogous graph to the unemployment rate graph above, except this time it is the monetary base (since that is the denominator on the right hand side of the model) with the data in blue and the model calculation using the interest rate in black:

And here is the analogous plucking shock graph (again, recessions in red -- I look at the difference in the log of the MB since the base grows exponentially over the time domain and a fractional difference shows the exact same behavior):

This one is less conclusive; if you use an ordinary fit, you get what could easily be random fluctuations around a trend:

However, there is some encouraging news if you look at both sets of plucking shocks (interest rate, black and unemployment rate, green) on the same graph (the former normalized to the latter):

The shocks seem to be somewhat correlated except the 1980s and the early 2000s. Maybe those are different kinds of recessions? Both were Fed-induced (aren't they all, asks the monetarist), the former to curb inflation, the latter to create a "soft landing" for the dot-com bubble. One thing that is cool is that we could use graph above to use the unemployment rate to solve for the interest rate.

One final note is that the difference shown in the graph above represents the difference between IS ≤ ID where the former is the information received by the supply and the latter is the information transmitted by the demand. We do not actually know where the zero point should be -- it is possible we reached 100% efficiency (IS = ID) in the late 1960s or the late 1990s, but I doubt it.  They likely only represent the peak efficiency and that might represent only 50% efficiency (IS = 0.5 × ID) as we don't know e.g. the maximum theoretical information transfer efficiency of the market mechanism. However we can say the Bernanke era represents the lowest efficiency of the interest rate market of the past half century. 

Microfoundations are not necessarily necessary

Nick Rowe has an interesting post about his desire for microfoundations in economics:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2013/12/microfoundations-for-peoples-sake.html

I have two issues with it:

1. If he wants to know why people do what they do he should study psychology.

2. The microfoundations he describes completely eliminates whole classes of models. His formulation would capture e.g. traffic models where traffic jams come from following distances and reaction times, propagating backwards through the vehicles on the road, but it would never capture things like the ideal gas law or any other theory where the underlying degrees of freedom become irrelevant (thermodynamics) or are replaced with composite degrees of freedom (quarks forming hadrons).

It bothers me particularly because it eliminates my theory where the trends are described by information theory (the deviations may be described by random fluctuations, human behavior or some combination of the two).

Useful ratios

One of the great things about the "first law" of information transfer economics being a simple ratio is that it can serve as a good guide to economically useful ratios. Matthew Yglesias talks about the employment-population ratio being so broad as to be problematic. Of course the correct ratios to look at are NGDP/U and NGDP/L where U and L are the total number of people unemployed and the total number of people employed. These ratios are proportional to the price level, with the former showing far more of the "business cycle" than the latter:

The former is also related to the Phillips Curve.

A delicate balance, part 2

Can the interest rate r and the inflation rate i be balanced, maintaining a steady state equilibrium condition r ~ i?

This is an accepted piece of economic theory with proponents from Nick Rowe to Scott Sumner to Paul Krugman to Steve Williamson. Absent external shocks or mistakes by the central bank, you should be able to keep the (nominal) interest rate constant given constant inflation. In an earlier post, I wrote down some of my thoughts about the big controversy that happened last week that all seemed to follow from one unconventional interpretation of an equilibrium condition. Part 1 of this pair of posts discussed what that equilibrium was and how it worked. In this post, I will show that the equilibrium does not exist.

The information transfer model says r ~ i is impossible. Effectively any pairing of constant rates r and i require accelerating nominal GDP (and monetary base), hence accelerating real GDP growth (since inflation is constant).

Using the information transfer model, I constructed a path of the economy with a constant nominal interest rate and a constant inflation rate. On the graph above, the lines of constant interest rate are shown as dotted red, the ∂P/∂MB = 0 line is solid red, the actual path of the economy is black and counterfactual constant r,i path is blue.

Here is the price level (black = model, green = data, blue = counterfactual model):

And here are the NGDP and MB paths:

We can start to see the problem if we look at the RGDP growth rate:

You can see a steady increase in RGDP growth rate. In fact, a constant interest/constant inflation path requires not just a constant increase in NGDP and MB, but an accelerating increase in NGDP and MB that is more apparent in this longer time series:

There are actually no stable paths where the interest rate and inflation rate are constant -- therefore there is no delicate balance enabling the equilibrium to exist in the first place (unless it is a dynamic equilibrium). Of course, there has never been a time with constant inflation rate and interest rate; in recent US history (since 1960) the inflation rate and interest rate have climbed up to ~10% or more and fallen back down.

A delicate balance, part 1

Nick Rowe puts forward an interesting analogy using a car's speed and speedometer to make his point about equilibrium conditions and causality. If the car's speed is S and the location of the needle is N, then in equilibrium, aS = bN. In the analogy S is assigned to inflation and N is assigned to the interest rate. He goes on to say that increases/decreases in S cause increases/decreases in N, but not the other way around -- at least not in the intuitive way. Rowe points out: "if I grab the speedometer needle, and rotate it clockwise, this will not cause the speed to increase and the gas pedal to go down". In fact, he goes on to say "that when [the Bank of Canada] wants the car to increase speed it turns the speedometer needle counterclockwise, which is the opposite direction that the equilibrium relationship would suggest."

My immediate response was: what kind of equilibrium is this?

One type to check is thermodynamic equilibrium. Effectively, if macroscopic variables S and N are related by aS = bN in thermodynamic equilibrium, the set of different microstates with macrostate S must be equivalent to the set of different microstates with macrostate N. The different microstates include situations where everyone's financial situations are re-assigned to different people, for example (much like trading the positions and speeds among identical particles). However if we change N to N', changing the microstate, S would have to change to S': if the new microstate had been already in the equivalence class S, then N' would have to be equal to N. And vice versa. This is how, e.g. entropic forces work. To maintain equilibrium, a delicate balance of changes in microstates has to be occurring.

So the aS = bN equilibrium can't be a thermodynamic equilibrium if it doesn't work both ways.

Another type to check is mechanical equilibrium where the balance of forces on an "object" cancel, leaving no net force and therefore no acceleration. Non-conservative forces like friction exist and can produce outcomes like the kind Rowe mentions above. It could also be the case that aS = bN represents an unstable mechanical equilibrium, per Paul Krugman. I believe an unstable equilibrium is consistent with Rowe as well, but I am not sure because he doesn't say what happens if you turn the needle clockwise. In this picture, an increase in N leads to a decrease in S and vice versa. In this way, your object starts to move away from the unstable equilibrium where aS = bN. 

But!

Where does it move to?

In Rowe's picture, S gets bigger as N gets smaller, making aS' > aS = bN > bN'. Now the entire point of Rowe's argument is that the economy at (S', N') doesn't experience a force to return to (S, N) (which is what Steve Williamson is saying). That leaves only three possibilities: 

1. (S', N') represents a new stable equilibrium cS' = dN'
2. There is a force directing (S', N') to a new stable equilibrium (S'', N'') such that cS'' = dN''
3. The economy never reaches an equilibrium (wheeeeeeee!!!!)

Interestingly, none of these situations are aS = bN which Rowe (and Scott Sumner) say the economy should return to in the long run. Now their explanations give me reason to believe that they really saying the economy will actually go towards cS'' = dN''. Switching back to the underlying economics for a minute, there are two ways economic agents could potentially get rid of unwanted cash:

• Somehow the agents make holding cash look more attractive by lowering inflation (this is what Williamson is saying and is consistent with considering aS = bN a stable equilibrium). Krugman says he needs to see the "somehow" story to believe it.
• Agents buy goods and services with the cash which should cause inflation (this is what everyone else besides Williamson is saying and is consistent with considering aS = bN an unstable equilibrium)

The second choice, if it stops, represents the path to the new stable equilibrium cS'' = dN'' in 2) mentioned above; there will be a new inflation rate S'' and a new nominal interest rate N'' which can't be (S, N) because then S, N would have been stable we would have used the first choice to get rid of the cash.

One way out of this conundrum is that the economy is actually a different economy in the future (for one thing, it's larger) and the condition aS = bN at time t1 is equivalent in some way to cS'' = dN'' at time t2. That's entirely possible, but I prefer a different way out: the equilibrium aS = bN does not exist.

That is the subject of part 2.