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.

Deflationary monetary expansion?

It is nice to know that my claim that I would get laughed out of an economics conference for proposing that monetary expansion could be deflationary (at least 4 months ago) was off the mark. Apparently I'd just be called an idiot and be asked to return my economist union card. I'm glad I don't have one.

I think Noah Smith does a good job summing up and crystallizing the differences of opinions in his post and walks away with what I think is the best take on the whole affair. The idea seems to be is that there are models with two "scenarios" -- I will refrain from using the word equilibria (I mean what does a low inflation equilibrium mean? Isn't inflation a time derivative?) -- one of which is a high inflation scenario and one of which is a low inflation scenario. There is a difference of opinion on which is the stable scenario. Smith points out that models exist where either scenario is the stable one. Additionally, Smith appears to be of the opinion that only one case is sensible while Paul Krugman seems to be of the opinion that only one case exists. Steve Williamson's original opinion that set the controversy off is that the other case exists and is sensible.

Scott Sumner chimes in with some explanation from a monetarist perspective. He provides a graph that allows for two scenarios where 1) the future MB is higher and future NGDP is higher and 2) The future MB is higher but future NGDP is lower. More on this below.

Nick Rowe adds some pretty existential stuff in this post. Does a representative agent know he or she is a representative agent? Maybe it's all just a representative dream. Can I take the blue pill now? 

The problem seems to lie in the facts that 

1. You can construct a economic model to show just about anything 
2. You can construct a plausible sounding human behavior explanation for just about anything
3. Rational expectations consists entirely of the statement that the economic model in 1) is the plausible explanation in 2)

Taking all of the above with a large grain of salt, what does your humble blogger have to say about this?

Well, it is nice to see the existence of two scenarios where a given interest rate (monetary base) results in two different inflation rates depending on other economic indicators. This happens in the information transfer model (the price level is dependent on the monetary  base and nominal GDP). However, these are not "equilibria" in the information transfer model in the traditional sense; they are simply locations. It would be like saying you can be anywhere along the Taylor rule + ZLB curve, not just at the intersection with the Euler equation condition (which in a sense defines your rational expectations/human behavior). The reason it seems like you are restricted derives from the degree of control over NGDP and the lack of large moves by the central bank or national government.

The information transfer picture analgous to Krugman's or Sumner's is of a price level curve at constant RGDP shown in this blog post. I did my best to put their views in terms of that diagram below. 

First we'll focus on the picture that started all this off. The Euler condition (green line) shows two "equilibria" at different interest rates r and inflation rates i in the graph on the top right. In the information transfer picture, we have the graph of the price level vs monetary base (blue) at constant NGDP which increases for small MB and decreases for large MB. The dashed curve represents an increase in RGDP. Depending on your starting location, the same monetary expansion (red arrows/rightward shifts, which lower interest rates) can result in different amounts of inflation. There is a low interest rate regime (right side of the blue graph) and a high interest rate regime (left side of the blue graph). This basically recovers the Krugman/Williamson picture. 

The picture that Sumner provides is similar, except the axes are the change in MB and the change in NGDP instead of interest rate and inflation rate. Again we have a situation where the same increase in the monetary base can lead to different changes in the price level, which for a constant increase in RGDP (going from the solid to dashed blue line) means different increases in NGDP. One thing to note is that Australia is not at the bottom of the U-shaped curve as Sumner suggests, but is actually on the right side. Additionally Zimbabwe is actually described by a different model (accelerating inflation), but can be approximated by the right side of Sumner's curve in the short run.

In both cases, both "equilibria" are "stable" in the sense that small changes in MB leave you near the equilibrium you started at.

The information transfer framework does not depend on "expectations" adaptive, rational or otherwise to produce these "equilibria" -- it doesn't even care about human behavior. Your economy is in one scenario or the other based on the values of macroeconomic aggregates; it is the tendency for central banks and national governments to only make small changes that keeps you near one scenario or the other. The economy is not stuck at the zero lower bound because of natural rates of interest or expected inflation, but rather because you've already added enough money to the economy to capture the information from the aggregate demand (and are on the right side of the peak in the blue curve). During what are considered "normal times", say 1960-1980, the US economy had a "reserve" of unrealized NGDP. It was halfway up the incline as shown by point 1 in the graph below. Monetary policy could control the economy by adding small amounts of money to realize that reserve. The current US economy (as of 2013) does not have much of that reserve available and it cannot be created by expecting it to appear. It is in the location shown by point 2 in the graph. The same size changes in monetary policy result in smaller changes than point 1.

I'm sad I missed the excitement as it was happening. Lousy vacation.