Walras' law, information theory edition

Nick Rowe has a new post up and it inspired me to take up his challenge (entering as a non-economist). Rowe is probably one of the best economist bloggers out there if you want to get more technical than the typical post from Scott Sumner or Paul Krugman. His question is this:

Q. Assume an economy where there are (say) 7 markets. Suppose 6 of those markets are in equilibrium (with quantity demanded equal to quantity supplied). Is it necessarily true that the 7th market must also be in equilibrium (with quantity demanded equal to quantity supplied)?

I've looked at Walras' law before (e.g. this post). I'm going to answer this using information theory with progressively more complexities, but I'll start with some notation.

Define $I(D_{k})$ to be the source (demand) information in the $k^{\text{th}}$ market and $I(S_{k})$ to be the received information (supply). Define aggregate source information (aggregate demand, AD) and aggregate received information (aggregate supply, AS) as

$$
I(AD) =  I(\sum_{k} D_{k}) \;\;\text{and}\;\; I(AS) =  I(\sum_{k} S_{k})
$$

If the information in each market is independent, this becomes:

$$
I(AD) = \sum_{k}  I(D_{k}) \;\;\text{and}\;\; I(AS) = \sum_{k} I(S_{k})
$$

And lastly, define excess information in the $k^{\text{th}}$ market as

$$
\Delta I_{k} \equiv I(D_{k}) - I(S_{k})
$$

Rowe's question becomes

$$
\text{If } \Delta I_{k = 1 .. 6} = 0 \text{ then what is } \Delta I_{7} \text{ ?}
$$

First is the "Walras' law is correct" version [1] ...

We assume that the information in each market is independent and that $I(AD) = I(AS)$, so that

$$
0 = I(AD) - I(AS) = \sum_{k} I(D_{k}) - \sum_{k} I(S_{k}) = \sum_{k} \Delta I_{k}
$$

rearranging the terms

$$
0 = \Delta I_{7} + \sum_{k = 1}^{6} \Delta I_{k} = \Delta I_{7} + 0
$$

Therefore, $\Delta I_{7} = 0$.

Now the thing is that all we can really say is that $I(AS) \leq I(AD)$ (the market doesn't necessarily transfer all the information), so that brings us to the non-ideal information transfer version [2] ...

We assume that the information in each market is independent and that $I(AS) \leq I(AD)$, so that

$$
0 \leq I(AD) - I(AS) = \sum_{k} I(D_{k}) - \sum_{k} I(S_{k}) = \sum_{k} \Delta I_{k}
$$

rearranging the terms

$$
0 \leq \Delta I_{7} + \sum_{k = 1}^{6} \Delta I_{k} = \Delta I_{7} + 0
$$

Therefore, $\Delta I_{7} \geq 0$.

That means Walras' law doesn't pin down that last market, and says that there can be excess demand. But it's even worse than that, which brings us to the non-independent (i.e. mutual) information version [3] ...

As I keep mentioning, we're assuming the information in each market is independent, i.e.

$$
I(D_{j} + D_{k}) = I(D_{j}) + I(D_{k})
$$

But this isn't necessarily true and in general (e.g. Shannon joint entropy)

$$
\text{(1) } I(D_{j} + D_{k}) \leq I(D_{j}) + I(D_{k})
$$

This says for practical purposes that some of the information in the source in one market may be the same as the information in the source in another, hence they do not necessarily add to yield more information. So that all we really know is that

$$
I(\sum_{k = 1}^{6} D_{k}) \geq I(\sum_{k = 1}^{6} S_{k})
$$

based on the fact that you can't get more information out than you put in. This means that knowing the six markets clear doesn't necessarily even tell us about the aggregate demand of the 6 markets (ignoring the seventh).

Nick Rowe basically arrives at this last version -- he says there can be excess demands/supplies of money in each of the six markets so Walras' law can't really tell us anything about the seventh. The information theory argument presented here does not require money, which is consistent with Rowe. He says that the same result could hold in a barter economy because some good could effectively operate as money and there would be excess demands for various barter goods in each of the individual markets. Rowe says that:

Walras' Law is true and useful for the economy as a whole only if there is only one market in the whole economy, where all goods are traded for all goods.

This appears to be saying that if you can't decompose $AD = D_{1} + D_{2} + \cdots$ (or the decomposition is trivial), then you get Walras' law back -- and it's true. If you can't decompose the markets, then there are no "joint entropies" that can be formed from their decomposition, so there is no information loss in equation (1) above. This doesn't rule out non-ideal information transfer in version [2] above, but assuming markets work, saying you can't decompose the markets (or the decomposition is trivial) gets you back to version [1] where Walras' law holds.

So is Nick's post essentially re-deriving the sub-additivity of joint entropy?

Was the Fed's quantitative easing serious overkill?

In an earlier post I tried to make a play for the null hypothesis in saying that David Beckworth's claim that the Fed is achieving its inflation target is hard to justify since the currency component of the monetary base (M0) seems to describe inflation over the past 50 years -- and thus the Quantitative Easing (QE) (or large scale asset purchases LSAPs) appearing in the monetary base (MB) is irrelevant. Tom Brown asks a great question in a comment: did the rounds of QE/LSAPs cause both MB and M0 to go up?

Here is a graph of the monetary base -- reserves in dotted red, currency in solid blue:

The rounds of QE appear as vertical lines. There is a hint that QE1 may have caused a jump in M0, but little evidence that subsequent rounds did anything. It will help to look at this data in another way. Here are the (logarithmic) derivatives of the data, scaled to the maximum value between 2007 and 2014:

The rise in M0 coinciding with QE1 jumps right out in this graph, but the other two rounds show little (obvious) impact. This becomes even clearer if we look at the data in yet another way:

In this graph, I show the blue line as the x-axis and the red line as the y-axis. If changes in MB reserves affected M0, then the dots should all appear along the line y = x. For QE1 (shown as blue dots) this is a reasonable model (ok, reasonable is a stretch given the data -- let's try plausible). For QE2 (red dots) and QE3 (green dots), the data seem to fall along the line y = constant with the rest of the data (gray dots) -- again, implying zero influence.

Interpretation?

It is possible that QE1 helped cause M0 to rise, but subsequent rounds of QE didn't do much of anything (i.e. maybe QE1 already did as much as could be done). Imagine M0 as a partially filled glass of water. QE1 filled it up; QE2 and 3 simply sloshed over the side. This view seems to be somewhat supported by data.

Before 2008, M0 (and MB) started to fall below trend. The crisis hit and the first round of QE starts sending M0 back to the trend. Subsequent rounds of QE do less because M0 is closer to trend (in a sense, this is a model where the impact of MB is proportional to the difference between M0 and the trend). Here is a graph that illustrates this point (M0 in blue, the pre-crisis trend of M0 is dashed blue and MB -- M0 including reserves -- is red):

The remaining slow return to trend may have more to do with waiting for NGDP and unemployment to return to normal than QE2 and QE3. The brings up the question: was QE1 enough? Actually, that question might not be strong enough: was QE1 serious overkill? Did we only need, say, 300 billion dollars worth of QE1 rather than 3 trillion over the course of three rounds?

The data is too limited at this point to make solid conclusions. QE1 seems to have been concurrent with a rise in M0 (causality is difficult to determine without a model -- perhaps QE lowered interest rates and caused output to increase via the ISLM model which caused M0 to go up?), but there is no evidence other rounds of QE did anything to influence inflation in the model where M0 determines inflation. (Again, maybe MB lowered interest rates and caused output to increase through the IS-LM mechanism, causing inflation to increase.)

Improved estimate of pre-Depression currency in circulation

I found this interesting data set from FRED on the currency in circulation in the US from 1875 to 1914 [1] , which allows me to improve the estimate I used here to look at the pre-Great Depression trend. Here is the updated graph (the spike in currency in WWI was much sharper than previously shown):

[1] Other Fed data on currency goes back to 1918, leaving 1914-1918 still undetermined, hence it's still an estimate in the region shown in the graph.

Smooth move

Sometimes you make interesting mistakes. I wanted to address David Beckworth's claim that the Fed is hitting its inflation target where the evidence consists of looking where the core PCE inflation data is, defining that as the target, and saying therefore the Fed must be on target [1]. This involved me switching over from core CPI data to core PCE data. I also made the change from fitting the model to price level data to fitting to inflation data. That's not the interesting part.

In the process I accidentally over-smoothed the money supply and NGDP data (FYI, I normally don't do any smoothing at all) and found a pretty awesome result. This graph reveals, for the first time ever [2], trend inflation:

Here's the same result for CPI inflation:

For completeness, here is the PCE price level model fit and the error distribution for PCE inflation:

Pretty Gaussian!

Of course, these results are based on the monetary base minus reserves (aka currency in circulation, aka "M0"), which means the large scale asset purchases (LSAP) Beckworth claims are influencing core PCE inflation are irrelevant to describing PCE inflation [2]. In fact, the information transfer model explains the inflation trend outside even before financial crisis (the ostensible onset of Beckworth's "corridor" of 1-2% core PCE inflation) [3].

[1] For many market monetarists, the Bayesian prior probability of the model that the central bank can achieve its target is P = 1, therefore whatever inflation is measured to be, that must be the target (or measurement error).

[2] Assuming the information transfer model is right :)

[3] I am calculating inflation by the instantaneous derivative of the logarithm (the local slope on a log scale), so it's a bit noisier than Beckworth's graph.

Looking at the foundations of money

My copy of Money and the Early Greek Mind: Homer, Philosophy, Tragedy arrived the other day; I'll be looking for insights into the information transfer picture. In the meantime, here's a nice discussion. I like the way one of the properties of money is described there "[money] facilitates transitive relations between objects".

This makes some sense of the evolution of money [what I say here isn't novel] -- one way to facilitate transitive relations between objects is to choose one of those objects and look at all of the relationships between every other object and that object. This happens when e.g. cigarettes become a form of currency in POW camps or prisons (or high school). Everything gains a price in cigarettes and the relative value of e.g. chocolate and bacon can be related to each other via cigarettes.

Money becomes money when it loses its own intrinsic value and becomes only valuable for its ability to facilitate these transitive relations -- a medium of communicating information.

Fisher's proto-information transfer economics

Irving Fisher's 1892 thesis and an information equilibrium relationship with information transfer index k = 1.

One of Irving Fisher's thesis advisors was Willard Gibbs (of thermodynamics fame, which I mention because of the connection between information theory and thermodynamics). Here's a link to his 1892 thesis; I was struck by how close some of the equations are to the information transfer model.

Fisher looks at the exchange of some number of gallons of $A$ for some number of bushels of $B$ and states: "the last increment $dB$ is exchanged at the same rate for $dA$ as $A$ was exchanged for $B$". Fisher writes this as an equation on page 5 (see picture above, added 7/13/15):

$$
\text{(1) } \frac{A}{B} = \frac{dA}{dB}
$$

The argument seems to have been introduced by both Jevons and Marshall. Of course it's generally false. Many goods exhibit economies of scale (i.e. buying in bulk) or other effects so that either the last increments of $dA$ and $dB$ are cheaper or more expensive than the first increments. A somewhat less restrictive assumption is that if we scale the total amount of $A$ and $B$ then the relationship between the rate $\alpha A$ was exchanged for $\alpha B$ and the rate the last increment $d(\alpha A) = \alpha dA$ is exchanged for the last increment $\alpha dB$ is unchanged. This property is called homogeneity of degree zero, and you can think of it as what would happen if we doubled the price of everything along with how much money we make: i.e. nothing.

Equation (1) is not the most general equation consistent with homogeneity of degree zero, but rather

$$
\text{(2) } \frac{A}{B} = k \frac{dA}{dB}
$$

This is identical to the result from this argument and the basis for the information transfer model. What is actually equilibrating in the market is the information the market is moving around when $A$ is exchanged for $B$.

Is policy relevant?

In my ongoing series of posts ([1], [2]) removing all human agency from the realm of economics, I am now going to question whether government policy (fiscal or monetary) has any effect at all.

However, I probably shouldn't try to work out a novel explanation for something in the comments at Scott Sumner's blog. Here's what I wrote:

[The situation in the EU] makes me think of 1937 in the US (on which you [Sumner] have far more expertise than I). Sure, maybe in both cases raising rates killed the recovery. But maybe in both cases the recovery returned to trend and ended on its own, independent of monetary (or fiscal) policy?

Of course I had zero evidence of this before I blurted it out. But I got lucky in this case: the US situation in the Great Depression looks almost exactly like the EU in the Great Recession.  I grabbed the monetary base data post-1918 from FRED and estimated a few points from 1900 to 1918 using data from a 1910 report to Congress by the Comptroller of the Currency. Here is the result (data/estimate is blue and the information transfer model trend is in gray):

Which looks eeriely similar to the EU, at least, until the onset of WWII (again data is in blue, ITM trend is in gray):

So 1937 in the US looks like 2011 in the EU.

The purely-fiscal version of the Keynesian explanation is that Roosevelt tried to balance the budget in 1937 and that the EU implemented austerity in 2011. Monetarists and more monetary-leaning Keynesians also point to the central bank raising interest rates (the Fed in 1937 and the ECB in 2011).

But what if policy (monetary or fiscal) was irrelevant? After the initial bust (in 1929 or 2008), the economies fell from a point far above trend to one far below. The economies rapidly recovered until they returned to the (stagnating) trend at which point the recovery ended -- regardless of monetary or fiscal policy.

In the US a major monetary (and fiscal) policy regime change took place at the onset of WWII, so policy does have some sort of impact. It appears, though, that the scale of impactful policy change would have to be on the order of creating a bout of wartime hyperinflation.

You don't need to understand how people make choices ...

... to get 80% of the way there.

Given what I've been doing with this blog, this is not the way to go for the future of macroeconomics:

The classical economic of choice is therefore far too simple as it does not capture what goes on in people’s brain when they make choices. “It is also much too static to capture the sensitivity and dynamics of the process,” [Daniel McFadden] said.

Maybe microeconomics might benefit from the study of human behavior, but macro seems to follow optimal information transfer. Optimal doesn't necessarily mean perfect, however. There is a microeconomic behavioral experiment at the beginning of the linked piece that shows a lot of information doesn't get through the market mechanism:

[McFadden] highlighted an experiment he carried out some time ago at his university where half of the students were given a chit saying they were entitled to a pencil and half did not. The two groups could trade as buyers and sellers.

While traditional economic theory said the market should clear with half the pencils sold at close to a median value. In fact less than a fifth were traded. “One answer is that people have agoraphobia – they don’t like markets and that influences resource allocation,” he said.

This brings up an interesting point about information transfer I've mentioned before (see the last paragraph). I've said ideal information transfer is the condition that the information transmitted by the demand is equal to the information received by the supply, I(S) = I(D). In real life, human rationality and behavior factors might put a limit on this so that I(S) = α I(D) for some α < 1. The thing is, α is completely unknown (at least right now). Maybe, according to the experiment mentioned, α = 2/5. This may seem like a problem for the theory, but in fact has no particular effect in any of the calculations and is essentially captured by the fitted value of "kappa" (the information transfer index). Another way to say this is that ideal information transfer might only refer to the ideal practically realizable information transfer.

To bring in an analogy with thermodynamics: there is a maximum efficiency of a Carnot cycle but this never reaches 100%. When we say we have ideal information transfer we are saying something analogous to saying we have a maximum efficiency Carnot cycle, taking that maximum efficiency to be an unknown parameter (it is fit to empirical data).

The most nihilistic way to put it is like this: if we only ever see (through a market mechanism) 2/5 of the total information available (at peak efficiency), what does it matter if that other information exists? The remaining 3/5 of the information is like an event outside of one's light cone. Saying I(S) = I(D) where I(D) is the accessible information is not mathematically different from saying I(S) = α I(D) for some α < 1.

Are interest rates a good indicator of recessions?

Vincent Cate commented on my earlier post asking if I had looked into Austrian business cycle theory. My earlier post had been about whether you could predict recessions by looking at whether the economy had gone off the information transfer model (ITM) "theoretical NGDP path" shown in gray (data is shown in blue and recessions marked in red):

Austrian business cycle theory makes the prediction that printing money and/or low interest rates lead to malinvestment that turns into credit crunch (recession) when it becomes unsustainable. Since the theoretical NGDP path (gray, above) can be plugged into the interest rate formula to show how interest rates "should" behave, I thought that we could test the Austrian model against the ITM. The result was inconclusive for the long term interest rates (10-year Treasury):

The data is shown in green with recessions highlighted in red. The theoretical interest rate path is shown in gray (based on the theoretical NGDP path in the first graph). If we turn to the short term interest rate (effective Fed funds or 3-month Treasury rate), we do get a pretty good indicator a recession is coming:

Almost every time the actual effective Fed funds rate (green) is above the theoretical curve (gray) a recession follows. This is actually a better indicator than NGDP in the earlier post.

However, the ITM result is in direct conflict with the Austrian picture where low interest rates cause malinvestment and recessions. In the ITM, unusually high interest rates (relative to the theoretical curve) are precursors to recessions. This does conform with the earlier result that says NGDP above its theoretical level is a precursor -- interest rates r are given by:

log r ~ log NGDP/MB

so that if log NGDP is above where it "should" be, then log r will be above where it "should" be.

Answering the question in the title: yes, short run interest rates seem to be a good indicator. There are still some other questions here. Are rates high because the central bank is trying to cause a recession? Are rates low afterward because of the response from the Fed? To help answer these questions, here are the theoretical long and short run interest rates on their own:

Note that these two lines include fluctuations of the monetary base (and currency) --  only a small fraction of the remaining fluctuations in the interest rate come from changes in NGDP. That means most of the fluctuations in the interest rate (the ones that become indicators of a recession two graphs above) are deviations from the model. Therefore the market seems to be doing the indicating.

Markets desiring high interest rates could be a sign of many things (never reason from a price [change]), one of which is expected future NGDP growth (why put money in Treasuries when there are higher returns in the stock market or other investments -- a mechanism that lowers demand). That is interesting: it is an indication that over-optimism can lead to a recession (as I've discussed before). Another possibility is that high rates result from an excess supply of Treasuries (too much government deficit spending outside of a recession leads to a recession). There are other possibilities; feel free to add your own in comments!

[I should be extra careful I got that right: high rates mean a low price for a Treasury bond. Low prices mean a demand deficit or excess supply.]

Will I be labeled an ECB apologist, too?

Scott Sumner is upset with Paul Krugman going easy on the ECB. Sumner says:

Over the years Krugman and I have both bashed the ECB for their almost unbelievable incompetence. The ECB that has repeatedly raised interest rates in the midst of the biggest recession since the 1930s.

Was this interest rate policy justified? In the light of my last post, let's have a look at the EU NGDP vs monetary base trend:

It appears ECB policy was less competent than the US in the initial fall after the 2008 financial crisis. In the US, the fall in NGDP is accompanied by an increase in currency, causing the path to fall downwards, but to the right. ECB policy may have lead to the path (blue) overshooting the trend (gray). This overshoot may have lead to the rapid rise in NGDP after 2010 (a "rebound") that likely contributed to the ECB's decision to raise interest rates in 2011.

Did the decision to raise rates in 2011 head off an over-correction in the opposite direction? That hypothesis is, of course, premised on interest rate policy having any traction at all. That brings us to the question: did policy have any impact at all? Maybe the EU just returned to the trend on its own and stopped when it got there. Regardless of the effect of monetary policy in 2011, it appears in the graph above that the EU is back on trend, albeit a stagnating trend.

As a bonus graph, here is EU inflation right up to the most recent GDP data:

I'm still expecting sub-2% inflation for the foreseeable future.

Can information theory predict recessions?

I'm back from a way-too-short vacation and catching up on my reading. Noah Smith has a post about developments in Real Business Cycle (RBC) theories. He sums up with this:

... aggregate shocks are sometimes really hard to identify. There were the oil price shocks in the 1970s and Volcker's tightening in the early 1980s, but a lot of recessions don't seem to be externally provoked. So maybe that means there is some kind of random mass-psychological sentiment thing going on, or maybe it means that recessions are sunspots caused by the interaction of a whole bunch of frictions, and thus completely unknowable. But this network/linkage idea seems like a promising alternative to those unhappy possibilities. ... I suspect they're on to something that many have expected for a long time - the idea that economic fluctuations are the result of the complexity of economic systems.

The information transfer model (ITM) says that there is a trend based on information theory. However, we observe deviations from that trend; what causes those deviations seems to be up for grabs as far as the ITM goes right now. I personally like the idea of mass-psychological sentiment (deviations from "rational expectations") per the quote from Noah Smith. I've actually put that hypothesis forward in different forms before, but allow me to wade into some RBC theory.

A couple months ago I showed how a long run trend in NGDP vs M0 (monetary base) arises from a collection of random markets. Now identifying that long run trend is key to business cycle theory: it tells you what the cycle is! If we fit the general function in that post to the long run time series data and highlight the recessions (in red), something interesting pops out. Recessions appear to happen when NGDP gets high relative to M0 (or M0 is low relative to NGDP):

The cleanest example is the great recession, shown here zoomed in:

NGDP climbs well above the trend curve in gray and snaps back to the trend line (which it has been roughly following ever since). This picture also works for the recessions of the 1980s and 1990s (right graph below) and the "Great Moderation" (left graph below):

In each of the recession cases, the data go above and to the left of the trend line. The Great moderation can be seen as a sustained period where the data were below and to the right of the trend offering little opportunity for a recession to occur (in this picture). The only recession that appears out of place in this picture is the early 2000s recession (where there is some dispute as to its actual recession status).

How do we interpret this mechanism? Well, one way is with the monetary sand pile analogy I wrote up several months ago where recessions are like avalanches due to the sand pile being too high for the volume of sand. Another analogy would be earthquakes: stress builds up over time (being to the left and above the trend) and suddenly snaps back into a low stress state (being on trend).

Both of these analogies point to recessions being random events (they happen at some time we can't really calculate), but not totally unpredictable. If we see that we are away from the trend for a long period, we can say that a recession may be on its way soon. If we are at or below trend, then a recession is less likely. This could be the basis for a "stress" index.

Interestingly, Robert Shiller has recently pointed to CAPE being above 25 as a sign of a future fall in the stock market. There was some additional discussion of this by Brad DeLong and Scott Sumner (both seeming skeptical). As far as the picture presented here goes, while there may be a bubble in the stock market, there currently doesn't seem to be much danger of a recession in the US. That could mean there is no bubble in the market or that a popped bubble won't impact the economy at large.

However, this model would prune my olive branch to the human-centric economics community: even the large deviations from the ITM appear to be the result of a "natural" process that depends mostly on the amount of money carrying information and the size of the economy. I guess human behavior could trigger these avalanches or earthquakes.

Rationality and entropic forces

An anonymous commenter hit upon what may be a key point in a unifying picture of human behavior and market forces. Entropic forces would be indistinguishable from some plausible set of human decisions. You could say that atoms experience an entropic force to diffuse into a volume or you could say that the atoms want to diffuse into the volume (they derive utility from expanding into the volume, but it is a diminishing marginal utility that reaches equilibrium when the atoms are uniformly distributed across it).

For atoms, that seems silly. For humans, it doesn't. It gives us a sense of agency when we are being led (individually) by forces beyond our control, at least in the information theory picture.

Scott Sumner says that people underestimate rationality of economic agents:

Economists and non-economists also underestimate just how rational people really are, at least in aggregate. How much they understand about the world. And how efficiently markets aggregate information. And they do so because people don't seem very smart, or very rational. As always in economics, appearances can be deceiving.

However, we can interpret this as people's irrationality being irrelevant (the core common sense is right -- people don't seem hyper-rational), and their apparent rationality with respect to economic theory as being the result of information-theoretic entropic forces beyond their control.

We appear rational in economic theory only because we are adrift and carried along by the rational invisible hand.

The meaning of economies without humans

There were a lot of emails, comments (here and elsewhere), forum discussions as well as a couple of blog posts that were generated by my post "Against human-centric macroeconomics". Widespread opposition is an apt description. Part of that was probably because I framed the question in a particularly antagonistic way (the title started with "against", for instance). I recommend reading these discussions because they bring up many excellent points.

I was actually surprised to see that the underlying idea was controversial. If a market is an information processing system, then it has a peak information processing capacity (assume a single-good market for simplicity). That capacity doesn't usually change day to day -- the price mechanism (not a given price, but the mechanism itself) continues to be capable of sending us the same amount of information.

One way we know this is that we as humans know how to work with prices we see in the market. We don't think that because the price of bacon went up a dollar to 6 dollars per pound that we won't be able to fork over 6 dollars and get a pound of bacon. The new price of 6 dollars doesn't tell us that there is a new disease affecting pigs or that a new fad bacon diet has started. It just tells us that something happened to supply or demand. Sure a new tax on bacon or subsidy could affect the processing capacity, but new taxes don't happen every day.

Humans, however, will behave differently from day to day. Maybe you decided you should hold off on the bacon from now on. Maybe you think the apocalypse is coming and need to stock up. Whatever you decide, the peak information processing capacity of that single good market for bacon is still the same. With the exception of abolishing money or outlawing bacon, most changes will leave the peak information processing capacity of the bacon market unaffected.

In the information transfer model, we call operating at peak capacity "ideal information transfer" and identify it with the condition I(D) = I(S): the information transmitted by the demand is equal to the information received at the supply. No more information than I(D) can be processed by the market and transferred to the supply; I(D) is all the information that exists.

The thing that I thought would be uncontroversial is the statement that I(D) = I(S) is independent of human decision-making. Sure, human decision-making could change the value of D to D' so that I(D) becomes I(D'), but that just means that I(D') = I(S') if we are operating at peak information processing capacity. The peak is still the case where the market transfers all the information.

The information transfer model assumes I(D) = I(S) most of the time in a functioning market. Human decision-making can make I(S) < I(D) as I have observed. I have some intuition that this is the mechanism behind recessions and unemployment. For those of you out there more familiar with information theory, the maximum channel capacity is independent of the encoding you use (there exists an encoding that produces peak capacity, and many encodings will leave you with less information getting through your channel than the maximum capacity). The general idea is that assuming a market is operating at peak information processing capacity most of the time leads to some pretty good empirical success (see for example here).

I came up with a pretty good analogy for all this. Imagine a race with jockeys and horses. To a first approximation, I would say that the speed around the track is that of the maximum speed of the horse, independent of the jockey. That is the analogy of I(D) = I(S). The (human) jockey can affect that speed -- based on his expectations of the other jockeys' strategies, he or she may hold some speed in reserve for the home stretch or may abort the race completely if the horse seems injured (recession). Humans also bred horses to be faster and may introduce a new breed from time to time -- but not in the middle of a race! These new breeds would be analogous to new institutions or monetary policy regimes. No jockey can make a horse go faster than its top speed -- and the horse's top speed is independent of the jockey (but can be influenced through human directed breeding over time).

Before I go, I have a question for the integral human view of economics: what is gained by having the market consist of only human decision-making?

One benefit to economic theorists is wide latitude in modeling. We don't fully understand human behavior, so anything that is plausible can be allowed. The central bank can set inflation expectations at x% and achieve x% inflation through the market alone without performing any open market operations. Wide latitude in modeling is hard to distinguish from just-so stories, though.

Integral humans also produces two major problems for money: the value of money and indeterminacy. There is no reason in human-based theories of economics for fiat currencies to have any value, and there are an infinite number of future paths for monetary policy that are consistent with the current state. In the information transfer model, the human independent peak information processing capacity of the market sets the value of money (in terms of its information carrying capacity) and there is a pretty solid anchor for the future path of e.g. inflation.

What have I missed that makes adding humans to the problem so attractive?

On taking the people out of economics

I have recently returned from business travel and haven't had much time until now to properly respond to this riposte from Mike at his blog Free Radical and the comments below from myself and Tom Brown. This post originated as a response in the comments at the blog, but has since became so unwieldy that I decided to post it here.

In general, I am am aware that the ideas I am presenting on this blog are outside the mainstream and may well be totally wrong (or worse, trivially obvious) when translated into more traditional economics language. That's why I try to make contact with more traditional economics as much as possible. Some examples: the quantity theory of money, the IS-LM model, some other pieces of Keynes' General Theory, equilibrium in a two-good market, and the behavior of interest rates with monetary policy.

In the following, I cut some quotes from the comments on Mike's post and respond to them below (so please excuse the lack of narrative flow). Most of the points Mike makes are valid criticisms of the theory (or are simply differences of opinion), so I'll mostly focus on answering explicit or implicit questions in the comments and the things that I disagree with.

Mike said: "If [the information transfer model] functional form doesn’t come from the data, I can’t tell where it comes from since, seemingly by his own description, it doesn’t come from some kind of logical analysis of human decision making."

It follows from information theory. The foundational hypothesis is that human decisions are demand information communicated through a channel (monitored by the price mechanism) to the supply. But like information theory, the meaning of the information in the channel is largely irrelevant, only the amount matters. The price moves to bring the information coming from the demand and the information received by the supply into equilibrium under ideal circumstances. Additionally, the total information flowing through a particular market is proportional to the size of that market. In most cases, I assume that the information flowing through a market is equal to the maximum amount of information that can flow through that market. That seems to work remarkably well. Human decisions do appear to have influence -- most of the time reducing the amount of information flowing through a market.

Mike: [questioning whether “dD/dS” is meaningful]

The mathematical object dD/dS is like an exchange rate of demand for supply and is equal (in the model) to the price of that specific good demanded/supplied. It is the infinitesimal change in demand that comes with an infinitesimal change in supply. A really good analogy is that it's a definition of the force due to the invisible hand. Demand must in general be allowed to change with changing supply so any theory of economics that involves supply and demand should state some relationship that includes dD/dS. It may be couched in terms of money -- so that we'd see dS/dm and dD/dm, which come together via (dD/dm) x (dm/dS) = dD/dS.

Mike: "if you have one model with a lot of “shortcuts” built mainly on statistical observations ... "

The information transfer model isn't built on statistical observations. It's built on information theory, and the information theory is consistent with statistical observations. It's more of a standard "posit some axioms and see what results" kind of approach (things are never that simple in practice, but that's kind of what I'm going for). Newton posited some axioms about force and momentum conservation. The predictions from that theory appeared to "explain" statistical observations.

Mike: " ... relationships with no logical justification related to individual decision making ..."

The idea is that humans can't violate information theory regardless of what they decide. Now that idea might not be useful (from looking at the data, it has some success but applicability limited to broad trends), but it can't be wrong -- e.g. humans cannot send more information through a channel than its capacity no matter how hard they try. Information theory represents a boundary of any possible economic theory. Boundaries are only useful if the system pushes up against them, though. Knowing the total available wind power available on Earth for extraction by windmills is a fairly useless number because neither wind power nor human power consumption are anywhere near that limit. Economic systems appear to be operating fairly close to their information processing limits and those limits are independent of the information being processed (e.g. human decisions).

Mike: " ... There is no logical reason why it makes sense for gravity to exist. It just does. It can only be identified through observation."

If you observe the effects of special relativity (a specific symmetry of the universe), then gravity (general relativity) has to exist unless you can produce a good reason it shouldn't exist. I only metion this because it carries over into the discussion here: if you observe long run neutrality of money (an approximate "symmetry" of economics), then functions of the supply and demand functions must obey homogeneity of degree zero -- if D → α D and S → α S, then f(αS,αD)  → α^0 f(S, D) -- which means that the simplest relationship between supply and demand you can write down is (1) dD/dS = c D/S. In the information transfer model, this equation gives you supply and demand curves (alternately holding S and D constant). If this equation (1) isn't true of your micro theory based on human decision-making, then your human theory is going to violate long run neutrality. What is interesting is that the information transfer model gives us long run neutrality as a symmetry without a lot of effort.

Another related point: people frequently say things like such and such result proves Newton was wrong about gravity and Einstein was right. Any theory of gravity must reduce to Newton's theory in some limit because it can't violate some basic mathematics of three dimensional space (and of course, Einstein's theory does). In this same way, any economic theory that at least approximately has long run neutrality must be approximated by the information transfer model -- and that is independent of human decision making. It is possible long run neutrality isn't even approximately true and the reason could well be due to human decision-making (or it could be true because of human-decision making). However, those are empirical points that don't come down to whether or not you think human decision-making is relevant to economics. Either there is an approximate long run neutrality of money or there isn't.

Mike: "For instance inflation runs at 5% forever but people continue to expect it to run at 1%. This is precisely that kind of thing is easy to overlook if you aren’t paying any attention to what makes sense for people to do and that’s what happened with early versions of the Phillips curve and that is why we now have rational expectations."

Yes, I completely agree that observing the Phillips curve and using the regularity as a basis for a theory is not always good methodology (it can be an interesting thing to try, but then your theory becomes vulnerable to whatever unknown effects you aren't modeling, not even restricted to changes in human behavior). However in the information transfer model, we start with ideas that must be true regardless of what human behavior is. Information transmitted through the price mechanism cannot be less than the information received at the other end, for example. If information in equals information out (the assumption in the information transfer model), you can say more -- especially about changes in the information on one side of the transaction vs the other. This approach doesn't have to result in anything useful (in physics, the information approach results in a rather simplified view of the expansion of the universe that doesn't say much more than "the universe can be expanding"); it is a happy accident if it does (in physics, the approach can be used to derive the ideal gas law). In economics, it seems to have some useful results.

Mike: "I’m not exactly sure what you mean by “analog.” Maybe a single molecule can’t “evaporate” but there is still something happening on the molecular level regarding the behavior of molecules relative to each other or something like that which you wouldn’t notice if you had no concept of a molecule to work with."

Entropic forces do not exist (have no analog) at the micro level (they are weakly emergent per Tom Brown's comment).  At the molecular level there is nothing happening that isn't captured at the macro level by the boiling point and heat of vaporization for a general fluid. Most of the laws of thermodynamics were worked out (correctly!) before anyone knew of the existence of atoms. Atomic theory allows you to predict the numbers (boiling points, heat of vaporization). In this sense, microeconomics and human behavior should allow you to e.g. predict the coefficients in the information transfer model and the deviations from it.

Mike: "I can’t remember who ([Tom Brown] or Jason) said it or where and I think I already said this once before but demand does exist at the individual level."

It was me. I was saying demand curves for single markets don't exist at the individual level, at least not in any way that could be captured mathematically. Nor does "aggregate demand". I'm not challenging the idea that an individual person will want to purchase something they want, though. They just won't necessarily have a smooth curve vs price that is independent of other goods. In a sense, a statement of demand for an individual is something like this:

I'd buy an android tablet if one is available soon between 100 and 200 dollars, or an iPad if the price drops to 350 dollars**. Maybe my wife will buy me an iPad for my birthday, so I'll stop shopping around when my birthday gets closer. I don't need more than one tablet, so regardless of price, I won't buy more than one.

This is just one piece of the information that is being transferred to the tablet supply when this person buys a Nexus tablet on sale for 200 dollars the next day. It strongly depends on more than a single good (iPad, Nexus tablet), has a non-linear dependence on price (constant between 100 and 200, but also totally different for an iPad), the preferences of other economic agents (his wife), and time (the birthday). If you aggregate every such desire in the information transfer model, you get something like a demand curve (the total quantity demanded vs price) even though most individuals will buy only one tablet (this is not critical and doesn't apply to all goods, but is at least one way demand curves don't really exist for individuals). The reason you can aggregate these desires in the information transfer model is because you really don't care about the content of the desire -- you just know it exists.

This is my opinion, but I'd say any micro theory that purports to model the content of that indented block of text above is, in a word, hubristic. (Maybe the ITM is just a different kind of hubris.) It's even more difficult than it seems because humans exhibit many cognitive behaviors that would render that block of text totally irrelevant. The preference may change in an Apple store (framing effects) or may not even represent an accurate description of the preference at the time (affective forecasting and rationalization). And even if you were successful, the details of the human model can't have any impact at the macro level if macro can be reduced to a small set of variables like the price level, NGDP, monetary base, etc (an kN dimensional space of N heterogeneous agents with k attributes cannot be collapsed into an M-dimensional space of M macro variables with M << kN unless human behavior doesn't matter except as small collection of parameter values). If macro is tractable as a finite dimensional model, then individual behavior can't matter as anything more than a coefficient.

Mike: "The only way to be able to predict [changes in economic relationships] is to try to identify that structure which is based on human decision making."

This may be more of a yin-yang or figure-ground thing. I'd say the only way to know how to incorporate human decision-making in economics is to first understand what is independent of human decision-making (or another way: what is true regardless of human decision-making). Maybe that isn't the best approach, but I'm giving it a try.

...

Tom Brown did a great job giving an accurate account from a different perspective of what I've been saying; if you don't want to take my word for it, Tom's account is great. Therefore I have only a couple of things to say about his comments (and an answer to a question):

Tom: "“If [microfoundations survive aggregation to into a macroeconomic model as anything other than a coefficient], the resulting model is likely intractable.” ... I take the idea to be one of his hypotheses, not necessarily a statement of absolute fact."

I'd liken it more to intuition (based on my experience with complex systems), but I guess statements of intuition are essentially hypotheses. And actually, I was quoting something I said that a commenter had an issue with.

Tom: "I wonder if Jason would describe his theory as looking at macro as a weakly emergent phenomena, rather than a strongly emergent one?"

Yes, weakly emergent is a good description. It should be possible to compute the macro theory from the micro theory, but the behavior of the macro theory doesn't necessarily follow directly from the micro theory.

** I don't write dollar signs because they mess up the LaTeX stuff on the blog via mathjax.

Testing an animation

I never quite got around to putting up my youtube video about information transfer economics, but I'm going to test the gif capability of blogger with a piece of it. Here is a animation of how the "unit of account (information)" and "medium of (information) exchange" relate:

One thing that is not quite correct: at constant N, increasing M should keep the height of the the stack the same by adding more units to it (making them shrink). Otherwise, this is a pretty good picture to have in your head.

In which I agree with John Cochrane

When I first read what John Cochrane said about interest rates, it was in a quote at Noahpinion:

[S]tandard theory makes a pretty clear prediction about [QE's] effects [on interest rates]: zero. OK, then we dream up "frictions," and "segmentation," and "price pressure" or other stories.

I thought this was ridiculous; anyone can see that the 3 month interest rate dropped precipitously (to the zero lower bound) at the onset of QE:

Maybe he was talking about real interest rates or some other measure. I checked the original post and sure enough Cochrane was talking about long term interest rates like the 10-year treasury. And that's something I agree with. The 10-year rate appears to be controlled by the currency component of the monetary base ("M0"). QE, which involved asset purchases, has full monetary base MB -- including reserves -- rising to over 4 trillion dollars. The full monetary base appears to control short term interest rates.

Let's say we have two markets rl:NGDP→M0 and rs:NGDP→MB with the same information transfer index (κ) where rl is the long run interest rate (10 year treasury rate) and rs is the short run interest rate (3 month rate). I fit κ to the 10 year treasury rate rl:NGDP→M0 and then looked at how well the MB data fit the 3-month rate in the same function:

The M0 model result is the darker blue line, while the MB result is the lighter blue one. The 10 year rate is darker green, while the 3-month rate is lighter green. The fit for both uses the function 

log r = c log NGDP/(κ Mx)

with κ = 10.4 and c = 2.8, and Mx being either the currency component ("M0") or currency + reserves (MB).

Now if the Treasury were to print more currency (or the Fed somehow caused banks to request more currency which would then grant Fed grants), then that would bring down long term interest rates. At least it would in today's economy with the "liquidity effect" dominating because log M0/log NGDP is close to ~ 1. If we were back in the 1970s, the extra currency would cause rates to rise via the inflation/income effect (log M0/log NGDP was closer to 0.5). See here for more details.

While I agree that QE has no impact on  long term interest rates, I don't agree with the just-so argument behind it. This has nothing to do with Wallace neutrality or Modigliani-Miller. It has to do with the information exchanged in the treasury markets in an economy and monetary base of a given size. The result requires ideal information transfer (information transmitted from the demand is equal to the information received by the supply), which is the information theory equivalent of "complete frictionless markets", but it is independent of government policy (except inasmuch as it affects NGDP, the monetary base or level of currency). The fluctuations around the theory likely come from non-deal information transfer and could be anything -- and likely stems from (irrational) human behavior.

Against human-centric macroeconomics

"Gravity might not be explainable in terms of any broader, more general phenomenon. But we know for a fact that macroeconomics is the result of a whole bunch of little economic decisions by individuals and companies."

Noah Smith

Do we really know this? For a fact? To be specific, I'm not questioning the idea that an economy is made up of humans making decisions with money (of course it is) -- I'm questioning the idea that observed macroeconomic relationships (price level and money supply, RGDP and employment) are the result of humans making decisions with money. This blog posits that macroeconomics is just about the large quantity of things (money, people in the labor force, goods and services) and human thought has a peripheral role. In that list we don't care what goods or money think, so why are humans so special?

I also have another question: is the idea of including human decision-making in economics a byproduct of our own human sense of agency rather than, say, solid reasoning or empirical justification? I think the answer is yes, it's a byproduct. Modern economics grew out of ethical philosophy and morality (it's all over everything: utilitarianism, the "Puritan work ethic" and macroeconomic "austerity", Adam Smith's The Moral Theory of Sentiments, the perceived morality of debt) and has therefore always been about human thought. It didn't grow out of natural philosophy (or science as it is known today) or accounting where it might have arisen from observation ("I noticed that everything seems to get more expensive on our books with each passing year, but also prices don't rise when business isn't good and we aren't hiring more people.").

In one of my early posts, I mention that my approach to economics has been that of an alien observer who has good enough instruments to see how nighttime lights are increasing and CO2 is increasing on Earth, land is cleared and posits the idea of a "civilization" on Earth that operates under a theory of "economics" ... analogous to Boltzmann positing atoms operating under a theory of statistical mechanics.

Macroeconomics does not take this approach, but instead started at the very beginning assuming that human behavior was important. Modern economics has some of its origins in physiocracy, and in that economic theory sits the 17th century analog of expectations (quoting from wikipedia, emphasis mine):

Pierre Le Pesant ... advocated less government interference in the grain market, as any such interference would generate "anticipations" which would prevent the policy from working. For instance, if the government bought corn abroad, some people would speculate that there is likely to be a shortage and would buy more corn, leading to higher prices and more of a shortage.

and incentives:

Le Pesant asserted that wealth came from self-interest and markets are connected by money flows (i.e. an expense for the buyer is revenue for the producer). Thus he realized that lowering prices in times of shortage – common at the time – is dangerous economically as it acted as a disincentive to production.

All that can be said (from the information-theoretic point of view) is that a price control changes the information transfer capacity of a particular channel detected by that price (relative to other channels). An incentive (i.e. the knowledge people are willing to pay way more for your goods and/or services than it costs to produce them) is just one piece of information transferred from demand to supply. So is irrational fear. So are speculative hedges ("I may pay top dollar for your goods now, but by the time you build the capacity, I won't"). So is random lack of knowledge ("I guess bacon just costs 50 dollars per pound"). The correctness (or content) of these ideas are not (to first order) important. This was part of the revolution of information theory (see the overview at that link) -- the human meaning of the message is irrelevant. It may be that human thought is very important to a description deviations from the underlying trend:

But in not looking at a human-independent baseline, we don't really know what the real economic fluctuations are (see here and here). Have we been led astray because we as humans were too close to the problem?

Tom Brown asks in a comment (that I can't seem to find right now, update: Tom found it, thanks!) if the information transfer model would apply to non-human economies. I don't know the answer to that. However, macroeconomics as conducted today has an answer: no, it doesn't apply. There are no economic laws that are independent of human thought. Even supply curves depend on expectations of future prices and demand curves depend on consumers' tastes and preferences (and diminishing marginal utility). Recessions might not happen among Klingons and the Ferengi Phillips curve might be perfectly stable. It would be an act of hubris to think an alien civilization thinks the same way we do ... e.g. would risk premia be the same? Some economists do this already with different human cultures, and I touched on it in an earlier post. Maybe this is the correct approach. If it is, then economists should completely abandon any pretense to universality and instead become a sub-discipline of history (the chapter on the period when economists thought universal laws existed would be entertaining).

I believe there is at least a major component that is independent of human behavior, not only because of the successes of the information transfer model but because one of the most successful economic predictions ever assumed that human behavior, on average, summed up to noise:

Suppose that there are a whole ton of different behavioral biases, and that these vary across time, across people, and across situations so much that even with a billion lab experiments we couldn't find them all. Only once in a while will the forces be aligned to make one behavioral bias dominate; most of the time, the net effect of all the biases will be unpredictable by the outside observer. When you have an unpredictable mishmash like that, you have to model it as a stochastic process. In other words, if it's too complicated to explain deterministically, then you treat it as randomness.

So what if psychology usually just ends up injecting randomness into our decisions?

That was from Noah Smith again. It is exactly the theory behind information transfer economics.

PS (added 8/4/2014, 9:28pm MDT): The thrust of this post is opposition to human-centric macro, and not saying human behavior has no role whatsoever (the third sentence says "of course" an economy is made of people, the graph in the middle is quite literally me calculating the impact of expectations, and I kept the piece of the quote of Noah's post that says "once in a while will the forces be aligned to make one behavioral bias dominate").

Monetary regime change

I'd previously noticed that monetary policy can undergo a "phase transition" (or regime change) where the information transfer model changes from one set of fit parameters (or even particular solution to the differential equation) to another. Both cases I've seen (US and UK) involved a period with pegged interest rates during WWII (see the previous links for the US and see this link for the UK). Working with Swiss data I recently found going back to the 1980s from my last post, I noticed another case:

This new case didn't involve a wartime economy or pegged interest rates (however, Switzerland may have imported interest rates through this mechanism, obscuring any information there). Switzerland changed their monetary target from a 1-year monetary base target to a 5-year target in 1990, and then transitioned to a completely new target that involved base growth, interest rates and price stability (inflation) more reminiscent of the US dual mandate (except without employment) that became official in 1999.

In this transition, Switzerland rapidly changed from an almost perfect "quantity theory of money" economy to an almost perfect "liquidity trap economy":

The Swiss economy also rapidly transitioned from relatively high NGDP growth to more modest growth (the smooth black line is LOESS curve):

It is a good sign that these two calculations are consistent with each other at the transition.

So now we have three examples of monetary phase transitions/regime changes. Two (US and UK) are likely due to the wartime pegging of interest rates and in the US may have been accompanied by a bout of hyperinflation. The third (Switzerland) is a more ordinary change in the monetary policy target (it may have been accompanied by a brief bout of hyperinflation as you can see in the first graph, but the period is too short to be conclusive [1]).

In Switzerland, the information transfer index was slightly below 0.5 before the 1990s -- that would cause the price level to increase more rapidly than the monetary base. Because of that, the SNB would see their monetary base target was producing too much inflation and therefore they would think they needed to change to a different target. In the monetary economics paradigm, it probably would have been thought that "expected" inflation was at a higher level than the concrete steps the SNB was taking, so they had to influence expectations. That's why the SNB changed from a 1-year to a 5-year base growth guidance.

[1] I have no doubt the hyperinflation solution would fit to the period 1990-1995, it just wouldn't be very illuminating. In the US, the solution operates over almost 20 years from 1940 to 1960; 5 years in Switzerland is just not long enough to be conclusive.