Tuesday, January 11, 2011

The Statistical Mechanics of Money

Yesterday I listened to a talk by Victor Yakovenko of the University of Maryland about the physics of money and it was quite interesting. I think that after this talk I am finally beginning to understand economics while at the same time I suspect that most economists don't.

In his talk he said that back in 2000 he published a paper on how to apply statistical mechanics to free market economics. He did it as a hobby and did not consider it to be very important, meaning he still did his normal research involving condensed matter physics. A few years later he realized that his paper on the statistical mechanics of money was getting more citations than any of his other papers, so he switched his focus of research to Econophysics.

One of the most interesting results that he found was that in a free (random) system of monetary exchanges the natural distribution of money will converge, over time, on a Boltzmann-Gibbs distribution. The argument is simple. In stat mech you have a system where small, discreet amounts of energy are passed from one particle to another. This system leads to a probability distribution characterized by the equation [all images come from Dr. Yakovenko unless otherwise stated]:
Where we have the probability of finding a particle at a particular energy level on the left and the energy, represented by ε, and temperature on the right. If we change the energy to money (or income) and temperature to average income we have:
So what does this mean in reality? Well consider a system where everyone is given $10 to start out and they can engage in economic transactions worth $1 each (they can either pay $1 or receive $1 per transaction), with the requirement that if anyone reaches $0 they can no longer pay money (i.e. they can't go into debt) but they can receive money, it's just that the other person does not receive money in return (think of it as work done). Over time this system moves to maximum entropy and achieves a Boltzmann-Gibbs distribution. An undergrad at Caltech named Justin Chen, who worked with Dr. Yakovenko, made an animation of this system [you can find more information on this animation here]:
As we see, the system starts out as a delta function but changes to a Gaussian distribution, but because of the hard boundary at m=0 (think of it as the ground state) we end up with a Boltzmann distribution. So as soon as they thought about this they started to look into other things, such as allowing for debt, taxes, and interest on the debt (this is of course assuming conservation of money). When they allowed for debt, but only down to a certain level (i.e. the ground state was shifted to a negative value), they found that it did not change the shape of the distribution but it did raise the temperature of the money (raised the average money).
The way this works is that every time someone goes into debt (negative money) an equivalent amount of positive money is created. Thus while the total amount of money has not changed, the amount of positive money goes up, raising the temperature and thus creating the illusion that there is more money. While this does put some people into debt, it does raise the number of people with more money, thus it makes more people rich (note: the rich do get richer under this system, but the main effect is that there are more rich people. So it is not just that the same people get richer because of the debt, it's that more people have more money). But you may say that this is not very physical because in the real world there is no low limit of debt. In theory someone could have infinite debt (infinite in the sense of very, very large). And in their models, if they remove the debt floor they have precisely this effect, the number of people who go into deep debt grows until we have a flat line fore the distribution.

This problem was solved in 2005 by Xi, Ding and Wang when they considered the effect of the Required Reserve Ratio (RRR), which to explain simply limits the amount of money that lenders can lend. In terms of the simulations this means that every time someone goes into debt the corresponding positive money that is created cannot all be used to finance more debt. A certain amount must be held in reserve. This limits the total debt that can be issued by the system based on the total amount of money that was in the system to begin with. The end effect is that the distribution becomes something like this [From Xi, Ding and Wang]:
This effectively creates a double exponential distribution that is centered about the average money amount. Now we can start to look at some of the effects that things like taxes and interest have on this system. First, interest. If we add interest the system does not have a stable state and without any form of check the system will expand to infinity, in both directions. Second, taxes. Assuming a tax on all transactions, either a fixed amount or a percentage, and then assuming the tax is then evenly redistributed, this will shift the peak slightly to the right, though it will not change the edges that much. So a redistributive tax will raise the income of people near 0 slightly, away from 0, but it will have very little effect on the overall shape or the edges (the very rich or the most indebted). If we allow for bankruptcies then that acts as a stabilizing force on the system, that is, it keeps it from spreading to infinity. Thus interest creates no stationary state, but bankruptcy creates a stable state. He noted that what the Fed is currently doing is in response to the economic crisis, the negative end of the money (the debt) has shrunk considerably because of foreclosures, and other problems, but to keep the positive end from shrinking as well they pumped more money into the system to maintain the size and shape of the distribution. While this does prevent the positive side from collapsing, the trade off is that there is more money put into the system, creating the potential of inflation. (As a side note: part of the reason we got into this mess in the first place is because banks found a way around the RRR requirement, allowing them to issue more debt than they should have, which gave the impression that the economy was booming because there was more positive money, but this ignored the fact that there was a corresponding amount of debt being created. The debt was limited by personal bankruptcy and thus created a domino effect that removed both the debt, but also a significant amount of the positive money. We have yet to even come close to recovering to the level we were at before. If you hear talk about the Fed keeping interest rates low to spur lending, they are trying to create more positive money through lending, by creating negative money, but because that is not working they have begun printing more money (buying government bonds) to create more positive money without having to create more negative money.)

Now let's look at some real data. Using US census data from 1996 they plotted income and found that it fits nicely to a Boltzmann distribution, as long at you do not get too close to $0 (above $10,000 works fine).
But this only works up to a point. They found that up to some point the income of people fits very nicely to a Boltzmann Distribution but above a certain point income behaves differently. If you plot all they way up into the highest levels of income you see a clear point where the income changes from Boltzmann-Gibbs to a Pareto power law distribution.
They noted that while the lower Boltzmann distribution does not change much over the years, despite recessions and boom times, the top "super-thermal" tail changes a lot depending on the economy. This super-thermal tail represents only 3% of the people, but holds anywhere between 10-20% of the money. The total amount of the inequality depends on the economy at the time.

So what happens when this distribution gets upset by programs such as social welfare or other effects. In a paper published in 2006, Banerjee, Yakovenko, and Di Matteo give the following graph showing the distribution of income in Australia.
The spike on the lower end is the welfare limit. People below this limit are brought up to the limit though welfare, but people just above it are also brought down to the level because of taxes. Hence the spike but also the dips on both sides. The result is more equality, but at the expense of lowering the entropy of the system and creating this deviation from the normal distribution. The desirability of doing something like this is debatable (as one professor asked after the talk, "What's wrong with inequality?")

When asked about the predictive power of this his response was, "None." He explained that this data takes a long time to collect and to analyze, which creates a lag of several years in the data. But by the time the data has been collected and analyzed and we are able to see a bubble (like the housing bubble) the bubble has already burst and we have moved on. But this also means that modern free market economics behaves in a very statistical manner and can thus be controlled (in theory), which is what the Fed is currently trying to do. Dr. Yakovenko did not get too much into the politics of it, but he did make it clear that he did not agree with the way the bailouts were handled, and the current tactic of the Fed. But it was also clear that he preferred a free market, though a heavily regulated one. He was also very concerned about the inequality in the system, and noted the existence of two classes of people and made reference to Karl Marx at that point.

On the whole I was rather impressed with what he was doing and like I said, I think I am finally beginning to understand economics. Someone just had to explain it to me in terms of physics and it made perfect sense.

XI, N., DING, N., & WANG, Y. (2005). How required reserve ratio affects distribution and velocity of money☆ Physica A: Statistical Mechanics and its Applications, 357 (3-4), 543-555 DOI: 10.1016/j.physa.2005.04.014


  1. Interesting post. I'm curious why you say that economists do not understand the economy (whereas, you are beginning to)? That seems like an unnecessary and unwarranted dig at economists.

    Question. What happens (e.g. in the Caltech simulation) if you assume that getting a dollar at time t increases your chances of getting a dollar at time t+1 and giving up a dollar at time t increases your chances of giving up another dollar at time t+1?

  2. Quantumleap42,

    Really nice description of all of this. And thanks for the links to the animations. I am sure statistical mechanics is really be a great description of economics. Lot's of interactions taking place all the time.

  3. Perhaps I am a little harsh in my assessment of economists, but it seems like every time I listen to one speak I have to wonder if they really understand what in the world they are talking about. And even if they do understand it, I feel like they are intentionally being hard to understand, so that no one will bother them with real questions or comments like, "Um, I don't think that will work, because your theories are a bunch of bunk." They also tend to make something so simple, so complex.

    Anyway, so Jonathan you are asking what happens when the transfer of money is non random? Or what happens when you have "intelligent" and "non-intelligent" ("stupid" may be too harsh, "unlucky"?) agents. If that were the case then you would not have a "free market economy", unless the number of "intelligent" and "unintelligent" agents were small compared to the overall population. The reason why the Boltzmann statistics work in the first place is because it appears that overall we do not have a significant number of "intelligent" agents who always make money and "unintelligent" agents who always loose it.

    I did mention briefly about the "super-thermal" tail in the income data. This high income segment of people represent 3% of the total population but control 10-20% of the money. These do not follow normal Boltzmann statistics and I assume that these people could be considered to be "intelligent" agents who always make money. But part of the point of Dr. Yakovenko's talk was that even these people are subject to a statistical law, just a different one from the rest of the population. So their behavior, even though it is different, can be well modeled. He talked about this much more than I did. He also mentioned that this may be the origin of Carl Marx's two classes. Dr. Yakovenko viewed this as a problem and talked about the inequality of the system, which prompted one of the professors in the audience to ask, "What's wrong with inequality?"

    So to recap, the point of Dr. Yakovenko's current research is to look at this high end and model them according to Pareto's statistics and see how that affects the system as a whole.

  4. Oh, no. I didn't mean to say that some people should *always* make or lose money. What I meant to indicate was a dependency over time. (That is what usually shows up in time series, for example.) The specific thing that I had in mind was something like lucky people get luckier. So, suppose everyone starts with a probability 0.5 of making a dollar at time t and a corresponding probability of 0.5 of losing a dollar at time t. (I actually don't know how the probabilities work in the simulations you discuss.) Now, suppose that the player Bob is lucky at time t and wins a dollar. What I am proposing is to let the win affect the probability with which Bob wins or loses at time t+1. Let a winner like Bob win at the next time with probability p(t+1) = p(t) + delta. Just for concreteness, let delta be 0.01. So, at time t+1, Bob has a probability of 0.51 of winning and a corresponding probability of 0.49 of losing. Do the same thing, but with a penalty term for the losers.

    Then my question is -- what would such a system look like?

    I am completely unsurprised that human behavior can be modeled statistically. That's been assumed (and at various times even demonstrated) since the 19th century. One way of reading Darwin is as an application of statistical mechanics to biology.

    As to economists ... well, I don't know. I still think you're being a little uncharitable. Maybe your lack of comfort with the economists is just a feature of differing cultures. I wonder how often people go into their respective fields because they are comfortable with the way people talk in those fields? I consider myself a philosopher, and I'm comfortable with philosophical babbling (most of the time). On the other hand, I'm sensitive to the fact that many people find philosophy painfully obscure and often irrelevant/useless. I also have a healthy appetite for statistics, but when I take statistics classes or read stats books, I always feel like things are being made intentionally unclear.

  5. So here's my problem with trying to use this as a model for policy decisions: people are not engaging in random interactions with a fixed amount of value in the world. This is a fascinating model and these types of models are exactly how I would attack the question, but the economists aren't stupid and their methods at least have some predictive power.

  6. That being said, I think there is a lot of power in applying these sorts of cross-disciplinary approaches to difficult problems. This is why a general education is a good idea, and why it's good to talk to someone in a totally different field every now and then.

  7. Jonathan: Your 'rich get richer' scenario looks to me to be mathematically the same as interest. The article notes that interest makes the entire system unstable unless some check (bankruptcy law) is in place.

  8. Great post. First of all, it would seem to me that interest would be more like P(t+1) ~ total accumulated wealth, rather than what Jonathan was saying with P(t+1) ~ P(t) +/- delta(t). (i.e. The rich get richer because they have more money already, not because they are inherently more lucky nor learn to become more lucky.)

    Second, I know that this is a really big field. There are A LOT of statistical physicists here that go to work on Wall Street. I would also like to assume that we are all "intelligent" investors, but the experimental data seems to show that, for at least the vast majority of the population, most of us do no better than making decisions at random. It would be interesting to try and build in some sort of "intelligence" (or graded intelligence, or work ethic, etc) into the system and see whether or not you get qualitatively different results. I don't know if we see the results we see because our total intelligence is small enough that it doesn't make a difference against the statistics (i.e. whether you're rich or poor really is just a matter of "luck"), or if intelligence does make a difference, but it gives the same macroscopic behavior (i.e. factoring in intelligence would still give a Boltzmann distribution, but the more "intelligent" would rise to the top -- or maybe not?).

    As far as the question "What's wrong with inequality?" -- I guess that just depends on what the people collectively want. I would tend to agree that a (heavily regulated) free market society is probably about our best option at the moment. However, as a Christian and someone who cares about the poor, I can't help but find the heavily populated ground and close-to-ground states somewhat distressing. Freedom and Charity are two extremely vital principles that seem to be at odds here (or maybe we just haven't found the right knobs to tweak yet -- what would happen if you built in some sort of "Charity" or "Charitable organizations" into the mix, rather than merely "intelligence"?).

    The really great thing that I take away from all of this is that it is good to know that the system is controllable. (Australia, though the results may not be what many of us would consider ideal, is nonetheless a good example of this.) Maybe if we can speed up our data taking and analyzing, we can respond quickly enough to prevent economic depressions and other problems. The real power is in knowing how the system works and in following correct principles.

    Once again, great post.

  9. Very interesting!
    I'd be interested to see the same plots for wealth rather than income.


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