## Saturday, March 23, 2019

### The statistical mechanics of money

Slow posting recently because of many real-life things going on after the March Meeting.  We had a very engaging colloquium this week by Victor Yakovenko, a condensed matter theorist from the University of Maryland.   A number of years ago, he got into "econophysics", applying insights from physics to the economy.  A great review is here

A classic example is in his highly cited paper, with the same title as this post.  Make some simple assumptions:  Money is a conserved quantity, and the rates of transactions don't depend on the financial direction of the transactions.  Take those assumptions, start with everyone having the same amount of money, and allow randomized transactions between pairs of people.  The long-time result is an exponential (Boltzmann-like) distribution of wealth - the probability of having a certain amount of money $m$ is proportional to $\exp(-m/\langle m \rangle)$, where $\langle m \rangle$ is the average wealth, a monetary "temperature".  The take-away:  complete equality is unstable just because of entropy, the number of possible transactions.

Apparently similar arguments can be applied to income, because it would appear that you can describe the distribution of incomes in many countries as an exponential distribution (more than 90% of the population).  Basically, for a big part of the population, it seems like income distribution is dominated by these transactional dynamics, while the income distribution for the top 3-ish% of the population follows a power-law distribution, likely because that income comes from returns on investments rather than wages.  The universality is quite striking, largely independent of governmental policies on managing the economy.

Yakovenko would be the first to say not to over-interpret these results, but the power of statistical arguments familiar from physics is impressive.  Now all we have to do is figure out the statistical mechanics of people....

Don Monroe said...

I'm confused about the point here. The Boltzmann wealth distribution for those assumptions seems plausible at first glance, and some intrinsic inequality seems tolerable. But when CEOs are getting paid hundreds of times what the rank and file are getting, doesn't that the model is missing something important? It is those income inequalities, and the similar long tails on the wealth distribution, that strike many people as unfair, inappropriate, and even immoral. If econophysics doesn't capture them, it doesn't seem very helpful.

Douglas Natelson said...

Don, I’m not making a value judgment. The observation seems to be that the distribution of incomes can be divided into two regimes. The “lower” class, which makes up something like 95-97% of the populace, has income distributed in a Boltzmann way, which seems to be a consequence of economic trade offs and the fact that there is (roughly) a floor at zero. The “upper” class has a long power-law tail, apparently because their income is based on returns on accumulated wealth. It is surprising to me that even in Scandinavian countries with much different tax structures, this seems to be true. All that seems to vary, and even then not by much, is the crossover point. I think the set of super wealthy CEOs is small compared to the set of super wealthy other people who make a lot of money off investments.

Don Monroe said...

Thanks, Doug. I was also not intending to endorse a value judgement, although I probably tipped my hand. What I was trying to say was that the exponential part (which I see as the null hypothesis) does not address what many people would see as the most important aspect of inequality, which is the tail. Indeed, from what I gather in a little googling, that upper-class tail encompasses more than half of wealth in the U.S. and in the world. (In the U.S. in the mid teens, the top 1%, well into the tail, held about 40%.) The Boltzmann model doesn't seem to say anything interesting about things like the position of the crossover and the power-law exponent, so it seems like a spherical cow.

DanM said...

I'm also concerned about the starting assumption: that money is a conserved quantity. This is manifestly not true. I can imagine that this could make a big difference in the analysis, including the details of where that cross-over point occurs (which, as Don has noted, is really a key point). If you can print your own money, then you're going to end up in the 'tail', not in the 'exponent'. There's a lot of that sort of thing going on, these days.

Douglas Natelson said...

DanM, I actually asked exactly this question in the talk. To zeroth order, money is a conserved quantity - only nation-state central banks can alter the total money supply. Now, I’m really not sure that I understand why income acts like a conserved quantity....

Anonymous said...

It actually seems that if you assume money is conserved, econophysics suggests a Boltzmann-Gibbs like exponential distribution of money. It seems that in fact you have to account for non-monetary conservation, or broken time-reversal symmetry (asymmetric transactions) to get the infamous tails...

Douglas Natelson said...

Anon, that's right. From the talk, my sense is that investment income (rather than simple transactions) effectively breaks time-reversal: You're more likely to make money if you already have money.

J said...

Who says that only central banks can create money? The econ 101 theory of money creation doesn't involve central banks at all but rather ordinary lending.
I have $100 that I deposit in a bank. That bank makes$90 in loans based on my deposits (choose your own leverage ratio for the bank to modify the number)
So I now believe and behave like I have $100, but someone else (the loan recipient) behaves like they have$90.
They deposit that amount into a bank, which causes a further $81 in loans to be issued, making some other loan recipient behave as though they have$81. As this process is repeated, in this example the initial $100 of mine turns into$1000 of apparent money (though you can add more friction to decrease this multiple).

It's true that on a balance-sheet accounting, there is still $100 in initial assets and$900 in loans-payable to cancel out the $900 in all of the borrowers' checking accounts. But when you ask "how much money is there in the country/world," it's not like you're just asking to total up the paper/metal money that has been printed by the mint. If you ask me how much money I have in the bank, I don't tend to subtract my mortgage before giving you an answer. The banking system relies on the fact that all of the depositors will not ask for their money back at the same time (because the bank has lent out most of the money), which makes us all think that we *have* that money, while in fact the total amount of deposits exceeds the total amount of printed money that has ever been deposited. Anonymous said... Your assumption is incorrect. You don't behave like you have$100 when it is locked in a CD; you're not buying whatever it is that you want and that you can get for $100. So economically, you are not multiplying the money. Same as when you own a car; you are not behaving like you have$5000; it's tied up in the car.
With the bank you bought a return on investment for lending your money.

J said...

It's not really personally about me, Anonymous. But who said I locked up the money in a CD? When I put money into a checking account, the bank still uses that money to make loans, and I certainly treat my checking account like it's ready cash.

In any case, this whole thing isn't my idea but rather part of econ 101. That doesn't mean it's correct, of course, but you can find aa simple introduction here: https://www.investopedia.com/terms/m/multipliereffect.asp

The relevance of this idea to Doug's original point is that only the M0 money supply is controlled by central banks. But when people engage in the exchanges described in the model, in which they eventually arrive at the exponential distribution, is M0 the correct monetary base to choose? When I engage in transactions, I probably think about them including my checking account and money market balances (M2). I don't know if that larger (and not fixed) monetary supply would change their conclusions at all.

Douglas Natelson said...

The author would agree immediately that the “money is conserved” case plus time symmetric transactions is a toy model. The surprising thing is that the resulting functional form of the distribution describes the lower 9x% of the income distribution so well in such great generality. Perhaps the fact that it’s income rather than wealth per se has something to do with why multiplier effects as J describes them do not show up in an obvious way.