The most interesting scientific projects are those that surprise, when the mathematics, or the code, tells us something we didn’t expect. In our study of US wealth dynamics that’s what happened. We wrote it up in a paper, but that’s only the end product not the curious route by which we got there. Hence this post.

The equation of life

We started thinking about wealth dynamics some time in 2010 or 2011. We had been studying ensembles of growth processes, and that naturally led to thinking about ensembles of people and their growing wealths. Here’s what we did (“we” being Yonatan Berman, Alex Adamou, and I): we started with a ridiculously simple model for personal wealth, namely geometric Brownian motion (GBM).

(1) dx=x(\mu dt + \sigma dW)

I like to call this the equation of life. Why? Because life can be (and has been) defined as the thing that self-reproduces, and that’s what the equation describes. A quantity x that produces more of itself in a noisy way. It describes what happens to the biomass of an embryo in its early stages of development, or to the population of some species growing in a rich environment.

Once we’ve got self-reproduction in an environment with some fluctuations, evolution gets going, and beautiful structures like the ones we see around us follow sooner or later.

Equation (1) doesn’t just model biomass or populations but is also quite good at describing stock price dynamics. So we thought that it may be good at describing personal wealth too. After all, in one way or another both the stock market and our monetary fortunes reflect something that is happening in the economy. Let’s actually name the thing: we’re talking about capitalism. The genius of capitalism is precisely its multiplicative nature. Unused resources — capital — can be deployed to produce more of themselves. In this way a capitalist economy resembles the basic dynamic of evolution.

Our model does resemble wealth in a capitalist structure, but we were aware of its simplifying assumptions. It pretends that any changes in wealth are proportional to current wealth, whereas I could be poor and nonetheless boost my wealth through earned income. We treat everyone the same and pretend that differences in skill or earnings potential are random and not persistent etc. Nonetheless, we were curious about what would happen in a world where people’s wealth simply followed GBM.

Re-allocation: stability, Pareto tail, middle class

The first observation is this: under GBM the distribution of wealth never stabilizes, not even relative wealth stabilizes (that’s personal wealth divided by total population wealth). If we wait for long enough, essentially one person ends up with all the wealth. That struck us as unrealistic: we don’t live under feudalism. But we used to live under feudalism, so the real dynamic must be less extreme than GBM. That makes some sense — after all, the government collects taxes, and there are institutions that fund all sorts of social programs. We decided to make the model a little more realistic and included re-allocation of wealth. Surely the poor are helped by the rich in some way. So we changed the equation to

(2) dx=x([\mu-\tau] dt + \sigma dW) + \tau \langle x \rangle_N dt.

The new terms say this: every year everyone in the economy contributes a proportion \tau of his wealth to a central pot, and then the pot is split evenly across the population ( \langle x \rangle_N is per-capita wealth). Again, this is very simplistic — \tau represents a lot of different effects: collective investment in infrastructure, education, social programs, taxation, rents paid, private profits made… The equation can be re-written, which is very neat.

(3) dx=x(\mu dt + \sigma dW) - \tau (x- \langle x \rangle_N) dt.

This shows that it’s just like GBM (the first term) plus a mean-reversion process that attracts wealth to the population average. If I’m richer than the average, I’m likely to become a little poorer (relative to the average — my wealth can still grow); if I’m poorer I’m likely to become a little richer. The strength of the reversion is \tau, which can be thought of as a social cohesion parameter.

This equation is great! Whereas GBM leads to a diverging (unstable) log-normal distribution of relative wealth, equation (3) leads to a stationary inverse-gamma distribution. I mean if you let the equation run for a while, the number of people with a given wealth will follow an inverse gamma distribution. That distribution has a power-law tail, similar to what has been observed many times since Pareto‘s first studies. So it’s already pretty good, on a coarse-grained level.

What else did we know? Under GBM, wealth cannot become negative. Since the poor are always better off under equation (2), this is also true here.

Enter the computer

Thanks to tremendous efforts by many authors, including Tony Atkinson, Thomas Piketty, Emmanuel Saez, Gabriel Zucman, Wojciech Kopcuk, Jesse Bricker, Alice Henriques, Jacob Krimmel, and John Sabelhaus, we have a fairly good idea of the US wealth distribution over the past 100 years. So we took those observed distributions, created 100,000,000 individuals on a computer, fixed \mu directly from the wealth data and set \sigma roughly to the values observed in the stock market, and let the computer tune \tau each year so as to reproduce the real distributions.

Just for fun, we then looked at the individual wealths that had been produced by this procedure, and we noticed something strange. Many of them were negative. So back to the code, what did we do wrong? An error in the discretization scheme? Some other bug? No, the effect was real.

Negative re-allocation

Here’s what happened: in order to reproduce the data, towards the end of the analyzed period the algorithm had to make \tau negative, see figure 2 below. But what happens under those conditions to equation (3)?

Well, it describes negative re-allocation. Everyone pays the same dollar amount into a central pot, and then everyone receives from the pot an amount in proportion to how much he already has. That means if I have nothing, then I receive nothing but I still have to pay. That can make my wealth negative.

Look at equation (3) again, imagining \tau to be negative. The second term now describes mean repulsion. Whereas before wealth was attracted to the population mean, which generates a middle class, now wealth is repelled from it. If I’m a bit richer than the average, I’ll be boosted up even further; if I’m a little poorer, I’ll be pushed down even further. Run this equation for a little while and a large class of negative-wealth individuals arises.

At is turns out, something like that exists in reality. The cumulative wealth of the poorer half of the American population is roughly zero, meaning there must be a large class of negative-wealth individuals.

Falling interest rates

This is a blog post, so let me be speculative and push the story a little further than in the paper. How do those who have less than nothing keep giving to the rich? Simple: they go deeper into debt, deeper into negative wealth. But how can that be sustained over a long time? Debts don’t need to be paid off, but they do need to be serviced. To service growing debt with stagnant income (the situation in the US roughly since 1980), we need to lower interest rates.

Interest rates have been falling since about 1980, see Figure 4, precisely the time when the re-allocation rate became negative (c.f. Figure 1). What if there’s a causal link?

Now it gets interesting: interest rates have hit zero. What do we do? How can the poor keep paying the rich? Sure, let’s have some quantitative easing, but can that go on forever? Or will it break at some point? Is redistribution from poor to rich a threat to our monetary system? Is it a threat to our democracy? Where does the system go from here?

Let’s be clear about what we’ve done. We built a simple model and fitted its one main parameter. This wraps everything that’s actually happening into this one parameter. There are loose ends — the model may be fooling us, but we’re certainly not in a regime where we can comfortably rely on stabilization. We don’t claim that the world really works like equation 2, but that’s not the point of the exercise. Instead we say “pretend that equation 2 describes the dynamics of wealth; what parameter values would then best resemble what really happens?” The model is no more than a model and as such brushes over many details. For example, we don’t explicitly treat inheritance or income tax or some specific welfare program. Rather, this is all treated implicitly: our \tau summarizes everything that affects the wealth distribution beyond the null model of GBM. It reflects the overall trend in the complete economic system.

Ergodicity

That the model produced behavior beyond our (initial) imagination is encouraging. It means we didn’t accidentally constrain our study to confirm our beliefs. We wanted to know by how much we need to slow down the increase in wealth inequality implied by GBM to get to a realistic model. The model said: no, you’re asking the wrong question. GBM actually understates the increase in wealth inequality, and you need to correct the other way. Under GBM relative wealth is non-ergodic. The ergodic hypothesis as it is made in studies of wealth inequality thus excludes GBM as too extreme. Now it turns out that real wealth dynamics are better described by correcting GBM to make it even more strongly non-ergodic. None of us had expected that.

We should have written down our guesses for \tau before we started the study. We didn’t do this, but we certainly thought we would find a positive value. In a private correspondence, from the time before we looked at the data, Jean-Philippe Bouchaud set \tau =5\% p.a. in an example calculation, and we all felt that was the right order of magnitude. It could be 2% but obviously not as small as 0% (which would be GBM, equation 1).

Time scales

The connection to interest rates is speculative, but here’s one rock solid message about time scales that may hint at how we got here. A change in the effective re-allocation rate, \tau, takes decades to feed through. These processes operate on time scales of generations, not election cycles. That means it’s easy to oversteer because the consequences of policy changes only become visible after 30 or 50 years, long after whoever made the policy changes has left office, and at a time when the reasons for making the changes may no longer be valid. We certainly mustn’t assume rapid equilibration. However, rapid equilibration — the ergodic hypothesis — is a standard assumption in studies of wealth distributions.

The basic dynamic of a multiplicative-wealth economy — capitalism — seems underappreciated to me. If we “do nothing” ( \tau =0), inequality increases indefinitely. If we re-distribute fast enough ( \tau>0), inequality will stabilize at some level. If we actively destabilize ( \tau<0) as we seem to have done in recent decades, the middle class vanishes and we create a division between rich and poor — a poor person behaving reasonably is as unlikely to become middle class as a rich person behaving reasonably.

——

p.s. we can make the model arbitrarily complex. One aspect we later singled out is the effect of earnings, by including observed earnings in equation (2). Usually earnings have a stabilizing effect (meaning the process that describes only wealth must be less stable when earnings are treated explicitly). In the last 10 years or so, that stabilizing effect has been absent because of earnings inequality. Consequently, the values we find for \tau with this version of the model are smaller (more negative) up until about 2000 and then unchanged, see figure 5 below.