Robin Hood Reversed: A New Model of American Wealth Inequality

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Photo by Adam Nir on Unsplash

Why do we model complex social phenomena?

Good social science research not only yields insightful information that expands our knowledge of society, but it also enables us to solve some of its problems. Wealth inequality — this particular phenomenon captures public attention like few others. With a growing number of working households struggling to afford their basic cost of living, the widening gap between America’s richest — and everyone else — is becoming increasingly hard to ignore.

The breach has drawn scholars of all stripes to explain its causes. And so they collect data, make models, and draw up distributions. In a given country, household wealth should follow something like a log normal or inverse gamma distribution: it’s like if you took the bell curve and stretched out its right-hand tail. This so-called ‘Pareto’ tail represents the decreasing proportion of people with increasingly large amounts of wealth. In research, these distributions are conventionally confined to steady states, meaning they are held stationary in time. Particularly for social science research, observations of social phenomena entail a level of noise that make models necessarily simplistic. For example, economists will use stationary distributions to model the equilibrium shift from point A to point B after a shock to the system, like a tax cut.

Building informative models requires making trade-offs. Measuring market and policy shocks that shift the equilibrium is less complicated when distributions are stationary. Making a distribution stationary, however, requires ignoring real-world volatility. The alternative is to move away from analyzing stationary distributions in isolation, or to construct dynamic models. Dynamic models, however, are notoriously difficult to calibrate and make the final analysis incredibly complex: model parameters are sensitive and statistical properties fluctuate. Stationary distributions are comparatively straightforward. Researchers must therefore decide what is safe to assume, what to include, and what to ignore.

In light of these trade-offs, it follows that no model will be perfectly right. The practical question is how wrong they ought to be in order to not be useful. Emergent research suggests that, in the case of wealth inequality, the use of stationary distributions makes economic models miss the mark. The authors conduct a novel quantitative inquiry into the practice and find that, for the last several decades, stationary distributions of wealth do not exist.

In light of these trade-offs, it follows that no model will be perfectly right. The practical question is how wrong they ought to be in order to not be useful.

Researchers Yonatan Berman, Ole Peters, and Alexander Adamou from the London Mathematical Laboratory caution against the practice in their 2020 paper, “Wealth Inequality and the Ergodic Hypothesis: Evidence from the United States”. They present two provocative conclusions: First, wealth inequality in America is accelerating because wealth is being redistributed from poor to rich, and not the other way around. Second, even during the times when wealth was being redistributed from rich to poor, the time it took for the distribution to stabilize ranged from between a couple of decades to a couple of centuries.

In order to be convinced by the findings, there are three key concepts you should know: ergodicity, geometric Brownian motion, and reallocating geometric Brownian motion.

Let us say that we are interested in analyzing wealth in the United States, a quantity that varies both over time and across households. We take averages of this quantity using a stochastic process: the observation of n systems at time t. There are two averages we can take using this process: the ensemble average, and the time average.

The ensemble average — the average value between many systems at a fixed point in time — would be the average wealth of all American households in a given year. The time average — the average value of a single system over a long period of time — would be the average wealth of one household as the years go by. The ensemble average is analogous to the expected value (the population mean).

Ergodicity is when the expected value and its long-time average are equivalent. That is, when the population average and the individual average behave the same way in the long run. If they do, the stochastic process is ergodic. The validity of the ergodic hypothesis is of crucial importance: it allows economists to believe in the existence of unique, long-run equilibria independent of initial conditions. In other words, if the ergodic hypothesis is valid, a stationary distribution exists.

Studies of wealth inequality treat individual wealth as a growing quantity, which is non-ergodic. A neat trick to circumvent this problem is to divide individual wealth by the population average. This produces a rescaled wealth, and it has the potential to be ergodic. The issue that Berman, Peters, and Adamou find in the literature is that most models of wealth presume (or impose assumptions that guarantee) the ergodic hypothesis is valid. In order to test whether or not it makes sense to use the ergodic hypothesis for studies of wealth inequality, they use a model of the economy which does not take it for granted from the outset.

A basic geometric Brownian motion — GBM — is a continuous-time stochastic process. In GBM, wealth is explicitly not ergodic. The population average always shakes out as a positive quantity. Conversely, the time average is either zero, or diverges to positive/negative infinity, depending on whether individual trajectories grow or decay over time. Statistically, they are much more likely to decay.

Rescaling wealth in GBM does not make wealth ergodic: though mean rescaled wealth always remains the same, the distribution will become increasingly unequal over time without ever stabilizing. The gap between mean and median wealth widens, skewing the distribution further right, and the threshold needed to be considered part of the richest percentiles increases indefinitely.

Distribution behaviour under a GBM regime though time. Courtesy of Ergodicity TV on Youtube.

GBM has the opposite problem of most models. Since wealth in GBM is explicitly non-ergodic, it can’t be used to test if rescaled wealth is ergodic or not. Stabilizing the distribution requires the addition of a reallocation parameter, making it a reallocating geometric Brownian motion, RGBM. Practically, this means that at the end of every year, each household contributes a percentage of their wealth to a common pool which is redistributed evenly between everyone. Households with a net worth above the mean are net contributors, and households below the mean are net receivers.

Distribution behaviour under an RGBM regime through time. Courtesy of Ergodicity TV on Youtube.

The rate at which wealth is reallocated is denoted by tau, 𝜏. A positive tau value means redistribution from rich to poor; the larger the tau value, the faster the distribution becomes stationary between shocks. When tau equals zero there is no redistribution, and we are back to GBM. The reallocation parameter is a practical summarization of market dynamics, representing the net effect of all interactions in a socioeconomic network: employment, trade in goods and services, paying taxes, receiving welfare, the use of centrally-organized infrastructure such as roads, bridges, schools, legal systems, hospitals, etc.

The United States economy can be modelled by RGBM. It is large, socioeconomically complex, growing, and reallocating. Its tau threshold is effectively zero; rescaled wealth is ergodic if tau is positive, and follows an RGBM regime with the distribution of rescaled wealth converging over time. The thresholds needed to be considered a part of a particular wealth percentile are stable, and remain largely the same.

Plugging in U.S wealth distribution data yields a remarkable result. Berman, Peters, and Adamou find that the reallocation rate was largely positive between the 1930s and 1980s, and has been consistently negative since then. The ergodicity of rescaled wealth therefore cannot be validated, and there is no stationary wealth distribution that exists.

Fitted (dotted black) and smoothed (red) tau reallocation rates. Berman, Peters, and Adamou (2020), p12.

Tau’s negative value for most recent history has an especially severe implication: in these time periods, the reallocation of American wealth has effectively been from poor to rich, and not the other way around. For the last several decades, starting from about the 1980s to the present day, wealth in the socioeconomic network has aggregated in the hands of the richest few.

Implicitly, research using stationary distributions assumes that the convergence times between old and new equilibria are fast, but in the years with positive reallocation — from rich to poor — the tau value is low, and the rate of convergence is slow.

Time series of wealth distribution convergence times (black line). Grey blocks indicate periods where the distribution never stabilizes. Asymptotes indicate points of inflection and divergence (dotted line). Berman, Peters, and Adamou (2020), p14.

In practical terms, the time it takes the wealth distribution to reach a steady state after market or policy shocks is between a couple of decades and a couple of centuries (noted on the Y axis). This timeframe does not allow for the distribution to stabilize before subsequent shocks, and is not well-suited for politics.

Increasing inequality will emerge in a society that does not actively guard against it, and we possess the freedom to set the parameters governing our socioeconomic relations. This includes the ability to increase wealth redistribution through equitable democratic policies. They are, after all, the products of intentional human choices.

“There can be no progress without head-on confrontation.”

— Christopher Hitchens