As a third refinement, in 2019 we included negative wealth—one of the more disturbing aspects of modern economies—in our model. In 2016, for example, approximately 10.5 percent of the U.S. population was in net debt because of mortgages, student loans and other factors. So we introduced a third parameter, κ (or “kappa”), which shifts the wealth distribution downward, thereby accounting for negative wealth. We supposed that the least wealth the poorest agent could have at any time was –S, where S equals κ times the mean wealth. Prior to each transaction, we loaned wealth S to both agents so that each had positive wealth. They then transacted according to the extended yard sale model, described earlier, after which they both repaid their debt of S.

The three-parameter (χ, ζ, κ) model thus obtained, called the affine wealth model, can match empirical data on U.S. wealth distribution to less than a sixth of a percent over a span of three decades. (In mathematics, the word “affine” describes something that scales multiplicatively and translates additively. In this case, some features of the model, such as the value of Δω, scale multiplicatively with the wealth of the agent, whereas other features, such as the addition or subtraction of S, are additive translations or displacements in “wealth space.”) Agreement with European wealth-distribution data for 2010 is typically better than a third to a half of a percent [see box below].

To obtain these comparisons with actual data, we had to solve the “inverse problem.” That is, given the empirical wealth distribution, we had to find the values of (χ, ζ, κ) at which the results of our model most closely matched it. As just one example, the 2016 U.S. household wealth distribution is best described as having χ = 0.036, ζ = 0.050 and κ = 0.058. The affine wealth model has been applied to empirical data from many countries and epochs. To the best of our knowledge, it describes wealth-distribution data more accurately than any other existing model.

Credit: Jen Christiansen; Source: Bruce M. Boghosian; European Central Bank (country data)


We find it noteworthy that the best-fitting model for empirical wealth distribution discovered so far is one that would be completely unstable without redistribution rather than one based on a supposed equilibrium of market forces. In fact, these mathematical models demonstrate that far from wealth trickling down to the poor, the natural inclination of wealth is to flow upward, so that the “natural” wealth distribution in a free-market economy is one of complete oligarchy. It is only redistribution that sets limits on inequality.

The mathematical models also call attention to the enormous extent to which wealth distribution is caused by symmetry breaking, chance and early advantage (from, for example, inheritance). And the presence of symmetry breaking puts paid to arguments for the justness of wealth inequality that appeal to “voluntariness”—the notion that individuals bear all responsibility for their economic outcomes simply because they enter into transactions voluntarily—or to the idea that wealth accumulation must be the result of cleverness and industriousness. It is true that an individual’s location on the wealth spectrum correlates to some extent with such attributes, but the overall shape of that spectrum can be explained to better than 0.33 percent by a statistical model that completely ignores them. Luck plays a much more important role than it is usually accorded, so that the virtue commonly attributed to wealth in modern society—and, likewise, the stigma attributed to poverty—is completely unjustified.

Moreover, only a carefully designed mechanism for redistribution can compensate for the natural tendency of wealth to flow from the poor to the rich in a market economy. Redistribution is often confused with taxes, but the two concepts ought to be kept quite separate. Taxes flow from people to their governments to finance those governments’ activities. Redistribution, in contrast, may be implemented by governments, but it is best thought of as a flow of wealth from people to people to compensate for the unfairness inherent in market economics. In a flat redistribution scheme, all those possessing wealth below the mean would receive net funds, whereas those above the mean would pay. And precisely because current levels of inequality are so extreme, far more people would receive than would pay.

Given how complicated real economies are, we find it gratifying that a simple analytical approach developed by physicists and mathematicians describes the actual wealth distributions of multiple nations with unprecedented precision and accuracy. Also rather curious is that these distributions display subtle but key features of complex physical systems. Most important, however, the fact that a sketch of the free market as simple and plausible as the affine wealth model gives rise to economies that are anything but free and fair should be both a cause for alarm and a call for action.