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Self-Fulfilling Prophecies, Quasi-Non-Ergodicity and Wealth Inequality

Why is Jeff Bezos worth 0,000,000,000 while the median American has net assets of just 1,000? In a new working paper, co-authored with Jean-Philippe Bouchaud of the Collège de France, we argue that the vast inequalities we see in the world distribution of wealth are deeply connected to a somewhat esoteric concept from the theory ...

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Self-Fulfilling Prophecies, Quasi-Non-Ergodicity and Wealth Inequality

Why is Jeff Bezos worth $200,000,000,000 while the median American has net assets of just $121,000? In a new working paper, co-authored with Jean-Philippe Bouchaud of the Collège de France, we argue that the vast inequalities we see in the world distribution of wealth are deeply connected to a somewhat esoteric concept from the theory of stochastic processes. The world is quasi-non-ergodic. Our paper is published as CEPR Discussion Paper 15573. It is also available on my website linked [here].

Our paper has two intertwined themes. The first is that it is impossible to accurately predict the future by averaging over what has occurred in the past.  The second is that the inability to learn from experience implies that reasonable people will continue to disagree forever. Together, these two observations explain why, in the real world, wealth is so much more unequally distributed than income.

Our taking off point is the Pólya urn model reviewed in Pemantle (2007). In this model, an urn contains M red balls and (N−M) black balls. A ball is chosen at random.  If it is red (respectively black) then 2 red balls (respectively black balls) are re-introduced in the urn, which now contains N+1 balls.  The probability of drawing a ball with a specific colour therefore increases with the number of times this colour was selected in the past.

The long-term fate of the Pólya urn is surprising. As the number of draws tends to infinity, the probability to draw a red ball converges to a limiting value; but the value of this asymptotic probability is itself random.  Starting from the same urn with N red balls and N black balls, two different runs of the dynamics will lead to two limiting probabilities.   To  completely  characterize  the  behaviour  of  this  process one must introduce probabilities over probabilities. The dynamics of the Pólya urn are non-ergodic.

In my paper with JP, we build a model in which beliefs are described by a process, similar to the Pólya urn, but where there is no convergence to a number; instead, the probability of a given outcome is a random variable that converges to a limiting distribution. Stochastic processes of this form, where probabilities are themselves random variables, are referred to in physics as quasi-non-ergodic. In these models the averages of very long time series have the same mean as the mean of the invariant probability measure, but the length of time for that result to hold is astronomically long.

In our model, people trade assets contingent on an observable signal that reflects public opinion. The agents in our model are replaced occasionally and each person updates beliefs in response to observed outcomes. Interestingly, people continue to disagree forever, even though they are able to trade with each other in a complete set of financial markets. Trade continues because everyone believes that they know more than the market. 

Our model generates large  wealth inequalities that  arise from the multiplicative nature of wealth dynamics which  makes successful bold bets highly profitable. The flip side of this statement is that unsuccessful bold bets are ruinous and lead the person who makes such bets into poverty. People who agree with market prices have a low expected subjective gain from trading.  People who disagree may either become spectacularly rich, or spectacularly poor.

Self-Fulfilling Prophecies, Quasi-Non-Ergodicity and Wealth Inequality

In the paper, we simulate 300 years of weekly data for an economy with a million people and we show that equity prices in the model mimic equity prices in the real world. The outcome of this process is the wealth distribution in Figure 1. The Gini coefficient, a measure of inequality, is 0.7 in these simulated data, close to the wealth Gini’s we see in Western economics.

Why do people continue to bet with each other when these bets are highly risky?  The answer we propose is that everyone in our economy thinks that the market is wrong and that by betting, they will be able to make money on average.  They do not use the implied probability revealed by the markets to improve their estimates of true probabilities since this trading strategy is, in their opinion, sub-optimal.  

I started this post with a question: Why is Jeff Bezos rich? One answer is that Jeff Bezos created a company that has enormous social value and, as a consequence, his income is much greater than that of the average person. But although income is unequally distributed in the real world, income dispersion is not great enough to explain wealth inequality. In our model, everyone has the same income: But wealth is still highly concentrated because some people are luckier than others. 

Why are there no Warren Buffets who invest for the long run?  Our answer is that for any reasonable time period, the future is better approximated by averaging the frequency of recent observations than by assuming that the next draw of the stock market price is drawn from its unconditional long-run distribution. In the long run the market is correct.  But, as Keynes famously quipped: “In the long run we are all dead”.  

Roger Farmer
ROGER E. A. FARMER is a Distinguished Professor of Economics at UCLA and served as Department Chair from July 2008 through December 2012. He was the Senior Houblon-Norman Fellow at the Bank of England, January-December 2013.

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