On the uneven distribution of wealth

by William Silvert

Abstract

Almost all societies exhibit an uneven distribution of wealth, and this is usually ascribed to different levels of productivity or entrepreneurship by its members. However, even if all of the people in a society function equally, it is hard to see how they can all end up with the same financial status given that there will always be chance events that enrich some and impoverish others.

Introduction

If we were to construct a society in which all members were the same – possibly even clones of each other – could we reasonably expect that they would all end up equally wealthy? This seems highly unlikely unless the social structure contains some feedback mechanisms to cancel the effects of good and bad luck. Maybe one hunter finds a kills a nice fat deer while another is home with the flu while his family goes hungry. One author gets panned while another makes it on Oprah. Perhaps equality is possible in a Polynesian island paradise, but this would be a rare situation indeed.

While almost all societies are unequal in terms of wealth and power, and many studies have been carried out to document the degree of stratification, there seems to be little interest in understanding how these inequalities came to be. It is always assumed that if someone is very rich it must reflect great financial acumen and industriousness. However when we look at the lives of the very wealthy we also learn about how they took great risks, and often their stories reflect both making a lot of money and going bankrupt one or more times. While it may take great courage to gamble, it does not take any skill to win or lose, so did these rich folks really deserve their riches, or were they just the lucky ones?

If we look around the great cities where the rich hang out we also find homeless bums and working drudges who are barely getting by. Some of these were also enterprising young businessmen who lost out in the luck of the draw. What happens to the owners of all those crowded shops along a busy highway when a high-speed bypass is constructed? To the producers of LP records and turntables? To Enron investors?

Although chance cannot explain all of the transfers of wealth in society or provide a full explanation of how it becomes unequally distributed, it certainly is a factor and should be taken into consideration. This paper explores the role that random effects alone play as a starting point for a more complete assessment of the dynamics of wealth distribution in different societies.

The Gangster Model

Suppose that we set up an experimental community of eight players and give each of them $100 with which to gamble. Every month they bet all they have on the toss of a coin. After the first month half will be broke and the other half will each have $200, assuming that the probabilities work out exactly this way. The next month six of them will have lost everything and the two remaining players will have $400 each. Finally on the next and final month there will be one grand winner with $800 and his 7 fellows will be broke. There is no element of skill in this outcome, just simple random luck.

Can such a simple model have any relationship to what goes on in the real word? It certainly doesn’t seem much like the workings of normal society, but consider the underground world of gangsters. Suppose that each of the players represents a criminal mob, and they fight for control of the rackets just as they do in real life (or at least in the movies). The outcome of each gang battle is that one of the mobs loses and the other gains controls of its turf and other assets. This looks a bit more like the results of the model, so maybe even the simplest random model has some application.

Of course the gangs are not identical and victory reflects some degree of military superiority, but that does not really affect the applicability of the model. It also seems that the fights are zero-sum games (unless both gangs are wiped out in the fighting), which is not a constraint on the model although the assumption that the coin tosses always come out even implements this.

The main point of this model is not that it is a complete depiction of the economic dynamics of the underground, but that it illustrates that inequalities of wealth can arise through totally random factors, and that skill or industriousness is not necessary in order to get rich.

Risky Investments

A more plausible model is one where the players do not gamble everything at each turn, although of course that does happen occasionally in real life. Suppose that at each turn the players bet a fixed fraction of their money on the toss of a coin – what happens then?

A general theory of this kind of financial system will be presented shortly, but a very basic version shows the general properties of the model. Suppose that we start with 80 individuals, each of whom starts with $800, and each month everyone bets 50% of his cash on the flip of a coin. After the first month there will be (on the average) 40 players with $400 and 40 with $1200. The following month there will be 20 with $200, 40 with $600 and 20 with $1800. Next month we get 10 with $100, 30 with $300, 30 with $900 and 10 with $2700. Following that the distribution becomes:

5 with $50

20 with $150

30 with $450

20 with $1350

5 with $4050

 The total amount of money in the system remains roughly constant at $64,000, so the average wealth is still $800, but the median has fallen to $450 and will continue to decline the longer the game goes on. Of the original 80 players, 55 have less than they started with, 20 have increased their holdings by about 60%, and 5 have become extremely wealthy. There is no skill involved in their success, just blind luck.

There is an interesting counter-intuitive aspect to this model. It is commonly believed that “the rich get richer and the poor get poorer” and these results seem to confirm this. However the reality is that the final wealth of any player depends only on the total balance of wins and losses, and not on the order in which they occur.

Address all correspondence to the editor, william@silvert.org. As for copyright, who would steal a simple idea? Just be sure to acknowledge the source.