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Follow the Money

Brian Hayes

Molecular Economics

I first began experimenting with the yard-sale simulation after reading an article, "Wealth distributions in asset exchange models," by Slava Ispolatov, Paul L. Krapivsky and Sidney Redner of Boston University. The computer models described there seemed both intriguing and easy to re-create, and so I wrote a quick-and-dirty program to play with some of them. I was perplexed when my results were quite different from those reported in the article. A second look revealed that I had misread a key equation, so that my model differed from theirs in a small but crucial way. Later I found a paper by Anirban Chakraborti of the Saha Institute of Nuclear Physics in India that describes essentially the same model I had accidentally created.

At least two other groups of physicists have recently published work on related themes. In France, Jean-Philippe Bouchaud of the Centre d'etudes de Saclay and Marc Mézard of the Ecole normale supérieure have described "wealth condensation" in a somewhat different model. And Adrian Dragulescu and Victor M. Yakovenko of the University of Maryland have written on "the statistical mechanics of money."

A source of ideas for most of these models is the analogy between market economics and the kinetic theory of gases. The molecules of a gas are continually colliding with one another and exchanging energy, in much the way that randomly chosen buyers and sellers in an economic model exchange sums of money. Yet gases do not follow the evolutionary path of the yard-sale economy. An economic collapse, where one person sucks in all the money, corresponds to a gas where one molecule has all the kinetic energy, and the rest are standing still. Don't hold your breath waiting for that to happen.

Where the yard-sale model departs from the kinetic theory of gases is in the details of the exchange of wealth or energy. When two gas molecules collide, they can reapportion their energy in any way that leaves the total unchanged. If the molecules have energies a and b just before they collide, afterward they can have any combination of energies that add up to a+b. Translating this energy-redistribution process into financial terms yields a market in which the parties to a transaction combine their wealth and then randomly divide the total. A simulated economy based on this rule does not collapse the way the yard-sale model does; wealth remains spread throughout the population, although not uniformly so. The distribution follows an exponential curve: The number of people with wealth w is proportional to e–w/T, where T is the temperature. (In the economic context, Dragulescu and Yakovenko identify the temperature with the average amount of money available to the participants.)

Figure 2. Distribution of wealth . . .Click to Enlarge Image

An exponential distribution crowds most of the people into the lower economic strata, but compared with the lopsided outcome of the yard-sale model, the degree of inequity is fairly mild. At least it's not an all-or-nothing economy. Furthermore, although the shape of the distribution is stable, individuals do not remain stationary within it: There are many rags-to-riches-to-rags stories in such a society. The gap between rich and poor seems less unfair if people have a reasonable chance of moving between these categories.

An exponential distribution of wealth is clearly preferable to a winner-take-all outcome, and an economic model based on the kinetic theory of gases may also have a certain aesthetic appeal—at least to physicists. Nevertheless, the interpretation of the model is problematic. There is no obvious reason to expect economic agents to act like colliding molecules, and indeed the random repartitioning of kinetic energy is a fairly strange template for mercantile transactions. Applied in the yard-sale context, it suggests that when Bill Gates comes to browse among my lawn ornaments, he and I will pool all our assets and then randomly split up the pot.

One kind of financial transaction that might fit the pattern of the kinetic gas theory is marriage followed by divorce: This is a case where the parties do combine their holdings and later redivide them, although perhaps not quite randomly. In the corporate world, mergers and spin-offs might produce similar results.

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