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Follow the Money

Brian Hayes

Fair Trade

Models of the market economy may lead to some cute mathematics, but do they have the slightest connection with the price of peas in the real world? Can they predict the actual distribution of wealth observed in human societies?

As it happens, the shape of the actual distribution is uncertain and controversial. Most of the available data concern the distribution of income, which is not quite the same as the distribution of wealth. Pareto, 100 years ago, argued that the income distribution obeys a power law, so that the proportion of people whose income is at least x varies as x–a; Pareto believed that the exponent a is a universal constant with a value of about 2.5. Other economists have proposed a log-normal income curve, meaning that the distribution of the logarithm of income is Gaussian.

Figure 5. Economic mobility . . .Click to Enlarge Image

The model of Bouchaud and Mézard (which includes investment earnings as well as trade) yields a Pareto-like power law for the wealth distribution. Some of the "greedy" models of Ispolatov, Krapivsky and Redner also appear to fit a power-law curve. But the models drawn most directly from the kinetic theory of gases predict an exponential distribution of wealth. Dragulescu and Yakovenko argue that the middle part of the actual wealth distribution is indeed exponential, with a "Pareto tail" in the highest tax brackets. All the computational models are so crude, however, and the empirical measurements are so uncertain, that curve-fitting inspires little confidence.

Also unclear is whether events comparable to the collapse of the yard-sale model can happen in a real economy. Societies where a small elite controls almost all the property, while the rest of the people are destitute, are all too common. But does this situation result from a mathematical instability in the system of trade, or is there a simpler explanation, such as mere malice and greed? In any case, economic collapse seems never to go to completion in the real world, as it does in the models. Tycoons amass immense fortunes, but no ever one goes home with all the marbles. (Bill Gates holds much less than 1 percent of the world's wealth.)

Rather than trying to match the output of the models to economic statistics, it might be more fruitful to examine real-world economic practices for signs of the basic mechanisms that underlie the models. In particular, the fatal feature of the yard-sale model is the rule limiting the size of a transaction to the wealth of the lesser trading partner. The rule appears to be perfectly fair and symmetrical, and yet it has the effect that the farther you fall through the economic strata, the harder you'll find it to climb back up.

Is such a rule likely to be enforced in everyday commerce? Not always. It is clearly violated in many forms of gambling and speculation, where the whole point of the transaction is the hope of gaining more than you put at risk. Doubtless there are other exceptions as well. For the most part, though, those of us with less money are limited to smaller-scale buying and selling. And the lower the ceiling on your economic activity, the slower your progress up through the ranks. When I buy a new car, I have little chance—no matter how shrewdly I bargain—of significantly altering the balance of assets between me and General Motors.

Explaining the distribution of wealth among individuals is not the only possible application of the trading models. They might in fact be better suited to describing relations among companies, where a sudden consolidation of wealth could be interpreted as the emergence of a monopoly.

Beyond the corporate world, there is the question of whether the models might have anything to say about commerce among nations, and the ongoing debate over free markets, fair trade and a "level playing field." If some mechanism like that of the yard-sale model is truly at work, then markets might very well be free and fair, and the playing field perfectly level, and yet the outcome would almost surely be that the rich get richer and the poor get poorer. You've heard that before.

© Brian Hayes

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