That’s the way the money goes

Life’s so unfair. The rich get richer, while the rest of us just scrape by. Is society to blame or are deeper forces at work, asks Mark Buchanan

Life’s so unfair. The rich get richer, while the rest of us just scrape by. Is society to blame or are deeper forces at work, asks Mark Buchanan

WHY do rich people have all the money? This may sound like the world’s silliest question, but it’s not. In every society, most of the wealth falls into the hands of a minority. People often write this off as a fact of life–something we can do nothing about–but economists have always struggled to explain why the wealthy take such a big slice of the pie.

If Jean-Philippe Bouchaud and Marc Mézard are right, it is more than a fact of life: it’s a law of nature. These two scientists have discovered a link between the physics of materials and the movements of money, a link that explains why wealth is distributed in much the same way in all modern economies. Their theory holds out a scrap of hope to the poor of the world: there may be some surprising ways to make society a bit more equal. And it promises a new fundamental science of money. Economic theory is about to grow up.

In the 19th century, economists were certain that each society would have a unique distribution of wealth, depending on the details of its economic structure. But they were dumbfounded in 1897 by the claim of a Paris-born engineer named Vilfredo Pareto. The statistics, he insisted, prove otherwise. Not only do a filthy-rich minority always hog most of the wealth, but the mathematical form of the distribution is the same everywhere.

To get a feel for Pareto’s law, suppose that in Germany or Japan or the US you count up how many people have, say, $10 000. Next, repeat the count for many other values of wealth (W), both large and small, and finally plot your results on a graph. You will find that there are only a few extremely rich people, and that the number of people increases as W gets smaller–at least until you get down to those with almost no wealth at all. This is exactly what Pareto found: the number of people having wealth W is proportional to 1/WE. Pareto found that the exponent E was always between 2 and 3 (see Diagram) for every European country he looked at, from agrarian Russia to industrial England. And up-to-date statistics show the same thing.

This distribution means that most of the wealth gathers in the pockets of a small fraction of the people. In the US, for example, 20 per cent of the people own 80 per cent of the wealth. In Britain and in the nations of Western Europe the numbers are similar. The shape of the graph seems to be universal.

For over a century, this universal law of wealth distribution has defied explanation, many economists simply putting it down to the inherent distribution of people’s abilities. The truth may be simpler.

Economic theories have for years been founded on all sorts of dubious assumptions: that markets are in equilibrium, for example, or that people behave with perfect rationality. These assump- tions simplify economists’ intricate equations, but they often lead to rather peculiar conclusions. There is even a “no trades” theorem, says Bouchaud, that in an economy where all the participants are perfectly rational, no trade should ever take place. So Bouchaud and Mézard are pioneering a totally different approach. They are going back to basics, trying to get by with an absolute minimum of assumptions.

“Ten years ago,” says Mézard, speaking from his office at Paris-Sud University, “Jean-Philippe was one of the first physicists to get interested in finance.” Questioning many traditional ideas, he at first met resistance. Since then, however, Bouchaud has built a company dealing in risk management that has won the attention and respect of the financial industry. Now at the Centre d’Etudes de Saclay in Paris, he and Mézard are setting their sights on a more ambitious goal: to build a theory of economics from the ground up.

An economy is just a large number of people who can trade with each other. Each individual has a certain amount of money he or she can invest or use to buy the services or goods of others. This is all beyond argument. Things get more contentious when you try to turn these words into precise equations. Who trades with whom? Which investments pay off and which do not?

Bouchaud and Mézard start from ground zero, with only one assumption: life is unpredictable. Buy some stocks and you might get a healthy return or a devastating loss; returns on investments are random. The trade network is also haphazard. Each person trades with a few others chosen at random from the population. “Our idea,” says Bouchaud, “is to see how much we can explain on the basis of little more than pure noise.”

With these few ingredients, the model seems to contain almost nothing at all. “In its basic points,” says Mézard, “it’s really trivial.” There are many ways to build these basic elements into some equations, but fortunately the researchers had another clue.

A guiding principle of physics is the notion of invariance. Rotate a circle about its centre, and its shape remains unchanged. This is what makes a circle an especially simple and important shape in geometry. Similarly, the fundamental equations of physics are invariant under the action of certain mathematical operations, making them special cases in the space of all possible equations. Newton’s laws of mechanics don’t change if you alter the velocities of every body by an equal amount; otherwise, the physics you saw would change depending on how fast you were moving.

The economic equivalent of this is that a theory should produce the same results if you change the units of currency. “This is what we try to explain to our children,” he says, “when they complain that their pocket money will go down when we shift to euros.” Consequently, Bouchaud and Mézard wrote down the simplest equations they could find that were invariant with changes in currency.

Getting equations is one thing; solving them, another. There are millions of people in an economy, and that means millions of equations, which is why economists have tended to shun this “bottom up” approach. Bouchaud and Mézard made their task easier by keeping the ingredients of their model so simple, but they were still left with a daunting task. Then, earlier this year, they became the beneficiaries of a miraculous mathematical coincidence.

As “condensed matter” physicists, the pair have for two decades been investigating the properties of solids and liquids, substances in which the atoms or molecules are crammed together. The traditional subjects of this field are materials such as pure metals and water, whose particles settle into a well-defined state such as an ordered crystal. But since the 1970s, researchers have been increasingly intrigued by “ill-condensed” matter in which competing forces frustrate this condensation. In this class of materials–which includes glass, dirty alloys and polymers–the particles can end up in a vast number of disordered but more or less equivalent configurations.

Two competing forces

To overcome some of the mathematical difficulties in the theories of these materials, physicists have invented a simple “toy model” called the directed polymer. Imagine a long wire (the polymer) lying on a landscape that undulates up and down at random (click on thumbnail below for diagram). The wire is tethered at one end to a post. Gravity will tend to pull it down into the valleys, but as the landscape is random, the wire will have to bend to do so. So two forces–gravity and the wire’s desire to stay straight–compete with one another.
As a result, the wire has to compromise: running through the valleys, so long as that doesn’t entail too much bending, and, whenever the path becomes too tortuous, arching up over a pass to seek a straighter route. There is no obvious “best path” for it to follow.

Many physical systems behave in a similar way. Take a magnetic field line, for example, as it tries to slip through a high-temperature superconductor. Left alone, it would follow a straight path. But these materials contain defects–analogous to the valleys and peaks–that attract or repel the lines. So the path they take is some compromise between going straight through and swinging by attractive defects.

In such real, physical problems, working out the details of the compromise is difficult. In 1988, however, physicists Bernard Derrida and Herbert Spohn of the École Normale Supérieure in Paris solved a version of the directed-polymer problem exactly. There is one extra crucial element in the problem: temperature. In seeking some path across the random landscape, for example, the wire also puts up with a continual buffeting from air molecules, which knock it about from one path to another. The buffeting grows more vigorous with increasing temperature, and, as it turns out, the strength of this storm determines how the wire manages its compromise between going straight across and staying in the valleys.

“When the temperature is high,” says Mézard, “the peaks and valleys have little effect.” The buffeting is so violent that the polymer largely ignores the landscape, and flaps about all over the place. As the temperature falls, however, there comes a point at which the buffeting is no longer strong enough to drive the polymer over the landscape’s peaks. Suddenly, the peaks and valleys become far more influential, and the polymer gets stuck in place, trapped along one irregular path. This sudden condensation is like the freezing of glass, or the pinning in place of a magnetic field line.

Mézard and Bouchaud have now discovered that the equations for this directed polymer model are identical to those for their economy (Physica A, vol 282, p 536; So to solve their equations they need do no more than pluck out some gems from the physics literature. And what these equations show is that under normal conditions, their economy follows Pareto’s law.

To see how the model economy and a directed polymer are related takes a little imagination. Start with the irregular landscape, and throw a whole bunch of polymers across it. Let them settle down, and take a snapshot. This is now a picture of the economy over time.

Think of the people in the economy occupying positions on the y-axis of the landscape, and progressing to the right over time (see Diagram, p 24). A polymer plots the path of some quantity of money as it moves from person to person. So at any point, the wealth of a particular person is determined by the number of polymers that cross over their y-value.

The irregular returns on investments are reflected in the ruggedness of the landscape: deep valleys are places where there tends to be more money, the returns in investments being high; peaks are where investors fare badly and money is rare. The vigour of trade–how easily money flows between people–is analogous to the temperature. “The wealth follows a kind of random walk,” says Bouchaud.

There is, however, more than one kind of random walk. Which kind wealth follows depends on how “hot” the economy is. When trading is easy, and the irregularity of returns on investment not too severe, the economy behaves like a polymer at high temperature. Just as the polymer flaps up and down with ease, adopting almost any configuration without being too strongly affected by the underlying landscape, so does vigorous trading enable wealth to flow easily from one person to another, tending to spread money more evenly.

But because the returns on investment are proportional to the amount invested, rich people tend to win or lose larger amounts than poorer. Over time, even if all changes are random, wealth ends up following Pareto’s law with an exponent E between 2 and 3. How much money an individual has need have nothing to do with ability. Chop off the heads of the rich, and a new rich will soon take their place.

This is not to say that the distribution of wealth cannot be influenced. The model offers what might be the first lesson of economics to be firmly founded in mathematics: that the way to distribute wealth more fairly is to encourage its movement. Taxation, for example, tends to increase E. This is still a Pareto law, but with the wealth distributed somewhat more equitably, the rich own a smaller fraction of the overall pie. With an exponent of 3, for example, the richest 20 per cent would own 55 per cent of the wealth. It’s still not fair, but it’s better than the US today.

The model makes it clear, however, that taxes only work when they are redistributed evenly: if the rich get a disproportionate share–because of lucrative corporate contracts with the government, for example–then the social effect of the tax is wiped out. And according to economist Anthony Atkinson of Oxford University, economic texts have long assumed that there should be some kind of “trade-off” between equality and efficiency: that while spreading the wealth more evenly, higher taxes will also slow economic growth.

Fairer and freer trade

Yet there may be many ways besides taxes to help wealth move about, for instance by widening the number of people with which any one person tends to trade. In other words, Bouchaud and Mézard’s model implies that a more equal society could come from encouraging fairer and freer trade, exchange and competition. Happily for economists, this idea dovetails with their experience and expectations, but the model gives these expectations a robust mathematical foundation.

If hot means vigorous trading and low volatility in investments, cold means restricted trading and highly irregular returns. As Bouchaud and Mézard reduce the ease of trade and increase the degree to which investments are random, they find a sudden change in their distribution of wealth: in a cold economy, wealth freezes.

Just as the polymer gets trapped into one irregular valley, and so follows a path dictated almost entirely by the random landscape, so wealth finds itself unable to flow easily between people. In this case, the natural diffusion of wealth provided by trading is overwhelmed by the disparities kicked up by random returns on investment. In Bouchaud and Mézard’s model, the economy falls out of the Pareto phase into something much nastier. Now wealth becomes even less fairly distributed, condensing into the pockets of a handful of super-rich “robber barons”.

Might this be the case today in some developing or troubled nations? It has been estimated, for example, that the richest 40 people in Mexico have nearly 30 per cent of the money. According to economist Thomas Lux of the University of Kiel in Germany, “most economists would anticipate that wealth concentration will be higher in economies with limited exchange opportunities–such as Russia, for example”. Unfortunately, he adds, “these are usually the economies for which we have poor data or no data at all.” Mézard suspects that such societies may have been more common in the past, but again the economic data are sparse. So testing this prediction of the model won’t be easy.

Having illuminated Pareto’s law of wealth, Bouchaud and Mézard’s approach is pointing towards a deeper theoretical perspective on economics. They hope to build more realistic models by moving away from the assumptions of complete randomness, and that every economic agent is identical. Their work offers the promise of understanding not only how the economy behaves now, but also how things might conceivably change.

Could political instability throw an economy out of the Pareto regime? Or might wealth condensation be a generic risk if there is too much central planning in an economy? And are there any other hidden variables, changes that might tumble an economy over the precipice and into the depths of inequality? With the global economy becoming more and more tightly knit, these are questions that should concern the whole planet.

Mark Buchanan writes from the village of Notre Dame de Courson in northern France. He can be contacted at

From New Scientist magazine, 19 August 2000.

Author: Mark Buchanan

News Service: New Scientist Magazine


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: