From: Eugene Leitl (eugene.leitl@lrz.uni-muenchen.de)
Date: Thu Aug 24 2000 - 07:00:19 MDT
http://www.newscientist.com/features/features.jsp?id=ns225217
That's the way the money goes
Illustration: Brett
Rider
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; xxx.lanl.gov/abs/cond-mat/0002374). 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
mark.buchanan@wanadoo.fr
From New Scientist magazine, 19 August 2000.
This archive was generated by hypermail 2.1.5 : Fri Nov 01 2002 - 15:30:35 MST