# Inequality in Society

By now, just about everyone will have heard about Thomas Piketty, who has claimed that the world inequality is getting worse and inevitably it will get much worse, on the grounds that wealth generates wealth. He has been attacked from various quarters, usually on relatively irrelevant details, for example that some of the data in his statistics are not quite right, but so far nobody has provided a knockout blow. Now, in

a recent edition of Science, where a number of articles addressed this issue of inequality of income in the world, one particular item took my attention: it arose from some physicists who argued such inequality is natural and arises from considerations similar to those of the second law of thermodynamics. Very specifically, they consider the statistical origin of entropy, and argue that a distribution of wealth where everyone has the same is just one of very many distributions, so it is extremely improbable when one considers how wealth evolves.

One way to illustrate the concept is to construct a simple model, and this is instructive (in my opinion, anyway) because it also shows something about models. Consider a game with these rules. There are 128 participants, and they play in rounds, and every round the players earn one credit. At the end of the round they may spend any of what they have, or they can save. If they get four credits, on the next round they get a bonus credit (return on investment) and they also have the choice of borrowing a further four credits. To further simplify, assume there is a fifty per cent chance of taking a specific option from the choice of two, and if the option is to spend, the choices of how much is evenly divided amongst the options. Now, watch how this game evolves.

At the end of round 1, each player has 1 credit, and half elect to spend it, which gives 64 with 1 credit and 64 on zero. Following round two, half of the first 64 continue to save, and half of those who choose to spend use one credit and the other half both credits. So we now have 32 with two credits, 48 with 1 credit and 48 with none. The reader can keep this going for himself, but it soon becomes apparent that 8 soon reach the 4 credit mark, at which point they get their bonus, then two will further invest, and of these, 1 will take the option of borrowing, and that one gains two each round, even though by borrowing he effectively has to repay at some stage. So, after five rounds, out of 128 originals, 1 has got ahead of everyone else, and only one other is close behind.

The analogy with entropy is as follows. In statistical thermodynamics, the entropy of a state is proportional to the logarithm of the number of ways of forming it, and the more ways, the higher the entropy. The second law says a system tries to maximize entropy. There is only one way to get to maximum wealth, while there are many ways to get a low wealth.

You may protest that this game is too crude, and you would be right, but it shows something about models. The first point about models is you have to get all the equations (a numerical statement of the rules) down and you have to accurately fix all the constants and functions. In this example, all earnings are in units of 1 (a constant) but in practice, it will be a distribution. Similarly with investment returns, and there are a number of other problems. Nevertheless, this simple model gives a qualitative result that matches reality: the distribution of wealth will always be unequal because different people make different decisions on what to do with what they earn, and the effects become very pronounced quite quickly. What this model has really done is not to predict social behavior, but rather to show the effects of a proposition, and that is where models are strong.