Most Likely Wealth Distribution

In the last post, I gave vent to all my left-wing righteousness against the growing income disparity. Then it occurred to me — a totally uniform wealth distribution is stochastically unlikely. In fact, it is over seven billion times less likely than one person in the whole world holding all the wealth in the world. That brings me to the topic of this post – what is the most likely wealth distribution?

How do we attempt to answer this question? Mind you, it is a mathematical question, not a moral one. The most moral distribution, one might think, would be to take all the wealth in the world and divide it equally among the 7.4 billion people. Borrowing the concept of phase space from statistical mechanics, this particular distribution will correspond to just one point in a 7.4 billion dimensional space. The other extreme situation – any one person holding all the wealth in the world, and the rest with nothing at all – corresponds to 7.4 billion points, and is so much more likely, which is the statement I made in the first paragraph. But there should be a particular distribution in between these two extremes that is most likely. So I sat down in front of my trusted iMac, trying to figure it out.

Answering this question boils down to figuring out the number of ways of dividing n objects into k groups (the so-called stars and bars problem), and then sorting and counting the number of points in the phase space, over and over. Not a very hard problem, except that n and k are quite large, and the combinatorics involves factorials of large numbers. So you have to use Stirling’s approximation. Well, I will spare you the mathematical details and get straight to what I found.

What I found was quite surprising. The most likely situation seems to be the one in which the wealth distribution is highly skewed. The richest person would own more than 60% of the wealth, and the top 50 people would own practically everything!

So any wealth redistribution effort will have to fight the tremendous stochastic pressure in addition to our innate reluctance to part with the green stuff. Does this constitute a mathematical proof that socialism cannot work? I guess the right-wingers might take it that way. To me, it shows a couple of other things: As the total wealth grows, the inequality will grow faster. Secondly, those who find themselves on the right side of the income divide should thank their stars more than their so called talent, hard work etc. Sure, it takes all that to burrow your way to the right side (unless you were born there, which is statistically quite unlikely), but talent, proclivity for hard work, soft skills, hard skills etc. are all accidents of circumstances. By the same token, if you do find yourself on the right side, there is not too much reason to walk around with a troubled conscience — hey, somebody had to be here. Better you than somebody else, right?