Most Likely Wealth Distribution

En el último mensaje, I gave vent to all my left-wing righteousness against the growing income disparity. Entonces se me ocurrió — a totally uniform wealth distribution is stochastically unlikely. De hecho, 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? Eso sí, 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, y el resto con nada en absoluto – corresponds to 7.4 billion points, and is so much more likely, which is the statement I made in the first paragraph. Pero debe haber una distribución particular entre estos dos extremos que es más probable. So I sat down in front of my trusted iMac, trying to figure it out.

Responder a esta pregunta se reduce a averiguar el número de maneras de dividir n objetos en a grupos (la llamada estrellas y barras problema), and then sorting and counting the number of points in the phase space, una y otra. Not a very hard problem, excepto eso n y a son bastante grandes, and the combinatorics involves factorials of large numbers. So you have to use Stirling’s approximation. Bueno, 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. A mí, it shows a couple of other things: As the total wealth grows, the inequality will grow faster. En segundo lugar, 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. Seguro, se necesita todo eso para excavar su camino a la derecha (unless you were born there, which is statistically quite unlikely), but talent, proclivity for hard work, habilidades blandas, habilidades duras, etc.. son todos los accidentes de circunstancias. By the same token, if you do find yourself on the right side, no hay demasiada razón para caminar con un conciencia atribulada — hey, somebody had to be here. Better you than somebody else, derecho?

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