Summary: The folk saying, “the rich get richer, the poor get poorer”, implies that wealth flows from poorer to richer hands, a concentrating diffusion. Its academic statement is the surplus theory of social stratification. The author [Soc. Forces 65, 293-326 (1986)] abstracted this theory as an interacting particle system model of wealth distribution, the inequality process. In it, random pairs continually compete for each other’s wealth. The richer party has a greater chance of winning. The looser gives up a proportion of wealth. The author (1986) found the equilibrium distribution of the inequality process numerically.
This paper derives this distribution algebraically, showing that the inequality process implies that personal wealth is distributed as a mixture of negative binomial probability functions (NB pf’s). The NB pf is closely related to the two parameter gamma probability density function (pdf), what the author (1986) fitted to the equilibrium distribution of the process. The solution is given in terms of the parameters of the inequality process. Making wealth less fugitive by putting it into the form of human capital has the same effect on the shape of the wealth distribution as reducing the advantage of being richer in an encounter.