Maximum Entropy and the Entropy of Mixing for Income Distributions

Abstract

Over the last 100 years, a large number of distributions has been proposed for the modeling of size phenomena, notably the size distribution of personal incomes. The most widely known of these models are the Pareto, log-normal, generalized log-normal, generalized Gamma, generalized Beta of the first and of the second kind, the Dagum, and the Singh-Madala distributions. They are discussed as a group in this note, as general forms of income distributions. Several well-known models are derived from them as sub-families with interesting applications in economics. The behaviour of their entropy is what is here under study. Maximum entropy formalism chooses certain forms of entropy and derives an exponential family of distributions under certain constraints. Finding constraints that income distributions have maximum entropy is another direction of this note. In economics and social statistics, the size distribution of income is the basis of concentration on the Lorenz curve. The difference between the tail of the Lorenz function and the Lorenz function itself determines the entropy of mixing. In the final section of this note, theoretical properties of well-known income distributions are also derived in view of the entropy of mixing.