Income-Share Elasticity and the Size Distribution of Income

Abstract

The main purpose of this paper is to introduce the notion of Income-Share Elasticity as a convenient description of the size distribution of income. This concept, as an alternative to density functions, offers many advantages from both the theoretical and the applied point of view. On the one hand, it can help to capture some common features in seemingly unrelated density functions. On the other hand, it leads to a rather different approach to the description of the size distribution of income. Empirical research on the size distribution of income has been mainly confined to the descriptive task of fitting density functions to data. Most contributions consist of introducing a new density function and showing that it fits better than the functions previously considered. But there seems to be no clear rationale with regard to the search for new functions. This situation is essentially due to the lack of a theory on the size distribution of income, which is bound to persist unless we change the way of analysing empirical data. We must admit that the knowledge that, for instance, the Gamma function gives a better fit than the Log-Normal is not very inspiring. An alternative approach could be that of the identification of stylized facts about the distribution of individual incomes. Indeed, if the size distribution is to be analyzed, the data must show some sort of stable structure. Therefore, it would seem natural to identify these regularities for applied work. Once these stylized facts have been well established, we shall have a smaller set of density functions from which to choose. As an illustration of this approach, we consider a simple case in which we hypothesize three stylized facts and show that the three-parameter Generalised Gamma is the only density function that behaves accordingly. These three hypotheses are: a) distributions satisfy a modified version of Mandelbrot's [1960] Weak Pareto Law, which we call the Weak-WPL; b) they have a mode; and c) the Income-Share Elasticities have a constant rate of decline. These hypotheses can be subject to direct, independent tests. However, it is not our purpose to claim that they do satisfactorily correspond to observed facts. We cannot expect to have determined the class of functions that fit actual income distributions. The aim of the paper is to introduce an alternative way of approaching the description of the size distribution of incomes. We begin by defining the Income-Share Elasticity and showing that there is a