The dependence of the height of a Lorenz curve of a Zipf function on the size of the system

Abstract

The Lorenz curve of a Zipf function describes, graphically, the relation between the fraction of the items and the fraction of the sources producing these items. Hence it generalizes the so-called 80/20-rule to general fractions. In this paper we examine the relation of such Lorenz curves with the size of the system (expressed by the total number of sources). We prove that the height of such a Lorenz curve is an increasing function of the total number of sources. In other words, we show that the share of items in function of the corresponding share of sources increases with increasing size of the system. This conclusion is opposite (but not in contradiction) to a conclusion of Aksnes (studied in an earlier paper of Egghe) but where “share of sources” is replaced by “number of sources”.