Inequality, quasi-concavity, and gradual population shifts

An income distribution is a mixture of two given income distributions if the relative frequency it associates with each income level is a convex combination of the relative frequencies associated with it by the given two income distributions——e.g., the income distribution of a country is obtained as a mixture of the income distributions of its regions. In this article, it is established that all inequality measures commonly considered in the literature——the class of decomposable inequality measures and the class of normative inequality measures based on a social welfare function of the rank-dependent expected utility form——satisfy quasi-concavity properties, which imply, loosely speaking, that mixing income distributions increases inequality. These quasi-concavity properties are then shown to greatly reduce the possible patterns describing the evolution of inequality in the overall income distribution (a mixture) during a process in which population gradually shifts from one of its constituent income distributions to another over time.