Income 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.