Inequality and quasi-concavity

We discuss a property of quasi-concavity for inequality measures. Defining income distributions as relative frequency functions, this property says that a convex combination of any two given income distributions is weakly more unequal than the least unequal income distribution of the two. The quasi-concavity property is not essential to the idea of inequality comparisons in the sense of not being implied by the fundamental, i.e., Lorenz type, axioms on their own. However, it is shown that all inequality measures considered in the literature—i.e., 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 the property and even a stronger version). The quasi-concavity property is then shown to greatly reduce the possible inequality patterns over a much studied type of income growth process.