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.Inequality, Quasi-Concavity, Growth, Rank-Dependent Expected Utility

Equality preference in the claims problem: A questionnaire study of cuts in earnings and pensions

Many distributional conflicts are characterized by the presence of acquired rights. The basic structure of these conflicts is that of the so-called claims problem, in which an amount of money has to be divided among individuals with differing claims and the total amount available falls short of the sum of the claims. We describe the results of a questionnaire in which Belgian and German students were confronted with nine claims problems. In the "Firm" version, respondents had to divide revenue among the owners of a firm who contribute to the activities of the firm in different degrees. In the "Pensions" version, they had to divide tax money among pensioners who have paid different contributions during their active career. Responses in the Pensions version were more egalitarian than in the Firm version. For both versions, the proportional rule performs very well in describing the choices of the respondents. Other prominent rules in particular the constrained equal awards and constrained equal losses rules fail to capture some basic intuitions. A substantial part of the respondents tend to become more progressive as the amount to be distributed decreases other things equal, and tend to become more progressive as the inequality in the distribution of claims becomes more unequal other things equal. All of these conclusions are robust with respect to the difference in home-country of the respondents.claims problem, acquired rights, proportional rule, constrained equal awards rule, constrained equal losses rule, inequality

Lorenz comparisons of nine rules for the adjudication of conflicting claims.

Consider the following nine rules for adjudicating conflicting claims: the proportional, constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, adjusted proportional, random arrival, and minimal overlap rules. For each pair of rules in this list, we examine whether or not the two rules are Lorenz comparable. We allow the comparison to depend upon whether the amount to divide is larger or smaller than the half-sum of claims. In addition, we provide Lorenz-based characterizations of the constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, and minimal overlap rules.Rules; Claims problem; Bankruptcy; Constrained equal awards rule; Constrained equal losses rule; Talmud rule, Piniles’ rule; Random arrival rule; Minimal overlap rule;

An axiomatic approach to the measurement of envy

We characterize a class of envy-as-inequity measures. There are three key axioms. Decomposability requires that overall envy is the sum of the envy within and between subgroups. The other two axioms deal with the two-individual setting and specify how the envy measure should react to simple changes in the individuals’ commodity bundles. The characterized class measures how much one individual envies another individual by the relative utility difference (using the envious’ utility function) between the bundle of the envied and the bundle of the envious, where the utility function that must be used to represent the ordinal preferences is the ‘ray’ utility function. The class measures overall envy by the sum of these (transformed) relative utility differences. We discuss our results in the light of previous contributions to envy measurement and multidimensional inequality measurement

Lorenz comparisons of nine rules for the adjudication of conflicting claims

Consider the following nine rules for adjudicating conflicting claims: the proportional, constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, adjusted proportional, random arrival, and minimal overlap rules. For each pair of rules in this list, we examine whether or not the two rules are Lorenz comparable. We allow the comparison to depend upon whether the amount to divide is larger or smaller than the half-sum of claims. In addition, we provide Lorenz-based characterizations of the constrained equal awards, constrained equal losses, Talmud, Piniles’, constrained egalitarian, and minimal overlap rules.Claims problem, Bankruptcy, Taxation, Lorenz dominance, Progressivity, Proportional rule, Constrained equal awards rule, Constrained equal losses rule, Talmud rule, Piniles’ rule, Constrained egalitarian rule, Adjusted proportional rule, Random arrival rule, Minimal overlap rule

The Class of Absolute Decomposable Inequality Measures

We provide a parsimonious axiomatisation of the complete class of absolute nequalityindices. Our approach uses only a weak form of decomposability and does not require a priori that the measures be differentiable.inequality measures, decomposability, translation invariance