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The purpose of this paper is to analyze the advantages and disadvantages of several intermediate inequality measures, paying special attention to the unit-consistency axiom proposed by Zheng (2007). First, we demonstrate why one of the most referenced intermediate indices, proposed by Bossert and Pfingsten (1990), is not unit-consistent. Second, we explain why the invariance criterion proposed by Del Río and Ruiz-Castillo (2000), recently generalized by Del Río and Alonso-Villar (2008), leads instead to inequality measures that are unaffected by the currency unit. Third, we show that the intermediate measures proposed by Kolm (1976) may also violate unit-consistency. Finally, we reflect on the concept of intermediateness behind the above notions together with that proposed by Krtscha (1994). Special attention is paid to the geometric interpretations of our results.

Various indicators of income inequality and social welfare can be obtained simply by using the Gini index and the mean income of the population. This paper reviews existing indicators and presents several new indicators of this kind. While contemporary researchers seem to be preoccupied with relative inequality, this paper advocates for using intermediate inequality views and supplementing inequality rankings of countries with rankings based on social welfare. Empirical analysis, performed for 36 European countries, demonstrates such an approach’s advantages.

What change in the distribution of a population’s health preserves the level of inequality? The answer to this analogous question in the context of income inequality lies somewhere between a uniform and a proportional change. These polar positions represent the absolute and relative inequality equivalence criterion (IEC), respectively. A bounded health variable may be presented in terms of both health attainments and shortfalls. As a distributional change cannot simultaneously be proportional to attainments and to shortfalls, relative inequality measures may rank populations differently from the two perspectives. In contrast to the literature that stresses the importance of measuring inequality in attainments and shortfalls consistently using an absolute IEC, this chapter formalizes a new compromise concept for a bounded variable by explicitly considering the two relative IECs, defined with respect to attainments and shortfalls, to represent the polar cases of defensible positions.

We use a surplus-sharing approach to provide new insights on commonly used inequality indices by evaluating the underpinning IECs in terms of how infinitesimal surpluses of health must be successively distributed to preserve the level of inequality. We derive a one-parameter IEC that, unlike those implicit in commonly used indices, assigns constant weights to the polar cases independent of the health distribution.

This paper introduces a unit-consistent Lorenz dominance criterion that allows ranking income distributions according to centrist measures à la Seidl and Pfingsten (1997). In doing so, it defines α-Lorenz curves that generalize the absolute Lorenz curve. These curves allow implementing unanimous rankings for a broad set of centrist inequality notions, whereas they become closer and closer to the absolute curve when α approaches equity. In addition, this paper provides an empirical illustration of these tools using Australian income data. The results suggest that despite the reduction of relative inequality for Australian-born people between 1999 and 2003, their inequality increased for most centrist value judgments.

This paper examines the notion of intermediate inequality and its measurement. Specifically, we investigate whether the intermediateness of an intermediate measure can be preserved through repeated (affine) inequality-neutral income transformation. For all existent intermediate measures of inequality, we show that the intermediateness cannot be preserved through the transformation; each intermediate measure tends to either a relative measure or an absolute measure. This observation is then generalized to the class of unit-consistent inequality measures. An inequality measure is unit-consistent if inequality rankings by the measure are not affected by the measuring units in which incomes are expressed. We show that the unit-consistent class of intermediate measure of inequality consists of generalizations of an existent intermediate measure and, hence, the intermediateness also cannot be retained in the limit through transformations.

When health is measured by a bounded variable, differences in health can be presented as levels of attainment or shortfall. Measurement of heath inequality then usually involves the choice of either the attainment or the shortfall distribution, and this choice may affect comparisons of inequality across populations. A number of indices have been introduced to overcome this problem. This chapter proposes a framework in which attainment and shortfall distributions can be jointly analyzed. Joint distributions of attainments and shortfalls are defined from points of view consistent with concerns for relative, absolute or intermediate inequality. Inequality measures invariant according to the corresponding ethical criterion are then applied. A dominance criterion that guarantees unanimous rankings of the joint distributions is also proposed.

Theil's approach to the measurement of inequality is set in the context of subsequent developments over recent decades. It is shown that Theil's initial insight leads naturally to a very general class of decomposable inequality measures. It is thus closely related to a number of other commonly used families of inequality measures.

We investigate the relationship between the notion of progressive taxation and inequality reduction under a general version of the concept of inequality equivalence. We consider a two-parameter formalization of the concept of inequality equivalence that both includes, as special cases, the intermediate inequality equivalence and the path-independent/unit-consistent inequality equivalence. Both criteria could range from relative to absolute inequality views as the parameters in the formulation change. For the path-independent/unit-consistent inequality equivalence the condition of nondecreasing average tax rate is necessary and sufficient to guarantee the inequality-reducing effect of taxation for all the inequality views in between the relative and the absolute.

The chapter investigates inequality reducing taxation for various inequality views. Using the general definition of an inequality concept (Ebert, 2004), corresponding definitions of Lorenz dominance, inequality reduction, and measures of tax progression are provided. The framework allows us to simplify and clarify the different approaches found in the literature, to extend this analysis, and to present brief and transparent proofs.