Abstract:This paper is concerned with the problem of ranking and quantifying the extent of deprivation exhibited by multidimensional distributions, where the multiple attributes in which an individual can be deprived are represented by dichotomized variables. To this end we first aggregate deprivation for each individual into a "deprivation count", representing the number of dimensions for which the individual suffers from deprivation. Next, by drawing on the rank-dependent social evaluation framework that originates from Sen (1974) and Yaari (1988) the individual deprivation counts are aggregated into summary measures of deprivation, which prove to admit decomposition into the mean and the dispersion of the distribution of multiple deprivations. Moreover, second-degree upward and downward count distribution dominance are shown to be useful criteria for dividing the measures of deprivation into two separate subfamilies. To provide a normative justification of the dominance criteria we introduce alternative principles of association (correlation) rearrangements, where either the marginal deprivation distributions or the mean deprivation are assumed to be kept fixed. Keywords

In economic literature, the quality of life (QoL) in a city is usually assessed through the standard revealed-preference approach, which defines a QoL index as the monetary value of urban amenities. This paper proposes an innovative methodology to measure urban QoL when equity concerns arise. The standard approach is extended by introducing preferences for even accessibility to amenities throughout the city into the QoL assessment. The QoL index is then reformulated to account for the unequal availability of amenities across neighbourhoods. The more unbalanced the distribution of amenities across neighbourhoods, the lower the assessment based on the new index. This methodology is applied to derive a QoL index for the city of Milan. The results show that the unequal distribution of amenities across neighbourhoods significantly affects the assessment of QoL for that city.

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We study majority voting over a bidimensional policy space when the voters' type space is either uni-or bidimensional. We show that a Condorcet winner fails to generically exist even with a unidimensional type space. We then study two voting procedures widely used in the literature. The Stackelberg (ST) procedure assumes that votes are taken one dimension at a time according to an exogenously speci ed sequence. The Kramer-Shepsle (KS) procedure also assumes that votes are taken separately on each dimension, but not in a sequential way. A vector of policies is a Kramer-Shepsle equilibrium if each component coincides with the majority choice on this dimension given the other components of the vector.We study the existence and uniqueness of the ST and KS equilibria, and we compare them, looking e.g. at the impact of the ordering of votes for ST and identifying circumstances under which ST and KS equilibria coincide. In the process, we state explicitly the assumptions on the utility function that are needed for these equilibria to be well behaved. We especially stress the importance of single crossing conditions, and we identify two variants of these assumptions: a marginal version that is imposed on all policy dimensions separately, and a joint version whose de nition involves both policy dimensions.

Choice" and the "Groupe de Travail THEMA" for useful comments. This paper is part of the research program: "Fondements éthiques de la protection sociale: nouveaux développements" supported by the M.I.R.E. The usual caveat applies.† Corresponding author. 1 AbstractConsider an income distribution among households of the same size in which individuals, equally needy from the point of view of an ethical observer, are treated unfairly. Individuals are split into two types, these who receive more than one half of the family budget and those who receive less than one half. We look for conditions under which welfare and inequality quasi-orders established at the household level still hold at the individual one. A necessary and sufficient condition for the Generalized Lorenz test is that the income of dominated individuals is a concave function of the household income: individuals of poor households have to stand more together than individuals of rich households. This property also proves to be crucial for the preservation of the Relative and Absolute Lorenz criteria, when the more egalitarian distribution is the poorest. Extensions to individuals heterogeneous in needs and more than two types are also provided.