A general rank-dependent approach for distributional comparisons

This paper provides a normative framework for distributional comparisons within a rank-dependent and bidimensional setting where individuals' well-being is characterized by a monetary and a non-monetary dimension (income and needs for instance) and when the focus is on inequality at the bottom as well as at the top of distribution in both dimensions. To this end, we develop third order inverse stochastic dominance conditions for classes of social welfare functions satisfying: i) Threshold Dependent Positional Transfer Sensitivity with respect to the monetary dimension (TDPT); ii) TDPT combined with downside inequality aversion with respect to the non-monetary dimension; iii) TDPT combined with upside inequality aversion with respect to the non-monetary dimension. Our results emerge along with the existing one, formulated by Zoli (2000) supporting downside inequality aversion both with respect to income and needs and Aaberge (2009) supporting upside inequality aversion with respect to income. Keyword: sequential stochastic dominance; social welfare; inequality JEL Cassification: D31, D63, I31


Introduction
In recent years a growing effort has been devoted to expand the classical notion of income inequality towards a multidimensional space of achievements. Building upon a finding due to Kolm (1977), who noted the possibility of extending the results deriving from choice under uncertainty to the field of multidimensional evaluation, 1 Atkinson and Bourguignon Such sign restrictions allow to introduce a degree of substitutability between attributes, expressing aversion to increasing correlation. The more sign-restrictions are imposed, the more complete the ordering becomes, but this comes at the price of making stricter assumptions on the shape of the SWF, which conveys a harder interpretability to the results.
In their work, Atkinson and Bourguignon show that it is possible to assess the dominance according to each one of their SWFs by means of easily implementable statistical tests borrowed from portfolio theory and corresponding to first and second order stochastic dominance techniques extended to a bidimensional setting. It is worth noticing that this formulation encompasses a symmetric treatment of the attributes considered in the analysis, contrary to what happens in a subsequent work by the same authors where the attributes play an asymmetric role (Atkinson and Bourguignon, 1987). In particular, in this last work they develop a framework for ranking bivariate distributions of household income and 'needs', where needs reflect some non-income equity-relevant characteristics, such as the household's composition. Due to the transferability typical of income, it is used to compensate for deficiencies concerning the other dimension. They divide the populations to be compared into subgroups, each one of them being characterized by a different level of needs, and impose certain sign restrictions on the utility functions reflecting some desirable properties, requiring essentially that the needier is a household the higher is her marginal evaluation of income at any income level, and that such marginal evaluation decreases as income increases. In this way, it is possible to take into account, in the social evaluation, the interrelations existing between welfare components and, consequently, to embed aversion to cumulative deprivation. Even in this case, the welfare dominance for the class of SWFs introduced is shown to be equivalent to readily implementable tests, namely first and second order sequential stochastic dominance conditions, suitable for a framework characterized by different subgroups of the population and consisting essentially in checking the dominance by sequential aggregation of the subgroups. In particular, conditions of first degree dominance are associated to restrictions imposed on the sign of the cross-derivative, reflecting the way in which the marginal valuation of income varies with needs, whereas conditions of second degree dominance are obtained by making assumptions about higher order derivatives. This approach, which represents an alternative to the use of equivalence scales to compare household incomes, has the advantage of allowing to rank family groups in terms of needs without, however, specifying how much a group is needier than another.
Later on, these techniques have been extensively adopted in the field of multidimensional inequality measurement. Atkinson (1992) extended the previous results within the context of poverty measurement, by developing dominance conditions to compare income distributions of individuals with different needs, when poverty is measured by means of the class of additively decomposable poverty indices. Also Jenkins and Lambert (1993) and Chambaz and Maurin (1998)   The profile of inequality matters as inequality experienced at different parts of the distribution can play a different role in the economy. Top and bottom inequality can have different implications when it comes to evaluate the effect of inequality on the future prospects of growth of a society. High levels of top inequality often reflect the phenomenon of 'social separatism' and anti-social behaviours might arise as a consequence, especially when income inequality is reflected by political polarization. This is a situation in which the rich get involved in lobbying activities in order to force the introduction of policies that benefit themselves but that result into hampering the growth opportunities of the poor, for instance, preventing the implementation of pro-poor and other productive polices, like spending on human capital or infrastructure, appropriate the country's resources and subvert the legal and political institutions by rent-seeking and corruption (see Easterly 2001; Glaeser et al. 2003). This thesis is supported by the empirical evidence showing that it is mostly top inequality that is holding back growth at the bottom (Van der Weide and Milanovic 2018). At the same time, the evolution of top incomes is acquiring a central role in the academic and public debate (see Atkinson and Piketty, 2007;2010;Piketty 2013, Piketty 2020).
Inequality at the top is relevant when observed from the perspective of relative concerns theory, according to which people have social preferences, that is, their utility also depends on the consumption or utility levels of others. Some theories of relative concerns predict negative welfare effects when friends and neighbours become better-off. Models of 'envy' assume that any improvement benefited by richer individuals acts as a negative externality on own utility (Friedman and Ostrov 2008), by contrast models of 'compassion' assume that a welfare improvement experienced by poorer individuals has a positive effect on own utility (Bolton and Ockenfels 2000). In more referenced models, envy and compassion coexist but they are combined such that the negative effect of an income increase of a richer individual more than outweigh the positive effect of an income increase of a poor individual (Fehr and Schmidt 1999).
We should also observe that an increasing number of contributions in the literature adopt a rank-dependent and bidimensional approach to make comparisons between distributions of variables other than income and needs. A notable application can be found in the equality of opportunity literature, in which the downside and upside inequality aversion can be usefully implemented to define the attitude of a social planner with respect to individual efforts when an opportunity equalizing policy needs to be implemented (see, Thus, it becomes compelling to consider alternatives to the standard attitude toward downside inequality aversion, which is encompassed by a social welfare function satisfying the principle of diminishing transfer sensitivity within the utilitarian approach and the principle of downside positional transfer sensitivity within the rank-dependent setting. In this vein, Aaberge (2009)  In the present work, we draw together the arguments introduced above, namely the concern for the multidimensional evaluation of social welfare, the interest in inequalities affecting the top of the distribution and the application of the framework in other contexts.
To this end, we develop third-degree inverse stochastic dominance conditions suitable for a rank-dependent and bidimensional framework and able to encompass different attitudes towards inequality aversion. In particular, a greater attention may be devoted to inequalities arising among the poorest as well as the richest individuals of the population, both in a monetary and a non-monetary sense. To be more precise, we introduce the principle iii) TDPT combined with upside inequality aversion with respect to the non-monetary dimension.
We show that our results can be considered as a generalization of the existing one supporting downside inequality aversion both with respect to income and needs and upside inequality aversion with respect to income.
Notice that in all these cases the standard Pigou-Dalton transfer principle is preserved, what differs from one approach to the other is whether one puts more emphasis on differences among the richest or the poorest individuals, having in mind that, within our setting, each individual is characterized by a monetary and a non-monetary attribute.
A remark is in order at this point. The standard practice adopted in sequential dominance tests consists in dividing the population into subgroups depending on family size, age, type of housing. However, the method may also cover cases where this information is replaced by any other variable that is helpful for identifying and classifying individuals, such as, for instance, health, education, area of residence. Throughout the paper, we use the generic term 'needs', having in mind that it has a quite flexible and wide content.
This contribution represents an additional instrument in the researcher's toolbox to help evaluating bidimensional distributions that can account for the whole profile of inequality and accommodate alternative declinations of the principle of inequality aversion.
The remainder of the paper is organized as follows: Section 2 outlines the framework, Section 3 presents the theoretical results, Section 4 concludes. 6 2 Theoretical model

Notation
Let individuals s, belonging to population S, be described by an income level y and by an additional non-monetary dimension of well-being, that we indicate with the generic term 'needs'. Consider the population S as partitioned into groups S i , i = 1, ..., n, characterized by a decreasing level of these needs, in such a way that we can consider group i to be needier and consequently poorer in a non-monetary sense than group i + 1. We assume that the needs level of individuals gives an extra information about their identity to be read together with the traditional information about the monetary dimension, represented by income in this specific case. If F i (y) is the cumulative income distribution of group i and we denote by q F i the share of individuals belonging to group i, we have that the overall cumulative income distribution is F (y) = n i=1 q F i F i (y) . Let µ(F ) be the mean income of distribution F . Moreover, let be the set of all such cumulative distributions and the left continuous inverse of F i (y) denoting the income y of an individual at the p th percentile of the distribution of group i.
We are interested in a formulation of the rank-dependent social welfare function that is able to capture not only the extent of monetary wealth, but also its non-monetary side, proxied by the membership of individuals to different need-based groups. A Yaarri (1988) type SWF coherent with our framework can be defined as follows:

Properties
In this section, we discuss different restrictions on the weight function. By imposing a greater number of such restrictions, although we make stricter assumptions about the properties of the social welfare function, we will be able to perform more comparisons between intersecting distribution functions. In Property 1 we impose a standard monotonicity assumption, requiring that social welfare is a nondecreasing function of individuals' incomes. This amounts to specifying non-negative weights.  In Property 2, we are interested in a welfare representation that gives priority to individuals exhibiting higher needs, that is, if two individuals belonging to different groups are ranked at the same percentile in the income distribution of their respective group, the social planner should give higher weight to the income of the needier individual. An interesting application of this property is in the context of equality of opportunity, as it captures the essence of this theory, that is, inequalities due to exogenous factors (i.e. the non-monetary variable) are inequitable and needs to be compensated by society.

Property 2. (Between Groups Inequality Aversion
A class of social welfare functions satisfying both Properties 1 and 2 displays weights function v i (p) is equivalent to impose that a progressive income transfer from individuals belonging to group i and ranked at the (p + π) th percentile of the income distribution to individuals of the same group i but ranked at the p th income percentile, with π ≥ 0, is welfare enhancing (see Mehran, 1976and Yaari, 1987, 1988).

Property 4. (Between Groups Diminishing Inequality Aversion
This property states that the same progressive transfer of income is evaluated differently if it takes place in different need groups. Namely, W increases more the higher is the needs level of the group within which the progressive transfer takes place. Following Zoli (2000), we consider two identical progressive transfers δ > 0 from individuals ranked at the (p + π) th percentile of the income distribution to individuals belonging to the same group, 9 but ranked at the p th income percentile, with π ≥ 0. These transfers take place within two different groups: the former is within group i, while the latter is within group i + 1, that is less needy. W obeys Between Groups Diminishing Inequality Aversion if the impact on social welfare of the transfer within group i is greater than the impact of the same transfer applied to group i + 1, that is: f or any p, π, δ, i which corresponds to: Simplifying for δ and for small π it becomes:

or any p, i
Properties 3 and 4, taken together, require that: In other words, the social concern for within group inequality decreases with the level of needs. We are introducing a diminishing sensitivity to horizontal equity concerns, by prescribing that the lower is the level of inequality in the individual income within a group, the higher is social welfare, conditionally on individuals' need. Property 3 and 4 are conditions linking vertical and horizontal equity, saying that horizontal equity is more important within worst-off groups. To better understand the meaning of the DPTS, consider a fixed progressive transfer δ taking place between individuals belonging to group i and showing equal difference in ranks, but located in different positions within their group's distribution. We require the transfer taking place at lower ranks to be more equalizing, and thus more welfare improving, than the transfer taking place at higher ranks. In particular, we consider two progressive transfers -within group i -of the same amount, δ, one from individuals at rank q + π to individuals at rank q, and another from rank p + π to rank p, with q ≤ p and π expressing the equal difference in ranks, and denote the change in social welfare associated with them, respectively, by ∆ i W (q, π, δ) and ∆ i W (p, π, δ). W satisfies the DPTS if and In formal terms, this means that: Hence, we may interpret this condition as the requirement that a combination of a progressive and a regressive transfer of the same amount δ but taking place respectively from the (q + π) th to the q th percentile and from the p th to the (p + π) th percentile of the income distribution, with q ≤ p, does not lead to a welfare loss. Such combination has been called in Zoli (1999)  We now proceed by refining the above expression in order to obtain a condition in terms of the weight function. Simplifying for δ and for π small enough we get that: Having in mind that p = q + ε and for small ε this could be written as: inequality aversion with respect to income.
To better understand the meaning of the UPTS, consider a fixed progressive transfer δ taking place between individuals with equal difference in ranks. We require the transfer taking place at lower ranks to be less equalizing, and thus less welfare improving, than the transfer taking place at higher ranks.
We consider two progressive transfers -within group i -of the same amount, δ, one from an individual at rank q + π to an individual at rank q, and another from rank p + π to rank p, with p ≥ q and π expressing the equal difference in ranks, and denote the change in social welfare associated with them, respectively, by ∆ i W (q, π, δ) and ∆ i W (p, π, δ). W satisfies UPTS if and only if ∆ i W (q, π, δ) ≤ ∆ i W (p, π, δ) f or all q ≤ p, with p = q + ε. In formal terms, this means that: Hence, we may interpret this condition as the requirement that a combination of a progressive and a regressive transfer of the same amount δ, but taking place respectively from the (p + π) th to the p th percentile and from the q th to the (q + π) th percentile of the income distribution, with p ≥ q, does not lead to a welfare loss. We call such combination an 'Elementary Upside Favorable Composite Positional Transfer' (EUFCPT).
We now proceed by simplifying the above expression in order to obtain a condition in terms of the weight function.
Simplifying for δ and for π small enough, we get that: Having in mind that p = q + ε and for small ε, this could be written as: Therefore, in terms of the weight function, the principle of within-group upside positional transfer sensitivity is equivalent to impose negative second derivatives, namely: To better understand the meaning of the TDPT, consider two fixed progressive transfers δ taking place between individuals belonging to group i and showing equal difference in ranks, but located in different positions within their group's distribution. We require -for all individuals in a group ranked belowp -the transfer taking place at lower ranks to be more equalizing, and thus more welfare improving, than the transfer taking place at higher ranks, for all individuals in a group ranked belowp. We also require -for all individuals in a group ranked above or equalp -the transfer taking place at higher ranks to be more equalizing, and thus more welfare improving, than the transfer taking place at lower ranks.
In particular, we consider two progressive transfers -within group i for individuals ranked belowp -of the same amount, δ, one from individuals at rank q + π to individuals at rank q, and another from rank p + π to rank p, with q ≤ p and π expressing the equal difference in ranks. Then we consider other two progressive transfers -within group i for individuals ranked abovep -of the same amount, δ, one from individuals at rank q + π to an individual at rank q, and another from rank p+π to rank p, with p ≥ q and π expressing the equal difference in ranks, and denote the change in social welfare associated with them, respectively, by ∆ i W (q, s, π, δ) and ∆ i W (p, t, π, δ). W satisfies the TPTS if and only if In formal terms, this means that: ∆ i W (q, s, π, δ) ≥ ∆ i W (p, t, π, δ) for all i and all q ≤ p ≤p and for all s ≥ t ≥p, which corresponds to saying that: for small π and for p = q + and To be more precise, we consider two similar EDFCPT that differ only in the group within which they are applied, since one of them takes place within group i, whereas the other takes place within a less needy group, i + 1, and impose that the former is more welfare improving than the latter, which means that, for any p, π, δ, i: For small π and having in mind that p = q + ε with small ε, combining Property 5, this is equivalent to: Alternatively, we may be interested in upside inequality aversion with respect to the income component, thereby assuming the validity of Property 6; we might still endorse downside inequality aversion with respect to the non-income component, thus requiring that a EUFCPT be more welfare enhancing if applied to needier groups. To say it in other words, this perspective reflects the idea that we give priority to a progressive transfer at higher ranks in the income distribution rather than at lower ranks, hence imposing that a EUFCPT does not lead to a welfare loss, but, at the same time, we want such EUFCPT to be more effective the higher is the needs level of the group within which it takes place.
This implies that a EUFCPT has a greater effect if applied to group i rather than group i + 1, that is, for any p, π, i: For small π and having in mind that p = q + ε with small ε, combining with property It follows that Property 8 is compatible with Property 7 as it is shown in the following derivation. From Property 7 we have: if p = q + and s = t + , for small and combining Property 8 and Property 7 we have that the above relation is true if we might still be more sensitive to this distinction the better ranked is the needs-based group considered, as expressed in Property 9. Imposing Property 5, a EDFCPT does not lead to a welfare loss, but, at the same time, we want such EDFCPT to be more effective the lower is the needs level of the group within which it takes place. In more formal terms, this amounts to imposing that, between two similar EDFCPT taking place respectively within group i and within group i + 1, the former is less welfare improving than the latter, that is, for any p, π, i: For small π and having in mind that p = q + ε for small ε and combining Property 5 with Property 9: Now, we endorse the alternative view, encompassing upside inequality aversion both with respect to income and needs, thereby allowing for Property 6 and 9 to coexist. In particular, the interest in the inequalities affecting the upper part of the distribution would require to devote a greater attention not only to individuals at higher ranks of the income distribution, thus endorsing the view contained in Property 6, but also to groups characterized by a lower level of needs. In this case, the welfare effect of the same EUFCPT applied over different groups, should not be lower the lower is the needs level of the group, that is for any p, π, i: For small π and having in mind that p = q + ε for small ε, combining Property 6 with 9, this is equivalent to: Similarly, Property 9 is also compatible with property 7, if p = q + and s = t + , for small and combining with Property 7, the above relation ≤ 0 for all i and q ≤p and t ≥p.
The following families of social welfare functions can be identified on the basis of the properties introduced above:

Results
In this section we present the sequential inverse stochastic dominance conditions for the classes of social welfare functions previously defined. We start by reviewing the dominance conditions introduced in the literature for the bidimensional case (section 3.1). We then move to the main results of our paper (section 3.2) that define different versions of the third order sequential inverse stochastic dominance and emerge along with the existing 18 sequential dominance tests proposed in the literature.

Existing dominance conditions
That is, the necessary and sufficient condition for welfare dominance for social welfare functions belonging to the class W 1 is a sequential inverse stochastic dominance condition of the first order. This requires to carry out a comparison between the two distributions That is, the necessary and sufficient condition for welfare dominance for social welfare functions belonging to the class W 2 is a sequential inverse stochastic dominance condition of the second order. To interpret this condition, we define the Generalized Lorenz curve for F following the formulation by Gastwirth (1971): We rewrite Ψ i (p) explicitly in terms of inverse distributions to have that: where GL F i (p) and GL G i (p) corresponds to the Generalized Lorenz curves associated to distributions F and G. Hence, the dominance of F on G for all SWF in W 2 , requiring that: means that we have to compare, at every percentile p and every stage k of the sequential procedure, the Generalized Lorenz curves of every subgroup, weighted according to the relative population share q i .

Proposition 3 Given two distributions
That is, the necessary and sufficient condition for welfare dominance for social welfare functions belonging to the class W 3 is a sequential upward inverse stochastic dominance condition of the third order. 4 In order to check whether such dominance criterion holds, we have to perform two different tests. As far as the first test is concerned, recall that, generally speaking, GL(1) = µ. Therefore k i=1 Ψ i (1) ≥ 0 ∀k = 1, ..., n means that we have to compare, between the two distributions F and G, the weighted averages of mean incomes of every subgroup, sequentially aggregated starting from the neediest group, then adding the second, and so on, As regards the second test, checking whether k i=1 Γ i (p) ≥ 0 ∀k = 1, ..., n, ∀p ∈ [0, 1], amounts to comparing, at every percentile p and every stage k of the sequential procedure, the integrated Generalized Lorenz curves of every subgroup, integrated from below, weighted according to the relative population share q i . This could be expressed also as: Notice that, endorsing downside inequality aversion both with respect to income and needs, leads us to: i) integrate Generalized Lorenz curves starting from the poorest income percentile, that is p = 0; ii) aggregate needs groups in the sequential procedure starting from the neediest one, that is i = 1.
Proof. We want to find a necessary and sufficient condition for In order to prove sufficiency, we will make use of Lemma 1, known as Abel's Lemma, that we now state.
We now turn to the sufficiency proof.
Two conditions of the third order and two sequential conditions of the second order are characterized by this proposition. They are combined such that one third order test and one sequential test are applied on the bottom part of the distribution withing each group and the other two are applied on the top part of the distribution withing each group. The distinction between the two parts of the distribution depends on the value of the threshold p that we assume it is defined exogenously. 5 In particular, given a value for the threshold p, within each group we have to check that Γ i (p) ≥ 0 ∀i = i, ...n, ∀p ∈ [0,p], that is, we have to compare the integrated GL, integrated from below, and weighted according to q i , at every group, i = 1, ..., n and every p up to the thresholdp. This could be expressed as: Then, for the same portion of the distribution within each group, we have to compare, between the two distributions F and G, the weighted averages of mean incomes, sequentially aggregated starting from the lowest ranked group, then adding the second, and so on, that .., n, where µ i (p, F i ) = p 0 F −1 i (p)dp. The second part of the proposition introduces two tests to be applied on the upper part of the distribution within each group. The first imposes to check whether Ω i (p) ≥ 0 ∀i = i, ...n, ∀p ∈ [p, 1], which is a third order downward inverse stochastic dominance test to be applied to each group of the population. It amounts to compare at every p the integrated GL, integrated from above, and weighted according to q i . This could be expressed also as: Last step requires to compare, between the two distributions F and G, the weighted av-5 It could be for instance the poverty line.
erages of mean incomes of the top part of the distribution, sequentially aggregated starting from the lowest ranked group, then adding the second, and so on, that is dp. Special cases of Proposition 4 are gathered in the following two corollaries and are obtained by selecting two particular values for the thresholdp.
Proof. We want to find a necessary and sufficient condition for In order to obtain the necessary and sufficient conditions for equation (14), we start from equation (9) of the Proof of Proposition 4, reversing the order of integration and summation.
-By Properties 1 and 2, a sufficient condition for -By Properties 7 and 8 and application of the Abel's Lemma a sufficient condition for -Using similar arguments, by Properties 1 and 2, a sufficient condition for .., n. -By Property 7 and 8 and application of the Abel's Lemma, a sufficient condition for  1]. We can apply the following decompositions to equation (9), reversing the order of integration and summation, and rewrite it as follows Suppose for a contradiction that ∆W ≥ 0, but ∃ h ∈ {1, ..., n − 1} and ∃ h = n and an Given that {ω i (p) ≥ 0} i∈{1,...,n−1} and {ς i (p) ≥ 0} i∈{1,...,n−1} by Lemma 1 in Chambaz and Maurin (1998) k=1 Ω k (p)dp < 0 ∀p ∈ Z and given that ε n (p) ≥ 0 and τ n (p) ≥ 0, by Lemma 1 in Atkinson and Bourguignon (1987) we have that p 0 ε n (p) n k=1 Γ k (p)dp < 0 and k=1 Ω k (p)dp and Q(p) = τ n (p) n i=1 Ω i (p)dp, we have If we choose T (p) and R(p) such that T (p) → 0 for all p ∈ [0,p] \I R(p) → 0 and all p ∈ [p, 1] \Z, we have that: Furthermore, suppose for a contradiction that there exists some j ∈ {1, ..., n} for which a T (p)dp < 0 and b a R(p)dp < 0 and that p 0 S(p)dp < 0 and 1 p Q(p)dp < 0, we can always choose a combination of v i (p)Ψ i (p) and v i (1)X i (1) 0 and a combination 27 for all i = j, then ∆W < 0, a contradiction.
Also Proposition 5 characterizes two tests to be applied to the bottom part groupspecific distribution and two tests to be applied to the top part group-specific distribution.
The second and fourth conditions are identical to the second and fourth conditions of Proposition 4, while the first and third conditions are second order sequential versions of the first and third tests presented in Proposition 4. In particular, the first test of Proposition 5 is identical to Proposition 3, except for the fact that the check of the dominance is restricted to part of group-specific distribution with upper-boundp. Hence, first we have to compare, at every p up top, integrated GL, integrated from below, and weighted according to q i starting from i = 1, the lowest ranked group, than adding the second lowest ranked group and so on up to the highest ranked group and perform the same check at every step. This could be expressed as follows: The third test of Proposition 5 requires an upward sequential aggregation of the third order downward dominance conditions presented in the third test of Proposition 4. We start by comparing, at every p beginning formp, integrated GL, integrated from above, and weighted according to q i starting from i = 1, the lowest ranked group, than adding the second lowest ranked group and so on up to the highest ranked group and perform the same check at every step. This could be expressed as follows: As stated in Section 1 and 2, the framework developed in this paper is flexible enough to be applied for performing different kinds of comparisons. For instance, a social planner could implement it to evaluate the welfare effects of two alternative policies when the relevant characteristic for identifying individuals, in addition to income, is the area of residence, for instance urban or rural. In this case, the group of individuals living in rural areas can be considered to be more 'needy' than those living in urban areas. If the social planner is more concerned with fighting extreme poverty and inequality in the more disadvantaged areas (i.e. rural) of the country, he could base its decision on the result of testing Proposition 5. (2000) is a special case of Proposition 5 that is obtained when p = 1. A second special case is obtained whenp = 0 and is formalized in the next corollary.

Proposition 3 in Zoli
Corollary 3 boils down to an upward sequential second order condition (across groups) of the third order downward inverse stochastic dominance (within groups) to be applied to the whole distribution.
In this case, we are assuming upside inequality aversion with respect to income and downside inequality aversion with respect to needs. This attitude is suitable for a framework in which, in order to check for third order inverse sequential stochastic dominance, we have to: i) integrate Generalized Lorenz curves starting from the richest income percentile, i.e. p = 1; ii) aggregate needs groups in the sequential procedure starting from the neediest one, i.e. i = 1. Proof.
We want to find a necessary and sufficient condition for Before finding these conditions, we propose Lemma 2, representing an alternative formulation of Abel's Lemma: In order to obtain these conditions, we start from equation (9) of the Proof of Proposition 4. For the sufficiency part, notice that: -By Properties 1 and 2, a sufficient condition for -By Property 7 and 9, we can apply Lemma 2 to get that a sufficient condition for . Now, since W 6 satisfies also properties 3 and 4, k i=n Γ i (p) ≥ 0 ∀k = n, ...1, ∀p ∈ [0,p] is sufficient also for n i=1 v i (p)Γ i (p) ≥ 0. -By Property 7 and 9 and application of Lemma 2 a sufficient condition for n i=1 1 p v i (p)Ω i (p)dp ≥ 0 is that k i=n Ω i (p) ≥ 0 ∀k = n, ..., 1, ∀p ∈ [p, 1]. Now, since W 5 satisfies also properties 3 and 4, k i=n Ω i (p) ≥ 0 ∀k = n, ..., 1, ∀p ∈ [p, 1] is sufficient also for n i=1 v i (p)Ω i (p) ≥ 0.
For the necessity part, using the same reasoning as in the Proof of Proposition 5. Let ε 1 (p) = v 1 (p) ∀p and π i (p) = (v i+1 (p) − v i (p)) ∀i, ∀p ∈ [0,p] and let τ 1 (p) = −v 1 (p) ∀p and ς i (p) = −(v i+1 (p) − v i (p)) ∀i, ∀p ∈ [p, 1] , we rewrite equation (9) as follows A social planner could, for instance, implement the test proposed in Proposition 6 to evaluate the welfare effects of two alternative policies when the relevant characteristic for identifying individuals, in addition to income, is the level of education, so that it is possible to distinguish the group of individuals with middle or low level of education and the group of individuals with high level of education. In this case, the group of individuals having a high level of education can be considered to be less 'needy' than those having a middle or low level of education. Nevertheless, the social planner could be more concerned with fighting poverty and inequality especially among the highly educated individuals, for instance, to make higher education more attractive, he could base his decision on the result of testing Proposition 6.
Two special cases of Proposition 6 are formalized in the following two corollaries.
If we endorse downside inequality aversion with respect to income and upside inequality aversion with respect to needs, then we come up with a third order procedure in which we: i) integrate Generalized Lorenz curves starting from the poorest income percentile, that is p = 0; ii) aggregate needs groups in the sequential procedure starting from the least needy one, that is i = n.
This is the last possible attitude towards inequality aversion, supporting upside inequality aversion both with respect to income and needs. In this case, to check whether third order inverse sequential stochastic dominance holds, we have to: i) integrate Generalized Lorenz curves starting from the richest income percentile p = 1; ii) aggregate needs groups in the sequential procedure starting from the least needy one i = n.

Concluding remarks
Several contributions in the economic literature show the need to modify the standard framework to rank income distributions, in order to take into account non-income aspects of well-being. At the same time, an increasing interest has been shown for inequalities affecting the upper part of the income distribution. In this context, the choice between upside and downside inequality aversion specifies whether, in the evaluation of social welfare, one should give priority to equalizing transfers between poorer vis-a-vis richer individuals.
This work represents an attempt to bring together these issues, by introducing upside inequality aversion considerations within a bidimensional evaluation of social welfare. In particular, we adopt a rank-dependent and needs-based social welfare function and develop third order dominance conditions to rank bidimensional distributions when a concern for upside inequality aversion is embedded.
To this end, we propose some properties that our SWF should satisfy. These properties We show that our results represent a generalization of existing dominance conditions in the literature and discuss a number of special situations.