Abstract
We study the construction of Paretian egalitarian social ordering functions in a distribution model with multiple goods and heterogeneous preferences. Given the impossibility to combine equality of resources with efficiency (Fleurbaey and Trannoy, Soc Choice Welf 21:243–263, 2003), we define weaker axioms of equality and efficiency relying on the mean bundle in the allocation. We identify a leximin social ordering function that satisfy these weaker axioms by relying on welfare measures representing the fraction of the mean bundle that would leave an agent indifferent with her bundle.
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Notes
 1.
For a similar approach applied to a model with one public and one private good, see Fleurbaey and Van der Linden (2017).
 2.
In Schokkaert and Konow’s experiments, respondents are given no information on whether agents have otherregarding preferences.
 3.
Such comparisons to the mean bundle can be made sense of in a purely ordinal fashion. Let \(\bar{z}\) and \(\bar{z}'\) be the mean bundles before and after the fair efficiency improvement, respectively. Bundles \(z_i\) and \(z_i'\) are agent i’s bundles in the corresponding allocations. Let B be the set of bundles i prefers to \(z_i\) but likes less than \(\bar{z}\), and \(B'\) the set of bundles i prefers to \(z'_i\) but likes less than \(\bar{z}'\). We then have \(B' \subset B\), indicating that there was more room for improving the welfare of i with respect to the mean bundle before than after the fair efficiency improvement.
 4.
This is a corollary of Proposition 2 below, under the constant weighting function \(1(z,R_N) := 1\) for all \(z\in Z\) and all \(R_N \in \mathcal {R}\).
 5.
The weighted mean welfare \(v_i^w\) is not a representation of agent i’s preferences. Function \(v_i^w\) is only a tool in the construction of \(\varvec{R}^{w{\text {lex}}}\) and cannot be given an obvious interpretation.
 6.
The proof that \(\varvec{R}^{\bar{z}{\text {lex}}}\) satisfies Mean DominanceReducing Priority follows the same logic as the proof that \(\varvec{R}^{w{\text {lex}}}\) satisfies Mean DominanceReducing Transfer.
 7.
As we show in Lemma 1 in the Appendix, \(z_j \gg z_j' \gg \bar{z}' \gg z_k' \gg z_k\) implies \(z_j \gg \bar{z} \gg z_k\), so there is no need to add the latter requirement to the conditions for \(z_N'~\varvec{R}(R)~z_N\) in Mean DominanceReducing Priority.
 8.
The same is true of the SOFs introduced by Sprumont (2012).
 9.
Although the priority axioms defined in Fleurbaey and Maniquet (2011, Chapter 3) are independent from Mean DominanceReducing Priority, it can be shown that \(\varvec{R}^{w{\text {lex}}}\)’s violations of Mean DominanceReducing Priority when w is strictly decreasing also imply violations of these other priority axioms.
 10.
Weighted mean leximins also satisfy Strong Fair Efficiency, a stronger version of Fair Efficiency that parallels Strong Pareto. Under Strong Fair Efficiency, \([\bar{z}' \le \bar{z}, \text {and} z_i'~R_i~z_i \text { for all } i\in N]\) \(\Rightarrow \) \([ z_N'~\varvec{R}(R)~z_N]\), and if in addition, \(z_j~P_j~z_j\) for some \(j\in N\), then \(z_N'~\varvec{P}(R)~z_N\). Because weighted mean leximins also satisfy Egalitarian Consensus, adding the latter axiom to Mean DominanceReducing Transfer, Fair Efficiency, and UnchangedContour Independence does not either force SOFs to satisfy Mean DominanceReducing Priority.
 11.
Weak Separability is weaker than the axiom Fleurbaey and Maniquet (2011) call Separability which requires that \([ z_N~R(R)~z_N' ]\) \( \Leftrightarrow \) \([ (z_{N\backslash \{i\}}, z_i'')~\varvec{R} (R_i',R_{N\backslash \{i\}}) ~(z'_{N\backslash \{i\}}, z_i''), ]\), even when \(R_i' \ne R_i\).
 12.
 13.
As Fleurbaey and Van der Linden (2017) show, a similar situation occurs under the “reference preference” approach developed by Sprumont (2012). In this approach, for every preference profile \(R_N\), the SOF applies the leximin criterion based on some reference preference \(R^{R_N}\) (see the Appendix). It can then be the case that an agent is betteroff when the reference preference disagree with her preference.
 14.
The definition in Sprumont (2012) does not use a utility representation. The definition we provide is formally equivalent to that in Sprumont (2012) and helps the comparison with the other SOFs in this paper. Also, the definition in Sprumont (2012) is for a single profile. Here we present a natural extension of the singleprofile social ordering in Sprumont (2012) to domains of profiles.
 15.
Comparing the three SOFs requires a little bit of a stretch in the models, as only Fleurbaey and Maniquet (2006) formally account for an exogenous social endowment.
References
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M. Van der Linden is grateful to the editor and the referee for their valuable comments and remarks. M. Van der Linden also thank the attendants at presentations at Vanderbilt University and Université SaintLouis for their questions and suggestions.
Appendix
Appendix
Additional SOFs and axioms from Fleurbaey and Maniquet (2011) and Sprumont (2012)
In this Appendix, we formally define \(\varvec{R}^{\Omega \text {}{\text {lex}}}\) and \(\varvec{R}^{{\text {cons}}}\), the main SOFs from Fleurbaey and Maniquet (2006) and Sprumont (2012), as well as the axioms used in Table 1 that are not defined in the text.
Social ordering functions We start with the main SOF from Sprumont (2012). Every SOF in the class of consensual leximin \(\varvec{R}^{{\text {cons}}}\) is based on a list of reference preference relations \((R^{R_N})_{R_N \in \mathcal {R}_N}\). The only constraint on \(R^{R_N}\) is that it be consensual, i.e., x R y whenever everyone prefers x to y. Formally, \(\bigcap _{i\in N} R_i \subset R^{R_N}\). For any \(R_N \in \mathcal {R}_N\), select an arbitrary utility representation of \(R^{R_N}\), say \(u^{R^{R_N}}\). The social ordering \(\varvec{R}^{\text {cons}}(R)\) ranks allocations by applying the leximin criterion to the list of utility levels \((u^{R^{R_N}}(z_i))_{i\in N}\).^{Footnote 14}
Social Ordering Function 3
(consensual leximin \((\varvec{R}^{\text {cons}}))\) A social ordering function \(\varvec{R}^{\text {cons}}\) is a consensual leximin if and only if, for any \(R\in \mathcal {R}_N,\) there exists a reference preference \(R^{R_N}\) satisfying \(\bigcap _{i\in N} R_i \subset R\) such that, for any \(z_N, z_N' \in Z^N,\)
The SOF from Fleurbaey and Maniquet (2006) applies the leximin criteria to a utility index \(u_i^\Omega \) that, as opposed to \(u^{R^{R_N}}\), represents i’s preferences. For every agent \(i \in N\),
for some \(\Omega \gg 0\) representing the social endowment to be distributed.^{Footnote 15}
Social Ordering Function 4
\((\Omega \)equivalent leximin \((\varvec{R}^{\Omega \text {}{\text {lex}}}))\) A social ordering function \(\varvec{R}^{\Omega \text {}{\text {lex}}}\) is the \(\Omega \)equivalent leximin if and only if, for any \(R\in \mathcal {R}_N\) and any \(z_N, z_N' \in Z^N,\)
Efficiency axiom Consensus is the weakening of efficiency adopted by Sprumont (2012). The axiom says that allocation \(z_N'\) is socially preferred to allocation \(z_N\) if every agent \(i \in N\) not only prefers her new bundle \(z_i'\) to \(z_i\), but also prefers \(z_j'\) to \(z_j\) for all other agent \(j \in N\backslash \{i\}\).
Fairness Axiom 9
(Consensus) For all \(R\in \mathcal {R}_N,\) and all \(z_N, z_N'\in Z^N,\)
Strong Pareto is the strengthening of Weak Pareto used by Fleurbaey and Maniquet (2011) to prove the necessity for an SOF satisfying EqualSplit Transfer (see below) and UnchangedContour Independence to satisfy a priority version of EqualSplit Transfer similar to Mean DominanceReducing Priority.
Fairness Axiom 10
(Strong Pareto) For all \(R\in \mathcal {R}_N,\) and all \(z_N, z_N'\in Z^N,\)
If in addition, \(z_i'~P_i~z_i \text { for some } i\in N\) then \(z_N'~\varvec{P}(R)~z_N.\)
Equality axioms EqualSplit Transfer was described in the text.
Fairness Axiom 11
(EqualSplit Transfer) For all \(R\in \mathcal {R}_N\) and all \(z_N,z_N'\in Z^N,\) for any \(k,j \in N,\) and any \(\Delta \gg 0,\)
Balanced EqualSplit Transfer is EqualSplit Transfer limited to balanced allocations.
Fairness Axiom 12
(Balanced EqualSplit Transfer) For all \(R\in \mathcal {R}_N\) and all \(z_N,z_N'\in Z^N,\) for any \(k,j \in N,\) and any \(\Delta \gg 0,\)
NestedContour Transfer says that a transfer from j to k is desirable if, after the transfer, k agrees that she receives a better bundle than j, no matter how j and k are moved along their indifference curves. This is true when the indifference curves of j and k at the post transfer allocation are nested, i.e., they do not cross. Let us define the upper and lower contour sets at bundle \(z_i\) given preference \(R_i\) as

\(U(z_i,R_i) = \{x\in X ~~ x~R_i~z_i\}\), and

\(L(z_i,R_i) = \{x\in X ~~ z_i~R_i~x\}\).
Fairness Axiom 13
(NestedContour Transfer) For all \(R\in \mathcal {R}_N\) and all \(z_N,z_N'\in Z^N,\) for any \(k,j \in N,\) and any \(\Delta \gg 0,\)
Lemma 1
The next lemma shows that \(z_j \gg z_j' \gg \bar{z}' \gg z_k' \gg z_k\) implies \(z_j \gg \bar{z} \gg z_k\), which means there is no need to add the latter requirement (\(z_j \gg \bar{z} \gg z_k\)) to the conditions for \(z_N'~\varvec{R}(R)~z_N\) in the definition of Mean DominanceReducing Priority.
Lemma 1
For any \(z_N,z_N' \in Z^N\) and any \(k,j \in N,\) if \(z_i = z_i' \text { for all } i \in N\backslash \{j,k\},\) then
Proof
We prove that \(z_j' \gg \bar{z}'\) for any \(z_j' \gg z_j\). The proof that \(\bar{z}' \gg z_k'\) for any \(z_k'<< z_k\) is analogous.
Let \(\Gamma _j := z_j'  z_j \ge 0\) and \(\Gamma _k := z_k'  z_k \le 0\). Also let \(\bar{z}_{j,k}\) and \(\bar{z}'_{j,k}\) be the mean bundles among agents other than j and k in allocations \(z_N\) and \(z_N'\), respectively. Observe that because \(z_i = z_i' \text { for all } i \in N\backslash \{j,k\}\), we have
Also,
We then have
\(\square \)
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Van der Linden, M. Egalitarianism with a dash of fair efficiency. Econ Theory Bull 6, 219–238 (2018). https://doi.org/10.1007/s4050501701343
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Keywords
 Fairness
 Social orderings
 Equality
 Efficiency
JEL Classification
 D63
 D71