Resource egalitarianism with a dash of efficiency

Journal of Economic Theory 147 (2012) 1602–1613 www.elsevier.com/locate/jet

Resource egalitarianism with a dash of efficiency Yves Sprumont 1 Département de Sciences Économiques and CIREQ, Université de Montréal, C.P. 6128, succursale centre-ville, Montréal QC, H3C 3J7, Canada Received 28 December 2007; final version received 9 June 2010; accepted 17 July 2010 Available online 29 July 2010

Abstract We study the problem of defining inequality-averse social orderings over allocations of commodities when individuals have different preferences. We formulate a notion of egalitarianism based on the axiom that any dominance between consumption bundles should be reduced. This Dominance Aversion requirement is compatible with Consensus, a version of the Pareto principle saying that an allocation y is better than x whenever everybody finds that everyone’s bundle at y is better than at x. We characterize a family of multidimensional leximin orderings satisfying Dominance Aversion and Consensus. © 2010 Elsevier Inc. All rights reserved. JEL classification: D63; D71 Keywords: Multidimensional egalitarianism; Leximin; Efficiency

1. Introduction This paper is an attempt to define a notion of egalitarianism that would be adequate for guiding collective choices in a multi-commodity context. Egalitarianism, as understood here, is the view that an unequal treatment of individuals is socially desirable only if it is justified by some relevant difference between them. Equal treatment should prevail by default.

E-mail address: [email protected]. 1 I am grateful to two referees for helpful remarks and suggestions. I also thank F. Maniquet, H. Moulin and

W. Thomson for comments and discussions. This research was supported by the Social Sciences and Humanities Research Council of Canada and the Fonds Québécois pour la Recherche sur la Société et la Culture. 0022-0531/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jet.2010.07.009

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

1603

The specific problem we analyze is that of allocating commodities among individuals with no property rights, objectively equal needs, but possibly different preferences. Our goal is to construct a full ordering of all conceivable allocations, including those that are not efficient or even feasible, as a function of the preference profile. This is essential in order to make consistent social choices in an environment where constraints are changing. The issue is to identify which consumption inequalities are justified by differences in preferences. We want our social ordering to be averse to all unjustified inequalities in consumption. When preferences are monotonic, all individuals in society agree that consuming more of all commodities is better. Since society has no reason to overrule this consensus, there is no ethical justification for any physical dominance between individual consumption bundles. Any such dominance should therefore be reduced. If x = (x1 , x2 , x3 , . . . , xn ) is a multi-commodity allocation where individual 1 consumes less of all goods than 2, x1 < x2 , Dominance Aversion says that (y1 , y2 , x3 , . . . , xn ) is better than x whenever x1 < y1  y2 < x2 . A weaker condition, the Dominance-Reducing Transfer Principle states that a transfer of commodities from 2 to 1 is desirable as long as 2 keeps consuming as much as 1: (x1 + t, x2 − t, x3 , . . . , xn ) is better than x whenever x1 < x1 + t  x2 − t < x2 . We submit that these principles are the adequate basis for defining multidimensional egalitarianism, in a weak and a strong sense respectively, when individual preferences may differ but are monotonic. Unfortunately, Fleurbaey and Trannoy [7] showed that even the Dominance-Reducing Transfer Principle is incompatible with the Pareto principle. Consider two individuals with linear preferences over apples and bananas: one apple is worth two bananas to individual 1, two apples are worth one banana to individual 2. According to the Dominance-Reducing Transfer Principle, allocation x = ((11, 2), (12, 3)), where individual 1 consumes 11 apples and 2 bananas and individual 2 consumes 12 apples and 3 bananas, is better than allocation y = ((9, 0), (14, 5)) because the former is obtained from the latter by transferring 2 apples and 2 bananas from the richer individual to the poorer. The Pareto principle implies that society should be indifferent between allocation y and allocation z = ((3, 12), (2, 11)). But the Dominance-Reducing Transfer Principle tells us that z is better than w = ((5, 14), (0, 9)), which the Pareto principle deems equivalent to the allocation x we started from.2 One reaction to the incompatibility just described is that bundle dominance is, after all, a justifiable form of inequality: it cannot be avoided if we insist on the Pareto principle. Fleurbaey [4,5] and Fleurbaey and Maniquet [6] propose weakened versions of the Dominance-Reducing Transfer Principle (obtained by restricting its application either to pairs of agents with identical preferences or to particular regions of the commodity space) that are compatible with the Pareto principle. Fleurbaey and Maniquet use such weak versions of the transfer principle to defend the so-called Pazner–Schmeidler egalitarian social ordering and the Walrasian egalitarian social ordering. Alternatively – and this is the route we take – one may argue that the Dominance-Reducing Transfer Principle and Dominance Aversion are fundamental criteria of fairness. The natural question is then whether they can be combined with weaker versions of the Pareto principle. Many authors have stressed that an individual’s “actual preferences” – those revealed by his choices – sometimes differ from his “informed preferences”, which, in Harsanyi’s [10] words, 2 To keep the argument simple, we used an example involving indifferences. The incompatibility persists under the version of the Pareto principle based on strict preferences only.

1604

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

are “the hypothetical preferences he would have if he had all the relevant information and had made full use of this information”. The preferences we use in the formal analysis below must be interpreted as actual preferences. These preferences constitute the only information practically available to construct a social ranking of the allocations but are not perfectly suitable for that exercise because they do not always reflect individual welfare in a reliable way. As a consequence, the Pareto principle, applied to these actual preferences, is not necessarily compelling. A prime example illustrating this difficulty is the problem of allocating commodities under uncertainty. If objective probabilities cannot be attached to the possible states of the world, individual (actual) preferences reflect beliefs as well as tastes. In that case, as Gilboa, Samet and Schmeidler [8] argue, “society should not necessarily endorse a unanimous choice when it is based on contradictory beliefs”. A fortiori, actual preferences are not adequate for welfare analysis if they are based on false beliefs: see, e.g., Hausman and McPherson [11]. A related important example is that of addictive behavior, where the actual preferences of an individual may seriously conflict with his informed preferences. Even in such cases, however, we may still be reasonably confident that an individual is better off at an allocation y than at x if all individuals in society find that the bundle he consumes at y is better than the bundle he consumes at x. We therefore propose to replace the Pareto principle with a condition that we call Consensus, which says that an allocation y is better than an allocation x whenever everyone finds everybody’s bundle at y better than at x. The Consensus axiom is compatible with Dominance Aversion. The contribution of this paper is to offer a detailed study of a multidimensional generalization of the classic leximin ordering satisfying both axioms. This is but a modest first step towards a better understanding of egalitarianism in a multi-commodity context where individual preferences matter. Perhaps the most important problem that remains open is to identify and defend less extreme social orderings satisfying Consensus and the Dominance-Reducing Transfer Principle. 2. The model and the main conditions There is a fixed finite set of individuals, N = {1, . . . , n}, n  2, and a fixed finite number of goods, m  2. The commodity space is X = Rm + . Each individual i ∈ N has a continuous and strictly monotonic preference ordering Ri over X. Given the fixed preference profile (R1 , . . . , Rn ), we seek to construct a social ordering R over the set of conceivable allocations X N . The properties of a social ordering we are chiefly concerned with are Consensus and Dominance Aversion. Consensus is a Paretian axiom. If Pi denotes the strict preference relation associated with Ri and P the strict social preference associated with R, the standard Pareto principle, in its weak form, states that (y1 , . . . , yn )P(x1 , . . . , xn ) whenever yi Pi xi for all i ∈ N . Consensus is a weaker condition saying that an allocation is better than another if all individuals find that everyone gets a better bundle in the former than in the latter. Consensus. For all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N ,   [yi Pj xi for all i, j ∈ N ] ⇒ (y1 , . . . , yn )P(x1 , . . . , xn ) . A slightly stronger condition requires, in addition, that (y1 , . . . , yn )R(x1 , . . . , xn ) whenever yi Rj xi for all i, j ∈ N . We call this variant Full Consensus.

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

1605

As we explained in the Introduction, Consensus is motivated by the fact that the actual preferences of an individual do not necessarily coincide with his informed preferences. The interpretation of the axiom is fundamentally resourcist. Commodities have intrinsic value which individual preferences reflect approximately. A good social ordering should use these preferences to estimate the “correct” relative value of commodity bundles. If all individuals like a bundle better than another, then society should be confident that the former is indeed more valuable than the latter – it has no reason whatsoever to support the opposite view. If each bundle composing an allocation (y1 , . . . , yn ) is deemed more valuable than the corresponding component of another allocation (x1 , . . . , xn ), then society should prefer (y1 , . . . , yn ) over (x1 , . . . , xn ). Consensus does limit the scope of egalitarianism. For instance, it implies that more for all is always better, even if resources are (perhaps much) more unequally distributed. Monotonicity. For all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N ,   (y1 , . . . , yn )P(x1 , . . . , xn ) , [yi > xi for all i ∈ N ] ⇒ where we use the notation , >,  for vector inequalities. Observe that Monotonicity, which is found in the literature on multidimensional inequality measurement (see, e.g., Weymark [13]), abstracts from preferences altogether. The strength of the Consensus axiom depends on the diversity of preferences in society. At one extreme, Consensus would reduce to Monotonicity if the intersection of individual preferences were just the dominance ordering on X N . This could happen, for instance, if m  n and each individual i  m valued only good i.3 At the other extreme, Consensus is equivalent to the weak Pareto axiom when all preferences are identical. Next we turn to our main distributional axiom. Dominance Aversion says that reducing bundle dominance is always desirable. Dominance Aversion. For all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N and all i, j ∈ N ,     xi > yi  yj > xj and yk = xk for all k ∈ N \{i, j } ⇒ (y1 , . . . , yn )R(x1 , . . . , xn ) . Let us repeat that preferences are the only source of heterogeneity in our model. Individuals are otherwise identical. They cannot be distinguished according to claims (as individual endowments are unspecified) or needs (as utilities are absent). In such a simplified context, we believe that Dominance Aversion is a radical but ethically correct corollary to the view that bundle dominance is unjustified when all preferences are strictly monotonic. It is the proper extension of the one-dimensional inequality-aversion principle saying that it is desirable to reduce the difference between two individuals’ consumption levels even if this involves destroying resources.4 The Dominance-Reducing Transfer Principle, which restricts the application of Dominance Aversion to those cases where no resources are lost when switching from (x1 , . . . , xn ) to (y1 , . . . , yn ), will be briefly discussed in the concluding section. 3 This limit case is actually forbidden by the assumption that preferences are strictly monotonic. The intersection of individual preferences cannot be exactly equal to the dominance ordering but it can be arbitrarily close to it. 4 If x = (x , x , x , . . . , x ) is an allocation of a single commodity (say, wealth) such that x < x , then n 1 2 3 1 2 (y1 , y2 , x3 , . . . , xn ) is better than x whenever x1 < y1  y2 < x2 . Hammond [9], who states this principle for utility rather than wealth, calls it equity. He shows that equity, the standard anonymity condition and the Pareto principle jointly characterize the leximin ordering. See d’Aspremont [1] for a detailed discussion.

1606

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

3. Consensual egalitarianism This section describes a class of social orderings (which we sometimes just call orderings, for brevity) that satisfy the conditions just defined. Recall that the profile of individual preferences (R1 , . . . , Rn ) is fixed. Let R be a continuous ordering over the commodity space which agrees with i∈N Ri in the sense that, for any two bundles x, y ∈ X, [yPi x for all i ∈ N ] ⇒ [yP x].5 This ordering R is not a social ordering in the technical sense – it does not rank allocations. Rather, it expresses society’s evaluation of the relative value of commodity bundles. This evaluation is consensual in the sense that it respects individual preferences whenever they do not conflict. There is a simple way to construct such an ordering: if u1 , . . . , un are arbitrary numerical representations of the individual preferences, a function x → u(x) = f (u1 (x), . . . , un (x))  represents an ordering R agreeing with i∈N Ri if f is strictly increasing in all its arguments. For any allocation (x1 , . . . , xn ) ∈ X N , denote by (x1R , . . . , xnR ) any allocation obtained by R R . . . Rx R . rearranging the bundles x1 , . . . , xn from worst to best according to R, so that xnR Rxn−1 1 Definition 1. A social ordering R is a consensual  Rawlsian ordering if and only if there is a continuous ordering R on X which agrees with i∈N Ri such that, for all allocations (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N , [y1R P x1R ] ⇒ [(y1 , . . . , yn )P(x1 , . . . , xn )]. We say that R is based on R. In words: a consensual Rawlsian social ordering evaluates an allocation on the basis of its worst component according to a (continuous) preference that agrees with individual preferences. When there are more than two individuals, not every consensual Rawlsian ordering satisfies Dominance Aversion. This is because such an ordering need not pay attention at all to the components of an allocation that are not minimal according to the preference it is based on. The consensual leximin orderings avoid this difficulty. The leximin extension to X N of an ordering R on X is the ordering Rlex (R) on X N defined as follows: (y1 , . . . , yn )Rlex (R)(x1 , . . . , xn ) if and only if either there exists j ∈ N such that yiR I xiR for all i < j and yjR P xjR (in which case we write (y1 , . . . , yn )Plex (R)(x1 , . . . , xn )) or else yiR I xiR for all i ∈ N (in which case we write (y1 , . . . , yn )Ilex (R)(x1 , . . . , xn )). Definition 2. A social ordering R is a consensual leximin ordering if and only if there is  a continuous ordering R agreeing with i∈N Ri such that, for all allocations (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N , [(y1 , . . . , yn )Plex (R)(x1 , . . . , xn )] ⇒ [(y1 , . . . , yn )P(x1 , . . . , xn )]. A social ordering R is a simple consensual  leximin ordering if and only if it is the leximin extension of some ordering R agreeing with i∈N Ri , that is, R = Rlex (R). Clearly, every consensual leximin ordering is a consensual Rawlsian ordering. While a consensual leximin ordering need not respect the indifference relation associated with the ordering R it is based on, a simple consensual leximin ordering necessarily does. The rest of this section is an axiomatic study of the orderings introduced in Definitions 1 and 2. While the basic axioms we use are Consensus and Dominance Aversion, we do rely on additional 5 Because R is continuous and R , . . . , R are strictly monotonic, it follows that, for all x, y ∈ X, [yR x for all n i 1 i ∈ N ] ⇒ [yRx].

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

1607

properties. We do not regard these auxiliary properties as fundamental aspects of a definition of preference-sensitive multidimensional egalitarianism. They are disputable conditions that help us select salient examples of egalitarian orderings. We begin with two requirements of internal coherence of the social ordering which do not impose any relationship between the social ordering and the profile of individual preferences. The first is what we call Intrinsic Dominance. From a resourcist perspective, comparing two fully egalitarian allocations involves no issue of distributive justice at all. Therefore a statement such as (y, . . . , y)R(x, . . . , x) should reflect society’s evaluation of the relative intrinsic values of the bundles x and y. The axiom says that if society deems each of the bundles x1 , . . . , xn intrinsically at least as valuable as bundle x, then the allocation (x1 , . . . , xn ) should be at least as good as (x, . . . , x). An intrinsically richer society is preferable to a poorer one. Intrinsic Dominance. For all (x1 , . . . , xn ) ∈ X N , x ∈ X, [(xi , . . . , xi )R(x, . . . , x) for all i ∈ N ] ⇒ [(x1 , . . . , xn )R(x, . . . , x)]. This axiom combines two ideas. First, like Consensus, it limits society’s concerns for equality. This is best seen by considering the contraposition of the axiom. If an egalitarian allocation (x, . . . , x) is judged superior to another allocation (x1 , . . . , xn ), it cannot be just because the former is egalitarian: the common bundle x received by all individuals should be deemed intrinsically more valuable than at least one of the bundles composing the other allocation. Second, in keeping with the resourcist view, Intrinsic Dominance assumes that the intrinsic value of consumption bundles is enough, in specific cases, to determine the social ranking of allocations: if each of the bundles x1 , . . . , xn is judged intrinsically at least as valuable as x, then any allocation of these bundles to the individuals is at least as good as (x, . . . , x) – society need not worry about who gets what. Indeed, if (xi , . . . , xi )R(x, . . . , x) for all i ∈ N and π is an arbitrary permutation on N , then (xπ(i) , . . . , xπ(i) )R(x, . . . , x) for all i ∈ N and Intrinsic Dominance implies (xπ(1) , . . . , xπ(n) )R(x, . . . , x). Intrinsic Dominance is clearly incompatible with the standard Pareto principle. Consider for instance our earlier two-individual, two-commodity example where R1 , R2 are the linear preferences represented by u1 (α, β) = 23 α + 13 β and u2 (α, β) = 13 α + 23 β. Let x1 = (8, 2) and x2 = (2, 8). If (x1 , x1 )R(x2 , x2 ), then Intrinsic Dominance implies (x2 , x1 )R(x2 , x2 ) while the Pareto principle requires (x2 , x2 )P(x2 , x1 ). On the other hand, if (x2 , x2 )R(x1 , x1 ), then Intrinsic Dominance implies (x2 , x1 )R(x1 , x1 ) while the Pareto principle requires (x1 , x1 )P(x2 , x1 ). This does not mean that individual preferences cannot be taken into account. In particular, we emphasize that Intrinsic Dominance is compatible with the standard version of the weak Pareto principle. For instance, define for each individual i a numerical representation ui of Ri and let (y1 , . . . , yn )R(x1 , . . . , xn ) if and only if mini∈N ui (yi )  mini∈N ui (xi ). The social ordering R satisfies Intrinsic Dominance and the weak Pareto principle: (y1 , . . . , yn )P(x1 , . . . , xn ) whenever yi Pi xi for all i ∈ N . Our second auxiliary condition is a weak form of continuity of the social ordering. Weak Continuity requires that the social ranking of fully egalitarian allocations be continuous. Weak Continuity. For any x, y ∈ X and any sequence (y k ) in X converging to y,  k     y , . . . , y k R(x, . . . , x) for all k ⇒ (y, . . . , y)R(x, . . . , x) .

1608

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

Since egalitarian allocations are in a simple one-to-one correspondence with bundles, a discontinuous social ranking of such allocations would seem at odds with continuous individual preferences. We are now ready to state our first result. Proposition 1. If a social ordering satisfies Consensus, Dominance Aversion, Intrinsic Dominance, and Weak Continuity, then it is a consensual Rawlsian social ordering. Every simple consensual leximin social ordering satisfies these four axioms. The proof, which is in Appendix A, is constructive. Given  the individual preferences R1 , . . . , Rn , we build a continuous preference R agreeing with i∈N Ri and show that R must rank allocations on the basis of their worst bundle according to R. We stress that the “consensual” preference R is endogenous. Proposition 1 is not a complete characterization of the simple consensual leximin orderings. In order to offer such a characterization, we introduce two further properties. First, observe that the social evaluation of an allocation by a simple consensual leximin ordering depends only on the bundles that compose it, not on who consumes them. Strong Symmetry. For all (x1 , . . . , xn )∈X N and every permutation π on N , (x1 , . . . , xn )I(xπ(1) , . . . , xπ(n) ). This axiom is a purely resourcist variant of the familiar property of anonymity. While Strong Symmetry is clearly a very demanding property – it rules out anonymously preference-sensitive orderings –, it is important to observe that all consensual Rawlsian orderings do satisfy a very closely related condition. If, according to R, the worst bundle in (y1 , . . . , yn ) is strictly better than the worst bundle in (x1 , . . . , xn ), then any consensual Rawlsian ordering R based on R deems (y1 , . . . , yn ) strictly better than (x1 , . . . , xn ), irrespective of who in each allocation receives this worst bundle. This means that for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N and every permutation π on N , (y1 , . . . , yn )P(x1 , . . . , xn ) if and only if (y1 , . . . , yn )P(xπ(1) , . . . , xπ(n) ). Extending this property to social indifference implies Strong Symmetry. Simple consensual leximin orderings also satisfy a separability property. If (x1 , . . . , xn ) ∈ X N and i ∈ N , denote by x−i ∈ X N \{i} the sub-allocation (xj )j ∈N \{i} and, if yi ∈ X, write (yi ; x−i ) for the allocation obtained from (x1 , . . . , xn ) by replacing xi with yi . Internal Separability. Let i ∈ N and let xi , xi , yi , yi ∈ X be such that (xi , . . . , xi )I(xi , . . . , xi )  ∈ X N , (x ; z )R(x  ; z ) ⇔ (y ; z ) × and (yi , . . . , yi )I(yi , . . . , yi ). Then, for all zN , zN i −i i −i i −i   R(yi ; z−i ). Because society deems bundle xi intrinsically just as valuable as xi , it may ignore individual  ). Likewise, it may ignore individual i when comparing i when comparing (xi ; z−i ) to (xi ; z−i    ) should (yi ; z−i ) to (yi ; z−i ). Therefore the social ranking of the allocations (xi ; z−i ) and (xi ; z−i   be the same as the ranking of (yi ; z−i ) and (yi ; z−i ) since, ignoring i, the sub-allocations to be  . compared are the same in both cases, namely z−i and z−i The axiom is related to Fleming’s [3] separability axiom. The essential difference is that individual i is ignored not because he is indifferent between xi and xi and between yi and yi , but because society judges that the bundles he receives are equally valuable in both cases. If preferences are not well-informed, Fleming’s axiom is logically too demanding. The following proposition characterizes the simple consensual leximin social orderings.

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

1609

Proposition 2. A social ordering satisfies Consensus, Dominance Aversion, Weak Continuity, Strong Symmetry, and Internal Separability if and only if it is a simple consensual leximin social ordering. Notice that Intrinsic Dominance does not appear in this proposition; it is implied by the combination of Consensus, Weak Continuity and Internal Separability. Appendix A contains the proof of Proposition 2 as well as examples showing that all five axioms are independent. We conclude this section with three remarks. First, simple consensual leximin orderings remain arguably too crude. In particular, because they satisfy Strong Symmetry, they violate the following condition. Permutation Pareto Principle. For all (x1 , . . . , xn ) ∈ X N and every permutation π on N , [xπ(i) Pi xi for all i ∈ N ] ⇒ [(xπ(1) , . . . , xπ(n) )P(x1 , . . . , xn )]. This axiom is a restricted form of the standard weak Pareto principle. It says that if permuting bundles results in an allocation where every individual prefers his new bundle to the old, this new allocation should be regarded as better. The important point is that the new allocation is generated from the old through a very specific type of exchange: a permutation of consumption bundles. Such a permutation is distributionally neutral in the sense that it preserves any consumption inequalities originally present. Since there is no reason to favor either allocation on distributive grounds, a dash of efficiency should tip the balance in favor of the new allocation. There is a simple way to generate a social ordering that satisfies this condition along with our two fundamental axioms, Consensus and Dominance Aversion. As before, fix a continuous  ordering R on X agreeing with i∈N Ri and let R be the leximin extension of R to X N . Next, let R be a standard weakly Paretian social ordering on X N in the sense that [yi Pi xi for all i ∈ N] ⇒ [(y1 , . . . , yn )P (x1 , . . . , xn )]. Construct R by lexicographic application of R and R : (y1 , . . . , yn )R (x1 , . . . , xn ) if and only if (y1 , . . . , yn )P(x1 , . . . , xn ) or [(y1 , . . . , yn )I(x1 , . . . , xn ) and (y1 , . . . , yn )R (x1 , . . . , xn )]. The social ordering R is a consensual leximin ordering. It satisfies Consensus and Dominance Aversion (as well as Weak Continuity) because R does: see the proof of Proposition 1. To check the Permutation Pareto Principle, note that (xπ(1) , . . . , xπ(n) )I(x1 , . . . , xn ) for all (x1 , . . . , xn ) ∈ X N and every permutation π on N . Therefore (xπ(1) , . . . , xπ(n) )P (x1 , . . . , xn ) if and only if (xπ(1) , . . . , xπ(n) )P (x1 , . . . , xn ). The claim now follows from the fact that R is a standard weakly Paretian social ordering. Our second remark bears on both Propositions 1 and 2. In either result, the only axiom linking the social ordering of allocations to individual preferences is Consensus. Replacing Consensus with Monotonicity yields preference-free versions of our earlier results. In particular, we obtain the following variant of Proposition 2. We omit the proof, which is a straightforward modification of the proofs of Propositions 1 and 2. Proposition 3. A social ordering R on X N satisfies Monotonicity, Dominance Aversion, Weak Continuity, Strong Symmetry, and Internal Separability if and only if it is the leximin extension of a continuous and strictly monotonic preference ordering R on X. Our third remark is that the strength of Propositions 1 and 2 obviously depends on the preference profile (R1 , . . . , Rn ). If all preferences coincide, then there exists a unique simple consensual leximin social ordering and the axioms in those propositions force us to use that

1610

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

ordering. When preferences are more diverse, society is left with a nontrivial choice between possibly many consensual leximin social orderings. 4. Conclusion We have proposed a preference-sensitive definition of egalitarianism respecting the principle that dominance between consumption bundles should always be reduced. Two important problems remain unsolved. (1) Each of the simple consensual leximin orderings characterized in Proposition 2 is built upon a different “social preference” over the commodity space which agrees with the intersection of individual preferences. What should this social preference be? An answer to this question would likely necessitate a multiprofile framework and conditions specifying how the social ordering of allocations varies in response to changes in individual preferences. This is beyond the scope of this paper. Let us just remark, however, that rather standard conditions such as anonymity and neutrality (stating that the social ordering of the allocations should not vary with the identity of the individuals or with the names attached to the goods) may play a useful role. Consider again our two-individual, two-commodity example where Ru , Rv are the linear preferences represented by u(α, β) = 23 α + 13 β and v(α, β) = 13 α + 23 β. Anonymity and neutrality imply that the preference R used to define the simple consensual leximin ordering at profile (R1 , R2 ) = (Ru , Rv ) must be symmetric in the two goods. This is because the profile (Rv , Ru ) can be obtained from (Ru , Rv ) either by renaming the individuals or by renaming the goods. If R is chosen to be linear (which seems natural but does not follow from anonymity and neutrality), then it must be the preference represented by u(α, β) = 12 α + 12 β. (2) The leximin orderings identified in this paper are egalitarian in a strong sense: they satisfy the very demanding axiom of Dominance Aversion. The Dominance-Reducing Transfer Principle (studied, among others, by Fleurbaey and Trannoy [7]) is just the restriction of Dominance Aversion to those cases where no resources are lost when transferring commodities. Formally, define the relation on the set of allocations by (y1 , . . . , yn ) (x1 , . . . , xn ) if and only if there exist i, j ∈ N and t ∈ X such that (i) xi > yi = xi − t  xj + t = yj > xj and (ii) yk = xk for all k ∈ N\{i, j }. The Dominance-Reducing Transfer Principle requires that for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N , [(y1 , . . . , yn ) (x1 , . . . , xn )] ⇒ [(y1 , . . . , yn )R(x1 , . . . , xn )]. It is a natural extension of the classic Transfer Principle of Pigou [12] and Dalton [2] to our multidimensional context with monotonic preferences. The Dominance-Reducing Transfer Principle and Consensus do not imply Dominance Aversion. To see this, let R be a strictly monotonic preference ordering on X having a continuous numerical representation u with decreasing increments in the sense that, for all x, y, t ∈ X, x t) − u(x)  u(y + t) − u(y). Define (y1 , . . . , yn )R(x1 , . . . , xn ) if and only y ⇒ u(x +  if i∈N u(yi )  i∈N u(xi ). The social ordering R satisfies the Dominance-Reducing Transfer Principle butviolates Dominance Aversion. To guarantee Consensus, all we need is that R agrees with i∈N Ri . This causes no difficulty if at least one individual preference ordering has a continuous representation with decreasing increments6 : just let R coincide with it. 6 This is not a stringent restriction. For instance, any preference ordering represented by a twice differentiable function u such that, for some β > 0, ∂hk u/∂h u∂k u  β for all commodities h, k would qualify. Such a preference ordering need not be convex.

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

1611

Describing the class of all social orderings satisfying the Dominance-Reducing Transfer Principle and Consensus remains an open problem. More work is also needed to identify in that class interesting alternatives to the rather extreme simple consensual leximin orderings. Appendix A. Proofs Proof of Proposition 1. In order to prove the first statement, let R be a social ordering satisfying Consensus, Dominance Aversion, Intrinsic Dominance, and Weak Continuity. Define the binary relation R on X as follows: for all x, y ∈ X, yRx

⇔

(A.1)

(y, . . . , y)R(x, . . . , x).

Since R is an ordering and satisfies Weak Continuity, R is a continuous ordering on X. Moreover,  R agrees with i∈N Ri : if yPi x for all i ∈ N , Consensus implies (y, . . . , y)P(x, . . . , x), hence yP x by (A.1). Now, fix two allocations (x1 , . . . , xn ), (y1 , . . . , yn ) such that y1R P x1R .

(A.2)

We claim that (y1 , . . . , yn )P(x1 , . . . , xn ). Suppose, by way of contradiction, that (A.3)

(x1 , . . . , xn )R(y1 , . . . , yn ).

Without loss of generality, assume that = x1 . By (A.2), yi P x1 for all i ∈ N . Because R is continuous, there exists a bundle a > 0 such that x1R

yi P (x1 + a)

for all i ∈ N.

(A.4)

Let x > xi + a for all i ∈ N and choose a bundle b such that 0 < b < a. By Consensus, (x1 + b, x, . . . , x)P(x1 , . . . , xn ), hence, by (A.3), (x1 + b, x, . . . , x)P(y1 , . . . , yn ). By repeated application of Dominance Aversion, (x1 + a, x1 + a, . . . , x1 + a)R(x1 + b, x, . . . , x), hence, (x1 + a, x1 + a, . . . , x1 + a)P(y1 , . . . , yn ). By definition of R, (yi , . . . , yi )R(y1R , . . . , y1R ) for all i ∈ N . (y1 , . . . , yn )R(y1R , . . . , y1R ). Hence, by (A.5), (x1 + a, x1 + a, . . . , x1 by definition of R means (x1 + a)P y1R , a contradiction to (A.4).

(A.5) By Intrinsic Dominance, + a)P(y1R , . . . , y1R ), which

In order to prove the second statement  in Proposition 1, let R be the leximin extension of some continuous ordering R agreeing with i∈N Ri . To check Consensus, suppose yi Pj xi for all i, j ∈ N . Then y1R Pj x1R for all j ∈ N and since  R agrees with i∈N Ri , y1R P x1R. . By definition of R, (y1 , . . . , yn )P(x1 , . . . , xn ). To prove Dominance Aversion, suppose  xi > yi  yj > xj for some i, j ∈ N and yk = xk for all k ∈ N\{i, j }. Because R agrees with i∈N Ri , we have xi P yi Ryj P xj and yk I xk for all k ∈ N\{i, j }. Since R is the leximin extension of R, it follows that (y1 , . . . , yn )P(x1 , . . . , xn ). To prove Intrinsic Dominance, fix (x1 , . . . , xn ) ∈ X N , x ∈ X, and suppose (xi , . . . , xi )R(x, . . . , x) for all i ∈ N . By definition of R, xi Rx for all i ∈ N . Hence, by definition of R, (x1 , . . . , xn )R(x, . . . , x). Weak Continuity of R follows directly from continuity of R. 2

1612

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

Proof of Proposition 2. We have shown in the proof of Proposition 1 that all simple consensual leximin social orderings satisfy the first three axioms in Proposition 2. It is straightforward to check the last two, Strong Symmetry and Internal Separability. Conversely, let R be a social ordering satisfying the five axioms in Proposition 2. We begin by showing that R satisfies Intrinsic Dominance. Let (x1 , . . . , xn ) ∈ X N and x ∈ X be such that (xi , . . . , xi )R(x, . . . , x) for all i ∈ N . By Weak Continuity, there exist bundles a1 , . . . , an ∈ X such that (xi , . . . , xi )I(x + ai , . . . , x + ai )

for all i ∈ N.

(A.6)

Using (A.6), repeated applications of Internal Separability yield (xn , xn , . . . , xn )I(x + an , x + an , . . . , x + an ) ⇒ (x1 , xn , . . . , xn )I(x + a1 , xn , . . . , xn ) ⇒ (x1 , x2 , xn , . . . , xn )I(x + a1 , x + a2 , xn , . . . , xn ) ⇒ (x1 , . . . , xn )I(x + a1 , . . . , x + an ). By Consensus, (x + a1 , . . . , x + an ) × R(x, . . . , x). Hence (x1 , . . . , xn )R(x, . . . , x) by transitivity of R. This establishes Intrinsic Dominance. Next, define the binary relation R on X as in (A.1). The proof of Proposition 1 shows that R  is an ordering agreeing with i∈N Ri and that R is a consensual Rawlsian ordering based on R: for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N , y1R P x1R

⇒

(y1 , . . . , yn )P(x1 , . . . , xn ).

(A.7)

We claim that, in fact, R is the leximin extension of R, that is, R = Rlex (R). To show this, we fix (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N and proceed in two steps. Step 1. We show that (y1 , . . . , yn )Ilex (R)(x1 , . . . , xn ) ⇒ (y1 , . . . , yn )I(x1 , . . . , xn ). If (y1 , . . . , yn )Ilex (R)(x1 , . . . , xn ), then yiR I xiR for all i ∈ N . By (A.1), (yiR , . . . , yiR )I(xiR , . . . , R xi ) for all i ∈ N . Therefore, starting from the fact that (y1R , . . . , ynR )I(y1R , . . . , ynR ) and applyR , y R )I(x R , ing Internal Separability n times, (y1R , . . . , ynR )I(x1R , y2R , . . . , ynR )I . . . I(x1R , . . . , xn−1 n 1 . . . , xnR ). Hence, (y1R , . . . , ynR )I(x1R , . . . , xnR ). Invoking Strong Symmetry, (y1 , . . . , yn )I(x1 , . . . , xn ). Step 2. We show that (y1 , . . . , yn )Plex (R)(x1 , . . . , xn ) ⇒ (y1 , . . . , yn )P(x1 , . . . , xn ). If (y1 , . . . , yn )Plex (R)(x1 , . . . , xn ), there exists j ∈ N such that yiR I xiR

for all i < j

and yjR P xjR .

If j = 1, (A.8) means that y1R P x1R , and (A.7) implies (y1 , . . . , yn )P(x1 , . . . , xn ). If j > 1, (A.8) implies that    R yi , . . . , yiR I xiR , . . . , xiR for i = 1, . . . , j − 1.

(A.8)

(A.9)

Choose bundles z1 , . . . , zj −1 ∈ X such that zi P yjR

for i = 1, . . . , j − 1.

(A.10)

Given (A.9) and since trivially (zi , . . . , zi )I(zi , . . . , zi ) for i = 1, . . . , j − 1, applying Internal Separability j − 1 times yields     R y1 , . . . , ynR R x1R , . . . , xnR     ⇔ z1 , . . . , zj −1 , yjR , . . . , ynR R z1 , . . . , zj −1 , xjR , . . . , xnR . (A.11)

Y. Sprumont / Journal of Economic Theory 147 (2012) 1602–1613

1613

Defining (a1 , . . . , an ) = (z1 , . . . , zj −1 , xjR , . . . , xnR ) and (b1 , . . . , bn ) = (z1 , . . . , zj −1 , yjR , . . . , ynR ), (A.8) and (A.10) imply that a1R = xjR , b1R = yjR , and b1R P a1R . Therefore, by (A.7), (b1 , . . . , bn )P(a1 , . . . , an ), that is, (z1 , . . . , zj −1 , yjR , . . . , ynR )P(z1 , . . . , zj −1 , xjR , . . . , xnR ). By (A.11), (y1R , . . . , ynR )P(x1R , . . . , xnR ). By Strong Symmetry, (y1 , . . . , yn )I(y1R , . . . , ynR ) and (x1 , . . . , xn )I(x1R , . . . , xnR ), hence (y1 , . . . , yn )P(x1 , . . . , xn ). 2 The following examples show that the axioms in Proposition 2 are independent. An ordering violating only Consensus is the universal indifference relation: (y1 , . . . , yn )R(x1 , . . . , xn ) for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ X N . An example violating only Dominance Aversion isthe following simple consensual leximax ordering: let R be a continuous ordering agreeing with i∈N Ri and define (y1 , . . . , yn )R(x1 , . . . , xn ) if and only if either there is j ∈ N such that yjR P xjR and yiR I xiR for all i > j or yiR I xiR for all i ∈ N . For an example violating only Weak Continuity, choose a discontinuous ordering R agreeing with i∈N Ri and let R be the leximin extension of R. For instance, R could be a lexicographic refinement of, say, R1 : in the case X = R2+ for instance, define (α  , β  )R(α, β) if and only if [(α  , β  )P1 (α, β)] or [(α  , β  )I1 (α, β) and α   α]. For an example violating only Strong Symmetry, let R be a continuous preference agreeing with i∈N Ri . Let (y1 , . . . , yn )R(x1 , . . . , xn ) if and only if either (y1 , . . . , yn )Plex (R)(x1 , . . . , xn ) or (y1 , . . . , yn )Ilex (R) (x1 , . . . , xn ) and y1 Rx1 . For an example  violating only Internal Separability, choose again a continuous preferences R agreeing with i∈N Ri . If n  3, let R be the leximin extension of R. If n = 2, however, let R be the “Rawlsian extension” of R, that is, (y1 , y2 )R(x1 , x2 ) if and only if y1R Rx1R . References [1] C. d’Aspremont, Axioms for social welfare orderings, in: L. Hurwicz, D. Schmeidler, H. Sonnenschein (Eds.), Social Goals and Social Organization: Essays in Memory of Elisha Pazner, Cambridge University Press, Cambridge, 1985, pp. 19–76. [2] H. Dalton, The measurement of the inequality of incomes, Econ. J. 30 (1920) 348–361. [3] M. Fleming, A cardinal concept of welfare, Quart. J. Econ. 66 (1952) 366–384. [4] M. Fleurbaey, The Pazner–Schmeidler social ordering: a defense, Rev. Econ. Design 9 (2005) 145–166. [5] M. Fleurbaey, Two criteria for social decisions, J. Econ. Theory 134 (2007) 421–447. [6] M. Fleurbaey, F. Maniquet, Fair social orderings, Econ. Theory 38 (2008) 25–45. [7] M. Fleurbaey, A. Trannoy, The impossibility of a Paretian egalitarian, Soc. Choice Welfare 21 (2003) 243–264. [8] I. Gilboa, D. Samet, D. Schmeidler, Utilitarian aggregation of beliefs and tastes, J. Polit. Economy 112 (2004) 932–938. [9] P. Hammond, Equity, Arrow’s conditions and Rawls’ difference principle, Econometrica 44 (1976) 793–804. [10] J. Harsanyi, Utilities, preferences, and substantive goods, Soc. Choice Welfare 14 (1997) 129–145. [11] D. Hausman, M. McPherson, Preference, belief, and welfare, Amer. Econ. Rev. 84 (1994) 396–400. [12] A. Pigou, Wealth and Welfare, Macmillan, London, 1912. [13] J. Weymark, The normative approach to the measurement of multidimensional inequality, in: F. Farina, E. Savaglio (Eds.), Inequality and Economic Integration, Routledge, London, 2006, pp. 303–328.