A Fair Solution to the Compensation Problem

A Fair Solution to the Copensation Proble G. Valletta Discussion Paper 2007-38 Départeent des Sciences Éconoiques de l'université catholique de Louvain

CORE DISCUSSION PAPER 2007/77 A FAIR SOLUTION TO THE COMPENSATION PROBLEM Giacoo Valletta 1 Septeber 2007 Abstract We study equity in econoies where agents are endowed with different, non transferable, personal talents. To copensate the for such differences a given aount of oney needs to be shared aong the. We axioatize a faily of social orderings over allocations based on efficiency, fairness and robustness properties. Taking into account incentive constraints we derive the optial policy: individuals with the sae talent need to be equally copensated and only people whose level of talent is below a certain threshold should receive a positive copensation. Keywords: Copensation, talents, fairness, axios, social orderings, incentive-copatibility. JEL Classification: D63; D71 Acknowledgents: I wish to thank Francois Maniquet for any helpful discussions and detailed suggestions. I also thank Efthyios Athanasiou, Marc Fleurbaey, Juan D. Moreno-Ternero and the participants to workshops and conferences in Maastricht, Istanbul, Pavia, Louvain la Neuve, Haburg and Marseille for their coents. This text presents research results of the Belgian Progra on Interuniversity Poles of Attraction initiated by the Belgian State, Prie Minister s Office, Science Policy Prograing. The scientific responsibility is assued by the authors. 1 CORE, Université Catholique de Louvain, Belgiu

We study the fair allocation of a given aount of a one-diensional, divisible resource (e.g. oney) aong a finite population of individuals in order to copensate the for their differences in a non-transferable, personal endowent. Such a personal endowent can be either an innate talent or a handicap and can be exeplified by features of huan capital like health condition, bodily characteristics or social background. For instance, one could think of the proble of a social planner willing to provide assistance to people with disabilities by distributing aong the a given aount of available resources. %disability 33% 46% 66% 74% 99% 100% benef its Figure 1: disabilities. The Italian social security schee of benefits for people with In several countries, doctors evaluate the ental and physical condition of individuals with handicaps. An index is thus coputed which deterines to which extent the applicant is able to work. Soe benefits are then assigned as a function of this paraeter. For exaple, the copensation schee used in Italy is illustrated in figure 1 1. The vertical axis represents the degree of inability to work, whereas the horizontal axis represents bundles of benefits. The shape of the curve gives us a clear idea of how the copensation policy is designed: the bundle of benefits grows cuulatively as the level of handicap increases. A significant part of the applicants, people with a level of disability between 0 and 32%, do not receive any kind 1 The website http://www.disabili.it gives a precise description of the copensation schee, the figure is ine and it has only a descriptive purpose. 1

of copensation. People with a level of disability between 33% and 73% are entitled to soe onetary and non-onetary benefits that cuulate depending to the severity of their condition (i.e., specific edical supplies, priority in the uneployent, free drugs). People with a disability between 74% and 99% also receive a onthly onetary subsidy but only until their retireent age and provided their earnings are below a certain threshold. Individuals with a disability of 100% receive the sae aount of oney peranently and independetly of their wealth condition. The figure raises several questions. Is it fair to leave a relatively big share of the population without any copensation? Is the copensation individuals with severe disabilities receive enough? We will show that norative arguents and incentive-copatibility constraints give a partial justification to such a copensation schee. In particular, it ight indeed be optial to give a onetary copensation only to people whose level of disability is above a certain threshold. Such a threshold depends not only on the relative scarcity of the resources to share, but also on the willingness of the social planner to give priority to individuals with higher disability. On the other hand the strong discontinuities in such a schee, naely, the fact that two individuals with a non-negligible difference in their level of talent ight end up receiving the sae copensation, does not find any support in our analysis. Recent theories of justice agree that inequalities in agents outcoes ight arise both fro differences in characteristics they should not be held responsible for and fro differences in characteristics they should be held responsible for. They all arrive at the coon conclusion that society should only try to reduce inequalities deriving fro differences of the forer type 2. We will assue that each individual is endowed with two characteristics: her talent t, for which she is not responsible and her preferences R for which she is, since they ebody her subjective evaluation of her condition (naely the aount of resources she receives and her talent). The purpose of our analysis is to look, on one side, for a distribution echanis that would allocate a given aount of available external resources M in order to correct inequalities deriving fro differences in talent solely. This idea goes under the nae of principle of copensation (Fleurbaey [7], [8]). Notice that in perforing such a copensation we cannot disregard individual preferences: if all the individuals in a society dee equivalent being endowed with a given level of talent, rather than another 2 See for instance Rawls [20], Dworkin [6], Arneson [1], Cohen [4], van Parijs [24], Roeer [21] aong others. 2

one, then differences in talents do not need to be copensated. On the other side, we endorse the idea that differences due to characteristics that fall into the doain of responsibility should be treated neutrally: this eans that, in our case, the distribution echanis should give the sae aount of resources to individuals with the sae talent (but possibly different preferences). This is the so called principle of natural reward (Fleurbaey [7], [8]) To solve this proble, the literature has so far proposed allocation rules (for a coplete survey of the subject see Fleurbaey and Maniquet [11]). An allocation rule specifies the set of optial allocations as a function of the paraeters of the proble: the population s profile of talents, the population s profile of preferences and the available resources. Such allocation rules can be enforced only on the basis of a full knowledge of the population s profiles of characteristics. Typically allocation rules are not perfectly ipleentable because of incentive constraints. Alternatively, one ay look for social ordering functions specifying a coplete ranking of all the allocations as a function of the paraeters of the proble. This approach has been already applied to several econoic environents: for instance Maniquet and Spruont [17], [18] construct orderings of allocations in econoies with one private good and one partially excludable nonrival good. Fleurbaey and Maniquet [13], [14] consider a odel where agents have unequal production skills and different preferences and they characterize purely ordinal social ordering functions which satisfy properties of copensation for inequalities in skills and equal access to resources for all preferences. Fleurbaey and Maniquet [12] construct social preferences in the canonical econoic proble of dividing a bundle of coodities aong individuals with different preferences. In this paper, we follow the sae approach by proposing a way to rank alternatives based on efficiency, fairness and robustness properties. It is explained below how it is possible to ebody the principle of copensation and the principle of natural reward into a variety of specific, precise axios. It will be clear soon that it is not possible to find a social ordering function that satisfies both the principles. Let us iediately ephasize that such an incopatibility does not represent a dead end for our analysis. It is a purely introductory result telling us that we have to ake a choice between these two incopatible principles: we have to chose which one should have priority over the other. We will show how axios ebodying the principle of copensation and axios weakly ebodying the principle of natural reward together with efficiency and separability conditions (with soe vari- 3

ants, separability conditions state that indifferent agents should not atter in the social evaluation of two alternatives) single out a specific welfare representation of agents preferences and a specific way of ranking alternatives. Once an ordering of the allocations is available, it is always possible to produce a precise policy recoendation, even if the social planner faces inforational constraints (or any other kind of constraint). We will show in fact, that axiizing the ordering, under the relevant incentive constraints, gives us, as entioned earlier, a specific suggestion about the way resources should be allocated. The paper is organized as follows. Section 1 introduces the odel and the relevant notation. Section 2 describes the requireents iposed on the social preferences and describes logical relations between the. Section 3 deals with social orderings. Section 4 studies the optial allocation of resources under inforational constraints. Section 5 concludes. The Appendix provides the proofs. 1 The Model The odel we analyze is derived fro Fleurbaey [7], [8]. We consider a set of econoies with a finite set of agents N N. A given aount M R ++ has to be shared aong the through onetary transfers of aount i R +. Each agent in the econoy is characterized by a fixed, non transferable talent t i T. Let T = [t L, t H ] R be the set of possible talents. Moreover each agent is endowed with a personal preference ordering R i over pairs (, t) R + T. A pair (, t) will be called hereafter an extended bundle. This eans that individuals are assued to be able to copare extended bundles of individuals with a talent different fro their own. In this fraework one can think of preferences as to valuation functions: each individual is able to give a valuation of her own condition and copare it with soeone else s. The individual preference ordering R i is assued to be continuous and strictly onotonic with respect to i and t i, strict preference and indifference will be respectively denoted by P i and I i. Let R denote the set of such preferences. We will say that individual preferences R R exhibit a higher aversion to lack of talent than R R (R A R ) if they satisfy the following property: for any t, t T and for any, R N + we have t > t and (, t )R (, t) = (, t )P (, t) t < t and (, t )R (, t) = (, t )P (, t) 4

that is, any pair of indifference curves, one for R and one for R cross at ost once in the (, t) space. An econoy is denoted by e = (t N, R N, M), where t N = (t i ) i N is the population s profile of talents, R N = (R i ) i N is the profile of individual preferences and M is the available social endowent. The doain of all econoies satisfying the above assuptions will be denoted by D. An allocation is a vector N = ( i ) i N R N +. The first concern of this paper is to construct a social ordering of (feasible and non feasible) allocations for all econoies in the doain. A social ordering, for any given e D, is a coplete and transitive ranking defined over allocations. A social ordering function R associates every adissible econoy e = (t N, R N, M) D with a social ordering R(e). So for an econoy e = (t N, R N, M) D and two allocations N and N RN + we write N R(e) N to denote that N is (socially) at least as good as N. Strict social preference and indifference will be respectively denoted by P (e) and I(e). t H t j 2 j 1 j t t 1 k 2 j 2 k 1 j t k 1 k 2 k t L Figure 2: To illustrate our approach consider the siple econoy depicted in figure 2. We only have two agents, j and k, with different talents, t j and t k. We consider two allocations, ( 1 j, 1 k ) and (2 j, 2 k ) and we want to rank the. The social ordering function we propose in this paper works in the following way. First, a reference talent t is chosen. Then, for each i {j, k}, q {1, 2} we consider the individual level of resource q i 0 which would ake the agents accept the reference talent t instead of their current situation ( q i, t i). As we can see fro the figure, such a value does not exist for agent k at 5

( 1 j, 1 k ). In order to evaluate the situation of agent k, we then consider the level of talent t 1 k that would ake her accept a null transfer instead of her current situation. More forally, t 1 k is the level of talent such that (0, t 1 k )I k( 1 k, t k). To suarize, for each agent i we can have a continuous nuerical representation of her preferences over pairs (, t) R + T : { Ê Ri,et ( i if i, t i ) = i exists t i t otherwise. Finally the vectors of such welfare representations are copared using the axiin rule: in our exaple we have that ( 2 j, 2 k ) is strictly better than ( 1 j, 1 k ) since in{ê2 j, Ê2 k } > in{ê1 j, Ê1 k }. We will prove in Section 4 that both this particular welfare representation and the way of ranking alternatives are singled out by a specific cobination of axios. More precisely this is what we obtain if we cobine axios ebodying the principle of copensation, axios weakly ebodying the principle of natural reward together with efficiency and robustness requireents. Indeed, if we easured welfare differently, keeping the sae ranking criterion, we would not be able to obtain a copensation inded social ordering function. Consider, for exaple, the following way of ranking alternatives. Take two allocations N and N and order the extended bundles according to a reference preference R R in such a way that ( i, t i ) R( i+1, t i+1 ) and ( i, t i) R( i+1, t i+1) for all i (1,..., n 1) then, N is socially (strictly) preferred to N if ( n, t n ) P ( n, t n ) 3. This social ordering function fails to satisfy the principle of copensation (except for agents whose individual preference ordering is equal to the reference one) and on the contrary gives priority to the principle of natural reward. This duality is actually due to a basic incopatibility between the two ethical principles that was already pointed out by Fleurbaey [8] in the first best fraework and, as will be proven in section 3, it is still present in the second best fraework. On the other hand, suppose we keep the welfare representation proposed in the first place but we drop the axiin criterion. We could, for exaple, take the welfare-level vectors underlying two different allocations and calculate their Gini index (we will consider only the striclty positive eleents). The allocation with the sallest Gini index will be socially preferred to the other one. This ranking criterion is, to soe extent, copensation inded 3 notice that the agents with the worst-off extended bundles respect to e R, at the two allocations, are not necessarily the sae. 6

but it is not necessarily consistent with an infinite degree of inequality aversion. As we will see later this is a feature we are forced to accept if we cobine together copensation and robustness requireents. 2 Axios This section defines the properties we want to ipose on our social ordering function. We start with a basic efficiency condition. The Strong Pareto axio takes a very intuitive for here since talents are fixed and resources are one-diensional. Notice oreover that all allocations exhausting the resources are efficient. Strong Pareto. For all e D and N, N RN +, if i i for all i N, then N R(e) N ; if, in addition, i > i for soe i N, then N P (e) N. The second axio we introduce is a property of robustness of the social ordering with respect to changes in the set of agents. We require that adding or reoving agents who receive the sae bundle in two different allocations should not odify the social ordering. This axio was introduced by Fleurbaey and Maniquet [12] and is reiniscent of the Separability condition, quite failiar in social choice, due to d Aspreont and Gevers [2] and of the Consistency condition, widely used in the theory of fair allocation (see Thoson [23]), except that it does not require to delete the resources consued by the reoved agents fro the social endowent. Separation. For all e D and N, N RN +, if there is i N such that i = i, then N R(e) N N\{i}R(t N\{i}, R N\{i}, M) N\{i}. We can now turn to the fairness conditions expressing respectively the ideas of copensation and natural reward. Assue first that there are two agents, j and k, with equal preferences and a different talent. Assue also that they both consider agent j s extended bundle strictly better than agent k s. Assue finally that the onetary copensation received by agent j is reduced by a given aount and the one received by agent k is increased by the sae aount but they both still consider j s extended bundle to be better than k s. The allocation we obtain ust be, according to the principle of copensation, socially preferred to the forer one since it reduces 7

inequalities exclusively deriving fro differences in talent. Equal Preferences Transfer. For all e D and N, N RN +, if there exist j, k N and R + such that R j = R k, j = j, k = k + and ( j, t j)p j ( k, t k), with i = i for all i j, k, then N P (e) N. We also consider a quite natural strengthening of this axio: instead of requiring the preferences of the agents to be identical, we siply require that agents indifference curves trough their extended bundles in N and N are nested. For each i N let L(( i, t i ), R i ) and U(( i, t i ), R i ) denote respectively the (closed) lower contour set and the (closed) upper contour set of R i at ( i, t i ). Nested Preferences Transfer. F or all e D and N, N RN + if there exist j, k N and R + such that U(( j, t j), R j ) L(( k, t k), R k ) =, j = j and k = k +, with i = i N P (e) N. for all i j, k, then The rationale of this axio is that, in both allocations, one of the two agents envies the other (not only at her actual talent but also at any hypothetical level of talent she ay have), so perforing the transfer actually reduces resources inequality. Let us turn now to the principle of natural reward. Assue there are two agents with the sae talent (and possibly different preferences) who receive a different onetary copensation: perforing an equalizing transfer that reduces such an inequality should be considered a social iproveent. Equal Talent Transfer. For all e D and N, N RN +, if there exist j, k N and R + such that t j = t k and k + = k < j = j, with i = i for all i j, k, then N P (e) N. The fairness conditions presented so far are an adaptation to this fraework of the Pigou-Dalton principle of transfer, widely used in incoe inequality 8

theory. They are all biased toward equality but consistent with any degree of inequality aversion. Our first result shows that the copensation and natural reward requireents we have presented so far are incopatible. Lea 1 On the doain D, no social ordering function satisfies Equal Preferences Transfer and Equal Talent Transfer. We propose now an alternative way of capturing the ethical goals of copensation and natural reward that is inspired by Suppes grading principles (Suppes [22]) and was introduced in the literature by Maniquet [16]. The first property, called Equal Preferences Perutation, requires that peruting the outcoes of two agents having the sae preferences but possibly different talents gives us two socially equivalent allocations. Indeed, according to the principle of copensation there is no reason why society should prefer one of the two allocations: agents with the sae responsibility characteristics should be treated anonyously. Equal Preferences Perutation. For all e D and N, N RN +, if there exist j, k N such that R j = R k, ( j, t j )I j ( k, t k) and ( k, t k )I j ( j, t j), with i = i for all i j, k, then NI(e) N. The next axio requires, according to the principle of natural reward, that peruting the bundles of two agents having the sae talent but possibly different preferences does not alter the value of the allocation in the social ranking. Again the justification is clear: as those agents have the sae copensation paraeters, they should be treated anonyously. By peruting the transfers they receive, we obtain an equally good, or equally bad, allocation. Equal Talent Perutation. For all e D and N, N RN +, if there exist j, k N such that t j = t k, k = j and j = k, with i = i for all i j, k, then NI(e) N. The next result shows that the incopatibility between copensation and responsibility extends beyond Lea 1 since it has nothing to do with the 9

degree of inequality aversion exhibited by the axios that were involved there: also requireents like Equal Preferences Perutation and Equal Talent Perutation, which are consistent with any degree of inequality aversion (even a negative one), are in fact incopatible if cobined with Strong Pareto. Lea 2 On the doain D, no social ordering function satisfies Strong Pareto, Equal Preferences Perutation and Equal Talent Perutation. Siilar incopatibilities between copensation and natural reward, in a second best fraework, have already been pointed out by Fleurbaey and Maniquet [13] in a different econoic environent. Let us stress again that these negative results are purely introductory ones, they iply that we have to chose which of the two principles ust have priority. As we shall see, this choice has an iportant consequence concerning the the degree of inequality aversion that will be exhibited finally by the social ordering function. In fact, if we decide to give priority to the principle of copensation, then, cobining together a ild egalitarian requireent like Nested Preferences Transfer, with axios that are essentially neutral with respect to inequality aversion, deterines an extree for of inequality aversion described by the following axio. Nested Preferences Priority. For all e D and N, N RN +, if there exist j, k N such that ( j, t j )P j ( j, t j)p j ( k, t k)p k ( k, t k ) U(( j, t j), R j ) L(( k, t k), R k ) =, with i = i for all i j, k, then N P (e) N. Let us ephasize that this property represents an extree strengthening of Nested Preferences Transfer. It states that an inequality reduction is always socially desirable, provided that indifference curves are nested at the bundles, even when the gain fro k to k is very sall and the loss fro j to j is quite large. Lea 3 On the doain D, if a SOF satisfies Nested Preferences Transfer, Equal Preferences Perutation and Separation then it satisfies Nested Preferences Priority. Results siilar to this one can already be found in Fleurbay [9] and Maniquet and Spruont [17], [18]. They all show, in different econoic 10

environents, that cobining together fairness conditions that exhibit a liited aversion to inequality with robustness and efficiency conditions leads to an infinite aversion to inequality. The rearkable aspect here is that we reach a siilar conclusion dealing with a one-diensional resource 4. This akes the contrast with the incoe inequality easureent fraework, where cobining the Pigou- Dalton transfer principle with robustness requireents does not necessarily lead to such extree consequences (Chakravarty [5]), even stronger. It is oreover interesting to notice that if we decide to give priority to the principle of natural reward, we cannot find anything siilar. In this case, in fact, we would fall in a purely one-diensional fraework were we are eventually free to choose the degree of inequality aversion of the social ordering function 5. The fact that we have decided to give priority to the principle of copensation does not ean that we want to disregard copletely the principle of natural reward. In the last part of this section we will propose two alternative weakenings of Equal Talent Transfer that represent soehow the strongest natural reward axios that would not clash with Equal Preferences Transfer. The ipossibility pointed out by Lea 1 akes it clear that, if we do not want to violate Equal Preferences Transfer, then it is possible to perfor a redistribution between two agents with the sae talent t who receive a different onetary copensation at ost for one t T. A first option then, is to ipose the full transfer condition only aong agents with a talent equal to soe reference talent t fixed arbitrarily. Alternatively, if there are two agents j, k such that t j = t k < t and one of the does not receive a onetary copensation, while the other receives a positive one, then we propose a inial reward requireent: it ust be always possible to slightly reduce such a difference and the allocation we obtain is socially preferred to the forer one. 4 Interestingly a siilar result involving Equal Preference Transfer (and its Priority version) could be obtained if we replace Separation with a different inforational siplicity axio: Hansson Independence (Fleurbaey and Maniquet [14]). It says that the social ranking of two allocations should reain unaffected by a change in individual preferences which does not odify agents indifference curves at the bundles they receive in these two allocations. Indeed if a social ordering function satisfies Equal Preferences Transfer, Equal Preferences Perutation and Hannson Independence then it also satisfies Equal Preferences Priority. 5 We could, for exaple, copare allocations just by coputing their Gini index. This would satisfy Strong Pareto and Equal Talent Transfer (but it would violate Equal Preferences Transfer). 11

t-equal Talent Transfer. For all e D, N, N RN +, for at least one t T if there exist j, k N and R + such that t j = t k = t and j = j > k = k +, with i = i for all i j, k, then N P (e) N or if there exist j, k N such that t j = t k < t, j = 0, k > 0 with i = i for all i j, k, then there exists at least one ε R ++ such that ε = j < k = k ε and N P (e) N. The next axio does not involve the choice of a reference talent. We still consider a couple of agents with the sae talent who receive a different aount of resources. We say that it is socially desirable to perfor a transfer between the only if the recipient s preferences exhibits a higher degree of aversion to lack of talent than the donor s preferences. Alternatively, if there are two agents j, k such that t j = t k and one of the does not receive a onetary copensation, while the other receives a positive one, then it ust be always possible to slightly reduce such a difference while obtaining a social iproveent. Equal Talent Transfer to the Unhappy. for all e D and N, N R N +, if there exist j, k N and R + such that t i = t j, R j A R i and k + = k < j = j, with i = i for all i j, k, then N P (e) N or if there exist j, k N such that t j = t k, j = 0, k > 0 with i = i for all i j, k, then there exists at least one ε R ++ such that: ε = j < k = k ε and N P (e) N. 3 Social Orderings Before we show how cobining together the axios presented in the previous section deterines a specific way of ranking the social alternatives, we need 12

to introduce soe piece of notation. Assue a reference talent t is chosen. For each agent i N and for each i R + consider i 0 such that ( i, t i )I i ( i, t). As pointed in the exaple in section 2, such a value does not necessarily exist. If this is the case, consider the hypothetical level of talent t i such that (0, t)i i ( i, t i ). Finally, for each i N consider the function { i if Ê( i, t i, R i, t) = i exists t i t otherwise It is worth to stress again that such a value is a continuous nuerical representation of the preferences of the agents over extended bundles. Hence for each allocation we will have a vector of such welfare levels. It turns out that if we pursue the objective of giving priority to the principle of copensation then such vectors need to be copared using the axin rule. Theore 1 Let a Social Ordering Function satisfy Strong Pareto, Nested Preferences Transfer, Equal Preferences Perutation, t-equal Talent Transfer and Separation. Then, on the doain D, for any N, N RN +, one has N P (e) N if in i Ê( i, t i, R i, t) > in i Ê( i, t i, R i, t) The ain drawback of Theore 1 is that the evaluation function of each agent depends on the value of the reference talent t that is exogenously fixed by the social planner. So, a different choice of t ight lead to a different social ordering. We can overcoe this difficulty by substituting t-equal Talent Transfer with Equal Talent Transfer to the Unhappy. Using this particular weakening of Equal Talent Transfer we are able to endogenize the choice of the reference talent. More precisely, the introduction of Equal Talent Transfer to The Unhappy leads to select t H as the reference talent. Theore 2 Let a Social Ordering Function satisfy Strong Pareto, Nested Preferences Transfer, Equal Preferences Perutation, Equal Talent Transfer to the Unhappy and Separation. Then, on the doain D, for any N, N R, one has N P (e) N if in i Ê( i, t i, R i, t H ) > in i Ê( i, t i, R i, t H ) 13

The fact that, both for Theore 1 and Theore 2, no axio is redundant, will be shown in the appendix. Siilar results can be found in Fleurbaey and Maniquet [14] and Maniquet and Spruont [17], [18]. They propose axiin criteria that like ours rely only on ordinal, non-interpersonally coparable, individual preferences. No ex-ante inforation about individual utilities is needed in order to build the social ranking we propose: on the contrary, our axios single out a particular welfare representation of individual preferences. The corresponding welfare-level vectors need then to be ranked according to the axiin criterion: a way of ranking alternatives that gives priority to agents with a low Ê(.), that is, either agents with a low i or agents who dislike their level of talent. None of the theores provides a full characterization of social preferences because they do not tell us how to rank two allocations for which in i Ê( i, t i, R i, t) = in i Ê( i, t i, R i, t). Nonetheless, as we will see in the next section, they both give us a criterion to evaluate different copensation policies and, in particular, to single out the optial one. Moreover, in both cases, the refineent of the proposed social ordering obtained by applying the lexiin criterion 6 to the welfare-level vectors gives us a coplete ranking of the social alternatives and, as can be easily checked, satisfies all the involved axios. 4 Incentive Copatible Allocations With a social ordering function like the one we have introduced in the previous section, we would obtain, at the optiu, an allocation such that each agent is indifferent between the extended bundle she receives and a hypothetical extended bundle which is the sae for everyone (assuing that everyone receives a strictly positive transfer). Such an allocation would be achievable if the social planner knew the talent and the preferences of each agent. In our fraework we can reasonably assue that the forer is known (doctors can establish the degree of disability of an individual) while individual preferences are typically not observable 7. 6 Let L denote the usual lexiin ordering of R n +: for any w, w R n +, w L w if and only if the sallest coordinate of w is greater than the sallest coordinate of w or they are equal but the second sallest coordinate of w is greater than the second sallest coordinate of w and so on 7 If the talent was not observable either, the only incentive-copatible allocation would trivially be the one that gives an equal split of the available resources to each agent. 14

t H R a = R b R c t b = t c b c t (, t) t a a t L Figure 3: Agent b has an incentive to isreport her preferences. Consider, for instance, the siple three-agent econoy depicted in figure 3. At the optiu, we have full copensation between agent a and agent b. On the other hand, agent b and agent c, who have different preferences but equal talent, receive a different transfer: c > b. This clearly gives an incentive to agent b to isreveal her preferences. We can however still use the ranking criterion we have proposed in order to choose the second-best policy. Interestingly, we will see that the incentive copatibility constraint considerably shapes our choice between the principle of copensation and the principle of natural reward. Actually, fro the previous exaple it is easy to understand that, if the social planner knows the distribution of individual preferences, an allocation N R N is incentive-copatible if and only if i = j for all i, j N with t i = t j that is, agents with the sae talent should receive the sae aount of resources regardless of their preferences and this is indeed what we should do if we want to give priority to the principle of natural reward. For any given econoy e D, let T e and R e denote, respectively, the set of the different levels of talents and the set of individual preference orderings we can find in e. Before we show the shape of the optial incentive copatible allocation we need to introduce two ore assuptions. The first one is nothing but the adaptation to this fraework of the Mireless-Spence single crossing condition: for any couple of preferences ordering R and R in R e we either have that R exhibits a higher degree of inequality aversion to lack 15

of talent than R or vice versa. This eans that it is possible, within each econoy, to rank all the preference orderings with respect to the aversion to lack of talent they exhibit. Assuption 1: For each R, R R e, either R A R or R A R. Let R H and R L denote the preference orderings in R e with, respectively, highest and lowest aversion to lack of talent. The second assuption we introduce is a richness requireent that looks quite reasonable specially if the nuber of agent in the econoy is considerably large: for any class of agents with a given talent t T e there is at least one agent with preferences R, for each R R e. Assuption 2: For each (t, R) T e R e there exists i N such that R i = R and t i = t N et Given any reference talent t chosen fro T let N + = {i N t et i t} and = {i N t i < t}. Consider an incentive-copatible allocation. By Assuption 2, for each t t, there is at least one agent with such a talent and with preferences equal to R L. It is easy to check that, aong agents with the sae talent, she is the one with lowest Ê(.). Syetrically, for each possible level of talent t, the agents with lowest Ê(.) are those with preferences R H. Since we are using the axiin criterion, we should equalize, whenever possible, the lowest Ê(.) values across different classes. Theore 3 In any econoy e = (t N, R N, M) D, under Assuptions 1 and 2, if a Social Ordering Function satisfies Strong Pareto, Nested Preferences Transfer, Equal Preferences Perutation, t-equal Talent Transfer and Separation then, the optial incentive copatible allocation ( e t N ) distributes the available resources in the following way: and for all i, j N + et, (e t i, t i )I L ( e t j, t j ) or e t i = 0 and (0, t i )R L ( e t j, t j ). for all i, j N et, (e t i, t i )I H ( e t j, t j ) or e t i = 0 and (0, t i )R H ( e t j, t j ) Proof. By Theore 1 we know that if a Social Ordering Function satisfies the listed axios then, for any N, N R N +, in i Ê( i, t i, R i, t) > in i Ê( i, t i, R i, t) = N P (e) N 16

t H R H t in{ e t i } t L R L Figure 4: Proof of Theore 3. At e t N, given assuption 1 and 2, it holds true that the worst-off agents are (for a graphical exaple see figure 4, for siplicity we represent a situation in which each agent receives a strictly positive transfer) all j N + et such that R j = R L and j > 0 and all k N such that R et k = R H and k > 0: in i {Ê(e t i, t i, t, R i )} i N = Ê(e t j, t j, t, R L ) = Ê(e t k, t k, t, R H ). Assue, by contradiction, that there exists an incentive copatible allocation N such that iniê( i, t i, R i, t) > in i Ê( e t i, t i, R i, t). This iplies that, for all j N + such that R et j = R L and j > 0, j > e t j and, analogously, for all k N such that R et h = R H and k > 0, k > e t k. By incentive copatibility we should then increase by the sae aount the transfer received by all the other agents having the sae level of talent but different preferences. This contradicts that, by construction, e t N is an allocation exhausting the available resources. The resulting allocation is strongly affected by the incentive copatibility constraint: people with the sae talent receive the sae onetary transfer. The schee we obtain is hence natural-reward inded, nonetheless it shows us the highest level of priority that we can give to the principle of copensation given the inforational constraints we have to face. 17

t H R L t H t R L t H = t t R H R H R H t L (a) t L (b) t L (c) Figure 5: A different reference talent deterines a different copensation policy. A consequence of Theore 3 is that the optial copensation path crucially depends on the choice of t. So, for exaple, the higher is the reference talent chosen by the social planner the bigger is the copensation received by people with a severe disability (see figure 5 (a) and (b)). Moreover, the proportion of individuals with a low disability is typically higher than the proportion of agents with a severe disability and the equality of extended bundles (with respect to the references preferences R H and R L ) is perfored here in such a way that it ay be ipossible to fully copensate soe inequalities in personal characteristics with the available resources. In that case, soe inequalities persist and the better-off agents are left with no external resource, while only disadvantaged agents receive a strictly positive onetary copensation. The higher t the ore this is likely to happen: this, in fact, deterines a threshold above which = 0. It is clear that this gives justification to several copensation schees used in reality that choose to leave with no transfer people with a relatively high talent. Moreover it is also interesting to note that the copensation schee is relatively sooth if copared with copensation schees we can observe in reality: the aount received varies with the level of talent and it is never the case that people with a different t receive the sae transfer (except, of course, those who are not copensated at all). At this point it is quite siple to understand the policy iplications of Theore 2. As we have seen in this case the reference talent is endogenously deterined and is equal to t H. This eans that the copensation policy can no longer be arbitrarily changed by varying the reference talent but fol- 18

lows a path that is endogenously deterined. Corollary 1. In any econoy e = (t N, R N, M) D, under Assuptions 1 and 2, if a Social Ordering Function satisfies Strong Pareto, Nested Preferences Transfer, Equal Preferences Perutation, Equal Talent Transfer to the Unhappy and Separation then the optial incentive copatible allocation ( th N ) distributes the available resources in the following way: for all i, j N, ( th i, t i )I H ( th j, t j ) or i = 0 and (0, t i )R H ( j, t j ) Such a copensation path gives axial priority to agents with a lower talent leaving a relatively big share of the population without any copensation (see figure 5 (c)). 5 Conclusions In the first part of this paper we have exained how axios ebodying the principle of copensation and axios weakly ebodying the principle of natural reward lead to a particular way of evaluating individual welfare and single out particular social preferences. Such social preferences grant absolute priority to agents who dislike their level of talent and to agents who receive a low copensation. The easure of individual situations obtained here is derived fro the fairness principles and the analysis did not require any other inforation about individual welfare than ordinal non-coparable preferences. It ust be stressed again that alternative easures of individual welfare and alternative social preferences, both responding to different ethical principles, could be found. The purpose of our analysis was to see how one could derive specific policies recoendations fro given fairness principles. The second part of this paper has studied the iplications of such social preferences for the evaluation of incentive copatible-policies. The fact that incentive constraints prevent the full expression of priority given to the principle of copensation does not render the principle irrelevant. On the contrary it tells us to what extent an ipleentable policy can take this ethical requireent into account. The ain lesson we can draw fro this paper is in fact that leaving a certain share of the population without a onetary copensation can be defended on norative grounds, given the scarcity of the available resources 19

and the inforational constraints. In particular this share is endogenously deterined if we chose a particular weakening of Equal Talent Transfer. Finally let us briefly discuss the assuptions underlying the analysis. In Fleurbaey [7] [8] no particular structure is assued on the set T. Here Lea 1,2 and 3 would still hold under the sae assuption too. In our analysis assuing that T has at least soe ordinal structure plays a fundaental role in the way the social ordering function we propose is built. As we have seen, any copensation schees used in reality are based on indexes that induce a structure on individual talents and handicaps. Appendix: proofs Proof of Lea 1. Take the econoy e = (t, t, t, t, R, R, R, R, M) D, the extended bundles ( 1, t), ( 2, t), ( 3, t), ( 4, t), ( 5, t ), ( 6, t ), ( 7, t ), ( 8, t ) and R ++ with: t H t 1 2 3 4 t 5 6 7 8 t L Equal Preferences Transfer and Equal Talent Transfer are in- Figure 6: copatible. 1. t t 2. ( 1, t)p ( 8, t ) 3. ( 5, t )P ( 4, t) 4. 4 = 3 > 2 = 1 + 20

5. 8 = 7 > 6 = 5 + We give a graphical exaple of such an econoy in figure 6. By Equal Preferences Transfer and conditions 2, 4 and 5, ( 1, 8, 5, 3 ) P (e) ( 2, 7, 5, 3 ). By Equal Talent Transfer and condition 5, ( 1, 7, 6, 3 ) P (e) ( 1, 8, 5, 3 ). By Equal Preferences Transfer and conditions 3, 4 and 5, ( 1, 7, 5, 4 ) P (e) ( 1, 7, 6, 3 ). By Equal Talent Transfer and condition 4, ( 2, 7, 5, 3 ) P (e) ( 1, 7, 5, 4 ). Finally, by transitivity, ( 1, 7, 6, 3 ) P (e) ( 1, 7, 6, 3 ), which yields the desired contradiction. Proof of Lea 2. Take the econoy e = (t, t, t, t, R, R, R, R, M) D, the extended bundles ( 1, t), ( 2, t), ( 3, t), ( 4, t), ( 5, t), ( 6, t), ( 1, t ), ( 2, t ), ( 3, t ), ( 4, t ), ( 5, t ), ( 6, t ) with: t H t 6 5 4 3 2 1 t t L 6 5 4 3 2 1 Figure 7: Strong Pareto, Equal Preferences Perutation and Equal Talent Perutation are incopatible. 1. t t 2. 1 > 2 > 3 > 4 > 5 > 6 3. 1 > 2 > 3 > 4 > 5 > 6 4. ( 1, t)i( 2, t ) and ( 6, t)i( 3, t ) 5. ( 2, t)i ( 1, t ) and ( 3, t)i ( 6, t ) 21

A graphical exaple of such an econoy is given in figure 7. By Strong Pareto and conditions 2 and 3, ( 1, 4, 3, 5 ) P (e) ( 2, 5, 4, 6 ). By Equal Talent Perutation and condition 3, ( 4, 1, 3, 5 ) I(e) ( 1, 4, 3, 5 ). By Equal Preferences Perutation and condition 4, ( 4, 6, 2, 5 ) I(e) ( 4, 1, 3, 5 ). By Equal Talent Perutation and condition 3, ( 4, 6, 2, 5 ) I(e) ( 4, 6, 5, 2 ). By Strong Pareto and conditions 2 and 3, ( 3, 5, 4, 1 ) P (e) ( 4, 6, 5, 2 ). By Equal Preferences Perutation and condition 5, ( 2, 5, 4, 6 ) I(e) ( 3, 5, 4, 1 ). Finally by transitivity, we have ( 2, 5, 4, 6 ) P (e) ( 2, 5, 4, 6 ), a contradiction. Proof of Lea 3. Let R satisfy Nested Preferences Transfer, Equal Preferences Perutation and Separation. Let e = (t N, R N, M) D, N, N R N +, j, k N be such that: ( j, t j )P j ( j, t j)p j ( k, t k)p k ( k, t k ), U( j, R j) L( k, R k) =, and for all i j, k, i = i. We want to prove that N P (e) N. Let j = ( j j ) and k = ( k k). Let b, c N ++ \ N, R b, R c R, t b, t c, 1 b, 2 b, 1 c, 2 c R + and q N ++, be defined in such a way that: R b = R c ; ( 1 b, t b)i b ( 1 c, t c ) and ( 2 b, t b)i b ( 2 c, t c ); 1 b = 2 b + k q and 1 c = 2 c + j q ; (2 b, t b)p b ( k, t k), U(( 2 b, t b), R b ) L(( k, t k), R k ) =, ( j, t j)p j ( 1 b, t b) and U(( j, t j), R j ) L(( 1 b, t b), R b ) =. Let e = (t N, t b, t c, R N, R b, R c, M) D (an exaple of such a construction is given in figure 8). By Nested Preferences Transfer, ( N\{k}, k + k q, 2 b, 1 c)p (e )( N, 1 b, 1 c). By Equal Preferences Perutation, ( N\{k}, k + k q, 1 b, 2 c)i(e )( N\{k}, k + k q, 2 b, 1 c). By Nested Preferences Transfer, ( N\{k,j}, k + k q, j j q, 1 b, 1 c)p (e )( N\{k}, k + k q, 1 b, 2 c). By transitivity, ( N\{k,j}, k + k q, j j q, 1 b, 1 c)p (e )( N, 1 b, 1 c). 22

t H t j t b 1 b 2 b j j t c t k 2 1 c c k k t L Figure 8: Fro Nested Preferences Transfer to Nested Preferences Priority. Replicating the arguent q ties yields ( N\{k,j}, k + k, j j, 1 b, 1 c)p (e )( N, 1 b, 1 c). Since j j = j and k + k = k, by replacing into the forer equation one obtains Finally, by Separation, the desired result. ( N, 1 b, 1 c)p (e )( N, 1 b, 1 c). N P (e) N, Proof of Theore 1. Let R be a social ordering function that satisfies Strong Pareto, Nested Preferences Transfer, Equal Preferences Perutation, t-equal Talent Transfer and Separation. Choose any t R +. By Lea 3, the social ordering function R also satisfies Nested Preferences Priority. For the sake of clarity the reaining of the proof is divided in two steps. Step 1. Consider two allocations N and N and two different agents j and k such that for all i j, k, i = i. Let Ê = in{ê( j, t j, R j, t), Ê( j, t j, R j, t), Ê( k, t k, R k, t).} 23

Without loss of generality, let us assue that Ê( k, t k, R k, t) < Ê. We want to prove that N P (e) N. If we also have j j then by Strong Pareto we iediately get the desired result. Consider now the case j > j and assue, by contradiction, that NR(e) N. We have to exaine two possible cases. Case 1 : Ê( k, t k, R k, t) < 0. Let b, c N ++ \ N, R b, R c R, t b, t c T, b, c, k R + be defined in such a way that: (a graphical exaple of the following construction is given in figure 9): t H t j j j t t b = t c ε j b c c ε k j t k t k t L k k k Figure 9: Case 1. 1. t k < t b = t c < t if exists, t k < t b = t c < t otherwise 2. R k = R b and R j = R c 3. 0 = b < c and ( j, t j)p j ( c, t c ) 4. k < k < k and ( b, t b )P k ( k, t k). Let e = (t j, t k, R j, R k, M), e = (t j, t k, t b, t c, R j, R k, R b, R c, M) D. Since we have assued N R(e) N then, by Separation, we have ( j, k ) R(e ) ( j, c k ). Take any ε such that 0 < ε < 2. By Separation again ( j, k, b +ε, c ε) R(e ) ( j, k, b+ε, c ε). By Nested Preferences 24

Priority and conditions 2 and 3, ( j, k, b + ε, c ) P (e ) ( j, k, b + ε, c ε). By Nested Preferences Priority and conditions 2 and 4, ( j, k, b, c ) P (e ) ( j, k, b + ε, c ). By Strong Pareto and condition 4, ( j, k, b, c ) P (e ) ( j, k, b, c ). By Transitivity, ( j, k, b, c ) P (e ) ( j, k, b + ε, c ε). This is true for each ε such that 0 < ε < c 2 hence, given condition 1 and 3, it represents a violation of t-equal Talent Transfer and yields the desired contradiction so that N P (e) N Case 2 : Ê( k, t k, R k, t) 0. Let b, c N ++ \ N, R b, R c R, t b, t c T, b, b, c, c, k R +, R ++ be defined in such a way that: (for a graphical exaple of the following construction see figure 10): t H t j j j t k b b j c c k j t k t L k k k Figure 10: Case 2. (5) t b = t c = t (6) R k = R b and R j = R c (7) Ê( k, t k, R k, t) < b < c < Ê and b + = b < c = c (8) k < k and ( b, t b )P k ( k, t k). 25

Let e = (t j, t k, R j, R k, M), e = (t j, t k, t b, t c, R j, R k, R b, R c, M) D. Since we have assued N R(e) N then, by Separation we have respectively, ( j, k ) R(e ) ( j, k ) and ( j, k, b, c ) R(e ) ( j, k, b, c ). By Nested Preferences Priority and conditions 6 and 7, ( j, k, b, c) P (e ) ( j, k, b, c ). By Nested Preferences Priority and conditions 6,7 and 8, ( j, k, b, c) P (e ) ( j, k, b, c). By Strong Pareto and condition 8, ( j, k, b, c) P (e ) ( j, k, b, c). By Transitivity, ( j, k, b, c) P (e ) ( j, k, b, c ), which, given conditions 5 and 7, is a violation of t-equal Talent Transfer and again yields the desired contradiction so that N P (e) N. Step 2. Take now two allocations N and N such that in i Ê( i, t i, R i, t) > in i Ê( i, t i, R i, t). Then, by onotonicity of preferences, one can find two allocations N, N such that for all i, i < i, i < i, and there exists i 0 such that for all i i 0 Ê( i, t i, R i, t) > Ê( i, t i, R i, t) > Ê( i 0, t i0, R i0, t) > Ê( i 0, t i0, R i0, t). Let Q = N\{i 0 } and consider a sequence of allocations ( q N ) 1 q Q +1 such that while q i = i for all i Q such that i < q q i = i for all i Q such that i q i 0 = Q +1 i 0 This entails that for all q Q > Q i 0 >... > 1 i 0 = i0. Ê( q q, t, R, t) > Ê(q+1 q, t, R k, t) > Ê(q+1 i 0, t i0, R i0, t) > Ê( q i 0, t i0, R i0, t), while for all q Q, and all i i 0, q, q i = q+1 i. By Step 1, q+1 N P (e)q N for all q Q. By Strong Pareto, N P (e) Q +1 N and 1 N P (e) N. By transitivity, N P (e) N. Exaples of social ordering functions satisfying all the axios but: 1. Strong Pareto. Copare the welfare-level vectors using the lexicographic iniax criterion: a distribution is at least as good as another one if its axiu is lower, or they are equal and the second highest value is lower, or... or they are all equal. 26

2. Nested Preferences Transfer. Let R be defined by: for all e D, N, N R+ N, if (Ê( i, t i, R i, t)) i N > L (Ê( i, t i, R i, t)) i N and there exist k, k N with t k t k such that 0 < Ê( k, t k, R k, t) < Ê( k, t i, R i, t) < Ê( k, t k, R k, t) < Ê( k, t i, R i, t), with Ê( i, t i, R i, t) = Ê( i, t i, R i, t) for all i k, k, then N I(e) N. In all other cases N P (e) N. If then N I(e) N. (Ê( i, t i, R i, t)) i N = L (Ê( i, t i, R i, t)) i N 3. Equal Preferences Perutation. Let R be defined by: for all e D, N, N R+ N, if or (Ê( i, t i, R i, t)) i N > L (Ê( i, t i, R i, t)) i N (Ê( i, t i, R i, t)) i N = L (Ê( i, t i, R i, t)) i N and there exist k, k N with t k t k such that Ê( k, t k, R k, t) < Ê( i, t i, R i, t) for all i k; Ê( k, t k, R k, t) < Ê( i, t i, R i, t) for all i k, then N P (e) N. In all other cases N I(e) N. 4. t-equal Talent Transfer. Let R be defined by: for all e D, N, N R + N, if (Ê( i, t i, R i, t)) i N > L (Ê( i, t i, R i, t)) i N and there exist k, k N with R k R k such that Ê( k, t k, R k, t) < Ê( k, t i, R i, t) < Ê( k, t k, R k, t) < Ê( k, t i, R i, t), with Ê( i, t i, R i, t) = Ê( i, t i, R i, t) for all i k, k and U(( k, t k ), R k ) L(( k, t k ), R k ), then N I(e) N. In all other cases N P (e) N. If then N I(e) N. (Ê( i, t i, R i, t)) i N = L (Ê( i, t i, R i, t)) i N 27