## Abstract

This paper explores how to rank social allocations when individuals have other-regarding preferences (ORPs). Unlike the few existing studies on this issue, we focus on two different private goods, only one of which generates ORPs concerns. Specifically, individuals exhibit other-regarding views about the social health state but have standard self-centered preferences over other goods, namely consumption. Our social evaluation also incorporates a fairness view that aims to reduce inequalities that originate from factors for which individuals should not be deemed responsible. By resorting to a *non-resourcist* approach, we derive social preferences that seek to reduce individual well-being inequalities. Such differences are assessed by means of an interpersonal comparable measure that is related to an ideal situation which involves neither externalities nor unfair inequalities. We obtain that the use of the state of perfect health as the reference value leads society to give a higher priority to those who exhibit more altruistic preferences.

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## Notes

- 1.
- 2.
Winkelmann and Winkelmann (1995) and Mendolia (2014) obtain similar subjective well-being losses for the case of partner’s unemployment, and moreover that they clearly exceed the pecuniary losses. These authors also argue that such a gender asymmetry shows the traditional role distribution within the household.

- 3.
A similar way of introducing relative views in a framework in which consumption is the only good in the economy is presented by Treibich (2019).

- 4.
Although different, this property bears some resemblance to the principles that impose well-being lower and upper bounds (e.g., Maniquet and Sprumont 2005).

- 5.
The \(\lambda \)-relative equivalent for such agents would be a positive value that may go to \(+\infty \).

- 6.
Where \(Z^{n}(h^{*})\) denotes the set of allocations in which \(h_{i}=h^{*}\) for all \(i\in N\). These allocations can always be constructed since any individual’s level of health, that is \(h^{*}\), cannot be larger than the aggregate state in society.

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## Acknowledgements

I would like to thank Rafael Treibich, Mar Calo and the participants to seminars and conferences in Seville, Odense, Bilbao and Alicante for their comments. Financial support from the Spanish Ministry of Economy and Competitiveness (ECO2017-83069-P and ECO2014-57413-P) is gratefully acknowledged. The usual disclaimer applies.

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## Appendix

### Appendix

### Proof of Proposition 1

For any \(e\in {\mathcal {E}}\), let us assume an allocation \(z_{N}\in Z^{n}\). Because of *Consumption Desirability*, each individual \(i\in N\) prefers a strictly positive level of consumption than no consumption at all, whatever the others’ situation is. Therefore, \(((h_{i},c),\overline{h}_{z})P_{i}\left( (h^{*},0),h^{*}\right) \) for any \(c>0\). Hence, due to monotonicity in one’s own consumption \(\lambda _{i}(z_{N})\) cannot be negative.

As regards the existence of \(\lambda _{i}(z_{N})\), because of continuity, monotonicity and the previous result there exists \(\widehat{\lambda }\in {\mathbb {R}}_{+}\) such that for all \(\lambda \le \widehat{\lambda }\) we have that \(z_{N}R_{i}\left( (h^{*},\lambda c^{*}_{e}(\widehat{R}_{i})),h^{*} \right) \). Moreover, because of *Altruism*, monotonicity and the choice of \(h^{*}\) as the reference value it is always possible to find \(\widetilde{\lambda }\in {\mathbb {R}}_{++}\) such that \(\left( (h^{*},\lambda c^{*}_{e}(\widehat{R}_{i})),h^{*} \right) R_{i}z_{N}\) for all \(\lambda \ge \widetilde{\lambda }\). Since both \(\widehat{\lambda }\) and \(\widetilde{\lambda }\) exist for any \(i\in N\), and given that \({\mathbb {R}}_{+}\) is a connected space, by continuity there exits \(\lambda _{i}(z_{N})\in {\mathbb {R}}_{+}\) such that \(z_{N}I_{i}\left( (h^{*},\lambda _{i}(z_{N})c^{*}_{e}(\widehat{R}_{i})),h^{*} \right) \).

Finally, let us assume that for some individual \(i\in N\) there exist \(\lambda _{i}(z_{N}),\lambda '_{i}(z_{N})\in {\mathbb {R}}_{+}\) such that \(\lambda _{i}(z_{N})>\lambda '_{i}(z_{N})\). Because of monotonicity in one’s own consumption \(\left( (h^{*},\lambda _{i}(z_{N})c^{*}_{e}(\widehat{R}_{i})),h^{*} \right) P_{i} \left( (h^{*},\lambda '_{i}(z_{N})c^{*}_{e}(\widehat{R}_{i})),h^{*} \right) \). Therefore, the individual is not indifferent between these two allocations, and hence \(\lambda _{i}(z_{N})\) and \(\lambda '_{i}(z_{N})\) cannot both at the same time represent her preferences over the allocation \(z_{N}\in Z^{n}\).

Consequently, the \(\lambda \)-relative equivalent \(\lambda _{i}(z_{N})\) is well-defined for all agents, that is, this value exists and it is positive and unique.

### Proof of Theorem 1

The proof of Theorem 1 is based on the characterisation of the leximin criterion proposed by Hammond (1976), and it is split in four steps. The version of the proof that we present here relies on the assumption that the population can vary.

*Step 1: Indifference of *\({\mathbf{R }}^{\lambda }_{\mathrm{lex}}\) For any economy \(e\in {\mathcal {E}}\), let us assume two allocations \(z_{N},z'_{N}\in Z^{n}\) and two individuals \(j,k\in N\) such that, without loss of generality, \(\lambda _{k}(z'_{N})=\lambda _{j}(z_{N})<\lambda _{k}(z_{N})=\lambda _{j}(z'_{N})<1\), and \(\lambda _{i}(z_{N})=\lambda _{i}(z'_{N})\) for all \(i\ne j,k\). When the values of the \(\lambda \)-relative equivalents are different to the ones proposed here the same proof can be induced by means of *Perfect Consumption Scale Independence*. We will show that under such conditions \(z_{N}\mathbf{I }(e)z'_{N}\). Opposite to the desired result, let us assume that \(z_{N}\mathbf{P }(e)z'_{N}\).

Due to the properties of the ORPs it is possible to define two perfect health allocations \(\widehat{z}_{N},\widehat{z}'_{N}\in Z^{n}(h^{*})\)^{Footnote 6} such that for all \(i\in N\):

Note that since \(\lambda _{i}(z_{N})\) is a valid measure of the individual well-being, one has that \(\widehat{z}_{i}=\widehat{z}'_{i}\) for all \(i\ne j,k\). Hence, if we apply *Strong Pareto* (which implies *Pareto Indifference*) we have that \(\widehat{z}_{N}\mathbf{P }(e)\widehat{z}'_{N}\).

Let us now introduce two additional individuals *a* and *b* such that \(R_{a}=R_{j}\in {\mathcal {R}}\) and \(R_{b}=R_{k}\in {\mathcal {R}}\). Let us also assume, with \(\beta >1\), the following levels of consumption (see Fig. 3a):

According to the assumption \(z_{N}\mathbf{P }(e)z'_{N}\), by *Strong Pareto* and *Consistency* we obtain that \((\widehat{z}_{N},(h^{*},c_{a}),(h^{*},c_{b}))\mathbf{P }(e)(\widehat{z}'_{N},(h^{*},c_{a}),(h^{*},c_{b}))\). By *Priority Among Equals* twice we obtain the social relation \((\widehat{z}'_{N},(h^{*},c'_{a}),(h^{*},c'_{b}))\mathbf{I }(e)(\widehat{z}_{N},(h^{*},c_{a}),(h^{*},c_{b}))\). By *Transitivity* and *Consistency*\(((h^{*},c'_{a}),(h^{*},c'_{b}))\mathbf{P }(e)((h^{*},c_{a}),(h^{*},c_{b}))\). Finally, by *Perfect Consumption Scale Independence* we conclude that \(((h^{*},c'''_{a}),(h^{*},c'''_{b}))\mathbf{P }(e)((h^{*},c''_{a}),(h^{*},c''_{b}))\). However, using *Well-being Bound Transfer* we obtain that \(((h^{*},c''_{a}),(h^{*},c''_{b}))\mathbf{I }(e)((h^{*},c'''_{a}),(h^{*},c'''_{b}))\), which yields the desired contradiction.

*Step 2: Strict preference of*\({\mathbf{R }}^{\lambda }_{\mathrm{lex}}\) For any \(e\in {\mathcal {E}}\), let us assume two allocations \(z_{N},z'_{N}\in Z^{n}\) and two individuals \(j,k\in N\) such that, without loss of generality, \(\lambda _{j}(z'_{N})<\lambda _{k}(z_{N})<\lambda _{j}(z_{N})<\lambda _{k}(z'_{N})\), and \(\lambda _{i}(z_{N})=\lambda _{i}(z'_{N})\) for all \(i\ne j,k\). First, we need to prove that it must be the case that \(z_{N}\mathbf{P }(e)z'_{N}\). Opposite to the desired result, let us assume that \(z'_{N}\mathbf{R }(e)z_{N}\).

Due to the basic axioms of the ORPs it is possible to define two perfect health allocations \(\widehat{z}_{N},\widehat{z}'_{N}\in Z^{n}(h^{*})\) such that for all \(i\in N\):

Once again, \(\widehat{z}_{i}=\widehat{z}'_{i}\) for all \(i\ne j,k\), and hence \(\widehat{z}'_{N}\mathbf{R }(e)\widehat{z}_{N}\) by *Strong Pareto*.

To obtain our result we distinguish between two possible scenarios according to the value of \(\lambda _{j}(z'_{N})\). We first assume \(\lambda _{j}(z'_{N})<1\). Let us introduce two individuals *a* and *b* with ORPs \(R_{a}=R_{j}\) and \(R_{b}=R_{k}\). Let us also assume, with \(\beta ,\varepsilon \in {\mathbb {R}}_{++}\), the following levels of consumption:

Because of continuity and *Perfect Consumption Scale Independence*, such bundles can always be found (see Fig. 3b). Given the initial assumption \(z_{N}\mathbf{R }(e)z'_{N}\), by *Strong Pareto*\(\widehat{z}'_{N}\mathbf{R }(e)\widehat{z}_{N}\). By *Perfect Consumption Scale Independence*\(((h^{*},\beta \widehat{c}'_{i})_{i\ne j,k},(h^{*},c''_{j})\), \((h^{*},c''_{k}))\mathbf{R }(e)((h^{*},\beta \widehat{c}_{i})_{i\in N\setminus \{j,k\}},(h^{*},c'''_{j}),(h^{*},c'''_{k}))\). By *Consistency* it is straightforward to check that \(((h^{*},c''_{j}),(h^{*},c''_{k}),(h^{*},c_{a}),(h^{*},c_{b}))\)\(\mathbf{R }(e)((h^{*},c'''_{j}),(h^{*},c'''_{k}),(h^{*},c_{a}),(h^{*},c_{b}))\). If we apply *Priority Among Equals* twice we have \(((h^{*},c''''_{j}),(h^{*},c'''_{k}),(h^{*},c'_{a}),(h^{*},c'_{b}))\)\(\mathbf{R }(e)((h^{*},c''_{j}),(h^{*},c''_{k}),(h^{*},c_{a}),(h^{*},c_{b}))\). By *Strong Pareto* we obtain that \(((h^{*},c'''_{j}),(h^{*},c'''_{k}),(h^{*},c'_{a}),(h^{*},c'_{b}))\)\(\mathbf{P }(e)((h^{*},c''''_{j}),(h^{*},c'''_{k}),(h^{*},c'_{a}),(h^{*},c'_{b}))\). Finally, by *Transitivity*\(((h^{*},c'''_{j}),(h^{*},c'''_{k}),(h^{*},c'_{a}),(h^{*},c'_{b}))\)\(\mathbf{P }(e)((h^{*},c'''_{j}),(h^{*},c'''_{k}),(h^{*},c_{a}),(h^{*},c_{b}))\). However, by *Well-being Bound Transfer* axiom we obtain that \(((h^{*},c'''_{j}),(h^{*},c'''_{k}),(h^{*},c_{a}),(h^{*},c_{b})\)\(\mathbf{R }(e)((h^{*},c'''_{j}),(h^{*},c'''_{k}),(h^{*},c'_{a}),(h^{*},c'_{b}))\), which yields the desired contradiction.

Let us now derive a similar contradiction for \(\lambda _{j}(z'_{N})\ge 1\). To do so we assume, once again, two individuals *a* and *b* with ORPs \(R_{a}=R_{j}\) and \(R_{b}=R_{k}\). Let us also assume, with \(\beta ,\varepsilon \in {\mathbb {R}}_{++}\), the following levels of consumption:

One again, by *Strong Pareto*\(\widehat{z}'_{N}\mathbf{R }(e)\widehat{z}_{N}\). By *Consistency*\(((h^{*},\widehat{c}'_{j}),(h^{*},\widehat{c}'_{k}),(h^{*},c''_{a}),(h^{*},c''_{b}))\)\(\mathbf{R }(e)((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}_{k}),(h^{*},c''_{a}),(h^{*},c''_{b}))\). By *Priority Among Equals* we obtain that \(((h^{*},c''_{j}),(h^{*},\widehat{c}'_{k}),(h^{*},c'''_{a}),(h^{*},c''_{b}))\)\(\mathbf{R }(e)((h^{*},\widehat{c}'_{j}),(h^{*},\widehat{c}'_{k}),(h^{*},c''_{a}),(h^{*},c''_{b}))\), and moreover by *Strong Pareto* we also have \(((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}'_{k}),(h^{*},c'''_{a}),(h^{*},c''_{b}))\)\(\mathbf{P }(e)((h^{*},c''_{j}),(h^{*},\widehat{c}'_{k}),(h^{*},c'''_{a}),(h^{*},c''_{b}))\). Applying *Priority Among Equals* once more we obtain that \(((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}_{k}),(h^{*},c'''_{a}),(h^{*},c'''_{b}))\)\(\mathbf{R }(e)((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}'_{k}),(h^{*},c'''_{a}),(h^{*},c''_{b}))\). Finally, by Transitivity \(((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}_{k}),(h^{*},c'''_{a}),(h^{*},c'''_{b}))\)\(\mathbf{P }(e)((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}_{k}),(h^{*},c''_{a}),(h^{*},c''_{b}))\). However, by *Well-being Bound Transfer* axiom we obtain that \(((h^{*},c_{a}),(h^{*},c_{b}))\mathbf{R }(e)((h^{*},c'_{a}),(h^{*},c'_{b}))\), which by *Perfect Consumption Scale Independence* leads to \(((h^{*},c''_{a}),(h^{*},c''_{b}))\mathbf{R }(e)((h^{*},c'''_{a}),(h^{*},c'''_{b}))\). By *Consistency* we get the desired contradiction \(((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}_{k}),(h^{*},c''_{a}),(h^{*},c''_{b}))\)\(\mathbf{R }(e)((h^{*},\widehat{c}_{j}),(h^{*},\widehat{c}_{k}),(h^{*},c'''_{a}),(h^{*},c'''_{b}))\).

In the final part of this second step of the proof we show that for any pair of allocations \(z_{N},z'_{N}\in Z^{n}\) such that \(\min _{i\in N}\lambda _{i}(z_{N})>\min _{i\in N}\lambda _{i}(z'_{N})\), social preferences over these two allocations are characterised by \(z_{N}\mathbf{P }(e)z'_{N}\). Let us then assume \(z_{N},z'_{N}\in Z^{n}\) with \(\min _{i\in N}\lambda _{i}(z_{N})>\min _{i\in N}\lambda _{i}(z'_{N})\). Due to the properties of the ORPs it is possible to define two perfect health allocations \(x_{N},x'_{N}\in Z^{n}(h^{*})\) such that for all \(i\in N\) we have \(\lambda _{i}(z_{N})>\lambda _{i}(x_{N})\) and \(\lambda _{i}(x'_{N})>\lambda _{i}(z'_{N})\). Additionally, let \(i_{0}\in N\) be such that \(\lambda _{i_{0}}(z'_{N})=\min _{i\in N}\lambda _{i}(z'_{N})\), where

Let \(Q=N\setminus \{i_{0}\}\), and let us define a sequence of allocations \((x^{q}_{N})_{1\le q\le \mid Q\mid +1}\) such that for all \(i\ne i_{0}\)

and

This sequence implies that \(\lambda _{q}(x^{q}_{N})>\lambda _{q}(x^{q+1}_{N})>\lambda _{i_{0}}(x^{q+1}_{N})>\lambda _{i_{0}}(x^{q}_{N})\), with \(\lambda _{i}(x^{q}_{N})=\lambda _{i}(x^{q+1}_{N})\) for all \(i\ne q,i_{0}\). Consequently, and as we have previously proved, it must be the case that \(x^{q+1}_{N}\mathbf{P }(e)x^{q}_{N}\) for all \(q\in Q\). Moreover, by construction of *x* and \(x'_{N}\) we have both \(z_{N}\mathbf{P }(e)x^{\mid Q\mid +1}_{N}\) and \(x^{1}_{N}\mathbf{P }(e)z'_{N}\), and hence by *Transitivity* we finally get that \(z_{N}\mathbf{P }(e)z'_{N}\).

*Step 3: Lexicographic order of*\({\mathbf{R }}^{\lambda }_{\mathrm{lex}}\)We extend the previous result in order to meet the lexicographic criterion, that is, we show that whenever \((\lambda _{i}(z_{N}))_{i\in N}\ge _{\mathrm{lex}}(\lambda _{i}(z_{N}))_{i\in N}\), with at least one strict inequality, then \(z_{N}\mathbf{P }(e)z'_{N}\). Without loss of generality, let us assume that there exist \(j,k\in N\) such that \(\min _{i\in N}\lambda _{i}(z_{N})=\lambda _{j}(z_{N})=\lambda _{k}(z'_{N})=\min _{i\in N}\lambda _{i}(z'_{N})<1\). If the value of this minimum equivalent is different to the one proposed here, we can construct the allocations by means of *Strong Pareto* and *Perfect Consumption Scale Independence*. Additionally, let us assume that \(\min _{i\in N\setminus \{j\}}\lambda _{i}(z_{N})=\lambda _{k}(z_{N})>\lambda _{j}(z'_{N})=\min _{i\in N\setminus \{k\}}\lambda _{i}(z'_{N})\). The fact that these values are given by agents *j* and *k* can always be generated by Step 1 and *Strong Pareto*. To obtain the desired result we need to distinguish between four different cases.

Case 1: \(\lambda _{j}(z'_{N})<\lambda _{k}(z_{N})<1\). By *Strong Pareto* we have that \(z^{*}_{N}\mathbf{I }(e)z_{N}\), where \(z^{*}_{N}\in Z^{n}(h^{*})\) is the allocation in which for every individual *i* we have both \(h_{i}=h^{*}\) and \(z_{N}^{*}I_{i}z_{N}\). Likewise, we obtain \(z^{*'}_{N}\mathbf{I }(e)z'_{N}\). Let us now include one additional agent *a* who shares ORPs with individual *j*, that is \(R_{a}=R_{j}\in {\mathcal {R}}\), and who enjoys a health-consumption bundle \(z_{a}^{*}=(h^{*},c^{*}_{a})\in Z\) such that \(c^{*}_{a}>c^{*}_{e}(\widehat{R}_{j})\). According to *Consistency* and *Well-being Bound Transfer* we establish that \(({z}^{*'}_{N\setminus \{k\}},{z}^{*''}_{k},{z}^{*'}_{a})\mathbf{I }(e)({z}^{*'}_{N},{z}^{*}_{a})\), where \({z}^{*''}_{k},{z}^{*'}_{a}\in Z\) are the (perfect health) bundles that satisfy, respectively, \(c^{*''}_{k}/c^{*}_{e}(\widehat{R}_{k})=c^{*}_{a}/c^{*}_{e}(\widehat{R}_{j})>1\) and \(c^{*'}_{k}/c^{*}_{e}(\widehat{R}_{k})=c^{*'}_{a}/c^{*}_{e}(\widehat{R}_{j})<1\). By *Priority Among Equals* we get that \(({z}^{*'}_{N\setminus \{j,k\}},{z}^{*}_{j},{z}^{*''}_{k},{z}^{*''}_{a})\mathbf{I }(e)({z}^{*'}_{N\setminus \{k\}},{z}^{*''}_{k},{z}^{*'}_{a})\), where \({z}^{*''}_{a}={z}^{*'}_{j}\in Z\). If we apply *Well-being Bound Transfer* once again we have that \(({z}^{*'}_{N\setminus \{j,k\}},{z}^{*}_{j},{z}^{*'''}_{k},{z}^{*}_{a})\mathbf{I }(e)({z}^{*'}_{N\setminus \{j,k\}},{z}^{*}_{j},{z}^{*''}_{k},{z}^{*''}_{a})\), where \({z}^{*'''}_{k}\in Z\) is the bundle that satisfies \(c^{*'''}_{k}/c^{*}_{e}(\widehat{R}_{k})=c^{*''}_{a}/c^{*}_{e}(\widehat{R}_{j})<1\). According to the values of the \(\lambda \)-relative equivalents and *Strong Pareto*, we know from Step 2 that \({z}^{*}_{N\setminus \{j\}}\mathbf{P }(e)({z}^{*'}_{N\setminus \{j,k\}},{z}^{*'''}_{k})\). If we apply *Consistency* and *Transitivity* we get that \(({z}^{*}_{N},{z}^{*}_{a})\mathbf{P }(e)({z}^{*'}_{N\setminus \{j,k\}},{z}^{*}_{j},{z}^{*'''}_{k},{z}^{*}_{a})\mathbf{I }(e)({z}^{*'}_{N},{z}^{*}_{a})\). By means of *Consistency* and *Transitivity* we reach the desired result \(z_{N}\mathbf{P }(e)z'_{N}\).

Case 2: \(\lambda _{k}(z_{N})=1>\lambda _{j}(z'_{N})\) or \(\lambda _{k}(z_{N})>1>\lambda _{j}(z'_{N})\). The final result can be easily obtained by using the same line of reasoning that we have applied in the previous case.

Case 3: \(\lambda _{k}(z_{N})>\lambda _{j}(z'_{N})>1\). By *Strong Pareto* we have both \(z^{*}_{N}\mathbf{I }(e)z_{N}\) and \(z^{*'}_{N}\mathbf{I }(e)z'_{N}\), where \(z^{*}_{N}\) and \(z^{*'}_{N}\) are constructed exactly as before. According to the values of the \(\lambda \)-relative equivalents and *Strong Pareto* we have \({z}^{*}_{N\setminus \{j\}}\mathbf{P }(e)({z}^{*'}_{N\setminus \{j,k\}},{z}^{*'''}_{k})\), where \({z}^{*'''}_{k}\in Z\) is the bundle that satisfies \(c^{*'''}_{k}/c^{*}_{e}(\widehat{R}_{k})=c^{*'}_{j}/c^{*}_{e}(\widehat{R}_{j})>1\), and by *Consistency* we get that \({z}^{*}_{N}\mathbf{P }(e)({z}^{*'}_{N\setminus \{j,k\}},{z}^{*}_{j},{z}^{*'''}_{k})\). Given that \(\lambda _{j}(z^{*}_{N})=\lambda _{k}(z^{*'}_{N})<1\) and \(\lambda _{j}(z^{*'}_{N})=\lambda _{k}(z^{*'''}_{N})>1\), by *Well-being Bound Transfer* we obtain that \(({z}^{*'}_{N\setminus \{j,k\}},{z}^{*}_{j},{z}^{*'''}_{k})\mathbf{I }(e){z}^{*'}_{N}\). Again, by means of *Strong Pareto* and *Transitivity* we reach the desired result \(z_{N}\mathbf{P }(e)z'_{N}\).

Case 4: \(\lambda _{k}(z_{N})>\lambda _{j}(z'_{N})=1\). Once more, we construct \(z^{*}_{N},z^{*'}_{N}\in Z^{n}\). Next, let us assume alternative allocations \(z^{*''}_{N},z^{*'''}_{N}\in Z^{n}\) such that for all \(i\in N\) we have that \(h_{i}=h^{*}\), and moreover \(c^{*''}_{i}=\beta c^{*}_{i}\) and \(c^{*'''}_{i}=\beta c^{*'}_{i}\), with \(\beta \in (0,1)\). Because of continuity it is possible to find such allocations with \(\lambda _{k}(z^{*''}_{N})>1\) and \(\lambda _{j}(z^{*'''}_{N})<1\). From previous results we know that \(z^{*''}_{N}\mathbf{P }(e)z^{*'''}_{N}\). Using *Perfect Consumption Scale Independence* we get that \(z^{*}_{N}\mathbf{P }(e)z^{*'}_{N}\), and then by *Strong Pareto* and *Transitivity* we reach again the desired result \(z_{N}\mathbf{P }(e)z'_{N}\).

*Step 4: Independence of the SO axioms* To conclude the proof we additionally show that the five normative requirements are independent, and hence if one of them is dropped it is possible to find a SOF which does not satisfy Theorem 1.

- (1)
Drop

*Strong Pareto*Take \(\mathbf{R }\) defined as the application of the lexicographic minimax criterion over \((\lambda _{i}(z_{N}))_{i\in N}\). This rule establishes that a group of objects \(a_{N}\in {\mathbb {R}}^{n}\) dominates any other group \(b_{N}\in {\mathbb {R}}^{n}\) if the highest value in \(a_{N}\) is lower than the highest value in \(b_{N}\). If they are identical, then society eliminates these maximal elements and compares the highest values in the reduced allocations, and so on. The fact that \(\mathbf{R }\) satisfies*Consistency*and*Perfect Consumption Scale Independence*is straightforward. According to*Well-being Bound Transfer*, social welfare does not decrease when we move from allocation \(z'_{N}\) to \(z_{N}\) such that \(\lambda _{j}(z'_{N})>\lambda _{j}(z_{N})>1>\lambda _{k}(z_{N})>\lambda _{k}(z'_{N})\). Therefore, this axiom is also satisfied by \(\mathbf{R }\). The case of*Priority Among Equals*is shown in a similar way. - (2)
Drop

*Consistency*Consider \({\mathbf{R }}^{\lambda ^{+}}_{\mathrm{lex}}\) defined exactly as \({\mathbf{R }}^{\lambda }_{\mathrm{lex}}\) but assuming, for all \(i\in N\), that \(c^{*}_{e}(\widehat{R}_{i})\) is constructed using \(m^{+}\) instead of \(m^{-}\). Let \(\mathbf{R }\) now coincide with \({\mathbf{R }}^{\lambda ^{+}}_{\mathrm{lex}}\) if there exists \(i\in N\) such that \(m_{i}=m^{+}\), and with \({\mathbf{R }}^{\lambda }_{\mathrm{lex}}\) otherwise. With the exception of*Consistency*, all axioms are satisfied by \(\mathbf{R }\) since it is constructed exactly as \({\mathbf{R }}^{\lambda }_{\mathrm{lex}}\). - (3)
Drop

*Perfect Consumption Scale Independence*Take \(\mathbf{R }\) defined as the application of the lexicographic minimax criterion over \((\rho _{i}(z_{N}))_{i\in N}\), where \(\rho _{i}(z_{N})=c^{*}_{e}(\widehat{R}_{i})(1-\lambda _{i}(z_{N}))\), for all \(i\in N\). The fact that \(\mathbf{R }\) satisfies*Strong Pareto*and*Consistency*is straightforward. According to*Well-being Bound Transfer*, social welfare does not decrease when we move from allocation \(z'_{N}\) to \(z_{N}\) such that \(c^{*}_{e}(\widehat{R}_{j})(1-\lambda _{j}(z'_{N}))>c^{*}_{e}(\widehat{R}_{j})(1-\lambda _{j}(z_{N}))>0>c^{*}_{e}(\widehat{R}_{k})(1-\lambda _{k}(z_{N}))>c^{*}_{e}(\widehat{R}_{k})(1-\lambda _{k}(z'_{N}))\). Therefore, this axiom is also satisfied by \(\mathbf{R }\). The case of*Priority Among Equals*is shown in a similar way. - (4)
Drop

*Priority Among Equals*Let \(\mathbf{R }\) be defined such that for all \(e\in {\mathcal {E}}\) and \(z_{N},z'_{N}\in Z^{n}\), \(z_{N}\mathbf{R }(e)z'_{N}\) if \(\sum _{i\in N}\mu _{i}(z_{N})\ge \sum _{i\in N}\mu _{i}(z'_{N})\), where \(\mu _{i}(z_{N})=\lambda _{i}(z_{N})c^{*}_{e}(\widehat{R}_{i})\). The fact that \(\mathbf{R }\) satisfies*Strong Pareto*,*Consistency*and*Perfect Consumption Scale Independence*is straightforward. Note that \(\mu _{i}(z_{N})\) is the individual*i*’s level of consumption when she has perfect health. Therefore, since*Well-being Bound Transfer*is defined by assuming balanced transfers this axiom is also satisfied by \(\mathbf{R }\). In other words, no transfer reverses the social preferences. - (5)
Drop

*Well-being Bound Transfer*Take \(\mathbf{R }\) defined such that for all \(e\in {\mathcal {E}}\) and \(z_{N},z'_{N}\in Z^{n}\), \(z_{N}\mathbf{R }(e)z'_{N}\) if \((\mu _{i}(z_{N}))_{i\in N}\ge _{\mathrm{lex}}(\mu _{i}(z'_{N}))_{i\in N}\), where \(\mu _{i}(z_{N})\) is defined as in the previous case. The fact that \(\mathbf{R }\) satisfies all the four remaining axioms is straightforward. Note that*Perfect Consumption Scale Independence*changes \(\mu _{i}(z_{N})\), that is, the individual*i*’s level of consumption when she has perfect health, in the same proportion for all individuals, and hence the lexicographic ranking does not change.

Consequently, any SOF that satisfies *Strong Pareto*, *Consistency*, *Perfect Consumption Scale Independence*, *Priority Among Equals* and *Well-being Bound Transfer* is grounded on the application of the leximin criterion over the \(\lambda \)-relative equivalents. Other social rankings may also satisfy all these requirements, although the combination of the five axioms single out \({\mathbf{R }}^{\lambda }_{\mathrm{lex}}\).

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### Cite this article

Calo-Blanco, A. Health and fairness with other-regarding preferences.
*Rev Econ Design* (2020). https://doi.org/10.1007/s10058-020-00235-2

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### Keywords

- Health
- Fairness
- Other-regarding preferences
- Social ordering function

### JEL Classification

- D62
- D63
- D71
- I14