Marc Fleurbaey

Gregory Ponthiere
email: gregory.ponthiere@ens.fr

The Value of a Life-Year and the Intuition of Universality

Keywords: value of life, quality of life, social evaluation, universality, complementarity value of life, quality of life, social evaluation, universality, complementarity. JEL codes: I30, I38, J17, J18

The Value of a Life-Year and the Intuition of Universality

Marc Fleurbaey, Grégory Ponthière

Introduction

As highlighted by [START_REF] Sen | On the development of basic income indicators to supplement GNP measures[END_REF] in his early criticism of national accounts statistics, the measurement of the achievements of a society can hardly abstract from how long the members of that society live. Individual lifetime is a key dimension to be taken into account in the measurement of social achievements, whatever one wants to measure the degree of development of a society, or to measure social welfare. That intuition motivated the inclusion of indicators of human lifetimesuch as period life expectancy at birthin the construction of indexes of development, such as the Human Development Index (see UNDP 1990UNDP , 2010)), and in the construction of inclusive indexes of well-being, such as the equivalent income (see [START_REF] Fleurbaey | Beyond GDP. Measuring Welfare and Assessing Sustainability[END_REF]. Moreover, since various public policies modify the production and the distribution of life-years within a population (e.g. health care programs, transportation policies and environmental policies), the lifetime dimension can hardly be ignored when considering the design of policies (see [START_REF] Sen | Mortality as an indicator of economic success and failure[END_REF], Broome 1999[START_REF] Broome | Weighing Lives[END_REF].

Taking human lifetime into account requires the social evaluator or the government to find a way to weight the quantity of life-years against other relevant dimensions of life. In other terms, the social evaluator needs to find a way to assign a value to life-years. Assigning a value to life-years is necessary not only for the measurement of human development or well-being, but, also, to be able to solve policy dilemmas involving various implications in terms of the production and distribution of life-years within a population. This necessity to assign a value to life-years has given rise, in economics, to the increasingly large literature on the value of a statistical life (see Jones-Lee 1989). 3 Empirical estimates of the value of a statistical life have become key parameters in cost-benefit analyses, as well as in the study of optimal policies, in particular in the context of climate change.

The empirical literature on the value of a statistical life has a purely positive nature: it aims at quantifying how individuals tend, in real life, to solve trade-offs involving risk about the duration of life, or, in other words, how individuals are willing to exchange money against variations of the risk of death. Empirical studies of various kinds (wage-risk studies, contingent valuation methods, etc.) show that the value of a statistical life is varying with several factors, such as the age or the occupation, and is increasing with income. For instance, [START_REF] Miller | Variations between countries in values of statistical life[END_REF] provides, in his meta-analysis of empirical studies, a rule of thumb for the valuation of a life, which is a simple linear transformation of the GDP per capita. 4Obviously, it does not logically follow from those empirical studies that, when measuring social achievements or designing policy objectives, the social evaluator should make the valuation of life or the valuation of a life-year dependent on income or on any of those factors. Empirical studies on the value of life may well show that the value of a life-year is increasing with income, but such a positive premise cannot, if taken separately, lead to any normative corollary. Drawing such a conclusion would be nothing else than an occurrence of a naturalistic fallacy. There is thus a need to examine, at the normative level, how a social evaluator should value an extra life-year, that is, the principles that should govern such a valuation of life-years.

From a normative perspective, the valuation of a life-year leads to two conflicting intuitions: on the one hand, the intuition of universality; on the other hand, the intuition of complementarity.

According to the intuition of universality, the value of a life-year should be universal. The value of a life-year should be the same whatever the contexts considered; in particular, a life-year should have exactly the same value, whatever we consider a poor or a rich country. Thus, from that perspective, the fact that life-years are more valued in richer countries than in poorer countries should be totally irrelevant when considering the social valuation of those life-years: universalism requires to value life-years in the same way independently from the context, and, in particular, independently from the associated quality of life.

Such an intuition of universality concerning life-year valuations was defended, among others, by [START_REF] Anand | Recasting Human Development Measures[END_REF], who criticized the new HDIbased on a geometric average rather than on an arithmetic average across income, lifetime and education dimensionson the grounds that it violates this intuition of universal valuation of life-years. 5 According to [START_REF] Anand | Recasting Human Development Measures[END_REF], the value of an extra life-year should be the same, whatever one considers a rich or a poor country. That intuition of universality is satisfied by the standard HDI, but not by the multiplicative HDI.

This intuition of universality conflicts with another intuition, which can be called the intuition of complementarity. According to that intuition, the quantity of life cannot be valued independently from the quality of life. The reason lies in the singular nature of lifetime in comparison to other goods. Lifetime is not a good like a standard good, which could be enjoyed on its own. On the contrary, lifetime is like a "container", whose value depends on what it will allow for, that is, on what lies "within the container" (life-events, activities, lifeprojects, etc.). As a consequence, the valuation of life-years cannot be made independently from the associated quality of life. A corollary of this is that, when the quality of life varies, the value of the quantity of life cannot remain the same, and, hence, cannot be universal, in opposition to what the intuition of universality recommends.

The intuition of complementary can take two forms: on the one hand, a weak version, according to which the value of a life-year depends, among other things, on the quality of that life-year; on the other hand, a strong version, according to which the value of a life-year depends only on the quality of that life-year. Although the latter version is much stronger than the former, it finds some support in several simple thought experiments. For instance, if one could artificially slowdown life, but without modifying the number of life-events that take place in that life, then one could hardly regard this lengthening of life as valuable: enlarging the size of the "container" without modifying its content could hardly make a life better. Alternatively, consider another thought experiment, where one can shutdown life during one hour, and, by this, shorten all lives by that amount, without anyone being aware of that shutdown. As long as this temporary shutdown was unnoticed, and did not affect events among humans, it is hard to see how this reduction of the size of the "container" could have reduced the value of life. Those thought experiments provide some support for the strong version of the intuition of complementarity. But it is important to stress that even the weak version of that intuition is in conflict with the intuition of universality.

In order to better present the differences between the intuition of universality and the intuition of complementarity, it can also be useful to refer to the concept of intrinsic value of life, that is, the value of life per se, independently from the characteristics of that life. According to the intuition of universality, the value of a life-year is composed exclusively of its intrinsic value, and, thus, does not depend on the quality of that life-year (only the "container" has value). On the contrary, the weak version of the intuition of complementarity states that the value of a life-year may or may not have an intrinsic component, but, for sure, includes a component that is related to the quality of that life-year (what is inside the "container" matters, and possibly the "container" as well). Finally, the strong version of the intuition of complementarity states that the value of a life-year only includes a component that is related to its quality, and includes no intrinsic value component (only what is inside the "container" is valued, not the "container" itself).

The incompatibility between the intuition of universality and the intuition of complementarity raises deep challenges for the valuation of life-years. Although this is not often explicitly acknowledged, a large part of the literature on the valuation of life relies on the intuition of complementarity, and, as such, violates the intuition of universality. 6 Those violations are problematic only to the extent that the intuition of universality is worth being pursued. But the ethical appeal of the intuition of universality is hard to evaluate, simply because no precise account of that intuition has been given so far. As a consequence, it is difficult to have an idea of the precise implications of adopting such an intuition.

The goal of this paper is precisely to provide a more accurate account of the intuition of universality. In order to better understand what the intuition of universality is, we propose to study its implications for the valuation of life-years. For that purpose, this paper will proceed in two stages. In a first stage, we provide three distinct definitions of the intuition of universality, in terms of the constraints this intuition imposes on the form of value functions aimed at valuing life-years. That first approach informs us about the formal constraints that the intuition of universality imposes on the valuation of the quantity and the quality of life, but does not inform us, in general, about the priority to be given in allocation problems. Then, in a second stage, we consider a more general approach, in terms of social preferences, and reformulate those three accounts of the intuition of universality, in order to explore their consequences in terms of priority when considering problems of life-years allocations.

Anticipating on our results, we show that the three distinct accounts of the intuition of universality lead to counterintuitive implications from a normative perspective. One of these accounts is shown to be in conflict with a basic property of monotonicity, whereas two other accounts of the intuition of universality lead to indifference with respect to how life-years are distributed within the population, which is a quite negative result. Thus adopting a universal perspective on life-years valuationsand, thus, abstracting from the associated quality of those life-yearsleads to quite questionable consequences. Those negative results support the abandon of the intuition of universality. However, we show that abandoning the intuition of universality, and adopting instead the intuition of complementarity, does not prevent a social evaluator from giving priority, when allocating life-years, to individuals with the lowest quality of life.

This paper is related to several branches of the literature. First of all, this paper is connected to the philosophical literature on the normative foundations of the valuation of life. There is, in philosophy, an old tradition of thought dealing with the goodness of life and the evil of death, which dates back, at least, to the works of [START_REF] Epicurus | Letter to Menoeceus[END_REF]. In the 20 th century, Epicurus's viewdeath is not bad, since all good and evil consist in sensationwas rejected by [START_REF] Nagel | Death[END_REF], according to whom many sorts of evils are unexperienced, and, also, by [START_REF] Mcmahan | Death and the value of life[END_REF], who argued that the badness of a death depends on what would have happened if the person had not died. More recently, the philosophical foundations for the valuation of life-years have been reexamined in the works of [START_REF] Broome | Goodness is reducible to betterness: the evil of death is the value of life[END_REF]Broome ( , 1999[START_REF] Broome | Weighing Lives[END_REF]. Our paper is strongly connected with Broome's accounts of the value of life and life-years. Actually, there is a formal similarity between some arguments developed by [START_REF] Broome | Weighing Lives[END_REF] against the intuition of neutrality in the context of valuing the life of a person, and some of our arguments against the intuition of universality in the context of valuing a life-year. Secondly, this paper is also related to the economics literature on the measurement of human development, such as [START_REF] Ravallion | Troubling trade-offs in the Human Development Index[END_REF] and [START_REF] Anand | Recasting Human Development Measures[END_REF], who criticized the geometric HDI on the grounds of an intuition that is strongly related to the intuition of universality as explored in this paper. We actually show how some accounts of the intuition of universality can lead to a measure of development that has a similar structure to the standard (additive) HDI. Our paper is also related to multidimensional indicators of well-being inclusive of the lifetime dimension, such as the equivalent income (see [START_REF] Costa | Long term trends in health, welfare and economic growth in the United States[END_REF][START_REF] Nordhaus | The health of nations: the contributions of improved health to living standards[END_REF][START_REF] Becker | The quantity and the quality of life and the evolution of world inequality[END_REF][START_REF] Fleurbaey | Beyond GDP. Measuring Welfare and Assessing Sustainability[END_REF]. Thirdly, this paper is also related to the normative literature on fairness in the context of life and death, such as [START_REF] Fleurbaey | Prevention against equality[END_REF] and Adler et al (2014). The design of optimal policies is not independent from how lives are valued. This makes the distinction between the intuition of universality and the intuition of complementarity most relevant for policy purposes.

The rest of the paper is organized as follows. Section 2 present three distinct accounts of the intuition of universality and explores their consequences on the structural form of value functions. Then, Section 3 reformulates the three accounts of the intuition of universality in an alternative frameworkbased on social preference orderingsand explores their implications for the social ranking of distributions along the quantity and quality of life dimensions. Section 4 proposes simple ways to reconcile the abandon of the intuition of universality with the priority given to individuals with low qualities of life in problems of lifeyears allocation. Concluding remarks are left to Section 5.

Three accounts of the intuition of universality (1): an axiological approach

According to the intuition of universality, the valuation of a life-year is independent from the context under study, and, thus, independent from the associated quality of life. When stated in that way, that intuition is quite general, and is definitely in need of a more exact formulation, since the independence to which it refers may take various distinct forms. In this section, we propose three distinct accounts of the intuition of universality.

For that purpose, we adopt an axiological approach: our main object of study is a value function V(.), which is defined on a life, which is itself defined as a vector (a, b, …, k) whose entries a, b, …, k correspond to the quality of each life-year. This section explores the formal constraints that the intuition of universality imposes on the structure of the value function V(.). As such, this constitutes a first step towards a better understanding of the intuition of universality.

At the very outset, it should be stressed that a first, basic reading of the intuition of universality consists of stating that the value of a given life (x,…,x) in a country C should be exactly equal to the value of the same life in another country C'. That basic conception of the universality of the valuation of life-years is presented below, as the intuition U0.

The intuition of universality (U0)

V((x, …, x) country C ) = V((x, …, x) country C' )
The formulation U0 of the intuition of universality is quite intuitive: it makes a lot of sense to assume that the value of a life does not depend on the country where that life takes place. U0 is thus a quite intuitive property that a value function V(.) should satisfy.

Note, however, that the property U0 is not really original, because it coincides merely with a standard anonymity condition (see [START_REF] Blackorby | Population Issues in Social Choice Theory, Welfare Economics and Ethics[END_REF]. Anonymity being widely used in social evaluationswithout an explicit reference to the intuition of universalitywe believe that the property U0 does not exhaust what the intuition of universality is about, and, in some sense, does not suffice to do justice to the intuition of universality. Therefore we will, in the rest of this paper, take U0 for grantedbut as an anonymity conditionand explore the consequences of alternative formulations of the intuition of universality.

The intuition of universality can be formulated in terms of the variation of value induced by the addition of an extra life-year. One possible formulation of the intuition of universality consists of stating that the variation of value associated to the addition of an extra life-year should depend neither on the quality of the added life-year, nor on the quality of previous life-years, nor on the number of previous life-periods (duration of initial lives). That account of the intuition of universality can be formulated as follows.

The intuition of universality (U1)

V((a,…, g) + x) -V((a, …, g)) = V((h, …, m) + y) -V((h, …, m))
The left-hand side of the above equality is the variation in value when a life-year with a quality x is added to a life of quality (a, …, g), whereas the right-hand side is the variation in value when a life-year with quality y is added to a life of quality (h, …, m).

The formulation U1 of the intuition of universality states that the value of a life-year is universal, in the sense that it involves a triple independence: (i) independence with respect to the quality of the added life-year; (ii) independence with respect to the quality of previous life-years; (iii) independence with respect to the duration of initial lives to which a life-year is added (since the left-hand side and the right-hand-side of the above equality may involve initial lives of unequal lengths). As such, the formulation U1 captures a strong conception of universality.

To see how strong that conception of universality is, let us take the case of the addition of a life-year either in the United States, where average standards of living and life expectancy are high, or, alternatively, in Ghana, where standards of living and life expectancy are lower. The condition U1 states that the addition of a life-year in the United States (with U.S. standards of living) has exactly the same value as the addition of a life-year in Ghana (with Ghana's standards of living). That condition states also that adding an extra life-year in Ghana with U.S. standards of living has the same value as adding an extra-life year in the U.S. with Ghana's standards of living.

Although it may seem appealing at first glance, the formulation U1 of the intuition of universality has implications that are not attractive. In particular, the formulation U1 enters in conflict with the monotonicity condition, stating that the value of a life is increasing in its quality. To see that conflict, let us take, here again, the example of the addition of a life-year in the United States or in Ghana, with either U.S. life quality (equal to 2) or with Ghana's quality (equal to 1). Here are four alternative options:

A: Adding a life-year in Ghana with Ghana's standards of living. B: Adding a life-year in the U.S. with U.S.'s standards of living. C: Adding a life-year in Ghana with U.S.'s standards of living. D: Adding a life-year in the U.S. with Ghana's standards of living.

The conception U1 of the intuition of universality implies that the value assigned to the addition of a life-year in case A must be equal to the value assigned to the addition of a life-year in case B, that is:

V((1,…,1) + 1) -V((1,…,1)) = V((2,…,2) + 2) -V((2,…,2)).
However, by monotonicity, we have also that the value assigned to the addition of a life-year in case C exceeds the value assigned to the addition of a life-year in case A (where the added life-year has a lower quality, while everything else is left unchanged), that is:

V((1,…,1) + 2) -V((1,…,1)) > V((1,…,1) + 1) -V((1,…,1)).
Still by monotonicity, we have also that the value assigned to the addition of a lifeyear in case B exceeds the value assigned to the addition of a life-year in case D (where, here again, the added life-year has a lower quality, everything else remaining unchanged), that is:

V((2,…,2) + 2) -V((2,…,2)) > V((2,…,2) + 1) -V((2,…,2)).
Given that V((1,…,1) + 2) > V((1,…,1) + 1), and V((2,…,2) + 2) > V((2,…,2) + 1), one obtains, from the first equality, the following inequality:

V((1,…,1) + 2) -V((1,…,1) > V((2,…,2) + 1) -V((2,…,2)).
This inequality means that adding a life-year with U.S. standards in Ghana leads to a higher gain in value compared to adding a life-year with Ghana's standards in the U.S.

That inequality is in contradiction with the formulation U1 of the intuition of universality. According to that conception of universality, for sure adding a life-year with U.S. standards in Ghana should be equally good to adding a life-year with Ghana's standards in the U.S. Hence, we reach here a contradiction, which implies that the formulation U1 of universality is not logically compatible with the monotonicity condition.

That proof by contradiction is formally close to the argument developed by [START_REF] Broome | Weighing Lives[END_REF] concerning the logical incompatibility of the intuition of neutrality for the addition of a life with the principle of personal good. Note that another way to prove that the property U1 is incompatible with monotonicity consists of examining the constraints that that formulation of the intuition of universality imposes on the structural form of the value function V(.).

The property U1 implies that the variation in value associated to an extra life year, i.e. V((a,…, g) + x) -V((a, …, g)), depends neither on the quality and quantity of other life-years, nor on the quality of the added life-year. The variation in value associated to the added lifeyears is thus a constant. Denoting that constant by c, one can deduce, by repeated substitutions, that:

V((a, …, g) + x) = V((a, …, g)) + c [L years + 1] [L years] ↔ V((a, …, g) + x) = V((a, …, f)) + c + c [L years + 1] [L -1 years] ↔ V((a, …, g) + x) = (L + 1)c [L years + 1]
Thus the formulation U1 of the intuition of universality implies that the value function V(.) has a simple form: it is equal to the number of life-years multiplied by a constant. Hence, the formulation U1 of the intuition of universality implies that only the total lifetime matters, independently from the quality of life. Thus one can see, here again, that the formulation U1 of the intuition of universality is not compatible with the monotonicity property.

Given the natural appeal of the monotonicity condition, the logical incompatibility of the U1 formulation of the intuition of universality with monotonicity is quite problematic. This suggests that this formulation of the intuition of universality is too demanding, or too strong, which leads to incompatibilities with a property as simple as monotonicity. Therefore, in the rest of this section, we will propose to depart from the U1 formulation of the intuition of universality, and reformulate that intuition in different ways.

As we emphasized above, the formulation U1 of the intuition of universality was quite strong, since it involved a triple independence of the value of the extra life-year, with respect to the quality of the added life-year, with respect to the quality of previous life-years, and with respect to the number of previous life-years. In the remaining of this section, we will focus on weaker formulations of the intuition of universality, which relaxes some of those independence requirements.

The intuition of universality (U2)

V((x,…, x) + x) -V((x, …, x)) = V((y, …, y) + y) -V((y, …, y))
The formulation U2 of the intuition of universality states that the value of an extra lifeyear whose quality is equal to the quality of previous life-years should be universal, that is, independent from the quality of (previous and added) life-years, and, also, independent from the initial lengths of lives (since the left-hand side and the right-hand side may involve initially lives of unequal lengths). Thus the formulation U2 involves, as the formulation U1, a triple independence of the value of an extra life-year.

However, the conception U2 of universality is weaker than the conception U1, since in U2 the equality of the value of an extra life-year is conditional on its quality being equal to the quality of previous life-years, unlike what prevailed under U1. Thus U2 states the triple independence requirement only for lives of constant quality, and not for lives of non-constant quality. This limits the scope of the triple independence, and, as such, makes the conception U2 of the intuition of universality weaker than the conception U1.

In order to understand the implications of the formulation U2 of the intuition of universality on the form of the value function V(.), it is useful to notice that the variation in value due to the addition of a life-year of constant quality is the same on both sides of the above equation, despite the fact that the (constant) quality of life-years is not the same on the left-hand side and on the right-hand side, and despite also the fact that the lives being compared may initially differ also in terms of sizes. As a consequence, the variation in value due to the addition of a life-year must be independent from the (constant) quality of life, and from the length of the initial life. Therefore, this variation must be equal to a constant. Writing that constant with the letter c, we have,

V((x, …, x) + x) -V((x, …, x)) = c
Hence, we obtain, by successive substitutions:

V((x, …, x) + x) = V((x, …, x)) + c [L years + 1] [L years] ↔ V((x, …, x)) = V((x, …,x)) + c + c [L years + 1] [L -1 years] ↔ V((x, …, x)) = V(x) + (L -1)c [L years + 1] [1 year]
The value function V(x, …, x) thus takes the form of a linear combination of the value of a single life-year V(x), and of the length of that life.7 Note that this formulation of the intuition of universality leads to a value function that is increasing in the quality of the added life-year (i.e. x), and, as such, satisfies the monotonicity condition, unlike thestrongerformulation U1 of the intuition of universality.

Quite interestingly, the United Nations' Human Development Index, in its initial form (UNDP 1990) is additive in an index of life expectancy achievements, and in an index of GDP per capita achievements (and, also, in an index of education achievements). Hence, the initial HDI has a functional form that is compatible with the conception U2 of the intuition of universality, in the sense that it values the addition of a life-year with constant quality independently from the (constant) quality, and independently from the length of initial lives. The standard HDI thus captures the formulation U2 of the intuition of universality.

Whereas the above discussion assumes a constant quality of life profile, it may be useful to generalize that discussion to the case where the lifetime quality profile is not constant, and is, for example, equal to {a, b, …, h). One could then consider a value function V(.) that is, as above, additive, but takes the (more general) form: V(X) + (L -1)c, where X denotes the generalized mean of life quality.

If the generalized mean of the quality of life X is equal to the quality of the added lifeyear (i.e. x), then that value function satisfies the property U2. Note also that, provided the generalized mean of the quality of life X is increasing in the quality of the added life-year (i.e. x), this value function satisfies also the monotonicity condition.8 Moreover, the value function V(.) allows for a certain degree of complementarity between lifetime quantity and lifetime quality, something that was not possible under the formulation U1 of the intuition of universality. Clearly, adding a life-year with a quality above the average quality (i.e. x > X) is here regarded as good (i.e. increases value), whereas adding a life-year with a quality inferior to the average quality (i.e. x < X) is bad (i.e. reduces value). Finally, adding a life-year with a quality exactly equal to the average quality (i.e. x = X) is neutral.

Those positive results hold thanks to the fact that U2 is a weaker formulation of the intuition of universality than U1. It is thus compatible with monotonicity and also with some degree of complementarity, but at the cost of weakening the requirement of universality (with respect to formulation U1).

Let us now consider an alternative formulation of the intuition of universality, which imposes not an independence of the value of the extra life-year with respect to the quality of the added life-year, but an independence only with respect to the quality of previous life-years (which may not be constant, unlike under U2), and with respect to the number of previous life-years. This is the formulation U3 of the intuition of universality.

The intuition of universality (U3)

V((a, b, …, g) + z) -V((a, b, …, g)) = V((h, i, …, m) + z) -V((h, i, …, m))
The formulation U3 of the intuition of universality involves a double independence of the valuation of an extra life-year: (i) independence with respect to the quality of previous life-years, and (ii) independence with respect to the quantity of previous life-years (since the number of previous life-years involved on the left-hand side and on the right-hand side may differ).

To understand the implications of the formulation U3 of the intuition of universality on the structure of the value function V(.), let us notice that, since the variation of value due to the addition of a life-year does not depend on the quality and quantity of previous life-years, it can only depend on the quality of the added life-year, that is, on z. Hence, we have:

V((a, b, …, g) + z) -V((a, b, …, g)) = v L (z)
where v L (z) is a function of z.

Hence we obtain, by repeated substitution:

V((a, b, …, g) + z) = V((a, b, …, g)) + v L (z) [L years + 1] [L years] ↔ V((a, b, …, g) + z) = V((a, b, …)) + v L-1 (g) + v L (z) [L years + 1] [L -1 years] ↔ V((a, b, …, g) + z) = V(a) + v 2 (b) +… + v L-1 (g) + v L (z) [L years + 1] [1 year]
where v 2 (.), …, v L (.) are functions of the quality of each life-year.

The formulation U3 of the intuition of universality has thus a precise implication on the structure of the value function V(.). Actually, it imposes that the value function V(.) is a sum of the transformed qualities of all life-years involved, with a number of terms equal to the number of life-years in the life under study.

Quite interestingly, the functional form for V(.) implied by the property U3 satisfies the monotonicity condition, unlike the property U1. Moreover, it allows also for some degree of complementarity between the quality and the quantity of life. As for the formulation U2 of the intuition of universality, those positive results arise thanks to the fact that the property U3 is a weaker formulation of the intuition of universality. That more limited universality requirement allows for some compatibility with monotonicity and with the intuition of complementarity, but at the cost of weakening the universality requirement.

In sum, this section showed that the intuition of universality for the valuation of a lifeyear can be formulated in quite distinct ways, which all have their particular implications for the structure of the value function that measures the value of a life as a whole. Note, however, that although this section allowed us to provide precise accounts of the intuition of universality, and to explore the consequences of those formulations on the structure of value functions, this section had little to say, in general, about how priorities should be given when allocating life-years within the population. Actually, the variations in value associated to the addition of a life-year do not have direct implications in terms of priority, except if one adopts the social objective of maximizing the sum, across all individuals, of values V(.).

If one adopts that particular social objective, then an interesting thing to notice is that the formulations U1, U2 and U3 of the intuition of universality share an important direct implication in terms of social priority: these all imply social indifference regarding how lifeyears are allocated within the population. Thus those three conceptions of the intuition of universality lead the social evaluator to be indifferent with respect to how those life-years are distributed. That result is not particularly appealing: from an egalitarian perspective, one may prefer to give social priority to individuals whose lives are of low quality or of limited quantity. This view is clearly not compatible with the formulations U1, U2 and U3 of the intuition of universality, at least if the goal is to maximize the sum of values V(.) across individuals.

It should be stressed, however, that there is no obvious reason why the social evaluator should take, as an objective, the maximization of the sum of individual values V(.). Many other social goals could be adopted instead, and in those cases the above formulations of the intuition of universality do not have direct implications in terms of priority. The goal of the next section is to develop an alternative approach, in terms of social preference orderings, in order to explore, under more general social objectives, the implications of the three conceptions of the intuition of universality developed above for the allocation of life-years within a population.

Three accounts of the intuition of universality (2): a social preference approach

Let us now examine what are the implications of the intuition of universality in terms of priority in the context of a problem of life-years allocation. For that purpose, let us define an allocation as a vector of quality of life for each individual in the population, whose size is supposed to be constant and equal to N. Formally, an allocation can be written as a vector q = (q i ) i = 1, …, N , where q i = (q i1 , …, q ili ) is the life of individual i, who enjoys a life of length l i . We denote by Q the set of all such allocations. We denote by Q C the subset of Q that includes all allocations with constant quality along the life. Regarding individual longevities, we define by l the vector of individual durations of life (number of life-years). We have that l = (l i ) i = 1,…,N.

Let us now reformulate the three conceptions of the intuition of universality studied in Section 2 in terms of their logical consequences concerning the social ranking of allocations. In this section, we denote that social pre-ordering as ≤ S . That social preference relation is assumed to be reflexive, transitive and complete. As usual, strict social preference is denoted by < S , whereas social indifference is written as ~S.

Throughout this section, the intuition of universality will be formulated in terms of whether adding an extra life-year to a person i is equivalent to adding an extra life-year to a person j, which is, from a formal perspective, equivalent to stating that transferring a life-year from individual j to individual i leads to social indifference. Thus, even if the formulations of the intuition of universality developed below look like properties about "transfers" of lifeyears across individuals, these are only a formal way to formulate conditions of social indifference about who receives the extra life-year.

In terms of the pre-ordering on allocations, the formulation U1 of the intuition of universality states that a change in who receives an additional life-year, everything else being left unchanged, leads to an allocation that is regarded, from a social perspective, as equally good as the initial allocation, independently from the quality of the added life-year, and independently from the quantity and the quality of previous life-years.

The intuition of universality (U1)

For all q, q' in Q, if q it = q it ' for all i, t in {1, 2, …, min(l i , l i ')}, and if there exists i, j such that: l i ' = l i + 1 and l j ' = l j -1 and for all k ≠ i, j, l k ' = l k , then q ~S q'. From the perspective of U1, it does not matter whether an additional life-year is given to a person with a more or less long life, or with a life of more or less high quality: changing the recipient of the extra life-year leads neither to a social improvement, nor to a social worsening, but is just neutral. As such, the conception U1 captures some idea of universality in the valuation of life-years. It states that one is socially indifferent between allocating an extra life-year to a given life or to another life.

Note that this social indifference associated to who receives the extra life-year amounts to assume that the social valuation of life-years satisfies a triple independence: (i) independence with respect to the quality of the added life-year; (ii) independence with respect to the quality of previous life-years; (iii) independence with respect to the quantity of previous life-years.

Although the formulation U1 may seem intuitive at first glance, it has implications that are not so attractive regarding the allocation of life-years. In particular, it is incompatible with a basic monotonicity property. The monotonicity property can be stated as follows.

Monotonicity

For all q, q' in Q, if l = l', if q it ' > q it for some i, t in {1, …, l i } and q it ' = q it for all other i, t in {1, …, l i }, then q' > S q.

The monotonicity property is quite weak: it states that if some allocation q' involves a higher quality of life-years for some individuals in comparison to the allocation q, everything else remaining the same in q' and q, then the allocation q' is, from a social perspective, strictly better than the allocation q.

In order to see why the formulation U1 of the intuition of universality is logically incompatible with the monotonicity property, let us consider two allocations q and q' satisfying the conditions described in U1, and let us add a third allocation, denoted by q'', which is the same as the allocation q', except that it involves a strictly higher quality of life for the extra life-year enjoyed by individual i, that is, that q ili '' > q ili '.

It is easy to see that, when comparing allocations q and q'', the formulation U1 of the intuition of universality implies that there must be social indifference between q and q'', for the same reasons as there is social indifference between q and q'. We thus have, by property U1, that: q ~S q'' and q ~S q' This implies, by transitivity, that: q' ~S q'' However, the monotonicity property requires that: q'' > S q' Thus we reach here a contradiction. That contradiction implies that the social ranking of allocations cannot satisfy both the formulation U1 of the intuition of universality and the monotonicity property. A choice is to be made between those properties.

Given the natural appeal of the monotonicity condition for the social ranking of allocations, this negative result supports to give up the intuition of universality, at least under its U1 formulation. Actually, if being universalist regarding the valuation of life-years implies to violate monotonicity, and thus to be socially indifferent between allocations that are not equivalent at all, then the attractiveness of such a universalism can be questioned.

It should be stressed, however, that the above negative result only concerns the formulation U1 of the intuition of universality. As such, this cannot be generalized to all conceptions of the intuition of universality.

Let us now consider the implications of the second formulation of the intuition of universality on the allocation of life-years. Translated in terms of requirements regarding the social preference relation, the formulation U2 can be rewritten as:

The intuition of universality (U2)

For all q, q' in Q C , if q i1 = q i1 ' for all i, and if there exists i, j such that: l i' = l i +1 and l j ' = l j -1 and for all k ≠ i, j, l k ' = l k , then q ~S q'. Thus U2 states that, when comparing allocations with constant quality along lives, a change in the recipient of an extra life-year from person j to person i (while keeping everything else unchanged) leads to social indifference. Whatever the durations of life for the individuals i and j are, and whatever the qualities of their previous life-years are, whether it is person i or person j that receives the extra life-year is neutral. Note that this conception of universality is weaker than conception U1, because it is here restricted to the subset of allocations in which the quality of life is constant along a given life.

In order to explore the implications of the formulation U2 of the intuition of universality in terms of priority, a first, important, step, consists of examining the constraints that U2 imposes on the form of a social welfare function. Actually, it is shown in the Appendix that the account U2 of the intuition of universality has precise implications regarding the structure of the social welfare criterion to be pursued.

Characterization Theorem (formulation U2 of the intuition of universality)

A social welfare function W(.) satisfies the formulation U2 of the intuition of universality if and only if it takes the following form: W(q) = F(q i , q j , …, q N , ∑ l i ) where q i denotes the (constant) quality of life enjoyed at all his life-periods by individual i under the allocation q.

What is stated here is a representation result that takes the form of a logical equivalence: any social welfare function that satisfies the property U2 must have that particular form, and, also, any social welfare function that satisfies that form must also satisfy the formulation U2 of the intuition of universality. Quite interestingly, the form taken by the social welfare function is simple: it is a function of the (constant) qualities associated to the life-years of all individuals, and, also a function of the total lifetime in the population. 9An important corollary of this representation result is that, under the conception U2 of universalism, the particular distribution of life-years within the population does not matter; the only thing that matters concerning lifetime is the total amount of life-years that are lived. Whether the lifetime is shared more or less equally within the population does not matter.

That corollary is particularly counterintuitive. When considering the allocation of lifeyears within a population, a social planner may prefer, on the grounds of social justice, that life-years are distributed more equally across individual lives. Such an egalitarian perspective is clearly incompatible with the formulation U2 of the intuition of universality. Being universalist under the U2 conception implies being socially indifferent between allocations that keep the total number of life-years constant, independently from how those life-years are distributed in the population.

To put it in different terms, the formulation U2 of the intuition of universality leads to be indifferent with respect to the distribution of life-years across individuals, and, as such, this is incompatible with the idea of giving priority to the poor, who can be here represented as individuals with shorter lives and lower qualities of life. The formulation U2 of the intuition of universality prevents giving priority to those disadvantaged individuals.

Note that this result only presupposes the formulation U2 of the intuition of universality, and is not based on a particular assumption concerning the way in which the social welfare function aggregates value functions V(.). Clearly, if the social ordering on allocations were based on the sum of individual value functions V(.), as in Section 2, one would also obtain social indifference with respect to the distribution of life-years within the population. This section provides a more general argument, according to which the formulation U2 of the intuition of universality leads inevitably to social indifference with respect to the distribution of life-years, whatever the precise way (additive or not) in which value functions V(.) enter the social objective.

That corollary of the formulation U2 of the intuition of universality tends to question the attractiveness of universality when formulated in that particular way. If being universalist implies being indifferent with respect to inequalities in the length of life, then such a universalist perspective looks far from attractive. Ideally, we would like universalism to lead towards priority given to the disadvantaged, and, hence, towards more equality, and not to lead to indifference towards more inequality. We reach, here again, a negative result, but this negative result is relative to a particular formulation of the intuition of universality. 10Let us now turn to the third conception of universality developed in Section 2. When reformulated in terms of its implications on the social preference ordering over allocations, the formulation U3 of the intuition of universality is defined as follows.

The intuition of universality (U3)

For all q in Q, for all z in R + , for all i, j, we have: (…(q i1 , …, q ili , z)…(…(q j1 , …, q jlj )…) ~S (…(q i1 , …, q ili )…(…(q j1 , …, q jlj , z)…)

The property U3 states that changing the recipient of an extra life-year with quality z leads to social indifference, whatever the quality and the quantity of the life-years lived by the possible recipients. As such, it captures some intuition of universality, in the sense that the social evaluator is indifferent between giving an extra life-year to one person or to another, whatever the lives of those persons are.

Although that conception of universality may seem appealing, it faces the same problem as the conception U2 studied above: by valuing the allocation of life-years indifferently from the lives of the persons (in terms of their quantity and quality), the conception U3 of universality goes against the idea of giving priority to the disadvantaged.

To see this, let us take a simple 2-person example, involving persons i and j. The initial allocation is: ((q i1 , q i2 , …, q ili ), (q j1 , q j2 , …, q jlj ))

The formulation U3 of the intuition of universality states that changing the recipient of a life-year, from, let us say, person j to person i, leads to social indifference. If the last lifeyear of person j is reallocated to person i (with the associated quality q jlj ), we thus have: ((q i1 , q i2 , …, q ili ), (q j1 , q j2 , …, q jlj )) ~S ((q i1 , q i2 , …, q ili, q jlj ), (q j1 , q j2 , …, q jlj-1 ))

Then, by repeating reallocations of life-years successively, from person j to person i, one finally obtain: ((q i1 , q i2 , …, q ili ), (q j1 , q j2 , …, q jlj )) ~S ((q i1 , q i2 , …, q ili, q jlj , …, q j2 ), (q j1 ))

Thus property U3 leads to social indifference between two allocations that are extremely different: whereas in the initial allocation, the lifetime is divided between persons i and j, in the final allocation, almost the entire lifetime is concentrated on the person i, whereas only a single life-year remains for person j. That highly unequal distribution of lifetime does not seem to be as socially desirable as the initial allocation, but this is however what the formulation U3 of the intuition of universality implies. Repeated use of the universality property U3 lead to social indifference between allocations that are characterized by quite different degrees of inequality in the distribution of lifetime among persons.

From the perspective of social justice, one would prefer, on the contrary, to give priority, in the allocation of life-years, to disadvantaged individuals, who have either shorter lives, or lives of worse quality. The intuition of universality is hardly compatible with giving priority to the disadvantaged. On the contrary, it leads to social indifference with respect to how life-years are allocated between persons. Here again, as for the formulation U2, the intuition of universality goes against this ideal of giving priority to the disadvantaged.

Note also that, as for the conception U2, the argument provided here does not rely on a particular functional form for the social objective. Obviously, if the social goal is, as in Section 2, to maximize the sum of value functions V(.), then we would also obtain social indifference with respect to who receives the extra life-year. But the argument developed here is more general, since this does not postulate any particular social objective, that is, the social ordering does not need to be based on the mere sum of value functions V(.). We reach thus a robust result on the conflict of conception U3 of universality with giving priority to the disadvantaged.

In sum, this section leads to quite negative results concerning the implications of the intuition of universality. We showed that either the intuition of universality is incompatible ∂W / ∂l i = (∂W/∂V(q i )) u i (q ili ) The left-hand side of that equation is the variation in social welfare associated to a minor change in the duration of the life of individual i. This variation is equal to the product of two factors.

First, it depends on the degree of priority of the individual from a social perspective, which is captured by the factor (∂W/∂V(q i )). This degree of priority clearly depends on the degree of inequality aversion exhibited by the social welfare function. If the person i is particularly disadvantaged, an inequality-averse social planner assigns a high weight to improving the well-being of that person. This first effect is the social weighting effect.

Second, the marginal social welfare associated to a change in the duration of the life of individual i depends also on the value of this extra life-year for the individual, based on the quantity and the quality of his past life, and, also, based on the quality of the extra life-year itself. Clearly, one can expect that a life profile with a higher quality of life will generally imply a higher value for an increase in the duration of life. That second effect is the individual valuation effect (which may depend on individual subjective preferences or some other objective approach to the valuation of individual lives).

In the case of increasing the lifetime of an individual whose life has low quality, the social weighting effect and the individual valuation effect go in opposite directions when the social planner is inequality-averse. In that case, the social weighting effect is strong, while the individual valuation effect is low. On the contrary, when considering the marginal social welfare from increasing the duration of life of a person whose life has high quality, the opposite arises: the individual valuation effect is large, while the social weighting effect is low.

At the end of the day, whether a higher marginal social value is assigned to increasing the length of life of the person with a low or a high life quality depends on the degree of inequality aversion exhibited by the social welfare function, and, also, on individuals' valuations of life. It is quite possible that a higher marginal social value is assigned to increasing the duration of life of a person with a low life quality, despite the individual valuation effect. This is definitely the case when the social welfare function exhibits a high degree of inequality aversion.

To show this, let us take a simple analytical example, where the function u i (q it ) takes the following form:

u i (q it ) = [q it 1-a -q 0 1-a ] / (1 -a)
Moreover, let us suppose that the social welfare function takes a standard Atkinson-Stiglitz form (see [START_REF] Atkinson | Lectures on Public Economics[END_REF]:

W = ∑ (V(q i )) 1-e / (1 -e)
where the ethical parameter e captures the sensitivity of social valuations to inequalities in well-being across individuals. When e equals 0, there is a zero degree of inequality aversion.

In that analytical example and supposing a constant quality of life q it = q i , the marginal social welfare gain from increasing the duration of life of individual i is equal to: ∂W / ∂l i = (l i u i (q i )) -e u i (q i ) Note that, when the ethical parameter e equals 0, the marginal social welfare from increasing the duration of life of individual i is equal to u i (q i ), and, hence, is increasing with the quality of life enjoyed by the person i. In that case (zero inequality aversion), the social weighting effect is dominated by the individual valuation effect, and so a larger priority is given to individuals with a higher quality of life.

But that case is not the only possible one. Actually, under a large interval of values for the ethical parameter e, the opposite will take place, and the social weighting effect will dominate the individual valuation effect, leading to priority to the disadvantaged individuals.

It is straightforward to see that, when the ethical parameter e equals 1, the marginal social welfare from increasing the duration of life of individual i is equal to merely 1 / l i , that is, to the inverse of person i's duration of life. Hence, in that case, a higher priority will be given to individuals with a shorter life, and a lower priority to individuals with a longer life.

Alternatively, when e is superior to 1, an even larger priority is given to the disadvantaged individuals, since the marginal social welfare gain from increasing the length of life of a person i is then not only decreasing with the duration of life of that person, but, also, decreasing with the quality of his life. Thus priority is here given to individuals with shorter lives and lives of lower quality.

Those examples suffice to illustrate that it is possibleand actually quite easyto accommodate the intuition of complementarity with the ideal of giving priority to the disadvantaged. The intuition is that the marginal social value of increasing the duration of life of a person depends not only on the quality of that life (through the individual valuation effect), but, also, on the weight that is given to improving the situation of that person in the social welfare function (the social weighting effect). When the latter dominates the former, priority is given to individuals with a shorter life and with a lower quality of life.

Concluding remarks

Given that various policieshealth policies, safety policies, development policiesinfluence mortality, and, hence, individual lifetimes, the valuation of life-years has become a necessary stage in the design of optimal policies. The definition of optimal policies in lifeaffecting domains requires governments to be able not only to weight live-years against resources, but, also, life-years enjoyed by some persons against life-years enjoyed by other persons. Moreover, at the descriptive level, the measurement of economic development requires to be able to weight achievements in terms of longevity in comparison to achievements on other dimensions of life, and, also, to make longevity achievements in some countries comparable with longevity achievements in other countries.

When considering the valuation of life-years, two basic intuitions arise: on the one hand, the intuition of universality, according to which the value of a life-year should be universal, and, hence, independent from the duration and the quality of lives considered, and, on the other hand, the intuition of complementarity, according to which the value of a lifeyear should depend on what that life-year allows for, and, hence, on its quality.

Those two intuitions are plausible, but these are hardly compatible: the intuition of universality requires that the value of a life-year is universal, and, hence, does not depend on its quality, which goes against the intuition of complementarity, which makes the valuation of a life-year dependent on its quality. Thus a choice is to be made between those two intuitions concerning the valuation of life-years.

In order to cast original light on that ethical dilemma, this paper proposed to provide several distinct accounts of the intuition of universality, and to explore their logical implications in terms of the valuation of life-years, and, also, in terms of the priority to be given to the disadvantaged when considering the allocation of lifetime within a population.

Our results suggest that the intuition of universality, whatever the precise formulation considered, leads to implications that are far from appealing. Our accounts of the intuition of universality lead either to a conflict with a basic principle of monotonicity (i.e. the conception U1), or lead to a conflict with giving priority to the disadvantaged (i.e. conceptions U2 and U3). Those conflicts are quite problematic: imposing a universal valuation of life-years would lead to social indifference with respect to the distribution of lifetime within the population. Such social indifference would go against the ideal of equality, and, as such, is quite counterintuitive, and hard to justify.

On the contrary, the intuition of complementarity can be compatible with the idea of giving priority to the disadvantaged, and, as such, does not imply a social indifference with respect to how life-years are distributed within the population, unlike the intuition of universality. The underlying intuition is that the dependence of the valuation of a life-year on the quality of life is only one aspect of the social valuation of life-years, which depends also on how individual interests are weighted in the social welfare function. It is thus possible, when the social welfare function exhibits a sufficiently high degree of inequality aversion, to conciliate the intuition of complementarity with the ideal of giving priority to individuals with low qualities of life.

All in all, this paper suggests that the intuition of universality, although it may seem appealing at first glance, leads, at the end of the day, to the opposite of what it aims at: by valuing all life-years in a uniform way, the intuition of universality is not compatible with giving priority to the disadvantaged, and, hence, tends to play against equality. On the contrary, the intuition of complementarity can be made compatible with the ideal of giving priority to the disadvantaged, and, hence, is more compatible with equality.

Adler, M., Hammitt, J., Treich, N. (2014). The social value of mortality risk reduction: VSL versus the social welfare function approach. Journal of Health Economics,35,[82][83][84][85][86][87][88][89][90][91][92][93] That is, the social welfare function takes the form W(q) = F(q i , q j , …, q N , ∑ l i ). A similar proof could be provided for any N > 3.

Finally, it should be stressed that, whereas the above proof assumes the existence of a representation of the social ordering ≤ S , such an assumption is not necessary for the purpose at hand. Actually, it can be shown that there exists another preorder ≤* defined on ((q i ) i , ∑ l i ), which is such that: ((q i ) i , ∑ l i ) ≥* ((q i ') i , ∑ l i ') ↔ (q 1 . (∑ l i -N + 1), q i . 1) ≥ (q 1 '. (∑ l i ' -N + 1), q i '. 1)

The value of a statistical life is defined as the value that x individuals assign to a reduction of the risk of death from 1/x to 0, leading to saving one life.

According to Miller (2000), the value of a statistical life lies between 120 and 180 times GDP per capita.

The treatment of lifetime within the multiplicative HDI is also criticized by[START_REF] Ravallion | Troubling trade-offs in the Human Development Index[END_REF].

This is true for indicators of well-being relying on a lifecycle perspective, such as the equivalent income (see[START_REF] Costa | Long term trends in health, welfare and economic growth in the United States[END_REF][START_REF] Nordhaus | The health of nations: the contributions of improved health to living standards[END_REF][START_REF] Becker | The quantity and the quality of life and the evolution of world inequality[END_REF][START_REF] Fleurbaey | Beyond GDP. Measuring Welfare and Assessing Sustainability[END_REF]. This is also the case for normative studies on compensation for unequal lifetime (see[START_REF] Fleurbaey | Prevention against equality[END_REF].

Indeed, it is possible to add an extra term c on the right-hand side while still respecting the condition U2, which leads to V(x, …, x) = V(x) + cL.

Note that the generalized mean quality is not necessarily increasing in x. One could have X = min{a, b, …, h, x} = a ≠ x, or X = max{a, b, …, h, x} = d ≠ x.

Given that we consider populations of constant sizes, the social welfare function W(.) can also be regarded as a function of the average lifetime in the population.

Note that this negative result is here reached while assuming a representativity of the social preference ordering by means of a social welfare function W(.). However, as shown in the Appendix, our result is actually more general, and does not necessarily require assuming the existence of such a representation.

with the monotonicity property (conception U1), or leads to social indifference with respect to how life-years are allocated within the population, which goes against the ideal of giving priority to the disadvantaged (conceptions U2 and U3).

Whereas this section reached some negative results concerning the intuition of universality, one may wonder whether abandoning that intuition in favor of the intuition of complementarity would allow us to obtain more appealing implications. In particular, one may be curious to see whether adopting the intuition of complementarity would allow us to better meet the ideal of giving priority to the disadvantaged. That question is explored in the next section.

The intuition of complementarity and the priority to the worst-off

Under the intuition of complementarity, the value of a life-year depends on what that life-year allows for, that is, on the quality of that life-year. At first glance, one may believe that the intuition of complementarity, by leading to assign a higher value to life-years characterized by a higher quality (unlike the intuition of universality), could favor the allocation of life-years towards more life-years given to individuals who enjoy a high quality of life.

But that belief is actually wrong: when allocating life-years, the valorization of those years is only one aspect of the problem. Another, crucial, aspect concerns the priority that the social evaluator assigns to the well-being levels of the different individuals, and, in particular, his aversion to inequality. When the aversion to inequality is large, it can offset the valorization dimension, and lead to assign more life-years to individuals with low life quality. It is actually quite simple to combine the intuition of complementarity with giving priority to the disadvantaged.

To see that, let us assume that the value of an individual life takes a standard, timeadditive, form, that is:

where u i (q it ) represents the temporal utility associated to the life-year t for individual i.

For the sake of simplicity, we will assume here that lifetime is continuous rather than discrete, and thus consider the equivalent form in continuous time:

The social welfare function takes the general form:

W(q) = W(V(q 1 ), …, V(q N ))

Within that framework, the marginal social welfare from increasing the duration of the life of individual i is given by the derivative:

Appendix

The characterization result takes the form of an equivalence between a social welfare function W(q) satisfying property U2 and a social welfare function taking the form W(q) = F(q i , q j , …, q N , ∑ l i ).

To prove that equivalence result, we proceed in two steps.

Let us first prove that a social welfare function taking the form W(q) = F(q i , q j , …, q N , ∑ l i ) satisfies the property U2.

To see this, let us take a three-person case, with (constant) qualities of life a, b and c, and durations of life m, n, and o. Let us denote by a.m a life of m years with constant quality a. The allocation q is thus written (a.m, b.n, c.o). Let us now compare that allocation with another allocation, q', where a life-year is transferred from the second person to the first person. The allocation q' is thus written as (a.(m + 1), b.(n -1), c.o).

The property U2 requires that there is social indifference between allocations q and q', that is:

It is easy to see that this equality is satisfied by any function taking the form W(q) = F(q i , q j , …, q N , ∑ l i ). Indeed, in our 3-person case, the function takes the form: W(a.m, b.n, c.o) = F(a, b, c, m + n + o). It does satisfy the equality mentioned above, since the transfer or a life-year maintains the total number of life-years unchanged. Indeed, we have:

as required by property U2.

The same argument could be formulated for any case with N > 3, with any transfer of life-years. Thus we have that any social welfare function taking the form W(q) = F(q i , q j , …, q N , ∑ l i ) satisfies the property U2.

Let us now prove that any social welfare function satisfying the property U2 takes the form W(q) = F(q i , q j , …, q N , ∑ l i ).

To prove this, let us turn back to our 3-person case. We have, by repeated use of the property U2: