P A Jouvet

P Pestieau

G Ponthiere
email: gregory.ponthiere@ens.fr

The Spaceship Problem Re-Examined

Keywords: environmental congestion, fertility, longevity, population ethics, utilitarianism D63, Q56, Q57

This paper re-examines the spaceship problem, i.e. the design of the optimal population under the environmental constraint of a …xed area available for life, by focusing on the dilemma between adding new beings and extending the life of existing beings. For that purpose, we characterize, under the assumption that individual lifetime welfare depends positively on the length of life but negatively on population density, the preference ordering of a utilitarian planner over lifetime-equal histories, i.e. pairs of initial population size and survival conditions yielding an equal number of life periods. It is shown that a Benthamite planner is not necessarily indi¤erent between lifetime-equal histories, and that a Millian utilitarian planner prefers lifetime-equal histories yielding the smallest population with the longest life. The solutions under Critical-level and Number-dampened utilitarianisms are also shown to di¤er signi…cantly.

Introduction

Given that the Earth is of …nite size -and, thus, can be compared, following [START_REF] Boulding | The Earth as a spaceship[END_REF], to a 'spaceship'-, the question of the optimal population size can be formulated as the 'spaceship problem': are there too few, or, on the contrary, too many people living on our bounded, resources-…nite, spaceship? Undoubtedly, that question is an old issue, to which various answers were provided over time.

While Mercantilism was, during the 16th and 17th centuries, promoting a population as large as possible by means of various policies, it should not be deduced from this that the spaceship constraint was ignored by Mercantilist thought. 1 Actually, it is quite the opposite: a standard Mercantilist argument was that the …niteness of the living space, if coupled with a large population, would favour migrations, and, hence, the colonization of other countries.

The Italian philosopher Giovanni [START_REF] Botero | Della Ragione di Stato[END_REF] is generally regarded as one of the …rst thinkers who argued against the Mercantilist populationism, and asked the question of the optimal population size. Botero argued that men have a tendency to multiply themselves as much as nature allows them. However, natural resources are limited, so that, according to Botero, one cannot escape the following adjusments: either people will modify their behaviours, or there will be some adjustment in numbers through famines, diseases or wars.

Within economic thought, Richard [START_REF] Cantillon | Essay on the Nature of Trade in General[END_REF] provided another early study of the spaceship problem, by highlighting the existence of a quantity of life versus quality of life trade-o¤, which restraints feasible population sizes. According to Cantillon, Man's subsistance requires living space, whose amount depends on lifestyles, so that an arbitrage is to be made between the quantity and quality of life.2 While Cantillon did not solve that trade-o¤, he showed that, contrary to Mercantilists'beliefs, more people is not always better. 3Another contribution to the spaceship problem was made by Thomas [START_REF] Malthus | Essay on the Principle of Population[END_REF], who argued that the population size is necessarily limited within some boundaries. A population would follow a geometric progression if left unchecked (i.e. in the absence of resources constraints), but the production of means of subsistance follows, at best, an arithmetical progression (because of the …niteness of land), so that the population must, at some point, be 'checked', either by a positive check (deaths), or by a preventive check (fewer children). 4Given that Botero, Cantillon and Malthus's positive theories of population, by underlining the constraints imposed by the …niteness of land, only restrict the set of feasible population sizes, these leave open the normative question of the optimal population size.

That issue was widely studied within utilitarianism. While Jeremy Bentham's (1789) Classical utilitarianism recommends a number of people producing 'the greatest happiness of the greatest number', John Stuart [START_REF] Mill | Utilitarianism[END_REF], in his re…ned utilitarianism, recommends the maximization not of total welfare, but of welfare on average (i.e. average utility), which di¤ers from Benthamite utilitarianism in di¤erent numbers choices. However, as shown by [START_REF] Par…t | Reasons and Persons[END_REF], those two criteria of population ethics are unsatisfactory. Classical utilitarianism su¤ers from the Repugnant Conclusion, whereas Mill's utilitarianism faces the Mere Addition Paradox. 5 Hence, other population ethics criteria were developed, such as Critical-level utilitarianism [START_REF] Blackorby | Social criteria for evaluating population change[END_REF] and Number-dampened utilitarianism [START_REF] Ng | Social criteria for evaluating population change: an alternative to the Blackorby-Donaldson criterion[END_REF]. But although (partly) immunized against Par…t's criticisms, these su¤er from other weaknesses. Imposing a low critical level still implies the Repugnant Conclusion, while a high critical level leads to what [START_REF] Arrhenius | Interpersonal compensation and moral duties to future generations: moral aspects of energy use[END_REF] call the Sadistic Conclusion. 6Moreover, Number-dampened utilitarianism might lack intuitive support.

Whereas those discussions focused on the optimal number of people to be brought to life, another dimension of the problem was recently explored by [START_REF] Broome | Weighing Lives[END_REF]. In his Weighing Lives, Broome argued that the issue of the optimal number of people is not independent from the issue of the optimal length of life. True, making a person exist and making an existing person live more are, from a moral point of view, two distinct things, but these tend both to in ‡uence the number of existing people at a given point in time, so that population ethics could hardly focus on births and ignore deaths. Hence, in order to study the spaceship problem, the two ends of the 'demographic chain' -births and deaths -must be taken into account.

Although that survey suggests that much has been writen on the optimal population of our spaceship, this paper aims at casting a new light on that old problem. The speci…city of this paper with respect to the existing literature is precisely to consider simultaneously the two ends of the demographic chain: births and deaths. Our motivation lies in the fact that longevity is not …xed, but endogenous, so that the spaceship problem is not exclusively about bringing new people to life. The maintenance of existing people in life is also relevant for the spaceship problem. This is why longevity will also be considered here.

Therefore, we propose a re-examination, from a utilitarian perspective, of the spaceship problem, in which both births and deaths are endogenous. The question raised is the following. Suppose that a population, which enjoys a long life in a large living space, has to share a space of …nite size. What do the optimal population and the optimal longevity look like?

To answer that question, we shall make here some signi…cant simpli…cations, so that the present study should only be regarded as a …rst step towards a characterization of the solution of the spaceship problem.

First, we shall leave aside the production of goods, which, since Malthus's work, has occupied a central place in the study of the spaceship problem. Actually, given the observed growth of production per head despite the large population growth, it is not obvious that production limitations due to the …niteness of land can still be justi…ed. 7 Thus, in our model, the numbers's pressure will come only from population density, which a¤ects welfare negatively through a direct e¤ect (see [START_REF] Cramer | Quality of life in a city: the e¤ect of population density[END_REF], as well as through a limitation of individual longevity due to congestion.

Moreover, in order to formalize the longevity versus fertility trade-o¤s, we shall study the solutions of the spaceship problem by characterizing the preferences of a social planner on lifetime-equal histories, de…ned as pairs of initial population size and survival conditions yielding an equal number of life periods. Note that reasoning on the basis of a …xed total number of life periods is only an analytically convenient way to account for longevity versus fertility trade-o¤s, which will be complemented, in future works, by an explicit modelling of the transformation function characterizing such trade-o¤s.

Finally, given the intuitive weaknesses faced by the various criteria of population ethics (see [START_REF] Blackorby | Population Issues in Social Choice Theory, Welfare Economics, and Ethics[END_REF], it makes sense to re-examine the spaceship problem not on the basis of a single criterion, but, rather, by comparing the social optima under several ethical criteria. Because of space constraints, we shall here focus only on some widely used normative criteria, such as Classical utilitarianism, Millian (or Average) utilitarianism, Critical-level utilitarianism and Number-dampened utilitarianism. 8 Thus, this paper shall only compare the social planner's orderings on lifetime-equal histories under those ethical criteria, and neglect other criteria, which is also a signi…cant simpli…cation.

The rest of the paper is organized as follows. Section 2 formalizes the problem. Section 3 characterizes the preferences of a Benthamite utilitarian social planner on lifetime-equal histories. A similar task is carried out in Section 4 under a Millian social planner. Sections 5 and 6 concentrate on how Criticallevel utilitarianism and Number-dampened utilitarianism solve the spaceship problem. Concluding remarks are drawn in Section 7.

The spaceship problem 2.1 Assumptions

The spaceship problem can be formulated as follows. Suppose that a planet of …nite surface Q is available for life.

Axiom 1 A surface of …nite surface Q is available for the life of the population.

For simplicity, it is assumed that only living persons occupy a signi…cant space on that planet.9

Axiom 2 Only living persons occupy a part of the available space Q.

Moreover, all living persons at a particular point in time t = 0; 1; 2; :::; 1, whatever their age is, are assumed to occupy an equal amount of space, de…ned as the total space divided by the number of people alive at that time.

Axiom 3 Each person alive at time t enjoys an equal share of the total available space:

q t = Q L t
where L t denotes the population size at time t, whereas q t denotes the space available per person.

Thus we shall abstract here from problems of intragenerational distribution of space.

Here are the assumptions we shall make on individual welfare.

Axiom 4 At each period, the welfare of a person who is alive depends on the space available per person, according to the function:

u t = u(q t ) = Q L t 8t < T
where 0 1 and T denotes the time of death of the person.

Axiom 5 At each period, the welfare of a person who is dead equals 0:

u t = 0 8t > T Axiom 6
The lifetime welfare w of a person is the sum of the utilities associated to each period:

w = T X t=0 u (q t )
where T denotes the time of death of the person.

The …rst assumption -exclusive focus on space -is made for analytical conveniency. The second assumption allows us to avoid di¢ culties associated with the possibility of in…nite utility. The third assumption is, although largely questioned by [START_REF] Bommier | Uncertain lifetime and intertemporal choice: risk aversion as a rationale for time discounting[END_REF], still a standard one. This is used here for analytical conveniency.

In the present model, a high number of people -and thus a low space per head -can come not only from a large number of births, but, also, from a large length of life. To model this, we shall, for simplicity, assume that all births are concentrated on the …rst period.

Axiom 7 All births are concentrated on the …rst period, t = 0, and form the initial population L of …nite size.

By doing so, we avoid discussions on whether all agents should be born at an early point in time or, on the contrary, should all be brought to life at the latest possible moment. This simpli…cation is not problematic for the purpose at hand, as our goal is to focus on trade-o¤s between births and deaths, whatever the precise timing of births is.

Regarding the modeling of the survival process, we shall assume that, whereas the whole initial population L will enjoy the …rst period of life, only a proportion S of that population will survive up to the second period, and, then, only a proportion S of the surviving population will enjoy a third period of life, etc.

Axiom 8 At each passage of time, a fraction S of the population alive at a period survives to the next.

Thus, under those assumptions, the population at time t can be written as:

L t = S t L
where 0 < S < 1 is the proportion of survivors within a cohort after the passage of a unitary period of time. Actually, Axiom 8 amounts to assume that the strength of mortality is constant over time.

Note that, under Axiom 8, the total number of periods P lived by a population of initial size L is:

P = L + SL + S 2 L + ::: = L 1 + S + S 2 + ::: = L 1 1 S
Thus, if S tends towards 0, P tends towards L: the number of periods lived equals the number of persons. However, if S tends towards 1, P tends towards +1: as there is no death, the number of periods lived by a group of strictly positive size is in…nite.

Finally, regarding the de…nition of space per head q t , it follows from the previous formulae that, if the proportion of survivors S tends to 1, the space available per person is constant over time, and equal to Q=L, as:

lim S!1 q t = lim S!1 Q S t L = Q 1 t L = Q L
In that case, the population that is alive initially will remain forever in the spaceship, and will enjoy the same space conditions during their whole life.

On the contrary, if S tends towards 0, the space available per person q t tends to be in…nite for any t > 0, as:

lim S!0 q t = lim S!0 Q S t L = Q 0 t L = Q 0 = +1
Indeed, as S tends towards 0, the survivors can enjoy an in…nitely large space at each period of survival.

Problem setting

In order to study how a social planner would solve the spaceship problem, de…ned here as the choice of a number of persons and a proportion of survivors under the space constraint, we shall, in the rest of this paper, assume that the social planner has to choose among a set of histories, de…ned as follows.

De…nition 9 A history is a pair fL; Sg, where L is the size of the population at time 0 (0 < L), and S is the proportion of survivors from a period t to a period t + 1 (0 < S < 1).

In order to describe the planner's solution, we shall concentrate on a particular subset of the set of histories: lifetime-equal histories, de…ned as follows.

De…nition 10 Two histories fL; Sg and fL 0 ; S 0 g are lifetime-equal if and only if these exhibit an equal total number of periods lived (i.e. P = P 0 ).

Thus, we shall concentrate on histories fL; Sg and fL 0 ; S 0 g such that:

L 1 1 S = L 0 1 1 S 0
Obviously, under P = P 0 , L T L 0 () S S S 0 . In other words, for a given total number of periods lived, if one lifetime-equal history exhibits a higher initial population than another lifetime-equal history, it must also necessarily exhibit a lower survival probability, et vice versa.

Note that comparing two lifetime-equal histories fL; Sg and fL 0 ; S 0 g amounts to compare two histories for which there is an equality of the ratios:

L L 0 = 1 S 1 S 0
that is, there is an equality of the ratios of initial populations (LHS) and strengths of mortality (RHS) for histories fL; Sg and fL 0 ; S 0 g having an equal number of periods lived (i.e. for which P = P 0 ). Intuitively, if one history exhibits a larger initial population than another history with the same total number of periods lived (i.e. L > L 0 ), it must also be characterized by a larger mortality (i.e. 1 S > 1 S 0 ).

The Classical utilitarian solution

According to Classical utilitarianism, as …rstly stated by [START_REF] Bentham | Introduction to the Principles of Morals and Legislation[END_REF], all actions -at the individual and institutional levels -should be chosen in such a way as to produce the 'greatest happiness of the greatest number', in conformity with the Principle of Utility. That principle must also govern the choice of histories in general, and, in particular, of lifetime-equal histories. 10 De…nition 11 A social planner is Benthamite if and only if, when facing two histories fL; Sg and fL 0 ; S 0 g, he prefers the history yielding the highest total welfare, that is, fL; Sg fL 0 ; S 0 g ()

P 1 t=0 L t u(q t ) P 1 t=0 L 0 t u(q 0 t ) () P 1 t=0 LS t u( Q LS t ) P 1 t=0 L 0 S 0t u( Q L 0 S 0t )
What will be the attitude of a Benthamite social planner in front of the spaceship problem? Actually, it is tempting to believe that a Classical utilitarian social planner will be indi¤erent between lifetime-equal histories. The intuition behind that belief is that, from a Benthamite perspective, whether there exist a large number of people with short lives, or, on the contrary, a small number of people with long lives, should not matter that much. But such an intuition is untrue, as there is an additional dimension to the problem here: the impact of population density on individual welfare.

Contrary to what one may believe, a Benthamite social planner is not necessarily indi¤erent between lifetime-equal histories. His preferences are characterized in the following proposition.

Proposition 12 Consider two lifetime-equal histories fL; Sg and fL 0 ; S 0 g, with L > L 0 and S < S 0 . The preferences of the Benthamite social planner can be characterized as follows:

fL; Sg fL 0 ; S 0 g () 1 S 1 S 0 1 1 S 1 1 S 01
Proof. The Benthamite social planner prefers fL; Sg on fL 0 ; S 0 g if and only if total welfare is larger under fL; Sg than under fL 0 ; S 0 g : Social welfare is equal to

L Q L + SL Q SL + S 2 L Q S 2
L + ::: + :::

= L Q L h 1 + S 1 + S 2
(1 ) + ::: + :::

i = L Q L 1 + s + s 2 + ::: + ::: = L Q L 1 1 s where s S 1 .
Hence, social welfare is larger under fL; Sg than under fL 0 ; S 0 g i¤:

L Q L 1 1 S 1 L 0 Q L 0 1 1 S 0 1 L 1 1 1 S 1 L 01 1 1 S 0 1 Given that L 1 S = L 0 1 S 0 , we have L L 0 1 = 1 S 1 S 0 1
: Thus social welfare is larger under fL; Sg than under fL 0 ; S 0 g i¤:

1 S 1 S 0 1 1 S 1 1 S 0 1
Thus, contrary to what one expects a priori, the Benthamite social planner is not necessarily indi¤erent between lifetime-equal histories. Note also that the Benthamite planner's preferences on lifetime-equal histories are independent from the size of the available space Q. The only parameter that determines the planner's preferences on two lifetime-equal histories fL; Sg and fL 0 ; S 0 g is the parameter describing individual taste for space.

In the general case where 0 6 = 6 = 1, it is not obvious to see whether the Benthamite social planner prefers a high population with low survival or the opposite. This indeterminacy comes from the in ‡uence of population density on individual welfare. Initially, individual welfare per period is larger in fL 0 ; S 0 g, as the initial population is less numerous. However, after some periods, the higher mortality in fL; Sg makes population density fall there much more than under fL 0 ; S 0 g, so that the time pro…les of utility per period di¤er signi…cantly between fL; Sg and fL 0 ; S 0 g. Those di¤erent temporal utility paths explain why the preference parameter plays such an important role. Actually, the parameter , by governing the concavity of temporal welfare, is a major determinant of the spaceship problem regarded as a problem of intertemporal allocation of persons.

While it is not obvious at all to see whether the Benthamite social planner prefers lifetime-equal histories that exhibit a larger population and a lower survival or the opposite, it is possible to see what his preferences would be in the extreme case where = 0.

Corollary 13 Under a constant temporal welfare -i.e. = 0 -, the Benthamite social planner is indi¤ erent between any two lifetime-equal histories.

Proof. This corollary follows from the above condition. Fixing = 0 in

1 S 1 S 0 1 1 S 1 1 S 0 1 yields: 1 S 1 S 0
1 S 1 S 0 which is a strict equality, so that fL; Sg fL 0 ; S 0 g for any lifetime-equal histories fL; Sg and fL 0 ; S 0 g :

The intuition behind that corollary is the following. If temporal welfare is constant, the quality of life periods does not matter at all: only the number of periods lived matters, so that lifetime-equal histories are regarded as equivalent. In other words, when = 0, the e¤ect of population density is completely neutralized, so that having many people with short lives or few people with long lives is the same.

Finally, note also that, if = 1, the social planner cannot rank lifetime-equal histories, as the above condition does not yield any conclusion.

Corollary 14 Under a linear temporal welfare -i.e. = 1 -, there can be no ranking of the Benthamite social planner over any two lifetime-equal histories.

Proof. Fixing = 1 in the above condition yields:

1 0 0
which cannot be said to be true or false, as the RHS is indeterminate.

Thus, if individual utility is linear in space, the spaceship problem has no solution under Classical utilitarianism.

Millian utilitarianism

Although accepted as an ethical basis in various issues, the classical utilitarian doctrine faces intuitive di¢ culties in the …eld of population ethics. A major critique of Classical utilitarianism in the context of population choices consists of [START_REF] Par…t | Reasons and Persons[END_REF] Repugnant Conclusion: for any population with a signi…cant welfare per head, there exists a larger population of individuals with a very low welfare level, but which is ranked as better by Classical utilitarianism (the rise in the quantity of lives compensating the fall in the quality of lives).

One alternative criterion, which avoids the Repugnant Conclusion, is Millian utilitarianism, which recommends to maximize not total welfare, but average welfare. As this is well-known, this makes a di¤erence when considering population issues. In the present context, the problem of a Millian social planner is to choose a history fL; Sg in such a way as to maximize the average utility within the population.

De…nition 15 A social planner is Millian if and only if, when facing two histories fL; Sg and fL 0 ; S 0 g, he prefers the history yielding the highest average welfare (i.e. the highest welfare per existing person), that is, fL; Sg fL 0 ; S 0 g ()

1 L P 1 t=0 L t u(q t ) 1 L 0 P 1 t=0 L 0 t u(q 0 t ) () P 1 t=0 S t u( Q LS t ) P 1 t=0 S 0t u( Q L 0 S 0t )
How would a Millian social planner rank lifetime-equal histories? Is there here some indeterminacy as in the Benthamite case? As stated below, this is clearly not the case, as a Millian social planner, when facing lifetime-equal histories, chooses necessarily the history with the lower population and the higher survival.

Proposition 16 Consider two lifetime-equal histories fL; Sg and fL 0 ; S 0 g, with L > L 0 and S < S 0 . The preferences of the Millian social planner can be characterized as follows:

fL; Sg fL 0 ; S 0 g Proof. Under Millian utilitarianism, social welfare is equal to average welfare

L Q L + SL Q SL + S 2 L Q S 2 L + ::: + ::: L = Q L h 1 + S 1 + S 2
(1 ) + ::: + :::

i = Q L 1 + s + s 2 + ::: + ::: = Q L 1 1 s where s S 1 .
Hence, social welfare is larger under fL; Sg than under fL 0 ; S 0 g i¤:

Q L 1 1 S 1 Q L 0 1 1 S 0 1 L 0 L 1 S 1 1 S 0 1 Given that L 0 L = 1 S 0 1 S ; we have 1 S 0 1 S 1 S 1 1 S 0 1
which is always false, as the LHS is smaller than 1, whereas the RHS exceeds 1. Hence we have always fL; Sg fL 0 ; S 0 g. Thus, contrary to what prevailed under Benthamite utilitarianism, the Average utilitarian criterion leads to the unambiguous selection of histories with few people and extremely long lives. That result is independent from the total surface available, and from the preference parameter .

The intuition behind that result goes as follows. Under a given number of periods lived, it is always better, for a Millian social planner, to concentrate these on the smallest number of individuals. True, the high survival implies that population density will hardly fall over time, so that temporal welfare per period will hardly grow. But the few existing persons will enjoy this quasiconstant temporal welfare during a long period of time, which is, on average, much better than the situation where more people would enjoy it over a shorter period.

Critical-level utilitarianism

Whereas Millian utilitarianism avoids the Repugnant Conclusion, it su¤ers from another weakness, which was called by [START_REF] Par…t | Reasons and Persons[END_REF] the Mere Addition Paradox. Clearly, Average utilitarianism regards as undesirable the addition of a group of people with an individual welfare that is slightly lower than the initial average welfare, even if the additional people have a life worth living, and even if the added people do not in ‡uence in any way the lives of the initial group.

In order to avoid that counter-intuitive result -but without facing the Repugnant Conclusion -, one solution proposed by [START_REF] Blackorby | Social criteria for evaluating population change[END_REF] consists of summing, instead of absolute utilities, utilities net of the utility level that makes a life neutral. This yields Critical-level utilitarianism: a rise in the population is desirable if the welfare of additional people exceeds the critical level, and undesirable when it is lower than it. 11 Under Critical-level utilitarianism, the problem of the social planner is to choose a history fL; Sg in such a way as to maximize the sum of all cohorts's net welfares. In conformity with Blackorby and Donaldson's intuition, an agent's welfare is here de…ned not absolutely, but with respect to some critical level of welfare û (û 0) making life -as a whole -neutral. A life with a welfare higher than û is regarded as socially desirable, whereas a life with a lower welfare level is regarded as undesirable. A life with u = û is 'neutral'.

De…nition 17 A social planner is Blackorbian if and only if, when facing two histories fL; Sg and fL 0 ; S 0 g, he prefers the history yielding the highest net welfare, that is, fL; Sg fL 0 ; S 0 g ()

P 1 t=0 L t u(q t ) Lû P 1 t=0 L 0 t u(q 0 t ) L 0 û () L h P 1 t=0 S t u( Q LS t ) ûi L 0 h P 1 t=0 S 0t u( Q L 0 S 0t
) ûi where û is the critical welfare level, making a whole life neutral.

It is crucial here to distinguish the introduction of that neutral level for existence from the introduction of what [START_REF] Broome | Weighing Lives[END_REF] calls a 'neutral level for continuing existence' (i.e. a critical level de…ned for each period of life). The introduction of such a critical level for continuing existence would not alter the Benthamite condition for fL; Sg fL 0 ; S 0 g, unlike what is the case under the introduction of a lifetime critical level û: 12 The preferences of a Blackorbian social planner on lifetime-equal histories are characterized by the following proposition.

Proposition 18 Consider two lifetime-equal histories fL; Sg and fL 0 ; S 0 g, with L > L 0 and S < S 0 . The preferences of the Blackorbian social planner can be characterized as follows:

fL; Sg fL 0 ; S 0 g ()

Q h L 1 1 S 1 L 01 1 S 0 1 i û (L L 0 )
Proof. The Blackorbian social planner prefers fL; Sg on fL 0 ; S 0 g if and only if social welfare is larger in the former.

Social welfare is here equal to

L Q L + S Q LS + S 2 Q LS 2 + ::: Lû = L Q L h 1 + S 1 + S 2
(1 ) + ::: + :::

i Lû = L Q L 1 + s + s 2 + ::: + ::: Lû = L Q L 1 1 s Lû where s S 1 .
Hence, social welfare is larger under fL; Sg than under fL 0 ; S 0 g i¤:

L Q L 1 1 S 1 Lû L 0 Q L 0 1 1 S 0 1 L 0 û L Q L 1 1 S 1 L 0 Q L 0 1 1 S 0 1 û (L L 0 )
1 2 To see this, note that social welfare under such a critical level ũ is:

L Q L 1 + S 1 + S 2
(1 ) + ::: + ::: Lũ 1 + S + S 2 + :::

= L Q L 1 1 s Lû 1 1 S : Hence, fL; Sg fL 0 ; S 0 g () L 1 1 S 1 L 01 1 S 0 1 0 as under Benthamite utilitarianism.
Thus social welfare is larger under fL; Sg than under fL 0 ; S 0 g i¤:

Q L 1 1 S 1 L 01 1 S 0 1 û (L L 0 )
Note that the condition for fL; Sg fL 0 ; S 0 g collapses to the Benthamite condition under û = 0. Moreover, under û equal to average lifetime welfare, the condition for fL; Sg fL 0 ; S 0 g vanishes to

Q L 1 1 S 1 L 01 1 S 0 1 1 X t=0 S t Q LS t (L L 0 ) Q L 1 1 S 1 L 01 1 S 0 1 Q L 1 1 S 1 (L L 0 ) " L 0 1 S 1 S 0 1 L 01 1 S 0 1 S L L 0 L 0 # L 01 1 S 1 1 S 0 1 1 S 0 1 S 1 S 1 1 S 0 1
which is never satis…ed, as under Average utilitarianism. One can thus interpret Critical-level utilitarianism as a generalization of both Benthamite and Millian utilitarianism.

Note also that, in comparison with the optimality condition under Benthamite utilitarianism, the RHS of the condition for fL; Sg fL 0 ; S 0 g is likely to be larger than zero, so that the Blackorbian social planner tends to exhibit a preference ordering on lifetime-equal histories that di¤ers signi…cantly from the one of the Benthamite social planner. More precisely, whereas the Benthamite social planner prefers fL; Sg over fL 0 ; S 0 g if and only if

L 1 1 S 1 exceeds L 01 1 S 0 1
, the Blackorbian social planner requires L 1 1 S 1 to exceed L 01 1 S 0 1 to a larger extent, which makes the preference for histories with large populations and low longevity less likely. This is the natural corollary of the introduction of a critical utility level.

Thus, without surprise, the critical utility level û a¤ects the planner's ordering of lifetime-equal histories: the larger û is, the more likely is the preference for histories with a small number of persons and a long life. This dependency suggests that how the spaceship problem is solved depends crucially on the selection of an adequate critical utility level. Recent debates on how the critical utility level should be …xed (see [START_REF] Crisp | Neutrality and pleasure[END_REF][START_REF] Broome | Replies[END_REF] are thus not neutral as far as the spaceship problem is concerned.

One should also notice here that the available space Q does, under 6 = 0, in ‡uence the social planner's ordering on lifetime-equal histories, unlike what used to be the case under Classical and Millian utilitarianisms. Clearly, for a given critical utility level, the larger the available space is, the more likely the preference for a lifetime-equal history with a larger population is. This dependency on the available space is quite surprising, because both L and S a¤ect the space per person, so that one does not expect Q to in ‡uence the ranking between fL; Sg and fL 0 ; S 0 g, which exhibit, by construction, the same number of periods lived. But contrary to the intuition, the critical level does in ‡uence the solution of the spaceship problem.

Number-dampened utilitarianism

While Critical-level utilitarianism has become widely used as a criterion in population ethics, it can only avoid the Repugnant Conclusion and the Mere Addition Paradox for extremely-well selected critical levels û. If the critical level is too low, the Repugnant Conclusion arises, whereas if it is too high, the Mere Addition Paradox or the Sadistic Conclusion prevails. This observation -and the di¢ culties to select a single value of û -encouraged the development of other criteria not relying on a critical utility level. 13 One of those alternative criteria is [START_REF] Ng | Social criteria for evaluating population change: an alternative to the Blackorby-Donaldson criterion[END_REF] Number-dampened utilitarianism. 14 The value function under Number-dampened utilitarianism is equal to the average utility multiplied by a positive-valued function of the population size. When that function is a multiple of the population size, that criterion is equivalent to Benthamite utilitarianism, whereas, if that function is a constant, number-dampened utilitarianism is equivalent to Millian utilitarianism.

Under number-dampened utilitarianism, the problem of a social planner is to choose fL; Sg in such a way as to maximize the sum of all cohorts's welfares. Following Ng's intuitions, the welfare of a cohort is de…ned as the product of the average utility in that cohort and a concave transform of its size.

De…nition 19 A social planner is Ngian if and only if, when facing two histories fL; Sg and fL 0 ; S 0 g, he prefers the history yielding the highest average welfare multiplied by a concave transform of the population brought into existence, that is, fL; Sg fL 0 ; S 0 g ()

L 1 L P 1 t=0 L t u(q t ) L 0 1 L 0 P 1 t=0 L 0 t u(q 0 t ) () L P 1 t=0 S t u( Q LS t ) L 0 P 1 t=0 S 0t u( Q L 0 S 0t ) where 0 1.
Let us now consider how an Ngian social planner would rank lifetime-equal histories fL; Sg and fL 0 ; S 0 g. As stated in the proposition below, the preferences of an Ngian social planner on lifetime-equal histories can be characterized in a simple way.

Proposition 20 Consider two lifetime-equal histories fL; Sg and fL 0 ; S 0 g ;with L > L 0 and S < S 0 . The preferences of the Ngian social planner can be characterized as follows: fL; Sg fL 0 ; S 0 g ()

1 S 1 S 0 1 S 1 1 S 0 1 .
Proof. The Ngian social planner prefers fL; Sg on fL 0 ; S 0 g if and only if social welfare is larger in the former.

Social welfare is here equal to

L " 1 X t=0 S t Q LS t # = L Q L h 1 + S 1 + S 2 (
1 ) + ::: + :::

i = L Q L 1 + s + s 2 + ::: + ::: = L Q L 1 1 s
where is lower than 1. In conformity with Ng's intuitions, the marginal e¤ect of each additional person in terms of welfare is decreasing with the number of persons.

Hence, social welfare is larger under fL; Sg than under fL 0 ; S 0 g i¤:

L Q L 1 1 S 1 L 0 Q L 0 1 1 S 0 1 L 1 1 S 1 L 0 1 1 S 0 1 1 S 1 S 0 1 S 1 1 S 0 1
Note that, under equal to 1, we are back to the condition prevailing under Benthamite utilitarianism. Alternatively, under equal to 0, we are back to the Average utilitarian condition for fL; Sg fL 0 ; S 0 g. Number-dampened utilitarianism is thus an alternative generalization of standard utilitarian criteria, along with Critical-level utilitarianism.

However, a fundamental di¤erence with respect to the latter lies in the fact that the ordering of an Ngian planner on lifetime-equal histories is independent from the total space available Q, unlike what used to be the case under Criticallevel utilitarianism. Such an independence is surprising, as one may expect that, as a generalization of standard utilitarian criteria alternative to Critical-level utilitarianism, Number-dampened utilitarianism should exhibit a dependency on the available space, exactly as Critical-level utilitarianism does. But this is not the case, and this suggests that the dependency of the planner's ranking on Q is speci…c to the introduction of a critical utility level.

That signi…cant di¤erence between Critical-level utilitarianism and Numberdampened utilitarianism can play in favour of one or the other criterion, depending on whether one regards the dependency on the available space as a desirable property or not.

Concluding remarks

The goal of this paper was to cast a new light on the spaceship problem, by paying a particular attention to the trade-o¤s between adding new persons and extending the life of existing persons. For that purpose, we assumed that individual lifetime welfare is additive over time, and that temporal welfare depends negatively on population density, and we characterized, under those assumptions, the preference orderings of a utilitarian social planner on lifetime-equal histories, which all exhibit, by construction, an equal number of periods lived.

We showed that, contrary to the intuition, a Classical utilitarian planner is not necessarily indi¤erent between lifetime-equal histories, because of the non-constancy of population density over time, which, under non-linear utility in space per head, makes the Classical utilitarian planner prefer one history over another even though in each case the number of life periods is exactly the same. We showed also that an Average utilitarian planner would always opt for the history with the smallest number of people and the largest longevity. The Critical-level utilitarian solution and the Number-dampened utilitarian solution were also contrasted, and, quite surprisingly, were shown to di¤er in their treatment of the available space, the former being sensitive to it, while the later is insensitive to whether there is a large or a small space available for life.

While our re-examination of the spaceship problem casts a new light on that old issue, it should be stressed, however, that some re…nements of this analysis could be made, to explore the sensitivity of our conclusions to the postulates on which these rely. Assumptions at the individual level, such as the zero utility from death, and the additivity of lifetime welfare, may be questioned, and one may want to know how the spaceship problem is solved once those postulates are relaxed. Hence this work is only a …rst stage in the re-examination of the spaceship problem, which invites many others.

On this, seeSchumpeter (1954, 1, pp. 352-356).

See Cantillon's comparison of peasants living modestly in the South of France with grownup bourgeois living in abundance (1755, I, 15 p. 25). Cantillon argued also that land-owners, by deciding the allocation of land between di¤erent uses, in ‡uence the set of feasible population sizes.

3 SeeCantillon (1755, I, 15, p. 30):

Note that, in later writings, Malthus (1830) will add a third type of check: the moral restraint.

The Repugnant Conclusion is de…ned as follows: for any population with some welfare per head, there exists a larger population of individuals with a very low welfare per head, but which is regarded as better. The Mere Addition Paradox consists of regarding as undesirable the addition of a group of people with an average welfare that is only slightly lower than the initial average welfare, even if additional people have a life worth living and do not a¤ect the welfare of the initial group of people.

The Sadistic Conclusion consists of the mere fact that …xing a high critical utility level may make one prefer a situation where a small population exhibits an extremely low utility level to a situation where a very large population is just below the critical level, which is counter-intuitive under a high critical level. See also[START_REF] Bykvist | The good, the bad and the ethically neutral[END_REF].

Actually, as Heilig (1994) rightly argues in the light of (mistaken) past demographic studies on the maximum carrying capacity of the Earth in terms of population, the …niteness of space does not seem to be a problem for production, so that we can abstract from that constraint.

Naturally, this paper has no pretension to exhaustiveness, and will concentrate on some major normative criteria under varying population size. See[START_REF] Blackorby | Population Issues in Social Choice Theory, Welfare Economics, and Ethics[END_REF] for a synthesis of population ethics.

In other words, the dead are supposed to occupy no space.

0 Note that the Benthamite objective function does not involve any discounting here, in conformity withRamsey's (1928) views. Note also that the above formula presents social welfare by aggregating across time periods rather than persons (as usually done), but this is equivalent to aggregating across people, because of the additivity of lifetime welfare.

1 As this is well-known, Benthamite utiltiarianism amounts to Critical-level utilitarianism under a zero critical level, while Millian utilitarianism equals Critical level utilitarianism when the critical level equals the average welfare.

3 One solution is the adherence to Critical-band utilitarianism (see Balckorby et al,

2005), which allows some interval of individual welfare levels between which a life is neutral.1 4 On Number-dampened utilitarianism, see also[START_REF] Hurka | Value and population size[END_REF].