Soc Choice Welf DOI 10.1007/s00355-016-0953-4 ORIGINAL PAPER

Infinite-horizon social evaluation with variable population size Kohei Kamaga1

Received: 4 July 2014 / Accepted: 11 February 2016 © Springer-Verlag Berlin Heidelberg 2016

Abstract We present an infinite-horizon extension of the framework of variablepopulation social choice. Our first main result is the welfarism theorem using the axiom of intratemporal anonymity. By this theorem, the ranking of social alternatives is determined by an intratemporally anonymous and finitely complete quasi-ordering [which we call social welfare relation (SWR)] defined on the set of all streams of utility vectors of generations. We introduce three SWRs: the critical-level generalized utilitarian (CLGU) SWR, the critical-level generalized overtaking (CLGO) SWR, and the critical-level generalized catching-up (CLGC) SWR. They are infinite-horizon extensions of the critical-level generalized utilitarianism. We characterize (in terms of subrelation) the CLGU SWR with five axioms: Strong Pareto, Finite Anonymity, Weak Existence of Critical Levels, Restricted Continuity, and Existence Independence. Further, the CLGO and the CLGC SWRs are characterized by adding consistency axioms. We also present infinite-horizon reformulations of some population ethics axioms. In particular, we characterize the CLGO and the CLGC SWRs associated with a positive critical level by using the axiom of avoidance of the repugnant conclusion.

1 Introduction Since the seminal works of Koopmans (1960) and Diamond (1965), the social evaluation of the well-being of infinitely many generations has been studied using the framework of ranking infinite utility streams.1 In this framework, the well-being of each generation is represented by a single utility value. This simplification is useful

1 See Asheim (2010) and Lauwers (2014) for reviews of the literature.

B 1

Kohei Kamaga [email protected] Faculty of Economics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo 102-8554, Japan

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for focusing on the conflict of interest between generations. Further, the evaluation relations established in this framework can be applied to economic growth models to examine intergenerational sustainability of an optimal growth path (e.g., Asheim 1991; Asheim et al. 2001; Asheim and Mitra 2010; Asheim et al. 2012; Zuber and Asheim 2012). For complicated intergenerational problems, however, this framework has two restrictive aspects. First, the diverse levels of well-being of individuals within a generation are not taken into account. Some intergenerational problems, such as prevention of global warming, require a consideration of the conflict of interest within a generation as well as between generations. Second, it cannot deal with demographic changes across generations. For intergenerational problems where demographic changes across generations matter, such as for the design of a population policy to reverse a declining birthrate, the population size of each generation needs to be considered. The relationship between population growth and economic growth is one of the issues studied in economic growth theory, and an optimal population size is analyzed through economic growth models with endogenous population growth (e.g., Boucekkine and Fabbri 2013; Boucekkine et al. 2011, 2014; Palivos and Yip 1993; Razin and Yuen 1995). These models define an optimality criterion for infinite streams of finite- and variabledimensional utility vectors and are, therefore, not amenable to evaluation relations for infinite utility streams. In social choice theory, social evaluation with variable population size has been studied in the framework so-called variable-population social choice, where evaluation relations for finite- and variable-dimensional utility vectors are analyzed. The literature can be traced back to Blackorby and Donaldson (1984), who propose the critical-level generalized utilitarian (CLGU) ordering. The CLGU ordering ranks utility vectors by comparing the total sums of the differences between transformed utilities and a transformed fixed critical level of utility. Several works propose more generalized orderings that include the CLGU ordering as a special case.2 Among them, recent work by Asheim and Zuber (2014) introduces the rank-discounted critical-level generalized utilitarian (RDCLGU) ordering. The RDCLGU ordering compares the discounted sums of the differences between transformed utilities and a transformed fixed critical level of utility by using the geometric discounting that depends on the ranking of individuals’ utility levels. The RDCLGU ordering coincides with the CLGU ordering if the discount factor is unity. Further, it approaches the critical-level leximin ordering suggested by Arrhenius (2011) as the discount factor goes to zero.3 Although many evaluation relations have been proposed in variable-population social choice, they cannot be used, at least directly, to rank infinite-horizon utility distributions and, consequently, cannot be applied to economic growth models with endogenous population growth to examine intergenerational sustainability.4

2 See Blackorby et al. (2002, 2005) for reviews of the literature. 3 A variant of the critical-level leximin ordering is proposed by Blackorby et al. (1996). 4 Boucekkine et al. (2011) apply the CLGU ordering to an endogenous growth model by using time discounting and assuming that a transformation of utility levels is the identity mapping. However, the axiomatic foundation of their evaluation relation is not discussed.

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The purpose of this paper is to (1) present a framework for the analysis of infinitehorizon social evaluation with variable population sizes; (2) establish two versions of the welfarism theorem that justify our focus on streams of utility vectors in the social evaluation; and (3) introduce and axiomatize some infinite-horizon reformulations of the CLGU ordering. The framework we present is an infinite-horizon extension of the finite-horizon framework of variable-population social choice in Blackorby et al. (2005). In our extended framework, social alternatives are paths of temporal social states that do not necessarily include the same populations or population sizes at each time period. Each temporal social state is described by using the finite-horizon framework of variable-population social choice. For each path of temporal social states, an infinite-horizon profile of sets of individuals alive, interpreted as a profile of generations, is associated. Different paths of temporal social states may generate different profiles of generations. In the extended framework, we start by examining the aggregation rule called the social welfare functional, which associates a finitely complete quasi-ordering on the set of paths of temporal social states with each profile of the utility functions of individuals, each of whom is alive in some path of temporal social states. We consider a finitely complete quasi-ordering instead of an ordering satisfying full completeness because of the impossibility of explicit construction of Paretian and intergenerationally anonymous orderings for infinite utility streams (Dubey 2011; Lauwers 2010; Zame 2007)—an impossibility carries over to our framework.5 Our two versions of the welfarism theorem are infinite-horizon counterparts of those presented by Blackorby et al. (1999, 2005) in the finite-horizon framework of variable-population social choice. The first version of the welfarism theorem shows that a social welfare functional defined on the unlimited domain satisfies the axioms of Binary Independence of Irrelevant Alternatives (BIIA) and Pareto Indifference (PI) if and only if its ranking of the paths of temporal social states is determined by a single finitely complete quasi-ordering on the set of infinite-horizon profiles of pairs (listed from the first generation to future generations) of a set of individuals alive and a vector of their utility levels. In our second version of the welfarism theorem, we strengthen PI to Intratemporal Anonymity (IA). IA requires that any two paths of temporal social states be declared equally good if, at each generation, the corresponding utility vectors coincide by rearranging them. That is, it asserts that the identities of the individuals alive in each generation do not matter for the evaluation. The second version of the welfarism theorem shows that a social welfare functional on the unlimited domain satisfies BIIA and IA if and only if its ranking of the paths of temporal social states is determined by a single intratemporally anonymous and finitely complete quasiordering, which we generically call social welfare relation (SWR), on the set of streams of utility vectors (one vector for each generation). Based on the second version of the welfarism theorem, we examine an SWR for streams of utility vectors. We introduce three infinite-horizon variants of the CLGU ordering: the critical-level generalized utilitarian (CLGU) SWR, the critical-level generalized overtaking (CLGO) SWR, and the critical-level generalized catching-up (CLGC) SWR. The CLGU SWR ranks streams of utility vectors by applying the 5 The impossibility of explicit construction of Paretian and anonymous orderings for infinite utility streams was first suggested by Fleurbaey and Michel (2003).

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CLGU ordering to the heads of streams and the Suppes–Sen grading principle to each generation in the tails of streams.6 The CLGO and the CLGC SWRs consecutively apply the CLGU ordering to the heads of the streams of utility vectors in the same way as the overtaking criterion in von Weizsäcker (1965) and the catching-up criterion in Atsumi (1965) and von Weizsäcker (1965). We provide axiomatic characterizations of these three SWRs in terms of subrelation.7 The CLGU SWR is characterized by Strong Pareto (SP), Finite Anonymity (FA), Weak Existence of Critical Levels (WECL), Restricted Continuity (RC), and Existence Independence (EI) axioms. The first two axioms are reformulations of the standard Pareto and intergenerational anonymity axioms used in the framework of ranking infinite utility streams. The last three are reformulations of the axioms used in the finite-horizon frameworks of fixed- and variable-population social choice. As we will discuss in detail in Sect. 3.1, a notable point in our characterization result is that while none of the five axioms by itself implies an intergenerational application of the CLGU ordering, its intragenerational application is extended to the intergenerational one by FA and EI. The CLGO and the CLGC SWRs are characterized by using additional axioms of Weak Preference Consistency (WPC), Strong Preference Consistency (SPC), and Indifference Consistency (IC). These axioms formalize consistency between the evaluation of the general streams of utility vectors and that of those with a common tail. The CLGO SWR is characterized by the additional use of WPC and IC. The CLGC SWR is characterized by strengthening WPC to SPC. In this paper, we also present a reformulation of the repugnant conclusion due to Parfit (1976, 1982, 1984). Parfit points out, in the context of the evaluation of social states with finite and variable population sizes, that classical utilitarianism leads to the repugnant conclusion that some social state in which each member of the population has a high level of utility is considered worse than some social state with a much larger population in which each member has a utility level corresponding to a life barely worth living. This argument is based on the convention in population ethics that utilities are normalized so that utility levels above zero represent lives worth living (see Sect. 2 for further details). We employ this convention and reformulate the repugnant conclusion in the current framework. To do this, we define the dominance relation for streams of population sizes of generations by strict dominance in population size at each generation. Using this dominance relation, we reformulate the repugnant conclusion and consider the axiom of avoidance of the repugnant conclusion (ARC). We then show that additionally imposing ARC, the CLGO and the CLGC SWRs associated with a positive critical level are characterized (in terms of subrelation). In addition to this result, we show some negative results that the CLGO and the CLGC SWRs associated with a positive critical level are incompatible with infinite-horizon reformulations of the established population ethics axioms called Avoidance of the Weak Repugnant Conclusion (AWRC), Avoidance of the Very Sadistic Conclusion (AVSC), and Mere 6 The Suppes–Sen grading principle was introduced by Suppes (1966) and further analyzed by Sen (1970) in the finite- and fixed-population framework. 7 A characterization in terms of subrelation means a characterization of the class of all SWRs that include

the SWR considered as a subrelation. The notion of subrelation is explained in Sect. 3.1.

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Addition Principle (MAP). Then, we discuss the appropriate choice of positive critical levels for the CLGO and the CLGC SWRs. The rest of the paper is organized as follows. Section 2 presents the infinite-horizon extension of the framework of variable-population social choice. We establish two versions of the welfarism theorem in Sect. 2.2. Section 3 introduces the CLGU, the CLGO, and the CLGC SWRs and provides their axiomatic characterizations. In Sect. 4, we reformulate the repugnant conclusion in the extended framework and present the characterizations of the CLGO and the CLGC SWRs associated with a positive critical level. We also evaluate the CLGO and the CLGC SWRs associated with a positive critical level using other population ethics axioms. Section 5 concludes the study.

2 Framework 2.1 Social welfare functional Let R be the set of all real numbers and N be the set of all positive integers. The sets of all positive and negative real numbers are denoted by R++ and R−− , respectively. For all n ∈ N, 1n is the vector consisting of n ones. For any set A, |A| is the cardinality of A. For any sets A and B, we write A ⊆ B to mean that A is a subset of B and A ⊂ B to mean A ⊆ B and A = B. The empty set is denoted by ∅. The notation for the vector inequality is as follows: for all n ∈ N and all (u 1 , . . . , u n ), (v1 , . . . , vn ) ∈ Rn , (u 1 , . . . , u n ) ≥ (v1 , . . . , vn ) if and only if u i ≥ vi for all i = 1, . . . , n; and (u 1 , . . . , u n ) > (v1 , . . . , vn ) if and only if (u 1 , . . . , u n ) ≥ (v1 , . . . , vn ) and (u 1 , . . . , u n ) = (v1 , . . . , vn ). Further, for all (u 1 , u 2 , . . .), (v1 , v2 , . . .) ∈ RN , (u 1 , u 2 , . . .)  (v1 , v2 , . . .) if and only if u i > vi for all i ∈ N. We extend the finite-horizon framework of variable-population social choice in Blackorby et al. (2005) to an infinite-horizon setting by using their framework to describe a temporal social state and considering an infinite-horizon path of temporal social states. Let us begin by describing a temporal social state. We generically use t ∈ N to denote the time period of the society. For all t ∈ N, X t is a set of temporal social states x t at t and x t is a complete description of all aspects of the society at t including the identities of the individuals alive, their population size, and their actions such as their consumption and fertility choice. The cardinality of X t will be explained later. Different sets of individuals may be alive in different states x t , y t ∈ X t . For all t ∈ N, we assume that there are infinitely many individuals who will potentially alive in some x t ∈ X t at t and that each of the potential individuals lives one period. For all t ∈ N, we denote an individual potentially alive at t by the ordered pair (i, t) ∈ N×{t}. This means that any individual who is potentially alive at t is never alive at different time periods t  = t. For all t ∈ N, the set of all individuals potentially alive at t is N × {t} and the set of all non-empty and finite subsets of N × {t} is denoted by N t . For all t ∈ N and all x t ∈ X t , the set of all individuals alive in x t is denoted by N(x t ), and n(x t ) = |N(x t )| is the population size. We assume that, for all t ∈ N and all x t ∈ X t , 0 < n(x t ) < ∞, and thus, N(x t ) ∈ N t . For all t ∈ N and all N ∈ N t , let

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K. Kamaga t ⊆ X t be the set of all temporal social states x t that satisfy N(x t ) = N. We assume XN t | ≥ 3 for all t ∈ N and all N ∈ N t . This assumption is needed to establish that |X N the two versions of the welfarism theorem in Sect. 2.2. Since |N t | = |N|, |X t | ≥ |N| for all t ∈ N. X ∞ = t∈N X t is the set of all infinite-horizon paths of temporal social states x = {x t }t∈N . To simplify the analysis, we ignore the feasibility of paths of temporal social states—the linkage that a particular temporal social state (including a set of individuals alive and their consumption levels) will arise from a particular temporal social state—and X ∞ is the set of alternatives to be ranked. For all x ∈ X ∞ and all t ∈ N, we refer to N(x t ) as the t-th generation in x. For all x ∈ X ∞ , N(x) = (N(x t ))t∈N is the profile of generations in x, and n(x) = (n(x t ))t∈N is the stream of t  = ∅ for all t ∈ N and all N ∈ N t , the set of all their population sizes. Since X N possible generation profiles is t∈N N t . For all t ∈ N and all (i, t) ∈ N×{t}, Uit : X it −→ R is a utility function of individual (i, t), where X it = {x t ∈ X t : (i, t) ∈ N(x t )} is the set of all temporal social states at t in which individual (i, t) is alive. Any individual’s life is neutral if it is as good as a life without any experience, and thus, an individual’s life is worth living if her/his utility level is above neutrality. We employ the convention in population ethics that for any individual, a utility level of zero represents neutrality.8 For all t ∈ N, a profile of utility functions of all potential individuals at t is denoted by U t = (Uit )i∈N = (U1t , . . . , Uit , . . .). A profile of the utility functions of all individuals who are potentially alive at some t ∈ N is a list of infinitely many components, one for each profile U t of utility functions at t, and is denoted by U = (U t )t∈N = (U 1 , . . . , U t , . . .). Henceforth, we refer to U as the profile of utility functions. The set of all logically possible profiles of utility functions is denoted by U. A binary relation B on X ∞ is finitely complete if, for all x, y ∈ X ∞ , x B y or   yB x whenever there exists t ∈ N such that x t = y t for all t  ≥ t. Let Q be the set of all logically possible finitely complete quasi-orderings on X ∞ .9 A social welfare functional is a mapping F : D −→ Q that associates a finitely complete quasiordering on X ∞ with each profile of utility functions in D, where D is the domain of F that satisfies ∅ = D ⊆ U. For all U ∈ D, we write F(U ) = RU , and PU and IU denote, respectively, the asymmetric and symmetric parts of RU . For all U ∈ D, all x ∈ X ∞  , and all  t ∈ N, the utility vector of the t-th generation in x is denoted by U(x t ) = Uit (x t ) (i,t)∈N(x t ) , where the individuals alive in x t are ordered from the smallest to the largest with respect to the first component of (i, t). t t t t t For example, if N(x t ) = {(3, t), (1, t)}, U(x  ) 1= (U1 (x ), tU3 (x )). For all U ∈ D ∞ t and all x ∈ X , let U(x) = (U(x ))t∈N = U(x ), . . . , U(x ), . . . denote the stream of the utility vectors of all generations in x. Let  = ∪n∈N Rn . Note that U(x) ∈ N for all U ∈ D and all x ∈ X ∞ . A typical element of N is denoted by u = (ut )t∈N = ((u 11 , . . . , u 1n(u1 ) ), . . . , (u t1 , . . . , u tn(ut ) ), . . .), where ut is the t-th vector in u, u it is the i-th element of ut , and 8 For a discussion of neutrality and its normalization to zero, see Broome (1993). 9 A quasi-ordering is a reflexive and transitive binary relation.

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n(ut ) denotes the number of components in ut for all t ∈ N. For any u, v ∈ N and t any t ∈ N, we write (ut , v t ) as (ut , v t ) = (u t1 , . . . , u tn(ut ) , v1t , . . . , vn(v t ) ) ∈ . For

any u ∈ N and any t ∈ N, let u−t = (u1 , . . . , ut ) and u+t = (ut+1 , ut+2 , . . .). Thus, u = (u−t , u+t ) = (u−(t−1) , ut , u+t ). For any t ∈ N and any u = (u−t , u+t ) ∈ N , we refer to u−t as the head of a stream of utility vectors and u+t as the tail of a stream of utility vectors. 2.2 Welfarism

In this section, we establish two versions of the welfarism theorem. The first shows that, assuming a social welfare functional F has an unlimited domain, F satisfies the axioms of Binary Independence of Irrelevant Alternatives and Pareto Indifference if and only if the ranking F(U ) of the paths of temporal social states is determined   by a single finitely complete quasi-ordering defined for profiles (N(x t ), U(x t )) t∈N of pairs of a set of individuals alive and their utility vector of all generations irrespective of the profile of utility functions U considered. The second shows that F additionally satisfies the axiom of Intratemporal Anonymity if and only if F(U ) is determined by a single intratemporally anonymous and finitely complete quasi-ordering defined for streams U(x) of utility vectors irrespective of the profile of utility functions U considered. All the axioms we use for the two versions of the welfarism theorem are reformulations of those used in the finite-horizon framework of variable-population social choice (e.g., Blackorby et al. 1999, 2005). The assumption of the unlimited domain of F is stated as the following axiom. Unlimited Domain (UD) D = U. Binary Independence of Irrelevant Alternatives requires that the ranking of any two paths of temporal social states depends only on the utilities of the individuals alive. Binary Independence of Irrelevant Alternatives (BIIA) For all x, y ∈ X ∞ and all U, V ∈ D, if U(x) = V(x) and U( y) = V( y), then x RU y ⇔ x R V y. Pareto Indifference requires that any two paths of temporal social states be declared equally good if the individuals alive and their utilities are the same. Pareto Indifference (PI) For all x, y ∈ X ∞ and all U ∈ D, if N(x) = N( y) and U(x) = U( y), then x IU y. Intratemporal Anonymity asserts that any two paths of temporal social states are equally good if the corresponding streams of the utility vectors of generations coincide by using a permutation in each generation. This requirement means that even if the individuals alive are different in two paths of temporal social states, these paths are declared equally good if the corresponding streams of the utility vectors coincide by using a permutation in each generation. In this sense, this requirement is stronger than the standard requirement of anonymity. Intratemporal Anonymity (IA) For all x, y ∈ X ∞ and all U ∈ D, if, for all t ∈ N, there exists a bijection μt : N(x t ) −→ N(y t ) such that Uit (x t ) = U tj (y t ) for all (i, t) ∈ N(x t ) and all ( j, t) = μt ((i, t)), then x IU y.

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Note that IA implies the following property: for all x, y ∈ X ∞ and all U ∈ D, x IU y if U(x) = U( y). This property implies PI. Thus, IA implies PI. Let us present some further notation and definitions for the two versions of the welfarism theorem. Let  = {((N(x t ), U(x t )))t∈N : x ∈ X ∞ and U ∈ U} be the set of all possible profiles of pairs of a set of individuals alive and their utility vector of all generations. A binary relation R on  is finitely complete if and only if, for all ((Mt , ut ))t∈N , ((Nt , v t ))t∈N ∈ , ((Mt , ut ))t∈N R((Nt , v t ))t∈N or   ((Nt , v t ))t∈N R((Mt , ut ))t∈N whenever there exists t ∈ N such that (Mt , ut ) =   (Nt , v t ) for all t  ≥ t, where Mt and Nt denote an element of N t for all t ∈ N. Next, we define some properties of a binary relation R ∗ on N . For any binary relation R ∗ on N , let P ∗ and I ∗ respectively denote the asymmetric and symmetric parts of R ∗ . A binary relation R ∗ on N is intratemporally anonymous if and only if, for all u, v ∈ N , t ∈ N, there exists a bijection μt : {1, . . . n(ut )} −→ {1, . . . n(v t )} uI ∗ v if, for all   such that ut = vμt t (1) , . . . , vμt t (n(ut )) . A binary relation R ∗ on N is finitely complete if and only if, for all u, v ∈ N , uR ∗ v or v R ∗ u whenever there exists t ∈ N   such that ut = v t for all t  ≥ t. In the following theorem, we present the two versions of the welfarism theorem. They are infinite-horizon variants of those obtained by Blackorby et al. (1999, 2005) in the finite-horizon framework of variable-population social choice.10 We omit the proof of the theorem since it is a replication of their proof method (available in Kamaga 2016). Theorem 1 Suppose that a social welfare functional F satisfies UD. (i) F satisfies BIIA and PI if and only if there exists a finitely complete quasi-ordering R on  such that for all x, y ∈ X ∞ and all U ∈ D,     x RU y ⇔ (N(x t ), U(x t )) t∈N R (N(y t ), U(y t )) t∈N .

(1)

(ii) F satisfies BIIA and IA if and only if there exists an intratemporally anonymous and finitely complete quasi-ordering R ∗ on N such that for all x, y ∈ X ∞ and all U ∈ D, x RU y ⇔ U(x)R ∗ U( y).

(2)

Hereafter, we call an intratemporally anonymous and finitely complete quasiordering R ∗ on N a social welfare relation (SWR). In the rest of the paper, we assume that a social welfare functional F satisfies the axioms in Theorem 1 (ii), and we will analyze an SWR R ∗ on N instead of F. Here, let us make a remark on the analysis of R ∗ . The analysis of R ∗ on N can be interpreted as that of the ranking F(U ) of the paths of temporal social states, which is consistently applicable to any profile U of utility functions in the sense of BIIA. Instead, if one wants to interpret the analysis of R ∗ on N as that of F(U ) obtained for a single profile U , then for some U , the set Uit (X it ) of attainable utilities needs to be R for each (i, t). This is guaranteed 10 See also d’Aspremont (1985, 2007) and d’Aspremont and Gevers (1977) for similar results in the finite-

and infinite-horizon fixed-population frameworks.

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Infinite-horizon social evaluation with variable. . . t | = |R| for all t ∈ N and all N ∈ N t since it implies |X t | = |R| by assuming that |X N i for all (i, t). Note that Theorem 1 holds even under this assumption.

3 Infinite-horizon critical-level generalized utilitarianism 3.1 Critical-level generalized utilitarian SWR In this section, we introduce and provide axiomatic characterizations of three infinitehorizon variants of the critical-level generalized utilitarian ordering in Blackorby and Donaldson (1984): the critical-level generalized utilitarian (CLGU) SWR, the critical-level generalized overtaking (CLGO) SWR, and the critical-level generalized catching-up (CLGC) SWR.11 Let us begin with the CLGU SWR. It is associated with a given parameter α ∈ R called the critical level of utility, which is the utility level such that the addition of an individual with that utility level at some generation does not change the goodness of a stream of utility vectors. Formally, for any u ∈ N and any t ∈ N, we say that α ∈ R is a critical level for u at the t-th generation if uI ∗ (u−(t−1) , (ut , α), u+t ). To define the CLGU SWR, we need to first define the quasi-ordering on  from Suppes (1966) and Sen (1970), which we call the intratemporal Suppes– Sen grading principle. It is defined as the following binary relation  on : for all (u 1 , . . . , u m ), (v1 , . . . , vn ) ∈ , (u 1 , . . . , u m )  (v1 , . . . , vn ) if and only if m = n and there exists a permutation μ on {1, . . . , n} such that (u 1 , . . . , u n ) ≥ (vμ(1) , . . . , vμ(n) ). We denote the asymmetric and symmetric parts of  by  and ∼, respectively. It is easy to check that  is a quasi-ordering. Further, for all (u 1 , . . . , u n ), (v1 , . . . , vn ) ∈ , (i) (u 1 , . . . , u n )  (v1 , . . . , vn ) if and only if there exists a permutation μ on {1, . . . , n} such that (u 1 , . . . , u n ) > (vμ(1) , . . . , vμ(n) ), and (ii) (u 1 , . . . , u n ) ∼ (v1 , . . . , vn ) if and only if there exists a permutation μ on {1, . . . , n} such that (u 1 , . . . , u n ) = (vμ(1) , . . . , vμ(n) ). We are now ready to introduce the CLGU SWR. Let G be the set of all possible continuous and strictly increasing functions g : R → R with g(0) = 0. Given g ∈ G and α ∈ R, the CLGU SWR associated with g and α ranks the streams of utility vectors by (i) comparing the total sum of the gains of individuals’ transformed utilities over g(α) in the heads of streams and (ii) applying the intratemporal Suppes–Sen grading principle to each generation in the tails of streams. Formally, the CLGU SWR associated with g and α is defined as the following binary relation RU∗ on N : for all u, v ∈ N , uRU∗ v ⇔ there exists T ∈ N such that ut  v t for all t > T and T n(u T n(v  )  ) [g(u it ) − g(α)] ≥ [g(vit ) − g(α)]. t

t

t=1 i=1

t=1 i=1

(3)

The total-sum comparison in the heads of streams in (3) is a direct application of the CLGU ordering. It is easy to check that, for any choice of g ∈ G and α ∈ R, RU∗ 11 See also Blackorby et al. (1995, 2002, 2005) for the critical-level generalized utilitarian ordering.

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is an SWR. Further, by (3), α is the unique critical level for all u ∈ N at any t-th generation. According to the choice of g ∈ G and α ∈ R, the CLGU SWR represents different evaluation relations. If it is associated with g ∈ G and α = 0, it is an infinitehorizon variant of generalized utilitarianism, and if g is the identity mapping, it is an infinite-horizon variant of classical utilitarianism. Further, if the CLGU is associated with a (strictly) concave function g ∈ G, the application of the CLGU ordering to the heads of streams with the same total population represents (strict) inequality aversion. The CLGU SWR is related to an evaluation relation for infinite utility streams according to its restriction to RN ⊂ N . For any g ∈ G and any α ∈ R, the restriction of the associated RU∗ to RN coincides with the generalized utilitarian quasi-ordering for infinite utility streams (which uses the Pareto dominance in the tails) introduced by d’Aspremont (2007). Further, if g is the identity mapping, its restriction to RN coincides with the utilitarian quasi-ordering presented by Basu and Mitra (2007). The asymmetric and symmetric parts of RU∗ are characterized respectively as follows: for all u, v ∈ N , u PU∗ v ⇔ there exists T ∈ N such that ut  v t for all t > T and T n(u  ) t

g(α)] >

t=1 i=1

uIU∗ v

T n(v  ) t

[g(u it ) −

[g(vit ) − g(α)];

(4a)

t=1 i=1 t

⇔ there exists T ∈ N such that u ∼ v for all t > T and t

T n(u  ) t

t=1 i=1

T n(v  ) g(α)] = [g(vit ) − g(α)]. t

[g(u it ) −

(4b)

t=1 i=1

The derivation of (4a) and (4b) is easy and we omit it for the sake of brevity (available in Kamaga 2016). To provide an axiomatic characterization of the CLGU SWR, we consider five axioms. Imposing an axiom on R ∗ means imposing a similar requirement on a social welfare functional F satisfying the axioms of Theorem 1 (ii). We begin with the axioms of Strong Pareto and Finite Anonymity. Both are reformulations of the corresponding axioms used in the framework of ranking infinite utility streams. Strong Pareto requires the evaluation to be positively sensitive to individuals’ utilities as long as every generation has the same population size. Strong Pareto (SP) For all u, v ∈ N such that n(ut ) = n(v t ) for all t ∈ N, if ut ≥ v t   for all t ∈ N and there exists t  ∈ N such that ut > v t , then u P ∗ v. Finite Anonymity formalizes the equal treatment of finitely many generations by asserting that the relative ranking of any two streams of utility vectors must be invariant with respect to a transposition of generations. Finite Anonymity (FA) For all u, v, w, z ∈ N , if there exist t1 , t2 ∈ N such that ut1 = wt2 , ut2 = w t1 , v t1 = z t2 , v t2 = z t1 , and, for all t ∈ N\{t1 , t2 }, ut = w t and v t = z t , then uR ∗ v ⇔ w R ∗ z. FA is the direct reformulation of the finite relative anonymity axiom in Asheim et al. (2010) and is weaker (for a transitive relation) than the direct reformulation of the

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finite anonymity axiom defined as follows: for all u, v ∈ N , if there exist t1 , t2 ∈ N such that ut1 = v t2 , ut2 = v t1 , and ut = v t for all t ∈ N\{t1 , t2 }, then uI ∗ v. However, FA and this stronger property are equivalent for a finitely complete quasi-ordering. Since an SWR is a finitely complete quasi-ordering, we will consider this stronger property when referring to FA in the proofs of our results. The other three axioms are reformulations of the axioms used by Blackorby et al. (2005) in the finite-horizon framework of variable-population social choice.12 First, we present an axiom regarding the existence of a critical level of utility. Weak Existence of Critical Levels requires that a critical level of utility exist for at least one stream of utility vectors at at least one generation. Weak Existence of Critical Levels (WECL) There exist t ∈ N, α ∈ R, and u ∈ N such that uI ∗ (u−(t−1) , (ut , α), u+t ). The next axiom is Restricted Continuity, which basically requires that small changes in individuals’ utilities of a single generation do not lead to large changes in the evaluation. Our axiom formulates this property in a restricted form, dealing with only the same dimensional utility vectors of a generation. To state the axiom, we need additional notation and definitions. For all n, t ∈ N and all w ∈ N , let N t,n,w =   t}. For all n, t ∈ N and all w ∈ N , {u ∈ N : n(ut ) = n and ut = wt for all t  = n t t N we define the metric d on N t,n,w by d(u, v) = i=1 |u i − vi | for all u, v ∈ t,n,w . Note that, for all n, t ∈ N and all w ∈ N , the pair (N t,n,w , d) constitutes a metric space. Restricted Continuity (RC) There exist t ∈ N and w ∈ N such that for all n ∈ N N ∗ N ∗ and all u ∈ N t,n,w , {v ∈ t,n,w : v R u} and {v ∈ t,n,w : uR v} are closed in (N t,n,w , d). Finally, we present the Existence Independence axiom. It basically requires the evaluation to be independent of any addition of individuals at all generations. We formulate this axiom in a weak form such that the independence property is required only for the streams of utility vectors that have a common tail. Thus, the axiom is interpreted to assert that we can focus on a conflict of interest among the present and near future generations, if only we know that each of the distant future generations has the same utility vector. Existence Independence (EI) For all u, v, w ∈ N , ifthere exists  T ∈ N such that ut = v t for all t > T , then uR ∗ v ⇔ (ut , w t ) t∈N R ∗ (v t , w t ) t∈N . To state the axiomatic characterization of the CLGU SWR, we define the notion of subrelation. We say that an SWR R1∗ on N is a subrelation of an SWR R2∗ on N if I1∗ ⊆ I2∗ and P1∗ ⊆ P2∗ . The following theorem shows that the CLGU SWR is characterized (in terms of subrelation) by the five axioms we presented.

12 Their axioms also appear in Blackorby et al. (1995, 1998a, 1999).

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Theorem 2 An SWR R ∗ on N satisfies SP, FA, WECL, RC, and EI if and only if there exist g ∈ G and α ∈ R such that RU∗ associated with g and α is a subrelation of R∗. Proof The proof of the if-part is easy and we omit it (available upon request). We prove the only-if-part. First, we prove the following claim. Claim 1 There exist g ∈ G and α ∈ R such that, for all t ∈ N and for all u, v ∈ N   with ut = v t for all t  = t, ∗

uR v ⇔

t n(u )

i=1

[g(u it ) −

g(α)] ≥

t n(v )

[g(vit ) − g(α)].

i=1 

N N : ut = Given t ∈ N and w ∈ N , we define N t,w by t,w = {u ∈   t  t w for all t = t} and define the binary relation Rw on  by, for all u, v ∈ N t,w , t t v ⇔ uR ∗ v. u t Rw

(5)

t is an ordering on  for all t ∈ N Since R ∗ is a finitely complete quasi-ordering, Rw N t = R t for all w, z ∈ N . and all w ∈  . Choose t ∈ N arbitrarily. We show that Rw z ¯ v¯ ∈ N ¯ t = ut Let ut , v t ∈  and w, z ∈ N . By (5), we obtain that, for u, t,w with u t t and v¯ = v , t t ¯ ¯ ∗ v. v ⇔ uR u t Rw

˜ v˜ ∈ N ˜ t = ut and v˜ t = v t . By EI, we obtain Let u, t,z with u ¯ ∗ v¯ ⇔ (u¯ t , z t )t∈N R ∗ (v¯ t , z t )t∈N ⇔ uR ˜ ˜ ∗ v. uR By (5), we obtain ˜ ∗ v˜ ⇔ ut R tz v t . uR t = Rt . By combining the above equivalence assertions, Rw z t = R t for all w, z ∈ ∞ , we can define the ordering R t on  by, for all Since Rw z   u, v ∈ N with ut = v t for all t  = t,

ut R t v t ⇔ uR ∗ v.

(6)

We write P t and I t as the asymmetric and symmetric parts of R t . Since R ∗ is intratemporally anonymous and it satisfies SP and EI, R t satisfies the following corresponding properties. Property 1 For all n ∈ N and all ut , v t ∈ Rn , if there exists a permutation μ on t t , . . . , vμ(n) ), then ut I t v t . {1, . . . , n} such that ut = (vμ(1)

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Infinite-horizon social evaluation with variable. . .

Property 2 For all n ∈ R and all ut , v t ∈ Rn , if ut ≥ v t and ut = v t , then ut P t v t . Property 3 For all ut , v t , w t ∈ , ut R t v t ⇔ (ut , w t )R t (v t , w t ). Further, since R ∗ is transitive and it satisfies FA, WECL, and RC, Rt satisfies the following properties (corresponding to WECL and RC). Property 4 There exist ut ∈  and α ∈ R such that ut I t (ut , α). Property 5 For all n ∈ N and all ut ∈ Rn , {v t ∈ Rn : v t R t ut } and {v t ∈ Rn : ut R t v t } are closed in Rn . Theorem 6.10 in Blackorby et al. (2005) shows that Rt satisfying the five properties must be the critical-level generalized utilitarian ordering on , that is, there exist g ∈ G and α ∈ R such that, for all ut = (u t1 , . . . , u tm ), v t = (v1t , . . . , vnt ) ∈ , ut R t v t ⇔

m n   [g(u it ) − g(α)] ≥ [g(vit ) − g(α)]. i=1

i=1 

Since R ∗ is transitive and it satisfies FA, R t = R t for all t, t  ∈ N. Thus, the proof of the claim is completed. Let g and α be the function and the utility level in Claim 1. Next, we prove the following claim. Claim 2 Let T ∈ N. For all u, v ∈ N with u+T = v +T , uI ∗ v if T n(u  ) t

T n(v  ) t

[g(u it ) −

g(α)] =

t=1 i=1

[g(vit ) − g(α)].

t=1 i=1

We prove the claim by induction on T . By Claim 1, Claim 2 holds for T = 1. Let T ∈ N and assume that Claim 2 holds for T . To show that Claim 2 holds for T + 1, let u, v ∈ N with u+(T +1) = v +(T +1) and suppose t T +1 n(u  )

[g(u it ) −

g(α)] =

t=1 i=1

t T +1 n(v  )

[g(vit ) − g(α)].

(7)

t=1 i=1

Define  by T n(u  ) t

=

T n(v  ) t

[g(u it ) −

g(α)] −

t=1 i=1

[g(vit ) − g(α)].

t=1 i=1

Note that, by (7), =

T +1 n(v )

i=1

[g(viT +1 ) −

g(α)] −

T +1 n(u )

[g(u iT +1 ) − g(α)].

i=1

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K. Kamaga

We distinguish two cases: (i)  = 0 and (ii)  = 0. First, consider case (i). By the assumption of induction, uI ∗ (v −T , u+T ). Further, by Claim 1, (v −T , u+T )I ∗ v. Since R ∗ is transitive, we obtain uI ∗ v. Next, consider case (ii). Without loss of generality, we assume  > 0. Let γ ∈ R with γ > α. Since g is increasing, g(γ ) > g(α). Let m ∈ N be a sufficiently large number satisfying g(γ ) > g(α) +  m . Since g is continuous on [α, γ ], it follows from the intermediate-value theorem that there exists β ∈ (α, γ ) such that g(β) = g(α)+  m, N ¯ v¯ ∈  by or equivalently, m[g(β) − g(α)] = . We define u, u¯ = (v −(T −1) , (v T , β1m ), u+T ) and v¯ = (v −T , (u T +1 , β1m ), v +(T +1) ). Since m[g(β) − g(α)] = , we obtain u¯ ) T n(   t

T n(u  ) t

[g(u¯ it ) −

g(α)] =

t=1 i=1

[g(u it ) − g(α)]

t=1 i=1

and n( v¯ T +1 )

[g(v¯iT +1 ) −

i=1

g(α)] =

T +1 n(v )

[g(viT +1 ) − g(α)].

i=1

¯ ∗ u by the assumption of induction and v¯ I ∗ v by Claim 1. Since R ∗ is We obtain uI transitive, we obtain ¯ ¯ ∗ v. uR ∗ v ⇔ uR

(8)

ˆ vˆ ∈ N by Next, we define u, uˆ = ((u¯ t , β))t∈N and vˆ = ((v¯ t , β))t∈N . By EI, ˆ ∗ v. ˆ ¯ ∗ v¯ ⇔ uR uR

(9)

Next, we define uˇ ∈ N by uˇ T = uˆ T +1 , uˇ T +1 = uˆ T , and uˇ t = uˆ t for all t = T, T +1. By FA and the transitivity of R ∗ , ˇ ∗ v. ˆ ˆ ∗ vˆ ⇔ uR uR

(10)

Note that  uˇ T = (u T +1 , β), uˇ T +1 = (v T , β1m+1 ), and uˇ t = (v t , β) for all t = T, T + 1; vˆ T = (v T , β), vˆ T +1 = (u T +1 , β1m+1 ), and vˆ t = (v t , β) for all t = T, T + 1.

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Infinite-horizon social evaluation with variable. . .

ˇ wˆ ∈ N by Define w, 

wˇ T = u T +1 , wˇ T +1 = v T , wˆ T = v T , wˆ T +1 = u T +1 ,

and wˇ t = v t for all t = T, T + 1; and wˆ t = v t for all t = T, T + 1.

By EI, ˆ ˇ ∗ vˆ ⇔ wˇ R ∗ w. uR

(11)

ˆ Thus, by (8), (9), (10), and (11), we obtain uI ∗ v, and the proof of the By FA, wˇ I ∗ w. claim is completed. We now show that RU∗ associated with g and α in Claim 1 is a subrelation of R ∗ . To show that IU∗ ⊆ I ∗ , let u, v ∈ N and suppose that uIU∗ v. By (4b), there T n(ut ) t exists T ∈ N such that ut ∼ v t for all t > T and t=1 i=1 [g(u i ) − g(α)] = T n(vt ) t −T , v +T ). Since R ∗ is intratemporally anonyt=1 i=1 [g(vi )−g(α)]. Let w = (u ∗ ∗ mous, uI w. By Claim 2, w I v. Since R ∗ is transitive, we obtain uI ∗ v. Next, to show that PU∗ ⊆ P ∗ , let u, v ∈ N and suppose that u PU∗ v. By (4a), T n(ut ) t there exists T ∈ N such that ut  v t for all t > T and t=1 i=1 [g(u i ) − g(α)] > T n(vt ) t −T , v +T ). Since R ∗ is intratemporally anonyt=1 i=1 [g(vi )−g(α)]. Let w = (u T n(ut ) t mous and it satisfies SP, uR ∗ w. We show that w P ∗ v. Let  = t=1 i=1 [g(u i ) − T n(v t ) g(α)] − t=1 i=1 [g(vit ) − g(α)] and γ ∈ R with γ > α. By the same argument as in the proof of Claim 2, there exists β ∈ (α, γ ) such that m[g(β) − g(α)] =  for ¯ v¯ ∈ N by w¯ 1 = (v 1 , β1m ), v¯ 1 = (v 1 , α1m ), and sufficiently large m ∈ N. Define w, t t ¯ Since β > α, we obtain w¯ = v¯ = v t for all t > 1. By Claim 2, w I ∗ w¯ and v I ∗ v. w¯ P ∗ v¯ by SP. Since R ∗ is transitive, we obtain w P ∗ v, and thus, u P ∗ v follows.   Theorem 2 means that the class of all SWRs that satisfy all the axioms stated in the theorem coincides with the class of all SWRs that include the CLGU SWR RU∗ associated with g ∈ G and α ∈ R as a subrelation. Thus, RU∗ is the least element of this class with respect to set inclusion. By Arrow’s (1963) variant of Szpilrajn’s (1930) lemma, there exists an ordering extension of RU∗ in this class.13 However, any ordering extension of RU∗ cannot be explicitly described since (i) the restriction of an ordering extension of RU∗ to RN ⊂ N is an ordering on RN that satisfies the axioms of strong Pareto and finite anonymity defined on RN and (ii) any ordering on RN that satisfies these axioms involves the use of non-constructive mathematics (Dubey 2011; Lauwers 2010; Zame 2007). It is worth noting the interpretation of Theorem 2 based on the argument in the proof. As we have seen in the proof of Claim 2, the intragenerational application of critical-level generalized utilitarianism must be extended to the intergenerational one if we require the evaluation to satisfy FA and EI. Theorem 2 can be interpreted as follows

13 An ordering extension of a given binary relation is an ordering that includes it as a subrelation.

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K. Kamaga

to emphasize this point: if we apply critical-level generalized utilitarianism within a generation and accept FA and EI, then we must also apply it between generations. 3.2 Critical-level generalized overtaking criteria We next introduce the CLGO and the CLGC SWRs. They are different from the CLGU SWR in that they can compare (not all but some) streams of utility vectors with different population sizes in the tails. Further, as we will see below, they include the CLGU SWR as a subrelation. We provide their axiomatic characterizations by using additional consistency axioms. The CLGO SWR consecutively applies the CLGU ordering to the heads of streams of utility vectors in the manner of the overtaking criterion of von Weizsäcker (1965). Given g ∈ G and α ∈ R, the CLGO SWR associated with g and α is defined as the following binary relation R ∗O on N : for all u, v ∈ N , u PO∗ v ⇔ there exists T ∗ ∈ N such that for all T ≥ T ∗ , T n(u  ) t

T n(v  ) t

[g(u it ) −

g(α)] >

t=1 i=1

[g(vit ) − g(α)];

(12a)

t=1 i=1

uI O∗ v ⇔ there exists T ∗ ∈ N such that for all T ≥ T ∗ , T n(u T n(v  )  ) [g(u it ) − g(α)] = [g(vit ) − g(α)]. t

t

t=1 i=1

t=1 i=1

(12b)

It is easy to check that, for any g ∈ G and any α ∈ R, the associated R ∗O is well defined as a binary relation on N and it is an SWR. Given g ∈ G and α ∈ R, the associated RU∗ is a subrelation of the associated R ∗O since we see that PU∗ ⊂ PO∗ and IU∗ ⊂ I O∗ by comparing (4a) and (4b) with (12a) and (12b). For any α ∈ N, if R ∗O is associated with the identity mapping g and α, the restriction of it to RN coincides with the overtaking quasi-ordering presented by Asheim and Tungodden (2004) in the framework of ranking infinite utility streams (see also Basu and Mitra 2007). The CLGC SWR also consecutively applies the CLGU ordering to the heads of streams of utility vectors, but does it in the manner of the catching-up criterion of Atsumi (1965) and von Weizsäcker (1965). Given g ∈ G and α ∈ R, the CLGC associated with g and α is defined as the following binary relation RC∗ on N : for all u, v ∈ N , uRC∗ v ⇔ there exists T ∗ ∈ N such that for all T ≥ T ∗ , T n(u  ) t

t=1 i=1

T n(v  ) t

[g(u it ) −

g(α)] ≥

[g(vit ) − g(α)].

(13)

t=1 i=1

By (13), RC∗ is an SWR for any g ∈ G and any α ∈ R. It is easy to check that the asymmetric and symmetric parts of RC∗ are characterized as follows: for all u, v ∈ N ,

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Infinite-horizon social evaluation with variable. . .

⎫ ⎪ ⎪ ⎪ ⎪ t) t) ⎪ n(u n(v T T ⎪   ⎪ ⎪ t t [g(u i ) − g(α)] ≥ [g(vi ) − g(α)] ⎪ ⎪ ⎪ ⎬

u PC∗ v ⇔ there exists T ∗ ∈ N such that for all T ≥ T ∗ ,

t=1 i=1

t=1 i=1

and for all T  ∈ N, there exists T > T  such that

⎪ ⎪ ⎪ ⎪ ⎪ t t ⎪ ) ) n(u n(v T T ⎪   ⎪ ⎪ t t [g(u i ) − g(α)] > [g(vi ) − g(α)];⎪ ⎪ ⎭ t=1 i=1

(14a)

t=1 i=1

uIC∗ v ⇔ there exists T ∗ ∈ N such that for all T ≥ T ∗ , T n(u  ) t

t=1 i=1

T n(v  ) t

[g(u it ) −

g(α)] =

[g(vit ) − g(α)].

(14b)

t=1 i=1

Given g ∈ G and α ∈ R, the associated R ∗O is a subrelation of the associated RC∗ since, by comparing (12a) and (12b) with (14a) and (14b), we see that PO∗ ⊂ PC∗ and I O∗ = IC∗ . For any α ∈ R, if RC∗ is associated with the identity mapping g and α, its restriction to RN coincides with the catching-up quasi-ordering for infinite utility streams presented by Svensson (1980) (see also Asheim and Tungodden 2004; Basu and Mitra 2007). To present the axiomatic characterizations of the CLGO and the CLGC SWRs, we consider three consistency axioms. The first two are the weak and strong axioms of preference consistency, which are reformulations of the axioms presented by Kamaga and Kojima (2010) in the framework of ranking infinite utility streams.14 Both assert that the strict preference relation of the evaluation must be consistent with the evaluations obtained when the tails of streams of utility vectors are replaced with any common one. Weak Preference Consistency (WPC) For all u, v ∈ N , if (u−t , w +t )P ∗ (v −t , w +t ) for all t ∈ N and all w ∈ N , then u P ∗ v. Strong Preference Consistency (SPC) For all u, v ∈ N , if, for all w ∈ N , (u−t , w +t )R ∗ (v −t , w +t ) for all t ∈ N and, for all t  ∈ N, there exists t > t  such that (u−t , w +t )P ∗ (v −t , w +t ), then u P ∗ v. Clearly, SPC is stronger than WPC since the former allows weak preference relations in its premise. The next consistency axiom postulates a requirement similar to that of WPC for the indifference relation of the evaluation.15 Indifference Consistency (IC) For all u, v ∈ N , if (u−t , w +t )I ∗ (v −t , w +t ) for all t ∈ N and all w ∈ N , then uI ∗ v. 14 See also Asheim and Banerjee (2010), Asheim and Tungodden (2004), and Basu and Mitra (2007) for similar axioms in the framework of ranking infinite utility streams. 15 An axiom similar to IC is presented by Asheim and Tungodden (2002) in the framework of ranking infinite utility streams.

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K. Kamaga

In the following theorem, we present axiomatic characterizations (in terms of subrelation) of the CLGO and the CLGC SWRs. It shows that (i) by adding WPC and IC to the set of axioms in Theorem 2, we obtain the characterization of the CLGO SWR, and (ii) if we strengthen WPC to SPC, the CLGC SWR is characterized. Theorem 3 (i) An SWR R ∗ on N satisfies SP, FA, WECL, RC, EI, WPC, and IC if and only if there exist g ∈ G and α ∈ R such that R ∗O associated with g and α is a subrelation of R ∗ . (ii) An SWR R ∗ on N satisfies SP, FA, WECL, RC, EI, SPC, and IC if and only if there exist g ∈ G and α ∈ R such that RC∗ associated with g and α is a subrelation of R∗. Proof The proof of the if-parts of (i) and (ii) is easy and we omit it (available upon request). We prove the the only-if-parts of (i) and (ii). [Only-if-part of (i)] By Theorem 2, there exist g ∈ G and α ∈ R such that RU∗ associated with g and α is a subrelation of R ∗ . We show that R ∗O associated with g and α is a subrelation of R ∗ . To show that PO∗ ⊆ P ∗ , let u, v ∈ N and suppose u PO∗ v. By (12a), there exists T ∗ ∈ N such that for all T ≥ T ∗ , T n(u  ) t

t=1 i=1

T n(v  ) t

[g(u it ) −

g(α)] >

[g(vit ) − g(α)].

(15)

t=1 i=1 ∗

Let us define u˜ ∈ N by u˜ 1 = (u1 , . . . , u T ), u˜ t = α for all t ∈ {2, . . . , T ∗ }, and ∗ u˜ t = ut for all t > T ∗ . Similarly, define v˜ ∈ N by v˜ 1 = (v 1 , . . . , v T ), v˜ t = α for all t ∗ ∗ t ∗ ˜ Since t ∈ {2, . . . , T }, and v˜ = v for all t > T . By (4b), we obtain uIU u˜ and v IU∗ v. ˜ Since R ∗ is transitive, it suffices to show RU∗ is a subrelation of R ∗ , uI ∗ u˜ and v I ∗ v. ˜ we obtain that u˜ P ∗ v˜ to prove PO∗ ⊆ P ∗ . By (4a), (15), and the definitions of u˜ and v, −T −T ∗ ∗ N +T +T for all T ∈ N and all w ∈  , (u˜ , w )PU (v˜ , w ). Since RU is a subrelation of R ∗ , we obtain that for all T ∈ N and all w ∈ N , (u˜ −T , w +T )P ∗ (v˜ −T , w +T ). By ˜ Applying the same argument, we can show that I O∗ ⊆ I ∗ by using (12b), WPC, u˜ P ∗ v. (4b), and IC in place of (12a), (4a), and WPC. We omit the detailed proof of it. (Only-if-part of (ii)) By Theorem 3 (i), there exist g ∈ G and α ∈ R such that R ∗O associated with g and α is a subrelation of R ∗ . We show that RC∗ associated with g and α is a subrleation of R ∗ . Since IC∗ = I O∗ , it suffices to prove PC∗ ⊆ P ∗ . Since RU∗ associated with g and α is a subrelation of R ∗O , we can prove this by the same argument as the proof of the only-if-part of Theorem 3 (i) (using (14a), (3) and (4a), and SPC instead of (12a), (4a), and WPC). Thus, we omit the detailed proof of it.   Recall that, for any g ∈ G and any α ∈ R, PU∗ ⊂ PO∗ ⊂ PC∗ and IU∗ ⊂ I O∗ = IC∗ hold for RU∗ , R ∗O , and RC∗ associated with g and α. From a comparison of Theorems 2 and 3, the set inclusion is explained by WPC and SPC for the asymmetric parts and by IC for the symmetric parts. Finally, every axiom is independent of the other axioms in Theorems 2 and 3 (the proof is available in Kamaga 2016).

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4 Population ethics and permissible critical levels 4.1 Avoidance of the repugnant conclusion In this section, we present some population ethics axioms and examine permissible critical levels for the SWRs presented in the preceding section to be compatible with those axioms. We first reformulate Parfit’s (1976, 1982, 1984) repugnant conclusion in the current framework and show that the CLGO and the CLGC SWRs associated with a positive critical level are characterized by using the axiom requiring avoidance of the repugnant conclusion. In Sect. 4.2, we will show that the CLGO and the CLGC SWRs associated with a positive critical level are incompatible with some other population ethics axioms, and we will evaluate the CLGO and the CLGC SWRs associated with a positive critical level by using these results. The repugnant conclusion is the problem of classical utilitarianism (the comparison of utility sums) pointed out by Parfit (1976, 1982, 1984) in the context of the evaluation of social states with finite and variable population size. It is the following ethically unacceptable conclusion: some social state in which each member of the population has a high level of utility is declared to be worse than some social state with a much larger population in which each member has a utility level corresponding to a life barely worth living (which he sets as a positive but close to zero utility level). To reformulate the repugnant conclusion in the current framework, first recall that, as Parfit does, we employ the convention of population ethics and utility is normalized so that the zero utility level is neutrality and a utility level above zero represents a life worth living. A difficulty we face in reformulating the repugnant conclusion is that we need to define the way to compare streams of population sizes since the time horizon in the current framework, and therefore the total population, is infinite. There are some possible ways of comparison, including those based on aggregation of population sizes of different generations (e.g., the overtaking criterion). However, we employ the vector dominance of streams of population sizes as a comparison method since only one generation exists at each time period: thus, the issue of overpopulation should be treated generation by generation. The vector dominance we use is a strict dominance in each generation’s population size. By using it, we define the repugnant conclusion in the current framework as follows. An SWR R ∗ on N implies the repugnant conclusion if and only if, for any stream of population sizes (n t )t∈N ∈ NN and for any stream of positive utility levels of generations (ξt )t∈N , ( t )t∈N ∈ RN ++ satisfying (ξt )t∈N  ( t )t∈N , there exists a stream of population sizes (m t )t∈N ∈ NN with (m t )t∈N  (n t )t∈N such that ( t 1m t )t∈N P ∗ (ξt 1n t )t∈N . In the finite-horizon framework of variable-population social choice, the axiom requiring avoidance of the repugnant conclusion is formalized by taking its negation under the assumption that the evaluation is complete (e.g., Blackorby et al. 2005, 1998b). Although we do not assume the full completeness of an SWR, as Arrhenius (2000) discusses in the context of the finite-horizon framework, there is no plausible reason that we cannot compare those streams of utility vectors considered in the repugnant conclusion. Thus, we define the axiom of avoidance of the repugnant conclusion

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by not only taking its negation but also excluding the case of non-comparability of the streams of utility vectors considered. Avoidance of the Repugnant Conclusion (ARC) There exist (n t )t∈N ∈ NN and N (ξt )t∈N , ( t )t∈N ∈ RN ++ with (ξt )t∈N  ( t )t∈N such that for all (m t )t∈N ∈ N with (m t )t∈N  (n t )t∈N , (ξt 1n t )t∈N R ∗ ( t 1m t )t∈N . Note that ARC implies the negation of the repugnant conclusion. The following theorem shows that if we add ARC to the set of axioms in Theorem 3, the CLGO and the CLGC SWRs associated with a positive critical level are characterized. Theorem 4 (i) An SWR R ∗ on N satisfies ARC in addition to the axioms in Theorem 3 (i) if and only if there exist g ∈ G and α ∈ R++ such that R ∗O associated with g and α is a subrelation of R ∗ . (ii) An SWR R ∗ on N satisfies ARC in addition to the axioms in Theorem 3 (ii) if and only if there exist g ∈ G and α ∈ R++ such that RC∗ associated with g and α is a subrelation of R ∗ . Proof (i) To prove the if-part, suppose that R ∗O associated with g ∈ G and α ∈ R++ is a subrelation of R ∗ . By Theorem 3 (i), R ∗ satisfies all the axioms in Theorem 3 (i). To show that R ∗ satisfies ARC, let ξt = α and t ∈ (0, α) for all t ∈ N. Then, for any (m t )t∈N , (n t )t∈N ∈ NN with (m t )t∈N  (n t )t∈N , we obtain (ξt 1n t )t∈N PO∗ ( t 1m t )t∈N since g(ξt ) − g(α) = 0 and g( t ) − g(α) < 0 for all t ∈ N. Since R ∗O is a subrelation of R ∗ , we obtain (ξt 1n t )t∈N P ∗ ( t 1m t )t∈N . Thus, R ∗ satisfies ARC. Next, we prove the only-if-part. By Theorem 3 (i), there exist g ∈ G and α ∈ R such that R ∗O associated with g and α is a subrelation of R ∗ . By way of contradiction, suppose N α ≤ 0. Let (ξt )t∈N , ( t )t∈N ∈ RN ++ with (ξt )t∈N  ( t )t∈N . For any (n t )t∈N ∈ N , ∗ we obtain ( t 1m t )t∈N PO (ξt 1n t )t∈N if we consider sufficiently large m t ∈ N for all t ∈ N satisfying (m t )t∈N  (n t )t∈N and m t [g( t ) − g(α)] > n t [g(ξt ) − g(α)]. Since R ∗O is a subrelation of R ∗ , ( t 1m t )t∈N P ∗ (ξt 1n t )t∈N follows. This means that R ∗ implies the repugnant conclusion, and R ∗ violates ARC. (ii) To prove the if-part, suppose that RC∗ associated with g ∈ G and α ∈ R++ is a subrelation of R ∗ . By Theorem 3 (ii), R ∗ satisfies all the axioms in Theorem 3 (ii). Since R ∗O associated with g and α is a subrelation of RC∗ , it follows from Theorem 4 (i) that R ∗ satisfies ARC. Next, we prove the only-if part. By Theorem 3 (ii), there exist g ∈ G and α ∈ R such that RC∗ associated with g and α is a subrelation of R ∗ . Since R ∗O associated with g and α is a subrelation of RC∗ , if α ≤ 0 then we obtain a contradiction to Theorem 4 (i). Thus, α > 0.   Every axiom is independent of the other axioms in Theorem 4 (the proof is available in Kamaga 2016). It is impossible to obtain an analogous result for the CLGU SWR by adding ARC to the set of axioms in Theorem 2 since it violates ARC due to the non-comparability of streams of utility vectors with different population sizes in the tails. Although the CLGU SWR associated with a positive critical level violates ARC, it is an important SWR since ARC is violated because of its incompleteness: thus, we can use it as a base relation to explore other more comparable SWRs in order to satisfy ARC.

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4.2 Other population ethics axioms We next evaluate the CLGO and the CLGC SWRs by using other population ethics axioms. The axioms we use are reformulations of those used to criticize the CLGU ordering with a positive critical level in the finite-horizon variable population social choice. Three main criticisms are raised against it (the names of the properties noted below are based on Arrhenius (2011).16 First, it implies the weak repugnant conclusion that, for any population with a utility level above a given positive critical level, there is a larger population with a lower utility, though above the given critical level, declared to be worse (i.e., the special case of the repugnant conclusion that occurs for utility distributions above a positive critical level) (Broome 1992).17 Second, it implies the very sadistic conclusion that, for any population with negative utility levels, there is a population with positive utility levels declared to be worse (Arrhenius 2000). Third, it violates Parfit’s (1984) mere addition principle that if an individual with a positive utility level is added to a given population, the resulting population is not worse than the original one. In what follows, we reformulate the axioms requiring the weak repugnant and the very sadistic conclusions to be avoided and the axiom of the mere addition principle in the current framework. We then show that any SWR that includes the CLGO SWR associated with a positive critical level as a subrelation violates these axioms. We first reformulate the weak repugnant conclusion. An SWR R ∗ on N implies the weak repugnant conclusion if and only if, for any (n t )t∈N ∈ NN and any (ξt )t∈N , ( t )t∈N ∈ RN ++ with (ξt )t∈N  ( t )t∈N  (α, α, . . .), there exists (m t )t∈N ∈ NN with (m t )t∈N  (n t )t∈N such that ( t 1m t )t∈N P ∗ (ξt 1n t )t∈N , where α ∈ R++ is a critical level for all u ∈ N at any t ∈ N. As with ARC, the axiom of avoidance of the weak repugnant conclusion is defined as follows. Avoidance of the Weak Repugnant Conclusion (AWRC) There exist (n t )t∈N ∈ NN and (ξt )t∈N , ( t )t∈N ∈ RN ++ with (ξt )t∈N  ( t )t∈N  (α, α, . . .) such that for all (m t )t∈N ∈ NN with (m t )t∈N  (n t )t∈N , (ξt 1n t )t∈N R ∗ ( t 1m t )t∈N . Note that AWRC implies ARC. Next, to reformulate the very sadistic conclusion in the current framework, we define ++ = ∪n∈N Rn++ and −− = ∪n∈N Rn−− . An SWR R ∗ on N implies the very sadistic conclusion if and only if, for any stream of negative utility vectors u ∈ N −− , ∗ v. As with ARC, there exists a stream of positive utility vectors v ∈ N such that u P ++ we define the axiom of avoidance of the very sadistic conclusion by not only taking its negation but also not allowing the evaluation to conclude non-comparability. Avoidance of the Very Sadistic Conclusion (AVSC) There exists u ∈ N −− such that v R ∗ u for all v ∈ N . ++ Finally, we reformulate the mere addition principle by not allowing the evaluation to conclude non-comparability. It requires that, for any stream of utility vectors, if an 16 See also Carlson (1998) for other related criticisms. 17 Blackorby et al. (1997, footnote 35) note that Thomas Hurka also made this point.

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individual with a positive utility level is added to each generation, then the resulting stream of utility vectors is at least as good as the original one. t ∗ Mere Addition Principle (MAP) For all u ∈ N and all (ξt )t∈N ∈ RN ++ , (u , ξt )t∈N R u.

The following proposition shows that any SWR that includes the CLGO SWR associated with a positive critical level as a subrelation violates all of the three axioms presented above. Proposition 1 Let R ∗ be an SWR on N that includes R ∗O associated with g ∈ G and α ∈ R as a subrelation. (i) R ∗ violates AWRC if α > 0. (ii) R ∗ satisfies any of AVSC and MAP if and only if α ≤ 0. Proof (i) Suppose α > 0 and let ξt > t > α for all t ∈ N. By the same argument as in the proof of the only-if-part of Theorem 4 (i), we can show that R ∗ implies the weak repugnant conclusion, and we omit the detailed proof of it. [If-part of (ii)] Suppose α ≤ 0. To show that R ∗ satisfies AVSC, let u = (u, u, . . .) ∈ ∗ N R−− ⊂ N −− with u < α. Since g is strictly increasing, we obtain v PO u for all ∗ N ∗ ∗ N v ∈ ++ . Since R O is a subrelation of R , v P u for all v ∈ ++ . Next, to show that R ∗ satisfies MAP, let u ∈ N and (ξt )t∈N ∈ RN ++ . Since g is strictly increasing, we obtain (ut , ξt )t∈N PO∗ u. Since R ∗O is a subrelation of R ∗ , (ut , ξt )t∈N P ∗ u follows. [Only-if-part of (ii)] By way of contradiction, suppose α > 0. First, we assume N that R ∗ satisfies AVSC. Let u ∈ N −− and define v ∈ ++ as follows: v = (vt 1n t )t∈N and, for all t ∈ N, vt ∈ (0, α) and n t ∈ N is sufficiently large so as to satisfy nt (u) n t [g(vt ) − g(α)] < i=1 [g(u it ) − g(α)]. Then, we obtain u PO∗ v, and thus, u P ∗ v since R ∗O is a subrelation of R ∗ . This is a contradiction to that R ∗ satisfies AVSC. Next, we assume that R ∗ satisfies MAP. Let ξ ∈ (0, α). Since g is strictly increasing, we obtain u PO∗ (ut , ξ )t∈N for any u ∈ N . Since R ∗O is a subrelation of R ∗ , u P ∗ (ut , ξ )t∈N follows. This is a contradiction to that R ∗ satisfies MAP.   We briefly note the relationships between the CLGU SWR and the axioms in Proposition 1. Since AWRC implies ARC and, for any critical level, the associated CLGU SWR violates ARC, it does AWRC as well. Further, for any critical level, the associated CLGU SWR violates AVSC and MAP due to the non-comparability of streams of utility vectors with different population sizes in the tails. Let us now evaluate the CLGO and the CLGC SWRs associated with a positive critical level by using Theorem 4 and Proposition 1. First, by Theorem 4 and Proposition 1 (i), they violate AWRC while they satisfy ARC. However, if the critical level is set to be suitably high, the violation of AWRC is not so problematic since the weak repugnant conclusion is, we believe, not repugnant in that case. Thus, the CLGO and the CLGC SWRs associated with a suitably high positive critical level should be used in ranking streams of utility vectors. On the other hand, by Proposition 1 (ii), the CLGO and the CLGC SWRs associated with a positive critical level violate AVSC and MAP. Further, this violation is more problematic when the critical level is suitably high since the very sadistic conclusion and the violation of MAP are implied in more cases. This is a drawback of the CLGO

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and the CLGC SWRs. However, this drawback may not be a serious problem when we apply them to an economic growth model with endogenous population growth to examine a maximal stream of utility vectors. To discuss this point, it should be first noted that, when the CLGO associated with a positive critical level implies the very sadistic conclusion, the heads of the considered stream of positive utility vectors must have a larger total population than those of the considered stream of negative utility ∗ N vectors. To be precise, for any u ∈ N −− , if there exists v ∈ ++ such that u PO v, ∗ then there exists T ∈ N such that T  t=1

n(v t ) >

T 

n(ut )

for all T ≥ T ∗ ,

(16)

t=1

and the same is true for RC∗ . In economic growth models with endogenous population growth, each generation faces a tradeoff between higher consumption levels and population growth because child-rearing is costly. For example, in the model of Boucekkine et al. (2011, 2014), the output of the consumption good is partly devoted N to raising children. Thus, in such a model, if both u ∈ N −− and v ∈ ++ are feasible, then, by (16), the total resources used to yield the heads of v are larger than those for u, and thus, we may find another feasible stream w of utility vectors such that w PO∗ u (or w PC∗ u). Consequently, the CLGO and the CLGC SWRs associated with a positive critical level would never imply the very sadistic conclusion when comparing a maximal stream of utility vectors (if exists) and any other feasible stream of utility vectors. This argument would more likely apply to a violation of MAP because, by the definition of MAP, the stream that should be declared weakly better has one additional individual in each generation. Whether our argument holds in economic growth models with endogenous population growth is an issue to be addressed, but we leave it for future research.

5 Conclusion In this paper, we presented an infinite-horizon framework of variable-population social choice. Our second version of the welfarism theorem provides an axiomatic foundation for the analysis of SWRs on the set of streams of utility vectors. As SWRs for streams of utility vectors, we introduced three infinite-horizon variants of critical-level generalized utilitarianism, namely, the critical-level generalized utilitarian (CLGU), the critical-level generalized overtaking (CLGO), and the critical-level generalized catching-up (CLGC) SWRs. Further, we presented an axiomatic characterization of each of them. The characterization of the CLGU SWR (and the characterizations of the CLGO and the CLGC SWRs as well) shows that the intragenerational application of CLGU ordering in at least one generation, together with the intergenerational anonymity and the independence axioms (FA and EI), implies its intergenerational application. This implication of the intragenerational value judgment for the intergenerational one is worth noting. We also presented a reformulation of the repugnant conclusion in our framework by using the dominance between streams of population sizes of generations. We have

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seen that by adding the axiom of the avoidance of the repugnant conclusion (ARC), the CLGO and the CLGC SWRs associated with a positive critical level are characterized. On the other hand, it is also shown that these SWRs are incompatible with some infinitehorizon reformulations of population ethics axioms, namely, the axioms requiring the weak repugnant and the very sadistic conclusions to be avoided (AWRC and AVSC) and the axiom of the mere addition principle (MAP). As noted in the last section, future research should examine whether the CLGO and the CLGC SWRs associated with a suitably high positive critical level imply the very sadistic conclusion and the violation of MAP when comparing a maximal stream of utility vectors and any feasible one in models of economic growth with endogenous population growth. Further, since the optimal population size implied by our SWRs cannot be discussed without any reference to resource constraints, we should examine it in those models. Future research should explore two more issues. First, we should explore other possible SWRs for streams of utility vectors. In the large body of literature on variablepopulation social choice and ranking infinite utility streams, many evaluation relations have been proposed. Using these, we can formulate many other SWRs for streams of utility vectors. In particular, it is of interest to reformulate the rank-discounted critical-level generalized utilitarian (RDCLGU) ordering proposed by Asheim and Zuber (2014) (which we mentioned in Sect. 1) as an SWR since, as shown by the authors, the RDCLGU ordering is remarkable in that it satisfies the finite-horizon versions of ARC, AWRC, and AVSC. Second, we should also examine the issue of intergenerational sustainability by applying our three SWRs to economic growth models with endogenous population growth. To do this, we need to reformulate the standard notion of intergenerational sustainability used in an economic growth model with a single representative agent in each generation. In such a model, Asheim et al. (2001) present the following notion of sustainability: an infinite utility stream is sustainable if it is attained by the action of each generation (e.g., choice of utility level (or consumption level) and capital bequest) which guarantees that the same utility level as in one’s own generation is attainable in each subsequent generation. If we assume identical individuals in each generation, as is usually done in economic growth models with endogenous population growth, and consider a per capita consumption level, the above notion of sustainability can be reformulated for a stream of utility vectors by considering the utility level of a representative individual in each generation (and adding the choice of the number of children to the action of each generation). This reformulation focuses on utility levels of generations and does not (at least directly) deal with population sizes of generations. This is because the population size of each generation is determined by the action of each preceding generation and, thus, cannot be chosen by the current generation. We should examine, by incorporating the issue of the existence of a maximal stream of utility vectors, whether our SWRs imply the reformulated notion of intergenerational sustainability in economic growth models with endogenous population growth. Acknowledgments I am grateful to two anonymous referees and an editor of this journal. Their detailed comments and suggestions have greatly improved the current version of this paper. I also thank the participants at the 11th Meeting of the Society for Social Choice and Welfare for their comments. All remaining errors are my own. This study is partly supported by a Grant-in-Aid for Young Scientists (B) (No. 23730196) from the Ministry of Education, Science, Sports, and Culture, Japan.

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