Utilitarianism

Utilitarianism and the Theory of Justice* by Charles Blackorby, Walter Bossert and David Donaldson

August 1999

revised August 2001

Prepared as Chapter 11 of the Handbook of Social Choice and Welfare K. Arrow, A. Sen and K. Suzumura, eds., Elsevier, Amsterdam

Charles Blackorby: University of British Columbia and GREQAM Walter Bossert: Universit´ de Montr´al and C.R.D.E. e e David Donaldson: University of British Columbia

* We thank Don Brown, Marc Fleurbaey, Philippe Mongin, John Weymark and a referee for comments and suggestions. Financial support through a grant from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.
August 20, 2001

Abstract
This chapter provides a survey of utilitarian theories of justice. We review and discuss axiomatizations of utilitarian and generalized-utilitarian social-evaluation functionals in a welfarist framework. Section 2 introduces, along with some basic deﬁnitions, socialevaluation functionals. Furthermore, we discuss several information-invariance assumptions. In Section 3, the welfarism axioms unrestricted domain, binary independence of irrelevant alternatives and Pareto indiﬀerence are introduced and used characterize welfarist social evaluation. These axioms imply that there exists a single ordering of utility vectors that can be used to rank all alternatives for any proﬁle of individual utility functions. We call such an ordering a social-evaluation ordering, and we introduce several examples of classes of such orderings. In addition, we formulate some further basic axioms. Section 4 provides characterizations of generalized-utilitarian social-evaluation orderings, both in a static and in an intertemporal framework. Section 5 deals with the special case of utilitarianism. We review some known axiomatizations and, in addition, prove a new characterization result that uses an axiom we call incremental equity. In Section 6, we analyze generalizations of utilitarian principles to variable-population environments. We extend the welfarism theorem to a variable-population framework and provide a characterization of critical-level generalized utilitarianism. Section 7 provides an extension to situations in which the alternatives resulting from choices among feasible actions are not known with certainty. In this setting, we discuss characterization as well as impossibility results. Section 8 concludes. Journal of Economic Literature Classiﬁcation Numbers: D63, D71.

Keywords: Social Choice, Utilitarianism, Welfarism.

1. Introduction In A Theory of Justice, Rawls (1971) describes justice as “the ﬁrst virtue of social institutions” (p. 3) and identiﬁes “the primary subject of justice” as “the basic structure of society, or more exactly, the way in which the major social institutions distribute fundamental rights and duties and determine the division of advantages from social cooperation” (p. 7).1 The view of justice investigated in this chapter asserts that a just society is a good society: good for the individual people that comprise it. To implement such an approach to justice, the social good is identiﬁed and used to rank social alternatives. Of the alternatives that are feasible, given the constraints of human nature and history, the best is identiﬁed with justice. Even if the best alternative is not chosen, however, better ones are considered to be more just than worse ones. If societies are not perfectly just, therefore, social improvements can be recognized. Social choices are not made in isolation, however. Decisions made in a particular society aﬀect people in other parts of the world and people who are not yet born. In addition, both the number and identities of future people are inﬂuenced by choices made in the present. For that reason, principles that identify the social good are typically extended to rank complete histories of the world (or the universe if necessary) from remote past to distant future.2 The principle that asserts that a just society is a good society must be qualiﬁed, therefore, with a ceteris paribus clause. In this chapter, we investigate a particular conception of the social good, one that is based exclusively on individual good or well-being. Principles that reﬂect this view are called welfarist (Sen, 1979) and they treat values such as freedom and individual autonomy as ‘instrumental’—valuable only because of their contribution to well-being. Because of this, it is important to employ a comprehensive notion of well-being such as that of Griﬃn (1986) or Sumner (1996). We therefore focus on lifetime well-being and include enjoyment, pleasure and the absence of pain, good health, length of life, autonomy, liberty, understanding, accomplishment and good human relationships as aspects of it. Welfarist principles are sometimes criticized as taking a narrow view of being a person, seeing them as “locations of their respective utilities” only (Sen and Williams, 1982, p. 4). The use of comprehensive accounts of lifetime well-being, such as those of Griﬃn (1986) and Sumner (1996), which attempt to take account of everything in which individual
1 See Murphy (1998) for a discussion of the relationship between the principles of social justice and the principles that guide individual conduct. 2 Because some non-human animals are sentient—capable of having experiences—their interests are often included. Sidgwick (1907, 1966, p. 414) argues that we should “extend our concern to all the beings capable of pleasure and pain whose feelings are aﬀected by our conduct”. Throughout the chapter, however, we assume that only human well-being counts in social evaluation, a simpliﬁcation that makes our presentation simpler. Readers who are interested in the extension of welfarist principles to non-human sentient creatures are referred to Blackorby and Donaldson (1992).

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people have an interest, is, in our view at least, suﬃcient to answer it. Welfarism is mainly a consequence of Pareto indiﬀerence which is in turn implied by an axiom that we call minimal individual goodness. If one alternative is ranked as better than another, the axiom requires it to be better for at least one individual (Goodin, 1991). Any non-welfarist principle must, therefore, run the risk of claiming that a social change is good even though no one beneﬁts. Popularized by Bentham (1789, 1973), utilitarianism is a welfarist principle that can be used to rank social alternatives according to their goodness.3 Utility is an index of individual lifetime well-being and, for a ﬁxed population, utilitarianism declares alternative x to be better than alternative y if and only if total utility is greater in x than in y. Although this principle is unconcerned with the distribution of any ﬁxed total utility, it is not unconcerned with income inequality or social provision for special needs. The utilitarian indiﬀerence to inequality of well-being has prompted the complaint that “persons do not count as individuals in this any more than individual petrol tanks do in the analysis of the national consumption of petroleum” (Sen and Williams, 1982, p. 4). This criticism does not apply to welfarist principles that are averse to utility inequality, however: they rank more equal distributions of utility as better than less equal ones. A family of principles whose value functions have the same additively separable mathematical structure as the utilitarian value function is the generalized-utilitarian family of principles, which includes utilitarianism as a special case. Each of these principles employs transformed utilities and some exhibit aversion to utility inequality. The generalizedutilitarian principles satisfy an important property: if a social change aﬀects the utilities of a particular group of individuals only, the ranking of such changes is independent of the utility levels of others. This means that independent subprinciples exist for subgroups (including generations) and are consistent with the overall principle. Both the utilitarian and generalized-utilitarian families can be extended to cover changes in population size and composition. A social-evaluation functional assigns a ranking of all possible social alternatives to every admissible proﬁle of utility functions where a proﬁle contains one utility function for each member of society. Arrow’s (1951, 1963) seminal contribution to social-choice theory employs a domain that uses individual preference information only. His impossibility result can be avoided if the domain is changed to include proﬁles of utility functions together with conditions that ensure that utilities are, to some degree, interpersonally comparable.4 There are, of course, other ways of avoiding Arrow’s impossibility result by removing or weakening one or more of his axioms but, in our opinion, the most natural way to proceed is to allow for interpersonal comparisons. Although the principles discussed in this chapter
3 See also Mill (1861, 1969) and Sidgwick (1907, 1966) for other early formulations of utilitarianism. 4 See Sen (1970).

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all require a certain amount of interpersonal comparability, numerically meaningful utility functions are not necessary for most of them. For example, utilitarianism requires only cardinal measurability of individual utilities—with utility functions that are unique up to increasing aﬃne transformations—and interpersonal comparability of utility gains and losses between pairs of alternatives (cardinal measurability and unit comparability). In a two-person society, if one person gains more in moving from y to x than the other loses, utilitarianism declares x to be socially better than y. In this chapter, we survey a set of results in social-choice theory that provides an axiomatic basis for generalized utilitarianism and, in some cases, utilitarianism itself. Although we are concerned, for the most part, with comparisons of alternatives according to their goodness, we include a brief discussion of induced rankings. Utilitarianism and generalized utilitarianism can be used to rank combinations of institutions (including legal and educational ones), customs and moral rules, taking account of the constraints of history and human nature. If each of these combinations leads with certainty to a particular social alternative, they can be ranked with any welfarist principle. If, however, consequences are uncertain, the problem is more diﬃcult. One way of doing it is to attach probabilities (possibly subjective) to a set of ‘states of nature’ and use them to rank prospects: lists of corresponding alternatives. There are diﬀerent ways to make use of the resulting rankings. One is to take an uncomplicated maximizing approach and recommend to governments and individuals alike that they choose the best feasible action, a position taken by act utilitarians such as Sidgwick (1907, 1966). Other utilitarians realize that it may be impossible or unwise for individuals to engage in complicated assessments of consequences and, instead, follow general rules which are “constantly evolving, but on the whole stable, such that their use in moral education, including self-education, and their consequent acceptance by society at large, will lead to the nearest possible approximation to archangelic thinking” (Hare, 1982, p. 33), a position sometimes called rule utilitarianism. It is true, in addition, that the best actions may require an individual, state or generation to make very great sacriﬁces. Because concern for others is bounded, it may be important to choose rules that limit the sacriﬁces of agents or declare some actions to be supererogatory: beyond the call of duty.5 Mill (1861, 1969), a utilitarian who took such considerations seriously, was not a simple maximizer (see, for example, Brown, 1972). Section 2 introduces social-evaluation functionals. They make use of some or all available utility and non-utility information to rank alternatives according to their goodness. In addition, the section provides a formal account of information environments
5 See also Blackorby, Bossert and Donaldson (2000) for a discussion of limited altruism.

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and information-invariance conditions. Information environments are described by partitions of proﬁles of utility functions into equivalence classes. They determine which statements involving both intrapersonal and interpersonal utility comparisons are meaningful. Information-invariance conditions specify the amount of utility information that a socialevaluation functional may make use of.6 Section 3 turns to welfarism and it is shown that, given an unlimited domain of utility proﬁles, welfarism is characterized by the axioms binary independence of irrelevant alternatives and Pareto indiﬀerence. Section 4 contains a set of results that characterize generalized utilitarianism. Section 5 focuses on utilitarianism itself and presents characterization theorems for it. The axioms employed in Sections 4 and 5 are of two types. Some are information assumptions and they require the social-evaluation functional to make use of meaningful utility information only. Other axioms, such as strong Pareto and anonymity, are ethical in nature. Anonymity, for example, captures the idea of impartiality in social evaluation, an essential feature of many welfarist principles. Population issues are the concern of Section 6 which discusses extensions of utilitarianism and generalized utilitarianism to environments in which alternatives may diﬀer with respect to population size and composition. The axioms used in that section strengthen the case for the utilitarian and generalized-utilitarian principles and, in addition, characterize families of principles that extend the ﬁxed-population principles in an ethically attractive way. These families are known as critical-level utilitarianism and critical-level generalized utilitarianism. They require the speciﬁcation of a ﬁxed ‘critical level’ of lifetime utility above which additions to a population are, ceteris paribus, valuable. Section 7 makes use of subjective probabilities to rank prospects. We present a multi-proﬁle version of Harsanyi’s (1955, 1977) social-aggregation theorem. Instead of using lotteries as social alternatives, we employ prospects and assume that probabilities are ﬁxed and common to all individuals and the social evaluator. Both individual exante utilities and social preferences are assumed to satisfy the expected-utility hypothesis (von Neumann and Morgenstern, 1944, 1947) and individual ex-ante utilities are equal to the expected value of von Neumann – Morgenstern utilities (this is called the Bernoulli hypothesis by Broome, 1991a). Given that, we show that any welfarist ex-ante socialevaluation functional satisfying anonymity and the weak Pareto principle is utilitarian. Harsanyi (1953) presents another argument for utilitarianism in his impartial-observer theorem. It is discussed in detail in Mongin (2001) and is omitted in this survey. Welfarist social evaluation is an attractive option but it is not the only one. Many people who reject welfarism do not believe that welfare considerations are completely irrelevant to social decision-making, however. Accordingly, the results of this chapter should be of interest to most people who are concerned with social evaluation. Other
6 For a more complete guide to these requirements, see d’Aspremont and Gevers (2001).

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theories, such as Sen’s (1985) treatment of functionings and capabilities, can be modiﬁed to ﬁt our framework. Functionings are the things that people can do, and the idea can be used to provide an account of (some aspects of) well-being. Capabilities are opportunities and include freedoms. If the two are aggregated into a single index of ‘advantage’ for each person, welfarist social-evaluation functionals can be used to rank alternatives. On the other hand, it is possible to use welfarist principles to aggregate functionings into an index of social functioning, which would be only one factor in overall social evaluation.7 A major challenge to welfarism has appeared in recent years. It replaces concern for well-being with concern for opportunities for well-being on the grounds that individual people are responsible for their choices (in certain circumstances they may be thought to be responsible for their preferences as well).8 In practice, welfarists often agree that the provision of opportunities is socially warranted, but their concern is with actual well-being. If autonomy is a signiﬁcant aspect of well-being, people must be free to make important choices for themselves, and this provides a constraint which restricts the feasible set of social possibilities. By way of analogy, parents typically provide opportunities to their children, but that does not mean that opportunities are what they care about.

2. Social-Evaluation Functionals Social-evaluation functionals use information about the members of a set of possible social alternatives to rank them according to their social goodness. An alternative is a complete history of the world from remote past to distant future. Let X be a set of alternatives that contains at least three members (some slightly stronger requirements on the minimal number of alternatives are employed in Sections 6 and 7). No other restrictions are imposed on X: it may be ﬁnite, countably inﬁnite, or uncountable. The set of individuals in an n-person society is {1, . . . , n} where n ∈ Z++ .9 Except for our discussion in Section 6, we consider only comparisons of alternatives with the same population. For any i ∈ {1, . . . , n}, Ui : X → R is i’s utility function10 and ui = Ui (x) is the utility level of individual i ∈ {1, . . . , n} in alternative x ∈ X. Utilities are interpreted as indicators of lifetime well-being and measure how good a person’s life is from his or her own point of view. This does not mean that the utility function Ui is a representation
7 Suzumura (1999) considers the value of social procedures in addition to the value of individual wellbeing. 8 See, for example, Arneson (1989, 2000), Roemer (1996) and, for an unsympathetic critique, Anderson (1999). See also Fleurbaey and Maniquet (2001) and Foster (2001). 9 Z ++ is the set of positive integers and Z+ is the set of nonnegative integers. 10 R, R and R + ++ are the sets of all real numbers, nonnegative real numbers and positive real numbers respectively. In addition, 1n = (1, . . . , 1) ∈ Rn .

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of person i’s actual preferences. Preferences and utility functions may be inconsistent because of individual non-rationalities, altruism or insuﬃcient information.11 A proﬁle of utility functions is an n-tuple U = (U1 , . . . , Un ) with one utility function for each individual in society. The set of all possible proﬁles is denoted by U. For U ∈ U and x ∈ X, we write U(x) = (U1 (x), . . . , Un (x)). This vector represents the welfare information for alternative x given the proﬁle U. In addition to welfare information, nonwelfare information may be available, and we assume that each x in the set X contains a full description of all non-welfare aspects of alternatives that may be considered relevant for social evaluation. A social-evaluation functional is a mapping F : D → O, where ∅ = D ⊆ U and O is the set of all orderings on X.12 D is the domain of admissible utility proﬁles and it may consist of a single proﬁle or many proﬁles. In the latter case, the social-evaluation functional can cope with diﬀerent proﬁles of utility functions and inter-proﬁle consistency conditions such as binary independence of irrelevant alternatives or various informationinvariance conditions (see below) may be imposed on it. The social-evaluation functional may make use of non-welfare information in addition to welfare information contained in the proﬁle U. For simplicity of notation, we write RU = F (U); IU and PU are the symmetric and asymmetric components of RU . For any x, y ∈ X, xRU y means that x is socially at least as good as y, xIU y means that x and y are equally good, and xPU y means that x is socially better than y. General social-evaluation functionals may make use of both welfare and non-welfare information, but welfarist functionals ignore non-welfare information and completely nonwelfarist functionals ignore welfare information, making use of non-welfare information only. In the latter case, a single ordering of the alternatives in X is produced because only a single set of non-welfare information is available. In a multi-proﬁle environment, it is possible to restrict the welfare information that the social-evaluation functional F may make use of. This is done by partitioning D into subsets of informationally equivalent proﬁles called information sets. Usable information in a proﬁle in D is that which all informationally equivalent proﬁles in the corresponding information set have in common. If utilities are ordinally measurable and interpersonally noncomparable, for example, two proﬁles U, V ∈ D are informationally equivalent if and only if there exist increasing functions φ1 , . . . , φn with φi : R → R for all i ∈ {1, . . . , n} such that (V1 (x), . . . , Vn (x)) = (φ1 (U1 (x)), . . . , φn (Un (x))) for all x ∈ X. The utility comparison U1 (x) > U1 (y) is meaningful in such an environment because it is true in all
11 See Broome (1991a) and Mongin and d’Aspremont (1998) for discussions of individual well-being and its relationship to preferences, information and self-interest. Hammond (2001) oﬀers a very diﬀerent account, interpreting “individual welfare as a purely ethical concept”. 12 An ordering is a reﬂexive, transitive and complete binary relation. Social-evaluation functionals are also referred to as social-welfare functionals and were introduced by Sen (1970).

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informationally equivalent proﬁles or false in all of them. On the other hand, if D = U, the interpersonal comparison U1 (x) > U2 (x) is not meaningful because it is true in some informationally equivalent proﬁles and false in others. Restrictions on social-evaluation functionals imposed by the available information regarding the measurability and interpersonal comparability of individual utilities can be summarized using information-invariance conditions. In order to represent the informational environment, the set of admissible proﬁles is partitioned into information sets and an information-invariance condition requires F to be constant on each of them. That is, if two proﬁles U and V are informationally equivalent, RU and RV are identical. A partition of D into information sets can be deﬁned using an equivalence relation ∼ on D with an information set given by an equivalence class of ∼.13 That is, for U, V ∈ D, U ∼ V if and only if U and V are informationally equivalent. A social-evaluation functional F satisﬁes information invariance with respect to the information environment described by ∼ if and only if it assigns the same social ordering to all proﬁles in an equivalence class of ∼. Information Invariance with Respect to ∼: For all U, V ∈ D, if U ∼ V , then RU = RV . The most commonly used approach to formalizing various types of informational assumptions identiﬁes the equivalence relation ∼ by specifying a set of admissible transformations of utility proﬁles that lead to informationally equivalent proﬁles.14 An invariance transformation is a vector φ = (φ1 , . . . , φn ) of functions φi : R → R for all i ∈ {1, . . . , n} whose application to a proﬁle U results in an informationally equivalent proﬁle. Let Φ denote the set of invariance transformations used to generate the equivalence relation ∼. That is, for all U, V ∈ D, U ∼ V if and only if there exists φ ∈ Φ such that V = φ ◦ U, where ◦ denotes component-by-component function composition.15 Various information assumptions that can be expressed in terms of admissible transformations have been considered in contributions by Blackorby and Donaldson (1982), Blackorby, Donaldson and Weymark (1984), d’Aspremont and Gevers (1977), De Meyer and Plott (1971), Dixit (1980), Gevers (1979), Roberts (1980b) and Sen (1970, 1974, 1977a, 1986) among others. We restrict attention to the information assumptions that are relevant for the purposes of this chapter and refer the interested reader to Bossert and
13 An equivalence relation is a reﬂexive, transitive and symmetric binary relation. 14 This approach was developed in contributions such as d’Aspremont and Gevers (1977), Roberts (1980a,b) and Sen (1974). See Basu (1983), Bossert (1991, 2000), Bossert and Stehling (1992, 1994), Falmagne (1981), Fishburn, Marcus-Roberts and Roberts (1988), Fishburn and Roberts (1989) and Krantz, Luce, Suppes and Tversky (1971) for discussions of information-invariance assumptions in terms of meaningful statements and their relations to uniqueness properties of measurement scales. 15 We only consider sets Φ of invariance transformations such that the resulting relation ∼ is an equivalence relation. See Bossert and Weymark (2001) for a discussion and for conditions guaranteeing this.

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Weymark (2001) or d’Aspremont and Gevers (2001) for more detailed treatments. Each of these assumptions is deﬁned by specifying the set of invariance transformations Φ which induces the equivalence relation ∼ that partitions D into sets of informationally equivalent utility proﬁles. For each information assumption listed below, we implicitly assume that the domain D contains all proﬁles that are informationally equivalent to U for each U ∈ D. This is true, in particular, for the unrestricted domain D = U on which we focus for most of the chapter. Additional information assumptions are introduced in Section 6 in a variable-population context. If utilities are cardinally measurable, individual utility functions are unique up to increasing aﬃne transformations, thereby allowing for intrapersonal comparisons of utility diﬀerences. If, in addition, some comparisons of utility are meaningful interpersonally, these transformations must be restricted across individuals. An example is cardinal unit comparability. In that information environment, admissible transformations are increasing aﬃne functions and, in addition, the scaling factor must be the same for all individuals. This information assumption allows for interpersonal comparisons of utility diﬀerences but utility levels cannot be compared interpersonally because the intercepts of the aﬃne transformations may diﬀer across individuals. Cardinal Unit Comparability (CUC): φ ∈ Φ if and only if there exist a1 , . . . , an ∈ R and b ∈ R++ such that φi (τ ) = ai + bτ for all τ ∈ R and for all i ∈ {1, . . . , n}. An information environment that provides more information than CUC is one in which the unit in which utilities are measured is numerically signiﬁcant. In this case, we say that utilities are translation-scale measurable. Utility diﬀerences are interpersonally comparable and, in addition, their numerical values are meaningful. Because the functions φ1 , . . . , φn may be diﬀerent for each person, utility levels are, again, not interpersonally comparable. Translation-Scale Measurability (TSM): φ ∈ Φ if and only if there exist a1 , . . . , an ∈ R such that φi (τ ) = ai + τ for all τ ∈ R and for all i ∈ {1, . . . , n}. If utilities are cardinally measurable and fully interpersonally comparable, both utility levels and diﬀerences can be compared interpersonally. In this case, utility functions are unique up to increasing aﬃne transformations which are identical across individuals. Cardinal Full Comparability (CFC): φ ∈ Φ if and only if there exist a ∈ R and b ∈ R++ such that φi (τ ) = a + bτ for all τ ∈ R and for all i ∈ {1, . . . , n}. CFC deﬁnes a ﬁner partition of the set of admissible utility proﬁles than CUC and, therefore, places weaker invariance requirements on the social-evaluation functional. If all the information in a proﬁle is meaningful, we say that utilities are numerically measurable and fully interpersonally comparable. In this case, each information set consists of a singleton. 8

Numerical Full Comparability (NFC): φ ∈ Φ if and only if φi (τ ) = τ for all τ ∈ R and for all i ∈ {1, . . . , n}. In general, increases in available information reduce the restrictions on F implied by the information-invariance condition. For example, information invariance with respect to TSM is a weaker restriction than information invariance with respect to CUC, and invariance with respect to NFC provides no restriction at all.

3. Welfarism The orderings on X generated by welfarist social-evaluation functionals compare any two alternatives x, y ∈ X solely on the basis of the individual utilities experienced in x and in y. All non-welfare information is ignored when establishing the social ranking. Welfarism is a consequence of three axioms, the ﬁrst of which is an unrestricteddomain assumption. This axiom requires the social-evaluation functional F to be deﬁned on the set of all possible utility proﬁles. Unrestricted Domain (UD): D = U. The next axiom is an independence condition which links the orderings associated with diﬀerent proﬁles. It requires the social ranking of any two alternatives to be independent of the utility levels associated with other alternatives. Binary Independence of Irrelevant Alternatives (BI): For all x, y ∈ X, for all U, V ∈ D, if U(x) = V (x) and U(y) = V (y), then xRU y if and only if xRV y. The above independence axiom for social-evaluation functionals is weaker than the corresponding independence axiom for social-welfare functions (see Arrow, 1951, 1963, and Sen, 1970). Arrow’s independence axiom requires the social ordering of a pair of alternatives to depend only on the individual rankings of the two alternatives. BI is equivalent to Arrow’s binary-independence axiom if the social-evaluation functional satisﬁes information invariance with respect to ordinally measurable, interpersonally noncomparable utilities. As formulated above, binary independence is compatible with any assumption concerning the measurability and interpersonal comparability of individual utilities. The ﬁnal axiom used to generate welfarism is Pareto indiﬀerence. If all individuals are equally well oﬀ in two alternatives, it requires the social-evaluation functional to rank them as equally good. Pareto Indiﬀerence (PI): For all x, y ∈ X, for all U ∈ D, if U(x) = U(y), then xIU y. Pareto indiﬀerence is an attractive axiom if utility functions measure everything that is of value to individuals. For that reason, we endorse comprehensive accounts of lifetime

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utility such as the ones provided by Griﬃn (1986) and Sumner (1996).16 Griﬃn includes enjoyment, pleasure and the absence of pain, good health, autonomy, liberty, understanding, accomplishment and good human relationships as aspects of well-being. He argues, in addition, that there is a moral dimension to well-being. Sumner also discusses the role that individual attitudes can play. Pareto indiﬀerence is a consequence of the fundamental welfarist view that, if one alternative is ranked as better than another, it must be better for at least one individual (see Goodin, 1991). We call this axiom minimal individual goodness, and it is deﬁned formally as follows. Minimal Individual Goodness (MIG): For all x, y ∈ X, for all U ∈ D, if xPU y, then there exists k ∈ {1, . . . , n} such that Uk (x) > Uk (y). Minimal individual goodness is equivalent to the conjunction of Pareto indiﬀerence and the following Pareto-weak-preference axiom. Pareto Weak Preference (PWP): For all x, y ∈ X, for all U ∈ D, if Ui (x) ≥ Ui (y) for all i ∈ {1, . . . , n} with at least one strict inequality, then xRU y. We obtain Theorem 1: A social-evaluation functional F satisﬁes MIG if and only if F satisﬁes PI and PWP. Proof. Suppose F satisﬁes MIG. We ﬁrst prove by contradiction that PI is satisﬁed. Suppose not. Then there exist x, y ∈ X and U ∈ D such that U(x) = U(y) and not xIU y. Because RU is complete, we must have either xPU y or yPU x. In either case, we obtain a contradiction to MIG. Now suppose F violates PWP. Then there exist x, y ∈ X and U ∈ D such that Ui (x) ≥ Ui (y) for all i ∈ {1, . . . , n} with at least one strict inequality and not xRU y. By the completeness of RU , we must have yPU x, again contradicting MIG. Finally, suppose F satisﬁes PI and PWP but violates MIG. Then there exist x, y ∈ X and U ∈ D such that xPU y and Ui (y) ≥ Ui (x) for all i ∈ {1, . . . , n}. If Ui (y) = Ui (x) for all i ∈ {1, . . . , n}, we obtain a contradiction to PI, and if there exists k ∈ {1, . . . , n} such that Uk (y) > Uk (x), we obtain a contradiction to PWP. In the presence of unrestricted domain, BI and PI together imply that non-welfare information about the alternatives must be ignored by the social-evaluation functional. If, in one proﬁle, utility numbers for a pair of alternatives are equal to the utility numbers
16 See also Broome (1991a) and Mongin and d’Aspremont (1998) for accounts based on self-interested preferences under conditions of full information.

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for another pair in a possibly diﬀerent proﬁle, the rankings of the two pairs must be the same. This property is called strong neutrality. Strong Neutrality (SN): For all x, y, z, w ∈ X, for all U, V ∈ D, if U(x) = V (z) and U(y) = V (w), then xRU y if and only if zRV w. We obtain the following theorem (see, for example, Blau, 1976, d’Aspremont and Gevers, 1977, Guha, 1972, and Sen, 1977a, for this and related results). Theorem 2: Suppose that a social-evaluation functional F satisﬁes UD. F satisﬁes BI and PI if and only if F satisﬁes SN. Proof. First, suppose that F satisﬁes UD and SN. That BI is satisﬁed follows immediately by setting x = z and y = w in the deﬁnition of SN. Setting U = V and y = z = w, SN implies that xRU y if and only if yRU y when U(x) = U(y). Because RU is reﬂexive, this implies xIU y, which demonstrates that PI is satisﬁed. Now suppose that F satisﬁes UD, BI and PI. Suppose that U(x) = V (z) = u and ¯ ˆ ˜ U(y) = V (w) = v. By UD, there exists an alternative x ∈ X and proﬁles U, U, U ∈ D ¯ ¯ ¯ x ¯ ˆ ˆ x ˆ ˜ x such that U(x) = U (¯) = u and U(y) = v, U(z) = U (¯) = u and U(w) = v, and U(¯) = u ˜ ˜ and U(y) = U(w) = v. By BI, xRU y if and only if xRU y. By PI and the transitivity ¯ of RU , it follows that xRU y if and only if xRU y. A similar argument implies that xRU y ¯ ¯ ¯ ¯ ¯ ¯ ¯ ˜ if and only if xRU y if and only if xRU w. Applying the same argument once again, we ¯ ˜ have xRU w if and only if xRU w if and only if zRU w. By BI, zRU w if and only if zRV w. ¯ ˜ ¯ ˆ ˆ ˆ Therefore, xRU y if and only if zRV w which proves that F satisﬁes SN. ∗ Given unrestricted domain, SN is equivalent to the existence of an ordering R on Rn which can be used to rank the alternatives in X for any utility proﬁle U. The social betterness (strict preference) relation and the equal-goodness (indiﬀerence) relation corre∗ ∗ ∗ ∗ sponding to R are denoted by P and I respectively. We refer to R as a social-evaluation ordering.17 Combined with Theorem 2, this observation yields the following welfarism theorem (see d’Aspremont and Gevers, 1977, and Hammond, 1979).18 Suppose that a social-evaluation functional F satisﬁes UD. F satisﬁes BI ∗ and PI if and only if there exists a social-evaluation ordering R on Rn such that, for all x, y ∈ X and for all U ∈ D, ∗ xRU y ⇔ U(x)RU(y). (3.1)
∗ 17 Gevers (1979) calls R a social-welfare ordering. 18 Bordes, Hammond and Le Breton (1997) and Weymark (1998) prove variants of this theorem with speciﬁc domain restrictions, that is, with weaker domain assumptions than UD.

Theorem 3:

11

∗ Proof. If there exists a social-evaluation ordering R such that, for all x, y ∈ X and all U ∈ D, (3.1) is satisﬁed, BI and PI are satisﬁed. Now suppose that F satisﬁes UD, BI and PI. By Theorem 2, F satisﬁes SN. Deﬁne ∗ ∗ the relation R as follows. For all u, v ∈ Rn , uRv if and only if there exist a proﬁle U ∈ D and two alternatives x, y ∈ X such that U(x) = u, U(y) = v, and xRU y. By SN, the relative ranking of any two utility vectors u and v does not depend on the proﬁle U or on ∗ the alternatives x and y used to generate u and v and, therefore, R is well-deﬁned. That ∗ R is reﬂexive and complete follows immediately because RU is reﬂexive and complete for ∗ all U ∈ D. It remains to be shown that R is transitive. Suppose u, v, w ∈ Rn are such ∗ ∗ that uRv and v Rw. By UD and the maintained assumption that X contains at least three elements, there exists a proﬁle U ∈ D and three alternatives x, y, z ∈ X such that ∗ ∗ U(x) = u, U(y) = v, and U(z) = w. Because U(x)RU(y) and U(y)RU(z), it follows that ∗ xRU y and yRU z by deﬁnition of R. Transitivity of RU then implies that xRU z. Hence, ∗ ∗ ∗ U(x)RU(z) or, equivalently, uRw which shows that R is transitive. ∗ Note that the social-evaluation ordering R in the statement of Theorem 3 is proﬁleindependent. Pairs of alternatives whose utility vectors are the same are ranked in the same way, regardless of the utility proﬁle. If D consists of a single proﬁle, the result is true but BI is not needed (Blackorby, Donaldson and Weymark, 1990). ∗ For notational convenience, we concentrate on the social-evaluation ordering R in most of the remainder of the chapter. All axioms and results regarding this ordering can be reformulated in terms of the social-evaluation functional F by deﬁning the properties ∗ analogous to those deﬁned for R and adding the welfarism axioms. Given a set of invariance transformations Φ (and, hence, an equivalence relation ∼) and the welfarism axioms UD, BI and PI, the information-invariance axiom for the socialevaluation functional F is equivalent to an analogous condition formulated in terms of the ∗ corresponding social-evaluation ordering R. Information Invariance with Respect to Φ: For all u, v, u , v ∈ Rn , if there exists ∗ ∗ φ ∈ Φ such that u = φ(u) and v = φ(v), then uRv if and only if u Rv . Next, we introduce some axioms that are commonly required in welfarist social evaluation. Continuity is a regularity condition. It ensures that ‘small’ changes in individual utilities do not lead to ‘large’ changes in social rankings. ∗ ∗ Continuity (C): For all u ∈ Rn , the sets {v ∈ Rn | v Ru} and {v ∈ Rn | uRv} are closed in Rn .

12

∗ Anonymity ensures that the ordering R treats individuals impartially, paying no attention to their identities. That is, any permutation of a given utility vector must be indiﬀerent to the utility vector itself. Note that this is a strengthening of Arrow’s (1951, 1963) condition that prevents the existence of a dictator. Anonymity (A): For all u ∈ Rn , for all bijective mappings π: {1, . . . , n} → {1, . . . , n}, ∗ uI uπ(1) , . . . , uπ(n) . (3.2)

The weak Pareto principle requires an increase in everyone’s utility to be regarded as a social improvement. ∗ Weak Pareto (WP): For all u, v ∈ Rn , if ui > vi for all i ∈ {1, . . . , n}, then uP v. A strengthening of both weak Pareto and Pareto indiﬀerence is the strong Pareto principle. In addition to Pareto indiﬀerence, it requires that if no one’s utility has decreased and at least one person’s utility has increased, the change is a social improvement. ∗ Strong Pareto (SP): For all u, v ∈ Rn , (i) if ui = vi for all i ∈ {1, . . . , n}, then uI v; ∗ and (ii) if ui ≥ vi for all i ∈ {1, . . . , n} with at least one strict inequality, then uP v. Note that (i) in the deﬁnition of SP is redundant in a welfarist framework—this restriction ∗ is implied by the reﬂexivity of R. We have chosen to include it in the deﬁnition of strong Pareto order to follow the conventional terminology. Finally, we introduce an axiom which prevents the social ordering from exhibiting a strong version of inequality preference. Minimal Equity (ME): There exist i, j ∈ {1, . . . , n} and u, v ∈ Rn such that uk = vk ∗ for all k ∈ {1, . . . , n} \ {i, j}, vj > uj > ui > vi , and uRv. See d’Aspremont (1985), d’Aspremont and Gevers (1977) and Deschamps and Gevers (1978) for this axiom and Hammond (1976) for a related condition. Continuity and the weak Pareto principle ensure the existence of a continuous repre∗ sentation of the social-evaluation ordering R. We obtain ∗ Theorem 4: If a social-evaluation ordering R satisﬁes C and WP, then, for each u ∈ Rn , ∗ there exists a unique ξ = Ξ(u) ∈ [min{u1 , . . . , un}, max{u1 , . . . , un }] such that uI ξ1n .

13

∗ Proof. If n = 1, WP and reﬂexivity imply that, for all u, v ∈ R, uRv if and only if u ≥ v. Consequently, the result follows from letting ξ = Ξ(u) = u (which is equal to both maximum and minimum utility) for all u ∈ R. Suppose n ≥ 2, let u ∈ Rn be arbitrary and suppose, by way of contradiction, that ∗ min{u1 , . . . , un}1n P u. By C, there exists a neighborhood of u such that min{u1 , . . . , un }1n ∗ is preferred to all points in that neighborhood according to R. Because this neighborhood contains points that strictly dominate u and, thus, min{u1 , . . . , un}1n , this contradicts ∗ ∗ WP. Therefore, uR min{u1 , . . . , un }1n . Analogously, it follows that max{u1 , . . . , un }1n Ru. ∗ By C, it follows that there exists ξ ∈ [min{u1 , . . . , un }, max{u1 , . . . , un }] such that uI ξ1n . WP implies that ξ must be unique for each u and thus can be written as a function Ξ: Rn → R. The representative utility ξ is analogous to the equally-distributed-equivalent income used in ethical approaches to income-inequality measurement.19 The function Ξ is the ∗ representative-utility function corresponding to R, and it is easy to see that it is a repre∗ sentation of R—that is, for all u, v ∈ Rn , ∗ uRv ⇔ Ξ(u) ≥ Ξ(v). (3.3)

∗ Furthermore, Ξ is continuous because R is, and WP implies that Ξ is weakly increasing. We conclude this section with some examples of welfarist social-evaluation orderings. The utilitarian social-evaluation ordering uses the sum of the individual utilities to make social comparisons. According to utilitarianism, for all u, v ∈ Rn , ∗ uRv ⇔ n n

ui ≥ i=1 i=1

vi .

(3.4)

The class of social-evaluation orderings that respect all strict rankings of utility vectors according to utilitarianism is the class of weakly utilitarian orderings (see Deschamps ∗ and Gevers, 1978). R is weakly utilitarian if and only if n n

ui > i=1 i=1

∗ vi =⇒ uP v

(3.5)

for all u, v ∈ Rn . The members of the class of generalized-utilitarian orderings perform social comparisons by adding the transformed utilities of the members of society. If the transformation applied to individual utilities is (strictly) concave, the resulting ordering represents (strict)
19 See, for example, Atkinson (1970), Dalton (1920), Kolm (1969) and Sen (1973).

14

∗ aversion to utility inequality. Formally, a social-evaluation ordering R is generalized utilitarian if and only if there exists a continuous and increasing function g: R → R such that, for all u, v ∈ Rn , n n ∗ g(ui ) ≥ g(vi ). (3.6) uRv ⇔ i=1 i=1

All generalized-utilitarian social-evaluation orderings satisfy C, A, and SP (and thus WP). If g is concave, ME is satisﬁed as well. Utilitarianism is a special case of generalized utilitarianism in which the transformation g is aﬃne. An example of a class of generalized-utilitarian orderings is the class of ∗ symmetric global means of order r. R is a symmetric global mean of order r if and only if there exist β, r ∈ R++ such that, for all u, v ∈ Rn , ∗ r uRv ⇔ ur − β |ui|r ≥ vi − β |vi |r . (3.7) i i∈{1,...,n}: ui ≥0 i∈{1,...,n}: ui j and u(j) > v(j) 15

for all u, v ∈ Rn . Maximin is continuous and violates the strong Pareto principle (but satisﬁes weak Pareto) and leximin satisﬁes strong Pareto but not continuity. Both orderings satisfy A and ME. The extremely equality-averse counterpart of leximin is the leximax ordering, which is deﬁned by ∗ uRv ⇔ u is a permutation of v or there exists a j ∈ {1, . . . , n} such that u(i) = v(i) for all i < j and u(j) > v(j) for all u, v ∈ Rn . Leximax satisﬁes A and SP (and therefore WP) but violates C and ME. The class of single-parameter Gini social-evaluation orderings provides another possibility for a generalization of utilitarianism—their level sets are linear in rank-ordered ∗ subspaces of Rn . A social-evaluation ordering R is a single-parameter Gini ordering if and only if there exists a real number δ ≥ 1 such that, for all u, v ∈ Rn , ∗ uRv ⇔ n n

(3.10)

i − (i − 1) u(i) ≥ δ δ i=1 i=1

iδ − (i − 1)δ v(i).

(3.11)

The single-parameter Ginis are special cases of the generalized Ginis introduced by Weymark (1981) and discussed in Bossert (1990a) and Donaldson and Weymark (1980) in the context of ethical inequality measurement. For δ = 1, we obtain utilitarianism, and as δ approaches inﬁnity, maximin is obtained in the limit. The case δ = 2 yields the social-evaluation ordering corresponding to the Gini index of inequality (Blackorby and Donaldson, 1978). All single-parameter Ginis satisfy C, A, SP and ME.

4. Generalized Utilitarianism A distinguishing feature of generalized-utilitarian social-evaluation orderings is that they possess an additively separable structure. This separability property is closely related to several plausible independence conditions which deal with the inﬂuence of the well-being of unconcerned individuals on the social ordering, and those conditions can, together with some of our earlier axioms, be used to provide characterizations of generalized utilitarianism. Suppose that a social change aﬀects only the utilities of the members of a population subgroup. Independence of the utilities of unconcerned individuals requires the social assessment of the change to be independent of the utility levels of people outside the subgroup.

16

Independence of the Utilities of Unconcerned Individuals (IUUI): For all M ⊆ {1, . . . , n}, for all u, v, u , v ∈ Rn , if ui = vi and ui = vi for all i ∈ M and uj = uj and vj = vj for all j ∈ {1, . . . , n} \ M, then ∗ ∗ uRv ⇔ u Rv . (4.1)

In this deﬁnition, the individuals in M are the unconcerned individuals—they are equally well oﬀ in u and v and in u and v . IUUI requires the ranking of u and v to depend on the utilities of the concerned individuals—those not in M—only. In terms of a realvalued representation, this axiom is referred to as complete strict separability in Blackorby, Primont and Russell (1978). The corresponding separability axiom for social-evaluation functionals can be found in d’Aspremont and Gevers (1977) where it is called separability with respect to unconcerned individuals. d’Aspremont and Gevers’ separability axiom is called elimination of (the inﬂuence of) indiﬀerent individuals in Maskin (1978) and Roberts (1980b). In the case of two individuals, this axiom is implied by strong Pareto. Therefore, its use is typically restricted to societies with at least three individuals. In that case, we obtain the following characterization of generalized utilitarianism. ∗ Theorem 5: Suppose that n ≥ 3. A social-evaluation ordering R satisﬁes C, A, SP and ∗ IUUI if and only if R is a generalized-utilitarian social-evaluation ordering. Proof. Applying Debreu’s (1959, pp. 56–59) representation theorem, continuity implies that there exists a continuous function f: Rn → R such that, for all u, v ∈ Rn , ∗ uRv ⇔ f(u) ≥ f(v). (4.2)

By SP, f is increasing in all arguments, and A implies that f is symmetric. IUUI requires that {1, . . . , n} \ M is separable from its complement M for any choice of M ⊆ {1, . . . , n}. Gorman’s (1968) theorem on overlapping separable sets of variables (see also Acz´l, 1966, p. 312, and Blackorby, Primont, and Russell, 1978, p. 127) implies e that f is additively separable. Therefore, there exist continuous and increasing functions H: R → R and gi : R → R for all i ∈ {1, . . . , n} such that n f(u) = H i=1 gi (ui )

(4.3)

17

for all u ∈ Rn . Because f is symmetric, each gi can be chosen to be independent of i, and ∗ we deﬁne g = gi for all i ∈ {1, . . . , n}. Therefore, because f is a representation of R, ∗ uRv ⇔ H n n n

g(ui ) ≥ H i=1 n i=1

g(vi ) (4.4)

⇔ i=1 g(ui ) ≥ i=1 g(vi )

for all u, v ∈ Rn . See also Debreu (1960) and Fleming (1952) for variants of this theorem. Due to the presence of A, IUUI could be weakened by suitably restricting the possible sets of unconcerned individuals. An alternative to independence of the utilities of unconcerned individuals is the population substitution principle. It considers a sequence of social-evaluation orderings ∗ {Rn }n∈Z++ , one for each population size in Z++ . Given, for each n ∈ Z++ , anonymity and the axioms guaranteeing the existence of the representative-utility functions {Ξn}n∈Z++ (see Theorem 4), the population substitution principle requires that replacing the utilities of a subgroup of the population with the representative utility of that subgroup is a matter of indiﬀerence (see Blackorby and Donaldson, 1984). Population Substitution Principle (PSP): For all n ≥ 3, for all u ∈ Rn , for all M ⊆ {1, . . . , n}, ∗ uI n Ξ|M | (ui )i∈M 1|M |, (uj )j∈{1,...,n}\M . (4.5) ∗ As is the case for IUUI, PSP implies that for all n ≥ 3, the representation Ξn of Rn must be additively separable and, therefore, it provides an alternative way of characterizing generalized utilitarianism in the presence of continuity, anonymity and strong Pareto. The proof of this result is analogous to the one of Theorem 5 and is thus omitted. ∗ Theorem 6: A sequence of social-evaluation orderings {Rn }n∈Z++ satisﬁes C, A, SP and PSP if and only if there exists a continuous and increasing function g: R → R such ∗ that, for all n ∈ Z++ , Rn is a generalized-utilitarian social-evaluation ordering with ∗ uR n v ⇔ for all u, v ∈ Rn . n n

g(ui ) ≥ i=1 i=1

g(vi )

(4.6)

18

See Blackorby and Donaldson (1984) for a related result in a variable-population framework. Generalized utilitarianism can also be characterized in an intertemporal framework. In that case, a very weak and natural separability axiom can be employed. This condition, the variable-population version of which was introduced in Blackorby, Bossert and Donaldson (1995), requires social evaluations to be independent of the utilities of individuals whose lives are over in both of any two alternatives and who had the same birth dates, lengths of life, and lifetime utilities in both. Consider a model where each alternative x ∈ X contains (among other features that may be considered relevant for social evaluation) information about individual birth dates and lengths of life. We assume that no one can live longer than L ∈ Z++ periods (L may be arbitrarily large). For i ∈ {1, . . . , n} and x ∈ X, let si = Si (x) ∈ Z+ be the period before individual i is born in alternative x, and let li = Li (x) ∈ {1, . . . , L} be i’s lifetime in x (in periods). Thus, individual i is alive in periods si + 1 to si + li, and ui = Ui (x) is i’s lifetime utility in alternative x. Let s = (s1 , . . . , sn ), l = (l1 , . . . , ln ) and, as before, u = (u1 , . . . , un ). In order to extend our model to this intertemporal framework, instead of a social∗ ◦ n evaluation ordering R, we employ an ordering R on A = Z+ × {1, . . . , L}n × Rn and the objects to be ranked are vectors (s, l, u) of the birth dates, lifetimes, and lifetime utilities of everyone in society. It is straightforward to reformulate intertemporal versions of the axioms continuity, anonymity and strong Pareto in this framework. We use the following deﬁnitions, each of which is stated for an arbitrary n ∈ Z++ . ◦ Intertemporal Continuity (IC): For all (s, l, u) ∈ A, the sets {v ∈ Rn | (s, l, v)R(s, l, u)} ◦ and {v ∈ Rn | (s, l, u)R(s, l, v)} are closed in Rn . Intertemporal Anonymity (IA): For all (s, l, u) ∈ A, for all bijective mappings π: {1, . . . , n} → {1, . . . , n}, ◦ (s, l, u)I (sπ(1) , . . . , sπ(n) ), (lπ(1) , . . . , lπ(n) ), (uπ(1) , . . . , uπ(n) ) . (4.7)

Intertemporal Strong Pareto (ISP): For all (s, l, u), (r, k, v) ∈ A, (i) if ui = vi for all ◦ i ∈ {1, . . . , n}, then (s, l, u)I (r, k, v); and (ii) if ui ≥ vi for all i ∈ {1, . . . , n} with at least ◦ one strict inequality, then (s, l, u)P (r, k, v). At any time, all of the possible alternatives have a common past: the history that has actually obtained. This means that we can think of history as having a branching structure. Decisions taken at a particular time have the eﬀect of selecting the branch along which events will unfold. At any time, some individuals’ lives may have ended in all possible alternatives or their lives may be over in some and not in others. 19

The axiom independence of the utilities of the dead requires that, in any period t ∈ Z++ , the relative ranking of any two alternatives is independent of the utilities of those individuals whose lives are over in t and who had the same birth dates, lifetimes, and lifetime utilities in both alternatives. To deﬁne this axiom formally, we need more notation. Let, for all (s, l, u) ∈ A and all t ∈ Z++ , Dt (s, l, u) = {i ∈ {1, . . . , n} | si +li < t} and Bt (s, l, u) = {i ∈ {1, . . . , n} | si + 1 < t}. The individuals in Dt (s, l, u) are those individuals whose lives are over before period t, and Bt (s, l, u) contains the individuals who are born before t. We can now deﬁne our intertemporal independence condition. Independence of the Utilities of the Dead (IUD): For all (s, l, u), (r, k, v), (s , l , u ), (r , k , v ) ∈ A, for all t ∈ Z++ , if Bt (s, l, u) = Bt (s , l , u ) = Bt (r, k, v) = Bt (r , k , v ) = Dt (s, l, u) = Dt (s , l , u ) = Dt (r, k, v) = Dt (r , k , v ) = Mt , (4.8)

(si , li , ui ) = (ri , ki , vi) and (si , li , ui ) = (ri , ki , vi ) for all i ∈ Mt , and (sj , lj , uj ) = (sj , lj , uj ) and (rj , kj , vj ) = (rj , kj , vj ) for all j ∈ {1, . . . , n} \ Mt , then ◦ ◦ (s, l, u)R(r, k, v) ⇔ (s , l , u )R(r , k , v ). (4.9)

IUD is a very weak separability condition because it applies to individuals whose lives are over only and not to all unconcerned individuals. That is, if all generations overlap, it does not impose any restrictions. However, when combined with the intertemporal version of the strong Pareto principle, this axiom has important consequences. In particular, independence of the utilities of the dead and intertemporal strong Pareto together imply an intertemporal version of independence of the utilities of unconcerned individuals, which is deﬁned as follows. Intertemporal Independence of the Utilities of Unconcerned Individuals (IIUUI): For all M ⊆ {1, . . . , n}, for all (s, l, u), (r, k, v), (s , l , u ), (r , k , v ) ∈ A, if ui = vi and ui = vi for all i ∈ M and uj = uj and vj = vj for all j ∈ {1, . . . , n} \ M, then ◦ ◦ (s, l, u)R(r, k, v) ⇔ (s , l , u )R(r , k , v ). We obtain ◦ Suppose that n ≥ 3 and an intertemporal social-evaluation ordering R ◦ ◦ satisﬁes ISP. R satisﬁes IUD if and only if R satisﬁes IIUUI. Theorem 7: (4.10)

20

◦ Proof. Clearly, IIUUI implies IUD. Now suppose R satisﬁes ISP and IUD. Let M ⊆ {1, . . . , n}, and suppose (s, l, u), (r, k, v), (s , l , u ), (r , k , v ) ∈ A are such that ui = vi and ui = vi for all i ∈ M and uj = uj and vj = vj for all j ∈ {1, . . . , n} \ M. Let si = ri = 0 and li = ki = 1 for all i ∈ M, and sj = rj = 1 and lj = kj = 1 for all ◦ ◦ ◦ j ∈ {1, . . . , n} \ M. By ISP , (s , l , u)I (s, l, u), (s , l , u )I (s , l , u ), (r , k , v)I (r, k, v), ◦ and (r , k , v )I (r , k , v ). Therefore, ◦ ◦ (s, l, u)R(r, k, v) ⇔ (s , l , u)R(r , k , v) and ◦ ◦ (s , l , u )R(r , k , v ) ⇔ (s , l , u )R(r , k , v ). (4.11)

(4.12)

Furthermore, by deﬁnition, B2 (s , l , u) = B2 (s , l , u ) = B2 (r , k , v) = B2 (r , k , v ) = D2 (s , l , u) = D2 (s , l , u ) = D2 (r , k , v) = D2 (r , k , v ) = M2 = M, and IUD implies ◦ ◦ (s , l , u)R(r , k , v) ⇔ (s , l , u )R(r , k , v ). (4.13)

(4.14)

Together with (4.11) and (4.12), this implies that IIUUI is satisﬁed. The conclusion of Theorem 7 remains true if ISP is weakened by requiring part (i), an intertemporal version of Pareto indiﬀerence, only; note that part (ii) of ISP is not used in the proof. As an immediate consequence of Theorem 7, a result analogous to Theorem 5 can be obtained in this intertemporal setting. Thus, the characterization result for generalized◦ utilitarian principles is remarkably robust. R is an intertemporal generalized-utilitarian social-evaluation ordering if and only if there exists a continuous and increasing function g: R → R such that, for all (s, l, u), (r, k, v) ∈ A, ◦ (s, l, u)R(r, k, v) ⇔ n n

g(ui ) ≥ i=1 i=1

g(vi ).

(4.15)

The proof of the following theorem is an immediate consequence of Theorem 7 and a change in notation that allows us to adapt Theorem 5 to the intertemporal model, and its proof is therefore omitted. ◦ Theorem 8: Suppose that n ≥ 3. An intertemporal social-evaluation ordering R satisﬁes ◦ IC, IA, ISP and IUD if and only if R is an intertemporal generalized-utilitarian socialevaluation ordering.

21

In addition to providing further support for generalized utilitarianism, the results of the intertemporal model discussed above illustrate an alternative way of obtaining fully welfarist social-evaluation functionals. Instead of imposing binary independence of irrelevant alternatives in an atemporal model, a limited version of welfarism that includes, in addition to lifetime utilities, birth dates and lengths of life as the available data, can be used to obtain welfarism by means of the strong Pareto principle alone. Weakenings of the strong Pareto principle that allow for birth dates or lifetimes to matter in intertemporal social evaluation are discussed in Blackorby, Bossert and Donaldson (1997a, 1999b).

5. Utilitarianism The ethical appeal of generalized utilitarianism rests, in part, on its separability properties. Utilitarianism is but one possibility within that class of social-evaluation orderings, and it is appropriate to ask whether it should have special status. The arguments for utilitarianism that we present in this section are based, for the most part, on information-invariance properties. It is easy to verify that all of the information-invariance assumptions introduced formally in Section 2 are compatible with utilitarianism. In an informational environment that allows for cardinal unit comparability at least, the utilitarian social-evaluation ordering can be employed. This is not the case for generalized utilitarianism: many generalizedutilitarian orderings do not satisfy information invariance with respect to cardinally measurable and fully comparable utilities (CFC) or with respect to translation-scale measurable (TSM) utilities.20 The inequality aversion that generalized utilitarianism permits has, therefore, an informational cost. The application of generalized utilitarianism is restricted to informational environments that allow (at least) for the comparability properties described by the set of admissible transformations in the following theorem. Theorem 9: Suppose that n ≥ 2 and Φ contains n-tuples of continuous and increasing functions only. Generalized utilitarianism with a continuous and increasing function g satisﬁes information invariance with respect to Φ if and only if, for each φ ∈ Φ, there exist a1 , . . . , an ∈ R and b ∈ R++ such that φi (τ ) = g −1 ai + bg(τ ) for all τ ∈ R and for all i ∈ {1, . . . , n}.
20 In addition, some generalized-utilitarian principles fail to satisfy other information-invariance conditions such as ratio-scale full comparability or translation-scale full comparability which are not discussed in this chapter. See, for example, Blackorby and Donaldson (1982).

(5.1)

22

Proof. That generalized utilitarianism satisﬁes information invariance with respect to Φ if (5.1) is satisﬁed can be veriﬁed by substitution. Now suppose generalized utilitarianism, generated by a function g, satisﬁes information invariance with respect to Φ and Φ contains continuous and increasing transformations only. Information invariance requires that an admissible transformation φ = (φ1 , . . . , φn ) ∈ Φ must satisfy the condition n n n n

g φi (ui ) ≥ i=1 i=1

g φi (vi ) ⇔ i=1 g(ui ) ≥ i=1 g(vi )

(5.2)

for all u, v ∈

Rn .

This is equivalent to the functional equation n n

g (φi (ui ) = H i=1 i=1

g(ui )

(5.3)

for all u ∈ Rn , where H is increasing. Letting zi = g(ui ) and Gi = g ◦ φi ◦ g −1 for all i ∈ {1, . . . , n}, (5.3) can be rewritten as n n

Gi (zi ) = H i=1 i=1

zi ,

(5.4)

a Pexider equation, and it follows that Gi (τ ) = ai + bτ with b ∈ R++ and ai ∈ R for all i ∈ {1, . . . , n}.21 Substituting back, we obtain φi (τ ) = g −1 ai + bg(τ ) for all i ∈ {1, . . . , n}. Condition (5.1) says, in eﬀect, that the information environment must support cardinal unit comparability of transformed utilities (g(u1 ), . . . , g(un )). It is diﬃcult to justify such an information environment unless the function g is aﬃne, in which case we are back to utilitarianism. Therefore, the informational diﬃculties involved in applying generalizedutilitarian principles other than utilitarianism suggest that the utilitarian social-evaluation functional has an important advantage over its competitors within that class. The ﬁrst characterization of utilitarianism we present is new. It does not require an information-invariance assumption but, as demonstrated below, it can be used to provide an alternative proof of a characterization result that does. We employ an axiom that we call incremental equity. In its deﬁnition, 1j is the vector x ∈ Rn with xj = 1 and xi = 0 n for all i ∈ {1, . . . , n} \ {j}. Incremental Equity (IE): For all u ∈ Rn , for all δ ∈ R, for all j, k ∈ {1, . . . , n}, ∗ u + δ1j I u + δ1k . n n (5.5)

21 See Acz´l (1966, Chapter 3) for a detailed discussion of Pexider equations and their solutions. Because e the functions φi and g (and thus the inverse of g) are continuous, the domains of H and the Gi are nondegenerate intervals, which ensures that the requisite functional-equations results apply. Pexider equations are also discussed in Eichhorn (1978).

23

IE requires a kind of impartiality with respect to utility increases or decreases. If a single individual’s utility level changes by the amount δ, IE requires the change to be ranked as equally good, no matter who receives the increment (see Hare, 1982, p. 26). Incremental equity and weak Pareto together characterize utilitarianism. ∗ ∗ Theorem 10: A social-evaluation ordering R satisﬁes WP and IE if and only if R is the utilitarian social-evaluation ordering. Proof. That the utilitarian social-evaluation ordering satisﬁes WP and IE is easily checked. If n = 1, WP alone implies the result. Now let n ≥ 2. Applying IE to (u − δ1j ), (5.5) n implies that ∗ (5.6) uI u − δ1j + δ1k . n n For any u ∈ Rn , (5.6) implies ∗ 1 uI n ∗ 1 I n . . . ∗ 1 I n n n n

ui , u2 , . . . , un + u1 − i=1 n i=1

1 n

n

ui i=1 n

1 ui , n

n i=1

2 ui , . . . , un + u1 + u2 − n

ui i=1 (5.7) n ui , . . . , i=1 i=1

ui −

n−1 n n n

ui i=1 =

1 n

ui 1n . i=1 Using WP, this implies ∗ uRv ⇔ for all u, v ∈ Rn . n ui ≥ i=1 i=1

vi

(5.8)

Equation (5.6) in the proof shows that IE requires social indiﬀerence about transfers of utility from one individual to another. Consequently, all distributions of the same total must be regarded as equally good and utilitarianism results. There is an interesting link between Theorem 10 and a well-known characterization of utilitarianism by means of A, WP and information invariance with respect to TSM. The following theorem illustrates this relationship. ∗ If a social-evaluation ordering R satisﬁes A and information invariance ∗ with respect to TSM, then R satisﬁes IE. Theorem 11:

24

∗ Proof. Suppose R satisﬁes A and information invariance with respect to TSM. By information invariance with respect to TSM, we have ∗ ∗ u + δ1j R u + δ1k ⇔ δ1j Rδ1k n n n n (5.9)

∗ for all u ∈ Rn , for all δ ∈ R, for all j, k ∈ {1, . . . , n}. By anonymity, δ1j I δ1k and, thus, n n (5.9) implies ∗ u + δ1j I u + δ1k , (5.10) n n which establishes that IE is satisﬁed. As can be veriﬁed easily, it is also the case that IE implies A, and WP and IE together imply information invariance with respect to TSM. By combining Theorems 10 and 11, we obtain an alternative proof of the abovementioned characterization result, stated in the following theorem. It is a strengthening of a result due to d’Aspremont and Gevers (1977) who use the stronger axiom information invariance with respect to cardinal unit comparability instead of information invariance with respect to translation-scale measurability (see also Blackwell and Girshick, 1954, Milnor, 1954, and Roberts, 1980b). Theorem 12: ∗ A social-evaluation ordering R satisﬁes A, WP and information invari∗ ance with respect to TSM if and only if R is the utilitarian social-evaluation ordering. Proof. That the utilitarian social-evaluation ordering satisﬁes the required axioms is ∗ easily veriﬁed. Conversely, suppose R satisﬁes A, WP and information invariance with ∗ ∗ respect to TSM. By Theorem 11, R satisﬁes WP and IE. By Theorem 10, R must be utilitarian. An alternative characterization of utilitarianism can be obtained for the case n ≥ 3 by employing information invariance with respect to CFC and the separability axiom IUUI together with C, A and SP. This theorem is due to Maskin (1978)—see also Deschamps and Gevers (1978). ∗ Suppose that n ≥ 3. A social-evaluation ordering R satisﬁes C, A, SP, ∗ IUUI and information invariance with respect to CFC if and only if R is the utilitarian social-evaluation ordering. Theorem 13:

25

Proof. That utilitarianism satisﬁes the required axioms is easy to verify. Now suppose a ∗ social-evaluation ordering R satisﬁes C, A, SP, IUUI and information invariance with re∗ spect to CFC. By Theorem 5, R is generalized utilitarian with a continuous and increasing function g. Because any increasing aﬃne transformation of g leads to the same ordering of utility vectors, we can without loss of generality assume that g(0) = 0 and g(1) = 1. It remains to be shown that, given this normalization, g must be the identity mapping. Information invariance with respect to CFC requires n n n n

g(a + bui) ≥ i=1 i=1

g(a + bvi) ⇔ i=1 g(ui ) ≥ i=1 g(vi )

(5.11)

for all u, v ∈ Rn , a ∈ R, b ∈ R++ . This is equivalent to the functional equation n n

g(a + bui ) = Ha,b i=1 i=1

g(ui )

(5.12)

for all u ∈ Rn , a ∈ R, b ∈ R++ , where Ha,b is increasing. Letting zi = g(ui ) and Ga,b (zi ) = g a + bg −1 (zi ) , (5.12) can be rewritten as n n

Ga,b (zi ) = Ha,b i=1 i=1

zi .

(5.13)

Our continuity and monotonicity assumptions ensure that all solutions to this Pexider equation are such that Ga,b (zi ) = A(a, b) + B(a, b)zi. Substituting back, we obtain the equation g(a + bτ ) = A(a, b) + B(a, b)g(τ ) (5.14) for all a, τ ∈ R and all b ∈ R++ , where we use τ instead of ui for simplicity. Setting τ = 0 and using the normalization g(0) = 0, we obtain A(a, b) = g(a), and choosing τ = 1 in (5.14) yields, together with the normalization g(1) = 1, B(a, b) = g(a + b) − g(a). Therefore, (5.14) is equivalent to g(a + bτ ) = g(τ ) g(a + b) − g(a) + g(a) for all a, τ ∈ R and all b ∈ R++ . Setting a = 0, we obtain g(bτ ) = g(b)g(τ ) for all τ ∈ R and all b ∈ R++ . Analogously, choosing b = 1 in (5.15) yields g(a + τ ) = g(τ ) g(a + 1) − g(a) + g(a) (5.17) (5.16) (5.15)

26

for all a, τ ∈ R. This is a special case of Equation 3.1.3(3) in Acz´l (1966, p. 150)22 and, e together with the increasingness of g, it follows that either there exists a c ∈ R++ such that ecτ − 1 (5.18) g(τ ) = c e −1 for all τ ∈ R, or g(τ ) = τ for all τ ∈ R. Because (5.18) is incompatible with (5.16), this completes the proof. Continuity plays a crucial role in Theorem 13. Deschamps and Gevers (1978) examine the consequences of dropping C from the list of axioms in the above theorem. Among ∗ other results, they show that if a social-evaluation ordering R satisﬁes A, SP, IUUI and ∗ information invariance with respect to CFC, then R must be weakly utilitarian, leximin or leximax. It is remarkable that these axioms narrow down the class of possible socialevaluation orderings to that extent. When minimal equity is added, only weakly utilitarian principles and leximin survive because leximax obviously violates ME. Therefore, we obtain the following theorem, which is due to Deschamps and Gevers (1978). Because the proof is very lengthy and involved, we state the theorem without proving it and refer interested readers to the appendix of their paper. ∗ Suppose that n ≥ 3. If a social-evaluation ordering R satisﬁes A, SP, ∗ ME, IUUI and information invariance with respect to CFC, then R is the leximin socialevaluation ordering or a weakly utilitarian social-evaluation ordering. Theorem 14: It should be noted that the above theorem is not a characterization result because its statement is an implication rather than an equivalence. The reason is that not all weakly utilitarian orderings satisfy all the required axioms.

6. Variable-Population Extensions Utilitarian and generalized-utilitarian social-evaluation orderings may be extended to a variable-population framework in diﬀerent ways. As an example, average and classical utilitarianism, which use the average and total utility of those alive to rank alternatives, coincide on ﬁxed-population rankings but may order alternatives with diﬀerent population sizes diﬀerently. As a full description of the corresponding state of aﬀairs, an alternative contains, in particular, information regarding the number and the identities of those who are alive in the state.
22 To see this, set f(x) = k(x) = g(τ ) and h(y) = g(a + 1) − g(a) in Acz´l (1966, Equation 3.1.3(3)). e

27

X is the set of possible alternatives. For each x ∈ X, let N(x) = N denote the set of individuals alive in x, where N ⊆ Z++ is ﬁnite and nonempty.23 Furthermore, let Z++ be the set of potential people and deﬁne Xi = {x ∈ X | i ∈ N(x)} to be the set of all alternatives in which individual i ∈ Z++ is alive. Individual i’s utility function is Ui : Xi → R and a proﬁle of utility functions is U = (Ui )i∈Z++ . We follow the standard convention in population ethics and normalize lifetime utilities so that a lifetime-utility level of zero represents neutrality. A life, taken as a whole, is worth living for an individual if and only if lifetime utility is above neutrality. Consequently, a fully informed self-interested and rational person whose lifetime-utility level is below neutrality would prefer not to have any of his or her experiences.24 We assume that, for each ˆ ˆ nonempty and ﬁnite N ⊆ Z++ , the set {x ∈ X | N(x) = N} contains at least three elements. This assumption, which is analogous to the ﬁxed-population assumption that X contains at least three elements, ensures that a variable-population version of the welfarism theorem is valid. The vector of lifetime utilities of those alive in alternative x ∈ X is Ui (x) i∈N(x) = (ui )i∈N . U E is the set of all possible utility proﬁles (Ui )i∈Z++ , which extends the domain U employed in earlier sections to a variable-population framework. A variable-population social-evaluation functional is a mapping F E : DE → O, where DE ⊆ U E is the set of admissible proﬁles. For all U ∈ DE , the social no-worse-than relation E E is RE = F E (U) and IU and PU denote its symmetric and asymmetric components. As U in the ﬁxed-population case, variable-population welfarism is the consequence of three axioms. Population Unrestricted Domain (PUD): DE = U E . Population Binary Independence of Irrelevant Alternatives (PBI): For all x, y ∈ X, for all U, V ∈ DE , if Ui (x) = Vi (x) for all i ∈ N(x) and Ui (y) = Vi (y) for all i ∈ N(y), then xRE y if and only if xRE y. U V Population Pareto Indiﬀerence (PPI): For all x, y ∈ X such that N(x) = N(y), for E all U ∈ DE , if Ui (x) = Ui (y) for all i ∈ N(x), then xIU y. Note that population Pareto indiﬀerence is a ﬁxed-population axiom (it applies to comparisons of alternatives with the same people alive in each only), whereas population binary independence of irrelevant alternatives imposes restrictions on the comparison of alternatives that may involve diﬀerent populations and population sizes. PBI requires the social ranking of any pair of alternatives to be the same if two proﬁles coincide on the pair.
23 If the empty set were included as a possible population, all results in this section would still be valid. See, for example, Blackorby, Bossert and Donaldson (1995) for details. 24 It is also true, of course, that such a person would want any change that increases his or her lifetime utility. See Broome (1993, 1999) for discussions of neutrality and its normalization to zero.

28

Results analogous to Theorems 2 and 3 are valid in this variable-population model— see Blackorby, Bossert and Donaldson (1999a) for details. Because we restrict attention to anonymous variable-population social-evaluation functionals in this section, we do not provide formal statements of the corresponding generalizations and, instead, state a related result that incorporates a variable-population anonymity condition. Population Anonymity (PA): For all U, V ∈ DE , for all bijective mappings π: Z++ → Z++ such that Ui = Vπ(i) for all i ∈ Z++ , RE = RE . U V (6.1)

∗ Let Ω = ∪n∈Z++ Rn . An ordering RE on Ω is anonymous if and only if the restriction of ∗ RE to Rn satisﬁes A for all n ∈ Z++ . We now obtain the following anonymous variable-population version of the welfarism theorem. Since the proof of this theorem is analogous to its ﬁxed-population version, it is omitted. See Blackorby, Bossert and Donaldson (1999a) and Blackorby and Donaldson (1984) for details. Theorem 15: Suppose that a variable-population social-evaluation functional F E satisﬁes PUD. F E satisﬁes PBI, PPI and PA if and only if there exists an anonymous ordering ∗ RE on Ω such that, for all x, y ∈ X and for all U ∈ DE , xRE y ⇔ Ui (x) U i∈N(x) ∗ RE Ui (y)

i∈N(y)

.

(6.2)

∗ ∗ ∗ We call RE a variable-population social-evaluation ordering, and we use I E and P E to denote its symmetric and asymmetric components. ∗ We say that the ordering RE satisﬁes the ﬁxed-population axioms continuity, weak Pareto, strong Pareto and independence of the utilities of unconcerned individuals respec∗ tively if and only if, for all n ∈ Z++ , the restriction of RE to Rn satisﬁes the appropriate ﬁxed-population axiom. If C and WP are satisﬁed, population size and representative utility are all the information that is needed to rank alternatives. No variable-population axioms are required for this result, which is due to Blackorby and Donaldson (1984). The existence of a value function representing this ranking is not guaranteed by the ﬁxed-population axioms alone. For that, continuity must be strengthened to extended continuity. Extended Continuity (EC): For all n, m ∈ Z++ and for all u ∈ Rn , the sets {v ∈ Rm | ∗ ∗ v RE u} and {v ∈ Rm | uRE v} are closed in Rm .

29

The following representation theorem is due to Blackorby, Bossert and Donaldson (2001). In the theorem statement, Ξn is the representative-utility function for the restriction of ∗ RE to Rn . ∗ Theorem 16: If an anonymous variable-population social-evaluation ordering RE satisﬁes EC and WP, then there exists a value function W : Z++ × R → R, continuous and increasing in its second argument, such that, for all n, m ∈ Z++ , for all u ∈ Rn , and for all v ∈ Rm , ∗ uRE v ⇔ W n, Ξn(u) ≥ W m, Ξm(v) . (6.3) See Blackorby, Bossert and Donaldson (2001) for a proof of Theorem 16.25 In the theorems that follow, extended continuity is not assumed and the existence of representations is derived from other axioms. A natural generalization of IUUI requires its separability property to hold for all comparisons, not only those that involve utility vectors of the same dimension. See Blackorby, Bossert and Donaldson (1998) for a discussion. Extended Independence of the Utilities of Unconcerned Individuals (EIUUI): ∗ ∗ For all u, v, w ∈ Ω, uRE v if and only if (u, w)RE (v, w). The next axiom establishes a link between diﬀerent population sizes. It requires, for each possible alternative, the existence of another alternative with an additional individual alive that is as good as the original alternative, where the individuals alive in both alternatives are unaﬀected by the population augmentation. This assumption rules out social orderings that always declare an alternative with a larger population better (worse) than an alternative with a smaller population, thereby ensuring nontrivial trade-oﬀs between population size and well-being. ∗ Expansion Equivalence (EE): For all u ∈ Ω, there exists c ∈ R such that (u, c)I E u. The number c in the deﬁnition of EE is a critical level of lifetime utility for u ∈ Ω. If ∗ RE satisﬁes SP and EE, the critical level for any u ∈ Ω is unique and can be written as c = C(u), where C: Ω → R is a critical-level function. For most of the results in this section, it is not necessary to impose EE—a weaker version which requires the existence of a critical level only for at least one u ∈ Ω is ¯ suﬃcient. Therefore, we deﬁne Weak Expansion Equivalence (WEE): There exist u ∈ Ω and c ∈ R such that ¯ ∗E (¯, c)I u. u ¯
25 Blackorby and Donaldson (1984) falsely claims that the existence of a representation is guaranteed by C and WP (without EC). See Blackorby, Bossert and Donaldson (2001) for a discussion.

30

∗ ∗ If a variable-population social-evaluation ordering RE satisﬁes EIUUI and WEE, then RE satisﬁes EE. Moreover, if c is a critical level for u, then c is a critical level for all u ∈ Ω. ¯ See Blackorby, Bossert and Donaldson (1995) for an analogous result in an intertemporal model. Theorem 17: ∗ If an anonymous variable-population social-evaluation ordering RE sat∗ isﬁes EIUUI and WEE, then RE satisﬁes EE and there exists α ∈ R such that α is a critical level for all u ∈ Ω.

∗ Proof. By WEE, there exist u ∈ Ω and c ∈ R such that (¯, c)I E u. Let u ∈ Ω be arbitrary. ¯ u ¯ ∗ ∗E ¯ By EIUUI, (u, u, c)I (u, u), and applying EIUUI again, it follows that (u, c)I E u. Letting ¯ α = c, the theorem is established. If SP is added to EIUUI and WEE, Theorem 17 implies that the critical-level function C is constant.26 There are several possibilities for extending utilitarianism to a variable-population framework. For example, average utilitarianism is deﬁned as follows. For all n, m ∈ Z++, for all u ∈ Rn , for all v ∈ Rm , 1 ∗ uRE v ⇔ n n i=1

1 ui ≥ m

m

vi . i=1 (6.4)

Average utilitarianism satisﬁes EE with average utility as the critical level for any utility vector u ∈ Ω. In addition, average utilitarianism satisﬁes IUUI but not EIUUI.27 Critical-level utilitarianism (Blackorby, Bossert and Donaldson, 1995, Blackorby and Donaldson, 1984) uses the sum of the diﬀerences between individual utility levels and a ﬁxed critical level of utility as its value function. If a person is added to a population that is unaﬀected in terms of utilities, the critical level is that level of lifetime utility that ∗ makes the two alternatives equally good. RE is a critical-level utilitarian social-evaluation ordering if and only if there exists α ∈ R such that, for all n, m ∈ Z++, for all u ∈ Rn , for all v ∈ Rm , n m ∗E uR v ⇔ ui − α ≥ vi − α . (6.5) i=1 i=1

Critical-level utilitarianism satisﬁes EE with a constant critical level α and, in addition, satisﬁes EIUUI. Classical utilitarianism is a special case of critical-level utilitarianism with α equal to zero, the level of lifetime utility representing neutrality.
26 Hammond (1988, 2001) and Dasgupta (1993) argue for constant critical levels and normalize them to zero. Both set the critical level above neutrality, which implies that the utility level that represents neutrality is negative. 27 See Blackorby and Donaldson (1984, 1991), Bossert (1990b,c) and Hurka (1982) for discussions of average utilitarianism.

31

We believe that a positive value (that is, a value above neutrality) should be chosen for the critical level α. If α is at or below neutrality, the ‘repugnant conclusion’ results (Parﬁt, 1976, 1982, 1984). A variable-population social-evaluation ordering leads to the repugnant conclusion if, for any level of utility experienced by each individual in a society of a given size (no matter how far above neutrality this utility level is) and for any utility level µ above neutrality but arbitrarily close to it, there exists a larger population size m so that an alternative in which each of m individuals has a utility of µ is considered better than the former alternative. Therefore, a situation with mass poverty is superior to an alternative in which every person leads a good life. Because we consider the repugnant conclusion ethically unacceptable, we recommend a level of α above neutrality. Positive critical levels are incompatible with the Pareto plus principle (Sikora, 1978), which requires the ceteris paribus addition of an individual above neutrality to a given population to be desirable. Classical utilitarianism satisﬁes the Pareto plus principle and leads to the repugnant conclusion. On the other hand, average utilitarianism does not lead to the repugnant conclusion and it does not satisfy the Pareto plus principle. Although it performs well in this regard, it possesses the ethically unattractive property of declaring the ceteris paribus addition of a person below neutrality desirable if the average utility of the existing population is even lower. The incompatibility between avoiding the repugnant conclusion and the Pareto plus principle is not restricted to utilitarian principles, however. More general impossibility results are reported in Arrhenius (2000), Blackorby, Bossert, Donaldson and Fleurbaey (1998), Blackorby and Donaldson (1991), Carlson (1998) and Ng (1989). The variable-population versions of generalized utilitarianism use the same transformation of utilities g for all population sizes. Moreover, because this function is unique up to increasing aﬃne transformations, we can, without loss of generality, assume that g(0) = 0; this ensures that the utility level that represents neutrality is preserved when applying the transformation. ∗ RE is the average generalized-utilitarian social-evaluation ordering if and only if there exists a continuous and increasing function g: R → R with g(0) = 0 such that, for all n, m ∈ Z++, for all u ∈ Rn , for all v ∈ Rm , ∗ 1 uRE v ⇔ n n i=1

1 g(ui ) ≥ m

m

g(vi ). i=1 (6.6)

∗ RE is a critical-level generalized-utilitarian social-evaluation ordering if and only if there exist α ∈ R and a continuous and increasing function g: R → R satisfying g(0) = 0 such that, for all n, m ∈ Z++ , for all u ∈ Rn , for all v ∈ Rm , ∗ uRE v ⇔ n m

g(ui ) − g(α) ≥ i=1 i=1

g(vi ) − g(α) .

(6.7)

32

Setting α = 0 yields classical generalized utilitarianism. As is the case for average utilitarianism, average generalized utilitarianism satisﬁes EE with average utility as the critical level and satisﬁes IUUI but violates EIUUI. Criticallevel generalized utilitarianism satisﬁes EE with the constant critical level α and EIUUI. Classical generalized utilitarianism satisﬁes the Pareto plus principle and leads to the repugnant conclusion but average generalized utilitarianism and critical-level generalized utilitarianism with a positive critical level avoid the repugnant conclusion and violate Pareto plus. The representative-utility function for generalized utilitarianism is given by Ξn (u) = g −1 1 n n g(ui ) i=1 (6.8)

for all n ∈ Z++ , u ∈ Rn . The value function for average generalized utilitarianism can be written as WAGU (n, ξ) = g(ξ) (6.9) and the value function for critical-level generalized utilitarianism can be written as WCLGU (n, ξ) = n g(ξ) − g(α) . (6.10)

All members of the two families satisfy extended continuity and the value functions are continuous and increasing in their second arguments. Analogously to Theorem 5, EIUUI can be used to characterize critical-level generalized utilitarianism in the variable-population case. The following theorem is due to Blackorby, Bossert and Donaldson (1998). ∗ Theorem 18: An anonymous variable-population social-evaluation ordering RE satisﬁes ∗ C, SP, EIUUI and WEE if and only if RE is a critical-level generalized-utilitarian socialevaluation ordering. Proof: By Theorem 5, ﬁxed-population comparisons for population sizes n ≥ 3 must be made according to ﬁxed-population generalized utilitarianism with continuous and increasing functions g n . Because each g n is unique up to increasing aﬃne transformations only, we can without loss of generality assume gn (0) = 0 for all n ≥ 3. EIUUI requires n n n n

g i=1 n+m

(ui ) ≥ i=1 g

n+m

(vi) ⇔ i=1 g (ui ) ≥ n i=1

g n (vi)

(6.11)

for all n, m ≥ 3, for all u, v ∈ Rn , which implies that the g n can be chosen independently of n, and we deﬁne g = g n for all n ≥ 3. By Theorem 17 and SP, there exists a unique

33

constant critical level α ∈ R. Consider u ∈ Rn and v ∈ Rm with n ≥ 3 and, without loss of generality, n ≥ m. Because α is a critical level for all utility vectors, it follows that ∗ ∗ uRE v ⇔ uRE (v, α1n−m ). Because u and (v, α1n ) are of the same dimension n ≥ 3, it follows that ∗ ∗ uRE v ⇔ uRE (v, α1n) n m

(6.12)

⇔ i=1 n

g(ui ) ≥ i=1 g(vi ) + (n − m)g(α) m (6.13)

⇔ i=1 g(ui ) − g(α) ≥ i=1 g(vi ) − g(α) .

∗ If n < 3, the deﬁnition of a critical level can be used again to conclude uI E (u, α13−n ), and the above argument can be repeated with u replaced by (u, α13−n ). If the requirement that the repugnant conclusion be avoided is added to the axioms of Theorem 18, it follows immediately that the critical level must be positive. As in the ﬁxed-population case, critical-level generalized utilitarianism can be characterized in an intertemporal model with a variable-population version of independence of the utilities of the dead. This extended version of IUD is obtained from IUD in the same way EIUUI is obtained from IUUI; see Blackorby, Bossert and Donaldson (1995) for details. Alternative intertemporal consistency conditions are explored in Blackorby, Bossert and Donaldson (1996). In the intertemporal setting, individuals are assumed to experience utilities in each period which aggregate into lifetime utilities. Forward-looking consistency requires that, in any period, future utilities are separable from past utilities. In Blackorby, Bossert and Donaldson (1996), it is shown that consistency between forward-looking social evaluations and intertemporal social evaluations implies, together with some other axioms, classical generalized utilitarianism and thus the repugnant conclusion. The same results are obtained for a full intertemporal consistency requirement which is stronger than forward-looking consistency by itself but is equivalent to it in the presence of other axioms. The consequences of weakening the intertemporal strong Pareto principle are examined in Blackorby, Bossert and Donaldson (1997a,b), where versions of critical-level generalized utilitarianism and classical generalized utilitarianism that allow for discounting are characterized. We conclude this section with a discussion of information-invariance assumptions in the variable-population framework. Let ∼E denote an equivalence relation deﬁned on the domain DE of a variable-population social-evaluation functional F E . Information invariance with respect to ∼E is deﬁned as follows. 34

Information Invariance with Respect to ∼E : For all U, V ∈ DE , if U ∼E V , then RE = RE . U V As in the ﬁxed-population case, one possible way to deﬁne an information assumption is to specify a set of admissible vectors of utility transformations. In the variablepopulation case, the elements of such a set Φ can be written as φ = (φi )i∈Z++ , where each φi is a function φi : R → R that transforms individual i’s utility ui into φi (ui ). The following information assumptions are used in this section. Ordinal Full Comparability (OFC): φ ∈ Φ if and only if there exists an increasing function φ0 : R → R such that φi = φ0 for all i ∈ Z++ . Cardinal Measurability (CM): φ ∈ Φ if and only if there exist ai ∈ R and bi ∈ R++ for each i ∈ Z++ such that φi (τ ) = ai + bi τ for all τ ∈ R and all i ∈ Z++ . Cardinal Full Comparability (CFC): φ ∈ Φ if and only if there exist a ∈ R and b ∈ R++ such that φi (τ ) = a + bτ for all τ ∈ R and all i ∈ Z++ . Numerical Full Comparability (NFC): φ ∈ Φ if and only if φi (τ ) = τ for all τ ∈ R and all i ∈ Z++ . In the presence of welfarism, information invariance can alternatively be deﬁned in ∗ terms of RE . For information-invariance assumptions deﬁned with sets of admissible transformations, we obtain the following deﬁnition. Information Invariance with Respect to Φ: For all φ ∈ Φ, for all n, m ∈ Z++, for all u, u ∈ Rn , for all v, v ∈ Rm , if ui = φi (ui ) for all i ∈ {1, . . . , n} and vi = φi (vi ) for all i ∈ {1, . . . , m}, then ∗ ∗ uR E v ⇔ u R E v . (6.14) In the variable-population framework, information assumptions are considerably more restrictive than in the ﬁxed-population case. In the presence of anonymity and the weak Pareto principle, for example, the possibility of level comparisons is necessary for the existence of a variable-population social-evaluation functional. This observation, which is stated in the following theorem, implies that stronger information-invariance requirements than information invariance with respect to OFC cannot be satisﬁed. We use ΦOF C to denote the set of admissible vectors of utility transformations according to ordinal full comparability. ∗ Theorem 19: If an anonymous variable-population social-evaluation ordering RE satisﬁes WP and information invariance with respect to Φ, then Φ ⊆ ΦOF C .

35

∗ Proof. Suppose RE is anonymous and satisﬁes WP and information invariance with respect to Φ. By way of contradiction, suppose that Φ \ ΦOF C = ∅. Then there exist φ ∈ Φ, γ ∈ R, and i, j ∈ Z++ such that φi (γ) = φj (γ). Consider the one-dimensional ∗ utility vectors (ui ), (uj ) ∈ Ω such that ui = uj = γ. Because RE is reﬂexive, we must have ∗ ∗ (ui )I E (uj ). By information invariance with respect to Φ, we obtain (φi (ui ))I E (φj (uj )). ∗ Let vi = φj (uj ). This implies (φi (ui ))I E (vi ). Because φi (ui ) = vi , this contradicts weak Pareto. Some information-invariance assumptions impose signiﬁcant restrictions on ethical parameters such as critical levels, which clearly is undesirable. It can be shown that information invariance with respect to CFC leads to average utilitarianism (for comparisons involving at least three individuals) in the presence of some other axioms including IUUI, and if IUUI is strengthened to EIUUI, an impossibility result is obtained. These results are proved and discussed in Blackorby, Bossert and Donaldson (1999a). Because of these negative observations, we suggest an alternative way of formulating information invariance in a variable-population framework. The fundamental diﬃculty appears to be that the standard welfarist framework with an unrestricted domain is inadequate to deﬁne norms, such as the utility level associated with a neutral life, which permit interpersonal comparisons of utility at a single utility level. Therefore, we suggest the use of a systematic procedure for incorporating such norms (see Blackorby, Bossert and Donaldson, 1999a). In particular, we propose the use of norms to restrict the domain of admissible utility proﬁles.28 In addition to avoiding the diﬃculties associated with extending the traditional taxonomy of comparability and measurability assumptions to a variable-population framework, we think that this approach using norms is more intuitive. For U ∈ U E and i ∈ Z++ , let ηi (Ui ) denote the level of utility individual i assigns to a neutral life, given the utility function Ui . Suppose, in addition, that a second norm denotes a life above neutrality at some satisfactory or ‘excellent’ level (not necessarily a critical level). It is possible, given these norms, to represent the value of a neutral life with a utility level of zero and the value of an excellent life with a utility level of one. Letting εi (Ui ) denote the utility level representing an excellent life according to i’s utility function Ui , the restricted domain that respects both normalizations is given by
E DE = Uηε = U ∈ U E | ηi (Ui ) = 0 and εi (Ui ) = 1 ∀i ∈ Z++ .

(6.15)

These normalizations allow us to start with very demanding information-invariance assumptions on the unrestricted domain U E and yet have remarkable ﬂexibility in designing social-choice rules if we restrict attention to the proﬁles respecting our normalizations. We obtain the following theorem (see Blackorby, Bossert and Donaldson, 1999a).
28 See Tungodden (1999) for a discussion of a single norm in combination with ordinally measurable utilities. For a diﬀerent approach using normalized utilities, see Dhillon (1998).

36

Theorem 20: If a variable-population social-evaluation functional F E satisﬁes information invariance with respect to CM and the utility levels representing a neutral life and an excellent life are normalized to zero and one respectively, then the restriction of F E to E DE = Uηε satisﬁes information invariance with respect to NFC.
E Proof. Suppose F E satisﬁes information invariance with respect to CM. Let U ∈ Uηε . By deﬁnition, ηi (Ui ) = 0 and εi (Ui ) = 1, and it follows that φi (0) = 0 and φi (1) = 1 for all i ∈ Z++. Consequently, ai = 0 and bi = 1 for all i ∈ Z++ .

Note that only cardinal measurability is required in Theorem 20; full interpersonal comparability is provided by the two norms. Thus, the theorem shows that, if utilities are cardinally measurable and two norms are employed, utilities on the resulting restricted domain are numerically measurable and fully interpersonally comparable. Therefore, the norms generate suﬃcient additional information to apply any social-evaluation functional. Similar results involving a single norm can be found in Blackorby, Bossert and Donaldson (1999a).

7. Uncertainty Suppose that a government or an individual must take an action, which might be a simple one or a more complex one that leads, for example, to the establishment of an institution, custom or moral rule, from a set of feasible actions. If the agent knows with certainty the alternative that results from each action, a social-evaluation functional can be used to rank them. In that case, a function f maps actions into alternatives and, for any two actions a and b, action a is at least as good as action b if and only if f(a) is socially no worse than f(b).29 In most cases, the consequences of actions are not known with certainty at the time a choice of action has to be made. It may be possible, however, to attach probabilities to the outcomes that may materialize and, in that case, actions can be ranked by ranking prospects. Prospects can be identiﬁed with vectors of social alternatives if probabilities are ﬁxed. For such an approach to make normative sense, probabilities may be subjective but must be based on the best information available at the time decisions are taken. If probabilities represent uninformed individual beliefs, the normative force of this approach is weakened substantially. In any normative investigation, rationality plays an important role and this suggests that both social and individual preferences should satisfy the expected-utility hypothesis (von Neumann and Morgenstern, 1944, 1947). Given that, two diﬀerent versions of
29 See Broome (1991b) for a discussion.

37

welfarism are possible. Ex-ante welfarism bases social evaluations of prospects on individual valuations while ex-post welfarism orders alternatives after the uncertainty has been resolved and aggregates these judgements into a social ordering of prospects. Harsanyi (1955, 1977) investigates ex-ante welfarism and shows that it has surprising consequences for social evaluation. In his formulation, individuals have ex-ante utility functions that satisfy the Bernoulli hypothesis (Broome, 1991a), a condition that is stronger than the expected-utility hypothesis (see the discussion below).30 The Bernoulli hypothesis requires that individual ex-ante utilities are equal to the expected value of von Neumann – Morgenstern (vNM) utilities. There are m ≥ 2 ‘states of nature’ with probabilities p = (p1 , . . . , pm ) ∈ Rm , m pj = 1, and they are agreed upon by individuals and + j=1 by the social evaluator. Individual ex-ante utilities are given by m m

uA i

= j=1 pj uj i

= j=1 pj Ui (xj )

(7.1)

for all i ∈ {1, . . . , n}, where uj is individual i’s utility in state j and xj is the social i alternative that occurs in state j. There is a single proﬁle of utility functions and xj is ﬁxed for all j ∈ {1, . . . , m}. Consequently, the utility level uj is ﬁxed for all i ∈ {1, . . . , n} i and all j ∈ {1, . . . , m}. Each probability vector p is called a lottery and all lotteries p ∈ Rm with m pj = 1 are permitted. Social ex-ante preferences are represented by + j=1 m m

uA 0

= j=1 pj U0 (x ) = j=1 j

pj uj 0

(7.2)

where uj is social utility in state j and U0 : X → R is a social utility function. Harsanyi re0 quires social preferences over lotteries to satisfy ex-ante Pareto indiﬀerence, which requires society to rank any two lotteries as equally good whenever they are equally valuable for each individual. This axiom alone has the consequence that there exist γ ∈ Rn and δ ∈ R such that social utilities are weighted sums of individual utilities with uj = n γi uj + δ 0 i=1 i n A = A + δ. If p and q are any two lotteries, then for all j ∈ {1, . . . , m} and u0 i=1 γi ui n m n m

p

q⇔ i=1 γi j=1 pj uj i

≥ i=1 γi j=1 q j uj i

(7.3)

where p q means that p is socially at least as good as q. This result is called Harsanyi’s (1955) social-aggregation theorem.31 The weights (γ1 , . . . , γn ) in (7.3) are, in general, not
30 Arrow (1964) provides an account of the expected-utility hypothesis that is consistent with our approach. 31 Border (1981) presents an elegant proof which is reproduced in expanded form in Weymark (1994). See also Blackorby, Bossert and Donaldson (1998), Blackorby, Donaldson and Weymark (1999, 2001), Broome (1990, 1991a), Coulhon and Mongin (1989), Domotor (1979), Fishburn (1984), Hammond (1981, 1983), Mongin (1994, 1995, 1998) and Mongin and d’Aspremont (1998).

38

unique and need not be positive. The imposition of stronger Pareto conditions implies some restrictions on their signs, however. If strong Pareto is satisﬁed, (7.3) can be satisﬁed with positive weights32 and if weak Pareto is satisﬁed, (7.3) can be satisﬁed with non-negative weights, at least one of which is positive.33 Because the above-described model employs a single proﬁle of utility functions and, thus, diﬀers from the approach of this survey, we do not include a proof. Instead, we present a variant of Harsanyi’s theorem that uses the basic model employed in the rest of this chapter. It is a multi-proﬁle model which allows for interpersonal comparisons of utilities and permits the application of the anonymity axiom.34 X is a set of alternatives with at least four elements. A prospect is a vector x = (x1 , . . . , xm) ∈ X m with m ≥ 2 and the prospect xc = (x, . . . , x) ∈ X m is one in which x ∈ X occurs for certain. The vector of positive probabilities is ﬁxed at p = (p1 , . . . , pm ) ∈ Rm with m pj = 1.35 ++ j=1 As in Harsanyi (1955, 1977), we assume that individual utilities satisfy the Bernoulli hypothesis with the ex-ante utility function UiA : X m → R given by m UiA (x)

= EUi (x) = j=1 pj Ui (xj )

(7.4)

for all x ∈ X m . UiA (x) is the value of the prospect x to person i, Ui : X → R is individual i’s vNM utility function and EUi (x) is i’s expected utility for prospect x. (7.4) implies that, for all x ∈ X, UiA (xc ) = EUi (xc ) = Ui (x). A proﬁle of ex-ante utility functions is A A U A = (U1 , . . . , Un ) and a proﬁle of vNM utility functions is U = (U1 , . . . , Un ). Writing A A U A (x) = (U1 (x), . . . , Un (x)) = EU(x) = (EU1 (x), . . . , EUn (x)) for all x ∈ X m and U(x) = (U1 (x), . . . , Un (x)) for all x ∈ X, m U (x) = EU(x) = j=1 A

pj U(xj )

(7.5)

for all x ∈ X m . The functions U1 , . . . , Un do double duty in this formulation: they are the individuals’ vNM utility functions and they measure individual well-being. An ex-ante social-evaluation functional F A: DA → OA is a function which maps each proﬁle of ex-ante utility functions into an ordering on X m . We say that the domain DA is A the Bernoulli domain DB if and only if it consists of all proﬁles of ex-ante utility functions A A U A = (U1 , . . . , Un ) such that (7.5) is satisﬁed for some U = (U1 , . . . , Un ) ∈ U.
32 See Domotor (1979), De Meyer and Mongin (1995), Weymark (1993) and Zhou (1997). 33 See Weymark (1993, 1994, 1995) and Zhou (1997). 34 Multi-proﬁle models are presented in Blackorby, Donaldson and Weymark (2001), Hammond (1981, 1983), Mongin (1994) and Mongin and d’Aspremont (1998). 35 Because probabilities are assumed to be ﬁxed, any state of nature with a probability of zero may be dropped.

39

∗ F A is (ex-ante) welfarist if and only if there exists an ordering RA on Rn such that ∗ A A A A xRA y ⇔ U1 (x), . . . , Un (x) RA U1 (y), . . . , Un (y) U (7.6)

for all x, y ∈ X m , where RA = F A(U A ) is the social ordering of prospects. On the U Bernoulli domain, ex-ante welfarism is a consequence of the assumptions binary independence of irrelevant alternatives and Pareto indiﬀerence (applied to ex-ante utilities).36 Social preferences satisfy the expected-utility hypothesis if and only if there exists a function U0 : X × DA → R such that, for all x, y ∈ X m , m m

xRA y U

⇔ j=1 pj U0 x , U

j

A

≥ j=1 pj U0 y j , U A .

(7.7)

Note that this is somewhat weaker than (7.4) because there is no need to measure a social ex-ante utility level. The social vNM function is allowed be proﬁle-dependent. In our multi-proﬁle setting, if the social vNM utility function were written without U A , an imposed social ranking would result. In Harsanyi’s lottery problem, there is only a single proﬁle of vNM utility functions and the second argument of U0 is not needed. If the value of the social-evaluation functional satisﬁes the expected-utility hypothesis for every A U A ∈ DA we say that the range of the functional is OEU . A Now suppose that the domain of the welfarist social-evaluation functional is DB , so A that individual utilities satisfy the Bernoulli hypothesis, and its range is OEU , so that social preferences satisfy the expected-utility hypothesis. Thus, we consider a social-evaluation A A A functional FBEU : DB → OEU . Then it must be true that, for all x, y ∈ X m , m m

xRA y U

⇔ j=1 pj U0 x , U

j

A

≥ j=1 pj U0 y j , U A (7.8)

∗ ⇔ EU1 (x), . . . , EUn (x) RA EU1 (y), . . . , EUn (y) ∗ ⇔ EU(x)RA EU(y)
A where RA = FBEU (U A ). Setting x = xc and y = yc in (7.8) results in U

xcRA yc ⇔ U0 x, U A ≥ U0 y, U A U ∗ ⇔ U1 (x), . . . , Un (x) RA U1 (y), . . . , Un (y) ∗ ⇔ U(x)RA U(y).

(7.9)

This implies that there is a single social-evaluation ordering which is the same for all states, ∗ ∗ and it is the ex-ante social-evaluation ordering RA. The ordering RA orders prospects and
36 See Blackorby, Donaldson and Weymark (2001), Mongin (1994) and Mongin and d’Aspremont (1998).

40

it also orders alternatives once the uncertainty has been resolved. Such a social-evaluation functional is both ex-ante and ex-post welfarist. Next, we prove a theorem that shows that any welfarist ex-ante social-evaluation functional on the Bernoulli domain whose social preferences satisfy the expected-utility ∗ hypothesis must possess a property that is equivalent to the requirement that RA satisfy information invariance with respect to translation-scale measurability (TSM).37 In order to ﬁnd the largest class of functions satisfying our axioms, no information-invariance restriction is placed on the social-evaluation functional.
A A Theorem 21: Suppose that |X| ≥ 4. If an ex-ante social-evaluation functional FBEU : DB → A OEU is welfarist, then, for all u, v, a ∈ Rn ,

∗ ∗ uRA v ⇔ (u + a)RA (v + a).

(7.10)

A Proof. For any u, v, a ∈ Rn , choose a proﬁle U A ∈ DB such that there exist x, y, z, w ∈ X with U(x) = u/p1 , U(y) = v/p1 , U(z) = 01n and U(w) = a/ m pj where U is the vNM j=2

proﬁle corresponding to U A . Consider x, y ∈ X m with x1 = x, y 1 = y and xj = y j = z for all j ∈ {2, . . . , m}. Because EU(x) = u and EU(y) = v, (7.8) implies ∗ uRAv ⇔ p1 U0 x, U A + m m

pj U0 z, U j=2 A

≥ p1 U0 y, U

A

+ j=2 pj U0 z, U A (7.11)

⇔ p1 U0 x, U A ≥ p1 U0 y, U A . Now consider w, z ∈ X m with w1 = x, z 1 = y and wj = z j = w for all j ∈ {2, . . . , m}. Because EU(w) = (u + a) and EU(z) = (v + a), (7.8) implies ∗ (u + a)RA(v + a) ⇔ p1 U0 x, U A + m m

pj U0 w, U A ≥ p1 U0 y, U A + j=2 j=2

pj U0 w, U A

⇔ p1 U0 x, U A ≥ p1 U0 y, U A . (7.12) Because the second lines of (7.11) and (7.12) are identical, (7.10) is immediate. The property described by (7.10) is the same as information invariance with respect to translation-scale measurability and we use the result of Theorem 12 to show that, given ∗ anonymity and weak Pareto, RA must be the utilitarian ordering.
37 Mongin (1994) and Mongin and d’Aspremont (1998) prove a similar theorem for lotteries.

41

A A Theorem 22: Suppose that |X| ≥ 4. An ex-ante social-evaluation functional FBEU : DB → A OEU is welfarist and satisﬁes A and WP if and only if, for all u, v ∈ Rn ,

∗ uRA v ⇔

n

n

ui ≥ i=1 i=1

vi

(7.13)

A and, for all x, y ∈ X m and all proﬁles U A ∈ DB , n n

xRA y ⇔ U ⇔

EUi (x) ≥ i=1 n m i=1 j=1 i=1 j

EUi (y) n m

(7.14) pj Ui (y ). j pj Ui (x ) ≥ i=1 j=1

Proof. Necessity follows from Theorems 12 and 21. Suﬃciency is immediate. ∗ Note that continuity of RA is not needed in Theorem 22. A variant can be proved by adding continuity and dropping anonymity. In that case, (7.13) becomes ∗ uRA v ⇔ n n

γi ui ≥ i=1 i=1

γi vi

(7.15)

∗ where γ ∈ Rn \ {01n }. The social-evaluation ordering RA is weighted utilitarian and the + weights are nonnegative with at least one that is positive. Because anonymity is such an important axiom in welfarist social ethics, however, we have presented the theorem that uses it. An objection that is sometimes made to Harsanyi’s theorem is that vNM utility functions are not unique (increasing aﬃne transformations represent the same preferences) and, thus, equal weights on utilities are meaningless. Our framework does not suﬀer from this diﬃculty. Because of our assumptions regarding the measurability and comparability of individual utilities, a particular vNM function is selected for each person. The result of the theorem implies that the information structure (for both vNM utility functions and A A the ex-ante utility functions (U1 , . . . , Un )) must support cardinal unit comparability. An interesting question of interpretation is whether Theorem 22 provides a convincing argument for utilitarianism. If the answer is ‘yes’, it should be noted that the theorem requires the Bernoulli hypothesis to be satisﬁed, a stronger requirement than the expectedutility hypothesis. Thus, the utility functions (U1 , . . . , Un ) must represent people’s good, free of the irrationalities of compulsive gambling, for example. It is not usual to present arguments in favour of the Bernoulli hypothesis, over and above the requirements of the expected-utility hypothesis, but it has been done by Broome (1991a).

42

The theorem requires individual and social probabilities to coincide. If, however, individual probabilities are subjective and can diﬀer across individuals, impossibility theorems emerge.38 The same-probability requirement is a demanding one but it might be justiﬁed by regarding probabilities as ‘best-information’ probabilities. Suppose that, instead of the Bernoulli hypothesis, individual ex-ante utilities satisfy the expected-utility hypothesis. In that case, writing UiN M as individual i’s vNM utility function, m m

UiA (x)

≥

UiA (y)

⇔ j=1 pj UiN M (xj )

≥ j=1 pj UiN M (y j )

(7.16)

for all x, y ∈ X m and for all i ∈ {1, . . . , n}. It follows that there exist increasing functions h1 , . . . , hn with hi : R → R for all i ∈ {1, . . . , n} such that, for all x ∈ X m and for all i ∈ {1, . . . , n}, m UiA (x)

= hi j=1 pj UiN M (xj ) .

(7.17)

If x = xc in (7.17), UiA (xc) = hi (UiN M (x)) and this utility level can be regarded as person i’s actual utility level when alternative x is realized. Writing Ui as i’s actual utility function, Ui (x) = hi (UiN M (x)) for all x ∈ X and (7.17) becomes m m

UiA (x)

= hi j=1 pj UiN M (xj )

= hi j=1 pj h−1 Ui (xj ) i

.

(7.18)

A DEU if and only if, for each UiN M and some increasing

The domain of an ex-ante social-evaluation functional is the expected-utility domain i ∈ {1, . . . , n}, (7.18) is satisﬁed for some vNM utility function function hi . On such a domain, no welfarist ex-ante socialevaluation functional exists.

Theorem 23: Suppose that |X| ≥ 4. There exists no ex-ante social-evaluation functional A A A FEU : DEU → OEU that is welfarist and satisﬁes A and WP.
A A A Proof. First consider the subdomain DB of DEU and assume that FEU is welfarist and satisﬁes A and WP. Theorem 22 implies that, for all u, v ∈ Rn

∗ uRA v ⇔

n

n

ui ≥ i=1 i=1

vi .

(7.19)

A A Now consider the subdomain Dh of DEU in which, for all i ∈ {1, . . . , n}, m

UiA (x)

=h j=1 pj UiN M (xj )

(7.20)

38 See Hammond (1981, 1983) and Mongin (1995).

43

∗ where h: R → R is increasing but not aﬃne. Deﬁne the ordering RA on Rn by h ∗ ∗ uRA v ⇔ h(u1 ), . . . , h(un ) RA h(v1 ), . . . , h(vn ) h for all u, v ∈ Rn . Writing EUiN M (x) = i ∈ {1, . . . , n}, we know that ∗ U A (x)RA U A (y)
N N ⇔ h EU1 M (x) , . . . , h EUn M (x) m N M (xj ) j=1 pj Ui

(7.21)

for all x ∈ X m and for all

∗ N N RA h EU1 M (y) , . . . , h EUn M (y)

(7.22)

∗ N N N N ⇔ EU1 M (x), . . . , EUn M (x) RA EU1 M (y), . . . , EUn M (y) . h ∗ Because h is the same for each individual, RA is anonymous and Theorem 22 implies that, h n, for all u, v ∈ R n n ∗A uRh v ⇔ ui ≥ vi . (7.23) i=1 i=1

Because h is increasing, (7.21) can be rewritten as ∗ ∗ uRA v ⇔ h−1 (u1 ), . . . , h−1 (un ) RA h−1 (v1), . . . , h−1 (vn ) , h and (7.19), (7.23) and (7.24) together imply ∗ uR A v ⇔ n n n

(7.24)

ui ≥ i=1 i=1

vi ⇔ i=1 h

−1

n

(ui ) ≥ i=1 h−1 (vi )

(7.25)

for all u, v ∈ Rn . (7.25) can be satisﬁed if and only if h−1 is aﬃne. This requires h to be aﬃne, and a contradiction results. The proof of Theorem 23 shows that, when the transforms h1 , . . . , hn in (7.18) are identical, prospects must be ranked by using the sums of expected utilities. This means, ∗ however, that the social-evaluation ordering RA must depend on h, and welfarism, which requires a single social-evaluation ordering, is contradicted. A variant of Theorem 23 can be proved by dropping anonymity and adding continuity.39 A way out of this impossibility result can be provided by restricting the domain of the social-evaluation functional to a single utility proﬁle or information set. If the expected-utility hypothesis is satisﬁed, ∗ Blackorby, Donaldson and Weymark (2001) prove that, on a single information set, RA must be generalized utilitarian if the Bernoulli hypothesis is not satisﬁed and utilitarian if it is satisﬁed.
39 See also Blackorby, Donaldson and Weymark (1999, 2001), Roemer (1996), Sen (1976) and Weymark (1991).

44

One possible escape from Harsanyi’s theorem is to embrace ex-ante welfarism without requiring social preferences to satisfy the expected-utility hypothesis, a move that has been suggested by Diamond (1967) and Sen (1976, 1977b, 1986). They argue that social preferences that satisfy the expected-utility hypothesis cannot take account of the fairness of procedures by which outcomes are generated (see also Weymark, 1991). An ex-ante social-evaluation functional of the type they advocate is given by xRA y ⇔ ΞA U A (x) ≥ ΞA U A (y) U (7.26)

for all x, y ∈ X m and all proﬁles U A ∈ DA . In the equation, ΞA : Rn → R is a strictly concave and, therefore, inequality averse, ex-ante representative-utility function. If the social-evaluation functional satisﬁes anonymity, ΞA must be symmetric. See Weymark (1991) for a discussion. An important question to consider, however, is whether it is appropriate to require ex-ante welfarism. This form of welfarism is not applied to actual well-being, and that suggests that ex-post welfarism may be more appropriate and ethically more basic. It is true of course that, given the Bernoulli hypothesis, ex-ante welfarism implies ex post, but the converse is not true. A second way out of the result of Theorem 22, therefore, is provided by requiring ex-post welfarism only. Suppose, for example, that ΞP : Rn → R is an ex-post representative-utility function, which expresses a social attitude toward utility inequality. Then ex-post welfarism is satisﬁed by a principle given by m m P pj U0 j=1

xRP y U

⇔

Ξ

P

U(x )

j

≥ j=1 P pj U0 ΞP U(y j )

(7.27)

P for all x, y ∈ X m and all proﬁles U ∈ D. In (7.27), U0 is a social vNM utility function which expresses a social attitude toward representative-utility uncertainty. Even if society P is neutral toward such uncertainty (U0 is aﬃne), such a principle is not consistent, in general, with ex-ante Pareto indiﬀerence if individual ex-ante utilities satisfy the expectedutility hypothesis. This means that x may be regarded as better than y even though the same standard of rationality that is used socially ranks prospect y as better for each person. With such a principle, therefore, social rationality trumps individual rationality. As an example of this last claim, let there be two individuals, two states with equal P probabilities, let U0 be the identity map, and let ΞP (u1 , u2 ) = 1/4u(1) + 3/4u(2) . In

prospect x, each individual’s utility level is 20 in both states, so ΞP (U(x1 )) = ΞP (U(x2 )) = 20 and expected social value is 20. In prospect y, utilities are (u1 , u2 ) = (40, 4) in state 1 and (4, 40) in state 2. Consequently, ΞP (U(y 1 )) = ΞP (U(y 2 )) = 13, expected social value is 13, and society ranks x as better than y. Each individual’s expected utility is 20 in x and 22 in y, however, and each is better oﬀ, ex ante, in y.

45

8. Conclusion The idea that a just society is a good society can be an attractive one if the good receives an adequate account. Welfarist social-evaluation functionals are capable of performing well as long as the notion of well-being that they employ captures everything of value to individual people. Given that, principles for social evaluation that are non-welfarist run the risk of recommending some social changes from which no one beneﬁts. This is the lesson of Theorems 1 and 3. If a welfarist principle is to be used to rank alternatives that are complete histories of the world, a single social-evaluation ordering is suﬃcient to do the job for every proﬁle of utility functions. Although such orderings can be used to rank changes which aﬀect a population subgroup, such as the citizens of a single country or the people in a particular generation, the induced ordering over their utilities is not, in general, independent of the utilities of others. Independence is guaranteed by the axiom independence of the utilities of unconcerned individuals and, in conjunction with continuity, anonymity and strong Pareto, that axiom leads to generalized utilitarianism. In a dynamic framework, the same result is the consequence of independence of the utilities of the dead together with intertemporal versions of continuity, anonymity and strong Pareto. Generalized-utilitarian social-evaluation functionals are ethically attractive, but some of them may require utility information that is diﬃcult to acquire. In parsimonious information environments, utilitarianism itself may prove to be more attractive than the other members of that family of orderings. The only generalized-utilitarian social-evaluation ordering that satisﬁes information invariance with respect to cardinal full comparability is utilitarianism. And if individual utilities are translation-scale measurable, anonymity and weak Pareto alone imply that the social-welfare ordering must be utilitarian. Information restrictions are not the only axioms that generate utilitarianism, however. Incremental equity is an axiom that requires a kind of impartiality with respect to utility increases or decreases. If one person’s utility increases or decreases, the axiom requires the change to be equally good no matter who the aﬀected person is. This axiom, together with weak Pareto, characterizes utilitarianism. The utilitarian and generalized-utilitarian social-evaluation functionals can be extended to environments in which population size and composition may vary across alternatives. Two properties can be considered particularly desirable in this framework: extended independence of the utilities of unconcerned individuals and avoidance of the repugnant conclusion. Given continuity, anonymity and strong Pareto, extended independence of the utilities of unconcerned individuals implies that the social-evaluation ordering must be critical-level generalized utilitarian. The critical level is a parameter that represents the smallest utility level above which additions to a utility-unaﬀected population

46

have value. The repugnant conclusion is avoided if and only if the critical level is above neutrality. If utilities are cardinally measurable, interpersonal comparisons at any two norms are suﬃcient to produce numerical full comparability. We might choose norms, for example, at neutrality and at a utility level that represents a satisfactory or excellent life. Utility numbers such as zero and one may be chosen for these, and NFC results. It follows that cardinal measurability and two norms are suﬃcient to employ any welfarist socialevaluation functional. Social-evaluation functionals can be extended to rank prospects as long as probabilities can be attached to the various states of nature. If individual ex-ante utilities satisfy the Bernoulli hypothesis, social preferences satisfy the expected-utility hypothesis and all subjective probabilities coincide, the only ex-ante social-evaluation functional that satisﬁes anonymity and (ex ante) weak Pareto is the utilitarian one. If, however, the domain is expanded to include all individual ex-ante utility functions that satisfy the expectedutility hypothesis or if subjective probabilities can be diﬀerent for diﬀerent people, an impossibility results. If the von Neumann – Morgenstern utility functions, in addition to representing people’s good, express an attitude toward uncertainty that is rational and has some normative standing, it can be argued that the Bernoulli hypothesis is a reasonable assumption. In that case, Harsanyi’s social-aggregation theorem provides support for utilitarianism. Together, these results make a strong case for utilitarian and generalized-utilitarian social evaluation. When these social-evaluation functionals are coupled with an adequate account of lifetime well-being, the resulting principles are ethically attractive and perform well in environments in which other principles perform poorly. Social-contract theories, for example, are not able to give an adequate account of justice between generations when the existence of people in one generation is contingent on decisions made by another. On the other hand, the critical-level generalized-utilitarian principles can cope with fully dynamic environments in which history has a branching structure and the identities of those alive, their numbers, quality of life and length of life can vary across alternatives.

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