Critical levels and the (reverse) repugnant conclusion

Vol. 67 (1998), No. 1, pp. 1-15 Journal of Economics Zeitschrift for National6konomie

Q Springer-Verlag 1998 - Printed in Austria

Critical Levels and the (Reverse) Repugnant Conclusion

Charles Blackorby, Walter Bossert, David Donaldson, and Marc Fleurbaey

Received May 26, 1997; revised version received September 22, 1997

It is well-known that there is a trade-off among the properties of population principles that are used to make social evaluations when the number of people in the society under consideration may vary. The commonly used principles either lead to the repugnant conclusion (which is the case for classical utilitarianism), or they violate the Pareto-plus principle and related properties (average utilitarianism is an example of such a principle). This paper examines the na- ture of this trade-off and shows that the incompatibility between avoiding the repugnant conclusion and the Pareto-plus principle is fundamental and not restricted to the commonly used population principles.

Keywords: population ethics, repugnant conclusion, critical levels.

JEL classification: D63.

1 Introduction

Variable-population social-evaluation orderings can be used to make welfare comparisons in situations where the number of individuals alive is not fixed. These principles are needed because issues such as the al- location of resources to population-control programs, prenatal care, and intergenerational distribution cannot be addressed in a fixed-population framework. The most commonly used principles for variable-population social evaluation are classical utilitarianism and average utilitarianism. Both are special cases of welfarist principles - principles that use the identities of those alive and their lifetime utilities as the only determi- nants of social rankings. Classical utilitarianism compares states of the world on the basis of the sum of the individual utility gains over the utility level representing neutrality. The life of an individual is neutral if it is as good for the person as a life without any experiences. Thus, a life is worth living if and only if lifetime utility is above neutrality.

2 C. Blackorby et al.

The common convention in the literature is to assign a utility level of zero to a neutral life and we follow it in this paper. 1 Average utilitarianism uses the average of the lifetime utilities of those alive to make ethical comparisons. Both of these social-evaluation orderings can be generalized by allowing for the application of a transformation to individual utilities. This makes it possible to have principles that exhibit strict inequality aversion (in utilities).

Parfit (1976, 1982, 1984) pointed out a shortcoming of classical utilitarianism - the repugnant conclusion. A population principle leads to the repugnant conclusion if, for every state of the world where everyone alive has an arbitrarily high lifetime utility, there exists another state with a suitably large population where everybody experiences a utility level above but arbitrarily close to neutrality such that the sec- ond state is considered better than the first. This means that population size can always be used to substitute for quality of life, no matter how close to zero (neutrality) everyone's utility in the larger population is. Average utilitarianism does not lead to the repugnant conclusion, nor do principles such as critical-level (generalized) utilitarianism with any positive critical level. 2

On the other hand, neither average utilitarianism nor any of the above-mentioned critical-level principles satisfies the Pareto-plus principle (Sikora, 1978). The Pareto-plus principle requires the addition of an individual above neutrality to a utility-unaffected population to be socially preferred. A closely related condition - the zero-critical-levels axiom - requires the ceteris paribus addition of a person at neutrality to be a matter of social indifference. Zero critical levels is violated by the above rules as well.

The incompatibility between avoidance of the repugnant conclusion and the zero-critical-levels axiom goes well beyond these examples. Ar- rhenius (1997), Blackorby and Donaldson (1991), Parfit (1976, 1982, 1984), and Ng (1989) provide more general impossibility results show- ing that, under some standard assumptions, a principle that satisfies the Pareto-plus principle and a weak form of inequality aversion or non-antiegalitarianism must lead to the repugnant conclusion.

The purpose of this paper is to examine the dividing line between possibility and impossibility in that respect. We work in a general

1 See Broome (1993), for a discussion of neutrality and its normalization. Dasgupta (1993, 1994) uses a negative utility to represent a neutral life.

2 Critical-level utilitarianism compares two states of the world on the basis of the individual utility gains over a fixed critical level. Critical-level generalized utilitarianism uses transformed utilities. See Blackorby et al. (1995, 1996a, b, 1997a-c) and Blackorby and Donaldson (1984) for details.

The (Reverse) Repugnant Conclusion

framework without assuming any additional properties such as separability, continuity, or anonymity. We show that, given a weak mono- tonicity property (which is weaker than the usually imposed Pareto principle), a weak form of inequality aversion is sufficient to generate the above-described incompatibility. Furthermore, we discuss the im- plications of a property which is "dual" to the repugnant conclusion - the reverse repugnant conclusion. 3 A principle implies the reverse repugnant conclusion if every state in which each member of the population experiences a utility level below neutrality is declared to be better than a state in which a sufficiently large population has a utility level that is below neutrality but arbitrarily close to it. It can be shown that, under analogous assumptions to those employed for the repugnant conclusion, avoidance of the reverse repugnant conclusion implies that there exist states of the world where the ceteris paribus addition of a person below neutrality is considered desirable.

2 Social-evaluation Orderings

The set of positive integers is denoted by Z, and Tr (~++, TO=_) is the set of (positive, negative) real numbers. Let 7-r z be the set of all vectors (Ui)ic Z such that ui c ~ for all i o Z . . A / i s the set of all non- empty and finite subsets of Z. For N c N', T~ x is the subspace of T~ z corresponding to the coordinate labels in N. Tr u is the subset of ,]'~N

such that, for all u c ~ U and all i 6 N , ui > O. Furthermore, for N 6 A/', 1N is the element in j-~U the components of which are all equal to one, and we use n to denote the number of elements in N.

X is a set of social states of affairs, and N = N(x) c N" is the set of individuals alive in state x c X. For each i c N(x), ui = U i ( x ) is i 's lifetime utility in state x. In a welfarist framework, knowledge of the sets of individuals alive in the states x and y and their lifetime utilities is sufficient to rank x and y from a social point of view. Therefore, given welfarism, all that needs to be established is a social-evaluation ordering R on the set

~D:= {(N,u) I N c A / a n d u c ' T ~ N } . (1)

That is, for (N, u) and (M, v) in D, (N, u )R (M, v) means that (N, u) is considered at least as good as (M, v). The strict preference relation corresponding to R is P, and I denotes indifference. Given R, x c X

3 This was suggested by Frank Jackson.


is at least as good as y 6 X if and only if (N(x), (Ui(x))ieN(x)) R (N(y), (Ui (Y))icN(y)). For one of our results, it will be useful to restrict attention to the set

79+ := { (N, u) I N E A/" and u ~ ~ U } . (2)

The most commonly discussed variable-population social-evaluation orderings are classical utilitarianism and average utilitarianism. 4 Clas- sical utilitarianism is defined by letting, for all (N, u), (M, v) c 79,

"( ',. Z H i >" y ~ l ) i . (3) (N,u) R(M,v) ioN icM

According to average utilitarianism,

1 ( N , u ) R ( M , v ) ,( '~ - ~ - ~ u i > vi (4)

n ~. m icM

for all (N, u), (M, v) ~ 79, where n = I NI and m = [MI.

3 Fixed-population Axioms

First, we state the strict Pareto condition.

Strict Pareto (SP): For all N ~ A/', for all u, v c ~N , if Ui >_ Vi for all i 6 N with at least one strict inequality, then (N, u) P (N, v).

Strict Pareto is weaker than the strong Pareto axiom because it does not encompass Pareto indifference.

The following axiom is a weakening of strict Pareto. It requires that, if the same individuals are alive in two alternatives and utilities are distributed equally within each state, a higher lifetime utility for everyone leads to a better alternative.

Equal-utility increasingness (EUI): For all N c A/', for all u, v E 7~, if u > v, then (N, ulN) P (N, VlN).

4 Alternative population principles are discussed in Blackorby et al. (1995, 1996a, b), Blackorby and Donaldson (1984), Hurka (1983), and Ng (1986), for example.


S-concavity (see, e.g., Marshall and Olkin, 1979) is an axiom which is commonly employed in discussions of inequality- measurement. It is satisfied by any fixed-population social-evaluation ordering that re- spects the Lorenz quasi-ordering.

S-Concavity (SC): For all (N, u) E 7?, for all doubly stochastic n x n matrices B, (N, Bu) R (N, u).

A weaker property requires that an equal-utility distribution must be considered at least a~s good as another distribution with the same total utility. Unlike S-concavity, this axiom does not impose a restriction on the relative ranking of two unequal distributions. We call this axiom weak equality preference.

Weak Equality Preference (WEP): For all (N, u) ~ 7?, (N, ( ( l /n) x ZirX Ui)IN) R (N, u).

The following property is implied by equal-utility increasingness and weak equality preference.

Positive-limit Property (PLP): For all N E N', for all u 6 ~++, for all e E (0, u), there exists M D N such that (M, elM) R (M, (ulN, 01M\N)).

We obtain

Lemma 1: If an ordering R on 7? satisfies equal-utility increasingness and weak equality preference, then R satisfies the positive-limit property.

Proof" Let N c N ' , n = INI, u c ~++, and e c (0, u). Consider M D N such that m = ]M] > nu/e. Furthermore, let 3 = nu/m. By definition, 3 < e. By WEP, it follows that (M, 31M) R (M, (ulN, 01M\N)). By EUI, (M, elM) P (M, 8IM). Therefore, (M, elM) P (M, (UlN, 01M\N)) and hence (M, elM) R (M, (UlN, 01M\N)). []

We do not impose anonymity (requiring that people are treated im- partially, irrespective of their identities), continuity (which ensures that "small" changes in utilities do not lead to "large" changes in the social ranking), or any separability condition. Furthermore, we do not require the strong Pareto principle - the weaker EUI is sufficient to prove all results other than Theorem 7.


4 Critical Levels

One natural way to examine the properties of population principles is to analyze critical levels of lifetime utility. For a given alternative, adding an individual at a critical utility level is a matter of social indifference provided no member of the existing population is affected in utility terms. In general, such a utility level need not exist, and if it exists, it need not be unique (note that we do not impose the strong Pareto principle). The following axioms deal with the existence and properties of such critical levels.

Existence of Critical Levels (ECL): For all (N, u) e 7), for all j ~ Z \ N , there exists c E 7~ such that

(NU{j}, (u,c)) I (N,u) . (5)

Zero Critical Levels (ZCL): For all (N,u) e 7), for all j c I \ N ,

(N U {j}, (u, 0)) I(N, u) . (6)

Zero-augmentation Improvement (ZAI): For all (N, u) ~ 7), for all j ~27\N,

(N U {j}, (u, 0)) R (N, u) . (7)

Zero-augmentation Deterioration (ZAD): For all (N, u) ~ 7), for all j e I \ N ,

(N, u) R (N U {j}, (u, 0)) . (8)

Zero critical levels is the conjunction of zero-augmentation improvement and zero-augmentation deterioration. The conjunction of the (fixed-population) strict or strong Pareto principle and zero critical levels implies the Pareto-plus principle, which requires that the ceteris paribus addition of an individual above neutrality is desirable (see Sikora, 1978).

Pareto Plus (PP): For all (N, u) 6 7), for all j E Z \ N, for all u e 7-r

(N U {j}, (u, u)) P (N, u) . (9)


5 The Repugnant Conclusion and Inequality Aversion

Consider a society with n people alive where everyone in N experiences the same level of lifetime utility u 6 ~++ and u can be arbitrarily high. If a social-evaluation ordering leads to the repugnant conclusion, it is possible to find, for any e c (0, u) (no matter how close to zero), an expanded population M D N such that everyone in M has a lifetime utility of e, and the latter alternative is considered better than the former.

Repugnant Conclusion (RC): For all N 6 N', for all u 6 ~++, for all e 6 (0, u), there exists M D N such that (M, elM) P (N, ulN).

Avoidance of the repugnant conclusion is the negation of RC.

Avoidance of the Repugnant Conclusion (ARC): There exist N c .N', u c 7~++, e E (0, u) such that (N, ulN) R (M, ElM) for all M D N.

Arrhenius (1997) shows that equal-utility increasingness, weak equality preference, zero-augmentation improvement, and avoidance of the repugnant conclusion are incompatible (see also Blackorby and Do- naldson, 1991).

Theorem 1: There exists no ordering R on D satisfying equal-utility increasingness, weak equality preference, zero-augmentation improvement, and avoidance of the repugnant conclusion.

Theorem 1 can be strengthened by weakening weak equality preference to the positive-limit property. We obtain

Theorem 2: There exists no ordering R on 7? satisfying equal-utility increasingness, the positive-limit property, zero-augmentation improvement, and avoidance of the repugnant conclusion.

Proof." Suppose R satisfies all of the above axioms. Let N E N', u ~++, and e ~ (0, u). By repeated application of ZAI, (M, (ulN,

01M\N)) R (N, ulN) for all M D N. Let 3 c (0, e). By PLP, there exists M D N such that (M, 31M) R (M, (ulN, 01M\N)). By EUI, (M, elM) P (M, 31M). Therefore, by transitivity, (M, elM) P (M, (ulN, 01M\N)) R (N, ulN), contradicting ARC. []


The positive-limit property is needed in Theorem 2. To see this, let, for any N E A/" and u E Tr N, fi be a permutation of u such that fii >/~j for all i, j E N such that i < j . Define the ordering R by letting, for all (N, u), (M, v) E 79, (N, u) R (M, v) if and onlY if

Z Ctiq- Z ui/2i-l>- ~ ~)i+ Z vi/2i-1 �9 (10) iEN:~ti<O icN:fii>_O iEM:~i<O iEM:~)i>O

R satisfies all axioms stated in Theorem 2 except PLP. In addition, it satisfies zero critical levels and all standard fixed-population-size axioms that are usually imposed (such as the strong Pareto principle, anonymity, and continuity).

PLP cannot be weakened further if zero-augmentation improvement is strengthened to zero critical levels. First, we show

Lemma 2: If an ordering R on 79 satisfies zero-augmentation deterioration and the repugnant conclusion, then R satisfies the positive-limit property.

Proof." Suppose PLP is violated. Then there exist N 6 .iV', u 6 ~++, e 6 (0, u) such that (M, (ul~, 01M\N)) P (M, elM) for all M D N. By repeated application of ZAD, (N, ulN) R (M, (ulN, 01M\N)) for all M D N. Therefore, (N, u ly) R (M, elM) for all M D N, contradicting RC.

[]

Combining Theorem 2 and Lemma 2, we obtain

Theorem 3: Suppose an ordering R on 79 satisfies equal-utility increasingness and zero critical levels. R satisfies the repugnant conclusion if and only if R satisfies the positive-limit property.

Among the orderings satisfying equal-utility increasingness, zero critical levels, and S-concavity, those orderings that are classical utilitarian for nonnegative utilities perform best in terms of "minimizing" the instances where we have the repugnant conclusion. Define the relation >'-ARC on the set of all orderings on 79 by letting R ___ARC R' if and only if, for all N E A/', for all u E T~++, for all e E (0, u), for all M D N ,

(M, elM) P (N, ulN) > (M, elM) P' (N, ulN) . (1 1)


According to this definition, an ordering R is at least as good as an ordering R I if and only if R t produces an instance of the repugnant conclusion whenever R does. Therefore, ->ARC ranks social orderings with respect to their ability to limit the extent to which the repugnant conclusion occurs.

We obtain

Theorem 4: Let .4 be the set of orderings on D satisfying equal-utility increasingness, zero critical levels, and S-concavity. R* is a best element in .4 according to ->ARC if and only if R* coincides with classical utilitarianism on D+.

Proof." i. Let R* be classical utilitarian on D+, and let R 6 .4 be arbi- trary. Suppose that, for some N E A/', u c ~++, e ~ (0, u), and M D N, (M, elM) P* (N, ulN). By definition of R*, me > nu and hence, e > nu/m. By ZCL, (M, (ulN, 01M\N)) I (N, UlN) for all M D N. By SC, (M, (nu/m)lM) R (M, (ulN, 01M\N)). By EUI, (M, elM) P (M, (nu/m)lM). Therefore, by transitivity, (M, elM) P (N, ulN), which implies R* -->ARC R and, therefore, R* is a best element in .4.

ii. Conversely, suppose R is best in .4. By (i), this implies R ~"ARC R* whenever R* coincides with classical utilitarianism on D+. There- fore, for all N ~ N', u c ~++, e c (0, u), and M D N,

(M, elM) P (N, ulN) .'. ',. (M, elM) P* (N, UlN) . (12)

By definition of R*, (12) is equivalent to

(M, elM) P(N, ulN) .', ~ . m e > n u . (13)

Let e = nu/m. By (13),

(N, ulN) R (M, (nu/m)lM) . (14)

By ZCL, (M, (ulN, 01M\N)) I (N, UlN). Therefore, using (14), we obtain

(M, (ulN, O1M\N)) R (M, (nu/m)lM) . (15)

By SC, (M, (nu /m) lM)R (M, (ulN, 01M\N)), which, together with (15), yields

(M, (nu/m)lM) I (M, (ulN, 01M\N)) �9 (16)

For fixed-population comparisons of societies with a single individ-


ual, all orderings satisfying EUI are equivalent to utilitarianism. Now consider N E A/" such that n _> 2. Let u E 7-r N. If u = 01N, it follows trivially that

ui 1N �9 (N,u)I(N,(li~cN ) ) (17,

If u r 01N, let v E 7-r N be such that Vk = ~ i c N ui for some k e N and vi = 0 for all i E N\{k}. By (16), (N,v)I(N,((1/n)Y~4eNUi)IN). By SC, it follows that

(N, ( I ~NU~)IN) R (N, u) R (N, v) . (18)

Therefore,

(17) and (19) imply that, for all (N, u), (N, v) E D+,

(19)

-( Vi 1y t t ) 20, which, by EUI and reflexivity of R, is equivalent to

1 1

(N,u)R(N,v) < ?'--i~N b l i>-~y i n iEN ' ZUi>~l)i'icN iEN (21)

This means that fixed-population comparisons are made according to utilitarianism for nonnegative utility vectors of any population size. By ZCL, it follows that R must coincide with classical utilitarianism on 7)+.

[]

Theorem 4 demonstrates that, given the axioms in the theorem state- ment, the repugnant conclusion occurs in the fewest instances if there is no strict inequality aversion. The relationship between inequality aversion and the repugnant conclusion is also illustrated in Blackorby et al. (1996a): if social evaluations are made according to the leximin ordering (which represents extreme inequality aversion) with zero critical levels, augmenting the population by a single person is sufficient to obtain the repugnant conclusion.

The (Reverse) Repugnant Conclusion 11

6 The Reverse Repugnant Conclusion

A property which is "dual" to the repugnant conclusion says that, for any alternative in which each person experiences the same negative utility u and for any e 6 (u, 0), no matter how close to zero, there exists an alternative with an expanded population in which everyone experiences a lifetime utility of e which is considered worse. This "reverse" repugnant conclusion is defined as follows.

Reverse Repugnant Conclusion (RRC): For all N E A/', for all u E ~__ , for all s E (u, 0), there exists M D N such that (N, ulN) P (M, elM).

Avoidance of the reverse repugnant conclusion is defined as

Avoidance of the Reverse Repugnant Conclusion (ARRC): There exist N c A/', u E ~__ , e E (u, 0) such that (M, elM) R (N, u ly) for all M DN.

A result analogous to Theorem 1 cannot be obtained if we replace avoidance of the repugnant conclusion with avoidance of the reverse repugnant conclusion and zero-augmentation improvement with zero- augmentation deterioration. Let, for any N ~ N" and u 6 7~ N, fi be a permutation o fu such that/~i 5 t~j for all i, j c N such that i < j . Define the ordering R by letting, for all (N, u), (M, v) E D, (N, u) R (M, v) if and only if

1 21{icN:~i<O}l Z {ti'+- Z ui/2i-1

icN:Fti>O iEN:Fti<O 1

->2rli~M:~<o/I ~ ~+ ~ ~i/2~-~" iEM:vi>O icM:f3i<O

(22)

R satisfies zero critical levels, equal-utility increasingness, weak equality preference, and avoidance of the reverse repugnant conclusion. In addition, R satisfies the usual fixed-population axioms (except for continuity) and S-concavity.

To obtain results analogous to Theorems 2 and 3, the positive-limit property must be replaced with a dual condition that applies to negative values of u.


Negative-limit Property (NLP): For all N 6 N', for all u 6 ~__ , for all e E (u, 0), there exists M D N such that (M, (ulN, 01M\N)) R (M, elM).

It is interesting to note that this condition is as appealing as the positive- limit property, whereas the dual of weak equality preference that would be needed in the adaptation of Theorem 1 to negative values is not appealing at all because it favors inequality.

We obtain the following theorem, the proof of which is analogous to the proof of Theorem 2.

Theorem 5: There exists no ordering R on 7? satisfying equal-utility increasingness, the negative-limit property, zero-augmentation deterioration, and avoidance of the reverse repugnant conclusion.

The following theorem parallels Theorem 3.

Theorem 6: Suppose an ordering R on 7? satisfies equal-utility increasingness and zero critical levels. R satisfies the reverse repugnant conclusion if and only if R satisfies the negative-limit property.

Theorem 5 implies that, if critical levels exist and the reverse repugnant conclusion is avoided, any ordering R satisfying strict Pareto and the negative-limit property must be such that some critical levels are negative. This, in turn, implies that some additions of persons below neutrality are considered desirable. We obtain

Theorem 7." If an ordering R on 7? satisfies strict Pareto, the negative- limit property, existence of critical levels, and avoidance of the reverse repugnant conclusion, then there exist (N, u) ~ 77, j E 37 \ N, and c

7-4__ such that (N U {j}, (u, e)) P (N, u) . (23)

Proof" Suppose R satisfies the properties stated in the theorem. Theo- rem 5 implies that R must violate ZAD. Therefore, there exist (N, u) E 7? and j ~ Z \ N such that

(N U {j}, (u, 0)) P (N, u) . (24)

By ECL, there exists c ~ ~ such that (N U {j}, (u, c) ) I (N, u). SP and


(24) imply c < 0. Using SP again, it follows that (N U {j}, (u, c')) P (N, u) for all c I > c. Choosing c' 6 (c, 0) yields the desired conclusion.

[]

Given the negative-limit property, Theorem 7 illustrates that avoiding the reverse repugnant conclusion implies that the addition of some people whose lives are below neutrality is desirable. Note that the reverse repugnant conclusion only applies to situations where everyone alive is below neutrality. It can therefore be argued that, for most prac- tical purposes, the reverse repugnant conclusion does not seem to pose as serious a problem as the repugnant conclusion.

7 Concluding Remarks

This paper provides a discussion of the incompatibility between avoidance of the repugnant conclusion and the Pareto-plus principle (or weaker conditions). It is shown that this incompatibility is fundamental: only a very weak notion of inequality aversion is needed in order to prove the result. We conclude that at least one of the above properties should be abandoned.

Theorem 4 may provide some guidance to those who opt for zero critical levels but regard the repugnant conclusion as ethically unappeal- ing (but necessary). 5 It shows that, given equal-utility increasingness, S-concavity, and zero critical levels, classical utilitarianism performs better than other principles in the sense that it minimizes instances of the repugnant conclusion. The cost of adopting classical utilitarianism is, of course, that it is not inequality averse.

An alternative approach is to abandon zero critical levels and make some allowance for positive ones. Blackorby etal. (1995, 1996a, b, 1997a-c) have examined principles that do this and they argue that the critical-level utilitarian and critical-level generalized utilitarian principles with positive (fixed) critical levels outperform all the other options.

Acknowledgements

Financial support through a grant from the Social Sciences and Humanities Re- search Council of Canada is gratefully acknowledged. We thank three referees and seminar audiences at Namur and ESEM97, Toulouse, for comments.

5 See Ng (1989), for an argument in favor of the position that the repugnant conclusion is not really repugnant.


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Critical levels and the (reverse) repugnant conclusion