Social criteria for evaluating population change

Journal

of Public Economics

SOCIAL

CRITERIA Charles

25 (1984) 13-33. North-Holland

FOR

EVALUATING

BLACKORBY

POPULATION

and David

CHANGE

DONALDSON*

The University cf British Columbia, Vancouver, British Columhia, Canada V6T I Y2 Received

February

1982, revised version

received November

Many applications of economic analysis require social different numbers of people. In this paper we use the tools axiomatic formulation of this problem. It yields a class generalized utilitarianism’. This class and its implications are found to yield intuitively appealing results.

1983

evaluations of alternatives involving of social-choice theory to provide an of social criteria called ‘critical-level are compared with other criteria and

1. Introduction Many applications of economic analysis require social evaluations of alternatives involving different numbers of people. For example, Wicksell, building on the earlier analysis of the relationships among population, subsistence and land provided by Condorcet, Godwin and Malthus,’ argued that the best population size was the one that maximized per capita output. And, more recently, optimal population policies have been investigated jointly with optimal savings programmes [Dasgupta (1969), Lane (1975, 1977), Meade (1955), Pitchford (1972), Samuelson (1975)]. In each of these cases it was necessary to pick an objective function that compared social states with different populations. The main candidates for the task of (completely) ordering states of affairs with different populations are classical utilitarianism, average utilitarianism, and (Rawls’) maximin principle. The classical utilitarian principle prefers social outcomes with more utility to outcomes with less, regardless of the number of people that enjoy it; the average utilitarian principle prefers outcomes in which average utility (the ‘standard of living’) is greater; while Rawls’ principle evaluates outcomes by using the utilities (or real incomes) of the least advantaged only. In this paper we use the tools and methods of social choice theory (where *We are indebted to John Broome, Partha Dasgupta, Amartya Sen and John Weymark for extensive comments and criticisms. We wish to thank the Social Sciences and Humanities Research Council of Canada for research support. ‘Spengler (1965) provides a good survey of the positions taken by economists on population questions. 0047~2727/84/$3.00

0

1984, Elsevier Science Publishers

B.V. (North-Holland)

14

C. Biuckorhy

and D. Donaldson,

Evaluating population

change

interpersonal comparisons of utilities are allowed) to provide an axiomatic discussion of the problems involved in choosing a social value function when the population size can change. We use specific axioms to generate a particular class of such value functions that we think represents the most reasonable solutions to the problem. These rules proclaim that state of affairs x is socially at least as good as y if and only if

iE;x,MWX))-id41 2 iE;,,)cs(wY))-_g(41~

(1)

where N(x) is the population alive in x, U’(x) is i’s lifetime utility in X, g is a concave function, and IX is a non-negative utility level. We adopt the convention that life is worth living if and only if the utility index is positive. The function g and the critical utility level cxare chosen by the evaluator. We call this principle ‘critical-level generalized utilitarianism’, and it contains, as a special case, ‘critical-level utilitarianism’. In the latter case, g is the identity map, and x is socially at least as good as J’ if and only if

(2) The critical level CI disappears from (1) and (2) when population size is the same in x and y. In this case, (2) is ordinary utilitarianism and (1) is a generalized version, allowing for distributional preferences. For differentnumber choices, (Yis a parameter that determines how average or representative utility is to be traded off against population size. The discussion is set in a ‘welfarist’ framework [Sen (1977)]; welfarism is the view that individual welfares provide sufficient information for social choices. We assume as well that the interests of different people must be given equal weight in social decisions (anonymity). We first show (section 2) that if there are well-behaved fixed-numbers social-evaluation functions, that there must be a social value function that depends on population size and a measure of the social standard of living (‘representative’ utility)2 only. This representation theorem provides a test for proposed rules (if they cannot be written in this form, then they do not generate social orderings), and offers a method for generating social rules. In section 3 we turn to the fixed-numbers social-evaluation functions, and argue that if the ethics represented by the members of this family are to be the same, additive separability is a requirement. We then propose (section 4) an axiom for comparing populations in a simple hypothetical situation. This axiom contains the critical level CI in (1) *In the case that the fixed-numbers utility is average utility.

social-evaluation

functions

are utilitarian,

representative

C. Blackorby and D. Donaldson, Evaluating population change

15

and (2). We show that (1) is a necessary consequence of our previous axioms and this new one. Then, in section 5, we argue that our proposed rules perform well in situations involving population-planning problems. We discuss the questions of anonymity over present and possible populations in historical situations, the ‘pure’ population problem of dividing a fixed resource among a variable number of people, and Parlit’s ‘repugnant conclusion’ [a positive critical level (a) avoids it]. In section 6 we compare our rules with others, including generalizations of the average utilitarian principle. Section 7 concludes. 2. A variable-population

value function

In this section we suggest a general procedure for evaluating social outcomes with different populations. First, we assume that all the relevant information in a given social state is contained in the list of (named) utilities of the individuals who are alive in that state. This assumption is called welfarism by Sen (1977) and is equivalent, in the usual fixed-population social-choice model, to the conjunction of the assumptions of unlimited domain, independence of irrelevant alternatives and Pareto indifference. Welfarism permits the characterization of a social state by a list of named utilities; the social evaluation of alternative vectors of utilities is the same as the social evaluation of the states which they represent. In appendix B we extend this result to the variable-population case. Furthermore, we assume that utilities are perfectly measurable and fully interpersonally comparable. Weaker measurability and comparability assumptions would serve only to restrict the class of social-evaluation functions under discussion.3 Parlit (1982) distinguishes three kinds of policy options: same-people choices, same-number choices, and different-number choices. The first kind affects neither the number of people4 alive nor their personal identities, the second affects the identities of those who are alive but not their number, and the third affects both. Ordinary social-choice theory is concerned with the first of these. sPerfect measurability and full interpersonal comparability mean that the utility numbers are significant both inter- and intra-personally. Weaker measurability and comparability assumptions are discussed in Blackorby -and Donaldson (1982), Blackorby, Donaldson and- Weymark (1983). Maskin (19781. Roberts (1980a. 1980b). and Sen (1974. 1977). As an examole of this possibility, we note that critical-value utilitarianism (2), requires only cmdinally measurable fully comparable utilities. %r this paper we shall write as if sentient beings other than people do not ‘count’. The rules developed here can, however, be used over larger populations. In that case, the anonymity assumption we employ may be too strong. It can be relaxed to apply to groups such as people and various sorts of animals. These groups can then be treated asymmetrically. We are unsure about the ethical appropriateness of this asymmetry, but it is a technical possibility.

16

C. Blackorhy

and D. Donaldson,

Evaiuuting population

change

Anonymity is an axiom for same-people choices. It requires that the names of people should not count in ranking social states. Writing N(z) as the set of people alive in z, anonymity requires that, if N(x) = N( y) and the utilities in x are the same set of numbers as the utililities in y, then x and y are socially indifferent. This axiom can be extended to same-number anonymity, which requires that if n(x) =n( y) [where n(z) is the number of people alive in z] and the utilities in x are the same set of numbers as the utilities in y, then x and y are socially indifferent. We assume same-number anonymity throughout the paper, and note it seems reasonable and natural on ethical grounds, at least if social states are described in a ‘timeless’ way. We therefore require a social state to be a description of the whole history of humanity, with utilities being indexes of lifetime welfares. Same-number anonymity in conjunction with welfarism means that a state of affairs is completely characterized, for social-evaluation purposes, by the number of people alive in the state, and by the set of utilities that they enjoy. Given the welfarism theorem of appendix B, we may write a state as a utility vector u = (ur , . , u,) E 52” (Euclidean n-space), where n is the population size, and it is ranked (by same-number anonymity) as indifferent to any permutation of itself. We postulate the existence of an ordering R which serves to rank these utility vectors. For tin KY, ZIE R”, CRC means that U is at least as good as zL5 R also orders social states. If U=(U’(x), . . ., Uncx)(x)) and are the vectors of utilities in x and y, respectively, u==(U’(y),...,U”‘Y’(y)) then x is at least as good as y if iiRE. For each population size, n= 1,2,. . , we assume the existence of an ordinal6 fixed-population social-evaluation function W”:FY’-+lQ. Thus, if U, U=ER”, iiRik+W”(ii)

2 W”(G).

(3)

Anonymity requires that W” is symmetric. We further assume that W” is continuous (continuity), and increasing along the ray of equal utilities (minimal increasingness). This latter assumption is a weaker monotonicity assumption than any of the Pareto (preference) conditions. To construct a variable-population social-evaluation function, we need a particular numerical representation of these social-evaluation functions. The one we use is an adaptation of a similar notion in the income-distribution literature. For every UE R”, we define the representative utility I) to be that level of utility which, if given to each individual, is socially indifferent to CL’ ‘Preference and indifference are defined in the usual way, with 1TPfi++GRtTr\ -iiRti, and tilLbuRt= A CRC. ‘Since W” is an ordinal function, any increasing transform of it produces the same social ordering. ‘in the income-distribution literature the analogue is called ‘equally-distributed-equivalent income’ [Atkinson (1970) Kolm (1969). and Sen (1973)]. For population subgroups we have referred to it as ‘representative income’ [Blackorby. Donaldson and Auersperg (1981)].

17

C. Blackorhy and D. Donaldson, Eoaluating population change

More formally, implicitly by

the representative

utility

u associated

with UE R” is defined

IV(Ul”) = IV(U),

(4)

where 1, is an n-dimensional vector of ones. We assume that u always exists. That is, for each UE R”, n= 1,2,. . . , there exists UE iw such that (4) holds. It rules out the possibility of a UE R”, where U is either inferior or superior to all vectors of equally distributed utilities. Because of this assumption and minimal increasingness, we can solve (4) for the unique representative utility u= r(u).

(5)

function. By continuity of W”, Y” is a continuous specific numerical representation of W”. That is, W”(U) 2

wyq+-+ r-(U)

2 r(a)-u~

Furthermore,

Y” is a

IT.

(6)

Thus, one state is preferred to another with the same population size if and only if its representative utility is higher. The indifference surfaces of Y’ are numbered so that yn(xl,)=x,

VXER.

u is an inequality-adjusted (agreeing with the Lorenz

“,J i

average utility. If W” and quasi-ordering),8 then

(7) P

are

S-concave

ui,

?li=l

(8)

the average utility of the population. We are now in a position to compare states involving populations of different sizes. Consider two states, UE R” and U=E R” (where m is not necessarily equal to n). If 0 and I? are the corresponding representative utilities, then ulul, and U’lrY,. Hence, knowledge of (n,~?) and (m, I?) is sufficient to rank U and 5. This suggests that it is possible to rank U and 6 by using a value function that depends on population sizes and representative utilities alone.

‘See Berge (1963), Dasgupta, Sen and Starrett (1973) and Blackorby and Donaldson W” and Y” are quasi-concave, this, plus symmetry, guarantees S-concavity.

(1977). If

C. Blackorby and D. Donaldson, Evaluating population change

18

Theorem

2.1.

There

second argument,

is a value function

W: Z, + x [w-+[w,9 increasing

in its

such that, for all ii E KY, 17E IX”‘, n, m E L + + ,

iiRtkW(n,

I?)2 W(m, q,

(9)

where V= Y(U) and U==Y’“(G). Proof

See appendix

A.

W may, in general, be sensitive or insensitive to its first argument. Eq. (9) provides a test for rules that are to provide (complete) welfarist orderings of social alternatives. If the proposed rule can be represented as a function of population size and representative utility, then it passes the test. For example, Singer’s (1976) rule, based on fixed-numbers utilitarianism, was shown by Parlit (1976) not to be transitive. It does not satisfy (9). On the other hand, a rule suggested by Hurka (1982) (discussed in section 6 below) does satisfy (9). If the same-number rules are utilitarian, then

(10) where

s signifies

U=

ordinal

equivalence.

It follows that

“(U)=~ _~l

(11)

Ui)

l

or average utility. Classical utilitarianism utilities. and can be written as W(n, U)= no = jJ

ranks

states

(9). Average W(n,“)=v=i,$Iui.

total

(12)

Ui ,

i=l

satisfying

by comparing

utilitarianism

can be written

as

(13)

I

3. Complete strict separability The set of social-evaluation 9L++ is the set of positive integers.

functions

{WI,. . . , W”, . . . > and the attendant

C. Blackorby and D. Donaldson, Evaluating population change

19

set of representative utility functions (Y’, , Y’, . . . } are, at the moment, quite arbitrary and not related to each other. We need a principle that links these functions together so that they represent a single set of ethical judgements. Since each I” is a specific numerical representation of W”, we may work with either set. We adopt a single axiom that ensures additive separability of each Y” and W” when the strong Pareto principle (W” and Y increasing in each person’s utility) is satisfied. Consider a subgroup of an existing population, and calculate its representative utility on the assumption that it is the whole population. Substitution of this representative utility for the utilities of the subgroup would be a matter of social indifference if the subgroup existed on its own. The population substitution principle requires that this substitution is a matter of indifference in the larger population as well. Population

substitution

principle.

For

all n, rnE Z, +,

if u=(u’, u”), U’E KY,

u” E R”, then

PCrn(u)=r+m(u’,UN)= r+yg )...)

L(u”),

(14)

where u’ = Y’(u’).

This principle implies that replication of a population, utility by utility, does not change representative utility. For example, if Y”(u)=u, then Y2”(u,u) = T2”(ol,,u)= Y2”(u1,,u1,)=u, which is closely related to Dalton’s (1920) principle of population. In addition, if a population is expanded without affecting existing utilities, and added individuals receive the representative utility, then the population substitution principle implies that representative utility is not affected. Thus, the principle is ethically attractive. The population substitution principle, when combined with the strong Pareto principle, has important consequences for the family {Y}. Theorem principle nEz++

3.1. If, in addition to the basic assumptions, the strong Pareto and the population substitution principle are satisfied, then, for all and for all UE R”,

(15) where g is an increasing, Proof

See appendix

A.

continuous function.

20

C. Blackorhy

und D. Donaldson,

Emluating

The strong Pareto requirement is necessary requirement, we may have, for all n E Z + + ,

populution change

for theorem

3.1. Without

that

1~(~)=min{u,,...,u,},

(16)

the maximin rule. This family satisfies the population substitution principle. In (15) g is an arbitrary continuous and increasing function. I”’ and W” are quasi-concave (strictly quasi-concave) if and only if g is concave (strictly concave). Given additive separability and symmetry, S-concavity and quasiconcavity (along with their strict counterparts) are equivalent [Berge (1963)]. S-concavity of g means that unambiguous reductions in utility inequality are not dispreferred. An example of a concave function for g might be

g(t) =

1, y’

t

5 0,

(17)

t 10,

where /I> 1. Thus, miserable individuals are given more calculation. Since /I= 1 results in fixed-numbers utilitarianism, generalization of utilitarianism. Another example is

g,(t)= i

f7 _e-“,

where 7 is a non-negative family

Y =Q

(18)

y>o, parameter.

weight in the this is a simple

This makes

r” into the Kolm-Pollak’”

y=o, (19)

4. A critical-level value function We now turn to different-number choices, and propose an axiom that determines [when (15) holds] the value function Ii! Following Sikora (1978), we examine a hypothetical situation where two states, x and y, are compared. The population in y consists of the population in x and one added person; the utilities of the common population are the same in x and y. Sikora has proposed that y should be regarded as socially indifferent to x ‘%e Blackorby

and Donaldson

(1980).

C. Blackorby and D. Donaldson, Evaluating population change

21

if the added life is ‘neutral’, and better than x if the added life is worth living. He calls this the Pareto-plus principle. Adopting the convention that a positive utility (for an individual) signifies a life worth living, with a utility of zero indicating a neutral life, we weaken Sikora’s principle slightly, and call it the Pareto population principle: Pareto population

principle.

For any n EL + + , u E R”,

(20)

(ll,O)Iu.

This principle states that y is indifferent to x if the added life is neutral. The second part of the Pareto-plus principle follows if the strong Pareto principle holds. The Pareto population principle contains a critical level of utility (zero) for the added person which is independent of the utilities of the smaller population. This property can be preserved by selecting a critical level of utility, denoted by tl, that can be different from zero. Thus, in the hypothetical situation, if the added person’s utility is exactly CI, x and y should be socially indifferent. We call this principle the critical-level population principle: Critical-level

population

principle.

(2.4, a)Zu.

For all n E Z + + , u E R”

(21)

We believe that a should normally be positive. Given the strong Pareto principle, the added person must enjoy a standard of living above a before y proves to be socially better than x. Part3 (1982) has introduced a principle that is related to the critical-level population principle. It specifies a range of utilities for the added individual, allowing some indeterminacy to enter. This rule violates the ordering requirement of this paper, but may be used to generate social quasi-orderings (rules that are not necessarily complete). Narveson (1976) has proposed a rule that generates quasi-orderings as well. When combined with (15), the critical-level population principle generates a specific functional form for II! Theorem 4.1. Ix in addition to the basic assumptions, the strong Pareto principle, the population substitution principle, and the critical-level population principle hold, then

Wk 4 ~5W4 -g(41 (22)

22

C. Blackorhy and D. Donaldson, Evaluating population change

Proof:

See appendix

Corollary principle

4.1.

A.

Fixed-numbers

are satisfied

utilitarianism

and

the

critical-level

population

if and only if

W(n, 0) A n[v - Z]

=i$

C”i

These two principles

-al.

(23)

into formulae (1) and (2) and we call them [( 1) and (22) and critical-level ut[(2) and (23)]. Critical-level utilitarianism is implemented by subtracting cx from the utility of each person alive in a state to find its value. Critical-level generalized utilitarianism requires that transformed utilities be used and that the transformed value of c1be subtracted from each. Before concluding this section, we demonstrate that there is no value function W satisfying the critical-level population principle for any value of a when the family of representative-utility functions is maximin [eq. (16)].

critical-level ilitarianism

Theorem satisfying (fixed Proof

translate

generalized

4.2. the

numbers)

There

utilitarianism

is no value function

critical-level

population

representative-utility

See appendix

W(increasing principle

functions

for

in its second any

value

argument)

of a when

the

are maximin.

A.

That the maximin rule for fixed numbers is not flexible enough to satisfy the critical-level population principle for any a must be regarded as a serious deficiency in that rule. This does not mean, however, that we must give up the distributional ethics of maximin completely. Very concave g’s provide strong distributional preferences. For example, the Kolm-Pollak functions (19) approximate maximin when y-+ E.

5. Critical-level

principles in practice

In this section we argue that the critical-level principles conform to our intuitions about population problems quite well, and solve many of the problems that have concerned writers on population questions. First, we must say that we intend the utilities employed in the value functions for the critical-level principles to be indexes of lifetime welfare, and we intend the calculations to be timeless, thus measuring the value of a whole history of the planet. This guarantees that no counter-intuitive results on killing occur; members of the present generation whose lifetime utilities are

C. Blackorby and D. Donaldson, Evaluating population change

23

below a still count positively because they are alive in every feasible state of affairs. For example, using critical-value utilitarianism with a = 3, a population of one with utility equal to 4 is preferred to a population of two with utilities 4 and 2. This does not mean that is is preferable to kill the second person, since that would, by shortening the second person’s life, yield utilities of 4 and /I, 8~2, which is worse than 4 and 2. The calculations of (1) and (2) are at odds with the practice of many writers. For example, Dasgupta (1983) has distinguished between two classes of population problems, Genesis problems and actual problems. In Genesis problems, a planner chooses a population when none exists at the time the decision is made; in actual problems, a given group of people decides on its future size, thus raising the possibility of distinguishing between actual and potential people. In actual prbblems, Dasgupta suggests that potential people be given a smaller weight than actual people, even though this may result in incoherent preferences. For example, suppose that, in state x, the population consists of two people with utilities of 5 and 10, and they consider adding a person. In y the utilities are 5 and 10 for the original population with 2 for the added person, and in state z, utilities are 3 and 8 for the original population with 6 for the added person. If a weighted utilitarian value function is used, with a weight of one for actual people and one-half for potential people, then, from the point of view of x, y is better than x and z is worse. But if y is chosen, then the added person becomes an actual person, and z is better than y! This raises a real difficulty, since no single ordering of the alternatives is produced. Dasgupta is motivated, however, by an example that appeals strongly to our moral intuition. Suppose that the present population consists of an adult whose utility is 10 and a disabled child whose utility is 0. Two alternatives are possible. In x, resources are devoted to improving the child’s welfare resulting in utilities of 8 for the adult and 5 for the child; in y, a new child is born, and the same resources are devoted to it, resulting in utilities of 8, 0, and 8. Classical utilitarianism tells us that y is better than x, but our intuitions tell us that something is wrong. Critical-level utilitarianism offers us an alternative to Dasgupta’s solution, however, As long as a> 3, x is preferred to y. Thus, a positive value of CI serves a similar function to a smaller weight on the utilities of potential people without the awkward consequences. The effect of c( on population choices can be shown in a simple problem the pure population problem. A fixed amount z of a single resource ‘income’ is to be divided among n people. Each person has the strictly concave, differentiable, and increasing utility function U with a positive subsistence income s [given by U(s)=O]. Critical-level utilitarianism requires that each person get z/n (because of concavity of U). Treating n as a continuous variable, n 10, it should be chosen to maximise

24

a[(:)-11,

(24)

or

(25) where I = z/n is each person’s

u(z*) -z*---

!Y. =

income.

The optimal

value of I = I* is given by

U’( I*),

When x = 0, this is the SidgewickkMeade formula, with average and marginal utility equated. It is easy to show that I* is an increasing function of cx.” For example, if U(I) = 1 - l/Z, then

(27) Subsistence income is 1, classical utilitarianism (x = 0) yields I* = 2, and as x approaches 1 (its supremum since U is bounded above by l), I* becomes infinite and the population approaches zero. Dasgupta’s solution to the pure population problem with a weighted utilitarian value function and a weight of fl(O
p= 1 is the utilitarian solution (r^=2), and r^ increases without limit as /3 moves toward 0. It is clear from this example that the weight j3 mimics the effect of our critical level c(. Another important criticism of classical utilitarianism is that it leads to what Part3 (1982) calls the ‘repugnant conclusion’. It is this. Consider any two positive representative utilities, 6 and I?, where fi>E. Then there exist “Note that I* is independenl with a larger population making

of 1. A larger 2 leaves optimal up the difference.

income

and utility

unchanged,

C. Blackorby

and D. Donaldson,

Evaluating

population

change

25

UE R” and CE R”, 5> fi, such that O= P(U), O= F(G) and U’PU. That is, the classical utilitarian value function allows population size to substitute for any positive representative utility, no matter how close to zero it is. This property does not hold for the critical-level principles with a>O, since CIrepresents a floor on this kind of trade-off. In this case,

EPu+G[g(q

-g(a)]

> @g(G) --g(U)].

(29)

This equation can be satisfied with E> fi if and only if 17and U are greater than CLIndifference curves for critical-level utilitarianism are shown in fig. 1. The value function W(n, v)=~[I,-CC] is quasi-concave for positive values (i.e. for II> CC)and quasi-convex for negative ones. It is increasing in n for u> z and decreasing in n for 11
W (n,

u) = constant

W(n,u)=O

4

Fig. 1

V)

=

> 0

w (n,

Li)

constant

<

26

C. Blackorhy and D. Donaldson, Evaluating population change

6. Other value functions We believe that the critical-level value functions (22) and (23) and the corresponding principles (1) and (2) are the most promising choices to date for variable population problems. It is true, however, that there are other possibilities, and we investigate several of them here. The Pareto population principle considers the expansion of a population by one person a matter of indifference if the extra person receives a utility of zero and no one else is affected by the change.” This could easily be modified to make the critical level for the added person the representative utility of the smaller population, and we call this principle the Wicksell population principle: Wicksell

principle.’ 3 For any n E Z + + , u E W,

population

(UTVu)Pu. This principle

(30)

gives rise to the average

principle.

Theorem 6.1. If; in addition to the basic axioms, the principle is satisfied, then for all n E L + + and .for all IIE R, W(n, I)) p Proof

Wicksell

population

(31)

Il.

See appendix

A.

This result does not depend on the population substitution principle or strong Pareto being satisfied, so it is a very general result. It might be interesting to discover whether the Wicksell population principle could be successfully generalized. It is possible to replace l”(u) in (30) with (al”(u)), where r~ is a given fraction between zero and one. In the case that the fixedpopulation rules are utilitarian, this results in the value function:

[iDjL&)]t~.

W(n, 11)z

unwieldy

rule

1.

(32)

n= 1.

I 4 This

n> l,n=

(it cannot

be extended

to

continuous

n) has

some

“Note that preferences about crowding and other external diseconomies are included in the individual utilities. “We use Wicksell’s name because he was the earliest user of an average principle in the discussion of optimal population size. Wicksell chose population to maximize real income per person. Many writers name Mill as the earliest average utilitarian. Sumner (1978) argues that this attribution is false.

C. Blackorby and D. Donaldson, Evaluating population change

21

unfortunate properties. The chief one is that, although the optimal population in the pure population problem declines as CJ increases, it may ‘skip over’ certain values on its way down. This, together with the other standard complaints about average principles [see, for example, Hurka (1982)] and the ethical unattractiveness of the generalized Wicksell principle itself, make us suggest that it be rejected as a useful rule.r4 Hurka (1983) has suggested that a useful value function might approximate the average principle for large n, and the classical principle for small n. Such a value function is fV(n,u)=u(l

-eey”),

(33)

where y >O is a free parameter determining the ‘speed’ of transition as n increases. The limit as ~z-+co is clearly u, and for small n, using a Taylor series expansion of W: W(n,

u) k W(0, u) +

awo,0)

n an

= ynu

the classical value function. This function lacks the separability properties of the critical-level principles, and so it may prove to be very difficult to use in practice. Also, y has no obvious interpretation. For that matter, it is very hard to think of choosing y without reference to the carrying capacity of the planet, or some other consideration that belongs properly in the domain of constraints rather than the domain of objectives.

7. Concluding remarks We hope that we have presented a convincing argument that principles for the social evaluation of states of affairs with different populations may be derived from plausible axioms. Furthermore, we suggest that the critical-level population principle leads to rules which are ethically attractive whenever a positive critical level of utility is assumed. It produces coherent choices and complete orderings.

“+The value function (32) can be generalized to the case of transformed utilities by replacing u with R(u) in the formula. A modification of the generalized Wickwell principle is satisfied in this case.

C. Blackorhy and D. Donaldson, Eoaluatinx population change

28

Appendix A Proof' of theorem

From

2.1.

(4) and (3),

iiRu=t*tA, Rbl m. Define R*, an ordering of Z+ + x IX, by (n,iJ) R*(m, t3++Ul,,R 51,. It follows that, for any UE KY’,GE I%“‘,iiRG++(n,rS) R*(m,tT. For any value of ~1,R* may be represented by a function whose range is an open interval of length one. Thus, R* can be represented by a function whose range is, at most, a countable number of these intervals. Hence, there exists a function W such that (n,t?) R*(m,E)++W(n,G)~ W(m,E). Eqs. (3) and (6) require that W is 0 increasing in its second argument. substitution principle Proof of theorem 3.1. The population bility of 1” in each partition, since, for any U’E [w”, U”E [w”, p +m(U’, u”)

zz

}”

+m( Yyu’), . . , I”, i__

l’yu”),

n

= : F( ryu’),

Since

t

[ i=l

gnt”i)

)

1 .

separa-

l”(u”))

m

I”( Cl”)).

This, together with increasingness of 1” (strong additively separable for any II 2 3,” so that

y”(“)=4n

.

requires

(35) Pareto)

implies

that

1” is

(36)

Yn(~l,) =.x, (37)

or, with t:=ng,(x),

(38) Thus:

“This result is a consequence of Gorman’s (1968) theorem on overlapping separable sets of variables. See also Blackorby, Primont and Russell (1978, Theorem 4.7) and Blackorby, Donaldson and Auersperg (1981, Theorem 1).

29

C. Blackorby and D. Donaldson, Eualuating population change

Direct application of the population substitution for any mz3, km, and for any z?EIJ@:

g,’ ; [i

principle

to (39) yields,

II

Ig,(r(u))+i=~+~g,(ui)

(40)

so that

(41)

Therefore results.16

any g,, [7

n 2 3, may

be chosen

for all population

sizes, and

Let UE R”, ZIE R”, n, m E Z, +, n >m. principle,

Proof of theorem 4.1. critical-level population u=Z(u=, c(,. . .) a).

Then,

(15)

by the

(42)

(X) By (15) and transitivity

++uR(ii, cc,. . . , a)

Hi$l

d”i)

eitl

Mui)

2

i$l

+(n-mM4

-g(cc)l L i$l

++nCg(u) -g(a)1 Consequently,

Ani)

(22) holds.

M”‘i)

-g(a)l

2 mCg(o) -g(a)l.(43) Cl

“It can be shown that gI and g, must be cardinally 3a>O,b JPE

B

such that g,(t)=ag,(t)+b

for all te350R.

equivalent.

That

is, for any m, [EZ+ +,

30

C.

Blackorby and D. Donaldson, Evaluating population change

Proqf’ of theorem 4.2. Let iZ=(fl)~[W, ti=(fi+2)~iW, G=(/I,cc)ELP, ti=(fi + 2, a) E LQ’, where fi > a. By fixed numbers maximin, Jlti, and by the criticallevel population principle, JlU= and tilii. Transitivity yields u’lti, contradicting fixed numbers maximin. 0 Proof of’ theorem principle, lzal,r(al,+

6.1.

Let UE R’“, EE R”, n >m. By the Wicksell

,)I.. . I(El”),

where I?= L”(E). Writing

population

(44)

I?= Y”(U), it follows that

tiRu=tttiR(lY .) ++(l?l”)R(61,) (45) where the last step follows from minimal

increasingness.

0

Appendix B In this appendix we extend the welfarism theorem [Guha (1972), d’Aspremont and Gevers (1977), Hammond (1979)]. Let X be a set of social states (finite or infinite) and let N(x), x EX, be the set of all individuals alive in state x. n(x) is the cardinality of N(x), and is assumed finite and greater N(x) is the set of people alive in any state than two for each XEX. N=uxEx in X. If N(x) = N( y), then the number alive in x is equal to the number alive in y and the individuals are the same people. A social-evaluation functional F:Q-+R is a mapping from the set of all admissible (see below) real-valued utility functions on S: = {(x, i)lx E X, iE N(x)} to the set of all orderings of X. A typical element of S is (x, i), iE N(x), and it can be thought of as a social ‘station’, ‘being person i in state x’. For U E%, U(x, i) is the utility that person i gets in state x. Thus, the station (x,i) is weakly preferred to (y,j) if and only if U(x,i)L U(y,j). U(.,i) represents an ordering for person i of the states he or she is alive in, and U(x, .) represents an ordering of the individuals alive in state x (from best to worst-oft). The functional F admits interpersonal comparisons of utility. These may be ruled out, however, by requiring that, for any two utility functions for which the orderings of social states for each person are the same, the social orderings must coincide. This is an Arrow (195 1) social welfare function and his impossibility result holds under the usual assumptions. Our assumption

31

C. Blackorby and D. Donaldson, Evaluating population change

that U is real-valued requires each individual’s ordering to be representable, but that it is always true if the number of alternatives in X is finite or countable. Three conditions on F are: (1) unlimited domain. U consists of all real-valued utility functions on S; (2) Pareto indifference. Vx, y E X 3 N(x) = N(y),

W(x, i) = U(Y, i) Vi E N(x)W( y))l+xI,

Y,

where I, is the indifference relation corresponding to F(U); (3) independence of irrelevant alternatives. VU, 0 E %!, if U, U coincide S,:={(x,i)IxEACX,

on

iEN(x

then F(U) and F(U) coincide on A. Pareto indifference requires only that if the same individuals are alive in x and y, and they are all indifferent between x and y, then x is socially indifferent to y. Independence requires that the social ordering on any subset of X depends on the utilities of the corresponding social stations only. Theorem.

F satisfies unlimited domain, Pareto indifference,

and independence

of

irrelevant alternatives zf and only if there is an ordering R on UXEX [WLnCx)l . the subspace of [w” corresponding (where [w[“(“)1 zs to N(x) and n is the cardinality

of N) such that Vx, YEX,

where

u’

R[“‘““,

R,=

F(U).

E

u”

E

[Wb(Y)l,

VUE%,

ui = U(x, i) Vi E N(x).

u:’ = U( y, i) Vi E N(y)

and

Proof

(a) If F satisfies unlimited domain, Pareto indifference and independence, it is straightforward to establish [Hammond (1979)] that Vx, y, w, ZEX~ N(x) = N(w), N(y) = N(z), VU, U E@ 3 U(x, i) = U(w, i) VIE N(x), U( y, i) = U(z, i) Vz, i) Vi E N(y) [xR,y-wR,z]

A [yR,x++zRDw].

Define R by u’Ru”++[3a,

b E X 3 u’ E l@“(“)l, u” E [WLnCbJ1,

3U~%3U(a,i)=u;Vi~N(a),

U(b,i)=uj’ViEN(b)

and

aR,b].

32

C. Blackorby and D. Donaldson, Evaluating population change

R, is reflexive and complete, u’

E

and xRUy~u’Rd’

where

u” E &@fl(Y)l,

@“‘“)I,

U(x, i) = u; Vi E N(x), U( y, i) = u:’ Vi E N(y)

[Hammond

Transitivity of R follows from Hammond’s argument that the cardinality of each N(x) is a least three.17 (b) Given R, R, is defined by [xR,y++u’Ru”]

and

(1979)]. the assumption

A [yR,x+w”Ru’],

where u’

E

@“‘“)I,

u;=U(y,i)

R, is an ordering Binary independence follows from lemma

u” E @n(y)],

ui = U(x, i),

Vi E N(x),

KEN(y). satisfying unlimited domain and Pareto indifference. is implied by the binariness of R,. Independence 1 in d’Aspremont and Gevers (1977). 0

“hammond (1979) assumed that X (for fixed population) contains at least However, his proof appears not to work unless X contains at least three.

two alternatives.

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