Charles Blackorby,

David Donaldson,

Walter Bossert

English

Research Papers in Economics

Price-Independent Welfare Prescriptions
and
Population Size*
Charles Blackorby, Walter Bossert, David Donaldson
December 1997
Charles Blackorby: University of British Columbia and GREQAM
Walter Bossert: University of Nottingham
David Donaldson: University of British Columbia
* Financial support through a grant from the Social Sciences and Humanities Research
Council of Canada is gratefully acknowledged.
December 9, 1997Abstract
This note investigates the extension of Roberts' price-independent welfare prescriptions to
alternatives in which population size and composition can vary. We show that ethically
unsatisfactory orderings result. Suppose that a single person is to be added to a population
that is unaected in utility terms. Either all such additions must be regarded as bad or
some expansions in which the added person's life is not worth living must be ranked as
social improvements.
Journal of Economic Literature Classication Number: D63.
Keywords: Price-Independent Welfarism, Population Ethics.Price-Independent Welfare Prescriptions and Population Size
Charles Blackorby, Walter Bossert, David Donaldson
This note investigates the extension of price-independent welfare prescriptions (Roberts
[1980a]) to dierent populations sizes. Price independence requires that there exist a single
ordering of nominal incomes that is consistent with a Bergson-Samuelson social-welfare or-
dering for each price vector. Such orderings can be used to justify inequality measures based
on nominal incomes and are also needed for consistency of cost-benet tests (Blackorby and
Donaldson [1984a], Roberts [1980a]). Cost-benet analysis must often deal with projects
that change the number of people in the population. This is most obviously the case for
population-control projects, but population changes are also the result of public-health and
infrastructure projects.
Given welfarism, policies can be evaluated by means of the corresponding alternatives.
An alternative A is described by a set of people who are alive and a corresponding vector of
utilities, and we write
A =

N;

ui	
i2N

(1)
where N is the set of named individuals alive with jNj nite and ui is person i's utility level
for all i 2 N. To allow for the largest class of Bergson-Samuelson social-welfare orderings,
we assume that utilities are numerically measurable and fully interpersonally comparable.1
A is the set of all possible alternatives,2 and R is an ordering (a reﬂexive, complete, and
transitive binary relation) of A.  AR ^ Ameans that alternative  A is socially at least as good
as alternative ^ A, and strict preference (P) and indierence (I) are dened in the usual way.
We assume that R satises the strong Pareto condition.
Strong Pareto: For all  A; ^ A 2Asuch that  N = ^ N = N:
(i) if  ui =^ u ifor all i 2 N, then  AI ^ A;
(ii) if  ui  ^ ui for all i 2 N with at least one strict inequality, then  AP ^ A.
1 See Blackorby, Donaldson, and Weymark [1984], Bossert [1991, 1997], d'Aspremont and Gevers [1977],
Roberts [1980b,c], and Sen [1974, 1977] for discussions of information assumptions in xed-population social
choice. Blackorby, Bossert, and Donaldson [1996] examine information invariance in a variable-population
framework.
2 A may or may not contain the null alternative in which is no one is alive. Whether or not this alternative
is included has no consequences on our results.
1R (R+, R++) is the set of (non-negative, positive) real numbers, and Z++ is the set of
positive integers. Person i's direct utility function is Ui:Rm
+ 7−! R and his or her indirect
utility function is V i:Rm
++ R +7−! R,w h e r em2Z ++ and m  2. That is,
V i(p;yi)=m a x
x i
n
U( x i)jpx iy i
o
(2)
where xi 2R m
+ is person i's consumption vector, p 2R m
++ is a price vector (faced by all
individuals who are alive)3 and yi is person i's income, wealth, or lifetime consumption. We
assume that each Ui is such that a solution to the maximization problem in (2) exists for all
p 2R m
++ and all yi 2R +. Furthermore, Ui is assumed to be monotonic which guarantees
that V i is increasing in yi.
Following standard practice in the population-ethics literature, we introduce the utility
level that represents neutrality, and call it u.4 If utility is above neutrality, life is worth
living, and if it is below, life is not worth living|a rational, fully informed, self-interested
person would prefer not to have his or her experiences. We assume that neutrality requires
a positive level of income for every price vector5 and that such a level of income exists. That
is, for each i 2Z ++ and each p 2R m
++, there exists si 2R ++ with
V i(p;s i)=u : (3)
Consequently,
V i(p;yi)<u  for all yi <s i (4)
because Ui is monotonic and, therefore, V i is increasing in yi.
Income-alternatives correspond to utility alternatives and we write
Ay =

N;

yi	
i2N

: (5)
Ay is the set of all possible income-alternatives. Price-independent welfarism requires that
there exist an ordering Ry of Ay such that

 N;

 yi	
i2 N

Ry

^ N;

^ yi	
i2 ^ N

()

 N;

Vi(p; yi)
	
i2 N

R

^ N;

Vi(p;^ yi)
	
i2 ^ N

(6)
for all  Ay, ^ Ay 2A yand all p 2R m
++. Because R satises strong Pareto, price independence
implies that Ry must satisfy the analogous condition applied to incomes.
3 See Slivinsky [1983] for a discussion of xed-population price independence in situations where dierent
individuals may face dierent prices.
4 See Broome [1993]. Neutrality is usually normalized to zero. For an introduction to population ethics,
see Blackorby, Bossert, and Donaldson [1995, 1997a,b], Blackorby and Donaldson [1984b], Bossert [1990],
Broome [1992], Hammond [1988], Heyd [1992], Hurka [1982, 1983], McMahan [1981], Narveson [1967], Part
[1976, 1982, 1984], and Sen [1991].
5 The alternative to our assumption would imply the implausible claim that any positive level of con-
sumption, no matter how small, would make life worth living.
2Lemma 1 shows that the ordering Ry satises extended homotheticity: common scaling
of the incomes in any two income-alternatives preserves their ranking. This is a straightfor-
ward generalization of Roberts' [1980a] corresponding xed-population result, stated in his
Proposition 3.
Lemma 1: Price-independent welfarism implies that Ry satises extended homotheticity.
That is,

 N;

 yi	
i2 N

Ry

^ N;

^ yi	
i2 ^ N

()

 N;

 yi	
i2  N

Ry

^ N;

^ yi	
i2^ N

(7)
for all  Ay, ^ Ay 2A yand all  2R ++.
Proof: For any  Ay, ^ Ay 2A y,2R ++,

 N;

 yi	
i2 N

Ry

^ N;

^ yi	
i2^ N

()

 N;

Vi(p; yi)
	
i2 N

R

^ N;

Vi(p;^ yi)
	
i2 ^ N

()

 N;

Vi(p; yi)
	
i2 N

R

^ N;

Vi(p;^ yi)
	
i2^ N

()

 N;

 yi	
i2 N

Ry

^ N;

^ yi	
i2 ^ N

;
(8)
where the third line of (8) follows from homogeneity of degree zero of the V i and the second
and fourth lines follow from price-independent welfarism.
Suppose that a single individual is to be added to an alternative without aecting the
utilitiesof any of the existingpeople. A critical levelof utilityfor alternativeA and individual
j= 2Nis the utility level which, if experienced by j, would make the two alternatives equally
good. If the critical level is uc,t h e n

N[f jg ;
 
u i	
i 2 N;u c

IA : (9)
If uc exists, it must be unique because of strong Pareto, and it may depend on any or all of
the identities of the existing people (N), the utility levels of the existing people (fui	
i2N),
and the identity of the added person (j).6
If a person-j critical level of utility exists for A, then price independence implies

N [f jg ;
 
y i	
i 2 N;y c

I yA y; (10)
6 Except for Theorem 2, strong Pareto is only used to guarantee the uniqueness of critical levels.
3where V i(p;yi)=u ifor all i 2 N,a n dV j ( p ;y c)=u cfor all p 2R m
++. yc is a person-j
critical level of income for the income-alternative Ay. Because the reverse implication is also
true, a critical level of utility exists for A and j if and only if a critical level of income exists
for Ay and j. Both critical levels must be unique because of strong Pareto. We write the
person-j critical level of income for Ay as
yc = CN
j
 
fyigi2N

: (11)
Lemma 1 implies that

N[fjg;

yi	
i2N;y c

I y

N;

yi	
i2N

()

N [f jg ;
 
yi	
i2N;y c

I y

N;

yi	
i2N
 (12)
for all  2R ++. This implies that if the person-j critical level of income exists for an
income-alternative, it exists when incomes are scaled; in addition, the critical level is scaled
by the same factor. This observation is sucient for Corollary 1.
Corollary 1: If price-independent welfarism is satised and the person-j critical level of
income CN
j
 
fyigi2N

exists for Ay =
 
N;
 
fyi	
i2N

and for j= 2N , then, for all  2R ++,
the person-j critical level exists for the alternative
 
N;
 
fyi	
i2N

,a n d
C N
j
 
f yigi2N

= CN
j
 
fyigi2N

: (13)
If all critical levels exist, the functions fCN
j g are dened for all fyigi2N, and are homo-
geneous of degree one. Theorem 1 demonstrates that, in this case, there must be critical
levels of utility that are below neutrality.
Theorem 1: If price-independent welfarism holds and all critical levels exist, there exist
critical levels of utility that are below neutrality.
Proof: For any alternative Ay 2A y ,l e ty c=C N
j
 
f y i g i 2 N

be the person-j critical level,
where j= 2N . Now consider the alternative A
y =
 
N;
 
fyi	
i2N

. Its critical level is yc
by Corollary 1. Consequently, for any p 2R m
++,t h ep e r s o n - jcritical level of utility for the
alternative A =
 
N;
 
fV i(p;yi)
	
i2N

must be V j(p;y c).  c a nb ec h o s e ns ot h a tyc
is arbitrarily close to zero, with
V j(p;y c)<u ; (14)
and the critical level is below neutrality.
4An example of a price-independent welfare prescription is given by the utility functions
ui = V i(p;yi)=a ( p ) y ifor all i 2Z ++, and a principle that requires
 AR ^ A ()
X
i2  N
 u i 
X
i2 ^ N
^ u i; (15)
so that
 Ay Ry ^ Ay ()
X
i2  N
 y i 
X
i2 ^ N
^ y i: (16)
This principle is not the same as classical utilitarianism, which uses the value function of
(15) only when utilities are normalized so that zero represents neutrality. In this example,
all critical levels of income and utility are zero and neutrality is above zero. Consequently,
all critical levels are below neutrality.
Another example is provided by
 AR ^ A ()
1
j  N j
X
i2  N
 u i 
1
j ^ N j
X
i2 ^ N
^ u i (17)
and by the utility functions used in the rst example. This is the average-utilitarian principle
and
 Ay Ry ^ Ay ()
1
j  N j
X
i2  N
 y i 
1
j ^ N j
X
i2 ^ N
^ y i: (18)
In this example, critical levels are average utility and average income. Consequently, for low
enough levels of average utility, critical levels of utility are below neutrality.
Although critical levels exist for all commonly used population principles, it is possible
that critical levels may not exist for some or all alternatives and added individuals. Given
strong Pareto, three possibilities exist. Dening
A+ =

N [f jg ;
 
u i	
i 2 N;u j

; (19)
the rst case has A+ PAfor all uj in the image of Uj|the expanded population is always
regarded as better. This is the strong pro-natalist position and it favours the creation of
people whose lives would be below neutrality. The second case is strongly anti-natalist, with
APA + for all possible uj. The third case is slightly more complex. It allows A+PAfor
some values of uj and APA + for the rest. Because of strong Pareto, all values of uj for
which A+PAmust be greater than all the values for which APA +.
The rst is clearly unsatisfactory: it regards the ceteris paribus addition of people below
neutrality as good. The second regards population expansion as bad no matter how well o
t h ea d d e dp e r s o ni s .
5Price-independent welfarism has implications even when critical levels do not exist. The-
orem 2 indicates that price-independent welfare prescriptions must either be strongly anti-
natalist or must regard the addition of some person below neutrality to a utility-unaected
population as a social improvement.
Theorem 2: If price-independent welfarism holds, then either
(i) there exist an alternative A 2A , an individual j= 2N , and a utility level uj such that
uj is below neutrality (uj <u )and

N [f jg ;
 
u i	
i 2 N;u j

PA (20)
or
(ii) for all A 2A ,a l lj= 2N , and all uj,
AP

N[f jg ;
 
u i	
i 2 N;u j

: (21)
Proof: Clearly, (i) and (ii) are mutually exclusive. Suppose that (ii) does not hold. Then
there exist A 2A ,j= 2N ,p2R m
++, fyigi2N,  yj 2R +such that

N [f jg ;
 
Vi( p ;yi)
	
i2N;Vj(p; yj)

R

N;

V i(p;yi)
	
i2N

: (22)
By monotonicity and strong Pareto,

N [f jg ;
 
Vi( p ;yi)
	
i2N;Vj(p;yj)

P

N;

V i(p;yi)
	
i2N

(23)
for all yj >  yj.L e t y jbe any income level with that property. Consequently, by price-
independent welfarism,

N [f jg ;
 
y i	
i 2 N;yj

P y

N;

yi	
i2N

: (24)
By extended homotheticity (see Lemma 1),

N [f jg ;
 
yi	
i2N;yj

P y

N;

yi	
i2N

(25)
for all  2R ++. Using price-independent welfarism again,

N [f jg ;
 
Vi( p ;yi)
	
i2N;Vj(p;yj)

P

N;

V i(p;yi)
	
i2N

(26)
6for all p 2R m
++ and all  2R ++.  can be chosen to be small enough so that
V j(p;yj)<u ; (27)
and the theorem is proved.
The results of our theorems lead us to conclude that sensible price-independent welfare
prescriptions are not possible when population size and composition can change across al-
ternatives. If critical levels exist, some of them must be below neutrality and the principles
must judge the ceteris paribus addition of at least some people whose lives are not worth
living to be a good thing. If critical levels do not exist, either the principle must possess
the same ethically unsatisfactory property or all additions to a utility-unaected population
must be regarded as bad.
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8

1980a, \Price-Independent Welfare Prescriptions,&quot;

1980b, Possibility theorems with interpersonally comparable welfare levels,

1980c, Interpersonal comparabilityand social choice theory,

Average utilitarianism,

Birth-date dependent population ethics: critical-level principles,

Consequentialist demographic norms and parenting rights,

Consumers' surpluses and consistent cost-bene tests,

Counting the Cost of Global Warming,

Equity and the informational basis of social choice,

Future generations, further problems,

Genethics: Moral Issues in the Creation of People,

Goodness is reducible to betterness: the evil of death is the value of life,

Income distribution evaluation and the law of one price,

Information invariance in variablepopulation social-choice problems, Working Paper 9616,

Informational bases of alternative welfare approaches: aggregation and income distribution,

Intertemporal population ethics: criticallevel utilitarian principles,

On doing the best for our children, in Ethics

On intra- and interpersonal utilitycomparisons,

On weights and measures: informational constraints in social welfare analysis,

Problems of population theory,

Reasons and Persons,

Social choice with interpersonal utility comparisons: a diagrammatic introduction,

Price-Independent Welfare Precriptions and Population Size

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