ECONOMY

412 interpersonal comparisons of well-being
common to restrict the individual utilities by assuming, for instance, that consuming
more of all goods is always better than consuming less.
We assume that a social ranking is an ordering—a reflexive, transitive, and complete
social goodness relation. A social evaluation functional is welfarist if and only
if there exists a social evaluation ordering (referred to as a social welfare ordering
by Gevers 1979) on the set of all possible utility vectors such that, for any profile
U =(U 1 ,...,U n ) and for any two alternatives x and y, x is socially at least as
good as y for the profile U if and only if the utility vector u =(u 1 ,...,u n )=
(U 1 (x),...,U n (x)) = U (x) is at least as good as the utility vector v =(v 1 ,...,v n )=
(U 1 (y),...,U n (y)) = U (y) according to the social evaluation ordering. Welfarism is
a consequence of three axioms: unlimited domain, Pareto indifference, and binary
independence of irrelevant alternatives. We now provide formal definitions of these
axioms.
Unlimited domain requires the social evaluation functional to produce a social
ordering for every logically possible utility profile. That is, no individual utility function
and no combination of such functions is excluded as a possible way of assigning
individual well-being to the alternatives.
Unlimited domain: The social evaluation functional generates a social ordering for
every logically possible utility profile.
Pareto indifference demands that if, according to a utility profile U , the individual
utilities for two alternatives x and y are the same, then x and y must be equally good
according to the social ranking generated by U .
Pareto indifference: For all alternatives x and y and for all profiles U ,ifU (x) =
U (y), then x and y are equally good for the profile U .
Binary independence of irrelevant alternatives is a consistency condition that imposes
restrictions across different profiles. If the utilities for two alternatives x and y
are the same in two profiles U and V, then the social rankings of x and y resulting
from the two profiles should be the same.
Binary independence of irrelevant alternatives: For all alternatives x and y and for
all profiles U and V, ifU (x) =V(x) andU (y) =V(y), then the ranking of x and
y for the profile U is the same as the ranking of x and y for the profile V.
For a social evaluation functional that satisfies unlimited domain, Pareto indifference
and binary independence of irrelevant alternatives together are equivalent
to welfarism. This result, which is implicit in d’Aspremont and Gevers (1977) and
explicit in Hammond (1979), is referred to as the welfarism theorem. It requires our
maintained assumption that there are at least three alternatives.
Theorem 1: Suppose a social evaluation functional satisfies unlimited domain. The
social evaluation functional satisfies Pareto indifference and binary independence of
irrelevant alternatives if and only if there exists a social evaluation ordering of all
utility vectors such that, for all alternatives x and y and for all profiles U , x is at least