ECONOMY

418 interpersonal comparisons of well-being
If utilities are cardinally measurable and no requirements regarding the interpersonal
comparison of utilities are imposed, we obtain the information assumption
cardinal non-comparability.
Information invariance with respect to cardinal non-comparability: For all utility vectors
u and v, for all positive constants a 1 ,...,a n and for all constants b 1 ,...,b n ,
u is at least as good as v if and only if (a 1 u 1 + b 1 ,...,a n u n + b n ) is at least as good
as (a 1 v 1 + b 1 ,...,a n v n + b n ).
Information invariance with respect to ordinal non-comparability and information
invariance with respect to cardinal non-comparability are equivalent (see Sen 1970
and, for a diagrammatic illustration, Blackorby, Donaldson, and Weymark 1984).
Intuitively, this is the case because, for any two utility vectors u and v and for any
vector of increasing transformations, the values of the transformations can be replicated
by the values of a vector of increasing affine transformations because a straight
line is determined by two points on that line. As an immediate consequence of this
equivalence, it follows that, among the social evaluation orderings of the previous
section, strong dictatorships are the only ones satisfying information invariance with
respect to cardinal non-comparability.
If utility levels are comparable both intrapersonally and interpersonally but no
further information is available, we obtain ordinal full comparability. In that case,
only common increasing transformations can be applied to the utilities without
changing the information relevant for social evaluation.
Information invariance with respect to ordinal full comparability: For all utility
vectors u and v and for all increasing functions 0 , u is at least as good as v if
and only if ( 0 (u 1 ),..., 0 (u n )) is at least as good as ( 0 (v 1 ),..., 0 (v n )).
Ordinal full comparability implies that utility levels can be compared interpersonally.
The inequality u i ≥ v j is preserved even for different individuals i and j if
a common transformation 0 is applied to all utility values: we have u i ≥ v j if
and only if 0 (u i ) ≥ 0 (v i ) for all increasing functions 0 . Comparisons of utility
differences are not possible in this informational environment, either intrapersonally
or interpersonally. For example, suppose that u i =2,v i =0,w i =3,and
t i =2.Wehaveu i − v i =2> 1=w i − t i . However, this inequality is not preserved
if an arbitrary increasing transformation is applied to the individual utilities. Letting
i (u i )=ui 3, it follows that i (u i ) − i (v i )=8< 19 = i (w i ) − i (t i )and
the inequality is reversed. Information invariance with respect to ordinal full comparability
is satisfied by the strongly dictatorial social evaluation orderings, the
strong positional dictatorships, and leximin. Utilitarianism does not satisfy this
invariance property.
If utilities are cardinally measurable and fully interpersonally comparable, both
utility levels and utility differences can be compared interpersonally. In this case,
the only admissible transformations are increasing affine transformations which are
identical across individuals.