ECONOMY

414 interpersonal comparisons of well-being
with anonymity, weak Pareto also implies that if everyone is better off in a permutation
of a utility vector u than in v, thenu is better than v even if u and v
cannot be compared according to the weak Pareto principle alone. For example, let
u =(−2, 1, 4) and v =(−1, −3, 0). Weak Pareto alone is silent about the ranking of
u and v. However, anonymity implies that u and (1, −2, 4) are equally good and, as
mentioned earlier, (1, −2, 4) is better than v by weak Pareto. By transitivity, u must
be better than v. (SeealsoSuppes1966 for a discussion.)
The strong Pareto requirement extends weak Pareto to cases in which no one’s
utility decreases and at least one individual’s well-being increases.
Strong Pareto: For all utility vectors u and v, ifu i ≥ v i for all i with at least one
strict inequality, then u is better than v.
To illustrate the difference between strong Pareto and weak Pareto, consider the
two utility vectors u =(2, −2, 0) and v =(2, −3, 0). Strong Pareto requires that u is
better than v but weak Pareto does not.
Continuity is a condition that prevents the goodness relation from exhibiting
“large” changes in response to “small” changes in the utility distribution.
Continuity: For all utility vectors u and v and for all sequences of utility vectors
〈u j 〉 j =1, 2,... where u j =(u j 1 ,...,u j n) for all j ,
(a) if the sequence〈u j 〉 j =1, 2,... approaches v and u j is at least as good as u for all
j ,thenv is at least as good as u;
(b) if the sequence〈u j 〉 j =1, 2,... approaches v and u is at least as good as u j for all
j ,thenu is at least as good as v.
To illustrate the continuity axiom, consider the following example. Let u =
(1, −1, −1) and v =(0, 0, 0). Suppose that a sequence of utility vectors 〈u j 〉 j =1, 2,... is
given by u j =(0, 0, 1/j ) for all j and that u is at least as good as u j for all j . Because
the sequence 〈u j 〉 j =1, 2,... approaches v, continuityrequiresthatu is at least as good
as v.
The next axiom is an equity requirement; see d’Aspremont and Gevers (1977) and
Deschamps and Gevers (1978). It is called minimal equity and, loosely speaking, it
requires that the social ordering does not always favor inequality. More precisely,
the axiom requires the following. Consider two utility vectors u and v such that the
utilities of all but two individuals i and j are the same in u and in v and the utilities of
i and j are closer together in u than in v in the sense that v j > u j > u i >v i ;thatis,
in moving from v to u, thebetter-off individual j loses and the worse-off individual
i gains, without reversing their relative ranks. The pair given by u =(3, 4, 7) and
v =(3, 2, 12) is an example for such a situation. Individual 1 has the same utility in
both, individual 2 is worse off than individual 3 in both vectors, individual 2 gains and
individual 3 loses when moving from v to u. An extremely equality-averse ordering
would always declare v better than u in these circumstances, and the minimal equity
axiom requires that this is not the case: there must be at least one pair of utility vectors
u and v with these properties such that u is at least as good as v.