ECONOMY

charles blackorby & walter bossert 421
ordering ranks utility vectors which have the same sum. However, it shows that
there do not exist many orderings other than utilitarianism and leximin satisfying
the axioms of the theorem statement. The following two theorems illustrate how
utilitarianism or leximin can be obtained by modifying one or another of the axioms.
We begin with utilitarianism, which has received a considerable amount of attention
in the literature on social choice. The following theorem is due to Maskin (1978;
see also Deschamps and Gevers 1978). It is obtained by replacing minimal equity with
continuity in Theorem 6.
Theorem 7: Suppose there are at least three individuals. A social evaluation
ordering satisfies anonymity, strong Pareto, continuity, independence of the utilities
of unconcerned individuals, and information invariance with respect to cardinal full
comparability if and only if it is utilitarian.
Theorem 7 illustrates the power of the continuity axiom. Leximin clearly is not
continuous and neither is leximax. Consequently, neither of these principles survives
if this property is added. Because minimal equity is required in Theorem 6 only
to exclude leximax, it is no longer needed in Theorem 7 because this principle is
already ruled out by continuity. Continuity also forces equal goodness according to
utilitarianism to be respected if the corresponding betterness relation is respected
and, therefore, utilitarianism rather than the weakly utilitarian rules are obtained.
D’Aspremont and Gevers (1977) provide an alternative characterization of utilitarianism
that does not require the independence axiom (see also Blackwell and Girshick
1954;Milnor1954;andRoberts1980b).
A characterization of the leximin ordering due to d’Aspremont and Gevers (1977)
replaces information invariance with respect to cardinal full comparability by information
invariance with respect to ordinal full comparability in Theorem 6.
Theorem 8: Suppose there are at least three individuals. A social evaluation ordering
satisfies anonymity, strong Pareto, minimal equity, independence of the utilities of
unconcerned individuals, and information invariance with respect to ordinal full
comparability if and only if it is leximin.
Clearly, ordinal full comparability rules out all weakly utilitarian principles (including
utilitarianism itself) and, as a consequence, Theorem 8 follows immediately from
Theorem 6.
Hammond (1976) provides an alternative characterization of leximin. Because it
does not employ an information invariance condition, we do not state it here.
6 Concluding Remarks
.............................................................................
This chapter provides a brief introduction to welfarist social choice theory. It is argued
that the most promising route of escape from the negative conclusion of Arrow’s