Essentially lexicographic aggregation

Abstract

In the paper a new approach to lexicography is developed by which in the general framework of ordered blocks with a monotonic basis it is shown that a nontrivial ordering is translation-invariant if and only if it is essentially lexicographic of degree n. Here, the latter means that the ordering can be represented by an ordinary lexicographic ordering in n dimensions. As an application it is shown that a nontrivial social welfare ordering on Euclidean space possesses a useful invariance property (cardinality and non comparability) if and only if the ordering is essentially lexicographic of a strong kind in that it can be obtained from ordinary lexicography by permutation, cutting-off and order reversal with respect to components. This result generalizes the characterization of lexical individual dictatorship obtained by Gevers and d'Aspremont and it provides, within the social welfare approach, a strong version of Arrow's impossibility theorem by not invoking any Pareto principle at all.