Essential Data, Budget Sets and Rationalization

According to a minimalist version of Afriat's theorem, a consumer behaves as a utility maximizer if and only if a feasibility matrix associated with his choices is cyclically consistent. An essential experiment consists of observed consumption bundles (x_1,..., x_n) and a feasibility matrix \alpha. Starting with a standard experiment, in which the economist has access to precise budget sets, we show that the necessary and sufficient condition for the existence of a utility function rationalizing the experiment, namely, the cyclical consistency of the associated feasibility matrix, is equivalent to the existence, for any budget sets compatible with the deduced essential experiment, of a utility function rationalizing them (and typically depending on them). In other words, the conclusion of the standard rationalizability test, in which the economist takes budget sets for granted, does not depend on the full specification of the underlying budget sets but only on the essential data that these budget sets generate. Starting with an essential experiment (x_1,..., x_n; alpha) only, we show that the cyclical consistency of alpha, together with a further consistency condition involving both (x_1,..., x_n) and alpha, guarantees the existence of a budget representation and that the essential experiment is rationalizable almost robustly, in the sense that there exists a single utility function which rationalizes at once almost all budget sets which are compatible with (x_1,..., x_n; alpha). The conditions are also trivially necessary.