Revealed Preference and Nonparametric Analysis – Continuous Extensions and Recoverability

This paper shows how revealed preference relations, observed under general budget sets, can be extended using closure operators which impose certain assumptions on preferences. Common extensions are based on the assumption that preferences are convex and/or monotonic, but we also consider satiated single-peaked preferences. For the obtained extended relations, the paper provides necessary and sufficient conditions for the existence of continuous complete extensions of the revealed preference relation. These results lead to a nonparametric analysis of revealed preference data which allows to recover all that can be said about a decision maker's preferences. The approach makes explicit what additional assumptions are imposed on the revealed preference relation. For example, Varian's (1982) revealed preferred set imposes monotonicity and convexity. The approach focuses strictly on what is observable. In particular, it does not assume that we observe all bundles on a budget among which the decision maker is indifferent.