Revealed Preference with Stochastic Demand Correspondence

We unify and expand the theory of consumer’s behavior, based on Samuelson’s Weak Axiom of Revealed Preference, to permit simultaneously both random choice and non-singleton choice sets. We provide a consistency postulate for demand behavior when such behavior is represented in terms of a stochastic demand correspondence. When the consumer spends her entire wealth, our rationality postulate is equivalent to a condition we term stochastic substitutability. This equivalence generates: (i) Samuelson’s Substitution Theorem, (ii) the central result in Bandyopadhyay, Dasgupta and Pattanaik (2004) and (iii) a version pertinent to deterministic demand correspondences (which independently yields Samuelson’s Substitution Theorem), as alternative special cases. Relevant versions of the non-positivity of the own substitution effect, the demand theorem and homogeneity of degree zero in prices and wealth for the consumer’s demand behavior, also follow as corollaries in every case.