Revealed Preference and Stochastic Demand Correspondence: A Unified Theory

We expand the theory of consumer’s behavior, based on the Weak Axiom of Revealed Preference, to permit simultaneously both random choice and choice sets, which are not singletons. We provide a consistency postulate for demand behavior when such behavior is represented in terms of a stochastic demand correspondence. We show that, when the consumer spends her entire wealth, our rationality postulate is equivalent to a condition we term stochastic substitutability. This equivalence generates the following results as special cases: (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 and the demand theorem also follow as corollaries in each case.