REVEALED INDEPENDENCE AND QUASTILINEAR CHOICE

Abstract

Revealed Independence requires that revealed preferences satisfy the independence property from expected utility theory. This condition is essentially the same as the quasi-linearity of the choice function; i.e., it is a linear operator on convex combinations of budget sets. We demonstrate the equivalence of the Strong and Weak Axioms of Revealed Preference for quasi-linear choice functions with convex domains. IN GENERAL terms, revealed preference theory derives utility from demand, complementing subjective preference theory, which derives demand from utility. This reversal of procedure has profound operational significance, because it places neoclassical economics upon a more satisfactory behavioral foundation, where all axioms become testable hypotheses about observable variables. Indeed, the utility function is unobservable to the social scientist, but the typical demand function is at least partially observable, lending itself to statistical estimation. Samuelson [15] originally developed revealed preference theory in the context of commodity markets. He unsuccessfully sought to recover the utility function from his Weak Axiom of Revealed Preference (WARP) by integrating an appropriate system of partial differential equations derived from the Hicks-Slutsky substitution matrix. This integrability problem was eventually solved by Houthakker [7], who introduced the Strong Axiom of Revealed Preference (SARP) to replace WARP. Nevertheless, a social scientist still needs a constructive procedure that verifies whether or not a particular collection of choice data satisfies these behavioral axioms. In this regard, WARP would clearly provide a much simpler foundation for consumer demand theory, and Samuelson demonstrated its sufficiency in a world with only two commodities. Unfortunately, Gale [4] constructed an important counterexample, verifying that WARP is insufficient to recover the utility function from the Marshallian demand function in general. The exact characterization of WARP and SARP in terms of the Hicks-Slutsky substitution matrix thoroughly exposes the logical distinction between WARP and SARP in the context of the neoclassical model of the competitive consumer [9]. Yet this matter is not finished for other types of decision problems. We consider here Uzawa's extension of revealed preference theory to a logical setup that encompasses social choice theory [19]. The essential step is the generalization of the concept of a demand function to a choice function,