Revealed Preference and the Number of Commodities

This work consists of two parts: First, it is shown that for a two-dimensional commodity space any homothetic utility function that rationalizes each pair of observations in a set of consumption data also rationalizes the entire set of observations. The result is stated as a pairwise version of Varian's Homothetic Axiom of Revealed Preference and is used to provide a simplified nonparametric test of homotheticity. In the second part a unifying proof technique is presented to show that the Weak Axiom of Revealed Preference (WARP) implies the Strong Axiom of Revealed Preference (SARP) for two commodities yet not for more commodities. It also shows that preference cycles can be of arbitrary length.While these results are already known, the proof here generalizes and unifies the existing ones insofar as it gives necessary and sufficient conditions for preference cycles to exist. It is then shown that in two dimensions the necessary condition cannot be fulfilled, whereas in more than two dimensions the sufficient conditions can always be met. The proof admits an intuitive understanding of the reason by giving a geometric interpretion of preference cycles as paths on indifference surfaces.