On the Recoverability of Preferences and Beliefs

We examine the extent to which an investor's tastes and beliefs can be jointly recovered from knowledge of his/her consumption choice. More precisely, we assume that the investor's preferences admit an expected utility representation, but with subjective (unknown) probabilities, and investigate what joint restrictions can be placed on utility functions and beliefs. If the investor draws utility from intertemporal consumption, we show that the set of utility functions and beliefs that are consistent with a given consumption choice can be characterized by a martingale condition. In the Markovian case, this characterization can be restated in terms of a Riccati differential equation that must be satisfied by the investor's relative risk aversion function. To each solution of this differential equation is associated a unique utility function and a unique set of beliefs supporting the given consumption choice. Moreover, we show that the differential equation has at most one solution in the class of utility functions displaying infinite absolute risk aversion at the origin. Thus, preferences (and associated beliefs) can be uniquely recovered within this class