On the Recoverability of Preferences and Beliefs in Financial Models

We examine the extent to which an investor's preferences and beliefs are uniquely determined from knowledge of the equilibrium prices and of his/her consumption strategy. More precisely, we assume that the investor's preferences admit an expected utility representation, but with subjective probabilities, and investigate what joint restrictions can be placed on utility functions and beliefs. If the investor has preferences for wealth or consumption at a single future date, then the problem is indeterminate. In fact, for any given ''well behaved'' assets' price dynamics, utility function and consumption choice, we can construct investor's beliefs that would support the given consumption choice. On the other hand, if the investor draws utility from intertemporal consumption, we show that the set of utility functions and beliefs that are consistent with a given price and consumption process can be characterized by a martingale condition. In the Markovian case, this characterization can be reexpressed in terms of a partial 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 set of beliefs. Some general implications of time-homogeneous price and consumption processes are discussed.