Intertemporal asset pricing under knightian uncertainty

Abstract

Modern asset pricing theory typically adopts strong assumptions about agents’ beliefs.According to the rational expectations hypothesis, for example, there exists an objective probability law describing the state process, and it is assumed that agents know this probability law precisely. More generally, even if existence of the latter is not assumed, each agent’s beliefs about the likelihoods of future states of the world are represented by a subjective probability measure or Bayesian prior, in conformity with the Bayesian model of decision-making and, more particularly, with the Savage (1954) axioms. As a result, no meaningful distinction is allowed between risk, where probabilities are available to guide choice, and uncertainty, where information is too imprecise to be summarized adequately by probabilities. In contrast, Knight (1921) emphasized the distinction between risk and uncertainty and argued that uncertainty is more common in economic decision-making.1 Particularly, in the context of asset prices, Keynes emphasized the importance of “animal spirits” when, because of Knightian uncertainty, individuals cannot estimate probabilities reliably and so cannot make a good calculation of expected values. (SeeKeynes (1936) and (1921: Ch. 6); see alsoKoppl (1991) for discussion and additional references.) This chapter provides a formal model of asset price determination in which Knightian uncertainty plays a role. Specifically, we extend the Lucas (1978) general equilibrium pure exchange economy by suitably generalizing the representation of beliefs. Two principal results are the proof of existence of equilibrium and the characterization of equilibrium prices by an “Euler inequality.” The latter represents the appropriate generalization of the standard Euler equation to the context of uncertainty. A noteworthy feature of our model is that uncertainty may lead to equilibria that are indeterminate; that is, there may exist a continuum of equilibria for given fundamentals. That leaves the determination of a particular equilibrium price process to “animal spirits” and sizable volatility may result.