Social States of Belief and the Determinant of the Equity Risk Premium in a Rational Belief Equilibrium

Abstract

We review the issues related to the formulation of endogenous uncertainty in Rational Belief Equilibria (RBE). In all previous models of RBE, individual states of belief were the foundation for the construction of the endogenous state space where individual states of belief were described with the method of assessment variables. This approach leads to a lack of “anonymity” where the belief of each individual agent has an impact on equilibrium prices but as a competitor he ignores it. The solution is to study a replica economy with a finite number of types but with a large number of agents of each type. This gives rise to “type-states” which are distributions of beliefs within each type. The state space for this economy is then constructed as the set of products of the exogenous states and the social states of belief which are vectors of distributions of all the types. Such an economy leads to RBE which do indeed solve the problem of anonymity. We then study via simulations the implications of the model of RBE with social states for market volatility and for the determinants of the equity risk premium in an RBE. Under i.i.d. assessments one uses the law of large numbers to induce a single social state of belief and we show that the RBE of such economies have the same number of prices as in rational expectations equilibrium (REE). However, the RBE may exhibit large fluctuations if agents are allowed to hold extreme beliefs. Establishing 5% boundary restrictions on beliefs we show that the model with a single social state of belief cannot explain all the moments of the observed distribution of returns. We then introduce correlation among beliefs and this leads to the creation of new social states. We next show that under correlation among beliefs the model simulations reproduce the values of four key moments of the empirical distribution of returns. The observed equity premium is then explained by two factors. First, investors demand a higher risk premium to compensate them for the endogenous increase in the volatility of returns. Second, at any moment of time there are both rational optimists as well as rational pessimists in our financial markets and such a distribution leads automatically to a decrease in the riskless rate and to an increase of the risk premium. We show that correlation among beliefs of agents leads to fluctuations over time in the social distribution of beliefs and such fluctuations add to endogenous volatility and lead to a higher equilibrium equity risk premium.

This research was supported by Fondazione ENI Enrico Mattei, Milano, Italy. The author thanks Stanley Black, William Brock, Mark Garmaise, Kenneth Judd, Michael Magill, Maurizio Motolese, Carsten Nielsen, and Martine Quinzii for many helpful suggestions regarding the research reported in this paper. He is grateful to Stanley Black and Maurizio Motolese for dedicated assistance in carrying out the computations. This paper is a drastic revision of [26].