A Note on Heterogeneous Beliefs with CRRA Utilities

Abstract

This note will extend the research presented in Brown & Rogers (2009) to the case of CRRA agents. We consider the model outlined in that paper in which agents had diverse beliefs about the dividends produced by a risky asset. We now assume that the agents all have CRRA utility, with some integer coefficient of relative risk aversion. This is a generalisation of Brown & Rogers (2009) which considered logarithmic agents. We derive expressions for the state price density, riskless rate, stock price and wealths of the agents. This sheds light on the effects of risk aversion in an equilibrium with diverse beliefs. * Wilberforce Road, Cambridge CB3 0WB, UK (phone = +44 1223 337969 , email = A. A.Brown@statslab.cam.ac.uk)

Diverse Beliefs

“…The Figures 6,7,8,9,10,11, show the corresponding results for 50 agents, with similar qualitative features; notice particularly the dramatic crash when no agent is diligent! What we see in the simulations are qualitative features of bubbles and crashes, which one might also try to explain by models of herding, or behavioural effects.…”

“…Now consider the price that agent j is willing to pay at time s for a contingent claim which pays amount Y t at time t > s. Denote this price by π j s (Y t ) 8 . By considering the change in agent j's objective from buying this (marginal) contingent claim, the first order conditions give:…”

“…An interesting extension of the log agent is[8], where agents are supposed to be CRRA with an integer coefficient of relative risk aversion.…”