Diverse beliefs

Abstract

Agents can differ in many ways, but differences in beliefs or information are perhaps the most interesting. In this paper, we adopt a general diverse beliefs framework that makes such models relatively easy to analyse. Our first result proves that a finite-horizon discrete-time dynamic stochastic general equilibrium (DSGE) where agents have diverse information is observationally equivalent to the one where the agents have diverse beliefs. This is important because diverse beliefs models are quite easy to study, whereas diverse information models are not. The solution to a continuous-time central-planner equilibrium problem is easy to characterize in the framework adopted, and we develop various properties of the solution, including expressions for equilibrium interest rates and stock price dynamics for an economy of constant relative risk aversion (CRRA) agents, and an expression for the volume of trade.

Keywords:

• diverse beliefs
• private information
• probability
• equilibrium
• beauty contest
• mistaken beliefs

Acknowledgements

We thank seminar participants at the Cambridge-Wharton meeting, June 2009, particularly our discussant.

Notes

^1. A.A. [email protected]

^2. This specific set-up is not essential to the argument presented, but formulating a very general statement would be clumsy and obscure the main ideas.

^3. The restriction to a single asset is for notational convenience only; the entire analysis works also for multi-assets situations.

^4. If agent j has probability measure , we could use the average of as a reference measure.

^5. This data set can be downloaded from http://www.econ.yale.edu/∼ shiller/data.htm.

^6. In practice, the potential minor notational clash with (3.2) will be avoided by using roman letters exclusively in the single-agent context, and greek letters exclusively in the central-planner context.

^7. Thus under the measure the process X becomes a Brownian motion with drift (by the Cameron–Martin–Girsanov Theorem; see [36], IV.38 for an account).

^8. Compare with Kurz et al. [28].

^9. In the case where all the agents have the same belief, we have that

hence there is no volatility in the evolution of .

^10. In the context of a finite-horizon Lucas tree model…

^11. And (in the log case) the stock price and the volatility of the stock price…

^12. An interesting extension of the log agent is Ref. [5], where agents are supposed to be CRRA with an integer coefficient of relative risk aversion.

^13. These empirical values are calculated by Kurz and are based on the Shiller data set. They are based on monthly data from the S&P 500 between 1871 and 1998. See Refs [27,28] for further details.