New Economist discusses Robert Barro's latest contribution to the equity premium literature over here, and Brad DeLong weighs in as well. As in Marty Weiztman's recent paper "A Unified Bayesian Theory of Equity 'Puzzles'" (pdf) (discussed by Brad DeLong here), the main result rests on a returns density with fat tails - that is, one that assigns larger probabilities to extreme events than what the usual Gaussian 'thin-tailed' distribution would predict. Barro's model (building on Reitz's (1998) article) generates this fat-tailed density by supposing - as in Reitz (1988) - that there is a non-negligible probability of a catastrophic event. In Weitzman's model, agents are continually learning about their environment, and this extra uncertainty generates predictive densities that are fat-tailed, even if the world is Gaussian.

I'm sceptical of these approaches. In the standard expected-utility framework (Arrow, 1971; Lucas, 1978), utility is *bounded* - as far as I know, no-one has yet extended expected utility analysis to the case where utility is unbounded. The way I see it, the best way to interpret the unbounded utility functions and the densities defined over the entire real line that we use in applied work is that they are convenient approximations to what the theory requires. But if the approximation is to be a good one, then we need to use distributions such that the model's expected utility over the entire real line is a good approximation for the true expectation over [u_{0} , u_{1}]. In other words, the contributions of the tails in the calculation of expected utility should be 'small.'

Since utility is non-decreasing in consumption, the only way that we can be certain that the tails' contribution will be small is if the density for utility goes to zero fairly quickly outside [u_{0} , u_{1}]. In this context, it would seem to me that thin-tailed distributions would be preferred to those with fat tails: for a given interval [u_{0} , u_{1}]. , as the tails get fatter, the distortions generated by using unbounded utility functions and by using densities that are strictly positive for all realisations of utility will get larger and larger.

Since expected utility theory hasn't yet been extended to the case where utility functions are unbounded, it's not clear just how we can interpret results that are based on the behaviour of distributions out in the tails, or how robust these results will be to how we specify marginal utility at u_{0.}

Of course, my judgment might be clouded by the fact that my co-author Pascal St-Amour and I have a couple of articles where we present a competing explanation of the equity premium puzzle, one based on state-dependencies in preferences toward risk:

Gordon, Stephen and Pascal St-Amour, "A preference regime model of bull and bear markets'', *American Economic Review*, **90** (4), (2000), pp 1019-1033.

________, "Asset returns and state-dependent risk preferences'', *Journal of Business and Economic Statistics*, **22** (3), (2004), pp 241-252.

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