Abstract
The more risk-averse of two individuals need not have the smaller certainty equivalent for a risk \~x if another risk or combination of risks w is present. It is shown that he must, however, if either individual's conditional certainty equivalent for x is increasing in w. For independent risks, this condition follows immediately if either individual is decreasingly risk-averse, giving a natural proof of a known result. Another short proof of this result and necessary and sufficient conditions in the independent case are give. For multivariate utilities, the corresponding results do not hold, but it is proved simply that any mixture of decreasingly risk-averse utilities is decreasingly risk-averse. Also touched upon are risk aversion's relation to generalized means, concave composition, risk sharing, and interest rates, the application of the results to discounting under uncertainty and selection of investment level, and their connection to singly crossing distributions, noise, and dominance.
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Pratt, J.W. Aversion to one risk in the presence of others. J Risk Uncertainty 1, 395–413 (1988). https://doi.org/10.1007/BF00117643
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DOI: https://doi.org/10.1007/BF00117643