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Proper Risk Aversion

John W. Pratt
Richard J. Zeckhauser

Abstract

We introduce and investigate a significant behavioral condition on utility functions for wealth, which we call proper risk aversion, namely that an undesirable lottery can never be made desirable by the presence of an independent, undesirable lottery. (Independent random background wealth is allowed.) One consequence is that offering insurance and other forms of hedging to proper investors can only encourage them to accept other, independent risks. Properness implies decreasing risk aversion and, much less immediately, several stronger intuitive conditions on certainty equivalents and risk premiums. We prove properness for all mixtures of risk-averse exponential functions and hence (by complete monotonicity) of all risk-averse power and logarithmic functions. We derive analystical necessary and sufficient conditions, which seem to be unavoidably complicated, and local necessary conditions, alas not sufficient, which are tractable.