Mixed risk aversion and preference for risk disaggregation: a story of moments

In a recent article entitled “Putting Risk in its Proper Place,” Eeckhoudt and Schlesinger (2006) established a theorem linking the sign of the n-th derivative of an agent’s utility function to her preferences among pairs of simple lotteries. We characterize these lotteries and show that, in a given pair, they only differ by their moments of order greater than or equal to n. When the n-th derivative of the utility function is positive (negative) and n is odd (even), the agent prefers a lottery with higher (lower) n + 2p-th moments for p belonging to the set of positive integers. This result links the preference for disaggregation of risks across states of nature to the complete structure of moments preferred by mixed risk averse agents. It can be viewed as a generalization of a proposition appearing in Ekern (1980) which focused only on the differences in the n-th moments.