On the Economic Meaning of Machina's Frâ„chet Differentiability Assumption

This note shows that Machina's (1982) assumption that preferences over lotteries are smooth has some economic implications. We show that Frâ„chet differentiability implies that preferences represent second order risk aversion (as well as conditional second order risk aversion). This implies, among other things, that decision makers buy full insurance only at the absence of marginal loading. We also show that with constant absolute and relative risk aversion, expected value maximization, second order risk aversion, and Frâ„chet differentiability are equivalent.