On the Economic Meaning of Machina's Fréchet Differentiability Assumption

This note shows that Machina's (1982) assumption that preferences over lotteries are smooth has some economic implications. We show that Fr echet di erentiability 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 Fr echet di erentiability are equivalent. Journal of Economic Literature Classi cation Number: D81. We thank Edi Karni and especially Mark Machina for their comments. Uzi Segal thanks the Social Science and Humanities Research Council of Canada for nancial support.