First-order risk aversion and non-differentiability

Labora tory experiments have repeatedly shown that decision makers do not satisfy the expected utility hypothesis. The evidence did not discourage the use of expected utility, partly because it was a useful paradigm and partly due to the lack of convenient alternatives. Furthermore, it was not clear that the results obtained by expected utility theory do not hold for more general models. Indeed, Machina [7] argues that given a Fr6chet differentiability condition, nonexpected utility functionals can be locally approximated by expected utility. Hence, many results of expected utility may be extended to nonexpected utility, especially comparative statics analysis, where local changes are examined (see Machina [-8] and Chew and Nishimura [,-1]). Expected utility analysis of decision making under risk strongly depends on the differentiability of the utility function. For example, consider a disturbance g whose expected value is zero. Multiply its outcomes by a positive number t and let t tend to zero. The risk premium ~(t) that a risk averse decision maker is willing to pay out of his present wealth level x* to avoid the risk t .g also declines to zero. For smooth expected utility, Prat t [,10] showed that 7c(t)= [-u"(x*)/u'(x*)]a2t2+ o(t 2) which tends to zero at the rate t 2.