JoHN W. PRATT 6. CONSTANT RISK AVERSION If the local risk aversion function is constant, say r(x)=c, then by (12) (25)u(x)~eifr(x)=c<0 These utilities are, respectively, linear, strictly concave, and strictly convex. If the risk aversion is constant locally, then it is also constant globally, that a change in assets makes no change in preference among risks. In fact, for any k u(k +x) u(x) in each of the cases above, as is easily verified Therefore it makes sense to speak of"constant risk aversion"without the qualification""or Similar remarks apply to constant risk aversion on an interval, except that global consideration must be restricted to assets x and risks i such that x +2 is certain to stay within the interva 7. INCREASING AND DECREASING RISK AVERSION Consider a decision maker who(i)attaches a positive risk premium to any risk, but(i)attaches a smaller risk premium to any given risk the greater his assets Formally this means (1 x ane (i i) I(x, i) is a strictly decreasing function of x for all Restricting i to be actuarially neutral would not affect or (ii), by (2) with u=e(z) We shall call a utility function(or a decision maker possessing it) risk-averse if the weak form of ( i) holds, that is, if T(x, 2)20 for all x and 2; it is well known that this is equivalent to concavity ofu, and hence tou"soand to r20. A utility function is strictly risk-averse if (i) holds as stated; this is equivalent to strict concavity of u and hence to the existence in every interval of at least one point where u"<o, r>0 We turn now to(i). Notice that it amounts to a definition of strictly decreasing risk aversion in a global(as opposed to local)sense. On would hope that decreasing global risk aversion would be equivalent to decreasing local risk aversion r(x) The following theorem asserts that this is indeed so. Therefore it makes sense to speak of“ decreasing risk aversion” without the qualification" local”or“"gobl:” What is nontrivial is that r(x) decreasing implies I(x, 2)decreasing, inasmuch as r(x) pertains directly only to infinitesimal gambles. Similar considerations apply to the probability premium p(x, h) THEOREM 2: The following conditions are equivalent (a)The local risk aversion function r(x)is [strictly] decreasing