128 OHN W. PRATT premiums for the same probability distribution of risk but for two different utilities This does not mean that when Theorem 1 is applied to two decision makers, they must have the same personal probability distributions, but only that the notation is imprecise. The theorem could be stated in terms of T (x, i,)and I2(x, i2) where the distribution assigned to i, by the first decision maker is the same as that assigned to i2 by the second decision maker. This would be less misleading, but also less onvenient and less suggestive, especially for later use. More precise notation would be, for instance, I,(x, F)and I2(, F), where Fis a cumulative distribution nction THEOREM 1: Let r(x), I(x, 2), and(x) be the local risk aversion, risk premium, ponding to the utility function ui,i=1, 2 following conditions are equivalent, in either the strong form(indicated in brackets) or the weak form(with the bracketed material omitted) (a) rI(x)2r2(x) for all x [and >for at least one x in every intervall (b) T,(x, 2)2[>]I2(x, 2) for all x and i (c) PI(r, h)2[>lp2(x, h) for all x and all h>0 (d) u1u2(t))is a [strictly] concave function of n4ts[<23()-n2(x) (e)l1(y)-1(x) u2(w)-u,vor anD, w,x, y with v<wsx<y The same equivalences hold if attention is restricted throughout to an interval, that is, f the requirement is added that x, x+i, x+h, x-h, u2(O), D, w, and y, all lie in a PROOF: We shall prove things in an order indicating somewhat how one might discover that(a) implies(b)and(c) To show that(b)follows from(d), solve(1)to obtain (13)xx,2)=x+E()-u2(E{u(x+2)}) (14)x1(x,2)-x2(x,2)=H2(E{u2(x+2)})-11(E{u1(x+2(}) where i=u2(x+2). If u(u2(t) is [strictly] concave, then(by Jensen's inequality) (15)E{u1(u2(D)}≤[<]u1(u21(E{}) Substituting(15)in(14), we obtain(b)