STOCHASTIC

128 JOHN 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 jr^x.z,) and TT2(A:,Z2) where
the distribution assigned to z, by the first decision maker is the same as that assigned
to z2 by the second decision maker. This would be less misleading, but also less
convenient and less suggestive, especially for later use. More precise notation
would be, for instance, n^x.F) and it2(x,F), where Fis a cumulative distribution
function.
THEOREM 1: Let rt(x), nt(x,z), and p:(x) be the local risk aversion, risk premium,
and probability premium corresponding to the utility function u,, i=l,2. Then the
following conditions are equivalent, in either the strong form (indicated in brackets),
or the weak form (with the bracketed material omitted).
(a) r1(x)7ir2(x)for all x [and > for at least one x in every interval].
(b) n 1(x,z)^.[>]n2(x,z) for all x and z.
(c) pi(x,h)^[>]p2(x,h) for all x and all h>0.
(d) "i("j'(0) < s a [strictly] concave function oft.
,s "i(v)-Ui(x) „ r -,Ui(y) — u2(x) r „ ... „
00 —r^ i 7-c^[