STOCHASTIC

S. L. BRUMELLE AND R. G. VICKSON
This example is of interest because of its relation to the Pratt-Arrow index
of risk aversion, r = —/"//'• If/is a concave increasing function such that r
is decreasing, then necessarily /'" ^ 0. However, the converse does not hold.
Note that/'" ^ 0 is equivalent to convexity of/'. Thus, let U be the set of
functions/defined on (—00, +00) such that there exists a convex decreasing
nonnegative function h with/(x) = $%h(y) dy.
Theorem 2.4 v(/)g^(/) for all /e U with n(f + ) < 00, if and only if
EX ^ £7 and v(g) ^ /t(#) for each g e G, where (7 is the set of functions of
the form — (w — x) 2 I^x&wl.
Note that if X is bounded below by a, then
£[(Z-w) 2 /[Xgw]] = a - w + 2J\l-F(x))(x-w) dx
follows by an integration by parts. Thus if X and Y are bounded below by a,
then v(g) ^ nig) for each g e G if and only if
(l-F(x))(w-x)rfx^ (l-G(x))(iv-x)^
a Ja
for each w^fl, and EX^EY.
Proof Without loss of generality, it can be assumed that /(0) = 0. If
f(x) = c-x, then v (/) ^ n if) if and only if EX k EY.
If f e C/is not linear, then there exists some convex, nonnegative, decreasing
function h which is not identically constant, such that/(x) = \ x „ hiy) dy. As in
Example 2, h can be arbitrarily well approximated by a positive weighted sum
of functions of the form (w — x) + . Formally, there exists some function p such
that hix) = l x 0piy) dy. Let L = sup{x:pix) — 00. If L = 00, then define
where
If L < oo, then define
*i = ^_£±i, ;=i,2,...,i+2-,
hnlix) = l-jnix-hiL)) + hiL)T and
A.i = IpiKdix-KJ + hiKdT,
106 PART II QUALITATIVE ECONOMIC RESULTS