STOCHASTIC

48 NEAVE
where s" = s"(A), s' = s'(A), and CT = s/A, then UN is increasingly relative
risk averse.
Proof. Several cases must be considered, and it will be noted that some
hypotheses of the theorem can be relaxed in each case.
(i) Vs = 0 or if s = A, the conclusion that rv* is increasing follows
immediately using the reasoning employed in Theorem 1.
(ii) Suppose je (0, A). Then from Eq. (3.9),
so that"
ru*(A) = Arv{c) c' = Arz(s)s', (3.10)
ru*(A) = Vc*(c) = V/(s), where v, = xf\x)lf(x). (3.11)
The reasoning of Theorem 1 can be applied whenever it can be shown
that T)C and r)s are not both decreasing. Such a demonstration requires
examining four cases, some preliminary discussion of which is useful.
For given A and for e > 0, Taylor's theorem may be used to write
and
s(A + «) i=» s(A) + cs'(A) = oA + e