STOCHASTIC

50 NEAVE
Finally, although the result is implied by the statement of Theorem 2,
it is useful to present the next lemma because of the alternative method of
proof employed.
LEMMA 7. Let the functions vN and zN be defined as in Eqs. (3.3) • „ and rv are nonincreasing functions, then U„ is a
strictly concave decreasing!)' absolute risk averse function.
Proof. The proof is exactly the same as for Theorem 1.
We turn now to the task of relating relative risk aversion in periods
n + 1 and n.
THEOREM 4. Suppose that the functions v„ and £/„+1 of Eqs. (3.12) are
strictly concave, and that r,,* and r.* are nondecreasing functions. 1 '
Let e > 0. If
s"(s' — CT) < 0,
12 As noted before, extensions to intinite-honzon problems will not be dealt with
in the present investigation.
13 As indicated in footnote 9, no ambiguity arises from our omitting to write the
functions as cv and s„ .
14 To be precise, it should be stipulated that rt* is nondecreasing for x ; xY ; see the
remarks following Lemma 5. This complication is omitted to simplify the proof.
OPTIMAL CAPITAL ACCUMULATION AND PORTFOLIO SELECTION