STOCHASTIC

46 NEA.VE
I —- stochastic rate of return on wealth invested in period
n; I s [Ji, yi] C (0, oo);
A„ = total wealth available at beginning of period n.
Discount factors or time preference are assumed to be reflected in the
transformation factor /? e (0, 1).
To continue the analysis, redefine Eq. (3.1) recursively as
Un(A) = max.} E{v„(A - a) + pu„+1(k -~ &)), n == 1, 2,..., N,
UK+l(A) = »'N(^1), (3.2)
where subscripts discernible from the context of the recursive formulation
have been dropped, and the equality constraints are incorporated in
Eqs. (3.2). The functions £/„ now represent the expected utility of following
an optimal policy from the beginning of period n to the end of period TV,
given that the wealth at the beginning of period n was A.
A set of conditions sufficient for the functions U„ to exhibit decreasing
absolute and increasing relative risk aversion for all n = 1, 2,..., N will
now be presented. 8 These characteristics of the multiperiod return functions
are obtained inductively and proceed from a consideration of the
one-period problem.
Let
0\(A) = max E\vy(A — a) -- BwN(k — Ca)}, (3.3)
ii6(0./l]
and write
Uy(A) = i'.v(c) + zN(s), c(A) = A - a'(A). s(A) = a'(A), (3.4)
where a"(A) satisfies Eq. (3.3).
THEOREM 1. If the functions vN and ws of Eq. (3.3) are strictly concave
and if re and ru. are nonincreasing functions," then UN is a strictly concave
decreasingly absolute risk averse function.
Proof. It will first be shown that rv is nonincreasing, thus establishing
that Uy is decreasingly absolute risk averse. Three cases are considered in
this part of the proof.
(i) i = 0. In this case US(A) — vy(A) + const; hence
ru'{A) < 0 o rv'(A) sC 0.
8 An extension of the above results to infinite-horizon problems using techniques
developed in Ref. [4] can be effected.
9 No ambiguity arises from our writing r„ and ru. rather than rv and /-,„ , because
the time subscripts can be inferred from the context of the discussion.
2. OPTIMAL CAPITAL ACCUMULATION AND PORTFOLIO SELECTION 507