STOCHASTIC

The function t/,(C,_,, wt|a,, ft) in (9)
provides the maximum expected utility of
lifetime consumption and bequests if the
consumer is alive in state 0t at period t, his
wealth is w,, his past consumption was
Ct_i, and optimal consumption-investment
decisions are made at the beginning of
period t and all future periods at which he
is alive.
Since (7)-(9) are just a special case of
the model summarized by (4), Proposition
1 applies directly to the probabilistic
death model. In this case the proposition
implies that for all t and &, Ut(Ct-i,
v)t Zt, &), £/t(C«_i, wt\at, $»), and l/t(Ct-i,
z"t dt, (3t) are monotone increasing and
strictly concave in (Ct_i, wt).
Finally, expressions (7)-(9) suggest an
alternative to the "sure death" interpretation
of the horizon T+1. For given
wealth uii, and state of the world |3i, the
consumer's problem at period' 1 is to
choose ci and Hj, which
max I U,(Ci, BR' | s,, fodF^lfi,)
- max I [xtUACi, HR' | a,, &)
FAMA: MULTIPERIOD DECISIONS 171
+ (1 - x,)Ut(Cu HR' | dt, fofldFutft).
Using (8) and (9) to expand this expression,
it can be shown that the decision
problem of period t has weight
*t*t_i •••*, = x(at | a,)
in the expected utility for the decision of
period 1. The probability *(at|oi) of being
alive at t will decrease with t. Thus in
general for some t=T+l, the decisions of
periods T+1 and beyond will have negligible
weight in the expected utility for the
decision at period 1, so that in the decision
at period 1 it is unnecessary to look beyond
T+1.
More simply, since the consumer is
likely to be dead, the effects of distant
future decisions, which are unlikely to be
made, can be ignored. And this result does
not arise from discounting of future consumption,
though the effect is the same. At
period 1, consumption in period T+1 may
be regarded as equivalent to consumption
in period 1. But in the decision of period 1
the decision of T+1 is weighted by the
probability that the consumer will be alive
at that time, which reduces the importance
of the future consumption in the current
decision.
V. Conclusion
In sum, assuming only that markets for
consumption goods and portfolio assets
are perfect and that the consumer is risk
averse in the sense that his utility function
for lifetime consumption is strictly concave,
it has been shown that though he
faces a multiperiod problem, in his consumption-investment
decision for any
period the consumer's behavior is indistinguishable
from that of a risk averter who
has a one-period horizon. It was then
shown how this result can be used to provide
a multiperiod setting for the more
detailed hypotheses about risk averse
consumer behavior that are traditionally
derived in a one-period framework.
APPENDIX
Proposition J. If t/t+^Ct, a>l+i|/3t+i) is
monotone increasing, and strictly concave
(henceforth m.i.s.c.) in (Ct, i«t+i), then
Ut(Ct-u wt\pt) is m.i.s.c. in (Ct-i, ait).
Proof: The proof of the proposition relies
primarily on straightforward applications of
well-known properties of concave functions
(cf. Iglehart). We first establish:
Lemma 1. If U^.i(Ct, u>t+i|0tfi) is m.i.s.c.
in (Ct, tt>t+i), the expected utility function
j tf„.,(C„ w,+l | ft+O'WutCft+i)
•'d+i
(10) - f £/«+i(Ct, HnR{fi,+1)' I /3l+1)
•''t+l
is strictly concave in (Ct, Hn).
• (WfltOSt+i)
2. RISK AVERSION OVER TIME IMPLIES STATIC RISK AVERSION 397