STOCHASTIC

MULT1PERI0D RISK PREFERENCE
between consumption and an income-generating asset. He is then assumed
to receive an income from sale of labor services and to realize gains or
losses on his risky investment, after which he is required to make another
allocation decision.
It will be assumed that utilities are additive in time, and that the singleperiod
functions exhibit decreasing absolute and increasing relative risk
aversion; with these conditions and some additional qualifications it will
be shown that the consumer's multiperiod decisions are made in a manner
which exhibits both decreasing absolute and increasing relative risk
aversion. Thus, sufficient conditions for single-period risk preferences to
be reflected in the derived utility functions of tne sequential maximization
problem will be exhibited.
To proceed formally, define the problem
where
( N )
max E £ j3 n - 1 un(a„ , bn)l subject to a„ + b„ = An ,
°- b («=i ) C3-1)
u,,(a„ . b„) = »„(*„);
a„e[0,A„],Aa+l = k + b„,
n = 1,2,..., tf- 1;
"N(".\ . bN) = vN(bN) + fiwf,(k + £aN);
v„( ) = utility of consumption in period 5 n;
b„ = dollar value of consumption in period n\
? W N{ ) = beginning-period utility of wealth 6 possessed at end of
period .V;
k = nonnegative cash income during each period 7 ;
a„ = amount of wealth invested in period n\
'" Economists differ concerning the importance of assuming these functions to be
bounded. Reference [2] discusses the types of paradox which can arise if functions are
not bounded; Ref. [13] maintains that qualifications respecting boundedness are not
of prime importance. For simplicity we.omit the boundedness assumption.
6 Wealth is assumed to have utility at the problem's horizon because the individual
is supposed to regard it either as a bequest (which provides him satisfaction) or as a
proxy for consumption at points in time beyond those included in his planning problem.
The need for (and desirability of) including the bequest motive disappears in infinitehorizon
formulations of the problem. For discussions of this point, see Refs. [3, 8, 131.
7 It is assumed that transactions between cash and the risky asset are costless; hence,
it becomes unnecessary to distinguish between the forms in which wealth may be held
at the beginning of any period.
506 PART V. DYNAMIC MODELS
45