STOCHASTIC

world. This weak sufficiency test may be checked for any particular project
by solving a linear program. Projects that do not meet this stringent criterion
may still be acceptable if there is a funding program that leads to an increase
in expected utility. One may verify this "stronger" sufficient condition for
project inclusion by solving a concave program. A relatively simple calculation
can be made to determine if a project can be excluded from consideration in
the present diversified package of projects. In Exercise CR-4, the reader is
asked to provide direct rules for the inclusion and exclusion of projects in
terms of a simple expected utility evaluation. Exercise ME-1 illustrates a
two-period consumption-investment allocation problem that can be analyzed
via a static analysis. Of particular interest is the relationship between the
optimal investment decision that results when the random return takes on its
mean value and the optimal investment decision resulting from the stochastic
problem.
II. Risk Aversion over Time Implies Static Risk Aversion
The paper by Fama considers a multiperiod consumption-investment
problem. The consumer's objective is to make investment allocations between
several random investments and consumption withdrawals in each period to
maximize the expected utility of lifetime consumption. The consumer's utility
function is assumed to be a strictly increasing, strictly concave function of his
lifetime consumption. It is not necessary in the development to assume that the
utility function is intertemporally additive. It is assumed that the markets
for consumption goods are perfect. Fama then shows that the consumer's
behavior is indistinguishable from that of a risk-averse expected utility
maximizer who has a one-period horizon. That is, the process of backward
induction may be employed to reduce the multiperiod problem into an
equivalent static problem. Moreover, the maximization and expectation
operations preserve the concavity of the derived utility function at each stage
of the induction. Fama's presentation is in a quite general setting and the
results hold if the utility function is only concave (Exercise CR-1) or if the
consumer's lifetime is uncertain. The results also provide a partial multiperiod
justification for the static two-parameter portfolio models described in the
papers by Lintner and Ziemba in Parts II and III, respectively. The basic idea
that concavity properties of dynamic models are often preserved under
maximization and expectation operations is a very useful one and dates back
at least to early results by Bellman and Dantzig in the 1950s concerned with
stochastic dynamic programming and linear programming under uncertainty,
respectively. [See Wets (1966, 1972a, b) and Olsen (1973a, b) for recent results
and additional references in this area.] Such reductions allow one to develop
368 PART IV DYNAMIC MODELS REDUCIBLE TO STATIC MODELS