STOCHASTIC

some qualitative characteristics of the optimal first-period decision in many
dynamic models. However, the equivalent static program cannot generally be
determined in explicit analytic form; hence quantitative properties of the
optimal first-period decision cannot generally be ascertained.
In certain special cases, one can determine an explicit functional form for
the derived utility function. Exercise CR-2 illustrates such a calculation when
the utility function is quadratic and there are two risky assets. Conditions are
developed that lead to optimal investment entirely in one of the risky assets.
Exercise ME-2 illustrates the classic Simon (1956)-Theil (1957) certainty
equivalence results. Preferences for state and/or decision variables are assumed
to be quadratic. The constraints that link the state and decision variables are
linear functions whose uncertainty enters additively. It is also assumed that
all relevant maxima and expectations exist and that the maxima occur at
interior points of any additional constraints. The backward induction process
then yields a sequence of induced quadratic programs. It also develops that
one can replace uncertain random variables by their conditional means to
obtain a certainty equivalent. The certainty equivalent is an explicit deterministic
quadratic program involving only variables from period 1 and whose
optimal solution is an optimal first-period decision for the multiperiod
problem. Exercise ME-3 develops similar results when it is not assumed that
the constraints on the decision variables are nonbinding. The discussion
proceeds by considering the following questions: (1) When is the optimal
decision in each period independent of all random vectors and all other
decisions; (2) when is the optimal decision in each period independent of all
other decisions; and (3) when is it possible to develop an explicit deterministic
static program whose solution provides an optimal first-period decision?
Conditions are presented that provide answers to each of these questions.
They generally involve the assumption that the state vectors appear linearly
in the preference function and in the period-by-period linkage constraints.
For the first two questions, it is also necessary to assume that certain intertemporal
separability conditions are satisfied by the decision and random
vectors. All of the conditions provide for the optimality of zero-order decision
rules so that optimal decisions for each period can be determined before any
random variables are observed. The results are used to construct a multiperiod
stochastic capital budgeting model in Exercise ME-4.
Some dynamic investment problems have the property that they possess
simple optimal policies that are stationary in time. Exercise CR-5 considers
the dynamic portfolio problem when the utility function is logarithmic and
the investment choice in each period is between a risk-free asset and a lognormally
distributed asset. Conditions are developed such that it is optimal in
each period to invest totally in the risky asset. A similar result obtains if the
utility function is a power function (Exercise CR-6) or if the investor receives
INTRODUCTION 369