STOCHASTIC

The Hakansson paper studies a generalization of the problem treated by
Samuelson. Hakansson's assumptions regarding utility for consumption are
identical to Samuelson's, except for an assumed infinite lifetime. The investment
possibilities facing Hakansson's investor are, however, more general. The
individual is assumed to possess a (possibly negative) initial capital position
plus a noncapital steady income stream which is known with certainty. In
each period, borrowing and lending at a known, time-independent rate of
interest, and investment in a number of risky assets with known joint distribution
of returns, are possible. As usual, the risky asset returns are assumed to
be independent identically distributed (iid) over time. It is further assumed
that a subset of the risky assets may be sold short. In each period, initial wealth
to be consumed and invested consists of the certain income plus gross return
on the previous period's investment. The objective is maximization of the
expected utility of consumption over an infinite lifetime.
Hakansson introduces two important and apparently realistic constraints
on the risky returns and the investor's decision variables. The capital market
is assumed to impose a so-called no-easy-money restriction on the risky asset
returns. This restriction ensures that no mix of risky assets exists which provides
with probability 1 a return exceeding the risk-free rate, and that no mix of
long and short sales exists which guarantees against loss in excess of the riskfree
lending rate. The consumption-investment decision variables are assumed
to satisfy the so-called solvency constraint. This requires that the investor
always remain solvent with probability 1, i.e., that in any period t, the investor's
initial capital position plus the capitalized value of the future income stream
be nonnegative with certainty.
Hakansson uses the familiar backward induction procedure of dynamic
programming to set up a functional equation for the optimal return function
(i.e., the maximum expected utility of present and future consumption, given
any value of initial wealth). Since the risky returns are iid and the lifetime
infinite, the functional equation obtained is stationary over time: The optimal
decision variables in period t depend only on initial wealth (in period t), on
the single-period asset returns, and on the relative risk-aversion index of
consumption. Hakansson's treatment of the infinite horizon dynamic programming
problem is intuitively plausible, but not wholly satisfactory from a
rigorous standpoint. (The results are correct, however.) First, he assumes that
the optimal return function exists and satisfies the functional equation; in a
totally rigorous development, these facts need proof. Next, he verifies the
form of the optimal solution by direct substitution, that is, by showing that
the assumed form of solution actually solves the functional equation. Here
the question of uniqueness arises, and is not treated satisfactorily in the
development. One way of treating the problem rigorously would be to deal
first with the finite horizon problem (as Samuelson does) and subsequently
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