STOCHASTIC

2. EXISTENCE AND DIVERSIFICATION
OF OPTIMAL PORTFOLIO POLICIES
JOURNAL OF ECONOMIC THEORY 4, 35-44 (1972)
On the Existence of Optimal Policies
under Uncertainty
HAYNE E. LELAND*
Stanford University, Stanford, California 94305
Received August 10, 1970
I. INTRODUCTION
Many economic theories presume that a feasible policy set contains
an optimal policy. In portfolio theory, for example, theorists often
presume that first order maximizing conditions are satisfied by a feasible
portfolio, without consideration of whether the existence of an optimal
policy is implied by, or even consistent with, the basic set of assumptions.
Theorists who have taken care to ensure the existence of solutions
commonly have resorted to introducing explicit bounds on a closed
policy or action set (in R"), thereby rendering it compact. Given a continuous
objective function, Weierstrass' Theorem can then be used to
prove the existence of an optimal action. Hakansson [2], for example,
bounds the feasible set of his portfolio model by requiring that terminal
wealth be nonnegative with probability one.
But costs may be incurred by bounding the action set. Bounds often
are difficult to justify on economic grounds. 1 Or they are so loose (e.g.,
"use of resources must not exceed world supply") that the question of
existence is transformed to the question of whether the bounds are tight
or slack at the optimum. But the answer to the latter question conditions
the predictions of the theory as critically as the answer to the former. To
escape Scylla and Charybdis, some theorists introduce constraints to
ensure existence, and then assume these constraints are slack at the
optimum! Clearly, this is little better than to have ignored the original
existence questions.
* This work was supported by National Science Foundation Giant GS-2530 at the
Institute for Mathematical Studies in the Social Sciences at Stanford University.
•Hakansson's requirement that final wealth be nonnegative, for example, seems
rather restrictive. If an asset had normally distributed returns, no investment would
occur no matter how large the expected return and how small the variance, as there
always is a nonzero probability of an arbitrarily large loss.
© 1972 by Academic Press, Inc.
2. EXISTENCE AND DIVERSIFICATION OF OPTIMAL PORTFOLIO POLICIES 267