STOCHASTIC

ON THE EXISTENCE OF OPTIMAL POLICIES UNDER UNCERTAINTY 39
Substituting for xx from (2) gives an unconstrained maximization problem
max E{U[{\ +r)BB + (e2 - r) xj}. (3)
jr2
Differentiating w.r.t. x2 and writing out the expected value gives
dElUffll/dxt = (1X« - r) UW) + (JXA - r) WW), (4)
where Bf = 31 when e2 = a, 2^* = Bt when e2 = A.
Suppose now U(B^j satisfies (7.1 and U.2, and, furthermore,
lim £/'CBi) = *!; lirn^ U'(B,) = fc2,
where Ar2 > A:, > 0. Note that the first R.H.S. term of (4) is positive for
all x2, while the second term is negative for all x2 • It follows from U.2
that
But if
dElWBjydxi > (J)[(o - r) h + (b - r) *J for all x2. (5)
*i/*s > t-(* - '•)]/(« - O. implying [(a - r) fc, + (6 - r) Ar2] > 0,
it follows from (5) that dE[U(B^i\ldx2 > 0 for any x2. Therefore, X in
this case does not possess a maximal element, for larger positions in x%
are always preferred. As none of the assumptions on G, X, and U are
violated in this example, we conclude that these assumptions are not
sufficient for X to possess a maximal element.
In the Introduction, it is suggested that the approach of bounding
the feasible set X has certain intuitive and operational drawbacks. A
second approach, which we shall develop in subsequent sections, is to
introduce further assumptions on the nature of the preference ranking
over X. These assumptions, which seem to have considerable attraction
in other areas of economic theory, will be shown to guarantee the existence
of optimal policies.
IV. SUFFICIENT CONDITIONS FOR THE EXISTENCE OF OPTIMAL POLICIES
We shall now show that, if the expected returns of all gambles are
finite, the further condition
(7.3a lim U'(c) = 0
EXISTENCE AND DIVERSIFICATION OF OPTIMAL PORTFOLIO POLICIES