STOCHASTIC

40 LELAND
is sufficient for X to have a maximal element. If we do not assume finite
expected returns, the slightly stronger condition
U.3b U(c) bounded above
is a sufficient condition. To prove these results, several preliminary
concepts and propositions are required.
DEFINITION 1. Z = {* e X | V(x) > K(0)}.
Comment. Clearly, X will have a maximal element if and only if Z
has a maximal element. Because V is continuous, Z is closed. If it is
bounded, it will have a maximal element, as a continuous function reaches
a maximum over a compact set. The object of the following analysis is
to show that Z is bounded.
PROPOSITION I. X is convex.
Proof. As X is convex, it remains only to show that for x 1 , x % e X with
Vix 1 ), V(x*) > K(0), then V[ocx l + (1 - K(0) for all a e (0, 1).
But this follows immediately from the concavity of V(x).
DEFINITION 2. Z* = {x s Z | \\x || > H).
Comment. Z = Z*VJZ~X*, where X — X* = {xeX\\\x\\ < H}.
Clearly, Z — Z* is bounded. If Z* is bounded, Z will be bounded, as the
union of bounded sets is bounded.
DEFINITION 3. Y = {y e R" 11| y \\ = 1, Hy e Z}.
Comment. Y is the (compact) set of directions from the origin such
that, when x = Hy, x e X, and V(Hy) > K(0).
PROPOSITION II. V(\y) is a strictly concave function of \ for y e Y.
Proof Vu(Xy) = d*[E[U(\e'y)]]/d\* = E[(e'yf U"(Xe'y)} < 0, as
U"(c) < 0 for all c by U.2, and pr[(e'yf = 0] < 1 by X.3, since, by
assumption, Ay e X, when || Ay || = A = //.
DEFINITION 4. 7* = {x = \y | y e y, A > //, K(Ay) > K(0)}.
PROPOSITION III. Z* C Y*.
Proof. Choose any z e Z*. Define y(z) = z/|| z ||, and A(z) = || z ||.
Therefore, z = A(z)j(z), where ||y(z)||= 1. To show zeY*, we must
show A(z) > H and y(z) e Y. The first follows directly, as A(z) = \\z\\ ^ H.
PART III. STATIC PORTFOLIO SELECTION MODELS