STOCHASTIC

Now df/dXi is homogeneous of degree 0, hence
W. T. ZIEMBA
and (ii) is satisfied.
If x;* = 0, (iii) is trivially satisfied.
Ifx;*>0, (iii) is equivalent to (ii(a)) which is satisfied for ex* iff x*
satisfies (ii(a)).
By Theorem 2 an optimal solution to (2') must solve (1); hence £"=1 xf* = w
and the x;* are independent of the u.
The proof and statement of the separation theorem given here are similar
in spirit to that given by Breen [2]. Breen considered an efficiency problem
in which one minimizes a-dispersion given that expected return is a stipulated
level as well as some alternative assumptions regarding the risk-free asset.
For the analysis indicated in Fig. 1 to be valid it is necessary that the adispersion
measure/(x) = {Z"=i Sixi*} l/ '' be convex and that q> be a concave
function of /?. The function / is actually strictly convex as we now establish
using
Theorem 3 (Minkowski's inequality) Suppose
K(y) = {(!/») .1 y'} \ n^r>\, n < oo,
and that a and b are not proportional (i.e., constants qx and q2 not both
zero do not exist such that q1a = q2b); then
M,(a) + Mr(b) > Mr(a + b).
Proof See, e.g., Hardy et al. [13, p. 30].
Let Ps {x|x^0, e'x = l}.
Lemma 1 f(x) s {£"=1 StX*} 11 " is a strictly convex function of x on P if
2 ^ a > 1 and Sf > 0, i = 1, ...,n.
Proo/ Let flj = A^X; 1 and ^ = (1 -A^x, 2 , j = 1,...,«, where x 1 # x 2 ,
and 0 < A < 1. Then by Minkowski's inequality
(d/«) .1 ««"] + ((!/») .1 V) > ((I/") .1 («« + *i)j
=>
( i \l/« (• n ll/a / n U/a
(.Z [AS; x; 1 ]"j + J£ [(1 - A) S, x, 2 ]" J > (£ [AS, x,. 1 + (1 - A) S, x, 2 ]* J .
252 PART III STATIC PORTFOLIO SELECTION MODELS