STOCHASTIC

R. G. VICKSON
for distinct i, j, and /. Assuming that 0 < kt < u'(0) for / = 1, ...,n (which is
always possible by suitable choice of p), it follows that y (k) will actually be an
interior solution for all k g 1. Since (22) holds as an identity in k and kt, it
is permissible to set k = 1, to fix kj and kt at arbitrary values in the interval
(0, K'(0)), and to let z = k{ be a variable. Then (22) becomes
rx z 2 g"(z) + r2 zg'(z) + r3g(z) = 0 (23)
for real constants r1; r2, and r3. It is easy to verify that, if a and /? are the
roots of the quadratic equation
then the general solution to (23) is
rlx{x-\) + r2x + r3 = 0, (24)
(Az" + Bz f if a*P,
9(z) = (25)
\z*(A + Blogz) if a = p,
with constant A and B. Note that g(z) must have the form (25) in each interval
within which g" is continuous. Although it might seem that g could be made up
from different "pieces" of the form (25), with constants A, B, a, and /?
changing, this is, in fact, not true. The reader can verify that separation,
together with the assumed continuity of g and g', implies that the single set
of constants A ,B, a, fi applies throughout the entire subdomain {z\z< u'(0)}.
In terms of u, this means that the following functional representation holds:
_ (Au'ffl + Bu'tfY (a # P), (26a)
" (u'(Zr IA + B log «'(0] (« = P) (26b)
for all £ e (0, oo). Note that separation forces u to be three times continuously
differentiable on (0, oo), even though this was originally assumed to be true
only in a piecewise sense.
For problem Pk with 0 g k < n, the reader can show that Eqs. (26a) and
(26b) are valid throughout the entire domain £ e (u, oo) of u, even if u < 0.
In fact, it is always possible to construct a nonsingular matrix p such that all
relevant nonnegativity constraints are satisfied strictly, while the unconstrained
assets are sold short, so that g(z) is "probed" for ze(u,0). However, if
k = n, the nonnegativity constraints form a natural boundary which always
restricts £ in (26) to the set (0, oo). In this case, even if u < 0, no information
about the behaviour of M(^) for £ < 0 can be obtained from the portfolio
problem. The following fundamental theorem of Cass and Stiglitz has thus
been proved.
162 PART II QUALITATIVE ECONOMIC RESULTS