STOCHASTIC

30 THE REVIEW OF ECONOMICS AND STATISTICS
First, we note that if a single project j is added
to an existing body of assets So°, we have
(340) ARi - y&Au = S/"' - r*Hi°
- y [H„+ 26 „] = X,= X,
when
(366, C, d, e) Oy°& O, ?y°S O,
My°£ o, i,y«& o
and
(36/,g) «°o;°= o, uy°?/=°,
where
j = 1, 2 . . . n
in each set (360) — (36^).
Once again, these equations can be readily solved
by the Wilson Simplicial Algorithm [23] on
modern computing equipment. It may be observed
that this formulation in terms of independent
investment projects can readily be
generalized to cover mutually exclusive, contingent,
and compound projects" with no difficulty.
It is also apparent that the absence of a financing
constraint (due principally to our assumption
H The proof that the indicated solution satisfies the Kuhn-
Tucker conditions with respect to the variables fly 0 andyiy 0 is
identical to that given above footnote 28 upon the substitution
of 2[ for 2, °J f° r **> ami m for MV, and need not be
repeated. The two additional necessary conditions are
(3') \J-L- jo ^ 0( which is satisfied, since from (35) we have
—^-1° = 9? ^ o by virtue of (36c); and this latter relation
*K J
shows that the corresponding complementary slackness condition,
(4') M°y \-^— |° = o, may be written ji? jy° = o, and is there-
fore satisfied because of (36^).
Ail three sufficiency conditions are also satisfied because
the variance-covariance matrix ^ is positive definite, making
>p (fl, £°> 8°) a concave function on a and ^ (a, u°, »°) a
convex function on both u and n.
" See Weingartner [22], n and 32-34.
148 PART II. QUALITATIVE ECONOMIC RESULTS