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332 THE JOURNAL OF BUSINESS
subject to
and
*iM > 0 i>.>. + rM > o\ = 1 n = 1, . . . , iVi+1 (36)
' 1-2 '
have maxima for all / = 1, ...,/ — 1 and all m. Moreover, the maximizing vectors,
v*„, are finite and unique. The proof may be found in Hakansson (1968)."
Let us now assume that
//„(*) = * w all m ; (37)
and let kjm denote the maximum of (34) subject to (35) and (36) when u(x) = x 112 ;
that is,
kjm = 2J PiE\ I E (0Hm„ — rjm)v*im + rM\ > j = 1, . . . , 7 — 1 all m . (38)
By the theorem, we know that kjm exists.
Let us now determine fj-i,m(xj-i)- From (31) we obtain for all m
subject to
and
"j
fj-i.m(xj-i) = mai^H,^l(*,|M»)'«l, (39)
V-i.« "- 1
Zi.j-i,m > 0 *€5j_i,m (40)
"j-i
Pr | E C8iV_,,„. - fw,)^,/.,,. + >V-i,mz.7-i > o| = 1 » = 1, . . . , Nj . (41)
By (29) and (41), fj-\(xj-i) does not exist for Xj^i < 0. For Xj-i > 0, (39) may be
written, since (32) and (33) are equivalent to (35) and (36) when xj-\ > 0,
fj-l,m(xj-l) = Xjli max ^pj-l,mn
*'->.- ""' (42)
subject to
and
— r/_l.m)»i,j-l.m + »7-l.mJ J ,
»iv-i,« > 0 t€5j_i.„ (43)
M J-l.m
Pr I E (/3,-,/_i.», - f;.,,)ri,,.i,„ + r/.i,. > o| = 1 « = 1 iV.,. (44)
By the theorem, we now obtain that/J_1,„(»/_i) exists for all m and av-i > 0 and
is given by, using (38),
h-i.m(x) = ifej.,..* 1 " all m . (45)
11 Nils Hakansson, "Optimal Entrepreneurial Decisions in a Completely Stochastic Environment,"
Management Science: Theory 17 (March 1971): 427-49.
3. MYOPIC PORTFOLIO POLICIES 409