STOCHASTIC

596 NILS H. HAKANSSON
The proof is immediate from the observation that u(c) is monotone increasing
and that r, {/}f}, and the feasible tif are bounded.
We are now ready to prove the theorem. The method of proof will be to verify
that (22H25) is the (only) solution to (11). 8
PROOF OF THEOREM 1 FOR MODELS I-II: Denote the right side of (11) by T(x) upon
inserting (22) for f(x). This gives, for all decision points ;',
f 1 Y\l M
(35) T(x) = max^c' + a(l - (a/cy)" 1 '-'')'-^ - Y (ft - r)zt
subject to
(12) c > 0,
(13) Zi>Q, its,
and
+ r(x - c) + y + Y
(14) Prj J i», ~ r)z{ + i[x - c) + y + (y/(r - 1)) > oj = 1.
Since (14) may be written
Pr { S (ft - r)z, + r(x + Y - c) > 0 J = 1,
it follows from the "no-easy-money condition" (4) that (14) is satisfied if and only if
either
(36)
and
(37)
or
(38)
and
(39)
s - c = 0
zf = 0
s - c > 0
P r C u
Pr {.I(ft-'-W(s-c) l(P
+ r>oJ = l)
where s = x + Y.
Under feasibility with respect to (14), we then obtain
maxUs', T{x)\, 0