STOCHASTIC

w*t = glW,;Z,-1,...,Z0]
= gT_i[W>] if the Z's are independently
distributed.
Our problem is now solved for every case
but the important case of infinite-time horizon.
For well-behaved cases, one can simply let
T -» oo in the above formulas. Or, as often
happens, the infinite case may be the easiest of
all to solve, since for it C*t = j{Wt), w*t =
g(W,), independently of time and both these
unknown functions can be deduced as solutions
to the following functional equations:
0= U'[f(W)] - (l+p)- 1
/•
>[(W - j(W)) {(1+r)
-g(W)(Z- l-r)}][(l-t-r)
-g(W)(Z-\-r))dP(Z) (17')
0= f"v'[{W-j(W)}
{l+r-g(W)(Z- 1-r)}]
[Z-l-r]. (17")
Equation (17'), by itself with g(W) pretended
to be known, would be equivalent to equation
(13) of Levhari and Srinivasan [4, p. f]. In
deriving (17')-(17"), I have utilized the envelope
relation of my (13), which is equivalent to
Levhari and Srinivasan's equation (12) [4,
P- 5].
Bernoulli and Isoelastic Cases
To apply our results, let us consider the interesting
Bernoulli case where U = log C. This
does not have the bounded utility that Arrow
[1] and many writers have convinced themselves
is desirable for an axiom system. Since
I do not believe that Karl Menger paradoxes
of the generalized St. Petersburg type hold any
terrors for the economist, I have no particular
interest in boundedness of utility and consider
log C to be interesting and admissible. For this
case, we have, from (12),
Ji(W) = Max logC
{CM
+ E(l+p)-*\og[(W-C)
{(l-Bi)(l+r)+i»Z}]
= MaxlogC+ (l-HO-Hogr.W-C]
{C}
LIFETIME PORTFOLIO SELECTION 243
+ Max /"log[(l-si)(l+r)
•'o
M
+ wZ)dP(Z). (18)
Hence, equations (12) and (16')—(16") split
into two independent parts and the Ramsey-
Phelps saving problem becomes quite independent
of the lifetime portfolio selection problem.
Now we have
0= (l/C) - (l+P)- 1 (W r -C)- 1 or
CT-i = (l+p){2+P)- 1 Wr-.1
(19')
• /
(Z-l-r)[(l-u.)(l+r)
+ wZ]~ l dP(Z) or
wT~i = w* independently of WT-i- (19")
These independence results, of the CT_i and
wT_l decisions and of the dependence of wT-i
on WT_lt hold for all U functions with isoelastic
marginal utility. I.e., (16') and (16")
become decomposable conditions for all
V(C) = l/yCv, y