STOCHASTIC

OPTIMAL INVESTMENT 605
Let us now turn to Models I—III and let, as before, s, = x, + Y. From (7), (23),
and (25) we now obtain
(69) SJ.+ 1= S J(1-B)
£ (ft - r)v' + A
= SjW 0=1,2,...)
where Wis a random variable. By (28), W ^ 0. Attaching the subscript n to Wtor
the purpose of period identification, we note that since
(70 Sj = s, n w„,
n = 1
(70) verifies that
Sj ^ 0 for ally whenever s, ^ 0 (Models I-III).
Moreover, since Pr {W > 0} = 1 in Models II and III by Corollary 1, it follows
that
(71) Sj > 0 whenever S, > 0 for all finite j (Models II-III).
From (70) we also observe that Sj = 0 whenever sk = 0 for ally > k. Consequently,
v = — Y is a trapping state which, once entered, cannot be left. In this state; the
optimal strategies in each case call for zero consumption, no productive investments,
the borrowing of Y, and the payment of noncapital income y as interest on
the debt. In Models II and III, it follows from (71) that the trapping state will
never be reached in a finite number of time periods if initial capital is greater
than - Y.
Equation (70) may be written
The random variable Z^~ , log W„ is by the Central Limit Theorem asymptotically
normally distributed; its mean is (j - l)£[log W\ By the law of large numbers,
Z' '°8 W -
'^ j • £[log W] as j -» oo.
Thus, since st > s, if and only if .Vj > x,, it is necessary and sufficient for capital
growth to exist that £[log W] > 0.
11 is clear that ft given by ft = ? £ "" 8 "' may be interpreted as the mean growth rate
of capital. By (69), we obtain
£[logW] = log(l - B) + E log] I