STOCHASTIC

606 NILS H. HAKANSSON
For Model III, this becomes, by (30) and (26),
£[log W] = log a + max E
f M
'og{ Kft-rfo + r
subject to (27) and (28). Thus, a person whose one-period utility function of
consumption is logarithmic will always invest the capital available after the allotment
to current consumption so as to maximize the mean growth rate of capital
plus the present value of the noncapital income stream.
7. GENERALIZATIONS
We shall now generalize the preceding model to the nonstationary case. We
then obtain, by the same approach as in the stationary case, for all;',
(72)
subject to
(73)
(74)
and
(75)
Cj S5 0,
c
*o > °.
Pr{*m
max < u(cj) + djE
i£S />
> -
^+i}>
/j+il % (fa ~ nK + r /*j - C J) + y,)
where the patience factor a, the number of available investment opportunities M
and S and their random returns /?, — 1, the interest rate r, and the noncapital income
y may vary from period to period; this, of course, requires that they be time
identified through subscript j. Time dependence on the part of any one of the
preceding parameters also requires that f(x) be subscripted.
As shown in [7], the solution to the nonstationary model is qualitatively the same
as the solution to the stationary model.
In the case of a finite horizon, the problem again reduces to (72H75) with
/„+ ,(xn + l) = 0 if the horizon is at decision point n + 1. In this case, /(x), x, c, z{,
and Y must clearly be time identified through subscript j even in the stationary
model. Under a finite horizon, a solution always exists even for Model I. Again, the
solution is qualitatively the same as in the infinite horizon case except that the
constant of consumption proportionality Bj increases with time j, B„ = 1, and
z» = 0 for alii."
University of California, Berkeley
Manuscript received September, 1966; revision received January, 1969.
1 ' The implications of the results of the current paper with respect to the theory of the firm may be
found in [6].
544 PART V. DYNAMIC MODELS