178 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -

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(v+d). Thus, if the rate of increase of the population is
v =0.02 per year and if the desired rate of increase of per capita
income is 0.06 per year, the limit of the savings ratio is of
the order of 25%.
When the v’s and d’s are not constant over time, we have
to substitute their values directly into (5.1). It is seen that
the influence of future v’s and d’s is of the decreasing exponential
 type. But it is impossible to compute x? when the future
v's are unknown. We have then to rely on maximizing expected
 utility and on the first-period certainty equivalence theorem,
 provided of course that the relevant expectations are
known. Note that these expectations are conditional expectations,
 given the information available at the moment when the
decision must be made. Suppose, e.g., that the rate of increase
of the population fluctuates around a mean v and that it satisfies
the following stochastic difference equation:

5-7)

+

—_— yy —

I
— (y — WV +e,

where €, is a random variable with zero mean and zero correlations
 over time. Then the expectation of v, - v, given the
information available at the beginning of the first year, is
I _— _
3 (Yo-v); that of v,-v (under the same condition) is
[ — . .
i (vo-v); and so on. By substituting these expectations in
the right-hand side of (5.1) we obtain the first-period decision
of the maximizing strategy (under the assumption d,=d):

5-8)

7 : Me (v + d) —
TEN NE Trg)
A (b - 1/p) (vo —v) ;
7 {4 = 2%) (6? + b/a + 9)

71 Theil - pag. 14