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 expo- 
nential type. But it is impossible to compute x? when the future 
v's are unknown. We have then to rely on maximizing ex- 
pected utility and on the first-period certainty equivalence theo- 
rem, provided of course that the relevant expectations are 
known. Note that these expectations are conditional expecta- 
tions, 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 cor- 
relations 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