476 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 
28 
effectively regarded as infinite. The derivations are given in 
the Appendix and the result for the optimal first-period de- 
cision (x9) is as follows: 
5.1) 
2b 
x? — he, — bE+ be +9 0 + d,) + 
(1 —X) à +I/p eu 
"bg à (ve +d), 
where 
5.2) 
, — [I +42°(0*+b/p+g)li—1 
[t + 40° (D? + b/o +9)l# + 1 
(0 <n <1) 
The result (5.1) shows that the optimal savings ratio in 
the first year (x{) consists of. two parts, one of which deals 
with the savings ratio in the preceding year (x,) and the other 
with population increases and desired consumption increases 
in the first year and later (v,, d;, v,, d,, ...). We find that 
the effect of v, and d, on the optimal savings ratio is negative 
but that the effect of later v’s and d’s is positive. This is the 
natural consequence of the fact that society has to spend for 
immediate welfare but to save for future welfare. 
As an example we take the special case v,=v, d,=d for 
all #; then (5.1) is simplified to 
| . DE , 
(5-3) x) = My TBE Bar +d). 
which means that this year’s savings ratio is a certain fraction 
of last vear’s savings ratio (Ax,) plus a constant (k, say). For 
71 Theil - pag. 12