234 
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 
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the labor force L. These assumptions imply full employment 
of labor and capital, constant returns to scale, and nonincreas- 
ing returns to an increase in only one factor of production. 
Since capital is treated as a stock of the single producible com- 
modity, output is at any time # to be allocated to a positive 
rate of consumption X,, and to a positive, zero, or even ne- 
gative rate of net investment Y,. Hence, if we use a continuous 
time concept, and denote derivatives with respect to time by 
dots, we have 
5) 
(6) 
X, Y= F(Z, L,) ( 
Y.=7. . 
F(Z, L) is defined for all Z>o0, L=0. We assume further 
that both labor and capital are essential to production, that 
either factor has a positive marginal productivity, and that 
returns to increases in only one factor are strictly decreasing, 
“ (7a, 8) 
' (7e, d 
tye, * 
21s, 
Vt 
OF 
tt 
). 
~ 
Sd, 0)—=« 
OF 
> 0 . 
Lu 
ô?F 
=> < O 
SL? 
Finally, we assume that the labor force increases at a 
constant positive exponential rate À, from a given initial magni- 
tude L, , 
ro 
(u a, ob 
L, 
L Nn © : 
J 
"41 Koopmans - pag. 10