234

PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -

2K

the labor force L. These assumptions imply full employment
of labor and capital, constant returns to scale, and nonincreasing
 returns to an increase in only one factor of production.
Since capital is treated as a stock of the single producible commodity,
 output is at any time # to be allocated to a positive
rate of consumption X,, and to a positive, zero, or even negative
 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 magnitude
 L, ,

ro
(u a, ob

L,

L Nn © :

J

"41 Koopmans - pag. 10