236 ‘PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA 
The per-worker production function f(z) is therefore represented 
by a curve such as is shown in Figure 6. The curve rises from 
the value f(0)=0 with a decreasing slope. In particular, any 
line Az through the origin and of slope A such that O<A<f(0) 
will ultimately intersect the curve and continue above it 1) 
oy oan. 
x > o such that o 
> such that 
- 
: f' (0) there is 
- 
3 
(') To obtain (11) suppose that, for some such À, f(z)=)\z for arbitrarily 
large values of z. Then, by f(o)=o0 and the concavity of f, f(z)=\z for 
all z>o0. Bat then, for any Z>o, (7b) and the continuity of F(Z, L) imply 
7 . 
‘he contradiction o=F(Z, o)=lim F(Z, L)=Ilim LF(_, 1)=Z lim Haz 
>Z)>o. ts 1 —0 r=» Z 
4] Koopmans - pag. 12