238 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
tion that z, is given, thus making initial capital a free good. 
Restrict the attainable set instead by an arbitrary stipulation 
that consumption per worker and capital per worker are to 
be held constant over time. 
x. =x, 2,=z for all =o 
[he new « attainable » set then is given by 
{12 a, b, ¢) x=f(2)- Az, x>o, z>o0. 
Finally, choose z so as to maximize x, the permanent level of 
consumption per worker. This leads to the choice of that value 
Zz of z for which. 
(13 a, b) f(8)=), so =f) 2% 
where f(z) denotes the derivative of f(z). 
Figure 7 shows the construction. Because, of the essen- 
tiality of labor to production, i.e., assumption (#7 b) as reflected 
in (11), there is for any given slope A such that o<<A<f'(0) a 
point 2 for which the tangent to the production function per 
worker has that slope. To interpret the condition (13 a) note 
that, if we hold L fixed, then by the homogeneity of F, 
(2, 
SF(Z/L, 0 SF Z, 
3 ZL) 7 
"41 Koopmans - pag. 14