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 essentiality
 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