256 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28

Now if T is increased, the benefitted generation becomes
a more and more distant one. If T=o0, there is no benefitted
generation, and the limiting position of the curve in Figure 10,
while mathematically well-defined, merely describes a path of
indefinite and fruitless sacrifice.
The problem appears in even sharper light if technological
progress is also recognized. A study by INAGAKI [1963] uses
a CoBB-DOUGLAS production function

F(Z, L, t) = const. ef Z*1.1-¢

subject to exogenous technological progress at the constant proportional
 rate 3, an instantaneous utility function

(53)

wr) ~ logs -- log x

as

exhibiting suitable behavior for large values of x, and a labor
force growing exponentially at the rate A. Among other results,
 INAGAKI finds that, for the integral V(p) as defined in (19)
to converge on the counterpart of our path (x,, z,) =(£(p), #(e)),
it is necessary that

-— Œ-“4]

 Koopmans - pag. 32