278 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
the quadrant x>0, zZ0 according to the signs of à, 2, that 
‘follow from (65). Figure 17 sketches the trajectories of the 
point (x, z,) starting from arbitrary initial values (x,, z,). 
Each trajectory is defined as the solution x(z) with x(z,)=x,, 
or z(x) with z(x,)=z,, of the corresponding differential equation 
dz _ u(x) g@)—x dr w@) g'(&—e 
( regs I esis ET me, JT ts mele, Bore 
66) de u(x) g@@—o" RE æ) g(2)—æ 
respectively, obtained from (65) by elimination of #. Any seg- 
ment of any trajectory defines a path optimal on a suitable 
time interval with prescribed initial and terminal values z,, zr 
of z,. If we prescribe only z, and examine the trajectories for 
various x, We find that there is one unique value £, of x, 
which together with z, identifies a trajectory (shown in the 
diagram as a heavier line) that meets condition (x) of Propo- 
sition (I) of an asymptotic approach, for f—oo, to (£(p), 2(p)) 
as defined in (60), Proposition (H). 
We now denote the path resulting from that particular 
choice à, of x, by £,, 2). If z,<2(p), the initial consumption 
low £, leaves room for growth in the capital stock per worker, 
and both #, and 2, increase with ¢ to approach their asymptotic 
values £(p), 2(p), respectively, as t—oo. If z,>%, both £,, £, 
decrease, and approach the same asymptots from above. Fi- 
nally, if z,=2(p) we must have £, = £(p), #,=2(p) for all #=o. 
Since #, approaches the positive number £(p) as t—oo, p, 
is by its definition (21) asymptotic to e=°* u(£(p)). Hence, in 
30), lim pr(zr- 25)=0 by (40), and (£,, %,) is optimal. 
Too 
Moreover, if (x, z,) differs from (£, #,), we must have x, #$, 
for some à, because in the contrary case (36) and z,=2, would 
imply z,=#, for all £. But then, by the attainability condi- 
Hon (35 ¢), we have a strict inequality in (24) and, by (25), 
a strict inequality in (30), hence (x,, z,) is not optimal. There- 
"41 Koopmans - pag. 54