21% 
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA 
18 
Figure 15 shows the determination of x = £(z) for the two cases 
z<%Z and z>%. It is easily seen from the diagram or analytic- 
ally, using Assumptions (c), (d), (e), that a function #(z) can 
be uniquely determined from (61) for all values of z on 0<z<Z, 
so as to be independent of z, continuous and increasing for 
all z, and differentiable for 2-24. In particular, 
(> 
9 
lim 22) =o, af =2=gF) 
Moreover, since any feasible x, is by (35 c) continuous to the 
right, and since for any superior path #(z) is continuous and 
monotonic, #*(z) must be continuous to the right if z,<2, to 
the left if z,>2. Hence £*(z)=£(z) for every value of z in its 
domain, and the asterisk can now be omitted from X*(2). 
Once %(z) has been determined in the manner indicated, 
one reintroduces the time variable #=?(z) by 
HE pt 
37 
Way =i) 
= gy) — x(y) 
The function 2(z) and its inverse #, are monotonic and differen- 
tiable with the proper range and domain in each case because, 
by (61) and the monotonicity of £(2), 
g(2) — 2(2) =. 
u 
-ux(z 
° 
< 
k 
A 
& 
4] Koopmans - pag. 50