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 analytically,
 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 differentiable
 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