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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -

28

Ty) — u —T) gz" — x) < o
‘u(r, à) dt <u'(T*—T) :9(2"— x) <
7;

by (38 a). It follows that the path

(xt, 2) = (Xs 2p) for ot <T ,
(4 prop ; Zean*_g) for T = t

is likewise attainable, and indeed eligible and preferable to
(x,, 2,), because it achieves a utility

(=U +. U*=Ur+ UUs; + +Urs +. U=U .

Therefore (59) cannot occur in an optimal path.
It follows that, if z,#%, an optimal path shows a strictly
monotonic approach of 2, to the value zr =% for 0<¢<'T, where
I'=co. We shall call any eligible path with that property a
superior path. To complete the proof of the second and third
sentences of (C) we only need to show that for an optimal path
T=oo. This is best obtained as a corollary of the proof of (D).
The proof of the first sentence of (C) will also be combined
with that of (D).
Proof of (D). For all superior paths we can now make a
useful change of the variable of integration in (58) from ¢ to z.
Since, by (36), z,=2% for £=T implies x,=X, u(x,) =14, we have
for all superior paths, using (36),

mn.

) u(æ(z)) —u a
12) — X(2)

Koopmans - pag. 4.