SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 27.

is a nonincreasing function of T. Hence G: either possesses a
limit for T—oo or diverges to — oc. ln view of (57), the same
must then be true for Ur.
This completes the proof of statement (B). In addition.
we have found

LEMMA 3: If o=o0, a necessary condition for eligibility o:
the bath (x,, z,) is that (57) is satisfied.

Proof of (C). An optimal path (x,, 2,) is now defined as

one that maximizes

(58)

u(æ,;) — u) dt

on the attainable-and-eligible set. A beautifully simple procedure
 used by RAMSEY in his slightly different problem can be
adapted to the present problem as long as g=o.
From Lemmas 1 and 3 we conclude that, in any optimal
path, 2, exhibits a nondecreasing, constant, or nonincreasing
approach to lim 2,=2 according as z,<2, =2 or >2 is
t=>00
establishes the second and third sentences of statemen.
with the term « weakly monotonic » substituted for « stric’
monotonic ». Now consider an attainable-eligible path (a
for which

(59)

tor

; NVST*, where

Then. along the lines of (56),

4} Koobmans - pag. 47