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 proce- 
dure 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