SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 26s 
it ny denotes the number of completed bulges in [0, T'? FE i 
lim #np.=oc because there are infinitelv manv bul se = 
T’>00 
[0, oc]. Hence the choice of Ty such that ny >N/«- estap- 
lishes Lemma 2 in case (49 c) holds. The proof from (4G 5) i. 
similar. 
A 6. PROOFS FOR A ZERO DISCOUNT RATE (. 
Proof of (A). In (43 a) take 
Then if we write u(x)Za, u’. 
(54) 
Uu 
di =U + |. 
y wu 
. À 
Aw 
by (38), (40), regardless of T, T*, hence also for . 
Proof of (B). We distinguish three cases regarding 
asymptotic range [§, ©] of the given path (x, z,). 
Case (1), {<Ç. In this case we have from Lemma 2 and 
from (54) applied to (7%, 27), for anv N c 
for an 
r— 
T> 1 
waa — uly) dr - 
ULX + » - 
In this case. therefore. . 
u. a 
N +. 
+ diverges to - \ a 
a 
. Koopmans - pag. 45