3EMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIOUE ETC. 
OTF 
Hence #, is monotonic and differentiable. In order to see that 
T=œ whenever z 72, one readily computes from TAYLOR 
expansions of g(z) — g(£) with respect to z - 2, and of #27. N- 
with respect to Rs 
|; 
/ 
|. 
1 
u 
a negative real number. 
A 
{2 <2 
AN 
i i) 
_1 .ollows that 
- ~ . 
t zy, >2 lm 
He 
— 
Therefore T=oc. The proofs of (C) and (D) are thereby com- 
plete. In addition. we note that £(z) is differentiable also for 
= 
A 7. PROOFS FOR A POSITIVE DISCOUNT RATE (0< g<_f (0) 
Proof of (E). Let (x,, 2z,, be a feasible path with : = 
for all &. In (4°) we insert}: ..< — z <2 such that gi. 
Then, if ..f uw YY x we have #<u(x, and hence 
o<T 
hence lim 1» 
T Tar 
+ 
rf 
—-u whenever 
1: } 
yes 
+1 Koopmans - pag. 5.