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. Nwith
 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 complete.
 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.