SEMAINE D ETUDE SUR LE ROLE DE L ANALYSE ECONOMETRIOUE ETC. 
27C 
fore (£,, £,) is uniquely optimal for the given 2,. This comple- 
tes the proof of Propositions (H), (I), (J). 
For completeness we consider two additional questions. LI 
(x,, z,) is eligible and attainable, we have from (2) 
F. 
mn 
*F} 
r 
9, 
q(z4)- 
“ 
dl 2 
ar. 
Pp, G(2) + psy) dt — py 2, 
Examining the behavior of p, and br for t> one finds from 
this formula that all the integrals occurring in statements (i 
(ii), (iii) interpreting (F) and (I) converge for T—oc, 
Finally, what rules out the trajectories in Figure 17 for 
which x,#2%,? Those with x,>% reach the boundary z=o0 at 
some finite time, making it impossible to satisfy both (65) and 
(35 a) for all £Zo0. For each x; with o<x' <£y there is a 
unique attainable path (x7, Z}) satisfying (65) for all #Zo, but 
in such a way that lim x;=0. This must entail either the 
t>o 
ineligibility of (x} z;), or the unboundedness of p* associated 
with that path by (21), because otherwise (30) with (xt. «+ #, 
substituted for (£, 2,, p,) would imply the optimalitv of : 
path (x3, 23) already proved nonoptimal. 
A 8. 
PROOFS FOR A NEGATIVE DISCOUNT RATE 1 
We shall need the following lemma. 
LEMMA 4. If ¢(x) ts a positive and nonincreasing function 
of x defined for all x>o, and if x, is a positive integrable func- 
tion of t on the interval [T'. T2Y TY<T2. such that 
+1 Koopmans - pag. 55