SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 
26. 
Figure 11 shows z, for a path with two bulges, both denoted 
[T, T*]. The effect on the utility integra! «= strai<htening 
out» a bulge is found from (43) by taking Giz#, 
and satisfies 
(46) 
ve 
| 
ee! 
(tx 
— Wl 
if z*#%(p), because in that case g'(z*)-¢ and z,-z* are 
opposite in sign. If 2*=2(p), and if for instance fo. 
for T<¢<T* as in the second bulge in Figure 11, we can 
by suitable choice of a number z*- <{3(g) write the left hand 
member of (46) as the negative sum of two such integrals, 
one comparing (x,’, z; ) defined by z;=max {z**, z} with 
(x*, 2*)=(£(¢), #(e)) on [T, T*], the other comparing (x, z,) 
with (x**, 2**), where x**=g(z**), on an interval [T**, T*#+] 
such that TT**<{T***<T*. Since of these integrals the 
former is nonpositive, the latter negative, (46) is valid also if 
z*=2(c). We thus have 
LEMMA 1: For any ç, a path (x,, z,) optimal on any finite 
or infinite time interval cannot contain a bulee. 
This conclusion, and the inequality (46) on which it is 
based, remain valid for T° æ and çZo if the definition of a 
bulge is extended to read « (44 b) and either (44 a) or (44 a”, ». 
(44 a) o< 1 
- 
Tc 
, and ui ; =o then! . , 
Foy A 
as illustrated in Figures 12 (¢=o0) and 1, vu 
4] Koopmans - pag. 4