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