# Full text: Study week on the econometric approach to development planning

```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
``` ## Note to user

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