252 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 
28 
is uniformly bounded from above for all feasible paths (x}, =) 
and all values of T. The following statement says that no 
such path exists. 
(K) For each 0>>o0, for each attainable path (x,, z,), where 
O<Z, SZ, and for each number N>o, there exist another attain- 
able path (Xi, z;) and a number T* such that 
2) 
W*(-0)>N for all T>T* 
This says, essentially, that there is no upper bound to the 
range, on the attainable set, of a utility function of the type 
we are seeking to define. The case p<o is therefore analogous 
to the case in ordinary linear programming illustrated by Fi- 
gure 4. The same difficulty was noticed and discussed by 
TINBERGEN [1960] and by CHAKRAVARTY [1962] in connection 
with the case p=o0 for a model with constant returns to increases 
in the amount of capital alone. 
In the present case, the reasons for the absence of an 
optimal path for p<o can be illustrated in terms of the path 
(Xp 2;)=(%, 2), optimal if p=0 and z,=2. From (21) we see 
that the implicit price of the unit of consumption good per 
worker, associated with this path would have to be a constant, 
p,=u'(à) for all 
t 
This means that a sacrifice of one unit in per capita con- 
sumption, now made for a short period as a slight departure 
from this path, can be taken out by any future generation in 
the form of an equal augmentation of per capita consumption 
beyond that provided by the path, for a period of the same 
short duration. Now if either the discount rate p<<o, or if o=0 
4] Koopmans - pag. 28