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 attainable
 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 Figure
 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 consumption,
 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