230 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA -

22

utility functions that on a priori grounds appear quite plausible
and reasonable do not permit determination of an optimal
growth path even in a constant technology. Tentative and
intuitive explanations for this finding are offered.
Section 8 discusses in a tentative way, and without proofs,
possible extensions of the analysis to a changing technology
and/or a variable rate of population growth, with none, one,
or both of these regarded as policy variables.

3. PERTINENT ASPECTS OF LINEAR AND OF CONVEX PROGRAM-T
 MING

Let linear programming be applied to an allocation problem
in terms of the quantities x,, j=1, ..., » of a finite number #
of commodities. Then the feasible set D is given by a finite
number of linear inequalities

n
2 aj; XL; = b; ,
fo

2 1, .….. M

The objective function, or maximand, is a linear form in the x,

<

i
U =
2 i x;

The feasible set D is always closed, and may be bounded
(as in Figure 1) or unbounded (Figure 2).
The range R of the objective function on the feasible set
{the set of values assumed by the maximand on the points of
the set D) is an interval. If D is bounded (contained in some
hypercube), then R is necessarily also bounded. If D is un-"41

 Koopmans - pag. 6