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