A 
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
(il) « consumption expenditure at implicit prices » 
I 
| 2 x, dt 
. 
reaches its minimum, on the set of paths with utility 
Uz(p) equal to or exceeding that of the path (£, 2), if 
(x,, 2,)= (%X,, Z,). 
(G) If (x, 5), (x, z,) ave two eligible and attainable paths, 
5) 
r I 
Joue — dt = [late — 3) — bé, — 21) dt = 
9 5 
= lea, +5) (2,— 2,) ar — plan — 2, 
for all finite T, and for T=oo if the integrals converge and 
the last term has a limit. 
Again, all three members vanish if (x,, 2,) is itself the path 
(#,, 2,). The inequality in (25), rewritten as 
I 
[puta + 2, — Ly — 2. di 
1 
[ae —2ndt=o ; 
4] Koopmans - pag. 22