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245 
(DY) The optimal bath satisfies the condition 
(18) 
wx) 2.=14 — ulx. 
that at any time the net increase in capital per worker mul- 
tiplied by the marginal utility of consumption per worker equals 
the net excess of the maximum sustainable utility level over 
the current utility level. 
This condition is similar to the KEyNES-RAMSEY condition 
[RAMSEY, 1028, equation (5)] formulated in terms of absolute 
amounts of consumption, and reverts to it for A=0. KEYNES 
intuitive reasoning in support of this condition carries over 
with only slight reinterpretation. 
A number of analogous results can be obtained if the uti- 
lity of a consumption path is defined as an integral over the 
instantaneous utility flow discounted at a positive instantaneous 
rate p. 
(EY The utility function 
(19) Va) = 
20 
[ oot ulx,) dt. where 5 > o 
is defined for all feasible paths for which x,>>x for all t, when- 
ever X >0 
RAMSEY’s device is therefore unnecessary in this case. 
We shall however obtain an economy of notation if instead 
of V(p) we use the utility function 
20) 
“lo. 
U.a 
~~ 0 
.| Koopmans - pag. 19