SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 255 
but some weight is explicitly given to population size, it will 
always increase utility to delay still further the time at which 
the fruit of the initial sacrifice is reaped. 
In the proofs of Propositions (A) - (K), given in the Ap- 
pendix, one common characteristic of the problems considered 
is repeatedly used without explicit mention. At any time in 
an optimal path (%,, £,), the capital stock #, is the only link 
between the past and the future. This is due, on the one hand, 
to the utility function being an integral over time of instan- 
taneous utilities (discounted or not). On the other hand, it 
arises from the fact that the feasibility constraint (10 a) restricts 
z, but not %,. Hence the function x, is in principle free to vary 
discontinuously (even though it is found optimal for it not to 
do so). However, Z, is bounded by (10 a, b), hence z, can only 
vary continuously. The resulting property can be expressed. 
formally as follows: If (£,, #,) is an optimal path for given 
then, for any T, the path (-"* ” defined +- 
is optimal for <- 
7. 
ADIUSTING PREFERENCES TO OPPORTUNITIES 
What have we learned from our « logical experiments »? 
We have confronted a simple model of production with a utility 
function representing a sum of future per-capita utilities, dis- 
counted by a positive, zero, or negative instantaneous rate of 
discount p. We have found that g=o0 is the smallest rate for 
which an optimal path exists 
41 Koopmans - pag. 29