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{(« adjusting preferences to opportunities »). Prof. KoopMANs has 
found that the traditional approach to optimal growth (which con- 
sists of maximizing utility over time, by accepting a certain rate of 
discount of utility, called p, as given by individual preferences) can- 
not always be applied. More precisely, he has found that it can be 
applied only when the time horizon considered is finite. When time 
is allowed to run from o to oo, then p<70 becomes impossible 
calthough p>o still remains possible) because there simply would not 
exist a utility function to be maximized, 
Thus — Prof. Koopmans concludes — the open-endedness of the 
future imposes limits on individual preferences. He seems to be 
so surprised and even so afraid of this result as to prefer, at this 
point, to begin to speculate on the meaning of all this. 
I would suggest that the mathematical exercise should be com- 
pleted, by allowing time to run from —oo to +oo (and not only 
from 0 to +00). I may add perhaps that to consider time as running 
from —oo to +oo does not mean allowing time to run in reverse. 
It simply means putting ourselves in a slightly different position with 
respect to the one Prof. KooPMANS has chosen. Instead of saying, 
as he does: suppose we begin our process of maximization at time 
zero, whatever happened before; we say: suppose that optimization 
has been taking place since the beginning of time. (This, by the 
way, appears to me a more logical approach to take in the context 
of Prof. KooPMANS’ stationary society). Now, if we allow time to 
run from —oo to +00, it is easy to see that, in a stationary economic 
system, also p>>0 becomes impossible. The only value of p that 
makes any process of utility maximization over infinity possible is 
0=0. 
Prof. KooPMANS might be even more surprised. For, by following 
his arguments, we should conclude that individuals have not even 
a limited inter-temporal preference choice: they have no choice at all. 
But is it so? This conclusion — it seems to me — is fallacious, 
although of course the mathematical results are correct. And the 
fallacy stems from not bringing out explicitly the implications of the 
following theorem: on the optimum growth path (by which I mean 
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