204 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA 
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arbitrary, and has no justification; and I would interpret the results 
of his mathematical analysis as simply showing the impossibility of 
such an arbitrary restriction. I should conclude, therefore, that the 
misunderstanding has arisen from having introduced a p into the 
analysis at all. For, this has meant introducing exactly what Prof. 
KooPMANS has been afraid of, namely restrictions on consumers’ 
preferences. 
KOOPMANS 
I do not understand the operational meaning of Prof. PASINETTT’s 
suggestion to maximize utility over a period from —oo to +oo. The 
following comments apply therefore to maximization from o to oo, 
although I may thereby fail to do justice to PASINETTI’S thought. 
In the sentence in which Prof. PASINETTI refers to the « golden 
rule path », he uses the term optimum in a sense different from mine. 
[f an optimal path is defined as one that maximizes a utility function 
of the type I have discussed, the golden rule path is optimal only if 
both (a) the initial ratio of capital stock to labor force happens to 
coincide with that characteristic of the golden rule path, and (b) the 
chosen utility function has no discounting (p=0). If both these 
conditions are satisfied, the golden rule path is optimal in my sense 
as well, and as PASINETTI observes the interest rate p+ A equals the 
exogenously given growth rate \ of the labor force. However, if even 
only one of these conditions fails to hold, the optimal path, if one 
exists, differs from the golden rule path, and the interest rate differs 
from À most or all of the time, and is determined by the interplay 
of preferences and production possibilities I have analyzed. 
Finally, Prof. PASINETTT’s analogy with the one-period problem 
of utility maximization misses the main point of my paper. In the 
one-period problem with a finite number of commodities, an optimal 
consumption choice is bound to exist if the utility function is conti- 
nuous (a slight restriction on preferences!) and the opportunity set 
closed and bounded In the infinite-horizon case. there is a new 
4] Koopmans - pag. 70