254 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
Let us assume for the sake of argument that the present 
model is representative enough to be looked on as a tentative 
test of the applicability of the ethical principles under consi- 
deration. Then we have just managed to avoid discriminating 
against future generations on the basis of remoteness of the 
time at which they live. However, this close escape for virtue 
was possible only by making welfare comparisons on a per 
capita basis. If instead we should want to weight per capita 
welfare by population size, then we are forced to discriminate 
on the basis of historical time by positive discounting. There 
seems to be no way, in an indefinitely growing population, 
to give equal weight to all individuals living at all times in the 
future. 
This dilemma suggests that the open-endedness of the future 
imposes mathematical limits on the autonomy of ethical thought. 
The suggestion may come as a shock to welfare economists, 
because no such logical obstacles have been encountered in 
the more fully explored problems of allocation and distribution 
for a finite population. It is true that the mere fact that we 
are considering an infinite number of people does not fully 
explain the dilemma. For Ramsey was able, albeit by artifi- 
cial assumptions, to indicate a fair solution to the problem 
for the infinite future of a population of constant size. Our dif- 
ficulty is therefore connected with the assumption of an inde- 
finite growth in the population. 
The following reasoning may further illuminate the reasons 
for the nonexistence of an optimal path with negative p. As- 
sume that 0>>p>f (2) - A. (Of course, p= - À would correspond 
to equal weights given to the utilities of all individuals. How- 
ever, f(z) -A>-X, and our illustration is simpler if we do 
keep 2(p)<<z by taking p>f(z) - ». Consider now an optimal 
path for the finite time period 0<¢<T, defined by initial and 
terminal per-worker capital stock levels z,=2r=2 both equal 
to that level £ which, if maintained at all times, would secure 
the maximum maintainable consumption per head. The analy- 
F4] Koopmans - pag. 30