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
