226 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 98 
Contrasting with this view is the position, expressed among 
others quite explicitly by Professor ALLAIS [1947, Ch. VI, X], 
that the balancing of the interests of different generations is an 
sthical or political problem, in which the competitive market 
solution has no valid claim to moral superiority over other 
solutions that depend for their realization on action by the 
state. À more specific optimality concept is implied in the 
strictures of Professor HARROD [1048, p. 40] and of FRANK 
RAMSEY [1928, p. 543] against any discounting of future uti- 
lites. These authors leave little doubt that they regard only 
equal weights for the welfare of present and future generations 
as ethically defensible. 
The purpose of the present paper is to do some « logical 
experiments », in which various mathematical forms of the 
optimality criterion are confronted with a very simple model 
of technology and of population growth, to see what their 
maximization leads to. Our study is similar in purpose to 
RAMSEY’s classical paper, and to TINBERGEN’s recent explo- 
ration [1960] of the same problem. The underlying idea of 
this exploratory approach is that the problem of optimal 
growth is too complicated, or at least too unfamiliar, for one 
to feel comfortable in making an entirely a priori choice of 
an optimality criterion before one knows the implications of 
alternative choices. One may wish to choose between prin- 
ciples on the basis of the results of their application. In order 
‘0 do so, one first needs to know what these results are. This 
is an economic question logically prior to the ethical or political 
choice of a criterion. 
What is a suitable mathematical formalization of the idea 
of an optimality criterion? The most basic notion is that of a 
preference ordering of growth paths. Such an ordering states 
for each pair of alternative growth paths whether they are 
equally good, and if not, which is preferred. Indifference, pre- 
ference and preference-or-indifference are usually required to 
be transitive. 
4] Koopmans - pag. 2