306 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - oR 
cause it gives a higher value for C, at all times. However, if 
the total period considered were restricted to a horizon T as 
shown in the diagram, programs 1, 2 and 3 would appear as 
optimal, each one with respect to an appropriate terminal con- 
dition on Ki. 
If the social interest rate €, used in the definition of the 
utility function, is positive, then condition 1 is satisfied by 
the regular program @” that gives the highest value to Co. This 
program #7 is therefore optimal. Its optimality may still be 
proved directly for the case e =0. But, when the social rate of 
interest € is negative, no program is optimal. 
For instance, program #2 is preferred to the program #! 
exhibited on the diagram. However a better program can be 
found as follows when e<o : up to time T, select a regular 
program close to # but allowing a little higher value of K,, 
and therefore a little smaller value of C,; at time T take an 
extra consumption by reducing Kr to its value in #2, there- 
after continue with program #. The program thus defined is 
not optimal either, because one would prefer to postpone ever 
farther in the future the time T at which one switches back 
to 4? 
In section 5, I consider the case in which the production 
function would simply imply a fixed capital-output ratio. The 
determination of regular programs then boils down to the so- 
lution of a recurrence equation on c,. 
Choosing a type of utility function proposed by R. Frise 
and more recently used by J. TINBERGEN, I find that an optimal 
program exists only when the social rate of interest ¢ exceeds 
a positive minimum that may be of some 10% per annum. 
This suggests that, in the programming of future development, 
one can hardly avoid taking into account the decrease in the 
marginal productivity of capital, unless one discount heavily 
against the future. 
5] Malinvaud - pag. 6