304 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
traints. Condition 1 is specific to the consideration of unlimited 
programs. We shall have occasions to see in the following 
examples that it is not vacuous. 
We may also observe that the equations «,=o0, §,=0 imply 
together with the equilibrium condition at time # a set of dif- 
ference equations which may usually be solved recursively for 
all C,, K,, N, starting from any given allocation of S, between 
Co and K,. Thus, the regular programs generate a one pa- 
rameter family of programs, the parameter being C, (or equi- 
valently Ko, S, being given). Typically, one and only one 
program of this family will meet condition 1 and therefore be 
optimal. A possible computational procedure would be first to 
determine the regular program corresponding to any fixed 
initial allocation, and then to find by trials and errors the initial 
allocation that leads to fulfilment of condition 1. 
Before turning to specific examples, we may look at the 
economic interpretations of equations (22), stating that the «,’s 
and §3,’s are all zero, and of condition 1. 
The first equation (22) may be written as: 
34) I+fæ = (1+e) (1+T,) (I+U,) 
f= being the marginal productivity of capital, or the equili- 
brium rate of interest, e the social rate of interest used for the 
discount factor of the utility function, =, the rate of increase 
of the population, and u, the rate of decrease of the marginal 
ut'ity of consumption. This equation exhibits in a suggestive 
way the three components which explain the equilibrium rate 
of interest. It is similar to relations recently put forwards by 
Sir Roy HARROD and Professor RAGNAR FRISCH. 
Condition 1 implies that the equilibrium rate of interest be 
larger than the rate of increase in the stock of good S,, and the- 
refore that the present value of S, decreases to zero with £. 
5] Malnvaud - pag.