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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
multiple d - d, has to be added. This multiple is obviously 
positive, since the increasing desires in later years require more 
saving in the beginning. 
6. THE SIMULATION TECHNIQUE 
We shall now apply the ideas set forth numerically by 
means of a simulation technique. Thus we consider a number 
of countries whose economies are taken care of by an equal 
number of decision makers. These decision makers control the 
savings ratios of their respective countries and do so, year 
after year, on the basis of the decision rule (5.10). That is, 
each of them maximizes the expectation of the utility function 
(3.7) subject to the random constraint (3.4), where it is assumed 
that the desired values of the log-changes in per capita con- 
sumption are of the form (5.9) and that the rate of change of 
the population satisfies the stochastic difference equation (5.7). 
This equation supplies the random element of the process, 
which is of course the rationale of the simulation technique. 
Fifty countries have been considered, each during a period 
of 50 years. All start with an initial savings ratio (x,) of 10%. 
The first-year desired increase (d,) of per capita consumption 
is 2% %, the long-run desired increase (d) is 74%. The coef- 
ficient 7 of (5.9) is put equal to 0.95, which implies that after 
about #=15 years there is a desired increase d, half-way be- 
tween the first-year and the long-run desire, d,= (4, +d)=5%- 
The expected value of the rate of increase of the population, 
v, is taken equal to 0.015. The random variables e, of (5.7) 
have been generated as normal variates with zero mean and 
three alternative standard deviations: 0.005, o.or and 0.02. 
These alternatives are chosen to illustrate the importance of 
7] Theil-- pag. 16