470 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
|*; = *,_,| in Section 3. Therefore, we shall approximate the 
expression (2.8) by -b(x,-x,_,), where b is regarded as a 
fixed coefficient (about equal to 1%). 
Equation (2.7) is now written as 
(2.9) À [log (C,/N,)] = - bx, + (b+1/e)x,_, - v,, 
which is the form with which we shall work in the remainder 
of this paper. 
3. THE PREFERENCE FUNCTION 
Our decision maker is supposed to formulate a quadratic 
preference function which he wishes to maximize. We shall 
consider a very simple quadratic function, which amounts to 
a sum of squares as far as the successive log-changes in per 
capita consumption is concerned. This should not be regarded 
in the sense that we really believe that the decision maker’s 
desires are quadratic; it means only that we try to ap- 
proximate the decision maker’s preferences by a quadratic 
function in the relevant range (in the same way as the cons- 
traints of the economy are approximated linearly in the rele- 
vant range). 
Specifically, let d, be the « desired rate » of increase of the 
logarithm of per capita consumption in year £. For example, 
d,=0.1 (10%); then, given the quadratic character of the 
preference function, an actual rate of increase of 5% will have 
a disutility of (5 — 10)”=25, a 4% increase will have a disuti- 
lity of (4 - 10)?=36, and so on. Let us write y, for the discre- 
pancy between the actual and the desired rate of increase in 
year t: 
(3.1) 
y,=ATlog (C,/N,)1 - d, 3 
271 Theil - pag. 6