SEMAINE D'ÉTUDE SUR LE ROLE DE L ANALYSE ECONOMETRIQUE ETC. 
469 
Now K, ,/K, ,=1+1, /K, ,=1+x%x,_; Y,_/K,_,from (2.2, 
and (2.3). Since Y,_,/K, ,=1/p from (2.1), we conclude that 
the ratio of K,_; to K,_, is equal to 1+x,_,/¢. Taking loga- 
rithms in (2.6), we then find the following expression for the 
first difference of the logarithm of per capita consumption: 
(2.7) — A [log (C;/N,)] = € 
log (1 + x,_1/2) — 
log (1 + v,) . 
This equation is the only aspect of the model that will be 
used in the sequel. We shall consider a decision maker who is 
interested in two things: the savings ratio x,, which he controls, 
and the logarithmic rate of change of per capita consumption. 
which he does not control. The approach to be followed requi- 
res that the latter (uncontrolled) variable be expressed linearly 
in the former. Eq. (2.4) is nonlinear and it will therefore be 
linearized. Taking all logarithms as natural logarithms, we 
shall approximate the log of 1+x,_,/¢ by x,_,/p. We shall 
also approximate the log of 1+v, by v, (although this is not 
strictly necessary, since the expression does not involve the 
savings ratio). For the first term on the rirht we use- 
(2.8) 
But this is still nonlinear in x,_; and we will therefore apply 
the following (crude) approximation (3). We should expect 
that the savings ratio will be of the order of 15 to 25%, so that 
1/(1-x,_,) is then of the order of 1.2 or 1.3. This range of 
ancertainty is not very sizable; moreover, we multiply 
1/(x-x,_,) by x,- x,_,, which is generally close to zero, par- 
ticularly since we shall put a penalty on large values of 
‘?) But see footnote 5 below for a more accurate approximation 
71 Theil - pag. 
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