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 logarithms
 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 requires
 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, particularly
 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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