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An introduction to log-linearizations(宏观经济理论-英文版)
An introduction to log-linearizations
Fall 2000
One method to solve and analyze nonlinear dynamic stochastic models is to
approximate the nonlinear equations characterizing the equilibrium with log-
linear ones. The strategy is to use a first order Taylor approximation around
the steady state to replace the equations with approximations, which are linear
in the log-deviations of the variables.
Let Xt be a strictly positive variable, X its steady state and
xt ≡ log Xt − log X (1)
the logarithmic deviation.
First notice that, for X small, log(1 + X ) X , thus:
Xt
x ≡ log(X ) − log(X ) = log( ) = log(1 + %change) %change.
t t
X
1 The standard method
Suppose that we have an equation of the following form:
f (X , Y ) = g (Z ). (2)
t t t
where Xt , Yt and Zt are strictly positive variables.
This equation is clearly also valid at the steady state:
f (X, Y ) = g (Z). (3)
To find the log-linearized version of (2), rewrite the variables using the iden-
tity X = exp(log(X ))1 and then take logs on both sides:
t t
log(X ) log(Y ) log(Z )
log(f (e t , e t )) = log(g (e t )). (4)
Now take a first order Taylor approximation around the steady state (log(X ),
log(Y ), log(Z)). After some calculations, we can write the left hand side as
1
log(f (X, Y )) + [f (X, Y )X (log(X ) − log(X )) + f (X
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