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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 ≡ log(Xt) − log(X) = log(
Xt X
) = log(1 + %change) ' %change.
1 The standard method
Suppose that we have an equation of the following form:
f (Xt, Yt) = g(Zt). (2)
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 Xt = exp(log(Xt))^1 and then take logs on both sides:
log(f (elog(Xt), elog(Yt))) = log(g(elog(Zt))). (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
log(f (X, Y )) +
f (X, Y )
[f 1 (X, Y )X(log(Xt) − log(X)) + f 2 (X, Y )Y (log(Yt) − log(Y ))]. (5) (^1) This procedure allows us to obtain an equation in the log-deviations.
Similarly, the right hand side can be written as
log(g(Z)) +
g(Z)
[g^0 (Z)Z(log(Zt) − log(Z))]. (6)
Equating (5) and (6), and using (3) and (1), yields the following log-linearized equation:
[f 1 (X, Y )Xxt + f 2 (X, Y )Y yt] ' [g^0 (Z)Zzt]. (7)
Notice that this is a linear equation in the deviations! Generalizing, the log-linearization of an equation of the form
f (x^1 t , ..., xnt ) = g(y^1 t , ..., ynt )
is:
X^ n
i=
fi(x^1 , ..., xn)xixit '
X^ m
j=
gj (y^1 , ..., ym)yj^ yjt.
2 A simpler method
However, in the large majority of cases, there is no need for explicit differenti- ation of the function f and g. Instead, the log-linearized equation can usually be obtained with a simpler method. Let’s see. Notice first that you can write
Xt = X(
Xt X
) = Xelog(Xt^ /X)^ = Xext
Taking a first order Taylor approximation around the steady state yields
Xext^ ' Xe^0 + Xe^0 (xt − 0) ' X(1 + xt)
By the same logic, you can write
XtYt ' X(1 + xt)Y (1 + yt) ' XY (1 + xt + yt + xtyt)
where xtyt ' 0 , since xt and yt are numbers close to zero. Second, notice that
f(Xt) ' f(X) + f^0 (X)(Xt − X) ' f(X) + f^0 (X)X(Xt/X − 1) ' f(X) + f(X)η(1 + xt − 1) ' f(X)(1 + ηxt)
Notice that at the steady state
Rσ−^1 βσ^ = 1 − Π.
and
Π = 1 − Rσ−^1 βσ.
Using (8) and (9) we can write the nonlinear difference equation as
Rσ−^1 βσ(1 + (σ − 1)rt+1 + πt − πt+1) ' 1 − (1 − Rσ−^1 βσ)(1 + πt).
Canceling out constants yields
Rσ−^1 βσ[(σ − 1)rt+1 + πt − πt+1] ' −(1 − Rσ−^1 βσ)πt.
Rearranging, we obtain
Rσ−^1 βσ^ − 1 Rσ−^1 βσ^
πt ' (σ − 1)rt+1 + πt − πt+
and, finally,
πt ' Rσ−^1 βσ^ [(1 − σ)rt+1 + πt+1].
2.1.3 The Euler equation
The consumption Euler equation is
1 = Rt+1β(Ct+1/Ct)−γ^.
Using (9) and (10) we can write it as
1 ' Rβ(1 + rt+1 − γ(ct+1 − ct)).
Canceling out constants yields
0 ' rt+1 − γ(ct+1 − ct)
and, rearranging,
ct ' −σrt+1 + ct+
where σ = 1/γ is the intertemporal elasticity of substitution.
2.1.4 Multiplicative equations
If the equation to log-linearize contains only multiplicative terms, there is a faster procedure. Suppose we have the following equation:
XtYt Zt = α
where α is a constant. To log-linearize divide first by the steady state variables:
( X Xt )( Y Yt ) ( Z Zt )
α α
Now take logs:
log( Xt X
) + log( Yt Y
) − log( Zt Z
) = log(1) = 0.
Using (1) we arrive then easily to the log-linearized equation:
xt + yt − zt = 0.
Notice that in this case the log-linearized equation is not an approximation!
