An introduction to log-linearizations(宏观经济理论-英文版).pdf

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