204A-noteLogLin

204A-noteLogLin - Note on Linearized Solutions to the...

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1 Note on Linearized Solutions to the Optimal Growth Model Econ 204A - Prof. Bohn This note reviews the linearized dynamics of the optimal growth model and derives log-linearized solutions. General Problem: Linearization Linearization is a common approach in macroeconomics to obtain approximate solutions. The basic principle is the Taylor series approximation. Consider first the univariate case. Suppose ) ( x h y = is a twice differentiable function that you want to linearize around point x . Taylor’s Theorem says that 2 2 1 ) ( ) ( " ) ( ) ( ' ) ( ) ( x x h x x x h x h x h ! " + ! " + = # where ! is between x and x . The last term is known as the error term, and it is often written as ] ) [( 2 x x O ! to highlight that the error grows with the square of the distance between x and x . A linear approximation is obtained by omitting the error term: ) ( ) ( ' ) ( ) ( x x x h x h x h ! " + # Note that this relationship not an equation, but an approximation. Taylor’s theorem applies similarly to multivariate functions. Suppose ) ,..., ( 1 n x x h y = is a twice differentiable function in n ! that you want to linearize at a point ) ,..., ( 1 n x x x = . Taylor’s theorem says ) ( ) ( ) ( ' ) ( ) ,..., ( 1 1 x x O x x x x h x h x x h i i i n i n ! + ! " # # \$ + = = where the last term an unspecified error that is a bounded multiple of the Euclidian distance between ) ,..., ( 1 n x x and x (that is has the same order of magnitude “O”). As approximation, this is written as ) ( ) ( ' ) ( ) ,..., ( 1 1 i i i n i n x x x x h x h x x h ! " # # \$ + % =

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2 Application to Optimal Growth In the optimal growth model, the non-linear differential equations for k and c are k g n c k f k ) ( ) ( ! + + " " = ! c g k f c ! " " " = ] ) ( ' [ 1 # \$ % ! The steady state conditions are f '( k * ) = + " + g and c * = f ( k * ) ! ( n + g + ) k * In logarithms, one can write )) ln( ), (ln( ) ( ) ( ) ( / / ) ( / 1 ) ln( ) ln( ) ln( ) ln( ) ln( c k h g n e e e f g n k c k k f k k k c k k dt k d = + + ! ! = + + ! ! = = ! ! ! )) (ln( ) ( ) ( ' / 2 1 ) ln( 1 ) ln( k h g e f c c k dt c d = ! ! ! = = ! The Taylor series approximation at ) , ( ) , ( * * k c k c = simplifies because 0 )) ln( ), (ln( * * 1 = c k h and 0 )) (ln( * 2 = k h . The relevant partial derivatives are: k c k f k c k k k k c k k k k h f e e f e e f e e e f + ! = + " ! " = ! = ! ! ! ! ! # # # # ' ' ] ) ( [ ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( 1 k c k c k c c c h e e = ! = ! = ! ! " " " " ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( ] [ 1 , and k k f e e f e f k k k k k h ) ( " ) ( " )] ( ' [ 1 ) ln( ) ln( 1 ) ln( 1 ) ln( ) ln( 2 = = = " " " " Evaluating them at ) , ( * * k c one obtains ) ln( ) ln( ) ln( ) ln( ] ' [ * * * * * * * * * * * c c k c k k c c k c k k k c k f k k f ! " = ! " + ! # ! and ) ln( * * * ) ( " k k k k f c c ! " ! . Collecting c- and k-terms, the log-linearized equations can be written in matrix form as: ! ! " # \$ \$ % & = ! ! " # \$ \$ % & ( ( ! ! " # \$ \$ % & ( ( = ! ! " # \$ \$ % & ) / ln( ) / ln( ~ ln ln ln ln 0 / / * * * * * * c c k k A c c k k c c k k k c ) * ! ! for ˜ A = " c * k * " 0 \$ % & & ( ) ) where = " 1 f "( k * ) k * > 0 . This is a system of homogenous linear differential equations with constant coefficients.