Notes on Log-Linearization

7/27/2019 Notes on Log-Linearization

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Organization Introduction

Basic Idea

Log-Linearization

Examples

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The solutions to many discrete timedynamic economic problems take theform of a system of non-lineardifference equations.

Many modern economic models are

hard to solve due to non-linearities.

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1. Introduction



One method to solve and analyze nonlineardynamic stochastic models is to approximatethe 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.

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2. Basic Idea



Consider an arbitrary univariate function,Taylor’s theorem tells us that this can be

expressed as a power series about a particular

point x, where x belongs to the set of possible x values:

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Taylor Series

( ) f x

2( ) ( )( ) ( ) ( ) ( )

1! 2!

f x f x f x f x x x x x

3( )( ) ...

3!

f x x x



For a function that is sufficiently smooth,the higher order derivatives will be small,and the function can be well approximated(at least in the neighborhood of the point of evaluation, x) linearly as:

For an arbitrary multivariate function,

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( ) ( ) ( )( ) f x f x f x x x

( , ) ( , ) ( , )( ) ( , )( ) x y f x y f x y f x y x x f x y y y



One particularly easy and very commonapproximation technique is that of log-linearization.

First, take natural logs of the system of non-linear difference equations.

Linearize (using Taylor series) the loggeddifference equations about a particularpoint (usually a steady state)

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3. Log-Linearization



Simplify until we have a system of

linear difference equations where the variables of interest are percentagedeviations about a point (again, usually a steady state).

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Cobb-Douglass production function:

Take logs

Do the first order Taylor series

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4. Examples

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t t t t y a k n

ln ln ln (1 ) lnt t t t y a k n

1 1ln ( ) ln ( )t t y y y a a a

y a

(1 )ln ( ) (1 ) ln ( )t t k k k n n n

k n



Simplify as percentage deviations

Note that,

Therefore,

Using “hat” variables being percentage deviationsfrom steady state, we have

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1 1 (1 )( ) ( ) ( ) ( )t t t t y y a a k k n n

y a k n

ln ln ln (1 ) ln y a k n

ˆˆ ˆ ˆ(1 ) y a k n



Note that

Given the fact that

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then ˆlog logt t z z z

log( )t z

z

log(1 ) log(1 % )

t z z

change z

%change

In words, log-deviations can be interpreted as

percentage deviations from the steady state

ˆ log logt t z z z

log(1 ) z z z for a small



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The economy resource constraint:

Take logs


Simplify as percentage deviation

t t t y c i

ln ln( )t t t y c i

1 1 1ln ( ) ln( ) ( ) ( )

( ) ( )t t t y y y c i c c i i

y c i c i

ˆˆ ˆ

c i y c i

y y



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Capital accumulation equation:

Take logs



1 (1 )t t t k i k

1ln ln( (1 ) )t t t k i k

1

1ln ( ) ln( (1 ) )t k k k i k

k

1ˆ ˆˆ (1 )t t t

ik i k

k

1 1( ) ( )

( (1 ) ) ( (1 ) )t t i i k k

i k i k



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Consumption Euler equation:

Take logs



1( ) (1 )t t

t

cr

c

1ln ln ln ln(1 )t t t c c r

1ln ( ) ln ( )t t c c c c c cc c

1ln ln(1 ) ( )

(1 )t r r r

r

1

1ˆ ˆ ˆ

t t t c c r

with ˆ ( )t t r r r

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(1 )r

and



Note that Since interest rate (r) is already a percent, it is common

to leave it in absolute (as opposed to percentage)deviations. Therefore,

We approximate the term

The equation says that the growth rateof consumption is approximately proportional to the

deviation of the real interest rate from steady state interpreted as the elasticity of inter-temporal

substitution of consumption

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ˆ ( )t t r r r

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(1 )r

1ˆ ˆ ˆ1t t t c c r

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Notes on Log-Linearization

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