Upload
cadeau01
View
222
Download
0
Embed Size (px)
Citation preview
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 1/15
1
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 2/15
Organization Introduction
Basic Idea
Log-Linearization
Examples
2
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 3/15
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.
3
1. Introduction
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 4/15
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.
4
2. Basic Idea
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 5/15
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:
5
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
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 6/15
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,
6
( ) ( ) ( )( ) 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
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 7/15
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)
7
3. Log-Linearization
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 8/15
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).
8
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 9/15
Cobb-Douglass production function:
Take logs
Do the first order Taylor series
9
4. Examples
1
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
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 10/15
Simplify as percentage deviations
Note that,
Therefore,
Using “hat” variables being percentage deviationsfrom steady state, we have
10
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
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 11/15
Note that
Given the fact that
11
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
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 12/15
12
The economy resource constraint:
Take logs
Do the first order Taylor series
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
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 13/15
13
Capital accumulation equation:
Take logs
Do the first order Taylor series
Simplify as percentage deviation
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
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 14/15
14
Consumption Euler equation:
Take logs
Do the first order Taylor series
Simplify as percentage deviation
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
11
(1 )r
and
7/27/2019 Notes on Log-Linearization
http://slidepdf.com/reader/full/notes-on-log-linearization 15/15
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
15
ˆ ( )t t r r r
11
(1 )r
1ˆ ˆ ˆ1t t t c c r
1