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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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.