ECN/APEC 7240
Spring 2010
Homework 4 Solutions
1)
We augment the Consumption RBC Model with CobbDouglas production and
CRRA utility by including government spending financed by lumpsum taxes.
Let
T
t
be
the tax at
t
with
τ
t
= ln
T
t
=
τ
t
0
+
τ
t
1
.
We assume that
τ
t
0
=
τ
0
+
gt
and
τ
t
1
is an AR(1)
process such that
,
1
1
1
1
τ
ε
φτ
τ
+
+
+
=
t
t
t
where
φ
∈
(

1,1) and
ε
t
τ
is i.i.d and uncorrelated with
ε
t
a
.
ζ
=
T
0
/
Y
0
is exogenous.
a)
The Bellman equation for this model is
]
,

)
'
,
'
),
,
(
'
,
'
(
[
)
(
max
)
,
,
,
(
'
,
T
A
T
A
A
K
K
K
V
E
C
u
T
A
K
K
V
a
a
K
C
a
β
+
=
subject to
.
)
,
(
)
,
(
'
K
A
K
R
A
K
W
T
K
C
a
a
+
=
+
+
Note that the factor prices will still be
.
1
)
,
(
)
1
(
)
,
(
1
1
δ
α
α
α
α

+
=

=


A
A
K
A
K
R
A
K
A
K
W
Thus in equilibrium, where
K
a
=
K
, the budget constraint implies the incomeexpenditure
identity:
.
)
1
(
)
,
(
)
,
(
1
K
A
K
K
A
K
R
A
K
W
δ
α
α

+
=
+

K
Y
T
K
C
)
1
(
'
δ

+
=
+
+
.
The Lagrangian for the household’s problem is
]
'
)
,
(
)
,
(
[
]
,

)
'
,
'
),
,
(
'
,
'
(
[
)
(
T
K
C
K
A
K
R
A
K
W
T
A
T
A
A
K
K
K
V
E
C
u
L
a
a
a
a



+
+
+
=
λ
β
.
The firstorder conditions are