Noah Williams
Economics 714
Department of Economics
Macroeconomic Theory
University of Wisconsin
Spring 2010
P
ROBLEM
S
ET
3
Solutions by Tim Lee
1.
(a) Although households take prices, taxes and transfers as given, it must be able
to project next period’s prices and taxes in order to solve its problem. So we
need to introduce a
perceived law of motion
, which we do by assuming rational
expections. To do this, define the aggregate state as the vector
X
= [
K z
]
.
D
EFINITION
1
A recursive competitive equilibrium (RCE) with a government is a
value function V
(
k
;
X
)
, a set of decision rules
{
c
(
k
;
X
)
,
k
0
(
k
;
X
)
,
n
(
k
;
X
)
}
, a set of
prices
{
r
(
X
)
,
w
(
X
)
}
, transfers
{
T
(
X
)
}
, and a perceived law of motion for capital
K
0
(
X
)
, given the tax rate
τ
and transistion function P
(
z
0
,
z
)
and N
=
1
, such that
i. Given prices, taxes, transfers and the law of motion, the decision rules solve the
household’s problem:
V
(
k
;
X
) =
max
c
,
k
0
u
(
c
) +
β
Z
z
0
V
(
k
0
;
K
0
(
X
)
,
z
0
)
dP
(
z
,
z
0
)
c
+
k
0
= (
1

τ
)(
w
(
X
)
n
+
r
(
X
)
k
) + (
1

δ
)
k
+
T
(
X
)
ii. For all X
= [
K z
]
,
(
K
,
N
)
solves the representative firm’s problem given prices:
zF
K
(
K
,
N
)
=
r
(
X
)
zF
N
(
K
,
N
)
=
w
(
X
)
iii. For all X, the government balances budget:
T
(
X
) =
τ
(
w
(
X
)
N
+
r
(
X
)
K
)
iv. Markets clear, i.e. k
0
(
k
;
X
) =
K
0
(
X
)
and n
(
k
;
X
) =
N
=
1
for all
[
k
;
X
]
,
v. k
=
K, i.e. aggregate state equals individual state. This is required due to the
representative agent setting.
vi. The law of motion is induced by
K
0
(
X
) =
zF
(
K
,
N
) + (
1

δ
)
K

c
(
K
;
X
) =
k
0
(
K
;
X
)
,
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
(b) Given today’s aggregate state
X
and the law of motion
K
0
(
X
)
, the household
rationally projects next period’s aggregate state as
X
0
= [
K
0
(
X
)
,
z
0
]
. Since there
is no preferences for leisure,
n
(
k
;
X
) =
1 for all
k
,
X
. To write a FE in terms of
k
0
(
k
;
X
)
, it will be easier to let the agent’s state variable be his wealth, which in
turn is a function of
[
k X
]
:
a
(
k
;
X
)
≡
(
1

τ
)[
w
(
X
) +
r
(
X
)
k
] + (
1

δ
)
k
+
T
(
X
)
,
so from the perspective of the agent who chooses
k
0
but forecasts
X
0
, it faces the
individual law of motion:
a
(
k
0
;
X
0
) = (
1

τ
)[
w
(
X
0
) +
r
(
X
0
)
k
0
] + (
1

δ
)
k
0
+
T
(
X
0
)
.
(1)
Now let
V
(
a
(
k
;
X
)
,
z
)
≡
V
(
k
;
X
)
, the Bellman equation for the HH is
V
(
a
(
k
;
X
)
,
z
) =
max
k
0
u
(
a
(
k
;
X
)

k
0
) +
β
Z
z
0
V
(
a
(
k
0
;
X
0
)
,
z
0
)
dP
(
z
,
z
0
)
subject to the law of motion (
1
). The f.o.c. at the solution is
u
0
(
a
(
k
;
X
)

k
0
(
k
;
X
))
=
β
Z
z
0
(
1

τ
)
r
(
G
(
X
)
,
z
0
) +
1

δ
·
V
a
(
a
(
k
0
(
k
;
X
)
;
X
0
)
,
z
0
)
dP
(
z
,
z
0
)
and combining with the envelope condition
V
a
(
a
(
k
;
X
)
;
X
) =
u
0
(
a
(
k
;
X
)

k
0
(
k
;
X
))
we get the Euler equation
u
0
(
a
(
k
;
X
)

k
0
(
k
;
X
))
=
β
Z
z
0
(
1

τ
)
r
(
X
0
) +
1

δ
·
u
0
a
(
k
0
(
k
;
X
)
;
X
0
)

k
0
(
k
0
(
k
;
X
)
;
X
0
)
dP
(
z
,
z
0
)
.
Now being explicit about rational expectations and the individual’s state vari
ables, we rewrite
u
0
(
a
(
k
;
K
,
z
)

k
0
(
k
;
K
,
z
)) =
β
Z
z
0
(
1

τ
)
r
(
K
0
(
K
,
z
)
,
z
0
) +
1

δ
×
(2)
u
0
a
(
k
0
(
k
;
K
,
z
)
;
K
0
(
K
,
z
)
,
z
0
)

k
0
(
k
0
(
k
;
K
,
z
)
;
K
0
(
K
,
z
)
,
z
0
)
dP
(
z
,
z
0
)
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Economics, Inflation, Monetary Policy, Interest, Trigraph, mixed monetary equilibrium

Click to edit the document details