Timing of Taxation
1
Ricardian Equivalence
Based on Barro (1974, JPE) and LS Ch. 10.
•
Question: Does the timing of (lump sum) taxes matter?
1.1
Environment
•
Unit measure of identical in
f
nitely lived agents as well as a govt.
•
Prefs:
P
∞
t
=0
β
t
u
(
c
t
)
•
Endowments
y
t
.
•
Household borrowing or lending
a
t
at price
Q
t
where
a
0
=0
•
Exogenous sequence of government expenditure
g
t
f
nanced by lump sum
taxes/transfers
τ
t
or one period government debt
D
t
at price
q
t
where
D
0
.
1.2
Equilibrium
•
Household problem
max
c
t
,a
t
+1
,D
t
+1
≥
0
∞
X
t
=0
β
t
u
(
c
t
)
(1)
s.t.
c
t
+
Q
t
a
t
+1
+
q
t
D
t
+1
=
y
t
−
τ
t
+
D
t
+
a
t
,
∀
t.
•
Starting in period
t,
can rewrite sequence of budget constraints as
Q
t
a
t
+1
+
c
t
+
q
t
D
t
+1
−
y
t
+
τ
t
−
D
t
=
a
t
Q
t
+1
a
t
+2
+
c
t
+1
+
q
t
+1
D
t
+2
−
y
t
+1
+
τ
t
+1
−
D
t
+1
=
a
t
+1
....
Substituting
a
t
+1
from the second line into the
f
rst yields
Q
t
Q
t
+1
a
t
+2
+
Q
t
q
t
+1
D
t
+2
+
Q
t
[
c
t
+1
−
(
y
t
+1
−
τ
t
+1
)]
+
c
t
−
(
y
t
−
τ
t
)
+(
q
t
−
Q
t
)
D
t
+1
=
a
t
+
D
t
1
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Since arbitrage requires the price of government and private debt be equal
ized
q
t
=
Q
t
,
∀
t
, we can rewrite this as
∞
X
j
=0
Ã
j
Y
i
=1
Q
t
+
i
−
1
!
[
c
t
+
j
−
y
t
+
j
+
τ
t
+
j
]=
a
t
+
D
t
,
(2)
where we have imposed the transversality condition
lim
j
→∞
j
Y
i
=1
Q
t
+
i
−
1
a
t
+
j
+1
=0
as well as the normalization
Ã
0
Y
i
=1
Q
t
+
i
−
1
!
=1
.
•
The sequence of govt. budget constraints are given by
g
t
+
D
t
=
q
t
D
t
+1
+
τ
t
,
∀
t
(3)
given initial debt
D
0
.
•
Using a transversality condition, (3) implies an intertemporal govt. budget
constraint
D
t
=
∞
X
j
=0
Ã
j
Y
i
=1
q
t
+
i
−
1
!
(
τ
t
+
j
−
g
t
+
j
)
(4)
where again
Ã
0
Y
i
=1
q
t
+
i
−
1
!
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 Spring '07
 CORBAE

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