The Optimal Mix of
Distortionary Taxation
with Commitment
Draws on Atkeson, Chari, and Kehoe (1999, FRBMinn QR). See also p.478—
88 of LS.
Question: Should the government tax capital?
1
First Best
•
Planner solves
max
{
c
t
,n
t
,k
t
+1
}
∞
t
=0
∞
X
t
=0
β
t
U
(
c
t
, n
t
)
subject to
c
t
+
g
+
k
t
+1
=
F
(
k
t
, n
t
) + (1
−
δ
)
k
t
(1)
where
U
c
>
0
, U
n
<
0
,
and
F
is CRS.
•
First order conditions
n
t
:
−
U
n,t
=
U
c,t
F
n,t
(2)
k
t
+1
:
U
c,t
=
βU
c,t
+1
[
F
k,t
+1
+ 1
−
δ
]
(3)
•
First best has no distortion to the MB of working
F
n,t
or saving
F
k,t
+1
+
1
−
δ
.
2
Statement of the Competitive Equilibrium Prob
lem
•
Govt expenditure
g
is
fi
nanced through proportional taxes on income from
capital (rate
θ
t
) and labor (rate
τ
t
).
•
The government’s intertemporal budget constraint is
∞
X
t
=0
p
t
g
=
∞
X
t
=0
p
t
(
τ
t
w
t
n
t
+
θ
t
(
r
t
−
δ
)
k
t
)
(4)
1
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•
HH problem
max
{
c
t
,n
t
,k
t
+1
}
∞
t
=0
∞
X
t
=0
β
t
U
(
c
t
, n
t
)
subject to
∞
X
t
=0
p
t
(
c
t
+
k
t
+1
) =
∞
X
t
=0
p
t
((1
−
τ
t
)
w
t
n
t
+
R
kt
k
t
)
(5)
where
R
kt
= 1 + (1
−
θ
t
)(
r
t
−
δ
)
and
p
0
= 1
.
•
The
fi
rst order conditions for a HH are
c
t
:
β
t
U
c,t
=
λp
t
(6)
n
t
:
−
β
t
U
n,t
=
λp
t
(1
−
τ
t
)
w
t
(7)
k
t
+1
:
p
t
=
R
k,t
+1
p
t
+1
(8)
where
λ
is the multiplier on (5).
•
The
fi
rm’s problem is
max
k
t
,n
t
F
(
k
t
, n
t
)
−
r
t
k
t
−
w
t
n
t
•
The
fi
rst order conditions for the
fi
rm are:
r
t
=
F
k,t
(9)
w
t
=
F
n,t
(10)
•
From these sets of conditions it is possible to see “tax wedges”. In partic
ular, (6), (7), (8), (9), (10) imply that for the houshold:
−
U
n,t
U
c,t
= (1
−
τ
t
)
F
n,t
and
U
c,t
=
βU
c,t
+1
[1 + (1
−
θ
t
+1
)(
F
k,t
+1
−
δ
)]
These two conditions di
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 Spring '07
 CORBAE
 Government, Trigraph, Rkt

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