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Assume a person can choose her hours of work. How ine
ﬃ
cient are taxes on
earnings?
Suppose
W
isthewagerate
,
P
is the price of consumption,
T
is the time endow
ment,
l
is time spent at leisure,
n
≡
T
−
l
is time at work, and
c
is consumption.
With no taxes her budget constraint is
Wl
+
Pc
=
WT.
With a proportional tax on earnings
(
W
(
T
−
l
))
at rate
t
this becomes
(1
−
t
)
Wl
+
Pc
=(1
−
t
)
WT.
To keep the calculations manageable assume she has CobbDouglas preferences:
U
=
l
a
c
1
−
a
,
0
<a<
1
.
From our earlier work with CD utility functions we know
l
=
a
(1
−
t
)
WT
(1
−
t
)
W
=
aT
c
=(
1
−
a
)
(1
−
t
)
WT
P
V
((1
−
t
)
W, P,
(1
−
t
)
WT
)=
a
a
(1
−
a
)
1
−
a
((1
−
t
)
W
)
−
a
P
−
(1
−
a
)
(1
−
t
)
WT
e
(
W, P, U
)=
a
−
a
(1
−
a
)
−
(1
−
a
)
W
a
P
1
−
a
U.
Tax revenue is
tWn
=
tW
(
T
−
l
)=
tW
(
T
−
aT
)=(1
−
a
)
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 Fall '08
 Burbidge,John

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