Econ 387L: Macro II
Spring 2004, University of Texas
Instructor: Dean Corbae
Answers  Final Exam
Please answer each question on separate sheets. Remember to adjust the time you spend
on each question in proportion to what they are worth.
(1) 30 points. Consider the following two period problem where the government can
tax capital gains and/or bequests to fund an exogenous amount of government expenditure.
There is a unit measure of households (HHs) who make a portfolio choice at
t
=0
and
a bequest decision at
t
=1
.
Their preferences are given by
u
(
c
)+
αu
(
b
)
where
c
≥
0
is
consumption at
t
,
b
≥
0
are bequests for one’s children chosen at
t
,
and
α>
0
measures the relative importance of bequests vs own consumption. Assume.
u
0
(
·
)
>
0
,
u
00
(
·
)
<
0
,
and
u
0
(0) =
∞
.
Each HH receives an endowment
ω>
0
at
t
andatthattime
must either make a storage decision in taxfree assets
a
≥
0
yielding gross return equal to
1
at
t
or a productive asset
k
≥
0
yielding
R>
1
which is proportionately taxed at rate
δ
∈
[0
,
1]
in period
t
.
Assume that if HHs are indifferent between storing in
a
or
k,
they
save in
k.
Then at
t
,
HHs choose how much to consume
c
≥
0
or leave bequests
b
≥
0
.
At
t
,
HH’s have to pay
τb
for any bequests they leave where
τ
∈
[0
,
1]
.
Taxes are used
to
f
nance government expenditure
G
at
t
.
a) 20 points. Assume government announces a tax package
(
δ,τ
)
at
t
and can
commit to it. De
f
ne a Ramsey equilibrium. Be explicit about the HH’s choice problem.
What are the optimal capital gains
δ
taxes? How do bequests react to changes in
τ
?
Do HHs
leave bequests? Are bequest taxes
0
or
100%
with commitment?
Answer: HH problem
max
c,k,a,b
u
(
c
αu
(
b
)
s.t.a
+
k
=
ω
c
+
b
=
R
(1
−
δ
)
k
+
a
−
As long as
R
(1
−
δ
)=1
or
δ
=
R
−
1
R
,
HHs will choose
k
=
ω
and
a
.
In that case, HHs
solve
max
b
u
(
R
(1
−
δ
)
ω
−
(1 +
τ
)
b
αu
(
b
)
satisfy
(1 +
τ
)
u
0
(
ω
−
(1 +
τ
)
b
)=
αu
0
(
b
)
(1)
To see bequest taxes
τ
decrease bequests, use implicit function theorem:
F
(
τ,b
)
≡
(1 +
τ
)
u
0
(
ω
−
(1 +
τ
)
b
)
−
αu
0
(
b
)=0
⇐⇒
db
dτ
=
−
F
τ
F
b
db
dτ
=
−
u
0
(
c
)+(1+
τ
)
u
00
(
c
)
b
−
(1 +
τ
)
2
u
00
(
c
)
−
αu
00
(
b
)
<
0
We also must check that the government budget constraint is satis
f
ed. To get a suf
f
cient
1
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View Full Documentcondition invert (1) to yield
ω
−
(1 +
τ
)
b
=
u
0
−
1
μ
α
(1 +
τ
)
¶
b
and take
τ
=1
b
(
τ
=1)=
ω
£
u
0
−
1
¡
α
2
¢
+2
¤
In that case
(
R
−
1)
ω
+
τb >
(
R
−
1)
ω
+
b
(
τ
=1)
,
since the max of
τb
could always be
τ
.
(b) 10 points. Assume that there is no commitment on the part of the government with
respect to its announced tax package. That is, after households choose their portfolio in
t
=0
,
the government chooses taxes in
t
,
and then households choose their bequests.
De
f
ne and solve for a subgame perfect equilibrium. What are equilibrium
(
δ,τ
)
? Compare
tax revenues in part (a) and (b).
Answer: If the government chooses
(
)
after the household moves at
t
,
then it will
choose
δ
.
Given this the household chooses
k
and
a
=
ω.
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