Noah Williams
Economics 714
Department of Economics
Macroeconomic Theory
University of Wisconsin
Spring 2009
Problem Set 4 Solutions (Sketch)
1.
(a) Under consumption taxes, household’s FOC are
β
u
c
(
c
t
+1
,
1

N
t
+1
)
u
c
(
c
t
,
1

N
t
)
=
q
t
+1
(1 +
τ
c
t
+1
)
q
t
(1 +
τ
c
t
)
u
c
(
c
t
,
1

N
t
)
u
l
(
c
t
,
1

N
t
)
=
1 +
τ
c
t
w
t
q
t
q
t
+1
=
r
t
+ (1

δ
)
Under a labor and capital income taxes, we have
β
u
c
(
c
t
+1
,
1

N
t
+1
)
u
c
(
c
t
,
1

N
t
)
=
q
t
+1
q
t
u
c
(
c
t
,
1

N
t
)
u
l
(
c
t
,
1

N
t
)
=
1
w
t
(1

τ
N
t
)
q
t
q
t
+1
=
r
t
(1

τ
k
t
) + (1

δ
)
¿From this, you can derive the conditions for the tax schemes to yield equivalent
marginal utilities.
(b) The equations describing the transitional dynamics are
u
c
(
c
t
+1
)
u
c
(
c
t
)
=
[1

δ
+
f
k
(
k
)](1 +
g
)(1 +
ρ
)
c
t
+
k
t
+1

(1

δ
)
k
t
+
G
=
f
(
k
t
)
The steady state is
c
*
=
f
(
k
*
)

δk
*

G
and 1 = [1

δ
+
f
k
(
k
*
)](1 +
g
)(1 +
ρ
).
Taking
G
and
g
as given without worrying about the government’s BC, increasing
them has the effect of decreasing both steady state consumption and capital stock.
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 Spring '08
 Staff
 Economics, Trigraph, HHBC

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