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Review Questions: Money in discrete time
Prof. Lutz Hendricks. August 6, 2009
1
Money and leisure
Consider the following version of a Sydrauski model. A representative household solves
max
1
X
t
=0
t
u
(
c
t
; m
t
+1
=p
t
;
1
n
t
)
subject to the budget constraint
k
t
+1
+
m
t
+1
=p
t
+
c
t
=
F
(
k
t
; n
t
) +
m
t
=p
t
+
x
t
where
x
t
is a (money) transfer from the government and
n
t
is labor time. The government
hands out money according to the rule
x
t
=
g
t
m
t
=p
t
where
f
g
t
g
is an exogenous sequence. Assume
that
F
(
k; n
)
(a) State the household±s Bellman equation. Derive a system of equations that solves the house
hold problem.
(c) Now assume that
g
t
=
g
in all periods. Derive a system of equations that characterizes the
steady state.
(d) Prove that money is superneutral, if the utility function is of the form
u
(
c; m=p;
1
n
) =
U
(
c;
1
n
) +
W
(
m=p
)
.
1.1
Money and leisure
(a) The household±s Bellman equation is given by
V
(
k; m
) =
u
(
c; m
0
=p;
1
n
) +
(
k
0
; m
0
) +
±
f
F
(
k; n
) +
m=p
+
x
m
0
=p
c
k
0
g
u
c
(
t
)
u
m
(
t
)
=
(
p
t
=p
t
+1
)
u
c
(
t
+ 1)
(1)
u
n
(
t
)
=
u
c
(
t
)
F
n
(
k
t
; n
t
)
u
c
(
t
)
=
c
(
t
+ 1)
F
k
(
k
t
; n
t
)
A solution to the household problem is a set of sequences
f
c
t
; m
t
; n
t
; k
t
g
that solves the 3
FOCs and the budget constraint. The transversality conditions are
lim
t
!1
t
u
c
(
t
)
k
t
= 0
and
lim
t
!1
t
u
c
(
t
)
m
t
=p
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 '09
 LUTZHENDRICKS

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