Review Questions: In°nite Horizon Models in Discrete Time
Prof. Lutz Hendricks. August 6, 2009
1
Taxes and government spending
Consider the following growth economy, extended to include government spending and a tax on
output at rate
°
t
. A representative household chooses
f
c
t
; k
t
+1
g
1
t
=0
to solve the problem
max
X
1
t
=0
±
t
u
(
c
t
)
subject to
c
t
+
k
t
+1
= (1
°
°
t
)
f
(
k
t
)
k
0
given.
Capital depreciates completely each period. The government °nances exogenous spending,
g
t
,
each period by levying taxes at rate
°
t
.
Government spending does not a/ect private utility or
production possibilities. The government budget constraint is
°
t
f
(
k
t
) =
g
t
You may °nd it convenient to de°ne government spending as a share of output by introducing
the variable
s
g
t
=
g
t
=f
(
k
t
)
Further, note the two budget constraints imply the aggregate resource constraint
c
t
+
g
t
+
k
t
+1
=
f
(
k
t
)
or
c
t
+
k
t
+1
= (1
°
s
g
t
)
f
(
k
t
)
The household takes the time paths of the policy variables as given.
Assume the functional
forms:
u
(
c
) = ln(
c
)
and
f
(
k
) =
k
°
, 0
< ² <
1.
(a) Show that there is a maximum sustainable capital stock,
k
max
, for this economy.
(b) Assuming that
k
0
2
(0
; k
max
)
°nd the steady state level of the capital stock. Assume that
°
is constant over time. Note that there are no °rms; households produce and consume.
1
(c) Write down the Bellman equation for the household.
(d) Solve for the equilibrium consumption and investment decision rules as functions of the
current state. Hint: What are reasonable guesses given log utility?
(e) Why doesn±t expected future policy a/ect current consumption and investment decisions?
Consider the same economic environment as above, but now allow the government to sell real
oneperiod bonds to °nance any discrepancies between spending and revenues.
Let
b
t
+1
denote
bonds sold in period
t
at price
q
t
, which pay one unit of consumption goods in period
t
+1
. Agents
enter each period
t
with capital,
k
t
, and bonds,
b
t
.
The representative household now chooses
f
c
t
; k
t
+1
; b
t
+1
g
1
t
=0
1
You can convince yourself, however, that it would not make a di/erence if °rms were added to the model.
1
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to maximize utility subject to
c
t
+
k
t
+1
+
q
t
b
t
+1
= (1
°
°
t
)
f
(
k
t
) +
b
t
k
0
given. Assume the same functional forms before. The household takes as given the time paths
of policy variables. The government chooses paths for
f
°
t
; b
t
+1
g
1
t
=0
to °nance
s
g
t
according to:
q
t
b
t
+1
+
°
t
f
(
k
t
) =
g
t
+
b
t
b
0
given.
(f) De°ne the state of the economy and write down Bellman±s equation for the household.
(g) Assuming that in a steady state the tax rate and government spending share are given and
constant, derive expressions for steady state consumption, capital, and government debt.
1.1
Answer: Taxes and government spending
(a) It su¢ ces to show that k is bounded, even if c and g both equal zero forever. Since
k
t
+1
=
k
°
t
the corresponding steady state has
k
= 1
. If
k
t
>
1
then
k
t
+1
< k
t
.
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 '09
 LUTZHENDRICKS

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