Econ 387L: Macro II
Spring 2008, University of Texas
Instructor: Dean Corbae
Answers Problem Set #6
Consider the following version of the Lucas asset pricing model. Preferences are given
by
U
(
c
t
)=ln
(
c
t
)
.
Each household is endowed with project that generates dividends
y
t
∈
D
⊂
R
++
where
D
is a finite set. The dividends follow a Markov process with
π
(
y
t
+1

y
t
)=
prob
(
y
t
+1
=
y
0

y
t
=
y
)
.
The government spends amount
g
t
=
ε
t
y
t
per capita
where
ε
t
∈
[0
,
1)
.
Government spending follows a Markov process
φ
(
ε
t
+1
=
ε
0

ε
t
=
ε
)
.
The only assets traded are shares (or titles) to trees.
1. Assume that all government expenditures are financed by a lump sum tax
τ
t
per
household which is independent of the shares owned by the household. Calculate the
equilibrium share price function.
Answer.
We first note that regardless of the tax regime, the market clearing condition in
the goods market will be:
c
t
+
g
t
=
y
t
∀
t
(1)
⇒
c
t
=
y
t
(1
−
ε
t
)
∀
t
(2)
In this case the problem of the HH is:
max
E
t
∞
X
t
=0
β
t
ln (
c
t
)
(3)
s.t.
c
t
+
p
t
s
t
+1
=
s
t
(
y
t
+
p
t
)
−
τ
t
(4)
We can compute the F.O.C. with respect to
s
t
+1
:
p
t
1
c
t
=
βE
t
∙
1
c
t
+1
(
y
t
+1
+
p
t
+1
)
¸
⇒
p
t
=
βE
t
∙
c
t
c
t
+1
y
t
+1
+
c
t
c
t
+1
βE
t
+1
∙
c
t
+1
c
t
+2
y
t
+2
+
c
t
+1
c
t
+2
p
t
+2
¸¸
⇒
p
t
=
E
t
∙
β
c
t
c
t
+1
y
t
+1
+
β
2
c
t
c
t
+2
y
t
+2
+
β
2
c
t
c
t
+2
p
t
+2
¸
using the law of iterated expectations
p
t
=
E
t
⎡
⎣
T
X
j
=1
β
j
c
t
c
t
+
j
y
t
+
j
+
β
T
c
t
c
t
+
T
p
t
+
T
⎤
⎦
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 Spring '07
 CORBAE
 Public Finance, per capita, price function, market clearing condition, F.O.C.

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