Econ 387L: Macro II
Spring 2005, University of Texas
Instructor: Dean Corbae
M
idtermExamSo
lut
ion
1. Consider the following version of a Lucas asset pricing model. There are
two kinds of assets. The
f
rst asset, call it stocks, gives no direct utility but
yields a stream of strictly positive dividends
{
y
t
}
given by a
f
nite state
Markov process. The dividends are paid at the beginning of the period, can be
consumed, but are nonstorable. The second asset, call it art, yields direct
utility but no dividends. That is, art, can be traded in the same way as stocks
but yields no dividends. Both assets are in unit supply. There is one unit of
each type of asset for each household in the economy. Preferences are given by
u
(
c
t
,a
t
)=ln
c
t
+
γ
ln
a
t
,where
a
t
denotes the household’s holdings of art.
a. 5 points. Let
p
t
be the price of stocks, denoted
s
t
,and
q
t
be the price of
art. De
f
ne a competitive equilibrium (be complete).
ANSWER.
A competitive equilibrium is a sequence of prices
{
p
t
,q
t
}
∞
t
=0
and
allocations
{
c
t
,s
t
t
}
∞
t
=0
such that:
1. Given prices, Households maximize. Hence,
{
c
t
t
+1
t
+1
}
∞
t
=0
solve:
max
E
t
"
∞
X
t
=0
ln
c
t
+
γ
ln
a
t
#
(1)
s.t.
c
t
+
p
t
s
t
+1
+
q
t
a
t
+1
=
s
t
(
y
t
+
p
t
)+
q
t
a
t
s
0
given
a
0
given
(2)
2. Markets clear:
c
t
=
y
t
Market for goods
(3)
s
t
=1
∀
t
Market for stocks
(4)
a
t
∀
t
Market for art
(5)
b. 15 points. Find pricing functions mapping the state of the economy at
t
into
p
t
and
q
t
.
ANSWER.
Using (1) and (2) we can get the F.O.C. with respect to
s
t
+1
and
1
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t
+1
:
s
t
+1
:
1
c
t
p
t
=
βE
t
·
1
c
t
+1
(
y
t
+1
+
p
t
+1
)
¸
(6)
a
t
+1
:
1
c
t
q
t
=
t
·
1
c
t
+1
q
t
+1
+
γ
a
t
+1
¸
(7)
Then, iterating (6) one period we
f
nd that:
p
t
=
t
·
c
t
c
t
+1
y
t
+1
+
c
t
c
t
+1
t
+1
·
c
t
+1
c
t
+2
(
y
t
+2
+
p
t
+2
)
¸¸
We can continue iterating the price function up to any period
T
and using the
law of iterated expectations we get:
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
Finally, under the regular assumptions, using (3) and taking the limit as
T
→∞
we get the price function for stocks:
p
t
=
E
t
β
∞
X
j
=0
β
j
y
t
y
t
+
j
y
t
+
j
=
β
1
−
β
y
t
(8)
Similarly, using (7) and following the same steps we get:
q
t
=
E
t
·
β
c
t
c
t
+1
μ
E
t
+1
·
β
c
t
+1
c
t
+2
q
t
+2
¸
+
βγ
a
t
+2
c
t
+1
¶¸
+
a
t
+1
c
t
q
t
=
β
2
E
t
·
c
t
c
t
+2
q
t
+2
¸
+
β
2
γ
a
t
+2
c
t
+
a
t
+1
c
t
q
t
=
β
T
E
t
·
c
t
c
t
+
T
q
t
+
T
¸
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