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Review Questions: Asset pricing (deterministic)
Prof. Lutz Hendricks. August 6, 2009
1
Deterministic Fruit Tree Prices
of identical, in²nitely lived households. Each receives an endowment of
e
t
in each period. In
addition, each household owns one tree at the beginning of time (date 1) which yields
d
t
units of
the consumption good in each period. Goods cannot be stored, but trees live forever. Trees can be
bought or sold at a price of
p
t
, which is of course endogenous. The number of trees in the economy
cannot be altered; it is always 1 per household.
Hence, the household solves
max
X
1
t
=1
t
u
(
c
t
)
subject to
p
t
k
t
+1
=
k
t
(
d
t
+
p
t
) +
e
t
c
t
with
k
1
= 1
. In the budget constraint, the household receives as income the endowment
e
t
and
capital income proportional to the number of trees owned
(
k
t
)
. This is spent on consumption ct
and the purchase of new trees
(
k
t
+1
)
.
(a) State the household problem as a Dynamic Program.
(b) State the conditions that de²ne a solution to the household problem.
(c) De²ne a competitive equilibrium.
(d) Assume that
e
t
and
d
t
are constant over time. Derive the steady state price of a tree
(
p
)
and the steady state rate of return of holding a tree.
1.1
Answer: Deterministic Fruit Tree Prices
(a) Bellman equation
V
(
k
) = max
u
(
e
+
k
(
d
+
p
)
pk
0
) +
(
k
0
)
(b) A solution consists of sequences
(
c
t
;k
t
)
that satisfy the budget constraint and the Euler
equation
u
0
(
c
) =
&u
0
(
c
0
)(
d
0
+
p
0
)
=p
(c) A CE consists of sequences
(
c
t
;k
t
;p
t
)
that satisfy the 2 household optimality conditions and
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 '09
 LUTZHENDRICKS

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