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Recursive Competitive
Equilibrium
Up to now, every problem has been formulated as an in
f
nite sequence prob
lem. Next we will focus on the equivalence of a recursive equilibrium formulation
to the in
f
nite sequence competitive equilibrium and characterize properties of
the recursive equilibrium. Mehra and Prescott (1980, Econometrica) did this.
•
The sequence problem is primitive, but it involves in
f
nite sequences.
•
Instead of working with in
f
nite sequences, a recursive formulation de
f
nes
a
f
nite number of functions (remember, however, that a function is in
f
nite
dimensional).
1
Stochastic Growth Environment
•
Technology shocks:
z
t
follow a
f
nite state Markov Process
π
(
z
t
+1
)=
π
(
z
t
+1
=
z
0

z
t
π
(
z
t
+1
=
z
0

z
t
=
z
)
π
(
z
t
=
z

z
t
−
1
=
b
z
)
···
•
The process
{
z
t
}
is assumed to be stationary with bounded ergodic set
Z
, i.e., if
z
t
∈
Z
then
z
t
+1
∈
Z
with probability one. The transition
probabilities
π
:
Z
×
Z
→
[0
,
1]
.
•
Productive Technology:
Y
t
=
z
t
F
(
K
t
)where
F
is CRS.
•
Investment Technology:
K
t
+1
=(
1
−
δ
)
K
t
+
I
t
where
K
0
is given and
K
t
∈
X
⊂
R
+
where
X
is assumed to be closed and convex.
•
Unit measure of exante identical agents with preferences:
P
∞
t
=0
P
z
t
β
t
π
(
z
t
)
u
(
c
t
)
where
u
is continuous, strictly increasing, strictly concave, and di
f
eren
tiable.
2S
e
q
u
e
n
t
i
a
l
C
E
•
At the beginning of each period in history
z
t
, households rent capital
k
t
∈
X
to
f
rms, consume
c
t
and invest
i
t
.
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 Spring '07
 CORBAE

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