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Algebraic Addendum to second part slide 36
.
The algebraic details are as follows:
Take the expectation at t=1 of both right and left hand side of b.c.
E
1
"
T
X
t
=1
C
t
#
=
E
1
"
B
0
+
T
X
t
=1
Y
t
#
Since
B
0
is known at t=1, this can be rewritten as
T
X
t
=1
E
1
(
C
t
)=
B
0
+
T
X
t
=1
E
1
(
Y
t
)
Now observe that the left hand side of this equality can be rewritten as
E
1
(
C
1
)+
E
1
(
C
2
E
1
(
C
3
.....
+
E
1
(
C
T
)
Observe that
E
1
(
C
1
C
1
.
Moreover, from the Euler condition
C
t
=
E
t
(
C
t
+1
)
,
we
can write
E
1
(
C
2
C
1
.
Consider now the terms
E
1
(
C
3
)
,E
1
(
C
4
)
1
(
C
T
)
.
For
example, take
E
1
(
C
3
):
we know that
E
1
(
C
3
E
1
[
E
2
(
C
3
)]
by the "Law of it
erated expectations". But
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This note was uploaded on 07/25/2008 for the course ECON 813A taught by Professor Minetti during the Spring '08 term at Michigan State University.
 Spring '08
 MINETTI
 Macroeconomics

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