Spring 2004, Econ 702
Problem 1
We have de
f
ned the Sequence of Markets Equilibrium (SME) as an
allocation
{
c
∗
t
,k
∗
t
+1
,n
∗
t
,b
∗
t
+1
}
∞
t
=0
and a sequence of prices
{
R
∗
t
,
w
∗
t
,i
∗
t
}
∞
t
=0
such
that:
·
Given
{
R
∗
t
,w
∗
t
,i
∗
t
}
∞
t
=0
,
{
c
∗
t
,k
∗
t
+1
,n
∗
t
,
0
}
∞
t
=0
solves the following representa
tive consumer’s problem:
max
{
b
t
+1
}
,
{
c
t
,k
t
+1
}
≥
0
∞
X
t
=0
β
t
u
(
c
t
,
1
−
n
t
)
(1)
subject to
c
t
+
k
t
+1
+
b
t
+1
=
R
∗
t
k
t
+
w
∗
t
n
t
+(1+
i
∗
t
)
b
t
∀
t
(2)
n
t
∈
[0
,
1]
∀
t
(3)
and
k
0
given,
b
0
=0
(4)
and Noponzi condition
lim
T
→∞
T
Y
s
=1
b
T
+1
(1 +
i
∗
s
)
=0
(5)
·
Given
{
R
∗
t
,w
∗
t
}
,
{
k
∗
t
,n
∗
t
}
solves the following representative
f
rm’s prob
lem: for
∀
t
max
{
y
t
,k
t
,n
t
}
{
y
t
−
w
∗
t
n
t
−
(
R
∗
t
−
1+
δ
)
k
t
}
(6)
subject to
y
t
≤
F
(
k
t
,n
t
)
(7)
·
Goods market clear:
y
t
+
k
t
(1
−
δ
)=
c
t
+
k
t
+1
∀
t
(8)
We have also de
f
ned the Valuation (ArrowDebreu) Equilibrium (VE) as
an allocation
{
x
∗
it
,y
∗
it
}
∞
t
=0
,i
=1
,
2
,
3
and a valuation function (assuming it has
an inner product representation)
b
p
(
x
)=
∞
P
t
=0
3
P
i
=1
p
∗
it
x
it
such that given
b
p
(
x
)
,
1
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 Spring '08
 Staff
 Recursion, Bellman equation, following representative ﬁrm

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