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Macroeconomics 702, Second Midterm
Suggested solution by Ahu and Vivian
Question 1
(a)
(10 points: 3 for consumption possibility set and production set, 3 for
consumer’s problem and
f
rm’s problem, 4 for de
f
ning AD equilibrium)
Let
ε
t
=(
s
A,t
,s
B,t
)
∈
Z
2
≡
Ξ
and
H
=
Ξ
×
Ξ
×···
where
Ξ
is a support of
mood shocks to agents. And an element of
H
,
h
t
ε
0
,ε
1
, ..., ε
t
)
∈
Ξ
t
is a
history of shocks up to period t. We denote the probability of
h
t
conditional on
information at 0 is
π
(
h
t
)
.
Then, the commodity space is:
L
=
{
x

x
t
(
h
t
)=(
x
1
t
(
h
t
)
,x
2
t
(
h
t
)
3
t
(
h
t
))
∈
R
3
,
∀
t, h
t
and
k
x
k
<
∞
a.s.
}
Consumption possibility set for agent of type i is
X
i
=
{
x
i
∈
L

∃
{
c
i
t
(
h
t
)
,a
i
t
+1
(
h
t
)
}
∞
t
=0
≥
0
such that
x
i
1
t
(
h
t
)=
a
i
t
+1
(
h
t
)+
c
i
t
(
h
t
)
,
x
i
2
t
(
h
t
)
∈
[0
,
1]
i
2
t
(
h
t
)
≤
a
i
t
(
h
t
−
1
)
∀
t
,
h
t
and
a
0
given
}
where
x
i
1
t
(
h
t
)
is the produced
f
nal goods,
x
i
2
t
(
h
t
)
is the capital service input,
for agent of type i at time t after history
h
t
.
Production set for
f
rm is
Y
i
=
{
y
i
∈
L

0
≤
y
i
1
t
(
h
t
)
≤
F
[
y
i
3
t
(
h
t
)
,y
i
2
t
(
h
t
)]
,
∀
t, h
t
.
}
For a given price system
p
(
x
)
,
the type i consumer’s optimization problem is:
max
x
∈
X
i
∞
X
t
=0
X
h
t
∈
H
t
β
t
π
(
h
t
)
(
c
t
)
1
−
σ
1
−
σ
(1)
such that
3
X
j
=1
X
t
X
h
t
p
j,t
(
h
t
)
x
i
j,t
(
h
t
)
≤
0
And
f
rm’s optimization problem:
max
y
∈
Y
i
3
X
j
=1
X
t
X
h
t
p
j,t
(
h
t
)
y
j,t
(
h
t
)
(2)
An AD competitive equilibrium is a triad
(
p
∗
A
∗
B
∗
∗
)
that satis
f
es
1. Given
p
∗
i
∗
solves type i consumer’s problem,
i
=
A, B.
2.
y
i
∗
solves
f
rm’s problem.
3. market clears
x
A
∗
1
+
x
B
∗
1
=
y
A
∗
1
+
y
B
∗
1
(3)
x
A
∗
3
+
x
B
∗
3
=
y
A
∗
3
+
y
B
∗
3
(4)
x
A
∗
2
=
y
A
∗
2
B
∗
2
=
y
B
∗
2
(5)
.
1
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View Full DocumentRemark 1
Because goods can be transported freely across islands, people’s in
vestments have the same rate of return. Therefore, there is only one variable,
x
i
3
(
h
t
)
for saving in each agent’s consumption possibility set. And
f
rms uses
the capital that people invest to. Market clearing condition for capital is (4).
But since labor cannot move, market clearing condition is (5).
(b)
Theorem 2
First Basic Welfare Theorem: If the preferences of consumers are
nonsatiated (
∃
{
x
n
}
∈
X
that converges to
x
∈
X
such that
U
(
x
n
)
>U
(
x
)
), an
allocation
(
x
∗
,y
∗
)
of an ADE
(
p
∗
,x
∗
∗
)
is PO.
Theorem 3
Second Basic Welfare Theorem: If (i) X is convex, (ii) preference
is convex (for
∀
x, x
0
∈
X,
if
x
0
<x
,then
x
0
<
(1
−
θ
)
x
0
+
θx
for any
θ
∈
(0
,
1)
),
(iii)
U
(
x
)
is continuous, (iv) Y is convex, (v)Y has an interior point, then with
any PO allocation
(
x
∗
∗
)
such that
x
∗
is not a satiation point, there exists a
continuous linear functional
p
∗
such that
(
x
∗
∗
,p
∗
)
is a QuasiEquilibrium ((a)
for
x
∈
X
which
U
(
x
)
≥
U
(
x
∗
)
implies
p
∗
(
x
)
≥
p
∗
(
x
∗
)
and (b)
y
∈
Y
implies
p
∗
(
y
)
≤
p
∗
(
y
∗
)
)
Lemma 4
If, for
(
x
∗
∗
∗
)
in the theorem above, the budget set has cheaper
point than
x
∗
(
∃
x
∈
X
such that
p
(
x
)
<p
(
x
∗
)
), then
(
x
∗
∗
∗
)
is a ADE.
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 Spring '08
 Staff
 Macroeconomics

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