Growth Model
Question 1
This is a twosector growth model. There is an investment sector and a con
sumption sector.
Commodity Space:
L
=
{
x

x
t
(
h
t
) = (
x
1
t
(
h
t
)
, x
2
t
(
h
t
)
, x
3
t
(
h
t
))
∈
R
3
∀
t, h
t
and
x
<
∞}
Consumption possibility set: —–
X
=
{
x
A
, x
B
∈
L
:
∃{
c
A
t
(
h
t
)
, c
B
t
(
h
t
)
, k
t
+1
(
h
t
)
}
∞
t
=0
≥
0 such that
c
A
t
(
h
t
) +
k
t
+1
(
h
t
)
=
x
A
1
t
(
h
t
) + (1

δ
)
k
t
(
h
t

1
)
∀
t,
∀
h
t
c
B
t
(
h
t
)
=
x
B
1
t
(
h
t
)
∀
t,
∀
h
t
x
A
2
t
(
h
t
) +
x
B
2
t
(
h
t
)
∈
[0
,
1]
∀
t,
∀
h
t
x
A
3
t
(
h
t
) +
x
B
3
t
(
h
t
)
≤
k
t
(
h
t

1
)
∀
t,
∀
h
t
k
0
=
given
}
where
x
A
1
t
(
h
t
)=received goods in terms of apples,
x
A
2
t
(
h
t
)=labor supply to tech
nology A,
x
A
3
t
(
h
t
)=capital service to technology A,
x
B
1
t
(
h
t
)=received goods in
terms of bananas,
x
B
2
t
(
h
t
)=labor supply to technology B,
x
B
3
t
(
h
t
)=capital ser
vice to technology B.
Production possibility sets:
Y
A
=
{
y
∈
L
:
y
A
1
t
(
h
t
)
≤
F
A
(
y
A
3
t
(
h
t
)
, y
A
2
t
(
h
t
))
}
Y
B
=
{
y
∈
L
:
y
B
1
t
(
h
t
)
≤
F
B
(
y
B
3
t
(
h
t
)
, y
B
2
t
(
h
t
))
}
Consumer’s problem:
max
t
h
t
u
[
c
A
t
(
h
t
)
, c
B
t
(
h
t
)
,
1

x
A
2
t
(
h
t
) + 1

x
B
2
t
(
h
t
)]
s.t.
t
h
t
3
i
=1
[
p
A
it
(
h
t
)
x
A
it
(
h
t
) +
p
B
it
(
h
t
)
x
B
it
(
h
t
)]
Producer’s problem:
max
y
A
∈
Y
A
,y
B
∈
Y
B
3
i
=1
[
p
A
it
(
h
t
)
y
A
it
(
h
t
) +
p
B
it
(
h
t
)
y
B
it
(
h
t
)]
An ArrowDebreu Competitive Equilibrium is (
p
A
*
, x
A
*
, y
A
*
, p
B
*
, x
B
*
, y
B
*
) such
that
1
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1.
x
A
*
, x
B
*
solves the consumer’s problem.
2.
y
A
*
, y
B
*
solves the firm’s problem.
3. Markets clear,
x
A
*
=
y
A
*
x
B
*
=
y
B
*
Question 2
Theorem 1
(FBWT) 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
*
, y
*
) is PO.
Theorem 2
(SBWT) If (i) X is convex, (ii) preference is convex (for
∀
x, x
∈
X,
if
x < x
, then
x <
(1

θ
)
x
+
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
*
, y
*
) such
that
x
*
is not a satuation point, there exists a continuous linear functional
p
*
such that (
x
*
, y
*
, 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
*
))
Contracting Frictions
Question 10
Note that the agent does not care about leisure. Therefore, he will allocate all
of his endowment of efficiency units of labor to fishing. Let
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 Spring '08
 Staff
 Supply And Demand, ht, max Γ

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