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.
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
*
))
Question 3
First write down the problem of the household for future reference.
The ag
gragate state variables are technology shock vector
z
= (
z
A
, z
B
) capital stock
vector of two sectors
K
= (
K
A
, K
B
) and individual state variable is asset hold
ings of the household. Note that the stochastic processes are independent. The
2