Theorem 4.1
Suppose that
X
i
is a polyhedral set for every
i
∈
N.
Then there exists a set
¯
Θ
⊂
Θ
such that
(i) for each nonstandard dividend equilibrium
¯
x
there exists
q
∈
¯
Θ
such that
q
is an equilibrium hierarchic price for
¯
x
(or, which is equivalent, such
that
q
represents nonstandard equilibrium prices
p
corresponding to the
equilibrium
¯
x
).
(ii)
¯
Θ
is a union of manifolds of dimension less than or equal to
l

1
.
Proof.
We continue to use the convention
w
i
= 0 for every
i
∈
N.
Suppose
that ¯
x
is a nonstandard dividend equilibrium and
p
∈
*
IR
l
and
d
∈
*
IR
n
+
are corresponding prices and dividends.
It is possible to classify consumers
according to the structure of their budget sets
¯
B
(
p, d
i
)
.
Proposition 3.6 implies
that each consumer
i
faces (for some
m
=
m
(
i
)
∈ {
1
, . . . , k
+ 1
}
)
m

1 budget
constraints in the form of the equalities:
q
t
x
= 0
,
t
∈ {
1
, . . . , m

1
}
,
and (possibly) one budget constraint in the form of the inequality:
q
m
x
≤
0 or
q
m
x
≤
μ
j
,
where
μ
i
=
◦
(
d
i
/λ
j
(
p,d
i
)
). Let
N
m
⊆
N
be the set of all agents for
whom the last budget restriction involves the vector
q
m
.
For all such agents we
have
¯
B
i
(
p

X
t>m
λ
t
q
t
, d
i
) =
¯
B
i
(
p, d
i
)
.
Put
i
∈
N
1
if agent
i
faces no restrictions at all, that is if
¯
B
i
(
p, d
i
) =
X
i
.
Thus,
given nonstandard prices
p
and dividends
d
i
we partition the set of agents
N
into
k
subsets
N
1
, . . . , N
k
.
Consider an arbitrary agent
i
;
i
∈
N
m
for some
m.
His budget set is a subset of
the set
X
i
(
q
1
, . . . , q
m

1
) =
{
x
∈
X
i

q
1
x
= 0
, . . . , q
m

1
x
= 0
}
.
Each vector
q
t
, t
= 1
, . . . , m

1
,
supports
X
i
(
q
1
, . . . , q
t

1
)
,
(though it does not
have to support
X
i
), which implies that the sets
X
i
⊃
X
i
(
q
1
)
⊃
. . .
⊃
X
i
(
q
1
, . . . , q
m

1
)
form a finite sequence of faces of
X
i
contained in each other. Denote the face
X
i
(
q
1
, . . . , q
t

1
) by
F
t
i
(
p
) (note that the superscript
t
specifies that there are
t

1 equalities).
Let us construct a set Θ
p
which contains a hierarchic price
q
p
= (
q
1
, . . . , q
k
)
representing
p.
It will be done in
k
steps, and Θ
p
will finally be obtained as a
product of
k
spheres in some linear space.
20