(
i
)
¯
B
(
p, γ
) =
X
(
q
1
, . . . , q
j

1
)
and
μ
= +
∞
;
(
ii
)
¯
B
(
p, γ
) =
{
x
∈
X
(
q
1
, . . . , q
j

1
)

q
j
x
≤
μ
}
and
μ <
+
∞
;
(
iii
)
¯
B
(
p, γ
) =
B
m
(
p
)
for some
m < j
and there exists
y
∈
¯
B
(
p, γ
)
such that
q
m
y <
0
.
Here
q
= (
q
1
, . . . , q
k
)
is a hierarchic price representing
p, j
=
j
(
p, γ
)
is the
infinitesimality level of
γ.
Proof.
Consider an (
l
+1)dimensional set
X
*
=
X
×{
1
}
,
and a “budget” set
{
x
∈
*
X
*

p
*
x
≤
0
}
,
(11)
where the price vector
p
*
∈
IR
l
+1
is given by
p
*
=
(
p
0
,

γ
)
,
if
γ/λ
j
≈
+
∞
,
p
0
=
p

∑
t
≥
j
λ
t
q
t
,
(
p
0
,

λ
j
μ
)
,
if
γ/λ
j
<
+
∞
,
p
0
=
∑
t
≤
j
λ
t
q
t
.
By construction

p

p
0

/γ
≈
0 and
λ
j
μ/γ
≈
1
.
Since 0
∈
X,
Lemma 3.5 implies
that the projection of the standardization of the set defined in (11) onto the
first
l
components of the set
X
*
coincides with
¯
B
(
p, γ
)
.
At the same time,
p
*
=
X
t<j
λ
t
q
*
t
+
λ
*
j
q
*
j
,
where
q
*
t
= (
q
t
,
0) if
t < j,
q
*
j
=
‰
(0
,

1)
,
if
γ/λ
j
≈
+
∞
,
(
q
j
,

μ
)
,
if
γ/λ
j
<
+
∞
,
and
λ
*
j
=
‰
γ,
if
γ/λ
j
≈
+
∞
,
λ
j
,
if
γ/λ
j
<
+
∞
.