Example
4.10
.
(
X,

) = (
R
n
,
≤
)
with
x
≤
y
iff
x
i
≤
y
i
,
i
= 1
, . . . , n
is a lattice with
x
∧
y
= (min(
x
1
, y
1
)
, . . . ,
min(
x
n
, y
n
))
and
x
∨
y
= (max(
x
1
, y
1
)
, . . . ,
max(
x
n
, y
n
))
.
X
is not
a complete lattice, but
([0
,
1]
n
,
≤
)
is, with
inf
S
= (inf
{
x
1
:
x
∈
S
}
,
inf
{
x
2
:
x
∈
S
}
)
and
sup
S
= (sup
{
x
1
:
x
∈
S
}
,
sup
{
x
2
:
x
∈
S
}
)
.
35
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Chapter 4.5
Example
4.11
.
Suppose that
X
6
=
∅
, let
P
(
X
)
denote the class of subsets of
X
, and for
A, B
∈ P
(
X
)
, define
A

B
if
A
⊂
B
.
(
P
(
X
)
,

)
is a complete lattice, with
inf
S
=
T
{
E
:
E
∈ S}
for
S ⊂ P
(
X
)
and
sup
S
=
S
{
E
:
E
∈ S}
.
We can turn lattice orderings around and keep the lattice structure.
Example
4.12
.
Suppose that
(
X,

)
is a lattice.
Define
x


y
if
y

x
.
Show that
(
X,


)
is a lattice.
[
Thus,
(
R
n
,
≤

)
, i.e.
(
R
n
,
≥
)
, is a lattice, as is
(
P
(
X
)
,
⊃
)
.
]
Example
4.13
.
(
P
(
N
)
,
⊆
)
is a lattice with many noncomparable pairs of elements, but the
class of sets
{{
1
}
,
{
1
,
2
}
, . . . ,
{
1
,
2
, . . . , n
}
, . . .
}
is a chain in
P
(
N
)
.
Problem
4.2
.
Show that if
A
⊂
B
and
B
is totally ordered, then
A
is totally ordered.
As an application, show that, in
(
R
2
,
≤
)
, any subset of the graph of a nondecreasing function
from
R
to
R
is a chain.
There are partially ordered sets that are
not
lattices.
Example
4.14
.
Let
T
1
⊂
R
2
be the triangle with vertices at
(0
,
0)
,
(1
,
0)
and
(0
,
1)
.
(
T
1
,
≤
)
is partially ordered, but is not a lattice because
(0
,
1)
∨
(1
,
0)
is not in
T
1
. On the other hand,
(
T
2
,
≤
)
is a lattice when
T
2
is the square with vertices at
(1
,
1)
,
(1
,
0)
,
(0
,
1)
, and
(0
,
0)
.
5. Monotone comparative statics
We now generalize in two directions:
(1) we now allow for
X
and
T
to have more general properties, they are not just linearly
ordered in what follows, and
(2) we now also allow for the set of available points to vary with
t
, not just the utility
function.
5.1. Product Lattices.
Suppose that (
X,

X
) and (
T,

T
) are lattices. Define the order

X
×
T
on
X
×
T
by (
x
0
, t
0
)
%
X
×
T
(
x, t
) iff
x
0
%
X
x
and
t
0
%
T
t
. (This is the unanimity order
again.)
Lemma
4.15
.
(
X
×
T,

X
×
T
)
is a lattice.
Proof
:
(
x
0
, t
0
)
∨
(
x, t
) = (max
{
x
0
, x
}
,
max
{
t
0
, t
}
)
∈
X
×
T
.
(
x
0
, t
0
)
∧
(
x, t
) = (min
{
x
0
, x
}
,
min
{
t
0
, t
}
)
∈
X
×
T
.
5.2. Supermodular Functions.
Definition
4.16
.
For a lattice
(
L,

)
,
f
:
L
→
R
is
supermodular
if for all
‘, ‘
0
∈
L
,
f
(
‘
∧
‘
0
) +
f
(
‘
∨
‘
0
)
≥
f
(
‘
) +
f
(
‘
0
)
,
equivalently,
f
(
‘
∨
‘
0
)

f
(
‘
0
)
≥
f
(
‘
)

f
(
‘
∧
‘
0
)
.
Example
4.17
.
Taking
‘
0
= (
x
0
, t
)
and
‘
= (
x, t
0
)
recovers Definition
4.2
.
Problem
4.3
.
Show that a monotonic convex transformation of a monotonic supermodular
function is supermodular.
36
Chapter 4.5
Problem
4.4
.
Let
(
L,

) = (
R
n
,
≤
)
.
Show that
f
:
L
→
R
is supermodular iff it has
increasing differences in
x
i
and
x
j
for all
i
6
=
j
. Show that a twice continuously differentiable
f
:
L
→
R
is supermodular iff
∂
2
f/∂x
i
∂x
j
≥
0
for all
i
6
=
j
.
5.3. Ordering Subsets of a Lattice.
Definition
4.18
.
For
A, B
⊂
L
,
L
a lattice, the
strong set order
is defined by
A

Strong
B
iff
∀
(
a, b
)
∈
A
×
B
,
a
∧
b
∈
A
and
a
∨
b
∈
B
.
Recall that interval subsets of
R
are sets of the form (
∞
, r
), (
∞
, r
], (
r, s
), (
r, s
], [
r, s
),
[
r, s
], (
r,
∞
), or [
r,
∞
).