Monotone Comparative Statics
Let
X
and
T
be subsets of
R
and let
f
:
X
×
T
→
R
. We consider how the set of maximizers
of
f
(
x,t
) on
X
vary with the parameter
t
.
Deﬁnition.
f
satisﬁes the
strict singlecrossing property
(SSCP) in
(
x,t
)
if for every
x
00
,
x
0
in
X
and
t
00
,
t
0
in
T
, with
x
00
> x
0
, and
t
00
> t
0
f
(
x
00
,t
0
)
≥
f
(
x
0
,t
0
) implies
f
(
x
00
,t
00
)
> f
(
x
0
,t
00
)
.
The following simple Comparative Statics Theorem has many applications.
Theorem
(Easy Corollary to Shannon, 1995, Theorem 4)
.
Let
f
satisfy the strict single
crossing property (SSCP) in
(
x,t
)
and let
t
1
,
t
0
be any two points in
T
with
t
1
> t
0
. If
x
i
maximizes
f
(
x,t
i
)
on
X
for
i
= 0
,
1
, then
x
1
≥
x
0
.
Note that the only assumption we made on
f
is the SSCP.
X
and
T
can be ﬁnite or inﬁnite;
if they are intervals,
f
need not be concave or diﬀerentiable or even continuous in
x
.
Exercise 1.
Prove the Comparative Statics Theorem. (Hint: just write down what it means
for
x
i
to solve the problem for
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 Spring '10
 schlee
 Microeconomics, single crossing condition, single crossing property, Comparative Statics Theorem

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