Economics 201B–Second Half
Lecture 11
Tranversality Theorem, and Generic Regularity
Theorem 1 (2.5’, Transversality Theorem)
Let
X
×
Ω
⊆
R
n
+
p
be open
F
:
X
×
Ω
→
R
m
∈
C
r
with
r
≥
1+max
{
0
,n
−
m
}
If
F
(
x, ω
)=0
⇒
DF
(
x, ω
)
has rank m
then for all
ω
except for a set of Lebesgue measure zero,
F
(
x, ω
⇒
D
x
F
(
x, ω
)
has rank m
In particular, if
m
=
n
, there is a local implicit function
x
∗
(
ω
)
characterized by
F
(
x
∗
(
ω
)
,ω
and
x
∗
∈
C
r
.
Interpretation of Tranversality Theorem
•
Ω: a set of parameters.
In our case, Ω =
R
LI
++
,th
es
e
to
f
strictly positive endowment proFles,
p
=
LI
.
•
X
: a set of variables.
In our case,
X
=
R
L
−
1
++
e
f
strictly positive prices normalized by
p
L
=1
.
•
R
m
is the range of
F
.Inou
rca
s
e
,
F
(
x, ω
)=ˆ
z
(
x
), when the
endowment proFle is
ω
,
m
=
n
=
L
−
1.
1
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F
(
x, ω
)=0s
a
y
st
h
a
t
x
is an equilibrium price when the
endowment profle is
ω
.
•
rank
DF
(
x, ω
)=
m
=
L
−
1 says that, by adjusting either
the prices
x
or the endowments
ω
, it is possible to move
F
=ˆ
z
in any direction in
R
L
−
1
•
rank
D
x
F
(
x, ω
m
=
L
−
1saysdet
D
x
F
(
x, ω
)
6
=0
,wh
ich
says the economy is regular and is the hypothesis oF the Im
plicit ±unction Theorem. This will tell us that the equilibrium
prices are given by a fnite number oF implicit Functions oF the
parameters (endowments).
•
Parameters oF any given economy are fxed. However, we want
to study the
set
oF parameters For which the resulting economy
is wellbehaved.
•
Theorem says the Following:
“IF, whenever ˆ
z
(ˆ
p
∗
) = 0, it is possible by perturbing the
endowments and adjusting the prices to move ˆ
z
in any
direction in
R
L
−
1
, then For almost all endowments, the
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 Spring '06
 ANDERSON
 Economics, Empty set, Open set, Topological space, Lebesgue measure, RL1

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