•
X
: a set of variables (price vectors, or strategies).
•
R
m
is the range of
F
(excess demand, or best-response strategies).
•
F
(
x, ω
) = 0 is equilibrium condition, given parameter
ω
.
•
Rank
DF
(
x, ω
)=
m
says that, by adjusting either the variables or parameters, it is possible to move
F
in any direction.
•
When
m
=
n
,Rank
D
x
F
(
x, ω
m
says det
D
x
F
(
x, ω
)
6
= 0, which says the economy is regular and
is the hypothesis of the Implicit Function Theorem. This will tell us that the equilibrium prices are
given by a ±nite number of implicit functions of the parameters (endowments), and the equilibrium
correspondence is thus lower hemicontinuous.
•
Parameters of any given economy are ±xed. However, we want to study the
set
of parameters for
which the resulting economy is well-behaved.
•
Theorem says the following:
“If, whenever
F
(
x, ω
) = 0, it is possible by perturbing the parameters and variables to move
F
in any direction, then for almost all parameter values, all equilibria are regular, and hence
there are ±nitely many equilibria, the equilibria are implicitly de±ned
C
r
functions of the
parameters, and the equilibrium correspondence is lower hemicontinuous.”
•
If
n<m
D
x
F
(
x, ω
)
≤
min
{
m, n
}
=
. Therefore,
(
F
(
x, ω
)=0
⇒
DF
(
x, ω
)hasrankm)
2