Chemistry Reference
In-Depth Information
and
r
kx
D
.f
00
x
k
/=.2f
0
C
0
k
/:
All eigenvalues are inside the unit circle, if any norm of the Jacobian is less than
unity. By selecting the
k
:
k
1
,
k
:
k
1
and Frobenius norms, we get three sufficient
conditions, namely
ma
k
˚
j
1
C
a
k
.r
kx
1/
jC
.N
1/a
k
j
r
kQ
j
<1
(5.72)
or
n
o
<1
j
1
C
a
k
.r
kx
1/
jC
X
l
max
k
a
l
j
r
lQ
j
(5.73)
¤
k
or
n
.1
C
a
k
.r
kx
1//
2
C
.N
1/a
k
r
kQ
o
<1:
X
N
(5.74)
kD1
Example 5.8.
Assume linear cost functions and (subjective) best response dynam-
ics, then C
0
k
1
2
D
0,sor
kx
r
kQ
D
and a
k
D
1 for all k. In this special case
f
00
x
k
2f
0
1
C
a
k
.r
kx
1/
D
r
kx
D
(5.75)
at the equilibrium, and
f
0
C
f
00
x
k
2f
0
a
k
r
kQ
D
;
so (5.72) reduces to
x
k
j
f
00
. Q/
jC
.N
1/
j
f
0
. Q/
C
x
k
f
00
. Q/
j
<2
j
f
0
. Q/
j
(5.76)
at the equilibrium for all k. Assume next quadratic cost functions C
k
.x
k
/
D
d
k
C
e
k
x
k
and (subjective) best response dynamics. Then C
k
.x
k
/
D
2e
k
x
k
and
C
0
k
.x
k
/
D
2e
k
,so
f
00
x
k
2.f
0
e
k
/
1
C
a
k
.r
kx
1/
D
r
kx
D
and
f
0
C
f
00
x
k
2.f
0
e
k
/
;
therefore condition (5.72) can be rewritten as
a
k
r
kQ
D
x
k
j
f
00
. Q/
jC
.N
1/
j
f
0
. Q/
C
x
k
f
00
. Q/
j
<2
j
f
0
. Q/
e
k
j
;
(5.77)
for all k. The other two matrix norms can be used to obtain similar conditions, the
details are left as easy exercises for the reader.
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