Chemistry Reference
In-Depth Information
@R
k
@x
k
@R
k
@Q
k
where R
0
kx
D
;R
0
kQ
D
.So
f
0
C
x
k
f
00
2f
0
C
x
k
f
00
C
0
k
R
0
kQ
D
;
(4.69)
K
0
k
and
K
0
k
2f
0
C
x
k
f
00
C
0
k
R
0
kx
D
:
(4.70)
K
0
k
It is easy to see that as in the concave case the derivatives of the reaction function
of firm k satisfy
1<R
0
kQ
0
R
0
kx
<1
and
1<R
0
kQ
R
0
kx
0:
Consider first the dynamic process (1.28)-(1.29) with adaptive expectations,
which here assumes the form
0
0
@
X
l¤k
1
1
@
Q
k
.t/
C
˛
k
x
l
.t/
Q
k
.t/
A
;x
k
.t/
A
;
x
k
.t
C
1/
D
R
k
(4.71)
0
@
X
1
A
:
Q
k
.t
C
1/
D
Q
k
.t/
C
˛
k
x
l
.t/
Q
k
.t/
(4.72)
l¤k
The Jacobian of this system at the equilibrium has the special form
J
11
J
12
J
21
J
22
where
0
@
1
A
0
@
1
A
r
1x
r
1Q
a
1
r
1Q
a
1
r
1Q
.1
a
1
/
r
2Q
a
2
r
2x
r
2Q
a
2
r
2Q
.1
a
2
/
J
11
D
;
J
12
D
;
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
r
NQ
a
N
r
NQ
a
N
r
Nx
r
NQ
.1
a
N
/
0
@
1
A
0
@
1
A
0a
1
a
1
1
a
1
a
2
0
a
2
1
a
2
J
21
J
22
D
; and
D
;
:
:
:
a
N
a
N
0
:
:
:
:
:
:
:
:
:
1 a
N
@R
k
@Q
k
@R
k
@x
k
and r
kQ
evaluated at the equilibrium. If is an eigen-
value and .
u
1
;:::;
u
N
;
v
1
;:::;
v
N
/ is an associated eigenvector, then the eigenvalue
D
and r
kx
D
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