Chemistry Reference
In-Depth Information
r
k
a
k
.h
k
1/
u
k
C
r
k
a
k
h
k
X
l
u
l
C
r
k
.1
a
k
/
v
k
D
u
k
;
(5.8)
¤
k
a
k
.h
k
1/
u
k
C
a
k
h
k
X
l
u
l
C
.1
a
k
/
v
k
D
v
k
.k
D
1;2;:::;N/: (5.9)
¤
k
Subtract the r
k
-multiple of the second equation from the first one to obtain
.
u
k
r
k
v
k
/
D
0:
We may assume that
¤
0,so
u
k
D
r
k
v
k
. If we substitute this relation into (5.8)
we see that
.r
k
a
k
.h
k
1/
C
.1
a
k
//
u
k
C
r
k
a
k
h
k
X
l¤k
D
u
k
;
u
l
which is the eigenvalue equation of the N
N matrix
0
@
1
A
r
1
a
1
.h
1
1/
C
1
a
1
r
1
a
1
h
1
:::
r
1
a
1
h
1
r
2
a
2
h
2
r
2
a
2
.h
2
1/
C
1
a
2
r
2
a
2
h
2
H
D
: (5.10)
:
:
:
:
:
:
:
:
:
:
:
:
r
N
a
N
h
N
r
N
a
N
h
N
:::r
N
a
N
.h
N
1/ C 1 a
N
Observe that this matrix is the Jacobian of system (5.6), the model with partial
adjustment towards the best response. Therefore if local asymptotic stability is our
concern, then the stability conditions the systems (5.4)-(5.5) and (5.6) are again
equivalent, and the eigenvalues of the matrix (5.10) determine whether or not an
equilibrium is stable.
Assume that functions f , f
k
and C
k
for all k are twice continuously differen-
tiable, furthermore,
(A) f
0
.Q/ < 0, f
k
.Q/ < 0,
(B) x
k
f
0
k
.Q/
C
f
k
.Q/
0,
(C) f
k
.Q/
C
0
k
.x
k
/<0,
for all feasible values of x
k
and Q (see Chap. 2, Sect. 2.1, for an economic interpre-
tation of these conditions).
The main stability result of this section is given in the following theorem.
Theorem 5.1.
Assume that
a
k
>0
for all
k
, and conditions (A)-(C) are satisfied.
(i) The equilibrium is locally asymptotically stable if for all
k
,
a
k
.1
C
r
k
/<2
(5.11)
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