Chemistry Reference
In-Depth Information
The characteristic equation of this Jacobian is given by
2
C
p
C
q
D
0,where
p
D
2
C
a
1
C
a
2
and q
D
.1
a
1
/.1
a
2
/
a
1
a
2
B
2
=.4.B
C
e
1
/.B
C
e
2
//.
The necessary and sufficient conditions for the eigenvalues to be located inside the
unit circle, which are conditions for the local asymptotic stability of the interior
Nash equilibrium E, are given by the inequalities (see Appendix F, Lemma F.1)
1
C
p
C
q>0 ; 1
p
C
q>0; q<1:
(1.40)
These inequalities, respectively, reduce to
B
2
4.B
C
e
1
/.B
C
e
2
/
<1;
B
2
4.B
C
e
1
/.B
C
e
2
/
<1
C
2
2
a
1
a
2
;
a
1
a
2
B
2
4.B
C
e
1
/.B
C
e
2
/
>1
a
1
C
a
2
a
1
a
2
:
Observe that the first stability condition coincides with condition (1.37) under which
this is the only equilibrium and so is globally asymptotically stable. The other
conditions do not affect the stability properties, because the second condition is
implied by the first one (since 0<a
k
1) and the last condition is always satis-
fied (since the left hand side is positive, whereas the right hand side is negative).
If B
2
>4.B
C
e
1
/.B
C
e
2
/, then the interior equilibrium is unstable. This is the
situation in case (ii) of Example 1.2, where we might have three equilibria with an
unstable interior equilibrium.
Consider now the case shown in Fig. 1.3 and the monopoly equilibrium .0;x
2
/.
In the neighborhood of this equilibrium x
2
<x
2
<L
2
,soR
1
.x
2
/
D
0.Furthermore
x
1
D
0 or a small positive value. Notice that the segments where R
1
.x
2
/
D
L
1
,or
R
2
.x
1
/
D
L
2
are empty, which implies that the sets
.k/
for k
D
3;2;9;8;7 are also
empty. Therefore any point in a small neighborhood of the equilibrium .0;x
2
/ is
in the region
D
.4/
D
where the Jacobian matrix is
1
a
1
!
:
0
(1.41)
a
2
B
2.B
e
2
/
1
a
2
C
Let
x0
01
P
D
(1.42)
be a diagonal matrix with x>0. Then the row norm generated by this matrix is
bounded by the row norm of the matrix
x0
!
1
a
1
!
x
0
01
!
0
(1.43)
a
2
B
2.B
e
2
/
1
a
2
01
C
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