Chemistry Reference
In-Depth Information
Models including externalities can be discussed similarly, the final conclusions
remain similar to those to be presented in this section. First the local asymptotic
stability of the equilibrium is discussed. The Jacobian of the system (2.39) has the
special form
0
@
1
A
C
0
1
/
1
C
a
1
.x
1
f
00
C
2f
0
a
1
.x
1
f
00
C
f
0
/
:::a
1
.x
1
f
00
C
f
0
/
C
0
2
/:::a
2
.x
2
f
00
C
a
2
.x
2
f
00
C
f
0
/
a
2
.x
2
f
00
C
2f
0
f
0
/
1
C
:
:
:
:
:
:
:
:
:
H
D
a
N
.x
N
f
00
C
f
0
/
a
N
.x
N
f
00
C
f
0
/
a
N
.x
N
f
00
::: 1
C
2f
0
C
0
N
/
C
where all derivatives are taken at the equilibrium and a
k
D
˛
0
k
.0/ for all k. Notice
that this matrix has the special structure (E.4) introduced in Appendix E, therefore
(E.5) can be used to write the characteristic polynomial as
"
1
C
#
:
Y
N
X
N
a
k
.x
k
f
00
C
f
0
/
1
C
a
k
.f
0
C
0
k
/
.1
C
a
k
.f
0
C
0
k
/
/
kD1
kD1
Similarly to best response dynamics with adaptive expectations , the eigenvalues
are 1
C
a
k
.f
0
C
0
k
/ and the roots of the equation
X
N
a
k
.x
k
f
00
C
f
0
/
1
C
a
k
.f
0
C
0
k
/
D
0:
1
C
(2.41)
k
D
1
Since assumptions (A)-(C) hold, the graph of the function on the left hand side of
(2.41) is the same as shown in Fig. 2.1, so all eigenvalues are inside the unit circle
if and only if for all k,
a
k
.C
0
k
f
0
/<2;
(2.42)
and
X
N
a
k
.x
k
f
00
C
f
0
/
2
C
a
k
.f
0
C
0
k
/
>
1:
(2.43)
k
D
1
Notice that these conditions are very similar to conditions (2.21) and (2.22) given
for the best reply dynamics with adaptive expectations. In order to compare the two
cases substitute relation (2.5) (from which r
k
is calculated) into conditions (2.21)
and (2.22) to obtain
C
0
k
f
0
a
k
.2f
0
C
x
k
f
00
C
0
k
/
<2
(2.44)
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