Chemistry Reference
In-Depth Information
to fully developed chaos, as it corresponds with
D
4. Hence, stability is obtained
for a limited number of oligopolists, namely N
5, and as usual instability occurs
as the number of players increases.
5.2.2
Dynamics Under Adaptive Adjustment
If we assume that in a neighborhood of the equilibrium the left hand side of (5.52)
is strictly decreasing in x
k
, then a unique best response (under the assumptions of
LMA) is obtained, and it is a continuously differentiable function R
k
.Q
k
.t/;x
k
.t//.
By implicit differentiation of (5.52) and noticing that Q.t/
D
Q
k
.t/
C
x
k
.t/ we
have
f
0
C
2f
00
x
k
C
2f
0
R
0
kQ
f
00
x
k
.t/
C
0
k
R
0
kQ
D
0
and
f
0
C
2f
00
x
k
C
2f
0
R
0
kx
f
00
x
k
.t/
f
0
C
0
k
R
0
kx
D
0;
@ R
k
@Q
k
.t /
@ R
k
whereweusethenotationR
0
kQ
and R
0
kx
D
D
@x
k
.t /
.Sowehave
f
0
C
2f
00
x
k
f
00
x
k
.t/
2f
0
C
0
k
R
0
kQ
D
(5.69)
and
2f
00
x
k
f
00
x
k
.t/
2f
0
C
0
k
R
0
kx
D
:
(5.70)
The discrete dynamical process with adaptive adjustments based on the above
subjective best responses has the usual form
x
k
.t
C
1/
D
x
k
.t/
C
˛
k
R
k
.Q
k
.t/;x
k
.t//
x
k
.t/
;
(5.71)
where ˛
k
is a sign preserving function. It is easy to see that the interior equilibria
under full information are steady states of this system.
The local asymptotic stability of the equilibrium depends on the location of the
eigenvalues of the Jacobian of the system, which has the form
0
@
1
A
1
C
a
1
.r
1x
1/
a
1
r
1Q
:::
a
1
r
1Q
a
2
r
2Q
1
C
a
2
.r
2x
1/:::
a
2
r
2Q
J D
;
:
:
:
:
:
:
:
:
:
a
N
r
NQ
a
N
r
NQ
:::1
C
a
N
.r
Nx
1/
where a
k
D
˛
0
k
.0/,andr
kQ
and r
kx
are the partial derivatives of the subjective best
response function at the equilibrium, given by
D
.f
0
C
f
00
x
k
/=.2f
0
C
0
k
/
r
kQ
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