Chemistry Reference
In-Depth Information
D
R
0
k
. Q
k
/ and a
k
D
˛
0
k
.0/:
r
k
(2.16)
If we use .
u
1
;
u
2
;:::;
u
N
;
v
1
;
v
2
;:::;
v
N
/ to denote the typical eigenvector and
the associated eigenvalue of the matrix (2.15) then it is relatively straightforward
to see that the
u
k
;
v
k
(for k
D
1;2;:::;N/are given by the two sets of equations
r
k
a
k
X
l
u
l
C
r
k
.1
a
k
/
v
k
D
u
k
; k
D
1;2;:::;N/;
(2.17)
¤
k
a
k
X
l¤k
C
.1
a
k
/
v
k
D
v
k
; k
D
1;2;:::;N/:
u
l
(2.18)
By subtracting the r
k
-multiple of (2.18) from (2.17) we find that
.
u
k
r
k
v
k
/
D
0:
(2.19)
Since a zero eigenvalue does not destroy the asymptotic stability of the system,
we will consider only nonzero eigenvalues of the Jacobian. If
¤
0, then (2.19)
implies that
u
k
D
r
k
v
k
and if we substitute this condition into (2.17) we see that the
u
k
values are determined by
r
k
a
k
X
l¤k
u
l
C
.1
a
k
/
u
k
D
u
k
; 1
k
N/;
which is readily shown to be the eigenvalue equation of the N
N matrix
0
@
1
A
1
a
1
r
1
a
1
::: r
1
a
1
r
2
a
2
1
a
2
::: r
2
a
2
:
:
:
:
:
:
:
:
:
:
:
:
r
N
a
N
r
N
a
N
:::1
a
N
H D
:
(2.20)
Observe that this matrix coincides with the Jacobian of the partial adjustment
dynamics (1.30). Therefore, if local asymptotic stability is our concern, then the
conditions for the process (1.28)-(1.29) of best reply dynamics with adaptive expec-
tations is equivalent to the process (1.30) of partial adjustment towards the best
response with naive expectations. This means that best reply dynamics with adap-
tive expectations and best reply dynamics with partial adjustments share the same
local asymptotic stability properties, and the eigenvalue structure of matrix (2.20)
determines whether an equilibrium is locally asymptotically stable or not. In the case
of N
D
2 (duopoly) or very special response functions with arbitrary value of N,the
two processes are even equivalent as was shown earlier in Sect. 1.2. The following
theorem presents conditions for the local asymptotic stability of the equilibrium. It
allows us to assert that if the initial outputs of the firms are sufficiently close to the
equilibrium, then as t
!1
; the outputs converge to the equilibrium.
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