Chemistry Reference
In-Depth Information
This weighting function has been frequently used in the analysis of time lagged
dynamical system (see the Appendix D and Cushing (1977)) since as we shall see, it
affords a great deal of mathematical tractability. The main properties of this weight-
ing function are summarized in Appendix D, here we simply point out that T may
be interpreted as the average time delay (in fact for m>0the weighting func-
tion peaks at t
s
D
T ), whilst the parameter m plays the role of “squeezing” the
weights around this average value. If we interpret
w
as a distribution then T is the
mean and m is related to the inverse of the standard deviation. We see from (2.50)
that firm k may apply a different average time lag .T
k
/ and “squeezing” factor .m
k
/
to information about output of the rivals than to information about its own output
(denoted by S
k
and l
k
respectively). This reflects the fact that a firm should be better
informed about its own production process than that of its rivals.
Equation (2.50) is a Volterra-type integro-differential equation, and as is also
shown in Appendix D, it is equivalent to a system of ordinary differential equations.
Therefore all known tools from the stability theory of ordinary differential equa-
tions can be used to analyze the asymptotic behavior of system (2.50), including
linearization.
For k
D
1;2;:::;N,letx
kı
.t/ denote the deviation of x
k
.t/ from its equilibrium
level, then the linearized system has the form
8
<
9
=
r
k
Z
t
0
Z
t
w
.t
s;T
k
;m
k
/
X
l¤k
x
kı
.t/
D
a
k
x
lı
.s/ds
w
.t
s;S
k
;l
k
/x
kı
.s/ds
;
:
;
0
(2.52)
where 1
k
N, a
k
D
˛
0
k
.0/ and r
k
D
R
0
k
P
l¤k
x
l
as before. The character-
istic equation of this linear system can be obtained with the same technique that is
usually used in the case of linear differential equations (see Miller (1972)). We seek
the solution in the form
D
v
k
e
t
x
kı
.1
k
N/;
substitute it into (2.52) and let t
!1
. The resulting equation becomes
C
a
k
Z
1
0
w
.s;S
k
;l
k
/e
s
ds
v
k
a
k
r
k
Z
1
0
w
.s;T
k
;m
k
/e
s
ds
X
l¤k
v
l
D
0:
By using the limiting values of the integral (D.3) derived in Appendix D we can
further simplify this equation to
A
k
./
v
k
C
B
k
./
X
l
v
l
D
0;
(2.53)
¤
k
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