Chemistry Reference
In-Depth Information
for k
D
1;2;:::;N,sincefirmk observes the market price f.Q/ with delay.
Similarly to the full information case we linearize this equation around the equi-
librium to have
r
k
h
k
Z
t
0
x
lı
.s/ds
x
kı
.t/
!
x
kı
.t/
!
;
X
N
x
kı
.t/
D
a
k
w
.t
s;T
k
;m
k
/
l
D
1
where x
kı
denotes the deviation of x
k
from its equilibrium level, and a
k
;r
k
and h
k
are the same as in Sect. 5.1.1. By seeking the solution as x
kı
D
v
k
e
t
, substituting
it into the linearized equation and letting t
!1
, we obtain the equation
a
k
r
k
h
k
Z
1
0
w
.s;T
k
;m
k
/e
s
ds
N
X
.
C
a
k
.r
k
C
1//
v
k
v
l
D
0:
l
D
1
We can further simplify this equation by using the limiting values of the integrals
(D.3) derived in Appendix D to obtain
A
k
./
v
k
C
B
k
./
X
l
v
l
D
0.k
D
1;2;:::;N/;
(5.37)
¤
k
where
A
k
./
D
C
a
k
.r
k
C
1/
a
k
r
k
h
k
1
C
.m
k
C1/
T
k
p
k
and
B
k
./
D
a
k
r
k
h
k
1
C
.m
k
C1/
T
k
p
k
;
with
(
1 if m
k
D
0;
m
k
if m
k
>0;
as before. System (5.37) has a non-trivial solution if its determinant is zero. Notice
that the determinant has the same structure as (2.54) in the full information case,
in addition its characteristic polynomial can also be expressed similarly to (2.55),
which here has the form
p
k
D
2
3
Y
N
X
N
a
k
r
k
h
k
.
C
a
k
.r
k
C
1//
1
C
4
5 D
0:
.
C
a
k
.r
k
C
1//
1
m
k
C1
T
k
p
k
kD1
kD1
In the concave and isoelastic cases r
k
>
1 for all k,soifa
k
>0for all k, then the
values
a
k
.r
k
C
1/ are negative, so in order to examine stability we have only to
analyze the locations of the roots of the equation
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