Chemistry Reference
In-Depth Information
, then the row norms of the Jacobians around E
1
generated by the matrix
P
are bounded by the row norm of the matrix
x0
x0
01
As before, let
P
D
!
1
a
1
a
1
!
x
0
01
!
1
a
1
a
1
Bx.N
1/
2.B
!
B.N
1/
2.B
C
e
1
/
D
C
e
1
/
(1.45)
0
1
a
2
0
1
a
2
01
which is below one if
1
a
1
C
a
1
Bx.N
1/
2.B
C
e
1
/
<1;
that is, when
x<
2.B
C
e
1
/
B.N
1/
:
Hence the equilibrium E
1
is locally asymptotically stable for all values of N.
Similarly, E
2
is locally asymptotically stable if there is a positive x such that
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
a
2
B
2x.B
C
e
2
/
C
1
a
2
a
2
B.N
2/
2.B
C
e
2
/
<1
which occurs if
1<1
a
2
1
C
<1:
B.N
2/
2.B
C
e
2
/
Therefore, E
2
is stable provided that
0<a
2
B.N
2/
C
2.B
C
e
2
/
2.B
C
e
2
/
<2:
From this stability condition we can now derive several interesting results. First,
as already shown before, in the case of duopoly .N
D
2/ the boundary equilibrium
E
2
is also always stable, like E
1
. Moreover, the boundary equilibrium E
2
is stable
provided that a
2
is sufficiently small, which means that firms 2;:::;N haveahigh
inertia in adjusting their quantities toward the best responses. Finally, increasing the
number of firms has a destabilizing role. In fact the stability condition can be written
as
2.2
a
2
/.B
C
e
2
/
Ba
2
N<2
C
;
so that for given cost parameters and adjustment speeds asymptotic stability is lost
when the number of firms reaches a certain size.
To conclude this section, we study the global dynamics of the semi-symmetric
model. Consider again the parameter values A
D
450, B
D
30 and c
1
D
c
2
D
:::
D
c
N
D
275, e
1
D
e
2
D D
e
N
D
17. For the adjustment speeds of the two
firms we select a
1
D
0:6 and a
2
D D
a
N
D
0:45. For these parameter values
the stability condition derived in the previous paragraph tells us that the boundary
equilibrium E
2
is asymptotically stable if N<4. In Fig. 1.12a we depict the basins
Search WWH ::
Custom Search