Chemistry Reference
In-Depth Information
In the following part of this section we will show that the characteristic
polynomial of the Jacobian can be derived in a simple form, and therefore the eigen-
values can be easily located. Therefore a more accurate stability condition can be
derived which is “almost” sufficient and necessary.
Notice that this Jacobian has the special structure (E.4), so its characteristic
polynomial can be written in the special form given by (E.5), namely
"
1
C
#
:
N
N
Y
X
1
C
a
k
.r
kx
r
kQ
1/
a
k
r
kQ
1
C
a
k
.r
kx
r
kQ
1/
'./
D
k
D
1
k
D
1
(5.78)
Consider first the concave oligopolies of Chap. 2, where we have assumed that
(A) f
0
<0;
(B) x
k
f
00
C
f
0
0;
(C) f
0
C
0
k
<0,
for all k and feasible output levels. Then
r
kQ
0 and 0<r
kx
r
kQ
<1
1
for all k. Note that in the special case of linear cost functions, r
kx
r
kQ
2
.The
eigenvalues are 1
C
a
k
.r
kx
r
kQ
1/ together with the roots of the equation
D
X
N
a
k
r
kQ
1
C
a
k
.r
kx
r
kQ
1/
D
0:
g./
D
1
C
(5.79)
kD1
The values 1
C
a
k
.r
kx
r
kQ
1/ are inside the unit circle if and only if
2
1
.r
kx
r
kQ
/
:
a
k
<
(5.80)
Since 0<a
k
1 is assumed, this inequality always holds.
The poles of the left hand side of (5.79) are between
1 and
C
1 under assump-
tion (5.80), its derivative is negative (unless all r
kQ
D
0). The graph of the left hand
side is the same as the one shown in Fig. 2.1. Under condition (5.80) all roots are
real, and they are between
1 and
C
1 if and only if
X
N
a
k
r
kQ
2
C
a
k
.r
kx
r
kQ
1/
>0:
g.
1/
D
1
C
(5.81)
kD1
If this inequality is violated with strict opposite inequality, then the equilibrium is
unstable.
Search WWH ::
Custom Search