Chemistry Reference
In-Depth Information
where we can assume that all
j
values are nonzero. This last equation is equivalent
to a polynomial equation of degree s,sothereares real or complex roots. Clearly,
g./
D
1;
g./
D˙1
lim
lim
!˙1
!
1
a
j
.1
C
r
j
/
˙
0
and
X
s
j
.1
a
j
.1
C
r
j
/
/
2
<0:
g
0
./
D
j D1
Using these properties we can graph g./ as shown in Fig. 2.1. This figure indicates
that the structure of the roots is such that there is one root before 1
a
1
.1
C
r
1
/,
and one root between each pair of poles 1
a
j
.1
C
r
j
/ and 1
a
j C1
.1
C
r
j C1
/ for
j
D
1;2;:::;s
1. So all roots have been found and they are real. Furthermore all
are inside the unit circle if and only if
1
a
1
.1
C
r
1
/>
1; and g.
1/ > 0:
At least one eigenvalue is outside the unit circle if either 1
a
1
.1
C
r
1
/
1 or
g.
1/ < 0.
In the case of constant speeds of adjustment we usually assume that 0<a
k
1
for all k, and from relation (2.7) we know that
1<r
k
0. So condition (2.21) is
usually satisfied in this case.
1-
a
1
(1+
r
1
)
1-
a
2
(1+
r
2
)
1-
a
3
(1+
r
3
)
a
s
-1
(1+
r
s
-1
)
1-
a
s
(1+
r
s
)
-1
1
λ
Fig. 2.1
Graph of g./, the roots of which are eigenvalues of the Jacobian of the system describing
the dynamics of the discrete time oligopoly under best reply dynamics
Search WWH ::
Custom Search