Chemistry Reference
In-Depth Information
4.2.3
Continuous Time Models
The dynamic model with continuous time scales has the same form (1.31) as the
one introduced in Sect. 1.2 and applied in Chaps. 2 and 3 for concave and isoelastic
oligopolies. Therefore the Jacobian and its eigenvalue equation are also the same as
the one given in (2.48), which we repeat here for the sake of convenience:
2
3
Y
s
X
s
a
j
.1
C
r
j
/
m
j
j
a
j
.1
C
r
j
/
C
4
1
5
D
0;
(4.20)
j D1
j D1
where a
1
.1
C
r
1
/>a
2
.1
C
r
2
/>
>a
s
.1
C
r
s
/ are the different a
k
.1
C
r
k
/ values
and
j
is the sum of all r
k
a
k
values such that a
k
.1
C
r
k
/
D
a
j
.1
C
r
j
/.Wealso
assume that a
k
>0for all firms. Since in general r
k
does not have a definite sign,
the same holds for
j
.If
j
D
0 or m
j
2,then
a
j
.1
C
r
j
/ is an eigenvalue of
the Jacobian. Notice that they are all negative, since r
k
>
1. All other eigenvalues
are the solutions of the equation
X
s
j
a
j
.1
C
r
j
/
C
D
0:
1
j
D
1
If g./ denotes again the left hand side of the above equation, then similarly to
the discrete time case we have
lim
!˙1
g./
D
1;
(
˙1
; if
j
<0;
1
; if
j
>0:
lim
g./
D
!
a
j
.1
C
r
j
/
˙
0
Since g
0
./ has no definite sign, no monotonicity property of g can be estab-
lished. Notice that all poles are negative. For the sake of mathematical simplicity
assume again that there is at most one sign change in the sequence
1
;
2
;:::;
s
,
and it is from “
”to“
C
”. Under this condition we have the same three possibilities
as in the discrete case (see Figs. 4.3-4.5), and the graph of function g is the same as
in the discrete case with the only difference being that all poles are now negative.
Case 1. All
j
>0. Local asymptotic stability occurs if g.0/>0.
Case 2. All
j
<0. Then the equilibrium is always locally asymptotically stable.
Case 3. There is a sign change in the sequence
1
;
2
;:::;
s
. Local asymptotic
stability occurs if g.0/>0:
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