Chemistry Reference
In-Depth Information
A
c
1
C
c
2
Ac
k
.c
1
C
c
2
/
2
:
Q
D
Q
k
and
D
From Example 1.5 we know that
s
AQ
k
c
k
R
k
.Q
k
/
D
Q
k
;
and so
c
1
C
c
2
2c
k
R
0
k
. Q
k
/
D
1;
therefore
c
2
c
1
2c
1
c
1
c
2
2c
2
r
1
D
and
r
2
D
:
Assume that c
1
¤
c
2
, and that the firms select identical adjustments, that is,
a
1
D
a
2
a. The characteristic equation of the Jacobian of the dynamic process
with partial adjustment towards the best response is given in general by (2.23),
which simplifies to
.1
a.1
C
r
k
/
/
1
C
2
Y
r
1
a
1
a.1
C
r
1
/
C
r
2
a
1
a.1
C
r
2
/
D
0:
k
D
1
This equation reduces to the quadratic
2
C
.2a
2/
C
.1
2a
C
a
2
a
2
r
1
r
2
/
D
0;
with roots
1;2
D
.1
a/
˙
ia
p
r
1
r
2
;
since
.c
1
c
2
/
2
4c
1
c
2
r
1
r
2
D
<0:
By an appropriate choice of the parameters c
1
and c
2
the quantity r
1
r
2
can take any
negative value. Clearly if c
1
¤
c
2
, then both roots are complex, and since
2
D
1
2a
C
a
2
.1
r
1
r
2
/;
j
1;2
j
the roots can be both inside and outside the unit circle. The equilibrium is locally
asymptotically stable if
a.1
r
1
r
2
/<2;
and unstable if this condition is violated with strict inequality. An analogous condi-
tion for the stability of the equilibrium in the duopoly case with constant adjustment
speeds has been derived by Puu (2003, Chap. 7). With fixed r
1
and r
2
, stability
occurs if the value of a is sufficiently small. With a fixed value of a
2
.0;1 we
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