Chemistry Reference
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two further preimages, O
.2/
1
and O
.3/
1
, are located on the line
1
of the equation
5
1
:
1
1
a
x
1
C
x
2
D
1
C
in symmetric positions with respect to (see Fig. 3.14b). Hence
a.
C
1/
1
C
p
a
2
2
C
2a.1
a/
3.a
2
C
1/
C
6a
O
.2/
D
;
1
2a
!
a.
C
1/
1
p
a
2
2
C
2a.1
a/
3.a
2
C
1/
C
6a
(3.28)
2a
and the symmetric point O
.3/
1
is obtained from O
.2/
1
by swapping the two coordi-
nates.
To conclude this subsection, we would like to reflect on several issues. First, the
occurrence of the bifurcation which transforms the basins from simply connected to
disconnected sets causes a loss of predictability concerning the long-run outcome of
the adjustment process. The presence of many disjoint components of both basins
causes a sensitivity with respect to the initial production quantities, in the sense that
a small perturbation may lead to a crossing of the boundary which separates the two
basins and, consequently, the trajectory may converge to a different Nash equilib-
rium. Second, for increasing values of the adjustment coefficient a, as the line
1
in Fig. 3.14b moves upwards, certain connected parts of the basins of the equilibria
come closer to the corresponding other equilibrium. That is, initial production quan-
tities which eventually lead to convergence to E
i
are located close to the equilibrium
E
j
;i
ยค
j, and vice versa. In contrast to a global analysis, a study based only on
the local properties of the process around the equilibria would not have been able
to provide us with information on the size of the neighborhood from which conver-
gence to the corresponding equilibrium is achieved. Finally, our global analysis also
reveals that for .;a/
2
s
.E
i
;C
2
/ three coexisting attractors are present
6
. Hence
the outcome of the oligopoly game is highly path dependent and could end up at any
of the attractors depending on the initial conditions.
3.2.2
Non-Identical Speeds of Adjustment
We now turn to the case of different speeds of adjustment. In contrast to the previous
situation, a rigorous mathematical analysis cannot be provided. However, guided by
5
This can be seen by setting x
1
D
x
2
in (3.26) and adding or subtracting the two symmetric
equations.
6
We remind the reader that the stability region of E
1
, E
2
and C
2
is defined in Proposition 3.1.
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