Chemistry Reference
In-Depth Information
are sufficiently close to the true slope B. This local analysis leaves open several
questions. First of all, what are the necessary conditions for local stability, such
that an exact delineation of the stability region in the space of the parameters can
be obtained? What kinds of bifurcations occur when the boundaries of such stabil-
ity regions are crossed? How does the steady state lose stability and what kind of
disequilibrium asymptotic dynamics should be expected when the steady state is
unstable? Finally, what are the extent and the shape of the basin of attraction of the
steady state (when it is stable) or of other attractors when the steady state is unsta-
ble? To answer these questions is, in general, not easy if one considers a nonlinear
N-dimensional dynamical system. Therefore, we will try to gain some insight into
these problems by considering some simple situations, such as the symmetric case
of an oligopoly with N identical firms starting from identical initial guesses, and
a duopoly with two heterogeneous firms that start from arbitrary initial guesses for
the scale factors "
k
.0/, k
D
1;2.
Example 5.12.
Consider first an oligopoly with identical firms such that c
k
D
c;
a
k
D
a for each k, and assume that they also have identical initial conditions,
"
k
.0/
D
".0/:
So we have "
k
.t/
D
".t/ for each t
0. The dynamics of ".t/ are governed by the
following one-dimensional difference equation
1
".t/
C
a.A
Nc/
N
C
1
;
(
5
.102)
which can be derived easily from (5.91). At the unique positive equilibrium "
D
B
the derivative of the function g becomes
Aa
.N
C
1/B
aBcN
N
C
1
1
".t/
C
".t
C
1/
D
g.".t//
D
aA
.N
C
1/B
acN
.N
C
1/B
;
g
0
.B/
D
1
and it is easy to realize that the condition for the local asymptotic stability
1<g
0
.B/ < 1 is fulfilled for
a.Nc
C
A/
B.N
C
1/
<2;
(5.103)
which can be rewritten as
a<
2B.N
C
1/
Nc
C
A
:
The stability condition (5.103) illustrates the stabilizing role of small values of the
speed of adjustment a. The role of the number of firms is also clear, since the left
hand side of the stability condition (5.103) is a decreasing function of N if c<A
(note that the reservation price has to be larger than the unit cost in order to make
production profitable). Hence, in our case a higher number of identical firms helps
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