Chemistry Reference
In-Depth Information
In order to illustrate our general approach to piecewise differentiable dynamic
models, we reconsider the simplest oligopoly model introduced in Example 1.1
and further studied in Examples 2.1 and 2.2. We assume a linear inverse demand
function, p
D
f.Q/
D
A
BQ, and linear cost functions, C
k
.x
k
/
D
c
k
x
k
,
k
D
1;::;N, where for simplicity we let the fixed costs be zero. The presence
of non-negativity and capacity constraints makes the resulting dynamical system
non-differentiable. In what follows we investigate the asymptotic dynamics for an
increasing number of firms in the industry and we explore the role of the capacity
constraints, the presence of which leads to non-differentiability of the dynamical
system considered. Moreover, we will explain some peculiar dynamic properties of
piecewise linear maps, consider a particular type of bifurcation which causes the loss
of stability of the unique equilibrium, and illustrate which kind of non-equilibrium
dynamics might occur.
Example 2.3.
As a first step let us consider the symmetric case of N identical firms
with c
1
D
c
2
D
:::
D
c
N
D
c, linear adjustment functions with a
1
D
a
2
D
:::;a
N
D
a,andL
1
D
L
2
D
:::
D
L
N
D
L. We assume that A>cand that firms
use partial adjustment towards their best responses. Then, given that firms start from
the identical initial condition x
1
.0/
D
x
2
.0/
D
:::
D
x
N
.0/
D
x.0/, the dynamics
are captured by the one-dimensional model
x.t
C
1/
D
T.x.t//
D
.1
a/x.t/
C
aR..N
1/x.t//;
where
8
<
Ac
.N
2L
N
L
if x
1
;
1/B
A
c
2B
N
1
2
x if
Ac
.N
2L
N
Ac
.N
R..N
1/x/
D
x
:
1/B
1
1/B
c
.N 1/B
:
A
0
if x
Obviously, the function T is a piecewise linear map, characterized by three regions
where it is differentiable. These regions are separated by two kinks (points of non-
differentiability), so that the map can be written in detail as
8
<
Ac
.N
2L
N
.1
a/x
C
aL
if x
1
;
1
x
C
a
A
c
2B
1/B
a.N
C
1/
2
c
.N 1/B
A
2L
N 1
c
.N 1/B
;
A
T.x/
D
if
x
:
A
c
.1
a/x
if x
1/B
:
.N
Figure 2.3 depicts a typical graph of the map T.x/ together with the graph of
the reaction function (dashed). It should be clear that the exact shape of the graph
and the locations of the kinks depend on the market and cost parameters A;B;c,
on the capacity level L of the firms and the number of firms N in the industry,
and in particular on the adjustment speed a. For larger values of a the graph of
T.x/is closer to that of R.x/, for smaller values of a the graph of T.x/is closer
to the diagonal. Furthermore, as we now show, these parameters determine if the
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