Chemistry Reference
In-Depth Information
Let L
k
denote the capacity limit of firm k, and since the payoff of firm k is strictly
concave in x
k
, the best response of firm k is
8
<
z
k
0;
0
if
R
k
.Q
k
;S/
D
z
k
L
k
if
L
k
;
:
z
k
otherwise.
In the following discussion we consider the symmetric case of N identical firms,
so that c
k
D
c, d
k
D
d, L
k
D
L for all k, and we assume adaptive output adjust-
ments with identical speeds a
k
D
a. If all firms are assumed to start with the same
initial output x.0/ then their outputs remain the same for all future periods, and
therefore Q
k
D
.N
1/x for all k. Due to the presence of the state variable S,by
assuming partial adjustment towards the best response the dynamic model obtained
is a two-dimensional discrete time dynamical system given by
x.t
C
1/
D
.1
a/x.t/
C
aR..N
1/x.t/;S.t//;
S.t
C
1/
D
ˇ
S
S.t/
C
Nx.t
C
1/
D
ˇ
S
S.t/
C
NŒ.1
a/x.t/
C
aR..N
1/x.t/;S.t//;
(4.47)
where
8
<
0 if
z
<0;
L if
z
>L;
R..N
1/x.t/;S.t//
D
:
z
otherwise;
with
r
A
c
..N
1/x
C
ˇ
S
S/
.N
1/x
ˇ
S
S:
All parameters are non-negative, with the constraints 0<a
z
D
ˇ
S
<1. Notice
that in the limiting case ˇ
S
D
0 the best response coincides with the best response
in Example 1.5 and in Example 3.4 for the one-dimensional symmetric case of N
identical firms. In the following we are mainly interested in the role of the parameter
ˇ
S
, which measures the inertia of the effects of past sales, on the global dynamical
properties of the model. Again, the presence of non-negativity and capacity con-
straints makes the dynamical system piece-wise differentiable, and the phase space
D D
Œ0;L
Œ0;
C1
can be divided into subregions. In each of these subregions,
the dynamical system is differentiable and these regions are separated by lines (or
borders) of non-differentiability:
D
1, 0
.2/
D
f
.x;S/
W
.N
1/x
C
ˇ
S
S>A=c
g
where
z
is negative,
D
D
f
.x;S/
W
z
1
<.
N
1/x
C
ˇ
S
S<
z
2
g
with
z
1;2
D
.3/
A=c
2L
˙
p
A=c.A=c
4L/
2
,where
z
>L,
D D n
D
.3/
:
.1/
.2/
D
[ D
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