Chemistry Reference
In-Depth Information
N
k
for all k and let .x
0
1
;:::;x
0
N
/ be the new equilibrium. Let Q
D
P
kD1
x
k
N
k
and Q
0
D
P
kD1
x
0
k
be the total production levels of the industry in the two cases.
Then the following theorem shows that any increase in cooperation levels results in
a decrease of the industry output.
Q
Q
0
.
Theorem 4.5.
If conditions
.A/;.B
0
/
and
.C
0
/
hold, then
Q<Q.Then
Proof.
Assume on the contrary that
X
N
X
N
Q
D
R
k
. Q;
N
k
/
R
k
. Q
0
;
N
k
/
kD1
kD1
X
N
R
k
. Q
0
;
N
k
/
D
Q
0
;
k
D
1
which is an obvious contradiction.
4.5.1
Local Stability Analysis
We will consider only continuous time scales since the discrete time case can
be examined analogously to the model discussed in Sect. 4.4. In this subsection
we consider the local stability of the equilibria and in the next subsection the
global dynamics. In the current context the continuous time dynamical model (1.31)
becomes
x
k
.t/
D
˛
k
.R
k
.
X
l
x
l
;
X
l
kl
x
l
/
x
k
/;
(4.100)
¤
k
¤
k
where R
k
.Q
k
;S
k
/ is the best response function of firm k.
The Jacobian of this system at an equilibrium has the structure
0
@
1
A
a
1
.R
0
1Q
C
12
R
0
1S
/ ::: a
1
.R
0
1Q
C
1N
R
0
1S
/
a
1
a
2
.R
0
2Q
C
21
R
0
2S
/
::: a
2
.R
0
2Q
C
2N
R
0
2S
/
a
2
:
:
:
:
:
:
:
:
:
:
:
:
:
a
N
.R
0
NQ
C
N1
R
0
NS
/a
N
.R
0
NQ
C
N2
R
0
NS
/ :::
a
N
(4.101)
In the following analysis we will consider this matrix at an interior equilibrium.
In the general case unfortunately, this Jacobian does not have any special struc-
ture that makes it possible to express its eigenvalue equation in a simple form. In
some important special cases however it is possible to do so. In the general case
computational methods are available to compute the eigenvalues and to check the
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