Chemistry Reference
In-Depth Information
where k
D
1;:::;N. Notice that the constraint
z
k
D
L
k
is ineffective if L
k
A=.4c
k
/, otherwise we have R
k
D
L
k
for
h
A
2c
k
L
k
2c
k
p
A.A
4c
k
L
k
/;
A
2c
k
L
k
1
Q
k
2
2c
k
p
A.A
4c
k
L
k
/
i
1
C
(see Fig. 1.9). In the duopoly case, N
D
2, already considered in Example 3.2, partial
adjustment towards the best response is governed by the discrete time dynamical
system
x
1
.t
C
1/
D
.1
a
1
/x
1
.t/
C
a
1
R
1
.x
2
/;
(3.13)
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
a
2
R
2
.x
1
/;
and the unique Nash equilibrium is given by
Ac
2
.c
1
C
c
2
/
2
I
.c
1
C
c
2
/
2
:
Ac
1
x D
.x
1
I
x
2
/
D
(3.14)
The local stability properties of
x
in the duopoly case have already been derived in
Example 3.2. For identical adjustment coefficients, a
1
D
a
2
D
a, the equilibrium is
locally asymptotically stable if a.1
r
1
r
2
/<2,wherer
k
D
R
0
k
.Q
k
/
D
.c
1
C
c
2
2c
k
/=.2c
k
/. Inserting these expressions for the derivatives of the best replies
allows us to express the stability condition in terms of the cost ratio
D
c
2
=c
1
(cf.
also Example 3.3 for the semi-symmetric case). Hence, in this case local asymptotic
stability of the equilibrium given in (3.14) is ensured if
a.1
C
/
2
4
<2:
Consequently, for any given a
2
.0;1, as long as
4
a
2
p
4
2a
a
!
;
4
a
C
2
p
4
2a
a
2
holds, the equilibrium is stable. Note that since
D
1 is always inside this interval
for all adjustment coefficients a
2
.0;1, the equilibrium is always stable if firms
have identical marginal costs. It is also worth pointing out that the cost difference
between the firms has to be quite strong in order to render the equilibrium unstable.
To demonstrate this, we look at a particular case of the best reply dynamics, namely
a
1
D
a
2
D
1. Here th
e
Nash e
qu
ilibrium (3.14) is stable if and only if the cost ratio
D
c
2
=c
1
2
.3
2
p
2;3
C
2
p
2/
'
.0:17;5:83/ (see also Puu (1991, 2003)). If, for
example, c
1
D
1; this result shows that the unit cost of firm 2 has to be either at least
almost 6 times higher than firm 1's unit cost or less than about 1=6 of it in order that
instability occurs. If the cost ratio c
2
=c
1
exits this interval, then the Nash equilibrium
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