Chemistry Reference
In-Depth Information
@'
2
=@x
2
D
0 assuming that the second order conditions are satisfied. Note, how-
ever, that fixed points which are not Nash equilibria may exist. Furthermore, it
should be mentioned that the functional form of the speeds of reaction
i
.
/ are
inconsequential for the computation of the Nash equilibrium.
To keep our analysis simple, we will assume that
1
.x
1
/
D
v
1
x
1
and
2
.x
2
/
D
v
2
x
2
. In economic terms, the dynamical system then incorporates the idea that the
relative
change in marketing efforts is proportional to the marginal profits, where the
positive parameters
v
1
and
v
2
are the proportionality factors. Using the expressions
for the market shares s
1
and s
2
given in (4.1) and the profits in (4.2), the resulting
dynamic market share attraction model (4.4) can be written as
8
<
x
1
.t/
ˇ
1
x
2
.t/
ˇ
2
x
1
.t/
ˇ
1
C
kx
2
.t/
ˇ
2
2
;
x
1
.t
C
1/
D
.1
v
1
c
1
/x
1
.t/
C
v
1
ˇ
1
Ak
T
W
(4.5)
:
x
1
.t/
ˇ
1
x
2
.t/
ˇ
2
x
1
.t/
ˇ
1
C
kx
2
.t/
ˇ
2
2
;
x
2
.t
C
1/
D
.1
v
2
c
2
/x
2
.t/
C
v
2
ˇ
2
Ak
where k
D
˛
2
=˛
1
. The two-dimensional map
T
W
.x
1
.t/;x
2
.t//
!
.x
1
.t
C
1/;x
2
.t
C
1//
generates the sequences of marketing efforts resulting from the decisions of the two
competitors. The corresponding market shares are then obtained via (4.1).
4.1.1
Local Stability
Although the Jacobian matrix for our dynamical system can be easily derived, the
fact that the Nash equilibrium for the general case cannot be given in closed-form
makes a standard stability analysis intractable. Here we have to rely on numerical
methods. However, for the symmetric case, where ˇ
1
D
ˇ
2
D
ˇ; c
1
D
c
2
D
c;
˛
1
D
˛
2
D
˛,and
v
1
D
v
2
D
v
, an analytic characterization of the local stability
properties of the symmetric equilibrium (4.3) is possible. In this case, the Jacobian
matrix computed at the Nash equilibrium E
becomes .1
v
c/
I
,where
I
is the
identity matrix. Therefore, in the symmetric case the unique Nash equilibrium (4.3)
is locally asymptotically stable if 0<
v
c<2(see Bischi and Kopel (2003
b
), for
more details).
4.1.2
The Feasible Set and Global Stability
We now turn to the question as to whether the Nash equilibrium is globally stable
and if so, under which conditions. Obviously, it only makes sense to consider sit-
uations where both firms expend positive efforts. That is, mathematically the map
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