Chemistry Reference
In-Depth Information
Example 5.6.
We consider now the duopoly model with the same isoelastic price
function p
D
1=Q as before and with quadratic cost functions C
k
D
d
k
C
e
k
x
k
.
The best reply dynamics with complete knowledge of the demand function cannot
be expressed by a simple dynamical system. In fact, the profit of player k is '
k
D
x
k
=.x
1
C
x
2
/
d
k
e
k
x
k
, and the first order conditions for profit maximization
give rise to third degree algebraic equations. For example, the condition for the
reaction function of player 1 becomes
2e
1
x
1
C
4e
1
x
2
x
1
C
2e
1
x
2
x
1
x
2
D
0:
Since the left hand side strictly increases in x
1
, it is easy to see that a unique positive
solution x
1
D
R
1
.x
2
/ exists, however its precise form is not easily obtained. On
the other hand, if we consider the dynamics with LMA, a simple two-dimensional
dynamical system is obtained based on the two-dimensional iterated map
2x
1
.t/
C
x
2
.t/
2.1
C
e
1
.x
1
.t/
C
x
2
.t//
2
/
;
x
1
.t
C
1/
D
(5.60)
x
1
.t/
C
2x
2
.t/
2.1
C
e
2
.x
1
.t/
C
x
2
.t//
2
/
;
x
2
.t
C
1/
D
which can be derived from (5.54). The equations for the determination of the fixed
points, obtained by setting x
i
.t
C
1/
D
x
i
.t/ in (5.60), become
2e
1
x
1
.x
1
C
x
2
/
2
D
x
2
;
(5.61)
2e
2
x
2
.x
1
C
x
2
/
2
D
x
1
:
q
e
e
1
By dividing the first equation by the second we have x
1
D
x
2
. Substituting this
into (5.61) we calculate the unique non-vanishing equilibrium
p
e
2
p
e
1
C
p
e
2
1
p
2
p
e
1
e
2
x
1
D
;
(5.62)
p
e
1
p
e
1
C
p
e
2
1
p
2
p
e
1
e
2
x
2
D
.
This equilibrium is always locally asymptotically stable. In fact, sufficient condi-
tions for its stability are easily obtained from the computation of the Jacobian matrix
J
D
J
ij
of (5.60) at the equilibrium. The diagonal entries are
x
1
f
00
Q
2
f
0
. Q/
e
1
D
3e
1
p
e
2
3e
1
p
e
2
C
e
1
p
e
1
C
2e
2
p
e
1
;
J
11
D
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