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8
<
x
1
.t
C
1/
D
x
1
.t/
C
a
1
x
1
.t/ŒA
c
1
2.B
C
e
1
/x
1
.t/
Bx
2
.t/;
T
g
W
:
x
2
.t
C
1/
D
x
2
.t/
C
a
2
x
2
.t/ŒA
c
2
2.B
C
e
2
/x
2
.t/
Bx
1
.t/:
(1.46)
It is easy to see that the interior steady state of the adjustment process based on
marginal profits coincides with the unique interior Nash equilibrium E
D
.x
1
; x
2
/
given in (1.12). To study the local asymptotic stability of E, we consider the
Jacobian matrix of (1.46). Since the Nash equilibrium is located at the intersec-
tion of the two reaction functions given in (1.34) and (1.35), we have B x
i
D
A
c
j
2
B
C
e
j
x
j
(i;j
D
1;2, i
¤
j). Therefore, the Jacobian matrix
evaluated at the interior equilibrium E can be written as
1
2a
1
.B
C
e
1
/x
1
:
a
1
B x
1
(1.47)
a
2
B x
2
1
2a
2
.B
C
e
2
/x
2
We can check the stability conditions by use of the relations (1.40) with
q
D
.1
2a
1
.B
C
e
1
/x
1
/.1
2a
2
.B
C
e
2
/x
2
/
a
1
a
2
B
2
x
1
x
2
;
and
p
D
2
C
2a
1
.B
C
e
1
/x
1
C
2a
2
.B
C
e
2
/x
2
:
By assuming that B
C
e
k
>0for k
D
1;2, clearly q<1. Notice that
p
C
q
C
1
D
4a
1
a
2
.B
C
e
1
/.B
C
e
2
/x
1
x
2
a
1
a
2
B
2
x
1
x
2
;
which is positive if B
2
<4.B
C
e
1
/.B
C
e
2
/. Similarly,
p
C
q
C
1
D
4
4a
1
.B
C
e
1
/x
1
4a
2
.B
C
e
2
/x
2
C
4a
1
a
2
x
1
x
2
.B
C
e
1
/.B
C
e
2
/;
so this is positive, if
.4.B
C
e
1
/.B
C
e
2
/
B
2
/x
1
x
2
a
1
a
2
4.B
C
e
1
/x
1
a
1
4.B
C
e
2
/x
2
a
2
C
4<0:
(1.48)
If B
2
<4.B
C
e
1
/.B
C
e
2
/ and the equilibrium E is positive, then this additional
condition can be used to determine a region of stability in the .a
1
;a
2
/-plane. In con-
trast to the adjustment process where firms partially adjust their quantities towards
the best reply, here the speeds of adjustment are crucial for local asymptotic stabil-
ity of the Nash equilibrium. As remarked earlier, the stabilizing role of sufficiently
small values of the adjustment speeds has been observed before by many authors
(see for example Fisher (1961), McManus and Quandt (1961), and Flam (1993)). In
Fig. 1.13 we depict the stability region (shaded) in the .a
1
;a
2
/ plane obtained for the
parameter values A
D
450, B
D
30, c
1
D
c
2
D
275, e
1
D
e
2
D
11.Forvalues
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