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and
f
0
C
x
k
f
00
.2f
0
C
x
k
f
00
C
0
k
/
a
k
X
N
>
1;
(2.45)
C
0
k
f
0
.2f
0
C
x
k
f
00
C
0
k
/
2
a
k
k
D
1
from which it is clear that if a
k
is replaced by a
k
=.
.2f
0
C
x
k
f
00
C
0
k
// for all
firms, then (2.44) is the same as (2.42), and (2.45) is identical to (2.43).
If 2f
0
C
x
k
f
00
C
0
k
D
1; then the cases are exactly the same. If 2f
0
C
x
k
f
00
C
0
k
>
1 for all k, then (2.44) implies (2.42) and (2.45) implies (2.43), so the best
reply dynamics are more stable. Notice that
2f
0
C
x
k
f
00
C
0
k
D
.f
0
C
x
k
f
00
/
C
.f
0
C
0
k
/;
where the first term is non-positive and the second term is negative. The condition
that this quantity is larger than
1 requires that the absolute values of both terms be
sufficiently small. For example, in the case of linear price and cost functions,
f
00
D
C
0
k
D
0;
and the condition requires that f
0
>
1=2, so price cannot decrease very fast with
increasing total output of the industry in order to guarantee stability. If 2f
0
C
x
k
f
00
C
0
k
<
1; then we reach the opposite conclusion.
The Jacobian of the continuous time system (2.40) also has the special form,
H
I
,where
H
is the Jacobian of the discrete system and
I
is the identity matrix.
It is very easy to show that all eigenvalues of
H
I
are real and negative, so the
gradient adjustment process is always locally asymptotically stable. Therefore, for
continuous time scales there is no difference between best reply dynamics and gradi-
ent adjustments as long as our concern is only local asymptotic stability. The global
asymptotic properties of gradient adjustment and adaptive adjustment processes are
usually different.
2.5
Continuous Time Oligopolies and Local Stability
Consider now the continuous time model (1.31) of the dynamics of partial adjust-
ment towards the best response with naive expectations. A vector ( x
1
;:::; x
N
)isa
steady state of the system if and only if
D
R
k
.
X
l
x
k
x
l
/;
ยค
k
in which case ( x
1
;:::; x
N
) is a Nash equilibrium.
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