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and on the construction of special Lyapunov functions. Hahn (1962) has shown with
the special choice of
X
N
1
2
˛
k
x
k
V
.
x
/
D
kD1
that the equilibrium of the continuous time system (1.31) is globally asymptotically
stable with symmetric firms and linear cost functions. This result has been gener-
alized to non-symmetric firms by Okuguchi (1964). We also mention that Sect. 6.4
of Okuguchi and Szidarovszky (1999) discusses the multi-product case with special
Lyapunov function selections and derives particular stability conditions.
2.6
Continuous Time Oligopolies with Continuously
Distributed Time Lags
In examining the dynamic model (1.31) we assumed that at each time period t,
each firm knew the simultaneous output levels x
l
.t/.l
¤
k/ of the competitors, so
it was able to apply the adjustment scheme represented by the right hand side of
the governing differential equation. This assumption is however unrealistic in real
economic situations, since there is an inevitable time lag because of information
collection and decision implementation. A similar situation occurs when the firms
want to react to certain averaged past information rather than reacting to sudden
market changes. In both cases the output of the rest of the industry as well as the
firm's own output levels have to be replaced by averaged values of corresponding
past information.
Therefore the differential equations (1.31) are modified to the form
R
k
Z
t
x
l
.s/ds
!
w
.t
s;T
k
;m
k
/
X
l¤k
x
k
.t/
D
˛
k
0
w
.t
s;S
k
;l
k
/x
k
.s/ds
!
:
Z
t
(2.50)
0
In the first term of (2.50) the firm reacts to a time weighted average (back to the
beginning of the process) of the output of the rest of the industry. In the second term
the firm computes its reaction to a time weighted average of its own output. In the
ensuing analysis we select the weighting function given by
8
<
1
T
exp
f
.t
s/=T
g
if m
D
0;
w
.t
s;T;m/
D
m
T
mC1
:
1
mŠ
.t
s/
m
exp
f
m.t
s/=T
g
if m
1:
(2.51)
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