Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 162

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where Q k D P l¤k x l as before. If R k Q k .t C 1/ denotes the best response

function of firm k without intertemporal demand interaction, then R k Q k .t C 1/

C ˇ S S.t/ is the best response when it is taken into account. At time period t C 1;
when firm k makes its decision on its production level, the simultaneous decisions
of the competitors are not known, so instead of the true output Q k .t C 1/ of the rest
of the industry, firm k uses some expectation Q k .t C 1/ of this value. If we assume
best reply dynamics with the adaptive expectations scheme (1.18), then the resulting
dynamical system becomes

0 0

@ X
l¤k

1 1

A C ˇ S S.t/ A ; 1

@ Q k .t/ C ˛ k

x l .t/ Q k .t/

x k .t C 1/ D R k k N/

(4.23)

0
@ X
l

1

A ; 1

Q k .t C 1/ D Q k .t/ C ˛ k x l .t/ Q k .t/ k N/ (4.24)

¤ k

0 0

@ X
l¤k

1 1

X

N

@ Q k .t/ C ˛ k

x l .t/ Q k .t/

A C ˇ S S.t/ A ;

S.t C 1/ D ˇ S S.t/ C R k

kD1

(4.25)

where ˛ k is a sign-preserving function for all k.
Clearly .x 1 ;:::; x N ; Q 1 ;:::; Q N ; S/ is a steady state of the dynamical system
(4.23)-(4.25) if and only if for all k,

D X
l¤k

Q k

x l ;

(4.26)

D R k . Q k

S/;

x k C ˇ S

(4.27)

and

N

X

.1 ˇ/ S D

1 x k :

(4.28)

k D

Assume next continuous time scales. If we rewrite the discrete equation (4.21) as

X

N

S.t C 1/ S.t/ D x k .t C 1/ .1 ˇ S /S.t/;

k D 1

we see that S.t/ in the continuous case is driven by the differential equation

N

X

S D

x k S S; (4.29)

k D 1

where S D 1 ˇ S >0. For the market saturation example this equation can also
be interpreted as expressing the fact that during each time period the value of S.t/