Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 163

increases by the new sales, however a certain proportion of the new sales has to be
used for replacement of goods which are not in usable condition anymore. Assume
that the price decreases if either the total output of the industry or the value of S
increases. For the sake of simplicity we will assume that the unit price is a function
of a linear combination of these two factors, so it is given as
f N C ˇ S S ! ;

X

x l

l D 1

with some ˇ S >0. Therefore the profit of firm k can be written as
x k f N C ˇ S S !

X

x l C k .x k /: (4.30)

lD1

If R k .Q k / denotes the best response function of firm k without intertemporal mar-
ket interaction, then its best response becomes R k .Q k C ˇ S S/ when taking it into
account, and so the dynamical system in continuous time becomes

R k X

C ˇ S S !

! ; 1

x k D ˛ k x l x k k N/ (4.31)

l ¤ k

X

N

S D

x l S S; (4.32)

lD1

where ˛ k is a sign-preserving function for all firms k. Clearly .x 1 ;:::; x N ; S/ is a
steady state of this system if and only if

X

C ˇ S S ! x k D R k x l

(4.33)

l¤k

and

N

X

S:

1 x k

D S (4.34)

k D

4.3.1 Discrete Time Models and Local Stability

The main result of this section on the stability of the discrete time intertemporal
demand interaction dynamical system (4.23)-(4.25) is the following:
Theorem 4.1. Assume that for all k , a k >0 , 1<r k

0 and .a k C ˇ S /.1 C r k /<1 ,

k D 1

r k
1 a k .1 C r k / 1 is satisfied. Then the equilibrium of the
system (4.23)-(4.25) is locally asymptotically stable if

N

furthermore