Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 61

a 2

2
1Cr 2

2

1 r 1 r 2

2

a 1

2

2
1r 1 r 2

2
1Cr 1

Fig. 2.2 Stability region of a discrete time duopoly under best reply dynamics in the .a 1 ;a 2 / plane

2.3 Discrete Time Oligopolies and Global Stability

We start this section with a simple discussion of global asymptotic stability. Our
analysis will be based on the sufficient condition presented in Appendix B which
states that if there exists a matrix norm such that the norms of the Jacobians of a dis-
crete time dynamical system in all regions are less than some q<1everywhere in
the phase space, then the system is globally asymptotically stable. Here we present
the case of partial adjustment towards the best response, the case of best reply with
adaptive expectations can be discussed in a similar way. The feasible output set is
divided into subregions depending on the different cases in the best response func-
tion (2.3). In each subregion the Jacobian of the partial adjustment dynamics (1.30)
(that we shall denote by H ) has the special structure (2.20), where

0
@ X
l¤k

1 0 0

@ X
l¤k

1 1

D R 0 k

A

D ˛ 0 k

@ R k A x k A ;

r k x l and a k x l

where r k is either given by (2.5) or equals zero. Assume that 0<a k 1 for all k and
for all feasible output levels x 1 ;x 2 ;:::;x N . Under conditions (A)-(C), inequality
(2.7) holds for all k and all feasible output levels. Therefore with the choice of the
row norm, we have

k H k 1 D max
k

fj 1 a k jC .N 1/ j r k a k jg

D max
k

f 1 a k .1 C r k .N 1// g : (2.30)