Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 73

Therefore the equilibrium is globally asymptotically stable if

max 1

<x<

a 2 < 4
N

a 2
4 Na 2

2
N 1 :

and

2 I

Notice that a feasible x exists if

max 1

<

a 2 < 4
N

a 2
4 Na 2

2
N 1 :

and

2 I

The second relation implies that N 4. fN D 2,thena 2 <2 and
a 2 =.4 2a 2 /<2are the stability conditions, which can be rewritten as a 2 <1:6.
This condition always holds since we assume that a 2 1.IfN D 3, then the con-
ditions reduce to a 2 <4=3. Finally, if N D 4,thena 2 <1and a 2 =.4 2a 2 /<2=3
are the conditions which can be summarized as a 2 <1. As in the earlier examples,
the Jacobian matrix assumes different forms in the different regions (see Fig. 2.10),
where the different regions refer to the different regions of the best response func-
tions based on the non-negativity and capacity constraints, similarly to Sect. 1.3.1.
Therefore, whenever a variation of parameters causes a displacement of the equilib-
rium point (or of a periodic point of a cycle) into a different region by crossing the

x 2

A − c 2
B(N-2)
A − c 1
B(N-1)

D (3)

D (2)

D (4)

x

A − c 2
B(N-2)

2L 2

D (1)

N-2

D (7)

A − c 1
B(N-1)

2L 1

N-1

D (5)

D (6)

x 1

Fig. 2.10 Example 2.4; linear inverse demand and cost functions, the case of semi-symmetric
capacity constrained firms. The regions for the different expressions for the quantity dynamics
map