Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 199

f 0 .1 C k / C .x k C k Q k /f 00
2f 0 C .x k C k Q k /f 00 C 0 k

D R 0 kQ C k R 0 kS r k D ;

then the Jacobian has exactly the same form as (2.46) for concave oligopolies.
Notice that under conditions (A), (B 0 )and(C 0 ), there holds 1<r k 0, similar
to the case of concave oligopolies. Hence one can assert the following theorem:

Theorem 4.9. Assume that a k >0 for all k . Under conditions (A), (B 0 ), (C 0 ) and
with identical kl ( l ¤ k ) values for all k , the equilibrium is locally asymptotically
stable.

The analysis of global asymptotic stability of the equilibrium with continuous time
adjustment as well as the introduction of continuously distributed time lags can be
carried out in a similar fashion to the cases considered in the previous chapters.

4.5.2 Global Dynamics

In this section we will illustrate the type of global dynamics that can arise in
oligopolies with partial cooperation under a discrete time adjustment process by
considering a specific example.

Example 4.15. We consider the hyperbolic price function f.Q/ D A=Q and linear
cost functions C k .x k / D d k C c k x k .k D 1;2;:::;N/. Assume that the firms
have identical cooperation levels toward their competitors, so kl k for all l ¤

k. Then the payoff function (4.89) of firm k can be written as

.d k C c k x k / X
l¤k

.x k C k Q k /A
x k C Q k

‰ k .x 1 ;:::;x N / D k .d l C c l x l /: (4.105)

Assuming an interior optimum, the first order conditions imply that

AQ k .1 k /
.x k C Q k / 2

c k D 0;

from which we have the solution

s AQ k .1 k /
c k

z k D Q k : (4.106)

Let L k denote the capacity limit of firm k, then the strict concavity of the payoff
function (4.105) implies that the best response of firm k can be obtained as