General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 103

Hence, we obtain

.N 1/ 2 Ac k
. P l c l / 2

Q k Q x k

D D :

In order to guarantee that x k 0 we have to assume that

P l ¤ k c l
N 2 :

P l c l
N 1

c k or c k (3.5)

We can also find conditions such that Q 2 Q k for all k implying that 1<R 0 k 0
at the equilibrium, so the local asymptotic properties of the equilibrium become the
same as in the concave case. This condition has the special form

2.N 1/ 2 Ac k
. P l c l / 2

.N 1/A
P l c l

;

which can be rewritten as

P l c l

2.N 1/ :
Notice that this lower bound is the half of the upper bound given in (3.5). The upper
bound guarantees the non-negativity of the equilibrium outputs and the lower bound
guarantees that the derivatives of the best responses at the equilibrium are between
1 and 0 as in the concave case. If N D 2, then this is true if c 1 D c 2 , otherwise it
holds for one firm and does not hold for the other. If N 3, then this condition is
certainly satisfied if none of the firms has very low marginal costs compared to its
competitors.

c k

3.1.1 Discrete Time Models and Local Stability

The local asymptotic behavior of the best reply dynamics with adaptive expectations
and partial adjustment towards the best response with naive expectations (1.28)-
(1.30) are equivalent to each other as has been shown earlier. So similar to the
concave case we will discuss only system (1.30). The Jacobian of this dynamic
system was derived in (2.20), where we did not use any special form of the best
response functions, therefore the nonzero eigenvalues of the Jacobian of the isoelas-
tic case are also the eigenvalues of the matrix

H . Its characteristic equation is also

given by (2.23), or equivalently by (2.24).
In the case when all r k D R 0 k .Q k / values are non-positive, all local stabil-
ity results remain the same as demonstrated for the concave case. However in
the general case the local asymptotic behavior of the equilibrium becomes more
complicated.
Assume now that for a firm k 0 , r k 0 >0.Then Q>2 Q k 0 or equivalently,
x k 0 > Q k 0 . This condition means that firm k 0 produces more than the total output