Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 223

X

N

a k r k h k
. C a k .r k C 1// 1 C

1 D 0: (5.38)

m k C1

T k
p k

k D 1

In the general case computational methods can be used to locate the roots. In order
to obtain analytic results we will consider the case of symmetric firms, when a k
a;r k r;h k h;T k T;m k m and so p k p. In both concave and isoelastic
cases 1<r 0;a > 0 and h>0. Notice that in the isoelastic case at most one
firm can have a positive r k value, but in the symmetric case all firms would have
positive derivatives r k , which is impossible. Then (5.38) becomes the polynomial
equation

mC1

. C a.r C 1// 1 C

T
p

Narh D 0:

(5.39)

Our results can be summarized in the following theorem.

Theorem 5.3. Assume symmetric firms and that h 1=N: The equilibrium with
information lag is locally asymptotically stable if T D 0; or T>0 and m D 0: If
T>0 and m D 1; then

(i) The equilibrium is locally asymptotically stable, if

Nhr C 8.r C 1/ > 0 .

8

(ii) The equilibrium is locally asymptotically stable for all aT ¤ 1 C

Nh ,when

Nhr C 8.r C 1/ D 0 .

(iii) Otherwise the equilibrium is locally asymptotically stable if
aT < .aT/ 1 or aT > .aT/ 2 ;
where .aT/ 1 and .aT/ 2 ..aT/ 1 <.aT/ 2 / are given in (5.43), and the equilibrium
is unstable, if

.aT/ 1 <aT<.aT/ 2 :
At the critical values aT D .aT/ 1 , and aT D .aT/ 2 Hopf bifurcations occur giving
the possibility of the birth of limit cycles around the equilibrium.

Proof. Assume first that T D 0, that is, there is no time lag. Then (5.39) becomes

C a.1 C r.1 Nh// D 0;

with the only root D a.1 C r.1 Nh//.If f k is a reasonable approximation
of f ,thenh ' 1,soNh 1 implying that <0and the equilibrium is locally
asymptotically stable.
Assume next that T>0and m D 0. Then (5.39) is reduces to the quadratic
equation

2 T C .1 C a.r C 1/T/ C a.1 C r.1 Nh// D 0: