Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 231

After adding these two equations, we obtain

2 ˛
c 1 C c 2 :

.x 1 C x 2 / ˛

D

This equilibrium condition shows us that a realistic non-vanishing steady state exists
only if ˛<2. We can use this equation to substitute, for example, x 2 D x 1 C

2 ˛
c 1 C

1=˛

in one of equations (5.57), from which we get the unique non-vanishing
equilibrium x D . x 1 ; x 2 /, with

c 2

2 ˛
c 1 C c 2

˛ c 2 c 1 .1 ˛/
c 1 C c 2

;

1
˛

x 1 D (5.58)

2 ˛
c 1 C c 2

˛ c 1 c 2 .1 ˛/
c 1 C c 2

:

1
˛

x 2 D

This equilibrium is positive if ˛>1 min f c 1 =c 2 ;c 2 =c 1 g . The study of the stability
of this equilibrium is particularly easy, because the Jacobian matrix for the map
(5.56) given by

2 ;

2˛ Œ.˛ C 1/c 1 .x 1 C x 2 / ˛

1

2˛ Œc 1 .˛ C 1/.x 1 C x 2 / ˛

1

1 1

J.x 1 ;x 2 / D

2˛ Œc 2 .˛ C 1/.x 1 C x 2 / ˛

1 1

2˛ Œ.˛ C 1/c 2 .x 1 C x 2 / ˛

1

1

2

1

computed at the equilibrium becomes

0

2˛ h .˛ C 1/c 1 c 2 1 i 2˛ h c 1 .˛ C 1/ 2 ˛ c 2 1 i

1

1
2

1 2 ˛ 1

c 1 C c 1 C

@

2˛ h c 2 .˛ C 1/ 2 ˛ c 2 1 i 2˛ h .˛ C 1/c 2 c 2 1 i

A

J.x 1 ; x 2 / D

1 1

2

1 2 ˛

c 1 C c 1 C

and has the simple characteristic equation

1 C ˛
2

˛
4 D 0:

2 C

Hence the eigenvalues are 1 D 1=2 and 2 D ˛=2. This implies that the equilibrium
.x 1 ; x 2 / is locally asymptotically stable for each ˛ in the range 0<˛<2.
This contrasts with the results obtained for the best reply dynamics with complete
knowledge of the demand function as discussed by Puu (1991), in Example 3.4 with
˛ D 1. It has been shown there that the unique Nash equilibrium is given by (5.58)
with ˛ D 1,thatis

.c 1 C c 2 / 2 .

c 2
.c 1 C c 2 / 2 ;

c 1

x D .x 1 ; x 2 / D

(5.59)

Furthermore, as we have demonstrated in this example the local stability of this
equilibrium under the best reply dynamics depends on the ratio between the marginal