The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 28

x

L

2

(3)

D

(4)

D

(5)

D

Ac
B

−

1

(2)

D

(1)

D

(6)

D

Ac BeL
B

−− 2( + )

1 1 1

(9) (8) (7)

D D D

x

Ac BeL
B

−− 2( + ) Ac

B

−

L

2 2 2 2

1

Fig. 1.10 Phase space regions for the Cournot duopoly game where firms use partial adjustment
towards the best response

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 .A c 1 Bx 2 /= 2.B C e 1 / ;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 L 2 ;

T a j D .2/ W

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 0;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 L 2 ;

T a j D .3/ W

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 0;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 .A c 2 Bx 1 /=.2.B C e 2 //;

T a j D .4/ W

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 0;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 0;

T a j D .5/ W

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 .A c 1 Bx 2 /= 2.B C e 1 / ;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 0;

T a j D .6/ W

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 2 L 1 ;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 0;

T a j D .7/ W

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 L 1 ;

x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 .A c 2 Bx 1 /= 2.B C e 2 / ;

T a j D .8/ W

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 L 1 ;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 L 2 :

T a j D .9/ W

The derivative of the best response function of firm k is either zero or B=.2.B C
e k //, or does not exist in the cases when Q k D .A c k /=B and Q k D .A c k

2.B C e k /L k /=B.