The dynamics of a Bertrand duopoly with differentiated products and bounded rational firms revisited

The dynamics of a Bertrand duopoly with differentiated products and

bounded rational firms revisited

Fanti, Luciano and Gori, Luca

Department of Economics, University of Pisa, Department of Law and Economics "G.L.M. Casaregi", University of Genoa

9 September 2011

Online at https://mpra.ub.uni-muenchen.de/33268/

MPRA Paper No. 33268, posted 09 Sep 2011 14:51 UTC

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A

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[ ( ) ]

(

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5 .

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(

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(

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d!" d_1,F" d_2,F" d_3,F d_4,F"

2 A1) A d_1,H d_2,H *

" * ; " " d_1,F =−0.5 "

1403 .

, 0

2F =−

d " d_3,F =0.3596 d_4,F =0.8903 " d_1,H =−0.2623 d_2,H =0.7623

' "

& ! 3 * 4 *4 4 2 2 2 5 E₃

2 % 1 " # - %2 d =0 " ( ( 1 2 - 2

% # 2 5 * % 2 2 6 4

;

2 5 % * - 4 4 d ( (

F

F d

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2 % 1 " # - %2 d =0 2 *

- % * 1 2 ( ( 2 d 0

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2 4*2 1 4- * - 4 d_2,F < d_1,H % 2 (

'" ( F

'

5 " "

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F J0

; ! "

1 2" A d

5 .

=0

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;;

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9 = α =0.5" a=3 w=0.5!

% '" d L 0.585<d <0.598

46 . 1 65

.

0 < p^* <

;

% '" d L 0.592<d <0.5946

937 . 0 67

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0 < p^*<

% '" d L 0.624<d <0.662

46 . 1 3

.

0 < p^*<

) d =0" * +

" 0 " 5

;+

" d 0 1!" ; !

5 −0.1403<d <0.3596 ) "

0 ; ! " * + E₃

1403 .

, 0

1F =−

d d "

" ! d_2,F =0.3596

d " " !

K " d=−0.1403" d"

1 2" =

1403 . 0 2823

.

0 < <−

− d 1 8 −0.3207<d <−0.2823 1

8 −0.36<d <−0.3207 ; ! 0

" −0.415<d <−0.36

" 5 8 " d <−0.415

; ! K " d =0.3596"

d" 1

2 " = 0.3596<d<0.588 1 8

65 . 0 588

.

0 <d < ; ! 0 "

7021 . 0 65

.

0 <d< " 5

8 d >0.7021 ; !

* + 1+ + "

0 1 1

" " //! * + + ! 5

* + !

F * , / d"

* + " = ! 5

1 2

d ! 5

1 2 d

!

;,

!

!

;/

!

!

;:

!

!

!

;G

!

D!

;- A!

!

% ) ( 0<d <1 9 d

1 2 != ! d =0.59" ! d =0.595"

! d =0.64" ! d =0.6491 ! d =0.65" ! d =0.655" ! d =0.675" ! d =0.685" ! 69

.

=0

d " D! d =0.695" A! d=0.696 ! d=0.701 9 = α =0.5" a=3

5 .

=0

w !

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:! D * G G !"

60 1 7

J0 ; !

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7 "

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= ;! 1 8 ! 5 !8 +!

6 1 A 7!8 ,! 0 1 ' "

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" M

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1 "

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& A " ; GG " " " *

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GC " $1 1 A

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;

* , 0<d <1

1 2

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595 .

=0

d d =0.64 ) "

M +" 1

5 E₃

3596 .

=0

d ! d =0.589 1 2 A1) A

5 D !

F " " 1

9 N

0 " A 5

' 2 A1) A ^-'

0 1 1 0 !

9 N '

* , " $1

9 N

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* , " d =0.6491

! d =0.65"

* , ;- L d

" ;-

695 .

=0

d * , D!

L

* "

1 2" * / 1 "

1 1 A !"

! d : 1 "

! F " d "

: 1 :

6 7 * / !" 6 7 0

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1 0 !"

"

= ;! 0 2 A1

) A 8 ! 0 0 +1^!

8 +! 0 8 ,!

1 0 8 /! "

−1^"

!8 :! 2 A1) A "

51 D "

1 2 5 !

0 1 +$ $ 9 N !" !

C 12 A1) A " L ; G;8 ) A " ;!

L O ; :" ,+!"

5 "

M ;1+"

!

:

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A !

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4 5 Le1! d F Le1 !"

60 1 7 ! * : "

1 2"

! d 1 ; !

" 0 1 2

* : * +" , /

L A "

F ; !"

p1

; * G G p₁ p_1,0 =0.2

3 .

0 0

,

2 =

p " p_1,0 =0.200001 p_1,0 =0.300001 d =0.62

0 1 !" d =0.701

!" L 5 " d =0.62

" d =0.701

" 5

* "

" O 1& A

D " "

' "

0

* "

0 $ 0 ;! .

. 1 !"

0 O ; :! *

$ 0 !"

" " " J D ) " -" ; !^;;

; L A " 1 A !

0 C

0 1 d ^'

A1 O " ; :" ,/!

;;) ; -!" ; ! ? !

;

!

!

!

!

+

!

!

,

!

!

!

/ D!

A!

:

!

% * ( −1<d <0 9 d

1 2 != ! d =−0.32" !

325 .

−0

=

d " ! d =−0.33" ! d =−0.33326348" ! d =−0.34" ! d =−0.355" ! d =−0.362"

! d =−0.37" ! d =−0.39" D! d =−0.395" A! d =−0.405 ! d =−0.414 9

= α =0.5" a=3 w=0.5!

% + ' 4 5 0.585<d <0.705 !

G

% ," ) p₁ ! F

= p_1,0 =0.2 p_2,0 =0.3 p_1,0 =0.200001 p_2,0 =0.300001

9 = α =0.5" a=3" w=0.5 d =0.622!

% ," ) p₁ ! F

= p_1,0 =0.2 p_2,0 =0.3 p_1,0 =0.200001 p_2,0 =0.300001

9 = α =0.5" a=3" w=0.5 d =0.701!

- ( ( 92 % 2 84

F 5 "

" " *

3 ;;!" 5 '

i i q_i = L_i " L_i =q_i²

i '

0 " =

( )

i i i²

i q wL wq

C = = " +!

" "

>0 qi

9 i =

2 i i i

i = p q −wq

π ,!

* ) " 5

( ) [ ( ) ] (

=

) ( )

(

²

)

^{2 1}

2 2

2 1

2 1 1

1

1 2 2 1

1 ,

d

p w d w

d dp d a p

p p

−

+

−

− +

− +

= −

∂

∂π " / ;!

( ) [ ( ) ] ( ) ( )

(

²

)

^{2 2}

2 2

1 2

2 1 2

1

1 2 2 1

1 ,

d

p w d w

d dp d a p

p p

−

+

−

− +

− +

= −

∂

∂π / !

' " 1 1 1 2

0 J0 / ;! / ! p₁ p₂" "

=

( ) ( ) [ ( ) ] ( )

(

^{dw w}

)

^{d w}

dp d p a

p p p p

+

−

+

− + +

= −

⇔

∂ =

∂

2

2 2

2 1 1

2 1 1

1 2

2 1

0 1

π , _{" : ;!}

( ) ( ) [ ( ) ] ( )

(

^{dp dw w}

)

^{d w}

d p a

p p p p

+

−

+

− + +

= −

⇔

∂ =

∂

2

2 1

1 2 2

2 1 2

1 2

2 1

0 1

π , _{: !}

E 5 " 1

0 =

( ) { [ ^{( )} ] ( ) ( ) }

( ) { [ ⁽

⁻

⁾

⁺

] (

^{− +}

) (

^{− − +}

) }

+ −

=

+

−

− +

− +

− − +

=

+ +

t t

t t

t

t t

t t

t

p w d w

d dp

d a d p p

p

p w d w

d dp

d a d p p

p

, 2 2

2 ,

2 1 2 , 2 ,

2 1 , 2

, 1 2

2 ,

2 2 2

, 1 ,

1 1 , 1

1 2 2 1

1 1

1 2 2 1

1 1

α α

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' 5 1 G!

1 , 1 1 ,

1 p p

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(

^{p*1, p*2}

)

^G!

1 =

( ) { ^{[ ( ) ]} ( ) ( ) }

( ) { ^{[ (}

⁻

⁾

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^{− +}

) (

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) }

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−

− +

− +

− −

0 1

2 2 1

1 1

0 1

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1 1

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2 2 1

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1 2

2 2 2

2 1

p w d w

d dp d a d p

p w d w

d dp d a d

p

α α

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)

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+

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0 ,

0 ₂

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w d d E a

w d

w d d E a

E " ;!

( )

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(

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1 1

2

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(

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q = + + −

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2

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( )

( ) ( )

[ ]

²

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1 2

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+

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' "

& ) 3 * 4 *4 4 2 2 2 5

E3 - - 8 ( " !( # " 0( # 2 % 1 " # " :#

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1 2 ( ( 2 d 0 1 2 2 , 17 2

84 - 4 E₃ ; 4 * 1 4- * - 4 4

84 2 4 (

& * 3 * 4 *4 4 2 2 2 5

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1 2 ( ( 2 d 0 −1 2 2 , 1

7 2 84 - 4 E₃ ; 4 * 1 4- * - 4

4 84 2 4 (

+;

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2 .

0 0

,

1 =

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2 .

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,

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L 3 ( ;, " -,+.-:

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;, .;:

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L $ ? L " /

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/-/

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J J

M ,-" :-;.: :

; 6 7 6 7

!

++

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# J ; " .+

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G" ; G.;

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F # P ( " ;/:G.;/GG

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= 9 $" +G .+-:

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J 4 ; " G .G,

CH A " # " L 0

# J ' +" ++. +

C " C " ; ) J # + " ,;./G

% " 4 $ " ; G K ( 0

# J ' G/" ;+.

3 " L " ; ( $ ' L J

( EO!= ( E 9

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9 M 4 G " , +., :

M " $ " ' A " * " ; G; K (

3 9 " ;:G.;

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) " 2 " @ " Q " ; -, 9 0

ML2$ # J ;/" /,:.//,

@ " Q " ; -/ K ( 0

# J ' +:" ;::.;G/

' " * " C

J 3 G" +/ .+/G

" # " $ " % " " & " G L

J 3 ," ;+-.;,-

" # " $ " % " " & " '

( " ) * + " ,-. //