Luciano Fanti and Luca Gori

Discussion Papers

Collana di

E-papers del Dipartimento di Scienze Economiche – Università di Pisa

Luciano Fanti and Luca Gori

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

Discussion Paper n. 120

2011

Discussion Paper n. 120: presentato Settembre 2011

Luciano Fanti

Department of Economics, University of Pisa Via Cosimo Ridolfi, 10, I–56124 Pisa (PI), Italy e-mail address: [email protected]

tel.: +39 050 22 16 369 fax: +39 050 22 16 384

Luca Gori

Department of Law and Economics “G.L.M. Casaregi”, University of Genoa Via Balbi, 30/19, I–16126 Genoa (GE), Italy

e-mail address: [email protected]

tel.: +39 010 209 95 03 fax: +39 010 209 55 36

© Luciano Fanti e Luca Gori

La presente pubblicazione ottempera agli obblighi previsti dall’art. 1 del decreto legislativo luogotenenziale 31 agosto 1945, n. 660.

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max 2 a d p dp

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( # =

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1 2 1

,

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p p

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( )

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( )

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[

^a

(

^d

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]

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∂

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1 1

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π , _{# ! ;"}

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( )

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[

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1 1

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5

5 # !"

G

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p

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1

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>0

αi ^{D iB}

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=

∂ + ∂

=

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=

+ +

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t

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2 , 2 ,

2 1 , 2

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=

( )

[ ]

( )

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^{− − + +}

]

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=

+ +

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p p p

w dp p d d a

p p p

t t t

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(

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1 1

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−

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; 5

1

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#

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5

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* A A

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5 E₃ 1 ; "

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=1 d "

−1

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d " ( #

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A

L 5 # = α =0.5# a=3 w=0.5 ^:

d #

[ ( ) ]

(

^{aa ww}

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d d zF

+ +

+

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= αα

2 2 : 2

( )

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(

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^w

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: # zH

F

z d

d < ^{( # 0 2 0}

5 # !# //"

2 A1* A # J0 ;/"

$ 5 ; " # # !# / "

:C # 5 5 #

F H "# + ; #

# 5 E₃

=0 d

;;

% ! + F d 9 = α =0.5# a=3

5 .

=0 w "

% $ 2 A1* A H d 9 =

5 .

=0

α # a=3 w=0.5"

+ ; " 2 A1* A " d

(

^−1,1

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^{5 " d}

( )

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(

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)

^"

* d"

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

d_1, < _2, % - 4 4 d ( ( d_3,F <d_4,F (

& $ 3 * 4 *4 4 2 2 2 5 E₃

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

4- 4 - - % * 1 2 ( ( 2 d 0

1 2 2 , 17 2 84 - 4 E₃ - 5 4 2 4*2

1 4- * - 4 d_3,F <d_2,H % 2 (

& ' 3 * 4 *4 4 2 2 2 5 E₃

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

- % * 1 2 ( ( 2 d 0

−1 2 2 , 17 2 84 - 4 E₃ - 5 4

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

'" ( F

(

5 # #

#

# 4 5 #

F J0

; " #

1 2# A d

5 .

=0

α # a=3 w=0.5

;

% '" d F = p_1,0 =0.2 p_2,0 =0.3

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

.

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 "

L # #

+ 1 "# + ,1/"# 4 5 +

:" D + G G "#

60 1 7

J0 ; "

5 N #

;!-:8 3 # ;!! " F 6 1

7 #

" # A #

= ;" 1 8 " 5 "8 "

6 1 A 7"8 ," 0 1 ( #

( 0 1

# M

( A ;!G;"# 1 0 0 1

1 #

( # # # )

' A # ;!GG # # # +

# ;!- "

1 A "#^G A

#

GC # %1 1 A

# 3 # ;!!G"

;

+ , 0<d <1

1 2

( d + , #

, , d =0.59 #

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

( #

+ , # 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

-3 # 1 1

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 #

"

:

+ / / "^;

* + , # , A# , / 1/1

A "

F #

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 # ;!!:# ,/"

;;* ;!!-"# ;!!!" ? "

"

"

,

"

"

/

"

"

:

"

"

"

G 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 "

!

% ," * 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 p #₂ #

=

( ) ( ) [ ( ) ] ( )

(

^{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 =

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

( ) { [ ⁽

⁻

⁾

⁺

] (

^{− +}

) (

^{− − +}

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