Stability analysis in a Bertrand duopoly with different product quality and
heterogeneous expectations
Fanti, Luciano and Gori, Luca
Department of Economics, University of Pisa, Department of Law and Economics "G.L.M. Casaregi", University of Genoa
17 September 2011
Online at https://mpra.ub.uni-muenchen.de/33480/
MPRA Paper No. 33480, posted 17 Sep 2011 18:55 UTC
$ % &'($()( * + ' , - . / 0 ' "' #
! " "
#
! "
!
!
" $
%&'( )
* + $ + $ + , +
-
! .(/+ $01+ 2%1+ 2%3
4 1 5lfanti@ec.unipi.it fanti.luciano@gmail.com+ 5 61& 737 // %( 1(&+ 5 61&
737 // %( 1'0
44 . 1 5luca.gori@unige.it dr.luca.gori@gmail.com+ 5 61&
7%7 /7& &3 71+ 5 61& 7%7 /7& 33 1(
/
"# $
: ) )
# )
: )
; " <
% %&(7 ,
$ %&'(
= =
> )
>
- %&&% %&&'+ ? %&&(+ @ /77A /77&+
/7%7 B
: C
D /771 /770 @ /77A /7%7
@ /77& 8 9 /7%%
, )
) B
5 !
" + !
! "
B
5 )
E
C 5 %
+ /
#
!
%: ) )
B
. %'1'
)
1
! " !
!
! "
F /
F 1
! ! F 0
%# & ' /
5 % ) H L
+ / H
L+ 1 C
. φ∈
[ ]
a,b 0≤a<b1 - U φ
(
p)
u p i 5{ }
H LUi φ, i =φ i− i, = , %
uH uL uH >uL
H L
pi
i #
51
∈ 0,2b
a /
2 φ
H L pH
pL F UH
(
φ,pH)
=UL(
φ,pL)
L L H
H p u p
u − =φ −
φ 1
φ 5
L H
L H
u u
p p
−
= −
φ 0
H L 5
( )
L H
L H L
H
H u u
p b p
b p p
D −
− −
=
−
= φ
, 3 %
/ /7%%
) ) ,
) )
1 D %1 /(
0
( )
au u
p a p
p p D
L H
L H L
H
L −
−
= −
−
=φ
, 3 /
5
( )
= − −−Π
L H
L H H
L H
H u u
p b p
p p
p , ( %
( )
= −− −Π a
u u
p p p
p p
L H
L H L L H
L , ( /
C D ( % ( / pH pL
5
L H
L H H
H
u u
p b p
p −
− −
∂ = Π
∂ 2
A % u a
u p p
p H L
L H L
L −
−
= −
∂ Π
∂ 2
A /
! ! H L
D A % A / C pH pL 5
( )
L[
L(
H L) ]
H H
H p p p b u u
p = ⇔ = + −
∂ Π
∂
2
0 1 ' %
( )
H[
H(
H L) ]
L L
L p p p a u u
p = ⇔ = − −
∂ Π
∂
2
0 1 ' /
B H L
"
(# '
.( ( ,2/3$
B
H *, L "
2
H t=0,1,2,...
+1
t $ %&'( 5
t H
t H t H H t H t
H p p p
p
, , , ,
1
, ∂
Π + ∂
+ = α &
>0
αH > H
t
pH, pH,t+1
G D & ) L " !
) 5
3
=
∂ Π + ∂
=
+ +
t L t L
t H
t H t H H t H t
H
p p
p p p
p
, 1 ,
, , , ,
1
, α
%7
G D A % ∂ΠH,t/∂pH,t %7
D ' / ! %7
5
( )
[
− −]
=
−
− − +
=
+ +
L H t
H t
L
L H
t L t H t
H H t H t
H
u u a p p
u u
p b p
p p
p
, 1
,
, , ,
, 1 ,
2 1
α 2
%%
D pH,t+1 = pH,t = pH pL,t+1= pL,t = pL
%% 5
( )
[
− −]
=−
− =
− − 2 0 1
2 0
L H H
L
L H
L H H
H
u u a p p
u u
p b p
α p
%/
! EBH/NL =
(
p*H,p*L)
D %% ! D %/ 5
( )
=(
−)(
−) (
−)(
−)
= H L H L H L
NL
BH p p b a u u b a u u
E 2
3 ,1 3 2
, * 1
*
/ %1 %
8 %1 % p H − p L =
(
a+b)(
uH −uL)
3
* 1
*
3 b a+
φ = 8 5
3 , 2
3
2 *
* b a
a D
D H = b− L = − %1 /
( ) ( ) ( ) ( )
9 , 2
9
2 * 2
2
* H L
L L H H
u u a b u
u a
b− − Π = − −
=
Π %1 1
5 uH −uL
b a
B %1 %
! %% H J
NL
EBH/ 5
(
−) (
−)
= −
=
2 0 1
3 2 2 1
3 1 2
22 21
12 11 /
a b a
b J
J J J J
H H
NL BH
α α
%0 5
(
J)
J J(
b a)
Tr
T = BH NL = + = − H 2 − 3
1 2
: / 11 22 α %3
(
J)
J J J J(
b a)
Det
D = BH NL = − =− H 2 −
6
: / 11 22 12 21 1α %(
%0 5
(
( )
T DG λ =λ2− λ+ %A
( ) (
2)
06 2 1
3 1 2 4 :
2
2− = − − + − >
=T D b a b a
Q αH αH F
NL
JBH/
: )
B
%%
5
( )
( )
(
−)
>= +
−
=
>
−
= +
−
=
− >
= − + +
=
6 0 2 1 6
: ) (
0 2 2
1 1 : ) (
6 0 2 5 1 12
: ) (
a D b
H iii
a b D
T TC
ii
a D b
T F
i
H H
H
α α
α
%'
%'
5 !
−1 F=0+
+1 TC =0+ )!F )
1
=0
H Det
( )
J =1 Tr( )
J <2[
0 b, /2)
%'a∈ F
> α
* C %&&&+ : C
D /771+ @ /77A+ /77&+ /7%7
B a B
αH b a
D / EBH/NL
H NL H
BH
F b
a αα
5 12 10
/
= − %&
b
F
5 6
1 =
α /7
b
F
15 24
2 =
α /%
a *,= 2 aFBH NL b
2 limα→+∞ / =
> α αF2 >αF1
A
α D %& C b/2
a %' ! /%
) " " # $ 0<α <αF1 4 ( 54 4 3 4 67 - 7 EBH/NL 4 % 1
" # - a∈
[
0 b, /2)
( " # $2
1 F
F α α
α ≤ < 4 ( 54 * 7 %4 4 4 3 4 67 - 7
NL
EBH/ - - 4 7*4 1
7- * - 7 %4 4 a 7 FBH NL
a
a= / ( ".# $ α ≥αF2
4 ( 54 4 3 4 67 - 7 EBH/NL 7 - (
) F aFBH/NL<0 0<α <αF1 F >0 a∈
[
0 b, /2)
% F 0≤aFBH/NL <b/2 αF1≤α <αF2 F >0
NL BH
aF
a
b/2> > / F =0 FBH NL
a
a= / F <0 aFBH/NL >a>0 / F aFBH/NL ≥b/2 α ≥αF2 F <0 a∈
[
0 b, /2)
1 *# #+#
.( ( 32/,$
B H " , L
*2
+1 t
> 5
t L
t L t L L t L t
L p p p
p
, , , ,
1
, ∂
Π + ∂
+ = α //
>0
αL !
5
∂ Π + ∂
=
=
+ +
t L
t L t L L t L t L
t H t
H
p p p
p
p p
, , , ,
1 ,
, 1 ,
α /1
G D A / ∂ΠL,t/∂pL,t /1
D ' % ! /1 5
( )
[ ]
− − + −
=
− +
=
+ +
u a u
p p p
p p
u u b p p
L H
t L t H t L L t L t L
L H t
L t
H
, ,
, ,
1 ,
, 1
,
2 2
1
α
/0
D pH,t+1= pH,t = pH pL,t+1= pL,t = pL
/0 5
( )
[ ]
=
− −
−
=
− +
−
2 0 2 0 1
u a u
p p p
u u b p p
L H
L H L L
L H L
H
α
/3
'
! ENH/BL =
(
p*H,p*L)
D /0 ! D /3 5
( )
=(
−)(
−) (
−)(
−)
= H L H L H L
BL
NH p p b a u u b a u u
E 2
3 ,1 3 2
, * 1
*
/ /(
B
H L
B D %1 % /( EBH/NL =ENH/BL
B /(
! /0 H J
BL
ENH/ 5
(
−)
−(
−)
=
=
a b a
J b J
J J J
L L
BL NH
3 2 1 2 3 2
1
2 0 1
22 21
12 11
/ α α /A
5
(
J)
J J(
b a)
Tr
T NH BL L 2
3 1 2
:= / = 11+ 22 = − α − /'
(
J)
J J J J(
b a)
Det
D NH BL L 2
6
:= / = 11 22 − 12 21 =−1α − /&
/A 5
( )
T DZ λ =λ2− λ+ 17
( ) (
2)
06 2 1
3 1 2 4 :
2
2 − = − − + − >
=T D b a b a
P αL αL F
BL
JNH/
! /0 5
( )
( )
(
−)
>= +
−
=
>
−
= +
−
=
− >
= − + +
=
6 0 2 1 6
: ) (
0 2 2
1 1 : ) (
6 0 2 5
1 12 : ) (
a D b
H iii
a b D
T TC
ii
a D b
T F
i
L L
L
α α
α
1%
L BL L
NH
F b
a α α
10 12 5
/ = − 1/
b
F
5 12
3 =
α 11
a ,=*2
> α α D 1/ C
2 /
/ b
aFNH BL → α →+∞ αF3 >αF2 >αF1 1% ! 11
&
) % " # $ 0<α <αF3 4 ( 54 4 3 4 67 - 7 ENH/BL 4 % 1
" !# - a∈
[
0 b, /2)
( " # $3
αF
α ≥ 4 ( 54 * 7 %4 4 4 3 4 67 - 7 ENH/BL
- - 4 7*4 1 7- *
- 7 %4 4 a 7 FNH BL
a
a= / (
) F aFNH/BL <0 0<α <αF3 F >0 a∈
[
0 b, /2)
% F 0≤aFBH/NL<b/2 α ≥αF3 F>0 FNH BL
a a
b/2> > /
=0
F FNH BL
a
a= / F <0 aFNH/BL >a>0
/
*# #+#
%'
1% > αL αH
a b
L
H u
u −
#
- % /
H L
! *,= 2 ,=*2
α α αH = L =
) ( 54 " # - * 4 3 4 67 - 7
BL NH NL
BH E
E / = / * %4 4 32/,$ 8 4 %4 4 4
,2/3$ 8 (
) F FBH NL
BL NH
F a
a / < / α >αF1 D
%& 1/ - 1 *# #+#
8 %
( )
α,a ,=*2*,= 2 B
> >
*,= 2 ; < ,=*2
>
C )
"
%7
" F !
( )
α,a *,= 2 ,=*2)
, ' - " 54 - 4 7 67 3 4 67 - 7
BL NH NL
BH E
E / = / 7 4 9 4 7 : 7 - 4
a %4 - * - 4 7 67 4 -
7 - 4 uH uL( 54 4 7 :
* 7 67 " % 7 a# 4 ; 7 ; " #
7 - (
, ' - % -7 * 4 <7 α 4 4
7 : * 7 67 " % 7 a# 4 ;
7 ; " # 7 - %4 4 ,2/3$ 8 4 %4
4 4 32/,$ 8 (
.# !
* /7%7
)
%%
! ,
* 5
#
B ! "
!
!
! "
" $ %&'(
) )
! "
8 ! !
! >
,
: C , D : : /771 .
- : 1/7 3%/I3/0
: C , D : : /770 .
: # . %0& '01I'(7
* H %''1 J J H
F 0' 0&&I37'
* 9 B C : %&&& 9
: $ 9 : 3
* ) * . /7
. : %'1' K - # J J
K - 5 ,
$ : ? %&'( . B D K
/A %7AI%//
8 2 9 2 /7%% *
#-K: ) - 11/('
? # %&&( F > .
. F 8 %/ /71%I/70'
- %&&% . . F 8 % 3A1I3'%
%/
- %&&' H D * L
M C 11 1'3I1&0
M 9 C C H H 2 $ /7%% E :
H B .
K $ %&(7 M .
K D F /A %11I%10
8 /7%7 ,
D # /A 137I13A
@ H $ N O /77A :
D # /0 %1'I%0'
@ H $ N O /77& *
. F 8 1& /70'I/733