• Keine Ergebnisse gefunden

Cross-ownershipandstabilityinaCournotduopoly Fanti,LucianoandGori,Luca MunichPersonalRePEcArchive

N/A
N/A
Protected

Academic year: 2022

Aktie "Cross-ownershipandstabilityinaCournotduopoly Fanti,LucianoandGori,Luca MunichPersonalRePEcArchive"

Copied!
8
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

Munich Personal RePEc Archive

Cross-ownership and stability in a Cournot duopoly

Fanti, Luciano and Gori, Luca

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

7 November 2011

Online at https://mpra.ub.uni-muenchen.de/34574/

MPRA Paper No. 34574, posted 07 Nov 2011 18:36 UTC

(2)

$ % &'($()( * + ' , -

. / 0 ' "' #

1 2

1 2

! " # $ %

2

1 # % 1 2

# %

# %

& ' ' ' (

)*' (+,' -.,' -+

/ 1 0lfanti@ec.unipi.it fanti.luciano@gmail.com' 0 1,2 343 ** .) ,)2' 0 1,2 343 ** .) ,5+

// 1 0luca.gori@unige.it dr.luca.gori@gmail.com' 0 1,2

3.3 *32 24 3,' 0 1,2 3.3 *32 44 ,)

(3)

- 6 - 7

*

6 "

8 9 #.22.%

&: ; $ < $ #*33=%

&: ; $ #*3..%

" 7 #*33)%

> #*3.3% $

6 #*3..%

.

(

# & *3.3%

# -

.222' ? $ *33*' ? $ @ *33, *33+' A *33='

*3.3% ?

# ( .25)% !

B C

0

# %

!

D

# %

E

8 *

$ 8 ,

8 +

!

1 2

$

# D .2*2' .2,,

. "

! # ? #.22=% ?

& #.225% > ! ; #.22=% ? & #*33*% #*33=%

%

(4)

,

Α 2

2 E Β

E h #0<h<1/2%

Α 2 8

E

Α Β 0

2

1+ Π

Π

=

ΠΑ h #. .%

(

1−

)

Π2

=

ΠΒ h #. *%

i 0

(

)

, =

{ }

1,2

=

Πi pi w qi i #*%

1 0<w<

# % !

# 8 9 .22+' 6 7 *3.. %0

2 1

1 1 q dq

p = − − #, .%

1 2

2 1 q dq

p = − − #, *%

d $ F

0 1< <

d #0<d <1% 1 2 # %

−1

d #d →1% # %

=0 d

!

8 E

1 2

@ #. .% #. *% q1 q2 0

( )

w q dq

(

h

)

q

( )

q

[

w dq

(

h

) ]

q q

q = − − − + = ⇔ = − − +

∂ Π

Α

1 2 1

0 1 1

2 , 1

2 2

1 2

1 1

2

1 #+ .%

( ) ( ) ( )(

2 2 1

)

2

( ) (

1 1

)

2 2

1 1

2 0 1

1 1 , 1

q d w q

q q

d q w h q

q h q

q =− − + − − − − = ⇔ = − −

∂ Π

Β #+ *%

F

A #*33=%

#*3.3% 0

1 #2% # %

7

0

=

∂ Π + ∂

=

+ + Α

t t

t t t t

t

q q

q q q

q

, 2 1 , 2

, 1

, , 1 , 1 1 ,

1 α

#4 .%

>0

α E 1G

+1

t Α q1

t

H @ #+ .% #+ *% @ #4 .%

0

(5)

- 6 - 7

+

( )

[ ]

(

)

=

=

+

− +

=

+ +

t t

t

t t

t t

t

q d w q

q

h q

d q w q

q q

, 1 ,

2 1 , 2

, 2 , 1 ,

1 , 1 1 , 1

2 1 1

1 2

α 1

#4 *%

6 @ #4 *% h

Α 2

# % 0<d <1 #−1<d <0% F

1 2 # %

E α h

t # % # 1

+1

t % 1 t

# % ! # %

" # $

@ q1,t+1 =q1,t =q1 q2,t+1 =q2,t =q2

@ #4 *% 0

( )

[ ]

=

− −

= +

− 2 0 1

0 1

2 1

2 1

2 1 1

q q d w

h q d q w αq

#)%

(

q*1, q*2

)

E 0

( ) [ ( ) ]

( ) ( )( )

(

+

)

− +

+

= −

h d

d w h

d

h d E w

1 4

2 , 1

1 4

1 2 1

2

2 #=%

2

* 1

* q

q < # *2 1

* q

q = % 0<h<1/2 # h=0 % H

1 2

B C

F E

I

( ) ( ) [ ( ) ]

( ) ( )( ) [ ( ) ]

( )

− − +

+

− + +

+

− +

=

=

2 0

1 4

1 2 1 1 1

4

1 2 1 2 1

4

2 2

2

22 21

12 11

d

h d

h d w h d h

d

h d w h

d J

J J J J

α α

#5%

Jii Jij

@ #=% J 0

( ) ( ) ( ) [ ( ) ]

(

h

)

d

h d w h

J d J J Tr

T − +

+

− +

= − +

=

= 4 1

1 2 1 2 1

: 4 2

2 22

11

α #2%

( ) ( )( ) [ ( ) ]

( )

[

4 1

]

0

2

1 2 1

: 1 2

2 21

12 22

11 <

+

+

− +

= −

=

= d h

h d w h J d

J J J J Det

D α #.3%

#5% 0

( )

T D

F λ =λ2 − λ+ #..%

6 @ #4%

λa λb #..%

0

(6)

4

( )

[ ]

( ) [ ( ) ]

( )( ) [ ( ) ]

( )

[

+ +

]

+ >

+ −

=

>

+

=

+ >

= −

1 0 4 2

1 2 1 1 1

) (

0 1

2 2 1

) 1 (

1 0 4 ) 2

(

2 2

2

h d

h d h w H d

iii

h d w TC

ii

h F d

i

α

α #.*%

#.*%

0 # % # −1%

=0

F ' # % # +1%

=0

TC ' # % ! 8 ! #

1% H =0 D=1 T <2 6 @

#.*% # % # % # %

E

@ #4 *% - Β

(

α,h,d,w

)

@ #=% #F =0% 0

(

, , ,

)

:=4

[

4 2

(

1+

) ]

(

1

) [

4+ 2

(

1+

) ] [

2

(

1+

) ]

=0

Βα h d w d h α w d h d h #.,%

E 0 # %

#0<h<1/2%

#h=0% 1 2

2 * 3 #d =1% # %

1 2

#0<d<1% #−1<d <0%

( ) [ ( ) ]

(

w

) [

d

(

d h

) ] [

h d

(

h

) ]

w d

F h

+

− + +

+

= −

1 2 1 4 1

1 4 , 4

, 2

α 2 #.+ .%

( ) ( )

(

w

)(

hh

)(

h

)

w

F h

+

= −

5 1 1

3 , 4

1

α , #.+ *%

# % α 1

2

% $ −1<d ≤1 ( 42 " # 0<α <αF

(

h,d,w

)

2 3 15 2

63 - 3 E 2 % 1 " ( # - 7 " #

- 3 * α =αF

(

h,d,w

)

7 ".# α >αF

(

h,d,w

)

2 3 15 2

63 - 3 E 3 - (

% 8 Β

(

α,h,d,w

)

>0 0<α <αF

(

h,d,w

)

Β

(

α,h,d,w

)

=0 α =αF

(

h,d,w

) (

, , ,

)

<0

Βα h d w α >αF

(

h,d,w

)

> . & '

% ! 89 * 3 3 d =1:( 42 - * 3

1 % 2 % 2 %2 % - * 2 2

(7)

- 6 - 7

)

2 2 2 2 2 Α 2 2 -

2 ; 63 - 3 (

% 8 αF

(

h,1,w

)

>αF

(

0,1,w

) ( ) ( )( )

(

1 47

)(

1

) (

1 5

)

0

, 1 ,

2

2 >

+

+

= −

h h

w

h h h

w

F h α

> * & '

% " 8 3 −1<d <1:( < 2

2 2 2 2 Α 2 2 " # - %2

3 " # 3- 3 (

% 8

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

(

1161

) [

4

(

1 1

) ] [

28

(

11

) ]

0

4 , ,

2 2 2

2

2 <

+

− +

+

+

− + +

= −

h d h

d w

h d h d d d

h w d

F h

α #

>0 % 0

1< <

d #0<d <1% > , & '

2 Α

F

h

(

!

!

D

2 1

# %

! 0

E

! # %

)

? $ D D $ ? 8 @ ? ? *33*

$ "

8 45 .,,J.+)

? $ D @ ? ? *33,

> ? ,*3 4.*J4*+

? $ D @ ? ? *33+

? " .+2 5+,J5)3

? & .225 ? +5

(8)

=

? @ 8 & - *33* !

! K @ I *, =,J24

&: ; $ I < $ *33=

@ & .4 .J5

&: ; $ I " - *3.. >

FL@;-? ?L ! > *3..+= H > M

9 ( 6 ? : @ N F

& 7 F L " 8$ $! 6 *3.3 <

8 & & 0 8 9

@ .2,, " #"?%0

D H >

( ? L .25) F @ ;

*= .3=J.**

6 - *3.. ( > .*=

( @ H >

6 - 7 - *3..

">;? ! > ,,+==

7 ( " O 8 O *33) >

;? ( I @ ,= 5.J22

D D .2*2 8 @ I ,2 +.J4=

- ( L .222

? < ; 52 .)4J.=,

" 8 F 9 .22. 8

0 I @ 4, *54J*2=

> ; *3.3 D K $ F

7 F ( ; " 8

0PP P P P > *3.3 3.4

> ! > " ; - D .22= 0 !

;? ( I @

*5 ,3+J,**

8 9 Q .25+ >

;? ( I @ .4 4+)J44+

6 *3.3 D

@ " *= ,43J,4=

6 *33= ( K @

F ! I F 6 " ! F

" .= =2J.3.

A I ( R O *33= ?

@ " *+ .,5J.+5

Referenzen

ÄHNLICHE DOKUMENTE

Department of Economics, University of Pisa, Department of Law and

In contrast to an economy without public health spending – which is always stable with monotonic trajectories –, an economy with tax-financed health care services

When endogenous mortality, based on public health investments, is introduced in the overlapping generations (OLG) model of neoclassical growth with exogenous fertility

Moreover, a large enough increase in the child tax shifts upward the phase map G ( ) k in such a way to cause the loss of the lowest stable equilibrium, thus allowing

Therefore, a rise in µ increases unemployment and reduces both the interest rate and skilled wage, and thus it also decreases the ratio of the skilled wage to the unskilled one.

Therefore, although it is well known that OLG economies with myopic expectations and elasticity of substitution in the production and in the utility functions

This analysis has shown the remarkable complicated role of either a public health policy, through their effects on longevity, or behaviours as regards the “gift” from

We show – with respect to the previous models with exogenous longevity – that the role played by the preference for old-aged services on economic growth is