Adoptionofacleantechnologyusingarenewableenergy BenYoussef,Slim MunichPersonalRePEcArchive

Adoption of a clean technology using a renewable energy

Ben Youssef, Slim

Ecole Supérieure de Commerce de Tunis

September 2010

Online at https://mpra.ub.uni-muenchen.de/25576/

MPRA Paper No. 25576, posted 05 Oct 2010 13:57 UTC

, $

- ! ! .# !

! ! # # ! !

! / . # # #

# ! # .

. ! - # # #

.# ! # .# !

0 ! # ! # # ! # .#

1 2 . # 1 ! # .#

# ! # ! # #

. # # 1 3 # #

! ! # ! #

# ! # ! #

Keywords$ ! # ! . !

JEL classification$ 4& 2+) 56 5++

! "

! ! # . ! # ! . !

! # ! #

0 # ! ! #

. / ! # # !

# ! # # . #

. !

4 % (()' # ! .# # . #

! # ! ! # # !

. # / # . !

! # ! . # # 1 7 8

! # # # ! # #

! # ! #

# . # ! # !# % +'

9 # / : # !

.# # .# # ! 1 # !

.# # 2 # #

! # # 3 ; % ('

! 1 # # 9 # 9 !

. ! # # $

! # # 9 # ! 1 3 #

: < = % '

# # ! . # !

2 # . # # 1

. # #

-# # # % )' # # !

. # ! ! ! 3 # > " 7 #

<# % (' # . # #

# ! ? et al. % (' # . # # # .

. ! ! @A4 !

% (' # 7

. ! # / + # 7

# .# "# %">'. 7

# # ! #

!# ! # !

7 # 7 ! # # ! . #

! # ! " < B % (' . #

. % / ' 0 # #

! !

# # . 7 # # # ! # .

# # ! . # 1

# # ! 1 !

- ! ! ! % !'

# ! - # 7 7

8 ! 9 % C' .# # # . !

! # ! ! #

# ! . # !

! / # # . !

# ! # 9 .

. ! - # .#

# ! # 1 / .# #

# ! .# # ! # ! -

# .# # !

! # ! # ! # !

# .# # ! # #

# # 1

3 # # 1 ! # # !

# .#

0 .# # ! . # # ! # ! !

/

: # 8 .#

# # ! 0 # #

. # !

. 2 . # # !

# ! # # # , # # #

# ! # .# -# # 1 !

# ! # ! :

# #

# . # # # 1 !

# !# # .#

D . # # < = % ' .#

# ! # #

! # 8 .# # # # !

E 4 % (()' #

! B # # ! # !

# ! F8 88 .# # # !

/ 9 # . # #

# .# # !

# . # *

# ! 6 # 1 !

0 + . # # &

E /

#!

- ! ! : q #

7 p(q)=a bq, a,b>0

# # ! ! :

2

0 ( ) ( ) 2

)

( bq

q q p dz z p q

CS =

∫

E # ! ! # ! 0 # !

# ! ! # 9

G e>0

# # # E=eq .# # ! #

: D=αE .# α>0 # ! ?

# # . # ! #

# . # 7

- # # ! # ! # d>0 # #

1 Π_d = p(q)q−dq

# # # 9 .

# ! . # τ # # !

. ! % ! ' # #

. c ! 0<c<d # # #

. ) (q q cq

c = p −

Π

E ! # . # ! #

# @A4 G # : #

! # ! # . . #

# !

# ! # # ! τ 9 0 $

θ τ

τ ^{e mr}

V( )= ⁻ % '

.# θ>0 # # ! # ! r>0 #

m>1 # #

.# m !

, V ! # / 1

# ! # # .#

/ # @A4 .# #

# ? 8 7 # τ=+∞ # #

0 .# . # d c . # % !' # !

$!

0 # . # .# # ! #

@ # # 7 ! # 1 ! .

. # 1 : 3 / . #

!

-# # # # ! # . $

) ( ) ( )

(q D q q

CS

S_d = − +Π_d % '

/ 9 ! # / ! % ' . # q ! # 1

. # # ! # ! $

b e d

qˆ_d = a− −α %*'

- # . ! # : $

e d

a> +α %6'

# # / . ! # ! # !# # #

! # ! !

# ! ! E 1 / 1

t # 1

0 # # $

) ( ) (q tE q

U_d =Π_d − %+'

# 1 1 1 / # # qˆ_d $

e q b d

t = a− −2 ˆ^d %&'

! # / qˆ_d . # . # $

e d a

t >⁰⇔ < +²α %)'

# # 1 / .# # ! !

# !# !# D # . ! ! # # ! 9

. # #

-# # # # ! # . $

) ( )

(q q

CS

S_c = +Π_c %H'

/ 9 ! # / ! %H' . # q ! # 1

. # # ! # ! $

b c

qˆ_c = a− %('

0 # qˆ_c >qˆ_d # # # !

. #

# # 1 !

1 s # 1

0 # # $

sq q

U_c =Π_c( )+ % '

# 1 1 # # qˆ_c $

a c q b

s=2 ˆ_c + − % '

! # / qˆ_c . # . # s>0

%! & '

0 # . . # .# # !

.# # # !

-# # # ! # / 9 Π_d ! #

$ ²

b d q_dⁿ a

2

= − % '

-# # ! # ! # / 9 Π_c ! #

$

b c q_cⁿ a

2

= − % *'

0 # # . # # # !

.

(

)

cn q q >

- # # $

# n # 1 !

e d a q

q^ˆ_d > _dⁿ ⇔ − >²α % 6'

0 . # # ! # ! # 1 7

# ! # # !#

# # .# # ! ! . !# 2 .

. # # # ! # . . # qˆ_c >q_cⁿ

7 .

(! )

# . # !

1 ! $

τ τ

τ^S_d ^q_d ^{e rtdt S}_c ^q_c ^{e rtdt} θ^{e mr}

IS ⁻

+∞ −

− + −

=

∫

∫

τ τ

τ^U_d ^q_d ^{e rtdt U}_c ^q_c ^{e rt dt} θ^{e mr}

IU =

∫

∫

τ

τ _{d d}^{n rt} _{c c}^{n rt} θ ^mr

n q e dt q e dt e

IU =

∫

∫

0 # . # . ! .# #

. # !θ G m# !# !#$

mr q

U q

U_c(ˆ_c)− _d(ˆ_d)<θ % H'

# ! # / 9 # . #

τ ! # 0 # E / . #

1 # # !

# 1 ! .# # $

) 0 (ˆ ˆ )

ln ( ) 1 (

ˆ 1 >



 



 −

= −

mr q S q S r m

d d c c

τ θ

(19)

) 0 (ˆ ˆ )

ln ( ) 1 (

* 1 >



 



 −

= −

mr q U q U r m

d d c c

τ θ

(20)

0 ) ( ) ( ) ln

1 (

* 1 >









Π −Π

= −

mr q q

r m

n d d n c n c

τ θ

(21)

We have the following ranking for the optimal adoption dates:

*

* ˆ

0<τ <τ <τⁿ (22) Therefore, the optimal adoption date for the regulated firm is earlier than the socially optimal one. However, the optimal adoption date for the non regulated firm is later than the socially optimal one.

Proof. See the Appendix.

# # # #

! # ! # # 1 ! #

# # : % (' 0 .# # ! #

# ! !

/ # .

: # ! # ! # ! , #

# # ! !

# 8 .# # 1 ! #

# : ! # . # ! #

3 # # 1 ! # !

# ! . #

" / # ! # # !

# 1 # # # #

#

0 # # ! ^IU(τ^*) IU⁽τ^ˆ).# #

τ* τ^ˆ # #

% ' $

0 ˆ) ( )

( ^* − >

=^IU τ ^IU τ

g % *'

The regulator can push the regulated firm to delay its adoption of the clean technology by giving it an innovation subsidy that compensates the firm for the losses it incur when it delays its optimal adoption date to the socially optimal one.

*!

0 # . ! ! !

# ! 2 . # # ! . #

! ! / #

# ! # ! # 9

. . ! - #

.# # ! # 1 / .#

# # ! .# # ! -

# .# # !

-# # ! . # # ! # ! !

/

: ! 0 #

# . #

! . 2 . #

# ! # ! # # # -# #

1 ! # ! # !

:

# # # . , # .

# . !

# 1 ! # # ! #

.# 1 0 ! # ! # !

# ! # .# # !

# # # # #

# 1 2 . # !

! # ! # #

# ! #

+

, "

I _{c c d d} b q_d q_c c q_c q_d d c q_d eq_d a

q S q

S (ˆ ˆ ) (ˆ ˆ ) ( )ˆ ˆ

) 2 ˆ ( ) ˆ

(  − + − +α



− +

−

=

−

< ! # / qˆ_d qˆ_c # 7 / :

2 . e c d− +α

# . # $

0 ˆ ) (ˆ ) 2

(ˆ ˆ )

( _c − _{d d} = − + _c + _d >

c d c e q q

q S q

S α

% 6' IUc(qˆc)−Ud(qˆd)=

[

]

< # ! ! # /t # s # /

qˆd qˆ_c . $

0 ˆ ) (ˆ ˆ )

( ˆ )

( _c − _{d d} = _c²− _d² >

c q U q b q q

U % +'

I

[ ]

n d cn n d cn n

d d cn

c q −Π q = a−b q +q q −q +dq −cq

Π ( ) ( ) ( )( )

< ! q_cⁿ q_dⁿ . # 7 # . ! $

0 ) 2 (

) ( )

( −Π = − + >

Π_{c c}ⁿ _{d d}ⁿ d c q_cⁿ q_dⁿ q

q % &'

# # # ! # . .#

# 1 0 #

# # ! 1 !

,

I< ! / % 6' % +' . # $

(

) (

)

(

)

(

)

2 ˆ

) ˆ (ˆ ˆ )

( ˆ )

( ˆ )

(  +

 − − − +

=

−

−

− α

< ! # / qˆ_d qˆ_c # 7 / . $

( ) ( ) (

)

) 2 (ˆ ˆ )

( ˆ )

( ˆ )

( _c − _{d d} − _{c c} − _{d d} = − + _c+ _d >

c d c e q q

q S q S q

U q

U α

% )' I< ! / % 6' % &' . ! $

( ) ( ) ^{( )} ( )

[

] ⁽

⁾

cn d c

n d cn d

n c d d cn c d

d c c

q e q

q q q c q

d

q c q

q d e q

c q d

q q

S q S

ˆ 2 ˆ

ˆ 2 ˆ

ˆ 2 2 ˆ

) ( ) ( ˆ )

( ˆ )

(

+ +

−

−

− +

=

− +

− + +

= − Π

− Π

−

−

α α

< ! qˆ_d qˆ_c, q_cⁿ q_dⁿ # # 7 . $

( ) (

) ⁽

⁾

c c d

d c

c e q q

b e d c a c q d

q q

S q

S ˆ ˆ

2 2

2 2

) 2 ( ) ( ˆ )

( ˆ )

( + +

 − − −

= − Π

− Π

−

− α α

! %6' # 7 ! $

(

)

(

)

⁽

⁾

c c d

d c

c e q q

b c q d

q q

S q

S α

% H'

# . # # . ! 7 !$

ˆ ) ( ˆ )

( ˆ )

( ˆ )

( ) ( ) (

0<Π_c q_cⁿ −Π_d q_dⁿ <S_c q_c −S_d q_d <U_c q_c −U_d q_d % ('

# ! ! # # ! ! # !

# # .# # # # 1 !

, )

I ! # 1 # ! / 9 #

. ! % +' . # τ$

(

)

∂ =

∂ _{− τ} θ _{− τ}

τ ^{Sd qd Sc qc e r mre mr}

IS %* '

: %* ' : $

⇔

= +

− (ˆ ) ⁻ 0

ˆ )

( _{d c c} θ ^{(1 m)rτ}

d q S q mre

S 



 



 −

= −

mr q S q S r m

d d c

c θ

τ ^{ln (ˆ ) (ˆ )}

) 1 (

ˆ 1 %* '

< m>1 % H' : % (' τ^ˆ>⁰

- # $

( )

τ

mr d r

d c

c q S q e mr e

S

IS r _{− −}

−

−

∂ =

∂ 2

2 2

) ( ˆ )

( ˆ )

(

! # ! %* ' . ! $

0 )

1 ) (

(ˆ 2 ˆ

2

2 = − <

∂

∂ θ _{− τ}

τ

τ m m r e mr

IS

#

I # ! / 9 ! % &' . # τ$

(

)

∂ =

∂ _{− τ} θ _{− τ}

τ ^{Ud qd Uc qc e r mre mr}

IU %* '

: %* ' : $

⇔

= +

− (ˆ ) ⁻ 0

ˆ )

( _{d c c} θ ^{(1 m)rτ}

d q U q mre

U 



 



 −

= −

mr q U q U r m

d d c c

τ ^{ln (ˆ )}θ ^{(ˆ )} )

1 (

* 1

%**'

< m>1 : % H' τ^* >⁰

- # $

( )

τ

mr d r

d c

c q U q e mr e

U

IU r _{− −}

−

−

∂ =

∂ 2

2 2

) ( ˆ )

( ˆ )

(

! # ! %* ' . $

0 )

1 ) (

( ₂ ^*

2

*

2 = − <

∂

∂ θ _{− τ}

τ

τ m m r e mr

IU

# #

I # 1 ! / 9 ! % )' . #

τ$

(

)

∂ =

∂ _{− τ} θ _{− τ}

τ ^{d dn c cn r mr}

n

mre e

q IU q

%*6'

# : $

⇔

= +

Π

−

Π_d(q_dⁿ) _c(q_cⁿ) θmre^(1−m)rτ 0 











Π −Π

= −

mr q q

r m

n d d cn n c

τ ^{ln (} θ^{) ( )}

) 1 (

* 1

%*+'

< m>1 : % H' % (' τ^n* >⁰

- # $ _τ

(

)

c c n

e mr e

q q

IU r _{− −}

− Π

− Π

∂ =

∂ 2

2 2

) ( )

( ) (

! # ! %*6' . $

0 )

1 ) (

( ₂ ^*

2

*

2 = − <

∂

∂ n n ₋mr ⁿ

e r m

IU m θ _τ

τ τ

#

-,

0 : % (' # # m>1 7 # . ! 7 !$

*

* ˆ

0<τ <τ <τⁿ

# ! # .# .# # 1

!

< = % ' F8 E # !

0 0 88 Economics Bulletin * )*61 )6*

C % (' F8C ! ! # , @ . !

88 Energy Economics * 66*166(

4 % (()' F8 " E , 0

" 0 # ! # ! " < 88

Environmental & Resource Economics H+1*

? 2 2 J 7 1 # ? @ " < K C 2 % (' F8"

@ ! @ , , $ 0 ! ! 2 #

- ! 1 1" ! @ # 4 88 Ecological Economics &H )* 1)6

3 , C ; % (' F8 E / C 1 # E <

C # 88 Energy Policy *) 6+ 616+

" 0 @ @ < B % (' F8@ . ! 0 $ " 88

Energy *6 () 1(H

4 " > % +' F8 # 0 " 0 # !

E ! 1 ! # ! 88 Resource and Energy Economics ) *+1

6)

> @ " 7 # 0 K <# % (' F8 !

0 @ . ! 88 Renewable and Sustainable Energy Reviews *

) &1 )

-# # J ? # % )' F8 - ! " C !

" ! $ E /1E /1" 2 # < !

E # 88 Resource and Energy Economics ( 6)1 &