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Discussion Papers No. 505, June 2007 Statistics Norway, Research Department

Geir B. Asheim and Bjart Holtsmark

Pareto-efficient climate agreements

Abstract:

Recent contributions show that climate agreements with broad participation can be implemented as weakly renegotiation-proof equilibria in simple models of greenhouse gas abatement where each country has a binary choice between cooperating (i.e., abate emissions) or defecting (no abatement).

Here we show that this result carries over to a model where countries have a continuum of emission choices. Indeed, a Pareto-efficient climate agreement can always be implemented as a weakly renegotiation-proof equilibrium, for a sufficiently high discount factor. This means that one need not trade-off a “narrow but deep” treaty with a “broad but shallow” treaty.

Keywords: Climate, non-cooperative game-theory, repeated games, weakly renegotiation-proof agreements

JEL classification: C72, F53, Q54.

Acknowledgement: We thank Jon Hovi for many helpful comments.

Address: Geir B. Asheim, Department of Economics, University of Oslo.

E-mail: g.b.asheim@econ.uio.no

Bjart Holtsmark, Statistics Norway, Research Department.

E-mail: bjart.holtsmark@ssb.no

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). D;<= D7- =1;8>01 $;898<2=287 =1. ,87-2=287 =1*= .7<>;.< =1*=

=12< <=;*=.0B 9;8D5. 2< * <>+0*6.9.;/.,= .:>252+;2>6 *7- =1.7 9;8,..-

=8 9;8?2-. =1;8>01 $;898<2=287 =1. ,87-2=287 =1*= .7<>;.< =1*= <>,1 *

<>+0*6.9.;/.,= .:>252+;2>6 2< @.*45B ;.7.08=2*=2879;88/ $;898<2=287 2< * -2;.,= ,87<.:>.7,. 8/ $;898<2=287< *7-

&"#% $$

.= -.78=. =1. *?.;*0. -2<,8>7=.- 9*B8L8/ .*,1 <207*=8;B @1.7 2< /8558@.- 24.@2<. 5.= -.78=. =1. *?.;*0. -2<,8>7=.- 9*B8L8/ *

<207*=8;B @1.7 2< /8558@.-

#!" )" #" ' " # "$ "

% " " #!

#!! B 27<.;=270 27=8 *7- @. 8+=*27

(10)

B 27<.;=270 27=8 *7- @. 8+=*27

'1. 5.66* 2< 8+=*27.- +B <>+=;*,=270 /;86

.66* 02?.< =1. <2C. 8/ =1. />=>;. 9>72<16.7= 27E2,=.- 87 * <20 7*=8;B @1.7 2= -.?2*=.< /;86 =1. <2695. <=;*=.0B 9;8D5. -.=.;627.- +B '12< 6><= +. ,869*;.- =8 =1. <18;==.;6 0*27 =1*= * <207*=8;B ,*7

;.*9 +B -.?2*=270 /;86 =1. *+*=.6.7= 9;.<,;2+.- +B =12< <=;*=.0B 9;8D5.

'1. <2C. 8/ =12< <18;==.;6 0*27 2< 9;8?2-.- +B =1. /8558@270 5.66*

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=1.;68;. 27-.9.7-.7=5B 8/ *7- =1. +.1*?28; 8/ =1. 8=1.; ,8>7=;2.<

,8>7=;B 6*A262C.< 2=< <18;==.;6 9*B8L+B ,188<270 .7,.

2< =1. 6*A26*5 <18;==.;6 0*27 =1*= ,8>7=;B ,*7 ;.*9 =1;8>01 * >72 5*=.;*5 -.?2*=287

). ,*7 78@ ,1*;*,=.;2C. <>+0*6.9.;/.,=7.<< /8; =1. <.= 8/ <=;*=.0B 9;8D5.< ,87<2-.;.-

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#!! " .= *7- <*=2</B *7- ). 875B 7..- =8 ,1.,4 /8; 87.9.;28- -.?2*=287< <27,. 2= /8558@< /;86

(11)

=1. =1.8;B 8/ ;.9.*=.- 0*6.< @2=1 -2<,8>7=270 +;.> 9

=1*= * 95*B.; ,*778= 0*27 +B * 6>5=29.;28- -.?2*=287 2/ 1. ,*778= 0*27 +B <86. 87.9.;28- -.?2*=287

'1;8>018>= 787<207*=8;2.< *;. 9;.<,;2+.- +B =1. <=;*=.0B 9;8D5. =8 ,188<.*< =1.2; *+*=.6.7= 5.?.5 .7,. .?.7 =18>01 *7B 87.9.;28- -.?2*=287 +B * 787<207*=8;B 2< 78= 9>72<1.- 2= /8558@< /;86 .66*

=1*= =1.B 1*?. 78 27,.7=2?. =8 -.?2*=.

&207*=8;2.< *;. 9;.<,;2+.- =8 ,188<. *5870 = /8558@< /;86 .66*< *7- =1*= =1.;. 2< 78 9;8D=*+5. -.?2*=287 2/

@12,1 ,*7 +. ;.@;2==.7 *<

'1. <207*=8;2.< 27,5>-270 ,8>7=;B2=<.5/ 78= 27E2,=270 9>72<16.7=

87 ,8>7=;B27 =1. D;<= <=*0. 8/ *7- *55 <207*=8;2.< 27 5*=.; <=*0.< 8/

*;. *5<8 9;.<,;2+.- =8 ,188<. /8558@.- +B 2/ =1.;. 2< * >725*=.;*5 -.?2*=287 +B * <207*=8;B *7- +B2/ =1.;. 2< 78 <>,1 -.?2*=287 .7,.

*5<8 27 =1.<. ,*<.< =1.;. 2< 78 9;8D=*+5. -.?2*=287 2/ 2< <*=2<D.- 27*55B =1. <207*=8;2.< 27E2,=270 9>72<16.7= 87 ,8>7=;B *;. 9;.

<,;2+.- =8 ,188<. 27 =1. D;<= <=*0. 8/ /8558@.- +B 2/ =1.;.

2< * >725*=.;*5 -.?2*=287 +B * <207*=8;B *7- +B 2/ =1.;. 2< 78 <>,1 -.?2*=287 B .66*< *7- =1.;. 2< 78 9;8D=*+5. -.?2*=287 2/

@12,1 ,*7 +. ;.@;2==.7 *<

&27,. 27.:>*52=2.< *7- *;. .:>2?*5.7= =8 27.:>*52=B ' " &>998<. &27,. 2= /8558@< /;86 =1*=

=1.;. 2< * 9;8D=*+5. -.?2*=287 /;86

<<>6. =1*= &>998<. =1*= 2< 78= <*=2<D.- '1.7 2= /8558@<

/;86 =1*= =1.;. 2< * 9;8D=*+5. -.?2*=287 /;86 &>998<. =1*= 2<

78= <*=2<D.- '1.7 2= /8558@< /;86 =1*= =1.;. 2< * 9;8D=*+5. -.?2*=287 /;86 =1. D;<= <=*0. 8/ .*,1 9>72<16.7= 9*=1

&27,. *7- *;. .:>2?*5.7= =8 @. 1*?. =1*= *7- *;. 7.,.<<*;B ,87-2=287< /8; =1. <>+0*6.9.;/.,=7.<< 8/ =1. <2695.

<=;*=.0B 9;8D5. -.=.;627.- +B @2=1 *7-

(12)

# !%%! "#!! $$

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=8;2.< 27E2,=270 9>72<16.7= 87 ,8>7=;B @1.7 2< 2695.6.7=.-

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-.=.;627. @1.7 =1.;. -8 78= .A2<= =@8 ,87=27>*=287 .:>252+;2* <>,1 =1*=

*55 95*B.;< <=;2,=5B 9;./.; =1. 87. =8 =1. 8=1.; 2?.7 =1. <=;>,=>;. 8/

=1. <2695. <=;*=.0B 9;8D5. -.=.;627.- +B =1.;. .A2<= -2L.;.7=

,87=27>*=287 .:>252+;2* 2695.6.7=270 =1. 95*B 8/*7- /8; *55

&27,. =1. <=;*=.0B 9;8D5. 2< <>+0*6.9.;/.,= 2= /8558@< /;86 $;898

<2=287 =1*= 2695B270 =1*= >;=1.;68;. *55 787

<207*=8;2.< *< @.55 *< *55 <207*=8;2.< 78= 27E2,=270 9>72<16.7= <=;2,=5B 9;./.; =8 *7B 9>72<16.7= 9*=1 / =1.7 *55 ,8>7=;2.<

27,5>-270 =1. 9>72<1270 <207*=8;2.< <=;2,=5B 9;./.; =1. ,87=27>*=287 .:>2 52+;2>6 27 =1. G.69=BH 12<=8;B =8 =1. ,87=27>*=287 .:>252+;2>6 /8558@270

* >725*=.;*5 -.?2*=287 +B ,8>7=;B / =1.7 =1. ,87=27>*=287 .:>252+;2>6 /8558@270 * >725*=.;*5 -.?2*=287 +B ,8>7=;B2< * +.<= ,87=27

>*=287 .:>252+;2>6 /8; .*,1 <207*=8;B 27E2,=270 9>72<16.7= 87 ,8>7=;B

*7- * @8;<= ,87=27>*=287 .:>252+;2>6 /8; ,8>7=;B 2=<.5/ 2695B270 =1*=

*55 95*B.;< 7.?.; <=;2,=5B 9;./.; 87. ,87=27>*=287 .:>252+;2>6 =8 *78=1.;

.7,. =1.;. -8 78= .A2<= =@8 ,87=27>*=287 .:>252+;2* <>,1 =1*= *55 ,8>7=;2.< <=;2,=5B 9;./.; =1. 87. =8 =1. 8=1.; 2/ *7- 875B 2/

B 27<.;=270 27=8 *7- 2= /8558@< =1*=

B ,869*;270 @2=1 @. 8+=*27

2695B270 =1*= 2< .:>2?*5.7= =8

(13)

&$%#%!

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.:>252+;2>6 2695.6.7=270 *7 .K,2.7= ,526*=. *0;..6.7=6*B 5884 524.

+B *995B270 =1. ;.<>5= 8/ $;898<2=287 =8 * =@8;.0287 @8;5- .*,1 @2=1 ,8>7=;2.< %./.; =8 =1. ;.0287< *< *7- &27,. =1. =8=*5 7>6 +.; 8/ ,8>7=;2.< 2< .?.7 @2=1 =@8 .:>*55B <2C.- ;.0287< @. 6*B <*=2</B

=1. ;.:>2;.6.7=< 87 =1. <>+<.= 8/ ,8>7=;2.< 9>72<1270 * -.?2*=270 ,8>7=;B7*6.5B *7- =1. 7>6+.; 8/ ,8>7=;2.< 27 .:>*5< +B 1*?270 * >725*=.;*5 -.?2*=287 +B * ,8>7=;B 27 ;.0287 +.

9>72<1.- +B *55 ,8>7=;2.< 27 ;.0287 *7- ?2,. ?.;<*

'12< 6.*7< =1*= * >725*=.;*5 -.?2*=287 +B * ,8>7=;B 27 ;.0287

=;200.;< * 87.9.;28- ;.->,=287 27 *+*=.6.7= +B *55 ,8>7=;2.< 27 ;.0287 =12< 27E2,=< *7 .:>*55B 1*;- 9>72<16.7= 87 *55 ,8>7=;2.< 27 #7 =1.

8=1.; 1*7- <27,. 2= /8558@< /;86 $;898<2=287 =1*= *55 ,8>7=;2.< 27 ;.0287 <=;2,=5B +.7.D= +B ,*;;B270 8>= =1. 9>72<16.7=

'12< *;;*70.6.7= ,*7 +. ,87=;*<=.- @2=1 =1. ;.0287*5 *0;..6.7=<

,87<2-.;.- +B <1.26 .= *5 @1.;. * >725*=.;*5 -.?2*=287 +B * ,8>7=;B 27 ;.0287 =;200.;< * 87.9.;28- ;.->,=287 27 *+*=.6.7= +B

*55 =1. 8=1.; ,8>7=;2.< 27 ;.0287 .7,. =1. ,8>7=;2.< =1*= +.7.D=

->;270 =1. 87.9.;28- 9>72<16.7= 91*<. *;. 27 =1. <*6. ;.0287 *< =1.

-.?2*=270 ,8>7=;B @125. =1. ,8>7=;2.< 27 ;.0287 *;. 1*;6.- =@2,. "

+B =1. 272=2*5 >725*=.;*5 -.?2*=287 8/ * ,8>7=;B 27 ;.0287 +B =1.

<>+<.:>.7= 9>72<16.7= +B =1. 8=1.; ,8>7=;2.< 27 ;.0287

). @*7= =8 9827= 8>= =1*= =1. *5=.;7*=2?. 9;898<.- 27 =1. 9;.<.7=

9*9.; 1*< =1. *99.*5270 /.*=>;. 8/ 27E2,=270 =1. 9>72<16.7=< @2=127 =1.

;.0287 @12,1 2< =8 +5*6. /8; =1. =.698;*;B +;.*4-8@7 8/ ,889.;*=287

*7- ;.@*;-270 =1. 2778,.7= ,8>7=;2.< 8/ =1. 8=1.; ;.0287

# $

+;.> #7 =1. =1.8;B 8/ 27D72=.5B ;.9.*=.- 0*6.< @2=1 -2<,8>7=270

,8786.=;2,* I

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!*7*0.6.7= I

& *;;.== =1.8;B 8/ />55 27=.;7*=287*5 ,889.;*=287 8>;7*5 8/ '1.

8;.=2,*5 $852=2,< I

(14)

& *;;.== 87<.7<>< =;.*=2.< 8>;7*5 8/ 7<=2=>=287*5 *7- '1.8;.=2,*5 ,8786862,< I

*;;.55 *7- !*<427 %.7.08=2*=287 27 ;.9.*=.- 0*6.< *6.< *7- ,87862, .1*?28; I

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<8>;,. ,87862,< I

;8B7 *7- 8?2 ,526*=. *0;..6.7= @2=1 />55 9*;=2,29*=287 626.8 %# *7- .9*;=6.7= 8/ $852=2,*5 &,2.7,. (72?.;<2=B 8/ #<58

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