International Environmental Agreements with Uncertainty, Learning and Risk Aversion

Transcription

1 Internatona Envronmenta Agreements wt Unertanty, Learnng and Rsk Averson Abstrat Mae Fnus (Unversty of Bat) Pedro Pntassgo (Unversty of Agarve) and Astar Up (Unversty of Manester) Trd Draft: November 2012 Unertanty and earnng pay an mportant roe for te formaton of nternatona envronmenta agreements (IEAs). Sentf unertanty about mate damages and tenooga mtgaton optons s st arge despte ongong resear. It as been sown n te strateg ontext of vountary partpaton but strong free-rder nentves tat earnng may ave a negatve mpat on te suess of IEAs. Ts paper extends te mode of Kostad (2007) and Kostad and Up (2008) by onsderng rsk averson. Ts seems suggestve as unertantes n mate ange are gy orreated and ene poong rsks may be mted. It s sown tat te negatve onuson wt respet to te roe of earnng derved for rsk neutraty as to be quafed. JEL-Cassfaton: C72, D62, D80, Q54 Keywords: nternatona envronmenta agreements, unertanty, earnng and rskaverson, game teory An earer verson of ts paper was presented at a worksop on Vountary Approaes to Envronmenta Proteton at te Bren Soo, Unversty of Caforna Santa Barbara, Apr We are gratefu to te partpants and n partuar to Larry Karp for epfu omments. Te usua dsamer appes.

2 1. Introduton Envronmenta ssues su as mate ange pose four key aenges for eonom anayss: () te proess of mate ange s effetvey rreversbe; () tere are onsderabe unertantes about te key future osts of bot abatement, but more espeay envronmenta damages; () our understandng of tese unertantes anges over tme as a resut of earnng more about mate sene, possbe tenooga responses and beavoura responses by ouseods, frms and governments; (v) te probem s goba, but sne tere s no snge goba ageny to take mate ange, poes need to be negotated troug nternatona envronmenta agreements (IEAs). Te frst tree ssues ave been studed qute extensvey n te ontext of a snge goba government, espeay weter te possbty of future earnng n a probem wt unertanty and rreversbty eads to more or ess urrent abatement. 1 Te preautonary prnpe argues for more urrent abatement, but te teoreta and empra anayss s more ambguous. Tere as aso been an extensve terature on te fourt ssue, bot teoreta and empra, but many n te ontext of ertanty about te net benefts of takng mate ange. 2 Te onusons ave been rater pessmst, n te sense tat we tere are substanta benefts to a ountres oaboratng to take mate ange, reatve to ountres atng non-ooperatvey, f ountres dede ndependenty weter to jon an IEA, te reatve gans from su an agreement are sma. More reenty, tese two strands of terature ave begun to be ntegrated. Up and Up (1996), Up and Maddson (1997) ompare te fuy ooperatve and te nonooperatve senaros wen ountres fae unertanty about damage osts. Tey sow tat te vaue of earnng about damage osts may be negatve wen ountres at nonooperatvey and damage osts are orreated aross ountres. Na and Sn (1998), Up (2004), Kostad (2007), Kostad and Up (2008, 2009) ave onsdered ow te 1 2 See, e.g. Arrow and Fser (1974), Epsten (1980), Kostad (1996a,b), Goer, Juen and Tre (2000), Up and Up (1997) as we as Naran, Fser and Hanemann (2007). Cass papers are Barrett (1994), Carraro and Snsao (1993) we for nstane Fnus (2001, 2003) and Barrett (2003) provde surveys of te terature. 1

3 prospet of future resouton of unertanty affets te nentves for ountres to jon an IEA. Agan, te resuts ave been rater pessmst. Kostad and Up (2008) onsder a mode were ountres fae ommon unertanty about te eve of envronmenta damage osts. 3 Tey onsder tree senaros of earnng: wt fu earnng, unertanty about damage osts s resoved before ountres dede weter or not to jon an IEA; wt parta earnng, unertanty s resoved after ountres dede weter or not to jon an IEA, but before tey oose ter emssons eves; wt no earnng, unertanty s not resoved unt ountres ave deded weter or not to jon an IEA and set ter emsson eves. Kostad and Up (2008) sow tat te prospet of earnng, eter fu or parta, generay redues te expeted wefare n stabe IEAs. A tese modes ave assumed tat ountres are rsk neutra. However, n te mate ontext, rsks are gy orreated and ene possbtes for rsk-sarng are mted so tat te assumpton of rsk averson may be qute reevant. Endres and O (2003) sow n a two-payer prsoners demma tat rsk averson an nrease te prospets of ooperaton one t reaes a ertan tresod. Bramoué and Tre (2009) ntegrate rsk averson n a goba emsson mode tat ompares te non-ooperatve wt te fuy ooperatve souton. Tey sow tat equbrum emssons are ower wt unertanty and ts dfferene nrease wt te degree of rsk averson as part of a edgng strategy but te effet on goba wefare s ambguous. Bouer and Bramoué (2010) use te mode of Kostad and Up (2008) to anayze te effet of rsk averson on oaton formaton as we do n ts paper. In ontrast to us, tey use te expeted utty approa wereas we empoy te mean-standard devaton approa as expaned n more deta beow. Our approa s smper, w aows us to onsder aso te senaro of parta earnng tat appears to be partuary reevant n atua treatymakng. Moreover, ke n Endres and O (2003), we an apture a regme sft for suffenty g eves of rsk averson, addng te aspets of oaton formaton and varous senaros of earnng. 3 By ommon unertanty we mean tat ea ountry faes te same ex-ante dstrbuton of possbe damage osts, and wen unertanty s fuy resoved tey fae te same ex-post eve of damage osts,.e. te rsks tey fae are fuy orreated aross ountres. Kostad and Up (2009) extend ts mode to onsder te ase were te rsks ea ountry faes are unorreated. Unorreated unertanty s aso onsdered n a sgty dfferent mode n Fnus and Pntassgo (2009) and empray nvestgated n a mate mode wt tweve word regons n Denk et a. (2008). 2

4 As ponted out above, we apture rsk atttudes usng te mean-standard devaton (MS) deson rteron, ntrodued by Markowtz (1952) and Tobn (1958), w s a wdey used aternatve to te expeted utty (EU) rteron for deson-makng under unertanty, ntrodued by von Neumann and Morgenstern (1944). Tere ave been a arge number of studes omparng te reatve advantages of ea approa, askng under w ondtons are tey onsstent.e. tey produe te same resuts. In a pat-breakng arte Meyers (1987) sowed tat under a gven parameter restrton (LS oaton and sae) te two approaes are onsstent. Ts ods n partuar f te payoff funton s near on te unertan parameter, w s te ase n our mode (see equaton 1 beow). Saa (1997) argues tat, despte te onssteny of te two rtera, te MS s generay more fexbe n representng aternatve rsk preferenes and s smper n terms of empra appatons. One man advantage s tat t nudes expty te two moments of te payoffs, as for a arge ass of preferenes tey ontan a te reevant nformaton of a deson probem under unertanty. Aordng to Saa, ts expans wy te MS rteron s so wdey used, bot n teoreta and empra studes. Te paper proeeds as foows. In seton 2, we set out te teoreta mode. We frst summarse te benmark ase of ertanty and ten we move on to ntrodue te ase of unertanty and rsk averson. In seton 3, we present our resuts and seton 4 summarses our man onusons. 2. Te Mode 2.1 No Unertanty Tere are N denta ountres, ndexed 1,..., N. Ea ountry produes emssons x w we assume an take one of two vaues: x 1 (poute) or x 0 (abate). N Aggregate emssons are denoted by X 1 x. Aggregate emssons ause goba envronmenta damages. Te ost of envronmenta damages per unt of goba emssons s and te beneft per unt of ndvdua emssons s normazed to 1. (Tus, essentay measures te ost-beneft rato.) Te payoff to ountry s gven by 3

5 (x,x) x X. (1) In order to make ts mode nterestng, we requre tat te ndvdua beneft exeeds te ndvdua unt damage ost from pouton,.e. 1 (ene ountres poute n te Nas equbrum) but not te goba unt damage ost,.e. 1 N (ene ountres abate n te soa optmum), w togeter mpes: Assumpton 1: 1 1 N. In order to study IEA formaton, we sa use te two-stage mode of Barrett (1994), borrowed from te terature on arte formaton (d Aspremont et a. 1983), w s soved bakwards. In te seond stage, te emsson game, for any arbtrary number of IEA members n, 1 n N, te members of te IEA (w we denote by te symbo for oaton ountres) and te remanng ountres (w we denote by te symbo f for frnge ountres) set ter emsson eves as te outome of a Nas game between te oaton and te frnge ountres. Tat s, te oaton members togeter maxmze te aggregate payoff to ter oaton wereas frnge ountres maxmze ter own f payoff. As we assumed 1, x 1; oaton members w ose x 0 provded f 1 n, and so (n) 1 (N n) and (n) (N n) ; owever f 1 n, f ten oaton members w aso poute, x 1 and so (n) (n) 1 N. 4 Knowng te payoffs to oaton and frnge ountres for any arbtrary number of IEA members, we ten determne te equbrum n te frst stage, te membersp game, agan, as a Nas equbrum: no oaton ountry oud beome better off by defetng from te oaton, and no frnge ountry oud be made better off by jonng te oaton: 5 Interna stabty: C: (n) (n 1) (2) f Externa stabty: C: (n) (n 1). (3) f 4 5 It s now evdent wy we need Assumpton 1: t avods trva outomes were a ountres eter abate or poute no matter weter tey are oaton members or frnge ountres. Wtout oss of generaty, te strong nequaty oud be repaed by a weak nequaty sgn for externa stabty. Our assumpton avods knfe-edge ases were a frnge ountry s ndfferent between stayng outsde and jonng a oaton. 4

6 Consder a oaton wt n members tat abates beause 1 n. Now f one member eft te oaton and t st pad te remanng n 1 to abate,.e. 1 (n 1), nterna f stabty (n) (n 1) woud requre 0 (N n) 1 (N n 1), or equvaenty 1, w voates Assumpton 1. Hene, we requre tat t does not pay te remanng ountres to abate one a member eaves,.e. 1 (n 1), and ten nterna stabty requres 0 (N n) 1 N w mpes tat n 1 /. Tus, te nternay stabe oaton n * s te smaest nteger I( ) no ess tan 1 / w, as an be easy eked, s aso externay stabe and ene stabe. It s stragtforward to see tat I(γ) s a non-nreasng funton of γ. 6 Sne ex-ante a ountres are denta, tere s no expt proess for determnng w ountres get seeted as IEA members and w as frnge ountres. We sa assume, foowng Kostad and Up (2008) and Rubo and Up (2007) tat tere s random proess for determnng w ountres beome IEA members. Tus, we * defne te average or expeted payoff per ountry by n / N * f (N n )/ N w s a strty dereasng funton of. 7 Tus, ts smpe mode provdes a reatonsp between te unt damage ost and te equbrum number of oaton members. Te equbrum s a knfe-edge equbrum wt * n ountres formng te oaton, w de fato dssoves one a member eaves te oaton as no ountry woud abate anymore. Te equbrum oaton sze weaky dereases n te ost-beneft rato from emssons te arger te ess ountres are needed to make ooperaton proftabe. 6 7 Kostad and Up (2008) use te approxmaton I( ) 1 / w gnores te nteger nature of I( ). Ten I( ) an be onsdered a strty dereasng and onvex funton of γ. However, as Karp (2009) ponted out, te orgna funton I (γ) s neter onvex nor onave. f Usng and from above, notng n * I( ), ten 1 N I 1/ N. Consder an nfntesma varaton 0, su tat I does not ange. Ten, N I 0, I N. However, f 0 mpes I 0, ten N I I 1/ N 0, w ompetes te proof. Usng te approxmaton n footnote 6 (gnorng te nteger nature of I( )), ten s a strty dereasng onave funton of γ. See Kostad and Up (2008). 5

7 2.2 Unertanty Now assume tat te unt damage ost of goba emssons s unertan and equa for a ountres, bot ex-ante and ex-post. We denote te vaue by s n te state of te word s and ene (1) beomes: (x,x) x X. (4),s s For smpty, we assume tat s an take one of two vaues: ow damage osts,, wt probabty p, and g damage osts, wt probabty ( 1 p) were and 0 p 1. We denote by p ( 1 p) te expeted vaue of unt damage osts, and by ( ) ( ) p( 1 p) te standard devaton of unt damage osts. s To assess ow ountres evauate payoffs aross states of te word, we assume tat ea ountry s atttude to rsk an be represented by a mean-standard devaton (MS) utty funton, w s te same for a ountres: V(x,X) E (x,x) (x,x) (5) were 0 s te oeffent of rsk averson 8 ;.e. te utty to a ountry s te expeted payoff mnus te standard devaton of payoffs wegted by te fator. 0 orresponds to rsk neutraty. For ater purposes, t w be usefu to defne: ˆ ( ) (6) s as te rsk-adjusted expeted unt damage ost. We ex-ante ountres fae unertanty about te true vaue of unt damage osts, we want to aow for te possbty tat ountres may earn nformaton durng te ourse of te game w anges te rsk tey fae. We sa foow Kostad and Up (2008) n onsderng tree very smpe senaros of earnng. Wt No Learnng (NL) ountres make ter desons about membersp and emssons wt unertanty about te true vaue of unt damage osts. Wt Fu Learnng (FL) ountres earn te true vaue of 8 We oud onsder as n Endres and O (2003) tat an be negatve f payers are rsk-ovng. However, we dsard ts possbty n order to keep te dsusson as bref as possbe: a resuts for rsk-averson are just reversed for rsk-ovng. 6

8 unt damage osts before tey ave to take ter desons on membersp and emssons. Wt Parta Learnng () ountres earn te true vaue of damage osts after tey ave made ter membersp desons but before tey make ter emsson desons. Tus, n ts smpe anayss, earnng takes te form of reveang perfet nformaton. As n te mode wtout unertanty, we ave to ntrodue some parameter restrtons. Moreover, te equbrum sze of te oaton an be reated to unt damage osts. We defne n I( ), n I( ), n I( ), ˆn I( ˆ ) and nn N. It w turn out tat stabe IEAs w take one of tese vaues. For sensbe resuts, we make te foowng assumpton. Assumpton 2: () 1/N 1 () n n n nn. Assumpton 2() s essentay Assumpton 1 n te ontext of unertanty. Assumpton 2() basay states tat tere are dfferenes (at east 1) between te szes of te stabe IEAs under unertanty. For te teoreta anayss, t s epfu to onsder two parameter onsteatons, w we sa a ase 1 and 2. Cases: () Case 1: ˆ 1 0 ( 1 )/ ( s ) and () Case 2: ˆ 1 ( 1 )/ ( s ) Case 1 s essentay Assumpton 1 n te ontext of unertanty; abatement never pays for a snge ountry. However, now we want to aow wt ase 2 for te possbty tat f payers are suffenty rsk averse, ten even a snge payer may dede to abate. Note tat ase 1 nudes te ase of rsk neutraty ( 0 ). It w turn out tat gong from ase 1 to ase 2 eads to a regme sft. From a statsta pont of vew, t s ear tat as ong as te degree of rsk averson s suffenty ow, te restrton ˆ 1 s not reay bndng. Ony for arger degrees of rsk averson w te keood of ˆ 1 beome sgnfant and for arge enoug eves of te keood of ˆ 1 may even exeed tat of ˆ 1. 7

9 Note tat now wt unertanty te expeted utty per ountry from an ex-ante V n/ N V (N n)/ N V were n may take on one of te perspetve s f vaues mentoned above (.e. n, n, n, ˆn or n N ). 3. Resuts 3.1 Anayta Resuts In ts sub-seton, we generaze te resuts of Kostad and Up (2008) wo assume rsk neutraty ( 0 ). 9 In terms of te rsk averson parameter, we dstngus between ase 1 and ase 2 as speed out n sub-seton 2.2. We start wt te rater unnterestng ase of Fu Learnng (FL). Payers know te reazaton of te damage parameter at te outset of te oaton formaton game. Tus, resuts foow drety from wat we know from a game wt ertanty and ene rsk averson does not pay a roe. However, t as to be ponted out tat even wt FL an evauaton as to take an ex-ante perspetve n order to aow for a sensbe omparson of oaton szes and utty eves, aross dfferent senaros of earnng, for a gven eve of rsk averson. Proposton 1: Fu Learnng If state s, as been reveaed at te outset, ten n te emsson game, frnge members aways poute, and oaton members abate f n ns I( s ). In te membersp game, te stabe IEA as n s members; te utty to a oaton member s,fl s s f,fl s s V (s) (N n ) and to a frnge ountry t s V (s) 1 (N n ). Tus, n state s, te expeted utty per ountry s V,FL(s) (N n s )(1/ N s ). From an exante perspetve, te expeted sze of an IEA s nfl pn (1 p)n and te expeted utty per ountry s: V,FL pv,fl(s ) (1 p)v,fl(s ) (1 N ) pn ( 1/ N ) (1 p)n ( 1/ N ). Hene, rsk averson neter affets te oaton sze nor expeted utty. Wt FL, te outome n ea state s n terms of te sze of a stabe IEA, te utty to oaton and frnge ountres and ene te expeted utty per ountry s obvousy 9 Beause te proofs of Propostons 3 and 4 are very ong, a proofs are avaabe as a tena appendx from te autors upon request. 8

10 just te same as n an IEA game were te eve of damage osts s s known wt ertanty. Tus, by takng expetatons aross te two states of te word, we obtan te expeted sze of an IEA and te expeted utty per ountry. Proposton 2: No Learnng Case 1: ˆ 1 ( 0 (1 )/ ( s ) ) In te emsson game, frnge ountres aways poute, we oaton members abate f n nˆ and poute oterwse. In te membersp game, te unque stabe IEA as n nˆ I( ˆ ) members abatng and (N n) ˆ frnge ountres poutng, wt ˆn n. NL Te expeted utty to a oaton member s frnge ountry t s,nl ˆ ˆ V N n 1/ N. f ˆ,NL ˆ ˆ,NL ˆ V (N n) and to a V 1 (N n) and ene expeted utty per ountry s Te sze of te stabe oaton n NL s weaky dereasng and te expeted utty per ountry s strty dereasng n te rsk averson parameter. Case 2: ˆ 1 ( (1 )/ ( s ) ) In te emsson game, a payers abate and ene n NL {1,..., N } and V,NL 0 for a eves of rsk averson above tresod. Wt respet to ase 1, for 0, te resut dspays ertanty equvaene n te sense tat te outome s te same as woud be obtaned f ountres faed unt damage osts wt ertanty and no rsk. Any mean-preservng nrease n rsk reated to te unertan damage ost parameter as neter an effet on te expeted oaton sze nor on expeted wefare under rsk neutraty. Ts s dfferent wt rsk averson..e. 0 (toug ): pereved unt damage ost nrease wt te degree of rsk averson and sne te sze of te stabe oaton dereases n unt damage osts, rsk averson eads to smaer equbrum oatons. Note for No Learnng (NL), oaton sze and expeted oaton sze are te same. 9

11 Aso te mpat of rsk averson on expeted per ountry utty s negatve due to two effets. 10 Frst for a gven oaton, by assumpton, expeted utty of oaton members and frnge ountres dereases n rsk averson, w s te dret effet. Seond, expeted utty per ountry dereases n te oaton sze and as just ponted out rsk averson owers te equbrum oaton sze, w may be vewed as an ndret effet. Wt respet to ase 2, f te degree of rsk averson reaes a tresod su tat rsk-adjusted unt damage ost parameter exeeds 1, ˆ 1, even for frnge payers abatement pays. Ts may be nterpreted as a oatons beng stabe or tat tere s no need for ooperaton as even n te non-ooperatve equbrum a payers abate. Taken togeter, rsk-averson as a negatve mpat on oaton sze and expeted per ountry utty, but one a tresod s reaed, ts eads to a regme sft wt fu ooperaton. Proposton 3: Parta Learnng Tere exsts a stabe oaton. Case 1: ˆ 1 ( 0 (1 )/ ( s ) ) Tere are two possbe stabe IEAs: 1 2 n n n n. () Tere s a frst stabe IEA of sze 1 n n were oaton members poute n te ow damage ost state and abate n te g damage ost state and frnge ountres poute n bot states. Te expeted utty to oaton and frnge ountres s gven by: V (n ) ˆN p (1 p) n [1 n ] p(1 p) and f 2 V (n ) 1 N (1 p) n p(1 p) ( )N n. 10 Note tat strty speakng utty annot be ompared for dfferent degrees of rsk averson. Terefore, ts resut w ony be used ater wen omparng utty aross dfferent senaros of earnng for te same degree of rsk averson. 10

12 () Tere s a seond stabe IEA of sze n 2 n n w oaton members abate n bot states and frnge ountres poute n bot states of te word. Te expeted utty to oaton and non-oaton ountres s gven by: V ˆ (N n ) and V 1 ˆ (N n). f () If te seeted IEA as n members, ten te expeted payoff per ountry s gven by: n (N n ) f V, V (n ) V, (n ) N N (v) If 0, 1 n n s aways an equbrum wereas n n s an equbrum f 1 2 and ony f p p wt p and n 1/. If equbrum n n 1 1 exsts, t Pareto-domnates n n. (v) If 0, te keood of equbrum n 1 n dereases, te keood of equbrum n 2 n nreases and ene te expeted oaton sze n nreases wt te degree of rsk averson. Te expeted per ountry utty n ea equbrum dereases wt te degree of rsk averson. Case 2: ˆ 1 ( (1 )/ ( s ) ) 2 Tere are two possbe stabe IEAs, 1 2 N n n n n. Tere may be a frst equbrum n 1 n wt te features as desrbed under () above. Tere s a seond equbrum, te grand oaton, n 2 nn N, w s aways an equbrum: a ountres abate and ene te expeted utty per ountry s V 0. Te seond equbrum Pareto-domnates te frst equbrum and ene te expeted oaton sze s n N. As t appears from te resuts above, Parta Learnng () s te most ompated of te tree earnng senaros. In te seond stage, te emsson game, te vaue of te damage parameter s reveaed to te payers. Hene, te emsson game s ke te game under FL. Importanty, and very dfferent from NL, ts mpes tat frnge payers or snge payers never abate, regardess of te degree of rsk averson. Tus, te grand oaton n ase 2 omes ony about due to te membersp game were desons dfferent from FL ave to be based on expeted utty. Sne te grand 11

13 oaton aways Pareto-domnates te smaer equbrum oaton, t seems obvous to expet fu ooperaton as te outome n ase 2. In ase 1, te prospets for ooperaton are ess brgt. Tere are agan two equbra were te seond and arger equbrum typay Pareto-domnates te frst equbrum, toug te seond equbrum does not aways exsts. In partuar for no rsk averson, te probabty of te arger seond equbrum s very ow. Ts s evdent from te defnton of te tresod p and, reang tat n s te nteger vaue of 1 / and ene tends to be very sma. Consequenty, p s ose to 1. Hene, ony f te probabty of te ow damage ost p s ose to 1 w te arger seond equbrum, n 2 n, materaze. Ts woud normay be regarded as an unnterestng parameter onsteaton. However, one we depart from te benmark 0, wt nreasng, te keood of n 2 n nreases even for vaues of p sgnfanty ower tan 1. Fnay, part (v) of Proposton 3 suggests n ase 1 tat ke NL, for expeted per ountry utty dereases wt rsk-averson n ea equbrum, n and n. 11 Ts s te dret negatve effet w rsk as on expeted per ountry utty. Sne rsk nreases te keood of te arge seond equbrum n 2 1 n, wt typay ger expeted per ountry utty tan te smaer equbrum n n, te overa mpat of rsk on expeted utty annot be teoretay dedued. Te smuaton runs on w we report n te next sub-seton sow tat te overa mpat s negatve. We now ompare te outomes n terms of expeted sze of te IEA and expeted utty per ountry aross te tree modes of earnng. In order to benmark ts wt prevous resuts, we onsder under ase 1 te spea ase of 0. Proposton 4: Comparson of Outomes under te Tree Senaros of Learnng Case 1: ˆ 1 ( 0 (1 )/ ( s ) ) a) 0 () 1 2 NL n n n n n n 11 Te same quafaton as mentoned n footnote 9 appes. 12

14 () 1 2 FL n n n n n () nfl nnl 1 (v) V,(n ) V,NL(n ) V,(n ) (v) V,(n ) V,FL V,(n ) b) 0 () Tere s a tresod of above w nfl nnl nˆ. () 1 2 n n n n aways ods. Moreover, tere s a tresod of above w mn 1 2 and ene, 1 NL ˆ n n n n n n,n n. NL () Tere s a tresod of above w n nfl. (v) Tere s a tresod of above w V ˆ,FL V,NL (n). (v) V,FL V,(n ) aways ods. Moreover, tere s a tresod of above w V,FL V,(n ) and ene * 1 2,,, V (n ) max V (n ),V (n ). (v) V,(n ˆ ) V,NL(n). *, FL, V V (n ) were Case 2: ˆ 1 ( (1 )/ ( s ) ) () n nnl N nfl () V,(N ) V,NL V,FL In a frst step, we fous on te man resuts for rsk-neutraty (Case 1, a)). Te ow membersp equbrum for (w s aways an equbrum; see Proposton 3) as ower membersp and expeted utty tan eter NL or FL, and te opposte ods for te g membersp equbrum for. However, we may rea tat te g membersp equbrum for s ony an equbrum f p p w s very unkey. Karp (2012) as sown tat tere s no genera rankng of te outomes for NL and FL tat appes to any possbe set of parameter vaues, toug t s possbe to exude te outome were FL as bot smaer expeted membersp and ger expeted utty tan NL. However, for any parameter vaues for w te approxmaton n footnote 6 13

15 ods, t woud be te ase tat FL resuted n at east as great expeted membersp and at east as ow expeted utty as wt NL (see Kostad and Up 2008). In summary, wt rsk neutraty, f we vew te parameter vaues p n 2 p generatng n as beng rater unnterestng (.e. we are nterested n ases were te rsk of g damage osts s qute sgnfant), ten we onude tat yeds a ower expeted oaton sze and utty tan eter NL or FL, we for te approxmaton n footnote 6 FL aso yeds ower expeted utty tan NL. In oter words, n terms of membersp, we typay expet te rankng FL>NL> and n terms of expeted per ountry utty te rankng NL>FL>. Sne te sze of an agreement as su s of ess mportane tan te wefare mpaton, te major onuson from prevous work s tat earnng an be bad n a strateg ontext of oaton formaton. In a seond step, we onsder wat anges wt rsk-averson, assumng tat te rskadjusted damage ost parameter s beow 1 and ene no regme sft ours (Case 1, b) n Proposton 4). Consder frst membersp. From Propostons 1 and 2 we know tat te sze of te oaton s not affeted by rsk averson under FL and dereases n under NL. From Proposton 3 we know tat te oaton szes n te two possbe equbra are unaffeted by rsk averson under, toug te keood of te arger equbrum nreases and te keood of te smaer equbrum dereases and ene te expeted oaton sze nreases wt rsk averson. Consequenty, t foows mmedatey, tat above a tresod of rsk averson, te rankng FL>NL s aways true (Case 1, b), resut ()), >NL wt respet to te sma -equbrum n 1 n and ene wt respet to any -equbra (Case 1, b), resut ()) and >FL ods n terms of expeted membersp (Case 1, b), resut ()). Tus, nreasng eves of rsk averson eary anges te rankng n terms of membersp, gvng and FL an advantage over NL. In terms of expeted utty, we know rsk averson as not mpat for FL (Proposton 1), but a negatve mpat for NL (Proposton 2) and on ea of te two equbra for (Proposton 3). Consequenty, above a tresod of rsk averson, ts eads to te rankng FL>NL (Case 1, b), resut (v)), FL> n terms of te arger -equbrum and ene aso f we pk te Pareto-superor -equbrum n ase tere are two (Case 14

16 1, b), resuts (v)). Tat, s FL mproves ompared to NL wt nreasng rsk averson ontnuousy, and s ertany superor to above a ertan tresod. Ony te reaton between and NL s ess stragtforward beause bot are negatvey nfuened by rsk averson. Aordng to resut Case 1, b), (v), te rankng estabsed for rsk neutraty for te arge -equbrum, n 2 n, namey, >NL, remans vad but for te sma -equbrum, n 1 n, NL> may or may not be true. Hene, overa onusons are dffut, toug our smuatons n te next sub-seton sow tat wt nreasng rsk averson >NL eventuay. Hene, te man message to take away ere s tat te more payers are rsk averse, te more tey earn te better t s. Puttng t smpy, earnng s good f payers are rsk-averse. Fnay, n a trd step, we address te possbty of a regme sft (Case 2). Above a tresod of w eads to ˆ 1, NL and aeve te soay optma outome wereas ts s not te ase under FL. Tus, te man onuson from step 2 above, namey tat earnng s good for expeted utty te more rsk averse payers are, s now reversed one a tresod of rsk-averson as been passed. In oter words, earnng an be bad n a strateg ontext of oaton formaton above a tresod of rsk averson, very mu ke te resut obtaned for rsk neutraty. 3.2 Smuaton Resuts In ts seton, we present some smuaton resuts. Te Appendx desrbes ow we onduted te Monte Caro smuatons. In a frst step, we ook at tose few ssues for w we oud not estabs anayta resuts. In a seond step, we present some resuts w ompaty summarze te overa onuson of our paper, reatng t to te queston about te keood of ase 1 and 2. Step 1 Proposton 3 argued tat, n ase 1, rsk averson as a negatve mpat on expeted per ountry utty n ea of te two oaton equbra under, but tat te keood of te arger equbrum wt ger expeted utty nreases wt rsk averson. Hene, t was not ear ow expeted utty anges wt rsk averson, assumng tat n ase 15

17 of mutpe equbra, te Pareto-superor s osen. Tabe 1, oumn, sows tat te overa effet of nreasng rsk averson s negatve. Proposton 4 argued tat, n ase 1, t s not obvous ow te reatve rankng n terms of expeted utty between and NL s affeted by nreasng rsk averson. Tabe 1, oumn and NL onfrms tat for rsk neutraty NL> on average. However, wt nreasng rsk averson, ts rankng s reversed. Step 2 Te anayta resuts dstngused between ase 1 and 2. Now we ask te queston ow key tese ases are and wat ts means for overa onuson. Tabe 2 ustrates tat te keood of ase 1 onstanty dereases wt te degree of rsk averson and te opposte s true for ase 2. Hene, under, te reatve oaton sze onstanty nreases wt rsk averson (Tabe 3). Frst beause n ase 1 te keood of te arge equbrum nreases and te keood of te sma equbrum dereases. Seond beause te keood of ase 2 nreases ompared to ase 1, wt te grand oaton beng te outome n ase 2 wt a regme sft. Under NL, te reatve oaton sze frst dereases and ten nreases (Tabe 3). Wt nreasng rsk averson, Proposton 2 sowed tat te oaton sze dereases n ase 1. However, wt nreasng rsk averson aso te keood of ase 2 nreases wt te outome tat te grand oaton forms wen tere s a regme sft. In terms of expeted utty, a smar pattern an be observed (Tabe 3). It dereases under and NL frst beause rsk as a negatve mpat on expeted utty but ts s reversed for very g eves of rsk averson beause te proporton of ases n w te grand oaton forms approaes fnay 100%. 4. Summary and Conusons Kostad and Up (2008) sowed tat wt rsk neutraty te possbty of earnng more nformaton about envronmenta damage osts n te future generay ad rater pessmst mpatons for te formaton of IEAs. Exept for a reatvey sma set of parameter vaues for w parta earnng woud seet a g IEA membersp, earnng resuted n ower expeted membersp for parta earnng and ower expeted 16

18 utty for bot fu and parta earnng. Hene, n a strateg ontext, earnng redues expeted utty for a wde range of parameter vaues. Ts resut was quafed f we take rsk averson nto aount. As rsk averson nreases, fu and parta earnng mprove ter rankng ompared to no earnng n terms of te sze of stabe oatons and n terms of expeted utty. However, above a tresod of rsk averson, tere s a regme sft, w eads to fu ooperaton under parta and no earnng. In terms of te roe of earnng, te onusons from prevous papers (e.g. Kostad 2007, Kostad and Up 2008 and Na and Sn 1998) tat earnng s usuay bad n a strateg ontext of envronmenta treaty formaton s reversed for rsk averson beow some tresod, but onfrmed above ts tresod. In terms of te roe of regme sft, te onuson from Endres and O (2003) tat n a word wtout fu nformaton f rsk averson passes some tresod, ts an ave a postve mpat on te suess of envronmenta treates s onfrmed. We an sow tat ts does not ony od for no but aso parta earnng and extends to an N-ountry prsoners demma wt oaton formaton. However, dfferent from tem, n our mode beow te tresod, nreasng eves of rsk averson makes tngs worse. Ts s ertany an mportant aspet; t suggest tat te mate ange probem annot smpy be soved by makng governments more aware tat tey soud be more rsk averse n order to avod dsasterous mate ange. As ong as governments are not extremey rsk averse, nvestment n ganng more nformaton appears to be a more suessfu strategy. 17

19 Referenes Arrow, K. and A. Fser (1974), Envronmenta preservaton, unertanty and rreversbty. Quartery Journa of Eonoms, 88: d Aspremont, C., A. Jaquemn, J.J. Gabszewz and J.A. Weymark (1983), On te stabty of ousve pre eadersp. Canadan Journa of Eonoms, 16: Barrett, S. (1994), Sef-enforng nternatona envronmenta agreements. Oxford Eonom Papers, 46: Barrett, S. (2003), Envronment and Stateraft: Te Strategy of Envronmenta Treatymakng. Oxford Unversty Press, New York. Bouer, V. and Y. Bramoué (2010), Provdng goba pub goods under unertanty. Journa of Pub Eonoms, 94: Bramoué, Y. and N. Tre (2009), Can unertanty aevate te ommons probem? Journa of te European Eonom Assoaton, 7(5): Carraro, C. and D. Snsao (1993), Strateges for te nternatona proteton of te envronment. Journa of Pub Eonoms, 52: Denk, R., M. Fnus and N. Oeman (2008), Te stabty keood of an nternatona mate agreement. Envronmenta and Resoure Eonoms, 39: Endres, A. and C. O (2003), Internatona envronmenta ooperaton wt rsk averson. Internatona Journa of Sustanabe Deveopment, 6: Epsten, L. (1980), Deson-makng and te tempora resouton of unertanty. Internatona Eonom Revew, 21: Fnus, M. (2001), Game Teory and Internatona Envronmenta Cooperaton. Edward Egar, Cetenam, UK. Fnus, M. (2003), Stabty and desgn of nternatona envronmenta agreements: te ase of goba and transboundary pouton. In: Fomer, H. and T. Tetenberg (eds.), Internatona Yearbook of Envronmenta and Resoure Eonoms 2003/4. Edward Egar, Cetenam, UK,. 3, Fnus, M. and P. Pntassgo (2009), Te roe of unertanty and earnng for te suess of nternatona mate agreements. Strng Dvson of Eonoms Dsusson Paper, , Unversty of Strng, UK. 18

20 Goer, C., B. Juen and N. Tre (2000), Sentf progress and rreversbty: an eonom nterpretaton of te Preautonary Prnpe. Journa of Pub Eonoms, 75: Karp, L. (2012), Te effet of earnng on membersp and wefare n an nternatona envronmenta agreement. Cmat Cange, 110: Kostad, C. (1996a), Fundamenta rreversbtes n stok externates. Journa of Pub Eonoms, 60: Kostad, C. (1996b), Learnng and stok effets n envronmenta reguatons: te ase of greenouse gas emssons. Journa of Envronmenta Eonoms and Management, 31: Kostad, C. (2007), Systemat unertanty n sef-enforng nternatona envronmenta agreements. Journa of Envronmenta Eonoms and Management, 53: Kostad, C. and A. Up (2008), Learnng and nternatona envronmenta agreements. Cmat Cange, 89: Kostad, C. and A. Up (2009), Unertanty, earnng and eterogenety n nternatona envronmenta agreements. Mmeo. Markowtz, H. (1952), Portfoo seeton. Journa of Fnane, 7: Meyers, J. (1987), Two-moment deson modes and expeted utty maxmzaton. Ameran Eonom Revew, 77: Na, S.-L. and H.S. Sn (1998), Internatona envronmenta agreements under unertanty. Oxford Eonom Papers, 50: Naran, U., A. Fser and M. Hanemann (2007), Te rreversbty effet n envronmenta desonmakng. Envronmenta and Resoure Eonoms, 38: von Neumann, J. and O. Morgenstern (1944), Teory of Games and Eonom Beavor, Prneton Unversty Press. Rubo, S. and A. Up (2006), Sef-enforng nternatona envronmenta agreements revsted. Oxford Eonom Papers, 58: Saa, A. (1997), Rsk Preferene Estmaton n te Nonnear Mean Standard Devaton Approa. Eonom Inqury, 35: Tobn, J. (1958), Lqudty preferene as beavor towards rsk. Revew of Eonom Studes, 67:

21 Up, A. (2004), Stabe nternatona envronmenta agreements wt a stok poutant, unertanty and earnng. Journa of Rsk and Unertanty, 29: Up, A. and D. Maddson (1997), Unertanty, earnng and nternatona envronmenta poy oordnaton. Envronmenta and Resoure Eonoms, 9: Up, A. and D. Up (1996), Wo gans from earnng about goba warmng? In: van Ierand, E. and K. Gorka (eds.), Te Eonoms of Atmosper Pouton. Sprnger, Hedeberg,. 3, Up, A. and D. Up (1997), Goba warmng, rreversbty and earnng. Eononom Journa, 107:

22 Appendx: Desrpton of te Monte Caro Smuatons ) Te tota number of smuatons s osen; 100,000 n a reported resuts. ) Te eve of rsk averson s osen, {0,0.5,1,2,5,10,100,1000,10000 }, ) toug not a vaues are reported ere. For ea smuaton, te number of payers N s generated from a dsrete unform dstrbuton over te set {4,, 200}. v) Te nverse of te expeted damage ost, 1/, s generated from a ontnuous unform dstrbuton n te range 3,N 1. v) Te nverse of te g damage ost, 1/, s generated from a ontnuous unform dstrbuton n te range 2,1/ 1. Ts ensures tat 2 n n 1. v) Te nverse of te ow damage ost, 1/, s generated from a ontnuous unform dstrbuton n te range 1/ 1,N. Ts ensures tat n 1 n N. We obtan te probabty of ow damages from sovng p v) and ene get: ˆ p 1 p. v) In ase 1, wt restrton of ˆ, f ˆ / x). 1 1 s, ten go bak to step (), oterwse go to te next step. Wtout restrton of ˆ, go to te next step. Te oaton szes and expeted utty foow from te equatons presented n seton 3. In te ase of mutpe equbra under parta earnng te Paretosuperor equbrum s osen. A numbers ave been rounded to te trd dgt. Sma dfferenes aross entres n dfferent tabes were same numbers soud be expeted are due to dfferent smuaton runs. 21

23 Tabes Tabe 1: Expeted Utty n te Tree Learnng Senaros n Case 1 FL NL Tabe 2: Probabty of Rsk-adjusted Damage Parameter to be Larger or Smaer tan 1 ˆ 1 ˆ Tabe 3: Expeted Reatve Coaton Sze and Utty n te Tree Learnng Senaros wtout Restrton on ˆ * nfl n n NL FL NL N N N In ase of two equbra under, te Pareto-superor s osen. 22