Uncertainty, Learning and International Environmental Agreements The Role of Risk Aversion

Transcription

1 Unertainty Learning and Internationa Environmenta Agreements Te Roe of Risk Aversion Aistair Up (University of Manester) Pedro Pintassigo (University of Agarve) and Miae Finus (University of Bat) Abstrat Tis paper anayses te formation of internationa environmenta agreements (IEAs) under unertainty earning and risk aversion. It bridges two strands of te IEA iterature: (i) te roe of earning wen ountries are risk neutra; (ii) te roe of risk aversion under no earning. Combining earning and risk aversion seems appropriate as te unertainties surrounding many internationa environmenta probems are arge often igy orreated (e.g. imate ange) but are graduay redued over time troug earning. Te paper anayses tree senarios of earning. A key finding is tat risk aversion an ange te ranking of tese tree senarios of earning in terms of wefare and membersip. In partiuar te negative onusion about te roe of earning in a strategi ontext under risk neutraity is quaified. Wen ountries are signifianty risk averse ten it pays tem to wait unti unertainties ave been argey resoved before joining an IEA. Tis may suggest wy it as been so diffiut to rea an effetive imate ange agreement. JEL-Cassifiation: C72 D62 D8 Q54 Keywords: internationa environmenta agreements unertainty earning and risk aversion game teory We are gratefu to Larry Karp for epfu omments on an earier version of tis paper. Te usua disaimer appies.

2 1. Introdution Environmenta issues su as imate ange pose tree key aenges for eonomi anaysis: (i) tere are onsiderabe unertainties about te ikey future osts of environmenta damages and abatement; (ii) our understanding of tese unertainties anges over time as a resut of earning more about imate siene possibe tenoogia responses and beaviora responses by ouseods firms and governments; (iii) te probem is goba but sine tere is no singe goba ageny to take imate ange poiies need to be negotiated troug internationa environmenta agreements (IEAs). 12 Reenty tese tree issues ave begun to be integrated in one framework. Two stands of iterature an be distinguised. Te first strand of iterature studies unertainty and IEA formation wit te fous on te roe of earning but under te assumption of risk neutraity. Up and Up (1996) and Up and Maddison (1997) ompare te fuy ooperative and te non-ooperative senarios wen ountries fae unertainty about damage osts. Tey sow tat te vaue of earning about damage osts may be negative wen ountries at non-ooperativey and damage osts are orreated aross ountries. Na and Sin (1998) Up (24) Kostad (27) Kostad and Up (28 211) ave onsidered ow te prospet of future resoution of unertainty affets te inentives for ountries to join an IEA. Kostad and Up onsider a mode were ountries fae ommon unertainty about te eve of environmenta damage osts. 3 Tree senarios of earning are onsidered: wit fu earning unertainty about damage osts is resoved before ountries deide weter to join an IEA; wit partia earning unertainty is resoved after ountries deide weter to join an IEA but before tey oose teir emissions eves; wit no earning unertainty is neiter resoved before stage 1 nor stage 2. Tey sowed tat te prospet of earning eiter fu or partia generay redues te expeted tota 1 On te first two issues see for instane Arrow and Fiser (1974) Epstein (198) Kostad (1996ab) Up and Up (1997) Goier Juien and Trei (2) as we as Narain Fiser and Hanemann (27). 2 On te tird issue see for instane te assi papers by Carraro and Sinisao (1993) and Barrett (1994). Te most infuentia papers ave been oeted in a voume by Finus and Caparros (215) wo aso provide a survey. 3 By ommon unertainty we mean tat ea ountry faes te same ex-ante distribution of possibe damage osts and wen unertainty is fuy resoved tey fae te same ex-post eve of damage osts i.e. te risks tey fae are fuy orreated aross ountries. Kostad and Up (211) extend tis mode to onsider te ase were te risks ea ountry faes are unorreated. Unorreated unertainty is aso onsidered in a sigty different mode in Finus and Pintassigo (213) and empiriay investigated in a imate mode wit tweve word regions in Deink and Finus (212). 1

3 payoff in stabe IEAs. In partiuar Kostad and Up (28 211) sowed tat partia earning woud yied te igest tota payoff for ony a sma proportion of parameter vaues. For a signifiant majority of parameter vaues te igest expeted tota payoff arose under no earning. Hene it is better to form an IEA before waiting for better information: removing te vei of unertainty seems to be detrimenta to te suess of internationa environmenta ooperation. A tese modes ave assumed tat ountries are risk neutra. However in te imate ontext risks are igy orreated and ene possibiities for risk saring are imited so tat te assumption of risk aversion may be quite reevant. Terefore we extend te two-stage oaition formation setting by Kostad and Up (28) by departing from te assumption of risk neutraity. In tis paper we aow for ountries to be risk averse and sow tat if ountries ave a reativey ig degree of risk aversion ten for a majority of parameter vaues fu earning yieds iger expeted tota utiity tan no earning. Tis may ep to expain wy it as taken su a ong time between te start of te proess of taking imate ange (te Kyoto Protoo) to rea a more substantia agreement in Paris ountries are risk averse and so needed to wait unti tey ad more information about te risk of imate ange before ommitting to signifiant ation to take imate ange. Te seond strand of iterature studies unertainty and IEA formation wit te fous on te roe of risk aversion toug under te assumption of no earning. Endres and O (23) sow in a simpe two-payer prisoners diemma using te mean-standard deviation approa to apture risk aversion tat risk aversion an inrease te prospets of ooperation one it reaes a ertain tresod. Te reason is tat te benefits of mutua ooperation inrease reative to te payoffs of uniatera ooperation and no ooperation beause ooperation redues te variane of payoffs. Te more risk averse payers are te more attrative ooperation beomes ompared to free-riding. In teir mode tere is a first tresod above wi te prisoners diemma turns into a iken game and a seond tresod above wi te game turns into an assurane game. Compared to teir paper we aow for an arbitrary number of payers mode ooperation as a two-stage oaition formation game and onsider expiity te roe of earning. Bramoué and Trei (29) onsider risk-averse payers in a goba emission mode in wi a payers beave non-ooperativey as singetons. Tey sow tat equiibrium emissions are ower under unertainty tan under ertainty as part of a edging strategy but 2

4 te effet on goba wefare is ambiguous. Te autors aso find tat emissions derease wit te eve of risk aversion. Unike our paper Bramoué and Trei are not onerned wit earning and oaition formation. Bouer and Bramoué (21) onsider te effets of risk aversion on oaition formation but ony wit no earning. Tey anayze te formation of an internationa environmenta treaty using a simiar oaition game and payoff funtion as adopted in tis paper. Using an expeted utiity approa teir anaysis fouses on te effet of unertainty and risk aversion on signatories efforts te partiipation eve in an agreement and tota expeted utiity. Tey sow tat if additiona abatement redues te variane of ountries payoffs ten under risk aversion an inrease in unertainty tends to inrease abatement eves and may derease equiibrium IEA membersip wie te reverse is true if additiona abatement inreases te variane of ountries payoffs. 4 In tis paper our mode of no earning satisfies te first ondition but we extend te anaysis of Bouer and Bramoue (21) by onsidering aso partia earning and fu earning. Tus taken togeter in our paper we generaize te anaysis of Kostad and Up (28) by aowing for risk aversion and te anaysis of Bouer and Bramoue (21) and Endres and O (23) by onsidering te roe of earning. Te key findings are tat as ountries beome more risk averse it is no onger te ase tat for most parameter vaues te senario of No Learning yieds te igest expeted aggregate utiity but inreasingy it is te senario Fu Learning. Moreover te set of parameter vaues for wi te senario Partia Learning yieds te igest expeted aggregate utiity wi is a sma subset of su vaues wen ountries are risk neutra beomes even smaer as ountries beome more risk averse. Tus we quaify te negative onusion about te roe of earning in a strategi ontext if payers are suffiienty risk averse. In our mode emissions ast for just one period wi may seem restritive in te ontext of dynami environmenta probems su as imate ange. However it as been sown in te iterature on IEA formation under unertainty tat one period modes produe simiar resuts to muti-period modes. For instane Kostad and Up (28) using a one-period mode and 4 Hong and Karp (213) sow tat it does not matter weter one anayses te provision of a pubi good or te ameioration of a pubi bad. Wat matters is weter payers ations inrease or derease te voatiity of payoffs. In our mode as in Endres and O (23) and te emission game in Bouer and Bramoue (21) abatement (emissions) redues (inrease) te voatiity of payoffs. 3

5 Up (24) using a two-period mode found simiar resuts regarding te impiations of different senarios of earning for te size of an IEA and expeted wefare. 5 Taken togeter te tension we seek to apture in our modeing is between No Learning were ountries deide to join an IEA and base teir deisions for ever on expeted damage osts ignoring any ater information Fu Information were ountries deay making any deision to join an agreement unti (amost) a unertainty about damage osts as been resoved or Partia Learning were ountries start te proess of joining an IEA knowing tat as tey get better information tey wi be abe to use tat to adjust teir emissions poiies. Tis woud seem to be partiuary reevant to te kind of situation to wi Weitzman (29) as drawn attention a sma probabiity of atastropi imate ange. Te paper proeeds as foows. In setion 2 we set out te teoretia mode and present our teoretia resuts in setion 3. Setion 4 presents some simuation resuts wie Setion 5 summarizes our main onusions and impiations for future resear. 2. Te Mode 2.1 No Unertainty To estabis te basi framework we set out te mode wit no unertainty. Tere are N identia ountries indexed i 1... N. Country i produes emissions x i wit aggregate emissions denoted by X N i1 x. Aggregate emissions ause goba environmenta i damages. Te ost of environmenta damages per unit of goba emissions is and te benefit per unit of individua emissions is normaized to 1. (Tus essentiay measures te ost-benefit ratio.) Te payoff to ountry i as a funtion of own and aggregate emissions is given by x X x X (1) i i i 5 In many papers wit a dynami payoff struture but fixed membersip resuts are quaitativey simiar to te one period emission game (e.g. Rubio and Casino 25). Te extension to fexibe membersip woud be more interesting but is teniay very aenging. See Rubio and Up (27). 4

6 wit a positive onstant. In tis simpe mode wit a inear payoff funtion foowing te x 1 iterature te (ontinuous) strategy spae an be normaized to mode interesting we make te foowing assumption: Assumption 1: i) 1 1 N ; (ii) N 1. i. 6 To make tis Te individua benefit exeeds te individua unit damage ost from poution i.e. 1 (so ountries poute in te Nas equiibrium) but does not exeed te goba unit damage ost i.e. 1 N (so ountries abate in te soia optimum); te seond ondition is a suffiient ondition su tat (.) i wi we wi need wen we onsider expeted utiity.7 In order to study oaition formation we empoy te widey used two-stage mode of IEA formation (Carraro and Sinisao 1993 and Barrett 1994) wi is soved bakwards. In stage 2 te emission game for any arbitrary number of IEA members n 1 n N te members of te IEA (wi we denote by te symbo for oaition ountries) and te remaining ountries (wi we denote by te symbo f for fringe ountries) set teir emission eves as te outome of a Nas game between te oaition and te fringe ountries. 8 Tat is te oaition members togeter maximize te aggregate payoff to teir oaition wereas fringe ountries maximize teir own individua payoff. Given 1 x 1 foows; oaition members ose x if 1 n and so f f n N n and ( n ) N n ; owever if 1 n ten oaition members wi aso poute x 1 n n N. 9 and so 1 f 1 Knowing te payoffs to oaition and fringe ountries for any arbitrary number of IEA members we ten determine te stabe (Nas) equiibrium in stage 1 te membersip game. 6 Eiter benefits from emission are ower tan damage osts in wi ase equiibrium emissions are zero or te reverse is true in wi ase equiibrium emissions woud obtain teir maximum. Tus te upper bound is set to 1 ere. 7 Te owest possibe payoff is derived if a ountry abates and a oter ountries free-ride wi is given by i 1 ( N 1) and ene given tat 1 N 1 is a suffiient ondition for (.) i. 8 A sequentia Stakeberg game in te seond stage as an aternative assumption (e.g. Barrett 1994) woud make no differene ere as payers ave dominant strategies. Tis aso appies to Bouer and Bramoué (21). 9 It is now evident wy we need Assumption 1: it avoids trivia outomes were a ountries eiter abate or poute in te Nas equiibrium and te soia optimum (and ene aso for partia ooperation). 5

7 No member soud ave an inentive to eave te oaition e.g. te oaition is internay stabe ( n) ( n1) and no fringe ountry soud ave an inentive to join te oaition f e.g. te oaition is externay stabe ( n) ( n1). 1 It is now easy to sow tat te f stabe IEA is of size n * ( ) I(1/ ) wi is te smaest integer no ess tan 1 /. Consider interna stabiity and onsider te non-trivia situation were n members do not poute beause 1 n. If after one member eft and 1 ( n 1) was sti true so te remaining oaition members ontinue not to poute te gain from eaving woud be positive: te additiona benefit is 1 and te additiona damage is wit 1 by Assumption 1. Tus a oaition of n members an ony be stabe if and ony if 1 ( n 1) is true after one member eft as te remaining oaition members woud swit from x ( n ) to x ( n 1) 1. Ten te additiona benefit from poution of 1 fas sort of te additiona damage n as by assumption 1 n in te initia situation wit n members. It is easiy eked tat su an equiibrium is aso externay stabe. Te tota payoff in a stabe oaition is given by: * * * * f ( ) n ( ) ( n ( )) ( N n ( )) ( n ( )) N ( N n ( ))(1 N) *. (2) Tus tis simpe mode provides a reationsip between te unit damage ost and te equiibrium number of oaition members. Te equiibrium is a knife-edge equiibrium wit * n ( ) ountries forming te oaition wi de fato dissoves one a member eaves te oaition as no ountry woud abate anymore. Te equiibrium oaition size weaky dereases in te ost-benefit ratio from emissions te arger is te smaer is te number of ountries in a stabe IEA. 2.2 Unertainty Risk Aversion and Learning Now assume tat te unit damage ost of goba emissions is unertain and equa for a ountries bot ex-ante and ex-post. We denote te vaue by s in te state of te word s and ene (1) beomes: x X x X. (3) is i i s 1 Witout oss of generaity te weak inequaity for externa stabiity oud be repaed by a strong inequaity sign. Our assumption avoids knife-edge ases were a fringe ountry is indifferent between staying outside and joining a oaition. 6

8 Foowing Kostad and Up (28) and Bouer and Bramoue (21) we assume for simpiity tat s ig damage osts an take one of two vaues: ow damage osts wit probabiity 1 p were and wit probabiity p 1 p and. We denote by p 1 p te expeted vaue of unit damage osts and by.5( ) te median vaue of damage osts. We define n I(1/ ) n I(1/ ) n I(1/ ) n I(1/ ) and make te foowing assumptions: Assumption 2: (i) 1/ N 1; (ii) 2 n n n N. Assumption 2(i) is just te anaogue to Assumption 1(i). Assumption 2(ii) means tat unertainty matters in te sense tat it impies signifiant differenes in te size of te stabe IEAs tat woud arise if we knew for ertain wi state of te word prevaied. To aow for risk aversion we assume tat ea ountry as an identia utiity funtion over payoffs: u u u Wie ex-ante ountries fae unertainty about te ' '' ( i) ( i) ( i). true vaue of unit damage osts we want to aow for te possibiity tat ountries may earn information during te ourse of te game wi anges te risk tey fae. We sa foow Kostad and Up (28) in onsidering tree very simpe senarios of earning denoted by m m{ PL FL }. Wit No Learning (m=) ountries make teir deisions about membersip and emissions wit unertainty about te true vaue of unit damage osts. Wit Fu Learning (m=fl) ountries earn te true vaue of unit damage osts before tey ave to take teir deisions on membersip (stage 1) and emissions (stage 2). Wit Partia Learning (m=pl) ountries earn te true vaue of damage osts at te end of stage 1 tat is after tey ave made teir membersip deisions but before tey make teir emission deisions (stage 2). Tus in tis simpe anaysis earning takes te form of reveaing perfet information. 11 We wi ompare te outomes of te tree senarios of earning in terms of te expeted size of IEAs and expeted aggregate utiity from an ex-ante perspetive i.e. before stage We use te term partia earning in ine wit te IEA-iterature atoug tis may be miseading; partia earning is usuay referred to as deayed but perfet earning so not partia earning in te sense of Bayesian updating. 7

9 3. Anaytia Resuts In tis setion we set out te equiibrium of te IEA mode for ea of te tree modes of earning wit risk aversion generaizing te resuts of Kostad and Up (28) wo assumed risk neutraity. Te proofs are provided in te Appendix. 3.1 Fu Learning We start wit te benmark senario of Fu Learning (FL). Payers know te reaization of te damage parameter at te outset of te oaition formation game i.e. before stage 1. Tus te resuts foow direty from wat we know from te game wit ertainty in setion 2.1 above. Proposition 1: Fu Learning If state s as been reveaed before stage 1 ten te subsequent membersip and emission deisions are as in te mode wit ertainty wit damage ost parameter γs. In ea state s= te size of te stabe IEA is ns=i(1/γs) and te expeted membersip is E(n FL )=pn+(1 p)n. Te expeted aggregate utiity is given by: E( U ) pn u( ) p( N n ) u( ) (1 p) n u( ) (1 p)( N n ) u( ) FL FL FL FL FL f f FL FL FL wit ( N n) 1 ( N n ) ( N n ) and FL 1 ( N n ). f f Note tat wit Fu Learning wie te degree of risk aversion does not affet te expeted size of te IEA it wi affet expeted utiity. Importanty te size and utiity are omputed from an ex-ante perspetive to make a omparison wit te oter modes of earning meaningfu. 3.2 No Learning In tis setion we address te senario of No Learning in wi payers take teir membersip (stage 1) and emission (stage 2) deisions under unertainty. 12 We begin by soving for optima emissions of ountries for any number of IEA members n. Sine te benefit of one unit of emissions exeeds te damage ost in bot states of te word it is 12 Our anaysis of No Learning is simiar to te anaysis provided by Bouer and Bramoue (21). 8

10 straigtforward to see tat fringe ountries wi aways poute. To sove for te optima emissions for a oaition member for any n wi we denote by x(n) we need to introdue some notation. We define: E( u ( x ( n) n))) pu( ( x ( n) n)) (1 p) u( ( x ( n) n)) (4a) were ( x ( n) n) ( N n) x ( n)(1 n) s s s s (4b) E( u ( x ( n) n ))) is te expeted utiity to an IEA ountry wen tere are n IEA members wo set emissions x and a fringe ountries set emissions equa to 1. Ten: Eu ( ) ' ' p(1 n) u [ ( x n)] (1 p)(1 n) u [ ( x n)] (5a) x 2 Eu ( ) 2 '' 2 '' p(1 ) [ 2 n u ( x ( n) n)] (1 p)(1 n) u [ ( x ( n) n)] (5b) ( x ) From (5a) it is ear tat if n 1 ten Eu ( ) x and ene it is optima for IEA ountries to ompetey abate wie if n 1 ten Eu ( ) x and ene it is optima for IEA ountries to ompetey poute. To get tigter bounds on wen IEA ountries ompetey poute or abate we define n as te argest vaue of n su tat: p n u n p n u n (6a) ' ' (1 ) [ (1 )] (1 )(1 ) [ (1 )] and n as te smaest vaue of n su tat: p n u n p n u n (6b) ' ' (1 ) [ ( )] (1 )(1 ) [ ( )] We summarise te resuts on emissions in te foowing Lemma: Lemma 1: Emission Deisions wit No Learning (i) x ( n ) 1 n 2 n N ; f (ii) n 1 n n n (iii) p n n n n ; p 1 n n n n 9

11 (iv) n n x ( n) 1; n n x ( n) ; (v) nnn As aready noted for any size of an IEA n fringe ountries aways oose te upper imit of emissions. For IEA members tere is a ritia range of vaues for n [ n - 1 and n emissions for su tat IEA members wi ompetey abate if n n ~ n ~ n ~ n n~ ; but if tere are vaues of n wi ie witin te range ] wi ies between and oose te upper imit of nn ten oaition members oose a eve of emissions beow te upper imit. Note tat as in Bramoue and Bouer (21) wit bot risk neutraity and risk aversion nnn. But for nnn wit risk neutraity x ( ) n x ( ) 1 n for n n and x ( ) 1 n for ; wie wit risk aversion x ( n) 1for n n n and x ( ) n are ower wit risk aversion tan wit risk neutraity. for n n n. So for n n n aggregate emissions Proposition 2: No Learning Wit No Learning for a parameter vaues tere exists a stabe IEA wit membersip n wi is te same in bot states of te word wit n [ n 1 n]. Tis is aso expeted membersip i.e. E( n ) n [ n 1 n ] wi is (weaky) ower tan under risk neutraity wit E( n ) n as n n. Emissions of fringe and IEA members are given in Lemma 1. Expeted aggregate utiity is given by: E( U ) pn u( ) p( N n ) u( ) (1 p) n u( ) (1 p)( N n ) u( ) f f wit ( N n ) (1 n ) x ( n ) ( N n ) 1 n x ( n ) f ( N n ) (1 n ) x ( n ) and ( N n ) 1 n x ( n ). f So te expeted equiibrium oaition size is (weaky) smaer under risk aversion tan risk neutraity. Wit unertainty ountries are unsure about te state of te word. Wit risk and ene onave utiity ountries sy away from te ommitment to be a member in an IEAs members aways ave ower expeted utiity tan fringe ountries. Tis is in ine wit te findings in Bouer and Bramoue (21). 1

12 3.3 Partia Learning In te senario of Partia Learning ountries ave to make teir deision on weter to join an IEA witout knowing te true damage ost of emissions but an make teir subsequent emission deisions based on tat knowedge. We ave argued above tat tis is te one out of te tree senarios of earning we present wi most osey represents te situation te word faes. It foows tat te emission deisions of ountries do not depend on risk aversion and so are te same as in Kostad and Up (28). Sine for one unit of emissions te benefit exeeds damage osts in ea state of te word a fringe ountry wi optimay set x 1 s=; for an IEA member optima emissions depend on te size of te IEA n; so n n x ( n ) s ; n n n x ( n ) 1 and x ( n ) ; n n x ( n ) 1 s. Tat is fringe ountries aways poute; if tere are at east n IEA members ten IEA members aways abate; if tere are ess tan n IEA members ten IEA members aways poute; oterwise IEA members poute in te ow damage ost state and abate in te ig damage ost state. As in Kostad and Up (28) for ertain vaues of p tere may be more tan one stabe IEA wit Partia Learning. In our mode a seond stabe IEA exists iff p p 1 were p 1 u[ ( N n ) (1 )] u[ ( N n ) ] wit. It is straigtforward to sow u[ ( N n ) ] u[ N 1] 1 tat wen ountries are risk neutra p as in Kostad and Up (28). However n more generay we ave not been abe to determine anaytiay ow s p f s s varies wit te degree of risk aversion. In setion 4.2 we report our findings on tis from our simuation resuts. Proposition 3: Partia Learning Wit Partia Learning for a parameter vaues tere aways exists a stabe IEA wit n PL 1 n members. A ountries poute in te ow damage ost state wie in te ig damage ost state oaition members abate and fringe ountries poute. Expeted aggregate utiity is given by: 11

13 E( U ) pn u( ) p( N n ) u( ) (1 p) n u( ) (1 p)( N n ) u( ). PL 1 PL 1 PL 1 PL 1 PL 1 f f 1 wit PL PL1 PL1 ( N n ) and 1 N PL1 f 1 ( N n ). f 1 N If p p1 ten tere is a seond stabe IEA wit PL2 n n members. In bot states of te word oaition members abate and fringe ountries poute. Expeted aggregate utiity is given by: E( U ) pn u( ) p( N n ) u( ) (1 p) n u( ) (1 p)( N n ) u( ) PL 2 PL 2 PL 2 PL 2 PL 2 f f PL2 PL2 PL1 wit ( N n ) 1 ( N n ) ( N n ) and N n. PL2 f 1 ( ) f Sine te seond equiibrium Pareto-dominates te first equiibrium if it exists expeted PL1 membersip is eiter E( n ) n if p p PL2 or E( n ) n if p. p1 As te degree of risk aversion affets p it as an effet on te ikeiood of a seond oaition wit iger membersip n being stabe. Tis effet is furter expored in setion 4.2 were we sow tat te ikeiood of te arger equiibrium dereases wit risk aversion. 3.4 Comparison Aross te Tree Senarios of Learning In tis sub-setion we investigate wat we an say about expeted IEA membersip payoffs and expeted utiity aross te four possibe equiibria FL PL1 and PL2. PL1 In terms of expeted membersip of an IEA it is ear tat sine E( n ) n tis equiibrium PL2 as te owest expeted membersip wie sine E( n ) n tis equiibrium as te igest expeted membersip. Note aso tat: FL 1 1 p (1 p) E( n ) p( ) (1 p)( ). Moreover it is straigtforward to sow tat: p (1 p) 1 p p (1 p) 2 (1 p)( ) 12

14 Hene: FL En ( ) 1 p (1 p) (7a) From Proposition 2 and Lemma 1 we ave tat: 1 E( n ) n n I( ) (7b) p (1 p ) Taken togeter (7a) and (7b) woud suggest tat for a wide range of parameters: PL1 FL PL2 n n n n (8a) However beause FL 1 1 E( n ) pi( ) (1 p) I( ) (7a) and (7b) are not suffiient to ensure FL tat E( n ) E( n ) so as we sa see tere are parameter vaues for wi: E n E n E n E n (8b) PL1 FL PL2 ( ) ( ) ( ) ( ) is possibe. In terms of payoffs aross te four equiibria it is straigtforward to see from Proposition 1 2 and 3 tat: ; ; ; (9a) PL 2 FL PL 1 PL 2 FL PL 1 PL 2 FL PL 1 PL 2 FL PL 1 f f f f f f For in te ow damage ost state of te word te igest payoff to oaition members is wen x=1 wi is ess tan or equa to te payoff to oaition members in PL2 sine n 1; in te ig damage ost state of te word te igest payoff to oaition members is wen x = wi is ess tan te payoff to members in PL2 sine E( n ) n. So it must be te ase tat: ; ; ; (9b) PL 2 PL 2 PL 2 PL 2 f f f f Atoug (9a) and (9b) aow us to rank many of te payoffs aross te four possibe equiibria for bot members and fringe ountries in te ig and ow damage ost states of te word tis is not suffiient to aow us to rank expeted aggregate utiity at an anaytia and genera eve. Te next setion reports te simuations we ave arried out to ompare expeted IEA membersip and expeted wefare aross te different modes of earning. 13

15 4. Resuts from Simuations Tere are tree sets of issues we wis to expore using numeria simuations. (i) Wat is te expeted size of te IEA in te ase of No Learning En ( ) in reation to te teoretia imits n 1 and n and more importanty to te key parameters of our mode n and n and ow does tis vary aross different degrees of risk aversion? (ii) In te ase of Partia Learning wat is te ritia vaue of te ikeiood of ow damage state of te word p su tat for p p1 tere is seond stabe IEA ( PL n 2 n ) and ow does p vary aross different degrees of risk aversion? (iii) How does te expeted size of IEA and expeted aggregate utiity ompare aross te tree different modes of earning Fu Learning (FL) No Learning () and Partia Learning (PL1 PL2) and ow does tis omparison depend on te degree of risk aversion? To address tese questions assume first tat ea ountry as a CRRA utiity funtion 13 A u( i) (1 ) 1 i (1) were ρ measures te degree of reative risk aversion and wit A a onstant wi we wi set equa to Moreover we set N - 1 in payoff funtion (1) te smaest vaue required to ensure non-negative payoffs. We use te risk neutra ase (ρ = ) as a benmark and ten oose 1 vaues of ρ = and 2. to apture wat we beieve to be a reasonabe range of vaues for ountry-eve risk aversion. For ea of tese vaues of risk aversion we ondut 5 Monte Caro simuations aross te vaues of te remaining key parameters (N p γ γ). We first oose N as an integer in te range [41] and p as a rea number in te range [.1.999]. We ten oose and to 13 Meyer and Meyer (26) note tat te CRRA utiity funtion is widey used in empiria studies of risk aversion and tat empiria estimates of ρ vary between and 1. Tey note tat su estimates depend on te variabe tat enters te utiity funtion and for te tree most ommony used variabes weat inome and profits te appropriate empiria estimate inreases as one moves from weat to profits. In our one-period mode te reevant variabe is inome toug tere is no distintion between weat and inome. Hene we ave osen a range of vaues for ρ at te ower end of te range noted by Meyer and Meyer. See detais beow. Aso note tat quaitative onusions woud not ange for iger degrees of risk aversion. 14 Te onstant A is a mutipiative fator wi as no effet on te simuation resuts presented in tis setion. 14

16 ensure tat n and n satisfy Assumption 2 and are eveny distributed between 2 and N. 15 Tus our simuations basiay apture a arger parameter range. 4.1 Resuts for Size of Stabe IEA wit No Learning Rea tat from Kostad and Up (28) te expeted size of te stabe IEA wit No Learning wen ountries are risk neutra is E( n ) wen ρ >. 16 n. In Tabe 1 we present te resuts Tabe 1: Expeted Size of IEA under no earning for different degrees of risk aversion 1 ρ % interior Soution 3 % of ases reating n n 1 n 4 n 1 E( n ) n n 1 E( n ) n n 1 E( n ) n n 1 E( n ) n % of ases reating n n n : 9 n E( n ) n n E( n ) n n E( n ) n n E( n ) n We know from Proposition 2 tat n 1 E( n ) n and from Lemma 1(v) taking aount of te previous footnote tat if n E( n ) n ten < x < 1. Row 2 of Tabe 1 sows te perentage of ases for wi < x < 1 for different vaues of ρ wie rows 4-7 sow te perentage of ases for ow En ( ) reates to sub-intervas of [ n 1 n]. It is readiy eked tat for ea vaue of ρ te perentage in row 2 equas te sum of te perentages in rows 4 and 6 onfirming Lemma 1(v). Wen ρ=.1 for more tan 99.9% of te ases E( n ) n i.e. te resuts are very ose to te resut under risk neutraity. As risk aversion inreases te 15 Furter detais are provided in te Appendix. 16 In te proofs of Lemma 1 and Proposition 2 in te Appendix we note tat we annot rue out te teoretia possibiity of mutipe vaues of eiter n or stabe IEAs. In te simuations su outomes ourred in ess tan.1% of parameter ombinations and we ave just ignored su ases. 15

17 number of ases were E( n )n n fas to just over 78% and te number of ases were E( n ) n 1 n rises to just under 22%; in ess tan.2% of te ases do we ave n 1 E( n ) n. In terms of te key parameters of te mode n and n from Lemma 1 we aso know tat n E( n ) n and rows 9-12 in Tabe 4 sow for ea vaue of ρ te perentage of sub- ases tat an arise. Wen ρ=.1 for more tan 99.7% of a parameter vaues E( n ) n wi is very ose to te resut wit risk neutraity in Kostad and Up (28). As ρ inreases te perentage of ases were E( n )n n remains rougy onstant at about 16% refeting te fat tat p n n irrespetive of te degree of risk aversion. However for te remaining ases were n n ten as risk aversion inreases: (i) te perentage of ases were n E( n ) n fas sarpy from 84% to 29%; (ii) te perentage of ases were n E( n ) n rises sarpy from.2% to 3% and (iii) te proportion of ases were n E( n ) n rises steadiy from.4% to 25%. Hene wit No Learning inreasing risk aversion drives down te number of ountries joining an IEA as disussed in our teoretia anaysis in subsetion Seond stabe IEA wit Partia Learning We sowed in setion 3.3 tat wit Partia Learning tere aways exists a stabe IEA wit n members wo abate in te ig damage ost state and poute in ow damage ost state but tere is a ritia vaue of p wi we defined as p su tat for p p 1 tere exists a seond stabe IEA were members abate in bot states of te word. We aso said tat we ad been unabe to prove anaytiay ow p varies wit te degree of risk aversion. In rows 2 and 3 of Tabe 2 we sow for ea vaue of risk aversion between and 2 te average vaue of p and te perentage of su ases for wi p p 1 ours respetivey. From row 2 we see tat as te degree of risk aversion inreases te average vaue of p rises from.9437 to.9685 wie from row 3 te perentage of simuations for wi p p 1 fas from 5.66% to 3.14%. So inreasing risk aversion redues te ikeiood tat wit Partia Learning tere exists a seond stabe IEA wit iger membersip. 16

18 Tabe 2: Expeted membersip and aggregate utiity for tree senarios of earning 1 ρ Average 3 % Cases were p p p 4 % Cases for membersip PL En ( ) En ( ) FL En ( ) 8 % Cases for aggregate utiity: PL EU ( ) EU ( ) FL EU ( ) igest igest igest igest igest igest Comparison Aross te Tree Senarios of Learning In te remaining part of Tabe 2 we present for ea vaue of ρ between and 2 te perentage of simuations for wi we ave partiuar rankings for expeted membersip and expeted aggregate utiity aross our tree modes of earning. In te ase of PL if p p 1 and tere are two stabe IEAs ten we ose te seond Pareto-superior equiibrium. Te first resut we note is tat for bot expeted membersip and expeted aggregate utiity te perentage of simuations were PL performs best is equa to te perentage of simuations for wi p p 1 as disussed in setion 4.2. Tis sows tat Partia Learning is te preferred mode of earning ony in tose ases were p p 1 i.e. if te arge oaition equiibrium emerges. As we ave aready noted te perentage of su ases deines from just under 5.66% wen ρ= to just 3.14% wen ρ = 2. Fousing now on just FL and as ρ inreases from to 2 te perentage of simuations were FL as te igest expeted membersip rises from 79.27% to 87.42% wie te perentage of simuations were as te igest expeted membersip dereases from 15.7 % to 9.44%. Tis is onsistent wit te resut presented in Tabe 1 namey tat as risk aversion inreases te size of te IEA wit dereases from n towards n. More importanty as risk aversion inreases te perentage of simuations were expeted aggregate utiity is strity igest under dereases from just under 73% to just over 33% 17

19 wie te perentage of simuations were FL as igest expeted aggregate utiity inreases from just over 21% to just over 63%. Te impiation of tese resuts is tat as ountries beome more risk averse ten for an inreasing perentage of parameter vaues te form of earning wi eads to te igest aggregate utiity is Fu Learning. Te perentage of parameter vaues favouring No Learning dereases and for ig enoug vaues of risk aversion te perentage of ases for wi Fu Learning is te preferred senario is iger tan No Learning. Tis is aso true for te expeted utiity of an individua ountry from an ex-ante perspetive wi for symmetri payers is simpy te aggregate utiity divided by te tota number of ountries. Tus oud governments oose te senario of earning endogenousy in stage zero tey preferred fu earning for ig eves of risk aversion. So ountries woud be better off eaving te deision to form an IEA and set teir emissions unti tey ave Fu Information about te risks of imate ange. Te reativey sma perentage of parameter vaues for wi te preferred mode of forming an IEA is Partia Learning i.e. deiding weter to join an IEA before getting better information about te risks of imate ange but aowing emissions to be set after better information is avaiabe aso deines as ountries beome more risk averse. Tus te pessimisti onusion about te roe of earning in a strategi ontext derived in previous papers is quaified in partiuar if te degree of risk aversion is suffiienty arge. 5. Summary and Conusions Tis paper bridges two strands of iterature on te formation of IEAs under unertainty by addressing te ombined roes of earning and risk aversion. Tis approa aowed us to expore te impat of earning for any given eve of risk aversion as we as te impat of anging risk aversion under various senarios of earning. We generaized te mode of Kostad and Up (28) wo sowed tat wit risk neutraity te possibiity of earning more information about environmenta damage osts generay ad rater pessimisti impiations for te suess of te formation of IEAs. Exept for a reativey sma set of parameter vaues for wi partia earning woud seet a ig IEA membersip earning resuted in ower or equa expeted membersip for partia earning and ower expeted aggregate and individua payoff for partia and fu earning ompared to no earning. Moreover tis parameter range required tat te probabiity of ow damage ost is very ig a rater uninteresting parameter onsteation in te ontext of imate ange. Hene in a strategi ontext earning redues expeted aggregate and individua payoffs for a 18

20 wide range of parameter vaues. Aross te different modes of earning tey sowed tat for a arge set of parameter vaues te senario of earning wi yieded igest expeted aggregate and individua payoff was No Learning wi woud suggest tat ountries are better off forming an IEA rater tan waiting for better information. In tis paper we ave aowed ountries to be risk averse using an expeted utiity approa wi maps payoffs into utiity. We first derived te teoretia resuts for ea of our tree senarios of earning wit risk aversion onfirming te main findings of Bouer and Bramoue (21) for te No Learning ase. For No earning risk eads to smaer stabe oaitions and iger goba emissions. In terms of equiibrium oaitions and goba emissions we sowed tat Fu Learning remains unaffeted by risk and anges for Partia Learning are sma. However even wit speia funtiona forms for te underying utiity funtions tere was imited sope for deriving anaytia omparisons aross our tree senarios of earning primariy beause wefare effets oud differ for signatory and nonsignatory ountries. Our simuation resuts sowed tat ontrary to te finding wit risk neutraity wen ountries beome signifianty risk averse te set of parameter vaues for wi ountries are better off wit No Learning ompared to Fu Learning srinks signifianty and tose ases for wi tis is reversed inreases aordingy. Tis may expain wy it as taken so ong for a proper imate agreement to be reaed ountries are risk averse and waited ti tey ad mu better information about te risks of imate ange. In terms of future resear it woud be desirabe to use a mode wit asymmetri ountries toug it is unikey to be possibe to derive anaytia resuts; so it may be more usefu to introdue different modes of earning into Integrated Assessment Modes of imate ange. It woud aso be interesting to endogenise te proess of earning by aowing ountries to invest in resear in order to obtain better information. 19

21 Referenes Arrow K. and A. Fiser (1974) Environmenta preservation unertainty and irreversibiity. Quartery Journa of Eonomis 88: Barrett S. (1994) Sef-enforing internationa environmenta agreements. Oxford Eonomi Papers 46: Bouer V. and Y. Bramoué (21) Providing goba pubi goods under unertainty. Journa of Pubi Eonomis 94: Bramoué Y. and N. Trei (29) Can unertainty aeviate te ommons probem? Journa of te European Eonomi Assoiation 7(5): Carraro C. and D. Sinisao (1993) Strategies for te internationa protetion of te environment. Journa of Pubi Eonomis 52: Deink R. and M. Finus (212) Unertainty and imate treaties: does ignorane pay? Resoure and Energy Eonomis 34: Endres A. and C. O (23) Internationa environmenta ooperation wit risk aversion. Internationa Journa of Sustainabe Deveopment 6: Epstein L. (198) Deision-making and te tempora resoution of unertainty. Internationa Eonomi Review 21: Finus M. and A. Caparrós (215) Game Teory and Internationa Environmenta Cooperation: Essentia Readings. Edward Egar Cetenam UK. Finus M. and P. Pintassigo (213) Te roe of unertainty and earning for te suess of internationa imate agreements. Journa of Pubi Eonomis 13: Goier C. B. Juien and N. Trei (2) Sientifi progress and irreversibiity: an eonomi interpretation of te Preautionary Prinipe. Journa of Pubi Eonomis 75: Hong F. and L. Karp (214) Internationa environmenta agreements wit exogenous and endogenous Risk. Journa of te Assoiation of Environmenta and Resoure Eonomis Kostad C. (1996a) Fundamenta irreversibiities in stok externaities. Journa of Pubi Eonomis 6: Kostad C. (1996b) Learning and stok effets in environmenta reguations: te ase of greenouse gas emissions. Journa of Environmenta Eonomis and Management 31:

22 Kostad C. (27) Systemati unertainty in sef-enforing internationa environmenta agreements. Journa of Environmenta Eonomis and Management 53: Kostad C. and A. Up (28) Learning and internationa environmenta agreements. Cimati Cange 89: Kostad C. and A. Up (211) Unertainty earning and eterogeneity in internationa environmenta agreements. Environmenta and Resoure Eonomis Markowitz H. (1952) Portfoio seetion. Journa of Finane 7: Meyer J. (1987) Two-moment deision modes and expeted utiity maximization. Amerian Eonomi Review 77: Meyer D. and J. Meyer (26) Measuring risk aversion. Foundations and Trends in Miroeonomis 2: Na S.-L. and H.S. Sin (1998) Internationa environmenta agreements under unertainty. Oxford Eonomi Papers 5: Narain U. A. Fiser and M. Hanemann (27) Te irreversibiity effet in environmenta deisionmaking. Environmenta and Resoure Eonomis 38: Rubio S. and B. Casino (25) Sef-enforing internationa environmenta agreements wit a stok poutant. Spanis Eonomi Review 7: Rubio S. and A. Up (26) Sef-enforing internationa environmenta agreements revisited. Oxford Eonomi Papers 58: Rubio S. and A. Up (27) An Infinite-orizon mode of dynami membersip of internationa environmenta agreements. Journa of Environmenta Eonomis and Management 54: Up A. (24) Stabe internationa environmenta agreements wit a stok poutant unertainty and earning. Journa of Risk and Unertainty 29: Up A. and D. Maddison (1997) Unertainty earning and internationa environmenta poiy oordination. Environmenta and Resoure Eonomis 9: Up A. and D. Up (1996) Wo gains from earning about goba warming? In: van Ierand E. and K. Gorka (eds.) Te Eonomis of Atmosperi Poution. Springer Heideberg Up A. and D. Up (1997) Goba warming irreversibiity and earning. Eonomi Journa 17:

23 Weitzman M. (29) On modeing and interpreting te eonomis of atastropi imate ange. Review of Eonomis and Statistis 91:

24 Appendix: Proofs of Resuts Proposition 1: Fu Learning Sine te true state of te word is reveaed before ountries deide weter to join an IEA te resuts for stabe IEAs probabiities and payoffs in te four states of te word sown in Proposition 1 foow immediatey from Kostad and Up (28). Te differene from Kostad and Up is tat we now auate aggregate expeted utiity using te expeted utiity approa wi aptures attitudes to risk rater tan expeted payoff. Tis ompetes te proof of Proposition 1. Lemma 1: Emission Deisions wit No Learning. Fringe Country Output Deision In stage 2 fringe ountry i takes as given te output of a oter ountries X-i and ooses xi to maximise pu[ x ( x X )] (1 p) u[ x ( x X )]. Sine f x ( n ) 1 n 2 n Member Country Emission Deision i i i i i i N. Tis proves resut (i). (i) Properties of Payoff Funtion From (4b) we obtain te foowing: 1 ( x n) ( x n) n 1 x 1; (A1a) x s 1 n ; (A1b) s ( n) N n ; (1 n) N 1 (1) n (A1) s s s s s s and ene (1) ( n) ( n) (1) if 1 n 1 (A1d) As stated in setion 3.2 for a given n ea oaition member ooses x(n) to maximise E( u ( x ( n) n))) pu( ( x ( n) n)) (1 p) u( ( x ( n) n)) wi eads to te foowing first and seond order ondition: 23

25 Eu ( ) ' ' p(1 n) u [ ( x n)] (1 p)(1 n) u [ ( x n)] (A2a) x 2 Eu ( ) 2 '' 2 '' p(1 ) [ 2 n u ( x ( n) n)] (1 p)(1 n) u [ ( x ( n) n)] (A2b) ( x ) Boundary Vaues for x(n) From (A2a): 1 1 Eu ( ) n x x( n) x 1 1 Eu ( ) n x x( n) 1 x We want to see if we an get tigter bounds for n tat guarantees wen x ( n) x. So we now fous on te range ; x x ( n) 1 and 1 1 n 1 n n. From (A1b) in tis range To make progress we treat n as if it was a rea vaue z. To save notation define: u[ (1)] u[ (1)] ( z) u[ ( z)] and ( z) u[ ( z)] were from (A1d): ( z) ( z) (A3) We first define z as te unique vaue of z su tat: E( u(1 z)) p (1 p) p(1 z) (1 p)(1 z) z. (A4) x p (1 p) From (A4) we get: 1 1 z and E( u(1 z)) z z. Tus x ( z) 1 z z. x We now define z su tat: 24

26 E( u ( z)) p ( z) (1 p) ( z) p(1 z) ( z) (1 p)(1 z) ( z) z x p ( z) (1 p) ( z) (A5) and again 1 z 1. Note we ave not been abe to prove tat tere is a unique vaue of wi soves (A5). We wi disuss te impiations sorty but te next steps appy to any wi soves (A5). We first sow tat: z z z z [ p ( z) (1 p) ( z)][ p (1 p) ] ( z) [ p (1 p) ][ p ( z) (1 p) ( z)] p(1 p)( ) ( z)[ ] ( z) Using (A3) it an be sown tat: (A6) ( z) sign z z signp(1 p)( ) ( z)[ ]. So any ( z) must be at east as arge as z. z wi soves (A5) Next we sow tat 1 z. Define ( z)/ ( z) 1. Ten: p (1 p) p (1 p) z 1/ p (1 p) p (1 p) p(1 p)( ) p(1 p)( ) ( 1)( ) (A7) So any z wi soves (A5) must be no greater tan 1/. As z if z is an integer n I z 1 if z is not an integer and z if z is an integer n I z 1 if z is not an integer ten n 1 n n n - resut (ii); n n x ( n) 1; n n x ( n) - resut (iv). Finay from (A4) and (A5) it is straigtforward to see tat: p n n n n ; p 1 n n n n - resut (iii). We ave not been abe to prove anaytiay tat tere is a unique vaue of z so in wat foows we wi treat z and ene n as te argest su vaue. 25

27 (ii) Interior vaues for x (n) It foows from te definitions of Hene n and n n n x n tat 1 and n n x n n n n x n 1 - resut (v). Tis ompetes te proof of Lemma 1.. Proposition 2: No Learning From Lemma 1 we know te vaues of xf(n) and x(n) for a n. We now sove Stage 1. Define: E( u ( n)) pu( 1 [ N n nx ( n)]) (1 p) u( 1 [ N n nx ( n)]) f E( u ( n)) pu( x ( n) [ N n nx ( n)]) (1 p) u( x ( n) [ N n nx ( n)]) and ( n) E( u ( n)) E( u ( n 1)) noting tat n E( n ) is a stabe IEA iff ( n ) and ( n 1). f To save notation define s s N s. (i) nn 1 ( n) pu( n) (1 p) u( n) pu( n (1 )) (1 p) u( n (1 )) So no su n oud be stabe. (ii) n n A ountries set x =1 so Δ(n) =. We exude tese trivia ases. (iii) n 2 n n ( n) p[ n x ( n)(1 ) ( n 1) x ( n)] (1 p)[ n x ( n)(1 ) ( n 1) x ( n)] p[ n (1 ) ( n 1) x ( n 1)] (1 p)[ n (1 ) ( n 1) x ( n 1)] So if x ( n) x ( n 1) ten Δ(n)< and so n oud not be a stabe IEA. If x ( n) x ( n 1) ten te sign of Δ(n) depends on te preise vaues of x () n and x ( n 1). 26

28 (iv) n n1 From (ii) we know tat x ( n) x ( n) 1 E( u ( n)) E( u ( n)). So wen f f n n 1 IEA members oud ave set x ( n1) 1 but by definition of n tey did not so E( u ( n 1)) E( u ( n)) E( u ( n)) ( n 1). f To ompete te argument suessivey inrease n from n 1unti Δ(n) < in wi ase n = n-1 is a stabe IEA. Tere must exist su a stabe IEA sine we know from (iv) tat ( n 1) and from (i) tat ( n 1). We annot rue out te possibiity tat tere is more tan one stabe IEA. Tis ompetes te proof of Proposition 2. Proposition 3: Partia Learning Beause emission deisions in stages 2 are taken under ertainty te resuts are te same as in Kostad and Up (28) namey: Stage 2: (a) x ( n) 1 s n f s (b) n n x ( n) s s () n n n x x 1 (d) n n x ( n) 1 s s So using te notation s s N s introdued in te proof of Proposition 2 te payoffs are: (i) n n E( u ( n)) pu[ n 1] (1 p) u[ n 1] f E( u ( n)) pu[ n ] (1 p) u[ n ] (ii) n n < n E( u ( n)) pu[ 1] (1 p) u[ n 1] f E( u ( n)) pu[ 1] (1 p) u[ n ] (iii) n < n E( u ( n)) E( u ( n)) pu[ 1] (1 p) u[ 1] f 27

29 Stage 1: (i) nn 1 ( n) pu( n ) (1 p) u( n ) pu( n (1 )) (1 p) u( n (1 )) So no n in tis range an be a stabe IEA. (ii) n n ( n ) pu( n ) (1 p) u( n ) pu( 1])(1 p) u( n (1 )) As n 1 so: p u( n (1 )) u( n ) ( n ) (1 p) u( n ) u( 1) i.e. ( n ) p p 1 Sine from (i) ( n 1) n is stabe iff p p. (iii) n 1 n n ( n) pu( 1) (1 p) u( n ) pu( 1) (1 p) u( n (1 )) So no n in tis range an be stabe. (iv) n n ( n ) pu( 1) (1 p) u( n ) pu( 1) (1 p) u( 1) Sine n 1 ( n ) and sine from (iii) ( n 1) n is aways a stabe IEA. (v) n n ( n) ; as in Proposition 2 we ignore tese ases. 28

30 So tere aways exists a stabe IEA wit PL n n and if p p tere is a seond stabe IEA wit PL n n. Tis ompetes te proof of Proposition 3. Furter Detais about Simuations in Setion 4 We do four steps: (i) we oose randomy te inverse of te median 1/ ying in te range [2 N 1] and auate n I(1/ ) ; (ii) we seet te smaer of te two ranges 1 n 1 ( nn respetivey) and oose randomy 1/ (1/ respetivey} to ie in tat range and ene set n I(1/ ) ( n I(1/ ) respetivey); (iii) we next auate n (n respetivey) to satisfy n n 2n ; (iv) finay we oose randomy respetivey) to ie in te range auations is tat simuations of.571 and n 1 n ( n 1 n 1/ (1/ respetivey). Te resut of tese ies in te range [ ] wit an average vaue over te ies in te range [ ] wit an average vaue of 29