UNCERTAINTY, LEARNING AND HETEROGENEITY IN INTERNATIONAL ENVIRONMENTAL AGREEMENTS* October Abstract

Transcription

1 UNCERTAINTY, LEARNING AND HETEROGENEITY IN INTERNATIONAL ENVIRONMENTAL AGREEMENTS Cares D. Kostad # and Aistair Up + Abstract Tis paper concerns te ormation o Internationa Environmenta Agreements under uncertainty about environmenta damage under dierent modes o earning (compete earning, partia earning or no earning). Te resuts o te existing iterature are generay pessimistic: te possibiity o eiter compete or partia earning generay reduces te eve o goba weare tat can be acieved rom orming an IEA reative to no earning. Tat iterature regards uncertainty as a parameter common to a countries, so tat countries are identica ex ante as we as ex post. In tis paper we extend te iterature to te case were tere is no correation between damage costs across countries; eac country is uncertain about a particuar parameter (in our case te beneitcost ratio) drawn rom a common distribution but, ex post, eac country s reaized parameter vaue is independenty drawn. Consequenty, wie countries remain identica ex ante, tey may be eterogeneous ex post. We sow tat tis cange reinorces te negative concusions about te eects o partia earning on internationa environmenta agreements, but, under certain conditions, moderates te negative concusions about te eects o compete earning. An earier version o tis paper was presented at te conerence Learning and Cimate Cange, IIASA, Laxenburg, Austria, Apri 2006, at a seminar at te University o St. Andrews and at a May, 2008 worksop sponsored by te Frisc Centre in Oso. We are grateu to participants or comments. #Department o Economics and Bren Scoo o Environmenta Science and Management, University o Caiornia, Santa Barbara. +Facuty o Humanities, University o Mancester.

2 1. INTRODUCTION One o te major issues in cimate cange poicy is ow to dea wit ubiquitous uncertainty: toug muc is known, neary everyting is uncertain. Wat compicates tings or cimate poicy is tat uncertainty itse is canging we are earning about te science and economics o te probem as time passes. Furtermore, we are taking active steps to increase our knowedge o te processes troug signiicant R&D programs. In a decade we expect to be muc better inormed about te probem and in two decades, even better inormed. Tis process o earning raises an important timing question. On te one and it can be argued tat society soud deay taking action to reduce greenouse gas emissions, unti more is known. Ater a, i we act and subsequenty earn tat cimate cange is ess serious tan we ad tougt, we wi ave taken steps unnecessariy. On te oter and, i we do earn tat cimate cange is more serious tan we ad tougt, we can aways acceerate action ater. However it is oten argued tat tis earn-ten-act approac ony makes sense i te accumuation o greenouse gas emissions is reversibe, so tat i we make a poicy mistake, we can undo te eects o te decision. But te accumuation o greenouse gases is oten viewed as irreversibe 1, so tat by te time we earn tat cimate cange is a serious issue we may ave buit up suc concentrations o greenouse gases tat we are aced wit drastic consequences wic cannot be readiy undone. Tis irreversibiity in te cimate process eads to cas or impementing a precautionary approac tat, ar rom deaying taking steps to reduce greenouse gas emissions wie we wait or better inormation, we soud take more steps now, to guard against getting bad news in te 1 Te process o accumuating greenouse gases in te atmospere is not iteray irreversibe, but te residence time in te atmospere o most greenouse gases is on te order o decades or centuries. Kostad & Up 1

3 uture and inding it is too ate to do anyting about it 2. Te precautionary principe as a good dea o intuitive appea and is oten empoyed in debates on cimate-cange poicy to support cas or increased immediate action by governments to reduce greenouse gas emissions. However, tere is a substantia body o economics iterature wic sows tat tis simpistic version o te precautionary principe is not aways a correct approac to cimate cange poicy 3. In tis iterature on te precautionary principe it is assumed tat tere is a singe decisionmaker, so te anaysis is appicabe to eiter an individua nationa government, or a putative word government. Neiter is reevant to cimate cange wic is a goba poution probem tat must be soved troug negotiations among a arge number o individua nationa governments acting in teir own se-interest. In tis case we are interested in wat kinds o outcomes emerge rom mutipe independent countries negotiating in se-interest. Some ave argued (eg, Young 1994) tat uncertainty and earning aciitates agreement beore negotiating positions become ardened by knowing exacty wic particuar agents win or ose uncertainty is iberating. Yet oters (eg, Cooper, 1989) ave suggested tat it is ony ater uncertainty is argey resoved tat countries wi come togeter and agree to sove a goba commons probem. Te teoretica iterature on te impications o countries interacting strategicay wit uncertainty and earning is tin. Up and Up (1996) and Up and Maddison (1997) consider a two-country mode in wic countries coose teir emissions eiter non-cooperativey or cooperativey and eiter beore te true state o te word is known (wic we ca No Learning) or ater te state o te word is known (wic we ca Compete Learning). Tey sow tat in te co-operative equiibrium, te vaue o inormation is positive, but in te non-cooperative 2 Tere are many deinitions o te precautionary principe; see Foster et a (2000). 3 Key papers are Arrow and Fiser (1974), Henry (1974a,b), Epstein (1980), Kostad (1996a,b), Up and Up (1997), Goier et a (2000), Narain et a (2007). See Ingam and Up (2005) or a survey. Kostad & Up 2

4 equiibrium te vaue o inormation may be negative i te uncertainty about damage costs is negativey correated between countries 4. Wit ony two countries, te cooperative and non-cooperative equiibria are te ony two possibiities. Wit more tan two countries one can consider intermediate cases were countries take two decisions in two stages: irst, weter to join an Internationa Environmenta Agreement (IEA) te Membersip Game; second, ow muc to emit te Emissions Game - wit signatories determining teir emissions to maximize teir coective se-interest, wie nonsignatories act in teir own se-interest. Since te semina papers by Barrett (1994) and Carraro and Siniscaco (1993), a arge iterature, bot teoretica and empirica, as deveoped in tis ied 5. Atoug concepts o te ormation o cooperating bocs o countries (coaition ormation) ave become consideraby more sopisticated, te issue o uncertainty and earning as been argey ignored, and tere is an obvious additiona question to be posed: ow do uncertainty and te prospect o earning aect te incentives or countries to join an IEA? Note tat te two-stage game structure aows an additiona structure or earning: te true state o te word may be reveaed ater countries decide weter or not join an IEA, but beore determining teir emissions, wic we ca Partia Learning. Te irst paper to address tis issue is Na and Sin (1998). Tey consider a tree country mode in wic, ex ante, te countries ave te same expected net beneits rom emissions o a goba ow poutant. Tota goba net beneits are known wit certainty; uncertainty is about teir distribution across te tree countries. Tey compare Compete Learning and No Learning, and 4 Te intuition is tat te cooperative equiibrium is simiar to a singe decision-maker, so inormation is aways vauabe, but in te non-cooperative equiibrium a country tat earns tat it as damage costs beow teir expected vaue wi raise its emissions (reative to emissions wit expected damage costs), wie a country tat earns it as ig damage costs cuts emissions. But in te non-cooperative equiibrium tese responses trigger urter strategic reactions by te two countries: te ow cost country urter expands its emissions, te ig cost country urter cuts its emissions. 5 See Barrett (2002) and Finus (2001) or exceent overviews, wit Boringer and Finus (2005) appying suc anaysis to te Kyoto protoco. Kostad & Up 3

5 sow tat earning is bad or cooperation. A imitation o te Na and Sin mode is tat or te particuar mode tey empoy, te maximum number o countries wo woud join an IEA is tree (see Finus (2001)). So te act tat Na and Sin can obtain te grand coaition as an outcome in te membersip game is an artiact o teir assumption tat tere are ony tree countries. To assess te impact o earning on IEA membersip it woud seem more sensibe to empoy a mode in wic te tota number o countries and possibe signatories are greater tan tree. Up (2004), Kostad (2007) and Kostad and Up (2008) 6 empoy a specia case o te Barrett (1994) mode, in wic emission strategies are discrete and tere is an arbitrary number o signatories and were te size o a stabe IEA can take any vaue between 2 and te grand coaition. In tese papers countries are ex ante identica and ave a common uncertainty about teir damage costs in te sense tat ex post tey a ave te same damage cost. Tey sow tat bot Partia and Compete Learning generay reduce weare compared to No Learning. Te case o eterogeneous agents as received very itte attention. Barrett (2001) provides one o te ew anayses o tis case, toug not or te case o uncertainty. Tat is te contribution o tis paper. We extend te modes o Up (2004), Kostad (2007) and Kostad and Up (2008) to te case were uncertainty is individua to countries, in te sense tat ex so countries can end up wit dierent damage costs. We sow tat wit te resut tat Partia Learning reduces weare is reinorced, but Compete Learning may raise weare. Te next section sets out te mode and te anaytica resuts. Section 3 presents numerica resuts were it as not been possibe to derive resuts anayticay. 6 Kostad and Up (2008) summarises te dierences between te modes o Up (2004) and Kostad (2007), and provides an integrated and extended anaysis o teir resuts. Kostad & Up 4

6 2. THEORETICAL RESULTS 2.0 Te Mode Our genera ramework is tat o N identica countries, indexed i = 1, N. Country i s emissions are denoted q i, wic or simpicity we assume can take one o two vaues: q i = 0 (abate) or q i = 1 (poute). We denote byq j q j, te aggregate emissions o a countries and by Q i q j j i, te aggregate emissions o a countries oter tan i. Country i s net beneits are V = q γ q + Q ), (1) i i i ( i i were γ i denotes te amount o environmenta damage a unit o emissions causes to country i reative to te private beneit o emitting a unit. Countries are risk neutra. Trougout tota net beneit (or aggregate weare) wi be denoted by W: V summed over a countries. We assume tat γ can take just one o two vaues: γ ow damage costs; or γ ig damage costs, wit te oowing assumption: Assumption 1: 1/N < γ < γ < 1. (2) Tis assumption precudes suc sma vaues o γ tat even te uy cooperative soution woud invove no abatement and suc arge vaues o γ tat abatement is a dominant strategy or individua countries. Te probabiity tat damage costs are ow is p and tus te probabiity tat tey are ig is 1-p. Tis distribution is known to a. Deine te expected damage cost or eac country: γ pγ + (1 p)γ (3) We consider two possibe structures o uncertainty across countries: Common Uncertainty: tere are just two states o te word: wit probabiity p a countries ave parameter vaue γ and wit probabiity (1 p) a countries ave parameter vaue γ. Kostad & Up 5

7 Individua Uncertainty: For eac country i = 1,, N tere is an independent probabiity p tat γ i = γ and a probabiity (1 p) tat γ i = γ. Let N be te number o countries wic ave ig damage costs, and N (= N N ) be te number wit ow damage costs. So tere are now N + 1 states o te word depending on te vaue o N. Te probabiity tat te state o te word wit N ig damage cost countries wi arise is te probaby o exacty N successes (probabiity 1-p) in N Bernoui trias, and is given by:! ( N N ) N N p N!( N N )! (1 p). Kostad and Up (2008) present te resuts or No Learning, Partia Learning and Compete Learning wit Common Uncertainty. In sub-section 2.1 we summarise tese resuts, aong wit te resuts or No Uncertainty, as basis or comparison wit te resuts we present in sub-section 2.2 or No Learning, Partia Learning and Compete Learning wit Individua Uncertainty, te ocus o tis paper. 2.1 Summary o Previous Resuts or No Uncertainty and Common Uncertainty No Uncertainty In te context o our mode, no uncertainty simpy means tat γ = γ γ. Foowing Barrett (1994) and oters, we mode tis as a two-stage game. In stage 1, (membersip game) eac country decides weter or not to join te agreement (an IEA). We seek a Nas equiibrium in announcements (ie, in or out ) in wic no country wises to uniateray eave or join te coaition. In stage 2, (emission game) eac non-signatory, or ringe, country, denoted by superscript, takes as given te emissions o a oter countries and cooses its emissions to maximize its individua net beneit; te signatory countries, denoted by superscript s, coectivey coose teir emissions to maximize te aggregate net beneit o te signatory countries taking as given te emissions o te non-signatories. Te outcome o te stage 2 game is a Nas equiibrium Kostad & Up 6

8 in wic eac non-signatory acts non-cooperativey wie te signatories to te IEA act coectivey wit respect to eac oter, but non-cooperativey wit respect to te non-signatories. Deinition 1. Deine te unction I(x) as te smaest integer greater tan or equa to 1/x. Lemma 1 (Emission Game Equiibrium) Given N countries, o wic n 2 are signatories to an IEA, non-signatory countries aways poute. I n I(γ), a signatory countries abate, and te net beneits to a signatory and non-signatory country respectivey are: s V ( n) = ( N n) γ ; (4a) V ( n) = 1 ( N n) γ (4b) I n < I(γ), a signatory countries poute and te net beneit to a signatory and non-signatory country respectivey wi be V s ( n) = V ( n) = 1 Nγ. (4c) Tere are two ways o deining te equiibrium o te membersip game. Te irst, as presented in Barrett (1994) borrows te concept o a stabe coaition rom te iterature on oigopoy and deines a stabe IEA as oows: Deinition 2: An IEA wit n signatories is stabe i it satisies te two conditions: Interna Stabiity: V s ( n) V ( n 1) (5a) Externa Stabiity: V s ( n) V ( n + 1) (5b) i.e. no signatory country as any incentive to uniateray eave te IEA, and no non-signatory as any incentive to uniateray join te IEA, taking as given te membersip decisions o a oter countries. Tis deinition is equivaent to saying tat a stabe IEA is a Nas equiibrium o te membersip game. Lemma 2 (Membersip Game Equiibrium) Te unique stabe abating IEA o te membersip game as n = I(γ) signatory countries wit aggregate word net beneit o Kostad & Up 7

9 W(n) = (N-n)(1-Nγ) < 0. (6) To compete tis section it is wort noting te properties o I(γ)). I we approximate I(γ) 1/γ, and tus assume it is dierentiabe, ten it is straigtorward to see tat I(γ) is a decreasing and convex unction o γ, and W(γ)=W(n) W(1/γ), is a decreasing and concave unction o γ 7. O course, more precisey, I(γ) is an integer unction, and not dierentiabe, and so is not stricty a convex unction (and simiary or W). For te purposes o tis paper we sa assume tat we can treat I(γ) as convex and W(γ) as concave Common Uncertainty. We irst provide some sortand: n I ( γ ), n I( γ ), n I( γ ) (7) Eq. (7) indicates te number signatories tat woud arise rom an IEA i it were known or certain tat a countries ad damage costs equa to ow damage costs, expected damage costs and ig damage costs respectivey. We want to ensure tat uncertainty matters, i.e. tat γ, γ, γ are suicienty distinct rom eac oter tat te resuting stabe IEAs woud ave distincty dierent membersip sizes and dierent aggregate word net beneits; so we make te oowing assumption: Assumption 2: n >> n >> n. (8) From wic it oows tat te word net beneits rom dierent coaition sizes are W ( n ) >> W ( n ) >> W ( ). (9) n To introduce te possibiity o earning we assume tat earning takes te orm o perect earning i.e. te true state o te word is reveaed to a countries. Again tis is a very specia I = 1/ γ ; I = 2 / γ ; W = 2N N γ 1/ γ ; W = (1/ γ ) N = n N < 0; W = 2 / γ. Kostad & Up 8

10 mode o earning. Te crucia issue is te timing at wic suc inormation becomes avaiabe, and we deine tree possibe cases: No Learning: te true state o te word is reveaed to a countries ater a teir decisions (membersip and emissions) ave been taken. Compete Learning: te true state o te word is reveaed to a countries beore any decision is taken, in particuar beore te membersip decision is taken. Partia Learning: te true state o te word is reveaed to a countries ater te membersip game but beore te emissions game. It coud be argued tat te most reevant modes o earning or cimate cange just now are No Learning and Partia Learning. Lemma 3 (Common Uncertainty No Learning) Wit Common Uncertainty and No Learning, te unique stabe IEA as NL NL W n =n =I( γ ) signatories and aggregate word net beneits are = W = (N n)(1 γ N) < 0. Given our assumption o risk neutraity, certainty equivaence appies and te outcome is as i damage costs were known wit certainty to be equa to expected damage costs. Lemma 4 (Common Uncertainty Compete Learning) Wit Common Uncertainty and Compete Learning, te membersip and aggregate word net beneits o te unique stabe IEA in te ow and ig damage cost state are: ( n, W ( n )), ( n, W ( n )) respectivey, were, by Assumption 2, n > n > n and n W ( n ) > W ( n) > W ( ( ). Expected membersip is: n pn + 1 p) n and expected CL aggregate word net beneits are: W = pw ( n ) + (1 p) W ( n ). n > n ; W < W. Te ast resuts oow rom te convexity o I(γ) and te concavity o W(γ). CL CL NL CL NL Kostad & Up 9

11 Lemma 5 (Common Uncertainty and Partia Learning Emission Game) Wit Common Uncertainty and Partia Learning, non-signatory countries aways poute no matter wat te true state o te word or te number o signatories (n). Te emission strategies o signatories, and expected net beneits o signatories and non-signatories are as oows: (i) or n n, signatories abate in bot states o te word and expected net beneits are: V s ( n) = ( N n) γ, V ( n) = 1 ( N n)γ ; (ii) or n <, signatories poute in bot states o te word, and expected net beneits are: V s n ( n) = V ( n) = 1 Nγ ; (iii) or n n <, signatories abate in te ig damage cost state o te word and n poute in te ow damage cost state, and expected net beneits are: V s ( n) = N γ. γ + p + n(1 p) γ, V ( n) = Nγ + 1+ n(1 p) For te next resut we deine: p crit 1 γ γ I( γ ) γ. Lemma 6 (Common Uncertainty and Partia Learning Membersip Game Wit Common Uncertainty and Partia Learning, i p < p crit, ten tere is a unique stabe IEA wit membersip (1) = and aggregate word expected net beneits o: W n n (1) = p[ N(1 γ N )] + (1 p)[( N n )(1 γ N)]; wie i p pcrit tere is a second stabe IEA wit (2) = members and aggregate word expected net beneits o: W n n (2) = ( N n )(1 γ N). NL CL We can summarise te above resuts as oows: n (1) < n < n < n (2) ; CL NL W (1) < W < W < W (2). So wit Common Uncertainty, or a parameter vaues Partia Kostad & Up 10

12 Learning and Compete Learning ave outcomes wic yied ower weare tan No Learning. Wen p p crit tere may be a second equiibrium or Partia Learning wit iger weare tan No Learning. Note owever tat tere are many parameter vaue combinations or wic p p crit wi be excuded by Assumption 2. To provide a irmer base or comparison wit te case o Individua Uncertainty, we sa enceort restrict parameter vaues so tat p < p crit. We tereore ave a singe outcome or Common Uncertainty and Partia Learning: n = n (1); W = W (1), and te comparisons between te tree cases or Common Uncertainty can be summarized as : n < n < n NL CL CL NL ; W < W < W, so wit Common Uncertainty Partia Learning and Compete Learning yied ower weare tan No Learning. 2.2 Resuts or Individua Uncertainty We now turn to te main contribution o tis paper: te case o te ormation o IEAs wit Individua Uncertainty, were eac country aces te uncertainty o aving ig or ow damages independent o te damage costs o oter countries. As in te previous sub-section, we anayse in turn te tree cases o No Learning, Compete Learning and Partia Learning Individua Uncertainty and No Learning As wit Common Uncertainty, in bot te emissions game and te membersip game certainty equivaence appies and eac country acts as i known damage costs were equa to expected damage costs, γ, and so we obtain: Proposition 1. Wit Individua Uncertainty and No Learning, te unique stabe IEA as NL = I( γ ) signatories and aggregate word net beneits are W = (N n)(1 γ N) < 0. So te resuts are te same as wit Common Uncertainty. NL n =n Kostad & Up 11

13 2.2.2 Emissions Game or Compete and Partia Learning We now ave to aow or te act tat i inormation about payers is reveaed beore tey make teir emissions decisions, ten amost a o te N +1 states o te word wi invove a mix o ig and ow damage cost countries. We know rom Assumption 1 tat non-signatory countries wi aways poute, no matter wat teir damage costs are reveaed to be. But we need to anayse te emissions strategy o signatory countries or an arbitrary mix o ig and ow damage cost countries. Notationay, tings get a itte more compicated. Previousy emissions strategies or signatories depended ony on te size o te coaition. Now we are concerned not ony wit te size o te coaition but te mix o ig and ow cost countries in te coaition. Speciicay, te coaition is described by (n, n ), were n is te tota number o countries in te coaition and n is te number o ig damage cost countries in te coaition. For any number o signatories, n 2, o wom n ave ig damage costs (0 n n) deine: nγ + n ( γ γ ) γ ( n, n ) (10a) n n n nγ 1 ( ) (10b) γ γ γ n, n ) is te average damage cost o te signatories, and (n) is te minimum number o ig ( damage cost countries in an IEA o size n wic ensures tat te signatories wi a abate. Since te signatories coose emissions to maximize teir coective net beneits, tey act as i a signatories ad damage costs equa to γ n, n ) and so we get immediatey: ( n Kostad & Up 12

14 Lemma 7 Wit Individua Uncertainty and Compete or Partia Learning, a group o n signatories wit n ig damage cost members wi abate i nγ ( n, ) 1 i.e. n n (n) and wi poute oterwise. It is straigtorward to see tat n (n) is decreasing in n and tat (i) n < n n ( n) n n, (ii) n n n ( n) 0 ; (iii) n n < n 0 < n < n. Tus we can derive a more precise statement o te emission strategies o countries and te resuting net beneits: Lemma 8 Wit Individua Uncertainty and Compete or Partia Learning and a group o n signatories wit n ig damage cost members, non-signatories wi aways poute and: (i) i eiter (a) n < or (b) n n < and n n (n), ten signatories wi poute and n n te net beneits to signatory and non-signatory countries are: < V s i ( n, n ) = V ( n, n ) = 1 Nγ ; i i (ii) i eiter (a) n or (b) n n < and n n (n), ten signatories wi abate and n n te net beneits to signatory and non-signatory countries are: V s i ( n, n ) = γ ( N n); V ( n, n ) = 1 γ ( N n). i i i Outside te range n n < n te emission strategies o signatories are te same as wit Common Uncertainty. Inside te range n n < n we can tink o te strategy or signatories in te case o Common Uncertainty as a specia case o teir strategy wit Individua Uncertainty. Wit Common Uncertainty tere are ony two possibe states o te word: eiter a countries ave ig damage costs, in wic case n = n n (n), and te signatories abate, or a countries > ave ow damage costs, in wic case n = 0 n ( n) and te signatories poute. Tis concudes < te anaysis o te emission game wit Compete and Partia Learning. We now turn to te membersip games. Kostad & Up 13

15 2.2.3 Membersip Game wit Compete Learning Reca tat wie Assumptions 1 and 2 impy tat N > n > n, te distribution o te countries between ig and ow cost depends on te state o te word. Tere are N + 1 possibe states o te word, depending on te number o countries wit ig damage costs, N. Beore categorizing te stabe IEAs tat can arise in eac possibe state o te word, we need to introduce some new deinitions. First we need to extend te deinition o a stabe IEA to recognize bot tat in eac state o te word tere wi be a speciic number o ig and ow cost countries avaiabe, and tat in genera any potentia set o signatories wi contain bot ig and ow cost signatories. Deinition 3: An IEA wit n N ow cost signatories and n N ig cost countries is stabe i it satisies te two conditions: s s Interna Stabiity: (a) V n, n ) V ( n 1, n ) ; (b) V ( n, n ) V ( n, n 1) ; ( s Externa Stabiity: (a) V ( n, n ) V ( n + 1, n ), n N 1; s (b) V ( n, n ) V ( n, n + 1), n N 1. Note tat te externa stabiity condition now ony appies as ong as tere is anoter country o te appropriate cost type wic migt join. Tis as not been an issue wit Common Uncertainty as countries ave aways been identica at te membersip game stage: eiter in te ex ante sense o aving te same expected costs; or in te ex post sense o aving te same reveaed costs - Common Uncertainty, and Assumption 2 guarantee tat tere are aways suicient countries suc tat te Externa Stabiity condition can be appied witout restriction. In te next Proposition we identiy or any state o te word caracterised by N ig damage cost countries wat are te possibe stabe IEAs. In doing so we need to introduce te Kostad & Up 14

16 oowing deinition o a minimum number o ow cost countries necessary to generate a stabe IEA: Deinition 4: N n n ( n 1) Ten we ave: Proposition 2 Wit Individua Uncertainty and Compete Learning, or any state o te word wit N ig damage costs cost countries, consider te oowing two conditions on N : (A) N N N ; and (B) N. Ten depending on weter eiter or bot o tese conditions n od, te possibe stabe IEAs are as oows: (i) i condition A ods ten an IEA wit n signatories in tota, o wom N n n are ow cost and te rest, n = n n (wic may be zero), are ig cost, wi be stabe, te signatories wi abate and aggregate weare in tat state wi be: ˆ W ( N ) ( N n )[1 N ( γ γ ) Nγ ]; (ii) i condition B ods ten an IEA wit signatories, a o wom are ig cost, wi be n stabe, te signatories wi abate, and aggregate weare in tat state is ˆ ) ( = N n W ( N )[1 N ( γ γ ) Nγ ]; (iii) i neiter conditions A nor B od, ten tere is no stabe IEA, te outcome is te noncooperative equiibrium in wic a countries poute and aggregate weare or tat state is Wˆ ( N ) N[1 N ( γ γ ) Nγ ]. Te proo is in Appendix 1. Beore providing te intuition, it is wort noting te oowing impications o Proposition 2. Note irst tat i N N + n < tere wi some states o te word or wic te condition (iii) ods, so tere is no stabe IEA, and te outcome is te non-cooperative Kostad & Up 15

17 equiibrium, an outcome worse tan any in te case o Common Uncertainty. Second, i ( n 1) 1 ten (i) can ave mutipe stabe IEAs a o size, but wit between 0 and n n ( 1) ig cost countries and te rest ow cost. However weare is te same irrespective o n n te number o ig cost signatories. Tird, i N N + n ten tere are no states o te word or wic te condition or (iii) ods but tere wi be states o te word or wic te conditions or bot (i) and (ii) od, so again tere wi be mutipe stabe IEAs. Finay, note tat te possibiity o mutipe stabe IEAs means we cannot assign a unique outcome and ence a unique aggregate weare eve or eac state o te word, and ence we cannot compute expected weare or te Compete Learning case. We discuss ow we resove tis issue in Section 3. To understand te intuition beind Proposition 3, assume tat N N N n. No IEA wit n > n signatories can be stabe, because, by Lemma 8, i one country et, signatories woud continue to abate, and it woud tereore aways pay a country, o eiter type, to eave. I tere are n signatories, o wom at most n ( 1), possiby zero, are ig cost, ten, again by Lemma 8, n te deection o any country wi cause a remaining signatories to poute and tis is a suicient treat to make suc an IEA stabe. Any IEA o size n ying stricty between and cannot be n n stabe. I n > n n (n), so te signatories abate, ten any ow cost country wi eave, since even i tat triggers te remaining signatories to poute, te gain rom being abe to poute outweigs te damage cost, since n < n. I n = n, ten any ig cost country wi deect since it knows remaining countries wi abate. Finay i n n (n), so signatories poute, it wi pay a < ig cost country to join te IEA, since i tis means signatories continue to poute it is no worse o by joining, wie i it induces signatories to abate, it is stricty better o since n > n. Tis Kostad & Up 16

18 eaves an IEA wit IEA wit n countries, a ig cost, as a stabe IEA, by te arguments beind Lemma 2. So in tis case tere are mutipe stabe IEAs, as given in Proposition 3 (i) and (ii). I N > N N, tis eiminates te irst type o stabe IEA, given in Proposition 3(i). I N < n tis rues out te second type o stabe IEA, given in Proposition 3(ii). I bot are true, tere is no stabe IEA, and te outcome is te non-cooperative equiibrium. In comparison wit te case o Compete Learning wit Perect Correation, te ricer set o states o te word wit No Correation resuts in a ricer set o possibe stabe IEAs. However te possibiity o mutipe stabe IEAs or any given state o te word, makes te cacuation o te expected size o IEAs and expected weare impossibe witout some criterion or seecting between te dierent stabe IEAs. We return to tis issue in Section Membersip Game wit Partia Learning At te membersip game stage, countries do not know teir type, and need to cacuate expected payos or a typica signatory and non-signatory country or any given size o IEA, using Lemma 8 to identiy te emission strategies o signatories. From Lemma 8, countries know: (i) i n ten signatories wi abate; (ii) i n < ten signatories wi poute; (iii) i n n n < ten signatories wi abate i n n (n) and poute oterwise. In te irst two cases it n n is straigtorward or countries to work out wat te expected payos wi be i an IEA o size n is ormed. For te ast case we need a ew more deinitions to aow countries to compute te expected payos or signatories and non-signatories. Starting rom an IEA o n members, te random variabe tat is te number o members tat are ig cost is distributed as binomia. In particuar, or any n suc tat n n < n, te Kostad & Up 17

19 probabiity tat exacty i o tese countries wi be ig cost is te same as te probabiity o i successes in n independent Bernoui trias (two discrete outcomes), were a success is deined as a ig cost draw (probabiity 1-p). Notationay, we deine φ( i, n) as te probabiity o aving i out o te n countries turn out to be ig cost countries, or any i suc tat 0 i n ; next we deine te compementary cumuative distribution unction as n ϕ ( n) φ( i, n), wic is te i= n ( n) probabiity tat te signatories wi abate, i.e. te probabiity tat at east n (n) out o n countries turn out to be ig cost countries; and inay we deine n ~ γ ( n) φ( i, n) γ ( n, i) as te expected i= n ( n) damage cost tat wi be saved by eac o te n signatories wen tey do indeed abate. Ten we ave: Lemma 9 Wit Individua Uncertainty and Partia Learning, te expected payos to signatory and non-signatory countries or any number o signatories, n, are as oows: (i) s n n ; V ( n) = ( N n) γ ; V ( n) = 1 ( N n) γ ; s (ii) n < n ; V ( n) = V ( n) = 1 Nγ ; (iii) n n < n ; V s ~ ( n) = 1 Nγ ϕ( n) + nγ ( n); V ( n) = 1 Nγ + n ~ γ ( n) Note tat or te payo in a tree cases, te irst two terms, 1 Nγ, represent te expected payos in te non-cooperative equiibrium. Te remaining terms represent te net gains rom orming an IEA, reative to te payos in te non-cooperative equiibrium. Tus te tird term, or signatories and non-signatories, is te expected net savings to eac country in damage costs wen te n signatories abate: wit Individua Uncertainty signatories abate ony i tere are at east n (n) ig cost signatories and te expected savings are tereore nγ ~ ( n ). Finay, or signatories te ourt term represents te expected oss o output to eac signatory wen tey ave to abate, Kostad & Up 18

20 wic is given by te probabiity tat signatories abate, wic is φ(n) wit Individua Uncertainty. Tis eads to te oowing resut on te stabiity o an IEA: Lemma 10 Wit Individua Uncertainty and Partia Learning, no IEA wit n members suc tat n > n or n < n can be stabe. Tis just oows or te same reasons as wit Lemma 2: or n > n any signatory knows tat i it eaves te IEA te remaining signatories wi continue to abate, and so it pays to eave, wie or n < signatories wi poute, so a country is as we o joining an IEA as staying out. n However tis is as ar as we ave been abe to proceed anayticay. Lemma 10 cannot te us weter tere exists a stabe IEA, weter tere exist mutipe stabe IEAs, and ow any suc stabe IEAs compare wit te tree candidates or stabe IEAs so ar identiied in tis paper, n, n, n. In te next section we present resuts o some numerica simuations wic sow (i) or a o te sets o parameter vaues we study tere aways exists a stabe IEA, and in 99.5% o cases it is unique, but in 0.5% o cases tere are 2 stabe IEAs; (ii) in 90.1% o cases te stabe IEA is n, but in te remaining 9.9% o cases te stabe IEA ies stricty between n and n. As we discussed in te case o Compete Learning, te existence o mutipe stabe IEAs means we cannot compute expected aggregate weare or Partia Learning and Individua Uncertainty. We sa aso resove tis issue using te numerica simuations we report in te next section. 3. NUMERICAL SIMULATIONS Te anaysis in Section 2.2 was not abe to determine anayticay two issues: (i) wat are te stabe IEAs or Individua Uncertainty and Partia Learning; (ii) or Individua Uncertainty wat are te expected weare comparisons between No Learning, Partia Learning and Compete Kostad & Up 19

21 Learning. We ave tereore used numerica simuations to sed igt on tese issues, and we report on tese in turn. 3.1 Stabe IEAs or Individua Uncertainty and Partia Learning To tis point tere ave been tree candidates or a stabe IEA: n I ( γ ), n I ( γ ), n I( γ ) in wic signatories abate. Tat we ave not been abe to determine anayticay wat te stabe IEAs migt be or te case o Individua Uncertainty and Partia Learning is in part expained by te act tat te numerica simuations we ave conducted ave reveaed tat tere can be mutipe stabe IEAs, and tat in genera te stabe IEAs need not be any o te tree candidate vaues identiied in te previous cases. We now describe te resuts o tese numerica simuations. For te numerica anaysis o stabe IEAs wit Individua Uncertainty and Partia Learning te tree key parameters are p, γ, γ. We ave conducted a Monte Caro simuation over vaues o pε[0,1], γ ε[1/n,1] and γ ε[1/n, γ ), randomy coosing parameter vaues bearing in mind Assumption 2, te need to ave distinct vaues o n, n, n. As discussed at te end o Section 2.1, in randomy coosing parameter vaues, we discard tose in wic p pcrit. In addition, to rue out very ig numbers o countries, we excuded cases were n n + > 125. Tis reduced te number o combinations o parameter vaues we coud use to 26235, 82% o te u set o possibe combinations. Finay in appying our stabiity criteria (Deinitions 1 and 2), wic use strict inequaities between te reevant payos o signatories and non-signatories, we ave appied a trembing and criterion to rue out virtua equaity between te payos. So, or exampe, or Interna Stabiity we ave required tat s s [ V ( n) V ( n 1)] > τ {0.5[ V ( n) + V ( n 1) ]}, were τ is a very sma number (10-15 ); tat is, te dierence between te payo a signatory gets rom Kostad & Up 20

22 staying in te IEA and te payo it woud get i it et te IEA must be stricty positive in te sense o being greater tan τ times te average o te absoute vaues o te two payos. For eac o 100,000 randomy cosen sets o parameter vaues, excuding tose wic do not meet te criteria mention above, we ave cacuated te stabe IEAs wit Individua Uncertainty and Partia Learning. 8 Te resuts can be summarized as oows. In ess tan 1% o te cases tere was no stabe IEA. In te remaining cases, in 94% tere was a unique stabe IEA; in 5% o cases tere were 2 stabe IEAs. A very ew ad tree or our stabe IEAs, toug tose may be attributabe to numerica issues. In terms o te size o te stabe IEAs, in a o te cases, te number o members o stabe IEA was between n and n. For te cases wit a unique IEA, over two tirds o te cases were or a size o n. For te cases wit two stabe IEAs, in most cases one was o size and te oter was stricty between and n. Tus in our simuations, te size o te n n stabe IEA wit Individua Uncertainty and Partia Learning is aways ess tan wit Individua Uncertainty and No Learning. 3.2 Weare Comparison wit Individua Uncertainty and No, Partia and Compete Learning We noted in Section 2.2 tat wit Individua Uncertainty te existence o mutipe stabe IEAs or bot Partia Learning and Compete Learning made it impossibe to cacuate expected weare or tese two cases and compare expected weare wit No Learning. In tis section we 8 Wit 100,000 Monte Caro trias, 16,322 passed a o our criteria or parameter vaues and ranges. O tese, 0.95% ad no equiibria, 94.27% ad one equiibrium, 4.69% ad two equiibria, 0.09% ad tree equiibria and 0.01% ad our equiibria. O te cases wit one equiibrium, 72% o te equiibria are at n, 27% stricty between n and n and 0.1% at n. O te cases wit two equiibra, 42% o te equiibra are at, 56% are stricty between and n and 1.5% are at n. n n Kostad & Up 21

23 describe ow we resove te issue o mutipe stabe IEAs and ten use numerica simuations to compare word net weare under te tree dierent earning regimes. Wit No Learning we know tat te unique stabe IEA as NL NL aggregate expected weare = (N- n )(1-Nγ ). W n NL = n signatories and In te case o Partia Learning, i tere is more tan one stabe IEA we simpy seected te stabe IEA wit igest expected weare. Denote te size o tat stabe IEA by n wit associated aggregate expected weare, rom Lemmas 9 and 10, given by W ~ N[1 Nγ ] + n [ Nγ ( n ) ϕ( n )]. For Compete Learning, state-dependent weare is a unction o te number o ig cost countries (Prop. 3). Again, i or any vaue o N tere are mutipe stabe IEAs we ave seected te one wit iger word weare 9. Expected membersip and weare invoves taking an expectation over tese states, using te probabiity o N ig cost countries out o N countries, as te probabiity o N successes out o N Bernoui trias. We denote te resuting expected stabe IEA and expected weare by n, W CL CL respectivey. Tabe I sows te resuts o Monte Caro simuations, as described earier. 10 In tis case, we conduct te simuations or dierent N (tota numbers o countries). We are interested in te size o agreements or te tree dierent kinds o earning, as we as overa weare. Tabe I sows tat te resuts or Individua Uncertainty conirm te indings or Common Uncertainty wit respect to expected weare or Partia Learning: or neary a cases, expected 9 Reca aso tat rom Proposition 3(i), were te stabe IEA as signatories, tere coud be potentiay be a sma number o ig cost signatories, and so tere coud be mutipe stabe IEAs depending on precisey ow many ig cost signatories tere migt be. Since tis as no weare signiicance, we sa simpy assume tat a n signatories are ow cost. 10 For eac coumn sown in te Tabe, 10,000 Monte Caro runs were conducted, resuting in vaid combinations o parameter vaues (smaer number or smaer N). n Kostad & Up 22

24 weare wit Partia Learning is stricty ower tan wit eiter No Learning or Compete Learning. For Compete Learning, owever, in a majority o cases expected weare wit Compete Learning is iger tan wit No Learning or Partia Learning, wic is contrary to te case wit Common Uncertainty were expected weare wit Compete Learning is aways ess tan expected weare wit No Learning. Furtermore, as te number o countries increases, te ordering o weare or te tree types o earning appears to asymptote to compete earning giving igest expected weare and partia earning te owest. Te resuts or expected size o IEA wit Individua Uncertainty broady mirror te resuts or expected weare. Te rationae is tat IEAs do not necessariy bring signiicant weare gains over te Non-Cooperative equiibrium. Asymptoticay, it woud appear tat Compete Learning gives te argest IEA, wereas Partia Learning yieds te owest. 4. Concusions In tis paper we ave extended our earier work on te ormation o internationa environmenta agreements under uncertainty and dierent modes o earning to aow or inormation to revea dierences between countries. In te case o Partia Learning, wic, as indicated in Section 2, we consider to be te more reevant mode o earning just now or issues suc as cimate cange, our anaysis sows tat expected IEA membersip may be somewat iger wit Individua Uncertainty tan wit Common Uncertainty and, wit Individua Uncertainty, may be iger tan wit Compete Learning, toug aways ess tan wit No Learning. However, it remains te case tat, just as wit Common Uncertainty, wit Individua Uncertainty, expected weare wit Partia Learning is aways ess tan wit eiter No Learning or Compete Learning. For te more pausibe mode o earning, Partia Learning, our anaysis Kostad & Up 23

25 suggests tat our inding in te case o Common Uncertainty tat inormation can ave negative socia vaue is robust wen we aow inormation to revea dierences between countries. To noneconomists it may seem strange tat inormation can ave negative vaue, since or a singe decision-maker inormation cannot ave negative vaue, because it can aways be ignored. However economists ave ong been aware tat wen tere are strategic interactions between a number o decision-makers responding to inormation, tese strategic interactions can give inormation a negative vaue. O course te anaysis in our paper is extremey simpe. In a recent paper Deink, Finus and Oieman (2005), using a somewat dierent approac to ours (te stabiity ikeiood: ow ikey is it tat a particuar coaition o countries woud orm a stabe IEA) expore te impications o uncertainty and earning in an empirica mode o cimate cange using 12 word regions. Teir mode is rater ricer tan ours, or exampe aowing countries to cose rom a continuum o abatement strategies, and using a wider range o uncertainties, wit uncertainties about goba damage costs, te sare o goba damages or eac country (wic is modeed in a way quite simiar to our assumption o Individua Uncertainty), and abatement costs. Tey consider ony Compete Learning and No Learning. For teir base case tey sow tat expected weare wit Compete Learning is ower tan wit No Learning, toug in teir aternative speciication, wit a ower mean vaue o goba damage costs, tey sow tat expected weare wit Compete Learning is iger tan wit No Learning. Tis woud seem to conirm our inding tat wit Compete Learning te vaue o inormation may be positive or negative. However it is interesting to note tat te reason tat teir aternative speciication produces a positive vaue or inormation wit Compete Learning is tat in teir No Learning case tere is no stabe coaition. In our mode it is ony in te Compete Learning case tat tere is a possibiity tat tere is no Kostad & Up 24

26 stabe IEA, and in suc a case te vaue o inormation wit Compete Learning is unambiguousy negative. However we cite te Deink et a (2005) paper to suggest tat our concusions about te impact o earning on IEA are not due just to te rater specia eatures o our mode. Kostad & Up 25

27 Tabe I: Comparisons o Expected Weare and Size o IEA For Tree Dierent Modes o Learning Number o Countries (N) RESULTS Expected Weare W > W > W CL NL NL CL W > W > W Expected Size o IEA CL NL n > n > n n NL NL > n n CL n > n > n CL Resuts sow, or eac Case, proportions o Monte Caro runs (cosen by randomy seecting vaues o γ, γ and p, resuting in vaid sets o parameters, meeting criteria discussed in text). Sown are resuts or (i) expected weare (ii) expected size o IEA across tree dierent NL NL modes o earning: No Learning ( W, n ), Partia Learning ( W, n ) and Compete Learning ( W CL, n CL ). Kostad & Up 26

28 Reerences Arrow K. and A. Fiser, Environmenta preservation, uncertainty and irreversibiity, Quartery Journa o Economics, 88: (1974). Barrett, S., Se-enorcing internationa environmenta agreements, Oxord Economic Papers, 46: (1994). Barrett, S., Environment and Statecrat (Oxord University Press, Oxord, 2002). Barrett. S., Internationa Cooperation or Sae, Eur. Econ. Rev., 45: (2001). Boringer, C and M. Finus, An economic anaysis o te Kyoto Agreement, in D. Hem (ed.), Cimate Cange and Poicy Response, (Oxord University Press, Oxord, 2005). Carraro, C. and D. Siniscaco, Strategies or te internationa protection o te environment, J. Pubic Economics, 52: (1993). Cooper, R., Internationa cooperation in pubic eat as a proogue to macroeconomic cooperation, pp in R. N. Cooper et a (ed.), Can Nations Agree? (Brookings Institution, Wasington, DC, 1989). Deink, R., M. Finus, and N. Oieman, Te stabiity ikeiood o an internationa environmenta agreement, FEEM Working Paper , Venice (2005). Finus, M., Game Teory and Internationa Environmenta Cooperation (Edward Egar, Cetenam, UK, 2001). Foster, K. P. Veccia and M. Repacoi, Science and te precautionary principe, Science, 288, (2000). Goier C., B. Juien and N. Treic, Scientiic progress and irreversibiity: an economic interpretation o te Precautionary Principe, J. Pubic Economics, 75: (2000). Kostad & Up 27

29 Henry C., Investment decisions under uncertainty: te irreversibiity eect, American Economic Review, 64: (1974a). Henry C., Option vaues in economics o irrepaceabe assets, Rev. Econ. Stud., 41(S): (1974b). Ingam A. and Up A., Uncertainty, irreversibiity, precaution, and te socia cost o carbon, in D. Hem (ed.), Cimate Cange and Poicy Response, (Oxord University Press, Oxord, 2005). Kostad, C, Systematic uncertainty in se-enorcing internationa environmenta agreements, J. Environmenta Economics and Management 53:68-79 (2007). Kostad, C. and A. Up, Learning and internationa environmenta agreements, Cimatic Cange, 89: (2008). Na, S.-L. and H. S. Sin, Internationa environmenta agreements under uncertainty, Oxord Economic Papers, 50: (1998). Narain, U, A. Fiser, and M. Hanemann, Te irreversibiity eect in environmenta decisionmaking, Environmenta and Resource Economics, 38: (2007) Up, A. and D. Maddison, Uncertainty, earning and internationa environmenta poicy coordination, Environmenta and Resource Economics, 9: (1997). Up, A., and D. Up, Wo gains rom earning about goba warming?, pp in E. v. Ierand, and K. Gorka (eds.) Te Economics o Atmosperic Poution (Springer-Verag, Heideberg, 1996). Up A. and D. Up, Goba warming, irreversibiity and earning, Econ. J., 107: (1997). Up A., Stabe internationa environmenta agreements wit a stock poutant, uncertainty and earning, J. Risk and Uncertainty, 29:53-73 (2004) Kostad & Up 28

30 Young, O., Internationa Governance: Protecting te Environment in a Stateess Society (Corne University Press, Itaca, NY, 1994) Kostad & Up 29

31 Appendix 1 Proo o Proposition 2 Suppose to begin tat N N n + 1 n ( n 1), N n. Ten consider te oowing possibiities or a stabe IEA. (i) n >. Tis cannot be stabe. ( n) = 0, so i any country i (ig or ow cost) eaves it wi n n cause te remaining countries to continue to abate. Te gain to eaving is tus 1- γ i > 0. (ii) n = (a) 0 n n ( n 1) 1. Since n ( ) = 0, te signatories wi abate poution. I any n n country i (ig or ow cost) eaves, it wi cause remaining countries to poute. Since n γ n γ 1 tere is no gain to eaving, so tis is a stabe IEA. (b) > n = n <. Since ( n 1) n n 1, a ow cost country woud aways eave, since te remaining signatories woud continue to abate and te gain to te quitting country woud be 1 - γ > 0. (c) n ( n 1) + 1 n n same argument a ig or ow cost country woud ave an incentive to quit since te remaining signatories wi continue to abate. (iii) n < n < n, n n ( n). By te (a) Suppose n 1; ten a ow country wi aways eave, since eiter n n ( n 1), in wic case remaining signatories abate and te gain is 1 γ >0, or n ( n 1) > n n ( n), in wic case te remaining signatories poute, and te gain is 1 nγ > 0. So any ow cost country wi aways eave te IEA. (b) Suppose ten tat n = 0, so n = n n ( n) n 1 = n 1 n ( n 1). Ten i any ig cost country quits te IEA, te remaining signatories wi abate and te gain to eaving is 1 γ > 0. So ig cots country woud aways quit. Kostad & Up 30

32 (iv) n < n < n, n < n ( n) < n ( n 1). So a signatories poute. I any country eaves, ig or ow cost, signatories wi continue to poute. Since since staying in te IEA is no better tan quitting, an IEA o tis type is not internay stabe. From (iii) and (iv) no n suc tat n < n < n can be stabe (v) n = n = n = n ( n ). A countries abate. I one country quits, remaining signatories poute. Gain is 1 - n γ < 0 by deinition o n, so tere is no gain to quitting. So tis is a stabe IEA. (vi) n < n. Signatories aways poute, so by te same argument as in (iv) no IEA o tis size can be stabe. So i N N, N n, tere are two possibe stabe IEAs: (a) n n, 0 n < n ( n 1) ; = (b) n = n = n I N < N ten te stabe IEA o type (a) cannot exist. I N <, ten te stabe IEA o type (b) n cannot exist. I bot N n N < and N < tere is no stabe IEA and te outcome is te noncooperative equiibrium. Tis competes te proo o Proposition 2. Kostad & Up 31