Uncertainty in the emissions game : an analysis of the stability of international environmental agreements

Faculty of Economics and Business

Amsterdam School of Economics

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Uncertainty in the emissions game: An analysis of the stability

of international environmental agreements

Nina Floor van Ettekoven

10071369

MSc in Econometrics Track: Free Track

Date of final version: 24 March 2017 Supervisor: Dr. T.A. Makarewicz

Abstract

International cooperation to achieve global warming mitigation goals has proven difficult over the last decades. This study aims to determine the effect of uncertainty in emission levels and asymmetry among countries on the stability of international climate agreements. The global warming process is modeled as a dynamic public goods game in which the state variable is the current stock of pollutants. The implications of the theoretical results are studied in a realistic eight-region application example. I find that uncertainty in emissions limits the possibilities of reaching a successful agreement. Asymmetries among countries in terms of benefits from emission and environmental damage costs impede cooperation as well. Numerical analysis shows that a cooperative equilibrium is possible, though severe restrictions on the discount factor and punishment strategy are imposed.

Acknowledgements

First of all, I am grateful to Dr. Tomasz Makarewicz for supervising this thesis project, monitoring my progress and providing me with valuable insights and suggestions. Furthermore, I am thankful to Ernst de Kwaasteniet for revising the text. Lastly, I would like to thank my friends and family for their support during the writing process.

This document is written by student Nina Floor van Ettekoven who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work,

Contents

1 Introduction 4

2 The stage game 11

2.1 Two benchmarks . . . 12

2.2 Asymmetry . . . 13

3 The dynamic emissions game 14 3.1 The social optimum . . . 14

3.2 The Nash equilibrium . . . 16

3.3 Asymmetry . . . 17

4 A climate agreement under perfect monitoring 19 4.1 Stability conditions . . . 19

4.2 Results . . . 21

4.2.1 Full symmetry . . . 21

4.2.2 Two asymmetric regions . . . 22

4.2.3 Three asymmetric regions . . . 25

4.2.4 Application example: eight regions case . . . 28

5 A climate agreement under imperfect monitoring 32 5.1 Stability conditions . . . 32

5.2 Results . . . 36

5.2.1 Full symmetry . . . 36

5.2.2 Two asymmetric regions . . . 40

5.2.3 Three asymmetric regions . . . 44

5.2.4 Application example: eight regions case . . . 46

6 Conclusions 49

References 50

Appendices 52

Appendix A. Proofs 52

1

Introduction

In this thesis I will study the effect of uncertainty in countries’ emission levels on the stability of international climate agreements. I will show that uncertainty limits the possibility of a successful agreement. Asymmetries amoung countries in terms of benefits from emission and environmental damage costs hinder cooperation as well.

Climate change has emerged as a central issue in today’s world, and its consequences are widespread. 15 of the top 16 warmest years ever recorded have occurred since 2000 (National Centers for Environmental Information, 2016). The worlds ice-belts in the Arctic and Antarctic are melting and the Intergovernmental Panel on Climate Change (IPCC) predicts that if we continue on the current path, by 2100 global sea levels will probably have risen by 9 to 88 cm and average temperatures by between 1.5 and 5.5 degrees Celsius (Dutta and Radner, 2009). The major factor contributing to global warming is the increase in greenhouse gases (GHG) caused by the burning of fossil fuels. According to the IPCC, human activity is responsible for most of the global warming we have witnessed since the 1950s (Pacheco et al., 2014). In absence of a world government, the mitigation of climate change requires countries’ to coordinate their efforts, which must be achieved by means of an international agreement. However, similar to other public goods dilemmas of collective action, a country that curbs emissions pays an individual cost, while the benefits are shared between all the countries. This results in incentives to free ride: benefiting from other countries’ emission reduction without making the (costly) effort to curb emission oneself (Pacheco et al., 2014). Another complication for such an agreement is uncertainty. As will be explained soon, uncertainty can arise in different ways.

Game-theoretic literature has been studying the stability of international environmental agree-ments for well over two decades. The climate change issue is often modeled as a public goods game, wherein countries are assumed to be unitary actors making choices about their emission levels so as to maximize their expected payoffs. The prisoners’ dilemma is the most cited game in the literature that gives a simple explanation towards the difficulties of reaching a stable international environmental agreement. Finus (2001) gives the example of two countries suffering from transboundary emissions resulting from energy production. If both countries would switch from an old to a new technology, net benefits would increase for both countries as compared to the present situation. However, even though this switch is attractive to both countries, it is even more beneficial for a country not to invest in the new technology if the other country does. The free-rider enjoys a cleaner environment, whilst not having to bear the costs of the

new technology. Because of this free-rider incentive, in equilibrium both countries choose not to invest in the new technology, resulting in social inefficiency. Ward (1996) defines a simple static game with two players (nations, or blocks of nations) and reaches the same conclusion. The Nash equilibrium is where both players defect. This equilibrium is not Pareto efficient, as both players would be better off if both cooperated.

Ward (1996) extends his simple analysis to a dynamic supergame, which is a repeated one-shot game. When binding agreements are impossible, the key to stable, cooperative action is that a player’s cooperation is conditional on the past cooperation of the other player. There should be a threat to retaliation: if one player does not cooperate in one period, the other player should refuse to continue to cooperate from the next period on. Thus, players should use a grim trigger strategy. Whether conditional strategies deter free riding depends on the severity of punishment and credibility of the threat. Ward (1996) concludes that a stable cooperating equilibrium is possible if the gain from short-term free riding is low, the penalty per round of breakdown is high and future payoffs are not too heavily discounted. Finus (2001) also extends his analysis to a symmetric two-player dynamic game with an infinite time horizon, and emphasizes the importance of credible punishment in reaching a stable agreement. While the Folk Theorem ensures that the full cooperative outcome can be sustained as an equilibrium for any number of countries as long as the discount factor is sufficiently high, this result may not hold if the threat of punishment is not credible (Barrett, 1994). Finus (2001) therefore argues that an agreement should be renegotiation-proof: if one of the countries deviates, it must be rational for the other country to carry out the punishment. An agreement is unstable if players cannot suppress the temptation to renegotiate, as this makes the threat of punishment non-credible. Finus (2001) distinguishes two cases: a discount factor close to 1 and a discount factor strictly lower than 1. The latter case is motivated by possible impatience of political representatives, focusing mainly on short-term political success (Hahn, 1987). It is found that there exists punishment profiles for which a stable, renegotiation-proof agreement is possible in the infinite horizon case for both cases of the discount factor.

Dockner et al. (1996) also consider a symmetric two-player dynamic emissions game. Their study differs from the aforementioned ones in that the pollutant stock is allowed to accumulate. The ’global Pareto optimum’ is defined as the situation wherein players choose their emission levels such that joint welfare is maximized. The authors show that it can be sustained as an equilibrium if the discount factor is high enough and players use a trigger strategy of reverting

to competitive emission levels forever, once the other player deviates. Likewise, Chander and Tulkens (1995) argue that if players agree to abandon the agreement once any party defects, this is sufficient to keep the agreement stable. However, Wagner (2001) reaches a different conclusion and states that the trigger strategy is too unforgiving and therefore not renegotiation-proof. Why would countries harm themselves by reverting to Nash equilibrium emission levels forever? Views on punishment strategies are clearly divided. Barrett (1994) argues that there is a trade-off between the effectiveness and the credibility of punishment: on the one hand, the magnitude of punishment should be severe enough in order to deter free riding, while on the other hand it should not be too harsh as to remain credible. However, in a survey on public good experiments, Chaudhuri (2011) finds that conditional cooperators1 _{are often willing to punish free-riders, even}

if such punishment is personally costly and has no benefits in the long run. Furthermore, even in one-shot versions of the game, contributions to the public good (climate change mitigation in the present case) were much higher than predicted in the Nash equilibrium. This suggests that players might be willing to hand out harsher punishments in practice than would be predicted in theory.

Another extension to the emissions game regards the number of players. Barrett (1994) studies a static representation of a coalition formation model in a N -country world, focusing on the number of countries that would be willing to cooperate and sign an international environmental agreement. The situation considered is fully symmetric, implying that costs and benefits from emission are assumed to be equal for all countries. The author finds that many countries will sign the international environmental agreement, but only if the benefit-cost ratio from emissions is small. This is precisely the case where the global net benefits are only slightly increased in the agreement compared to the non-cooperative outcome. Conversely, cooperation would increase global benefits the most if the benefit-cost ratio from emission is large, but in this case the agreement is not sustainable with a large number of signatories. Thus, paradoxically, the agreement will achieve very little if the degree of participation is high. Furthermore, for some specifications of countries’ payoff functions, the degree of participation will be low for any value of the benefit-cost ratio. For example, under linear marginal benefits and constant marginal damage costs, only 2 or 3 countries would sign the agreement. If the payoff function implies constant marginal benefits and linear marginal damage costs, no equilibrium is found at all. Carraro and Siniscalco (1993) and P´ereau and Tazda¨ıt (2001) also study a static coalition formation model and draw a similar conclusion, namely that a successful coalition can only

1_{A conditional cooperator is defined as a player whose contribution to the public good is positively correlated}

consist of a small number of identical players.

Several studies consider a dynamic representation of the symmetric N -player game (Barrett, 1994; Barrett, 1999; Barrett, 2003). The climate agreement is modeled as an infinitely repeated game. As the agreement should be renegotiation-proof, the studies conclude that the full cooperative outcome can only be sustained if the difference between global benefits in the social optimum and the non-cooperative outcome is small. Froyn and Hovi (2008) offer a more optimistic view. The authors consider a dynamic game with N identical players and a binary choice between cooperation and defection, and demonstrate that it is possible to sustain a climate agreement with full participation. Asheim and Holtsmark (2009) show that this result carries over to the case where players choose their emission level from a continuum of choices if the discount factor is high enough. However, in either case only a few players are permitted to punish free riders.

In reality, countries are not identical. There exist substantial asymmetries regarding countries’ benefits from emission and damage costs from climate change. The effect of such country asymmetries on the stability of an agreement is not clear cut. A necessary condition for a climate agreement to be stable is that it is profitable for all signatories (Wagner, 2001). In the case of identical players, this condition is always fulfilled as the benefits from cooperation are shared equally. However, if players are strongly asymmetric this need not be the case. As will be shown later in this study, countries with relatively high benefits from emission and low damage costs might actually be better off in a non-cooperative outcome. Hence, asymmetry among countries might hinder cooperation. Hoel (1992) assumes a payoff function with symmetric linear marginal benefits and asymmetric constant damage costs, and concludes that the asymmetry in costs decreases the maximum number of signatories to an agreement compared to the full symmetry case. Bauer (1992) confirms this finding. In contrast, Carraro and Siniscalco (1993) and Barrett (1994) find that asymmetry may allow for larger coalitions than in the fully symmetric case, as countries with high abatement costs are likely to finance emission reductions in countries with lower abatement costs. As a result, more countries would be willing to commit themselves to cooperation. Finus (2001) considers the situation of N asymmetric countries in a supergame framework. Countries have equal benefits from emission but different, uniformly distributed, damage costs. Clearly, countries with higher damage costs have a greater interest in a reduc-tion of emissions than countries with low damage costs. They choose lower emissions in the non-cooperative equilibrium than countries with low costs, as these high damage cost countries suffer more from pollution. It is found that the socially optimal solution - where the sum of the

N countries’ payoffs is maximized - is stable only when the number of countries suffering from pollution is small. The problem for the case of many countries is that the optimal coordination of punishment may be difficult and entails high transaction costs. Therefore, Finus (2001) argues that if the number of countries is large, punishment profiles should be simple in order for a grand coalition to be stable. Heitzig et al. (2011) also study a model wherein costs from climate change are different across countries, while benefits from emission are the same. The payoff function considered implies linear marginal benefits from emission and linear damage costs. It is found that this type of asymmetry between players allows for a full cooperative equilibrium and that renegotiation can be avoided. However, the design of their agreement requires a social planner, which is politically unrealistic. Vasconcelos et al. (2014) find that if countries are heterogeneous in terms of wealth, environmental agreements are much more difficult to achieve.

A common feature of the aforementioned studies is that they are built on the notion of perfect information. That is, players know exactly the benefits from emission, environmental damage costs and past emission levels of all other players. However, the perfect information assumption is unrealistic. First, there might be information asymmetries between players. Statistics are rare and incomplete in some parts of the world, so that knowledge about emission benefits and damage costs in one region might be unavailable to the other (Caparr´os et al., 2004). Second, there is considerable uncertainty about the exact costs of climate change (Kolstad and Ulph, 2008; IPCC 2015). Third, measuring techniques of the pollution stock are not perfectly accurate, so that other players’ past emissions cannot be precisely observed (Laukkanen, 2009; IPCC, 2015). Several authors have taken imperfect information into account in their research. Caparr´os et al. (2004) consider an emissions game with two regions, North and South, where the North has incomplete information about the South’s capacity to reduce emissions. The authors show that information asymmetry complicates reaching a cooperative agreement. Cherry et al. (2015) focus on imperfect information regarding damage costs from climate change in an experimental setting. Participation in the environmental agreement was high, and contributions to the public good (the mitigation of climate change) were significant, even when costs from climate change were uncertain. Konrad and Thum (2012) find that uncertainty in the costs from emissions abatement reduces the probability of reaching an efficient international environmental agreement. Kolstad and Ulph (2008) study the effect of imperfect information about benefits from emissions as well as environmental damage costs, and find that uncertainty decreases the size of a cooperating coalition in an international environmental agreement. Also, Kolstad and Ulph (2008) study the effect of learning. The first result is negative: if countries learn about their benefit-cost

ratio after committing to a coalition but before deciding on their emission level, it is found that the size of a stable coalition is reduced. However, if the benefit-cost ratio from emission is low, learning actually tends to result in a larger stable coalition. Unfortunately, the benefits from cooperation are only modest in case of a low benefit-cost ratio. This discouraging result is in line with earlier results of Barrett (1994) that the number of countries willing to cooperate in an international environmental agreement tends to be smallest when the agreement is most needed. Athanassoglou (2009) focuses on another type of uncertainty, namely uncertainty in emission levels. The author constructs an N -player dynamic game with a finite time horizon and finds that the cooperative equilibrium requires only moderate monitoring capabilities. This result extends to an infinite time horizon. However, even though countries’ individual emissions levels are uncertain, it is not assumed that the total pollution stock is perfectly observable. Laukkanen (2009) takes a different perspective. The author studies how to construct a self-enforcing climate agreement in case of a stock pollutant with stochastic stock dynamics, while assuming N iden-tical countries who each observe the current stock of pollution, but not countries’ individual emissions or the stochastic pollutant shock. Hence, if a high pollution stock is detected, it is uncertain whether this is caused by high emission levels or simply a high environmental shock. Benefits from emissions and environmental damage costs are assumed to be quadratic functions of emissions and the pollutant stock respectively. Punishment is triggered once deviation from the agreement is suspected, meaning that the monitoring statistic observed is high. Punishments are harsh, in that they entail emissions levels higher than Nash equilibrium levels, and last a finite number of periods, after which the agreement is resumed. The author finds that a cooperative equilibrium exists wherein the payoff is higher than the Nash equilibrium payoff. The exact payoff depends on the definition of the monitoring statistic, i.e. the rule that determines when punishment goes into effect. Laukkanen (2009) concludes that uncertainty in the envi-ronmental process makes it more difficult to achieve cooperation in the case of N identical players.

While the present paper focuses on the area of environmental economics, the study of collusion and cartel sustainability has broad applications in the field of industrial organization. Several studies examined the effect of uncertainty and asymmetry on collusion stability. Green and Porter (1984) consider an oligopoly of N firms and examine the nature of cartel self-enforcement in the presence of demand uncertainty. As in the typical emissions game, firms are assumed to use a trigger strategy once a firm deviates from the agreement. The authors find that a collusive equilibrium wherein no firm ever defects from the cartel exists, though it requires restrictive assumptions on the industry structure. In a more recent paper, Ciarretta and Guti´errez-Hita

(2012) investigate the effect of production cost asymmetry on cartel formation. The authors find that cartel formation and collusion sustainability are hindered as cost asymmetry increases. Goto and Iizuka (2016) confirm this finding. However, the authors conclude that other types of firm heterogeneities, such as product differentiation, complicate collusion more.

The paper closest to this one is that of Dutta and Radner (2009), who model the global warming process as a dynamic game with N asymmetric players, a continuous emissions level choice set and infinitely many periods. The authors assume that there is an optimal level of energy use, so that countries’ benefits are a concave function of its own emissions. Environmental damage costs are assumed to be a linear function of the pollutant stock. The linearity assumption is made for the sake of simplicity. For a theoretical exercise to have any chance of informing policy-makers its conclusions have to be simple, Dutta and Radner state. Secondly, linear costs allow the derivation of closed-form solutions of equilibria. Finally, there is little consensus on what is the correct form of non-linearity in costs. Different than in the aforementioned studies, both benefits and damage costs are allowed to differ between countries. Like Dockner et al. (1996) and Laukkanen (2009), the authors deviate from other research in that the global pollutant stock is allowed to accumulate over time. Dutta and Radner (2009) restrict their analysis to the perfect monitoring case. Thus, the assumption is that each country observes the pollutant stock precisely in each period and all countries’ emissions histories are known. The central finding is that there exists a cooperative equilibrium that maximizes the sum of country payoffs under the threat of reversion to Nash equilibrium emissions levels. The only punishment strategy considered is the grim trigger strategy: once a country deviates, emission levels never return to globally optimal levels.

The present study adds to the paper by Dutta and Radner (2009) by incorporating the notion of imperfect monitoring. The assumption that each country knows exactly the other countries’ emissions levels will be relaxed. Furthermore, I will allow for a shorter punishment phase. As apparent from the discussed literature, one could question the renegotiation-proofness of a grim trigger strategy in an international environmental agreement. Therefore, this study will allow for any positive number of punishment periods. The main finding is that uncertainty in emissions and asymmetries among countries limit the possibility of a successful international environmental agreement. A cooperative equilibrium is possible, though it imposes severe restrictions on the discount factor and punishment strategy.

2

The stage game

Consider a public good game with N countries. A country’s utility consists of the difference between the benefits from its own emission and the environmental damage costs that accrue from the total pollutant stock. Environmental damage costs are the negative economic impacts of, for example, forest loss, air and water pollution and rises in the sea level. Benefits accrue through the country’s production function, which is assumed to increase with the emission level, as higher energy use allows for higher production. As energy use increases, the increase in production that can be attained becomes less, so that marginal benefits are decreasing (Dutta and Radner, 2009). Hence, the concave production function of country i in period t is defined as

yit= zilog(1 + ait),

where yit≥ 0 and ait≥ 0 represent the production and emission level of country i in period t

respectively and zi> 0 represents country i’s production technology.

The total emission in period t is

At= N

X

i=1

ait.

For the sake of simplicity explained in the previous section, the economic cost of climate change is assumed to depend linearly on the global pollutant stock. The pollutants are the greenhouse gases, and the stock in period t is denoted Gt. The utility of country i in period t therefore

reads as

Uit= zilog(1 + ait) − ciGt.

Note that this definition allows for asymmetry: both the benefits from emission and the damage costs of climate change are allowed to differ between countries.

The pollutant stock is assumed to evolve as follows:

Gt+1= ρGt+ At, t ≥ 0, (1)

where the starting stock, G0 ≥ 0, is given and 0 < ρ < 1 is a given parameter that indicates

the ’surviving’ part of the previous period stock. Hence, ρ can be interpreted as the pollution survival rate, while (1 − ρ) is the pollution decay rate. The law of motion for the pollutant stock is not relevant in the one shot game, but will be considered in the dynamic representation of the game.

2.1 Two benchmarks

In the one shot game, a country’s utility equals

Ui = zilog(1 + ai) − ci  N X i=1 ai  .

Two benchmark situations can be defined. The first one is the Nash equilibrium, wherein countries behave selfishly and choose their emission level to maximize their own utility, without taking into account the harm their emission causes to other countries. Countries solve

MaxaiUi

so that the first order condition is given by

0 = ∂Ui ∂ai = zi 1 1 + ai − ci

and the individually optimal emission, denoted aN E_i , that follows is

aN E_i = zi ci

− 1.

The second benchmark is the social optimum, in which countries do take account of the damage their emission causes to other countries, so that emission is chosen to maximize the sum of the countries’ utilities. To be solved is

MaxaiΠ = N

X

i=1

Ui

which gives the following i first order conditions:

0 = ∂Π ∂ai = zi 1 1 + ai − N X i=1 ci

and resulting individual emission levels, denoted aSO_i , of

aSO_i = zi P ci

− 1.

As ci > 0 for all i, emission levels in the social optimum are strictly lower than in the Nash

equilibrium. As a result, global welfare - defined as the sum of the countries’ utilities - is strictly higher in the social optimum than in the Nash equilibrium.

2.2 Asymmetry

The fact that global welfare is maximized in the social optimum does not necessarily mean that every country is best off in that case. Whether individual countries are better off in the social optimum or in the Nash equilibrium depends on their relative production benefits from emission zi and their damage costs of climate change ci. The difference in a country i’s utility between

the two benchmarks is given by

U_iSO− U_iN E = ci N X i=1 zi ci − PN i=1zi PN i=1ci ! − zilog PN i=1ci ci ! .

Consider first the fully symmetric case: zi = z and ci = c for all countries i. The difference then

becomes U_iSO− UN E i = c N z c − z c ! − z log N c c ! = (N − 1)z − z log(N ).

As N > 1, this expression is strictly positive. Hence, every country is better off in the social optimum than in the Nash equilibrium if countries are fully symmetric. This result is also obtained by Wagner (2001) and Dutta and Radner (2009).

Now consider two asymmetric countries (or regions). Suppose the damage costs and emission benefits of two countries i and j are given by

αci= cj,

βzi= zj

where α, β > 0. The case α = β = 1 corresponds to full symmetry. Values of α and β farther from 1 indicate larger asymmetries. For country i, the difference between the benchmark utilities can now be expressed as

U_iSO− U_iN E= ci zi ci +βzi αci −(1 + β)zi (1 + α)ci ! − zilog (1 + α)ci ci ! = zi  α2+ β α(α + 1) − log 1 + α
 . (2)

Thus, whether the difference between the benchmark utilities is positive or negative depends solely on the values of α and β. In other words, only relative costs and benefits determine whether a country is better off in the social optimum or in the Nash equilibrium. The difference is increasing

in β, meaning that a relatively lower benefit zi (thus, a less efficient production technology)

increases utility from the social optimum relative to the Nash equilibrium. Furthermore, the difference is decreasing in α, so that lower relative environmental damage costs leads a country to prefer the Nash equilibrium over the social optimum. As β tends to zero, α must become very small and close to zero for the difference to be positive. In other words, relatively high benefits from emission must be accompanied by relatively high damage costs for a country to prefer the social optimum over the Nash equilibrium. In addition, the higher α is, the higher β must be to have a positive difference. This means that low relative environmental damage costs must be accompanied by a relatively inefficient production technology for a country to prefer the social optimum. For example, if country i’s damage costs are half the damage costs of country j (α=2), its production benefits can be at most _β1 ≈ 38.5% of country j’s benefits (as β must be & 2.59). On the other hand, if country i’s damage costs are double the damage costs of country j (α=0.5), the restriction on β for the difference in utility to be positive is that β & 0.05, so that the high-cost country’s production benefit is allowed to be as much as 1_β ≈ 20 times as high as that of the low-cost country. Also notable from the expression is that the difference in the benchmark utilities does not depend on the environmental damage cost ci itself, while it does

depend on the value of zi: the higher the production benefits from emission, the larger is the

difference between welfare in the social optimum and the Nash equilibrium in absolute terms.

3

The dynamic emissions game

Consider an extension of the emissions game to an infinite number of time periods and assume players consider discounted utility. The two benchmarks defined in the static version carry over to this dynamic representation of the game.

3.1 The social optimum

Define global welfare as the sum of the N countries’ total discounted utilities. In the social optimum, global welfare is maximized subject to the law of motion for the total pollutant stock;

Maxait N X i=1 ∞ X t=0 δtUit(ait, Gt) s.t. Gt+1 = ρGt+ N X i=1 ait

where δ is a discount factor (0 < δ < 1) assumed to be equal for all countries. Substituting utilities, the Lagrange function for this problem reads as

Π = N X i=1 ∞ X t=0 δtzilog(1 + ait) − ci(Gt)  + ∞ X t=0 λt  Gt+1− ρGt− N X i=1 ait  .

In order to maximize total welfare, the triple (ait, Gt+1, λt) must satisfy the Lagrange conditions

0 = ∂Π ∂ait = δtzi 1 1 + ati − λt (3) 0 = ∂Π ∂Gt+1 = −δt+1Xci+ λt− ρλt+1 (4) 0 = ∂Π ∂λt = Gt+1− ρGt− N X i=1 ait. (5)

Solving the recurrence relation in (4) by substitution, the expression for λtbecomes

λt= lim x→∞ρ x_λ t+x+  δt N X i=1 ci  ∞ X t=0 ρtδt+1  .

Under the transversality condition the limit on the right hand side is zero, so that

λt= δt N X i=1 ci δ 1 − δρ. (6)

Substituting (6) into (3), it follows that the optimal emission level in the social optimum, denoted aSO_i , is

aSO_i = zi(1 − δρ) δPN

i=1ci

− 1. (7)

Note that the optimal emission level for country i is independent of t and hence constant over time. This is a result of the linearity of the damage costs of climate change in countries’ utilities. Constant emission of aSO_i results in the pollutant stock evolving as

GSO_t = ρtG0+

1 − ρt 1 − ρA

SO

where ASO =PN

i=1aSOi . GSOt converges to the steady state

GSO_∗ = lim t→∞G SO t = lim_t→∞  ρtG0+ 1 − ρt 1 − ρA SO  = 1 1 − ρA SO

because ρt→ 0 as t → ∞.

Global welfare in the social optimum, denoted as GWSO, can therefore be written as

GWSO = N X i=1 ∞ X t=0 δtzilog(1 + aSOi ) − ci(Gt)  = 1 1 − δ N X i=1 zilog(1 + aSOi ) − 1 1 − δρ N X i=1 ciG0− δ (1 − δ)(1 − δρ) N X i=1 ciASO. (8)

Country welfare of country i, denoted CW_iSO, equals

CW_iSO = ∞ X t=0 δtzilog(1 + aSOi ) − ci(Gt)  = 1 1 − δzilog(1 + a SO i ) − 1 1 − δρciG0− δ (1 − δ)(1 − δρ)ciA SO_{. (9)}

3.2 The Nash equilibrium

In the Nash equilibrium, countries optimize their own total discounted utility and ignore the damage their emission causes to other countries. The problem each country solves is

Maxait ∞ X t=0 δtUit(ait, Gt) s.t. Gt+1= ρGt+ N X i=1 ait.

The Lagrangian becomes

Π = ∞ X t=0 δtzilog(1 + ait) − ci(Gt)  + λt  Gt+1− ρGt− N X i=1 ait 

and the first order conditions are

0 = ∂Π ∂ait = δtzi 1 1 + ait − λ_t (10) 0 = ∂Π ∂Gt+1 = −δt+1ci+ λt− ρλt+1 (11) 0 = ∂Π ∂λt = Gt+1− ρGt− N X i=1 ait. (12)

From the recurrence relation in (11) it follows that

λt= δtci

δ

1 − δρ. (13)

Substituting (13) into (10), it follows that a country’s individually optimal emission level, denoted aN E_i , equals

aN E_i = zi(1 − δρ) δci

− 1. (14)

This emission level is again constant over time and is strictly higher than in the social optimum, as ci > 0 for all i. The pollutant stock evolves as

GN E_t = ρtG0+

1 − ρt 1 − ρA

N E_,

where AN E =PN

i=1aN Ei . Hence, the global pollutant stock converges to the steady state

GN E∗ = lim t→∞G N E t = lim_t→∞  ρtG0+ 1 − ρt 1 − ρA N E  = 1 1 − ρA N E_.

As AN E > ASO, the steady state of the pollutant stock is strictly higher than in the social optimum.

Global welfare in the Nash equilibrium, denoted GWN E, is given by

GWN E= N X i=1 ∞ X t=0 δt  zilog(1 + aN Ei ) − ci(Gt)  = 1 1 − δ N X i=1 zilog(1 + aN Ei ) − 1 1 − δρ N X i=1 ciG0− δ (1 − δ)(1 − δρ) N X i=1 ciAN E (15)

and country welfare of country i, denoted CW_iN E, equals

CW_iN E = ∞ X t=0 δt  zilog(1 + aN Ei ) − ci(Gt)  = 1 1 − δzilog(1 + a N E i ) − 1 1 − δρciG0− δ (1 − δ)(1 − δρ)ciA N E_{. (16)} 3.3 Asymmetry

In the static game, it was derived that whether a country is better off in the social optimum or in the Nash equilibrium depends on emission benefit and damage cost asymmetries. This result carries over to the dynamic game considered here.

The difference between the two benchmarks for country i is now defined as country welfare in the social optimum minus country welfare in the Nash equilibrium:

D = CW_iSO− CWN E

i .

Using (9) and (16), gives

D = 1 1 − δ δ 1 − δρci(A N E_{− A}SO_{) − z} ilog  1 + aN E i 1 + aSO i ! . As 1+aN Ei 1+aSO i = P ci ci and A N E_{− A}SO ₌ 1−δρ δ  Pzi ci − P zi P ci  , it follows that D = 1 1 − δ ci  Xz_i ci −P zi P ci  − z_ilog P ci ci ! 2_.

Note that D must be positive for countries to prefer the social optimum over the Nash equilibrium. In the case of full symmetry, where zi = z and ci = c for all countries i, the difference simplifies

to z 1 − δ  N − 1 − log(N )  .

As in the one shot game, this expression is strictly positive for N > 1. Hence, every country would prefer the social optimum to the Nash equilibrium in case of full symmetry.

Consider again the two-region example and define asymmetry in terms of α and β:

αci= cj,

βzi= zj

where α, β > 0. D now becomes

D = z 1 − δ  α2+ β α(α + 1) − log 1 + α
 . (17)

The only difference between (17) and (2) in the static game is the factor _1−δ1 on the right hand side. Hence, the impact of α, β, zi and ci on D is exactly the same. The difference is increasing

in β, decreasing in α and hence high production benefits must be accompanied by high damage costs in order for a country to prefer the social optimum. Likewise, low damage costs must be

2PN

i=1zi PNi=1ci

accompanied by low production benefits from emission. Moreover, the higher the discount factor is, the higher is the absolute value of the difference between the country welfare in the social optimum and the Nash equilibrium.

4

A climate agreement under perfect monitoring

Consider first a climate agreement under perfect information, meaning that countries are able to observe other countries’ past emissions precisely. The agreement is the following: countries agree to emit the socially optimal amount, aSO_i , in every period as long as all other countries do so. If a country deviates, a punishment phase goes into effect. All countries switch to their individually optimal emission level, the Nash equilibrium levels aN E_i , from the next period onwards. The punishment phase remains in effect for k periods, where k ∈ [1, ∞). The higher k is, the less forgiving countries are and hence the more sustainable the agreement would be. The case k → ∞ corresponds to a grim trigger strategy, where countries never return to socially optimal emission levels once a country deviates. Dutta and Radner (2009) focus on this type of agreement with infinite punishment only. The present study takes a more general approach by varying k, for reasons discussed in the introduction: several authors have argued that an agreement with an infinite number of punishment periods may involve an incredible threat, and is therefore not sustainable in the first place.

4.1 Stability conditions

At the beginning of each period, a country decides on its emission for the coming period. It could either choose to cooperate and emit the socially optimal amount aSO

i , or it could choose

to emit its individually optimal amount, aN E_i , and in doing so break the agreement 3. Several aspects need to be taken into account. The upside of deviating from the agreement is equal to the additional production benefit that results from the higher emission. A deviating country is best off by doing so at t = 0, as the additional production benefit is not discounted then. As the defection will result in k periods of punishment and the emission of individually optimal levels, the deviating country will enjoy the additional production benefit for k + 1 periods. The upside of defection for country i is therefore

U p = k X t=0 δtzilog 1 + aN E i 1 + aSO_i ! = 1 − δ k+1 1 − δ zilog P ci ci ! .

3_{Note that the individually optimal amount a}N E

i does not depend on other countries’ emissions. Therefore, a

The downside of defection that higher costs of climate change will be incurred as a result of the higher total emissions. First, country i emits aN E_i instead of aSO_i from period 0 to period k. Second, the other countries j, j 6= i, emit aN E_j instead of aSO_j from period 1 to period k (as it takes one period for them to notice and respond to the defection of country i). Compared to the case of no defection, the increase in total emission is

t = 0 : aN E_i − aSO i

t = 1, .., k : AN E− ASO_.

As apparent from the law of motion for the pollutant stock in (1), emission in one period affects Gt from the next period onwards. The (discounted) increase in costs for country i is therefore

Down = ci ∞ X t=1 δtρt−1 aN E_i −aSO i
+ k+1 X t=2 δt1 − ρ t−1 1 − ρ (A N E_−ASO₎₊ ∞ X t=k+2 δtρ t−k−1_{− ρ}t−1 1 − ρ (A N E_−ASO₎ ! .

The third term appears because once the punishment phase is over and emissions return to socially optimal levels, the extra emissions require more time to decay and hence keep affecting costs. Simplification leads to the following expression for the downside of defection:

Down = zi 1 − ci P ci ! +δ(1 − δ k₎ 1 − δ ci Xz_i ci −P zi P ci ! .

Country i will cooperate if the downside of defection is larger than the upside. This results in the following condition for stability of the agreement:

Down > U p =⇒ zi 1 − ci P ci ! +δ(1 − δ k₎ 1 − δ ci Xz_i ci −P zi P ci ! −1 − δ k+1 1 − δ zilog P ci ci ! > 0.

Rearranging terms gives

δ(1 − δk) 1 − δ > zilog  P ci ci  − z_i  1 − ci P ci  ci  Pzi ci − P zi P ci  − z_ilog  P ci ci  > 0. (18)

The stability condition can also be written in terms of the welfare difference between the two benchmarks for country i, CW_iSO− CWN E

i : CW_iSO− CW_iN E > zi δ(1 − δk₎ log  P ci ci  −  1 − ci P ci ! .

The agreement will be stable if this condition holds for all countries. As the right hand side of the inequality is strictly positive, a necessary condition for a stable agreement is that every country is strictly better off in the social optimum than in the Nash equilibrium. As ρ does not appear in the condition, stability is not affected by the decay rate of the pollutant stock under perfect monitoring. Hence, countries’ decisions to cooperate or not are not influenced by the length of time pollution stays in the atmosphere.

4.2 Results

4.2.1 Full symmetry

In the case of full symmetry (with ci = c and zi = z for all i), condition (18) for stability and

reaching the social optimum becomes

δ(1 − δk) 1 − δ >

N log N − N + 1 N N − log N − 1
.

The condition on the discount factor hence only depends on the number of countries N and the number of punishment periods k. The values of z and c have no effect on stability. Define δmin as the minimum discount factor for which the agreement would be stable. Figure 1 shows δmin as function of N for several values of k. The most forgiving form of punishment, playing the Nash equilibrium for 1 period after defection, imposes the highest constraint on the discount factor. For N = 2 and k = 1, δ should be at least 0.63 for the agreement to be stable. As N

2 10 20 30 40 50 60 N 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 δ min k=1 k=2 k=3 k=4 k= ∞

increases, the constraint on δ rapidly becomes less binding and δmin converges to as low as 0.04. Hence, an increase in the number of regions/countries has a strong positive effect on the stability of the agreement in the case of full symmetry. After all, as the number of countries increases, so does the global externality. For higher values of k δmin follows the same pattern as for k = 1, but is lower for every N . The graphs of k > 1 converge quickly to the grim trigger strategy case, where k → ∞. For k = 2, δ should be at least 0.44 when N = 2. For k = 3 and k = 4 the minimums are 0.40 and 0.39 respectively, while for k → ∞, δ should be at least 0.39. As N increases, δmin _{converges to the same value as for k = 2, namely 0.04.}

Clearly, the discount factor is allowed to take on extremely low values if countries are fully symmetric. This can be explained by the fact that the benefits from behaving socially and taking other countries’ costs of climate change into account are shared equally in this case. Each country is that much better off in the social optimum than in the Nash equilibrium, that is does not pay off to deviate from the agreement even for very low discount factors. Hence, the agreement will be stable in the case of full symmetry and perfect monitoring, unless countries care very little about their future welfare.

4.2.2 Two asymmetric regions

To see what happens when the assumption of full symmetry is relaxed, the asymmetric case will be studied for two regions first (say, North and South). As in the one shot game, asymmetry is defined in terms of α, β > 0, where values farther from 1 indicate larger asymmetries between the regions. Environmental damage costs and emission benefits are such that

αci = cj,

βzi= zj.

The condition for region i to be willing to cooperate becomes

δ(1 − δk) 1 − δ >

α (1 + α) log(1 + α) − α
α2_{− α(1 + α) log(1 + α) + β} > 0

asP ci = ci+ cj = (1 + α)ci and P zi = zi+ zj = (1 + β)zi.

For region j the condition is

δ(1 − δk) 1 − δ >

β (1 + α) log(1 + _α1) − 1
α2_{− (1 + α)β log(1 +} 1_{) + β} > 0,

as P ci = (1 +_α1)cj and P zi = (1 +_β1)zj. Once again it follows that stability does not depend

on the exact values of z and c, but on relative environmental damage costs and production benefits only. For the agreement to be stable, both regions should have no incentive to defect. Hence, both conditions must be satisfied.

Figure 2 shows the minimum discount factor required for stability, δmin, for both regions when emission benefits are equal (β = 1) while the cost distribution is varied. Figure 2a depicts the case of the agreement with the most forgiving form of punishment, k = 1. Figure 2b depicts the agreement involving grim trigger strategies (k → ∞).

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ min region i region j (a) k = 1 0.8 0.9 1 1.1 1.2 1.3 1.4 α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ min region i region j (b) k → ∞

Figure 2: Minimum discount factor as a function of cost asymmetry for equal production benefits and two regions

The area above the two curves represents combinations of the discount factor and cost asymmetry for which the agreement is stable. Note that the values on the horizontal axes differ for the two cases. The intersection of the curves is where α = β = 1, in which case the regions are fully symmetric. In the intersection points, δmin equals 0.63 for k = 1 and 0.39 for k → ∞. These values correspond to the δmin found under full symmetry for N = 2 in the previous subsection. From the figure it is clear that α should be between 0.91 and 1.10 if k = 1 and between 0.71 and 1.41 if k → ∞ for there to exist discount factors for which the agreement would be stable. Outside these ranges, the agreement would not last for any δ, as it would always be optimal for one of the regions to defect. Furthermore, the range of α’s for which a stable agreement is possible is larger for higher k, which means that a more severe punishment allows for higher levels of damage cost asymmetry.

(α = 1), but different emission benefits (varied β). Again, Figure 3a depicts the most forgiving form of punishment (k = 1) and Figure 3b the harshest (k → ∞).

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 β 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ min region i region j (a) k = 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 β 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ min region i region j (b) k → ∞

Figure 3: Minimum discount factor as a function of benefit asymmetry for equal damage costs and two regions

The intersection points are the cases α = β = 1 of full symmetry, so that for k = 1 (k → ∞) δmin is 0.63 (0.39) as before. In the case of k = 1 (one punishment period), β must be be-tween 0.78 and 1.29 for a potentially stable agreement, while for k → ∞, β must be in the interval (0.39, 2.55). Any β outside these ranges means that there is too much asymmetry between the two regions for a stable agreement. Moreover, a stable agreement allows for larger asymmetry in terms of production technologies (a less severe restriction on β) if the threat of punishment is more severe (k is higher). Furthermore, asymmetry in terms of emission benefits has a smaller effect on stability than asymmetry in costs, as seen in the ranges of α and β.

In Figure 4, variation in both environmental damage costs and production benefits is allowed and the discount factor is set to δ = 0.95. The blue area depicts combinations of α and β for which both countries have no rational incentive to defect, so that the agreement is stable and the social optimum is reached. The white area depicts combinations of α and β for which one of the regions is better off by defecting, so that the agreement is impossible. In Figure 4a the punishment length is set to 1, while in Figure 4b the punishment length is set to infinity, which allows us to compare the two most extreme forms of punishment.

Clearly, for both k the agreement is stable if α = β = 1. This is the full symmetry case considered before. As the blue area is quite flat in both cases, it is again clear that more asymmetry in

(a) k = 1 (b) k → ∞

Figure 4: Blue area: combinations of cost- and benefit asymmetries for which agreement is stable at δ = 0.95, N = 2

is increased from 1, the range of β’s allowed for a stable agreement gets larger. Furthermore, the lower limit of the β-range increases as α increases. This means that for the agreement to be stable, higher cost inequality must be accommodated by higher benefit inequality, where the region with the highest cost must also have the highest production benefits.

4.2.3 Three asymmetric regions

Now consider the case of three asymmetric regions, h, i and j. The costs of climate change and the production benefits from emission of the regions are given by the following relations:

1 αci = ch, αci= cj, 1 βzi= zh, βzi = zj.

where α > 0 and β > 0 are again the asymmetry parameters, and α = 1, β = 1 corresponds to the full symmetry case. I use this form of asymmetry with two parameters for the sake of tractability.

For region h, the condition to be willing to cooperate is

δ(1 − δk) 1 − δ > log(1 + α + α2) −_1+α+αα+α22 1 +β_α +β_α22 − 1+β+β2 1+α+α2 − log(1 + α + α2) > 0

For region i, the condition is δ(1 − δk₎ 1 − δ > log 1 + α + _α1
−_1+α+α1+α22 1 +β_α +α_β −1+β+ 1 β 1+α+_α1 − log 1 + α + 1 α
> 0,

as in terms of region i’s parametersP ci = (1 + _α1 + α)ci and P zi= (1 + 1_β + β)zi.

For region j, the condition is

δ(1 − δk) 1 − δ > log 1 +_α1 +_α12
−_1+α+α1+α 2 1 +α_β +α_β22 − 1+_β1+1 β2 1+_α1+ 1 α2 − log 1 + 1 α + 1 α2
> 0

as in terms of region j’s parametersP c_i = (1 + _α1 +_α12)cj and P zi= (1 + 1_β +_β12)zj.

All three conditions must be satisfied for the agreement to be stable. As in the two-region example, Figure 5 shows the minimum discount factor required for stability for the three regions as a function of α for β = 1. Thus, the regions’ production technology coefficients are equal and fixed while the cost distribution is varied. Figure 6 shows the opposite case where three regions have equal costs of climate change (α = 1), but different production benefits (varied β). Again, the two most extreme forms of punishment are considered: k = 1 and k → ∞. By definition, region i is always exactly in the middle of the three regions in terms of benefits and costs. Therefore, this region does not impose a binding constraint on the discount factor in any of the cases. In all four panels of Figures 5 and 6, the three curves cross at the point where α = β = 1, which is the full symmetry point. The minimum discount factors in these points are lower than in the corresponding cases with only two regions, as is also clear in the full symmetry graph in Figure 1. This is a result of the positive effect an increase in N was found to have on the stability of the climate agreement in the full symmetry case. Intuitively, total emission will increase if the number of polluting countries N increases, so that each country will suffer more from climate change and benefit more from the agreement.

In the case of equal production benefits and varying costs, the range of α allowed for stability is higher for both forms of punishment than in the case with only two regions. α should now be between 0.87 and 1.14 if k = 1 and between 0.69 and 1.44 if k → ∞. Thus, three regions allow for a higher level of cost asymmetry than two regions, even though the difference between the lowest and the highest cost region is bigger in the former case (|α −_α1| versus |α − 1|). Hence, the positive effect on stability of increasing N prevails.

α 0.9 0.95 1 1.05 1.1 δ min 0 0.2 0.4 0.6 0.8 1 region h region i region j (a) k = 1 α 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 δ min 0 0.2 0.4 0.6 0.8 1 region h region i region j (b) k → ∞

Figure 5: Minimum discount factor as a function of damage cost asymmetry for equal production benefits and three regions

β 0.8 0.9 1 1.1 1.2 1.3 δ min 0 0.2 0.4 0.6 0.8 1 region h region i region j (a) k = 1 β 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 δ min 0 0.2 0.4 0.6 0.8 1 region h region i region j (b) k → ∞

Figure 6: Minimum discount factor as a function of benefit asymmetry for equal damage costs and three regions

In the case of equal costs and varying production benefits, the range of β is wider in case of one punishment period, but narrower in the grim trigger case compared to when N = 2. For k = 1, β must now be in the interval (0.75, 1.34), while for k → ∞ it must be in the interval (0.45, 2.23). Thus, for relatively small ranges of β it is the positive effect on stability of the increase in N that prevails: countries suffer more from climate change and benefit more from the agreement if more countries pollute. However, a change in β has a bigger effect on asymmetry in the case of three regions than in the case of two regions (as in the case of cost asymmetry, the difference between the lowest and highest benefit region is now |β −_β1| instead of |β − 1|). For larger ranges of β, this negative effect on stability dominates the positive effect of increasing N .

four panels of Figures 6 and 5, but less visible in the other three as a result of the smaller ranges of α and β. The curved shape comes from the increase in P zi

P ci that results from moving α or

β away from 1. The increased asymmetry leads zi

ci to decrease relative to P zi

P ci. As i’s relative

production benefit decreases, it becomes less attractive to defect from the agreement and so the condition on δ imposed by region i becomes lower.

Figure 7 shows the combinations of α and β that allow for a stable agreement in the case of three regions at a discount factor of 0.95. Compared to the case of two regions, the blue area has become larger for k = 1 as well as k → ∞. Therefore, the addition of one region has led to an increase in the possibilities of reaching a successful agreement, despite the fact that the difference between the highest and lowest cost/benefit region has become larger compared to the two region case. Note that the shapes of the blue areas differ from the two region case. In particular, the slope of the upper limit on α decreases as α passes a certain threshold, say αT 4.

For k = 1, αT is close to 2, while for k → ∞ αT is approximately 4. This implies that for values

of α higher than αT, the asymmetry in costs becomes so large that a further increase in the

cost asymmetry must be accompanied by an even higher increase in benefit asymmetry than for lower values of α.

(a) k = 1 (b) k → ∞

Figure 7: Blue area: combinations of cost- and benefit asymmetries for which agreement is stable at δ = 0.95, N = 3

4.2.4 Application example: eight regions case

While the two previous subsections contain results on the stability of the agreement in the theoretical cases of full symmetry and two or three asymmetric regions, the present one con-tains an application using parameter estimates of the real world coefficients zi and ci based

on data from Nordhaus and Boyer (2000). The authors divide the world into eight regions,

4

namely the USA, Other High Income countries, Western Europe, Eastern Europe, Middle Income countries, Lower Middle Income countries, China and Low Income countries. For each of these regions data and estimates are available on GDP, CO2 emission and economic costs of climate change in 1995. The latter is based on the estimated economic impact of global warming on seven potential areas of concern, namely agriculture, sea-level rise, other market sectors, health, nonmarket amenity impacts, human settlements and ecosystems and catastrophes. From the data, estimates of ziand ci for the present model are constructed and shown in Figure 8.

0 50 100 150 200 250 300 350 z_i 0 20 40 60 80 100 120 140 ci USA Other high income

Western Europe

Middle income Low income China

Eastern

Europe Lower middle income

Figure 8: Estimates of production benefits and damage costs per region

Western Europe appears to be especially vulnerable to climate change. A major factor causing this is the high coastal vulnerability. Production benefits are also highest for Western Europe, meaning that this region is able to achieve the highest increase in GDP as a result of increasing emissions. Furthermore, China is the region that has both the lowest costs of climate change and the lowest production benefit from emission. Note that the values of zi and ci are meaningless

in isolation. Regions are less willing to cooperate and keep the agreement if their production benefit zi is high relative to their costs of climate change ci and if their costs ci are low relative

to those of other regions. Figure 9 therefore shows zi

ci per region, while Figure 10 shows ci P ci per

USA Other high income Western Europe Eastern Europe Middle income Lower middle income China Low income 0 2 4 6 8 10 12 zi /ci

Figure 9: Production benefit relative to costs of climate change per region

USA Other high

income Western Europe Eastern Europe Middle income Lower middle income China Low income c i /sum(c) 0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 10: Costs of climate change relative to the world total per region

Figures 9 and 10 imply that for China and the Middle Income region it will be most beneficial to deviate from the agreement. Their costs of climate change are very low compared to the other regions. In addition, their production benefits relative to cost are high and therefore the immediate profit from defecting will be high as well. China and Middle Income will therefore impose the highest constraints on the discount factor for stability of the agreement. In contrast, Western Europe imposes almost no constraint on the discount factor. For this region the costs

are extremely high and as a result the production benefits relative to their costs are low, even though Western Europe’s zi itself is the highest of all. It is therefore almost always optimal for

Western Europe to cooperate. The minimum discount factor required for regions to be willing to cooperate, δmin as a function of the number of punishment periods is shown in Figure 11.

4 5 10 15 20 25 30 k 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ min USA

Other high income Western Europe Eastern Europe Middle income Lower middle income China

Low income

Figure 11: Minimum discount factor required for cooperation per region as a function of the number of punishment periods

For k < 4 a stable agreement is impossible, as for at least one of the regions the stability condition is not satisfied for any δ < 1. A stable agreement requires the stability condition to be satisfied for all regions. Figure 12 therefore corresponds with the binding restriction in Figure 11, which is that of China. It shows the minimum discount factor that allows for full stability, again as a function of the number of punishment periods k.

4 5 10 15 20 25 30 k 0.7 0.75 0.8 0.85 0.9 0.95 1 δ min

Figure 12: Minimum discount factor required for stability as a function of the number of punishment periods

Despite the presence of substantial asymmetries between the eight regions, a stable agreement to achieve the social optimum would be possible for higher values of k. However, the restrictions on the discount factor are much more severe than in the full symmetry case. For k = 4, δ should be at least 0.96 for the agreement to be stable. As the number of punishment periods increases, the minimum discount factor required decreases and converges to 0.78. Comparing the cost and benefit distributions of the present example to the ranges of allowed α and β that were found in the cases of two and three asymmetric regions, it is clear that the case with eight regions allows for higher levels of asymmetry.

5

A climate agreement under imperfect monitoring

As discussed earlier, in reality information in the emission game is far from perfect. In particular, measuring techniques for the pollutant stock in the atmosphere are not perfectly accurate. Therefore, this section incorporates the notion of imperfect monitoring, meaning that countries are uncertain about other countries’ exact emission levels. As in the previous section, I study the stability of a climate agreement in which each country is supposed to emit the socially optimal level. Again, the agreement involves the threat of reversion to Nash equilibrium emission levels for k periods (where k could take any discrete value higher than zero) once one or more countries deviate from socially optimal emission levels. How does the imperfect monitoring of other countries’ emission affect the stability of such a climate agreement?

5.1 Stability conditions

Suppose that the pollutant stock is measured at the beginning of each period, just as in the dynamic game under the notion of perfect monitoring. However, different from Sections 2-5, it is now commonly known that the amount measured might differ from the actual pollutant stock, Gt. The amount measured at the beginning of period t is denoted Xt, and Xt is assumed to be

normally distributed with mean Gt and variance σ2:

Xt∼ N (Gt, σ2)

Denote the error in the measured amount as εt:

εt= Xt− Gt,

It follows that

εt= Xt− Gt∼ N (0, σ2).

If every country keeps the agreement and emits the socially optimal amount in every period, we have

Gt+1= ρGt+ ASO ∀t ≥ 0.

Replacing Gtfor Xt− εt, this reads as

Xt+1= ρXt+ ASO+ ξt+1,

where ξt= εt− ρεt−1 and ξt∼ N 0, (1 + ρ2)σ2
as εt and εt−1 are independent.

In contrast, if the agreement is broken in period t, we have

Gt+1> ρGt+ ASO

and therefore

Xt+1− ρXt− ASO> ξt+1.

Countries deal with uncertainty with a simple statistical test. Suppose that countries’ null hypothesis is that every country keeps the agreement so that socially optimal levels are emitted in every period. Then, in period t we have

H0 : Xt+1= ρXt+ ASO Ha: Xt+1> ρXt+ ASO. Under H0, Zt+1= Xt+1− ρXt− ASO (1 + ρ2_)σ2

has a standard normal distribution and can be used to test the null hypothesis with a simple one-sided test. Using a 95 percent confidence level, H0 will be rejected in period t if Zt> 1.645.

In this case, the agreement fails and the punishment phase will go into effect.

Just as in the perfect monitoring case, countries decide on their emission level at the beginning of each period. The difference is that once a country defects, it is uncertain whether other countries

will notice and the agreement will fail. On the other hand, there always exists a 5 percent probability that the agreement will fail even if all countries emit socially optimal amounts 5. However, the probability of a failure increases as a country emits more than the socially optimal amount, as the extra emission in period t increases Gt+1 and thereby the mean of Xt+1.

Country i’s choice of emission level can therefore be set out as follows. The upside and downside of defection, as stated in the perfect monitoring agreement, can now be split into a certain and an uncertain part, where the uncertain part depends on whether the defection leads to failure of the agreement or not. The certain upside of defection is equal to the immediate production profit resulting from the higher emission in the present period:

U pcertain= zilog 1 + aN E_i 1 + aSO i ! = zilog P ci ci ! .

The certain downside of defection is the increase in costs of climate change resulting from the higher emission in the present period:

Downcertain= ci ∞ X t=1 δtρt−1 aN E_i − aSO_i
= zi 1 − ci P c_i ! .

Defection increases the probability of the agreement to fail with

Pincrease,i= P agreement fails | At= ASO+ (aN Ei − aSOi )
− P agreement fails | At= ASO

= P Xt+1+ (a N E i − aSOi ) − ρXt− ASO (1 + ρ2_)σ2 > 1.645 ! − P Xt+1− ρXt− A SO (1 + ρ2_)σ2 > 1.645) ! = P  Zt+1> 1.645 − (aN E_i − aSO i ) (1 + ρ2_)σ2  − P Zt+1> 1.645
= Φ(1.645) − Φ(1.645 − γ), where γ = (aN Ei −aSOi )

(1+ρ2_)σ2 6. Remark that (aN E_i − aSO_i ) is country i’s excess emission. As

(aN E_i − aSO_i ) = 1 − δρ δ zi  1 ci − 1 P c_i  , (19)

excess emission is a decreasing function of both ρ and δ. Hence, as pollution stays in the

atmo-5

Note that the choice of the significance level will affect the stability of the agreement. Especially for harsh punishments, a 5 percent probability of failure if none of the countries defects might be considered high. However, a lower significance level implies that defecting countries are less likely to be caught, and increases incentives to deviate.

6_{Note that defection in period t affects the failure probability in the next period only, as Z}

t+1depends on the

sphere longer or as the discount factor increases, the difference between socially and individual optimal emission levels becomes smaller. An increase in either ρ or δ decreases aSO_i , but it decreases aN E_i even more. (After all, countries take into account the world total of costs of climate change P ci in the social optimum, and only their own costs ci in the Nash equilibrium.

A given change in ρ or δ is therefore relatively smaller in the social optimum than in the Nash equilibrium.) As a result, the increase in the failure probability caused by defection is decreasing in ρ and δ. It is also decreasing in σ2, so that for higher uncertainty defection is less likely to be noticed.

In case the agreement fails, the punishment phase goes into effect from the next period onwards. Hence, the Nash equilibrium of individual optimal levels will be played for k periods. This results in a benefit to production for country i of

U puncertain= k X t=1 δtzilog 1 + aN E_i 1 + aSO i ! = δ t_{(1 − δ}k₎ 1 − δ zilog P ci ci ! .

and an increase in costs of

Downuncertain= k+1 X t=2 ciδt 1 − ρt−1 1 − ρ A N E_{− A}SO_{) +} ∞ X t=k+2 ciδt ρt−k−1− ρt−1 1 − ρ A N E_{− A}SO₎ = δ(1 − δ k₎ 1 − δ ci Xz_i ci −P zi P ci ! .

Assuming risk neutrality, country i would choose to cooperate in a certain period if the expected downside of defection is larger than the expected upside:

Downcertain− U pcertain+ Pincrease,i Downuncertain− U puncertain
> 0.

The found expressions lead to the following condition:

zi  1 − ci P ci  + Pincrease,i δ(1 − δk) 1 − δ ci  Xz_i ci −P zi P ci  − z_ilog P ci ci ! − z_ilog P ci ci  > 0,

which must hold for every country. Rearranging terms gives

Pincrease,i· δ(1 − δk) 1 − δ > zilog  P ci ci  − z_i  1 − ci P ci  ci  Pzi ci − P zi P ci  − z_ilog  P ci ci  > 0. (20)

Remark that δ has a positive effect on δ(1−δ_1−δk), but a negative effect on Pincrease,i. An increase in

the discount factor decreases Pincrease,i, because it reduces a country’s excess emission (19) and

therefore defection will less likely be noticed. Thus, the discount factor now has two opposing effects on stability: on one hand it should be high for the agreement to be stable, but on the other hand it should not be too high, as than Pincrease,ibecomes so small that defection is likely

not to be noticed (and so defection would be optimal).

In the perfect monitoring case, stability was found to be independent of the pollution survival rate. In the present case of imperfect monitoring, ρ does appear in the stability condition. It affects Pincrease,i negatively, so that an increase in the pollution survival rate now has a negative

effect on stability.

In terms of the difference in country welfare, the stability condition implies

CW_iSO− CW_iN E > 1 Pincrease,i · zi δ(1 − δk₎ log  P ci ci  −  1 − ci P ci ! .

As the increase in the failure probability is smaller than 1, the difference between country welfare in the social optimum and the Nash equilibrium must now be larger than in the perfect monitoring case for the agreement to be stable.

5.2 Results

To assess the effect of uncertainty in emissions on climate agreements, I will consider the same cases as under the notion of perfect monitoring and compare the results. The variance of the observed GHG stock, σ2, can be expressed in terms of a percentage of the average benefit-cost ratio: σ2 = v 1 N N X i=1 zi ci ,

where v denotes the uncertainty level.

5.2.1 Full symmetry

Consider the case of full symmetry, so that zi= z and ci = c for all i. In the case of imperfect

monitoring, condition (20) for stability and reaching the social optimum becomes

Pincrease·

δ(1 − δk)