Endogenous learning in international environmental agreements: the impact of research spillovers and the degree of cooperation

Abstract

We consider an endogenous learning-by-doing process where countries can invest in research that reduces the systematic uncertainty about climate change damages. We analyse a coalition model in which countries decide whether to join a treaty and then choose their level of research and abatement. Countries can cooperate on research and abatement or only on one of these items. We consider the entire range of possible research spillovers. Cooperation on all issues and large research spillovers are generally welfare improving, but lead to smaller coalitions, as they encourage free-riding. However, on balance, in equilibrium, we find that cooperation should not be confined to research and should include abatement, and research findings should freely travel.

1. Introduction

Climate change is one of the most severe environmental threats that humankind is currently facing and one of the most difficult to solve. The global public bad nature of climate change, as well as the large uncertainties surrounding the relation between greenhouse gas emissions, temperature increase, and climate damages poses severe difficulties in tackling this challenge effectively. This pessimistic view is mainly shared by the game-theoretic literature on the formation of international environmental agreements (IEAs), going back to Barrett (1994) and Carraro and Siniscalco (1993), and more recent articles, of which the most important contributions have been collected in a volume edited by Finus and Caparrós (2015). Barrett (1994) coined this pessimistic view the ‘paradox of cooperation’, which may be summarized as follows: only small agreements are stable if the gains from cooperation are large; large stable coalitions may emerge, however, only if the gains from cooperation are small.

In reality, considering the entire array of IEAs (Kellenberg and Levinson, 2014), there has also been some success in overcoming the paradox of cooperation, which necessitates to extend the basic model in order to capture this grey area of reality. Moreover, from a normative point of view, it is important to consider alternative agreement designs and their contribution to mitigate the free-rider problem.¹ In this article, we focus on how uncertainty and endogenous learning through investment in research influences the formation of climate agreements. We extend previous models by considering an endogenous learning process and an active research process in the form of investment in research.

The first paper on exogenous learning is Na and Shin (1998). They consider uncertainty about the distribution of benefits of emission reduction and show that the veil of uncertainty can pay: with uncertainty all players are ex-ante symmetric, whereas with no uncertainty they know that the gains from cooperation will be asymmetrically distributed, leading to smaller stable coalitions. Kolstad (2007) and Kolstad and Ulph (2008) consider uncertainty about the level of the benefits from emission reduction, which they call systematic uncertainty. They also conclude that in a strategic context, ‘learning may be bad for the success of IEAs’. This negative conclusion about the role of learning has been later qualified by Finus and Pintassilgo (2012, p. 2013), who show that the pessimistic conclusions rest on special assumptions (e.g. linear payoff functions with binary boundary solutions), which do not generalize.

One extension of the papers above is the departure from risk-neutrality. Based on the model of Kolstad (2007) and Kolstad and Ulph (2008), Bramoullé and Treich (2009) include risk aversion in a non-cooperative model, which Boucher and Bramoullé (2010) and Hong and Karp (2014) extend to include IEA formation. Exogenous learning leads to less pessimistic conclusions, provided that the degree of risk aversion is sufficiently high.

A second extension introduces endogenous learning, though passive in the form of learning-by-doing. Ulph (2004) as well as Nkuiya et al. (2015) consider a simple two-period IEA model with exogenous uncertainty. Karp and Zhang (2006) and Masoudi et al. (2016) build on these models and consider a fully fledged dynamic climate model with a Bayesian learning-by-doing process. Due to the complexity of these models, Karp and Zhang (2006) do not test for the stability of IEAs but assume some form of partial cooperation, whereas Masoudi et al. (2016) consider only a Nash equilibrium without coalition formation. Since general conclusions are difficult to draw from this type of model, most results are based on simulations.

This article pursues a third extension by considering an endogenous learning process in the form of learning-by-investment. We are not aware of any other paper that considers intentional learning in an IEA setting. In reality, while countries decide about their climate change mitigation policies, they also decide about how much they should invest in research to improve their understanding about climate change and its consequences. We consider systematic uncertainty about the benefits from emission reduction, as in Kolstad (2007) and Kolstad and Ulph (2008), but unlike them, based on the insights in Finus and Pintassilgo (2012, 2013), we assume a strictly concave payoff function. More importantly, in our setting learning is intentional, as uncertainty can be reduced through costly investment in research.

The structure of our game is technically similar to R&D and output or price cartels (Kamien et al., 1992; Amir, 2000; Cabral, 2000) and R&D and IEA games (El-Sayed and Rubio, 2014; Rubio, 2017) although the idea is conceptually different. In R&D models, players invest to improve their technology (e.g. to reduce production costs) while, in our model, players (countries or a group of countries) invest to reduce their uncertainty in order to make better-informed abatement decisions. In the first stage of our game, players decide whether to join a coalition or remain an outsider. In the second stage, players decide on their investments in research about the benefits of emission reduction. In the third stage, players choose their level of abatement. This sequence follows the models on R&D and IEA games and appears to be the natural extension of models on uncertainty and exogenous learning. In order to make a difference, investment in research needs to take place before players chose their abatement strategies.

The choice of membership in the first stage determines whether research and/or abatement are conducted non-cooperatively or cooperatively. We consider three possible scenarios. The coalition conducts research and abatement cooperatively (full cooperation), or the coalition cooperates on only one of those decisions (partial cooperation of type 1 or 2). For simplicity, and in line with the literature on R&D and output or price cartels as well as on R&D and IEA games, we do not consider the choice of the scenario itself as an endogenous decision. We compare the outcomes of the three different scenarios, which already leads to some interesting results. We comment on possible extensions in the conclusions.

Similar to the R&D technology games (e.g. Amir, 2000; Cabral, 2000; Endres et al., 2015), in our setting, research is jointly conducted and its result is perfectly shared among coalition members, but not among outsiders, who conduct their research independently. Also, in line with this literature, research may disseminate to different degrees (which we call ‘research spillovers’) between coalition members and outsiders and among outsiders. This allows us to capture, in a simple and tractable way, the fact that learning is technically a complex process and information transmission may not be perfect, particularly, in the case of countries that do not share and coordinate their research activities.

The remainder of this article proceeds as follows. Section 2 introduces the three-stage game. Subsequently, we solve for equilibrium strategies according to the sequence of backward induction. Section 3 derives equilibrium abatement and Section 4 discusses equilibrium research. Before moving to the first stage about equilibrium membership in Section 6, we derive some positive and normative properties in Section 5 that help to understand and evaluate the formation of stable coalitions. In Section 7, we pull results of all stages together in order to evaluate the overall outcome and Section 8 considers the extension to asymmetric countries. Section 9 concludes. All proofs are provided in the Supplementary Appendix.

2. The model

Let there be a set $N$ with $n \geq 2$ countries that play a three-stage coalition formation game. The objective of each player $k$ (either a single country or a coalition) is to maximize its expected net payoff. In each stage, decisions are taken simultaneously by all players. The timing of the game is as follows:

Stage 1, Coalition Formation: each country decides whether to join coalition $S \subseteq N$ with $m$ members, $1 \leq m \leq n$ ⁠. Countries that do not join the coalition act as singletons.
Stage 2, Research: each player (either a single country or the coalition) chooses the level of investment $z_{k}$ ⁠. Research can be successful or not. Countries in coalition $S$ share information; hence, they have the same probability of being informed. Moreover, the probability of success depends on research spillovers.
Stage 3, Abatement: each player (either a single country or the coalition) chooses the level of abatement $q_{i}$ ⁠, conditional on its information after stage 2.

In stage 1, countries are called signatories or non-signatories depending on whether they join coalition $S \subseteq N$ or remain outside. Henceforth, if we talk about a coalition, we mean a non-trivial coalition, i.e. $2 \leq m$ ⁠. Comparing signatories and non-signatories only makes sense if $2 \leq m < n$ ⁠. In stages 2 and 3, non-signatories always decide non-cooperatively about research and abatement, i.e. they individually maximize their expected payoff. Regarding signatories, we consider three alternative scenarios of cooperation: they jointly maximize their aggregate payoff (i) in terms of research, (ii) in terms of abatement, or (iii) in terms of both strategies. Accordingly, we abbreviate the first scenario by CN, the second scenario by NC and the third scenario by CC.² Given the complexity of the game, we need to keep the model very simple in order to be able to derive analytical solutions.

The gross payoff

π_{i}

(excluding research costs) and net payoff

Π_{i}

(including research costs) of a country

i

are given, respectively, by

π_{i} = γ Q - \frac{q_{i}^{2}}{2}

(1)

Π_{i} = π_{i} - β z_{i} = γ Q - \frac{q_{i}^{2}}{2} - β z_{i}

(2)

where

γ

is the marginal benefit of abatement and

β

the unit cost parameter of research, both being strictly positive. The abatement activity of the country

i

q_{i}

⁠,

Q = \sum_{i = 1}^{n} q_{i}

is the total abatement, and

z_{i}

is the investment in research of country

i

⁠. Benefits from abatement depend on total abatement, abatement costs depend on individual abatement, and research costs depend on individual investment. Thus, abatement is a pure public good; the nature of the research will be discussed below. The corresponding gross and net payoffs of the coalition follow from the aggregation across all the signatories, i.e.

π_{S} = \sum_{i \in S} π_{i}

(⁠

Π_{S} = \sum_{i \in S} Π_{i}

⁠).

For analytical tractability, we make several simplifications. First, we assume symmetric countries (all have the same payoff function), even though they are ex-post different because signatories and non-signatories behave differently, i.e. they choose different levels of research and abatement. This allows us to derive almost all results analytically, including equilibrium research and abatement in stages 2 and 3 and the underlying forces and properties of agreement formation. (An extension to asymmetric countries is considered in Section 8.) Secondly, we have assumed linear benefits and quadratic costs from abatement. This is probably the simplest extension of the linear payoff function considered in most models on uncertainty and IEAs, which allows for interior instead of binary corner equilibrium mitigation strategies. Thirdly, we assume a linear research cost function. This is a quite common assumption that is in line when viewing research costs as monetary expenses.³

Regarding uncertainty, we assume that the value of $γ$ is unknown. That is, $γ$ is a random variable; the distribution is assumed to be public knowledge, with expected value $μ > 0$ and standard deviation $σ > 0$ ⁠. Thus, we consider what Kolstad (2007) called systematic uncertainty, i.e. ex ante all players have symmetric expectations and information about the distribution of the random variable and its ex-post realization is the same for all countries.⁴ We make this assumption for at least three reasons.

First, considering also the abatement and research cost parameters as uncertain would render the analysis complex and long. Secondly, we are interested in cooperation. Therefore, the incentive to conduct joint research is higher if it is about a common parameter and not an individual cost parameter. Thirdly, in climate change, the largest uncertainty concerns the benefits, in the form of reduced damages. This is because there are large uncertainties surrounding the connection between greenhouse concentration and temperature increase, between temperature increase and the harm of the environment and the economy, and between damages and their monetary values. In contrast, abatement costs are better known and research cost is simply the money spent on research. The fact that the success of research is uncertain will be captured by the assumptions below.

We assume that countries can reduce their uncertainty by investing in research, but learning is a complex and risky activity which does not always pay. Moreover, information transmission is imperfect in the sense that the results from research are not perfectly transmitted across researchers and across countries. Research conducted in different centers and laboratories can give rise to different, and sometimes contradictory results. Countries may be reluctant to take the results obtained in other countries at face value, in particular if they are not consistent with their own findings. This explains why, in practice, although basically, all research on climate change is publicly available (notably from the IPCC), not all countries have the same perceptions and beliefs about the severity of climate change and there are even some climate skeptics and deniers. In line with this evidence, we assume that information travels at the research stage rather than at the knowledge stage. That is, countries may benefit from others’ research but draw their own conclusions.

To capture these ideas in a tractable way, we follow Dahm et al. (2008, 2009) and assume that with a certain probability $p_{k}$ ⁠, research is successful for the player $k$ ⁠, with $k$ denoting either an individual country (⁠ $k = i$ ⁠) or the coalition (⁠ $k = S$ ⁠). This means that, with probability $p_{k}$ ⁠, player $k$ finds out the true value of $γ$ with certainty, and with probability $1 - p_{k}$ the research is unsuccessful and the player $k$ ends up with exactly the same information as before, which is the a priori distribution but not the true value of $γ$ ⁠.⁵

Denote by

z_{k}

the investment conducted by the player

k

⁠, with

z_{k} = z_{i}

if conducted by an individual country

i

and

z_{k} = z_{S} = \sum_{i \in S} z_{i}

if conducted jointly by for coalition

S

⁠. We capture the idea that a player may benefit to some extent from other players’ research, by defining the concept of ‘effective research’. Effective research of player

k

⁠, denoted by

Z_{k}

⁠, represents all the research that is relevant for the player

k

⁠, including own research

z_{k}

and spillovers received from others, denoted by

L_{k} (z_{- k})

⁠. Intuitively, we can expect such spillovers to be an increasing function of total research conducted by all players other than

k

⁠, i.e.

\partial L_{k} (z_{- k}) / \partial z_{- k} > 0

⁠. Nevertheless, at this stage, we do not need to make any assumptions about this function. Thus, effective research of player

k

is defined as

Z_{k} : = z_{k} + L_{k} (z_{- k}) .

(3)

We assume that the probability of player

k

’s research being successful depends on his/her effective research in the following way:

p_{k} (Z_{k}) = \frac{Z_{k}}{1 + Z_{k}} .

(4)

This specification appears to be the simplest form to capture the following sensible properties: $p_{k} (0) = 0$ ⁠, $\lim_{Z_{k} \to \infty} p_{k} (Z_{k}) = 1$ ⁠, and $p_{k} (Z_{k})$ is strictly increasing but strictly concave, capturing the idea that there is a positive but diminishing return on investment, i.e. $p_{k}^{'} (Z_{k}) > 0$ and $p_{k}^{″} (Z_{k}) < 0$ ⁠.

This formulation assumes perfect spillovers among signatories, which always have the same effective research $Z_{S} = Z_{i \in S}$ and thus the same probability to find out the true value of $γ .$ This could be interpreted for instance that all signatories engage in a research joint venture. For spillovers between signatories and non-signatories, and among non-signatories, the general formulation of (3) allows for the possibility that research is a pure public, impure public, or club good, depending on the specification of $L_{k} (z_{- k})$ ⁠. In Section 4, we will further elaborate on spillovers.

Anticipating the equilibrium in stages 2 and 3, countries decide in stage 1 whether to join coalition

S

⁠, or to remain a singleton player. Accordingly, a signatory will receive a payoff

Π_{i \in S}^{*} (m)

and non-signatory a payoff

Π_{i \notin S}^{*} (m) .

Employing the stability concept of internal and external stability, frequently applied in the literature on IEAs (e.g. Carraro and Siniscalco, 1993; Barrett, 1994), a coalition of size

m

is called stable if it is internally and externally stable:

\begin{matrix} Internal stability : E (Π_{i \in S}^{*} (m)) \geq E (Π_{i \notin S}^{*} (m - 1)) \\ External stability : E (Π_{i \notin S}^{*} (m)) > E (Π_{i \in S}^{*} (m + 1)) \end{matrix}

(5)

where

E

is the expectation operator. That is, given a coalition

S

with

m

members, no signatory has an incentive to leave, and no non-signatory has an incentive to join the coalition. (We assume that in case of indifference a non-signatory joins the coalition to avoid knife-edge cases.)

We determine the subgame-perfect Nash equilibrium of the game by backwards induction.

3. Third stage: abatement

Non-signatories always chose their abatement levels non-cooperatively. Signatories may choose them non-cooperatively (CN-scenario) or cooperatively (NC- and CC-scenarios). The choice of cooperative and non-cooperative players depends on whether they are informed (⁠ $I$ ⁠) or uninformed (⁠ $U$ ⁠). This depends on whether the research conducted in stage 2 was successful. An informed player (whether a single country or the coalition) acts with the objective of maximizing the realized value of its payoff whereas an uninformed player can only maximize its expected payoff. We assume that the information status of all coalition members is always the same, i.e. either all signatories are informed or all are uninformed. This implies that, regardless of whether coalition members cooperate on research, they will share the results of their research when choosing their abatement, i.e. they will make a joint assessment of $γ$ ⁠.

Summing up, an informed non-signatory maximizes (2), or equivalently (1), since $z_{i}$ is given in stage 3, whereas a uniformed non-signatory maximizes the expected value of the same function. An informed (uninformed) coalition maximizes the (expected) value of the same function multiplied by the coalition size. From the first-order conditions of these optimization problems, the following proposition follows.

Proposition 1.

Individual equilibrium abatement. The equilibrium abatement of country

i

, depending on whether it is a member of coalition

S

or not, and depending on whether it is informed (I) or uninformed (U), is given by

\begin{array}{l} Non - signatories : & all scenarios : & q_{i \notin S}^{* I} = γ, & q_{i \notin S}^{* U} = μ; \\ Signatories : & C N - scenario : & q_{i \in S}^{* I} = γ, & q_{i \in S}^{* U} = μ and \\ N C - & C C - scenario : & q_{i \in S}^{* I} = m γ; & q_{i \in S}^{* U} = m μ . \end{array}

In line with the literature on IEAs, we conclude that cooperative countries choose higher abatement levels than non-cooperative countries, as the former internalize the external effect of their actions on other signatories. This is true irrespective of their information status. Due to the linearity of the benefit function, being informed or uninformed does not affect expected total abatement levels from an ex-ante perspective, as stated in Corollary 1, which follows just by aggregating the expected contributions of all countries.

Corollary 1.

Expected aggregate equilibrium abatement. Given a coalition

S

of size

m \geq 2

⁠, and regardless of the number of informed countries, from an ex-ante perspective in stage 3, the expected aggregate abatement in equilibrium is given by

C N - scenario : E (Q^{*}) = n μ,

N C - & C C - scenario : E (Q^{*}) = (n + m^{2} - m) μ,

which increases in the coalition size only in the NC- and CC-scenarios.

Corollary 1 reveals that the size of coalition $S$ ⁠, $m$ ⁠, matters for expected total abatement only if there is cooperation on abatement (NC- and CC-scenarios), i.e. cooperation on abatement increases the expected total mitigation level.

4. Second stage: research

4.1 Some general observations

Corollary 1 showed that the number of countries being informed or uninformed does not matter for expected abatement from an ex-ante perspective. However, it matters for expected payoffs. If a player is informed, then she can optimally choose abatement considering the realized value of the uncertain parameter

γ

⁠. In contrast, an uninformed player can choose abatement only optimally ‘on average’. To illustrate this, notice that whether a country

i

is informed or not, its total expected benefit,

E (B_{i})

⁠, can be expressed as the sum of all the countries’ contributions:

E (B_{i}) = E (γ Q) = E (γ \sum_{j = 1}^{n} q_{j}) = \sum_{j = 1}^{n} E (γ \cdot q_{j}) = E (γ \cdot q_{i}) + \sum_{j = 1, j \neq ì}^{n} E (γ \cdot q_{j})

where

E (γ \cdot q_{i})

is country

i

’s own expected contribution and

E (γ \cdot q_{j})

is the expected contribution of each country

j

⁠, other than

i

⁠. Now, let us take into consideration the cooperation and information status of each country. The contribution of a country that does not cooperate on abatement is

E (γ \cdot μ) = μ^{2}

if it is uninformed and

E (γ^{2}) = μ^{2} + σ^{2}

if informed. The contribution of a country that cooperates on abatement is

E (m \cdot γ \cdot μ) = m \cdot μ^{2}

if it is uninformed and

E (m \cdot γ^{2}) = m (μ^{2} + σ^{2})

if informed. Therefore, due to the public good nature of abatement, other things being equal, each country’s payoff increases in the number of informed countries. That is, countries do not only directly benefit from their research, but also indirectly by generating knowledge spillovers to other countries, which, in return, will be more likely to become informed and, thus, make higher contributions.

In stage 2, there are two types of countries. Countries that do not cooperate on research (non-signatories, and signatories in NC-scenario), and countries that do cooperate on research (signatories in the CN- and CC-scenarios). We assume that each player $k$ chooses her/his investment in research, $z_{k}$ ⁠, in order to maximize her/his expected payoff, taking as given the amount of research and the information status (⁠ $I$ or $U$ ⁠) of other players. As noted above, all signatories have the same effective research and the same information status.

The investment decision in research is taken by considering the unit cost of research $β$ and the expected benefit, which emerges from the fact that learning increases the probability of being informed. We define the value of information (VI) of player $k$ ⁠, $V I_{k}$ ⁠, as the difference between the expected gross payoff in the event of being informed and the expected gross payoff in the event of being uninformed, other things being equal, i.e. $V I_{k} = E (π_{k} | I) - E (π_{k} | U)$ ⁠. Using the definition of gross payoffs given in (1) and the optimal values of abatement, as established in Proposition 1, we can compute the value of information (i.e. the benefit of research without considering costs) as follows (see Supplementary Appendix 1).

Proposition 2.

Value of information. The value of information of a non-signatory, a signatory and the coalition are given, respectively, by

Non - signatories : all scenarios : V I_{i \notin S}^{} = \frac{σ^{2}}{2},

Signatories : C N - scenario : V I_{i \in S}^{} = (m - \frac{1}{2}) σ^{2}, N C - & C C - scenario : V I_{i \in S}^{} = m^{2} σ^{2} .

Proposition 2 has the following implications. First, the value of information of all players depends positively on the variance of the uncertain parameter

γ

⁠,

σ^{2}

⁠. In the extreme case with no uncertainty (⁠

σ^{2} = 0

⁠), research would be meaningless while if uncertainty was high, the fact of being informed would make an important difference when it comes to making abatement decisions. Secondly, the value of information of a signatory, and, hence, that of the coalition, is increasing in the coalition size and is always higher than that of a non-signatory. Thirdly, the value of information of a signatory is higher if there is cooperation on abatement, which strengthens the incentives to invest in research. Taken together, we have:

V I_{i \in S} (C C) = V I_{i \in S} (N C) > V I_{i \in S} (C N) > V I_{i \notin S} .

In order to derive equilibrium investment in research, each player solves the following problem:

\max_{z_{k} \geq 0} E (Π_{k}) = p_{k} \cdot E (Π_{k} | I) + (1 - p_{k}) \cdot E (Π_{k} | U) = E (π_{k} | U) + p_{k} \cdot V I_{k} - β z_{k}

(6)

where the probability of being informed,

p_{k}

⁠, is given by (3) and (4). The objective function in (6) can be split in two parts. The first part,

E (π_{k} | U)

⁠, is the expected payoff in the absence of learning and can be viewed as the baseline payoff, as it corresponds to the (expected) payoff which player

k

would receive in a standard pure abatement game without research. For the maximization, this component is not relevant, as it is independent of

z_{k}

⁠. The second component,

p_{k} \cdot V I_{k} - β z_{k}

⁠, can be viewed as an additional informational payoff. It is the probability of success times the expected value of information that a player can obtain when conducting research minus research costs. Note that

V I_{k}

is the value of information of the entire coalition when signatories cooperate on research (⁠

V I_{k} = V I_{S}

in the CN- and CC-scenarios) or the individual value of information of a country otherwise (⁠

V I_{k} = V I_{i \in S}

for signatories in the NC-scenario and

V I_{k} = V I_{i \notin S}

for non-signatories). The first order condition of (6) is given by

\frac{1}{{(1 + z_{k} + L_{k} (z_{- k}))}^{2}} V I_{k} - β \leq 0 (with equality if z_{k}^{*} > 0)

(7)

from which we get the equilibrium value of research in an interior solution:

z_{k}^{*} = \sqrt{\frac{V I_{k}}{β}} - 1 - L_{k} (z_{- k}) \geq 0 .

(8)

The concavity of the objective function ensures that the second order condition is satisfied. Using the definition of effective research (3), $Z_{k}^{} = z_{k}^{*} + L_{k} (z_{- k})$ ⁠, and introducing the concept of the target level of effective research⁶, ${\hat{Z}}_{k}^{} : = \sqrt{\frac{V I_{k}}{β}} - 1$ ⁠, we can summarize our result in the following proposition.

Proposition 3.

Best response in terms of research. The best response of player

k

(whether a country or a coalition) in terms of research effort is given by

z_{k} (z_{- k}) = \max {{\hat{Z}}_{k}^{} - L_{k} (z_{- k}), 0}

(9)

where

{\hat{Z}}_{k}^{} : = \sqrt{\frac{V I_{k}}{β}} - 1

(10)

is the target level of effective research of player

k

⁠.

The best response of player $k$ in (9) is the difference between his/her target level of effective research, ${\hat{Z}}_{k}^{}$ ⁠, and the spillovers he/she receives from other players, which are taken as given. In other words, player $k$ will make up for the research that is not done by others in order to obtain his/her target level of effective research. Hence, if research spillovers are sufficiently large, own research will be zero and we have a boundary solution.

Note that the target value of effective research increases in the gross value of information and decreases in the unit cost of research. To avoid uninteresting solutions, we introduce the following assumption:

Assumption 1:

$β \leq \frac{σ^{2}}{2}$ ⁠,

which implies that the unit research cost does not exceed the value of information. This ensures that the target level of effective research ${\hat{Z}}_{k}$ is positive for all players.⁷ Nevertheless, individual equilibrium research can be zero if spillovers are sufficiently high.

Noting that $V I_{k}$ in the CN- and CC-scenarios in which signatories cooperate on research is $V I_{k} = m \cdot V I_{i \in S}$ in Equations (6)–(10), and that the target level of research is increasing in the value of information, as indicated in Proposition 2, we can state the following result.

Corollary 2.

Different cooperative scenarios and the target value of effective research. The target level of effective research of a signatory is higher than that of a non-signatory. The former depends on the cooperative scenario and the latter is constant across all three cooperative scenarios. Formally:

{\hat{Z}}_{i \in S}^{} (C C) > {\hat{Z}}_{i \in S}^{} (C N) > {\hat{Z}}_{i \in S}^{} (N C) > {\hat{Z}}_{i \notin S}^{} (C C) = {\hat{Z}}_{i \notin S}^{} (C N) = {\hat{Z}}_{i \notin S}^{} (N C) .

Thus, cooperation on research and abatement increases the value of information for signatories and, thus, their target level of effective research.

4.2 The role of research spillovers for equilibrium investment in research

Up to now, we have been working with a very general formulation of spillovers, $L_{k} (z_{- k})$ ⁠. For the sake of concreteness, we introduce the following linear specification, which we assume to hold subsequently.

Assumption 2.

The effective research of non-signatories is given by

Z_{i \notin S} = z_{i} + ω \sum_{j \in S} z_{j} + δ \sum_{l \notin S, l \neq i} z_{l}

(11)

and the effective research of the coalition is equal to the effective research of each coalition member and is given by:

Z_{S} = Z_{i \in S} = \sum_{j \in S} z_{j} + δ \sum_{l \notin S} z_{l}

(12)

with $ω, δ \in [0, 1]$ ⁠.

According to Assumption 2, $ω$ represents the proportion of signatories’ research that spills over to non-signatories and $δ$ determines the proportion of a non-signatory’s research that spills over to other countries (signatories and other non-signatories). Note that this specification is quite general and covers two classical cases, such as the ‘club good case’, which corresponds to $δ = ω = 0$ ⁠, and the ‘pure public good case’, which corresponds to $ω = δ = 1$ ⁠.

Given symmetric payoff functions of all countries, it is natural to focus on symmetric equilibria.⁸ Due to the linearity of the research cost function, only optimal total research is uniquely determined for a coalition that cooperates on research. In order to preserve symmetry, we make the following standard assumption.

Assumption 3.

When the coalition cooperates on research, the total research effort of the entire coalition is split evenly among all the coalition members, i.e. $z_{i}^{*} = z_{S}^{*} / m \forall i \in S .$

For equilibrium research levels, we can state the following result, which implies that in a corner solution, only signatories invest in research (see Supplementary Appendix 2).

Proposition 4.

Equilibrium levels of research. Let research spillovers be given by (11) and (12). Then equilibrium research of signatories and non-signatories in stage 2 are given by

z_{i \in S}^{*} = {\begin{matrix} \frac{Ψ {\hat{Z}}_{i \in S}^{} - δ (n - m) {\hat{Z}}_{i \notin S}^{}}{m [Ψ - δ ω (n - m)]} > 0 & i f ω < \tilde{ω} \\ \frac{{\hat{Z}}_{i \in S}^{}}{m} > 0 & i f ω \geq \tilde{ω} \end{matrix}

z_{i \notin S}^{*} = {\begin{matrix} \frac{{\hat{Z}}_{i \notin S}^{} - ω {\hat{Z}}_{i \in S}^{}}{Ψ - δ ω (n - m)} > 0 & i f ω < \tilde{ω} \\ 0 & i f ω \geq \tilde{ω} \end{matrix}

where

Ψ : = 1 + δ (n - m - 1)

⁠,

\tilde{ω} : = \frac{{\hat{Z}}_{i \notin S}^{}}{{\hat{Z}}_{i \in S}^{}}

. Moreover,

1 > {\tilde{ω}}^{N C} > {\tilde{ω}}^{C N} > {\tilde{ω}}^{C C} > 0

⁠.

Proposition 4 shows that the threshold value of the corner/interior solution ${\hat{Z}}_{i \notin S}^{} / {\hat{Z}}_{i \in S}^{}$ differs between the three scenarios. According to Corollary 2, ${\hat{Z}}_{i \notin S}^{}$ is constant across scenarios and ${\hat{Z}}_{i \in S}^{} (C C) > {\hat{Z}}_{i \in S}^{} (C N) > {\hat{Z}}_{i \in S}^{} (N C)$ ⁠. Thus, $1 > {\tilde{ω}}^{N C} > {\tilde{ω}}^{C N} > {\tilde{ω}}^{C C} > 0$ follows. Intuitively, the more ambitious signatories are in terms of research, the larger is the scope for corner solutions for which non-signatories can ‘relax on the backseat’ without making a contribution to research.

Corollary 3 summarizes how equilibrium research ranks across signatories and non-signatories and across the three cooperative scenarios (see Supplementary Appendix 3).

Corollary 3.

Ranking of equilibrium research. For any of the three cooperative scenarios and for any given coalition of size

m

⁠,

1 < m < n

, signatories invest more in research than non-signatories, i.e.

z_{i \in S}^{*} > z_{i \notin S}^{*}

⁠. Moreover, the equilibrium research of signatories and non-signatories is ranked across cooperative scenarios as follows:

Signatories : z_{i \in S}^{*} (C C) > z_{i \in S}^{*} (C N) > z_{i \in S}^{*} (N C),

Non - signatories : z_{i \notin S}^{*} (N C) \geq z_{i \notin S}^{*} (C N) \geq z_{i \notin S}^{*} (C C) .

Thus, the ranking of signatories’ and non-signatories’ equilibrium research efforts are inversely related because research efforts are strategic substitutes. Moreover, signatories will conduct more research than non-signatories because they internalize the positive externalities, resulting from joint research and/or joint mitigation.

5. Properties of the coalition formation game: a digression

Before moving to the membership stage, it is helpful to establish some positive and normative properties. The positive properties will be useful in understanding why stable coalitions tend to be small. It will also become clear why it is not straightforward to predict how the degree of spillovers and cooperation affect the size of stable coalitions. The normative properties are useful in demonstrating that global welfare increases with the size of coalitions, the degree of spillovers and degree of cooperation. As each player faces two possible states, informed or uninformed, we take an ex-ante perspective when evaluating expected payoffs. (For details, see Supplementary Appendix 4.)

Proposition 5.

Signatories’ versus non-signatories’ payoffs. Given a coalition of size $m$ ⁠, $1 < m < n$ , for any cooperation scenario (NC, CN, and CC), and any degree of spillovers, $ω, δ \in [0, 1]$ , the expected payoff of a non-signatory is strictly higher than the expected payoff of a signatory. That is, $E (Π_{i \in S}^{*} (m)) < E (Π_{i \notin S}^{*} (m))$ ⁠.

This result is quite common for models of public provision but also for output and price cartels (see Yi (1997) for an overview). In our model, all players have the same benefits but signatories have higher abatement and research costs in equilibrium than non-signatories due to the internalization of positive externalities among their members. This property already provides an indication why stable agreements tend to be small. The coalition formation game resembles a kind of $n$ -player chicken game, with signatories being the chickens. Nobody likes to be the chicken, even though if no coalition forms at all, some players may join the coalition.

The next property is called positive externality and captures the ‘push-out-effect’, i.e. the incentive to leave a coalition. It may be viewed as the ‘classical free-rider property’.

Proposition 6.

Positive externality. In all three cooperative scenarios (NC, CN, and CC) and for any degree of spillovers,

ω, δ \in [0, 1]

, the coalition formation game is characterized by a strict positive externality, i.e. for all

j \notin S

⁠,

1 < m < n

⁠, the expected payoff of non-signatories is strictly increasing in the size of the coalition:

E (Π_{j \notin S}^{*} (m)) > E (Π_{j \notin S}^{*} (m - 1)) .

The driving force of this property has four dimensions when analysed in detail (see Supplementary Appendix 5.) At the aggregate, if a non-signatory accedes to the coalition, all other non-signatories benefit from higher mitigation and research efforts of the coalition, while they keep their efforts constant or reduce their efforts, implying higher benefits and weakly lower costs.

The next property is called superadditivity and captures the ‘pull-in-effect’, i.e. the incentive to stay in a coalition or to enter it.

Proposition 7.

Superadditivity. In all three cooperative scenarios (NC, CN, and CC) and for any degree of spillovers,

ω, δ \in [0, 1]

, the coalition formation game is characterized by strict superadditivity, i.e. for all

i, j \in N

⁠,

2 \leq m \leq n

⁠:

m \cdot E (Π_{i \in S}^{*} (m)) > (m - 1) \cdot E (Π_{i \in S}^{*} (m - 1)) + E (Π_{j \notin S}^{*} (m - 1)) .

The driving force of this property derives from the fact that countries can achieve more if they cooperate by internalizing the externalities among signatories, even though the details are much more complicated. (For details, see Supplementary Appendix 6.)

Overall, the size of stable coalitions depends on the relative strength of the push-out- and the pull-in-effect. As a tendency, for smaller coalitions the pull-in-effect is stronger than the push-out-effect, but for larger coalitions this is reversed, which explains that, usually, the grand coalition is not stable. However, despite these three properties are interesting by themselves, they do not allow for a precise prediction of the size of stable coalitions.⁹ The only conclusion that we can draw at this level of generality is that $m^{*} \geq 2$ ⁠. This is because for the move from $m = 1$ to $m = 2$ the condition of superadditivity is identical to the condition of internal stability. Hence, as we know that superadditivity generally holds, the coalition of size $m = 2$ is internally stable. Consequently, a non-trivial (⁠ $m^{*} \geq 2$ ⁠) stable coalition always exists.

The two positive properties in Propositions 6 and 7 give rise to a normative property called full cohesiveness. That is, total welfare across all players increases with the coalition size. Consequently, global welfare is maximized in the grand coalition and there is a rationale to care for large stable coalitions.

Corollary 4.

Full cohesiveness. In all three cooperative scenarios (NC, CN, and CC) and for any degree of spillovers,

ω, δ \in [0, 1]

, the coalition formation game exhibits strict full cohesiveness, i.e. for all

i, j \in N

⁠,

1 < m \leq n

⁠:

\begin{matrix} m \cdot E (Π_{i \in S}^{*} (m)) + (n - m) E (Π_{j \notin S}^{*} (m)) > \\ (m - 1) \cdot E (Π_{i \in S}^{*} (m - 1)) + (n - m + 1) E (Π_{j \notin S}^{*} (m + 1)) . \end{matrix}

We now consider properties directly related to our two central themes, the degree of spillovers and the degree of cooperation.

Proposition 8.

Degree of spillovers and payoffs. In all three cooperative scenarios (NC, CN, and CC), and for any generic coalition of size $m$ ⁠, $1 < m < n$ , the effect of the spillover coefficients on individual and aggregate expected payoffs are given in Table 1:

The proof is given in Supplementary Appendix 7. In a corner solution, non-signatories make no contribution to research. Therefore, $δ$ is irrelevant for payoffs as no spillovers emerge from non-signatories. However, an increase in $ω$ increases the effective research of non-signatories and, thus, their probability of making informed decisions, which increases the expected payoff of all countries, as explained in Section 4.1.

Table 1

Open in new tab

Effect of spillover parameters on individual and global payoffs for a given coalition of size $m$ ⁠.

		$Π_{i \in S}^{*} (m)$	$Π_{i \notin S}^{*} (m)$		$\sum_{i = 1}^{n} Π_{i}^{*} (m)$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$ω ↑$	$↑$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$δ ↑$	$\leftrightarrow$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$ω ↑$	$↓$	$↑$		$↑$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$δ ↑$	$↑$	$↑$ if $m < n - 1$	$↓ i f ω > 0; \leftrightarrow i f ω = 0$ if $m = n - 1$	$↑$

		$Π_{i \in S}^{*} (m)$	$Π_{i \notin S}^{*} (m)$		$\sum_{i = 1}^{n} Π_{i}^{*} (m)$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$ω ↑$	$↑$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$δ ↑$	$\leftrightarrow$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$ω ↑$	$↓$	$↑$		$↑$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$δ ↑$	$↑$	$↑$ if $m < n - 1$	$↓ i f ω > 0; \leftrightarrow i f ω = 0$ if $m = n - 1$	$↑$

Note: All relations assume $1 < m < n$ so as to ensure that there are at least two signatories and one non-signatory. $↑$ = increase; $↓$ = decrease; $\leftrightarrow$ = no change.

Table 1

Open in new tab

Effect of spillover parameters on individual and global payoffs for a given coalition of size $m$ ⁠.

		$Π_{i \in S}^{*} (m)$	$Π_{i \notin S}^{*} (m)$		$\sum_{i = 1}^{n} Π_{i}^{*} (m)$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$ω ↑$	$↑$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$δ ↑$	$\leftrightarrow$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$ω ↑$	$↓$	$↑$		$↑$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$δ ↑$	$↑$	$↑$ if $m < n - 1$	$↓ i f ω > 0; \leftrightarrow i f ω = 0$ if $m = n - 1$	$↑$

		$Π_{i \in S}^{*} (m)$	$Π_{i \notin S}^{*} (m)$		$\sum_{i = 1}^{n} Π_{i}^{*} (m)$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$ω ↑$	$↑$
Corner solution (⁠ $ω \geq \tilde{ω}$ ⁠)	$δ ↑$	$\leftrightarrow$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$ω ↑$	$↓$	$↑$		$↑$
Interior solution (⁠ $ω < \tilde{ω}$ ⁠)	$δ ↑$	$↑$	$↑$ if $m < n - 1$	$↓ i f ω > 0; \leftrightarrow i f ω = 0$ if $m = n - 1$	$↑$

Note: All relations assume $1 < m < n$ so as to ensure that there are at least two signatories and one non-signatory. $↑$ = increase; $↓$ = decrease; $\leftrightarrow$ = no change.

In an interior solution, the effective research that all countries enjoy is their own target, which does not depend on the spillover coefficients, $ω$ and $δ$ ⁠. However, as detailed in Proposition 4, $ω$ and $δ$ affect equilibrium research and, thus, research costs. If $ω$ increases, signatories will increase and non-signatories will decrease their research effort. This is a substitution effect. Accordingly, signatories’ payoffs will decrease and non-signatories’ payoffs will increase. Increasing $δ$ means that signatories will be able to reduce their research efforts and, hence, their payoffs increase. Increasing $δ$ also increases the spillover among non-signatories and allows them to decrease their research efforts (except if $m = n - 1$ ⁠, when there is only one non-signatory).

Thus, research spillovers are always beneficial for those players receiving spillovers and may also be beneficial for the countries from which these spillovers emerge, with the exception for signatories in an interior solution. Overall, spillovers always increase the expected aggregate payoff for any generic coalition of size $m$ ⁠, $1 < m < n$ ⁠. Therefore, if we were able to conclude that spillovers weakly increase the size of stable coalitions $m^{*}$ ⁠, spillovers would have an unequivocally positive effect on global welfare for stable coalitions.

However, the effect of spillovers on $m^{*}$ is not straightforward for at least four reasons. First, in a corner (an interior) solution an increase in $ω$ (⁠ $δ$ ⁠) increases both signatories and non-signatories’ payoffs. Hence, the effect on $m^{*}$ depends on the relative strength of these effects. Secondly, in an interior solution, an increase in $ω$ increases non-signatories’ payoffs and decreases signatories’ payoffs, which, in principle, should lead to a smaller coalition. However, because $m^{*}$ must be an integer value, we can only conclude $m^{*} (ω_{1}) \geq m^{*} (ω_{2})$ if $ω_{1} < ω_{2}$ (not $m^{*} (ω_{1}) > m^{*} (ω_{2})$ ⁠). This is different in a corner solution considering a change of $δ$ ⁠. Such a change does not affect payoffs and, hence, also not $m^{*}$ ⁠. Thirdly, increasing $ω$ may move the equilibrium from an interior to a corner solution once the threshold value $\tilde{ω}$ has been passed. Passing the threshold will certainly benefit non-signatories but the effect on signatories is unclear. (Signatories have higher research costs because non-signatories make no contribution to effective research anymore but benefit from higher expected benefits as non-signatories make better-informed decisions.) Hence, the effect on $m^{*}$ is undetermined. Fourth, according to Proposition 4, the threshold is given by $\tilde{ω} = {\hat{Z}}_{i \notin s}^{} / {\hat{Z}}_{i \in s}^{}$ ⁠, where the target values of research depend on $β$ ⁠, $σ^{2}$ ⁠, and $m$ ⁠, and, we have $\partial \tilde{ω} / \partial m < 0$ ⁠. Hence, we may have an interior solution for smaller values of $m$ and a corner solution for larger values of $m$ ⁠, which makes analytical predictions about $m^{*}$ in the entire range $1 < m \leq n$ impossible.

Therefore, a comprehensive picture of how research spillovers affect the size of stable coalitions $m^{*}$ requires simulations. However, we are able to derive analytical solutions for two benchmark cases: (i) the club good case, i.e. $ω = δ = 0$ ⁠, which implies an interior solution for every $m$ and (ii) the public good case, i.e. $ω = δ = 1$ ⁠, which implies a corner solution for every $m$ ⁠. This concerns Section 6 in terms of first stage results and Section 7 about the overall results. Both sections are split into two subsections with analytical solutions (Sections 6.1 and 7.1) and simulations (Sections 6.2 and 7.2).

Proposition 9.

Degree of cooperation and payoffs. For any degree of spillovers, and for any generic coalition of size $m$ ⁠, $1 < m \leq n$ , moving from partial cooperation to full cooperation, i.e. moving either (i) from the CN- to the CC-scenario or (ii) from the NC- to the CC-scenario, (a) always benefits non-signatories, (b) benefits signatories if before and after the move the equilibrium is in a corner solution, and (c) always increases the aggregate payoff over all players.

The proof is given in Supplementary Appendix 8. Non-signatories always benefit if signatories cooperate more and, thus, internalize more externalities, which may be viewed as a variant of the positive externality property discussed above. Intuitively, abatement is a pure public good and research is an impure public good. Hence, non-signatories benefit from higher research efforts (cases (i) and (ii) in Proposition 9) and sometimes from higher abatement efforts (case i in Proposition 9).

For signatories conclusions are less straightforward. From Proposition 4, we have: $1 > {\tilde{ω}}^{N C} > {\tilde{ω}}^{C N} > {\tilde{ω}}^{C C} > 0$ ⁠. Hence, the move from partial to full cooperation, allows for three cases. Case 1: before and after the move, the equilibrium is in a corner solution. Case 2: before and after the move the equilibrium is in an interior solution. Case 3: the equilibrium is an interior before the move but a corner equilibrium after the move.

Only case 1 is straightforward because in a corner solution signatories act like in autarky, as non-signatories do not change their equilibrium abatement and research and more cooperation benefits signatories. In cases 2 and 3, equilibrium research by non-signatories declines if signatories move from partial to full cooperation. This research leakage may decrease signatories’ payoffs.

Nevertheless, at the aggregate level, we are able to conclude that the total payoff over all players increases with the degree of cooperation. Hence, if stable coalitions were to increase with the degree of cooperation, we could conclude that the degree of cooperation positively affects global welfare. However, the effect of the degree of cooperation on $m^{*}$ is not straightforward at all.

For instance, even in a corner solution, we only know that both signatories and non-signatories benefit from a higher degree of cooperation. Thus, the effect on $m^{*}$ depends on the exact payoff changes. Moreover, as mentioned above, we have $1 > {\tilde{ω}}^{N C} > {\tilde{ω}}^{C N} > {\tilde{ω}}^{C C} > 0$ and each $\tilde{ω}$ depends on the parameters of our model and on the coalition size. Thus, whether the equilibrium is in the interior or at a corner depends on the number of signatories, $m$ ⁠, and the cooperative scenario. Hence, analytical predictions about how the degree of cooperation will affect $m^{*}$ in the entire range of $1 < m \leq n$ are generally not possible, except for the benchmark cases, the club good and the public good case. In all other cases, we require simulations.

6. First stage: membership

6.1 Analytical results

As explained above, our analytical results focus on the club good case $(ω = δ = 0)$ and the public good case $(ω = δ = 1)$ (See Supplementary Appendix 9).

Proposition 10.

Equilibrium coalition size in the club good and public good case. In the three cooperative scenarios, assuming $n \geq 3$ , the equilibrium coalition size is given by:

Club good case
- In the NC- and CC-scenarios: $m^{*} = 3$ ⁠.
- In the CN-scenario: $m^{*} \geq 5$ (provided $n \geq 5$ ). A coalition of size $m > 5$ is stable if the ratio $\frac{β}{σ^{2}}$ is small enough. If $\frac{β}{σ^{2}} \leq 0.17$ , the grand coalition is stable.
Public good case
- In the NC- and CC-scenarios: $m^{*} \geq 3$ . A coalition of size $m > 3$ is stable if $n$ is sufficiently large. The grand coalition ( $m = n$ ) is never stable if $n > 3$ ⁠.
- In the CN-scenario: A coalition of size $m > 2$ is internally stable if $n$ is sufficiently large. The grand coalition ( $m = n$ ) is never stable.

It is well known from the literature (see the collections of articles and the survey in Finus and Caparrós (2015)) that in the pure abatement game with linear benefits and quadratic costs and no uncertainty the equilibrium coalition size is $m^{*} = 3$ ⁠. The same is true for exogenous learning and systematic uncertainty about parameter $γ$ as derived for this payoff function by Finus and Pintassilgo (2013). We can now ask what changes when we allow for endogenous active learning by means of research.

Since in most models in the literature signatories cooperate on abatement, the obvious comparison is with the NC- and CC-scenarios. Proposition 10 reveals that if research is a club good the standard result persists, with or without cooperation on research. Conversely, if research is a public good, larger coalitions may be stable. Hence, surprisingly, research spillovers do not aggravate but may mitigate the free-rider problem. In fact, $m^{*}$ does not decrease (because the externality effect is aggravated and free-riding encouraged with the number of players) but increases with the number of countries $n$ ⁠. As explained in Section 4, countries’ research increases the probability of learning, their own but also of other countries through spillovers, which increases the expected benefits from the contributions of other countries. This harvesting effect reinforces the superadditivity property and allows for larger $m^{*}$ with increasing $n$ ⁠, even though the grand coalition is never stable. This phenomenon also holds in the CN-scenario in the public good case. However, we have to issue a note of caution: even though $m^{*} (n)$ may increase in absolute terms, this may not alleviate the free-rider problem, as in relative terms, $\frac{m^{*} (n)}{n}$ may well decline with $n .$

It is also interesting that if signatories only cooperate on research but not on abatement (CN-scenario), and there are no spillovers (club good case), stable coalitions may be quite large, especially if the marginal cost of research $β$ is small and/or uncertainty is large (i.e. large $σ^{2}$ ⁠), implying that the value of information is large. The free-rider incentive is reduced, as signatories do not cooperate on abatement. The free-rider incentive is further reduced, as in the club good case the positive research externalities are exclusive.

6.2 Simulations

In order to gain more insights on how the degree of spillovers and the degree of cooperation affect the size of stable coalitions, we run extensive simulations, which are explained in detail in Supplementary Appendix 10. As $m^{*}$ in itself is not very meaningful if we vary the number of countries $n$ ⁠, from 4 to 100, we determine the coalition size in relative terms, ${\tilde{m}}^{*} = \frac{m^{*}}{n} \cdot 100$ ⁠. Figure 1 shows the average value (across different parameter combinations) of ${\tilde{m}}^{*}$ as a function of the spillover coefficients $ω$ and $δ$ ⁠, respectively, for the three cooperative scenarios, CN, NC, and CC.

Figure 1

Relative size of stable agreements and the degree of spillovers and cooperation.

Note: Source: Authors’ calculations.

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First, the spillovers from a non-signatory to all other countries, $δ$ ⁠, hardly affects ${\tilde{m}}^{*}$ in all scenarios. This is in line with Proposition 8, which concluded that $δ$ does not affect payoffs at all in a corner solution and increases both, signatories’ and non-signatories’ payoffs, in an interior solution. (Hence, $m^{*}$ may well be unaffected.) Secondly, the spillover from signatories to non-signatories, $ω$ ⁠, decreases ${\tilde{m}}^{*}$ in the CN-scenario. In the NC- and CC-scenarios the same is true, even though this is hardly visible in Figure 1 due to the scaling. Spillovers from signatories to non-signatories either only benefit non-signatories in an interior equilibrium or benefit both groups, but non-signatories more (see Proposition 8) and, thus, tend to reduce membership ${\tilde{m}}^{*}$ ⁠.

Third, ${\tilde{m}}^{*}$ is larger in the CN-scenario than in the NC-and CC-scenarios, where the latter two scenarios have almost the same value ${\tilde{m}}^{*},$ with ${\tilde{m}}^{*}$ in the NC-scenario slightly above the CC-scenario. Therefore, cooperating only on research provides little free-rider incentives to players. A departure from full cooperation, particularly a departure from cooperation on abatement, pays in terms of membership.

7. Overall results

7.1 Analytical results

By pulling results from the three stages together, we draw overall conclusions about global welfare in stable agreements.

Corollary 5.

Consider aggregate welfare in stable agreements of size $m^{*}$ ⁠.

In the NC- and CC-scenarios, global welfare is larger in the public good than in the club good case.
In the club good case, global welfare is larger in the CC- than in the NC-scenario.

Part (i) of Corollary 5 follows from Proposition 8, which states that global welfare weakly increases in $δ$ and strictly in terms of $ω$ ⁠, Proposition 10, which states that $m^{*}$ is weakly larger in the public good case than in the club good case and Corollary 4, which claims that global welfare increases in the number of signatories.

Part (ii) of Corollary 5 follows from Proposition 9, part c, which states that global welfare increases with the degree of cooperation and Proposition 10, which claims that $m^{*}$ is the same in the CC- and NC-scenarios in the club good case.

Even though the results refer to special cases, similar qualitative conclusions are derived below in the simulations, considering intermediate cases of the spillover coefficients. First, research spillovers may lead to smaller stable coalitions, but the effect on aggregate welfare is positive. These opposing effects may more or less balance or may be in favour of an overall higher performance, like in Corollary 5, Part (i). Secondly, partial cooperation may lead to larger stable coalitions, but full cooperation has a positive effect on aggregate welfare. These opposing effects may more or less balance or may be in favour of an overall higher performance due to cooperation, like in Corollary 5, Part (ii).

This exhausts what we can say analytically about the global welfare of stable coalitions.

7.2 Simulations

In order to evaluate the performance of coalitions for simulations, it seems suggestive to use a relative and not an absolute index. We define the closing-the-gap index as follows:

CGI = \frac{Π^{*} (m^{*}) - Π^{* N E}}{Π^{* S O} - Π^{* N E}} \cdot 100,

(13)

where

Π^{*} (m^{*})

is the aggregate payoff over all countries in a stable coalition of size

m^{*}

⁠,

Π^{* N E}

is the aggregate payoff in the Nash equilibrium with no cooperation at all and

Π^{* S O}

the aggregate payoff in the social optimum. Note that even if

m^{*} = n

⁠, only in the CC-scenario would

Π^{*} (m^{*} = n) = Π^{* S O}

hold. This index is bounded between 0 and 100. Figure 2 shows the average value (across simulations) of the CGI as a function of the spillover coefficients

ω

and

δ

for the three cooperative scenarios.

Figure 2

Relative performance in welfare terms of stable agreements and the degree of spillovers and cooperation.

Note: Source: Authors’ calculations.

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We highlight three results. First, on average, the CGI is rather insensitive to the degree of spillovers. A finer scaling would reveal that the CGI is at least not decreasing in the spillover coefficients. Thus, the opposing effects of spillovers on global welfare and stable coalitions more or less balance in the simulations on average. Secondly, in contrast, the degree of cooperation makes a difference. Even though cooperation on research only (CN-scenario) leads to much larger stable coalitions, the welfare loss from not cooperating on abatement is even larger compared to the NC- and CC-scenarios. This highlights that from the size of stable agreements, one cannot infer its success. Moreover, agreements on research cooperation are not a good substitute for cooperation on abatement. Hence, our model does not support the conjecture that due to high free-rider incentives, it would be better to give up the ambition to strike an agreement on mitigating emissions, but instead focus on research. Cooperation on mitigation is more important.

Thirdly, even though the CC-scenario performs best among all scenarios, even for this scenario the average CGI is relatively small. This is because the size of stable coalitions $m^{*}$ in relation to the total number of countries $n$ is simply too small to make a significant difference in closing the gap between full and no cooperation. This is a variant of the paradox of cooperation, as stated by Barrett (1994) and reiterated by many others since then.

8. Extension: asymmetric countries

8.1 Preliminaries

In this section, we test whether our main qualitative conclusions hold if we allow countries to be asymmetric and whether new interesting insights emerge. In principle, all parameters of our model could be asymmetric, but we focus on asymmetry on the benefit side. We assume each country has a share parameter

α_{i}

which we normalize such that

\sum_{i = 1}^{n} α_{i} = 1

⁠. (See, for instance, Finus and McGinty (2019) for a similar assumption.) Hence, the modified net payoff function is given by

{\tilde{Π}}_{i} = α_{i} γ Q - \frac{q_{i}^{2}}{2} - β z_{i}

(14)

instead of (2). Consequently, we can retrieve symmetry if

α_{i} = 1 / n

⁠. If we assume two types of countries with a low and high share

α_{i}

⁠, we can define

α_{L} = 1 / n - Δ

and

α_{H} = 1 / n + Δ

⁠.¹⁰ By varying

Δ

in the range

[0, 1 / n]

⁠, we can model different degrees of asymmetry. A coalition

S

may comprise symmetric (all low or high

α_{i}

-type countries), but also asymmetric countries (⁠

α_{L}

- and

α_{H}

-type countries). If there is cooperation on abatement, all signatories choose the same equilibrium mitigation level

q_{i \in S}^{* I} = γ \sum_{i \in S} α_{i}

if informed and

q_{i \in S}^{* U} = μ \sum_{i \in S} α_{i}

if uninformed, and all signatories choose symmetric research levels due to symmetric research costs (see Assumption 3). Consequently, signatories of the

α_{L}

-type will receive a lower payoff than those of the

α_{H}

-type. As it is well-known from the literature, such asymmetric payoffs imply that asymmetric stable coalitions are smaller than symmetric stable coalitions.¹¹ Hence, in order to render the analysis of asymmetric countries interesting, we follow the literature and assume an ‘optimal transfer scheme’: then the size of stable coalitions is at least as large with asymmetric than with symmetric players and the coalition with the highest global welfare among those coalitions, which can be ‘potentially internally stabilized’ is stable, provided a game is characterized by positive externalities and superadditivity.¹²

Each signatory’s payoff after transfers is given by ${\tilde{Π}}_{i}^{* T} = {\tilde{Π}}_{i}^{*} (S \ {i}) + λ_{i} Ω_{S}$ with $Ω_{S} = \sum_{i \in S} {\tilde{Π}}_{i}^{*} (S) - \sum_{i \in S} {\tilde{Π}}_{i}^{*} (S \ {i})$ ⁠, $\sum_{i \in S} λ_{i} = 1$ and $λ_{i} > 0$ for all $i \in S$ ⁠. That is, every signatory receives its free-rider payoff if they would leave coalition $S$ ⁠, ${\tilde{Π}}_{i}^{*} (S \ {i})$ ⁠, plus a share $λ_{i}$ of the surplus $Ω_{S}$ ⁠, which is defined as the total payoff of coalition $S$ ⁠, $\sum_{i \in S} {\tilde{Π}}_{i}^{*} (S)$ ⁠, minus the sum of free-rider payoffs $\sum_{i \in S} {\tilde{Π}}_{i}^{*} (S \ {i}) .$ Note that transfers are budget neutral, i.e. $\sum_{i \in S} {\tilde{Π}}_{i}^{*} (S) = \sum_{i \in S} {\tilde{Π}}_{i}^{T *} (S) .$

Even though shares $λ_{i}$ matter for the payoffs of signatories, they do not matter for the set of stable coalitions. A positive surplus $Ω_{S}$ implies that a coalition is potentially internally stable and can be stabilized with an appropriate transfer scheme. In our case, the $α_{H}$ -type countries, which benefit more from cooperation, will pay the $α_{L}$ -type countries, which benefit less than proportionally from cooperation, a transfer such they stay in the coalition. However, not all coalitions are potentially internally stable, i.e. $Ω_{S} < 0$ ⁠, if the free-rider incentives are too strong. This will be particularly the case for large coalitions.

If, among the potentially internally stable coalitions, coalition $S$ is the one which generates the largest aggregate payoff, then coalition $S$ is also externally stable.¹³ In the subsequent analysis, we display exactly this very stable coalition, even though there may be more than one stable coalition.

In order to keep the number of simulations tractable, we consider in the following our two benchmark cases, which we called the public good and club good case. We recall that the public good case implies full research spillovers among all countries and the club good case no spillovers among non-signatories and between the group of signatories and all non-signatories. Nevertheless, research knowledge is always shared among signatories. We consider our three cooperative scenarios (CC-, NC-, and CN-scenarios) and restrict the number of countries to $n = 10$ with five low and five high-share countries. Similar qualitative results are obtained for different values of $n$ ⁠. The parameter values and the simulation strategy are explained in Supplementary Appendix 10, which confirms that the results are representative, given the large number of simulations. In Table 2, we show the average values of the key variables across simulations. Specifically, we calculate the average stable coalition size (see Section 6.2), the composition of high and low $α_{i}$ -type countries in the stable coalition and the average closing-the-gap index (see Section 7.2), our relative global welfare index.

Table 2

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Effect of asymmetry on the size of stable coalitions $m^{*}$ and the relative welfare index $CGI$ ⁠.

	Public good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	12.82	1.5	1.5	3.0	12.51	2.0	3.0	4.0	1.25
0.5	4.0	0.0	4.0	23.81	4.0	0.0	4.0	23.49	2.0	3.0	5.0	1.24
1.4	2.0	2.0	4.0	23.01	2.0	2.0	4.0	22.79	4.0	0.0	5.0	1.12
2.4	2.0	2.0	4.0	22.93	2.0	2.0	4.0	22.77	1.0	5.0	4.0	1.00
3.3	0.0	5.0	5.0	26.81	0.0	5.0	5.0	26.46	3.0	1.0	6.0	0.88
4.3	1.0	4.0	5.0	28.97	1.0	4.0	5.0	28.90	2.0	4.0	4.0	0.76
5.2	1.0	4.0	5.0	27,30	1.0	4.0	5.0	27.26	2.0	4.0	6.0	0.64
6.2	1.0	5.0	6.0	34.82	1.0	5.0	6.0	34.80	2.0	5.0	6.0	0.51
7.1	1.0	5.0	6.0	31.91	1.0	5.0	6.0	31.91	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.28	2.0	3.0	5.0	31.26	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.11	2.0	5.0	7.0	48.10	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	3.0	7.0	0.00

	Public good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	12.82	1.5	1.5	3.0	12.51	2.0	3.0	4.0	1.25
0.5	4.0	0.0	4.0	23.81	4.0	0.0	4.0	23.49	2.0	3.0	5.0	1.24
1.4	2.0	2.0	4.0	23.01	2.0	2.0	4.0	22.79	4.0	0.0	5.0	1.12
2.4	2.0	2.0	4.0	22.93	2.0	2.0	4.0	22.77	1.0	5.0	4.0	1.00
3.3	0.0	5.0	5.0	26.81	0.0	5.0	5.0	26.46	3.0	1.0	6.0	0.88
4.3	1.0	4.0	5.0	28.97	1.0	4.0	5.0	28.90	2.0	4.0	4.0	0.76
5.2	1.0	4.0	5.0	27,30	1.0	4.0	5.0	27.26	2.0	4.0	6.0	0.64
6.2	1.0	5.0	6.0	34.82	1.0	5.0	6.0	34.80	2.0	5.0	6.0	0.51
7.1	1.0	5.0	6.0	31.91	1.0	5.0	6.0	31.91	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.28	2.0	3.0	5.0	31.26	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.11	2.0	5.0	7.0	48.10	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	3.0	7.0	0.00

	Club good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	11.92	1.5	1.5	3.0	11.76	4.2	4.2	8.4	1.25
0.5	3.2	0.8	4.0	22.73	3.4	0.6	4.0	22.60	0.0	5.0	5.0	1.24
1.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.14	0.0	5.0	5.0	1.12
2.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.18	0.5	5.0	5.5	1.00
3.3	0.0	5.0	5.0	26.36	0.0	5.0	5.0	26.13	1.0	5.0	6.0	0.88
4.3	1.0	4.0	5.0	28.57	1.0	4.0	5.0	28.53	1.0	5.0	6.0	0.76
5.2	1.0	4.0	5.0	26.97	1.0	4.0	5.0	26.95	1.0	5.0	6.0	0.64
6.2	1.0	5.0	6.0	34.62	1.0	5.0	6.0	34.60	1.1	5.0	6.1	0.51
7.1	1.0	5.0	6.0	31.76	1.0	5.0	6.0	31.76	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.14	2.0	3.0	5.0	31.13	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.07	2.0	5.0	7.0	48.06	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	5.0	7.0	0.00

	Club good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	11.92	1.5	1.5	3.0	11.76	4.2	4.2	8.4	1.25
0.5	3.2	0.8	4.0	22.73	3.4	0.6	4.0	22.60	0.0	5.0	5.0	1.24
1.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.14	0.0	5.0	5.0	1.12
2.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.18	0.5	5.0	5.5	1.00
3.3	0.0	5.0	5.0	26.36	0.0	5.0	5.0	26.13	1.0	5.0	6.0	0.88
4.3	1.0	4.0	5.0	28.57	1.0	4.0	5.0	28.53	1.0	5.0	6.0	0.76
5.2	1.0	4.0	5.0	26.97	1.0	4.0	5.0	26.95	1.0	5.0	6.0	0.64
6.2	1.0	5.0	6.0	34.62	1.0	5.0	6.0	34.60	1.1	5.0	6.1	0.51
7.1	1.0	5.0	6.0	31.76	1.0	5.0	6.0	31.76	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.14	2.0	3.0	5.0	31.13	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.07	2.0	5.0	7.0	48.06	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	5.0	7.0	0.00

Note: Total number of countries is $n = 10$ with five high and five low benefit countries, with $α_{L} = \frac{1}{n} - Δ$ and $α_{H} = \frac{1}{n} + Δ$ ⁠. Parameter values of the simulations are given in Supplementary Online Appendix 10. Source: Authors’ calculations.

Table 2

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Effect of asymmetry on the size of stable coalitions $m^{*}$ and the relative welfare index $CGI$ ⁠.

	Public good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	12.82	1.5	1.5	3.0	12.51	2.0	3.0	4.0	1.25
0.5	4.0	0.0	4.0	23.81	4.0	0.0	4.0	23.49	2.0	3.0	5.0	1.24
1.4	2.0	2.0	4.0	23.01	2.0	2.0	4.0	22.79	4.0	0.0	5.0	1.12
2.4	2.0	2.0	4.0	22.93	2.0	2.0	4.0	22.77	1.0	5.0	4.0	1.00
3.3	0.0	5.0	5.0	26.81	0.0	5.0	5.0	26.46	3.0	1.0	6.0	0.88
4.3	1.0	4.0	5.0	28.97	1.0	4.0	5.0	28.90	2.0	4.0	4.0	0.76
5.2	1.0	4.0	5.0	27,30	1.0	4.0	5.0	27.26	2.0	4.0	6.0	0.64
6.2	1.0	5.0	6.0	34.82	1.0	5.0	6.0	34.80	2.0	5.0	6.0	0.51
7.1	1.0	5.0	6.0	31.91	1.0	5.0	6.0	31.91	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.28	2.0	3.0	5.0	31.26	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.11	2.0	5.0	7.0	48.10	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	3.0	7.0	0.00

	Public good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	12.82	1.5	1.5	3.0	12.51	2.0	3.0	4.0	1.25
0.5	4.0	0.0	4.0	23.81	4.0	0.0	4.0	23.49	2.0	3.0	5.0	1.24
1.4	2.0	2.0	4.0	23.01	2.0	2.0	4.0	22.79	4.0	0.0	5.0	1.12
2.4	2.0	2.0	4.0	22.93	2.0	2.0	4.0	22.77	1.0	5.0	4.0	1.00
3.3	0.0	5.0	5.0	26.81	0.0	5.0	5.0	26.46	3.0	1.0	6.0	0.88
4.3	1.0	4.0	5.0	28.97	1.0	4.0	5.0	28.90	2.0	4.0	4.0	0.76
5.2	1.0	4.0	5.0	27,30	1.0	4.0	5.0	27.26	2.0	4.0	6.0	0.64
6.2	1.0	5.0	6.0	34.82	1.0	5.0	6.0	34.80	2.0	5.0	6.0	0.51
7.1	1.0	5.0	6.0	31.91	1.0	5.0	6.0	31.91	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.28	2.0	3.0	5.0	31.26	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.11	2.0	5.0	7.0	48.10	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	3.0	7.0	0.00

	Club good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	11.92	1.5	1.5	3.0	11.76	4.2	4.2	8.4	1.25
0.5	3.2	0.8	4.0	22.73	3.4	0.6	4.0	22.60	0.0	5.0	5.0	1.24
1.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.14	0.0	5.0	5.0	1.12
2.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.18	0.5	5.0	5.5	1.00
3.3	0.0	5.0	5.0	26.36	0.0	5.0	5.0	26.13	1.0	5.0	6.0	0.88
4.3	1.0	4.0	5.0	28.57	1.0	4.0	5.0	28.53	1.0	5.0	6.0	0.76
5.2	1.0	4.0	5.0	26.97	1.0	4.0	5.0	26.95	1.0	5.0	6.0	0.64
6.2	1.0	5.0	6.0	34.62	1.0	5.0	6.0	34.60	1.1	5.0	6.1	0.51
7.1	1.0	5.0	6.0	31.76	1.0	5.0	6.0	31.76	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.14	2.0	3.0	5.0	31.13	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.07	2.0	5.0	7.0	48.06	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	5.0	7.0	0.00

	Club good case
	CC-scenario				NC-scenario				CN-scenario
$Δ \cdot 100$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$	$m_{H}^{*}$	$m_{L}^{*}$	$m^{*}$	$CGI$
0.0	1.5	1.5	3.0	11.92	1.5	1.5	3.0	11.76	4.2	4.2	8.4	1.25
0.5	3.2	0.8	4.0	22.73	3.4	0.6	4.0	22.60	0.0	5.0	5.0	1.24
1.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.14	0.0	5.0	5.0	1.12
2.4	2.0	2.0	4.0	22.29	2.0	2.0	4.0	22.18	0.5	5.0	5.5	1.00
3.3	0.0	5.0	5.0	26.36	0.0	5.0	5.0	26.13	1.0	5.0	6.0	0.88
4.3	1.0	4.0	5.0	28.57	1.0	4.0	5.0	28.53	1.0	5.0	6.0	0.76
5.2	1.0	4.0	5.0	26.97	1.0	4.0	5.0	26.95	1.0	5.0	6.0	0.64
6.2	1.0	5.0	6.0	34.62	1.0	5.0	6.0	34.60	1.1	5.0	6.1	0.51
7.1	1.0	5.0	6.0	31.76	1.0	5.0	6.0	31.76	2.0	5.0	7.0	0.39
8.1	2.0	3.0	5.0	31.14	2.0	3.0	5.0	31.13	2.0	5.0	7.0	0.26
9.0	2.0	5.0	7.0	48.07	2.0	5.0	7.0	48.06	2.0	5.0	7.0	0.13
10.0	2.0	5.0	7.0	45.88	2.0	5.0	7.0	45.88	2.0	5.0	7.0	0.00

8.2 Results

We first check whether the general properties, as established in Section 5, hold.

Result 1.

Properties. In all simulations, the properties of positive externality, superadditivity and, hence, full cohesiveness hold weakly. For any generic coalition $S$ , the total payoff over all players $\sum_{i = 1}^{n} {\tilde{Π}}_{i}^{*} (S)$ in terms of research spillovers is weakly larger in the public good than in the club good case, irrespective of the degree of cooperation (i.e. CC-, NC-, and CN-scenarios). For any generic coalition $S$ , the total payoff $\sum_{i = 1}^{n} {\tilde{Π}}_{i}^{*} (S)$ for the three scenarios of cooperation, $C C$ ⁠, $N C$ and, $C N$ can be ranked as follows: $\sum_{i = 1}^{n} {\tilde{Π}}_{i}^{* C C} (S) >$ $\sum_{i = 1}^{n} {\tilde{Π}}_{i}^{* N C} (S) >$ $\sum_{i = 1}^{n} {\tilde{Π}}_{i}^{* C N} (M)$ , irrespective whether research spillovers are of the club or public good type case.

In Proposition 6, we stated that non-signatories benefit from the accession of another non-signatory to coalition $S$ and in Proposition 7 that the aggregate payoff of those involved in an accession increases. Consequently, the aggregate payoff continuously increases with the enlargement of the coalition (Corollary 4), implying that the largest payoff is obtained in the grand coalition. In the context of asymmetric players, these properties also hold, but only weakly. For instance, consider the cooperative scenario CN, where there is only cooperation on research, but not on abatement and the public good case where there are full research spillovers. Suppose a coalition comprises only $α_{L}$ -countries and assume that the asymmetry between types is sufficiently large. In this case, all research is conducted by non-signatories of the $α_{H}$ -type, as their value of information is higher than that of the coalition, despite signatories internalize their spillovers. That is, we are in a corner solution. Suppose that another $α_{L}$ -type non-signatory joins the coalitions, but equilibrium research and mitigation may not have changed, as countries do not cooperate on mitigation. Then, payoffs remain constant. This is different for the club good case where we always have interior solutions in research investment and/or if there is cooperation on mitigation. Then, the properties positive externality and superadditivity hold strictly.

Proposition 8 stated that research spillovers increase the aggregate payoff of any generic coalition and Proposition 9 remarked that cooperation on research and mitigation implies higher global payoffs than if signatories cooperate on only one of these issues, again, for any generic coalition. Result 1 confirms this conclusion for asymmetric countries. Further results are displayed in Table 2.

Result 2.

Degree of spillovers. In all simulations, the relative welfare index CGI of stable coalitions is higher in the public than in the club good case for all three scenarios of cooperation, though differences tend to decrease with the degree of asymmetry among countries. In the CC- and NC-scenarios, the size of the stable coalition $m^{*}$ is the same in the public and club case; in the CN-scenario, $m^{*}$ may be lower in the public than in the club good case.

Result 2 relates to research spillovers and confirms the main conclusions from Sections 6 and 7. Even though research spillovers may imply smaller stable coalitions, overall, they are welfare improving. There is a positive return of research spillovers for those conducting more research than others, as informed decisions by outsiders improve own benefits from mitigation, as explained in Section 4.1. This advantage diminishes with the size of stable coalitions, as there are less non-signatories. Therefore, as we find that the size of stable coalitions increases with the degree of asymmetry, the difference between the CGI in the public and club good case also decreases. Also, for symmetric countries, we found that in the CN-scenario, stable coalitions may be smaller in the public than in the club good case, despite in terms of CGI this relation is reversed.

Result 3.

Degree of cooperation. In all simulations, the relative welfare index CGI of stable coalitions is higher in the CC- than in the NC-scenario and higher in the NC- than in CN-scenario, irrespective of whether research spillovers are of the club or public good case type. Differences in the CGI between the CC- and CN-scenario as well as between NC- and CN-scenario tend to increase with the degree of asymmetry between countries.

Result 3 confirms our results in Sections 6 and 7. Even though the CN-scenario, with cooperation on research only, may lead to large stable coalitions than cooperation on mitigation (NC- and CC-scenarios), this does not pay in global payoff terms. The internalization of emission externalities is important and the size of stable agreements is not a good indicator for success. The larger the asymmetry between countries, the larger are stable agreements, and the more pronounced is the global welfare loss of a failure to cooperate on mitigation.

Result 4.

Degree of asymmetry. In all simulations, the size of stable coalitions tends to increase with the degree of asymmetry between countries. The relative welfare index CGI of stable coalitions tends to increase in the CC- and NC-scenarios, but tends to decrease in the CN-scenario with the degree of asymmetry, irrespective of whether research spillovers are of the club or public good case type.

Whereas Results 1, 2, and 3 qualitatively confirmed what we found for symmetric players, Result 4 highlights the new aspect that emerges for asymmetric players. The fact that asymmetry is not an obstacle, but may be an asset if asymmetries are well balanced through an ‘optimal transfer scheme’ is in line with the literature (see notes 10–12). This literature considered only a mitigation game without uncertainty and research where signatories cooperate on mitigation. In our setting, this assumption is closest to the NC- and CC-scenarios for which this conclusion is confirmed. However, in the CN-scenario with cooperation on research only, the opposite conclusion seems suggestive. This highlights once more the importance that climate agreements should not focus exclusively on research cooperation, as joint efforts to reduce global emissions come along with large welfare gains.

9. Conclusions

In this article, we have addressed the triple decision problem whether to join a climate agreement, whether and how much research should be conducted in order to reduce the systematic uncertainty about climate change damages and whether and how much mitigation should be undertaken. We contributed to the theoretical IEA literature by suggesting a tractable approach to model intentional learning about the consequences of climate change and the interactions of such type of learning with climate mitigation and coalition formation.

For this purpose, we analysed a three-stage coalition formation model, considering that signatories may cooperate on research and abatement (full cooperation) or may only cooperate with respect to one of these issues (two versions of partial cooperation). We analysed research spillovers from signatories to non-signatories and from non-signatory to all other countries, by allowing spillover coefficients to take on any value. Different from the previous literature, we modelled learning as an active endogenous process, i.e. learning-by-investment.

We stressed the complementarity between abatement and research. Members of an agreement will conduct more research than outsiders, and the level of research is higher if members also cooperate on mitigation. Research increases the probability of making better-informed decisions. If research travels, it generates two additional externalities. A direct effect for other countries, which benefit from research spillovers without bearing the cost, and which encourages free-riding, as this also happens with mitigation. An indirect effect, absent for mitigation. If non-members to an agreement make better-informed decisions, members also benefit, a kind of return on a positive externality. Thus, research spillovers lead to higher global welfare for any size of agreement. Therefore, our results suggest that investment in research on climate change should be shared as widely as possible, as this will generate positive welfare externalities at the aggregate. However, spillovers, in particular from signatories to non-signatories may lead to smaller stable coalitions. Either signatories benefit less than non-signatories from these spillovers or signatories are even disadvantaged because of research leakage. Hence, the overall positive effect of spillovers on global welfare is reduced, considering that agreements must be self-enforcing.

We also showed that even though cooperation on research only may lead to larger stable agreements than if countries either cooperate on abatement only or cooperate on both issues, abatement and research, this does not pay overall. In other words, our results do not support the view that a pragmatic and modest approach, which confines cooperation to research because cooperation on abatement faces strong free-rider incentives, will pay. We showed that modesty leads to larger stable agreements, but the welfare loss due to a failure of cooperation on mitigation is too large to render modesty a globally sensible strategy. Cooperation on mitigation should not be substituted by cooperation on research. Our results also suggested that from the size of an agreement, one should not infer its success.

Our article provides a first approach to model the formation of a climate change agreement with endogenous learning-by-investment. Naturally enough, there is scope for extensions by giving up some of the simplifying assumptions. First, instead of assuming a simple three-stage game with a flow pollutant, we could consider a dynamic payoff structure with a stock pollutant and a coalition formation process with flexible membership, i.e. countries can revise their membership decision over time. In such an extended framework, it would also be interesting to give up our simplifying assumption about learning and consider that probabilities are revised over time through Bayesian updating. Another extension, more in line with our setting, could be to endogenize the choice of the degree of cooperation. That is, countries decide whether to cooperate or not and if they cooperate on which issues.

Footnotes

Extensions include, for instance, a departure from the standard payoff functions typically considered in the literature on IEAs (Karp and Simon, 2013), the possibility of cooperation on R&D in order to discover new innovative abatement technologies instead of cooperating on abatement itself (Barrett, 2006; Hoel and de Zeeuw, 2010; El-Sayed and Rubio, 2014; Rubio, 2017), the possibility of a consensus treaty with modest emission reduction targets in contrast to a focal treaty with ambitious targets (Barrett, 2002; Finus and Maus, 2008).

CN means that signatories cooperate (C) on research (stage 2) but not (N) on abatement (stage 3) and so on.

See, e.g. Cabral (2000) or Endres et al. (2015).

Systematic uncertainty about the benefits from global abatement appears to be the most frequent assumption in the literature. See, for instance, Kolstad (2007), Kolstad and Ulph (2008), Boucher and Bramoullé (2010), Bramoullé and Treich (2009).

In a more sophisticated framework, countries could reduce their uncertainty only to a certain extent but not fully. Our approach allows us to have manageable analytical expressions and still captures the central idea that research is a risky investment in the sense that success is not guaranteed.

Note that this corresponds to what is sometimes called ‘autarky level’.

The lowest target level of research is that of non-signatories. Hence, Assumption 1 ensures that ${\hat{Z}}_{i \notin s}^{} \geq 0$ ⁠.

With ‘symmetric equilibrium’ we mean that countries that belong to the same group choose the same research investment. Thus, all signatories choose the same and all non-signatories choose the same investment. However, this does not mean that signatories chose the same investment as non-signatories.

This would be different in games with a negative externality and superadditivity for which Weikard (2009) has shown that the grand coalition is the unique stable coalition. In such games, there is always an incentive for players to join the coalition, until the grand coalition has been reached.

For the assumption of two types of countries see, for instance, Fuentes-Albero and Rubio (2010) and (Pavlova and De Zeeuw, 2013).

Finus and McGinty (2019) call this the ‘Coalitional Folk Theorem’. It relates to asymmetry either on the benefit or cost side (one-sided asymmetry).

See Eyckmans et al. (2012) and Carraro et al. (2006).

If this was not the case, a coalition $S \cup {j}$ would be internally stable with a higher aggregate payoff due to full cohesiveness.

Supplementary material

Supplementary material is available on the OUP website. These are the simulation files and the online appendix.

Funding

F. J. André acknowledges financial as well as logistic support from the University of Bath, UK as well as support from the Spanish Ministry of Science and Innovation (grants PID2019-105517RB-I00 and PID2022-138754OB-I00). M Finus acknowledges the financial and logistic support of the University Autonoma, Spain.

Acknowledgements

This article is a completely revised and extended version of a previous manuscript with Leyla Sayin. The authors have greatly benefitted from discussions with Alistair Ulph and Chuck Mason. This paper was partly written while F. J. André was visiting the Department of Economics at the University of Bath, UK, and M Finus was visiting the University Autonoma of Madrid, Spain. Both authors would like to acknowledge constructive comments by two reviewers which helped to improve the paper substantially.

References

Amir

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2000

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Modelling imperfectly appropriable R&D via spillovers

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Article Contents

Endogenous learning in international environmental agreements: the impact of research spillovers and the degree of cooperation

Abstract

1. Introduction

2. The model

3. Third stage: abatement

4. Second stage: research

4.1 Some general observations

4.2 The role of research spillovers for equilibrium investment in research

5. Properties of the coalition formation game: a digression

6. First stage: membership

6.1 Analytical results

6.2 Simulations

7. Overall results

7.1 Analytical results

7.2 Simulations

8. Extension: asymmetric countries

8.1 Preliminaries

8.2 Results

9. Conclusions

Footnotes

Supplementary material

Funding

Acknowledgements

References

Supplementary data

Citations

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