The international community has failed to significantly slow the pace of climate change, even though its causes are known and its effects are potentially devastating.Footnote 1 This failure is partially explained by incentives: individual governments are tempted to let other countries bear the costs of climate change abatement and so reap the benefits without taking on these costs themselves.Footnote 2 Thus scholars and practitioners have emphasized the importance of well-designed international institutions to help shape the incentives of national actors.Footnote 3 In so doing, an ideal institution would need to not only manage free riding but also reflect variability between countries, by asking countries that can reduce fossil fuel emissions at a lower cost to do more and requiring others to do less.Footnote 4 Agreements with nonuniform obligations are more cost-effective because member states have varying costs, as well as broader because variability ensures that each country's quota is appropriate given the country's relative capability.Footnote 5
The importance of accounting for individual differences has been recognized in major efforts to create a global climate agreement. First, the precursor to modern climate agreements, the 1987 Montreal Protocol on the ozone layer, set the standard by assigning nonuniform commitments in which developing countries were given less strict emissions limits.Footnote 6 This was echoed in later climate agreements such as the 1992 United Nations Framework Convention on Climate Change (UNFCCC), which recognized the “common but differentiated responsibilities” of all countries.Footnote 7 In due course, the 1997 Kyoto Protocol set emissions limits for only developed countries and even set country-specific limits wherein economies in transition were allowed to use a different base year to compare their reductions to.Footnote 8 Most recently, the 2016 Paris Agreement simply let each country determine its own quota.Footnote 9
Reflecting variability between countries is a desirable component of a climate agreement. First, a one-size-fits-all policy would be suboptimal because some countries are able to reduce fossil fuel emissions at a much lower cost than others. So in an agreement with uniform requirements, some countries bear disproportionate burdens while others that could do more are not required to do so.Footnote 10 Second, one-size-fits-all agreements suffer from a “broad versus deep” trade-off which limits their success. If international agreements require the same commitments of all members, then they can either have broad membership or require deep commitments, but not both.Footnote 11 By contrast, a variable agreement encourages both broader participation and deeper commitments for countries able to make them by setting each country's quota separately according to its capabilities. However, because countries’ leaders are likely to be better informed about the individual circumstances in their own countries than are leaders of other countries or of international organizations (IOs), an agreement must rely to some extent on self-reports to implement a variable agreement—relatedly, it could be the case that an international agency (such as the United Nations Environment Programme) gathers information transmitted by countries before drafting an agreement (as in the Montreal Protocol), or that an agreement relies on each country to determine the level of commitment it is able to make (as in the Paris Agreement).Footnote 12
Unfortunately, as Harrison and Lagunoff have argued, overcoming free riding while accounting for individual differences is difficult in the face of this asymmetric information.Footnote 13 To see why, consider an agreement involving only two countries and a mediator helping countries decide what commitments they must make under the agreement. Setting enforcement issues aside for the moment, assume the two countries have significant enough repeated interactions that they can make credible commitments to one another, knowing that they can mutually punish defection from their agreement.Footnote 14 The ideal agreement has a country commit to only a moderate reduction in emissions when it faces a hardship resulting from a change in economic or political conditions, and to a more significant reduction when it faces no such hardship.Footnote 15 Consistent with the free-riding problem, however, a country facing no hardship may still prefer to commit to only moderate reductions and let the other country take on more of the costs of climate action. This creates a problem for the mediator trying to decide what commitments each country should make under information asymmetry.Footnote 16 Each country is tempted to exaggerate its hardships so that it is allowed to make smaller commitments under the agreement.Footnote 17 Information asymmetry is significant enough that in some circumstances, the best agreement that can be implemented is one that has no variation in commitments.Footnote 18
In this paper we analyze how to design an optimal climate agreement when countries’ costs of climate action are private information. The problem the IO must solve is to maximize global welfare subject to the constraint that countries are willing to participate and also to submit accurate self-reports. To focus on the informational aspect of the problem, we assume that the IO has some (perhaps limited) ability to enforce the commitments member states make.Footnote 19 For instance, enforcement leverage may come from some positive linkage such as controlling access to a club good or other benefit that accrues to countries for being in good standing with the organization,Footnote 20 or even from some negative linkage such as punishments used to coerce countries into compliance.Footnote 21 Nonetheless, without any investigative resources, this leverage for inducing compliance does not help the organization design an optimal agreement.
Even as the IO has very high leverage for enforcement, it cannot prevent countries from exaggerating compliance costs. However, when the IO has the power to investigate the compliance costs of member states, it may be able to overcome misreporting while also allowing variability in commitments. Notably, the availability of investigative resources means that the IO's leverage for encouraging compliance can also help it incentivize truth telling to make it so that the agreement is able to assign realistic commitments for each member state. This could be a benefit of major climate agreements with a verification procedure to check countries’ reports, including the Montreal Protocol (for the ozone), UNFCCC, Kyoto Protocol, and Paris Agreement.Footnote 22 To increase their investigative capacity, IOs can also oversee investigations conducted by external bodies. Indeed, although most environmental organizations lack significant monitoring capabilities, they have historically been able to rely on nongovernmental organizations (NGOs) or other actors to conduct monitoring for them. For example, the 1979 Long-Range Transboundary Air Pollution Convention did not have the ability to monitor but was able to depend on environmental NGOs to do so.Footnote 23 Likewise, because free-riding behavior is so costly to all countries in an agreement, they will have an incentive to request or acquire extensive information on each other's true types.Footnote 24 So even if the organization itself lacks the ability to collect all of the information it needs, it can still expect entities such as NGOs and member states to conduct their own investigations.
We contribute to the literature in two main ways. First, we contribute to the literature on mechanism design and climate change cooperation. In general, we follow the literature in viewing the main function of climate agreements as attempting to overcome problems of free riding.Footnote 25 The closest papers are by Harrison and Lagunoff and Slechten.Footnote 26 Harrison and Lagunoff take a mechanism-design approach to constructing an international agreement to limit carbon consumption. In their model, the best implementable agreement subjects every country to the same requirements, even though the best possible policy would account for individual differences in costs and abilities. The problem is that if countries have private information about these differences, they may try to free ride by exaggerating their costs from reducing carbon consumption. We complement that paper by considering independent information gathering as a potential solution to this issue of misreporting.
Slechten also considers the possibility of verifying self-reports in a mechanism-design setting for environmental agreements.Footnote 27 In her model, states can voluntarily allow the IO to “certify” their private information, making it completely verifiable. By contrast, we consider probabilistic verification and allow the IO to freely allocate investigative resources determining the likelihood of verification. We also differ from Slechten by considering a multilateral agreement with no transfers, reflecting a weak IO that cannot freely move resources between countries.
Second, we contribute to the international relations literature on flexibility in the design of international institutions more broadly. Much of the work on flexibility comes from the literature on trade agreements such as the World Trade Organization.Footnote 28 In these papers, two factors driving the need for flexibility are changes over time that require renegotiation,Footnote 29 and unexpected situations within countries leading to a need for escape clausesFootnote 30 or compensatory payments.Footnote 31 We conceptualize flexibility in agreements differently than in these papers, and use the term variability instead to avoid conflating these ideas. Although these papers refer either to renegotiation or to forgiveness from punishment, we simply mean allowing the terms of the agreement to be different for each country. However, some of the basic insights about flexibility are similar, and we believe similar informational problems to those raised in this paper could apply to escape clauses, compensation, or renegotiation.Footnote 32 The flexible mechanism closest to the kind of variability we examine here is the escape clause, which enables countries to temporarily lower their commitments when their costs of abatement (or type) become higher due to an economic shock, military conflict, or another similar event.Footnote 33
In particular, we address the climate agreement literature's commonly noted broad versus deep trade-off between the strictness of actions required under an agreement and the ability to gain more signatories.Footnote 34 The basic intuition is that an agreement that requires more significant action on the part of its members will be more costly for those countries, so all countries will be less inclined to sign the agreement initially.Footnote 35 In other words, an agreement with uniform commitments can either be small in membership but assign a difficult quota (in which case only the most committed countries would be members), or large in membership but assign an easy quota (in which case only the least committed countries would curb their emissions). This problem is particularly difficult when it comes to climate change because nonmembers of climate agreements are not only refraining from action but also making the problem worse because maintaining business as usual actually results in further deterioration of the climate. Therefore, it is not evident whether it is better to try to craft a deeply committed agreement of only the most willing countries and risk letting less-willing countries pollute as normal, or to form a weak agreement that includes all countries but which does not require much of any of its members.Footnote 36
Variability in commitments may increase the number of willing signatories by ensuring that countries that face higher compliance costs receive some relief. An agreement that allows variation in the requirements made of less-well-off countries or in less secure times will be able to both require more of countries and get more countries to join because these countries know they will not be asked to do more than they are able at any given time.Footnote 37 Here, our conception of variation is the extent to which the agreement's requirements account for individual differences. An optimally varying agreement softens the trade-off between breadth and depth but requires independent investigations of member states’ claims to overcome the incentive of countries to exaggerate their costs of compliance. We now turn to the model.
The Game
In the benchmark model, the players comprise N countries, where N = {1, …, n}. We assume that each country i ∈ N chooses its own level of carbon emissions c i ≥ 0, and that all countries choose simultaneously.Footnote 38 However, we also assume that each country has a type θ i which takes a value in 
$[ {\underline{\theta } , \;\bar{\theta }} ] \subset ( {0, \;1} ) $. Higher types value carbon consumption more than other countries or would incur relatively large costs from reducing emissions, while lower types value abatement more than other countries or would incur relatively small costs from reducing emissions. For example, a high type of country could be one that will adapt easily to climate change or whose economic strategy entails continued investment in energy-intensive manufacturing sectors. A low type of country may be one invested in clean technologies or less energy-intensive industries. We may also think of types as related to internal political pressures within a country, given that the policy preferences of national leaders are influenced by the relative power of environmental interest groups, industry lobbyists, and so on. Types are private information but are commonly known to be independent and identically distributed along a continuous distribution F with density f and support in 
$[ {\underline{\theta } , \;\bar{\theta }} ] $.
Based on these assumptions, this model then focuses on the possible strategies that could be employed by an IO to guide countries’ compliance with global abatement goals. In particular, the IO is able to first collect information about countries’ types by asking countries to voluntarily disclose this information through their announcements 
$\hat{\theta }_i$, and then it can recommend an emissions quota for each country. We denote this quota by 
$\tilde{c}( \hat{\theta }_i) $ to show that it may depend on type announcements. The IO in our model is a utilitarian planner that seeks to maximize the sum of all countries’ payoffs.Footnote 39
Each country's payoff is a function of how much carbon is emitted and its type, which can be written as
$$u_i( c, \;\theta _i) = \theta _i\log ( c_i) + ( 1-\theta _i) \log ( \omega -C) $$In this expression, 
$C = \sum\nolimits_{i = 1}^n {c_i} $ is the aggregate level of carbon emissions and ω > 0 represents the carbon stock. If country i does not sign the agreement, it receives u i(c, θ i) for its chosen level of emissions but pays a fixed penalty K ≥ 0 for noncooperation. The first term in the utility function represents the countries’ direct benefits from carbon consumption, while the second term represents the benefits from conservation. The type θ i serves as a weight on these two components, so naturally countries value conservation less relative to consumption as θ i increases.
This form captures the idea that high types’ utility increases by more when they emit (because the utility function is written in a way such that emissions are valued relatively more and conservation relatively less when the type is high), while low types’ utility increases by less when they emit (because the utility function is written so conservation increases in value and emissions decrease in value when the type is low). For a country that has not signed the agreement, it has this same utility function related to emissions and abatement but also pays a fixed penalty K for noncooperation. This fixed cost could be seen as the loss of any club benefits,Footnote 40 alliances, or simply prestige as a result of not being in the agreement. For the moment, we will treat K as exogenous. However, in the online supplement we consider an extension in which K arises endogenously from issue bundling. One thing we wish to make clear is that the IO need not have enforcement power in a formal sense (as these organizations often do not). Instead, fairly large values of K can be sustained by the IO just helping countries to coordinate on self-enforcing equilibria that punish climate defectors.
Overall, we analyze an institution in which each country makes an announcement to the IO and the IO recommends an emissions level. Although this seems restrictive if it is taken literally, the revelation principle implies that the outcome of any equilibrium from any institution can be implemented by a direct mechanism such as this one in which all players have an incentive to truthfully report their types. That is, suppose an outcome (i.e., emissions levels for each type) is implemented as an equilibrium to some potentially complicated mechanism. The revelation principle tells us that the same outcome can be implemented by a direct mechanism in which countries are asked to reveal their types and then are assigned emissions levels. Therefore, our setup encompasses a very broad set of institutions.Footnote 41
The form of the utility functions and the choices of the IO in the benchmark model follow previous work by Harrison and Lagunoff.Footnote 42 The primary differences are that their model is dynamic in contrast to our static model, and that our model imposes a positive cost of noncooperation on countries that fall outside the agreement. These two choices are connected: the dynamic Harrison and Lagunoff model generates punishments for noncompliance endogenously, but our static model requires an exogenous punishment for noncooperation. Our setup enables simpler analysis and also lets us compare what happens when the IO can impose large costs for noncooperation versus when it cannot.Footnote 43
Complete Information Benchmark
First, we solve for the quotas the IO would suggest assuming that there was perfect information about countries’ types and that there was no need to deal with any perverse incentives held by the countries. This complete information benchmark has a socially optimal quota which is increasing in the global carbon stock and each country's type but decreasing in the number of countries, as suggested by the following lemma:
Lemma 1: Under complete information the socially optimal quota is 
$\tilde{c}( \theta _i) = {{\omega \theta _i} \over n}$
Lemma 1 implies that those countries that value their emissions more are able to emit more under this quota (because the optimal quota is increasing in θ i), but the quotas for all countries are also influenced by how many countries are a part of the agreement (because the extent of this allowance is decreasing in the total number of countries and increasing in the total available carbon stock).
Our complete information benchmark captures the idea that the optimal policy is variable in the sense of allowing for individual differences. In the next section, we introduce private information and reproduce the full compression result of Harrison and Lagunoff, which states that the best implementable quota is completely inflexible and thus requires the same actions of all countries.
Best Mechanism Under Constraints
To consider how an IO with some limited enforcement power but no investigative powers could attempt to implement an agreement, we examine the quotas the IO would suggest after imposing countries’ incentive constraints. The first of these two kinds of constraints is the incentive compatibility constraint, which says that each country must find it in its best interest to reveal its true type given the quotas associated with each announced type:
$$\int_{\Theta _{{-}i}} {u_i} ( \tilde{c}( \theta _i) , \;\tilde{{\boldsymbol c}}( \theta _{{-}i}) , \;\theta _i) dF_{{-}i}( \theta _{{-}i}) \ge \int_{\Theta _{{-}i}} {u_i} ( \tilde{c}( \hat{\theta }) , \;\tilde{{\boldsymbol c}}( \theta _{{-}i}) , \;\theta _i) dF_{{-}i}( \theta _{{-}i}) $$for all i ∈ N and 
$\hat{\theta }\in [ 0, \;1] $, where 
$\tilde{{\bf c}}( \hat{\theta }_{{-}i}) $ is the profile of all other emissions given that all other countries report their own types and comply with the recommended quota, and F −i is the joint probability distribution of θ −i. That is, given the quotas associated with each announced type, every type of every country weakly prefers to report its true type.
The second is the participation constraint, which states that each country must prefer joining the agreement under its recommended quota to leaving the agreement. This requires that the agreement must offer every type of country at least the utility it would get by not being in the agreement:
$$\int_{\Theta _{{-}i}} {u_i} ( \tilde{c}( \theta _i) , \;\tilde{{\vector {\boldsymbol {c}}}}( \theta _{{-}i}) , \;\theta _i) dF_{{-}i}( \theta _{{-}i}) \ge \int_{\Theta _{{-}i}} {} \mathop {\max }\limits_{c_i \ge 0} u_i( c_i, \;\tilde{{\vector {\boldsymbol c}}}( \theta _{{-}i}) , \;\theta _i) dF_{{-}i}( \theta _{{-}i}) -K$$for all i and θ i. That is, each country's payoff for going it alone is its payoff-maximizing emissions level minus the nonparticipation penalty K. The agreement must then offer every type of every country at least this payoff.
We solve for the best mechanism that satisfies the constraints (2) and (3). The incentive compatibility condition (2) represents the idea of the revelation principle: if an outcome is implementable at all, then it must be implementable by a direct mechanism in which all players are incentivized to reveal their true types. Thus, restricting ourselves to incentive-compatible mechanisms is natural in the sense that any achievable outcome can be implemented by one satisfying this constraint. The participation constraint (3) represents the idea that no country can be forced to sign an agreement. Thus, a feasible agreement must satisfy the property that every country would sign on to and comply with the agreement rather than going it alone. An agreement that nobody would join clearly cannot be effective. However, this constraint is limiting in a particular way: we look for the optimal mechanism among those that are fully inclusive, that is, that all countries would sign. As we discuss in the next section and in the online supplement, without investigations it may not be the case that the optimal mechanism overall is fully inclusive.
The IO must maximize the sum of all these utilities based on type:
$$\Sigma _{i = 1}^n [ \theta _i\log ( \tilde{c}( \theta _i) ) + ( 1-\theta _i) \log ( \omega -C) ] $$subject to (2) and (3). The full compression result is
Proposition 1: (Harrison and Lagunoff Reference Harrison and Lagunoff2017) Assume 
$\underline{\theta } > {{\bar{\theta }} \over {n + \bar{\theta }-n\bar{\theta }}}.$ The optimal mechanism under incomplete information in the baseline model is fully compressed: all countries must meet the same quota regardless of type announcements.
Under the incentive compatibility constraint, the IO can assign all types of countries only the same quota. Such a result is referred to as “fully compressed,” in that there is one quota for all countries despite the existence of different types. The reason this result is the same for every country is that all countries have the incentive to report a type higher than their true type to get a more lenient quota and therefore maximize their own utility. In other words, if quotas depend on countries’ private information, then each country can indirectly free ride by reporting a type higher than its true type. This agreement is inefficient compared to the optimal agreement because it treats all countries as equal even though their costs of carbon abatement differ.Footnote 44 Notably, Proposition 1 makes use of the truth-telling constraint only. We have not yet analyzed how the participation constraint affects the optimal quota.
Clearly, a fully compressed quota meets the incentive compatibility constraint because no country can have a strict incentive to lie. To find the optimal quota, we can therefore identify the best compressed quota that satisfies all countries’ participation constraints. This calculation boils down to finding the strictest quota that the highest type will be willing to accept. Given a compressed quota c*, the highest type's utility if it opts out of the treaty is
$$\mathop {\max }\limits_{c_H} {\bar{\theta }}\log ( c_H) + ( 1-\bar{\theta }) \log ( \omega -c_H-( n-1) c^\ast ) $$Solving this maximization problem for c H gives 
$c_H( c^\ast ) = \bar{\theta }( \omega -c^\ast ( n-1) ) $.
To get some intuition about the optimal compressed quota, it is useful to consider two extremes. The highest type incurs no cost from violating the agreement when K (the cost of not being in the agreement) is zero, and so the compressed quota c* cannot be lower than the value at which the highest type receives equal utility from remaining in or leaving the agreement. This indifference condition is met only when the quota is set to the amount the highest type of country would choose to emit were it not in the agreement:
$$c^\ast{ = } \displaystyle{{\bar{\theta }\omega } \over {1 + \bar{\theta }( n-1) }}$$This is the lower bound on the optimal quota of allowed emissions. In the literature, this issue of agreements’ commitment levels being dragged down by the least committed country is known as the law of the least ambitious program.Footnote 45 Essentially, in agreements that try to achieve broad membership but require the same quota of all countries, it is possible to assign every country only a “shallow” quota that is permissive enough for even the least committed country to meet (or else risk losing members).
If we assume that K is so large that the participation constraint no longer has an effect, then the compressed quota is simply the average quota under full information conditions:
$$\sum\nolimits_{i = 1}^n {[ {\theta_i\log ( c^\ast ) + ( 1-\theta_i) log( \omega -c^\ast n) } ] } $$Taking first-order conditions and solving for c* (again using 
$T = \sum\nolimits_{i = 1}^n {\theta _i} $) yields 
$c^\ast{ = } {{\left({{T \over n}} \right)\omega } \over n}$. Notice that this is a particularly intuitive mechanism: it simply asks every country to follow the average quota under the full information solution. Note also that this is generally a lower (more demanding) quota than the one when K is zero. This illustrates the following result.
Proposition 2: The optimal quota is decreasing in K with 
${\rm lim}_{K\to 0}c^\ast{ = } {{\bar{\theta }\omega } \over {1 + \bar{\theta }( n-1) }}$ and 
$\lim _{K\to \infty }c^\ast{ = } {{\left({{T \over n}} \right)\omega } \over n}$.
Proposition 2 gives us some insights about how an organization must design an agreement when it has no independent investigative power. Recall that the parameter K is meant to represent the IO's enforcement leverage. For instance, if the IO controls access to a club good as Nordhaus suggested,Footnote 46 or if recognition by the IO is very valuable to the countries, then K may be large in the sense that the IO has leverage to make countries comply with an agreement. An insight from Proposition 2 is that this power is useful but not good enough to implement an optimal agreement. The reason is that this enforcement leverage enters countries’ participation constraints but not their incentive compatibility constraints. In other words, an IO with a lot of leverage can induce countries to comply with the recommended quotas but cannot make them tell the truth about their types. As we show in the next section, introducing investigative powers helps the IO convert its leverage for compliance into leverage for honest reporting.
Noninclusive Mechanisms
Proposition 2 characterizes the best mechanism among those that are incentive compatible and fully inclusive. By fully inclusive, we mean that the participation constraint is satisfied for every country, so the agreement is a global one. However, the optimum among all possible mechanisms need not be fully inclusive. The central mechanism at play is the well-known broad versus deep trade-off from the literature on climate agreements. Since the optimal mechanism without investigations is fully compressed, IOs without investigative capacity can only be either broad (including many members) or deep (requiring strong commitments), given that all countries will be asked to meet the same quota despite inherent differences in their capabilities.
In our model, this trade-off is evident in the fact that the quota for every country is primarily a function of the participation constraint of the highest type. That is, all countries are asked to do only as much as the least environmentally friendly country is willing to do. Alternatively, a mechanism that is not fully inclusive lets signatories make deeper commitments but carries the disadvantage of leaving the countries with the highest carbon emissions completely unconstrained.
As we show by example in the online supplement, a noninclusive mechanism may be optimal in some cases. The structure of the optimal agreement is still similar to that in Proposition 2 in the sense that the mechanism should be fully compressed among those that sign the agreement to ensure incentive compatibility, but leaves nonsignatories (all countries above some type cutoff) free to choose their emissions levels outside of the agreement. This primarily depends on how much leverage the international community has over enforcement of an agreement (represented in our model by the parameter K). If K is near zero, meaning there is very little impetus to comply with any agreement, the participation constraints of the heaviest polluters become very salient, and it is likely that fully inclusive mechanisms will not be optimal. By contrast, as K becomes large, the optimal mechanism will be fully inclusive because all participation constraints are easily met. This also explains why the optimal mechanism in Harrison and Lagunoff is fully inclusive, because the dynamic model in that paper allows unbounded punishments, which closely corresponds to our model as K goes to infinity.Footnote 47
Mechanisms with Limited Investigations
We now extend the benchmark model in the following way: the IO has a fixed amount R of investigative resources available and can distribute these resources however it chooses after seeing the type announcements of each country. Let 
$r_i( {\hat{\theta }, \;{\hat{\theta }}_{{-}i}} ) \in [ {0, \;1} ] $ denote the amount of investigative resources devoted to country i when country i reports that its type is 
$\hat{\theta }$ and the other countries report 
$\hat{\theta }_i$.Footnote 48 We assume that i's type is verified with probability 
$r_i( {\hat{\theta }, \;{\hat{\theta }}_{{-}i}} ) $. The mechanism we consider is one where, if a country's verified type is different from its reported type, that country is removed from the agreement and incurs the cost K of noncompliance but is also free to choose its carbon emissions levels from then on.
The IO's budget constraint is that 
$\sum\nolimits_{i = 1}^n {r_i( {{\hat{\theta }}_i, \;{\hat{\theta }}_{{-}i}} ) } = R$ for any 
$\hat{\theta }\in [ {\underline{\theta } , \;\bar{\theta }} ] ^n$. Clearly if R ≥ n, this corresponds to a complete-information case where the first-best quota is implementable as long as K is large enough. We are interested in the conditions under which the first-best quota is implementable for smaller values of R and in characterizing the optimal allocation of investigative resources in this case. Proposition 3 states our result.
Proposition 3: Assume 
$\underline{\theta } > {{\bar{\theta }} \over {n + \bar{\theta }-n\bar{\theta }}}$. For any R > 0, there exists K*(R) such that the optimal quota under complete information is implementable by a mechanism with limited investigations when K ≥ K*(R).
This result shows that the optimal quota may be implementable when the IO can conduct investigations. It also shows that enforcement leverage magnifies the effect of investigative resources. If K is large, then the optimal quota can be implemented with minimal investigative resources. More specifically, given any level of investigative resources, there exists some value of K such that the optimal rule can be implemented with that amount of resources. This illustrates the intuition given before: the effect of investigations is to convert leverage over compliance into leverage over honest reporting of types.
In Proposition 3, unlike the result in Proposition 2, the restriction to fully inclusive mechanisms is without loss of generality. For a variable agreement, the quota can be tailored to each individual country and type. An interpretation of this is that the ability to implement an agreement with variability solves the broad versus deep trade-off because we need not assign shallower commitments to lower types just to accommodate more difficult cases. In fact, Proposition 3 involves implementing a quota that is optimal among all possible quotas.
Note that, in contrast to many regulatory models with investigative capacity, our IO is using resources to directly investigate countries’ costs of carbon abatement rather than to monitor emissions levels. So, like the model illustrating the broad versus deep trade-off in Barrett, our model assumes that emissions are perfectly observable.Footnote 49 Determining the right allocation of informational capacity between monitoring compliance and learning types may be an interesting area for future research in a version of this model in which emissions levels are not always observed. However, we expect that some investigative capacity must always be devoted to monitoring types. This is because, in contrast to the crime deterrence models such as that of Becker, the IO does not directly control the magnitude of the punishment.Footnote 50
Discussion and Conclusion
Initially, our model allows us to see that the ideal agreement involves an optimal quota which is increasing in countries’ true types, increasing in the total available carbon stock, and decreasing in the total number of countries. However, without investigations, the best incentive-compatible agreement is fully compressed, meaning all countries must meet the same quota regardless of type announcements. This is inefficient because those countries with high and low types are made to take the same actions, and thus those with low types are not contributing as much as they could, while those with high types are made to take on an unsustainable burden. The fully compressed agreement also gives rise to the familiar broad versus deep trade-off—because every country commits to the same quota, the behavior of all members of a fully inclusive agreement is determined by whether the least environmentalist country is willing to participate.
Only in the model with institutional oversight is variability possible. In this version of the model, we allow the institution limited resources to investigate countries’ claims about their types. As a result, countries become increasingly likely to tell the truth about their types as the amount of the institution's investigative resources (related to the likelihood of being caught) or the cost of not belonging to the agreement (related to the punishment from being caught) increases. This “truth-telling” equilibrium possible under institutional oversight then allows an optimal assignment of emissions quotas, which ultimately enables the agreement to achieve both greater membership and greater compliance because countries are more willing to join and make deeper commitments in agreements with variability in commitments. Thus an optimal agreement supported by sufficient investigations and punishments can be both broad and deep.
Ultimately, these results suggest that an IO without the ability to investigate countries’ claims can only inefficiently request the same quota of all countries, regardless of type. If the cost of not remaining in the agreement is very high, then this compressed quota is set at the average quota given all countries’ types. However, if the cost of not belonging to the agreement is close to zero, then the quota must be set at the least committed country's willingness to abate or else risk losing members. By contrast, an IO with some ability to investigate countries’ claims can incentivize countries to reveal their true types in an effort to maximize global welfare by assigning appropriate quotas for each country, through increasing either its investigative resources or the costs of not being part of the agreement. Furthermore, if countries are able to punish defectors very effectively, as is the case in the dynamic setting of Harrison and Lagunoff, then the optimal agreement can be implemented with minimal investigative resources.
Our analysis focuses on simple mechanisms in which the IO elicits reports about countries’ types and then suggests emissions quotas, but the revelation principle suggests that our analysis helps us understand what is possible in other, more familiar environments. In particular, a flexible interpretation of our model could motivate several examples from the literature of variability mechanisms paired with information collection by the international community. First, as Slechten pointed out, climate negotiations are often preceded by an umbrella convention which sets up institutions to gather countries’ information related to climate change efforts.Footnote 51 In this environment, we can interpret the investigations in our model in a similar way to the certifications in her paper; one role of the IO in this stage of the negotiation may be to help verify the information submitted by the parties to reduce information asymmetries. Second, escape clauses in international agreements introduce a type of variability in line with that emphasized here by permitting parties to be subject to different requirements based on individual circumstances. These are often paired with procedures by which countries must submit verification of reasons for exceptions from usual rules. For instance, Article 8 of the London Dumping Convention permits exceptions to some articles in certain circumstances but requires contracting parties to consult with other parties and submit documentation of reasons for the exception.Footnote 52 Our model suggests that credible implementation of situation-specific variability in obligations may depend in part on these reporting procedures.
We leave several issues for future research. For example, one promising avenue for research might involve further probing the bargaining-over-institutions problem. As part of this effort, researchers could investigate under what conditions powerful states would form institutions (on the climate or other issues) with both investigative and enforcement abilities. Though the revelation principle suggests that some game should implement agreements that are outcome-equivalent to the direct mechanisms analyzed here, this could help political scientists predict whether such institutions are likely to be adopted given a particular existing set of bargaining protocols. A second promising avenue for future research may be to consider alternative objective functions for the IO. For instance, some IOs appear to primarily seek the approval of a small set of powerful states.Footnote 53 This may limit the ability to implement good agreements to the extent that it makes the IO reluctant to investigate the powerful countries. If the IO has sufficient difficulty committing to investigate powerful countries, then it may not be possible to implement agreements that account for the private information of those countries.
Supplementary Material
Supplementary material for this research note is available at <https://doi.org/10.1017/S0020818321000400>.
Acknowledgments
We thank Amanda Kennard and David Victor as well as participants in the Washington University formal theory seminar for valuable comments.
