Estimating a Strategic Model of Treaty Formation:
The Case of the Montreal Protocol
Ulrich J. Wagner∗
February 2014
This paper develops an empirical framework for analyzing the timing of
multilateral agreements. A treaty is modeled as a repeated game among governments that take a (reversible) decision on participation in every period.
The taste for treaty membership increases over time. The treaty creates
spillover effects that result in participation being a strategic complement, a
strategic substitute or a dominant action. The predictions of the model can
be exploited for estimation of the structural parameters using a cross section
of treaty ratification dates. I apply this approach to the Montreal Protocol and find robust evidence of strategic complementarity in the ratification
process. Through a series of counterfactual experiments, I explore different
mechanisms behind this effect.
∗
Departamento de Economı́a, Universidad Carlos III de Madrid. This paper is based on the first
two chapters of my Ph.D. dissertation. I am grateful to my committee members Pat Bayer, Steve
Berry, and Chris Timmins for their advice and support. I would like to thank four anonymous
referees and the editor for comments that have substantially improved the paper. I have benefited
from discussions with Scott Barrett, Dirk Bergemann, Antonio Cabrales, Klaus Desmet, Hanming
Fang, Geoff Heal, Nat Keohane, Gerard Llobet, Pei-Yu Lo, Pedro Mira, Marc Möller, Juan Pablo
Rincón, Bernard Salanié, François Salanié, and Philipp Schmidt-Dengler. Seminar participants at
Alicante, Berkeley, Colby, Columbia, CORE, CSIC, Hunter, IESE, Maastricht, NYU, Toulouse, and
Yale have provided helpful comments. All errors are my own. I gratefully acknowledge receipt of an
ERP Fellowship from the German National Academic Foundation, a Fisher Dissertation Fellowship
from Resources for the Future, and a Georg Walter Leitner Grant in International and Comparative
Political Economy from Yale University. Research funded in part by the Spanish Ministry for Science
and Innovation, reference number SEJ2007-62908.
1
Electronic copy available at: http://ssrn.com/abstract=1730183
1. Introduction
What drives countries to cooperate in the pursuit of common policy goals? Ever since
World War II, cross-border policy coordination has been on the rise in many areas, including military defense, human rights, international trade and finance, public health,
and environmental protection. While the specific reasons behind such alliances differ
widely, as a rule they materialize in situations where coordinated efforts are better
suited to achieve the common good than unilateral policies. This suggests that collective rationality is a driving force behind international cooperation. However, there
are numerous international policy issues to which collective rationality would dictate
a cooperative approach and yet international cooperation fails badly. This failure to
provide global public goods or to manage common property resources efficiently is often
attributed to the adverse incentives faced by individual governments. If countries do
not take into account the external benefits of their contribution to the public good, the
aggregate provision level falls short of the social optimum (Samuelson 1954). Worse,
even though the provision of the public good is desirable for all countries, some of them
may prefer to free-ride by enjoying the external benefits without sharing the cost.
At the national level, such conflicts between individual and collective rationality can
be resolved by the intervention of the government (Demsetz 1967). At the international
scale, however, there is no supranational authority that could coerce states into adopting
efficient policies if they run counter to national interests. Filling the void are international agreements. Under the terms of the Vienna Convention On The Law of Treaties, a
state that ratifies a multilateral treaty chooses partially to surrender its sovereignty and
to subject its policies in a specific domain to the rules and prescriptions of the treaty.1 In
so doing, sovereign states agree to coordinate their policies in mutually beneficial ways.
By the very nature of sovereignty, however, the agreement is fundamentally non-binding
and states can always withdraw from it. Therefore, the fact that public good provision
is implemented through an international agreement should not change a country’s incentives to contribute per se – unless the treaty alters the country’s incentives to cooperate
in other ways.
This paper proposes a new empirical framework to shed light on these incentives. I
develop a structural econometric model that exploits the variation in ratification dates
1
The legal procedure of a state joining a multilateral agreement is the signature followed by ratification,
which marks the legal accession. In this paper, I will be using the terms “join”, “accede”, “ratify”,
“participate” interchangeably, referring to the legal act of accession as opposed to a mere signing of
a treaty.
2
Electronic copy available at: http://ssrn.com/abstract=1730183
to estimate the sign and magnitude of strategic interaction. A treaty is modeled as a
repeated game among governments that choose between two actions in every period:
to participate or not. The difference in payoffs to both actions – the relative payoff
to cooperation – is assumed to increase monotonically over time. This accommodates
the possibility that a country’s attitude towards treaty participation changes for reasons
unrelated to other countries’ actions, such as an exogenous decline in compliance cost or
an increasing valuation of the treaty’s benefits. When an econometric duration model is
used to explain the timing of ratification of an international treaty, independent behavior
is the maintained assumption.
My approach nests this hypothesis into a model where interdependent ratification behavior may arise as a consequence of strategic choices by forward-looking governments.
To allow for rich patterns of strategic interaction, I consider three types of spillovers.
First, treaty membership generates positive spillovers to everyone else, reflecting the fact
that public good provision is the key objective in a large class of international agreements.
Second, this spillover may differ between treaty participants and non participants. Consequently, the relative payoff to cooperation can either decrease or increase with the
number of treaty members. If ratification becomes less desirable as treaty membership
grows, it is a strategic substitute (Bulow et al. 1985). This parallels the free riding
incentive in a public good game with decreasing marginal benefits. In contrast, if the
relative benefit to cooperation increases with the number of treaty members, ratification
is a strategic complement. Strategic complementarity may arise for a variety of reasons,
including social preferences or concerns about reputation held by governments, discreteness of the public good the treaty seeks to provide, or aspects of the treaty design itself.
Third, when ratification is a strategic complement, the ratification game is supermodular and hence easy to solve even when spillovers vary across country pairs. I exploit this
feature to empirically test specific channels through which strategic complementarity
affects treaty formation.
The assumptions on the payoff function strike a balance between the need to accommodate complex dynamic interactions arising in a broad class of treaties and the necessity of keeping the empirical model simple enough to be viable in settings where data
availability is often limited to a single ratification history. The monotonicity assumption
ensures that all countries join the treaty eventually, which allows me to solve the dynamic
game via backward induction. In the unique strongly renegotiation-proof equilibrium
(roughly, the payoff-maximizing subgame perfect equilibrium), strategic substitutability
delays treaty ratification whereas complementarity has the opposite effect. If strategic
3
complementarity is sufficiently strong, ratification by one country triggers ratification by
another, generating clusters of ratification events in the data. These predictions identify the sign and magnitude of strategic interaction in a structural econometric model
of treaty ratification. Moreover, the structural model allows me to run counterfactual
experiments to explore the role of such channels with a view to optimal treaty design.
I apply this method to the Montreal Protocol on Substances that Deplete the Ozone
Layer which was opened for signature in 1987 and has since been ratified by 191 countries. Its member states have committed to phasing out emissions of chlorofluorocarbons
(CFCs) and other pollutants that, via the depletion of stratospheric ozone, cause severe
damages to human health and to the natural environment. The Montreal Protocol is
credited with accomplishing the provision of a pure public good at the global scale,
overcoming countries’ incentive to free-ride.
The success of the Montreal Protocol has been studied widely, not least because
it seemed to offer important lessons for the 1997 Kyoto Protocol whose targets and
timetables for greenhouse gas abatement closely followed the design of the Montreal
Protocol. Earlier analyses attribute much of the success of the Montral Protocol to
participation being a dominant strategy for the dominant producer – the US – and to the
trade ban which deterred free-riding by fringe producers (Benedick 1998, Parson 2003,
Barrett 2003, 1997b, Sunstein 2007). This literature reveals that incentives to participate
in the Montreal Protocol depend on a complex combination of factors, including the
economics of ozone depletion, the industrial organization of the polluting industries,
specific features of the treaty design and aspects of international diplomacy more broadly.
While this literature has followed a qualitative approach, I offer a strictly quantitative
perspective on this important policy issue. Moreover, taking a model-based approach
to estimation allows me to tease apart the moving parts of the mechanisms that drive
participation through a series of counterfactual explorations.
Relation to the literature
This paper extends the empirical literature on timing games in oligopolistic settings (e.g.
Einav 2010, Schmidt-Dengler 2006, Sweeting 2006) by solving and estimating a discrete
dynamic game of complete information where actions can be either strategic substitutes
or complements. Similar to independent work by de Paula (2009) it is shown that clustering occurs as a consequence of strategic complementarity in the payoff functions.2 His
2
In a ‘synchronization game’ with social interactions, de Paula (2009) derives a test for strategic
complementarity based on simultaneous stopping by multiple agents. Ackerberg and Gowrisankaran
4
contribution and the analysis in this paper are complementary in the sense that they
derive this result starting with fundamentally different assumptions about the information structure (incomplete vs. complete), the time structure (continuous vs. discrete),
the strategy space (irreversible vs. reversible actions), and the solution concept.
From an econometric point-of-view, the framework presented in this paper is an alternative to well-known duration models which rule out strategic behavior by definition.3
Recently, Honoré and de Paula (2010) have analyzed identification of an “interdependent” duration model, cast as the Nash equilibrium of a two-player game with irreversible
actions. This paper adds to their work by focusing on the subgame perfect equilibrium
of an N -player game with reversible actions.
Much progress has been made recently in the estimation of dynamic entry models in
oligopolistic markets (see Aguirregabiria and Mira 2010, for a survey). Applying these
methods requires a data set sufficiently rich to estimate state transition probabilities
and policy functions for all players, which is impractical in the context of a single treaty.
The advantage of the method proposed in this paper lies in the fact that it is feasible
in settings with limited data availability, including the case where only a single history
of the game is observed. This is achieved by allowing players actions to be reversible
which limits the set of payoff-relevant state variables to a time effect.
In applying the framework to the Montreal Protocol, I contribute to a sizable literature on the international political economy of environmental treaties (see Wagner
2001, for a survey). The workhorse in this literature has been the “self-enforcing” agreements model (Barrett 1994, Carraro and Siniscalco 1993, Hoel 1992) which models treaty
participation as a non-cooperative game played among governments. While the basic
version of this model is static, some authors have also considered a supergame to allow
for a richer strategies space (e.g. Barrett 1994, Finus and Rundshagen 1998, Asheim and
Holtsmark 2009) or the differential game to allow for richer dynamics (de Zeeuw 2008,
Rubio and Ulph 2007). This paper develops and estimates the first model to explain the
timing of ratification across countries. The model accommodates heterogeneous payoff functions with asymmetric strategic interactions – features that engender multiple
equilibria and may prevent an analytical solution in the self-enforcing agreements model
(Barrett 1997a, McGinty 2007). Since that model’s key prediction is the number of
treaty members, its empirical implementation is complicated given the lack of a sample of independent and identical treaties. Consequently, previous empirical work has
3
(2006) also exploit this for identification, even though not in the context of a timing game.
Fredriksson and Gaston (2000) and Neumayer (2002) fit duration models to data on ratification of
multilateral environmental agreements.
5
taken reduced-form approaches (Murdoch and Sandler 1997, Beron et al. 2003, Finus
and Tjøtta 2003, Murdoch et al. 2003, Bratberg et al. 2005, Wagner 2009, Bernauer
et al. 2010, Aakvik and Tjøtta 2011), limited the number of agents to two stylized
players (Auffhammer et al. 2005) or ignored strategic interaction altogether (Congleton
1992, Fredriksson and Gaston 2000, Neumayer 2002, Hathaway 2004). My paper bridges
the gap between theory and empirics by developing a strategic model of treaty formation which is estimable using a cross-section of ratification times. Since information is
complete, countries condition their actions on observed participation rather than on unobserved latent benefits as in Beron et al. (2003). Furthermore, since endogenous effects
are identified off the time intervals between subsequent ratification decisions, a priori
there is no need to specify a political or economic channel through which interactions
work – though the relevance of such channels can and will be tested.
The remainder of the paper is structured as follows. Section 2 develops a theoretical
model of the timing of treaty ratification and derives the key testable implications.
Section 3 explains how these implications are exploited for identification and estimation
of the model parameters. Section 4 applies the framework to study the ratification of
the Montreal Protocol. Section 5 concludes.
2. A timing game of treaty participation
The goal of this paper is to develop an empirical framework that allows researchers (i)
to infer the nature and channels of strategic interaction from a history of participation
decisions, and (ii) to conduct counterfactual experiments. The structural approach is
based on a game theoretical model that should be general enough to fit a broad class of
treaties, while also being specific enough to generate testable predictions on equilibrium
play that inform the estimation of the model parameters. This section introduces the
model and explains how these requirements condition the modeling choices.
2.1. Model setup
Treaty participation is modeled as an infinite-horizon game played among N countries.
Countries are assumed to have complete information about all primitives of the game. In
each period t ∈ T = {0, 1, . . . } all countries choose simultaneously whether to participate
in the treaty (ai = 1) or not (ai = 0).4 In line with the Vienna Convention on the Law
4
I adopt
set, and
Q standard notation, i.e. I denotes the set of countries, Ai = {0, 1} the action
Q
A = i∈I Ai the action space with generic element a = (ai , a−i ), where a−i ∈ A−i ≡ j6=i Aj is the
6
of Treaties, countries can costlessly withdraw from the treaty. Country i’s continuation
payoff in period t under strategy profile s ∈ St is given by
Vi (s, t) =
∞
X
πi (ai (s, τ ), a−i (s, τ ), τ )e−ri (τ −t)
(2.1)
τ =t
where πi (·) denotes the per-period payoff function of country i, a(s, t) is the action
profile induced at time t by strategy profile s and ri > 0 is country i’s discount rate.
Each country i chooses its strategy si so as to maximize Vi (si , s−i , 0) taking as given
the strategies s−i chosen by the other countries. The following transversality condition
ensures that this program has a well-defined solution as time goes to infinity
Assumption 1. limt→∞ e−ri t πi (a, t) = 0 ∀i ∈ I ∀a ∈ A
Assume further that the per-period payoff takes the form
πi (ai (t), a−i (t), t) =

γ P
j6=i wij aj (t)
0
f (φ , t) + γ
i
1
if ai = 0
(2.2)
P
j6=i wij aj (t) if ai = 1.
where γ0 , γ1 , {φi }i∈I are parameters and the weights wij ∈ [0, 1]∀i, j ∈ I measure the
intensity of country j’s participation decision on country is payoff.
This specification admits the possibility of both exogenous and endogenous changes
in the incentive to participate. This incentive is characterized by the relative payoff to
cooperation, defined as
∆πi (a−i , t) ≡ πi (1, a−i , t) − πi (0, a−i , t)
P
= f (φi , t) + γ j6=i wij aj (t)
(2.3)
where γ ≡ γ1 − γ0 . In what follows, I shall refer to the first and second terms of this
expression as the ‘private’ and the ‘strategic’ incentive to participate, respectively.
The following assumptions on the payoff function are designed to pin down three
common features that determine the level of participation in a broad set of international
treaties. First, I allow for positive externalities that arise in treaties that provide a
public good.
Assumption 2. ∀i ∈ I, ∀t ∈ T , ∀a, a0 ∈ A such that a−i ≤ a0−i , a−i 6= a0−i ;
πi (ai , a−i , t) < πi (ai , a0−i , t).
vector of actions taken by all countries other than i.
7
Next, to allow for an exogenous driver of treaty participation I assume that the relative
payoff to cooperation increases over time, i.e. ∆πi (a−i , t + 1) > ∆πi (a−i , t) ∀a−i ∈ A−i .
Assumption 3. The function f (φi , t) is differentiable and strictly increasing in its arguments.
This assumption nests the hypothesis of non strategic ratification behavior. If γ = 0
then the incentive to cooperate is invariant with respect to the number and identity of
other contributors, i.e.
∆πi (a−i , t) = ∆πi (a0−i , t)
∀a−i , a0−i ∈ A−i .
Alternatively, the relative payoff to cooperation may increase or decrease if an additional country joins the treaty. The sign and magnitude of this strategic effect depend
on the parameter γ. If γ > 0 then the relative payoff to cooperation exhibits increasing
differences
∆πi (a−i , t) ≤ ∆πi (a0−i , t)
a−i ≤ a0−i , a−i 6= a0−i
and participation in the treaty is a strategic complement as long as the inequality is
strict for at least one country. In turn, if γ < 0 the relative payoff to participation decreases when an additional country joins the treaty and hence participation is a strategic
substitute.
This simple yet general specification encompasses a broad set of models and settings
with rich patterns of strategic interaction that may affect the timing of treaty ratification.
For instance, in the case of a pure public good one would set wij = b to reflect nonexcludable external benefits and assume constant (γ0 = γ1 ) or decreasing (γ1 < γ0 )
marginal benefits of contributing to the public good. In contrast, if the treaty bans
member states from trading with non-member states, then the incentive to participate
increases as a country’s most important trading partners join the treaty.5 To reflect this,
one would calibrate wij on bilateral exports and assume γ1 > γ0 .
Notice that both strategic complementarity and substitutability also arises in more
sophisticated models of public good provision. In a differential game setting, contributions to a public good with decreasing marginal benefits are strategic substitutes when
the public good is continuous (Fershtman and Nitzan 1991) and complements when the
public good is discrete (Kessing 2007, Georgiadis 2013). The simple framework proposed
here gives up some of the rigor and elegance of those theoretical models in order to give
5
I explain this mechanism in more detail in Section 4 below, and in Section C of the online appendix.
8
empirical tractability to the estimation of strategic interaction based on readily available ratification data. As I will show next, the one-parameter specification of strategic
interaction is sufficient to generate sharp predictions on equilibrium play. These predictions allow me to estimate fundamental parameters of the model from the data, without
assuming the nature of strategy interaction a priori.
2.2. A graphical illustration of equilibrium with two countries
To gain some intuition for the solution of the dynamic game, consider the graphical
representation of a simple two-country version of the game in Figure 2.1. Panel (a)
depicts the time path of the relative payoff to cooperation (2.3) for both countries under
the assumption that γ = 0 (constant differences). It is easily seen that country is
dominant strategy is to join the treaty only from period t0i onwards.
The case of strategic substitutability (γ < 0) is depicted in panel (b). While the first
country joins the treaty at the same time as in panel (a), the negative payoff externality
means that the relative payoff to participation shifts downwards for country 2 and hence
delays ratification of the treaty from period t02 until t12 .
In contrast, if ratification is a strategic complement (γ > 0), panel (c) shows how the
relative payoff curve of country 2 shifts upward once country 1 joins the treaty in period t01 , so that cooperation is country 2’s best response from period t12 onwards. Hence
strategic complementarity accelerates treaty formation compared to the base case depicted in Panel (a). With strong strategic complementarity (γ 0), the payoffs can be
depicted as in Panel (d). Both countries start out as non signatories but now both of
them would be better off as treaty members even before period t01 is reached. In fact,
both cooperation and no cooperation are Nash equilibria at every stage in the interval
[t12 , t01 ). Because of positive externalities (assumption 2), both countries prefer the cooperative Nash equilibrium to the non-cooperative one. Below I show that the equilibrium
refinement of strong renegotiation-proof equilibrium selects a unique subgame perfect
equilibrium in which both players start to cooperate at the earliest possible period t12
and never revert to non-cooperative behavior. An important implication of the case
γ > 0 is that even asymmetric countries may begin to cooperate in the same period, as
part of a cluster of cooperation decisions. The empirical strategy adopted below exploits
all these strategic effects for parameter identification, i.e. the delayed vs. accelerated
ratification by late adopters depicted in Panels (b) and (c) as well as the feature of
clustered participation decisions depicted in Panel (d).
9
(a) Constant differences
(b) Strategic substitutability
(c) Weak strategic complementarity
(d) Strong strategic complementarity
Figure 2.1: Evolution of stage-game payoffs over time
2.3. Equilibrium
This section analyzes the equilibrium of the dynamic game under different assumptions
about the sign of γ. I start by characterizing the pure-strategy Nash equilibria of the
stage-game Γ(t) = (I, A, π(t)) played in period t, where π(t) is a vector of individual
payoff functions πi (t) for all countries. I then characterize the subgame perfect equilibria
of the infinite-horizon game G = {I, S, V (0)}, formed by the infinite sequence of stage
games Γ(t)∞
t=1 . All proofs are relegated to the Appendix.
For the case of γ ≥ 0, the theory of supermodular games has established that purestrategy Nash equilibria of the stage game exist, that they are Pareto-ranked and monotonic in time.
Theorem 1. If γ ≥ 0, the stage game Γ(t) has a smallest and a largest Nash equilibrium
in pure strategies for all t ∈ T . Both Nash equilibria are weakly increasing in t. All
10
players prefer the largest (in terms of the number of treaty members) Nash equilibrium
over any other Nash equilibrium.
Corollary 1. If γ = 0 the pure-strategy Nash equilibrium of stage game Γ(t) is in
dominant strategies and is induced by the profile
a∗ (t) = a∗i , i ∈ I|a∗i = 1{πi (1,0,t)≥πi (0,0,t)} .
Lest the special case described in Corollary 1, strategic complementarity engenders
multiple Nash equilibria in the stage game.6 This means that one can construct a great
many subgame perfect equilibria of the infinitely repeated game. To obtain sharper predictions on equilibrium play, I focus on the sub game-perfect equilibrium that maximizes
total payoffs across countries. Formally, this equilibrium is selected when applying the
refinement of strongly renegotiation-proof equilibrium.7
Definition 1. (Farrell and Maskin 1989a) A subgame-perfect equilibrium s is weakly
renegotiation-proof (WRP) if there do not exist continuation equilibria s1 , s2 of s such
that s1 strictly Pareto-dominates s2 . A WRP equilibrium is strongly renegotiation-proof
(SRP) if none of its continuation equilibria is strictly Pareto-dominated by another WRP
equilibrium.
Since the stage-game Nash equilibria are ordered, any continuation equilibrium containing dominated Nash equilibria gives countries an incentive to jointly renegotiate
towards one that does not contain dominated Nash equilibria. The SRP equilibrium
profile is simply given by the unique sequence of undominated Nash equilibria for every
stage of the game.
A straightforward algorithm to compute this profile starts in the first period in which
full participation in the treaty is a Nash equilibrium. Working backwards in time, a
6
Even when γ = 0, countries may be indifferent about treaty participation in a particular period.
I rule out multiple Nash equilibria in this case by assuming that countries choose to cooperate
whenever they are indifferent between both actions. This assumption is not very restrictive in that
it rules out equilibria that differ from the chosen equilibrium in at most N periods. This is because
assumption 3 implies that potential ties in payoffs cannot last for longer than one period. Moreover,
in the empirical framework below it is assumed that payoff functions are subject to a continuously
distributed random disturbance (assumption 4 below), so that ties occur with probability zero.
7
This is the definition of renegotiation-proofness commonly used in the literature on the stability of
international environmental agreements (e.g. Barrett 1994, Finus and Rundshagen 1998, Asheim and
Holtsmark 2009). Pearce (1990) reviews alternative definitions of renegotiation-proofness. In Section
A of the online appendix, I analyze the issue of multiple equilibrium with strategic complementarities
in more detail and show that renegotiation-proofness induces a coalition-proof Nash equilibrium at
every stage of the game.
11
country is dropped from the set of signatories if it has an incentive to defect given the
set of other signatories. Since the Nash equilibrium monotonically increases over time,
this procedure is guaranteed to find the largest treaty (the largest Nash equilibrium)
in every period. Theorem 2 below states that this profile indeed constitutes the unique
SRP equilibrium of the repeated game.
To obtain a formal representation of this algorithm, consider a Nash equilibrium am
in period t(m) . Suppose that am induces exactly m countries to cooperate. Denote by
Km ⊂ I the set of m cooperators and use the definition of Nash equilibrium to write
∆πi am
−i , t(m) ≥ 0
∆πj am
,
t
<0
(m)
−j
∀i ∈ Km
(2.4)
∀j ∈ I\Km
(2.5)
Further, let t(m) ∈ T denote the earliest period in which am is a Nash equilibrium of the
stage game. Then there is a country j(m) ∈ Km that has the least incentive to cooperate
∆π(m)
am
−j(m) , t(m)
≤ ∆πi am
−i , t(m)
∀i ∈ Km
(2.6)
and that prefers not to cooperate in the period just before t(m)
∆π(m) am
,
t
−
1
< 0.
−j(m) (m)
(2.7)
Starting with m = N and working backwards to m = 1, the algorithm uses eqs.
(2.4)-(2.7) recursively to define N sets K(m−1) ≡ Km \{j(m) }, along with the action
profiles inducing them am ≡ (aj , j ∈ I|aj = 1{j ∈ Km }), as well as N periods t(m) .
These elements are needed in the following theorem to characterize the SRP equilibrium.
Theorem 2. If γ ≥ 0, the game G has a unique strongly renegotiation-proof equilibrium
s∗ = {s∗ (t)}∞
t=0 where
s∗ (t) = s∗(i) (t), i ∈ I|s(i) (t) = 1{t≥t∗(i) } .
Country j(i) joins the treaty in period t∗(i) and never withdraws from it at any time after
∗
that. The sequence {t∗(i) }N
i=1 is given by t(N ) = t(N ) and
t∗(N −i) = min t∗(N −i+1) , t(N −i)
i = 1, . . . , N − 1,
where t(m) is defined in equations (2.4)-(2.7) for all m = 1, . . . , N .
12
(2.8)
Characterizing the equilibrium is particularly easy when γ = 0 because each country
always has a dominant action and this is weakly increasing in t, so that the country
never wants to withdraw.
Corollary 2. If γ = 0, the unique SRP equilibrium is given by the strategy profile
s∗ = {s∗ (t)}t∈T where
s∗ (t) = {s∗i (t), i ∈ I|s∗i (t) = 1{t≥t0i } }
and t0i is given by
t0i ≡ {t ∈ T |∆πi (0, t − 1) < 0 ≤ ∆πi (0, t)}.
(2.9)
For the case of γ < 0, the game is no longer supermodular and hence not much can be
said about the N -player game without imposing more structure. By assuming symmetric
spillovers, I obtain a framework that – while still complex – yields sharp predictions on
equilibrium play for the subsequent empirical analysis.
Theorem 3. If γ < 0 and wij = c ∀i, j ∈ I s.th. i 6= j the stage game Γ(t) has a Nash
equilibrium in pure strategies for all t ∈ T . The Nash equilibrium is unique, up to the
identity of treaty members, and is weakly increasing in t.
Multiplicity of equilibrium only arises if two countries have very similar payoffs to
cooperation so that either one – but not both – can be treaty members in a Nash
equilibrium. These equilibria are not necessarily Pareto-ranked, so that either one can
be part of the SRP equilibrium of the supergame.
Theorem 4. If γ < 0, wij = c ∀i, j ∈ I s.th. i 6= j, and countries get to decide on
ratification in descending order of their net benefit from treaty participation, f (φi , t),
then the strategy profile s∗ = {s∗ (t)}∞
t=0 induces a unique strongly renegotiation-proof
equilibrium of game G, where
∗
s (t) =
s∗(i) (t), i
∈ I|s(i) (t) = 1{t≥t(i) } .
Country j(i) joins the treaty in period t(i) and never withdraws from it at any time after
that, where t(i) is defined in equations (2.4)-(2.7) for all i = 1, . . . , N .
To establish uniqueness of the SRP equilibrium, I resolve multiplicity of equilibrium in
the stage game by imposing sequential moves in the stage game, paired with a particular
order of moves. This trick has been widely applied in the literature on entry games. It is
13
no longer needed below when I consider asymmetric countries and let the period length
go to zero.8
A sketch of the proof of SRP equilibrium is as follows. Since strategies are monotonically increasing with time, full cooperation is a dominant-strategy equilibrium of the
stage game from some period t̂ onwards (t̂ = t0N for γ ≥ 0 and t̂ = t(N ) for γ < 0).
Trivially, this continuation equilibrium is WRP because no other profile is played along
the equilibrium path. It is also the unique SRP continuation equilibrium because full
cooperation strictly dominates every other outcome of the stage game. Since no punishment of prior deviations is possible beyond period t̂, a Nash equilibrium of the stage
game must be played in all earlier periods. For γ = 0, this gives rise to a unique sequence
of stage-game Nash equilibria in dominant strategies. For γ > 0, the stage game may
have multiple Nash equilibria but they are Pareto-ranked. Again, only the largest Nash
equilibria in each stage game can be part of the SRP equilibrium. All that remains to
be shown is that the algorithm correctly identifies the largest Nash equilibrium in every
stage of the game. This step is proven by induction exploiting the monotonicity of Nash
equilibrium with respect to time. For γ < 0 and symmetric spillovers, there can be
multiple Nash equilibria of equal size, in which case the algorithm picks the one induced
by the assumed order of moves.
3. Empirical Implementation
3.1. Implications for empirical analysis
A comparison of the equilibrium outcomes characterized in the previous section reveals
the effects of strategic interaction on the timing of treaty participation. Strategic substitutability does not affect the first country to ratify the treaty, but it delays subsequent
ratification times. In contrast, strategic complementarity accelerates participation by
reinforcing the incentive to cooperate. Moreover, if this effect is sufficiently strong,
treaty ratification by one country triggers ratification by others, leading to clustered
ratification decisions, as depicted in Figure 2.1 above.
In order to exploit this prediction in an empirical application to learn about the sign
and strength of strategic interaction effects, one must rule out reasons for clustering
that may occur even in the absence of strategic complementarity. For example, as
the length of a time period increases from a day or week to a month or year, it is
8
See the remark following the proof of theorem (4) in the Appendix.
14
inevitable that ratification times of some countries are clustered. To abstract from
this effect, I characterize the equilibrium as time periods become infinitesimally small.
A first step into this direction is to write down the continuous-time analogues to the
equilibrium ratification times in the discrete-time game. For instance, define t̃0i = {t̃ ∈
R+ |∆πi (0, t̃) = 0} as the first calendar time at which cooperation is a dominant strategy
for country i, and let t̃∗(N ) = min{t̃ ∈ R|∆πi (1, t̃) ≥ 0 ∀i ∈ I} denote the first calendar
time at which full cooperation is a Nash equilibrium of the stage game. Computation of
the remaining calendar times t̃∗(N −1) , . . . , t̃∗(1) in a SRP equilibrium is by straightforward
application of the algorithm described in Section 2.3 above.9
To establish the link between ratification events in discrete and continuous time,
consider the condition
∆πi (a−i , t) ≥ 0
(3.4)
where a induces some set of member countries. For example, a = 0 pins down t0i and
a = 1 pins down t∗(N ) . The only difference between discrete periods t∗ ∈ T and calendar
time t̃∗ ∈ R+ is that the latter solves this condition with equality (due to the continuity
assumption 3 and by the intermediate value theorem) whereas in the discrete-time game
one needs to find the first period on the time grid satisfying the weak inequality. On
a time grid with period length one the relationship between the two is thus given by
t∗ = dt̃∗ e.
Consider now the sequence of discrete-time games {k G}, where k G is played over a
grid with period length 2−k , and k ∈ N0 . A node on this grid, k t, has the property
k
t = t2k . That is, an increase in k by 1 doubles the number of decision nodes and cuts
the period length in half. As k goes to infinity, the grid length 2−k goes to zero. The
SRP equilibrium in the limit game is characterized by the following theorem.
9
Some adjustements to the notation are needed. Denote by subscript (m) the player j(m) ∈ I who is
just indifferent between cooperating or not in the stage game played at time t̃(m) if all players in
Km cooperate. Compute this time as
m
m
t̃(m) = {t̃ ∈ R|∆π(m) (am
−j(m) , t̃(m) ) = 0 ∧ ∆πi (a−i , t̃(m) ) ≥ 0 ∀i ∈ Km ∧∆πj (a−j , t̃(m) ) < 0 ∀j 6∈ Km }.
(3.1)
Define
Km−1 ≡ Km \j(m) .
(3.2)
Let KN = I and compute t̃(m) , j(m) , Km by recursive application of (3.1) and (3.2) for m = N, N −
1, . . . , 1. The equilibrium provision times are given by t̃∗(N ) = t̃(N ) and
t̃∗(N −i) = min t̃(N −i+1) , t̃(N −i)
15
i = 1, . . . N − 1.
(3.3)
Theorem 5. As time periods become infinitesimally small, the SRP equilibrium of the
limit game ∞ G exists and equilibrium provision times converge to their continuous-time
analogues
∀i ∈ I.
lim k t∗i · 2−k = t̃∗i
k→∞
With this result in hand, it is straightforward to establish that clustering of ratification
times is a probability-zero event if countries have different private net benefits of treaty
ratification:
Assumption 4. For all i ∈ I, φi ∈ R is a continuous, i.i.d. random variable with
distribution function F (φ).
Theorem 6. Under assumptions 1-4, if time periods become infinitesimally small then
clustering of ratification decisions among asymmetric countries occurs only in the presence of strategic complementarity.
The theorem suggests an intuitive strategy to empirically identify strategic complementarity in the timing of treaty participation: Given a data set in which countries start
out as non signatories to a treaty and participate only at a later stage, the timing of the
ratification decision traces out the distribution of the relative benefits to cooperation.
Furthermore, if ratification times are recorded on a sufficiently fine time grid, clustering
must be attributed to strategic complementarity unless countries are identical.
3.2. Parameter identification
In line with assumptions 3 and 4, I assume that the private net benefit of joining the
treaty takes the form
f (φi , t) = −φi + eλt .
(3.5)
and that the term φi denotes an exponential index of net cost
φi = exp(x0i β + i )
(3.6)
where ∼ N (0, σ ) is a vector of iid disturbances that are observed by all players but
not by the econometrician. The vector xi contains country characteristics that shift the
private net benefit of ratification. Assumption 1 is satisfied for ri > λ.
Applying the results from Section 2, the equilibrium ratification times are determined
by the relative payoff to cooperation
∆πi (a−i , t) = −φi + eλt + γ
16
P
j6=i wij aj (t)
(3.7)
and depend on the private net benefit φi , on the rate at which the benefit increases λ,
and on the net strategic interaction parameter γ. I compute the sequence of ratification
times {t∗(i) }i∈I using the following algorithm:
Algorithm. Let K be a set of signatories including country i. Denote by t̃(K) ∈ RN
the vector of calendar times with elements

t̃i (K) =

1
log φi − γ 
λ

X
wij 
, i ∈ K.
(3.8)
j∈K\{i}
These are potential equilibrium ratification times of countries in K. Compute the identity
of the last country to ratify the agreement as
j(N ) ≡ arg max{t̃k (I)}
k∈I
and j(N ) s ratification time as t∗(N ) = dt̃(N ) (I)e.
Recursively, for m = N − 1, N − 2, . . . , 1, define the set Km ≡ Km+1 \j(m+1) and compute
the identity of the mth country to join the agreement as
j(m) ≡ arg max {t̃k (Km )}
k∈Km
i
h
with ratification time t∗(m) = min t∗(m+1) , dt̃(m) (Km )e .
The vector of model parameters to be estimated is given by (γ, β, λ). For fixed σ ,
the parameter vector β and λ are parametrically identified from the joint distribution
of country characteristics and ratification times unless all countries ratify on the same
day.10 For γ = 0, a country’s optimal ratification time is given by t0i = d λ1 ln(φi )e, i.e.
the smallest integer to satisfy −φi + eλt ≥ 0. This sweeps out the distribution of φi . For
γ 6= 0, the timing of ratification is also determined by the magnitude of the net spillovers
γwij between any two countries i and j. There are three sources of identification of the
spillover parameter γ. First, for a given parametric specification, the time elapsed between subsequent ratification events is informative about the sign and magnitude of γ.
Second, the presence of ratification clusters in the data identifies strategic complementarity regardless of the specification. Both these sources of identification are available
when testing for the sign of γ under the assumption that spillovers are homogenous.
10
Since choices are binary, γ0 and γ1 are not separately identified and the variance σ is not identified.
17
Third, under the assumption that γ ≥ 0, data on spillover intensities wij between country pairs (i, j) can be exploited to test whether strategic complementarity arises through
a particular bilateral channel, such as trade relationships, reputation effects, or equity
concerns. In sum, this suggests an analysis in two steps. In the first step, γ is estimated
under the assumption of homogenous spillovers wij = N1−1 , i 6= j and wii = 0. If this
yields a positive estimate γ̂, a second estimation substitutes heterogeneous spillovers for
wij and proceeds under the constraint that γ ≥ 0.
This identification strategy rests on two key assumptions. First, assumption 4 guarantees that players are not identical with probability one by introducing a random disturbance that shifts the payoff function in a strictly monotonic fashion. This assumption
is frequently made in applied research to prevent “over fitting” of the model. Second,
I assume a smooth time trend in ∆π (assumption 3). If, contrary to this assumption,
the relative payoff to cooperation were subject to a common, positive shock, then two
or more countries that are close to the threshold will ratify immediately after the shock
hits, even as the grid length goes to zero. This would lead to an overestimation of the
magnitude of strategic complementarity. Conversely, a sudden drop in relative benefits to cooperation would result in an underestimation of strategic complementarity or
overestimation of strategic substitutability. Assumption 3 rules out common shocks.
Whether or not this is reasonable depends on the particular application at hand and
will be further discussed in context below.
From a conceptual point-of-view, a smooth time trend is equivalent to the hazard rate
being a smooth function of time, which is the assumption underlying basic econometric
duration models. More recent duration models have made progress towards relaxing this
assumption, but they still yield inconsistent estimates when decisions are interdependent,
as Honoré and de Paula (2010) show in a two-player model. An advantage of the
framework presented here is that it explicitly accounts for forward-looking strategic
behavior of players who anticipate mutually beneficial deviations from dominated stagegame Nash equilibrium.
3.3. Econometric estimation
Since the expectation of equilibrium ratification times does not have a closed-form solution in the presence of strategic complementarity, I employ the method of simulated
moments (MAM) to estimate the parameter vector θ = (β, γ, λ). The estimation algorithm takes S random draws s on the distribution of and solves for the vector
of ratification times, t(s ; θ0 ), for a given vector of parameter values θ0 . A consistent
18
estimator of θ is given by
θ̂ = min g(θ)0 W g(θ)
(3.9)
θ
where
N
S
1 X
1X
µ(ti (s ; θ))
g=
µ(ti ) −
N i=1
S s=1
!
× f (Xi )
(3.10)
is a vector of sample analogues to the moment conditions µ of observed and simulated
ratification times ti and ti (s ; θ), respectively (see McFadden 1989, Pakes and Pollard
1989, Lee and Ingram 1991). The term f (X) denotes functions of the instruments X.
√
The estimator θ̂ converges in probability to θ and N (θ̂ − θ) converges in distribution
to a normally distributed random vector with mean zero and covariance matrix
1
1+
S
−1
−1
∂g 0 −1 ∂g
∂g 0 −1 ∂g
∂g 0 −1
0
−1 ∂g
E0
W
W E0 gg W
E0
W
E0
.
∂θ
∂θ0
∂θ
∂θ0
∂θ
∂θ0
(3.11)
Using the optimal weighting matrix W ∗ = E0 gg 0 the asymptotic covariance matrix of
√
N (θ̂ − θ) can be reduced to
1
1+
S
−1
∂g 0 ∗−1 ∂g
W
E0
.
∂θ
∂θ0
(3.12)
The estimation algorithm computes the simulated moments and evaluates the criterion
function (3.9) with W taken to be the identity matrix. Minimization of (3.9) in this
fashion yields an initial consistent estimate of θ that is used to compute a consistent
estimate of W ∗ . Standard errors are obtained via non-parametric bootstrap with resampling and 150 random samples.
The moments are chosen so as to capture relevant variation in the data that helps to
identify the parameters of the model. I match ratification times interacted with all other
co variates and with a set of dummies that splits the sample into five bins of (nearly)
equal size. In order to identify the spillover effect I match the absolute difference between
P
i’s ratification time and that of j, weighted by wij ; j6=i |ti − wij tj |. I also match the
frequency of countries ratifying on the same day, 1{ti = ti+1 }.
For the minimization of the MSM criterion function (3.9) I sequentially employ two
algorithms. First, I use Goffe et al.’s (1994) implementation of the simulated annealing
algorithm to avoid that the algorithm gets trapped in a local minimum. After the
simulated annealing algorithm has closed in on a minimum, the result is turned over as
the starting value to a Nelder-Mead simplex algorithm which provides the final vector
19
of parameter estimates.
4. Application to the Montreal Protocol
4.1. Background
The objective of the Montreal Protocol is to phase out emissions of CFCs and other
man made pollutants that deplete ozone molecules in the stratosphere.11 Since their
invention in the 1920’s, CFCs were widely used as freezing agents in air conditioning,
as aerosol propellants in spray cans, as plastic-foam-blowing agents and as solvents in
the manufacturing of microchips and telecommunications parts. The soaring demand
for these products fueled a very rapid growth in worldwide CFC production that lasted
until the 1970’s, when atmospheric chemists predicted that unaltered CFC emissions
would substantially reduce ozone concentrations in the stratosphere. This was a reason
for concern because stratospheric ozone acts as a shield against harmful UV B radiation
from space which increases the prevalence of skin cancer and eye cataracts in humans and
causes damage to crops and ecosystems. By one estimate, fatal cases of skin cancer in the
U.S. population would increase by more than 3.1 million between 1985 and 2075 in the
absence of further regulation of CFC emissions (EPA 1987). The empirical relevance of
stratospheric ozone depletion became very salient in 1985, when new data showed what
became known as the ‘ozone hole’ – a seasonal decline in ozone concentrations of about
50% compared to levels measured in the 1960s. Subsequently, an international expert
panel concluded that worldwide ozone losses were wholly or in part due to CFCs.
Since CFCs mix uniformly in the stratosphere, policy-makers recognized the publicgood nature of CFC abatement and that its efficient provision would require international
cooperation. An international treaty, the Montreal Protocol on Substances that Deplete
the Ozone Layer, was negotiated and opened for signature on September 16, 1987. This
treaty stipulated a 50% cutback in consumption of five CFCs from 1986 levels by 1998,
with a stabilization by 1989 and a 20% reduction by 1993. It also enacted a freeze of
three halons at 1986 levels. Under article 5 of the Protocol, developing countries with a
per-capita consumption of less than 0.3 kg were granted a grace period of 10 years to meet
these obligations. To prevent leakage, the treaty prohibited bulk imports of controlled
substances from non-member states and bulk exports from developing countries. Trade
11
Section B of the online appendix provides further details on ozone depletion and the Montreal Protocol.
20
between developed member countries was left unrestricted, but imports were counted
towards a country’s net consumption.12
The Montreal Protocol went into force on January 1, 1989, by which time it had been
ratified by most major developed countries. To encourage accession by large developing
countries such as India and China, the second meeting of parties held in London in June
1990 established a multilateral fund to pay for the incremental costs of compliance incurred by these countries. Contributions to the fund were the responsibility of developed
country parties according to the UN scale of assessment. Subsequent meetings brought
forward the time table for phasing out ozone-depleting substances, augmented the list of
controlled substances and adopted measures to curb illegal trade. The global phase-out
of CFCs has been proceeding swiftly within the parameters set by the Montreal Protocol,
making the treaty one of the most successful international environmental agreements so
far.
4.2. Assumptions
The empirical framework developed above nests the competing hypotheses of complementarity, substitutability and dominance of ratification decisions. This well-suited to
capture the multifaceted and fundamentally asymmetric incentives to participate in the
Montreal Protocol. The most fundamental aspect of the treaty is the public good aspect
of protecting the ozone layer. This is reflected in assumption (2). The payoff specification flexibly admits both constant marginal external benefits of abatement (γ0 = γ1 ) or
decreasing marginal external benefits (γ1 < γ0 ). The model also admits the possibility
of strategic complementarity (γ0 < γ1 ) which may arise as a consequence of the trade
ban, issue linkage, reputation effects and social preferences (discussed in Section (4.5)
below).
Since ozone depleting substances accumulate in the stratosphere, the underlying economic problem is one of controlling a stock pollutant with multiple agents. While it
would be desirable to fully account for the dynamics of stock pollution, to my best
knowledge, the theoretical literature on multilateral agreements over stock pollution has
not yet developed models that are amenable to econometric estimation.13
12
Art. 1 (6) of the Montreal Protocol defines a country’s net consumption as production plus imports
minus exports of controlled substances. Targets had to be met as a weighted average across all
controlled substances, with weights corresponding to their “ozone-depleting potential”.
13
This literature uses differential game theory to analyze multilateral agreements for controlling stock
pollutants among N countries. In order to be able to calculate a numerical solution to such a model,
Rubio and Ulph (2007) assume that all countries are identical. A recent strand of theoretical research
21
Therefore, I resort to simpler dynamics, which are governed by the function f (φi , t) =
−φi + eλt . This function is consistent with an exogenous process – such as GDP growth
– driving participation. Using the standard model of the voluntary provision of a public
good, Murdoch and Sandler (1997) have argued that income growth was the principal
driver of CFC abatement before the Montreal Protocol went into force. This income
effect predicts that, as countries grow richer over time, more of them are willing to
assume the cost of complying with the abatement targets mandated under the Montreal
Protocol. Modeling GDP growth as an exponential trend is standard. Additivity and
the uniform growth rate λ are simplifying assumptions made to ensure tractability of
the model.14
As was explained in Section 3.1 above, identification of γ hinges on the assumption
that it is not common shocks to f (φi , t). For example, a sudden cost drop that occurs
only in country i gets absorbed into the estimate of φi while a smooth decline in the
cost of CFC substitutes gets absorbed into the estimate of λ.15 Both leave the estimate
of γ unaffected. In contrast, an unanticipated drop in the cost of CFC abatement that
pushes several countries over the threshold causes clustering of ratification decisions and
hence increases the estimate of γ.
In principle, the assumption of a smooth trend in the net benefits to ratification is not
testable. In view of an analysis of CFC abatement cost in the U.S. – the world leader in
CFC production at the time – by Hammitt (2000) the possibility of unanticipated drops
appears rather unlikely. Hammitt documents that inflation-adjusted wholesale prices for
CFCs increased monotonically over time (his data cover the years from 1986 until 1994).
Had prices fallen during this time period, unanticipated drops in marginal abatement
cost would seem to be more of a concern. In fact, Hammitt shows that contemporary
estimates of marginal abatement cost published by EPA in December 1987 and August
1988 underestimated the realized cost. This suggests that, if anything, countries were
expecting actual abatement cost to be too low and would have had an incentive to
postpone ratification once they found out. Since this would lead to downward bias in
the estimated γ, the results obtained for γ̂ > 0 below can be interpreted as conservative
estimates of strategic complementarity.
blends contract theory and dynamic games to analyze the optimal design of agreements over stock
pollutants (Harstad 2009, 2012, Battaglini and Harstad 2012).
14
An alternative justification for the monotonicity assumption (3) is that, in a world with a monopoly
country producer, that country would eventually find it worthwhile to curb its use of CFC’s. I am
grateful to an anonymous referee for suggesting this argument.
15
Recall that the empirical model does not discriminate between smooth reductions in costs and smooth
growth in the benefits of abatement over time.
22
4.3. Data
The data are taken from various sources and summarized in Table 1. The endogenous
variable DAY is the day on which a country ratified the Montreal Protocol, which marks
the legal act of acceding to the agreement. Ratification dates are available at UNEP’s
ozone web site.16 The mean time to ratification in the sample of 140 countries is 5
years, 1 month, and 13 days. More than twenty years elapsed between the first and
last ratification (see also Figure B.1 in the online appendix). This suggests that the net
benefits of joining were highly dispersed and provides ample variation for the estimation
of the empirical model. Moreover, while ratification by most countries is spaced out over
time, some ratified the treaty in clusters, e.g. in the same month or week, or even on
the same day.
The choice of covariates in the vector x is subject to a tradeoff. Ideally, one would like
to include all variables that could possibly shift a country’s net benefits. In practice,
however, the need to estimate a highly non-linear model with a relatively small data set
dictates a parsimonious specification that conserves on degrees of freedom and minimizes
multi-collinearity.17 The most severe known consequence of the depletion of the ozone
layer is an increased risk of skin cancer. Per capita income and population (LNPCY,
LNPOP, respectively, both in logs) are thus expected to shift outward the net benefit of
abatement in per capita and in absolute terms, respectively. Since stratospheric ozone
is thinning more quickly in higher latitudes (LATDEG), the benefit of CFC abatement
should be higher there. Whether or not a country hosts a CFC producer (PRODUCER)
is expected to shift the cost of CFC abatement. The article 5 dummy (ART5) picks up
structural differences between developing and developed countries, as well as the effect
of the grace period granted to the latter. Furthermore, I include a dummy variable
(ART5xLONDON) for article 5 countries that joined after the London Amendments of
June 1990. This controls for unanticipated changes in the incentive to participate which
resulted from the offer of financial aid to developing countries that acceded to the treaty.
All covariates are evaluated at their 1986 levels – the year before the treaty was
opened for signature – in order to preclude simultaneity issues. Country characteristics
such as GDP in 1995 US$ and population size in millions were taken from the World
16
17
see http://ozone.unep.org
For example, the final specification includes a dummy variable for CFC producing countries and not
the country’s actual CFC consumption because the latter is highly collinear with income, population
and article 5 status. Table D.1 in the online appendix lists all included covariates along with the
rationale behind their inclusion and the expected signs of their coefficients.
23
Table 1: Summary statistics
Variable
(1)
Mean
(2)
Std. Dev.
(3)
Min
(4)
Max
DAY
LNPCY
LNPOP
LATDEG
PRODUCER
ART5
ART5xLONDON
1869.214
7.184
1.534
0.215
0.129
0.807
0.607
1456.187
1.385
2.070
0.156
0.336
0.396
0.490
197
4.891
-3.156
0
0
0
0
7588
9.986
6.972
0.650
1
1
1
DAY
LNPCY
LNPOP
LATDEG
PRODUCER
ART5
ART5xLONDON
1449.825
7.506
2.095
0.241
0.165
0.709
0.485
906.145
1.633
1.617
0.167
0.373
0.457
0.502
197
4.573
-1.415
0
0
0
0
3992
10.650
6.972
0.640
1
1
1
Description
Panel A (N = 140)
Day of ratification
Log per capita income
Log population (millions)
Degrees of latitude (100s)
Producer country (dummy)
Art. 5 country (dummy)
Art. 5 country post London (dummy)
Panel B (N = 103)
Day of ratification
Log per capita income
Log population (millions)
Degrees of latitude (100s)
Producer country (dummy)
Art. 5 country (dummy)
Art. 5 country post London (dummy)
Notes: Panel A: N = 140. Clustering (day): Denmark, Germany, Italy, Netherlands, Spain (457); France, Switzerland
(469); Greece, Malta (470); Iceland, Malaysia (713); Antigua and Barbuda, Paraguay (1905); Dominica, Jamaica, Peru
(2023)
Panel B: N = 103. Clustering (day): Italy, Netherlands, Denmark, Spain (457); France, Switzerland (469); Iceland,
Malaysia (713); Peru, Jamaica (2023)
24
Development Indicators.18 The PRODUCER dummy equals 1 for all countries that
report positive production of any CFC, where production data is taken from UNEP
(2004) and includes all five CFCs regulated under the Montreal Protocol. A list of
countries with article 5 status is available at UNEP’s web site. The variable LATDEG
refers to the latitude of the country’s capital and is taken from the CIA World Fact
Book.19
4.4. Global interaction effects
Table 2 reports estimates of the parameter vector (γ, β, λ) when symmetric weights
wij = (N − 1)−1 are imposed. The parameter γ is unrestricted in the sense that it
can take on both negative and positive values. It can thus be interpreted as a global
strategic interaction effect. The first three columns of the table report different the
estimates obtained for different sets of covariates. The estimated γ is positive and
statistically significant for all specifications, indicating that ratification decisions are
strategic complements. Columns 1 to 3 show that the estimated γ changes very little as
further covariates are added to the model.
18
19
available online at http://data.worldbank.org/indicator
available online at https://www.cia.gov/cia/publications/factbook/index.html. The variable is reported in hundreds of degrees for computational reasons.
25
Table 2: Parameter estimates: Global interaction effect
(1)
(2)
(3)
(4)
γ
5.753
(2.034)
3.225
(1.463)
4.031
(1.172)
10.001
(3.105)
Constant
3.784
(0.340)
3.056
(0.419)
4.007
(0.300)
3.876
(0.303)
LNPCY
-0.274
(0.049)
-0.224
(0.054)
-0.301
(0.043)
-0.256
(0.040)
LNPOP
-0.177
(0.042)
-0.168
(0.042)
-0.180
(0.041)
-0.177
(0.041)
0.001
(0.030)
0.000
(0.003)
0.063
(0.023)
-0.218
(0.099)
-0.024
(0.014)
PRODUCER
LATDEG
ART5
-0.223
(0.069)
-0.103
(0.113)
-0.355
(0.123)
-0.196
(0.068)
ART5xLONDON
1.320
(0.169)
1.370
(0.160)
1.332
(0.157)
1.319
(0.148)
λ · 104
10.258
(1.099)
9.763
(1.049)
10.044
(1.130)
10.720
(1.212)
∆ mean (days)
∆ median (days)
306
273
270
247
235
212
445
377
Coordination
yes
yes
yes
no
Non strategic counterfactual:
Notes: N = 140 countries. Spillovers are symmetric wij =
1
∀i
N −1
6= j and wii = 0 ∀i.
Bootstrapped standard errors are in parentheses.
The estimated coefficients on the variables shifting the net cost of accession are robust
across specifications and in line with intuition. For example, the negative coefficient on
per capita income suggests that richer countries place a greater value on environmental quality. Similarly, the negative coefficient on population indicates that populous
countries benefit more from global abatement efforts in absolute terms than less populous countries. The coefficient on the dummy variable for article 5 countries is negative
and suggests that the 10-year grace period for abatement lowered the cost of accession.
In contrast, the dummy variable ART5xLONDON must, by construction, enter with a
26
positive coefficient in the cost term. The coefficient on CFC production is positive but
small and not statistically significant. The negative coefficient on LATDEG implies that
countries in the higher latitudes – where ozone depletion is more pronounced – face a
lower net cost of complying which is consistent with higher private benefits from abatement. The 13 simulated moments used to estimate the most comprehensive specification
match their data counterparts very closely (see Figure D.1 in the online appendix for
more details).
Both the estimation results for γ and the observed clustered ratification events in the
data lend empirical support to the fundamental decision-making process proposed in
this paper. A crucial aspect for the plausibility of this mechanism is that countries have
complete information, which implies that they are not caught by surprise when another
country ratifies, and then take an instant decision to follow suit. To the contrary, a
country that correctly anticipates ratification by another can plan ahead and try to
coordinate the ratification date for both countries in mutually beneficial ways.20
In the real world, do countries – especially democratic ones – have the ability to
actually coordinate on the same ratification date, as posited by the model? Benedick’s
(1998, p.116f) describes a compelling case-in-point. Eight out of twelve member states
of the EU (then EC) ratified the Montreal Protocol simultaneously on December 16,
1988. Two other member states ratified two weeks later, and the other two countries
had ratified the Protocol already two months earlier. This coordination was achieved by
the European Commission asking its member states, many months before, to ratify the
Montreal Protocol simultaneously with the Commission. Obviously, this coordination
device was not perfect, but it worked for the majority of countries, and despite the fact
that the process had to be approved by eight national parliaments.
The coefficient estimates in columns 1 to 3 are based on the assumption that countries
coordinate on the Pareto-efficient outcome at every stage of the game – embodied in the
equilibrium refinement of strong renegotiation-proofness. In view of the EC example, one
might wonder what an overly optimistic assumption about coordination implies for the
point estimate for γ. To investigate this, I re-estimate the model based on the subgameperfect Nash equilibrium with complete coordination failure. In the simple two-player
game shown in Figure 2.1c, this means that players start to cooperate in the latest
possible period t01 rather than in the earliest possible period t12 . Imposing coordination
failure more than doubles the estimated parameter γ in column 4 of Table D.2. The
20
As illustrated in Figure 2.1d, when strategic complementarity is strong there is usually a window of
opportunity for coordinating ratification times.
27
reason is that, as coordination on the early period fails, strategic complementarity must
be stronger to rationalize a given speed of ratification in the data. This why the preferred
estimate in column (3) can hence be viewed as a lower bound on the magnitude of global
strategic complementarity.
As a robustness check, I re-estimate the model allowing for random deviations η ∼
N (0, ση ) from the optimal ratification times – in addition to the structural error term .21
This addresses the concern that the model might overestimate strategic complementarity
because it attributes clustered ratifications to strategic complementarity even when this
occurs purely by chance. Introducing optimization error in the simulation corrects for
this. The results (reported in Table D.2 of the online appendix) are virtually identical,
suggesting that random deviations from optimality do not have a major effect on the
estimation results.
4.5. The sources of strategic complementarity
Having established that global interaction effects in the ratification of the Montreal Protocol are strategic complements rather than substitutes, I now turn to studying the
sources of this effect. The economics and international relations literature on international treaties have discussed specific government interaction effects that engender strategic complementarity. It has been suggested that governments (or their constituents) have
preferences for fair and equitable sharing of the burden of abatement, that social norms
engender a reputation loss for non-signatories. It has also been pointed out that linkage
of environmental cooperation to other international policy issues, or to international
trade, can deter free-riding. In what follows, I briefly review how these effects induce
strategic complementarity in the context of the Montreal Protocol (a more detailed review of this is relegated to Section C in the online appendix) and use the empirical
framework developed above to test whether they matter empirically. This is accomplished by using actual data on bilateral spillovers to calibrate wij in eq. (2.2) before
estimating the parameter γ ≥ 0 associated with a particular mechanism of interaction
between governments.
21
Specifically, if t̃∗ (s ; θ) is the vector of optimal ratification times for a given parameter vector θ and
random draw s , the final vector of ratification dates is computed as t(s , η s ; θ) = bt̃∗ (s ; θ) + η s c.
28
Economic dependency and the trade ban
Previous research has conjectured that a country’s ratification decision may be influenced
by the decisions of countries that it depends upon economically. Using data on the early
phase of the Montreal Protocol, Beron et al. (2003) test whether a country i is more
likely to follow suit to another country j’s ratification if a large share of i’s exports go
to j. I adopt their “power matrix” by computing spillover weights as
wij =
exports from i to j
.
total exports from country i
(4.1)
I compute these weights using the NBER-United Nations Trade Data on bilateral flows
of total commodity exports, averaged over the years from 1962 until 1986.
Another reason for including trade-based measures of interdependencies is the treaty’s
ban of trade in controlled substances between parties and non-parties. Barrett (1997b)
argues that this ban transforms trade in controlled substances into a club good whose
benefits were exclusive to member states. Once the club reaches a critical size, trade
with member states gives higher benefits than trade with non-member states. To gauge
the effect of trade restrictions on ratification, one can thus compute the weights in eq.
(4.1) using data on bilateral trade in controlled substances before the Montreal Protocol
went into force. Using sectoral trade flows from the NBER-United Nations Trade Data,
I construct bilateral exports flows for a set of 4-digit industries that either produce
controlled substances or that rely heavily on those substances as inputs. In so doing, I
follow guidelines proposed by the UNEP Ozone Secretariat regarding the use of customs
data by member states to comply with their reporting requirements UNEP (1999).22
Again, exports are averaged over the years from 1962 until 1986.
Issue linkage and reputation effects
Another mechanism for creating complementarity is by linking different policy issues.
Diplomats may choose to negotiate different topics jointly in order to achieve more stable
outcomes (Tollison and Willett 1979, Sebenius 1983). Folmer et al. (1993) and Cesar
22
Figure D.2 in the online appendix lists HS codes of controlled substances and of the main products
that use them. I map the HS codes of controlled substances to 4-digit SITC (Rev 2) classification used in the NBER-United Nations Trade Data. SITC code 5113 (“Halogenated derivatives of
hydrocarbons”’) contains the main CFCs. In addition, SITC code 5989 (“Chemical products and
preparations, nes”) contains several mixtures of hydrofluorocarbons. From the products listed in
Figure D.2, I include 7415 (“Air conditioning machines and parts thereof, nes”), 7752 (“Domestic refrigerators and freezers”) and 7414 (“Non-domestic refrigerators and refrigerating equipment,
parts, nes”).
29
and de Zeeuw (1996) provide specific examples of how such interconnections can help to
stabilize an international environmental agreement. Essentially, the greater the number
of policy issues in which two countries are involved the better the prospects for linking
those issues in a mutually beneficial way. In order to measure this effect, I calibrate
wij to the degree to which country pair i, j is also involved in one of R pre-existing
international agreements,
R
wij =
X
1
1{iand jsigned treaty r}.
R(N − 1) r=1
(4.2)
Computation is based on data on membership in eleven eminent international agreements.23 I focus on pre-existing treaties and use membership status as of 1986 in order
to avoid potential simultaneity between the decision to join the Montreal Protocol and
to ratify other treaties.
Closely related to issue linkage is the notion that states ratify international agreements
out of a desire to conform with other countries. For instance, Hoel and Schneider (1997,
p. 155) argue that “a government may feel uncomfortable if it breaks the social norm of
sticking to an agreement of reduced emissions, even if in strict economic terms it may
benefit from being a free rider”. Applied to the present model, their argument implies
that country i’s reputation benefit from ratification is greater the more of its “peers”
are among the signatories.24 Hence, the treaty weights (4.2) can also be interpreted as
a plausible (though not the only conceivable) proxy for the reputation cost of breaking
a social norm.25
23
The agreements are: Charter of the United Nations; International Court of Justice; Law of Sea
Convention; International Tribunal for the Law of Sea Jurisdiction; International Convention on
the Elimination of Racial Discrimination; Convention Against Torture and Other Cruel, Inhuman,
or Degrading Treatment or Punishment; International Convention on Civil and Political Rights;
Convention on the Political Rights of Women; Optional Protocol to the International Convention on
Civil and Political Rights; Convention on the Prevention and Punishment of Genocide; Convention
on the Elimination of all Forms of Discrimination against Women. The data are available online at
http://untreaty.un.org. I thank Oona Hathaway for graciously providing me with her data set on
human rights treaties.
P
24
In a binary choice framework, a cost exclusive to non-members is equivalent to the benefit γ wij
exclusive to member states.
25
Fundamentally, it remains challenging to distinguish empirically between issue linkage and social
norms because it might well be the implicit threat of retaliation in various policy domains which
enforces social norms at the intergovernmental level.
30
Fairness and equity
Preferences for fairness and equity have always been pre-eminent in the public debate on
international environmental agreements and have shaped many such treaties in one way
or another. For example, most agreements on transboundary pollution stipulate uniform percentage reductions in emissions because they appear to be more equitable than
differential abatement targets. Fairness has received considerable attention in the recent
economics literature on social preferences (see,e.g., Fehr and Schmidt 1999, Bolton and
Ockenfels 2000), and has been given some consideration in the literature on environmental agreements as well (Hoel 1992, Lange and Vogt 2003). Here I allow for the possibility
that the decision to ratify the Montreal Protocol is at least partially driven by concerns
about fairness and equity. In particular, I conjecture that a treaty is perceived as more
equitable the more large polluters have joined it, where the size is measured as the share
in global CFC emissions
(CFC emissions)j
.
wij = wj = PN
k=1 (CFC emissions)k
(4.3)
I use data on CFC consumption from UNEP (2004) to compute these weights. Consumption is calculated as production plus imports minus exports of controlled substances and
is reported by the member states to the treaty secretariat which monitors compliance.
Consumption is measured in metric tons and comprises all five CFCs that were regulated
under the Montreal Protocol, weighted by their ozone-depleting potential.
A positive coefficient γ means that accession of large emitters to the Montreal Protocol
accelerates ratification by other countries. In order to distinguish the effect of social
preferences on ratification from other size effects, I control for GDP, using weights defined
analogously
(GDP)j
wij = wj = PN
.
(4.4)
k=1 (GDP)k
Results
The sample for which all the weights can be computed contains 103 countries and is
summarized in Panel B of Table 1. Compared to the larger sample used above, countries
in are larger, richer, more likely to produce CFCs and tend to ratify earlier.
Table 3 displays the results when the different weights are used in the estimation of
the model. The first column shows estimates of the benchmark case with equal weights.
As before, I find a positive and significant global interaction effect γ̂ > 0. While the
31
coefficient estimates differ in magnitude, the signs are identical to those obtained in the
larger sample (cf. column 3 of table 2), and the coefficient estimates on the covariates
are robust across specifications except for the coefficient on LATDEG.
The remaining columns of Table 3 report results from specifications with two interaction coefficients: The first one, γ, corresponds to the different weights indicated in
the row header. The second one, γGDP , is obtained for the GDP weights (4.4) that are
included to control for general size effects that might otherwise confound the estimate
of γ. The estimated γGDP is statistically significant in all but one specification.
The interaction parameter obtained when using bilateral exports of all commodities
as weights is positive and significant (column 2). This means that, on average, countries
were more likely to ratify the Montreal Protocol when their principal trading partners
were among the members, conditional on the weight of the trading partners in the world
economy. Following Beron et al. (2003) one could interpret this result as the effect of
economic dependencies on ratification.
Measured in this way, economic dependency is necessarily a very broad concept because bilateral trade flows proxy for a number of unobservable country pair effects –
notably distance, but presumably also many other variables that enter a gravity equation of bilateral trade flows with a significant coefficient. To narrow down the channel of
interaction, the weights used in column 3 are calibrated on bilateral exports of controlled
substances and products. The parameter γ is virtually zero, suggesting that the ban of
trade in controlled substances had no effect on ratification.26
Column 4 of Table 3 reports the results obtained when using treaty membership as
weights. The estimate of the complementarity parameter γ is positive and statistically
significant. This suggests that the strong positive association between ratification of
international environmental agreements and membership in international organizations
found in reduced-form analyses (Bernauer et al. 2010) is in fact driven by local interactions between peer groups of countries that are concerned about their reputation.
The γ-coefficient on CFC weights reported column 5 is also positive and statistically
significant. This can be interpreted as evidence of social preferences: Governments are
more willing to abate pollution when large emitters have also joined the agreement, as
this makes the the agreement appear more equitable to their political constituency. For
this interpretation to be credible, it is reassuring to control for confounding size effects,
26
Neumayer (2002), in a duration analysis of the Montreal Protocol, obtains the seemingly opposed
result that export intensive countries ratified earlier. It appears problematic, however, to attribute
this finding to the effectiveness of trade sanctions because the duration model rules out the very
strategic interaction that makes trade sanctions work in theory.
32
which I do by including γGDP .
The magnitude of the estimates γ̂ is not directly comparable across columns because
the weighting matrices are calibrated in different ways. The next section conducts a
series of counterfactual experiments that shed light on this issue.
4.6. Counterfactual simulations
How strong are the strategic complementarities implied by the estimated γ parameters
in Tables 2 and 3? The structural model allows me to answer this question by comparing
the predicted ratification sequence that arises with strategic interaction to the sequence
of ratification that results when the strategic interaction effect is removed. The latter is
obtained in a counterfactual experiment that sets γ = 0 and simulates the ratification
times based on the parameter estimates for the remaining model parameters.
An obvious metric for the strength of strategic complementarity is by how many days
it brings forward the ratification time compared to the non strategic counterfactual.
Specifically, I calculate the mean and the median difference in ratification times. As this
metric allows for direct comparison across specifications, it is reported right below the
estimation results in Tables 2 and 3.
Regarding the magnitude of the global effect interaction effect, in the most general
specification (reported in column 3 of Table 2 ) strategic complementarity accounts for a
reduction of 235 days (34 weeks) in the mean ratification time and of 212 days (30 weeks)
in the median ratification time. This effect is significant as it corresponds to about one
eighth of the mean and one sixth of the standard deviation of the time to ratification.
Assuming perfect coordination failure as the alternative equilibrium selection rule almost
doubles the reduction in the time to ratification.
To report these differences in Table 3, I conduct counterfactual experiments that quantify the partial effect of a given channel through which strategic complementarity affects
ratification times. For instance, the partial effect of size is computed by subtracting
the ratification times of the fitted model with (γ, γGDP ) = (b
γ , γ[
GDP ) from the fitted
ratification times in a counterfactual scenario where (γ, γGDP ) = (γ̂, 0). Analogously,
I compute the partial effect of the interaction effect measured by γ as the difference
between ratification times in the counterfactual model (γ, γGDP ) = (0, γ[
GDP ) and the
fitted model.
In all specifications, the size effect captured by γGDP accounts for an acceleration
in ratification times that is comparable in magnitude to the one captured in the more
basic specification with symmetric weights (reported in column 1). However, when local
33
interactions are identified using data on total exports, joint treaty membership and
CFC shares, I find an effect on top of that, which accounts for a mean acceleration in
ratification times by 45, 18 and 70 weeks, respectively. The acceleration at the median
is lower, especially when fully bilateral weights give rise to highly asymmetric effects.
To put these numbers in perspective, it is helpful to consider the pace at which the CFC
phase out proceeded at the time. The second meeting of parties held in 1990 brought
forward the timetable for halving CFC emissions by 156 weeks (3 years). Compared to
this, the strategic effects found here are significant, in particular the ones attributed to
CFC shares.
In sum, the estimation results and counterfactual explorations presented provide robust evidence that strategic complementarity accelerated ratification of the Montreal
Protocol in a significant manner. The empirical model attributes this to the frequency
with which countries collaborate in other international agreements, to their relative
emission intensity, and to bilateral trade relations – though not to bilateral trade in
controlled substances.
The results give rise to a number of conclusions pertaining to the optimal design
of global environmental treaties and multilateral agreements more broadly. First, the
finding that ratification by large polluters enhanced broad participation in the treaty
suggests that a fair distribution of the compliance cost can help balance free-riding
incentives. Second, reputation effects that operate in peer groups of countries that
have ratified pre-existing agreements were also found to be important, albeit to a lesser
extent. This suggests that targeting the negotiations to the key players in such networks
is a more efficient strategy to encourage participation than by trying to please every
single country. Third, the finding that bilateral trade in controlled substances did not
affect participation lends empirical support to anecdotal evidence that the trade ban was
inconsequential. As Barrett (2003) points out, the ban was never actually challenged
before the World Trade Organization even though it violated multilateral trading rules
on two counts (non-discrimination and production and processing methods). Moreover,
during the early years of the protocol, the trade ban was thwarted by a thriving illegal
trade in controlled substances (Benedick 1998). This highlights the necessity for trade
sanctions to be both credible and enforceable in order to effectively promote participation
in an international treaty.
34
Table 3: The sources of strategic complementarity
Weights wij
(1)
Equal
(2)
(3)
Bilateral exports
Total
(4)
Treaty
(5)
CFC
11.395
(4.439)
6.196
(3.753)
8.206
(2.966)
5.059
(1.303)
21.682
(7.018)
8.792
(3.153)
4.289
(1.433)
Controlled
Substances
0.001
(0.000)
14.548
(5.587)
Constant
4.058
(0.312)
3.591
(0.335)
3.689
(0.348)
3.160
(0.496)
4.211
(0.425)
LNPCY
-0.162
(0.044)
-0.084
(0.043)
-0.099
(0.032)
-0.080
(0.046)
-0.121
(0.057)
LNPOP
-0.115
(0.046)
-0.074
(0.041)
-0.090
(0.043)
-0.090
(0.047)
-0.109
(0.054)
PRODUCER
-0.376
(0.123)
-0.467
(0.175)
-0.424
(0.153)
-0.321
(0.236)
-0.227
(0.087)
LATDEG
0.435
(0.202)
-0.330
(0.285)
0.390
(0.134)
-0.003
(0.003)
-0.665
(0.351)
ART5
-0.433
(0.129)
-0.462
(0.187)
-0.314
(0.134)
-0.394
(0.206)
-0.738
(0.295)
ART5xLONDON
1.781
(0.341)
1.783
(0.545)
1.756
(0.432)
1.733
(0.768)
1.780
(0.612)
λ · 104
18.562
(2.024)
18.570
(3.540)
19.511
(2.705)
16.813
(4.718)
18.118
(3.997)
∆mean
γ
γGDP
309
454
321
276
561
0
517
488
123
438
637
487
345
∆median
γ
γGDP
504
205
42
174
517
0
561
449
61
418
584
427
298
yes
yes
yes
yes
yes
γ
γGDP
Non strategic counterfactual
Coordination
Notes: N = 103 countries. Bootstrapped standard errors are in parentheses.
35
5. Conclusion
This paper has developed a general framework for empirically analyzing the formation of international agreements. Based on a dynamic ratification game, the framework
nests strategic and non-strategic mechanisms driving countries to ratify an agreement
and is amenable to econometric estimation using a single history of ratification dates.
The principal benefits are twofold. Unlike conventional duration models, my framework
explicitly takes into account strategic, forward-looking behavior on the part of a government that decides on participation in an international treaty. Moreover, by estimating
the behavioral parameters of a structural model, the framework can be used to conduct
counterfactual experiments that explore specific channels of interaction.
I apply this framework to study ratification of the Montreal Protocol on Substances
that Deplete the Ozone Layer, the most successful global environmental treaty to date.
Much has been said about how this treaty achieved broad participation, but so far there
is little quantitative evidence on this important issue. Based on the structural model, I
test for global interaction effects and find robust evidence that ratification was a strategic complement. The fitted model predicts that strategic complementarities reduced
the average time to ratification by at least 34 weeks. An analysis of different sources
of strategic complementarity points to concerns about reputation and a preference for
equity. By contrast, I find no evidence that banning trade in controlled substances engendered strategic complementarity in the ratification decision. This suggests that the
design of an effective treaty must focus on equitable burden sharing, prioritize participation by the key players in blocs of peer countries, and refrain from trade restrictions
if they are not credible or difficult to enforce.
Applying the framework developed here to analyze ratification of other environmental
treaties is straightforward and an interesting topic for future research. In addition to the
environmental field, the framework is suitable to study trade agreements and standard
setting organizations. For instance, according to the ‘domino theory of regionalism’,
regional trade agreements divert trade flows between members and non-members in ways
that makes accession to the agreement a strategic complement. The empirical framework
here lends itself to an empirical test of this theory with only minor modifications to the
model.27 These and other applications are left as a topic for future research.
27
Specifically, assumption 2 can be dropped, which requires an alternative equilibrium selection rule.
36
Appendix: Proofs
Q
Proof of Theorem 1. The action space A = N
i=1 Ai is a complete lattice because it is
the direct product of N finite chains, Ai . By assumption γ ≥ 0, the payoff function
πi (a, t) is supermodular on A for each t ≥ 0, i ∈ I (see Corollary 2.6.1 in Topkis 1998)
and hence Γ(t) is a supermodular game ∀t ∈ T . Under these conditions, Topkis (1979)
and Vives (1990) have shown that the set of Nash equilibria is non-empty and contains
a greatest and a least element. Milgrom and Roberts (1990) have proven that these
elements are pure-strategy Nash equilibria and that they are non-decreasing functions
of the parameter t. Moreover, they have shown that the largest Nash equilibrium is the
most preferred one for all players.
Proof of Corollary 1. In a dominant strategy equilibrium, the best response correspondence
R(a−i , t) ≡ arg max πi (ai , a−i , t).
(5.1)
ai ∈Ai
is constant on A. Suppose first that R(·, t) is not constant. Then there exists some i ∈ I
and a−i , a0−i ∈ A−i such that a−i ≤ a0−i and Ri (a0−i , t) 6= Ri (a−i , t). From assumption
γ = 0 we have that
πi (Ri (a0−i , t), a0−i , t) − πi (Ri (a−i , t), a0−i , t) = πi (Ri (a0−i , t), a−i , t) − πi (Ri (a−i , t), a−i , t).
(5.2)
By the definition of the best response function (5.1), the LHS of this expression is
non-negative and the RHS is non-positive. Hence both sides of the equation must be
equal to zero. Since ties for different actions are ruled out, eq. (5.2) can only hold if
Ri (a0−i , t) = Ri (a−i , t), a contradiction. Thus, each player has a strictly dominant action
and the profile of dominant actions constitutes the unique Nash equilibrium in period
t.
Proof of Theorem 2. I first use backward induction and the dominance property to prove
that the SRP induces the largest Nash equilibrium in every stage of the game. Next,
I establish that the profile s∗ defined above selects the largest Nash equilibrium of the
stage game in every period t ∈ T .
Step 1: SRP equilibrium induces the largest Nash equilibrium in every stage game
Denote by t0N the first period in which cooperation is a dominant strategy for all players.
Full cooperation is the unique NE of the stage game Γ(t0N ) and, by the monotonicity
of equilibrium stated in Theorem 1, in all subsequent periods. Indefinite repetition of
this profile constitutes a subgame-perfect Nash equilibrium of the continuation game
G(t0N ). The equilibrium is WRP because it is not dominated by any of its continuation
equilibria. Since the action profile aN = 1 dominates every other profile in the stage
game, there can be no other WRP equilibrium that dominates indefinite cooperation.
Hence full cooperation is the unique SRP equilibrium of the continuation game G(t0N ).
It follows that there is no credible “punishment” that could be inflicted on a player
who deviates in period t0N − 1 and hence that any subgame-perfect equilibrium induces
a Nash equilibrium of the stage game at node t0N − 1. Backward induction yields that
37
a stage-game Nash equilibrium must be played in all previous periods, too. Moreover,
the refinement of strong renegotiation-proofness requires that none of these stage-game
equilibria be Pareto-dominated, for any such WRP profile t̃ would be dominated by
another WRP profile s̃0 that induces the largest Nash equilibrium in stage t0 and is
otherwise identical to s̃. Hence the SRP equilibrium induces the largest Nash equilibrium
in every stage game.
Step 2: The profile s∗ induces the largest Nash equilibrium in every stage game By
the monotonicity of the Nash equilibrium vector in t, a player who cooperates in some
period t0 will not revert her decision in any later period t00 > t0 . For instance, the last
player j(N ) to cooperate in period t∗(N ) strictly prefers to defect in period t∗(N ) − 1. Thus
j(N ) cannot be part of the largest (and hence: any) Nash equilibrium in the periods
t ≤ t∗(N ) − 1. Solving for the equilibrium path hence boils down to finding, recursively
for each player i ∈ N , the period in which i cooperates for the first time in the largest
Nash equilibrium.
The principle of induction is invoked to prove that, for each k = 1, . . . , N − 1 the
strategy profile s∗ induces the largest Nash equilibrium in the periods from t∗(N −k) until
(but not including) t∗(N −k+1) , i.e. until max[t∗(N −k) , t∗(N −k+1) − 1]. Consider the base case
k = 1. Suppose first that t(N −1) ≥ t∗(N ) . For any such t(N −1) the strategy profile s∗
induces full cooperation, i.e. the largest Nash equilibrium of the stage game from period
t∗(N ) onwards. Next, suppose that t(N −1) < t∗(N ) and the algorithm sets t∗(N −1) = t(N −1)
according to eq. (2.8). The definition of t(N −1) implies that cooperation by the players
in KN −1 is a Nash equilibrium of the game Γ(t(N −1) ) and that full cooperation is not.
By monotonicity, this is true for all t(N −1) ≤ t < t∗(N ) .
The inductive hypothesis maintains that, for some k such that 1 ≤ k ≤ N − 1, the
strategy profile s∗ induces
the largest Nash
h
i equilibrium in the stage game from period
t∗(N −k) through max t∗(N −k) , t∗(N −(k−1)) − 1 . The inductive step is to prove that the profile
h
i
s∗ induces the largest NE in all periods from t∗(N −(k+1)) until max t∗(N −(k+1)) , t∗(N −k) − 1 .
For t(N −(k+1)) ≥ t∗(N −k) the algorithm assigns t∗(N −(k+1)) = t∗(N −k) and the inductive step
follows trivially from the inductive hypothesis.
Suppose now that t(N −k−1) < t∗(N −k) . Then t∗(N −k−1) = t(N −k−1) and s∗ induces the
profile aN −k−1 with the set of cooperators KN −k−1 = KN −k \{j(N −k) }. From equations
(2.4) and (2.6), the profile aN −k−1 is a Nash equilibrium of the game in period t(N −k−1) .
For any larger Nash equilibrium ã ≥ aN −k−1 in the stage game Γ(t∗(N −k) −1) monotonicity
requires that ã ≤ s∗ (t∗(N −k) ). Without loss of generality assume that s∗ (t∗(N −k) ) = aN −k+l ,
i.e. N − k + l players cooperate in period t∗(N −k) . The “cluster size” l ∈ {0, . . . , k} is
given by the value of l that satisfies
t∗(N −k) = t(N −k+l) .
To see that ã 6= aN −k+l let j = N − k + l and write
∆π(j) (aj−j , t∗(N −k) − 1) = ∆π(j) (aj−j , t(j) − 1) < 0
38
(5.3)
where we have used eqs. (5.3) and (2.7). For ã ≤ aN −k+l player j(N −k+l) ’s deviation
incentive is equally strong (γ = 0) or stronger (γ > 0). Thus any profile in which j(N −k+l)
cooperates cannot be a Nash equilibrium of the stage game Γ(t∗(N −k) − 1).
For l > 0, the logic of this argument applies to all other players who cooperate under
aN −k+l but not under aN − k − 1. For all j = N − k + l − 1, . . . , N − k it is true that
∆π(j) (aj−j , t∗(N −k) − 1) ≤ ∆π(j) (aj−j , t(j) − 1) < 0
where the first inequality follows from assumption 3 and the fact that t∗(N −k) < t(j) and
the second inequality follows from eq. (2.7).
Thus, it must be the case that ã = aN −k−1 and s∗ induces the largest Nash equilibrium
in period t∗(N −k) − 1. Monotonicity implies that this is true for all periods t such that
t(N −k−1) ≤ t < t∗(N −k) . By the principle of induction, the claim must be true for all
k = 1, . . . , N − 1.
From eq. (2.7) recall that player j(1) has no incentive to join the coalition at any time
before t∗(1) . Therefore, the largest Nash equilibrium in periods t = 0, . . . , t∗(1) − 1 has no
player cooperate. This concludes the proof that s∗ induces the largest Nash equilibrium
in every stage game. From step 1, this profile constitutes the unique SRP equilibrium
of the game.
Proof of Theorem 3. Existence is proven by Berry (1992) in the context of a static entry
game. The proof is by arranging countries in decreasing order of f (φi , t) and finding the
equilibrium number of treaty members m that satisfies
f (φm , t) + γ(m − 1) ≥ 0 > f (φm+1 , t) + γ(m − 1).
(5.4)
Since ∆πi (a−i , t) is strictly decreasing in a−i the equilibrium number of treaty participants m is unique. Multiplicity arises when f (φm , t) and f (φm+1 , t) are close so that
f (φm+1 , t) + γ(m − 1) ≥ 0 > f (φm , t) + γ(m − 1).
(5.5)
also holds.
Suppose that m is the Nash equilibrium number of signatories in period t, and that
the Nash equilibrium is not monotone. Then there must be a period t0 > t and a k such
that f (φk , t)+γ(m−1) ≥ 0 and f (φk , t0 )+γ(m−1) < 0. This implies f (φk , t0 ) > f (φk , t),
which contradicts Assumption 3.
Proof of Theorem 4. Monotonicity implies that full participation is the unique Nash
equilibrium of the stage game from period t(N ) onwards. Since no punishment of deviations is possible beyond this point, a stage-game Nash equilibrium must be played in
all periods prior to t(N ) . From Theorem 3, a stage game equilibrium in pure strategies
exists and is unique up to the identity of players. When multiple equilibria arise in the
stage game, this leads to multiple SRP equilibria as the stage-game equilibria are not
ranked. That is, while it is clear from the proof of theorem (2) that the algorithm used
39
to compute SRP equilibrium selects the largest Nash equilibrium in each stage game,
this does not resolve the multiplicity issue because all equilibria have the same number
of treaty members. Consider the a period t with two equilibria that have m member
states, as described by eqs. (5.4) and (5.5) above. Suppose that the algorithm has
selected the equilibrium where country (m + 1) is a member and not country (m) , as
in eq. (5.5). This means that in some period t0 > t, the algorithm dropped country
(m) from the set of treaty members but not (m + 1). From eq. (2.6), this only occurs
if f (φm , t0 ) < f (φm+1 , t). However, this contradicts f (φm , t0 ) ≥ f (φm+1 , t) which is true
because we have ordered countries in decreasing order of their relative payoff to cooperation. Hence, to resolve ties, the algorithm always picks the country with larger f (·).
This is precisely how the ambiguity is resolved by the order of moves assumed in the
theorem.
Remark. As the length of the time periods goes to zero, the order of moves assump˜ =
tion is no longer needed. From theorem 5, country m will ratify at time t(m)
−1
−1
f (−γ(m − 1), φm ) which is strictly earlier than t̃(m+1) = f (−γ(m − 1), φm+1 ) given
˜ .
that f is decreasing in the index to φ and given the definition of (3.1) t(m)
Proof of Corollary 2. Under assumption γ = 0 eq. (2.4) is equivalent to ∆πi (0, t) ≥ 0
and condition (2.7) is equivalent to ∆π(m) (0, t(m) − 1) < 0. From this we have that candidate equilibrium times satisfy ∆π(m) (0, t(m) − 1) < 0 ≤ ∆π(m) (0, t(m) ) which coincides
with the definition of t0i in eq. (2.9). To see that t∗(m) always equals t(m) , suppose that
there is some l > m such that t(l) < t(m) . Hence we have that
π(m) (0, t(l) ) ≤ π(m) (0, t(m)−1 ) < 0
where the first inequality follows from assumption (3) and the fact that t(l) < t(m) . The
second inequality follows from eq. (2.7). By monotonicity, am ≤ al and eq. (2.4) implies
that
π(m) (al , t(l) ) = π(m) (0, t(l) ) ≥ 0
– a contradiction. Hence it must be the case that t(1) ≤ t(1) ≤ t(2) ≤ · · · ≤ t(N ) which
completes the proof.
Proof of Theorem 5. Suppose that the SRP equilibrium provision time of player i in
game G is given by t∗i = dt̃∗i e. Note that G is equivalent to 0 G. Hence, the equilibrium
provision time of player i in the game k G can be obtained by simply relabeling the
d2k t̃∗ e
decision nodes of the game. This gives k t∗i = d2k t̃∗i e. The sequence xk ≡ 2ki converges
to t̃∗i because, for any > 0, there exists an integer N such that xk − t̃∗i < for all
k ≥ N . To see this, notice that, by definition of the ceiling function,
2k t̃∗i + 1
d2k t̃∗i e
∗
−
t̃
<
− t̃∗i .
i
k
2k
2
|
{z
}
=2−k
40
ln 0
and let N = − ln
. By the same token, xk0 − t̃∗i < 2−k < 2−k for all k 0 > k.
2
Proof of Theorem 6. Consider a pair of heterogeneous players i and j with (limit) equilibrium provision times t̃∗i , t̃∗j , respectively. From Theorem 5, provision times converge
to t̃∗i and t̃∗j as the grid length goes to 0. In the limit, clustering occurs if and only
if t̃∗i = t̃∗j . To see this, consider the relative payoff to cooperation in the absence of
strategic complementarity, given by ∆πi (a−i , t) = f (φi , t). The equilibrium conditions
(3.4), f (φl , t̃0l ) = 0 ∀l ∈ I have the solution t̃0l = f −1 (0, φl ) where assumptions 3 and 4
have been invoked to invert f . Therefore, t̃0i = t̃0j is equivalent to f −1 (0, φi ) = f −1 (0, φj ).
Since φ is a continuous random variable and f is strictly monotonic in φ, this event has
probability zero. In contrast, if payoff functions exhibit increasing differences (γ > 0)
then it follows from the recursive definition of t̃∗(m) in eq. (3.3) that limit provision times
can be identical even among asymmetric players.
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45
Online Appendix
A. Equilibrium selection with strategic
complementarity
Multiple equilibria are a well-known property of games with strategic complementarities
(see e.g. Cooper and John 1988, Topkis 1998). In the infinite-horizon game, multiplicity
of equilibrium is further exacerbated because any individually rational payoff vector
can be sustained in subgame-perfect Nash equilibrium (Fudenberg and Maskin 1986).
Multiple equilibria are an undesirable feature of the game theoretical model because they
limit its predictive power. In addition, they complicate econometric estimation of the
model because there is no one-to-one mapping from the data into the parameter space
if different equilibria are played across observed histories. To circumvent this problem,
researchers often impose additional assumptions that result in sharper predictions on
equilibrium play. For example, it is common to assume that players move in a particular
order (see, e.g. Berry 1992, Schmidt-Dengler 2006) or to select a Nash equilibrium with
certain properties, such as joint profit maximization (Sweeting 2006) or robustness to
perturbations in the information structure (de Paula 2009). Jia (2008) estimates an
entry game under different equilibrium selection rules and examines their effect on the
parameter estimates.
In contrast, observing a single game with many players – which is the focus of this
paper – will not lead to problems in the estimation as the econometrician can condition
on the equilibrium actually played in the data. The unique SRP equilibrium is arguably
the most plausible outcome, since any other equilibrium gives players an incentive to
renegotiate.
It has been pointed out that “renegotiation-proofness is to some extent a cooperative
requirement” (Farrell and Maskin 1989a, p. 355) in games with more than two players,
because it disregards the possibility that a subset of players K ⊂ I has a mutually
beneficial deviation. In an environment with unconstrained communication it would
seem more plausible to require the outcome to be robust to such collective deviations – an
idea underlying the equilibrium refinement of coalition-proof Nash equilibrium (CPNE)
by Bernheim et al. (1987).1 While there is no definition of coalition-proof equilibrium
for infinite-horizon games, it is easy to show that the Nash equilibrium induced by SRP
equilibrium coincides with the unique CPNE at every node of the game.2
Theorem 7. The largest Nash equilibrium in every stage game Γ(t), t ∈ T corresponds
to the unique coalition-proof Nash equilibrium of Γ(t).
1
The defining characteristic of CPNE is that a deviation is not relevant unless it is self-enforcing, i.e.
robust to further deviations. What is more, CPNE is Pareto-efficient in the class of self-enforcing
profiles.
2
Bernheim et al. (1987) define perfectly coalition-proof Nash equilibrium for games in extensive form,
but its recursive definition limits the applicability of this concept to the study of finite-horizon games
(Bernheim and Ray 1989).
i
Proof. By definition, every CPNE is also a Nash equilibrium, but the converse is not
true. From Theorem 1, only the largest Nash equilibrium a∗ can be a CPNE because it
dominates all other Nash equilibria. It remains to be shown that a∗ is self-enforcing, i.e.
that there is no profitable deviation by any proper subset of players K ⊂ I that is itself
robust to further deviations.
Denote by K ∗ the set of cooperating players induced by a∗ . Consider first the deviation
a0 ≤ a∗ by a subset C ⊆ K ∗ , |C| ≥ 2 of players who change their actions from 1 to 0.
The deviation is not self-enforcing since it gives every i ∈ C a strictly lower payoff
πi (0, a0−i , t) < πi (0, a∗−i , t) ≤ πi (1, a∗ , t)
where the first inequality follows from assumption 2 and the second inequality from the
definition of Nash equilibrium.
Next, consider the deviation a00 ≥ a∗ where a subset of players D ⊆ I\K ∗ change their
action from 0 to 1. A necessary condition for the deviation to be self-enforcing is that
no player in D must have an incentive to revert back to 0 unilaterally, i.e.
∆πi (a00 , t) ≥ 0 ∀i ∈ D.
Moreover, none of the players in K ∗ wants to withdraw upon accession of D since
∆πi (a00 , t) ≥ ∆πi (a∗ , t) ≥ 0 ∀i ∈ K ∗ ,
where the first inequality follows from γ ≥ 0 and the second inequality from the definition
of Nash equilibrium. Since best response functions are weakly increasing, the best
response of the remaining players I\(K ∗ ∪ D) does not change the optimal action by all
players in K ∗ and D. Consequently, if the deviation by D is self-enforcing, there exists
another Nash equilibrium a∗∗ ≥ a∗ (where a∗∗ = a00 or a∗∗ ≥ a00 ). This contradicts the
fact that a∗ is the largest Nash equilibrium of the stage game and hence the deviation
by D cannot be self-enforcing.
Finally, consider a simultaneous deviation ã ≡ a∗ + ∆a0 + ∆a00 (where ∆a ≡ a − a∗ ) by
any two coalitions C and D. All players i ∈ C benefit from reverting back to cooperation
πi (1, a00−i , t) − πi (0, ã−i , t) > πi (1, a00−i , t) − πi (0, a00−i , t) ≥ ∆πi (a∗−i , t) ≥ 0 ∀i ∈ C ⊆ K ∗ .
(A.1)
where the first inequality follow from assumption 2. The second inequality follows from
the assumption that γ ≥ 0, and the third inequality from the definition of Nash equilibrium. The reversion by C is a self-enforcing deviation since any subsequent deviation
by proper subsets of C gives each member a strictly lower payoff. This is true because
the inequality (A.1) holds for any subset C 0 of K ∗ and, hence, for any C 0 ⊂ C.
Hence a rigorous application of coalition-proofness at each node of the game yields
the same outcome as SRP equilibrium applied to the supergame.
ii
Table B.1: Average annual percentage growth rates of worldwide CFC production
(1)
Substance
1931-1977
CFC-11
CFC-12
CFC-113
CFC-114
CFC-115
20.63
14.25
n.a.
n.a.
n.a.
(2)
(3)
Time period
1978-1986 1986-2003
1.0
0.4
10.7
4.0
3.9
-27.7
-20.4
-34.1
-23.8
-33.9
Source: AFEAS (2006), own calculations.
B. More background on ozone depletion and the
Montreal Protocol
The objective of the Montreal Protocol is to stop the depletion of stratospheric ozone
caused by emissions of CFCs and other man made pollutants.3 CFCs were invented
in 1928 and commercially applied as working fluids in refrigerators. Thanks to their
stable chemical structure and low production cost, CFCs were soon used in a range of
different areas, e.g. as freezing agents in air conditioning, as aerosol propellants in spray
cans, as plastic-foam-blowing agents and as solvents in the manufacturing of microchips
and telecommunications parts. The soaring demand for these products fueled a very
rapid and sustained growth in worldwide CFC production. Table B.1 displays average
annual growth rates in worldwide consumption of five common CFCs, namely CFC-11
(CCl3F) , CFC-12 (CCl2 F2 ), CFC-13 (CCl2 FCClF2 ), CFC-14 (CClF2 CClF2 ), and CFC113 (CClF2 CCl2 F). The time periods correspond to different regimes in the regulation
of CFC emission. Column 1 contains the years without regulation, column 2 contains
the years with unilateral regulation by some countries and column 3 contains the years
following the negotiation of the Montreal Protocol up until 2003. Between 1931 and
1977, production of CFC-11 and CFC-12 grew at an annual rate of 20.6% and 14.3%,
respectively, meaning that CFC production doubled every four to five years during this
time.
The trend break occurred in the 1970’s when atmospheric chemists predicted that
ozone concentrations in the stratosphere would fall substantially as a consequence of
unaltered CFC emissions.4 According to the scientists’ theory, CFC molecules remain
intact for several decades after being released and slowly travel up to the stratosphere
where solar radiation gradually breaks them apart. Chlorine atoms set free in this
process trigger a catalytic chain reaction that interferes with the natural conversion
3
4
The exposition in this section draws on the accounts by Benedick (1998) and Parson (2003).
The theory was developed by Paul Crutzen, Mario Molina and Sherwood Rowland who were awarded
the Nobel prize in 1995.
iii
cycle between ozone and oxygen, causing ozone concentrations to fall. Scientists were
alarmed because stratospheric ozone acts as a shield against harmful UV B radiation
from space which increases the prevalence of skin cancer and eye cataracts in humans and
causes damage to crops and ecosystems. By one estimate, fatal cases of skin cancer in the
U.S. population would increase by more than 3.1 million between 1985 and 2075 in the
absence of further regulation of CFC emissions (EPA 1987). The empirical relevance of
stratospheric ozone depletion was disputed until 1985 when the British Antarctic Survey
(BAS) published measurement data showing a seasonal decline in ozone concentrations
of about 50% compared to levels measured in the 1960s (the ‘ozone hole’) . Subsequently,
an international expert panel organized by NASA concluded that worldwide ozone losses
were wholly or in part due to CFCs.
Starting in the late 1970s, some countries adopted unilateral regulation to curb CFC
emissions.5 However, since CFCs mix uniformly in the stratosphere, policy-makers recognized the public-good nature of CFC abatement and that its efficient provision would
require international cooperation. Several years of intense negotiations culminated in
the first international treaty to establish binding targets and timetables for emission reductions, the Montreal Protocol on Substances that Deplete the Ozone Layer. Opened
for signature on September 16, 1987, the protocol stipulated a 50% cutback in consumption of five CFCs (CFC-11, CFC-12, CFC-113, CFC-114, CFC-115) from 1986 levels by
1998, with a stabilization by 1989 and a 20% reduction by 1993. It also enacted a freeze
of three halons at 1986 levels. Under article 5 of the Protocol, developing countries with
a per-capita consumption of less than 0.3 kg were granted a grace period of 10 years
to meet these obligations. To prevent leakage, the treaty prohibited bulk imports of
controlled substances from non-member states and bulk exports from developing countries. Trade between developed member countries was left unrestricted, but imports
were counted towards a country’s net consumption.6 In addition, the member states
were required to provide UNEP with data on production, consumption and trade of
controlled substances, and to promote technical assistance to developing countries.
The Montreal Protocol went into force on January 1, 1989, by which time it had been
ratified by most major developed countries. To encourage accession by large developing
countries such as India and China, the second meeting of parties held in London in June
1990 established a multilateral fund to pay for the incremental costs of compliance incurred by these countries. Contributions to the fund were the responsibility of developed
country parties according to the UN scale of assessment.
Subsequent meetings brought forward the time table for phasing out ozone-depleting
substances. During the London negotiations, the parties agreed on a complete phaseout of CFCs by 2000, with interim goals of 50% in 1995 and 85% in 1997. Two years
later, the goals for CFC were revised to a 75% reduction in 1994 and a phase-out
5
The U.S. implemented a ban on non-essential aerosol uses of CFCs (essentially in spray cans) in 1978.
The European Community (EC) committed to a 30% reduction of such uses from 1976 levels by
1981 and capped the production capacity at its 1980 level.
6
Art. 1 (6) of the Montreal Protocol defines a country’s net consumption as production plus imports
minus exports of controlled substances. Targets had to be met as a weighted average across all
controlled substances, with weights corresponding to their “ozone-depleting potential”.
iv
by 1996.7 The schedule for developing countries remained unchanged, stipulating the
phase-out of most controlled substances by 2010. Further meetings of parties at Vienna
(1995), Montreal (1997) and Beijing (1999) augmented the list of controlled substances
to currently include 98 items and adopted measures to curb illegal trade. Meetings of the
parties to the Vienna convention also gave rise to further, more ambitious protocols, that
were ratified by a subset of the members of the Montreal Protocol. The global phase-out
of CFCs has been proceeding swiftly within the parameters set by the Montreal Protocol,
making the treaty one of the most successful international environmental agreements so
far.
Figure B.1 depicts the pattern of ratification over time for a sample of 140 countries
(summarized in Table 1). The fact that more than 20 years (7391 days) went by between
the first and last ratification suggests that the net benefits of joining the treaty are highly
dispersed. Moreover, while ratification by most countries is spaced out over time, some
ratified the treaty in clusters, e.g. in the same month or week, or even on the same day.
C. Strategic Interaction in the Montreal Protocol
Standard models of the private provision of a pure public good yield the prediction that
equilibrium contributions are (weak) strategic substitutes, meaning that a player’s incentive to cooperate is (weakly) decreasing in the number of other players who cooperate.
In the context of the mitigation of global public bads such as pollution, this prediction
is consistent with the failure of the Kyoto Protocol to curb greenhouse gas emissions,
but it contrasts starkly with the success of the Montreal Protocol at phasing out the
production of ozone-depleting substances around the globe.8 In the main body of the
paper I find evidence that higher levels of cooperation in this treaty were possible due to
strategic complementarity. In what follows, I provide an analytical survey of plausible
mechanisms driving this. The structure is as follows:
1. As the base case, I consider a one-shot game of the private provision of a public
good and show that the incentive to cooperate is either invariant to the number
of other players who cooperate – i.e. defection is a dominant strategy – or strictly
decreasing in it – i.e. cooperation is a strategic substitute.
2. Drawing on the economics and international relations literatures, I review several
extensions to the base case which are relevant for the Montreal Protocol. For each
of them,9 I show that the incentive to cooperate increases with the number of
other players that cooperate (γ > 0) i.e. cooperation is a strategic complement.
In particular, I consider the effects of
7
Moreover, a deadline was set for the phase-out of halons by 1994 and a time table for the phase-out
of HCFCs – an early substitute for CFCs later found to be ozone depleting – by 2030.
8
In a similar vein, economists have long been trying to reconcile high levels of cooperation in public
goods experiments with the much more modest contributions predicted by standard models (e.g.
Isaac and Walker 1988).
9
See also the discussion in Section 4.5 of the paper
v
vi
5
6
7
8
9
10
0
Switzerland
Libya
Bahrain
Israel
Saudi Arabia
Bahamas, The
Kuwait
Qatar
Suriname
Oman
Malawi
Togo
Burkina Faso China
Uganda
Nigeria
Gambia,
The
Zambia
Bangladesh
Ghana
Sri Lanka
Kenya
2000
Niger
Guinea
Pakistan
IndiaKiribati
Benin
Madagascar
Nepal
Lao PDR
Haiti
4000
day of ratification
Congo, Dem. Rep.Burundi
Ethiopia
Lesotho
Mali
Chad
Samoa
Philippines
Senegal
Solomon Islands
Indonesia
Vietnam
Mauritania
Comoros
Central African
RepublicLiberia
Mozambique
Belize
Mauritius
St. Vincent and the Grenadines
Tunisia
Syrian Arab Republic
Jamaica
Ecuador
Botswana
Colombia
Guatemala
Cameroon
Paraguay
Dominican
Republic
PeruHonduras
VanuatuRep.
Thailand
Cote d’Ivoire Congo,
Nicaragua
El Salvador
Morocco
Maldives
Tonga
Albania
Papua New Guinea
Egypt, Arab Rep.
Guyana Bolivia
Zimbabwe
Angola
Sudan
Swaziland
Greece
Barbados
Cyprus
Iran, Islamic Rep.
Gabon
Trinidad and Tobago
Antigua and Barbuda
Malta
Argentina
Venezuela, RB
Seychelles
Korea,Algeria
Rep.
PanamaSouth Africa
Bulgaria
Hungary
Jordan
St. Kitts and Nevis
Poland
Uruguay
Brazil
Fiji
St. Lucia
Malaysia
Mexico
Costa Rica
Dominica
Turkey
Chile
Singapore
Spain
New Zealand
United
States
Norway
Denmark
Japan
Sweden
Iceland
Luxembourg
United Arab Emirates
Finland
Canada
France
Germany
Austria
Netherlands
Australia
Italy
6000
Somalia Guinea−Bissau
Sierra Leone
Rwanda
Cape Verde
Sao Tome and Principe
Bhutan
Equatorial Guinea
8000
Iraq
Figure B.1: Ratification dates in the Montreal Protocol
Notes: The Figure plots the number of days it took a country to ratify the Montreal Protocol after it was opened for signature on September 16,
1987. The vertical axis shows the country’s per capita GDP in 1986, expressed in 1995 US$ and displayed on a logarithmic scale. The sample
comprises 140 countries (cf. Panel (a) of Table 1).
log per capita income
a) A preference for fair and equitable sharing of the burden of abatement. I
employ two widely used definitions of social preferences, namely
i. Inequality aversion (Fehr and Schmidt 1999)
ii. Equity, reciprocity and competition (Bolton and Ockenfels 2000)
b) Social norms that engender a reputation loss for non-signatories.
c) Linkage of environmental cooperation to other international policy issues.
d) The trade ban implemented under the Montreal Protocol to deter free-riding.
C.1. The Base Case: Cooperation as a Strategic Substitute
Consider a one-shot game with N identical players who can choose between two actions:
cooperate (C) and defect (D). For a given number of cooperators n, denote by B(n) and
C(n) the benefits and costs of cooperation, respectively. We can think of these functions
as the benefits from avoided environmental damage and abatement cost, respectively.
Assume that B(n) ≥ 0, B 0 > 0, B 00 ≤ 0, and C(n) > 0. For a given number of other
players k who cooperate, a player’s payoffs are as follows
cooperate: πC (k) ≡ B(k + 1) − C(k + 1)
defect: πD (k) ≡ B(k)
The following additional assumptions on the payoffs ensure that the private provision of
a public good is an N-player prisoners dilemma (e.g. Lange and Vogt 2003).
1. Cooperation is strictly dominated: πC (k) < πD (k)
B(k + 1) − C(k + 1) − B(k) < 0 ∀k
(C.1)
2. The public good is socially desirable
N B(k + 1) − (k + 1)C(k + 1) ≥ N B(k) − kC(k) > 0
k = 1, . . . , N − 1 (C.2)
3. The public good is individually desirable
B(k + 1) − C(k + 1) ≥ B(k) − C(k)
(C.3)
4. The total cost of public good provision kC(k) increases with k.
Assumption 1 states that the relative payoff to cooperation
∆π(k) = B(k + 1) − C(k + 1) − B(k)
(C.4)
is strictly negative. If ∆π(k) does not vary with k then defection is a dominant strategy. In contrast, if ∆π(k) increases (decreases) with k, then cooperation is a strategic
vii
complement (substitute). That is, whether cooperation is a strategic complement or
substitute depends on the sign of the expression
∆π(k + 1) − ∆π(k) = B(k + 2) − C(k + 2) − B(k + 1) − [B(k + 1) − C(k + 1) − B(k)] .
(C.5)
It is reasonable to assume that the cost of abatement is independent of the number
of countries that cooperate, i.e. C(k) = c ∀k.10 Then the cost terms drop out and
expression (C.5) boils down to
∆π(k + 1) − ∆π(k) = B(k + 2) − B(k + 1) − [B(k + 1) − B(k)]
(C.6)
It is straightforward to see that with strictly concave benefits of abatement (B 00 < 0), this
expression is strictly negative and hence cooperation is a strategic substitute. This is the
classical free-riding case: As contributions to the public good from other players increase,
the marginal benefits to cooperation decrease, thus further weakening the incentive to
cooperate. With linear benefits (B 00 = 0), defection is a dominant strategy.
C.2. Extensions: Cooperation as a Strategic Complement
C.2.1. Equity and Fairness
Fairness and equity are always prominent elements of the public debate about global
environmental agreements such as the Montreal Protocol, and the decisions by democratically elected governments are likely to reflect the preferences of their constituency.
Using two widely used definition of social preferenes, I show that concern about fairness
induces strategic complementarity in the abatement game.
Inequality Aversion According to Fehr and Schmidt (1999), much of the experimental evidence at odds with neoclassical utility theory can be reconciled with a utility
function of the form
N
N
βi X
αi X
max {xj − xi , 0} −
max {xi − xj , 0}
Ui (x) = xi −
N − 1 j=1
N − 1 j=1
(C.7)
where αi ≥ β and 0 ≤ βi < 1. That is, for any given payoff x, players also care about
the equality of the payoff distribution. The second and third terms in (C.7) measure the
utility loss from disadvantageous and advantageous inequality, respectively. Given his
own payoff xi , a player’s utility function obtains a maximum at xj = xi ∀j.
For ease of notation, I drop the subscript on equity preferences as I compute the
incentive to cooperate facing a country i. From assumption (C.1), πC (k) < πD (k) ∀k,
10
Trivially, cooperation can be shown to be a strategic substitute if C(k + 1) < C(k), i.e. if abatement
cost is subject to a network externality. Then expression (C.5) is positive provided that the benefits
of abatement are not too concave, |B 00 | < |C 0 |. Cost complementarities were first discussed by Heal
(1994) and have been revisited in a series of recent papers on climate treaties with “breakthrough
technologies” (Barrett 2006, Hoel and Zeeuw 2010, Narita and Wagner 2011).
viii
so that the utility of cooperation and defection can be written as, respectively,
α (N − k − 1)
[πD (k) − πC (k)]
N −1
βk
[πD (k) − πC (k)]
UD (k) = πD (k) −
N −1
UC (k) = πC (k) −
The incentive to cooperate ∆U (k) = UC (k) − UD (k) is given by
∆U (k) = [πC (k) − πD (k)]
N − 1 + α(N − k − 1) − βk
N −1
(C.8)
To check whether this is increasing in k, consider the difference
∆U (k+1)−∆U (k) = [∆π(k + 1) − ∆π(k)]
{z
}
|
≤0
N − 1 + α(N − k − 1) − βk α + β
∆π(k + 1)
−
| {z }
N
−
1
N
−
1
|
{z
}
<0
>0
(C.9)
+ 1) is positive. If it is large enough, expression (C.9) is
For α > 0 the term
positive and cooperation a strategic complement.
α+β
∆π(k
−N
−1
Equity, Reciprocity, and Competition Bolton and Ockenfels (2000) propose an
alternative form of social preferences to account for equity, reciprocity and competition
(ERC). These preferences depend both on player i ’s individual payoff xi and on her
share in the aggregate payoff, σi ≡ Pxixj :
j
Ui (x) = ui (xi ) + ωi r (σ)
(C.10)
where u0i > 0, ui ” ≤ 0, and ωi ≥ 0. The function r(·) is differentiable, strictly concave
and obtains its maximum at σ = N1 . Without loss of generality,11 I assume linear utility
u(xi ) = xi .
Lange and Vogt (2003) have analyzed international environmental protection with
ERC preferences. The incentive to cooperate is given by the difference between the
utility of cooperating
B(k + 1) − c
UC (k) = πC (k) + ωr
N B(k + 1) − (k + 1)c
and the utility of defecting
UD (k) = πD (k) + ωr
B(k)
N B(k) − kc
for given number k of other countries that cooperate, where I have dropped the subscript
11
The case of strictly concave utility u is isomorphic to a strictly concave benefit function B(k) and
constant C(k) = c.
ix
on ωi for ease of notation. This yields
B(k + 1) − c
B(k)
∆U (k) = ∆π(k) + ω r
−r
N B(k + 1) − (k + 1)c
N B(k) − kc
Linear benefits
(C.11)
In the special case of linear benefits, B(k) = bk, eq. (C.11) becomes
c b − k+1
b
∆U (k) = b − c + ω r
−r
(C.12)
Nb − c
Nb − c
The incentive to cooperate is strictly increasing in k
c c b − k+1
b − k+2
−r
>0
∆U (k + 1) − ∆U (k) = ω r
Nb − c
Nb − c
(C.13)
where the sign follows from the fact that r is maximized at b−c/N
= N1 and strictly
N b−c
increasing to the left of it. That is, cooperation is a strategic complement.
Strictly concave benefits Differentiating eq. (C.11) yields
B(k + 1) − c
d∆U (k)
0
0
0
= B (k + 1) − B (k) +ω r
·
|
{z
}
dk
N B(k + 1) − (k + 1)c
|
{z
}
<0
>0
0
B (k + 1) [N B(k + 1) − (k + 1)c] − [B(k + 1) − c] · [N B 0 (k + 1) − c]
(C.14)
[N B(k + 1) − (k + 1)c]2
|
{z
}
>0
B(k)
B 0 (k) [N B(k) − kc] − B(k) [N B 0 (k) − c]
0
·
− ωr
R0
N B(k) − kc
[N B(k) − kc]2
|
{z
} |
{z
}
·
<0
R0
To determine the sign of the derivative, recall that r(·) is maximized at
1
N
, and that
1
B(k + 1) − c
1
1
B(k)
B(k)
B(k + 1) − c
=
·
<
<
·
=
k+1
k
N B(k + 1) − (k + 1)c
N B(k + 1) − N c
N
N B(k) − N c
N B(k) − kc
Further, from assumption C.2, we have that N B(k)−kc > 0. Given that full cooperation
is the efficient outcome, the first derivative of the social planner’s objective function
N B(k) − kc is positive, N B 0 (k) − c > 0 ∀k < N since B 00 < 0. Since the last term is
ambiguous, the incentive to cooperate can be increasing or decreasing in k.
For cooperation to be a strategic complement, the free riding incentive arising from
the strictly concave benefit function – i.e. the first difference in eq. (C.14) – must be
more than offset by a positive effect arising from the preference for equitable payoff
distributions. Sufficient conditions for this are that (i) B is not too concave, (ii) ω is
x
0
0
(k)
B (k)−c
sufficiently large and (iii) BB(k)
> NNB(k)−kc
. The latter is true, for instance, when the
γ
benefit function takes the form B(k) = k with 0 < γ < 1.12
C.2.2. Social Norms and Reputation
Hoel and Schneider (1997) posit that countries that refuse to participate in a treaty
to protect the global environment incur a “non-environmental cost” α(k), which they
think of as the penalty for not complying with the social norm of protecting the environment. Hoel and Schneider assume that this penalty strictly increases with the number
of countries that comply with the social norm, α(k + 1) > α(k). This property seems
plausible when α(k) is interpreted as the reputation loss suffered by a country not willing
to share in the cost of providing the global public good.13 If the non-environmental cost
is high enough, cooperators have the ability to “shame” a free rider into cooperating.
The relative payoff to cooperation (C.4) becomes
∆U (k) = πC (k) − πD (k) + α(k)
(C.15)
and (C.5) changes to
∆U (k + 1) − ∆U (k) = B(k + 2) − c − B(k + 1) + α(k + 1) − [B(k + 1) − c − B(k) + α(k)]
= B(k + 2) − B(k + 1) − [B(k + 1) − B(k)] + α(k + 1) − α(k) (C.16)
{z
} |
{z
}
|
≤0
>0
Unlike eq. (C.5) this expression can be strictly positive if B 00 is not too concave.
C.2.3. Issue Linkage
According to the international relations literature, “issue linkage” – i.e. the linking of
multiple policy issues in a single negotiation – can lead to mutually beneficial outcomes
(Tollison and Willett 1979, Sebenius 1983, Raiffa 1982). Inspired by Bernheim and
Whinston (1990), economists have modeled the linkage of international environmental
protection to other policy issues as an infinitely repeated game where the underlying
stage game is the combination of several asymmetric prisoner’s dilemmas. These games
are linked in such a way as to ensure that the cooperative outcome can be sustained as
a subgame perfect equilibrium once the linked game is repeated infinitely often (Folmer,
v. Mouche and Ragland 1993, Cesar and de Zeeuw 1996).14
12
While I consider a simple one-shot game of pollution abatement here, equity preferences have also been
shown to engender strategic complementarity in multi-stage games of international environmental
protection where countries first decide on accession to a treaty and then on abatement (cf. Barrett
1994). See Lange and Vogt (2003) for the case of ERC preferences and Kosfeld, Okada and Riedl
(2009) for the case of inequality aversion.
13
For example, one can think of a fixed reputation loss ρ > 0 which is incurred vis-a-vis each of the
cooperating countries. Then α(k) = ρk and α(k + 1) − α(k) = ρ > 0.
14
Cesar and de Zeeuw (1996) consider a two-player prisoner’s dilemma where the stage-game payoff
to cooperation is not individually rational for one of the players, so that the folk theorem does
xi
Clearly, the repeated-games framework that gives rise to this result is very different
from the one-shot game considered thus far. A simple way of accounting for mutually
beneficial issue linkage is by adding the present-value payoffs of the linked supergame
θj (k), j ∈ {C, D}. The relative payoff to cooperation then becomes
∆U (k) = πC (k) − πD (k) + θC (k) − θD (k).
(C.17)
As shown above, a necessary condition for strategic complementarity is that θC (k + 1) −
θD (k + 1) > θC (k) − θD (k). This is true, for example, if adding countries facilitates
issue linkage. In a two-player game, Blonski and Spagnolo (2003) show that adding
policy issues will always improve the efficiency of issue linkage.15 To the extent that new
countries bring more issues to the bargaining table, and provided that negotiations do
not become too complex, this result might hold up as the number of players grows.
While this is a promising strand of research, the assumptions of asymmetric countries
and infinite repetition of the stage game constitute material departures from the model
used so far. To show the benefits of issue linkage within the paradigm of the symmetric,
static game of public good provision, it suffices to link the N-player prisoners dilemma
to a coordination game. This is exemplified in the next section, where public good
provision is linked to the issue of free trade.
C.2.4. Trade Restrictions
The use of trade sanctions to deter free riding in the provision of global public goods
is appealing for two reasons. First, a ban of trade with free riders prevents a shift
of polluting activities to countries that free ride (“leakage”). Second, the possibility
to exclude free riders from the gains from trade creates a club good that can render
cooperation a strategic complement.
I illustrate this using Barrett’s (1997b, 2003) model of the trade ban implemented
in the Montreal Protocol. Barrett considers a global public bad (e.g. CFC emissions)
which is set free as a joint output of a homogeneous and tradeable good x. There are a
N symmetric countries and each one of them hosts a single firm. Firms act as Cournot
competitors in segmented markets i = 1, . . . , N (Brander 1981, Brander and Krugman
1983).
Public good provision is a game in two stages.16 In the first stage, governments simultaneously choose whether to cooperate (abate: qj = 1) or to defect (pollute: qj = 0) so
not apply. By linking this stage game to its mirror image game, cooperation becomes individually
rational for both players and hence cooperation can be sustained as a subgame perfect equilibrium
when the linked game is repeated infinitely often.
15
Intuitively, adding issues increases the space of punishment strategies needed to enforce cooperation.
16
Barrett (1997b, 2003) considers a richer model setup where countries first decide whether they want to
accede to a ‘self-enforcing agreement’. In keeping with the analysis in the remainder of this document
and in the paper, I restrict the action space of the government to {cooperate,defect}. Doing so allows
me to show that strategic complementarity does not depend on any of the additional assumptions
going into Barrett’s model, namely that signatories maximize joint welfare and that they get to
move before non-signatories.
xii
as to maximize the sum of domestic firm profits and consumer surplus lest environmental
damage. In the second stage, firms simultaneously choose their segmented outputs.
Stage 2: Firms Solving this game backwards, in stage two firm j maximizes its
profits taking the abatement standard qj and the output of the N − 1 other firms as
given. Inverse demand in country i is linear p(xi ) = 1 − xi where xi denotes the total
consumption of the good in country i. The firm solves
max vj =
x1j ,...,xN
j
N
X
(1 − xi − ϕqj )xij
(C.18)
i=1
where ϕqj xj denotes firm j’s total abatement cost and ϕ > 0. xij is the quantity of the
good that firm j ships to market i at zero transportation
cost.PEmissions are given by
PN i
i
i
xj (1 − qj ). Markets clearance requires that i=1 xj = xj and N
j=1 xj = x .
The first-order conditions17 for an interior solution are given by
1 − xi − ϕqj − xij = 0 ∀i, j.
and give a unique solution vector (x1 , . . . , xN ) that results in aggregate emissions
qi ). The solution depends upon the trade regime in place.
(C.19)
PN
i=1
xi (1−
Free trade Assume that M countries have chosen to cooperate in stage 1. Using
symmetry, the first-order conditions of the firms in these countries boil down to
1 − M xc − (N − M )xd − xc − ϕ = 0
(C.20)
and the first-order conditions of firms in countries that defect are given by
1 − M xc − (N − M )xd − xd = 0.
(C.21)
Equations (C.20) and (C.21) pin down xc and xd as the optimal quantities supplied to
each market by the firms located in cooperating and defecting countries, respectively.
These optimal quantities are conditional on the government choices in stage 1 and on
the free trade regime. We can solve the system to get
xd =
ϕM + 1
N +1
and xc =
ϕM + 1
−ϕ
N +1
(C.22)
Thus, the equilibrium quantity in market i is given by xi = M xc + (N − M )xd = NN−ϕM
+1
1 N −ϕM 2
2
and consumer surplus amounts to 2 N +1 . Profits are given by N xc in cooperating
countries and by N x2d in defecting countries. Total emissions correspond to total output
by firms in defecting countries and are given by E = N (N − M )xd .
17
The conditions are also sufficient for a maximum since
xiii
∂ 2 vj
∂xij ∂xk
j
< 0 ∀i, j, k = 1, . . . , N .
Trade ban The Montreal Protocol prohibits trade in controlled substances between
signatories and non-signatories. In this model, this means that firms located in defecting countries cannot export to cooperating countries and vice versa. The first-order
conditions thus become
whence
x̃c =
1 − M x˜c − x˜c − ϕ = 0.
1 − (N − M )x̃d − x̃d = 0
(C.23)
(C.24)
1−ϕ
M +1
(C.25)
and
x̃d =
1
N −M +1
It follows that profits in cooperating and defecting countries are given by M (x˜c )2 and
(N −M ) (x˜d )2 , respectively, and consumer surplus is given by 21 (M x˜c )2 and 12 [(N − M )x˜d ]2 ,
(N −M )2
.
N −M +1
respectively. Total emissions amount to Ẽ =
Stage 1: Governments In stage one, governments choose between cooperate and
defect so as to maximize the sum of profits and consumer surplus lest environmental
damage, taking as given firm’s reaction functions and play by other governments:
N
N
X
X
1 j 2
i
i
xi (1 − qi )
max Wj =
(1 − x − ϕqj )xj + (x ) − b
qj ∈{0,1}
2
i=1
i=1
(C.26)
Substitution of firm choices into (C.26) yields the welfare functions under free trade
2
2
ϕM + 1
1 N − ϕM
1 + ϕM
WC (M ) = N
−ϕ +
(C.27)
− bN (N − M )
N +1
2
N +1
N +1
2
2
ϕM + 1
1 N − ϕM
1 + ϕM
WD (M ) = N
(C.28)
+
− bN (N − M )
N +1
2
N +1
N +1
and with the trade ban
WCT (M )
WDT (M )
=
=
1−ϕ
M +1
2
M
2
1
N −M +1
(N − M )2
−b
(C.29)
N −M +1
N −M +2
(N − M )2
(N − M )
−b
(C.30)
2
N −M +1
M +2
2
Figure C.1 plots these functions for N = 10, b = 0.01 and ϕ = 0.0125. These
numbers are identical to those used in the simulations by Barrett (1997b), and they
imply that abatement is a prisoner’s dilemma game. Cooperation is not individually
rational because b < ϕ, but N b ϕ. Figure C.1a shows that introducing free trade
does little to change this. In fact, cooperation is a strictly dominated action as it gives
a country strictly lower welfare than defection, regardless of the number of signatories.
xiv
(a) Free Trade
(b) Trade Ban
Figure C.1: Welfare under cooperation and defection
(a) Incentive to cooperate
(b) ∆(Incentive to cooperate)
Figure C.2: Incentives: Free trade vs. Trade ban
In contrast, when the trade ban is implemented, cooperation becomes a strategic complement. Figure C.1b hows that the welfare gap narrows as the number of cooperators
increases, and cooperation becomes a best response once at least half of the countries
cooperate. For both trade regimes, Figure C.2a shows the government’s relative payoff
to cooperation ∆W (k) = W (k + 1) − W (k) as a function of k, the number of other countries that cooperate. Under free trade, this difference is negative and does not change
with k. With the trade ban in place, ∆W (k) starts out negative but increases in a
strictly monotonic fashion with k.
The reason for this is that imperfect competition creates an additional externality
that counteracts the incentive to free ride. For cooperating countries, the gains from
trade are strictly increasing in k because the number of export markets grows, and
because increasing competition reduces the deadweight loss associated with oligopoly
at home. Conversely, the welfare of countries that defect decreases with k because
xv
Table D.1: Expected signs for the parameters of the net cost of complying φ
Variable
Description
Sign
LNPCY
Log per capita income
–
shifts demand for environmental quality
LNPOP
Log population (millions)
–
shifts total benefits from abatement
LATDEG
Degrees of latitude (100s)
–
stronger ozone depletion in higher latitudes
PRODUCER
Producer country (dummy)
ART5
Art. 5 country (dummy)
ART5xLONDON
Art. 5 country post London (dummy)
+/–
–
+/–
Rationale
shrinking market for CFCs/profitable substitutes
10-year grace period to meet targets
control for financial incentives
the number of export markets shrinks and due to decreasing competition in the home
market. Therefore, free trade can be regarded as a club good in both groups and its
provision is a coordination game with two equilibria: Either all countries cooperate or
all countries defect. Upon linking this game to the environmental problem, the former
equilibrium is strictly preferred by all countries because it improves upon environmental
quality.
D. Additional Tables and Figures
xvi
Table D.2: Robustness: Global interaction effects
(1)
(2)
(3)
(4)
γ
4.924
(1.621)
5.022
(1.463)
4.085
(1.189)
9.506
(3.593)
constant
3.584
(0.382)
4.324
(0.419)
3.676
(0.349)
3.896
(0.255)
LNPCY
-0.260
(0.054)
-0.337
(0.054)
-0.250
(0.045)
-0.258
(0.037)
LNPOP
-0.174
(0.039)
-0.193
(0.042)
-0.163
(0.042)
-0.177
(0.040)
0.059
(0.030)
-0.070
(0.031)
0.065
(0.024)
-0.532
(0.311)
-0.072
(0.022)
PRODUCER
LATDEG
ART5
-0.204
(0.065)
-0.338
(0.113)
-0.365
(0.151)
-0.233
(0.082)
ART5xLONDON
1.334
(0.159)
1.306
(0.160)
1.356
(0.153)
1.335
(0.149)
λ
10.083
(1.060)
10.290
(1.049)
9.973
(1.096)
10.682
(1.097)
279
246
268
245
242
217
445
377
Non strategic counterfactual
∆ mean (days)
∆ median (days)
Coordination
yes
yes
yes
no
Notes: N = 140 countries. Bootstrapped standard errors are in parentheses.
The model includes an optimization error in addition to the structural error.
xvii
N=140
N=103
16
14
14
12
simulated moments
simulated moments
12
10
8
10
8
6
6
4
4
2
2
4
6
8
10
data moments
12
14
2
16
0
2
4
6
8
data moments
10
12
14
(a)
(b)
Notes: The graphs plot data moments against simulated moments (on a log scale) for the preferred
specification with symmetric spillover weights, reported in col. 3 of Table (2) and col. 1 of Table 3.
The correlation coefficient between the 13 data moments and 13 simulated moments is log moments is
0.9861 in panel (a) with 140 countries and 0.9945 in panel (b) with 103 countries. The simulation gives
rise to an average of 33.4 clustered ratifications in the large sample vs. 32.7 in the smaller sample.
Figure D.1: Data moments vs. simulated moments
xviii
Source: United Nations Environment Programme. Avialable online at www.unep.org/ozonaction
Figure D.2: HS codes of controlled substances and main products containing them
xix
Chemical name
Dichlorotetrafluoroethanes
Chloropentafluoroethane
CFC-113
CFC-114
CFC-115
Bromotrifluromethane
Dibromotetrafluroethanes
Halon-1301
Halon-2402
Dichlorofluoroethanes
1,1-dichloro-1-fluoroethane
Chlorodifluoroethanes
1-chloro-1,1-difluoroethane
Dichloropentafluoropropanes
HCFC-141
HCFC-141b
HCFC-142
HCFC-142b
HCFC-225
Group III
Bromodifluoromethane
Group I
CH3Br
CH2BrCl
CHF2Br
C3HF5Cl2
Chemical name
HFC-1,2,3,4yf
R-1,2,3,4yf
R-245fa
R-23
1,1,1,3,3-Pentafluoropropane CF CH CHF
3
2
2
CH2=CFCF3
2,3,3,3-Tetrafluoropropene
CHF3
Trifluoromethane
R-32
HFC-245fa
R-143a
CH2F2
HFC-23
R-125
CF3CH3
Difluoromethane
HFC-32
CF3CHF2
Pentafluoroethane
1.1.1-trifluoroethane
HFC-143a
R-134a
R-152a
CF3CH2F
only
A1
A1
A2
A2
A1
A2
A1
ASHRAE
1
ASHRAE
#
safety
for
refrigerants
group
CHF2CH3
Formula
HFC-125
1,1,1,2-Tetrafluoroethane
1,1-Difluoroethane
HFC-134a
HFC-152a
Hydroflourocarbons (HFCs)
Name/Group
Selected Non-Ozone Depleting Substances
4
R-418A
R-408A
R-409A
R-415B
HCFC-22/HFC-143a/HFC-125
HCFC-22 / HCFC-124/HCFC-142b
R-22/R-152a (25/75)
HC-290/HCFC-22/HFC-152a
R-408A (FX 10)
R-409A (FX 56)
R-415B
R-418A
R-500
R-502
R-401A
CFC-12 / HFC-152a
HCFC-22 / CFC-115
HCFC-22/HFC-152a/HCFC-124
R-22/R-600a/R-142b (55/04/41)
R-5005
R-5025
R-401A (MP-39)
R-406A
1973
1062
1018
2831
1864
1009
1974
1020
1958
1028
1017
811-97-2
460-73-1
75-46-7
75-10-5
420-46-2
354-33-6
75-37-6
3159
3824.74
3824.74
3824.74
3824.74
3824.71
3824.71
3824.74
3824.74
2903.39
2903.79
2903.79
2903.75
2903.74
2903.74
2903.73
2903.73
2903.79
2903.72
2903.71
2903.19
2903.14
2903.77
2903.76
2903.76
2903.76
2903.77
2903.77
2903.77
2903.77
2903.77
3824.74
3824.74
3824.74
3824.74
3824.71
3824.71
3824.74
3824.74
2903.39
2903.49
2903.49
2903.49
2903.49
2903.49
2903.49
2903.49
2903.49
2903.49
2903.49
2903.19
2903.14
2903.45
2903.46
2903.46
2903.46
2903.44
2903.44
2903.43
2903.42
2903.41
HS code
until 31
Dec 2011
2903.39
2903.39
2903.39
2903.39
2903.39
2903.39
2903.39
2903.39
HS code
HS code
since 1 Jan
UN3 #
2012
2
3
CAS # UN #
**
**
**
**
**
**
**
**
74-83-9
74-97-5
1511-62-2
75-68-3
CH3CF2Cl
R-142b
75-68-3
C2H3F2Cl
1717-00-6
2837-89-0
306-83-2
75-45-6
71-55-6
56-23-5
75-72-9
124-73-2
75-63-8
353-59-3
76-15-3
76-14-2
1717-00-6
R-124
R-141b
B1
A1
A1
A1
75-71-8
75-69-4
76-13-1
C2H3FCl2
C2HF4Cl
A1
A1
A1
CAS2 #
CH3CFCl2
R-22
R-123
C2HF3Cl2
The most popular ODS-containing mixtures
(Refrigerants)
Methyl bromide (or Bromomethane)
Annex E,
Bromochloromethane
Annex C,
HBFC-22B1
Group II (HBFCs)
Chlorotetrafluoroethanes
HCFC-124
Annex C,
Chlorodifluoromethane
Dichlorotrifluoroethanes
HCFC-123
Annex C, Group I (HCFCs)
HCFC-22
R-140a
R-13
CHF2Cl
C2H3Cl3
Annex B, Group III
1,1,1-trichloroethane or methyl chloroform
CF3Cl
CCl4
Chlorotrifluoromethane
R-114B2
C2F4Br2
R-115
CCIF2CF3
R-13B1
R-114
C2F4CI2
R-12B1
R-113
CF3Br
R-12
C2F3CI3
CF2BrCl
R-11
CF2CI2
only
CFCI3
Formula
Tetrachloromethane or carbon tetrachloride
Annex B, Group II
CFC-13
Annex B, Group I (Other CFCs)
Bromochlorodifluromethane
Halon-1211
Annex A, Group II (Halons)
Trichlorofluoromethane
Dichlorodifluoromethane
Trichlorotrifluoroethanes
CFC-11
CFC-12
Annex A, Group I (CFCs)
Name/Group
ASHRAE
1
ASHRAE
#
safety
for
refrigerants
group
The most common Ozone Depleting Substances (ODS)
CUSTOMS OFFICER'S QUICK TOOL FOR SCREENING ODS
Chemical name
Formula
R143a/125/134a
R143a/125
R32/125/134a
R32/125/134a
R32/125/134a
R32/125
R23/116
R23/116
Iso-Butane
Propane
R-600a
R-290
74-98-6
75-28-5
R-290
C3H8
A3
R-600a
C4H10
A3
106-97-8
7664-41-7
124-38-9
B2
**
**
**
**
**
**
**
**
CO2
R-717
A1/A1
A1
A1/A1
A1/A1
A1/A1
A1/A1
A1/A1
A1/A1
1978
1969
1005
CAS2 # UN3 #
CH3CH2CH2CH3
NH3
R-404A
R-507A
R-407A
R-407B
R-407C
R-410A
R-508A
R-508B
only
ASHRAE
ASHRAE1
#
safety
for
refrigerants
group
2711.12
2901.10*
2901.10*
2811.21
2814.10
3824.78
3824.78
3824.78
3824.78
3824.78
3824.78
3824.78
3824.78
HS code
4 - Their HS codes may be used to disguise ODS
5 - International trade not allowed (contains CFCs)
B1 Higher Toxicity & No Flammability
B2 Higher Toxicity & Lower Flammability
B3 Higher Toxicity & Higher Flammability
Pressurised Gases
Hazardous to Health
Irritating substances
Oxidizing substances
Flammable substances
Countries that produce ODS
China, Japan
China, Republic of Korea (2009 data)
China, India, Russian Federation
Producing Countries
Product
HS code/codes
3909.50
3814.00
Quality reviewed and revised by: Dr. Janusz Kozakiewicz
Prepared by: Compliance Assistance Programme (CAP)
Regional Office for West Asia
Composite solvents
39.17, 39.20, 39.21, 39.25, 39.26
8424.10
CHAPTER 87
8414.30
84.18, 84.19, 85.09
8415.90
All codes under 84.15
Polyurethanes
Insulating boards & pipe covers
Fire Extinguishers
Vehicles
Compressors
Refrigerators & Freezers
AC components
AC systems
China, Israel, Japan, United States of America
HS codes for selected products that may contain ODS
(list is not exhaustive)
Methyl Bromide
Argentina, Canada, China, Democratic People's Republic of
Hydrochlorofluorocarbons Korea, France, Germany, India, Japan, Mexico, The Netherlands,
Republic of Korea, Russian Federation, Spain, United States of
(HCFCs)
America, Venezuela
Carbon tetrachloride (CCl4)
Halons
Chlorofluorocarbons
(CFCs)
Group
Source: Article 7 data for 2010 reporting year, only countries with positive production figures.
United Nations Environment Programme
Environmentally
dangerous substances
Corrosive substances
Explosive substances
Toxic substances
DANGER SYMBOLS
ARCTON - ASAHIFRON - ASAHIKLIN - FORANE - FREON - GENETRON - ISCEON - SOLKANE - SUVA - FLORON
Most popular refrigerants trade names
** CAS # for mixtures is a combination of the CAS # of its components
(Example: R-500 CAS # is: 75-71-8 / 75-37-6 which are the CAS # for CFC-12 & HFC-152a)
2- CAS #: Chemical Abstract Service Number
3- UN #: United Nations Number for some Chemicals
A1 Lower Toxicity & No Flammability
A2 Lower Toxicity & Lower Flammability
A3 Lower Toxicity & Higher Flammability
1- ASHRAE Safety Groups (ASHRAE: American Society for Heating Refrigeration & Air-conditioning Engineers):
* The HS Code applies only if the concentration of butane or Iso-butane is higher than 95% . Otherwise, the substance should be classified in the specific provision of
subheading 2711.13 for “Butanes”.
Carbon dioxide
Butane
R-600
Ammonia
R-744
R-717
Halogen-free Refrigerants
R-404A
R-507A
R-407A
R-407B
R-407C
R-410A
R-508A
R-508B
Hydroflourocarbons blends
(HFCs) (HFCs mixtures)
Hydrofluorocarbons
Name/Group
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xxii