HIAS-E-92

A Dynamic Analysis of Climate Change Mitigation with
Endogenous Number of Contributors: Loose vs Tight
Cooperation
Luca Colombo(a) , Paola Labrecciosa(b) and Ngo Van Long(c),(d)(a) Deakin University(b) Monash University(c) McGill University(d) Hitotsubashi Institute for Advanced Study, Hitotsubashi University

December, 2019

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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A dynamic analysis of climate change mitigation
with endogenous number of contributors: loose vstight cooperation
Luca Colombo (Deakin University)
Paola Labrecciosa (Monash University)
Ngo Van Long (McGill University, and Hitotsubashi Institute for Advanced Study,
Hitotsubashi University)
Abstract
We propose a dynamic model of climate change abatement in which the number ofcontributors is endogenous and thus may differ between two modes of cooperation,
namely, loose vs tight. In the tight mode of cooperation, each member is prescribeda specific target, whereas in the loose one, members choose their own abatementlevels as Nash players. Conditions exist such that the incentive to free ride is lower
and the number of contributors is higher in the loose cooperation framework, andthis can lead to higher welfare, both in the steady state and along the transition
path. Our theoretical results suggest that the loose coalition mode, such reflected in
the spirit of the Paris International COP21 Conference on Climate Change, byattracting more participants, could turn out to be more effective in reducing
emissions than the Kyoto Protocol.
JEL Classification: Q2, Q52, C73.
Keywords: differential games; pollution abatement; climate change; mode ofcooperation.

1

Introduction

The literature on international environmental agreements (IEAs) assumes that a target is set foreach country as part of the agreement - tight cooperation (see, e.g., Carraro and Siniscalco, 1993;
Barrett, 1994; Rubio and Ulph, 2006; Eichner and Pethig, 2013). A robust theoretical result underthat assumption is that the equilibrium number of signatories to a tight IEA is very small.1 In
practice, proposed agreements that prescribe tight cooperation often fail to attract a su¢cientnumber of truly committed signatories. For example, the Kyoto Protocol, which was adopted
in 1997 and entered into force in February 2005, was a failure. Although it was signed by 178
countries, only a small number of countries were required to reduce emissions in the Örst phase.
The second phase speciÖed targets for 37 countries, but so far only seven have ratiÖed. Anticipationthat no country will meet their prescribed targets has led to the Paris Agreement in 2015, whichused a distinctly di§erent approach from the Kyoto Protocol. This new approach may be termeda loose mode of cooperation. At the Paris International COP21 Conference on Climate Change,
countries have agreed on an overall objective of limiting global warming to 2 degrees C relative tothe pre-industrial temperature, but no country is required to set a speciÖc target by a speciÖc date.
Signatories are free to determine their own target, and there is no penalty if a target is not met.
There were some indications that the this form of loose agreement, by attracting more participants,
could turn out to be more e§ective in reducing emissions than the Kyoto Protocol.2
A reason why the IEAs with a tight mode of cooperation do not attract many truly committed
signatories is that the incentive to free ride is very strong. In this paper, we develop a model of loose
cooperation, where countries/members agree in principle to aggregate targets but do not commit tocountry-speciÖc targets. One of the advantages of loose targets is that more potential contributors
are willing to join. To the best of our knowledge, this paper represents the Örst theoretical attempt
to show that, compared to tight IEAs, a looser form of cooperation can work better in terms ofimproving welfare.
We propose and analyze a dynamic (continuous time) model of voluntary abatement of a public
bad that departs from the existing literature in two ways: (i) the number of contributors is endogenous and dependent on the mode of cooperation; (ii) in addition to the standard cost which isincreasing in the amount of contribution, abating agents must also incur a Öxed participation cost.3
1

Barrett performed numerical simulations. The results obtained were proven analytically in Rubio and Ulph(2006).
2
Indeed, the reactions of the stock markets after the Paris Agreement was reached could be one such indication.
Renewable energy share prices rose after the Paris Agreement. The iShare Global Clean Energy Exchange Trade Fundrose by 1.4% and the MAC Global Solar Energy index rose by 1.9%. Stock prices of coal companies fell sharply (11.3%
for Peabody Energy, 4.9% for Consol Energy Inc.). The U.S. oil and gas index dropped by 0.5%. See Mukanjari and
Sterner (2018), van der Ploeg and Rezai (2019).
3
Classical references on the voluntary provision of public goods include Chamberlin (1974), Bergstrom, Blume andVarian (1986), Bernheim and Douglas (1986), Cornes and Sandler (1986), and Andreoni (1988). On the voluntaryprovision of public goods in dynamic settings see Fershtman and Nitzan (1991), Wirl (1996), Marx and Matthews(2000), Itaya and Shimomura (2001), Yanase (2006), Long and Shimomura (2007), Benchekroun and Long (2008),

2

As to (i), we take a similar approach to that taken by Rubio and Ulph (2007) and de Zeeuw (2008),
who study the dynamic incentives for countries to sign international environmental agreements,
with one important di§erence.4 Rubio and Ulph (2007) and de Zeeuw (2008) assume that membersof an IEA fully coordinate on their contributions. We, instead, do not restrict our attention to thatcase, but consider also the scenario in which active contributors decide non-cooperatively on how
much to contribute. As to (ii), the existence of a Öxed participation cost, besides its theoreticalinterest, makes the model applicable to a wide range of real-world problems. Consider, for example,
the Paris Accord mentioned above. There are Öxed costs of administration for each country, e.g.,they have to set up a bureaucratic machinery for the operation of the emission-permits market
and/or the collection of carbon taxes, assessment of Örmsí emission levels, auditing Örmsí reporton emissions of carbon, etc. Besides administrative costs, the Öxed cost may include the political
economy cost, for instance, the cost of overcoming opposition from pressure groups to pass thelegislation and the cost of redistributing from winners to losers within the country.5
We compare the welfare outcomes across two scenarios of private contributions to abatementof a public bad (such as the stock of pollution) with Öxed participation cost, such that in equilibrium some agents choose to be free riders, thus avoiding the Öxed cost. In the non-coordination
scenario, after paying the Öxed cost, contributors choose their contribution levels as Nash players.
In the coordination scenario, those who have chosen to be contributors must coordinate on theircontribution levels. The main result of our analysis is that there exist parameter regions in which
non-coordination results in an increase in both the number of contributors and the aggregate contribution, and consequently, higher social welfare. Loose cooperation can be welfare-improving not
only in the steady state equilibrium but also in terms of present value of net utility.
Another advantage of loose cooperation is that it allows individual countries the áexibility torespond to idiosyncratic shocks. Many of these shocks are related to political economy considerations, such as discontents from a powerful segment of the electorate. Shocks of this nature arequite often private information (i.e., not veriÖable by third parties) and therefore state-contingent
transfer payments are not feasible (Bagwell and Staiger, 2005; Amador and Bagwell, 2013). These
are important considerations. However, for simplicity, as a Örst step, we do not model idiosyncratic
shocks and asymmetry among countries.
There are many examples of successful loose cooperation in world history. Of note is the
Fujiwara and Matsueda (2009), Battaglini et al. (2014), Georgiadis (2015, 2017), and Bowen et al. (2019), inter alia.
All of these papers suppose that there are no Öxed participation costs.
4
Rubio and Ulph (2007) and de Zeeuw (2008) build on earlier works on cartel stability in oligopolistic markets bydíAspremont et al. (1983), and on earlier works on IEAs by Carraro and Siniscalco (1993) and Barrett (1994). On
the stability of oligopolistic cartels and IEAs see also Diamantoudi (2005) and Diamantoudi and Sartzetakis (2015),
respectively.
5
Another example of contributions to a pure public good with Öxed cost of participation is R&D with complete
spillovers. The Öxed participation cost arises because a Örm that does R&D must incur the cost of setting up a
research lab. Similarly, those who want to contribute articles to Wikipedia must at Örst incur the Öxed costs oflearning the technical side of navigating that website.

3

Hanseatic League, a Central European loose confederation of merchant guilds and market cities thatcame to dominate Baltic maritime trade for over 300 years, reaching its peak in the 15th century,with over 100 member cities (Atat¸re, 2008). A more recent example is the Association of South
East Asian Nations (ASEAN) which promote free trade among member countries without requiringthem to abide by strict rules. In modern democracies, political parties are notable instances of loose
cooperation. In the USA, for example, Republicans and Democrats can move in and out of their
parties without penalties, and while political donations are encouraged, party members are notasked to commit to speciÖc donations. Similarly, religious organizations typically do not insist
on tight cooperation. It seems that our theory of loose cooperation can be useful to understandphenomena outside the domain of environmental economics.
The remainder of this paper is organized as follows. The model is speciÖed in Section 2. Sections
3 and 4 analyze the non-cooperative and the cooperative case, respectively. A welfare comparison
is provided in Section 5. Section 6 concludes.

2

The Set-up

Consider an economy populated by n ! 2 inÖnitely-lived agents. Time is continuous and denotedby t 2 [0; 1). At t = 0, each agent decides whether or not to contribute to the abatement of astock of a public bad (henceforth, the stock of pollution). Contributors are denoted by the index
i = 1; :::; m, where m $ n. The remaining n % m individuals are called free riders, and we use theindex j to refer to free riders. We assume that to be a contributor, an agent must incur a Öxedparticipation cost F > 0.
All agents are ex-ante identical. Each agentís maximum productive capacity of the Önal consumption good is a positive number a, which we refer to as their ìbusiness-as-usualî level of output.
Each unit of output generates a unit of emission of a pollutant. The stock of pollution is a publicbad. Individuals realize that if they produce the consumption good at their maximum productive
capacity, they will each add a units of pollutant to the accumulation of the public bad. Cutting
output below the maximum capacity is referred to as ìabatement.î If an individual i chooses theabatement level xi (t), where 0 $ xi (t) $ a, he will have only a % xi (t) units of output to consume.
We assume that the direct utility derived from consumption is quadratic:
U (a % xi (t)) & [a % xi (t)] % [a % xi (t)]2 ,where we assume that a < 1=2, so that any xi > 0 constitutes a sacriÖce of direct utility. The stockof pollution is denoted by k(t) ! 0. At any time t ! 0, the stock k(t) ináicts a damage áow bk(t)2
to each agent, where b > 0. The instantaneous (net) utility function of a contributor is deÖned asui (k(t); xi (t)) & U (a % xi (t)) % b [k (t)]2 & [a % xi (t)] % [a % xi (t)]2 % b [k (t)]2 ,4where xi (t) 2 [0; a].6 The instantaneous utility function of a non-contributor is given by
uj (k(t)) & ui (k (t) ; 0) & a % a2 % b[k(t)]2 .
The evolution of the stock of pollution is governed by the following di§erential equation:mX
dk (t)= na %xi (t) % 3k (t) ,
dt

(1)

i=1

where 3 is the natural decay rate of the stock of pollution, with 3 > 0. By (1), the addition to thePstock of pollution is increasing in na and it decreases as the sum of abatement levels ( mi=1 xi (t))
increases.
Let r be the positive rate at which future payo§s are discounted. Once an agent has decided to
be a contributor, his objective functional is
JiC &

Z1
n
o
e"rt %b [k (t)]2 + a % xi (t) % [a % xi (t)]2 dt.
0

The payo§ of a non-contributor (i.e., a free rider) is
JjN &

Z1
n
o
e"rt %b [k (t)]2 + a % a2 dt.

(2)0

In our model, all agents are identical ex-ante, but they behave di§erently ex-post. We focus on
the case where in equilibrium there are exactly two groups: a group GC consisting of m contributors(where m is endogenously determined) who, in equilibrium, contribute the same amount, x#i > 0
for all i 2 GC , and a group GN consisting of n % m free riders, whose contribution is nil, xj = 0
for all j 2 GN .7
At the beginning of the game (t = 0), individuals decide on whether they are members of groupC
G or of group GN . We will refer to t = 0 as Stage 1 and to the time period (0; 1) as Stage
2. Accordingly, we will refer to the decision at t = 0 as Stage 1 decision, and to the decisions at
t 2 (0; 1) as Stage 2 decisions. In Stage 1, each member of group GC must incur a participation
cost, F > 0. In Stage 2, members of group GC , fully informed about the size of GC , choose theircontribution levels, while members of group GN (who by choice did not incur the participation cost
F in Stage 1) are free riders: their contribution is nil.
Let ViC and VjN denote the value functions of a contributor and a free rider, respectively. (These
functions are obtained by solving Stage 2 game). Given ViC and VjN , for a (Stage 1) equilibriumwith m contributors to exist it must be that each i 2 GC has no unilateral incentive to deviate and6

We will impose parameter restrictions which guarantee that ui is decreasing in k and increasing in a, and that
consumption net of the abatament, a ! xi , is nonnegative.
7
We do not consider asymmetric equilibria where abatement levels di§er across the m contributors.

5join GN , and each j 2 GN has no unilateral incentive to deviate by joining GC . Formally, for eachi 2 GC , it must hold that8
ViC (k0 ; m) % F ! VjN (k0 ; m % 1) ,(3)
and, for each j 2 GN , it must hold that
VjN (k0 ; m) ! ViC (k0 ; m + 1) % F .

(4)Inequality (3) states that no contributor would gain by deviating, i.e., it must be individually
rational to contribute (in which case the total number of contributors is m) rather than free riding(in which case, the total number of contributors becomes m % 1).9 We will refer to condition (3)as the contributor-rationality condition.
Inequality (4) states that, for a free rider, it must be better to contribute nil rather than joining
group GC (in which case it assumes that the number of contributors becomes m + 1, and that all ofthem contribute at their new symmetric equilibrium level with m + 1 contributors). We will refer
to condition (4) as the free-rider-rationality condition.
As is well-known in the literature, if a plan is optimal at time t=0 it does not necessarily meanthat it will remain optimal at time 9 > 0: time consistency at t = 0 and k(0) = k0 requires that,at any future date 9 2 (0; 1), each contributor (resp. free rider) will Önd it individually rational
to continue to contribute (resp. not contribute). Formally, for each i 2 GC , it must hold that
ViC (k(9 ; m); m) ! VjN (k(9 ; m); m % 1) ,(5)

and, for each j 2 GN , it must hold that
VjN (k(9 ; m); m) ! ViC (k(9 ; m); m + 1) % F ,(6)where k(9 ; m) denotes the equilibrium state trajectory with m < n contributors. If (5) and (6)are both satisÖed then the decisions made at t = 0 by the contributors and the free riders will betime-consistent, i.e., they will remain optimal (payo§-wise) at any future date.10
In what follows, we will consider two scenarios of contributions to the abatement of the stock ofpollution with Öxed cost of participation, namely, a non-coordination scenario and a coordination
scenario. In the non-coordination scenario, in Stage 2, contributors choose their contribution levels
taking the contributions of the others as given (i.e., as Nash players). We assume that contributors
8

These inequalities are standard in the tight coalitions literature (see DíAspremont et al., 1983; Barret, 1994) but
they have never been applied to loose coalitions.
9
The agent assumes that if he leaves group GC to join group GN , then (i) the number of contributors will becomem ! 1, i.e., the other m ! 1 contributors stay in GC ; and (ii) the remaining contributors will adjust their contribution
level to the new equilibrium level with m ! 1 contributors.
10
Note that in (5) and (6) the value functions are evaluated under the requirement that the number of contributors
is equal to m < n and constant from time t = 0 until the comparison time ) . Note also that the Öxed cost (paid atthe comparison time ) ) appears only in (6). This is so because at time ) the Öxed cost has to be paid only by a freerider who decides to join the group of contributors; a contributor who at time ) > t decides to join the group of freeriders cannot recover the Öxed cost that he paid at time t = 0.

6

use (stationary) feedback strategies, i.e., they condition their contributions at time t on the current
level of the stock of pollution, exclusively. In the coordination scenario, those who have chosen to
be contributors collectively decide on their common contribution level with the aim of maximizing
the discounted sum of their utilities.

3

Non-coordination Behavior

We consider Stage 2 decisions Örst. Under non-coordination, each contributor i takes the strategies
of other contributors as given. We must then solve for a non-cooperative feedback equilibriumof the dynamic game. Given a number of contributors m ! 1, non-cooperative feedback equilibrium strategies of a generic contributor have to satisfy the following Hamiltonian-Jacobian-Bellman(HJB) equation, where ViC (k; m) denotes the value function of a contributor, given that there arem contributors:nrViC (k; m) = max %bk 2 + a % xi % (a % xi )2
xi 2[0;a]&dViC (k; m)+(na % xi % X"i (k) % 3k) ,(7)dkP
where X"i (k) = m
h6=i xh (k), together with the usual transversality condition. Maximization of theright-hand side of (7) gives (for an interior solution):11'(
1
dViC (k; m)#xi (k) = a %
1+
.
(8)
2
dk
Proposition 1 The feedback equilibrium strategy of a contributor under non-cooperative behavior
is given by (i = 1; 2; :::; m)
1 + AC k + B C
x#i (k) = a %2#CCfor k such that xi (k) 2 (0; a), where A ; B < 0 are constants (given in the Appendix) that dependon the parameters of the model.
Proof. See Appendix A.
Corollary 1 The vector of strategies (x#1 (k); x#2 (k); :::; x#m (k)) induces a trajectory of k given by##k # (t) = kss+ e"-t (k0 % kss)# is the steady state of kwhere = > 0 is the speed of convergence (given in the Appendix) and kss
under non cooperative behavior, given by)*
m 1 + B C + 2a (n % m)#
kss =23 % AC mwhich is stable.

For an interior solution x!i (k) 2 (0; a) it must be that dViC (k)=dk 2 (!1; 2a ! 1). We will verify later thatconditions on the parameters of the model exist such that dViC (k)=dk 2 (!1; 2a ! 1) at any point in time, implyingthat the equilibrium trajectory of x!i remains between 0 and a throughout the entire planning horizon.
11

7

Proof. See Appendix B.
Two remarks are in order. First, the equilibrium strategy given in Proposition 1 is for aninterior solution, x#i (k) 2 (0; a). Given the purpose of our analysis, corner solutions are clearly notinteresting: when x#i (k) = 0 the distinction between contributors and free riders vanishes; whenx#i (k) = a abatement levels become state-independent and private consumption turns out to be nil.
Second, the equilibrium strategy given in Proposition 1 is linear in k, with a positive slope equal
to %AC =2 > 0. Thus x#i (k) is increasing in k: the higher the stock of pollution, the higher thecontribution to the abatement by agent i = 1; 2; :::; m, for any given m ! 1.12
Figure 1 below depicts the abatement strategy x#i (k) for the case in which x#i (0) > 0.13
[Insert Fig. 1 about here]
Figure 2 below illustrates the steady-state equilibrium. (The aggregate emission is measured
along the vertical axis.)[Insert Fig. 2 about here]
Let us now turn to the following question: does an increase in the number of non-cooperative# iscontributors lead to a lower long-run stock of pollution? Interestingly, the impact of m on kss
ambiguous, depending on the parameters of the model. For instance, if we evaluate the derivative# w.r.t. m at m = 1 we Önd that the sign depends on a. SpeciÖcally, if a is below a certainof kss# increases in m), otherwise it is negative.
threshold, b
a, then the derivative is positive (therefore kss
Surprisingly, having more contributors to the abatement of the public bad can make things worse(i.e., the long-run stock of the public bad is bigger when there is an additional contributor). The
intuition is as follows. We note that the smaller a is, the greater is the marginal loss of direct utility
caused by a given abatement level xi (because, for any given xi , the marginal utility of consuming
a % xi is higher when a is lower). Therefore, when a is small, a contributorís best reply to anincrease in the sum of contributions by others tends to be a big reduction in her own contribution.
This can lead to the result that an increase in m leads to a decrease in total abatement, and hence# .
an increase in kss
The impact of an increase in m (starting from m = 1) on the steady-state stock of pollution
is illustrated in Figure 3. As shown in Figure 3, when the number of contributors increases from# moves to the right (respectively, left) if a < b
1 to a higher number, the steady-state stock kss
a(respectively, a > b
a).
[Insert Fig. 3 about here]12

As is well-known in the di§erential game literature, the linear feedback strategy is only one of the inÖnitely many
feedback strategies that satisfy the di§erential equation resulting from di§erentiating the maximized HJB equation
w.r.t. the state variable. However, value functions associated with nonlinear feedback strategies can be obtained only
implicitly, whereas value functions associated with linear feedback strategies are polynomials of degree two, and canbe easily used for the derivation of the equilibrium number of contributors.
13
All the pictures in the paper are drawn for parameter values such that x!i (0) > 0. The deÖnition of k is given in
the Appendix (see Appendix A).

8

Remark 1 The above surprising result (i.e., an increase in m leads to a decrease in total abatement) cannot be obtained in a corresponding static version of the model. In the static version, withm contributors and n agents, contributor i chooses xi to maximize(a % xi ) % (a % xi )2 % b(k0 + na % X"i % xi )2
The FOC is%1 + 2(a % xi ) + 2b(k0 + na % X"i % xi ) = 0
Let x#i;m denote the symmetric Nash equilibrium, then
%1 + 2(a % x#i;m ) + 2b(k0 + na % mx#i;m ) = 0
It follows that
x#i;m =

2b(k0 + na) + 2a % 1
2b(k0 + na) + 2a % 1
and x#i;(m+1) =(2bm + 2)(2b(m + 1) + 2)

Clearly, an increase in m will always increase total abatement and therefore decreases the pollution:
mx#i;m =2b(k0 + na) + 2a % 1
2b(k0 + na) + 2a % 1<= (m + 1)x#i;(m+1) :2
2
2b + m2b + m+1

In view of the above Remark, one may ask why there is no static counterpart to the result(obtained in our dynamic game model) that a bigger m can lead to increased pollution. The key
to the answer is that in a dynamic model, each agent expects that if he increases his emission
today, the pollution stock will be bigger tomorrow, which would in turn induce other agents toemit somewhat less tomorrow than otherwise; this dynamic strategic consideration may give him
an incentive to undertake less abatement today when he learns that the number of contributors has
increased. In a static model, by deÖnition, such dynamic strategic considerations do not exist.
The impact of m on x#ss is also ambiguous: the derivative of x#ss w.r.t. m evaluated at m = 1
is positive if a is below a certain threshold, ea, otherwise it is negative. One can verify that ba>ea.
# and x# increase in m; if e
# increases
Starting from m = 1, if a < ea then both kss
a
<
a
<
b
a
then
k
ss
ss
# and x# decrease in m (see Appendix C).
and x#ss decreases in m; if a > b
a then both kss
ss
How does m impact on the speed of convergence, =? We Önd that = is increasing in m for everym ! 1: more contributors imply faster convergence to the steady state.
Using x#i given in Proposition 1 we can compute the value function of a free rider, which has tosatisfy the following HJB equation:N

rVjN (k; m) = %bk 2 + a(1 % a) +dVj (k; m)(na % mx#i (k) % 3k) .
dk

It can be checked that VjN (k; m) = AN k 2 =2 + B N k + E N , with AN < 0, B N < 0, and E N given in
the Appendix (see Appendix D).
9Let us now turn to the Stage 1 decision and the investigation of the existence and uniqueness(or otherwise) of the equilibrium number of contributors under non-coordination, m# . Clearly, the(m)contributor-rationality condition (3) is satisÖed if and only if F $ F , where
F

(m)

$ F
& ViC (k0 ; m) % VjN (k0 ; m % 1)
- C.
.
= A (m) % AN (m % 1) k02 =2 + B C (m) % B N (m % 1) k0 + E C (m) % E N (m % 1) , (9)and the free-rider-rationality condition (4) is satisÖed if and only if F ! F (m) , whereF! F (m) & ViC (k0 ; m + 1) % VjN (k0 ; m).
.
= AC (m + 1) % AN (m) k02 =2 + B C (m + 1) % B N (m) k0 + E C (m + 1) % E N (m) .(10)Interestingly, the curve F (m) is graphically a horizontal left-ward displacement of the curve
F , because of the following property:(m)Lemma 1 F(m)

= F (m"1) .

Proof. By deÖnition.
(m)
If both F
and F (m) are monotone decreasing in m (in the Örst quadrant) then the graph of(m)(m)
the curve F (m) lies below that of the curve F , with F> F (m) > 0 for all m 2 [0; me % 1] and
(m)
(m)
e
(m)
F
>F
< 0 for all m 2 (me % 1; m),e with me being the unique real number such that F=0(m)(assuming that Fintersects the horizontal axis). Let bmc
e denote the largest integer that is lessthan or equal to m.
e Assume that n > bmc.
eDenote by S the set of bmc
e critical values of Öxed costat which the constraint (3) is binding:
n
o
(1)
(2)
(bmc)
e
S & F ; F ; :::; F
.
(s)

(s+1)

Clearly, if the Öxed participation cost F is such that F> F > Ffor some integer s 2
f1; 2; :::; bmcg,
ethen the unique equilibrium conÖguration of contributors and free riders (m# ; n%m# )
(s)
(s+1)
(s)
is (s; n%s). To see this, note that, by Lemma 1, F > F > F
is equivalent to F > F > F (s) ,which means that both rationality constraints (3) and (4) are satisÖed, and therefore (s; n % s) is
an equilibrium conÖguration. Uniqueness follows from the fact that if F satisÖes the condition(s)(h)
F > F > F (s) , then F cannot satisfy F! F ! F (h)h for any otheri integer h 6= s.
(1)
(bmc)
e
It follows that for almost every F in the real interval F ; F, there is a unique equilibrium
conÖguration of contributors and free riders. The only cases in which non-uniqueness of equilibrium
could arise are when F happens to coincide with one member of the set S deÖned above. In such(s)cases, there are exactly two equilibria. More precisely, for s = 1; 2; :::; bmc,
e if F = F , thenthere are exactly two equilibrium conÖgurations, (s; n % s) and (s % 1; n % (s % 1)). At the former
equilibrium, the rationality constraint of contributors is binding. At the latter one, the rationality
constraint of free riders is binding.
The above discussion can be summarized by the following proposition:
10

!

(m! )

Proposition 2 In the non-cooperative scenario, if F (m ) < F < Fthen there exists a unique##equilibrium in which m agents contribute and n % m agents free ride, with n > bmc,
e bmc
e being
the largest integer that is less than or equal to m.
e(m)Corollary 2 If F
> 0 for all m ! 0 or n < bmc,
e conditions on the parameters of the model#
exist under which m = n, i.e. there are no free riders.

The following numerical example illustrates Proposition 1: Let the parameter values be a =0:4825, b = 0:012, n = 10, 3 = 0:5, k0 = 0, r = 0:1 and F = 0:045. Can m = 10 be the equilibrium(10)
number of contributors? As F (10) = 0:0423 and F= 0:0499, clearly if F 2 (0:0423; 0:0499) then#14m = 10.
One can also check that m# = 10 satisÖes the time-consistency condition (5), since ViC (k(9 ); 10)%N
Vj (k(9 ); 9) = 0:0616 + 0:0029e"1:1877/ % 0:0146e"0:5939/ > 0 for all 9 2 [0; 1). Therefore, the decisions made at t = 0 by all agents to join group GC are time-consistent.

4

Cooperative Behavior

We now turn our attention to the cooperative scenario. We need to derive ViC (k0 ; mjcoop) andVjN (k0 ; mjcoop) under the assumption that those who contribute must act cooperatively, i.e., theymust coordinate their contributions. Under cooperation among contributors, the objective functional of contributor i is given by
(
'
( '
( )
Z1
X (t)
X (t) 22C"rt
Ji (k0 ; mjcoop) & e%b [k (t)] + a %
% a%
dt,
m
m
0

and that of non-contributor j is given by (2), where X (t) =m denotes the coordinated contributionof the representative contributor i.
As in the previous section, we consider Stage 2 decisions Örst. Given a number of contributors
m ! 1, equilibrium strategies of a generic contributor under cooperative behavior have to satisfy
the following HJB equation:
(
)
3
4
X
X 2 dViC (k; mjcoop)C2
rVi (k; mjcoop) = max%bk + a %% a%+(na % X % 3k) ,mm
dk
X2[0;na](11)
together with the usual transversality condition. Maximization of the right-hand side of (11) gives(for an interior solution):15'
3
4(
1
dV C (k; mjcoop)
X =m a%
1+m i
.
(12)
2
dk
One can check that x! (0) = 0:0428, and limt!1 x! (t) = 0:1123, thus a ! x! (t) > 0 for all t 2 [0; 1).
C
For an interior solution x!!
i (k) 2 (0; a) it must be that dVi (k; mjcoop)=dk 2 (!1=m; (2a ! 1)=m). We will verify
later that conditions on the parameters of the model exist such that dViC (k; mjcoop)=dk 2 (!1=m; (2a ! 1)=m) at any
point in time implying that the equilibrium trajectory of x!!
i remains between 0 and a troughout the entire planninghorizon.
14
15

11

Proposition 3 The feedback equilibrium strategy of a contributor under cooperation is given by(i = 1; 2; :::; m))*
1 + m EC k + F C##
xi (k) = a %2CCfor k such that x##
i (k) 2 (0; a), where E ; F < 0 are constants (given in the Appendix) that dependon the parameters of the model.

Proof. See Appendix E.
####
Corollary 3 The vector of strategies (x##1 (k); x2 (k); :::; xm (k)) induces a trajectory of k given by####
k ## (t) = kss+ e"2t (k0 % kss)## is the steady state of k underwhere G is the speed of convergence (given in the Appendix) and kss
cooperative behavior, given by)*
m 1 + F C + 2a (n % m)##
kss =23 % EC mwhich is stable.
Proof. See Appendix F.
Since EC < 0, clearly x##
i (k) is increasing in k: the higher the stock of public bad, the higher thecontribution to the abatement. This is the same as under non cooperation. It can be veriÖed that#mEC < AC for all m > 1, implying that x##
i (k) is steeper than xi (k). As the public bad increases,
the response of contributors in the cooperative scenario is stronger than that of their counterpartsin the non-cooperative scenario.
Figure 4 below shows that, if the number of contributors is greater than 1 and is the same undercooperation and under non-cooperation, then the long-run pollution stock under non-cooperation# > k ## .16
is higher than under cooperation, i.e., kss
ss[Insert Fig. 4 about here]
Would an increase in the number of cooperative contributors result in a lower long-run stockof pollution? Surprisingly, the answer is: ìIt depends on parameter values.î Indeed, the derivative## w.r.t. m evaluated at m = 1 is positive or negative, depending on whether the followingof kssexpression is positive or negative:
4(a) & % f2ab (2n % 1) + 3 (2a % 1) (3 + r) + bg .
Notice that 4(a) is decreasing in a and is equal to zero at a = a, where
a=
16

1
n
%
> 0 if (3=b) (3 + r) > 1.
2 2n + [(3=b) (3 + r) % 1]

The deÖnition of k is given in the Appendix (see Appendix E).

12

## increases (resp. decreases) in m (starting from m = 1). Similar
Thus, if a < a (resp. > a) then kss
to the non-cooperative case, we Önd that in the cooperative case, a surprising result can arise: if
a and b are small, having more contributors can lead to a greater stock of public bad in the longrun. It is instructive to compare the threshold a of the cooperative case with the threshold b
a ofthe non-cooperaive case. One can verify that b
a > a. This implies that there exists an interval of a,(i.e. a < a < ba) where an increase in m (starting from = 1) leads to an increase in the steady-state
stock of public bad under non-cooperation, but a decrease in the steady-state stock of public badunder cooperation.
The comparative steady-state analysis w.r.t. m is summarized in Figure 5.

[Insert Fig. 5 about here]
As in the case of non-cooperation, the speed of convergence, G, increases in m for every m ! 1.
By comparing G with =, we see that, in general, G can be either higher or lower than =. However,
if r ! 0, we have that = > G. By continuity, = > G for r small enough. If agents are su¢ciently
patient then convergence is faster under non-cooperation than under cooperation.
Using x##i given in Proposition 3 we can compute the value function of a free rider, which has
to satisfy the following HJB equation:()
dVjN (k; mjcoop)N2##(na % mxi (k) % 3k) .
rVj (k; mjcoop) = %bk + a(1 % a) +dk
We can show that VjN (k; mjcoop) = EN k 2 =2+F N k+HN , with EN , F N , and HN given in the Appendix(see Appendix G).
Let us now turn to the investigation of the existence and uniqueness (or otherwise) of theequilibrium number of contributors under cooperation.
The following inequalities are the counterparts of inequalities (9) and (10) for the case of cooperation among contributors:
(m)

F

$ F
& ViC (k0 ; mjcoop) % VjN (k0 ; m % 1jcoop)- C.
.
= E (m) % EN (m % 1) k02 =2 + F C (m) % F N (m % 1) k0 + HC (m) % HN (m % 1) , (13)F

! F (m) & ViC (k0 ; m + 1jcoop) % VjN (k0 ; mjcoop).
.
= EC (m + 1) % EN (m) k02 =2 + F C (m + 1) % F N (m) k0 + HC (m + 1) % HN (m) . (14)The following Lemma is the counterpart of Lemma 1 for the case of cooperation among contributors.
Lemma 2 F(m)

= F (m"1) .

Proof. By deÖnition.
13

If both F(m)and F (m"1) are monotone decreasing in m (in the Örst quadrant), and if there

exists an mb > 0 such that FF(m), with F

(m)

(m)
b

= 0, then the graph of the curve F (m) lies below that of the curve> F (m"1) > 0 for all m 2 [0; mb % 1] and F!!

(m)> F (m"1) < 0 for all m 2 (mb % 1; m).
b(m!! )

Proposition 4 In the cooperative scenario, if F (m ) < F < Fthen there exists a unique####equilibrium in which m agents contribute and n % m agents free ride, with n > bmc,
b bmc
b being
the largest integer that is less than or equal to m.
b(m)Corollary 4 If F
> 0 for all m ! 0 or n < bmc,
b conditions on the parameters of the model
exist under which m## = n, i.e. there are no free riders.

As in the non-cooperative scenario, there are two cases in which the equilibrium number of!!
!!
contributors is non-unique: (i) F = F (m ) , with F (m ) > 0, in which case there exist two equilibria,(m!! )(m!! )

m## and m## + 1; (ii) F = F, with F> 0, in which case there exist two equilibria, m##
and m## % 1.
The following numerical example illustrates Proposition 2: a = 0:4825, b = 0:012, n = 10,
3 = 0:5, k0 = 0, r = 0:1, and F = 0:045. Can m = 2 be the equilibrium number of contributors in(2)

the cooperative case? We get F (2) = 0:0289 and F = 0:4518, implying that if F 2 (0:0289; 0:4518)
then m## = 2.17 One can also check that m## = 2 satisÖes the time-consistency condition (5),since ViC (k(9 ); 2) % VjN (k(9 ); 1) = 0:4678 + 0:0008e"2:3515/ % 0:0167e"1:1758/ > 0, and (6), sinceF = 0:045 > 0:0313+0:0002e"2:3515/ %0:0026e"1:1758/ for 9 2 [0; 1). Therefore, the decisions madeat t = 0 by two agents to join group GC and by eight agents to join group GN are time-consistent.
Figure 6 below depicts the four thresholds of F , namely F , F , F and F (for the parametervalues used in the previous numerical examples).
[Insert Fig. 6 about here]
As can be seen in Fig. 6, suppose we draw a horizontal line representing a give Öxed cost level F
and suppose this line intersects the vertical line (representing a given integer m, say m = 2) at apoint in the region CDGF then we can deduce that the number of contributors in the cooperative
and the non-cooperative scenarios is the same m (see for instance point H in Fig. 6).18 Outside
this region, the numbers of contributors in the two scenarios are di§erent from each other: if thehorizontal line representing F is above C then m## > m# ; if F is below E then m## < m# . For
F = 0:045, the number of contributors is 10 under non-cooperation (see point B in Fig. 6) and 2
under cooperation (see point A in Fig. 6).
17
One can check that x!! (0) = 0:1415, and limt!1 x!! (t) = 0:3176, so that a ! x!! (t) > 0 for all t 2 [0; 1).
Moreover, limt!1 k!! (t) = 8:3796. Note that, compared with the non-cooperative case, where m! = 10, we havelimt!1 x!! (t) > limt!1 x! (t) but, because m!! = 2 < 10, limt!1 k!! (t) > limt!1 k! (t).
18

For a = 0:4825, b = 0:012, n = 10, 5 = 0:5, k0 = 0 and r = 0:1 we get F (2) = 0:167 and F

F = 0:2 then m! = m!! = 2 (recall that F (2) = 0:0289 and F

14

(2)

= 0:4518).

(2)= 0:2023. Hence, if

The trajectories of k # (t) and k ## (t) (for the parameter values used in the previous numericalexamples) are depicted in Figure 7.
[Insert Fig. 7 about here]
As shown in Figure 7, under the parameter values stated above, the stock of public bad turns
out to be higher under cooperation than under non-cooperation not only at the steady state but
also at any point in time during the transition phase. This is so because the free riding incentive isstronger when potential free-riding agents know that the active contributors will coordinate theircontributions rather than behaving as Nash players.
Finally, Figures 8a and 8b below plot the trajectories of individual and total contributions in thecooperative and non-cooperative scenarios (for the parameter values used in the previous numericalexamples).
[Insert Fig. 8 about here]
Individual contributions are higher under cooperation than under non cooperation at any
point in time (see Fig. 8a). However, the number of contributors in the non cooperative scenario(m# = 10) exceeds that in the cooperative scenario (m## = 2), and total contribution turns out tobe higher in the former than in the latter (see Fig. 8b).

5

Welfare Comparison between Cooperation and Non-cooperation

Welfare with m contributors and n % m free riders is given by
W # (k0 ; m; n) = mViC (k0 ; m) + (n % m)VjN (k0 ; m) % mF ,(15)

if the contributors do not coordinate their contributions, and by
W ## (k0 ; m; n) = mViC (k0 ; mjcoop) + (n % m)VjN (k0 ; mjcoop) % mF ,(16)

if the contributors act cooperatively. Equations (15) and (16) make use of the value functions andtherefore measure the discounted sum of (net) utilities. From (15) and (16), using Corollaries 1 and
2, we can derive the steady-state welfare levels in the cooperative and non-cooperative scenarios,## and W # , respectively, which we will use later in this section.
W1
1
While the equilibrium number of contributors under non-coordination is in general not thesame as that under coordination, it turns out to be useful to deÖne, as an intermediate step in our
analysis, for the same common m, the di§erence 6W & W ## % W # , with.
6W = m ViC (k0 ; mjcoop) % ViC (k0 ; m).
+(n % m) VjN (k0 ; mjcoop) % VjN (k0 ; m) .
It can be checked that the term between the Örst set of square brackets is positive. The term
between the second set of square brackets can be rewritten as
Z 1
n
o
b
e"rt [k # (t)]2 % [k ## (t)]2 dt,
015

since the consumption level of the non contributors is the same irrespective of whether the contributors cooperate or not. Under the hypothesis that the number of contributors is the same undercooperation and under non-cooperation, it is easy to verify that k # (t) > k ## (t) for any t, implyingthat the above integral is positive. We can then state the following (unsurprising) proposition:
Proposition 5 For any given common m ! 1, it holds that welfare is higher under cooperation
than under non cooperation.
The above proposition establishes that if the equilibrium m is the same, then the welfare
outcome when the contributors coordinate their contributions is superior to the outcome under
non-cooperation. However, when the endogenous number of contributors under non-cooperation islarger than that under cooperation, it is possible that non-cooperation may lead to higher welfare
than cooperation. The following example illustrates this interesting result.
Numerical Example: a = 0:4825, b = 0:012, n = 10, 3 = 0:5, k0 = 0, r = 0:1. Under theabove parameter values, we determined before that m# = 10 and m## = 2. If F = 0:045 we Öndthat welfare when the contributors cooperate is smaller than under non-cooperation:
W ## (2; 10jF = 0:045) = %42:1706 < W # (10; 10jF = 0:045) = %28:842.
Going beyond numerical examples, let us formalize a key result of our paper:
Proposition 6 (Cooperation can reduce welfare when the number of free riders is endogenous) Take parameter values such that m# > m## . If m# % m## is large enough then there
exists an interval of F such that W ## (m## ; n) < W # (m# ; n).
Proof. See Appendix H.
For the parameter values in the numerical example above, the set of admissible Öxed costs suchthat m# = 10 and m## = 2 is given by [0:0289; 0:0499]. Any F 2 [0:0289; 0:0499] would lead to
W ## (2; 10) < W # (10; 10).
In the remainder of this section, we will compare the steady-state welfare level under cooperation
with that under non-cooperation. DeÖne
.
##
#
##
#
6W1 & W1
% W1
= m ViC (k1; mjcoop) % ViC (k1
; m)
.
##
#
+(n % m) VjN (k1; mjcoop) % VjN (k1; m) .

It can be checked that 6W1 > 0. Hence, for the same number of contributors, we obtain the same
qualitative result as that previously obtained with the discounted welfare measures. In general,
the equilibrium number of contributors under non-cooperation di§ers from that under cooperation.
Using the parameter values of the numerical example above we Önd that##
W1
(2; 10jF = 0:045) = %61:6167 < W # (10; 10jF = 0:045) = %42:9171,

which is in line with the discounted welfare comparison.
In conclusion, we have shown that there exists parameter values such that cooperation is welfarereducing, both at the steady state and along the transition path.
16

6Concluding Remarks

We have analyzed a dynamic game of voluntary abatement of a public bad with Öxed cost ofparticipation, in which the number of contributors is endogenously determined. One of our striking
results is that cooperation among contributors can worsen social welfare and lead to a higher
steady-state level of public bad compared to Nash behavior.
When the number of contributors is the same in the cooperative and the non-cooperative settings, social welfare turns out to be higher if contributors coordinate on their contribution levels
rather than choosing their contribution levels as Nash players. However, our analysis has shownthat the number of contributors in the cooperative and non-cooperative settings may di§er significantly. Conditions on the parameters of the model exist such that the incentive to free ride ishigher and, consequently, the number of contributors is lower in the cooperative setting, comparedto the non-cooperative one, leading to a lower social welfare in the former than in the latter. A
policy implication of this Önding is that insisting on a tight coalition is not necessarily welfare
improving. In the context of international environmental agreements, our analysis suggests thatan accord in the spirit of COP 21, i.e., the Paris Accord, might be able to bring more countries
on board, leading to higher total contribution to the abatement of the public bad compared to anagreement imposing speciÖc targets for each country, which is what the Kyoto Protocol was trying
to achieve.
Possible extensions of our framework include: (i) relaxing the assumption of full cooperationby considering a coe¢cient of cooperation ranging from zero to one (see, for instance, Vives, 2008;
Colombo and Labrecciosa, 2018); (ii) allowing for agents heterogeneity (see McGinty, 2007; Pavlova
and de Zeeuw, 2013); (iii) allowing for asymmetric information about agentsí characteristics (seeBagwell and Staiger, 2005; Amador and Bagwell; 2013); (iv) accounting for uncertainty and ambiguity about the evolution of the stock of public bad (see, for instance, Lemoine and Traeger, 2014,2016).
Acknowledgment: Part of this research was carried out at Hitotsubashi Institute for Advanced
Study, Hitotsubashi University.

17

Figuresax Di
2a?1?B C
2

0

k

k

Figure 1: Equilibrium strategy of a contributor

Nk

an ? mx Di

a›n ? mﬁ

k Dss

0

k

k

Figure 2: Steady-state equilibrium

Â
a<a

Nk
an ? x Di |m=1

Âa>aa›n ? 1ﬁa›n ? mﬁ

0

k Dss

k

k

#
Figure 3: The impact of an increase in m on kss

18

Nk
an ? mx Di
an ? mx DD
ia›n ? mﬁDk DD
ss k ss

0

k

k

k

Figure 4: Steady-state comparison for a given m

/k DD
ss
/m

m=1

/k Dss
/m

m=1

/k DD
ss
/m
/k Dss
/m

>0
>0

m=1
m=1

a

0

<0

/k DD
ss
/m

>0

/k Dss

Â
a

/m

m=1
m=1

<0
<0

a

Figure 5: Comparative steady-state analysis w.r.t. m

0. 247

F

C

0. 2

D

H
G

E
F

F

0. 045

A
1

2

F

B
10 m

3

Figure 6: determination of m# and m##

19

k DD
ss

k Dss
k DD ›tﬁ

k D ›tﬁ

t0

Figure 7: state trajectories comparison (m# = 10, m## = 2)

x DD
ss

0. 3176m D x Di ›tﬁ

x iDD ›tﬁ

x Dss

0. 1123m DD x DD
ss

0. 635 2mDD x DDi ›tﬁ

x Di ›tﬁ

0D
m D x ss

1. 1230tFigure 8a: individual contributionstFigure 8b: total contributions

20

Appendix
Appendix A: Proof of Proposition 1
Inserting (8) into (7) yields (under symmetry)
(
' C
(2 )
1
dV
(k;
m)
i
rViC (k; m) =1 % 4bk 2 %
4
dk
5
'
(
&
dViC (k; m)mdViC (k; m)+an +1 % 2a +% 3k .
dk
2
dk

(A.1)

Given the linear quadratic structure of the game at hand we guess a value function of the form
k2
+ BC k + EC ,
(A.2)
2
q
where AC , B C , and E C are constants. Let 6 = 4b (2m % 1) + (23 + r)2 . It can be checked that
(A.2) with
23 + r % 6
AC =
< 0,
2m % 1
ViC (k; m) = AC

BC =
and

AC (2a (n % m) + m)
< 0,
AC (1 % 2m) + 2 (3 + r).
B C 2m(1 % 2a) + 4an + B C (2m % 1) + 1
E =
,
4r
C

satisÖes (A.1) for any k 2 (k; k), with k = (2a % 1 % B C )=AC and k = %(1 + B C )=AC . For k 2= (k; k)####we have a corner solution, either xi = 0 or xi = a. SpeciÖcally, xi = 0 for k < k and xi = a fork > k. Note that k $ 0 if a is su¢ciently close to 1=2; in that case x#i (k) > 0 for all k > 0.

Appendix B: Proof of Corollary 1
The equilibrium trajectory of the stock of pollution under no cooperation, k # (t), is the solution of
the following Örst-order di§erential equation:dk(t)= an % mx#i (k(t)) % 3k(t),dtwith x#i (k) given in Proposition 1. It can be checked that##k # (t) = kss+ e"-t (k0 % kss
),

where
==

23 (1 % m) % m (6 % r)
>0
2 (1 % 2m)# , with k # given in Corollary 1 (the expression of 6 is given inis the speed of convergence to kssss# .
Appendix A). It is immediate to verify that limt!1 k # (t) = kss

21

Appendix C: Thresholds of a
# w.r.t. m evaluated at m = 1 is decreasing in a and nil at a = b
The derivative of kss
a, withba=%2b2 + b (3 + r) ( 6jm=1 % 23 % r) % 3 (3 + r)2 ( 6jm=1 % r)no.
2 2b2 (n % 1) + b (3 + r) [(1 % 2n) ( 6jm=1 % r) + 23 (n % 1)] % 3 (3 + r)2 ( 6jm=1 % r)

The derivative of x#ss w.r.t. m evaluated at m = 1 is decreasing in a and nil at a = e
a, withea=

3 (43 + 3r % 6jm=1 ).
2 f2bn % 3 [23 (n % 2) + r (n % 3) % (n % 1) 6jm=1 ]g

Appendix D: Coe¢cients of VjN (k; m)
The coe¢cients of the value function of a non contributor under non cooperation are given by:
AN =

B

N

2b (1 % 2m)
m6 + (23 + r) (m % 1)'("1m(6 % (23 + r))=+3+r
2(2m % 1)'(m
N m(2a(m % n) % m)(2b(2m % 1) % (3 + r)(6 % (23 + r))).A+ a(n % m) +4(2m % 1)(b(2m % 1) + 3(3 + r))
2
EN

h
m
= r"1 a(1 % a) + aB N (n % m) + B N
2(mB N (2a(m % n) % m)(2b(m % 1) % (3 + r)6 % (23 + r))+
4(2m % 1)(b(2m % 1) + 3(3 + r))

with 6 given in Appendix A.

Appendix E: Proof of Proposition 2
Inserting (12) into (11) yields
rViC (k; mjcoop)=5
dV C (k; mjcoop)
1 % 4bk 2 + i
dk
'
(&
C
2 dVi (k; mjcoop). 2m (1 % 2a) % 4k3 + 4an + m
.
dk
1
4

(E.1)

As in the non cooperative case, we guess a value function of the form
ViC (k; mjcoop) = EC

k2
+ F C k + HC ,
2

(E.2)

q
where EC , F C , and HC are constants. Let 9 = 4bm2 + (23 + r)2 . It can be checked that (E.2)
with
23 + r % 9
EC =
< 0,
m2
22

FC =
and
HC

EC [2a(m % n) % m]
< 0,
EC m2 % 2 (3 + r)hm
= r"1 a(1 % a) + aB N (n % m) + B N
2(mB N (2a(m % n) % m)(2b(m % 1) % (3 + r)6 % (23 + r))+,
4(2m % 1)(b(2m % 1) + 3(3 + r))

satisÖes (E.1) for any k 2 (k; k), with k = ((2a % 1)=m % F C )=EC and k = %(1=m + F C )=EC . For####
k2= (k; k) we have a corner solution, either x##
i = 0 or xi = a. SpeciÖcally, xi = 0 for k < k and##x##
i = a for k > k. Note that k $ 0 if a is su¢ciently close to 1=2; in that case xi (k) > 0 for all
k > 0.

Appendix F: Proof of Corollary 2
The equilibrium trajectory of the stock of pollution under cooperation, k ## (t), is the solution of thefollowing Örst-order di§erential equation:dk(t)= an % mx##i (k(t)) % 3k(t),dtwith x##i (k) given in Proposition 2. It can be checked that####k ## (t) = kss+ e"2t (k0 % kss
),

where

9%r
>0
2
## , with k ## given in Corollary 2 (the expression of 9 is given inis the speed of convergence to kssss## .
Appendix E). It is immediate to verify that limt!1 k ## (t) = kss
G=

Appendix G: Coe¢cients of VjN (k; mjcoop)
The coe¢cients of the value function of a non contributor under cooperation are given by:
EN = %
FN = %2b< 0,
9

EN (3 + r) [m (2a % 1) % 2an] (9 % r)
< 0,
2 [bm2 + 3 (3 + r)] (9 + r)

and
HN

h
m
= r"1 a(1 % a) + aF N (n % m) + F N
278#
B N [2a(m % n) % m] 2bm2 % (3 + r) [9 % (23 + r)]+,
4 [bm2 + 3(3 + r)]

with 9 given in Appendix E.
23

Appendix H. Proof of Proposition 6
The di§erence W ## (m## ; n) % W # (m# ; n) is increasing in F (for m# > m## ) and equal to zero atFe, whereFe =1

m# % m##7

m# ViC (k0 ; m# ) + (n % m# )VjN (k0 ; m# ).8
% m## ViC (k0 ; m## jcoop) + (n % m## )VjN (k0 ; m## jcoop) ,which is positive if m# % m## is large enough. If F 2 F \ F # \ F ## , with F & fF : 0 $ F $ Feg,
!

(m! )

!!

(m!! )

F # & fF : maxf0; F (m ) g < F < F
g, and F ## & fF : maxf0; F (m ) g < F < Fg,then the equilibrium number of contributors is m# in the non-cooperative scenario and m## in thecooperative scenario, and W ## (m## ; n) < W # (m# ; n). For the parameter values in the numericalexample in Section 5, we obtain Fe = 1:7110. Therefore, any F 2 [0:0289; 0:0499] would lead to
W ## (2; 10) < W # (10; 10).

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