A SIMPLE DYNAMIC MODEL OF INTERNATIONAL
ENVIRONMENTAL AGREEMENTS WITH A STOCK POLLUTANT
SANTIAGO RUBIO
(University of Valencia)
AND
ALISTAIR ULPH
(University of Southampton)
Very Preliminary Version
November 2001
ABSTRACT
Much of the literature on international environmental agreements uses purely static models of
transboundary or global pollution, despite the fact that the important problems to which this
literature is addressed involve stock pollutants. The few papers that study IEAs in the context
of dynamic models of stock pollutants do not allow for the possibility that membership of the
IEA may change endogenously over time in response to the dynamics of the stock pollutant. In
this paper we set up a very simple two-period model of a stock pollutant with symmetric
countries, but where countries are limited to two actions (pollute or abate). We study an openloop equilibrium, where IEA membership is determined at the outset and then countries
choose their emissions over the two periods, and a feedback equilibrium, where at the start of
each period, countries have to decide both whether to join an IEA for that period and whether
to abate or pollute. In this simple context we show that: (i) the open-loop model may have two
stable IEAs, the most common of which IEA members abating only in the first period; (ii) the
feedback equilibrium involves membership rising sharply over the two periods; (iii) one needs
to model carefully how membership decisions get linked over time; (iv) the feedback
equilibrium may have no stable IEA in period 1 but significant membership in period 2;
generally welfare is higher in the feedback equilibrium than in the open-loop. We suggest that
some of these results may not be robust to extension beyond two periods, in particular that the
feedback equilibrium may involve IEA membership falling over time. Nevertheless the
important point is that the dynamics of IEA membership may be closely linked to the
dynamics of the stock pollutant.
Key words: international environmental agreement, stock pollutant, stable
agreements, open-loop equilibrium, feedback equilibrium
JEL Classification: F02, F18, Q20
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1. Introduction.
There is now an extensive literature on international environmental agreements (see Barrett
(2001) and Finus (2001) for excellent recent books summarising this literature). Yet for the
most part this literature works with simple static models of pollution despite the fact that the
important problems (climate change, ozone depletion, acid rain) which this literature seeks to
shed light on involve stock pollutants. Does this matter? The only way to find out is to see
what happens when one works with models of dynamic pollutants, and there are now some
papers which do this.
Rubio and Casino (2001) extend the simple Barrett (1994) model of IEA formation to allow
for a stock pollutant. The model is analysed in two stages: countries first of all decide whether
or not to sign an IEA; then countries choose their paths of emissions. These emission paths are
calculated by solving a differential game in either open-loop or feedback strategies assuming
that the IEA signatories act to maximise their joint welfare while non-signatories just
maximise individual welfare. Unlike the static model they argue that there is no difference
between the case where the IEA signatories act in Cournot or Stackelberg fashion with respect
to the non-signatories. Having solved for the emission paths and hence evaluated payoffs to
signatories and non-signatories, Rubio and Casino then ask how many countries will want to
sign the IEA, using the stability analysis familiar from the work of Barrett (1994), Carraro and
Siniscalco (1993) etc. Rubio and Casino derive a rather more pessimistic result than the static
framework. In the static framework Barrett (1994) showed that when gains to cooperation are
high few countries will join an IEA. Rubio and Casino show that the number of signatories
will be 2 no matter what the gains to cooperation are. However, the reason for this more
pessimistic result is that they impose a restriction on parameter values to ensure that the
signatories will produce strictly positive emissions in steady state for any size of IEA, and
these parameter values drive down the size of the stable IEA. But it may be appropriate that for
some numbers of signatories, the optimal policy would involve signatories producing zero
emissions in steady-state, and it would be interesting to check whether relaxing this constraint
would allow a bigger range of stable IEAs.
Germain, Toint, Tulkens and de Zeeuw [GTTZ] (2000) extend the framework of Chander and
Tulkens (1995) to a dynamic model of a stock pollutant. As is now well understood the
Chander and Tulkens approach, based on core concepts, is able to get much more optimistic
conclusions about IEA formation, with the grand coalition being formed, because they assume
that if one country defects from the grand coalition all countries will revert to non-cooperative
behaviour. This punishment is sufficiently severe to deter defections. By contrast the stability
analysis of Barrett and others assumes that if one country defects, the remaining countries will
act to optimise their joint interests; this is less damaging to a defect and so encourages fewer
countries to sign. With asymmetric countries it will also be important to use income transfers
to ensure the stability of the grand coalition and Chander and Tulkens calculated the transfers
needed to ensure this stability. GTTZ(2000) extend this analysis by calculating the transfers
that will be needed to ensure stability of the grand coalition when there is a stock pollutant.
In both these cases, however, it would seem that the extension of the analysis from a static
model to a dynamic model of pollution is not really generating any fundamental new insights.
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So maybe it does not matter very much if our analysis of IEAs draws heavily on an essentially
static model. In this note we argue that this conclusion may be premature, because the papers
cited above do not really allow the dynamics of the pollution problem to affect the formation
of the IEA. This is seen most clearly in the Rubio and Casino model where countries
essentially have to commit at the outset to being a signatory or a non-signatory. The dynamics
of the pollution problem affects the calculation of the payoffs to signatories and nonsignatories but the game of whether or not to join the IEA is just as static as in the Barrett
framework. Moreover, while they analyse both open-loop and feedback strategies, with the
latter allowing countries to set their emissions as a function of the stock, countries are not
allowed to condition their decision on whether to be a signatory or a non-signatory on the
stock. So one cannot ask questions such as would the number of signatories increase or
decrease as the pollution stock increases. Equally in the GTTZ model, transfers are designed to
maintain the grand coalition, so again the question of whether the number of signatories may
change over time due to the pollution dynamics does not arise. At least in this case the fact that
the number of signatories does not change over time is a conclusion from the analysis rather
than an a priori assumption.
Both Rubio and Casino and GTTZ analyse models in which an IEA operates over the whole
life of a stock pollutant. A paper which is closer to the spirit of the issues we seek to address is
the paper by Karp and Sacheti (1997). They consider a two-period model of a stock pollutant
but assume that an IEA will only form in one of the periods – either just in the first, because
the IEA will fall apart at the end of one period, or in the second, because there may be
substantial delays in forming an IEA – and ask how the dynamics of the pollution problem
affect the incentives to join an IEA1 as well as by differences in the extent to which the
pollutant is local or global and the extent to which planners discount the future. They argue
that static models of pollution may overstate or understate the difficulties of forming an IEA
and that it is important to consider the dynamics of the pollution problem itself. This is in the
same spirit as our approach, except that we do not restrict an IEA to only last one period but
rather ask how the number of signatories may vary over time. However, we shall show that in
one formulation of the feedback equilibrium the most likely outcome is that there will be no
signatories to an IEA in the first period but significant numbers in the second period, but this is
determined endogenously. We shall also see that for the most likely equilibrium of the openloop model, although, by construction, there will be signatories to the IEA in both periods, in
the second period the IEA is ineffective – signatories do not abate pollution.
In this note we shall use the Rubio and Casino framework but now allow for the possibility
that in the feedback solution the number of signatories each period is determined as a function
of the stock of the pollutant in existence at the start of the period. Hence the number of
signatories may vary over time. This reflects the key point emphasised by Barrett (2001) that
the fundamental problem of IEAs is the essentially anarchic nature of international relations,
since national sovereignty allows countries the freedom to choose their actions, and if it is
their interest to join or leave an IEA they will do so. As Barrett shows, most IEAs do allow for
countries to join at different dates, and any punishment for leaving has to be built into the IEA
itself, i.e. IEAs must be self-enforcing. However the model we use is a drastically simplified
They also use a rather different approach to modeling an IEA as a “modest” perturbation on a non-cooperative
Nash equilibrium.
1
3
version of the Rubio and Casino model in which there are just two periods and we restrict
countries to just two actions. The reason for simplification is to make the solution of the
feedback equilibrium tractable. One problem is that the stable IEA is defined as an integer
number of signatories, which makes the value function in the feedback equilibrium nondifferentiable. Even if one ignores this integer problem, the value function with continuous
actions and endogenously determined IEA membership becomes very non-linear. In the earlier
literature on static models of IEAs determination of the stable IEA often resorted to numerical
solutions because payoff functions were non-monotonic. To allow us to identify the different
factors at work and to minimize the reliance on numerical solutions we have chosen to work
with a model of discrete actions. However, we view this as a first step towards a more general
analysis which will provide some guidance on modeling issues in a more general model. We
hope to report results on a model with continuous actions in a later paper.
Despite the simplicity of the model, we find some interesting results: (i) there can be multiple
stable IEAs in the open-loop model; (ii) the open-loop equilibrium usually involves only
partial cooperation by IEA signatories, in the sense that they cut emissions only in the first
period and set high emissions in the second period; (iii) in the feedback equilibrium the
number of signatories rises over time; (iv) for a wide range of parameter values, the number of
signatories in the feedback equilibrium and the level of abatement is at least as high in each
period as in the open-loop equilibrium; (v) there are some equilibria of the feedback model
where there is no stable IEA in period 1; this depends in part on assumptions about how
countries are assigned to membership of the IEA in period 2; (vi) for most (but not all)
parameter values, the feedback equilibrium generates higher welfare than in the open-loop
equilibrium, contrary to the result in standard non-cooperative differential game models of
stock pollutants (see e.g. Hoel (1992), van der Ploeg and de Zeeuw (1992)).
In the next section we set out the static version of the model. In section 3 we analyse the
simplest possible extension of the static model to capture a stock externality by assuming that
second period damage costs depend on the total stock of pollution at the end of that period,
which includes the stock of pollution inherited from period 1. We assume that unit damage
costs are not dependent on the pollution stock and solve for the open-loop and feedback
equilibria of this model. While the feedback equilibrium has a higher number of signatories in
period 2 than period 1, the number of signatories in period 2 is independent of the stock of
pollution carried over from period 1. While this is unsatisfactory the simplicity of the model
allows us to get sharp results. In section 4 we make a further extension to allow the unit
damage costs of pollution in period 2 to depend on the stock of pollution inherited from period
1, and solve for the open-loop and feedback equilibria. Again the number of signatories in
period 2 of the feedback model will be higher than in period 1, but now the number of
signatories depends (negatively) on the stock of pollution carried over from period 1. In
section 5 we speculate on how our results might be affected by weakening two key
restrictions: countries have only two actions, and there are only two periods. We sketch an
argument to show that in an infinite horizon model the feedback equilibrium could have the
number of IEA members declining over time as the stock of pollution increases. However this
does not contradict a basic message of this paper that there could be an important link between
the dynamics of accumulation of a stock pollutant and dynamics of IEA membership. Section
6 concludes.
4
2. The Simple Static Model.
There are N identical countries, indexed i, each of whom can choose two possible emission
levels, qi, low or high, which we normalise to take values 0,1 respectively. We denote
aggregate emissions by all countries by Q, and aggregate emissions by all countries other than
i by Q-i. The net payoff benefit function of country i is:
 i (qi , Qi )  bqi  cQ  bqi  c(Qi  qi )
Thus we are assuming constant unit benefits from own emissions, b, and constant unit costs of
aggregate emissions, c.
It is straightforward to see that we will have a Prisoner’s Dilemma in which every country has
a dominant strategy to set high emissions no matter what other countries do if   b/c >1, i.e.
holding constant other emissions, the unilateral benefit of an extra unit of emissions exceeds
the additional damage cost. In that case the non-cooperative equilibrium involves all countries
setting high emissions and getting net payoff b - Nc. If all countries act cooperatively then they
will set low emissions provided b – Nc < 0, i.e. N > , i.e. the benefit to each country of all
countries abating pollution exceeds the gain each would get from emitting a unit of pollution.
So in the cooperative outcome all countries set low emissions and get payoff 0. Thus the gains
from cooperation, the difference between payoff in the cooperative and non-coooperative
outcomes, are G = Nc – b, and so the gains are increasing in N and c and decreasing in b.
We now consider an IEA. Suppose n countries sign an IEA. We assume that the remaining N –
n countries act non-cooperatively and, given the dominant strategy, set high emissions. The n
signatories know this and will act to maximise their joint payoff. They will choose to each set
low emissions iff –c(N-n) > b – cN i.e n > . The interpretation is the similar to that above:
IEA members will agree to jointly abate pollution provided the benefit to each country from
reduced damage costs exceeds the benefit from a unit of emission. This requires that the IEA
exceed a critical minimum size: n*, defined as the smallest integer greater than . Note that
since  > 1, n*  2. So, if n  n*, IEA members will set low emissions and signatories and
non-signatories will get payoffs:  s (n)  c( N  n) and  f (n)  b  c( N  n) respectively; if n
< n*, signatories will set high emissions, and both signatories and fringe countries will get
payoffs:  s (n)   f (n)  b  cN . Note that for all n  f (n)   s (n) .
We now look for a self-enforcing or stable IEA. An IEA of size n is said to be stable iff:
 s (n )   f (n  1) and  s (n  1)   f (n ) . The first condition ensures that any IEA with fewer
than n signatories would not be stable because there would be incentives for fringe countries
to join the IEA to get a higher payoff. The second condition ensures that any IEA with more
than n signatories would not be stable since a signatory country would have an incentive to
leave the IEA. Then it is straightforward to see that:
Proposition 1: The unique stable IEA has n* signatories.
Proof:
The first stability condition is satisfied because:
5
 s (n*)  c( N  n*)   f (n * 1)  b  Nc, i.e. n*   . While the second stability
condition is satisfied because:  s (n * 1)  c( N  n * 1)  b  c( N  n*)   f (n*) , i.e.  > 1.
The first stability condition derives from the condition for the minimum size of IEA at which it
pays IEA members to abate pollution; this simply says you cannot have a stable IEA less than
because if any country left the IEA it would make an IEA worthless – all countries would
pollute – and it is this threat which stops any country leaving an IEA of size n*. On the other
hand the second stability condition is just the condition for unilateral free-riding, and says that
any IEA larger than n* is vulnerable to free-riding. So taken together what Proposition 1 tells
us is that, as long as countries realize that it will pay an IEA to continue to collectively abate
pollution, then individual free-riding pays. This acts to reduce any potential IEA from N to n*.
But having got the IEA down to n*, any further free-riding makes collective action to abate
pollution worthless, and this deters any further free-riding. So IEA stability is determined by
the minimum size of IEA at which collective action to abate pollution is worthwhile.
We have spelt out this argument at some length because it is this basic argument that we will
see applied in later analysis: the conditions for IEA stability are two-fold: the condition for
upward stability (what prevents the IEA getting bigger) is just the condition that makes
unilateral free-riding pay; the condition for downward stability (what prevents the IEA getting
smaller) is just the minimum size condition that ensures that it will pay the IEA to collectively
abate. Free-riding drives the size of the stable IEA down to its minimum viable size.
We now turn to comparative statics – how does the size of the stable IEA vary with
parameters? Note that n* is increasing in β and hence increasing in b and decreasing in c, and
hence stable IEAs will be smaller the greater are the gains from co-operation. In particular note
that since β can take any value between 1 and N, the stable IEA can take any value between 2
and N, so depending on the relative benefits and costs of emissions we can get any size of IEA
up to the grand coalition of all countries. This is the basic result from Barrett(1994).
3. IEA with a Stock Pollutant – Unit Damage Costs Independent of Stock
To extend the model in the simplest way that allows for stock pollution we assume that there
are two periods, indexed t = 1,2, and that pollution damages in period t depend on the total
stock of pollution at the end period t. The stock of pollution at the end of period t is denoted
by Z t . Assuming there is no natural decay of the pollution stock, the dynamics of the stock
pollutant are simply: Z t  Z t 1  Qt . We normalise and assume that Z 0  0 . Country i’s payoff
in period t is:  it (qit , Qit , Z t 1 )  bqit  c( Z t 1  Qit  qit ) . Note that this implies that while
total damage costs depend on the stock of pollution, unit damage costs do not We assume
there is no discounting.
3.1 Open-Loop Equilibrium.
In the open-loop equilibrium we assume that at the start of period 1, each country i announces
its emission strategy (qi1 , qi 2 ) . The first question we consider is whether playing (1, 1) is a
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dominant strategy for country i given any pattern of emissions by other countries ( Qi1, Qi 2 ).
The payoffs to all possible strategies are:
V (1,1)  b  c(Qi1  1)  b  c(Qi1  1  Qi 2  1)  2b  3c  2cQi1  cQi 2
V (1,0)  b  2c  2cQi1  cQi 2
V (0,1)  b  c  2cQi1  cQi 2
V (0,0)  2cQi1  cQi 2
Clearly V(0,1) > V(1,0), so if a country is only going to emit in one period, with a stock
pollutant it is always better to do it in the second period. Next note that V(1,1)> V( 1,0) and V
(0,1) > V(0,0) iff  > 1. This just says that, irrespective of what the country did in the first
period, it will pay to pollute in the second period iff the one-period benefit from and extra unit
of pollution exceeds the one-period damage cost. Not surprisingly, this is just the condition we
imposed in the static model to justify unilateral free-riding. Similarly note that V(1,1) > V(0, 1)
and V(1,0) > V(0,0) iff  > 2; i.e. irrespective of what the country does in period 2, it will
always pay to pollute in period 1 if the one-period benefit from emitting one unit of pollution
exceeds the additional damage that will cause over two periods. Putting these two elements
together we have:
Lemma 1 A sufficient condition for a country to unilaterally emit high in both periods is  >
2.
Turning to the gains from cooperation, the payoffs to each country if all N players play the
same strategy are: V(1,1) = 2b – 3Nc; V(1,0) = b – 2Nc; V(0,1) = b – Nc; V(0,0) = 0. Thus if N
>  > 2 the optimal cooperative strategy is to emit low in both periods. The gains from
cooperation are G = 3Nc – 2b which again are increasing in N and c and decreasing in b.
We now consider an IEA. Suppose n countries sign an IEA and maximise their joint payoff,
recognising that the non-signatories will emit high in both periods. The payoffs to each
signatory country from the four possible strategies are: V(1,1) = 2b – 3Nc; V(1,0) = b – 3Nc +
nc; V(0,1) = b – 3Nc +2nc; V(0,0) = -3Nc +3nc. We again define n* as the smallest integer
greater than , where now  > 2 implies n*  3. We also define n̂ as the smallest integer
greater than /2, so n̂  2. Then we have the following:
Lemma 2 The optimal strategy of the signatories and associated payoffs for signatories and
fringe countries for any size of IEA, n, are as follows:
n n* ; strategy: (0,0); payoffs:  s (n)  3Nc  3nc;  f (n)  2b  3Nc  3nc ;
n* > n  n̂ ; strategy: (0,1); payoffs:  s (n)  b  3Nc  2nc;  f (n)  2b  3Nc  2nc
/2 > n; strategy: (1,1); payoffs:  s (n)   f (n)  2b  3Nc.
Note the important new feature of the open-loop model is that there are two critical minimum
sizes: n̂ is the minimum size of IEA which justifies partial abatement – i.e. abatement only in
period 1, because the period 1 benefit of a unit of emissions is outweighed by the benefit over
two periods each member gets by the IEA collectively reducing period 1 emissions; n* is the
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minimum size which justifies full abatement, i.e. the one period gain of emissions in period 2
is outweighed by the benefit to outweighed by the benefit each member gets by the IEA
collectively reducing its period 2 emissions. Again, not surprisingly, this second minimum size
is exactly the same as in the static model. Because of the higher cost of pollution emitted in
period 1, it takes a smaller critical size to justify an IEA taking collective action in period 1
than in period 2. Note again, for all n,  f (n)   s (n) .
We now turn to stability analysis. n* and n̂ are the obvious candidates for stable IEAs. We
have the following:
Proposition 2 For the open-loop equilibrium, if 2 <  < 2.5, the stable IEA is n* =3 and the
optimal strategy of the signatories is to set low emissions in both periods; if  > 2.5, the stable
IEA is n̂ and the optimal strategy of the signatories is to set low emissions only in period 1
Proof:
To check whether n* is stable, the first stability condition requires that:
 s (n * 1)   f (n*)  3Nc  3(n * 1)c  2b  3Nc  3n * c    1.5 , which is true by
Lemma 1. The second stability condition requires:
 s (n*)   f (n * 1)  3Nc  3n * c  2b  3Nc  2(n * 1)c  n*    (   2). Since
n* is defined as the smallest integer greater than  this second stability condition will only be
satisfied if 2.5 >  >2.
We now check whether n̂ is stable. The first stability condition requires that:
 s (nˆ  1)   f (nˆ ) . Now for 2 <  < 3, n̂ = 2, n̂ +1 = 3 > ; otherwise n̂ +1 < . In the first
case, the first stability condition requires:
 3Nc  3(nˆ  1)c  2b  3Nc  2nˆc  nˆ  3  2     2.5 . In the second case, the first
stability condition requires b  3Nc  2(nˆ  1)c  2b  3Nc  2nˆc    2 , which holds by
Lemma 1. The second stability condition requires:
 s (nˆ )   f (nˆ  1)  b  3Nc  2nˆc  2b  3Nc  nˆ   / 2 , which is satisfied by the
definition of n̂ . So n̂ is stable provided  > 2.5. QED
So, upward stability of n* is guaranteed by the sufficient condition for free-riding, and
downward stability of n̂ is guaranteed by the definition of the minimum size to ensure at least
partial abatement by the IEA. But n* may no longer be downwardly stable because if one
member defects this may not trigger the IEA to abandon abatement completely, but only to
abandon it in period 2 and to carry on abating in period 1. The narrow range of values over
which n* is downwardly stable are those where one defection would trigger complete
abandonment of abatement. Similarly n̂ is not always upwardly stable – for exactly the same
reason – in this narrow range of values adding another member gets the IEA to move from
partial abatement to full abatement. Note, however, that for this model we get a unique IEA.
There are three interesting corollaries. First, for  < 2.5, the stable IEA is n* =3; for 2.5 <  <
4, the stable IEA is n̂ = 2; while for  > 4, the stable IEA is n̂  3. Thus, contrary to the static
model, over an initial range of parameter values, an increase in  may reduce the size of the
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stable IEA. Second, note that, since β can take any value between 2 and N, n̂ can take any
value between 2 and N/2. So with an open-loop equilibrium and a two-period stock pollutant
we can never get an IEA with more than half the countries as members, unlike the simple static
model where, as β tended to N we could get an IEA consisting of the grand coalition of all
countries. Finally, the most likely equilibrium, n̂ , involves only partial abatement – there is no
abatement by signatories in period 2. So the IEA becomes ineffective in the second period.
This has some resemblance to the Karp and Sacheti (1997) model, except that where they
imposed the restriction that the IEA only last for the first period, we have determined
endogenously an outcome in which an IEA becomes ineffective after one period.
3.2 Feedback Equlibrium.
In the open-loop equilibrium countries selected their membership of the IEA and their
emission strategies over the two periods at the start of the game. In the feedback equilibrium,
countries select their emissions in each period as function of the stock of pollution at the start
of each period. It seems natural then to allow them to also determine their membership of an
IEA in each period as a function of the stock of pollution at the start of that period. Thus
membership may change endogenously over time. As usual in solving a feedback equilibrium
we work backwards.
Period 2.
At the start of period 2 there will be an initial stock of pollution Z1  Q1 . Each country thus
faces a period 2 payoff function:  i 2  bqi 2  c(Q1  Qi 2  qi 2 ) . Given the linearity of our
payoff function, the initial stock of pollution at the start of period 2 enters the payoff function
as a fixed cost, and so cannot affect any decisions in period 2. So period 2 is effectively isomorphic to the simple static model set out in section 2. We then get immediately:
Proposition 3 In period 2 of the feedback equilibrium the unique stable IEA has n* members,
who set low emissions, and N-n* non-signatories who set high emissions.
Note that an immediate implication is that since β can take any value between 2 and N, it will
now be possible in the second period of the feedback equilibrium to get a stable IEA which is
as large as the grand coalition. Recall that in the open-loop equilibrium there can never be
more than half the countries in a stable IEA.
The value functions of signatories and non-signatories at the start of period 2 will be:
V2s (Q1 )  c(Q1  N  n*); V2f (Q1 )  b  c(Q1  N  n*) . Clearly these value functions
decrease linearly in the stock of pollution inherited from period 1, for the simple reason that
there is now a bigger stock of pollution in period 2 to cause damages.
Note, that, as elsewhere, signatories in period 2 are worse off than non-signatories in period 2.
This raises an interesting modeling issue. Since it will matter to countries in period 1 whether
they will be signatories or non-signatories in period 2, what is the process for determining
which countries will be signatories in period 2? We shall consider two possible assignment
rules. First we consider a random assignment rule in which each country has the same
9
probability of being assigned as a signatory country in period 2 independent of what happened
in period 1. We then consider a status quo assignment rule, which assumes that at the start of
period 2 there is an existing IEA inherited from period 1; if the number of signatories in period
2 is higher than in period 1, than all the existing signatories remain signatories and some
previous non-signatories now join the agreement; on the other hand if the number of
signatories in period 2 is less than in period 1, all the original non-signatories remain nonsignatories and some of the period 1 signatories now leave the agreement. In both cases there
is a random process for determining which countries switch.
Period 1 – Random Assignment Rule.
We begin by assuming that there is a random process for determining whether a country is a
signatory or non-signatory in period 2, independent of whether the country was a signatory or
not in period 1. Thus each country has the same expected payoff in period 2:
N  n*
V2  b
 c(Q1  N  n*) .
N
So at the start of period 1 all countries face the same payoff function:
N  n*
Vi1  bqi1  c(qi1  Qi1 )  b
 c(qi1  Qi1  N  n*)
N
N  n*
 bqi1  2c(qi1  Qi1 )  b
 c( N  n*)
N
Note that again the last two terms on the RHS are independent of any decisions taken in period
1, and so the effective payoff function is just the first two terms on the RHS. Again the first
period problem is iso-morphic to the static one period problem of section 2, except that now
the unit cost of damages is 2c, not c. So, provided  > 2, we can just apply the results of that
section to get:
Proposition 4 The stable IEA in period 1 of the feedback equilibrium with random assignment
rule has n̂ members, who set low emissions and N - n̂ non-signatories who set high emissions.
To summarise, the feedback equilibrium with random assignment rule says that for all  > 2,
in period 1 there will be n̂ signatories, with total emissions N - n̂ , and in period 2 there will
be n* > n̂ signatories with total emissions N – n*. So the number of signatories rises in the
feedback equilibrium. By contrast in the open-loop equilibrium, if 2 <  < 2.5, there will be n*
signatories in both periods with total emissions N – n* in both periods, while if  > 2.5 the
open-loop equilibrium will involve n̂ signatories in both periods with emissions N - n̂ in
period 1 and N in period 2. In the first case the open-loop equilibrium produces a better
outcome than the feedback equilibrium, while in the second case, involving a much greater
range of parameter space, the feedback equilibrium is significantly better than the open-loop
equilibrium. Note also that in the open-loop equilibrium there can never be more than half the
countries in a stable IEA; that is also true of the period 1 of the feedback equilibrium but in the
second period the number of signatory countries can be as large as the grand coalition for a
large enough value of β.
10
Period 1 – Status Quo Assignment Rule.
We now turn to a different assumption about how countries are assigned their roles as
signatories or non-signatories in the second-period. The relevant case is where the number of
signatories in period 1, n1 , is less than the number of signatories in period 2, n*. We suppose
that all signatories in period 1 remain signatories in period 2, while N-n* of the N – n1 nonsignatories in period 1 remain non-signatories in period 2; n * n1 non-signatories in period 1
will join the IEA in period 2. In that case the period 1 value functions for signatories and nonsignatories can be written as:
V1s  bqi1  2c(Qi1  qi1 )  c( N  n*)
N  n*
b
N  n1
Compared to the assumption of a random assignment of signatories and non-signatories in
period 2, first-period signatories are worse off, non-signatories better off.
V1 f  bqi1  2c(Qi1  qi1 )  c( N  n*) 
From the analysis of section 2, it is straightforward to see that signatories in period 1 will be
better off setting low emissions iff n1 > /2. So the period 1 value functions for signatory and
fringe countries are:
n1   / 2 : V1s (n1 )  2c( N  n1 )  c( N  n*);
V1 f (n1 )  b  2c( N  n1 )  c( N  n*) 
N  n*
b
N  n1
N  n*
b
N  n1
The obvious candidate for a stable IEA is n̂ , the smallest integer greater than /2, but given
the value functions set out above, is it stable? We have the following:
n1   / 2 : V1s (n1 )  b  2cN  c( N  n*); V1 f (n1 )  b  2cN  c( N  n*) 

N  n*
) then the stable IEA in period 1 of the feedback
2
N  nˆ  1
equilibrium with status quo assignment rule has size n̂ . If the condition is not satisfied then
there is no stable IEA.
Proposition 5 If nˆ 
(1 
Proof:
The first stability condition requires:
V1s (nˆ  1)  V1 f (nˆ )  2c( N  nˆ  1)  c( N  n*)  b  2c( N  nˆ )  c( N  n*) 
N  n*
b
N  nˆ
N  n*
)
N  nˆ
This stability condition is satisfied since we are assuming 2 < . So we cannot have a stable
IEA larger than n̂ . The second stability condition requires:
 2   (1 
11
V1s (nˆ )  V1 f (nˆ  1)  2c( N  nˆ )  c( N  n*)  b  2 Nc  c( N  n*) 
 nˆ 

2
(1 
N  n*
b
N  nˆ  1
N  n*
). QED
N  nˆ  1
Since n̂ is the smallest integer greater than /2, the condition for n̂ to be stable will only be
satisfied for values of N close to n*. As N gets larger, this inequality will be violated, and there
will be no stable IEA in period 1. As noted in the Introduction, this assignment rule leads to an
endogenous determination of an outcome in which an IEA will not form until the second
period of a dynamic stock pollutant problem, in contrast to the approach of Karp and Sacheti
who just assumed that an IEA would only last for one period, either the first or the second.
So making the alternative assumption about how signatories are chosen in period 2 may have
quite a dramatic effect on the durability of an IEA. It will still be the case that in this feedback
equilibrium, the number of signatories rises over time, from 0 to n*. Clearly the outcome of
the feedback equilibrium under this choice of period 2 signatories is worse than under the
previous assignment rule, and so if  < 2.5 must be unambiguously worse than open-loop. If 
> 2.5 it will depend on parameter values whether this feedback equilibrium is better or worse
than open-loop.
4. A Model of a Stock Pollutant with Stock-dependent Unit Damage Costs
The model in the previous section is the simplest possible extension of the static model to
allow for a stock pollutant. Its simplicity has the merit of allowing us to set out clearly the
modeling strategy, in particular the importance of how one assigns period 2 signatories.
However, it has the somewhat unsatisfactory feature that while total damage costs in period 2
depend on the stock of pollution accumulated at the end of period 1, the unit damage cost is
independent of the period 1 stock. This allowed us to effectively decompose the feedback
equilibrium of the dynamic problem into two sequential static models. More importantly for
our purposes, the model has the property that the number of signatories in the second period of
the feedback equilibrium is independent of the stock of pollution at the end of period 1. Given
that we want to ask how the dynamics of the pollution problem affect the incentives to join an
IEA this is an undesirable feature of the model. In this section we overcome these objections
by extending the model to allow unit pollution costs in a period to depend on the accumulated
pollution inherited from previous periods. Thus we now assume that the payoff function for
country i in period t is given by:
 it  bqit  (c  kZt 1 )( Z t 1  qit  Q it )
and we assume again that Z 0 = 0. For later notation purposes we define:   b / c,   k / c .
4.1 Open-Loop Equilibrium
As in previous sections we start by analysing the non-cooperative equilibrium, and in
particular we ask under what conditions setting high emissions in both periods is a dominant
12
strategy for every country. Thus we consider the payoffs to a country i from its four possible
strategies for any given (Q i1 , Q i 2 ) . We define cˆ  c  kQ i1 . Then the four payoffs are:
V (1,1)  2b  c  cQ i1  2(cˆ  k )  k (Q i1  Q i 2 )  cˆ(Q i1  Q i 2 )
V (1,0)  b  c  cQ i1  (cˆ  k )  k (Q i1  Q i 2 )  cˆ(Q i1  Q i 2 )
V (0,1)  b  cQ i1  cˆ  cˆ(Q i1  Q i 2 )
V (0,0)  cQ i1  cˆ(Q i1  Q i 2 )
As in the previous model V(0,1) > V(1,0), so if the country is only going to set high emissions
in one period it is better to do it in the second period. Clearly V(0,1) > V(0,0) iff b > ĉ ; while
V(1,1) > V(1,0) iff b  cˆ  k . In both cases the interpretation is that free-riding in period 2
requires that the one-period benefit from emitting a unit of pollution must exceed the one
period damage cost of an extra unit of pollution, and this cost is higher if country i has also
polluted in period 1. Thus a sufficient condition for country i to free-ride in period 2 for any
value of Q i1 and hence ĉ , and irrespective of what it did in period 1, is b > c + kN or
  1  N . Finally, V(1,1) > V(0,1) iff   2   (3N  1), while V(1,0) > V (0,0) iff
  2   (3N  3) . Again the interpretation is that a country will free-ride in period 1 if the
one-period benefit from emitting an extra unit of pollution exceeds the additional costs over
two periods, where this cost is higher if the country is also emitting pollution in period 2. So a
sufficient condition for country i to free-ride in period 1, no matter what it does in period 2 is
  2   (3N  1) . Putting all this together we get:
Lemma 3 A sufficient condition for a country to free ride in both periods in the open-loop
equilibrium is   2   (3N  1) .
Turning to a cooperative equilibrium and the gains from cooperation, if each country plays the
same strategy then the payoffs to each country are:
V (1,1)  2b  3cN  2kN 2 ; V (1,0)  b  2cN  kN 2 ; V (0,1)  b  cN ; V (0,0)  0
So, if  < N, V(0,0) > V(0,1) > V(1,0) > V(1,1), and the optimal cooperative strategy is to set
low emissions in both periods. Hence, to ensure both that there is a free-riding problem in both
periods and that the grand coalition would choose to abate in both periods we need to assume
that N >  > 2 + (3N – 1). A necessary condition for this pair of inequalities to hold for is  <
(N – 2)/(3N – 1) < 1/3. Note that the gains to cooperation are: G  3cN  2kN 2  b , which are
increasing in c, k, N and decreasing in b.
Finally we consider a stable IEA. Suppose that n countries join an IEA. We know that all nonsignatories will play their dominant strategy of setting high emissions in each period. What is
the optimal strategy for the IEA to maximize the joint welfare of its members? Assuming that
each signatory plays the same strategy, the payoffs to each signatory are:
13
V (1,1)  2b  3cN  2kN 2 ; V (1,0)  b  n(c  kN )  3cN  2kN 2 ;
V (0,1)  b  2cn  3knN  kn2  3cN  2kN 2 ; V (0,0)  3cn  4knN  2kn2  3cN  2kN 2
Note that the last two terms in each payoff are the same for all four payoffs; for ease of
notation, in what follows we shall ignore these common terms. It is straightforward to see that,
again, V(1,0) < V(0,1). It is then readily checked that:
V (0,0)  V (0,1)   (n)     n(1  N )  N 2  0
V (0,1)  V (1,1)   (n)     n(2  3N )  N 2  0
Note that for all n  0,  (n)   (n)
Define n1 , n4 and n2 , n3 as the two positive roots of  (n)  0 and  (n)  0 respectively. It is
straightforward to show that 1  n1  n2    N  n3  n4 ; n2  2; n1   / 2 . Note that as
  0, n1   / 2, n2   . Then, we have the following:
Lemma 4 For all values of n ≤ N, the optimal strategy of the IEA and payoffs to signatories
and fringe countries are as follows:
(i) n  n1;  (n)   (n)  0;  V (1,1)  V (0,1)  V (0,0)  V s (n)  V f (n)  2b
(ii)
n1  n  n2 ;  (n)  0; (n)  0;  V (0,1)  V (0,0); V (0,1)  V (1,1) 
V s (n)  b  2cn  3knN  kn2 ; V f (n)  2b  2cn  3knN  kn2
n2  n  N ;  (n)   (n)  0;  V (0,0)  V (0,1)  V (1,1) 
(iii) s
V (n)  3cn  4knN  2kn2 ; V f (n)  2b  3cn  4knN  2kn2
For j =1,2 define n̂ j as the smallest integer greater than n j ; clearly nˆ1  2, nˆ 2  3 . These are
the natural candidates for a stable IEA. We have:
Proposition 6 There cannot be a stable IEA greater than n̂2 or less than n̂1 ; either or both
nˆ1 or nˆ 2 can be a stable IEA depending on parameter values.
Proof: see section A1 of Appendix.
Numerical calculations show that: (i) if nˆ1  2, nˆ 2  3 (which only occurs rarely, and for low
values of N ,  ,  ) then n̂2 =3 is the unique stable equilibrium; (ii) if nˆ1  2, nˆ 2  4 (which,
for given N, is more likely to occur for higher values of β and γ) then nˆ1  2 is always stable,
and, for higher values of γ, n̂2 = 4 may also be stable; (iii) otherwise the only stable
equilibrium is n̂1 , and this is increasing in N and β but decreasing in γ. This latter finding
makes the important point that as we increase the impact of the stock of pollution on unit
damage costs, we decrease the size of the stable open-loop IEA. n̂1 is increasing in β and
14
decreasing in γ; in the limiting case   0,   N nˆ1  N / 2 , so, as in the model of the
previous section, with the open-loop equilibrium we can never get an IEA with more than half
the countries as members.
These results have similarities with those of the previous section. At low values of β and γ an
increase in parameter values can cause the size of the stable IEA to fall from 3 to 2 (as the
equilibrium switches from nˆ 2 to nˆ1 ) and then rise as β rises. What is new here is the possibility
of two stable IEAs and the fact that increasing the impact of the stock pollutant on unit damage
costs (γ) reduces the size of the stable IEA.
4.2 Feedback Equilibrium.
Period 2.
At the start of period 2, there will be a stock of pollution Z1  Q1 , and each country faces a
second period payoff function:  i 2  bq12  (c  kQ1 )(Q1  Q i 2  qi 2 ) . Defining
c2  c  kQ1 we can rewrite the payoff function as:
 i 2  bqi 2  c2 (Qi 2  qi 2 )  c2 Q1
Since the last term is fixed as far as period two decisions are concerned the payoff function is
effectively iso-morphic to that for the simple static model in section 2, except that the unit
damage costs are now c2 . Define  2  b / c2   /(1  Q1 ) and m2 as the smallest integer
greater than
 2 . Note that  2 > 1, since we are assuming that
  2  3N    1  N  1  Q1 . Then we know from section 2:
Proposition 7 In the second period of the feedback equilibrium, the stable IEA will have m2
members each emitting low while the non-signatories emit high, so that aggregate emissions in
period 2 are N - m2 .
We can thus define the period 2 value functions for signatories and non-signatories, and the
expected period 2 value function across signatories and non-signatories respectively as:
V2 s (Q1 )  c 2 ( N  m2 )  c 2 Q1
V2 f (Q1 )  b  c 2 ( N  m2 )  c 2 Q1
V2 (Q1 ) 
N  m2
b  c 2 ( N  m2 )  c 2 Q1
N
A change in the stock of pollution inherited from period 1 affects these value functions in four
ways. First, there is a higher stock of pollution in period 2, and that increases total damage
costs and hence reduces the period 2 value function. Second, the unit damage cost of pollution
rises, and that further increases damage costs and reduced the value function. These are
standard results in stock pollution models. The third effect, which is new, is that an increase in
period 1 pollution reduces the size of the stable IEA in period 2, and this increases the stock of
pollution at the end of period 2, increasing period 2 damage costs and further reducing the
15
payoff. All three effects apply to each of the value functions. The final effect applies only to
the expected value function. The reduction in size of the period 2 stable IEA increases the
number of period 2 non-signatories, and since they are better off than signatories, this effect
increases the expected value function. We shall show shortly that the first three effects
dominate.
It is immediately clear that we have overcome the objection we raised to the model of section
3, where the size of the stable IEA in period was independent of the stock of pollution
generated in period 1. For now the size of the stable IEA in period 2 decreases as the stock of
pollution inherited from period 1 increases (  2 and hence m2 are decreasing in Q1 ) .
It will be important to understand a bit more about the properties of the value functions. The
definition of m2 as an integer means that the value functions are not differentiable, but for
simplicity we shall ignore the integer problem and set m2   2 . We show in section A2 of the
Appendix that the three value functions have negative first and second derivatives. It will be
useful for later purposes to work with the following scaled period 2 value functions:
1
1
V2s (Q1 )  V2s (Q1 ); V2 (Q1 )  V2 (Q1 ) which clearly also have negative first and second
c
c
derivatives.
Period 1 – Random Assignment Rule
We now turn to the analysis of the period 1 equilibrium. As in section 3, we need to specify
how the m2 signatories in period 2 get selected. We begin by assuming that it is decided
randomly whether a country is a signatory or a non-signatory in period 2, and so the expected
payoff to each country in period 2 is given by the period 2 value function V2 (Q1 ) irrespective
of whether it is a signatory or non-signatory in period 1. Thus the period 1 value function for
country i is given by:
Vi1 (qi1 , Qi1 )  bqi1  c(qi1  Qi1 )  V2 (qi1  Qi1 )
We first check whether setting high emissions is a dominant strategy, i.e. is
Vi1 (1, Qi1 )  Vi1 (0, Qi1 )  Qi1 . This is true iff   1  V2 (Qi1 )  V2 (Qi1  1) , which we shall
dV2
. Again this has the standard interpretation: free-riding will pay
dQ1
in period 1 if the one period benefit of an extra unit of emissions exceeds the additional
damage cost, defined as the immediate additional damage cost incurred in period 1 plus the
effect of the additional stock of period 1 pollution in reducing the period 2 value function,
including the effect of the stock of pollution on period 2 IEA membership. Since we want this
to be true for all values of Qi1 , and since we know that the second derivative of V2 is
negative, we need to evaluate the derivative at Qi1  N  1 . A little manipulation shows that:
approximate by:   1 
16
Lemma 5 A sufficient condition for free-riding in period 1 of the feedback equilibrium with
random assignment rule is:   2  3N  2 
 2
N (1   ( N  1))
Note that this condition is already satisfied since, for the open-loop equilibrium, free-riding
required that   2  3N   .
We next check the gains from cooperation. We want to ensure that when all countries play the
same strategy, Vi1 (0,0)  Vi1 (1, N  1) , i.e. V2 (0)  b  cN  V2 ( N ) i.e V2 (0)  V ( N ).    N ,
which is satisfied since the LHS is positive and the RHS is negative, since, for the open-loop
equilibrium we need to assume β < N.
Finally, we look for a stable IEA. Suppose there are n signatories in period 1. We know that
the non-signatories will set high period 1 emissions. What is the optimal strategy of the
signatories? They will set low emissions in period 1 iff
 c( N  n)  V2 ( N  n)  b  cN  V2 ( N )
  ( n )  n  V2 ( N  n )  V2 ( N )  
This is just the condition that benefit each member country gets from a collective action of
reducing period 1 abatement, taking account of the effect this has on period 2 value function,
exceeds the benefit each member would get from emitting an extra unit of pollution. From the
properties of V2 it is clear that  (0)  0;  (n)  1;  (n)  0 ; so there exists
n , 0  n    N s.t. n  V2 ( N  n )  V2 ( N )   . Define m1 as the smallest integer greater
than n . So again, m1 is the minimum size of IEA which makes collective period 1 abatement
worthwhile. Then the period 1 value functions for signatories and non-signatories are:
 n1  m1 : V1s (n1 )  c( N  n1 )  V2 ( N  n1 ); V1 f (n1 )  b  c( N  n1 )  V2 ( N  n1 )
 n1  m1 : V1s (n1 )  V1 f (n1 )  b  cN  V2 ( N )
It is now straightforward to show that:
Proposition 8 m1 is the stable IEA for period 1 of the feedback equilibrium with random
assignment rule.
Proof: The first stability condition requires that:
V1s (m1  1)  V1 f (m1 )  c( N  m1  1)  V2 ( N  m1  1)  b  c( N  m1 )  V2 ( N  m1 )
   1  V2 ( N  m1  1)  V2 ( N  m2 )
which, again, is just the condition for free-riding for all Qi1  N  1 . The second stability
condition requires that:
V1s (m1 )  V1 f (m1  1)  c( N  m1 )  V2 ( N  m1 )  b  cN  V2 ( N )
 m1  V2 ( N  m1 )  V2 ( N )   (m1 )    m1  n
17
which, again, is true by the definition of m1. QED
To summarise: the feedback equilibrium with random assignment rule will have m1
signatories in period 1, resulting in period 1 emissions N  m1 ; in period 2 the number of
signatories will be m2 , the smallest integer greater than  /[1   ( N  m1 )] with period 2
emissions N  m2 . What is the relationship between m1 and m2 ? In section A3 of the
Appendix we prove:
Proposition 9 In the feedback equilibrium with random assignment rule, there are at least
twice as many signatories in period 2 as in period 1: 2 m1  m2 .
So, as in the model of section 3, the dynamics of the stock pollution lead to more countries
joining the IEA over time, and more precisely there will be at least twice as many signatories
in period 2 as in period 1.
Period 1 – Status Quo Assignment Rule
We now analyse the implications of using the status quo assignment rule, whereby assuming
there are more signatories in the second period, all period 1 signatories remain signatories in
period 2 and some period 1 non-signatories switch to becoming signatories in period 2. Then,
assuming we have n1 < m2 signatories in period 1, the relevant period 2 value function for
N  m2
~
signatories is V2s (Q1 ) while for non-signatories it is: V2 f (Q1 )  V2s (Q1 ) 
b . Note that
N  n1
bm2
m2
~f
s
s
V2  V2 
 0  V2 
 0 . We shall assume that this set of inequalities
N  n1
N  n1
holds for all relevant values of n1 .
We can now define the following period 1 value functions for signatories and non-signatories:
~
V1s  bqi1  c(qi1  Qi1 )  V2s (Q1 ); V1 f  bqi1  c(qi1  Qi1 )  V2f (Q1 )
Then we have:
Lemma 6 A sufficient condition for non-signatory countries to free-ride in period 1 of the
 m2
1
feedback equilibrium with status quo assignment rule is:   1  V2s 
N  n1
 m2
~
~
~ 
 1 .QED
Proof: b  c  V2f (Q1  1)  V2f (Q1 )  b  c  V2f    1  V2s 
N  n1
Again we have the usual interpretation: non-signatories will free-ride in period 1 if the one
period benefit from an extra unit of emissions exceeds the additional damage cost which
18
equals the immediate additional damage cost incurred in period 1 plus the reduction in period
2 value function caused by an increase in the stock of period 1 pollution.
We now turn to the behaviour of signatories. Suppose there are n1 IEA signatories in period 1.
It will be optimal for them to set low emissions if and only if:
 c( N  n1 )  V2s ( N  n1 )  b  cN  V2s ( N )   (n1 )  n1  V2s ( N  n1 )  V2s ( N )  
Again this is just the condition that the benefit to each member from collective abatement in
period 1, which is the sum of the immediate benefit of reducing damage costs in period 1 plus
the gain in the value function in period 2, exceeds the one period benefit of emitting an extra
 (0)  0;       0;    0
unit of pollution. Now
so there must exist
~
~
~
~
~
~
~ as the smallest integer greater than n~
~ is
~ . Again m
n : 0  n  n   s.t.  (n )   . Define m
1
1
the minimum size of period 1 IEA which makes collective abatement in period 1 worthwhile.
~  m . Then the value functions for signatories and non-signatories are:
Clearly m
1
1
~ : V s  c( N  n )  V s ( N  n ); V f  b  c( N  n )  V s ( N  n )  N  m2 b
n1  m
1
1
1
2
1
1
2
1
1
N  n1
~ : V s  b  cN  V s ( N ); V f  b  cN  V s ( N )  N  m2 b
n1  m
1
1
2
2
1
N  n1
where of course m2 is also a function of the stock of pollution inherited from period 1.
~ is stable. We have:
Finally we check whether m
1
~ is likely to be upwardly stable but unlikely to be downwardly stable.
Proposition 10 m
1
Proof: The first stability condition requires that:
~  1)  V~ f (m
~ )
V1s (m
1
1
1
~  1)  V s ( N  m
~  1)  b  c( N  m
~ )  V s (N  m
~ )  N  m2 b
 c ( N  m
1
2
1
1
2
1
~
N m
1

N  m2
  (1 
)  1  V2s
~
N  m1
Now the condition for non-signatories to want to set low emissions was that

m2 
 [1 
]  1  V2s  n1 . So a sufficient condition for the stability condition to hold is that
N  n1

N  m2  m2  N  m2 [1 
~ ) ] . For a wide range of parameter values this
1   (N  m
1
condition is likely to be satisfied.
~ )  V f (m
~  1) i.e.
The second stability condition requires that : V1s (m
1
1
1
19
~ )  V s (N  m
~ )  b  cN  V s ( N )  N  m2 b
 c( N  m
1
2
1
2
~
N m
1
~  V s (N  m
~ )  V s ( N )   [1  N  m2 ]
m
1
2
1
2
~
N m
1
~ )   [1  N  m2 ]
  (m
1
~
N m
1
Since m2 on the RHS corresponds to the situation where Q1  N , and hence will be rather
small, the last inequality is unlikely to hold. QED.
Thus the feedback equilibrium with the status quo assignment rule is likely to have no stable
equilibrium in period 1, with consequent emission levels N, but a stable IEA in period 2 with
~ signatories, defined as the smallest integer greater than  /(1  N ) , and hence period 2
m
2
~ . Clearly this implies that membership of the IEA will rise over time.
emissions N  m
2
4.3 Some Numerical Results.
We have characterized the open-loop and feedback equilibria of the model where unit damage
costs depend on the stock of pollution, but there are two sets of issues we have not been able to
resolve analytically. First, for the open-loop equilibrium we have shown that there two
candidates for a stable IEA and either one of these is the unique stable IEA or both might be;
how likely are these different equilibria to be. Similarly in the feedback equilibrium with status
quo assignment rule all our analysis could conclude was that the obvious candidate for a stable
~ , was likely to be upwardy stable but not downwardly – again just what is
IEA in period 1, m
1
meant by “likely”. Second, the way we have characterized the period 1 stable IEA membership
size for the feedback equilibrium with both random assignment and status quo assignment
rules makes it difficult to compare analytically the membership size in the open-loop and
feedback equilibria, and hence to compare welfare levels. To get a bit more insight inot these
issues we have conducted some numerical experiments. In this section we report the results of
those experiments.
There are three key parameters in our model: N, , . For the first set of numerical calculations
we took values of N between 25 and 150 in steps of 25. For , we took 150 values lying
between the two limits 0 and (N – 2)/(3N – 1); while for  we took 150 values lying between
the limits 2+3N -  and N. This guarantees that the sufficient conditions for free-riding and for
an IEA to want to collectively abate are met. Note that for each value of N we conduct 22500
simulations. Table 1 presents the results. The columns show the results values of N; for each
N we present two sets of results. Because for some parameter values we can get two stable
open-loop equilibria, the parameter OLEQ indicates which equilibrium we select:
1  nˆ1; 2  nˆ 2 . Obviously this matters only when it comes to a comparison between openloop and feedback.
20
Turning to the rows of Table 1, the first three rows tell us what proportions of the 22500 cases
have nˆ1 , nˆ 2 individually or both being stable. What this shows is that by far the most likely
outcome (between 70% and 80% of cases) has n̂1 as the unique stable equilibrium. If N is less
than or equal to 50, then in a tiny proportion of cases (less than a quarter of 1%), n̂2 is the
unique stable equilibrium; and in the remaining cases, between 20% and 30%, both are stable.
It is for these cases that the equilibrium selection parameter OLEQ is relevant.
~ , the period 1 IEA in the feedback equilibrium
The next row tells us about the stability of m
1
with status quo assignment rule. Proposition 10 tells us that this is likely to be upwardly stable
but not downwardly stable. This is confirmed by the numerical results. The figures show the
~ is stable, and as N rises it falls from about 9% to 1.5%; in all
proportion of cases for which m
1
other cases it is upwardly stable but not downwardly, so there will be no period 1 stable IEA in
those cases.
We now turn to a comparison of feedback and open-loop equilibria, and obviously this
depends on which of the open-loop equilibria we select if both are stable, and which of
assignment rules we use for the feedback equilibrium – the random assignment rule (RAR) or
the status-quo assignment rule (SQAR). In each case we first compare the number of
signatories in the open-loop equilibrium and the two periods of the feedback equilibrium.
There are five possible relationships, as shown at the bottom of Table 1, and the Table shows
what proportion of the 22500 simulations falls into each of the five cases. What this shows is
the following: with RAR, the proportion of cases 4 and 5 (where the number of signatories in
the open-loop equilibrium is at least as great as the number of signatories in period 2 of the
feedback equilibrium) is equal to the proportion of cases for which n̂2 is the open-loop
equilibrium, either because it is the unique equilibrium or because it is selected when both are
stable. Cases 1 – 3 arise when n̂1 is the unique or selected open-loop equilibrium, and by far
the most likely outcome is that this will also equal the number of signatories in period 1 of the
feedback equilibrium. With SQAR, the outcome for Cases 4 and 5 is exactly the same as with
RAR, but now we never get case 1, and we only have the number of signatories in period 1 of
the feedback equal to the number of signatories in open-loop when the period 1 feedback is
~  1 , we get Case 3.
stable; otherwise, since m
1
As a final indicator on number of signatories we calculated the maximum number of
signatories in the open-loop and feedback equilibria. Now we know that theoretically the upper
limit on signatories in the open-loop and first-period feedback is N/2 and the upper limit on
signatories in period 2 of the feedback is N, with these being approached as   N ,   0 .
We see that for N = 25 we attain these upper limits, but fall increasingly short as N increases.
But this just indicates that our grid of search points gets a bit coarser as N increases. Not
surprisingly the feedback with SQAR has lower maximum signatories than with RAR.
Finally we compare welfare in the open-loop and feedback equilibria. We first ask in what
proportion of simulations is welfare higher in the feedback equilibrium than in the open-loop
equilibrium. The answer is that this is just the same as the proportion of simulations in Cases 1
–3 which, as we have seen is just equal to the proportion of cases when n̂1 is the open-loop
21
equilibrium. And this is true for both RAR and SQAR. So what this says is that as long as the
feedback equilibrium has more period 2 signatories than the open-loop, then welfare will be
higher in the feedback, even if the feedback equilibrium has no signatories in period 1. On the
other hand if the number of open-loop signatories is at least as high as the second-period
feedback, then, not surprisingly, the open-loop will have higher welfare than feedback. Next
we computed the average level of welfare in the open-loop and feedback equilibria across
22500 simulations and look at the ratio of average welfare in open-loop to average welfare in
feedback (AWOL/AWFB). When we use the SQAR rule for feedback equilibrium and select
n̂2 when there are multiple stable open-loop equilibria, the combination which is most
favourable to the open-loop equilibrium, then, for N > 25 we get average welfare in open-loop
very slightly higher than in feedback; otherwise average welfare is lower in open-loop than
feedback, sometimes quite significantly so.
Table 1 essentially presents summary statistics for the very large number of simulations we
have run, and obviously we could not present the detailed results for each set of parameter
values we have searched across. In Table 2 we present the number of signatories and welfare
levels in the stable IEAs of the open-loop and feedback equiulibria for a much coarser grid of
parameter values: we have chosen low, medium and high values for each of N, , and . Not
surprisingly, for none of these cases was n̂2 the unique stable IEA for the open-loop model,
~ was stable for feedback with SQAR. The results in Table 2
but we did get one case where m
1
confirm those we have given in Table 1. In particular welfare in open-loop will be higher than
in feedback when n̂2 is a stable equilibrium – otherwise feedback is higher than open-loop,
even when we have the SQAR and no stable IEA in period 1 of the feedback equilibrium.
5. Further Extensions of the Models.
The models we have developed in sections 3 and 4 demonstrate our contention that taking
account of the dynamics of stock pollution can have important implications for the analysis of
IEAs. In particular those models showed that one could get a significant increase over time in
the number of countries joining an IEA. However the models from which this conclusion is
drawn are extremely simple, in two key respects: they restrict countries to just two actions, and
they have just two periods. In this section we comment briefly on how we believe our results
are likely to be affected by weakening these restrictions.
One reason for thinking that allowing countries to choose any non-negative value for
emissions might change our results comes from the analysis of Rubio and Ulph (2001). For the
static model with quadratic benefit and cost functions and continuous actions, they show that
when the IEA signatories take as given the emissions of non–signatories (Nash behaviour),
then the unique stable IEA has 2 members, no matter what the cost and benefit parameters are.
These results were confirmed by the extension to a differential game. This would seem to
suggest that with a feedback model where membership is determined each period, there will be
little scope to get significant variation in membership over time. However, they showed that
when the IEA signatories act in Stackelberg fashion with respect to non-signatories then, in the
static model, any size of membership up to the grand coalition can be a stable IEA for an
22
appropriate choice of benefit and cost parameters2. Our conjecture would be that, if one starts
from a static model where, by appropriate choice of parameters, it is possible to generate a
wide variation in the size of a stable IEA, then when one extends such a model to a stock
pollutant, with membership determined in a feedback fashion, then membership will vary over
time as the stock of pollution varies.
By contrast, we believe that our results are likely to be significantly affected by extension to
more than two periods. We sketch a simple argument which shows that if we extend our model
with discrete actions to an infinite horizon we can reverse the conclusion that membership will
rise over time. We take the random assignment rule and suppose that, in an infinite horizon
model, at the start of period t the expected value function, V ( Z t 1 ) for each country (taking
expectations over whether a country will be a signatory or non-signatory from period t
onwards) depends solely on the stock of pollution at the start of that period, Z t 1 . In other
words, the expected value function is autonomous – all the dynamic aspects of the problem,
including the determination of the size of the stable IEA in each period, are captured through
the stock variable and the functional form. We further suppose, as our analysis has suggested,
that V   0, V   0 . Let ct  C ( Z t 1 ) be the unit damage cost in period, which is an increasing
function of the stock of pollution at the start of period t. Finally, to establish notation, it will be
useful to use a different form of scaling from earlier sections. Now define:
  c / b; V  V / b; which clearly have the properties    0; V   0, V   0 .
t
t
t
We now go through the same steps of analysis as we have done in sections 3 and 4 for the
feedback model with the random assignment rule. We first need a condition to ensure that in
any period t, when countries act non-cooperatively they free-ride, i.e. set high emissions, for
any given choice of emissions by other countries in period t, Qit . We have:
Lemma 7 A sufficient condition for free-riding in period t of the infinite horizon feedback
model is 1   t  V  .
Proof: Country i will set high emissions in period t iff:
b  ct [ Z t 1  Q it  1]  V ( Z t 1  Q it  1)  ct [ Z t 1  Q it ]  V ( Z t 1  Q it )
i.e. 1   t  V  QED
.
2
Diamantoudi and Sartzetakis (2001) working with a very similar model claim that with the static Stackelberg
model there can never be more than 4 countries in a stable IEA. They argue that the numerical results of Barrett
(1994), which showed that one could get much larger stable IEAs for an appropriate choice of parameter values,
were wrong because Barrett did not respect the constraint that abatement cannot exceed emissions (or emissions
cannot be negative) and once this restriction is imposed the size of the stable IEA falls. However, as Rubio and
Ulph (2001) argue, Diamantoudi and Sartzetakis deal with the requirement that emissions must be non-negative
by imposing restrictions on parameter values, and it is these restrictions which generate the small size of stable
IEA. The more appropriate approach is to directly impose the restriction that emissions must be non-negative,
using Kuhn-Tucker conditions, and allow for the possibility of corner solutions where emissions of either or nonsignatories may be zero. These restrictions can be imposed without limiting parameter values, and Rubio and
Ulph show that for certain parameter values it is still possible to get the grand coalition as a stable IEA despite
requiring emissions to be non-negative.
23
This has the same interpretation as in previous sections: free-riding in period t will pay if the
one-period benefit of an extra unit of pollution exceeds the additional damage cost, defined as
the additional damage cost incurred immediately in period t plus the reduction in the next
period value function through the change in the stock of pollution passed on from period t,
including any change in IEA membership.
Now consider an IEA of nt members in period t. We have:
Lemma 8 A sufficient condition for an IEA with nt members to collectively abate in period t
is: nt  t  V ( Z t 1  N  nt )  V ( Z t 1  N )  1 .
Proof: Members of the IEA will be better off abating iff:
 ct ( Z t 1  N  nt )  V ( Z t 1  N  nt )  b  ct ( Z t 1  N )  V ( Z t 1  N )
i.e. nt  t  V ( Z t 1  N  nt )  V ( Z t 1  N )  1 QED
Again the sufficient condition has the interpretation that the gain to each IEA member from
collectively cutting their emissions, defined as the immediate benefit in terms of reduced
damage costs plus the longer term benefit of an increase in the value function through a
reduction in the stock of pollution passed on to the next period, must exceed the one-period
benefit from not emitting an extra unit of pollution.
Define nt by:
nt  t  V ( Z t 1  N  nt )  V ( Z t 1  N )  1
(1)
and nt* as the smallest integer greater than nt . nt* is the smallest IEA in period t for which it
is worthwhile for IEA to collectively abate pollution. Then the value functions for signatories
and non-signatories at the start of period t for any size of IEA are:
nt  nt* : Vts (nt )  ct ( Z t 1  N  nt )  V ( Z t 1  N  nt ); Vt f (nt )  b  Vts (nt )
nt  nt* : Vts (nt )  Vt f (nt )  b  ct ( Z t 1  N )  V ( Z t 1  N )
We now get the following result:
Proposition 11 nt* is the unique stable IEA in period t of the feedback equilibrium with
infinite horizon.
Proof:
Upward stability requires:
24
Vts (nt*  1)  Vt f (nt* )
 ct ( Z t 1  N  nt*  1)  V ( Z t 1  N  nt*  1)  b  ct ( Z t 1  N  nt* )  V ( Z t 1  N  nt* )
 t V   1
which of course is satisfied as long as the sufficient condition for free-riding is satisfied
(Lemma 7). Downward stability requires:
Vts (nt* )  Vt f (nt*  1)  ct ( Z t 1  N  nt* )  V ( Z t 1  N  nt* )  b  ct ( Z t 1  N )  V ( Z t 1  N )
 nt* t  V ( Z t 1  N  nt* )  V ( Z t 1  N )  1
which is true by the definition of nt* . QED
Finally we derive the following result:
Proposition 12 nt ( and hence nt* ) decline over time as the stock of pollution increases.
Proof: Differentiate (1) with respect to the stock of pollution Z t 1 to get:
dnt
n    V ( Z t 1  N  nt )  V ( Z t 1  N )
 t t
 0 QED
dZ t 1
( t  V )
What this result essentially says is that because an increasing stock of pollution drives up the
marginal cost of pollution, in terms of both the immediate impact on current damage costs and
the impact in reducing the present value of expected future net benefits that reduces the
incentives to join an IEA. This just generalizes to a stock pollution context the finding of the
simple static model that IEA membership is decreasing in the unit damage cost.
So why does this general argument not hold in the two-period model? The obvious reason is
that with a stock pollutant the relevant concept of marginal damage cost is the present value of
additional damage costs in the current and all future periods; with a finite horizon model, even
if the unit damage cost is an increasing function of the stock of pollution, as time goes by there
are simply fewer future periods over which one needs to take account of the impact of an
additional unit of pollution. So the steady reduction in the time horizon in a finite period model
makes pollution less damaging over time, and, as we saw, in a two-period model it is this
effect which dominates. This might suggest that in a model of a stock pollutant with a long but
finite time horizon one might get a U-shaped pattern to IEA membership in a feedback
equilibrium.
Of course these arguments are suggestive rather than rigorous. We have yet to establish the
existence of an autonomous value function with negative first and second derivatives. Another
factor which would need to be taken into account in a rigorous analysis of an infinite horizon
model is that the increase in damage costs over time may at some stage cause the sufficient
condition for free-riding to be violated (Lemma 7).
25
The analysis in this section has focused on the question of whether IEA membership might be
expected to increase or decrease over time as the stock of pollution changes. It would also be
interesting to check how robust are other results from the two-period model – that the openloop may involve only partial abatement by IEA members, and that the feedback equilibrium
generally yields higher welfare than the open-loop.
Finally we would note that another key simplification of our model is the symmetry
assumption, and this should also be relaxed. Introducing asymmetries may help to resolve the
question of how to determine which countries should be members of an IEA in any time
period. We do not underestimate the difficulties of doing this.
6 Conclusions
We have used a very simple model of an international environmental agreement to manage a
stock pollutant to ask how the dynamics of a stock pollutant might affect membership of the
IEA over time. Using a simple two-period, two-action model we have shown the following: (i)
an open-loop equilibrium can have a stable equilibrium which involves only partial abatement
(i.e. abatement only in the first period); (ii) there can be multiple stable IEAs in the open-loop
equilibrium; (iii) in the feedback equilibrium the number of signatories rises over time. (iv) if
the number of signatories changes over time, it matters what assumptions one makes about
how countries get assigned to be signatories or non-signatories in later periods. (v) there is a
wide range of parameter values for which the feedback equilibrium generates higher welfare
than the open-loop equilibrium. One could interpret this last finding as meaning that, for those
parameter values one will certainly not want to prevent new countries joining the agreement at
later dates, and also one should not try to get countries to agree to very long-term
commitments. It makes more sense to have the agreements evolve over time with the stock of
pollution.
Of course our model is extremely simple. Some simplifications we have made such as no
discounting and no natural decay of the polluting stock were made just to reduce parameters
and notation and have no substantive effects (indeed one could think of these factors as being
captured in the parameters of the unit cost function). Other simplifications matter, and the
discussion in the previous section showed that one needs to be cautious about the claims one
can make on the basis of the two period model in sections 3 and 4. However we assert that the
central message of this paper – that there may be important links between the dynamics of a
stock pollutant and the dynamics of IEA membership – remains unaffected, and indeed is
reinforced, by the discussion in section 5. We also claim that the model has brought to light the
important question of what to assume about how countries might get assigned to be IEA
members over time, which is partly an issue about how analysts of IEAs might formulate their
models, but also has real implications for how would design IEAs if one wanted to allow
membership to vary over time. Thus to return to the question we posed at the beginning of this
paper – does it matter that most of the models that have been used to analyse international
environmental agreements to manage stock pollutants have ignored the stock dynamics of the
pollutant – we believe that this paper suggests that the answer is that it does matter. But the
paper also shows there is much more that needs to be done to test this answer more rigorously.
26
Table 1: Characteristics of Open-Loop and Feedback Equilibria
N
OLEQ
25
1
50
2
1
75
2
1
100
2
1
125
2
1
150
2
1
2
n̂1
n̂2
Both
0.778 0.778 0.722 0.722 0.706 0.706 0.699 0.699 0.695 0.695 0.691 0.691
0.002 0.002 0.001 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.220 0.220 0.278 0.278 0.294 0.294 0.301 0.301 0.305 0.305 0.309 0.309
~ stable
m
1
0.088 0.088 0.045 0.045 0.030 0.030 0.021 0.021 0.017 0.017 0.015 0.015
RAR
Case 1
Case 2
Case 3
Case 4
Case 5
n̂1 max
m1 max
m2 max
WFB > WOL
AWOL/AWFB
0.059
0.835
0.104
0.002
0.000
12
12
25
0.998
0.816
0.059
0.713
0.006
0.214
0.008
12
12
25
0.778
0.914
0.050
0.869
0.080
0.001
0.000
22
23
47
0.999
0.907
0.050
0.655
0.016
0.275
0.003
22
23
47
0.722
0.968
0.046
0.890
0.065
0.000
0.000
31
33
69
1.000
0.938
0.046
0.644
0.017
0.292
0.002
31
33
69
0.706
0.981
0.041
0.903
0.056
0.000
0.000
39
42
89
1.000
0.953
0.041
0.642
0.016
0.300
0.001
39
42
89
0.699
0.986
0.038
0.914
0.048
0.000
0.000
46
50
107
1.000
0.962
0.038
0.642
0.015
0.305
0.001
46
50
107
0.695
0.989
0.035
0.922
0.043
0.000
0.000
52
57
124
1.000
0.968
0.035
0.642
0.015
0.309
0.000
52
57
124
0.691
0.991
0.000
0.088
0.909
0.002
0.000
12
12
25
0.998
0.880
0.000
0.000
0.778
0.214
0.008
12
12
25
0.778
0.973
0.000
0.045
0.954
0.001
0.000
22
2
45
0.999
0.941
0.000
0.000
0.722
0.275
0.003
22
2
45
0.722
1.001
0.000
0.030
0.970
0.000
0.000
31
2
65
1.000
0.961
0.000
0.000
0.706
0.292
0.002
31
2
65
0.706
1.003
0.000
0.021
0.979
0.000
0.000
39
2
82
1.000
0.971
0.000
0.000
0.699
0.300
0.001
39
2
82
0.699
1.003
0.000
0.017
0.983
0.000
0.000
46
2
98
1.000
0.977
0.000
0.000
0.694
0.305
0.001
46
2
98
0.695
1.003
0.000
0.015
0.985
0.000
0.000
52
2
113
1.000
0.981
0.000
0.000
0.691
0.309
0.000
52
2
113
0.691
1.003
SQAR
Case 1
Case 2
Case 3
Case 4
Case 5
n̂1 max
m1 max
m2 max
WFB>WOL
AWOL/AWFB
Case 1 : n1  m1  m2
Case 2 : m1  n1  m2
Case 3 : m1  n1  m2
Case 4 : m1  n1  m2
Case 5 : m1  m2  n1
27
Table 2 Calculation of Open-Loop and Feedback Equilibria
Parameters
N
25
25
25
25
25
25
25
25
25
75
75
75
75
75
75
75
75
75
125
125
125
125
125
125
125
125
125
Open-Loop Equilibrium
Feedback Equilibrium
Feedback Equilibrium
Random Assignment Rule
Status Quo Assignment
Rule


n̂1
V (nˆ1 )
n̂2
V (nˆ 2 )
m1
m2
V (m1, m2 )
m1
m2
V (m1 , m2 )
.03
.03
.03
.15
.15
.15
.27
.27
.27
.03
.03
.03
.15
.15
.15
.27
.27
.27
.03
.03
.03
.15
.15
.15
.27
.27
.27
6.30
14.61
22.92
14.29
19.05
23.81
22.28
23.49
24.70
15.35
41.86
68.37
39.54
55.30
71.06
63.73
68.74
73.75
24.40
69.11
113.82
64.79
91.55
118.31
105.18
114.00
122.80
2
4
6
2
2
2
2
2
2
2
5
9
2
2
3
2
2
2
2
6
9
2
2
3
2
2
2
-97
-79
-62
-226
-216
-207
-354
-352
-350
-526
-465
-405
-1809
-1779
-1736
-3094
-3084
-3074
-1255
-1149
-1050
-4894
-4841
-4769
-8534
-8517
-8500
4
4
4
4
4
4
4
4
4
-
-171
-264
-262
-1650
-2814
-2805
-4630
-8066
-8048
-
2
4
7
2
2
2
2
2
2
2
5
9
2
2
3
1
2
2
2
6
9
2
2
2
2
2
2
4
9
15
4
5
6
4
4
4
5
14
23
4
5
7
4
4
4
6
16
26
4
5
7
4
4
4
-86
-60
-35
-193
-182
-170
-301
-299
-297
-500
-403
-310
-1718
-1676
-1591
-2993
-2923
-2914
-1211
-1035
-883
-4742
-4671
-4582
-8265
-8248
-8231
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
9
14
4
5
6
4
4
4
5
13
22
4
5
6
4
4
4
6
15
25
4
5
7
4
4
4
-90
-69
-52
-205
-193
-181
-301
-318
-316
-508
-436
-367
-1752
-1710
-1669
-2993
-2983
-2974
-1224
-1099
-977
-4799
-4728
-4638
-8365
-8348
-8331
28
Appendix: Proofs of Results in Section 4
A1. Stability of Open-Loop IEA.
A. Suppose that nˆ 2  nˆ1  2
(i) Consider stability conditions for n̂2 .
The first stability condition requires V s (nˆ 2  1)  V f (nˆ 2 ) , i.e.
3c(nˆ 2  1)  4kN (nˆ 2  1)  2k (nˆ 2  1) 2  2b  3cnˆ 2  4kNnˆ 2  2knˆ 22    1.5  N    2knˆ 2
which is true since   2  3N   . Thus there cannot be a stable IEA greater than n̂2 . The
second stability condition requires V s (nˆ 2 )  V f (nˆ 2  1) i.e.
3cnˆ 2  4kNnˆ 2  2knˆ 22  2b  2c(nˆ 2  1)  3kN (nˆ 2  1)  k (nˆ 2  1) 2
nˆ 2  2  Nnˆ 2  3N  2   nˆ 22  2nˆ 2  
[   2  3N   ]  [ (nˆ 2  2)]  [nˆ 2 ]  [( nˆ 2  n2 )( (nˆ 2  n2 )  (1  N ))]  0
The first 3 terms in square brackets on the LHS are all positive; the last term is negative. So it
depends on parameter values whether or not n̂2 is stable. As n̂2 increases, the last term gets
smaller in absolute value, the second and third terms get larger in absolute value, and so it is
more likely that the LHS will be strictly positive and hence n̂2 will be unstable.
(ii) Consider stability conditions for n̂1 .
The first stability condition requires V s (nˆ1  1)  V f (nˆ1 ) , i.e.
b  2c(nˆ1  1)  3kN (nˆ1  1)  k (nˆ1  1) 2  2b  2cnˆ1  3kNnˆ1  knˆ12
   2  3N    2nˆ1
which is satisfied by the restriction on β. The second stability condition requires
V s (nˆ1 )  V f (nˆ1  1) i.e. b  2cnˆ1  3kNnˆ1  knˆ12  2b  0  2nˆ1  3Nnˆ1  nˆ12  
which is true since nˆ1  n1 . So n̂1 is stable.
B Now suppose that nˆ 2  nˆ1  1 .
The first stability condition for n̂2 and the second stability condition for n̂1 are unaffected, so
there can be no stable IEA greater than n̂2 or less than n̂1 . We now check the remaining
conditions.
(i) Second stability condition for n̂2 now becomes: 3cnˆ 2  4kNnˆ 2  2nˆ 22  2b which is
satisfied since: 3nˆ 2  4Nnˆ 2  2nˆ 22  2   2nˆ 2  2Nnˆ 2  2nˆ 22  2   0 since nˆ 2  n2 . So
n̂2 is now stable.
(ii) First stability condition for n̂1 now becomes:
29
3c(nˆ1  1)  4kN (nˆ1  1)  2k (nˆ1  1) 2  2b  2cnˆ1  3kNnˆ1  knˆ12
 [nˆ1 (1  N )  nˆ12   ]  [2  3N     ]  [ ( N  4nˆ1 )]  [1   ]  0
Now the first term in square brackets on the LHS will be positive if nˆ1  nˆ 2 but negative if
nˆ1  nˆ 2 -1. The second term is negative, the third term could be positive or negative and the
fourth term is positive. So it is ambiguous whether n̂1 is stable or not.
To resolve these ambiguities we carried out some numerical calculations. We took values of N
between 25 and 150, in steps of 25. For each N, we took 150 values of γ between 0 and (N –
2)/(3N – 1). And for each value of N and γ we took 150 values of β between 2 + γ(3N – 1) and
N. These gave the following results. (i) The case where n̂2 is the only stable equilibrium is
rare and occurs when n̂2 = 3; when N = 25 there were only 19 out of 10000 such cases, and
for N greater than 50 there are no such cases. (ii) A necessary, but not sufficient, condition for
both n̂1 and n̂2 to be stable is nˆ1  2, nˆ 2  4 ; the proportion of such cases rises with N, from
22% when N = 25 to 30% when N = 150. (iii) In all other cases, the most common, the only
stable equilibrium is n̂1 . n̂1 increases with N and  and decreases with γ, so that increasing
the extent to which the stock of pollution drives up costs over time, reduces the size of a stable
IEA.
A2. Feedback Equilibrium: Derivatives of Period 2 Value Functions
Since the value functions for signatories and non-signatories differ only by a constant we shall
work only with the value function for signatories; since we are interested only in the sign of
the derivatives we shall also divide the value functions by c. Then we have:
m2 


2 


; m2  
; m2 
2
1  Q1
(1  Q1 )
(1  Q1 ) 2
1
V2 s (Q1 )  V s (Q1 )  (1  Q1 )(Q1  N  m2 );
c
 ( N  m2 )
1
V2 (Q1 )  V (Q1 ) 
 V s (Q1 )
c
N

V2 s  (1  Q1 )   (Q1  N  m2 )  (1  Q1 )m2   0
In this first-derivative, the first term is just the effect of having a bigger period 1 stock of
pollution carried forward to period 2, for a given unit damage cost; the second term is the
effect of the bigger period 1 pollution stock increasing the unit cost of damage; the third term
is the effect of the bigger period 1 pollution stock reducing the size of the period 2 IEA.
 
V2   V2 s  m2  <0
N
The first-derivative of the expected value function adds an additional term – decreasing the
size of the stable IEA increases the number of non-signatories and, since they are better off
30
than signatories by an amount β, this increases expected benefits; however since we are
assuming that β < N, the absolute size of this effect is smaller than the absolute size of the third
term in the first derivative of the signatories value function, and so this first-derivative is also
negative.
Turning to second derivatives we have:

V2 s  2  2m2   (1  Q1 )m2   2 
V2   V
s


N
2  2
2  2

 2  0
(1  Q1 ) 2 (1  Q1 ) 2
m2   0
A3. Feedback Equilibrium with Random Assignment Rule: 2 m1  m2
Ignoring integer values we have:
m2 

[1   ( N  m1 )]
  m1  V2 ( N  m1 )  V2 ( N )
 m2 [1   ( N  m1 )]  m1  [1   ( N  m1 )][ 2 N  m1  m2 ]  [1  N ][ 2 N  m2 ]
m2 (1  N )  m1  2 N (1  N )  [1   ( N  m1 )][ 2 N  m1 ]
m2 (1  N )  m1 (2  3N  m1 )
 m2  2m1  m1  N
31
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32