Cooperation on Climate-Change Mitigation†
Charles F. Mason
Department of Economics and Finance, University of Wyoming
Stephen Polasky
Department of Applied Economics, University of Minnesota
Nori Tarui
Department of Economics, University of Hawaii
This draft: 17 November, 2008
Abstract
We model greenhouse gas (GHG) emissions as a dynamic game. Countries’ emissions increase
atmospheric concentrations of GHG, which negatively affects all countries' welfare. We analyze
self-enforcing climate-change treaties that are supportable as subgame perfect equilibria. A
simulation model illustrates conditions where a subgame perfect equilibrium supports the
first-best outcome. In one of our simulations, which is based on current conditions, we explore the
structure of a self-enforcing agreement that achieves optimal climate change policy, what such a
solution might look like, and which countries have the most to gain from such a agreement.
†
The authors thank Stephen Salant, Akihiko Yanase, and the seminar participants at the University of Hawaii, the
ASSA Meetings in 2007, Doshisha, Japan Economic Association Meeting, Hitotsubashi, Tokyo Tech, Tokyo,
Tsukuba, Kobe, Keio University, and the Occasional Workshop on Environmental and Resource Economics. The
authors are responsible for any remaining errors.
Cooperation on Climate-Change Mitigation
Abstract
We model greenhouse gas (GHG) emissions as a dynamic game. Countries’ emissions increase
atmospheric concentrations of GHG, which negatively affects all countries' welfare. We analyze
self-enforcing climate-change treaties that are supportable as subgame perfect equilibria. A
simulation model illustrates conditions where a subgame perfect equilibrium supports the
first-best outcome. In one of our simulations, which is based on current conditions, we explore the
structure of a self-enforcing agreement that achieves optimal climate change policy, what such a
solution might look like, and which countries have the most to gain from such a agreement.
1. Introduction
Global environmental problems such as climate change, depletion of the ozone layer and loss of
biodiversity have risen to the top of the world’s environmental agenda. For each of these problems
there is a large scientific literature warning of the dangers of failing to successfully address the
issue and continuing business-as-usual. For climate change, the Intergovernmental Panel on
Climate Change (IPCC) concluded that continued emissions of greenhouse gases (GHG) would
likely lead to significant warming over the coming centuries with the potential for large
consequences on the global environment (IPCC 2007). Climate change and other global
environmental problems, however, are particularly difficult to address because actions that address
these problems impose private costs and generate a global public good. Successfully addressing
global public goods require concerted action by numerous sovereign countries. Sovereignty of
nations implies that any international environmental agreement must be self-enforcing for every
country. But designing self enforcement agreements is problematic given the nature of global
public goods because the self interest of each country is best served by having other countries bear
the cost of addressing the problem while they free-ride on these efforts.
A large number of prior studies have analyzed the benefits and costs of reducing greenhouse gas
emissions (e.g., Cline 1992, Fankhouser 1995, Manne and Richels 1992, Mendelsohn et al. 2000,
Nordhaus 1991, 1994, Nordhaus and Yang 1996, Nordhaus and Boyer 2000, Tol 1995; see Tol
2005 and 2007 for recent summaries). The release of the Stern Review (Stern et al. 2006), which
argued for much swifter and deeper cuts in GHG emissions than the dominant view in the
published economics literature, ignited a new round of discussion about optimal climate change
policy (e.g., Dasgupta 2007, Mendelsohn 2007, Nordhaus 2007, Tol and Yohe 2006, Weitzman
2007, Yohe et al. 2007). For the most part these studies do not analyze equilibrium in which
countries can choose emissions strategically.1
Analyzing countries' strategic interactions is crucial for understanding climate-change policy. As
the negotiations over the Kyoto Protocol illustrate, countries can choose to participate or stay on
the sidelines (e.g., US and China). Even if a country chooses to participate, there are limited
1
An exception is Nordhaus and Yang (1996) which solved for an open-loop Nash equilibrium emissions allocation by
countries as well as a Pareto optimal GHG emissions allocation.
-1-
sanctions available to punish countries that do not meet their climate change treaty obligations.
Addressing questions of whether a country will choose to participate in a climate change
agreement or will choose to comply with an agreement requires an analysis of the strategic
interests of each country involved in climate change negotiations. A number of studies apply static
or repeated games to consider countries' strategic choice of GHG emissions (Barrett 2003, Finus
2001). Bosello et al. (2003), de Zeeuw (2008) and Eyckmans and Tulkens (2002) incorporate the
dynamics of GHG stock to analyze an international agreement on climate change. However, these
studies focus on the stability of an environmental treaty by a subset of countries where the treaty
members are assumed to cooperate even when cheating may improve a treaty member's welfare.
Nordhaus and Yang (1996) investigate a dynamic game but assume that countries adopt open-loop
strategies (where countries commit to future emissions at the outset of the game, and so are not
necessarily subgame perfect equilibria). Yang (2003) studies a dynamic game allowing for
closed-loop strategies, but without including the role of punishment for potential defectors.
A desirable model of international environmental agreements as applied to problems such as
climate change would include stock effects (so that the interaction is properly modeled as a
dynamic, rather than a repeated, game), and would allow for countries to credibly punish defector
states. Such a model would by necessity focus on closed-loop or feedback strategies. To our
knowledge, only a few papers include these ingredients. Dockner et al. (1996) and Dutta and
Radner (2000, 2005) find conditions under which cooperative equilibrium can be supported as a
subgame perfect equilibrium through use of a trigger strategy. In these models, once some country
cheats on the agreement by over-emitting, punishment begins and continues forever. In the climate
change application, however, such a trigger-strategy profile would involve mutually assured
over-accumulation of GHGs if punishment were ever called for. A legitimate criticism of such
strategies is that they are not robust against renegotiation upon a country's deviation because the
countries restart cooperation once a temporary sanction is completed. In addition, most
international sanctions are temporary in nature, calling into question the empirical relevance of
strategies involving perennial punishment.2
2
Based on 103 case studies of economic sanctions between World War I and 1984, Hufbauer et al. (1985) find that the
average length of successful and unsuccessful sanctions were 2.9 and 6.9 years. Success of a sanction is defined in
-2-
In this paper, we reconsider the problem of designing a self-enforcing international environmental
agreement for climate change. Our model presents a dynamic game in which each country in each
period chooses its level of economic activity. Economic activity generates benefits for the country
but also generates emissions that increase atmospheric concentrations of GHG, which negatively
affect the welfare of all countries. Atmospheric concentrations evolve over time through an
increase of concentrations from emissions of GHG and the slow decay of existing concentrations.
We analyze a strategy profile in which each country initially chooses emissions that generate a
Pareto optimal outcome (first best or cooperative strategy) and continues to play cooperatively as
long as all other countries do so. However, our strategies entail temporary punishment: If a
country deviates from the cooperative strategy, all countries then invoke a two-part punishment
strategy. In the first phase, countries inflict harsh punishment on the deviating country by
requiring it to curtail emissions.
In the second phase, all countries return to playing the
cooperative strategy. The design of the two-phase punishment scheme guarantees that the
punishment is sufficiently severe to deter cheating, and that all countries will have an incentive to
carry out the punishment if called upon to do so.
We identify conditions under which this strategy can support the first-best outcome as a subgame
perfect equilibrium, i.e., when a self-enforcing international environmental agreement can
generate an efficient outcome. We provide a simulation model to illustrate conditions when it is
possible for a self-enforcing agreement to support an efficient outcome. We also parameterize the
simulation model to mimic current conditions to show whether a self-enforcing agreement that
achieves optimal climate change policy is possible, the structure of what such a solution might
look like, and which countries have the most to gain from such a agreement (or to lose from failure
to agree).
The two-part punishment scheme that we use here to find subgame perfect equilibrium that
supports an efficient solution is similar to that in previous studies that analyze cooperation in a
dynamic game of harvesting a common property resource (Polasky et al. 2006, Tarui et al. 2008).
However, unlike a harvesting game in which a player can always guarantee non-negative payoffs
terms of the extent to which the corresponding foreign policy goal is achieved (p.79).
-3-
(by simply not harvesting), the GHG emissions game can have arbitrarily large negative payoffs.
Damages increase with the stock of GHGs and the stock of GHGs is outside the control of any
single country.
In addition, and again in contrast to Dutta and Radner’s model, we assume nonlinear damage
effects of GHG stock on each country.3 Though there is a large degree of uncertainty about the
economic effects of future climate change, scientists predict that the effects may be nonlinear in
the atmospheric greenhouse gas concentration. Studies predict nonlinear effects of climate change
on agriculture (Schlenker et al. 2006, Schlenker and Roberts 2006). Nonlinearity may also arise
due to catastrophic events such as the collapse of the thermohaline circulation (THC) in the North
Atlantic Ocean: climate change may alter the circulation, which would result in significant
temperature decrease in Western Europe. Our numerical example with quadratic functions
captures this nonlinearity.
In what follows, section 2 describes the assumption of the game and a two-part strategy profile
with a simple penal code to support the cooperative outcome. Using an example with quadratic
functions, section 3 discusses the condition under which the two-part strategy profile is a subgame
perfect equilibrium. In Section 4, we choose the parameter values of the quadratic functions based
on previous climate-change models in order to illustrate the implication to climate-change
mitigation. Section 5 concludes the paper.
2. Basic model
2.1 Assumptions
There are N  2 players in a dynamic game, each representing a country. In each period
t = 0,1, each country i = 1,…,N chooses a GHG emission level xit  0 . For each country i, we
assume there is a maximum feasible emission level xi  0 . The transition of the GHG stock in the
atmosphere is given by
3
Dockner et al. (1996) assumes nonlinear damage functions and linear emission reduction costs. Our model assumes
both nonlinear damages and nonlinear emission reduction costs.
-4-
S t 1 = g ( S t , xt ) = S   ( S t  S )  X t ,
where X t  ixit , 1   represents the natural rate of decay of GHG per period ( 0   < 1 ), S is
the GHG stock level prior to the industrial revolution, and  is the retention rate of current
emissions ( 0    1 ).4 Let xi be a vector of emissions by all countries other than i and let
X i   j = ix j , the total emissions by all countries other than i .
We denote the period-wise return of country i with emission xi when the GHG stock is S by
 i ( xi , S ) . Each country’s emission level is linked to its output, and hence generates net benefits
from consumption. On the other hand, each country suffers flow damages associated with its
emissions. The function i summarizes the combination of these effects. Because emissions are
linked to net benefits, reducing emissions is costly for country i. Lastly, each country suffers
damages from GHG concentration; these damages are increasing in the GHG stock.
We assume that each country i's period-wise return equals the economic benefit from emissions Bi ,
which is a function of its own emissions, minus the climate damage Di , a function of the current
GHG stock:
 i ( xit , St ) = Bi ( xit )  Di (St ).
We assume that Bi is a strictly concave function with Bi (0) = 0 , that it has a unique maximum
xib  0 , and that Bi ' ( x) > 0 for all x  (0, xib ) . We call x ib the “myopic business-as-usual”
(myopic BAU) emission level of country i . This level of emissions maximizes period-wise returns,
without taking into account any future implications associated with contributions to the stock of
GHGs. The damage function is increasing and convex ( Di ' > 0, Di ' ' > 0 ), which captures the
nonlinear effects of climate change.
Countries have the same one-period discount factor   (0,1) . We assume that the economic
benefit and the damage for all countries grow at the same rate. The discount factor δ incorporates
4
Many studies have used this specification of GHG stock transition (Nordhaus and Yang 1996, Newell and Pizer
2003, Karp and Zhang 2004, Dutta and Radner 2004).
-5-
the growth rate of benefits and damages (see section 3.2 for more discussions about the discount
factor). All countries' return functions are measured in terms of a common metric. There is no
uncertainty (i.e., countries have complete information). In each period, each country observes the
history of GHG stock evolution and all countries' previous emissions.
2.2 First best solution
The first best emissions path solves the following problem.

max  t  i ( xit , S t )
t =0
i
s.t. S t 1 = g (S t , xt ) for t = 0,1, given S 0 .
The solution to this problem generates a sequence of emissions {xt*}t= 0 where xt* = {xit* }iN=1 . The
corresponding value function solves the following functional equation.
V ( S ) = max  i ( xi , S )  V ( S )
x
i
s.t. S  = g ( S , x).
We assume the solution is interior. The optimal emission profile given S ,
x* ( S ) = {x1* ( S ), x2* ( S ), , x*N ( S )} , satisfies the following.
 i ( xi* ( S ), S )
 V ( g ( S , X * ( S )))  = 0
xi
for all i . The first term represents the marginal benefit of emissions in country i while the second
term is the discounted present value of the future stream of marginal damages in all countries from
the next period. Thus, under the first best allocation, the marginal abatement costs of all countries
in the same period must be equalized, and they equal the shadow value of the stock.
The unique steady state S * satisfies the following equation:
 i ( x ( S ), S )


xi
1  
*
i
*
*
  j ( x *j ( S ), S )
j
S
= 0.
Given S 0 < S * , the stock increases monotonically to the steady state S * . For the rest of the paper
we assume S 0 < S * . In what follows, we describe a strategy profile that supports x* as a subgame
-6-
perfect equilibrium.
2.3 A strategy profile to support cooperation
Consider the following strategy profile  * , which may support cooperative emissions reduction
with a threat of punishment against over-emissions.5
Strategy profile  *

Phase I: Countries choose { xi*} . If a single country j chooses over-emission, with
resulting stock S  , go to Phase II j (S ) . Otherwise repeat Phase I in the next period.

Phase II j (S ) : Countries play x j = ( x1j ,, xij ,, xNj ) for T periods. If a country k
deviates with resulting stock S  , go to Phase II k (S ) . Otherwise go back to Phase I.
The idea of the penal code x j is to induce country j (that cheated in the previous period) to
choose low emissions for T periods while the others enjoy high emissions. Each sanction is
temporary, and the countries resume cooperation once the sanction is complete. The punishment
for country j in Phase II j works in two ways, one through its own low emissions (and hence low
benefits during Phase II) and the other through increases in its future stream of damages due to an
increase in the other countries' emissions during Phase II. Under some parameter values and with
appropriately specified penal codes {x j } , each country's present-value payoffs upon deviation
will not exceed the present-value payoffs upon cooperation. We now turn to a discussion of the
condition under which  * is a subgame perfect equilibrium.
Sufficient conditions for first best sustainability
Let V jC (S , I) and V jD (S , I) be country j 's payoff upon cooperation and the maximum payoff
upon deviation in Phase I given current stock S . Similarly, let V jC (S , IIi ), V jD (S , IIi ) be j 's payoff
5
The design of the penal code to support cooperation is similar to those discussed in Abreu (1988).
-7-
upon cooperation and an optimal deviation starting in Phase II i given stock S . With these
notations, the above strategy profile is a subgame perfect equilibrium if the following conditions
are satisfied for country j , j = 1,  , N :
Condition (1) Country j has no incentive to deviate in phase I: V jC (S , I)  V jD (S , I) ;
Condition (2) Country j has no incentive to deviate in phase II j : V jC (S , II j )  V jD (S , II j ) ; and
Condition (3) Country j has no incentive to deviate in phase II k for all k = j :
V jC (S , II k )  V jD (S , II k ) ;
for all possible stock levels given initial stock S 0 .6 Because each player's periodwise return is
bounded from above and the discount rate is positive, the principle of optimality for discounted
dynamic programming applies to this game. Hence, in order to prove that  * is subgame perfect, it
is sufficient to show that any one-shot deviation cannot be payoff-improving for any player
(Fudenberg and Tirole 1991). Because this is a dynamic game, we need to verify that no player has
an incentive to deviate from the prescribed strategy in any phase and under any possible stock
level .7
Because combined emissions are nonnegative and bounded by the maximum feasible level
X  i xi , the feasible stock levels lie between 0 and S  0 where S satisfies
S = S   ( S  S )  X .
We can exploit a few properties to simplify the above three conditions for first best sustainability.
Let xiD (S ) be the optimal deviation that maximizes country i’s payoff upon deviation in
either phase I or II. Under a reasonable assumption on xiD , the following propositions hold.
(Proofs are relegated to the appendix.)
Proposition 1 Suppose x jj  z  0 is independent of the post-deviation stock level, and
x
i i
j
( S ) = X * ( S ) given any S. Condition (2) implies conditions (1) and (3) provided
6
Notice the focus is on a single deviation. The reader may wonder what happens if more than one player deviates; in
keeping with the usual tradition in dynamic games, a simple way to avoid the complication of considering multiple
defections is to assume the game remains in Phase I if more than one player deviates; see Fudenberg and Tirole (1991,
pp. 157-160) for details.
7
See Dutta (1995a) for a similar analysis in a dynamic game context.
-8-
xiD ( S ) > xi* ( S ) holds.
Proposition 1 shows that, for a particular form of the penal code where the country that has
deviated emits a fixed level z and where other countries’ emissions are designed to render
combined emissions equal to the first best level given the post-deviation stock, it is sufficient to
evaluate condition (2). As such, the proposition greatly simplifies the task of verifying that the
strategy profile  * yields a subgame perfect equilibrium.
Proposition 2 Let t  T be the timing of deviation in phase II j where country j deviates from
( z, ) . The gains from deviation are largest when t = 1 for all j given any stock level at the
beginning of phase II j .
Proposition 2 extends Proposition 1 to penal strategies with multiple periods of punishment. Since
country j may cheat in any period during Phase IIj in this context, the concern is that there are now
many conditions to check. The proposition shows that it is sufficient to check condition (2) for the
first period of Phase IIj.
Proposition 3 Let Sm be the stock at which gains from deviation in phase IIj are maximized. If
these gains are non-negative, the penal strategy described in Proposition 1 yields a subgame
perfect equilibrium.
Because there is an upper bound on stock, there are at most three candidate values for stock that
could yield maximal deviation gains during phase II. If gains are monotonic in stock, the argmax
is either the lowest possible value8 or the largest possible value. If gains are non-monotonic, then
either there is an interior maximum (in which case the relevant stock value is that value which
deliver the interior maximum) or there is an interior minimum, in which case the argmax is one of
the two corners. In any event, it becomes a simple matter to check whether deviation gains in
8
Certainly pollution stocks cannot be negative. Plausibly, there is a subsistence level of economic activity (that
induces an associated subsistence level of emissions). Any country emitting less than that amount for an indefinite
period of time would preclude its survival (though it might be able to survive for short periods, as during a punishment
-9-
phase IIj are never positive.
When  * is not a subgame perfect equilibrium, there may be another strategy profile that supports
the first best as a subgame perfect equilibrium outcome. A punishment is most effective as
deterrence against over-emitting if it induces the over-emitter’s minmax (i.e. the worst perfect
equilibrium) payoff. Though such punishment supports cooperation under the widest range of
parameter values, a two-part punishment scheme inducing the worst perfect equilibrium may be
too complicated to generate useful insights about self-enforcing treaties. Previous dynamic game
studies have analyzed cooperation with worst perfect equilibria in the context of local commonproperty resource use (Dutta 1995b, Polasky et al. 2006). With local common-property resource
use, the minmax level is defined by outside options for resource users—the payoffs that they
would receive if they quit resource use. With a global commons problem such as climate change,
there are no outside options: a country can never escape from changed climate (without spending
potentially large amounts of resources for adaptation). With linear damage functions, Dutta and
Radner (2004) find that the worst perfect equilibria take a simple form (constant emissions by all
countries). With nonlinear damage functions, the worst perfect equilibria will be more
complicated because they may depend nonlinearly on the state variable. In this study, we restrict
our attention to  * , a strategic profile with a simple penal code, in order to gain insights about
countries’ incentives to cooperate in a treaty.
3. Homogeneous countries
Assume that each of the N countries' period-wise return functions are quadratic:
 i ( xi , S ) = axi  bxi2  dS 2 ,
where a, b, d > 0. The negative of the derivative with respect to emissions,  (a  2bxi ) ,
represents the marginal abatement cost associated with emissions xi . Country i 's myopic BAU
emission that maximizes the period-wise return is xˆi  xˆ 
a
. As in the appendix, the value
2b
function is quadratic and a unique linear policy function exists for the first best problem. The
phase). If so, the lower bound on stock is strictly positive, and conceivably equal to the initial value.
- 10 -
values 2b and 2d represent the slopes of the marginal costs of emission reduction and the marginal
damages from pollution stock.
We suppose that the initial stock is smaller than the potential steady state stock: S 0  S * . The
maximum feasible emission is given by B( xi )  0 , so xi  x  a / b for all i. Thus, S 
Na
.
b(1   )
Finally, we simplify the stock transition by setting S  0 and  = 1.
The strategy profile we investigate takes a particularly simple form: x jj (S )  z > 0 , a constant
smaller than x* ( S * ) , for all S  0 and all j ; xij ( S )  y ( S ) 
X * (S )  z
for all i = j ; and T=1.
N 1
With this penal code, all countries i = j choose the optimal aggregate emissions X * (S )
collectively for all S .
An example where a treaty works
Figure 1 illustrates a case where  * is a subgame perfect equilibrium. In each panel, the solid
curve represents the payoff upon cooperation while the dotted curve represents the payoff upon
optimal deviation. The optimal steady state is around 1.85 while the maximum feasible stock S
equals 4. Under the assumed combination of parameter values, the payoffs upon cooperation
exceed the payoffs upon optimal deviation under all relevant stock levels in each phase.
- 11 -
Cooperation vs. deviation in phase I
2.5
cooperate
deviate
2
1.5
Cooperation vs. deviation in phase IIj for j
2.5
cooperate
deviate
2
0
1
2
3
4
stock
Cooperation vs. deviation in phase IIk for j
2.5
cooperate
deviate
0
1
1.5
0
1
2
stock
3
4
2
1.5
2
stock
3
4
Note: The figure assumes a = 10, b = 1,000, d = 0.001, T = 1, z = 0, N = 4,  = .99,  = .99 .
Figure 1: An example where  * is a subgame perfect equilibrium.
Supportability of the penal code
b (slope of marginal benefits of GHG emissons)
1000
900
Shaded area: penal code is supportable
A
800
700
600
B
500
400
300
200
C
100
1
2
3
4
5
6
7
8
d (slope of marginal damages of GHG stock)
9
10
-3
x 10
Note: The figure assumes a = 10, T = 1, z = 0, N = 4,  = .99,  = .99 . The vertical and horizontal axes
measure b  [1,1,000] and d  [.00001,.01] respectively.
Figure 2: Supporting cooperation. (I)
Supportability of cooperation, marginal abatement costs, and marginal damages
- 12 -
Figure 2 illustrates that  * is a subgame perfect equilibrium when the ratio of the slopes of
marginal abatement costs and marginal damages, b/d is neither too large (as point A indicates) or
too small (as point C indicates). At point A , the magnitude of damages from pollution stock is
small relative to the magnitude of the costs of reducing emissions. In this case, the difference
between the optimal emissions and noncooperative emission levels are small. This implies that the
gains from cooperation might be too small for each country to find cooperation a best response.
For a smaller value of b/d , the marginal damages increase faster than the marginal abatement
costs as pollution stock increases. This fact implies that the difference between the optimal
emissions and noncooperative emission levels becomes larger. Because the optimal emission
control calls for larger emission reduction to each country, both the gains from cooperation and
temptations to deviate increase. At a point like B , the former exceeds the latter and  * supports
cooperation. However, at a point like C , the temptation to deviate exceeds the gains from
cooperation and hence  * is not a subgame perfect equilibrium.
Supportability of the penal code
0.995
0.99
A
delta (discount factor)
0.985
B
0.98
0.975
0.97
0.965
0.96
0.955
Shaded area: penal code is supportable
0.95
1
2
3
4
5
6
7
8
d (slope of marginal damages of GHG stock)
9
10
-3
x 10
The figure assumes a = 10, b = 500, T = 1, z = 0, N = 4,  = .99 . The vertical and horizontal axes measure
  [.950,.999]
and d  [.00001,.01] , respectively.
Figure 3: Supporting cooperation. (II)
Supportability of cooperation and discount factor
While in repeated games cooperative outcomes are only supportable if players are sufficiently
- 13 -
patient (an implication of the folk theorem), this need not be true in dynamic games (Dutta, 1995b).
Indeed, supportability of  * as a subgame perfect equilibrium is not necessarily monotonic in the
discount factor (see the arrow B in Figure 3). A key factor leading to this non-monotonicity result
is that the first best, optimal emissions and the associated optimal stock transition change as the
discount factor changes while the optimal actions would stay constant in repeated games. When
 is very low, cooperation is not supportable because the future payoff associated with
cooperation is discounted too heavily. As  increases, the payoff associated with cooperation
increases while the first-best emission level decreases. Therefore, both the future payoff associated
with cooperation and the payoff associated with optimal deviations increase. Movement along
arrow B in Figure 3 indicates that the latter may increase by a larger amount than the former when
the discount factor is sufficiently large.
Gains from deviation at different stock levels
(a) Gains from deviation decreasing in stock?
(b) Supportability of the penal code
100
100
Shaded area: Yes
80
90
b (slope of marginal benefits of GHG emissons)
b (slope of marginal benefits of GHG emissons)
90
A
70
60
B
50
40
30
20
10
80
A
70
B
60
50
40
30
20
10
0.5
1
1.5
2
2.5
3
3.5
d (slope of marginal damages of GHG stock)
Shaded area: penal code is supportable
f
4
e
0.5
1
1.5
2
2.5
3
3.5
d (slope of marginal damages of GHG stock)
-4
x 10
4
-4
x 10
Note: The figure assumes N = 4, δ=0.99, a=10, λ = 0.99, b between 1 and 100, d between .00001, .0004.
Figure 4 Supportability of cooperation and gains from deviation
(c) Xopt nonnegative?
(d) xd nonnegative?
100
100
Shaded area: Yes
Shaded area: Yes
Figure
4 illustrates the relation between the gains from
optimal deviation and the stock level.
90
90
rginal benefits of GHG emissons)
rginal benefits of GHG emissons)
Panel
(a) describes the set of combinations of parameter
values (b and d) where the gains from
80
80
deviation for player j in phase IIj are non-increasing in stock for levels. Panel (b) illustrates the set
70
60
- 14 50
40
70
60
50
40
of b and d where the penal code is a subgame perfect equilibrium. { this paragraph is weak: it
basically says we are wasting everyone’s time… I suggest augmenting panel b) to highlight
the range of values where a) applies, and perhaps labeling the graph in some way – maybe
with a named line segment from the y-axis to the top of the graph… then explain there are
values where both a) and b) apply… after that we can say there are also values where they
don’t both apply } We observe from (a) and (b) that (i) the gains from deviation can be
non-increasing when the penal code is not subgame perfect (such as at point A) and (ii) the penal
code can be subgame perfect when the gains from deviation are non-increasing (such as at point
B).
- 15 -
Minimum threshold stock level for cooperation
Minimum threshold stock level for cooperation (%)
100
90
GHG stock (% of the maximum feasible level
350
maximum feasible stock level
optimal steady state
minimum threshold for cooperation
GHG stock level
300
250
200
150
100
50
0
optimal steady state
minimum threshold for cooperation
80
70
60
50
40
30
20
10
50
100
150
b (slope of marginal benefits of GHG emissons)
200
0
50
100
150
b (slope of marginal benefits of GHG emissons)
200
Note: “Maximum feasible stock level” is the steady state associated with the maximum feasible emissions a/b.
The figure assumes a = 10, N =4, δ = .99, λ = .99, d = 0.00003, and b = 10, 11, …, 200.
Figure 5 Minimum threshold stock for cooperation
When the gains from deviation are non-increasing in stock, the treaty may become supportable as
stock increases (if we disregard the possibility that countries choose emissions lower than the
specified levels). Figure 5 illustrates the maximum feasible steady-state stock, the optimal steady
state, and the minimum stock level above which the penal code is supportable as a subgame perfect
equilibrium under a range of values of b. The line segment ef in Figure 4 represents the range of
parameter values in Figure 5. As b increases, the minimum threshold for cooperation (i.e. the stock
level at which the gains from deviation are non-positive) increases. When b is about 110, the
minimum threshold coincides with the optimal steady state. With any b larger than 110, the
minimum threshold exceeds the steady state stock and hence the penal code (with z=0, T=1) is not
subgame perfect starting with any stock level. At b=183, the gains from deviation become
negative at all possible stock levels. In order for cooperation to become self-enforcing, the
marginal damage (or the marginal benefit from emission reduction) must become large enough
(relative to the marginal cost of emission reduction).
- 16 -
4. Illustration with plausible parameter values for climate change
As we have seen so far, there are a few parameter values that play a crucial role in determining
whether a two-part penal code is a subgame perfect equilibrium: the discount factor, the ratio of
the slope of MAC to the slope of MD (b/d), and the number of countries N. What does our model
predict regarding the supportability of a simple treaty if we adopt to our study the parameter values
from other economic studies on climate change?
First, define B/D as the ratio of the slopes of MAC and MD at the global level. Given the number
of countries N, it turns out that B = b/N and D = Nd.9 Therefore, b/(N2d) = B/D. We consider a
range of the values of B/D including those used in the previous studies (e.g. Nordhaus and Boyer
2000, Newell and Pizer 2003, Karp and Zhang 2005). We set the annual retention rate of CO2
emissions at λ=.9917 (Nordhaus and Yang 1996, Newell and Pizer 2003).
Next, we extend the earlier discussion by allowing z to vary with S: z(S) = αx*(S), where 0  α <1
and x*(S) is the first best emission per country given stock S. The parameter α represents the
severity of punishment: if country j cooperates in phase IIj, it would reduce the emissions by 1- α
percent relative to the first best level. We retain the assumption that the countries other than j
choose X*(S) - z(S) collectively in phase IIj; they each therefore choose [X*(S) - z(S)]/(N-1). We
also retain the assumption that T = 1.
Choosing the discount factor
If we assume that country i’s net benefit from emissions and the damage from stock are
proportional to country i’s national income, and if income grows at the rate 1+g+n (where g is the
per capita income growth rate, and n the population growth rate), then the periodwise return to
country i is given by
( xit , St )  y0 (1  g  n)t {axit  bxit }  dy0 (1  g  n)t St2 ,
2
where y0 is the income in period 0. The payoff will be
9
When N identical countries choose the same emission level x, the global periodwise return is given by
N (ax  bx 2  dS 2 )  aNx  (b / N )( Nx) 2  NdS 2  AX  BX 2  DS 2 , where B  b / N and
D  Nd .
- 17 -


 t  ( xit , S t ) 
t 0


t 0

 t y 0 (1  g  n) t {axit  bxit 2 }  dy 0 (1  g  n) t S t2



y 0 t 0  (1  g  n) {axit  bxit }  dS t2 .

t
2

If δ = 1/(1+r) where r is the consumption discount rate, then the appropriate discount rate in our
context would be
(1+g+n)/(1+r).
Under optimal growth, r is the sum of the rate of time preference, , and a term equaling the
product of the growth rate of consumption, g, and the elasticity of marginal utility, . Then the
discount factor in our model would be
(1+g+n)/(1+  + g) .
A number of recent studies discuss what values should be used for these parameters when we
analyze climate change (Stern et al. 2006, Nordhaus 2007, Dasgupta 2007). As IPCC AR4 (2007)
illustrates, the damages caused by climate change in the distant future are uncertain. In particular,
uncertain growth rate of consumption implies that the discount factor is also uncertain. In the
presence of such uncertainty, the lowest possible discount rate should be used when we evaluate
the benefits and costs deterministically (Weitzman 1999, 2007). We illustrate whether our penal
code constitutes an equilibrium under a range of values of discount rates discussed in the
economics of climate change.
The number of countries
Though there are over190 countries in the world, a small number of countries (and regional
economic unions) have large shares of greenhouse gas emissions. According to the Netherlands
Environmental Assessment Agency (2008), in 2007 China’s CO2 emission was 24% of the world
total emission while the US, EU, India, the Russian Federation and Japan generated 22%, 12%, 8%,
6% and 4.5% of the total. Together, the emissions in these countries and regions constitute over
70% of the total world emissions. In our simulation, we consider numbers of countries relatively
smaller than the total number of countries and close to the number of large-scale GHG emitters.
- 18 -
gains from deviation <=0?
100
% stock levels at which gains <=0
90
80
70
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
alpha
0.6
0.7
0.8
0.9
1
Note: The figure assumes a = 10, b=184,000, N =6, δ = 1/1.01, λ = .9917, d =1/N2.
Figure 6 Penal code and supportability of cooperation. (I)
Figure 6 is based on the parameter values B/D = 1.84E+05 (Newell and Pizer 2003), N=6, and δ =
1/1.01 (i.e., a discount rate of 1%). With these values, the gains from deviation are nonpositive at
all stock levels for a range of values of α (between roughly 0.67 and 0.91). For other values of α,
however, there are some stock levels at which the gains from deviation are positive. Further, the
penal code becomes unsupportable at larger values of N when the other parameters are held fixed.
Importantly, the ability to sustain cooperation is linked to the discount factor. For larger values,
the penal code described above does not yield a subgame perfect equilibrium. Figure 7 assumes the
same parameter values as for Figure 6 except for δ, which is set at 1/1.02; here we see the penal
code is not supportable for any value of α. The two figures imply that, with N=6, the discount rate
must be quite small in order for the penal code to support cooperation.
While the two preceding figures suggest a non-monotonic relation between deviation gains and α,
this does not hold in general. Figure 8 displays the relationship when N = 20 and δ = 1/1.001.
Here, cooperation is supportable as long as α is lower than 0.51.
- 19 -
gains from deviation <=0?
100
90
% stock levels at which gains <=0
80
70
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
alpha
0.6
0.7
0.8
0.9
1
Note: The figure assumes a = 10, b=184,000, N =6, δ = 1/1.02, λ = .9917, d =1/N2.
Figure 7 Penal code and supportability of cooperation. (II)
gains from deviation <=0?
100
% stock levels at which gains <=0
90
80
70
60
50
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
alpha
0.6
0.7
0.8
0.9
1
Note: The figure assumes a = 10, b=184,000, N =20, δ = 1/1.001, λ = .9917, d =1/N2.
Figure 8 Penal code and supportability of cooperation. (III)
- 20 -
The slopes of MC of emission reduction and MD of stock
Though virtually all integrated assessment models assume nonlinear emissions abatement costs
and climate damages due to CO2 emissions, these studies use different values of B/D. Figure 9
illustrates supportability of the penal code for a range of the values of B/D. The red lines indicate
two values of B/D used in the literature. The figure has a couple of implications regarding the
economics of climate change. First, non-monotonicity regarding the discount factor (recall Figure
3) is not only a theoretical possibility but may be relevant in the context of climate change. We
observe that non-monotonicity might occur (i.e. an increase in discount factor makes the penal
code non-supportable) for some values of B/D between the two used in Newell and Pizer (2003)
and Karp and Zhang (2005). Secondly, the penal code is not supportable for low values of B/D
even though a lower value of B/D would certainly justify an aggressive climate-change mitigation
as an optimal policy. Therefore, a decrease in B/D—due to more optimistic estimates of the slope
of the marginal abatement costs or more pessimistic estimates about the slope of the marginal
damages—does not necessarily imply that cooperation becomes more supportable.
Supportability of the penal code
0.998
0.996
delta (discount factor)
0.994
0.992
0.99
B/D = 1.85*105 (used in Newell and Pizer 2003)
0.988
0.986
0.984
B/D = 0.123*105 (used in Karp and Zhang 2005)
0.982
Shaded area: penal code is supportable
0.98
0.5
1
1.5
2
2.5
3
3.5
4
4.5
B/D (ratio of slope of MC of emission reduction to slope of marginal damages of GHG stock)
Note: The figure assumes z=0, N=5, λ = 0.9917, a = 100, T=1, z=0.
- 21 -
5
5
x 10
Figure 9 Supportability of cooperation.
Number of countries
Figure 10 describes the relation between supportability of the penal code and the number of
countries. For a given value of the global ratio B/D, we set b=B/D and d = 1/N2. As expected, the
penal code tends to be supportable for lower numbers of countries. But we also note that the penal
code is only supportable when B/D is neither too small nor too large.
Supportability of the penal code
12
11
10
N (number of countries)
9
8
7
6
5
4
3
Shaded area: penal code is supportable
2
0.5
1
1.5
2
2.5
3
3.5
4
4.5
B/D (ratio of slope of MC of emission reduction to slope of marginal damages of GHG stock)
5
5
x 10
Note: The figure assumes z=0, δ= .995, λ = 0.9917, a = 100, T=1, z=0.
Figure 10 Supportabiliy of cooperation (II).
5. Heterogeneous countries
How does heterogeneity across countries influence supportability of  * ? In this section we sketch
out a variation of our model that allows investigate of this question. To this end, we assume the
punishment phase is characterized by x jj (S )  0 for all S  0 and all j and xij ( S )  xib (the
myopic BAU emission) for all i = j . We retain the earlier assumption that S 0  S * .
- 22 -
We allow transfers among countries. Let  it be the net transfer to country i in period t where

i it
= 0 for all t . Country i 's net one-period return in period t is given by  i ( xit , St )   it . In the
context of climate-change mitigation, the transfers would be determined based on cost burden
sharing agreed on by the countries.
Suppose the value to country i is given by  iV where  i  0 and

i i
= 1 . Define a transfer  *
where
 i* ( S )   i  j ( x*j ( S ), S )   i ( xi* ( S ), S )
j
for all S and all i . By choosing x* and  * , the countries realize the first best outcome with the
shares induced by  .
Table 1 Condition (2) with heterogeneous countries.
Note: In all cases we assume
(Low) cases, and
 = .99 and vi = V/N
i . ai =
 = 0 . Conditions (1) and (3) hold for all S  S *
NL: Condition (2) does not hold when
hot hold for any
for all
S
is low relative to
10.2 (10.0),
bi = 5.0001 (5), di
= .00002 (.000025) for High
for all countries in all cases. Y: Condition (2) holds for all
S * , NH: Condition (2) does not hold when S
is close to
S  S* ,
S * , N: Condition (2) does
S  S* .
Table 1 describes whether condition (2) holds under different discount factor values for a game
with 8 heterogeneous countries. We assume two values---high and low---for each of the benefit
and damage function parameters {ai , bi , di } and the eight countries have different combinations of
- 23 -
these parameter values. The table assumes vi = V/N for all i , i.e. the payoffs upon cooperation are
divided equally across countries. With this example, conditions (1) and (3) under which  * is a
subgame perfect equilibrium holds given all discount factor values considered (2.5 – 10%).
Condition (2) also holds (and hence  * is a subgame perfect equilibrium) when the discount rate is
2.5%. For a larger discount rate, condition (2) is violated – first for the countries with larger BAU
emissions (i.e. larger
ai
) and low marginal damages di , then for those with low BAU emissions
2bi
and high marginal damages. Given equal sharing of V , a country with a higher BAU emission and
lower marginal damages has less to lose by deviation than countries with lower BAU emissions
and higher marginal damages. This example illustrates different incentives for controlling
emissions by countries with different benefits and damages.
6. Discussion
Climate change mitigation is a global public good where reducing GHG reduction is costly for
each country while the GHG stock accumulation in the atmosphere is likely to cause damages to
many countries. In order for sovereign countries to cooperate through an international agreement
to control GHGs, the agreement must be self-enforcing for each country. We applied a dynamic
game to illustrate an international agreement with a simple rule of sanctions in order to support the
first best, cooperative climate-change mitigation. Instead of a trigger strategy where all countries
choose over-emissions forever upon some country's cheating, we considered a two-part penal code
where the sanction against an over-emitter is temporary and where countries resume cooperation
upon completion the sanction. With numerical examples and illustration using a simple
climate-change model, we examined the conditions under which such a simple two-part sanction
scheme---where the country being sanctioned chooses a low emission and the others choose
over-emissions for one period---is a subgame perfect equilibrium. In particular, we studied how
heterogeneous countries' incentive for cooperation may change over time given GHG stock
dynamics and nonlinear effects of GHG on each country's payoff.
Our numerical example confirmed that each country's incentive to cooperate may change as the
- 24 -
stock level changes. We might expect that it may become easier for countries to avoid free riding
and cooperate as GHG stock increases; however, we found that a larger stock level does not
necessarily imply that the sanction scheme is more likely to be a subgame perfect equilibrium.
Because sanctions are more severe when the number of countries is larger, the sanction scheme
that is not an equilibrium for a given total number of players can be an equilibrium given a larger
number of players.
Our linear-quadratic climate-change model with parameter values from existing studies illustrates
different incentives to support cooperation held by countries with different benefits from GHG
emissions and different potential damages from increases in the GHG stock. Given heterogeneous
benefits from emissions, countries such as US and Europe, which have larger baseline GHG
emissions than the other countries, have larger incentive to deviate from the first best emissions
reduction than the others. Considering heterogeneity in potential damages from climate change,
we found that lower-middle income countries and Western European countries will have the most
to gain from cooperation due to relatively larger vulnerability to climate change. This finding
about heterogeneity is similar in spirit to Mason and Polasky (2003), who found that an
oil-producing country's OPEC membership is significantly associated with the country's larger oil
reserves (implying larger benefits from cooperation) and smaller domestic oil consumption
(implying smaller benefits in consumer surplus from non-cooperation).
Our climate-change model also suggested the weak link for cooperation---those countries that
have the most to gain from deviation---depends on how the burden of GHG emission control is
distributed across countries. In international negotiation and under Kyoto Protocol, policymakers
argue that richer countries including US or those countries with large GHG emissions in the past
should bear larger cost burden. Our simulation suggests that US is unlikely to cooperate with the
simple sanction scheme if the cost burden is proportional to GDP (because of its relatively large
benefits from GHG emissions and relatively moderate potential damages from US). In contrast,
Western Europe may have incentive to cooperate under the same cost sharing rule despite its
relatively large GDP because Western Europe is likely to be more vulnerable to climate change
than US. These findings imply that the cost sharing rule must be correctly specified in order for
climate-change mitigation to be self-enforcing to all countries and that the self-enforcing cost
- 25 -
sharing rule may not coincide with a rule perceived to be fair in an international context.
Further sensitivity analysis will be necessary for our linear-quadratic climate-change model.
Future research should also address a number of assumptions that we made to keep our analysis
simple. We assumed that each country's periodwise return function is time-invariant. However, the
benefit from GHG emissions and damages from climate change will change over time due to
changes in population, economic growth and technological progress. In addition, the benefits and
damages may change at different rates for different countries. Our analysis on heterogeneity does
not consider these possibilities.
A natural extension of our model would be to incorporate uncertainty regarding climate change.
Another useful extension would be to consider sanctions through means other than increased
emissions such as trade (Barret 2003). A temporary trade sanctions may be less costly for each
country than sanctions with increased emissions, the effect of which will last for a long time
because of the nature of GHG as a stock pollutant. Future research may study the extent to which
the availability of trade sanctions increases the likelihood of a self-enforcing treaty.
Our findings, despite their tentativeness, imply that dynamic-game formulation provides a useful
framework for analyzing a self-enforcing treaty for climate-change mitigation and a useful insight
that may not be available from static-game or repeated-game analysis.
Appendix 1
Conditions for first-best sustainability
Suppose T = 1 . Consider countries' incentive to deviate in Phase I. In period t , given current
stock St , country j 's payoff upon cooperation is V j ( S t ) . Country j 's payoff upon deviation, with
over-emission x Dj in period t , is given by


 j ( x Dj , St )  max 0,  j  i ( xi* ( S ), S )   j ( x*j ( S ), S )   j ( x jj ( St 1 ' ), St 1 ' )   2V j ( St  2 ' )


i
where St 1 ' = g (St , xj j , x Dj ) and St  2 ' = g ( St 1 , x* ( St 1 )) . This is a discounted sum of a current gain
by over-emitting in period t , a low return in period t  1 due to punishment, and continuation
- 26 -
payoffs with a larger GHG stock due to its own over-emission. Therefore, no country deviates
from Phase I if


V j ( S )   j ( x Dj , S )  max 0,  j  i ( xi* ( S ), S )   j ( x*j ( S ), S )
i


(1)
  j ( x jj ( g (S , x Dj , x* j (S ))), g (S , x Dj , x* j (S )))   2V j ( g ( g (S , x Dj , x* j (S )), x j ( g (S , x Dj , x* j (S )))))
for all x Dj  0 , S  S0 and all j .
In Phase II j , country j does not deviate from cooperation if
 j ( x jj (S ), S )  V j ( g (S , x j (S )))   j ( x Dj , S )   j ( x jj ( g (S , x Dj , xj j (S ))), g (S , x Dj , xj j (S ))) (2)
  2V j ( g ( g (S , x Dj , xj j (S ))), x j ( g (S , x Dj , xj j (S ))))
for all x Dj  0 and all possible stock levels given S 0 (i.e. for all S  g (S0 , ( x* j (S0 ), x Dj )) ).
Similarly, in Phase II k , country j has no incentive to deviate if
 j ( x kj (S ), S )  V j ( g (S , x k (S )))   j ( x Dj , S )   j ( x jj ( g (S , xk j , x Dj )), g (S , xk j , x Dj ))
(3)
  2V j ( g ( g (S , xk j , x Dj ), x j ( g (S , xk j , x Dj ))))
for all x Dj  0 and all S  g ( S 0 , ( x* k , xkD )) .
To summarize, the strategy profile is a subgame perfect equilibrium if conditions ((1)), ((2)), and
((3)) (for all k = j ) hold for all possible deviations, all S  S0 and all j .
Proof of Proposition 1
Let x̂ be the optimal deviation by country i in phase II i , v the optimal value function of country
i , and X * the optimal total emission of N countries (all of which are functions of stock S .) Let
G ( z , St ) be the gain from devaition for country i in Phase II i ( S t ) :



G ( z , S t ) = B( xˆ )  D( S t )   B( z )  D( S td1 )   2 v( S td 2 )  B( z )  D( St )  v( St*1 )


= B( xˆ )  B( z )   B( z )  D( S td1 )   2 v( S td 2 )  v( S t*1 ),
where S td1  S t  X * ( S t )  z  xˆ , Std 2  Std1  X * ( Std1 ), and St*1  St  X * ( St ).
G
= B ( xˆ )  (1   ) B ( z ) .
Step 1. To show
z
- 27 -

The partial derivative of G with respect to z is given by
G
xˆ
xˆ 
xˆ 


= B ( xˆ )  B ( z )  B ( z )  D ( S td1 )  1     2 v ( S td 2 )   X *' ( S td1 )  1  
z
z
z 
z 



=


xˆ
B ( xˆ )  D ( S td1 )   2 v ( S td 2 )   X *' ( S td1 )
z




 B ( z )  B ( z )  B ( xˆ )  B ( xˆ )  D ( S td1 )   2 v ( S td 2 )   X *' ( S td1 )
where the first term equals zero due to the envelope theorem. Therefore,
G
=  B ( xˆ )  D ( S td1 )   2 v ( S td 2 )   X *' ( S td1 )  B ( xˆ )  (1   ) B ( z ).
z
Applying the envelope theorem once again, we have
G
= B ( xˆ )  (1   ) B ( z ).
z
Let xˆ I ( S t ) be the optimal deviation by country i in phase I ( St ) and GI ( z, St ) the gain from


deviation for country i in Phase I ( St ) :




G I ( z , S t ) = B( xˆ I )  D( S t )   B( z )  D( S td,1I )   2 v( S td,2I )  B( x* )  D( St )  v( St*1 )



= B( xˆ I )  B( x * )   B( z )  D( S td,1I )   2 v( S td,2I )  v( S t*1 ),
 S t  X ( S t )  x ( S t )  xˆ I , S  S  X * ( Std1 ), and St*1  St  X * ( St ).
GI
= B( z ) .
Step 2. To show
z
The partial derivative of G with respect to z is given by
GI
xˆ
xˆ
xˆ
= B( xˆ I ) I  B( z )  D(S td1 ) I   2 v(S td 2 )   X *' ( S td1 ) I
z
z
z
z
where S
d ,I
t 1
*
*
d ,I
t 2
d
t 1




xˆ I
B( xˆ I )  D( S td1 )   2 v(S td 2 )  B( z ) = B( z ),
z
where the last equality follows from the envelope theorem.
Step 3. To show G( z, S )  GI ( z, S ) for all S .
=
Note that G( x* ( S ), S )  GI ( x* ( S ), S ) = 0, i.e. the gains from deviation are the same in
Phases I and II when z equals x* ( S ) . Steps 1 and 2 imply

[G ( z , S )  G I ( z , S )] = B ( xˆ ) < 0
z
for all z [0, x* (S )] given any stock level as long as xˆ > z . It follows from the last two equations
that G  GI for all z . ▄
The first best solution of a quadratic example with heterogeneous countries
Suppose country i 's periodwise return in period t is given by
ai xit  bi xit2  ( g i St  d i S 2  f i ),
- 28 -
given emissions xit and stock St where ai , bi , d i > 0 for all i . (The parameters {gi , f i } may take
positive or negative values.) Let a = (a1 , a2 ,, aN ) , b = (b1 , b2 ,, bN ) , and d = (d1 , d 2 ,, d N ) be
column vectors. Let G = igi and F = i f i . Let B be an N by N diagonal matrix whose
diagonal entries are b . Given stock S and an emissions profile x , The total periodwise return of
N countries is
ax  xBx  (GS  DS 2  F ),
where D = idi . The state transition is given by
St 1 = S   x  h
where   (0,1) ,  a column vector where 1  i represents the short-run decay rate, and
h  (1   )S is a constant. Let V be the total value function of all N countries:
V ( S ) = max ax  xBx  (GS  DS 2  F )  V ( S )
x
s.t. S  = S   x  h . Because V is quadratic, let V ( S ) = PS 2  QS  R where P, Q, R are
scalars. We have
V ( S ) = (S   x  h) P(S   x  h)  Q(S   x  h)  R
= (S  h) P(S  h)  (S  h) P x  xbP(S  h)  xP x  QS  Q x  Qh  R.
So

V ( S ) = P(S  h)  P(S  h)  2P x  Q
x
= 2P(S  h)  2P x  Q .
The first order condition is
a  2 Bx   [2 P(S  h)  2P x  Q ] = 0.
Hence [2 B  2P ]x = a  2P(S  h)  Q , i.e.
x = (1/2)[ B  P ]1[a  2P(S  h)  Q ].
Substitute this expression into the functional equation and we obtain the following expression:
P =  D  2 P   22 P 2  ( P)  ,
where ( P)  [ B  P ]1 . Solve this function for P , and we can solve for the remaining
unknown Q and R .
 Pa( P)   G  2 2 P 2h ( P)  2hP
Q=
,
1   2 P ( P)   
1
1
R=
{ (a  Q )( P)( a  Q )  (a  Q ) ( P)Ph
1 
4
  2 P 2  ( P)h 2  Qh  F  h 2 P}.
Value functions with no transfers
Here we compute each country's value function under the first best outcome when there are no
transfers among countries. Let
xi* ( S ) =  i S   i
- 29 -
(4)
be country i 's optimal emission given stock S  0 and X * ( S ) = S   the optimal aggregate
emission given stock S  0 , where   i i and   i i .
S1* = S 0  X * ( S 0 )  h = (  ) S 0    h,
S 2* = S1  X * ( S1 )  h = (  ) S1    h = (  ) 2 S 0  (  )(   h)    h,
,
t
St* = (  ) t S 0  (  ) i 1 (   h)
i =1
= (  )t S0 
  h
{1  (  )t }.
1  (  )
Plug this into country i 's payoff under the first best outcome (with no transfers) and obtain

vi ( S 0 )   t i ( xit , St* ) =  t [ai xi ( St* )  bi ( xi ( St* )) 2  {g i St*  d i St*2  f i }]
t =0
t
=  t [ai (i St*   i )  bi (i2 St*2  2i i St*   i2 )  {gi St*  di St*2  f i }]
t
=  t [ J i St*2  K i St*  Li ],
t
where J i  b   d i , K i  {aii  2bii i  gi , and Li  ai i  bi i2  f i . Note that
2
i i
St*2 = (  ) 2t S02 
2(  )t S0 (   h){1  (  )t }
1  (  )
2
   h 
t
2t

 {1  2(  )  (  ) }.
1  (  ) 
Plug this into the expression of vi , and we have
J i St*2  Ki St*  Li = J i (  ) 2t S02 
2 J i (  )t S0 (  h) 2 J i (  )t S0 (  h)(  )t

1  (  )
1  (  )
2
   h 
t
2t
 Ji 
 {1  2(  )  (  ) }
1

(




)


  h
 K i (  )t S0  K i
{1  (  )t }  Li .
1  (  )
So

vi ( S0 ) = S02   t J i (  ) 2t

t
 2 J i (  )t (   h) 2 J i (  )t (   h)(  )t

 S0   

 Ki (  )t 
1  (  )
1  (  )
t


2
    h 

  h
t
2t
  t  J i 
{1  (  )t }  Li 
 {1  2(  )  (  ) }  K i
1  (  )
t
 1  (  ) 

t
- 30 -
Hence, country i 's value function vi satisfies
vi ( S ) = pi S 2  qi S  ri ,
where
Ji
,
1   (  ) 2
2 J (    h)
2 J (    h)
Ki
1
1
qi  i
 i

2
1  (  ) 1   (  ) 1  (  ) 1   (  ) 1   (  )
pi 
2
2
   h  1
   h 
L
1
  h
1
  h
1
ri  J i 

 Ki
 Ki
 i ,


1  (  ) 1  
1  (  ) 1   (  ) 1  
1  (  )  1   1  (  )  1   (  )
for all S . Note that
p
i
i
= P,
q
i i
= Q , and
r = R
i i
hold so that
v
i i
= V , the aggregate
value function.
Optimal deviation
Assume T = 1 , x jj (S )( z j (S )) = x*j (S ) where 0   < 1 , and
xij ( S ) = xi* ( S ) 
xi* ( S )
(1   ) x*j ( S ) . Note that
*
k = jxk (S )
x jj ( S )  xkj ( S ) = x*j ( S )  xi* ( S )  (1   ) x*j ( S ) = X * ( S ),
k = j
k = j
for all S . Hence the total emission in Phase II j (S ) equals the socially optimal aggregate emission
given stock S .
The optimal deviation given S soves the following problem.


2
2
*
*
2
2
2
max ai x  bi x  (d i S  g i S  f )   aixi (S1 )  bi {xi (S1 )}  (d i S1  g i S1  f )   vi (S 2 ),
x 0
where S1 = S   ( X i  x)  h = S  X i  h  x ,
S2 = S1  X * ( S1 )  h = S1   (S1  )  h = (  )(S  X i  h)  (  ) x    h .
The first-order condition is

x* ( S ) S1
x* ( S ) S1 
S
S 
0 = ai  2bi x   ai i 1
 2bi 2 xi* ( S1 )  i 1
  2di S1 1  gi 1 
S1 x
S1 x 
x
x 

S
  2 2 pi {(   )(S  X i  h)  (  ) x    h}  qi  2
x
2
= ai  2bi x   aii   2bi ( i (Ci  x)   i )  i   (2d i (Ci  x)   g i  )

  2 2 pi {(   )Ci  (  ) x    h}  qi (  )  ,
where Ci  S  X i  h .
Arrange the terms and we have
0 = ai   aii   2bi 2 ( i Ci   i ) i   (2d i Ci   g i  )




  2 pi {(   )Ci    h}  qi (  )   2bi x    2bi ( i x) i   (2d i x )
2
  2 2 pi {(   ) x}(  )  .
- 31 -
2

Solving the condition for x , we obtain country i 's optimal deviation x d :
xid ( S ) =


ai   aii   2bi 2 ( i Ci   i ) i   (2d i Ci   g i  )   2 2 pi {(   )Ci    h}  qi (  ) 
.
2bi   2bi 2 ( i  ) i   (2d i  2 )   2 2 pi {(   )  }(  ) 


Note that C i depends on the stock level as well as the phase in which country i deviates:



S   { x * ( S )}  h,
in Phase I ( S );

j

j = i

Ci = S   {(1   ) xi* ( S )  x *j ( S )}  h,
in Phase II i ( S );
j = i




*
x
(
S
)



*
*
*
i
S   x j ( S )  xi ( S )  x * ( S ) (1   ) xk ( S )  h, in Phase II k ( S ).

j
 j = i


j = k



- 32 -
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