Optimal Contracts and Transboundary Pollution

The Case of International Rivers

Master’s Thesis

Submitted by

Caroline E. Abraham

In partial fulfilment for the

Degree of Master of Science in Economics

Submitted to:

Department of Policy Studies

TERI University

Plot # 10, Institutional Area, Vasant Kunj

New Delhi - 110070

INDIA

May 2012

Acknowledgement

I would like to thank Prof. Krishnendu Ghosh Dastidar, without whose unfailing kindness,

support and patience this paper would not have been possible. I shall always remain indebted

to him for helping me at every stage, answering every one of my panicked emails with

perfect equanimity and promptness, and gently encouraging me to believe in myself and this

paper at all times.

I would also like to thank Prof. Badal Mukherji, in whose office the ideas for this paper were

first discussed, for always giving me the strength to pursue the road less travelled.

I am also extremely indebted to Dr. Shubhro Bhattacharya for providing many hours of

insightful discussion, support and encouragement, all free of charge.

I would also like to acknowledge the invaluable help of Mr. Soumendu Sarkar, who has

played a significant role in raising the quality of this paper with his extremely useful and

incisive comments.

I would like to thank my friends, with special mention to Ankita Rathi, Akansha Nath,

Naveen J. Thomas and Ayush Pant for helping and supporting me through all the countless

sleepless nights and stressful days.

Last, but certainly not the least, I would like to thank my family – I could never have reached

this point without all of you.

i

Table of Contents

Abstract ii

Chapter 1: STATEMENT OF THE PROBLEM 1

Chapter 2: LITERATURE REVIEW 4

Chapter 3: THE MODEL 8

3.1 Social Optimum 10

3.2 Gains to Cooperation 11

3.3 A Principal-Agent Formulation of the Transboundary Pollution

Problem 12

3.3.1 Case (a): D can observe q1 and q2 individually 13

3.3.2 Case (b): D can only observe the total Q2 = q1 + q2 14

Chapter 4: PERFECT BAYESIAN EQUILIBRIUM AS A SOLUTION

CONCEPT 23

Chapter 5: CONCLUSION 28

REFERENCES

ii

Abstract

This paper attempts to model the problem of transboundary pollution in international rivers

using the framework of a Principal-Agent model. With 3 countries located in sequence along

a river, the lack of observability of countries‟ individual polluting actions (except in the

instance of the first player) introduces the problem of imperfect information. Downstream

countries being affected by the pollution of those upstream will devise contracts in order to

enjoy the gains possible from overall cooperation. The paper looks at two particular

contracting structures: In the first case, the country furthest downstream (D) attempts to

contract with both upstream agents individually in the presence of such asymmetry in

information, and in the second, I propose a contracting structure which overcomes the issue

of imperfect information by building in incentives for each downstream country to make

transfers to the country immediately upstream, or its immediate predecessor. The results

show that in the former setup, unilateral transfers from D to the upstream states fail to

achieve the cooperative outcome, to the extent that countries will actually be incentivized to

pollute more than their Nash levels in response to the offers. In the latter situation, the

cooperative outcome can be achieved provided that the parameters of the model fall within an

appropriate range. The paper also examines the feasibility of applying the concept of Perfect

Bayesian Equilibrium to the problem of transboundary pollution and finds that optimal

transfers will achieve the cooperative outcome, but the beliefs necessary to sustain such an

outcome may very likely be unrealistic.

1

Statement of the Problem

The objective of this paper is to model a specific form of global pollution, namely, the

problem of transboundary pollution in international rivers. Countries sharing a river may

utilize it for various purposes, not least of which includes the environmental service offered

by the river as a repository for industrial waste and other pollutive agents. The geographical

character of a river ensures that countries are located sequentially along the flow of the river.

This immediately creates a situation where the actions of countries located upstream of the

river affect the payoffs of downstream parties. Specifically, the use of the river for dumping

waste by upstream states immediately poses a negative externality to the states located

downstream. I intend to examine this scenario in the general framework of a Principal-Agent

model, where countries located downstream play the role of the Principal and attempt to

achieve a cooperative outcome by means of contracting with upstream Agents.

The problem is then one of designing an optimal contract or agreement between the two types

of parties to achieve the socially optimal aggregate level of pollution, where by definition, the

requirements of incentive compatibility and individual rationality are satisfied. While

frameworks that deal with global pollution problems do exist, the unique nature of this

problem rests in that the damages caused by the pollutive behaviour of one party are

unidirectional in nature, i.e. they do not add to overall damages (as in the instance of carbon

emissions contributing to global warming) but directly impact only downstream parties. The

contribution made by this paper relates to the fact that much formal study of the

transboundary river pollution problem has not yet been done using the Principal-Agent

framework.

For the purpose of keeping the analysis tractable while maintaining the spirit of the original

theoretical problem (involving an international river flowing through a number of countries),

I restrict my model to include just 3 countries located in sequence alongside an international

2

river, namely: an upstream country (U), a country located midway (M) along the river and

downstream of U, and the country located farthest downstream (D), i.e. third in the line after

U and M. Each riparian state derives a benefit as well as suffers a cost from pollution. I

assume that riparian states possess the right to pollute the river. In addition, I assume that

pollution is observable. Each country can always observe the total level of pollution in the

river before it makes its own choice to pollute.

The unidirectional flow of negative externalities allows us to represent the pollution problem

in the form of a game tree. The basic structure of the problem can be illustrated as follows:

While the above diagram, coupled with the assumption that each country knows the total

level of pollution flowing into its own territory, suggests a sequential game, in this paper I

begin by assuming that each country chooses its level of pollution independent of the actions

of the others; in other words, I use the game-theoretic concept of simple Nash equilibrium as

opposed to that of Sub-game Perfect Nash Equilibrium. This assumption is made in order to

reflect what appears to be the usual case: Countries usually make their own decisions

regarding pollution without taking into consideration the impact of their choice on other

states.

Regardless of the above assumption, writing the problem in the form of the above figure

allows us to highlight two pertinent features of the following analysis:

3

(1) As U is located farthest up-stream and so is the first to exercise the option to either

pollute or not pollute, M always knows whether U has made the decision to pollute or

not, and more importantly, can observe the amount that U has chosen to pollute.

(2) Country D on the other hand, being located last in the sequence, knows whether the

river has been polluted or not, but is unable to tell whether U has polluted, M has

polluted or whether both have contributed to pollution.

I argue that D being affected by the actions of both M and U will find it in its interest to

attempt to effect cooperation between the three parties, and so will offer a contract to the

upstream party(s) such that it is incentive compatible to produce the cooperative level of

pollution. The complication in designing such a contract arises because D is unable to

observe and/or verify how much U and M individually pollute, as D can only observe the

total amount of pollution. The rest of the model attempts to illustrate the problem in

contracting with all upstream parties in the presence of imperfect information. I then suggest

a particular contract structure that contains in-built incentives for further contracting as a

solution to the above problem.

4

Literature Review

Transboundary pollution has certain distinguishing characteristics. The polluting behaviour of

one country can have negative externalities which spill outside the boundaries of their source

of origin. Popular examples include acid rain, ozone depletion, global warming and the

environmental degradation of lakes and rivers shared by more than one country. This type of

pollution poses certain unique institutional challenges: the difficulty in designing regulatory

instruments and furthermore, in enforcing these pollution-control mechanisms. The absence

of a world government or an independent third-party arbiter vested with the necessary

resources and authority to carry out these tasks then implies that the solution to this problem

must lie within. As the management of transboundary pollution cannot be based on

regulation, pollution abatement must become a product of negotiations and voluntary

agreements amongst states, which evolve a commonly acceptable legal framework for

resolving the issue.

International environmental agreements (IEAs) are one example of such an effort undertaken

by the states party to a particular pollution problem. In practice, however, IEAs have found

themselves extremely vulnerable to the problem of free-riding, a natural consequence of the

public nature of transboundary pollution. Each member of an IEA will find it profitable to

reap the benefits of pollution abatement accruing from the abating activities of the other

members, whilst simultaneously refraining from such behaviour in an effort to reduce her

own costs (Xepapadeas, 1997). In essence, the possibility of capturing large gains through

free-riding will erode the sustainability of the IEA, rendering the environmental problem

effectively unresolved.

While global pollution in general presents institutional challenges which are a result of its

nature as a public „good‟, one particular instance of international pollution presents certain

unique difficulties for agreement design: the problem of transboundary pollution in

5

international rivers. The geographical character of a river immediately creates upstream-

downstream relationships amongst the countries located along the river, or through whose

territories the river flows. The immediate implication of this feature is that the negative

externalities posed by the polluting behaviour of upstream states are unidirectional in nature:

Owing to the unidirectional flow of a river, upstream countries do not suffer the

consequences of their polluting actions; these are invariably borne by downstream parties

(Moellenkamp, 2007). Historically, efforts to solve such problems have been incredibly time-

consuming and, unsurprisingly, unsuccessful. One exception is the case of the Rhine river in

Europe, which was a rare (and famous) example of how international cooperation amongst

the concerned parties succeeded in evolving various mechanisms to solve the pollution

problem and monitor the actions of the polluting offenders (Huisman et al, 2000).

The challenge posed by the upstream-downstream relationship is therefore the necessity to

devise sufficient incentives so that the upstream country alters its pollutive behaviour and

equally if not more importantly, to find an acceptable means of monitoring and enforcing the

agreement between the two types of parties. The difficulties involved in designing and

achieving such a solution are intrinsically related to two basic issues: (a) the question of how

the gains to cooperation are to be divided amongst the riparian states and (b) issues of trust

and cooperation amongst the concerned parties.

The division of the gains to cooperation amongst countries is essentially a question of

property rights. The Coase theorem tells us that under general conditions, the allocation of

property rights is irrelevant for the achievement of the efficient level of pollution, but

determines the final allocation of wealth between the involved parties (Coase, 1960). In the

context of international rivers, there are two relevant concepts of property rights: the principle

of unlimited territorial sovereignty and the principle of unlimited territorial integrity. The

former principle decrees that countries have “exclusive rights to the use of waters within its

territory,” while the latter states that “the quantity and quality of water available to a country

6

cannot be altered by another country” (Barrett, 1994a). Adherence to either doctrine will

determine the outcome of the bargaining process, as each doctrine implies something

different for property rights prior to negotiation between the countries. In the parlance of

environmental economics, unlimited territorial sovereignty then implies the right to pollute

and unlimited territorial integrity the right to a clean environment. In the instance where the

former principle operates, cooperation may be achieved via payments made by downstream

states to those upstream, whereas when the latter principle is adopted, upstream states will

have to compensate those downstream in order to pollute the river.

Cooperative game theory developed the concept of self-enforcing IEAs as a way of dealing

with issues of trust and cooperation amongst countries. Self-enforcing agreements contain

inbuilt mechanisms that ensure that nations find it in their self-interest to adhere to the

agreement and hence have no incentive to renege from it. Research on the study of such

coalitions, however, has been largely pessimistic. Cooperation between a large number of

countries cannot be sustained by IEAs if the difference in global net benefits between the

non-cooperative and full-cooperative outcomes is large, i.e. if the free-riding incentives are

too high (Barrett, 1994b; McGinty, 2007).

With reference to the subject of this paper, coalition analysis has also been used to examine

the ability to achieve cooperation in a transboundary pollution game with agents located

along a river and exerting a unidirectional pollutant flow. The number of upstream agents in a

coalition exerts a positive influence on overall welfare owing to the greater number of

downstream members who enjoy the benefits of the internalization of the externality

(Gengenback, Weikard and Ansink, 2010). van der Laan and Moes (2012) have also

formally modelled the transboundary river pollution problem and devise different rules for

sharing the gains to cooperation between upstream and downstream agents.

7

Despite the substantial work done in the framework of coalition analysis and with respect to

the distribution of the gains to cooperation, formal study of the transboundary river pollution

problem in the non-cooperative game theoretic framework has not received much attention. It

is the intention of this paper to make a small inroad to this end. Where the results of non-

cooperative game theory are usually accused of being overly sensitive to functional form,

parameter values and order of play, the failure of cooperative game theory to provide a

convincing solution to the free-riding problem suggests that perhaps it is time to revisit a non-

cooperative perspective of the problem. Huber and Wirl (1998) have established that in a

situation of unidirectional externalities and asymmetric information, the “pollutee pays

principle” (in the context of this paper, a situation where the victims of pollution make

transfers to upstream states) does indeed encourage voluntary arrangements (or contracts) and

therefore makes it possible to internalize at least part (if not all) of the externality. Empirical

work has also shown that transfers from downstream parties to those upstream can succeed in

minimizing costs, environmental damage as well as the total stock of pollution present in

river water itself (Fernandez, 2009).

To this end, this paper intends to examine the possibility of achieving cooperation amongst

riparian states through such voluntary arrangements. For the sake of expositional ease, the

number of countries is restricted to three, but the paper attempts to retain the spirit of the

general nature of the problem by looking at a situation where a downstream country may only

be able to observe the total amount of pollution present in a river, but not the individual

polluting actions of its upstream states.

8

The Model

As mentioned earlier, I consider 3 countries located alongside a river: U (the country furthest

upstream), M (the country lying midstream between U & D) and D (the country furthest

downstream). Assume that each country discharges a pollutant q that is a by-product of

producing some composite commodity x1.

Each country derives a benefit from discharging the pollutant into the river, denoted by

)( iq , for i = 1, 2, 3, where subscripts 1, 2 & 3 denote countries U, M & D respectively, and

)( iq is given by:

(A1) )2/()(2

iii qqaq

Each riparian state also suffers damages from the total amount of pollution entering that

country, i.e. the social cost of pollution to each country is a function of the sum of pollutants

discharged by all the countries upstream to it as well as its own pollution.

I further assume that the damages imposed by a unit of pollution are the same in each country

and therefore assume a uniform damage function for each country. This cost is denoted by:

(A2) )(1

i

j jq = )(1

i

j jqc with ac 0 , for i = 1, 2, 3.

For example, the cost of pollution to country U (or 1) is )( 1q and similarly, that of M

is )( 21 qq .

For notational ease, assume that the total amount of pollution facing each country is denoted

by Qi (=

i

j jq1

) . The social cost function of country i is then written as )( iQ . I also make

the standard assumptions that the benefit function of every country is strictly concave and

that the social cost function is strictly increasing in its argument.

1 In this model I will not consider the main activity of each country (the production of x) but will deal solely

with the externality generated, i.e. q.

9

(A3) 0'i 0

''i , 0

' i and 0

'' i .

(A4) All functions are common knowledge.

(A5) Property rights are assumed to be vested in the upstream party, i.e. upstream riparian

states possess the right to pollute the river.

(A6) Pollution is assumed to be observable. Each country is assumed to be capable of

monitoring and verifying the level of pollution present in the river at the point of its entry into

its own territory. In other words, each country can observe the collective polluting decision of

all the agents upstream to it by measuring the total level of pollution at the river‟s point of

entry into its own territory.

The implication of (A6) is that country D, being located last in the sequence, can observe the

totality of the polluting decisions of its upstream states, U and M, but cannot know their

individual levels of pollution.

Each country i solves:

iqmax iiii UQq )()(

where Ui is the utility of country i.

(A7) I further assume that in the absence of any agreement or third-party arbiter, each country

decides to pollute their Nash equilibrium in the above-specified 3-country case, given by

[NNN

qqq 321 ,, ].

The first-order conditions of the above maximization problem are given by:

:1q )()( 1

'

1

'

1 Qq

:2q )()( 2

'

2

'

2 Qq

10

:3q )()( 3

'

3

'

3 Qq

The solution to the above system is given by 1321

NNNqqq where ac / and

therefore, )1(3 NQ .

3.1 Social optimum:

Suppose we now consider the socially optimal level of total pollution Q* to be the summation

of the pollution levels of each country when they cooperate, i.e. the cooperative or socially

optimal outcome is defined as the solution to the following problem:

321 ,,max qqq ))()(())()(())()(( 333222111321 QqQqQqUUU

The solution is given by 31*1 q , 21*2 q and 1*3q (and therefore

)21(3* Q and the relevant first order conditions by:

:1q )(')()()( 32

'

1

'

1

'

1 QQQq

:2q )()()( 3

'

2

'

2

'

2 QQq

:3q )()( 3

'

3

'

3 Qq

We can see that the total pollution level in the cooperative case is less than that when

countries pollute their Nash equilibrium levels, i.e. NQQ * .

We can also observe that the cooperative solution will involve each country equating its

marginal benefit from pollution to the aggregate marginal social costs to all the countries

suffering from its pollution. In other words, each upstream country internalizes the social cost

of its own pollution suffered by all the parties downstream of it.

11

3.2 Gains to cooperation

The gains to cooperation in the above problem may be described for each country as the

difference in utility of that country between the Nash and cooperative outcomes. Hence, the

gains to cooperation may be defined as: N

ii UU * for all i =1, 2, 3.

Then, comparing the utilities of each country in the Nash and socially optimal cases, we

have:

Country U: N

UU 11 * = 242 ac )0(

Country M: N

UU 22 * = 2/)3(3 2 ac )0(

Country D: N

UU 33 * = c3 )0(

Hence, as can be verified above, the maximal gains from cooperation accrue to D, followed

by lower, but positive, gains to M, and finally, negative gains (or loss) to U from cooperation.

This is intuitively sensible; D will gain the most from cooperation since the cooperative

outcome ensures that each country accounts for the negative impact of its pollution on all

parties downstream. While M also internalizes its negative externality, in the cooperative

outcome it does not suffer from the costs associated with U‟s pollution. U, naturally, does not

gain by cooperating as it does not suffer any externality from the others, being situated first

along the river.

The question that naturally arises next is: how can this cooperative outcome be achieved? As

shown above, it is clear that D can gain substantially from cooperation (and to a lesser extent,

M) and therefore, D will have an incentive to effect such an outcome. Traditional

environmental economics tells us that in the presence of an externality, if trade is allowed

between the concerned parties, the efficient level of the externality can be achieved regardless

of the assignation of property rights to the environmental resource. As I have assumed that

property rights belong to the polluter, this will imply that payments made by downstream

12

parties to upstream countries will achieve the cooperative solution. But such side-payments

are effective in establishing the efficient outcome only if pollution is observable to the party

making the payment. When there are only two countries located along a river, such payments

will achieve the socially optimal levels of pollution, but as the number of countries exceeds

two, problems of monitoring crop up.

In our problem comprising 3 riparian countries, each country can always observe the total

amount of pollution present in the river when it enters its territory. This will mean that D,

located farthest downstream, will be able to observe the total amount of pollution resulting

from U‟s and M‟s actions, but not their individual actions. A system of payments in this

situation will not achieve the social optimum as in the instance of a violation D will be unable

to hold the offending party(s) accountable for his actions owing to D‟s lack of information

concerning the violator(s)‟ actions.

3.3 A Principal-Agent Formulation of the Transboundary Pollution

Problem

As discussed earlier, when pollution is observable and there are only two parties, the

downstream country has an incentive to induce the upstream party to cooperate and pollute

the efficient level by means of side payments. Such a potential resolution exists in the

relationship between U and M. This paper intends to look at the more interesting question of

payments between 3 countries when pollution is not necessarily observable. Broadly, we can

divide this problem into two parts:

(a) When D can observe q1 and q2 individually

(b) When D can only observe the total 212 qqQ

D must now determine how to compensate U and M such that *11 qq and *22 qq are

achieved. We examine each of the above cases separately in sections 3.3.1 and 3.3.2.

13

3.3.1 Case (a): D can observe q1 and q2 individually

In this instance there is no problem of asymmetric information; D can observe how much U

and M each pollute, thereby enabling D to detect any violations and enforce the contract.

Suppose he decides to pay z1 per unit of reduction in pollution to U and z2 per unit of

pollution abatement to M, where z1 and z2 are constant. The total payment made to U would

then be TU = z1.q1 and to M, TM = z2.q2.

Then D will maximize:

321 ,,max qqq MU TTqqqq )()( 32133

subject to N

U UTqq 1111 )()( (IRU)

N

M UTqqq 22122 )()( (IRM)

The individual rationality constraints of both U and M will be satisfied as equalities at the

optimum, implying then that the above problem reduces to the cooperative optimization

problem, giving us *11 qq , *22 qq and *33 qq .

Therefore, D can achieve the first-best outcome by offering a contract. One example of such

a contract would be the simple forcing contract:

0

*. 11 qzTU

*,

*,

11

11

qq

qq

0

*. 22 qzTM

*,

*,

22

22

qq

qq

Essentially, when D can observe and/or verify each agent‟s pollution level, their

compensation becomes contingent on their individual levels of pollution, i.e. )( 1qTT UU

and )( 2qTT MM .

14

3.3.2 Case (b): D can only observe the total Q2 = q1 + q2

When D cannot observe U or M‟s individual polluting activity, then compensation becomes

contingent on the total level of pollution faced by D, ie. Q2 = q1 + q2. I examine two types of

contracting structures, one where D attempts to contract with U and M separately in the

presence of this asymmetric information and the other where D contracts only with the party

immediately upstream, i.e. M, making the offer contingent upon observing the cooperative

total Q2* (= q1* + q2*), thereby incentivizing M to contract with U to undertake pollution

abatement.

The following sections show that the former setup fails to achieve the first-best outcome and

furthermore, incentivizes the two upstream countries to pollute more than their Nash

equilibrium levels to take advantage of the offer made by D. However, the second contracting

structure achieves the first-best solution by creating incentives for further contracting

amongst upstream states.

3.3.2.1 Suppose D attempts to contract with U and M individually when he can only

observe the total Q2* (= q1* + q2*).

As this is a situation of imperfect information, D will place beliefs on U and M having

polluted a certain part (q1 and q2 respectively) of the total level Q2, upon observing some

given Q2.

To define these beliefs, let q1 and q2 be two independent, jointly continuous random variables

with probability density functions (.)1qf and (.)

2qf respectively. Let Q2 = q1 + q2 be a

continuous random variable with density (.)2Qf . For the purpose of analytical ease, assume

that Q2 is independent of and jointly continuous with both q1 and q2.

15

Suppose D believes that the probability that M has contributed some q2 to any observed Q2 is

given by the conditional probability density function of q2 given Q2 = z, denoted

by (.)22 |Qqf (for all

22 Qq ).

As q2 and Q2 are independent, )|(22 | zyf Qq

)(

),(

2

22 ,

zf

yzf

Q

qQ = )(

2yfq .

Let )|(22 | zyF Qq denote the conditional cumulative distribution function of q2 given Q2. Then,

)|(22 | zyF Qq )(

2yFq .

Similarly for U, let the probability that U has contributed some q1 to any observed Q2 be

given by the conditional probability density of q1 given Q2 = z, denoted by (.)21|Qqf , for all

21 Qq .

Let )|(21| zxF Qq denote the conditional cumulative distribution function of q1 given Q2. Then

(as q1 and Q2 are independent), )|(21| zxF Qq )(

1xFq .

D wishes to implement the cooperative pollution levels of U and M, i.e. q1 = q1* and q2 =

q2*. He then makes the following offers to U and M,

(A8)

0

. 22 qzTU

*,

*,

22

22

qq

qq

0

. 11 qzTM

*,

*,

11

11

qq

qq

As D places certain beliefs on the values of q1 and q2, the expected payment made by D to U

is given by:

E [TU] = ).].(|*[ 11211 qzQqqP )0].(|*[ 211 QqqP

= ).).(|*( 1121| 21qzQqF Qq

= ).*).(( 1111qzqFq

Similarly, the expected payment made by D to M is given by:

16

E [TM] = ).).(|*( 2222| 22qzQqF Qq

= ).*).(( 2222qzqFq

Now let the timing of the game be as follows: D announces the offers TU and TM, which are

known to both U and M. M does not place any beliefs on the wage offers and optimizes

taking the expected payment, E [TM], as given. U observes M‟s optimal strategy 2q and then

optimizes taking E [TM] and 2q as given to get

1q .

Proposition 1: Given assumptions (A1) to (A8) and the conditional cumulative distribution

functions, )|(21| zxF Qq and )|(

22 | zyF Qq , if the agent farthest downstream contracts with each

upstream state whose actions affect its payoff with a view to achieving the socially optimal

outcome, he will not be able to achieve the first-best outcome. Furthermore, such a

contracting structure will not even be able to improve on the Nash equilibrium (by obtaining

a total level of pollution less than that in the Nash outcome), but will instead succeed in

incentivizing the upstream states to pollute more than their Nash levels.

Proof: D‟s optimal contracting problem can be written as:

ji zq ,max *)(..*)(..)()( 222111333 21qFqzqFqzQcq qq 2,1;3,2,1 ji

subject to the following constraints:

(i) *)(..)()(maxarg 1111111 1qFqzQcqq q (ICU)

(ii) N

q UqFqzqcq 1111111 *)(..)()(1

(IRU)

(iii) *)(..)()(maxarg 2222222 2qFqzQcqq q (ICM)

(iv) N

q UqFqzqqcq 22222122 *)(..)()(2

(IRM)

17

Constraints (i) and (iii) effectively give us the optimal strategies 1q and

2q ,

*)(.)1(ˆ1

11 1

qFa

zq q

*)(.)1(ˆ2

22 2

qFa

zq q

The individual rationality constraints (ii) and (iv) may be expressed as N

U UTEU 11 ][

andN

M UTEU 22 ][ , which give us the following equations:

1)( zca + 2

*)(. 1

2

1 1qFz q

*)(

)1(

1

2

1qF

a

q

(IRU)

2)( zca + 2

*)(. 2

2

2 2qFz q

*)(

*)(..

2

1

1

2

1

qF

qFzc

q

q (IRM)

As in equilibrium IRU and IRM are satisfied as equalities, we can substitute the above

expressions for1q ,

2q , IRU and IRM into D‟s maximization problem and solve to get:

1ˆ3q

1z

*)(

.

*)(

)(1

*)(2

1

211 211qF

c

qF

ca

a

qFqqq

2z

*)(

)(1

*)(2

1

22 22qF

ca

a

qFqq

where and are the Lagrange multipliers associated with the IRU and IRM constraints

respectively.

18

It is clear now that z1 and z2 depend on and , assuming that

a

qFq *)(2 11 and

a

qFq *)(2 22 are non-zero.

While we are unable to say anything meaningful about and at this point, looking at the

equations for 1q and

2q , it is clear that if z1 and z2 are positive in sign, then 1q

Nq1 and

2qN

q2 and that the above contract in fact produces an outcome that is worse than the Nash

equilibrium one (i.e. the total level of pollution obtained in this situation is higher than QN).

But if z1 and z2 are negative, then 1q

Nq1 and

Nqq 22

ˆ , implying that the Nash outcome can

be improved upon only through negative transfers, or in other words, by penalizing U and M,

which contradicts the purpose of entering into a voluntary agreement with D, namely that of

improving each party‟s payoffs upon entering the contract.

Therefore we see that when only the sum total of countries‟ pollution levels can be observed,

contracting with countries individually will not even achieve a second-best outcome (any

outcome where NQQQ * ).

3.3.2.2 Suppose D contracts only with his immediate predecessor, M.

Let us now consider a contract where D only attempts to contract with M in order to achieve

the cooperative outcome, instead of all the parties who pose a negative externality to D by

way of their pollutive actions.

Hence, D now makes an offer to M, conditioning his payment upon observing the cooperative

total Q2*. Essentially, D wishes to incentivize M to cooperate with U. This would involve

changing M‟s payoff such that M will be in a position to choose the cooperative total, Q2*, as

his payment depends on the sum of the pollution levels of U and M.

19

Suppose D offers the following contract to M,

(A9) TM =

0

)()( 111 qcq

*,

*,

22

22

QQ

QQ

The choice of (A9) by D is not arbitrary – note that if M decides to pollute *22 QQ and

thereby inserts TM into his payoff function, his utility will then be exactly the same as the case

of a social planner choosing to maximize the joint welfare of countries U and M. The crucial

point, however, is essentially this: by conditioning M‟s payment on the aggregate levels of

pollution of both U and M, D creates an incentive for M to control the total level of pollution

faced by D.

Claim: In response to a contract such as TM, M will calculate the optimal Q2 = 2

~Q with a view

to contracting with U in order to achieve this 2

~Q (This is not an unreasonable claim as (i) D‟s

payment to M is made contingent on the value of Q2 and (ii) q1 is readily observable to M and

so any contract between U and M is enforceable).

Proposition 2: Given assumptions (A1) to (A7) and a contract between D and M defined by

(A9), the country farthest downstream will be able to achieve the first-best outcome by

incentivizing M to undertake further upstream contracting with U. D will achieve the above

by making the payment to M conditional upon observing the total of the cooperative levels of

pollution of U and M, namely: *** 212 qqQ .

Proof:

M now has a choice between choosing some *22 QQ or *22 QQ .

If he chooses *22 QQ , his maximization problem will then be as follows:

22 ,max Qq )(]2/)([)(]2/)([ 1

2

1121

2

22 qcqqaqqcqqa (= U1 + U2)

20

= )(]2/)()[()(]2/)([ 22

2

22222

2

22 qQcqQqQaQcqqa

(as q1 = Q2 – q2)

which gives us )1(~2 q and )32(

~2 Q , implying that )21(~

1 q .

But as *~

22 QQ )52( , and M is subject to the restriction that *22 QQ , the solution for

the choice of Q2 will be *22 QQ .

If on the other hand M chooses *22 QQ , he will optimize his payoff with respect to q1 and

q2 (choosing q1 is mathematically equivalent to choosing Q2; I adopt this approach for the

sake of analytical ease):

21 ,max qq U2 = )(]2/)([ 21

2

22 qqcqqa

This gives us )1(2 q and 01

2 cdq

dU, implying that 01 q and that )1(2 Q .

If 25.0 , *22 QQ , which implies that for these values of , M will not choose

*22 QQ . On the other hand, if 25.0 , then *22 QQ and )1(2 Q will be chosen.

The choice between *22 QQ and *22 QQ will be made as follows: M will choose

*22 QQ provided that 22

~UU and that 25.0 .

In other words, we must have 02

122 and that 25.0 . (A)

Therefore, given that M chooses *22 QQ , he must now choose that optimal division of

)52(*2 Q between U and M such that his payoff is maximized. Let us suppose that M

must choose a fraction ]1,0[ of Q2* and correspondingly determine U‟s pollution level q1

as *)1( 2Q .

21

M then solves:

max *)1(2

*]).1[(*)).1((*.

2

*).(*).( 2

2

222

2

22 Qc

QQaQc

QQa

The above problem gives us the solution )52(

)21(

.

This gives us the optimal division of Q2* between U and M as follows: *)31( 11 qq

and *)21( 22 qq

.

Requiring that 1 gives us the following parameter restriction: 33.0 (B)

M will now wish to make a transfer payment TU to U to induce him to choose q1*. This

implies that TU should be sufficiently high so as to make it rational for U to choose q1*

instead of N

q1 and that M should gain relative to his Nash payoff after entering into such a

contract with (or making such a payment to) U.

In other words, N

UM UTTU 22

and N

U UTU 11

. M‟s individual rationality constraint

will be satisfied provided that –

TU 2

66 2 acac and that 0UT (C)

Assuming for the sake of simplicity that the above equation is satisfied as an equality, U‟s

participation constraint N

U UTU 11

would then require that .02

12 2 (D)

Finally, in order to ensure that D will find it worthwhile to initiate such a contracting setup,

we must have N

M UTU 33

. (E)

Conditions (A), (B), (C), (D) and (E) then allow us to characterize the parameter values of

for which the above contracting structure will achieve the cooperative outcome {q1*, q2*,

q3*}:

29.024.0

22

Therefore, we can see that in the presence of asymmetric information, when externalities are

one-sided and agents are situated sequentially, a contracting structure where agents enter into

agreements with only their immediate predecessor, conditioning their offers on the sum total

of the pollution facing them, will be more effective in achieving the cooperative level of

pollution as opposed to an agent simultaneously contracting with all the parties whose actions

pose a negative externality.

23

Perfect Bayesian Equilibrium as a Solution Concept

I now examine the question of whether Perfect Bayesian Equilibrium can be applied as a

solution concept to the transboundary river pollution problem. To review the facts of the case,

our model deals with 3 agents who move strategically in response to the assignation of beliefs

by a player. Furthermore, these agents move in sequence, thereby requiring that their moves

be sequentially rational. In the presence of the abovementioned conditions, the concept of

Perfect Bayesian Equilibrium would appear to be a natural choice as a predictor of possible

equilibria that would achieve the cooperative outcome. However, while the existence of

Perfect Bayesian Equilibria for finite games is assured, no such general existence theorem has

yet been stated for the case of infinite games. As we have assumed that the choice of

pollution, q, is a continuous variable which has a probability density attached for every value,

our problem is then one of an infinite game, where the set of possible actions available to

each player is infinite. Therefore, in this problem, Perfect Bayesian Equilibria may or may

not exist.

These general considerations in mind, I will now propose an example of a Perfect Bayesian

Equilibrium that will allow us to characterize the optimal contracts necessary to achieve the

cooperative outcome. A Perfect Bayesian Equilibrium can be defined as “a set of strategies

and beliefs such that, at any stage of the game, strategies are optimal given the beliefs, and

the beliefs are obtained from equilibrium strategies and observed actions using Bayes‟ rule.”2

As in our problem beliefs are not updated, the above definition will then merely require that

countries U and M choose their optimal strategies given the beliefs assigned by D.

Suppose {1q ,

2q , 3q , 1z ,

2z , (.)21|QqF , (.)

22 |QqF } constitutes a Perfect Bayesian Equilibrium,

where the players U and M optimize their payoffs given the beliefs of player D, and the fact

2 Fudenberg & Tirole (1991)

24

that U takes into account M‟s optimal behaviour while maximizing his payoffs ensures that

subsequent play is optimal. Thus we see that when D assigns beliefs to the possible actions

taken by upstream states, U and M behave sequentially in their optimizing behaviour.

Let us conjecture an equilibrium such that q1* and q2* are achieved, and that the

corresponding belief structure of D regarding U‟s behaviour is as follows:

0

1*)( 11

qFq *,

*,

22

22

QQ

QQ

In other words, the probability that q1 is less than or equal to q1* is 1 if the sum total observed

by D is less than or equal to Q2*.

Similarly, for M:

0

1*)( 22

qFq *,

*,

22

22

QQ

QQ

Then M will optimally determine 2q keeping in mind D‟s beliefs, and so will essentially

have to choose between either *22 qq or q2 > q2*.

If M chooses *22 qq , then he will maximize the following problem:

2max q 2221

2

22 .)(]2/)([ qzqqcqqa

with the solution a

zq 2

2 )1(

As 2q *22 qq

N and M is subject to the restriction *22 qq , we have M‟s optimal choice

being q2 = q2* with payoff 2U .

If M chooses q2 > q2*, then he will solve the following problem:

2max q )(]2/)([ 21

2

22 qqcqqa

which gives us nothing but the Nash solution N

qq 22~ and the Nash payoff,

NU 2 .

25

Claim: There exists a 2z such that in equilibrium M will choose q2 = q2* if

22 zz .

Proof: M will always choose q2 = q2* instead of N

qq 22~ if 2U

NU 2 .

This implies that if)21(

.2

3 2

2

ca

z = 2z , M will choose q2 = q2*.

U now observes M‟s optimal response and chooses q1. Again, reasoning in exactly the same

way as in M‟s case, U will maximize his payoffs with respect to D‟s beliefs about q1 upon

observing Q2. Then, going by D‟s belief structure, U will have to choose between either

*11 qq and *11 qq .

If U chooses *11 qq , then *)( 11qFq = 1 and U solves his maximization problem to obtain

a

zq 1

1 )1( , which exceeds q1* and so has the optimal choice q1 = q1*.

On the other hand, if U chooses *11 qq , he has the same choice as his Nash equilibrium

level of pollution, q1 = q1N.

Again, we can make the following claim: There exists a 1z such that in equilibrium U will

choose q1 = q1* if 11 zz .

U will always choose the cooperative level of pollution, q1* if N

UU 11 , or:

1

2

1)31(

24z

caz

D, in order to maximize his payoff, will choose the minimum level of z1 and z2 required to

implement q1* and q2*, and therefore will choose 11

ˆ zz and 22

ˆ zz .

D will now solve:

3max q *.ˆ*.ˆ)**()]2/([ 2211321

2

33 qzqzqqqcqqa

with the solution *ˆ33 qq .

26

In order that the above contract be individually rational for D, it is necessary to have

NUU 33

ˆ .

3U = ccaa

31162

2

> ccaa

3322

2

= N

U 3

Therefore, the above contract is incentive-compatible for all parties, and we have the

following Perfect Bayesian Nash Equilibrium: *)(*),(,,*,*,*, 2121321 21qFqFzzqqq qq

with:

0

1*)( 11

qFq *,

*,

22

22

QQ

QQ

and

0

1*)( 22

qFq *,

*,

22

22

QQ

QQ

While the above is an example of a Perfect Bayesian Equilibrium that achieves the

cooperative outcome, note that the definition of PBE does not place any restrictions on

beliefs specified for off-the-equilibrium paths. This implies that for “strong” enough beliefs,

we will be able to obtain our objective, namely that of the cooperative outcome3. Even in the

example presented earlier, the equilibrium belief systems *)( 11qFq and *)( 22

qFq imply that

D holds a belief where if he observes a total *22 QQ , it means that both U and M have

exceeded their respective cooperative levels of pollution, q1* and q2*. But this need not be

the case as there may be instances where *22 QQ and only one of the parties (U or M) has

exceeded their cooperative level of pollution. Hence, we may be able to find Perfect Bayesian

Equilibria where the cooperative outcome is supported by such strange beliefs – this in turn

raises doubts as to the relevance of PBE as applied to our particular problem. The lack of

3 I would like to thank Ms. Debdatta Saha for bringing this point to my attention.

27

restrictions requiring that off-the-equilibrium-path beliefs be reasonable then suggests that

Perfect Bayesian Equilibrium may not be a realistic solution concept in our model.

28

Conclusion

This paper addresses the issue of transboundary pollution in international rivers and the

possibility of using voluntary arrangements or contracts to achieve an efficient outcome. We

find that in the presence of one-sided externalities, when agents move sequentially,

contracting individually with all upstream parties whose actions pose negative externalities

will not achieve the first-best outcome. Furthermore, it will not even achieve a second-best

outcome, producing instead equilibrium pollution levels greater than those of the Nash

outcome.

However, when unidirectional externalities accumulate sequentially, it is possible to achieve

the first-best situation if the party furthest downstream contracts only with his immediate

predecessor, rather than all the countries upstream of it. This is because conditioning

payments on observing the cooperative total level of pollution will create incentives for

further contracting upstream.

Finally, allowing for the optimal formation of beliefs may allow us to achieve the cooperative

outcome via offers, but the lack of restrictions on beliefs off the equilibrium path renders the

idea of Perfect Bayesian Equilibrium an unrealistic approach to the pollution problem.

The above results hold intriguing implications for policy and implementation – it may be

possible to achieve a lasting solution to the pollution problem through voluntary

arrangements between the concerned parties. Building trust and cooperation between

countries, not to mention the international public opinion necessary to propel environmental

cleanup in international rivers, is a costly affair – it delays collective action on restoring the

quality of the river which may prove to be disastrous, both for the countries dependent on the

shared resource as well as the existence of the water resource itself. It is the contention of this

paper that non-cooperative game theory, in assuming that agents always act rationally and in

29

their own self-interest, may well be better suited to addressing this problem, by looking at the

set of incentives necessary to motivate each party to behave in a desirable manner.

Possible ideas for future research include a generalization extending the model to N

countries, which is still yet to be undertaken, as well as an exploration of how sensitive the

results are to functional form. I have assumed identical benefit functions and a constant

marginal cost of pollution for all the countries involved– it may be interesting to see how the

results behave when asymmetry in both benefits and costs are introduced into the problem.

30

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