Sharing a Polluted River

Transcription

1 Sharing a Polluted River Debing Ni School of Management University of Electronic Science and Technology of China Chengdu, Sichuan, P.R. China, Yuntong Wang Department of Economics University of Windsor Windsor,Ontario,Canada, N9B 3P4 January 12, 2006 Debing Ni is a visiting scholar at the Department of Economics, University of Windsor, Windsor, Ontario, Canada. Wang thanks the Social Sciences and Humanities Research Council of Canada. Corresponding author. Tel: (519) ext.2382; Fax: (519) ; yuntong@uwindsor.ca 1

2 ABSTRACT: A river carries pollutants to people living along it if it is polluted. To make the water in the river clean, some costs are incurred. This poses a question of how to split the costs of cleaning the whole river among the agents located along it. To answer this question, we resort to the two main advocated doctrines in international disputes: the theory of Absolute Territorial Sovereignty (ATS) and the theory of Unlimited Territorial Integrity (UTI). Applying these two doctrines, we accordingly propose two methods: the Local Responsibility Sharing (LRS) method and the Upstream Equal Sharing (UES) method. For each method, we provide an axiomatic characterization. Interestingly, both the LRS method and the UES method coincide with the Shapley value solutions to the corresponding (cost) games that are naturally induced according to the ATS and the UTI doctrines respectively. Moreover, both the LRS solution and the UES solution are the core allocations of the corresponding games. Thus, both the LRS method and the UES method can be considered as fair or reasonable solutions to the pollution cost allocation problem. JEL classification: D61; D62; C71 Keywords: Externality; Fair allocation of pollution costs; Shapley value. 2

3 1 Introduction There are 148 rivers in the world flowing through two countries, 30 through three, 9 through four and 13 through five or more (See, Ambec and Sprumont 2002, and Barret 1994). Transborder rivers provide people in different countries with water resources, but they also bring pollutants to those people if they are polluted. The same problem also occurs to the people who live within a national border but in different regions through which a river flows. A polluted river, which is polluted to some degree but not so seriously that people cannot make any use of it, has both a beneficial aspect and a harmful aspect to people who live along it. On the beneficial side, Ambec and Sprumont (2002) develop a model to study how agents (e.g. countries, regions, or cities) living along a river share the water resources. Since the property rights over flowing water are not welldefined, the Coase (1960) theorem cannot be applied. Instead, they base on two main doctrines advocated in international disputes, namely the theory of Absolute Territorial Sovereignty and the theory of Unlimited Territorial Integrity (for short, ATS and UTI, respectively) to define rights owned by the agents. The ATS theory says that a country has absolute sovereignty over the area of any river basin on its territory, while the UTI theory says that a country shouldn t alter the natural conditions on its own territory to the disadvantage of a neighboring country. 1 With their welfare interpretation of rights, ATS implies a core-like constraint (the core lower bounds) while UTI implies the constraint of the aspiration upper bounds on welfare allocation. Interestingly, Ambec and Sprumont (2002) show that these two bounds uniquely determine a method, called the downstream incremental distribution, to allocate the optimal total welfare among the agents. This paper focuses on the harmful side, namely the pollution. Consider a river which is divided into n segments. Within each segment there are some agents who dump certain amount of pollutants of some kind into the river. The pollutants usually do harm to people so that some costs are incurred for people to prevent them from the harmfulness. 2 This raises two questions: who are responsible for the costs? How should the costs be shared? It 1 For more detailed discussion on ATS and UTI, see Godana (1985) and Kilgour and Dinar (1996). 2 In practice, the costs, for example, may be incurred by firm s installing a filter or household s installing a swimming bath. 3

4 is relatively easy to answer the first question. It is generally accepted that those, who discharge pollutants in the river, should take the responsibility for the costs. However, it is not straightforward to answer the second question, i.e., how to divide fairly the total river-polluting responsibility among the polluters? This research on the pollutant cost sharing problem is inspired by Ambec and Sprumont (2002). However, our problem is different and we look at the theories of ATS and UTI from a different perspective. We argue that there is a dual relationship between rights and responsibilities (or duties). We regard responsibilities as the counterpart of rights. We therefore interpret the ATS and the UTI doctrines in terms of responsibilities in our pollution cost allocation problem. Specifically, the ATS doctrine can be read as a statement that people living the jth segment have an absolute sovereignty to ask any polluter located within the segment j to pay the costs of cleaning pollutants. Under the ATS, the responsibility for the costs of cleaning river pollutants in the jth segment should be assigned to the polluters located in that segment. We call this translation of the ATS the principle of local responsibility (or LR for short). On the other hand, the responsibility version of the UTI extends the scope of a polluter s responsibilities for pollutant-cleaning. It says that people in segment j have the rights to ask polluters in the jth segment as well as all upstream polluters to pay the pollutant-cleaning costs. This means that an upstream polluter bears some responsibilities for all downstream pollutant-cleaning costs. We call this interpretation of the UTI the principle of downstream responsibility (or DR for short). To make the problem well defined, we shall first make clear what is the relevant amount of the pollutant-cleaning costs to be divided among the polluters? We assume that each segment uses the most efficient way to cleaning pollutants in order that the water falls in line with the environmental standard specified by the environmental supervision authority in that segment. 3 Thus the pollutant-cleaning costs of all the segments are at their lowest levels 3 Here, efficient way means both technologically and economically. That is, the most advanced technology is adopted and arranged efficiently to do the job of cleaning. Moreover, the authorities in different segments may set different environmental-supervision standards. For example, the U.S. Water Quality Act of 1965 permits states to set their own standards of water quality (Boyd 2003). 4

5 determined by the corresponding environmental requirements. 4 Now our problem can be specified as follows. For the n pollutant-cleaning costs, how do we split them among the polluters according to the LR and/or the DR principles? The LR tells us a clear-cut structure of cost-sharing which implies that if the costs in segment j are zero, all agents in segment j should have zero cost shares. Accordingly, we propose an axiom, called No Blind Cost, to capture this implication of the LR principle. As for the DR principle, it is less straightforward. First of all, it is clear that for the costs incurred in segment j, all downstream polluters should not take any responsibility for that costs. This can be captured by an axiom, called Irrelevance of Upstream Costs. On the other hand, the DR principle says that all upstream polluters are responsible for downstream costs. But it does not specify how to allocate them among all upstream polluters. Because of the fact that upstream pollutants go downstream with the water and that the cross-pollutant interactions (such as chemical reactions) often happen in a polluted river, it is difficult to distinguish each upstream polluter s contribution to the downstream costs. We take a viewpoint of equality and assume that for any given downstream costs, all upstream polluters share them equally. This assumption is captured by an axiom called Upstream Symmetry. Accordingly, we propose two methods to share the total pollutant-cleaning costs. The first is called Local Responsibility Sharing (or LRS for short) method which corresponds to the LR principle. It simply charges the agent in a given segment his own local costs (see Section 2 or 3). The second is called Upstream Equal Sharing (or UES for short) method which corresponds to the DR principle. The UES method charges an agent the sum of the equal divisions of all downstream costs including his own local costs (see Section 2 or 4). We provide axiomatic characterizations for both the LRS method and the UES method, respectively. We show that the LRS method is characterized by No Blind Cost, Additivity and Efficiency (Theorem 1), and the UES method by Irrelevance of Upstream Costs, Upstream Symmetry, Additivity and Efficiency (Theorem 2). Interestingly, both the LRS method and the UES method coincide with the Shapley value of the associated cost games generated from the problem (Propositions 1 and 3). Moreover, these two 4 We assume that the pollutant-cleaning costs in segment j are zero if the water quality meets the environmental standard. 5

6 solutions are in the cores of the corresponding games since both games are convex (to be exact, concave cost games) (Propositions 2 and 4). The rest of this paper is organized as follows. Section 2 develops a model for the problem of sharing the total pollutant-cleaning costs. Section 3 provides a characterization of the LRS method and shows that the LRS coincides with the Shapley value and is in the core of the corresponding game generated from the problem. Section 4 provides a characterization of the LRS method and shows that the UES coincides with the Shapley value and is in the core of a different game generated from the problem. 2 A Model Consider a river which is divided into n segments indexed in a given order i = 1, 2,..., n from upstream to downstream. There are n household-firm pairs (agents) located along a river, each of which is located in one of segments according to the above order. We assume that each firm generates a certain amount of pollutants of some kind which all households try to avoid and that in each segment, firm i is located immediately before the household i. In every segment i (i = 1, 2,..., n), an environmental authority sets a standard of the degree of pollution which requires agent i (i.e. a household-firm pair) spending c i to clean the pollutants in segment i so that the quality of the waterbody satisfies the environmental standard. We want to find meaningful methods to allocate the total pollutant-cleaning costs (c c n ) among all the household-firm pairs. From the viewpoint of responsibility, this cost allocation problem can be viewed as how to split the total costs among the n firms because they are responsible for the pollution of the river. 5 Formally, a pollution cost sharing problem is a pair (N, C) where N = {1,..., n} and C = (c 1,..., c n ) R+. n A solution to a problem (N, C) is a vector x = (x 1,..., x n ) R+ n such that i x i = i c i. A method is a mapping x that assigns to each problem (N, C) a solution x(n, C). When N is fixed, 5 In this paper, we are not concerned with the allocation within a household-firm pair. There exist various approaches in the literature to solve this problem. For example, theoretically it may be solved by the Coase theorem if the property rights of any given segment of the river is well-defined and other conditions, such as no transaction cost, perfect competition, complete information, etc., are satisfied (Canterbury and Marvasti 1992). 6

7 we simply call vector C a problem. Under the LR and the DR principles mentioned in the introduction, we propose the following two methods. The Local Responsibility Sharing (LRS) method corresponds to the LR principle while the Upstream Equal Sharing (UES) method corresponds to the DR principle. Definition 1 For any C R+, n The Local Responsibility Sharing method is given by x LRS i (C) = c i, i = 1,..., n (1) Definition 2 For any C R+, n the Upstream Equal Sharing method is given by x UES i (C) = 1 i c i + 1 i + 1 c i n c n, i = 1,..., n (2) In order to provide a game theoretic analysis on these two methods (e.g., their connections with the Shapley value and core), we define the following two games that are related to the LR and the DR principles, respectively. Let N = {1, 2,..., n} be the set of firms (or household-firm pairs, hereafter we refer to firms only). Suppose that all the n households keep their locations unchanged no matter how the firms form their coalitions. Let S N be any coalition of the n firms. Denote by min S the smallest element in S, i.e. the most upstream firm in the coalition S. Under the LR principle, each member of S is responsible only for the pollutant-cleaning costs in its own segment, and the total responsibility of the coalition S is simply the sum of its members local responsibilities. Thus, for any given C R+, n the total costs of the coalition S can be written as v C (S) = c i (3) i S While under the DR principle, each member of S takes the responsibility not only for the pollutant-cleaning costs in its own segment but also for all the costs in its downstream segments. Thus the total costs of the coalition S should be w C (S) = i=min S c i (4) 7

8 Assume that v C ( ) = w C ( ) = 0. Now, for any given C R+, n we have generated two games, (N, v C ) and (N, w C ), respectively. In the next two sections, we will show that the cost allocations according to the LRS method and the UES method are consistent with the Shapley value of the game (N, v C ) and (N, w C ), respectively, and that they are in the core of the corresponding games. 3 A Characterization of the LRS Method In this section, we first provide an axiomatic characterization of the LRS method. Then we investigate its relationship with the Shapley value as well as the core of the game (N, v C ). We now introduce the following axioms. Additivity: For any C 1 = (c 1 1,..., c 1 n) R n + and C 2 = (c 2 1,..., c 2 n) R n +, we have x j (C 1 + C 2 ) = x j (C 1 ) + x j (C 2 ) for all j N. No Blind Cost: For any i N and any C R n +, if c i = 0, then x i (C) = 0. Efficiency: n j=1 x j = n j=1 c j. Theorem 1 The LRS method is the only method satisfying Additivity, No Blind Cost, and Efficiency. Proof: It is clear that the LRS method satisfies the above three axioms. In the following we show that the LRS is the only method satisfying these axioms. Consider C k = (0,..., 0, 1, 0,..., 0) where 1 is the k-th component of the n-dimensional vector C k, k = 1, 2,..., n. By No Blind Cost, x j (C k ) = α if j = k and x j (C k ) = 0 otherwise. By Efficiency, we have x j (C k ) = α = c k j = 1 j=1 j=1 Thus, x j (C k ) = 1 if j = k and x j (C k ) = 0 if j k. 8

9 Note that the cost vectors, C k (k = 1, 2,..., n), form a basis of R n. Thus, for any C R+, n we can write C = n k=1 c k C k = (c 1, c 2,..., c n ). Then Additivity implies x j (C) = x j ( c k C k ) k=1 = c k x j (C k ) k=1 = c j for all j N. The theorem is proved. = c j = x LRS j (C) (5) Remark 1: The LRS method clearly indicates that all firms are treated fairly in the sense that no cost is imposed on a firm which bears no local responsibility at all, and that the cost distribution is anonymous: even if a firm changes its location, its cost share does not change. These two features together with Additivity and Efficiency remind us that there would be some connections between the LRS method and the Shapley value of certain game induced from the problem. Indeed, the following proposition shows that the LRS solution exactly coincides with the Shapley value ϕ of the game (N, v C ) for all C R n +. Recall that the Shapley value of a game (N, v) is defined by ϕ i (v) = 0 s n 1 s!(n s 1)! n! We have the following proposition. ( S N\i, S =s v(s {i}) v(s)). Proposition 1 For all C R n + and v C defined by (3), we have x LRS i (C) = ϕ i (v C ), i N. Proof: For any C R n + and any i N, it is obvious that firm i s marginal contribution is v C (S {i}) v C (S) = c i for all S N \ {i} (including ). 9

10 Let s = S be the size of S, then the Shapley value of the game (N, v C ) for firm i is ϕ i (v C ) = = 0 s n 1 0 s n 1 = c i ( = c i ( 0 s n 1 0 s n 1 = c i = x LRS i (C) s!(n s 1)! n! s!(n s 1)! n! ( ( s!(n s 1)! n! 1 n ) for all i N. The proposition is proved. S N\i, S =s S N\i, S =s v C (S {i}) v C (S)) c i ) (n 1)! s!(n s 1)! ) Furthermore, the following proposition shows that the Shapley value, and therefore the LRS solution, is in the core of the game (N, v C ). It is well known that a convex game always has a nonempty core and, in particular the Shapley value is in the core (see Moulin 1988). For cost sharing games, the corresponding convexity is instead defined by the concavity of the game. A game v is called concave if, for all i N, all S, T N \ i, S T, we have v(s {i}) v(s) v(t {i}) v(t ) (6) Proposition 2 For all C R+, n the Shapley value of the game (N, v C ) is in the core, i.e. for any S N, i S ϕ i (v C ) = i S x LRS i (C) v C (S). Proof: It suffices to show that the game (N, v C ) is concave, i.e. for all i N and all S, T N \ i, if S T, then v C (S {i}) v C (S) v C (T {i}) v C (T ) (7) By (3), both the left hand and the right hand of (7) are equal to c i. Thus the game (N, v C ) is concave. The proposition is proved. 10

11 Remark 2: Propositions 1 and 2 show that the LRS method satisfies the stand-alone or subsidy-free principle. 6 That is, the costs charged to any firm or any coalition of the n firms do not exceed the costs incurred by the firm or the coalition. Thus, the LRS method ensures that no firm or coalition has any incentive to change locations. In this sense, the LRS method ensures the stability of the cooperation on sharing a polluted river. 4 A Characterization of the UES Method In this section, we first provide an axiomatic characterization of the UES method. Then we examine its relationship with the Shapley value and core. First, we introduce the following two axioms. Independence of Upstream Costs: For any i N, any C, C R n + such that c l = c l, l > i, for all j > i, we have x j (C) = x j (C ). Upstream Symmetry: For any i N, for all j, k i, we have x j (0,..., 0, c i, 0,..., 0) = x k (0,..., 0, c i, 0,..., 0). Now we have the following theorem. Theorem 2 The UES method is the only method satisfying Additivity, Independence of Upstream Costs, Upstream Symmetry, and Efficiency. Proof: It is easy to check that the UES satisfies the above four axioms. Below, we show that it is the only method satisfying these four axioms. For any k = 1, 2,..., n, consider C k = (0,..., 0, 1, 0,..., 0) where 1 is the k-th component of the n-dimensional vector C k. By Independence of Upstream Costs, x j (C k ) = x j (0,..., 0) = 0 for all j > k. By Upstream Symmetry, x j (C k ) = α for all j k. By Efficiency, we have x j (C k ) = kα = 1. j=1 Thus, x j (C k ) = 1 k if j k, x j(c k ) = 0 if j > k. 6 For detailed exposition of the principle, see Faulhaber (1975). 11

12 Since the cost vectors, C k (k = 1, 2,..., n), form a basis of R n, for any C R n +, it can be written as C = n k=1 c k C k = (c 1, c 2,..., c n ). Then Additivity implies that, for all j N, The theorem is proved. x j (C) = x j ( c k C k ) k=1 = c k x j (C k ) k=1 = 1 j c j + 1 j + 1 c j n c n = x UES j (C) Remark 3: The UES solution requires that, for a cost component c i (1 < i n), all upstream firms should bear equal responsibility for that cost. For example, all firms have the same share (1/n) of c n. If there is a firm j which does not need to pay any cost to clean the waterbody in segment j according to the standard of its local environmental supervision authority (i.e. c j = 0), it would argue that it should not share any cost for cleaning the river. This seemingly challenges the fairness of the cost allocation according to the UES method. However, Firm j s discharge of pollutants has a social impact on its downstream households even if it passes the test of its local authority. If the DR principle is accepted as a guideline to divide the total pollution costs, it should take some responsibilities for its downstream pollutant-cleaning costs. Indeed, the following proposition shows that, under the DR principle, the UES method splits the total costs of cleaning the pollutants in the river in the same way as suggested by the Shapley value. Proposition 3 For all C R n + and w C defined by (4), we have x UES i (C) = ϕ i (w C ), i N. Proof: Consider C k = (0,..., 0, 1, 0,..., 0) where 1 is the k-th component of the n-dimensional vector C k, k = 1, 2,..., n. The games corresponding to C k (k = 1, 2,..., n) is given by w Ck (S) = 0 if min S > k; and w Ck (S) = 1 otherwise. 12

13 Clearly, for the game (N, w Ck ), all agents i > k are dummies and all agents i k are symmetric. The axiomatization of the Shapley value (Shapley 1953) implies that the Shapley value of the game (N, w Ck ) is ϕ i (w Ck ) = 0 if i > k ; and ϕ i (w Ck ) = 1 k otherwise. (8) Note again that for any C R n +, write C = n k=1 c k C k. By the definition of the game (N, w C ), we have, for all S N, w C (S) = = = = c j j=min S ( [c k C k ] j ) j=min S k=1 c k ( [C k ] j ) k=1 j=min S c k w Ck (S) k=1 where [C] j is the jth component of the vector C. By Additivity and equation (8), we have ϕ i (w C ) = c k ϕ i (w Ck ) k=1 = 1 i c i + 1 i + 1 c i n c n = x UES i (C) for all i N. The proposition is proved. The following proposition shows that the Shapley value of the game (N, w C ) or the UES solution is a core allocation. Proposition 4 For all C R+, n the Shapley value of the game (N, w C ) is in the core, i.e. for any S N, i S ϕ i (w C ) = i S x UES i (C) w C (S). 13

14 Proof: It suffices to show that the game (N, w C ) is concave. For any S, T N, S T, any i / T, denote 1 = w C (S {i}) w C (S) 2 = w C (T {i}) w C (T ) 3 = w C (S {i}) w C (T {i}) 4 = w C (S) w C (T ). Clearly, the concavity of the game (N, w C ) is equivalent to or Since S T N \ i, we have min S min T. Then for any given i N, i / T, there are three possibilities: i min T min S, min T < i min S and min T < min S i. Now we check the concavity of (N, w C ) in each of these three cases. Case 1. Suppose that i min T min S. In this case, min T {i} = min S {i} = i. By the definition of w C, we have 3 = 0 and 4 0. Thus, Case 2. Suppose that min T < i min S. In this case, min T {i} = min T and i = min S {i} min S. The definition of w C implies that 1 0 and 2 = 0. Thus, Case 3. Suppose that min T < min S i. In this case, min T {i} = min T and min S {i} = min S. Again by the definition of w C, we have 1 = 0 and 2 = 0. Thus, 1 2 = 0. To summarize, we show that for any S, T N, S T, any i / T, w C (S {i}) w C (S) w C (T {i}) w C (T ), thus the game (N, w C ) is concave. The proposition is proved. Remark 4: As in Remark 2, Propositions 3 and 4 show that the UES method, as the Shapley value of certain game, embodies certain equity properties and, as core allocation, meets the stand-alone tests, in the problem of sharing a polluted river if the DR principle is accepted. On the other hand, 14

15 the UES method seems treating upstream firms unfairly. For instance, firm 1 s cost share is generally very much greater than its local costs c 1. We argue that the UES allocation method can be justified by Tiebout s (1956) equilibrium theory of locations, which says that people voluntarily choose locations to best satisfy their preferences even if the (living) costs may be higher. The same is true for firms. For example, the technologies (production functions) of upstream firms must be operated with more clean water and they are willing to pay a greater share of pollutant-cleaning costs. References [1] Ambec, S. and Sprumont, Y. Sharing a River. Journal of Economic Theory 107 (2002), [2] Barret, S. Conflict and cooperation in managing international water resources, Working Paper 1303, World Bank, Washington [3] Boyd, J. Water pollution taxes: A good ideal doomed to failure. Discussion paper 03-20, Research for the Future, Washington [4] Canterbury, E.R. and Marvasti, A. The Coase theorem as a negative externality. Journal of Economic Issues 26 (1992), [5] Coase, R. The problem of social cost. Journal of Law and Economics 1 (1960), [6] Faulhaber, G. Cross-subsidization: Pricing in public enterprises. American Economic Review 65 (1975), [7] Godana, B. Africa s shared water resources. France Printer, London, [8] Kilgour, M. and Dinar A. Are stable agreements for sharing international river waters now possible? Working Paper 1474, World Bank, Washington [9] Moulin, H. Axioms of Cooperative Desion Making, Cambridge University Press,

16 [10] Shapley, L. S. A Value for n-person Games. In Contributions to the Theory of Games II. edited by H. W. Kuhn and A. W. Tucker. Annals of Mathematics Studies 28 (1953), [11] Tiebout, C. A Pure Theory of Local Expenditures. Journal of Political Economy 64 (1956),