Water-Quality Trading: Can We Get the Price of Pollution Right?

Water-Quality Trading:Can We Get the Price of Pollution Right? 1Yoshifumi KonishiFaculty of Liberal ArtsSophia UniversityJay S. CogginsDepartment of Applied EconomicsUniversity of MinnesotaBin WangDepartment of Public HealthPennsylvania State UniversityDraft: September 24, 2011AbstractA substantial challenge has loomed in designing water-quality trading (WQT) mechanisms:getting the prices right. By this we mean providing participants with price signals that accountproperly for spatially explicit damage relationships in a watershed. This paper extends recent workby Hung and Shaw (2005) and Farrow et al. (2005) by incorporating two important features that arecharacteristic of many watersheds: (i) branching rivers; and (ii) nonlinear pollution damages. Evenunder first-best conditions, the mechanism of Hung and Shaw fails to achieve the social optimumwhen there are critical zones in a branching river. It is robust to nonlinear damages. The mechanismof Farrow et al. fails when damages are nonlinear, but is robust to branching. Under the second-bestcondition in which the initial distribution of permits is not optimal, neither mechanism dominates.The former precludes efficient trades across branches while the latter encourages inefficient trades.We also find, however, that the efficiency loss due to not getting the total supply of permits rightis substantially larger than that from not getting prices right.Keywords: Water-quality trading; trading-ratio system;1 We gratefully acknowledge financial support from a U.S. Environmental Protection Agency 2005 Targeted WatershedGrant and a Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists.

1 IntroductionThe idea of using water-quality trading (WQT) to aid in protecting water quality is appealing. TheU.S. experience with its SO 2 allowance market proved that markets for pollution can work for air.Should they not work for water pollution too? The U.S. Environmental Protection Agency (EPA)seems to think they can. It actively encourages states to establish rules for water-quality trading(WQT). 2 Currently, there are a total of 54 water-quality trading programs in the United States,with eleven states having a state-wide trading policy in place or in development and three moreadopting watershed-specific state trading programs (EPA, 2011). The results from these programshave, for the most part, been disappointing. Several barriers to trading have emerged (see, forexample, EPA, 2008; King and Kuch, 2003; Morgan and Wolverton, 2005; Woodward and Kaiser,2003).One such barrier is the difficulty of getting the prices of pollution right for WQT (Farrow etal., 2005; Hung and Shaw, 2005). By this we mean, following Muller and Mendelsohn (2009) thateach source faces a set of permit prices that reflect correctly the marginal damages caused over thelandscape by its own emissions and those of its trading partners. The spatial relationship betweenthe location of air emissions and the location of resulting damages is well known (Mauzerall et al.,2005). Spatial dependence is likely even more prominent for water pollution, where the attenuationand transport characteristics of numerous water pollutants are often critically dependent upon localhydrogeographic conditions at and downstream from each source (Todd and Mays, 2005; Schnoor,1996).Thirty years ago or so a lively literature arose in which various permit-trading schemes wereproposed. Montgomery’s (1972) ground-breaking ambient pollution system (APS) establishes aseparate permit market for every receptor point. This system is impractical, as firms would haveto know the impacts of their emissions on all relevant receptors and participate in a number ofdownstream markets. Another early contribution is by Atkinson and Tietenberg (1982), who considereda system of pollution offsets (POS) in which each new or expanding source is required tobuy offsets from existing sources if their emissions violate an ambient environmental standard atany receptor point. The emissions must be traded at the ratio of the two sources’ transfer coefficients.A version of the POS has been used in water-quality trading, but has often resulted insizable transaction costs, because each bilateral trade must undergo intensive scientific evaluationand ad hoc negotiations with potential trading partners. 3The papers by Montgomery and by Atkingson and Tietenberg belong to a large literature inwhich a host of competing arrangements for trading systems were proposed. Most were concerned,either explicitly or implicitly, with trading for air quality.A more recent literature, aimed specifically at water trading, focuses on designing tradablepermitsystems that address characteristics specific to water. Our focus is upon two of the recent2 The Agency’s “Water-Quality Trading Policy” (EPA 2003) and “Water-Quality Trading Assessment Handbook”(EPA 2004) are meant to help guide state and local environmental policy. Improving water quality in a cost-effectivemanner has become a top EPA priority, in part due to a series of litigations concerning Section 303(d) of the CleanWater Act (CWA) since the 1980’s. The CWA requires all states, territories, and authorized tribes to develop listsof “impaired waters” every two years and to develop the total maximum daily load (TMDL) for every impairedwaterbody/pollutant. By the early 2000’s, EPA was placed under court order, agreeing in a consent decree to enforcea TMDL in 27 litigated cases. A waterbody is designated as “impaired” for a pollutant when it violates ambientwater-quality standards for that pollutant. As of September 2010, 39,988 waters were listed as “impaired.” A TMDLis the maximum amount of a pollutant that a waterbody can receive and still meet water-quality standards. It alsoallocates that load among the various sources of the controlled pollutant.3 In this connection, the following comment on Oregon’s thermal-trading initiative makes the point: “[T]he tradetook considerable resources on the part of both CWS and DEQ to develop. The effort would have been neitherpractical nor worthwhile for a source much smaller than CWS to undertake” (Oregon DEQ, 2007).1

contributions, both of which restrict attention to point-source problems. Hung and Shaw’s (2005)trading-ratio system (TRS) takes into account an important feature of rivers: water flows unidirectionallyfrom upstream to downstream. The TRS transforms ambient environmental standardsinto ambient zonal discharge constraints according to physical transfer coefficients. Firms then participatein a single watershed-wide market in which they can trade with each other (with certainrestrictions) at the predetermined trading ratios subject to the zonal discharge constraints. Anessential feature of the TRS is that the trading ratios are based on physical transfer coefficientsrather than marginal damages, which can be more difficult to estimate. Though it offers severaladvantages over APS or POS, the TRS has an important drawback. The unidirectional nature ofthe transfer characteristics means that the TRS might not work as advertised for branching riversystems.Farrow et al. (2005) proposed an alternative trading system, which was later applied successfullyin the study of air pollution markets by Muller and Mendelsohn (2009). Like the TRS, Farrowet al.’s system allows firms to trade freely in a single watershed-wide market at predeterminedexchange rates, but the rates are based on the ratios of marginal damages (hence, we refer to it asa damage-denominated trading ratio system or DTRS). Because the DTRS does not rely on theunidirectional nature of the transfer characteristics, it is robust to branching. Its disadvantage,however, is that it may not be robust to damages that are nonlinear in pollution levels. Nonlineardamages may result from either nonlinear pollution-transport processes (Todd and Mays, 2005) ora nonlinear biological response of aquatic species to water pollution (Anderson et al., 2002; VanKirk and Hill, 2007) or both, even if economic agents’ marginal (dis)utility from water pollution isapproximately constant. 4This paper extends the work of Hung and Shaw (2005) and Farrow et al. (2005) by incorporatinginto a single model the two important features noted above: (i) branching rivers and (ii) nonlineardamages. We investigate the efficiency and cost-effectiveness properties of the two systems in aframework similar in nature to that of Muller and Mendelsohn (2009). In Section 2, we start bydefining a social planner’s efficient decision program in water-quality management for a genericwatershed. Our planner minimizes the sum of abatement costs and pollution damages by choosinga vector of emissions from stationary point sources distributed across space in a watershed drainedby a branching river. The model generalizes Farrow et al. (2005) and Muller and Mendelsohn (2009)in a non-trivial manner by accounting for nonlinear damages. We show that under some regularityconditions, there exists a cost-effectiveness program that implements the efficient outcome. Theresult thus allows us to discuss the TRS and DTRS on the same efficiency grounds.Sections 3 and 4 consider in turn the problems with the TRS and the DTRS. First, in Section 3,we show that the TRS fails to achieve a cost-effective optimum and, hence, the social optimum,when a critical zone (that is, a hot spot at which the pollution arriving from upstream excedesthe zone’s concentration constraint) exists at a confluence of a branching river. In this case thedischarge constraint in the critical zone must be set to zero and the constraint upstream of thecritical zone must be tightened. At a confluence, though, the required adjustment to the upstreamdischarge constraints becomes indeterminate. Thus, Hung and Shaw’s main result, that the TRSequilibrium achieves the cost-effective outcome, is not robust to branching.Second, in Section 4, we show that Farrow et al.’s DTRS is robust to branching, but failsto achieve the cost-effective optimum (hence, the social optimum) in the presence of nonlineardamages. Under the DTRS, the trading ratios across space must be fixed prior to trading, whichcan give false incentives for trading participants at the margin. Mathematically, this result is due4 It appears that the effects of nonlinear damages have not yet been given the attention they deserve in the contextof water trading, though their existence and importance have long been well recognized in the economics literature(Helfand and House, 1995; Larson et al., 1996; Segerson, 1988).2

to the multiplicity of emissions vectors that can satisfy the necessary conditions for the socialoptimum, which occurs precisely because of the nonlinearity of pollution damages.The zest of our paper lies in Section 5. We show that the TRS fails precisely where the DTRSsucceeds, and vice versa. This suggests that the relative performance of the two systems maydepend on the distribution of sources in a watershed featuring both branching rivers and nonlineardamages. We investigate this question by constructing a small numerical model and perturbing thegeographic distribution of pollution sources in a watershed. Our simulation model is an illustrativeadaptation of the National Water Pollution Control Assessment (NWPCA) model developed forEPA. That model was used in Farrow et al. (2005) as well as in other regulatory applications. Themodel is well grounded in hydrology and so is suitable for estimation of the water-quality impactsof pollution in a complex watershed.Using our parameterized model, we consider a second-best scenario in which the total numberof permits available at the initial allocation is optimal, but their distribution among sources is not.This assumption helps us disentangle the sources of inefficiency if either TRS or DTRS fails toachieve the social optimum. Because the total amount of permits is optimal, any inefficiency mustbe attributed to problems with the trading ratios. That is, sources are faced with incorrect pricesignals. Our simulation results demonstrate that neither system dominates; each stumbles in itsown way. On one hand, the TRS can result in welfare loss because the trading ratios based ontransfer characteristics may preclude some efficient trades across branches. On the other hand,the DTRS can result in welfare loss because the fixed trading ratios based on marginal damagesat some emissions vectors may, due to incorrect marginal incentives, encourage inefficient trades.Under the DTRS, sources may either over-abate or under-abate relative to the optimum. In thissense, our results suggest the impossibility of getting the spatially explicit prices right for WQT.Alas, we do not offer a magic bullet to solve these problems.More encouraging, though, is the fact that the deadweight losses associated with either systemare relatively modest so long as the total number of permits is optimal. Put another way, theefficiency loss from failing to issue the correct number of permits is much greater than the efficiencyloss from failing to set the correct trading ratios. This last is what is meant by getting prices right.Moreover, somewhat paradoxically, even with perfectly competitive markets we show that issuingthe correct number of permits can be also essential for getting the prices of pollution right. We deferto the concluding section a brief discussion of the implications of our results for nonpoint-sourcepollution.2 A theoretical model of water-quality managementIn this section we develop a static model of water-quality management in a generic river basin.We show how the TRS and the DTRS can both be derived directly from this more generic model.Thus, the two alternatives can be compared on the same efficiency grounds within our framework.Let e = (e 1 , . . . , e i , . . . , e N ) be a vector of emissions, where e i represents emissions from source i,and let ē be a vector of baseline or uncontrolled emissions. Index i serves the dual purposeof denoting a source and also its geographic location. Clearly, e i ≤ ē i for every i. Let x =(x 1 , ..., x m , . . . , x M ) be a vector of ambient pollution levels, where x m denotes concentration atreceptor m. Assume that there exists a linear mapping T : R N → R M describing the scientificrelationship between e and x. This linearity assumption has a long heritage in the economicsliterature (see Montgomery, 1972; Krupnick et al., 1983; McGartland and Oates, 1985; and Hungand Shaw, 2005). Let T be given by x = T e ′ , where T is a M × N matrix of nonnegative transfer3

coefficients. 5 Let S : R M → R, given by S(x), be a differentiable function that describes totaleconomic damages as a function of the vector of ambient pollution levels. Assume that ∂S/∂x m > 0for all m. It follows that total economic damages as a function of emissions are differentiable andare given by D(e) = S(T e ′ ). Define a vector of abatement levels a = ē − e, where by definitiona i ∈ [0, ē i ]. Each source i is assumed, here and throughout the paper, to have a twice-differentiableabatement cost function C i (a i ), with C i ′ > 0 and C′′ i > 0.Under these standard assumptions, an efficient program minimizes the sum of abatement costsand damages:mina∑ Ni=1 C i(a i ) + D(ē − a). (1)Given that the C i ’s and D are differentiable (and so continuous) and that a i ∈ [0, ē i ] for all i, theWiestaurass theorem ensures that a solution to (1) exists. Denote this optimum a eff .In the earlier literature (Montgomery, 1972; Krupnick et al., 1983; McGartland and Oates,1985; Hung and Shaw, 2005), it was often assumed that a social planner solves, instead of (1), anauxiliary cost-effectiveness program of the form:mina∑ Ni=1 C i(a i ) s.t. x m ≤ ¯X m ∀m and x = T e ′ , (2)where ¯X = ( ¯X 1 , . . . , ¯X M ) is a vector of environmental constraints on ambient pollution levels, onefor each receptor. Denote the solution to (2), by a HS . As we will see in Section 3, the TRS attemptsto solve the cost-effective program (2) rather than the efficient program (1).And we will see in Section 4 that Farrow et al. (2005) considered a different program still.Their DTRS is aimed at minimizing the sum of abatement costs subject to a constraint on totaldamages, TD. Assuming that D is an additively separable, linear damage function of emissions,D(e) = ∑ i d ie i , Farrow et al. solvemina∑ Ni=1 C i(a i ) s.t. D(ē − a) ≤ TD. (3)Muller and Mendelsohn (2009) observe that in order for the solution to (3) to coincide with thesolution to (1), the planner must set the constraint on total damages, TD, at the efficient level.Let a FSCH denote the solution to (3).Let us now establish a preliminary result, which links Hung and Shaw’s TRS and Farrow etal.’s DTRS.Proposition 1. Provided that the C i ’s and D are continuous, the following are true:(i) Given the efficient solution a eff , there exists a constraint vector ¯X eff in terms of pollutionconcentrations such that the solution a HS to the auxiliary program (2) subject to ¯X eff is theoptimal solution a eff ;(ii) Given the efficient solution a eff , there exists a constraint value TD eff of total damages suchthat the solution a FSCH to the auxiliary program (3) subject to T D eff is the optimal solutiona eff ; and5 The literature on groundwater hydrology suggests that the mapping T may not be linear (see Todd and Mays,2005). Thus, damage functions can be nonlinear for two different reasons. First, environmental harm may be anonlinear function of concentrations (our S). Second, concentrations at receptors may themselves be a nonlinearfunction of emissions (our T ). Our results apply to linearity of either type, but we maintain the assumption oflinearity in T throughout the paper.4

(iii) The social planner requires no more information to implement program (2) than to implementprogram (3) in order to achieve the efficient solution a eff .Proof. To see (i), given the efficient solution a eff , let the efficient constraint vector ¯X eff be definedas¯X eff = T (ē − a eff ) ′ .Suppose, by way of contradiction, that a HS solves (2) subject to ¯X = ¯X eff , but a HS ≠ a eff . BecauseD is originally an increasing function of concentrations x, and because x eff = T (ē − a eff ) ′ = ¯X eff ,we have:∑C i(a effii ) + D(x eff ) < ∑ C i(a HSii ) + D(x HS ) ≤ ∑ C i(a HSii ) + D(x eff ).The first inequality follows from a HS ≠ a eff and the second inequality follows because x HS ≤ x effimplies D(x HS ) ≤ D(x eff ). But this last inequality implies that there exists a eff ≠ a HS such that∑i C i(a eff i ) < ∑ i C i(a HS i ) with x eff = ¯X eff . This contradicts the assumption that a eff minimizesthe sum of abatement costs subject to the environmental constraint ¯X eff .The proof of (ii) is analogous, with the constraint value TD eff defined as TD eff = D(ē − a eff ).To establish (iii), note that in order for the solution to (3) to achieve a eff , the planner must setTD at the efficient level, which requires that the planner knows the two mappings T : R N → R Mand D : R M → R. But this information is all that is required for the regulator to find the optimalconstraint vector ¯X eff . This completes the proof.Proposition 1 establishes the practical equivalence of (2) and (3), the two alternative costeffectivenessprograms. In practice, so long as the social planner has perfect knowledge of T andD, it does not matter whether environmental policy is set based upon ¯X m or upon T D. Thisdoes not mean, however, that the two trading mechanisms are equivalent. As we shall see, theTRS and the DTRS present different informational requirements: the TRS requires that transfercoefficients be estimated while the DTRS requires that ratios of marginal damages be estimated.More importantly, we will describe a set of conditions under which the TRS equilibrium maynot achieve the solution to (2). We will also describe a (different) set of conditions under whichthe DTRS equilibrium may not achieve the solution to (3). We show, therefore, that one cannotguarantee that the equilibrium under the TRS is equivalent to that under the DTRS. In the followingsections, we investigate these questions by incorporating (i) branching of a river in the mapping Tand (ii) nonlinear damages in the mapping S.A word of caution is in order when interpreting our Proposition 1. This is that the equivalencebetween the two cost-effectiveness programs assumes that the planner knows a eff . This in turnrequires that she has complete information regarding the abatement cost functions. A primaryappeal of permit trading is that in many cases the policy can be put in place, and an optimaloutcome thence achieved, by a planner who has no information regarding individual abatementcost functions. That may not be true here. The point is important because the apparent advantageof DTRS over TRS is that DTRS can achieve the efficient optimum provided that T D is setoptimally. This advantage of DTRS disappears, though, in view of our proposition, if the TRSequilibrium can itself achieve the optimum of the alternative program (2). For then, the sameoptimum can be achieved either by TRS or by DTRS. In determining under which conditions oneis to be preferred over the other, then, it is important for us to re-evaluate the equivalence betweenthe TRS equilibrium and the optimum of (2) under general conditions.5

3 The Trading-Ratio System (TRS)The Hung-Shaw TRS allocates tradable discharge permits beginning at the zone (and thus thesource) that is furthest upstream. Allocation proceeds from there on down the stream, ensuringalong the way that the concentration standard is met at each zone. This means that for somesources low on the river few permits, or even none, will be recieved in the initial allocation. 6 TheHung-Shaw system indexes zones so that m = 1 indicates the most upstream source and M themost downstream source. 7 For simplicity, Hung and Shaw assume that there is one discharger ineach zone. This means that, in our notation, the set of zones {m} coincides with the set of pollutingsources {i}. As they observe, this does not jeopardize the generality of their results. Thus, in thissection we shall use i to denote both sources and zones (or receptors). Given the unidirectionalflow of a river, the transfer matrix T has a special characteristic: for any m and n with m > n,τ mn = 0, where τ mn is the element of T that measures the water-quality impact of of pollutionfrom zone m upon concentration at zone n. As do Hung and Shaw, we assume that each sourceinfluences its own zone in a unitary fashion: τ ii = 1 for all i.Given the ambient zonal pollution standards ¯X from program (2), the TRS regulator usesthe transfer coefficients in T to allocate zonal tradable discharge permits (TDPs) ¯Z so that thestandards are met if no trade occurs. Starting from the most upstream zone, define ¯Z 1 = ¯X 1 and,for j > 1, define ¯Z j = ¯X j − ∑ j−1i=1 τ ij ¯Z i . It is possible that, for a given j, we might find thatτ (j−1)j ¯Xj−1 > ¯X j . That is, the level of pollutant arriving from upstream when the standard isexactly met there exceeds zone j’s standard even when no pollution is emitted in zone j. In thiscase, zone j is called a critical zone. The Hung-Shaw allocation scheme sets ¯Z j = 0 and, in turn,reduces the allocation of permits to the upstream zone (or, possibly more than one upstream zone)to the point at which zone j is no longer critical: ¯Zj−1 = ( ¯X j /τ (j−1)j ) − ∑ j−2k=1 τ kj ¯Z k . (See Hungand Shaw, p. 88).The TRS allocation scheme ensures that the water-quality impacts of all upstream zonal standardson a given zone are accounted for via the upstream transfer coefficients. Note that in usingthe TRS procedure, the regulator takes as given the set of zones {i}, the zonal environmentalstandards ¯X, and the transfer coefficients T . Each discharger is then allowed to trade freely in awatershed-wide permit market according to the transfer coefficients T , so long as its emissions donot exceed the permits it holds.Formally, each source i solves:minr ki ,r si ,r sjC i (a i ) − p i r si + ∑ j p jr ji (4)s.t.¯Zi ≥ (ē i − r ki ) − ∑ i−1jir jij=1(5)a i = r ki + r si (6)r si = ∑ nijj=i+1(7)0 ≤ r ki , r si , r sj , (8)where p i and p j are the market prices of permits from sources i and j, r ji is the amount of pollution6 The TRS allocation scheme, by design, privileges upstream sources over downstream sources. This might createa certain amount of political resistence in practice, but it makes good economic sense. An efficient outcome should“fill the river” with pollution up to the standard at each receptor. Failing to do this will lead to higher aggregateabatement costs.7 Indexes along two branches above their confluence, though important for bookkeeping purposes, have no ordinalrelationship to each other.6

control purchased from source j to offset pollution at source i, r ki is the amount of pollution controlfrom source i that is kept by source i to meet the zonal standard ¯Z i , and r si is the amount ofpollution control sold by source i. As Hung and Shaw observe, the TRS possesses two advantagesover other trading schemes. The first is that each discharger must participate in only a singlewatershed-wide permit market, so that transaction costs are low. The second is that the regulatorallocates initial zonal discharge permits ¯Z in such a way that the ambient environmental constraints¯X are satisfied exactly at the initial allocation.One can rewrite constraint (5) to obtain Hung and Shaw’s trading constraint (their equation(5)):e i ≤ ¯Z i + ∑ i−1τ jir ji − ∑ nr ij, (9)j=1 j=i+1where r ij is the net amount of zonal discharge permits sold by source i to source j. This constraintmeans that any discharger can buy permits only from upstream zones and sell permits only todownstream zones. Because sources can trade permits at exchange rates τ, in any TRS equilibrium(including the boundary case), for any j > i, the spatially explicit prices of permits must satisfyτ ij p j = p i . (10)The economic implications of this equality are substantial. Even if a high-cost source is locatedupstream of, or on a different branch from, a low-cost source, this constraint strictly prohibits anycost-minimizing trade between them: τ ij = 0 for i > j. This might seem justifiable at first on thegrounds that water flows downstream, so that any downstream pollution reduction or a reductionon a different branch has no effect on the concentration at the upstream location. However, themarginal damages of pollution from dischargers located in the different branch can be higher thanthose located in the upstream, particularly when damages are nonlinear. If so, then more abatementby low-cost firms in the different branch in exchange for less abatement at high-cost firms in theupstream would be Pareto improving. The TRS, then, disallows some trades that would increasewelfare. We shall return to this point in Section 5 when presenting the results of our numericalwork.According to Proposition 1, the solution to program (2) also solves program (1), regardless ofbranching or nonlinear damages. The question is whether Hung and Shaw’s TRS equilibrium isguaranteed to achieve the solution to program (2). Our next result, Proposition 2, shows that theanswer is no. 8 There are situtions, not unusual in actual practice, in which the outcome of theTRS is either indeterminant (the permit-allocation scheme breaks down) or inefficient (it fails tosolve program (2)).We first turn our attention to an important property of the transfer coefficients. This propertyis satisfied in Hung and Shaw’s numerical example, but is not otherwise noted in their paper. Thecoefficients must be associative. Intuitively, this means that the amelioration or degradation of aunit of pollutant between zone i and zone i + 1 is the same whether that unit was emitted at zone ior arrived from upstream. Formally, associativity is defined as follows.Definition. Given a matrix T = {τ ij } of transfer coefficients, say that T is associative if for alli, m, and k, τ im τ ki = τ km . Say that T is non-associative if there exist i, m, and k for whichτ im τ ki ≠ τ km .8 We have also shown that when there are multiple adjacent critical zones, the original TRS allocation schemebreaks down. For this situation we have derived a modified version of the TRS in which permits are allocated startingat the downstream-most source and proceeding upstream. Our modified version achieves the optimal outcome in theface of adjacent critical zones. The proof of this claim is available upon request. It appears that the TRS cannot besalvaged in the case of a critical zone at the confluence of branches.7

Proposition 2. The equilibrium under the Hung-Shaw TRS does not achieve the cost-effectivesolution to program (2) if (i) transfer coefficients are non-associative or (ii) there exists a criticalzone at the confluence of upstream branches.Proof. To prove (i), suppose that T is non-associative and let i, m, and k be such that τ im τ ki > τ km .(The argument is similar if this inequality is reversed.) Using the transfer coefficients and thedefinition of x i , the constraint in program (2) can be rewritten as∑i τ ime i ≤ ¯X m for all m.Let Ω eff be the set of emissions vectors that satisfy this constraint. Let Ω trs be the set of emissionsvectors that satisfy the TRS trading constraint (9). Because each polluter must obey this constraint,the TRS equilibrium solves program (2) only if the constraint sets Ω eff and Ω trs are equivalent.We shall show that there exists an emissions vector e ∈ Ω trs that is not in Ω eff .For any e ∈ Ω TRS , (9) is satisfied. Thus, let A m = ¯Z m − e m + ∑ m−1i=1 τ imr im − ∑ ni=m+1 r mi ≥ 0.Using the definition ¯Z m = ¯X m − ∑ m−1i=1 τ im ¯Z i , we havem−1∑e m − τ im r im +i=1n∑i=m+1m−1∑r mi + A m + τ im ¯Zi = ¯X m .Using the trading constraint (9) for ¯Z i and rearranging terms, we obtaini=1∑ m τ ime m + ∑ nr mi + A m − ∑ m−1i=1 i=m+1+ ∑ m−1τ im(− ∑ i−1τ kir ki + ∑ ni=1 k=1i=1 τ imr imk=i+1 r ik)≤ ¯X m .The last two terms of the left-hand side can be further rearranged to yield:− ∑ m−1τ imr im + ∑ m−1τ im(− ∑ i−1τ kir ki + ∑ m−1r ik + r im + ∑ ni=1 i=1 k=1 k=i+1= ∑ m−1i=1∑ i−1k=1 (−τ imτ ki + τ km )r ki + ∑ m−1i=1 τ imk=m+1 r im∑ nk=m+1 r ik,where we have used the fact that indexes i and k are anonymous and are therefore interchangeable.Thus we obtain:∑ m τ ime m + ∑ nr mi + A m + ∑ m−1τ ∑ nim r iki=1 i=m+1 i=1 k=m+1+ ∑ m−1i=1∑ i−1k=1 (−τ imτ ki + τ km )r ki ≤ ¯X m . (11)Note that because A m ≥ 0 and r ik , r mi ≥ 0, if the last term on the left side of (11) is non-negativethen it must be the case that ∑ i τ ime i ≤ ¯X m (that is, e ∈ Ω eff ). But if transfer coefficients arenon-associative with τ im τ ki > τ km , as we have assumed, then the sum of the last four terms in (11)can be negative. It follows then that there exists e ∈ Ω trs with e /∈ Ω eff .To see (ii), suppose that e hs is the solution vector for program (2). By assumption, we musthave a critical zone at the confluence receptor m: ∑ m−1 iτ (m−1i )m ¯X m−1i > ¯X m where {m − 1 i } i isthe collection of indices immediately upstream of zone m, along two or more branches. We knowthat at the optimum,¯X m ≥ ∑ m−1 iτ (m−1i )me hsm−1 i,)8

with equality if abatement costs are strictly increasing. Suppose, without loss of generality, thatthere are two zones upstream of the critical confluence, say, m − 1 i = a, b. By assumption,¯X m = τ am e hsa+ τ bm e hsb ,On the other hand, the TRS must allocate zonal discharge load standards ¯Z’s such that ¯Z m = 0and¯X m = τ am ¯Za + τ bm ¯Zb ,without knowledge of e hs . Because there is only one constraint equation for two zonal standards,the allocation is indeterminate. It is trivial to see that if the TRS allocates, for example, e hsa > ¯Z aand ¯Z b = ( ¯X m − τ am ¯Za )/τ bm > e hsb, the trading equilibrium can never achieve e hs . Similararguments apply when there are more than two upstream zones. This completes the proof.Proposition 2 implies that if either condition (i) or condition (ii) holds, the TRS cannot berelied upon to deliver the efficient outcome even if the ambient environmental constraints ¯X areset optimally. One might ask whether the conditions are likely to be met in practice. Nonassociativityis unlikely to be a serious concern. In many cases, perhaps most cases, a linear T is agood approximation and so associativity is guaranteed. We return to this point in Section 5. 9We believe that the second condition, in which a critical zone lies at a confluence of branches,is not at all unusual. In a branching river, confluence zone m is critical if ∑ m−1 iτ (m−1i )m ¯X m−1i >¯X m , where {m − 1 i } i is the collection of indices directly upstream of zone m, along all contributingbranches. Economic activity and population both tend to concentrate around the confluence ofrivers. The water quality there is often important for both aquatic species and people livingnearby. Thus a zone of confluence might be more likely than others to be critical.Moreover, the TRS mechanism also has a practical disadvantage. Consider a branchless riversystem. Here the TRS equilibrium achieves the efficient outcome, if it achieves it at all, with notrade. To see this, note that as in the proof of Proposition 1, the efficient environmental constraintsare found by setting ¯X eff = T (ē − a eff ) ′ . Then as Hung and Shaw show, in a branchless watershedthe constraint set arising from ¯Z is equivalent to that arising from ¯X eff and the TRS equilibriumachieves the cost-effective outcome. But because the cost-effective outcome must coincide with theefficient outcome, which also coincides with the initial allocation, and because it is assumed thatthere is only one discharger in each zone, this implies that discharging pollution so as to satisfy¯Z exactly, without engaging in any trade, is also cost-minimizing. Put another way, the regulatorcannot implement the efficient optimum in a decentralized manner. This claim is stated in thefollowing result, which we state without proof.Proposition 3. Suppose that in program (2), zonal environmental constraints ¯X are set at theefficient levels and that there is only one discharger in each zone. Then if the TRS trading achievesthe cost-effective optimum of program (2), it is achieved with no trade.9 Associativity is violated in the following hydrological model of pollutant flow in a groundwater aquifer. Toddand Mays (2005) model the concentration of a pollutant at distance δ and time t from a point source as:andτ(δ) = 1 2{1 − erf( d − vt2 √ tDX(d) = X 0τ(δ),) ( ) [ dv+ exp 1 − erfD( d + vt2 √ tD)]},where erf(·) is the Gauss error function, D the dispersion coefficient, v the average linear velocity, and X 0 the pollutionconcentration at the point source. The transfer coefficients τ(δ) derived from this model are not associative. See alsoSado et al. (2010), who apply the TRS to a set of point sources on the Passaic River in New Jersey. Their transfercoefficients do not quite satisfy the associativity property.9

4 The Damage-denominated Trading-Ratio System (DTRS)The DTRS of Farrow et al. is similar to the TRS of Hung and Shaw in that both are innovativeschemes for controlling water quality through trade in permits to emit pollution. 10 Both requirethat trades between sources satisfy a set of trading ratios. There the similarity ends.The fundamental regulatory constraint in the DTRS is a single limit on aggregate monetarydamages, here denoted TD, rather than a set of physical environmental standards. The tradingratios are themselves based upon marginal damages, rather than upon physical transfer coefficients.Each source i’s marginal damage d i is calculated by integrating its contribution to monetary damagesover that source’s “zone of influence.” Having calculated marginal damages for each source, theregulator distributes permits ¯L i (in terms of emissions at the point of discharge) in such a way thataggregate damages meet the overall monetary constraint at the initial allocation: ∑ i d i ¯L i = TD.Trade is allowed between any two sources, but only according to the ratio of their marginal damages.The aggregate limit on damages will continue to be satisfied in the face of any permissibletrade at these ratios.Given the vector d of marginal damages and a vector e of emissions, Farrow et al. (and alsoMuller and Mendelsohn 2009) assume that aggregate damages are linear: D(e) = ∑ ni=1 d ie i . It isthis quantity that must not exceed TD. The assumed linearity of the damage function means thatthe d i ’s do not depend upon emissions from other sources.Each source i solves the following cost-minimization program:min C i (a i ) − p i r si + ∑ p jr sj (12)r ki ,r si ,r sj js.t. (ē i − r ki ) − ∑ d jr sj ≤j d ¯L i (13)ia i = r ki + r si (14)r ki , r si , r sj ≥ 0, (15)where p i and p j are the market prices for a permit from source i and j, r sj is the amount of pollutioncontrol purchased from source j to offset pollution at source i, r ki is the amount of pollution controlfrom source i that is kept by source i to meet the emissions standard ¯L i , and r si is the amount ofpollution control sold by source i.Note that substituting e i = ē i − a i and r ki = a i − r si into (13), one obtains an analogue of (9),the Hung-Shaw trading constraint:e i ≤ ¯L i + ∑ jd jd ir sj − r si . (16)This constraint means that each polluting source can trade with any source, according to themarginal damage ratios, so long as the level of its discharge does not exceed the sum of theoriginal discharge limits ¯L i and the net purchase of damage-denominated permits ∑ j (d j/d i )r sj −r si .Because sources can trade permits at the exchange rates d j /d i , the spatially explicit prices of permitsin the equilibrium (including the boundary case) satisfy the analogue of (10):d jd ip j = p i . (17)10 In fact, the DTRS can be said to be the more powerful of the two because it does not rely upon the strictlydownstream flow of a river. Muller and Mendelsohn (2009) build directly upon the DTRS in their model of air-qualitytrading.10

Note that unlike in the TRS, one can be sure that d i ≠ 0 in practice for all i: a source for whichthis is not true would not be part of the trading system. Therefore, each source can trade with anyother source, including those located upstream or downstream or on other branches of the river.Farrow et al. (2005) derive the first-order necessary (and sufficient) conditions for each source’soptimization problem, from which the following interior equilibrium condition is derived:∂C i /∂a i∂C j /∂a j= d id j= p ip j. (18)This condition is, however, an incomplete characterization of a market equilibrium. First, therewill be n − 1 equations for (n × 2) + n 2 unknowns {a i , r ki , r sj } i,j . Second, as Farrow et al. observe,this condition holds only when source i is a net buyer and source j is a net seller (and vice versa).Proposition 4 confirms that equation (17) must hold for any equilibrium, interior or boundary. Italow establishes conditions that must be satisfied at an interior equilibrium.Proposition 4. Suppose that marginal abatement cost C ′ i (a i) is strictly increasing for every source i.Then for given baseline emissions {ē i }, initial permits {¯L i }, and trading ratios {d i }, an interiormarket equilibrium of Farrow et al.’s DTRS mechanism is a vector {p ∗ i , a∗ i , r∗ ki , r∗ si , r∗ sj } i,j that solvesthe following system of equations:a ∗ i = R i (p ∗ i ), (19)∑di[ēi − R i (p ∗ i ) − ¯L i]= 0, (20)p ∗ id i= p∗ jd jfor alli, j, (21)where R i (p i ) is an abatement decision function given the price p i . The vector {rki ∗ , r∗ si , ∑ jis uniquely determined by the following response functions:d jd ir ∗ sj } i,jr ∗ ki = { ¯Li if ē i − R i (p ∗ i ) ≤ ¯L i0 if ē i − R i (p ∗ i ) > ¯L i(22){ ¯Lirsi ∗ − ē=i + R i (p ∗ i ) if ē i − R i (p ∗ i ) ≤ ¯L i0 if ē i − R i (p ∗ i ) > ¯L i(23)∑{d jr ∗ 0 if ēi − Rjsj =i (p ∗ i ) ≤ ¯L id i ē i − R i (p ∗ i ) − ¯L i if ē i − R i (p ∗ i ) > ¯L i(24)Proof. First, we show a stronger version of condition (18): under the DTRS, prices must satisfy(21) in any equilibrium regardless of whether each source is a net buyer or a net seller. To see this,note that if (21) does not hold, say ifp i> p j,d i d jthen unlimited arbitrage profits are available to any source k ≠ i, j who buys permits from j andsells them to i. Market demand for permits from j is infinite while the number of permits availablefrom j is finite at ¯L j . Note that whether they can actually buy and sell permits or not is irrelevant:they simply demand permits while taking prices as given. Thus equilibrium prices must adjust insuch a way that (21) is satisfied. Therefore, each source i faces the same effective price in all zonalmarkets j: p i = p j (d i /d j ). It is irrelevant, then, from which sources i buys permits or to whichsources it sells. It then follows that i abates to the point at which ∂C i /∂a i = p i . Because ∂C i /∂a i11

is strictly increasing, the optimal abatement level a ∗ i = MC−1 d(p i ) is unique. Thus r = p i = p i j d jfor all j.Now, let us construct an excess demand function. Given an arbitrary price r of permits, sourcei would choose abatement R i such that MC i = r. Thus, R i is a well-defined function. The excessdemand for permits from source i is z i (p i ; ē i , ¯L i ) = ē i − R i (p i ) − ¯L i . If z i > 0, then i must buypermits from elsewhere. If z i < 0, it sells its excess permits in the market. All permits sold toand purchased from i must be exchanged at the ratio d i /d j with permits from any source j. Thismeans that the common units of exchange are d i z i . Thus, the market clears in equilibrium only ifequations (19)–(24) holds. For a given vector {ē i , ¯L i , d i } i and n sources, this gives us n equationsfor n unknown prices {p ∗ i } i. Thus the equilibrium is exactly identified. The proof of the expressionsfor {r ki , r si , ∑ d jj d ir sj } i,j is obvious and thus omitted. The completes the proof.This characterization of market equilibrium turns out to be useful for the simulations in Section5. There may be non-trivial boundary equilibria in which a ∗ i = 0 or a∗ i = ē i. These boundarycases can be dealt with by defining R i (p ∗ i ) = 0 if ∂C i/∂a i > p ∗ i for all a i ∈ [0, ē i ] and R i (p ∗ i ) = ē iif ∂C i /∂a i < p ∗ i for all a i ∈ [0, ē i ]. The rest of the equilibrium conditions are intact. We now askwhether the equilibrium of the DTRS achieves the cost-effective solution of program (3).Our next result is directly analogous to Proposition 2. According to Proposition 1, the solutionto program (3) also solves program (1), regardless of branching or nonlinear damages. The questionis whether the DTRS equilibrium is guaranteed to achieve the solution to program (3). Proposition 5shows that the answer is no. If the aggregate damage function is nonlinear in concentrations, thenthe DTRS equilibrium does not minimize costs.Proposition 5. Suppose that aggregate environmental damages are a nonlinear function of pollutionconcentration, such that for all e we have:D(e) ≠ ∑ i (∂D(e)/∂e i)e i .Then the DTRS equilibrium does not achieve the cost-effective solution of program (3).Proof. We offer a proof for the case of an interior optimum of (3). Note that at the interioroptimum, the emission vector e eff must satisfy the necessary (but not sufficient) condition∂D(e eff )/∂e i∂D(e eff = ∂C i(a eff i )/∂a i)/∂e j ∂C j (a eff)/∂ajjfor all i, j,where e eff i= ē i − a eff i . On the other hand, under Proposition 4, at an interior equilibrium we haved i= ∂C i(a eff i )/∂a id j ∂C j (a eff)/∂ajjfor all i, j.Thus, in order for the equilibrium to achieve the cost-effective solution, the regulator must evaluatethe exchange rates (the d’s) at the optimum: d eff i = ∂D(e eff )/∂e i . Under the DTRS, the regulatorallocates ¯L in such a way that ∑d effii¯L i = T D = D(e eff ). (25)We now ask whether there exists some initial allocation ¯L, satisfying (25), such that the resultingequilibrium would achieve the cost-effective solution. Suppose, by contradiction, that there existssuch an allocation ¯L and that the resulting equilibrium is also efficient: e dtrs = e eff . Because the12

equilibrium must satisfy the market-clearing condition (21), we have∑d effii¯L i = ∑ i d effiDT RSei . (26)However, because the aggregate damage function is nonlinear, we have∑d effii e effi ≠ D(e eff ). (27)Combining (25), (26), and (27), we see that∑d effii¯L i = ∑ d effii e dtrsiwhich contradicts that e dtrs = e eff . This completes the proof.= D(e eff ) ≠ ∑ d effii e effi .As is evident from the proof, the DTRS breaks down because the initial allocation of permits ¯Lfollows Farrow et al.’s original allocation rule (25). A natural question arises: what would happenif one were to use a different allocation rule? For example, the regulator could allocate permitsso that S(¯L 1 , . . . , ¯L n ) = TD. Here one encounters an insuperable difficulty: there is no allocationrule the regulator could rely upon in this case. Indeed, the problem arises under the TRS too.To see this, suppose that the regulator agreed upon the desired level of aggregate damage TD.Because the damage function is nonlinear, there will inevitably exist many vectors ¯L such thatS(¯L 1 , . . . , ¯L n ) = TD. The regulator’s problem is indeterminate. (Recall that D is the compositionfunction D(e) = S(T e).)In the following section, we investigate how the TRS and DTRS perform, relative to the efficientsolution as well as to each other, if the initial allocation of permits follows such an arbitrary rule.5 A numerical modelThe results of the previous two sections imply that in numerous watersheds of practical interest,water-quality trading based on either TRS or DTRS may not achieve the efficient outcomes. Thenumerical exercise reported here is designed to quantify the welfare losses associated with theshortcomings of the two systems, relative to each other and also relative to the social optimum.We do this by constructing a numerical model. Though the model is fairly sophisticated in itsconstituent parts, incorporating the relevant scientific aspects of a realistic watershed, in order toilluminate our key question we have kept it relatively small. The specific parameter values arehypothetical but realistic.Consider a second-best scenario in which the regulator has imperfect information regarding polluters’abatement costs, but has perfect information regarding environmental damages. Together,these conditions mean that the fundamental constraints ¯X (for the TRS) or TD (for the DTRS)cannot be set at the efficient levels. There can be infinitely many ways to allocate initial permitsunder such a scenario, all meeting the constraint in the absence of trade. We assume that the totalnumber of permits in the initial allocation is equal to the socially optimal level. This assumptionenables us to disentangle the sources of inefficiency, as it implies that TRS and DTRS could potentiallyachieve the social optimum. Put another way, if either the TRS or the DTRS fails to achievethe social optimum, it is not because the supply of permits is incorrect relative to the optimum.Indeed, the theoretical premise of ideal permit trading is that, in the absence of market failures orother distortions, the trading outcome does not depend on the initial distribution of permits. Herewe have no market imperfections, but the premise is violated anyway.13

5.1 The water-quality modelThe Environmental Protection Agency has developed a theoretically acceptable and practicalmethod of modelling water-quality impacts, the NWPCAM. Farrow et al. (2005) consider a waterqualityimpact model based on the hydrologic assumptions used in NWPCAM. We follow the samestrategy.Water pollutants, such as phosphorus, nitrogen, and heat, decay as water carrying them flowsdownstream through a river system. For instance, a pound of phosphorus discharged at one point inthe river contributes to phosphorus concentrations at all points downstream of the discharge pointuntil it is entirely assimilated. Let x mi be source i’s contribution to the pollution concentration atlocation m downstream. Then x mi is a function of the emissions e i at source i, stream flow Q, andan exponential decay term:{0 if m is not downstream of ix mi =( )e iQ exp −ˆkδ mi if m is downstream of i, (28)where ˆk is the decay parameter and δ mi is a distance in river miles between source i and location m. 11Note that (28) implies that in the absence of other pollution sources, the pollution concentrationat location i is simply x i = x ii = e i /Q. The pollution concentration at location m is the sumof contributions to the pollution concentrations across sources: x m = ∑ i x mi. This hydrologicalmodel incorporates both the unidirectional flow and the natural decay as important characteristicsof a river system. Furthermore, the model can also incorporate branching in the river. 12It is clear that the impact of changes in pollution concentrations at source i’s location onpollution concentrations at any downstream location j is linear:τ ijdef= dx jdx i= exp(−ˆkδ ij ), (29)where the τ ij are the transfer coefficients. Note that we can easily establish the associative propertyof the transfer coefficients: τ ik = τ ij τ jk for any sources i, j, k where source j is located downstreamof source i but upstream of source k. 13Given the hydrological relationship between emissions and ambient pollution concentrations,11 In Farrow et al., the function takes the form{ ( )}e iQ exp −kθ t−20 δmi,Uwhere k is the nominal decay rate, θ is a coefficient reflecting sensitivity of k to the mean temperature t, and U isthe stream velocity. Because units do not have any significant meaning in this simulation, we have simplified thisrelationship by replacing kθ (t−20) /U with ˆk, which is assumed to be constant and is evaluated at average hydrologicalparameters.12 To see this, suppose two sources, i and j, are located along two different branches upstream of a confluence. Theimpacts of emissions from i and j on pollution concentrations at location k below the confluence are expressed as x kiand x kj .13 Using (29), we can write:))τ ijτ jk = exp(−ˆkδ ij exp(−ˆkδ jk ,()= exp −ˆk[δ ij + δ jk ] ,)= exp(−ˆkδ ik = τ ij,where the last equality follows because δ ik = δ ij + δ jk by assumption.14

an important question remains as to how we should model the relationship between pollutionconcentrations and (monetary) damages. In this regard, Farrow et al. assumed that marginaldamages D are constant at each location m: ∂D/∂x m = W T P × H m where W T P is the constantper-capita marginal damages from changes in water quality and H m is the population size atlocation m. According to this specification, damages from each source’s emissions are given byD i (e i ) = d i e i , where d i is constant and independent of emissions levels e i :d i =M∑W T P × H m × τ mi × 1 Q .m=1To justify the constant marginal willingness to pay (WTP), Farrow et al. argue that water qualityis inversely related to pollution concentrations (that is, ∂W m /∂x m = −1) and that “the householdmarginal willingness to pay for a small improvement in water quality, WTP, is constant . . . over therange of water-quality conditions considered in this study.” (2005, p. 197).In the non-market valuation literature, however, it is often found that individuals obtain utilityfrom recreational and aesthetic values of water quality rather than directly from pollution concentrationlevels. That is, individuals care if the river water is swimmable, drinkable, and fishable,and if the river water provides habitat for important aquatic species. The quality of water andaquatic habitat at any location in the river is typically a nonlinear function of concentrations ofthese pollutants in the area. Numerous examples exist: brown trout may cease growth at watertemperatures above 18.7-19.5 celsius degrees (Elliott and Hurley, 2001) and may not survive sevendays above 24.7±0.5 celsius degrees (Elliott, 2000). The population size of cutthroat trout is ahighly nonlinear function of selenium exposure and was estimated to experience 90% declines atmean selenium concentrations exceeding 17 µg/g of dry weight (Van Kirk and Hill, 2007). Andalgae growth is often modeled as a logistic function of nutrient concentrations (Anderson et al.,2002). Biological and chemical responses to pollution levels may not be linear.Willingness to pay may not be linear either. In many watersheds in the U.S., water qualityis already impaired, so that a further increase in pollution concentrations might cause seriouswater-quality degradation. Even if WTP for a small change in quality is constant over a rangeof concentration levels, it might be much bigger for the same small change above a particularconcentration threshold. Indeed, our communications with water practitioners reveal that theirprimary concern is closely related to this nonlinear biological response around hotspots. We stresshere that this concern is one of the important political factors that has plagued water-qualitytrading in many watersheds.To model the nonlinear biological responses that allow for the type of threshold effects highlightedabove, we consider a logistic damage response to pollution concentrations at each locationm:bS m (x m ) =for all m = 1, . . . , M, (30)1 + exp(−a(x m − c))where a is a damage-sensitivity parameter, b a scale parameter, and c a concentration threshold.The total economic damages are then given by D(e) = ∑ m S m(x m ). Logistic models are commonlyused in biology and ecology for modelling the response of species’ mortality or population size topollution. Though in biology, the parameters a and c often depend on a variety of environmentalfactors, we treat them as constants that do no vary by time or location for simplicity of analysis.The parameter b is a scaling parameter that transforms biological damages into monetary economicdamages, which we also assume are constant. With these assumptions, marginal aggregate damageswith respect to emissions from any source i depend not only upon emissions from that source but15

Parameter Units Valuek Decay rate mile −1 0.005Q Stream flow ft 3 /s 10a Damage parameter none 5b Damage scale parameter none 6.7c Concentration threshold mg/L 5Table 1: Parameters for water-quality modelalso∑upon emissions from other sources, both downstream and upstream of i (that is, ∂D/∂e i =m (∂S m/∂x m ) (∂x m /∂e i )). When there is a branch in a river system, marginal damages alsodepend on the emissions from sources located along the other branches.The parameter values of the model can vary widely by pollutant and watershed. Becauseour goal is to obtain generic efficiency properties of the two trading systems, we decided to chooserepresentative parameter values for the water-quality model (28) and then choose a set of parametersa, b, and c that generate interior optima for at least two out of three sources given the assumedcost parameters (see below). The value of ˆk = 0.005 is chosen based on three parameters: themean of the decay rates for seven representative water pollutants (U.S. EPA, 2002), the averagewater temperature of 20 Celsius degrees, and the stream velocity of 1.5 miles per hour. The scaleparameter b and the threshold parameter c are important in generating interior solutions. We thusstarted with arbitrary values a = 5 and c = 5 and then searched for the associated value of b. Theparameters used for the simulation are summarized in Table 1.5.2 Simulation ScenariosWe assume that the river has a main stem M and a single branch B. The river has a maximumlength of 200 river miles along the main stem and 150 river miles along the branch: m M ∈ [0, 200]and m B ∈ [0, 150]. The confluence occurs at m M = 100 (or equivalently, m B = 50). There arethree polluting sources. Source 1 is located in the most upstream point of the main stem (m M = 0),source 2 at the confluence (m M = 100 or m B = 50), and source 3 at the most upstream point ofthe branch (see Figure 1). The firms (or sources) have quadratic abatement cost functions of theform C i (a i ) = (1/α i ) × a 2 i , with a i ∈ [0, ē i ] and α i > 0.For a given initial allocation of permits, we first compute the social optimum, and then allocatepermits in an equal amount to each polluting source: ¯L1 = ¯L 2 = ¯L 3 = ∑ e eff /3. Under the TRS,this means that zonal environmental standards, the ¯X’s, are allocated so that ¯X 1 = ¯L 1 , ¯X3 = ¯L 3 ,¯X 2 = ¯L 2 + τ 12 ¯X1 + τ 32 ¯X3 . Once again, we do not assume that the regulator knows e eff i or ∑ e eff i .We chose this allocation rule because we are interested in the relative performance of the twotrading systems under conditions that can be compared to the social optimum. Under the TRS,the correct transfer coefficients are known to the regulator and are announced to the polluters.Under the DTRS, the regulator does not know the social optimum, and so she evaluates the d i ’sat the initial allocation. 14 In each of these setups, we simulate the trading outcomes for two setsof cost parameters: (A) α 1 = 7.5, α 2 = 15, α 3 = 7.5 and (B) α 1 = 15.0, α 2 = 7.5, α 3 = 15.0.14 We also experimented with various choices of d i’s. The results were not sensitive to these values.i16

Figure 1: Hypothetical riverCase A Outcome e 1 e 2 e 3 Damage Cost TotalTRS 23.7 23.7 23.7 511 1,942 2,454DTRS 48.9 0.0 34.4 530 1,589 2,119Optimum 42.0 0.0 29.0 60 1,787 1,848Case B Outcome e 1 e 2 e 3 Damage Cost TotalTRS 18.5 32.0 0.0 10 1,710 1,720DTRS 20.5 31.4 0.0 9 1,716 1,725Optimum 21.0 34.5 0.0 42 1,655 1,6975.3 Simulation Results5.3.1 Case A: α 1 = 7.5, α 2 = 15, α 3 = 7.5Table 2: Simulation resultsIn this case, a low-cost firm is located downstream of two high-cost firms. At the socially optimaloutcome, source 2 abates all of its emissions while source 1 emits more than source 3 despite thefact that they have the same marginal costs and the same baseline emissions (Table 2). This occursbecause marginal damages at the optimum increase more in source 3’s emissions than in source 1’semissions. The TRS mechanism, as noted above, precludes upstream firms from buying permitsfrom downstream firms. It also precludes firms located on different branches above a commonconfluence from engaging in any trades. Because the potential seller (the low-cost firm) is locateddownstream, no trade can occur between the downstream firm and the upstream firms. Moreover,at a social optimum efficient trades should occur between the two upstream firms. Yet again notrade between them is allowed under the TRS. As a result, firms incur higher abatement costsunder the TRS than at the optimal outcome. On the other hand, the DTRS does allow tradesamong any of the three sources.The problem with the DTRS, however, is that in this case the damage-denominated tradingcoefficients turn out to be poor approximations to the true marginal damages at the optimum. This17

Figure 2: Marginal damages, marginal costs, and trading coefficientspoint is demonstrated in Figure 2. The figure plots marginal damages as a function of each source’semissions, holding other sources’ emissions at the optimum. Figure 2 also shows each source’smarginal cost and trading coefficient. The social optimum occurs where marginal damages fromeach source are equated with its marginal cost and the overall constraint is satisfied. Interestingly,the equilibrium does not occur where each source’s marginal cost equals its trading coefficient d i .This is because each source makes its abatement/trading decision so that its marginal cost equalsthe spatially explicit price it faces, p i = (d j /d i )p j . Thus, when the d i ’s are poor approximationsto actual marginal damages, the trading outcome under DTRS does not equate marginal damageswith marginal costs.Another problem with the DTRS∑is that, in equilibrium, the sum of emissions can exceed thesum of initial emissions permits: e dtrsi > ∑ e eff i . This occurs because neither the individualpolluters nor the equilibrium market-clearing condition are constrained by ∑ e dtrsi ≤ ∑ e eff i . Themarket clears instead with ∑ d i e dtrsi ≤ ∑ d i ¯Li where ∑ d i ¯Li∑can be greater than or less thandi e eff i . Because sources can emit more than the socially efficient amount, environmental damagesare higher, but abatement costs are lower, at the DTRS equilibrium than at the social optimum.Lastly, note that Figure 2 shows that the first-order conditions are not sufficient in two ways.First, for source 3, its marginal cost curve intersects with the marginal damage curve at two points.Second, though not plotted, each source has infinitely many marginal damage curves correspondingto different emissions levels by other sources.18

5.3.2 Case B: α 1 = 15.0, α 2 = 7.5, α 3 = 15.0In this case, a high-cost firm is located downstream of two low-cost firms. At the social optimum,source 3 abates all of its emissions while source 2 emits the most (Table 2). Efficient trades shouldoccur under the TRS, because this system allows the high-cost downstream source to buy permitsfrom the two low-cost upstream sources. Indeed, this is exactly what happened in the simulation.Each firm is allocated 18.5 units of discharge permits-+ initially in both the TRS and the DTRS.In the TRS equilibrium, firm 2 bought 14.4 units from source 3 to increase its emissions to 32.9while firm 3 sold 18.5 units at the trading ratio τ 32 ≈ 0.78 (i.e. 18.5×τ 32 = 14.4 units for source 2).Note that under the TRS, the downstream firm has a choice of buying permits from either source1 or source 3. Therefore, if the effective price of permits from source 1 is higher than that fromsource 3, then source 2 would buy permits from source 3 and vice versa.It follows then that the effective equilibrium prices are equalized across space: p 1 = τ 12 p 2 =τ 32 p 2 = p 3 . At this equilibrium price, source 1 has no incentive to sell its permits to source 2and thus ends up emitting exactly at the initial allocation. The trading outcome in the DTRS issimilar. Source 3 abates its emissions down to zero and sells its permits mostly to source 2. Adifference occurs, because source 3 also sold its permits to source 1. As we have noted, under theDTRS firms located in different branches are allowed to trade, and the equilibrium prices satisfyp 1 = (d 1 /d 2 )p 2 = (d 1 /d 3 )p 3 . It turns out that at these equilibrium prices, it is cheaper for source 1to buy permits from source 2. As a result, source 1’s equilibrium emissions are slightly higher underthe DTRS than under the TRS while source 3’s equilibrium emissions are lower under the DTRSthan under the TRS. The extra trade between source 1 and source 3, however, decreased efficiencyslightly, compared to the TRS equilibrium. This is because the damage-denominated tradingcoefficients, the d i ’s, are poor approximations of the true marginal damages at the optimum, asshown in Figure 2. In this case, therefore, the DTRS encouraged inefficient trades. Note, however,that both TRS and DTRS closely approximate the social optimum in this case.5.4 Prices vs. quantitiesOn one hand, our numerical analysis suggests the impossibility of getting prices right in watershedshaving certain characteristics. Neither the TRS nor the DTRS succeeds in providing the correctprice signals for water-quality trading. On the other hand, our analysis also indicates that in somecases, the equilibria approximate the social optimum quite closely. We obtained these results bysetting the total supply of permits equal to the socially optimal level. A natural question thenis, which type of inefficiency is larger: not getting the prices of pollution right or not getting thequantity of permits right? We investigate this question by simulating the trading outcomes forvarying levels of the supply of permits (in percentage reduction from the baseline discharge level)and allocating permits in equal number to each discharger. The result is shown in Figure 3.First, the relative performance of the two systems varies, in an unsystematic way, with thesupply of permits. In Case A (α 1 = 7.5, α 2 = 15, and α 3 = 7.5) where a low-cost source is locateddownstream of two high-cost sources, the DTRS performed substantially better than TRS when thetotal supply of permits was kept to the socially optimal level. This is because the TRS prohibitedany trade from taking place. However, when the total supply of permits is reduced to 60-70% of thebaseline discharge level, the TRS performs better than the DTRS despite the fact that no tradingstill takes place under the TRS. This occurs because the efficiency loss due to the TRS precludingtrading was outweighed by the efficiency loss due to the DTRS encouraging inefficient trades,which increased environmental damages substantially relative to no trade. In contrast, in Case B(α 1 = 15.0, α 2 = 7.5, and α 3 = 15.0), in which a high-cost source is located downstream of two19

low-cost sources, the DTRS performed slightly better than the TRS for all levels of initial permitsupplies. In this case, TRS and DTRS provide similar price signals so that the magnitude of theefficiency loss due to environmental damages is similar between the two systems (see the left panelof Case B in Figure 3). However, because the DTRS offers more flexibility in trading, it reducesabatement costs a bit more than does the TRS. This effect dominates the relative performance ofthe two systems.Second, total economic costs C + D do not exhibit a simple convex relationship with respectto the total supply of permits under the two systems. This is because neither environmentaldamages nor abatement cost has a simple relationship to the supply of permits. Despite thefact that environmental damages are defined as a decreasing function of emissions or pollutionconcentrations, environmental damages in the trading equilibrium are not necessarily a decreasingfunction of the reduction in the total supply of permits, and analogously, despite the fact thatabatement costs are a convex function of abatement levels, total abatement costs are not necessarilya convex function of the reduction in the total supply of permits. These effects are especiallystrong in the DTRS, because the damage-denominated trading coefficients are based on marginaldamages at the initial allocation, and thus depend endogenously on the initial supply of permits.Somewhat counter-intuitively, these trading coefficients can either decrease or increase efficiencyrelative to the TRS. On one hand, the trading coefficient can decrease efficiency by providing wrongtrading margins, which adversely affects environmental damages. On the other hand, however, thetrading coefficients can improve efficiency by providing flexibility for trading partners, which reducesabatement costs.Lastly, at least in the current model, getting the quantity of permits right appears to be moreimportant than getting the prices of permits right. In Case A, the estimated efficiency losses areonly 18.2% and 0.1% of the total economic damages, respectively, for TRS and DTRS when thesocially optimal number of permits are distributed. (In Case B the corresponding results are 3.6%for TRS and 0.03% for DTRS.) In contrast, the maximum efficiency losses due to mis-specifying thetotal supply of permits are 80.9% and 97.8% of the total economic damages, respectively, for TRSand DTRS. (In Case B the corresponding results are 80.3% for TRS and 78.1% for DTRS.) It isimportant to emphasize that this result is not a direct consequence of the logistic damage responsewe assumed in (30). Rather it stems from the multiple effects of mis-specifying the quantity ofpollution, including the incorrect price signals that arise from it.6 DiscussionThis paper examined the efficiency properties of two recently developed water-quality tradingmodels, the trading ratio system (TRS) proposed in Hung and Shaw (2005) and the damagedenominatedtrading ratio system (DTRS) proposed in Farrow et al. (2005). These two modelsemerged as potential water-quality trading models that address both spatially explicit damagesand transaction costs. We showed that both trading systems fail to achieve the efficient optimum(and the cost-effectiveness optimum) under general conditions that are likely to hold in numerouswatersheds. More specifically, the TRS fails when the river has critical zones in a branching riverwhereas the DTRS fails when the pollution damages are nonlinear in either emissions levels orpollution concentrations. We derived these results under the first-best scenario in which the regulatorknows the efficient vector of environmental constraints (for TRS) and the efficient damageconstraint (for DTRS).Furthermore, in a second-best scenario where the regulator cannot set these constraints atthe efficient levels, neither system dominates in terms of efficiency, because the TRS excludes20

Figure 3: Total supply of permits and relative performance of TRS and DTRSefficient trades while the DTRS promotes inefficient trades. These results indicate, in this sense, theimpossibility of getting the spatially explicit prices of pollution right under either system. However,our computational results do indicate the possibility that the two systems may still approximatethe socially efficient optimum sufficiently closely if the total allowable permits are set initially atlevels sufficiently close to the optimum. The (maximum) efficiency loss due to mis-specifying thetotal supply of permits was much larger than that from mis-specifying the spatial prices of permits.Interestingly, the magnitude of inefficiency due to incorrect signals may also depend on the totalsupply of permits. Thus our paper suggests the importance of getting the quantity of pollutionright even while striving to get the spatial prices of pollution right.Though we kept the assumptions of our model as general as possible with respect to waterpollution and watershed characteristics, like Hung and Shaw and also Farrow et al. we ignored theproblem of nonpoint-source pollution (NSP). Nonpoint sources can of course play an importantrole in a watershed. It is usually difficult and costly to identify and monitor the level of NPSpollution-causing activity (or discharge levels), because land-use practices (for example, fertilizerapplication) or land use itself (for example, buildings and parking lots) are the major sourcesof such pollution. If the discharge from each source is difficult to identify, neither trading nordirect control would achieve the efficient outcome. However, in recent years substantial efforts havebeen devoted to transforming nonpoint sources into point sources. Scientists of various disciplinescontinue to improve their ability to identify and monitor pollution levels from nonpoint sources.As our understanding of NPS improves, the present study could have important implications forthe optimal management of nonpoint-source pollution.In the case of nonpoint pollution, the business of getting the spatial prices of pollution rightbecomes even more difficult for two reasons. First, because water pollution can travel through mul-21

tiple nonpoint sources before it reaches the river, and because the number of nonpoint sources isoften quite large, the spatial dependence of marginal damages from each source’s pollution is likelyto be exacerbated. Second, each nonpoint source’s discharge is likely to affect pollution concentrationlevels at multiple receptors. In such a case, allowing sources to trade at the exchange ratesbased on the transfer coefficients between the receptor points along the river, as in the TRS, wouldlikely result in substantial deadweight loss for two reasons. First, because each source’s emissionscan affect pollution concentrations at multiple receptors for some of which transfer coefficients canbe zero (across branches, for example), trading based only on the non-zero transfer coefficients mayresult in inefficient trades. Second, just as with point-source pollution, the TRS can also precludeefficient trades from taking place by restricting the exchange among sources whose emissions affectpollution concentrations at receptors located on separate branches. On the other hand, the DTRSwould be relatively robust to nonpoint-source pollution. As long as the discharge level from eachnonpoint source can be identified, pollution damages still being the function of pollution concentrationlevels at receptor points, the marginal damage from each nonpoint source can be calculated.Because trading ratios based on the marginal damages are the correct exchange rates for the nonpointsources, the DTRS could potentially offer the correct trading incentives. The problem arises,however, when damages are highly nonlinear. As we have demonstrated, evaluating the marginaldamages at any allocation (including the optimum) and fixing the trading ratios at that evaluationpoint would encourage inefficient trades to take place by giving incorrect trading incentives. Thenumber of trading sources is likely to be large in the case of NSP, and so the error from pre-fixingthe trading ratios might result in substantial deadweight loss.22

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