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

More documents

Recommendations

Info

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
Page 1 and 2: Water-Quality Trading:Can We Get th
Page 3 and 4: contributions, both of which restri
Page 5 and 6: coefficients. 5 Let S : R M → R,
Page 7: 3 The Trading-Ratio System (TRS)The
Page 11 and 12: 4 The Damage-denominated Trading-Ra
Page 13 and 14: is strictly increasing, the optimal
Page 15 and 16: 5.1 The water-quality modelThe Envi
Page 17 and 18: Parameter Units Valuek Decay rate m
Page 19 and 20: Figure 2: Marginal damages, margina
Page 21 and 22: low-cost sources, the DTRS performe
Page 23 and 24: tiple nonpoint sources before it re
Page 25: [16] Morgan, Cynthia and Ann Wolver

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

Create successful ePaper yourself

Delete template?

Save as template?