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

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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
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Page 5 and 6: coefficients. 5 Let S : R M → R,
Page 7 and 8: 3 The Trading-Ratio System (TRS)The
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Page 11 and 12: 4 The Damage-denominated Trading-Ra
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Page 17 and 18: Parameter Units Valuek Decay rate m
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Page 25: [16] Morgan, Cynthia and Ann Wolver

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

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