Cost effective policies for alternative distributions of stochastic water pollution

Journal of Environmental Management (2002) 66, 145±157doi:10.1006/jema.2002.0569, available online at http://www.idealibrary.com on

1

Cost effective policies for alternativedistributions of stochastic water pollution

Ing-Marie Gren* ², Georgia Destouni ³ and Raul Tempone §

² Department of Economics, Swedish University of Agricultural Sciences (SLU),Box 7013, 75005 Uppsala, Sweden³ Department of Land and Water Resources Engineering, Royal Institute of Technology (KTH),100 44 Stockholm, Sweden§ NADA, Royal Institute of Technology, 100 44 Stockholm, Sweden

Received 4 December 2000; accepted 30 January 2002

This study investigates the role for cost effective coastal water management with regard to different assumptions ofprobability distributions (normal and lognormal) of pollutant transports to coastal waters. The analytical results indicate adifference in costs for a given probability of achieving a certain pollutant load target whether a normal or lognormaldistribution is assumed. For low standard deviations and con®dence intervals, the normal distribution implies a lower costwhile the opposite is true for relatively high standard deviations and con®dence intervals. The associated cost effectivecharges and permit prices are higher for lognormal distributions than for normal distributions at relatively high con®denceintervals and probabilities of achieving the target. An application to HimmerfjaÈ rdenÐan estuary south of Stockholm,SwedenÐshows that the minimum costs of achieving a 50 per cent reduction in nitrogen load to the coast varies more fora lognormal than normal probability distribution. At high coef®cient of variation and chosen probability of achieving thetarget, the minimum cost under a lognormal assumption can be three times as high as for a normal distribution.

# 2002 Elsevier Science Ltd. All rights reserved.

Keywords: cost effectiveness, economic policy instruments, stochastic water pollution, probabilitydistributions.

Introduction

In spite of decades of research on water resourcesprotection and remediation, many water systemssuffer from different types of pollution, such ascoastal areas that are damaged from excessivenutrient loads (Elmgren et al., 1997). A scienti®cchallenge for the assessment of ef®cient mitiga-tion methods for such pollution problems is posedby the quanti®cation of pollutant loads from thehydrological catchments, due to the complex andinherently uncertain pollutant pathways fromemission sources to water recipients. Recent ®eldobservations of chemical interactions at theland±sea interface and of stream water chemistryfurther emphasise our knowledge limitations and

0301±4797/02/$ ± see front matter

* Corresponding author. Email: [email protected]

uncertainties regarding the dynamics and actualpathways of pollutants within catchments (Moore,1996, 1999; Church, 1996; Kirchner et al., 2000;Stark and Stieglitz, 2000). In the case of coastalzones, pollutants may also be transported withinand between different coastal water basins, beforeentering the marine water, which has been shown tohave important implications for ef®cient manage-ment of coastal water quality (Gren et al., 2000a).

Since the pollutant transport in water catch-ments and coastal zones can be described onlyunder conditions of uncertainty, this uncertaintymust be accounted for when assessing the resourcesneeded for mitigating pollution damages and sug-gesting appropriate policy schemes. The role ofuncertainty in the identi®cation of cost effectiveand ef®cient solutions to water pollution problemshas been analysed in several papers (Beavis andWalker, 1983; Shortle, 1990; Malik et al., 1993;

# 2002 Elsevier Science Ltd. All rights reserved.

146 I.-M. Gren et al.

Segerson, 1988; Horan et al., 1998; Shortle et al.,1998; BystroÈm et al., 1999; Gren et al., 2000a). Inspite of the theoretical insights provided by thesestudies, however, we ®nd relatively few applicationsof them. One reason might be the dif®culty inquantifying the pollutant transport variabilityand the resulting prediction uncertainty due tosuch natural variability in both space and time(Cvetkovic et al., 1992; Destouni, 1992, 1993;Destouni and Graham, 1997; Graham et al., 1998;Foussereau et al., 2000, 2001).

A common assumption in the few applied studiesof cost effectiveness with regard to stochastic coastalwater pollution abatement is that of a normal pro-bability distribution of pollutant loads into the sea(BystroÈm et al., 1999; Gren et al., 2000a). However,this assumption has mostly been made for simplicityand lacks an empirical, or mechanistic basis insupport of its validity, implying that alternativepollutant load distributions may in fact be equally,or more relevant. Xu et al. (1996) have investigatedalternative (normal and lognormal) distributionassumptions for sediment yields that affect farmingsystem returns, showing that these assumptionsmay have important impacts on cost-effective abate-ment solutions.

The main purpose of this paper is to investigateand quantify the impacts of alternative distributionassumptions for coastal pollution loads on thecost-effective allocation of abatement measuresand policy instruments for reducing these pollutantloads. Similar to Xu et al. (1996), this paper specif-ically investigates the alternative assumptions ofnormal and lognormal probability distributionsof pollutant loads. These two distributions are ofinterest also for the coastal pollution problem,because the former is a symmetric probability dis-tribution that is often assumed in applied economicstudies (Gren et al., 2000a), whereas the latter is acommon asymmetric distribution that may also behydrologically relevant (Andersson and Destouni,2001). The paper begins by presenting a generaltheoretical analysis of the alternative distributionassumptions, which is then applied to the speci®cproblem of cost-effective reduction of nitrogen loadsto HimmerfjaÈrden, an estuary located about 60 kmsouth of Stockholm.

Stochastic water pollution loads

The transport of pollutants from a land area tothe coast may take place along any, or all three,of the following pathways: (a) groundwater ¯ow

discharging into the catchment stream networkthat in turn discharges into the coastal water;(b) direct groundwater ¯ow into the coastal water;and (c) overland water ¯ow to streams that dis-charge into the coastal water. All of these differenttransport pathways are dif®cult or impossible todescribe and predict deterministically. Even thoughvarious modelling methods are possible, such asseparation of hydrographs into different runoffcomponents and estimation of the pollutant contentof each component, all observation data available forthe model construction, calibration and testing havelimited support scales in time and space. Theextrapolation that is required in both time andspace, from site and time speci®c observations tolarge scale, long-term predictions relevant for coast-al areas and their water quality management, willalways be subject to uncertainty.

Recent ®eld observations, for instance, challengeprevious views on the relative importance of thedifferent possible pathways (a)±(c) discussed above(Moore, 1996, 1999; Church, 1996; Kirchner et al.,2000; Stark and Stieglitz, 2000). These observationsshow that the groundwater ¯ow and transportcontributions involved in pathways (a) and (b)may commonly be considerably underestimated, ormisrepresented, thus demonstrating the uncer-tainty about even the expected, main physicalmechanisms for pollutant transport within a catch-ment. Additional uncertainty is furthermoreimplied by the generally high and irregular vari-ability of natural water systems in both timeand space. Such variability is evident in temporalweather patterns that drive water ¯ow and pollu-tant transport dynamics (Foussereau et al., 2000,2001), in spatial ¯ow and pollutant transport pat-terns through different subsurface (soil, aquifers;Cvetkovic et al., 1992; Destouni, 1992, 1993;Destouni and Graham, 1997; Graham et al., 1998)and surface (overland ¯ow, streams; Rinaldo et al.,1989, 1991; Marani et al., 1991) water systems, andin spatio-temporal variations of biogeochemicalproperties that govern pollutant mass transferprocesses and reactions (Eriksson and Destouni,1997; MalmstroÈm et al., 2000). This random naturalvariability in space and time implies predictabilitylimits and associated modelling/extrapolation uncer-tainties, which, in turn, imply that pollutant trans-port through natural water systems should beconsidered as a stochastic variable in studies ofcost-effective water management.

The randomness of pollutant loads in naturalwater systems thus needs to be quanti®ed in a pre-dictive sense, in terms of resulting statistical distri-butions and possible correlation structures for

Alternative distributions of stochastic water pollution 147

different water management options, as input tocost-effectiveness analysis (Gren et al., 2000a;Andersson and Destouni, 2001). However, predic-tive quanti®cation of pollutant loads in differentwater systems is commonly limited to yielding onlythe ®rst two statistical moments, mean and variance(Cvetkovic et al., 1992; Destouni, 1992, 1993;Destouni and Graham, 1997; Graham et al., 1998;Foussereau et al., 2000, 2001). Such limited infor-mation is generally not suf®cient for discriminatingbetween different possible statistical distributionsand correlations of pollutant loads, which thenhave to be assumed, rather than independently

Figure 1. The normal (solid line) and lognormal (dashed linto unity) random variable with (a) CV� 0�5, (b) CV� 1, and (c

determined, subject to the constraints imposed bythe predicted mean and variance values.

The normal distribution is often assumed valid invarious applications, for instance, for describingstochastic pollutant loads in studies of cost-effectivewater management (Gren et al., 2000a). This dis-tribution is fully determined by the mean andvariance (or standard deviation) values, as illus-trated in Figure 1 (solid lines) for an arbitrary,normalised stochastic variable (mean value 1) withdifferent coef®cients of variation (CV, de®ned as theratio between standard deviation and mean value).Figure 1 shows that an increasing CV value in

e) distribution for an arbitrary normalised (mean value equal) CV� 2.


the symmetric, normal distribution considerablyincreases the probability of negative values of theconsidered stochastic variable. The stochastic vari-ables of interest in the following, however, areannual pollutant loads from land to sea. Theseshould generally be positive, to be consistent withthe net annual input of freshwater into the sea thatis required for maintaining annual hydrologicalbalance; failure to maintain this balance is highlynotable, such as in the Aral Sea case (Glantz, 1999;Lindahl Kiessling, 1999).

As an alternative to the normal distribution, ithas been suggested in the literature that the asym-metric (de®ned only for positive values), lognormaldistribution may represent a more relevant, yetequally simple distribution of pollutant loads(Andersson and Destouni, 2001). Figure 1 illus-trates the comparison between the normal (solidlines) and the lognormal (dashed lines) distributionfor an arbitrary normalised stochastic variable,showing that the difference between the two distri-butions increases considerably with increasing CVvalue. In the following, it is investigated how such adifference between alternative distribution assump-tions may affect cost effective policies for reducingstochastic pollutant loads to coastal waters.

Even though these pollutant loads may in thefuture be shown to follow other types of probabilitydistribution than the normal or the lognormalconsidered here, comparison between the effects ofthese two particular distribution assumptions isrelevant at this point for two reasons. First, theyrepresent two distinctly different, symmetrical andasymmetrical, types of distributions, comparison ofwhich is expected to shed light on some generaleffects of distribution asymmetry on cost-ef®cientsolutions. Furthermore, these distributions haveboth previously been used for similar applicationproblems (Gren et al., 2000a; Andersson andDestouni, 2001), and are fully determined by thepredictive statistical information (mean and vari-ance) commonly presented in the literature for suchproblems (Cvetkovic et al., 1992; Destouni, 1992,1993; Destouni and Graham, 1997; Graham et al.,1998; Foussereau et al., 2000, 2001).

A model of cost effectivepollutant load reductions

In general, the catchment area of a speci®c watersystem, such as a lake, or a river, can be dividedinto smaller sub drainage basins. We then have

i� 1, . . . , m different drainage basins of a waterrecipient. Two types of pollutant sources are identi-®ed in each drainage basin: the deposition of pollu-tants on land, with the pollutants being transportedby groundwater and/or surface water to the recipi-ent under study, NiL, and the direct discharges intothe water recipient, Di. An example of direct deposi-tion is sewage treatment plant discharges into thecoastal water.

The deposition on land, which in the following isregarded as non-point source pollution, may besubjected to retention by wetlands before enteringthe water recipient. The removal of pollutant bywetlands, Niw, is then simply written as

Niw � Niw�NiL, Wi; Hi� �1�

where Wi is the area of wetlands in the drainagebasin, and Hi is a vector of climatic, hydrological andbiogeochemical factors in¯uencing the retentioncapacity of the wetland measure. It is assumedthat qNiw/qW i and qNiw/qN iL are nonnegative.The role of wetlands for nitrogen abatement hasbeen debated among natural scientists for a periodof about 20 years, see BystroÈm (1998) for a discus-sion of this in an economic context.

In order to focus on the role of stochastic landtransport, it is assumed that, on the one hand, thedirect discharges incur no uncertainty with respectto their impact on the water recipient. On the otherhand, pollutant transport on land minus wetlands'removal, Ni, can be predicted only under conditionsof uncertainty, which is written as

Ni � Ni�NiL ÿNiw, ei� �2�

where Ni is the stochastic pollutant transport term,which we assume is distributed with zero mean andvariance ei.

A simpli®cation is made in (2) by disregardingdifferent locations of abatement measures withineach drainage basin. Since the location of a non-point emission source is quite likely to in¯uence theload to the coastal water recipient, this simpli®ca-tion implies inef®ciency in the allocation of mea-sures, which is a well-known dif®culty in theliterature on non-point source regulation (Shortleet al., 1998). A simplifying assumption is madeon uniform regulation of non-point sources withineach drainage basin, the seriousness of which isdetermined by the relation between marginalcosts of measures at different locations in thedrainage basins and their marginal impacts onthe coastal water recipients (BraÈnnlund andGren, 1999). Total pollutant load to the water


recipient from drainage basin i is then

Ti � Ni�NiL ÿNiw, ei� �Di �3�It is also assumed that there exists one pollutionreduction measure for each type of load, and furtherthat the reductions in pollutant loads can be asso-ciated with a continuous, increasing, and convexcost function in emission reductions from eachsource. The cost functions are then CiW(Wi), CiL(Li)and CiR(Ri) where Li� (NiL

0ÿNiL), Ri� (Di0ÿDi),NiL

0and Di0 are initial, or unregulated, pollution

emission levels. A further simpli®cation is made bydisregarding limited pollutant reduction capacityfor each measure. This is accounted for in thenumerical calculations in Chapter 5.

The cost minimisation problem is then formu-lated as choosing the allocation of W i, Li, and Ri,which minimises total costs, according to

MinX

i

�CiW�Wi� � CiL �Li� � CiR �Ri�� 4�

s:t: �1�ÿ�3�

ProbX

i

Ti �M*

!� a

The solution to (4) is much simpli®ed by replacingthe constraint by its deterministic equivalent(see Charnes and Cooper, 1964). Disregardingco-variances between coastal basins, the constraintsfor normally distributed pollutant loads arerewritten asX

i

mi �Ka�Var M�1=2 � M* �5�

where mi�E{Ti} denotes the expected value of therandom variable Ti, Var M�Var

Pi Ti is the

variance of the (random) sum,P

i Ti, of T i, and Ka

is de®ned by the following con®dence intervalrelation:

PX

i

Ti4X

i

mi �KaÿVarX

i

Ti�1=2

!� a;

05a51:

In Appendix 1, it is shown that equation (5) mayin fact also be used for lognormally distributedpollutants loads. The value of Ka is then Kn(a) orKlog n(a), for an underlying normal or lognormalprobability distribution, respectively. For the stand-ard normal distribution the value Kn(a) can befound in many introductory statistics textbooks.For the lognormal distribution, the computationof Klog n(a) can be reduced to the standard normalcase by means of a suitable transformation, yielding

the relation (Appendix 1)

Klog n�a�� E�Y�STD�Y�

6exp Kn�a�

��ln 1� STD�Y�=E�Y�� 2� �r� �

��1� STD�Y�=E�Y�� 2

q ÿ 1

0BB@1CCA�6�

Thus, Klog n(a) gives the con®dence interval param-eter Ka in equation (5) for the lognormal distribu-tion in terms of the con®dence interval parameterfor the normal distribution, Kn(a), and the dimen-sionless coef®cient of variation CV(Y). In Figure 2,values of Klog n(a) are illustrated for different levelsof a and coef®cients of variation. Once the value ofKlog n(a) is known, the deterministic equivalent (5)to the probabilistic constraint in (4) can be directlyapplied also for lognormally distributed pollutantloads, (see Gren et al., 2000b for numerical results ofKlog n(a, CV(Y )), where Y may be any lognormallydistributed random variable).

If Kn(a) and Kln(a) were the same for given valuesof mi and M*, the costs would be the same underthe two distributions. Not only would the costs bethe same, but also the allocation of measures andeffective charges at pollutant source, which is mosteasily seen by differentiating (1)±(5) with respect to

Figure 2. Calculated Klog n(a) for different coef®cient ofvariation CV(Y)�STD(Y)/E(Y), for an arbitrary randomvariable Y, and for different Kn(a). Speci®cally, the variouscurves correspond to alternative values of Kn(a), rangingfrom 0�2 to 3�0, according to the Klog n(a) value at CV(Y)� 0.The relevant curve is thus found by selecting the correctordinate at the origin, where the limit of Klog n(a) is Kn(a),when CV(Y) goes to zero. For example, if a� 0�96,Kn(a)� 1�8, and CV(Y)� 1�4, which implies that Kln(a)CV(Y ))� 2�0.


the different pollutant load reduction measures. Inorder to simplify the calculations, it is assumed thatall co-variances are zero which implies thatVar(M)�SiVar(Ti)�Sim

i. The cost minimisinglevels of the reduction measures are then given bythe ®rst-order conditions

CiLLi � mi

Ligÿ 1

2��sip si

LigKa �7�

CiWWi � mi

Wigÿ 1

2��sip si

WigKa

CiRRi � mi

Rig

where

miLi � E�Ni

Li ÿNiwNiL N

iLLi � � 0,

miW � E�Nwi

Wi � � 0,

miRi � ÿ1,

sub-indexes are partial derivatives, and g denotesthe Lagrange multiplier of the constraint in (5).The latter can be interpreted as the change in totalcosts associated with a marginal change in theconstraint. Allowing for non-zero co-variancesamong measures would either increase, or decreasethe variance terms on the right-hand sides of (7)depending on their sign (BystroÈm et al., 1999).

The left-hand sides of (7) are the marginal costs ofeach measure. The right hand sides measure theimpacts on the pollutant targets. The higher theimpact, the more is used of the measure in questionsince the cost functions are increasing and convex innutrient reductions. The impact on the water targetfrom marginal changes in Li and W i are divided intotwo main components: expected changes in coastalpollutant load targets, and changes in variance inpollutant loads. The uses of Li and W i are increasedrelative to that of Ri when the variance in pollutantload decreases from marginal increases in Li and W i

respectively. That is, if a measure has the ability notonly to reduce expected coastal load but also the loadvariance, this measure has a cost advantage relativeto a measure that has no, or an increasing impact onthe variance. One example is provided by BystroÈmet al. (1999), who showed the potential of wetlandsto reduce the variance by acting as a nitrogen sinkfor nitrogen from non-point sources, mainly agri-culture, where the variance in the nitrogen out-loadfrom the wetland is lower than that of the in-load.This impact of measures changing the variance ofpollutant loads is enhanced for higher probabilitiesof achieving a certain target, i.e. for higher Ka.

Note also from the ®rst condition in (7) that amarginal change in Li affects negatively the impactsof wetland measures. The reason is that pollutant

removal effectiveness of wetlands is determined byupstream pollutant emission. Given a certain costper ha of wetlands creation, a reduction in pollutantloads to the wetland implies an increase in costs ofwetlands pollutant removal (see BystroÈm, 1998, forderivation of wetland nitrogen abatement cost func-tions). At a given value of g, the impact of coastalload from a marginal change in Li is reduced, and,hence, this measure becomes less attractive ascompared to when wetlands are excluded as anabatement measure.

Since the con®dence intervals and variancesaffect the cost effective allocation of measures, thedesign of ef®cient polity instruments is affected aswell. In this paper two types of policy instrumentsare analysed; pollution charges and market forpollution permits. The latter implies that permitsare distributed to all involved ®rms, which then areallowed to trade permits. Under a charge system,each pollutant emission source is charged accordingto its pollution impact on the target. The larger theimpact the higher is the charge, which can be seenby rearranging condition (7) according to

tiL � g miLi ÿKa 1

2��sip si

Li

� �� CiL

Li �8�

tiR � gmiRi � CiR

Ri

tiw � g miWi ÿKb 1

2��sip si

Wi

� �� CiW

Wi

where tiL is the charge of the non-point source emis-sion, tiR is the charge of the point source emissionsand tiw corresponds to the charge of wetland emis-sions. Expression (8) thus states that the costeffective charges on emissions from the non-point,tiL, wetland, tiw, and point sources, tiR, correspond tothe effective charge at the water recipient target,ti� g, multiplied by the respective impact of emis-sion reductions on the coastal zone.

Under a permit market system each emissionsource is distributed permits, which in total corres-ponds to the pollutant load targets. If a competitivepermit market is created, permit prices are esta-blished which re¯ect the marginal costs and impactsin the same way as the determination of chargesin (8). However, in practice, the establishment ofcompetitive permit markets with cost effective equi-librium permit price for each source and drainagebasin is far from a trivial matter (Tietenberg, 1995).Instead, a permit market including the entire catch-ment region can be created. However, since dif-ferent sources have different impacts on the waterrecipient, a single permit price will not generateef®cient allocation of measures unless trading ratiosbetween sources are created. These trading ratios


re¯ect relations among emission source impacts onthe water recipient and are derived from the ®rst-order conditions for optimum in (7). Choosingmeasures that reduce direct deposition into thewater as the common denominator, the tradingratios between the pollutant mitigation measuresare determined by their relative impacts on targetsaccording to

CiLLi

CiRi

� miLi ÿKa �1=2

��sip�siL

Li

miRi

�9�

Ciwwi

CiRi

� miwi ÿKa �1=2

��sip�siW

Wi

miRi

�10�

The left-hand sides of (9) and (10) re¯ect therelation between marginal costs and the right-handsides that between impacts on the pollutant loadtargets. Similar results are obtained in Malik et al.(1993) where only nonpoint and point sources areincluded. The addition of wetlands, the effective-ness of which depends on the load from upstreamnon-point sources, implies a decrease in the tradingratio since a marginal reduction in their emissionsreduces wetlands' impacts on the water targets.

The right-hand sides of (9) and (10) reveal thedifference in impacts on pollutant load targetsbetween the measures changing non-point sourceemissions and point source emissions respectively.The measures Li and wetlands Wi affect expectedload to the water recipient and also the variance.Both these impacts are determined by the hydro-logical and biogeochemical processes affectingpollutant transports in the drainage basin i. Theimpact of the measure reducing the ef¯uents dir-ectly into the coastal water, Ri, occurs only throughthe change in expected load. Under deterministicspeci®cation of the water pollution target the impacton the variances is not included and, for givenmarginal costs, the allocation of these measuresis determined by their impacts on the expectedloads only.

Impacts of probabilitydistributions on costs andpolicy instruments

The impacts on choice of costs and policy instru-ments of choosing lognormal instead of normalprobability distributions of pollutants are classi-®ed into: (i) differences in when the probability

distributions affect costs and policy design, and (ii)the magnitude of differences in cases where costsand policy design are affected. Starting with the ®rstpoint, (i), it can be seen from eqs. (5) and (7) thatuncertainty does not affect cost effective solutionswhen s and/or Ka are zero. Statistical tables ofcon®dence intervals for the standard normal distri-bution show that Kn(a) equals zero when a� 0�5.From (6) it can be calculated that Klog n(a) is zerowhen

Kn�a� �ln

��1� CV�Y�2

q��ln�1� CV�Y�2�

q �11�

where Y may represent any lognormally distri-buted random variable with coef®cient of variationCV(Y ). Obviously, the level where Klog n(a) is zerovaries with a and the coef®cient of variation. FromFigure 2 it can be seen that Klog n(a) is close to zerofor relatively low values of a. At higher coef®cient ofvariations, however, Klog n(a) may be negative, andthus smaller than Kn(a), which is always nonnega-tive. For example, Klog n(a)5 0 for a5 0�76 in thecase of CV(Y )� 3.

The fact that Klog n(a) may attain negative valuescan be understood by recalling that the arithmeticmean value of a lognormal variable, E(Y )� e(m�s2/2),is always greater than the geometric mean em, with mand s2 being the mean and variance of lnY, respect-ively. In the lognormal distribution, it is the smallergeometric mean that corresponds to the probabilitylevel a� 0�5, whereas the arithmetic mean corre-sponds to higher probability levels. For low values ofa and suf®ciently large variance, a lognormal vari-able that it is mapped onto a normal distributionthrough equations (5) and (6), requires negativeKlog n(a) values because the variable value thatyields the right probability level, a, is smaller thanthe arithmetic mean value; a negative Klog n(a) valuethus points out that it is a variable value smallerthan the arithmetic mean that ful®ls the constraint(5). For high probability levels a, however, Klog n(a)is greater than Kn(a) (see Figure 2). For example,when a� 0�998 and CV(Y )� 2�6, Klog n(a) is 3�2times as large as Kn(a).

When s (or CV ) and Ka differ from zero, thechosen probability distribution assumption doesaffect costs for pollutant load reduction and costeffective allocation of abatement measures. In orderto address issue (ii) in the above classi®cation, fourdifferent impact cases are identi®ed:

� s4 0. The higher the value of Ka, the tighter isthe restriction (5) and the larger is the cost ofthe given target M*. For Klog n(a)4 (5) Kn(a) the


costs are higher(lower) in the lognormal than inthe case of normal probability distribution.

� qs/qL� 0, and qs/qW� 0. The ®rst order condi-tions are reduced to the deterministic case whereonly expected impacts on coastal nutrient loadsare included. Then, the choice of probabilitydistribution and associated Kn(a) and Klog n(a)respectively, has no effect on the allocation ofmeasures, but only on total costs.� qs/qL4 0, and qs/qW4 0. In this case, a

marginal increase in the use of the measuresincreases the variance in the load of pollutants tothe coast. This implies a decrease of the right-hand sides of (7), and thus a decrease in the useof the measure as compared to the deterministiccase. For Klog n(a)4 (5), Kn(a), the use of the mea-sures are lower(higher) in the lognormal thanin the case of normal probability distribution.

� qs/qL5 0, and qs/qW5 0. A marginal increasein the use of the measures decreases the variancein this case. This means that the use of themeasures increase as compared to the determin-istic case, and the increase is larger for higherlevels of Ka. Hence, optimal levels of Li and W i

are higher for Klog n(a)4Kn(a).

The actual magnitude of differences implied by thetwo alternative distribution assumptions is deter-mined by the relation between Kn(a) and Klog n(a),which varies for different a and CV(Y ). Figure 3illustrates the quotient Klog n(a)/Kn(a) (a derivationis found in Gren et al., 2000b).

The quotient Klog n(a)/Kn(a) is increasing in a, thatis, for a given level of CV(Y), Klog n(a)/Kn(a) increasesfor higher chosen probabilities of achieving thewater quality target (see proof in Gren et al.,2000b). This means that the cost of achieving acertain target under Klog n(a) (assuming lognormalpollutant loads) increases as compared to the cost ofthe same target under Kn(a) (assuming normalloads) for higher probability levels. Further, thecharge levels and the permit prices and trading

Figure 3. The quotient Klog n(a)/Kn(a) at different levelsof a, (0�58±0�998) and coef®cient of variation CV.

ratios are raised for pollution measures that reducethe variance in the load to coastal waters. However,the impact on Klogn(a) from changes in CV(Y ) cannot be determined without numerical simulations(see proof in Gren et al., 2000b), a speci®c example ofwhich we present in the following section.

Application to HimmerfjaÈ rden

The drainage basin of HimmerfjaÈrden is locatedabout 60 km south of Stockholm. It covers an area ofabout 1286 km2 which corresponds to 5�5 timesthe water area. It constitutes a connection betweenthe third largest lake of Sweden, MaÈlaren, and theBaltic Sea. Mean depth of the estuary is 17 m and itcan be divided into four coastal basins as de®ned bysea bottom thresholds (Enqvist, 1997).

Like many other estuaries, HimmerfjaÈrdensuffers from ecological damages due to eutrophica-tion, i.e. excessive loads of nutrients. According toElmgren et al. (1997), the limiting nutrient isnitrogen. Decreases in nitrogen load thus reducedamages from eutrophication. This has been recog-nised, not only for the HimmerfjaÈrden estuary, butalso for the entire Baltic Sea since the 1970s.A ministerial declaration was made to reduce theload to the Baltic Sea by 50% (Helcom, 1993), anda cost-bene®t study of this target was made by Grenet al. (1997). In the following analysis, costs aretherefore calculated for 50% nitrogen reductions ofannual nitrogen loads. Nitrogen emissions fromsources in the catchment are assumed to followeither a normal or a lognormal probability distri-bution for load reaching the coastal water. Since theHimmerfjaÈrd catchment region and loads of nitro-gen are described in detail by Gren et al. (2000b) andin Scharin (1999), only a very brief presentation isgiven here.

The drainage basin water transport of nitrogen,or the non-point source pollution, consists of leach-ing from arable and forestland, where the ®rsttype of leaching is dominating. However, due tothe HimmerfjaÈrd sewage treatment plant locatedin Basin 2, direct discharges constitute the majorsource (about 70%) of coastal nitrogen loads,the main components of which are summarised inTable 1 based on calculations by Scharin (1999) forannual loads in 1994.

In Table 1, the numbers for the non-point nitro-gen transport from the drainage basin implicitlyaccount for the retention during the transport fromthe emission source to the coastal water (Johnssonand Hoffman, 1997). Due to lack of additional data,

Figure 4. Costs of 50% N reduction for the normal (n) andlognormal (ln) distributions for different levels of CV anda�pr.

Table 1. Annual nitrogen loads to the HimmerfjaÈ rd, intons of N, for the year 1994

Drainagebasin

Non-pointdrainage basin

transport

Directdischarge

Total

1 53 0 532 42 572 6143 114 16 1304 16 0 16

Total 225 588 813

Source: Scharin (1999).


these speci®c numbers for 1994 are used as generalestimates of mean values of annual nitrogen loads,around which we assume the existence of inter-annual random ¯uctuations. The random variabilityof resulting stochastic annual loads is quanti®ed bydifferent possible CV values and follows either anormal or a lognormal probability distribution. It isfurther assumed that retention rates are the samein all four drainage basins, and that annual directdischarge of nitrogen is deterministically knownand equal to the amounts listed in Table 1.

When estimating minimum costs for a 50%nitrogen reduction, i.e. 407 tons of N, three typesof nitrogen removal options are included; pointsources, changes in agriculture practices, and con-struction of wetlands. One point source mitigationmeasure is included, improvement of the nitrogencleaning capacities at the sewage treatment plants.It is assumed that the cost corresponds to SEK 13/kg(9�38 SEK� 1 Euro, November 19, 2001) N reduc-tion from sewage plants (BjoÈrklund Ltd., 1998).

Two types of non-point source emission reductionmeasures are considered: reductions in the use ofnitrogen fertilisers and cultivation of catch crops.Catch crops are sown at the same time as theordinary crop but continues to grow and, thereby,make use of residual nitrogen in the soil after theordinary crop is harvested. The unit cost per kgN reduction for catch crops is assumed to amount toSEK 20 (Gren et al., 2000a). Costs of nitrogen ferti-liser reductions in each drainage basin are calcu-lated as associated losses in pro®ts. These pro®tsare, in turn, calculated as changes in producersurplus from estimated nitrogen demand functions(see Gren et al., 2000b).

The cost of wetlands is estimated as the oppor-tunity costs of land for wetland purposes. It is thenassumed that wetlands are located only on arableland and the cost then corresponds to foregonepro®ts, which varies between SEK 1000 and 5000per ha in the region depending on cultivated crop(SLU info., 1995). Here it is assumed that the cost

amounts to SEK 3000/ha wetland. Another impor-tant factor is the nitrogen sink capacity of wet-lands, which varies considerably between differentSwedish regions. It is then simply assumed that theretention capacity corresponds to 0�5 of the nitrogenload to the wetland. The area of land suitable forwetland restoration is assumed to cover at the most10% of the arable land.

Given these assumptions, minimum costs arecalculated for achieving the 50 per cent targetunder the alternative assumptions of the two prob-ability distributions and for different values of thecoef®cient of variation, CV, see Figure 4.

The minimum cost under deterministic condi-tions amounts to approximately SEK 5�5 million.Figure 4 shows that the cost for obtaining a 50%nitrogen reduction can thus increase by 5 timeswhen we have a lognormal probability distribution.This occurs for a high probability, a� pr� 0�975,and high coef®cient of variation, CV� 1. The cost forthe lognormal distribution is then twice as high asthe normal distribution. On the other hand,when the probability is relatively low, a� pr� 0�8,the cost under a lognormal distribution can beapproximately 30% lower than under the normalassumption.

Similarly to the estimated costs, the cost effectivecharge and permit price (corresponding to g inequation (8)) increase for higher standard devia-tions and probabilities for achieving the targetunder both probability distributions. These chargesand permit prices correspond to the effective levelsas shown in eq. (8) for emission sources with directdischarges into the coastal water, i.e. the sewagetreatment plants. As can be seen from equation (8),the charge/permit price paid by the non-pointsources depends also on leaching, retention andimpact on variance in the load to the coastal water.In Table 2, we present calculated charges/permitprices for non-point and point source nitrogenpollution for the two probability distributionswhen CV� 1, and also for the deterministic case.


As shown by the results in Table 2, the emissioncharges can vary considerably for all emissionsources depending on the treatment of uncertaintyin nitrogen loads to the coastal water. In the deter-ministic case, emission charges vary between SEK2/kg N and 15 depending on emission source. Understochastic speci®cations, charges are higher forlognormal than for the normal distribution whenthe chosen probability of achieving 50% nitrogenreduction is 0�975, and they are lower when theprobability is 0�8. This is explained by the differentvalues of Kln(a) and Kn(a) as shown in Sections 3and 4.

However, charges on nitrogen application on landcovered by catch crops and on nitrogen input towetland would not give incentives to the cultivationof catch crops and creation of wetlands respectively.Therefore, for these abatement measures, thenumbers in Table 2 can be interpreted as thecost-effective subsidies for these measures.

Table 2. Emission charges per unit N for a 50SEK/kg N

Fertiliser* C

Deterministic case 2

Stochastic cases, CV� 1:a� 0�8:

Normal 6Lognormal 4

a� 0�975:Normal 9Lognormal 67

* Charge per kg N in fertiliser applied on land.yCharge per kg N applied on land covered by catczCharge per kg N in load to wetland.§ Charge per kg discharge into coastal water.Source: Calculations are presented in Gren et al., 2

Table 3. Trading ratios under a permit markeHimmerfjaÈ rden

Fertiliser1 Ca

Deterministic case 7�5Stochastic cases, CV� 1:a� 0�8:

Normal 2�8Lognormal 3�8

a� 0�975:Normal 2Lognormal 1�7

1 fertiliser N applied on arable land.2 N applied on land covered by catch crops.3 N in load to wetland.4 N discharge into coastal water.Source: Calculations are presented in Gren et al., 2

Provided the cost per unit of nitrogen reductionto the coastal water is lower than the subsidy, themeasures will be undertaken.

Under a permit market system different prices forthe emission sources would be required. This is, inpractice, a dif®cult issue to solve, (Tietenberg,1995). An alternative is then a single permit marketwith trading ratios between the sources. Assuming apermit market for the entire region, the tradingratios between nitrogen emission sources re¯ect theimpacts of different emission sources as shown inSection 3. In Table 3 such trading ratios are pre-sented where sewage treatment plant is chosenas the common denominator. That is, eachnumber in the Table 3 shows how many permitsa sewage treatment plant requires in exchangefor one permit.

The trading ratios measure how many permitsthat can be substituted for 1 point source permitwithout changing the impact on the target. For

% nitrogen reduction to the HimmerfjaÈ rden,

atch cropy Wetlandz Sewagetreatment plant§

5 7�5 15

10 15 177 11 15

15 24 18105 167 115

h crops.

000b.

t system for a 50% nitrogen reduction to the

tch crop2 Wetland3 Sewagetreatment plant4

3 2 1

1�7 1�1 12�1 1�4 1

1�2 0�8 11�1 0�7 1

000b.


example, in the deterministic speci®cation, 7�5permits for fertiliser emission must be used inexchange for 1 sewage plant emission. Or, equi-valently, a fertiliser emission permit corresponds to0�13 sewage plant emission. Assuming that 1 permitcorresponds to 1 kg N emission, a sewage treatmentplant then needs to exchange 7�5 fertiliser permitsin order to increase its own emission by 1 kg N.

The number of permits required for a sewagetreatment plant when trading with fertilizer isreduced for the stochastic speci®cation and amountto 2 or 1.7 when the chosen probability is 0�975. Thevalue of permits for these sources thus increases forthe stochastic speci®cations as compared to thedeterministic case. If the trading option is wetlands,less than 1 permit is needed. The reason is therelatively high impact on the variance in coastalnitrogen load from wetlands.

A permit market system with trading ratios alsocreates incentives for the establishment of catchcrops and wetlands from their lower ratios ascompared to fertiliser nitrogen. For example, forthe lognormal speci®cation and a� 0�975, catchcrops and wetlands are less expensive options thenfertiliser nitrogen reduction since fewer permitsare required for replacing one nitrogen fertiliserpermit. If the marginal cost of cultivating catchcrops amounts to approximately 60% of the cost ofreducing fertilisers, catch crops are a cheaper mea-sure. Wetland creation is also less costly and corres-ponds to about 40% of nitrogen fertiliser reduction.

Conclusions

The main purpose of this paper has been to inves-tigate the implications on costs of alternativeassumptions of probability distributions (normalor lognormal) for pollutant loads to a water reci-pient. A chance constraint model was applied wherecosts for achieving a certain water quality targetare minimised. In this model, choice of probabilitydistribution enters as a con®dence interval in thedeterministic equivalence of a probabilistic waterquality target. This means, theoretically, thatthe choice of probability distribution in¯uences thecost-effective solution in three different ways:the total cost of achieving the determined target,the cost-effective allocation of abatement measures,and the design of economic instruments.

In principle, for a non-zero variance in the load ofpollutants to the water recipient, the impact on totalcosts varies with the chosen probability of achievingthe target and the assumed coef®cient of variation

of pollutant loads. For relatively low probabilities(less than 0�95) the normal speci®cation implieshigher costs than the lognormal and vice versa athigher probability levels. This means that optimalcharges/permit prices are higher (lower) under thelognormal speci®cation than under the normalwhen the chosen probability is relatively high(low), exceeding (being below) 0�95. Another impli-cation is that measures for reducing the variance inpollutant load to the water recipient become more(less) attractive under the lognormal assumptionfor relatively high (low) probabilities and coef®cientof variations.

The application to HimmerfjaÈrden, a eutrophi-cated estuary 100 km south of Stockholm, showsthat, depending on coef®cient of variation andchosen probability level, the variation in costs ofreducing nitrogen load by 50% to the estuary ishighest under the lognormal assumption. Thecosts then range between SEK 6±36 million. Therange is smaller under the normal assumption,since the con®dence interval is higher for smallprobabilities and coef®cient of variation, and lowerfor high. Associated emission charges also showlarger variation among abatement measures underthe lognormal case, between SEK 4 and 167 per kg Nemission. At given coef®cient of variation, emissioncharges can be about three times higher under thelognormal than the normal assumption.

Needless to say, both the theoretical and empir-ical analyses were carried out under simplifyingassumptions. The theoretical analysis assumed sto-chastic water pollutant transport, but could equallywell include uncertainty in polluters' behaviour byassuming a joint probability distribution of pollu-tant load to the water recipient. Further theoreticalassumptions were the existence of one abatementmeasure in each catchment and zero co-variationamong catchments and abatement measures. Theanalysis is easily extended to the case with non-zero co-variation by expanding the total varianceexpression (BystroÈm et al., 1999; Horan et al., 1998;Shortle et al., 1999). An alternative theoreticalspeci®cation of the reliability constraint couldapply Tchebyshev's inequality, which does notassume any speci®c probability distribution, (Eltonand Gruber, 1992). But, this inequality constraintis also expressed in terms of mean and variance ofa variable. Thus, the analysis presented in thispaper can be carried out also for the Tchebyshev'sinequality constraint, which then uses the meanand the variance of a variable with underlying eithernormal or lognormal probability distribution.

However, although the framework of the theor-etical analysis is relatively robust to changes in

C

D

D

D

E

E

E

E

F

F

G

G

G

G

G

H


assumptions, results from the empirical demonstra-tion are sensitive to speci®cations of costs and pollu-tant transport functions. For example, it is quitelikely that uncertainty in behaviour of pollutersaffects the magnitude of costs and allocation amongabatement measures. The reason is that asymmet-ric information among polluters and regulatingagencies implies larger costs if the ®rms try togain pro®ts from receiving larger subsidies thantheir abatement costs, or to pay less chargepayments than their pollution levels (so calledinformational rents) (Laffont and Tirole, 1993;Horan et al., 1998). Additional costs may alsooccur from the need for monitoring and supervisingregulated ®rms (Segersson, 1988). If these addi-tional costs are signi®cant, analytical tools from theprincipal agent theory can be applied for theirminimisation (Laffont and Tirole, 1993). An inter-esting topic for future research is the developmentof the analysis in this paper with regard to thetreatment of asymmetric information.

Acknowledgements

We are much indebted to three anonymous referees forvaluable comments, and to the Swedish Fund for StrategicEnvironmental Research for ®nancial support.

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Appendix 1: Proof

Recall that if X*N(m,s2) is a normal randomvariable then Y� eX is lognormal. Under theseassumptions both the expected value of Y, E(Y),and its variance, Var(Y), can be expressed in termsof m and s as follows:

E�Y� � e�m�s2�, Var�Y� � e�2m�s

2��es2 ÿ 1�Conversely, the values m, and s can be written in

terms of E(Y), STD(Y) as

m � lnE�Y��

1� STD�Y�=E�Y�� 2q264

375 �A1�

s2 � ln 1� STD�Y�E�Y�

� �2" #

Consider now a con®dence interval for X of theform P(X4m�sKn(a))� a, 05a5 1. Then, sinceby de®nition Y� eX, we have P(Y4 em�sKn(a))� a.Now de®ne the value Kln(a) by the relation

E�Y� � STD�Y�Klog n�a� � em�sKn�a� �A2�Thus, by virtue of (A1) and (A2) we arrive at

Klog n�a� � E�Y�STD�Y�

6exp Kn�a�

��ln 1� STD�Y�=E�Y�� 2� �r� �

��1� STD�Y�=E�Y�� 2

q ÿ 1

26643775

�A3�

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Cost effective policies for alternative distributions of stochastic water pollution