The Economics of Safe Drinking Water

The Economics of Safe Drinking Water

Robert Innes and Dennis Cory

ABSTRACT This paper studies a drinking watermarket in which a water company faced withrandom contamination chooses a treatment sys-tem treatment levels and whether to notify con-sumers that they should drink bottled water ratherthan risk exposure to contaminants in the tap wa-ter The paper describes efreg cient practices in-cluding protocols which prescribe when a com-pany should notify customers to drink bottledwater and regulatory standards on post-treat-ment water quality that depend upon system sizethe extent of contamination and the customer no-tireg cation decision Implications for contemporarysafe drinking water law are discussed (JEL L51Q25)

I INTRODUCTION

The provision of safe drinking water is es-sential in maintaining the quality of life in allcommunities Unfortunately ground and sur-face sources of drinking water can be con-taminated by bacteria from human or animalsources over owing storm sewers defectivestorage tanks leaking hazardous landreg llspesticides and fertilizers from agriculturalrun-off underground injection of hazardouswastes decay products of radon and ura-nium and industrial solvents (US EPA1994) Once contamination has occurred theability of communities to provide safe andaffordable drinking water is determined bycost-effective treatment and delivery to con-sumers Treatment and delivery are largelythe concern of public water systems legallydereg ned as systems which serve piped waterto at least 15 service connections or regularlyserve an average of at least 25 people eachday at least 60 days per year (US EPA1993)1 In the United States the Safe Drink-ing Water Act (SDWA) governs public watersystems by prescribing drinking water stan-dards for contaminants treatment techniquessampling regimens record keeping proce-dures and public notireg cation protocols whenSDWA requirements have been violated

The success of the SDWA in preventing

Land Economics middot February 2001 middot 77 (1) 94plusmn 117

disease is a subject of some debate (NRDC1994) Between 1986 and 1992 the Centersfor Disease Control and Prevention reporteda total of 102 drinking water disease out-breaks linked directly or indirectly to micro-scopic bacteria viruses or parasites striking34155 people in 35 states In 1993 approxi-mately 400000 people in Milwaukee alonebecame ill and over 100 died when cryp-tosporidium slipped through water-reg ltrationplants In fact some experts believe that asmany as 25 outbreaks go unreported for ev-ery 1 that is documented because symptomsare often confused with other illnesses If soas many as 1 in 8 Americans are being ex-posed to potentially harmful contaminants(Friedman 1996) On the other side of thiscoin are potential illnesses avoided by theSDWA For example the EnvironmentalProtection Agency (EPA) estimates that theSurface Water Treatment Rule alone helpsavoid 90000 cases annually of acute gastro-enteritis and that the Lead and Copper Ruleprotects 140 million people including 18million children from exposure to unsafelevels of lead (US GAO 1992) Such bene-reg ts do not come without cost The GeneralAccounting Ofreg ce (US GAO 1992) has es-timated that full compliance with the SDWAacross all community water systems wouldcost $14 billion annually with many systemshaving to install new equipment

The authors are professors in the Department of Ag-ricultural and Resource Economics at the University ofArizona Both are indebted to an anonymous reviewerand Professor Bromley for very helpful comments onearlier drafts as well as instructive conversations withRulon Pope Erik Lichtenberg Dave Sunding andBrian Wright at various stages of this research The au-thors are also grateful for the comments of seminar par-ticipants ar the University of Arizona the agriculturalrisk meetings Monash University and the AustralianNational University As always responsibility for theviews expressed here and for any remaining macr aws inthe paper resides solely with the authors

1 Over 85 of Americans receive their drinking wa-ter from public water systems (Friedman 1996)

77(1) Innes and Cory The Economics of Safe Drinking Water 95

Economic logic can provide insights intoboth the appropriate settings for drinking wa-ter standards and the design of regulatorypolicy Although economists have exten-sively studied the process and regulation ofwater pollution relatively little attention hasbeen paid to the process and regulation ofwater treatment and delivery given contami-nation that has already occurred2 Notable ex-ceptions are recent papers that have recog-nized the importance of victim mitigation inthe water context3 In this work victims ofwater contaminationETH the water consum-ers ETH can undertake costly measures that re-duce the harm to which they are exposed Inpractice however there are signireg cant econ-omies of scale in both the process of watertreatment and in the acquisition and pro-cessing of information about the nature of thecontamination the health risks it poses andthe appropriate and available method of treat-ment Due to these economies of scale watertreatment is conducted by companies who donot themselves suffer the harm from ingest-ing any contaminated water that they deliverOf course mitigation may also be done byconsumers themselves who can switch fromdrinking tap water to drinking bottled water4

The multiple decisions involved in this pro-cessETH and the regulatory need to elicit be-havior from water companies that remacr ects so-cietal interests in safe drinking waterETH areour central focus in this paper

Specireg cally in view of the importance ofsafe drinking water regulation in practiceour purpose here is to present and analyze arealistic but tractable model of water treat-ment and use in the presence of (1) randomcontamination (2) a water company treat-ment process that involves both reg xed invest-ments in treatment systems and variabletreatment inputs that can be chosen ex-postafter the level of contamination has been ob-served and (3) a consumer alternative to thedrinking of contaminated tap water bottledwater

We are reg rst concerned with optimality intreatment and use (in Section 3) For exam-ple how does the level of water contamina-tion and the size of a companyrsquo s customerservice population affect optimal treatmentactivity and the optimal decision on whether

to substitute bottled water for tap water indrinking And how does a switch to bottledwater affect optimal tap water treatmentUnder an optimal policy we reg nd that post-treatment water quality falls with higher lev-els of contamination when contamination issufreg ciently great (and treatment costs arecorrespondingly high) consumers are ad-vised to drink bottled water and the tap wateris treated to a lower ``non-drinking-usersquo rsquoquality standard that imposes a low cost oftreatment In addition we reg nd that larger wa-ter systems optimally invest more in thetreatment system and treat to higher stan-dards of quality

In Section 4 we turn to the regulatory sideof the problem how to prompt efreg cient be-havior from water suppliers using either lia-bility or regulatory standards enforced withsufreg ciently stiff reg nes Our main policy con-

2 The importance of water treatment as an alterna-tive to pollution prevention is widely recognized in theeconomics literature In a general setting Polinsky andShavell (1994) examine the incentive effects of differ-ent liability rules in inducing reg rms to employ optimallevels of care and mitigation Sunding et al (1995)study optimal treatment in the context of CaliforniaDBCP contamination (see also Lichtenberg Zilbermanand Bogen 1989) Lichtenberg and Penn (1996) studythe choice between preventing and treating nitrate con-tamination of Maryland waters Barrett and Segerson(1997) examine the choice between prevention andtreatment when policymakers pursue objectives otherthan Paretoefreg ciency Conceptually these analyses dif-fer from ours because they do not focus on the treat-ment process and they abstract from the consumermiti-gation and regulatory standard setting that are centralto our analysis of drinking water markets

3 See Segerson (1990) Miceli and Segerson (1993)and Wetzstein and Centner (1992) Also see Oates(1983) and Shibata and Winrich (1983) for studies onvictim mitigation in pollution control and Shavell(1983) and Grady (1988) for victim mitigation in liabil-ity determination

4 An alternative consumer mitigation strategy is toinstall water treatment devices in onersquo s home Such de-vices generally employ the same technologies as docentralized systems Due to economies of scale in thesetechnologies and in their operation and maintenancewater company treatment is almost always more cost-effective for all but the tiniest systems (NRC 1997) Forexample in a cost comparison between home treatmentand a company granular activated carbon technologyGoodrich et al (1992) reg nd that the home devices areonly cost-effective when there are fewer than 20 serviceconnections In this paper we assume that water treat-ment is done at the company level

96 Land Economics February 2001

clusion is that economic efreg ciency is servedby a regulatory regime that has two compo-nents (1) protocols that prescribe both whena company should notify customers to drinkbottled water and often also when it shouldnot notify and instead treat to safe drinkingwater standards and (2) post-treatment waterquality standards that rise with the size of theservice population fall with initial levels ofwater contamination fall when consumersare efreg ciently notireg ed to drink bottled waterand give the company free reign to design ef-reg cient treatment systems

However if the regulatory regime is en-tirely ``performance-basedrsquo rsquo ETH tying all stan-dards and notireg cation protocols to levels ofpost-treatment water qualityETH then it will of-ten prompt an inefreg cient company aversionto notifying customers that they should drinkbottled water As a result customer notireg ca-tion will occur less frequently than is efreg -cient the company will treat water to an inef-reg ciently high standard of quality (when itinefreg ciently fails to notify) and the companywill overinvest in the treatment system in or-der to lower its ex-post costs of supplyinghigh quality water The reason that reg rms mayinefreg ciently fail to notify is as follows Whenwater contamination is sufreg ciently greatthere can be health benereg ts associated withconsumers switching from drinking opti-mally treated (but imperfect) tap water todrinking bottled water these benereg ts are off-set by higher costs of bottled water deliver-ies net of cost savings from lower treatmentstandards that apply when consumers donrsquo tdrink the tap water A public water companythat is subject to efreg cient public utility regu-lation must pay the net costs of switching toa bottled water regime but does not reap thehealth benereg ts of the switch The resultingincentive to undernotify can be cured by ty-ing the notireg cation protocol to the initiallevel of water contamination rather than thecompany-controlled choice of post-treatmentwater quality

These conclusions suggest some tentativeprescriptions for the design of the SafeDrinking Water Act As we discuss morefully in our concluding Section 5 some ofthese prescriptions are heeded in the recentreauthorization of the SDWA others are not

II THE MODEL

We consider a risk neutral economy inwhich a water company treats and deliverstap water to a homogeneous population ofsize N The company obtains its water fromsurface or ground sources Untreated waterfrom these sources has a quality level of XThe company treats the water to bring it upto a quality level of u $ X Higher levels ofwater quality improve the health and safetyof water use in a sense made precise below

The pre-treatment (incoming) water qual-ity X has the probability density g(X) on thesupport (XX) The probabalistic nature of Xremacr ects inherent randomness in the dispersalof contaminants in the water stream andortemporal macr uctations in water quality due toseasonal patterns in the weather and produc-tion activities that cause contamination

The Water Consumers

The water companyrsquo s customers use waterfor two purposes (1) drinking and (2) otheruses that entail a much reduced exposure toany contaminants The `òther usesrsquo rsquo includeboth ``residual riskrsquo rsquo uses such as showeringand washing hands and ``no riskrsquo rsquo uses suchas macr ushing toilets and watering gardens

Consumers are assumed to drink a given(reg xed) amount of water w5 However fortheir non-drinking uses consumers may varytheir water consumption in response to theprice that they are charged for water A con-sumerrsquo s total water consumption for bothdrinking and non-drinking uses will be de-noted by w Excluding any health effectsfrom their water consumption each con-sumer obtains the gross monetary benereg tB(w) where Bcent(w) 0 and Bsup2(w) 0 (non-drinking water consumption yields positivemarginal benereg ts that decline as the con-sumer proceeds to less valuable uses)

For their drinking water needs consumershave an alternative to tap water they can buybottled water Bottled water is assumed to betreated to a reg xed and high level of quality uwith background liability and regulatory

5 In Arizona for example households drink aboutone half of one percent of their water use (Gelt 1996)


standards sufreg cient to ensure that bottled wa-ter always meets this quality standard6 Con-sumers will want to drink bottled water whenthe quality of the tap water is sufreg ciently lowand hence the health risk associated withdrinking the tap water is sufreg ciently highFor simplicity however we assume that thecost of using bottled water for ``residualriskrsquo rsquo uses exceeds any expected health bene-reg ts from doing so and hence consumers al-ways use tap water for non-drinking pur-poses This assumption will be valid whenthe volume of residual risk water use is rela-tively large the cost of bottled (vs tap) wateris sufreg ciently large andor health risks asso-ciated with residual risk use are sufreg cientlysmall7

When a consumer uses tap water for bothdrinking and non-drinking purposes she suf-fers an expected health ``damagersquo rsquo or costequal to DA(u) where DAcent(u) 0 and DAsup2(u)$ 0 (higher water quality reduces healthcosts at a non-increasing rate) If a consumerinstead buys bottled water to drink she willincur the purchase cost of bottled water cand an expected health cost of DB(u) whereDBcent(u) 0 (higher tap water quality may re-duce the health costs of ``residual riskrsquo rsquo use)and DBsup2(u) 5 0 (a normalization of the waterquality measure u)8

Figure 1 depicts the relationship betweenthe health damage functions DA(u) andDB(u) Because the drinking of water yieldsa much greater consumer exposure to anywater contaminants marginal health benereg tsof increased tap water quality are higherwhen consumers drink the tap water2DBcent(u) 2DAcent(u) However if tap waterhas the same high quality level as does bot-tled water expected health damages are thesame to the consumer whether she drinks tapwater or bottled water DA(u) 5 DB(u)

Water Treatment

The local water supply company makestwo choices in the water treatment processFirst it chooses a treatment capacity y ex-ante before observing any realization of Xthe incoming water quality When making itstreatment capacity choice the companyknows the probability distribution of X

Higher levels of treatment capacity lead tolower ``variable costsrsquo rsquo of treating water to agiven quality level u (for any given X) Sec-ond the water company selects the level oftreatment u ex-post after observing X9

6 In the United States approximately one in reg fteenUS households buy bottled drinking water spending$27 billion annually on almost 700 brands (Gelt 1996)In our model and in practice the function of bottledwater is to provide an alternative to tap water albeit athigh cost that ``tastes better is safe and healthy or isfree of contaminantsrsquo rsquo (conclusion of the GAO re-ported in CNI 1991) A suitably high level of bottledwater quality u is thus an optimal property of this prod-uct which for simplicity and without loss in generalitywe take to be exogenous in this paper Our analysis isnonetheless robust to uncertainty in bottled water qual-ity with u representing an exogenous certainty equiva-lent Although bottled water has not always been freeof contaminants in practice (US Water News 1996)the recent reauthorization of the SDWA strengthensgovernment regulation of bottled water quality by theUS Food and Drug Administration requiring that itmeet all contaminant standards set by the EPA underthe SDWA

7 Casual empiricism supports this conjecture withbottled water as much as one thousand times as expen-sive as tap water In California for example Allen andDarby (1994) report tap water prices ranging between$45 and $285 per thousand gallons compared with av-erage US bottled water prices of $90 per single gallon(in 1990)

8 Implicit in our specireg cation of the health damagefunctions is the realistic premise that the volume of aconsumerrsquos ``residual riskrsquo rsquo water use does not affectthe expected health costs associated with this use Notealso that the health damage function DB( ) will dependupon the quality standard for bottled water we ignorethis dependence for notational simplicity

9 There are a variety of technologies that can be usedfor water treatment including (1) disinfection (2) cor-rosion control (3) membrane reg ltration systems (4) re-verse osmosis (5) electrodialysis electrodialysis re-versal (6) adsorption (7) lime softening and (8) ozonesystems Given any of these treatment processes (whichwe call capacity investments) water companies make avariety of decisions on the extent to which they treatwater after observing the level of contamination (ourtreatment level choice) among these decisions are theset and extent of chemical additives the scheduling ofreg lter and membrane cleansing and replacement mea-sures to monitor and remove residual concentrations ofdisinfectants and contaminants (including testing andrecirculation) reliance on alternative sources and wastestream disposal (NRC 1997) In this paper we are envi-sioning a given ground andor surface source for waterthat is subject to random and exogenous contaminationIn practice a water company may have multiple possi-ble sources of water When one source becomes con-taminated one strategy to reduce the impact of the


FIGURE 1Consumer Expected Health Costs from Water Use

The companyrsquo s choices of y and u lead tothe following ex-post costs of treatment

Treatment Costs 5 F(y) 1 v(u X y W)

where W 5 wN is the total amount of watersupplied The investment cost function F(y)has the following properties Fcent( ) 0 andFsup2( ) 0 (more treatment capacity is in-creasingly costly) The ``variable costrsquo rsquo func-tion v( ) includes costs of water collectiontreatment and delivery and is assumed to sat-isfy vuu 0 and vyy $ 0 (weak convexity)vy 0 vuy 0 and vWy 0 (more treatmentcapacity reduces variable costs of treatment)vu 0 and vuX 0 (treating water to a higherquality standard is costly but less costlywhen the incoming water quality is higher)vW 0 (marginal costs of water delivery arepositive) vX 0 and vWX 0 (costs of sup-plying water of a given quality level declinewith higher levels of initial quality X) Notethat these cost function constructs implicitlyembed reg xed costs costs that are invariant towater quantities delivered and that imply thelarge economies of centralized (rather thanhome) treatment described in the introduc-

tion Note also that with a higher level of ini-tial water quality X it is likely to be at leastas costly to achieve a given incremental im-provement in water quality the foregoingproperties of our cost function v( ) are con-sistent with this property

To convey the intuitive content of ourmodel as simply as possible we assume10

Assumption 1 The marginal cost of de-livering treated water is constant vWW 5vuW 5 0

III OPTIMALITY IN WATERTREATMENT AND USE

Ex-post after X has been observed by thewater company the envisioned sequence ofevents is as follows (1) the company simul-taneously selects (i) its treatment level(which determines the quality of the tap wa-

contamination on the quality of delivered tap water isto draw more from an uncontaminated source and lessfrom the contaminated one The costs of switchingaway from the contaminated source are implicitly em-bedded in the treatment costs modeled here

10 At the cost of complexity our expanded paper de-rives our results without Assumption 1


ter) (ii) the price that consumers will becharged for the tap water and (iii) whetheror not to notify consumers that the tap wateris of a lower quality than is suitable fordrinking and (2) consumers then decide (i)whether to drink tap water or buy bottled wa-ter and (ii) how much water to consume intotal Because the measurement of waterquality and the potential effects of water con-taminants on consumer health are very com-plex we will assume that consumers defer tothe advice of their water suppliers (and theexpert regulators who oversee them) to buybottled water11 Therefore with the com-panyrsquo s water pricing decision isomorphic tothe consumerrsquo s water quantity decision it isas if all of the ex-post choices (of u w anddrinking water source) are made simulta-neously by the water company

At this juncture we are not concernedwith how these choices are made or elicited(a topic to which we return in the next sec-tion) rather we are concerned with proper-ties of efreg cient decisions To characterizethese properties it is convenient to beginwith the choice of treatment level u for eachof the two possible cases of consumer drink-ing behavior (bottled vs tap water) Buildingon this choice an efreg cient rule for the drink-ing water source can be characterized fol-lowed by the determination of an efreg cient ex-ante treatment capacity y

The Choice of Treatment Level u and WaterUse w When Consumers Drink Tap Water

In this reg rst case (Case A) net market-wide benereg ts of water consumption after de-ducting costs of treatment and expectedhealth damages are

(B(w) 2 DA(u))N 2 v(u X y Nw) [1]

Optimal levels of u and w will maximize thenet benereg ts in (1) subject to u $ X Assum-ing interior solutions to this problem12 theywill solve the following reg rst order condi-tions

u 2vu (u X y Nw) 2 DAcent(u) N 5 0 [2a]

w Bcent(w) 2 vW(u X y Nw) 5 0 [2b]

An optimal treatment level equates marginalbenereg ts of treatment in reduced health dam-ages 2DAcent( )N with the marginal cost oftreatment vu( ) Similarly optimal per-capitawater use equates marginal consumer bene-reg ts Bcent(w) with marginal costs of water de-livery vW( ) These optima will be denotedby uA(X y N) and wA(X y N)

By directly lowering marginal costs oftreatment (vu) increases in initial water qual-ity (X) and treatment capacity (y) prompthigher optimal levels of post-treatment waterquality (u) Similarly an increase in the ser-vice population N increases the total healthbenereg ts of marginal treatment 2DAcentNwhich also prompts an increase in the opti-mal treatment level

Proposition 1 uA(X y N) increases withX y and N13

The Choice of u and w When Consumers DrinkBottled Water

In this second case the net market-widebenereg ts of water consumption are as follows

(B(w) 2 DB(u))N

2 v(u X y N(w 2 w)) 2 cN [1cent]

The objective function in [1cent] differs from itsanalog in [1] in three respects First healthdamages from tap water of quality u are nowlower DB(u) rather than DA(u) Second thewater company now supplies water only for

11 The US Safe Drinking Water Act for exampleregulates over 90 contaminants today Understandingthe health implications of contaminant measurementsand converting such measurements into costs is argua-bly possible only for experts on the chemistry epidemi-ology and economics of drinking water Our analysisdoes not rely on consumer cognizance of this informa-tion based uponwhichconsumers could themselves de-cide whether to drink bottled water or tap water How-ever in Section 4 below we show that our results arerobust to a consumerrsquos ability to independently test thequality of her tap water and in turn decide whether topurchase bottled water

12 Some positive treatment will be optimal so longas marginal health benereg ts of treating untreated waterare at least as high as corresponding marginal treatmentcosts This will be true for example if the marginalcosts of treating untreated water is sufreg ciently small

13 The Appendix provides proofs for all Proposi-tions


consumersrsquo residual needs after their de-mand for drinking water has been methence total water deliveries are now N(w 2w) And third consumers must now pay thecosts of bottled drinking water c per person

Here optimal levels of u and w will max-imize the net benereg ts in [1cent] subject to u $X Solutions to this problem will be denotedby uB(X y N) and wB(X y N) Again as-suming interior solutions these optima willsolve the following reg rst order conditions

u 2vu (u X y N(w 2 w)) 2 DBcent(u) N 5 0 [3a]

w Bcent(w) 2vW(u X y N(w 2 w)) 5 0 [3b]

Comparative static properties of the opti-mal treatment level uB( ) are the same as de-scribed in Propositions 1 for uA( ) Moreoverwhen consumers drink bottled water thebenereg ts of treating tap water in reducinghealth risks are lower than when consumersdrink the treated tap water Therefore wehave

Proposition 2 When consumers drinkbottled water optimal levels of tap waterquality are lower than when they drink thetap water uB(X y N) uA(X y N)

The Drinking Decision Bottled vs Tap Water

Consumers should drink bottled waterwhen the resulting level of overall economicwelfare given optimal treatment and wateruse choices is higher than when they drinktap water

Economic welfare when consumers

drink bottled water

JB(X y N)

5 (B(wB( )) 2 DB(uB( )))N

2 v(uB( ) X y N(wB( ) 2 w)) 2 cN

(B(wA( )) 2 DA(uA( )))N

2 v(uA( ) X y NwA( ))

JA(X y N)

5 Economic welfare when consumers

drink tap water [4]

To determine when [4] holds we beginwith a compelling premise

Assumption 2 Economic Benereg t ofDrinking Tap Water The marginal cost ofdelivering untreated tap water is less than theper-unit cost of bottled water

When initial water quality X is as high asthe quality of bottled water u Assumption 2implies that the water company can provideconsumers with untreated tap water at lowercost than bottled water that has the samequality Hence we have

JA(u y N) JB(u y N) [5]

As X falls however the net economic bene-reg ts of using tap water also fall a decline inX leads to a greater increase in treatmentcosts when the water company treats to thehigher drinking water standard uA( ) thanwhen it treats to the lower standard uB( )14

paraJA(X y N)paraX

5 2vX(uA( ) X y NwA( ))

2vX(uB( ) X y N(wB( ) 2 w)) [6]

5 paraJB(X y N)paraX

When incoming water quality is sufreg cientlypoor X 5 X costs of treating all of the tapwater to levels desirable for ``safe drinkingrsquo rsquomay exceed costs of the relevant alterna-tiveETH drinking bottled water and treating tapwater to a lower standard of quality that isappropriate for non-drinking uses In suchcases bottled water will be more econom-ical

JA(X y N) JB(X y N) (7)

Condition [7] need not always hold but ismore likely when (1) the two health damagefunctions are ``further apartrsquo rsquo at the mini-mum safe drinking water standard uA(X yN) and (2) marginal costs of treatment arerelatively large (for u uA(X y N)) Whenswitching to bottled water under these cir-cumstances the treatment level can be low-

14 By Assumption 1 (vuW( ) 5 0) equation [2b] andequation [3b] we have that wA( ) 5 wB( ) Inequality[6] thus follows from uA( ) uB( ) (Proposition 2)vXu 0 and vXW 0 By the same logic we also havethat 2vy(uA( ) X y NwA( )) 2vy(uB( ) X y N(wB( )2 w))


FIGURE 2Economic Welfare and the Optimal Drinking Decision

ered dramaticallyETH yielding large cost sav-ingsETH without causing health damages anyhigher than those experienced when drinkingtap water Greater maximal levels of watercontamination (lower X) lower treatment ca-pacity investments and smaller customerservice populations all favor condition [7] byraising treatment costs lowering the mini-mal safe drinking standard uA(X y N) andthereby widening the disparity betweenhealth damages under the alternative drink-ing source strategies

As illustrated in Figure 2 equations [5] plusmn[7] will imply that there is a critical value ofinitial water quality X 5 X(y N) belowwhich bottled water should be used for drink-ing and above which tap water should beused for drinking When this critical qualitylevel is above X (because [7] holds) it can beshown to decline with increases in treatmentcapacity y and the service population N Ahigher level of treatment capacity reduces thecost of raising the treatment level from thelower ``drink bottled waterrsquo rsquo standard uB( )to the higher drinking water standard uA( )as a result the ``drink tap waterrsquo rsquo alternativebecomes relatively more economical and isoptimally invoked more frequently Simi-

larly a larger N reduces the per-capita costof switching to the higher drinking waterstandard again implying that the ``safedrinking waterrsquo rsquo regime is optimally invokedmore often

Formally let us dereg ne the switch-pointquality level X(y N) min X Icirc [XX)JA(X y N) $ JB(X y N) If condition (7)does not hold then X(y N) 5 X If condi-tion [7] holds then X(y N) is above X anduniquely solves15

JA(X y N) 5 JB(X y N) at X5 X( y N) [8]

This construct gives us the following ``drink-ing water rulersquo rsquo

Proposition 3 The optimal rule for theconsumersrsquo choice between drinking tap wa-ter and bottled water is either (a) alwaysdrink tap water regardless of the initial waterquality X or (b) drink tap water when the in-

15 If condition [7] does not hold then X(y N) 5Xby the dereg nition of X( ) and equation [6] If [7] holdsthe existence of X(y N) X as dereg ned in equation[8] follows from equation [5] (Assumption 2) and theIntermediate Value Theorem Uniqueness of X( ) fol-lows from equation [6]


FIGURE 3Ex-P ost Optimal P olicy The Treatment and Drinking Decisions

coming water quality X is at or above a crit-ical value X(y N) Icirc (X X) and drink bot-tled water when X is below this critical valueThe optimal frequency with which bottledwater is consumed falls when treatment ca-pacity (y) or system size (N) is higherparaX(y N)paray 0 and paraX(y N)paraN 016

Propositions 1 to 3 together imply the op-timal water treatment policy depicted in Fig-ure 3 When the incoming water quality X ishigh the cost of treating water to make itsuitable for drinking are sufreg ciently smallthat the water company should do so As Xfalls the post-treatment quality of watershould also be lowered Below some criticalX level consumers should often be advisedto drink bottled water not tap water and thetap water should be treated to a substantiallylower standard of quality that is appropriatefor non-drinking uses and that imposes a cor-respondingly low cost of treatment

The Treatment Capacity Decision y

Because the treatment capacity is chosenbefore X is observed optimality requires ex-pected economic welfare to be maximized

given efreg cient choices that result from thechosen y level ex-post

maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)

JA(X y N) g(X)dX [10]

The corresponding necessary condition foran optimal y y(N) is

2Fcent(y)

2 X(y N)

Xvy(uB( ) X y N(wB( ) 2 w))g(X)dX

2 XAring

X(y N)

vy(uA( ) X y NwA( ))g(X)dX 5 0

[11]

16 A higher level of bottled water cost c will alsolead to a reduced optimal frequency of bottled waterconsumptionFor example if logistical rigidities lead toa relatively high cost of responding to a high temporarydemand for bottled water then c will remacr ect this highcost of switching customers to bottled water and willreduce the economic desirability of the switch


Equation (11) equates the ex-ante marginalcost of y with the expected variable cost re-ductions achieved with marginal investmentsin treatment capacity

With vyu 0 investments in treatment ca-pacity yield higher cost-reduction benereg tswhen ex-post treatment levels are higherTherefore a larger customer service popula-tion N will lead to a higher optimal y ifhigher levels of N are also associated withhigher optimal treatment levels ex-post Nowrecall that a higher N leads to a higher levelof treatment given a consumer decision todrink either tap water or bottled water (fromProposition 1) Moreover a higher N leadsto a higher optimal frequency of the highertreatment (Case A) consumer drinking deci-sion (recalling Propositions 2 and 3) Thusfor all possible incoming water quality levelsX a higher N yields a higher optimal treat-ment level and thereby prompts a higher op-timal investment in treatment capacity17

Proposition 4 The optimal treatment ca-pacity level y(N) increases with the cus-tomer service population N

In sum we reg nd that larger water systemsshould (all else equal) make larger treatmentcapacity investments meet higher ex-posttreatment standards (by Proposition 1) andinvoke the bottled water option less fre-quently (by Proposition 3)

IV OPTIMAL REGULATION HOWTO ACHIEVE EFFICIENCY

Drinking water markets may fail becausethe health costs of contaminated water areexternal to the water companyrsquo s optimizationcalculus Policy measures to control healthrisks may apply before (or independently of)the occurrence of harm or they may only betriggered by the occurrence of harm (Shavell1987) Using tort law to impose liability forharm done is an example of the latter ap-proach whereas regulatory regimes such asbest management practices injunctions andcorrective taxes are examples of the formerIn this section we explore possible avenuesto control health risks from water contamina-tion beginning with a strict liability bench-mark then turning to ``negligence rulesrsquo rsquothat may be implemented with either liability

or regulatory standards and reg nishing withregimes of reg xed standards and variable reg nesWe will be interested in how these rules canbe designed to elicit the efreg cient water com-pany practices described in Section 3 aboveAlthough our initial analysis presumes thatconsumers only buy bottled water when ad-vised to do so by the water company we willlater show (in Section 4D below) that our re-sults are robust to a consumer ability to inde-pendently test the tap water and based uponthis test to choose between drinking tap andbottled water

When making its decisions a water com-pany may be subject to public utility regula-tion For conceptual clarity we do not wantto motivate water treatment law by any pre-sumed inefreg ciencies in the public utility reg-ulatorrsquo s restraints on the companyrsquo s exerciseof monopoly power We therefore assumethat in the absence of reg nes or liability as-sessments the company obtains proreg ts or``rentsrsquo rsquo that include all non-health-relatedcosts and benereg ts of water treatment deliv-ery and use less a per-capita consumer ``res-ervation benereg trsquo rsquo (before health costs) of BThis specireg cation is consistent with a varietyof actual regulatory settings including when(1) the company is a private price-discrimi-nating monopoly which is subject to either aregulatory constraint that consumers eachobtain net benereg ts of B or a market constraintthat consumers obtain at least the net benereg tavailable from an alternative water source

17 We assume here that the distribution of waterquality X is invariant to the service population N How-ever if a higher N requires the water company to seekout less pristine water sources andor yields more pol-luted downstream sources then the distribution of Xwill be worsened This would increase the frequencywith whichcustomers are notireg ed to drink bottled water(ceteris paribus) which would tend to lower marginalbenereg ts of treatment capacity However it may also in-crease the frequency with which incoming water qualityis rather poor but not sufreg ciently poor to require theswitch to bottled water In these circumstances the ad-ditional treatment that is called for (ie u plusmn X) tends tobe greaterETH and hence the cost-reducing benereg ts oftreatment capacity tend to be higherETH than when X ishigher Overall it seems unlikely that the net impact ofa worsenedXdistributionon capacity investment incen-tives would be so strongly negative that it would domi-nate the effects that underpin Proposition 4


(such as delivered water)18 (2) the companyis municipal with managers who maximizethe economic rents available to them andtheir employers subject to a political con-straint that consumers each obtain the B netnon-health benereg t or (3) the company issubject to ``price caprsquo rsquo regulation whichspecireg es a maximum price that the utility cancharge19

Strict Liability

Under a strict liability rule the water com-pany pays for realized health damages andthereby faces the true expected health costsDB(u) when it ``notireg esrsquo rsquo and DA(u) when itdoes not notify Strict liability confronts thecompany with all of the societal costs andbenereg ts of its choices and therefore elicitsefreg cient behavior However for a variety ofreasons this rule is difreg cult and perhapsimpossible to implement Particularly forhealth damages that are realized only aftermany years both causation and damage lev-els are difreg cult to prove20 Moreover regu-lated water companies often have ``shallowpocketsrsquo rsquo that prevent the assessment of fullhealth damages even when such damagescan be established Critics of the tort systemargue that toxic tort litigation is very costlyand that courts lack the technical competenceto deal effectively with health and safety is-sues when designated regulatory agencies arebetter positioned to perform well21

Negligence

Under a negligence liability rule the wa-ter company is subject to liability for healthdamages or another high penalty if and onlyif it behaves ``negligentlyrsquo rsquo by not exercisingsufreg cient ``carersquo rsquo Similarly regulatory stan-dards of ``due carersquo rsquo can be enforced withsufreg ciently high reg nes for non-complianceAs is well known such negligence rules canelicit efreg cient decisions with prospectivepenalties for inefreg cient negligent behaviorthat are smaller than true expected damagesThe judgment proof problem (for reg rms withshallow pockets) can thereby be mitigated

Using regulatory approaches to enforce non-negligent behavior has added advantageseconomizing on the damage assessment pro-cess exploiting regulatory economies in ac-quiring the specialized expertise needed toevaluate the toxic properties of contaminantsand their health impacts and further mitigat-ing the judgement-proof problem that ariseswhen realized damages are high relative to

18 In this context B might also be thought of as theminimum customer benereg t needed to forstall voter en-actment of price regulation In either case the companycan maximize proreg ts using two-part prices that extractall non-health-related economic rents For example thecompany can charge a high price on a reg rst traunch ofa consumerrsquos water consumption (the `èssential usersquo rsquobelow which consumption never falls) and a marginalcost price on additional use The high price can be setto extract either the per-capita net revenue B(w) 2 B(when consumers drink tap water) or B(w) 2 c 2 B(when consumers buy bottled water)

19 So long as the price cap is set above the marginalconsumer value of optimal water use (for all X) themonopolist will maximize proreg ts by (i) charging theprice cap on an initial traunch of water use (ie up toa w at which Bcent(w) equals the price cap) and (ii) onfurther use levying the maximum of marginal cost anddiscriminatory prices (Bcent(w) or slightly less)

20 In practice defendants have brought suit againstwater companies for delivering contaminated drinkingwater under both breach of warranty and strict productliability theories of liability (See for example SaltRiver Project Agricultural Improvement and PowerDistrict v Westinghouse Electric Corp 143 Ariz 368694 P 2d 198 1984 Zepp v Mayor amp Council of Ath-ens 180 Ga App 72 348 SE 2d 673 1986 Moodyv City of Galveston 524 SW 2d 583 Tex Ct App1975)) Once held liable damages include loss of earn-ings and earning capacity pain and suffering loss ofconsortium medical expenses fear of disease medicalmonitoring and diminution in quality of life (Froder-man Karnas and Lucia 1996) Obtaining full recoveryfor this wide array of damages is onerous for plaintiffsIn addition claimants must demonstrate that the con-duct of the defendant has been a substantial factor inbringing about the harm suffered and overcome a vari-ety of defenses to product liability defenses that them-selves present complex efreg ciency issues (Boyd and In-gberman 1997) For further discussion of theseproblems see Kaplow and Shavell (1999)

21 The strengths and limitations of using tort law asa means to control environmental and other risks havebeen discussed extensively in the literature (eg Rose-Ackerman 1991 Shavell 1984 Viscusi 1984) as haslimited liability (eg Innes 1999 Kaplow and Shavell1999)


average damages22 For all of these reasonsa negligence rule enforced with regulatoryreg nes is less subject to the economic draw-backs of strict liability

In this section we focus on regulatory re-gimes that elicit non-negligent (compliant)behavior23 Fines or liability for violations of``due carersquo rsquo standards are assumed to belarge enoughto deter violations The relevantquestion is How must negligence standardsbe dereg ned in order for the companyrsquo s opti-mal non-negligent choices to be efreg cientFor example must treatment capacity be reg-ulated with a minimum standard Must ex-post treatment levels be regulated with vari-able standards that depend upon the extentof initial contamination (X) and the size ofthe customer service popoulation (N) or canreg xed standards sufreg ce Can negligence bedereg ned using standards that are only tied topost-treatment water quality u How mustcustomer notireg cation about the quality of thetap water be regulated To address thesequestions three alternative dereg nitions ofnegligence will be examined24

Rule 1 Due care (non-negligence) requires(i) u $ uA(X y(N) N) uSA(X N) when the

company does not notify consumers(ii) u $ uB(X y(N) N) uSB(X N) when

the company notireg es consumers that theyshould drink bottled water

(iii) notifying when X X(y(N) N) X(N) and

(iv) not notifying when X $ X(N)Rule 2 Due care requires only (i) (ii) and (iii)

of Rule 1 (not (iv))Rule 3 Due care requires (i) and (ii) of Rule

1 and(iiicent) notifying when u uA(X(N) y(N)

N) uSN(N)

None of these rules regulates treatment ca-pacity at all Unlike Rule 1 Rule 2 does notpenalize a company when it inefreg ciently noti-reg es consumers in order to lower the standardof treatment to which it is held Howeverboth rules set a consumer notireg cation stan-dard that depends upon incoming water qual-ity X rather than post-treatment water qual-ity u Rule 3 requires notireg cation wheneverthe water companyrsquo s treatment level falls be-low the minimum treatment level that is efreg -cient when a no-notireg cation strategy is also

efreg cient (see Figure 3) Rule 3 thus ties allstandards to post-treatment water quality asdoes contemporary US policy Unlike re-cent safe drinking water law (see Section 5)all three rules dereg ne treatment standards thatare variable not reg xed We turn now to theeffects of the three negligence rules on com-pany incentives to treat water notify con-sumers and invest in treatment capacity

Treatment levels Given (X y) and a notireg -cation decision the company lowers its costsby lowering the treatment level u Thereforethe company will exactly meet its treatmentstandards and will not `òver-complyrsquo rsquo Bythe same token because efreg cient treatmentlevels are variable (depending upon X and Nby Proposition 1) the treatment standardsmust also be variable in order to prompt ef-reg cient behavior

Consumer notireg cation In the optimum de-scribed in Section 2 above consumers arenotireg ed that they should drink bottled water

22 Historically damages in privately-initiated tortactions must be set equal to the actual damages visitedupon the defendant due to the plaintiffrsquo s negligent actnot based upon the expected average typical or rea-sonably anticipated damages resulting from the act (apractice known as the ``thin skullrsquo rsquo doctrine in tort the-ory) For a discussion of the advantages of liabilitybased on expected damages in the natural resource con-text see Phillips and Zeckhauser (1995)

23 Negligence rules designed in this way may alsobe advantageous because they never require the actualassessment of liability or reg nes as a result the adminis-trative and legal costs of such assessments are neverborne Generally speaking an offsetting disadvantageof such negligence rules is that by confronting reg rmswith no liability for harm caused they lead to excessiveentry (Polinsky 1980 Shavell 1980) However in thepresent context one water company efreg ciently serves agiven customer population and entry is not an issue

24 The question posed hereETH how alternative negli-gence rules perform in eliciting desired conductETH issimilar to that posed by Shavell (1992) in a somewhatdifferent context In Shavell (1992) potential injurerschoose both their level of care (accident prevention ef-fort) and whether to obtain information about their riskof a harmful accident (before care is undertaken) If thenegligence rule stipulates standards for both efreg cientcare and the efreg cient acquisition of information efreg -cient conduct is elicited however Shavell (1992) alsoshows that efreg cient conduct results from an appropri-atly designed negligence rule that only regulates careSimilarly in the present analysis we are interested inwhether and how negligence rules can elicit efreg cientwater company behavior without regulating all deci-sions


if and only if incoming water quality is suf-reg ciently low X X(N) For the sake ofexposition let us suppose that such customernotireg cation is sometimes efreg cient X X(N) Although Rule 1 dictates a reg rst-bestnotireg cation policy Rule 2 permits over-noti-reg cation (notifying when X is above X(N))and Rule 3 permits both over-notireg cation andunder-notireg cation25

To determine when such deviations fromthe reg rst-best policy can occur we need reg rstto dereg ne company proreg t functions under``notireg cationrsquo rsquo and ``no notireg cationrsquo rsquo strate-gies assuming that the company meets thetreatment standards (i) and (ii) (of Rules 1plusmn3) and is not subject to any negligence penal-ties26

pA(X y N)

5 ``no-notireg cationrsquo rsquo proreg t

5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)

5 ``notireg cationrsquo rsquo proreg t

5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]

Now suppose that for all X above the no-tireg cation standard X(N) pA( ) is at least ashigh as pB( ) as depicted in Figure 4 Thenthe company prefers not to notify consumerswhenever notireg cation is not required underRule 2 hence no over-notireg cation occursHowever if there is any X above X(N)such that pA( ) is less than pB( ) then Rule 2will sometimes prompt the company to no-tify consumers when it is inefreg cient to do soEfreg ciency of Rule 2 is thus upset if given anefreg cient treatment capacity choice y(N)pA( ) is strictly less than pB( ) at the notireg ca-tion standard X(N)27

pA(N) pA(X(N) y(N)N)

pB(X(N) y(N) N)

pB(N) Ucirc DA(uSA(X(N) N))

DB(uSB(X(N) N)) [13]

Conversely the efreg ciency of Rule 2 is notupset if the inequality in (13) is reversed

pA(N) pB(N) Ucirc DA(uSA(X(N) N))


Equation [14] requires that under optimaltreatment policies at the critical water qualitylevel X( ) health damages are higherwhen consumers drink tap water than whenthey drink bottled water This condition re-macr ects the intuitive notion that switching fromtap water to bottled water (at the X( )switch point) has the benereg t of reducedhealth risk and the cost of increased waterexpenditures However if a switch to bottledwater is instead motivated by cost savingsthat it permits by lowering the standard towhich tap water must be treated then condi-tion [13] will hold (not (14]) and Rule 2 willbe inefreg cient

Under Rule 3 the companyrsquo s payoffs un-der no-notireg cation and notireg cation strategiesare exactly the same as under Rule 2 (iepA( ) and pB( ) in (12)) when X is above theefreg cient notireg cation threshold X(N)Therefore if condition [13] holds Rule 3will also be inefreg cient providing incentivesfor over-notireg cation

Let us suppose instead that inequality [14]holds as depicted in Figure 5 In this casethe company will sometimes want not to no-tify when notireg cation is efreg cient X X(N) Under Rule 3 the company can in-efreg ciently fail to notify and still escape neg-ligence by treating to the notireg cation stan-dard uSN(N) and thereby obtain proreg ts of

25 Under-notireg cation should not be confused withfalsely failing to notify consumers that their water isunsafe to drink Rather it implies that the companysometimes fails to notifyETH and treats its water to thehigher ``safe drinking waterrsquo rsquo standard ETH when itshould not in such instances consumers generally haveevery reason to heed the companyrsquo s advice to drinktheir tap water (see Section 4 below) Similarly over-notireg cation should not be confused withfalsely notify-ing consumers that their tap water is unsafe to drink

26 Equation (12) remacr ects our earlier premise that thewater company obtains all non-health-related economiccosts and benereg ts of its decisions less each consumerrsquosreservation benereg t B

27 The implication in equation [13] follows from theequivalent representation of the reg rst inequality pA( ) 2pB( ) 5 (JA( ) 2 JB( )) 1 DA( ) 2 DB( ) 0 wherewith an interior X(N) (above X) JA( ) 2 JB( ) 5 0at X 5 X(N) (by the dereg nition of X(N))


FIGURE 4Water Company P ayoffs Under a Negligence Rule and Alternative Notification

Decisions

FIGURE 5Water Company P ayoffs Under ``Treatment Negligence rsquo rsquo (Rule 3) and Alternative

Treatment and Notification P olicies


p1A(X y N)

5 company proreg t when u is setequal to uSN(N)

5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSN(N) X y(N) wN)

for X X(N) [15]

When X is less than X(N) p1A( ) differs

from pA( ) the no-notireg cation proreg t functionin equation (12a) because the notireg cationstandard binds the company to a higher treat-ment level uSN(N) uSA(X N) and therebylowers company proreg ts p1

A(X y N) pA(Xy N) for X X(N) However becauseuSN(N) equals uSA(X N) at the switchpointX(N) p1

A(X y N) 5 pA(X y N) at X 5X(N)

In view of these properties Figure 5shows that there is a range of X levels belowthe switchpoint X( ) wherein p1

A( ) isgreater than pB( ) so that the company prefersto set its treatment level to uSN(N) and notnotify rather than notify and set its treatmentlevel to the lower standard uSB( ) Since noti-reg cation is efreg cient for these X values Rule 3yields inefreg cient under-notireg cation In intu-itive terms condition [14] implies that thecompany strictly prefers not to notify at theoptimal switch point X( ) because it doesnot obtain the net health benereg ts producedbythe shift to bottled water Hence as X fallsbelow X( ) the company will still prefernot to notify even though under Rule 3 itcannot lower its treatment level

To develop a positive efreg ciency propertyof Rule 2 it is instructive to consider the fol-lowing simplifying restriction

Assumption 3 When zero the net com-pany benereg t of customer notireg cation

pB( ) 2 pA( ) is non-increasing in X

para(pA( ) 2 pB( ))paraX $ 0 if pA( ) 5 pB( )

Assumption 3 is generally realistic be-cause a higher incoming water quality levelX reduces treatment costs in the higher treat-ment (Case A) regime more than in the lowertreatment (Case B) regime28 Given thispremise company proreg t functions are as de-picted in Figures 4 and 5 with a switchpointbelow which notireg cation is preferred andabove which it is not X1 (y N) min X

pA(X y N) $ pB(X y N) With an efreg cienttreatment capacity choice Rule 2 thus yieldsefreg cient notireg cation decisions so long ascondition (13) is violated (see Figure 4)

Table 1 summarizes the efreg ciency proper-ties of the alternative rules including thetreatment capacity effects to which we nowturn

Treatment Capacity

With any of the three negligence rules de-scribed above societal costs and benereg ts oftreatment capacity decisions are exactly thesame as those which confront the water com-pany viz marginal costs of capacity invest-ments Fcent(y) and benereg ts of attendant reduc-tions in variable costs of treatment vy( ) 0 Because treatment and notireg cation stan-dards are invariant to the companyrsquo s actualtreatment capacity potential indirect effectsof the capacity choice on company proreg tsdue to affects on standards are absentHence given the notireg cation and treatmentpolicies that a treatment capacity choice elic-its the company will select y efreg cientlyHowever the resulting treatment capacitychoice need not be reg rst-best if as can be trueunder Rule 2 or Rule 3 resulting notireg cationdecisions are not themselves efreg cient

If Rule 2 or Rule 3 prompts over-notireg cation (because condition [13] holds)a reg rst-best treatment capacity choice yieldsless-than-reg rst-best treatment levels whenconsumers are inefreg ciently notireg ed and thecompany is thereby held to a lower treatmentstandard uSB uSA Because the cost-reduc-ing benereg ts of treatment capacity are lowerwhen ex-post treatment levels are lower thecompanyrsquo s privately optimal policy of over-notireg cationETH and undertreatment will alsoprompt a lower (less-than-reg rst-best) treat-ment capacity investment y y(N)

Similarly when Rule 3 prompts under-notireg cation (with condition [14] andAssump-tion 3 holding as in Figure 5) ex-post treat-ment levels are never lower than reg rst-bestand they are strictly higher when the com-

28 For example if variable costs of water treatmentand delivery depend upon the treatment innovation v(uX y W) 5 vo (u 2 X y W) then Assumption 3 holds(see our expanded paper)


TABLE 1

Features and Efficiency P roperties of Three Negligence Rules

Rule 1 Rule 2 Rule 3

Rule FeaturesEx-Post Efreg cient Variable Treatment Standards x x xStandard Requiring Consumer Notireg cation When X Is Small x x ETHStandard Requiring No Notireg cation When X Is Large x ETH ETHStandard Requiring Consumer Notireg cation When u is Small ETH ETH x

Efreg ciency Properties of Alternative Rules Rule 1 Rule 2 Rule 3

With Assumption 3Efreg cient Over-notify Over-notify

y y(N) y y(N)pA(N) pB(N)

Without Assumption 3per eqn [13] a

Efreg cient Over-notify Inefreg cienty y(N) (some over-

notireg cation)

With Assumption 3Efreg cient Efreg cient Under-notify

y y(N)pA(N) pB(N)

Without Assumption 3per eqn [14] b andX(N) X

Efreg cient May Be Efreg cient Inefreg cient(some un-der notireg -cation)

a If notireg cation is sometimes efreg cient X(N) X condition [13] will hold if the optimal switch to bottled water is motivatedby treatment cost savings that offset higher health risks DA(uSA(X(N) N)) DB(uSB(X(N) N))

b With X(N) X condition [14] will hold if the optimal switch to bottled water lowers health risks DA(uSA(X(N) N)) DB(uSB(X(N) N))

pany inefreg ciently refrains from customer no-tireg cation and sets treatment to the high noti-reg cation standard uSN(N) uSA(X N) uSB(X N) for X X(N) Because higherex-post treatment levels yield higher cost-saving benereg ts from investment in treatmentcapacity the companyrsquo s privately optimalpolicy of under-notireg cationETH and overtreat-mentETH will also prompt it to invest morein treatment capacity than would otherwisebe efreg cient y y(N)29 Hence our Rule 3the negligence rule that most closely resem-bles contemporary policy can only elicit ef-reg cient reg rm choices if either notireg cation isnever optimal (X(N) 5 X) or by happen-stance health damages are exactly the sameat the optimal switch point regardless ofwhether consumers drink tap water or bottledwater DA(uSA(X( ) N)) 5 DB(uSB(X( )N) If instead notireg cation is sometimes opti-mal (X( ) X) and the optimal switch tobottled water is motivated by the lowering

of health costs DA(uSA(X( ) N)) DB(uSB(X( ) N) then Rule 3 generallyprompts the company to (a) notify too infre-quently (b) treat water to a higher standardthan is efreg cient when it inefreg ciently fails tonotify and (c) overinvest in treatment ca-pacity

In contrast when treatment and notireg ca-tion decisions are reg rst-bestETH as they are un-der Rule 1 and sometimes under Rule 2(when condition [14] and Assumption 3hold)ETH expected company proreg ts are max-imized with a reg rst-best treatment capacitychoice thus prompting fully efreg cient companybehavior In sum (referring again to Table 1)

Proposition 5 In order for a negligencerule to prompt efreg cient non-negligent behav-ior it need not dereg ne a ``due carersquo rsquo standard

29 The effects of Rules 1plusmn 3 on the water companyrsquo streatment capacity choice as stated here are derivedformally in our expanded paper


for the investment in (or design of) the treat-ment system but must (i) dereg ne standardsfor minimum post-treatment water qualitythat are tied to the incoming water qualitylevel X the size of the customer service pop-ulation N and the decision on whether or notto notify consumers that their tap water is un-safe to drink (ii) almost always dereg ne a stan-dard for consumer notireg cation that is tied tothe incoming water quality level X and notthe post-treatment water quality u and (iii)sometimes require a company not to notifyconsumers when notireg cation (and treatmentto a lower standard) are inefreg cient

Fixed Standards

With negligence rules designed to promptnon-negligent compliant behavior variablestandards are needed to achieve efreg cient out-comes However reg xed standards combinedwith a regimen of reg nes for non-complianceand ``bonusesrsquo rsquo for over-compliance canbe designed to elicit efreg cient non-compliantand over-compliant behavior Consider thefollowing notireg cation-contingent treatmentstandards (I) uSA without notireg cation and (II)a lower standard with notireg cation uSB uSAwhere

DB(uSB) 5 DA(uSA) 5 D

expected health costs with optimal behavior30

and a violation or exceedance of a standardprompts a net reg ne that charges (or rewards)the reg rm for the difference between true ex-pected damages and damages with exactcompliance

Net Fine 5 DA(u) 2 DA(uSA)

when there is no notireg cation

Net Fine 5 DB(u) 2 DB(uSB)

when there is notireg cation

Here when damages are greater than D(which will occur for example when thecompany sets u below uSA and does not no-tify) the company will pay a reg ne howeverwhen damages are less than D (which willoccur for example when the company sets

u above uSA and does not notify) the com-pany gets a bonus (a negative reg ne)

Because this rule is equivalent to strict lia-bility (with a reg ne equal to health damagesless the constant D) it elicits efreg cientchoices However the reg nes are based on ex-pectations (not realizations) of damages andare zero on average thus avoiding many pit-falls of strict liability

Bilateral Care Consumer Water Testing

Despite costs and complexities associatedwith testing for water contaminants a con-sumer might be able to test her tap water inorder to determine its quality based uponsuch a test a consumer could make an inde-pendent judgement about whether to drinktap or bottled water In the prior analysis weassume that consumers do not test their waterand instead defer to the water companyrsquo s ad-vice about whether to buy bottled waterHowever this premise is not essential for ourresults Specireg cally we have

Proposition 6 If regulatory policyprompts efreg cient water company treatmentand notireg cation decisions then (i) any con-sumer who tests her tap water quality willmake the same (tap vs bottled water) drink-ing decision as is recommended by the com-pany and (ii) consumers will not test theirtap water if the test involves a positive cost

Consider a consumerrsquo s incentive to``switchrsquo rsquo away from company adviceETH forexample drinking bottled water when thecompany recommends drinking the tap wa-ter The consumer gains less from the unilat-eral purchase of bottled water than would begained per-capita from a full blown ``soci-etal switchrsquo rsquo to bottled water With a ``soci-etal switchrsquo rsquo the company optimally lowersits post-treatment water quality level andlowers its marginal cost price in tandemboth of these actions raise the potential bene-reg t to be enjoyed from the bottled water alter-

30 Formally

D X( )

X

DB(uB(X y(N) N)g(X)dX

1 XAring

X( )

DA(uA(X y(N) N)g(X)dX


native These optimal adjustments are notmade when a consumer unilaterally pur-chases bottled water hence the consumerhas less incentive to make the switch thandoes society as a whole31 Because society asa whole loses with a ``switchrsquo rsquo away fromefreg cient company advice a consumer mustalso lose if she unilaterally rejects this ad-vice Hence an independent water qualitytest reveals information that a consumer willnever use clearly consumers will not pay forsuch worthless information

Now suppose that the water company isnot behaving efreg ciently but rather is inefreg -ciently notifying consumers to drink bottledwaterETH or inefreg ciently failing to notify ETH aswe argued could occur under the negligenceRules 2 and 3 Then will a consumer some-times have an incentive to ``switchrsquo rsquo Andif so will this behavior affect the companyrsquo snotireg cation incentives Because the watercompany charges a marginal price equal tomarginal cost a consumerrsquo s decision to``switchrsquo rsquo does not affect either the watercompanyrsquo s proreg ts or therefore its incentivesto notify (or not) However when the com-pany is not notifying efreg cientlyETH and soci-etal gains from a full blown ``societalswitchrsquo rsquo are therefore positive ETH consumersmay conceivably have an incentive to rejectthe companyrsquo s advice given the informationneeded for them to do so Of course any pos-itive consumer benereg ts from ``switchingrsquo rsquomust be high enough and frequent enough tojustify the consumerrsquo s cost of water qualitytesting Moreover if a consumer can raiseher welfare by testing her tap water societalwelfare will also be raised (with producerwelfare unaffected) which will mitigate (butnot eliminate) the efreg ciency costs of overand under-notireg cation32

In sum a consumerrsquo s ability to test her tapwaterETH and use the test results to decidewhether or not to purchase bottled waterETHdoes not alter any of our qualitative conclu-sions If an efreg cient regulatory regime is inplace consumers will have no incentive toengage in independent water testing How-ever under the inefreg cient regulatory regimesdiscussed in this paper independent con-sumer testing and switching could occur inprinciple and if it did such testing would re-duce the extent of inefreg ciency

V CONCLUSION IMPLICATIONSFOR SAFE DRINKING WATER LAW

In this paper we present a simple modelof drinking water markets in which a watercompany faced with random contaminationof its source chooses a treatment systemtreatment levels and whether to notify con-sumers that they should drink bottled waterrather than risk exposure to contaminants inthe tap water In closing we consider whatimplications this theory may have for recentexperience and practice with the Safe Drink-ing Water Act (SDWA)

Under the SDWA water suppliers are re-sponsible for ensuring that drinking watermeets EPA standards and for complying withestablished monitoring operation and main-tenance protocols Each supplier of watermust collect samples from the water systemtake them to a certireg ed laboratory for analy-sis and send the results to the regulatory au-thority Any time there is a violation of a re-quirement the public must be notireg ed Ingeneral public notices must include a discus-sion of the violation the potential for adverseeffects the population at risk such as chil-dren or pregnant women the steps taken to

31 Implicit in this argument is the following observa-tion When post-treatment water quality u and water usew remain unchanged a consumerrsquos ``switchrsquo rsquo (to bot-tled water for example) yields the same net benereg ts tothe individual consumer as it does to society With con-stant u and w there are three effects of a ``switchrsquo rsquo onboth societal and individual benereg ts (1) health dam-ages change (for example from DA to DB) (2) the bot-tled water cost c is borne (or saved) and (3) the costof drinking tap water wvw( ) is saved (or borne)

32 In principle the water company could be requiredto notify customers about the post-treatment water qual-ity u essentially making consumerwater testing unnec-essary If such notireg cations were costless they couldpotentially reduce prospective inefreg ciencies of over andunder-notireg cation for the reasons described above Inpractice however we believe that such water qualityinformation is unlikely to be of practical use to consum-ers absent the specireg c recommendations to drink tap orbottled water that we model in this paper There are tworeaons First translating the multitude of contaminantmeasurements into health-based prescriptionsfor action(see note 11) requires expertise that most consumerslack Second prospective consumer gains from re-jecting the companyrsquo s (tap vs bottled water) advice aresubstantially less than corresponding per-capita gainsfrom a ``societal switchrsquo rsquo hence even with inefreg cientregulations these gains can be small and infrequent ifthey are ever positive at all


TABLE 2

Distribution of System Size (N) and Contamination (X) in Arizona

A) System Size (1994) a

Number of Systems Percent of PercentService Population in Arizona Population Served of Systems

50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

Rates of Groundwater Contamination in Arizona Wells (in Samples from 1990plusmn 1993)c

Number of StandardContaminant Observations Minimum Maximum Mean Median Deviation

Metalsd 23 00 531 216 219 139Nitrates 22 25 10 683 692 261VOCe 19 00 172 017 00 039

Median Levels of Groundwater Contamination Across 10 Arizona Groundwater Basins f


Nitratesg 8 4 13 49 25 5032Sulfatesh 10 143 345 3307 714 110366

a Source ADEQ (1994a)b Source ADEQ (1994b)c For a given active management area (AMA) the rate of contamination is calculated as the proportion of samples that contained

positive levels of the contaminant Sample observations were obtained in all nine groundwater planning regions of Arizona for allcontaminants but not in all active management areas within each region

d Metals include heavy metals and semi-metals such as arsenic and seleniume VOCs are volatile organic chemicalsf This table presents statistics for the set of median contaminant measurements from groundwater samples in each of ten ground-

water basins (For nitrates sample observations were only obtained in eight basins) Levels are measured in mglg The maximum contaminant level for nitrates (under the SDWA) is 10 mglh The (secondary) maximum contaminant level for sulfates is 250 mg l

correct the problem and recommended pre-cautions

Currently states regulate more than186000 public water systems These systemsare heterogeneous in terms of the size of thecustomer populations that they serve and thecontaminant circumstances that they con-front sources of heterogeneity that this paperhas stressed in its modeling For example forthe State of Arizona where 60 of drink-ing water supplies come from groundwater(ADEQ 1993) Table 2 illustrates the con-siderable variability across the state in both(1) rates at which different contaminantsETHincluding metals (such as arsenic and sele-nium) nitrates and volatile organic chemi-

calsETH are found in groundwater suppliesand (2) levels of observed contamination

Table 2 also indicates a wide disparity insystem size This disparity is mimiced na-tionwide where 96 of public water systemsare ``smallrsquo rsquo serving fewer than 3300 per-sons each and collectively serving 11 ofthe US population In addition 85 of thesystems are considered very small servingfewer than 500 persons Overall 66 of com-munity water systems are in compliance withthe SDWA Of the remaining systems in non-compliance either for direct contaminationviolations or for failure to monitor contami-nants 90 percent are small (US EPA 1995)As drinking water regulations have become


more complex small systems have found itincreasingly difreg cult to comply The 1996reauthorization of the SDWA attempts toaddress cost and compliance issues facingsmall systems33

A principal thrust of the reauthorization isrisk assessment and benereg t-cost analysis inregulatory decision making (Clark 1997 Tie-man 1996) Under the reformed law the EPAmay consider overall risk reduction benereg tsin setting new drinking water standards andmust determine whether the benereg ts of astandard justify its costs The SDWA reau-thorization also revokes the requirement thatthe EPA regulate an additional 25 contami-nants every three years and establishes aprocess for EPA to select contaminants forregulatory consideration based on occur-rence health effects and meaningful oppor-tunity for health risk reduction Theseamendments are consistent with the implicitprescription of this paperrsquo s analysis that stan-dards of post-treatment water qualityETH asmeasured by both the level of contaminantstandards and the set of contaminants that areregulatedETH remacr ect tradeoffs between costsand health benereg ts

The reauthorization also adds some macr exi-bility to the enforcement of standards Forsmall systems the reauthorization enablesstates to grant technology and treatment vari-ances with EPA approval based on cost andhealth-risk reduction considerations an ex-plicit recognition that optimal treatment lev-els must vary with the size of the customerpopulation In addition the new law requiresStates to establish programs for water sys-tems to attain the technical reg nancial andmanagerial capacity to comply with theSDWA These capacity development provi-sions attempttoavoiddetailedcommand-and-control regulations by enabling small watersystems and their state regulators to tailortreatment practices to individual circum-stances (Shanaghan 1996) Our analysis sug-gests that these changes are likely to enhanceeconomic welfare Under plausible condi-tions in our model larger water systems opti-mally invest more in the treatment systemtreat to higher standards of quality (becausehealth benereg ts of treatment are higher) andresort less frequently to bottled water ``warn-

ingsrsquo rsquo (Proposition 4) Eliciting such optimalbehavior requires regulatory standards onpost-treatment water quality that dependupon the size of the system the extent ofcontamination and whether or not consum-ers are notireg ed to drink bottled water (Propo-sitions 1 and 5)

Despite its added macr exibility the reautho-rized SDWA retains two regulatory rigiditiesthat this analysis suggests may impede eco-nomic efreg ciency The reg rst concerns notireg -cation The SDWA amendments clarify pub-lic notireg cation requirements for violationsof maximum contaminant levels treatmenttechniques testing procedures monitoringrequirements and violations of a variance orexemption If violations have the potentialfor ``serious adverse effectrsquo rsquo consumers andthe state must be notireg ed within 24 hoursof the violation However notably absentfrom the notireg cation reforms is a recognitioncombined with efforts to inform water com-panies and the public they serve that therecan sometimes be efreg ciency advantages ofnotifying customers to drink bottled waterWhen contamination is sufreg ciently great(and treatment costs are correspondinglyhigh) we reg nd that consumers are optimallyadvised to drink bottled water not tap waterand the tap water is optimally treated to asubstantially lower standard of quality that isappropriate for non-drinking uses and thatimposes a correspondingly low cost of treat-ment Efreg cient notireg cation protocols pre-scribe both when a company should notifycustomers to drink bottled water and oftenalso when it should not notify and instead

33 According to the National Research Council(NRC 1997) the number of small community watersystems has increased substantially in the United Statesin the last three decades from 16700 in 1963 to morethan triple that number 54200 by 1993 with approxi-mately 1000 new systems formed each year The smallwater systems will incur nearly 70 of the total cost ofcomplying with drinking water requirements (Raucher1994) Beyond other reforms describedabove the 1996SDWA amendments will promote compliance of smallsystems by (i) requiring operator certireg cation programsto be in place by the year 2001 and (ii) creating a re-volving fund for states to make direct grants and low-interest loans to public water systems The loans maybe used to upgrade water system facilities equipmentand piping or for compliance activities operator certi-reg cation programs and source water protection


treat to safe drinking water standards Con-trary to contemporary practice this analysissuggests that efreg ciency gains can be securedby explicitly recognizing that notireg cationshould sometimes occur and basing notireg ca-tion protocols on the level of initial contami-nation not the post-treatment water quality(Proposition 5)34

Secondly the reauthorized SDWA pre-serves a regulatory preoccupation with thedesign of a companyrsquo s treatment system re-quiring government approval of the treat-ment technology that is used approval that isinformed by government technological bulle-tins and other sources of regulatory guidanceAlthough the government can exploit econo-mies of scale in providing information tocompanies about treatment technologies thisanalysis suggests that it need not regulatetreatment design decisions either indirectly(with design-contingent standards for exam-ple) or directly (Proposition 5) Rather it canrely upon water quality standards that do notspecify (or change with) a companyrsquo s choiceof treatment system and that provide compa-nies with an unfettered ability to adopt cost-saving design innovations By doing so thegovernment can also free up regulatory re-sources from treatment system oversight toimprove both SDWA enforcement35 and theinformation used to design drinking waterregulations that efreg ciently tradeoff healthdamages averted and costs incurred

APPENDIX

Proof of Proposition 1 Differentiating the reg rstorder conditions [2a] and [2b] and exploiting sec-ond order conditions yields

parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s

(vuW w 1 DcentA)(Bsup2 2 vWWN)

1 vWW wN vuW (reg rst form)

5 (Bsup2N)(vuWW 2 vu)

1 vWW vu (second form) [A1c]

where ``5s

rsquo rsquo denotes `èquals in signrsquo rsquo and thesecond equality in [A1c] follows from substitu-tion for DcentA 5 2vuN from condition [2a] Thereg rst bracketed term in [A1a] is positive (withvuX 0 and from second order conditions Bsup2 2vWW N 0) and the second bracketed term is zero(by Assumption 1) Identical logic applies to[A1b]Turning to [A1c] note that constant returns toscale (vWW( ) 5 0 from Assumption 1) imply thefollowing variable cost function

v 5 a(u X y) 1 b(u X y) W [A2]

Furthermore with vu 0 everywhere au must bepositive Differentiating v( ) to evaluate the sec-ond form for parauA( )paraN above yields

parauAparaN 5s

2(Bsup2N) au 0

where the inequality results from Bsup2 0 andau 0 QED

Proof of Proposition 2 Appealing to Assump-tion 1 (vuW( ) 5 0) we can evaluate our reg rst ordercondition for uB( ) [3a] at u 5 uA( ) as follows

2vu( ) 2 DcentB (u) N 5 (DcentA(u)) 2 DcentB(u)) N 0

where the inequality follows from DcentA(u) DcentB(u) 0 and (by convexity of v( ) in u) establishes theProposition QED

Proof of Proposition 3 The reg rst part of theProposition follows from equation [5] equation[6] and the dereg nition of X( ) To establish the

34 In this paper we assume that the population ofconsumers is homogeneous If instead there are ``highriskrsquo rsquo and ``low riskrsquo rsquo consumers (with consumersknowing what they are) there will be added benereg ts ofmore complicated notireg cationETH and attendant treat-mentETH regimens When initial levels of water contami-nation are neither too great nor too small optimal cus-tomer notireg cations will need only to warn high riskgroups that bottled water should be substituted withsuch notices issued (and high risk consumers presumedto respond) optimal tap water treatment may fallslightly or not at all For cases of greater water contami-nation optimal notices will warn all consumers as de-scribed in the foregoing analysis This paperrsquo s qualita-tive conclusions about the nature of optimal regulationwill extend to such an environment although the formalanalysis would clearly become more complicated

35 Resources to support monitoring and enforcementactivities are severely limited in most states The GAOconcluded in 1993 ``Severe resource constraints havemade it increasingly difreg cult for many states to effec-tively carry out the monitoring enforcement and othermandatory elements of EPArsquo s drinking water program The situation promises to deteriorate furtherrsquo rsquo (USGAO 1993)


derivative properties of X( y N) differentiateequation [8] with respect to y

paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]

where vB( ) 5 v(uB( ) X y N(wB( ) 2 w)) andvA( ) 5 v(uA( ) X y NwA( )) denote cost functionvalues with optimal treatment and water usechoices for Cases B and A respectively vB

z (X ) 5vz(uB( ) X y N(wB( ) 2 w)) and vA

z (X ) 5vz(uA( ) X y NwA( ) denote corresponding deriv-atives for z 5 y X The inequality in (A3) followsfrom equation (6) and its counterpart 2 vA

y 2vB

y (see note 14)Differentiating [8] with respect to N gives

paraX(y N)

paraN

5s

[paraJB(X( ) y N)paraN

2 paraJA(X( ) y N)paraN]

5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]

where the second equality is obtained by per-forming the indicated differentiation and substi-tuting from equation [8] By Assumption 1 vB

W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5

v(uB( ) X y NwB( )) 2 w vBW(X ) Substituting

into [A4] we thus have

paraX(y N)

paraN

5s

N21v(uB( ) X y NwA( ))

2 v(uA( ) X y NwA( )) 0 [A4cent]

where the inequality is due to uA( ) uB( ) (Prop-osition 2) and vu( ) 0 QED

Proof of Proposition 4 Differentiating equa-tion (11) and appealing to the second order neces-sary condition for an optimum we have

dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB

yW(X)5(wB 2 w) 1 NparawB

paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA

paraN 64dX1 vBy (X( )) 2 vA

y (X( ))paraX(y N)

paraN )y5y(N)

[A5]

where vA and vB are as dereg ned in [A3] With parawAparaN 5 parawBparaN 5 0 (by Assumption 1 equation[2b] and [3b] ) vyu 0 vyW 0 and vB

y (X( ))2 vA

y (X( )) 0 (note 14) we see that the follow-ing conditions are sufreg cient for dy(N)dN in[A4] to be positive (1) parauAparaN 0 for all X $X( y(N) N) (2) parauBparaN 0 for all X X( y(N) N) and (3) paraX(y N)paraN|y5y(N) 0Propositions 1 and 3 imply these three inequali-ties and hence Proposition 4 QED

Proof of Proposition 6 It sufreg ces to show thatwith marginal cost pricing of marginal (drinkingwater) consumption a consumerrsquo s gain from``switchingrsquo rsquo away from efreg cient company ad-viceETH about whether to drink tap water (Case A)or bottled water (Case B)ETH is negative (In CaseB marginal cost pricing is implied by optimalpublic utility regulation ``switchingrsquo rsquo by drink-ing from the tap adds water consumption w thatis marginal cost priced In Case A if switching(to bottled water) occurs then marginal cost pric-ing of the unconsumed drinking water is impliedby optimal public utility regulation hence if suchmarginal cost pricing implies no switching (as wewill show) we have a contradiction) To this endwe reg x (X y N) in the background (without lossof generality) and dereg ne Cj

k as the individual con-sumer welfare obtained when kIcirc A B is theefreg cient advice offered by the company and theconsumer chooses to drink tap water ( j 5 A) orbottled water ( j 5 B) Formally

Ckk 5 B 2 Dk(uk) for k Icirc A B [A6a]

CBA 5 B 2 DB(uA) 1 b(uA)w 2 c [A6b]

CAB 5 B 2 DA(uB) 2 b(uB)w 1 c [A6c]

where b( ) is the marginal cost of water (from[A2]) When a consumer ``switchesrsquo rsquo she con-sumes ths same total amount of water (by ourpremise of marginal cost pricing) and thereby ob-tains benereg ts that differ due to alterred healthdamages saved (or added) cost of drinking the wtap water and added (or saved) costs of drinkingbottled water c as indicated in [A6b] plusmn [A6c] Toprove the Proposition we must now show that


CBA 2 CA

A 0 when JA 2 JB $ 0 [A7a]

CAB 2 CB

B 0 when JA 2 JB 0 [A7b]

where Jk (for k Icirc A B) is the maximal societalbenereg t in case k as dereg ned in equation (4) Nownote (I) with JO

k (u w) dereg ned as the societal ob-jective function given in (1) (for k 5 A) and (1cent)(for k 5 B) we have Jk 5 maxuw JO

k (uw) and(II) from [A6] and the dereg nitions of JO

k ( ) and Jk

CBA 2 CA

A 5 JOB (uA wA) 2 JA JB 2 JA [A8a]

CAB 2 CB

B 5 JOA(uB wB) 2 JB JA 2 JB [A8b]

where the inequalities follow from (uA wA) sup1 (uBwB) and observation (I) [A8] implies [A7] QED

References

Allen L and J Darby 1994 ``Quality Controlof Bottled and Vended Water in Californiarsquo rsquoJournal of Environmental Health 56 (Apr)17plusmn 22

Arizona Department of Environmental Quality(ADEQ) 1993 Groundwater Protection inArizona Phoenix ADEQ

ETH ETH ETH 1994a Economic Impact Statement forSafe Drinking Water Act Requirements Phoe-nix ADEQ

ETH ETH ETH 1994b Arizona Water Quality Assess-ment 1994 Phoenix ADEQ

Barrett J and K Segerson 1997 ``Preventionand Treatment in Environmental Policy De-signrsquo rsquo Journal of Environmental Economicsand Management 33 (2) 196plusmn 213

Boyd J and D Ingberman 1997 ``Should `Rel-ative Safetyrsquo Be a Test of Product Liability rsquo rsquoJournal of Legal Studies 26 (2) 433plusmn 73

Clark S 1997 `Òverview of the Safe DrinkingWater Act Amendments of 1996rsquo rsquo Water Sci-ence and Technology Board Newsletter 14 (1)1plusmn 3

Community Nutrition Institute (CNI) 1991``FDA Not Enforcing Rules on Bottled WaterGAOrsquo rsquo Nutrition Week (Apr) 6

Friedman M 1996 ``Troubled Watersrsquo rsquo Parents(March) 50plusmn 54

Froderman T D Karnas and A Lucia 1996Toxic Tort Law in Arizona Eau Claire WisNational Business Institute Inc

Gelt J 1996 ``Consumers Increasingly Use Bot-tled Water Home Water Treatment Systems toAvoid Direct Tap Waterrsquo rsquo Arroyo 9 (Mar)1plusmn 12

Goodrich J J Adams B Lykins and R Clark1992 ``Safe Drinking Water From Small Sys-temsrsquo rsquo Journal of the American Water WorksAssociation 84 (1) 49plusmn 55

Grady M 1988 ``Common Law Control of Stra-tegic Behavior Railroad Sparks and theFarmerrsquo rsquo Journal of Legal Studies 17 (1) 15plusmn42

Innes R 1999 `Òptimal Liability With Stochas-tic Harms Judgement Proof Injurers andAsymmetric Informationrsquo rsquo International Re-view of Law and Economics 19 (2) 181plusmn203

Kaplow L and S Shavell 1999 `ÈconomicAnalysis of Law`` Discussion Paper 251Cambridge Harvard Law School

Lichtenberg E and T Penn 1996 ``Groundwa-ter Quality Policy under Uncertaintyrsquo rsquo Depart-ment of Agricultural and Resource EconomicsUniversity of Maryland Working Paper

Lichtenberg E D Zilberman and K Bogen1989 ``Regulating Environmental HealthRisks under Uncertainty Groundwater Con-tamination in Californiarsquo rsquo Journal of Environ-mental Economics and Management 17 (1)22plusmn 34

Miceli T and K Segerson 1993 ``RegulatingAgricultural Groundwater Contamination ACommentrsquo rsquo Journal of Environmental Eco-nomics and Management 25 (2) 196plusmn 200

National Research Council (NRC) 1997 SafeWater From Every Tap Washington DCNational Academy Press

Natural Resources Defense Council (NRDC)1994 Danger on Tap Protect Americarsquo sDrinking Water Madison Wis Sierra ClubGreat Lakes Program

Oates W 1983 ``The Regulation of Externali-ties Efreg cient Behavior by Sources and Vic-timsrsquo rsquo Public Finance 3 (2) 362plusmn 74

Phillips C and R Zeckhauser 1995 ``Confront-ing Natural Resource Damages The Econo-mistrsquo s Perspectiversquo rsquo In Natural ResourceDamages A Legal Economic and PolicyAnalysis ed R Stewart Washington DCNational Legal Center for the Public Interest

Polinsky M 1980 ``Strict Liability vs Negli-gence in a Market Settingrsquo rsquo American Eco-nomic Review 70 (2) 363plusmn 67

Polinsky M and S Shavell 1994 `À Note onOptimal Cleanup and Liability After Environ-mentally Harmful Dischargesrsquo rsquo Research inLaw and Economics 16 (1) 17plusmn 24

Raucher R 1994 ``Cost-effectiveness of SDWARegulationsrsquo rsquo Journal of the American WaterWorks Association (August) 28plusmn 36

Rose-Ackerman S 1991 ``Tort Law as a Regu-


latory Systemrsquo rsquo American Economic Review81 (May) 54plusmn 58

Segerson K 1990 ``Liability for GroundwaterContamination from Pesticidesrsquo rsquo Journal ofEnvironmental Economics and Management19 (3) 227plusmn 43

Shanaghan P 1996 ``SDWA Amendments of1996 Whatrsquo s the Impact on Small Systems rsquo rsquoOn Tap 5 (Winter) 1 14plusmn 15

Shavell S 1980 ``Strict Liability Versus Negli-gencersquo rsquo Journal of Legal Studies 9 (1) 1plusmn 25

ETH ETH ETH 1983 ``Torts in Which Victim and In-jurer Act Sequentiallyrsquo rsquo Journal of Law andEconomics 26 (Oct) 589plusmn 612

ETH ETH ETH 1984 ``Liability for Harm Versus Regu-lation of Safetyrsquo rsquo Journal of Legal Studies 13(June) 357plusmn 74

ETH ETH ETH 1987 Economic Analysis of AccidentLaw Cambridge Harvard University Press

ETH ETH ETH 1992 ``Liability and the Incentive toObtain Information About Riskrsquo rsquo Journal ofLegal Studies 21 (June) 259plusmn 70

Shibata H and J Winrich 1983 ``Control ofPollution When the Offended Defend Them-selvesrsquo rsquo Economica 50 (Nov) 425plusmn 37

Sunding D D Zilberman G Rausser and AMarco 1995 ``Flexible Technology and theCost of Improving Groundwater Qualityrsquo rsquoNatural Resources Modeling 9 (2) 177plusmn 92

Tieman M 1996 Safe Drinking Water ActAmendments of 1996 Overview of PL 104plusmn182 Washington DC Congressional Re-search Service

US Environmental Protection Agency (EPA)1993 The Safe Drinking Water Act A PocketGuide to the Requirements for the Operatorsof Small Water Systems San Francisco EPARegion 9

ETH ETH ETH 1994 The Quality of Our NationsWater 1992 Washington DC Ofreg ce ofWater

ETH ETH ETH 1995 The National Public Water SystemSupervision Program FY 1994 ComplianceReport Washington DC Ofreg ce of Enforce-ment and Compliance Assurance

US General Accounting Ofreg ce (GAO) 1992Drinking Water Widening Gap BetweenNeeds and Available Resources ThreatensVital EPA Program Washington DCGAO

ETH ETH ETH 1993 Drinking Water Program StatesFace Increased Difreg culties Meeting Basic Re-quirements Washington DC GAO

US Water News Inc and the Freshwater Foun-dation 1996 ``Municipal Water Holds ItsOwn Against Bottled Water Labelsrsquo rsquo USWater News 12 (Apr) 24plusmn 25

Viscusi K 1984 ``Structuring an Effective Oc-cupational Disease Policy Victim Compensa-tion and Risk Regulationrsquo rsquo Yale Journal onRegulation 2 (1) 53plusmn 81

Wetzstein M and T Centner 1991 ``Regulat-ing Agricultural Contamination of Groundwa-ter through Strict Liability and NegligenceLegislationrsquo rsquo Journal of Environmental Eco-nomics and Management 22 (1) 1plusmn 11


Economic logic can provide insights intoboth the appropriate settings for drinking wa-ter standards and the design of regulatorypolicy Although economists have exten-sively studied the process and regulation ofwater pollution relatively little attention hasbeen paid to the process and regulation ofwater treatment and delivery given contami-nation that has already occurred2 Notable ex-ceptions are recent papers that have recog-nized the importance of victim mitigation inthe water context3 In this work victims ofwater contaminationETH the water consum-ers ETH can undertake costly measures that re-duce the harm to which they are exposed Inpractice however there are signireg cant econ-omies of scale in both the process of watertreatment and in the acquisition and pro-cessing of information about the nature of thecontamination the health risks it poses andthe appropriate and available method of treat-ment Due to these economies of scale watertreatment is conducted by companies who donot themselves suffer the harm from ingest-ing any contaminated water that they deliverOf course mitigation may also be done byconsumers themselves who can switch fromdrinking tap water to drinking bottled water4

The multiple decisions involved in this pro-cessETH and the regulatory need to elicit be-havior from water companies that remacr ects so-cietal interests in safe drinking waterETH areour central focus in this paper

Specireg cally in view of the importance ofsafe drinking water regulation in practiceour purpose here is to present and analyze arealistic but tractable model of water treat-ment and use in the presence of (1) randomcontamination (2) a water company treat-ment process that involves both reg xed invest-ments in treatment systems and variabletreatment inputs that can be chosen ex-postafter the level of contamination has been ob-served and (3) a consumer alternative to thedrinking of contaminated tap water bottledwater

We are reg rst concerned with optimality intreatment and use (in Section 3) For exam-ple how does the level of water contamina-tion and the size of a companyrsquo s customerservice population affect optimal treatmentactivity and the optimal decision on whether

to substitute bottled water for tap water indrinking And how does a switch to bottledwater affect optimal tap water treatmentUnder an optimal policy we reg nd that post-treatment water quality falls with higher lev-els of contamination when contamination issufreg ciently great (and treatment costs arecorrespondingly high) consumers are ad-vised to drink bottled water and the tap wateris treated to a lower ``non-drinking-usersquo rsquoquality standard that imposes a low cost oftreatment In addition we reg nd that larger wa-ter systems optimally invest more in thetreatment system and treat to higher stan-dards of quality

In Section 4 we turn to the regulatory sideof the problem how to prompt efreg cient be-havior from water suppliers using either lia-bility or regulatory standards enforced withsufreg ciently stiff reg nes Our main policy con-

2 The importance of water treatment as an alterna-tive to pollution prevention is widely recognized in theeconomics literature In a general setting Polinsky andShavell (1994) examine the incentive effects of differ-ent liability rules in inducing reg rms to employ optimallevels of care and mitigation Sunding et al (1995)study optimal treatment in the context of CaliforniaDBCP contamination (see also Lichtenberg Zilbermanand Bogen 1989) Lichtenberg and Penn (1996) studythe choice between preventing and treating nitrate con-tamination of Maryland waters Barrett and Segerson(1997) examine the choice between prevention andtreatment when policymakers pursue objectives otherthan Paretoefreg ciency Conceptually these analyses dif-fer from ours because they do not focus on the treat-ment process and they abstract from the consumermiti-gation and regulatory standard setting that are centralto our analysis of drinking water markets

3 See Segerson (1990) Miceli and Segerson (1993)and Wetzstein and Centner (1992) Also see Oates(1983) and Shibata and Winrich (1983) for studies onvictim mitigation in pollution control and Shavell(1983) and Grady (1988) for victim mitigation in liabil-ity determination

4 An alternative consumer mitigation strategy is toinstall water treatment devices in onersquo s home Such de-vices generally employ the same technologies as docentralized systems Due to economies of scale in thesetechnologies and in their operation and maintenancewater company treatment is almost always more cost-effective for all but the tiniest systems (NRC 1997) Forexample in a cost comparison between home treatmentand a company granular activated carbon technologyGoodrich et al (1992) reg nd that the home devices areonly cost-effective when there are fewer than 20 serviceconnections In this paper we assume that water treat-ment is done at the company level





II THE MODEL



The Water Consumers









Water Treatment
























(B(w) 2 DA(u))N 2 v(u X y Nw) [1]









(B(w) 2 DB(u))N
















drink bottled water

JB(X y N)

5 (B(wB( )) 2 DB(uB( )))N

2 v(uB( ) X y N(wB( ) 2 w)) 2 cN

(B(wA( )) 2 DA(uA( )))N

2 v(uA( ) X y NwA( ))

JA(X y N)


drink tap water [4]






paraJA(X y N)paraX

5 2vX(uA( ) X y NwA( ))

2vX(uB( ) X y N(wB( ) 2 w)) [6]























maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References



















































II THE MODEL



The Water Consumers









Water Treatment
























(B(w) 2 DA(u))N 2 v(u X y Nw) [1]









(B(w) 2 DB(u))N
















drink bottled water

JB(X y N)

5 (B(wB( )) 2 DB(uB( )))N

2 v(uB( ) X y N(wB( ) 2 w)) 2 cN

(B(wA( )) 2 DA(uA( )))N

2 v(uA( ) X y NwA( ))

JA(X y N)


drink tap water [4]






paraJA(X y N)paraX

5 2vX(uA( ) X y NwA( ))

2vX(uB( ) X y N(wB( ) 2 w)) [6]























maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References



















































Water Treatment
























(B(w) 2 DA(u))N 2 v(u X y Nw) [1]









(B(w) 2 DB(u))N
















drink bottled water

JB(X y N)

5 (B(wB( )) 2 DB(uB( )))N

2 v(uB( ) X y N(wB( ) 2 w)) 2 cN

(B(wA( )) 2 DA(uA( )))N

2 v(uA( ) X y NwA( ))

JA(X y N)


drink tap water [4]






paraJA(X y N)paraX

5 2vX(uA( ) X y NwA( ))

2vX(uB( ) X y N(wB( ) 2 w)) [6]























maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References
































































(B(w) 2 DA(u))N 2 v(u X y Nw) [1]









(B(w) 2 DB(u))N
















drink bottled water

JB(X y N)

5 (B(wB( )) 2 DB(uB( )))N

2 v(uB( ) X y N(wB( ) 2 w)) 2 cN

(B(wA( )) 2 DA(uA( )))N

2 v(uA( ) X y NwA( ))

JA(X y N)


drink tap water [4]






paraJA(X y N)paraX

5 2vX(uA( ) X y NwA( ))

2vX(uB( ) X y N(wB( ) 2 w)) [6]























maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References




















































(B(w) 2 DA(u))N 2 v(u X y Nw) [1]









(B(w) 2 DB(u))N
















drink bottled water

JB(X y N)

5 (B(wB( )) 2 DB(uB( )))N

2 v(uB( ) X y N(wB( ) 2 w)) 2 cN

(B(wA( )) 2 DA(uA( )))N

2 v(uA( ) X y NwA( ))

JA(X y N)


drink tap water [4]






paraJA(X y N)paraX

5 2vX(uA( ) X y NwA( ))

2vX(uB( ) X y N(wB( ) 2 w)) [6]























maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

























































drink bottled water

JB(X y N)

5 (B(wB( )) 2 DB(uB( )))N

2 v(uB( ) X y N(wB( ) 2 w)) 2 cN

(B(wA( )) 2 DA(uA( )))N

2 v(uA( ) X y NwA( ))

JA(X y N)


drink tap water [4]






paraJA(X y N)paraX

5 2vX(uA( ) X y NwA( ))

2vX(uB( ) X y N(wB( ) 2 w)) [6]























maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References
































































maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References






















































maxy

2 F(y) 1 X(y N)

XJB(X y N) g(X) dX

1 XAring

X(y N)



2Fcent(y)

2 X(y N)


2 XAring

X(y N)


[11]














Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References



























































Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

















































Strict Liability


Negligence

















N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

























































N) uSN(N)











pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References


















































pA(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)

2 v(uSA( ) X y wN) [12a]

pB(X y N)


5 maxw

(B(w) 2 c 2 B)N 2 F(y)

2 v(uSB( ) X y(w 2 w)N)

[12b]



pB(X(N) y(N) N)














Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

















































Decisions




p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References
















































p1A(X y N)


5 maxw

(B(w) 2 B)N 2 F(y)


for X X(N) [15]














Treatment Capacity






TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References
















































TABLE 1









notireg cation)


y y(N)pA(N) pB(N)












Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

















































Fixed Standards
















30 Formally

D X( )

X


1 XAring

X( )












TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

























































TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References
















































TABLE 2




50000 9 59 5

3300 amp 50000 79 17 44

1000 amp 3300 204 16 1131000 1519 8 839

B) Contaminationb

























APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

























































APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References


















































APPENDIX


parauA( )paraX 5s

vuX (Bsup2 2 vWW N)

1 (vWX N) vuW [A1a]

parauA( )paray 5s

vuy (Bsup2 2 vWW N)

1 (vWy N) vuW [A1b]

parauA( )paraN 5s





where ``5s


v 5 a(u X y) 1 b(u X y) W [A2]


parauAparaN 5s

2(Bsup2N) au 0










paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References

















































paraX( y N)paray

5 2[vBy (X( )) 2 vA

y (X( ))]

[vBX(X( ) ) 2 vA

X(X( ))] 0 [A3]




y 2vB


paraX(y N)

paraN

5s



5 N21[vB(X )

2 vBW(X ) N(wB( ) 2 w)]

2 [vA(X ) 2 vAW(X )wA( )N] [A4]


W

(X ) 5 vAW(X ) wA( ) 5 wB( ) and vB( ) 5



paraX(y N)

paraN

5s





dy(N)dN5s

X(y(N) N)

X 3vByu(X)

parauB

paraN1 vB


paraN64dX

1 XAring

X(y(N) N)3v I

yu(X)parauA

paraN1 vB

yW(X) 5wA 1 NparawA


y (X( ))paraX(y N)

paraN )y5y(N)

[A5]


y (X( ))2 vA









CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References
















































CBA 2 CA


CAB 2 CB





k ( ) and Jk

CBA 2 CA


CAB 2 CB



References



































































Documents

The Economics of Safe Drinking Water