Derived Willingness-To-Pay For Water: Effects Of Probabilistic Rationing And Price

Transcription

1 Derved Wllngness-To-Pay For Water: Effects Of Probablstc Ratonng And Prce Roberto Garca Alcublla Abstract A two stage lnear programmng approach s used to estmate the wllngness to pay WTP of ndvdual households or groups of households for changes n a combnaton of probablstc water supply relablty and retal prce of water. By modelng the costs fnancal and perceved of mplementng long and short-term conservaton measures and assumng ratonal expected value cost mnmzng behavor demand curves for water and expected water use curves can be estmated. Derved demand curves for conservaton measures can also be calculated. Next Monte Carlo smulaton technques are used to represent household varablty n the model parameters and derve estmates of aggregate WTP for water supply relablty water demand curves and demand curves for conservaton measures. Several examples are provded to llustrate the approach. Introducton The goal of water managers s to delver a relable water supply at a reasonable cost. Indeed any ncrease n relablty nvolves a cost that has to be balanced aganst the benefts assocated wth the resultng reducton n frequency of water scarcty. The economcally optmal water supply relablty wll be such that the margnal cost of ncreased relablty equals the margnal cost of ncreased shortage Howe and Smth 994 Hoagland 998. Snce early modern engneerng and plannng studes n the water resources a ey queston has been whether the users would be wllng to pay the costs of ncreasng supply capacty Duput 844. Provdng decson maers wth a probablstc supply valuaton based on wllngness-to-pay WTP becomes a ey tool n relablty plannng. If the cost of a relablty enhancement proect water recyclng extra capacty water transfers etc s below the consumer s WTP the proect s economcally vable Abrahams et al. Conversely n hghly relable systems consumers mght be wllng to accept a greater frequency of shortages n exchange for reduced water blls Howe and Smth 994. The extra water made avalable could be mareted by the water agency and transferred to other sectors Hoagland 998. On a regonal perspectve where multple water uses compete for constraned resources ncreasng water servce relablty to urban areas s lely to shft the rs of shortage and the cost to other sectors Howe and Smth 994 Grffn and Melde. Tradtonally the cost of shortages has been borne by n-stream users however ths s becomng less lely Grffn and Melde and the focus s slowly shftng to agrcultural users often wth compensaton from urban users. Snce the 99 drought n Calforna and the resultng severe water ratonng for urban users ctes have started to argue that the costs of water shortages to the urban/resdental sector should be reevaluated upwards and that the value of servce relablty to urban users out-weghts agrcultural losses. In Calforna n lght of the ncreasng populaton pressure the allocaton of water to envronmental uses more strngent regulatory requrements and ncreasng margnal costs of new proects water shortage management wll become more mportant Wlchfort and Lund 997. However as Lund 995 noted lttle effort has been devoted to valung urban water supply relablty. Most of the approaches have been emprcal; ether usng prce elastcty Howe 967 and 98 Greene et al. 998 or contngent valuaton technques Carson and Mtchell 987 CUWA 994 Howe and Smth

2 994 Grffn and Melde. These emprcal studes typcally gnore much of the nteracton between long-term and short-term conservaton measures and loo only at a sngle shortage event defned by a gven level of shortage wth a certan frequency.e. a % shortage once every 5 years Lund 995. Yet there s a necessty for estmatng the shortage losses over the whole range of possble shortages Howe and Smth 994. Indeed nvestments n water supply relablty enhancement alter the frequency of all shortage levels so estmatng the value of an entre probablty dstrbuton of shortages s desrable. Several studes Grffn and Melde CUWA 994 have shown that consumers have dffculty nterpretng probablstc nformaton whch leads to nconsstent results. The CUWA 994 study concluded that these lmtatons mae t dffcult to apply CV data to a real world hydrology that produces a mx of shortages. Thus tradtonal emprcal methods used for valung the benefts of urban water supply relablty are n some ways ll-suted for probablstc settngs. The approach developed by Lund 995 and Wlchfort and Lund 997 addresses some of these lmtatons. These authors proposed a two-stage optmzaton model to estmate WTP to avod shortages that consders the user economc response to an entre shortage probablty dstrbuton dfferent levels and frequences. Ths study extends ther approach to nclude the retal prce of water and allows for household varablty n the model parameters. Ths allows dervaton of demand curves for water and for conservaton measures as well as probablstc estmates of WTP for water supply relablty. The paper begns wth a smple analytcal treatment of long- and short-term conservaton optons n the context of a probablty dstrbuton of water ratonng levels and varyng retal prces for water. A twostage lnear program s then formulated to numercally estmate the WTP of a sngle household to avod a probablty dstrbuton of shortages for dfferent retal water prces. Ths two-stage optmzaton program s appled to derve ndvdual consumer water demand curves wth and wthout shortage. Input demand curves for conservaton measures are also presented. The method s then extended to develop aggregate demand curves for both water and conservaton measures for a group of households. General Formulaton for Indvdual Consumers The formulaton of Lund 995 can be extended to nclude the prce of water for each shortage event p Q. The household s obectve s to mnmze the expected value of total annualzed costs necessary to meet each level of probablstc shortage subect to household water ratonng and management constrants. Ths expected cost mnmzaton assumpton appears reasonable snce water costs are usually a small part of most total household expendtures. Ths s formulated as a mathematcal program: Mnmze Z f [ g + p Q ] n + P Q Subect to: h Q 3 Q r Where: long-term conservaton measures avalable short-term conservaton measures avalable for each shortage event f cost of mplementng permanent water conservaton efforts g cost of mplementng short-term conservaton efforts h water use n event for mplementaton of short and long-term conservaton optons p Q retal prce of water for each shortage event Q water use for each shortage event

3 3 P probablty of occurrence of each shortage event r raton amount for event Here Z s the expected value of total household water costs wth component costs for long and short-term water conservaton efforts and the purchase cost of water for each ratonng event ncludng no ratonng. The latter two terms are weghted by the probablty of each ratonng event to account for the hydrologc uncertantes n water supply see Lund 995 or Wlchfort and Lund 997 for detals. Constrants and 3 state that water use s a functon of the permanent and short-term reducton efforts by functon h and that the total water use must be less than the raton amount r for each ratonng event. Ths problem can be examned analytcally usng the method of Lagrange multplers. Substtutng Equaton nto Equatons and 3 the Lagrangan functon s: The resultng frst-order condtons are: Rearrangng Equaton 5 results n Equaton 7 g h p P P f n Q 7 λ Ths result shows the effect of prce on the optmal margnal cost of mplementng any permanent conservaton opton. The mplementaton of the permanent conservaton optons s encouraged to the extent of the expected value of decreased household water expendtures over the n possble shortage events the entre shortage probablty dstrbuton. Thus prce effects on conservaton mplementaton are proportonal to the effectveness of the conservaton opton. If prce does not vary between shortage events p Q p Q then the prce effect s drectly proportonal to the expected value of water savngs from a gven conservaton opton. The condton n Equaton 6 s easly rearranged to: 8 p P h g Q λ Ths condton holds that the margnal mplementaton costs-effectveness should be equal across shortterm conservaton actons for each dfferent event. Ths result s very smlar to Lund 995 except that consderng prce lowers the margnal level of mplementaton for short-term measures. Hgher water prces mae t optmal to use short-term conservaton measures that are less mplementaton-cost-effectve for water use reducton. [ ] n n Q r h h p g P f L λ h p p g P f L n Q λ h P p g P L Q 6 + λ

4 4 Puttng Equaton 8 nto Equaton 5 yelds a modfed frst-order condton for permanent conservaton optons. Ths frst-order condton no longer explctly ncludes the retal water prce; prce s mplctly ncluded through Equaton 8 on the optmal magntude of short-term mplementaton-cost/effectveness. 9 h g g P f n + If mplementaton of long term permanent conservaton measures does not affect event-specfc short term mplementaton costs δg / δ and Equaton 9 becomes: h g P f n Further f the mplementaton of permanent optons has the same water-conservaton effectveness for each ratonng event then δh / δ constant not varyng wth. In ths case Equaton becomes: h g P h f n Where these condtons hold the margnal mplementaton-cost-effectveness should be equal across all permanent conservaton optons mplemented and should also equal the expected value of the margnal mplementaton-cost-effectveness for each mplemented short-term conservaton opton. However ths smplfed result does not consder that mplementng long-term conservaton measures often reduces the effectveness of short-term optons demand hardenng. Therefore δ /δ > and the RHS wll have to be adusted to account for ths mang the mplementaton of permanent conservaton optons less lely. Whle these analytcal solutons provde some nsght n practce more specfc numercal formulatons are more useful. Ths problem can be ncely formulated nto lnear programs for several cases. Lnear Program Formulaton for a Sngle Household For an ndvdual household undertang a gven water conservaton measure mples a unt cost to the household of c. Economc motvatons for water conservaton measures are the retal prce of water and lmts placed on water avalablty due to shortage event ratonng levels or outages. The overall mpact of water shortages depends on the probablty of occurrence of each event as specfed by a shortage probablty dstrbuton smlar to those generated from water resource models Hrsch 978. Ths means that once a long-term conservaton measure s mplemented the household bears ts full cost c whereas the cost c of short-term water conservaton optons has to be weghted by the probablty

5 of each shortage event P. The same apples to the water bll p Q Q allowng prce to vary wth shortage event. Ths stochastc optmzaton problem can be represented as a two-stage lnear program Wagner 975 as shown below. The lnear program extends earler formulatons Lund 995; Wlchfort and Lund 997 by ncludng the retal prce of water. The frst stage decsons concern long-term conservaton measures whch must be mplemented before the shortage and normally have a long lfe span and fxed annualzed costs. The second stage decsons concern the mplementaton of short-term conservaton measures to reduce demand/water use for each shortage event. n m + n MnZ c P c + pqq Subect to: 3 u 4 u n 5 d q q Q 6 Q r + n 7. Where: level of mplementaton of long-term measure level of mplementaton of short-term measure n shortage event c cost of long-term measure annualzed c annual cost of short-term mesure n event q unt annual water saved by long-term a measure q unt annual water saved by short-term measure durng shortage u upper lmt of long-term conservaton measure u upper lmt of short-term conservaton measure under event p Q retal prce of water for each shortage event Q water use for each shortage event P probablty of occurrence of shortage event r raton amount for event d full servce demand/water use for event m number of shortage levels events n number of long-term measures n number of short-term measures The total expected value cost of household water management efforts s represented n Equaton ncludng the costs for long-term water conservaton measures the cost of short-term water conservaton efforts for each of the m shortage events and the payments for water use the water bll for each event allowng the prce of water p Q to vary wth shortage event. Each long and short term water conservaton measure has a maxmum level of mplementaton as shown n Equatons 3 and 4. The lower lmt of applcaton no applcaton at all s represented by the nonnegatvty constrant n Equaton 7. If more detaled water use data s avalable ths lmts could also be a functon of the water use affected by each conservaton measure.e. % of water used for tolet flushng. More elaborate equatons representng the upper bounds of conservaton optons exst where long-term 5

6 conservaton optons such as xerscapng affect the effectveness of short-term conservaton such as restrctng lawn waterng and vceversa Lund 995; Wlchfort and Lund 997. Equaton 5 calculates the quantty of water purchased for each event full servce water use d mnus the effectveness of water conservaton efforts. Several authors Dzegelews et al 993 Vcers have suggested that consderng effectveness as a percentage reducton of the water use relevant to each opton mght be more approprate. Indeed f a household uses twce as much water as another one for clothes washng t s lely that a more water-effcent clothes-washer wll conserve more water n the frst case. Equaton 6 defnes the water avalablty for each event and requres that total water use not exceed a gven water raton r. Often t s dffcult to reduce the water bll to a unt cost of water tmes the quantty consumed p Q Q as requred n ths formulaton. Many utltes charge sewer rates and other charges based on water use so the model s prce parameter should be adusted accordngly. The fxed part of the bll that utltes commonly charge can be gnored. If the consumer conserves water he wll beneft by payng less for water use and consequent wastewater charges but wll stll have to pay the fxed part of the water bll. Specfc rate structures can be accommodated n the model. If consumers are operatng under an ncreasng convex bloc rate structure ths can be accommodated by separaton of Q nto peces for each bloc. A decreasng concave bloc rate structure can smlarly be accommodated wth the addton of sutable bnary nteger varables as an nteger-lnear program. If we are concerned about the effect of water conservaton on water utlty revenues a constrant can be set on net utlty revenues by mang the prce parameter p Q nto a decson varable p Q and addng m a constrant on revenue P pq Q desred revenue. Varable drought prcng can be examned by replacng the prce parameter p Q wth a decson varable and a revenue constrant and even a constrant lmtng revenue varablty. In both these cases the lnear program above becomes non-lnear. The same approach apples to a bloc rate structure. A more dffcult aspect to ntegrate n the model s to now exactly what prce varable consumers respond to f any at all. An abundant and controversal body of lterature has consdered how consumers respond to the prce of water Howe and Lnaweaver 967 Wong 97 Young 973 Nordn 976 Bllngs and Agthe 98 Howe 98 Neswadomy and Molna 99. Yet prce and technology conservaton optons are only some of a complex set of varables that affect the consumer s demand for water. Household s characterstcs locaton or ncome are some others. One mght want to capture these effects by restatng the problem usng dfferent coeffcents for dfferent locatons and classes of household. Ths possblty wll be dscussed at length later n ths paper. In any case the cost coeffcents must be annualzed and nclude both fnancal and perceved costs. For tolet retrofttng t s lely that provded that low flush tolets wor correctly the consumer wll see no nconvenence n the change. However some consumers mght prefer to use a standard rather than a low-flow showerhead and therefore mplementaton of such and opton wll ental both fnancal and nconvenence cost. It s therefore assumed that the model s cost coeffcents are estmates of consumer s wllngness to pay to avod mplementng specfc water conservaton measures as determned by fnancal and contngent valuaton or other valuaton technques. The approach and method s llustrated by a seres of examples. Integer constrants are neglected n ths problem snce many of the measures could be partally mplemented or mplemented only by a fracton of the households Lund 995. For smplcty the retal prce of water and the cost and effectveness of conservaton measures wll be fxed over the range of shortage events for these examples. A sngle-famly household s assumed to have the followng long and short term conservaton optons avalable : long-term : retroftng tolet from 3.5 gallons 3.5 lters per flush gpf to.6gpf 6.lpf- TR- erscape mplement xerscapng n a part of ther garden-- 3 erscape mplement xerscapng n all of ther garden area --. Short-term optons : TD nstallng a 6

7 dsplacement devce n the tolet DR stop waterng the lawn but not the shrubs and trees 3 DS stop waterng the shrubs and trees n the garden. The costs and water conservaton effectveness for each water conservaton measure are gven n Table and have values representatve of the lterature Schulman and Ber 994. Formulaton for a Sngle Household Wthout Ratonng We frst examne the effects of prce on a sngle household s water use n the absence of ratonng. Ths can be smulated by settng m n the problem formulaton only one event and elmnatng constrant 6 the only ratonng event has no ratonng. The lnear program s then solved for dfferent prce levels. As the p Q coeffcent ncreases n equaton the optmal mx of conservaton measures wll change and reduce the quantty of water used Q. Results are represented n Fgure as a thc sold lne. As llustrated n Fgure the common sense result s that as the retal prce of water ncreases households face an ncentve to mplement more conservaton measures and reduce ther water use. When faced wth an ncrease n water prce the household has to decde whether to contnue consumng as much water as before and pay a hgher water bll or mplement conservaton measures to reduce ther water use at some cost. When water s nexpensve the household has lttle ncentve to ncur a cost n money tme etc to reduce ts water use. At some pont the prce of water gets hgh enough that a gven conservaton opton becomes cost-effectve and s mplemented; then the reducton n the water bll from conservaton exceeds the costsfnancal and perceved- of conservaton. For each of the sx conservaton measures these ponts are gven by steps n the water demand curve. At those turnng ponts the prce of water equals the cost per unt of water saved by mplementng a gven conservaton measure n $/ gallons. Conservaton measures wll be mplemented n order of ther relatve cost-effectveness. For example nstallng a tolet dam or dsplacement devce brcs can save a reasonable amount of water has vrtually no fnancal cost provdes the same flushng performance as before and does not tae a lot of tme perceved costs. On the other hand to stop waterng garden shrubs can ental great fnancal expenses for replacement and may have large perceved cost n lost aesthetc benefts. Yet water mght be so expensve that havng a garden s an unaffordable luxury.e. n some ard countres. A household s demand for conservaton measures can be derved ndrectly from the soluton to ths LP problem. As prce ncreases so does use of conservaton measures. In ths LP formulaton once the prce hts the level where a specfc conservaton measure becomes cost-effectve t s fully mplemented. In a way we could say that the household s substtutng water use for conservaton measures. Ths tradeoff between conservaton costs and savngs on the water bll s represented n the obectve functon. Therefore the obectve functon provdes the least costly way of coverng the household s water needs for a gven retal prce of water. Under the assumpton that the model s cost coeffcents are estmates of household s wllngness-to-pay to avod mplementng specfc water conservaton measures and that they capture the household s utlty preferences ths means that the obectve functon mnmzes the loss of consumer surplus CS well-beng or welfare due to an ncrease n the retal prce of water where free water serves as the baselne. A prce ncrease wll lead to some loss n consumer surplus by: payng more for the water the household uses and payng to reduce water use by mplementng conservaton measures. The optmzaton program mnmzes the sum of those losses and hence mnmzes the loss of consumer surplus. Graphcally the CS s gven by the area between the demand curve and a horzontal lne drawn at a gven prce level. Ths area s maxmal when prce s zero and wll dmnsh as prce ncreases. The change loss n CS due to an ncrease n prce from zero to p s therefore gven by the shaded area n Fgure. Ths fgure also shows what the demand curve for water would loo le f no conservaton measures where mplemented. In ths case the loss of CS due to an ncrease n prce would be hgher confrmng that mplementng conservaton measures though costly contrbutes to mnmzng the loss n CS. Table compares the loss n CS calculated usng the obectve functon and usng the graphcal method. A converson factor has to be appled to the area calculatons to translate them to the same unts as the obectve functon $/year.the results show the correspondence between the obectve functon and the 7

8 loss of CS. The slght dscrepances that arse are due to dscretzaton errors that nfluence the calculaton of the areas under the demand curve. Formulaton for a Class of Consumers Wthout Ratonng Even wthn a class of customers respondng to the same prces ndvdual users wll have varyng perceved conservaton mplementaton costs c effectveness of water conservaton optons q and normal levels of water use d Lund 995. The types of conservaton measures avalable to each household also may vary. Varablty n water demand s wdely recognzed even for households wth smlar characterstcs. Household occupancy rates can vary but more mportantly lfestyle characterstcs and water use patterns vary between households n ways that are not well understood. For example studes have shown that tolet flushng by males and females have dfferent patterns and there s a wdespread varablty n the duraton of showers Vcers. Such factors are lely to affect the household s total water use and can enhance or reduce the effectveness of lets say a low flow showerhead. The cost of mplementng a gven conservaton measure can be decomposed nto a fnancal expendture that accounts for the cost of materals and nstallaton of conservaton devces and a perceved cost accountng for all other non maret values such as the consumer s preference for one or another conservaton measure the value of the tme the consumer wll spend mplementng a conservaton opton resstance to bother wth conservng water nconvenences of tang shorter showers or flushng the tolet less often etc. Fnancal cost may seem easer to estmate; yet the choce of conservaton devces for a specfc water use s rapdly ncreasng and costs vary wdely. Vcers reports ranges of 75-5 to 65 $ for low flush tolets. The perceved costs assocated wth the mplementaton of a specfc conservaton measure cannot be easly ascertaned. The CUWA 994 contngent valuaton study reported that the maor cost of conservaton was havng to spend tme and effort to montor water use. It should be expected that the range of varablty n perceved costs would be larger than for fnancal costs. Contngent valuaton studes of wllngness-to-pay to avod mplementaton of a specfc conservaton measure would be requred to gan nsght on ths matter. Such specfc contngent valuaton studes should represent an mprovement over the more general and vague drect contngent valuaton studes of wllngness-to-pay to avod probablstc shortages snce they would only solct evaluatons of drect water conservng actons that consumers mght tae n response to shortages Lund 995. To the author s nowledge no such study has been undertaen up to date. As noted by Lund 995 a lower bound for the cost coeffcents could be estmated by consderng only fnancal costs. Though Lund 995 focused hs dscusson on the dffculty of estmatng actual and perceved costs of conservaton measures great varablty also exsts n the effectveness of conservaton measures. In ther manual for evaluatng urban water conservaton programs Dzegelews et al. 993 note: An underlyng shortfall nvolvng the mplementaton of water conservaton as a demand management tool s the lac of relable nowledge of actual water savngs maret penetraton and nteracton effects between conservaton measures. Further: For many conservaton measures data are non exstent or non relable Dzegelews et al Indeed a revew of the abundant studes on ths subect shows that for the same type of conservaton measure reported effectveness values can range from smple to double or trple Dzegelews et al. 993 Vcers CUWCC Maddaus 987. All these sources agree that the effectveness of a conservaton measure wll depend on the applances beng replaced on the new ones nstalled.e. from a 5 gpflush tolet to a.6 or a gpf on techncal consderatons such as the water pressure of the servce area and on behavoral aspects of water use patterns. Varablty n the effectveness of water conservaton measures should be consdered because of ts effect on the relatve cost effectveness of the dfferent conservaton optons the rato of cost to conservaton; $/cost per unt water use reducton. Some models that calculate the effectveness of conservaton measures exst Wals 984 but they are data ntensve and account only for varablty n engneerng parameters. Ths maes them of lmted usefulness. 8

9 To gve some ntuton about the effects of parameter varablty n the models results the no ratonng case dscussed above s solved for dfferent sets of parameter values costs effectveness full servce demand and results are presented n Fgure. These demand curves are not only shfted along the axs due to changes n full servce demand but the steps that mar mplementaton of a partcular conservaton measure also vary wdely. The average curve that results from solvng the LP usng the average of the parameters of the other runs features a full servce demand smlar to the orgnal case yet the pattern of mplementaton of conservaton measures s very dfferent. A more rgorous approach would be to consder the model s parameters probablstc and to assgn them a probablty dstrbuton based on the best avalable nowledge of household varablty n model parameters. Monte Carlo smulatons can then be used to examne the mportance of ths varablty wthn a class of customers. By solvng the LP model for a set of random parameters usng Monte Carlo smulatons confdence ntervals can be calculated for the loss of consumer surplus and water use. The Monte Carlo approach has the advantage of accountng for varablty n the model s parameters wth mnmum nformaton requrements. Usng maxmum entropy ME estmaton we can assgn a probablty dstrbuton to a specfc parameter wth as lttle as one observaton truncated exponental dstrbuton. Generally more nformaton s lely to be avalable from the lterature or can be generated relatvely easly. Water utltes can provde extensve nformaton about demand/water use levels that can be used to ft a probablty dstrbuton. Informaton about lower and upper bounds for the fnancal costs can be drawn from the lterature. Informaton about perceved costs can be obtaned by conductng contngent valuaton studes and should generate enough nformaton mean standard devaton to estmate probablty dstrbutons. New developments n measurng and modelng end uses of water and determnng the effectveness of conservaton efforts usng flow trace analyss technques mae t now possble to gather emprcal evdence of the effectveness of specfc conservaton measures at the local level qucly and at a relatvely low expense Weber 993 Mayer DeOreo et al 999 Trace Wzard. Ths technque also can help n studyng the nteracton between dfferent conservaton measures. If we wshed to account for such nteractons the same Monte Carlo approach would apply. For ths wor we assumed the model parameters to be normally dstrbuted wth nown mean and standard devaton as shown n Table. Fgure 3 shows the results for the No raton case for dfferent numbers of Monte Carlo teratons. By accountng for varablty n parameters the ponts at whch dfferent conservaton measures become cost effectve shft to gve a smooth aggregate demand curve. The water use curve stablzes after a hundred teratons. Ths aggregate curve can be used to estmate the prce elastcty of water wthn dfferent prce ranges as shown n Table 3. These elastcty values should not be taen lterally because the model s parameters used for ther estmaton are not emprcally based yet ther relatve values and varaton along the demand curve are consstent wth what the theory of resdental water use predcts. The almost perfectly nelastc response that we observe n the low ranges of water prce -6 $/ g can be attrbuted to the lmted range of low cost conservaton optons consdered for ths model. It can also be argued that when the prce of water s low the effects of a prce ncrease on the household s ncome wll be small and lead to very lttle change n water use. For such low prce levels the effort of botherng about water use montorng esceeds any foreseeable beneft. As the prce of water gets hgher 6-3 $/ g water use becomes more responsve to changes n prce water untl water becomes essental water used for drnng coong so water use cannot be curtaled any further and barely responds to prce ncreases. Formulaton for a Sngle Household Wth Ratonng Water demands n ard and sem-ard regons are sgnfcantly affected by perceptons of water avalablty. Households expectng frequent epsodc reductons n water avalablty water ratonng are lely to change ther water-related nvestments n landscapng and plumbng to reduce normal water use and allow them to more easly reduce water demands/use durng perods of shortage. 9

10 The response of a sngle household n the presence of water ratonng can be studed by solvng the LP problem for several shortage probablty dstrbutons P. The soluton output provdes the mx of conservaton measures that mnmzes total cost to the consumer whle ensurng that water use n each of the shortage events Q s wthn the raton for that event r. To llustrate the most relevant aspects of the ratonng case let s focus on a very smple case wth a 5% probablty of no shortage event- and a 5% probablty of a % shortage event - a mandatory reducton of % compared to the full servce water use. The relevant results for ths case appear n Table 4. Focusng on water use for event we can see that for low prce levels demand s drven by the water raton. Because ratonng s mandatory there s a need to mplement conservaton measures at prce levels where these are not cost-effectve. For those prces no conservaton would have been mplemented n the absence of ratonng. Wthn ths range of prces ratonng mposes hgher total costs on the household column V on Table 4. However as prce ncreases the effect of the raton s dluted because the mplementaton of conservaton measures becomes cost-effectve even n the absence of ratonng. Conservng water s now at those water prces economcally nterestng for the expected value costmnmzng household; the effect of the raton s no longer felt and the water use curves for all events converge to the no raton water use curve see Fgure 5. At each prce level the hgher costs that ratonng mposes on the household result n an addtonal loss of consumer surplus compared wth the no ratonng case that s gven by the dfference n the value of the obectve functon. Ths extra loss can alternatvely be calculated as the area between the demand curves wth and wthout ratonng. Note that for the shortage probablty dstrbuton we are consderng the water use curve for Event no shortage s supermposed to the orgnal no raton curve. Snce the level of water use specfed by Event only occurs wth a certan probablty.5 ths factor has to be ncluded n the area calculaton column I n table 4. Table 4 shows that the two approaches are equvalent and clearly establshes dscretzaton as the source of error. The dvergence observed between columns VII and I can be traced to the ponts where there s a shft on the demand curve. The same result can be obtaned usng the expected value curve for ths shortage probablty dstrbuton the weghted average of water use for each event. Ths extra loss provdes an estmate of the consumer s maxmum wllngness to pay WTP to avod a partcular shortage dstrbuton hence ndrectly establshng the value of water supply relablty for the household. If by payng $Y those shortages could be avoded the household should be wllng to ncur that cost as long as t s less than the loss of CS mposed by a partcular shortage dstrbuton P. The household s WTP to avod a specfc shortage probablty dstrbuton decreases as the retal prce of water ncreases column VII n Table 4 because hgher prce levels provde an economc ncentve to mplement conservaton measures and reduce water use voluntarly thus neutralzng the effect of mandatory ratonng. When water s very expensve households would rather reduce ther water use than eep ther current consumpton level and pay extremely hgh water blls. These results are consstent wth fndngs from contngent valuaton studes Grffn and Melde. In practce there s some level of unrelablty assocated wth every water supply system and therefore n most cases the relevant analyss would be for the ncremental value of movng from one dstrbuton of unrelablty to another CUWA 994. Ths can be smulated n ths model by comparng two probablty dstrbutons that ental dfferent levels of unrelablty where not all shortages are avoded. For the analyss of WTP to avod shortage two aspects of the relablty of a water supply system are relevant. The relablty of a system s characterzed by the probablty of havng no shortages the hgher the better and by the levels and probabltes of shortage assocated wth gven levels of unrelablty. A system that has 7% chance of havng no shortages and 3% probablty of a % shortage s potentally Note that a unt converson s necessary to translate the area under the curves n $/day to the same unts as the obectve functon $/year.

11 less damagng than one that has the same probablty of no shortages but where there s a 3% probablty of a % shortage and should therefore be preferred. Ths s llustrated by comparng the consumer s WTP to avod shortage probablty dstrbutons A B and C descrbed n Table 5. Though dstrbutons B and C appear to be more relable than A f we consder not only the probablty of experencng no shortage but the whole probablty dstrbuton they turn out to be probablstcally equvalent. The three of them have an expected value shortage of 8 gallons per day gpd. However these shortage probablty dstrbutons mpose wdely dfferent costs on the consumer as s shown by the dfferent WTP to avod them Fgure 5. More severe shortages encourage more conservaton efforts and costs so the consumer s wllng to pay more to avod these events. To cope wth the shortages mposed by dstrbuton C long-term LT conservaton optons have to be mplemented boostng the consumer s WTP to avod ths stuaton. The cost of LT optons s somewhat a fxed cost as opposed to the short-term optons that are only pad for when there s actually a shortage. Therefore the optmal soluton would often be to mplement short-term ST optons frst and turn to LT optons only f the former do not acheve the levels of water conservaton needed. Ths very smple example provdes groundng to the commonly-accepted noton that frequent small shortages should be preferred to bg but nfrequent shortages and s consstent wth results reported by other authors CUWA 994 Koss and Khawaa. Ths noton s the man ustfcaton for hedgng n reservor management. As dscussed before WTP to avod ratonng decreases as the prce of water ncreases and the expected value EV demand curves converge wth the no ratonng case see Fgure 6. Note also that the EV demand correspondng to dstrbutons B and C are not perfect step functons as one would expect n a lnear optmzaton model. Ths s the result of a shft from short-term conservaton to long-term conservaton n the most extreme event as prce rses. When prce reaches a certan level n ths case around $/ gallons those ST measures that where used because of the water avalablty constrant but where not cost-effectve Dry Shrubs are replaced by permanent measures Tolet Retrofttng or erscape II. The benefts of payng the hgh cost of Dry Shrubs only durng shortages are less than the benefts of extra water conservaton made possble by LT optons reducng both the water bll and the need for the Dry Shrubs measure. Ths example llustrates the relevance of consderng the whole probablty dstrbuton of shortages n estmatng the costs mposed by shortage and hence the value placed n relablty. The dstrbuton of unrelablty between dfferent shortage levels s an aspect of the consumer s WTP that contngent valuaton studes of relablty cannot fully grasp because they only consder one shortage level at a tme. To further llustrate the nteracton between the shortage probablty dstrbuton and the costumer s WTP to avod those shortages lets analyze the examples provded n Tables 67 and 8. Table 6 shows a seres of shortage probablty dstrbutons where the frequency of small shortages gradually ncreases mang each dstrbuton less relable than the prevous. As seen from Fgure 7 the WTP to avod shortages ncreases lnearly wth the EV of shortage and decreases also lnearly wth prce. Therefore for small shortages the cost of movng from no shortage to % shortage s the same as the cost of movng from 4% shortage to 5% shortage. In many cases small shortages can be handled by mplementng only short-term conservaton measures so ncreasngly frequent shortages wll mpose costs that are drectly proportonal to the frequency of occurrence. In contrast for very severe shortages Table 7 WTP does not respond lnearly to ncreases n the level of shortage Fgure 8. In ths case the cost and hence the consumer s WTP of movng from no shortage to % shortage s bgger than cost of gong from to 3 % shortage. Ths non-lnear response s trggered by the need to mplement LT conservaton and ncur fxed costs to cope wth the shortage. When movng from a stuaton wth no shortage to a dstrbuton such as I the household s forced to mplement long-term conservaton measures. After that f the shortage becomes more frequent dstrbutons J and K LT optons wll already be n place but because shortages are more frequent short-term conservaton wll be pad for more often. Ths however represents a relatvely mnor cost when compared to the fxed expendture for long-term optons. As prce ncreases there s decay n ths non-lnear response because LT conservaton becomes cost-effectve and s mplemented regardless of the level of ratonng. For hgh levels

12 of shortage there s hgh value assocated wth avodng severe events and there s comparatvely less value from reducng the probablty of those events. Severe events mae t necessary to mplement LT conservaton that s pad for ndependently of the frequency of occurrence of the event. Fgure 8b shows the very drastc reducton n EV water use mposed by dstrbuton I wth respect to the non-raton case compared to the more mld curtalment n demand needed to absorb the effects of ncreased frequency of shortage. Yet to emphasze the complex relatonshp between the WTP and the shortage probablty dstrbuton the example n Table 8 llustrates that for a partcular consumer there s no WTP to go from dstrbuton L to the more bengn M whch represents a % reducton n the EV of shortage! Ths s because the possblty of havng a very severe event 4% shortage maes LT conservaton measures necessary. For ths partcular set of parameters these LT conservaton measures alone are more than enough to cope wth small shortages of % and no extra short-term measures are requred. Therefore changng the probablty of havng a small % shortage has no effect on costs. LT conservaton wll have to be pad for n any case and there would be no need to pay for short-term conservaton more frequently. Formulaton for a Class of Costumers wth Ratonng The analyss presented above can be extended to a class of costumers. We llustrate the more relevant aspects for ths formulaton by studyng dstrbuton M n the context of a class of costumers. Fgure 9 presents the average demand curves for a class of customers confronted by shortage probablty dstrbuton M. The demand curve for event no shortage clearly llustrates the permanent effects of long-term conservaton reducng water use for all events. The curves correspondng to the no raton case wth both determnstc and probablstc parameters are provded for comparson. By mposng the need for conservaton measures ths ratonng scheme reduces the consumer s prce-responsveness for the ntermedate prce ranges. Statstcal analyss of the Monte Carlo smulaton results of household response to ratonng can provde useful nformaton for the water utlty and help gan understandng about the structure of demands for water and conservaton measures wthn a partcular class of customers. Fgure provdes the derved demand for long-term conservaton measures and shows that as the prce of water ncreases so does the demand for conservaton measures. Fgure shows how mplementaton of a specfc conservaton measure nstallng a tolet dam responds to prce for each of the fve shortage events. Smlar demand analyses can be done usng the reduced costs of non-mplemented conservaton measures whch provdes nformaton about the reducton n cost necessary for a conservaton measure to be mplemented. Fgure shows the hgh degree of varablty n the reduced cost of the erscape II opton from the Monte Carlo results. Ths fgure can be used to answer questons such as how the cost of a gven measure affects ts maret penetraton n the user populaton. Reduced cost of dfferent conservaton measures also can be used to compare the relatve subsdes/refunds needed to mae customers shft from one conservaton measure to another to better accommodate to water supply condtons. Some combnatons of short-term measures mght be more benefcal than others from the water utlty s standpont. For example a water utlty mght want to encourage costumers to shft from Tolet Dams to Dry Lawn n order to reduce pea weeend demand on the system. Curves such as Fgures to can be generated for each conservaton measure provdng the water utlty wth nformaton for the desgn of a water conservaton campagn. Perhaps the most relevant nformaton concerns the range of varablty n wllngness to pay to avod shortage wthn a class of costumers. Fgure 3 shows that for any gven retal prce of water $6/ g n ths case some costumers are wllng to pay more than $947 per year to avod the shortage probablty dstrbuton M whle others are only wllng to pay $67. These dfferences n WTP are derved from the structure of preferences of each costumer costs full servce demand effectveness of the conservaton optons and should be consdered when plannng to as the costumers for funds to ncrease the relablty of the system. The mplcatons of varablty n WTP are very nterestng. Frst t shows that many people would be wllng to pay a premum each bll for a preferental level of relablty. Ths would be le buyng nsurance for protecton durng shortage Flory and Panella 994. The funds generated by the premum

13 rates to fnance long term conservaton or relable supply optons desalnaton plants dry year contracts wth the agrcultural sector etc. One could envson a system of contracts that offered ncreased hgher than standard relablty for customers that are wllng to pay for t and dscount rates for lower relablty.e. nterruptble servce durng drought events. Such servce optons or prorty prcng s already n operaton n the electrc ndustry Flory and Panella 994. In the water ndustry such a prcng scheme would probably be easer to mplement by ntegratng dfferent water use sectors. Industral users or some emergency servce facltes could be offered hgher relablty by establshng dfferent levels of prorty durng drought but at a hgher prce. Other possblty would be establshng a water maret where water ratons could be transferred from the low value users to those who value them the most. Shortages would be allocated to those customers wth lower costs from ratonng creatng some gans n economc effcency. Yet from an operatonal standpont any of these systems mght be dffcult to handle and mportant socal and equty ssues would need to be studed. Formulaton for a Water Servce Area Prce and technology conservaton optons are only some of a complex set of varables that affect the consumer s demand for water. Household characterstcs locaton or ncome are some others Howe 967 Ber 993 Dzegelews et al. 993 Grffn and Melde. The approach can be extended to a whole water servce area by restatng the problem usng dfferent coeffcents for dfferent locatons and classes of consumers. The range of conservaton optons avalable could also be adapted as needed. The servce area can be dscretzed nto groups as homogeneous as possble and the partal results can be aggregated to study the global effect of dfferent parameters and shortage dstrbutons on consumers wllngness-to-pay. Alternatvely we could account for dfferences n the water servce area by ncreasng the degree of varablty n model parameters. Dscusson The approach presented n ths paper enables the dervaton of water demand curves that are consstent wth our current understandng about resdental water use and management. Demand curves for dfferent shortage events can be generated easly and used to study the effects of ratonng on customer s water use. Ths approach should provde valuable nformaton to understand and perhaps predct the effects of the retal water prce and the nteracton of long-term and short-term conservaton measures n water demands. These nteractons are mportant for water utltes because long-term conservaton measures can severely affect water utlty revenues and operaton. Implementaton of long-term conservaton measures entals sgnfcant and permanent reductons n water use that reduce utlty revenue. Ths would lead to prce ncreases lely to mae water conservaton even more attractve to costumers. If approprately planned t also can lead to the mplementaton of emergency puntve water prce agreements to sell the surplus of saved water or to expand the utlty s operatons to new costumers. The man contrbuton of ths model s to provde estmates of the consumer s WTP for changes n relablty n ways consstent wth economc theory and that elmnates the nconsstent results sometmes obtaned n contngent valuaton studes CUWA 994. Another advantage over contngent valuaton studes s ts capablty to examne a complete shortage probablty dstrbuton and the ablty to account for prce effects. However the mathematcal formulaton of the problem s unable to capture the emprcally observed asymmetres n wllngness to pay for ncreased relablty and wllngness to accept for decreased relablty. As Grffn and Melde put t: the change n value for an ncrease n relablty can be expected to be less n absolute value than the change n value for an equvalent relablty fall. Ths asymmetry of ncrease and decrease n relablty may be due to the fxty of durables Grffn and Melde not accounted for n the optmzaton model. It s mportant to emphasze that the wllngness to pay nterpretaton of the results rests on the assumpton that the model s costs coeffcents are estmates of the wllngness to pay of customers to avod 3

14 mplementng specfc water conservaton measures Lund 995 and that the household optmzaton process s costless. The Monte Carlo probablstc optmzaton approach presented has the advantage of provdng nformaton about consumer s varablty n wllngness to pay as well as some relevant nformaton for explorng nnovatve management optons n the resdental sector such as prorty prcng or urban water marets. Ths approach could also contrbute to the desgn of cost-effectve water conservaton programs by usng the nformaton provded by the model maret penetraton estmates mplementatons for dfferent events and prces reduced costs for each measure senstvty analyss to adapt conservaton programs to the characterstcs of each group of customers. The model can be used for example to estmate the effects of a water conservaton campagn launched by a local water agency. Ths can be done by fxng to the extent possble the materal costs of a conservaton measure and lettng the varaton nvolve only the perceved or nconvenence costs to people of mplementng t. Inconvenence costs would bascally nvolve tme spent mplementng the conservaton measure whch can be reduced f the maretng strategy s approprate. The resultng reducton n the model s cost coeffcents s however very dffcult to determne. The approach should also provde a way of ntegratng retal water prce nto studes of the economc mpact of shortage or water resources plannng models that explctly consder urban shortage management Jenns and Lund Hoagland 998. It could be argued that the model s parameters wll vary for dfferent events. Indeed t s lely that the effectveness of short-term conservaton measures wll ncrease wth the severty of the shortage event. Smlarly the cost coeffcents mght be reduced n very extreme events because ncreased customer concern about drought s lely to drve down perceved costs assocated wth conservaton. Approprate long-term montorng of conservaton effectveness and contngent valuaton studes mght provde estmates on ths. There s a long-standng lterature regardng behavoral problems wth cost-mnmzng expected-value decson-mang assumed by ths method Khanemann and Tversy 979. Problems wth the lnear programmng formulaton are dscussed n Lund 995. Fnally the formulaton presented constrans the household to meet the raton level for all events ndependently of ther probablty of occurrence. Ths constrant results n an extremely rs-adverse behavor by the household because even a very small probablty of occurrence.e..% of a shortage event wll force the household to reduce water use and ncur conservaton costs. Further t assumes that households can perfectly montor ther water use and that the water agency can cut water supply to the household when t has consumed ts raton. An alternatve approach would be to ntroduce penaltes for exceedng the raton level perhaps as excess use prces. The approach taen here reles on mcro-scale modelng of consumer demand decsons. Ths requres a great deal of model calbraton and computatonal effort for demand studes of classes of consumers and servce area studes. An alternatve approach would be to use a more sem-emprcal approach such as postve mathematcal programmng Howtt 995; Pars and Howtt 998. Here the quadratc matrx n the obectve functon of a quadratc program or two-stage quadratc program mght be calbrated based on aggregate consumer decsons and water uses ether by customer class or by water servce area. Conclusons The two-stage lnear programmng approach presented by Lund 995 s extended to nclude the retal prce of water and to allow for varablty n the model s parameters for an urban populaton. Ths allows for an easy dervaton of demand curves for water wth or wthout ratonng that are consstent wth the current nowledge about resdental water demands. Derved demands for conservaton measures also can be obtaned usng ths approach. The model provdes nformaton about the nteracton between long and short-term conservaton the retal prce of water and water use n the resdental sector that should prove valuable for water agences to desgn water conservaton programs. Under the assumptons that the model s cost coeffcents represent the consumer s wllngness-to-pay to avod mplementaton of specfc 4