UC Davis UC Davis Previously Published Works

Transcription

1 UC Davis UC Davis Previously Published Works Title Nopoit source pollutio cotrol uder icomplete ad costly iformatio Permalik Joural Evirometal & Resource Ecoomics, 8(4) ISSN Authors Farzi, Y. Hossei Kapla, J D Publicatio Date Supplemetal Material Peer reviewed escholarship.org Powered by the Califoria Digital Library Uiversity of Califoria

2 Nopoit Source Pollutio Cotrol uder Icomplete ad Costly Iformatio Y. H. Farzi* Uiversity of Califoria, Davis Departmet of Agricultural ad Resource Ecoomics ad J. D. Kapla Califoria State Uiversity, Sacrameto Departmet of Ecoomics Abstract We aalyze the efficiet maagemet of opoit source pollutio (NPS) uder a limited pollutio cotrol budget ad icomplete iformatio. We focus o the tradeoff betwee data collectio ad pollutio abatemet efforts by icorporatig iformatio acquisitio ito a NPS pollutio cotrol model. Comparative static results show coditios uder which (i) a favorable chage i the abatemet costs at oe source may lead to a icrease i the treatmet level at all sources, ad vice versa, (ii) a icrease i data collectio cost leads to a icrease i data collectio level, ad (iii) a icrease i the efficiecy of iformatio acquisitio leads to a decrease i the level of data collectio. More importatly, the model simulatios illustrate that acquirig ad eploitig iformatio o heterogeeity of sedimet loadig distributios across pollutig sources leads to a more efficiet budget allocatio ad hece a greater reductio i pollutio damage tha would be the case without such iformatio. JEL Classificatio: D61, D81, D83, Q8 Key Words: opoit source pollutio, ucertaity, costly iformatio, costraied pollutio cotrol budget *Correspodeces to: Y. H. Farzi, Departmet of Agricultural ad Resource Ecoomics, Uiversity of Califoria, Davis, Oe Shield Aveue, Davis, CA 95616, U.S.A., Phoe (530) , Fa (530) , farzi@primal.ucdavis.edu We thak the two aoymous referees of this oural for very helpful commets ad suggestios ad also the participats at aual cofereces of the Europea Associatio of Evirometal ad Resource Ecoomists (EAERE), Oslo, Norway, ad the America Agricultural Ecoomics Associatio (AAEA), Nashville, U.S.A. Farzi thakfully ackowledges a research grat from Giaii Foudatio. Fiacial support for Kapla was provided, i part, by a grat (Proect No. W-887) from the Uiversity of Califoria Water Resources Ceter, ad the Uited States Evirometal Protectio Agecy "Sciece to Achieve Results" Graduate Fellowship Program.

3 Nopoit Source Pollutio Cotrol Uder Icomplete ad Costly Iformatio 1. Itroductio This paper eamies the role of iformatio acquisitio i efficiet maagemet of NPS pollutio. By icorporatig iformatio acquisitio ito a NPS pollutio cotrol model, we focus o the tradeoff betwee abatemet effort ad abatemet effectiveess, a questio that has ot received adequate attetio i the literature. We eplicitly cosider the heterogeeity amog the pollutig sources i the maager s decisio to reduce the pollutio-related damage for a give epediture o abatemet activity. We aalyze how abatemet cost, data collectio cost, ad efficiecy of etractig iformatio from collected data affects the efficiet budget allocatio betwee iformatio acquisitio ad abatemet effort across pollutig sources, ad also elaborate o the policy implicatios. I the aalysis NPS pollutio is defied as pollutio from diffuse sources where the iformatio o the likage betwee pollutig sources ad ambiet load is icomplete. The pollutio maager observes total ambiet load or the cosequet damage but is uable to detect with certaity the pollutio from idividual sources. This icomplete iformatio (ucertaity) about the pollutio loadig creates iefficiecies i allocatio of abatemet effort across the sources. The pollutio cotrol maager depicted i this model reduces pollutio loadig ucertaity by obtaiig iformatio through data collectio. The maager updates her subective prior distributio about the pollutio loadig with the acquired iformatio, resultig i a posterior distributio that improves abatemet effectiveess by allowig the maager to reallocate abatemet effort to sources with relatively larger epected pollutio loadig, all else the same. However, the maager evetually faces a eplicit tradeoff 1

4 betwee the scale of abatemet effort ad abatemet effectiveess because data collectio is costly ad the maager is fiscally costraied. We cosider, for eample, the case of sedimet loadig from forestlad i Redwood Natioal Park, located i orthwester Califoria. Sedimet loadig mostly occurs durig high storm evets, (i.e., whe raifall itesity is high ad storm duratio is short), whe storm ruoff overflows stream chaels at road crossigs, causig sedimet to eter tributaries as the ruoff returs to the chael dowstream. 1 The sedimet that eters the waterways i Redwood Natioal Park fills i salmo spawig pools, thus reducig the umber of available spawig sites. This sedimet also fills i the stream chael upstream ad adacet to the Tall Trees Grove, home of the world s tallest trees, icreasig the icidece of floodig, bak erosio ad saturatio of the root zoe, which all cause the tall trees to topple (Sprieter, Frake ad Steese, 1981). If perfect iformatio was available o sedimet loadig attributable to each pollutio sources (the loggig roads), park maagers could allocate their etire sedimet cotrol budget to abatemet effort. However, with icomplete iformatio, the maagemet of sedimet loadig requires a eplicit allocatio of resources betwee iformatio collectio ad abatemet. The results draw from this aalysis ca, i geeral, shed light o the role of iformatio ad budgetary costraits o the efficiet maagemet of NPS pollutio problems such as groudwater cotamiatio, greehouse gas emissios ad acid rai, which are characterized as pollutio geerated from diffuse sources. This research has implicatios for other targetig programs as well. Babcock et al. (1996, 1997) cosider targetig optios uder the USDA Coservatio Reserve Program. However, i these aalyses, there is o eplicit provisio of

5 iformatio that allows the budget maagers to improve the allocatio of limited resources, thereby icreasig overall evirometal beefits. The pollutio cotrol literature typically looks at market-based approaches to cotrollig NPS pollutio. However, market-based approaches are ot relevat whe a private idividual or public maager must decide where o their property (or the property they maage) to cocetrate pollutio abatemet efforts (i.e., target abatemet resources). A idividual polluter may ot face a eplicit budget costrait but a public maager is usually limited i her abatemet decisios by a fied aual budget. Previous research o NPS pollutio cotrol has also focused o the social welfare optimizatio problem without regard for fiscal costraits. For istace, Cabe ad Herriges (199) cosider the ucostraied social plaer problem of NPS pollutio cotrol, icorporatig ucertaity ad iformatio acquisitio, where iformatio reduces the social cost of settig a cotrol mechaism through a ambiet ta, as proposed by Segerso (1988). Elsewhere, Xepapedeas (1995) eamies the ucostraied social plaer s use of a effluet ta, i couctio with a ambiet ta, as a icetive for idividual polluters to reveal iformatio useful i ucoverig the coectios betwee geerator ad ambiet emissios. I the specific case of sedimet loadig, the budget-costraied maager caot impose fees agaist ature to lear more about each source's cotributio to the total ambiet load. The maager must eped limited resources to obtai the iformatio that otherwise could have bee obtaied by the social plaer through the use of effluet fees. Furthermore, the optimal decisio 1 The primary focus of erosio cotrol i Redwood Natioal Park is prevetig or reducig erosio from loggig roads withi the Park (DOI 1981). It is well kow that loggig roads are the mai cotributors to sedimet loadig (Mout 1995; GAO 1999; EPA 1999). The Europea Evirometal Agecy (000, 001) ad the US-GAO (1999) report that a lack of fiacial resources limits the ability of public evirometal maagers to achieve their core obectives. 3

6 rules derived i a social welfare framework differ from those derived from the costraied maagemet approach ad lead to differet policy prescriptios (Barrett ad Segerso 1997). Garvie ad Keeler (1994) cosider a fiscally costraied regulator who miimizes ocompliat pollutio geeratio by allocatig a limited budget betwee data collectio ad eforcemet. Data collectio provides evidece ecessary to prosecute, ad, as such, improves eforcemet effectiveess. Our focus is o the role of acquired iformatio i reducig pollutio loadig ucertaity, which improves abatemet effectiveess. We also evaluate the miimizatio of epected cost of evirometal damages whereas Garvie ad Keeler cosider miimizig pollutio irrespective of the related damage. Uless the damage fuctio is liear the result will vary betwee these two obectives. The rest of the paper is structured as follows. Sectio develops a model to aalyze sedimet cotrol whe iformatio is icomplete ad data collectio is costly. Sectio 3 provides the comparative static results for key parameters of the model. Sectio 4 presets the results of simulatig the model for a case resemblig the public maagemet of sedimet loadig i Redwood Natioal Park, located i orthwester Califoria. There, we compare the allocatio decisios of a perfectly iformed maager, a completely uiformed maager, ad a imperfectly iformed maager who acquires iformatio through data collectio. Sectio 6 cocludes.. A Model of NPS Pollutio Cotrol with Costly Iformatio Acquisitio Faced with budget ad iformatio costraits, the maager chooses betwee the level of abatemet effort at each source ad data collectio, where the iformatio acquired through data collectio reduces pollutio loadig ucertaity. We model the decisio to acquire iformatio 4

7 ad abate as a sequetial problem sice each activity occurs durig separate periods throughout the year. For the Redwood Natioal Park case, data (stream flow ad ambiet sedimet load measures) are collected durig the rai seaso betwee October ad April. The abatemet proects begi i late summer ad ed before the rai seaso begis agai. These decisios are liked by a sigle budget, which is allocated over both periods. 3 There is o discoutig of the budget give the short duratio of time that epires betwee allocatig epeditures for iformatio acquisitio ad abatemet effort. I this formulatio, we look directly at the tradeoff the maager faces betwee abatemet effort ad abatemet effectiveess, where the latter depeds o iformatio about pollutio loadig across the sources. The iformatio acquisitio or sequetial updatig process requires the maager to make a decisio e ate (prior to the realizatio) o the data collectio frequecy. 4 This e ate decisio is made usig a prior epectatio o the iformatio cotet of a give data collectio frequecy. The epected iformatio cotet is simply the epected reductio i ucertaity about the pollutio geerated by each source. 5 Iitially, the maager chooses the frequecy of data collectio o total sedimet loadig ad stream flow ad updates the prior subective sedimet loadig distributio for each of the sources. I practice, the maager, prior to the begiig of the rai seaso, determies a fied umber of daily samples to collect at each data collectio poit throughout the rai seaso. We 3 The separability of the iformatio acquisitio ad abatemet effort decisios is ot uique to the eample of sedimet loadig i Redwood Natioal Park. Take for eample, self-reportig provisios for the USDA EQIP policy (Cattaeo, 001). Here, the govermet solicits iformatio from agricultural producers o farmig practices that coveys the evirometal beefits derived from implemetig such practices. This iformatio acquisitio caot occur simultaeously with the decisio to allocate resources toward evirometal beefits. 4 Sequetial updatig typically results i sub-optimal decisio-makig whe observed e post. That is, if we kew yesterday what we kow today, the the optimal decisio we made yesterday could have bee improved upo. 5 The model preseted here eamies a sigle roud of decisio-makig process i a two-period sequece. Although iformatio acquisitio ad learig are dyamic pheomea, the two-period model we preset ca capture the iter-temporal decisio-makig process by repeatig the two-period model. The sigle roud of decisio makig demostrates the theoretical uderpiigs of the problem. 5

8 deote the frequecy of data collectio by. A icrease i the frequecy of data collectio implies that a greater umber of samples are collected each day throughout the rai seaso. Sice the maager is costraied by a fied budget, B, the maimum umber of possible data collectio frequecies is equal to B/m, where m is the per frequecy cost of data collectio. This costrait ( B / m ) ever bids sice damages caot be cotrolled without abatemet at some sources. Note, however, that the reverse is ot true. Net, havig collected data, abatemet effort levels X (,,, ) = for the N 1 L pollutig sources are determied so as to miimize the epected cost of evirometal damage, give the updated posterior sedimet loadig distributio. The damage cost fuctio D (Q) is twice cotiuously differetiable, icreasig ad cove i Q, the ambiet pollutio (i.e., N D Q > 0 D ad > 0 ). Q Let, q be the uobservable toage of pollutio loadig from the th source, where = 1,, L, N. We defie q = q ; w, ( )) as a fuctio of abatemet effort, stochastic ( raifall ( w ) ad site-specific characteristics ( ) that defie the relatioship betwee abatemet ad raifall o pollutio loadig at that source. 6 Followig Shortle ad Alber (1997), ucertaity is itroduced ito the problem by takig these site-specific characteristics to be ucertai. The icomplete iformatio about, is eplicitly icorporated ito the defiitio of sedimet loadig by allowig site-specific characteristics to deped o, where low values of are less certai tha high values. This icomplete iformatio does ot directly affect the sedimet 6 This characterizatio of pollutio loadig is similar to prior models of stochastic opoit source pollutio cotrol (see Beavis ad Walker; Shortle ad Du; Shortle; ad Hora, Shortle ad Abler amog others). However, i our iterpretatio of the model is the maager s iformatio or kowledge about site-specific characteristics for the th source rather tha the th firm s private kowledge as depicted i Shortle ad Alber (1997). 6

9 loadig put does affect the margial productivity of abatemet effort (i.e., the margial abatemet effectiveess). The pollutio loadig fuctio has the followig properties q q < 0, > q > 0, w q > 0, = 0 < ad > 0. The observable ambiet pollutio load, Q, is defied such that Q q. 7 Whe decisios are made o allocatig resources to abatemet effort, the maager uses ) the post-data or posterior coditioal distributio for ( w stochastic raifall give the ucertaity about pollutio loadig. We shall fully icorporate data collectio ad iformatio acquisitio ito the maagemet model shortly. For ow, assume kowledge is fied so that we ca derive the maager's optimal abatemet decisio. The total abatemet cost epediture, C, is defied as C = c. Recall that the per frequecy cost of data collectio, m, is also assumed to be costat so that M = m represets the total data collectio epediture. These liear cost specificatios allow us to focus attetio o the tradeoff betwee abatemet effort ad abatemet effectiveess. The budget costrait is c + m B (1) Give a fied level of data collectio, ad subect to equatio (1), the maager chooses abatemet effort across the sources to Miimize ED( Q), L, 1, N () where E is the epectatio operator for the posterior coditioal distributio for stochastic raifall. The first order coditios for a iterior abatemet effort allocatio are (1) ad 7 We assume a liear specificatio sice iteractios betwee sources are egligible i our eample. However, this simplificatio caot be maitaied as a rule (Liter ad Weersik, 1999). 7

10 8 N c q Q D E,.., 1,, 0 = = + (3) where is a Lagragea multiplier (the shadow price of the budgeted resources). 8 From (1) ad (3), we obtai the optimal abatemet effort allocatios ))) ( ( ( ~ 1 N, w,m,,c,,c c L =, where ~ maps the parameters of the model ito the optimal abatemet effort. Coditios (3) simply state that, at the optimum, the maager chooses abatemet effort at each source such that the epected margial reductio i the cost of damages (i.e., the epected margial beefit from abatemet effort) is equal to the margial cost of abatemet effort. We ca rewrite (3) as N c c q Q D E q Q D E, = 1,,.., = (4) or N c c q Q D q E Q D E q Q D q E Q D E 1,,..,,, cov, cov = = + + (4') which recasts the optimality coditio i the form of the familiar requiremet that the epected margial rates of trasformatio across ay two sources should equal the relative margial abatemet costs (the poit at which the budget costrait ad the iso-epected damage curve are taget). Equatio 4 illustrates that the optimality coditio with oliear damage costs differs 8 Give desirable curvature properties of the damage cost fuctio, we assume the secod order coditio holds at the miimum.

11 from the liear damage fuctio ad the solutio to miimizig a evirometal goal such as pollutio loadig, where the optimality coditio simplifies to q E q E c = c, = 1,,.., N (5) Before formally derivig the optimizatio problem whe iformatio is acquired through data collectio, let us eamie the mechaism by which abatemet effort allocatios are affected by data collectio ad acquired iformatio. If, for eample, source s actual cotributio to the ambiet load is greater tha epected ad s cotributio is less tha epected, give abatemet effort ad raifall, the data collectio chages ) such that for a give abatemet effort ( w the epected loadig from the th source icreases while it decreases at the th source. I this eample sources ad were selected arbitrarily from amog the N sources but i actuality, loadigs from ay two sources may be greater tha or less tha epected loadigs, a priori. However, sice the total ambiet load is fied for a give abatemet allocatio ad raifall evet, it must be the case that if oe source s load is greater tha previously epected the at least oe other source s load must be less tha previously epected. We have simply characterized this eample. A reeamiatio of equatio 4 reveals that data collectio icreases the umerator ad lowers the deomiator o the LHS, so that the LHS of (4') rises. To restore the equilibrium associated with a larger value of, we must have d d > 0 ad < 0 d d (these q coditios follow from the assumptio > 0 ). So, there will be a reallocatio of the abatemet efforts from source to source. But this is for a uchaged abatemet budget. Sice 9

12 a icrease i reduces the budget for abatemet activity, oe has to cosider the et effect of a chage i by differetiatig both (1) ad (3) with respect to. To formally model iformatio acquisitio, we assume the iformatio acquired through data collectio allows the maager to update ˆ, = 1,, L N, the prior subective probability, distributio for pollutio loadig at each source ad derive ( ; ); ˆ ), the posterior ( probability distributios for pollutio loadig at each source that is closer to the true uderlyig distributio. I this cotet, "closer" refers to the otio that the iformatio cotet of the posterior distributio is closer to the iformatio cotaied i the true, yet ukow, distributio. The parameter reflects the efficiecy of iformatio acquisitio. Whe data is collected, the rate at which the epected pollutio loadig at each source is updated toward the true uderlyig loadig values icreases as icreases. I essece, represets the etet of the maager's skill ad ability i utilizig the collected data to etract iformatio. This otio of ability is i keepig with Arrow (1974, pp. 37) who states that each idividual has the ability to receive a sigal from atural ad social eviromets. However, it is the limited capacity ad scarcity of iformatio-hadlig ability that sets idividuals apart. Learig capacity may be ehaced through eogeous meas such as techological advaces i data collectio or educatio programs ad thus we are iterested i how chages i capacity may affect the optimal data collectio strategy. Returig to the pre-data collectio problem, the maager chooses the level of data collectio that miimizes the epected damage cost from pollutio loadig. Substitutig the optimal abatemet fuctios ( ~ ) derived above ito (1) ad (), the e ate optimizatio problem is formally writte as 10

13 Miimize ED q( ~, w, ( ( ; ))) (6) s.t. c ~ + m B (7) It should be oted agai that implicit i Equatio (6) is the assumptio that the maager has a epectatio about how data collectio affects her subective probability distributios about pollutio loadig, ad uses this prior kowledge to choose the data collectio frequecy that miimize the epected cost of pollutio related damage. The first order coditios for the optimal level of data collectio are (7) ad D E Q q ~ ~ = + ~ m c (8) where is the Lagragea multiplier o the e ate allocatio of the pollutio cotrol budget. Notig that iformatio acquisitio is aki to provisio of a collective good, coditio (8) has a straightforward iterpretatio. That is, the maager optimally allocates resources to data collectio so that the epected margial reductio i the cost of damages over all sources (the LHS of (8)) equals the total margial opportuity cost of acquirig iformatio (the RHS of (8)). The total margial opportuity cost of acquirig iformatio cosists of a direct cost of a additioal uit of iformatio, m, ad a idirect cost (or beefit) give by the epressio ~ c which reflects the effect of a additioal uit of iformatio o the abatemet epediture by causig a reallocatio of abatemet effort amog the various sources. Solvig equatio (8) yields = ˆ ( m,, B), the optimal allocatio of data collectio, where ˆ maps the parameters ito the optimal data collectio allocatio. Now substitutig this optimal allocatio of data collectio ito the optimal abatemet allocatio yields 11

14 = ˆ ( ˆ( m,, B), c, c, m, B, ( w, ( ˆ( m,, B)); )) where ˆ is the optimal abatemet effort that maps the parameters of the data collectio problem ito the optimal level of abatemet effort. 3. Comparative Static Aalysis The comparative static results for the costly iformatio acquisitio model are derived i the stadard maer. Because of the sequetial ature of this problem, the timig of ay cost chage determies the relevat comparative static result. For eample, if a chage i the state of abatemet costs occurs after data collectio, the the maager ca oly chage the optimal allocatio of abatemet across the sources. I this sectio we first cosider the effect of a chage i abatemet cost ( c ) o the optimal abatemet levels. We the cosider the effect of a chage i the data collectio cost (m) ad the iformatio efficiecy () o the optimal data collectio frequecy (ˆ ). Note that ˆ is uaffected by c as data collectio precedes abatemet activity. The sig of the comparative static ~ c is ambiguous ad depeds o how abatemet epediture at source chages i respose to a chage i c. The margial cost of abatemet is likely to eperiece favorable chage with advaces i techology ad ufavorable chage due to such evets as ladslides withi the abatemet area, or regulatio o abatemet effort to preserve edagered species for istace. The chage i abatemet epediture at source c ~ depeds o the magitude of the ow cost elasticity of abatemet effort (!, c = ). c 1

15 Differetiatig the budget costrait (7) with respect to a chage i the per uit abatemet cost at source ( c ), we derive: ~ sg = sg(!, " 1) c c (9) Propositio I: If! 1, a icrease i the per uit abatemet cost at source will result i a c, < icrease i abatemet epediture at that source ad hece a decrease i the abatemet epeditures at the remaiig sources ad vice versa. The ituitio for Propositio I is straightforward ad eeds o further eplaatio. This respose to a chage i the per uit abatemet cost at oe source may occur at oe other source or be distributed across multiple sources. Propositio I implies that regulatio aimed at protectig a species i oe area ca, by raisig the cost of pollutio abatemet i that area, also egatively affect cotrol efforts i other areas withi the watershed, with the uiteded cosequece of raisig epected damage costs from pollutio. To capture the trade-off betwee data collectio ad abatemet effort we tur to the comparative static result for a chage i the data collectio cost, m. The et effect o from a chage i m ca be decomposed ito two separate effects. A icrease i m causes a reductio i e ate. I tur, this reductio i geerates a chai of secodary effects by chagig the epected sedimet loadig for give abatemet effort levels at each source. Sice this secodary effect occurs before abatemet efforts are actually chose the abatemet effort levels ca chage to eploit the gais i abatemet effectiveess geerated by the icrease i iformatio. Chages i abatemet effort i tur chages the abatemet epediture, ad hece the resources available 13

16 for iformatio acquisitio, ad therefore. This chai of effects is depicted below, where $ deotes chage i a variable: ( + ) (") $ m'$ (, ): $ [ E( q ), E( q )]'$(, ) & $ m + m $ = $ ( m) = % "$ ( c + c ) 1 1 We formally epress this result by ˆ sg = " sg( + m ˆ c )( m+ m ˆ c ) (10) The sig of this comparative static derivative is ambiguous. The secod parethesized term o the right-had side (RHS) of equatio (10) is the margial cost of data collectio, which is positive from the first order coditio (8). The sig of the first parethesized term o the RHS of (10), which we term the "sequetial effect," is however ambiguous. The sequetial effect represets the tradeoff betwee higher data epediture, through the chage i data collectio level, ad lower abatemet epediture, through chages i the abatemet effort levels, s. I geeral, the effect of the chage i data collectio o each idividual source s abatemet level is ambiguous, but total abatemet epediture icreases (decreases) with a icrease i data collectio cost, m, if period 1 data collectio is cost elastic (ielastic). Thus, from (10) we ca state the followig propositio: Propositio II: If data collectio is sufficietly cost ielastic, so that with a higher cost of data collectio (m), the decrease i abatemet epediture is larger tha the icrease i data collectio epediture, the the sequetial effect is egative, thus iducig a icrease i the level of data collectio. 14

17 At first, this result seems couterituitive. We would epect less data collectio whe the cost of data collectio icreases. But, because of the sequetial effect of data collectio, i the form of a more efficiet abatemet allocatio, there will be efficiecy gais i the form of et savigs o abatemet costs, which allow the maager to icrease data collectio efforts e ate. To better appreciate the result stated i Propositio II, we should bear i mid that the chage i abatemet levels at various sources whe data collectio chages i respose to a chage i its cost (m), depeds o the size ad directio of chages respectively i the abatemet budget ad i abatemet productivity effects. The budget effect is a declie i abatemet levels at all sources because the resources spet o data collectio must be take from the same give budget. I other words, resources spet o data collectio are uavailable for abatemet. The abatemet productivity effect is a declie i the productivity at sources with lower posterior epected sedimet loadig ad a rise i the productivity at sources with higher posterior epected sedimet loadig. Thus, Propositio II is more likely to hold whe, assumig liear costs, (i) the posterior distributio is highly sesitive to data collectio (i.e., farther away is the prior distributio from the uderlyig true distributio), ad (ii) the margial abatemet productivity is very sesitive to sedimet loadig. Propositio II has a importat policy implicatio. It cautios us, for eample, that a policy subsidizig data collectio to reduce the ucertaity about pollutio flows from various sources, ad thereby ehacig the efficiecy of abatemet programs, may lead to the opposite result by shiftig resources away from data collectio to more abatemet activity. We have also eamied the effect of a chage i the productivity of iformatio acquisitio () o data collectio. This is give by 15

18 ˆ sg = " sg( c ˆ )( m+ c ˆ ) (11) We agai obtai a ambiguous result. We refer to the first term o the RHS of (11) as the "iformatio efficiecy effect" sice it reflects the chage i abatemet epediture resultig from a chage i the efficiecy of iformatio acquisitio, give the frequecy of data collectio ( ). Whe the efficiecy of iformatio acquisitio icreases, the curvature coditios o the abatemet fuctios esure that, over a iterval of abatemet levels (, ), the abatemet epediture at the th source icreases while it decreases at the th source. Recall that we have assumed that the sedimet loadig from the th source is greater tha epected e ate ad vice versa for the th source. This leads to the followig propositio, which highlights the trade-off betwee the level ad efficiecy of iformatio acquisitio. Propositio III: If a higher iformatio acquisitio efficiecy () raises the abatemet epediture at source by less tha it reduces the abatemet epediture at source, the the iformatio efficiecy effect is egative, thus iducig a icrease i the level of data collectio. Oe might ormally epect, give dimiishig margial productivity of iformatio, a maager who is more efficiet i etractig iformatio from collected data would take advatage of that skill ad, everythig else equal, opt for less data collectio. Propositio III idicates the coditio uder which the opposite occurs. I such cases there is a tradeoff betwee iformatio acquisitio efficiecy ad the itesity of data collectio, with a possible cosequece of shiftig resources from abatemet activity to data collectio. A policy implicatio of Propositio III is that a program aimig to improve pollutio maagers 16

19 kowledge ad skills i iformatio acquisitio may come at the cost of reduced abatemet activity. 4. Model Simulatio: Heterogeeity of Pollutig Sources ad the Value of Iformatio To provide a umerical illustratio, we simulate a simplified pollutio cotrol model based o Redwood Natioal Park s sedimet cotrol program for Redwood Creek. Overall three separate models are simulated. Model I presets the case of a perfectly iformed maager (PI), Model II is that of a uiformed maager (UI) who is assumed to believe that the sedimet loadig is uiform across all pollutig sources, ad Model III is the case of a data-collectig, imperfectly iformed maager. These simulatios highlight the value of iformatio ad the ecoomic tradeoff betwee abatemet effort ad abatemet effectiveess whe iformatio is acquired. To evaluate the value of iformatio we compare the optimal abatemet decisio i model I ad II over a rage of scearios about the heterogeeity of sedimet loadig across sources. I each model two sources geerate sedimet loadig. For each heterogeeity sceario, the damage costs attributable to the hypothetical assumptio of a prior uiform distributio o sedimet loadig is compared with the damage costs uder a perfect iformatio assumptio i order to derive the value of perfect iformatio. The secod set of simulatios evaluates Model III to determie the optimal level of data collectio ad abatemet ad reveals the ecoomic tradeoff betwee abatemet effort ad abatemet effectiveess whe iformatio is acquired. I these latter simulatios we solve the model for the optimal data collectio level that miimizes epected cost of damages ad illustrate the respose of optimal data collectio to a chage i the cost of data collectio ad the iformatio acquisitio efficiecy. Give the assumptio of dimiishig margial returs to 17

20 iformatio, complete resolutio of ucertaity about the sedimet loadig geeratio by source may etail a ifiite amout of data collectio, which is prohibitively costly. Thus with data collectio, it must be the case that damages are less tha those resultig from a uiformed maager's abatemet decisio but greater tha those arisig from the decisio of a perfectly iformed maager. To facilitate the simulatio, the model is calibrated usig estimates derived i Kapla (1999). First, the damage cost fuctio is l( Damage($)) = " l(Q) (1) The fuctioal form for sedimet loadig is adapted from a kow physical relatioship relatig ambiet sedimet loadig with stream flow such that 9 q q = ( flw ) Q = (13) s where (flw ) is the average stream flow measure geerated from withi each of the pollutig sources, which is a proy for stochastic raifall, ad s is the sedimet loadig parameter for the th source. For illustrative purposes ad based o empirical evidece from Kapla, Howitt ad Farzi (003), we defie the relatioship betwee the sedimet loadig parameter ad abatemet effort as follows: s = 1" To costruct the uderlyig stream flow values for each source for each heterogeeity sceario, we applied the followig formulas: flw 1 = (1+0.1*h)*140,768, ad flw = (1-0.1*h)*140,768 where h = 0,1,,9, ad give that the average stream flow for Redwood Creek is approimately 140,768 cubic yards. For computatioal coveiece ad without alterig the qualitative results, we use h to be the sceario umber. We vary the heterogeeity of stream flow from each source to reflect a few of the possible sedimet loadig distributios that ature imposes o the system. 18

21 I costructig q, the uobservable sedimet loadig at each source ad the ambiet sedimet load Q we substitute the source specific sedimet loadig parameter ad stream flow equatios ito (13). Net, the abatemet cost fuctio was obtaied from Kapla (1999), where the abatemet cost fuctio for each source is estimated as a liear fuctio of abatemet level C = c e, + ad abatemet is measured as the umber of haul roads removed. To focus attetio o the role of data collectio i icreasig the efficiecy of abatemet effort ad thus the trade-off betwee data collectio ad abatemet epeditures we assume the abatemet cost coefficiet is idetical across sources ad estimated the coefficiet with least squares without a itercept term for obvious reasos. The value of the estimated cost coefficiet is with t-statistic of ad a R of The aual budget is fied at $00,000, which is the average aual abatemet budget for removig haul roads i Redwood Natioal Park. Table 1 presets the optimal abatemet levels for the perfectly iformed maager uder the various heterogeeity scearios. I this case the maager kows q, the actual sedimet loadig from each source. To derive the optimal level of abatemet uder perfect iformatio we miimize (1) subect to (13) ad the abatemet cost fuctio defied above. For the uiformed maager (UI) who assumes a prior uiform distributio over all heterogeeity scearios, the optimal abatemet level for each source, for all scearios is ( 1 = 57.7, = 57.7). This correspods to the optimal abatemet levels chose by the perfectly iformed maager if the pollutig sources were i fact homogeeous. This is because, give all the same iformatio o costs ad damages ecept the true heterogeeity of the sources, the maager will make the same decisio whe the true sedimet loadig distributio is uiform. 9 This fuctioal relatioship ca be foud i Kapla ad Howitt (00), ad Sigh ad Krstaovic (1987). 19

22 Table 1. Optimal Abatemet uder Perfect Iformatio Sceario Table presets the resultig damage costs uder PI ad UI cases. Colum i Table shows the damage cost correspodig to the optimal abatemet levels whe the maager has perfect iformatio about the distributio of ambiet load across sources. Colum 3 shows the damage cost resultig from a uiformed budget allocatio. I both cases the reductio i damage costs whe abatemet is udertake, compared to the o abatemet case, eceeds the $00,000 spet to cotrol sedimet loadig. The greater damage cost for the uiformed model is a result of the maager's lack of iformatio with respect to true sedimet loadig. Whe the maager has perfect iformatio less damage costs results because the maager eploits the kowledge about the degree of heterogeeity to allocate the budget more efficietly. This case of a perfectly iformed maager is aalogous to poit source pollutio cotrol sice there is o ucertaity about the pollutio geerated from each source. Colum 4 shows the value of perfect iformatio (VPI) for each sceario as the differece betwee damage costs uder UI ad PI. These costs are respectively the upper ad lower bouds for possible damage costs whe iformatio is optimally acquired. Furthermore, the last colum of Table shows that the margial value of perfect iformatio icreases as the differece betwee the true distributio 0

23 ad the prior (uiform) distributio grows, i.e., as heterogeeity rises, ecept i the last two scearios due to the corer solutio result. 10 Table. Damages for Model I ad II Sceario PI UI VPI=UI-PI Margial VPI 0 657, , ,70 657,185 5,483 5, , ,886 1,98 16, , ,689 49,65 7, , ,541 88,765 39, , , ,74 50, , ,000 03,196 63, ,33 634,49 80,016 76, ,96 65, ,774 87, ,47 614,45 443,05 75,431 I the secod set of simulatios, where costly iformatio acquisitio is evaluated, we limit the heterogeeity to sceario 9; that is, we assume that this sceario represets the true sedimet loadig heterogeeity at the two sources. Selectio of ay other sceario does ot alter the results. I this set of simulatios, we stipulate differet values for m, the data collectio cost, ad for, the iformatio efficiecy parameter (see Table 3 for the assumed values). Ay data collectio epediture also comes from the iitial budget of $00,000 thereby reducig the resources available for abatemet effort. To icorporate iformatio acquisitio ito the maager's obective fuctio, we substitute the sedimet loadig parameter i equatio (13) with the epected sedimet loadig parameter fuctio E ( s ) = 1" ( A( )), where A ( ) = 1" (1 " ), 0 A ( ) < 1 ad 0 < < 1. Abset ay prior fuctioal forms for A ( ) we costructed this correspodece from the desired curvature properties (i.e., 10 Whe the pollutio load reaches higher levels of heterogeeity the abatemet budget is allocated to abatemet at oe source oly. Above this level of heterogeeity the margial returs to perfect iformatio declie because the maager, ca o loger reallocate abatemet effort ad thus caot take advatage of the better iformatio. 1

24 A( ) A( ) A( ) > 0, < 0, > 0 A( ) ad < 0 ). This ew epected sedimet loadig parameter equatio asymptotically approaches certaity (i.e., A ( ) ' 1 as ' ) ). This epected sedimet loadig parameter ecapsulates both the abatemet effort ad the ucertaity about the relatioship betwee stream flow ad abatemet effort as discussed i the theoretical sectio. Table 3 presets the optimal data collectio, ad epected ad actual damages for the low ad high ( =0.6, 0.8) iformatio efficiecy scearios. The divergece betwee the epected ad actual damage, reported i Table 3, is a result of the maager ot havig perfect iformatio whe choosig abatemet epeditures across sources. If the maager were perfectly iformed, the the epected ad actual damages would coicide. As Colum 4 ad 7 show, whe it is optimal to acquire iformatio, the actual damage lies i betwee the etreme cases of UI ad PI abatemet, where actual damages are $614,45 ad $171,47 respectively (see sceario 9 i Table ). These results show the importat role iformatio acquisitio ca play i improvig the budget allocatio ad hece reducig the epected damage whe compared with the case of the e ate, uiform prior distributio. Whe, uder heterogeeity sceario 9, data is optimally collected, the actual damage cost is always lower tha the actual damage (D = $614,45) that would result from the allocatio of the budget oly to abatemet efforts ( = 0 ) uder the uiformed maagemet. It is should be oted that this reductio i damage costs uderstates the actual beefits of iformatio acquisitio sice, i practice, the beefit of iformatio spills over to a much loger time period, a cosideratio ot accouted for i this eample. Comparig the optimal levels of data collectio ( ) uder the two iformatio acquisitio efficiecy scearios, we see that a icrease i the iformatio acquisitio efficiecy

25 has a ambiguous effect, which is cosistet with Propositio III. I particular, we see that whe m = $90,000, a icrease i from 0.6 to 0.8 leads to a higher level of data collectio ( icreases from 0 to 1). This ca be eplaied ituitively by otig that the effect of higher iformatio efficiecy is like lowerig the cost of iformatio. Coupled with a high margial retur o the first uit of data collectio, this reders it optimal to collect data. O the other had, whe cost of data collectio is as low as m = $15,000, so that it is optimal to collect iformatio with relatively high frequecy ( = 3 for = 0.6), the margial retur o iformatio acquisitio is relatively low. Together with high efficiecy of iformatio acquisitio, this makes it optimal to reduce the frequecy of data collectio to = whe = 0.8. Reducig data collectio to = allows the maager to sped more o abatemet effort where the margial retur (i terms of lowerig the epected damage cost) is relatively high. Table 3. Optimal Data Collectio, ad Epected ad Actual Damages ($) =0.6 =0.8 m E(Damage) Damage E(Damage) Damage $90,000 0 $657,615 $614,453 1 $84,576 $599,97 $50,000 1 $816,53 $33,650 1 $515,09 $33,650 $15,000 3 $356,754 $310,063 $76,776 $5, Cocludig Remarks This paper has eamied the problem of NPS pollutio cotrol uder icomplete ad costly iformatio. We have aalyzed the problem withi a costraied maagemet framework to brig to light a more realistic settig for studyig NPS pollutio cotrol. The comparative static results showed the coditios uder which (i) the maager may lower abatemet efforts at all sources whe a ufavorable chage (e.g., stricter evirometal regulatios or adverse atural evets) causes the abatemet cost to go up at oe specific source, ad vice versa (Propositio I), (ii) data collectio effort may icrease despite a rise i data collectio cost (Propositio II), ad 3

26 (iii) a higher iformatio efficiecy ca lead to less data collectio (Propositio III). The model simulatio results showed that by eploitig the kowledge of sedimet loadig heterogeeity across the pollutig sources, the maager ca improve the overall efficiecy of budget allocatio to abatemet efforts ad thereby further reduce pollutio damages. Of may possible etesios of the preset study, we believe that studyig the problem i a dyamic settig ca be particularly isightful. Over a fiite time horizo, the maager chooses ivestmet paths for both iformatio acquisitio ad abatemet efforts. Durig this time horizo, several factors will ifluece the dyamics of each path. Pricipal amog these factors is the declie of the productivity of iformatio acquisitio as ucertaity about the degree of heterogeeity is reduced. This suggests that the maager may fid it optimal to decrease data collectio ad iformatio acquisitio over time. Secodly, as abatemet at the source with the largest sedimet load occurs early i the time horizo, the system will become icreasigly less heterogeeous. With a decreasig heterogeeity of sedimet loadig over time, we epect that at some future date the abatemet policy will chage from a heterogeeous abatemet strategy to a homogeeous oe. Future research should shed light o these issues. 4

27 Refereces Arrow, K.J., The Limits of Orgaizatio. New York: W.W. Norto ad Co., Babcock, B.A., Lakshmiaraya, P.G., Wu, J., ad D. Zilberma (1996), The Ecoomics of Public Fud for Evirometal Ameities: A Study of CRP Cotracts, America Joural of Agricultural Ecoomics 78: (1997), Targetig Tools for the Purchase of Evirometal Ameities, Lad Ecoomics 73(3): Barrett, J., ad K. Segerso (1997), Prevetio ad Treatmet i Evirometal Policy Desig, Joural of Evirometal Ecoomics ad Maagemet 33: Beavis, B., ad M. Walker (1983), Achievig Evirometal Stadards with Stochastic Discharges, Joural of Evirometal Ecoomics ad Maagemet 10: Cabe, R., ad J.A. Herriges (199), The Regulatio of Nopoit Source Pollutio uder Imperfect ad Asymmetric Iformatio, Joural of Evirometal Ecoomics ad Maagemet, Cattaeo, A. 001, EQIP: Coservig While Farmig, Agricultural Outlook Spetember/AO- 84, 6-7. Europea Evirometal Agecy (000), EEA Aual Work Programme 000, Europea Evirometal Agecy, Doc EEA/053/99fial. Europea Evirometal Agecy (001), Chapter. Curret Evirometal Policy i The State of Actio to Protect the Eviromet i Europe: Epert Corer Report. No 1, Garvie, D., ad A. Keeler (1994), Icomplete Eforcemet with Edogeous Regulatory Choice, Joural of Public Ecoomics 55: Hora, R.D., Shortle, J.S., ad D.G. Abler (1998), Ambiet Taes Whe Polluters Have Multiple Choices, Joural of Evirometal Ecoomics ad Maagemet 36: Kapla, J.D. (1999), Nopoit Source Pollutio, Icomplete Iformatio ad Learig: A Etropy Approach, Ph.D. Dissertatio, Departmet of Agricultural ad Resource Ecoomics, Uiversity of Califoria at Davis. Kapla, J.D. ad R.E. Howitt (00) Estimatig Nopoit Source Pollutio: A Applicatio of a Sequetial Etropy Filter, Water Resources Research 38(3): /000WR Kapla, J.D., Howitt, R.E. ad Y.H. Farzi (003), A Iformatio-Theoretical Aalysis of Budget-Costraied Nopoit Source Pollutio Cotrol, Joural of Evirometal Ecoomics ad Maagemet 46:

28 Liter, A.M. ad A. Weersik (1999), Edogeous Trasport Coefficiets: Implicatios for Improvig Water Quality from Multi-Cotamiats, Evirometal ad Resource Ecoomics 14: Segerso, K. (1988), Ucertaity ad Icetives for Nopoit Source Pollutio Cotrol, Joural of Evirometal Ecoomics ad Maagemet 15: Shortle, J.S., ad D.G. Alber (1997), Nopoit Pollutio, i The Iteratioal Yearbook of Evirometal ad Resource Ecoomics 1997/1998 (H. Folmer ad T. Tieteberg, Eds.), Edward Elger, Chelteham, UK, ad Northhampto, MA, USA; pp Shortle, J.S., ad J.W. Du (1986), The Relative Efficiecy of Agricultural Source Water Pollutio Cotrol Policies, America Joural of Agricultural Ecoomics 68: Shortle, J.S. (1990), The Allocative Efficiecy Implicatios of Water Pollutio Abatemet Cost Comparisos, Water Resources Research 6(5): Sigh, V. P. ad P.F. Krstaovic (1987), "A Stochastic Model for Sedimet Yield Usig the Priciple of Maimum Etropy." Water Resource Research 3: Spreiter, T.A., Frake, J.F. ad D.L. Steese (1995), 'Disturbed Lads Restoratio: The Redwood eperiece. Paper Presetatio at the Eight Bieial Coferece o Research ad Resource Maagemet i Parks ad Public Lads, April 17-1, 1995, George Wright Society, Hacock, Michiga. Uited States Geeral Accoutig Office (1999), Water Quality: Federal Role i Addressig-ad-Cotributig to-nopoit Source Pollutio, GAO/RCED-99-45, US Govermet Press, Washigto DC. Uited States Departmet of the Iterior (1981), Watershed Rehabilitatio Pla: Redwood Natioal Park, Uited States Natioal Park Service, April, US Govermet Press, Washigto DC. Xepapedeas, A.P. (1995), Observability ad Choice of Istrumet Mi i the Cotrol of Eteralities, Joural of Public Ecoomics 56: