Emission versus Input Taxes for Diffuse Nitrate Pollution Control in the Presence of Transaction Costs

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1 Joural of Evirometal Plaig ad Maagemet, 45(1), , 00 Emissio versus Iput Taxes for Diffuse Nitrate Pollutio Cotrol i the Presece of Trasactio Costs ATHANASIOS KAMPAS* & BEN WHITE *Macaulay Lad Use Research Istitute, Aberdee, UK Departmet of Agricultural Ecoomics ad Food Marketig, Uiversity of Newcastle upo Tye, Newcastle upo Tye NE1 7RU, UK. be.white@ewcastle.ac.uk (Received February 001; revised July 001) ABSTRACT The most importat obstacle to solvig diffuse pollutio problems is that emissios are either uobservable or caot be observed at a reasoable cost. Biophysical models may provide suf ciet iformatio to set a cost-effective emissio tax. However, evidece from recet studies has show that trasactio costs for emissio-based policies are higher per hectare tha for iput-based policies. A ecoomic model of agriculture for the Keet catchmet i south-east Eglad shows that, whe trasactio costs are accouted for, a iput tax is more ef ciet tha a emissio tax over a rage of emissio stadards. This result has policy implicatios i that it idicates, rst, that ecoomists policy recommedatios should accout for trasactio costs, ad, secodly, that the stadard advice that emissio-based policies are superior may be wrog where trasactio costs differ substatially betwee emissio- ad iput-based policies. Itroductio The regulatio of diffuse pollutats from agricultural activities is complicated by their uobservability at a reasoable cost. This has led regulators to cosider idirect cotrol measures which regulate iput use ad/or maagemet practices. For istace, iput taxes have log bee cosidered alteratives to emissio taxes (Holterma, 1976; Commo, 1977; Grif & Bromley, 198; Segerso, 1990). Such policies are based upo a relatioship betwee emissios ad observed iputs. This relatioship ca be de ed through biophysical models (Bouzaher et al., 1993). A umber of studies that combie biophysical models with a ecoomic aalysis have show that a emissio tax is more ef ciet tha a iput tax (see, for example, Miltz et al., 1988; Johso et al., 1991). However, these studies take o accout of how trasactio costs vary betwee alterative policies ad ca affect a policy s relative ef ciecy (Vat, 1998). We compare the total costs, that is abatemet costs plus trasactio costs, of establishig itrate emissio stadards for the ha Keet catchmet i souther Eglad. Nitrate pollutio is critical to the catchmet, as levels of itrate i drikig water i the regio are ofte close to, or i excess of, the Europea Uio (EU) limit of 50 mg/l for drikig water. I the ext sectio, a theoretical model is developed that de es the optimal itrate pollutio cotrol problem. The followig sectio ad the Appedix describe the empirical model Prit/ Olie/0/ Ó 00 Uiversity of Newcastle upo Tye DOI: /

2 S 130 A. Kampas & B. White The ext sectio compares the relative abatemet ad trasactio costs of emissio ad iput taxes. The al sectio cocludes the paper. Theoretical Model Cosider a agricultural river catchmet with a xed agricultural lad area ad a xed umber of farms. Nitrates are assumed to be a ow pollutat which imposes market costs o water cosumers through icreased costs of water puri catio ad o-market costs due to the degradatio of aquatic ecosystems. We ca de e the emissio fuctio as a strictly icreasig fuctio of the rm s iput vector x i. Thus the emissio level e i, of the ith producer is give by: e i 5 e i (x i ; y) with e xi. 0, e xi x i. 0 (1) where variable subscripts refer to partial derivatives ad the vector y icludes the i uece of weather ad other site-speci c factors o emissios, such as elevatio ad soil type, etc. Sice these factors are beyod the producer s cotrol, y is dropped. The assumptio of a icreasig ad covex emissio fuctio is commo i the literature (Requate, 1998). We assume that the regulator has two differet policy optios to cotrol itrate pollutio, amely emissio-based i the form of a emissio tax ad iput-based i the form of a iput tax. A output-based policy is rejected, sice Spulber (1985) shows that output regulatio is ief ciet i the log ru. Emissio Tax I much of the regulatio literature, it is assumed that the regulator wishes to achieve ecoomic ef ciecy by maximizig some measure of ecoomic welfare (Tomasi et al., 1994). Therefore, it is assumed that the regulator aims to miimize the total social costs i terms of the rm s compliace costs, trasactio costs ad damage costs (Stavis, 1995). Thus the regulator s problem ca be writte as: Miimize e i i 5 1 (c i (e i ) 1 ce i (e i ) 1 d e i ) () The cost of emissios, c i (e i ), is de ed as: c i (e i ) 5 p i (e* i ) p i (e i ); c i (e* i ) 5 0; c i e, 0; c i ee. 0 (3) where p i (e i ) is the pro t of rm i as a fuctio of emissios. This speci catio of the pro t fuctio was itroduced by Porter (1974), ad is referred to as the rm s bee t fuctio from egagig i a pollutat-geeratig activity (Malik, 1991; Xepapadeas, 1991; Chambers & Quiggi, 1996). Pro t-maximizig emissios are give by e* i ; at this level of emissios the cost of compliace is zero. Trasactio costs, ce i (e i ), have a xed cost elemet per hectare associated with providig moitorig equipmet ad permaet staff, processig applicatios ad observig farm activities. The variable cost elemet relates to tax collectio ad some forms of moitorig. This compoet declies iversely with the strigecy of the emissio restrictios. The reaso for this is that, as the emissio level is icreased, the icetive to exceed it falls, ad thus the regulator eeds a lower level of moitorig to eforce the policy; hece the form: ce i (e i ) 5 cef i 1 cev i (e i ) (4)

3 g 6 S Emissio versus Iput Taxes 131 where cef i is the xed cost of admiisterig a scheme for rm i ad cev i (e i ) is the variable cost. The variable cost is decliig with respect to emissios; assume cev i e, 0 ad cev i ee $ 0. The last term i equatio () is the damage cost, give here as a liear fuctio of emissios, where d is the margial damage per uit of emissios. The socially optimal level of emissios is give by: c i e 5 d 1 ce i e 5 s e The stadard solutio, igorig trasactio costs, requires the margial cost of emissios to be equal to the margial damage (Krupick et al., 1998), ad this is implemeted by settig the emissio tax equal to the margial damage. From equatio (5) the optimal emissio tax, s e, equals the margial bee t of emissios plus the margial trasactio costs. A similar result is obtaied by Vat (1998) usig a utility-maximizig framework. Of course, xed trasactio costs do ot eter the margial coditio, but they do affect the choice betwee policies. (5) Iput Tax If, as Grif & Bromley (198) assume, the regulator has perfect iformatio about the emissio fuctio (equatio (1)), the emissio restrictios ca be traslated ito iput restrictios. The cost of compliace fuctio ca be de ed as follows (Ribaudo et al., 1999): c i [e(x i )] 5 p i [e(x* i )] p i [e(x i )]; c i [e(x* i )] 5 0; c i x, 0; c i xx, 0 (6) where x i are iputs. The pro t fuctios are ow de ed i terms of iputs. For a iput-based policy the regulator maximizes: Miimize x i i 5 1 {c i [e(x i )] 1 cx i (x i ) 1 d e i (x i )} (7) where x i (x i ) is the trasactio cost. The iput tax, s x, is set so that: c i e e i x 5 d e i x 1 cx i x 5 s x Comparig equatio (5) with equatio (8) results i: c i e e i x 5 s x c i e 5 s e Þ s x 5 s e e i x Û d e i x 1 cx i x 5 e i x (d 1 ce i e) (9) Simplifyig equatio (9) gives the ecessary coditio uder which the iput tax is equivalet to emissio tax: cx i x x i ce i ee(x i ) 5 This coditio would hold, iter alia, if the elasticity of emissio with respect to x, g, equals the ratio of the relative variable trasactio costs. Ituitively, the two policies are the same if their relative effectiveess i reducig emissios, the right had side of equatio (10), approximately equals their trasactio cost ratio. This result will hold exactly if the trasactio cost fuctio are liear. The ecessary coditios give i equatios (5) ad (8) give the optimal emissio tax ad iput tax, but to compare the policies we must also take ito (8) (10)

4 13 A. Kampas & B. White accout the xed costs of a policy, which may be quite differet for a emissio tax compared with a iput tax. Thus, oce optimal tax rates have bee xed, the best policy is chose by comparig equatios () ad (7) at their miimum poits. A Brief Descriptio of the Empirical Model ad Data Model Structure The applicatio of the theoretical model to the Keet catchmet required the represetatio of itrate emissio fuctios ad abatemet costs withi a ecoomic model. The approach adopted was to divide the catchmet ito lad classes characterized by differet productio ad emissio possibilities ad to use a aggregate o-liear programmig model to simulate producers resposes to differet itrate pollutio cotrol measures. The empirical model proceeded i three stages. First, a geographical iformatio system (GIS) was used to de e lad classes o the basis of their soil properties ad to idetify the total area of the catchmet. Secodly, itrate emissio fuctios for all the major soil/crop combiatios were estimated ecoometrically from itrate emissio estimates derived from the simulatio models: SUNDIAL, for the arable lad; ad N-CYCLE, for the grasslad. SUNDIAL (SimUlatio of Nitroge Dyamics I Arable Lad) is a persoal computer-based versio of the Rothamsted Nitroge Turover Model ad was used to estimate itrate losses, give a set of agroomic parameters ad weather coditios (Smith et al., 1996). Similarly, N-CYCLE is a empirical mass balace which simulates the itroge cycle i grasslad systems. N-CYCLE was developed at the Istitute of Grasslad ad Evirometal Research (Lockyer et al., 1995). I additio, a simple hydrological model, TOPCAT (Qui & Atoy, 1996), was used to estimate the cocetratio of itrates i the draiage water. The patter of itrate emissio estimates for the Keet catchmet was cosistet with the rage of values foud i the literature (Bradbury et al., 1993; Whitehead, 1995). Thirdly, the productio ad pollutio iformatio was assembled withi a o-liear optimizatio framework. A o-liear programmig model was used, sice both the productio ad the emissio fuctios are o-liear. Crop productio fuctios were take from the literature (Eglad, 1986). The whole catchmet was represeted as a sigle pro t-maximizig rm, which is a typical speci catio of sector models withi a mathematical programmig framework (Hazell & Norto, 1986). Output ad iput prices refer to the productio period. Figure 1 depicts the model structure of the empirical applicatio. Similar itegrated modellig aalysis for diffuse pollutio cotrol is followed by Bouzaher & Shogre (1995), Vat et al. (1997) ad Wu et al. (1995). The mathematical form of the model is give i the Appedix. Trasactio Costs The trasactio costs associated with the two policies examied comprise the costs of implemetig the schemes. The reported moitorig costs of the Nitrate Sesitive Areas (NSAs) ad the Evirometal Sesitive Areas (ESAs) are take as a proxy for xed trasactio costs associated with emissio taxes ad iput taxes, respectively. NSAs are the product of a EU policy that ivolves periodic

5 Emissio versus Iput Taxes 133 Figure 1. Modellig framework for itrate pollutio cotrol. itrate emissio moitorig i order to assess whether the scheme is beig successful i reducig itrate emissios (Miistry of Agriculture, Fisheries ad Food (MAFF), 1998). ESAs are the product of a EU scheme that ivolves iput moitorig (MAFF, 1997). The xed per hectare costs of the ESA scheme were 5.19 for the period , while the relevat costs for the NSA scheme were (House of Commos Agriculture Committee, 1997). These are treated as the xed costs of the alterative policies. Variable trasactio costs associated with policies that ivolve taxatio are assumed to be a speci c proportio of the relevat tax paymet (Polisky & Shavell, 198). For example, Koh (1991) uses the value of 0.5%, while sources for the UK suggest a rage of values betwee 0.5% ad 1% (Falcoer, 1999). I terms of our case-study, the gure of 1% was used. This represets the variable trasactio costs, as they vary, depedig upo the tax rate. These estimates of trasactio costs are subject to differet iterpretatios, ad the fact that they are estimated from schemes o the basis of differet policies led us to iclude a sesitivity aalysis i the followig sectio.

6 134 A. Kampas & B. White Results Baselie Case O the basis of the model described above, we examie the relative ef ciecy of emissio versus iput taxes i achievig speci c reductios i emissio levels. These reductios are chose to be 10%, 0% ad 30% of the baselie solutio, where the baselie solutio correspods to urestricted pro t maximizatio. The reductios are equivalet to a sesitivity aalysis for the margial damage cost of emissios, i the sese that a tighter emissio stadard implies a higher margial damage cost. The istrumet that has the lowest total cost is the most ef ciet. The model i the Appedix is used to estimate the resource costs of the alterative policies. Additioal calculatios are required to accout for the trasactio costs. Table 1 gives a breakdow of the total costs of emissio ad iput taxatio for differet emissio reductios. Sice the emissio fuctio is foud to be a covex fuctio i iputs ad exhibits icreasig returs to scale, a emissio tax outperforms a iput tax, as Steves (1988) has show. For all the differet levels of emissio reductios examied, a emissio tax resulted i lower abatemet costs tha a iput tax. Abatemet costs represet the resource costs from society s poit of view i meetig the speci c emissio reductio targets. Strictly speakig, sice markets are distorted because of the various forms of govermet itervetio i agriculture, these costs are ot the true resource costs. I additio, sice trasfer paymets do ot cotribute to ecoomic welfare, they are excluded from the social welfare cost (Prato, 1998). The relative superiority of emissio taxes over iput taxes as show is altered whe the compariso betwee emissio ad iput taxes is based o the total cost, which comprises the resource, the moitorig ad the admiistratio costs. It is oteworthy that total costs do ot iclude the tax paymets. Emissio taxes are ot the most ef ciet istrumet whe the relevat trasactio costs are icluded. This result cotradicts the covetioal ecoomic wisdom, which recommeds emissio taxes as the most ef ciet policy. However, accordig to Vat (1998), statemets that propose emissio taxes as ef ciet assume zero trasactio costs, ad ca lead to serious errors i policy choice. Sesitivity Aalysis I this sectio we explore the rage of values over which the coclusios relatig to the ef ciecy of the policies hold. To do this we take the baselie give i Table 1. I this case a iput tax is optimal for all iput reductios. The trasactio costs are the varied to assess the rage of values over which this baselie result holds; thus the trasactio costs for emissio taxes are reduced i steps dow to 40% of their baselie level ad the trasactio costs of iput taxes are icreased by 40%. Table gives the trasactio cost rages icluded i the sesitivity aalysis ad Table 3 summarizes the results of sesitivity aalysis. It shows that the superiority of iput taxes holds over a wide rage of trasactio costs. For istace, the xed trasactio costs for a iput tax would have to rise by 40% ad the xed trasactio costs for a emissio tax fall by 30% for the optimal policy to switch from a iput tax to a emissio tax.

7 Emissio versus Iput Taxes 135 Table 1. Cost of emissio tax (ET) ad iput tax (IT) for differet emissio reductios ( ) Emissio reductio 10% 0% 30% ET IT Rak ET IT Rak ET IT Rak Abatemet costs ET ET ET Tax paymet ET ET ET Resource costs (A) ET ET ET Trasactio costs Variable costs (B) ET ET ET Fixed costs (C) IT IT IT Total costs (A B C) IT IT IT

8 136 A. Kampas & B. White Table. Differet values of xed trasactio costs used for sesitivity aalysis Fixed costs ( ) Chage (%) Emissio tax Iput tax Coclusios This paper examies, both theoretically ad empirically, the icorporatio of trasactio costs ito the ecoomic aalysis of a diffuse pollutat. As was show i the secod sectio, icludig trasactio costs determies the relative ef ciecy of emissio ad iput taxes. The results show that, o the basis of trasactio costs estimated from existig agri-evirometal schemes, a iput tax policy is more ef ciet tha a emissio tax. A sesitivity aalysis shows that this result holds for a relatively wide rage of trasactio costs. Assumig the results are ot cotext- or site-speci c, it is legitimate to argue that the exclusio of the relevat trasactio costs ca result i Table 3. Optimal policy for differet combiatios of xed trasactio cost chages: emissio tax (ET) ad iput tax (IT) Chage i xed trasactio cost IT policy (%) ET policy (%) ET ET ET ET ET IT IT IT 35 ET ET ET IT IT IT IT IT 30 ET IT IT IT IT IT IT IT 5 IT IT IT IT IT IT IT IT 0 IT IT IT IT IT IT IT IT 15 IT IT IT IT IT IT IT IT 10 IT IT IT IT IT IT IT IT 5 IT IT IT IT IT IT IT IT

9 Emissio versus Iput Taxes 137 misleadig policy recommedatios. This paper idicates that, whe trasactio costs are accouted for, it is ef ciet to pursue iput-based policies to regulate itrate emissios. Refereces Bouzaher, A. & Shogre, J. (1995) Modelig opoit source pollutio i a itegrated system, i: W. Marti (Ed) Evirometal Policy Modelig (Bosto, MA, Kluwer Academic Publishers). Bouzaher, A., Lakshmiarya, P., Cabe, R., Carriouiry, A., Gassma, P. & Shogre, J. (1993) Metamodels ad opoit pollutio policy i agriculture, Water Resources Research, 9, pp Bradbury, N., Whitmore, A., Hart, P. & Jekiso, D. (1993) Modelig the fate of itroge i crop ad soil i the years followig applicatio of 15N-labelled fertilizer to witer wheat, Joural of Agricultural Sciece, 1, pp Chambers, R. & Quiggi, J. (1996) No-poit source pollutio regulatio as a multi-task pricipalaget problem, Joural of Public Ecoomics, 59, pp Commo, M. (1977) A ote o the use of taxes to cotrol pollutio, Scadiavia Joural of Ecoomics, 79, pp Eglad, R. (1986) Reducig the itroge iput o arable farms, Joural of Agricultural Ecoomics, 37, pp Falcoer, K. (1999) Trasactio costs (persoal commuicatio). George, B. (1984) Desig ad iterpretatio of itroge respose experimets, i: P. Needham, J. Archer, R. Sylvester-Bradley & G. Goodlass (Eds) The Nitroge Requiremet of Cereals (Lodo, HMSO). Grif, R. & Bromley, D. (198) Agricultural ruoff as a opoit exterality: a theoretical developmet, America Joural of Agricultural Ecoomics, 64, pp Hazell, P. & Norto, R. (1986) Mathematical Programmig for Ecoomic Aalysis i Agriculture (New York, Macmilla). Holterma, S. (1976) Alterative tax systems to correct for exteralities, ad the ef ciecy of payig compesatio, Ecoomica, 43, pp House of Commos Agriculture Committee (1997) ESAs ad Other Schemes uder Agri-evirometal Regulatios (Lodo, HMSO). Johso, S., Adams, R. & Perry, G. (1991) The o-farm costs of reducig groudwater pollutio, America Joural of Agricultural Ecoomics, 73, pp Koh, P. (1991) Trasactio costs ad the optimal istrumet ad itesity of air pollutio cotrol, Policy Scieces, 4, pp Krupick, A., McCoell, V., Austi, D., Stoessell, T. & Morto, B. (1998) The Chesapeake Bay ad the Cotrol of NOx Emissios: A Policy Aalysis, Report No (Washigto, Resource for the Future). Lockyer, D., Schole eld, D. & Dawso, B. (1995) N-CYCLE (Aberdee, MERTaL Courseware). Malik, A. (1991) Markets for pollutio cotrol whe rms are ocompliat, Joural of Evirometal Ecoomics ad Maagemet, 18, pp Miltz, D., Brade, B. & Johso, G. (1988) Stadards versus prices revised: the case of agricultural o-poit source pollutio, Joural of Agricultural Ecoomics, 39, pp Miistry of Agriculture, Fisheries ad Food (MAFF) (1997) Evirometally Sesitive Areas Scheme ( MAFF (1998) Nitrate Sesitive Areas (htttp:// Polisky, A. & Shavell, S. (198) Pigouvia taxatio with admiistrative costs, Joural of Public Ecoomics, 19, pp Porter, R. (1974) The log-ru asymmetry of subsidies ad taxes as ati-pollutio policies, Water Resources Research, 10, pp Prato, T. (1998) Natural Resource ad Evirometal Ecoomics (Ames, Iowa, Iowa State Uiversity Press). Qui, P. & Atoy, S. (1996) TOPCAT: MS-Excel Based Freeware Versio of TOPMODEL, (Wolverhampto, ADAS R&D). Requate, T. (1998) Icetives to iovate uder emissio taxes ad tradeable permits, Europea Joural of Political Ecoomy, 14, pp

10 S S b S d b S 138 A. Kampas & B. White Ribaudo, M., Hora, R. & Smith, M. (1999) Ecoomics of Water Quality Protectio from Nopoit Sources: Theory ad Practice, Report No. 78 (Washigto, DC, US Departmet of Agriculture, Resource Ecoomics Divisio). Segerso, K. (1990) Icetive policies for cotrol of agricultural water pollutio, i: J. Brade & S. Lovejoy (Eds) Agriculture & Water Quality (Boulder, Colorado, Lye Rieer Publishers). Smith, J., Bradbury, N. & Addiscott, T. (1996) SUNDIAL: a PC-based system for simulatig itroge dyamics i arable lad, Joural of Agroomy, 88, pp Spulber, D. (1985) Ef uet regulatio ad log-ru optimality, Joural of Evirometal Ecoomics ad Maagemet, 1, pp Stavis, R. (1995) Trasactio costs ad tradeable permits, Joural of Evirometal Ecoomics ad Maagemet, 9, pp Steves, B. (1988) Fiscal implicatios of ef uet charges ad iput taxes, Joural of Evirometal Ecoomics ad Maagemet, 15, pp Tomasi, T., Segerso, K. & Brade, J. (1994) Issues i the desig of icetive schemes for opoit source pollutio cotrol, i: C. Dosi & T. Tomasi (Eds) Nopoit Source Pollutio Regulatio: Issues ad Aalysis (Dordrecht, Kluwer Academic Publishers). Vat, A. (1998) Iput versus emissio taxes: evirometal taxes i a mass balace ad trasactio costs perspectives, Lad Ecoomics, 74, pp Vat, A., Bakke, L., Ludeby, H., Romstad, E., Rorstad, P. & Vold, A. (1997) Regulatig opoit source pollutio from agriculture: a itegrated modellig aalysis, Europea Review of Agricultural Ecoomics, 4, pp Whitehead, D. (1995) Grasslad Nitroge (Walligford, CAB Iteratioal). Wu, J., Teague, M., Mapp, H. & Berardo, D. (1995) A empirical aalysis of the relative ef ciecy of policy istrumets to reduce itrate water pollutio i the US souther High Plais, Caadia Joural of Agricultural Ecoomics, 43, pp Xepapadeas, A. (1991) Evirometal policy uder imperfect iformatio: icetives ad moral hazard, Joural of Evirometal Ecoomics ad Maagemet, 0, pp Appedix: The Catchmet Model Miimize a, u, xx H p * F S S m i 5 1 j 5 1 (p j q ij 1 c j u ij w a ij x ij w f x )G J (A1) Crop productio fuctio: y ij 5 b 0 ij 1 b 1 ij (b ij) xij 1 b 3 ij x ij "i, j (A) Crop output: q ij 5 y ij a ij "i, j (A3) Grass productio fuctio: g ij 5 b 0 ij 1 1 ij x ij 1 3 ij x ij "i, j (A4) Grass output: c ij 5 g ij a ij "i, j (A5) Metabolizable eergy: i 5 S m 1 j u ij z # i 5 S m 1 j c ij x 1 x "i, j (A6) Livestock stockig rates: u ij 5 s j a ij "i, j (A7) Livestock uits: Lad: i 5 1 u ij # ū j "i, j (A8) m S a ij # b i "i, j (A9) j 5 1 Rotatios: S i a ij r ij # 0 "i, j (A10) Nitrate load from crops (kg/ha): e c ij 5 exp(d 0 ij 1 1 ij x ij) "i, j (A11) Nitrate load from grasslad (kg/ha): e g ij 5 h 0 ij 1 h 1 ij x ij 1 h ij x ij "i, j (A1) Average itrate leachig from crops (kg/ha): l c ij 5 d 0 ij 1 d 1 ij e c ij "i, j (A13) Average itrate leachig from grasslad (kg/ha): l g ij 5 d 0 ij 1 d 1 ij e g ij "i, j (A14) Total average itrate leachig: e 5 S S m l c ij a ij 1 S m l g ij a ij "i, j (A15) 1j 5 1 1j 5 1 i 5 i 5

11 Emissio versus Iput Taxes 139 The regulator s objective fuctio miimizes the reductio of agricultural pro t for the catchmet as whole, where pro t is de ed as a retur to producers maagemet, lad ad other xed resources over the cost of itroge fertilizer, w a ijx ij, ad livestock feedig costs, w f x ; p j stads for the price of j crop, q ij the amout of j crop produced i ith soil type lad, w is the price of itroge fertilizer, x ij is the amout of itroge fertilizer applied i j activity to the ith soil type ad a ij stads for the amout of i soil type lad devoted to j activity. Differet livestock grazig uits located o differet soil types are deoted with u ij, while q j idicates the gross margi of j livestock uit. The urestricted pro t p * is a costat ad plays o part i the optimizatio. I terms of the theoretical models, equatio (A1) ca represet objective fuctio equatios () ad (7) as special cases, depedet upo which of the variables is targeted by the policy. Lad productivity is de ed by crop ad livestock productio. Takig the rst of these, crop productio is represeted by a set of productio fuctios, which give the respose of the crop to itroge. These productio fuctios may vary, depedet upo the soil type. Equatio (A) gives the itroge respose fuctio where y ij is the yield per hectare ad the coef ciets b 0 ij, b 1 ij, b ij, b 3 ij are estimated coef ciets. This liear plus expoetial form of productio fuctio has bee foud to give the best predictio of the itrate respose (George, 1984). The crop output is give by equatio (A3), where q ij is the output of crop i o lad class j ad a ij is the area. Usig a similar framework, grasslad productio ad output are give by equatios (A4) ad (A5). Equatio (A4) gives grass yield g ij as dry matter per hectare ad equatio (A5) gives the amout of grass produced of differet types i each lad class c ij, where b ij, b ij, b ij, b ij are coef ciets. The metabolizable eergy costrait balaces the eergy required by a livestock uit, u ij, with eergy provided by grass plus a extra eergy x provided either by cor silage ad/or barley, where z is the eergy required by a livestock uit ad x is the eergy cotet of grass. Livestock, which icludes sheep, beef cattle ad dairy cows, are subject to quotas (equatio (A7)), where u j is the quota level for lad class j ad s j is the average stockig rate. Lad allocatios are costraied by the total availability of lad, b j, of differet types i the catchmet (equatio (A9)). A set of rotatioal costraits imposes restrictios o the order i which crops ca be grow (equatio (A10)). Equatios (A11) (A1) give the itrate emissio compoets of the model. Nitrate load from crops, e c 0 1 ij, is give as a semilog fuctio i equatio (A11) ad coef ciets d ij, d ij. Equatio (A1) gives the itrate load from grasslad, e g ij ad the terms h 0 ij, h 1 ij, h ij are coef ciets. Equatios (A13) ad (A14) are the average itrate leachig per hectare for crop, i c ij, ad grasslad, i g ij, where d 0 ij, d 1 ij are coef ciets.