Modeling Steady-State Irrigated Production

Transcription

1 Abstract Modelng Steady-State Irrgated Producton Rchard Howtt 1, Duncan MacEwan 2, and Swa Msang 3 In ths paper we evaluate the ntensve, extensve, and dynamc margn response to sustanable agrcultural practces. We defne sustanable agrculture as a system n whch the resource and polluton stocks assocated wth producton have a steady-state soluton consstent wth a resource-use path resultng n long run utlty maxmzaton. As an emprcal applcaton we consder two state varables, fertlty levels and groundwater stocks, and a two-crop rotaton system, alfalfa-cotton. Agrcultural producton leads to a fundamental tradeoff represented by the rotaton between net-ntrogen usng crops and net-ntrogen fxng crops. In addton, ntrogen usng crops generally have hgher net returns than ntrogen fxng crops, but ntrogen fxng crops have a hgher water use per unt revenue. It follows that rotaton shfts that favor ntrogen fxng crops result n greater groundwater depleton. To acheve sustanablty n ths context, one must smultaneously strke a steady-state balance, or at least a repeated cycle, between the rates of fertlty change and changes n stocks of groundwater. We estmate the sustanablty problem n three stages. Frst we estmate the rotaton dynamc frst order condtons. Next we ncorporate the estmated rotaton parameters nto a calbrated economc producton model that reflects the mplct crop producton costs and calbrates to the base observed soluton. Fnally, we use the outputs from the calbrated producton model to drve the equatons of moton for groundwater use and fertlty changes. We estmate the model wth 13 years of geo-referenced feld data n Calforna's Central Valley and smulate the effect of changng rotatons on the ablty to acheve sustanable groundwater and ntrogen use, and the mplct socal costs of achevng sustanablty. 1 Professor Emertus, Department of Agrcultural and Resource Economcs at UC Davs; Prncpal, ERA Economcs 2 Prncpal, ERA Economcs 3 Senor Research Fellow, Internatonal Food Polcy Research Insttute

2 Introducton Agrculture by ts very nature seeks to ncrease the productvty of specfc speces by reducng the ecologcal entropy of the natural ecosystem. Ths reducton n entropy can be measured n terms of bodversty or energy flows whch are changed by the process of agrculture, a process that can be vewed as the addton of both energy and nformaton to change an ecosystem. Early agrculture was ntated by crop selecton, cultvaton, rrgaton, and rotaton. Over the past 80 years the frst three agrcultural practces have bloomed and developed beyond all recognton, whle the fnal aspect of rotatons, tradtonally used to mantan fertlty and control weeds and dseases, has been steadly reduced n ts mportance to hghly productve rrgated agrculture by the substtuton of energy, chemcals and nformaton. The remarkable productvty that ths process has engendered s well known. However, there s a growng concern over the depleton of natural resources and contamnaton of the envronment by external effects of conventonal hgh-nput agrculture. The queston arses as to whether the current system s sustanable. It s our vew that the queston of sustanablty rests on whether an equlbrum steady-state natural resource use and economc feasblty exsts. If such feasblty does not exst wth the current technology, what technologcal shfts would enable an rrgated agrcultural system to move towards the steadystate of water use, water and land qualty mantenance, and economc vablty? The central tenant of ths paper and the resultng model s that sustanable rrgaton systems can only be acheved by substtutng rotatonal fertlty and pest controls for some of the current level of appled chemcal technology. In smple economc terms we propose that sustanablty n rrgated agrcultural systems can only be acheved by movng back around the soqant producton curve towards a greater relance on crop and probably lvestock rotatons. For purposes of smplcty and emprcal convenence, we locate our emprcal example n an agrcultural regon of Southern Calforna called Kern County, and dentfy felds rotated wth two major crops, namely, alfalfa and cotton. Smlarly we characterze the effect on the agrcultural resource base n terms of water quantty, measured n ths case by depth to groundwater, and water qualty measured by the prncpal polluton n the regon, namely the concentraton of ntrates n the groundwater. We focus on these two effects because we feel that they domnate the resource base used n rrgated agrculture n many parts of the world. 2

3 The contrbutons of ths paper are three-fold, () quantfy the dynamcs of agrcultural rotatons, () ncorporate the dynamcs of agrcultural producton nto a calbrated optmzaton model, and () use the model to estmate the dynamc steady-state and mplct costs of sustanablty. We consequently offer contrbutons n appled modelng, estmaton and calbraton methods, and polcy applcaton. The paper s structured as follows. Frst, we dscuss and defne sustanablty to put the polcy applcaton n ths paper nto context. Next, we present our three-step modelng and estmaton framework. We conclude wth a polcy example for the study regon of Kern County and dscuss applcaton to other regons and externaltes. Global Costs of Unsustanable Producton Sustanable practces requre that resource use follows a steady-state dynamc soluton whch results n long run utlty maxmzaton. In the absence of ths sustanable path, we expect to deplete or degrade the avalable resource. We see ths trend n a number of areas and applcatons. In many regons of the world wth rrgated agrculture current levels of water use are unsustanable. We can measure sustanablty by the current rate of overdraftng of groundwater supples. A recent study Gleeson et al.(2012) used a worldwde hydrologc model to estmate that almost a quarter of the world's populaton, 1.7 bllon people, lve n regons where groundwater s beng overdrafted and used n an unsustanable manner. They also show that many of the overdrafted regons are those n whch water supples are most crtcal and urban and agrcultural water stress s the hghest. In Calforna, the Unted States Geologcal Survey has recently released a study usng satellte mages that shows the rate of subsdence n Calforna s Central Valley s greater than prevously estmated (Sneed et al, 2013). The USGS estmates subsdence at more than 1 foot per year n some areas, creatng substantal stran on Calforna s state and federal water conveyance nfrastructure. In addton to overuse, surface and groundwater global supples are threatened by qualty degradaton due to unntended salnty or chemcal polluton. In the Central Valley of Calforna Harter et al. (2012) show that ntrate polluton of groundwater stocks has rsen to the level where t poses a severe health hazard to several agrcultural communtes. Salnzaton s the oldest water qualty problem facng rrgated agrculture and has been responsble for removng several establshed rrgaton systems and cvlzatons. A recent study by Schoups et al. (2005) estmates 3

4 the scope of the global problem salnty affects between 20 and 30 mllon hectares. They show that salnty s a growng concern for rrgated agrculture along the Westsde of Calforna s Central Valley. Groundwater overdraft and water qualty degradaton are dffcult to quantfy on a broad scale. Ptman and Lauchl (2002) estmate that at least 20% of the 227 mllon hectares of worldwde rrgated land suffers from reduced yelds due to secondary salnzaton. Ghassem et al. (1995) estmate the cost, n constant 1995 dollars, to be at least $12 bllon. In addton to the costs of salnzaton, contamnaton by heavy metals and pestcdes may be a more serous longterm threat to groundwater. These contamnants move slowly through the aqufer and are essentally rreversble once establshed. Other pollutants such as ntrogen and salnty can be treated f groundwater s pumped and used on crops whch are tolerant to salts and heavy metals and enable some removal of these contamnants by the crop harvest. We thnk the contrbuton of rrgated agrculture towards world food producton and the threat of unsustanable producton on the long-term prospects for rrgated agrculture warrant a formal and emprcally-based modelng approach. Such an approach wll help encourage future research on what the process of achevng sustanable rrgated agrculture nvolves, and may show what technologcal shfts would be needed to acheve sustanablty n both water quantty and qualty whle mantanng vable economc producton. Defnng and Modelng Sustanable Irrgated Producton The lack of mportance of rotatons to most agrcultural economsts s shown by the standard mcroeconomc analyses of agrcultural producton that defne ther classfcaton of response n terms of the ntensve and extensve margns producton. MacEwan and Howtt (2013) argue that there s a thrd margn of adjustment n agrculture that reflects the postve or negatve ntertemporal effects of contnuous or rotatonal croppng. They term ths dynamc effect on agrcultural productvty the dynamc margn, and propose that a full ntertemporal mcroeconomc analyss should nclude both the ntensve and extensve statc margns, n addton to the dynamc margn. They argue that only n the context of all three dmensons can one correctly reflect productvty and the user cost of the resources used n agrculture. To evaluate the ntensve, extensve, and dynamc margn response to sustanable agrcultural practces, we defne sustanable rrgated agrculture as a system n whch the resource and 4

5 polluton stocks assocated wth producton have a steady-state soluton that s consstent wth a resource-use path that results n long run utlty maxmzaton. There s a substantal body of lterature whch defnes sustanable systems n a more general economc equlbrum. Our approach defnes sustanablty as only dependng on the sustanablty of the utlty of the consumers. Ths defnton of sustanablty has several attractve features, most notably t s consstent wth patterns of resource use that may be systematcally depletng the natural resource stock, as long as t s nvested suffcently well n alternatve captal assets that mantan or grow the utlty of consumers. Essentally, the utlty approach treats renewable resources as extractve resources wth fnte horzons f ther productvty, compared wth alternatve uses of the captal, s low. A classc paper showng the logcal concluson of ths approach s by Spence (1974) who long ago noted that the reproductve rate of Blue Whales s below the rate of return of treasury blls, and thus the optmal long-term equlbrum s to drve Blue Whales nto extncton and turn them all nto treasury blls. We refne the strct utlty-maxmzaton approach and take a more ecologcal vewpont of sustanablty and defne t n terms of a path that results n steady-state of water quantty and qualty, whle maxmzng net returns and presumably long run utlty from an rrgated agrculture system. Agrcultural producton leads to a fundamental tradeoff whch, snce the advent of the Norfolk Four Course rotaton n the 18 th century, s represented by a rotaton between net-ntrogen usng crops and net-ntrogen fxng crops. In addton, ntrogen usng crops generally have hgher net returns than ntrogen fxng crops, but ntrogen fxng crops have a hgher water use per unt revenue. It follows that rotaton shfts that favor lower appled ntrogen requrements wll result n reduced leachng of ntrates but greater groundwater depleton. In most cases, the adopton of crop ntensfcaton technologes results n greater use of both ntrogen and water. Sustanable ntensfcaton recognzes that, to offset these, we must have a change n the rotaton to preserve the physcal steady-state between groundwater and ntrogen use. The sustanablty problem s, n ths context, one of smultaneously strkng a steady-state balance, or at least a repeated cycle, between the rates of ntrogen polluton and changes n stocks of groundwater. We can thnk of ths problem n several dmensons. The exstence of steady-state equlbrum of groundwater and ntrate levels can be vsualzed usng a phase dagram. The exstence of steady-state depends on the startng levels of ntrates and groundwater and n the, yet unknown, rates-of-change and use of these resources n producton. The problem s further complcated by 5

6 the statc ntensve and extensve margn, whch means the grower has the ablty to shft land use and nput use ntensty. Shfts around the farm-level producton surface, even along the same soquant, presumably lead to a dfferent dynamc system and steady-state equlbrum. As an emprcal applcaton we consder two state varables, water qualty n terms of ntrate levels n the groundwater, and groundwater stocks measured by the depth of groundwater n the aqufer. We model a two crop rotaton system, alfalfa and cotton, observed over a set of felds on a sngle farm where felds are assumed to have a unform physcal captal endowment. Whle fallow s often ncluded n the rotaton for short perods, t s excluded n ths example for emprcal smplcty. We use ths framework to smulate the effect of changng rotatons on the ablty to acheve sustanable groundwater and ntrogen use, and the mplct socal costs of achevng sustanablty. In addton, we conjecture that dfferent ntensfcaton technologes wll notably change the ablty to acheve steady-state resource use. We develop a sequental-estmaton framework for the agrculture sustanablty problem that ncludes three stages. Frst, we estmate the rotaton dynamc frst-order condtons usng georeferenced feld-level data n Kern County and the method developed by MacEwan and Howtt (2013). Ther approach uses the dynamc frst order condtons to nfer the expected yeld change above or below the expected average, adjusted for sol type salnty and water avalablty, whch explans the observed actons of the farmer and swtchng crops. Next, we ncorporate the estmated rotaton parameters nto a farm-level calbrated economc producton model that reflects the mplct crop producton costs and calbrates to the rotatonal path that s observed n the base data. Ths stage uses standard calbraton methods such as n Howtt et al (2012) that utlze pror econometrc estmates of the elastctes of supply, elastctes of substtuton and elastctes of the demand. In the last stage, we use the outputs from the calbrated producton model to drve the equatons of moton for groundwater use and fertlty changes. Dynamc Swtchng Crop Rotaton Problem To quantfy the dynamc margn of adjustment we need to frst understand the nter-temporal effects of crop rotaton. We apply a method proposed by MacEwan and Howtt (2013) that mposes the dynamc frst-order condtons of the observed rotaton n order to estmate the dynamc swtchng costs. Ther approach uses a three-step procedure where rotaton systems are frst dentfed usng a sequence dentfcaton algorthm, the correspondng dynamc frst-order 6

7 condtons are derved, and then they estmate the correspondng rotaton parameters. We are able to leverage our emprcal framework and dentfy felds n an alfalfa-cotton based rotaton, thereby crcumventng the frst step. Alfalfa s treated as a fxed four-year perennal crop. We specfy the rotaton estmaton problem as follows. Consder alfalfa as a four year crop, and defne crops alfalfa and cotton as and k a, c. We denote the crop-specfc prces as p, yeld as y, and varable costs of producton as F. Equaton (1) defnes the profts generated from the feld planted n crop for year t, n the absence of rotaton effects. Average yelds and producton costs are assumed constant over tme. p y F (1) t, t, Annual crop yelds vary due to a range of stochastc factors ncludng weather shocks and management effects. We estmate the average yeld varance from a tme-seres of countyspecfc average yelds,, and allow stochastc yeld shocks to follow a statonary normal 2 2 dstrbuton, ~ N(0, ). Equaton (2) represents feld profts wth stochastc average yelds. t, t, 7 p y F (2) In practce, crop rotaton affects both crop yeld and producton costs. For emprcal smplcty we allow for nter-temporal crop yeld effects and fx average producton costs. We allow for a one-perod rotaton effect lag n the model. In other words, the prevous crop s the only factor that affects today s crop yeld. We ntroduce the parameter k for the yeld carry-over effect and we measure the rotaton effects as devaton from the mean, y. The, k entry of the matrx represents the yeld adjustment from plantng crop today gven that the farmer planted crop k n the prevous year. We allow the parameters to represent both postve and negatve agronomc effects from rotatng crops. We can now defne the current perod profts, t, t, k p y F (3) Rotaton effects may be postve or negatve, dependng on the relatonshp between the crop planted n the current and prevous perod. In our example, cotton extracts a relatvely large amount of nutrents from the sol, whereas alfalfa replaces sol nutrents. Cotton-cotton rotaton decreases average yelds and requres addtonal ntrogen applcaton, whereas alfalfa-cotton requres lower ntrogen applcaton. We assume that sol qualty and salnty are constant across

8 all felds n our dataset. As we dscuss below, we estmate our model usng data from a subset of felds on a sngle farm and consequently feel that ths assumpton s a reasonable emprcal smplfcaton. We can derve the correspondng set of Euler condtons that must hold for the observed base alfalfa-cotton rotaton. Varatons on the base rotaton system result from economc factors ncludng changes n relatve prces, costs, resource constrants, or changes n feld-specfc physcal captal. We consder the effect of changes n some of these factors n the polcy smulatons n the followng sectons of ths paper. Gven n felds we can wrte the correspondng set of Euler equatons whch must hold at any tme t, where we have suppressed the tmesubscrpt. (4) n, a c n, a c (5) n, c a n, a a Equatons (4) and (5) descrbe the frst-order swtchng condtons of the feld-level crop rotaton problem. We dscuss estmaton of these condtons n a subsequent secton. Dynamc Calbrated Optmzaton Model The rotaton model tells us the dynamc swtchng cost of the cotton-alfalfa rotaton. To estmate sustanablty and grower response to polces, we requre a framework that allows for smulaton of polcy alternatves. We specfy a calbrated optmzaton model of the alfalfacotton farm-wde operaton. We specfy the calbrated optmzaton farm model usng the method of Postve Mathematcal Programmng (PMP), after the procedure n Howtt et al (2012). We make a modfcaton to the calbraton procedure to account for the dynamcs of crop rotaton. Standard PMP uses avalable data n the form of supply elastctes, average producton costs, and observed land use n order to calbrate model parameters. The calbrated model reproduces the observed base year soluton n nputs and outputs. Ths procedure works as expected so long as the observed producton process s n steady-state equlbrum. We argue that the standard PMP procedure omts the dynamc effects of crop rotatons. We consequently adjust the calbraton process. 8

9 Cotton and alfalfa producton, n the absence of rotaton effects, follows a Constant Elastcty of Substtuton (CES) producton relatonshp. The CES allows for lmted substtuton between nputs. We allow four nputs to producton: land, labor, other supples (fertlzer), and water. Water s avalable from two sources, groundwater and surface water. We can wrte the CES as, /, land, land, labor, labor, supply, supply, water, water, Y x x x x and we use exponental PMP land cost functons, (6) x, land ( ) e. C x (7) The calbraton procedure follows the standard three-step PMP procedure detaled by Howtt et al (2012). To account for the observed rotaton we nclude the observed lagged crop acreage n the frst-step calbraton procedure. We specfy the rotaton problem wth a one-perod lag, thus the prevous season observed acreage allocaton nfluences current season yelds. We assume that each feld on the farm has homogenous producton condtons (absent rotaton effects). Consequently, t s the average proportonal allocaton of acreage that contans necessary rotaton nformaton and we can calbrate the farm-wde model by takng a rotaton-adjusted average yeld for the two crops. We defne XBt, as observed land use n tme perod t for crop. We calbrate the model to perod t, takng nto account lagged crop acreage n tme perod t-1 and the estmated rotaton swtchng costs. Let denote the proporton of felds n frst, second, or thrd year alfalfa and we can defne the rotaton yeld adjustment, R t,, for alfalfa, R ta, XB (1 ) XB a, a t 1, a a, c t 1, a, (8) XB t1, a and for cotton, R tc, XB (1 ) XB (1 ) XB c, c t 1, c t 1, a c, a t 1, a. (9) XB t1, c Equaton (8) shows the farm-wde average alfalfa yeld after accountng for the observed rotaton. Some proporton of felds,, are alfalfa stands n frst, second, or thrd year of producton. These felds wll stay n alfalfa for perod t, gven our assumpton of a four-year alfalfa rotaton. We can let 0.75 f we assume an ntal unform dstrbuton of felds across the farm. Equaton (9) shows the farm-wde average cotton yeld. Agan, we observe some 9

10 proporton of felds rotated from alfalfa nto cotton (1 ) and t follows that the other felds were rotated as contnuous cotton. Let yt, denote observed average yeld n perod t, then we calculate the rotaton-adjusted yeld for cotton and alfalfa n the calbraton perod t as, YR y R (10). t, t, t, The resultng calbraton problem s what we term quas-dynamc PMP. We proceed wth the three-step calbraton procedure and sequental dagnostc process to frst derve the shadow values on the resource and calbraton constrants, calbrate the exponental PMP cost functons gven observed average response elastctes, and fnally calculate the CES parameters. The quas-dynamc aspect of ths calbraton procedure s that the observed yelds n the base-year t are adjusted to account for lagged observed plantng decsons. Consequently, the dynamc rotaton effects are brought nto the resource and calbraton shadow values and carred through the PMP calbraton procedure. We can wrte the calbrated farm proft maxmzaton problem for any perod t (suppressng the tme subscrpt) as max p Y C ( x ) x subject to x x, j, land j, j jland B, water j GW SW (11) We have defned x, jas amount of nput j used to produce crop at cost j, and resource avalablty constrant. We let water 10 B j as the be equal to the weghted average water cost between surface and groundwater n the regon. Fnally, GW and SW correspond to groundwater and surface water use on the farm. The grower manages the farm to maxmze profts subject to resource avalablty, resource costs, and average crop prces. In ths framework we are able to estmate the statc ntensve and extensve margn response to polces. Addtonally, we have ncorporated the dynamc margn of adjustment through the lagged observed crop plantng decsons and yeld swtchng costs. The model wll calbrate to any year t observed land and nput use, gven the observed lagged crop acreages.

11 Dynamc Ntrate and Groundwater Model Fnally we lnk the calbrated farm proft maxmzaton model, wth rotaton swtchng costs, to a dynamc framework for ntrates and groundwater. Ntrogen-based fertlzers are essental for modern croppng rotaton systems and help ensure that we are able to meet global food demand. However, when fertlzer applcaton exceeds plant demand and dentrfcaton capacty of the sol ths can lead to leachng nto the groundwater. Pont and non-pont sources of ntrate contamnaton nclude fertlzer and manure, dssolved ntrogen n rrgaton and septc tanks and dary lagoons. In ths paper we model a smplfed dynamc framework for ntrate concentratons n groundwater and solate the effect of fertlzer applcaton on the Kern County farm over an assumed closed groundwater basn. We model a system where ntrogen s appled as fertlzer to the crops n the rotaton as a fxed average requrement n pounds per acre. We assume zero runoff and a fxed proporton for plant uptake, leavng the remanng proporton avalable for leachng. Dentrfcaton, mmoblzaton, and mneralzaton are modeled as a functon of appled water and depth to groundwater. Increased rrgaton water applcaton ncreases leachng, and ncreased depth to groundwater leads to greater dentrfcaton. Alfalfa ncreases and cotton decreases avalable sol ntrogen and thus potental ntrate leachng. As the rato of alfalfa to cotton changes across the farm there wll be a correspondng shft n the total amount of ntrates leached nto the groundwater. The rate of change depends on the alfalfa-cotton rato and relatve dfferences n appled ntrogen and appled water. Defne ANt, as appled ntrogen to crop n perod t, and nf and nl as the crop use adjustment and leachng factors specfc to the farm. Recall that x, js amount of nput j used for crop and we can defne the ntrate dynamc equaton as: N t1 N t nf AN x N x t,, land t t,, water nl TDL RC Alfalfa has hgher evapotranspraton than cotton and thus requres more rrgaton water. We nclude a dynamc feedback between depth to groundwater and pumpng cost. We calculate the change n depth to groundwater as, t t (12) 11

12 1 (13) AF RC DP x t 1 t t,, water Where AFt 1 s acre-feet of recharge n perod t+1, RCt s recharge n perod t, DP s the deep-percolaton factor for each crop, and xt,, water s water appled to crop n perod t. Fnally, the change n depth depends on the area of the groundwater basn, BAS, and we can calculate the change n depth as AFt CHG t1. BAS (14) Pumpng costs wll vary wth changes n depth to groundwater. We calculate the total cost of groundwater as the sum of fxed, O&M, and energy costs. Energy costs are based on a blend of agrcultural power rates n the area. We can wrte average groundwater cost per acre foot n year t+1, as 1.02ce ct 1,, gw fxed OM TDLt, eff where fxed s the fxed cost of groundwater pumpng per acre-foot representng the (15) amortzed fxed costs of the pump and well. The parameter ce s the cost of electrcty per klowatt-hour. Well effcency ( eff ) captures the effectveness of wells to yeld water accountng for energy loss due to frcton, etc. OM represents operaton and mantenance costs. Fnally, TDLt s the total dynamc lft n feet n year t, whch ncludes both statc lft and dynamc drawdown. As the rato of alfalfa to cotton ncreases we expect the farm to use more water and apply less ntrogen. Consequently, ntrate contamnaton wll decrease and the groundwater table wll decrease. The converse s also true. For a sustanable polcy we seek to reach a steady-state balance between the rates-of-change of these two factors consstent wth long-run utlty maxmzaton. Data and Estmaton At ths pont we have specfed a seres of models used to estmate agrcultural sustanablty. The frst stage s the estmaton of a rotaton swtchng model. Ths model, whch uses the lagged crop as an ndcator varable for the current fertlty level, can be estmated ether by drect 12

13 observaton of crop swtchng behavor of optmzng producers, or ndrectly usng crop growth models to smulate the effects of lagged crops. In the second model stage, parameters descrbng the effect of the lagged crop are ncorporated nto a calbrated economc producton model that reflect the mplct crop producton costs, and calbrates to the base observed soluton. In the thrd stage of the model, the outputs from the calbrated producton model are used to drve the equatons of moton for groundwater use and fertlty changes. The fertlty equaton of moton s approxmated by ntrates leached nto the groundwater. The groundwater state varable s measured by the depth to groundwater and the equaton of moton s drven by natural recharge the system, aggregate pumpng levels, and the amount of deep percolaton to the groundwater basn. We estmate the models usng a set of geo-referenced land use data for Kern County, Calforna. Agrculture n the regon s rrgated wth water comng from State and Federal surface water projects and groundwater n addton to local surface supples. The data nclude all rrgated agrcultural land n Kern County between 1997 and We exclude years pror to 2000 due to a change n county reportng methods. On each feld and year we observe the crop grown, feld sze n acres, farm owner, and farm manager of the feld. We are able to unquely dentfy and track felds across tme. We extract a subset of felds n an alfalfa-cotton based rotaton n a regon of the Western-edge of Kern County. The estmaton procedure used n the dynamc rotaton problem s descrbed n MacEwan and Howtt (2013) and summarzed here. The dynamc frst-order condtons result n an ll-posed estmaton problem and we use the method of Generalzed Maxmum Entropy (GME) to estmate the parameters (Golan et al 1996). Gven that one has ncomplete observatons about a statstcal process, an nformaton-theoretc consstent method to recover parameters for nference s to mpose probablstc structure on the model n such a way that t s consstent wth observed data and mposes as lttle addtonal nformaton as possble. Followng ths prncple, we specfy a truncated unform support space for the parameters of plus or mnus 100 percent of average yeld and maxmze the Entropy measure of the system. Table 1 summarzes the estmated parameter values for the dynamc crop rotaton problem and shows the margnal yeld effect mpled from the observed rotaton when crop (row) follows crop (column). For example, cotton-cotton results n an average reducton of tons/acre whch s a 4.9% decrease n average yelds. When cotton follows alfalfa, average yelds ncrease by tons/acre, or 3.4% above average yeld. 13

14 Table 1. Crop Rotaton Parameters Yeld Adjustment (tons/acre) Alfalfa Cotton Alfalfa Cotton We estmate the calbrated optmzaton model usng the method of PMP as descrbed n Howtt et al (2012). The calbraton procedure nvolves three steps: a lnear program to calculate resource and calbraton dual values, a least-squares parameters calbraton program that ncludes exogenous acreage response elastcty estmates, and a seres of parameter calculatons used n the fully-calbrated CES model. Rotaton effects are ncorporated as a fourth-step where we calculate the average adjusted crop yelds, gven the observed rotaton, to ncorporate nto the frst-step of the PMP procedure. The end result s a fully-calbrated CES model that ncludes dynamc crop rotaton effects and satsfes the economc and numercal calbraton checks proposed n Howtt et al (2012). Table 2. Optmzaton Model Parameters Optmzaton Model Parameters Value Unts Alfalfa Acreage Response Elastcty Cotton Acreage Response Elastcty Alfalfa average AW 4.48 af/ac/year Cotton average AW 3.26 af/ac/year The ntrate and groundwater equatons of moton are parameterzed usng exogenous nformaton avalable from county data. The dynamc equatons work n a feed-back loop wth the calbrated optmzaton model and track ntrates and groundwater levels across tme. The combned framework allows us to estmate the statc ntensve and extensve margn, n addton the dynamc margn. Tables 3 and 4 summarze the ntrate and groundwater parameters used n the model. Table 3. Ntrate Parameters Ntrate Dynamcs Parameters Value Unts Ntrate Crop Use Factor 0.6 factor Ntrate Leachng Factor 100 factor Alfalfa Appled Ntrogen 0 lb/ac Cotton Appled Ntrogen 180 lb/ac Appled Ntrogen Cost $/lb 14

15 Table 4. Groundwater Parameters Groundwater Dynamcs Parameters Value Unts Deep Percolaton Factor Alfalfa 0.13 factor Deep Percolaton Factor Cotton 0.1 factor Natural Recharge 0.02 af/year Groundwater Basn Sze 1000 ac Aggregate Rotaton Dynamcs and Sustanablty Before we proceed wth polcy smulatons we want to emphasze the nteracton between rotatons, producton, and externaltes. In ths secton we fx the alfalfa cotton rato on the farm and consequently fx the average farm rotaton. Conceptually, we can smplfy our set of control varables to changng the rato of alfalfa to cotton acres over a gven unt of land. As the rato of alfalfa to cotton ncreases, the level of ntrogen appled decreases along wth the excess ntrates that percolate to the groundwater for two reasons. Frst, on average, cotton acreage s smaller and thus the total burden of ntrogen appled s reduced. Second, the ncreased alfalfa area wll fx ncreasng quanttes of ntrogen whch then can be utlzed by the cotton. At some pont n the alfalfa to cotton rato wll reach equlbrum where the quantty of ntrogen beng removed from the groundwater n the rrgaton water pumped s equal to the quantty percolatng down from the cotton acreage, and the net ntrogen appled to cotton through artfcal fertlzer s removed n the crop tself. Essentally when the cotton crop that was grown to make your jeans requres no more mported ntrogen than that emboded n the fber of your jeans, the system wll be n ntrate, and thus water qualty, steady-state. Fgure 1 llustrates the trade-off between ntrate polluton and groundwater levels, holdng other factors constant. As the rato of alfalfa to cotton ncreases there s a reducton n fertlzer applcaton and a correspondng decrease n ntrates leached nto the groundwater. However, alfalfa s also more water-ntensve than cotton and ths requres the grower to deplete the groundwater resource at an ncreased rate. As the depth to groundwater ncreases, the total dynamc lft and pumpng costs ncrease. A smlar logc apples to the use of water to grow crops, but nterestngly, the water use functon has the opposte slope to the ntrate functon for gven shft n the alfalfa to cotton rato. As n the ntrogen example, thnk of an ncrease n the alfalfa to cotton rato whch reduces the 15

16 Change n Groundwater Depth Change n Ntrates Leached ntrogen burden on groundwater, but wll, n ths case, ncrease the pumpng burden on groundwater due to the hgher evapotranspraton requrements of alfalfa. Alfalfa s typcally rrgated twce a month for seven months per season, as opposed to cotton whch has a shorter growng season and s delberately stress rrgated at the end of the season to mprove the yeld n terms of the rato of boll to vegetatve growth. Fgure 1. Change n Ntrates and Groundwater Depth as Rotaton Shfts Rato of Alfalfa to Cotton Rato of Alfalfa to Cotton We want to strke a steady-state balance between groundwater depleton, ntrates, and farm proftablty. As the grower shfts to a more alfalfa-dependent rotaton ths leads to lower 16

17 Ntrate Concentraton (mg/l) proftablty. Although cotton s the hgher-value crop, alfalfa monoculture wll further decrease yelds. Polcy Smulatons In ths secton we smulate the dynamcs of the rotaton system. We smulate the model for the years usng observed acreage decsons for as ntal condtons for the rotaton dynamcs. In all smulatons we hold relatve crop prces and producton costs constant to solate the effect of water qualty and quantty. Surface water supples are held constant at the calbrated baselne average. In future teratons of ths model we ntend to nclude stochastc surface water delveres, reflectng the uncertan nature of dstrct delveres and seasonal precptaton. Smlarly, average crop yelds are assumed known and held constant n the analyss. Ntrate Dynamcs An acceptable level of ntrates n the groundwater s n the range of 3 5 mg/l. Concentratons above ths level are known to ncrease the rsk of dsease and cancer. We normalze the level of ntrates ntally n the groundwater basn to be 5 mg/l to smulate a groundwater basn wth exstng hgh levels. Fgure 2 llustrates ntrate levels n the groundwater under baselne condtons from As shown, there s a generally ncreasng trend n ntrates n the groundwater basn. We can see the cyclcal rotaton underlyng the smulaton and the system trends toward a level of just under 10 mg/l concentraton, above acceptable levels. Fgure 2. Groundwater ntrates under no polcy acton

18 Ntrate Concentraton (mg/l) It s clear from fgure 2 that the level of ntrates n the groundwater exceeds acceptable levels under the current producton system. A commonly proposed strategy to reduce ntrates s to dscourage excess applcaton of fertlzers. We can smulate ths effect wth a tax per pound appled ntrogen. To llustrate ths stuaton we mpose a tax of $30/lb of appled ntrogen. Fgure 3 llustrates ntrate levels under the baselne smulaton and under a tax of $30 per pound of appled ntrogen. Fgure 3. Groundwater ntrates under no polcy and fertlzer tax Baselne Tax The fertlzer tax reduces the relatve proftablty of cotton and encourages a shft toward alfalfa. If the grower were only able to adjust along the statc ntensve and extensve margns then we would expect a correspondng decrease n cotton yelds and farm proftablty. The presence of a rotaton, ncluded n our model framework, allows for adjustment along the dynamc margn and substtuton of alfalfa for cotton as part of a mult-year rotaton. Ths may partally or totally offset some of the yelds losses, thereby allowng the grower to move toward more sustanable groundwater ntrates at a lower cost to farm profts and the regonal economy. Fgures 4 and 5 llustrate the ntensve margn response to a fertlzer tax. Baselne average appled ntrogen per acre for cotton s 180 pounds per acre. Under a fertlzer tax we see a substantal ntensve margn response as the grower reduces appled ntrogen. Ths has a 18

19 Appled Ntrogen (lb/ac) Appled Ntrogen (lb/ac) correspondng cotton yeld effect (movement around the CES surface) whch can be partally offset by ncreased alfalfa rotaton. We see agan the cyclcal effect of crop rotaton and reducton n fertlzer use, decreasng from an average of 171 lb/ac to 121 lb/ac. Fgure 4. Cotton appled ntrogen per acre under no polcy acton Baselne Fgure 5. Cotton appled ntrogen per acre under fertlzer tax Tax Groundwater ntrate levels are reduced under a fertlzer tax. The grower substtutes away from cotton toward ncreased alfalfa n the rotaton and there s a correspondng effect on both cotton and alfalfa yelds. Importantly, alfalfa has hgher water requrements than cotton, whch ncreases groundwater use and may further deplete the aqufer. A steady-state soluton requres balancng grower proftablty, ntrate levels, and groundwater use. 19

20 Change n depth (ft) Total Dynamc Lft (ft) Groundwater Dynamcs In addton to ntrate levels we are also nterested n management of the underlyng aqufer. The farm receves annual surface water delveres that are suffcent to meet less than 40 percent of observe base-year crop demand. The grower makes up the remanng water requrements by pumpng groundwater. Increased pumpng leads to drawdown of the aqufer and ncreased dynamc lft, whch translates nto hgher pumpng costs and lower farm profts. Fgure 6 llustrates the (dynamc) depth to groundwater under the baselne polcy smulatons. Fgure 6. Baselne aqufer dynamc depth In the absence of polcy nterventon we see the aqufer trendng toward an apparent steadystate of around 300 feet of dynamc lft. Wth pump effcency, mantenance costs, and electrcty prces ths translates nto a cost of $123 per acre foot of rrgaton groundwater. By comparson, surface water delveres are avalable n the regon between $20 and $70 per acre foot, we use a cost of $30 per acre foot n the model. Fgure 7 llustrates the annual change n total dynamc lft. Fgure 7. Annual change n depth

21 Total Dynamc Lft (ft) Agan, we see the cyclcal pattern drven by the underlyng alfalfa-cotton rotaton. There s large varaton n year-to-year change n dynamc lft and a correspondng change n pumpng costs. Of course, the farm n our example s assumed to le over a closed groundwater basn whch s why we see such large seasonal varablty. To prevent drawdown we can mpose a groundwater extracton charge on the grower. Here we smulate the model wth a charge of $25 per acre foot. Fgure 8 llustrates the results. Fgure 8. Groundwater depth under baselne and groundwater charge A groundwater pumpng charge of $25 per acre foot reduces the depth to groundwater. The grower reduces appled water per acre and substtutes toward the less water-ntensve cotton n the rotaton. The average total dynamc lft over the smulaton perod decreases from 294 feet to 269 feet. It s of note that some of the more stable groundwater basns n Calforna mpose a groundwater charge on agrcultural users. In Santa Clara County, the Santa Clara Valley water dstrct mposes a charge of just under $20 per acre-foot. Ths combned wth other management efforts has worked to stablze groundwater levels n a regon hstorcally plagued by substantal subsdence. 21

22 Total surface water supples are held constant n the analyss. If nstead we were to smulate addtonal surface water avalable we could estmate the correspondng change n groundwater depth and the mplct benefts of the addtonal surface water. As wth the ntrogen tax, there are ntensve and extensve margn adjustments underlyng the polcy smulatons. Summary and Steady-State Soluton and Socal Costs One defnton of a steady-state sustanable soluton requres that the rate of change n ntrates and depth to groundwater are constant and consstent wth long-run utlty maxmzaton for the grower. We can alternatvely conceptualze sustanablty from a physcal perspectve and mpose threshold levels of groundwater levels and ntrates. We wll extend ths work n the future to model the formal dynamc steady-state soluton. Other areas for further refnement nclude ncorporatng a sequental dynamc PMP procedure, ncorporatng a dynamc feedback between groundwater qualty and crop yelds and producton costs, and expandng the model from the farm to a larger basn. In summary, we have proposed a three-step estmaton framework to evaluate sustanable polces. The framework allows for estmaton of the ntensve, extensve, and dynamc margn response. We estmate grower response to polces desgned to encourage a more sustanable agrculture and dscuss further steps to estmate the full steady-state sustanable soluton. In the framework presented n ths paper, to reach steady-state of fertlty can only be acheved by rotatng net ntrogen usng crops, whch may or may not need addtonal fertlzer, wth net ntrogen ncreasng crops such as legumes. In general, net ntrogen usng crops such as cotton are characterzed by hgher profts per unt land, and lower water use per unt land. Conversely, net ntrogen ncreasng crops such as dry beans or alfalfa, have lower net revenues per acre than cotton, but a hgh water requrement per acre than cotton. We have shown ths explct trade-off between the rates of change of fertlty stocks and resource stocks and we have presented the sustanablty problem as one of smultaneously strkng a steady-state balance between the rates of fertlty change and changes n stocks of groundwater. We feel that the framework presented n ths paper wll be of value for evaluatng sustanable ntensfcaton strateges, for example n developng countres. In most cases the adopton of crop ntensfcaton technologes results n greater use of ntrogen and water. Sustanable ntensfcaton recognzes that, to offset these, we must have a change n the rotaton to preserve 22

23 the physcal steady-state between groundwater and ntrogen use. Intensfcaton that s also sustanable must balance both the rotaton and the productvty of new technology. 23

24 References Ghassem, Feredoun A., A.J. Jakeman, and H.A. Nx Salnsaton of land and water resources: Human causes, extent, management and case studes. UNSW Press, Sydney, Australa, and CAB Internatonal, Wallngford, UK. Gleeson, T., Yoshhde Wada, Marc F.P. Berkens, and Ludovcus P.H. van Beek Water balance of global aqufers revealed by groundwater footprnt. Nature (488) (August): Golan, Amos, George Judge, and Douglas Mller. (1996), Maxmum Entropy Econometrcs: Robust Estmaton wth Lmted Data, Chchester, England: John Wley & Sons. Harter, Thomas and Jay R. Lund (PI s) Addressng Ntrate n Calforna s Drnkng Water. Prepared for the Calforna State Water Resources Board. Calforna Ntrate Project, Implementaton of Senate Bll X2 1. Center for Watershed Scences Unversty of Calforna, Davs. Howtt, Rchard. E., Josué Medellín-Azuara, Duncan MacEwan, Jay R. Lund Calbratng dsaggregate economc models of agrcultural producton and water management. Envronmental Modellng & Software (38) (December): MacEwan, Duncan. and Rchard E. Howtt An Emprcal Model of Crop Rotatons. Unversty of Calforna. Department of Agrcultural and Resource Economcs. Davs, CA. Ptman, Mchael G., & André Läuchl. (2002). Global mpact of salnty and agrcultural ecosystems. Salnty: envronment plants molecules. Dordrecht: Kluwer, Schoups, Gerrt, Jan W. Hopmans, Chuck A. Young, Jasper A. Vrugt, Wesley W. Wallender, Ken K. Tanj, Sorab Panday Sustanablty of rrgated agrculture n the San Joaqun Valley, Calforna. Proceedngs of the Natonal Academy of Scences, USA (102): Sneed, Mchelle, Justn Brandt, Mke Solt Land Subsdence along the Delta-Mendota Canal n the Northern Part of the San Joaqun Valley, Calforna, U.S. Geologcal Survey Scentfc Investgatons Report , 87 p. Spence, Mchael A Blue Whales and Appled Control Theory, n System Approaches and Envronmental Problems, H. W. Gottnger (ed.), Vandenhoeck and Ruprecht (Gottngen). 24