The economics of sheep farming at northern latitudes. Potential effects of climate change


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1 The economics of sheep farming a norhern laiudes. Poenial effecs of climae change Absrac The paper sudies he economy and ecology of sheep farming when climae change and uncerainy may affec vegeaion quaniy. The analysis is a he farm level and includes wo differen caegories of he animals, ewes (adul females) and lambs. The model is formulaed in a Nordic economic and biological seing wih he crucial disincion beween he oudoors grazing season and he winer indoors season. During he oudoors grazing season, animals may face limied grazing resources so ha he animal densiy in addiion o emperaure, precipiaion and oher weaher condiions deermine he weigh and hence he value of lambs. The model is analyzed in wo seps, wihou aking uncerainy ino accoun and wih uncerainy. Because empirical evidence suggess ha climae changes, e.g. increased spring emperaure, would have conrasing effecs on lamb weighs, we compare he impac of such changes across specific sies. Keywords: sheep farming, weaher condiions, climae change, vegeaion growh, sage model, uncerainy 1
2 1. Inroducion IPCC projecions indicae ha emperaures will increase and he increase will be sronges a higher laiudes (IPCC 2007). These projecions indicae ha Nordic sheep farmers will face novel climae condiions in he fuure. Nielsen e al. (2012) showed ha e.g. increased spring emperaure would have conrasing effecs on lamb auumn body mass, depending on he area under sudy. This indicaes ha any aemp o include weaher condiions and climae change in opimizaion models for individual farmers has o be sie specific. To illusrae he effec of he spaially inconsisency in climae effec we include in our reasoning and models wo areas where he effec of increased spring emperaure has conrasing effec. Our aim is o illusrae how poenial climaic effecs may aler he opimal socking rae, he per animal slaugher values and profiabiliy of he farmers. Our sheep farming sudy is conduced wih a crucial disincion made beween he oudoors grazing season (spring, summer and fall) and he indoor winer feeding period, and differen caegories of animals (lambs and ewes) are included. Lambs are born in early spring, jus before he oudoor grazing season sars which is he ypical siuaion found in many srongly seasonal environmens a norhern laiudes such as in he Nordic counries and a high aliudes in coninenal Europe (e.g., mounainous areas in France and Spain). The analysis essenially relaes o he economic and biological seing found in Norway, bu should also have relevance for sheep framers in Iceland and Greenland. The problem analyzed here is o find he opimal number of animals o be fed and kep indoors during he winer season for a given farm capaciy (i.e., farm size). A corollary of his problem is o find he effec ha summer grazing sheep densiy has on vegeaion produciviy and hence on peranimal mea producion. The problem is analyzed under he assumpion ha he farmer aims o do i as well as possible, represened by expeced presenvalue profi maximizaion. The animal growh model presened in his paper builds on Skonhof (2008), bu is exended o include he effec on animal weigh gain from oudoors grazing condiions. Balancing he number of animals and weigh of animals is indeed seen as a crucial managemen problem in he Nordic counries (e.g., Olafsdoir and Juliusson 2000, Myserud and Ausrheim 2005, Thomson e al. 2005). However, here is considerable variaion among years in he weigh gain experienced by he animals during he oudoors grazing season, suggesing ha weaher condiions are affecing he qualiy and quaniy of he vegeaion as well as he behavior and physiology of he animals. Skonhof e al. (2010) exended he Skonhof s model (2008) o 2
3 include a relaionship beween vegeaion availabiliy and lamb weigh. Here we aim o develop his relaionship furher o include weaher condiions and uncerainy affecing he vegeaion. The uncerainy canno be conrolled by he farmer, bu increased knowledge on he relaionship beween paricular weaher variables and weigh gain migh enable he farmers o adap o changing climae condiions. In he naural resource and agriculural economics lieraure, here is an increased focus on he poenial effecs of climae changes. Dieker e al. (2010) analyzing he Barens Sea cod fishery assume ha climae changes are channeled hrough a emperaure variable affecing he recruimen of he cod sock, and where a higher emperaure improves he recruimen. There is no uncerainy included in his analysis, and he recruimen funcion simply shifs for a given value of he emperaure. Hannesson (2007) also sudies a siuaion where climae changes are maerialized hrough sea emperaure. His analysis is dealing wih poenial effecs on he migraion paern of fish beween differen jurisdicions. Quaas and Baumgärner (2012) sudy opimal livesock managemen in semiarid rangelands wih uncerain rainfall. They solve for he opimal socking rae (raio of livesock populaion o vegeaion biomass) under rainfall uncerainy when assuming ha increased rainfall increases vegeaion biomass. The presen sudy differs from he above conribuions in wo ways. Firs, we consider climaic facors (i.e., emperaure) as having no direc impac on animal recruimen bu as derimenal o lamb weighs. Furhermore, we presen an agespecific model consising of adul animals and lams. Second, along wih empirical findings, we consider increased spring emperaure as having a posiive or negaive effec on lamb weighs depending on he specific sie of consideraion. We sudy how emperaure changes effecs may aler he opimal slaughering composiion (lamb and aduls), he socking rae, and profiabiliy of he farmers. This paper is organized as follows. Secion 2 describes briefly he Nordic sheep farming sysem. Secion 3 provides informaion abou sheep animal growh and presens he biological model. While animal populaion growh is unaffeced by poenial climae effecs and uncerainy, weigh growh per animal is affeced and his relaionship is discussed in Secion 4. The revenue and cos funcions follow in Secion 5. The socking problem of he farmer is solved under he assumpion of no uncerainy in Secion 6 while Secion 7 provides a 3
4 numerical illusraion. The socking problem in presence of uncerainy is analyzed in Secion 8. Secion 9 finally summarizes our findings. 2. The Nordic sheep farming sysem There are approximaely 16,000 sheep farms in Norway, all family farms. Because here are around 2.1 million animals during he oudoors grazing season, he average farm size only accouns for some 130 animals during he summer. Norwegian farms are locaed eiher close o mounain areas and oher sparsely populaed areas or along he coas, wih a means o ranspor sheep o more disan alpine areas for summer grazing. The main produc is mea, which accouns for abou 80% of he average farmer s income. The remainder comes from wool, because sheep milk producion is virually nonexisen oday (Nersen e al. 2003). In Iceland, here are abou 450,000 winerfed animals oday. Mea is also he mos imporan produc from sheep farming here. In Greenland, he land for sheep grazing is much more resriced, and he populaion of ewes in 2007 was esimaed a 25,000 (Ausrheim e al. 2008a). Housing and indoor feeding is required hroughou winer because of snow and harsh weaher condiions (Figure 1). In Norway, winer feeding ypically consiss of hay grown on pasures close o farms (80%), wih he addiion of concenrae pelles provided by he indusry (20%).. The spring lambing scheme is conrolled by he farmers because of he In Viro Ferilizaion proocol used o ime he lambing o fi curren climaic condiions (REF??). In lae spring and early summer, he animals usually graze on fenced land close o he farm a low elevaions, ypically in he areas where winer food for he sheep is harvesed during summer. When weaher condiions permi sheep are released ino rough grazing areas in he valleys and mounains. Plan spring phenology is driven by he amoun of snow he previous winer, bu also emperaure and precipiaion in spring (Nielsen e al. 2012). In Norway, mos sheep (abou 75% of he oal meabolic biomass) graze in he norhern boreal and alpine region (Ausrheim e al. 2008b). The oudoors grazing season ends beween lae Augus and he middle of Sepember. The lengh of oudoor he grazing season is relaively fixed, parly because of local climaic condiions bu also, a leas in cerain areas, local radiions and hisorical reasons play a role in he iming. In general he oudoor grazing season does no exceed 130 days. During he rough grazing period, flocks may be vulnerable o accidens and disease and in some regions 4
5 also o large predaors (Aunsmo e al. 1998, Nersen e al. 2003). A leas in heory he sheep are aken down from he mounains when he animals growh curves are flaening ou (Svalheim e al. 2004). Consequenly, among years variaion in local weaher induces addiional uncerainy for boh he exen of he grazing season and he naure of he animals growh curve on he oufield pasures. Afer he grazing season, he animals are musered and he wool is shorn. Slaughering akes place immediaely or afer a period of grazing on he farmland (more deails are provided in Ausrheim e al. 2008a). The seasonal subdivision is similar in Iceland and Greenland. Figure 1 abou here 3. Biological model The sheep animal growh model is formulaed a a discree ime wih a seasonal subdivision beween he oudoors grazing period (spring, summer and fall) and indoors winerfeeding period. The sheep populaion is srucured (e.g., Caswell 2001) as ewes and lambs. The farmers are in full conrol of he socking rae, as feriliy and he number of animals released in spring are unaffeced by weaher condiions. All naural moraliy is supposed o occur during he grazing season and is also assumed o be independen of summer grazing and weaher condiions. Accordingly, he growh in he number of animals is independen of grazing condiions and no subjec o uncerainy. Naural moraliy differs beween aduls and lambs, and is considered fixed and densiy independen. Lambs no slaughered, eners he adul populaion afer he slaughering period (i.e., Sepember Ocober). All male lambs are assumed o be slaughered because very few (or none when arificial inseminaion is praciced) are kep for breeding. Therefore, only female aduls are considered. Demographic daa on sheep are available in Myserud e al. (2002). The number of adul females in year ( 1) afer he slaugher, consiss of he previous year s aduls and female lambs ha have survived naural moraliy and have no been slaughered. This is wrien as 1 (1 ) (1 s h s h ), where is he number of female lambs, s and and s are he naural survival fracions of adul females and lambs, respecively, and h are he fracions slaughered. Wih he fecundiy rae b (lambs per adul female) and as he fracion of female lambs recruied ( is usually close o 0.5), b yields he h 5
6 number of female lambs. Therefore, when ignoring he possibiliy of addiional animals from ouside, he ewe populaion growh is governed by: (1) 1 (1 ) (1 b s h s h ). Because he populaion growh equaion (1) is linear for number of animals, here are infinie combinaions of harvesing fracions ha susain a sable populaion. Therefore, for a consan number of animals 1, we have: (1 ) bs (1 h ) s (1 h ), or simply 1 bs (1 h ) s (1 h ) when 0 (see Figure 2). This isocline inersecs wih he, which may be above or below 1. Therefore, he h axis a [1 (1 bs ) / s ] highes adul slaugher rae compaible wih zero animal growh is min{1,[1 (1 bs ) / s ]}. For all realisic parameer values, i is below 1 (see numerical secion), and his is assumed o hold in he subsequen analysis. The isocline inersecs wih he h axis a [1 (1 s ) / bs ] 1 and is hence he highes lambslaughering rae compaible wih equilibrium. Figure 2 abou here 4. Weaher condiions, herbivore performance, and weigh gain High grazing pressure may cause a reducion in plan qualiy and/or quaniy (Myserud and Ausrheim 2005). However, moderae grazing is expeced o faciliae plan biomass producion in producive habias and hus he fodder availabiliy for moderae grazing as compared wih no grazing (McNaughon 1979). In low produciviy ecosysems in paricular, such as he one considered here, heavy grazing may favor heavily defended, nonpalaable plan species o he derimen of palaable species (Ausrheim e al. 2007). Increased dominance of such species affecs he srengh of densiydependen effecs on he weigh growh of sheep in he long erm. The farmer may hus increase curren sock numbers a he expense of growh rae in subsequen years. Indeed, wih increasing densiy of sheep on pasure, a higher proporion of lowqualiy plan species (Kausrud e al. 2006) and vegeaion ypes (Mobæk e al. 2008) may resul. 6
7 In he Norwegian sheep farming sysem, he major growh season of he animals is when hey roam freely in he mounains and hence, he per animal produciviy is affeced by environmenal condiions (e.g. emperaure). I has previously been shown ha local weaher condiions during winer, in spring (before he animals are released o he mounains) and during summer (he oufield grazing season) affec lamb weighs (Nielsen e al. 2012). However, which weaher variables (snow deph, precipiaion or emperaure) ha is of imporance varies among Norwegian mounain ranges; no only in srengh bu also in direcion. Increased precipiaion in spring and summer on he wes side of he mounains of Souhern Norway (high precipiaion) is negaive for lamb growh, while he effec of precipiaion is posiive on he drier eas side of he mounains. In he norh, increased emperaure in spring and summer implies increased lamb growh while he effec of emperaure is negaive in he souh and souhwes (Nielsen e al. 2012). Since he effec of cerain changes in weaher condiions are sie specific we model wo paricular areas where he effec of change differ. We focus here on emperaure, bu he exercise could be done on any measure of local weaher condiions where is relaionship o lamb growh is known. The relaionship beween lamb weigh gain during he grazing season year w, he spring/summer socking rae (1 b ) spring emperaure ΔT is formulaed as: (2) w w ((1 b), T) and he effec of a cerain change in ΔT = 0 represens he siuaion as i is oday while ΔT > 0 indicae a posiive and ΔT < 0 a negaive effec of a cerain fuure climae change. We focus on wo mounain ranges; Forollhogna (norh) and Hardangervidda (souhwes), where increased spring emperaure has been shown o have a posiive and negaive effec, respecively, on lamb growh. Nielsen e al. (2012) found ha for an increase in average spring emperaure of 1 C he average lamb weigh would increase wih 0.37kg (ΔT = 0.37) in he norh and decrease wih 0.69kg (ΔT = ) in souhwes. Though hey modeled lamb auumn body mass, we use he same esimaes o illusrae he weigh gain hrough he summer. A negaive relaionship beween he socking rae and lamb auumn weigh is wellesablished (Myserud e al. 2011), also in our focal areas (Nielsen e al. 2012); ha is, we ypically have w / ((1 b) ) 0. However, his funcional form may also be concave and increasing for a small socking rae, reach a peak value and hen decline. This migh be he case if a cerain 7
8 level of grazing pressure faciliaes plan growh and increases he qualiy and/or quaniy of he vegeaion available o he animals (e.g. Myserud e al. 2011). The effec of ΔT is defined such ha a higher value (ΔT > 0) shifs he weigh funcion upwards, while a negaive value (ΔT < 0) shifs i downwards irrespecively of he form he funcion akes (more deails numerical secion 7). The weigh gain of lambs during he grazing season coincides wih he weigh a he end of he season; ha is he slaugher weigh. Therefore, equaion (2) represens he lamb slaugher weigh (kg per animal). In he weigh growh model (2), ΔT is assumed o be consan across ime. In he nex version of he model, uncerainy is included, and we hen have: (2 ) w w ((1 b), T, ) where is a sochasic variable, assumed o be independen and idenically disribued (i.i.d.) 2 over ime wih mean zero and variance. For he aduls, here is generally no weigh change during he grazing season on producive pasures while here may be some loss in low produciviy areas (Myserud and Ausrheim 2005). However, as a reasonably good approximaion, we simply neglec any possible connecion beween he amoun of vegeaion and weigh, and herefore also weaher facors and weigh. The adul slaugher weigh is herefore simply fixed and deermined ouside he model and given as: (3) w w. 5. Revenue and coss We disregard income from wool producion, and mea sales are hen he only revenue componen for he farmer. Because slaughering akes place afer naural moraliy, he number of ewes and female lambs removed are s h and b s h, respecively. As menioned above, he enire male lamb subpopulaion (1 ) bs is slaughered. The number of animals removed year is hen H b s ( h 1 ) s h. Wih p as he ne (of slaughering coss) ewe slaughering price (NOK per kg) and p he lamb ne slaughering price, boh assumed o be fixed and independen of he number of animals supplied a he farm level, he curren mea income of he farmer is given by R [ pwbs( h 1 ) p w sh ]. 8
9 The cos srucure differs sharply beween he oudoor grazing season and he indoor feeding season, and he indoor coss are subsanial higher. Throughou his analysis, we assume a given farm capaciy. Therefore, he coss of buildings, machinery and so forh are fixed (see also below). The indoor season variable coss include labor (ypically an opporuniy cos), elecriciy, and veerinary coss in addiion o fodder. I depends on he indoors sock size and is given as C C( ). The cos funcion is assumed go be increasing and convex; ha is, C ' 0 and C '' 0. As indicaed, during he grazing period he sheep may graze on communally owned lands ( commons ) or privae land. Wihin he Nordic sheep farming sysem, such land may be available cos free, or he farmer may pay a fixed yearly renal (Ausrheim e al. 2008a). There may be some ransporaion and mainenance coss, bu such coss are negleced because hey are generally raher low. The oal yearly variable cos is hence simply assumed o be he indoor season cos. Therefore, when ignoring discouning wihin he year, he curren profi of he farmer is described by: (4) ( 1 ) R C p w bs h p w s h C( ). 6. The opimal program when no uncerainy Opimaliy condiions We sar o analyze he siuaion wihou uncerainy, and hence equaion (2) represens he per lamb weigh relaionship. We assume ha he farmer is well informed and raional, and aims o maximize he presen value of profi over an infinie ime horizon, 0 under he biological consrain in (1). 1/(1 ) is he discoun facor wih 0as he (yearly) fixed discoun rae. Insered from (1), (2), and (4), he Lagrange funcion may be wrien as 0 L p w ((1 b), T) bs ( h 1 ) p w s h C( ) 1 11s (1 h ) bs (1 h ) where 1 0 is he animal shadow price. Following he KuhnTucker heorem he firsorder necessary condiions of his problem (assuming 0 ) are: (5) L/ h ( p w 1) 0 ; 0 h 1, 9
10 (6) L/ h [ p w ((1 b), T) 1] 0 ; 0 h 1 and (7) L/ p bs ( h 1 ) w ((1 b), T) ( w / )(1 b) p w s h C'( ) 1 s (1 h ) bs (1 h ) 0. The conrol condiion (5) saes ha he slaughering of he aduls should ake place up o he poin where he per animal value is below, equal or above he cos of reduced growh in animal numbers, evaluaed a he shadow price. The lamb conrol condiion (6) is analogous. Equaion (7) is he porfolio condiion, which saes ha he number of female aduls is deermined so ha he immediae ne reurn on adul females equals he shadow price of naural growh. The firs erm in he firs bracke reflecs ha increased animal numbers increases he oal mea weigh, whereas he second erm accouns for he marginal cos of increased animal numbers due o reduced weigh per lamb. From he conrol condiions (5) and (6) i is eviden ha he per animal slaugher value seers he slaugher composiion and no oher facors, e.g., feriliy, play a direc role. If he demand condiions are in favor of lambs, hen p p (see he numerical analysis). If, in addiion, he climaic condiions are favorable so ha weigh of he lambs w ((1 b), T) is high, hen he per animal slaugher value of he lambs exceeds ha of he ewes. Tha is p w ((1 b), T) p w. The conrol condiions hen indicae a higher harvesing fracion of he lambs han he ewes. This can be saisfied in hree ways: i) h 1and 0 1, ii) h 1and h 0 and iii) 0 1and h 0. On he conrary, wih h unfavourable climaic condiions and low lamb weigh, such ha h p w p w ((1 b), T) and hence a more aggressively harvesing of he aduls, we find ha he conrol condiions can be saisfied eiher as iv) h 1and 0 1, v) h 1and h 0, or as vi) 0 1and h 0. We may also find ha he lamb weigh is such ha h he value per lamb equals he value per ewe, p w ((1 b), T) p w. This is possibiliy vii) and can be saisfied by a cerain combinaion of slaugher raes in he domain 0 1 and 0 1(more deails below). When having deermined he slaughering raes from (5) h h h 10
11 and (6), he conrol condiion in (7) deermines he number of animals which hen mus be consan over ime. Seady sae analysis In a possible (inerior) seady sae where all variables are consan over ime wih a high lamb weigh and p w ((1 b), T) p w (he ime subscrip is dropped when considering seady sae), we find ha he above conrol condiions can be saisfied only as possibiliy iii), 0 h 1and h 0 because slaughering all he lambs is no an opion in a possible seady sae. See equaion (1 ) and Figure 2. A corollary of * h 0 is ha (female) lamb slaughering should ake place a he highes level compaible wih he sheep populaion * equilibrium; ha is, h 1 (1 s x )/ bs y 1. The opimal slaughering rae depends herefore on biological condiions only (naural moraliy and feriliy). Single sage slaughering was he opimal harvesing sraegy derived by Skonhof e al. (2010). They argued ha his resul occurred because hey considered harves benefi as linear in boh harves conrols and because of he lack of any densiydependen effecs in he animal growh funcion. In he presen, he addiional condiion is ha he slaugher value per lamb is above ha of he slaugher value per ewe slaugher (see above). * Lambs no slaughered ener he ewe populaion nex spring. When insering h 0, * 1 (1 )/( * h s bs ), and for pw((1 b ), T) / from (6) ino (7) and rearranging, he opimal number of animals o keep during he indoor season is deermined by he equaliy p ( bs s 1 ) w ((1 b), T ) C '( ) p ( bs s 1)(1 b) ( w / ). The lef hand side is he marginal benefi of saving animals for nex season lamb slaughering ne of he discoun rae, and reflecs ha saving an addiional animal increases he oal number of lambs available for slaughering nex year. The righ hand side is he marginal cos of keeping animals for he nex season, and equals he sum of he cos of keeping an addiional animal indoor plus he weigh loss an addiional animal impose on all lambs. Noe ha economic parameers, in addiion o biological parameers, influence he opimal number of adul animals. 11
12 In he opposie case of a low lamb weigh and more valuable ewes han lambs, he conrol x condiions in a possible seady sae can generally be saisfied eiher as case iv) wih h 1 x y x y and 0 h 1, case v) wih h 1and h 0, or case vi) wih 0 h 1and h 0. However, as already indicaed, seady sae slaughering of all aduls is no a possible opion due o he acual demographic parameer values (again, see numerical secion). Therefore case vi) wih * h x 1 (1 bs y )/ s x 1and y* h 0 will be he only seady sae possibiliy when aduls are more valuable han lambs. Tha is, female lamb slaughering equals zero whereas adul slaughering should ake place a he highes level compaible wih he susainable sheep populaion equilibrium, cf. equaion (1 ) and figure 2. When insering for he opimal seady sae slaughering values ino (7) and rearranging, he opimal animal populaion is deermined by he equaliy pbs (1 ) w ((1 b ), T) p w ( s bs ). C'( ) p bs (1 )(1 b) ( w / ). The lef hand side equals he marginal benefi of saving animals for nex season male lamb and adul slaughering, while he righ hand side equals he marginal cos of keeping animals for he nex season. The second erm on he righ hand side indicaes ha keeping an addiional animal imposes a weigh loss on male lambs. Compared o above, he firs erm on he lef hand side reflecs ha saving an addiional adul increases he oal number of male lambs available for slaughering nex year. The second erm reflecs ha saving an addiional adul increases he adul populaion available for slaughering nex year. On he righ hand side, he second erm is he weigh loss imposed on fuure male lambs of keeping an addiional animal for he nex season. As already menioned opion vii) is me when p w ((1 b), T) p w, meaning ha he slaugher value per lamb equals he slaugher value per ewe. Then, his condiion alone deermines he opimal number of animals. In case of a peak valued concave weigh funcion (see numerical secion 8) here are generally wo socking raes giving he lamb weigh ha fulfills his equaion. However, a his weigh he higher socking rae is for obvious reasons he only candidae for represening a maximum value. The animal shadow price is hen given by p w / p w ((1 b), T)/. When insering his ino (7) and rearranging, we find p ( bs ) w ((1 b), T ) C '( ) p bs ( h 1 )(1 b) ( w / ). Tha is, once he opimal number of animals is deermined by he condiion above, hen he opimal 12
13 slaughering rae of lambs is deermined such ha he marginal benefi of saving animals for nex season lamb slaughering equalizes he marginal cos of keeping animals for he nex season. The second erm on he righ hand side is he weigh loss imposed on fuure female and male lambs of keeping an addiional animal for he nex season. When having deermined he opimal animal populaion and lamb slaughering, he opimal adul slaugher follows from equaion (1 ). To sum up, he opimal seady sae slaugher composiion depends only on he per animal slaugher values. If he demand and climaic condiions drive he slaughering value per lamb above ha of aduls, hen slaughering lambs only is opimal. Slaughering aduls only is, however, opimal if he slaughering value per adul exceeds ha of lambs. Finally, a combinaion of lamb and adul slaughering is opimal if he prevailing demand and climaic condiions resul in equalized per animal slaughering values. In our example from wo mounain ranges in Norway an increase in emperaure would imply more favourable condiions in he norh, bu unfavourable condiions in souhwes. Assume iniially ha all farmers are faced wih equal prices and climaic condiions favouring lamb slaughering only. Increased emperaure will hen have no impac on he slaughering composiion among farmers. However, he marginal benefi of saving animals for nex season lamb slaughering increases for hese farmers, and hence, he sheep populaion increases in norh. In souh, on he oher hand, farmers less likely o slaugher lambs only when faced wih a emperaure increase. Furhermore, he marginal benefi of saving animals for nex season lamb slaughering reduces, which simulaes souhern farmers o reduce he sheep populaion. Dynamic analysis Above some properies of a possible seady sae wih a consan number of animals hrough ime was sudied. We now ask if a dynamic soluion may be sable. From he animal growh equaion (1) 1 (1 ) (1 b s h s h ), or 1 Q( ), we find Q/ bs (1 h ) s (1 h ) which equalizes 1 in any seady sae. Therefore, from his we canno conclude wheher any seady sae of our sysem is sable or no. The dynamics is furher considered in he numerical secion Numerical illusraion 13
14 Daa and specific funcional forms To shed some furher ligh on he above analysis, he model is illusraed numerically, applying he abovespecified vegeaion naural growh funcion and animal consumpion curve. The sheep biological daa are based on a large se of observaions from Norwegian sheep farming, and he baseline parameer values are shown in Table 1. The ewe weigh is 30 (kg/animal) wih a mea marke slaugher value of 35 (NOK/kg). Therefore, he fixed ewe x x slaugher value is pw ,050 (NOK/animal) (Table 1). The lamb mea value is p 60 (NOK/kg). We assume furher a sricly concave mainenance cos funcion, C ( ) ( c/2), wih c 12 (NOK/animal 2 ). 2 Table 1 abou here In he baseline calculaions, he lamb weigh funcion is specified downward sloping as w w b T k k b T wih 0 1 ((1 ), ) (1 ) 0 k 22 and 1 k 0.1. As menioned in secion 4 above, evidence show ha local weaher condiions affec lamb weighs, bu he effec varies boh in srengh and direcion depending on he area under sudy (Nielsen e al. 2012). Therefore, he following numerical analysis is based on esimaed effecs of increased spring emperaure in wo paricular Norwegian mounain ranges, Norh and Souh: In he Norh case increased emperaure in spring increases he lamb weighs, while in he Souh case he relaionship is negaive. As menioned in secion 4, a emperaure increase of 1 C increases he average lamb weigh wih 0.37kg in he norh and decreases he lamb weigh wih 0.69kg in he souh (Nielsen e al. 2012). These values are used for he Norh and Souh case, respecively, in he numerical analysis. Hence, he wo cases illusrae how he possible biological and economic impac of increased emperaure on sheep farming varies beween he norhern and souhern pars of Norway. In he following, he opimal managemen policy is found for he baseline parameer values. Then we sudy he effecs of emperaure changes, as well as changes in he discoun rae and he cos parameer. If slaughering lambs only is opimal prior o a emperaure change, hen farmers experiencing increased lamb weigh will no aler he opimal slaughering combinaion in seady sae. However, he dynamics owards seady sae will be affeced. On he oher hand, farmers experiencing reduced lamb weigh may, evenually, change he 14
15 opimal slaughering combinaion. Also in his case, we will illusrae he effec on he dynamics owards seady sae. Resuls We firs presen he basic dynamic resuls. As already indicaed, because of he srong degree of lineariy in he model ogeher wih densiy dependen regulaion of he value gain of he animals hrough he weigh socking relaionship, he model approaches a sable equilibrium. As he profi funcion is linear in he harves when here is no uncerainy, economic heory suggess ha harves should be adjused such as o lead he populaion he seady sae as fas as possible (for proof of he Mos Rapid Approach Pah Theorem (MRAP), see Spence and Sarre 1975). Hence, if he number of ewes is below he opimal seady sae i will be no harves unil i has grown o he seady sae. On he oher, if i is above he seady sae will be slaughered down o he opimal level he firs year. Figure 3 demonsraes firs he socking rae dynamics wihou wih he baseline parameer values (see Table 1) bu when allowing for changes in he discoun rae. The resul here seems parly o confirm MRAP where he seady sae socking rae in baseline ( 0.03) of 123 animals is reached in year 3. Because he value per lamb exceeds he value per adul, i is opimal o slaugher only lambs in all years and he seady sae opimal lamb harvesing rae is See Table 2. No surprisingly, increasing he discoun rae resuls in progressively smaller populaions wih corresponding higher harvesing raes of lambs, while he dynamics do no change qualiaively 1. I is hence beneficial for he farmer, o inves less in he animal sock, and insead pu he process in he bank. We have also sudied he effecs of changing iniial sock size, and all he ime we find ha he sock size and harves approach he same seady sae (ergodic dynamics). Figure 3 abou here 1 Because he seady sae harvesing composiion is independen of he discoun rae, adul harvesing in seady sae is zero also in all hree scenarios. 15
16 Table 2 repors seady sae sock size, slaughering raes, lamb weigh, and he presen value profi over all periods for differen parameer values. In he baseline scenario mea prices and climaic condiions are in favour of lamb slaughering only (see also Figure 3). An increase in he price of lamb mea ( 70 NOK) will herefore have no impac on he slaughering composiion and hence, no impac on he slaughering raes. The laer is because he opimal slaughering raes depend on biological condiions only. See he seadysae analysis in secion 6. However, a higher price of lamb mea increases he marginal benefi of saving animals for nex season lamb slaughering and hence, he animal sock increases and he number of lambs slaughered increases compared o he baseline scenario. Therefore, alhough lamb weighs reduce, an increase in he lamb mea price increases he presen value profis. If, on he oher hand, he price of lamb mea is reduced ( 50 NOK) so ha he mea value per lamb falls below he mea value per ewe, hen slaughering ewes only is opimal. Now he marginal benefi of saving animals for nex season lamb slaughering reduces compared o he baseline scenario and hence, he seadysae animal sock reduces. The laer has a negaive effec on he number of animals slaughered and herefore reinforces he direc negaive effec of reduced price of lamb mea on presen value profis. A sufficien increase in he price of ewes works in he direcion of slaughering ewes only ( 45 NOK). Then, a higher price of ewes simulaes farmers o keep more animal for fuure ewe slaughering compared o he baseline scenario. The presen value profis increases bu less compared o he case of a higher lamb mea price ( 70 NOK). The reason is ha when increases aduls are slaughered a a higher price while male lambs are slaughered a he baseline price, whereas when increases all slaughered animals (male and female lambs) are sold a a higher price. An increase in he cos of keeping animals indoors over he winer season (increased ) simulaes farmers o save less animals for nex season slaughering, and hence, he sheep populaion reduces. The increase in lamb weigh canno compensae for a reduced animal sock and hence, he presen value profis reduces. Table 2 abou here 16
17 Nex we sudy he effec of increased emperaure. See he Table 3. Consider firs he Norh case where a higher emperaure increases he lamb weighs. Increased weighs increase he ne income of lamb slaughering for a fixed sock size. Furhermore, higher emperaures simulae farmers o keep more animals over he winer for nex season lambing and slaughering, which in urn increases he number of lambs and adds an addiional posiive effec on he profi. This means ha increased emperaure involves a double dividend for sheep farmers in he sense ha i increases he slaughering value per lamb and increases he number of lambs slaughered. A a emperaure increase of 1 C, he direc effec increased weighs adds NOK o he slaughering income, whereas he indirec effec working hrough a changing sock size is sronger and increases he slaughering income by NOK. Finally we sudy he Souh case where increased emperaures affec lamb weighs negaively. A small emperaure increase (e.g., 1 C) reduces he lamb weigh bu no enough o aler he opimal slaughering composiion. Reduced lamb weigh reduces he ne income of lamb slaughering for a fixed sock size. Furher, farmers will keep fewer animals over he winer for nex season lambing and slaughering, which in urn reduces he number of lambs and shrinks he profi even more. A a emperaure increase of 1 C, he direc effec reduces he slaughering income by NOK, whereas he indirec effec working hrough a changing sock size reduces he slaughering income by NOK. A furher increase in he emperaure may drive he slaughering value per lamb below he slaughering value per adul animal and hence, shif he opimal slaughering composiion from lamb slaughering only o adul slaughering only. A firs his simulaes farmers o increase he winer sock. The reason is ha he marginal cos of keeping a larger sock in erms of reduced lamb weighs is of less imporance when (female) lamb slaughering is zero. However, all male lambs are sill slaughered and hence, furher emperaure increases simulae reducions in he winer sock because he marginal benefi of saving animals for fuure male slaughering reduces as heir weigh reduces. Table 3 abou here 8. Inroducing uncerainy 17
18 Equaion (2) says ha he lamb weigh can be prediced exacly from he curren socking rae and weaher condiions. These changes are in fac parly random. Equaion (2 ) says ha he curren socking rae and curren weaher condiions are known o he farmer wih cerainy, bu fuure weaher condiions and hence, fuure lamb weigh is (in par) random. The nex sep is o analyze how such uncerainy affecs he opimal slaughering composiion and he animal sock. In he nex version of he model, uncerainy is included, and we hen have: (8) w w ((1 b), T, ) where is a sochasic variable, assumed o be independen and idenically disribued (i.i.d.) 2 over ime wih mean zero and variance. Equaion (8) says ha climaic condiions and heir impac on weighs are random o he farmer. In wha follows we solve for he opimal slaughering program in presence of climaic uncerainy. We now depar from secion 6 above by assuming ha farmers are risk averse 2. Tha is, we assume ha farmer uiliy increases wih he profi a a decreasing rae i.e., U '( ) 0 and U ''( ) 0. Therefore, he farmer now aims o maximize he expeced presen value uiliy over an infinie ime horizon, E0 U( ) 0, under he biological consrain (1). E 0 is expecaion given informaion a ime 0. Insered from (1), (2 ), and (4), he Lagrange funcion of his problem may be wrien as L E 0 0 U p w ((1 b), T, ) bs ( h 1 ) p w s h C( ) 1 11s (1 h ) bs (1 h ) The firs order condiions are now given by: (9) L/ h E[ U'( )] p w 1 0; 0 h 1, (10) L/ h E[ U'( ) w ((1 b), T, )] p 1 0; 0 h 1 and 2 I can be verified ha uncerainy and risk neuraliy yields he same soluion as in secion 6 where climaic uncerainy is negleced. 18
19 (11) L/ p bs ( h 1 ) E[ U'( ) w ((1 b), T, )] (1 b ) ( w / ) EU [ '( )] p ws h EU [ '( )] C'( ) EU [ '( )] 1 s (1 h ) bs (1 h ) 0 Because he weaher condiions a ime are known when h and h are o be deermined, he expecaion operaor in (9)  (11) a period is E. The srucure of his problem is hence parallel o ha of price uncerainy in he fisheries economic lieraure (see, e.g., Andersen 1982). The conrol condiions (9) and (10) can be given similar inerpreaions as he conrol condiions (5) and 6), excep ha he marginal gain now is given as expeced values. Equaion (11) saes ha he socking rae is deermined such ha he immediae expeced marginal uiliy of ewes equals he shadow price of naural growh. Seady sae analysis The firs erm in he bracke in (10) can be rewrien as p E[ U'( )] E[ w ((1 b), T, )] cov( U'( ), w ((1 b), T, )). The covariance erm is negaive as higher lamb weigh, and hence higher profi, yields reduced marginal uiliy for a risk averse farmer. The expeced marginal uiliy of lamb slaughering is herefore smaller he larger is he covariance erm (in absolue value). When combining his expression wih (9), we find ha he farmer in presence of uncerainy will slaugher a higher fracion of lambs han aduls only if pew [ ((1 b ), T, )] pw cov( U'( ), w ((1 b ), T, ))/ EU [ '( )] 0. Tha is, wih he risk aached o lamb weigh, he expeced slaughering value per lamb should exceed he slaughering value per adul by more han required in he deerminisic case for a higher fracion of lamb slaughering o be opimal. More precisely, he difference in he expeced slaughering values should exceed he absolue value of he covariance erm divided by he expeced marginal uiliy of income, i.e. he sensiiviy rae of he marginal uiliy o lamb weigh changes. If his condiion is fulfilled, hen a higher harvesing fracion of lambs han aduls can only be saisfied wih 0 h 1and h 0 because, as already seen, slaughering all he lambs is no a possible opion a seadysae. Hence, as in he deerminisic case, opimal slaughering rae hen equals h 1 (1 s )/ bs 1. However, wih uncerainy, he likelihood for lamb slaughering only o be opimal is smaller. 19
20 * * When insering h 0, h 1 (1 s )/ bs *, and EU [ '( ) w ((1 b), T, )] / from (9) ino (10), insering he covariance, and rearranging, he opimal number of animals is deermined by he equaliy p ( bs s 1 ) E[ w ((1 b), T, )] cov( U'( ), w ((1 b), T, )) / EU [ '( )] C'( ) p ( bs s 1)(1 b) ( w / ). The lef hand side is he expeced marginal benefi of keeping lambs for nex season slaughering ne of he discoun rae. The righ hand side is he marginal cos of saving animals for he nex season when aking he loss weigh of lambs ino accoun. Consequenly, a larger covariance (in absolue value) reduces he expeced marginal benefi of keeping animals for he nex season relaively o he marginal cos, and hence, reduces he opimal number of animals. Tha is, he more sensiive he marginal uiliy of income is o lamb weigh changes, he smaller is he opimal sheep sock. If insead he expeced slaughering value per lamb exceeds he slaughering value per adul by less han he covariance erm, hen slaughering a larger fracion of adul animals is opimal. As in he deerminisic case, his slaughering sraegy can only be saisfied as 0 h 1and h 0. See again (1 ) and he numerical secion. Hence, he opimal slaughering rae equals h 1 (1 s )/ bs 1. Therefore, again uncerainy affecs he possibiliy for adul slaughering only o be opimal, bu no he size of he slaughering rae. * When insering h 0, x * 1 (1 y )/ x * h bs s 1, and EU [ '( )] w p / from (8) ino (10) and rearranging, he opimal number of animals is deermined by he equaliy pbs (1 ) Ew [ ((1 b ), T, )] cov( U'( ), w ((1 b ), T, )) / EU'( ) p w ( bs s 1 ) C'( ) p bs (1 )(1 b) w /. The lef hand side is he expeced marginal benefi of saving animals for male lamb and adul slaughering nex season, whereas he righ hand side is he marginal cos of keeping animals for he nex season. The second erm on he righ hand side reflecs he weigh loss on male lambs of keeping an addiional animal for he nex season. Because male lambs are slaughered, a larger covariance (in absolue value) reduces he expeced marginal benefi of keeping animals for fuure male lamb slaughering, and hence, reduces he opimal number of animals in seady sae. 20
21 Finally, if he expeced marginal uiliy of lamb slaughering equals he slaughering value per adul, i.e. p EU [ '( )] Ew [ ((1 b), T, )] cov( U'( ), w ((1 b), T, )) p w, hen i is opimal o slaugher boh adul animals and female lambs. The shadow price is hen given by EU [ '( )] p w /. When insering his ino (10) he sae condiion reads pbs ( h 1 ) Ew [ ((1 b ), T, )] cov( U'( ), w ((1 b ), T, )) / EU'( ) p w ( bs (1 h ) s 1 ) C'( ) p bs ( h 1 )(1 b) ( w / ). Hence, he conrol condiion and he sae condiion ogeher wih (1 ), deermine simulaneously he opimal slaughering raes and he number of animals in seady sae. For more deails, see he numerical secion below. Dynamic analysis When here is uncerainy, and hence he harves is no longer linear in he objecive funcion, we should expec a more gradually approach o he seady sae hrough a saddle pah approach (Clark 1990). To sum up, wih climae uncerainy presen here is a risk aached o saving animals for fuure lamb slaughering. The opimal seady sae slaugher composiion depends on he expeced per animal slaugher values and he covariance beween he marginal uiliy of profi and he lamb weighs. The sronger he marginal uiliy of profis covariaes wih he lamb weigh, he smaller is he likelihood for lamb slaughering only o be opimal. Insead, slaughering boh lambs and aduls, or even only aduls, are possible beer sraegies if he covariance is srong. 9. Concluding remarks This paper has analyzed he economics of sheep farming in a wosage model of lambs and adul females (ewes). The analysis is a he farm level in a Nordic conex wih a crucial disincion beween he oudoor grazing season and he winer indoor feeding season. The weigh gain of lambs during he oudoor grazing season is subjec o poenial climae change effecs and uncerainy. The farmer is assumed o be raional and well informed, and aims o find he level of animal slaugher ha maximizes profi, or he expeced uiliy of profi when uncerainy is inroduced, he accompanying number of summer grazing animals and he number of animals o be kep indoors during he winer. 21
22 In his wosage model of lambs and ewes, he seady sae harvesing decision is basically shaped by economic facors alone when here is no uncerainy. For he given price and marke condiions whereby he value of lambs is higher han ha of ewes, lambonly slaughering a he highes possible level is he opimal sraegy. On he oher hand, he opimal lamb slaugher fracion is deermined by sheep biological facors alone. The reason for his sharp disincion beween he effecs of economic and biological forces is he lack of any densiydependen facors regulaing sheep populaion growh. The excepion is when he socking rae and hence he grazing condiions is such ha he per animal value is similar for ewes and lambs. Wih uncerainy he seady sae slaugher composiion depends on he expeced per animal slaugher values and he covariance beween he marginal uiliy of profi and he lamb weighs. The sronger he marginal uiliy of profis covariaes wih he lamb weigh, he smaller is he likelihood for lamb slaughering only o be opimal. The opimal slaugher fracion is, however, deermined as in he deerminisic case. The numerical illusraions indicae ha shifing economic condiions for he farmer have small effecs on he socking rae and vegeaion uilizaion. Such shifs, a leas wihin he acual range of parameer values, spill over o changing farm profiabiliy. On he oher hand, we find vegeaion produciviy o have crucial allocaion effecs. For example, when comparing wo equally sized farms locaed in areas wih differing produciviy, he farmer ha benefis from high produciviy will find i rewarding o keep a significanly higher socking rae han he oher one. The high produciviy farmer will receive subsanial higher economic benefis as well. The opimal sheep farming decision may hence be more sensiive o changes in pasure qualiy and produciviy han changes in economic condiions. 22
23 References Andersen, P Commercial fisheries under price uncerainy. Journal of Environmenal Economics and Managemen 9: Aunsmo, L.G. (Ed.) Saueboka ( The Sheep Book ). Landbruksforlage Oslo. Ausrheim, G., A. Myserud, K. Hassel, M. Evju and R.H. Økland Ineracions beween sheep, rodens, graminoids and bryophyes in an oceanic alpine ecosysem of low produciviy. Ecoscience 14: Ausrheim, G., L.J. Asheim, G. Bjarnason, J. Feilberg, A.M. Fosaa, Ø. Holand, K. Høegh, I.S. Jónsdóir, B. Magnússon, L.E. Morensen, A. Myserud, E. Olsen, A. Skonhof, G. Seinheim, and A.G. Thórhallsdóir. 2008a. Sheep grazing in he Norh Alanic region: A long erm perspecive on managemen, resource economy and ecology. Repor Zoology Ser. 2008, 3 Vienskapsmusee NTNU. Ausrheim e al. 2008b. Caswell, H Marix Populaion Models. Sinauer Boson. Clark, C Mahemaical Bioeconomics. Wiley Inerscience, New ork. Dieker, F. K., D. Ø. Hjermann, E. Nævdal, and N. C. Senseh Spare he young fish: Opimal harvesing policies for NorhEas Aric Cod. Environmenal and Resource Economics 47: Gaillard, M. and N.G. occoz Temporal variaion in survival of mammals: a case of environmenal canalizaion? Ecology: 84: Hannesson, R Global warming and fish migraions. Naural Resource Modeling 20:
24 Mobæk, R., A. Myserud, L. Loe, Ø. Holand, and G. Ausrheim Densiy dependen and emporal variabiliy in habia selecion by a large herbivore: An experimenal approach. Oikos 118: Myserud, A., G. Seinsheim, N. occoz, O. Holand, and N.C. Senseh Early onse of reproducive senescence in domesic sheep. Oikos 97: Myserud, A.,. Rekdal, L. E. Loe, M. Angeloff, R. Mobæk, and Ø. Holand A Landscape Level Evaluaion of Grazing Capaciy for Domesic Sheep on Alpine Ranges. Rangeland Ecology and Managemen. In press. Myserud, A., Hessen, D.O., Mobæk, R., Marinsen, V., Mulder, J., and G. Ausrheim Plan qualiy, seasonaliy and sheep grazing in an alpine ecosysem. Basic and Applied Ecology 12: Nersen, N., A. Hegrenes, O. Sjelmo, and K. Sokke Saueholde i Norge ( Sheep Farming in Norway ). Repor Norwegian Agriculural Economic Research Insiue Oslo. Nielsen A., N. G. occoz, G. Seinheim, G. O. Sorvik,. Rekdal, M. Angeloff, N. Peorelli, Ø. Holand, and A. Myserud Are responses of herbivores o environmenal variabiliy spaially consisen in alpine ecosysems? Global Change Biology. In press. Ólafsdóir, R. and Á.D. Júlíusson Farmers percepion of landcover changes in NE Iceland. Land Degradaion & Developmen 11: Pinheiro, J. C. and D. M. Baes Mixedeffec models in S and SPLUS. Springer, New ork. Quaas, M. and S. Baumgärner Opimal grazing managemen rules in semiarid rangelands wih uncerain rainfall. Naural Resource Modeling 25: Skonhof, A Sheep as capial and farmers as porfolio managers: A bioeconomic model of Scandinavian sheep farming. Agriculural Economics 38:
25 Svalheim, E. J., A. Grødum, and M.B. Søbe Kvaliaive undersøkelser av umarksbeier i Aus  Agder. Fylkesmannens Landbruksavdeling i AusAgder, Arendal. 25
26 Tables and figures winer spring summer and early auumn lae auumn Indoor feeding and lambing Lambs released and field grazing Rough grazing period Field grazing, slaughering, shearing Figure 1: Seasonal subdivision in he Nordic sheep farming sysem. 1 1 Figure 2: Equilibrium (consan animal populaion) harvesing relaionshipr p (Eq. 1 ). h, female lamb slaughering fracion; h, ewe (adul female) slaughering fracion. 26
27 Socking rae ear Figure 3: Baseline socking rae dynamics, 0. Table 1: Baseline ecological and economic parameer values Parameer Parameer descripion Value s Naural survival fracion lambs 0.91 s Naural survival fracion ewes 0.95 b Feriliy rae 1.53 (lamb/ewe) (Myserud e al. 2002) Proporion of female lambs k Ineracion erm lamb weigh funcion 22 (kg/animal) 1 k Slope erm lamb weigh funcion, socking rae 0,01 (kg/animal) w Adul (ewes) slaugher weigh 30 (kg/animal) p Adul (ewe) slaugher price 35 (NOK/kg) p Lamb slaugher price 60 (NOK/kg) c Cos coefficien 10 (NOK/animal) Table noe: Exchange rae: 1 Euro = 7.90 NOK (Jan. 2012). 27
28 Table 2: Seadysae sensiiviy analysis, 0. Winer # lambs w sock Baseline 1) ) See Table 1 for baseline parameer values. Table 3: Seadysae sensiiviy analysis, changing emperaure. Case # C Winer # lambs increase sock Baseline 1) Norh Souh ) See Table 1 for baseline parameer values. 28