Livestock management at northern latitudes. Potential economic effects of climate change in sheep farming

Transcription

1 Livesock managemen a norhern laiudes. Poenial economic effecs of climae change in sheep farming Anne B. Johannesen Deparmen of Economics Norwegian Universiy of Science and Technology (NTNU) Anders Nielsen Cenre for Ecological and Evoluionary Synhesis (CEES) Deparmen of Biology Universiy of Oslo Anders Skonhof(*) Deparmen of Economics Norwegian Universiy of Science and Technology (NTNU) Absrac We have sudied he economy and ecology of sheep farming under fuure climae change scenarios. The analysis is a he farm level and includes wo differen caegories of he animals, ewes (adul females) and lambs wih a crucial disincion beween he oudoors grazing season and he winer indoors season. The model is formulaed in a Nordic economic and biological seing. During he oudoors grazing season, animals may experience growh consrains as a resul of limied grazing resources. The available grazing resources are deermined by animal densiy (socking rae) and weaher condiions poenially affecing he weigh, and hence, he value of lambs. Because empirical evidence suggess ha climae changes, e.g. increased spring emperaure, have conrasing effecs on lamb weighs depending on he locaion of he farm, he spaial effecs of such changes are analyzed. Keywords: sheep farming, weaher condiions, climae change, vegeaion growh, sage model.. (*) Correspondence: 7491 Dragvoll-Trondheim, Norway. Anders.skonhof@sv.nnu.no 1

2 1. Inroducion IPCC projecions indicae ha emperaures will increase and he increase will be sronges a higher laiudes (Solomon e al. 2007). These projecions indicae ha Nordic sheep farmers will face novel climae condiions in he fuure. Nielsen e al. (2012) showed ha in souhern Norway increased spring emperaure would have conrasing effecs on lamb auumn body mass, depending on he locaion of he farm. 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 effecs, we include in our reasoning and model wo areas where he effec of increased spring emperaure has been shown o have conrasing effec. Our aim is o show how climae change may aler he opimal socking rae and sock composiion, he per animal slaugher values and profiabiliy of he farmers. Our sheep farming sudy is carried ou wih a crucial disincion made beween he oudoors grazing season (spring, summer and fall) and he indoor winer feeding period, and beween differen caegories of animals (lambs and ewes). 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 such as 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 farmers in Iceland and Greenland, and possible also in mounainous areas in France and Spain. 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 per-animal mea producion. The problem is analyzed under he assumpion ha he farmer aims o do i as well as possible, represened by presen-value profi maximizaion. The animal growh model presened in his paper builds on Skonhof (2008). Skonhof e al. (2010) exended he Skonhof (2008) model o include a relaionship beween vegeaion availabiliy and lamb weigh. Here we aim o develop his relaionship furher by allowing in our model lamb weighs o be affeced by weaher and oudoors grazing condiions. Balancing he number of animals and weigh of animals is indeed seen as a crucial managemen problem 2

3 in he Nordic counries as well as oher places (e.g., Olafsdoir and Juliusson 2000, Myserud and Ausrheim 2005, Thomson e al. 2005). In he naural resource and agriculural economics lieraure, here is an increased focus on he poenial effecs of climae changes and weaher uncerainy. 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. 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 he exclusive economic zones of differen counries. Quaas and Baumgärner (2012) sudy opimal livesock managemen in semi-arid rangelands wih uncerain rainfall. Rainfall has no direc effec on livesock growh in heir model bu affecs he grazing capaciy of he rangeland. They solve for he opimal socking rae (raio of livesock populaion o vegeaion biomass) and demonsrae how i is influenced by he degree of risk aversion and amoun of rainfall. 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 as in Dieker e al. (2010), bu as derimenal o lamb slaugher weighs. Furhermore, we presen and analyze an age-specific model consising of adul animals and lambs. Second, along wih empirical findings, we consider increased spring emperaure as having a posiive or negaive effec on lamb slaugher weighs depending on he specific sie of consideraion; ha is, he spaial paern and he locaion of he farm play a role. We focus on wo mounain ranges and wo scenarios; he Norhern scenario, exemplified by Forollhogna in Trøndelag and he Souhwesern scenario, exemplified by wesern side of Hardangervidda, where increased spring emperaure has been shown o have a posiive and negaive effec, respecively, on lamb growh over summer (Nielsen e al. 2012). See Figure 1. We analyze how emperaure changes may aler he opimal slaughering composiion (lamb and ewes), he socking rae, and profiabiliy of he farmers. No climae uncerainy is considered in he main modeling, bu some possible effecs of including uncerainy and risk aversion is included in he Appendix. Figure 1 abou here 3

4 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, 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 hen solved in Secion 6 while Secion 7 provides numerical resuls. Secion 8 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). On Iceland, here are abou 450,000 winerfed and 1.2 million oudoor grazing animals oday. Mea is also he mos imporan produc from sheep farming here. On Greenland, he available land for sheep grazing is much more resriced, and he populaion of ewes and oudoor grazing animals in 2007 was esimaed a 25,000 and 65,000, respecively (Ausrheim e al. 2008). Housing and indoor feeding is required hroughou winer because of snow and harsh weaher condiions (Figure 2). 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%) (Skonhof e al. 2010). 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. 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. In Norway, mos sheep (abou 75% of he oal meabolic biomass) graze in he norhern boreal and alpine region (Ausrheim e al. 2008). The oudoors grazing season ends beween lae Augus and he middle of Sepember. The lengh of he oudoor grazing season is relaively fixed, parly because of local climaic 4

5 condiions bu also, a leas in cerain areas, because local radiions and hisorical reasons play a role in he iming. In general he oudoor grazing season does no exceed 130 days. Throughou he oudoor grazing season, lamb growh is affeced by climae condiions, boh direcly, bu also indirecly hrough climae effecs on he vegeaion (Nielsen e al. 2012). However, also weaher condiions in winer and spring, before he lambs are released o heir grazing areas, have been shown o affec lamb auumn weighs. In paricular, winer condiions affec lamb auumn weighs indirecly hrough snow mel effecs on he vegeaion (Myserud e al. 2011, Nielsen e al. 2012), while spring emperaure and precipiaion has an indirec effec hrough heir effec on plan spring phenology (Nielsen e al. 2012). 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. 2008). The seasonal subdivision is similar in Iceland and Greenland. Figure 2 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 winer-feeding 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 grazing and weaher condiions. Accordingly, a change in he number of animals is independen of grazing and weaher condiions. Naural moraliy differs beween aduls and lambs, and is considered fixed and densiy independen. Lambs no slaughered, eners he adul (ewe) populaion afer he slaughering period (i.e., Sepember Ocober). All male lambs are 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 s (1 h ) s (1 h ), where is he number of female lambs, 5

6 s and and s are he naural survival raes (fracions) of adul females and lambs, respecively, h 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 number of female lambs. Therefore, he ewe populaion growh is governed by: (1) 1 b s (1 h ) s (1 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 3). This isocline inersecs wih he h axis a [1 (1 bs ) / s ], which may be above or below 1. Therefore, he 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 lamb-slaughering rae compaible wih equilibrium. Figure 3 abou here 4. Weaher condiions, herbivore performance, and weigh gain High grazing pressure may cause a reducion in plan qualiy and/or quaniy which in urn migh affec herbivore growh (Myserud and Ausrheim 2005). Experimenal sudies show lower auumn weigh of lambs a high sheep densiy as compared wih low sheep densiy (Myserud and Ausrheim 2005). In he Norwegian sheep farming sysem, he major growh season of he animals is when hey roam freely in he mounains. Consequenly, he per animal produciviy (auumn weigh) is subjec o among years variaion in environmenal condiions (e.g., emperaure and precipiaion) ha influence vegeaion qualiy and quaniy. 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 6

7 lamb weighs (Nielsen e al. 2012). However, which weaher variable (snow deph he previous winer, precipiaion or emperaure in spring or summer) ha is mos significan varies among Norwegian mounain ranges; no only in srengh, bu also in direcion. Increased precipiaion in spring and summer on he wes side of Hardangervidda (high precipiaion area) is found o be negaive for lamb auumn weighs, while he effec is posiive on he drier Hardangervidda eas (see map Figure 1). In Forollhogna in Trøndelag, increased spring emperaure implies increased lamb auumn weighs while he effec is negaive in Seesdal in he souh and on he wes side of Hardangervidda (Nielsen e al. 2012). Since he effec of cerain changes in weaher condiions are sie specific, we choose o model wo paricular areas where he effec differs. We focus here on spring emperaure (more precisely, mean emperaure in May), bu he exercise could be done on any measure of local weaher condiions where is influence on lamb auumn weigh is known. We assume ha he socking rae in spring indicaes he grazing pressure during he grazing season. When in addiion assuming similar grazing pressure among lambs and aduls, he grazing pressure year is hence defined by he number of animals (1 b ). The relaionship beween he socking rae, a cerain change in mean spring emperaure ΔT, and lamb weigh gain during he grazing season year (2) w w ((1 b), T). w is herefore formulaed as: As already indicaed, a negaive relaionship beween he socking rae and he lamb auumn weigh is well-esablished (Myserud and Ausrheim 2005, Myserud e al. 2011), also in our,1 focal areas (Nielsen e al. 2012); ha is, w ((1 b), T) / ((1 b) ) w 0. ΔT = 0 defines he siuaion as i is oday and ΔT > 0 hence indicaes a posiive shif in emperaure in he fuure. As menioned, he effec of ΔT > 0 is sie specific and can be posiive, w b T T w, which will be he Norhern scenario, exemplified by,2 ((1 ), ) / 0 Forollhogna in Trøndelag, or negaive, exemplified by wesern side of Hardangervidda.,2 w 0, which will be he Souhwesern scenario, 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 7

8 connecion beween he amoun of vegeaion and weigh, and herefore also any effecs of weaher facors on ewe weigh. The ewe 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, so mea sales are he only revenue componen for he farmer. Slaughering akes place in he fall afer he oudoors grazing season. Therefore, he number of ewes and female lambs removed are defined as s h and b s h, respecively. As menioned above, he enire male lamb subpopulaion (1 ) b s 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 as 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 ]. The cos srucure differs sharply beween he oudoor grazing season and he indoor feeding season, he indoor coss being subsanially higher. Throughou his analysis, we assume a given farm capaciy. Therefore, he coss of buildings, machinery and so forh are fixed. 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 o 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. 2008). 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 (yearly) profi of he farmer is described by: 8

9 (4) ( 1 ) R C p w bs h p w s h C( ). 6. The opimal program Opimaliy condiions 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, given he biological growh consrain (1). 1/(1 ) is he discoun facor wih 0as he (yearly) fixed discoun rae. The Lagrange funcion may be wrien as 0 L p w bs ( h 1 ) p w s h C( ) 1 11s (1 h ) bs (1 h ) where 0 is he animal shadow value. Following he Kuhn-Tucker heorem, he firsorder necessary condiions of his problem (when 0 ) are: (5) L/ h ( p w 1) 0 ; 0 h 1, (6) L/ h [ p w 1] 0 ; 0 h 1 and (7) L p bs h w b w,1 / ( 1 )( (1 ) ) p w s h C 1 s h bs h ' (1 ) (1 ) 0. The conrol condiion (5) indicaes ha 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 and saes ha he number of female aduls is deermined such 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. These condiions are also sufficien when he Lagrangean is concave in he sae and conrol variables. Since he Langrangean is linear in he conrols, he sufficiency condiions boil down o 2 2 L/ 0 (he weak Arrow sufficiency condiion). Wih sricly convex cos funcion, C '' 0, and 9

10 concave, decreasing lamb weigh gain funcion, i.e.,,1 w 0 and w,1 / ((1 b) ) 0, we find his condiion o be saisfied. From he conrol condiions (5) and (6) i is eviden ha he per animal slaugher value seers he opimal slaugher composiion. If he demand and marke condiions are in favor of lambs, which is he ypical siuaion (see numerical secion 7), hen p p. If, in addiion, he climaic condiions are favorable so ha he weigh of he lambs w w ((1 b), T) is high, we find ha he per animal slaugher value of he lambs will exceed ha of he ewes, pw((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, if he h h demand condiions are in favour of ewes and/or he climae condiions are unfavourable so ha lamb weigh is low, hen p w p w ((1 b), T). In his siuaion a more aggressively harvesing of he aduls is opimal, and he conrol condiions (5) and (6) can be saisfied eiher as iv) h 1and 0 1, v) h 1and h 0, or as vi) 0 1and h 0. h h Seady sae analysis In a seady sae where all variables are consan over ime wih a high lamb weigh and hence p w ((1 b), T) p w (he ime subscrip is omied when considering seady sae), we find he above conrol condiions o be saisfied only as possibiliy iii) wih 0h 1and h 0 because slaughering all he lambs is no an opion in a possible seady sae. See equaion (1 ) and Figure 3. 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 )/ bs 1. Therefore, he opimal slaughering rae depends on biological condiions only and such ha higher feriliy b and higher survival raes indicae ha i is beneficial o slaugher a higher fracion of he lambs. Lambs no slaughered ener he ewe populaion nex spring. When insering h 0, h 1 (1 s )/( bs ), and addiionally pw((1 b ), T) / from condiion (6) ino equaion (7) and rearranging, he opimal equilibrium number of animals o be kep during he 10

11 ,1 indoor season is deermined by p ( bs s 1 ) w C ' p ( bs s 1)(1 b) w. The lef hand side is he marginal benefi of keeping 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 equalizes he cos of an addiional animal indoor plus he weigh loss an addiional animal imposes on all lambs. Noe ha economic as well as biological parameers influence he opimal seady sae number of adul animals. When a higher emperaure yields higher lamb weigh,2 w 0, we find / T 0 by using he sufficiency condiion and in addiion assume ha he marginal lamb weigh loss funcion is non-decreasing in he emperaure effec, w,1 / T 0. Because he seady sae harvesing fracion is deermined by biological parameers alone, we hence also find ha more lambs should be slaughered. In his case increased emperaure hus represens a double dividend for sheep farmers; i increases he value per lamb slaughered and i increases also he opimal number of lambs slaughered. In he opposie case when a higher emperaure yields lower lamb weigh, i will be beneficial for he farmer o reduce he socking rae. Oher comparaive saic resuls may also be deduced. For example, wih a higher slaugher price he farmer will find i rewarding o keep more animals, / p 0. As he summer socking rae hen also increases, he lamb weigh reduces accordingly. The effec of a higher discoun rae is a smaller socking rae and higher lamb value. 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 1and 0h 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 can be ruled ou as an opion because of he acual demographic parameer values (again, see numerical secion 7). Therefore case vi) wih h 1 (1 bs )/ s 1and 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 1. Also now only biological parameers 1 The Appendix demonsraes ha wih uncerainy abou fuure lamb weigh, he likelihood for lamb slaughering only (adul slaughering only) o be opimal is smaller he sronger (less) he marginal uiliy of 11

12 influence he opimal harvesing rae. When insering for he opimal seady sae slaughering values ino (7) and rearranging, he opimal animal populaion is now deermined by he equaion (1 ) (,1 pbs w p w s bs (1 )) C' p bs (1 )(1 b) w. The inerpreaion is similar o he above lamb only slaughering case, alhough now animals kep over winer add o fuure male lamb and adul slaughering. When a higher emperaure yields lower lamb weigh, and we in addiion have w,1 / T 0, we now find / T 0. We also find ha a higher slaugher price, his ime of he ewes, means ha i is beneficial for he farmer o increase he socking rae and hence also he number of animals slaughered. In our example from wo mounain ranges in Norway an increase in emperaure implies more favourable vegeaion growh condiions in he Norhern scenario and less favourable vegeaion growh condiions in he Souhwesern scenario. If all farmers iniially face marke and climae condiions favouring lamb slaughering only, hen increased emperaure will have no impac on he slaughering composiion for farmers in norh. However, as demonsraed, he sheep populaion increases. In souh, on he oher hand, farmers are less likely o slaugher lambs only when faced wih a emperaure increase. Furhermore, increased emperaure moivaes souhern farmers o reduce he sheep populaion. The dynamics Above some properies of a possible seady sae wih a consan number of animals hrough ime was sudied. As he profi funcion is linear in he conrols, economic heory suggess ha harves should be adjused such as o lead he populaion o seady sae as fas as possible; ha is, Mos Rapid Approach Pah (MRAP) dynamics, bu no necessarily exacly a MRAP-pah as wo conrols are included. Hence, if he iniial socking rae is below he opimal seady sae, and he per lamb value is above ha of he ewes, i will for sure be no ewe harvesing, bu some (small) lamb harvesing such ha he opimal conrol condiions (5) and (6) are saisfied. On he oher hand, if he iniial sock is above he seady sae and sill wih he per lamb value above ha of he ewes, he sock should be slaughered down o he opimal sae level as fas as possible. This sraegy may include slaughering all lambs as well as some ewe slaughering, or i may include a high lamb slaughering while no ewe profi covariaes wih he lamb weigh. The opimal slaughering raes are, however, jus as in he above deerminisic case. 12

13 slaughering. The seady sae may be reached he firs year, bu i can also ake a somewha longer ime. The complexiy of analyzing he approach pahs in muli-dimensional models is exemplified by he predaor prey model of Meseron-Gibbons (1996). The dynamics is furher considered in he numerical secion Numerical resuls Daa and specific funcional forms We now presen some numerical resuls. 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 se o 30 (kg/animal) wih a mea marke slaugher value of 35 x x (NOK/kg). Therefore, he fixed ewe slaugher value is pw ,050 (NOK/animal). The lamb mea value is p 60(NOK/kg). We assume a sricly concave mainenance cos 2 funcion, C ( ) ( c/2), wih c 10 (NOK/animal 2 ). Table 1 abou here As already indicaed, several aspecs of climae condiions have been shown o affec lamb weighs in auumn (Nielsen e al. 2012). We use mean emperaure in spring as he projecion for he climae variable because i is more spaially synchronous as compared o e.g. precipiaion and ha he emperaure change is expeced o be larger in spring han in summer (Chrisensen e al and Hanssen-Bauer e al. 2003). Prediced fuure changes in climae condiions are based on oupu from global climae models (e.g. Chrisensen e al. 2007). The simulaed annual mean warming from 1980 o 1999 o 2080 o 2099 in Norhern Europe varies from 2.3 C o 5.3 C, wih he larges warming occurring in winer (Chrisensen e al. 2007). These models are, however, raher imprecise in predicing exac changes in e.g. seasonal average emperaures in paricular areas. A few aemps have been made o down scale global climae projecions o Norwegian condiions (e.g., Hanssen-Bauer e al and Benesad 2011). These sudies esimaed mean spring emperaure o increase approximaely 1 C in he period as compared o he period They found no significan difference in emperaure increase beween he wo areas included in our sudy. As discussed, we focus on wo mounain ranges; he Norhern and he Souhwesern areas (see Figure 1), where increased spring emperaure has been shown o have a posiive and negaive 13

14 effec, respecively, on lamb auumn weigh. In he baseline calculaions wih no climae change and T 0 C, he lamb slaugher weigh funcion (2) is specified as w w b k k bwih 0 1 ((1 ),0) (1 ) 0 k 22 and 1 k Accordingly, wih a number of grazing animals of, say, (1 b ) 250, we find w (kg/animal) and p w ,170 (NOK/animal) and herefore a subsanial higher slaugher value of he lambs han he ewes (see above). Wih climae change we assume a uniform shif of he weigh funcion such ha equaion (2) now reads w w ((1 b), T) k k (1 b) k T. Under his assumpion climae change hus has no effec on he marginal weigh socking rae relaionship; ha is, w,1 / T 0.This simple shif is no necessarily realisic, as climae effecs migh be sronger a higher socking raes (addiive effecs). We do, however, find i as a reasonable simplificaion. Nielsen e al. (2012) found ha for an increase in average spring emperaure of 1 C ( T 1) he average lamb auumn weigh would increase wih 0.37 kg in he norh and decrease wih 0.69 kg in souhwes. Though hey modeled lamb auumn body mass, we use he same esimaes o illusrae he effecs on lamb slaugher weigh. Tha is, 2 k is assumed o be 0.37 and in he Norhern and Souhwesern scenario, respecively. However, we sill only model he curren condiions as compared o a down scaled projeced climae change scenario expeced o represen climae condiions in A more realisic approach would have been o use a dynamic T represening a coninuous change in emperaure over ime. We do no however, find his o be necessary o illusrae he poenial effecs of fuure climae change on he economy of he sheep farming. In he following, we firs calculae he opimal managemen policy for he baseline parameer values, including no climae change. We hen sudy he effecs of climae change hrough emperaure shifs given as T as 3 C in addiion o1 C, as well as changes in some of he key parameer values like he discoun rae and he mea prices. Resuls We sar wih presening he basic dynamic resuls. While we solve he model for a long ime horizon (50 years), we only repor he resuls for he firs 35 years. This long ime horizon ensures ha he repored soluions will be numerically indisinguishable from he infinie 14

15 horizon soluion over he repored period of 35 years. As already indicaed, because he profi funcion is linear in he conrols MRAP dynamics, bu necessarily exacly a MRAP-pah, is supposed o describe he opimal ransiional dynamics. Figure 3 seems parly o confirm his where he seady sae socking rae approaches he seady sae value of 123 animals afer abou 3 years wih he baseline parameer values and where he discoun rae is 3%, During he ransiional phase, as well as in he seady sae, he value per lamb exceeds he value per adul. In he firs year, all lambs are slaughered before i is gradually reduced o is opimal seady sae harvesing rae of h 1 (1 s )/( bs ) = See also Table 2. No ewes are slaughered. No surprisingly, we find ha increasing he discoun rae resuls in progressively smaller populaions wih corresponding higher harvesing raes of lambs during he ransiional phase, bu sill no ewes slaughered, while he dynamics do no change qualiaively. 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 4 abou here Nex we sudy he effec of climae changes exemplified by an increase in mean spring emperaure. Table 2 repors seady sae sock sizes, slaughering raes, lamb weigh and profi for he differen emperaure increase values. Consider firs he Norhern scenario where a higher emperaure is beneficially and increases he lamb weigh. Increased weigh increases he ne income of lamb slaughering for a fixed sock size. Furhermore, a he same ime higher emperaure means ha i is beneficial for he farmer o keep more animals over he winer for nex season lambing and slaughering, which in urn increases he number of lambs and yields an addiional posiive effec on farm profiabiliy. A a emperaure increase of 1 C, he direc effec of increased weigh (from kg o kg) adds 3,663 NOK o he yearly slaughering income, while he indirec effec adds 4,606 NOK and is due o he increased sock size and a hence he corresponding reducion in lamb weigh. See Figure 5. The remaining change in he yearly profi is due o increased mainenance cos. Thus, as indicaed, increased emperaure represens a double dividend for sheep farmers (secion 6 above), and he indirec economic effec of he increased socking rae is sronger han he direc effec. A a emperaure increase of 3 C, we also find ha he direc effec exceeds he indirec effec. 15

16 Figure 5 abou here The Souhwesern scenario, where increased emperaure affecs he lamb weigh negaively is hen considered. The low scenario emperaure increase of 1 C reduces he lamb weigh bu no sufficien o give a smaller per animal value of he lambs han ha of he ewes. Therefore, he opimal seady sae slaughering composiion is unchanged. However, he profi reduces due o he direc negaive effec of reduced lamb weigh and he indirec negaive effec working hrough a lower socking rae. Wih a emperaure increase of 1 C, he direc effec of reduced weigh reduces he yearly farm income by 6,732 NOK, whereas he indirec effec on slaughering income working hrough a smaller sock size reduces he yearly income by 4,531 NOK. Therefore, also in his Souhwesern scenario he indirec effec is srong. The remaining change in he yearly profi is due o reduced mainenance cos. A furher increase in he emperaure may drive he slaughering value per lamb below ha of he ewes and hence shif he opimal seady sae slaughering composiion from lamb slaughering only o adul slaughering only. Table 2 shows ha his happens when T is 3 C. Thus, in his case he less favorable vegeaion growh condiions mean ha he lamb slaugher weigh reduces such ha we find pw , 010 pw x x , 050 (NOK/animal). Table 2 abou here The resuls in Table 2 indicae ha emperaure changes have crucial spaial effecs. For example, when comparing wo equally sized farms locaed in our wo areas where lamb weighs (or produciviy) are affeced in an opposie manner by emperaure changes, he farmer ha benefis from high produciviy will find i rewarding o keep a significanly higher socking rae han he oher one. In case of a 1 C emperaure increase, he farmer ha gains from climae change (Norhern scenario) will earn some 10% higher profi per year (115,267 NOK vs. 104,596 NOK) han he farmer locaed in an area negaively affeced (Souhwesern scenario). Wih an even higher emperaure change, he profi discrepancy increases furher, and wih 3 C emperaure increase he difference becomes abou 30%. Table 3 repors seady sae effecs of changing economic condiions under he differen assumpions abou he emperaure change. Firs, we sudy he effec of reducing he discoun rae. Ignoring discouning and 0 wihou any emperaure change has no impac on he 16

17 seady sae slaughering composiion and hence, no impac on he slaughering raes which are deermined by biological facors only. However, as also seen in Figure 4, he farmer will find i beneficial o keep more animals. The profi also increases compared o he baseline scenario a he seady sae of presen-value maximizing wih zero discouning coincides wih maximizing profi in biological equilibrium. Ignoring discouning wih higher emperaure has he same qualiaive effecs, see also Table 2; a all levels of emperaure change i is beneficial o keep more animals in boh he Norh and Souhwes scenario wih a zero discoun rae. Increasing he price of lamb mea o p 70 NOK/kg, has no impac on he seady sae slaughering composiion compared o he baseline scenario. However, as also indicaed (secion 6), a higher lamb mea value increases he marginal benefi of saving animals for nex season and hence, he animal sock increases. Because higher emperaure increases he value of lambs hrough increased weigh in he Norhern scenario, he impacs on socking rae and profi of a higher lamb mea price are srenghened in his area wih 0. The opposie occurs in Souhwes where increased emperaure dampens he impac of a higher lamb mea price. Finally, Table 3 demonsraes he effecs of increasing he ewe slaugher value, and for p 40 NOK/kg i is beneficial for farmers in boh Norh and Souhwes o change he slaugher sraegy and only slaugher ewes. This sraegy is even beneficial wih 3 in he Norhern scenario because p w 4030 p w Table 3 abou here A higher lamb mea value (or lower discoun rae) increases he seady sae profi in boh Norh and Souhwes and, as indicaed in Table 3, he spaial effecs of emperaure changes in presence of higher lamb mea value are of similar srengh compared o he baseline economic scenario. Tha is, a a 1 C emperaure increase farmers in Norh earn some 10% more han farmers in Souhwes (149,659 NOK compared o 135,824 NOK). A a higher price of ewe slaugher, however, he disribuional effec of increased emperaure may be lower beween Norh and Souhwes because only ewe slaughering becomes beneficial and he slaugher value of his animal caegory is no relaed o emperaure changes. 8. Concluding remarks This paper has analyzed he economics of sheep farming in a wo-sage model of lambs and adul females (ewes). The analysis is a he farm level in a Nordic conex wih a crucial 17

18 disincion beween he oudoor grazing season and he winer indoor feeding season. The farmer is assumed o be raional and well informed, and aims o find he number of animals slaughered ha maximize presen value profi. The oudoor grazing season makes he auumn weigh of he lambs subjec o changes in environmenal condiions and possible climae change effecs. Several aspecs of climae condiions have been shown o affec lamb weighs in auumn (Nielsen e al. 2012), and we used mean emperaure in spring as he fuure projecions for he climae variable. According o IPCC, he simulaed annual mean warming from 1980 o 1999 o 2080 o 2099 in Norhern Europe varies from 2.3 C o 5.3 C (Chrisensen e al. 2007), while downscaling have indicaed an increase of ~1 C in spring emperaure in our focal areas (Hanssen-Bauer e al and Benesad 2011). In our modeling we focused on spring emperaure increases in he range of 1 C o 3 C. In he wo-sage model of lambs and ewes, he seady sae harvesing decision is basically shaped by economic facors alone. For he given price and marke condiions wih more valuable lambs han ewes, lamb only slaughering a he highes possible level is he opimal seady sae harvesing 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 densiy-dependen facors regulaing sheep populaion growh. We find ha higher emperaure represens a double dividend for he farmer experiencing increased lamb weigh; i increases boh he slaugher value per animal and he number of lambs he farmer will find i beneficial o slaugher. Boh he direc effec, represened by he increased lamb weigh and higher slaugher value, and he indirec effec, working hrough increased number of animals slaughered, may conribue significanly o increased profiabiliy for he farmer. The numerical illusraions also indicae ha shifing emperaure has crucial spaial effecs. For example, when comparing wo equally sized farms locaed in areas in which emperaure affec lamb weigh in differen direcions, he farmer ha benefis from higher emperaure will find i rewarding o keep a significanly higher socking rae han he oher one. The farmer experiencing increased lamb weigh will receive subsanial higher economic benefis as well. A a realisic emperaure increase of 1 C he farmer benefiing from increased lamb weigh will earn some 10% higher profi han he farmer facing reduced lamb weigh. Wih 3 C increase, he profi gain increases o 30%. The spaial effec of increased emperaure is of less imporance when adul slaugher is opimal. 18

19 References 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 Sheep grazing in he Norh Alanic region: A long erm perspecive on managemen, resource economy and ecology. Repor Zoology Ser. 2008, 3 Vienskapsmusee NTNU. Benesad R.E A New Global Se of Downscaled Temperaure Scenarios. Journal of Climae, 24, Caswell, H Marix Populaion Models. Sinauer Boson. Chrisensen J.H., B. Hewison and A. Busuioc Pp in Solomon e al. Dieker, F. K., D. Ø. Hjermann, E. Nævdal, and N. C. Senseh Spare he young fish: Opimal harvesing policies for Norh-Eas Aric Cod. Environmenal and Resource Economics 47: Hannesson, R Global warming and fish migraions. Naural Resource Modeling 20: Hanssen-Bauer I, E. J. Forland, J. E. Haugen and O. E. Tveio Temperaure and precipiaion scenarios for Norway: comparison of resuls from dynamical and empirical downscaling. Climae Research 25, Meseron-Gibbons, M A echnique for finding opimal wo-species harvesing policies. Ecological Modeling 92: Myserud, A., G. Seinsheim, N. occoz, O. Holand, and N.C. Senseh Early onse of reproducive senescence in domesic sheep. Oikos 97: Myserud, A. and G. Ausrheim Ecological effecs of sheep grazing in alpine habias, shor erm effecs. Umarksnæringen i Norge 1-05:

20 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 18: Ólafsdóir, R. and Á.D. Júlíusson Farmers percepion of land-cover changes in NE Iceland. Land Degradaion & Developmen 11: Quaas, M. and S. Baumgärner Opimal grazing managemen rules in semi-arid 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: Skonhof, A., G. Auserheim and A. Myserud A bioeconomic sheep vegeaion radeoff model: An analysis of he Nordic sheep farming sysem. Naural Resource Modeling 23: Solomon, S. D., Q. M. Manning, Z. Chen, M. Marquis, K. B. Avery, M. Tignor, and H. L. Miller (eds.) Climae change 2007 he physical science basis. Cambridge Universiy Press, Cambridge. 20

21 Appendix Equaion (2) in he main ex indicaes ha lamb weigh can be prediced exacly from he curren socking rae and climae condiions. However, hese changes are in fac parly random o he farmer, and in his Appendix i is shown how uncerainy and risk aversion may affec he opimal slaughering composiion and he animal sock. Therefore, we now specify he lamb weigh as sochasic: (A1) 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. I can be verified ha uncerainy ogeher wih he assumpion of risk neuraliy yields he same soluion as in secion 6. We herefore solve he model by assuming ha farmers are risk averse. Tha is, we assume ha farmer uiliy increases wih he curren profi a a decreasing rae, i.e., U '( ) 0 and U ''( ) 0. Under risk aversion, he farmer now aims o maximize, given he he expeced presen value uiliy over an infinie ime horizon, E0 U( ) 0 biological consrain (1) and equaions (A1) and (3). E 0 is expecaion given informaion a ime 0. The 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 1 1s (1 h ) bs (1 h ) The firs order condiions are now given by: (A2) L/ h E[ U'( )] p w 1 0; 0 h 1, (A3) L/ h E[ U'( ) w ((1 b), T, )] p 1 0; 0 h 1 and (A4) L/ p bs ( h 1 ) E[ U'( ) w ((1 b), T, )] (1 b ) ( w / ) EU [ '( )] pwsheu [ '( )] C'( ) EU [ '( )] 1 s (1 h ) bs (1 h ) 0 21

22 I is assumed ha he weaher condiions a ime are known when h and h are deermined. Therefore, he expecaion operaor in (A2) - (A4) a period is E.The conrol condiions (A2) and (A3) can be given similar inerpreaions as he conrol condiions (5) and (6) in he main ex, excep ha he marginal gain now is given as expeced values. Equaion (A4) saes ha he socking rae is deermined such ha he immediae expeced marginal uiliy of ewes equals he shadow price of naural growh. We only look a he seady sae soluion in his Appendix. The firs erm in he bracke in (A3) may be rewrien as p EU [ '( )] 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 he risk adverse farmer. The expeced marginal uiliy of lamb slaughering is herefore smaller he larger absolue value of he covariance erm. When combining his expression wih (A2), we find ha he farmer in presence of uncerainy will slaugher a higher fracion of lambs han ewes suggesed ha pew [ ((1 b ), T, )] pw cov( U'( ), w ((1 b ), T, ))/ EU [ '( )] 0. Tha is, wih 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 as he above case iii) in he main ex secion 6 wih 0h 1and h 0 because slaughering all he lambs is no a possible opion a seady-sae. 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. When insering h 0, h 1 (1 s )/ bs, and EU [ '( ) w ((1 b), T, )]/ from (A2) ino (A3), insering he covariance, and rearranging, he opimal number of animals is deermined by he condiion 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 22

23 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. The oher cases wih a higher slaugher value of he ewes han ha of he expeced value of he lambs can be analyzed in a similar manner. Figures and ables 23

24 Figure 1: Norway and he sudy area. The Norhern scenario (Forellhogna) and he Souhwesern scenario (Hardangervida are indicaed wih solid lines, while he oher areas referred o in he ex (Seesdal in he souh ana easern side of Hardangervidda) are shown wih doed lines. 24

25 25

26 Figure 2: Seasonal subdivision in he Nordic sheep farming sysem. winer spring summer and early auumn lae auum nn Indoor feeding and lambing Lamb released and field grazing Rough grazing period Field grazing, slaughering, shearing Figure 3: Equilibrium (consan animal populaion) harvesing relaionship (Eq. 1 )., female lamb slaughering fracion;, ewe (adul female) slaughering fracion

27 Figure 4: Sock dynamics for differen discoun rae values. Baseline parameer values and Socking rae ear Figure 5: The lamb weigh socking rae relaionship for differen emperaure changes, 0 and 1. The Norhern scenario

28 Table 1: Baseline ecological and economic parameer values Parameer Parameer descripion Value Source s s Naural survival fracion lambs Naural survival fracion ewes 0.91 Myserud e al. (2002) 0.95 Myserud e al. (2002) b Feriliy rae 1.53 (lamb/ewe) Myserud e al. (2002) 0 k 1 k 2 k w p Proporion of female lambs Ineracion erm lamb weigh funcion Slope erm lamb weigh funcion, socking rae Slope erm lamb weigh funcion, emperaure Adul (ewes) slaugher weigh Adul (ewe) slaugher price 0.50 Myserud e al. (2002) 22 (kg) Nielsen e al. (2012) 0.01 (kg/animal) Nielsen e al. (2012) 0.37 (kg/ C) (Norh) (kg/ C) (Souhwes) Nielsen e al. (2012) 30 (kg/animal) Skonhof e al. (2010) 35 (NOK/kg) Skonhof e al. (2010) p Lamb slaugher price 60 (NOK/kg) Skonhof e al. (2010) c Cos coefficien 10 (NOK/animal 2 ) Calibraed (scales he farm size) Discoun rae 0.03 Assumpion Table noe: Exchange rae: 1 Euro = 7.50 NOK (Aug. 2012). 28

29 Table 2: Seady-sae sensiiviy resuls. Changing emperaure Case T ( C) Winer sock #animals slaughered (NOK) Baseline 1) Norh Souhwes ) See Table 1 for baseline parameer values. 29

30 Table 3: Seady-sae sensiiviy analysis. Changing economic condiions Winer # animals w (NOK) sock slaughered Baseline 1) Norh Souhwes ) See Table 1 for baseline parameer values. 30