Local Common Property Exploitation with Rewards

Transcription

1 Loal Common Property Exploitation with Rewards Anne Borge Johannesen and Anders Skonhoft Department of Eonomis Norwegian University of siene and tehnology NO-7491 Trondheim Norway ABSTRACT This paper analyses oalition formation in a livestok-pasture system where livestok are privately owned and the pasture is a ommon property. While the standard models on oalition formation predit rather low prospets of ooperation, this paper introdues a ost advantage of ooperation based on Saami reindeer herding whih may explain higher oalition partiipation. In ontrast to the existing fishery literature on oalition formation, all players are assumed ex ante homogenous, but may differ ex post due to the ost advantage. A stable equilibrium with ooperation an be reahed and a moderate exploitation level an be sustained ompared to the tragedy of the ommons outome. The authors are, respetively, researher and professor at Department of Eonomis, Norwegian University of Siene and Tehnology. The authors want to thank two referees for onstrutive and ritial omments and the Norwegian Researh Counil and Styret for Forskningsmidler over Reindriftsavtalen for funding this projet.

2 I. Introdution For a long time it has been reognized that institutions play an important role in ommon property management (e.g., Bromley 1991), and that an effiient exploitation requires an integrated system of ooperation and ethial odes (e.g., Ostrom 1990). Privately owned livestok grazing on ommon land is a lassi example. When ooperation fails and the individual herdsman ignores the externality imposed on the other herdsmen by his grazing animals, and the vie versa, the possible result may be serious overgrazing and redued livestok produtivity; in short, the tragedy of the ommons (Hardin 1968). However, several studies have hallenged the tragedy of the ommons as a general haraterization of soial behaviour when applied to loal ommons suh as pastures, forests and inshore fisheries. Examples inlude, among others, grazing areas on the Alpine meadows in Switzerland (Ostrom 1990), irrigation systems in Nepal (Ostrom and Gardner 1993), and ommunity forests in Himalaya (Ostrom et al. 1997, hap 12). More examples are provided in Ostrom (1990). The above ited studies indiate that soial norms fostering ooperation an result in wellfuntioning ommon property management. This ontrast the rather grim prospets of ooperation presented in the early literature on oalition formation. This literature, analyzing international environmental agreements, demonstrates that stable oalitions are large if there is not muh to gain from ooperation, and typially very small otherwise (Barrett 1994). See also Hoel (1992). However, more reent theoretial studies have hypothesized that soial norms may promote ooperation. Hoel and Shneider (1997), for example, model soial norms as a ost on non-ooperators being soially punished (or disliked) by o-operators, while Osés-Eraso and Viladrih-Grau (2007), studying ooperation in the utilization of a ommon property pasture, onsider soial norms through a reward to ooperators when assuming that ooperative behaviour produes soial approval and reognition. On the other hand, Sethi and Somanathan (1996) assume that punishment imposes a ost on both nonooperators and ooperators enforing the punishment. Both punishment and reward reate inentives to ooperate as punishment redues the benefit of being a free-rider, while soial reward enhanes the benefit of ooperation. To what extent suh inentives sueed in internalizing the atual externalities hinges ruially on the resulting level of ooperation. Depending on the strength of these inentives and the systems being studied, the presene of soial norms may result in a stable equilibrium onsisting of a mix of ooperators and nonooperators (Osés-Eraso and Viladrih-Grau 2007), or ooperators only (Sethi and 1

3 Somanathan 1996). In the last ase, externalities are fully internalized beause the oalition of ooperators maximizes the olletive benefit of its members. In the former ase, on the other hand, externalities are partly internalized beause the oalition ignores the externalities imposed on non-ooperators. Still, however, deentralized management of ommon property resoures results in more effiient and sustainable resoure utilization ompared to a situation with no ooperation. Rewards for joining the oalition or punishment for defeting, may, however, be aused by other harateristis than soial norms, and in this paper an alternative explanation based on experiene from Saami reindeer herding in northernmost Norway (Finnmark ounty) is offered. See Figure 1. In interviews with more than forty management units (July 2007) a number of herdsmen emphasized that ooperation reates a ost advantage. The reindeer floks migrate aross an extensive area during the year. While the interior ontinental parts of Finnmark is used for grazing during the winter, the summer ranges are loated on the islands and peninsulas near the oast (Johansen and Karlsen 2005). The migration route is mainly determined by the reindeer herd itself, but the herdsman follows the flok to guard and keep it gathered. This is a highly time onsuming ativity. When the herdsmen ooperate, however, individual herds are merged together and herdsmen look after the flok in shifts. Herdsmen laim that this enables them to spend less time on the grazing ranges than they would have if operating alone. Hene, ooperation means less individual effort use. See also Paine (1970). In this way, ooperation redues their individual herding ost. Therefore, this paper assumes that there is a ost advantage, or reward, of being a ooperator. A somewhat similar approah is found in Yi (1998) who studies effiieny gains in researh oalitions in a Cournot oligopoly. Figure 1 about here The following model analyses ooperation in the above mentioned livestok-pasture system of reindeer herding, where livestok is privately owned while the pasture is a ommon property. Reindeer husbandry in Norway is by law reserved for Saamis from Saami husbandry families. That is, herdsmen must have parents or grandparents with reindeer herding as their main oupation (Austenå and Sandvik 1998). The system onsidered is therefore of loal ommon property type with a fixed number of herdsmen (more details on Saami reindeer herding in setion five below). As in Yi (1998), the model analysed is of the 2

4 open membership type; that is, membership of the oalition is open to all herdsmen who are willing to abide by its rules. Thus, any herdsman an hoose either to join the oalition, or not. The possibility to form multiple oalitions is ignored, meaning that non-members of the single oalition will at independently and in pure self-interest. Cooperation is primarily a way of internalizing grazing externalities, while the ost advantage of ooperation omes as an additional effet. Non-ooperators ignore the external effet they impose on others and benefit by free-riding on the pasture improvement indued by livestok restritions by ooperators. At the same time, however, ooperators gain a ost advantage of ooperating. Beause of this asymmetry, a oalition of partial, or full, ooperation may be a stable equilibrium outome. Our reasoning and model differ from oalition formation in high seas fisheries as modelled by Kaitala and Lindroos (1998). In their paper it is simply assumed that the oastal state fleet ex ante is more ost effiient than the distant water states fleets. See also Kaitala and Pohjola (1988), Pintassilgo (2003), and Pintassilgo and Lindroos (2007). As a onsequene, in a stable oalition, the most ost effiient nation is the only ative oalition member and stability is ensured by sharing the benefits with all other oalition members. 1 In ontrast to this, as indiated, we have an explanation why ooperation may result in a ost advantage. That is, in our model, players are ex ante ost idential, but may be different ex post. The paper is organized as follows. The livestok-pasture system is presented in setion two. This is obviously a dynami system, but is, just as the above mentioned fishery studies, analysed in eologial equilibrium only. Next, in setion three, the exploitation of the system is studied when there is no ost advantage of ooperation. Hene, the individual herding ost is ex ante and ex post idential among the herdsmen. This model is referred to as the homogeneous ase and serves as a benhmark when the model of ex post ost advantage is introdued in setion four. A numerial illustration of reindeer herding (Finnmark ounty) follows in setion five, while setion six onludes the paper. II. The livestok-pasture eologial system There are surprisingly few eonomi studies of livestok and ommon grazing land systems, and they all onsider the degree of ooperation exogenously; either as full ooperation, or no ooperation at all (e.g., Barrett 1989 and Brekke et al. 2007). Perrings (1993) studies the same eologial model as Barrett (1989), but is basially interested in how the system reovers 3

5 from eologial shoks related to adverse weather onditions suh as drought. Skonhoft (1999) assumes no ooperation and ompares the standard neolassial model of resoure rent maximization with a model of herdsmen maximizing their herd size, subjet to an inome onstraint. See also Bosted (2005). The eologial part of the model used in this paper is a modified version of the elebrated Noy-Meir biomass model (Noy-Meir 1975). Vegetation quantity (i.e., lihen) on the pasture grows aording to a logisti funtion and dereases due to onsumption by the grazing livestok (reindeer). Hene, growth in vegetation quantity X at time t (the time subsript is omitted) is governed by rx (1 X / K) with K as the vegetation arrying apaity, and r as the intrinsi (maximum speifi) vegetation growth rate. The vegetation onsumption is governed by a fixed number (see above) N ex ante idential herdsmen that utilizes the pasture. 0 n N of these herdsmen form a single oalition and eah of them keeps animals at time t. Eah of the remaining ( N n) non-ooperators holds y number of n y animals. The n total number of animals is aordingly Y = ny + ( N n) y. Noy-Meir assumes that the vegetation onsumption per animal inreases in the amount of vegetation, but at a dereasing rate. In order to keep the model traeable, however, the per animal onsumption is supposed to inrease linearly in the vegetation quantity as given by bx with b > 0. 2 Therefore, the n vegetation onsumption is bx ( ny + ( N n) y ) and the vegetation growth equals: 3 n dx / dt = rx (1 X / K) bx ( ny + ( N n) y ). [1] For a fixed vegetation quantity dx / dt = 0, we hene find the equilibrium animal-vegetation n relationship as X = Kr ( bny ( + ( N ny ) ) / r. The equilibrium vegetation quantity thus dereases linearly in the number of grazing animals. The livestok population hanges aording to natural growth and slaughtering. Following Noy-Meir (1975), the per animal growth is assumed to be proportional to the vegetation onsumption, i.e., qbx, where the parameter 0 < q < 1 measures the transformation of i vegetation biomass into meat biomass. With h as the animal offtake by herdsman i ( i =, n), the population growth of the individual flok size reads: 4

6 i dy dt qbxy h i i =, [2] so that the individual equilibrium slaughtering is h i qbxy i =. Total population growth adds n simply up to dy / dt = qbxy H, where H = nh + ( N n) h. The equilibrium slaughteringvegetation relationship may be found by eliminating livestok from the system when dx / dt = dy / dt = 0 and reads: H = qrx(1 X / K). [3] Therefore, this model indiates a standard logisti relationship between the amount of vegetation and slaughtering as well, with H = 0 when X = K aompanied byy = 0, or H = 0 when X = 0 together with the livestok population size as Y = r/ b. Furthermore, msy X = X = K /2 is the vegetation quantity assoiated with the maximum sustainable harvest. In line with traditional reasoning, eologial overgrazing is said to take plae if the equilibrium vegetation quantity is below assessment). msy X (but see Mysterud 2006 for a ritial III. Exploitation without ost advantage The eonomi motives for keeping animals and slaughtering are now introdued under the assumption that the herdsmen are homogeneous; that is, there is no ost advantage attahed to ooperation. 4 Therefore, the per animal herding ost w is assumed idential aross all herdsmen whether they ooperate (i = ), or not (i = n). With p as the per animal i slaughtering prie (net of slaughtering osts), assumed to be fixed, the individual profit, π, is: 5 ph wy i i i π = ; i, = n. [4] The prospets of oalition formation, where slaughtering effort is oordinated to maximize total harvesting profit of the (single) oalition, are then onsidered. This inludes a one-shot game with a two stage proess. In the first stage, eah herdsman onsiders two options. Those (if any) who initially ooperate deide whether to stay in the oalition or leave, while those who initially stay outside deide whether to remain playing singleton or to join the oalition. 5

7 In the seond stage, the harvest and livestok number, as well as the aompanying grazing pressure, are determined through a simultaneous Nash-Cournot game, where the oalition plays Nash against singletons, and singletons play Nash against all. This is the same set-up as in most oalition formation game models (e.g., Hoel 1992, Sethi and Somanathan 1996, Hoel and Shneider 1997, Yi 1997 and 1998, Finus et al. 2006, Osés-Eraso and Viladrih-Grau 2007). The game is solved by bakward indution. Given the hoie of non-ooperation, a singleton determines slaughtering and herd size in order to maximize own profit, subjet to the i eologial onstraints [1]-[2] in equilibrium, i.e., when dx / dt = 0 and dy / dt = 0 (i = n), while ignoring the negative impat upon the remaining ( N 1) herdsmen. When inserting for the steady state eologial onstraints (setion two above) into [4], the profit of a singleton n n n n reads π = pqbk( r bny b( N n) y ) y / r wy. Therefore, with ontrol variable, the first order ondition of this problem is given by: n y as the single pqbx pqb 2 ( K / r) y n w = 0. [5] The first term in this ondition represents the private marginal inome of adding another animal to the herd, for a given vegetation quantity. The seond and third terms are the marginal ost omponents, and where the seond term implies that the herdsman takes into aount his own impat only on the vegetation quantity. It reflets the loss of future slaughtering inome due to own herd size inrease. The optimal number of livestok kept by a singleton is thus determined by the equity between the private marginal inome and private marginal osts. When inserting for the steady state vegetation quantity ( ( n X = Kr bny + ( N ny ) )/ rinto equation [5], the seond order ondition reads pqb K r 2 2 ( / ) < 0. The oalition maximizes its total profit while taking the vegetation impat on all herdsmen within the oalition into aount. The individual number of animals is hene determined when aounting for the grazing externalities working between the n oalition members, while ignoring the impat on the ( N n) non-ooperators. Therefore, the deision problem of the oalition is to determine the individual number of animals maximizing the joint profit 6

8 nπ = n( ph wy ) subjet to the eologial onditions [1]-[2] (with dx / dt = 0 and i dy / dt = 0, see above). When inserting for these onditions, the profit of the oalition yields nπ = n pqbk( r bny b( N n) y n ) y / r wy.the first order ondition of this problem is then: pqbx pqb 2 ( K / r) ny w = 0. [6] The seond term equals now the soial marginal ost of the oalition of an additional animal in the individual herd flok and implies that the vegetation quantity impat upon the others in the oalition is taken into aount as well. The optimal number of animals kept by a oalition member is thus determined by the equity between the private marginal inome and the within oalition soial marginal ost. By substituting the steady state vegetation quantity (see above) into [6], we derive the seond order ondition 2 2 pqb ( K / r) n < 0. n Solving equations [5] and [6] when inserting X = Kr ( bny ( + ( N ny ) )) / r(setion two above), yields: (1 /( )) ( ) r y w pqbk n = bn( N n + 2) [7] and (1 /( )) ( ) r y w pqbk n n =, [8] bn ( n+ 2) and where the number n within the left hand side parenthesis implies that n herdsmen are inluded in the oalition. Beause the oalition internalizes the grazing externalities among it s members, the resulting herd size of a ooperator is below that of a non-ooperator for all 2 n N. In absene of ooperation, that is all herdsmen at as singletons, n = 1, we find n y (1) = y (1) = r(1 w /( pqbk)) /( b( N + 1)). 7

9 In the first stage of the game the individual herdsmen deide whether to ooperate or not. A stable oalition must fulfil the following two onditions of internal and external stability (e.g., Barrett 1994): n π ( n) π ( n 1) ooperators [9] and n π ( n) π ( n + 1) non ooperators. [10] That is, every oalition member should not be worse off by staying within the oalition with n herders than to beome a non-ooperator with n 1 ooperators left. By the same token, no non-ooperator should be worse off by staying outside the oalition with n herders than joining it so that the oalition size beomes n + 1. A oalition of n= n* N ooperators that simultaneously satisfies onditions [9] and [10] is stable. Beause the oalition members restrit their flok sizes, non-members are typially better off when more herdsmen join the oalition as free rider strategies an be adopted. The presene of free-rider inentives therefore reates low prospets of oalition formation in this model. 6 Apart from the situation when the loal ommon onsists of just two herdsmen (i.e., N = 2 ), a stable oalition simply does not exist (see Appendix A1). The same result is also the outome of the well known global pollution model by Barrett (1994), and in the (stati) Gordon- Shaefer fishery model by Pintassilgo and Lindroos (2007). 7 Indeed, the result in our model stems from the homogenous and linear shape of the livestok-pasture model as the eologial i i i equilibrium individual profits read π = pqbxy wy, whih has the idential struture as in the Gordon-Shaefer fishery model. Therefore, for N > 2, the system settles in a nonooperative equilibrium where all herdsmen at as singletons. By inserting n = 1into n equations [7] and [8] the herd size of eah singleton equals y = r [ 1 w /( pqbk) ]/[ b( N + 1) ] and the orresponding vegetation quantity is X = K(1 + Nw /( PqbK)) /( N + 1). IV. Cost advantage of ooperation Above it was demonstrated that the prospet of ooperation is bleak when the herdsmen are idential with onstant marginal herding osts. Exept when the number of players is two, 8

10 eah individual herdsman is better off by following his narrow self interests and play singleton. As mentioned (setion one above) reindeer herders in northernmost Norway report that ooperation means redued individual herding ost; that is, oalition members may earn effiieny gains by merging their herds, and hene exploit eonomies of sale advantages in animal herding and guarding. The model is now extended to take are of this. Again, it should be stressed that this possible type of effiieny gain is quite different from that in the above (introdutory setion) mentioned fishery models with heterogeneous agents. While ex ante ost heterogeneity promotes oalition formation in these fishery models, the herdsmen are assumed ex ante homogenous in our problem. If a oalition emerges, however, all members earn an endogenous ost advantage over the remaining non-ooperators. Hene, herdsmen may be ex post heterogeneous. n Let now w= w be the marginal herding ost of non-ooperators, again assumed fixed. On the other hand, for ooperators the marginal ost is no longer onstant, but dereases with the number of ooperators, i.e., w= w( n) with w ' < 0 and w'' 0 for all 2 n N. Therefore, n notie that w(1) = w. When using the same arguments as above (setion three), the profit maximizing number of animals kept by a member of the oalition and a non-ooperator is found as: n r ( N n+ 1) w( n) ( N n) w y ( n) = 1 + bn( N n + 2) pqbk pqbk [11] and n n r 2 w w( n) y ( n) = 1 +, [12] b( N n + 2) pqbk pqbk respetively. It is seen diretly from these two equations that reduing wn ( ) relatively to for a given oalition size, allows oalition members to keep more livestok, while nonooperators redue their stok. This last effet works through hanged vegetation quantity. For details, see Appendix A2.2. When inserting [11] and [12] into [1] (when dx / dt = 0) the orresponding vegetation quantity equals: n w 9

11 n K ( N n) w w( n) X( n) = [13] ( N n + 2) pqbk pqbk The definition of a stable oalition is again given by the onditions [9]-[10]. If the ost advantage is small and negligible, eah herdsman will again find it benefiial to follow his narrow self interests and at as singleton, i.e., the system settles with no ooperation. For a herdsman to join the oalition, the ost advantage must be suffiient to offset the free riding benefit. See Appendix A2.1. In this ase, as the size of the oalition inreases, the marginal herding ost of oalition members delines whih works towards an inrease in the inentives to join the oalition. However, a larger oalition aounts for a larger proportion of the grazing externalities, whih works towards inreased free riding inentives of staying outside the oalition. Herdsmen will join the oalition as long as the ost advantage of doing so exeeds the free rider inentive. A stable oalition emerges where oalition members have no inentives to break out and non-ooperators have no inentives to join the oalition. Therefore, depending on the shape of the ost funtion wn ( ), i.e., the ost sensitivity of being a oalition member, the possible oalition strutures are partial ooperation ( 2 n< N) and full ooperation (grand oalition stable). Equation [13] demonstrates that ooperation imposes two ontraditing effets on the vegetation quantity ompared to the homogeneous ase (setion three). The term before the braket indiates that a larger proportion of the grazing externalities is taken into aount in presene of ooperation. This effet works hene in the diretion of fewer total animals and redued vegetation pressure ompared to the homogeneous ase. On the other hand, ooperation implies on average redued marginal herding ost, as refleted by the seond and third terms within the braket and works in the diretion of more grazing animals and less vegetation ompared to the homogeneous model. Therefore, the net effet of ooperation on the vegetation quantity is ambiguous, i.e., ooperation driven by a ost advantage may result in higher or lower grazing pressure than in absene of ooperation. However, the stable oalition onditions imply that the vegetation quantity under partial ooperation stability exeeds what is found in absene of ooperation. Reall that the external stability ondition states that non-members should benefit more from free riding on the oalition than they would gain from joining in. Partial ooperation stability hene means that oalition members restrit their herd sizes ompared to a situation of no ooperation, otherwise there would be nothing 10

12 to gain from free riding on the oalition. Moreover, also grand oalition stability may result in improved vegetation quantity ompared to the homogeneous model as long as the vegetation effet of taking the grazing externalities into aount dominates the impat of redued marginal herding ost. In the opposite ase, however, grand oalition stability means less vegetation ompared to the homogeneous model. See Appendix A2.2 for more details. The various eonomi and eologial fores influene the oalition size, livestok number and vegetation quantity. We now take a brief look at this while the effets on profit and distribution are illustrated in the numerial example in setion five below. Consider first an exogenous redution in the marginal herding ost of oalition members wn ( ) relative to that of non-ooperators n w. The diret effet (i.e., for a fixed oalition size n) means that eah oalition member finds it benefiial to keep more animals, and this works in the diretion of redued vegetation quantity (equation [13]). The new equilibrium is, however, assoiated with more ooperation as the profit of being a oalition member improves relative to that of a nonooperator. This indiret effet works in the diretion of fewer animals beause those shifting from being a non-ooperator to a ooperator redue their herd sizes. The total effets on livestok numbers and grazing pressure are thus ambiguous. Inreased ost disrepany in favour of oalition members may therefore promote vegetation onservation, even though the average marginal herding ost delines. Next, onsider a more profitable prodution through a higher slaughtering prie p whih may happen through hanging demand onditions, a higher subsidy (see setion five below), or both. The diret livestok number effet ( n fixed) is positive. This is seen from equation [13] as the vegetation quantity effet is negative. Again, the indiret effet works through a hanging oalition size. Beause non-ooperators initially keep more animals than oalition members, the profit of being outside the oalition inreases relative to that of joining. Hene, the oalition number delines. As the oalition size shrinks, a smaller proportion of the grazing externalities is taken into aount, whih strengthens the positive effet on the total animal number. Therefore, in the new equilibrium the vegetation quantity redues. Consider finally a redution in the vegetation arrying apaity K whih may be aused by, say, enroahments suh as infrastruture expansions and ottage villages (Nelleman et al. 2001). Reindeer herdsmen have frequently laimed suh enroahments to have redued the 11

13 vegetation over over the past deades (Kitti et al. 2006). The diret vegetation quantity effet is obviously negative. The indiret effet works through a hanging oalition size. Redued vegetation means redued per animal value pqbx. Beause non-ooperators keep more animals than oalition members, we find the profit of a oalition member to inrease relative to that of a non-ooperator. Consequently, the degree of ooperation inreases as the ommon property resoure beomes more sare. A larger oalition aounts for a greater proportion of the grazing externalities and hene, the indiret effet works in the diretion of inreased vegetation quantity. The total effet on vegetation the quantity is therefore ambiguous. V. Reindeer herding in northernmost Norway The theoretial reasoning will now be illustrated by data that fits well with reindeer herding in Finnmark ounty, the main area of reindeer herding in Norway. Reindeer herding in Finnmark an be traed to the hunting of wild reindeer sine time immemorial. During the 15th entury, entire reindeer herds were domestiated and part of the Saami people beame herding nomads. This tradition has preserved until today (Johansen and Karlsen 2005). A similar pastoral system (while the grazing sheme is somewhat different) is found in Sweden (Parks et al. 2002, Bostedt et al. 2003, and Bostedt 2005). Other well-known pastoralist systems inlude attle herding in East Afria and entral Asia (e.g., Perrings 1993). On a national sale, reindeer herding in Norway is a small industry. The total industry omprises just 556 management units keeping in total reindeer (NRHA 2007). There is a restrition on entering the industry whih an be performed by Saami people only (NRHA 2007). A unit leader (i.e. the owner and manager of a management unit) must have herding as his main oupation (Austenå and Sandvik 1998). Very often, a management unit inludes reindeer belonging to the owner s spouse and hildren, as well as sisters and brothers. In total some 3000 persons own reindeer (NRHA 2007). Even though reindeer herding is a small industry on a national sale, it is of great importane to the Saami people both eonomially and, not at least, ulturally. In our survey of reindeer herders in Finnmark, 80 per ent of the unit leaders seem relutant to quit reindeer herding, even if given better inome alternatives (Johannesen and Skonhoft 2008). This indiates that an important ultural identity is attahed to being a reindeer herder. Vegetation studies indiate that the vegetation over of the pasture land in Finnmark has delined signifiantly during the past three deades or so (Johansen and Karlsen 2005). 12

14 Redued ooperation and higher reindeer flok sizes are regarded as the main explanations (Johansen and Karlsen 2005). Today, some 50 per ent of the management unit leaders laim that other herdsmen interrupt and enter in onflit with them over grazing land, espeially on winter ranges (Johannesen and Skonhoft 2008). At the same time, however, when spending time with herdsmen in Finnmark, we observed unit leaders to be in lose ollaboration with eah other. When asking herdsmen why they ooperate, it beame obvious that there is an advantage attahed to ooperation and this advantage is related to the migratory pattern of reindeer. In Finnmark reindeer migrate aross a huge area during the year (again, see Figure 1). The migratory pattern is related to food and snow onditions. During the summer reindeer graze on grass, herbs and sedges on the islands and peninsulas near the oast, while the winter ranges are found in the interior ontinental parts haraterized by vegetation types rih in lihens (Johansen and Karlsen 2005). The Reindeer Farming At gives the Saamis in Finnmark the right to graze their herds in pratially all non-private land areas in the ounty (Austenå and Sandvik 1998) to seure the migration between oast and inland. This migration route has been important to seure an appropriate balane between winter and summer ranges (Johansen and Karlsen 2005). During the migration, as well as while on the summer and winter pastures, the herdsmen follow the flok to guard it and keep it gathered (f. the introdutory setion). When the herdsmen ooperate, they merge the individual herds together and look after the flok in shifts. As argued in setion one, this sharing of responsibility reates a ost advantage, or effiieny gain, of ooperation. The following numerial analysis fouses on the prospets of ooperation in presene of suh a ost advantage. That is, whether a ost advantage an make herdsmen aount for grazing externalities. In the numerial analysis the vegetation over is speified as kilo vegetation (i.e., lihen) per km 2. The number of management units N (i.e. households) is fixed as 10 (but see Appendix A3.2). The herd sizes are measured as number of animals per management unit. The marginal n herding ost of a oalition member is speified as wn ( ) = w /( βn) for all 2 n N, and with β 1/ 2 as a parameter. 8 Table 1 presents the baseline eonomi and eologial parameter values. See also Appendix A3.1. Table 1 about here 13

15 Table 2 demonstrates the profit, herd size and vegetation level orresponding to eah possible n n in the baseline ase. Total profit is defined as Π= nπ ( n) + ( N n) π ( n). It is seen that the vegetation quantity inreases with the members of the oalition. The reason is that more ooperation implies that a higher proportion of the externalities is aounted for, whih in the baseline ase dominates the vegetation effet of redued marginal herding ost. Improved vegetation enables eah non-ooperator to keep more animals. Both effets inrease animal growth and slaughtering n h of non-ooperators. Therefore, in the baseline alulation, outsiders remaining non-ooperators are better off as the size of the oalition inreases. In ontrast, the profit of a ooperator redues along with inreased ooperation for 2 n 5. The reason is that aounting for a higher proportion of the externalities restrits the individual flok size of ooperators suffiient to ause a negative impat on animal growth and slaughtering h. This effet dominates the positive ost advantage effet on the profit of oalition members. For n > 5, however, a further inrease in vegetation quantity enhanes animal growth and hene, the animal offtake of oalition members inreases as well. Within this range, this auses a positive assoiation between the profit of a ooperator and the number of herdsmen joining the oalition. Table 2 about here In the baseline alulation we hene find that n * = 5 represents the stable equilibrium. Reall n n that n = 1 means no ooperation at all and ompare π (1) with π (2). As π (1) < π (2), it is profitable for a non-ooperating herdsman to form a oalition with another herdsman. By ontinuing in this way, Table 2 indiates that non-ooperators always do better by joining the oalition for n < 5. For all n 5, on the other hand, non-ooperators are better off by staying outside the oalition. Hene, a oalition onsisting of five herdsmen is the only stable equilibrium in the baseline ase. Not surprisingly, when ompared to a situation with no ost advantage (i.e., n = 1), all herdsmen are better off in the partial ooperation stable equilibrium. As explained in the above setion four, the remaining non-ooperators free ride on the oalition and hene, obtain a higher profit level than the oalition members. Table 3 demonstrates what happens to n *, as well as grazing pressure and profit, when the slaughter prie p shifts while the other parameters are kept fixed at their baseline values. As 14

16 indiated in the theoretial analysis (setion four), a lower prie inreases the oalition size, and when reduing the prie to p = 600 the system is even grand oalition stable. Furthermore, when omparing Table 3 and Table 4, whih reports the situation with no ost advantage (setion three) for some few values of the prie, it is seen that a oalition joined by many herdsmen (e.g., p = 600 ) inreases the vegetation quantity and ommunity profit signifiantly ompared to the non-ooperative outome. This result differs from Barrett (1994), who finds that if stable oalitions are large there is not muh to gain from ooperation. Tables 3 and 4 about here In order to redue the grazing pressure on winter pastures, Saami reindeer herdsmen reeive a subsidy per kilo slaughtered meat (NRHA 2007). This works in the diretion of higher slaughtering prie, and hene more valuable animals. The diret effet (n fixed) on the total reindeer number is therefore positive (f. equation [13]). 9 Furthermore, the indiret effet working through inreased free rider inentives and redued oalition size (setion four above) strengthens the diret animal number effet. Aordingly, this example indiates that the negative impat on vegetation onservation of a prie subsidy may even be stronger when the oalition size effet is taken into aount. The last three olumns of Table 3 report the profit effets of inreased slaughter prie. A higher prie aompanied with no oalition size impat (e.g., from p = 1346 to p = 1400 ) result in improved profit for all herdsmen. Whenever a prie inrease auses a redution in the oalition size, those shifting from being a oalition member to beoming a non-ooperator are always better off; otherwise they would not leave the oalition. On the other hand, the remaining ooperators are better off only if the diret positive profit effet dominates any indiret negative effet of possible lower slaughtering aused by a redution in the vegetation quantity. In ontrast, the initial non-ooperators experiene an eonomi loss beause the diret positive effet is dominated by the indiret effet of redued slaughtering. This is obviously a strange result, but an be explained by the non-ooperating behaviour of the singletons. Beause eah singleton imposes negative externalities on eah other, as well as on the oalition members, additional negative externalities are imposed on the initial singletons as the size of the oalition delines. Singletons may therefore be worse off with a higher slaughtering prie. This possible outome follows the logi of the lassi externality paper by 15

17 Lipsey and Lanaster (1956). The last olumn of Table 3 demonstrates that a more valuable harvest leaves the herding ommunity as a whole worse off whenever non-ooperating herdsmen are worse off. Table 5 illustrates what happens when the ost parameter n w shifts. A higher ost with no n n oalition size impats (e.g., from w = 225 to w = 250 ) results in redued ommunity profit, but inreased vegetation quantity. 10 However, inreased n w hits singletons more serious than those ooperating beause of the relative effiieny gains obtained by the oalition members. Consequently, the inentive to join the oalition improves. Whenever n n more herdsmen join the oalition (e.g., from w = 200 to w = 225 ), the new oalition aounts for a larger fration of the grazing externalities, whih, even with higher ost, enables the remaining singletons to inrease their herd size as well as the number of animals slaughtered. Atually, the inreased slaughtering inome dominates the ost effet, making the remaining singletons better off. In most instanes, the positive impat on vegetation utilization is strong enough to ensure enhaned slaughtering in the oalition as well (exept for the new members of the oalition). In sum, following our numerial example, inreased n w means lower vegetation utilization and higher ommunity profit whenever the oalition size inreases. Table 5 about here Finally, in Table 6, we onsider a redution in the vegetation arrying apaity K, whih, as mentioned (setion four), may be aused by infrastruture expansions. 11 It is seen that the vegetation quantity redues, but the effet is rather modest. The reason is that redued arrying apaity stimulates inreased ooperation, whih prevents a larger drop in the vegetation quantity. We also see that the herding ommunity as a whole may atually be better off from enroahments. This result is surprising, but may be explained as follows. First, onsider a redution in the arrying apaity from, e.g., K = 1200 to K = 900. Suh redution stimulates more ooperation whih enables the remaining singletons to inrease their animal numbers and slaughtering. The remaining singletons are therefore better off. This effet is, however, dominated by the negative impat on oalition members within this range, leaving the ommunity as a whole worse off. Seond, assume instead that vegetation arrying apaity is initially low ( K = 800 ). Then further enroahments make the initial oalition 16

18 members better off. The reason is that a larger oalition redues the marginal herding ost enough to stimulate oalition members to inrease their herd size. As a onsequene, eventually the number of animals slaughtered inreases and the individual profit within the oalition improves. As more herdsmen join the oalition, the free rider benefits per remaining singleton inreases as well. This happens beause the last member of the oalition redues its herd size more than the total inrease of the former members. Hene, every herdsman remaining outside the oalition is also better off. So is the ommunity as a whole. The equilibrium is full ooperation stable for K = 500. Certainly, enroahments reduing the vegetation arrying apaity below this level will redue the profit of the ommunity. Table 6 about here VI. Conluding remarks This paper analyses ooperation and oalition formation in a livestok-pasture system where livestok are privately owned while the pasture is a ommon property. The standard literature on oalition formation draws a rather pessimisti piture of the prospetive of ooperation (e.g., Hoel 1992, Barrett 1994). However, modifiations of the standard models to inlude soial approval (Osés-Eraso and Viladrih-Grau 2007), or santions (Hoel and Shneider 1997), demonstrate inentives to ooperate beause soial approval (santions) works as a reward to members of the oalition. This paper offers an alternative type of reward based on experiene from reindeer herding in northernmost Norway (Finnmark ounty). When herdsmen ooperate the individual herds are merged together and every individual herdsman benefit through less effort use. This reates a ost advantage of being a member of the oalition and implies that a oalition of partial, or full, ooperation may be stable equilibria. The model differs from oalition formation in high seas fisheries (Kaitala and Lindroos 1998) beause the herdsmen are assumed ex ante ost idential, but may be different ex post. While fishery models may imply that the most ost effiient oalition member is the only ative member, the present (endogenous) ost differene model allows for all oalition members to be ative. Hoel (2008) also analyzes a kind of ost advantage of ooperation, but in a model of international environmental agreements where the ost of adopting a new and more effiient abatement tehnology is lower when the oalition investing in the new tehnology is large. 17

19 Although the ost advantage of ooperation itself works in the diretion of more animals and inreased grazing pressure ompared to the non-ooperative senario, it has been shown that oalition formation may ompensate for redued herding ost so that the equilibrium vegetation utilization in fat delines. This has also a positive effet on the ooperators and non-ooperators utility, or profit. Only if the herding ost of the oalition members responds rather strongly to an inreased oalition size, ooperation may result in redued grazing pressure and more vegetation ompared to the senario of no ooperation at all. In line with existing models of oalition formation (Sethi and Somanathan 1996, Osés-Eraso and Viladrih-Grau 2007), this paper demonstrates that a higher resoure value redues the level of ooperation and leads to redued vegetation onservation. This result ontrasts the famous findings in Demsetz (1967) who argues that more valuable resoures inrease the benefit of reating institutions to internalize externalities. More preisely, Demsetz laims that institutions are promoted when the resoure value is high relative to transation and enforement osts. Furthermore, and in ontrast to the previous mentioned models of oalition formation, this paper examines the welfare effets of a higher resoure value. The numerial analysis demonstrates that a higher slaughtering prie may redue the ommunity welfare. This surprising result ours if the indiret effet working through redued vegetation quantity, and hene redued slaughtering inome, dominates the diret positive profit effet of a higher prie. As apposed to full ooperation equilibrium, only a proportion of the grazing externalities are taken into aount in ase of partial ooperation. The partial ooperation equilibrium is therefore of a seond-best type and hene the ommunity may be worse off with a higher slaughtering prie. This possible outome follows the lassi externality paper by Lipsey and Lanaster (1956). 18

20 Appendix A1. No ost advantage of ooperation (homogeneous herdsmen) When inserting equations [1] (with dx / dt = 0), [7] and [8] into equation [4] the profits read 2 2 π ( n) = pqbkr(1 w /( pqbk)) /( bn( N n + 2) ) and n 2 2 n π ( n 1) = pqbkr(1 w/( pqbk)) /( b( N n+ 3) ) and hene, π ( n) π ( n 1) = pqkr(1 w /( pqbk)) ( N n + 3) n( N n + 2) / n( N n + 2) ( N n + 3). A stable n oalition (equation [9] main text) requires π ( n) π ( n 1) 0 and hene, 2 2 ( N n 3) n( N n 2) Result 1 If N = 2, this ondition is fulfilled for n = 2. That is, in ase of two herdsmen, the stable equilibrium onsists of a grand oalition. For N 3, however, the grand oalition is not stable. 2 2 Proof Assume that N = n. Then ( N n+ 3) n( N n+ 2) = 9 4n whih is negative for all n n 3. Hene, π ( n) π ( n 1) < 0 for all oalitions of full ooperation when N 3. QED. Result 2 A self-enforing oalition of partial ooperation n< N does not exist. Proof Assume that n = 2 is a stable oalition. Then n + + = + < for N 3 and hene, π ( n) π ( n 1) < 0 for ( N n 3) n( N n 2) ( N 1) 2N 0 N 3. This means that a oalition of two herdsmen an not be stable when there are three or more herdsmen in the ommunity. From Result 1 it then follows that there is no ooperation when N = 3. Assume now that N 4. Again, a stable oalition requires ( N n 3) n( N n 2) = 2 (1 n)( N n) 2( N n)(3 2 n) (9 4 n) The first term on the right hand side is negative for all oalitions of partial ooperation. The seond term and third terms are negative for all oalitions of partial ooperation onsisting of n 3 herdsmen. Together with Result 1 this proves that a self-enforing oalition of partial ooperation does not exist. QED. A2. Cost advantage of ooperation 19

21 A.2.1 Stability When inserting [11], [12] and [13] into [4] the profits read n 2 2 ( n) pqbkr 1 ( N n 1) w( n)/ pqbk ( N n) w / pqbk /( bn( N n 2) ) π = and n n 2 2 ( n 1) pqbkr 1 w( n 1)/ pqbk 2 w / pqbk /( b( N n 3) ) π = + +. When omparing these expressions, the denominators reflet a positive free rider inentive reated by oalition members aounting for grazing externalities for all 2 n N and N > 2. This is idential the homogeneous ase in Appendix A1. What differs from the homogeneous model is the ost n advantage of ooperation found in the braket terms. When assuming wn ( ) = w / n(see setion five), the differene between the braket terms in π ( n) and π n ( n) equals, whih is positive for all 2 n N. That is, the n 2 w ( n 1) ( N n + 2) /( pqbkn( n 1)) brakets reflet a positive inentive to join the oalition due to the ost advantage. A herdsman will join the oalition if the ost advantage exeeds the free riding benefit. A2.2 Herd size and vegetation quantity It is here demonstrated how stok levels and vegetation quantity under ooperation may differ from the homogeneous ase of no ooperation. In the homogeneous ase the herd size is equal n n aross all herdsmen, y = y (1) = y (1) = r(1 w /( pqbk)) /( b( N + 1)) (equation [7] and [8] with n = 1). When subtrating this from [11] and [12] we find n n y ( n) y = r ( n 1) ( N + n) w /( pqbk) + ( N + 1) w( n)/( pqbk) /( b( N n + 2)( N + 1)) and n y ( n) y = r( N n + 1) ( n 1) + ( N + n) w /( pqbk) ( N + 1) w( n) /( pqbk) /( bn( N n + 2)( N + 1)), respetively. Hene, in presene of a stable ooperating oalition, non-ooperators keep more and ooperators keep fewer animals than in the homogeneous ase if n n 1 > ( N + n) w /( pqbk) ( N + 1) w( n)/( pqbk). The left hand side of this ondition reflets that a larger fration of the grazing externalities is aounted for in presene of ooperation, while the right hand side reflets that ooperation means redued marginal herding ost in the oalition. Hene, a dominating first effet implies that oalition members keep fewer animals than in absene of ooperation, while non-ooperators free ride on the oalition by keeping more animals. Of ourse, when omparing the vegetation quantity in equation [13] with the homogeneous model (setion three), it is seen that a stable ooperating oalition ( 2 n N) means improved vegetation quantity if this ondition holds. 20

22 n If instead n 1 < ( N + n) w /( pqbk) ( N + 1) w( n) /( pqbk) for 2 n N, then members of a stable oalition keep more animals than in the homogeneous model. This is beause the ost advantage is strong enough to stimulate oalition members to inrease their herd sizes. n Beause this ondition also implies y ( n) > y ( n) for 2 n< N, a oalition of partial ooperation an no longer be stable. The reason is obvious: the ost advantage dominates the free rider inentives, making outsiders better off by joining the oalition. So if n n 1 < ( N + n) w /( pqbk) ( N + 1) w( n)/( pqbk) for 2 n N, then the system settles in full ooperation stability with a vegetation quantity below that in the homogeneous model. A3. Numerial analysis A3.1 Numerial speifiation The eologial parameter values are based on Moxnes et al. (2003). They assume the vegetation arrying apaity K to be kilo per km 2 and the maximum vegetation growth (i.e. at X msy ) to kilo per km 2. Therefore, when utilizing the logisti vegetation growth funtion (setion two), the intrinsi growth rate r equals The eologial per animal vegetation onsumption parameterb is alulated from the n vegetation equilibrium X = Kr ( bny ( + ( N ny ) ) / r, when using the above K and r values. In addition, reasonable values for the vegetation quantity and reindeer stok are applied. Data from 2005/06 shows that the average management unit in West Finnmark kept 409 animals (NRHA 2007). This number is used as a proxy for all herds when alulating the baseline value of b. There exists, however, no aurate data on the vegetation over in this area. Analyses of some of the distrits in the area imply that during the vegetation over was redued to 20 per ent of the 1987 level (Johansen and Karlsen 2005). When alulating the parameter b a rough proxy for the vegetation quantity X is assumed, namely 0.2K = kilo per km 2. The result is b = Data from 2005/06 shows that 99 animals were slaughtered by an average management unit (NRHA 2007). When inserting this number, together with the above values for the vegetation quantity and the average number of reindeer, into the eologial equilibrium [2] (with i dy / dt = 0), q equals On average, this mans that about 2 perent of the animal vegetation biomass intake is onverted to meat biomass. This may seem as a small number, 21

23 but remember that we are onsidering a biomass model where the weight of adult animals is more or less fixed over the year yle. Following Moxnes et al. (2001) the per animal herding ost in absene of ooperation n w is set to 200 NOK. See also Bostedt et al. (2003). When aounting for the fration of alves, adult males and females, the average slaughtering weight is alulated to 21.2 kilo per animal (NRHA 2007). The slaughtering prie inludes the prie reeived at a registered slaughter and slaughtering subsidies. The prie reeived at a registered slaughter is NOK 52 per kilo (Labba et al. 2006), while subsidies amount to roughly NOK 244 per animal (NRHA 2007). The per animal baseline value slaughtering prie p is therefore assumed to be NOK A3.2 Sensitivity analysis Table A1 demonstrates the impat of inreased ommunity size N while keeping all other parameters fixed at their baseline levels. Columns six and ten report the gain of ooperation in terms of vegetation quantity and ommunity profit, respetively, ompared to the orresponding non-ooperating outomes. The results demonstrate that the fration of herdsmen ooperating inreases with the size of the ommunity. However, the gains of ooperation in terms of vegetation onservation are redued. Still, it takes a very large ommunity to eliminate the onservation gains of ooperation. In ontrast, the welfare gains inrease with the size of the ommunity. The reason is that large oalitions benefit more from the ost advantage. Table A1 about here 22