The economics of a stage-structured wildlife population model

Transcription

1 The econoics o a stage-structured wildlie population odel Anders Skonhot and Jon Ola Olaussen Departent o Econoics Norwegian University o Science and Technology N-7491 Dragvoll-Trondhei, Norway Abstract A siple three-stage odel o the Scandinavian oose (Alces alces) (young, adult eale and adult ale) is orulated. Fecundity is density dependent while ortality is density independent. Two dierent harvesting regies are explored; hunting or eat and trophy hunting. The paper gives an econoic explanation o the biological notion o eales as valuable and ales as non-valuable. The paper also deonstrates how this notion ay change under shiting econoic and ecological conditions. 1. Introduction The ai o this paper is two-old; irstly, to deonstrate the econoic content o a structured wildlie population odel, and secondly, to show how this econoic content ay change under dierent anageent scenarios. Analysing structured wildlie harvesting odels, i.e., odels where the species is grouped in dierent classes according to age and sex, has a long tradition within biology and Caswell (21) gives a recent in-depth overview. No econoic considerations are, however, present here. This is also so in e.g., Milner-Gulland et al. (24). Econoic analysis is introduced in Cooper (1993) who orulates a siulation odel that inds the econoically optial level o deer tags or hunting zones and where the deer population is structured in bucks and does. Skonhot et al. (22) analyses various anageent strategies or a ountain ungulate living in a protected area and a hunting area. Four stages are included here; eales and ales within and outside the protected area. Due to the coplexity o these odels, however, it is diicult to understand the various econoic echaniss inluencing harvesting and abundance. The present paper analyses such econoic echanis ore explicitly where a siple three stages odel, including young, adult eale an adult ale, is orulated. The present analysis is ost siilar to that o Clark and Tait (1982) who studied the optial harvest value in a sex-selective harvesting odel and where the population hence was grouped in two stages. As in Clark and Tait, we are analysing biological equilibriu where natural growth is balanced by harvesting. However, in contrast 1

2 to Clark and Tait, trophy hunting, in addition to eat value axiisation, is analysed. We also calculate the shadow values o the adult ales and eales. We are thus giving an econoic explanation o the biological notion o eales as valuable and ales as nonvaluable. The odel is applied or a oose population (Alces alces), and is studied within a Scandinavian ecological and institutional context. Moose is by ar the ost iportant gae species in Scandinavia, and in Norway and Sweden abut 4, and 1, anials, respectively, are shot every year. Moose hunting has traditionally been a local activity, and the landowners receive the hunting value. The hunters have basically been the local people; the landowners and their ailies and riends, and where the anageent goal has been to axiise the eat value or stable populations (ore details are provided in Skonhot and Olaussen 25). During the last years, however, a ore coercialised hunting and wildlie industry has eerged, and the Scandinavian oose hunting is gradually shiting ro a aily and riend activity to a gae hunting arket. The trophy value o old ales plays an iportant role here. Both the traditional exploitation schee and the new coercialised schee are studied, and the consequences or harvesting and the population coposition are analysed. The paper is organised as ollows. In the next section, the three stage oose population odel is orulated. Section three deonstrates what happens when the hunting is steered by the traditional landowner goal o axiising eat value. Next, in section our we study the sex and age coposition under the new exploitation regie o trophy hunting. A downward sloping inverse deand curve or hunting old ales is introduced. In addition, it is also assued that the deand ay shit due to a quality eect captured by the density o old ales. Section ive illustrates the odels by soe nuerical siulations, while section six suarises the indings. 2. The population odel The Alces alces is a large ungulate with ean slaughter body weight (about 55% o live weight) or adult oose in Scandinavia o about 17 kg or ales and 15 kg or eales. The non-harvest ortality rates are generally low due to lack o predators, and there is no evidence o density dependent survival. On the other hand, ecundity has proven to be 2

3 aected by the eale density while the nuber o ales, within the range o oose densities in Scandinavia, sees to be o negligible iportance (see, e.g., Nilsen et al. 25 or ore details). Usually, the oose population is structured in our stages; calves, yearlings, adult ales and adult eales. In what ollows, however, to obtain a traceable analytical odel, while still catching the ain ecological content, just three stages are considered as yearlings are let out. The population at tie (year) t is hence structured as calves and adult ales ( 1 yr) X t so that the total population is X t, adult eales ( 1 yr) X = X + X + X. Yearlings t t t t are thereore included aong the adult ale and eale. These three stages are henceorth called young, eale and ale. The population is easured in spring ater calving. All stages are generally harvested, and the hunting takes place in Septeber-October. All natural ortality is assued to take place during the winter, ater the hunting season as the natural ortality throughout suer and all is sall and negligible. The sae natural ortality rate is iposed or ales and eales. As indicated, natural ortality is ixed and density independent, while reproduction is density dependent. It is urther assued the sae sex ratio or the young when they enter the old stages (again, see Nilsen et al. 25). X t Neglecting any stochastic variations in biology and environent, and neglecting any dispersal in and out o the considered area, the nuber o young at tie ( t + 1) is irst governed by: (1) X t+ 1 = rx t t with r t > as the ertility rate (nuber o young per eale). The ertility rate is density dependent in the nuber o eales: (2) r = r( X t ) t with dr / dx = r ' < (when oitting the tie subscript) and where r () > is ixed. Cobing (1) and (2) gives the recruitent unction ( X = r X ) X and dierentiating yields dx / dx = ( r ' X + r). The recruitent unction is assued to be concave. For obvious reasons dx / dx should hold in an optial harvesting prograe. 3

4 The abundance o (old) eale ollows next as: (3) X h X h X t+ 1 =.5(1 )(1 t) t + (1 )(1 t) t where > and > are the density independent ortality ractions o young and eale (and ale), respectively, while h t and h t are the harvesting ractions. Hal o the young population enters the eale, ater harvesting and natural ortality. The nuber o (old) ales is inally given by: (4) X h X h X t+ 1 =.5(1 )(1 t) t + (1 )(1 t) t where h t is the ale harvesting raction. When cobining equations (1) - (3), the eale population dynaic reads X =.5(1 )(1 h ) r( X ) X + (1 )(1 h ) X. This is a second order non-linear t+ 1 t t t 1 t t dierence equation, and nuerical analyses deonstrate that the equilibriu is stable or ixed harvesting ractions (see, e.g., Gandolo 21 or a theoretical exposition). Oitting the tie subscript, the equilibriu reads: (5) X =.5(1 )(1 h ) r( X ) X + (1 )(1 h ) X. There are two equilibria; the trivial one o X = and X > given by 1.5(1 )(1 ) ( h r X ) (1 )(1 h ) = +. Because r ' <, the non-trivial equilibriu will be unique and ay be written as: (5 ) X = F( h, h ) where F(..) represents a unctional or. We ind F / h = F < and F <. The isopopulation eale lines slope thereore downwards in the closer to the origin yield a higher stock. (, ) h h plane, and where lines 4

5 By cobining equations (1), (2) and (4), the ale population growth reads X =.5(1 )(1 h ) r( X ) X + (1 )(1 h ) X. The dynaic o the ales is t+ 1 t t t 1 t t thereore contingent upon the eale growth (but not the vice versa as only eale abundance regulates ertility), and again nuerical analyses deonstrate that the equilibriu is stable. The equilibriu is: (6) X =.5(1 )(1 h ) r( X ) X + (1 )(1 h ) X. There are two equilibria or the ale population as well; the trivial one when X =, and X > when X >. Equation (6) ay also be written as: (6 ) X G h h r X X = (, ) ( ) where Gh (, h ) =.5(1 )(1 h) /[1 (1 )(1 h )]. Again, it ay be conired that higher harvesting rates ean ewer anials, G < and G <. The ale iso-population lines hence slope downwards in the (, ) h h plane, and lines closer to the origin yield a higher stock. On the other hand, a higher eale sub-population shits these iso-population lines away ro the origin (suggested that the slope o the recruitent unction is positive, see above) eaning that there is roo or ore ale harvesting or a given young sub-population harvesting, and the vice versa. Equation (6 ) also indicates that the equilibriu ale eale proportion decreases with ore eales. This holds, but the ale-eale proportion ay ore easily be recognized when cobing (5) and (6) which yields X / X = [1 (1 )(1 h )]/[1 (1 )(1 h )]. We thereore siply have X / X = 1 i h = h, as the ortality o the ale and eale are equal, and the sae raction o young enters the eale and ale sub-populations. 3. Exploitation. The traditional regie; hunting or eat As indicated, the traditional exploitation o the Scandinavian oose has been directed by axiising the eat value in ecological equilibriu. Because natural ortality takes place 5

6 ater the hunting season, the equilibriu nubers o anials reoved are siply H = hrx, H = h X and H = h X so that the total harvest equals H = H + H + H. The anageent goal o the landowner(s) in the given area is accordingly: (7) ax U = p( w H + w H + w H ) = p[ w h r( X ) X + w h X + w h X ] X, X, h, h, h subject to the ecological constraints (5 ) and (6 ). w < w < w are the (average) body slaughter weights (kg per anial) o the three stages while p is the eat price (NOK per kg). However, or obvious reasons, the eat price will not aect the optiization except or scaling the shadow price values (see below). The Lagrangian o this proble writes L = p w h r X X + w h X + w h X X F h h X G h h r X X with λ > and µ > as the shadow prices o the eale and ale population, respectively 1. [ ( ) )] λ [ (, )] µ [ (, ) ( ) ] The irst order conditions o this axiising proble are (the second order conditions are ulilled due to the concavity o the recruitent unction): (8) L/ X = p[ w h ( r' X + r) + w h ] λ + µ G( r' X + r) =, (9) L / X = pw h µ =, (1) L h = pw rx + F + G rx ; / λ µ h 1 <, (11) L/ h = pw X + λf ; h < 1 and 1 The interpretation o λ and µ as shadow prices are not obvious as the population sizes are deterined within the odel. However, when adding X, interpreted as an exogenous nuber o introduced eales, to the stock constraint (5 ) it can be shown that * * U / X = λ, and where U denotes the axiu value o U. In the sae anner, adding X as an exogenous nuber o introduced ales to (6 ), gives * U / X = µ. 6

7 (12) / L h = pw X + µ G rx ; < h 1. Conditions (8) and (9) steer the shadow price values, and (9) says that the ale shadow price should just be equal its arginal harvesting value. Equation (8) is soewhat ore coplex, but says basically that the eale shadow price should be equal the su o the arginal harvesting value o the eale and the young sub-populations, plus the indirect ale arginal harvesting value, evaluated at its shadow price. Rewriting equation (8) when using condition (9) yields λ = p( wh + w h G)( r' X + r) + pw h. As the slope o the recruitent unction is non-negative, ( r' X + r) (see above), λ pw h holds. While the shadow value o the ale population is exactly equal its arginal harvesting value, we thereore ind that the shadow value o the eale population is above its arginal harvesting value. In this sense, eales ay be considered as ore valuable than ales and is line with the biological notion o eales as valuable and ale as non-valuable. Conditions (1) (12) are the control conditions with the actual copleentary slackness conditions stated. Fro the ale condition (12), harvesting down the whole population is hence considered as a possibility as this is the biological end product. On the other hand, keeping the eale and young sub-populations unexploited are options as these stages represent the reproductive and potentially reproductive biological capital. Condition (1) indicates that the harvesting o the young should take place up to the point where the harvesting beneit is equal, or below, the cost in ters o reduced population o ales and eales evaluated at their respective shadow prices. When (1) holds as an inequality, we have that the arginal harvesting beneit is below its arginal cost and harvesting is thus not proitable, h =. The interpretation o the eale harvesting condition (11) is soewhat sipler. Due to the ecundity density eect, eaning that one ore eale on the argin yields a saller recruitent when the eale population is high than when being low, h = sees less likely. The ale harvesting condition (12) is parallel to that o the eale harvesting condition (11), but the cost-beneit ratio works generally in the opposite direction. It can be shown that this condition always will hold as an inequality. This is revealed when irst cobining conditions (9), (12) and (6 ) which yields ( G+ h G ). When next inserting or G (and G ) ro equation (6) this condition reads 7

8 {.5(1 )(1 h ) /[1 (1 )(1 h )]}{1 (1 ) h /[1 (1 )(1 h )]}. Ater soe sall rearrangeents, this condition boils down to. Accordingly, because >, we ind h = 1and the whole ale population should be harvested. Notice that this result holds irrespective o the values o ales and eales (as given by the body weights). The reason or harvesting down the whole biological end product as the best option is the lack o any trade-os when the eat value is axiised; there is neither any biological eedback eects ro the other stages nor any price deand response. The ale-eale proportion becoes accordingly X / X = [1 (1 )(1 h )] (section two above) in the optial progra while one ore ale (c. also ootnote 1) yields a beneit o µ = pw (NOK per anial). I the optial policy at the sae tie yields h =, the eale shadow price reads λ = pw GrX ( ' + r) + pwh. As G =.5(1 ) when h 1 = and h = (equation 6 ), and w < w, the eale shadow price ay be lower than the ale shadow price irrespective o the above notion o eales as ore valuable than ales. In addition, ro condition (8), it ay also be shown that i 1 Gr ( ' X + r) > wh then µ > λ. This shows ore directly that a low eale slaughter weight ay pull in the sae direction. 4. Exploitation. Modern ties; trophy hunting The oose harvesting regie in Scandinavia (like wildlie hunting other places) is gradually changing, and a hunting and wildlie industry is eerging (see e.g., Skogeierorbundet 24). Modern ties is odelled by introducing a arket or trophy hunting or ales while still having eat value hunting o the other two stages. The arket or trophy hunting is probably soething between a copetitive arket and onopoly. One o these extrees is chosen, and we assue that trophy hunting licences are supplied under onopolistic conditions. Following the practice in Norway, one licence allows the buyer to kill one anial, which is paid only i the anial is killed. In addition to price, the deand or trophy hunting licences ay also be contingent upon quality, expressed by the abundance o ales (Skonhot and Olaussen 25). The inverse arket deand or ale hunting licences is hence given as: (13) q= q( h X, X ). 8

9 The licence price q (NOK per anial) decreases with a higher otake, q = q/ ( h X ) <, while it shits up with ore anials available, q >. Supplying H trophy hunting licences is also costly and depends on the nuber o anials shot: X (14) C = C( h X ) with ixed cost C () >, and variable cost C ' > and C ''. The ixed coponent includes the cost o preparing and arketing the hunting, whereas the variable coponent includes the cost o organising the perit sale, the costs o guiding and various transportation services, and so orth. The landowner anageent goal is now accordingly to ind a harvesting policy that axiises the su o the eat value and trophy hunting proit, or (15) ax π = p[ w h r( X ) X + w h X ] + [ q( h X, X ) h X C( h X )], X, X, h, h, h again subject to the constraints (5 ) and (6 ). The irst order conditions o this proble are (where L again is reerring to the Lagrange unction): (16) L X = q h X + qh C h + q h X µ = 2 / H( ) ' X and (17) L h = q X h + qx C X + G rx ; < h 1, 2 / H( ) ' µ in addition to conditions (8), (1) and (11). The ale harvesting beneit is now expressed by a arginal proit ter plus a arginal stock eect through the deand quality eect and the interpretation o (16) and (17) is straightorward (see also above). Cobing these conditions and (6 ) yields ( ')( ) qhx h + q C G+ h G + qxh X G, and where the ter ( G+ h G ) still is strictly positive because > (see section three above). When irst disregarding the quality 9

10 eect, q X =, we thereore ind that h < 1and hence not harvesting all ales, will represent the optial solution i the arginal harvesting proit equalises zero, ( q X h + q C') =. Fro condition (16) (as well as ro equation 17), it is seen that this iplies a zero value ale shadow price. A zero value shadow value while not harvesting down the whole stage is a counterintuitive result, but hinges on the biological end product nature o the adult ales; the ertility is not contingent upon the nuber o ales. The zero arginal harvesting proit condition ay be et i the arginal cost is high and/or the inverse deand schedule is steep (or in-elastic). On the other hand, i the arginal revenue exceeds the arginal cost or h = 1, we obtain the sae solution type as above and where µ is positive. H When taking the deand quality eect into account, q X >, h < 1 ay still hold as an optial solution when the arginal revenue exceeds the arginal cost ( q X h + q C') >, as G ' < and ( G+ h G' ) > (see above). The econoic reason or this result is siple as constraining the harvest and keeping a high stock size works in the direction o a higher trophy hunting licence price. Fro equation (16) it is seen that this situation iplies µ >. The corner solution o h = 1 is also now a possibility, but the arginal harvesting proit ust then exceed a certain iniu, equal the shadow price. H While the irst order conditions or harvesting eale and young are the sae as in the traditional harvesting regie, the new conditions or ale harvesting will obviously spill over to these stages. With h < 1, we ay typically ind that the ale-eale proportion X / X increases copared to the traditional regie which ay be reinorced i h shits up at the sae tie. Moreover, while the eat price p had no eect on the optial harvesting policy in the traditional regie, it ay now inluence the optial harvesting policy o all three stages. This will generally occur when the quality eect is included and we have µ >. In line with the standard harvesting theory, one ay expect that ore harvest and ewer anials o ale and young then will go hand in hand with a higher price. On the other hand, with no quality eect and with a zero shadow price value o the ales it will have no eect as the equations (5), (8), (1) and (11) then alone deterine h, h, X and λ. 1

11 5. Nuerical illustrations Data and speciic unctional ors The two exploitation schees will now be illustrated nuerically. The reproduction rate, decreasing in the nuber o eales, is speciied as a sigoidal unction with an increasing degree o density dependence at high densities (Nilsen et al. 25): r (2 ) r = r( X ) = 1 + ( X / K) b with r > as the intrinsic growth rate (axiu nuber o young per eale) and K > as the eale stock level or which density dependent ertility equalises density independent ertility. Thus, or a stock level above K, density dependent actors doinate. The copensation paraeter b > indicates to what extent density independent eects copensate or changes in the stock size. (2 ) iplies a recruitent unction ( ) /[1 ( / ) ] which is o the so-called Sheperd-type. b X = r X X = rx + X K The trophy deand unction is speciied linear. In addition, it is assued that the quality eect as given by the nuber o ales, through the paraeterγ, shits the deand uniorly up: (13 ) X γ q= αe βh X. Accordingly, the choke priceα > gives the axiu willingness to pay with a zero quality eect, γ =, whereas β > relects the arket price response in a standard anner. The trophy cost unction is given linear as well: (14 ) = + C c ch X so that c is the ixed cost and c > is the constant arginal cost. Table 1 gives the baseline paraeter values. For these deand and cost unctions we ind that the optial nuber o hunted ales will be h X = ( α c) / 2β without deand quality eect and when µ = at the sae tie (equation 16). 11

12 Table 1 about here Results Table 2 reports the results with baseline paraeter values. As a benchark, a no-hunting scenario is also included (irst row). Since the young enters the (adult) ale and eale stages at the sae sex ratio, the nuber o ales and eales are here the sae. In the traditional regie with eat value axiisation, the eale harvest rate becoes.26 while no harvest o young represents an optial policy. The arginal harvesting beneit o young is hence below the arginal cost in ter o losses ro reduced harvesting o ales and eales. Notice that the nuber o young is lower in the no-hunting scenario than in the traditional regie. This is due to the act that the nuber o eales is above the value representing the peak value o the recruitent unction and we have dx / dx = ( r ' X + r) < without harvesting. The ale shadow value is about our ties above that o the eale shadow value. As deonstrated in section three, the ale shadow value is exactly equal its arginal h harvesting value µ = pw while the eale shadow value should be above its arginal harvesting value. However, due to the low eale harvesting raction and an optial harvesting policy close to the peak o the recruitent unction; that is ( r' X + r) is sall positive (see above), the eale shadow value becoes low. Table 2 about here The odern ties exploitation schee is irst studied when the quality eect is disregarded; that isγ = and the inverse deand unction (13 ) reads q = α βh X. Harvesting down all the ales is no longer an optial policy, and we now ind a substantial lower harvesting raction, h =.24. As expected, the ale-eale proportion increases, and at the sae tie the eale harvesting raction shits soewhat up copared to the traditional regie. However, it is still beneicial to keep the young population unexploited. When the quality eect is added, the ale harvesting rate, as expected, is urther reduced accopanied by a positive shadow value indicating that the arginal harvesting incoe exceeds the arginal cost in optiu. The dierence between the ale and eale shadow values is now quite sall. The eale harvesting rate decreases soewhat as well. As a 12

13 consequence, the total stock size is higher when the quality eect is included and substantially higher than that o the traditional harvesting schee o eat value axiisation. The table also deonstrates that the proit increases copared to the traditional regie, and it urther increases when the deand quality eect is added. However, or obvious reasons, the speciication o the deand unction and paraeterisation play a critical role here. Shiting up the eat price p, just scales up the shadow price values in the traditional regie. In the trophy hunting regie with no quality eect and with a zero shadow price value o the ales, the harvesting activity and stock sizes will be unaected as well. As indicated, the logic behind this result is that the equations (5), (8), (1) and (11) alone deterine h, X and λ when µ =.On the other hand, with the quality included and µ >, the ale harvesting activity interacts with the other stages and hence p has an allocation eect. However, sensitivity analyses show that the eale harvest rate increase only odestly even or a quite draatic price increase. The reason or this hinges on the act that eale stock is close to the peak o the recruitent unction. h, 6. Concluding rearks The paper has analysed a three stages odel o the Scandinavian oose with ertility as the only density dependent actor. Two exploitation schees have been studied and it is deonstrated that harvesting down the whole biological end product, i.e., the (adult) ale population in this odel, always represents the best option when eat value axiisation is the goal. In the nuerical exaples, this option is accopanied by zero harvesting o the young and odest eale harvesting. Within this regie the biological notion o eales as valuable and ales as non-valuable is easily recognized even i the shadow value o the ales ight be higher than that o the eales. The odern ties exploitation schee with a arket or trophy hunting, changes the optial harvesting condition o the ales. Hunting down the whole population will no longer be the best option suggested that a well developed arket or trophy hunting is present. In addition, the allocation in the trophy hunting arket also spills over to the conditions o eat value axiisation o young and eales. The aleeale population ratio will increase, and the nuerical exaples show that ore eale harvesting ay take place. 13

14 Although the odel is siple and thereore soewhat unrealistic, it encopasses soe general results that will survive in ore coplex stage-structured odels. Most iportant, we have highlighted the econoic orces inluencing harvest in three dierent stages that, in various degrees, are present in any structured population odels. In our odel there are two recruiting stages which recruit in dierent ways. The young represents a value through recruitent to the (old) ale and eale stages. As long as density dependent growth actors are weak, or non-existing (as here), harvesting young does not pay o. For the eales, on the other hand, a traditional trade-o between recruitent and harvest is present through the density dependent ertility echanis. This echanis will also be present in ore coplex odels. Finally, the (old) ale stage is considered as the biological end product, and thus inluences not the recruitent. It is thereore tacitly assued that there are always enough ales or reproduction. However, irrespective o this, our odel deonstrates that the ale optial harvest policy depends critically on econoic circustances. Literature Caswell, H. 21: Matrix population odels: Construction, analysis and interpretation (2th. Ed.). Sinauer, Boston Clark, C. and D. Tait 1982: Sex-selective harvesting o wildlie populations. Ecological Modelling 14, Cooper, J. 1993: A bioeconoic odel or estiating the optial level o deer and tag sales. Environental and Resource Econoics 3, Gandolo, G. 1996: Econoic dynaics. Springer, Berlin. Milner-Gulland, E.J., T. Coulson and T. Clutton-Brock 24: Sex dierences and data quality as deterinants o incoe ro hunting red deer Cervus elaphus. Wildlie Biology 1, Nilsen, E., T. Pettersen, H. Gundersen, A. Mysterud, J. Milner, E. Solberg, H. Andreassen and N.C. Stenseth 25: Moose harvesting strategies in the presence o wolves. spatially structured populations. Journal o Applied Ecology 42,

15 Skonhot, A., N. Yoccoz and N.C. Stenseth 24: Manageent o a chaois (Rupicapra rupicapra) oving between a protected area and a hunting area. Ecological Applications 12, Skonhot, A. and J. O. Olaussen 25: Managing a igratory species that is both a value and pest. Land Econoics 81, 34-5 Storaas, T., H.Gundersen, H. Henriksen and H. Andreassen 21: The econoic value o oose in Norway. Alces 37, Skogeierorbundet (Norwegian Forest Owners Association) 24: Utvikling av utarksbaserte reislivsbedriter (in Norwegian), Oslo 15

16 Table 1: Biological and econoic paraeter values Paraeters Description Baseline value Reerence/source r> ax. speciic growth rate 1.15 Nielsen et al (25) K eale stock level where density 1, Nielsen et al (25) dependent actors doinates density independent actors b density copensation paraeter 2 Nielsen et al (25) w average weight young 6 Kg SSB 24 w average weight eales 15 Kg SSB 24 w average weight ale 17 Kg SSB 24 natural ortality young.5 Nielsen et al (25) natural ortality eale and ale.5 Nielsen et al (25) p eat price 5 NOK Storaas og Henriksen (22) α choke price 3, NOK Calibrated γ quality paraeter deand.1 Calibrated β slope paraeter deand 6 NOK Calibrated c ixed harvest cost 5, NOK Calibrated c arginal harvest cost 2, NOK Calibrated 16

17 Table 2: Ecological and econoic equilibriu, dierent anageent regies. h harvest raction young, h harvest raction eale, h harvest raction ale,h total harvest raction, X nuber o young (in 1 anials),x nuber o eales (in 1 anials), X nuber o ales (in 1 anials), X total stock ( in 1, anials), λ eale shadow price (in 1, NOK per anial), µ ale shadow price (in 1, NOKper anial) and, π proit (in 1, NOK) Hunting regies h h h h X X X X λ µ π No harvest Traditional regie, hunting or eat Modern ties; trophy hunting. No quality eect Modern ties; trophy hunting. With quality eect , , ,594 (-- indicates value not calculated) 17