Perfec seleciviy vs. biomass harvesing in an age- srucured fishery populaion model. Wha is he gain? Anders Skonhof Deparmen of Economics Norwegian Universiy of Science and Technology Trondheim, Norway (Anders.skonhof@sv.nnu.no) Absrac A generic ype age srucured fishery populaion model consising of wo harvesable maure age classes and one juvenile age class is formulaed. Exploiaion wih i) similar (uniform) fishing moraliy and ii) perfec fishing seleciviy among he age classes is considered. Perfec seleciviy allows for differen fishing moraliy among he age classes when he harves is subjec o maximizaion in one or anoher respec, and he gain of his ype of exploiaion is compared wih he uniform, or biomass, exploiaion. Susainable yield biomass funcions are developed, and he maximum susainable yield (MSY) soluions are found under boh exploiaion schemes. Finally, he maximum economic yield (MEY) soluions are found and characerized. Keywords: Fishery economics, age-srucured model, biomass harvesing, perfec fishing seleciviy JEL code: Q
1.Inroducion The working horse model in fishery economics has for a long ime been he biomass model (or surplus producion model, or lumped parameer model, or he Clarkmodel, Clark 1990). The firs sudies using his model were published in he mid 1950 s, and were concerned wih unconrolled, or 'open access', fishing aciviy. Some weny years laer he analysis was exended o accoun for he opimal harvesing, or fishing, aciviy over ime and where fish was considered as capial. Wihin his capial heoreic framework, he fish could eiher be kep in he ocean, or i could be harvesed and sored in he bank as capial. I was shown ha hree facors played an imporan role deermining he degree of exploiaion; he price-cos raio of he yield, he naural growh rae of he fish sock and he opporuniy cos of he naural capial, i.e., he rae of ineres. Addiionally, when sudying he biomass model wih conrolled versus unconrolled harvesing, i was eviden ha he managemen srucure of he fish sock, or he propery righs scheme, played an imporan role for he exploiaion pressure and he economic ren. While he predicions and he driving forces from he Clark-model are in line wih real life experience (see, e.g., Brown 000 and he references herein), i is also clear ha he biomass model generally gives oo lile guidance in real life managemen siuaions. Wha ofen in needed in an acual managemen siuaion is a more fineuned, or disaggregaed, populaion model which have he capabiliy o say somehing abou fishing moraliy, as well as naural growh and spawning abiliy, among differen sub-groups, or age-classes, of he considered fish populaion. For example, in acual managemen siuaions, he manager is ofen confroned wih he quesion of deciding wha fishing gears ha should be used, which fishing grounds ha should be exploied, he iming of he fishing over he year cycle, and so forh. In mos siuaions hese decisions implicily, or explicily, are relaed o he quesion wheher he young or he old fish of he populaion should be argeed and subjec o fishing moraliy. In acual managemen siuaions in fisheries here are hence need for a more disaggregaed assessmen of he fish socks 1. 1 In an influenial aricle Jim Wilen (000, p.30) wries: Las, over much of his period (he las few decades, A.S.), economiss coninued o depic bioeconomic sysems in simple analyically racable ways, assuming homogeneous populaion processes capured by lumped parameer models. During he same period, biologiss were moving away from
Analysing age, or sage, srucured fishery models, i.e., models where he species are grouped in differen classes according o age and sex, has a long radiion wihin biology. Caswell (001) gives an in-deph overview; see also Gez and Haigh (1989). However, economics play a minor role in hese works. Colin Clark has a chaper on age srucured fishery models in he 1976 ediion of his milesone book. This chaper is more or less lef unchanged in he 1990 ediion (Clark 1990, Ch. 9) where he, among ohers, sudy he condiions for fishing a single age class, or cohor, independenly of oher cohors, and wihou aking he effecs on recruimen ino serious accoun. Anoher early conribuion is Reed (1980) who analyically solved he maximum susainable yield problem, and where he found ha opimal harvesing includes a mos wo age classes. Olli Tahvonen (009) has more recenly derived boh analyical and numerical resuls on opimal harvesing in a dynamic seing under various simplifying assumpions. Oher recen conribuions demonsraing analyical resuls in age-srucured fishery models include Dieker e al. (010), Skonhof e al. (01) and Quaas e al. (013). While Tahvonen analyzes he siuaion where one flee (or one agen) is argeing he various age classes of he fish sock wih similar, or differen, cachabiliy coefficiens, Skonhof e al. includes wo flees argeing wo differen age classes wih perfec as well as imperfec fishing seleciviy. In his paper a generic ype age srucured fishery populaion model consising of wo maure harvesable classes is considered. The exploiaion of his fish populaion wih perfec fishing seleciviy is compared o he siuaion where exploiaion akes place wih similar (uniform) harvesing raes. Perfec argeing generally allows for differen fishing moraliy among he age classes when he harves is subjec o maximizaion in one or anoher respec. Uniform harvesing wih no disincion beween he offake, a fish is a fish, is considered as biomass exploiaion. The aim of his paper is o compare he exploiaion of his fish sock under hese wo differen harvesing hese simple views by incorporaing size and age srucures and more realisic depicions of spaial processes, including heerogeneiy, pachy resource abundance, dispersal, and oceanographic linkages. While biologiss developed new and richer depicions of more realisic populaion processes wih simulaion modeling, calibraion, and saisical esimaion echniques, economiss mosly coninued o work wih simpler models ha could be analyically solved. 3
schemes. We derive susainable yield funcions, and find he maximum susainable yields (MSY). We herefore demonsrae ha MSY is no a well-defined concep in a model wih several harvesable age classes. No surprisingly, we find ha MSY wih perfec argeing exceeds MSY wih uniform harvesing, and hus represens a gain. Various facors affecing his gain are analyzed. Maximum economic yield (MEY) fishing is also included in our paper and is compared o he MSY soluion. The presen paper is relaed o Tahvonen (009), Lawson and Hilborn (1985), Gez (1980), and Walers (1986, p. 13-140) who all, in various ways, discusses possible relaionships beween age srucured models and biomass models. See also Flaaen (011). The paper is organized as follows. In he nex secion, he generic populaion model is formulaed. Secion 3 demonsraes wha happens when he fish sock is exploied as a biomass model wih uniform fishing moraliy among he wo harvesable age classes. We find he susainable yield funcion, and he harvesing is opimized giving he peak value of his funcion, he MSY. In secion 4 he susainable yield funcion and MSY are derived when he fish sock is exploied wih perfec harvesing seleciviy. In secion 5 prices and coss are inroduced, and maximum economic yield (MEY) harvesing is explored. Finally, secion 6 summarizes our findings.. Populaion model As already indicaed, our sudy of harves of differen age classes of a fish populaion is ackled by using a quie simple ( generic ) model wih jus hree cohors of he fish populaion, and is similar o ha of Skonhof e al. (01). The younges class is immaure fish which is no harvesed due o, say, gear regulaions (mesh size resricions), whereas he wo older classes are harvesable and conribue boh o he spawning sock. Recruimen is endogenous and densiy dependen, and he old maures are assumed o have higher feriliy han he young maures. Naural moraliies, or equivalenly he survival raes, are assumed o be fixed and densiy independen for all hree age classes. In he single period of 1 year, hree evens Flaaen (011, p.113) righly observes ha when working wih deailed and complex year class models we mus be aware ha even in such models we can find he maximum susainable yield (MSY) and he corresponding sock size, hough hese characerisics are no as apparen as in he aggregaed biomass models. 4
happen in he following order: Firs, recruimen and spawning, hen fishing which akes place insananeously of boh maure age classes, and finally naural moraliy. The resricion of naural moraliy o ake place only a cerain period of he year is cerainly one of our generic model assumpions. The fish populaion in number of individuals a ime (year) is srucured as recruis X ( 0, yr 1), young maure fish X 1, (1 yr ) and old maure fish X, ( yr ). The number of recruis is governed by he recruimen funcion: (1) X 0, R X1, X, (, ) where R() is assumed o be of he Beveron-Hol ype characerized by R(0, 0) 0 and R / X i, R' i 0, ogeher wih R'' 0 ( i 1, ) (see also below). As we assume higher feriliy of he old han he young spawning age-class, R' R' holds as well. The number of young maure fish follows nex as: 1 () X1, 1 s0x 0,, where s0 is he fixed naural survival rae. Finally, he number of old maure fish is described by: (3) X, 1 s1(1 f1, ) X 1, s(1 f, ) X, where f1, and f, are he oal fishing moraliies of he young and old maure sage, respecively, while s1 and s are he naural survival raes of hese maure sages, assumed o be fixed and densiy independen. Typically, hese survival raes do no differ oo much. Eq. (3) describing he relaionship beween he wo maure age classes is noified as he spawning consrain. When combining Eqs. (1) and () we find: (4) X1, 1 s0r X1, X, (, ), which in our model is he recruimen consrain. i The weigh of he young and old fish (kg per fish) are denoed by w 1 and w, respecively, and wih w w1. The biomass harvesed in year and he sanding biomass are hus defined hrough: H ( w X f w X f ), (5) 1 1, 1,,, 5
and (6) B w1 X1, w X, ( ), respecively, as fish abundance is measured in he beginning of he year, before fishing, and fishing akes place before naural moraliy. The populaion equilibrium for fixed fishing moraliies f, i f, 0,1,..., is defined i by Xi, 1 Xi, Xi ( 1,) i such ha Eqs. (3) and (4) hold as: (3 ) 1 1 1 and X s (1 f ) X s (1 f ) X X s R( X, X ). (4 ) 1 0 1 An inernal equilibrium is shown in Figure 1 where he recruimen is specified as he a( X1 X ) Beveron-Hol funcion R( X1, X ) wih a 0 as he scaling b ( X X ) 1 parameer, b 0as he shape parameer and 1as he parameer indicaing higher feriliy of he old maure fish. Higher fishing moraliies shif down he spawning consrain and hence lead o smaller equilibrium socks. A he same, he maure fish sock composiion changes such ha he abundance of young maure sock increases relaively o he old maure sock. Noice ha his happens also if only he fishing moraliy of he young sock shifs up. Noice also ha an inernal equilibrium is defined for 0 f1 1only; ha is, no all he young maure fish can be harvesed. Thus, wih f1 1 and 0 f 1he spawning consrain becomes fla and we find X 0 ogeher wih he recruimen consrain as X1 s0r X1 (,0) 0. Is his case, he equilibrium number of fish harvesed of he young maure caegory is f 1 X 1 X 1. Figure 1 abou here 3. Uniform harvesing and exploiing he fish sock as a biomass model 3.1 The susainable yield funcion We sar o analyze he exploiaion of our fish populaion as a biomass model in biological equilibrium wih similar, or uniform, fishing moraliies among boh age classes, i.e., f1 f f. Uniform exploiaion may ypically be inerpreed as (oally) imperfec harvesing seleciviy; ha is, one fishing flee, or agen, is 6
argeing boh socks wih idenical cachabiliy coefficiens (see also secion 5 below). Our firs ask hen is o solve for he biomass harves Eq. (5), defined as 1 1 H ( w X w X ) f, and for he sanding biomass Eq, (6), defined as B w1 X1 w X ( ), for he whole range of possibly fishing moraliies wih he populaion defined by Eqs. (3 ) and (4 ), and where Eq. (3 ) now reads 1 1 X ( s X s X )(1 f ). For he above specified Beveron Hol recruimen funcion, and our baseline parameer values (Table A1 in Appendix), H and B are depiced as funcions of he fishing moraliy in Figure. The unexploied biomass, implying f H 0, represens he carrying capaciy, B K. Moreover, we find he smalles sanding biomass when f 1, bu sill wih some harves of he young maure sock and also posiive biomass, H Bmin w1 X1 0. The reason for posiive biomass when harvesing all he maure fish is ha here is no harves of immaure fish. No surprisingly, because he recruimen funcion is no peak-valued (i.e., no of he Ricker-ype), we also find he size of he sanding biomass o decline monoonically wih he exploiaion pressure and higher fishing moraliies, while he harvesed biomass firs increases, reach a peak value and hen declines. Figure abou here In Figure 3 we have ploed he funcional relaionship beween H and B, H F( B), for idenical fishing moraliies. This peak valued curve FB ( ) hus represens he susainable (equilibrium) yield biomass relaionship, or he susainable yield funcion, in our age srucured fishery model when exploiaion is governed by uniform fishing moraliies. We find he maximum susainable yield (MSY) a H 117 (onne), reached a a biomass level B 53(onne) when f f 0,50 (see also Figure ). Wih f 1 we have H F( B ) B 800 (onne). The similariies as well as he differences min min compared o he sandard logisic growh biomass funcion are apparen. Our susainable yield funcion is lef-hand skewed, and he MSY biomass size is locaed well below half of he biomass carrying capaciy. The mos sriking difference is, 7
however, he resriced definiion se as FBis ( ) no defined below ha of he biomass level B governed by f 1. min Figure 3 abou here In Table A (Appendix), we have also included he size of he fish socks. A sriking feaure wih he sock evolvemen along H F( B) when moving from B K, is he rapid reducion in number of old maure fish wih increasing fishing moraliy. The raio of he old o he young maure fish is defined hrough Eq. (3 ) when f 1 f f as X X s f s f / 1 /[1 1 ]. For he unexploied fish sock; ha is, a he 1 1 biomass carrying capaciy, we find.33 ( X / X1 s1 / (1 s ) 0.7 / 0.3 1840 / 789 ) under he baseline parameer values scenario (Tables A1 and A). This raio reduces o 0.54 a B. Wih f 1, we find, as already indicaed, X / X1 0, and only juveniles no exploied enering he young maure class conribues o he sanding biomass. Lawson and Hilborn (1985) also consruced a susainable yield funcion based on uniform harvesing for a fish populaion model including several age classes. The recruimen funcion was of he Beveron Hol ype (bu hey also used he Ricker funcion), and found he maximum susainable yield funcion o have srong similariies wih he sandard logisic growh funcion. In an ad hoc manner hey also fied heir funcion o he sandard logisic funcion and calculaed he maximum specific growh rae. See also Tahvonen (009) who sudy uniform harvesing, as well as quasi uniform harvesing where he fishing moraliies are fixed hrough he raio of he cachabiliy coefficiens using he sandard Schaefer harvesing funcion. 3. Opimizing he harves We now proceed o characerize he maximum susainable biomass level maximum susainable harves H B and he (MSY) wih uniform fishing moraliies. The problem of finding he uniform fishing moraliy maximizing he equilibrium biomass harves is saed as H max[( w1 X1 w X ) f ] subjec o he spawning consrain (3 ) wih f1 f f, and he recruimen consrain (4 ). The lagrangian of his 8
problem may be wrien as L ( w1 X1 w X ) f [ X1 s0r( X1, X )] 1 1 [ X ( s X s X ) 1 f ] where 0 and 0 are he shadow prices of he recruimen consrain and spawning consrain, respecively. The firs order necessary condiions (wih X i 0, i 1, ) are: (7) L f w1 X1 w X s1 X1 sx / ( ) ( ) 0 ; 0 f 1, (8) L X w f s R s f and / ' 1 0 1 1 0 1 1 (9) L X w f s R s f / ' 1 0. 0 These equaions ogeher wih he biological consrains define he fishing moraliy f, he number of fish X ( i 1, ) he maximum susainable yield i H (MSY), and also B hrough Eq. (6). Conrol Eq. (7) indicaes ha fishing should ake up o he poin where he oal marginal biomass gain equalizes he oal marginal biomass loss due o reduced spawning of boh age classes, evaluaed by he spawning consrain shadow price. When rewriing his condiion ( w s ) X ( w s ) X 0 and aking ino accoun ha we have as 1 1 1 w / s w1 / s1, as w w1and he survival raes ypically do no differ oo much (secion ), we hus find ha he marginal biomass gain of fishing he old maure age class mus exceed is marginal loss given by he biological discouned spawning consrain shadow price, i.e., w s. The opposie holds for he young maure class, w1 s1. Rewriing he sock condiion (8) as w f s R' s 1 f, i 1 0 1 1 saes ha he young maure fish should be mainained so ha is shadow price equalizes is marginal biomass gain value plus he marginal reduced recruimen sr' loss plus he marginal reduced spawning s f loss, boh loss facors 0 1 evaluaed a heir respecive shadow pieces. Condiion (9) can be given a similar inerpreaion. 1 1 While our inerpreaion of he condiions characerizing H make good economic sense in erms of marginal gains and marginal losses, he above necessary condiions 9
are no paricularly helpful o indicae how he various biological forces influence he value of f and he locaion of B in his uniform harvesing siuaion. However, no surprisingly, numerical simulaions demonsrae ha higher weighs of he fish socks and higher feriliy increase H while higher naural moraliies, or lower naural survival raes, work in he opposie direcion. We also find ha he value of f is only modesly affeced when, say, boh fishing weighs are doubled. However, when he raio of he fish weighs shifs, subsanial changes may ake place. See Table 1below. 4. Harvesing he fish sock wih perfec seleciviy We now proceed o sudy harves of our fish populaion when boh age classes are perfecly argeed wih separae harves conrols, or fishing moraliies. We firs find and characerize he MSY harves and he associaed sanding biomass level. Nex, we consruc he susainable yield funcion. While consrucing he susainable yield funcion wih uniform fishing moraliy is sraighforward, he consrucion is more unclear when he fishing moraliies are differen simply because i is more o choose abou. More deails on his below. The problem of finding he maximum susainable yield (MSY) is now defined by H max( w X f w X f ) subjec o he spawning consrain (4 ) and recruimen 1 1 1 consrain (3 ). The Lagrangian of his problem is wrien as L ( w X f w X f ) [ X s R( X, X )] 1 1 1 1 0 1 1 1 1 [ X ( s X (1 f ) s X 1 f ]. Following he Kuhn-Tucker heorem he firs order necessary condiions (assuming X i 0, i 1, ) are: L / f X ( w s ) 0; 0 f1 1, (10) 1 1 1 1 / ( ) 0 ; 0 f 1, (11) L f X w s (1) L X w f s R s f and / ' 1 0 1 1 1 0 1 1 1 (13) L X w f s R s f / ' 1 0. 0 10
The conrol condiion (10) saes ha he fishing moraliy of he young maure sock should ake up o he poin where he marginal biomass gain is equal, or below, i s biological discouned marginal biomass loss, evaluaed a he spawning consrain shadow price. Because he fishing moraliy of his sock mus be below one o preven sock depleion (secion above), he marginal biomass gain can no exceed he discouned marginal biomass loss. Conrol condiion (11) is analogous for he old maure sock, excep ha he marginal biomass gain may exceed he marginal discouned biomass loss if fishing moraliy equals one. Eqs. (1) and (13) again seer he shadow price values and has more or less he same conen as in he uniform harvesing siuaion. As he maximum susainable yield problem wih perfec fishing seleciviy is a less consrained han wih uniform fishing, i follows from he very logic of opimizaion ha he value of H exceed H below. in his perfec seleciviy siuaion will found in he uniform fishing siuaion. See also numerical illusraion From he above conrol condiions, i is now clear ha differences in he biological presen-value biomass conen w / s ( i 1, ) seers he fishing prioriy. These i i facors were also prevalen in he uniform harvesing siuaion, bu now he implicaions are much clearer. Wih w / s w 1 / s 1 he above Kuhn-Tucker condiions indicae ha he fishing moraliy should be higher for he old han he young maure sock; ha is, f f1. There are hen generally hree possibiliies, all corner soluions, o mee he above conrol condiions (10) and (11); Case i) 0 f1 1 and f 1, Case ii) f1 0 and f 1, and Case iii) f 1 0 and 0 f 1. The size of boh maure socks will be higher in Case iii) han in Case ii), which again will be higher han in Case i). Therefore, again will be higher han in Case i). B will be higher in Case iii) han in Case ii), which For our baseline parameer values (again, see Table A1 Appendix), we find ha Case ii) wih old maure fishing only is opimal and hence f 1and f1 0. Therefore, he spawning consrain Eq. (3 ) as X s1 X1 ogeher he recruimen consrain (4 ) define X 1 and X in his opimum. For hese sock sizes, he maximum 11
susainable yield reads H wx, or H ws1 X1, while he associaed biomass level becomes B w1 X1 w X ( w1 ws1 ) X1. We hus find ha he maximum susainable yield biomass raio simply may be wrien as H / B w s / ( w w s ). 1 1 1 Having successfully been able o characerize he maximum susainable yield and he relaed susainable biomass level in his perfec seleciviy fishing siuaion (see also Skonhof e al. 01), he nex quesion is how o consruc he relaed susainable yield funcion, H F( B). There are various possibiliies, bu he following wo seps procedure is adaped. In he firs sep, saring a he carrying capaciy B wih f1 0 and f 0, f1 0 is kep fixed while f is increased gradually up o f 1 where H and he associaed biomass level B are reached. In sep saring a his poin wih f1 0 and f 1, f 1 is now kep fixed while f1increases gradually up o f 1 1. We hen have he smalles sanding biomass, bu sill wih some harves of he young maure sock and H Bmin w1 X1 0 (see above). Figure 4 depics he oucome where also he susainable yield funcion wih uniform fishing moraliies from Figure 3 is redrawn. While he wo end-poins of hese wo maximum susainable yield funcions are idenical, H F( B) will elsewhere in he perfec seleciviy case exceed ha of he uniform case. Therefore, for all biomass levels B min B B K, perfec fishing seleciviy represens a more efficien way o exploi he fish sock. See Appendix for a proof. K Figure 4 abou here Table 1 gives a numerical illusraion. The maximum susainable yield wih perfec argeing is 3 percen higher (1,378/1,17) han he uniform harvesing siuaion under he baseline parameer values scenario. In oher words, he maximum biomass offake gain is hus abou 3 percen when he fishing moraliies are composed in an opimal way. Somewha surprisingly, we find ha he size of he old maure sock is higher in he perfec seleciviy siuaion. In his able we have also included resuls from a scenario where he weigh of he old maure sock is reduced from is baseline 1
value of 3 o.3 (kg/fish). As a resul, he opimal harvesing scheme wih perfec seleciviy shifs from Case ii) o Case i) wih 0 f1 0.16 1 and f 1. Hence, he exploiaion becomes more aggressive. Fishing becomes also more aggressive in he uniform fishing siuaion, and where we now find f 0.55. The mos sriking change is, however, ha he H difference reduces quie dramaically, and he gain of perfec fishing seleciviy under his scenario is jus 5 % (1,061/1,016). Therefore, as he fish socks become more homogenous, here in erms of weighs, he gain of exploiing he fish populaion wih perfec fishing seleciviy reduces. Indeed, i can be shown ha if we hypoheically have w / s w1 / s1 he maximum susainable yield will be similar under hese wo harvesing schemes, and ha H wih perfec argeing can be me by an infinie number of combinaions of fishing moraliies. The inuiion is ha boh conrol condiions (10) and (11) in his siuaion mus hold as equaions, and hus give he same informaion. Accordingly, i will be one degree of freedom in our sysem of equaions deermining he fishing moraliies. On he oher hand, i can also be shown ha wih w / s w1 / s1 we sill find an unique soluion in he uniform fishing siuaion. Table abou here 5. The maximum economic yield (MEY) soluion We now proceed o sudy he maximum economic yield (MEY) problem, and compare his soluion o ha of he MSY problem. We firs consider he perfec seleciviy fishing siuaion, where wo flees, or wo agens, are argeing he fish populaion. Wih Ei, as he effor use of flee i a ime and assuming a sandard Schaefer harvesing funcion, he harves, or cach, of he wo flees is hus defined as: (14) hi, qi Ei, Xi, ; i 1,, where q i are he cachabiliy coefficiens. Therefore, fi, qi Ei, yields he fishing moraliies under our assumpion ha fishing akes place before naural moraliy. The spawning consrain Eq. (3) now becomes: X s (1 q E ) X s (1 q E ) X. (15), 1 1 1 1, 1,,, 13
Wih p 1 and p as he fish prices (Euro/kg), assumed o be fixed over ime and no influenced by he size of he cach, and where he old maure fish ypically is more valuable han he young fish ( larger files ) p p1, and ci as he uni effor cos (Euro/effor) ( i 1, ), also assumed o be fixed, p w q X c E p w q X c E describes he curren join profi of 1 1 1 1, 1 1,,, he wo flees wih perfec seleciviy. The MEY problem is hus saed as max subjec o he biological consrains (15) and (4) ogeher wih he effor E1,, E, 0 use consrain, 0 Ei, 1/ qi ( i 1, ). In addiion, he iniial sock sizes X i,0 have o be known. 1/ (1 ) 0 is he discoun facor wih 0 as he (consan) discoun ren. The lagrangian of his problem may be wrien as 0 L { p1w1 q1 X1, c1 E1, pwq X, c E, 1 X1, 1 s0r( X1,, X, ) 1 X, 1 s1 1 q1e 1, X1, s 1 qe, X, }. Following he Kuhn-Tucker heorem he firs order necessary condiions are now (when assuming Xi, 0, i 1, ): (16) L E1, p1w 1q1 X1. c1 1s1q 1X1. / ( ) 0 ; 0 E1, 1/ q1, 0,1,,..., (17) L E, pwq X. c 1sqX. / ( ) 0 ; 0 E, 1/ q, 0,1,,..., (18) L X1, p1w 1q1E 1, 1s0R 1 1s1 q1e 1, / [ ' 1 ] 0, 1,,3,... and (19) L X, pw qe, 1s0 R 1s qe, / [ ' 1 ] 0, 1,,3,.... Noice ha while he fishing moraliy of he young maure class mus be below one o susain he old maure populaion in biological equilibrium, or seady sae, his is no necessary so in a dynamic seing as E 1, 1/ q 1 may hold for some periods wihou depleion he old maure sock. In ligh of he previous secion, he inerpreaion of 14
he conrol condiions (16) and (17) should be clear. There are only wo differences; economic discouning,, and no only biological discouning plays a role, and he cos parameers are included. The dynamic sock Eqs. (18) and (19) seer he shadow price values, and have sraighforward inerpreaions (see also he above secion 3.). As he profi funcions are linear in he conrols, economic heory suggess ha fishing should be adjused o lead he fish socks o seady sae as fas as possible; ha is, he Mos Rapid Approach Pah (MRAP) dynamics. However, he MRAP is no a regular one in our age-srucured fish populaion, among ohers because he seady sae will be a corner soluion (see below). The age srucure also implies ha he populaion could be above ha of he opimal seady sae level for one age class and a he same ime lower han he opimal seady sae level for he oher age class. Since fishing is confined o wo age classes, he MRAP may imply a large harves in one period and small, or zero, harves in he nex (see Skonhof and Gong 014 for a parallel discussion). In wha follows we analyze seady saes where all variables are consan over ime. Omiing he ime subscrip and combining he conrol condiions (16) and (17) gives a possible soluion where boh conrols are inerior. This equaion, which is noified as a separaing condiion (see below), is wrien as: (1/ s )( p w c / q X ) (1/ s )( p w c / q X ), (0) 1 1 1 1 1 1 and is an increasing, concave curve in he ( X1, X) -plane saring from he origin. See Figure 1. Because pw / s p1w 1 / s1 we may suspec ha wih c / q c1 / q1 his inerior soluion can be an opimal opion. As demonsraed in Skonhof e al. (01) in a saic version of his model, however, his seady sae oucome is a saddle poin of he objecive funcion. Therefore, possible opimal seady sae soluions saisfying he above necessary condiions (16) - (19) indicae differen marginal profi (Euro/fish) among he wo fish socks, and in an opimum a leas one of he conrols mus be se o is lower, or upper, bound. Recognizing ha E1 canno be se o is maximum in seady sae, here are now four possible cases o be considered. We have he same hree cases as above (secion 4), now saed as Case i) wih 15
0 1/ E1 q1and E 1/ q E1 0 and 0 E 1/ q, Case ii) wih E1 0 and E 1/ q, Case iii) wih. Addiionally, Case iv) wih 0 E11/ qand E 0 is also a possible opimal opion. From he conrol condiions (16) and (17) we find ha he separaing condiion (0) holds as (1/ s ) p w c / q X (1/ s ) p w c / q X 1 1 1 1 1 1 in Cases i) iii) and he opposie in he las Case iv). Therefore, he soluion in Cases i) iii) is locaed wih an ineracion beween he recruimen consrain Eq. (4 ) and he spawning consrain (3 ) o he norh-wes of his separaing condiion Eq. (0). On he conrary, he soluion in Case iv) is locaed o he souh-eas of he separaing condiion. See Figure 1, which illusraes eiher Case i), Case ii) or Case iii) as he opimal opion. In our baseline parameer value scenario (Table A1) wih similar fishing coss of he wo flees, c1 / q1 c / q (Euro), we find ha fishing he old maure sock only and Case ii) is opimal. Tha is, he soluion of he MEY problem coincides wih he soluion of he MSY problem. This resul is no in accordance wih he sandard biomass model (he Clark-model). While discouning pulls in he direcion of an opimal biomass level below ha of B in he Clark-model, sock dependen harvesing coss pull in he oher direcion. In our age-srucured model he discoun ren plays a more indirec role as i generally influences which of he cases i), ii), iii) or iv) ha will be mos profiable, while he srucure of he harves coss may work in he opposie direcion of ha of he Clark-model. For example, as already indicaed, we find ha higher flee coss may change he opimal harves opion from Case ii) o Case iv) accompanied by smaller fish socks and a smaller sanding biomass level. Table illusraes. The effecs of changing discouning are also demonsraed, and where doubling he discoun ren does no change he opimal harves opion in his perfec fishing seleciviy MEY problem. Table abou here We hen consider he MEY problem wih one flee. The harves funcions are hen generally defined as: 16
(1) hi, qi E Xi, ; i 1,. Wih similar cachabiliy coefficiens, q1 q q and (perfec) uniform fishing moraliies, he spawning consrain reads: () X, 1 s1 (1 qe ) X 1, s(1 qe ) X, while he curren profi is described by 1 1,, p qx p qx c E. The MEY problem is hus now saed as max 1 1,, E p qx p qx c E subjec o 0 consrains (4) and () ogeher wih he resricions on he fishing moraliy and effor use. Again we assume ha i is economically beneficial o fish and also no o deplee he fish socks. We also now only consider seady sae wih all variables consan over ime. The numerical resuls wih he baseline parameer values are shown in Table (row four). The soluion here differs from he MSY soluion (Table 1), and he fishing moraliies become lower. Therefore, under his scenario we find ha he cos effec dominaes he discouning effec such ha his soluion is locaed o he righ hand side of he MSY soluion wih less aggressive harvesing and a larger sanding biomass. In his baseline parameer value scenario wih c1 c c, we also find ha he seady sae profi in his uniform harvesing problem is lower han in he perfec seleciviy siuaion. This comes no as a surprise as he parameer values are similar, and we again have ha he MEY problem wih perfec seleciviy is a less consrained maximizaion problem han wih uniform fishing. We also find ha he sock sizes and yield becomes lower wih uniform harvesing. Increasing he harvesing cos works now jus as in he Clark model wih reduced effor use, smaller fishing moraliies and a higher sanding biomass. On he oher hand, a more myopic fishing policy; ha is, increasing he discoun ren, gives a more aggressive harvesing compared o he baseline scenario. Therefore, wih uniform harvesing in our age srucured model, discouning works also in he same direcion as in he Clark-model. 6. Concluding remarks 17
This paper has compared he exploiaion of an age srucured fish populaion under perfec fishing seleciviy wih similar (uniform) fishing moraliies for he harvesable socks. We find ha he maximum susainable yield (MSY) will be highes in he perfec seleciviy case. This follows from he very logic of opimizaion, as his siuaion represens a less consrained problem. We also find ha he susainable yield funcion in his siuaion will exceed he susainable yield funcion wih uniform harvesing in he whole range of possible fishing moraliies. Wih our illusraive baseline daa, we find he MSY o be subsanial higher (3 percen) in he perfec seleciviy siuaion. This difference reduces when he harvesable fish socks become more homogeneous hrough, say, smaller weigh discrepancies. In conras o he sandard biomass model, or he Clark model, our analysis demonsraes ha MSY is no a well-defined concep in a real world siuaion where he fish sock is composed of differen age classes. The MSY, as well as he susainable yield funcion, will depend on he degree of fishing seleciviy and he prioriizing of he harves among he harvesable age classes. The European Communiy (EU) wih is new Common Fishery Policy (CFP) saes ha Fishing susainably means fishing a levels ha do no endanger he reproducion of socks and a he same ime maximises caches for fishermen. This level is known as he 'maximum susainable yield' (MSY). Under he new CFP, socks mus be fished a hese levels The new CFP shall se he fishing levels a MSY levels by 015 where possible, and a he laes by 00 for all fish socks (hp://europa.eu/rapid/pressrelease_memo-13-115_en.hm). However, wihou furher specificaions his goal is of resriced value. Maximum economic yield fishing MEY is also sudied in our age srucured fishery model. Wih perfec seleciviy we find ha he locaion of he MEY soluion compared o he MSY generally will differ from wha we find in he Clark model, and ha he driving forces may be differen. For example, in our numerical illusraion, we find ha higher fishing coss may lead o more aggressive fishing and a higher exploiaion pressure, while increasing he discoun rae may no change he opimal MEY policy. On he oher hand, wih uniform fishing moraliies our numerical illusraions indicae similar driving forces as in he Clark model. For example, a higher discoun rae yields a more aggressive seady sae harvesing policy. 18
Lieraure Brown, G., 000. Renewable naural resource managemen and use wih and wihou markes. Journal of Economic Lieraure 37, 875-914 Caswell, H., 001. Marix Populaion Models: Consrucion, analysis and inerpreaion (h. Ed.). Sinauer, Boson. Clark, C., 1990. Mahemaical Bioeconomics (h. Ed). Wiley, New York. Dieker, F., D. Hjermann, E. Nævdal and N. C. Senseh, 010. Non-cooperaive exploiaion of muli-cohor fisheries-he role of gear seleciviy in he norh-eas arcic cod fishery. Resource and Energy Economics 3, 78-9. Flaaen, O., 011, Fishery Economics and Managemen. hp://munin.ui.no/bisream/handle/10037/509/book.pdf?sequence=3 Gez, A. and R. G. Haigh, 1989. Populaion Harvesing. Princeon Universiy Press, Princeon. Lawson, T. and R. Hilborn, 1985. Equilibrium yields and yield isoplehs from a general age-srucured model of harvesed populaions. Canadian Journal of Aquaic Science 4, 1766-1771 Quaas, M., T. Requae, K. Ruckes, A. Skonhof, N. Vesergaard and R. Voss, 013. Incenives for opimal managemen of age-srucured fish populaions. Resource and Energy Economics 35, 113-134 Skonhof, A., N. Vesergaard and M. Quaas, 01. Opimal harves in an age srucured model wih differen fishing seleciviy. Environmenal and Resource Economics 51, 55-544 Skonhof, A. and P. Gong, 014. Wild salmon fishing. Harvesing he old or he young? Resource and Energy Economics 36, 417-435 Tahvonen, O., 009. Economics of harvesing age-srucured fish populaions. Journal of Environmenal Economics and Managemen 58, 81-99. Walers, C., 1986. Adapive Managemen of Renewable Resources. Macmillian Publishing, London. Wilen, J., 000. Renewable Resource Economiss and Policy: Wha Differences Have We Made? Journal of Environmenal Economics and Managemen 39, 306-37 Appendix 19
Daa and addiional resuls The numerical simulaions are merely an illusraion and he parameer values are no relaed o any paricular fisheries. Table A1 shows he baseline parameer values. The same naural survival raes for he young and old maure age class is assumed while he weigh of he old age class is assumed 50 % higher han he young maure age class. Similar cachabiliy coefficiens and uni effor coss of he wo flees are assumed. The fishing prices are assumed o be similar as well. Table A1 abou here Table A gives deailed resuls from he uniform susainable yield harvesing siuaion wih he baseline parameer values. Table A abou here The susainable yield funcions The biomass Eq. (6) (when omiing he ime subscrip) may be rewrien as X B / w ( w1 / w ) X1. For a fixed value of B his equaion represens an isobiomass line, and is inersecion wih he recruimen consrain Eq. (4 ) hus gives he unique sock size combinaion condiioned upon he fixed biomass level B. Therefore, every biomass level B along he horizonal axis in Figure 4 in he domain B min B B K define a unique combinaion of X1 and X. In oher words, for he same sanding biomass level he number of fish will be idenical a he susainable yield funcions in he uniform and he perfec seleciviy fishing siuaion. When he harves is locaed o he righ hand side of he peak value wih perfec seleciviy and Case ii) is he opimal harves opion ( sep 1, secion 4), he susainable yield difference beween perfec seleciviy and uniform harvesing wries: w X f ( w X w X ) f. (A1) 1 1 Wih uniform harvesing, he spawning consrain Eq. (3 ) may be wrien as X s (1 f ) X 1 1 1 1 while i is defined as X (1 s(1 f)) s f sx (1 (1 )) wih perfec seleciviy 0
when f 1 0. Therefore, he relaionship beween he fishing moraliies is described f by f s (1 ). Insered ino (A1) yields: f f w f X w X f s (1 f) (A) ( ) 1 1 (1 s(1 f )) X afer some small rearrangemens. When nex insering for X1, we s (1 f) find: 1 s(1 f ) (1 s(1 f )) (A3) [ w ( 1) w1 ] fx 0 s (1 f ) s (1 f ) 1 wih s s 1 and w w1. Therefore, he susainable yield is higher wih perfec selecive fishing han wih uniform fishing when being o he RHS of he MSY wih perfec seleciviy. Similar oucome can be demonsraed when being o he lef hand side of MSY of he susainable yield funcion ( sep secion 4). 1 1
* X Figure 1. Biological equilibrium wih fixed fishing moraliies ( 0 f1 1 and 0 f 1). Beveron-Hol recruimen funcion. X Recruimen consrain Eq. (4 ) Spawning consrain Eq. (3 ) Separing condiion Eq. (0) ( s a b) 0 X 1
Figure. Susainable biomass (serie1) and susainable yield (serie ). In onne. Uniform fishing moraliies. Baseline parameer values 8000 7000 6000 5000 4000 3000 Serie1 Serie 000 1000 0 0 0, 0,4 0,6 0,8 1 Figure 3. Susainable yield funcion, H F( B). Uniform fishing moraliies. Biomass B (onne) horizonal axis and harves H (onne) verical axis. Baseline parameer values 100 1000 800 600 400 00 0 0 1000 000 3000 4000 5000 6000 7000 8000 3
Figure 4: Susainable yield funcions. Uniform harvesing (serie, lower curve) and perfec seleciviy (serie 1, upper curve). In onne. Baseline parameer values 1600 1400 100 1000 800 600 Serie1 Serie 400 00 0 0 000 4000 6000 8000 Table 1. Maximum susainable yield (MSY) uniform harvesing and perfec seleciviy. H (in onne) and associaed biomass level B (in onne) and fish socks (in 1,000 # of fish) f B X Uniform harvesing. Baseline parameer values f 1 H X 1 0.50 0.50,53 1,17 63 336 Uniform harvesing, w.3 (kg/fish) 0.55 0.55 1,847 1,016 604 78 Perfec seleciviy. Baseline parameer values 0 1.00,690 1,378 656 459 Perfec seleciviy, w.3 (kg/fish) 0.16 1.00,19 1,061 635 374 4
Table. Seady sae maximum economic yield (MEY) soluion perfec fishing seleciviy and uniform harvesing. H (in onne) and associaed biomass level mey mey B (in onne) and fish socks X1 and X (in 1,000 # of fish) mey mey mey mey f 1 f X 1 X mey B mey H Perfec seleciviy. Baseline parameer values Perfec seleciviy. Flee harvesing cos up, c 30 Perfec seleciviy. Discoun ren doubled, 0.10 Uniform harvesing. Baseline parameer values Uniform harvesing. Harvesing cos up, c 30 Uniform harvesing. Discoun ren doubled, 0.10 0 1.00 656 459,690 1,378 0.84 0 577 11 1,786 973 0 1.00 656 459 690 1378 0.48 0.48 633 368 369 115 0.40 0,40 660 478 755 110 0,50 0,50 63 336 53 117 5
Table A1. Biological and economic baseline parameer values Parameer Descripion Baseline value s Naural survival rae recruis 0.6 0 s 1 Naural survival rae young maure 0.7 s Naural survival rae of maure 0.7 a Scaling parameer recruimen funcion 1,500 (1,000 # of fish) b Shape recruimen funcion 500 (1,000 # of fish) Feriliy parameer 1.5 w Weigh young maure (kg/fish) 1 w Weigh old maure 3 (kg/fish) q 1 q Cachabiliy coefficien flee one Cachabiliy coefficien flee wo 0.01 (1/effor) 0.01 (1/effor) p Fish price young maure 1 (Euro/kg) 1 p Fish price old maure 1 (Euro/kg) c Effor cos flee one 10 (Euro/effor) 1 c Effor cos flee wo 10 (Euro/effor) Discoun ren 0.05 6
Table A: Uniform fishing. Susainable yield (in onne), biomass (in onne) and fish socks (1,000 # of fish). Baseline parameer values X B H f X 1 0 789 1840 7099 ( B K ) 0 0.1 759 193 5397 540 0, 78 96 436 847 0.3 695 668 3394 1018 0.4 660 478 555 110 0.5 63 336 53 ( B ) 117 ( H ) 0.6 584 7 1850 1110 0.7 54 144 1517 106 0.8 489 81 140 99 0.9 451 34 1003 90 1.0 400 0 800 ( B min ) 800.. 7