RENT-SEEKING AND QUOTA REGULATION OF A RENEWABLE RESOURCE. Harald Bergland*, Derek J. Clark# and Pål Andreas Pedersen$ March 2001

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1 ISSN: RENT-SEEKING AND QUOTA REGULATION OF A RENEWABLE RESOURCE Harald Bergland*, Derek J. Clark# and Pål Andreas Pedersen$ March 200 Abstract The paper deals wth rent-seekng behavour among agents competng for future shares of a common renewable natural resource. Rent-seekng mght become proftable when the agents expect that the dstrbuton of the natural resource n future perods wll be dependent on the agents extracton of the resource n the past, even though hgh explotaton mght reduce the stock that future ndvdual quotas wll be based upon. Whether aggressve rentseekng behavour by one agent wll encourage other agents to rent-seek more, however, s generally found to be ambguous. JEL Classfcaton: Q2, D7, C7. Keywords: Rent-seekng, quota regulaton, renewable resources. Address for Correspondence: Pål Andreas Pedersen, Department of Economcs, Keynes College, Unversty of Kent at Canterbury, Canterbury, Kent, CT2 7NP, UK. Tel: (+44) ; Fax: (+44) ; Emal: P.A.Pedersen@ukc.ac.uk. * Harstad College, Norway # Department of Economcs and Management, Unversty of Tromsø, Norway and Norwegan School of Management, Tromsø, Norway. $ Bodø Graduate School of Busness, Bodø, Norway and Department of Economcs, Unversty of Kent at Canterbury, UK

2 RENT-SEEKING AND QUOTA REGULATION OF A RENEWABLE RESOURCE. Introducton It s well known that agents who share a common-pool resource wll not take account of the full socal cost of ther actons. Ths means that the non-cooperatve soluton leads to a suboptmal management of the resource (Gordon, 954). A usual response to ths problem s to attempt to regulate extracton of the resource n some way n order to enforce optmal management. Indrect regulaton can be mplemented through a system of taxes, whlst drect regulaton nvolves a quantty constrant on producton (see for nstance the overvews wrtten by Bohm and Russell, 985 and Munro and Scott, 985). In ths paper we examne how ratonal actors expectatons of a drect regulatory regme can nfluence ts effcency. Ths type of regulaton mples a reducton n the explotaton of the resource for some or all of the actors, accordng to some pre-specfed crteron. Equal rate of reducton s an often-used prncple; for nstance, a fshery may be regulated by the use of quotas whch specfy a total allowable catch for each naton. The szes of the natons quotas are very often based upon observable and verfable varables such as hstorc catch or the number of vessels partcpatng n the dfferent fsheres. These magntudes can be freely chosen by the natons n an unregulated (free) fshery. When the partcpants know (or suspect) that resource extracton wll be drectly regulated, they may have an ncentve to adjust ther behavour n antcpaton of ths, even before regulaton s mplemented. Imagne that quotas wll be awarded on the bass of hstorc catch, defned over a certan perod. The naton can then attempt to secure a larger quota ex post by ncreasng the sze of ther catches n the perods leadng up to the mplementaton of regulaton. The ncentve to over-fsh s an example of rent-seekng behavour whch can counteract the effcency of the regulatory regme. It s precsely ths mechansm whch s the focus of ths paper. We concentrate our analyss on two

3 2 aspects: frst we dscuss under whch condtons the regulaton wll lead to over-fshng compared to a welfare optmal allocaton and dfferent non-cooperatve solutons (stuatons wthout any explct regulaton); secondly, we examne the strategc nteracton between actors whch may arse wthn the regulatory regme. For nstance, one may ask whether one naton s over-fshng may stmulate or moderate other natons ncentves to adopt such a strategy. Rent-seekng has been dscussed systematcally n appled economc theory snce the semnal work of Tullock (967). The term rent-seekng appears to have been coned by Krueger (974), but the same type of actvty was dscussed and analysed formally as early as n 954; see Haavelmo (954) who called the actvty grabbng and ntroduced an allotment functon where the value obtaned by an actor from unproductve actvtes s a functon of hs capacty for such actvty relatve to others. The quota allocaton mechansm that we wll focus on n our model s smlar to the allotment functon ntroduced by Haavelmo. Game-theoretc resource modellng s prmarly concerned wth strategc aspects of externaltes arsng from the common explotaton of natural resources. Mesterton-Gbbons (993) surveys resource games n fsheres and other renewable resources. Most of the lterature s concerned wth comparng dfferent Nash equlbra wth welfare solutons (or global optma). Many papers deal wth the case of two agents (see for nstance Levhar and Mrman, 980 and Fsher and Mrman, 996). However, the mportance of the number of agents s consdered by e.g. Clark (980) and Hannesson (997). Wthn the context of an nfntely repeated game, Hannesson consders how the number of agents who share a fsh Professor Tore Thonstad, Unversty of Oslo, made us aware of ths contrbuton n an open lecture at Unversty of Tromsø, 4 February 2000.

4 3 stock affects the possblty of fndng a self-enforcng soluton that yelds the welfaremaxmzng allocaton. In our paper we wll use game-theoretc resource modellng to focus on a rent-seekng mechansm whch can result from regulaton. The analyss s smplfed by consderng a model wth only two agents and two tme perods, n whch there s no regulaton n the frst perod, and where a quota regme s mplemented n the second perod. Two-perod models have also been used to fsheres n a game-theoretc approach by e.g. Nato and Polasky (997). The model whch we use to dscuss the phenomenon of rent-seekng n antcpaton of regulaton may have other applcatons than fsheres, e.g. the extracton of other renewable natural resources or the emsson of polluton. However, n order to motvate the type of regulaton consdered n our model, we wll now present some evdence from nternatonal fsheres. For resources that are exploted by more than one naton, t s common that a total allowable catch (TAC) s negotated by the nterested partes; the sze of the TAC s often determned under advsement from marne researchers. The actual allocaton of ths TAC between the partcpatng natons often depends upon past harvests. An early example of such a quota allocaton between natons s the whale quota agreement from 962. Accordng to ths agreement, the fve nvolved natons (Norway, Japan, Unted Kngdom, Netherlands and Sovet Unon) shared the total quota accordng to the relatve catch dstrbuton before the agreement (Tønnesen 970) 2. A thorough study of quota allocaton between natons s found n Underdal (980), who examnes the quota regulaton of North East Atlantc fsheres wthn the North East Atlantc Fshery Commsson. The study ncludes dfferent fsheres 2 One excepton was that the Sovet Unon receved a consderably hgher share than ther hstorc share of catches. Ths was explaned by the fact that the Sovet Unon, at the tme t entered nto ths agreement, was buldng up ts whalng ndustry.

5 4 nvolvng dfferent numbers of natons. Underdal examnes to what extent the actual quota allocaton devates from the pre-regulaton catch dstrbuton. He also studes the arguments and the prncples used n the negotaton process. One mportant concluson s that the redstrbuton mpled by the ntroducton of quota regulatons s n most cases very small 3. A fnal example of an nternatonal quota allocaton s the agreement between the members of the European Unon establshed n 983. Accordng to ths agreement, the TAC - manly based on bologcal crtera - s shared by the members accordng to a rule whch s fxed from one year to the next. One of the determnants of the shares s the hstorc catch dstrbuton between the EU natons, especally for the perod (Holm 990). Our examples suggest that, n the regulaton of nternatonal fsheres, the dstrbuton of hstorc catches play an mportant role n the dvson of the TAC among the nvolved natons. Ths leads us to the followng conjecture: gven that past catch performance wll play a promnent role n the allocaton from any future regulatory regme, there s an ncentve to ncrease current catches n order to secure a large share of the TAC n the future,.e. there s an ncentve to rent-seek. Possble rent-seekng mechansms n natural resource quota allocaton have earler been mentoned by Hannesson (99) and formally studed by Boyce (998). When searchng for optmal allocaton prncples and evaluatng dfferent possble allocaton rules whch mght be used to dstrbute the common resource to several partcpants n a competton, Boyce fnds that the allocaton where the wnner takes all, combned wth full compensaton from the wnner to the losers, gves a frst-best soluton. Unlke Boyce, we 3 When comparng the last year wth free fshng and the frst year of quota regulaton, he fnds that n 3/5 of the cases the changes are less than +/-2% of the total. In only approxmately 5% of the cases are the changes +/-5% or more - see table 4.3 n Underdal (980).

6 5 do not go through any normatve analyss searchng for optmal allocaton mechansms. As commented on above we restrct ourselves to dscussng allocatons where all partcpants obtan a level of the resource after regulaton has been mplemented whch s a functon of the actors hstorcal explotaton of the resource. The model on whch the analyss s based s ntroduced n Secton 2. There we present wdely known solutons to the resource allocaton problem, whch we compare wth our rent-seekng soluton n Secton 3. Conclusons are offered n Secton The Model We consder a model wth two perods and two actors (countres). In order to smplfy our analyss, the actors explot a common renewable natural resource. At the begnnng of perod we have a gven hstorc stock volume. The natural growth of the stock n each perod s dependng on the stock volume at the begnnng of the perod. Furthermore, because we consder only two perods, t s reasonable to ntroduce a restrcton on the stock volume at the end of perod 2. Ths means that the stock cannot be lower than a certan level at the end of perod 2. In ths secton we wll consder the resource constrant and establsh the welfare optmum soluton and two non-cooperatve solutons. Let S t be the stock volume at the end of perod t and h t be the total harvest n perod t. Then we can defne the resource dynamcs n the model by: S = S0 + G( S0) h, S2 = S+ G( S) h2, S2 = S, ()

7 6 where S 0 s a gven stock sze at the start of perod, and S s a gven stock volume whch s requred at the end of perod 2. 4 Furthermore, GS ( t ) s the natural growth of the stock n perod t; the functon has frst dervatve denoted by G ( S t ). Normalzng the product prce n both perods to unty, we can wrte actor s rent n perod t as: r = h C ( S, h ), =, 2, t =, 2 (2) t t t t C C C C C > 0, > 0, < 0, 0, 0, S S 2 2 t t t St t t where h s actor s harvest n perod t andc s the cost functon for actor, assumed to be t dentcal for both perods. The cost s supposed to be strctly ncreasng and convex n the harvest volume and decreasng n the stock volume. Furthermore, we assume that the cost s convex n stock volume and that the margnal cost n harvest s non-ncreasng n the stock volume. We can now defne the present value of actor s two-perod proft as: V = h C ( S0, h) +δ h2 C ( S, h2), =,2, (3) where δ 0 s a common dscount factor. Usng the three equatons n (), we may wrte the resource dynamc constrant as: S = S + G( S ) h + G( S + G( S ) h) h (4) where h = h + h 2, t =,2. t t t 4 We may thnk of two types of hstory for the resource stock we consder: (a) Hgh hstorc explotaton and therefore a requrement of buldng up the stock,.e. S 0 < S. (b) A resource whch s dscovered n perod and therefore hgh hstorc stock volume compared to the requred future stock sze,.e. S > S 0.

8 7 2. Three Allocatons In the followng we wll defne three standard soluton concepts for the allocaton problem. As they are standard, we gve a bref defnton of each and then collect the frst-order condtons whch govern ther solutons n a table to facltate comparson wth the rentseekng approach adopted n Secton 3. Welfare Soluton (WS) We defne the welfare soluton (WS) as the catch allocaton whch maxmses the sum of the actors present value 2 W = V + V subject to the resource constrant n (4). The soluton to ths problem s gven by equatons (5) and (6) n Table, where λ s the ncrease n the welfare (total present value) from a small reducton n the requred stock volume at the end of perod 2 ( S ). Non-cooperatve Solutons Wthn the standard neo-classcal framework, each actor wll explot the resource n order to maxmze hs own resource rent. The soluton concept s of the Nash-Cournot type meanng that each actor takes nto account the effect that hs own actons have on the common resource stock but dsregards the effects emanatng from others. We wll consder two cases of ths non-cooperatve soluton dependng upon the actors atttude towards the resource constrant. NS(I) The non-cooperatve soluton wth a bndng common resource constrant The non-cooperatve soluton wth a common resource constrant, denoted by NS(I), s the catch allocaton whch maxmses V subject to the resource constrant n (4). In ths case t s assumed that the actors are ndvdually ratonal, and that they consder the common future stock requrement gven by (4) to be a real constrant on ther behavour. The soluton n ths

9 regme s gven by equatons (7) and (8), where 8 µ s the ncrease n agent s present value from a small reducton n the requred stock volume at the end of perod 2 ( S ). NS(II) The non-cooperatve soluton wthout a bndng resource constrant Here the actors are assumed to dsregard the resource constrant ( S ), so that (4) does not constran ther behavour (.e. µ = 0, =, 2). Ths s equvalent to free maxmzaton of equatons (9) and (0) dctate the soluton to ths problem. V ; In Table below we have collected the frst order condtons for these three standard regmes. 5 One of our man concerns wll be how regulaton affects the allocaton of the harvest between tme perods. 6 Table 2 presents the condtons whch govern ths allocaton n the three regmes presented so far. 5 We have dropped the presentaton of the second order condtons whch here, and for all other regmes analysed throughout the paper, are supposed to be satsfed. 6 It should be notced that n the NS(I) regme the actual catch dstrbuton between the actors n perod 2 s not determned n the model. One could possbly thnk that the catch volumes n perod 2 are decded upon n a negotaton between the agents, or by publc authortes. For our purpose here, the man thng s that the agents under ths regme want to meet the resource constrant n (4) and that the decsons taken n perod do not nfluence the actual catch dstrbuton n perod 2.

10 Allocaton WS NS(I) NS(II) 9 Table Frst-Order Condtons for the Standard Solutons Frst-order condtons j C C C +δ ( G( S)) 0,, 2, j, 2, j + λ + = = = S S C δ 0,,2 λ = = 2 C C +δ µ ( + G ( S)) = 0, =,2 S C δ 0,,2 µ = = 2 C C +δ = 0, =,2 S C δ 0,,2 = = 2 (5) (6) (7) (8) (9) (0) Allocaton Table 2 Allocaton of Harvest between Perods 7 Frst-order condtons j C C C C WS +δ ( G( S)),, j,2, j + =δ + = S S 2 () C C C NS(I) +δ =δ ( G( S)),,2 + = S 2 (2) NS(II) C C C 0= +δ =δ,,2 = S 2 (3) 7 Equaton () s derved from (5) and (6); (2) comes from (7) and (8), whlst (3) follows from (9) and (0).

11 0 2.2 Comments on the Allocatons Welfare Soluton (WS) From (5) and (6) t follows that: 2 C C =, t =,2, 2 t t so that the margnal costs for each of the perods should be the same for both actors. Equaton () corresponds to the standard golden rule equaton for the optmal equlbrum bomass. 8 However, ths condton for optmalty n the harvest allocaton between the perods can also be drectly nterpreted from (). The two frst terms of the left hand sde express the margnal rent n perod. The thrd term s the dscounted margnal cost effect followng from reduced stock volume n perod 2. Ths margnal ncrease n the present value for an extra unt catch n perod, measured by the left hand sde of (), should be equal to the rght hand sde. We see that the rght hand sde s a product of three factors. The two frst factors are the dscounted margnal rent for perod 2, whle the thrd factor s the harvest volume effect n perod 2 caused by one unt harvest n perod, where the resource constrant s taken nto account 9. 8 In order to deduce the standard golden rule equaton (see Clark and Munro, 975) from equaton (), one has to formulate bologcal equlbrum harvest, ntroduce the connecton between the dscount factor and the rate of nterest and specfy the cost functon as C = c ( S) h. 9 It s reasonable to assume that + G ( S ) > 0 for an exploted resource (.e. n the case where S s not too hgh), and ths condton s assumed to hold n the rest of ths paper.

12 Non-cooperatve Soluton NS(I) In ths case, µ s the ncrease n agent s present value from a small reducton n the requred stock volume at the end of perod 2 ( S ). If the two agents have dfferent costs, these resource shadow prces wll also be dfferent. If the stock requrement represents a true constrant on actor, t follows that µ >0. Furthermore, by comparng the NS(I) and WS solutons n Table, t s seen that n the second perod, the NS(I) soluton gves a sub-optmal allocaton of harvest between the agents when ther resource shadow prces dffer. It also follows that the allocaton between actors n the frst perod s not optmal, both because of possble dfference n resource shadow prces and because the cost effects of the stock volume mght be dfferent 2 for the two actors,.e. C / S C / S. Equaton (2) shows that the NS(I) soluton wll not gve an optmal harvest allocaton between perods. The rght hand sde of (2) s the same qualtatve expresson as the rght hand sde of (). However, by comparng the left hand sde of these two equatons, we see that the non-cooperatve actors, although beng aware of the stock requrement n the future, wll not take account of the full socal cost of ther actons. Each actor takes account of the cost ncrease only for hmself n perod 2 caused by lower resource stock n perod 2, but dsregards the same effect for the other actor. Because of the assumpton that the cost s strctly ncreasng and convex n the harvest volume, we fnd that the harvest volume n perod s too hgh n the NS(I) case compared to the welfare soluton (WS). Non-cooperatve Soluton NS(II) Here t follows drectly from (0) that the margnal rents for both actors n the second perod wll be dentcal, meanng an optmal allocaton of the total harvest n perod 2. However, the

13 2 total harvest n perod 2 wll be hgher than for NS(I) and WS. Furthermore, t follows from (9) that each agent only takes account of the prvate cost effect on reducng stock n perod 2, meanng that the harvest volume for both actors also wll be too hgh n perod. From equaton (3), t s seen that the relatve catch dstrbuton between the perods wll be dfferent n ths case compared to both NS(I) and WS. Ignorng the resource constrant means a relatvely more ntensve fshery n perod compared to the harvest volume n perod 2 than n NS(I) or WS. 3. Quota Regulaton and Rent-seekng So far our model fndngs correspond to the standard results. Agents who share a commonpool resource wll not take account of the full socal cost of ther actons, whch means that the non-cooperatve solutons lead to a sub-optmal management of the resource. Ths s well known n the lterature back to Warmng (9) and Gordon (954). A natural response to ths problem s to attempt to regulate extracton of the resource n some way n order to enforce optmal management. For fshery resources where more than one naton are nvolved n the explotaton, t s common that a total allowable catch (TAC) are determned and dstrbuted among the nvolved natons. Past harvest s often one mportant crteron when t comes to the allocaton between natons as mentoned n Secton. When partcpants know (or suspect) that the resource extracton wll be drectly regulated n ths way, they may have an ncentve to adjust ther behavour n antcpaton of ths. Imagne that quotas wll be awarded on the bass of hstorc catch, defned over a certan perod. The country can then attempt to secure a larger quota ex post by ncreasng the sze of ther catch n the perods leadng up to the mplementaton of regulaton. In ths secton we wll focus on ths possble rent-seekng mechansm and compare ths to the allocatons n Secton 2.

14 3 In our analyss of a quota regulaton regme based on hstorc catch, we wll consder two dfferent stuatons: n the frst case, both actors choose ther harvest n perod smultaneously, denoted by RS(A); the second soluton s charactersed by sequental choces where one of the natons observe the other natons harvest n perod before t chooses ts own harvest n perod. Ths soluton s denoted by RS(B). In both cases, the actors are supposed to be unregulated n perod, whlst drect regulaton s ntroduced n the second perod. Ths means that the countres are free to choose the level of economc actvty (harvest) n the frst perod, but face a quantty constrant n the second va the ntroducton of a quota scheme whch lmts the total amount of resource extracton. Countres ndvdual shares of the total quota n perod 2 are decded by the sze of ther harvest n perod one. The ntroducton of ths quota regme s known to both actors n the frst perod. In order to smplfy the analyss, we assume that these allocatons n perod 2 are non-tradeable. We begn our analyss of ths regulated case by notng that the resource constrant n (4) defnes the total quota for perod 2,.e. h = S + G( S ) h + G( S + G( S ) h) S = h ( h), (4) where h = h + h and h 2 s the ndvdual quota for actor n perod 2. Furthermore, we assume that the actors catch volumes n perod determne the dstrbuton of the total quota n perod 2. Ths can be wrtten as: h = g ( h, h ) h ( h), =,2, (5) where the ndvdual share functons sum up to one, g ( h, h ) + g ( h, h ) =. Furthermore, t s supposed that actor s share s ncreasng and strctly concave n own harvest n perod and decreasng n the other actor s harvest n perod,.e.

15 4 g ( h, h ) > 0, =,2 2 and g ( h, h ) < 0,, j =,2, j. 2 j As mentoned above, a specal case of ths allocaton rule wll be that the relatve amounts of the total harvest n perod 2 n the regulaton case are the same as the catch dstrbuton before regulaton s mplemented,.e. 0 j h 2 h + h g ( h, h ) =, =, 2, j =, 2, j. (6) The problem facng actor n perod 2 s to maxmse ts proft n perod 2, subject to the gven quota. Ths means that actor maxmses (2), nterpreted for perod 2, subject to: h,,2. 2 h2 = (7) The frst-order condton for maxmum s then gven by: C =β, =,2, (8) 2 where β s the ncrease n actor s rent from a small ncrease n ts ndvdual quota. We wll assume that the drect regulaton represents a true constrant on actor, whch means that β >0 and (7) holds as an equalty. When the quota allocaton n perod 2 s non-tradeable, as we have assumed, these quota shadow prces mght be dfferent because the actors may have dfferent costs and because the quota obtaned by the actors n perod 2 may vary. 0 Accordng to Underdal (980) ths has been a common prncple n the quota allocaton between natons n the North East Atlantc Fshery Commsson (NEAFC). In ths case we have that the welfare optmum condton from (6) s not fulflled. However, f the ndvdual quotas n perod 2 are tradeable n an atomstc quota market, the margnal rent n perod 2 would be equal for all actors (dentcal β s), and the condton n (6) s fulflled. However, the model and ts results concernng the allocaton between actors n perod and the allocaton between perod and perod 2 would not be affected f the quotas were supposed to be tradeable.

16 5 Let us now consder the actors choces of harvest n the frst perod. The present value of the actors rents can now be wrtten as: V = h C ( S0, h) +δ h2 C ( S, h2), =,2. (9) (a) Smultaneous Moves The actors who are unregulated n the frst perod can freely choose the harvests whch maxmse (9). Ths means that the frst-order condtons n the case of smultaneous moves, are gven by: C C C 2 +δ = δ,,2. = S 2 (20) Furthermore, t follows from (4) and (5) that: g = h + g, =,2, (2) where = = ( + G ( S)) < 0,, j =, 2. j Let us take a closer look at the expressons n (2). In general, when an actor ncreases hs harvest n perod, he may experence hgher or lower ndvdual quota n perod 2. The frst term on the rght hand sde s clearly postve because the actor s share of the total quota s ncreasng n hs harvest volume n perod, see (5). However, the second term measures the decrease n the total quota n perod as a consequence of ncreasng harvest n perod. Rearrangng (2) gves us that: 2 > 0 f g h g h = El g > El h =, =,2. (22) 2 h 2 h h2 Equaton (22) tells us that actor s quota n perod 2 wll be ncreasng n hs harvest volume n perod f the relatve ncrease n actor s share of the quota n perod 2, caused by a

17 6 margnal ncrease n harvest n perod, s hgher than the relatve decrease n the total quota n perod 2, caused by the same margnal ncrease harvest n perod. Usng the allocaton j rule n (6), the condton n (22) becomes gh2 > h( + G ( S)), j, =, 2, j, where the left hand sde s the quota for actor j n perod 2, and the rght hand sde s the reducton n total quota n perod 2 caused by a margnal ncrease n the catch of actor n perod. In the case of the allocaton rule n (6), ths means that (22) holds for both actors when the ndvdual catch quotas for perod 2 are hgher than the reducton on the stock n perod 2 whch one unt catch n perod causes. By nsertng (20) n (2), we obtan the frst order condtons n ths RS(A) case: C C C g +δ =δ g ( G( S)) h2,,2. + = S 2 (23) Let us now compare the condton n (23) wth the condton n (2). Generally, we know that g g ( + G ( S )) h < + G ( S ) 2 because g < and g > 0. Ths means that the rght hand sde of (23) s always less than the rght hand sde of (2). Because the cost functon s strctly ncreasng and convex n the harvest volume, we fnd that the harvest volume n perod wll always be hgher for both actors n ths regulated soluton, (RS(A)), than n the non-cooperatve soluton wth a bndng resource constrant (NS(I)). However, f we compare ths regulated soluton to the case where the actors do not feel any responsblty to meet the common future stock constrant (NS(II)) t follows that harvest for the two actors n perod under RS(A) wll be hgher only f (22) holds. Ths means that more ntensve harvestng s seen n the RS(A) case than n the NS(II) case f the ndvdual quotas n perod 2 are ncreasng n the actors harvest volume n perod. The conclusons are summed up n result below.

18 7 Result : (I) The ndvdual harvests n perod under the regulated soluton wth smultaneous moves, RS(A), are always hgher than the harvests n perod under the noncooperatve soluton when actors feel responsble for meetng the common future stock requrements, NS(I). (II) Moreover, the actors harvests n perod under the quota regme wth smultaneous moves, RS(A), wll also be hgher than the actors harvests n perod under the noncooperatve soluton wthout a bndng resource constrant, NS(II), f the ndvdual 2 quotas are ncreasng n the harvest volume n perod,.e. > 0, mplyng that condton (22) holds. To gve a better understandng of Result, we may thnk of what s gong on n our model both as a game aganst the other actor, and a stock game from the resource constrant. To see ths, we dvde the total game nto a pure strategc game and a pure stock game. Isolatng the strategc game by excludng the stock game means that we gnore the common future stock constrant n the non-cooperatve soluton,.e. we have the NS(II) case where µ = 0, =, 2. The analogous soluton n the RS(A), where the stock game s excluded, wll be found by dsregardng that the ndvdual harvest volume n perod has any quota effect 2 n perod 2,.e. assumng = 0, =,2. In ths pure strategc game we have that the left hand sde of (2) s equal to zero, whle the rght hand sde of (23) s negatve. Ths means that n the pure strategc game caused by the quota allocaton mechansm n (5) leads to over-fshng compared to the non-cooperatve soluton,.e. the allocaton rule results n a race for quotas between the two actors.

19 8 The pure stock game s found by excludng the strategc game. The relevant non-cooperatve soluton s now NS(I). In order to obtan the solated effects caused by the stock game n g RS(A), we look on the case where the quota shares are gven n advance,.e. = 0. The C rght hand sde of (23) now becomes δ g ( + G ( S)) h, whch s less than the rght 2 hand sde of (2) because g <. Hence, the regulaton quota regme leads to over-fshng n perod compared to the relevant non-cooperatve soluton. The reason s qute smple: the ntroducton of an ndvdual quota means that the actors wll feel less responsble for the future of the resource stock n the regulaton case than n the non-cooperatve case wth a bndng common resource constrant. Ths s because n the regulaton case, the quota effect n perod 2 of ncreased harvest n perod s only multpled by the prvate share whch the actor obtans (equal to g < ), whle t s multpled wth the whole effect on the future stock n the NS(I) case (equal to). (b) Non-smultaneous Moves The followng regulaton soluton s symbolsed by RS(B). We now assume that actor s takng hs acton after actor j s harvest volume n perod s known,.e. that actor j s a leader and actor s a follower n a non-smultaneous game gong on n perod. Ths means that the optmal behavour of actor s stll defned by equaton (23). However, when actor j chooses hs catch volume n perod, he wll take nto account the reacton whch he causes on actor s behavour n the same perod. Generally, ths reacton wll be mplctly defned by (23) and mght be wrtten as: h = h ( h ) (24) R j

20 9 where dfferentaton of (23) wth regard to the leader s harvest n perod, n general terms, gves: dh dh 2 V R j = j 2 V 2 ( h ) The denomnator s negatve when the second order condtons are fulflled, so the sgn of the reacton s determned by the sgn of the numerator. By usng the specfcatons n our model, the numerator can be wrtten as: 2 2 C g g C 2 2 ( ( )) + j h g + G S 2 S2 ( S) I II 2 2 V C g g 2 g g 2 2 =δ ( h 2 2) h2g ( G( S)) ( g ) ( G ( S)) j + j j + (25) ( h2) III 2 C g g g h2 ( + G ( S)) + + g G ( S) j j IV When takng a closer look at (25), t s seen that our general assumptons regardng the cost functon and the dstrbuton functon gve ambguty regardng the sgn of the sum of the four terms of the rght hand sde. Only the sgn of the second term II s clearly non-postve. Furthermore, t s seen that the frst, the thrd and the fourth term consst of combnatons of what we above have called strategc effects, where the actor s share changes, and stock effects whch nfluence on actor s margnal present value. One way to examne the sgn of the expresson n (25) further, wll be to separate the total effect nto a pure stock game (where g g 2 = = 0 ) and a pure strategc game (where j = 0 ), see the Appendx. However, as seen n the appendx, the sgn of (25), and thereby the sgn of the slope of the reacton functon n (24), s generally ambguous n both the pure strategc and pure stock

21 20 game. Ths s a consequence of many dfferent knds of mechansms workng smultaneously n our model. As an llustraton let us consder two effects whch work n opposte drectons. For nstance, a postve effect follows from term I as a consequence of the stock game. The margnal cost for actor n perod 2 s lowered because actor s quota s reduced as a consequence of reduced stock when the leader ncreases hs harvest n perod. An example of a negatve effect that reduces the margnal present value for actor s seen n term II. Ths effect measures the ncrease n margnal cost when the stock level s reduced as a consequence of more ntensve harvestng n perod. Let us now turn to the leader s optmal choce of harvest n perod. Gven the reacton j j j functon n (24), actor j, as the frst mover, maxmses the present value V ( h, h ( h )) wth regard to hs harvest volume n perod. The frst order condton for maxmum s then gven by: R V V dh + = 0 h dh j j R j j (26) j j V 2 C C where =δ ( G ( S)) 0 j + + < 2 S (27) In (27) t s seen that the present value for actor j s a decreasng functon of the harvest volume for actor n perod. Increased harvest n perod from actor causes reduced present value for actor j n two dfferent ways: The frst term n (27) s the effect of reduced quota for actor j, both as a consequence of a lower share of the total quota and lower total quota n perod 2. The second term measures the ncreased costs because of reduced stock n the regulated perod.

22 2 When we now use (27) n (26), t s seen that the sgn of the second term on the left hand sde wll be condtonal on the sgn of the slope of the reacton functon. If the harvest volumes n perod are strategc substtutes (complements),.e. that the slope s negatve (postve), the sgn of ths second term n (26) wll be postve (negatve); see Bulow et al (985). Rememberng that the present value for actor j s assumed to be concave n actor j s harvest volume n perod, ths leads to a hgher (lower) catch volume for frst mover actor j n perod n ths non-smultaneous case compared to the stuaton where the actors move smultaneously f the harvests are strategc substtutes (complements). Furthermore, t follows that second mover actor chooses lower harvest n case RS(B) than n RS(A), no matter what the slope of the reacton functon. Ths means than n the case where the harvest volumes n perod are strategc complements, we know for sure that the total harvest n perod n the stuaton of non-smultaneous moves s less than the total harvest n the stuaton of smultaneous moves. However, when the fsh levels n perod are strategc substtutes, the total harvest n perod n the RS(B) case can be both hgher, the same and lower than n the RS(A) case, condtonal on the szes of ncrease and decrease n actor j s and s harvest respectvely, when gong from the smultaneous to the non-smultaneous case. These fndngs are summed up n Result 2 below. Result 2: In a non-smultaneous game (denoted RS(B)), the frst mover s harvest n perod wll be hgher (lower) and the second mover s harvest n perod wll be lower than n a smultaneous game (RS(A)) when the these harvests are strategc substtutes (complements). The total harvest n perod wll be lower n case RS(B) than n RS(A) when the harvest volumes n the unregulated perod are strategc complements, whle t

23 22 s not clear whether RS(B) gves lower or hgher total harvest n perod than RS(A) when the harvest levels n perod are strategc substtutes. Hence, whether the non-smultaneous game gves less or more rent-seekng than the smultaneous game s an open queston. Based on our analyses above, t s seen that n stuatons where the harvest levels n perod are strategc complements, and n stuatons where the harvests are strategc substtutes, but the ncrease n actor j s harvest s less than the reducton of actor s harvest when movng from the smultaneous to the non-smultaneous case, we would see a less ntensve rent-seekng behavour n the RS(B) compared to the RS(A) case. 4. Conclusons Fsheres have often been regulated by agreements whch determne total allowable catches for dfferent fsh stocks. Furthermore, the total quota s often dstrbuted to agents based on actual choces of ndvdual catches n the past. Based on a smple model n whch the agents know ths allocaton mechansm and the dynamc growth n the renewable resource, we have seen that t mght be ndvdually advantageous for agents to attempt to nfluence the allocaton through rent-seekng behavour. Ths ndvdual quota regme mght be worse than a stuaton wthout any regulaton for two reasons: frstly, f the actors feel responsble for meetng a future resource constrant n a non-regulated stuaton, the frst perod harvests wll be lower than when the fshery s regulated by gvng ndvdual quotas n perod 2 f the actors move smultaneously. Ths s because the ndvdual quota regulaton weakens the actor s ndvdual responsblty for meetng the future constrant on the level of the resource. Secondly, ndvdual quota regulaton where the actors shares are based on hstorcal catches,

24 23 unlke the non-regulated stuaton, opens the possblty for the actors to nfluence the level of the ndvdual shares of the total quota whch nduces a race for quotas (Result ). Furthermore, we have focused on whether the order of moves taken by the actors nfluences the rent-seekng actvty wthn a quota regulaton wthout fndng unambguous conclusons. Among other thngs, the results are condtonal on whether the frst mover, by choosng an aggressve strategy nduces the second mover to choose an aggressve reply,.e. whether the harvests n perod are strategc complements or substtutes (Result 2). Our conclusons are based on a model where we have made several smplfyng assumptons. Fnally, let us comment on some of the most crtcal assumptons. Frstly, n practcal polcy the total quota may be dstrbuted to agents based on actual choces of other ndvdual varables than only catch volume n the past (e.g. nputs). Ths aspect, together wth the case of an endogenous number of agents, s dscussed n Bergland et al (2000). Furthermore, we have assumed that the agents have perfect nformaton about the future total quota (.e. the resource constrant), the actual quota allocaton rule chosen by the regulaton authorty, and the exact pont of tme when regulaton s mplemented. However, n practce agents wll often have only mperfect nformaton concernng possble future regulaton, whch means that there wll be uncertanty as to the possble future gans from rent-seekng. Ths would very possbly lead to less rent-seekng compared to a stuaton where the relevant nformaton concernng the future publc regulaton was known for sure. For nstance, we could thnk of a stuaton where the authortes shared the total allowable quota n the future on ndvdual average producton and ndvdual average levels of nputs, estmated on the bass of statstcs for some unknown years n the past. Then the agent s current ablty to nfluence future

25 24 ndvdual quotas s weakened, and the possble advantage of adoptng rent-seekng behavour for a sngle year s reduced. Even though our analyss s based on several smplfyng assumptons, also others than those explctly commented on here, we beleve that ths paper ponts to an mportant ssue whch occurs n the practcal regulaton of common natural resources. However, further theoretcal research concernng the type and sze of possble rent-seekng mechansms s necessary, as are emprcal studes from dfferent common natural resources. For nstance, t s stll an open queston whether the over-fshng seen n many nternatonal fsheres today can be attrbuted, at least n part, to the type of rent-seekng mechansm that we have examned. As we have demonstrated, the regulaton of a renewable resource s consderably complcated by the exstence of rent-seekng behavour. Furthermore, there are many relatonshps that need to be determned and quantfed n order before one can be sure that quota regulaton wll have the desred effect.

26 25 APPENDIX Here we analyse the sgn of the components of (25) by lookng at a stock game and a pure strategc game (as defned n the text). Frstly, by gnorng the strategc game n order to solate the nfluence from the stock game, (25) becomes: V C C C 2 2 C =δ 2 ( ( )) ( ) ( ( 2 2 )) ( ) j g + G S g + G S + g G S S h 2 ( S) ( h2) h I II III IV (A) On the rght hand sde of (A), I s supposed to be non-negatve, whle the other terms (II, III and IV) are supposed to be ether negatve or non-postve when G ( S) 0 2. Ths means n the pure stock game, we know that (A) s negatve f the margnal cost s ndependent of the stock volume, 2 C S 2 = 0. Ths means that n ths case the margnal present value for actor w.r.t. actor s harvest n perod wll be decreasng n actor j s harvest n perod. Hence, t follows from (24) that n ths specal case actor would choose to reduce hs harvest volume as the harvest volume of actor j ncreases, mplyng a negatvely sloped reacton functon so that the ndvdual harvest levels are strategc substtutes n the model. Moreover, when 2 C S 2 < 0, we cannot exclude the possblty that the reacton functon s postvely sloped n ths pure stock game,.e. that the ndvdual harvest levels n perod 2 are strategc complements. Ths case mght occur f the absolute value of 2 C S 2 s suffcently hgh such that the frst term domnates the three followng terms n (A). 2 Ths means that the growth n the stock ncreases less the hgher the stock becomes, whch s a usual assumpton n bo-economc models descrbng the dynamcs of a fsh stock.

27 26 When we solate the effects followng from a pure strategc game,.e. when gnorng the stock game, (25) becomes: 2 2 C g g C + j h2 2 S2 ( S) I II V C g g g g =δ 2 ( h2) hg ( G ( S) j + j ) + j + ( h2) III 2 C g g g j h ( + G ( S) ) + j IV (A2) As n the case of the pure stock game, the slope of the response curve s generally ambguous n ths pure strategc game because the sgn of (A2) can be both postve, zero and negatve. In order to obtan some more nformaton about the sgn of the rght hand sde of (A2), we may take a closer look at the case where the dstrbuton functon s gven by (6). 3 As n the general case, where both the stock and the strategc game are gong on, we have that the second term n (A2) s non-postve. Furthermore, the frst and the fourth term are nonpostve (negatve) when h j j ( < ) h, whle the thrd term s clearly postve when h h. Ths means that we are unable to draw any unambguous conclusons regardng the slope of the reacton functon n (24), both n the pure stock and pure strategc game, and of course n the case where both games are workng smultaneously. 3 In ths case we have g h g h g h h h h + h h h + h h h h + h j 2 j =, =, = j 2 j j 2 j j 3 ( ) ( ) ( ).

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