Sabotage in a Fishery

Transcription

1 Sabotage i a Fishery Ngo Va Log Departmet of Eoomis, MGill Uiversity, Motreal H3A 2T7, Caada ad Stephaie F. MWhiie Shool of Eoomis, Uiversity of Adelaide, SA 5005, Australia Prelimiary ad Iomplete - Commets Welome 1

2 SABOTAGE IN A FISHERY Abstrat: This paper presets a simple model of a ommo aess fishery where fisherme egage i sabotage ativities i additio to produtive fishig ativity. We model sabotage as reduig the athability oeffiiet of rival fisherme. We show that the steady-state fish stok is higher, the greater is the effetiveess of the sabotage tehology: it seems that sabotage a mitigate the over-exploitatio assoiated with the tragedy of the ommos. We also osider the effet of sabotage o steady-state profit ad fid some iterestig results: i the presee of sabotage, idustry profit at first ireases as the umber of fisherme ireases but later dereases, this o-mootoiity aot happe i the stadard model. Ireasig the effetiveess of sabotage redues sabotage effort but has a o-mootoi effet o levels of fishig effort. 2

3 1 Itrodutio Sabotage is a fat of life. Reports of sabotage have bee made i a variety of fisheries iludig gear tamperig, stealig ad itimidatio. A fisherma may wat to sabotage the fishig effort of his rivals for various reasos. Oe possible motive for sabotage i a oligopolisti market is to raise rivals osts, thus reduig their outputs. Alteratively, i a perfetly ompetitive market (where the fisherme are prie-takers), there is a ietive for a fisherma to sabotage if he shares the fishig groud with a fixed umber of rivals, beause his sabotage would redue their athes, thus ireasig ext period s stok (relative to what it would be without his sabotage) ad hee ireases his future ath at ay give effort level. A third motive for sabotage is evy: a idividual may are about his relative iome, i additio to the usual oer for absolute iome. I this paper we ivestigate whether sabotage may be helpful to a fishery: by reduig athes it may mitigate the tragedy of the ommos. We preset a simple model of a ommo aess fishery where, i additio to produtive fishig ativity, prie-takig fisherme also egage i sabotage. We ivestigate how the steady-state fish stok is depedet o the effetiveess of the sabotage tehology. I partiular, we ask if sabotage a result i a higher steady-state stok tha the soially optimal stok ad evaluate whether other model parameters suh as the prie/ost ratio lead to higher or lower steady-state levels of sabotage. I additio, we osider the effet of sabotage o steady-state profits relative to the o-sabotage profit level. I our model, agets at i their ow best iterests whe harvestig from a ommo-pool resoure. What differs from the stadard Clark-Muro model is that, istead of simply hoosig effort to maximise profits, they a also hoose to egage i sabotage ativities to redue the athability oeffiiet i the produtio futio of their rivals. We show that the greater the effetiveess of the sabotage tehology, the higher the steady-state fish stok: it seems that sabotage a mitigate the over-exploitatio assoiated with the tragedy of the ommos. We also show that lower sabotage effort does ot eessarily imply higher athability if the sabotage 3

4 tehology has improved ad that ay hage i the prie/ost ratio that results i a higher steady-state stok will result i a lower steady-state level of sabotage effort. Our umerial aalysis o the effet of sabotage o profits gives some iterestig results: the respose to a irease i sabotage effetivess is o-mootoe: idustry profit at first ireases as the umber of fisherme ireases but the it reahes a peak ad afterward further ireases i the umber of fisherme lead to lower idustry profit, this o-mootoiity aot happe i the stadard model without sabotage. There is a large literature o sabotage i the otexts of ret-seekig ad touramet, see, for example, Gozalez (2007), orad (2000, 2009), Amegashie ad Rukel (2007). I otrast, to our kowledge, there has bee o formal aalysis of sabotage i a fishery. By osiderig sabotage we aim to highlight the effet of o-profit-maximisig behaviour o fisheries ad, reewable resoures more geerally. I additio, by extedig the aalysis to allow for ooperative produtio ad the ability to egage i protetio (as is urretly i progress) we hope to shed light o the effet of ooperative behaviour amogst fisherme. 2 The Model The basis for our model is Clark ad Muro s (1975) dyami, sigle speies model. That is, we osider a fish stok exploited by ymmetri fisherme who live i the same ommuity. They do ot oordiate their harvestig deisio. We assume that their outputs are sold i a large market, so that aggregate quatity of fish they ath does ot ifluee the market prie, whih we assume to be a ostat p. Let x t deote the stok size ad L it the effort level of fisherma i at time t. Followig Shaefer (1957) we assume that the amout harvested by eah aget is h it = q it x t L it, where q it is the athability oeffiiet speifi to aget i, ad let the stok grow at a atural rate mius the total harvest. The ost per uit of effort is > 0. We ow diverge from the stadard model ad suppose that, i additio to produtive fishig ativity, fisherme may udertake sabotage ativities agaist their 4

5 rivals. We model sabotage by rivals as affetig the athability oeffiiet of eah aget. Let s i jt deote the sabotage effort of aget j direted agaist aget i with ost per uit of sabitage effort > 0 (the same as the ost of fishig effort). Let S i t be the sum of sabotage efforts (by all other fisherme) direted agaist i: S i t = j i The athability, q it, depeds o sabotage efforts by all rivals direted at i: q it = s i jt q 1 + (θ/σ)(s i t) σ q i(s i t) where 0 < σ 1 (1) where q is a positive ostat, ad θ > 0 is the effetiveess of the sabotage tehology. The restritio 0 < σ 1 esures that q i(s i ) < 0 ad q i (S i ) > 0. Iorporatig sabotage i this way meas that eah aget s harvest is diretly affeted by the sabotage efforts of his rivals, ad this affets both idividual profits ad the growth futio of the stok. I additio, eah uit of effort that i puts ito sabotage redues his ow profits. That is, aget i s profit, et of his fishig effort ost ad the ost of his ow sabotage effort direted agaist his rivals, is: π it = (pq it (S i t)x t )L it j i ad the et rate of growth of the stok is ( x t = rx t 1 x ) t q it (St)x i t L it i=1 s j it (2) where r > 0 is the itrisi growth rate ad > 0 is the arryig apaity. We assume that i the absee of sabotage, the idustry is potetially profitable. More preisely, we make the followig assumptio: Assumptio 1: There exists a stok level x t < suh that p qx t > for x t > x t. That is, p qx t > iff x t > x t, where x t <. We restrit the sum of aget i s fishig effort ad sabotage effort direted agaist other agets to ot exeed L it : L it + j i s j it L it (3) 5

6 I what follows, we will fous o the symmetri equilibrium where eah aget ats idetially ad sabotages all rivals equally. The we a defie: s it s j it for all j i ad aget i thus faes 1 idetial agets, eah alloatig L jt uits of effort to fishig, ad s jt uits of effort to sabotagig eah rival. The effort ostrait (3) therefore redues to: L it + ( 1)s it L it We assume that eah aget s goal is to maximize the itegral of disouted utility of et profit, where utility is: where α 0. 1 u it (π it ) = π1 α it 1 α Player i takes the atural growth rate of the fish stok (x t ) ad others fishig ad sabotage effort levels (L jt ad s jt, j i) as give. His problem is to hoose his ow fishig ad sabotage effort levels (L it ad s it ) to maximise the disouted stream of utility: ˆ 0 e δt 1 [ ] (pqit (S i 1 α 1 α t)x t )L it ( 1)s it dt Takig ito aout the o-egativity ad time ostraits o fishig ad sabotage effort ad deotig aget i s shadow prie of the fish stok at time t as ψ t we get the Hamiltoia for aget i as: H = e δt [ ] (pqit (S i 1 α 1 α t)x t )L it ( 1)s it [ ( +e δt ψ t rx t 1 x ) t ( 1)q jt (S jt )x t L jt q it (S it)x ] t L it +e δt η t L it + e δt µ t s it + e δt ν t ( L it L it ( 1)s it ) (4) 1 I the steady-state, the parameter α plays o role but we retai this formulatio to aid future osideratio of the motives for sabotage, that is, purely profit drive or behavioural motives suh as evy. 6

7 Assumig a iterior solutio, the first order oditios with respet to L it, s it, x t ad ψ t are: 2 π α it (pq it (S i t)x t ) ψ t q it (S i t)x t = 0 (5) π α q jt S j t it ( 1) ψ t ( 1)x t L jt S j = 0 (6) s t it ( π α it pq it (St)L i it + ψ t r 1 2x ) ( ) t ψ t ( 1)q jt (S j t )L jt ψ t q it (S i t)l it = ψ t δψ t (7) ( rx t 1 x ) t ( 1)q jt (S j t )x t L jt q it (S i t)x t L it = 0 (8) Coditio (5) says that at the aget s optimum, today s margial utility obtaied by exerisig fishig effort is equated to the imputed ost of havig a lower stok tomorrow. Provided that 1 0, oditio (6) says that today s margial ost of sabotage effort agaist 1 rivals is equated to the beefit of reduig their athes (i terms of iduig a higher stok tomorrow). If 1 = 0 the of ourse there will be o sabotage; i what follows, we assume 1 > 0. At the symmetri steady state, L it = L jt = L, s jt = s it = s, ad q it = q jt = q(s ) where S = ( 1)s. This meas (5) ad (6) a be rewritte as: ad ψ = π α (pq(s ) ) q(s ) (9) ψ = π α L q S S s 2 We have igored orer solutios for simpliity. To take ito aout orer solutios, we ote that (i) fishig effort would be zero wheever the stok level is so low that the value margial produt of fishig effort is lower tha the margial effort ost, ad that (ii) sabotage effort s i would be zero if the margial beefit of sabotage (evaluated at s = 0) is lower tha the margial effort ost,. Clearly, if 0 < σ < 1 the sabotage is always positive whe fishig effort is positive. If σ = 1 the for sabotage to be positive, θ must be suffiietly large. I what follows, we fous o the iterior solutio, ad assume that θ is suffiietly large. 7 (10)

8 Equatig (9) ad (10) ad fidig q S S s from (1) gives: pq(s ) q(s ) ad substitutig i for ˆq from (1) gives: = ( 1 + θ S σ) 2 σ L ˆqθS σ 1 pq(s ) = ( 1 + θ σ S σ) L θs σ 1 (11) whih is the arbitrage oditio that the margial value from a extra uit of effort used for fishig must equal the margial value of effort used for sabotage. Substitutig (9) ito (7) ad settig ψ = 0 gives the steady-state relatioship: δ = r ) [ ] (1 2x + pq(s )L q(s ) pq(s ) Further, (8) i steady-state a be rearraged to give: L = r q(s ) whih a be substituted ito (12) to give: δ = r ) (1 2x r ) ( ) ] (1 [ x pq(s ) pq(s ) q(s )L (12) (13) (14) Equatio (14) is the usual fisheries modified golde rule exept that the athability oeffiiet is a futio of sabotage. 3 However, as there are two edogeous variables i (14), the equilibrium requires a additioal equatio. The other steadystate relatioship we a use is the the itratemporal arbitrage oditio (11) with (13) substituted i, that is: pq(s ) = ( 1 + θ S σ) q(s ) σ θs σ 1 r ( ) (15) 3 The ituitio of (14) is as usual: at the margi, the retur from harvestig aother fish today must equal the retur from leavig it i the oea to grow for tomorrow, et of the hage that a rival will take it first. 8

9 This meas that the steady-state equilibrium of this model is defied by both the itertemporal arbitrage equatio (14) ad the itratemporal arbitrage equatio (15). I subsequet setios we use these oditios to evaluate the effet of sabotage o equilibrium stok levels ad the effet of sabotage o steady-state profits. Lemma 1: Equilibrium steady state profit per fisherma is: ( ) π 1 = + S θs σ 1 σ 1 where S is the Nash equilibrium level of sabotage. 4 Proof: I a symmetri equilibrium, we a see from (2) ad (11) that: (16) ad π = (pq(s ) )L S (17) (pq(s ) )L = ( 1 + θ S σ) σ (18) θs σ 1 Substitutig (18) ito (17) gives (16). 3 The ase where 0 < σ < 1 I the ase where 0 < σ < 1, we have a system of two equatios i two ukows, ad S, equatios (14) ad (15) whih a be rewritte as: ) δ = r (1 2x r ) [ ] ( ( 1) pq(s ) ) r ( θs σ 1 = q pq(s ) (19) (20) It turs out that these two equatios a be solved i two steps. I the first step, S a be expressed as a expliit futio of ad of other parameter values, S = φ( ; q, θ, σ, δ, ). I the seod step, this futio is substituted ito eq (19) to obtai a equatio i. We state these results as Lemma 2 ad Lemma 3. 4 I the speial ase where σ = 1, we get π = /θ, whih is rather surprisig: it depeds oly o ad θ. 9

10 Lemma 2: Give that 1 > 0, the steady-state stok uiquely determies the steady-state sabotage effort by the followig equatio: S = ( ) ( 1 θ 1 σ ˆq r ( ) 2 2 [ δ r + ( + 1) rx ] ) 1 1 σ (21) Proof: Equatio (19) a be re-writte as: δ r ( ) 1 2x ( ) + ( 1) = r 1 pq(s ) (22) whih gives the same right-had-side as (20). Therefore, equatig (22) ad (20) gives: ad rearragig: δ r ( ) 1 2x ( ) + ( 1) = r r 1 ) ( θs σ 1 q [ S σ 1 = ˆq ( ) ] δ r 1 2 ( θ r ( )) ( 1 ) r 1 [ ] ˆq = θ ( ( )) r 1 2 δ r + rx ( + 1) (23) From (23) we a see that S σ 1 will be positive if ad oly if > r δ ( + 1)r (24) We have already assumed that profits are positive whih tells us that the right-hadside of (22) must be positive. This implies the left-had-side must also be positive, that is: 1 + δ r ( ) 1 2x ( ) > 0 r 1 10

11 whih a be rearraged to show that > r δ ( + 1)r Therefore, as profits are positive, S σ 1 must be positive so we a deote S = φ( ) suh that: 5 S = ( ) ( 1 θ 1 σ ˆq r ( ) 2 2 [ δ r + ( + 1) rx ] ) 1 1 σ = φ( ; δ, r,,, θ, q) (25) It a be see from equatio (21) that the derivative of S with respet to is egative for all x, provided that oditio (24) holds. Thus we obtai the followig propositio: Propositio: Give that 0 < σ < 1, ay hage i a parameter (other tha ) that results i a higher steady-state stok will result i a lower steady-state sabotage effort, S. Note, however, that a lower steady-state sabotage effort does ot eessarily imply a higher steady-state athability oeffiiet. For example, ireasig the sabotage effetiveess parameter θ may lead to less sabotage effort beig eessary to ahieve the same or a lower athability oeffiiet for rivals ad hee higher stoks. Lemma 3: Assume that 1 > 0. The there exists a steady-state stok of fish <. It is a solutio of the followig equatio: 6 (θ/σ) ( ) [ σ/(1 σ) θ q r ( ) 2 2 ( δ r + ( + 1)r x (δ r + ( + 1)r( /)) (p/) q 1, where δ + r( /) )] σ/(1 σ) = (r δ) r( + 1) < < (26) 5 Note that (25) does ot otai the parameter p/, we will eed aother equatio to determie the effet of hagig this ratio later. 6 Equatio (26) determies the steady-state stok i the feasible iterval (0, ). It does ot seem possible to prove uiqueess aalytially; it is oeivable that there are several solutios, i.e. multiple steady states). 11

12 Proof of Lemma 3: Substitute ito the rearraged modified golde rule equatio (22) the equatio for q(s ) (1) ad subsequetly substitute S = φ( ;...) [ ( )] δ r 1 2 ( ) + ( 1) = r 1 p qx 1+ θ σ S σ whih a be rearraged to: p qx r 1 + θ σ [φ(x ;...)] σ 1 = δ r ( 1 2x = 1 p qx 1 1+ θ σ [φ(x ;...)] σ ( ) 1 ) + ( 1) r ( ) p qx 1 + θ σ [φ(x ;...)] σ = 1 + r 1 δ r + ( + 1) rx ( ) 1 > 1 (27) The right-had-side of (27) must be greater tha oe whe < ad profits are positive, that is, from oditio (24). Further rearragig gives: Thus p qx 1 + θ σ [φ(x ;...)] σ = δ + rx δ r + ( + 1) rx [ ] θ σ [φ(x ;...)] σ = p δ r + ( + 1) r qx 1 (28) δ + rx Substitutig (21) for φ( ;...) i (28) gives us a oditio for : θ σ ( ) [( σ θ 1 σ q r ( ) 2 2 [ δ r + ( + 1) rx ] )] σ 1 σ [ = p δ r + ( + 1) rx qx δ + rx ] 1 (29) where from (24) (r δ) < <. r(+1) The graph of the left-had side of eq (29) agaist x i the iterval (r δ) < r(+1) < is a otiuous urve, whih approahes ifiity whe x is arbitrarily lose 12

13 to (r δ), ad whih approah zero as x teds to. r(+1) The graph of the righthad side of eq (29) agaist x i the iterval (r δ) r(+1) < < is a otiuous urve, whih approahes 1 whe is arbitrarily lose to (r δ), ad approahes r(+1) (p/) q 1 > 0 as teds to. Therefore the two urves must iterset at least oe. This ompletes the proof of Lemma 3. 7 Remark: I the speial ase where σ = 1, we obtai from (29) the followig 2 ubi equatio: 2 θ2ˆq ( r2 ) [ (δ r + ( + 1)r x pˆqx Defiig z = x, this ubi equatio is: ) 2 ) (δ + r x = ) (δ r + ( + 1)r x )] (δ + r x (30) 2θ 2 r 2 q [ (1 z) 2 (δ+rz) (δ r + ( + 1)rz) (δ r + ( + 1)rz) pˆq ] z (δ + rz) = 0 (31) 4 Numerial aalysis for the ase of σ = 1 2. I this setio we explore the effet of hagig various parameters o the level of sabotage, the fish stok, ad idividual ad idustry profit uder the assumptio that σ = 0.5. For our base parameters we use: r = 1, δ = 0.05, = 10, σ = 0.5, q = 1, = 1, p = 222.2, = 1, θ = 100 Substitutig these iitial parameter values ito (31) gives a uique positive root: z = 0.5. Thus the steady-state stok is = 0.5, whih is at the maximum sustaiable yield level. We a ow use equatios (21), (1), (13) ad (16) to alulate the equilibrium sabotage (S = 0.25), athability oeffiiet (q(s ) = ), fishig effort (L = 5.05), ad idividual (π = 0.255) ad idustry profit (π = 2.55). 7 Equatio (29) determies the steady-state stok i the feasible iterval (0, ). It does ot seem possible to prove uiqueess aalytially. It is oeivable that there are multiple steady states. 13

14 4.1 Chages i theta 0.58 x * 0.28 S * 0.28 π * θ 2.6 π * θ 10 x 10 3 q * θ 5.1 L * θ θ θ If we raise the effetiveess of sabotage, we a see that the steady-state stok rises while effort devoted to sabotage falls. Both idividual ad idustry profits fall. Iterestigly, effort devoted to fishig has a o-mootoi respose. 14

15 4.2 Chages i the umber of fisherme 1 x * 0.8 S * 0.8 π * π * q * L * As we raise the umber of fisherme, we a see that, as usual, the steady-state stok ad idividual profits fall. The effort devoted to sabotage ad to fishig falls. The o-mootoi respose of idustry profits that ours here is ot possible i the stadard model without sabotage: usually idustry profits are stritly dereasig i the umber of fisherme. 15

16 4.3 Chages i the prie 0.52 x * 0.3 S * 0.35 π * p π * p q * p L * p p p As the prie rises (or equivaletly osts fall), the steady-state stok falls while fishig effort ad idividual ad idustry profits rise. Sabotage efforts rise. 16

17 5 Prelimiary Colusios This paper models the effet of sabotage o stok status ad profits i a dyami, reewable framework. I otrast to stadard fisheries models, we allow fisherme to udertake sabotage ativities agaist their rivals as well as odut produtive fishig. The effet of sabotage mitigates the usual tragedy of the ommos result of overexploitatio of stoks. We show that ireasig the sabotage effetiveess parameter has a o-mootoi effet o fishig effort. I additio, i the presee of sabotage, idustry profits are o-mootoi i the umber of fisherme, whih is i otrast to the model without sabotage. While sabotage protets agaist the tragedy of the ommos, we do ot propose that eouragig sabotage is a optimal poliy respose. Our ext step is to osider the effets o stoks ad profits of ooperative produtio with profit sharig, as i Se (1966), ad allowig fisherme to egage protetio. 17

18 APPENDIX Cosider the speial ase where σ = 1. Rewrite the modified golde rule (14) slightly: δ = r ) (1 2x r ) [ ( ( 1) ad also rewrite the (11) with (1) ad σ = 1: pq(s ) = θˆq r ) ( ] pq(s ) Substitutig (33) ito (32) we get a sigle equatio that determies the steady-state stok: δ = r = r ) (1 2x r ( + 1)r ( + θˆq r 2 2 ) ( r ( 1) + ( (32) (33) )) 2 ( θ q ) 2 (34) If this equatio has a uique solutio, it will be the uique adidate iterior steady-state stok. Notie that (34) does ot otai terms suh as p ad. We must also use eq (33), whih (with (1)) a be writte i the followig form : S qp = q θr(1 x ) ad verify that S is ideed positive at the value that satisfies (34). (35) Proof of equatio (35). Let us fid S as a futio of θ i the ase where σ = 1. From (33), with σ = 1, we get pq(s ) pq(s ) = = q θr ( q θr ( ) + 1 = q + θr ( ) θr ( ) 18 )

19 i.e. i.e. θs = qpx θr ( ) q + θr ( ( ) 1 q + θr 1 = 1 + θs qp θr ( ) (36) ) 1 = qpx θr ( ) ( ) q θr 1 ) q + θr ( S = r ( ) [ qp ] ( q/θ) ( q/θ) + r ( ) (37) (So for S to be positive, it is eessary that qp > 0.) Simplify S r ( ) qpx = 1 + r ( ) + ( q/θ) The, for S to be positive, we eed qp = qp > + q θr(1 x ) q θr ( This oditio is satisfied if θ is suffiietly great. We ow state the followig result: Lemma A1: Assume σ = 1. ) The there exists a uique positive adidate steady-state stok < if ad oly if the sabotage tehology is suffiietly effetive, so that the followig iequality is satisfied: θr 2 q 2 > δ r (38) This adidate steady-state stok is admissible iff S as give by (35) is also positive. Proof: Let us osider the graph of the left-had side of (34) where x is measured alog the horizotal axis. It is a straight lie with positive slope, uttig the vertial axis at δ (r/) ad uttig the vertial lie x = at the value δ + r. The graph 19

20 of the right-had side of (34) is dowward slopig urve, uttig the verial axis at θr 2 /( q 2 ) ad uttig the horizotal axis at x =. Therefore the two urves have a uique itersetio at some < if ad oly if the iequality (38) holds. (Note: if the iequality (38) is reversed, the there are two roots, both of whih are o-admissible: oe root is egative, ad the other is greater tha.) Remark: havig foud the steady-state stok, we a fid the steady-state sabotage S from ad verify that S > 0. pq(s ) = ( G( ) Propositio A1: A irease i the effetiveess of the sabotage tehology will irease the steady-state fish stok. Proof: A irease i θ shifts the graph of the right-had side of (34) upwards, ad does ot affet graph of the left-had side of (34). Hee we obtai a higher From (16), i the ase σ = 1, steady-state profit is π = (pq(s ) )L S = ) θ q + θ(s S ) σ 1 ( ) 1 σ 1 π = θ (39) It is rather surprisig that π is idepedet of all other parameter values. Let us alulate the effet of a irease i θ o the steady-state stok. Defie ad The (34) beomes where z x D q2 θr 2 z 2 + βz + γ = 0 (40) β rd 20

21 where γ 1 δd + r D The roots of the quadrati equatio are real, beause z 1,2 = β ± 2 ( ) β 2 4γ = r 2 D 2 + 4D(δ + r) > 0 Assumptio (38) implies that γ > 0, ie. both roots are positive, with the smaller oe i the iterval (0, 1). The smaller positive root is z = β 2 (It is ot lear that a irease i θ will lead to a irease or a derease i S : if is kept uhaged, the from (35) the diret effet of a irease i θ is a irease i S ; however, from Propositio 1, ireases whe θ ireases.) Steady-state fishig effort is L = r q(s ) Upo substitutio, usig (36), L = r q ( ) ( qp θr ( ) q + θr ( = r ) q (1 + θs ) ( ) ) ( ) We a ompare the steady-state profit uder sabotage, π = /θ, with the steady-state profit whe sabotage is ot preset beause of a low θ. I the formerase, profit is a ostat: π = /θ I the latter ase, the steady-stok is give by the usual equatio [ ] δ = G ( ) G(x ) ( 1) p q 21

22 ad the steady-state profit is π = (p q ) G(x ) q (p qx ) L 22

23 REFERENCES: Amegashie, A., Rukel, M., Sabotagig Potetial Rivals. Soial Choie ad Welfare 28, Clark, C.W., Muro, G.R., The eoomis of fishig ad moder apital theory. Joural of Evirometal Eoomis ad Maagemet 2, Gozalez, F.M., Effetive property rights, oflit ad growth. Joural of Eoomi Theory 137, orad,.a., Sabotage i Ret-seekig Cotests. Joural of Law, Eoomis, ad Orgaizatio 16, orad,.a., Strategy ad Dyamis i Cotests. Oxford Uiversity Press. Se, A., Labor Alloatio i a Cooperative Eterprise, Review of Eoomi Studies, 33, Shaefer, M.B., Some Cosideratios of Populatio Dyamis ad Eoomis i Relatio to the Maagemet of Marie Fisheries. The Fisheries Researh Board of Caada 14,