SEQUENTIAL MOVEMENT IN THE GREAT FISH WAR: A PRELIMINARY ANALYSIS

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1 Mjlh Bdn Pengkjin dn Penerpn Teknologi No. 79, Feruri 997 ISSN SEQUENTIAL MOVEMENT IN THE GREAT FISH WAR: A PRELIMINARY ANALYSIS Budy P. Resosudrmo Direktort Pengkjin Sistem Sosil, Ekonomi dn Pengemngn Wilyh Astrt This pper investigtes slightly different prolem of joint fishery eploittion s in the gret fish wr y Levhri Mirmn. The prolem in this pper rises from the ft tht some fish migrte etween two or more ountries. A non-oopertive gme model is pplied to nlyze this joint fishery eploittion prolem. Key Words: Environmentl Resoure Eonomis Gme Theory. INTRODUCTION Conflit interests mong different ountries in joint-eploittion of fishery hs long een prolem. In the ook, the Future of Interntionl Fishery Mngement (975), L. Anderson emined this prolem using stti fishery model. G.R. Munro in 979 set up the joint-eploittion of fishery s dynmi oopertive gme model eplined tht the prolem risen from differenes in pereptions of the soil rte of disount, fishing effort osts, onsumer preferenes. But it ws D. Levhri L.J. Mirmn in 980 who were properly le to pply dynmi non-oopertive gme model in the prolem of joint fishery eploittion. Using simple funtionl forms they derived dynmi Cournot-Nsh Stkelerg solutions for this prolem. More thn tht, their pper ttrted eonomists to roder lss of prolem, nmely the prolem of produtive ssets. J. Cve in 987 proposed thret strtegy to seure oopertive ehvior of the ountries involved. By this thret strtegy, Cve found trigger strtegy equiliri. In 99, J. Benhi R. Rodner ontinued nlyzing the trigger strtegy equiliri when the informtion out one ountry s defetion from the oopertive strtegy is not diretly reeived y the other ountry, i.e. there is ertin time period elpses efore one ountry s defetion from the oopertive strtegy is reveled. R. Sundrm in 989, working with generl form of Levhri Mirmn's model, ws le to define the eistene equilirium theorem for lss of symmetri gmes. P. K. Dutt R. Sundrm (989) emined the hrteristis of sttionry dynmi gmes for this prolem. The depth of the nlysis in this pper is muh less thn ny of the ppers ove, ut nevertheless this pper tries to rouse n importnt issue in the prolem of joint renewle resoure eploittion, nmely when the tions of ountries involved re sequentil. A good emple would e the eploittion of fish tht migrte etween ountries. Eh ountry hs its own time spot to eploit the fish, ut eh tion will ffet other ountry s strtegies. In this pper, the prolem is simplified to one type of fish tht migrte etween two ountries. The model in Levhri Mirmn's pper will e used to nlyze this prolem to mke it omprle to the solution in Levhri Mirmn's pper where the two ountries eploit the fishery t the sme time the sme spot. The etended prt of this pper is to put some restritions in this sequentil fishery gmes model to mke the model more interesting to nlyze. BASIC MODEL A stok of fish migrtes etween ountry. The fish spwn in ountry the juvenile will join the stok when the fish is k in ountry. Define one period s yle migrtion of the fish. Let it e the level of fish stok in ountry i t time t it e the fish onsumption of ountry i t time t. For simpliity, let's ssume tht there isn't nturl mortlity when the fish migrte from ountry to ountry, then: () t t t The fish popultion will grow ording to the iologil rule: 6

2 Mjlh Bdn Pengkjin dn Penerpn Teknologi No. 79, Feruri 997 ISSN t t t () where 0 < < (normliztion of fish popultion model), when the fish move from ountry to ountry. The utility of eh ountry is logrithmi funtion of onsumption, i.e.: u it log ; for i =, (3) it Hving disount rte s, the prolem of eh ountry is t V M log i t0 i t (4) sujet to equtions () (). Consider the prolem in the ontet of finite horizon. Suppose there is only one-period horizon, without loss of generlity, let ountry e the first ountry eploiting the fish stok. Clerly ountry will tke ll the fish leve nothing for ountry. In the two-period horizon, ountry will lso tke ll the fish in the first period. The eplntion is tht ountry knows tht if ountry hs the hnge to fish, then ountry will tke ll the fish for ountry knows tht ountry will tke ll the fish in the lst period. Bering this logi, then the solution of eqution (4) is tht ountry will tke ll the fish now, ountry is left with nothing. Suppose the two ountries pln to ooperte, where ountry will reeive prt of totl onsumption t eh yer (0 < < ). For two-period horizon, the mimiztion prolem of oth ountry is: M log log log log (5) where is the initil popultion is the totl onsumption in the first period. The first-order ondition for this prolem is: whih gives: (6) (7) Following Levhri Mirmn's tehnique, we define "vlution" funtion for the two-period horizon prolem s: or, V log log log log (8) V log A o (9) where: A o log (0) Net, onsider three-period horizon prolem when oth ountries ooperte. The ojetive funtion for the three-period horizon is: M log log log A o () From the first-order ondition of the mimiztion prolem ove, we n find oopertive strtegy to onsume only s muh s: ~ () in the first yer. The proess n e repeted for infinite time horizon yielding solution: * (3) s the totl onsumption eh yer. Under this poliy, the fish popultion will onverge to stedystte fish popultion level: * (4) By doing repetedly the proess of vlution suh s equtions (8) (9), we find tht the form of the vlue funtion of ertin stok of fish for infinite time horizon is V() = A + log. Using the Bellmn's eqution s V() = M [log()+log((- ))+ßV((-) )], sustituting the form of V() the optiml *, we find the vlue of ertin fish stok for oth ountries s: V log (5) 6

3 Mjlh Bdn Pengkjin dn Penerpn Teknologi No. 79, Feruri 997 ISSN log From this vlue, ountry will reeive: V log (6) log ountry : V log (7) log If V () is greter thn log V () is greter thn log(- ), oth ountries would proly wnt to ooperte. But, trigger strtegy n not e imposed in this model. No thret my work in this model. If one of the ountries defets from the oopertive strtegy y tking ll the fish, the other n not pply ny sntion. EXTENDED MODEL To mke the prolem ove more interesting, let the pility of eh ountry to th the fish depends on the fish popultion oming to eh ountry; i.e. 0 it i it 0 < i <, for i =,. For the two-period horizon the ojetive of ountry is: M log log while ountry is: (8) M log (9) log The first-order onditions of the mimiztion prolems ove re: (0) () The solution of eqution (0) () is the Cournot-Nsh solution for two-period horizon. From this solution we n find vlution funtion for twoperiod horizon. We use the vlution funtion of two-period horizon to define the prolem of threeperiod horizon. The first-order onditions of the three-period horizon re () (3) The proess n e repeted for infinite time horizon yielding equtions s (4) (5) The first se of this Etended model is where re "lrge enough", so tht the solutions of equtions (4) (5) re possile. Then, the Cournot-Nsh solutions re: (6) (7) This strtegy will eventully led the fish popultion to stedy-stte level s: (8) Using the sme proedure s in finding the vlue of fish popultion in ooperte strtegy with the Bsi model, we find tht the vlue reeived y ountry from ertin level fish stok with this Cournot-Nsh strtegy is: 63

4 Mjlh Bdn Pengkjin dn Penerpn Teknologi No. 79, Feruri 997 ISSN V V log (9) log The seond se pplies when one of the ountries or oth n not ehve s the solutions (6) (7) ove, then the Cournot-Nsh solutions will e: (30) (3) tht will ring the fish popultion to stedy-stte level: (3) With this strtegy, the vlues reeived y ountry re: V log (33) log V log log (34) Coopertive strtegy in this Etended model, if possile, will e the sme s in the Bsi model. In the first se of the Etended model, the totl onsumption of the first period oopertion is smller thn totl onsumption of the first se nonoopertion, i.e. < +. But, the stedy-stte fish popultion of ooperte strtegy will e higher thn tht in the first se non-ooperte strtegy. If we tke the seond se of the Etended model, oopertion is not lwys possile, i.e. there is possiility tht > + ( - ). Either in se one or two, we re interested to nlyze when oopertive strtegy is possile to impose, i.e. < + ( - ). Country will e willing to ooperte if the vlue of nonoopertion reeived y eh of them is less thn the vlue reeived when they ooperte. In this se, the trigger strtegy n e pplied, mening if one of them deeives from the oopertive strtegy hs een greed y oth ountries, the other will impose sntion y thing the fish s muh s possile. DISCUSSION In the Bsi model, where there is no restrition on the mount of fish tht n e onsumed eh period, we n see how different the strtegy of eh ountries in the fish wr when the tion of ountries involved is sequentil ompred to tht when the tion of ountries involved is simultneous s in the Levhri Mirmn's pper. If sequentil, the ountry tht moves first will onsume ll of the fish immeditely, leve nothing for the other. The oopertive greement n possily inrese the vlue of the fish for oth ountries, ut no thret strtegy n e imposed to gurntee tht oth ountries will oey the greement. The Bsi model n e improved y pplying restrition in the pility to th the fish y eh ountry in ertin period of time. The Etended model is set up sed on this restrition, tht eh ountry is only le to onsume the fish stok prtilly within its region. We see tht when the pility of oth ountries to th the fish is "lrge enough", then the Cournot-Nsh equilirium strtegy of oth ountries is the sme s if the tion is simultneous (s in Levhri Mirmn's model). In this Etended model, we lso see the possiility to pply the trigger strtegy to gurntee the oopertion of oth ountries. The enhnement of this pper might e in the pplition of more generl form of equtions in the Etended model, in the nlysis of the eistene equilirium solution of the hrteristi of this equilirium, suh s the works of Sundrm (989) of Dutt Sundrm (989). REFERENCES. Anderson, L. G. "Criteri for Mimizing the Eonomi Yield of n Interntionlly Eploited Fishery" In H.G. Knight e.d., The Future of Interntionl Fishery Mngement, St. Pul: West, 975. Benhi, J., Rdner, R. "The Joint Eploittion of Produtive Asset: Gme-theoreti Approh" Eonomi Theory, Vol. (99), pp

5 Mjlh Bdn Pengkjin dn Penerpn Teknologi No. 79, Feruri 997 ISSN Cve, J. "Long-term Competition in Dynmi Gme: the Cold Fish Wr" R Journl of Eonomis, Vol. 8 (987), pp Dutt, P. K., Sundrm, R. "Mrkovin Equilirium in Clss of Stohsti Gmes: Eistene Theorems for Disounted Undisounted Models" Eonomi Theory, Vol. (99), pp of Opertion Reserh. He ompleted his mster s degree in 99. He then enrolled in dotorl progrm t Cornell University in the field of Agriulturl Eonomis. He ompleted his dotorl s degree in 996. He is now t the BPPT gin. Dutt, P. K., Sundrm, R. "The Trgedy of the Commons? A Chrteriztion of Sttionry Perfet Equilirium in Dynmi Gmes" Deprtment of Eonomis-Rohester University Working Pper No. 7, 989. Fudenerg, D., Tirole, J. Gme Theory, MIT Press, Cmridge, MA, 99. Mjumdr, M., Sundrm, R. "Symmetri Stohsti Gmes of Resoure Etrtion: The Eistene of Non-romized Sttionry Equilirium" Deprtment of Eonomis-Rohester University Working Pper No. 58, 988. Munro, G. "The Optiml Mngement of Trnsoundry Renewle Resoures" Cndin Journl of Eonomis. Vol. (979), pp Reingnum, J. F., Stokey, N. L. "Oligopoly Etrtion of Common Property Nturl Resoure: The Importne of the Period of Commitment in Dynmi Gmes" Interntionl Eonomi Review, Vol. 0 (985), pp Rustihini, A. "Seond Best Equilirium for Gmes of Joint Eploittion of Produtive Asset" Eonomi Theory, Vol. (99), pp Sundrm, R. "Perfet Equilirium in Nonromized Strtegies in Clss of Symmetri Dynmi Gmes" Journl of Eonomi Theory, Vol. 47 (989), pp AUTHOR Budy P. Resosudrmo ws orn on My 6, 96 in Jkrt. In 986, he reeived Srjn degree in the field of Eletril Engineering from the Bung Institute of Tehnology. In tht sme yer, he joined the Indonesin Government Ageny for the Assessment Applition of Tehnology (BPPT). In 989, Resosudrmo pursued grdute progrm t the University of Delwre in the field 65