Suppose you have a bank account that earns interest at rate r, and you have made an initial deposit of X 0

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1 IOECONOMIC MODEL OF A FISHERY (ontinued) Dynami Maximum Eonomi Yield In ou deivation of maximum eonomi yield (MEY) we examined a system at equilibium and ou analysis made no distintion between pofits in the pesent vesus pofits in the futue. oth types of pofits wee teated as being equally valuable. This MEY is sometimes desibed as the stati MEY. In ontast, the dynami MEY onsides the fat that we plae geate value on goods and sevies that we eeive today than on those that we might eeive in the futue. Fo example, if I offe you the hoie of eeiving $ today o $ in a month, hanes ae you will hoose to take the money now. ut, you might be indiffeent between $ today vesus $ in a month. The exta dolla eflets the time-value of the money. Suppose you have a bank aount that eans inteest at ate, and you have made an initial deposit of X. You balane: afte time peiod would be X X ( ) and afte 2 peiods it would be X X ( ) X ( ) 2 and afte n peiods it would be X X ( ) n We an manipulate this last equation to deive the elationship between money in the pesent and money in the futue. X n X > ( ) n Pesent_Value Futue_Value Disount_Fato We an wite the disount fato in an equivalent exponential fom as n ( ) ( ) exp n whee ln( ) is the disount ate. Fo small values of, the disount ate and inteest ate ae almost equal,.. ( ) ln( ) ( ) ( ).4879 This is analogous to the elationship between the motality fation µ and. ( ).953 the instantaneous motality ate M. Rathe than limiting ou analysis of maximum eonomi yield to a fishey at equilibium, we will allow the fish stok and fishing fleet to hange ove time. The objetive of dynami MEY is to find the onditions that will maximize the disounted pesent value of the steam of pofits flowing fom the fishey as the fishey hanges ove time. In mathematial tems the poblem is to maximize e t ( py( t) f( t) ) dt with Y( t) qf( t) ( t) FW43/53 Copyight 28 by David. Sampson ioeon2 - Page 49

2 subjet to the onditions d dt G( ( t) ) Y( t) and ( t) and f( t) f max This type of poblem is sometimes desibed as an optimal ontol o dynami optimization poblem. Clak (985), on the Supplemental Reading list, shows that a neessay ondition fo obtaining the maximum net pesent value fom ou model fishey is that the fishing effot f should be ontolled aoding to the following poliy f max... if > opt f opt G( opt )/(q opt )... if opt... if < opt whee G() is the latent gowth funtion fo the fish stok and opt, whih is a taget biomass level towads whih we should foe the stok, satisfies the following equation. G' ( opt ) ( ) G( opt ) p ( ) C' opt C opt Hee C() is the aveage fishing ost pe unit of ath, and the patial deivatives G' and C' ae with espet to. A poof of this esult is given in Clak (985) on pages 34 and 35. The esult applies fo any of the standad suplus-podution models (whee thee ae no time lags o time-vaying paametes). If fishing effot is ontolled aoding to the ules pesented above, then the disounted steam of pofits fom the fishey will be maximized. This type of poliy, in whih we swith between no fishing (when the stok is less than the theshold opt ) and full fishing (when the stok is geate than the theshold), is sometimes desibed as a bang-bang havest poliy. Fo the Gaham-Shaefe suplus podution model the following funtions apply. G( ) G' ( ) 2 C( ) Cost / unit effot q Cath / unit effot C' ( ) q 2 The ondition fo ahieving the maximum net pesent value fom the fishey is 2 q 2 p q If we divide both sides by and multiply the ight-hand tem by (q )/(q ), we get FW43/53 Copyight 28 by David. Sampson ioeon2 - Page 5

3 2 pq Divide top and bottom by (p q). 2 pq pq oae / (p q) ( ) 2 oae oae ( ) oae 2 2 oae 2 oae oae oae oae 2 2 oae oae Multiply by oae oae opt 4 oae oae 2 8 oae... to get a quadati equation in. y the way, the negative oot to the quadati equation podues a negative value fo opt beause we have b b 2 a_positive_numbe Notie that opt, whih is the theshold stok biomass that govens whethe we fish o not, depends on the disount ate. In the exteme ases of an infinite disount ate o a disount ate of zeo, we get the following fo opt. lim opt ( ) oae opt ( ) ( 2 oae ) MEY In wods this means that a fishey opeating at the open aess equilibium is an optimal fishey only if the disount ate is infinite, in whih ase futue pofits have a pesent value of zeo. Pesent_Value Futue_Value Futue_Valueexp n ( ) ( ) lim exp( n) FW43/53 Copyight 28 by David. Sampson ioeon2 - Page 5

4 A fishey poduing the maximum eonomi yield is an optimal fishey only if the disount ate is zeo, in whih ase futue pofits have the same value as pesent ones. f MEY f oae Fo values of geate than zeo, the optimal biomass level will fall somewhee between MEY and oae. It might even fall at MSY. Revenues & Costs Effot MEY oae /q Optimal Havesting and Extintion Sometimes in the liteatue you will find statements to the effet that the optimal havest poliy (the poliy that geneates the geatest disounted pofits) may be to fish a stok to extintion (with the impliation that fishing fo pofits is a bad thing beause it will ause fishes will wipe out all the fish). Fishing a stok to extintion will esult in the lagest net pesent value fom a fishey only unde the following onditions:. 2. if the intinsi gowth ate G'() is less than the disount ate, and if the pie fo the last fish is geate than the ost fo havesting it. When both these onditions hold, you will make moe money taking the last fish and investing the pofits in the bank, than you will by tying to fish on a sustainable basis. oth onditions must be satisfied fo extintion to be optimal. It has been agued that beause whales ae slow gowing ( % to 5% pe yea) elative to inteest ates, it makes eonomi sense to ath all the whales and invest the evenues in othe podutive assets. This agument ignoes the fat that it is likely to be pohibitively expensive to find and ath the last whale. With the Shaefe model, it is neve optimal to fish a stok to extintion unless the fishing osts ae zeo beause osts ae geate than evenues wheneve biomass is less than oae. Reall that OAE oesponds to an infinite and theefoe the ondition that is less than is always satisfied at OAE. The akwad ending Supply Cuve The models that we've examined so fa all assume that the pie fo fish is onstant. We an exploe the influene of fish pie on the system by witing the equations fo evenue and ost in tems of the quantity of fish landed athe than as funtions of fishing effot. We will only onside the ase of an open aess fishey. We will stat by ewiting the equation fo yield, Y quantity of fish biomass landed pe unit time (usually pe yea). oae Y G( oae ) oae and oae pq FW43/53 Copyight 28 by David. Sampson ioeon2 - Page 52

5 Y pq pq q p q p This equation has the fom Y a b. p p This is a quadati equation in (/p). Notie that Y when pb and Y as p. The yield is zeo whee p (q ). The quantity (q ) is the CPUE that would be expeiened when havesting was initiated fom a peviously unexploited stok. Unless evenue pe unit effot fo the vigin fishey is geate than the osts pe unit effot, no fishing on this stok will eve ou. q MSY Yield Fish Pie If we swith the hoizontal (pie) and vetial (Yield) axes, we get the longun (equilibium) supply uve fo this fishey, whih shows how the landings will be vay as fish pies hange. MSY This uve is sometimes desibed as a bakwad-bending supply uve. It has a vey diffeent appeaane than typial supply uves. Eonomists usually assume that podues inease thei podution in esponse to highe pies; supply uves, theefoe, ae typially shown as a simple ineasing uve o line. With a fishey the podution of fish yield an only inease if the stok biomass is less than MSY. Fish Pie q Yield FW43/53 Copyight 28 by David. Sampson ioeon2 - Page 53

6 The bakwad-bending supply uve also epesents the longun aveage ost uve fo the fishey. Total_Cost( Yield) Aveage_Cost The aveage ost hee is with espet to Yield the ath athe than to the fishing effot. Peviously we defined total ost as TC f. Hee we need to expess TC as a funtion of Y. One solution is to wite f as a funtion of Y and then multiply by. q Y qf f q 2 > f 2 qf Y A quadati equation in f. f( Y) q ( q) 2 4 q2 2 q2 Y 2 q 2 4 Y AC( Y) Y 2 q 2 4 Y Anothe fom of the bakwadbending supply uve. Notie that the aveage ost uve will take on eal values only if the quantity unde the squae oot sign is positive. Y 2 4 > > Y < MSY 4 Fo any yield between zeo and MSY we an podue that level of yield at two diffeent ost levels, depending on whethe the stok biomass is above o below the MSY level. Suppose that the pie fo fish vaies, but is independent of the quantity landed. Pie p is just the aveage evenue (aveaged with espet to the quantity landed). An eonomi equilibium ous wheneve AR AC. The intesetion of the pie line (AR) with the supply uve (AC) epesents suh an equilibium point, whih in this ase oesponds to the open aess equilibium point. Fish Pie p 2 p Quantity Landed FW43/53 Copyight 28 by David. Sampson ioeon2 - Page 54

7 The equilibium points in the diagam above ae stable. Fo example, when landings ae less than Y(p ) the evenues ae geate than the fishing osts so moe boats ente the fishey and inease the landings. When landings ae geate than Y(p ) the evenues ae less than the fishing osts so some boats lose money, leave the fishey, and the landings deease. Suppose that fish pie vaies with the quantity landed, and that lage landings tend to depess the pie. If the slope of the demand uve is geat enough, then we an get a system with thee bioeonomi equilibia, two that ae stable and one that is unstable. Stable In the diagam to the ight the middle equilibium point is unstable. If yield is slightly lage than this equilibium level (lose to MSY), osts ae lage than evenues so boats leave the fishey, whih auses landings to inease futhe. Fish Pie Unstable Stable Quantity Landed FW43/53 Copyight 28 by David. Sampson ioeon2 - Page 55