Should we internalize intertemporal production externalities in the case of pest resistance?
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- Alexandrina Fitzgerald
- 6 years ago
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1 Should we nernalze neremporal producon exernales n he case of pes ressance? Elsa MARI AgroSup Djon, CESAER (UMR 04 of IRA), elsa.marn@djon.nra.fr Paper prepared for presenaon a he EAAE 04 Congress Agr-Food and Rural Innovaons for Healher Socees Augus 6 o 9, 04 Ljubljana, Slovena Copyrgh 04 by E. Marn. All rghs reserved. Readers may make verbam copes of hs documen for non-commercal purposes by any means, provded ha hs copyrgh noce appears on all such copes.
2 Should we nernalze neremporal producon exernales n he case of pes ressance? Elsa MARI CESAER, AgroSup Djon Summary: Pescdes effcency decreases wh her global applcaon by farmers. Whn a sraegc dynamc framework, hs resuls n a classc neremporal producon exernaly. We analyze he fscal scheme ha can be mplemened n order o nernalze hs exernaly. We show ha s able o resore socally opmal pahs bu ha fnal me of pescde use dffers. Wh hs scheme, farmers have a endency o swch o alernave pes-conrol echnology, as negraed pes managemen, earler han s opmal. Furhermore, he socally opmal swchng me can be laer han he one obaned under a suaon whou conrol. Keywords : sock exernaly, pes ressance, echnology change. J.E.L. classfcaon: Q0, Q3, H3, C73. Elsa Marn, AGROSUP Djon, CESAER (UMR 04), 6 Boulevard Dr Pejean, BP 87999, 079 Djon, France. el. +33(0) , Fax. +33(0) E mal: elsa.marn@djon.nra.fr I would lke o hank he French envronmenal mnsry for fnancal suppor whn he framework of he research program GESSOL. Cenre d Econome e Socologe applquées à l Agrculure e aux Espaces Ruraux, UMR 04 of IRA and AgroSup Djon
3 Inroducon Pess are well-known for beng dffcul o manage because hey ofen develop ressance o pescdes. Ressance has been documened for a long me n several ways. For nsance, Georghou (986) enumeraed pes speces found o be ressan a one or more locaons for a leas one crop season for varous pescdes. Carlson (977) aggregaed ressan pess for major crop areas a varous pons over me and measured farmer pescde use choces over me for varous pescde classes. he desgn of ressance managemen programs has consequenly been a queson of huge neres. For nsance, Roush (989) crcally revewed a ls of ressance managemen accs o brng o he fore he mos promsng accs for general use n ressance managemen. In a dynamc framework, effecveness of a pescde can be consdered as a sock ha s declnng over me because of pes ressance (lke a non-renewable resource). Regev e al. (983), n an exenson of Regev e al. (976), developed a heorecal model n order o deermne opmal pescde use n hs framework. hey compared he opmal soluon wh he compeve soluon n order o brng o he fore neremporal producon exernales a work and o derve some polcy recommendaons. hese recommendaons were que basc snce hey conssed n ax per un of pescde use or n resrcons on pescde use. he major purpose of our work s o refne he polcy recommendaons ha can be formulaed wh respec o he nernalzaon of such neremporal producon exernales, especally by explcly akng no accoun he possbly of farmers o swch o oher echnologes han pescde use. he que recen developmen of ransgenc agrculure has nduced a wde leraure ha focus on refuge polces. Whn he framework of refuge polces, farmers can culvae rangenc crops ressan o pes upon he condon ha some refuge areas are culvaed wh convenonal crops. he man dea s o avod he developmen of pess ha are ressan o ransgenc crops and o bologcal conrol (specally for B crops). Some auhors followed a que smlar dynamcal heorecal framework as Regev e al. (983) n order o sudy he opmal managemen of pes ressance wh refuge polces. For nsance, Laxmnarayan and Smspon (00) derved he opmaly condons for he refuge sze n a sylzed dynamc model adaped from epdemology; Ambec and Desqulbe (0) added a spaal seng no he basc dynamc seng and compared he performance of refuge areas and axes on pescde varees. Oher auhors proposed o assess economcally he effes of refuge polces. For nsance, Hurley e al. (00) assessed her mpac on agrculural producvy, on convenonal pescde use, and on pes ressance; Frsvold and Reeves (008) focused on her welfare mpacs by akng no accoun producer surplus, consumer surplus, seed suppler profs, and commody program coss; Qao e al. (009) proposed an assessmen appled o a developng counry framework. All hese papers manly focus on a sngle echnology ha allows farmers o sruggle agans pess: ransgenc crops. he purpose of our work s o consder he possbly of farmers o swch o echnologes ha consue an alernave o a sngle pescde use, whou specfyng hs echnology. Anoher par of he recen leraure concerned wh pes ressance focus on R&D argeed owards echnologes ha can sruggle agans pess. Some auhors focused on nnovaon process of he boechnology secor. For nsance, O Shea and Ulph (008) suded he role of pes ressance n boechnology R&D nvesmen sraegy; Yerokhn and Moschn (008) suded he
4 mpac of nellecual propery rghs on he nvesmens n bologcal nnovaons whch value can be reduced o zero because of pes ressance. Here he new echnology s manly undersood as an alernave molecule of pescde or as ransgenc crops. Oher auhors focused on new echnologes ha rely on farmers sraeges. For nsance, Mullen e al. (005) descrbed he role of publc nvesmens n pes-managemen R&D and emphaszed he developmen of negraed pes managemen (IPM) sraeges. he IPM concep was frs developped by Sern e al. (959). I s a muldscplnary concep whch man prncple s o combne chemcal, mecancal and bologcal conrol sraeges agans pess. Mullen e al. (005) summed up ha he anhess of IPM s applyng broad-specrum pescdes on a fxed schedule relaed o he physologcal developmen of he crop, rrespecve of pes populaons. Kngh and oron (989) explaned ha one of he movaons for he developmen of IPM sraeges had been he problem of pes ressance. IPM s hus an alernave echnology ha s a farmers dsposal n order o sruggle agans pess. For nsance, Llewellyn e al. (007) suded he case of negraed weed managemen sraeges. hey nvesgaed he deermnans of he adopon of hese sraeges by Wesern Ausralan gran growers. We seek o desgn a framework ha s able o consder alernave pes-conrol echnologes as IPM n order o sudy no more deals he polcy ha can be mplemened n order o nernalze neremporal producon exernales lnked wh pes ressance. We propose o develop a sngle pes and sngle crop managemen model. We assume, lke n he non-renewable resources leraure 3, ha here s a backsop echnology wh respec o he pescde,.e. a echnology ha does no ncrease he pescde ressance of he pes populaon. hs echnology can be new molecule of pescde, ransgenc crop ressan o pess or IPM sraeges. In order o do so, we manly rely on Regev e al. (983) work. her compeve soluon reled on he assumpon ha farmers were non-sraegc: hey were sac mzers ha dd no ake no accoun effecs of her acons on he dynamcs of he sysem. hs assumpon was a plausble one n he 980s. owadays, progress of knowledge allowed farmers o be perfecly aware of he phenomena of pes ressance (see for nsance Benley and hele, 999, for a leraure revew appled o developng counres). Furhermore, pes ressance s a spaal phenomena (see for nsance Georghou and Mellon, 983 or Peck e al., 999 or Peck and Ellner, 997) ha can be a work a a local level. As a consequence, he number of farmers concerned wh he phenomena of pes ressance can be small. Whn such a framework, he farmers perfecly know he effecs of her acons on he dynamc of he sysem. One queson of neres s hus o hnk of farmers ha consder he mpac of her choces on he sock of he pescde effecveness. In hs work, we propose o exend he queson of Regev e al. (983) o sraegc farmers. For hs purpose, we have o resor o dfferenal games heory. Whn he framework of dfferenal games, s necessary o make src assumpons on he specfcaons of he funcons n order o oban closed form soluons boh for he seady saes and for he pahs. Closed form soluons are essenal o brng o he fore neremporal producon exernales n a que general case. Indeed, o do so, soluons obaned n he MPE case have o be compared o he soluons obaned n he socally opmal case. In mos works concerned wh dfferenal games, funcons are assumed lnear quadrac. Here, we explore he resuls of Long and Shmomura (998) by choosng homogeneous funcons ha are more general han lnear quadrac funcons. In a companon paper of Long and Shmomura (998), Cornes e al. (00) developed non-cooperave dfferenal games based on conceps of 3 See for nsance Am (986). 3
5 open-loop ash equlbrum and Markov perfec ash equlbrum n order o nvesgae he queson of pes ressance n a dfferenal game seng. hese conceps of equlbrum dffer by he fac ha players comm hemselves a he begnnng of he game for he enre horzon of me n he open-loop case. In he Markov perfec case, hey revse her decson a each perod of me accordng o he sae of he sock. Appled o he pescde case, an open-loop srucure means ha farmers are able o comm hemselves wh respec o her pescde applcaon for an enre me horzon. Snce farmers can observe he sae of he pescde effecveness a each perod of me, we focus on he Markov perfec equlbrum concep n whch farmers are able o revse her decson a each perod of me accordng o he sae of he pescde effecveness. Cornes e al. (00) developed a dscree me model wh hree perods and a connuous me one wh an nfne number of perods. For clary of presenaon of resuls, we concenrae on a connuous me model. Cornes e al. (00) assumed ha he ermnal me was fxed (o hree n he dscree me model and o nfne n he connuous me one). As Example of Long and Shmomura (998), we assume a free fne ermnal me ha s endogenous: he planners (farmers and cenral planner) choose he fnal me of he pescde use. Furhermore and conrary o Cornes e al. (00) and Example of Long and Shmomura (998), we assume ha here s a backsop echnology wh respec o he pescde,.e. a echnology ha does no ncrease pescde ressance of he pes populaon. In order o consder hs possbly of an alernave pes-conrol echnology whou a pror knowledge on he exac form of hs echnology, we nroduce a scrap value funcon ha reflecs he value aached by he decson-maker o he pescde effecveness ha s lef over a he ermnal me. hs assumpon s lnked wh echnologes evocaed prevously: new molecule of pescde, ransgenc crops and IPM. A new molecule of pescde s generally used n combnaon wh old one ha mus sll be effecve. In order o avod ha pess develop ressance o ransgenc crops, s generally mposed by governmen o keep radonal crops on whch pescde could sll be effecve. IPM sraeges conss n combnaon of mechancal, chemcal and bologcal conrol ha can be effecve only f chemcal conrol works. In secon, we wll presen he model and assumpons. Secon 3 wll be devoed o he dervaon of he Markovan perfec ash equlbrum. hs wll help us o brng o he fore n secon 4 he neffcences ha can occur n such a framework. Secon 5 wll be devoed o he schemes ha can be used n order o oban a socally opmal swch o an alernave pes-conrol echnology. Fnally, we wll conclude. All echncal proof wll be relegaed o an appendx. he model We consder n symmercal farmers ha use pescdes n order o produce a homogenous good. Each farmer apples a quany a, of pescde a me. he effecveness of hs applcaon s gven by a,. e where e denoes he effecveness of he pescde under consderaon. We know ha pess develop ressance o pescde over me. We assume ha ressance s a negave lnear funcon of he pescde applcaon. herefore, he moon of he pescde effecveness over me s gven by: n e ba, () where b measures he sensvy of he pescde effcency o he amoun of pescde appled; 4
6 s a consan ha s exogenously gven. We consder ha he pescde effecveness s lke a nonrenewable sock ha declnes over me. he pescde effecveness before pess develop ressance s exogenously gven: e0 e. We dsregard oher npus han pescde for smplcy and assume ha each farmer s prof ncreases wh he effecve amoun of he pescde ha he apples, a a decreasng rae: /, a, e, 0< < () where / s he elascy of he prof o he effecve amoun of he pescde appled. hs assumpon does no seem excessve n vew of he hgh use of pescde by farmers despe he effecs on her healh for nsance (see Mullen e al., 005 for nsance). o sum up, he curren pes-conrol echnology s used by farmer unl some fuure me when he wll use an alernave pes-conrol echnology. A me, snce we do no make any assumpon on he form of he alernave pes-conrol echnology, we assume ha he alernave pes-conrol echnology brngs n a ne reurn of: e exp( ( )) E e d where he effecveness of he pescde a me, e, s a proxy of he ne benef from he sock of he pescde effecveness lef over a he ermnal me and s a fxed dscoun rae. s also named a scrap curren value funcon. I represens he mum curren value of an negral of fuure uly flow sarng from me wh an nal sock of he pescde effecveness e. E he economc problem s defned as he choce of he amoun of pescde o apply, a,, and he swchng me o he new echnology, whch mze:, de 0 exp( ) exp( ) (3) subjec o () and e 0 e and a, 0. For he sake of clary he las consran wll be omed n he connuaon. Fnally, our model s manly a developmen of Example of Long and Shmomura (998) n whch we add a scrap value funcon n he same way as Regev e al. (983). 3 he Markovan Perfec ash Equlbrum (MPE) We look frs a he suaon n whch he n farmers play a dfferenal game. Farmer looks for a sraegy of pescde applcaon gnorng he sraegy chosen by he oher farmers. Each farmer sraegy s a funcon of he pescde effecveness, e, ha s easly observable: ( e ). herefore, e sums up he nfluence of he pas a me. Each farmer s perfecly aware of he fac ha he ohers selec her sraegy on he bass of he curren pescde effecveness. hs s why we assume ha he farmers sraegy are Markovan. 4 4 See Chaper 4 of Dockner e al. (00) for more deals on Markovan equlbrums. 5
7 For he purpose of he resoluon, s convenen o ransform varables. We defne a new / conrol varable c, a, e ha nduces modfcaon of equaons () and () as, respecvely, n c, e b and, c, e. More precsely, he MPE of he game modfed s he soluon of problem: 0 c, de c, j( e) j exp( ) exp( ) s..: e b e e e0 e We use he Ponryagn s mum prncple n order o solve hs problem. We focus on a symmercal soluon. We defne he Hamlonan of hs problem n curren value: c, j( e ) Hc (,, e, ) c,, b e j e If ( e) s a soluon o hs problem and he opmal swchng me (from farmer s pon of vew), hen here exss a connuous funcon, ha sasfes he condons:, b ( e) ( e) (4) e, H, e ( ( e ), e,, ) (5) bn ( e) e (6) e, bn ( e) ( e) e e 0 (7) e Based on Long and Shmomura s (998) resul, we assume ha farmer s sraegy s of he form ( e ). e f all oher farmers use sraeges ha are homogeneous of degree one: j ( e) j. e, j. Equaon (4) s he usual margnal condon of long-run prof mzaon. I saes ha, for each perod of me, he margnal prof of pescde applcaon s equal o s margnal cos. he cos s a measuremen of fuure losses mpled by ncreasng ressance. I s gven by farmer s prvae shadow prce,. Equaon (5) defnes he dynamcs of he co-sae varable,. When equaon (6) s also vald, we can conclude, as n Example of Long and Shmomura (998), ha: Remark If n >0 and farmers sraeges are of he form ( e ). e, hen here 6
8 exss a unque MPE for whch he equlbrum sraegy of farmer s. he resulng me pah of he pescde effecveness s b n e e exp( nb ). a, e where We see ha when n approaches, he amoun of pescde appled ends o nfny because farmers know ha hey wll no be able o apply pescde he me afer: her effecveness wll be oo low. Furher compuaons sae ha: n e n a, exp 0 b( n) n ne n e exp 0 n n hs means ha he effecveness and he applcaon of pescde declne over me bu unl when? o answer hs queson, we need o acheve he resoluon of he problem by deermnng he opmal swchng me (from farmer s pon of vew). o do so, we can drecly apply heorem 7.6. of Leonard and Long (99). he condon s vald for an Hamlonan funcon n presen value. We work wh an Hamlonan n curren value. Sandard compuaons lead o equaon (7) ha s he necessary ransversaly condon. Equaon (7) conrols he swch o a new pes-conrol echnology and saes ha he opmal swchng me (from farmer s pon of vew) s such ha he value of he opmal Hamlonan evaluaed a equals he margnal scrap value. We oban: Proposon he opmal swchng me (from farmer s pon of vew) from he pescde use o an alernave pes-conrol echnology s nb ln ln. nb e n 5 I ncreases wh e. Snce s dfferen from nfny, some effecveness s lef over before swchng o an alernave pes-conrol echnology. hs effecveness reflecs he need of pescde effecveness of he alernave pes-conrol echnologes. he opmal swchng me (from farmer s pon of vew) from he pescde use o an alernave pes-conrol echnology s a funcon of all parameers. We check ha he opmal swchng me (from farmer s pon of vew) ncreases wh he nal effecveness of he pescde, e.. 5 We assume ha he se of parameers s such ha: 0 7
9 4 he neremporal producon exernaly o value he effcency of he prevous MPE, we need o specfy he socally opmal soluon. Indeed, he MPE should be neffcen snce farmers do no ake no accoun boh he mpac of her own pescde applcaon on he pescde effecveness and he mpac of he oher farmers pescde applcaon on he effecveness. he mplcaon s ha hey may overuse pescde n order o benef from her effecveness before he oher farmers apply hem. In order o check hs, we frs characerze he socally opmal soluon and hen compare o he MPE. 4. he socally opmal soluon We assume here ha an agrculural auhory seeks o mze he aggregaed prof of he n farmers. hs auhory hus chooses he amoun of pescde appled by each farmer a me, ( a, ) n, and he swchng me o an alernave pes-conrol echnology,, ha mzes he presen value of he fuure aggregaed profs. o make comparsons easly, we follow a smlar resoluon mehod as for he MPE, we make he same varable ransformaon as prevously and we agan assume symmercal soluons. he agrculural auhory solves he followng problem: nc exp( d ) ne exp( ) 0 nbc s..: e e e0 e he Hamlonan of hs problem n curren value s: nbc Hc (, e, ) nc e where reflec he socal shadow prce of he sock of pescde effecveness. We know from he concavy of hs Hamlonan and from he Mangasaran condons ha he opmal soluon of he problem sasfes: nbc nc (8) e nbc (9) e bnc e (0) e 8
10 bnc n nc e ne e 0 () In order o be able o easly compare hs soluon wh he MPE, we choose o assume agan a rule such ha c e and we show ha s possble o fnd a > 0 such ha condons (8), (9) and (0) are checked. We fnally show ha: Proposon here exss a unque such ha he decson rule e s a socally opmal rule of pescde applcaon ha mzes he prof of farmers. () he socally opmal pescde applcaon s a e exp( nb ) where. he resulng me pah of he pescde effecveness s e e exp( nb ). bn( ) () he socally opmal swchng me from he pescde use o an alernave pesconrol echnology s nb ln ln. nb e 6 I ncreases wh e. We see from sandar compuaons ha he socally opmal pahs of pescde applcaon and of pescde effecveness declne over me as for he MPE pahs. he socally opmal swchng me from he pescde use o an alernave pes-conrol echnology s a funcon of all parameers. We check ha he varaons of he socally opmal swchng me s he same as n he MPE case wh respec o he nal effecveness of he pescde, e. 4. he neffcences of he MPE soluon he expressons of and are oo complex o be compared on analycal grounds. We can only conclude ha hey are dfferen. We have run a se of smulaons n order o derve more conclusons. 7 he resuls are summarzed n able,, 3, 4 and 5 of Appendx A.4. he man concluson s ha he swchng me from pescde use o an alernave pes-conrol echnology s laer n he socally opmal case han n he MPE: In order o explan hs resul s mporan o compare he pescde applcaon and effecveness n boh cases. We now urn o hs comparson. For clary sake, we frs reason on he same me for he MPE case and for 0. 6 We assume ha he se of parameers s such ha: 7 he dea of he smulaons s o pu values on our parameers. he parameerzaon s no made on emprcal bass because we do no wan o resrc ourselves o a specfc case. As a consequence, we run a se of smulaons for dfferen feasble values of he parameers. 9
11 he socally opmal case. Equaons (4) and (8) are usual margnal condons of long-run prof mzaon. hey sae ha, for each perod of me, he margnal prof of pescde applcaon s equal o s margnal cos. he cos, ha s a measuremen of fuure losses mpled by, b ( e) ncreasng ressance, s no he same n he MPE case,, and n he socally opmal e case, nbc. Indeed, s a prvae cos n he MPE case and s a collecve cos n he e socally opmal case. Furhermore, he shadow prce ( ) dffers n boh cases. he dfference beween boh can easly be undersood by comparng he dynamc of he co-sae varable n he MPE case:,,, b ( n) wh he one n he socally opmal case: bn In he MPE case, farmers value he mpac of her own pescde use on he pescde effecveness, b. Whereas he socal planner values he mpac of each farmer pescde use on he pescde effecveness: bn. In addon, n he MPE case, ( n ) reflecs he assumpon accordng o whch farmers use Markovan sraegy: each one knows perfecly ha he ohers use a rule ha s a funcon of he pescde effecveness a each perod of me. hs erm mples ha n he MPE case each farmer s gong o use more pescde han would have been socally opmal o use. he dea s ha n order o benef from effecve pescde, s opmal for each ndvdual farmer o use pescde before he oher farmers. hese nuons are analycally confrmed by Proposon 3. Proposon 3 he comparson beween he MPE and he socally opmal pah whou consderng he swchng me from he pescde use o an alernave pes-conrol echnology allows us o conclude ha: () he amoun of pescde appled n he MPE case s hgher han he amoun appled n a n n( ) he socally opmal case unl he me, ln >0, afer whch ( n) ( n) he amoun appled n he MPE case can be lower han he amoun appled n he socally a opmal case: a > a <. () he effecveness of he pescde s always lower n he MPE case han n he socally opmal case: e < e. Concluson () confrms he prevous nuons: n he MPE case, farmers overexplo he common ha, here, s he effecveness of he pescde. he mplcaon s ha he effecveness of he pescde s lower n he MPE case han n he socally opmal case. A frs glance, one can be que surprsed by he second par of resul () ha saes ha he amoun of pescde appled n he MPE case can be lower ha he amoun appled n he socally opmal case. hs resul s vald afer a frs regme durng whch he amoun of pescde appled n he MPE case s hgher han he amoun appled n he socally opmal case, whch fs more closely o he nuons. hs can help us explan he surprsng par of he resuls. Indeed, n a frs phase, farmers apply more pescde n he MPE case han n he socally opmal case. By 0
12 dong so, hey conrbue o he decrease of he sock of pescde effecveness unl a me a whch hey begn o apply less pescde han n he socally opmal case because of pescde reducng drascally n effcency: hs s he second phase. Anoher nerpreaon of a can be o consder ha hs s he me of overexploaon of he sock of pescde effecveness n he MPE case. he me a s a funcon of he number of farmers, n, of a proxy of he elascy of he prof o he effecve amoun of pescde appled,, and of he dscoun rae,. Basc comparave sac allows us o conclude ha: Remark he me a decreases wh he number of farmers, n, and wh he dscoun rae,. hese resuls are drec when one keeps n mnd he nerpreaon of a as he me of overexploaon of he sock of pescde effecveness. Indeed, when he dscoun rae ncreases, he value of he fuure decreases and he sock of pescde effecveness s exhaused early. In he same way, when he number of farmers ncreases, he sock of pescde effecveness s exhaused early. o acheve our comparsons, we now need o compare he pahs of boh he pescde applcaon and effecness afer. Beween and, boh he pescde effecveness and applcaon end o zero n he MPE case because of he swch o alernave pes-conrol echnology. he consequence s ha, beween and, boh he pescde effecveness and applcaon mus be hgher n he socally opmal case han n he MPE case. Fnally, we can conclude from smulaons presened n able,, 3, 4 and 5 of Appendx A.4 ha () he fnal (a ) pescde applcaon s lower n he socally opmal case han n he MPE case, a a, and () ha he fnal pescde effecveness s lower n he MPE case han n he socally opmal case, e e. o sum up, we have shown ha he swchng me from he pescde use o an alernave pes-conrol echnology s earler n he MPE case han n he socally opmal case. hs can be explaned by he fac ha he over-applcaon of pescde by farmers n he MPE case consderably reduces he pescde effecveness and her profs. hs means ha, n he MPE case, s more neresng for farmers o swch o alernave pes-conrol echnologes sooner han n he socally opmal case snce he common consued by he pescde effecveness s already exhaused. 5 oward he resoraon of he socally opmal soluon Once he neffcences ha are a work have been brough o he fore, he queson of her nernalzaon remans. In hs secon, we propose o ry some sraeges n order o resore effcency. We have wo neffcences here: he neremporal producon exernales ha are brough o he fore by he dfference beween he MPE and socally opmal pahs of pescde applcaon
13 and effecveness, he swchng me from he pescde use o an alernave pes-conrol echnology ha s laer n he socally opmal case han n he MPE case. Snce here s wo neffcences, one can hnk of wo nsrumens n order o correc hem. We wll frs check f a subsdy desgned n order o nernalze exernales s able o reach he opmal swchng me from he pescde use o an alernave pes-conrol echnology. We wll show ha s no he case and we wll look for a polcy ha s able o resore boh he socally opmal pahs and swchng me. 5. Obanng he socally opmal pahs of pescde applcaon and effecveness he exernaly leads o subopmal pahs boh for he amoun of pescde appled and for he pescde effecveness. A subsdy on amben pescde effecveness can nernalze hs exernaly hrough leadng o he socally opmal pahs of pescde applcaon and effecveness. In hs subsecon, we propose o consder hs by gnorng he horzon of me ha wll be he subjec of he nex subsecon. he nuon s ha snce farmers know perfecly how her pescde applcaon has an mpac on he pescde effecveness, subsdzng he amben level of pescde effecveness by wll gve hem ncenves o reduce her pescde applcaon. Such a subsdy scheme assumes ha, a each me, he agrculural auhory perfecly knows he pescde effecveness. he Hamlonan n curren value becomes: c, j( e ) Hc (,, e, ) c, e, b ( n) e e Only equaons (5) and (7) are modfed wh respec o he MPE case. he dynamc of he cosae varable s now reduced by he value of he subsdy whch means ha farmers are gong o gve more value o he pescde effecveness snce he co-sae varable eners n a negave way no he Hamlonan funcon:, H, e ( ( e ), e,, ) () I s precsely he man am of hs subsdy. Proposon 4 he opmal amben subsdy, ( n ) e >0, s such ha here exss a MPE for whch he equlbrum sraegy of farmer s equal o he socally opmal one. he resulng me pah of he pescde effecveness s also equal o he socally opmal one. he opmal subsdy,, s a funcon of he number of farmers, n, and of a proxy of he elascy of he farmers prof o he effecve amoun of pescdes appled,. In addon, because he subsdy s proporonal o he pescde effcency, s a funcon of he dscoun rae,, of he me,, of he sensvy of he pescde effcency o he amoun of pescde appled, b, and of he nal effecveness of he pescde, e. Remark 3 he opmal amben subsdy on he pescde effecveness ncreases wh, n
14 and and decreases wh b and e. he subsdy ncreases wh me because s proporonal o he pescde effecveness ha decreases wh me. hs decrease has o be compensaed by he subsdy n order o slow down he pescde applcaon. I s for he same reason ha he subsdy decreases wh he nal effecveness of he pescde and wh he sensvy of he pescde effcency o he amoun of pescde appled. he subsdy also ncreases wh he dscoun rae. Equaon () helps o undersand hs. We see ha he subsdy nervenes n he defnon of he shadow prce of he sock of pescde effecveness. Snce hs shadow prce ncreases over me wh he dscoun rae, he subsdy mus decrease wh he dscoun rae o compensae he effec of he dscoun rae ncrease on he shadow prce. he fac ha he opmal subsdy s a funcon of me can be a problem for mplemenaon purposes. Indeed hs means ha has o be adjused a each perod of me by he regulaor. 5. Obanng he socally opmal swchng me Once he opmal subsdy has been derved, wha remans o check s f he socally opmal swchng me can be obaned wh such a scheme. Inuvely, we canno expec o reach hs resul. hs can be seen wh he modfed ransversaly condon:, bnc, c, e ee 0 (3) e for he subsdy case. he fscal schemes seem o have an mpac on he swchng me from pescde applcaon o an alernave pes-conrol echnology. Proposon 6 confrms hs nuon. Proposon 5 () he swchng me from he pescde use o an alernave pes-conrol echnology s nb ln ln. nb e ( n ) 8 As before, ncreases wh e. ()he swchng me n he subsdy case s always lower han he swchng me n he socally opmal case:. If we recall, nb ln ln, he swchng me n he nb e socally opmal case, and 8 We assume ha he se of parameers s such ha: >0. 3
15 ln ln, he swchng me n nb he MPE case, we see ha hey are dfferen from he swchng me obaned n he subsdy cases. he equaons of he swchng mes are oo complex o conclude n a general way on he rankng of swchng mes. We can only conclude ha a subsdy decreases he swchng me wh respec o he socally opmal case. nb e n he comparson beween he swchng mes n he dfferen cases s unclear. he general conclusons ha we can derve are relaed o he sensvy of he dfferences o he parameer values. he conclusons are summarzed n he followng remark. Remark 4 he dfferences beween all swchng mes are ncreasng wh or ndependan of parameers e and b. In order o oban more neresng resuls, we have run a se of smulaons. For he followng se of parameers, b, e 000, 0.04, 0.00 and n 50 we oban he followng rankng: < (6) In order o es he robusness of hs rankng, we have run a se of smulaons n whch parameers n, and vary. he resuls are summarzed n able 6, 7, 8, 9 and 0 of Appendx A.4. Havng n mnd Remark 6, we have also run a se of smulaons n whch parameers e and b are lower han he prevous se. he smulaons show ha he dfferences beween all swchng mes decrease wh he dscoun rae,, and are ndependan of he sensvy of he pescde effcency o he amoun of pescde appled, b. hs las resul mus come from he low values aken n he smulaons bu we know from Remark 6 ha hese value are he only one for whch he rankng n (6) can be reversed. Furhermore, he dfference beween he swchng me n he socally opmal case and he swchng me n he subsdy case,, s he lowes one. hs means ha he amben subsdy leads o a soluon very close o he socally opmal case. Fnally, he man mplcaon of all hese smulaons s ha he rankng n s robus/ 6 Concluson and dscusson hs paper s concerned wh he queson of he nernalzaon of neremporal producon exernales. We concenrae on pes ressance: s well known ha he amoun of pescde appled by farmers decreases her effecveness over me. Developmens n dynamc game heory leraure show how s possble o assume ha, on he one hand, each farmer s perfecly aware of he mpac of hs decsons on he pescde effecveness. On he oher hand, we assume ha each farmer does no know how he oher farmers ac. We hus model he neremporal producon exernales n he pescde case as a dfferenal game. Furhermore, 4
16 we add o he pcure a scrap value of he swch from he pescde use o an alernave pesconrol echnology. he frs sep s o brng o he fore he neremporal producon exernales ha are a play. In order o do hs, we solve and compare wo problems for he complee me horzon (and no only a he seady sae as s commonly he case): he MPE one and he socally opmal one. We show ha he effecveness of he pescde s always (over me) lower n he MPE case han n he socally opmal case. We hus propose o assmlae he sock of pescde effecveness o a common ha s overexploed by farmers because of exernales. he second sep concerns he swchng me from he pescde use o an alernave pes-conrol echnology. We show ha s laer n he socally opmal case ha n he MPE case. We hen look for he fscal schemes ha can be mplemened n order o nernalze hese neremporal producon exernales. We show ha a subsdy s able o nernalze hem. We hen explore he full poenal of a dynamc model by sudyng he mes of swch from pescde applcaon o an alernave pes-conrol echnology. We show ha he subsdy does no lead o he opmal swchng me. he lack of effecveness of fscal schemes n he pescde case was recenly confrmed emprcally by Skevas e al. (0). Conrary o hs las resul, we show ha a subsdy can be effcen. However, he subsdes ha hey suded are no he same as he one suded n our work. hey consdered a subsdy on he use of low-oxcy pescdes, subsdes on research and developmen of low-oxcy alernaves, and subsdes on R&D of more envronmenal frendly pescdes. We raher consder an amben subsdy: a subsdy ha s a funcon of he amben effecveness of pescde. he sudy of he swchng me o an alernave pes-conrol echnology can have neresng mplcaons from a publc polcy perspecve. Indeed, f he am of he polcy-maker (an agrculural auhory n our work) s smply o nernalze neremporal producon exernales, s necessary o mplemen basc nernalzng schemes. However, f he polcymaker s also concerned by he socally opmal swch from he pescde use o an alernave pes-conrol echnology, s no he case. Furhermore, f he polcy-maker has envronmenal goals n mnd, mplcaons are dfferen. If we assume ha alernave pes-conrol echnologes o pescde applcaon are more envronmenally frendly, he nernalzng polcy could be consdered as no beng necessary. Indeed, we show ha a subsdy pospones he swchng me o alernave pes-conrol echnologes. hs s rue f a huge quck applcaon of pescdes does no harm envronmen more han a low long applcaon. One neresng exenson of our work would be o add an envronmenal damage lnked wh he applcaon of pescde n order o conrol hese effecs. Such an exenson s beyond he scope of our work ha concenrae on producon exernales. I s why s lef for a fuure work on envronmenal exernales of pescde use. Appendx A. Proof of remark In order o solve he modfed game, we frs observe from he frs order condon (4) 5
17 ha:, Ae, ( ) Ae e where A s a new consan. We hen have:, ( ) Ae bn. We also know from equaon (5) and from he assumpon on symmery of soluons ha:,,, bn ( ). We replace by Ae, n hs equaon and we oban:, Ae b( n). We now look for A ha solves: ( ) Ae bn Ae b( n ) Ae Ae b ( n ) b ( n ) (7) Before gong furher, we express order condon (4) n whch we pu Ae : c Ae bc 0,, c, Ae b c, e( ba) wh respec o, We assume ha c, e. he mplcaon s ha expresson n equaon (7) and we oban: b n b n ( ba ) ba b n A : A b I drecly resuls from hs ha: b n A. o do so, we depar from he frs ( ba ). We now pu hs We can now look for he equlbrum pahs of he modfed game. We begn wh equaon ne (6) ha can now be wren as: e wh e0 e n he soluon of hs dfferenal equaon s: 6
18 ha gves: and n e e exp e exp nb n n c, eexp eexp nb b n n n a, eexp eexp nb b( n) n A. Proof of proposon We depar from frs order condon (4): pu no ransversaly condon (7): c, n c, e e 0 c c e c b ha we e b,,,, 0, c, e e n exp e nb e nb exp nb n e nb exp nb n e nb exp nb n e nb nb ln n ln ln : nb Our assumpon on symmercal farmers nsures ha here s no possble devaon from hs swchng me. Some basc compuaons hen lead o: n e >0 e n nb e n 7
19 A.3 Proof of proposon We use a smlar mehod as he one used for he proof of Remark and Proposon. () We frs observe from he frs order condon (8) ha: Ae ( ) Ae e where A s a new consan. We hen have: ( ) Ae bn We also know from equaon (9) ha:. We replace by Ae n hs equaon and we oban: Ae bn. We now look for A ha solves: ( ) Ae bn Ae bn Ae Ae b n( ) nb ( ) (8) Before gong furher, we express wh respec o A. o do so, we depar from frs order condon (8) n whch we pu Ae : n c Ae bc 0 c Ae b c e ( ba) We assume ha c e. he mplcaon s ha expresson n equaon (8) and we oban: bn bn ( ba) ba bn A : A b I drecly resuls from hs ha: bn ( ba). We now pu hs We can now look for he equlbrum pahs of he modfed game. We begn wh equaon 8
20 e (0) ha can now be wren as: e wh e0 e he soluon of hs dfferenal equaon s: e e exp e exp nb ha gves: exp c e e exp nb bn n and a e exp e expnb nb( ) c c e () We depar from frs order condon (8): c b 0 ha we e b pu no ransversaly condon (): c n n nc e ne 0 c e e e nb e exp nb exp nb e nb exp nb e exp nb nb e nb nb ln ln ln : nb nb e Some basc compuaons hen lead o: e >0 e 9
21 A.4 Resuls of he smulaons comparng he soluons n he socally opmal case and n he MPE case We have run some smulaons n order o see how he dfferences beween he socally opmal soluons and he MPE soluons vary accordng o parameer values. We oban he resuls n able,, 3, 4 and 5. n 5 n 50 n 500 n a a e e a a able : Dfferences for he se of parameers: b, e 000, 0.00 and 0, a a e e a a able : Dfferences for he se of parameers: b, e 000, 0.00 and n 50. 0
22 a a e e a a able 3: Dfferences for he se of parameers: b, e 000, n 50 and e 0000 e 000 e 00 e a a e e a a able 4: Dfferences for he se of parameers: b, 0.00, n 50 and b 000 b 00 b b a a e a e a able 5: Dfferences for he se of parameers: 0.00, e 000, n 50 and A.5 Proof of proposon 3 Frsly, we know from he assumpons ha: n > b( n ) < b( ) > b( n ) b( ) > () We look for a crossng pon: a a exp n e e exp b( n ) n nb( )
23 n( ) n exp ( n ) n n ( ) ( ) exp n n ( n) n n( ) ( n) exp ( n) n n n( ) a ln : ( n) ( n) We hen look for he sgn of hs crossng pon. Frsly, we know from he assumpons n >0. Furhermore, we know ha: ( n ) n < nn< n n( )<( n ) n( ) > ( n ) ha: n( ) ln > 0 ( n ) a We can hen conclude ha: >0. We now have o compare a a before and afer hs crossng pon. 0 occurs a before. Frsly, we know ha a 0 b( n ) know ha : n( )<( n ) > n( ) ( n ) e < e nb( ) b( n) a 0 < a 0 a We can hus conclude ha: a > a <. nb( ) e and a e 0. We also I s more complcaed o conclude analycally for > a. ha s why we use smulaons presened n ables,, 3, 4 and 5 of Appendx A.4. We observe from hese smulaons ha a a > a. We can hus conclude ha: a a. () We know from () ha: > b( n ) b( )
24 > b( n ) b( ) > nb < nb e exp( nb ) < e exp( nb ) e < e A.6 Proof of remark n a a n ( ) nn( )ln ( ) n <0 n( n) n n( )ln ( ) n <0 ( n) A.7 Proof of proposon 4 he begnnng of he proof s he same as n he proof of Remark. We hen know from equaon (3) and from he assumpon on he symmery of soluons ha:,,, bn ( ). We replace, by Ae n hs equaon and we oban:, Ae b( n). he nex sep s o look for A ha solves: ( ) Ae bn Ae b( n) (9) Remark 5 Le us observe ha: (3) (9) ( n )Ae b Le us se and go furher: (9) ( ) Ae bn Ae ( bn) Ae ( ) Ae bn b n( ) (0) We know from he proof of Remark ha equaon (0) and we oban: bn ( ba ) ( ba ). We now pu hs expresson n 3
25 bn ba bn A : A b I drecly resuls from hs ha: bn We can now look for he equlbrum pahs of he modfed game. We begn wh equaon e (0) ha can now be wren as: e wh e0 e he soluon of hs dfferenal equaon s: e e exp e Remark 6 Snce ( n ) e, we hen drecly have ( n) e exp >0 bn and We fnally have: c e exp c bn a e exp a nb( ) A.0 Proof of remark 3 ( )( n) e exp bn >0 ( n) e exp exp ( ) bn >0 e ( ) 4
26 snce <0 e exp exp ( n n) bn n >0 e n ( n) e exp exp bn b <0 be e ( )( n) e exp exp bn <0 e A. Proof of proposon 5 c, c, e () We depar from frs order condon (4): c,, b 0, ha we e b, bnc, pu ransversaly no condon (5): c, e ee e c, n c, e 0 e e n c, ( n ) e e e n e exp nb ( n) e expnb nb e exp nb 0 nb e nb e nb n exp exp e exp nb nb ( n ) e nb ln nb ( n ) 5
27 ln ln : nb ( ) nb e n Some basc compuaons hen lead o: e >0 e nb nb () ln ln nb ( n ) nb ( n ) ln ln nb nb nb ( n ) We know ha n > n > ( n ) ( n ) ln > 0 >0 snce <0 A. Proof of remark 4 Some basc compuaons lead o: n e >0 e ne ( n ) b b 0 bn ( ) LS LS e 0 e e b 0 A.3 Resuls of he smulaons comparng he swchng mes n he dfferen cases We have run some smulaons n order o see how he dfferences beween he swchng 6
28 mes vary accordng o parameer values. We oban he resuls n able 6, 7, 8, 9 and 0. n 5 n 50 n 500 n able 6: Dfferences for he se of parameers: b, and 0, able 7: Dfferences for he se of parameers: b, and n able 8: Dfferences for he se of parameers: b, and e 500 e 00 e 75 e able 9: Dfferences for he se of parameers: b, 0.00, n 50 and b 0.9 b 0.5 b 0. b able 0: Dfferences for he se of parameers: 0.00, n 50 and
29 References Ambec S., M. Desqulbe (0) "Regulaon of a Spaal Exernaly: Refuges versus ax for Managng Pes Ressance", Envronmenal and Resource Economcs 5(), Am R. (986) "Peroleum reservor exploaon: swchng from prmary o secondary recovery", Operaons Research 34(4), Carlson G.A. (977) "Long-run producvy of nseccdes", Amercan Journal of Agrculural Economcs 59, Cornes R.,. Long, K. Shmomura (00) "Drugs and pess: neremporal producon exernales", Japan and he World Economy 3 (3), Dockner E.J., S. Jorgensen,. Long, G. Sorger (00) Dfferenal Games n Economcs and Managemen Scence, Cambrdge Unversy Press, Cambrdge Frsvold, G.B. and Reeves, J.M. (008) "he coss and benefs of refuge requremens: he case of B coon", Ecologcal Economcs 65(), Georghou G.P.(986) "he magnude of he ressance problem" n Pescde Ressance: Sraeges and accs for Managemen. aonal Academy Press, Washngon, DC Georghou G.P., R.B. Mellon (983) "Pescde Ressance n me and Space", n Georghou G.P.,. Sao (Eds) Pes ressance o pescdes, Plenum Press, London, -46 Hurley M, Babcock B, Hellmch RL (00) "B crops and nsec ressance: an economc assessmen of refuges", Journal of Agrculural and Resource Econonomcs 6, Kngh A.L., G.W. oron (989) "Economcs of agrculural pescde ressance n arhropods", Annual Revew of Enomology 34, Laxmnarayan R., R. Smpson (00) "Refuge Sraeges for Managng Pes Ressance n ransgenc Agrculure" Envronmenal and Resource Economcs (4), Leonard D. and. Long (99) Opmal Conrol heory and Sac Opmzaon n Economcs, Cambrdge Unversy Press, Cambrdge Llewellyn R.S., R.K. Lndner, D.J. Pannell, S.B. Powles (007) "Herbcde ressance and he adopon of negraed weed managemen by Wesern Ausralan gran growers", Agrculural Economcs 36, 3-30 Long., K. Shmomura (998) "Some resuls on he Markov equlbra of a class of homogenous dfferenal games", Journal of Economc Behavor and Organzaon 33, Mullen J.D., J.M. Alson, D.A. Summer, M.. Kreh,.V. Kumnoff (005) "he payoff o publc nvesmens n pes-managemen R&D: general ssues and a case sudy emphaszng nergraed pes managemen n Calforna", Revew of Agrculural Economcs 7(4), O'Shea, L. and Ulph, A. (008) "he role of pes ressance n boechnology R&D nvesmen sraegy", Journal of Envronmenal Economcs and Managemen 55 (), 3-8 Peck, S.L., S.P. Ellner (997) "he Effec of Economc hresholds and Lfe-Hsory Parameers on he Evoluon of Pescde Ressance n a Regonal Seng", he Amercan aurals 49 (), Peck, S.L., F. Gould, S.P. Ellner (999) "Spread of Ressance n Spaally Exended Regons of ransgenc Coon: Implcaons for Managemen of Helohs vrescens (Lepdopera: ocudae)", Journal of Economc Enomology 9(), -6 Qao F., J. Wlen, J. Huang, S. Rozelle (009) "Dynamcally opmal sraeges for managng he jon ressance of pess o B oxn and convenonal pescdes n a developng 8
30 counry," European Revew of Agrculural Economcs 36(), Regev U., A.P. Guerrez, G. Feder (976) Pess as a Common Propery Resource: A Case Sudy of Alfalfa Weevl Conrol, Amercan Journal of Agrculural Economcs 58 (), Regev U., H. Shal, A.P. Guerrez (983) "On he opmal allocaon of pescdes wh ncreasng ressance: he case of Alfalfa Weevl", Journal of Envronmenal Economcs and Managemen 0, Roush R.. (989) "Desgnng ressance managemen programs: How can you choose?", Pescde Scence 6(4), Skevas., S.E. Sefanou, A.O. Lansnk (0) "Can economc ncenves encourage acual reducons n pescde use and envronmenal spllovers?", Agrculural Economcs 43, Sern V.M., R.F. Smh, R. van den Bosch, K.S. Hagen (959) "he negraed conrol concep" Hlagarda 9, 8-0 Yerokhn O., G.C. Moschn (008) "Inellecual Propery Rghs and Crop-Improvng R&D under Adapve Desrucon," Envronmenal and Resource Economcs 40(),
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