A Differential Game Approach to Adoption of Conservation Practices

Transcription

1 A Dffrnal Gam Aroach o Adoon of Consrvaon Praccs Jo L. Parcll Profssor Darmn of Agrculural and Ald Economcs Unvrsy of Mssour Columba, MO arcll@mssour.du and Halu Gdoglu Asssan Profssor of Agrculural Economcs Coorav Rsarch Programs Lncoln Unvrsy of Mssour Slcd Par rard for rsnaon a h Agrculural & Ald Economcs Assocaon s 213 AAEA & CAEA Jon Annual Mng, Washngon, DC, Augus 4-6, 213 Coyrgh 213 by [Jo L. Parcll and Halu Gdoglu]. All rghs rsrvd. Radrs may ma vrbam cos of hs documn for non-commrcal uross by any mans, rovdd ha hs coyrgh noc aars on all such cos.

2 A Dffrnal Gam Aroach o Adoon of Consrvaon Praccs Agrculural roducon can dgrad war sourcs hrough lachng of nrogn and hoshorus from agrculural land o surfac and ground war sourcs. To mnmz h olluon from agrculural roducon, h U.S. Darmn of Agrculur romos adoon of consrvaon raccs. Prvous suds ha analyzd adoon of nw chnologs dd no ncorora h wo moran faurs of chnologs ha ar rmarly usd o consrv h nvronmn; common rsourc and nracon bwn farmrs. Th currn sudy dvlos a concual framwor usng a dffrnal gam o analyz adoon of nw chnologs ha mac h war qualy. Th rsuls of h currn sudy show ha h sngl agn omzaon modls of h rvous suds would no lad o h omal soluon of h dffrnal gam. Currn sudy also shows ha f farmrs ac cooravly, hy dvo mor caal o consrv h nvronmn. Kywords: Tchnology Adoon, War Qualy, Omal Conrol, Dffrnal Gam 1

3 Good qualy war for drnng, aquac lf, and agrculural roducon s crucal for lf. Accordng o h Naonal War-Qualy Assssmn NAWQA Program of h U.S. Gologcal Survy, agrculural roducon s a sgnfcan sourc of olluan for undrground and surfac war sourcs U.S. Darmn of Inror Accordng o NAWQA srams and ground war n basns wh xnsv agrculural roducon hav hgh concnraons of nrogn and hoshorus. Nrogn and hoshorous lvls n h srams ar mosly hghr han h lvls ha caus ovrgrowh of lans n srams U.S. Darmn of Inror Anohr moran fndng of h NAWQA s ha 9 rcn of nrogn and 75 rcn of hoshorus ar smad o orgna from non-on sourcs, whch ar dffcul o dnfy and monor. To addrss h concrns abou war qualy, h U.S. Darmn of Agrculur romos adoon of consrvaon raccs ha can mnmz h lachng of nrogn and hoshorus from agrculural land o surfac and ground war sourcs. Th U.S. Darmn of Agrculur also rovds cos shar rograms, such as Envronmnal Qualy Incnvs Program EQIP ha rovds fundng o farmrs o ado consrvaon raccs. Adoon of nw chnologs by farmrs has bn analyzd n h lraur. Ths suds dvlod h hors for adoon of nw chnologs focusd rmarly on rof ornd raccs Gdoglu and McCann 212. Howvr, chnologs, such as consrvaon raccs, ha ar argng h nvronmnal qualy and h rof ornd chnologs show dffrn srucurs Gdoglu and McCann 212. Th nvronmn ornd chnologs nvolv a common rsourc such as war qualy. Hnc, bsds a caal and m consran of h farmrs, a common rsourc consran should also b ncludd n h analyss. Anohr facor ha s moran n adoon of nvronmn-ornd chnologs ha was no nvolvd n rvous suds s h nracon among farmrs Coor and Km Snc h acons 2

4 of farmrs mac h sa of h common rsourc, adoon dcson of a farmr macd from h acons of h ohr farmrs Docnr al. 2. Acons of farmrs also mac h wllbng of ach ohr drcly hrough h mac on h common rsourcs. Snc h common soc volv ovr m, a dynamc omzaon framwor nds o b usd for h adoon of nvronmn-ornd chnologs. Prvous suds mnond abov ncludd only sac omzaons modls. Snc war qualy volvs n connuous m, omal conrol hory wll b usd n h currn ar. Th sandard omal conrol roblm s srucurd for a rrsnav or a sngl agn omzaon roblm Goz and Zlbrman 2. Hnc, h soluon dos no nvolv h nracon among agns. Inracon among farmrs wll b ncororad no h modl hrough h gam hory. Omal conrol roblms ha nvolv h gam hory ar calld dffrnal gams Docnr al. 2. Ths gams ar calld dffrnal gams from h fac ha h sa varabl of h omal conrol roblm volvs accordng o a dffrnal quaon. Hnc, dffrnal gams nclud agns ha nrac o ach ohr and solv an omal conrol roblm, and h soluon o h roblm s n h form of a Nash Equlbrum. Dffrnal gams hav bn wdly usd n conomcs and managmn Ulh 1989; Khalabar 1977; Km and Long 198; Snn 1984; Boll 198; Docnr al. 2. To our nowldg hs s h frs ar ha uss dffrnal gams n chnology adoon. Th conrbuon of hs ar s, by usng h dffrnal gam framwor, o rovd alrnav concual framwors o analyz h farmrs adoon of nvronmnornd chnologs. Th framwors rovdd n h currn ar wll valua h mac of h nracon bwn farmrs. Th rsuls of h currn ar can b usd by olcy mars o dsgn ffcv rograms o romo h adoon of consrvaon raccs. 3

5 Concual Framwor In h currn modl, farmrs can sar som ar of hr caal or ncom for adoon of nvronmn-ornd chnologs, whch hav osv mac on h common rsourc or h war qualy n hs cas. Howvr, h caal usd for adoon of consrvaon chnologs dcrass h caal avalabl for agrculural roducon. W wll us a lnar convrson bwn consumon and roducon. Hnc a m for farmr c, whr c s h consumon, s h caal usd n agrculural roducon and s h caal sard for adoon of consrvaon raccs. W wll assum ha hr xss a grou of farmrs 1,2... N whos acons mac h war qualy and who ma adoon dcsons. Th ranson funcon for war qualy hn can b wrn as; N N w whr w s h chang n war qualy, whch s also h sa varabl for h omzaon roblm. Th ranson funcon shows ha h caal dvod for agrculural roducon mac h war qualy ngavly and h caal sard for adoon of consrvaon raccs macs h war qualy osvly. Th obcv funcon for farmr can b wrn as; max r u c d r u whr farmr maxmzs dscound sram of uly and h uly s gand from consumon, whch s a lnar funcon of caal as mnond abov. In h quaon abov r dnos h consan m rfrnc ra or h dscoun ra. d 4

6 Th Coorav Cas Th bnchmar wll b h cas whr all h farmrs ac cooravly. Ths s vry smlar o h sngl agn omzaon roblm. In hs cas, assumng ha h all h farmrs hav h sam uly funcon, h rsulng omal conrol roblm can b rrsnd as; max r Nu d s.. w N,, w w >, lm w Th soluon o hs roblm wll b found usng Ponyagn s maxmum rncl Docnr al. 2. Th omal soluon wll b n rms of h nal valu of h sa varabl and no h currn valu of h sa varabl. Ths nd of soluon s calld h on-loo qulbrum and h soluon ha ncluds h currn valu of h sa varabl s calld h Marow-Prfc qulbrum. In gnral boh soluons should b sam gvn h nal valu of h sa varabl s nown Docnr al. 2. Th Marow-Prfc qulbrum rqurs farmrs o obsrv h sa of h war qualy n ach m rod o form h omum dcson. Snc hs s dffcul for farmrs o do, w would us h on-loo qulbrum, whch only rqurs h nal valu of war qualy b nown. Usng Ponyagn s maxmum rncl, h Hamlonan can b wrn as; H w,, λ Nu λn Th condons for h omaly ar; u λ, λ rλ, Soluons o hs omaly condons wh usng a consan lascy of margnal uly; 5

7 u c lads o Ac 1η 1η A 1η 1η r λ λ and [ ] r /η Th valu of can b calculad nong ha an omal ah mus xhaus h war qualy n h sns ha lm w, whch shows ha any ah ha dos no xhaus h war qualy vnually wll b domnad by a fasbl ah ha xhauss, gvng rs o mor consumon ovr som m nrval. For hs rason, w N ηn d r, whch lads o omum soluon; * rw r / η and h m ah for h sa varabl, war qualy s ηn * r /η w w. Also, no ha * * rw, whch s h Marovan sragy ha rqurs farmrs o ηn now h valu of h sa a m o form h omal consumon, whras on-loo sragy dos no rqur farmrs o now h valu of h sa varabl a m, rahr rqurs farmrs o now h nal valu of h sa varabl. Anohr moran on s for any N 1, h sngl agn omzaon modls usd n rvous suds would no lad o h omum soluon for adoon of a chnology ha nvolvs a common rsourc. 6

8 7 Th Noncoorav Cas In h noncoorav cas, ach layr mus choos a h ous a m ah of caal for adoon of consrvaon raccs and hs chocs ar mad smulanously and noncooravly. Hnc, ach farmr s mang h omal dcson ndvdually, bu ang no accoun h acons of h ohr farmrs. Th oucom n h noncoorav suaon or gam wll b analyzd for wo cass 1 η and 1 < η for h uly funcon; η η η η A Ac c u Cas 1: 1 η In hs cas, lm c u c hnc zro m consumon ovr any m rod would gv a ayoff of mnus nfny. For ha rason, ach layr wll ma sur ha war qualy wll no b xhausd n fn m. For gvn m ahs,., layr mus choos a m ah of caal for consrvaon raccs. such ha o w w and w and lm w. Th obcv funcon of h farmr can b wrn as d u r and h Hamlonan s + u w H,, λ λ Th omaly condons can b oband as, r u λ λ λ

9 * rw / η Th soluon s oband as r rw r / η for any for ηn ηn * /η all, oghr wh r w w. Hnc, h coorav soluon s oband as h Nash qulbrum. Cas 2: η < 1 In hs cas lmc u c s fn, hnc hr wll b ncnv for ach layr o xhaus h war qualy n fn m, as all layrs ry o g a bggr oron of h common soc. Th consumon ahs for layr and for all wll b ha c d + c d d + d w In hs cas layr s rly o n-1 m ahs, s calld srcly fasbl onloo rls. In hs cas h coorav oucom s oband as h Nash qulbrum. Th ohr ossbly for consumon ahs o b, T T c d + c d d + T whr T nf { s s for all s } T d w, whch shows ha h consumon or h xracon of war qualy s zro n all das afr T. In hs cas farmr s allowd o xhaus h war qualy n fn m and frusra h oonn s lans by xhausng h war qualy. To s hs on, assum hr xss a valu T such ha farmr ss c for all > T. If for som, farmr s srcly osv a som > T, hn farmr s lan frusrad by as a ha m war qualy s alrady xhausd by layr. Hnc, whn farmr s oonns choos h coorav oucom rw * r / η, ηn 8

10 h omal rsons for farmr s o dva from coorav oucom and xhaus h war qualy n fn m. Farmrs whch hav shorr m horzon durng agrculural roducon may b lss concrnd abou h susanably of h war qualy. Accomlshmn of h coorav oucom would bnf h olcy mars as farmrs would sar hghr amoun of caal for adoon of consrvaon raccs. Succss of cos shar rograms such as Envronmnal Qualy Incnvs Program would also b macd from h accomlshmn of h coorav oucom. Soc Dndn Uly Ths scon of h ar focuss on h modfcaon of h rvous modl. For som farmrs, h uly may no b only gand from consumon, bu also from h currn sa of h war qualy. Ths mgh rrsn h nvronmnally concrnd farmrs. Assum now h farmrs oban uly from boh h consumon and h currn sa of h war qualy. For a uly funcon n h form of α α u c, w cw [ w], whr < α < 1, h coorav roblm s o maxmz, whch s smlar o h uly funcon n h rvous scon; r α N[ w] d s.. w N,, w w >, lm w h oonal consumon ah for hs roblm s * rw N 21 α and for h noncoorav cas, h omal consumon ah s gvn by 9

11 * rw 21 Nα Whch consus a symmrc Marov rfc Nash qulbrum, rovdd ha N < 1/ α. Th ra of xhauson s grar n h noncoorav han h coorav cas. As rvous scon, for any N 1, h sngl agn omzaon modls n rvous suds would no lad o h omum soluon for adoon of a chnology ha nvolvs a common rsourc. Concluson As h focus of h olcs bcom war and ar qualy, adoon of nw chnologs nd o b analyzd n a sng dffrn han h ons suggsd n rvous suds. Currn sudy rovds h concual framwor usng h dffrnal gam schm. Th rsuls of h currn sudy showd ha sngl agn omzaon modls, whch wr usd n rvous suds, would no lad o h omum soluon of h dffrnal gam whn hr s mor han on farmr macng h common rsourc. Currn sudy also shows ha farmrs would xhaus h war qualy fasr f hy do no ac cooravly. Ths rsul s vald also for h cas whr farmrs ar nvronmnally concrnd or gs uly from nvronmnal qualy. As h concrns abou h nvronmnal qualy ncrass and cos shar rograms such as Envronmnal Qualy Incnvs Program ar dvlod, h olcy mars wll bnf from farmrs acng oghr o dvo mor caal o consrv h nvronmn. Exnson rograms ar ndd o b dvlod ha show farmrs h bnf of acng cooravly. 1

12 Rfrncs Boll, F Th Effcn Us of a Non-Rnwabl Common-Pool Rsourc s Possbl bu Unlly. Journal of Economcs, 4, Coor, J. C., and Km, R. W Incnv Paymns o Encourag Farmr Adoon of War Qualy Procon Praccs. Amrcan Journal of Agrculural Economcs, 78, Docnr, E., Jorgnsn, S., Long, N. V., and Sorgr, G. 2. Dffrnal Gams n Economc and Managmn Scnc, Cambrdg Unvrsy Prss, Nw Yor. Envronmnal Procon Agncy. 26. Ag 11. h:// Fdr, G., Jus, R. E., and Zlbrman, D Adoon of Agrculural Innovaons n Dvlong Counrs: A Survy. Economc Dvlomn and Culural Chang, 332, Fdr, G., and O Mara, G. T On Informaon and Innovaon Dffuson: A Baysan Aroach. Amrcan Journal of Agrculural Economcs, 64, Fdr, G., and Slad, R Th Acquson of Informaon and Tchnology Adoon. Amrcan Journal of Agrculural Economcs, 663, Gdoglu, H. and L. McCann. Adoon of Wn-wn, Envronmn-ornd and Prof-ornd Praccs. Journal of Sol and War Consrvaon, May/Jun Issu, 212. Goz, R. U., and Zlbrman, D. 2. Th Dynamcs of Saal Polluon: Th Cas of Phoshorus Runoff from Agrculural Land. Journal of Economc Dynamcs & Conrol, 24,

13 Grlchs Z Hybrd Corn: An Exloraon n h Economcs of Tchnologcal Chang. Economrca, 254, Hbr, L. D Rs, Larnng, and h Adoon of Frlzr Rsonsv Sd Vars. Amrcan Journal of Agrculural Economcs, 56, Huffman, W. E Dcson mang: Th Rol of Educaon. Amrcan Journal of Agrculural Economcs, 56, Huffman, W. E Farm and Off-Farm Wor Dcsons: Th Rol of Human Caal. Th Rvw of Economcs and Sascs, 621, Km, M. C., and Long, N. V Rsourc Exracon undr Condons of Common Accss. n Km, M. C., and Long, N. V. ds., Exhausbl Rsourcs, Omaly, and Trad, Norh-Holland, Amsrdam, Khalabar, F Mar Imrfcons and h Omum Ra of Dlon of Naural Rsourcs. Economca, 44, Mansfld, E Tchncal Chang and h Ra of Imaon. Economrcs, 29, Ocobr, Snn, H. W Common Prory Rsourcs, Sorag Facls, and Ownrsh Srucur: a Courno Modl of h Ol Mar. Economcs, 51, Sundng, D., and Zlbrman, D. 21. Th Agrculural Innovaon Procss: Rsarch and Tchnology Adoon n a Changng Agrculural Scor. Handboo of Agrculural Economcs, B. L. Gardnr and G. C. Raussr, ds., Elsvr, Amsrdam, Ulh, A A Rvw of Boos on Rsourc and Envronmnal Economcs. Bulln of Economc Rsarch, 41,

14 U.S. Darmn of Inror Th Qualy of Our Naon s Wars: Nurns and Pscds. U.S. Gologcal Survy Crcular U.S. Darmn of Agrculur and Envronmnal Procon Agncy Unfd Naonal Sragy for Anmal Fdng Oraons. Wozna, G. D Th Adoon of Inrrlad Innovaons: A Human Caal Aroach. Th Rvw of Economcs and Sascs, 661,