Forest Rotations and Stand Interdependency: Ownership Structure and Timing of Decisions****

Transcription

1 Ggoy S. Amch* - Ekki Koskl** - Mkku Ollikinn*** ost Rottions nd Stnd Intdpndncy: Ownship Stuctu nd iming of Dcisions**** Dptmnt of Economics, Univsity of Hlsinki Discussion Pps No 584:3 ISSN ISBN X Dcm, 3 othcoming in: Ntul Rsouc Modling * Dptmnt of osty, Collg of Ntul Rsoucs, 37 Chthm Hll, Vigini Polytchnic Institut nd Stt Univsity, Blcksug, VA 46, USA. Emil: gmch@vt.du. ** Dptmnt of Economics, P.O. Box 7 Akdinktu 7, IN-4 Univsity of Hlsinki, inlnd, nd th Rsch Dptmnt of th Bnk of inlnd, P.O. Box 6, Hlsinki, inlnd. Emil: kki.koskl@hlsinki.fi. *** Dptmnt of Economics nd Mngmnt, P.O. Box 7, IN-4 Univsity of Hlsinki, inlnd, Emil: mkku.ollikinn@hlsinki.fi **** W would lik to thnk th Gust Associt Edito Jffy P. Pstmon nd th nonymous fs, s wll s Alix Ptson Zwn fo vy usful commnts. An li vsion of th pp ws psntd t th nd Wold Congss of Envionmntl nd Rsouc Economists, in Monty, Clifoni, Jun 4-7,. his pp is pt of th pojct Studis in Envionmntl nd Rsouc Economics finncd y th Acdmy of inlnd. h funding is gtfully cknowldgd. Koskl lso thnks th Rsch Unit of Economic Stuctus nd Gowth RUESG t th Univsity of Hlsinki fo finncil suppot, nd CESifo t th Univsity of Munich nd th Bnk of inlnd fo thi hospitlity.

2 ost Rottion und Intdpndnt Stnds: Ownship Stuctu nd iming of Dcisions Astct: his pp xtnds th Htmn modl to study th optiml ottion g of two intdpndnt stnds t th stdy stt, whn th ottion gs of th two stnds qul nd th stm of mnitis poducd fom th two stnds my complmnts o sustituts, oth in spc nd ov tim. In th psnc of stnd intdpndnc oth th ownship stuctu nd th squnc of dcision mking mtts. Rottion g choics xmind nd compd und vity of quilii, including Nsh, Stcklg, nd sol own css. W show tht th sol own s ottion g is long thn th ottion g und oth Nsh nd Stcklg ssumptions if th stnds sptil complmnts, ut shot if thy sustituts. h pcis ltionship twn th Nsh nd Stcklg ottion gs, nd th qulittiv poptis of ottion gs in tms of tim pics, gntion costs, nd intst ts, lso dpnd on how sptil sustitutility nd complmntity twn stnds volvs though tim. Ky wods: fost ottion, mnity pulic goods, stnd intdpndnc, nonindustil lndowns JEL clssifiction: Q3, H.

3 . Intoduction ost cosystms compis complx sit-spcific intctions twn plnt nd niml spcis. On spct of fost cosystms ly cknowldgd in conomics modls is th notion of stnd intdpndnc. h mjoity of modls focus on mngmnt of only singl stnd in isoltion, ut in pctic th mngmnt of ch stnd in givn gion should not undtkn indpndntly of oth stnds. Biologists hv long known this, guing tht ts of mny g clsss nd spcis mixs ncssy fo consvtion of iodivsity o contiguous hitt fo ctin niml spcis. Stnd intdpndnc my lso nthopognic in ntu. o instnc, th ctionl oppotunitis of lg fost s my dpndnt on th intction of svl stnds. It is wll known tht lndowns, spcilly non-industil ons, mng fosts with oth tim nd non-tim svics in mind Boyd nd Hyd 989. Mnging n intdpndnt multipl-stnd fost is chllnging tsk. Hvsting vn on stnd my somtims pos tht to th mintnnc of n nti cosystm. Whil th tsk is difficult nough fo on mng, it coms vn hd und th lity of nonindustil pivt lndownship. Lnd popty ights usully do not follow fost cov typs. Although cnt suvy wok vls tht fost lndowns my willing to jointly mng lnd with oth lndowns, und vious cosystm mngmnt sttgis such s psvtion of wildlif coidos, th dg of cooption nd coodintion diffs nd is likly to f fom complt.g., s Jcoson. In ltd suvy-sd study, Klosowski t l. shows tht lndowns in th Nothstn U.S. my willing to coopt nd mk dcisions jointly fo wildlif, ut only if conomic incntivs xist fo coodintd mngmnt. In fct, lndowns owning on pic of n cosystm my nglct, knowingly o unknowingly, th impct of thi pivt hvsting on th whol cosystm o on oth ny lndowns. h psnc of lndowns who my not l o willing to coodint ctions will socilly costly. h impct of on lndown s dcisions on th fost cosystm usd y noth lndown psnts typ of conomic xtnlity ssocitd with pivt fost mngmnt. Only socil plnn who mngs n nti fost cosystm Accoding to Smpl 996 cooption my difficult mong lndowns, spcilly whn th is htognity of th lndscp o lndowns divs in thi pfncs fo nontim svics.

4 3 s whol hs incntivs to solv fo th ottion g of ch stnd, conditionl on its impcts to ll oth stnds. In this pp w xmin svl issus not ddssd in th conomic mngmnt of intdpndnt fost stnds. W ssum lndowns civ tim nd non-tim mnity nfits tht dpnd on ottion gs of n djcnt stnd. By ssuming uniqu stdy stt quiliium xists, w xmin vious ssumptions fo th timing nd implmnttion of djcnt lndown dcisions, whn ottion gs qul. A if chctiztion of ou min findings s follows. If lndowns hv without gd to thi ffcts on noth lndown, thn th ffctivly mk fost mngmnt dcisions simultnously, i.., lndowns ply Nsh gm. h Nsh gm flcts pivt ownship in pctic, wh lndowns typiclly od smll num of nighouing lndowns nd mk dcisions without gd to th oth lndowns. An ltntiv stting is lso xmind wh on lndown movs fist, ut mks dcisions with th ction of noth lndown in mind. inlly, w xmin th ottion g dcision fo sol own who mks fost mngmnt dcisions tking into ccount th intdpndnc twn ll stnds. his is simil to sitution wh ll lndowns pfctly coodint, o coopt in, thi ctions. Compison of this outcom with th simultnous nd fist-mov outcoms will show th impotnc of coodintion, nd thy hint t th socil cost of not coodinting fost mngmnt ctions. h vy fw nlyticl ttmnts of th conomics polm hind stnd intdpndnc. Stnd intdpndnc ws oiginlly ddssd in Bows nd Kutill 985, 989. hy discuss nts ssocitd with multipl stnds und singl govnmnt own. Swllow nd W 993 nd Swllow t l. 997 w th fist to fomult xplicit sptil intctions fo non-tim mnity nfits twn two djcnt stnds, ut thy ly minly on numicl ppoximtions fo th cs of n xognous djcnt stnd. Koskl nd Ollikinn xmind nlyticlly th ottion g dcision fo singl lndown mking dcisions fo singl stnd, lso und th ssumption of puly xognous djcnt stnd. Moov, thi wok dos not focus on th diffnt lndown commitmnt ssumptions tht w xmin, no do thy xmin th impotnt sol own outcom. All of ths issus citicl to undstnding pivt lndown hvio und vious cicumstncs. h is lg littu on stnd intdpndnc in oth sttings, such s spcis consvtion. his wok, whn tkn in th contxt of fostd s, pomots th id tht multipl stnds ndd to sustin ctin spcis s.g. Csuti t l. 997,

5 4 Ando t l. 998, Polsky t l.. An incsing num of mpiicl studis now xist on consvtion, cosystm mngmnt, nd fost mngmnt. hs, howv, typiclly undtkn fom th viwpoint of nvolnt socil plnn s.g. Bvs t l. 995, Als 996, Bvs nd Hof 999, Hight nd vis 997, Montgomy 995. Unlik ou pp, this littu ith consids only th cs of th sol own, o it is sd on sit-spcific mpiicl dt. Hnc it stcts fom th intsting pcticl polms tht follow fom pivt lndowns who my not coodint th mngmnt of thi fosts. Yt coodintion my impcticl in th U.S., s mny lndowns do not liv clos to thi poptis, thus mking th sol own outcom vi cooption lss likly. h st of ou pp is ognizd s follows. In sction w intoduc n xtndd Htmn modl of fost mngmnt nd mk spcific th dfinition of sptil dpndnc twn stnds nd its volution ov tim, i.. tmpol dpndnc. W thn nlys ottion g dcisions und simultnous mov, fist-mov nd sol own timing ssumptions nd comp thi ltionships. Sction 3 chctizs th qulittiv dpndnc of ottion gs on impotnt pmts. inlly, in sction 4 w povid som concluding mks.. A modl of intdpndnt stnds W fist dsci sic fmwok fo th dtmintion of ottion gs fo two djcnt stnds, dnotd y stnd '' nd stnd ''. It is ssumd tht lndowns vlu nt hvst vnu nd th non-tim mnity svics poducd fom th stnd, just s in th convntionl Htmn modl of fost mngmnt Htmn ollowing Swllow nd W 993 nd Koskl nd Ollikinn, w ssum tht stnds intdpndnt in tms of mnitis ut indpndnt with gd to tim poduction. In ths modls, th gowth of stnds nd is n S-shpd function of ottion g. im volum t hvst is dnotd y f nd g, wh fs to th In Noth Amic, th hs n much wittn out snt lndowns who my not l to coodint ctions on thi lnd. Oths hv shown cntly tht lndowns lss intstd in pusuing joint mngmnt of fost lnd if thy not pivy to infomtion gding th nfits of coodintion Jcoson. 3 Pps including.g. Binkly 98 nd Kuuluvinn t l. 996 povid som indict mpiicl vidnc in fvo of th hypothsis tht pivt lndowns vlu non-tim mnity svics.

6 5 ottion g fo stnd nd τ fs to th ottion g of th stnd. im pics p nd q nd gntion costs c nd c fo stnd nd, spctivly, llowd to diff twn th stnds. hs ssumptions flct th typicl sitution in which stnds diff inhntly du to sit chctistics such s slop, t spcis, spct, o ccss. Pics, costs, nd th l intst t ssumd to constnt ov tim, s with th sic Htmn modl. h psnt vlus of tim poduction ov n infinit cycl of ottions fo ch stnd, spctivly, V V pf qg c c. W now intoduc mnity vlus in mnn tht flcts stnd intdpndnc. Lt s, dsci vlution of mnity nfits povidd y stnd t tim s nd s function of th djcnt stnd s ottion g of. Likwis,, x dnots vlution of mnity nfits of stnd t tim x whn stnd hs ottion g of. Using this nottion, th psnt vlu of mnitis ov n infinit sis of ottions of lngth nd fo oth stnds wittn, E E s s, ds x, x dx. Equtions nd flct th stdy-stt mnity vlution und ctin limiting ssumptions. As pointd out in Swllow nd W 993, th psnc of intdpndncy might imply tht th mnity vlu of ch stnd dpnds on th cunt clnd g of th djcnt stnd. In pticul, if th ottion gs of th two stnds diff, th dynmics of ny stdy-stt will dpnd not only on th ottion gs of th stnds, ut lso on th clnd g of th stnds. Und ths cicumstncs, on cnnot ppoximt th psnt vlu of mnity nfits using qutions nd. In Appndix w illustt n xtnsion of th modl y intoducing clnd g into. Using x nt givn nd diffnt ottion gs in numicl xmpl, Appndix

7 6 dscis how ottion g nd ctul g of stnds intct s th num of ottions is incsd. Equtions nd flct cs wh th intdpndnt stdy-stt ottion gs idnticl. Whil this pps to stictiv, it llows us to dvlop qulittiv thoy of stnd intdpndncy in its simplst fom. Such n ppoch is simil to on poposd in n impotnt contiution of oin nd Houthkk hy nlysd th ffcts of cdit tioning y sticting ttntion to mgin wh tiond nd f mkt solutions w ssumd idnticl. hn, thy pocdd with n ssssmnt of th mginl ffcts of tioning. W follow thi ppoch in spiit y ssuming ottion gs idnticl, nd thn xmining th hvio of th stdystt within this mgin. Most pvious fost conomics wok ssums th xistnc of stdy stt, lik w do. Vy fw studis lot on possil tnsitionl dynmics tht might is outsid of th stdy stt. An xmpl is Swllow nd W 993, who povid numicl simultion sults fo stnd intdpndncy in th cs of n xognous djcnt stnd. ht sid, fist nd ntul stp towds dvloping th nlytics of stnd intdpndncy tht dos not qui numicl simultion is to dtct th mginl ffcts of xognous pmts nd gnts hviou fo ny fom of stnd intdpndncy. 4 Pocding und th ssumptions of uniqu stdy stt nd qul ottion gs, qutions nd dfin vlution function of mnity nfits fo ch stnd in tms of its own nd th djcnt stnd ottion gs. h choic of ths ottion gs is md fo n infinit sis of ottions gun t tim t =, following th Htmn nd ustmnn modls. In ths modls, givn tht th choic of ottion g is md und ctinty nd xognous vils min constnt ov tim, ll futu ottions will simil in tms of th pth of tim nd mnity flows fov. hus, in this cs, th ottion g choic is th ppopit indicto of th pth of mnity svics consumd y th lndown. o susqunt nlysis w must lso chctiz how th mnity vlus in qutions nd hv in tms of chngs in thi own ottion g nd chngs in th djcnt stnd s ottion g. In dsciing ths ffcts, w will us th ll own- 4 Poviding sufficint conditions fo th xistnc of stdy stt o spcifying th tnsitionl dynmics govning th possil quilii, which my includ stdy stt, cyclicl, chotic o oth solutions, gos yond th scop of this pp ut is ctinly n impotnt futu sch topic.

8 7 stnd to f to th stnd in qustion, nd djcnt stnd to f to th oth stnd. Nglcting fo momnt th psnt vlu tms in nd, nd diffntiting th intgls fo ch stnd with spct to th own ottion g, w cn otin mginl mnity vlution function dfind t hvst tims fo oth stnd nd : Dnot ths s, nd, spctivly. W now psnt two dfinitions fom th littu, which chctiz sptil dpndnc of stnds, s wll s th volution of this ov tim, i.., tmpol dpndnc. Diffntiting ch stnd s mginl mnity vlution function with spct to th ottion g of th djcnt stnd indicts how th mginl mnity vlution chngs with spct to chngs in th ottion g of th oth stnd. hs divtivs dfin sptil dpndnc in th littu nd summizd in: 5 Dfinition Koskl nd Ollikinn. Sptil Dpndnc, nd, if stnds sustituts wt mnitis indpndnts wt mnitis complmnts wt mnitis Dfinition is consistnt with ALEP complmntity/sustitutility fist fomlizd y Smulson 974 nd oths in diffnt contxt. 6 If th stnds sptil sustituts, thn th mginl mnity vlution of ch stnd dcss with th ottion g of th djcnt stnd. If th stnds sptil complmnts, thn th opposit is tu, i.., mginl mnitis of ch stnd incs with th ottion g of th djcnt stnd. It is lso impotnt to know how sptil dpndnc twn stnds is ffctd y ottion g choics. his is otind y diffntiting th functions in Dfinition with spct to own-stnd ottion gs. h sulting scond divtivs dfin how sptil dpndnc twn stnds volvs with own ottion g. his is clld tmpol dpndnc in th littu. ht is, 5 In wht follows, divtivs of functions will dnotd y suscipts unlss othwis notd. 6 o th concpt of th Auspitz-Liig-Edgwoth-Pto ALEP complmntity/ sustitutility, s Smulson 974 nd futh discussions in Chipmn 977, Knni 98 nd W gding its implictions fo th poptis of dmnd functions.

9 8 Dfinition Koskl nd Ollikinn. mpol Dpndnc, ;, if stnd dpndnc dcss with stnd g stnd dpndnc unchngd with stnd g stnd dpndnc incss with stnd g om Dfinition, tmpol intdpndnc twn two stnds my constnt, incsing o dcsing dpnding on how sptil dpndnc twn th stnds chngs with incss in th ottion g of ch stnd. mpol indpndnc sults whn,,. his is th cs if sptil sustitutility o complmntity fom Dfinition is mly ssocitd with sit-spcific poptis tht min th sm gdlss of own-stnd ottion g. Incsing tmpol dpndnc twn th stnds mns tht, fo sptil complmnts, th complmntity twn stnds incss with own-stnd ottion g. But fo sptil sustituts, th sustitutility twn stnds dcss with own-stnd ottion g. Dcsing tmpol dpndnc implis just th opposit: complmntity wkns whil sustitutility coms stong fo incss in own-stnd ottion g. Ecologists hv shown tht mnity poduction dpnds on int-stnd ltionships lik th ons ov in Dfinitions nd s, fo xmpl nklin nd omn 987 nd Gils 978. As fosty xmpl of sptil sustituts nd complmnts consid th cs of lndown who vlus tim nd fog poduction consistnt with ig gm poduction, wh ig gm quis oth fog nd cov s.g. Swllow nd W 993. h own stnd nd th djcnt stnd function in this cs xist s sustituts in thi poduction of this mnity, s long s thy simultnously povid oth fog nd cov. h stnds complmnts if instd, fo xmpl, th own stnd povids only fog whil th djcnt stnd povids only cov. his typ of xmpl hs spcil lvnc to cosystm mngmnt in th U.S. his modis th holistic mngmnt of lnd in coidos dvoctd fo wildlif hitt potction Gumin 994, Moot nd Cotn 994. H, coidos function s complmnty stnds twn th stnds thy joining. Anoth xmpl lts to poduction of wt nd fost goods discussd in Bows, Kutill nd Shmn 984 nd Bows nd Kutill 989. In thi xmpls, th spcis composition of ts in fost i.., stnds compising ith conifous o lf ts s wll s thi loction on th lndscp,

10 9 impotnt oth to wt qulity nd th liklihood of floods nd oth hydologicl poptis. hus, stnds my complmnts o sustituts in th poduction of wt. h cs of incsing o dcsing tmpol dpndnc is mo complx. Consid th fist xmpl. Suppos tht th stnds oiginlly sptil sustituts nd th djcnt stnd ottion g incss. If this dcss incss fog nd cov fo th own stnd, du to chngs in undstoy vgttion, thn w hv th cs of dcsing incsing tmpol dpndnc twn stnds. On thfo xpcts tht tmpol dpndnc is impotnt in th cs of iodivsity consvtion. ypicl oldgowth spcis, such s d-cockdd woodpck, spottd owl o cpgll, qui significnt stnds of old tim fo thi popgtion. Whil som old gowth spcis highly spcilizd to old gowth fosts, oths to lss xtnt. 7 hfo, incsing th g of old gowth stnds mks th fost mo lss suitl to highly lss spcilizd spcis thus inducing stong wk intdpndnc twn stnds. In ou modl ths situtions would llld s incsing dcsing tmpol dpndnc spctivly.. h modls fo ottion g W now dpt fom th xisting littu nd consid ottion g solutions fo diffnt ownship stuctus nd timing of dcisions. h fist ottion g solution cosponds to Nsh gm, wh th two diffnt lndowns who own stnds nd, nd ths lndowns mk thi hvsting dcisions simultnously tking th oth s ction s givn. his mimics th pivt mkt solution wh lndowns tk th hvio of oths s givn. h scond ottion g solution follows whn th two lndowns, ut on is fist-mov, i.., lndowns ply tditionl two-stg Stcklg gm. h lndown moving fist is ssumd l to cdily commit to hvsting dcision fo th oth lndown movs, so tht ld-follow ltionship is stlishd. inlly, th thid ottion g solution is divd und th ssumption of sol own of oth stnds nd. his cn intptd s th cs wh lndowns coodint thi dcisions. 7 As mtpopultion thoy hs shown, ths spcis do not occupy ll stnds, ut still sufficintly high mount of old gowth stnds ndd s.g. Hnski 998.

11 In ll css, lndowns ssumd to pic tks nd, s such, do not ccount fo pic-inducd dmnd chngs whn mking ottion g choics. Ech lndown dos, howv, mk us of mnitis poducd y th oth stnd. hfo, th sol own modl, y yilding th fficint solution, povids hint t th socil costs ssocitd with uncoodintd hvsting if th no xtnlitis outsid th lndowns contol... Rottion gs in th Nsh gm H ch lndown chooss ottion g tking th oth lndown s ottion choic s givn y mximizing N V E nd N V E. h solution to th Nsh gm cn otind fo ch lndown y solving th following polms simultnously: Mx { } N V E 3 Mx { } N V E, 3 wh th tms in th ojctiv functions dfind in nd. h lls nd gin f oth to th stnd nd lndown, nd N dnots th Nsh gm. h following fist-od conditions chctiz th optiml ottions nd : N : pf, pf V E 4 N : qg, qg V E. 4 hs suggst tht oth lndowns qut thi pivt mginl nfit of dlying hvst LHS to th mginl oppotunity cost of dlying hvst RHS. Notic th is n xtnlity vidnt in th fist-od conditions ffcting th lst tms on th LHS nd RHS of 4 nd 4. his iss cus lndowns do not ccount fo th ffct of thi ottion g choic on th oth lndown s utility nd hvio. h scond-od conditions fo oth lndowns givn y N pf pf, 5

12 N qg qg, 5 W ssum tht th scond-od conditions 5 nd 5 hold. hy hold utomticlly whn th lndown s mginl mnity vlution dcss o mins constnt with incsing own-stnd ottion g, i.., fo nd. hy my not lwys hold whn th lndown s mnity vlution incss with own-stnd ottion gs, i.., fo nd s Stng 983. As fo th dynmics of th Nsh quiliium, w ssum tht lndowns djust thi ottion gs in od to incs th sum of nt psnt vlu of hvst vnu nd mnitis, tking th hvio of th oth lndown s givn. hs dynmics psntd y th following divtivs, N nd N 5c wh th pmts, indict th spd of djustmnt, nd th dots f to tificil tim, flcting dynmic djustmnts to ottion gs ov tim s.g. kym 988, pp h uniqunss nd stility condition fo th Nsh gm cn xpssd s N N N N N, 5d wh N s, s, ds nd N x,, x dx, so tht th dtminnt of th scondod divtivs mtix in 5d must positiv. 8 h scond od conditions 5 5 imply tht th fist pt of 5d is positiv. hfo, th uniqunss nd stility of th Nsh gm dpnds on th poduct of coss-divtivs N N, which jointly with th scond-od conditions dfin th slops of th ction functions fo th lndowns. 8 o futh dtils out uniqunss nd stility, s Dixit 986 nd Vivs 999, pp

13 h ction functions fo lndown nd spctivly cn otind fom th fist-od conditions y totlly diffntiting thm with spct to th ottion g of djcnt stnds, d d N d,. 6 N d N N Lmm chctizs how th coss divtivs of th ction functions dpnd on poptis of th mnity vlution functions. N N Lmm. if nd if Poof. S Appndix.. Accoding to Lmm nd qution 6, th ction functions of lndowns nd hv diffnt slops dpnding on tmpol stnd intdpndnc Dfinition. h th css, which w summiz in: Rsult. Poptis of th ction functions Und tmpolly indpndnt stnds, th ction functions vticl lins in, spc nd th quiliium is stl. Und incsing tmpol dpndnc, th ction functions incsing in, spc. Stility of th quiliium quis tht th ction function fo stnd is stp thn th ction function fo stnd. c Und dcsing tmpol dpndnc, th ction functions dcsing in, spc. Stility of th quiliium quis tht th ction function fo stnd is stp thn th ction function fo stnd. Rsult is illusttd in igus -4 fo oth dcsing nd incsing tmpol dpndnc. Dwn in th figus th ction functions fo th lndown of stnd nd, nd, ch of which is function of th ottion g choic of th oth lndown. In igus th ction functions downwd-sloping, flcting dcsing tmpol dpndnc twn th stnds. h upwd sloping ction

14 3 functions in igus 3 nd 4 flct incsing tmpol dpndnc twn th stnds. As w xplin lt, igus nd 3 dwn ssuming th stnds sptil sustituts, whil igus nd 4 dwn ssuming th stnds sptil complmnts. h uniqu nd stl Nsh quiliium solution stisfying 4 nd 4 occus t th point wh th ction functions fo th lndowns coss..3. Rottion gs in th Stcklg gm In th Stcklg gm, th ld movs with knowldg of how th following lndown sponds; th follow tks th ottion g of th ld s givn. Whil this modl my mimic som pivt mkt situtions, th ld might lso intptd s govnmnt fomlly stting long un hvst policy, ffctivly lding. Assum th ld is th lndown holding stnd. his lndown mximizs th following ojctiv function, Mx { } S V E 7 s. t. S S, q,, c, 7 wh th supscipt S fs to th Stcklg gm, nd S S, q,, c dscis th ction function of th follow who holds stnd. Utilizing th Nsh fist-od condition 4 fo th follow s ction function, th ld nd th follow fist-od conditions, spctivly, S N s. s, ds, 8 S qg, qg V E. 8 W ssum tht th scond-od condition holds fo this polm. 9 In lity, whth it holds dpnds gin on th mnity vlution function nd on th poptis of th follow s ction function S S, q,, c. 9 hus w ssum tht S N pf pf, s s s, ds s, ds,.

15 4 Consid fist th follow s hvio givn in 8. his condition is qulittivly th sm s th ncssy condition fo lndown in th Nsh gm; tht is, givn S, th follow chooss th ottion g. his is not so fo th ld qn 8. Compd to th Nsh gm, th is n dditionl tm in th Stcklg fist od condition, flcting th impct of th follow s ottion choic on th ld s mginl mnity nfits. h psnc of this dditionl tm implis tht th ld ptly ccounts fo th xtnlity tht iss fom th ffct of th follow s ottion g on th mnitis of th ld s stnd. his diffnc twn Stcklg nd Nsh outcoms will com impotnt lt whn w study comptiv sttics ffcts. Whth th ld hs long o shot ottion g compd to th Nsh quiliium ottion g dpnds on th lst tm in 8, i.., on th slop of th follow s ction cuv nd th intgl tm. o sign this intgl tm nd mk th nlysis tctl, w ssum tht th mnity function hs qudtic shp, nd w lso us scond od ppoximtion fo /. h qudtic mnity function, hs th following poptis:, nd, s, so tht it xhiits ll lvnt css dscid in Dfinitions nd. W now hv, Lmm. Und th qudtic mnity vlution function,,, w hv s s, ds s,. Poof. S Appndix 3. Accoding to Lmm th sign of th intgl tm dpnds on how th ottion g of th follow s stnd ffcts th mginl mnity vlution of th ld s stnd t th mgin, whn. Rtuning to igus -4, consid now th iso-nt-psnt-vluof-vnu cuvs. h iso-nt-psnt-vlu-of-vnu cuvs lins long which th nt psnt vlu is constnt fo givn intst ts, tim pics nd gntion costs. A fmily of ths thfo xist fo ch st of constnt pmts. W cn show tht high od ppoximtions fo th discount fcto will not chng th ntu of th sults in Lmm low.

16 5 Whn th stnds sptil sustituts, th iso-nt-psnt-vlu-of-vnu cuvs dcsing in th ottion g of th oth stnd, whil sptil complmnts imply th iso-nt-psnt-vlu-of-vnu cuvs incsing in th ottion g of th oth stnd. his mns tht, fo complmnts sustituts th nt psnt vlu of pofits fo th fost lndown is incsing whn moving up down th ction functions. In th figus, th Stcklg ottion g fo ld nd follow cosponding to 8 8 ov is dfind y th point wh th ld s highst iso-nt-psnt-vlu-of-vnu cuv is tngnt to th follow s ction function,. Using igus -4, Lmm, Lmm, nd Rsult, w cn now xmin th ltionship twn th Nsh nd th Stcklg ottion gs. Consid fist th ld. If th stnds tmpolly indpndnt, thn. nd th ction functions vticl lins. H, th sign of, dos not mtt, thfo, th Stcklg ottion g coincids with th Nsh ottion g. Und dcsing incsing tmpol dpndnc nd sptil sustitutility twn th stnds, w hv N N, cus now th lst tm in 8 is positiv ngtiv. In this cs, th ld s ottion g, S, is long shot thn th Nsh ottion g, N. If th stnds sptil complmnts, thn und dcsing incsing tmpol dpndnc w hv N N ; now th ld s ottion g is shot long thn th Nsh ottion g. o sptil sustituts complmnts, th follow s ottion g is shot long thn th Nsh ottion g und dcsing tmpol dpndnc. h intpttion is s follows. h follow osvs th ottion g of th ld pio to moving. Whn th stnds sustituts, th follow s ottion g must shot cus, und dcsing tmpol dpndnc, th ld s ottion g will long thn th Nsh ottion g, nd th follow s ction cuv will downwdsloping igu. Intuitivly, th long ottion g of th ld llows th follow to hvst soon ut still div fogon mnity nfits fom th ld s stnd. o sptil complmnts, th ld s ottion g is shot thn th Nsh g, nd th st spons of th follow is to lngthn th ottion g ltiv to Nsh g, cus tht dcss stnd complmntity. Simil soning cn pplid to incsing tmpol dpndnc. W cn summiz th ov discussion in:

17 6 Poposition. h ltionship twn Nsh nd Stcklg ottion gs dpnds on th ntu of stnd intdpndnc: N S N S Und tmpol indpndnc, nd. Und dcsing tmpol dpndnc, N S nd N S fo sptil complmnts, whil N S nd N S fo sptil sustituts. c Und incsing tmpol dpndnc, N S nd N S fo sptil complmnts, whil N S nd N S fo sptil sustituts..4. Rottion gs fo th sol own h sol own chooss ottion gs of oth stnds to mximiz, Mx W, V V E E. 9 h fist-od conditions chctizing sol own ottion g choics cn xpssd using th following modifiction of th Nsh conditions 4 nd 4, x, x dx N W s s, ds N W h scond-od conditionsw, W, nd SO psntd in Appndix 4 nd ssumd to hold. Equtions nd imply tht th sol own chooss ottion gs fo oth stnds tking into ccount how mnitis ffctd y ottion g of oth stnd s in th Stcklg gm nd now lso stnd s th lst tm in nd. Hnc, ll potntil xtnlitis ising fom th ffcts of hvsting on stnd on th oth stnd s mnitis intnlizd. h sol own outcom is thfo th fficint solution fo ou polm.

18 7 h lst tms in qutions nd dtmin how th sol-own ottion g of oth stnds comps to Nsh nd Stcklg ottion gs. om Lmm w know tht ths lst tms positiv whn th stnds sptil complmnts nd ngtiv whn th stnds sptil sustituts. hus, ltiv to th Nsh ottion g, th sol own chooss long ottion gs fo oth stnds whn thy sptil complmnts, ut shot ottion gs whn thy sptil sustituts. How dos th sol own ottion g comp to th Stcklg ottion g? h sol own fist od conditions diff fom th Stcklg conditions y th lst tm comp with 8. Du to th symmtic sol own fist-od conditions nd, w cn gphiclly distinguish th sol own optimum in igus -4 s points wh th iso-nt-psnt-vlu-of-vnu-cuvs fom oth stnds tngnt to ch oth. Rfing to th figus, fo sptil complmnts sustituts th sol own ottion g is long shot thn th Stcklg g fo oth ld nd follow. h impotnt diving fcto in th compison of th sol own s ottion g with Nsh nd Stcklg ottion gs is th sptil complmntity o sustitutility of stnds. h sol own intnlizs ll xtnlitis ssocitd with mnitis. Whn stnds tmpolly indpndnt, it is ntul tht th sol own ottion g coincids with th oth ottion gs cus th is no xtnl ffct of hvsting on stnd on th oth stnd. Howv this is not th cs whn stnds tmpolly intdpndnt. Now th compison of ottion gs dpnds on how th two stnds ltd sptilly; if thy sptil complmnts, th sol own incss ottion gs, whil th opposit is tu und sptil sustituts. W cn xpss th ltionship twn Nsh, Stcklg nd sol own ottion gs s follows Poposition. Und tmpol indpndnc, th sptil complmntity o sustitutility twn th stnds dos not mtt nd Nsh nd Stcklg ottion gs coincid with th sol own ottion gs. Und tmpol dpndnc, th sol own ottion g is long shot thn Nsh nd Stcklg solutions whn stnds sptil complmnts sustituts.

19 8 h diffncs in ottion gs und ou vious solutions will undoutdly ld to diffncs in wlf fo fost lndowns. Oviously, lndowns y dfinition tt off t th sol own optimum ltiv to th oth outcoms. An intsting compison of wlf und th oth outcoms cn otind fom igus -4, y noting th position of th quilii on th iso-nt-psnt-vlu-of-vnu cuvs. igus 3-4 show tht whn tmpol dpndnc is incsing, th wlf of oth lndowns is high und th Stcklg solution thn und th Nsh quiliium. his sult occus cus in th Stcklg gm, th ld ptilly ccounts fo th ffcts of his ottion g dcision on th follow lndown, nd this incss th wlf of oth ld nd follow. In th cs of dcsing tmpol dpndnc, igus - show tht th wlf compison twn Nsh nd Stcklg solutions is miguous. hus, w hv n dditionl Coolly: Coolly. Und incsing tmpol indpndnc, oth lndowns tt off whn ottion gs chosn ccoding to th Stcklg gm, ltiv to th cs whn ottion gs solvd und th Nsh quiliium. 3. Comptiv Sttic Anlysis W hv shown tht th Nsh, Stcklg, nd sol own ottion gs diff fom ch oth in th psnc of mnity nfits whn th is stnd intdpndnc. Now w study th qulittiv poptis of ths ottion gs. An impotnt point to liz is tht mkt pmts fo oth stnds cn diff givn tht sit chctistics inhnt to oth Wht hppns if only on of th two owns vlus mnitis? Suppos only lndown vlus mnity svics. In th Nsh quiliium, lndown s ction function would thn vticl lin in, spc. If lndown is th Stcklg follow, thn th intgl tm in 8 would qul zo cus of this ction function. inlly, in th sol own cs stnd is mngd lik in th convntionl Htmn modl. Howv, stnd is not mngd lik in th ustmnn modl, cus th ffct of its ottion g on mnitis povidd y stnd tkn into ccount. Hmilton nd Slutsky 99 hv shown tht, in modls of ndognous timing gding th dcisions of fims, th incntivs fo fims to mov squntilly if th is on Stcklg quilii tht Pto domints ll oth simultnous mov Nsh quilii. With two fims, th ncssy condition fo th Stcklg to otin is tht th ld s pofits high whn moving fist, compd to its pofits in th Nsh gm, othwis no fim would choos to mov fist nd oth would, ffctivly, ply Nsh gm. A sufficint condition fo this is tht oth ld nd follow pofits in th squntil mov gm t lst s high s thi Nsh pofits.

20 9 stnds could diff, nd thi ottion gs not gnlly quivlnt, s w showd ov. h sults of this sction condnsd in l. 3. Nsh gm o th ffct of tim pic p of stnd on oth ottion gs w otin, though totl diffntition of th fist od conditions, N N N N p p s A, N p N N p N, N wh f f f, p A c, E nd, E s, s Koskl-Ollikinn. Givn tht th psnt vlu of gntion costs is lwys positiv, w hv th convntionl ffct of shot ottion g du to n incs in th own-stnd hvsting pic whn, i.., th mginl mnity vlution incss o mins constnt with th g of stnds. Eqution vls tht th ottion g of stnd my lso ffctd y chng in stnd s tim pic if th stnds not tmpolly indpndnt, i.., if N. Assuming tht th mginl mnity vlution dos not dcs with th g of th own stnd, th oth lndown will shotn lngthn his ottion g s sult of is in p whn tmpol dpndnc twn stnds is incsing dcsing. Not tht th signs of N N q nd q symmtic givn nd. o ssss th impcts of chng in gntion costs of stnd, c, on ottion gs, w otin

21 N N N 3 c N N N c s N. 3 hus, th ffct of n incs in th own-stnd gntion cost is qulittivly th sm s in th ustmnn nd Htmn modl-sd littu. Howv, with spct to th djcnt stnd, th Nsh solution ings nw sult, fom 3. h ction of th djcnt stnd s lndown to chng in th gntion costs of th oth lndown dpnds on th tmpol dpndnc twn stnds. Mo spcificlly, if th dpndnc twn th stnds incss dcss with long ottion g fo th own stnd, thn th own of th djcnt stnd lngthns shotns his ottion g. Agin, th signs of N c N nd c symmtic, givn 3 nd 3. inlly, fo chng in th l intst t w otin, N N N N N N N N N N N N 4, 4 wh N nd N. Und incsing tmpol dpndnc oth ottion gs will unmiguously shotn fom 4 nd 4. Howv, und dcsing tmpol dpndnc th ffct is pioi miguous. Ntully, sufficint condition fo shot ottion g h is tht th own-stnd dict ffct of th intst t domints ll oth ffcts. Summizing w hv Rsult. In symmtic Nsh quiliium with intdpndnt stnds, A high own-stnd pic shotns th ottion g und incsing tmpol dpndnc, ut my incs ottion g und stong dcsing tmpol dpndnc. h ffct of own-stnd gntion cost on ottion g is positiv. h ffcts of high djcnt-stnd tim pic nd gntion costs on th ownstnd ottion g dpnd on th ntu of tmpol dpndnc twn th stnds. c A high intst t dcss ottion gs und incsing tmpol dpndnc, ut is pioi miguous und dcsing tmpol dpndnc

22 3. Stcklg gm In th Stcklg gm w hv only to solv th comptiv sttics fo th ld s ottion g. 3 A chng in th tim pic fo stnd impcts th ld s ottion g s follows S p S p S s A s. s, ds, 5 S wh f f f nd p A c, E. Rcll fom Rsult tht th sign of th ction function. dpnds on tmpol dpndnc twn th stnds, nd tht fom Lmm th sign of th intgl tm dpnds on whth th stnds sptil complmnts o sustituts. W cn fist s tht if th stnds tmpolly indpndnt, thn th lst tm in 5 is zo. hus, th pic ffct on th Stcklg ottion g is idnticl to th pic ffct on ottion g in th singl stnd Htmn modl. Scond, comintion of incsing mginl vlution, sptil complmnts nd incsing tmpol dpndnc, o sptil sustituts with dcsing tmpol dpndnc, implis unmiguously th convntionl ffct of shot ottion g. Givn tht th Stcklg gm is not symmtic lik th Nsh gm, w nxt solv fo th ffct of th pic of stnd on th ld s ottion g. Diffntiting th fist-od condition 8 with spct to this pic q nd noting tht ll ffcts mg fom th follow s spons function, w otin, S q. 6 S q S W show in Appndix 5 tht, whil this pic ffct is pioi miguous fo oth dcsing nd incsing tmpol dpndnc, it is zo fo tmpolly indpndnt 3 Comptiv sttics fo th follow is idnticl to th cs of n xognous djcnt stnd; this

23 stnds. h miguity und non-constnt tmpol dpndnc sults fom th fct tht th dict nd indict ffcts of tim pic q offstting vi shift in th follow s ction function nd its slop. h ffct of high gntion cost of th ld s stnd on th ld s ottion g is givn y, S c. 7 S, 7 S c Hnc, th ld s ottion g unmiguously lngthns. h intpttion of this finding is th sm s fo. If th gntion cost of th follow s stnd incss, it ffcts th ld s fist-od condition only vi th ction function of th follow. Givn tht., th is no ffct on th ottion g of th ld s stnd. c inlly, th ffct of n intst t chng on th ld s ottion g cn xpssd s, wh S s S, 8 s, S S pf V s ds E s, s s, s ds s ds s, S Accoding to this xpssion fo, th intst t ffct iss though th chnnls: dictly i though th pofitility of th ld s stnd, nd indictly oth ii vi th slop of th ction function, nd iii though th position of th follow s ction function. In Appndix 5 w show vi Lmm 3 tht ths ffcts count ch oth, so tht th ovll impct of th intst t is miguous. W cn summiz ou findings in: s ds Rsult 3. In Stcklg quiliium with intdpndnt stnds, cs hs n nlyzd in Koskl-Ollikinn.

24 3 Und incsing mginl mnity vlution, n incs in th ld s own-stnd pic will shotn th ld s ottion g whn th stnds sptil complmnts with incsing tmpol dpndnc, o sptil sustituts with dcsing tmpol dpndnc. A high own-stnd gntion cost incss th ld s ottion g unmiguously. An incs in th follow s own-stnd pic hs n pioi miguous ffct on ottion g whn stnds not tmpolly indpndnt, whil high own-stnd gntion costs fo th follow hv no ffct on th lds ottion g. c A high intst t hs n pioi miguous ffct on th ld s ottion g Sol own solution inlly, w consid comptiv sttics in th sol own cs. o th ffct of high tim pic of stnd on oth ottion gs w hv, SO p SO p pw SO W 9 SO p W W, 9 wh W p f f f, nd s s, ds N W,. By using th fist-od conditions nd, w cn show tht: sgnw = - sgn x A, x dx, wh A c, E. h sufficint conditions fo W p <, nd th impct of own-stnd pic on th ottion g to ngtiv, tht mginl mnity vlutions non-dcsing with th ottion g of th stnd. nd th stnds sptil complmnts. s Lmm. Und th sufficint condition fo W p <, th ffct of th djcnt stnd pic on own- p

25 4 stnd ottion g is ngtiv und incsing tmpol dpndnc nd positiv und dcsing tmpol dpndnc; cus ccoding to Lmm, s th tmpol dpndnc twn stnds incss dcss. hus, th pic ffcts h sml th clssic cs of complmnts nd sustituts, xcpt in ou modl complmntity is spcifid in tmpol sns. h ffct of chng in th own-stnd gntion costs of stnd on th ottion g is givn y, W SO c SO W SO SO c W s W. A high own-stnd gntion cost lngthns own ottion g, whil its ffct on th oth stnd dpnds gin on th tmpol dpndnc twn stnds. h ottion g of th oth stnd lngthns und incsing tmpol dpndnc nd shotns und dcsing tmpol dpndnc. inlly, fo th ffcts of n incs in th l intst t w hv, wh SO SO SO SO W W W W W W W W, W W N N x, x dx x, x ds s s s s,, s x ds dx h fist tms in W nd W ngtiv. In Appndix 6 w show tht, und plusil ssumptions concning th intst t nd ottion gs, th cd tms ngtiv if th stnds sptil complmnts. hus, und incsing tmpol dpndnc nd sptil complmntity of stnds, oth ottion gs will unmiguously shotn; othwis th ffcts pioi miguous. W thfo hv,

26 5 Rsult 4. In th cs of sol own with intdpndnt stnds, A high own-stnd pic shotns ottion g if stnds ith sptil complmnts, nd tmpol dpndnc is ith unchngd o incsing. A high own-stnd gntion cost unmiguously lngthns th ottion g Und ctin sufficint condition, th ffct of th djcnt stnd pic on ownstnd ottion g dpnds on th ntu of tmpol dpndnc. h ffct of th djcnt stnd s gntion cost will incs dcs th own-stnd ottion g und incsing dcsing tmpol dpndnc. c If stnds sptil complmnts nd tmpol intdpndnc is incsing, thn high intst t will shotn th ottion gs Discussion nd Policy Implictions Sustining fost cosystms quis tht stnds mngd in conct th thn in isoltion. Most fost conomics modls, howv, consid only singl isoltd stnd o singl lndown. his is not ncssily th cs fo pivt lnd ownship, wh individul popty ights mk pop coodintion of mngmnt dcisions coss lg nums of lndowns vy difficult. h lck of coodintion mong lndowns cn dtimntl to mnitis tht dpnd on th cosystm s whol, such s thos divd fom ctionl xpincs o th xistnc of ctin wildlif spcis. Und th ssumption tht ottion gs idnticl in th stdy stt, w hv xmind th possiility tht stnds cn tmpolly o sptilly intdpndnt in diffnt wys gding th poduction of mnitis y llowing fo svl css of lndown dcision timing nd commitmnt, including lndowns mking simultnous dcisions, lndowns coodinting thi ctions, o on lndown cting s fist mov. Ou sults xtnd th sic Htmnn modl of fost mngmnt tht fist intoducd 4 Suppos tht in th cs of sol own solution oth stnd nd stnd hv th sm tim pic o gntion cost chng. hn ssuming tht W p high tim pic will dcs oth ottion gs whn mginl mnity vlutions non-dcsing with th ottion g of stnds nd tmpol dpndnc twn stnds is non-dcsing. h ffct of chngs in th unifom gntion cost is positiv miguous whn tmpol dpndnc twn stnds is incsing dcsing. Poofs vill upon qust.

27 6 mnitis fo th cs of n isoltd stnd, s wll s oth modls of multipl stnds sd on numicl simultions o th ssumption of only sol ownship. W dmonstt tht ottion g dcisions dpnd on how djcnt stnds sptilly nd tmpolly dpndnt with gds to mnity vlution, nd on th stuctu of lndown dcision timing. Sol ownship psnts th socil optimum in ou modl. Comping this with th Nsh nd Stcklg outcoms givs qulittiv indiction of th socil costs ssocitd with lndowns who do not sk to jointly mximiz totl vnu nd mnity nts fom owning fosts. h collctiv sults fom ou pp summizd in Popositions,, Coolly, nd l. h sol own s ottion g is long thn th Nsh nd Stcklg ottion gs if th stnds sptil complmnts, ut shot if thy sptil sustituts with gd to mginl mnity vlution. Additionlly, th ltionship twn th Nsh nd Stcklg ottion gs lso dpnds on how this sustitutility nd complmntity volvs ov tim. h diffncs twn ths solutions lgly flct th ility o inility of lndowns to ith nfit fom noth lndown s dcisions. Intstingly, w show tht und incsing tmpol intdpndnc, it cn th cs tht on lndown moving fist my mk oth lndowns tt off ltiv to th cs wh thy do not coodint t ll nd ply Nsh gm. As sis fo policy nlysis, w lso chctiz in ths nw modls how ottion gs dpnd on tim pics, intst ts, nd gntion costs. W find tht th ffcts of ths pmts, divd in singl stnd modls, dos not usully hold. Instd, th sults dpnd on th sptil nd tmpol dpndnc twn th stnds, th ility of lndowns to commit to hvsting, nd whth th pmts chnging fo th stnd in qustion o th djcnt stnd. By nd lg, most of th diffncs twn ou modls nd xisting modls occu fo two sons. ist, th possiility of incsing o dcsing tmpol dpndnc oftn dtmins th signs of comptiv sttics sults, fo instnc, th intst t ffct on ottion g fo instnc is not ncssily ngtiv lik in xisting modls with on fost stnd s l. Scond, stnd intdpndnc implis tht pmts fom on stnd cn ffct th choics md in th oth stnd, s xmplifid y ou pic nd gntion cost ffcts. Ou nw sults nd ppoch to studying lndown hvio suggsts tht xisting modls of policy dsign should visd. W dmonstt tht oth Nsh nd Stcklg quiliium ottion gs long thn sol own ottion gs whn stnds sptil sustituts, ut Nsh nd Stcklg gs shot thn sol own gs

28 7 whn stnds sptil complmnts. Clly th scop fo using txs o susidis to djust ottion gs towd thi fficint lvls dpnds on th ntu of stnd intdpndnc gding mnitis, nd lso on th ility of lndowns to commit to ottion g ctions. hs ids hv not n pviously uncovd, vn within th sptil fosty littu. In th nd, th dsign of pop Pigouvin tx systm which mimics th fficincy of th sol own solution will much mo complictd thn pviously thought. It would usful to xtnd th optiml fost txtion nlysis in th Htmn fmwok y Koskl nd Ollikinn 3 fo llow fo stnd intdpndncis. Wht w cn ln fom mpiicl wok gding this intdpndnc will lso cucil to pcticl policy wok. Ou wok h suggsts mny oth sch topics, most of which motivtd y th nd to continu gnlizing th modlling ppoch. As pointd out in th txt nd illusttd in Appndix, it would usful to undstnd intdpndncy whn djcnt stnd ottion gs diff. H, th mnity vlu of ch stnd might dpnd on th clnd g of th djcnt stnd, s wll s th dynmics of th stdy-stt involving th two stnds. Poviding n nlyticl solution fo this polm would dmnding nd intsting topic fo futu sch, ut if solvd it would povid n nlysis of tnsitionl dynmics nd idntifiction of possil outcoms tht might is s ottion gs djust twn vious quilii. o this son, th ppliction of ou modl to unvn-gd mngd fosts with djcnt lndowns would intsting nd poly ld to diffnt sults. In modls wh ottion gs of stnds diffnt, w xpct tht th symmtis w idntify twn lndown intctions will continu to xist nd com vn mo complictd. inlly, it is cl fom ou wok tht gins fom cooption mong lndowns could highly lvnt to th ppliction of ny fost policy, spcilly if th policy tks stnd intdpndncy s its stting point. Anlyzing th policy polm thoticlly, nd quntifying nd comping th gins fom diffnt policy instumnts in th diffnt quilii w find h would impotnt in ssssing th ffctivnss nd scop fo policy.

29 8 igu. Dcsing tmpol dpndnc whn th stnds sptil sustituts N S SO SO N S igu. Dcsing tmpol dpndnc whn th stnds sptil complmnts SO S N S N SO

30 9 igu 3. Incsing tmpol dpndnc whn th stnds sptil sustituts N S SO SO S N igu 4. Incsing tmpol dpndnc whn th stnds sptil complmnts SO S N N S SO

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34 33 ABLE. Comptiv sttics of th ottion g in Nsh, Stcklg, nd sol own css. Solutions fo stnd.* + indicts n incs in ottion g, whil - indicts dcs in ottion g. Mginl mnity function signs cospond to Dfinitions nd in th txt. Exognous Pmt Nsh Rottion Ag N Own stnd pic - if +/- othwis Stcklg Rottion Ag Ld S - if,, o if,, Sol Own Rottion Ag SO - If ; +/- othwis Own-stnd gntion cost Adjcnt stnd pic s if - if +/- + if ; - If ; ; ; Intst t Adjcnt stnd gntion cost - if +/- othwis + if - if +/- - if, +/- othwis + if - if *By symmty, th comptiv sttics sults in th tl would lso hold fo stnd, with th pmts dfind fo this stnd.

35 34 Appndix. Stnd intdpndnc, psnt vlu of mnity nfits nd diffnt ottion gs fo stnds. Consid th mnity function fo stnd in th cs of th ottions whn th ottion g of stnd is. Assum tht th fist ottion fo ch stnd gins with lnd. h mnity function cn wittn s follows, E i i t, t i t dt i i t, t i t dt 3 i i t, t i t dt A. wh i initilly nd t th nd of st ottion of stnd, i t th nd nd ottion of stnd, nd i t th nd 3 d ottion of stnd. W cn dsci th mnity function fo stnd in n nlogous wy. Diffntiting qution A.. with spct to ottion g yilds, E,,,, A.,3, On cn s fom qution A. tht, in th psnc of stnd intdpndnc, th dynmics of oth stnds should includd in th mnity vlu function fo ch stnd if ottion gs diffnt fom ch oth. But, if ottion gs qul, on nds up with th spcifiction psntd in th txt this cn sn fom A.. In od to xmin th mnity function with diffnt ottion gs futh, ssum tht nd 3. Now, A. nd A. cn xpssd spctivly s E nd E i i t, t i t dt 4 i i t, t i3 t dt,,,, 4.,, 6 i 4 i t 4, t i3 t A.3 A.4 dt

36 35 Appndix. Poof of Lmm N is nlogous. h coss- h poof is givn only fo divtiv N N cn -xpssd s, cus th poof fo N s, s, ds A.. s s, ds d mpol Indpndnc: d Poof. If,,,, N ducs to. h two possiilitis. If, thn tivilly. Und, implis N,, so tht d,,. Hnc,. d d Incsing mpol Dpndnc: d Poof. i If thn N, s s ds,. Now s s,, ds s, ds s, s ds, s, s ds. Hnc, N d so tht. d ii If thn N, s, s ds s s,, ds s, ds. Now s, s ds, s, s ds. Hnc, N d so tht. d

37 36 d Dcsing mpol Dpndnc: d Poof. i If thn N, s s ds,. Now s s,, ds s, ds s, s ds, s, s ds ii If thn N. Hnc, N d so tht. d, s, s ds. Now s s,, ds s, ds s, s ds, s, s ds. Hnc, N d so tht. Q.E.D. d Appndix 3. Poof of Lmm Intgting th tm s s, ds in 8 y pts nd ssuming tht th thid divtiv of th mnity function is zo, i.., w gt,,,, A3. Und th qudtic mnity vlution function,, A3. with,, w hv,, nd,. Moov, w dfin stnds complmnts, s A3.3 stnds sustituts Hnc, w cn xpss A3. s

38 37 A3.4 Using scond-od ppoximtion yilds / Now,, A3.5 o complmnts, nd und incsing tmpol dpndnc,, whil dcsing tmpol dpndnc implis tht,. hus A3.5 is positiv in oth css: th fist on utomticlly, th scond on, cus. o sustituts condition A3.5 is utomticlly ngtiv fo dcsing tmpol dpndnc; nd ngtiv lso und incsing tmpol dpndnc, cus. QED. Appndix 4: Scond od conditions fo th sol own h scond od conditions ly on th following divtivs, fom nd W N N, x x dx W N N s, s ds SO W W W W. Appndix 5. Comptiv sttics of djcnt stnd s tim pic nd intst t in Stcklg modl Pic q: By diffntition fo th Stcklg ld w otin, S q q s s., s, ds. s, ds A5.

39 38 s. s, ds, q wh th tm q is dfind s: q S x dx g g s. A5. S hus q consists of shift in th follow s ction function s wll s chng in its slop in tms of. h sign of ll tms dpnds on th ntu of tmpol dpndnc twn th stnds. On th sis of ou pvious nlysis th sum of th fist two tms of qn A5. in ckts ngtiv positiv fo incsing dcsing tmpol dpndnc. h thid nd fouth tms of opposit sign, ing positiv ngtiv S fo incsing dcsing tmpol dpndnc, indicting tht th sign of q is pioi miguous fo oth dcsing nd incsing tmpol dpndnc. o tmpolly indpndnt stnds w hv S q. Intst t : W fist dvlop th coss-divtiv of th follow s ction function,.. Rclling qution 8, th divtiv with spct to is, x x x dx x dx dx, A5.3. wh qg qg nd. h sign of th fist two tms in cs dpnds on th sign of y Lmm. Appoximting th tm using implis tht / 3. x x dx /. W dtmin th sign of th intgl tm y using th qudtic mnity vlution function givn in A.3. in Lmm Lmm 3. If th mnity vlution function is qudtic, thn x x dx s,.