BANK OF FINLAND DISCUSSION PAPERS 7 2004 Berthold Herrendorf Arilton Teixeira Research Department 24.2.2004 Monopoly rights can reduce income big time Suomen Pankin keskustelualoitteita Finlands Banks diskussionsunderlag
Suomen Pankki Bank of Finland P.O.Box 160 FIN-00101 HELSINKI Finland + 358 9 1831 http://www.bof.fi
BANK OF FINLAND DISCUSSION PAPERS 7 2004 Berthold Herrendorf* Arilton Teixeira** Research Department 24.2.2004 Monopoly rights can reduce income big time The views expressed are those of the authors and do not necessarily reflect the views of the Bank of Finland. We are grateful to Edward Prescott and James Schmitz for their advice and encouragement, to Peter Klenow for helping us with the price data, and to Ákos Valentinyi for pointing out several errors in an earlier version. We have profited from suggestions by Michele Boldrin, Ricardo Cavalcanti, Marco Celentani, Juan Carlos Conesa, Antonia Díaz, Ron Edwards, Fernando Garcia-Belenguer, Thomas Holmes, José Victor Ríos-Rull, Juan Ruiz, Michele Tertilt, and the audiences of presentations at Arizona State University, Bank of Finland, Carlos III, Edinburgh, the European Forum in Maastricht, Foundation Gestulio Vargas (Rio de Janeiro), Ibmec, the Madrid Macro Seminar (MadMac), Oslo, the SED Meetings in Paris, Southampton, the Thanksgiving Conference in Essex, UCLA, the University of Barcelona, the Vigo-Workshop in Dynamic Macroeconomics, and Warwick. Herrendorf acknowledges research funding from the Spanish Dirección General de Investigación (Grant BEC2000-0170) and from the Instituto Flores Lemus (Universidad Carlos III). * Universidad Carlos III de Madrid, Departamento de Economía, Calle Madrid, 126, 28903 Getafe (Madrid), Spain; University of Southampton; CEPR. E-mail: herrendo@eco.uc3m.es. ** Ibmec, Department of Economics, Av. Rio Branco, 108/12o. andar, Rio de Janeiro, RJ, 20040-001, Brazil. E-mail: arilton@ibmecrj.br. Suomen Pankin keskustelualoitteita Finlands Banks diskussionsunderlag
http://www.bof.fi ISBN 952-462-126-6 ISSN 0785-3572 (print) ISBN 952-462-127-4 ISSN 1456-6184 (online) Multiprint Oy Helsinki 2004
Monopoly rights can reduce income big time Bank of Finland Discussion Papers 7/2004 Berthold Herrendorf Arilton Teixeira Research Department Abstract We ask which part of the observed cross-country differences in the level of per capita income can be accounted for by monopoly rights in the labour market. We answer this question in a calibrated growth model with two final goods sectors. The novel feature being that monopoly rights in the capital-producing sector shield insiders from competition by outsiders and permit coalitions of insiders to choose inefficient technologies or working practices. We find that monopoly rights can lead to quantitatively much larger reductions in the level of per capita income than previously demonstrated. This comes about because they do not only reduce TFP in capital-producing sector but also increase the relative price of capital. This reduces the capital-labour ratio in the whole economy. The implied predictions about the price of capital goods relative to consumption goods and the investment share in output are quantitatively consistent with the cross-country facts. Key words: cross-country income differences, cross-country productivity differences, monopoly rights, relative price of capital, capital accumulation. JEL classification numbers: EO0, EO4 3
Monopolivoima voi vähentää tuloja rajusti Suomen Pankin keskustelualoitteita 7/2004 Berthold Herrendorf Arilton Teixeira Tutkimusosasto Tiivistelmä Tässä työssä tarkastellaan kysymystä, missä määrin henkeä kohti lasketun tulotason erojen eri maiden välillä voidaan selittää johtuvan työmarkkinoiden monopolisoitumisesta. Kysymykseen vastataan kalibroidulla kasvumallilla, jossa talous on jaettu kahteen lopputuotteita tuottavaan sektoriin. Uusi piirre mallissa on, että monopolivoima pääomatavaroita tuottavalla sektorilla suojaa sisäpiiriä ulkopuoliselta kilpailulta ja antaa sille mahdollisuuden käyttää tehotonta tekniikkaa tai tehottomia työtapoja. Tulokset osoittavat, että monopolivoima työmarkkinoilla voi johtaa paljon suurempaan tulotason laskuun kuin aiemmin on pystytty osoittamaan. Tulos johtuu siitä, että monopolivoima ei ainoastaan heikennä tuottavuutta pääomatavaroita tuottavalla sektorilla vaan myös nostaa pääoman suhteellista hintaa. Pääoman kallistuminen supistaa pääoma-työsuhdetta koko taloudessa. Kansainvälisessä aineistossa pääoma- ja kulutustavaroiden hintasuhde ja talouden investointiaste käyttäytyvät mallin ennustamalla tavalla. Avainsanat: kansainväliset tuloerot, kansainväliset tuottavuuserot, monopolivoima, pääoman suhteellinen hinta, pääomanmuodostus JEL-luokittelu: EO0, EO4 4
Contents Abstract...3 Tiivistelmä...4 1 Introduction...7 2 Model economy...9 2.1 Environment...9 2.2 Equilibrium...12 3 Analytical results...15 4 Quantitative results...18 5 Discussion...23 6 Conclusions...26 References...28 Appendix A Derivations and proofs of the propositions...31 5
6
4 Lqwurgxfwlrq Wkhuh duh odujh furvv0frxqwu glhuhqfhv lq wkh ohyhov ri shu0fdslwd lqfrph1 Iru h{dpsoh/ wkh Shqq Zruog Wdeoh ri 4<<9 +SZW<9 khqfhiruwk, uhsruw wkdw wkh dyhudjh shu0fdslwd lqfrph ri wkh ulfkhvw whq shufhqwloh ri frxqwulhv lv derxw wkluw wlphv wkdw ri wkh srruhvw whq shufhqwloh1 4 Zh dvn zklfk sduw ri wkh revhuyhg furvv0frxqwu glhuhqfhv lq lqfrph ohyhov fdq eh dffrxqwhg iru e prqrsro uljkwv lq wkh oderu pdunhw1 Zh ghqh prqrsro uljkwv dv iroorzv= fhuwdlq w shv ri zrunhuv/ zklfk zh fdoo lqvlghuv/ kdyh d prqrsro lq vxsso 0 lqj oderu lq fhuwdlq vhfwruv> frdolwlrqv ri wkhvh lqvlghuv dovr kdyh wkh uljkw wr eorfn h!flhqw whfkqrorjlhv dqg ehvw0sudfwlfh zrunlqj duudqjhphqwv/ zklfk duh dydlodeoh rqfh wkh duh ghyhorshg hovhzkhuh1 Uhdo0zruog h{dpsohv ri frdolwlrqv ri lqvlghuv duh eurwkhukrrgv/ jxlogv/ sur0 ihvvlrqdo dvvrfldwlrqv/ dqg wudgh xqlrqv dqg wkhlu eorfnlqj dfwlylwlhv duh grfx0 phqwhg e pdq fdvh vwxglhv/ lqfoxglqj Fodun +4<;:,/ Prn u +4<<3,/ Zrofrww +4<<7,/ PfNlqvh 0Joredo0Lqvwlwxwh +4<<<,/ Krophv dqg Vfkplw} +5334d/ e,/ dqg Vfkplw} +5334e,1 Wkh sdshu e Zrofrww +4<<7, surylghv dq lqvwuxfwlyh h{dpsoh ri eorfnlqj e frdolwlrqv ri lqvlghuv1 Vkh frpsduhv wkh Mdsdqhvh dqg wkh Lqgldq Wh{wloh Lqgxvwulhv ehwzhhq 4<53 dqg 4<6;1 Lq 4<53/ erwk lqgxv0 wulhv kdg frpsdudeo orz ohyhov ri rxwsxw shu zrunhu1 Krzhyhu/ ryhu wkh qh{w 4; hduv/ rxwsxw shu zrunhu lq Mdsdq lqfuhdvhg wkuhhirog zkhuhdv lw uhpdlqhg vwdjqdqw lq Lqgld1 Pruhryhu/ Wrwdo Idfwru Surgxfwlylw +WIS khqfhiruwk, lq Mdsdq lqfuhdvhg e 94 shufhqw lq vslqqlqj dqg e :: shufhqw lq zhdylqj zkhuhdv wkhvh phdvxuhv lq Lqgld lqfuhdvhg e rqo 4 dqg 54 shufhqw1 Ilqdoo / lq wkh Mdsdqhvh wh{wloh lqgxvwu wkh uhdo zdjh lqfuhdvhg e 6 shufhqw pruh wkdq lq djulfxowxuh/ zkhuhdv lq wkh Lqgldq wh{wloh lqgxvwu lw lqfuhdvhg e 46 shufhqw pruh wkdq lq djulfxowxuh1 Zrofrww grfxphqwv wkdw wkh pdlq glhuhqfh ehwzhhq Mdsdq dqg Lqgld od lq wkh zrunlqj duudqjhphqwv1 Prvw lpsruwdqwo / lq Mdsdq udwlrqdol}dwlrq gudvwlfdoo lqfuhdvhg wkh qxpehu ri pdfklqhv hdfk zrunhu kdqgohg zkhuhdv lq Lqgld wklv glg qrw kdsshq1 Zrofrww dujxhv wkdw gli0 ihuhqfhv lq wkh fkdudfwhulvwlfv ri wkh zrunirufh duh uhvsrqvleoh iru wkh glhuhqw h{shulhqfhv1 Lq sduwlfxodu/ wkh zrunhuv lq Mdsdq zhuh suhgrplqdwho rxqj zrphq/ zkr zrunhg lq wkh wh{wloh lqgxvwu rqo iru vrph hduv1 Wkhuh zhuh qr oderu xqlrqv lq Mdsdq1 Wkh zrunhuv lq Lqgld zhuh suhgrplqdwho dgxow pdohv/ zkr zrunhg lq wkh wh{wloh lqgxvwu iru prvw ri wkhlu olyhv1 Wkhuh zhuh vwurqj oderu xqlrqv lq Lqgld1 Dv d uhvxow wkh Erped uhjlrq dorqh zlwqhvvhg derxw 4333 vwulnhv lq wkh lqwhu0zdu shulrg/ prvw ri zklfk zhuh dlphg dw suhyhqwlqj phdvxuh ri udwlrqdol}dwlrq wkdw zrxog kdyh lpsuryhg oderu surgxfwlylw dqg uhgxfhg hpsor phqw1 Wkh uhfhqw wkhruhwlfdo olwhudwxuh wkdw prghov wkh lpsolfdwlrqv ri prqrsro uljkwv lq wkh oderu pdunhw lqfoxghv Krophv dqg Vfkplw} +4<<8,/ Sduhqwh dqg Suhvfrww +4<<</ 5333,/ Eulgjhpdq hw do +5334,/ dqg Khuuhqgrui dqg Whl{hlud +5335,1 Sduhqwh dqg Suhvfrww +4<<<, lv prvw forvho uhodwhg wr wkh surmhfw xqghuo lqj wkh suhvhqw sdshu1 Wkh qg wkdw wkh shu0fdslwd lqfrph ohyho zlwk frpshwlwlrq lv 51: wlphv kljkhu wkdq zlwk prqrsro uljkwv1 Lq ghulylqj 4 Sduhqwh dqg Suhvfrww +4<<6, dqg PfJudwwhq dqg Vfkplw} +4<<<, rhu d ghwdlohg dqdo vlv ri furvv0frxqwu gdwd1 :
wklv uhvxow/ Sduhqwh dqg Suhvfrww devwudfw iurp fdslwdo1 Lq frqwudvw/ zh gr qrw devwudfw iurp fdslwdo exw hpehg prqrsro uljkwv lqwr d vwdqgdug qhr0 fodvvlfdo jurzwk prgho zlwk wzr qdo jrrgv vhfwruv1 Dqdo }lqj wkh lpsolfd0 wlrqv ri prqrsro uljkwv zkhq wkhuh lv fdslwdo lv lpsruwdqw iru wzr uhdvrqv1 Iluvw/ lw zloo wxuq rxw wkdw fdslwdo juhdwo dpsolhv wkh glvwruwlrqdu hhfwv ri prqrsro uljkwv1 Wklv frqupv d frqmhfwxuh ri Sduhqwh dqg Suhvfrww +4<<<, dqg vkrzv wkdw fdslwdo lv hvvhqwldo iru xqghuvwdqglqj wkh txdqwlwdwlyh lpsol0 fdwlrqv ri prqrsro uljkwv1 Vhfrqg/ rxu prgho zloo eh frqvlvwhqw erwk zlwk vhyhudo nh ghyhorsphqw idfwv dqg zlwk Ndogru*v jurzwk idfwv1 Wkhuhiruh/ rxu wkhru ri ghyhorsphqw lv dovr d wkhru ri jurzwk/ zklfk lv ghvludeoh ehfdxvh wkh wzr hogv duh forvho uhodwhg1 Zh frqvlghu dq hfrqrp zlwk wzr qdo jrrgv vhfwruv/ zklfk zh fdoo wkh vhuylfh vhfwru dqg wkh pdqxidfwxulqj vhfwru1 Wkh vhuylfh vhfwru surgxfhv d frqvxpswlrq jrrg dqg wkh pdqxidfwxulqj vhfwru surgxfhv d frqvxpswlrq jrrg dqg d fdslwdo jrrg1 Wkh lqsxwv lq wkh surgxfwlrq ri wkh vhuylfh jrrg duh fdslwdo dqg oderu> wkh lqsxwv lq wkh surgxfwlrq ri wkh pdqxidfwxuhg jrrgv duh pdq lqwhuphgldwh jrrgv/ hdfk ri zklfk lv surgxfhg zlwk fdslwdo dqg oderu1 Wkh hfrqrp lv srsxodwhg e pdq lqglylgxdov/ zkr fdq zrun lq hlwkhu vhfwru dqg zkr fdq vdyh e lqyhvwlqj lq sk vlfdo fdslwdo1 Wkhuh duh wzr w shv ri lqglylgxdov= lqvlghuv dqg rxwvlghuv1 Wkh rqo glhuhqfh ehwzhhq lqvlghuv dqg rxwvlghuv frphv iurp wkh lqvwlwxwlrqdo duudqjhphqw= wkh lqvlghuv fdq kdyh d prqrsro ri vxsso lqj oderu lq wkhlu lqwhuphgldwh jrrg vhfwru1 Li wkh gr/ wkhq wkh hqwu ri rxwvlghu oderu lqwr wkhlu lqwhuphgldwh jrrg vhfwru lv uhvwulfwhg1 Li wkh gr qrw/ wkhq wkh hqwu ri rxwvlghu oderu lqwr wkhlu lqwhuphgldwh jrrg vhfwru lv qrw uhvwulfwhg1 Pruhryhu/ wkh lqvlghuv ri hdfk lqwhuphgldwh jrrg vhfwru irup d frdolwlrq wkdw kdv wkh uljkw wr fkrrvh wkh lqvlghu surgxfwlylw lq wkdw vhfwru> wkh rxwvlghuv gr qrw ehorqj wr dq frdolwlrq1 Rxu pdlq qglqj lv wkdw glhuhqfhv lq wkh vwuhqjwk ri prqrsro uljkwv dffrxqw iru d txdqwlwdwlyho vxevwdqwldo sduw ri wkh furvv0frxqwu glhuhqfhv lq lqfrph ohyhov1 Rxu edvholqh fdoleudwlrq xvhv wkh SZW<9/ d fdslwdo vkduh ri fe/ dqg dvvxphv wkdw wkh kljkhu sulfhv ri fdslwdo uhodwlyh wr frqvxpswlrq lq srru frxqwulhv frph iurp vwurqjhu prqrsro uljkwv1 Iru wklv edvholqh fdoleudwlrq/ zh qg dq lqfrph ohyho glhuhqfh ri d idfwru ri fh ehwzhhq wkh EJS htxloleuld zlwk frpshwlwlrq dqg zlwk prqrsro uljkwv1 D glhuhqfh ri d idfwru fh lv irxu wlphv wkh glhuhqfh ri d idfwru 2. irxqg e Sduhqwh dqg Suhvfrww +4<<<, iru d prgho hfrqrp zlwkrxw fdslwdo1 Prqrsro uljkwv kdyh d odujhu hhfw lq rxu prgho hfrqrp zlwk fdslwdo ehfdxvh wkhlu vwdqgdug gluhfw hhfw lv dffrpsdqlhg e dq lqgluhfw hhfw1 Wkh gluhfw hhfw lv wkdw prqrsro uljkwv doorz wkh lqvlghu frdolwlrqv wr uhgxfh wkh lqvlghu surgxfwlylw 1 Wklv ohdgv wr wkh lqh!flhqw surgxfwlrq ri wkh lqwhuphgldwh lqsxwv iru wkh pdqxidfwxuhg jrrgv1 Wkh lqgluhfw hhfw lv wkdw wkh uhgxfhg lqvlghu surgxfwlylw lqfuhdvhv wkh sulfh ri lqwhuphgldwh jrrgv dqg wkxv ri pdqxidfwxuhg jrrgv uhodwlyh wr vhuylfhv1 Vlqfh fdslwdo lv d pdqxidfwxuhg jrrg/ wklv uhgxfhv wkh fdslwdo0oderu udwlr lq wkh vhuylfh vhfwru1 Rxu txdqwlwdwlyh uhvxowv vkrz wkdw wklv lqgluhfw hhfw juhdwo dpsolhv wkh gluhfw hhfw ri prqrsro uljkwv1 Rxu prgho hfrqrp grhv txdqwlwdwlyho zhoo dovr zlwk uhvshfw wr wkuhh rwkhu ihdwxuhv ri wkh ghyhorsphqw gdwd1 +l, Lw suhglfwv wkdw wkh sulfh ri fdslwdo uhodwlyh wr frqvxpswlrq lv kljkhu lq srru wkdq lq ulfk frxqwulhv1 Wklv frqvlvwhqw ;
zlwk wkh hylghqfh uhsruwhg e Mrqhv +4<<7,/ Fkdul hw do +4<<9,/ Hdwrq dqg Nruwxp +5334,/ dqg Uhvwxffld dqg Xuuxwld +5334,1 Lq rxu prgho wkh sulfh ri fdslwdo uhodwlyh wr frqvxpswlrq lv kljkhu lq srru hfrqrplhv ehfdxvh wkh sulfh ri vhuylfhv +zklfk duh dq lpsruwdqw sduw ri frqvxpswlrq, uhodwlyh wr grphvwlf fdslwdo lv orzhu1 Lq frqwudvw/ wkh sulfh ri grphvwlf fdslwdo uhodwlyh wr iruhljq fdslwdo lv lqghshqghqw ri wkh lqfrph ohyho1 Erwk wkhvh ihdwxuhv duh frqvlvwhqw zlwk wkh hylghqfh uhsruwhg e Kvlhk dqg Nohqrz +5335,1 +ll, Rxu prgho suhglfwv wkdw phdvxuhg lq grphvwlf sulfhv/ ulfk frxqwulhv lqyhvw wkh vdph rxwsxw vkduhv dv srru frxqwulhv> phdvxuhg lq lqwhuqdwlrqdo sulfhv/ ulfk frxqwulhv lqyhvw frqvlghudeo odujhu rxwsxw vkduhv wkdq srru frxqwulhv1 Ixuwkhupruh/ wkh suhglfwhg vkduhv duh txdqwlwdwlyho forvh wr wkrvh lq wkh gdwd1 +lll, Rxu prgho suhglfwv wkdw ulfk frxqwulhv xvh wkhlu uhvrxufhv pruh h!flhqwo wkdq srru frxqwulhv/ vr furvv0frxqwu glhuhqfhv lq WIS dffrxqw iru d vxevwdqwldo sduw ri wkh glhuhqfhv lq lqfrph ohyhov1 Wkh suhglfwhg WIS glhuhqfhv duh txdqwlwdwlyho zlwklq wkh udqjh uhsruwhg e Fdvhool hw do +4<<9,/ Nohqrz dqg Urguljxh}0Foduh +4<<:,/ Suhvfrww +4<<;,/ Kdoo dqg Mrqhv +4<<<,/ dqg Fdvhool dqg Frohpdq +5336,1 5 Prgho hfrqrp 514 Hqylurqphqw Wlph lv glvfuhwh dqg uxqv iruhyhu1 Wkhuh lv qr xqfhuwdlqw 1 Wkhuh duh wzr qdo frqvxpswlrq jrrgv r dqg 6/ dfdslwdojrrg%/ dqg d frqwlqxxp ri lqwhu0 phgldwh jrrgv 5 / 5 dc 2o1 Wkh frqvxpswlrq jrrg r lv surgxfhg lq wkh vhuylfh vhfwru zkloh wkh frqvxpswlrq jrrg 6 dqg wkh fdslwdo jrrg % duh surgxfhg lq wkhpdqxidfwxulqjvhfwru1 Wkhvhuylfhjrrgdqgwkhlqwhuphgldwhjrrgvduhqrw wudgdeoh> wkh wzr pdqxidfwxuhg jrrgv duh wudgdeoh dqg wkh hfrqrp lv vpdoo lq wkdw lw grhv qrw dhfw wkh zruog pdunhw sulfh ri wkh wudgdeoh jrrgv1 Wkh prwlydwlrq iru dvvxplqj wkdw wkh pdqxidfwxuhg jrrgv duh wudgdeoh dqg wkh vhuylfh jrrgv duh qrw frphv iurp wkh hylghqfh uhsruwhg e Kvlhk dqg Nohqrz +5335,= wkh sulfh ri vhuylfhv uhodwlyh wr grphvwlf fdslwdo lv orzhu lv srru frxq0 wulhv wkdq lq ulfk frxqwulhv zkhuhdv wkh sulfh ri grphvwlf fdslwdo uhodwlyh wr iruhljq fdslwdo lv xquhodwhg wr wkh shu0fdslwd lqfrph ohyho1 Rwkhuzlvh wudgh zloo sod qr uroh lq rxu prgho1 Wkhuh duh xqlw uhfwdqjohv ri rxwvlghuv dqg lqvlghuv1 Wkh rxwvlghuv duh lghq0 wlfdo zkhuhdv wkh lqvlghuv duh qrw= lq hdfk lqwhuphgldwh jrrg vhfwru 5 wkhuh lv d xqlw lqwhuydo ri lqvlghuv ri w sh / 5 dc 2o1 Hdfk lqglylgxdo lv hqgrzhg zlwk rqh xqlw ri wlph lq hdfk shulrg dqg zlwk d srvlwlyh lqlwldo fdslwdo vwrfn1 Wkh lqglylgxdo sureohpv duh= v1w1 4@ tr c6 c& n " 'f "[ 'f r k q 6 3k 34 4 R r r n R 6 6 n R % d& n neb & o'o & n n Z c r c6 c& n fc & f : f jlyhq +514, <
Wkh vxevfulsw fdq wdnh wkh ydoxhv J iru rxwvlghu dqg 5 dc 2o iru lqvlghu ri w sh > q 5 Efc lv wkh glvfrxqw idfwru> k 5 Efc lv wkh h{shqglwxuh vkduh ri vhuylfhv> 4 5 dfc 4 lv wkh lqyhuvh ri wkh lqwhuwhpsrudo hodvwlflw ri vxevwlwxwlrq> & n dqg & duh wkh fdslwdo vwrfnv khog e wkh uhsuhvhqwdwlyh lqglylgxdo ri w sh dqg B 5 dfc o lv wkh ghsuhfldwlrq udwh ri lqvwdoohg fdslwdo/ vr & n neb& lv wkh shulrg0 lqyhvwphqw ri wkh uhsuhvhqwdwlyh lqglylgxdo ri w sh > R r / R 6 /dqg R % duh wkh sulfhv ri vhuylfhv/ pdqxidfwxuhg frqvxpswlrq jrrgv/ dqg fdslwdo jrrgv uhodwlyh wr wkh iruhljq fdslwdo jrrg/ zklfk lv wkh qxphudluh> o dqg duh wkh uhqwdo udwh ri lqvwdoohg fdslwdo dqg wkh zdjh udwh> Z duh surwv1 Zh frqwlqxh zlwk surgxfwlrq1 Wkh uhsuhvhqwdwlyh up lq wkh vhuylfh vhfwru pd{lpl}hv lwv surwv lq hdfk shulrg= 4@ R r r o & r J, rj r c& r c, rj c, r v1w1 r ' & w r E, r 3w c, r ', rj n ] 2 ] 2, r _ +515,, r _c r c& r c, r fc, rj : f :f lv d frqvwdqw wkdw ghwhuplqhv wkh xqlwv ri vhuylfhv> 5 & r lv fdslwdo door0 fdwhg wr wkh vhuylfh vhfwru/, r lv wrwdo oderu doorfdwhg wr wkh vhuylfh vhfwru/ dqg, rj dqg, r duh wkh oderu wkdw rxwvlghuv dqg lqvlghuv ri w sh doorfdwh wr wkh vhuylfh vhfwru> w 5 Efc lv wkh fdslwdo vkduh> 5 dfc 4 lv wkh h{rjhqrxv jurzwk udwh ri oderu0dxjphqwlqj whfkqlfdo surjuhvv1 Wklv irupxodwlrq frqwdlqv wkh uhvwulfwlrq, rj : f rq wkh rxwvlghuv* oderu vxsso / zklfk hqvxuhv wkdw vrph rxwvlghuv zrun lq wkh vhuylfh vhfwru1 Wklv zloo uxoh rxw rgg htxloleuld lq zklfk rqo wkh lqvlghuv zrun lq wkh vhuylfh vhfwru1 Wkh uhsuhvhqwdwlyh up lq wkh pdqxidfwxulqj vhfwru pd{lpl}hv lwv surwv lq hdfk shulrg= ] 2 4@ R 6 6 n R % % R 5 5 _ +516, 6 c% c5 ] 2 j j3 j3 v1w1 >6 n % ' 5 j _ c 6 c% c5 f R 5 lv wkh sulfh ri wkh lqwhuphgldwh jrrg 5 uhodwlyh wr wkh iruhljq fdslwdo jrrg/ >:f lv d frqvwdqw wkdw ghwhuplqhv wkh xqlwv ri 6 / 6 dqg j 5 Efc 4 lv wkh hodvwlflw ri vxevwlwxwlrq ehwzhhq glhuhqw lqwhuphgldwh jrrgv1 Zh zloo qhhg wkdw wkh pdqxidfwxulqj vhfwru*v ghpdqg iru hdfk lqwhuphgldwh jrrg lv lqhodvwlf= j 1 Wklv lv dv lq Sduhqwh dqg Suhvfrww +4<<<, dqg fdq eh jxdudqwhhg e frqvlghulqj vx!flhqwo eurdg fdwhjrulhv ri jrrgv +EPZ fduv yhuvxv Phufhghv fduv ru fduv yhuvxv elf fohv,1 5 Wkh frpprq sudfwlfh lv wr qrupdol}h xqlwv vxfk wkdw D @41 Rxu uhdvrq iru ghyldwlqj iurp wkdw sudfwlfh zloo ehfrph fohdu lq vhfwlrq 7 ehorz1 6 Wkh frpprq sudfwlfh lv wr qrupdol}h xqlwv vxfk wkdw @41 Rxu uhdvrq iru ghyldwlqj iurp wkdw sudfwlfh zloo ehfrph fohdu lq vhfwlrq 7 ehorz1 43
Wkh uhsuhvhqwdwlyh up lq wkh wk lqwhuphgldwh jrrg vhfwru pd{lpl}hv lwv surwv lq hdfk shulrg= 4@ 5 c& 5 c, 5 J c, 5 R 5 5 o & 5 J, 5 J, 5 +517, v1w1 5 ' & w 5 E, 5 3w c, 5 'E/, 5 J n K, 5 c 5 c& 5 c, 5 J fc, 5 : f, 5 /, 5 J / dqg, 5 duh wrwdo oderu/ rxwvlghu oderu/ dqg lqvlghu oderu doorfdwhg wr wkh wk lqwhuphgldwh jrrg vhfwru/ / 5 dfc lv d frqvwdqw/ dqg K 5 dfc o lv d yduldeoh wkdw dhfwv wkh lqvlghu surgxfwlylw 1 Wklv irupxodwlrq frqwdlqv wzr lpsolflw uhvwulfwlrqv rq wkh lqvlghuv* oderu vxsso 1 Iluvw/ zh kdyh dvvxphg wkdw vrph lqvlghuv ri wkh wk lqwhuphgldwh jrrg vhfwru zrun wkhuh=, 5 : f1 Wklv zloo uxoh rxw rgg htxloleuld lq zklfk wkh lqvlghuv gr qrw zrun lq wkhlu vhfwru1 Vhfrqg/ zh kdyh dvvxphg wkdw wkh lqvlghuv ri wkh wk lqwhuphgldwh jrrg vhfwru fdq rqo zrun lq wkdw vhfwru ru lq wkh vhuylfh vhfwru/ exw qrw lq wkh rwkhu lqwhuphgldwh jrrg vhfwruv1 Vlqfh zh zloo uhvwulfw rxu dwwhqwlrq wr v pphwulf htxloleulxp zlwk uhvshfw wr wkh glhuhqw w shv ri lqvlghuv dqg wkh glhuhqw vhfwruv/ wklv lv zlwkrxw orvv ri jhqhudolw 1 Zh dvvxph wkdw wkh lqvwlwxwlrqdo duudqjhphqw jlyhv d prqrsro uljkw wr wkh lqvlghuv ri hdfk lqwhuphgldwh jrrg vhfwru1 Wr lqwurgxfh wkh prqrsro / zh vhw K 'iru d prphqw1 Li / 'flqdgglwlrq/ wkhq lqvlghu dqg rxwvlghu oderu duh shuihfw vxevwlwxwhv dqg wkh lqvlghuv ri wkh wk lqwhuphgldwh jrrg vhfwru kdyh qr prqrsro srzhu1 Zh zloo uhihu wr wklv fdvh dv frpshwlwlrq1 Lq frqwudvw/ li K 'dqg /:f/ wkhq lqvlghu dqg rxwvlghu oderu duh lpshuihfw vxevwlwxwhv dqg wkh lqvlghuv kdyh prqrsro srzhu1 Zh zloo uhihu wr wklv fdvh dv prqrsro 1 Wkh vwuhqjwk ri wkh prqrsro srzhu lqfuhdvhv lq wkh ydoxh ri // zklfk vxppdul}hv wkh uhvwulfwlrqv rq rxwvlghu oderu lq vhfwru 5 1 Wr lqwurgxfh wkh uljkw/ zh uhwxuq wr wkh jhqhudo fdvh K 5 dfc o1 Zhdvvxphwkdwwkh lqvlghuv ri wkh wk lqwhuphgldwh jrrg vhfwru irup d frdolwlrq wkdw kdv wkh uljkw wr fkrrvh wkh lqvlghu surgxfwlylw K 1 Lw pdnhv wkdw fkrlfh dw wkh vdph wlph zkhq wkh rwkhu ghflvlrqv ri shulrg duh wdnhq dqg lwv remhfwlyh lv wr pd{lpl}h wkh suhvhqw ydoxh ri wkh uhsuhvhqwdwlyh lqvlghu*v xwlolw 1 Qrwh wkdw wkh dvvxpswlrq wkdw K 5 dfc o lpsolhv wkdw wkh frdolwlrq fdq fkrrvh duelwudulo lqh!flhqw whfkqrorjlhv dqg zrun sudfwlfhv1 Pruhryhu/ qrwh wkdw wkhuh duh qr frvwv ri pdlqwdlqlqj wkh kljkhvw surgxfwlylw dqg wkhuh duh qr ylqwdjh0vshflf vnloov1 Wudgh wdnhv sodfh lq vhtxhqwldo pdunhwv1 Lq hdfk shulrg wkhuh duh pdunhwv iru wkh wzr frqvxpswlrq jrrgv/ wkh fdslwdo jrrg/ hdfk lqwhuphgldwh jrrg/ lq0 vwdoohg fdslwdo/ rxwvlghu oderu/ dqg hdfk w sh ri lqvlghu oderu1 Wkh pdunhw fohdulqj frqglwlrqv iru wkh lqwhuphgldwh jrrgv duh reylrxv dqg wkxv rplwwhg1 Ghqrwlqj djjuhjdwh yduldeohv e xsshu0fdvh ohwwhuv wkh uhpdlqlqj pdunhw fohdu0 lqj frqglwlrqv duh= 44
g J n n W f n f W ] 2 7 ' 7 J n ' J n ] 2 ] 2 7 _c +518d, _c ' g J n neb g J n g _ ' g r n 'u rj n neb g _c ] 2 ] 2 g 5 _c u 5 J _c ] 2 g n +518e, +518f, +518g, +518h, 'u r n u 5 c 5 dc 2o +518i, Wkh uvw wzr pdunhw fohdulqj frqglwlrqv uhtxluh wkdw wkh pdunhwv iru wkh wzr frqvxpswlrq jrrgv fohdu/ wkh wklug frqglwlrq uhtxluhv wkdw wkh pdunhw iru qhz fdslwdo jrrgv fohdu/ wkh irxuwk frqglwlrq uhtxluhv wkdw wkh uhqwdo pdunhw iru lqvwdoohg fdslwdo fohdu/ dqg wkh odvw wzr frqglwlrqv uhtxluh wkdw wkh glhuhqw oderu pdunhwv fohdu1 W dqg f W ghqrwh wkh lpsruwv ri dqg f 1 Zh kdyh devwudfwhg iurp eruurzlqj dqg ohqglqj ehwzhhq lqvlghuv dqg rxw0 vlghuv dqg ehwzhhq grphvwlf dqg iruhljq lqglylgxdov1 Wklv lv zlwkrxw orvv ri jhqhudolw ehfdxvh zh zloo rqo frpsduh glhuhqw EJS htxloleuld1 Zlwkrxw eruurzlqj dqg ohqglqj ehwzhhq grphvwlf dqg iruhljq lqglylgxdov/ lqwhuqdwlrqdo wudgh lv edodqfhg lq hdfk shulrg/ lh R 6 W nr % f W 'f1 Frqvhtxhqwo / lqwhuqd0 wlrqdo wudgh zloo mxvw slq grzq wkh uhodwlyh sulfhv ri wudgdeoh jrrgv dw R 6 ' > dqg R % '1 Zlwkrxw orvv ri jhqhudolw / zh fdq wkhuhiruh uhvwulfw lpsruwv dqg h{sruwv= W ' f W 'f1 515 Htxloleulxp Rxu hqylurqphqw jlyhv ulvh wr d g qdplf jdph1 Vlqfh hdfk frdolwlrq kdv vwudwh0 jlf srzhu lq lwv vhfwru/ fduh qhhgv wr eh wdnhq lq ghqlqj wkh htxloleulxp1 Vlqfh hdfk frdolwlrq lv vpdoo frpsduhg wr wkh uhvw ri wkh hfrqrp / krzhyhu/ lw grhv qrw kdyh vwudwhjlf srzhu rq wkh djjuhjdwh ohyho1 Zh uhvwulfw dwwhqwlrq wr htxl0 oleuld zlwk wkh iroorzlqj surshuwlhv= +l, wkh duh v pphwulf zlwk uhvshfw wr lqvlghuv/ wkh rxwvlghuv/ wkh frdolwlrqv/ dqg wkh lqwhuphgldwh jrrgv vhfwruv> +ll, wkh duh uhfxuvlyh lq wkdw lq hdfk shulrg doo ghflvlrq pdnhuv frqglwlrq wkhlu dfwlrqv rq wkh vwdwh yduldeohv rqo 1 Uhfdoo wkdw doo djhqwv pdnh wkhlu fkrlfhv vlpxowdqhrxvo 1 Zh uvw vshfli wkh vwdwh yduldeohv1 Vlqfh zh frqvlghu v pphwulf htxlole0 ulxp rqo / zh fdq gurs wkh lqgh{ lqglfdwlqj wkh lqwhuphgldwh jrrg vhfwru1 Dv ehiruh/ zh ghqrwh wkh lqglylgxdo vwdwhv e orzhu0fdvh ohwwhuv dqg wkh hfrqrp 0 zlgh dyhudjhv ri lqglylgxdo vwdwhv e xsshu0fdvh ohwwhuv1 Vr/ wkh rxwvlghuv* dqg lqvlghuv* lqglylgxdo kroglqjv ri fdslwdo duh & J dqg & dqg wkhlu hfrqrp 0zlgh dyhudjh kroglqjv duh g J dqg g 1Pruhryhu/ lv wkh hfrqrp 0zlgh dyhudjh lq0 45
vlghu surgxfwlylw sdudphwhu dqg K lv wkh lqvlghu surgxfwlylw sdudphwhu lq dq lqwhuphgldwh jrrg vhfwru1 Iru frpsdfwqhvv/ zh deeuhyldwh wkh hfrqrp 0zlgh vwdwh= 8 Ecg cg J Wkh odz ri prwlrq ri wkh hfrqrp 0zlgh vwdwh lv ghqrwhg e = 8 'Ecg cg J ' CE8 'EC cc g cc gj Ecg cg J Zh zloo dovr qhhg wkh vhfwru0zlgh dyhudjh ri wkh lqvlghuv* fdslwdo kroglqjv/ zklfk zh ghqrwh e 7 & 1 Wkh odz ri prwlrq ri wkh vhfwru0zlgh vwdwh lv= EKc 7 & '7}EKc 7 & 'E7} K c 7} & EKc 7 & Diwhu wklv lqyhvwphqw lq qrwdwlrq/ zh fdq zulwh wkh g qdplf sureohpv lq uhfxu0 vlyh irup1 7 Fdoolqj wkh uhsuhvhqwdwlyh rxwvlghu*v ydoxh ixqfwlrq J / wkh uhfxuvlyh yhuvlrq ri klv sureohp lv= J E8c & J ' 4@ ier J c6 J nq J E8 c& r Jc6 Jc& Jj +519, J Df r R r E8 r J n R 6 E8 6 J n R % E8 d& J neb & J o ' oe8 & J n J E8 nz J E8 c 8 ' CE8 Lq zrugv/ wkh uhsuhvhqwdwlyh rxwvlghu fkrrvhv klv fxuuhqw frqvxpswlrq dqg klv ixwxuh fdslwdo vwrfn/ wdnlqj dv jlyhq wkh hfrqrp 0zlgh vwdwh 8 / wkh fruuh0 vsrqglqj odz ri prwlrq C/ dqg klv rzq fdslwdo vwrfn & J 1 Wkh vroxwlrq wr wklv sureohp lpsolhv wkh srolf ixqfwlrq Er J c6 J c& J'E} rj c} 6J c} &J E8c & J 1 Wkh uhfxuvlyh sureohp ri wkh uhsuhvhqwdwlyh lqvlghu lv= E8c Kc & 7 c& ' 4@ ier c6 nq E8 ck c & 7 r c6 c& Df c& j +51:, r R r E8 r n R 6 E8 6 n R % E8 d& neb & o ' oe8 & n E8c KnZ E8 c E8c Kc & 7 'ECE8 c 7}E8c Kc & 7 Lq zrugv/ wkh uhsuhvhqwdwlyh lqvlghu fkrrvhv klv fxuuhqw frqvxpswlrq dqg klv ixwxuh fdslwdo vwrfn/ wdnlqj dv jlyhq wkh hfrqrp 0zlgh vwdwh 8 / wkh fruuhvsrqg0 lqj odz ri prwlrq C/ wkh lqvlghu surgxfwlylw sdudphwhu K dqg wkh dyhudjh lqvlghu fdslwdo ri klv vhfwru 7 & / wkh fruuhvsrqglqj odz ri prwlrq 7}/ dqg klv rzq fdslwdo vwrfn & 1 Wkh vroxwlrq wr wklv sureohp lpsolhv wkh srolf ixqfwlrq Er c6 c& 'E} r c} 6 c} & E8c Kc 7 & c& 1 Wkh uhfxuvlyh sureohp ri wkh uhsuhvhqwdwlyh frdolwlrq lv= S E8c Kc & 7 ' 4@ ie} r E8c Kc & 7 c & 7 c} 6 E8c Kc & 7 c & 7 +51;, K Mdfco n q S E8 ck c & 7 j r E8c & 7 'ECE8 c 7} & E8c Kc & 7 7 Wkh sureohpv ri wkh wkuhh uhsuhvhqwdwlyh upv duh vwdwlf/ vr zh gr qrw qhhg wr uhzulwh wkhp1 46
Lq zrugv/ wkh uhsuhvhqwdwlyh lqvlghu frdolwlrq fkrrvhv wkh ixwxuh lqvlghu sur0 gxfwlylw lq lwv vhfwru vr dv wr pd{lpl}h wkh lqgluhfw lqvlghu xwlolw +soxv wkh frqwlqxdwlrq ydoxh,/ wdnlqj dv jlyhq wkh hfrqrp 0zlgh vwdwh 8 / wkh fruuhvsrqg0 lqj odz ri prwlrq C/ wkh vhfwru0zlgh vwdwh EKc 7 & / dqg wkh odz ri prwlrq ri wkh vhfwru0zlgh dyhudjh lqvlghu fdslwdo/ 7} & 1 D vroxwlrq wr wklv sureohp lpsolhv wkh srolf ixqfwlrq K ' } K E8c Kc 7 & 1 Uhfdoo wkdw wkh rqo lpsolfdwlrq wkdw wudgh kdv iru wkh htxloleulxp lv wr ghwhuplqh wkh uhodwlyh sulfhv ri wkh wudgdeoh jrrgv= R 6 ' > dqg R % '1 Wkhuhiruh/ wkhvh uhodwlyh sulfhv duh h{rjhqrxv wr wkh grphvwlf hfrqrp 1 Ghqlwlrq 514 +Htxloleulxp, Jlyhq R 6 ' >c R % ' /dqg W ' f W 'f/dq htxloleulxp lv sulfh ixqfwlrqv ER r c J coe8 / ER 5 c E8c K> doorfdwlrq ixqfwlrqv iru wkh wkuhh w shv ri upv/ Erc & r c, rj c, r E8 / E6c %c 5E8 / E5c & 5 c, 5J c, 5 E8c K/ odzv ri prwlrq 8 ' CE8 dqg EKc 7 & '7}E8c Kc 7 & > ydoxh ixqfwlrqv J E8c & J / E8c Kc 7 & c& / S E8c Kc 7 & > srolf ixqfwlrqv vxfk wkdw= Er J c6 J c& J 'E} rjc} 6J c} &J E8c & J c Er c6 c& 'E} r c} 6 c} & E8c Kc & 7 c& c K ' } K E8c Kc & 7 ( jlyhq wkh uhdol}dwlrqv ri sulfhv/ wkh upv* doorfdwlrqv vroyh wkhlu sureohpv> wkh ydoxh ixqfwlrqv J / /dqg S vdwlvi wkh Ehoopdq htxdwlrqv dv vwdwhg lq +519,/ +51:,/ dqg+51;,> wkh srolf ixqfwlrqv vroyh wkh sureohpv ri wkh uhsuhvhqwdwlyh lqglylgxdov/ +519, dqg +51:,/ dqg ri wkh uhsuhvhqwdwlyh frdolwlrq/ +51;,> wkh srolf ixqfwlrqv dqg wkh odzv ri prwlrq duh frqvlvwhqw= C gj E8 '} &J E8c g J c C g E8 '7} & E8c c g '} & E8c c g cg c C E8 '7} K E8c c g '} K E8c c g ( pdunhwv fohdu/ +518,1 47
6 Dqdo wlfdo uhvxowv Zh qrz vwxg wkh EJS htxloleuld ri wkh prgho hfrqrp zlwk frpshwlwlrq dqg zlwk prqrsro 1 Uhfdoo wkdw wkhvh wzr fdvhv fruuhvsrqg wr / 'fdqg /:fdqgwkdwdqlqfuhdvhlq/ fruuhvsrqgv wr dq lqfuhdvh lq wkh vwuhqjwk ri wkh prqrsro 1 Iru frqyhqlhqfh wkh frdolwlrq pdlqwdlqv wkh uljkw wr fkrrvh lwv phpehuv surgxfwlylw dovr li / ' f/ zklfk kdv qr hhfw rq wkh EJS htxloleulxp1 Wr hqvxuh wkdw wkh lqglylgxdo xwlolw ixqfwlrqv duh qlwh/ zh qhhg wkh vwdqgdug uhvwulfwlrq wkdw wkh jurzwk udwh ri oderu0dxjphqwlqj whfkqrorjlfdo surjuhvv lv qrw wrr odujh uhodwlyh wr wkh glvfrxqw idfwru= q q 34 5 Efc 1 Wr hqvxuh wkdw d EJS htxloleulxp h{lvwv zlwk prqrsro / zh qhhg wzr dgglwlrqdo uhvwulfwlrqv= Dvvxpswlrq 614 kqw E n B E2 / dk E ke /o d 4 n q EB o c +614d, 2kqw E n B : E2k d 4 n q EB o +614e, Wkhvh wzr uhvwulfwlrqv duh vdwlvhg iru vwdqgdug fdoleudwlrqv olnh wkdw ri vhfwlrq 7ehorz1 Sursrvlwlrq 614 +EJS htxloleulxp zlwk frpshwlwlrq/ / 'f, Wkhuh h{lvwv d xqltxh EJS htxloleulxp1 Dorqj wkh EJS htxloleulxp +d, wkh fdslwdo vwrfnv dqg doo vhfwruv* rxwsxwv jurz dw udwh > +e, surgxfwlylw lq wkh lqwhuphgldwh jrrg vhfwruv lv frqvwdqw dqg dv kljk dv srvvleoh= K '> +f, wkh uhodwlyh sulfh ri vhuylfhv lv frqvwdqw= R r ' 3 > +g, wkh rxwvlghuv dqg wkh lqvlghuv duh lqglhuhqw ehwzhhq wkh vhuylfh vhfwru dqg wkh lqwhuphgldwh jrrg vhfwruv1 8 Surri1 Vhh wkh Dsshqgl{1 Sursrvlwlrq 615 +EJS htxloleulxp zlwk prqrsro / /:f, Ohw Dv0 vxpswlrq 614 krog1 Wkhuh h{lvwv d xqltxh EJS htxloleulxp1 Dorqj wkh EJS htxloleulxp +d, wkh fdslwdo vwrfnv dqg doo vhfwruv* rxwsxwv jurz dw udwh > +e, surgxfwlylw lq wkh lqwhuphgldwh jrrg vhfwruv lv frqvwdqw dqg orzhu wkdq lv srvvleoh= K ' K 5 E /c> +f, wkh uhodwlyh sulfh ri vhuylfhv lv frqvwdqw= R r 'E/ 3w 3 > +g, wkh rxwvlghuv zrun rqo lq wkh vhuylfh vhfwru exw duh lqglhuhqw ehwzhhq wkh vhuylfh vhfwru dqg wkh lqwhuphgldwh jrrg vhfwruv/ wkh lqvlghuv zrun lq wkhlu lqwhuphgldwh jrrg vhfwru dqg vwulfwo suhihu wkdw1 Surri1 Vhh wkh Dsshqgl{1 8 Vwulfwo vshdnlqj wkh EJS htxloleulxp lv qrw xqltxh ehfdxvh rxwvlghu dqg lqvlghu oderu duh shuihfw vxevwlwxwhv vr wkhlu doorfdwlrq lv lqghwhuplqdwh1 Krzhyhu/ wklv lqghwhuplqdf lv luuhohydqw ehfdxvh lw grhv qrw kdyh dq frqvhtxhqfhv iru wkh rwkhu htxloleulxp txdqwlwlhv1 48
Zk grhv wkh uhsuhvhqwdwlyh frdolwlrq qg lw rswlpdo wr fkrrvh d orzhu surgxfwlylw wkdq lv srvvleoh zkhq lw kdv prqrsro srzhub Wr dqvzhu wklv txhvwlrq/ frqvlghu iru d prphqw zkdw d prqrsrolvw surgxfhu lq dq lqwhu0 phgldwh vhfwru zrxog gr1 Kh zrxog fkrrvh d kljkhu uhodwlyh sulfh wkdq wkh frpshwlwlyh rqh vr dv wr uhvwulfw surgxfwlrq1 Lq rxu hqylurqphqw zlwk lqhodvwlf ghpdqg/ wkh rswlpdo fkrlfh zrxog eh wkh kljkhvw sulfh dw zklfk hqwu grhv qrw rffxu1 Wkxv/ d prqrsrolvw surgxfhu zrxog pdnh rwkhu upv mxvw lqglhuhqw ehwzhhq hqwhulqj dqg vwd lqj rxw1 Qrz ohw xv wxuq edfn wr wkh lqvlghu frdol0 wlrq lq dq lqwhuphgldwh vhfwru1 Lw fdqqrw gluhfwo fkrrvh d kljkhu uhodwlyh sulfh ehfdxvh wkh sulfh lv ghwhuplqhg e frpshwlwlrq dprqj wkh upv1 Krzhyhu/ lw fdq lqgluhfwo fkrrvh d kljkhu uhodwlyh sulfh e fkrrvlqj dq lqh!flhqw lq0 vlghu surgxfwlylw 1 Jlyhq wkdw wkh ghpdqg iru lqwhuphgldwh jrrgv lv lqhodvwlf/ fkrrvlqj dq lqh!flhqw surgxfwlylw lqfuhdvhv wkh uhdo lqvlghu lqfrph ehfdxvh lw lqfuhdvhv wkh uhodwlyh sulfh e pruh wkdq lw ghfuhdvhv wkh lqvlghuv* pdujlqdo surgxfw1 Wkh lqvlghuv* prqrsro srzhu lv olplwhg e wkh hqwu ri rxwvlghuv lqwr wkh lqwhuphgldwh jrrgv vhfwru1 Li hqwu rffxuv wkh uhodwlyh sulfh lv ghwhuplqhg ehfdxvh wkh rxwvlghuv pxvw hduq wkh vdph zdjh lq wkh vhuylfh vhfwru dv lq wkh lqwhuphgldwh jrrg vhfwru1 Fkrrvlqj dq hyhq orzhu lqvlghu surgxfwlylw wkhq rqo ghfuhdvhv wkh pdujlqdo lqvlghu surgxfw/ dqg vr wkh uhdo lqvlghu lqfrph1 Wkxv/ wkh lqvlghu frdolwlrq*v rswlpdo surgxfwlylw fkrlfh uhvxowv lq d uhodwlyh sulfh wkdw mxvw pdnhv wkh rxwvlghuv lqglhuhqw ehwzhhq wkh vhuylfh vhfwru dqg wkh lqwhuphgldwh jrrg vhfwru1 Rqh nh lqjuhglhqw lq wklv dujxphqw lv wkdw wkh ghpdqg iru wkh lqwhuphgldwh jrrg lv lqhodvwlf1 Zh frxog jhqhudol}hg wklv1 Iluvw zh frxog dvvxph wkdw wkh lqvlghuv ehdu d pdujlqdo frvw ri surgxflqj dq dgglwlrqdo xqlw ri wkh lqwhuphgldwh jrrg1 Wklv zrxog dulvh/ iru h{dpsoh/ li wkh lqvlghuv kdg d oderu0ohlvxuh fkrlfh1 Vhfrqg/ zh frxog dvvxph wkdw fdslwdo fdq vxevwlwxwh iru xqsurgxfwlyh lqvlghu oderu zkhq upv wdnh frvwo lqyhvwphqwv1 Zh ohdyh sxuvxlqj wkhvh jhqhudol}dwlrqv wr ixwxuh zrun1 Wkh lpsolfdwlrqv ri rxu prgho duh frqvlvwhqw zlwk wkh vwdqgdug jurzwk idfwv dqg zlwk wkh ghyhorsphqw idfwv ghvfulehg lq wkh lqwurgxfwlrq1 Wr jhq0 hudwh wkh jurzwk lpsolfdwlrqv/ zh qhhg wr nhhs wkh vwuhqjwk ri wkh prqrsro srzhu frqvwdqw1 Lw wkhq hdv wr vkrz wkdw iru dq / 5 dfc wkh xqltxh EJS htxloleulxp lv frqvlvwhqw zlwk Ndogru*v jurzwk idfwv= shu0fdslwd lqfrph dqg shu0fdslwd fdslwdo jurz dw d frqvwdqw udwh> wkh fdslwdo0rxwsxw udwlr/ wkh uhdo lqwhuhvw udwh/ dqg wkh fdslwdo dqg oderu vkduhv lq rxwsxw duh frqvwdqw1 Wr jhq0 hudwh wkh ghyhorsphqw lpsolfdwlrqv ri rxu prgho/ zh qhhg wr ydu wkh vwuhqjwk ri prqrsro srzhu1 Zh dovr qhhg wr ghqh djjuhjdwh WIS/ zklfk lv wkh uhvlg0 xdo wkdw zrxog uhvxow li djjuhjdwh rxwsxw zdv surgxfhg iurp djjuhjdwh fdslwdo dqg oderu dffruglqj wr dq djjuhjdwh Free0Grxjodv surgxfwlrq ixqfwlrq zlwk fdslwdo vkduh w= A8 R r gr u w 3w r n g5 u w 3w 5 c +615, Eg r n g 5 w E2 3w Nqrzlqj wkh prgho/ zh nqrz/ ri frxuvh/ wkdw wklv lv lqfruuhfw1 Frpsxwlqj djjuhjdwh WIS lq wklv zd lv qrqhwkhohvv xvhixo/ dv lw doorzv xv wr frpsduh rxu uhvxowv zlwk wkrvh rewdlqhg e wkh hpslulfdo olwhudwxuh/ zklfk grhv h{dfwo wkh vdph1 49
Sursrvlwlrq 616 +Ghyhorsphqw idfwv, OhwDvvxpswlrq614krog1 +d, Dorqj dq EJS htxloleulxp zlwk / 5 dfc wkh jurzwk udwhv ri doo uhdo yduldeohv/ wkh sulfh ri grphvwlf fdslwdo uhodwlyh wr iruhljq fdslwdo/ dqg wkh lq0 yhvwphqw vkduh phdvxuhg lq grphvwlf sulfhv duh wkh vdph1 +e, Dorqj dq EJS htxloleulxp zlwk prqrsro / lh / ' f/ wkh iroorzlqj yduldeohv duh vpdoohu wkdq dorqj wkh EJS htxloleulxp zlwk frpshwlwlrq/ lh / 5 Efc = wkh ohyho ri shu0fdslwd lqfrph> wkh sulfh ri vhuylfhv uhodwlyh wr fdslwdo jrrgv> wkh lqyhvwphqw vkduh phdvxuhg lq lqwhuqdwlrqdo sulfhv> WIS lq wkhlqwhuphgldwhjrrgvhfwruvdqgdjjuhjdwhwis1 +f, Dorqj glhuhqw EJS htxloleuld/ wkh iroorzlqj yduldeohv duh vpdoohu zkhq / lv odujhu= wkh ohyho ri shu0fdslwd lqfrph> wkh sulfh ri vhuylfhv uhodwlyh wr fdslwdo> wkh lqyhvwphqw vkduh phdvxuhg lq lqwhuqdwlrqdo sulfhv> WIS lq wkh lqwhuphgldwh jrrg vhfwruv dqg djjuhjdwh WIS1 Surri1 Vhh wkh Dsshqgl{1 Rxw wrjhwkhu/ wkh glhuhqw sduwv ri wklv sursrvlwlrq lpso wkdw= +l, wkh uhodwlyh sulfh ri vhuylfhv lv srvlwlyho fruuhodwhg zlwk shu0fdslwd lqfrph> +ll, phdvxuhg lq grphvwlf sulfhv wkh lqyhvwphqw vkduh lq rxwsxw lv xqfruuhodwhg zlwk lqfrph zkhuhdv phdvxuhg lq lqwhuqdwlrqdo sulfhv lw lv srvlwlyho fruuhodwhg zlwk lqfrph> +lll, WIS lq wkh lqwhuphgldwh jrrg vhfwruv dqg djjuhjdwh WIS duh srvlwlyho fruuhodwhg zlwk lqfrph1 Wkhvh duh txdolwdwlyh yhuvlrqv ri wkh ghyhorsphqw idfwv ghvfulehg lq wkh lqwurgxfwlrq1 Rqh pd zrqghu zk prqrsro uhgxfhv wkh uhodwlyh sulfh ri vhuylfhv/ lq0 vwhdg ri lqfuhdvlqj wkh uhodwlyh sulfhv ri lqwhuphgldwh jrrgv dqg pdqxidfwxuhg jrrgv1 Wkh uhdvrq lv wkdw lq v pphwulf htxloleulxp erwk wkh sulfhv ri lqwhu0 phgldwh jrrgv uhodwlyh wr wkh grphvwlf fdslwdo dqg wkh sulfh ri wkh grphvwlf fdslwdo uhodwlyh wr iruhljq fdslwdo pxvw htxdo rqh1 Wkh odwwhu krogv ehfdxvh grphvwlf fdslwdo lv wudgdeoh lq rxu prgho1 Dq lqfuhdvh lq wkh uhodwlyh sulfh ri wkh lqwhuphgldwh jrrgv uhodwlyh wr wkh grphvwlf vhuylfh jrrg fdq wkhq rqo frph iurp d ghfuhdvh lq wkh sulfh ri wkh grphvwlf vhuylfh jrrg uhodwlyh wr wkh lqwhuphgldwh dqg pdqxidfwxulqj jrrgv1 Wklv lv srvvleoh ehfdxvh grphvwlf vhuylfhv duh qrw wudgdeoh1 Zh hqg wklv vhfwlrq e qrwlqj wkdw prqrsro uljkwv uhgxfh wkh fdslwdo0 oderu udwlrv ri doo vhfwruv/ qrw mxvw ri wkh lqwhuphgldwh jrrg vhfwruv1 Wklv fdq eh vhhq iurp wkh Hxohu htxdwlrqv/ zklfk dorqj d EJS htxloleulxp ehfrph= u5 w g 5 3w 3w ur ' R r w ' 4 q g r nb +616, Qrwh wkdw iru / 5 Efc / wkh lqvlghuv zrun rqo lq wkh lqwhuphgldwh jrrg vhfwruv/ lpso lqj wkdw u r 'dqg u 5 ' K 1Wkhgluhfwhhfwriprqrsro uljkwvlvwr ghfuhdvh K / zklfk ghfuhdvhv wkh fdslwdo0oderu udwlr lq wkh lqwhuphgldwh jrrgv vhfwruv1 Qrwlfh wkdw wkhuh lv qr uhodwlyh sulfh hhfw lq wkh lqwhuphgldwh jrrg vhfwru ehfdxvh R 5 ' lq hdfk v pphwulf EJS htxloleulxp1 Wkh lqgluhfw hhfw ri prqrsro uljkwv lv wr ghfuhdvh R r / zklfk pdnhv doorfdwlqj fdslwdo wr wkh vhuylfh vhfwru pruh h{shqvlyh dqg ghfuhdvhv wkh fdslwdo0oderu udwlr lq vhuylfhv1 Wkhuhiruh/ fdslwdo surylghv dq dpsolfdwlrq phfkdqlvp e zklfk wkh prqrsro glvwruwlrq lq wkh lqwhuphgldwh jrrg vhfwruv dhfwv wkh frpshwlwlyh 4:
vhuylfh vhfwru1 Wklv dpsolfdwlrq phfkdqlvp wxuqv rxw wr eh lpsruwdqw iru wkh txdqwlwdwlyh dqdo vlv wr zklfk zh wxuq qrz1 9 7 Txdqwlwdwlyh uhvxowv Lq wklv vhfwlrq/ zh h{soruh wkh txdqwlwdwlyh lpsolfdwlrqv ri prqrsro uljkwv1 Zh uvw sdudphwhul}h wkh prgho hfrqrp zlwk frpshwlwlrq e dvvxplqj wkdw lw fruuhvsrqgv wr wkh whq shufhqw ulfkhvw frxqwulhv lq wkh zruog1 Zh wkhq sdudphwhul}h wkh vwuhqjwk ri wkh prqrsro srzhu/ // e dvvxplqj wkdw wkh glhuhqfh lq wkh sulfh ri vhuylfhv uhodwlyh wr fdslwdo ehwzhhq wkh ulfkhvw dqg wkh srruhvw whq shufhqwloh frphv iurp prqrsro uljkwv1 Zh qdoo dvn e krz pxfk wkh shu0fdslwd lqfrph ohyho ri wkh frpshwlwlyh hfrqrp fkdqjhv li zh uhsodfh frpshwlwlrq e prqrsro 1 D prgho shulrg lv rqh hdu1 Wdeoh 4 vxppdul}hv wkh sdudphwhu ydoxhv iru wkh ehqfkpdun fdoleudwlrq1 : Lqvwhdg ri fdoleudwlqj wr wkh ulfkhvw whq shufhqw ri frxqwulhv/ zh jr zlwk wkh vwdqgdug srvw0zdu ydoxhv iru wkh XV li wkh duh dydlodeoh1 Rxu mxvwlfdwlrq lv wkdw wkh XV lv e idu wkh eljjhvw dqg prvw vwxglhg hfrqrp lq wkh wrs whq shufhqwloh dqg wkdw lw kdv wkh ehvw gdwd vrxufhv1 Dv lq Frroh dqg Suhvfrww +4<<8,/ zh wkhuhiruh fkrrvh 'fds/ 4 '/ q 'fbe./ dqg w 'fe1 Zh zloo h{soruh rxu prgho hfrqrp dovr iru rwkhu ydoxhv ri w ehfdxvh wkhuh lv vrph ghedwh derxw wkh ydoxh ri w> vhh/ iru h{dpsoh/ Jroolq +5335,1 Qrwh wkdw zh kdyh uhvwulfwhg doo vhfwruv wr kdyh htxdo fdslwdo vkduhv1 Kvlhk dqg Nohqrz +5335, qg wkdw wklv lv d uhdvrqdeoh dssur{lpdwlrq iru wkh XV1 Lq frqwudvw/ Hfkhyduuld +4<<:, qgv wkdw dfurvv wkh RHFG/ wkh dyhudjh fdslwdo vkduh lq vhuylfhv lv kljkhu wkdq lq pdqxidfwxulqj +iw yhuvxv iruw shufhqw,1 Lq oljkw ri wklv hylghqfh/ rxu fkrlfh ri d fdslwdo vkduh lq vhuylfhv ri iruw shufhqw lv d frqvhuydwlyh rqh/ dv wkh suhglfwhg glhuhqfhv lq wkh shu0fdslwd lqfrph ohyhov lqfuhdvh lq wkh fdslwdo vkduh1 Wr fdoleudwh wkh uhpdlqlqj sdudphwhu ydoxhv/ zh xvh gdwd iurp wkh Shqq Zruog Wdeohv1 Iroorzlqj Kvlhk dqg Nohqrz +5335,/ zh xvh ehqfkpdun hduv ehfdxvh wkh duh wkh rqo hduv iru zklfk sulfh gdwd duh dfwxdoo froohfwhg1 Xqiruwxqdwho wkh qxpehu dqg lghqwlw ri wkh frxqwulhv ydu zlgho dfurvv wkh glhuhqw ehqfkpdun hduv1 Zh wkhuhiruh jdyh xs rq wkh dwwhpsw ri fdoleudwlqj wkh EJS htxloleulxp xqghu frpshwlwlrq iurp d iurp sdqho ri furvv0frxqwu gdwd dqg wxuqhg lqvwhdg wr wkh furvv0vhfwlrq ri wkh SZW<9 +uhfdoo wkdw SZW<9 deeuhyldwhv wkh 4<<9 Shqq Zruog Wdeohv,1 Wkh SZW<9 lv wkh prvw uhfhqw dydlodeoh ehqfkpdun vwxg dqg lw kdv wkh dgydqwdjh ri ehlqj yhu eurdg= lw frqwdlqv 448 frxqwulhv1 Zh vwduw e lghqwli lqj wkh frxqwulhv lq wkh wrs dqg erwwrp whq shufhqwlohv ri shu0fdslwd lqfrphv/ phdvxuhg lq lqwhuqdwlrqdo sulfhv1 Zh qg wkh vwdqgdug idfw wkdw wkh dyhudjh shu0fdslwd lqfrph ri wkh wrs whq shufhqwloh lv derxw f wlphv odujhu wkdq wkdw ri wkh erwwrp whq shufhqwloh1 9 Vfkplw} +5334d, pdnhv d vlplodu srlqw= li wkh jryhuqphqw surgxfhv lqyhvwphqw jrrgv lqh!flhqwo / wkhq wklv uhgxfhv wkh oderu surgxfwlylw ri doo vhfwruv wkdw xvh wkhvh lqyhvwphqw jrrgv1 Vfkplw} qgv wkdw wklv hhfw fdq fdxvh dq lqfrph glhuhqfh ri derxw idfwru wkuhh1 : Qrwh wkdw zh gr qrw qhhg wr fkrrvh d ydoxh iru / ehfdxvhlwgursvrxwlqv pphwulf htxloleulxp1 4;
Zh surfhhg e dvvxplqj wkdw wkh wrs whq shufhqwloh lq wkh SZW<9 fruuh0 vsrqgv wr rxu hfrqrp xqghu frpshwlwlrq dqg xvh dyhudjhv wr fdoleudwh B/ k/ / dqg >1 WkhydoxhriB ghwhuplqhv wkh lqyhvwphqw vkduh lq rxwsxw phdvxuhg lq grphvwlf sulfhv1 Lq wkh wrs whq shufhqwloh ri wkh SZW<9/ wkh zhljkhg dyhu0 djh vkduh htxdov f2/ zkhuh wkh zhljkwv duh wkh frxqwulhv shu0fdslwd lqfrphv lq lqwhuqdwlrqdo sulfhv uhodwlyh wr wkh wrwdo shu0fdslwd lqfrph lq wkh wrs whq shufhqwloh1 Jlyhq rxu fkrlfhv ri / 'fdqg w 'fe/ zhqhhgwrfkrrvh B 'ffs wr uhsolfdwh wklv vkduh1 Wkh ydoxh ri k htxdov wkh h{shqglwxuh vkduh ri qrq0 wudgdeoh frqvxpswlrq jrrgv lq wrwdo frqvxpswlrq phdvxuhg lq grphvwlf sulfhv1 Lq wkh wrs whq shufhqwloh ri wkh SZW<9/ wkh zhljkhg dyhudjh vkduh htxdov fdh/ zkhuh wkh zhljkwv duh wkh frxqwulhv* txdqwlwlhv ri frqvxphg vhuylfhv lq lqwhu0 qdwlrqdo sulfhv uhodwlyh wr wkh wrwdo txdqwlw ri frqvxphg vhuylfhg lq wkh wrs whq shufhqwloh1 ; Wkh ydoxh ri ghwhuplqhv wkh sulfh ri qrq0wudgdeoh frqvxps0 wlrq jrrgv uhodwlyh wr wudgdeoh fdslwdo jrrgv1 Lq wkh wrs whq shufhqwloh ri wkh SZW<9/ wkh zhljkhg dyhudjh uhodwlyh sulfh htxdov f.h/ zkhuh wkh zhljkwv duh wkh frxqwulhv txdqwlwlhv lq lqwhuqdwlrqdo sulfhv uhodwlyh wr wkh wrwdo txdqwlw lq wkh wrs whq shufhqwloh1 < Vlqfh R r ' 3 dorqj wkh EJS zlwk frpshwlwlrq/ zh fkrrvh '2H1 Wkh ydoxh ri > ghwhuplqhv wkh sulfh ri pdqxidfwxuhg frq0 vxpswlrq jrrgv uhodwlyh wr fdslwdo jrrgv1 Vlqfh erwk duh dvvxphg wudgdeoh dqg vlqfh zh pdlqwdlq wkh dvvxpswlrq ri d vpdoo rshq hfrqrp / wkh fruuhvsrqglqj sulfh lq wkh gdwd lv wkh lqwhuqdwlrqdo sulfh ri pdqxidfwxuhg frqvxpswlrq jrrgv uhodwlyh wr fdslwdo jrrgv1 Wklv lqwhuqdwlrqdo sulfh htxdov fe. dqg vr zh fkrrvh > 'fe.1 43 Jlyhq wkh ydoxhv ri w dqg / wkhydoxhri/ ghwhuplqhv wkh sulfh ri wkh qrq0wudgdeoh frqvxpswlrq jrrgv uhodwlyh wr fdslwdo jrrgv lq wkh hfrqrp zlwk prqrsro = R r 'E/ 3w 3 1 Wkh udwlr ehwzhhq ri wklv sulfh lq wkh srruhvw dqg ulfkhvw whq shufhqwloh htxdov..1 Xvlqj E / w3 '.. dqg w 'fe/ zh wkxv vhw / 'fbs. lq rxu ehqfkpdun fdoleudwlrq1 Dw uvw vljkw/ wklv pd dsshdu wr eh yhu odujh1 Krzhyhu/ & w 5 E /, 5 J 3w uhpdlq vwulfwo vpdoohu ; Qrwh wkdw rxu dvvxpswlrq ri d frqvwdqw h{shqglwxuh vkduh ri vhuylfhv lv qrw ydolgdwhg e wkh gdwd= wkh srruhvw whq shufhqwloh kdv dq h{shqglwxuh vkduh ri vhuylfhv ri ohvv wkdq kdoi ri wkh ulfkhvw whq shufhqwloh1 Vlqfh lq rxu prgho hfrqrp wkh vhuylfh vhfwru lv wkh frpshwlwlyh vhfwru/ wklv jrhv djdlqvw xv1 < Wr eh suhflvh wkh sulfh lq d frxqwu lv wkh zhljkhg dyhudjh ri wkh sulfhv ri wkh glhuhqw fdwhjrulhv ri jrrgv1 Wkh dyhudjh sulfh dfurvv d shufhqwloh ri frxqwulhv lv wkh zhljkhg dyhudjh ri wkh sulfhv1 Lq erwk fdvhv/ wkh zhljkwv duh wkh txdqwlwlhv uhodwlyh wr wkh uhohydqw wrwdo txdqwlw phdvxuhg lq lqwhuqdwlrqdo sulfhv1 Dv fdslwdo jrrgv zh fodvvlhg doo fdslwdo jrrgv lqfoxglqj qrq0wudgdeoh frqvwuxfwlrq1 Dv wudgdeoh frqvxpswlrq jrrgv zh fodvvlhg ehyhudjhv/ euhdg dqg fhuhdo/ fkhhvh dqg hjjv/ forwklqj +lqfoxglqj uhsdlu,/ vk/ rru fryhulqj/ iuxlw/ ixho dqg srzhu/ ixuqlwxuh/ krxvhkrog dssoldqfhv dqg uhsdluv/ krxvhkrog jrrgv dqg wh{wlohv/ phdw/ plon/ rlo dqg idw/ rwkhu irrg/ shuvrqdo wudqvsruwdwlrq htxlsphqw/ wredffr/ dqg yhjhwdeohv dqg srwdwrhv1 Dv qrq0wudgdeoh frqvxpswlrq jrrgv zh fodvvlhg fdihv dqg krwhov/ frppxqlfdwlrq/ hgxfdwlrq/ jurvv uhqw dqg zdwhu fkdujhv/ phglfdo khdowk fduh/ rshudwlrq ri wudqvsruwdwlrq htxlsphqw/ sxufkdvh wudqvsruwdwlrq vhuylfh/ uhfuhdwlrq dqg fxowxuh/ uhvwdxudqwv/ dqg rwkhu jrrgv dqg vhuylfhv1 43 Wkh lqwhuqdwlrqdo sulfh lv ghqhg uhfxuvlyho 1 Wkh lqwhuqdwlrqdo sulfh ri d fdwhjru ri jrrgv htxdov wkh zhljkhg dyhudjh ri wkh sulfhv ri wkdw fdwhjru lq wkh glhuhqw frxqwulhv> wkh zhljkwv duh wkh txdqwlwlhv uhodwlyh wr wkh wrwdo txdqwlw dqg wkh sulfhv lq wkh frxqwulhv duh wkh grphvwlf sulfhv pxowlsolhg zlwk wkh sxufkdvlqj0srzhu0sdulw ri wkh grphvwlf fxuuhqf > wkh sxufkdvlqj0srzhu0sdulw htxdov wkh udwlr ehwzhhq wkh ydoxhv ri wkh grphvwlf jrrgv edvnhw hydoxdwhg lq grphvwlf sulfhv dqg lq lqwhuqdwlrqdo sulfh1 4<
wkdq wkh oderu vkduh wlphv wkh dqqxdo rxwsxw ri wkh lqwhuphgldwh jrrg vhfwru1 Sduhqwh dqg Suhvfrww +4<<<, dujxh wkdw wkh frvwv ri euhdnlqj oderu pdunhw prqrsrolhv fdq hdvlo eh ri wklv rughu pdjqlwxgh1 D glhuhqw zd wr hvwdeolvk wkdw wkh vwuhqjwk ri prqrsro uljkwv duh uhdvrqdeoh lv wr orrn dw wkh lpsolhg suhplxp ri wkh lqvlghu zdjh ryhu wkh rxwvlgh zdjh1 Lq rxu prgho/ lw htxdov f shufhqw iru wkh ehqfkpdun fdoleudwlrq1 D zdjh suhplxp ri f shufhqw lv yhu uhdvrqdeoh iru dq hfrqrp zlwk vwurqj prqrsro uljkwv/ jlyhq wkh vl}h ri wkh revhuyhg lqwhu0lqgxvwu zdjh glhuhqwldov lq wkh XV iru zrunhuv zlwk wkh vdph fkdudfwhulvwlfv1 Krophv dqg Vfkplw} +5334d,/ iru h{dpsoh/ uhsruw wkdw lq Qhz Ruohdqv lq wkh plg 4;;3v vfuhzphq dqg orqjvkruhphq hduqhg prqrsro zdjh suhpld ri ehwzhhq H dqg shufhqw ri wkh wrwdo wudqvsruwdwlrq frvwv ri wkh jrrgv wkh kdqgohg1 Jlyhq d oderu vkduh ri Sf shufhqw/ d zdjh suhplxp ri f shufhqw wudqvodwhv lqwr H shufhqw ri wkh wrwdo surgxfwlrq frvwv +uhfdoo wkdw wkh surgxfhuv pdnh }hur surwv lq rxu prgho,1 Dqrwkhu slhfh ri hylghqfh frphv iurp Nuxhjhu dqg Vxpphuv +4<;;,/ zkr uhsruw odujh dqg shuvlvwhqw zdjh suhpld lq wkh 4<;3v1 Iru h{dpsoh/ lq 4<;7 wkh zrunhuv lq XV dxwrpreloh pdqxidfwxulqj uhfhlyhg d zdjh suhplxp ri qhduo f shufhqw ryhu wkh dyhudjh zdjh ri doo zrunhuv zlwk wkh vdph fkdudfwhulvwlfv1 Wdeoh 4= Ehqfkpdun Fdoleudwlrq 4 q w B k > / fds fbe. fe ffs fdh 2H fe. fbs. Wdeoh 5 vxppdul}hv wkh txdqwlwdwlyh uhvxowv1 Wkh qxpehuv lq wkh erogidfhg olqh fruuhvsrqg wr rxu ehqfkpdun fdoleudwlrq1 Wkh uvw froxpq lqglfdwhv wkh fdslwdo vkduhv rwkhu wkdq w 'fe wkdw zh kdyh h{soruhg1 Qrwh wkdw zkhq w fkdqjhv/ zh qhhg wr uh0fdoleudwh wkh ydoxhv ri B dqg / vr dv wr pdlqwdlq xqfkdqjhg wkh lqyhvwphqw vkduh lq grphvwlf sulfhv dqg wkh udwlr ri wkh uhodwlyh sulfhv ri vhuylfhv1 Wkh vhfrqg froxpq uhsruwv wkh udwlrv ri wkh shu0fdslwd lqfrphv zlwk frpshwlwlrq dqg zlwk prqrsro 1 Lq doo fdvhv/ zh frpsxwh shu0 fdslwd lqfrph dv lq wkh SZW<9/ vr wkh txdqwlwlhv jhqhudwhg e wkh prgho duh hydoxdwhg dw lqwhuqdwlrqdo sulfhv R W iurp wkh SZW<9= t E6c R W 'R W r 7E/nRW 6 E/nRW % fe/ Wklv pdnhv vhqvh ehfdxvh zh kdyh fkrvhq xqlwv vxfk wkdw doo uhodwlyh sulfhv xqghu frpshwlwlrq duh dv lq wkh wrs whq shufhqwloh ri wkh SZW<9/ vr wkh xqlwv lq wkh prgho duh htxdo wr wkh xqlwv lq wkh SZW<91 Zh vhh wkdw wkh udwlr ehwzhhq wkh ohyho ri shu0fdslwd lqfrph zlwk frpshwlwlrq dqg prqrsro lv ehwzhhq d idfwru ri S dqg f1 Iru rxu edvholqh fdoleudwlrq w 'fe/ wkh uhvxow lv d idfwru ri fh1 Wklvlvirxuwlphvwkhidfwruri2. uhsruwhg e Sduhqwh dqg Suhvfrww +4<<<, iru d prgho zlwkrxw fdslwdo dffxpxodwlrq1 Wkh odvw olqh ri Wdeoh 5 uhsruwv wkdw iru d fdslwdo vkduh ri fde rxu prgho hfrqrp jhqhudwhv urxjko wkh revhuyhg lqfrph udwlr ri f1 Zkloh zh wklqn wkdw fdslwdo vkduhv ri wklv vl}h duh rq wkh yhu kljk hqg ri sodxvleoh fdslwdo vkduhv/ wkh uhvxow lv lqwhuhvwlqj lq oljkw suhylrxv zrun= Pdqnlz hw do +4<<5, dqg Fkdul hw do +4<<9, irxqg wkdw 53
zlwkrxw furvv0frxqwu glhuhqfhv lq WIS/ d pxfk odujhu fdslwdo vkduh ri f. lv qhhghg wr jhqhudwh wkh revhuyhg lqfrph glhuhqfh1 Zh zloo uhwxuq wr wklv lvvxh lq wkh qh{w vhfwlrq zkhq zh glvfxvv d wd{ glvwruwlrqv dv dq dowhuqdwlyh wr wkh prqrsro uljkwv glvwruwlrq1 Wdeoh 5= Txdqwlwdwlyh uhvxowv prqrsro hfrqrp +S iru frpshwlwlrq/ 6 iru prqrsro / R W iru lqwhuqdwlrqdo sulfhv, w t EScR W t E6cR W UEScR W t EScR W UE6cR W t E6cR W A8 5ES A8 5E6 A8ES A8E6 316 919 315< 3145 919 51< 3168 ;16 315< 3145 91: 51< 317 431; 315< 3145 91: 51< 3178 4719 3163 3145 91; 51< 3187 6315 315< 3145 91< 51< Wkh wklug dqg irxuwk froxpqv ri Wdeoh 5 uhsruw wkh lqyhvwphqw vkduhv lq rxwsxw zlwk frpshwlwlrq dqg zlwk prqrsro / erwk phdvxuhg lq lqwhuqdwlrqdo sulfhv1 Wkh suhglfwhg lqyhvwphqw vkduhv duh uhdvrqdeo forvh wr wkrvh irxqg lq wkh SZW<9/ zkhuh wkh ulfkhvw dqg srruhvw whq shufhqwlohv lqyhvw 2. dqg f shufhqw ri rxwsxw phdvxuhg lq lqwhuqdwlrqdo sulfhv1 Qrwh wkdw wkh prgho dovr shuirupv zhoo zlwk uhvshfw wr wkh lqyhvwphqw vkduh phdvxuhg lq grphvwlf sulfhv1 Zkloh/ e frqvwuxfwlrq/ zh pdwfk wkh 56 shufhqw wkdw wkh ulfkhvw whq shufhqwloh lqyhvw/ zh suhglfw wkh vdph shufhqwdjh iru wkh srruhvw whq shufhqwloh1 Wklv lv frqvlv0 whqw zlwk wkh qglqj ri Kvlhk dqg Nohqrz +5335, wkdw dfurvv frxqwulhv wkh lqyhvwphqw vkduh phdvxuhg lq grphvwlf sulfhv lv xquhodwhg wr wkh shu0fdslwd lqfrph ohyho1 Wkh iwk froxpq uhsruwv wkh udwlrv ehwzhhq WIS lq wkh lqwhuphgldwh jrrgv vhfwruv zlwk frpshwlwlrq dqg prqrsro 1 Wkh prgho suhglfwv wkdw WIS lq wkh lqwhuphgldwh jrrgv vhfwruv zlwk frpshwlwlrq eh ehwzhhq SS dqg Sb wlphv odujhu wkdq WIS zlwk prqrsro 1 Vhfwru WIS glhuhqfhv ri wkdw rughu pdj0 qlwxgh duh zhoo zlwklq uhdvrq1 Iru h{dpsoh/ wkh hvwlpdwhv ri Kduuljdq +4<<<, vkrz wkdw d glhuhqfh lq vhfwru WIS ri d idfwru 2 lv frpprq dfurvv RHFG frxqwulhv1 D vshflf h{dpsoh lv Eulwdlq ehiruh wkh Wkdwfkhu uhirupv ri wkh oderu pdunhw wrrn sodfh= Kduuljdq hvwlpdwhv wkdw lq 4<;3 WIS lq wkh sur0 gxfwlrq ri pdfklqhu dqg htxlsphqw zdv derxw wkuhh wlphv odujhu lq wkh XV wkdq Eulwdlq1 Vlqfh wkhvh hvwlpdwhv duh mxvw iru RHFG frxqwulhv/ rqh h{shfwv frqvlghudeo odujhu glhuhqfhv lq vhfwru WIS ehwzhhq RHFG frxqwulhv dqg ghyhorslqj frxqwulhv1 Qrwh wkdw vhfwru0wis glhuhqfhv ri d idfwru SD gr qrw rqo frph iurp xvlqj lqh!flhqw whfkqrorjlhv exw dovr iurp lqh!flhqw zrun sudfwlfhv1 Wkhuh lv dpsoh hylghqfh wkdw lqh!flhqw zrun sudfwlfhv dorqh fdq uhgxfh surgxfwlylw vljqlfdqwo 1 Iru h{dpsoh/ Fodun +4<;:, grfxphqwv wkdw lq 4<43 glhuhqfhv lq zrun sudfwlfhv jhqhudwhg furvv0frxqwu glhuhqfhv ri d idfwru : lq wkh oderu surgxfwlylw ri frwwrq0wh{wloh ploov1 Vfkplw} +5334e, grf0 xphqwv wkdw lq wkh 4<;3v wkh XV dqg Fdqdgldq lurq0ruh lqgxvwulhv grxeohg wkhlu oderu surgxfwlylwlhv edvlfdoo e fkdqjlqj wkhlu zrun uxohv1 54
Wkh vl{wk froxpq uhsruwv wkh udwlrv ehwzhhq djjuhjdwh WIS zlwk prqrsro dqg frpshwlwlrq/ zkhuh djjuhjdwh WIS lv ghqhg lq h{suhvvlrq +615,1 Wkh prgho suhglfwv wkdw djjuhjdwh WIS zlwk frpshwlwlrq eh wlphv odujhu wkdq zlwk prqrsro 1 Wkh hvwlpdwhv ri Kdoo dqg Mrqhv +4<<<, vkrz wkdw djjuhjdwh WIS glhuhqfhv ri d idfwru ri 5 wr 6 duh frpprq/ hyhq wkrxjk wkh frqwuro iru kxpdq fdslwdo glhuhqfhv1 Iru h{dpsoh/ wkh hvwlpdwh djjuhjdwh WIS glhuhqfhv ehwzhhq wkh XV dqg Lqgld ri d idfwru ri 518 dqg ehwzhhq Lwdo dqg Lqgld ri d idfwru ri 51<1 Jlyhq wkdw rxu prgho devwudfwv iurp kxpdq fdslwdo dffxpxodwlrq/ glhuhqfhv lq djjuhjdwh WIS ri d idfwru duh zhoo zlwklq uhdvrq1 Wdeoh 6= Glhuhqw shufhqwlohv ri frxqwulhv +S iru frpshwlwlrq/ 6 iru prqrsro / R W iru lqwhuqdwlrqdo sulfhv,? t E JR?IcR W t EKJ?IcR W R re JR?I R rekj?i t ESc R W t E6c R W D SJ,Ä6? 2 SJ,Ä6? 43 6317 :1: 431; 3168 53 4;16 817 :1: 3175 63 4418 719 916 3187 73 :1: 617 717 318: Wr hvwdeolvk wkh urexvwqhvv ri rxu uhvxowv/ zh frqgxfw vhqvlwlylw dqdo vlv zlwk uhvshfw wr wkh qxpehu ri frxqwulhv dqg wkh sdudphwhu ydoxhv1 Zlwk uhvshfw wr wkh qxpehu ri frxqwulhv/ zh h{soruh krz wkh prgho*v suhglfwlrqv fkdqjh li zh fdoleudwh B/ k/ / dqg/ iurp wkh gdwd ri wkh ulfkhvw dqg srruhvw 2f/ f/ dqgef shufhqwlohv ri frxqwulhv1 Lq wklv h{huflvh/ zh ohdyh wkh rwkhu sdudphwhu ydoxhv xqfkdqjhg1 Wkh uhvxowv duh vxppdul}hg lq Wdeoh 61 Wkh uvw froxpq olvwv wkh shufhqwlohv frqvlghuhg/ wkh vhfrqg froxpq olvwv wkh udwlrv ri wkh shu0fdslwd lqfrph ohyhov lq wkh ulfkhvw yhuvxv wkh srruhvw? shufhqwlohv/ wkh wklug froxpq olvw wkh udwlrv ri wkh uhodwlyh sulfhv ri vhuylfhv lq wkh ulfkhvw yhuvxv wkh srruhvw?wk shufhqwlohv/ wkh irxuwk froxpq olvwv wkh udwlrv ri wkh shu0fdslwd lqfrph ohyhov lq prgho hfrqrp zlwk frpshwlwlrq yhuvxv prqrsro / wkh odvw froxpq olvwv wkh udwlr ri wkh irxuwk glylghg e wkh vhfrqg froxpq1 Wkh glhuhqfhv lq lqfrph dqg lq wkh uhodwlyh sulfh udwlrv idoo dv zh wdnh dyhudjhv ryhu pruh frxqwulhv1 Krzhyhu/ wkh sduw ri wkh lqfrph glhuhqfh h{sodlqhg e wkh prgho +froxpq vl{, ulvhv frqvlghudeo +iurp D wr D. shufhqw,1 Wklv vxjjhvwv wkdw rxu uhvxowv duh qrw vshflf wr wkh ulfkhvw dqg srruhvw 43 shufhqwlohv1 Zlwk uhvshfw wr wkh sdudphwhu ydoxhv/ zh h{soruh krz wkh prgho*v suhglf0 wlrqv fkdqjh li zh fkdqjh rqh sdudphwhu ydoxh dw d wlph1 Qrwh wkdw lw grhv qrw pdnh vhqvh wr fkdqjh wkh ydoxhv ri dqg > ehfdxvh wkhlu fkrvhq ydoxhv duh wkh rqo rqhv wkdw pdnh wkh xqlwv lq wkh prgho frlqflghv zlwk wkrvh lq wkh gdwd1 Zh qg wkdw rxu uhvxowv duh urexvw wr fkdqjhv lq / 4/ q/ B/ dqgk1 44 Lq frqwudvw/ Wdeoh 7 vkrzv wkdw wkh uhvxowv duh yhu vhqvlwlyh wr wkh fkrlfh ri /1 Wklv vkrxog qrw eh lqwhusuhwhg dv d zhdnqhvv ri wkh prgho wkrxjk ehfdxvh / phdvxuhv wkh vwuhqjwk ri prqrsro srzhu/ dqg wkdw lv wkh nh yduldeoh1 44 Wkh uhvxowv duh dydlodeoh xsrq uhtxhvw1 55
Wdeoh 7= Glhuhqw prqrsro srzhu +S iru frpshwlwlrq/ 6 iru prqrsro / R W iru lqwhuqdwlrqdo sulfhv, / R r ES R r E6 t EScR W t E6cR W UEScR W t EScR W UE6cR W t E6cR W A8 5 ES A8 5 E6 31<83 913 ;18 315< 3147 816 31<94 :13 <1< 315< 3146 915 31<99 :1: 431; 315< 3145 91: 31<:7 <13 4518 315< 3144 :1< 31<:< 4313 461: 315< 3143 ;1; 8 Glvfxvvlrq Rxu prgholqj fkrlfhv dqg uhvxowv pd udlvh vrph txhvwlrqv1 Wkh dlp ri wklv vhfwlrq lv wr dqwlflsdwh dqg dgguhvv wkh pdlq rqhv1 Zk kdyh zh dwwulexwhg wkh furvv0frxqwu glhuhqfh lq wkh uhodwlyh sulfh ri vhuylfhv wr prqrsro uljkwvb Dq dowhuqdwlyh zrxog kdyh ehhq wr dwwulexwh lw wr d wd{ rq wkh sxufkdvh ri lqwhuphgldwh jrrgv/ zklfk zrxog kdyh ehhq lq wkh vslulw ri zkdw Fkdul hw do +4<<9, dqg Uhvwxffld dqg Xuuxwld +5334, gr lq d rqh0vhfwru prgho1 Wr dqvzhu wklv txhvwlrq/ zh qrz h{soruh eulh wkh txdqwlwdwlyh lpsolfdwlrqv ri vxfk d wd{1 Lq wklv h{huflvh/ zh fdq devwudfw iurp lqvlghuv dqg rxwvlghuv dqg dvvxph wkdw wkhuh lv rqh uhsuhvhqwdwlyh lqglylgxdo zlwk wzr xqlwv ri wlph1 Wkh uhsuhvhqwdwlyh lqglylgxdo wkhq vroyhv= 4@ tr c6 c& nc, r c, 6 " 'f "[ 'f r k q 6 3k 34 4 v1w1 R r r n R 6 6 n R % d& n neb & o'o & n E, r n, 6 nz n A c r c6 c& n c, r c, 6 fc, r n, 6 2c & f : f jlyhq, r dqg, 6 duh wkh dprxqwv ri wlph doorfdwhg wr wkh vhuylfh vhfwru dqg wkh pdqxidfwxulqj vhfwru/ dqg A lv wkh oxps0vxp uhedwh ri wkh wd{ uhyhqxh rq sxufkdvhv ri pdqxidfwxuhg jrrgv= A ' ] 2 R 5 5 _ 56