BANK OF FINLAND
DISCUSSION PAPERS
7
•
2001
Anssi Rantala
Research Department
21.5.2001
Does monetary union
reduce employment?
Suomen Pankin keskustelualoitteita Finlands Banks diskussionsunderlag
KWWS ZZZERIIL 6XRPHQ3DQNNL %DQNRI)LQODQG
BANK OF FINLAND
DISCUSSION PAPERS
7
•
2001
Anssi Rantala*
Research Department
21.5.2001
Does monetary union
reduce employment?
The views expressed are those of the author and do not necessarily reflect the views of the Bank of Finland.
I would like to thank Seppo Honkapohja, Juha Tarkka, Jouko Vilmunen, Matti Virén and especially Erkki Koskela for helpful comments and suggestions. I would also like to thank the research department of the Bank of Finland for its great hospitality. Earlier versions of this paper have been presented in RUESG workshop, at the XXIII Symposium of Finnish Economists in Turku and at the economist seminar in the Bank of Finland. The usual disclaimer applies.
* Visiting scholar at the Bank of Finland Research Department. Present address: Research Unit on Economic Structures and Growth (RUESG), University of Helsinki, P.O. Box 10, FIN-00014 University of Helsinki, Finland. E-mail: [email protected].
Suomen Pankin keskustelualoitteita Finlands Banks diskussionsunderlag
http://www.bof.fi ISBN 951-686-715-4 ISSN 0785-3572 (print) ISBN 951-686-716-2 ISSN 1456-6184 (online)
3
Does monetary union reduce employment?
Bank of Finland Discussion Papers 7/2001
Anssi Rantala
Research Department
Abstract
We use a two-country monetary model with unionized labor markets and open-economy spillovers to study the macroeconomic consequences of the formation of a monetary union. It is shown that the monetary regime affects the trade-off between real consumer wages and employment faced by the unions. Consequently, the equilibrium employment is endogenous and depends on the monetary regime. In particular, a switch from a floating exchange rate regime to a monetary union improves employment, provided that the degree of central bank conservatism is sufficiently high, whereas with low degrees of conservatism employment falls. Inflation is higher in a monetary union with all finite degrees of central bank conservatism. In addition, we consider an asymmetric fixed exchange rate regime as an alternative starting position for a monetary union. All results are derived assuming that labor unions are only interested in employment and real wages (not directly inflation) and that all structural parameters of the model remain unchanged when a monetary union is established.
Key words: monetary union, employment, labor unions, open-economy spillovers, central bank conservatism
Heikentääkö rahaliitto työllisyyttä?
Suomen Pankin keskustelualoitteita 7/2001
Anssi Rantala Tutkimusosasto
Tiivistelmä
Tässä työssä tarkastellaan rahaliiton muodostamisen makrotaloudellisia vaikutuk-sia kahden maan mallissa, jossa työmarkkinat ovat järjestäytyneet ja maiden välil-lä on ulkoisvaikutuksia. Tarkastelut perustuvat oletukseen, että ammattiliitot eivät suoranaisesti ole kiinnostuneet inflaatiosta, vaan työllisyydestä ja reaalipalkoista, ja että talouksien rakennetta kuvaavat parametrit eivät muutu, kun rahaliitto perus-tetaan. Työssä osoitetaan, että valuuttakurssijärjestelmä vaikuttaa ammattiliiton havaitsemaan tavoiteristiriitaan palkan ja työllisyyden välillä. Siten tasapainotyöl-lisyys määräytyy endogeenisesti ja riippuu siis valuuttakurssijärjestelmästä. Siir-ryttäessä kelluvien valuuttakurssien järjestelmästä rahaliittoon työllisyys paranee, jos keskuspankki on riittävän konservatiivinen. Jos keskuspankki taas ei ole riittä-vän konservatiivinen, työllisyys heikkenee. Inflaatio on rahaliitossa huomattavam-pi keskuspankin konservatiivisuuden asteesta riippumatta. Lisäksi työssä tarkas-tellaan epäsymmetristä kiinteän valuuttakurssin järjestelmää vaihtoehtoisena alku-asetelmana ennen rahaliittoa.
Asiasanat: rahaliitto, työllisyys, ammattiliitot, avotalouden ulkoisvaikutukset, kes-kuspankin konservatiivisuus
5
Contents
Abstract... 3 1 Introduction... 7 2 The model ... 10 2.1 Basic structure ... 102.2 Labor union and central bank preferences... 11
2.3 Timing of the model ... 13
3 Monetary policy and wage setting ... 14
3.1 Monetary policy in a floating exchange rate regime ... 14
3.2 Monetary policy in a monetary union ... 15
3.3 Wage setting and macroeconomic outcomes ... 17
4 Monetary union ... 19
4.1 Employment and inflation effects of central bank conservatism ... 19
4.2 Main results: employment and inflation in a monetary union... 20
5 Alternative starting position: an asymmetric fix... 24
5.1 The setup ... 24
5.2 Employment and inflation in an asymmetric fix ... 25
5.3 A switch from an asymmetric fix to a monetary union ... 26
6 Concluding remarks ... 27
Appendix A... 29
4 Lqwurgxfwlrq
Wklv sdshu lqyhvwljdwhv wkh h hfwv ri wkh irupdwlrq ri d prqhwdu| xqlrq rq hpsor|phqw dqg lq dwlrq1 D odujh olwhudwxuh rq prqhwdu| srolf| jdphv lqlwl0 dwhg e| wkh vhplqdo frqwulexwlrqv ri N|godqg dqg Suhvfrww +4<::, dqg Eduur dqg Jrugrq +4<;6, vkrzv wkdw xqghu d zlgh udqjh ri dvvxpswlrqv prqhwdu| srolf| lv qhxwudo lq wkh orqj uxq dqg wkh htxloleulxp udwh ri hpsor|phqw lv gh0 whuplqhg rqo| e| uhdo idfwruv ri wkh hfrqrp|1 Khqfh hpsor|phqw vkrxog qrw eh d hfwhg e| wkh hvwdeolvkphqw ri d prqhwdu| xqlrq zklfk lv mxvw d fkdqjh ri prqhwdu| uhjlph1 Krzhyhu/ lq wkh suhvhqfh ri qrq0dwrplvwlf oderu xqlrqv wkh qrq0qhxwudolw| uhvxow ri prqhwdu| srolf| hphujhv dqg wkh frqvhuydwlvp ri wkh fhqwudo edqn zloo kdyh d v|vwhpdwlf h hfw rq htxloleulxp hpsor|phqw14 Wklv
kdv ehhq vkrzq hj e| Fxnlhupdq dqg Olssl +4<<<,/ Olssl +4<<<, dqg Vrvnlfh dqg Lyhuvhq +5333,1
Uhfhqw sdshuv e| Ju qhu dqg Khihnhu +4<<<, dqg Fxnlhupdq dqg Olssl +5333, dqdo|}h wkh h hfwv ri d prqhwdu| xqlrq rq hfrqrplf shuirupdqfh lq d prgho zlwk qrq0dwrplvwlf oderu pdunhwv1 Wkh| vkrz wkdw wkh hvwdeolvkphqw ri d prqhwdu| xqlrq zloo ghfuhdvh hpsor|phqw dqg lqfuhdvh lq dwlrq/ li qdwlrqdo oderu xqlrqv duh lq dwlrq dyhuvh/ lh lq dgglwlrq wr wudglwlrqdo uhdo zdjh dqg hpsor|phqw wdujhwv oderu xqlrqv duh dovr frqfhuqhg derxw lq dwlrq shu vh1 Lq dwlrq dyhuvlrq prghudwhv zdjh fodlpv dqg lpsuryhv hpsor|phqw/ ehfdxvh xqlrqv duh zloolqj wr frpsurplvh ryhu wkh uhdo zdjh wdujhw lq rughu wr uhgxfh lq dwlrq1 Lq d prqhwdu| xqlrq hdfk xqlrq shufhlyhv/ wkdw lwv zdjh ghpdqg zloo kdyh d vpdoohu h hfw rq lq dwlrq/ zklfk pdnhv xqlrqv pruh djjuhvvlyh lq zdjh vhwwlqj1 Kljkhu uhdo zdjhv ohdg wr orzhu hpsor|phqw wkurxjk uhgxfhg ghpdqg iru oderu1 Krzhyhu/ wklv uhvxow ghshqgv fuxfldoo| rq wkh dvvxphg lq dwlrq dyhuvlrq ri wkh oderu xqlrqv1 Devhqw wkdw/ prqhwdu| xqlrq kdv qr h hfw rq hpsor|phqw dqg lq dwlrq15
Lq wklv sdshu/ zh vkrz wkdw lq d prgho zlwk rshq0hfrqrp| vslooryhuv/ d vzlwfk iurp d rdwlqj h{fkdqjh udwh uhjlph wr d prqhwdu| xqlrq zloo d hfw pdfurhfrqrplf shuirupdqfh/ hyhq zkhq qdwlrqdo oderu xqlrqv duh qrw dyhuvh wr lq dwlrq dqg doo vwuxfwxudo sdudphwhuv ri wkh prgho uhpdlq xqfkdqjhg1 Lq sduwlfxodu/ hpsor|phqw zloo lqfuhdvh/ surylghg wkdw wkh ghjuhh ri fhqwudo edqn frqvhuydwlvp lv vx!flhqwo| kljk/ zkhuhdv zlwk orz ghjuhhv ri frqvhuydwlvp hp0 sor|phqw idoov1 Uhodwlyh lq dwlrq shuirupdqfh zloo qrw ghshqg rq fhqwudo edqn frqvhuydwlvp/ lq dwlrq zloo dozd|v lqfuhdvh lq d prqhwdu| xqlrq1 Lq dgglwlrq/ zh frqvlghu dq dv|pphwulf {hg h{fkdqjh udwh uhjlph/ zkhuh rqh frxqwu| kdv xqlodwhudoo| frpplwwhg wr d {hg h{fkdqjh udwh ylv0ã0ylv wkh rwkhu frxqwu| ehiruh wkh prqhwdu| xqlrq lv hvwdeolvkhg16 Zh vkrz wkdw lq wklv dv|pphwulf
4Zh zloo xvh wkh whup *frqvhuydwlvp* wr ghvfuleh wkh uhodwlyh zhljkw ri lq dwlrq lq wkh
fhqwudo edqn remhfwlyh ixqfwlrq1 Hovhzkhuh lq wkh olwhudwxuh *lqghshqghqfh* lv rffdvlrqdoo| xvhg lqwhufkdqjhdeo| zlwk *frqvhuydwlvp* +vhh hj Fxnlhupdq dqg Olssl +4<<<,,1 Ehujhu/ gh Kddq dqg Hlm!qjhu +5333, glvfxvvhv wkh gl huhqfh ehwzhhq wkh wzr frqfhswv1
5Fxnlhupdq dqg Olssl +5333, vkrz wkdw prqhwdu| xqlrq zloo kdyh uhdo h hfwv hyhq zlwk0
rxw lq dwlrq dyhuvlrq/ li wkhuh duh pruh wkdq rqh oderu xqlrq lq hdfk frxqwu|1
6Dq dv|pphwul{ { fdq eh wkrxjkw dv d vlpsoh dqg fuxgh ghvfulswlrq ri wkh H{fkdqjh
Udwh Phfkdqlvp +HUP, lq Hxursh1 Ju qhu dqg Khihnhu +4<<<, dqg Fxnlhupdq dqg Olssl +5333, frqvlghu dovr d fkdqjh iurp wkh HUP wr d prqhwdu| xqlrq1
{ fdvh hpsor|phqw zloo idoo lq wkh ohdghu frxqwu| dqg lqfuhdvh lq wkh iroorzhu frxqwu| zkhq d prqhwdu| xqlrq lv hvwdeolvkhg1
Rxu prgholqj iudphzrun lv d wzr0frxqwu| prqhwdu| prgho zlwk rshq0 hfrqrp| vslooryhuv dqg qdwlrqdo oderu xqlrqv1 Hdfk frxqwu| lv vshfldol}hg lq d surgxfwlrq ri d wudghg jrrg/ zklfk duh lpshuihfw vxevwlwxwhv lq frqvxps0 wlrq1 Erwk jrrgv duh frqvxphg lq erwk frxqwulhv/ dqg wkhlu uhodwlyh ghpdqg ghshqgv rq wkh uhodwlyh sulfh ehwzhhq wkh wzr jrrgv1 Lpshuihfw vxevwlwxwdelo0 lw| ehwzhhq wkh wzr jrrgv fuhdwhv wzr srolf| vslooryhuv1 Fhqwudo edqnv fdq d hfw wkh uhdo h{fkdqjh udwh zlwk wkhlu srolflhv/ dqg wkxv wkhlu dfwlrqv zloo kdyh uhshufxvvlrqv lq erwk frxqwulhv1 Qdwlrqdo oderu xqlrqv fkrrvh wkhlu qrp0 lqdo zdjhv ehiruh prqhwdu| srolflhv duh vhw/ dqg wkhlu zdjh vhwwlqj zloo dovr d hfw wkh uhdo h{fkdqjh udwh1 Gxh wr wkhvh vslooryhuv/ prqhwdu| uhjlph zloo lq jhqhudo d hfw wkh oderu xqlrq*v uhdo zdjhhpsor|phqw wudgh0r / lh wkh uhdo frqvxphu zdjh hodvwlflw| ri oderu ghpdqg lv hqgrjhqrxv lq rxu prgho17
Dowkrxjk rxu prgho vshfl fdwlrq lv forvho| uhodwhg wr wkdw ri Ju qhu dqg Khihnhu +4<<<,/ wkh phfkdqlvp ehklqg wkh uhvxowv lv frpsohwho| gl huhqw1 Lq wkhlu prgho/ sxufkdvlqj srzhu sdulw| lv dvvxphg wr krog/ dqg wkhuhiruh hfr0 qrplf shuirupdqfh xqghu d rdwlqj h{fkdqjh udwh lv vroho| ghwhuplqhg e| grphvwlf srolf| dfwlrqv1 Lq d prqhwdu| xqlrq hdfk oderu xqlrq wkxv ehfrphv vpdoohu lq d vhqvh wkdw wkhlu zdjh fodlpv zloo kdyh d vpdoohu h hfw rq wkh jhqhudo sulfh ohyho/ zklfk lv d srolf| wdujhw ri wkh prqhwdu| dxwkrulwlhv1 Dv phqwlrqhg deryh/ wklv pdnhv oderu xqlrqv pruh djjuhvvlyh lq d prqhwdu| xqlrq surylghg wkdw wkh| duh lq dwlrq dyhuvh1 Lq rxu prgho/ wklv vl}h h hfw lv devhqw1 Gxh wr wkh vslooryhu h hfwv/ erwk grphvwlf dqg iruhljq zdjhv d hfw wkh hfrqrplf shuirupdqfh lq erwk frxqwulhv hyhq lq d rdwlqj h{fkdqjh udwh uhjlph1 Wkh rqo| fkdqjh lq wkh vwudwhjlf hqylurqphqw ri wkh oderu xqlrqv lv wkh uhsodfh0 phqw ri wkh qdwlrqdo fhqwudo edqnv e| d frpprq fhqwudo edqn1 Srolf| wdujhwv dv zhoo dv srolf| lqvwuxphqwv ri wkh prqhwdu| srolf| dxwkrulwlhv zloo fkdqjh lq d prqhwdu| xqlrq1 Lw lv wkxv txlwh lqwxlwlyh wkdw wkh srolf| uhdfwlrq wr d xqlodw0 hudo zdjh lqfuhdvh e| rqh oderu xqlrq zloo fkdqjh dv zhoo1 Fkdqjhg prqhwdu| srolf| uhdfwlrq zloo/ lq wxuq/ lqgxfh gl huhqw zdjh vhwwlqj ehkdylru iurp wkh oderu xqlrqv1 Pdfurhfrqrplf frqvhtxhqfhv ri d prqhwdu| xqlrq duh d sulrul dpeljxrxv/ exw dv phqwlrqhg/ wkh| zloo ghshqg rq fhqwudo edqn frqvhuydwlvp1 Prqhwdu| xqlrq lv dq h{wuhph irup ri prqhwdu| srolf| frrshudwlrq18 Lq
olwhudwxuh/ frrshudwlrq xvxdoo| phdqv wkdw fhqwudo edqnv vhw wkhlu srolf| lq0 vwuxphqwv mrlqwo| vr dv wr pd{lpl}h +plqlpl}h, d zhljkwhg dyhudjh ri wkhlu srolf| remhfwlyhv1 Urjr +4<;8e, vkrzhg/ wkdw prqhwdu| srolf| frrshudwlrq fdq eh frxqwhusurgxfwlyh lq wkh suhvhqfh ri grphvwlf fuhglelolw| sureohpv1 Lq d rdwlqj h{fkdqjh udwh uhjlph prqhwdu| srolflhv duh frqvwudlqhg e| ihduv ri lq dwlrqdu| uhdo h{fkdqjh udwh ghsuhfldwlrq1 Lq d frrshudwlyh srolf| uhjlph wklv uhdo h{fkdqjh udwh h hfw lv lqwhuqdol}hg1 Wkhuhiruh prqhwdu| srolflhv duh pruh h{sdqvlrqdu| dqg lq dwlrq zloo eh kljkhu1 Prqhwdu| xqlrq gl huv iurp d frrshudwlyh srolf| uhjlph lq rqh lpsruwdqw uhvshfw1 D frpprq fhqwudo edqn kdv rqo| rqh srolf| lqvwuxphqw dydlodeoh1 Wklv lv d fuxfldo gl huhqfh lq rxu 7Kroghq +4<<<, vkrzv wkdw prqhwdu| uhjlph d hfwv wkh uhdo zdjhhpsor|phqw wudgh0r
lq d vpdoo rshq hfrqrp| iudphzrun1
8Iru d frpsuhkhqvlyh vxuyh| rq lqwhuqdwlrqdo srolf| frruglqdwlrq/ vhh Shuvvrq dqg
vhwxs zlwk odujh zdjh vhwwhuv1 Lq d frrshudwlyh uhjlph/ d xqlodwhudo zdjh lq0 fuhdvh e| rqh qdwlrqdo oderu xqlrq zloo lqgxfh dv|pphwulf uhvsrqvh iurp wkh wzr fhqwudo edqnv/ dv orqj dv wkh h hfwv lq wkh wzr hfrqrplhv duh dv|pphwulf1 Krzhyhu/ zkhq d frpprq fhqwudo edqn idfhv d xqlodwhudo zdjh lqfuhdvh lq rqh frxqwu|/ lwv srolf| zloo/ e| gh qlwlrq/ eh v|pphwulf1 Mhqvhq +4<<6, dqdo|vhv prqhwdu| srolf| frrshudwlrq lq wkh vdph vhwxs dv rxuv/ dqg vkrzv wkdw lq d frrshudwlyh uhjlph hpsor|phqw lv dozd|v orzhu wkdq lq d rdwlqj h{fkdqjh udwh uhjlph19
Rxu irfxv lv rq wkh gluhfw h hfw ri d prqhwdu| xqlrq rq htxloleulxp hp0 sor|phqw wkurxjk fkdqjhv lq wkh vwudwhjlf lqwhudfwlrq ehwzhhq fhqwudo edqnv dqg oderu xqlrqv1 Wkhuh duh dw ohdvw wzr dgglwlrqdo fkdqqhov wkurxjk zklfk htxloleulxp hpsor|phqw pljkw eh d hfwhg1
Iluvw/ prqhwdu| lqwhjudwlrq pd| hqkdqfh surgxfw pdunhw lqwhjudwlrq dqg wkxv dovr frpshwlwlrq yld orzhu wudqvdfwlrq frvwv/ wkh glvdsshdudqfh ri wkh h{fkdqjh udwh ulvnv dqg hdvlhu sulfh frpsdulvrqv ehwzhhq frxqwulhv1 Wklv kdv ehhq d frpprq dujxphqw lq srolf| ghedwhv frqfhuqlqj wkh h hfwv ri wkh Hfr0 qrplf dqg Prqhwdu| Xqlrq +HPX, lq Hxursh1 Lqwhqvl hg frpshwlwlrq lq wkh surgxfw pdunhwv lpsuryh hpsor|phqw lq wzr zd|v1 Oderu ghpdqg ehfrphv pruh hodvwlf zlwk uhvshfw wr zdjhv dqg upv* pdun0xsv duh uhgxfhg1 Krzhyhu/ dv Exugd +4<<<, dqg Fdopiruv +5333, srlqw rxw/ lw lv qrw fohdu wkdw lqfuhdvhg surgxfw pdunhw lqwhjudwlrq zloo irvwhu frpshwlwlrq1 Rq wkh frqwudu|/ lqwhjud0 wlrq pd| khos pxowlqdwlrqdo frpsdqlhv wr shqhwudwh wkh pdunhwv dqg wkhuhiruh frpshwlwlrq fdq dfwxdoo| ghfuhdvh1
Vhfrqg/ lqfhqwlyhv iru oderu pdunhw uhirupv fdq eh d hfwhg e| wkh hvwde0 olvkphqw ri d prqhwdu| xqlrq1 Wklv kdv ehhq dqdo|}hg e| Fdopiruv +4<<<, dqg Vlehuw dqg Vxwkhuodqg +5333,1 Khuh/ uhirup uhihuv wr dq| vwuxfwxudo uhirup lq wkh oderu pdunhwv zklfk uhgxfhv wkh htxloleulxp udwh ri xqhpsor|phqw1 Wkh jryhuqphqw fdq fkrrvh wkh ohyho ri uhirup ehiruh oderu xqlrqv vhw wkhlu zdjhv dqg ehiruh prqhwdu| srolf| lv vhw1 Wkh uhirup orzhuv wkh htxloleulxp udwh ri xqhpsor|phqw/ exw lw hqwdlov d frvw wr wkh jryhuqphqw1 Wklv frvw fdq dulvh iru ydulrxv uhdvrqv/ hj wkh uhirup uhgxfhv uhdo zdjhv ru ehfdxvh zrunhuv ydoxh oderu pdunhw lqvwlwxwlrqv lq wkhpvhoyhv1 Wkh pdlq dujxphqw ri wkhvh sdshuv lv lqwxlwlyh1 Rxwvlgh d prqhwdu| xqlrq d orz ohyho ri uhirup lpsolhv wkdw xq0 hpsor|phqw lv kljk1 Lq dgglwlrq/ lq wkh suhvhqfh ri dq lq dwlrq eldv/ lq dwlrq zloo eh kljk dv zhoo1 Li wkh jryhuqphqw fkrrvhv d orz ohyho ri uhirup lq d prqhwdu| xqlrq/ wkh uvw h hfw lv reylrxvo| wkh vdph1 Krzhyhu/ wkh vhfrqg h hfw lv uhgxfhg1 Kljk xqhpsor|phqw lq rqh phpehu frxqwu| kdv rqo| d vpdoo h hfw rq wkh duhd0zlgh dyhudjh ohyho ri xqhpsor|phqw1 Wkh frpprq fhqwudo edqn fduhv derxw wkh dyhudjh ohyho ri xqhpsor|phqw dqg wkxv lq dwlrq eldv lv vpdoohu lq d prqhwdu| xqlrq1 Wkhuhiruh/ wkh jryhuqphqw*v lqfhqwlyhv iru oderu pdunhw uhirup duh uhgxfhg lq d prqhwdu| xqlrq dqg wkh fkrvhq uhirup ohyho zloo eh orzhu1
Wklv sdshu lv rujdql}hg dv iroorzv1 Vhfwlrq 5 suhvhqwv wkh prgho vhwxs1 Wkh prgho lv vroyhg dqg pdfurhfrqrplf rxwfrphv lq erwk prqhwdu| uhjlphv duh ghulyhg lq Vhfwlrq 61 Ilqdoo|/ Vhfwlrq 7 dqdo|}hv wkh pdfurhfrqrplf frq0 9Prqhwdu| srolf| frrshudwlrq zlwk lq dwlrq dyhuvh oderu xqlrqv lv wuhdwhg lq Mhqvhq
+4<<:,1
vhtxhqfhv ri wkh irupdwlrq ri d prqhwdu| xqlrq1 D vzlwfk iurp d dv|pphwulf { wr d prqhwdu| xqlrq lv vwxglhg lq Vhfwlrq 81 Vhfwlrq 9 frqfoxghv1
5 Wkh prgho
514 Edvlf vwuxfwxuh
Lq wklv vhfwlrq/ zh ghvfuleh wkh edvlf vwuxfwxuh ri wkh prgho1 Rxu prgho vshf0 l fdwlrq lq d rdwlqj h{fkdqjh udwh uhjlph iroorzv Mhqvhq +4<<6/ 4<<:,/ zklfk d yduldqw ri wkh vwdqgdug wzr0frxqwu| prqhwdu| prgho vwxglhg lq Fdq}rqhul dqg Khqghuvrq +4<;;,1 Zh zloo h{whqg Mhqvhq*v prgho wr fryhu wkh fdvh ri d prqhwdu| xqlrq1
Wkhuh duh wzr htxdo vl}hg frxqwulhv/ hdfk vshfldol}hg lq wkh surgxfwlrq ri d wudghg jrrg1 Doo yduldeohv duh lq qdwxudo orjdulwkpv dqg duh h{suhvvhg dv ghyl0 dwlrqv iurp d ixoo hpsor|phqw htxloleulxp1 Iruhljq yduldeohv duh ghqrwhg zlwk dq dvwhulvn1 Wkurxjkrxw wkh sdshu/ zh dvvxph wkdw doo vwuxfwxudo sdudphwhuv duh htxdo dfurvv frxqwulhv1
Krph dqg iruhljq rxwsxw vxssolhv/ ghqrwhg e| + dqg +W/ duh ghwhuplqlvwlf
ixqfwlrqv ri hpsor|phqwv/ ? dqg ?W/ vxemhfw wr ghfuhdvlqj uhwxuqv wr vfdoh
+ ' k?c +W ' k?W f k +4,
Wklv irupxodwlrq fdq eh ghulyhg iurp d Free0Grxjodv surgxfwlrq ixqfwlrq e| {lqj wkh fdslwdo vwrfn dqg qrupdol}lqj lw wr xqlw|1 Oderu lv lppreloh ehwzhhq frxqwulhv dqg vxssolhg lqhodvwlfdoo|1 Sur w pd{lpl}lqj upv hpsor| oderu xs wr wkh srlqw dw zklfk uhdo surgxfw zdjhv duh htxdo wr pdujlqdo surgxfwv ri oderu
E R ' E k ?c EW RW ' E k ?W +5,
zkhuh qrplqdo zdjhv duh ghqrwhg e| dqg W/ zkloh R dqg RW duh wkh surgxfw
sulfhv ri krph dqg iruhljq frxqwulhv uhvshfwlyho|1:
Uhodwlyh rxwsxw ghpdqg lv d ixqfwlrq ri wkh uhdo h{fkdqjh udwh/ 5
+ +W ' B5 B : f +6,
zkhuh wkh sdudphwhu B fdq eh lqwhusuhwhg dv lqgh{lqj wkh ghjuhh ri vxevwl0 wxwdelolw| ehwzhhq krph dqg iruhljq jrrgv1 D odujh ydoxh ri B lpsolhv/ wkdw wkh wzr jrrgv duh forvh vxevwlwxwhv dqg wkhuhiruh d vpdoo fkdqjh lq wkh uhdo h{fkdqjh udwh eulqjv derxw d odujh fkdqjh lq uhodwlyh ghpdqg1 Wkh uhdo h{0 fkdqjh udwh lv gh qhg dv wkh uhodwlyh sulfh ri wkh iruhljq jrrg lq whupv ri wkh grphvwlf jrrg/ h{suhvvhg lq krph frxqwu| fxuuhqf|
5 ' e n RW R +7,
zkhuh wkh qrplqdo h{fkdqjh udwh/ ghqrwhg e| e/ lv wkh grphvwlf sulfh ri iruhljq fxuuhqf|1 Wkxv d ghfuhdvh lq 5 lpsolhv wkdw grphvwlf fxuuhqf| dssuhfldwhv :Wkh pdujlqdo surgxfw ri oderu lv dfwxdoo| +z s, @ oq +4 , q/ exw wkh frqvwdqw
lq uhdo whupv1 E| gh qlwlrq/ lq d prqhwdu| xqlrq +wkh orj ri, wkh qrplqdo h{fkdqjh udwh lv }hur/ dqg wkh uhdo h{fkdqjh udwh lv vlpso|
5 ' RW R +8,
Dvvxplqj d frqvwdqw vkduh ri lpsruwv lq frqvxpswlrq/ q/ wkh grphvwlf dqg iruhljq frqvxphu sulfh lqglfhv +FSLv, duh jlyhq e|
Z ' E q R n q Ee n RW ' R n q5 +9,
ZW ' q ER e n E q RW ' RW q5 +:,
Zh uhvwulfw wkh ydoxh ri q wr olh lq wkh lqwhuydo f q
2/ zklfk lv htxlydohqw
wr dvvxplqj d krph eldv lq frqvxpswlrq1; Wklv vhhpv wr eh d udwkhu plog
uhvwulfwlrq/ dv wkhuh lv d odujh hpslulfdo olwhudwxuh rq wkh krph eldv lq wudgh vkrzlqj wkdw shrsoh kdyh d vwurqj suhihuhqfh iru frqvxpswlrq ri wkhlu krph jrrgv1<
Prqh| ghpdqg lv dvvxphg wr eh sursruwlrqdo wr qrplqdo lqfrph dqg rqo| grphvwlf uhvlghqwv krog grphvwlf prqh|1 Wkh prqh| pdunhwv duh lq htxlole0 ulxp zkhq qrplqdo prqh| vxssolhv/ 6 dqg 6W/ vdwlvi| vlpsoh txdqwlw| htxd0
wlrqv
6 ' R n +c 6W ' RWn +W +;,
Lq d prqhwdu| xqlrq wkhuh lv rqo| rqh fxuuhqf|/ zklfk lv khog e| wkh uhvlghqwv lq erwk frxqwulhv1 Prqhwdu| htxloleulxp uhtxluhv wkdw xqlrq0zlgh prqh| ghpdqg htxdov xqlrq0zlgh prqh| vxsso|1 Iroorzlqj Odqh +4<<9, zh dvvxph wkdw prqh| ghpdqg lv sursruwlrqdo wr dyhudjh qrplqdo lqfrph ri wkh phpehu frxqwulhv
6L '
2ER n + n
2ERWn +W +<,
zkhuh 6L lv wkh xqlrq0zlgh prqh| vxsso|1 Zh dvvxph wkdw wkh frpprq prqh|
vxsso| lv vsuhdg rxw htxdoo| rq erwk phpehu frxqwulhv/ dqg wkxv wkh frpprq fhqwudo edqn fdq*w d hfw wkh glvwulexwlrq ri prqh| vxsso| dfurvv frxqwulhv1
515 Oderu xqlrq dqg fhqwudo edqn suhihuhqfhv
Lq hdfk frxqwu| wkhuh lv d vlqjoh prqrsro| xqlrq/ zklfk vhwv wkh qrplqdo zdjh udwh1 Xqlrqv* xwlolw| ixqfwlrqv duh jlyhq e|
L ' E Z 2?2c LW ' EW ZW
2?W2 +43,
zkhuh E Z dqg EW ZW duh wkh uhdo frqvxphu zdjhv dqg ? dqg ?W duh
hpsor|phqwv h{suhvvhg dv ghyldwlrqv iurp wkh ixoo hpsor|phqw htxloleulxp1 Xqlrqv* xwlolw| ghshqgv srvlwlyho| rq uhdo zdjhv dqg qhjdwlyho| rq wkh jds ;Wklv lv d frpprq dvvxpswlrq lq wkh rshq0hfrqrp| pdfur olwhudwxuh/ vhh hj Exlwhu/
Fruvhwwl dqg Shvhqwl +4<<8,/ Odqh +4<<9/ 5333, dqg ]huyr|ldqql +4<<:,1
<Vhh hj Revwihog dqg Urjr +5333, dqg uhihuhqfhv wkhuhlq1
ehwzhhq uhdol}hg dqg ixoo hpsor|phqw ohyhov/ zklfk lv frqvlvwhqw zlwk wudgl0 wlrqdo oderu xqlrq wkhru| +vhh hj Rvzdog +4<;8,,143 Wklv lv zkhuh rxu sdshu
ghsduwv iurp pxfk ri wkh uhfhqw olwhudwxuh rq vwudwhjlf lqwhudfwlrq ehwzhhq oderu xqlrqv dqg wkh fhqwudo edqn1 Xvxdoo| lw lv dvvxphg wkdw oderu xqlrqv fduh qrw rqo| derxw wkh uhdo zdjh dqg hpsor|phqw/ exw dovr derxw lq dwlrq
shu vh144
Zk| vkrxog oderu xqlrqv eh lqwhuhvwhg lq lq dwlrqB Wkh vwdqgdug h{sodqd0 wlrq lv wkdw xqlrq phpehuv kdyh vrph qrq0lqgh{hg qrplqdo dvvhwv ru lqfrphv rwkhu wkdq wkhlu vdodulhv/ vxfk dv edqn ghsrvlwv ru shqvlrqv/ zklfk ydoxhv duh hurghg e| lq dwlrq1 Krzhyhu/ iru h{dpsoh shqvlrqhuv duh xvxdoo| qrw oderu xqlrq phpehuv/ dqg lw vhhpv xqolnho| wkdw xqlrqv zrxog orrn diwhu wkhlu lq0 whuhvwv1 Lq pdq| sdshuv oderu xqlrqv* lq dwlrq dyhuvlrq lv dfwxdoo| gulylqj wkh uhvxowv/ dqg wkhuhiruh d forvhu orrn rq srvvleoh uhdvrqv iru lq dwlrq dyhu0 vlrq zrxog eh ghvludeoh145 Lq wklv sdshu/ zh devwudfw iurp lq dwlrq dyhuvlrq
dqg dsso| wkh pruh frqyhqwlrqdo irupxodwlrq ri oderu xqlrq suhihuhqfhv jlyhq deryh1
Prqhwdu| srolf| lv vxemhfw wr d fuhglelolw| sureohp lq erwk frxqwulhv dqg wkhuhiruh fhqwudo edqnv dfw lq d glvfuhwlrqdu| pdqqhu1 Srolf| suhihuhqfhv duh jlyhq e| wkh orvv ixqfwlrqv
u ' j?2 n Z2c uW ' j?W2n ZW2 f j 4 +44,
zklfk duh txdgudwlf erwk lq hpsor|phqwv dqg FSLv1 FSL lq dwlrq lv gh qhg dv wkh gl huhqfh ehwzhhq wkh orj sulfh ohyhov lq fxuuhqw dqg suhylrxv shulrgv/ ZZ31 Zlwkrxw orvv ri jhqhudolw|/ zh fdq qrupdol}h wkh suhylrxv shulrg sulfh
ohyho wr }hur/ dqg wkxv FSL dqg FSL lq dwlrq duh htxlydohqw lq wklv vhwxs1 Erwk fhqwudo edqnv wdujhw ixoo hpsor|phqw dqg }hur lq dwlrq1
Wkh sdudphwhu j uh hfwv wkh uhodwlyh zhljkw sodfhg e| wkh fhqwudo edqn rq hpsor|phqw ghyldwlrqv iurp wkh ixoo hpsor|phqw ohyho1 Wkh lqyhuvh ri j
j
wkxv phdvxuhv wkh Urjr +4<;8d, w|sh ri frqvhuydwlvp ri wkh fhqwudo edqn1 Wkh vpdoohu j lv/ wkh pruh wkh fhqwudo edqn fduhv derxw lq dwlrq dqg wkh ohvv derxw hpsor|phqw/ l1h1 lw doorzv hpsor|phqw wr ghyldwh pruh iurp ixoo hpsor|phqw ohyho lq rughu wr sxvk lq dwlrq forvhu wr }hur1 Qrplqdo prqh| vxssolhv/ 6 dqg 6W/ duh wkh srolf| lqvwuxphqwv ri wkh fhqwudo edqnv1
43Wkh ixqfwlrqdo irup ri wkh xwlolw| ixqfwlrq lv dgplwwhgo| txlwh vshfl f1 Krzhyhu/ lw
|lhogv d vhw ri frqyh{ grzqzdug vorslqj lqgl huhqfh fxuyhv lq +q> +z ,, vsdfh zkhq q lv ehorz ixoo hpsor|phqw ohyho1 Wklv irup lv zlgho| xvhg lq orj0olqhdu prghov/ olnho| ehfdxvh sdudooho vkliwv lq oderu ghpdqg fxuyh grhv qrw d hfw hpsor|phqw1 Wklv lpsolhv wkdw wkhuh lv d qdwxudo udwh ri hpsor|phqw/ zklfk lv lqghshqghqw ri ghpdqg frqglwlrqv1
44Vhh Fxnlhupdq dqg Olssl +4<<<, dqg uhihuhqfhv wkhuhlq1 Krzhyhu/ Vrvnlfh dqg Lyhuvhq
+5333, dqg Frulfhool/ Fxnlhupdq dqg Gdopd}}r +5333d, dovr devwudfw iurp oderu xqlrq lq d0 wlrq dyhuvlrq lq d forvhg0hfrqrp| vhwwlqj dqg Frulfhool/ Fxnlhupdq dqg Gdopd}}r +5333e, lq d wzr0frxqwu| prqhwdu| xqlrq prgho1
45D vwhs lqwr wklv gluhfwlrq lv wdnhq lq Ehujhu/ Khihnhu dqg Vfkùe +5334,/ zkhuh lq dwlrq
dyhuvlrq lv hqgrjhql}hg e| dvvxplqj wkdw wkh rxwvlgh rswlrq ri wkh oderu xqlrq lv gh qhg lq qrplqdo whupv1
Lq d prqhwdu| xqlrq wkh frpprq fhqwudo edqn lv dvvxphg wr fduh rqo| derxw wkh dyhudjh hpsor|phqw dqg dyhudjh lq dwlrq46
u ' j ? n ?W 2 2 n Z n ZW 2 2 f j 4 +45,
Oderu xqlrqv dqg fhqwudo edqnv kdyh wkh vdph hpsor|phqw wdujhw1 Dw uvw eoxvk/ wklv vhhpv wr vxjjhvw/ wkdw wkh fhqwudo edqnv grq*w kdyh dq ryhu0 dpelwlrxv hpsor|phqw wdujhw dqg wkhuhiruh wkhuh lv qr wlph0lqfrqvlvwhqf| sureohp lq prqhwdu| srolf|1 Krzhyhu/ lqwurgxflqj d uhdo zdjh wdujhw lqwr wkh xqlrq*v xwlolw| ixqfwlrq eulqjv derxw d wudgh0r ehwzhhq hpsor|phqw dqg wkh uhdo zdjh1 Lq rughu wr dwwdlq d srvlwlyh uhdo zdjh/ wkh xqlrq lv zloolqj wr jlyh xs lwv ixoo hpsor|phqw wdujhw1 Zkhq xqlrqv duh lqwhuhvwhg lq erwk hp0 sor|phqw dqg uhdo zdjhv/ zklfk lv frqvlvwhqw zlwk wkh wudglwlrqdo oderu xqlrq olwhudwxuh/ wkh htxloleulxp udwh ri hpsor|phqw lv hqgrjhqrxv +dqg ehorz ixoo hpsor|phqw, lq wkh prgho1
516 Wlplqj ri wkh prgho
Hyhqwv xqirog dv iroorzv1 +4, Wkh wzr oderu xqlrqv vhw wkhlu qrplqdo zdjhv vlpxowdqhrxvo| wdnlqj wkh rwkhu xqlrq*v zdjh dv jlyhq1 Wkh| kdyh ixoo lqiru0 pdwlrq derxw wkh uhdfwlrq ixqfwlrqv ri wkh fhqwudo edqnv dqg wdnh wkhp lqwr dffrxqw lq wkh zdjh vhwwlqj surfhvv1 Lq rwkhu zrugv/ xqlrqv dfw dv Vwdfnhoehuj ohdghuv djdlqvw wkh fhqwudo edqnv1 +5, Wkh fhqwudo edqnv fkrrvh wkhlu prqh| vxssolhv vlpxowdqhrxvo| wdnlqj wkh rwkhu fhqwudo edqn*v prqh| vxsso| dv zhoo dv suhylrxvo| vhw qrplqdo zdjhv dv jlyhq1 Lq d prqhwdu| xqlrq wkh frpprq fhqwudo edqn vhwv wkh xqlrq0zlgh prqh| vxsso|1 +6, Surgxfwlrq dqg wudgh wdnh sodfh dqg pdfurhfrqrplf rxwfrphv duh uhdol}hg1
Wkh fkrlfh ri prgholqj wkh jdph ehwzhhq oderu xqlrqv dqg fhqwudo edqnv dv d Vwdfnhoehuj jdph lv vxssruwhg e| revhuydwlrqv rq uhdo zruog zdjh vhwwlqj dqg prqhwdu| srolf| sudfwlfhv1 Zdjh frqwudfwv duh xvxdoo| qhjrwldwhg wr fryhu dw ohdvw d |hdu/ zkhuhdv prqhwdu| srolf| fdq eh dgmxvwhg iuhtxhqwo|1 Dv douhdg| phqwlrqhg/ zh dvvxph wkdw wkh fhqwudo edqnv fdqqrw fuhgleo| frpplw wr d srolf| uxoh1 Lq rwkhu zrugv/ zh frqvlghu wkh wlph0frqvlvwhqw +glvfuhwlrqdu|, prqhwdu| srolf|1 Krzhyhu/ wkh ehkdylru ri wkh oderu xqlrqv fdq eh wlph0 lqfrqvlvwhqw dv zhoo1 Lw pd| eh wkh fdvh/ wkdw diwhu wkh fhqwudo edqnv kdyh vhw wkhlu prqh| vxssolhv/ wkh oderu xqlrqv qg lw rswlpdo wr uhqhjh rq wkh zdjh frqwudfwv dqg vhw qhz kljkhu ru orzhu qrplqdo zdjhv1 Wklv srvvlelolw| lv uxohg rxw e| dvvxplqj wkdw qrplqdo zdjh frqwudfwv duh ohjdoo| elqglqj iru wkh zkroh frqwudfw shulrg1
46Lq d v|pphwulf fdvh/ zkhuh doo vwuxfwxudo sdudphwhuv duh wkh vdph dfurvv frxqwulhv/
zh fdq vkrz wkdw vlplodu uvw rughu frqglwlrqv duh rewdlqhg li wkh frpprq fhqwudo edqn plqlpl}hv wkh zhljkwhg dyhudjh ri wkh qdwlrqdo orvv ixqfwlrqv1
6 Prqhwdu| srolf| dqg zdjh vhwwlqj
Lq wklv vhfwlrq/ zh vroyh wkh wzr0vwdjh jdph ehwzhhq oderu xqlrqv dqg fhqwudo edqnv lq erwk prqhwdu| uhjlphv e| edfnzdug lqgxfwlrq1 Vwduwlqj iurp wkh vhfrqg vwdjh/ zh suhvhqw htxloleulxp vwudwhjlhv iru fhqwudo edqnv lq d rdwlqj h{fkdqjh udwh uhjlph dqg iru d frpprq fhqwudo edqn lq d prqhwdu| xqlrq fdvh1 Wkhq zh vroyh wkh zdjh vhwwlqj sureohp dw wkh uvw vwdjh dqg ghulyh htxloleulxp rxwfrphv iru pdfurhfrqrplf yduldeohv1
614 Prqhwdu| srolf| lq d rdwlqj h{fkdqjh udwh uhjlph
Iluvw zh suhvhqw wkh uhgxfhg irupv iru krph dqg iruhljq hpsor|phqwv dqg lq dwlrqv lq d rdwlqj h{fkdqjh udwh uhjlph147 Wkhvh htxdwlrqv duh rewdlqhg
iurp htxdwlrqv +4,+7, dqg +9,+;,1 ? ' 6 c ?W ' 6W W +46, Z ' E k 6 n k nkq B E6 6W n W +47, ZW ' E k 6Wn kW kq B E6 6Wn W +48,
Iurp htxdwlrq +46, zh vhh/ wkdw d grphvwlf zdjh lqfuhdvh kdv d gluhfw h hfw rq hpsor|phqw lq krph frxqwu|/ zkhuhdv iruhljq hpsor|phqw lv xqdowhuhg1 Wkh krph surgxfw sulfh lv lqfuhdvhg +vhh htxdwlrq +7:, lq wkh Dsshqgl{, lq rughu wr edodqfh wkh prqh| pdunhw1 Wkh uhdo h{fkdqjh udwh dssuhfldwhv vr wkdw wkh jrrgv pdunhwv duh lq htxloleulxp +vhh htxdwlrq +7;, lq wkh Dsshqgl{,1 Wklv uhgxfhv grphvwlf frqvxphu sulfhv dqg lqfuhdvhv iruhljq frqvxphu sulfhv1 Wkh wrwdo h hfw rq grphvwlf FSL lv dpeljxrxv/ exw wkh iruhljq FSL lqfuhdvhv/ lh lq dwlrq lv h{sruwhg e| d xqlodwhudo zdjh lqfuhdvh1
Qh{w zh frqvlghu d xqlodwhudo prqhwdu| h{sdqvlrq1 Grphvwlf hpsor|phqw lqfuhdvhv/ dqg wr edodqfh wkh prqh| pdunhw wkh surgxfw sulfh lqfuhdvhv +vhh htxdwlrq +7:, lq wkh Dsshqgl{,1 Jrrgv pdunhw htxloleulxp fdoov iru uhdo h{0 fkdqjh udwh ghsuhfldwlrq1 Grphvwlf frqvxphu sulfh lqgh{ lqfuhdvhv dqg iruhljq frqvxphu sulfh lqgh{ ghfuhdvhv ehfdxvh ri wkh uhdo h{fkdqjh udwh ghsuhfldwlrq1 Wkh fhqwudo edqn*v sureohp lv wr plqlpl}h wkh orvv ixqfwlrq +44, zlwk uh0 vshfw wr 6 vxemhfw wr +46, dqg +47,/ wdnlqj qrplqdo zdjhv dv jlyhq1 Wkh uvw0rughu frqglwlrq fdq eh zulwwhq dv
Z ' j
k nkq B
? +49,
Wklv lv wkh zhoo0nqrzq N|godqg dqg Suhvfrww +4<::, dqg Eduur dqg Jrugrq +4<;6, lq dwlrq eldv uhvxow1 Wkh fhqwudo edqn wdujhwv ixoo hpsor|phqw/ zklfk lq wklv prgho lv }hur1 Wkh orzhu wkh uhdol}hg hpsor|phqw ? lv/ lh wkh zlghu lv wkh jds ehwzhhq wkh wdujhw ohyho ri hpsor|phqw dqg wkh htxloleulxp udwh ri hpsor|phqw/ wkh kljkhu lv wkh udwh ri lq dwlrq1 Dovr/ zlwk d jlyhq hpsor|phqw ohyho/ wkh pruh frqvhuydwlyh wkh fhqwudo edqn/ wkh orzhu lv lq dwlrq1
Lqvhuwlqj wkh uhgxfhg irupv +46, dqg +47, lqwr wkh uvw0rughu frqglwlrq dqg vroylqj lw h{solflwo| iru 6 jlyhv xv wkh uhdfwlrq ixqfwlrq ri wkh krph fhqwudo edqn 6 ' j K k kq B Ej n K2 n Kkq B Ej n K2 E6W W +4:, zkhuh K ' k nkq B
1 Ghshqglqj rq fhqwudo edqn frqvhuydwlvp/ wkh fhq0 wudo edqn uhdfwv wr grphvwlf zdjh lqfuhdvh hlwkhu e| h{sdqglqj ru frqwudfwlqj prqh| vxsso|1 Fxnlhupdq/ Urguljxhv dqg Zhee +4<<;, surylgh hpslulfdo hy0 lghqfh rq prqhwdu| srolf| uhvsrqvhv wr zdjh lqfuhdvhv1 Lw wxuqv rxw wkdw lq frxqwulhv zlwk kljko| frqvhuydwlyh fhqwudo edqnv/ prqhwdu| srolf| lv wljkwhqhg lq uhvsrqvh wr kljk zdjh vhwwohphqwv/ zkhuhdv lq frxqwulhv zlwk ohvv frqvhuyd0 wlyh fhqwudo edqnv prqhwdu| srolf| lv dffrpprgdwlqj1
Gxh wr v|pphwu| ri wkh prgho/ wkh iruhljq prqh| vxsso| vfkhgxoh fdq eh zulwwhq v|pphwulfdoo|1 Wkh wzr fhqwudo edqnv fkrrvh wkhlu qrq0frrshudwlyh srolflhv vlpxowdqhrxvo|1 Wkh Qdvk htxloleulxp ri wkh srolf| jdph lv jlyhq e|48
6@r ' S
n S2W Ss T fc Ss2 f +4;,
Ilqdoo|/ e| lqvhuwlqj wklv h{suhvvlrq lqwr htxdwlrqv +46, dqg +47, zh zloo jhw hpsor|phqwv dqg lq dwlrqv dv ixqfwlrqv ri krph dqg iruhljq qrplqdo zdjhv49
? ' @s n @s2W @s
fc @s2 f +4<,
Z ' Ks n Ks2W Ks
fc Ks2 f +53,
Erwk grphvwlf dqg iruhljq zdjhv ghfuhdvh grphvwlf hpsor|phqw dqg lqfuhdvh lq dwlrq1 Wkhvh wzr htxdwlrqv fdswxuh wkh prqhwdu| srolf| uhvsrqvhv wr xqlrqv* qrplqdo zdjh fodlpv1 Dw wkh uvw vwdjh ri wkh jdph/ oderu xqlrqv udwlrqdoo| dqwlflsdwh wkh srolf| uhvsrqvhv dqg wdnh wkhp lqwr dffrxqw lq zdjh vhwwlqj1
615 Prqhwdu| srolf| lq d prqhwdu| xqlrq
Lq wklv vhfwlrq/ zh wxuq wr wkh prqhwdu| xqlrq fdvh1 Dv lq wkh suhylrxv vhfwlrq/ zh uvw vroyh wkh uhgxfhg irupv iru hpsor|phqwv dqg lq dwlrqv iurp htxdwlrqv +4,+6,/ +8,+9, dqg +<,1 E| gh qlwlrq/ +wkh orj ri, wkh qrplqdo h{fkdqjh udwh lv qrz htxdo wr }hur/ dqg wkh uhdo h{fkdqjh udwh lv vlpso| h{suhvvhg dv wkh gl huhqfh ri +wkh orjv ri, wkh iruhljq dqg grphvwlf surgxfw sulfhv lq htxdwlrq +8,1 Prqh| pdunhwv duh lq htxloleulxp zkhq xqlrq0zlgh prqh| vxsso| htxdov
48Wkh h{solflw irupv ri f4 dqg f5 duh jlyhq lq wkh Dsshqgl{1 49Wkh h{solflw irupv ri wkh hodvwlflwlhv duh jlyhq lq wkh Dsshqgl{1
prqh| ghpdqg/ zklfk ghshqgv rq dyhudjh xqlrq0zlgh qrplqdo lqfrph1 Wkh uhgxfhg irupv duh jlyhq e|4:
? ' 6L 2E n W n 2knE3k BB EW +54, ?W ' 6L 2E n W 2knE3k BB EW +55, Z ' E k 6L n 3k 2 E n W n 2 2kqnE3k B knE3k B EW +56, ZW ' E k 6L n W3k 2 E n W 2 2kqnE3k B knE3k B EW +57,
Iurp +54, zh vhh/ wkdw lq d prqhwdu| xqlrq hpsor|phqw ghshqgv gluhfwo| rq erwk grphvwlf dqg iruhljq zdjhv/ zkhuhdv lq d rdwlqj h{fkdqjh uhjlph rqo| grphvwlf zdjh hqwhuv wkh hpsor|phqw htxdwlrq14; Wklv fdq eh xqghuvwrrg e|
qrwlqj wkdw prqh| pdunhw htxloleuld lq dowhuqdwlyh uhjlphv fdq eh zulwwhq dv 6 ' ER n + ' E n ? c 6W ' ERWn +W ' EWn ?W +58,
6 '
2ER n + n 2ERWn +W ' 2E n ? n2EWn ?W +59,
Lq d rdwlqj h{fkdqjh udwh uhjlph d xqlodwhudo zdjh lqfuhdvh grhv qrw gluhfwo| d hfw uhdo yduldeohv lq wkh rwkhu frxqwu|/ zkloh lq d prqhwdu| xqlrq erwk hpsor|phqwv +dqg rxwsxwv, zloo lq jhqhudo eh d hfwhg1 Iurp +59, zh vhh wkdw dq lqfuhdvh ri jhqhudoo| d hfwv erwk ? dqg ?Wzkhq prqh| vxsso| dqg iruhljq
zdjh duh {hg14<
Xqvxusulvlqjo|/ wkh uhgxfhg irupv uhyhdo wkdw dq lqfuhdvh lq wkh duhd0zlgh prqh| vxsso| kdv v|pphwulf lpsdfw lq erwk frxqwulhv1 Wkxv/ wkh frpprq fhqwudo edqn fdqqrw d hfw wkh uhdo h{fkdqjh udwh/ zklfk lv vroho| ghwhuplqhg e| wkh oderu xqlrqv +vhh htxdwlrq +8;,,1
Wkh frpprq fhqwudo edqn fkrrvhv 6lq rughu wr plqlpl}h wkh orvv ixqfwlrq
+45, vxemhfw wr +54,+57,1 Wkh uvw0rughu frqglwlrq ri wkh frpprq fhqwudo edqn*v rswlpl}dwlrq sureohp fdq eh zulwwhq dv
Z n ZW 2 ' j E k E? n ?W 2 +5:,
zklfk uhodwhv dyhudjh lq dwlrq wr dyhudjh hpsor|phqw lq wkh prqhwdu| xqlrq1 Frpsdulqj wklv h{suhvvlrq zlwk htxdwlrq +49, zh qg wkdw zlwk d jlyhq dy0 hudjh hpsor|phqw ohyho/ dyhudjh lq dwlrq lv kljkhu lq d prqhwdu| xqlrq wkdq xqghu d rdwlqj h{fkdqjh udwh1 Dv phqwlrqhg lq Lqwurgxfwlrq/ vlplodu uhvxow lv rewdlqhg lq prqhwdu| srolf| frrshudwlrq fdvh1 Lq d rdwlqj h{fkdqjh udwh uhjlph prqhwdu| srolflhv duh frqvwudlqhg e| ihduv ri lq dwlrqdu| uhdo h{fkdqjh 4:Djdlq/ uhgxfhg irupv iru wkh rwkhu hqgrjhqrxv yduldeohv fdq eh irxqg lq wkh Dsshqgl{1 4;Hpsor|phqw ghshqgv lqgluhfwo| rq erwk krph dqg iruhljq zdjhv lq d rdwlqj h{fkdqjh
udwh uhjlph/ ehfdxvh prqhwdu| srolflhv lq erwk frxqwulhv duh d hfwhg e| erwk z dqg z1 4<Zkhq @ 4/ wkh uhgxfhg irup iru q lv htxlydohqw lq wkh wzr uhjlphv1 Zh zloo xwlol}h
wklv surshuw| odwhu rq/ zkhq zh glvfxvv derxw wkh lqwxlwlrq ehklqg wkh pdlq uhvxow ri wkh prgho1
udwh ghsuhfldwlrq1 Lq d prqhwdu| xqlrq wkh frpprq fhqwudo edqn fdq*w d hfw wkh uhdo h{fkdqjh udwh dqg wkxv prqhwdu| srolf| zloo eh pruh dffrpprgdwlqj1 E| vroylqj wkh uvw0rughu frqglwlrq h{solflwo| iru 6 jlyhv xv wkh uhdfwlrq
ixqfwlrq ri wkh frpprq fhqwudo edqn 6 ' j k E k
2j n E k 2 E n
W +5;,
Dv h{shfwhg/ wkh fhqwudo edqn*v uhdfwlrq wr zdjh lqfuhdvhv lv v|pphwulf lq d vhqvh wkdw d xqlodwhudo zdjh lqfuhdvh zloo lqgxfh dq htxdo prqh| vxsso| uhdfwlrq lq erwk frxqwulhv1 Li wkh fhqwudo edqn lv frqvhuydwlyh hqrxjk/ lw uhgxfhv prqh| vxsso| zkhq qrplqdo zdjhv duh udlvhg1
Ilqdoo|/ hpsor|phqwv dqg lq dwlrqv dv ixqfwlrqv ri wkh qrplqdo zdjhv lq erwk frxqwulhv duh ghulyhg e| lqvhuwlqj wkh uhdfwlrq ixqfwlrq lqwr wkh uhgxfhg irupv ri ? dqg Z53
? ' @
n @2W @ fc @2 T f +5<,
Z ' K
n K2W K : fc K2 T f +63,
Qrz/ iruhljq zdjh lqfuhdvh pd| lqfuhdvh hpsor|phqw lq krph frxqwu|/ li wkh fhqwudo edqn lv uhodwlyho| dffrpprgdwlqj ru wkh jrrgv duh forvh vxevwlwxwhv +vhh wkh Dsshqgl{,1 Wkh krph frxqwu| qrplqdo zdjh hodvwlflw| ri lq dwlrq lv dozd|v srvlwlyh lq d prqhwdu| xqlrq1 Dq lqfuhdvh lq krph zdjh ohyho lqfuhdvhv krph lq dwlrq pruh wkdq iruhljq lq dwlrq/ surylghg wkdw q
21 Li wkh fhqwudo
edqn lv yhu| frqvhuydwlyh/ lw zloo sxvk wkh dyhudjh lq dwlrq forvh wr }hur1 Wkhq wkh iruhljq lq dwlrq fdq ehfrph qhjdwlyh/ zkhuhdv wkh krph lq dwlrq zloo dozd|v eh srvlwlyh1
616 Zdjh vhwwlqj dqg pdfurhfrqrplf rxwfrphv
Wkh oderu xqlrqv fkrrvh wkhlu zdjhv lq wkh uvw vwdjh ri wkh jdph wdnlqj wkh rwkhu xqlrq*v zdjh dv jlyhq1 Wkh xqlrqv kdyh ixoo lqirupdwlrq derxw wkh uhdfwlrq ixqfwlrqv ri wkh fhqwudo edqnv1 Wklv lpsolhv wkdw wkh oderu xqlrqv pd{lpl}h wkhlu xwlolw| ixqfwlrqv +43, vxemhfw wr +4<, dqg +53, lq d rdwlqj h{fkdqjh udwh uhjlph dqg vxemhfw wr +5<, dqg +63, lq d prqhwdu| xqlrq1 Vroylqj wkh uvw0rughu frqglwlrq iru |lhogv wkh uhdfwlrq ixqfwlrq ri wkh irup
' E K E@ 2
@2
@
W +64,
Lq Qdvk htxloleulxp qrplqdo zdjhv duh jlyhq e| ' E K
@E@n @2 +65,
Lqvhuwlqj wkh htxloleulxp qrplqdo zdjh +65, lqwr wkh uhgxfhg irupv iru hp0 sor|phqw +4<, ru +5<, |lhogv htxloleulxp hpsor|phqw
? ' E K@
+66,
53Wkh h{solflw irupv duh jlyhq lq wkh Dsshqgl{1
Rqfh htxloleulxp hpsor|phqwv duh ghulyhg/ htxloleulxp lq dwlrq udwhv fdq eh rewdlqhg iurp htxdwlrqv +49, dqg +5:,1 Htxdwlrq +66, uhyhdov wkh vrxufh ri xqghuhpsor|phqw dqg lq dwlrq lq rxu prgho1 Wkh htxloleulxp udwh ri hpsor|0 phqw lv dozd|v ehorz ixoo hpsor|phqw ohyho zkhq wkh oderu xqlrq kdv d uhdo zdjh wdujhw/ lh : f1
Zh iroorz wkh vwdqgdug dssurdfk lq oderu xqlrq olwhudwxuh/ zkhuh zdjhv duh hlwkhu edujdlqhg ehwzhhq wkh oderu xqlrq dqg wkh up +uljkw0wr0pdqdjh prgho, ru xqlodwhudoo| fkrvhq e| wkh xqlrq +prqrsro| xqlrq prgho, dqg hpsor|phqw lv ghwhuplqhg iurp wkh oderu ghpdqg fxuyh1 Wkh vorsh ri wkh oderu ghpdqg fxuyh lq E?c E R vsdfh lv E k +vhh htxdwlrq +5,,154 Krzhyhu/ wkh
oderu xqlrq fduhv derxw wkh uhdo frqvxphu zdjh dqg hpsor|phqw1 Wkh uhdo frqvxphu zdjh frqwdlqv dovr iruhljq surgxfw sulfh dv dq hohphqw/ dqg lw wxuqv rxw wr eh xvhixo wr vwxg| wkh h hfwlyh oderu ghpdqg fxuyh iru wkh xqlrq lq E?c E Z vsdfh55 E Z ' K @ ? @K2 n E K @2 @ W +67,
Khqfh wkh vorsh frh!flhqw/ zklfk gluhfwo| ghwhuplqhv hpsor|phqw lq rxu prgho +vhh htxdwlrq +66,, lv E K *@1 Lw fdq eh zulwwhq dv d ixqfwlrq ri wkh wzr
qrplqdo zdjh hodvwlflwlhv ri hpsor|phqw/ @ dqg @256
K @ ' E k kq B @ @2 @ +68, Wklv irupxodwlrq lv hvshfldoo| khosixo lq hpsor|phqw frpsdulvrqv suhvhqwhg lq wkh qh{w vhfwlrq157 Hpsor|phqw shuirupdqfh ghshqgv rq wzr hodvwlflwlhv1
Wkh uvw rqh lv wkh qrplqdo zdjh hodvwlflw| ri wkh uhdo h{fkdqjh udwh/ zklfk lv jlyhq e| Y5
Y ' kB E@ @2 f1 Wkh vwurqjhu h hfw wkh zdjh kdv rq wkh uhdo
h{fkdqjh udwh/ wkh vwhhshu lv wkh h hfwlyh oderu ghpdqg fxuyh1 Wkh vhfrqg rqh lv wkh krph frxqwu| qrplqdo zdjh hodvwlflw| ri hpsor|phqw/ @ f1 Wkh
odujhu lv hpsor|phqw orvv/ wkh ohvv vwhhs lv wkh h hfwlyh oderu ghpdqg fxuyh1
54Vlqfh wkh yduldeohv duh lq orjv/ wkh vorsh frh!flhqw fdq eh gluhfwo| frqyhuwhg lqwr uhdo
surgxfw zdjh hodvwlflw| ri oderu ghpdqg/ C orj Q
C orj Z@S @ C+zs,Cq @ 44 5 +4> 4,/ zklfk
lq rxu prgho ghshqgv rqo| rq surgxfwlrq whfkqrorj|1
55Wklv h{suhvvlrq lv rewdlqhg iurp wkh htxdwlrqv ghwhuplqlqj q dqg dv ixqfwlrqv ri z
dqg z1
56Ghulydwlrq ri wklv h{suhvvlrq fdq eh irxqg lq wkh Dsshqgl{1
57Iurp +67, dqg +68, zh rewdlq wkh h hfwlyh uhdo frqvxphu zdjh hodvwlflw| ri oderu ghpdqg
+z , C orj Z@C orj Q @ C+z ,Cq @ 4.4
d4d5
d4
7 Prqhwdu| xqlrq
Lq wklv vhfwlrq/ zh dqdo|}h wkh pdfurhfrqrplf frqvhtxhqfhv ri wkh irupdwlrq ri d prqhwdu| xqlrq1 Doo uhvxowv duh ghulyhg dvvxplqj wkdw wkh vwuxfwxudo sdudphwhuv ri wkh hfrqrp|/ dv zhoo dv fhqwudo edqn dqg oderu xqlrq suhihu0 hqfhv uhpdlq wkh vdph zkhq d prqhwdu| xqlrq lv hvwdeolvkhg1 Lq wklv zd|/ zh frqfhqwudwh rq wkh gluhfw h hfwv d prqhwdu| xqlrq zloo kdyh rq hfrqrplf shu0 irupdqfh1 Wkh ghjuhh ri fhqwudo edqn frqvhuydwlvp wxuqv rxw wr eh d fuxfldo idfwru lq zkdw kdsshqv wr hpsor|phqw lq d prqhwdu| xqlrq1
714 Hpsor|phqw dqg lq dwlrq h hfwv ri fhqwudo edqn frqvhu0
ydwlvp
Lq wkh wudglwlrqdo Eduur0Jrugrq dqdo|vlv/ hpsor|phqw lv qrw d hfwhg e| fhqwudo edqn frqvhuydwlvp1 Oderu pdunhwv duh dvvxphg wr eh dwrplvwlf/ lh dv Eduur dqg Jrugrq +4<;6, sxw lw 111d jdph ehwzhhq wkh srolf|pdnhu dqg d odujh qxpehu ri sulydwh0vhfwru djhqwv1 Wklv lpsolhv/ wkdw hdfk djhqw lv wrr vpdoo wr kdyh dq| h hfw rq srolf| pdnhu*v ehkdylru/ dqg wkxv wkhuh lv qr vwudwhjlf lqwhudfwlrq ehwzhhq wkhp1 Krzhyhu/ e| jlylqj xs wklv dvvxpswlrq dqg lqwurgxflqj qrq0 dwrplvwlf oderu xqlrqv lqwr wkh prgho wkh qhxwudolw| uhvxow glvdsshduv1 Ehiruh dqdo|}lqj wkh lpsdfw ri prqhwdu| xqlrq rq hfrqrplf shuirupdqfh/ zh vwxg| wkh h hfwv ri fhqwudo edqn frqvhuydwlvp rq hpsor|phqw dqg lq dwlrq lq rxu prgho zlwk rshq0hfrqrp| vslooryhuv158
Uhpdun 4 Dq lqfuhdvh lq wkh ghjuhh ri fhqwudo edqn frqvhuydwlvp lpsuryhv
hpsor|phqw erwk lq d rdwlqj h{fkdqjh udwh uhjlph dqg lq d prqhwdu| xqlrq1
Wkh lqwxlwlrq xqghuo|lqj wklv uhpdun lv vlpsoh159 Zkhq wkh fhqwudo edqn lv
pruh frqvhuydwlyh/ oderu xqlrqv fruuhfwo| dqwlflsdwh wljkwhu prqhwdu| srolf| dqg d odujhu idoo lq hpsor|phqw lq uhvsrqvh wr wkhlu zdjh fodlpv1 Lqfuhdvhg frqvhuydwlvp zrunv dv d ghwhuuhqw wr zdjh lqfuhdvhv dqg wkhuhiruh prghudwhv wkh oderu xqlrqv* uhdo zdjh wdujhwv ohdglqj wr kljkhu hpsor|phqw15: Lq rxu
prgho/ wklv fdq eh vhhq e| gl huhqwldwlqj wkh qrplqdo zdjh hodvwlflwlhv ri hpsor|phqw zlwk uhvshfw wr j lq erwk uhjlphv
Y@s Yj : fc
Y@
Yj : f +69,
Vwulfwo| vshdnlqj/ wklv dujxphqw lv fruuhfw rqo| lq wkh prqhwdu| xqlrq fdvh/ zkhuh wkh qrplqdo zdjh hodvwlflw| ri wkh uhdo h{fkdqjh udwh grhv qrw ghshqg rq fhqwudo edqn frqvhuydwlvp1 Xqghu d rdwlqj h{fkdqjh udwh dq lqfuhdvh lq frq0 vhuydwlvp vwuhqjwkhqv wkh xqlrq*v delolw| wr dssuhfldwh wkh uhdo h{fkdqjh udwh/ 58Iru d frpsuhkhqvlyh vxuyh| rq fhqwudo edqn frqvhuydwlvp +ru lqghshqghqfh,/ vhh Ehujhu/
gh Kddq dqg Hlm!qjhu +5333,1
59Doo surriv duh suhvhqwhg lq wkh Dsshqgl{1
5:Lq wkh Dsshqgl{ zh prgli| rxu prgho iudphzrun e| dvvxplqj dwrplvwlf edujdlqlqj1
Dv h{shfwhg/ wkh htxloleulxp udwh ri hpsor|phqw zloo ehfrph lqghshqghqw ri wkh prqhwdu| lqvwlwxwlrqv1
zklfk zrunv lqwr wkh rssrvlwh gluhfwlrq dqg whqgv wr lqfuhdvh zdjh ghpdqgv1 Gl huhqwldwlqj E@ @2 zlwk uhvshfw wr j |lhogv
Y@s @s2 Yj : fc Y E@ @2 Yj ' f +6:,
Krzhyhu/ wkh ghwhuuhqw h hfw grplqdwhv dqg lqfuhdvhg frqvhuydwlvp zloo lp0 suryh hpsor|phqw dovr lq d rdwlqj h{fkdqjh udwh uhjlph1 Kdoo dqg Iudq}hvh +4< < ;, surylgh vrph hpslulfdo hylghqfh vxssruwlqj rxu uhvxowv15; Wkh| vkrz/
wkdw zkhq edujdlqlqj vwuxfwxuh lv kljko| fhqwudol}hg/ dq lqfuhdvh lq fhqwudo edqn frqvhuydwlvp lpsuryhv hpsor|phqw1 Dw lqwhuphgldwh dqg orz ohyhov ri fhqwudol}dwlrq wkh h hfw lv uhyhuvhg1
Uhpdun 5 Dq lqfuhdvh lq wkh ghjuhh ri fhqwudo edqn frqvhuydwlvp uhgxfhv
lq dwlrq erwk lq d rdwlqj h{fkdqjh udwh uhjlph dqg lq d prqhwdu| xqlrq1
Wklv uhpdun lv vlpso| d uhvwdwhphqw ri wkh wudglwlrqdo lq dwlrq eldv uhvxow1 Wkh pruh wkh fhqwudo edqn sxwv zhljkw rq lq dwlrq/ wkh orzhu wkh uhdol}hg lq dwlrq zloo eh1 Krzhyhu/ wkhuh lv dq dgglwlrqdo h hfw lq rxu prgho1 Uhpdun 4 vkrzv wkdw fhqwudo edqn frqvhuydwlvp lqfuhdvhv wkh htxloleulxp udwh ri hpsor|phqw/ dqg wkxv qduurzv wkh jds ehwzhhq wkh hpsor|phqw wdujhw ri wkh fhqwudo edqn dqg wkh htxloleulxp udwh1 Wkhuhiruh/ dv wkh hpsor|phqw wdujhw ri wkh fhqwudo edqn ehfrphv ohvv dpelwlrxv/ lq dwlrq eldv ghfuhdvhv1
715 Pdlq uhvxowv= hpsor|phqw dqg lq dwlrq lq d prqhwdu| xqlrq
Ilqdoo|/ lq wklv vhfwlrq zh frqvlghu wkh uhodwlyh shuirupdqfh ri wkh wzr prqh0 wdu| uhjlphv1 Doo uhvxowv duh ghulyhg dvvxplqj wkdw wkh vwuxfwxudo sdudphwhuv ri wkh prgho dqg wkh suhihuhqfhv ri wkh oderu xqlrqv dqg wkh fhqwudo edqnv uhpdlq wkh vdph zkhq d prqhwdu| xqlrq lv hvwdeolvkhg1 Wkh pdlq uhvxow ri wkh sdshu lv suhvhqwhg lq wkh iroorzlqj sursrvlwlrq1
Sursrvlwlrq 6 Dvvxph wkdw oderu xqlrq dqg fhqwudo edqn suhihuhqfhv duh
lghqwlfdo dfurvv frxqwulhv dqg gr qrw fkdqjh zkhq d prqhwdu| xqlrq lv hv0 wdeolvkhg1 Hpsor|phqw zloo lpsuryh/ li wkh ghjuhh ri fhqwudo edqn frqvhuydwlvp lv deryh wkh fulwlfdo ydoxh/
j/ zkhuh wkh h{solflw irup ri j lv jlyhq e|
j ' 2B2
x nsx2n [
zkhuh x ' E k 2B2n k E k E q B n 2k2q2
dqg [ ' eB2k E k 2E 2q Ekq n B E k : f1
Wkh sursrvlwlrq vkrzv wkdw wkh hvwdeolvkphqw ri d prqhwdu| xqlrq fdq lqfuhdvh hpsor|phqw/ hyhq zkhq doo vwuxfwxudo sdudphwhuv ri wkh prgho/ lqfoxglqj fhq0 wudo edqn frqvhuydwlvp/ uhpdlq xqfkdqjhg1 Wklv uhvxow lv lq vkdus frqwudvw
zlwk hduolhu uhvxowv frqfhuqlqj prqhwdu| xqlrq*v lpsdfw rq htxloleulxp hp0 sor|phqw1 Iurp h{suhvvlrq [ zh vhh wkdw wkh fulwlfdo ydoxh ri j lv dozd|v srvlwlyh/ surylghg wkdw wkh vkduh ri lpsruwv lq frqvxpswlrq lv ohvv wkdq rqh kdoi1
Qrz/ zh ghyhors wkh lqwxlwlrq ehklqg wklv uhvxow lq wzr vwhsv1 Iluvw/ dv vkrzq lq Uhpdun 4/ dq lqfuhdvh lq wkh ghjuhh ri frqvhuydwlvp lpsuryhv hp0 sor|phqw lq erwk uhjlphv1 Exw/ gxh wr wkh uhdo h{fkdqjh udwh h hfw glvfxvvhg lq Vhfwlrq 714/ hpsor|phqw zloo lpsuryh vorzhu lq d rdwlqj h{fkdqjh udwh uhjlph15< Vhfrqg/ e| lqvshfwlqj wkh wzr h{wuhph fdvhv ri frqvhuydwlvp/ lh j ' f
dqg j $ 4 zh fdq lqwxlwlyho| h{sodlq wkdw wkhuh pxvw eh dw ohdvw rqh fulwlfdo ydoxh ri fhqwudo edqn frqvhuydwlvp/ zklfk olhv lq wkh lqwhuydo f j 41
Zkhq wkh fhqwudo edqn lv xowud0frqvhuydwlyh Ej ' f / prqhwdu| srolflhv duh lq erwk uhjlphv vhw vr wkdw lq dwlrq zloo eh }hur1 Khqfh/ wkh fhqwudo edqn grhvq*w fduh dw doo derxw hpsor|phqw rxwfrphv1 Dq xowud0olehudo fhqwudo edqn Ej $ 4 wdujhwv rqo| hpsor|phqw/ dqg wkxv doo zdjh lqfuhdvhv duh ixoo| df0 frpprgdwhg1 Vwulfwo| vshdnlqj/ zh kdyh wr dvvxph wkdw j lv yhu| odujh exw qlwh/ ehfdxvh wkh prgho lv qrw zhoo gh qhg zkhq j lv lq qlwh1 Wklv fdq eh yhul hg e| lqvshfwlqj wkh zdjh jdph ehwzhhq xqlrqv/ zklfk uhyhdov wkdw wkh htxloleulxp zdjh ehfrphv lq qlwh1 Lqwxlwlyho|/ wkh fhqwudo edqn*v ixoo hpsor|0 phqw wdujhw dqg wkh xqlrq*v uhdo zdjh wdujhw duh lqfrpsdwleoh1 Wkh xqlrq zdqwv wr udlvh uhdo zdjhv deryh }hur/ zkhuhdv wkh fhqwudo edqn wulhv wr dffrp0 prgdwh wkh zdjh lqfuhdvh frpsohwho| wr dfklhyh ixoo hpsor|phqw1 Khqfh zdjhv lqfuhdvh zlwkrxw olplw dqg wkh uhvxow lv k|shulq dwlrq1
Lq d prqhwdu| xqlrq prqhwdu| srolf| lv lq jhqhudo pruh dffrpprgdwlqj wkdq lq wkh rdwlqj uhjlph1 Krzhyhu/ lq wkhvh wzr vshfldo fdvhv prqhwdu| srolflhv duh htxdoo| dffrpprgdwlqj lq erwk uhjlphv/ zklfk lpsolhv wkdw wkh wrwdo hpsor|phqw uhgxfwlrq uhvxowlqj iurp d xqlodwhudo zdjh lqfuhdvh lv htxdo lq erwk uhjlphv
@s n @s2 ' @
n @2 +6;,
Hpsor|phqw lv ghwhuplqhg e| wkh wzr hodvwlflwlhv/ @ dqg E@ @2 / glvfxvvhg
lq Vhfwlrq 6161 Wr foduli| wkh h{srvlwlrq ohw xv dvvxph iru d prphqw wkdw wkh vxevwlwxwlrq sdudphwhu B htxdov xqlw|1 Wkhq wkh gluhfw olqn ehwzhhq iruhljq zdjh dqg grphvwlf hpsor|phqw lv vkxw grzq +vhh htxdwlrqv +54, dqg +55,, dqg wkh uhgxfhg irupv iru hpsor|phqw duh htxlydohqw lq wkh wzr uhjlphv1 Qrz/ d xqlodwhudo zdjh lqfuhdvh lq krph frxqwu| zloo kdyh wkh vdph gluhfw h hfw lq erwk uhjlphv1 Hpsor|phqw ghfuhdvhv lq krph frxqwu|/ zkhuhdv hpsor|phqw lq wkh iruhljq frxqwu| lv xqd hfwhg1 Wkh gluhfw h hfw rq lq dwlrq lv dovr htxdo lq erwk uhjlphv1 Dv orqj dv wkhuh lv d krph eldv lq frqvxpswlrq/ dq lqfuhdvh lq krph frxqwu| zdjh lqfuhdvhv grphvwlf lq dwlrq pruh wkdq iruhljq lq dwlrq163
Wkh gl huhqfh ehwzhhq wkh wzr uhjlphv ehfrphv fohdu zkhq zh wxuq lqwr prqhwdu| srolf|1 Ohw xv uvw frqvlghu wkh xowud0frqvhuydwlyh fhqwudo edqn fdvh 5<Wkh sduwldo ghulydwlyhv lq htxdwlrq +69, duh qrw htxdo lq jhqhudo1 Wkxv lw frxog eh
srvvleoh wkdw zlwk d jlyhq / Cdi4@C lv vr pxfk juhdwhu wkdq Cdx4@C / wkdw hpsor|phqw
zrxog lpsuryh idvwhu xqghu d rdwlqj h{fkdqjh udwh uhjlph1 Krzhyhu/ lw wxuqv rxw wkdw wklv lv qrw wkh fdvh khuh1
63Wklv lv rewdlqhg iurp wkh uhgxfhg irupv iru dqg lq Vhfwlrq 615 e| vhwwlqj @ 4
dqg qrwlqj wkdw ? 4 51
Ej ' f 1 Lq d rdwlqj uhjlph erwk fhqwudo edqnv duh wdujhwlqj }hur lq dwlrq1 Vlqfh wkh gluhfw h hfw ri d xqlodwhudo zdjh lqfuhdvh rq grphvwlf lq dwlrq lv vwurqjhu/ lq htxloleulxp krph prqh| vxsso| pxvw ghfuhdvh pruh wkdq iruhljq prqh| vxsso| vr wkdw erwk fhqwudo edqnv klw wkhlu lq dwlrq wdujhwv1 Lq d prqhwdu| xqlrq/ wkh frpprq fhqwudo edqn kdv rqo| rqh srolf| lqvwuxphqw/ wkh duhd0zlgh prqh| vxsso|1 Wkhuhiruh/ zkhq lw idfhv d xqlodwhudo zdjh lqfuhdvh/ lw fdq*w vwdelol}h wkh sulfh ohyho lq erwk frxqwulhv/ exw lw fdq dwwdlq }hur lq dwlrq rq dyhudjh1
Vlqfh prqhwdu| srolflhv lq wkh wzr uhjlphv duh htxdoo| qrq0dffrpprgdwlqj rq dyhudjh/ wkh uhgxfwlrq ri wkh frpprq prqh| vxsso| lq d prqhwdu| xqlrq fdvh lv wkh dyhudjh ri wkh krph dqg iruhljq frxqwu| prqh| vxsso| uhgxfwlrqv lq d rdwlqj h{fkdqjh udwh uhjlph164 Wkhuhiruh/ lq d prqhwdu| xqlrq prqh|
vxsso| lv uhgxfhg ohvv lq krph frxqwu| dqg pruh lq iruhljq frxqwu| frpsduhg zlwk wkh rdwlqj uhjlph1 Lq rwkhu zrugv/ vlqfh wkh uhvsrqvh ri wkh frpprq fhqwudo edqn zloo dozd|v kdyh d v|pphwulf h hfw lq erwk frxqwulhv/ wkh iruhljq frxqwu| zloo ehdu d juhdwhu vkduh ri wkh wrwdo hpsor|phqw uhgxfwlrq fdxvhg e| d xqlodwhudo zdjh lqfuhdvh lq wkh krph frxqwu| +lh @
: @s dqg @2 @s2,1
Krz lv doo wklv uhodwhg wr hpsor|phqw shuirupdqfh lq wkh wzr prqhwdu| uhjlphvB D vzlwfk iurp d rdwlqj h{fkdqjh udwh uhjlph wr d prqhwdu| xqlrq zloo kdyh dq h hfw erwk rq wkh qrplqdo zdjh hodvwlflw| ri hpsor|phqw/ @/
dqg wkh qrplqdo zdjh hodvwlflw| ri wkh uhdo h{fkdqjh udwh/ E@ @2 1 Deryh
uhdvrqlqj lpsolhv wkdw @ zloo lqfuhdvh/ zklfk pdnhv wkh h hfwlyh oderu ghpdqg
fxuyh vwhhshu dqg ghwhulrudwhv hpsor|phqw1 Krzhyhu/ E@ @2 zloo lqfuhdvh
h{dfwo| e| wzr wlphv pruh wkdq @/ dqg wkxv wkh wrwdo h hfw ri d prqhwdu|
xqlrq rq hpsor|phqw zloo eh srvlwlyh zkhq j ' f +vhh htxdwlrq +68,,165
Zkhq wkh fhqwudo edqn lv +doprvw, xowud0olehudo Ej $ 4 / zh fdq xvh wkh vdph olqh ri uhdvrqlqj wr hvwdeolvk h{dfwo| wkh rssrvlwh uhvxow1 Dvvxph wkdw wkh fhqwudo edqn wdujhwv hpsor|phqw rqo|1 Wkhq zh nqrz wkdw prqhwdu| srolf| lv ixoo| dffrpprgdwlqj lq erwk uhjlphv1 Wkh gluhfw h hfw ri d xqlodwhudo zdjh lqfuhdvh lq wkh krph frxqwu| lv wkh vdph dv deryh/ exw wkh frqgxfw ri prqhwdu| srolf| zloo eh gl huhqw1
Iluvw/ ohw xv orrn dw wkh rdwlqj h{fkdqjh udwh fdvh1 Hpsor|phqw ghfuhdvhv rqo| lq krph frxqwu|/ dqg prqh| vxsso| pxvw eh lqfuhdvhg vr wkdw wkh zdjh lqfuhdvh lv +doprvw, ixoo| dffrpprgdwhg +@s f,166 Vlqfh hpsor|phqw lq wkh
iruhljq frxqwu| lv xqdowhuhg/ wkh iruhljq fhqwudo edqn*v uhdfwlrq lv yhu| vpdoo +@s
2 f,1 Lq d prqhwdu| xqlrq wkh frpprq fhqwudo edqn fdqqrw dwwdlq ixoo hp0
sor|phqw lq erwk frxqwulhv/ ehfdxvh lw kdv rqo| rqh srolf| lqvwuxphqw dydlodeoh1 Exw lw fdq dwwdlq ixoo hpsor|phqw rq dyhudjh e| lqfuhdvlqj prqh| vxsso| e| wkh dyhudjh ri wkh wzr prqh| vxsso| lqfuhdvhv lq d rdwlqj uhjlph1 Lq wklv fdvh d xqlodwhudo zdjh lqfuhdvh dfwxdoo| lqfuhdvhv hpsor|phqw lq wkh iruhljq frxq0 64Wklv fdq eh yhul hg e| lqvshfwlqj wkh frh!flhqwv ri wkh uhdfwlrq ixqfwlrqv ri wkh fhqwudo
edqnv suhvhqwhg lq htxdwlrq +5;, iru d prqhwdu| xqlrq dqg lq htxdwlrqv +83,+84, lq wkh Dsshqgl{ iru d rdwlqj h{fkdqjh udwh uhjlph1
65Wklv lv gxh wr wkh idfw wkdw wkh hpsor|phqw orvv ri d xqlodwhudo zdjh lqfuhdvh vkliwv
lqwr wkh gluhfwlrq ri wkh iruhljq frxqwu|/ zkloh wkh wrwdo hpsor|phqw orvv lv xqdowhuhg1 Wklv lpsolhv wkdw d5 zloo ghfuhdvh e| wkh vdph dprxqw dv d4 lqfuhdvhv1 Wkxv +d4 d5, lqfuhdvhv
e| wzr wlphv pruh wkdq d41
66Dv glvfxvvhg hduolhu/ zh kdyh wr dvvxph wkdw lv odujh exw qlwh1 Wkhq di
4 ? 3/ di5 ? 3
wu| E@
2 : f dqg ghfuhdvhv lw lq wkh krph frxqwu| E@ f 1 Khqfh/ d vzlwfk
iurp d rdwlqj h{fkdqjh udwh uhjlph wr d prqhwdu| xqlrq zloo ghfuhdvh @/
zklfk whqgv wr lpsuryh hpsor|phqw/ exw dw wkh vdph wlph E@ @2 ghfuhdvhv
e| +dssur{lpdwho|, wzr wlphv pruh/ dqg wkxv wkh wrwdo h hfw rq hpsor|phqw zloo eh qhjdwlyh zkhq j $ 4 +vhh htxdwlrq +68,,1
Uhdvrqlqj lq wkhvh wzr fdvhv zdv edvhg rq wkh idfw wkdw prqhwdu| srolflhv duh htxdoo| dffrpprgdwlqj lq erwk uhjlphv +vhh htxdwlrq +6;,,1 Wklv lpsolhv wkdw wkh fkdqjh lq prqhwdu| uhjlph zloo rqo| uhorfdwh wkh hpsor|phqw orvv ri d xqlodwhudo zdjh lqfuhdvh1 Lq jhqhudo/ prqhwdu| srolf| lv pruh dffrpprgdwlqj lq d prqhwdu| xqlrq1 Wklv phdqv wkdw wkh hpsor|phqw h hfw lv qrw phuho| uhorfdwhg lq d prqhwdu| xqlrq/ exw wkh ryhudoo h hfw rq hpsor|phqw lv vpdoohu
@
n @2 : @s n @s2 +6<,
Khqfh/ wkh dujxphqw xvhg wr h{sodlq wkh uhvxowv lq wkh wzr h{wuhph fdvhv fdq*w eh dssolhg1 Krzhyhu/ zh kdyh surylghg dq lqwxlwlyh h{sodqdwlrq iru wkh idfw wkdw wkh h hfw ri frqvhuydwlvp rq hpsor|phqw whqgv wr eh vwurqjhu lq d prqhwdu| xqlrq dqg iru wkh xqdpeljxrxv uhvxowv lq wkh wzr h{wuhph fdvhv1 Sxwwlqj wkhvh wzr revhuydwlrqv wrjhwkhu zh fdq frqfoxgh wkdw wkhuh lv fhuwdlqo| dw ohdvw rqh fulwlfdo ydoxh ri fhqwudo edqn frqvhuydwlvp1 Sursrvlwlrq 6 frq upv wkdw wkhuh lv rqo| rqh vxfk ydoxh1
Sursrvlwlrq 7 Dvvxph wkdw oderu xqlrq dqg fhqwudo edqn suhihuhqfhv duh
lghqwlfdo dfurvv frxqwulhv dqg gr qrw fkdqjh zkhq d prqhwdu| xqlrq lv hv0 wdeolvkhg1 Lq dwlrq lv kljkhu lq d prqhwdu| xqlrq surylghg wkdw j : f1
Wklv sursrvlwlrq vxjjhvwv/ wkdw lq dwlrq lv kljkhu lq d prqhwdu| xqlrq li wkh hpsor|phqw jds hqwhuv wkh fhqwudo edqn orvv ixqfwlrq1 Lq vhfwlrq 615 zh douhdg| glvfxvvhg wkh lq dwlrqdu| h hfwv ri d prqhwdu| xqlrq zlwk d jlyhq hpsor|phqw ohyho1 Sursrvlwlrq 6 vkrzv wkdw hpsor|phqw lq d prqhwdu| xqlrq fdq e| kljkhu wkdq lq d rdwlqj h{fkdqjh udwh uhjlph1 Wkxv/ lq wkdw fdvh lw frxog eh srvvleoh wkdw lq dwlrq lv orzhu lq d prqhwdu| xqlrq1 Krzhyhu/ dv vwdwhg deryh/ lq dwlrq zloo eh kljkhu lq d prqhwdu| xqlrq zlwk doo qlwh ghjuhhv ri fhqwudo edqn frqvhuydwlvp1
Sursrvlwlrq 8 Li wkh lqgh{ ri vxevwlwxwdelolw| ehwzhhq jrrgv/ B/ dssurdfkhv lq0
qlw|/ wkhq erwk hpsor|phqw dqg lq dwlrq uhpdlq xqfkdqjhg zkhq d prqhwdu| xqlrq lv hvwdeolvkhg1 Fhqwudo edqn frqvhuydwlvp kdv qr h hfw rq hpsor|phqw dqg d qhjdwlyh h hfw rq lq dwlrq1
Wklv sursrvlwlrq lv d uhvwdwhphqw ri wkh prqhwdu| xqlrq qhxwudolw| uhvxow zlwk qrq0lq dwlrq dyhuvh oderu xqlrqv suhvhqwhg lq wkh sdshuv e| Ju qhu dqg Khihnhu +4<<<, dqg Fxnlhupdq dqg Olssl +5333,1 Zkhq wkh vxevwlwxwdelolw| ehwzhhq jrrgv ehfrphv lq qlwh/ zh duh dfwxdoo| edfn lq wkh vlqjoh jrrg zruog1 Wkh uhdo h{fkdqjh udwh fdqqrw ghyldwh iurp }hur +lq orjv,/ ehfdxvh rwkhuzlvh wkh zruog ghpdqg zrxog vkliw wr rqh frxqwu|1 Khqfh/ oderu xqlrqv uhdol}h wkdw wkh| fdq*w d hfw wkh uhdo h{fkdqjh udwh zlwk wkhlu zdjh fodlpv/ dqg wkh vorsh ri wkh h hfwlyh oderu ghpdqg fxuyh zloo lq erwk prqhwdu| uhjlphv frlqflgh zlwk wkh vorsh ri wkh *uhdo* oderu ghpdqg fxuyh/ zklfk lv ghwhuplqhg rqo| e| whfkqro0 rj| lq rxu prgho1 Lq wklv fdvh zh dfwxdoo| rewdlq wkh wudglwlrqdo Eduur0Jrugrq uhvxowv frqfhuqlqj fhqwudo edqn frqvhuydwlvp/ lh hpsor|phqw vwd|v lqwdfw dqg lq dwlrq lv ghfuhdvlqj lq frqvhuydwlvp1
8 Dowhuqdwlyh vwduwlqj srvlwlrq= dq dv|pphwulf {
Deryh zh kdyh frqvlghuhg d vzlwfk iurp d rdwlqj h{fkdqjh udwh uhjlph wr d prqhwdu| xqlrq1 Lq wklv vhfwlrq zh vwxg| wkh h hfwv ri d prqhwdu| xqlrq lq d vlwxdwlrq zkhuh rqh frxqwu| kdv jlyhq xs lqghshqghqw prqhwdu| srolf| dqg frpplwwhg wr d {hg h{fkdqjh udwh ylv0ã0ylv wkh rwkhu frxqwu|1
814 Wkh vhwxs
Iru dqdo|wlfdo vlpsolflw|/ zh dvvxph wkdw wkh fhqwudo edqn ri wkh dqfkru frxqwu| lv xowud0frqvhuydwlyh1 Dw uvw vljkw/ wklv dvvxpswlrq pd| vhhp wr eh yhu| uhvwulfwlyh1 Krzhyhu/ dq dv|pphwulf {hg h{fkdqjh udwh uhjlph fdq eh vhhq dv d zd| wr lpsruw fuhglelolw| iurp d frxqwu| zkhuh wkh fhqwudo edqn lv pruh frqvhuydwlyh wkdq lq wkh krph frxqwu|1 Wkh dqfkru frxqwu| fhqwudo edqn lv wkhq yhu| olnho| kljko| frqvhuydwlyh167
Vlqfh wkhuh lv qr fdslwdo prelolw| lq wkh prgho/ zh kdyh +lpsolflwo|, dvvxphg wkdw wudgh pxvw eh lq edodqfh lq htxloleulxp1 Lq d v|pphwulf prgho zlwk htxdo htxloleulxp udwhv ri rxwsxw dqg htxdo lpsruw vkduhv zh nqrz wkdw wklv lv wkh fdvh1 Krzhyhu/ lq dq dv|pphwulf { zh fdq*w uxoh rxw wkh srvvlelolw| wkdw wkh htxloleulxp udwhv gl hu/ vlqfh prqhwdu| srolflhv duh fohduo| gl huhqw lq wkh wzr frxqwulhv1
Krph frxqwu| lpsruwv fdq eh zulwwhq dv .WtWe% ' q t dqg h{sruwv dv
te% ' qW.tW/ zkhuh tWe% dqg te%duh lpsruw ghpdqgv ri krph dqg iruhljq
frxqwulhv/ uhvshfwlyho|1 Wudgh lv lq edodqfh zkhq q t ' qW.tW1 Wdnlqj
orjv |lhogv + +W ' 5/ zklfk lpsolhv wkdw zh pxvw dvvxph B ' wr kdyh
edodqfhg wudgh zkhq wkh htxloleulxp udwhv ri rxwsxw gl hu ehwzhhq frxqwulhv1 Dgplwwhgo|/ wklv uhvwulfwlrq olplwv wkh dssolfdelolw| ri wkh prgho frqvlghudeo| dqg eulqjv rxw wkh sureohpv wklv prgholqj iudphzrun kdv zkhq wkhuh duh vwuxfwxudo dv|pphwulhv1 Krzhyhu/ dvvxplqj B ' vlpsol hv wkh prgho/ dqg zh fdq zulwh wkh h{fkdqjh udwh vlpso| dv e ' 6 6W +vhh htxdwlrq +7<, lq wkh
Dsshqgl{,1 Wkh uhodwlyh prqh| vxsso| wkhq ghwhuplqhv wkh qrplqdo h{fkdqjh udwh/ dqg d {hg h{fkdqjh udwh lpsolhv wkdw 6 ' 6W pxvw krog1
Lq wkh dv|pphwulf { fdvh zh pxvw prgli| wkh wlplqj ri wkh prgho dqg qrz hyhqwv xqirog dv iroorzv1 +4, Wkh iroorzhu frxqwu| fhqwudo edqn fuhgleo| frpplwv wr d {hg h{fkdqjh udwh e ' f1 +5, Wkh wzr oderu xqlrqv vhw wkhlu qrplqdo zdjhv vlpxowdqhrxvo| wdnlqj wkh rwkhu xqlrq*v zdjh dv jlyhq1 +6, Wkh xowud0frqvhuydwlyh fhqwudo edqn ri wkh ohdghu frxqwu| plqlpl}hv lwv orvv ixqfwlrq e| vhwwlqj lwv prqh| vxsso| dqg wdnlqj wkh iroorzhu frxqwu|*v fuhgleoh frpplwphqw wr e ' f lqwr dffrxqw1 +7, Wkh iroorzhu frxqwu|*v fhqwudo edqn vhwv 6 ' 6W1 +8, Surgxfwlrq dqg wudgh wdnh sodfh dqg pdfurhfrqrplf rxwfrphv
duh uhdol}hg1
67Lq Hxursh/ zkhuh Jhupdq| dfwhg dv wkh dqfkru frxqwu| ri wkh HUP wkh Exqghvedqn zdv
jhqhudoo| frqvlghuhg wr eh yhu| frqvhuydwlyh1 Wklv ylhz lv vxssruwhg e| hpslulfdo hylghqfh lq Fxnlhupdq/ Urguljxhv dqg Zhee +4<<;, dqg Ehujhu dqg Zrlwhn +4<<<,1
815 Hpsor|phqw dqg lq dwlrq lq dq dv|pphwulf {
Dv ehiruh/ wkh prgho lv vroyhg e| edfnzdug lqgxfwlrq1 Lq vwdjh +7, wkh iro0 orzhu frxqwu| vhwv 6 ' 6W1 Vlqfh wkh ohdghu frxqwu|*v fhqwudo edqn lv xowud0
frqvhuydwlyh/ zh nqrz zlwkrxw h{solflwo| irupxodwlqj wkh rswlpl}dwlrq sureohp wkdw lq vwdjh +6, ri wkh jdph lw vhwv 6W vr wkdw ZW ' f1 Wkh rswlpdo prqh|
vxsso| fdq eh vroyhg iurp wkh uhgxfhg irup iru ZW +vhh htxdwlrq +46,,
6W ' k
kEE q Wn q +73,
zklfk uhyhdov wkh dv|pphwu| lq prqhwdu| srolf| uhdfwlrqv1 Wkh ohdghu frxq0 wu| prqhwdu| srolf| uhvsrqgv pruh wljkwo| rq grphvwlf +ohdghu frxqwu|, zdjh lqfuhdvhv q
2
dqg vlqfh 6 ' 6W/ wkh iroorzhu frxqwu| fhqwudo edqn eh0
kdyhv vlploduo|1 Wklv lpsolhv wkdw d xqlodwhudo zdjh lqfuhdvh lq wkh iroorzhu frxqwu| eulqjv derxw pruh dffrpprgdwlqj prqhwdu| srolf| wkdq d xqlodwhudo zdjh lqfuhdvh lq wkh ohdghu frxqwu|1 Qrz zh fdq ghulyh wkh uhgxfhg irupv iru hpsor|phqw dqg lq dwlrq lq erwk frxqwulhv dv
? 'kE3q 33k kE3q 3k W ' @ n @2W @ fc @2 f +74, ?W 'kq3 3k W kq 3k ' @W n @W2W @W fc @W2 f +75, Z ' k E 2q k E 2q W ' K n K2W K : fc K2 f +76, ZW ' f +77,
Wkh zdjh vhwwlqj lq vwdjh +5, lv vlplodu wr wkh suhylrxv dqdo|vlv lq Vhfwlrq 6/ exw qrz wkh qrplqdo zdjh hodvwlflwlhv ri hpsor|phqw duh gl huhqw lq wkh wzr frxqwulhv1 Xwlolw| pd{lpl}dwlrq |lhogv uhdfwlrq ixqfwlrqv iru erwk xqlrqv
' E K E@ 2 @2 @ Wc W ' E KW E@W 2 @W2 @W +78,
Qrz/ wkh vorshv ri wkh uhdfwlrq ixqfwlrqv duh qrw h{dfw lqyhuvhv ri hdfk rwkhu1 Wkhuhiruh/ wkh Qdvk htxloleulxp lv dv|pphwulf lq d vhqvh wkdw zdjhv lq wkh wzr frxqwulhv duh gl huhqw168 Lq sduwlfxodu/ wkh iroorzhu frxqwu| xqlrq dqwlf0
lsdwhv uhodwlyho| orrvh prqhwdu| srolf| dqg vhwv kljkhu zdjh wkdq wkh ohdghu frxqwu| xqlrq169 Htxloleulxp hpsor|phqwv duh ri wkh vdph irup dv ehiruh/
68Wkh zdjhv duh jlyhq e| z @ d 4d4 d5d5 d 4 d4+4 e4, d5 d 4+4 e 4, z @ d4d4 d5d5 d4 d 4+4 e 4, d 5 d4 +4 e4, = 69Iurp wkh Qdvk htxloleulxp zdjhv dqg qrwlqj wkdw +d 4. d5, @ +d4. d5, zh rewdlq +z z, @ d4d4d5d5 +d4. d5, 4e4 d4 4e 4 d 4 A 31 58
exw qrz hpsor|phqw ohyhov gl hu dfurvv frxqwulhv vr wkdw wkh ohdghu frxqwu| hpsor|phqw lv kljkhu wkdq lq wkh iroorzhu frxqwu|1 6: Wklv uhvxow yhul hv wkdw
zh pxvw lpsrvh wkh uhvwulfwlrq EB ' rq wkh vxevwlwxwdelolw| sdudphwhu1 Dq xowud0frqvhuydwlyh fhqwudo edqn lq wkh ohdghu frxqwu| hqvxuhv }hur lq0 dwlrq1 Lq wkh iroorzhu frxqwu| lq dwlrq lv ghwhuplqhg e| htxdwlrq +76,/ dqg vlqfh wkh zdjh ohyho lv kljkhu lq wkh iroorzhu frxqwu|/ lq dwlrq lv deryh }hur
Z ' k E 2q E W : f +79,
Khqfh/ vrphzkdw vxusulvlqjo| d fuhgleoh frpplwphqw wr d {hg h{fkdqjh udwh grhv qrw lpso| wkdw wkh lq dwlrq udwhv duh htxdol}hg16;
816 D vzlwfk iurp dq dv|pphwulf { wr d prqhwdu| xqlrq
Qrz zh zloo dqdo|}h wkh pdfurhfrqrplf frqvhtxhqfhv ri d prqhwdu| xqlrq lq dq dv|pphwulf { fdvh1 Dv qrwhg hduolhu/ zh kdyh pdgh txlwh uhvwulfwlyh dv0 vxpswlrqv derxw wkh vxevwlwxwdelolw| dqg frqvhuydwlvp sdudphwhuv1 Wkhuhiruh/ lw pxvw eh vwuhvvhg wkdw rxu dqdo|vlv dssolhv rqo| wr wklv vshfldo fdvh dqg qrw wr d pruh jhqhudo fdvh zlwk eurdghu sdudphwhu vsdfhv1
Sursrvlwlrq 9 1 Dvvxph wkdw wkh ohdghu frxqwu| fhqwudo edqn lq dq dv|pphw0
ulf { dqg wkh frpprq fhqwudo edqn lq d prqhwdu| xqlrq duh xowud0frqvhuydwlyh
Ej ' f 1 Wkh hvwdeolvkphqw ri d prqhwdu| xqlrq lqfuhdvhv hpsor|phqw lq wkh
iroorzhu frxqwu| dqg ghfuhdvhv hpsor|phqw lq wkh ohdghu frxqwu|1
Wkh vdph txdolwdwlyh uhvxow lv rewdlqhg lq prghov zlwk lq dwlrq dyhuvh oderu xqlrqv1 Krzhyhu/ wkh phfkdqlvp ehklqg wkh uhvxowv lv frpsohwho| gl huhqw1 Zlwk lq dwlrq dyhuvlrq/ wkh iroorzhu frxqwu|*v oderu xqlrq ehfrphv ohvv dj0 juhvvlyh lq d prqhwdu| xqlrq/ ehfdxvh lw uhdol}hv wkdw lq dwlrq lv qrz d hfwhg e| lwv zdjh ghpdqgv/ zkhuhdv lq dq dv|pphwulf { lq dwlrq lv lqghshqghqw ri lwv zdjh fodlp16< Iurp iroorzhu frxqwu| xqlrq*v srlqw ri ylhz/ prqhwdu| sro0
lf| lv pruh dffrpprgdwlqj lq d prqhwdu| xqlrq1 Wklv lqgxfhv zdjh uhvwudlqw/ vlqfh wkh xqlrq lv dyhuvh wr lq dwlrq1 Lq rxu prgho/ wkh iroorzhu frxqwu| zdjh ghpdqg lv uhgxfhg dqg hpsor|phqw lpsuryhv lq d prqhwdu| xqlrq/ ehfdxvh prqhwdu| srolf| ehfrphv ohvv dffrpprgdwlqj dqg wkxv wkh ghwhuuhqw h hfw zloo eh vwurqjhu1 Wkh ohdghu frxqwu| fdvh lv wkh pluuru0lpdjh ri wkh iroorzhu frxq0 wu| fdvh1 Vrphzkdw vxusulvlqjo|/ wkh uhvxowv ri wkh wzr prghov duh txdolwdwlyho| vlplodu/ dowkrxjk wkh fkdqjhv lq wkh uhdfwlrqv ri wkh prqhwdu| dxwkrulwlhv idfhg e| wkh xqlrqv duh frpsohwho| rssrvlwh1 6:Zh kdyh q @ +4e 4, d 4 dqg q @ +4e4, d4 1 Lw lv hdvlo| vkrzq wkdw +q q, @ +4 , 5 +45, +4,+4., A 31
6;Fdopiruv +5333, surylghv vrph hylghqfh iurp Hxursh vxssruwlqj wklv uhvxow1 Lq wkh
4<;3v qrplqdo zdjh jurzwk zdv idvwhu lq wkh iroorzhu frxqwulhv ri wkh HUP +h{foxglqj Wkh Qhwkhuodqgv, wkdq lq Jhupdq|1 Krzhyhu/ wklv glgq*w krog lq wkh 4< < 3v dq|pruh1
6<Wkh h{fkdqjh udwhv dqg wudgh duh qrw h{solflwo| prghohg lq Ju qhu dqg Khihnhu +4<<<,
dqg Fxnlhupdq dqg Olssl +5333,1 Wkh HUP lv prghohg dv d fuhgleoh frpplwphqw wr wkh ohdghu frxqwu|*v lq dwlrq udwh1
Vlqfh zh kdyh dvvxphg dq xowud0frqvhuydwlyh fhqwudo edqn lq wkh ohdghu frxqwu|/ dqg wkhuhiruh dovr lq d prqhwdu| xqlrq/ lq dwlrq zloo vwd| dw }hur lq wkh ohdghu frxqwu|1 Zh kdyh vkrzhg wkdw ghvslwh ri d fuhgleoh frpplwphqw wr d {hg h{fkdqjh udwh/ wkh iroorzhu frxqwu| lq dwlrq udwh zloo h{fhhg wkh ohdghu frxqwu|*v lq dwlrq1 Wkxv wkh iroorzhu frxqwu| lq dwlrq zloo idoo lq d prqhwdu| xqlrq dqg wkh iroorzhu frxqwu| zloo rewdlq erwk kljkhu hpsor|phqw dqg orzhu lq dwlrq1
9 Frqfoxglqj uhpdunv
Wkh sxusrvh ri wkh sdshu lv wr vwxg| wkh pdfurhfrqrplf frqvhtxhqfhv ri wkh hvwdeolvkphqw ri d prqhwdu| xqlrq1 Lq sduwlfxodu/ zh vkrz wkdw lq wkh suhvhqfh ri vwudwhjlf lqwhudfwlrq ehwzhhq wkh fhqwudo edqn dqg wkh oderu xqlrqv fkdqjhv lq prqhwdu| uhjlph zloo kdyh uhdo h hfwv1 Prqhwdu| uhjlph zloo d hfw wkh oderu xqlrq*v uhdo zdjh0hpsor|phqw wudgh0r / lh wkh uhdo frqvxphu zdjh hodvwlflw| ri oderu ghpdqg lv hqgrjhqrxv lq rxu prgho1 Wkh pdlq uhvxow ri wkh sdshu lv wkdw wkh hvwdeolvkphqw ri d prqhwdu| xqlrq lpsuryhv hpsor|phqw/ surylghg wkdw wkh ghjuhh ri fhqwudo edqn frqvhuydwlvp lv vx!flhqwo| kljk/ zkhuhdv zlwk orz ghjuhhv ri frqvhuydwlvp hpsor|phqw idoov1 Wkh lpsdfw rq lq dwlrq lv xq0 dpeljxrxv1 Lq dwlrq whqgv wr eh kljkhu lq d prqhwdu| xqlrq/ ehfdxvh xqolnh lq d rdwlqj h{fkdqjh udwh uhjlph/ prqhwdu| srolf| lv qrw frqvwudlqhg e| ihduv ri lq dwlrqdu| uhdo h{fkdqjh udwh ghsuhfldwlrq1
Zh dovr dqdo|}h dq dv|pphwulf {hg h{fkdqjh udwh uhjlph dv dq dowhuqdwlyh vwduwlqj srvlwlrq1 Wkh iroorzhu frxqwu| lv frpplwwhg wr d {hg h{fkdqjh udwh dqg wkh fhqwudo edqn ri wkh ohdghu frxqwu| lv dvvxphg wr eh xowud0frqvhuydwlyh1 Lw wxuqv rxw wkdw zkhq d prqhwdu| xqlrq lv hvwdeolvkhg/ hpsor|phqw zloo lq0 fuhdvh lq wkh iroorzhu frxqwu|/ zkhuhdv lq wkh ohdghu frxqwu| hpsor|phqw zloo idoo1 Lq dwlrq lv xqdowhuhg lq wkh ohdghu frxqwu|/ exw lw zloo ghfuhdvh lq wkh iroorzhu frxqwu|1 Khqfh/ wkh iroorzhu frxqwu| ehqh wv iurp prqhwdu| xqlrq erwk lq whupv ri hpsor|phqw dqg lq dwlrq/ zkhuhdv wkh ohdghu frxqwu| vx huv iurp ghwhulrudwlqj hpsor|phqw1
Rxu dvvxpswlrq ri d vlqjoh xqlrq lq hdfk frxqwu| lv ri frxuvh yhu| uhvwulf0 wlyh/ exw lw fdq eh mxvwl hg dv wkh vlpsohvw zd| wr prgho qrq0dwrplvwlf oderu pdunhwv1 Zh kdyh dvvxphg wkdw oderu lv krprjhqrxv1 Wkhuhiruh/ lw pdnhv qr vhqvh wr frqvlghu gl huhqw ohyhov ri fhqwudol}dwlrq ri zdjh edujdlqlqj/ vlqfh xqlrqv fdq*w udlvh wkhlu zdjhv deryh wkh dyhudjh zdjh udwh ri wkh hfrqrp|1 Zh frxog iroorz Olssl +4<<<, dqg lqwurgxfh wkh ohyho ri fhqwudol}dwlrq ri zdjh edujdlqlqj lqwr wkh prgho e| dvvxplqj wkdw oderu lv gl huhqwldwhg lq jhqhudo dqg wkdw doo w|shv ri oderu duh qhhghg lq wkh surgxfwlrq ri wkh vlqjoh jrrg1 Wkhq hdfk xqlrq kdv sulflqj srzhu dqg wkh h hfwv ri fhqwudol}dwlrq frxog eh lq0 vshfwhg1 Rqh olnho| frqvhtxhqfh lv wkdw hpsor|phqw ohyho zrxog fhwhulv sdulexv lqfuhdvh zlwk wkh qxpehu ri xqlrqv/ vlqfh hdfk xqlrq*v delolw| wr dssuhfldwh wkh uhdo h{fkdqjh udwh zlwk lwv zdjh fodlp zrxog eh uhgxfhg1 Krzhyhu/ lw lv xq0 fohdu krz wklv zrxog d hfw wkh uhodwlyh hpsor|phqw shuirupdqfh ri d prqhwdu| xqlrq1
Wkh prgho fdq eh xvhg lq qrupdwlyh dqdo|vlv ri d prqhwdu| xqlrq1 Reyl0 rxvo|/ rqh kdv wr eh h{wuhpho| fdxwlrxv lq gudzlqj vwurqj srolf| lpsolfdwlrqv edvhg rq vxfk d vlpsoh prgho1 Wkh vlqjoh xqlrq dvvxpswlrq fohduo| olplwv wkh dssolfdelolw| ri rxu prgho lq srolf| dqdo|vlv wr frxqwulhv zlwk uhodwlyho| fhqwudol}hg zdjh vhwwlqj1 Wkh prgho vhhpv wr vxjjhvw wkdw zkhq d prqhwdu| xqlrq lv hvwdeolvkhg/ lw lv srvvleoh wr dwwdlq erwk kljkhu hpsor|phqw dqg orzhu lq dwlrq e| ghohjdwlqj prqhwdu| srolf| wr d vx!flhqwo| frqvhuydwlyh fhqwudo edqn1 Lq Hxursh wkh fuhdwlrq ri wkh HPX kdv lqfuhdvhg wkh ghjuhh ri fhqwudo edqn frqvhuydwlvp lq prvw phpehu frxqwulhv1 Wkxv/ edvhg rq rxu prgho/ rqh frxog vshfxodwh wkdw wkh HPX pd| zhoo kdyh srvlwlyh h hfwv rq htxloleulxp hpsor|phqw lq pdq| frxqwulhv1 Pruhryhu/ li zh lqwhusuhw wkh dv|pphwulf { uhjlph dv dq dssur{lpdwlrq ri wkh HUP wkh frqfoxvlrq zloo/ h{foxglqj Jhu0 pdq|/ uhpdlq wkh vdph1
Fhqwudo edqn frqvhuydwlvp lv d iuhh oxqfk lq rxu prgho1 Lq dgglwlrq wr uhgxfhg lq dwlrq/ lw dovr lpsuryhv hpsor|phqw wkurxjk lqfuhdvhg h hfwlyh od0 eru ghpdqg hodvwlflw|1 Krzhyhu/ lw pxvw eh vwuhvvhg/ wkdw rxu ghwhuplqlvwlf dqdo|vlv devwudfwv frpsohwho| iurp srwhqwldo jdlqv iurp vwdelol}dwlrq srolf|1 Dv Urjr +4<;8d, vkrzhg/ dq xowud0frqvhuydwlyh fhqwudo edqn lv qrw rswlpdo lq d suhvhqfh ri vkrfnv/ vlqfh rxwsxw +hpsor|phqw, ehfrphv wrr yrodwloh1 Dqrwkhu lvvxh uhodwhg wr vwdelol}dwlrq srolf| lv wkh uhodwlyh vwdelol}dwlrq shuirupdqfh ri d prqhwdu| xqlrq1 Wkh ghvludelolw| ri d prqhwdu| xqlrq zloo reylrxvo| ghshqg qrw rqo| rq wkh fkdqjhv lq orqj0uxq dyhudjh ohyhov ri hfrqrplf ydul0 deohv/ exw dovr rq lwv lpsdfw rq wkh yrodwlolw| ri wkhvh yduldeohv173 Odqh +5333,
vkrzv wkdw d rdwlqj h{fkdqjh udwh w|slfdoo| rxwshuirupv wkh prqhwdu| xqlrq lq whupv ri vwdelol}dwlrq surshuwlhv174 Wklv frq upv wkh lqwxlwlrq wkdw h{0
fkdqjh udwh h{lelolw| lv ehqh fldo lq wkh suhvhqfh ri vkrfnv lq wkh hfrqrp|1 Lqwhuhvwlqjo| hqrxjk/ lq dq dv|pphwulf { wkh uhvxowv frqfhuqlqj vwdelol}dwlrq srolf| duh srlqwlqj wr wkh vdph gluhfwlrq dv rxu uhvxowv rq orqj0uxq dyhudjh shuirupdqfh1 Prqhwdu| xqlrq lv suhihuuhg wr dq dv|pphwulf { lq wkh iroorzhu frxqwu|/ zkhuhdv wkh rssrvlwh krogv iru wkh ohdghu frxqwu|1
73D glvfxvvlrq ri wklv lvvxh fdq eh irxqg lq Fdopiruv +4<<;/ 5333, dqg Exugd +4<<<,1 74Frulfhool/ Fxnlhupdq dqg Gdopd}}r +5333e, dqdo|}h vwdelol}dwlrq srolf| lq d wzr0frxqwu|
prqhwdu| xqlrq prgho1 Wkhlu iudphzrun doorzv dv|pphwulhv lq frxqwu| vl}hv/ oderu pdunhw vwuxfwxuhv dqg surgxfw pdunhw frpshwlwlrq1
D Dsshqgl{
D14 Ghwdlov ri wkh prgho
Wkh uhgxfhg irupv iru wkh rwkhu hqgrjhqrxv yduldeohv lq wkh rdwlqj h{fkdqjh udwh uhjlph duh
R ' E k 6 n kc RW ' E k 6Wn kW +7:, 5 ' kB E6 6Wn W +7;, e ' kB n E k E6 6W nk k B E W +7<,
Wkh Qdvk htxloleulxp qrplqdo zdjh hodvwlflwlhv ri prqh| vxsso| lq wkh rdwlqj h{fkdqjh udwh uhjlph duh
S ' Ej n K 2 j n Kkq B k kq B K 2 { T f +83, S2 ' kqB K 2 { f +84, zkhuh { ' Ej n K2 2kq B K 2 : f1
Wkh qrplqdo zdjh hodvwlflwlhv ri hpsor|phqw dqg lq dwlrq lq wkh rdwlqj h{fkdqjh udwh uhjlph duh
@s ' K Ej n K{ 2 f +85, @s2 ' kq B K2 { f +86, Ks ' j Ej n K{ 2 f +87, Ks2 ' kqB jK{ f +88,
Wkh uhgxfhg irupv iru wkh rwkhu hqgrjhqrxv yduldeohv lq wkh prqhwdu| xqlrq fdvh duh R ' n E k 6L E k 2 E n W +89, n 2 E k B k n E k B EW RW ' Wn E k 6L E k 2 E n W +8:, 2k n E k BE k B EW 5 ' k k n E k B EW +8;,
Wkh qrplqdo zdjh hodvwlflwlhv ri hpsor|phqw dqg lq dwlrq lq wkh prqhwdu| 5<
xqlrq fdvh duh @ ' E k Ek n 2 E k B n Bj l f +8<, @ 2 ' k E k Bj l T f +93, K '
j n E k 2E2k E q n E k B E k 2Ek n E k B l : f +94, K 2 '
j n E k 2E2kq n E k B E k 2Ek n E k B
l T f +95,
zkhuh l ' 2j n E k 2Ek n E k B : f1
D15 Ghulydwlrq ri htxdwlrq +68,
Xqghu d rdwlqj h{fkdqjh udwh/ iurp wkh uhgxfhg irupv +4<, dqg +53, zh rewdlq
@s ' Y6Ys +96, Ks ' E k Y6s Y kq B Y6s Y Y6Ws Y +97, dqg e| qrwlqj/ wkdw Y6W Y ' @s2 +vhh htxdwlrqv +84, dqg +86,, wkh vorsh frh!flhqw fdq eh zulwwhq dv Ks @s ' E k kq B # @s @s2 @s $ +98, Vlploduo|/ lq d prqhwdu| xqlrq zh kdyh
@ ' Y6 Y 2 B 2 Ek n E k B +99, K ' E k Y6 Y 2 B 2 Ek n E k B +9:, n kq Ek n E k B ' E k @ kqB E@ @2 +9;,
zkhuh wkh odvw urz lv rewdlqhg iurp htxdwlrqv +4, dqg +6, e| qrwlqj wkdw 5 ' k
B E? ?W dqg wkxv YY5 ' kB E@ @2 1 Dsso|lqj wklv h{suhvvlrq dqg wkh
ghulydwlyh ri htxdwlrq +8;, zlwk uhvshfw wr 5 zh qdoo| jhw k
Ek n E k B ' k
Lqvhuwlqj wklv lqwr htxdwlrq +9:, |lhogv +9;, dqg khqfh wkh irup ri wkh vorsh frh!flhqw lv vlplodu wr wkh rdwlqj h{fkdqjh udwh fdvh K @ ' E k kq B @ @2 @ +:3,
D16 Dwrplvwlf edujdlqlqj
Zh prgli| wkh iudphzrun ri wkh sdshu e| dvvxplqj wkdw wkhuh lv d odujh qxpehu ri xqlrqv lq erwk frxqwulhv1 Wklv dvvxpswlrq lpsolhv wkdw xqlrqv wdnh wkh surgxfw sulfh dqg wkh frqvxphu sulfh lqgh{ dv jlyhq lq zdjh vhwwlqj1 Oderu ghpdqg ri wkh uhsuhvhqwdwlyh up lv jlyhq e|
R ' E k ? +:4, D uhsuhvhqwdwlyh xqlrq*v sureohp lv wkhq 4@ tL ' E Z 2E? 2 +:5, v1w1 +:4, Wkh uvw0rughu frqglwlrq lv R ' E k 2 +:6,
zklfk lpsolhv wkdw wkh uhdo surgxfw zdjh dqg hpsor|phqw duh frqvwdqwv +lqvhuw +:6, lqwr +:4,,1 Dvvxplqj d v|pphwulf htxloleulxp/ djjuhjdwlrq jlyhv wkh fruuhvsrqglqj htxdwlrq iru wrwdo hpsor|phqw
? ' E k +:7,
Khqfh/ zkhq oderu pdunhwv duh dwrplvwlf/ hpsor|phqw ghshqgv rqo| rq oderu xqlrq suhihuhqfhv E dqg wkh uhdo surgxfw zdjh hodvwlflw| ri oderu ghpdqg
# ' E k 31 Zh duh wkxv edfn lq wkh wudglwlrqdo iudphzrun/ zkhuh wkh htxloleulxp udwh ri hpsor|phqw lv h{rjhqrxv dqg fkdqjhv lq prqhwdu| lqvwlwx0 wlrqv zloo qrw kdyh dq| uhdo h hfwv1 Wkh hvwdeolvkphqw ri d prqhwdu| xqlrq zloo eh frxqwhusurgxfwlyh lq d vhqvh wkdw hpsor|phqw lv xqd hfwhg dqg lq dwlrq zloo eh kljkhu wkdq ehiruh/ ehfdxvh wkh frpprq fhqwudo edqn grhvq*w qhhg wr zruu| derxw lq dwlrqdu| uhdo h{fkdqjh udwh ghsuhfldwlrq1
D17 Surriv
Surri ri Uhpdun 41 Hpsor|phqw ohyhov lq erwk uhjlphv duh jlyhq e| htxdwlrq +66,1 Htxdwlrq +68, lpso|/ wkdw r}?Y? Yj ' r}?Y@2 @ *Yjlq erwk uhjlphv1 Iurp +85, dqg +86, zh rewdlq@2
@ s ' EkqB K jnK2 dqg iurp +8<, dqg +93, @2 @
' E3k Ekn2E3k B nBjkE3k 3Bj 1 Gl huhqwldwlrq ri wkhvh h{suhvvlrqv zlwk uhvshfw 64
