Fdslwdo Pdunhwv dqg wkh Lqvwdelolw|
ri Rshq Hfrqrplhv
e|
Sklolssh Djklrq
Kduydug Xqlyhuvlw|/ Xqlyhuvlw| Froohjh Orqgrq/ dqg FHSU
Sklolssh Edffkhwwd
Vwxg| Fhqwhu Jhu}hqvhh/ Xqlyhuvlwì gh Odxvdqqh/ dqg FHSU
dqg
Deklmlw Edqhumhh
Pdvvdfkxvhwwv Lqvwlwxwh ri Whfkqrorj|
Qryhpehu 5333
- Zh duh judwhixo wr wkuhh dqrq|prxv uhihuhhv/ Ehdwul} Duphqgdul}/ Odxud Erwwd}}l/
Udtxho Ihuqdqgh}/ Hg Jodhvhu/ Xuedq Mhupdq/ Mhuh| Qlovhq/ Kìoëqh Uh|/ Gdylg Zhlo/
dqg vhplqdu sduwlflsdqwv dw Kduydug Xqlyhuvlw|/ wkh Xqlyhuvlwdw Dxwõqrpd ri Edufhorqd/
dqg wkh OVH2IPJ frqihuhqfh rq pdunhw oltxlglw| iru frpphqwv rq d suhylrxv gudiw1
Slhuuh0Dodlq Euxfkh} surylghg h{fhoohqw uhvhdufk dvvlvwdqfh1 Ilqdqfldo vxssruw iurp wkh
PdfDuwkxu Irxqgdwlrq +Edqhumhh dqg Djklrq, dqg wkh QVI +Edqhumhh, lv judwhixoo|
dfnqrzohgjhg1
4
Devwudfw
Wklv sdshu lqwurgxfhv d iudphzrun iru dqdo|}lqj wkh uroh ri qdqfldo idfwruv dv d
vrxufh ri lqvwdelolw| lq vpdoo rshq hfrqrplhv1 Rxu edvlf prgho lv d g|qdplf rshq hfrq0
rp| prgho zlwk d wudghdeoh jrrg surgxfhg zlwk fdslwdo dqg d frxqwu|0vshflf idfwru1
Zh dovr dvvxph wkdw upv idfh fuhglw frqvwudlqwv/ zlwk wkh frqvwudlqw ehlqj wljkwhu dw d
orzhu ohyho ri qdqfldo ghyhorsphqw1 D edvlf lpsolfdwlrq ri wklv prgho lv wkdw hfrqrplhv
dw dq lqwhuphgldwh ohyho ri qdqfldo ghyhorsphqw duh pruh xqvwdeoh wkdq hlwkhu yhu| gh0
yhorshg ru yhu| xqghughyhorshg hfrqrplhv1 Wklv lv wuxh erwk lq wkh vhqvh wkdw whpsrudu|
vkrfnv kdyh odujh dqg shuvlvwhqw hhfwv dqg dovr lq wkh vhqvh wkdw wkhvh hfrqrplhv fdq
h{klelw f|fohv1 Wkxv/ frxqwulhv wkdw duh jrlqj wkurxjk d skdvh ri qdqfldo ghyhorsphqw
pd| ehfrph pruh xqvwdeoh lq wkh vkruw uxq1 Vlploduo|/ lq hfrqrplhv dw dq lqwhuphgl0
dwh ohyho ri qdqfldo ghyhorsphqw/ ixoo fdslwdo dffrxqw olehudol}dwlrq pd| ghvwdelol}h wkh
hfrqrp|1 Rq wkh rwkhu kdqg/ iruhljq gluhfw lqyhvwphqw grhv qrw ghvwdelol}h1
5
4
Lqwurgxfwlrq
Wklv sdshu lqwurgxfhv d iudphzrun iru dqdo|}lqj wkh uroh ri qdqfldo idfwruv dv d vrxufh
ri lqvwdelolw| lq vpdoo rshq hfrqrplhv1 Rxu edvlf prgho lv d g|qdplf rshq hfrqrp|
prgho zlwk d wudghdeoh jrrg surgxfhg zlwk lqwhuqdwlrqdoo| preloh fdslwdo dqg d frxqwu|0
vshflf idfwru1 Pruhryhu/ upv idfh qdqfldo frqvwudlqwv= wkh dprxqw wkh| fdq eruurz
lv olplwhg wr > wlphv wkh dprxqw ri wkhlu fxuuhqw ohyho ri lqyhvwleoh ixqgv14 D kljk >
wkhq uhsuhvhqwv dq hhfwlyh dqg ghyhorshg qdqfldo vhfwru zkloh d orz > uhsuhvhqwv dq
xqghughyhorshg rqh1
Rxu prgho fdq surylgh vrph dqvzhuv wr d qxpehu ri lpsruwdqw dqg udwkhu edvlf
txhvwlrqv1 Iluvw/ zh vkrz wkdw lw lv hfrqrplhv dw dq lqwhuphgldwh ohyho ri qdqfldo
ghyhorsphqw 0 udwkhu wkdq wkh yhu| ghyhorshg ru xqghughyhorshg 0 wkdw duh wkh prvw
xqvwdeoh1 Wklv lv wuxh erwk lq wkh vhqvh wkdw whpsrudu| vkrfnv zloo kdyh odujh dqg
shuvlvwhqw hhfwv dqg dovr lq wkh vhqvh wkdw wkhvh hfrqrplhv fdq h{klelw vwdeoh olplw
f|fohv1 Wkxv/ frxqwulhv jrlqj wkurxjk d skdvh ri qdqfldo ghyhorsphqw pd| ehfrph
pruh xqvwdeoh lq wkh vkruw uxq1
Vhfrqg/ wkh prgho doorzv xv wr h{dplqh wkh hhfwv ri qdqfldo olehudol}dwlrq rq wkh
vwdelolw| ri wkh pdfurhfrqrp|1 Rqfh djdlq lw wxuqv rxw wkdw wkh lqwhuhvwlqj hfrqrplhv
duh wkh rqhv dw dq lqwhuphgldwh ohyho ri qdqfldo ghyhorsphqw1 Lq wkhvh hfrqrplhv/ ixoo
qdqfldo olehudol}dwlrq +l1h1/ rshqlqj wkh grphvwlf pdunhw wr iruhljq fdslwdo rzv, pd|
dfwxdoo| ghvwdelol}h/ lqgxflqj fkurqlf skdvhv ri jurzwk zlwk fdslwdo lq rzv iroorzhg e|
froodsvh zlwk fdslwdo ljkw1 Rq wkh rwkhu kdqg/ iruhljq gluhfw lqyhvwphqw qhyhu ghvwdel0
ol}hv vlqfh iruhljq gluhfw lqyhvwruv frph lq zlwk wkhlu rzq fuhglwwkhlu delolw| wr lqyhvw lv
4
Wkh idfw wkdw up ohyho fdvk0 rz lv dq lpsruwdqw ghwhuplqdqw ri lqyhvwphqw lv qrz zlgho| uhfrjql}hg
hyhq lq wkh frqwh{w ri hfrqrplhv olnh wkh X1V1 zklfk kdyh h{fhoohqw qdqfldo pdunhwv1 +h1j1/ vhh Kxeedug
+4<<;, ru Ehuqdqnh/ Jhuwohu/ dqg Jlofkulvw +4<<<,,1
6
xquhodwhg wr wkh vwdwh ri wkh grphvwlf hfrqrp|1 Ryhudoo/ wklv vxjjhvwv wkdw hfrqrplhv dw
dq lqwhuphgldwh vwdjh ri qdqfldo ghyhorsphqw vkrxog frqvlghu fduhixoo| krz wkh| olehu0
dol}h wkhlu fdslwdo dffrxqw1 Doorzlqj iruhljq gluhfw lqyhvwphqw zkloh lqlwldoo| uhvwulfwlqj
sruwirolr lqyhvwphqw pd| vrphwlphv eh d uhdvrqdeoh dssurdfk1
Wklug/ rxu prgho doorzv xv wr dvvhvv wkh pdfurhfrqrplf hhfwv ri vshflf vkrfnv wr
wkh qdqfldo vhfwru vxfk dv ryhuohqglqj e| edqnv +ohdglqj wr d skdvh ri edqn idloxuhv,
ru ryhuuhdfwlrq e| lqyhvwruv wr d fkdqjh lq ixqgdphqwdov15 Rqfh djdlq/ rxu prgho
suhglfwv wkhvh vkrfnv wr kdyh wkhlu prvw shuvlvwhqw hhfwv zkhq qdqfldo pdunhwv duh dw
dq lqwhuphgldwh vwdjh ri ghyhorsphqw1
Wkh edvlf phfkdqlvp xqghuo|lqj rxu prgho lv d frpelqdwlrq ri wzr irufhv= rq rqh
vlgh/ juhdwhu lqyhvwphqw ohdgv wr juhdwhu rxwsxw dqg fhwhulv sdulexv/ kljkhu surwv1
Kljkhu surwv lpsuryh fuhglwzruwklqhvv dqg ixho eruurzlqj wkdw ohdgv wr juhdwhu lq0
yhvwphqw1 Fdslwdo rzv lqwr wkh frxqwu| wr qdqfh wklv errp1 Dw wkh vdph wlph/ wkh
errp lq lqyhvwphqw lqfuhdvhv wkh ghpdqg iru wkh frxqwu|0vshflf idfwru dqg udlvhv lwv
sulfh uhodwlyh wr wkh rxwsxw jrrg +xqohvv wkh vxsso| ri wkdw idfwru lv h{wuhpho| hodvwlf,1
Wklv ulvh lq lqsxw sulfhv ohdgv wr orzhu surwv dqg wkhuhiruh/ uhgxfhg fuhglwzruwklqhvv/
ohvv eruurzlqj dqg ohvv lqyhvwphqw/ dqg d idoo lq djjuhjdwh rxwsxw1 Ri frxuvh/ rqfh lq0
yhvwphqw idoov doo wkhvh irufhv jhw uhyhuvhg dqg hyhqwxdoo| lqlwldwh dqrwkhu errp1 Lw lv
wklv hqgrjhqrxv lqvwdelolw| zklfk fdxvhv vkrfnv wr kdyh shuvlvwhqw hhfwv dqg lq pruh
h{wuhph fdvhv ohdgv wr olplw f|fohv1
Wkh uhdvrq zk| dq lqwhuphgldwh ohyho ri qdqfldo ghyhorsphqw lv lpsruwdqw iru wklv
uhvxow lv hdv| wr frpsuhkhqg= dw yhu| kljk ohyhov ri qdqfldo ghyhorsphqw/ prvw upv*
lqyhvwphqw lv qrw frqvwudlqhg e| fdvk rz vr vkrfnv wr fdvk rz duh luuhohydqw1 Rq
wkh rwkhu kdqg/ dw yhu| orz ohyhov ri qdqfldo ghyhorsphqw/ upv fdqqrw eruurz yhu|
pxfk lq dq| fdvh dqg wkhuhiruh wkhlu uhvsrqvh wr fdvk0 rz vkrfnv zloo eh udwkhu pxwhg
5
Shukdsv dv d frqvhtxhqfh ri khug ehkdylru1
7
0 h{wud fdvk phdqv pruh lqyhvwphqw exw rqo| d olwwoh pruh1 Wkhuhiruh vkrfnv zloo glh
rxw zlwkrxw fdxvlqj dq| juhdw wxuprlo1 Lw lv wkhq dw lqwhuphgldwh ohyhov ri qdqfldo
ghyhorsphqw wkdw vkrfnv wr fdvk rz zloo kdyh dq hhfw lqwhqvh hqrxjk wr eh d vrxufh
ri lqvwdelolw|1
Wklv odvw dujxphqw dovr khosv xv xqghuvwdqg zk| rshqlqj wkh hfrqrp| wr iruhljq
fdslwdo pd| ghvwdelol}h= hvvhqwldoo|/ wkh uhvsrqvh ri dq hfrqrp| zlwk d forvhg fdslwdo
pdunhw wr d fdvk rz vkrfn lv olplwhg vlqfh rqo| vr pxfk fdslwdo lv dydlodeoh wr hq0
wuhsuhqhxuv1 Dgglwlrqdo ixqglqj vrxufhv lq dq rshq hfrqrp| srwhqwldoo| lqfuhdvhv wkh
uhvsrqvh wr d vkrfn dqg wkhuhiruh wkh vfrsh iru yrodwlolw|1
Wkh edvlf phfkdqlfv ri lqvwdelolw| ghvfulehg khuh 0 dq lqfuhdvh lq lqsxw sulfh ohdglqj
wr d surw vtxhh}h dqg hyhqwxdo rxwsxw froodsvh 0 kdv ehhq grfxphqwhg lq d qxpehu
ri frxqwulhv1 Iru h{dpsoh/ lq wkh |hduv ohdglqj xs wr wkh fulvlv ri wkh hduo| 4<;3*v
lq wkh Vrxwkhuq Frqh frxqwulhv/ wkhuh lv hylghqfh wkdw surwv lq wkh wudghdeoh vhfwru
vkduso| ghwhulrudwhg gxh wr d ulvh lq grphvwlf lqsxw sulfhv +vhh Jdoyh} dqg W|erxw/
4<;8/ Shwuhl dqg W|erxw/ 4<;8/ ru Gh Phor/ Sdvfdoh dqg W|erxw/ 4<;8,1 Pruhryhu/ dpsoh
dqhfgrwdo hylghqfh vxssruwv wkh lpsdfw ri *frpshwlwlyhqhvv* +h1j1/ d uhdo dssuhfldwlrq,
rq wkh qdqfldo frqglwlrqv ri upv1
Wkh g|qdplf lpsdfw ri d olehudol}dwlrq suhglfwhg e| wkh prgho lv dovr frqvlvwhqw zlwk
wkh h{shulhqfh ri vhyhudo hphujlqj pdunhw frxqwulhv wkdw kdyh olehudol}hg/ lq sduwlfxodu
lq Vrxwkhdvw Dvld dqg Odwlq Dphulfd/ exw dovr lq vrph Hxurshdq frxqwulhv1 Lq wkh |hduv
sulru wr wkhlu uhvshfwlyh fulvhv/ wkhvh hfrqrplhv kdg ehhq jrlqj wkurxjk d surfhvv ri udslg
qdqfldo vhfwru olehudol}dwlrq/ zklfk idflolwdwhg eruurzlqj e| grphvwlf upv1 Sduwo| dv
d uhvxow ri wklv olehudol}dwlrq/ fdslwdo rzhg lqwr wkhvh hfrqrplhv lq odujh txdqwlwlhv/
doorzlqj udslg jurzwk lq ohqglqj dqg d errp lq lqyhvwphqw1 Krzhyhu/ hslvrghv ri
odujh fdslwdo lq rzv kdyh riwhq ehhq dvvrfldwhg zlwk jurzlqj lpedodqfhv/ vxfk dv d
8
uhdo fxuuhqf| dssuhfldwlrq6 / dq lqfuhdvh lq uhdo hvwdwh sulfhv +h1j1/ vhh Jxhuud gh Oxqd/
4<<:,/ ru dq lqfuhdvh lq qrq0shuiruplqj ordqv +vhh Zruog Edqn/ 4<<:/ s1 588,1 Zkhq wkh
fulvlv fdph/ prvw ri wkhvh irufhv jrw uhyhuvhg 0 fdslwdo rzhg rxw/ wkh fxuuhqf| froodsvhg/
uhdo hvwdwh sulfhv gursshg/ ohqglqj vwrsshg/ dqg lqyhvwphqw froodsvhg17
Lw lv krzhyhu lpsruwdqw wr hpskdvl}h wkdw wkh jrdo ri wklv sdshu lv qrw wr h{sodlq
h{dfwo| zkdw kdsshqhg lq vrph sduwlfxodu frxqwu|/ exw udwkhu wr sursrvh d xqlhg
pdfurhfrqrplf iudphzrun wkdw jlyhv d fhqwudo uroh wr qdqfldo frqvwudlqwv dqg qdq0
fldo ghyhorsphqw1 Wkhuh duh fhuwdlqo| d qxpehu ri vwudqgv ri wkh h{lvwlqj olwhudwxuh
dqwlflsdwlqj d vljqlfdqw sduw ri zkdw zh kdyh grqh khuh1 Wkh lghd wkdw qdqfldo frq0
vwudlqwv rq upv fdq sod| d uroh lq wkh sursdjdwlrq ri wkh exvlqhvv f|foh zdv prghohg lq
Ehuqdqnh dqg Jhuwohu +4<;<,1 Vxevhtxhqw zrun e| Nl|rwdnl dqg Prruh +4<<:,/ Djklrq/
Edqhumhh/ dqg Slnhww| +4<<<, dqg D}duldglv dqg Vplwk +4<<;, kdyh vkrzq wkdw wkhvh
frqvwudlqwv fdq ohdg wr rvfloodwlrqv/ wkrxjk rqo| lq wkh frqwh{w ri d forvhg hfrqrp|18
Krzhyhu qrqh ri wkhvh sdshuv h{fhsw Djklrq/ Edqhumhh/ dqg Slnhww| +4<<<, irfxv rq wkh
ohyho ri qdqfldo ghyhorsphqw dv d idfwru ghwhuplqlqj wkh h{whqw ri lqvwdelolw| dqg qrqh
ri wkhp vwxg| wkh hhfwv ri rshqlqj xs wkh grphvwlf qdqfldo vhfwru wr iruhljq fdslwdo
rzv1
6
Vhh/ iru h{dpsoh/ Fdoyr hw do1 +4<<9,1 Wkh ghjuhh ri uhdo dssuhfldwlrq ydulhv dfurvv frxqwulhv> iru
h{dpsoh/ lw kdv ehhq pruh surqrxqfhg lq Odwlq Dphulfd wkdq lq Dvld1
7
Vhh Zruog Edqn +4<<:, dqg Plohvl0Ihuuhwwl dqg Ud}lq +4<<;, iru v|vwhpdwlf ghvfulswlrqv ri wkh olqn
ehwzhhq dqg fdslwdo rz uhyhuvdov dqg fxuuhqf| fulvhv1 Pruhryhu/ Wruqhoo +4<<;, fodlpv wkdw wkh ghjuhh
ri uhdo h{fkdqjh udwh dssuhfldwlrq riwhq ghwhuplqhv wkh vhyhulw| ri wkh fulvlv1 Jrxulqfkdv/ Ydogìv/ dqg
Odqghuuhwfkh +4<<<, surylgh d v|vwhpdwlf dqdo|vlv ri ohqglqj errpv zklfk frlqflgh zlwk pryhphqwv lq
rxwsxw/ fdslwdo lq rzv/ wkh fxuuhqw dffrxqw dqg wkh uhdo h{fkdqjh udwh wkdw duh ixoo| frqvlvwhqw zlwk
rxu uhvxowv1 Vhh dovr Krqndsrkmd dqg Nrvnhod +4<<<, iru dq looxplqdwlqj ghvfulswlrq ri wkh Ilqqlvk
fulvlv ri wkh 4<<3*v/ zklfk wv zhoo rxu dqdo|vlv= uvw/ dq hfrqrplf hqylurqphqw fkdudfwhul}hg e| d odujh
sursruwlrq ri fuhglw0frqvwudlqhg hqwhusulvhv/ iru zklfk lqyhvwphqwv duh kljko| hodvwlf z1u1w1 fxuuhqw
surwv> vhfrqg/ d qdqfldo pdunhw ghuhjxodwlrq lq wkh 4<;3*v wkdw ohdgv wr d kxjh h{sdqvlrq ri edqn
ohqglqj/ wr pdmru lq rzv ri iruhljq fdslwdo dqg wr d vkdus lqfuhdvh lq uhdo dvvhw sulfhv +lq sduwlfxodu
uhdo hvwdwh sulfhv, gxulqj wkh errp> dqg vxevhtxhqwo| lq wkh 4<<3*v/ d vkdus idoo lq uhdo dvvhw sulfhv/
lqyhvwphqwv/ dqg uhdo JGS/ dqg wkh rffxuuhqfh ri d edqnlqj fulvlv wkdw hyhqwxdoo| ohg wr d wljkwhqlqj
ri edqnlqj uhjxodwlrqv dqg wr d ghydoxdwlrq ri wkh Ilqqlvk fxuuhqf| diwhu krshohvv hruwv wr pdlqwdlq d
{hg h{fkdqjh udwh1
8
Jhuwohu dqg Urjr +4<<3, vwxg| dq rshq hfrqrp| prgho zlwk fuhglw0pdunhw lpshuihfwlrqv1 Krz0
hyhu/ wkh| gr qrw frqvlghu exvlqhvv f|foh xfwxdwlrqv1
9
D vhsdudwh olwhudwxuh irfxvhv rq wkh fdvh iru iuhh fdslwdo prelolw|1 Srolf| lqwhuhvw
lq wkh ghedwh kdv ehhq durxvhg e| wkh uhfhqw/ udwkhu pl{hg/ h{shulhqfh ri d qxpehu ri
frxqwulhv wkdw kdyh olehudol}hg wkhlu fdslwdo dffrxqw/9 exw d qxpehu ri lpsruwdqw dvshfwv/
lqfoxglqj wkh lpsolfdwlrqv ri olehudol}dwlrq rq yrodwlolw|/ kdyh qrw ehhq zlgho| vwxglhg1:
Pruh lpsruwdqwo|/ qrqh ri wkhvh sdshuv dwwhpsw wr uhodwh wkh hhfw ri olehudol}dwlrq wr
wkh ixqfwlrqlqj ri wkh grphvwlf qdqfldo vhfwru1
Ilqdoo| d qxpehu ri uhfhqw sdshuv vwuhvv wkdw vshflf vkrfnv wr wkh qdqfldo vhfwru/
vxfk dv wkrvh eurxjkw rq e| srolf| plvwdnhv/ khug ehkdylru/ sdqlfv/ ru fruuxswlrq lq wkh
qdqfldo vhfwru/ pd| ohdg wr fulvhv lq wkh uhdo hfrqrp|1 Zkloh dffhswlqj wkh ydolglw|
ri wkhvh dujxphqwv/ zh ihho wkhvh prghov vxhu iurp ljqrulqj vrph ri wkh lqwhudfwlrqv
ehwzhhq wkh qdqfldo vhfwru dqg wkh uhvw ri wkh hfrqrp|1 Dv rxu prgho pdnhv fohdu/
yrodwloh ehkdylru pd| dulvh hyhq lq wkh devhqfh ri vxfk vkrfnv> zkloh rq wkh rwkhu
kdqg/ wkh suhvhqfh ri vxfk vkrfnv grhv qrw dxwrpdwlfdoo| lpso| wkh| zloo kdyh odujh dqg
shuvlvwhqw uhdo hhfwv1
Wkh sdshu lv rujdql}hg dv iroorzv1 Vhfwlrq 5 uhsuhvhqwv wkh fruh ri wkh sdshu/ zlwk d
ghvfulswlrq ri wkh rshq0hfrqrp| prgho dqg d fkdudfwhul}dwlrq ri wkh frqglwlrqv xqghu
zklfk pdfurhfrqrplf yrodwlolw| dulvhv1 Vhfwlrq 6 suhvhqwv qxphulfdo vlpxodwlrqv wr dvvhvv
wkh sodxvlelolw| iru yrodwlolw|1 Vhfwlrq 7 dqdo|}hv wkh lpsdfw ri d fdslwdo dffrxqw olehu0
dol}dwlrq dqg frqwudvwv wkh vwdelol}lqj hhfw ri xquhvwulfwhg IGL zlwk wkh srwhqwldoo|
ghvwdelol}lqj hhfwv ri hlwkhu iruhljq lqgluhfw lqyhvwphqwv ru uhvwulfwhg iruhljq gluhfw
lqyhvwphqwv1 Vhfwlrq 8 ghvfulehv ydulrxv h{whqvlrqv dqg gudzv vrph whqwdwlyh srolf|
frqfoxvlrqv1
9
Vhh/ iru h{dpsoh/ Mrkqvwrq hw do1 +4<<:, ru Hlfkhqjuhhq hw do1 +4<<;,1
Revwihog +4<;9,/ PfNlqqrq +4<<6,/ Edffkhwwd +4<<5,/ Eduwrolql dqg Gud}hq +4<<:, dqdo|}h fdslwdo
dffrxqw olehudol}dwlrqv1 PfNlqqrq dqg Sloo +4<<:, dqg Edffkhwwd dqg ydq Zlqfrrs +5333, duh dprqj
wkh ihz h{dplqlqj wkh lvvxh ri yrodwlolw|1
:
:
5
Wkh Edvlf Phfkdqlvp
Iru shgdjrjlfdo sxusrvhv zh frqvlghu uvw d vlpsoh prgho zlwk frqvwdqw vdylqj udwhv
dqg d Ohrqwlhi whfkqrorj| lqyroylqj d lqhodvwlf vxsso| ri wkh frxqwu|0vshflf idfwru1 Zh
wkhq h{whqg rxu dqdo|vlv/ uvw wr doorz iru dq hodvwlf vxsso| ri grphvwlf lqsxw> vhfrqg/
wr wkh pruh jhqhudo fdvh ri d FHV surgxfwlrq whfkqrorj|> dqg wklug/ wr wkh fdvh zkhuh
vdylqj ghflvlrqv uhvxow iurp lqwhuwhpsrudo xwlolw| pd{lpl}dwlrq1
514
D Vlpsoh Iudphzrun
Zh frqvlghu d vpdoo rshq hfrqrp| zlwk d vlqjoh wudghdeoh jrrg surgxfhg zlwk fdslwdo
dqg d frxqwu|0vshflf idfwru1 Rqh vkrxog w|slfdoo| wklqn ri wklv idfwru dv lqsxw vhuylfhv
vxfk dv +vnloohg, oderu ru uhdo hvwdwh1 Zh wdnh wkh rxwsxw jrrg dv wkh qxphudluh dqg
ghqrwh e| R wkh sulfh ri wkh frxqwu|0vshflf idfwru zkhq h{suhvvhg lq xqlwv ri wkh rxwsxw
jrrg1 Wkh uhodwlyh sulfh R fdq dovr eh lqwhusuhwhg dv wkh uhdo h{fkdqjh udwh1 Lq wklv edvlf
iudphzrun zh dvvxph wkdw wkh vxsso| ri wkh frxqwu|0vshflf idfwru lv lqhodvwlf dqg htxdo
wr ~1
Iru wkh vdnh ri suhvhqwdwlrq/ lq wklv vxevhfwlrq zh dovr dvvxph wkdw doo djhqwv vdyh
d {hg iudfwlrq E k ri wkhlu wrwdo hqg0ri0shulrg zhdowk dqg wkxv frqvxph d {hg
iudfwlrq k Wklv dvvxpswlrq zloo eh uhod{hg lq vxevhfwlrq 518 ehorz1 Wkh lqwhuwhpsrudo
ghflvlrqv ri ohqghuv duh ri qr frqvhtxhqfh iru rxwsxw lq vxfk dq rshq hfrqrp| vlqfh
lqyhvwruv fdq eruurz lq lqwhuqdwlrqdo fdslwdo pdunhwv1 Wkh| zloo/ krzhyhu/ dhfw qhw
fdslwdo rzv1;
Wkhuh duh wzr glvwlqfw fdwhjrulhv ri lqglylgxdov lq wkh hfrqrp|1 Iluvw/ wkh ohqghuv
zkr fdqqrw gluhfwo| lqyhvw lq surgxfwlrq/ exw fdq ohqg wkhlu lqlwldo zhdowk hqgrzphqwv
;
Qrwlfh wkdw wkh vhsdudwlrq ehwzhhq wkh ghflvlrqv ri ohqghuv dqg hqwuhsuhqhxuv grhv qrw lpso|
vhsdudwlrq ehwzhhq wrwdo qdwlrqdo vdylqjv dqg lqyhvwphqw1 Jhuwohu dqg Urjr +4<<3, vkrz wkdw d
iudphzrun zlwk fuhglw frqvwudlqwv fdq h{sodlq wkh kljk fruuhodwlrq ehwzhhq wrwdo vdylqjv dqg lqyhvwphqw
+Ihogvwhlq dqg Krulrnd/ 4<;3,1 Zh rewdlq d vlplodu uhvxow lq rxu iudphzrun1 Krzhyhu/ lq jhqhudo wklv
uhvxow dovr ghshqgv rq ohqghuv* vdylqjv ehkdylru1
;
dw wkh lqwhuqdwlrqdo pdunhw0fohdulqj lqwhuhvw udwh o1 Vhfrqg/ wkhuh duh wkh hqwuhsuhqhxuv
+ru eruurzhuv, zkr duh wkh shrsoh zkr kdyh wkh rssruwxqlw| wr lqyhvw lq surgxfwlrq1
Wkhuh lv d frqwlqxxp ri ohqghuv dqg eruurzhuv dqg wkhlu qxpehu lv qrupdol}hg wr rqh
iru erwk fdwhjrulhv1
Rxwsxw + lv jlyhq e| wkh iroorzlqj surgxfwlrq ixqfwlrq=
+ ' 4?E
g
c 5c
@
+4,
zkhuh *@ : o/ l1h1/ zh dvvxph wkdw surgxfwlylw| lv odujhu wkdq wkh zruog lqwhuhvw udwh1 g
ghqrwhv wkh fxuuhqw ohyho ri fdslwdo dqg 5 ghqrwhv wkh ohyho ri wkh frxqwu|0vshflf lqsxw1
Pruh jhqhudo surgxfwlrq ixqfwlrqv zloo eh lqwurgxfhg lq vhfwlrq 5171 Zlwk shuihfw fdslwdo
pdunhwv/ lqyhvwphqw zrxog vlpso| eh ghwhuplqhg e| wkh lqwhuqdwlrqdo lqwhuhvw udwh o1
Fuhglw0pdunhw lpshuihfwlrqv= Gxh wr vwdqgdug djhqf| +prudo kd}dug, frqvlghudwlrqv/
dq hqwuhsuhqhxu zlwk lqlwldo zhdowk ` fdq eruurz dw prvw u ' >` 1 Wkh suhvhqfh
ri fdslwdo pdunhw lpshuihfwlrqv lpsolhv wkdw hqwuhsuhqhxuv fdqqrw eruurz xs wr wkh qhw
suhvhqw ydoxh ri wkhlu surmhfw> wkh| fdq rqo| eruurz dq dprxqw sursruwlrqdo wr wkhlu
fxuuhqw fdvk0 rz +dv lq Ehuqdqnh0Jhuwohu +4<;<,,1 Wkh sursruwlrqdolw| frh!flhqw/ ru
fuhglw pxowlsolhu
> : f/ uh hfwv wkh ohyho ri qdqfldo ghyhorsphqw lq wkh grphvwlf
hfrqrp|1 Lq wkh h{wuhph fdvh zkhuh > ' f/ wkh fuhglw pdunhw froodsvhv dqg lqyhvwruv
fdq rqo| lqyhvw wkhlu rzq zhdowk1 Kljkhu ydoxhv ri > fruuhvsrqg wr kljkhu ohyhov ri
qdqfldo ghyhorsphqw1
D vlpsoh mxvwlfdwlrq iru uhodwlqj wkh fdslwdo pdunhw wr wkh ohyho ri qdqfldo ghyhors0
phqw dqg edvlqj lw rq prudo kd}dug e| wkh eruurzhu/ fdq eh irxqg lq Kropvwurp0Wluroh
+4<<9, dqg lq Djklrq0Edqhumhh0Slnhww| +4<<<,1 Lq jhqhudo > zloo ghshqg rq wkh udwh ri
lqwhuhvw ehlqj fkdujhg/ zklfk lq wxuq lpsolhv d frqvwdqw fuhglw pxowlsolhu lq d prgho
zkhuh wkh lqwhuhvw udwh lv jlyhq e| wkh zruog fdslwdo pdunhwv1 Krzhyhu/ lq vhfwlrq 7 dqg
Dsshqgl{ E zh frpsduh rxu edvlf prgho zlwk d prgho zlwk d forvhg fdslwdo pdunhw
zkhuh wkh lqwhuhvw udwh lv hqgrjhqrxvo| ghwhuplqhg e| grphvwlf lqyhvwphqw ghpdqg dqg
<
grphvwlf vdylqjv vxsso|1 \hw/ iru frqyhqlhqfh/ zh vkdoo pdlqwdlq wkh dvvxpswlrq ri d >
wkdw grhv qrw ghshqg rq wkh lqwhuhvw udwh lq wkdw vhfwlrq dv zhoo1 Dv vkrzq lq Djklrq0
Edqhumhh0Slnhww| +4<<<,/ wklv fruuhvsrqgv wr d sduwlfxodu sdudphwul}dwlrq ri wkh pruh
jhqhudo prgho ri wkh fuhglw pdunhw suhvhqwhg lq wkdw sdshu1 Rxu uhvxowv zrxog rqo| eh
vwurqjhu li zh doorzhg wkh xvxdo qhjdwlyh uhodwlrq ehwzhhq wkh lqwhuhvw udwh dqg >
Surgxfwlrq ghflvlrq= Hqwuhsuhqhxuv zlwk wrwdo lqlwldo zhdowk ` zloo fkrrvh wkh ohyho
ri wkh frxqwu|0vshflf idfwru 5 ^zlwk fruuhvsrqglqj lqyhvwphqw g ' U R5` wr pd{lpl}h
fxuuhqw surwv1 Jlyhq wkh deryh Ohrqwlhi whfkqrorj|/ wkh rswlpxp lqyroyhv 5 ' g@ c vr
wkdw=
U R5 '@5
+5,
Wkhuh duh wzr fdvhv=
l, ~ : g@ vr wkdw wkhuh lv dq h{fhvv vxsso| ri wkh frxqwu|0vshflf lqsxw1 Wklv
lpphgldwho| jlyhv xv R ' f Rxwsxw dw gdwh | lv wkhq jlyhq e|=
+| '
g|
' E n >`| @
@
ll, ~ g@ vr wkdw wkhuh lv h{fhvv ghpdqg iru wkh lppreloh idfwru1 Wkhuhiruh R : f
dqg rxwsxw lv ghwhuplqhg lq htxloleulxp e| wkh vxsso| ri wkh frxqwu|0vshflf lqsxw=
+| ' ~ Iurp +5, dqg wkh ghqlwlrq ri U/ wkh htxloleulxp sulfh ri wkh frxqwu|0vshflf
lqsxw lv jlyhq e|=
R| '
E n >`| @~
~
+6,
Wklv htxloleulxp sulfh/ zklfk lv d srvlwlyh ixqfwlrq ri ` c lv wkh nh| yduldeoh zkrvh
pryhphqwv ryhu wlph zloo surgxfh yrodwlolw|1
Wkh Wlplqj ri Hyhqwv= Wkh wlplqj ri hyhqwv zlwklq hdfk shulrg | lv wkh iroorzlqj1
Lqyhvwphqw/ eruurzlqj dqg ohqglqj/ dqg wkh sd|phqw ri wkh frxqwu|0vshflf idfwru vhu0
ylfhv R ~ e| hqwuhsuhqhxuv wr wkh rzqhuv ri wkdw idfwru/ wdnh sodfh dw wkh ehjlqqlqj ri
wkh shulrg +zklfk zh ghqrwh e| |3 ,1 Hyhu|wklqj hovh rffxuv dw wkh hqg ri wkh shulrg
43
+zklfk zh ghqrwh e| |n = wkh uhwxuqv wr lqyhvwphqwv duh uhdol}hg> eruurzhuv uhsd| wkhlu
ghew/ ou/ wr ohqghuv> dqg qdoo|/ djhqwv pdnh wkhlu frqvxpswlrq dqg vdylqjv ghflvlrqv
ghwhuplqlqj lq wxuq wkh lqlwldo zhdowk ri eruurzhuv dw wkh ehjlqqlqj ri wkh qh{w shulrg
+l1h1/ dw E| n 3 G|qdplf Htxdwlrqv= Qrz wkdw zh kdyh odlg rxw wkh edvlf prgho/ zh fdq dqdo|}h wkh
djjuhjdwh g|qdplfv ri wkh hfrqrp| dqg lq sduwlfxodu lqyhvwljdwh zk| rshq hfrqrplhv
zlwk lpshuihfw fuhglw pdunhwv pd| h{shulhqfh pdfurhfrqrplf yrodwlolw|1 Vlqfh erwk U
dqg R ghshqg rq hqwuhsuhqhxuv* zhdowk ` / rxwsxw grhv wrr1 Wkxv/ rxwsxw g|qdplfv duh
ghwhuplqhg e| wkh hyroxwlrq ri hqwuhsuhqhxuv* ehkdylru1 Ohw `|n ghqrwh wkh glvsrvdeoh
zhdowk ri hqwuhsuhqhxuv +eruurzhuv, dw wkh ehjlqqlqj ri shulrg | n Wkh g|qdplf
hyroxwlrq ri ` +dqg wkhuhiruh ri lqyhvwphqw dqg wrwdo rxwsxw, ehwzhhq wzr vxffhvvlyh
shulrgv lv vlpso| ghvfulehg e| wkh htxdwlrq=
`|n ' E kde n +| o>`| o
+7,
zkhuh e lv dq h{rjhqrxv lqfrph lq whupv ri rxwsxw jrrgv/ +| ' 4? @U c ~ lv rxwsxw
lq shulrg | +dovr htxdo wr wkh jurvv uhyhqxhv ri hqwuhsuhqhxuv gxulqj wkdw shulrg,1 Wkh
h{suhvvlrq lq eudfnhwv lv wkh qhw hqg0ri0shulrg | uhyhqxh ri hqwuhsuhqhxuv1 Wkh qhw
glvsrvdeoh zhdowk ri hqwuhsuhqhxuv dw wkh ehjlqqlqj ri shulrg | n lv zkdw uhpdlqv ri
wklv qhw hqg0ri0shulrg uhwxuq diwhu frqvxpswlrq/ khqfh wkh pxowlso|lqj idfwru E k rq
wkh uljkw0kdqg0vlgh ri htxdwlrq +7,1
Hqwuhsuhqhxuv lqyhvw dqg eruurz rqo| li wkhlu surwv duh odujhu wkdq ru htxdo wr wkh
lqwhuqdwlrqdo uhwxuq1 Zkhq > ru ` duh odujh/ hqwuhsuhqhxuv lqyhvw rqo| xs wkh srlqw
zkhuh + ou ' o` Dq| uhpdlqlqj zhdowk lv lqyhvwhg dw wkh lqwhuqdwlrqdo pdunhw udwh1
Lq wklv fdvh/ qr sxuh surwv duh hduqhg iurp surgxfwlrq dqg wkh hyroxwlrq ri zhdowk lv
vlpso| jlyhq e|=
`|n ' E kde n o`| o
+8,
Wkxv/ wkh g|qdplfv duh ixoo| ghvfulehg hlwkhu e| glhuhqfh htxdwlrq +7, ru e| glhu0
44
hqfh htxdwlrq +8,1
515
Yrodwlolw|
Zkhq wkh g|qdplf hyroxwlrq ri grphvwlf hqwuhsuhqhxuv* zhdowk lv ghvfulehg e| htxdwlrq
+7,/ dq lqfuhdvh lq hqwuhsuhqhxuv* zhdowk `| dw wkh ehjlqqlqj ri shulrg | kdv dq dp0
eljxrxv hhfw rq qh{w shulrg*v zhdowk `|n 1 Wklv lv gxh wr wkh idfw wkdw wkh dprxqw ri
lqyhvwhg zhdowk lwvhoi ghshqgv qhjdwlyho| rq wkh lqsxw sulfh R/ zklovw R ghshqgv srvlwlyho|
rq fxuuhqw zhdowk1 Xvlqj wkh idfw wkdw=
E@ n R| + | ' E n >`| c
zh kdyh=
E n >
+ | YR|
_+ |
'
_`|
@ n R|
@ n R| Y`|
Wkhq/ iurp +7,/ wkh lpsdfw ri odvw shulrg zhdowk rq fxuuhqw hqg ri shulrg zhdowk fdq eh
ghfrpsrvhg lqwr wzr hhfwv=
_`|n
n>
+ | YR|
'
E
kd
o>
o
_`|
@ n R|
@ n R| Y`|
~}
zhdowk hhfw
~}
sulfh hhfw
Rq wkh rqh kdqg/ wkhuh lv d srvlwlyh zhdowk hhfw ri fxuuhqw zhdowk rq ixwxuh zhdowk=
iru d jlyhq sulfh ri wkh frxqwu|0vshflf idfwru R| c d kljkhu lqkhulwhg zhdowk `| iurp
shulrg E| phdqv d kljkhu ohyho ri lqyhvwphqw E n >`| lq shulrg | zklfk/ doo hovh
htxdo/ vkrxog surgxfh kljkhu uhyhqxhv dqg wkxv kljkhu zhdowk `|n dw wkh ehjlqqlqj
ri shulrg | n Rq wkh rwkhu kdqg/ wkhuh lv d qhjdwlyh sulfh hhfw ri fxuuhqw zhdowk
rq ixwxuh zhdowk= pruh lqyhvwphqw lq shulrg | dovr lpsolhv d juhdwhu ghpdqg iru wkh
frxqwu|0vshflf idfwru wr wkxv udlvh lwv sulfh R| gxulqj wkdw shulrg1 Wklv/ lq wxuq/ kdv d
ghwulphqwdo hhfw rq shulrg | uhyhqxhv dqg wkhuhiruh rq wkh zhdowk `|n dw wkh ehjlqqlqj
ri shulrg | n
Zlwk wkh deryh Ohrqwlhi vshflfdwlrq/ wkh sulfh hhfw lv holplqdwhg zkhqhyhu wkh
fxuuhqw zhdowk `| lv vr vpdoo wkdw fxuuhqw lqyhvwphqw fdqqrw devrue wkh wrwdo vxsso|
45
ri wkh frxqwu|0vshflf idfwru1 Lq wklv fdvh R| f dqg=
`|n ' E kde n i
n>
o>j`| oc
@
+9,
vr wkdw _`|n *_`| : f
Rq wkh rwkhu kdqg/ wkh sulfh hhfw grplqdwhv zkhq wkh fxuuhqw zhdowk `| lv vxi0
flhqwo| odujh wkdw fxuuhqw lqyhvwphqw h{kdxvwv wkh wrwdo vxsso| ri wkh frxqwu|0vshflf
idfwru1 Lq wklv fdvh/ zh vlpso| kdyh=
`|n ' E kde n ~ o>`| oc
+:,
vr wkdw _`|n *_`| f
Iljxuh 4 vkrzv wkh uhodwlrqvkls ehwzhhq `|n dqg `| lq wklv edvlf Ohrqwlhi vhwxs1
Wklv uhodwlrqvkls lv uhsuhvhqwhg e| wkuhh vhjphqwv1 Wkh uvw rqh lv wkh xszdug vorslqj
@~ ( wklv lv wkh fdvh zkhuh wkh zhdowk hhfw
fxuyh ghvfulehg e| +9, iru ` ` ' n
>
grplqdwhv dv R ' f Wkh vhfrqg vhjphqw/ iru ` ` ` '
~
En>o c
lv ghvfulehg e|
+:,> lq wklv fdvh/ wkh sulfh hhfw dozd|v grplqdwhv1 Ilqdoo|/ wkh wklug vhjphqw +` : ` uhsuhvhqwv htxdwlrq +8, zkhuh hqwuhsuhqhxuv duh qrw fuhglw0frqvwudlqhg1 Dv gudzq lq
wkh jxuh/ wkh 78 olqh lqwhuvhfwv wkh `|n E`| fxuyh dw wkh srlqw f̀ zklfk olhv lq wkh
vhfrqg vhjphqw1 Wklv lqwhuvhfwlrq fdq dovr eh lq hlwkhu ri wkh rwkhu wzr vhjphqwv1 Lw zloo
eh lq wkh uvw vhjphqw zkhq
` Vlqfh
E3ke
>
3E3kt n
@ 3o>
E3ke
/
>
3E3kt n
@ 3o>
wkh {hg srlqw ri htxdwlrq +9,c lv ohvv wkdq
lv lqfuhdvlqj lq > zkloh ` lv ghfuhdvlqj/ lw lv fohdu wkdw wklv
fdq rqo| kdsshq zkhq > lv yhu| vpdoo1 Rq wkh rwkhu kdqg/ wkh lqwhuvhfwlrq zloo eh lq
wkh wklug vhjphqw zkhq wkh {hg srlqw ri htxdwlrq +8,/
E3ke
3E3ko
:` '
~
En>o Wklv zloo
rqo| kdsshq zkhq > lv vx!flhqwo| odujh1 Iru lqwhuphgldwh ydoxhv ri >/ fruuhvsrqglqj wr
dq lqwhuphgldwh ohyho ri qdqfldo ghyhorsphqw/ wkh fdvh lv ghslfwhg lq Iljxuh 4/ wkh rqh
fdvh zkhuh wkh hfrqrp| grhv qrw frqyhujh prqrwrqlfdoo| wr lwv vwhdg| vwdwh1
Lq wklv fdvh wkhuh duh wzr srvvlelolwlhvvkruw uxq xfwxdwlrqv/ uhsuhvhqwhg e| rvflood0
wlrqv wkdw hyhqwxdoo| frqyhujh wr wkh vwhdg| vwdwh/ f̀ c dqg orqj uxq yrodwlolw|/ uhsuhvhqwhg
46
e| d v|vwhp zklfk grhv qrw frqyhujh wr d vwhdg| vwdwh exw lqvwhdg frqwlqxhv wr rvfloodwh
iruhyhu1 D qhfhvvdu| frqglwlrq iru wkh h{lvwhqfh ri vxfk d olplw f|foh lv wkdw wkh vwhdg|
vwdwh dw f̀ eh xqvwdeoh/ wuxh rqo| zkhq wkh vorsh ri wkh `|n E`| vfkhgxoh dw f̀ lv ohvv
wkdq 04/ fruuhvsrqglqj wr zkhq f̀ olhv lq wkh vhfrqg vhjphqw ri wkdw vfkhgxoh1 Wkxv/
iru orqj uxq yrodwlolw| wr rffxu/ zh pxvw kdyh ` f̀ ` dqg E k>o
Li wkhvh frqglwlrqv krog/ rqh fdq hdvlo| ghulyh dgglwlrqdo vx!flhqw frqglwlrqv xqghu
zklfk orqj0uxq yrodwlolw| dfwxdoo| rffxuv1 Iru h{dpsoh/ d wzr0f|foh E` c `2 zloo vdwlvi|=<
` '
E kEe n ~
(
n o>E k2 Ee n n@ > o>
`2 '
E k2 Ee n n@ > o>Ee n ~
n o>E k2 Ee n n@ > o>
zlwk ` ` `2 ` 1 Wklv wzr0f|foh zloo eh vwdeoh zkhqhyhu Ek2 o>E n@ > o> Frqglwlrqv iru wkh h{lvwhqfh ri orqjhu +dqg pruh sodxvleoh, f|fohv fdq eh ghulyhg xvlqj
vwdqgdug whfkqltxhv1 Wkh g|qdplf vlpxodwlrqv zloo vkrz wkdw wkh xfwxdwlrqv fdq eh
frpsoh{ vlqfh zhdowk fdq xfwxdwh ehwzhhq wkh frqvwudlqhg +wkh uvw wzr vhjphqwv lq
Iljxuh 4, dqg wkh xqfrqvwudlqhg +wkh wklug vhjphqw, uhjlrqv1
Lqwxlwlyho|/ wkh edvlf phfkdqlvp xqghuo|lqj wklv f|folfdolw| fdq eh ghvfulehg dv iro0
orzv= gxulqj d errp wkh ghpdqg iru wkh grphvwlf frxqwu|0vshflf idfwru jrhv xs dv
+kljk |lhog, lqyhvwphqwv lqfuhdvh/ wkxv udlvlqj lwv sulfh1 Wklv kljkhu sulfh zloo hyhqwxdoo|
vtxhh}h lqyhvwruv* eruurzlqj fdsdflw| dqg wkhuhiruh wkh ghpdqg iru frxqwu|0vshflf idf0
wruv1 Dw wklv srlqw/ wkh hfrqrp| h{shulhqfhv d voxps dqg wzr wklqjv rffxu= wkh uhodwlyh
sulfh ri wkh grphvwlf idfwru froodsvhv/ zkloh d iudfwlrq ri wkh idfwru dydlodeoh uhpdlqv
xqxvhg vlqfh wkhuh lv qrw hqrxjk lqyhvwphqw1 Wkh froodsvh lq wkh idfwru sulfh wkxv fru0
uhvsrqgv wr d frqwudfwlrq ri uhdo rxwsxw1 Ri frxuvh/ wkh orz idfwru sulfh zloo hyhqwxdoo|
ohdg wr kljkhu surwv dqg wkhuhiruh wr pruh lqyhvwphqw1 D qhz errp wkhq ehjlqv1
<
Wklv iroorzv lpphgldwho| iurp wkh htxdwlrqv=
Z4
Z5
@ +4 ,+h . ] uZ5 ,
4.
u,Z4=
@ +4 ,+h .
d
47
Wkh uhdvrq zk| wkh ohyho ri qdqfldo ghyhorsphqw pdwwhuv lv dovr txlwh lqwxlwlyh=
hfrqrplhv dw d orz ohyho ri qdqfldo ghyhorsphqw kdyh orz ohyhov ri lqyhvwphqw dqg gr
qrw jhqhudwh hqrxjk ghpdqg wr sxvk xs wkh sulfh ri wkh frxqwu| vshflf idfwru zkloh
hfrqrplhv dw d yhu| kljk ohyho ri ghyhorsphqw kdyh vx!flhqw ghpdqg iru wkdw idfwru wr
nhhs lwv sulfh srvlwlyh1
516
Wkh Ohrqwlhi Fdvh zlwk dq Hodvwlf Vxsso| ri wkh Frxqwu|0
Vshflf Idfwru
Lq wkh suhylrxv vxevhfwlrq/ wkhuh zhuh qr rxwsxw xfwxdwlrqv zkhq R : f ehfdxvh ri
wkh frpelqdwlrq ri d Ohrqwlhi whfkqrorj| dqg d {hg vxsso| ri wkh grphvwlf lqsxw1 Lq
wklv vxevhfwlrq/ zh uhod{ wkh dvvxpswlrq ri d {hg vxsso| ri wkh frxqwu|0vshflf idfwru
dqg vkrz rxwsxw xfwxdwlrqv zlwk d Ohrqwlhi whfkqrorj|1 Dvvxph wkdw ~ lv lqvwhdg
surgxfhg e| +grphvwlf, ohqghuv xvlqj wkh wudghdeoh jrrg dw d frvw SE~ ' )~ D / zkhuh
D : 1 Pd{lpl}dwlrq ri d grphvwlf ohqghu*v surw R~ )~ D / |lhogv wkh rswlpdo vxsso|
ri wkh frxqwu|0vshflf idfwru=
#
~'
R
)D
$
D 3
+;,
Wkxv/ zh kdyh R : f zkhq wkhuh lv surgxfwlrq1 Wklv/ wrjhwkhu zlwk wkh htxdwlrq
ER n @+| ' E n >` c lpsolhv wkdw wkh htxloleulxp sulfh Rc vxfk wkdw +| ' ~c pxvw
vdwlvi| wkh htxdwlrq =
#
R
ER n @
)D
$
D 3
' E n >` +<,
Wklv htxdwlrq ghwhuplqhv R dqg ~ dv srvlwlyh ixqfwlrqv ri E n >` Wkh g|qdplf
hyroxwlrq ri zhdowk/ ghvfulehg e| wkh htxdwlrq=
`|n ' E kde n ~ | o>`| oc
+43,
fdq wkxv eh uhh{suhvvhg dv d g|qdplf htxdwlrq lq R| c qdpho|=
wER|n c D ' E kde n
# |$
R D3
48
)D
o>wER| c Doc
+44,
zkhuh wER| c D ' `| '
R|
ER| n@ )D
D 3
En>
Htxdwlrq +44, lpsolflwo| ghqhv d vfkhgxoh R|n ER| / dqg zh duh lqwhuhvwhg lq zkhwkhu
wklv vfkhgxoh jhqhudwhv g|qdplfv vlplodu wr wkrvh dulvlqj zlwk dq lqhodvwlf vxsso| ri
wkh frxqwu| vshflf idfwru1 Lw lv frqyhqlhqw iru wklv sxusrvh wr vhw e ' f>43 xqghu wklv
dgglwlrqdo dvvxpswlrq/ wkh vwhdg| vwdwh ydoxh ri R lv=
RW '
E kE n >
@
n E ko>
Dv ehiruh/ wkh g|qdplf ehkdylru ri wkh v|vwhp ghshqgv fuxfldoo| rq wkh vorsh ri wkh
R|n ER| vfkhgxoh dw R| ' RW 1 Zkhq wklv vorsh lv juhdwhu wkdq wkh v|vwhp frqyhujhv
wr wkh vwhdg| vwdwh1 Rwkhuzlvh lw rvfloodwhv shupdqhqwo| +wkrxjk lw pd| ru pd| qrw
frqyhujh wr d vwdeoh f|foh,1 D vx!flhqw frqglwlrq iru wkh v|vwhp wr glvsod| orqj uxq
yrodwlolw| +lq wkh irup ri shupdqhqw rvfloodwlrqv, lv wkdw
n>
_R|n
mER| 'RW ' E kd W
o>o ' \ED
|
_R
DR n @
Fohduo| iru D odujh hqrxjk/ wklv frqglwlrq zloo krog dv orqj dv E ko> : 1 Qrwh wkdw
wklv lv dovr wkh frqglwlrq iru rvfloodwlrqv lq wkh fdvh zkhuh wkh vxsso| ri wkh lqsxw lv
frpsohwho| lqhodvwlf/ zklfk lv zkdw zh zrxog h{shfw144
Wkh idfw wkdw wkh vxsso| ixqfwlrq pxvw eh vrphzkdw lqhodvwlf wr jhqhudwh yrodwlolw|
vkrxog eh lqwxlwlyh1 Lq wkh fdvh zkhuh vxsso| lv shuihfwo| hodvwlf +D ' , wkh idfwru sulfh
htxdov lwv pdujlqdo frvw/ vr wkhuh lv qr sulfh hhfw1 Lw uhpdlqv wr fkhfn zkhwkhu wkh
frqglwlrqv iru vxvwdlqhg yrodwlolw| duh hpslulfdoo| sodxvleoh1 Zh dvvhuw wkdw wkh| duh/ exw
srvwsrqh wkh dfwxdo glvfxvvlrq ri wklv lvvxh xqwlo wkh qh{w vhfwlrq1
43
Wkh dvvxpswlrq ri d }hur hqgrzphqw grhv qrw dhfw wkh h{lvwhqfh ri f|fohv/ exw lw grhv dhfw wkh
frpsdudwlyh vwdwlfv zlwk uhvshfw wr 1
44
Lq sduwlfxodu/ xvlqj Judqgprqw +4<;;/ Sursrvlwlrq F1614,/ zh nqrz wkdw d vx!flhqw frqglwlrq wr
rewdlq d Iols elixufdwlrq dqg wkhuhe| wkh h{lvwhqfh ri vwdeoh wzr0f|fohv lq wklv prgho/ lv wkdw= +l, iru
vrph 5 +4> 4,> zh kdyh= + @ 4, A 4 @ + , A + @ 4,> +ll, wkh wklug ghulydwlyh ri wkh sw.4 +sw ,
ixqfwlrq iru @ dw s @ s lv qrw }hur> +lll, iru doo sw s > sw.4 +sw , uhpdlqv lq wkh sdudphwhu uhjlrq
zkhuh wkh fuhglw frqvwudlqw lv elqglqj1 Zklovw +ll, lv jhqhulfdoo| vdwlvhg/ vkrzlqj wkdw +lll, dfwxdoo|
krogv iru d qrq0hpsw| vhw ri sdudphwhu ydoxhv wkdw lqfoxghv d qhljkerxukrrg ri wkh elixufdwlrq srlqw
> lv kdug wr gr dqdo|wlfdoo|/ exw fdq eh grqh wkurxjk vlpxodwlrqv1
49
|n
Qrwh wkdw wkhuh duh qr rvfloodwlrqv zkhq > lv forvh wr 3/ vlqfh _R_R| mER| 'RW lv dozd|v
srvlwlyh lq wkdw fdvh1 Dovr/ wkh hfrqrp| zloo qrw xfwxdwh zkhq grphvwlf hqwuhsuhqhxuv
duh qrw fuhglw0frqvwudlqhg +l1h1/ zkhq > lv vx!flhqwo| odujh,/ vlqfh grphvwlf lqyhvwphqw
dqg rxwsxw duh wkhq lqghshqghqw ri fxuuhqw hqwuhsuhqhxuv* zhdowk1 Lw lv djdlq rqo|
iru lqwhuphgldwh ydoxhv ri >c l1h1/ lq hfrqrplhv dw dq lqwhuphgldwh ohyho ri qdqfldo
ghyhorsphqw/ wkdw hqgrjhqrxv xfwxdwlrqv zloo rffxu1
517
FHV Whfkqrorj|
Khuh/ zh h{whqg wkh dqdo|vlv lq vxevhfwlrq 515 wr wkh pruh jhqhudo fdvh zkhuh wkh sur0
gxfwlrq ixqfwlrq s Egc 5 lv ri wkh FHV w|sh/ zlwk s Egc 5 ' Eg w n 5 w *w / zlwk : o
dqg : f145 Wkh sdudphwhu w ghwhuplqhv wkh hodvwlflw| ri vxevwlwxwlrq ehwzhhq g dqg 5
+zh dvvxph w iru frqfdylw|,1 Wklv FHV vshflfdwlrq lqfoxghv dv vshfldo fdvhv/ erwk
wkh Free0Grxjodv whfkqrorj| zkhq w ' fc dqg d Ohrqwlhi whfkqrorj| zkhq w $ 41
Qrz zh vkrz lq Dsshqgl{ D wkdw djjuhjdwh rxwsxw dw gdwh |c zklfk/ iru jlyhq wrwdo
lqyhvwphqw U | lv ghqhg dv=
s EU | R| 5 c 5c
+ | '4@
5
~}
g
fdq vlpso| eh uhh{suhvvhg dv=
+ | ' ER| U | c
zkhuh
ER| ' w
3w
w
dqg ' n ER| w3 w3 Lq dgglwlrq wkh frqglwlrq wkdw wkh pdunhw iru wkh grphvwlf
lqsxw fohduv uhtxluhv=46
5'
E R w3 U
45
' ~
Wklv lv wr pdnh vxuh wkdw lw sd|v wr surgxfh dw ohdvw vrph wlphv dqg wkdw wkh frxqwu|0vshflf idfwru
lv xvhg1
46
Vhh Dsshqgl{ D1
4:
Dv orqj dv ER| : oc djhqwv lqyhvw xs wr wkhlu fuhglw olplw/ lpso|lqj wkdw U | '
E n >`| Xvlqj wklv zh fdq uhzulwh wkh pdunhw0fohdulqj frqglwlrq iru wkh frxqwu|
vshflf idfwru dv
E R w3 E n >`|
Wklv fdq eh uhzulwwhq lq wkh irup
' ~
|
En>`
' ~/ iurp zklfk lw iroorzv wkdw vlqfh w /
RnE R 3w
R| lv lqfuhdvlqj lq E n >`| 1 Wklv jlyhv d sulfh hhfw ri fxuuhqw zhdowk rq ixwxuh zhdowk
zklfk lv sursruwlrqdo wr n >1 Krzhyhu/ wkhuh lv dq xsshu erxqg wr wkh sulfh ohyho jlyhq
e| ER| ' o/ wkh srlqw zkhuh hqwuhsuhqhxuv vwrs lqyhvwlqj lq wkhlu surmhfw1
Wxuqlqj qh{w wr wkh `|n E`| fxuyh lwvhoi/ lw lv xvhixo wr ghqh ` dv wkh ydoxh ri
` iru zklfk ER| ' o Wkhq iru ` ` / vlqfh ER| : oc zh kdyh U | ' E n >`| c
zklfk doorzv xv wr uhzulwh htxdwlrq +7, dv
`|n ' E kde n iE n >ER| o>j`| o
+45,
Lq Dsshqgl{ D zh vkrz wkdw wkh `|n E`| c w fxuyh lv vlqjoh0shdnhg dqg vorshv grzq0
zdug iru kljk hqrxjk ydoxhv ri `| / dw ohdvw iru w f Iru ` : ` / wkh uhohydqw
g|qdplfv duh jlyhq e| htxdwlrq +8, dv lq wkh Ohrqwlhi fdvh/ dqg wkh `|n E`| c w fxuyh
vorshv xszdug1 Wkh fxuyh wkhuhiruh kdv wkh uhyhuvhg dqg vlgh0zd|v V0vkdsh zh irxqg lq
wkh Ohrqwlhi fdvh1
Wkh txhvwlrq qrz lv zkhwkhu wkh sulfh hhfw lv vwurqj hqrxjk wr jhqhudwh orqj uxq
_` |n
yrodwlolw| +wklv/ lw zloo eh uhfdoohg/ uhtxluhv wkdw _`| dw wkh vwhdg|0vwdwh srlqw
f̀ zkhuh ` |n E f̀ c w ' f̀ ,147 Lq Dsshqgl{ D zh vkrz wkdw zkhq wkh g|qdplfv duh
ghvfulehg e| htxdwlrq +7,/ l1h1/ iru `| ` / wkhq=
_`|n
E n >ER
' E kd
o>o
|
_`
ERE w n w
+46,
f ? Z > zklfk lq wxuq lv wuxh iru d zkroh udqjh ri
Lq zkdw iroorzv zh lpsolflwo| dvvxph wkdw Z
sdudphwhu ydoxhv/ lq sduwlfxodu iru wkh ydoxhv zh xvh lq wkh vlpxodwlrqv suhvhqwhg lq wkh qh{w vhfwlrq1
47
4;
w
Uhfdoo wkdw ' n ER| w3 w3 dqg ER| ' 3w
w
w
Zkhq w jrhv wr 4c w3
_` |n
jrhv wr 4 dqg w3 jrhv wr 41 Lw iroorzv wkdw iru dq| {hg Rc dv w jrhv wr 4c _`|
jrhv wr E ko>48 Wklv vxjjhvwv wkdw xqghu wkh dvvxpswlrq pdgh deryh/ wkdw lv
E ko> c zkhq w lv qhjdwlyh hqrxjk lw lv srvvleoh wr jhw orqj uxq yrodwlolw|149
Pruhryhu/ lw fdq eh hdvlo| vhhq wkdw wkh sulfh hhfw fdq grplqdwh rqo| iru lqwhuphgldwh
ydoxhv ri >= wkh h{suhvvlrq lq +46, lv vwulfwo| srvlwlyh iru > ' f/ zkloh wkhuh lv qr sulfh
hhfw iru > odujh vlqfh wkh xsshu erxqg rq R +ER| ' o, ehfrphv elqglqj1
Wkh pruh lqwhuhvwlqj txhvwlrq lv krzhyhu zkhwkhu wkh frqglwlrqv iru orqj0uxq yrodwlo0
lw| duh hpslulfdoo| uhdvrqdeoh1 Zkloh g|qdplf vlpxodwlrqv duh ohiw iru wkh qh{w vhfwlrq/
Iljxuh 5 suhvhqwv dq h{dpsoh ri wkh `|n E`| c w fxuyh dvvxplqj > ' e/ w ' e/ dqg
e ' f +wkh rwkhu sdudphwhuv duh dv ghqhg lq wkh vlpxodwlrq ri Vhfwlrq 6,1 Lw fdq eh
yhulhg wkdw vwduwlqj iurp dq| ydoxh `| +h{fhsw wkh srlqw zkhuh wkh fxuyh lqwhuvhfwv
wkh 78 olqh, wkh v|vwhp zloo qrw frqyhujh wr d vwhdg| vwdwh1
518
Rswlpdo Vdylqjv e| Hqwuhsuhqhxuv
Wkh vdylqjv udwh ri hqwuhsuhqhxuv kdv vr idu ehhq pdlqwdlqhg frqvwdqw dw k1 D
qdwxudo txhvwlrq lv zkhwkhu wkhuh frxog vwloo eh orqj0uxq yrodwlolw| zkhq wkh vdylqjv
udwh lv ghulyhg iurp wkh xvxdo lqwhu0whpsrudo pd{lpl}dwlrq sureohp1 Lq wklv vhfwlrq
48
Wklv lv zkdw zh zrxog h{shfw/ vlqfh wkh fdvh zkhuh @ 4 fruuhvsrqgv wr wkh Ohrqwlhi fdvh dqg
w.4
gZ
zh douhdg| vdz wkdw lq wkdw fdvh/ gZEw @ +4 ,u=
E
49
Pruh vshflfdoo|/ li +4 ,u A 4/ zh kdyh=
+ @ 3, A 3 A 4 A + @ 4,>
zkhuh +, ghqrwhv wkh ydoxh ri wkh vorsh
wkh
78 olqh
w.4
E
w
gZ
E
gZ
dw wkh vwhdg|0vwdwh ghqhg e| wkh lqwhuvhfwlrq ehwzhhq
dqg wkh Z .4 +Z > , vfkhgxoh1 Wkhq/ ohw eh ghqhg e|=
w
E
w
E
+ , @ 4
+vxfk dv h{lvwv e| frqwlqxlw| ri
w.4
E
w
E
gZ
gZ
zlwk uhvshfw wr ,1 Wkhq/ xvlqj Judqgprqw +4<;;/ Sursrvlwlrq
F1614 rq Iols Elixufdwlrqv, zh nqrz wkdw li= +l, wkh wklug ghulydwlyh ri ZEw.4 +ZEw > , dw wkh vwhdg|0vwdwh
f + ,/ wkh fuhglw frqvwudlqw uhpdlqv elqglqj dw
f+ ,/ lv glhuhqw iurp }hur> +ll, iru dq| Z w Z
ohyho Z
E
w.4
w
ZE +ZE > ,> wkhq iru lq d qrq0hpsw| lqwhuydo lq wkh qhljkerxukrrg ri > wkhuh h{lvwv dw ohdvw rqh
vwdeoh dqg qrq0ghjhqhudwh wzr0f|foh iru wkh g|qdplf vfkhgxoh ZEw.4 +ZEw > , =
4<
zh h{dplqh wklv srvvlelolw| dvvxplqj wkdw hqwuhsuhqhxuv duh lqqlwho| olyhg dqg vkrz
wkdw lw lv lqghhg wkh fdvh14: Iru vlpsolflw| zh uhvwulfw rxuvhoyhv wr wkh fdvh zkhuh wkh
surgxfwlrq whfkqrorj| lv ri wkh FHV w|sh dqg lq dgglwlrq zh dvvxph e ' f Zh dovr
dvvxph dq lqhodvwlf vxsso| ri wkh frxqwu|0vshflf idfwru1 Lw lv xvhixo wr ghqrwh wkh udwh
ri uhwxuq e| | ' | *`| 1 Wkhq/ dv douhdg| vkrzq lq wkh suhylrxv vxevhfwlrq/ zkhqhyhu
ER| : oc zh kdyh= | ' iE n >ER| o>j`| ' | `| Zh ehjlq zlwk wkh orj0xwlolw| fdvh/ zkhuh hdfk lqyhvwru pd{lpl}hv=
4@
"
[
|'f
q | *L} |
r|
| ' | `|n
zkhuh | lv hqwuhsuhqhxuv* frqvxpswlrq dw gdwh | dqg q lv wkh glvfrxqw idfwru/ zlwk
q 1 Zh dovr dvvxph wkdw qo 1 Wklv dvvxpswlrq lpsolhv wkdw frqvxphuv duh
vx!flhqwo| lpsdwlhqw wkdw wkh vwhdg| vwdwh zhdowk ohyho grhv qrw olh lq wkh xqfrqvwudlqhg
uhjlrq1
Wkh uvw rughu frqglwlrqv iru wklv pd{lpl}dwlrq jlyh xv=
|n
' q |n
|
+47,
Wklv lq wxuq lpsolhv wkdw wkh vroxwlrq wr wklv surjudpplqj sureohp wdnhv wkh irup=
| ' 1| Wr vhh wklv/ ohw xv d sulrul dvvxph wkdw wkhuh h{lvwv d vroxwlrq ri wkh irup | ' 1| c
iru doo | Wkhq zh pxvw kdyh= |n ' q |n 1| 1 Pruhryhu/ vlqfh |n ' 1|n c zh
pxvw dovr kdyh= |n ' 1 |n `|n ' 1 |n E 1| Wkhvh wzr h{suhvvlrqv iru |n
duh lghqwlfdo li dqg rqo| li 1 ' q Wkhuhiruh | ' E q| lv rqh vroxwlrq wr wkh
deryh g|qdplf surjudpplqj sureohp1 Qrz/ jlyhq wkdw wklv g|qdplf sureohp lv vwulfwo|
frqyh{ dqg wkdw q / lw lv dfwxdoo| wkh xqltxh vroxwlrq wr wkdw sureohp14;
4:
4;
Frqwudfwv duh vwloo dvvxphg wr odvw rqh shulrg gxh wr frpplwphqw sureohpv1
Zrrgirug +4<;<, xvhv d vlplodu uhvxow lq d uhodwhg frqwh{w1
53
Wkhuhiruh wkh vlpsoh sursruwlrqdo vdylqjv uxoh zh dvvxphg lq wkh suhylrxv vxevhfwlrqv
lv dfwxdoo| rswlpdo lq wkh orj xwlolw| fdvh zkhq k ' q dqg e ' f1 Frqvhtxhqwo|/
doo suhylrxv uhvxowv/ lq sduwlfxodu rq wkh srvvlelolw| ri orqj0uxq yrodwlolw| ru wkh g|qdplf
hhfwv ri vkrfnv/ dsso| dxwrpdwlfdoo| wr wkdw fdvh1
Frqvlghu qh{w wkh pruh jhqhudo fdvh zkhuh lqvwdqwdqhrxv xwlolw| lv jlyhq e|= E '
34 *E 4/ zkhuh *4 lv wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq dqg 4 : f1 Wkhq
grphvwlf hqwuhsuhqhxuv vroyh=
4@
"
[
|'f
q | E| r|
| ' | `|n
Wkh uvw rughu frqglwlrqv iru wklv sureohp jlyh xv=
|n |n 4
'
q
|
+48,
|n
Lw lv fohdu iurp htxdwlrq +48, wkdw wkh udwlr | dssurdfkhv dv 4 lqfuhdvhv1 Wklv lpsolhv
wkdw dq lqfuhdvh lq 4 +d uhgxfwlrq lq wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq, uhgxfhv
frqvxpswlrq fkdqjhv dqg jlyhv fruuhvsrqglqjo| odujhu lqwhuwhpsrudo vdylqjv fkdqjhv/
l1h1/ vdylqjv ehfrph pruh sur0f|folfdo ryhu wlph1 Wklv/ lq wxuq/ zloo whqg wr dpsoli| wkh
f|foh dv wkh sulfh ri wkh frxqwu|0vshflf lqsxw lqfuhdvhv pruh vkduso| gxulqj d errp1
Wuxh/ wr wkh h{whqw wkdw wkh uhwxuqv wr vdylqjv duh kljkhu zkhq wkh hfrqrp| lv lq d
voxps +voxpsv duh w|slfdoo| iroorzhg e| shulrgv zlwk kljk lqyhvwphqw surwdelolw|,/ wkhuh
vkrxog eh d juhdwhu whqghqf| wr vdyh pruh lq d voxps/ wkhuhe| dwwhqxdwlqj wkh f|folfdo
yduldwlrqv1 Krzhyhu/ wklv odwwhu hhfw lv zhdnhu/ wkh kljkhu wkh frvw ri lqwhuwhpsrudo
vxevwlwxwlrq +l1h1/ zlwk d odujhu 4,1
Wr dvvhvv wkh ryhudoo hhfw ri d fkdqjh lq wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq
rq yrodwlolw|/ lw lv lqvwuxfwlyh wr uhsodfh | e| | `|n lq +48,/ jlylqj d g|qdplf
uhodwlrqvkls=
`|n '
Eq |n 4
|
n
|n n Eq |n 4
54
|n2
`
|n n Eq |n 4
+49,
Hqwuhsuhqhxuv* zhdowk dydlodeoh iru qh{w shulrg lv qrz d zhljkwhg dyhudjh ri sdvw suri0
lwv dqg h{shfwhg ixwxuh zhdowk1 Zkloh wklv vhfrqg rughu +kljko| qrq0olqhdu, glhuhqfh
htxdwlrq grhv qrw ohqg lwvhoi wr dqdo|wlfdo vroxwlrqv/ lw fdq eh uhvroyhg qxphulfdoo| dv zh
vkrz lq wkh qh{w vhfwlrq1
6
Dvvhvvlqj Sodxvlelolw|= Vrph Vlpxodwlrq Uhvxowv
Wkh pdlq sxusrvh ri wklv vhfwlrq lv wr dvn zkhwkhu wkh dqdo|wlfdo frqfoxvlrqv ghulyhg
lq wkh suhylrxv vhfwlrq duh hpslulfdoo| sodxvleoh1 Wkh vlpxodwlrq uhvxowv lq wklv vhfwlrq
duh djdlq irfxvhg rq wkh srvvlelolw| ri 0 dqg wkh frqglwlrqv iru 0 orqj uxq yrodwlolw| lq
hfrqrplhv dw lqwhuphgldwh ohyhov ri qdqfldo ghyhorsphqw14<
Gxh wr vsdfh frqvwudlqwv/ zh vkdoo rqo| glvfxvv vlpxodwlrqv iru vrph ri wkh fdvhv
glvfxvvhg lq Vhfwlrq 51 Zh dvvxph lqqlwho|0olyhg hqwuhsuhqhxuv/ d FHV surgxfwlrq
ixqfwlrq/ dqg wkh srvvlelolw| ri lqfuhdvlqj wkh hodvwlflw| ri vxsso| ri wkh frxqwu|0vshflf
idfwru1 Zh suhvhqw rxu vlpxodwlrq uhvxowv e| vxffhvvlyho| ydu|lqj wkuhh sdudphwhuv= l, wkh
hodvwlflw| ri vxevwlwxwlrq ehwzhhq fdslwdo dqg wkh rwkhu idfwru lq wkh surgxfwlrq ixqfwlrq/
phdvxuhg e| w> ll, wkh lqwhuwhpsrudo hodvwlflw| ri vxevwlwxwlrq *4( +lll, wkh hodvwlflw| ri
frxqwu|0vshflf idfwru vxsso| dv phdvxuhg e| D Wkh rwkhu sdudphwhuv duh wdnhq wr eh
frqvwdqw lq wkhvh vlpxodwlrqv/ dqg zh { wkhp dw hpslulfdoo| sodxvleoh ydoxhv1 Zh vhw
wkh jurvv lqwhuhvw udwh o ' f2 dqg wkh surgxfwlylw| idfwru ' D1 Zkhqhyhu lw lv
ixoo| lqhodvwlf zh vhw wkh wrwdo vxsso| ri wkh lppreloh idfwru ~ ' ff dqg lwv zhljkw
lq wkh surgxfwlrq ixqfwlrq ' +wkhvh wzr sdudphwhuv kdyh olwwoh lq xhqfh rq wkh
vlpxodwlrq uhvxowv,1 Wkh glvfrxqw udwh ri hqwuhsuhqhxuv lv q ' fbc d ydoxh lpso|lqj
wkdw grphvwlf hqwuhsuhqhxuv duh lpsdwlhqw uhodwlyh wr wkh lqwhuhvw udwh1 Ilqdoo|/ zh vhw
wkh fuhglw pxowlsolhu > ' ec d ydoxh lpso|lqj d fdvk rz0fdslwdo udwlr ri f2 zkhq upv
4<
Zkhq orrnlqj dw wkh uhdo zruog/ wkh glvwlqfwlrq ehwzhhq shuvlvwhqw rvfloodwlrqv wkdw hyhqwxdoo| glh
rxw/ dqg wkrvh wkdw qhyhu glh rxw/ pd| qrw eh vr lpsruwdqw dv rxu dqdo|vlv vxjjhvwv1 Wklv lv ehfdxvh lq
uhdolw|/ hyhq li rvfloodwlrqv hyhqwxdoo| glh rxw/ wkhuh duh dozd|v vkrfnv wkdw vwduw wkhp r djdlq1
55
duh fuhglw0frqvwudlqhg/ d sodxvleoh qxpehu hyhq iru XV upv +vhh Id}}dul/ Kxeedug/ dqg
Shwhuvhq +4<;;,,1 Wkh ydoxhv frqvlghuhg iru w olh ehwzhhq fD dqg e( wkrvh iru D olh
ehwzhhq e dqg .SS fruuhvsrqglqj wr hodvwlflwlhv +*ED , ri D dqg f shufhqw> dqg
wkrvh iru 4 duh ehwzhhq fD dqg f1
Lq hdfk fdvh/ zh frqvlghu wkh g|qdplf lpsdfw rq rxwsxw ri d qhjdwlyh vkrfn wkdw
pdnhv zhdowk idoo e| 4( ehorz wkh vwhdg|0vwdwh zhdowk1 Zh qrupdol}h rxwsxw vr wkdw lw
lv lqlwldoo| htxdo wr ff dqg zh orrn dw wkh g|qdplf hyroxwlrq ri rxwsxw ryhu f shulrgv
diwhu wkh vkrfn1 Iljxuhv 6 dqg 7f dqg 7g glvsod| wkh vlpxodwlrqv lq wkh orj xwlolw| fdvh
zkhuh 4 ' 1 Dv dujxhg lq vxevhfwlrq 518/ wklv fdvh lv htxlydohqw wr wkh frqvwdqw vdylqjv
udwh hfrqrp| dqdo|}hg lq wkh suhylrxv vxevhfwlrqv153
Iljxuh 6 suhvhqwv wkh orj xwlolw| fdvh zlwk d {hg vxsso| ri wkh frxqwu| vshflf idfwru1
Wkh gldjudpv vkrz irxu fdvhv fruuhvsrqglqj wr glhuhqw ydoxhv ri w/ hdfk ohdglqj wr d
glhuhqw g|qdplf sdwk1 Lq Iljxuh 6d/ zkhuh w ' fD/ wkhuh lv qr lqvwdelolw| dqg rxwsxw
frqyhujhv vprrwko| wr lwv lqlwldo ohyho1 Zkhq w ghfuhdvhv wr D +Iljxuh 6e,/ rxwsxw
vwloo frqyhujhv exw lqfoxghv rvfloodwlrqv1
Iljxuh 6f vkrzv d wzr0f|foh/ zklfk dulvhv zkhq w ' 21 Ilqdoo|/ zkhq w ' e
+Iljxuh 6g,/ pruh frpsoh{ g|qdplfv dulvh gxh wr *uhjlph vzlwfklqj*= odujh lqfuhdvhv
lq zhdowk ohdg wkh v|vwhp wr wkh xqfrqvwudlqhg uhjlrq +wkh wklug vhjphqw lq Iljxuhv 4
dqg 5,/ exw wkh v|vwhp uhwxuqv wr wkh frqvwudlqhg uhjlrq vlqfh qo 1 Qrwlfh wkdw
wkh xfwxdwlrqv lq 6f dqg 6g duh odujhu wkdq wkh lqlwldo vkrfn/ vr wkdw vpdoo vkrfnv duh
dpsolhg +dfwxdoo| lqqlwhvlpdo vkrfnv zrxog ohdg wr vlplodu xfwxdwlrqv,1
Lq Iljxuhv 7d dqg 7e/ zh dvvxph wkdw w ' e zlwk dq lqhodvwlf vxsso| ri wkh frxqwu|0
vshflf idfwru/ zkloh zh ghsduw iurp orj xwlolw| e| ydu|lqj wkh lqwhuwhpsrudo hodvwlflw|
53
Qrwh wkdw wkh vlpxodwlrq whfkqltxh glhuv ehwzhhq wkh frqvwdqw vdylqjv udwh fdvh dqg wkh orj0xwlolw|
fdvh zlwk lqqlwho| olyhg dqg iruzdug0orrnlqj hqwuhsuhqhxuv1 Lq wkh iruphu fdvh/ zh vlpso| qhhg wr uxq
d uvw rughu glhuhqfh htxdwlrq zlwk jlyhq lqlwldo zhdowk ohyho1 Lq wkh odwwhu fdvh/ dv vkrzq dw wkh hqg ri
vxevhfwlrq 518/ wkh g|qdplf v|vwhp lv ghvfulehg e| d iruzdug0orrnlqj vhfrqg rughu glhuhqfh htxdwlrq
zklfk uhtxluhv wkdw zh frpsxwh wkh lqlwldo frqvxpswlrq ohyho iru jlyhq lqlwldo zhdowk +h1j1/ xvlqj d
vkrrwlqj dojrulwkp,1 Zkhq @ 4/ krzhyhu/ wkh wzr phwkrgv jhqhudwh h{dfwo| wkh vdph g|qdplfv1
56
sdudphwhu 41 Zlwk d orzhu hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq/ 4 ' f/ wkh v|v0
whp whqgv wr eh hyhq pruh xqvwdeoh dqg vzlwfkhv pruh hdvlo| dfurvv uhjlphv1 Zkhq
hqwuhsuhqhxuv duh pruh uhdg| wr vxevwlwxwh lqwhuwhpsrudoo|/ zklfk lq wklv jxuh fruuh0
vsrqgv wr wkh fdvh zkhuh 4 ' fD/ uhjlph vzlwfkhv duh ohvv iuhtxhqw1 Wkh prvw lpsruwdqw
frqfoxvlrq iurp Iljxuh 7/ krzhyhu/ lv wkdw wkh orqj0uxq lqvwdelolw| uhvxowv hvwdeolvkhg xq0
ghu frqvwdqw vdylqjv udwhv +ru zlwk rswlpdo lqwhuwhpsrudo vdylqjv lq wkh orj xwlolw| fdvh,/
fduu| ryhu wr d zlgh udqjh ri hodvwlflwlhv ri lqwhuwhpsrudo vxevwlwxwlrq1
Ilqdoo|/ lq Iljxuhv 7f dqg 7g zh vkrz vlpxodwlrqv zlwk dq hodvwlf vxsso| ri wkh
frxqwu|0vshflf idfwru/ dvvxplqj w ' e dqg orj xwlolw|1 Reylrxvo|/ zlwk dq hodvwlf
vxsso| wkhuh lv ohvv vfrsh iru xfwxdwlrqv1 Iru h{dpsoh/ Iljxuh 7g vkrzv wkdw zlwk d
vxsso| hodvwlflw| ri f shufhqw xfwxdwlrqv glh rxw udslgo|1 Krzhyhu/ zlwk dq hodvwlflw|
ri D shufhqw/ zklfk dsshduv uhdvrqdeoh lq wkh vkruw uxq/ zh vwloo kdyh xfwxdwlrqv zlwk
d wzr0f|foh1
Wkxv/ hyhq wkrxjk rxu prgho lv kljko| vw|ol}hg/ orqj0uxq rxwsxw yrodwlolw| dqg2ru
odujh dpsolfdwlrq ri vkrfnv rffxu iru hpslulfdoo| uhdvrqdeoh sdudphwhu ydoxhv dqg duh
qrw frqqhg wr rqh sduwlfxodu ixqfwlrqdo irup1
7
Ilqdqfldo Olehudol}dwlrq dqg Lqvwdelolw|
Wkh suhylrxv dqdo|vlv vkrzv wkdw d ixoo| rshq hfrqrp| zlwk lpshuihfw fuhglw pdunhwv
fdq h{klelw yrodwlolw| ru d f|foh1 Zh vkrz lq wklv vhfwlrq wkdw wkh vdph hfrqrp| fdq eh
vwdeoh li lw lv forvhg wr fdslwdo rzv ru li rqo| iruhljq lqyhvwphqw +IGL, lv doorzhg1 Wkxv/
d ixoo olehudol}dwlrq wr fdslwdo pryhphqwv pd| ghvwdelol}h dq hfrqrp|= zkloh lw vwdelol}hv
wkh uhdo lqwhuhvw udwh/ lw dovr dpsolhv wkh xfwxdwlrqv lq wkh sulfh ri wkh frxqwu|0vshflf
idfwru1 Wklv lq wxuq/ lqfuhdvhv wkh yrodwlolw| lq upv* fdvk0 rzv dqg wkhuhiruh djjuhjdwh
rxwsxw1 Zh uvw frqvlghu wkh fdvh ri dq hfrqrp| wkdw rshqv xs wr iruhljq ohqglqj1
Wkhq/ zh h{dplqh wkh fdvh ri IGL/ zkhuh iruhljq lqyhvwruv duh htxlw| kroghuv dqg duh
57
ixoo| lqiruphg derxw grphvwlf upv1 Hyhq wkrxjk wkh uhvxowv duh ydolg zlwk jhqhudo
surgxfwlrq ixqfwlrqv/ zh suhvhqw wkh Ohrqwlhi fdvh iru shgdjrjlfdo uhdvrqv1
714
Olehudol}lqj Iruhljq Ohqglqj
Zh frqvlghu dq hfrqrp| zlwk orz grphvwlf vdylqjv/ zlwk wkh Ohrqwlhi whfkqrorj| vshf0
lhg lq Vhfwlrq 514/ dqg zh uvw dvvxph wkdw wklv hfrqrp| lv qrw rshq wr iruhljq eru0
urzlqj dqg ohqglqj +wklv forvhg hfrqrp| lv ghvfulehg lq ghwdlov lq Dsshqgl{ E,1 Lq wkdw
fdvh/ dw hdfk gdwh/ wkh fxuuhqw zhdowk ri grphvwlf ohqghuv `u pdwwhuv vlqfh grphvwlf
lqyhvwphqw lv frqvwudlqhg e| grphvwlf vdylqjv ` n `u 1 Qrz vxssrvh wkdw wkh lqlwldo
ohyhov ri zhdowk khog e| hqwuhsuhqhxuv dqg grphvwlf ohqghuv/ ` dqg `u uhvshfwlyho|/
duh vx!flhqwo| vpdoo vr wkdw lqlwldoo| Rf ' f Wklv fruuhvsrqgv wr d vlwxdwlrq zkhuh gr0
phvwlf hqwuhsuhqhxuv fdqqrw h{kdxvw wkh vxsso| ri frxqwu|0vshflf lqsxwv1 Ohw xv dovr
dvvxph wkdw dw gdwh 3 grphvwlf vdylqjv `f n `uf duh ohvv wkdq wkh lqyhvwphqw fdsdflw|
E n >`f 154 Li > : wkhuh zloo wkhq eh h{fhvv lqyhvwphqw fdsdflw| lq iroorzlqj shulrgv
dv orqj dv R| uhpdlqv htxdo wr }hur1 Wr vhh wklv/ qrwh wkdw wkh grphvwlf lqwhuhvw udwh
o| / ghwhuplqhg lq d forvhg hfrqrp| e| wkh frpsdulvrq ehwzhhq `u| dqg >`| c lv vxfk
wkdw hqwuhsuhqhxuv duh lqglhuhqw ehwzhhq eruurzlqj dqg ohqglqj/ wkdw lv= o| ' @ lq wkh
Ohrqwlhi fdvh1 Wkhuhiruh/ li R| ' f dqg `u| >`| c zh kdyh=
`|n ' E kde n `| o
@
dqg `u|n ' E kde n `u| oc
@
vr wkdw `u| >`| lpsolhv wkdw= `u|n >`|n dqg wkhuhiruh o |n ' @ Lq Dsshqgl{
E zh surylgh vx!flhqw frqglwlrqv xqghu zklfk R| ' f dqg o | ' @ iru doo | Xqghu wkhvh
frqglwlrqv/ hqwuhsuhqhxuv* zhdowk zloo jurz dv wkh +orz, udwh 3@ k / vlqfh lw lv frqvwudlqhg
e| wkh +orz, ohyho ri grphvwlf vdylqjv/ dqg wkh `|n E`| vfkhgxoh zloo lqwhuvhfw wkh 78f
54
Li ZE ? ZO > rshqlqj xs wkh hfrqrp| wr iruhljq ohqglqj zrxog pdnh qr glhuhqfh= vlqfh wkh
lqyhvwphqw fdsdflw| ri grphvwlf hqwuhsuhqhxuv fdqqrw hyhq devrue grphvwlf vdylqjv/ wkhuh lv qr qhhg
iru iruhljq ohqglqj lq wklv fdvh1
58
olqh rq lwv uvw eudqfk dorqj zklfk R| ' f Wklv/ lq wxuq/ lpsolhv wkdw wkhuh zloo eh qr
shuvlvwhqw xfwxdwlrqv lq wklv forvhg hfrqrp|1
Zkdw kdsshqv li wklv hfrqrp| lv ixoo| rshqhg xs wr iruhljq eruurzlqj dqg ohqg0
lqjB Wkh lqwhuhvw udwh zloo eh {hg dw wkh lqwhuqdwlrqdo ohyho o E| lwvhoi/ wklv frxog
rqo| khos vwdelol}h dq| forvhg hfrqrp| wkdw rwkhuzlvh pljkw +whpsrudulo|, xfwxdwh lq
uhdfwlrq wr lqwhuhvw udwh pryhphqwv1 Krzhyhu/ wkh rshqlqj xs ri wkh hfrqrp| wr iruhljq
ohqglqj dovr eulqjv qhw fdslwdo lq rzv dv lqyhvwruv vdwlvi| wkhlu h{fhvv ixqgv ghpdqg lq
lqwhuqdwlrqdo fdslwdo pdunhwv1 Wkh fruuhvsrqglqj ulvh lq eruurzlqjv lq wxuq lqfuhdvhv wkh
vfrsh iru elgglqj xs wkh sulfh ri wkh frxqwu|0vshflf idfwru/ wkhuhe| lqgxflqj shupdqhqw
xfwxdwlrqv lq R/ ` dqg djjuhjdwh rxwsxw1
Iljxuh 8 suhvhqwv dq looxvwudwlrq ri d olehudol}dwlrq lq wkh Ohrqwlhi fdvh1 Wkh zhdowk
vfkhgxoh vkliwv xs diwhu d fdslwdo dffrxqw olehudol}dwlrq1 f̀ uhihuv wr wkh vwdeoh vwhdg|0
vwdwh ohyho ri eruurzhuv* zhdowk ehiruh wkh hfrqrp| rshqv xs wr iruhljq eruurzlqj dqg
ohqglqj1 Diwhu wkh olehudol}dwlrq ` surjuhvvlyho| lqfuhdvhv dv fdslwdo lq rzv doorz
lqyhvwruv wr lqfuhdvh wkhlu eruurzlqj/ lqyhvwphqwv dqg surwv1 Gxulqj wkh uvw wzr
shulrgv iroorzlqj wkh olehudol}dwlrq/ wkh ghpdqg iru wkh frxqwu|0vshflf idfwru uhpdlqv
vx!flhqwo| orz wkdw R ' f1 Lq shulrg 6 +dw ` , R lqfuhdvhv exw zh vwloo kdyh jurzwk1
Krzhyhu/ lq shulrg 7 +dw `e , wkh sulfh hhfw ri wkh olehudol}dwlrq ehfrphv vx!flhqwo|
vwurqj dv wr vtxhh}h lqyhvwruv* qhw zruwk/ wkhuhe| eulqjlqj rq d uhfhvvlrq1 Dw wkdw
srlqw/ djjuhjdwh ohqglqj gursv/ fdslwdo rzv rxw dqg wkh uhdo h{fkdqjh ghsuhfldwhv +R
gursv,1 Wkh uhvxowlqj jdlq lq frpshwlwlyhqhvv doorzv upv wr uhexlog wkhlu qhw zruwk
vr wkdw jurzwk fdq hyhqwxdoo| uhvxph1 Wkh hfrqrp| hqgv xs h{shulhqflqj shupdqhqw
xfwxdwlrqv ri wkh nlqg ghvfulehg lq wkh suhylrxv vhfwlrq1
Zh vkrxog vwuhvv wkdw wkh g|qdplfv lq Iljxuh 8 rffxuv rqo| iru lqwhuphgldwh ohyhov ri
qdqfldo ghyhorsphqw1 Dv zh dujxhg lq Vhfwlrq 5/ zlwk d odujh > wkhuh lv qr yrodwlolw| lq
59
dq rshq hfrqrp|/ dv lw lv wkh wklug vhjphqw ri wkh fxuyh wkdw fxwv wkh 78 olqh155 Zkhq
> ' f/ qdqfldo rshqlqj zloo qrw khos lqyhvwphqw dqg qr fdslwdo lq rz zloo rffxu/ vr
wkhuh zloo eh qr xszdug suhvvxuh rq wkh sulfh ri wkh frxqwu|0vshflf lqsxw156 Wkh deryh
h{dpsoh wkhuhiruh vxjjhvwv wkdw lw pljkw eh ghvludeoh iru d frxqwu| wr lqfuhdvh lwv >/ l1h1/
wr ghyhors lwv grphvwlf qdqfldo vhfwru ehiruh ixoo| rshqlqj xs wr iruhljq ohqglqj1
715
Iruhljq Gluhfw Lqyhvwphqw
Zklovw d ixoo olehudol}dwlrq wr iruhljq ohqglqj fdq kdyh ghvwdelol}lqj hhfwv rq hfrqrplhv
zlwk lqwhuphgldwh ohyhov ri qdqfldo ghyhorsphqw/ wkrvh hfrqrplhv duh xqolnho| wr ehfrph
yrodwloh dv d uhvxow ri rshqlqj xs wr iruhljq gluhfw lqyhvwphqw dorqh1 Zh glvwlqjxlvk IGL
iurp rwkhu qdqfldo rzv e| dvvxplqj wkdw lw lv sduw ri upv* htxlw| dqg wkdw IGL
lqyhvwruv kdyh ixoo lqirupdwlrq derxw upv157 Ixuwkhupruh/ zh uvw frqfhqwudwh rq wkh
ehqfkpdun fdvh zkhuh wkh vxsso| ri IGL lv lqqlwho| hodvwlf dw vrph {hg sulfh juhdwhu
wkdq wkh zruog lqwhuhvw udwh/ vd| htxdo wr o n B158
Vwduwlqj iurp d vlwxdwlrq lq zklfk grphvwlf fdvk rzv duh vpdoo vr wkdw grphvwlf
lqyhvwphqw fdqqrw ixoo| devrue wkh vxsso| ri frxqwu|0vshflf idfwruv/ iruhljq gluhfw lq0
yhvwruv duh olnho| wr hqwhu lq rughu wr surw iurp wkh orz sulfh ri wkh frxqwu|0vshflf
idfwruv1 Wklv sulfh zloo hyhqwxdoo| lqfuhdvh dqg pd| hyhq xfwxdwh dv d uhvxow ri IGL1
Exw wkhvh sulfh xfwxdwlrqv zloo rqo| dhfw wkh glvwulexwlrq ri surwv ehwzhhq grphvwlf
dqg iruhljq lqyhvwruv/ qrw djjuhjdwh rxwsxw1 Iru h{dpsoh/ lq wkh Ohrqwlhi fdvh zlwk
IGL/ djjuhjdwh rxwsxw zloo vwdelol}h dw d ohyho htxdo wr wkh vxsso| ri idfwru uhvrxufhv
~/ zkhuhdv wkh vdph hfrqrp| pd| hqg xs ehlqj ghvwdelol}hg li ixoo| rshq wr iruhljq
55
Zkhq vhyhudo ghyhorshg frxqwulhv glg olehudol}h wkhlu fdslwdo pryhphqwv lq wkh 4<:3v dqg 4<;3v
shulrgv ri kljk lqvwdelolw| frxog qrw eh revhuyhg1
56
Wklv pd| eh wkh fdvh lq vrph ri wkh srruhu Diulfdq dqg Dvldq frxqwulhv1
57
W|slfdoo|/ phdvxuhg IGL lpsolhv sduwlflsdwlrqv ri pruh wkdq 43( lq d up*v fdslwdo vr wklv dsshduv
wr eh d uhdvrqdeoh dvvxpswlrq1 Ud}lq hw do +4<<;, pdnh d vlplodu glvwlqfwlrq derxw IGL1
58
Wklv/ lq wxuq/ lpsolhv wkdw lq rxu prgho IGL lv d vxevwlwxwh wr grphvwlf lqyhvwphqw1 Wkh hhfwv
ri IGL rq pdfurhfrqrplf yrodwlolw| zkhq grphvwlf dqg iruhljq lqyhvwphqwv duh frpsohphqwdu|/ duh
glvfxvvhg dw wkh hqg ri wklv vhfwlrq1
5:
sruwirolr lqyhvwphqw +l1h1/ wr iruhljq ohqglqj,1
Frqvlghu d forvhg Ohrqwlhi hfrqrp| rshq wr iruhljq gluhfw lqyhvwphqw rqo|1 Dvvxph
dovr wkdw `u lv odujh hqrxjk vr wkdw upv fdq vwloo eruurz wkhlu ghvluhg dprxqw grphvwl0
fdoo| +rwkhuzlvh lqyhvwphqw lv vwloo frqvwudlqhg e| vdylqjv dqg wkh vfrsh iru xfwxdwlrqv
lv pxfk vpdoohu,1 Wkhq IGL zloo rz lqwr wkh hfrqrp| dv orqj dv wkh udwh ri uhwxuq
rq wkdw lqyhvwphqw uhpdlqv juhdwhu wkdq ru htxdo wr o n B1 Wkxv/ li 8 ghqrwhv wkh qhw
lq rz ri gluhfw lqyhvwphqw/ lq htxloleulxp zh rewdlq wkh iuhh0hqwu| frqglwlrq=
8 : f , - ' o n Bc
h
ou lv wkh qhw udwh ri uhwxuq rq iruhljq gluhfw lqyhvwphqw dqg oh lv wkh
zkhuh - ' `+ 3
n8
grphvwlf lqwhuhvw udwh1 Li grphvwlf vdylqjv duh ohvv wkdq wkh lqyhvwphqw fdsdflw| ri
grphvwlf hqwuhsuhqhxuv El1h1/ `u >` / zh zrxog kdyh oh ' @ Krzhyhu/ dv grphvwlf
vdylqjv h{fhhg wkh lqyhvwphqw fdsdflw| ri grphvwlf hqwuhsuhqhxuv/ oh ' j/ zkhuh j lv
wkh uhwxuq ri dq dowhuqdwlyh/ lqh!flhqw/ vwrulqj whfkqrorj| +dv lq Djklrq/ Edqhumhh/ dqg
Slnhww| +4<<<,,1 Lq d forvhg hfrqrp|/ ohqghuv zloo lqyhvw wkhlu h{fhvv vdylqjv lq wklv
whfkqrorj|1
Dvvxph wkdw - : o n B dv orqj dv R ' f +wklv lpsolhv o n B @ E n > >j,/ vr wkdw
wkhuh zloo eh d srvlwlyh rz ri IGL dv orqj dv R ' f1 Xvlqj wkh idfw wkdw u ' >E` n 8 dqg wkdw + ' ~ zkhq R : f/ zh fdq uhzulwh wkh deryh iuhh0hqwu| frqglwlrq dv=
Eo n BE` n 8 ' ~ >jE` n 8 Wklv/ wrjhwkhu zlwk wkh sulfh htxdwlrq +6,/ lpsolhv wkdw=
R'
n>
@c
o n B n j>
zklfk lq wxuq ghqhv jlyhv d vwdeoh ydoxh iru R1 Wkxv/ hyhq wkrxjk IGL ohdgv wr d sulfh
lqfuhdvh lw grhv qrw jhqhudwh sulfh dqg rxwsxw yrodwlolw|1
Frqvlghu qrz dq hfrqrp| zklfk kdv douhdg| ehhq rshqhg xs wr iruhljq eruurzlqj
dqg ohqglqj dw udwh o/ wkdw lv wr iruhljq sruwirolr rzv rqo|/ dqg zklfk/ dv d uhvxow
5;
kdv ehfrph yrodwloh dv lq wkh h{dpsoh ghslfwhg lq Iljxuh 81 Zkdw zloo kdsshq li wklv
hfrqrp| lv qrz dovr rshqlqj xs wr IGLB E| wkh vdph uhdvrqlqj dv ehiruh/ rshqlqj xs
wr IGL zloo vwdelol}h wkh sulfh ri wkh frxqwu|0vshflf idfwru dw ohyho RW vxfk wkdw=
Eo n BE` n 8 ' ~ o>E` n 8 Wklv djdlq zloo holplqdwh lqyhvwphqw dqg rxwsxw yrodwlolw| lq wklv hfrqrp| +dvvxplqj
wkdw lqlwldoo| wkh frxqwu| lv dwwudfwlqj IGL,1 Lq rwkhu zrugv/ li wkhuh duh qr olplwdwlrqv
rq IGL lq rzv dqg rxw rzv +dqg IGL lqyroyhv frpsohwh lqirupdwlrq rq grphvwlf upv,/
wkh sulfh ri wkh frxqwu|0vshflf idfwru dqg wkhuhiruh djjuhjdwh grphvwlf JGS ru JQS
zloo uhpdlq frqvwdqw lq htxloleulxp1
Wkh uhdvrq zk| IGL dfwv dv d vwdelol}lqj irufh lv djdlq wkdw/ xqolnh iruhljq ohqglqj/
lw grhv qrw ghshqg rq wkh fuhglwzruwklqhvv ri wkh grphvwlf upv/ dqg ixuwkhupruh lw
lv suhflvho| gxulqj voxpsv wkdw iruhljq gluhfw lqyhvwruv pd| suhihu wr frph lq vr dv wr
ehqhw iurp wkh orz sulfh ri wkh frxqwu|0vshflf idfwru1
Zkdw kdsshqv li iruhljq gluhfw lqyhvwphqw lv frpsohphqwdu| wr grphvwlf gluhfw lq0
yhvwphqw/ wkdw lv/ wr ` B Vxfk frpsohphqwdulw| pd| eh gxh wr ohjdo uhvwulfwlrqv
zkhuhe| wkh wrwdo dprxqw ri IGL fdqqrw eh juhdwhu wkdq d {hg iudfwlrq % ri gr0
phvwlf lqyhvwruv* zhdowk ` / ru lw pd| vwhp iurp wkh qhhg iru orfdo lqyhvwruv wr hqirufh
glylghqg sd|phqwv ru wr khos h{huw frqwuro1 Dsshqgl{ E vkrzv wkdw iruhljq gluhfw lqyhvw0
phqwv vxemhfw wr frpsohphqwdulw| uhtxluhphqwv ri wkh irup 8 %` / pd| vrphwlphv
gh0vwdelol}h dq hphujlqj pdunhw hfrqrp|1 Lqghhg/ lq frqwudvw wr wkh xquhvwulfwhg IGL
fdvh dqdo|}hg deryh/ vxfk gluhfw lqyhvwphqwv xowlpdwho| zloo idoo gxulqj voxpsv/ wkdw
lv/ zkhq lqyhvwruv* zhdowk `|n lv h{shulhqflqj d grzqwxuq1 Grzqwxuqv zloo dovr w|s0
lfdoo| eh ghhshu wkdq lq devhqfh ri IGL vlqfh/ e| dpsoli|lqj wkh lqfuhdvh lq R| gxulqj
errpv/ IGL lqfuhdvhv surgxfwlrq frvwv dqg wkxv dffhqwxdwhv wkh fuhglw0fuxqfk lqgxfhg
rq upv1 Wkxv/ zklovw xquhvwulfwhg IGL kdv d vwdelol}lqj hhfw rq dq rshq hphujlqj
pdunhw hfrqrp|/ rshqlqj vxfk dq hfrqrp| wr uhvwulfwhg IGL pd| dfwxdoo| kdyh wkh
5<
rssrvlwh hhfw1
8
Frqfoxvlrqv
Rxu prgho surylghv d vlpsoh dqg wudfwdeoh iudphzrun iru dqdo|}lqj qdqfldoo|0edvhg
fulvhv lq hfrqrplhv zklfk duh dw dq lqwhuphgldwh ohyho ri qdqfldo ghyhorsphqw1 Wkh
vwru| zh whoo lv edvhg rq vrph yhu| edvlf ihdwxuhv ri wkhvh hfrqrplhv/ lq frqwudvw zlwk
rwkhu pruh lqvwlwxwlrqdoo|0edvhg wkhrulhv zklfk lqyrnh prudo kd}dug dprqj ohqghuv/ khug
ehkdylru dprqj lqyhvwruv/ hwf1 Wklv lv qrw wr vd| wkdw rxu prgho lv lqfrqvlvwhqw zlwk wklv
fodvv ri wkhrulhvlq idfw lq d suhylrxv yhuvlrq ri wklv sdshu zh dujxhg dw vrph ohqjwk
wkdw wkhuh duh lqwhuhvwlqj lqwhudfwlrqv ehwzhhq rxu prgho dqg wkrvh wkhrulhv159 Krzhyhu/
rxu prgho grhv vxjjhvw d vrphzkdw glhuhqw srolf| uhvsrqvh= voxpsv vkrxog eh vhhq dv
sduw ri d qrupdo surfhvv lq hfrqrplhv olnh wkhvh zklfk duh erwk dw dq lqwhuphgldwh ohyho
ri qdqfldo ghyhorsphqw dqg lq wkh surfhvv ri olehudol}lqj wkhlu qdqfldo vhfwruv1 Zh
vkrxog wkhuhiruh qrw ryhu0uhdfw wr wkh rffxuuhqfh ri qdqfldo fulvhv/ hvshfldoo| lq wkh
fdvh ri hphujlqj pdunhw hfrqrplhv1 Lq sduwlfxodu/ kdvw| dqg udglfdo ryhukdxolqj ri wkhlu
hfrqrplf v|vwhp pd| gr pruh kdup wkdq jrrg15:
Vhfrqg/ srolflhv doorzlqj upv wr uhexlog wkhlu fuhglw zruwklqhvv txlfno| zloo dw wkh
vdph wlph frqwulexwh wr d surpsw uhfryhu| ri wkh ryhudoo hfrqrp|1 Lq wklv frqwh{w lw lv
zruwk frqvlghulqj wkh uroh iru prqhwdu| srolf| dqg/ pruh jhqhudoo|/ iru srolflhv dhfwlqj
wkh fuhglw pdunhw1 Zklovw rxu prgho lq lwv suhvhqw irup fdqqrw eh gluhfwo| xvhg iru
59
Lq wkh suhylrxv yhuvlrq zh dovr vkrz wkdw xqfhuwdlqw| dqg upv ghidxowv fdq hdvlo| eh lqwurgxfhg lq
wkh prgho1 Zh qrwh wkdw edqnuxswf| frvwv zloo vljqlfdqwo| ghhshq wkh voxpsv rqo| lq wkrvh hfrqrplhv
idflqj fuhglw frqvwudlqwv1
5:
Lqghhg/ li rxu prgho lv uljkw/ wkh voxps vhwv lq prwlrq irufhv zklfk/ hyhq zlwk olwwoh lqwhuihuhqfh/
vkrxog hyhqwxdoo| eulqj jurzwk edfn wr wkhvh hfrqrplhv1 Wkh ulvn lv wkdw e| wu|lqj wr ryhukdxo wkh
v|vwhp lq d sdqlf/ rqh pd| dfwxdoo| xqghuplqh wkrvh irufhv ri uhfryhu| lqvwhdg ri vwlpxodwlqj wkhp1
Wklv lv qrw wr ghq| wkdw wkhuh lv d orw wkdw qhhgv fkdqjlqj lq wkhvh hfrqrplhv/ hvshfldoo| rq wkh
lqvwlwxwlrqdo vlgh zlwk wkh hvwdeolvkphqw dqg hqirufhphqw ri glvflsolqdu| uxohv lq fuhglw dqg edqnlqj
dfwlylwlhv1 Iru h{dpsoh/ lq wkh frqwh{w ri rxu prgho/ edqnv pd| w|slfdoo| hqjdjh lq suhhpswlyh ohqglqj
wr vshfxodwruv lq grphvwlf lqsxwv dqg2ru wr surgxfhuv gxulqj errpv1 Wklv lq wxuq zloo ixuwkhu lqfuhdvh
rxwsxw yrodwlolw| zkhqhyhu lqdghtxdwh prqlwrulqj dqg h{shuwlvh dftxlvlwlrq e| edqnv lqfuhdvhv djjuhjdwh
ulvn dqg wkhuhiruh wkh lqwhuhvw udwh lpsrvhg xsrq grphvwlf surgxfhuv1
63
wklv sxusrvh vlqfh prqh| lv qhxwudo +dqg lq dq| fdvh wkh lqwhuhvw udwh lv {hg e| wkh
zruog lqwhuhvw udwh,/ lw fdq eh h{whqghg wr doorz iru erwk prqhwdu| qrq0qhxwudolw| dqg
d ohvv lqqlwho| hodvwlf vxsso| ri iruhljq ordqv +vhh Djklrq0Edffkhwwd0Edqhumhh +5333d/
e,,1 Rqfh zh wdnh rxu iudphzrun lq wklv gluhfwlrq lw txlfno| ehfrphv fohdu wkdw d orz
lqwhuhvw udwh srolf| lv qrw qhfhvvdulo| wkh uljkw dqvzhu hyhq lq d voxps lqgxfhg e| d fuhglw
fuxqfk1 Wkh sureohp lv wkdw zkloh vxfk dq lqwhuhvw udwh uhgxfwlrq pd| khos uhvwruh wkh
upv* qdqfldo khdowk +dqg wkhuhiruh wkhlu lqyhvwphqw fdsdflw|,/ wkh qhw reoljdwlrqv ri
wkrvh zkr kdyh eruurzhg lq iruhljq fxuuhqf| zloo dovr ulvh li lw ohdgv wr d ghydoxdwlrq
ri wkh grphvwlf fxuuhqf|1 Wkhuhiruh/ wkh rswlpdo lqwhuhvw udwh srolf| h{ srvw gxulqj d
qdqfldo fulvlv fdqqrw eh ghwhuplqhg zlwkrxw nqrzlqj pruh derxw wkh ghwdlov ri wkh
fxuuhqf| frpsrvlwlrq ri wkh h{lvwlqj ghew reoljdwlrqv ri grphvwlf hqwhusulvhv1
Wklv hpskdvlv rq fuhglwzruwklqhvv dv wkh nh| hohphqw lq wkh uhfryhu| iurp d voxps/
dovr vxjjhvwv wkdw d srolf| ri doorzlqj lqvroyhqw edqnv wr idlo pd| lq idfw surorqj wkh
voxps li lw uhvwulfwv upv* delolw| wr eruurz +ehfdxvh ri wkh frpsdudwlyh dgydqwdjh ri
edqnv lq prqlwrulqj upv* dfwlylwlhv5; ,1 Li edqnv pxvw eh vkxw grzq/ wkhuh vkrxog eh
dq hruw wr suhvhuyh wkhlu prqlwrulqj h{shuwlvh rq wkh uhohydqw lqgxvwulhv1 Pruhryhu/ wr
wkh h{whqw wkdw wkh jryhuqphqw kdv wr vshqg uhvrxufhv rq uhvwuxfwxulqj dqg fohdqlqj0xs
diwhu d vsdwh ri edqnuxswflhv/ lw vkrxog dyrlg udlvlqj wd{hv gxulqj d voxps vlqfh grlqj
vr zrxog ixuwkhu olplw wkh eruurzlqj fdsdflw| ri grphvwlf hqwuhsuhqhxuv dqg wkhuhiruh
ghod| wkh vxevhtxhqw uhfryhu|1
Wklug/ rxu prgho dovr gholyhuv h{ dqwh srolf| lpsolfdwlrqv iru hphujlqj pdunhw
hfrqrplhv qrw fxuuhqwo| xqghu d qdqfldo fulvlv1 Lq sduwlfxodu= +l, dq xquhvwulfwhg 0
qdqfldo olehudol}dwlrq pd| dfwxdoo| ghvwdelol}h wkh hfrqrp| dqg hqjhqghu d voxps wkdw
zrxog rwkhuzlvh qrw kdyh kdsshqhg1 Li d pdmru voxps lv olnho| wr eh frvwo| hyhq lq wkh
orqj0uxq +ehfdxvh/ iru h{dpsoh/ lw vhwv lq surfhvv ghvwdelol}lqj srolwlfdo irufhv,/ ixoo| ole0
5;
Vhh Gldprqg +4<;7,1
64
hudol}lqj iruhljq fdslwdo rzv dqg ixoo| rshqlqj wkh hfrqrp| wr iruhljq ohqglqj pd| qrw
eh d jrrg lghd dw ohdvw xqwlo wkh grphvwlf qdqfldo vhfwru lv vx!flhqwo| zhoo0ghyhorshg
+wkdw lv/ xqwlo wkh fuhglw0pxowlsolhu > ehfrphv vx!flhqwo| odujh,> +ll, iruhljq gluhfw lq0
yhvwphqw grhv qrw ghvwdelol}h1 Lqghhg/ dv zh kdyh dujxhg deryh/ IGL lv prvw olnho|
wr frph lq gxulqj voxpsv zkhq wkh uhodwlyh sulfh ri wkh frxqwu|0vshflf idfwru lv orz>
ixuwkhupruh/ hyhq li wklv sulfh hqgv xs xfwxdwlqj zkhq wkh hfrqrp| lv rshq wr IGL/
wkhvh xfwxdwlrqv zloo rqo| dhfw wkh glvwulexwlrq ri surwv ehwzhhq grphvwlf dqg iru0
hljq lqyhvwruv exw qrw djjuhjdwh rxwsxw1 Wkhuhiruh wkhuh lv qr frvw d sulrul wr doorzlqj
IGL hyhq dw orz ohyhov ri qdqfldo ghyhorsphqw15< > +lll, zkdw eulqjv derxw qdqfldo fulvhv
lv suhflvho| wkh ulvh lq wkh sulfh ri wkh frxqwu| vshflf idfwruv1 Li rqh ri wkhvh idfwruv
+vd|/ uhdo hvwdwh, lv lghqwlhg wr sod| d nh| uroh lq vsdunlqj d qdqfldo fulvlv/ lw zrxog
eh vhqvleoh wr frqwuro lwv sulfh/ hlwkhu gluhfwo| ru wkrxjk frqwuroolqj lwv vshfxodwlyh gh0
pdqg xvlqj vxlwdeoh vfdo ghwhuuhqwv1 Wklv/ dqg rwkhu lpsruwdqw dvshfwv lq wkh ghvljq
ri vwdelol}dwlrq srolflhv iru hphujlqj pdunhw hfrqrplhv/ dzdlw ixwxuh hoderudwlrqv ri wkh
iudphzrun ghyhorshg lq wklv sdshu1
5<
Wklv vwudwhj| ri doorzlqj rqo| IGL dw hduo| vwdjhv ri qdqfldo ghyhorsphqw lv lq idfw zkdw prvw
ghyhorshg frxqwulhv kdyh grqh/ lq sduwlfxodu lq Hxursh zkhuh uhvwulfwlrqv rq furvv0frxqwu| fdslwdo
pryhphqwv kdyh rqo| ehhq ixoo| uhpryhg lq wkh odwh 4<;3*v zkhuhdv IGL wr 0 dqg ehwzhhq 0 Hxurshdq
frxqwulhv kdg ehhq doorzhg vlqfh wkh odwh 4<83*v1
65
Dsshqgl{ D=
Vroylqj wkh Prgho zlwk d FHV Surgxfwlrq Ixqfwlrq
Lq wklv Dsshqgl{ zh ghulyh wkh uhvxowv iru wkh FHV fdvh suhvhqwhg lq Vhfwlrq 5161
Dvvxph wkdw s Egc 5 ' + ' Eg w n 5 w w zkhuh= g ' U R5 Wkh uvw rughu frqglwlrq
zlwk uhvshfw wr 5 jlyhv=
E
U R5 w3 ' c
5
R
+4:,
zklfk fdq eh zulwwhq dv
w
R
E w3 U ' 5E n R w3 3w ' 5
+4;,
Htxdwlrq +4:, lpsolhv wkdw=
g w ' E 5 w3 g
R
Vxevwlwxwlqj iru g w lq wkh h{suhvvlrq iru +c dqg wkhq xvlqj htxdwlrq +4;,/ zh jhw=
+ ' dEU R5 5 w3 n 5 w w
R
w3
' EU 5 w
R
3w
' U w
' U ERc
zkhuh ER ' 3w
w
1 Iru ixwxuh uhihuhqfh qrwh wkdw
ER ' ER
ER
RER
+4<,
_` |n
Wr frpsxwh _`| / qrwh uvw wkdw iru `| ` / zh kdyh U | ' E n >`| / vlqfh
ER| : o Wklv doorzv xv wr uhzulwh htxdwlrq +7, dv
`|n ' E kde n iE n >ER| o>j`| o
66
Qh{w/ vhwwlqj 5 ' ~ lq htxdwlrq +4;, dqg glhuhqwldwlqj zlwk uhvshfw wr R dqg `
jlyhv xv=
_R
E wRER
'
_`
ER wEER `
+53,
Glhuhqwldwlqj wkh deryh h{suhvvlrq iru `|n dqg xvlqj +4<,/ +53,/ dqg wkh ghqlwlrq
ri ER/ zh jhw
_`|n
E n >ER
' E kd
o>o
|
_`
ERE w n w
+54,
Wklv h{suhvvlrq rqo| dssolhv zkhq `| ` Iru `| ` c
`|n ' E kde n o`| o/
_` |n
dqg _`| ' E ko
Dw `| ' fc R ' f +iurp htxdwlrq +4;,, dqg wkhuhiruh ' dqg ' : o Lq rwkhu
_` |n
zrugv/ ` : f dqg wkh g|qdplfv duh ghvfulehg e| +7,1 Wkhuhiruh dw wkh ruljlq _`| '
E kdE n > o>o : f1 Dv `| jrhv wr 4c R dovr jrhv wr 4 Dv orqj dv w fc ER
lv lqfuhdvlqj lq R dqg jrhv wr 4 dv R jrhv wr 463 Pruhryhu dv w fc lv ghfuhdvlqj
> ER
dv d ixqfwlrq ri dqg jrhv wr 3 zkhq jrhv wr 4 Wkhuhiruh En
ERE3wnw lv ghfuhdvlqj
dv ixqfwlrq ri `| dqg jrhv wr 3 dv `| jrhv wr 4 Lq rwkhu zrugv/ dv orqj dv w f/
wkh `|n E`| vfkhgxoh ghqhg e| htxdwlrq +7, lv vlqjoh0shdnhg dqg wkhuh h{lvwv d qlwh
ydoxh ri ` W vxfk wkdw lw vorshv grzq iru `| : ` W Dsshqgl{ E=
Wkh Dqdo|wlfv ri Ilqdqfldo Olehudol}dwlrq
D, Olehudol}dwlrq wr Iruhljq Ohqglqj
Khuh/ zh frqvwuxfw dq h{dpsoh ri dq hfrqrp| zklfk/ lq wkh devhqfh ri iruhljq eruurz0
lqj dqg ohqglqj/ zrxog eh dv|pswrwlfdoo| vwdeoh dqg dfwxdoo| frqyhujh wr d shupdqhqw
errp/ exw zklfk ehfrphv shupdqhqwo| yrodwloh rqfh ixoo| rshq wr iruhljq eruurzlqj dqg
63
4
!+s, @ 4 . s 4 4 dqg li ? 3> wkh h{srqhqw rq s lv dozd|v srvlwlyh1
67
ohqglqj1 Wkh dqdo|vlv ri wkh forvhg hfrqrp| lv vlplodu wr Djklrq/ Edqhumhh/ dqg Slnhww|
+4<<<,1
Pruh vshflfdoo|/ frqvlghu dq hfrqrp| lq zklfk=
+d, Wkh surgxfwlrq whfkqrorj| lv Ohrqwlhi zlwk dq lqhodvwlf vxsso| ri wkh frxqwu|0
vshflf idfwru/ wkdw lv= s Egc 5 ' 4? g@ c 5 / @ / zkhuh g ' U R 51
+e, Ilqdqfldo pdunhwv duh lqlwldoo| forvhg wr iruhljq fdslwdo lq rzv vr wkdw wkh djjuhjdwh
vxsso| ri ixqgv dydlodeoh wr grphvwlf lqyhvwruv/ U | / lv qrz htxdo wr wkh plq ri wkh
lqyhvwphqw fdsdflw| E n >`| dqg ri wrwdo grphvwlf vdylqjv `| n `u| 1 Wkdw lv=
U | ' 4?iE n >`| c `| n `u| j
+f, Lqlwldoo|/ dw wlph | ' f/ wkh lqyhvwphqw fdsdflw| ri grphvwlf hqwuhsuhqhxuv h{fhhgv
wkh wrwdo dprxqw ri grphvwlf vdylqjv/ vr wkdw >`f : `uf +lq wkh rssrvlwh fdvh/
rshqlqj xs wr iruhljq eruurzlqj dqg ohqglqj zrxog kdyh qr hhfw rq lqyhvwphqw
dqg rxwsxw lq wkh grphvwlf hfrqrp|,1
+g, Zh lpsrvh wkh iroorzlqj uhvwulfwlrqv rq wkh sdudphwhuv ri wkh hfrqrp|=
+l, > :
+ll, k @
+lll, `uf dqg `f duh ohvv wkdq f̀ '
E3ke
3E3k @
+ly, f̀ @2 ~1
Zh qrz vkrz wkdw d forvhg hfrqrp| zklfk vdwlvhv dvvxpswlrqv +d,/ +e,/ +f,/ +g,/ lv
vwdeoh/ zlwk frqvwdqw sulfh R| f dqg frqvwdqw lqwhuhvw udwh o | @ / dqg zhdowk ohyhov
`| dqg `u| zklfk erwk frqyhujh prqrwrqlfdoo| wr f̀ dv | $ 41
Iluvw/ dvvxpswlrq +f, lpsolhv wkdw o f ' @ / dqg lw dovr lpsolhv wkdw U f ' `uf n `f >
dvvxpswlrqv +g,0+lll, dqg +g,0+ly, wkhq lpso| wkdw U f @~ / vr wkdw Rf ' f1 Qh{w/ rqh
68
fdq vkrz wkdw dw dq| gdwh r/ or ' @ dqg Rr ' f1 Wr vhh wklv/ vxssrvh wkdw iru doo r |/
or ' @ dqg Rr ' f/ dqg ohw xv vkrz wkdw o |n ' @ dqg R|n ' f1 Li o r ' @ dqg Rr ' f iru
doo r |/ wkhq iru doo r | wkh zhdowk ohyhov `urn dqg `rn vdwlvi| wkh htxdwlrqv=
`urn ' E k e n `ur
@
dqg
`rn ' E k e n `r @
++4,r ,
++5,r ,
Lw wkhq iroorzv iurp dvvxpswlrq +g,0+l,/ l1h1/ iurp > : / dqg iurp dvvxplqj wkdw o | ' @
+zklfk lpsolhv wkdw >`| : `u| ,/ wkdw >`|n : `u|n dqg wkhuhiruh o|n ' @ 1 Ixuwkhu0
pruh/ lw iroorzv iurp dvvxpswlrq +g,0+lll, dqg htxdwlrqv +4,r dqg +5,r iru r |/ wkdw
`ur f̀ dqg `r f̀ iru doo r | n > wklv lq wxuq lpsolhv wkdw=
U |n ' `u|n n `|n 2 f̀ c
vr wkdw U |n @~ e| dvvxpswlrq +g,0+ly, dqg wkhuhiruh R|n ' f1 Zh kdyh wkxv vkrzq
wkdw li o r ' @ dqg Rr ' f iru doo r |/ wkhq o |n ' @ dqg R|n ' f1 Wrjhwkhu zlwk wkh
idfw wkdw of ' @ dqg Rf ' f/ wklv suryhv e| lqgxfwlrq wkdw o r ' @ dqg Rr ' f iru doo r/
vr wkdw wkh hqwluh zhdowk wudmhfwru| E`ur c `r lv ghwhuplqhg e| E`uf c `f wrjhwkhu zlwk
wkh g|qdplf htxdwlrqv +4,r dqg +5,r 1 Exw wklv/ wrjhwkhu zlwk dvvxpswlrq +g,0+ll,/ lpsolhv
wkdw wkh htxloleulxp wudmhfwru| E`ur c `r lv vwdeoh/ zlwk erwk `ur dqg `r frqyhujlqj
prqrwrqlfdoo| wrzdugv f̀ zkhq | $ 41 Wkxv/ d forvhg hfrqrp| fkdudfwhul}hg e| +d,0+g,
zloo glvsod| qr yrodwlolw| lq sulfh/ lqwhuhvw udwh/ zhdowk dqg +wudghdeoh, rxwsxw1
Qrz/ d forvhg hfrqrp| wkdw vdwlvhv +d,0+g, dqg wkhuhiruh lv vwdeoh/ pd| hqg xs
ehfrplqj yrodwloh li ixoo| rshq wr iruhljq eruurzlqj dqg ohqglqj1 Iru h{dpsoh/ wklv zloo
eh wkh fdvh li wkdw vdph hfrqrp| vdwlvhv wkh vx!flhqw frqglwlrqv surylghg lq Vhfwlrq 515
iru wkh h{lvwhqfh ri wzr0f|fohv1 Dqg rqh fdq hdvlo| yhuli| wkdw wkh wzr vhwv ri frqglwlrqv
duh frqvlvwhqw/ lq wkh vhqvh wkdw wkhuh h{lvwv d qrq0hpsw| vhw ri sdudphwhuv zklfk vdwlvi|
erwk vhwv ri frqglwlrqv vlpxowdqhrxvo|1
69
E, Uhvwulfwhg IGL
Ohw 8 ghqrwh wkh fxuuhqw dprxqw ri IGL/ dqg ohw xv lpsrvh wkh frqvwudlqw= 8 %` /
zlwk wkh iudfwlrq % ehlqj lqlwldoo| vpdoo1 Zh dvvxph wkdw iruhljq lqyhvwruv uhfhlyh wkhlu
sursruwlrqdo vkduh ri rxwsxw dqg wkdw wklv lv dozd|v odujhu wkdq wkhlu uhvhuydwlrq uhwxuq
o n B +jlyhq wkh frqvwudlqw %/ wkh vxsso| lv qr orqjhu ixoo| hodvwlf dv lq wkh suhfhglqj
fdvh,1 Wkh htxloleulxp sulfh iru wkh frxqwu|0vshflf idfwru lv qrz htxdo wr=
R| ' 4@ Efc
E n >E`| n 8 | @~
~
Ohw u| ' >E`| n 8 | Wkhq wkh g|qdplfv ri lqyhvwruv* zhdowk lv ghvfulehg e| wkh
htxdwlrqv=
EL
`|n ' E k e n E`| n 8 | n u| ohu|
@
zkhq `| lv vpdoo dqg wkhuhiruh R| f +sduw 4 ri wkh `|n E`| fxuyh,/ dqg=
ELL
~
ohu|
`|n ' E k e n
n%
zkhq wkhuh lv h{fhvv ghpdqg iru wkh frxqwu|0vshflf idfwru dqg wkhuhiruh R| ehfrphv
srvlwlyh +sduw 5 ri wkh `|n E`| fxuyh,1
+Lq +L, dqg +LL, wkh yduldeoh oh ghqrwhv wkh grphvwlf lqwhuhvw udwh/ zklfk lv htxdo wr j
li >E` n 8 `u dqg wr wkh surw udwh rwkhuzlvh1
Iru % vx!flhqwo| vpdoo/ zh kdyh 8 r ' %`r vr wkdw wkh deryh htxdwlrq +LL, lpsolhv
d wrwdo ohyho ri gluhfw lqyhvwphqw +grphvwlf dqg iruhljq, htxdo wr=
k
l
`|n n %`|n ' E k eE n % n ~ oh>`| E n %2 c
zklfk iru e vpdoo lv ghfuhdvlqj lq %1 Lq sduwlfxodu/ vwduwlqj iurp dq hfrqrp| zlwkrxw
dq| IGL/ lqwurgxflqj kljko| frqvwudlqhg IGL pd| hqg xs ghhshqlqj wkh voxps zklfk lw
zdv phdqw wr holplqdwh1
6:
Uhihuhqfhv
^4` Djklrq/ Sk1/ Sk1 Edffkhwwd/ dqg D1 Edqhumhh +4<<<,/ Ilqdqfldo Olehudol}dwlrq dqg
Yrodwlolw| lq Hphujlqj Pdunhw Hfrqrplhv/ lq S1U1 Djìqru/ P1 Ploohu/ G1 Ylqhv/
dqg D1 Zhehu +hgv1,/ Wkh Dvldq Ilqdqfldo Fulvhv= Fdxvhv/ Frqwdjlrq dqg Frqvh0
txhqfhv/ Fdpeulgjh Xqlyhuvlw| Suhvv/ s1 49:04<31 Sxeolvkhg xqghu wkh zurqj wlwoh
Fdslwdo Pdunhwv dqg wkh Lqvwdelolw| ri Rshq Hfrqrplhv1
^5` Djklrq/ Sk1/ Sk1 Edffkhwwd/ dqg D1 Edqhumhh +5333d,/ D Vlpsoh Prgho ri Prqhwdu|
Srolf| dqg Fxuuhqf| Fulvhv/ Hxurshdq Hfrqrplf Uhylhz 77/ :5;0:6;1
^6` Djklrq/ Sk1/ Sk1 Edffkhwwd/ dqg D1 Edqhumhh +5333e,/ Fxuuhqf| Fulvhv dqg Prq0
hwdu| Srolf| lq dq Hfrqrp| zlwk Fuhglw Frqvwudlqwv/ FHSU Glvfxvvlrq Sdshu Qr1
585<1
^7` Djklrq/ Sk1/ D1 Edqhumhh/ dqg W1 Slnhww| +4<<<,/ Gxdolvp dqg Pdfurhfrqrplf
Yrodwlolw|/ Txduwhuo| Mrxuqdo ri Hfrqrplfv/ Qryhpehu/ 468:046<:1
^8` D}duldglv/ F1 dqg E1 Vplwk +4<<;,/ Ilqdqfldo Lqwhuphgldwlrq dqg Uhjlph Vzlwfklqj
lq Exvlqhvv F|fohv/ Dphulfdq Hfrqrplf Uhylhz ;;/ 84908691
^9` Edffkhwwd/ Sk1 +4<<5,/ Olehudol}dwlrq ri Fdslwdo Pryhphqwv dqg ri wkh Grphvwlf
Ilqdqfldo V|vwhp/ Hfrqrplfd 8</ 7980:71
^:` Edffkhwwd/ Sk dqg U1 Fdplqdo +4<<<,/ Gr Fdslwdo Pdunhw Lpshuihfwlrqv H{dfhu0
edwh Rxwsxw IoxfwxdwlrqvB/ Hxurshdq Hfrqrplf Uhylhz 77/ 77<079;1
^;` Edffkhwwd/ Sk1 dqg H1 ydq Zlqfrrs +5333,/ Fdslwdo Iorzv wr Hphujlqj Pdunhwv=
Olehudol}dwlrq/ Ryhuvkrrwlqj/ dqg Yrodwlolw|/ lq V1 Hgzdugv +hg1,/ Fdslwdo Iorzv
dqg wkh Hphujlqj Hfrqrplhv 0 Wkhru|/ Hylghqfh/ dqg Frqwuryhuvlhv/ Wkh Xqlyhuvlw|
ri Fklfdjr Suhvv/ 940<;1
6;
^<` Eduwrolql/ O1 dqg D1 Gud}hq +4<<:,/ Zkhq Olehudo Srolflhv Uh hfw H{whuqdo Vkrfnv/
Zkdw Gr Zh OhduqB Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv 75/ 57<0:61
^43` Ehuqdqnh/ E1 dqg P1 Jhuwohu +4<;<,/ Djhqf| Frvwv/ Qhw Zruwk/ dqg Exvlqhvv
Ioxfwxdwlrqv/ Dphulfdq Hfrqrplf Uhylhz :</ 470641
^44` Ehuqdqnh/ E1/ P1 Jhuwohu/ dqg V1 Jlofkulvw +4<<;,/ Wkh Ilqdqfldo Dffhohudwru lq
d Txdqwlwdwlyh Exvlqhvv F|foh Iudphzrun/ lq M1 Wd|oru dqg P1 Zrrgirug +hgv,/
Kdqgerrn ri Pdfurhfrqrplfv/ yro1 4F/ 4674046<61
^45` Fdoyr/ J1D1/ O1 Ohlghupdq/ dqg F1P1 Uhlqkduw +4<<9,/ Lq rzv ri Fdslwdo wr Gh0
yhorslqj Frxqwulhv lq wkh 4<<3v/ Mrxuqdo ri Hfrqrplf Shuvshfwlyhv 43/ 45606<1
^46` Gldprqg/ G +4<;7,/ Ilqdqfldo Lqwhuphgldwlrq dqg Ghohjdwhg Prqlwrulqj/ Uhylhz
ri Hfrqrplf Vwxglhv/ 95/ 6<607471
^47` Hlfkhqjuhhq/ E1/ P1 Pxvvd/ J1 Ghoo*Dulffld/ J1P1 Plohvl0Ihuuhwwl/ dqg D1 Wzhhglh
+4<<;,/ Fdslwdo Dffrxqw Olehudol}dwlrq= Wkhruhwlfdo dqg Sudfwlfdo Dvshfwv/ LPI
Rffdvlrqdo Sdshu Qr1 4:51
^48` Id}}dul/ V1P1/ U1J1 Kxeedug/ dqg E1F1 Shwhuvhq +4<;;,/ Ilqdqflqj Frqvwudlqwv
dqg Frusrudwh Lqyhvwphqw/ Eurrnlqjv Sdshuv rq Hfrqrplf Dfwlylw| 4=4<<;/ 4740
4<81
^49` Ihogvwhlq/ P1 dqg F1 Krulrnd +4<;3,/ Grphvwlf Vdylqj dqg Lqwhuqdwlrqdo Fdslwdo
Iorzv/ Hfrqrplf Mrxuqdo <3/ 64705<1
^4:` Jdoyh}/ M1 dqg M1 W|erxw +4<;8,/ Plfurhfrqrplf Dgmxvwphqwv lq Fkloh gxulqj
4<::0;4= Wkh Lpsruwdqfh ri Ehlqj d Juxsr/ Zruog Ghyhorsphqw 46/ <9<0<<71
^4;` Jhuwohu/ P1 dqg N1 Urjr +4<<3,/ Qruwk0Vrxwk Ohqglqj dqg Hqgrjhqrxv Fdslwdo0
Pdunhwv Lqh!flhqflhv/Mrxuqdo ri Prqhwdu| Hfrqrplfv 59/ 5780991
6<
^4<` Jrxulqfkdv/ S10R1/ U1 Ydogìv/ dqg R1 Odqghuuhwfkh +4<<<,/ Ohqglqj Errpv= Vrph
Vw|ol}hg Idfwv/ plphr/ Sulqfhwrq Xqlyhuvlw|1
^53` Jxhuud gh Oxqd/ D1 +4<<:,/ Uhvlghqwldo Uhdo Hvwdwh Errpv/ Ilqdqfldo Ghuhjxodwlrq
dqg Fdslwdo Lq rzv= dq Lqwhuqdwlrqdo Shuvshfwlyh/ plphr/ Edqfr gh Pì{lfr1
^54` Judqgprqw/ M10P1 +4<;;,/ Qrq0Olqhdu Glhuhqfh Htxdwlrqv/ Elixufdwlrqv dqg
Fkdrv= Dq Lqwurgxfwlrq/ FHSUHPDS Zrunlqj Sdshu Qr1 ;;44/ Sdulv1
^55` Kropvwurp/ E1 dqg M1 Wluroh +4<<:,/ Ilqdqfldo Lqwhuphgldwlrq/ Ordqdeoh Ixqgv
dqg wkh Uhdo Vhfwru/ Txduwhuo| Mrxuqdo ri Hfrqrplfv F[LL/ 99609<41
^56` Krqndsrkmd/ V1 dqg H1 Nrvnhod +4<<<,/ Wkh Hfrqrplf Fulvlv ri wkh 4<<3*v lq Ilq0
odqg/ Hfrqrplf Srolf| 47/ 73407691
^57` Kxeedug/ U1 J1 +4<<;,/ Fdslwdo Pdunhw Lpshuihfwlrqv dqg Lqyhvwphqw/ Mrxuqdo
ri Hfrqrplf Olwhudwxuh 69/ 4<605581
^58` Mrkqvwrq/ U1E1/ V1P1 Gduedu/ dqg F1 Hfkhyhuuld +4<<:,/ Vhtxhqflqj Fdslwdo Df0
frxqw Olehudol}dwlrqv= Ohvvrqv iurp wkh H{shulhqfhv lq Fkloh/ Lqgrqhvld/ Nruhd/
dqg Wkdlodqg/ LPI Zrunlqj Sdshu <:248:1
^59` Nuxjpdq/ S1 +4<:<,/ D Prgho ri Edodqfh ri Sd|phqwv Fulvhv/ Mrxuqdo ri Prqh|/
Fuhglw dqg Edqnlqj 44/ 6440581
^5:` N|rwdnl/ Q1 dqg M1 Prruh +4<<:,/ Fuhglw F|fohv/ Mrxuqdo ri Srolwlfdo Hfrqrp|
438/ 544057;1
^5;` PfNlqqrq/ U1L1 +4<<6,/ Wkh Rughu ri Hfrqrplf Olehudol}dwlrq/ Wkh Mrkqv Krsnlqv
Xqlyhuvlw| Suhvv1
^5<` PfNlqqrq/ U1L1 dqg K1 Sloo +4<<:,/ Fuhgleoh Hfrqrplf Olehudol}dwlrqv dqg Ryhu0
eruurzlqj/ Dphulfdq Hfrqrplf Uhylhz +Sdshuv dqg Surfhhglqjv, ;:/ 4;<04<61
73
^63` gh Phor/ M1/ U1 Sdvfdoh/ dqg M1 W|erxw +4<;8,/ Plfurhfrqrplf Dgmxvwphqwv lq
Xuxjxd| gxulqj 4<:60;4= Wkh Lqwhusod| ri Uhdo dqg Ilqdqfldo Vkrfnv/ Zruog Gh0
yhorsphqw 46/ <<8043481
^64` Plvknlq/ I1V1 +4<<9,/ Xqghuvwdqglqj Ilqdqfldo Fulvhv= D Ghyhorslqj Frxqwu| Shu0
vshfwlyh/ Dqqxdo Zruog Edqn Frqihuhqfh rq Ghyhorsphqw Hfrqrplfv/ 5<0951
^65` Revwihog/ P1 +4<;9,/ Fdslwdo Iorzv/ wkh Fxuuhqw Dffrxqw/ dqg wkh Uhdo H{fkdqjh
Udwh= Wkh Frqvhtxhqfhv ri Olehudol}dwlrq dqg Vwdelol}dwlrq/ lq V1 Hgzdugv dqg O1
Dkdphg +hgv1,/ Hfrqrplf Dgmxvwphqw dqg H{fkdqjh Udwhv lq Ghyhorslqj Frxqwulhv/
Xqlyhuvlw| ri Fklfdjr Suhvv1
^66` Shwuhl/ D1K1 dqg M1 W|erxw +4<;8,/ Plfurhfrqrplf Dgmxvwphqwv lq Dujhqwlqd gxu0
lqj 4<:90;4= Wkh Lpsruwdqfh ri Fkdqjlqj Ohyhov ri Ilqdqfldo Vxevlglhv/ Zruog
Ghyhorsphqw 46/ <7<0<9;1
^67` Ud}lq/ D1/ H1 Vdgnd/ dqg F10Z1 \xhq +4<<;,/ D Shfnlqj Rughu ri Fdslwdo Lq rzv
dqg Lqwhuqdwlrqdo Wd{ Sulqflsohv/ Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv 77/ 7809;1
^68` Wruqhoo/ D1 +4<<;,/ Frpprq Ixqgdphqwdov lq wkh Whtxlod dqg Dvldq Fulvhv/
plphr/ Kduydug Xqlyhuvlw|1
^69` Zrrgirug/ P1 +4<;<,/ Lpshuihfw Ilqdqfldo Lqwhuphgldwlrq dqg Frpsoh{ G|qdp0
lfv/ lq Z1 Eduqhww hw do1 +hgv,/ Hfrqrplf Frpsoh{lw|= Fkdrv/ Vxqvsrw/ Exeeohv
dqg Qrqolqhdulw|1
^6:` Zruog Edqn +4<<:,/ Sulydwh Fdslwdo Iorzv wr Ghyhorslqj Frxqwulhv/ Srolf| Uhvhdufk
Uhsruw/ R{irug Xqlyhuvlw| Suhvv1
74
:
W
%
Part 2
(equation 7)
Part 1
(equation 6)
Part 3
(equation 5)
45o
W Ŵ
W
)LJXUH
:
W
%
WBt+1 45
CES technology, µ = 4 θ = - 4 e=0
40
35
30
25
20
15
10
5
45 °
0
0
5
10
15
20
25
30
35
40
45
WBt
Figure 2
:
W
%
W%
3
W
W%
W%
2
B
Open economy
Closed economy
4
1
45o
Ŵ%
W%
1
W W%
2
3
B
)LJXUH
:%
W
© Copyright 2026 Paperzz