Second-best Pricing for Im- perfect Substitutes in

TI 2003-085/3
Tinbergen Institute Discussion Paper
Second-best Pricing for Imperfect Substitutes in Urban
Networks
Jan Rouwendal
Erik T. Verhoef
Department of Spatial Economics, Faculty of Economics and Business Administration, Vrije
Universiteit Amsterdam, and Tinbergen Institute.
Tinbergen Institute
The Tinbergen Institute is the institute for
economic research of the Erasmus Universiteit
Rotterdam, Universiteit van Amsterdam, and Vrije
Universiteit Amsterdam.
Tinbergen Institute Amsterdam
Roetersstraat 31
1018 WB Amsterdam
The Netherlands
Tel.:
Fax:
+31(0)20 551 3500
+31(0)20 551 3555
Tinbergen Institute Rotterdam
Burg. Oudlaan 50
3062 PA Rotterdam
The Netherlands
Tel.:
Fax:
+31(0)10 408 8900
+31(0)10 408 9031
Please send questions and/or remarks of nonscientific nature to [email protected].
Most TI discussion papers can be downloaded at
http://www.tinbergen.nl.
6HFRQGEHVW SULFLQJIRULPSHUIHFWVXEVWLWXWHVLQXUEDQQHWZRUNV
-DQ5RXZHQGDO DQG(ULN79HUKRHI 'HSDUWPHQWRI6SDWLDO(FRQRPLFV
)UHH8QLYHUVLW\$PVWHUGDP
'H%RHOHODDQ
+9$PVWHUGDP
7KH1HWKHUODQGV
-(/&ODVVLILFDWLRQFRGHV 55'
.H\ZRUGVWUDIILFFRQJHVWLRQURDGSULFLQJVHFRQGEHVWSROLFLHV WZRPRGHSUREOHP
$EVWUDFW
7KLVSDSHUFRQVLGHUVVHFRQGEHVWSULFLQJDVLWDULVHVWKURXJKLQFRPSOHWHFRYHUDJHRIIXOO
QHWZRUNV7KHPDLQSULQFLSOHVDUHILUVWUHYLHZHGE\FRQVLGHULQJWKHFODVVLFWZRURXWHSUREOHP
DQGVRPHH[WHQVLRQVWKDWKDYHEHHQVWXGLHGPRUHUHFHQWO\,QPRVWRIWKHVHVWXGLHVWKH
FRPSHWLQJURXWHVDUHDVVXPHGWREHSHUIHFWVXEVWLWXWHVZKLFKLVSUREDEO\QRWWKHFDVHIRU
PRVWSDUDOOHOURDGVLQUHDOLW\DQGHYHQOHVVOLNHO\IRUWKHFDVHZKHUHFRPSHWLQJFRQQHFWLRQV
UHSUHVHQWGLIIHUHQWWUDQVSRUWPRGHV ,QWKLVSDSHUD PRGHOOLQJIUDPHZRUNLQZKLFKWKH
DOWHUQDWLYHVDUHLPSHUIHFWVXEVWLWXWHV LVGHYHORSHGDQGQXPHULFDOUHVXOWVIRUWZRURDGVDQGWZR
PRGHVDUHSUHVHQWHG,QWKHPRGHOWULSJHQHUDWLRQDQGWULSGLVWULEXWLRQDUHGLVWLQJXLVKHGLQD
ZD\WKDWLVFRQVLVWHQWZLWKHFRQRPLFWKHRU\7KHPRGHO LVXVHGWRFRQVLGHUVLWXDWLRQVLQ
ZKLFKRQHURXWHRUPRGHFDQQRWEHWROOHG6LPXODWLRQUHVXOWVVKRZWKDWIRUWKHFKRVHQ
SDUDPHWHUYDOXHVWKHUHLVDVXEVWDQWLDOGLIIHUHQFHEHWZHHQWKHHIIHFWLYHQHVVRISROLFLHVLQ
ZKLFKWKHFDSDFLWLHVKDYHWREHWDNHQDVJLYHQDQGWKRVHLQZKLFKFDSDFLW\RIDWOHDVWRQH
PRGHFDQEHFKDQJHG$VWULNLQJIHDWXUHRIWKHSROLF\LQZKLFKWKHFDSDFLWLHVRIERWK
PRGHVURXWHV DQGWKHUDLOZD\IDUHWROORQRQHURDG FDQEHXVHGDVSROLF\LQVWUXPHQWVLVWKH
H[LVWHQFHRIWZRHTXLOLEULDIRUDUDQJHRIYDOXHVRI,QRQHHTXLOLEULXPWKHUHDUHVXEVWDQWLDO
QXPEHUVRIXVHUVRIERWKPRGHVZKHUHDVLQWKHRWKHUXVHRIRQHPRGHLVQHJOLJLEOH
$OVRDW:DJHQLQJHQ 8QLYHUVLW\ (FRQRPLFV2I&RQVXPHUVDQG+RXVHKROGV +ROODQGVHZHJ
.1:DJHQLQJHQ
$IILOLDWHGWRWKH7LEHUJHQ,QVWLWXWH5RHWHUVWUDDW:%$PVWHUGDP
,1752'8&7,21
7KHEDVLFHFRQRPLFLQWXLWLRQEHKLQGPDUJLQDOFRVWSULFLQJLVGHFHSWLYHO\VLPSOH
:KHQSULFHVDUHHTXDOWRPDUJLQDOVRFLDOFRVWVFRQVXPHUVZRXOGH[SDQG
FRQVXPSWLRQEHLWURDGWULSVRUDQ\RWKHUJRRGXSWRWKHSRLQWZKHUHWKHEHQHILWVRI
WKHODVWXQLWFRQVXPHGKDYHEHFRPHHTXDOWRWKHVHPDUJLQDOVRFLDOFRVWV7KHLPSOLHG
HTXDOLW\RIPDUJLQDOEHQHILWVDQGPDUJLQDOVRFLDOFRVWVVHFXUHVWKDWWKHVRFLDOVXUSOXV
GHILQHGDVWKHGLIIHUHQFHEHWZHHQWRWDOEHQHILWVDQGWRWDOFRVWVLVPD[LPLVHG7KLV
VRFLDOVXUSOXVLVRIWHQFRQVLGHUHGDVDQDSSURSULDWHLQGLFDWRUIRUVRFLDOZHOIDUHDQGLWV
PD[LPLVDWLRQLVDFRQGLWLRQIRUHFRQRPLFHIILFLHQF\WRSUHYDLO
$VH[SODLQHGLQ&KDSWHURIWKLVYROXPHWKHµSULFH¶IRUWKHXVHRIDURDGZRXOG
LQWKHILUVWSODFHLQFOXGHWKHFRVWVERUQHGLUHFWO\E\WKHURDGXVHUVXFKDVIXHO
H[SHQVHVDQGWKHYDOXHRIWKHWUDYHOWLPHVSHQWRQWKHURDG±WKHPDUJLQDOSULYDWH
FRVWV%HVLGHVDWROOPD\EHFKDUJHGIRUWKHXVHRIWKHURDGZKLFKDOVRDGGVWRWKH
SULFH0DUJLQDOFRVWSULFLQJLQURDGWUDQVSRUWWKHUHIRUHPHDQVWKDWWKHWROOVKRXOG
UHIOHFWWKHPDUJLQDOH[WHUQDOFRVWVWKHPDUJLQDOFRVWVRIURDGXVHRWKHUWKDQWKH
PDUJLQDOSULYDWHFRVW)RUDFRQJHVWHGURDGWKLVLQFOXGHVWKHYDOXHRIWLPHORVVHVDQG
WUDYHOWLPHXQFHUWDLQW\LPSRVHGRQRWKHUURDGXVHUVDVZHOODVWKHYDOXHRI
HPLVVLRQVQRLVHDQGDFFLGHQWULVNVFUHDWHG6XFKDWD[HTXDOWRPDUJLQDOH[WHUQDO
FRVWVLVRIWHQUHIHUUHGWRDVDµ3LJRXYLDQWD[¶DIWHULWVVSLULWXDOIDWKHU$UWKXU3LJRX
%DVHGRQWKLVLQWXLWLRQWUDQVSRUWHFRQRPLVWVDQGHQJLQHHUVDOLNHKDYHIRUORQJ
DGYRFDWHGWKHXVHRIURDGSULFLQJVFKHPHVIRUWKHPDQDJHPHQWRIURDGWUDIILF
FRQJHVWLRQHJ 3LJRX.QLJKW:DOWHUV9LFNUH\HWF(YHQ
LIURDGFDSDFLWLHVFDQEHDGMXVWHGWRFRSHZLWKFRQJHVWLRQFDSDFLW\SROLFLHVDORQHZLOO
W\SLFDOO\QRWEHVXIILFLHQWWRUHDFKDQRSWLPDORXWFRPHDQGWKHXVHRIFRQJHVWLRQ
SULFLQJUHPDLQVZDUUDQWHG $VSULFLQJQRWRQO\VXFFHHGVLQUHGXFLQJGHPDQGSHUVH
EXWLQGRLQJVRDOVRLQPDLQWDLQLQJH[DFWO\WKRVHWULSVIRUZKLFKWKHEHQHILWVDV
UHIOHFWHGE\WKHZLOOLQJQHVVWRSD\DUHKLJKHVWLWZLOOW\SLFDOO\OHDGWRKLJKHUZHOIDUH
JDLQVWKDQZKDWFRXOGEHUHDOLVHGE\DOWHUQDWLYHQRQSULFHEDVHGGHPDQG
PDQDJHPHQWVFKHPHVVXFKDVQXPEHUSODWHSROLFLHVDVFXUUHQWO\LQRSHUDWLRQLQ
$WKHQVDQG0H[LFR&LW\7KLVH[SODLQVPXFKRIWKHDQDO\VWV¶SUHIHUHQFHIRUWKLVW\SH
RISROLF\0RUHRYHUWKHSULQFLSOHRIPDUJLQDOFRVWSULFLQJVHFXULQJHIILFLHQF\FDUULHV
RYHUWRPRUHFRPSOH[VLWXDWLRQVLQFOXGLQJFRQJHVWLRQSULFLQJRQIXOOQHWZRUNVUDWKHU
WKDQDVLQJOHURDGKHWHURJHQHRXVXVHUVDQGWLPHYDU\LQJFRQJHVWLRQ7KLVVXJJHVWV
WKDWWKHSUDFWLFDODSSOLFDWLRQRIWKHSULQFLSOHRIRSWLPDOFRQJHVWLRQSULFLQJLQUHDOLW\
DOWKRXJK\LHOGLQJDGGLWLRQDOSUDFWLFDOFRPSOLFDWLRQVZRXOGQRWFUHDWHDGGLWLRQDO
FRQFHSWXDOGLIILFXOWLHVFRPSDUHGWRWKHEDVLFPHFKDQLVPVDVLGHQWLILHGLQWH[WERRN
PRGHOV+RZHYHUWKLVZRXOG±XQIRUWXQDWHO\±EHWUXHRQO\XQGHUUDWKHUVWULQJHQWDQG
RIWHQXQUHDOLVWLFDVVXPSWLRQV7ZRRIWKHVHXQUHDOLVWLFDVVXPSWLRQVDUHGLVFXVVHG
QH[W
$ILUVWRQHFRQFHUQVRSWLPDOLW\RIWKHSULFLQJLQVWUXPHQWZKLFKVKRXOGDOORZWKH
URDGUHJXODWRUWRGLIIHUHQWLDWHSULFHVVXFKWKDWHYHU\LQGLYLGXDOURDGXVHULQGHHGIDFHV
DWROOH[DFWO\HTXDOWRWKHPDUJLQDOH[WHUQDOFRVWVKHRUVKHFDXVHV7KLVUHTXLUHVWROOV
WREHGLIIHUHQWLDWHGDWOHDVWE\WLPHRIGD\URXWHIROORZHGDQGKHQFHWKHOHQJWKRI
WKHWULSW\SHRIYHKLFOHHJVPDOOYHUVXVODUJHFDUVRUSDVVHQJHUFDUVYHUVXVWUXFNV
DQGLWVVWDWHRIPDLQWHQDQFHIRUSROOXWLRQGULYLQJVW\OHIRUSROOXWLRQQRLVHDQG
DFFLGHQWULVNVHWF ,WDOVRUHTXLUHVWKDWWKHVHLVVXHVFDQEHPRQLWRUHGSHUIHFWO\E\WKH
UHJXODWRU$OWKRXJKHPHUJLQJWHFKQRORJLHVIRUHOHFWURQLFFKDUJLQJDUHOLNHO\WRDOORZ
1RWHKRZHYHUWKDWWKHUHYHUVHLVDOVRWUXHZKHQFDSDFLW\FDQEHFKRVHQSULFLQJDORQHZLOO
W\SLFDOO\QRWUHVXOWLQDQRSWLPDORXWFRPHXQOHVVLQLWLDOFDSDFLWLHVµKDSSHQ¶WREHRSWLPDO
IRUDPRUHVRSKLVWLFDWHGWROOGLIIHUHQWLDWLRQLQWKHQHDUIXWXUHLWUHPDLQVWREHVHHQWR
ZKLFKH[WHQWVXFKFRPSOH[SULFLQJ ZLOOLQGHHGEHSXWLQWRSUDFWLFH
$VHFRQGDVVXPSWLRQFRQFHUQVHIILFLHQF\LQDOOPDUNHWVUHODWHGWRWKHWUDQVSRUW
PDUNHWXQGHUFRQVLGHUDWLRQZKHUHDPDUNHWLVµUHODWHG¶WRWKHWUDQVSRUWPDUNHWLILWV
HTXLOLEULXPLVLQGLUHFWO\DIIHFWHGE\WUDQVSRUWSROLFLHVDQGLWVHTXLOLEULXPLV
LQHIILFLHQWLIDJDLQSULFHVDUHQRWHTXDOWRPDUJLQDOVRFLDOFRVWV7KHH[LVWHQFHRI
VXFKUHODWHGPDUNHWVLVWKHUXOHUDWKHUWKDQWKHH[FHSWLRQEHFDXVHWUDQVSRUWGHPDQGLV
RIWHQDGHULYHGGHPDQGZKHUHWKHµFRQVXPSWLRQ¶RIWUDQVSRUWLVRIWHQQRJRDOLQLWVHOI
EXWVHUYHVWRHQDEOHDPDUNHWWUDQVDFWLRQEHWZHHQVSDWLDOO\VHSDUDWHGVXSSOLHUVDQG
GHPDQGHUVRIDFHUWDLQJRRGRUVHUYLFH)RUH[DPSOHFRPPXWLQJW\SLFDOO\GRHVQRW
\LHOGDQ\EHQHILWLQLWVHOIEXWVHUYHVWRDOORZSHRSOHWRVXSSO\ODERXUVHUYLFHVDWD
GLIIHUHQWORFDWLRQWKHZRUNSODFHWKDQWKHLUUHVLGHQFHV/LNHZLVHIUHLJKWWUDIILF
HQDEOHVDWUDQVDFWLRQEHWZHHQDVXSSOLHURIJRRGVDQGGHPDQGHUVILQDOFRQVXPHUVRU
ILUPVGHPDQGLQJLQWHUPHGLDWHVZKRDUHVHSDUDWHGLQVSDFH:KHQHYHUWKHVHUHODWHG
PDUNHWVGRQRWIXQFWLRQSURSHUO\WKHVLPSOHSROLF\UXOHRIVHWWLQJWD[HVHTXDOWR
PDUJLQDOH[WHUQDOFRVWVZLOOQRORQJHUEHDWUXO\RSWLPDOFKRLFH,QVWHDGWUDQVSRUW
WD[HVVKRXOGRSWLPDOO\GHYLDWHIURPWKLVUXOHVRDVWRFRUUHFWIRUWKHLQHIILFLHQF\LQ
WKHUHODWHGPDUNHWZLWKRXWRIFRXUVHVDFULILFLQJWRRPXFKHIILFLHQF\LQWKHWUDQVSRUW
PDUNHWLWVHOI,PSRUWDQWUHDVRQVZK\SULFHVLQUHODWHGPDUNHWVPD\GHYLDWHIURP
PDUJLQDOVRFLDOFRVWVLQFOXGHDPRQJVWRWKHUVWKHH[LVWHQFHRIPDUNHWSRZHU
XQSULFHGH[WHUQDOLWLHVVXFKDVHQYLURQPHQWDOSROOXWLRQRUGLVWRUWLRQDU\WD[HVIRU
LQVWDQFHRQWKHODERXUPDUNHWLQWKHFDVHRIFRPPXWHUWUDIILF ,QHIILFLHQFLHVDUH
WKHUHIRUHOLNHO\WREHWKHUXOHUDWKHUWKDQWKHH[FHSWLRQ
'HVSLWHWKHDWWHQWLRQWKDWILUVWEHVWSULFLQJKDVWUDGLWLRQDOO\UHFHLYHGLQPRVWRIWKH
OLWHUDWXUHRQWUDQVSRUWSULFLQJLWZLOOEHFOHDUIURPWKHDERYHWKDWVHFRQGEHVWSULFLQJ
LVLQIDFWRIWHQWKHPRVWUHOHYDQWFDVHIURPDSUDFWLFDOSHUVSHFWLYH,QUHFHQW\HDUV
WKHUHKDVFRQVHTXHQWO\EHHQDQXSVXUJHRIVWXGLHVLQVHFRQGEHVWFRQJHVWLRQSULFLQJ
/LQGVH\DQG9HUKRHISURYLGHDUHFHQWUHYLHZRIWKLVOLWHUDWXUHDQGGLVFXVV
VHFRQGEHVWSULFLQJUHVXOWLQJIURPQHWZRUNLVVXHVKHWHURJHQHLW\RIXVHUVG\QDPLF
FRQVWUDLQWVRQWROOIOH[LELOLW\µVWHSWROOV¶XQFHUWDLQW\LQWUDYHOWLPHVLQUHODWLRQZLWK
UHDOWLPHLQIRUPDWLRQSURYLVLRQUHODWLRQVZLWKRWKHUVHFWRUVQRWDEO\ODERXUPDUNHWV
WKHVLPXOWDQHRXVH[LVWHQFHRIPXOWLSOHH[WHUQDOLWLHVHJ FRQJHVWLRQDQGSROOXWLRQ
FRQJHVWLRQSULFLQJE\SULYDWHURDGRSHUDWRUVDQGWKURXJKLQWHUDFWLRQVZLWKVXE
RSWLPDOFDSDFLW\FKRLFH
6HFRQGEHVWFRQJHVWLRQSULFLQJWKURXJKQHWZRUNHIIHFWVZLOOEHHVSHFLDOO\
LPSRUWDQWLQXUEDQFRQWH[WVZKHUHGHQVHQHWZRUNVZLOORIWHQSUHYHQWWKHDSSOLFDWLRQ
RIRSWLPDOO\GLIIHUHQWLDWHGWROOVRQHDFKDQGHYHU\OLQNRIWKHQHWZRUN7KLVFKDSWHU
FRQVLGHUVWKHVHLVVXHV$ILUVWDLPLVWRSURYLGHDQRYHUYLHZRIWKHEDVLFHFRQRPLFLQV
DQGRXWVRIVHFRQGEHVWFRQJHVWLRQSULFLQJLQQHWZRUNVE\UHYLHZLQJWKHFODVVLFWZR
URXWHSUREOHPLQ6HFWLRQEHORZ0RVWRIWKHXQGHUO\LQJOLWHUDWXUHFRQYHQLHQWO\EXW
VRPHZKDWXQUHDOLVWLFDOO\WUHDWVGLIIHUHQWURXWHVEHWZHHQWZRSRLQWVµQRGHV¶LQD
QHWZRUNDVSXUHVXEVWLWXWHVDQGWKHUHIRUHDSSOLHVVRFDOOHG:DUGURSLDQHTXLOLEULXP
SULQFLSOHV:DUGURSLQWKHGHWHUPLQDWLRQRIQHWZRUNHTXLOLEULDZLWKDQG
ZLWKRXWSULFHV7KLVPHDQVWKDWHYHQWKHVPDOOHVWHTXLOLEULXPSULFHGLIIHUHQFHZRXOG
PDNHD XVHU FKRRVHWKHFKHDSHVWURXWH,QUHDOLW\SHRSOHPD\KDYHVRFDOOHG
LGLRV\QFUDWLFSUHIHUHQFHVRYHUGLIIHUHQWURXWHVPHDQLQJWKDWWKH\ZRXOGKDYH
GLIIHUHQWSUHIHUHQFHVIRUFRPSHWLQJURXWHVZKHQWKHVHZRXOGKDYHHTXDOWULSSULFHV,W
LVWKHUHIRUHRILQWHUHVWWRVWXG\KRZWKLVZRXOGDIIHFWWKHRSWLPDOGHVLJQDQGZHOIDUH
SURSHUWLHVRIVHFRQGEHVWFRQJHVWLRQSULFLQJ0RUHRYHUE\DOORZLQJGLIIHUHQWURXWHV
WREHLPSHUIHFWVXEVWLWXWHVDIUDPHZRUNLVGHYHORSHGWKDWFDQDOVREHXVHGWRVWXG\
VHFRQGEHVWFRQJHVWLRQSULFLQJIURPDPXOWLPRGDOSHUVSHFWLYH$GGUHVVLQJWKHVH
LVVXHVLVWKHVHFRQGDLPRIWKHSDSHU$WKHRUHWLFDOPRGHOLVGHYHORSHGIRUWKHVWXG\
RIVHFRQGEHVWFRQJHVWLRQSULFLQJLQQHWZRUNVZKHUHSDUDOOHOURDGVDUHLPSHUIHFW
VXEVWLWXWHURXWHVDQGLWVSURSHUWLHVDUHLOOXVWUDWHGZLWKWKHKHOSRIDVLPXODWLRQPRGHO
6(&21'%(67&21*(67,2135,&,1*,175$163257
1(7:25.6
2QHRIWKHPRVWZLGHO\VWXGLHGLQVWDQFHVRIVHFRQGEHVWFRQJHVWLRQSULFLQJFRQFHUQV
WKHFDVHZKHUHQRWDOOOLQNVLQDXQLPRGDOWUDQVSRUWQHWZRUNFDQEHSULFHG7KLVFDVH
KDVJUHDWSUDFWLFDOUHOHYDQFHIRUDFWXDOSROLF\PDNLQJ)RULQVWDQFHSD\ODQHVDV
FXUUHQWO\LQRSHUDWLRQDWYDULRXVSODFHVLQWKH86$EHORQJWRWKLVFDWHJRU\RIVHFRQG
EHVWSULFLQJSROLFLHVVLQFHXQSULFHGODQHVDUHDYDLODEOHDVDGLUHFWVXEVWLWXWHIRUWKH
SD\ODQHV,QRWKHUSODQQHGURDGSULFLQJVFKHPHVEHLWEDVHGRQPDUJLQDOFRVW
SULFLQJSULQFLSOHVRUPRWLYDWHGE\WKHGHVLUHWRJHQHUDWHUHYHQXHVSULFHVZLOORIWHQEH
FKDUJHGRQDOLPLWHGQXPEHURIOLQNVLQDQHWZRUNRQO\HJ WROOURDGVRUWROO
FRUGRQV)XUWKHUPRUHWKHFDVHLVRILQWHUHVWDVLWFDQEHXVHGWRLOOXVWUDWHVRPHPRUH
JHQHUDOSULQFLSOHVRIVHFRQGEHVWSULFLQJWKDWZLOOEHUHOHYDQWIRURWKHUW\SHVRI
VHFRQGEHVWSULFLQJWRR7KHVHSULQFLSOHVLQFOXGHWKHIDFWWKDWLQVHFRQGEHVWSULFLQJ
WKHQDwYHXVHRI3LJRXYLDQWD[HVVHW HTXDOWRWKHGLUHFWPDUJLQDOH[WHUQDOFRVWV ZLOO
JHQHUDOO\QRWEHRSWLPDODQGWKHIDFWWKDWVHFRQGEHVWSULFLQJLWVHOIFUHDWHV
GLVWRUWLRQVZKLFKDIIHFWWKHRSWLPDOVHFRQGEHVWSULFHLWVHOI
7KH FODVVLFWZRURXWHSUREOHP
7KHVLPSOHVWYHUVLRQRIWKHSUREOHPDWKDQGFRQFHUQVWKHWZRURXWHSUREOHPZKHUH
DQXQWROOHGDOWHUQDWLYHURDGLVDYDLODEOHSDUDOOHOWRDWROOURDG7KLVSUREOHPKDVIRU
LQVWDQFHEHHQVWXGLHGE\/pY\/DPEHUW0DUFKDQGDQGPRUHUHFHQWO\
DOVRE\%UDLG9HUKRHIHWDO DQG/LXDQG0F'RQDOG)LJXUH
LOOXVWUDWHVWKHEDVLFVHWXSRIWKHSUREOHP7ZRSDUDOOHOFRQJHVWHGURDGVRIJLYHQ
FDSDFLWLHVFRQQHFWDQRULJLQ$DQGDGHVWLQDWLRQ%2QRQHRIWKHVHURDG7D
FRQJHVWLRQWROOFDQEHVHWZKHUHDVWKHRWKHUURDG8UHPDLQVXQWROOHG7KHURDGVDUH
SXUHVXEVWLWXWHVPHDQLQJWKDWHTXLOLEULXPµJHQHUDOLVHGSULFHV¶S WKHVXPRI
PRQHWLVHGWUDYHOFRVWVFSOXVDWROOWLIOHYLHGVKRXOGEHHTXDOIRUERWKURDGVZKHQ
ERWKDUHXVHGWKLVLVWKHµ:DUGURSLDQ¶HTXLOLEULXPSULQFLSOHPHQWLRQHGHDUOLHU
'ULYHUVDUHQHDUO\LGHQWLFDOWKH\KDYHWKHVDPHYDOXHRIWLPHGULYHWKHVDPHW\SHRI
FDUVHWFZLWKWKHH[FHSWLRQRIWKHLUZLOOLQJQHVVWRSD\IRUDWULSZKLFKYDULHVRYHU
GULYHUVZKHQHYHUGHPDQGLVQRWSHUIHFWO\HODVWLF7KHWZRFHQWUDOTXHVWLRQVDUHWKHQ
DWZKDWOHYHOVKRXOGWKHVHFRQGEHVWWROOEHVHWDQGZKDWDUHWKHZHOIDUHHIIHFWV
FRPSDUHGWRWKHILUVWEHVWRSWLRQRIWROOLQJERWKURDGV"
$
)LJXUH
7ROOURDG
µ)UHH¶URDG
%
7KHFODVVLFWZRURXWHSUREOHP
$QLPSRUWDQWUHVXOWIURPWKHVWXGLHVORRNLQJDWWKHVHSUREOHPVLVWKDWWKHVHFRQG
EHVWWD[VHWVRDVWRPD[LPLVHVRFLDOZHOIDUHJLYHQWKHSHUVLVWHQFHRIWKHVHFRQGEHVW
GLVWRUWLRQRIOHDYLQJWKHRWKHUURDGXQWROOHGLVW\SLFDOO\EHORZWKHRSWLPDO3LJRXYLDQ
WD[WKDWZRXOGEHVHWRQWKHVDPHURDGXQGHUILUVWEHVWSULFLQJEXWDOVREHORZWKH
PDUJLQDOH[WHUQDOFRVWVRQWKHWROOURDGLQWKHVHFRQGEHVWRSWLPXP,QDGGLWLRQWKH
VHFRQGEHVWWROO\LHOGVFRQVLGHUDEO\VPDOOHUZHOIDUHJDLQVWKDQILUVWEHVWSULFLQJ
,WLVLPSRUWDQWWRXQGHUVWDQGZK\WKHVHFRQGEHVWWROOVKRXOGLGHDOO\EHEHORZWKH
PDUJLQDOH[WHUQDOFRQJHVWLRQFRVWVRQWKHWROOHGURDG7KHUHDVRQLVWKDWWKH
FRQJHVWLRQFKDUJHRQWKHWROOURDGEULQJVERWKJRRGDQGEDGQHZVIURPWKH
SHUVSHFWLYHRIHIILFLHQF\7KHJRRGQHZVLVWKDWLQLWLDOO\H[FHVVLYHIUHHPDUNHW
FRQJHVWLRQRQWKHWROOURDGLVUHGXFHGWKURXJKWKHWROO7KHEDGQHZVLVWKDWVRPH
SHRSOHZKRDUHSULFHGRIIWKLVURDGZLOOEHGLYHUWHGWRWKHDOUHDG\FRQJHVWHGXQWROOHG
URDGWKXVDJJUDYDWLQJWKHFRQJHVWLRQWKHUH7KHVHFRQGEHVWWROOWUDGHVRIIWKHJRRG
QHZVDJDLQVWWKHEDGQHZV7KHJRRGQHZVDORQHZRXOGDVNIRUDWD[HTXDOWR
PDUJLQDOH[WHUQDOFRVWVIRUWKHUHDVRQVJLYHQLQWKHLQWURGXFWLRQEXWDVVRRQDVWKH
EDGQHZVLVUHOHYDQWLHZKHQWKHUHLVFRQJHVWLRQRQWKHXQWROOHGURDGDQGZKHQD
E\SURGXFWRIWKHWROOLVLQGHHGWRGLYHUWWUDIILFWRWKLVXQWROOHGURDGDGRZQZDUG
DGMXVWPHQWRIWKHWROOLVRSWLPDO$VDFRUROODU\DSROLF\RIVLPSO\LJQRULQJWKH
VHFRQGEHVWGLVWRUWLRQDQGVHWWLQJWKHWROOHTXDOWRWKHPDUJLQDOH[WHUQDOFRQJHVWLRQ
FRVWVRQWKHWROOHGURDGZRXOGOHDGWRDVRFLDOVXUSOXVEHORZWKDWLQWKHVHFRQGEHVW
RSWLPXPH[DFWO\EHFDXVHWKHSROLF\LVµQDwYH¶ DQGLJQRUHVWKHVSLOORYHUVRQWKHIUHH
ODQH,QGHHGVXFKµTXDVLILUVWEHVWSULFLQJ¶PD\HYHQOHDGWRDZHOIDUHORVVFRPSDUHG
WRWKHQRWROORXWFRPHZKLFKZRXOGFHUWDLQO\RFFXUZKHQWKHVHFRQGEHVWRSWLPDOWD[
LVQHJDWLYHDVLQVRPHH[DPSOHVLQ9HUKRHIHWDO 7KHVHFRQGEHVWRSWLPDOWD[UXOHIRUWKLVFODVVLFWZRURXWHSUREOHPUHIOHFWVWKH
DERYHPHQWLRQHGWUDGHRIIEHWZHHQWKHJRRGDQGWKHEDGQHZVIURPWKHWROO9HUKRHI
HWDOH[SUHVVLWDV
W
PHF7 PHF8 ˜
'c
F8c ' c
ZKHUHPHF7 PHF8GHQRWHVWKHPDUJLQDOH[WHUQDOFRVWVRQWKHWROOHGXQWROOHGURDG
'c WKHVORSHRIWKHLQYHUVHGHPDQGIXQFWLRQDQGF8c WKDWRIWKHFRVWIXQFWLRQRQWKH
XQWROOHGURDGDOOHYDOXDWHGDWWKHVHFRQGEHVWHTXLOLEULXP7KHWD[UXOHFOHDUO\VKRZV
KRZWKHVHFRQGEHVWWROOLVHTXDOWRWKHZKDWZLOOEHFDOOHGWKHµGLUHFWPDUJLQDO
H[WHUQDOFRVWV¶RIGULYHUVURDG7 PHF7 PLQXVDIUDFWLRQRIWKRVHRQWKHXQWROOHG
URDG8 PHF8EHFDXVH'c d DQG F8c tWKHZHLJKWIRUPHF8 LVDOZD\VEHWZHHQ
DQG
)XUWKHULQWXLWLRQFDQEHREWDLQHGE\FRQVLGHULQJDIHZLQVWDQFHVZKHUHWKH
IUDFWLRQPHQWLRQHGZRXOGDSSURDFKLWVSRVVLEOHH[WUHPHYDOXHVRIPHDQLQJWKDWW
PHF7RUPHDQLQJWKDWW PHF7 ±PHF8,QWXLWLYHO\WKHIUDFWLRQVKRXOGDSSURDFK
]HURZKHQWKHUHLVQRGLYHUWLQJRIWUDIILFWRWKHXQWROOHGURDGLQWKHVHFRQGEHVW
RSWLPXPGXHWRPDUJLQDOFKDQJHVLQW7KLVLVWKHFDVHZKHQHLWKHU ' c o UHJDUGOHVVRIWWKHXQWROOHGURDGZLOOLQHTXLOLEULXPEHILOOHGZLWKGULYHUVXSWRWKDW
OHYHO18 WKDWHTXDWHVF818WRWKHJLYHQYDOXHRI'RUZKHQ F8c o f WKHFRVW
IXQFWLRQIRUURDG8 LVSHUIHFWO\LQHODVWLFDQG18 LVJLYHQ,QWKHVHFDVHVWKH
UHJXODWRUFDQLJQRUHVSLOORYHUVRIWROOLQJRQURDG7 XSRQURDG8VLPSO\EHFDXVH
WKHVHZLOOQRWRFFXU7KHIUDFWLRQLQFRQWUDVWDSSURDFKHVXQLW\ZKHQHLWKHU ' c o f
RUZKHQ F8c o ,QWKHVHFDVHVWKHVHFRQGEHVWWROORQO\WULHVWRRSWLPLVHURXWHVSOLW
E\LQWHUQDOL]LQJWKHGLIIHUHQFHEHWZHHQPDUJLQDOH[WHUQDOFRVWVIRUWKHWZRURXWHVDQG
IXOO\LJQRUHVWKHHIIHFWVRQRYHUDOOGHPDQG7KLVPDNHVVHQVHZKHQGHPDQGLV
SHUIHFWO\LQHODVWLF ' c o f LQZKLFKFDVHRYHUDOOGHPDQGZRXOGQRWUHVSRQGWR
PDUJLQDOFKDQJHVLQW,WDOVRPDNHVVHQVHLIHYHU\XVHUSULFHGRIIURDG7 LPPHGLDWHO\
VZLWFKHVWRURDG8LQWKHVLWXDWLRQZKHUH F8c o LPSOLHVWKDWSULYDWHFRVWVRIXVLQJ
URDG8 DUHLQGHSHQGHQWRIWKHURXWHVSOLWVRWKDWWKHRYHUDOOGHPDQGLVJLYHQDQG
GHILQHGE\WKHHTXDOLW\RI' DQGF8,WVKRXOGEHQRWHGKRZHYHUWKDW F8c o DWWKH
7KHLPSHUIHFWLRQRIWKHWD[LQVWUXPHQWFDXVHVWKHWHUPµPDUJLQDOH[WHUQDOFRVWV¶WREHFRPH
DPELJXRXVDVDGLVWLQFWLRQEHWZHHQµIXOO¶DQGµGLUHFW¶PDUJLQDOH[WHUQDOFRVWVFDQEHPDGH
7KHIXOOPDUJLQDOH[WHUQDOFRVWVIRUGULYHUVRQWKHWROOHGURDGLQWKLVH[DPSOHDUHJLYHQE\WKH
WD[UXOH7KHVHLQFOXGHWKHVHFRQGWHUPZKLFKLVQRQ]HUREHFDXVHRIWKHLPSHUIHFWLRQRIWKH
WD[LQVWUXPHQW'LUHFWPDUJLQDOH[WHUQDOFRVWVDUHWKXVGHILQHGKHUHDVDOOPDUJLQDOH[WHUQDO
FRVWVWKDWDUHQRWFDXVHGE\LQHIILFLHQW EHKDYLRXULQGXFHGE\WKHWD[LQVWUXPHQWLWVHOI
VDPHWLPHLPSOLHVWKDWPHF8 VRWKDWWKHWD[UXOHLQWKLVFDVHHIIHFWLYHO\EHFRPHVW
PHF76HFRQGEHVWSULFLQJWKHQEHFRPHVLGHQWLFDOWRILUVWEHVWSULFLQJEHFDXVHVSLOO
RYHUVGRQRWUHVXOWLQDGGLWLRQDOZHOIDUHORVVHV
7KHUHODWLYHVL]HRIWKHZHOIDUHORVVHVIURPVHFRQGEHVWSULFLQJFRPSDUHGWRILUVW
EHVWSULFLQJGHSHQGVRQYDULRXVHODVWLFLWLHV9HUKRHIHWDOIRUH[DPSOH
GHPRQVWUDWHWKDWLIURXWH8 LVSDUWLFXODUO\FRQJHVWHGWKHRSWLPDOVHFRQGEHVWWROOFDQ
EHQHJDWLYHZKLFKW\SLFDOO\LPSOLHVUHODWLYHO\ORZHIILFLHQF\JDLQVGXHWRSULFLQJ
HYHQ]HURZKHQWKHRSWLPDOVHFRQGEHVWFRQJHVWLRQWROOKDSSHQVWREH]HUR0RUH
JHQHUDOO\WKH\VKRZWKDWWKHHIILFLHQF\JDLQVGHSHQGRQWKHUHODWLYHIUHHIORZWUDYHO
WLPHVDQGFDSDFLWLHVRIWKHWZRURXWHVDQGLQFUHDVHLIWKHWROOHGURDGEHFRPHV
UHODWLYHO\PRUHDWWUDFWLYH)XUWKHUPRUHWKHJDLQVGHSHQGRQWKHSULFHHODVWLFLW\RI
WUDYHOGHPDQGLQFRPELQDWLRQZLWKWKHFRVWSDUDPHWHUV6SHFLILFDOO\LIGHPDQG
DSSURDFKHVSHUIHFWLQHODVWLFLW\WKHHIILFLHQF\JDLQVIURPVHFRQGEHVWSULFLQJPD\
DSSURDFKWKRVHIURPILUVWEHVWSULFLQJLIWKHURDGVGLIIHULQIUHHIORZWUDYHOWLPHV7KH
RQO\IDFWRUGHWHUPLQLQJHIILFLHQF\LQWKDWFDVHZRXOGEHWKHURXWHVSOLWZKLFKFDQEH
RSWLPLVHGZLWKRQHWROORQO\+RZHYHULIWKHURDGVKDYHLGHQWLFDOIUHHIORZWUDYHO
WLPHVDVZRXOGRIWHQEHWKHFDVHIRUSD\ODQHVWKHHIILFLHQF\JDLQVIURPVHFRQGEHVW
SULFLQJPD\QHYHUWKHOHVVIDOOWR]HURDVWKHIUHHPDUNHWURXWHVSOLWZLWKRXWWROOLQJ
PD\WKHQDOUHDG\EHRSWLPDO(TXDOLVDWLRQRISULYDWHFRVWVWKHQLPSOLHVWKHRSWLPDO
HTXDOLVDWLRQRIPDUJLQDOFRVWV7KHVHFRQGEHVWRSWLPDOWROODSSURDFKHV]HURDQG
WKHUHIRUHLQGXFHVQRZHOIDUHHIIHFWVZKDWVRHYHU7KHVHILQGLQJVLOOXVWUDWHWKDWIRU
VHFRQGEHVWSULFLQJRIWHQµDQ\WKLQJJRHV¶LQWKHVHQVHWKDWWKHUHODWLYHHIILFLHQF\RI
VHFRQGEHVWSROLFLHVLVRIWHQLPSRVVLEOHWRSUHGLFWZLWKRXWSULRUGHWDLOHGNQRZOHGJHRI
WKHFDVHDWKDQG/LXDQG0F'RQDOGFRQVWUXFWHGDPRGHOGHVFULSWLYHRIRQHRI
WKH&DOLIRUQLDURDGSULFLQJGHPRQVWUDWLRQSURMHFWV6WDWH5RXWHLQ2UDQJH&RXQW\
DQGIRXQGWKDWWKHHIILFLHQF\JDLQVIURPVHFRQGEHVWFRQJHVWLRQSULFLQJDUHLQWKH
RUGHURISHUFHQWRIWKHSRWHQWLDOJDLQVIURPILUVWEHVWSULFLQJ%HDULQJLQPLQGWKH
H[SRVLWLRQDERYHWKLVVKRXOGLQIDFWQRWEHWRRGLVDSSRLQWLQJJLYHQWKHVLPLODULW\RI
WKHSD\ODQHVDQGWKHIUHHODQHVDQGWKHLQHODVWLFLW\RIGHPDQG
%H\RQGWKHFODVVLFWZRURXWHSUREOHP
7KHFODVVLFWZRURXWHSUREOHPKDVUHFHLYHGFRQVLGHUDEOHDWWHQWLRQIRUDWOHDVWWZR
UHDVRQV)LUVWO\LWLVSUREDEO\WKHVLPSOHVWSRVVLEOHDQGPRVWWUDQVSDUHQWVHWWLQJIRU
LOOXVWUDWLQJVRPHRIWKHPDLQHFRQRPLFLQVLJKWVLQWRWKHLQVDQGRXWVRIVHFRQGEHVW
SULFLQJSROLFLHVZKLFKRIWHQDOVRFDUU\RYHUWRRWKHUPRUHFRPSOLFDWHGVHWWLQJV
6HFRQGO\LWDSSHDUVWREHD UHDVRQDEOHDSSUR[LPDWLRQIRUPDQ\FRQJHVWLRQSULFLQJ
GHPRQVWUDWLRQSURMHFWVWKDWDUHFXUUHQWO\LQRSHUDWLRQ7KHEDVLFPRGHOLVFHUWDLQO\
UDWKHUUHVWULFWLYHDQGDQXPEHURIH[WHQVLRQVKDYHEHHQPDGHWRLQYHVWLJDWHVRPH
IXUWKHUSURSHUWLHVRIWKHSUREOHPDQGLWVVROXWLRQ)RUH[DPSOHVWDWLFPRGHOVDV
GLVFXVVHGDERYHPD\XQGHUHVWLPDWHWKHHIILFLHQF\JDLQVIURPVHFRQGEHVWRQHURXWH
WROOLQJEHFDXVHWKH\LJQRUHVRPHDOWHUQDWLYHZD\VLQZKLFKVHFRQGEHVWSULFLQJFDQ
DIIHFWGULYHUEHKDYLRXU%UDLGDQG'H3DOPDDQG/LQGVH\DOORZIRUWULS
WLPLQJDGMXVWPHQWVE\FRQVLGHULQJWLPHYDU\LQJWROOVLQWKH9LFNUH\ERWWOHQHFN
PRGHODSSOLHGWRWKHVDPHWZRSDUDOOHOURXWHVQHWZRUN7KH\ILQGWKDWVHFRQGEHVW
WROOLQJ\LHOGVKLJKHUDEVROXWHHIILFLHQF\JDLQVWKDQLQWKHVWDWLFPRGHODQGDJUHDWHU
IUDFWLRQRIWKHILUVWEHVWHIILFLHQF\JDLQV7KLVLVEHFDXVHWKHWROOQRWRQO\FXUEV
H[FHVVLYHWRWDOXVDJHEXWDOVRHOLPLQDWHVTXHXLQJRQWKHWROOHGURXWH
8QGHUVHFRQGEHVWSULFLQJLWLVQRORQJHURSWLPDOWRHTXDWHWD[HVWRWKHGLUHFWPDUJLQDO
H[WHUQDOFRVWVEXWLQVWHDGPRUHFRPSOLFDWHGWD[UXOHVDSSO\ZKLFKDLPWRSDUWO\FRUUHFWIRU
WKHUHOHYDQWGLVWRUWLRQVDQGQDwYH3LJRXYLDQWD[DWLRQLVWKHUHIRUHQRORQJHURSWLPDO,Q
DGGLWLRQWRWKDWWKHUHODWLYHHIILFLHQF\RIDJLYHQW\SHRIVHFRQGEHVWSULFLQJRIWHQGHSHQGV
UDWKHUVWURQJO\RQFDVHVSHFLILFFLUFXPVWDQFHV
7KHHIILFLHQF\RIVHFRQGEHVWWROOLQJPD\DOVREHLPSURYHGZKHQGULYHUVKDYH
GLIIHUHQWYDOXHVRIWUDYHOWLPH8VHUVZLWKDKLJKYDOXHRIWLPHZRXOGWKHQFKRRVHWKH
WROOHGFRQQHFWLRQDQGEHQHILWUHODWLYHO\VWURQJO\IURPWKHUHGXFHGWUDYHOWLPHZKLOH
WKRVHZLWKDORZYDOXHRIWLPHZRXOGQRWVXIIHUWRRPXFKIURPWKHLQFUHDVHGWUDYHORQ
WKHXQWROOHGFRQQHFWLRQWKDWWKH\SUHIHU6PDOODQG<DQDQG9HUKRHIDQG6PDOO
ILQGWKDWWKHHIILFLHQF\RISD\ODQHVUHODWLYHWRWKHILUVWEHVWRSWLPXPLVLQGHHG
KLJKHUWKDQLQWKHHTXLYDOHQWPRGHOZLWKQRKHWHURJHQHLW\LQWKHYDOXHRIWLPH
)LQDOO\RSWLPDOVHFRQGEHVWWROOVDQGWKHDVVRFLDWHGZHOIDUHJDLQVKDYHDOVREHHQ
LQYHVWLJDWHGIRUODUJHUQHWZRUNVRQZKLFKQRWDOOOLQNVFDQEHWROOHG9HUKRHI
GHULYHVRSWLPDOVWDWLFWROOVRQDQ\VXEVHWRIOLQNVLQFOXGLQJSDUNLQJVSDFHVLQDQ
DUELWUDU\QHWZRUN7KHWROOIRUPXODHZKLFKDUHTXLWHFRPSOLFDWHGLQFOXGHWHUPV
UHIOHFWLQJPDUJLQDOH[WHUQDOFRVWVRQRWKHUOLQNVDQGZHLJKWVWKDWGHSHQGRQYDULRXV
GHPDQGDQGFRVWHODVWLFLWLHV7KHVROXWLRQIRUWKHFODVVLFWZRURXWHSUREOHPSUHVHQWHG
DERYHLVDVSHFLDO±EXWWUDQVSDUHQW±LOOXVWUDWLYHFDVHRIWKHVHJHQHUDOIRUPXODH
7+(7:25287(352%/(0:,7+,03(5)(&768%67,787(6
,QWKLVVHFWLRQWKHQHWZRUNRI)LJXUHLVIXUWKHUGLVFXVVHG$VDQQRXQFHGWKLV
QHWZRUNZLOOEHUHFRQVLGHUHGIRUWKHFDVHZKHUHWKHWZRURXWHVDUHLPSHUIHFW
VXEVWLWXWHV7KHUHDUHDVEHIRUHWZRQRGHV$DQG%FRQQHFWHGE\WZROLQNVZKLFK
DUHQRZGHQRWHGDQG%HVLGHVWROOVFDVHVZKHUHFDSDFLWLHVRQWKHVHOLQNVFDQYDU\
ZLOODOVREHFRQVLGHUHGDQGVHFRQGEHVWLVVXHVDULVHLIDWOHDVWRQHRIWKHVHIRXUSROLF\
YDULDEOHVLVUHVWULFWHG
7KHOLQNVPD\UHIHUWRURDGVDVLQWKHFODVVLFWZRURXWHSUREOHPDQGVHFRQGEHVW
SUREOHPVHPHUJHIRULQVWDQFHLIRQO\RQHOLQNLVXQSULFHGRULILWLVSULFHGVXE
RSWLPDOO\7KHOLQNVPD\DOVRUHSUHVHQWGLIIHUHQWPRGHVZKLFKLVZKDWZLOOEH
FRQVLGHUHGLQ6HFWLRQ)RUFRPSHWLQJPRGHVLWLVFRPPRQWRUHJDUGWKHVHDV
LPSHUIHFWVXEVWLWXWHV,QDGGLWLRQWKHW\SHRIFRVWIXQFWLRQVIRUXVHUVDVZHOODVWKH
FRVWRISURYLGLQJFDSDFLW\IRURSHUDWRUVPD\WKHQEHGLIIHUHQWEHWZHHQWKHWZROLQNV
$UQRWWDQG<DQKDYHUHFHQWO\SURYLGHGDUHYLHZRIWKHSUREOHPVWKDWDULVHLI
WUDYHOZLWKRQHPRGHLVVXEVLGLVHGZKLOHWKHSROLF\PDNHUFDQFKRRVHWKHSULFHRI
WUDYHOOLQJRQWKHRWKHUPRGHDQGFDSDFLW\RQERWKOLQNV7KLVµWZRPRGHSUREOHP¶
DSSHDUVWREHKDUGWRVROYHLQJHQHUDOWHUPV
,IWZRURDGVDUHSHUIHFWVXEVWLWXWHVDQ\SULFHGLIIHUHQFHLPSOLHVWKDWRQHURXWHZLOO
QRWEHXVHG+RZHYHULIWKH\DUHLPSHUIHFWVXEVWLWXWHVSULFHGLIIHUHQFHVGRQRW
QHFHVVDULO\LPSO\WKDWGHPDQGIRUWKHPRVWH[SHQVLYHURXWHIDOOVWR,QUHDOLW\WZR
URDGVWKDWFRQQHFWWKHVDPHSDLURIWRZQVQHHGQRWQHFHVVDULO\EHFRQVLGHUHGDV
SHUIHFWVXEVWLWXWHV)RULQVWDQFHFRQVXPHUVPD\SUHIHURQHURXWHWRWKHRWKHUEHFDXVH
WKHLUUHVLGHQWLDOORFDWLRQLQ$LVFORVHUWRWKHSRLQWZKHUHURXWHVWDUWVEHFDXVHWKHLU
GHVWLQDWLRQLQ%LVFORVHUWRZKHUHURXWHHQGVRUEHFDXVHWKH\KDYHDSUHIHUHQFH
RYHUKLJKZD\VHDVLHUGULYLQJRUVHFRQGDU\URDGVQLFHUVFHQHU\SHUVH7KHUDQGRP
WHUPLQWKHORJLWPRGHOSUHVHQWHGEHORZPD\EHLQWHUSUHWHGDVUHIOHFWLQJVXFK
LGLRV\QFUDWLFGLIIHUHQFHV
7KHJHQHUDOPRGHO
,QZKDWIROORZVDPRGHOWKDWFDQFRYHUDOOWKHDIRUHPHQWLRQHGFDVHVLVFRQVWUXFWHG,Q
WKHJHQHUDOPRGHOWKHVRFLDOSODQQHUDWWHPSWVWRPD[LPLVHWKHVRFLDOVXUSOXVZKLFKLV
GHILQHGDVFRQVXPHUVXUSOXVSOXVWROOUHYHQXHVPLQXVFRVWRILQIUDVWUXFWXUHFDSDFLW\
66
&6 75 .
7KH0DUVKDOOLDQQRWLRQRIFRQVXPHUVXUSOXVLVXVHG)RUDVLQJOHJRRGLWLVWKHDUHD
XQGHUWKHLQYHUVHGHPDQGFXUYHDQGDERYHWKHSUHYDLOLQJSULFHEHWZHHQWKHRULJLQ
DQGWKHHTXLOLEULXPTXDQWLW\,IWZRRUPRUH JRRGVDUHLQYROYHGWKHWRWDOFRQVXPHU
VXUSOXVLVQRWXVXDOO\WKHVXPRIWKHFRQVXPHUVXUSOXVHVIRUWKHLQGLYLGXDOJRRGVDV
WKH\FRXOGEHFDOFXODWHGRQWKHEDVLVRIWKHWZRRUPRUHGHPDQGIXQFWLRQV7KH
UHDVRQLVWKDWWKHGHPDQGFXUYHVDUHLQWHUUHODWHGDQGZKHQWKHSULFHRIRQHJRRGLV
LQFUHDVHGWKHGHPDQGFXUYHVIRUWKHRWKHUJRRGVDQGWKHDVVRFLDWHGFRQVXPHU
VXUSOXVHVFKDQJHV+RZHYHUWRWDOVXUSOXVFDQVWLOOEHGHILQHGXQDPELJXRXVO\LI
WKHGHPDQGIXQFWLRQVDUHLQGHSHQGHQWRILQFRPHDQGWKDWLVZKDWZLOOEHDVVXPHG
KHUH7KHVLWXDWLRQZLWKWZRJRRGVLQGLFDWHGE\VXIIL[HVDQGLVFRQVLGHUHG7KH
TXDQWLWLHVT GHPDQGHGFDQEHGHULYHGIURPWKHFRQVXPHUVXUSOXVE\WDNLQJWKHILUVW
GHULYDWLYHVDQGSXWWLQJDPLQXVVLJQLQIURQW
TL
w&6
L wS L
7ROOVDUHGHQRWHG2DQGKDYHWREHPXOWLSOLHGE\WKHQXPEHURIWULSVLQRUGHUWRILQG
WROOUHYHQXHV
75
75 75
W T W T &RVWGHSHQGVRQFDSDFLW\
.
. FDS . FDS &RQVWDQWUHWXUQVWRVFDOHDUHQRWLPSRVHG&DSDFLW\LVDOORZHGWREHYHFWRUYDOXHG
HJWKHIUHTXHQF\RIWUDLQVDQGWKHQXPEHURIVHDWVSHUWUDLQPD\EHUHOHYDQW
GLPHQVLRQVRIFDSDFLW\EXWLWZLOOXVXDOO\EHDVVXPHGWREHD VFDODUHJURDG
ZLGWK
2QO\FDVHVLQZKLFKDXVHUHTXLOLEULXPLVREWDLQHGZLOOEHFRQVLGHUHG8VHU
HTXLOLEULXPLVIRXQGZKHUHWKHLQYHUVHGHPDQGDVUHSUHVHQWHGE\WKHIXOOSULFHS IRU
ERWKURXWHVLVHTXDOWRWKHVXPRISULYDWHWUDYHOFRVWFZKLFKGHSHQGVRQFDSDFLW\
DQGQXPEHURIWULSVDQGWROO2
SL
FL FDSL T L W L L 7KHILUVWEHVWSUREOHPLVWKHPD[LPLVDWLRQRIVRFLDOVXUSOXVE\FKRRVLQJWKHWZRWROOV
DQGFDSDFLWLHVXQGHUWKHVLGHFRQGLWLRQWKDWDXVHUHTXLOLEULXPVKRXOGEHREWDLQHG
7KH/DJUDQJLDQLV
/
66 O S F W O S F W ¦ ³ T [G[ W T S . FDS O S
f
L
SL
L
L
L
L
L
L
L
L
FL TL S L FDS L W L 7KHILUVWRUGHUFRQGLWLRQVZLWKUHVSHFWWRSSWW DQGFDS DQGFDS DUH
§ wF wT ·
§ wF wT ·
wT
wT
W O ¨¨ ¸¸ O ¨¨ ¸¸ wS
wS
© wT wS ¹
© wT wS ¹
§ wF wT ·
§ wF wT ·
wT
wT
T W W O ¨¨ ¸¸ O ¨¨ ¸¸ wS wS © wT wS ¹
© wT wS ¹
w. L
wFDS L
T W TL
OL L O L
wFL
L wFDS L
,QWKHILUVWEHVWRSWLPXPDOOFRQGLWLRQVDQGDUHVDWLVILHG,QVHFRQGEHVW
VLWXDWLRQVRQHRUPRUHRIWKHIRXUHTXDWLRQVDQGDUHQRWVDWLVILHG(TXDWLRQV
DQGDUHDOZD\VYDOLGVLQFHWKH\DUHWKHFRQVHTXHQFHRIWKHEDVLFUHTXLUHPHQWWKDW
WKHUHVKRXOGEHDXVHUHTXLOLEULXP
(TXDWLRQVDUHWKHUHTXLUHPHQWVIRURSWLPDOWROOLQJ7KH\VWDWHWKDWXQGHU
RSWLPDOWROOLQJWKH/DJUDQJHPXOWLSOLHUVZLOOEHHTXDOWRWKHDVVRFLDWHGQXPEHUVRI
WULSV ,IHTXDWLRQVDUHYDOLGWKH/DJUDQJHPXOWLSOLHUVFDQHDVLO\EHHOLPLQDWHG
,QWUDQVSRUWDWLRQHFRQRPLFVRIWHQDQDOWHUQDWLYHEXWHTXLYDOHQWIRUPXODWLRQRIFRQVXPHU
VXUSOXVLVXVHGEDVHGRQWKHLQYHUVHGHPDQGIXQFWLRQ,QWKDWFDVHSDUWLDOGHULYDWLYHVRIWKH
/DJUDQJHIXQFWLRQZLWKUHVSHFWWRWKHYROXPHVRIWULSVKDYHWREHXVHGDQGWKH/DJUDQJH
PXOWLSOLHUVDUHWKHQHTXDOWR]HURXQGHURSWLPDOWROOLQJ,QWKHSUHVHQWFRQWH[WZLWKSRVVLEO\
LPSHUIHFWVXEVWLWXWDELOLW\EHWZHHQWKHWZRURXWHVPRGHVWKHIRUPXODWLRQEDVHGRQWKH
µRUGLQDU\¶GHPDQGIXQFWLRQLVPRUHFRQYHQLHQW
IURPWKHV\VWHPRIHTXDWLRQV7KLVLPSOLHVDVXEVWDQWLDOVLPSOLILFDWLRQ(TXDWLRQV
FDQWKHQEHZULWWHQDV
§
wF · wT §
wF · wT
¨¨W T ¸¸ ¨¨W T ¸¸ wT ¹ wS ©
wT ¹ wS
©
§
wF · wT §
wF · wT
¨¨W T ¸¸ ¨¨W T ¸¸ wT ¹ wS ©
wT ¹ wS ©
7KLVLPSOLHVWKDWERWKWROOVVKRXOGEHHTXDOWRWKHH[WHUQDOFRQJHVWLRQHIIHFW
(TXDWLRQVEHFRPH
w. wFDS
w. wFDS T
wF
wFDS
wF T wFDS (TXDWLRQVWDWHVWKDWWKHPDUJLQDOFRVWRIFDSDFLW\VKRXOGEHHTXDOWRWKHPDUJLQDO
EHQHILWLQWKHIRUPRIUHGXFHGWUDYHOFRVWV
,WLVVWDQGDUGWRDVVXPHWKDWSULYDWHWUDYHOFRVWVDUHKRPRJHQHRXVRIGHJUHH]HUR
LQFDSDFLW\DQGQXPEHURIXVHUV7KLVLPSOLHV
TL
wFL
wFL
FDS L
wT L
wFDSL
L 8VLQJ(TXDWLRQWKHILUVWRUGHUFRQGLWLRQVZLWKUHVSHFWWRFDSDFLW\FDQEH
UHZULWWHQDV
FDSL
w. L
wFDSL
OL TL
wFL
L wT L
,IWKHUHDUHFRQVWDQWUHWXUQVWRVFDOH w. L wFDS L LVDFRQVWDQWNLDQGWKHOHIWKDQG
VLGHRI(TXDWLRQLVHTXDOWRWKHWRWDOFRVWRIFDSDFLW\RIOLQNL7KHULJKWKDQGVLGH
LVHTXDOWRL WLPHVWKHH[WHUQDOFRQJHVWLRQHIIHFWRIWUDIILFRQOLQNL,QWKHILUVWEHVW
,IWKHPDWUL[RISDUWLDOGHULYDWLYHV wTLwSM KDVDQLQYHUVHWKLVLVWKHXQLTXHVROXWLRQ7KH
PDWUL[KDVQRLQYHUVHLIWRWDOGHPDQGLVIL[HGRULIWKHVKDUHVRIWKHWZRURXWHVPRGHVDUH
IL[HG
HTXLOLEULXPWKH/DJUDQJHPXOWLSOLHUHTXDOVWKHQXPEHURIXVHUVDQGWKHFRVWRI
FDSDFLW\LVWKHUHIRUHHTXDOWRRSWLPDOWROOUHYHQXHV7KLVVHOIILQDQFLQJUHVXOWZDVILUVW
REWDLQHGE\0RKULQJ DQG+DUZLW] IRUDVLQJOHOLQN<DQJDQG0HQJ
UHFHQWO\VKRZHGWKDWLWDOVRKROGV LQDJHQHUDOQHWZRUN+RPRJHQHLW\RIGHJUHHRI
WKHXVHUFRVWIXQFWLRQZLOOQRWDOZD\VEHLPSRVHGWKHIXQFWLRQXVHGIRUSXEOLF
WUDQVSRUWLQWKHWZRPRGHPRGHOGRHVQRWVDWLVI\WKLVUHTXLUHPHQW
7ULSJHQHUDWLRQDQGWULSGLVWULEXWLRQ
$ GLVWLQFWLRQEHWZHHQWULSJHQHUDWLRQDQGWULSGLVWULEXWLRQLVRIWHQPDGHLQ
WUDQVSRUWDWLRQSODQQLQJPRGHOV,WLVFRQYHQLHQWEHFDXVHLWDOORZVIRUDGHFRPSRVLWLRQ
RIWKHWRWDOSODQQLQJSUREOHPLQSDUWVWKDWFDQEHWUHDWHGUHODWLYHO\LQGHSHQGHQWRI
HDFKRWKHU DQGLWLVWKHUHIRUHXVHIXOWRLQWURGXFHLWLQHFRQRPLFPRGHOVDVZHOO ,Q
RUGHUWRLQWURGXFHWKLVGLVWLQFWLRQLQDZD\WKDWLVFRQVLVWHQWZLWKHFRQRPLFWKHRU\
DVVXPHWKDWFRQVXPHU¶VVXUSOXVFDQ EHZULWWHQDVDIXQFWLRQRIDFRPSRVLWHSULFH 3 RI
WKHSULFHVRIWKHWZRWUDQVSRUWVHUYLFHV
&6 S S ZLWK
3
&6 3 3 S S 3 FDQEHLQWHUSUHWHGDVDFRPSRVLWHSULFHIRUWUDQVSRUWVHUYLFHV DQGHTXDWLRQ
VWDWHVWKDWWKHFRQVXPHUVXUSOXVGHULYHGIURPERWKWUDQVSRUWVHUYLFHVFDQEHZULWWHQDV
WKHFRQVXPHU¶VVXUSOXVRIDVLQJOHWUDQVSRUWVHUYLFH
,WFDQWKHQEHHDVLO\YHULILHGXVLQJ(TXDWLRQWKDWIRUWKHVXPRIWKHGHPDQGIRU
ERWKOLQNV 4
4
w&6 w3
T T § w3 w3
¨¨
© wS wS ·
¸¸
¹
ZKHUHDVWKHGLVWULEXWLRQVKDUHVV RIWKHWULSVRYHUWKHWZRPRGHV DUH
VL
TL
4
w3
wS L
§ w3 w3
¨¨
© wS wS ·
¸¸ L ¹
&RQVLGHUDEOHVLPSOLILFDWLRQRIWKHODWWHUWZRHTXDWLRQVLVSRVVLEOHLILWLVDVVXPHGWKDW
WKHSDUWLDOGHULYDWLYHVRI3 DGGXSWR
w3 w3
wS wS $FFRUGLQJWR(TXDWLRQDVPDOOFKDQJHGS LQWKHSULFHVRIWULSVRQWKHWZRURDGV
OHDGVWRDQLGHQWLFDOFKDQJHLQWKHFRPSRVLWHSULFHRIWUDQVSRUWRQERWKURXWHVZKLFK
LVLQWXLWLYH
(TXDWLRQUHIHUVWRWULSJHQHUDWLRQDQG(TXDWLRQWRPRGDOVSOLW RUURXWH
FKRLFH7KHVH HTXDWLRQVVKRZ WKDW ZKHQERWKDVVXPSWLRQVDUHPDGHWULSJHQHUDWLRQ
GRHVQRWGHSHQGRQWKHVSHFLILFDWLRQRI3ZKHUHDVWULSGLVWULEXWLRQGRHVQRWGHSHQG
RQWKHVSHFLILFDWLRQRI &6 6SHFLILFDWLRQZLWK3 VDWLVI\LQJLVWKHUHIRUH
FRQVLVWHQWZLWKWKHXVHRIGLIIHUHQWVXEPRGHOVIRUWKHVHWZRDVSHFWVRIWUDYHOGHPDQG
ZKLFKDUHUHODWLYHO\LQGHSHQGHQWRIHDFKRWKHU
7KHHIIHFWRIDQLQFUHDVHLQWKHSULFHRIRQHURXWHRUPRGHFDQEHGHFRPSRVHG
LQWRDWULSJHQHUDWLRQHIIHFWDQGDWULSGLVWULEXWLRQHIIHFWDVIROORZV
wT L
wS M
§ w 4 V L ·
¨
¸
¨
¸
w
S
M
©
¹
VL
wV
w4
4 L
wS M
wS M
wV
w4
VL V M
4 L
w3
wS M
7KLVHTXDWLRQKROGVIRULM 7KHORJLWPRGHOZKLFKLVDVWDQGDUGVSHFLILFDWLRQIRUPRGHFKRLFHUHVXOWVLIWKH
IROORZLQJ3 LVFKRVHQ
3
§ D H ES D H ES
OQ¨¨
D D ©
·E
¸¸
¹
ZLWKDQG..!7KHVKDUHVFDQEHGHWHUPLQHGXVLQJ(TXDWLRQDV
VL
D L H E SL
D H ES D H ES
.L FDQEHUHZULWWHQDVH[S.LZLWK.L OQ.LDQGLWLVSRVVLEOHWKDW.L GHSHQGVRQ
PRGHRUURXWHFKDUDFWHULVWLFVDVLVXVXDOLQHPSLULFDOORJLWPRGHOV XVHGWRVWXG\WULS
GLVWULEXWLRQ
,QWKHOLPLWLQJFDVHLQZKLFK OLQNFKRLFHLVLQGHSHQGHQWRIWKHSULFHV,WFDQ
EHYHULILHGIURP(TXDWLRQWKDWLIS S S3 LVDOVRHTXDOWRSWKDW3 LV
LQFUHDVLQJLQERWKSULFHVDQGWKDW3 DOZD\VWDNHVRQDYDOXHLQEHWZHHQWKDWRIWKH
WZRSULFHV7KLVPRWLYDWHVWKHLQWHUSUHWDWLRQRI3 DVDFRPSRVLWHSULFHIRUWUDQVSRUWRQ
WKHWZRPRGHVRUURXWHV 3 LV KRPRJHQHRXVRIGHJUHHLQWKH SULFHV RIDOOJRRGVLIS
DQGS DUHLQWHUSUHWHGDVUHODWLYHSULFHV
7KHWZROLQNVRIWKHQHWZRUNDUHSHUIHFWVXEVWLWXWHVLIWKHXVHUVDOZD\VFKRRVHWKH
RQHZLWKWKHORZHVWFRVW,QWKLVVLWXDWLRQWKHVKDUHRIOLQNVKRXOGEHHTXDOWR]HURLI
S!SDQGHTXDOWRLQWKHRSSRVLWHFDVHZKHUHDVWKHVKDUHVDUHLQGHWHUPLQDWHZKHQ
ERWKSULFHVDUHHTXDO,IWKHORJLWVSHFLILFDWLRQIRUWKHVKDUHVLVXVHGWKLVVLWXDWLRQFDQ
EHDSSUR[LPDWHGDUELWUDULO\FORVHO\E\FKRRVLQJWDNLQJWKHOLPLWIRU E o f RI
7KLVLVHDVLO\YHULILHGE\UHZULWLQJ(TXDWLRQIRUL DV
(TXDWLRQLVQRWWKHRQO\SRVVLEOHVSHFLILFDWLRQRI 3 WKDWVDWLVILHV$Q\IXQFWLRQ
3 OQ>JH[SSH[SS@ZLWKJKRPRJHQHRXVRIGHJUHHLQLWVWZRDUJXPHQWVGRHV
7KHIXQFWLRQ3 LVHTXDOWRWKHVXPRIDFRQVWDQWOQ..DQGWKHORJVXPPHDVXUHWKDW
FDQEHXVHGIRUZHOIDUHHFRQRPLFDQDO\VHV6PDOODQG5RVHQ
V
D H E S S D H E S S D DQGWDNLQJWKHOLPLWIRU E o f :KHQ HTXDOVWKHUHLVQRSULFHVHQVLWLYLW\DWDOO
7KHPRGHOGHYHORSHGLQWKHSUHVHQWVXEVHFWLRQVHHPVDWWUDFWLYHDVWKHEDVLVRID
VLPXODWLRQPRGHO,QSDUWLFXODULWRIIHUVWKHSRVVLELOLW\WRFRPSDUHKRZVHFRQGEHVW
SUREOHPVFKDQJHLIWKHWZROLQNVEHFRPHFORVHUVXEVWLWXWHV
$VLPXODWLRQPRGHOZLWKWZRURXWHV
7KHSUHYLRXVWZRVXEVHFWLRQVSURYLGHWKHPDLQLQJUHGLHQWVIRUDVLPXODWLRQPRGHOIRU
WKHWZRURXWHDQGWZRPRGHVLWXDWLRQVWKDWZLOOEHXVHGLQWKHUHPDLQGHURIWKLV
VHFWLRQ7KHWZRURXWHFDVHLVGLVFXVVHGKHUHDQGWKHPRGHOLVFRPSOHWHGE\
LQWURGXFLQJVSHFLILFIRUPVIRUWKHWRWDOGHPDQGDQGWUDYHOWLPHIXQFWLRQV
$TXDGUDWLFVSHFLILFDWLRQIRUFRQVXPHU¶VVXUSOXVLVDGRSWHGDVDIXQFWLRQRIWKH
WUDQVSRUWSULFHLQGH[3
&6 D 3 E 3 ZLWKD! DQGE7RWDOGHPDQGIRUWULSVLVWKHUHIRUHDOLQHDUIXQFWLRQRIWKHSULFHRI
WUDQVSRUW
4
DE3
7KHORJLWIRUPXODWLRQLVXVHGIRU38VHUFRVWLVJLYHQE\ WKHFRPPRQO\XVHG
%XUHDXRI3XEOLF5RDGVIRUPXODWLRQ
FL
ª
§ T
YRW IIWW « FL ¨¨ L
«¬
© FDSL
·
¸¸
¹
FLH
º
»
»¼
ZKHUHYRWGHQRWHVWKHYDOXHRIWLPHDQGIIWW IUHHIORZWUDYHOWLPH7KLVIXQFWLRQLV
KRPRJHQHRXVRIGHJUHH]HURLQWKHQXPEHURIWULSVDQGFDSDFLW\
6HHHJ6PDOOS
)RUWKHFRVWRIFDSDFLW\LQWKHWZRURXWHSUREOHPWKHIROORZLQJVSHFLILFDWLRQLV
XVHG
. L FDSL N L FDS L
7KLVFRVWIXQFWLRQKDVFRQVWDQWUHWXUQVWRVFDOH
7DEOH
3DUDPHWHUYDOXHVIRUWKHWZRURXWHSUREOHP
3DUDPHWHU
'HPDQG
D
E
E
D
8VHUFRVW
YRW
IIWW
F
FH
&RVWRIFDSDFLW\
N
5RXWHPRGH
5RXWHPRGH
7KHSDUDPHWHUYDOXHVXVHGIRUWKHWZRURXWHSUREOHPVHH7DEOHUHIHUWRD
VLWXDWLRQLQZKLFKWZRFLWLHVDUHFRQQHFWHGE\URDGVWKDWGLIIHULQFDSDFLW\DQGIUHH
IORZWUDYHOWLPH7KHURXWHZLWKWKHKLJKHVWIUHHIORZWUDYHOWLPHKDVWKHORZHVWFRVW
RIFDSDFLW\ SHUURDGNLORPHWUH3UHIHUHQFHVZLWKUHVSHFWWRWKHXVHRIWKHWZRURXWHV
DUHV\PPHWULFZKLFKPHDQVWKDWFRQVXPHUVGRQRWV\VWHPDWLFDOO\SUHIHURQHURDG
RYHUWKHRWKHU
/LQNKDVDIUHHIORZWUDYHOWLPHRIKDOIDQKRXUDQGOLQNKDVDIUHHIORZWUDYHO
WLPHRIWKUHHTXDUWHUVRIDQKRXUDQGWKHYDOXHRIWLPHLVVHWDWWKH'XWFKDYHUDJHRI
¼7KH%35SDUDPHWHUVF DQGFH KDYHWKHLUFRQYHQWLRQDOYDOXHV
7KHXQLWSULFHRIFDSDFLW\ZDVGHWHUPLQHGDVIROORZV$YDOXHRIFDS HTXDOWR
IRUWKH%35FRVWIXQFWLRQLPSOLHVDGRXEOLQJRIWUDYHOWLPHVDWDIORZRIDURXQG
YHKLFOHVSHUKRXU7KLVLVURXJKO\WKHIORZDWZKLFKHPSLULFDOO\WUDYHOWLPHV
GRXEOHIRUDVLQJOHKLJKZD\ODQH7KHKRXUO\XQLWSULFHRIFDSDFLW\RI¼IRUOLQN
WKHIDVWFRQQHFWLRQZDVGHWHUPLQHGE\GLYLGLQJWKHHVWLPDWHGDYHUDJHDQQXDOFDSLWDO
FRVWRIRQHKLJKZD\ODQHNLORPHWUHLQ7KH1HWKHUODQGV¼PLOOLRQE\
ZRUNLQJGD\VPXOWLSOLHGE\SHDNKRXUVSHUZRUNLQJGD\DVVXPLQJWZRSHDNVDQG
WKHQE\WKHQXPEHURIXQLWVRIFDSDFLW\RIDVWDQGDUGKLJKZD\ODQHDQG
ILQDOO\E\PXOWLSO\LQJWKDWUHVXOWE\WKHQXPEHURINLORPHWUHVWKDWFDQEHWUDYHOOHG
DWIUHHIORZVSHHGLQKDOIDQKRXU7KHSULFHRIFDSDFLW\DWOLQNZDVVHWDWWKHVDPH
YDOXHLQRUGHUWRIDFLOLWDWHFRPSDUDELOLW\$OWKRXJKWKHURDGLVORQJHUZKLFKPLJKW
FDOOIRUDKLJKHUNFRQVWUXFWLRQPD\EHFKHDSHUZKLFKPD\FDOOIRUDORZHUN
%DVHFDVH
7KHEDVHFDVHUHIHUVWRDXVHUHTXLOLEULXPZLWKRXWWROOVDQGZLWKFDSDFLWLHVDVVKRZQ
LQ7DEOHZLWKDKLJKHUFDSDFLW\IRUOLQN7KHPDLQFKDUDFWHULVWLFVRIWKHEDVHFDVH
DUHJLYHQLQ7DEOH
7DEOH
7KHEDVHFDVHRIWKHWZRURXWHSUREOHP
9DULDEOH
FDS
T
4
F S
3
(ODVWLFLW\RIT ZLWKUHVSHFW
WRS
(ODVWLFLW\RIT ZLWKUHVSHFW
WRS
(ODVWLFLW\RI4 ZLWK
UHVSHFWWR3
&6
.
66
5RXWHPRGH
5RXWHPRGH
1RWHFDS T DQG4 DUHPHDVXUHGDVQXPEHUVRIWULSVSHUKRXUF S 3 &6 . DQG66 LQ
HXURV6RXUFHRZQFDOFXODWLRQV
,WLVLQWHUHVWLQJWRVHHZKDWKDSSHQVZKHQWKHWZRURXWHVEHFRPHFORVHUVXEVWLWXWHVRU
LQRWKHUZRUGVZKHQWKHDEVROXWHYDOXHRI LVLQFUHDVHG)LJXUHVKRZVWKH
HTXLOLEULXPYDOXHVRISULFHVDQGGHPDQGIRUWUDYHORQERWKOLQNVIRUVXFFHVVLYHO\
KLJKHUYDOXHVRI7KHSULFHVXVHUFRVWVIRUWKHWZROLQNVLQ)LJXUHDSSURDFKHDFK
RWKHUFORVHO\6LQFHFDSDFLW\LVIL[HGWKLVLPSOLHVWKDWWKHOLQNWKDWKDGWKHORZHUXVHU
FRVWLQWKHEDVHFDVHLVXVHGPRUHLQWHQVLYHO\,WVORZHUSULFHDWWUDFWVPRUHXVHUVZKHQ
SULFHVHQVLWLYLW\LQFUHDVHVDQGIRUWKLVUHDVRQXVHUFRVWLQFUHDVHVDQGWKHSULFH
GLIIHUHQFHGLVDSSHDUV&RQYHUJHQFHWRWKHFDVHRISHUIHFWVXEVWLWXWHVIRU:’KDV
EHHQFRQILUPHGE\PHDQVRIDVHSDUDWHPRGHOWKDWXVHVWKH:DUGURSHTXLOLEULXP
FRQGLWLRQV
7KLVFKHFNKDVDOVREHHQFDUULHGRXWLQRWKHU XQLPRGDOFDVHVFRQVLGHUHGEHORZ
)LJXUH
(IIHFWRILQFUHDVLQJVXEVWLWXWDELOLW\RQXVHUFRVWXSSHUSDQHODQG
URXWHFKRLFHORZHUSDQHOLQWKHEDVHFDVHRIWKHWZRURXWHPRGHO
SULFH
S
S
EHWDDEVROXWHYDOXH
QXPEHURIWULSV
T
T
EHWDDEVROXWHYDOXH
)LUVWEHVW
)LQGLQJWKHILUVWEHVWVROXWLRQLVIDFLOLWDWHGE\WKHDVVXPSWLRQVRIFRQVWDQWUHWXUQVWR
VFDOHRIWKHFDSDFLW\FRVWIXQFWLRQDQGKRPRJHQHLW\RIGHJUHHRIWKHWUDYHOFRVW
IXQFWLRQ(TXDWLRQVLPSO\
NL
§ T
YRW IIWW FLFLH ¨¨ L
© FDS L
·
¸¸
¹
FLH ·
¸¸
¹
FLH
DQGWKLVGHWHUPLQHVWKHUDWLREHWZHHQQXPEHURIXVHUVDQGFDSDFLW\7KHRSWLPDOWROO
LVDIXQFWLRQRIWKLVUDWLRRQO\
WL
§ T
YRW IIWW FL FLH ¨¨ L
© FDS L
DVLVWKHXVHUFRVWRIWUDYHOJLYHQLQ(TXDWLRQ7RJHWKHUWKH\GHWHUPLQHWKHYDOXHV
RIWKHSULFHVSL E\(TXDWLRQ7KHSULFHVSL LQWXUQDOORZWRWDOGHPDQGDQGURXWH
FKRLFHWREHFRPSXWHG&RQVXPHUVXUSOXVWROOUHYHQXHVFDSDFLWLHVDQGDVVRFLDWHG
FRVWVDUHSUHVHQWHGLQ7DEOH
7DEOH
)LUVWEHVWVLWXDWLRQRIWKHWZRURXWH SUREOHP
9DULDEOH
FDS
T
4
F
W
S
3
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RI4 ZUW3
&6
75
.
66
5RXWHPRGH
5RXWHPRGH
6RXUFHRZQFDOFXODWLRQV
1RWHFDS T DQG4 DUHPHDVXUHGDVQXPEHUVRIWULSVSHUKRXUF 2 S 3&6 . DQG66
LQHXURV
)RUWKHILUVWEHVWFDVHLWLVDOVRLQWHUHVWLQJWRVHHZKDWKDSSHQVLIWKH SULFH
VHQVLWLYLW\RIURXWHFKRLFHLQFUHDVHV7KHILUVWWKLQJWRREVHUYHLVWKDWLQWKLVFDVHWKH
SULFHVXVHUFRVWVRQWKHWZRURXWHVFDQQRWEHFRPHHTXDO$VH[SODLQHGDERYHWKHVH
SULFHVDUHGHWHUPLQHGE\WKHUDWLREHWZHHQQXPEHURIXVHUVDQGFDSDFLW\DQGWKLV
UDWLRLVIL[HGE\WKHILUVWRUGHUFRQGLWLRQIRUFDSDFLW\FKRLFH7KHFRQVWDQWSULFH
GLIIHUHQFHEHWZHHQWKHWZRURXWHVZLOOKDYHDODUJHUHIIHFWRQGHPDQGZKHQSULFH
VHQVLWLYLW\LQFUHDVHV6LQFHFDSDFLW\LVDGMXVWHGWRWKHQXPEHURIXVHUVEHFDXVHRIWKH
FRQVWDQWUDWLREHWZHHQWKHWZRWKLVLPSOLHVWKDWWKHPRVWH[SHQVLYHURDGZLOO
JUDGXDOO\GLVDSSHDUZLWKHQRXJKVXEVWLWXWDELOLW\)LJXUHVKRZVWKDWGHPDQGVKLIWV
WRZDUGVWKHILUVWURXWHZKHQWKHDEVROXWHYDOXHRI LQFUHDVHV)RUYDOXHVODUJHUWKDQ
GHPDQGIRUWULSVRQURXWHLVYLUWXDOO\HTXDOWR
)LJXUH
(IIHFWRILQFUHDVLQJVXEVWLWXWDELOLW\RQURXWHFKRLFHLQWKHILUVWEHVW
VLWXDWLRQRIWKHWZRURXWHPRGHO
QXP EHURIWULSV
T
T
EHWDDEVROXWHYDOXH
6HFRQGEHVWWKHWZRURXWHSUREOHP
,QRUGHUWRVROYHVHFRQGEHVWSUREOHPVDQLWHUDWLYHSURFHGXUHZDVXVHG$VDVWDUWLQJ
SRLQWDXVHUHTXLOLEULXPIRUJLYHQYDOXHVRIWKHSROLF\YDULDEOHVZDVWDNHQ7KHQD
OLQHDULVHGYHUVLRQRIWKHILUVWRUGHUFRQGLWLRQVZDVVROYHGIRUWKHVHFRQGEHVW
RSWLPXP,QWKHIROORZLQJVWHSDZHLJKWHGDYHUDJHRIWKHRULJLQDOYDOXHVDQGWKRVH
VXJJHVWHGE\WKHOLQHDULVHGILUVWRUGHUFRQGLWLRQVIRUWKHSROLF\YDULDEOHVWKDWFDQEH
IUHHO\FKRVHQZDVXVHG7KLVSURFHGXUHZDVUHSHDWHGXQWLOFRQYHUJHQFHRFFXUUHG
$ VHTXHQFHRIVHFRQGEHVWVLWXDWLRQV ZKHUHLQHDFKVWHSDQDGGLWLRQDOSROLF\
LQVWUXPHQWLVDGGHGZLOOEHGLVFXVVHG,QWKHEDVHFDVHQRLQVWUXPHQWVDUHXVHGDWROO
RQURDGWZRLVWKHQLQWURGXFHGDQGWKHFDSDFLWLHVRIURDGVDQGDUHVXEVHTXHQOW\
DGGHG,IILQDOO\DWROORQURDGFRPHVDYDLODEOHDVDSROLF\LQVWUXPHQWWKHILUVWEHVW
VLWXDWLRQUHVXOWV
2QHWROOIL[HGFDSDFLWLHV
)LUVWWKHVLWXDWLRQLQZKLFKWKHWROORQURXWHLVHTXDOWR]HURFDSDFLWLHVRQERWK
URXWHVDUHIL[HGDQGWKHSROLF\PDNHU¶VVLQJOHLQVWUXPHQWLVWKHWROORQURXWH LV
DQDO\VHG7KLVLPSOLHVWKDWRIWKHILUVWRUGHUFRQGLWLRQVRQO\DQGIRU
L DUHVDWLVILHG,QWKDWVLWXDWLRQRQO\DPRGHVWZHOIDUHJDLQLQFRPSDULVRQWRWKH
EDVHFDVHFDQEHDFKLHYHGVRFLDOVXUSOXVULVHVWR¼ 7KHLQGH[RIUHODWLYH
ZHOIDUHLPSURYHPHQW& LVHTXDOWR7KHRSWLPDOWROOLVHTXDOWR DQG
VHYHUHFRQJHVWLRQUHPDLQVRQERWKURXWHV:KHQSULFHVHQVLWLYLW\LQFUHDVHVXVHUFRVWV
RQERWKURXWHVFRQYHUJHZKHUHDVWKHQXPEHUVRIWULSVGLYHUJHVRPHZKDWMXVWDVLQ
WKHEDVHFDVH
7KHVHUHVXOWVDUHVXUSULVLQJO\FORVHWRWKHUHVXOWVIURPWKHPRGHOZLWKSHUIHFW
VXEVWLWXWDELOLW\W DQGZ ZLWK66 ¼ LQWKHEDVHFDVH
66 ¼LQWKHVHFRQGEHVWFDVHDQG66 ¼LQWKHILUVWEHVWFDVHZKHUH
URDGLVLQIDFWHOLPLQDWHG7KLVVXJJHVWVWKDWDOWKRXJKWKHDVVXPSWLRQRILPSHUIHFW
VXEVWLWXWDELOLW\EHWZHHQURXWHVPDNHVWKHPRGHOULFKHUDQGDOORZVIRUPRGHFKRLFHLW
7KHLQGH[RIUHODWLYHZHOIDUHLPSURYHPHQWLVWKHLQFUHDVHLQVRFLDOVXUSOXVFRPSDUHGWRWKH
EDVHHTXLOLEULXPDVDIUDFWLRQRIWKHLQFUHDVHLQWKHILUVWEHVWRSWLPXP 1RWHWKDWWKHYDOXHRI
WKLVLQGH[LVGHSHQGHQWRQWKHSDUDPHWHUYDOXHVFKRVHQLQWKHEDVHFDVH )RULQVWDQFHWKH
GRHVQRWVWURQJO\DIIHFWWKHSROLF\FRQFOXVLRQVRQWKHGHVLJQDQGGHVLUDELOLW\RI
VHFRQGEHVWSULFLQJ
2QHWROORQHFDSDFLW\
7KHVHFRQGSKDVHFRQFHQWUDWHGRQWKHDQDO\VLVRIWKHVLWXDWLRQLQZKLFKWKHWROODQG
FDSDFLW\RIURXWHFDQEHGHWHUPLQHGE\WKHSROLF\PDNHU)LUVWRUGHUFRQGLWLRQV
DUHERWKVDWLVILHGEXWFRQGLWLRQVDQGDUHRQO\YDOLGIRUL ,QWKHVHFRQGEHVW
RSWLPXPWKHFDSDFLW\RIURXWHLVH[WHQGHGVXEVWDQWLDOO\WRWULSVSHUKRXU
ZKHUHDVWKHWROOLVQHJDWLYH¼±&RQJHVWLRQRQURXWHUHPDLQVVLJQLILFDQWZLWK
WKHQXPEHURIWULSVDW7KHZHOIDUHJDLQLVVXEVWDQWLDOVRFLDOVXUSOXVLVQRZ
HTXDOWR¼ZLWK& :KHQSULFHVHQVLWLYLW\LQFUHDVHVWKHRSWLPDOWROO
LQFUHDVHV,WEHFRPHVSRVLWLYHIRUYDOXHVRI KLJKHUWKDQ$WWKDWOHYHOZ LVVWLOO
SRVLWLYHDQGHTXDOWRZKLFKUHIOHFWVWKHZHOIDUHJDLQIURPRSWLPL]LQJURXWH¶V
FDSDFLW\HYHQZKHQWKHVHFRQGEHVWRSWLPDOWROOLV]HUR:KHQE LQFUHDVHVWKH
RSWLPDOFDSDFLW\RIURXWHLQFUHDVHVEXWFRQJHVWLRQRQURXWHDOVRJURZV,QWKH
OLPLWLQJFDVHRISHUIHFWVXEVWLWXWDELOLW\WKHVHFRQGEHVWWROOLVHTXDOWR¼WKH
FDSDFLW\WRWULSVSHUKRXUDQGZ LQFUHDVHVWR7KHVHUHVXOWVVXJJHVWWKDW
ZKHQFDSDFLW\FKRLFHLVDSROLF\YDULDEOHWKHLPSDFWRIDOORZLQJLPSHUIHFW
VXEVWLWXWDELOLW\XSRQSROLF\GHVLJQDQGHYDOXDWLRQLQFUHDVHV
2QHWROOWZRFDSDFLWLHV
)LQDOO\WKHFDVHLQZKLFKURXWHFDQQRWEHWROOHGZKHUHDVWKHWROORQURXWHDQGWKH
FDSDFLWLHVRQERWKURXWHVFDQEHIUHHO\FKRVHQZDVFRQVLGHUHG,QWKLVVLWXDWLRQDOO
YDOXHRIWKLVLQGH[IRUWKHSUHVHQWFDVH³RQHWROOIL[HGFDSDFLWLHV´ZRXOGKDYHEHHQODUJHULI
WKHFDSDFLWLHVLQWKHEDVHFDVHKDGEHHQFORVHUWRWKHLUILUVWEHVWRSWLPDOOHYHOV
ILUVWRUGHUFRQGLWLRQV H[FHSWIRUL DUHVDWLVILHG7KHRSWLPDOYDOXHVRIWKHVH
SROLF\YDULDEOHVDUHSUHVHQWHGLQ7DEOH
7DEOH
6HFRQGEHVWWKHWZRURXWHSUREOHPZLWKRQHWROODQGWZRFDSDFLWLHV
9DULDEOH
FDS
T
4
F
W
S
3
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RI4 ZUW3
&6
75
.
66
5RXWHPRGH
5RXWHPRGH
6RXUFHRZQFDOFXODWLRQV
1RWHFDS T DQG4 DUHPHDVXUHGDVQXPEHUVRIWULSVSHUKRXUF2S3&6. DQG66 LQ
HXURV
,QFRPSDULVRQWRWKHILUVWEHVWVLWXDWLRQWKHFDSDFLW\RIURXWHLVLQFUHDVHGZKHUHDV
WKHWROORQURXWHLVGHFUHDVHG7KHSULFHRIWUDQVSRUWLVORZHUZKLFKLQFUHDVHV
FRQVXPHUVXUSOXV7KHFRVWVRIFDSDFLW\DUHQRORQJHUFRYHUHGE\WROOUHYHQXHVDQG
WKHGHILFLWGHFUHDVHVVRFLDOVXUSOXV7KHVHWZRFKDQJHVUHVXOWLQDVRFLDOVXUSOXVWKDWLV
DOPRVWDVKLJKDVLQWKHILUVWEHVWVLWXDWLRQ& :KHQORRNLQJDWWKHVHQVLWLYLW\RIWKHVHUHVXOWVWRFKDQJHVLQWKHSULFHVHQVLWLYLW\
RIPRGHFKRLFH LWZDVIRXQGGXULQJWKHVLPXODWLRQVWKDWWKHQXPEHURIWULSVRQURXWH
LQWKHVHFRQGEHVWRSWLPXPVXGGHQO\IDOOVWRDYDOXHYHU\FORVHWR]HURZKHQWKH
DEVROXWHYDOXHRIE LQFUHDVHVIURPWR )XUWKHULQYHVWLJDWLRQUHYHDOHGWKDW
IRUWKHLQWHUYDO>@WKHUHH[LVWWZRHTXLOLEULDDRQHLQZKLFKERWKURXWHV
KDYHDVXEVWDQWLDOQXPEHURIXVHUVDQGDQRWKHUELQZKLFKURDGLVYLUWXDOO\XQXVHG
)LJXUH
(IIHFWRILQFUHDVLQJVXEVWLWXWDELOLW\RQXVHUFRVW XSSHUSDQHODQG
URXWHFKRLFHORZHUSDQHOLQWKHWZRURXWHSUREOHPZLWKRQHWROODQGWZRFDSDFLWLHV
SD
SULFH
SD
SE
SE
EHWDDEVROXWHYDOXH
QXPEHURIWULSV
TD
TD
TE
TE
EHWDDEVROXWHYDOXH
7KHQXPEHURIWULSVRQURXWHIDOOVIURPWRWKHQXPEHURIWULSVRQURXWHLQFUHDVHV
IURPWR
)RUYDOXHVRIE VPDOOHUWKDQLQDEVROXWHYDOXHWKHHTXLOLEULXPLQZKLFK
URXWHLVYLUWXDOO\XQXVHGGLVDSSHDUV)LJXUHVKRZVWKHHIIHFWVRQXVHUFRVWVDQG
TXDQWLWLHV7KHSULFHVDQGTXDQWLWLHVHQGLQJZLWKµD¶UHIHUWRWKHHTXLOLEULXPLQZKLFK
ERWKURXWHVDUHXVHGDQGWKRVHHQGLQJZLWKµE¶WRWKHHTXLOLEULXPLQZKLFKRQO\URXWH
LVXVHG
7KHSULFHIRUWULSVRQURXWHLQHTXLOLEULXPµE¶LVRQO\LQGLFDWHGIRUYDOXHVRI
ORZHUWKDQVLQFHLWLQFUHDVHVUDSLGO\WRYDOXHVKLJKHUWKDQ8VHRIURXWHLV
DOZD\VFORVHWR2QO\IRUYDOXHVRI VPDOOHUWKDQLVWKHQXPEHURIWULSVRQ
URDGLQHTXLOLEULXPµE¶ODUJHUWKDQ7KHHTXLOLEULXPXVHUFRVWVRQWKHWZR
URXWHVLQHTXLOLEULXPµD¶GRQRWFRQYHUJHDQGLQIDFWWKHGLIIHUHQFHEHWZHHQWKHP
LQFUHDVHV$VDUHVXOWWKHQXPEHUVRIXVHUVRQWKHWZRURXWHVGLYHUJH(YHQWXDOO\LW
EHFRPHVLQHIILFLHQWWRPDLQWDLQURXWH:LWKSHUIHFWVXEVWLWXWDELOLW\WKHZ IRUWKLV
SROLF\LVHTXDOWR7KLVLPSOLHVWKDWHOLPLQDWLQJDURDGLQWKHILUVWEHVWRSWLPXPLV
QRWKDUPIXOWRHIILFLHQF\LILWVWROOLVUHVWULFWHGWREHHTXDOWR]HUR:KHQWKHWZR
HTXLOLEULDH[LVWVLPXOWDQHRXVO\WKHPRUHEDODQFHHTXLOLEULXPµD¶DOZD\VKDVWKH
KLJKHVWVRFLDOVXUSOXV:KHQ LQFUHDVHVWKHGLIIHUHQFHEHWZHHQVRFLDOVXUSOXVHV
GHFUHDVHV
7KHILQGLQJVMXVWUHSRUWHGVKRZWKDWWKHUHPD\EHWZRVROXWLRQVRIWKHUHOHYDQW
VXEVHW RIILUVWRUGHUFRQGLWLRQVLQWKHWZRPRGHSUREOHPZLWKRQHWROODQGWZR
FDSDFLWLHVDVSROLF\LQVWUXPHQWV7KHWZRVROXWLRQVFRUUHVSRQGZLWKGLIIHUHQWYDOXHV
RIWKHSROLF\LQVWUXPHQWV:KHQWKHUHDUHWZRHTXLOLEULDERWKDUHVWDEOHLQ WZRVHQVHV
)LUVWWKHWZRVROXWLRQVFRUUHVSRQGZLWKGLIIHUHQWYDOXHVRIWKHSROLF\LQVWUXPHQWVEXW
IRUERWKFRQILJXUDWLRQVWKHXVHUHTXLOLEULXPLQWKHQHWZRUNLVWDEOH6HFRQGERWK
VHFRQGEHVWHTXLOLEULDUHSUHVHQWORFDOZHOIDUHPD[LPDQRWPLQLPD
:KDWKDVEHHQ IRXQGLVWKDWDSROLF\PDNHUZKRVWDUWVIURPDQDUELWUDU\XVH
HTXLOLEULXPDQGDGMXVWVWKHYDOXHVRIWKHLQVWUXPHQWVVRDVWRUHDFKDVHFRQGEHVW
RSWLPXPPD\HQGXSLQWZRGLIIHUHQWVLWXDWLRQVLIWKHSDUDPHWHU OLHVLQDSDUWLFXODU
LQWHUYDO7KHLQLWLDOYDOXHVRIWKHSROLF\LQVWUXPHQWV GHWHUPLQHLQZKLFKHTXLOLEULXP
KHHQGVXS$OWKRXJKQRH[WHQVLYHPDWKHPDWLFDORUQXPHULFDODQDO\VLV ZDV
XQGHUWDNHQSUHYLRXVH[SHULHQFHLQGLFDWHVWKDWZKHQWKHUHH[LVWWZRHTXLOLEULDERWK
FDQEHUHDFKHGZLWK VXEVWDQWLDOO\GLIIHUHQWLQLWLDOYDOXHVRIWKHSROLF\LQVWUXPHQWV
$6,08/$7,2102'(/)257+(7:202'(352%/(0
6HWXSRIWKHWZRPRGHSUREOHP
7KHUHDUHWZRPDLQGLIIHUHQFHVEHWZHHQWKHWZRURXWHDQGWKHWZRPRGHSUREOHPV
7KHILUVWLVWKDWVWXGLHVRIWKHWZRURXWHSUREOHPXVXDOO\DVVXPHSHUIHFW
VXEVWLWXWDELOLW\7KLVZDVDQLPSRUWDQWPRWLYDWLRQ IRUFRQVLGHULQJWKHDOWHUQDWLYHFDVH
RILPSHUIHFWVXEVWLWXWHVLQWKHSUHYLRXVVHFWLRQ7KHVHFRQGGLIIHUHQFHLQYROYHVWKH
WUDQVSRUWWHFKQRORJ\,QWKHWZRURXWHSUREOHP FRQVWDQWUHWXUQVWRVFDOHLQFDSDFLW\
FRVWDQGKRPRJHQHLW\RIGHJUHH]HURLQFDSDFLW\DQGGHPDQGDUHXVXDOO\DVVXPHG
ZKHUHDVSXEOLFWUDQVSRUWLVRIWHQSURGXFHGXQGHULQFUHDVLQJUHWXUQVWRVFDOH DQGZLWK
DWHFKQRORJ\WKDWLVQRWKRPRJHQHRXV,QWKHSUHVHQWVHFWLRQDVLPXODWLRQPRGHOIRU
7KLVLVFRQILUPHGIRULQVWDQFHLI VRFLDOVXUSOXV LVPD[LPLVHG E\FKRRVLQJFDSDFLW\DQGWROO
IRUURDGZLWKYDULRXVH[RJHQRXVO\GHWHUPLQHGYDOXHVRI WKHFDSDFLW\RIURDG)RUE WKHUHH[LVWVDORFDOLQWHULRUPD[LPXPZKHQWKHFDSDFLW\RIURDGLVFKRVHQDWWKHOHYHORI
HTXLOLEULXPDDQGDQRWKHURQHDWWKHERXQGDU\ZKHQWKHFDSDFLW\RIURDGHTXDOV]HUR
:KHQ E LVFORVHWREXWODUJHUWKDQ WKHVHFRQGPD[LPXPDOVREHFRPHVDQLQWHULRU
PD[LPXP
WKHWZRPRGHSUREOHPWKDWLQFRUSRUDWHVDGLIIHUHQWWUDQVSRUWDWLRQWHFKQRORJ\IRUWKH
VHFRQGPRGH LVGHYHORSHG,W LVRWKHUZLVHFRPSDUDEOHWRWKHWZRURXWHSUREOHPRIWKH
SUHYLRXVVHFWLRQ
)RUWKLV WZRPRGHPRGHOGHPDQGXVHUFRVWRIURDGVPRGHDQGFRVWRIURDG
FDSDFLW\DUHDOOVSHFLILHGDVLQWKHWZRURXWHPRGHO)RUUDLOPRGH WZRDVSHFWVRI
FDSDFLW\DUHGLVWLQJXLVKHGIUHTXHQF\DQGQXPEHURIVHDWVSHUWUDLQ$SHDNSHULRGRI
RQHKRXULVFRQVLGHUHGGXULQJZKLFKWKHDUULYDOIORZRIFRPPXWHUVDWWKHWUDLQ
SODWIRUPLVDVVXPHGWREHFRQVWDQW7KHDYHUDJHZDLWLQJWLPHLVWKHUHIRUHHTXDOWR
1ZKHUH1 GHQRWHVWKHIUHTXHQF\RIWUDLQGHSDUWXUHV3DVVHQJHUVGLVOLNHZDLWLQJ
DQGWKHFRVWRIZDLWLQJWLPHLVHTXDOWRZYRWZKHUHYRW GHQRWHVWKHYDOXHRIWLPH
DQGZ JLYHVWKHH[WUDZHLJKWJLYHQWRWKHYDOXHRIZDLWLQJWLPH0RKULQJ
7KHGLVFRPIRUWDVVRFLDWHGZLWKWUDYHOOLQJE\WUDLQLVDVVXPHGWREHD
SURSRUWLRQDOWRWUDYHOWLPHWWDQGELQFUHDVLQJLQWKHUDWLREHWZHHQWKHQXPEHURI
SDVVHQJHUVDQGWKHQXPEHURIVHDWV7KHWRWDOQXPEHURISDVVHQJHUVLVT DQGWKHWRWDO
QXPEHURIVHDWVDYDLODEOHLV1VZKHUHV GHQRWHVWKHQXPEHURIVHDWVSHUWUDLQ7KH
FRVWIXQFWLRQIRUWUDYHOOLQJE\WUDLQLVGHILQHGDV
F
F H
ª
§ T · º
YRW « Z 1 F WW ¨
»
© 1V ¹̧ »¼
«¬
7KLVIXQFWLRQLVKRPRJHQHRXVRIGHJUHHLQWKHQXPEHURIWULSVDQGWKHQXPEHURI
VHDWV,WLVQRW KRPRJHQHRXVRIGHJUHHLQWKHQXPEHURIWULSVDQGWULSIUHTXHQF\
7KHFRVWRISURYLGLQJUDLOWUDQVSRUWLVGHWHUPLQHGE\WKHQXPEHURIWUDLQ
GHSDUWXUHVDQGE\WKHSURGXFWRIWKHQXPEHURIGHSDUWXUHVDQGWKHQXPEHURIVHDWVSHU
WUDLQ
7KHFDVHLQZKLFKWKHWZRFDSDFLWLHVDUHWKHRQO\SROLF\LQVWUXPHQWVERWKWROOVDUHHTXDOWR
]HURKDVDOVREHHQFRQVLGHUHGDQGIRXQGQRLQGLFDWLRQVRIPXOWLSOHSROLF\HTXLO LEULD :KHQ
JHWVODUJHLQDEVROXWHYDOXHURXWHJUDGXDOO\GLVDSSHDUV
.
N 1 N 1V
7KHILUVWRUGHUFRQGLWLRQVIRUWKHRSWLPXPDUHDQDORJRXVWRFRQGLWLRQV7KH
RQO\GLIIHUHQFHLVWKDWWKHUHDUHQRZWZRILUVWRUGHUFRQGLWLRQVIRUWKHFDSDFLW\RIWKH
VHFRQGPRGH
7DEOH
3DUDPHWHUYDOXHVIRUWKHWZRPRGHSUREOHP
3DUDPHWHU
'HPDQG
D
E
E
D
8VHUFRVW
YRW
IIWW WW
Z
F
FH
&RVWRIFDSDFLW\
N
5RXWHPRGH
5RXWHPRGH
N N 7KHSDUDPHWHUYDOXHVIRUWKHVLPXODWLRQPRGHODUHOLVWHGLQ7DEOH7KHGHPDQG
SDUDPHWHUVDUHOHIWXQFKDQJHG7UDYHOWLPHE\UDLOLVDVVXPHGWREHHTXDOWRIUHHIORZ
WUDYHOWLPHE\FDU7KHDGGLWLRQDOFRVWRIZDLWLQJLVDVVXPHGWREHSHUFHQWRIWKH
UHIHUHQFHYDOXHRIWLPHDVLVFRQYHQWLRQDO7KHH[SRQHQWFH KDVDYDOXHRI7KH
FRVWSDUDPHWHUVDUHEDVHGRQ0X&RQVXOWZKHUHWKHWRWDOFRVWVRISDVVHQJHU
WUDQVSRUWRI'XWFK5DLOZD\VDUHDSSUR[LPDWHGZLWKDOLQHDUIXQFWLRQRIVHDW
NLORPHWUHVWUDLQNLORPHWUHVDQGSDVVHQJHUNLORPHWUHV 7KHSDVVHQJHUNLORPHWUHV
FRPSRQHQWLVUHODWLYHO\VPDOODQGLVLJQRUHGKHUH7KHPDUJLQDOFRVWRIWUDLQ
$WKRURXJKGLVFXVVLRQRQWKHYDOXHRIWLPHLQFOXGLQJWKHYDOXHRIZDLWLQJWLPHIRU
FRPPXWHUVLVSUHVHQWHGLQ&KDSWHURIWKLVYROXPH
NLORPHWUHVZDVIRXQGWREHPXFKKLJKHUWKDQWKDWRIVHDWNLORPHWUHV7KHGLVWDQFH
EHWZHHQ$DQG%LVDVVXPHGWREHNLORPHWUHV
7DEOH
9DULDEOH
FDS
7KHEDVHFDVHRIWKHWZRPRGHSUREOHP
T
4
F
W
S
3
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RI4 ZUW3
&6
75
.
66
5RXWHPRGH
5RXWHPRGH
V
1
6RXUFHRZQFDOFXODWLRQV
1RWHFDSIRUWKHURDG 1T DQG4 DUHPHDVXUHGDVQXPEHUVRIWULSVSHUKRXUF 2 S
3&6 . DQG66 LQHXURV
,QWKHEDVHFDVHLWLVDVVXPHGWKDWWUDLQVZLWKDFDSDFLW\RIVHDWVGHSDUWHYHU\
PLQXWHV,QWKH1HWKHUODQGVWKHIDUHIRUDUHWXUQWULSRINLORPHWUHVLVDURXQG
¼*LYHQWKDWIRUSHDNKRXUVPRVWRIWKHWULSVDUHUHWXUQWULSVLWVHHPVDSSURSULDWH
WRXVHKDOIWKHYDOXHRIDUHWXUQWLFNHW0DQ\RIWKHVHWULSVKRZHYHUDUHPDGHE\
VHDVRQWLFNHWKROGHUVZKRSD\DORZHUSULFHSHUWULS)RUWKLVUHDVRQSHUFHQW
LQVWHDGRISHUFHQWZDVXVHGDVWKHDSSURSULDWHSULFHIRUDRQHZD\WULSRINP
7KHVHDUHRQO\RSHUDWLQJFRVWVQRWFDSLWDOFRVWV0RUHRYHUIL[HGFRVWDQGFRVWVGLUHFWO\
UHODWHGWRWKHQXPEHURISDVVHQJHUVZKLFKWXUQHGRXWWREHUHODWLYHO\VPDOODUHLJQRUHGKHUH
5HVXOWVIRUWKLVEDVHFDVHDUHSUHVHQWHGLQ7DEOH7KHSULFHRIWUDQVSRUWLVKLJKHU
WKDQLQWKHFDVHRIWKHWZRURXWHSUREOHP7KHPDLQUHDVRQIRUWKLVGLIIHUHQFHLVWKDW
WKHFRVWIXQFWLRQIRUSDVVHQJHUWUDQVSRUWLVPRUHVHQVLWLYHWRFRQJHVWLRQWKDQWKDWIRU
URDGWUDQVSRUW)RUWKHODWWHUWKHUDWLREHWZHHQWKHQXPEHURIXVHUVDQGFDSDFLW\FDQ
EHPDGHHTXDOWRWZR)RUSDVVHQJHUWUDQVSRUWDWOHDVWLQWKH1HWKHUODQGVLWLV
YLUWXDOO\LPSRVVLEOHWRFDUU\WZLFHDVPDQ\SDVVHQJHUVDVWKHUHDUHVHDWV7KLV
GLIIHUHQFHLVUHIOHFWHGLQWKHVSHFLILFDWLRQVRIWKHFRVWIXQFWLRQVIRUUDLODQGURDG
WUDQVSRUWDVGLVFXVVHGDERYH,QWKHEDVHFDVHWKHUDWLREHWZHHQXVHDQGFDSDFLW\LV
IRUWUDLQV7KLVILJXUHVHHPVUHDVRQDEOHIRUFRPPXWHUWUDLQVLQWKH5DQGVWDGDUHD
GXULQJSHDNKRXUV5HYHQXHVIURPWUDLQWLFNHWVDUHVXIILFLHQWWRFRYHUDOOFRVWVRIUDLO
SDVVHQJHUWUDQVSRUWDVVHHPVUHDOLVWLFIRUSHDNWUDIILFLQWKH5DQGVWDGDUHD
,QRUGHUWRVWXG\WKHHIIHFWRIFKDQJHVLQWKHVXEVWLWXWDELOLW\EHWZHHQWKHPRGHVWKH
VDPHUDQJHRIYDOXHVRIWKHFRHIILFLHQWE WKDWZDV XVHGIRUURDGWUDQVSRUWLQWKHWZR
PRGHYHUVLRQVLPXODWLRQPRGHO ZHUHVLPXODWHG2QO\PLQRUGLIIHUHQFHVZLWKWKHWZR
URXWHVLWXDWLRQZHUHIRXQG $JDLQWKHSULFHVIRUWKHWZRPRGHVZKLFKDUHDOUHDG\
FORVHWRHDFKRWKHULQWKHEDVHFDVHEHFRPHYLUWXDOO\LGHQWLFDO7KHSULFHHODVWLFLWLHV
RIGHPDQGIRUWKHVHSDUDWHPRGHVEHFRPHYHU\ODUJHLQDEVROXWHYDOXHVEXWWKH
RYHUDOOSULFHRIWUDQVSRUWDQGLWVSULFHHODVWLFLW\UHPDLQDOPRVWLGHQWLFDO,WPXVW
WKHUHIRUHEHFRQFOXGHGWKDWDWOHDVWLQWKLVEDVHFDVHYHUVLRQRIWKHWZRPRGHPRGHO
WKHGLIIHUHQFHVEHWZHHQWKHVLWXDWLRQVRILPSHUIHFWDQGSHUIHFWVXEVWLWXWHVDUHPLQRU
DQGFRPSDUDEOHWRWKHGLIIHUHQFHVLQWKHWZRURXWHFDVH
7KHILUVWEHVWVLWXDWLRQDQGWKHWZRPRGHSUREOHP
'HWHUPLQLQJWKHILUVWEHVWHTXLOLEULXPLVPRUHFRPSOLFDWHGWKDQLQWKHWZRURXWHFDVH
7KHUDWLRT1V LVGHWHUPLQHGE\WKHILUVWRUGHUFRQGLWLRQIRUVEXWWKLVGRHVQRW
GHWHUPLQH17KLVPHDQVWKDWWKHFRVWRIUDLOZD\WUDQVSRUWFDQQRWEHFRPSXWHGIURP
WKHILUVWRUGHUFRQGLWLRQZLWKUHVSHFWWRFDSDFLW\DORQHDQGDQLWHUDWLYHSURFHGXUHLV
WKHUHIRUHQHHGHGLQRUGHUWRFRPSXWHWKHILUVWEHVWHTXLOLEULXP7KHHTXLOLEULXPZDV
ILUVWFRPSXWHGFRQGLWLRQDOXSRQDJLYHQYDOXHRI1WKHYDOXHRI1 VXJJHVWHGE\WKH
UHOHYDQWILUVWRUGHUFRQGLWLRQLQWKLVHTXLOLEULXPZDVWKHQFRPSXWHGDQG1 DGMXVWHG
WRZDUGVWKLVRSWLPDOYDOXH7KLVSURFHGXUHZDVUHSHDWHGXQWLOFRQYHUJHQFHRFFXUUHG
7DEOH
9DULDEOH
FDS
)LUVWEHVWVLWXDWLRQRIWKHWZRPRGHPRGHO
T
4
F
W
S
3
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RI4 ZUW3
&6
75
.
66
5RXWHPRGH
5RXWHPRGH
V 1 6RXUFHRZQFDOFXODWLRQV
1RWHFDSIRUWKHURDG1T DQG4 DUHPHDVXUHGDVQXPEHUVRIWULSVSHUKRXUF2S
3&6. DQG66 LQHXURV
7KHVLPXODWLRQUHVXOWVDUHSUHVHQWHGLQ7DEOH,QFRPSDULVRQZLWKWKHEDVHFDVH
WKHQXPEHURIUDLOSDVVHQJHUVLQFUHDVHVVXEVWDQWLDOO\7KHTXDOLW\RIUDLOWUDQVSRUWLV
LPSURYHG VXEVWDQWLDOO\WKHQXPEHURIGHSDUWXUHVSHUKRXULQFUHDVHVIURPWR
DQG WKHQXPEHURIVHDWVSHUWUDLQDOPRVW GRXEOHVDQGEHFRPHVHTXDOWR7KHUDWLR
EHWZHHQSDVVHQJHUVDQGVHDWVGURSVWR7KHVHLPSURYHPHQWVUHVXOWLQDPXFK
ORZHUXVHUFRVWRIWUDYHOWLPHIRUUDLOWUDQVSRUW¼LQVWHDGRI¼(YHQWKRXJK
WKHUDLOZD\IDUHLQFUHDVHVFRQVLGHUDEO\WR¼SHURQHZD\WULSWKHJHQHUDOL]HG
SULFHRIWUDLQWUDQVSRUWIDOOVFRQVLGHUDEO\EHFDXVHRIWKHORZHUWUDYHOWLPHDQGLVQRZ
ORZHUWKDQWKDWRIFDUWUDYHO
5RDGFDSDFLW\LVDOVRH[SDQGHGVXEVWDQWLDOO\DQGWKHSULFHRIURDGWUDQVSRUWLV
KDOYHGLQFRPSDULVRQZLWKWKHEDVHFDVH7ROOUHYHQXHVDUHH[DFWO\HTXDOWRWKHFRVWRI
URDGFDSDFLW\)DUHUHYHQXHVIURPUDLOWUDQVSRUWDUHQRZLQVXIILFLHQWWRFRYHUDOOFRVWV
7KHVHUHYHQXHVH[DFWO\FRYHUWKHFRVWRIVHDWNLORPHWUHVLHWKH\DUHHTXDOWRN1V
LQ(TXDWLRQ7KHUHDUHHFRQRPLHVRIVFDOHDVVRFLDWHGZLWKLQFUHDVLQJWKH
QXPEHURIVHDWVSHUWUDLQDQGPDUJLQDOFRVWSULFLQJLVWKHUHIRUHXQDEOHWRFRYHUWRWDO
H[SORLWDWLRQFRVWV
:KHQWUDLQDQGURDGEHFRPHFORVHUVXEVWLWXWHVGHPDQGVKLIWVWRZDUGVWKH
FKHDSHVWPRGH:LWKWKHDVVXPHGSDUDPHWHUYDOXHVWKHFKHDSHVWPRGHLVWKHWUDLQ
:LWKHFRQRPLHVRIVFDOHLQWKHUDLOZD\VWKLVGHPDQGVKLIWFDXVHVDIXUWKHUGURSLQWKH
FRVWRIUDLOWUDQVSRUWDQGURDGDVDPRGHRIWUDQVSRUWHYHQWXDOO\GLVDSSHDUV
5DLOZD\IDUHIL[HGFDSDFLWLHV
:KHQWKHFDSDFLWLHVRIERWKPRGHVDUHIL[HGDQGWKHRQO\SROLF\LQVWUXPHQWLVWKH
WUDLQIDUHSRVVLEOHZHOIDUHJDLQVDUHVPDOOZKHQFRPSDUHGWRWKHEDVHFDVH
66 ¼& 7KHRSWLPDOUDLOZD\IDUHWXUQVRXWWREHFRQVLGHUDEO\KLJKHU
WKDQLQWKHILUVWEHVWVLWXDWLRQ¼7KHUHLVVXEVWDQWLDOFRQJHVWLRQRQWKHURDG
T 8VHUFRVWVDUHHTXDOWR¼IRUWKHURDGDQG¼IRUWKHUDLOZD\
:KHQVXEVWLWXWDELOLW\EHWZHHQWKHWZRPRGHVLVLQFUHDVHGXVHUFRVWVFRQYHUJHWR
¼7KHRSWLPDOWROOGHFUHDVHVVRPHZKDW
5DLOZD\IDUHIUHTXHQF\DQGVHDWFDSDFLW\
:KHQWKHUDLOZD\IDUHQXPEHURIVHDWVDQGIUHTXHQF\RIGHSDUWXUHFDQDOOEHFKRVHQ
E\WKHSROLF\PDNHUZHOIDUHJDLQVDUHFRQVLGHUDEO\KLJKHUWKDQZKHQRQO\FDSDFLW\
FDQEHXVHGDVDSROLF\LQVWUXPHQW6RFLDOVXUSOXVLVHTXDOWR¼& ZKLFKLVFORVHWRWKHILUVWEHVWYDOXH7KHRSWLPDOUDLOZD\IDUHQRZWXUQVRXWWREH
QHJDWLYH ¼8VHUFRVWLVHTXDOWR¼IRUURDGDQG¼IRUUDLOZD\WUDQVSRUW
LPSO\LQJWKDWWKHSULFHRIWKHODWWHUPRGHLVDOVRQHJDWLYH¼:KHQ LV
LQFUHDVHGWKHUDLOZD\IDUHEHFRPHVSRVLWLYHZKLOHXVHUFRVWVRIWKHWZRPRGHV
FRQYHUJHJUDGXDOO\
7DEOH
9DULDEOH
FDS
6HFRQGEHVWWKHWZRPRGHSUREOHP
T
4
F
W
S
3
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RIT ZUWS
(ODVWLFLW\RI4 ZUWWR3
&6
75
.
66
5RXWHPRGH
5RXWHPRGH
V 1 5DLOZD\IDUHIUHTXHQF\DQGVHDWFDSDFLW\DQGURDGFDSDFLW\
7KHUHVXOWVIRUWKHWZRPRGHSUREOHPZKHUHFDSDFLW\RIWKHURDGFDQDOVREHFKRVHQ
RSWLPDOO\DUHSUHVHQWHGLQ7DEOH$VFDQEHVHHQLQWKHWDEOHWKHVKDUHRISRWHQWLDO
ZHOIDUHJDLQVLVHYHQODUJHUWKDQLQWKHFDVHZKHUHRQO\WKHFDSDFLW\RIUDLOFRXOGEH
XVHGDVDSROLF\LQVWUXPHQW7KHVRFLDOVXUSOXVHTXDOV¼& %RWKURDG
DQGUDLOZD\FDSDFLWLHVDUHFRQVLGHUDEO\KLJKHUWKDQLQWKHILUVWEHVWVLWXDWLRQ7KH
UHYHQXHVIURPUDLOZD\IDUHVDUHOHVVWKDQRQHWKLUGRIWKHFRVWVRIUDLOZD\WUDQVSRUW
:KHQVXEVWLWXWDELOLW\EHWZHHQWKHWZRPRGHVLQFUHDVHVUHVXOWVVLPLODUWRWKRVH
IURPWKHWZRURXWHSUREOHPDUHREWDLQHG)LJXUH LOOXVWUDWHVWKLV
)LJXUH
(IIHFWRILQFUHDVLQJVXEVWLWXWDELOLW\RQXVHUFRVWXSSHUSDQHODQG
URXWHFKRLFHORZHUSDQHOLQWKHWZRPRGHSUREOHP
SD
SULFH
SD
SE
SE
EHWDDEVROXWHYDOXH
QXPEHURIWULSV
TD
TD
TE
TE
EHWDDEVROXWHYDOXH
)RUWKHLQWHUYDO>@WKHUHDUHDJDLQWZRHTXLOLEULD,QRQHRIWKHVHWKH
URDGUHPDLQVYLUWXDOO\XQXVHG)RUDEVROXWHYDOXHVRI KLJKHUWKDQ WKHUHLVRQO\
RQHHTXLOLEULXP LQZKLFKWKHUHLVQRURDG)RUDEVROXWHYDOXHVRI ORZHUWKDQ
WKHUHLVRQO\RQHHTXLOLEULXPLQZKLFKERWKURDGDQGUDLOZD\DWWUDFWDVXEVWDQWLDO
QXPEHURIWULSV )RUVPDOOYDOXHVRI WKHHTXLOLEULXPLQZKLFKWZRPRGHVDUHXVHG
KDVWKHKLJKHVWVRFLDOVXUSOXVIRUODUJHUYDOXHVWKHHTXLOLEULXPLQZKLFKRQO\RQH
PRGHLVXVHGKDVWKHKLJKHVWVRFLDOVXUSOXV
6XPPDU\
7KHPDLQSXUSRVHRIWKLVSDUDJUDSKZDVWRSURYLGHDFRPSDULVRQRIWKHUHVXOWVRIWZR
PRGHVLWXDWLRQVZLWKWKRVHRIWKHWZRURXWHVLWXDWLRQVGLVFXVVHGLQWKHSUHYLRXV
VHFWLRQ7KHPRVWLPSRUWDQWFRQFOXVLRQLVWKDWWKHVHUHVXOWVDSSHDUWREHVRVLPLODU
,QWURGXFWLRQRILPSHUIHFWVXEVWLWXWDELOLW\LQWRWKHWZRURDGPRGHOPDNHVLWEHKDYH
PXFKOLNHWKHDQDORJRXVWZRPRGHPRGHOGHVSLWHWKHVXEVWDQWLDOGLIIHUHQFHLQWKH
FRVWVWUXFWXUH
7KHVLPLODULW\EHWZHHQWKHWZRPRGHOVLVQLFHO\LOOXVWUDWHGE\WKHGHYHORSPHQWRI
WKHLQGH[RIUHODWLYHZHOIDUHLPSURYHPHQWDVDIXQFWLRQRIWKHSULFHVXEVWLWXWDELOLW\
SDUDPHWHUZKLFKLVVKRZQLQ)LJXUH ,QERWKFDVHVRQO\PRGHVWLPSURYHPHQWV
LQZHOIDUHDUHSRVVLEOHZKHQRQHURDGRUPRGHKDVWRUHPDLQXQWROOHGDQGFDSDFLW\
KDVWREHWDNHQDVJLYHQ,IFDSDFLW\RIWKHURDGRUPRGHWKDWFDQEHWROOHGFDQDOVREH
XVHGDVDSROLF\LQVWUXPHQWDPXFKODUJHUVKDUHW\SLFDOO\PRUHWKDQWKUHHTXDUWHUV
RIWKHSRWHQWLDOILUVWEHVWZHOIDUHJDLQVFDQEHUHDOL]HG,IWKHFDSDFLWLHVRIERWK
URDGVRUPRGHVFDQEHXVHGDVSROLF\LQVWUXPHQWVSHUFHQWRIWKHSRWHQWLDO
ZHOIDUHJDLQVLVUHDOLVHG IRUODUJHYDOXHVRI E 7KHUHDVRQLVWKDWLQWKHVH
:KHQWKHUHDUHWZRHTXLOLEULDWKHRQH\LHOGLQJWKHKLJKHVWVRFLDOVXUSOXVZDVXVHGLQ
)LJXUH FLUFXPVWDQFHVWKHXQSULFHGURDGRUPRGHGLVDSSHDUZKLFKLPSOLHVWKDWLWVSULFH
EHFRPHVLUUHOHYDQW
)LJXUH
(IIHFWRILQFUHDVLQJVXEVWLWXWDELOLW\RQXVHUFRVWDQGURXWHPRGH FKRLFH
LQWKHWZRURXWHXSSHUSDQHODQGWZRPRGHORZHUSDQHOPRGHOV
RQHWROO
RP HJD
RQHWROODQGRQH
FDSDFLW\
RQHWROOWZ R
FDSDFLWLHV
EHWDDEVROXWHYDOXH
RQO\IDUH
RP HJD
IDUHDQGFDSDFLW\
WUDLQ
IDUHDQGFDSDFLWLHV
EHWDDEVROXWHYDOXH
7KHVLPXODWLRQUHVXOWVFOHDUO\VXJJHVWWKDWWKHUHLVDTXDOLWDWLYHGLIIHUHQFHEHWZHHQ
VLWXDWLRQV LQZKLFK VXEVWLWXWLRQEHWZHHQURDGVRUPRGHVLVGLIILFXOWZLWK WKHDEVROXWH
YDOXHRI OHVVWKDQDQGVLWXDWLRQVLQZKLFKLWLVHDVLHUEXWWKDWWKHUHLVQRWD
TXDOLWDWLYHGLIIHUHQFHEHWZHHQSHUIHFWVXEVWLWXWDELOLW\DQGLPSHUIHFWEXWUHODWLYHO\
HDV\VXEVWLWXWDELOLW\DODUJHDEVROXWHYDOXHRI 7ZRLPSRUWDQWUHDVRQVRUWKLV
FRQFOXVLRQDUHWKDWRQHURDGRUPRGHGLVDSSHDUVFRPSOHWHO\LQWKHILUVWEHVWFDVH IRU
ILQLWHYDOXHVRI DQGWKDWLQWKHFDVHZLWKWZRFDSDFLWLHVDQGRQHWROO WKHWUDQVLWLRQ
IURPDQHTXLOLEULXPLQZKLFKERWKURDGVPRGHVDUHXVHGWRRQHLQZKLFKRQO\RQHLV
XVHGWDNHVSODFHIRUUHODWLYHO\VPDOOYDOXHVRI
&21&/86,216
7KLVSDSHUFRQVLGHUVVHFRQGEHVWSULFLQJDVLWDULVHVWKURXJKLQFRPSOHWHFRYHUDJHRI
IXOOQHWZRUNV7KHPDLQSULQFLSOHVZHUHILUVWUHYLHZHGE\FRQVLGHULQJWKHFODVVLFWZR
URXWHSUREOHPDQGVRPHH[WHQVLRQVWKDWKDYHEHHQVWXGLHGPRUHUHFHQWO\,QPRVWRI
WKHVHVWXGLHVWKHFRPSHWLQJURXWHVDUHDVVXPHGWREHSHUIHFWVXEVWLWXWHVZKLFKLV
SUREDEO\QRWWKHFDVHIRUPRVWSDUDOOHOURDGVLQUHDOLW\DQGHYHQOHVVOLNHO\IRUWKH
FDVHZKHUHFRPSHWLQJFRQQHFWLRQVUHSUHVHQWGLIIHUHQWWUDQVSRUWPRGHV
$PRGHOOLQJIUDPHZRUNLQZKLFKWKHDOWHUQDWLYHVDUHLPSHUIHFWVXEVWLWXWHVZDV
GHYHORSHG DQGQXPHULFDO UHVXOWVIRU WZR URDGVDQGWZR PRGHVZHUH SUHVHQWHG,QWKH
PRGHOWULSJHQHUDWLRQDQGWULSGLVWULEXWLRQDUHGLVWLQJXLVKHGLQDZD\WKDWLV
FRQVLVWHQWZLWKHFRQRPLFWKHRU\$OLQHDUGHPDQGHTXDWLRQZDVXVHGIRUWULS
JHQHUDWLRQ DQGWKHORJLWPRGHOIRUWULSGLVWULEXWLRQ7KHPRGHORIIHUVWKHSRVVLELOLW\
WRVWXG\WKHHIIHFWRIFKDQJHVLQWKHVXEVWLWXWDELOLW\EHWZHHQWKHWZRURXWHVRUPRGHV
3HUIHFWVXEVWLWXWDELOLW\LVDOLPLWLQJFDVHLQZKLFKWKHDEVROXWHYDOXHRIRQHSDUDPHWHU
EHFRPHVLQILQLWHO\ODUJH
7KHPRGHOZDVXVHGWRFRQVLGHUVLWXDWLRQVLQZKLFKRQHURXWHRUPRGHFDQQRWEH
WROOHG6LPXODWLRQUHVXOWVVKRZWKDWIRUWKHFKRVHQSDUDPHWHUYDOXHVWKHUHLVD
VXEVWDQWLDOGLIIHUHQFHEHWZHHQWKHHIIHFWLYHQHVVRISROLFLHVLQZKLFKWKHFDSDFLWLHV
KDYHWREHWDNHQDVJLYHQDQGWKRVHLQZKLFKFDSDFLW\RIDWOHDVWRQHPRGHFDQEH
FKDQJHG,IRQO\DWROORQURXWHRUPRGHWZRFDQEHXVHGW\SLFDOO\OHVVWKDQDTXDUWHU
RIWKHWRWDOSRVVLEOHZHOIDUHJDLQVLVUHDOLVHG:KHQWKHFDSDFLW\RIDWOHDVWRQHURXWH
RUPRGHFDQEHGHWHUPLQHGE\WKHSROLF\PDNHUW\SLFDOO\PRUHWKDQWKUHHTXDUWHUVRI
WKHPD[LPXPSRVVLEOHZHOIDUHJDLQVDUHUHDOLVHG7KHVHILJXUHVDUHQRWYHU\
GHSHQGHQWRQWKHVXEVWLWXWDELOLW\EHWZHHQWKHWZRURXWHVRUPRGHV
$VWULNLQJIHDWXUHRIWKHSROLF\LQZKLFKWKHFDSDFLWLHVRIERWKPRGHVDQGWKH
UDLOZD\IDUHFDQEHXVHGDVSROLF\LQVWUXPHQWVLVWKHH[LVWHQFHRIWZRHTXLOLEULDIRUD
UDQJHRIYDOXHVRI,QRQHHTXLOLEULXPWKHUHDUHVXEVWDQWLDOQXPEHUVRIXVHUVRIERWK
PRGHVZKHUHDVLQWKHRWKHUXVHRIRQHPRGHLVQHJOLJLEOH,QWKLVFDVHWKHUHLVD
UHJLPHVKLIWWKDWLVUHODWHGWRWKHSRVVLELOLW\WRVXEVWLWXWHXVHRIRQHPRGHIRUDQRWKHU
EXWLWGRHVQRWFRLQFLGHZLWKWKHGLIIHUHQFHEHWZHHQSHUIHFWDQGLPSHUIHFW
VXEVWLWXWDELOLW\LQWHUSUHWHGDVDILQLWHDQGDQLQILQLWHYDOXHRI UHVSHFWLYHO\
)RUWKHILUVWEHVWFDVHWKHUHLVQRUHJLPHVKLIWWKDWFRLQFLGHVZLWKWKHGLIIHUHQFH
EHWZHHQSHUIHFWDQGLPSHUIHFWVXEVWLWXWDELOLW\HLWKHU7KHVLWXDWLRQLQZKLFKRQO\RQH
PRGHLVXVHGLVQRZDSSURDFKHGLQDFRQWLQXRXVZD\DQGLWLVUHDFKHGIRUDILQLWHDQG
UHODWLYHO\VPDOOYDOXHRI
7KHUHVXOWVIRUWKHFODVVLFWZRURXWHSUREOHPZLWKLPSHUIHFWVXEVWLWXWHVLQZKLFK
RQO\DWROOFDQEHFKDUJHGRQRQHRIWZRSDUDOOHOURDGVDUHVXUSULVLQJO\FORVHWRWKH
UHVXOWVIRUWKHPRGHOZLWKSHUIHFWVXEVWLWXWHV7KLVVXJJHVWVWKDWDOWKRXJKWKH
DVVXPSWLRQRILPSHUIHFWVXEVWLWXWDELOLW\EHWZHHQURXWHVPDNHVWKHPRGHOULFKHUDQG
DOORZVIRUPRGHFKRLFHLWGRHVQRWVWURQJO\DIIHFWWKHSROLF\FRQFOXVLRQVRQWKH
GHVLJQDQGGHVLUDELOLW\RIVHFRQGEHVWSULFLQJ 7KLVPDNHVWKHUHVXOWVRISULRUVWXGLHV
DSSHDUUREXVWWRWKHDVVXPSWLRQRISHUIHFWVXEVWLWXWDELOLW\
7KLVSLFWXUHFKDQJHVZKHQFDSDFLW\FKRLFHLVDOVRLQFOXGHGDVDSROLF\LQVWUXPHQW
:KHQRQO\WKHFDSDFLW\RIWKHSULFHGURDGFDQEHFKRVHQWKHFKRLFHPD\PDNHWKH
GLIIHUHQFHEHWZHHQDQHJDWLYHVHFRQGEHVWWROOIRULPSHUIHFWVXEVWLWXWHVDQGD
SRVLWLYHRQHIRUFORVHURUSHUIHFWVXEVWLWXWHVDVLOOXVWUDWHGE\WKHUHVXOWV:KHQERWK
FDSDFLWLHVFDQEHFKRVHQWKHFKRLFHPD\PDNHWKHGLIIHUHQFHEHWZHHQNHHSLQJDURDG
IRULPSHUIHFWVXEVWLWXWHVDQGHOLPLQDWLQJLWLQWKHORQJUXQIRUFORVHURUSHUIHFW
VXEVWLWXWHV
5()(5(1&(6
$UQRWW5DQG<DQ$ 7KH7ZR0RGH3UREOHP6HFRQG%HVW3ULFLQJDQG
&DSDFLW\5HYLHZRI8UEDQDQG5HJLRQDO'HYHORSPHQW6WXGLHV
%UDLG503HDN/RDG3ULFLQJRID7UDQVSRUWDWLRQ5RXWHZLWKDQ8QSULFHG
6XEVWLWXWH-RXUQDORI8UEDQ(FRQRPLFV 'H3DOPD$DQG/LQGVH\5 3ULYDWH7ROO5RDGV&RPSHWLWLRQXQGHU9DULRXV
2ZQHUVKLS5HJLPHV$QQDOVRI5HJLRQDO6FLHQFH .QLJKW)6RPH)DOODFLHVLQWKH,QWHUSUHWDWLRQRI6RFLDO&RVWV4XDUWHUO\
-RXUQDORI(FRQRPLFV /pY\/DPEHUW+7DULILFDWLRQGHV6HUYLFHVj4XDOLWp9DULDEOH$SSOLFDWLRQ
DX[3pDJHVGH&LUFXODWLRQ(FRQRPHWULFD /LQGVH\&5DQG9HUKRHI(77UDIILF&RQJHVWLRQDQG&RQJHVWLRQ3ULFLQJ
,Q'$+HQVKHUDQG.-%XWWRQHGV+DQGERRNRI7UDQVSRUW6\VWHPVDQG
7UDIILF&RQWURO+DQGERRNVLQ7UDQVSRUW SS$PVWHUGDP(OVHYLHU
3HUJDPRQ
/LX/1DQG0F'RQDOG-)(IILFLHQW&RQJHVWLRQ7ROOVLQWKH3UHVHQFHRI
8QSULFHG&RQJHVWLRQD3HDNDQG2II3HDN6LPXODWLRQ0RGHO-RXUQDORI8UEDQ
(FRQRPLFV 0DUFKDQG0$1RWHRQ2SWLPDO7ROOVLQDQ,PSHUIHFW(QYLURQPHQW
(FRQRPHWULFD 0RKULQJ+2SWLPL]DWLRQDQG6FDOH(FRQRPLHVLQ8UEDQ%XV7UDQVSRUWDWLRQ
$PHULFDQ(FRQRPLF5HYLHZ 0RKULQJ+DQG+DUZLW]0 +LJKZD\%HQHILWV(YDQVWRQ1RUWKZHVWHUQ
8QLYHUVLW\3UHVV
0X&RQVXOW 9RRUEHUHLGLQJ3UHVWDWLHFRQWUDFW163UHSDUDWLRQ$FKLHYHPHQWV
&RQWUDFW'XWFK5DLOZD\V5HSRUWIRUWKH'XWFK0LQLVWU\RI7UDQVSRUWLQ'XWFK
3LJRX$&:HDOWKDQG:HOIDUH0DFPLOODQ/RQGRQ
6PDOO.$8UEDQ7UDQVSRUWDWLRQ(FRQRPLFV +DUZRRG&KXU
6PDOO.$DQG5RVHQ+6 $SSOLHG:HOIDUH(FRQRPLFVZLWK'LVFUHWH
&KRLFH0RGHOV(FRQRPHWULFD 9HUKRHI(71LMNDPS3 DQG5LHWYHOG36HFRQG%HVW&RQJHVWLRQ3ULFLQJ
WKH&DVHRIDQ8QWROOHG$OWHUQDWLYH-RXUQDORI8UEDQ(FRQRPLFV 9LFNUH\:63ULFLQJLQ8UEDQDQG6XEXUEDQ7UDQVSRUW$PHULFDQ(FRQRPLF
5HYLHZ 9LFNUH\:6&RQJHVWLRQ7KHRU\DQG7UDQVSRUW,QYHVWPHQW$PHULFDQ
(FRQRPLF5HYLHZ3DSHUVDQG3URFHHGLQJV :DOWHUV$$7KHWKHRU\DQGPHDVXUHPHQWRISULYDWHDQGVRFLDOFRVWRI
KLJKZD\FRQJHVWLRQ(FRQRPHWULFD :DUGURS-6RPH7KHRUHWLFDO$VSHFWVRI5RDG7UDIILF5HVHDUFK3URFHHGLQJV
RIWKH,QVWLWXWHRI&LYLO(QJLQHHUV,, <DQJ+DQG0HQJ4 $1RWHRQ+LJKZD\3ULFLQJ DQG&DSDFLW\&KRLFH
XQGHUD%XLOG2SHUDWH7UDQVIHU6FKHPH7UDQVSRUWDWLRQ5HVHDUFK$