Investment, Financing Constraints and the Euler Equation

#,BwW 5AN#W #W + Rw 5W,~
NWV,+5WA
#W+AW,A #W ,W NwW
`Lh!?} @Tih LDf
H1 Vdowdul 0 J1 Wudydjolql
Lqyhvwphqw/ Ilqdqflqj Frqvwudlqwv
dqg wkh Hxohu Htxdwlrq
Urpd/ oxjolr 5335
Devwudfw
Lq zkdw iroorzv/ zh vkrz wkdw qdqflqj frqvwudlqwv dhfw up*v
lqyhvwphqw ghflvlrqv ehiruh wkh frqvwudlqw lv elqglqj1 Wklv rffxuv eh0
fdxvh wkh up dqwlflsdwhv dw wkh fxuuhqw wlph wkh ixwxuh erxqg1 Zh
dujxh wkdw/ hyhq zlwk lpshuihfw fdslwdo pdunhwv/ lqyhvwphqw srolf|
pxvw eh vxfk wkdw wkhuh duh qr duelwudjh rssruwxqlwlhv dw dq| wlph1
Frqwudu| wr wkh xvxdo _h{shfwdwlrqv ylhz%/ wklv iruzdug0orrnlqj eh0
kdylru jxdudqwhhv wkh uhvshfw ri wkh Hxohu htxdwlrq hyhq zkhq wkh
uhvwulfwlrq lv elqglqj/ wkdw lv wkh rswlpdolw| ri wkh frqvwudlqhg lqyhvw0
phqw ghflvlrqv1
Nh|zrugv= Ilqdqflqj frqvwudlqwv> Ilup ehkdylrxu g|qdplfv1
MHO fodvvlfdwlrq= H55/ H77
Lqyhvwphqw/ Ilqdqflqj Frqvwudlqwv dqg wkh
Hxohu Htxdwlrq
H1 Vdowdul0 J1 Wudydjolql|
4
Lqwurgxfwlrq
Wkh dlp ri wklv sdshu lv wr lqyhvwljdwh krz qdqflqj frqvwudlqwv fdq
dhfw lqyhvwphqw ghflvlrqv e| upv1 Lq wkh hfrqrplf olwhudwxuh/ lw lv
riwhq dujxhg wkdw wkh rqo| wlph zkhq hfrqrplf djhqwv* rswlpdo ehkdylru
lv dhfwhg e| frqvwudlqwv rq vwdwh yduldeohv +vxfk dv wkh ohyho ri ghew, lv
zkhq wkh frqvwudlqw elqgv wkh ydoxh ri wkh yduldeoh1 Wkxv xsshu erxqgv
rq h{whuqdo qdqflqj ru oltxlglw| frqvwudlqwv fdq ohdg upv wr dgrsw
fruqhu vroxwlrqv iru wkh doorfdwlrq ri uhvrxufhv1 Krzhyhu/ prvw dqdo|vhv
jlyh/ dw ehvw/ dq lqfrpsohwh dffrxqw ri wkh lqwhuphgldwh skdvhv ehiruh wkh
frqvwudlqw ehjlqv wr elqg1 Wkh w|slfdo lpsolfdwlrq ri wkhvh prghov lv wkdw
zkhq frqvwudlqwv duh fxuuhqwo| vodfn/ dq hfrqrplf djhqw zloo ehkdyh dv
li lw zloo qhyhu vxhu frqvwudlqwv lq wkh ixwxuh1 Dffruglqjo|/ wkh djhqw*v
ehkdylru lv lqglvwlqjxlvkdeoh iurp wkdw ri dq lghqwlfdo djhqw dfwlqj lq wkh
devhqfh ri ixwxuh frqvwudlqwv1
Wkh uhodwlrqvkls ehwzhhq rswlpl}lqj ehkdylru dqg qdqflqj frqvwudlqwv
lv ri hvshfldo lpsruwdqfh lq dq lqwhuwhpsrudo frqwh{w zkhuh ixwxuh uhvwulf0
wlrqv fdq dhfw wkh fxuuhqw ehkdylru ri upv dqg frqvxphuv1 Zklwhg
+4<<5,/ Erqg dqg Phjklu +4<;7,/ Kxeedug/ Ndvk|ds dqg Zklwhg +4<<8,/
iru h{dpsoh/ kdyh dqdo|}hg wkh lqwhudfwlrq ehwzhhq frusrudwh lqyhvwphqw
dqg qdqflqj frqvwudlqwv lq glvfuhwh wlph prghov zlwk dq lpshuihfw pdu0
nhw iru fdslwdo1 Lq wkhvh prghov/ upv idflqj d ixwxuh eruurzlqj eduulhu
Glsduwlphqwr gl Hfrqrpld Sxeeolfd/ Xqlyhuvlwã Od Vdslhq}d gl Urpd1 H0pdlo=
vdowdulCghs1hfr1xqlurpd41lw
|
Lvwlwxwr gl Vflhq}h Hfrqrplfkh/ Xqlyhuvlwã gl Xuelqr1 H0pdlo= wudydjolqlChfr0
q1xqlxue1lw
4
dfw dv li wkhlu fxuuhqw pdujlqdo uhyhqxh zhuh glvfrxqwhg dw d kljkhu udwh
wkdq zrxog rwkhuzlvh eh wkh fdvh1 Kd|dvkl +4<;:,/ ]hoghv +4<;<, dqg
Ghdwrq +4<<4,*v uhvxowv rq frqvxpswlrq xqghu oltxlglw| frqvwudlqwv iro0
orz wkh vdph olqh ri wklqnlqj1 Wkh edvlf lghd ri doo wkhvh sdshuv lv wkdw
vr orqj dv wkh djhqw +d up ru d frqvxphu, grhv qrw frph xs djdlqvw
wkh frqvwudlqw/ lw zloo eh deoh wr vdwlvi| wkh Hxohu htxdwlrq e| uhdoorfdw0
lqj qdqfldo uhvrxufhv/ shulrg e| shulrg1 Lq wkhvh frqglwlrqv qdqflqj
frqvwudlqwv zklfk duh elqglqj lq d ixwxuh shulrg +ru kdyh vrph sured0
elolw| ri ehfrplqj elqglqj lq wkh ixwxuh, gr qrw fdxvh d ylrodwlrq ri wkh
lqwhuwhpsrudo uvw rughu frqglwlrq1
Zkhq/ rq wkh rwkhu kdqg/ wkh frqvwudlqw lv douhdg| dfwlyh/ wkh uhdoor0
fdwlrq ri uhvrxufhv lv lqwhuuxswhg dqg wkh xsshu erxqg fuhdwhv d rqh0vlgh
ylrodwlrq ri wkh Hxohu htxdwlrq1 Wkh pdlq lpsolfdwlrq ri wklv g|qdplf
surfhvv lv wkdw hyhq zkhq qdqflqj frqvwudlqwv duh lqhhfwlyh/ vr wkdw
wkh Hxohu htxdwlrq orrnv olnh wkh vwdqgdug rqh/ wkh rswlpdo ehkdylru/ iru
djhqwv dfwlqj xqghu odwhqw frqvwudlqwv/ zloo qrw jhqhudoo| eh wkh vdph dv
iru dq djhqw zkr zloo qhyhu eh frqvwudlqhg1 Wklv pd| vrxqg frpsolfdwhg>
lw lv/ qrqhwkhohvv/ d qdwxudo frqvhtxhqfh ri lqwhuwhpsrudo rswlpl}lqj eh0
kdylru1 Wkh nh| srlqw lq wkhvh dqdo|vhv lv wkdw wkh rqo| wlph zkhq wkh
vwdqgdug Hxohu htxdwlrq lv ylrodwhg lv zkhq dq hfrqrplf djhqw/ vxfk dv
d up/ idfhv d elqglqj qdqflqj frqvwudlqw1
Krzhyhu/ wklv lpsolfdwlrq/ zklfk kdv prwlydwhg d juhdw ghdo ri hp0
slulfdo lqyhvwljdwlrq +Kxeedug 4<<;,/ lv vrphzkdw frxqwhulqwxlwlyh= lq d
iruzdug0orrnlqj hqylurqphqw rqh zrxog h{shfw wkdw djhqwv* fxuuhqw eh0
kdylru zrxog dqwlflsdwh wkh hhfwv ri ixwxuh frqvwudlqwv dqg wkdw wkh|
zrxog lpphgldwho| vhohfw wkh rswlpl}lqj sdwk zklfk zrxog pd{lpl}h wkh
ydoxh ixqfwlrq ryhu wlph/ jlyhq wkh suhvhqfh ri wkh odwhqw frqvwudlqw1 Lq
rwkhu zrugv/ d udwlrqdo djhqw vkrxog prgli| lwv rswlpdo srolf| iurp wkh
vwduw/ fkrrvlqj d wudmhfwru| zklfk uhvshfwv wkh Hxohu htxdwlrq/ hyhq zkhq
wkh frqvwudlqw lv qrw elqglqj1 Lq whfkqlfdo whupv/ wkh frqvwudlqw fdq eh
wuhdwhg dv dq lqwhuqdo vroxwlrq wr wkh sureohp1 Lqwxlwlyho|/ rxu uhvxow
ghshqgv rq wkh idfw wkdw ixwxuh frqvwudlqw dowhuv wkh fxuuhqw lqyhvwphqw
ghflvlrqv1 Wklv lv ehfdxvh lqyhvwphqw ghflvlrqv vwhp qrw rqo| iurp wkh
fxuuhqw ydoxh ri wkh up/ exw dovr rq wkh ixwxuh rqh> wkhq/ vlqfh wkh frq0
vwudlqw fkdqjhv wkh ixwxuh ydoxh ri wkh up/ lw dovr prglhv wkh fxuuhqw
lqyhvwphqw srolf|1 Lw lv zruwk hpskdvl}lqj wkdw wklv uhvxow lv rewdlqhg xv0
lqj d vwdqgdug g|qdplf iudphzrun/ lq zklfk vshfldo hpskdvl}h lv sodfhg
5
rq wkh lqwhuuhodwlrq ehwzhhq lqyhvwphqw ghflvlrqv dqg h{shfwdwlrqv rq
wkh dydlodelolw| ri ixwxuh qdqflqj uhvrxufhv1 G*Dxwxpqh dqg Plfkho
+4<;8,/ Ploqh dqg Urehuwvrq +4<<9, dqg Vdowdul dqg Wudydjolql +5334,
surylgh lqvljkwv vxssruwlqj wklv ylhz1
Lq zkdw iroorzv zh surylgh d udwlrqdoh iru wkhvh uhvxowv/ vkrzlqj wkdw/
lq d frqwlqxrxv g|qdplf frqwh{w/ wkh suhvhqfh ri d qdqflqj frqvwudlqw
grhv qrw lqydolgdwh wkh Hxohu htxdwlrq xs wr wkh erxqg= wkh up ehkdyhv
dv li lw zhuh frqvwudlqhg hyhq lq shulrgv zkhuh wkh Hxohu htxdwlrq lv vdw0
lvhg1 Wklv rffxuv ehfdxvh wkh up*v fxuuhqw ehkdylru dqwlflsdwhv wkh
ixwxuh frqvwudlqw1 Pruh vshflfdoo|/ zh zloo vkrz krz/ lq wkh suhvhqfh ri
dq xsshu erxqg rq qdqflqj/ d up zloo fkdqjh lwv ryhudoo lqwhuwhpsrudo
lqyhvwphqw suroh/ hqvxulqj lw uhvshfwv wkh Hxohu htxdwlrq lq doo shulrgv1
Wklv vljqlhv wkdw upv zklfk zloo eh udwlrqhg lq wkh ixwxuh/ exw zklfk
duh fxuuhqwo| xqfrqvwudlqhg/ fdq vhohfw dq rswlpl}lqj lqyhvwphqw wudmhf0
wru| uljkw iurp wkh vwduw1 Dw wkh srlqw zkhuh wkh frqvwudlqhg dqg wkh
xqfrqvwudlqhg sdwkv phhw hdfk rwkhu wkh wudmhfwru| pxvw eh frqwlqxrxv
dqg vprrwk1 Ri frxuvh/ wklv lv d qhfhvvdu| frqglwlrq iru rswlpdo fdsl0
wdo dffxpxodwlrq/ hqvxulqj wkh devhqfh ri duelwudjh rssruwxqlwlhv dw dq|
wlph1 Lqghhg/ li lw zhuh ylrodwhg wkh up frxog rewdlq duelwudjh surwv
e| frqvwuxfwlqj sruwirolrv ri wudghg dvvhwv1 Lw iroorzv wkdw lqyhvwphqw
ghflvlrqv duh rswlpdo rqo| li wkh| suhfoxgh duelwudjh rssruwxqlwlhv1
Dq dgglwlrqdo lpsolfdwlrq ri wklv h{shfwdwlrqv ylhz lv wkdw wkh Hx0
ohu htxdwlrq surylghv xv zlwk uhodwlyho| olwwoh lqirupdwlrq1 Lq wkh suhvhqw
iudphzrun wkh lqwhuwhpsrudo uvw rughu frqglwlrq hhfwv wkh rswlpdo eh0
kdylru ri erwk frqvwudlqhg dqg xqfrqvwudlqhg upv1 Lw iroorzv wkdw wkh
Hxohu htxdwlrq fdqqrw glvfulplqdwh ehwzhhq upv dfwlqj xqghu glhuhqw
qdqflqj frqvwudlqwv1 Dv d uhvxow/ hpslulfdo dqdo|vhv ri lqyhvwphqw dqg
qdqflqj uhvwulfwlrqv zklfk xvh wkh htxdwlrq duh olnho| wr jlyh vsxulrxv
uhvxowv14 Lq vkruw/ qdqflqj frqvwudlqwv kdyh idu0uhdfklqj hhfwv qrw rqo|
rq djhqwv* rswlpdo lqwhuwhpsrudo ehkdylru/ exw dovr rq wkh txdqwlwdwlyh
phwkrgrorjlhv dssolhg wr wkh lqyhvwljdwlrq ri wkh sureohp15
Wkh fhqwudo lghd ehklqg d juhdw ghdo ri hpslulfdo lqyhvwljdwlrq lv wkdw wkh Hxohu
htxdwlrq vkrxog idlo iru glvwuhvvhg upv zkloh uhpdlqlqj ydolg iru xqglvwuhvvhg rqhv1
Lw lv dujxhg/ pruhryhu/ wkdw/ iru wkh iruphu jurxs ri upv/ qdqfldo yduldeohv/ vxfk
dv oltxlglw| ru qhw zruwk/ zloo sod| d vljqlfdqw uroh lq lqyhvwphqw ghflvlrqv +Zklwhg/
4<<5> Erqg dqg Phjklu/ 4<<7> Kxeedug/ Ndvk|ds dqg Zklwhg/ 4<<8,
5
Mdhh dqg Vwljolw} +4<<3, revhuyh wkdw dqwlflsdwhg ixwxuh fuhglw udwlrqlqj fdq
kdyh hhfwv rq fxuuhqw djjuhjdwh ghpdqg/ hyhq zkhq wkhuh lv qr fuhglw udwlrqlqj dw
4
6
Wkh sdshu lv rujdql}hg dv iroorzv1 Lq wkh qh{w vhfwlrq zh ghqh wkh
txdqwlw| frqvwudlqw dqg vroyh wkh lqwhuwhpsrudo lqyhvwphqw sureohp iru
wkh xqfrqvwudlqhg up1 Lq vhfwlrq 6/ zh frpsduh skdvh gldjudpv iru ihd0
vleoh wudmhfwrulhv/ h{sodlqlqj krz ixwxuh frqvwudlqwv dhfw d up*v fxuuhqw
rswlpdo srolf|/ hyhq zkhq frqvwudlqwv duh vodfn1 Ilqdoo|/ lq vhfwlrq 7 zh
h{sodlq wkh udwlrqdoh xqghuo|lqj rxu dqdo|wlfdo vroxwlrq1 Vhfwlrq 8 frq0
foxghv1
5
Wkh frqvwudlqw
Lq rughu wr vwxg| wkh lq xhqfh ri erxqgdu| frqglwlrqv rq d up*v rswlpdo
lqyhvwphqw ehkdylru/ zh dqdo|}h wkh lqyhvwphqw surfhvv lq d sduwldo htxl0
oleulxp frqwh{w1 Wkh fkdudfwhulvwlf ihdwxuh ri wkh prgho lv wkh suhvhqfh
ri d frqvwudlqw lq wkh irup=
g | 6g|
+4,
zkhuh g | ' _g
lv qhw lqyhvwphqw/ dqg g| lv wkh lqlwldo fdslwdo vwrfn1 Wkh
_|
sursruwlrq 6  ghqhv wkh xsshu erxqg rq lqyhvwphqw/ jlyhq g| / wkh
vwrfn ri fxuuhqw dvvhwv lq sodfh1 Wklv frqvwudlqw fdq kdyh dowhuqdwlyh hfr0
qrplf lqwhusuhwdwlrqv1 Iru h{dpsoh/ lw pd| eh vhhq dv wkh frqvhtxhqfh
ri ohjdo uhvwulfwlrqv rq wkh udwh ri dffxpxodwlrq lq d vshflf vhfwru ri
wkh hfrqrp|/ dlphg/ shukdsv/ dw suhyhqwlqj dq lqfuhdvh lq wkh up*v
prqrsro| srzhu/ ru shukdsv lq srooxwlrq1 Dowhuqdwlyho|/ wkh frqvwudlqw
frxog uh hfw dq lqhuwldo phfkdqlvp lq d wlph wr exlog prgho/ dhfwlqj
|
wkh pd{lpxp udwh g
6 dw zklfk wkh up fdq hqodujh lwv lqlwldo fdslwdo
g|
hqgrzphqw1 Ilqdoo|/ lw fdq eh lqwhusuhwhg dv d qdqflqj frqvwudlqw rq
lqyhvwphqw1 Wkh up qhhgv h{whuqdo qdqfh iru qhz surmhfwv/ exw fdslwdo
pdunhw lpshuihfwlrqv lqgxfh wkh ohqghu wr olplw wkh pd{lpxp ordq/ dydlo0
deoh dw wkh jrlqj lqwhuhvw udwh/ dv vrph sursruwlrq ri wkh up*v dvvhwv lq
sodfh1 Lq rwkhu zrugv/ g| lv wkh froodwhudol}hg dvvhw rhuhg e| wkh up
wr jxdudqwhh wkh fuhglw1 Uhjdugohvv ri krz wkh frqvwudlqw lv udwlrqdol}hg/
zkdw lv ri lqwhuhvw khuh lv wkdw wkh xsshu erxqg rq lqyhvwphqw dhfwv wkh
up*v srolf| dw doo wlphv ryhu wkh shulrg vxemhfw wr rswlpl}dwlrq1
u
suhvhqw1 Wkxv wkh lpsdfw ri wkh fuhglw udwlrqlqj fdqqrw eh dvvhvvhg mxvw e| orrnlqj
dw wkrvh shulrgv lq zklfk wkhuh lv gluhfw hylghqfh iru lwv suhvhqfh +s1;:7,1
7
514
Wkh prgho
Wr dqdo|}h wkh lqwhuwhpsrudo sureohp/ dvvxph wkdw wkh up pd{lpl}hv
wkh glvfrxqwhg ydoxh ri surwv ryhu wlph Khqfh/ wkh up kdv wr vhohfw
zklfk/ ri d vhw ri dgplvvleoh lqyhvwphqw sdwwhuqv/ zloo |lhog wkh pd{lpxp
ydoxh iru lwv remhfwlyh ixqfwlrq1
Ohw ZEg| lqglfdwh wkh uhyhqxh ixqfwlrq dw wlph |1 Zh dvvxph wkdw wkh
ixqfwlrq lv olqhdu zlwk uhvshfw wr fdslwdo/ wkdw lv ZEg| ' @g| 1 Ixuwkhu/
 2
ohw SEU| ' U| n 2/
U| eh wkh wrwdo frvw ri lqyhvwphqw/ htxdo wr wkh vxp ri
 2
wkh gluhfw frvw U| c dqg wkh dgmxvwphqw frvw 2/
U| zklfk lv lqfuhdvlqj dqg
vwulfwo| frqyh{ lq U| 1 Wklv phdqv wkdw fdslwdo lv d txdvl0{hg idfwru/ dqg
wkdw SEU| phdvxuhv wkh frvw dvvrfldwhg zlwk dgmxvwlqj wkh fdslwdo vwrfn
dw d udslg udwh shu xqlw ri wlph1 Wr vlpsoli|/ zh dvvxph wkdw doo rwkhu
yduldeoh lqsxwv kdyh ehhq pd{lpl}hg rxw ri @g| 6 Lq dq| rqh shulrg wkh
up pd{lpl}hv lwv lqvwdqwdqhrxv surw=
 2
@g| U| n
U
2/ |
Wkh up*v sureohp lv wr qg dq U|W wkdw pd{lpl}hv wkh suhvhqw glvfrxqwhg
ydoxh ri surwv=
4@
vxemhfw wr=
U|
]
A
f
 2 3o|
@g| U| U e _|
2/ |
g |
g |
g Ef
g EA '
'
'
U| Bg|
6g|
gf
gA
+5,
+6,
+7,
+8,
+9,
zkhuh o lv/ iru vlpsolflw|/ wkh frqvwdqw lqwhuhvw udwh/ g | ' U| Bg| lv wkh
dffxpxodwlrq htxdwlrq/ dqg B wkh ghsuhfldwlrq udwh1 Wkh lqwhuydo dfc A o
uhsuhvhqwv wkh sodqqlqj krul}rq/ zlwk gf dqg gA wkh ydoxhv ri fdslwdo dw
Wkh lpsolflw dvvxpswlrq khuh lv wkdw oderu lqsxw kdv ehhq rswlpl}hg rxw ri wkh
surw ixqfwlrq/ dqg wkdw lq wklv irup wkh surgxfwlrq ixqfwlrq kdv frqvwdqw uhwxuqv wr
vfdoh1 Wklv h{sodlqv krz surw fdq eh d olqhdu ixqfwlrq ri fdslwdo1
6
8
wkh ehjlqqlqj dqg dw wkh hqg ri wklv shulrg1 Wkh sureohp lv wr fkrrvh U|
lq rughu wr pd{lpl}h wkh up*v ydoxh1
Wr vroyh wklv lqwhuwhpsrudo sureohp/ zh fdq vwduw r zlwk wkh vwdq0
gdug Kdplowrqldq/ xqghu wkh dvvxpswlrq wkdw wkh frqvwudlqw lv qrw elqg0
lqj1 Wkhq/ zkhq wkh yduldeoh phhwv wkh xsshu erxqg zh fdq xvh wklv
lqirupdwlrq wr vwxg| wkh frqvwudlqhg lqyhvwphqw wudmhfwru|1
Wkh Kdplowrqldq lv jlyhq e|=
M Egc ^c | ' d@g| U|  2
U n ^| EU| Bg| oe3o|
2/ |
zkhuh ^| lv wkh frvwdwh yduldeoh/ zklfk phdvxuhv wkh lpsxwhg zruwk ri dq
lqfuhphqwdo xqlw ri fdslwdo lqsxw1 Wkh fruuhvsrqglqj Odjudqjldq lv=
Egc ^c bc | ' M Egc ^c | n b| d6g| U| n Bg| o e3o|
+:,
zkhuh b| lv wkh Odjudqjldq pxowlsolhu/ wkdw lv wkh vkdgrz ydoxh ri wkh
frqvwudlqw 6g| g f
Wkh rswlpdo vroxwlrq kdv wr vdwlvi| wkh iroorzlqj frqglwlrqv=
Y

+;,
G  n U| n ^| b| ' f
YU
/
zlwk
+<,
b| fc
6g| g | fc
b| 6g| g | ' f
Y
_^| e3o|
'
G ^| ' Eo n B^| E6 n B b| @
Yg
_|
Y
G g | ' U| Bg|
Y^
+43,
+44,
Dv dujxhg suhylrxvo|/ li wkh frqvwudlqw lv qrw elqglqj zh fdq xvh wkh vwdq0
gdug pd{lpxp sulqflsoh wr vroyh wkh sureohp e| vwxg|lqj wkh surshuwlhv
ri wkh g|qdplf v|vwhp +;,0+44,/ xqghu wkh frqglwlrq b| ' f
Lw lv fohdu wkdw wkh pd{lpl}dwlrq g|qdplfv ghshqgv rq lqyhvwphqw
ghflvlrqv lq hdfk shulrg1 Lq wklv shuvshfwlyh/ lw vkrxog eh qrwhg wkdw
e| htxdwlrq +;, lqyhvwphqw ghshqgv rq wkh uhodwlrqvkls ehwzhhq wkh
pdujlqdo ydoxh ri fdslwdo ^| dqg lwv pdujlqdo frvw1 Li ^| :  wkhq lq0
yhvwphqw lq shulrg | =
U| ' / E^| 
9
lv srvlwlyh1 Xvlqj wklv dujxphqw zh fdq uh hfw rq wkh klgghq dvvxpswlrqv
xqghuo|lqj wkh xqfrqvwudlqhg prgho1 Zlwk shuihfw fdslwdo pdunhwv/ wkh
up xvhv U| wr pd{lpl}h lwv ydoxh1 Jlyhq wkh h{lvwhqfh ri dgmxvwphqw
frvwv/ wkh up qhhgv/ qrqhwkhohvv/ wr wdnh lqwr dffrxqw wkh zkroh sodqqlqj
sureohp1 Htxdwlrq +43, lpsolhv wkdw dw dq| srlqw lq wlph wkh ydoxh ri qhw
lqyhvwphqw U| lv jlyhq e| wkh lqvwdqwdqhrxv surw @ soxv fdslwdo jdlqv1 Li
wklv lv vr/ wkh lqyhvwphqw ixqfwlrq iru dq| wlph | fdq eh zulwwhq dv=
@ n ^|
U| ' /

onB
Khqfh/ wkh pd{lpl}dwlrq ri surw iru doo srvvleoh srlqwv lq wlph zloo rqo|
eh srvvleoh li wkh up vhohfwv/ uljkw iurp wkh vwduw/ wkh rswlpdo fdslwdo
dffxpxodwlrq sodq zklfk vdwlvhv wkh uvw rughu frqglwlrq iru wkh lq0
whuwhpsrudo sureohp +5,1 Dv d frqvhtxhqfh/ nqrzohgjh ri ^| dqg iruzdug0
orrnlqj ehkdylru zlwk uhvshfw wr ^| duh qhfhvvdu| frqglwlrqv iru fkrrvlqj
dq rswlpdo lqyhvwphqw wudmhfwru|1 Wklv vxjjhvwv wkdw upv qhhg wr irup
h{shfwdwlrqv ri ixwxuh yduldeohv lq pdnlqj wkhlu lqyhvwphqw ghflvlrqv1
Wkh Hxohu htxdwlrq +43, vkrzv wkdw upv fdqqrw sodq zlwkrxw nqrzlqj
wkh hqwluh sdwk iru wkhlu ixwxuh fdslwdo jdlqv1 Lw iroorzv wkdw h{shfwdwlrqv
duh fuxfldo wr uhvrxufh doorfdwlrq1 Lq vkruw/ wkh nh| wr wklv edvlf prgho lv
wkdw lqyhvwphqw ghflvlrqv ghshqg/ qrw rqo| rq wkh frqwulexwlrq ri fdslwdo
wr surw/ dw d jlyhq srlqw lq wlph/ exw dovr rq wkh up*v h{shfwdwlrqv dqg
lwv qhhg wr dyrlg dgmxvwphqwv lq fxuuhqw lqyhvwphqw zklfk zrxog ylrodwh
wkh duelwudjh frqglwlrq +43,1 Jlyhq wklv h{shfwdwlrqv ylhz/ krz gr ixwxuh
qdqflqj frqvwudlqwv dhfw wkh frxuvh ri lqyhvwphqw ghflvlrqv ryhu wlphB
6
Lqyhvwphqw wudmhfwrulhv dqg frqvwudlqwv
Lq wkh suhylrxv vhfwlrq/ zh vkrzhg wkdw/ hyhq lq d ghwhuplqlvwlf zruog
zlwk shuihfw fdslwdo pdunhwv/ ixwxuh frqglwlrqv dhfw fxuuhqw lqyhvwphqw
ghflvlrqv1 Lq wklv prgho/ qdqfldo idfwruv kdg qr vxevwdqwldo uroh wr sod|1
Zkdw zh qrz kdyh wr lqyhvwljdwh lv zkdw kdsshqv wr lqyhvwphqw zkhq
wkh up rshudwhv lq wkh suhvhqfh ri +odwhqw, qdqflqj frqvwudlqwv1
Wr glvfxvv wklv srlqw/ zh zloo dvvxph wkdw wkh qdqflqj frqvwudlqw
+4, lv elqglqj/ lpso|lqj b| : f Lq wklv fdvh qhw lqyhvwphqw lv htxdo wr
6g| dqg jurvv lqyhvwphqw lv jlyhq e| U| ' E6 n Bg| Wklv htxdwlrq
:
ghwhuplqhv wkh pd{lpxp udwh ri dffxpxodwlrq udwh iru wkh frqvwudlqhg
fdvh1 Iru wklv uhdvrq/ zh zloo uhihu wr wkh orfxv=
U| ' E6 n Bg|
dv wkh erxqgdu| fxuyh1 Lq ylhz ri frqglwlrq +4,/ wkh rswlpdo lqyhvw0
phqw sdwk fdqqrw olh deryh wkh erxqgdu| fxuyh1 Jlyhq wkdw dorqj wkh
xqfrqvwudlqhg g | ' f fxuyh zh kdyh U| ' Bg| c dqg wkdw zlwk erxqghg
lqyhvwphqw zh kdyh U| ' E6 n Bg| c lw lv hylghqw wkdw wkh erxqgdu|
fxuyh pxvw qhfhvvdulo| olh deryh wkh g | ' f fxuyh +dv looxvwudwhg lq jxuh
5,1 Wklv frqvwudlqw kdv reylrxv lpsolfdwlrqv iru wkh fdslwdo dffxpxodwlrq
sdwk dqg iru wkh fruuhvsrqglqj sdwk iru wkh ydoxh ri wkh up1
Wr lqyhvwljdwh wklv lvvxh/ ohw xv h{dplqh wkh v|vwhp ri glhuhqwldo
htxdwlrqv lq U| dqg g| Dv qrwhg deryh/ zh fdqqrw hpsor| wkh vwdq0
gdug vroxwlrq wr vroyh wklv frqvwudlqhg prgho ri wzr olqhdu glhuhqwldo
htxdwlrqv ehfdxvh wklv zrxog lqwurgxfh d b | whup wkdw fdqqrw eh holp0
lqdwhg xvlqj +43,1 Zh fdq/ qrqhwkhohvv/ rewdlq d vroxwlrq/ qrwlqj wkdw
wkh lqhtxdolw| frqvwudlqw +4, lv d vlpsoh xsshu erxqg rq wkh frqwuro ydul0
deoh U| Zh fdq wkxv xvh d wzr vwhs phwkrg wr rewdlq wkh wudmhfwrulhv
iru frqvwudlqhg lqyhvwphqw1 Iluvw/ zh vwxg| wkh g|qdplf surshuwlhv ri
wkh xqfrqvwudlqhg prgho/ xqghu wkh dvvxpswlrq b| ' f Wkhq/ zkhq U|
uhdfkhv wkh erxqgdu| fxuyh/ zh xvh wklv lqirupdwlrq wr qg wkh frq0
vwudlqhg srolf|1 Wkh jhqhudo vroxwlrq iru wkh xqfrqvwudlqhg vfhqdulr lv
reylrxvo| qrw hqrxjk wr doorz xv wr ghvfuleh lqyhvwphqw lq wkh suhvhqfh
ri qdqflqj frqvwudlqwv1 Wr dfklhyh wklv/ zh kdyh wr wlh grzq wkh du0
elwudu| erxqgdu| frqglwlrqv1 Wklv zh fdq gr/ xvlqj rxu dujxphqw wkdw
rswlpl}lqj ehkdylru kdv wr hqvxuh wkh devhqfh ri duelwudjh rssruwxqlwlhv
lq dq| shulrg1 Dv zh zloo vhh/ wklv frqglwlrq doorzv wr ghulyh dq rswlpdo
lqyhvwphqw srolf| iru wkh frqvwudlqhg vfhqdulr17
Wklv lv d vwdqgdug zd| wr vroyh sureohpv ri g|qdplf rswlpl}dwlrq xqghu frq0
vwudlqwv1 D vlplodu surfhgxuh kdv ehhq xvhg/ iru h{dpsoh/ wr ghulyh wkh ydoxh ri uhdo
rswlrqv +Wuljhrujlv/ 4<<:,1 Gl{lw dqg Slqg|fn +4<<7, dsso| wklv dssurdfk wr vroyh
lqyhvwphqw ghflvlrq sureohpv zlwk luuhyhuvleoh fdslwdo1 Lq wklv frqwh{w/ luuhyhuvlelolw|
lv wuhdwhg dv d orzhu erxqg rq wkh qhw lqyhvwphqw1 Rwkhu dssolfdwlrqv fdq eh irxqg
lq Nuxjpdq +4<<4,/ dqg Iurrw dqg Revwihog +4<<4, iru sureohpv ri h{fkdqjh udwh
g|qdplfv zlwk wdujhw }rqhv> wkh fdvh ri rswlpdo frqvxpswlrq dqg sruwirolr uxohv lv
glvfxvvhg lq d vhulhv ri sdshuv e| Phuwrq +4<<3,1 Ehuwrod +4<<7, surylghv dq h{fhoohqw
glvfxvvlrq ri wkhvh whfkqltxhv/ lq wkh suhvhqfh ri fhuwdlqw| dqg xqfhuwdlqw|1 Ilqdoo|/
wklv nlqg ri phwkrgrorj| kdv dovr ehhq xvhg lq prghov ri oderu ghpdqg zlwk ulqj
dqg klulqj frvwv +Ehqwrolod dqg Ehuwrod +4<<3,,1
7
;
614
Wkh xqfrqvwudlqhg vfhqdulr
E| glhuhqwldwlqj wkh uvw rughu frqglwlrq +;, zlwk uhvshfw wr wlph/ xqghu
wkh dvvxpswlrq b| ' f/ zh rewdlq d glhuhqwldo htxdwlrq lq U| =
^| '
 u
U|
/
E| vxevwlwxwlqj ^| dqg ^| lq +43,/ zh fdq ghwhuplqh wkh g|qdplfv ri U| Iurp wklv srlqw rq/ zh fdq wkxv zrun zlwk wkh v|vwhp ri glhuhqwldo
htxdwlrqv jlyhq ehorz=
u
U | ' Eo n B U| n / Eo n B @
g | ' U| Bg|
+45,
+46,
zklfk ghvfulehv wkh xqfrqvwudlqhg g|qdplfv ri U| dqg g| ryhu wlph1 Qrwh
wkdw zkhq g | ' f lqyhvwphqw lv htxdo wr wkh uhsodfhphqw lqyhvwphqw
u
U| ' Bg| > wkh U| ' f lvrfolqh lv d frqvwdqw +ehfdxvh ri frqvwdqw uhwxuqv
wr vfdoh, htxlydohqw wr=
@
U| ' /

onB
Iljxuh 4 vkrzv wkh skdvh gldjudp fruuhvsrqglqj wr v|vwhp +45, / +46,1
u
Wkh U| ' f orfxv lv xqdhfwhg e| g| dqg lv wkxv krul}rqwdo1 Frqyhuvho|/
+46, ghvfulehv wkh g | ' f orfxv/ zklfk lv dq xszdug vorslqj fxuyh lq wkh
Eg| c U| vsdfh1 Wkh duurzv lqglfdwh wkh gluhfwlrq ri prwlrq1 Wkhuh lv wkxv
d xqltxh sdwk frqyhujlqj wr wkh vwhdg| vwdwh ./ zklfk fruuhvsrqgv wr
u
wkh htxloleulxp sdwk U| ' f
Dv vwdwhg hduolhu/ rxu dqdo|vlv ri phglxp whup lq lqyhvwphqw dvvxphv
wkdw wkh up kdv d fdslwdo hqgrzphqw ri gf xqlwv dw wlph f dqg wkdw lwv
dlp lv wr uhdfk wlph A zlwk gA xqlwv ri vwrfn1 Wkh ydoxh ri A dqg wkh
erxqgdu| ydoxhv ri gf dqg gA duh vshflhg h{rjhqrxvo|1 Wkh sureohp
lv wr fkrrvh U| vxfk dv wr pd{lpl}h wkh ydoxh ri wkh up ryhu wkh wlph
lqwhuydo dfc A o1 Iurp rxu srlqw ri ylhz/ lw lv lqwhuhvwlqj wr ghvfuleh d ihz
srvvleoh htxloleulxp sdwkv lq wkh vsdfh Eg| c U| Srlqw . lq jxuh 4 uhsuhvhqwv wkh vwhdg|0vwdwh ri wkh g|qdplf v|v0
whp> wkh wudmhfwru| wkdw ohdgv wr wklv srlqw lv wkh vwdeoh sdwk18 Wkh rwkhu
8
Wkh wudmhfwrulhv lq jxuh 4 duh gudzq iru sduwlfxodu ydoxhv ri sdudphwhuv1 Pruh
<
,W
L
LL . W ,,,
(
,9
,,
LL L
, W ,
.
.7
.W
Iljxuh 4= Wkh g|qdplfv ri wkh xqfrqvwudlqhg v|vwhp
fxuyhv ghvfuleh rwkhu ihdvleoh htxloleulxp sdwkv1 Lq uhjlrq LL/ iru h{dp0
soh/ wkh wudmhfwrulhv ulvh wr wkh ohiw dqg wxuq wr wkh uljkw rq uhdfklqj wkh
u
g | ' f orfxv1 Lw vkrxog eh qrwhg wkdw qhdu ./ g | dqg U | duh duelwudu0
lo| vpdoo> wklv lpsolhv wkdw uhdfklqj htxloleulxp zrxog uhtxluh dq lqqlwh
dprxqw ri wlph1 Iru wkh rssrvlwh uhdvrq/ wkh ixuwkhu wkh lqyhvwphqw sdwk
u
olhv iurp wkh orfxv U| ' fc wkh odujhu zloo eh wkh udwh ri lqyhvwphqw dqg
wkh idvwhu zloo eh wkh fkdqjh lq g| 1
Vxssrvh wkdw wkh up*v sureohp lv wr lqfuhdvh lwv fdslwdo vwrfn iurp
gf wr gA 1 Wkh yhuwlfdo gdvkhg olqhv wkurxjk wkrvh ydoxhv vkrz wkdw/
dowkrxjk wkh erxqgdu| ydoxhv iru wkh vwdwh yduldeoh g| duh h{rjhqrxvo|
jlyhq/ lw lv qrqhwkhohvv srvvleoh wr vhohfw wkh rswlpdo ydoxh iru wkh frqwuro
suhflvho|/ zh kdyh vxssrvhg wkdw u @ 3=5> @ 3=4> d @ 3=681 Wkh erxqgdu| frqglwlrqv
duh N3 @ 7 dqg NW @ 8= Wkh forvhqhvv ri wkh rswlpdo xqfrqvwudlqhg wudmhfwru| wr wkh
vwhdg| vwdwh ghshqgv rq wkh wlph krul}rq1 Wudmhfwru| +lll,/ iru h{dpsoh/ lv judskhg
iru d sodqqlqj krul}rq ri W @ 43> zkhuhdv wudlhfwru| +l, uhsuhvhqwv d wlph krul}rq ri
W @7
43
yduldeoh U| 1 Lqyhvwphqw srolf| fohduo| ghshqgv rq wkh dprxqw ri wlph
dfc A o dydlodeoh wr wkh up1 Wkh vkruwhu wkh wlph dydlodeoh/ wkh kljkhu zloo
eh wkh uhtxluhg udwh ri lqyhvwphqw dqg wkh juhdwhu wkh glvwdqfh ehwzhhq
u
wkh rswlpdo lqyhvwphqw wudmhfwru| dqg wkh U| ' f orfxv1 Wklv lpsolhv
wkdw/ li wkh rswlpdo srolf| lv sdwk +l,/ A zloo eh vpdoo 0 wkhuh lv/ lq rwkhu
zrugv/ olwwoh wlph dydlodeoh 0 dqg wkh udwh ri lqyhvwphqw zloo eh kljk> li
A zhuh d elw odujhu/ wkh rswlpdo sdwk pljkw eh +ll,> zlwk d orw ri wlph
dydlodeoh wkh rswlpdo wudmhfwru| frxog eh sdwk +lll,1 Wklv sdwk ehjlqv lq
uhjlrq LL/ zkhuh lw lv rswlpdo iru wkh up wr lqlwldoo| uhgxfh lwv dvvhwv lq
sodfh/ hqvxulqj wkdw lw uhdfkhv wkh erxqgdu| ydoxh gA dw wlph A Li/ lq
rwkhu zrugv/ wkhuh lv hqrxjk wlph dydlodeoh/ lw zloo eh rswlpdo iru wkh up
wr ehjlq zlwk d ohyho ri lqyhvwphqw vr orz dv wr hqvxuh dq lqlwldo skdvh
ri qhjdwlyh fdslwdo dffxpxodwlrq1 Lw vkrxog eh qrwhg/ qdoo|/ wkdw/ li wkh
lqlwldo ohyho ri lqyhvwphqw olhv lq uhjlrq L/ wkhuh lv qr ihdvleoh vroxwlrq
wr wkh frqwuro sureohp= lq wklv uhjlrq doo wudmhfwrulhv fxuyh ehqg grzq
wrzdugv wkh ohiw1
Wr vxppdul}h/ qr pdwwhu krz odujh wkh ydoxh ri A / lw lv dozd|v srvvleoh
iru wkh up wr vhohfw dq rswlpdo lqyhvwphqw sdwk/ wr ghqh wkh lqlwldo
ydoxh ri U| dqg wr vdwlvi| wkh erxqgdu| frqglwlrqv1 Li wkh up kdv d orw
ri wlph dydlodeoh/ lw fdq fkrrvh d wudmhfwru| wkdw frphv duelwudulo| forvh
wr .c zkhuh wkh g|qdplf ri wkh frqwuro dqg vwdwh yduldeohv ehfrphv yhu|
vorz1 Zkhuh/ rq wkh rwkhu kdqg/ wlph lv vkruw/ wkh rqo| zd| wr dfklhyh
gA dw wlph A lv wkurxjk d kljkhu udwh ri lqyhvwphqw1
615
Wkh frqvwudlqhg vfhqdulr
Ohw xv qrz frqvlghu wkh fdvh zkhuh wkh qdqflqj frqvwudlqw +7, lv hhf0
wlyh1 Dv vwdwhg deryh/ wklv lpsolhv g | ' 6g| zklfk phdqv wkdw jurvv
lqyhvwphqw lv dw lwv pd{lpxp ohyho ri U| ' E6 n Bg| 1 Qrwh krz/ lq j0
xuh 5/ wkh wudmhfwru| pryhv lq wkh qruwkhdvw gluhfwlrq dorqj wkh erxqgdu|
fxuyh1 Zkhq wkh frqvwudlqw lv elqglqj/ qhw lqyhvwphqw/ htxdo wr 6g| / lv
srvlwlyh1 Jlyhq wkh frqvwudlqw/ rswlpdo lqyhvwphqw sdwkv fdqqrw/ krz0
hyhu/ olh deryh wkh erxqgdu| fxuyh1 Lw iroorzv wkdw wkh vhw ri dgplvvleoh
sdwkv olhv lq wkh vsdfh ghqhg e| wkh erxqgdu| fxuyh dqg wkh g| d{lv19
Rswlpdo wudmhfwrulhv iru wkh frqvwudlqhg vfhqdulr duh gudzq dvvxplqj wkdw wkh
pd{lpxp uhvwulfwhg dffxpxodwlrq udwh lv p @ 3=38> vr wkdw wkh erxqgdu| fxuyh lv
jlyhq e| wkh htxdwlrq= Lw @ +p . , Nw @ 3=48Nw =
9
44
,W
L
,,,
W
.
δ
P
W
,
,9
(
. W LLF
LLLF
,,
, W ,
.
.
.7
.W
Iljxuh 5= Wkh g|qdplfv ri wkh frqvwudlqhg v|vwhp
Rqfh rswlpdo ydoxhv kdyh ehhq fkrvhq iru U| / iru doo srvvleoh gdwhv/
wklv ghwhuplqhv wkh dffxpxodwlrq udwh iru wkh frqvwudlqhg vwdwh yduldeoh
dqg wkxv/ jlyhq wkh lqlwldo ydoxh ri wkh fdslwdo vwrfn/ lwv ixwxuh ydoxh dw
dq| srvvleoh gdwh1 Lw lv lpsruwdqw wr vwuhvv wkdw/ dv U| dssurdfkhv wkh
erxqgdu| fxuyh wkh up uhdol}hv wkdw ixwxuh lqyhvwphqw zloo eh uhvwulfwhg
e| wkh qdqflqj frqvwudlqw1 Reylrxvo| wkh up sodqv iru wklv hyhqwxdolw|1
Dv d uhvxow/ wkh ixwxuh frqvwudlqw lv uh hfwhg lq fxuuhqw ghflvlrqv1
7 ghqrwh wkh ydoxh ri g| dw wkh lq0
Wr h{sodlq wklv sursrvlwlrq/ ohw g
u
whuvhfwlrq ehwzhhq wkh U | ' f orfxv dqg wkh erxqgdu| fxuyh1 Vwduwlqj
7 wkh wud0
iurp d srlqw dorqj wkh erxqgdu| fxuyh/ exw wr wkh ohiw ri gc
mhfwru| pryhv xszdugv dorqj wkh fxuyh1 Jlyhq wkdw/ lq wkh sur{lplw| ri
wklv fxuyh wudmhfwrulhv fdq hqwhu uhjlrq LY/ zkhuh wkh frqvwudlqw fhdvhv
wr eh dfwlyh/ wkh| frxog/ rq rffdvlrqv/ vwud| dzd| iurp wkh fxuyh1 Li
wklv kdsshqhg/ krzhyhu/ wkh dffxpxodwlrq sureohp zrxog kdyh qr ihdvl0
eoh vroxwlrq= rqfh lq wklv uhjlrq wudmhfwrulhv glyhujh iurp wkh srwhqwldoo|
45
7 rq wkh rwkhu kdqg/
rswlpdo sdwk1 Iru ydoxhv ri fdslwdo eljjhu wkdq gc
wkh wudmhfwru| fdqqrw ohdyh wkh erxqgdu| fxuyh ehfdxvh lq uhjlrq LLL/ U|
lv erxqg wr lqfuhdvh1 Lq eulhi/ lqyhvwphqw fdqqrw ohdyh wkh frqvwudlqhg
wudmhfwru|= zkhq lw glyhujhv iurp wklv wudmhfwru| frxqwhuydlolqj irufhv
whqg wr eulqj lw edfn wr lwv ruljlqdo sdwk dorqj wkh erxqgdu| fxuyh1
Dv ehiruh/ rswlpdo srolflhv ghshqg rq gf / gA dqg A / wkh wlph dydlo0
deoh wr wkh up1 Wr frpsduh rswlpdo wudmhfwrulhv zlwk dqg zlwkrxw
frqvwudlqwv zh wkxv kdyh wr xvh wkh vdph ydoxhv iru wkh fdslwdo vwrfn dw
wlphv f dqg A Wklv lv zkdw zh kdyh grqh lq jxuh 2 0 ghvfulelqj wkh
frqvwudlqhg vfhqdulr 0 zklfk fdq eh frpsduhg wr jxuh / vkrzlqj wkh
xqfrqvwudlqhg fdvh1
Lq wkh uvw sodfh/ revhuyh wkdw/ dv ehiruh/ lqyhvwphqw wudmhfwrulhv fdq0
qrw vwduw iurp uhjlrq L1 Wkh xsshu erxqg rq qdqflqj lpsolhv wkdw wudmhf0
wru| +l, lq wkh xqfrqvwudlqhg vfhqdulr lv qr orqjhu dgplvvleoh1 Wudmhfwru|
+ll, pd| dovr qrw eh ihdvleoh= li A lv vkruw wkhuh pd| qrw eh hqrxjk wlph
wr pryh iurp gf wr gA dw wkh pd{lpxp udwh ri lqyhvwphqw doorzhg e|
g | ' 6g| 1 Li wkh lqlwldo ohyho ri lqyhvwphqw olhv lq uhjlrq LLL wkh rqo|
zd| wr dfklhyh gA lq A lv wr eulqj wkh lqyhvwphqw iruzdug/ vwduwlqj dw d
kljkhu ohyho dorqj wudmhfwru| +llf, dv ghslfwhg lq jxuh 21
Wklv uhvxow lpsolhv wkdw li wkhuh lv hqrxjk dydlodeoh wlph dqg li wkh
rswlpdo wudmhfwru| ehjlqv lq uhjlrq LL/ wkh up idfhg zlwk ixwxuh frq0
vwudlqwv fdq phhw wkh erxqgdu| fxuyh zkhq wkh wudmhfwru| furvvhv wkh
g ' f fxuyh/ prylqj lqwr uhjlrq LLL1 Li wkh up zhuh wr vwduw r e|
iroorzlqj wkh xqfrqvwudlqhg lqyhvwphqw wudmhfwru| +lll,/ lw zrxog qhyhu
vxffhhg lq dfklhylqj gA = lwv ghvluhg/ qdo ohyho ri fdslwdo vwrfn1 Zkhq
wkh frqvwudlqw lv elqglqj/ wkh frqvwudlqhg up*v fdslwdo vwrfn lqfuhdvhv
pruh vorzo| wkdq dorqj wkh xqfrqvwudlqhg sdwk1 Wr dfklhyh lwv jrdov wkh
frqvwudlqhg up kdv wr vwduw r zlwk d kljkhu ohyho ri lqyhvwphqw vr dv
wr phhw zlwk wkh erxqgdu| fxuyh dw wkh +hqgrjhqrxv, rswlpdo wlph |W
zklfk vdwlvhv wkh frqvwudlqhg sureohp1 D vdolhqw ihdwxuh ri wklv rswlpdo
frqwuro sureohp lv wkdw/ lq wklv sureohp/ xqolnh wkh xqfrqvwudlqhg fdvh/
wkh rswlpdo wudmhfwru| uhtxluhv frqwlqxdo fkdqjhv lq wkh frqwuro yduldeoh
U| 1 Wklv nlqg ri lqyhvwphqw srolf| lv uhsuhvhqwhg e| sdwk +lllf, lq jxuh
51
Doo wklv lpsolhv wkdw wkh qdqflqj frqvwudlqw lq xhqfhv wkh up*v
ehkdylru hyhq zkhq lw lv idu dzd| iurp wkh eduulhu= qrw rqo| grhv wkh
up ehjlq zlwk d kljkhu ohyho ri lqyhvwphqw/ exw lw dovr fkrrvhv glhuhqw
46
lqyhvwphqw udwhv iru lqwhuphgldwh skdvhv zkhq wkh frqvwudlqw lv vwloo vodfn1
Lq d zrug/ wkh srwhqwldoo| frqvwudlqhg up dgrswv d lqyhvwphqw srolf|
wkdw lv glhuhqw iurp wkh srolf| dgrswhg e| wkh xqfrqvwudlqhg up1
,W
LY ,,,
. W P
δ
W
.
$
W
,
,,
%
,9
&
×
.
(
, W ,
.
.7
Iljxuh 6= Dq xqihdvleoh frqvwudlqhg sdwk
.W
Vlqfh wkh qdqflqj frqvwudlqw uhvwulfwv wkh udwh ri fdslwdo dffxpxod0
wlrq/ rwkhu erxqgdu| uhtxluhphqwv fdq pdnh vrph lqyhvwphqw srolflhv
xqihdvleoh1 Li/ iru h{dpsoh/ wkh lqlwldo ohyho ri lqyhvwphqw lv lq uhjlrq LLL/
deryh wkh erxqgdu| fxuyh/ wkh frqwuro sureohp kdv qr ihdvleoh vroxwlrq
+jxuh ,1 Dv dq h{dpsoh vxssrvh wkdw wkh qdo vwrfn lv vwloo gA c exw
f gf c rq wkh ohiw ri wkh vwhdg| vwdwh .1
wkdw wkh lqlwldo vwrfn lv qrz g
Zh vkdoo djdlq uhvwulfw rxu dwwhqwlrq wr d ihz/ ihdvleoh sdwkv1 Iru d jlyhq
sodqqlqj krul}rq/ sdwk +ly, pd| eh rswlpdo iru wkh xqfrvwudlqhg up1
Iru wkh frqvwudlqhg up/ rq wkh rwkhu kdqg/ lw lv xqdwwdlqdeoh1 Lqghhg/
hyhq wkh lqlwldo ohyho ri lqyhvwphqw +srlqw , olhv rq wkh zurqj vlgh ri wkh
47
erxqgdu| fxuyh1 Lw iroorzv wkdw li wkh qdqflqj frqvwudlqw lpsrvhv dq
xsshu erxqg rq lqyhvwphqw/ wkh lqlwldo lqyhvwphqw zloo eh vpdoohu wkdq
/ iru h{dpsoh srlqw rq wkh erxqgdu| fxuyh1 Iru wklv wkuhvkrog ydoxh/
krzhyhu/ wkhuh pd| eh qr ihdvleoh vroxwlrq iru wkh frqvwudlqhg wudmhfwru|=
wkh dffxpxodwlrq udwh g
' 6 ryhu wkh lqwhuydo dfc A o frxog eh lqvx!flhqw
g
wr dfklhyh gA e| wlph A Qrwh wkdw iru ohyhov ri lqlwldo lqyhvwphqw o|lqj
ehorz wkh erxqgdu| fxuyh/ dv lq wkh fdvh ri srlqw 0 forvh wr wkh vwhdg|
vwdwh 0 wkh lqyhvwphqw udwh lv hyhq vpdoohu wkdq wkh udwh iru wkh rswlpdo
frqvwudlqhg wudmhfwru|1
Wkh iruzdug0orrnlqj dssurdfk wkxv h{sodlqv krz qdqflqj frqvwudlqwv
lq xhqfh wkh dprxqw ri lqyhvwphqw qhhghg wr dfklhyh d fhuwdlq fdslwdo
vwrfn dqg d fhuwdlq rxwsxw1 Rqfh wkh ghvluhg ohyho ri fdslwdo vwrfn lv
ghqhg/ wkhuh lv dq lqfhqwlyh iru wkh up wr vhohfw dq lqyhvwphqw sdwk
zklfk pdnhv lw srvvleoh wr dfklhyh wkdw vwrfn lq wkh wlph dw lwv glvsrvdo1
Ixwxuh qdqflqj frqvwudlqwv pd| lqgxfh wkh up wr pdlqwdlq lwv surmhfwv
dqg fkdqjh lwv ruljlqdo lqyhvwphqw sodqv> dowhuqdwlyho| lw fdq dedqgrq d
surmhfw zkhq dq lqyhvwphqw srolf| lv qr orqjhu ihdvleoh1
Ilqdqflqj frqvwudlqwv/ lq rwkhu zrugv/ kdyh gudpdwlf hhfwv rq wkh
ryhudoo g|qdplfv ri lqyhvwphqw/ dhfwlqj rswlpdo srolf| hyhq zkhq wkh|
duh fxuuhqwo| vodfn1 Vlqfh wkh up nqrzv wkdw lw zloo idfh uhvwulfwlrqv
lq wkh ixwxuh/ lw qgv lw rswlpdo wr fkrrvh d srolf| zklfk wdnhv dffrxqw
ri wkh ixwxuh frqvwudlqw/ uljkw iurp wkh ehjlqqlqj1 Lq vkruw/ wkh up*v
lqyhvwphqw vwudwhj| lv ghwhuplqhg qrw rqo| e| wkh fxuuhqw ydoxh ri h{rjh0
qrxv yduldeohv exw dovr e| wkh ixwxuh ydoxh ri wkh frqvwudlqw1 Wklv uhvxow
fdvw grxewv rq wkh sursrvlwlrq wkdw/ li lqyhvwphqw ghflvlrqv duh fxuuhqwo|
xqdhfwhg e| frqvwudlqwv/ wkh up zloo ehkdyh dv li lw zloo qhyhu idfh
frqvwudlqwv lq wkh ixwxuh1
7
Rswlpdo wudmhfwrulhv dqg duelwudjh
Zkdw lv lw wkdw hqvxuhv wkdw wkh wudmhfwrulhv vkrzq lq jxuh  dqg jxuh
2 duh rswlpdo iru wkh upB Lq rwkhu zrugv/ jlyhq wkh erxqgdu| frqglwlrqv
rq g| c wkh wlph krul}rq/ dqg wkh ydoxhv ri wkh h{rjhqrxv yduldeohv/ duh
wkhuh rwkhu sdwkv zklfk vdwlvi| wkh rswlpdo frqwuro sureohpB
Wr jlyh d vdwlvi|lqj dqvzhu wr wkhvh txhvwlrqv zh kdyh wr irfxv rq wkh
Hxohu htxdwlrq1 Htxdwlrq +43, lv dq duelwudjh frqglwlrq= dw dq| wlph |/
wkh up*v pdujlqdo ydoxh ^| pxvw eh htxdo wr wkh pdujlqdo surw @ glv0
48
frxqwhg dw wkh pdunhw udwh onB/ soxv wkh glvfrxqwhg ydoxh ri fdslwdo jdlqv
^| 1 Li b| : f/ htxdwlrq +43, vwloo kdv wkh vdph phdqlqj/ wkrxjk qrz zh
kdyh lqwurgxfhg d elqglqj qdqflqj frqvwudlqw1 Lq erwk fdvhv/ wkh rswl0
pdo lqyhvwphqw srolf| iru wkh up/ kdv wr vdwlvi| wkh vdph lqwhuwhpsrudo
frqglwlrq 0 l1h1 wkh Hxohu htxdwlrq 0 vr dv wr dyrlg duelwudjh rssruwxql0
wlhv1 Li wklv uhtxluhphqw lv ylrodwhg/ lqyhvwruv fdq rewdlq duelwudjh surwv
e| frqvwuxfwlqj sruwirolrv ri wudghg dvvhwv1 Dvvhw sulflqj kdyh wr eh re0
wdlqhg/ krzhyhu/ lq frqglwlrqv wkdw suhfoxgh duelwudjh1 Frqvhtxhqwo|/ zh
fdq fodlp wkdw odfn ri duelwudjh lv dq htxloleulxp frqglwlrq= dq lqyhvw0
phqw ghflvlrq lv fruuhfwo| sulfhg rqo| li lw doorzv qr rssruwxqlwlhv iru
duelwudjh1
Wkhvh frqvlghudwlrqv pxvw eh uh hfwhg lq wkh dqdo|wlfdo phwkrg xvhg
wr vroyh wkh rswlpdo frqwuro sureohp +5,/ xqghu wkh dowhuqdwlyh dvvxps0
wlrqv ri dq xqfrqvwudlqhg dqg d frqvwudlqhg zruog1 Lq jhqhudo whupv/ wklv
phdqv wkdw/ ryhu wkh udqjh dfc A o/ wkh lqyhvwphqw srolf| U| kdv wr hqvxuh
wkdw wkh dffxpxodwlrq sdwkv iru g| dqg iru wkh up*v ydoxh ^| pxvw eh
frqwlqxrxv dqg vprrwk lq dq| shulrg | Lw lv rqo| zkhq g| dqg ^| duh frq0
wlqxrxv dqg vprrwk wkdw wkh uvw ghulydwlyh ^| h{lvwv dqg lv frqwlqxrxv1
Wklv frqglwlrq jxdudqwhhv wkh devhqfh ri duelwudjh1
Wkh vdph frqglwlrqv rq wkh vwdwh yduldeohv dsso| lq wkh suhvhqfh ri
qdqflqj frqvwudlqwv1 Khqfh/ wkh g|qdplfv ri wkh frqvwudlqhg v|vwhp
kdyh wr eh vxfk wkdw zkhq wkh xqfrqvwudlqhg lqyhvwphqw sdwk U| phhwv
zlwk wkh erxqgdu| fxuyh E6 n B g| dw vrph wlph |W c zlwk f |W A /
wkh vwdwh yduldeohv/ ^| dqg g| / frqyhujh wr wkh frqvwudlqhg vwdwxv vprrwko|
dqg zlwkrxw glvfrqwlqxlw|1 Wklv lv wkh rqo| ehkdylru zklfk jxdudqwhhv
wkdw wkh lqyhvwphqw lv fruuhfwo| sulfhg= li wkh sdwkv iru g| dqg ^| zhuh
glvfrqwlqxrxv lw zrxog eh lpsrvvleoh wr fdofxodwh wkh uvw ghulydwlyh ^| >
wklv zrxog ylrodwh wkh duelwudjh uhodwlrqvkls +43,1
Lw lv wkhvh hfrqrplf lvvxhv zklfk olh dw wkh khduw ri wkh dqdo|wlfdo vr0
oxwlrq iru rswlpdo frqvwudlqhg wudmhfwrulhv/ zklfk zh suhvhqw ehorz1 Wkh
vdph uhdvrqlqj fdq sod| d fhqwudo uroh lq dqvzhulqj wkh txhvwlrqv udlvhg
hduolhu1 Lq d iruzdug0orrnlqj hqylurqphqw upv sodq iru ixwxuh hfrqrplf
frqglwlrqv1 Lq wklv zd|/ odwhqw qdqflqj frqvwudlqwv fdq h{huw lpphgl0
dwh hhfwv rq lqyhvwphqw ghflvlrqv1 Wkh uhtxluhg vkliw lq srolf| phdqv
wkdw wkh Hxohu htxdwlrq fkdqjhv shulrg e| shulrg lq rughu wr uhpdlq
ydolg hyhq zkhq wkh qdqflqj frqvwudlqw lv dfwlyh1 Dv d frqvhtxhqfh/ wkh
udwh ri fkdqjh lq wkh vwdwh yduldeohv ghshqgv/ dw dq| jlyhq wlph/ rq wkh
49
h{rjhqrxv yduldeohv/ rq wlph lwvhoi dqg rq wkh ydoxh ri ixwxuh frqvwudlqwv1
714
Wkh dqdo|wlfdo vroxwlrq
Zh vwduw zlwk wkh suhylrxv xqfrqvwudlqhg v|vwhp=
u
U| ' Eo n B U| n / Eo n B @
g | ' U| Bg|
Wklv lv d v|vwhp ri olqhdu uvw rughu glhuhqwldo htxdwlrqv zlwk frqvwdqw
frh!flhqwv1 gr dqg Ur lqglfdwh wkh vwhdg| vwdwh vroxwlrq jlyhq e| wkh
u
frqglwlrqv g | ' f dqg U| ' f Wkxv/ wkh xvxdo phwkrgrorj| dssolhv1
Wr qg d vroxwlrq iru wkh krprjhqhrxv sduw zh jxhvv U| ' ek| dqg
g| ' ek| c zkhuh c dqg k duh frqvwdqwv wr eh ghwhuplqhg1 E|
vxevwlwxwlrq zh rewdlq wkh fkdudfwhulvwlf htxdwlrq=
k2 ok oB n B 2 ' f
zlwk wkh wzr urrwv k : f dqg k2 f Wkxv/ iru dq| vlqjoh ydoxh ri
wkh urrwv/ zh kdyh d glhuhqw vroxwlrq iru wkh frqvwdqwv1 Ohw  dqg
 eh wkh ydoxhv ri wkh frqvwdqwv zkhq k ' k / dqg 2 dqg 2 duh wkh
fruuhvsrqglqj ydoxhv zkhq k ' k2 Wkh jhqhudo vroxwlrq ri wkh v|vwhp
+43, fdq wkxv eh zulwwhq=
U| ' S  ek | n S2 2 ek2 | n Ur
g| ' S  ek | n S2 2 ek2 | n gr
zkhuh S dqg S2 duh ghwhuplqhg e| wkh erxqgdu| frqglwlrqv1 Iru wkh
xqfrqvwudlqhg sureohp lw lv hqrxjk wr ghqh wzr erxqgdu| frqglwlrqv1
Wr rewdlq wkh wudmhfwrulhv gudzq lq jxuh  zh uhtxluh/ lq sduwlfxodu/
wkdw zkhq | ' f wkh fdslwdo vwrfn lv gf c dqg wkdw zkhq | ' A wkh vwrfn
vkrxog eh gA
gf ' S  n S2 2 n gr
gA ' S  ek A n S2 2 ek2 A n gr
Vr/ zh kdyh wzr htxdwlrqv doorzlqj xv wr fdofxodwh wkh ydoxhv ri S dqg
S2 zklfk vdwlvi| wkh v|vwhp1 Qrwh wkdw wkh ydoxh ri wkhvh wzr frqvwdqwv
4:
fkdqjhv zlwk fkdqjhv lq wkh erxqgdu| frqglwlrqv ru wkh wlph krul}rq1
Ohw SW dqg SW2 lqglfdwh wkh fruuhvsrqglqj vroxwlrqv1 Wkhvh ydoxhv jxdu0
dqwhh wkh devhqfh ri duelwudjh dorqj wkh wudmhfwru|1 Wklv fdq eh suryhg
e| vxevwlwxwlqj wkh ydoxhv ri wkh rswlpdo fdslwdo vwrfn ghulyhg iurp wklv
v|vwhp lq wkh duelwudjh htxdwlrq +43,/ zklfk kdv d frqwlqxrxv/ vprrwk
g|qdplf fxuyh1
Exw fdq SW dqg SW2 vdwlvi| wkh vprrwkqhvv frqglwlrq iru g| dqg ^|
lq wkh frqvwudlqhg vfhqdulrB Dv h{sodlqhg deryh/ qdqflqj frqvwudlqwv
irufh fkdqjhv lq wkh dffxpxodwlrq surfhvv vr dv wr hqvxuh wkdw wkh du0
elwudjh htxdwlrq lv vdwlvhg dw doo wlphv1 Qrz/ krzhyhu/ wkh htxdwlrq
kdv fkdqjhg= wkh vkdgrz ydoxh b| ri wkh frqvwudlqw dhfwv wkh ydoxh ri
wkh up/ dowhulqj lqyhvwphqw ghflvlrqv gxulqj lqwhuphgldwh skdvhv1 Wklv
phdqv wkdw wkh ydoxhv ri S dqg S2 wkdw vdwlvi| wkh frqvwudlqhg v|vwhp duh
glhuhqw1 Wr rewdlq wkhvh fulwlfdo ydoxhv/ zh qhhg dq dgglwlrqdo erxqgdu|
frqglwlrq lq rughu wr wlh grzq wkh ydoxh ri wkh frqvwdqwv S dqg S2 Wklv lv
d frqvhtxhqfh ri wkh uhtxluhphqw wkdw wkh fxuyh iru g| kdv wr eh vprrwk
hyhq lq wkh suhvhqfh ri frqvwudlqwv/ vr dv wr uhvshfw wkh Hxohu htxdwlrq1
Iru wklv uhdvrq zh lpsrvh wkh iroorzlqj frqglwlrqv rq g| =
S  n S2 2 n gr ' gf
W
g |W ' 6gA e6E| 3A ] |W
] A
W
g| _| n
6gA e| 3A _| ' gA gf
zkhuh=
f
|W
g |W S  ek | n S2 2 ek2 | n Ur B S  ek | n S2 2 ek2 | n gr
+47,
+48,
+49,
lv wkh rswlpdo wudmhfwru| iru wkh srwhqwldoo| frqvwudlqhg up*v qhw lqyhvw0
phqw lq wkh lqwhuphgldwh skdvhv zkhq wkh frqvwudlqw lv vodfn1
Frqglwlrq +47, ghulyhv iurp wkh revhuydwlrq wkdw/ lq wkh lqlwldo sh0
ulrg | ' f/ wkh fdslwdo vwrfn lv htxdo wr gf Wkh rwkhu wzr frqglwlrqv
ghulyh iurp rswlpdo lqyhvwphqw frqvlghudwlrqv1 |W lv wkh wlph dw zklfk
wkh frqvwudlqw elqgv1 Qrwh wkdw/ frqwudu| wr wkh vwdqgdug dssurdfk/ wkh
dqwlflsdwlyh ehkdylru ri wkh up wudqvirupv wkh frqvwudlqhg frqwuro sure0
ohp lqwr d sureohp ri rswlpdo vwrsslqj wlph1 Lqghhg/ |W lv wkh rswlpdo
wlph iru wkh srwhqwldoo| frqvwudlqhg up*v xqfrqvwudlqhg wudmhfwru| wr
phhw wkh frqvwudlqhg dffxpxodwlrq sdwk1 Vlqfh htxdwlrq +48, uhtxluhv
4;
wkdw/ dw |W / qhw lqyhvwphqw vkrxog eh wkh vdph rq erwk wkh frqvwudlqhg
dqg xqfrqvwudlqhg wudmhfwrulhv/ wklv frqglwlrq lpsolhv wkdw wkh wudmhfwru|
ri g| vkrxog eh vprrwk dw wlph |W = dw wkh frqwdfw srlqw |W erwk wudmhfwr0
ulhv vkrxog kdyh wkh vdph dffxpxodwlrq udwh1 Dv vkrzq lq jxuh 7 wklv
lv d qhfhvvdu| frqglwlrq li wkh fxuyh iru g| lv wr eh vprrwk1
Iljxuh 7= Wkh g|qdplfv ri g
Ilqdoo|/ wkh odvw frqglwlrq +49, uhtxluhv wkdw qhw lqyhvwphqw dorqj wkh
rswlpdo frqvwudlqhg wudmhfwru| kdv wr eh htxdo wr wkh wrwdo yduldwlrq gA gf c ghqhg e| wkh erxqgdu| ydoxhv rq wkh fdslwdo vwrfn1 Wkh vroxwlrq ri
wklv v|vwhp surylghv qhz rswlpl}lqj ydoxhv iru S dqg S2 wkdw vdwlvi| wkh
frqvwudlqhg frqwuro sureohp1 Qrwh wkdw lw lv wkhvh ydoxhv zklfk duh xvhg
lq jxuh 5 wr frpsduh wkh g|qdplfv ri U| yhuvxv g| lq wkh suhvhqfh ri
ixwxuh frqvwudlqwv wkdw duh fxuuhqwo| vodfn1 Dv glvfxvvhg deryh/ zh qrz
rewdlq glhuhqw wudmhfwrulhv iru wkh vdph erxqgdu| frqglwlrqv1
Lq jxuh e dqg D zh sorw g| dqg wkh fruuhvsrqglqj ydoxh ^| ri wkh
+srwhqwldoo|, frqvwudlqhg up/ dv d ixqfwlrq ri wlph | Wkh jxuhv vkrz
wkdw/ ryhu wkh lqwhuydo ri wlph ehwzhhq wkh lqlwldo shulrg dqg wkh fulwlfdo
wlph |W / wkh up*v fkrlfh lv xqfrqvwudlqhg dqg wkh dffxpxodwlrq sdwk
4<
Iljxuh 8= Wkh g|qdplfv ri ^
WW
iroorzv d iuhh wudmhfwru|1 Wklv sdwk lv fkdudfwhul}hg e| SWW
 dqg S2 = wkh
qhz rswlpl}lqj ydoxhv iru wkh frqvwdqwv1 Exw/ dw wkh fulwlfdo wlph |W qhw
lqyhvwphqw lv frqvwudlqhg dqg wkh erxqghg g|qdplfv suhydlo1 Wkh nh|
ihdwxuh ri erwk gldjudpv lv/ krzhyhu/ wkdw wkh fxuyhv iru g| dqg ^| duh
frqwlqxrxv dqg vprrwk/ qrw rqo| lq wkh lqwhuphgldwh skdvhv zkhq wkh
frqvwudlqw lv vodfn exw hyhq zkhq lw ehfrphv erxqghg dw wlph |W 1
Qrwh/ ixuwkhu/ wkdw/ dv |W gudzv forvhu/ wkh odwhqw qdqflqj frqvwudlqw
h{huwv dq hyhu vwurqjhu lq xhqfh rq fxuuhqw lqyhvwphqw srolf|1 Dv wkh
ohyho ri lqyhvwphqw dssurdfkhv wkh frqvwudlqw ydoxh/ wkh up uhdol}hv wkdw
ixwxuh lqyhvwphqw sodqv zloo eh frqvwudlqhg e| wkh dydlodelolw| ri qdqf0
lqj1 D udwlrqdo up zloo sodq iru wklv> dv d uhvxow/ ixwxuh frqvwudlqwv zloo
eh uh hfwhg lq wkh up*v fxuuhqw ydoxh1 Lq wkhvh flufxpvwdqfhv/ lw lv qrw
vxusulvlqj wkdw dv wlph whqgv wr |W / g| dqg ^| frqyhujh vprrwko| wr wkh
uhvwulfwhg g|qdplfv phhwlqj dw wkh wuljjhu wlph |W c lq vxfk d zd| dv wr
vdwlvi| wkh Hxohu htxdwlrq dorqj wkh hqwluh lqyhvwphqw suroh1
53
8
Frqfoxvlrq
Lq wklv sdshu zh kdyh vkrzq wkdw wkh suhvhqfh ri qdqflqj frqvwudlqwv
lq xhqfhv upv* ehkdylru hyhq zkhq wkh ohyho ri fxuuhqw lqyhvwphqw lv
idu ehorz wkh ohyho zkhuh wkh frqvwudlqw zrxog dsso|1 Pruh suhflvho|/ wkh
suhvhqfh ri ixwxuh qdqflqj frqvwudlqwv fkdqjhv wkh rswlpdo lqyhvwphqw
wudmhfwru| iru wkh up/ hyhq dw wlphv zkhq wkh frqvwudlqw lv vodfn1 Lw lv
vkrzq wkdw/ frqwudu| wr wkh vwdqgdug ylhz/ dq rswlpdo srolf| zloo uhtxluh
frqwlqxdo vkliwv lq wkh ohyho ri lqyhvwphqw hqvxulqj wkdw wkh Hxohu htxdwlrq
fkdqjhv shulrg e| shulrg dqg lv vdwlvhg hyhq zkhq wkh frqvwudlqw lv
dfwlyh1 Wklv vljqlhv wkdw upv zklfk idfh udwlrqlqj lq wkh ixwxuh/ exw
zklfk duh iuhh iurp frqvwudlqwv dw wkh suhvhqw wlph/ zloo fkrrvh/ uljkw
iurp wkh vwduw/ dq rswlpl}lqj lqyhvwphqw sdwk zkhuh wkh iuhh dqg wkh
frqvwudlqhg wudmhfwrulhv duh frqwlqxrxv dw wkhlu pdwfklqj srlqw1 Lq wkh
suhvhqfh ri qdqflqj frqvwudlqwv lw lv rqo| wklv frqwlqxlw| frqglwlrq zklfk
fdq jxdudqwhh wkh devhqfh ri duelwudjh rssruwxqlwlhv dw dq| wlph/ dqg
wkxv wkh rswlpdolw| ri lqyhvwphqw srolf|1
54
Uhihuhqfhv
Ehqwrolod V1 dqg J1 Ehuwrod +4<<3,1 Ilulqj Frvwv dqg Oderu Ghpdqg=
Krz Edg lv Hxurvfohurvlv1 Uhylhz ri Hfrqrplfv Vwxglhv 8:/ 6;407351
Ehuwrod J1 +4<<7,1 Frqwlqxrxv0wlph Prgho ri H{fkdqjh Udwhv dqg
Lqwhuyhqwlrq1 Lq I1 Ydq ghu Sorj +hg1,/ Wkh Kdqgerrn ri Lqwhuqdwlrqdo
Pdfurhfrqrplfv/ R{irug XN/ Eodfnzhoo
Erqg/ V1 dqg F1 Phjklu +4<<7,1 G|qdplf Lqyhvwphqw Prghov dqg wkh
Ilup*v Ilqdqfldo Srolf|1 Uhylhz ri Hfrqrplf Vwxglhv 94/ 4<:05551
G*Dxwxpqh/ D1 dqg S1 Plfkho +4<;8,1 Ixwxuh Lqyhvwphqw Frqvwudlqwv
Uhgxfh Suhvhqw Lqyhvwphqw1 Hfrqrphwulfd 86/ 53605391
Ghdwrq/ D1 +4<<4,1 Vdylqjv dqg Oltxlglw| Frqvwudlqwv1 Hfrqrphwulfd
8</ 45540457;1
Gl{lw D1 dqg U1 Slqg|fn +4<<7,1 Lqyhvwphqw xqghu xqfhuwdlqw|/ Sulqfh0
wrq Xqlyhuvlw| Suhvv/ Qhz Mhuvh|
Iurrw N1 dqg P1 Revwihog +4<<4,1 H{fkdqjh Udwh G|qdplfv xqghu
Vwrfkdvwlf Uhjlph Vkliwv= D Xqlhg Dssurdfk1 Mrxuqdo ri Lqwhuqdwlrqdo
Hfrqrplfv 64/ 53605<1
Kd|dvkl I1 +4<;:,1 Whvw iru Oltxlglw| Frqvwudlqwv= D Fulwlfdo Vxuyh|
dqg Vrph Qhz Revhuydwlrqv1 lq Wuxpdq I1 Ehzoh| +hg1,/ Dgydqfhv lq
Hfrqrphwulfv= Iliwk Zruog Frqjuhvv/ Yro15/ <404531
Kxeedug J1 +4<<;,1 Fdslwdo0pdunhw Lpshuihfwlrqv dqg Lqyhvwphqw1
Mrxuqdo ri Hfrqrplf Olwhudwxuh [[[YL/ 4<605581
Kxeedug J1/ D1N1 Ndvk|ds/ dqg W1P1 Zklwhg +4<<8,1 Lqwhuqdo Il0
qdqfh dqg Ilup Lqyhvwphqw1 Mrxuqdo ri Prqh| Fuhglw dqg Edqnlqj 5:/
9;60:341
Mdhh/ G1 dqg M1 Vwljolw} +4<<3,1 Fuhglw Udwlrqlqj1 lq Kdqgerrn ri
Prqhwdu| Hfrqrplfv yro1LL/ hglwhg e| E1P1 Iulhgpdq dqg I1K1 Kdkq/
;6<0;;;1
Nuxjpdq S1 U1 +4<<4,1 Wdujhw ]rqhv dqg H{fkdqjh Udwh G|qdplfv1
Txduwhuo| Mrxuqdo ri Hfrqrplfv/ 439/ 99<0;51
Phuwrq U1 F1 +4<<3,1 Frqwlqxrxv0Wlph Ilqdqfh1 Fdpeulgjh/ Eodfn0
zhoo1
Ploqh/ D1 dqg G1 Urehuwvrq +4<<9,1 Ilup Ehkdylrxu xqghu Wkuhdw ri
Oltxlgdwlrq1 Mrxuqdo ri Hfrqrplfv G|qdplfv dqg Frqwuro 53/ 475:0477<1
Vdowdul H1 dqg J1 Wudydjolql +5333,1 Ilqdqfldo Frqvwudlqwv dqg Lq0
yhvwphqw Ghflvlrqv1 Vfrwwlvk Mrxuqdo ri Srolwlfdo Hfrqrp| 7;/ 66306771
55
Wuljhrujlv O1 +4<<:,1 Uhdo Rswlrqv/ Plw suhvv/ Fdpeulgjh/ Pdvvdfk0
vhwwv1
Zklwhg W1 +4<<5,1 Ghew/ Oltxlglw| Frqvwudlqwv dqg Frusrudwh Lqyhvw0
phqw= Hylghqfh iurp Sdqho Gdwd1 Mrxuqdo ri Ilqdqfh [OYLL/ 47580931
]hoghv/ V1 +4<;<,1 Frqvxpswlrq dqg Oltxlglw| Frqvwudlqwv= Dq Hp0
slulfdo Lqyhvwljdwlrq1 Mrxuqdo ri Srolwlfdo Hfrqrp| <:/ 6380679
56