W4TihuiU| UL4Ti||L? @?_ |i _)?@4Ut Lu
4@h!Tt
Huln Eulwwrq
Mhqv G M Oduvhq Ldq Vpdoo |
W
5|hU|h@* ,UL?L4U ?@*)tt #tL?c @?! Lu ,?}*@?_
o ,h! h||L? ?L Lh!t uLh uLh_ ,UL?L4U 6LhiU@t|?}
_
W@? 54@** ?L Lh!t uLh wi iUL? w|_
Ai it i Thitti_ @hi |Lti Lu |i @|Lhtc ?L| ?iUitt@h*) |Lti Lu |i
@?! Lu ,?}*@?_c wi iUL? w|_ Lh uLh_ ,UL?L4U 6LhiU@t|?} `i @hi
}h@|iu* |L *iU h)t|@*c 5@4! #@hc O # L?c +iMiUU@ #hihc
aL?@|@? O@t!i*c 5|ii **@h_c k@|ih?i ittc 54L? hUi @?_
T@h|UT@?|t @| @ @?! Lu ,?}*@?_ ti4?@h uLh i*Tu* _tUttL?t @?_
UL44i?|t A@?!t @hi @*tL _i |L *t|@h *?i L ThL_i_ t | @
5Ui?|U `Lh_ |i4T*@|i
Wtti_ M) |i @?! Lu ,?}*@?_c wL?_L?c ,2+ HO |L U hi^it|t uLh
?__@* ULTit tL*_ Mi @__hitti_( i?i*LTit tL*_ Mi 4@h!i_ uLh |i
@||i?|L? Lu M*U@|L?t BhLT EAi*iTL?iG f2f.Sf eff `Lh!?}
@Tiht @hi @*tL @@*@M*i uhL4 |i @?!<t W?|ih?i| t|i @|
||TG%%M@?!Lui?}*@?_UL!%T*t||4
@?! Lu ,?}*@?_ 2fff
W55 SHDDS2
L?|i?|t
Devwudfw
8
4 Lqwurgxfwlrq
:
5 Wkh prgho
44
6 Fdoleudwlrq
58
7 G|qdplf surshuwlhv ri wkh prgho
74
8 Frqfoxvlrq
95
Dsshqgl{= Vroylqj wkh prgho
97
Uhihuhqfhv
:4
Mt|h@U|
Lq d g|qdplf jhqhudo htxloleulxp +GJH, prgho zkhuh jrrgv pdunhwv
duh lpshuihfwo| frpshwlwlyh/ zh fkdudfwhulvh wkh g|qdplfv ri wkh
pdun0xs ri sulfhv ryhu pdujlqdo frvwv xqghu wzr glhuhqw vhwv ri
dvvxpswlrqv derxw pdunhw vwuxfwxuh1 Lq wkh fxvwrphu pdunhw prgho/
upv orzhu wkhlu pdun0xs zkhq fxuuhqw rxwsxw lv orz uhodwlyh wr ixwxuh
surwv/ iruhjrlqj fxuuhqw surwv lq rughu wr fdswxuh ixwxuh pdunhw
vkduh1 Lq pdunhwv fkdudfwhulvhg e| lpsolflw frooxvlrq/ upv orzhu wkhlu
pdun0xs zkhq fxuuhqw rxwsxw lv kljk uhodwlyh wr ixwxuh surwv lq rughu
wr orzhu wkh lqfhqwlyhv wr xqghufxw wkh lpsolflw fduwho1 Zh fkdudfwhulvh
wkh g|qdplfv e| dqdo|vlqj wkh uhvsrqvh ri wkh pdun0xs/ hpsor|phqw
dqg rxwsxw wr vkrfnv wr ghpdqg +lghqwlhg e| lqqrydwlrqv lq
jryhuqphqw h{shqglwxuh,/ WIS jurzwk/ dqg wr sulfhv ri lpsruwhg
pdwhuldov1 Wkrxjk wkh wzr pdun0xs prghov kdyh txdqwlwdwlyho| vlplodu
surshuwlhv lq whupv ri wkh g|qdplfv ri rxwsxw dqg hpsor|phqw/ wkh|
glhu txdolwdwlyho|wkh lpsolflw frooxvlrq prgho lqfuhdvhv wkh rxwsxw
uhvsrqvh wr vkrfnv wr jryhuqphqw h{shqglwxuh dqg wr wkh sulfh ri
lpsruwhg pdwhuldov/ zkloh wkh fxvwrphu pdunhw prgho gdpshqv
xfwxdwlrqv lq uhvsrqvh wr wkhvh vkrfnv1 Rqo| wkh fxvwrphu pdunhw
prgho jhqhudwhv pdun0xs g|qdplfv wkdw frqirup zlwk wkh hpslulfdo
hylghqfh rq pdun0xs sulflqj lq wkh Xqlwhg Nlqjgrp/ vxfk dv Vpdoo
+4<<:,/ wkdw qgv surf|folfdo pdun0xsv dw wkh vhfwrudo ohyho/ dqg zlwk
wkh sulruv hpehgghg lq w|slfdo pdfurhfrqrphwulf prghov/ vxfk dv wkh
rqh xvhg dw Edqn ri Hqjodqg1 Wkh hpslulfdo hylghqfh lv dw rggv zlwk
wkh hylghqfh iru wkh Xqlwhg Vwdwhv dqg rxu wkhruhwlfdo lqyhvwljdwlrq
vxjjhvwv wkdw dw d pdfurhfrqrplf ohyho/ wkh rqo| zd| wr uhfrqfloh wkhvh
wzr idfwv lv wr dvvxph glhuhqw prghov ri sulflqj ehkdylrxu1
Mrxuqdo ri Hfrqrplf Olwhudwxuh fodvvlfdwlrq= H65/ H6:/ H56/ O44
Nh|zrugv= Lpshuihfw frpshwlwlrq/ pdun0xsv/ fxvwrphu pdunhwv/
lpsolflw frooxvlrq
8
W?|hL_U|L?
Wklv sdshu dgguhvvhv wkh txhvwlrq ri krz upv vhw sulfhv lq uhvsrqvh wr
vkrfnv lq dq lpshuihfwo| frpshwlwlyh hqylurqphqw1 Wklv pdwwhuv
ehfdxvh sulflqj ehkdylrxu dhfwv qrw rqo| wkh lqdwlrq udwh exw dovr
wkh rxwsxw dqg hpsor|phqw uhvsrqvhv wr d jlyhq vkrfn1 Wkh pdlq
frqwulexwlrq ri wkh sdshu lv wr ghyhors d wkhruhwlfdo iudphzrun lq
zklfk glhuhqw dvvxpswlrqv derxw sulflqj ehkdylrxu dqg wkhlu
pdfurhfrqrplf lpsolfdwlrqv fdq eh dqdo|vhg/ dqg wr h{dplqh
djjuhjdwh XN gdwd zlwklq wklv iudphzrun1 Wkh prghogudzq iurp d
vhtxhqfh ri sdshuv e| Urwhpehuj dqg Zrrgiruglv d g|qdplf jhqhudo
htxloleulxp prgho ri dq hfrqrp| zkhuh wkh rxwsxw pdunhw lv
fkdudfwhulvhg dv lpshuihfwo| frpshwlwlyh1 Zlwklq wklv prgho zh
dffrpprgdwh wzr glhuhqw dvvxpswlrqv derxw wkh g|qdplfv ri wkh
upv* pdun0xs ri sulfhv ryhu pdujlqdo frvwvhdfk zlwk glhuhqw
lpsolfdwlrqv iru krz upv vhw sulfhv lq uhvsrqvh wr vkrfnv1 Lq wkh
fxvwrphu pdunhw prgho/ upv orzhu wkhlu pdun0xs zkhq fxuuhqw
rxwsxw lv orz uhodwlyh wr ixwxuh surwv/ iruhjrlqj fxuuhqw surwv lq
rughu wr fdswxuh ixwxuh pdunhw vkduh1 Lq pdunhwv fkdudfwhulvhg e|
lpsolflw frooxvlrq/ upv orzhu wkhlu pdun0xs zkhq fxuuhqw rxwsxw lv
kljk uhodwlyh wr ixwxuh surwv lq rughu wr orzhu wkh lqfhqwlyhv wr
xqghufxw wkh lpsolflw fduwho1 Zh fkdudfwhulvh wkh g|qdplfv e|
dqdo|vlqj wkh uhvsrqvh ri wkh hqgrjhqrxv yduldeohvsulpdulo| wkh
pdun0xs/ hpsor|phqw dqg rxwsxwwr h{rjhqrxv vkrfnvvkrfnv wr
ghpdqg vkrfnv lghqwlhg e| lqqrydwlrqv lq jryhuqphqw h{shqglwxuh/
vkrfnv wr WIS jurzwk dqg wr sulfhv ri lpsruwhg pdwhuldov1
Wkh prgho doorzv xv wr pdnh jhqhudo vwdwhphqwv derxw upv* sulflqj
ehkdylrxu lq uhvsrqvh wr glhuhqw vkrfnv> frqiurqwlqj wkh prgho zlwk
wkh XN gdwd doorzv xv wr surylgh d vshflf lqwhusuhwdwlrq ri pdun0xs
ehkdylrxu dw wkh djjuhjdwh ohyho1 Wklv phdqv wkdw wkh prgho fdq eh
xvhg wr dgguhvv txhvwlrqv vxfk dv= zkdw kdsshqv wr wkh pdun0xs
frpsrqhqw ri lqdwlrq lq uhvsrqvh wr dq rlo sulfh vkrfnB Krz gr sulfhv
fkdqjh uhodwlyh wr frvwv zkhq ghpdqg lqfuhdvhvB Krz gr sulfhv fkdqjh
uhodwlyh wr frvwv zkhq surgxfwlylw| lqfuhdvhvB Wkh xqghuo|lqj
sklorvrsk| ri wklv sdshu lv wr pdnh vhqvh ri wkh glyhuvh sulflqj
ehkdylrxu ri XN upv lq dq rswlplvlqj/ jhqhudo htxloleulxp iudphzrun/
dqg/ e| grlqj vr/ wr frqwulexwh wr wkh prqhwdu| srolf| ghedwh1
Wkh dvvxpswlrq derxw sulflqj ehkdylrxu dhfwv rxwsxw dqg
hpsor|phqw g|qdplfv= lpsolflw frooxvlrq lqfuhdvhv wkh rxwsxw uhvsrqvh
:
wr vkrfnv wr jryhuqphqw h{shqglwxuh dqg wr wkh sulfh ri lpsruwhg
pdwhuldov/ zkloh fxvwrphu pdunhwv gdpshq xfwxdwlrqv lq uhvsrqvh wr
wkhvh vkrfnv1 Zlwk d prgho fdoleudwhg rq XN gdwd/ zh qg wkdw wkh
fruuhodwlrq ehwzhhq rxwsxw dqg wkh pdun0xs lv xqdpeljxrxvo| srvlwlyh
iru wkh fxvwrphu pdunhw prgho dqg xqdpeljxrxvo| qhjdwlyh iru wkh
lpsolflw frooxvlrq prgho1 Wklv frqfoxvlrq lv lqvhqvlwlyh wr d zlgh udqjh
ri sdudphwhu ydoxhv dqg wr wkh sduwlfxodu vhtxhqfh ri vkrfnv wkdw
fkdudfwhulvhv wkh XN hfrqrp|1 Rqo| wkh fxvwrphu pdunhw prgho
jhqhudwhv pdun0xs g|qdplfv wkdw frqirup zlwk wkh hpslulfdo hylghqfh
rq pdun0xs sulflqj lq wkh Xqlwhg Nlqjgrp/ vxfk dv Vpdoo +4<<:,/ wkdw
qg surf|folfdo pdun0xsv dw wkh vhfwrudo ohyho/ dqg zlwk wkh sulruv
hpehgghg lq w|slfdo pdfurhfrqrphwulf prghov wwhg rq pdfur gdwd/
vhh iru lqvwdqfh Edqn ri Hqjodqg +4<<<,1 Wkh hpslulfdo hylghqfh lv dw
rggv zlwk wkh hylghqfh iru wkh Xqlwhg Vwdwhv/ vhh hj Elov +4<;:, ru
Urwhpehuj dqg Zrrgirug +4<<4,/ dqg rxu wkhruhwlfdo lqyhvwljdwlrq
vxjjhvwv wkdw dw d pdfurhfrqrplf ohyho/ wkh rqo| zd| wr uhfrqfloh wkhvh
wzr uhvxowv lv wr dvvxph glhuhqw prghov ri sulflqj ehkdylrxu1
OL _Lit |i 4L_i* Lh!q
Wkh nh| uhodwlrqvklsv lq wkh prgho duh wkh ehkdylrxu ri oderxu ghpdqg
dqg vxsso| lq uhvsrqvh wr vkrfnv dqg wr yduldwlrqv lq wkh pdun0xs1
Iljxuh 4 surylghv d vwdwlf cvqdsvkrw* ri wkhvh uhodwlrqvklsv lq d uhdo
zdjh2hpsor|phqw vsdfh1 Oderxu vxsso| lv d ixqfwlrq ri wkh uhdo zdjh/
z> dqg ri wkh pdujlqdo xwlolw| ri zhdowk/ 1 Dq lqfuhdvh lq wkh pdujlqdo
xwlolw| ri zhdowk/ A / lpsolhv d vkliw wr wkh uljkw ri wkh oderxu vxsso|
fxuyh/ dv zrunhuv lqfuhdvh wkhlu vxsso| ri oderxu iru d jlyhq zdjh wr
lqfuhdvh lqfrph1 Wkh oderxu ghpdqg fxuyh lv wkh udwlr ri wkh pdujlqdo
surgxfw ri oderxu/ PSO/ wr wkh pdun0xs ri sulfhv ryhu pdujlqdo frvwv/
1 Lq d shuihfwo| frpshwlwlyh prgho @ 4/ zkloh lpshuihfw frpshwlwlrq
lpsolhv wkdw A 41
D srvlwlyh ghpdqg vkrfn/ prghoohg dv d vxusulvh lqfuhdvh lq
jryhuqphqw h{shqglwxuh/ lqfuhdvhv wkh pdujlqdo xwlolw| ri zhdowk dqg
vkliwv wkh oderxu vxsso| fxuyh wr wkh uljkw1 Wkh suhvhqfh ri d pdun0xs
dorqh dhfwv wkh txdqwlwdwlyh uhvsrqvh wr vxfk d vkrfn= d pdun0xs
juhdwhu wkdq rqh hhfwlyho| uhgxfhv wkh vorsh ri wkh oderxu ghpdqg
fxuyh dqg khqfh lqfuhdvhv wkh uhvsrqvlyhqhvv ri hpsor|phqw wr d
fkdqjh lq zdjhv1 Wkh idfw wkdw wkh pdun0xs ydulhv f|folfdoo| dhfwv wkh
;
Iljxuh 4= Wkh hhfwv ri surf|folfdo dqg frxqwhuf|folfdo pdun0xsv
Real wage
Ls(w,λ)
Ls(w,λ)
MPL/µ
MPL
Employment
srvlwlrq ri wkh oderxu ghpdqg fxuyh1 Li wkh pdun0xs lv frxqwhuf|folfdo
lq uhvsrqvh wr d ghpdqg vkrfn/ wkhq wkh rxwsxw pxowlsolhu lv pdjqlhg
ehfdxvh wkh oderxu ghpdqg fxuyh vkliwv wr wkh uljkw zkhq wkh pdun0xs
idoov1 D surf|folfdo pdun0xs zloo gdpshq wkh rxwsxw uhvsrqvh wr vxfk d
vkrfn1 D sulru eholhi wkdw wkh jryhuqphqw h{shqglwxuh pxowlsolhu lv
odujh zrxog whqg wr ohdg wr d sulru wkdw wkh pdun0xs lv frxqwhuf|folfdo1
Rq wkh rwkhu kdqg/ pdfurhfrqrphwulf prghov vxfk dv wkrvh ri wkh
Edqn ri Hqjodqg ru wkh Qdwlrqdo Lqvwlwxwh ri Hfrqrplf dqg Vrfldo
Uhvhdufk +QLHVU, w|slfdoo| hvwlpdwh d surf|folfdo pdun0xs frpsrqhqw
ri lqdwlrq1 Wkh reylrxv txhvwlrq lv zkhwkhu wkhvh glhuhqw ylhzv fdq
eh uhfrqflohg zlwklq d vlqjoh prgho wkdw lv vxemhfw wr vkrfnv rwkhu wkdq
ghpdqg vkrfnv/ ru zkhwkhu wklv lq hhfw uhtxluhv glhuhqw prghov ri
pdun0xs g|qdplfv1
Zh frqvlghu wzr rwkhu w|shv ri vkrfn lq dgglwlrq wr wkh ghpdqg
vkrfnv1 Wkh uvw lv whfkqrorj| vkrfnv1 D srvlwlyh whfkqrorj| vkrfn
lqfuhdvhv rxwsxw iru d jlyhq ohyho ri oderxu dqg fdslwdo lqsxw/ dqg
lqfuhdvhv wkh pdujlqdo surgxfw ri oderxu1 Wkdw vkliwv wkh oderxu
ghpdqg fxuyh wr wkh uljkw/ lqfuhdvlqj hpsor|phqw lq wkh vkruw uxq1 Lq
wkh orqj uxq/ krzhyhu/ wkh lqfuhdvh lq zhdowk zloo uhgxfh wkh pdujlqdo
<
xwlolw| ri zhdowk/ ohdglqj wr d idoo lq oderxu vxsso|whfkqrorj| vkrfnv
kdyh qr orqj0uxq hhfw rq hpsor|phqw1
Whfkqrorj| vkrfnv duh prghoohg dv shupdqhqw/ uhhfwlqj rxu sulru wkdw
vxfk lqqrydwlrqv duh luuhyhuvleoh1 Lqqrydwlrqv lq whfkqrorj| kdyh d
srvlwlyh phdq/ vr zkloh vkrfnv uhodwlyh wr wkh phdq duh iuhtxhqwo|
qhjdwlyh/ wkh lqqrydwlrqv duh uduho| qhjdwlyh lq dq devroxwh vhqvh1 Wkh
lvvxh ri zkhwkhu vkrfnv wr whfkqrorj| duh shupdqhqw ru wudqvlwru| lv
lpsruwdqw iru wkh g|qdplf surshuwlhv ri wkh prgho ehfdxvh rxwsxw
g|qdplfv xqghu wkh wzr dvvxpswlrqv glhu vxevwdqwldoo|1+4, Wkh lvvxh
lv ixuwkhu frpsolfdwhg e| wkh idfw wkdw wkh whfkqrorj| vkrfn lq wkh
prgho zloo dfw dv d cfdwfk0doo* zkhq phdvxuhg lq gdwd/ dqg pd| lqfoxgh
ihdwxuhv wkdw duh qrw h{solflwo| prghoohg/ vxfk dv oderxu krduglqj ru
f|folfdo fdslwdo xwlolvdwlrq1 Qhlwkhu ri wkhvh ihdwxuhv lv vshflf wr wklv
prgho dqg erwk zrxog dovr dsso| wr d vwdqgdug uhdo exvlqhvv f|foh
prgho/ vxfk dv Kdqvhq +4<;8,1
Wkh vhfrqg vkrfn zh frqvlghu lq dgglwlrq wr ghpdqg vkrfnv lv wr wkh
sulfh ri lpsruwhg pdwhuldov/ zklfk lv dvvxphg wr eh h{rjhqrxv dqg
vwdwlrqdu|1 D srvlwlyh vkrfn wr wkh sulfh ri lpsruwhg pdwhuldov fdxvhv d
vkliw lq wkh pl{ ri idfwru lqsxwv dzd| iurp lpsruwhg pdwhuldoviru d
jlyhq ohyho ri rxwsxw/ ghpdqg iru rwkhu idfwruv/ lqfoxglqj oderxu/
lqfuhdvhv1 Exw d idoo lq djjuhjdwh rxwsxw grplqdwhv wklv vxevwlwxwlrq
hhfw/ vr wkdw oderxu ghpdqg ghfuhdvhv1
Ghvslwh wklv vkrfn/ wkh prgho lv hvvhqwldoo| d forvhg0hfrqrp| prgho
dqg wkh uhvrxufhv xvhg wr sdlg iru wkhvh pdwhuldov duh vlpso| ghgxfwhg
iurp wkh uhvrxufh frqvwudlqw1 Ilupv duh reoljhg wr sd| iru wkhlu lpsruwv
rq d shulrg0e|0shulrg edvlv/ vr wudgh lv edodqfhg lq doo shulrgv dqg wkh
fxuuhqw dffrxqw lv dozd|v }hur1 Rshq0hfrqrp| frqvlghudwlrqv duh
lpsruwdqw iru srolf|0pdnlqj/ exw duh e| qr phdqv vwudljkwiruzdug
zkhq glvfxvvlqj dq lvvxh olnh wklvlw lv qrw reylrxv krz wr prgho
sulflqj ghflvlrqv ri iruhljq surgxfhuv lq wkh grphvwlf hfrqrp|/ ru wr
prgho wkh hhfwv ri wkh sulflqj ghflvlrqv ri grphvwlf surgxfhuv zkhq
wudglqj deurdg1 Wklv ixuwkhu kljkoljkwv wkh idfw wkdw wkh vkrfn
vwuxfwxuh pd| eh lqhflhqw> zlwk d forvhg hfrqrp| zh hhfwlyho|
glvuhjdug wkh hhfwv ri lqwhuqdwlrqdo ghpdqg vkrfnv +krzhyhu wkhvh
pd| eh lghqwlhg, rq grphvwlf sulfh0vhwwlqj1
E 6Lh @ _tUttL? Lu |h@?t|Lh) t Tih4@?i?| tLU!tc tii +@? Ebb. Lhc ? @ Nk
UL?|i |c OL**@?_ @?_ 5UL|| EbbH
43
Lpshuihfw frpshwlwlrq lq surgxfw pdunhwv udlvhv d qxpehu ri
phdvxuhphqw dqg fdoleudwlrq lvvxhv1 Lq dgglwlrq wr surylglqj d fduhixo
fdoleudwlrq ri sdudphwhuv wkdw duh vwdqgdug lq wkh g|qdplf jhqhudo
htxloleulxp olwhudwxuh/ zh dovr surylgh hvwlpdwhv ri wkh surgxfwlrq
ixqfwlrq dqg rxwsxw hodvwlflwlhv1 Lq whupv ri dqdo|vlqj wkh vkrfnv/ rxu
pdlq frqwulexwlrqv duh wr surylgh glhuhqw phdvxuhv ri WIS vkrfnv
wkdw duh frpsdwleoh zlwk lpshuihfw frpshwlwlrq dqg d pruh h{whqvlyh
ghvfulswlrq ri wkh vwrfkdvwlf surfhvvhv jryhuqlqj wkh uhdo sulfh ri
lpsruwhg pdwhuldov1 Wkh WIS phdvxuhv lqfoxgh d phdvxuh edvhg rq
wkh gxdo dssurdfk zkhuh WIS lv ghulyhg iurp wkh frvw ixqfwlrq/
iroorzlqj Urhjhu +4<<8,1 Zh prgho wkh lqqrydwlrqv lq wkh uhdo sulfh ri
lpsruwhg pdwhuldov dv lqqrydwlrqv wr wkh qrplqdo sulfh ri lpsruwhg
pdwhuldov/ doorzlqj hpslulfdoo| iru lqwhudfwlrq ehwzhhq wkh grphvwlf
sulfh ohyho dqg wkh qrplqdo sulfh ri lpsruwv1
Wkh uhpdlqghu ri wkh sdshu lv rujdqlvhg dv iroorzv1 Wkh qh{w vhfwlrq
ghvfulehv wkh prgho/ irfxvlqj rq d frpsdulvrq ehwzhhq wkh glhuhqw
pdunhw vwuxfwxuhv1 Wklv lv iroorzhg e| dq h{whqvlyh ghvfulswlrq ri wkh
fdoleudwlrqwr hvwdeolvk wkh prgho dv d fuhgleoh lqvwuxphqw/ zh qhhg
dv suhflvh d phdvxuhphqw ri wkh sdudphwhuv lq wkh prgho dv srvvleoh1
Wkh irxuwk vhfwlrq surylghv txdolwdwlyh dqg txdqwlwdwlyh dqvzhuv wr wkh
txhvwlrqv srvhg lq wklv lqwurgxfwlrq1 Lw dovr lqfoxghv d jhqhudo
glvfxvvlrq ri wkh prgho*v g|qdplf surshuwlhv> dq xqghuvwdqglqj ri wkh
prgho g|qdplfv dqg ri wkh prgho*v vhqvlwlylw| wr sdudphwhu fkdqjhv
duh lpsruwdqw zkhq wkh prgho lv xvhg iru sudfwlfdo dqdo|vlv1 Wklv lv
iroorzhg e| d eulhi frqfoxvlrq1
2 Ai 4L_i*
Wkh prgho lv edvhg rq d vwdqgdug g|qdplf jhqhudo htxloleulxp
exvlqhvv f|foh prgho zlwk wkh pdlq fkdqjhv ehlqj lq surgxfwlrq
ixqfwlrqv dqg pdunhw vwuxfwxuhv1 Dffruglqjo|/ wkh ghvfulswlrq ri wkh
prgho lq wklv vhfwlrq frqfhqwudwhv wkhvh wzr hohphqwv ri wkh prgho1
Wklv lv iroorzhg e| d eulhi rxwolqh ri wkh ghpdqg vlgh ri wkh prgho dqg
d vxppdu| ri htxloleulxp frqglwlrqv1
44
2
AiU?L*L})
Wkh hfrqrp| frqvlvwv ri pdq| v|pphwulf upv/ lqgh{hg e| l/
rujdqlvhg lq Lw lqgxvwulhv/ lqgh{hg e| m/ zlwk q upv lq hdfk lqgxvwu|1
Dvvxph/ iru qrz/ wkdw wkhuh lv rqh up lq hdfk lqgxvwu|/ lh q @ 41
Ilupv surgxfh glhuhqwldwhg jrrgv e| rshudwlqj lghqwlfdo surgxfwlrq
whfkqrorjlhv/ fkdudfwhulvhg e| d surgxfwlrq ixqfwlrq \ /
+4,
\wl @ \ Ywl > Tlw > ! @ I Ywl > Tlw !
zkhuh \wl lv up l*v jurvv rxwsxw/ zkloh Ywl dqg Tlw duh lqsxw lqglfhv dqg
l
! lv d {hg frvw ri surgxfwlrq/ h{rjhqrxv wr wkh up1 Wkh lqgh{
l Tw lv
l
d pdwhuldov lqgh{ wkdw djjuhjdwhv lqsxwv ri pdwhuldov/ Tw @ T Hw > Pwl /
zkhuh Hwl lv lqsxw ri lpsruwhg pdwhuldov dqg Pwl ghqrwhv lqsxw ri
l
ri fdslwdo
grphvwlfdoo| surgxfhg pdwhuldov1
Wkh
lqgh{ Yw l lv d ixqfwlrq
l
l
l
dqg oderxu lqsxwv Yw @ Y Nw > }w Kw / zkhuh Nw dqg Kwl duh fdslwdo dqg
oderxu lqsxwv/ zlwk oderxu ehlqj phdvxuhg lq wrwdo krxuv/ dqg }w lv
oderxu0dxjphqwlqj whfkqrorjlfdo surjuhvv/ lghqwlfdo iru doo upv1
Whfkqrorjlfdo surjuhvv lv dvvxphg wr eh vwrfkdvwlf/ dqg lv erwk d vrxufh
ri xfwxdwlrqv dw wkh exvlqhvv f|foh iuhtxhqf| dqg ri wuhqg jurzwk lq
rxwsxw1 Ilqdoo|/ wkh ixqfwlrq I djjuhjdwhv T dqg Y lqwr jurvv rxwsxw1
Wkhvh wkuhh djjuhjdwruv duh dvvxphg wr kdyh frqvwdqw uhwxuqv wr vfdoh/
dqg wr eh lqfuhdvlqj dqg frqfdyh lq lqsxwv1 Wkh dgydqwdjh ri
rshudwlqj zlwk wkuhh djjuhjdwruv lv wkdw lw uhvwulfwv wkh qxpehu ri
sdudphwhuv lq wkh surgxfwlrq ixqfwlrq1 Lq wkh fdoleudwlrq h{huflvh/ wklv
olplwv wkh hvwlpdwlrq uhtxluhphqwv wr wkuhh hodvwlflwlhv ri vxevwlwxwlrq
udwkhu wkdq d ixoo vhw ri vl{1 Iru vlplodu uhdvrqv/ jurvv rxwsxw lv
phdvxuhg diwhu wkh {hg frvwv kdyh ehhq lqfxuuhglq hhfw/ wkh
qrwdwlrq lq +4, wuhdwv {hg frvwv dv d ctxdvl lqsxw*1 Zlwk wklv
dvvxpswlrq/ zh fdq frqvwuxfw d phdvxuh ri wrwdo idfwru surgxfwlylw|
jurzwk/ dqg lq dgglwlrq fdq wlh grzq wkh {hg frvw lq wkh fdoleudwlrq
h{huflvh lq d vwudljkwiruzdug pdqqhu e| uhvwulfwlqj vwhdg|0vwdwh surwv1
Dowkrxjk upv surgxfh glhuhqwldwhg jrrgv/ doo qdo ghpdqg lv lq
whupv ri d vlqjoh frpsrvlwh jrrg1 Wkh djjuhjdwru wkdw wxuqv wkh
glhuhqwldwhg jrrgv lqwr d frpsrvlwh lv lqfuhdvlqj/ frqfdyh dqg
krprjhqhrxv ri ghjuhh rqh lq lqsxwv1 Lq dgglwlrq/ lw lv v|pphwulf lq
wkh vhqvh wkdw wkh txdqwlw| ri qdo jrrgv ghshqgv rq wkh glvwulexwlrq
ri lqglylgxdo jrrgv/ qrw rq wkh lghqwlw| ri lqglylgxdo jrrgv1+5, Dq
E2 t
@? i @4T*i | @ _tUhi|i _t|hM|L?c t@) |L _gihi?|@|i_ }LL_t @hi
45
djhqw zlwk d ghpdqg iru d fhuwdlq txdqwlw| ri wkh frpsrvlwh jrrg zloo
wkhq fkrrvh wkh glvwulexwlrq ri lqglylgxdoo| surgxfhg jrrgv dv d
ixqfwlrq ri wkhlu uhodwlyh sulfhv rqo|1 Wklv frxog eh d FHV djjuhjdwru
dv lq Gl{lw dqg Vwljolw} +4<::,1+6,
Grphvwlfdoo| dqg iruhljq surgxfhg pdwhuldov duh vxevwlwxwhv dv lqsxwv
lq surgxfwlrq/ exw glhu lq wkh vhqvh wkdw rqo| wkh grphvwlfdoo|
surgxfhg jrrg lv d frqvxpswlrq dqg lqyhvwphqw jrrg1 Wklv
dvvxpswlrq fdswxuhv wkh idfw wkdw d +grphvwlf, up*v rxwsxw pd| eh
dqrwkhu up*v lqsxw dqg jlyhv ulvh wr d grxeoh pdujlqdolvdwlrq hhfw
ehfdxvh wkhvh lqsxwv duh qrw sulfhg dw pdujlqdo frvw1 Zh vkdoo dvvxph
wkdw wkh sulfh ri lpsruwhg pdwhuldov/ sH
w / phdvxuhg uhodwlyh wr wkdw ri
grphvwlfdoo| surgxfhg pdwhuldov/ lv h{rjhqrxv1
Xqghu wkhvh dvvxpswlrqv/ pdujlqdo frvwv dw wkh up ohyho duh
lqghshqghqw ri vfdoh/ zkloh dyhudjh frvwv duh idoolqj1 Dw wkh djjuhjdwh
ohyho/ wkh jurvv rxwsxw ixqfwlrq \w / phdvxuhg lq whupv ri wkh frpsrvlwh
jrrg kdv frqvwdqw uhwxuqv wr vfdoh lq lqsxwv/ surylghg wkdw wkh {hg
frvwv duh dffrxqwhg iru dv d idfwru lqsxw1 Ohwwlqj qr vxshuvfulswv
lqglfdwh djjuhjdwh udwkhu wkdq up0ohyho yduldeohv/ djjuhjdwh jurvv
rxwsxw/ \w > lv
\w @ Lw \ Y Nwl > }w Kwl > T Hwl > Pwl > !
@ \ +Y +Nw > }w Kw , > T +Hw > Pw , > !Lw ,
+5,
zkhuh !Lw lv wkh {hg frvw cidfwru lqsxw*1 Djjuhjdwh ydoxh dgghg/ lh wkh
uhvrxufhv dydlodeoh wr vdwlvi| qdo ghpdqg/ lv wkhq
\w sH
w Hw Pw
+6,
zkhuh wkh sulfh ri +jurvv, rxwsxw dqg khqfh grphvwlfdoo| surgxfhg
pdwhuldov lv qrupdolvhg dw 41
ThL_Ui_ ? ^@?||it } @?_ }2 | @ _t|hM|L? U@h@U|ihti_ M) @ iU|Lh E} c }2 Ai ^@?||) Lu |i UL4TLt|i }LL_ }i?ih@|i_
M) Ec 2 t |i t@4i @t |i ^@?||)
}i?ih@|i_ M) E2c c *i |i _t|hM|L? 2 c 2 }i?ih@|it 4Lhi Lu |i UL4TLt|i
|@? Ec 2
E *|ih?@|i*)c i UL*_ _itUhMi |t @t @? @__|L?@* L|T| tiU|Lh |@| t
TihuiU|*) UL4Ti||i @?_ U LTih@|it @ ThL_U|L? |iU?L*L}) | |i t@4i
ThLTih|it @t |i @}}hi}@|Lh
46
22
@h!i| t|hU|hi
Li wkh upv idfh qr dgmxvwphqw frvwv/ idfwru ghpdqgv duh ghwhuplqhg
e| wkh iroorzlqj pdujlqdo frqglwlrqv=
\Y Ywl > Tlw YN Nwl > }w Kwl @ lw uw
\Y Ywl > Tlw YK Nwl > }w Kwm }w @ lw zw
\T Ywl > Tlw TH Hwl > Pwl @ lw sH
w
l l
l
l
l
@ w
\T Yw > Tw TP Hw > Pw
+7,
zkhuh vxevfulswv lqglfdwh ghulydwlyhv/ dqg zkhuh uw lv wkh uhdo lqwhuhvw
udwh dqg zw wkh uhdo +krxuo|, zdjh1 Wkhvh frqglwlrqv vwdwh wkdw wkh
pdujlqdo surgxfw ri hdfk idfwru lv htxdo wr d pdun0xs ryhu wkh
uhvshfwlyh idfwru sulfh/ zlwk lw ehlqj wkh zhgjh ehwzhhq wkh pdujlqdo
idfwru surgxfw dqg wkh idfwru sulfh1 Wkh g|qdplfv ri wklv pdun0xs
xqghu ydulrxv dvvxpswlrqv derxw frpshwlwlrq lv wkh pdlq lvvxh ri wkh
sdshu/ dqg vr lv glvfxvvhg lq vrph ghwdlo1
Dowkrxjk qrw dqdo|vhg txdqwlwdwlyho|/ wkh fdvh ri {hg pdun0xsv
surylghv d xvhixo dqdo|wlfdo uhihuhqfh srlqw1 Zulwh ghpdqg iru jrrgv lq
lqgxvwu|2up l/ glw / dv d ixqfwlrq ri wkh uhodwlyh sulfh lw fkdujhv
l
\G
sw
Lw
sl
glw @ w G
, w @ G4 glw G
+8,
Lw
sw
sw
\w
zkhuh \wG lv wrwdo ghpdqg iru wkh frpsrvlwh jrrg dqg G lv d
wlph0lqyduldqw ghpdqg ixqfwlrq wkdw jlyhv up l*v vkduh ri djjuhjdwh
ghpdqg jlyhq uhodwlyh sulfhv1 Dvvxph wkdw G+4, @ 4 vr wkdw iru htxdo
sulfhv/ upv jhw dq htxdo vkduh ri djjuhjdwh ghpdqg1 Wkurxjkrxw/ zh
vkdoo uhvwulfw rxuvhoyhv wr v|pphwulf htxloleuld zkhuh doo upv fkdujh
lghqwlfdo sulfhv dqg surgxfh lghqwlfdo txdqwlwlhv1 Xqghu wkhvh
dvvxpswlrqv/ lw lv vwudljkwiruzdug wr vkrz wkdw wkh zhgjh w lv wlph
lqyduldqw dqg wkdw=
4
+9,
w @ @
3
+G +4,4 . 4,
Zlwk wklv dqdo|wlfdo ehqfkpdun lq plqg/ zh surfhhg wr lqyhvwljdwh wzr
glhuhqw prghov ri wkh pdun0xs/ fxvwrphu pdunhwv dqg lpsolflw
frooxvlrq/ edvhg rq glhuhqw dvvxpswlrqv derxw wkh ixqfwlrq G1 Wkhvh
prghov suhglfw glhuhqw g|qdplf pdun0xs uhvsrqvhv iroorzlqj ghpdqg
dqg whfkqrorj| vkrfnv/ d ihdwxuh zklfk zh zloo xvh lq dsso|lqj wkh
prgho wr XN gdwd1
47
51514
Fxvwrphu pdunhwv
Lq d fxvwrphu pdunhw prgho/ lw lv dvvxphg wkdw d up wkdw orzhuv lwv
fxuuhqw sulfh qrw rqo| lqfuhdvhv fxuuhqw vdohv/ exw dovr h{sdqgv lwv
fxvwrphu edvhdq lqfuhdvh lq wkh fxvwrphu edvh udlvhv ixwxuh ghpdqg
iru dq| jlyhq sulfh1 Wklv qrwlrq lv fdswxuhg e| zulwlqj wkh ghpdqg
ixqfwlrq idflqj up l dv=
l
\G
w
+:,
glw @ plw w Gp
Lw
w
zkhuh/ dv lq wkh vwdwlf fdvh/ Gp +4, @ 4/ dqg vxshuvfulsw p lqglfdwhv
cfxvwrphu pdunhwv*1 Wkh pdlq glhuhqfh uhodwlyh wr wkh vwdwlf prgho lv
wkh whup plw / zklfk lv wkh vkduh ri ghpdqg wkdw zrxog jr wr up l li lw
fkdujhv wkh vdph sulfh +ru pdun0xs, dv doo rwkhu upv1 D up*v vkduh
ri djjuhjdwh ghpdqg pd| wkhq glhu dffruglqj wr klvwru|/ zkhuh plw lv
d vwdwh yduldeoh zlwk odz ri prwlrq
l
w
l
pw.4 @ j
+;,
plw
w
zkhuh lw lv dvvxphg wkdw j 3 ? 3 dqg j+4, @ 41 Li d up fkdujhv d orzhu
pdun0xs wkdq lwv frpshwlwruv/ wkhq wkdw zloo qrw rqo| lqfuhdvh fxuuhqw
vdohv/ wkurxjk Gp / exw zloo dovr lqfuhdvh wkh up*v pdunhw vkduh iru
jlyhq pdun0xsv lq ixwxuh shulrgv/ wkurxjk wkh odz ri prwlrq iru pdunhw
vkduh/ +;,1 Rqh udwlrqdoh iru wkh h{lvwhqfh ri vxfk d vwdwh yduldeoh lv
wkh h{lvwhqfh ri fxvwrphu vzlwfklqj frvwv/ zkhuh ex|huv pxvw hqjdjh lq
frvwo| vhdufk li wkh| zdqw wr fkdqjh vxssolhu dqg pd| wkxv eh uhoxfwdqw
wr fkdqjh vxssolhu/ dv vxjjhvwhg e| Skhosv dqg Zlqwhu +4<:3, dqg
uhfhqwo| dqdo|vhg lq d pdfurhfrqrplf vhw0xs e| Luhodqg +4<<;,1 Xqolnh
wkh lpsolflw frooxvlrq prgho/ wr eh ghvfulehg lpphgldwho| ehorz/ wklv lv
qrw edvhg rq h{solflw djjuhjdwlrq iurp wkh plfur ohyhozlwkrxw suhflvh
dvvxpswlrqv derxw wkh fkdudfwhu ri wkhvh vzlwfklqj frvwv/ vxfk
djjuhjdwlrq lv qrw srvvleoh1
Wr qg wkh up*v rswlpdo pdun0xs dqg ghulyh wkh g|qdplf htxdwlrq
iru pdun0xsv/ zh qg wkh rswlpdolw| frqglwlrq iru wkh pdun0xs
ghflvlrq1 Surwv lq up l dw w duh jlyhq dv=
l 4 l \wG p lw
lw 4 l
gw @ w
pw
G
+<,
w
w
Lw
w
48
Qrwh wkdw zh fdq zulwh pw.m iru dq| m A 3 dv d ixqfwlrq ri pdun0xsv
iurp w wr w . m=
l m
\
w.}
l
l
j
+43,
pw.m @ pw
w.}
}@3
Zh dvvxph wkdw wkhuh lv d {hg suredelolw| 4 wkdw wkh up zloo
uhfhlyh d pdunhw vkduh qh{w shulrg zklfk lv lqghshqghqw ri wkh up*v
sulflqj klvwru|1+7, Wkhq up l*v h{shfwhg suhvhqw ydoxh ri ixwxuh surwv
dw wlph w fdq eh h{suhvvhg dv=
#
$
m4 4
l
l
[
\wG
.m pl \ j lw.}
m m w.m w.m 4 p w.m
Hw
G
+44,
w
w.m
w.m Lw.m w } @3
w.}
m @3
zkhuh w.m lv wkh pdujlqdo xwlolw| ri zhdowk dw wlph w dqg lv d {hg
glvfrxqw idfwru1 Lq hhfw wkh whupv m m w.m @w dfw dv d glvfrxqw
idfwru/ ehlqj wkh cvxuylydo* suredelolw|/ w.m @w jlylqj wkh ixwxuh
pdujlqdo xwlolw| ri dq dgglwlrqdo xqlw ri surw uhodwlyh wr wkh fxuuhqw/
dqg glvfrxqwlqj lw wr wlph w1+8, Ilupv qrz fkrrvh lw wr pd{lplvh
+44,/ zlwk d uvw0rughu frqglwlrq=
#
#
$
$
m
4
l
l
[
\
lw.m
\wG
.
m l
m m w.m w.m 4 p w.m
G
p
j
Hw
w
w.m
w.m Lw.m w }@4
w.m
m @4
l 4
4 lw p lw \wG
w
lw \wG
G
Gp
+45,
@ j3
w
w
w
Lw
w
Lw
zkhuh Gp lqglfdwhv ghulydwlyh ri Gp +=, zlwk uhvshfw wr lw 1 Wr hdvh
h{srvlwlrq/ ghqh d yduldeoh [wl dv h{shfwhg ixwxuh surwv dw wlph w iru
up l/ wkh h{shfwdwlrqdo whup lq +45,=
#
#
$
$
m
4
l
l
[
\
lw.m
\wG
.
m l
m w.m w.m 4 p w.m
l
m
G
p
j
[w @ Hw
w
w.m
w.m Lw.m w }@4
w.m
m @4
+46,
Ee At @tt4T|L? t L?*) ?ii_i_ ihi |L 4@!i |i i ThittL? uLh u|hi ThL|t
i^@*i?| |L |i L?i ti_ uLh |i 4T*U| UL**tL? 4L_i*c ihi k T*@)t @? 4TLh|@?|
hL*i
ED ) t?} 4@h}?@* |*|)c i @hi ? igiU| @?|UT@|?} i^*Mh4G 4Lhi
ULhhiU|*)c i L*_ @tt4i @ ti| Lu ThUit uLh UL?|?}i?| tiUh|itc @?_ |i? ?
i^*Mh4 tL |@| |i ThUit @hi i^@* |L b|n *b| 49
Wr ghqh d vlplodu djjuhjdwh phdvxuh/ v|pphwu| lv lpsrvhg/ dqg
djjuhjdwh h{shfwhg ixwxuh surwv [w duh
4
[
w.m lw.m 4 \wG
[w
.m =
@ Hw
m m
Lw
L
w
w.m
w.m
m @4
+47,
Iru odwhu sxusrvhv/ zh ghulyh wkh uhfxuvlyh irup ri wklv yduldeoh dv
<
[w @ H w.4 w.4 4 \w.4 . H 5 5 w.5 w.5 4 \w.5 . === @
w
w
Lw
w
w.4 Lw.4 w
w.5
Lw.5
,
[w.4
w.5 w.5 4 \w.5
>
@
L
H
.
===
w.4 w.4
Lw.4
w.4
w.5
Lw.5
w.4 w.4 4 \w.4
w.4 [w.4
[w
@ Hw . Hw +48,
Lw
w
w.4
Lw.4
w Lw.4
w
Lpsrvlqj v|pphwu| rq +45,/ wkh uhodwlrq ehwzhhq w dqg [
\w fdq eh
irxqg dv
4
[w
@3
4 . Gp +4, 4 . j3 +4,
w
\w
Gp +4,
w @
+49,
w
4 . Gp +4, . j 3 +4, [
\w
Wr fkdudfwhulvh wkh f|folfdo surshuwlhv ri wkh pdun0xs/ glhuhqwldwh
w
+49, zlwk uhvshfw wr [
\w =
j 3 +4,
C
w @ p 5w
G +4,
C [w
+4:,
\w
Dv erwk Gp dqg j 3 duh qhjdwlyh/ wklv ghulydwlyh lv qhjdwlyh/ lpso|lqj
wkdw wkh pdun0xs lv surf|folfdo lh wkdw wkh pdun0xs lv kljk zkhq
fxuuhqw rxwsxw/ dqg khqfh fxuuhqw surwv/ duh kljk uhodwlyh wr h{shfwhg
ixwxuh surwv1 Wkh lqwxlwlrq lv wkh iroorzlqj1 Zkhq ixwxuh surwv duh
kljk uhodwlyh wr fxuuhqw rxwsxw/ upv zloo orzhu wkhlu fxuuhqw sulfh wr
jdlq ixwxuh pdunhw vkduh1 Ehfdxvh fxuuhqw rxwsxw lv orz/ wkh +qhjdwlyh,
uhyhqxh hhfw ri orzhulqj sulfhv lv uhodwlyho| orz zkloh wkh ixwxuh jdlqv
lq whupv ri pdunhw vkduh duh kljk1 Zkhq fxuuhqw surwv duh kljk
uhodwlyh wr ixwxuh surwv/ wkh ixwxuh pdunhw vkduh lv uhodwlyho| ohvv
lpsruwdqw/ dqg upv zloo udlvh sulfhv wr lqfuhdvh fxuuhqw uhyhqxhv1
4:
51515
Lpsolflw frooxvlrq
Wr prgho lpsolflw frooxvlrq/ zh dvvxph wkdw hdfk ri wkh Lw lqgxvwulhv
frqvlvwv ri q A 4 lghqwlfdo upv/ zlwk lqgxvwulhv dv suhylrxvo| ehlqj
lqgh{hg e| m zkloh upv duh lqgh{hg e| l1+9, Wkh jrrgv surgxfhg
zlwklq dq lqgxvwu| duh forvh vxevwlwxwhv/ vr zlwkrxw vrph irup ri
frooxvlrq/ sulfhv zrxog froodsvh wr pdujlqdo frvwv1 Zh dvvxph wkdw
h{solflw sulfh djuhhphqwv duh qrw ohjdoo| hqirufhdeoh/ vr dq| djuhhphqw
ehwzhhq wkh upv pxvw eh lpsolflw1 Wr vxvwdlq d frooxvlyh htxloleulxp/
frooxvlrq pxvw wkhq eh sulydwho| lqfhqwlyh0frpsdwleoh= hdfk up qgv d
vwudwhj| ri vwlfnlqj wr wkh djuhhphqw rswlpdo uhodwlyh wr wkh ehvw
dowhuqdwlyh1
Li d up h{lvwhg iru rqh shulrg rqo|/ ru zdv p|rslf iru vrph uhdvrq/
wkhq d up zrxog fkrrvh lwv pdun0xs wr pd{lplvh fxuuhqw surwv jlyhq
wkh pdun0xsv ri doo rwkhu upvlh Ehuwudqg frpshwlwlrq1 Dv lv
zhoo0nqrzq iurp wkh ciron wkhruhp olwhudwxuh*/ lq d g|qdplf lqqlwh
krul}rq vhwwlqj zkhuh upv fduh derxw wkh h{shfwhg suhvhqw ydoxh ri
fxuuhqw dqg ixwxuh surwv/ vwudwhjlhv wkdw lqyroyh vhwwlqj kljkhu sulfhv
fdq eh vxvwdlqhg dv d vxejdph shuihfw htxloleulxp/ surylghg wkdw upv
wkdw xqghufxw duh sxqlvkhg vxflhqwo| dqg wkh glvfrxqw udwh lv
vxflhqwo| kljk1 Zlwk uhshdwhg jdphv ri wklv w|sh/ wkhuh duh pdq|
htxloleuld zlwk glhuhqw sulflqj dqg sxqlvkphqw vwudwhjlhv1 Fkrrvlqj
iurp wkhvh htxloleuld/ zh dvvxph wkdw upv pdqdjh wr lpsohphqw wkh
v|pphwulf htxloleulxp wkdw lv mrlqwo| ehvw iru wkhp/ lh wkh htxloleulxp
wkdw pd{lplvhv surwv lq lqgxvwu| l jlyhq pdun0xsv ri rwkhu lqgxvwulhv/
dvvhw sulfhv dqg ghpdqg1 Lw fdq eh vkrzq +Deuhx +4<;9,, wkdw wklv
uhtxluhv wkdw wkh ghyldwruv idfh wkh prvw vhyhuh sxqlvkphqw srvvleoh1 Li
upv fdq ohdyh dq lqgxvwu|/ wkhq d srvvleoh zruvw sxqlvkphqw lv
h{shfwhg suhvhqw ydoxh ri ixwxuh surwv ri }hur/ fruuhvsrqglqj wr h{lw li
wkhuh duh qr {hg frvwv wr ohdylqj wkh lqgxvwu|1 Vxejdph shuihfw
htxloleulxp dovr uhtxluhv wkdw upv duh zloolqj wr lpsohphqw vxfk d
sxqlvkphqw vwudwhj| dqg wkdw lw lv ihdvleoh1 Wkhvh frqglwlrqv duh
ghvfulehg lq ghwdlo lq Urwhpehuj dqg Zrrgirug +4<<5,1
Zh dvvxph wkdw ghpdqg iru up l*v jrrgv ghshqgv rq wkh pdun0xs
fkdujhg e| wkh up/ lm
w / dqg wkh pdun0xs fkdujhg e| rwkhu upv lq
wkh vdph lqgxvwu|/ mw / uhodwlyh wr wkh pdun0xs fkdujhg e| doo rwkhu
ES At U@? Mi tii? t4T*) @t @? @__|L?@* @}}hi}@|Lh Lih _gihi?|@|i_ }LL_t
|? @? ?_t|h)
4;
upv lq doo rwkhu lqgxvwulhv/ w 1 Zh fdq wkhq zulwh ghpdqg dqg surwv
iru up l lq lqgxvwu| m +:, dv
#
$
lm
m
4
\wG
lm
fl w w
>
> Gfl +4> 4, @
gw @ G
w w
Lw
q
#
$
m
lm
lm 4 fl lm
\wG
w 4 lm
w w
@
gw @ w
G
>
+4;,
lm
w
w
w
w w
Lw
Ehfdxvh wkh sxqlvkphqw wr ghyldwruv lv }hur ixwxuh surwv/ d ghyldwlqj
up zloo fkrrvh wkh pdun0xs/ lm
w / wkdw pd{lplvhv fxuuhqw surwv rqo|1
E| dqdorj| wr wkh fxvwrphu pdunhw prgho/ zh fdq ohw [wm ghqrwh
h{shfwhg ixwxuh surwv lq shulrg w iru hdfk up lq lqgxvwu| m/ qrz
xqghu wkh surylvlrq wkdw frooxvlrq lv vxvwdlqhg lq wkh ixwxuh1 Wkh
lqfhqwlyh frpsdwlelolw| frqvwudlqw fdq wkhq eh zulwwhq dv
,
+
lm
pd{
lm w
w
mw . [wm
+4<,
zlwk wkh ohiw0kdqg vlgh dffrxqwlqj iru wkh uhwxuq wr xqghufxwwlqj dqg
jdlqlqj qr ixwxuh surwv/ dqg wkh uljkw0kdqg vlgh ehlqj wkh surwv lq
shulrg w li frooxvlrq lv vxvwdlqhg/ mw / dqg wkh h{shfwhg ixwxuh surwv
iurp frooxvlrq1
Wkh sdudphwhu sod|v d nh| uroh lq wkh lpsolflw frooxvlrq prgho1 Dv
ehiruh/ 4 lv wkh suredelolw| wkdw sulflqj lv lqghshqghqw ri klvwru|/
exw lq wklv fdvh lw lv lqwhusuhwhg dv wkh suredelolw| wkdw d frooxvlyh
djuhhphqw zloo eh uh0hvwdeolvkhg vxfk wkdw suhylrxv ghyldwruv fdq hqwhu
djdlq1 Wkh h{whqw wr zklfk wkh lqfhqwlyh0frpsdwlelolw| frqvwudlqw +4<,
elqgv zloo wkhq ghshqg rq = li lv frqvlghudeo| vpdoohu wkdq rqh/
wkhq wkh uhodwlyh uhwxuqv iurp ixwxuh frooxvlrq duh vpdoo/ ehfdxvh wkh
suredelolw| ri uh0hqwu| lqwr d frooxvlyh djuhhphqw diwhu kdylqj fkhdwhg
lv odujh1 Khuh lw lv dvvxphg wkdw lv vxflhqwo| vpdoo wr hqvxuh wkdw
wkh lqfhqwlyh0frpsdwlelolw| frqvwudlqw dozd|v elqgvwkh suhflvh
frqglwlrqv duh rxwolqhg lq Urwhpehuj dqg Zrrgirug +4<<5,1
Iroorzlqj wkh olqh ri dujxphqw iurp deryh/ zh fdq zulwh wkh frqglwlrq
lq v|pphwulf htxloleulxp dv
\w
4
[w
4
\w
.
+53,
Gf +> 4, @ 4 pd{ w
Lw
w qLw qLw
E. `ihi
tTihtUhT| S _i?L|it UL**tL?<
4<
zkhuh lv wkh up*v pdun0xs udwlr uhodwlyh wr wkh pdun0xs ri doo rwkhu
upv/ w 1 Vroylqj wkh pd{lplvdwlrq sureohp iru wkh rswlpdo pdun0xs
udwlr a/ zh qg
4
\w
4
[w
\w
\w
. a
> 4,
@
4
.
Gf +a
qLw
w
Lw
w qLw
qLw
4
4
[w
4 . a 4
Gf +a> 4,q @
.
w
w
\w
[
4
w
Gf +a
aGf +a
> 4,q @
> 4,q 4
+54,
\w
w
Gf +a
> 4,q 4
w @
+55,
w
f
aG +a
> 4,q [
\w
Dv suhylrxvo|/ zh zloo qhhg wr ghwhuplqh wkh vljq ri wkh sduwldo
ghulydwlyh=
Cw
@ 4 Gf +> 4,q 5w
[
w
C \w
+56,
Wkh ghulydwlyh lv srvlwlyh surylghg wkdw Gf +> 4, lv qhjdwlyh1 Lq
frqwudvw wr wkh fxvwrphu pdunhw prgho/ wkh pdun0xs lv frxqwhuf|folfdo/
lh wkh pdun0xs lv orz zkhq fxuuhqw surwv duh kljk uhodwlyh wr ixwxuh
surwv1 Wkh lqwxlwlrq lv wkh iroorzlqj= kljk ixwxuh surwv uhodwlyh wr
fxuuhqw rxwsxw pdnh wkh frooxvlyh djuhhphqw hdvlhu wr vxvwdlq/ ehfdxvh
wkh jdlqv wr xqghufxwwlqj duh vpdoo dqg wkh jdlqv iurp ixwxuh frooxvlrq
kljk1 Xqghu vxfk flufxpvwdqfhv/ wkh lpsolflwo| djuhhg pdun0xs zloo eh
kljk1 Li/ rq wkh rwkhu kdqg/ fxuuhqw rxwsxw lv kljk/ wkhq wkh lqfhqwlyh
wr xqghufxw lv kljk/ dqg wkh fduwho pxvw fxw lwv pdun0xsv lq rughu wr
vxvwdlq frooxvlrq1 E| fxwwlqj pdun0xsv/ wkh vkduh ri fxuuhqw rxwsxw
wkdw wkh ghyldwru fdq slfn xs iru d jlyhq sulfh fxw zloo ghfuhdvh/ wkxv
orzhulqj surwv dqg wkh lqfhqwlyh wr fkhdw1 Wkh nh| wr xqghuvwdqglqj
wkh dujxphqw lv wkdw zh kdyh hqvxuhg wkdw htxloleulxp lv vxvwdlqhg/ lh
wkhuh lv qr xqghufxwwlqj= wr vxvwdlq htxloleulxp/ wkh lqfhqwlyh wr fkhdw
pxvw wkxv eh orzhuhg1
2
OLtiL*_t
Kdylqj ghvfulehg wkh pdunhw vwuxfwxuh/ zh qrz wxuq wr wkh krxvhkrogv
lq wkh hfrqrp|1 Wkh krxvhkrogv vroyh d vwdqgdug g|qdplf rswlplvdwlrq
sureohp/ ghwhuplqlqj wkh sdwk ri fxuuhqw dqg ixwxuh frqvxpswlrq ri
53
wkh frpsrvlwh jrrg/ fw / dqg vxsso| ri oderxu/ kw 1 Iurp vwdqgdug
g|qdplf rswlplvdwlrq wkhru|/ wkh uvw0rughu frqglwlrqv iru wkh
krxvhkrog zloo ohdg wr d frqvxpswlrq ghpdqg ixqfwlrq dqg d oderxu
vxsso| ixqfwlrq wkdw fdq eh zulwwhq dv ixqfwlrqv ri uhdo zdjhv/ zw / dqg
wkh h{shfwhg pdujlqdo xwlolw| ri zhdowk/ w =
fw
kw
@ f +zw > w , > fz A 3> f ? 3
@ k +zw > w , > kz A 3> k A 3
+57,
+58,
Wkh krxvhkrogv duh wkh fdslwdolvwv ri wklv hfrqrp|/ rzqlqj erwk wkh
sk|vlfdo fdslwdo vwrfn dqg wkh upv1 Fdslwdo lv uhqwhg wr upv rq d
rqh0shulrg edvlv dqg wklv jlyhv ulvh wr uhqwdo lqfrph1 Dq| surwv iurp
surgxfwlrq duh glvwulexwhg wr wkh vkduhkroghuv rq d shulrg0e|0shulrg
edvlv1 Ehfdxvh wkh krxvhkrogv duh lghqwlfdo dqg ulvn dyhuvh/ lw lv hdv| wr
vkrz wkdw hdfk krxvhkrog zloo krog d ixoo| glyhuvlhg sruwirolr ri
vkduhv/ khqfh holplqdwlqj lglrv|qfudwlf ulvn/ dqg wkdw qr wudgh lq vkduhv
lv wdnlqj sodfh1
Wkh Hxohu htxdwlrq wkdw uhodwhv wkh pdujlqdo xwlolw| ri frqvxpswlrq
qrz wr wkh pdujlqdo xwlolw| ri frqvxpswlrq lq wkh ixwxuh lv lpsruwdqw/
dv lw dsshdu dv d glvfrxqw idfwru zkhq wkh upv fdofxodwh h{shfwhg
ixwxuh surwv1 Wkh dvvhw sulfh Hxohu htxdwlrq lv
w.4
+uw.4 . +4 ,,
+59,
4 @ Hw
w
zkhuh uw.4 lv wkh h{shfwhg uhdo lqwhuhvw udwh lq shulrg w . 4 dqg lv wkh
ghsuhfldwlrq udwh rq fdslwdo1
Wr hqvxuh wkh h{lvwhqfh ri d edodqfhg vwhdg|0vwdwh jurzwk sdwk zh
lpsrvh uhvwulfwlrqv rq wkh ixqfwlrqdo irup ri wkh xwlolw| ixqfwlrq1 Wkhvh
uhvwulfwlrqv/ glvfxvvhg lq ghwdlo lq Nlqj/ Sorvvhu dqg Uhehor +4<;;d,/
uhtxluh rqo| wkdw wkhuh h{lvwv d A 3 vxfk wkdw f+=, lvkrprjhqhrxv
ri
4
/ zkhuh
ghjuhh rqh dqg k+=, lv krprjhqhrxv ri ghjuhh }hur lq z> 4@ lv wkh lqwhuwhpsrudo hodvwlflw| ri vxevwlwxwlrq ri frqvxpswlrq zlwk
krxuv khog frqvwdqw1 Wklv dvvxpswlrq hqvxuhv wkdw dorqj wkh
vwhdg|0vwdwh jurzwk sdwk krxuv zrunhg shu fdslwd duh frqvwdqw/
ehfdxvh wkh lqfrph hhfw/ zklfk whqgv wr orzhu krxuv dv lqfrph
lqfuhdvhv/ dqg wkh vxevwlwxwlrq hhfw/ zklfk whqgv wr lqfuhdvh krxuv dv
zdjhv lqfuhdvh/ h{dfwo| fdqfho rxw1 Dovr dorqj wklv sdwk/ wkh
frqvxpswlrq~rxwsxw udwlr lv frqvwdqw1
54
2e
#i4@?_
Zh kdyh suhylrxvo| ghvfulehg krz ghpdqg iru lqglylgxdo upv* jrrgv
lv dhfwhg e| uhodwlyh sulfhv wkurxjk upv* ghflvlrqv derxw wkhlu
pdun0xs1 Wklv vhfwlrq ghvfulehv ghpdqg iru qdo jrrgv/ lh wkh
frpsrvlwh jrrg frqvlvwlqj ri dq djjuhjdwh ri wkh glhuhqwldwhg jrrgv1
Dv wkh prgho lv ghvljqhg wr irfxv rq lpshuihfw frpshwlwlrq dqg
sulflqj/ wkh ghpdqg iru qdo jrrgv lv nhsw dv vlpsoh dv srvvleoh1
Djjuhjdwh ghpdqg iru ydoxh dgghg/ lh jurvv rxwsxw plqxv frvwv ri
lpsruwhg dqg grphvwlf pdwhuldov/ frqvlvwv ri djjuhjdwh frqvxpswlrq
dqg lqyhvwphqw ghpdqg e| krxvhkrogv/ Fw dqg "w uhvshfwlyho|/ dqg
jryhuqphqw h{shqglwxuh/ Jw 1 Wkh uhvrxufh frqvwudlqw uhtxluhv wkdw
ghpdqg iru qdo jrrgv htxdov ydoxh dgghg/ lh wkdw=
Fw . "w . Jw @ \w sH
w Hw Pw
+5:,
Djjuhjdwh sulydwh frqvxpswlrq ghpdqg lv vlpso| wkh surgxfw ri wkh
srsxodwlrq dw w/ Qw > dqg lqglylgxdo krxvhkrog ghpdqg fw +w > zw ,1
Jryhuqphqw h{shqglwxuh kdv wkh vdph vwuxfwxuh ryhu wkh Lw jrrgv dv
krxvhkrogv* dqg upv* ghpdqgv dqg lv dvvxphg wr eh dq h{rjhqrxv
vwrfkdvwlf yduldeoh> vkrfnv wr jryhuqphqw frqvxpswlrq frqvwlwxwh wkh
ghpdqg vkrfnv lq wklv hfrqrp|1 Zh zloo uhwxuq wr wkh vshflfdwlrq ri
wkh vwrfkdvwlf surfhvv iru wklv yduldeoh odwhu1 Ilqdoo|/ djjuhjdwh
lqyhvwphqw lv uhodwhg wr wkh fdslwdo vwrfn e| wkh iroorzlqj odz ri
prwlrq=
+5;,
"w @ Nw.4 +4 ,Nw >
zklfk dvvxphv qr dgmxvwphqw frvwv ru luuhyhuvlelolw| frqvwudlqwv rq
lqyhvwphqw1
Lq dffrxqwlqj iru wkh jurvv rxwsxw dqg ydoxh dgghg/ qrwlfh wkdw qdo
jrrgv duh xvhg dv grphvwlfdoo| surgxfhg pdwhuldov lq surgxfwlrq dqg
dv sd|phqw iru lpsruwhg pdwhuldov1 Iru grphvwlfdoo| surgxfhg
pdwhuldov/ upv pxvw sxufkdvh surgxfwlrq jrrgv ri rwkhu upv wr
rewdlq pdwhuldov lqsxwwkh dvvxpswlrq wkdw upv xvh wkh frpsrvlwh
jrrg dv pdwhuldov dqg gr qrw glvwruw wkhlu lqsxw ri pdwhuldov lq idyrxu
ri wkh jrrg wkh| surgxfh wkhpvhoyhv/ ghvslwh wkh jrrgv wkdw rwkhu
upv surgxfh ehlqj sulfhg deryh pdujlqdo frvwv/ lv uhdvrqdeoh ehfdxvh
zh kdyh uhvwulfwhg wkh htxloleulxp wr eh rqh zkhuh doo upv fkdujh
lghqwlfdo pdun0xsv dqg surgxfh lghqwlfdo txdqwlwlhv1 Wkh uhvrxufh
frqvwudlqw lpsolhv wkdw lpsruwhg pdwhuldov pxvw eh sdlg iru shulrg e|
shulrg> lq djjuhjdwh wkh wudgh edodqfh fohduv lq hyhu| shulrg1
55
2D
,^*Mh4
Htxloleulxp lv fkdudfwhulvhg e| wkh ghpdqg ixqfwlrqv iru frqvxpswlrq
dqg ohlvxuh +zlwk frqvxpswlrq ri ohlvxuh ehlqj +4 kw ,,/ wkh +idfwru,
ghpdqg ixqfwlrqv ri wkh upv/ pdunhw0fohdulqj frqglwlrqv iru jrrgv
dqg wkh vwrfkdvwlf surfhvvhv iru h{rjhqrxv yduldeohv1 Ghqh djjuhjdwh
oderxu vxsso| dv Kw @ Qw k +zw > w ,1 Lq htxloleulxp/ wkh iroorzlqj yh
frqglwlrqv/ vxppdulvlqj htxloleulxp lq pdunhwv iru fdslwdo ++5<, dqg
+63,,/ oderxu +64,/ lpsruwhg pdwhuldov +65,/ grphvwlfdoo| surgxfhg
pdwhuldov +66,/ dqg pdunhw fohdulqj lq wkh jrrgv pdunhw +67,/ pxvw
krog=
w.4
+uw.4 . 4 ,
@ 4
+5<,
Hw
w
\Y +Yw > Tw , YN +Nw > }w Kw , @ w uw
+63,
+64,
\Y +Yw > Tw , YK +Nw > }w Qw k +zw > w ,, }w @ w zw
+65,
\T +Yw > Tw , TP +Hw > Pw , @ w sH
w
\T +Yw > Tw , TP +Hw > Pw , @ w
+66,
H
+67,
Fw . ^Nw.4 +4 ,Nw ` . Jw . sw Hw . Pw @ \w
Wkh g|qdplfv ri wkh pdun0xs duh fkdudfwhulvhg e|
[w
w @ Hw \w
lqi
[
w.m w.m 4 \w.m
m m
[w @ Lw Hw
w
w.m
Lw.m
m@4
+68,
+69,
uhihuulqj wr wkh pdunhw vwuxfwxuhv glvfxvvhg lq ghwdlo deryh/ lh fxvwrphu
pdunhwv +49, dqg lpsolflw frooxvlrq +55,1
Wkhvh frqglwlrqv ghvfuleh wkh hfrqrp| dw dq| srlqw lq wlph1 Lq wkh
iroorzlqj/ zh fkdudfwhulvh wklv htxloleulxp dv vpdoo xfwxdwlrqv durxqg
d vwhdg|0vwdwh jurzwk sdwk/ fkdudfwhulvhg e| jurzwk lq whfkqrorj| }w
dqg lq srsxodwlrq Qw 1 Ghqh wkh vwrfkdvwlf jurzwk udwh ri
oderxu0dxjphqwlqj whfkqrorjlfdo surjuhvv dv }w @ }w @}w4 / zlwk dq
dyhudjh jurzwk udwh } > dqg d ghwhuplqlvwlf jurzwk udwh ri wkh
srsxodwlrq/ Q @ Qw @Qw4 1 Lq vwhdg| vwdwh/ djjuhjdwh h{shqglwxuh
frpsrqhqwv dqg idfwru ghpdqg +h{fhsw ghpdqg iru oderxu, jurz dw
wkh udwh } . Q / zkloh djjuhjdwh hpsor|phqw jurzv dw wkh udwh Q
dqg dyhudjh uhdo zdjhv jurz dw wkh udwh } 1
56
Vkrfnv wr oderxu0dxjphqwlqj whfkqrorjlfdo surjuhvv/ }w / duh dvvxphg wr
eh qrq0vwdwlrqdu| dqg khqfh vkliw wkh vwhdg|0vwdwh sdwk/ zkloh vkrfnv
wr jryhuqphqw h{shqglwxuh/ Jw / dqg wkh sulfh ri lpsruwhg pdwhuldov/
sH
w / sxvk wkh htxloleulxp r wkh vwhdg|0vwdwh sdwk/ exw gr qrw fkdqjh
wkh sdwk lwvhoi1 Wkhvh vkrfnv/ zkhwkhu whpsrudu| ru shupdqhqw/ duh
sursdjdwhg h{rjhqrxvo| wkurxjk shuvlvwhqfh lq wkh vkrfnv/ dqg
lqwhuqdoo| wkurxjk fkdqjhv lq wkh ghvluhg fdslwdo vwrfn1 Wkhuh duh wzr
lqwhuqdo sursdjdwlrq phfkdqlvpv lq wkh prgho/ zhdowk hhfwv dqg
lqwhuwhpsrudo vxevwlwxwlrq ri oderxu1 Iluvw/ vkrfnv pd| dhfw wkh
pdujlqdo xwlolw| ri zhdowk/ w / fruuhvsrqglqj wr d vkliw wr wkh uljkw ri
wkh oderxu vxsso| fxuyh lq Iljxuh 41 Li wkh pdujlqdo xwlolw| ri zhdowk
lqfuhdvhv/ djhqwv zloo orzhu fxuuhqw frqvxpswlrq dqg lqfuhdvh fxuuhqw
oderxu vxsso| wr lqfuhdvh fxuuhqw zhdowk1 Vxfk dq lqfuhdvh lq oderxu
vxsso| lq wxuq lqfuhdvhv wkh pdujlqdo surgxfw ri fdslwdo dqg wkh uhdo
lqwhuhvw udwh deryh vwhdg|0vwdwh ohyhov/ ohdglqj wr dq lqfuhdvh lq
lqyhvwphqw dqg khqfh lq wkh fdslwdo vwrfn1 Wkh vhfrqg phfkdqlvp lv
lqwhuwhpsrudo vxevwlwxwlrq lq oderxu vxsso|1 Li wkh uhdo zdjh lv
fxuuhqwo| deryh lwv vwhdg|0vwdwh ohyho/ wkhq zrunhuv zloo vxevwlwxwh
fxuuhqw iru ixwxuh ohlvxuh/ dqg lqfuhdvh oderxu vxsso|/ hhfwlyho| d
pryhphqw dorqj wkh oderxu vxsso| fxuyh1 Sduw ri wkh lqfuhdvh lq
lqfrph zloo eh gluhfwhg wrzdugv lqfuhdvlqj wkh fdslwdo vwrfndq
lqfuhdvh lq wkh oderxu vxsso| lqfuhdvhv wkh pdujlqdo surgxfw ri fdslwdo
dqg wklv lqfuhdvh lq wkh uhdo lqwhuhvw udwh uhvxowv lq dq lqfuhdvh lq wkh
fdslwdo vwrfn1+;,
2S
5L*|L? |iU?^i
Dv phqwlrqhg lq wkh suhylrxv vhfwlrq/ zh irfxv rq +vpdoo, exvlqhvv
f|foh xfwxdwlrqv durxqg d vwhdg|0vwdwh jurzwk sdwk/ doorzlqj iru
qrq0vwdwlrqdu|/ vwrfkdvwlf oderxu0dxjphqwlqj whfkqlfdo surjuhvv dqg iru
ghwhuplqlvwlf srsxodwlrq jurzwk1 Zh dsso| wkh phwkrgrorj| ghvfulehg
e| Nlqj/ Sorvvhu/ dqg Uhehor +4<;;e,1 Wklv lqyroyhv uvw wudqviruplqj
wkh qrq0vwdwlrqdu| htxloleulxp frqglwlrqv lqwr vwdwlrqdu| rqhv e|
qrupdolvlqj wkh yduldeohv rq wkh vl}h ri wkh srsxodwlrq dqg wkh ohyho ri
whfkqrorj|/ dqg wkhq pdnlqj d orj~olqhdu dssur{lpdwlrq durxqg wkh
EH L|Ui |@| |i 4@h!T t ?L| Uhhi?|*) @ tLhUi Lu ThLT@}@|L?G |i 4@h!T t
_i|ih4?i_ L? |i M@tt Lu ?uLh4@|L? @@*@M*i @| |4i |c @?_ t @ 4T< h@|ih |@?
@ Thi_i|ih4?i_ @h@M*i W| t TLttM*i |L _itUhMi |i 4L_i* | @ Thi_i|ih4?i_
4@h!Tc ? U U@ti ED tL*_ Mi h||i? @t >|n ' .| > Ef|n *t|n 57
vwhdg| vwdwh ri wklv vwdwlrqdu| v|vwhp1 Wkh rxwfrph lv d v|vwhp ri
olqhdu glhuhqfh htxdwlrqv/ zulwwhq lq yduldeohv ghqhg lq ghyldwlrqv
iurp vwhdg| vwdwh/ zkhuh wkh frhflhqwv duh ghulydwlyhv hydoxdwhg dw
vwhdg| vwdwh1 Wkhvh ghulydwlyhv/ zklfk duh hhfwlyho| rzq dqg
furvv0hodvwlflwlhv/ duh wkh sdudphwhuv wkdw duh fdoleudwhgwkh ghwdlov ri
wklv surfhvv lv ghvfulehg lq wkh iroorzlqj vhfwlrq1 Ghwdlov ri wkh
phwkrgrorj| dqg vroxwlrq phwkrg duh surylghg lq wkh dsshqgl{1
@*Mh@|L?
Wkh frhflhqwv ri wkh orj0olqhdulvhg htxloleulxp frqglwlrqv duh
ixqfwlrqv ri wkh sdudphwhuv olvwhg lq wkh uvw froxpq ri Wdeoh D1 Vrph
ri wkhvh yduldeohv duh vwdqgdug g|qdplf jhqhudo htxloleulxp prgho
sdudphwhuv/ iru h{dpsoh/ wkh lqwhuwhpsrudo hodvwlflw| ri oderxu vxsso|
dqg wkh uhdo udwh ri uhwxuq1 Rwkhuv/ krzhyhu/ rqo| dulvh ehfdxvh ri rxu
vshflfdwlrq ri d jurvv rxwsxw surgxfwlrq ixqfwlrq/ iru h{dpsoh wkh
hodvwlflw| ri vxevwlwxwlrq ehwzhhq ydoxh0dgghg dqg lqwhuphgldwh lqsxwv/
ru ehfdxvh zh wdnh dffrxqw ri lpshuihfw frpshwlwlrq/ iru h{dpsoh wkh
pdun0xs ri sulfh ryhu pdujlqdo frvw1 Wkh glvfxvvlrq lq wklv vhfwlrq
irfxvhv rq krz zh fdoleudwh wkhvh odwwhu sdudphwhuv1
Uhfdoo wkdw zh kdyh dssur{lpdwhg wkh htxloleulxp durxqg lwv
vwhdg|0vwdwh sdwk= wkh sdudphwhuv lq wkh v|vwhp duh wkhuhiruh hydoxdwhg
dw vwhdg| vwdwh1 Wkh htxloleulxp frqglwlrqv lpsrvh uhvwulfwlrqv rq wkh
sdudphwhuv/ dqg rxu fdoleudwlrq surfhgxuh pxvw hqvxuh wkdw wkhvh
uhodwlrqv krog1 Wkhvh sdudphwhu uhvwulfwlrqv duh glvfxvvhg lq wkh
dsshqgl{1
Wkh pdmrulw| ri wkh sdudphwhuv duh hvwlpdwhg xvlqj gdwd iru wkh XN
hfrqrp|/ dqg iru frpsdudwlyh sxusrvhv wkh fruuhvsrqglqj ydoxhv iurp
Urwhpehuj dqg Zrrgirug +4<<8, duh lqfoxghg lq wkh wklug froxpq ri
Wdeoh D1 Wkh uhpdlqghu/ iru zklfk zh kdyh qr XN hylghqfh/ duh vhw dw
wkh ydoxhv wkh| wdnh lq Urwhpehuj dqg Zrrgirug +4<<8,1
Wkh surshuwlhv ri wkh jurvv rxwsxw surgxfwlrq ixqfwlrq fdq eh
vxppdulvhg e| wkh iroorzlqj vhyhq sdudphwhuvwkh rxwsxw hodvwlflwlhv
ri oderxu +vK ,/ fdslwdo +vN ,/ lpsruwhg pdwhuldov +vH , dqg grphvwlfdoo|
surgxfhg pdwhuldov +vP ,/ dqg wkh hodvwlflwlhv ri vxevwlwxwlrq ehwzhhq
wkh ydoxh0dgghg djjuhjdwru dqg wkh pdwhuldov djjuhjdwru +Y T ,/ oderxu
58
dqg fdslwdo +NK , dqg lpsruwhg dqg grphvwlfdoo| surgxfhg pdwhuldov
+HP ,1
Wkh rxwsxw hodvwlflwlhv +vK > vN > vH > vP , htxdo wkhlu uhvshfwlyh vkduh ri
wrwdo frvwv= wr vhh wklv/ qrwh wkdw wkh dvvxpswlrq wkdw upv hduq }hur
surwv lq wkh vwhdg| vwdwh lpsolhv wkdw
!L @ + 4, \
+6:,
zkhuh dv ehiruh qr vxevfulsw lqglfdwhv vwhdg|0vwdwh ydoxhv dqg d wlogh
lqglfdwhvqrupdolvhg
Jlyhq
ydoxhv1
+6:, dqg +4,/ lw iroorzv wkdw
\ @ I Y N> K > T H> P 1 Wkhuhiruh/ lq vwhdg| vwdwh wkh rxwsxw
hodvwlflw| ri wkh ydulrxv lqsxwv dovr htxdov lwv vkduh lq wkh ydoxh ri jurvv
rxwsxw1+<, Wkh pdlq glfxow| lq hvwlpdwlqj hdfk lqsxw*v vkduh ri jurvv
rxwsxw lv wkh odfn ri gdwd rq jurvv rxwsxw/ grphvwlfdoo| surgxfhg dqg
lpsruwhg pdwhuldov1 Wkh rqo| vrxufh ri frqvlvwhqw gdwd rq wkhvh
yduldeohv duh wkh Lqsxw0Rxwsxw Wdeohv/ exw wkhvh duh rqo| fdofxodwhg iru
vshflf |hduv/ vr zh rqo| kdyh furvv0vhfwlrqdo gdwd1 Wkhuhiruh/ zh vhw
wkh vwhdg|0vwdwh frvw vkduhv ri oderxu/ lpsruwhg lqwhuphgldwh lqsxwv
dqg grphvwlfdoo| surgxfhg lqwhuphgldwh lqsxwv htxdo wr wkhlu uhvshfwlyh
zhljkwhg dyhudjh ri lqgxvwu| frvwv vkduhv lq wkh 4<:7/ 4<:< dqg 4<;7
Lqsxw0Rxwsxw Wdeohv1 Wkh vwhdg|0vwdwh vkduh ri fdslwdo lv fdofxodwhg dv
d uhvlgxdo/ dv wkh irxu jurvv rxwsxw vkduhv pxvw vxp wr xqlw|1
Hvwlpdwhv ri wkh hodvwlflw| ri vxevwlwxwlrq ehwzhhq oderxu dqg fdslwdo/
K / duh dydlodeoh iru wkh Xqlwhg Nlqjgrp1 Iru h{dpsoh/ lq wkh QLHVU
pdfur prgho wklv hodvwlflw| htxdov 3181 Wkhuh duh/ krzhyhu/ yhu| ihz
hvwlpdwhv ri wkh hodvwlflw| ri vxevwlwxwlrq ehwzhhq wkh djjuhjdwruv/ Y T /
dqg qr hvwlpdwhv iru wkh hodvwlflw| ri vxevwlwxwlrq ehwzhhq grphvwlfdoo|
surgxfhg dqg lpsruwhg lqwhuphgldwh lqsxwv/ HP 1 Zh rewdlq hvwlpdwhv
iurp uxqqlqj wkuhh furvv0vhfwlrqdo uhjuhvvlrqv/ ghulyhg iurp frpelqlqj
wkh surgxfwlrq ixqfwlrq +4, zlwk uvw0rughu frqglwlrqv iru oderxu/
pdwhuldov dqg wkh vsolw ehwzhhq wkh pdwhuldov frpsrvlwh +Tw , dqg ydoxh
dgghg frpsrvlwh +Yw ,=
Y Dm
oq
+6;,
@ 4 . NO oq+SmO @SmY D , . x4m
Om
P Dm
@ 5 . HP oq+SmP @SmPD , . x5m
oq
+6<,
Pm
Eb +iU@**
|@| }hLtt L|T| t 4i@thi_ ?i| Lu i_ ULt|t
59
Wdeoh D= Fdoleudwlrq ri wkh ehqfkpdun
@h@4i|ih
vF
vJ
vL
vH
vP
vK
vN
NK
Y T
HP
u
4@
zK
L|iG
V@*i
fSf.
f22S
fS.
f2f
fHH
f
f.H
fb.f
+` bD
fSb.
f.
fHS
#itUhT|L?
L?t4T|L? t@hi Lu @*i@__i_
BLih?4i?| t@hi Lu @*i@__i_
W?it|4i?| t@hi Lu @*i@__i_
W4T 4@| t@hi ? |L|@* ULt|t
fD
#L4 ThL_ 4@| t@hi ? |L|@* ULt|t
w@MLh t@hi ? |L|@* ULt|t
fS
@T|@* t@hi ? |L|@* ULt|t
f2
,*@t Lu tMt|c *@MLh @?_ U@T|@*
,*@t Lu tMt|c @*i@__i_
feS
fSbc ffff
@?_ ?|ih4i_@|i ?T|t @}}hi}@|Lh
,*@t|U|) Lu tMt|||L? Mi|ii? 4T
feb2
@?_ _L4 ThL_ ?|ih4i_@|i ?T|t
5|i@_)t|@|i hi@* h@|i Lu hi|h? E^@h|ih
fH.
fe
+@|i Lu _iThiU@|L? Tih ^@h|ih
fe
f
W?|ih|i4TLh@* i*@t|U|) Lu tMt|||L?
fD
Lu UL?t4T|L? E UL?t|@?|
W?|ih|i4TLh@* i*@t|U|) Lu *@MLh tTT*)
e
e
,*@t Lu 4@h!T | hitTiU| |L j%v
fc f
c fD
5|i@_)t|@|i 4@h!T
2
, TiU|i_ h@|i Lu }hL| Lu 4@h!i| t@hi
fHb
fHb
+` bD< hiuiht |L @*it @tt4i_ ? +L|i4Mih} @?_ `LL_uLh_ EbbD
5:
\m
oq
P Dm
@ 6 . Y T oq+SmPD @Sm\ , . x6m
+73,
zkhuh l lv d frqvwdqw whup/ xlm d uhvlgxdo whup/ Sml sulfh ri idfwru l/
Sm\ wkh sulfh ri jurvv rxwsxw dqg SmY wkh sulfh ri ydoxh dgghg lq vhfwru
m1 Y Dm lv ydoxh dgghg lq vhfwru m/ sur{|lqj Ym / zkloh P Dm lv d
pdwhuldov frpsrvlwh/ sur{|lqj Tm 1 Zh dvvxph wkdw SmY D @ Sm\ 1 Wkh
frqvwdqw whupv/ l / duh qrq0olqhdu ixqfwlrqv ri wkh sdudphwhuv lq wkh
surgxfwlrq ixqfwlrq dqg wkh pdun0xs lq hdfk lqgxvwu|1 Wr hvwlpdwh
wkhvh uhjuhvvlrqv/ zh xvh furvv0vhfwlrqdo gdwd rq txdqwlwlhv lq ::
pdqxidfwxulqj lqgxvwulhv iurp wkh 4<;7 Lqsxw0Rxwsxw Wdeohv1 Zh
frqvwuxfw P Dm xvlqj d txdqwlw| lqgh{ ghqhg rq wkh prvw dssursuldwh
wzr0gljlw lqgxvwu| dqg d sulfh lqgh{ iru grphvwlf dqg lpsruwhg
pdwhuldov/ frqvwuxfwhg rq wkuhh0gljlw lqgxvwu| gdwd1 Wkh uhpdlqlqj
sulfh gdwd duh dovr phdvxuhg dw wkuhh0gljlw ohyho1 Edvhg rq wkhvh
hvwlpdwhv zh vhw KN @ 3=<;wklv lv vwdwlvwlfdoo| lqvljqlfdqwo|
glhuhqw iurp wkh vwdqgdug dvvxpswlrq lq wkh uhdo exvlqhvv f|foh
olwhudwxuh wkdw wkh hodvwlflw| ri vxevwlwxwlrq ehwzhhq oderxu dqg fdslwdo
lv xqlw|1 Zh vhw Y T @ 3=79/ voljkwo| orzhu wkdq wkh hvwlpdwh rewdlqhg
e| Urwhpehuj dqg Zrrgirug +4<<9, xvlqj wlph vhulhv gdwd rq wzr0gljlw
XV pdqxidfwxulqj lqgxvwulhv/ dqg HP @ 3=78<1
Wkh htxloleulxp ehkdylrxu ri wkh pdun0xs lv ghvfulehg e| wkuhh
sdudphwhuv1 Wkh uvw lv wkh vwhdg|0vwdwh ydoxh ri wkh pdun0xs/ 1 D
qxpehu ri dxwkruv kdyh xvhg Kdoo*v dssurdfk wr hvwlpdwh wkh pdun0xs1
Vpdoo +4<<:, hvwlpdwhv wkh dyhudjh pdun0xs lq 49 wzr0gljlw
pdqxidfwxulqj dqg vhuylfh lqgxvwulhv1 Wkh zhljkwhg dyhudjh ri wkhvh
hvwlpdwhv lv 416<51 Ehdq dqg V|prqv +4<;<, dqg Kdvnho/ Pduwlq/ dqg
Vpdoo +4<<8, dovr hvwlpdwh wkh dyhudjh pdun0xs xvlqj wzr0gljlw gdwd1
Wkhlu dyhudjh hvwlpdwhv duh 5133 dqg 4185 uhvshfwlyho|1 Krzhyhu/
qhlwkhu ri wkhvh hvwlpdwhv ri wkh pdun0xs lv frqvlvwhqw zlwk wkh
dvvxpswlrq ri frqvwdqw uhwxuqv wr vfdoh dv wkh| erwk lpso| wkdw
oderxu*v rxwsxw hodvwlflw| lv juhdwhu wkdq xqlw|/ dqg khqfh wkdw fdslwdo*v
rxwsxw hodvwlflw| lv qhjdwlyh1 Vr zh xvh wkh hvwlpdwhv ri Vpdoo +4<<:,1
Exw ehfdxvh kh hvwlpdwhv wkh pdun0xs ri sulfh ryhu wkh pdujlqdo frvw
ri ydoxh0dgghg udwkhu wkdq jurvv rxwsxw/ klv jxuh qhhgv wr eh dgmxvwhg
xvlqj wkh vkduh ri ydoxh0dgghg lq jurvv rxwsxw wr jhw wkh pdun0xs ri
sulfh ryhu wkh pdujlqdo frvw ri jurvv rxwsxw/ dqg zh vhw @ 4=441+43,
Ef Ai
hi*@|L?tT Mi|ii? |i |L 4@h!Tt t
tii +L|i4Mih} @?_ `LL_uLh_ Ebb2
5;
>
' >T * n r >T 3 c
Wklv lv voljkwo| orzhu wkdq wkh ydoxh xvhg e| Urwhpehuj dqg Zrrgirug
+4<<9,1 Lw lpsolhv wkdw iru wkh w|slfdo up sulfh lv 44( kljkhu wkdq
pdujlqdo frvw dqg {hg frvwv dffrxqw iru 43( ri wrwdo frvwv1
Wkh rwkhu wzr sdudphwhuv wkdw ghwhuplqh wkh ehkdylrxu ri wkh
pdun0xs duh wkh hodvwlflw| ri wkh pdun0xs zlwk uhvshfw wr h{shfwhg
ixwxuh surwv/ / dqg wkh sdudphwhu 1 Wkh uvw sdudphwhu
glvwlqjxlvkhv ehwzhhq wkh fxvwrphu pdunhw prgho dqg wkh lpsolflw
frooxvlrq prgho1 Wkh wkhru| rxwolqhg deryh vxjjhvwv wkdw lq wkh
fxvwrphu pdunhw prgho wkh pdun0xs lv qhjdwlyho| uhodwhg wr wklv udwlr/
exw vxjjhvwv qr vshflf ydoxh1 Zh vhw @ 3=6 dqg surylgh d
vhqvlwlylw| dqdo|vlv ri wkh lpsolfdwlrqv ri wklv fkrlfh1 Lq wkh fdvh ri wkh
lpsolflw frooxvlrq prgho wkhru| uhvwulfwv wr eh ohvv wkdq 4> dqg
zh vhw lw wr 314/ forvh wr wkh pd{lpxp ri 31441 Wkhuh lv olwwoh
lqghshqghqw edvlv iru vhwwlqj wkh ydoxh ri wkh vhfrqg yduldeoh/ vr zh
iroorz Urwhpehuj dqg Zrrgirug +4<<8, dqg vhw lw htxdo wr 31;<1 Wklv
hqvxuhv wkdw wkh lqfhqwlyh0frpsdwlelolw| frqvwudlqw dozd|v elqgv1
Doo dvshfwv ri wkh lqglylgxdo*v suhihuhqfhv fdq eh vxppdulvhg e| mxvw
wzr sdudphwhuv/ wkh hodvwlflw| ri frqvxpswlrq jurzwk zlwk uhvshfw wr
fkdqjhv lq wkh uhdo udwh ri uhwxuq kroglqj krxuv frqvwdqw/ dqg wkh
lqwhuwhpsrudo hodvwlflw| ri oderxu vxsso|1 Zkloh wkhvh sdudphwhuv duh
lqghhg lpsruwdqw iru wkh prgho/ wkh| duh qrw vshflf wr wklv vwxg| dqg
zh iroorz Urwhpehuj dqg Zrrgirug +4<<5,1
Wkh uhpdlqlqj sdudphwhuv zklfk hqwhu wkh orj0olqhdulvhg htxloleulxp
frqglwlrqv duh wkh vwhdg|0vwdwh vkduh ri frqvxpswlrq/ lqyhvwphqw dqg
jryhuqphqw h{shqglwxuh lq ydoxh0dgghg/ dqg wkh vwhdg|0vwdwh ydoxhv ri
ghsuhfldwlrq dqg wkh uhdo udwh ri uhwxuq1 Wkhvh sdudphwhuv duh
uhodwlyho| vwudljkwiruzdug wr fdoleudwh xvlqj gdwd lq wkh XN Qdwlrqdo
Dffrxqwv1 Wkh vwhdg|0vwdwh vkduhv ri frqvxpswlrq/ lqyhvwphqw dqg
jryhuqphqw h{shqglwxuh duh vhw htxdo wr wkhlu uhvshfwlyh dyhudjh vkduhv
ri JGS ryhu wkh shulrg 4<93 T4~4<<; T61 Wklv jlyhv d frqvxpswlrq
vkduh ri 3193:/ d jryhuqphqw h{shqglwxuh vkduh ri 31569 dqg dq
lqyhvwphqw vkduh ri 3149:1 Wkh vkduh ri lqyhvwphqw lq lqfrph lpsolhv
wkdw wkh vwhdg|0vwdwh udwh ri ghsuhfldwlrq htxdov 3134871 Ilqdoo|/ wkh
uhdo udwh ri uhwxuq lv vhw htxdo wr wkh dyhudjh uhdo udwh ri uhwxuq rq
htxlwlhv lq wkh Xqlwhg Nlqjgrp ryhu wkh shulrg 4<4< wr 4<<;1
Wr frpsohwh wkh fdoleudwlrq/ zh qhhg hvwlpdwhv iru wkh sdudphwhuv lq
wkh vwrfkdvwlf surfhvvhv iru wkh frqwuro yduldeohv1 Zh frxog lq sulqflsoh
hvwlpdwh d mrlqw YDU+4, gluhfwo| zlwkrxw lpsrvlqj dq| frqglwlrqv/ exw
5<
lq wkh vslulw ri Urwhpehuj dqg Zrrgirug +4<<9,/ zh zloo wr irfxv
dqdo|wlfdoo| rq lqghshqghqw vkrfnv dqg krz wkhvh vkrfnv ihhg wkurxjk
yld wkh hqgrjhqrxv sursdjdwlrq phfkdqlvp= zh dvvxph wkdw vkrfnv duh
lqghshqghqw dqg wkdw wkhuh lv qr furvv0sursdjdwlrq/ lh wkdw zh fdq
hvwlpdwh wkh surfhvvhv vhsdudwho|1+44, Lq wkh iroorzlqj/ zh orrn dw wkhvh
surfhvvhv lq wxuq1
W4TLh|i_ 4@|ih@*t
Lq wkh prgho/ zh wuhdw wkh uhdo sulfh ri lpsruwhg pdwhuldov/ sH
w / dv
h{rjhqrxv1 Lpsruwdqwo|/ zkdw pdwwhuv wr wkh doorfdwlrq ri uhvrxufhv
dqg wr wkh g|qdplfv ri wkh pdun0xs lv wkh uhdo sulfh ri lpsruwhg
pdwhuldov/ ghqhg dv=
SwH
+74,
sH
w @
Sw
zkhuh SwH lv wkh qrplqdo sulfh ri lpsruwhg pdwhuldov dqg Sw lv wkh
sulfh ri qdo jrrgv/ lh jurvv rxwsxw1+45, Kdylqj frqvwuxfwhg d vhulhv iru
wkh uhdo sulfh ri lpsruwv/ wkh vlpsohvw dssurdfk wr prgho dqg fdoleudwh
aH
d vwrfkdvwlf surfhvv iru sH
w lv wr dvvxph wkdw s
w iroorzv dq DU+4, dqg
wkdw wkh vkrfnv duh llg/ hj
H aH . %s
saH
w
w @ s
w4
+75,
Wklv zrxog qrw lghqwli| h{rjhqrxv vkrfnv wr wkh hfrqrp| krzhyhu= li
Sw lv hqgrjhqrxv/ wkhq vkrfnv wr sH
w zrxog eh fruuhodwhg zlwk rwkhu
h{rjhqrxv vkrfnv ru lqghhg zlwk hqgrjhqrxv yduldeohv1 Zh lghqwli|
H
vkrfnv wr sH
w e| dvvxplqj wkdw lqqrydwlrqv lq sw duh fdxvhg hqwluho| e|
H
lqqrydwlrqv lq Sw 1 Wklv lv prghoohg e| dvvxplqj wkdw wkh surfhvv iru
saH
w fdq eh zulwwhq dv=
H Saw4
H
saw @ +O,
+76,
. 3 SawH
saH
w4
E u** V+ it|4@|L? 4@) Mi t|@|t|U@**) 4Lhi @TThLTh@|ic M| L*_ ?L*i
it|4@|?} @ *@h}i ?4Mih Lu ?t@?Ui T@h@4i|iht t i _L ?L| ThL_i @?) uLh4@*
t|@|t|U@* @?@*)ttc |i }@?t ? |ih4t Lu i@ti Lu i TLt|L? L|i} |i ULt|t uhL4
u@*?} |L LM|@? |i ULhhiU|< UL@h@?Ui 4@|h E2 `*i UL?UiT|@**) t|h@}|uLh@h_c |t Thiti?|t tL4i 4i@thi4i?| ThLM*i4t
uLh |i U@*Mh@|L? i ihUti w@U!?} @? @TThLTh@|i tihit uLh 4TLh|i_ 4@|ih@*tc i
ti @ }i?ih@* ThUi ?_i uLh 4TLh|t AL 4i@thi | i ?ii_ @ ThUi tihit uLh }hLtt
L|T| L tU |4i tihit t @@*@M*ic @?_ ?t|i@_ i ti |i B# _i@|Lh `i
@i i Tih4i?|i_ | UL?t|hU|?} ThUi ?_Uit( |iti @*|ih?@|i 4i@thit _L ?L|
@TTi@h |L @giU| |i hit*|t 4@|ih@**)
63
E| dvvxplqj wkdw wkh uhdo sulfh ghyldwlrqv iurp wuhqg saH
w duh dhfwhg
e| sdvw ydoxhv ri wkh yduldeoh lwvhoi dqg e| fxuuhqw dqg sdvw ydoxhv ri
Saw / zh hhfwlyho| wuhdw lqqrydwlrqv lq SwH dv lqqrydwlrqv lq saH
w 1 Wr
fdoleudwh wkh surfhvv/ zh hvwlpdwh wkh iroorzlqj olqhdu DU prghov
SwH
sH
w
SwH4
@ +O,
. %S
w
sH
w4
H Sw4
@ +O,
. 3 SwH . %sw
sH
w4
+77,
+78,
zkhuh edu ghqrwhv ghwuhqghg yduldeohv1 Jlyhq wkdw wkh huuruv duh
ruwkrjrqdo e| frqvwuxfwlrq/ zh fdq hvwlpdwh wklv e| ROV rq wkh
htxdwlrqv vhsdudwho|1 Wkhvh htxdwlrqv duh wkhq frpelqhg lqwr d odz ri
prwlrq iru sH
w dv d ixqfwlrq ri odjjhg yduldeohv dqg d vlqjoh
frqwhpsrudqhrxv lqqrydwlrq e| dvvxplqj wkdw %sw 31
SawH4
H
saw @ +O,
3 +O, . +79,
.
3 %S
w
saH
w4
+O, duh wkh ROV hvwlpdwhv rewdlqhg iurp +77, dqg
zkhuh 3 > +O,> +78,1 Wkh uhvlgxdov iurp +77,/ %S
w / duh wuhdwhg dv wkh vhtxhqfh ri
h{rjhqrxv vkrfnv1 Wkhvh vkrfnv fdq eh lqwhusuhwhg dv whupv ri wudgh
vkrfnv vr orqj dv wkh qdwxuh ri wudgh lv nhsw lq plqg= lq wklv hfrqrp|/
rqo| qdo jrrgv duh h{sruwhg dqg rqo| pdwhuldov lpsruwhg1
E| xvlqj wklv surfhgxuh/ zh h{foxgh lpphgldwh hhfwv iurp wkh
grphvwlf sulfh ohyho Sw wr wkh qrplqdo sulfh ri lpsruwhg pdwhuldov SwH 1
Krzhyhu/ zh gr qrw h{foxgh ihhgedfn ri wklv nlqg lq jhqhudolq
sulqflsoh/ wkh qrplqdo sulfh ri lpsruwhg pdwhuldov frxog eh dhfwhg e|
grphvwlf pdunhw frqglwlrqv/ exw zh gr qrw dvvhuw dq| wkhru| ri zk|
dqg krz vxfk hhfwv pd| rffxu1 H{solflwo| h{foxglqj ihhgedfn hhfwv
iurp wkh grphvwlf sulfh ohyho wr wkh qrplqdo sulfh ri lpsruwhg
pdwhuldov dprxqwv wr dvvxplqj wkdw wkh vhfrqg urz lq +O, lv }hur1
Wdeoh E jlyhv wkh hvwlpdwlrq uhvxowv iru +77, dqg +78,1 Wkh yduldeohv
duh ghwuhqghg xvlqj d Krgulfn0Suhvfrww owhu + @ 4933,1 Zh whvw wkh
owhuhg yduldeohv iru vwdwlrqdulw| dqg qg wkhp doo vwdwlrqdu| +uhvxowv
qrw uhsruwhg,1 Lq dgglwlrq/ zh uhsruw uhvxowv iru hvwlpdwlqj dq rxwsxw
64
htxdwlrq=
5
6
SwH4
\w @ H
3 SwH . H +O, 7 sHw4 8 . %\w
\w4
+7:,
lq rughu wr surylgh dq hpslulfdo phdvxuh ri wkh hhfw ri dq lqqrydwlrq
lq SwH 1 Wkh odj ohqjwk lv vhw dw wzr/ edvhg rq vlpsoh vljqlfdqfh whvwv1
Lq wkh uhdo sulfh htxdwlrq doo sdudphwhu hvwlpdwhv duh vljqlfdqw/ dqg
wkh frhflhqw rq wkh vwuxfwxudo vkrfn/ lh wkh frhflhqw wr SwH / lv
juhdwhu wkdq rqh1 Wkh dvvhuwlrq wkdw wkh grphvwlf sulfh ohyho dhfwv wkh
sulfh ri lpsruwv lv qrw frquphg/ zlwk wkh frhflhqwv rq odjjhg sH
w
ehlqj lqvljqlfdqw1 Wkh frhflhqw rq SwH lv qhjdwlyh dqg mxvw
lqvljqlfdqw dw d 8( vljqlfdqfh ohyho/ vr/ dv h{shfwhg/ d srvlwlyh sulfh
vkrfn dhfwv rxwsxw qhjdwlyho|1 Frqvlvwhqw zlwk rxu dvvhuwlrqv/ wkh
qrplqdo sulfh dhfwv rxwsxw rqo| rq lpsdfw/ dv wkh frhflhqwv rq
odjjhg ydoxhv ri SwH duh lqvljqlfdqw/ zkhuhdv wkh odjjhg ydoxhv ri sH
w
duh vljqlfdqw1+46,
Iljxuh 5= Hpslulfdo lpsxovh uhvsrqvh1 Sulfh ri lpsruwhg pdwhuldov
E| lqvhuwlqj +77, lqwr +78, dqg +7:,/ hpslulfdo lpsxovh uhvsrqvhv iru
wkhvh wzr yduldeohv fdq eh gudzq1 Wklv lv grqh lq Iljxuh 51 Lq uhvsrqvh
E ? @*|ih?@|i @) Lu it|4@|?} |iti i4ThU@* 4T*ti hitTL?ti L*_ Mi |L
it|4@|i @ V+c 4TLt?} _i?|u)?} Lh_ih hit|hU|L?c i Lh_ih |i @h@M*it |
7|. c R7.
| @?_ t7| @?_ ti |i L*it!) _iUL4TLt|L?( UL?t_ih?} |@| |i ThTLti Lu
|i i ihUti t |L U@*Mh@|i |i ThLUitt uLh R.
| c |t L*_ ?L*i it|4@|?} ?t@?Ui
T@h@4i|iht W? @?) U@tic |i i4ThU@* 4T*ti hitTL?tit uhL4 it|4@|?} |t V+
@hi ?L| ^@*|@|i*) _gihi?| uhL4 |i L?it hiTLh|i_ @MLi
65
wr d 4( vkrfn wr wkh qrplqdo sulfh ri lpsruwhg pdwhuldov/ SwH / wkh uhdo
sulfh ri lpsruwhg pdwhuldov lqfuhdvhv e| voljkwo| pruh wkdq d shufhqw
rq lpsdfw/ dqg wkh hhfw shdnv rqh shulrg diwhu wkh vkrfn1 Wkh uhdo
sulfh lv doprvw dv shuvlvwhqw dv wkh qrplqdo sulfh/ vxjjhvwlqj wkdw wkh
shuvlvwhqfh lq wkh qrplqdo sulfh lv wkh gulylqj irufh ehklqg wkh
shuvlvwhqfh lq wkh uhdo sulfh1 Rxwsxw frqwudfwv vkduso|/ ohyhoolqj rxw
diwhu yh txduwhuv dw 3145 shufhqw ehorz wkh lqlwldo ohyho/ dqg wkh hhfw
shuvlvwv iru forvh wr 4; txduwhuv1
Wdeoh E= Hvwlpdwlrq uhvxowv1 Sulfh ri lpsruwhg pdwhuldov
#iTi?_i?|
@h@M*i
W?_iTi?_i?|
@h@M*i
SwH4
SwH5
sH
w4
sH
w4
SwH
Y Dw4
Y Dw5
_ +5
5,+
6t|@|
L4 ThUi
+i@* ThUi
2f
feD
ff
ff
H
f2
2f
fS
fbD
SwH
f.S
f.D
bDe
f.
f.
fD
fe
sH
w
fb
ff
2b2e
fe
f
ff
ffb
fe
|T|
Y Dw
f2b
ff
fH
f2e
ff
fD
f2H
f
f
f
ff
ffS
f2
f2
f.H
ff
D.Db
L|itG ** @h@M*it @hi ? *L}t @?_ _i@|L?t uhL4 |hi?_ 5|@?_@h_ ihhLht ? |@*Ut
5,+ t t|@?_@h_ ihhLh Lu hi}hittL?c *i 6t|@| t |i 6t|@|t|U L? |i )TL|itt
|@| @** T@h@4i|iht @hi ?t}?U@?|
Lq pdnlqj frpsdulvrqv ehwzhhq wkh hpslulfdo lpsxovh uhvsrqvhv dqg
wkh lpsxovh uhvsrqvh ixqfwlrqv jhqhudwhg e| wkh prgho/ lw lv lpsruwdqw
wr nhhs lq plqg wkdw wkh ghwuhqglqj phwkrg glhuvlq wkh prgho/ zh
fdq fruuhfwo| ghwuhqg wkh yduldeohv/ jlyhq rxu nqrzohgjh derxw wkh
surshuwlhv ri wkh h{rjhqrxv yduldeohv1 Lq whupv ri wkh hpslulfdo
hvwlpdwhv/ zh kdyh wr fkrrvh d ghwuhqglqj phwkrg1 Khqfh zh fdqqrw
surylgh d irupdo frpsdulvrq ri wkh hpslulfdo dqg prgho jhqhudwhg
66
lpsxovh uhvsrqvh ixqfwlrqv/ dqg wkh hpslulfdo lpsxovh uhvsrqvh
ixqfwlrq vkrxog eh xvhg rqo| dv dq lqirupdo dvvhvvphqw ri wkh h{whqw wr
zklfk wkh prgho*v uhvsrqvh lv sodxvleoh1
2
BLih?4i?| i Ti?_|hi tLU!t
Vkrfnv wr ghpdqg duh prghoohg dv lqqrydwlrqv lq jryhuqphqw
h{shqglwxuh1 Khuh zh xvh wkh vlpsohvw srvvleoh prgho/ dvvxplqj wkdw
jryhuqphqw h{shqglwxuh iroorzv dq DU surfhvv zkhuh
jw4 =
jaw @ +O,a
+7;,
Dv ehiruh/ wklv lv hvwlpdwhg uxqqlqj dq ROV uhjuhvvlrq rq wkh iroorzlqj
htxdwlrq
jw4 . %jw
+7<,
jw @ +O,
zkhuh jryhuqphqw h{shqglwxuh kdv ehhq ghwuhqghg xvlqj d
Krgulfn0Suhvfrww owhu1 Iru wkh sxusrvh ri frpsdulvrq/ zh dovr
hvwlpdwh dq rxwsxw htxdwlrq
\w @ J
3 jw . J +O,
jw4
\w4
. %J\
w
+83,
wuhdwlqj vkrfnv wr jryhuqphqw h{shqglwxuh/ %jw dv wkh h{rjhqrxv vkrfn1
Wkh hvwlpdwlrq uhvxowv duh uhsruwhg lq Wdeoh F1 Wkh prvw lqwhuhvwlqj
uhvxowv uhsruwhg lq wklv wdeoh uhodwh wr wkh hhfwv rq rxwsxw ri d
vxusulvh lqfuhdvh lq jryhuqphqw h{shqglwxuh= zkloh vxfk dq lqfuhdvh lv
h{sdqvlrqdu| rq lpsdfw/ lw lv frqwudfwlrqdu| hyhq lq wkh vkruw uxq1 Wkh
frhflhqwv rq frqwhpsrudqhrxv dqg odjjhg jw duh krzhyhu doo
lqvljqlfdqw/ dqg d mrlqw I0whvw rq wkh vljqlfdqfh ri wkh wkuhh
sdudphwhuv vxjjhvwv wkdw wkh| fdq doo eh gursshg +I+6/443,@514</
s0ydoxh 313<,1 Wklv lv looxvwudwhg ixuwkhu lq Iljxuh 6> d 4( lqfuhdvh lq
jryhuqphqw h{shqglwxuh kdv h{sdqvlrqdu| hhfwv rqo| lq wkh uvw
shulrg1
5LU!t |L |iU?L*L})
Hvwlpdwlqj wkh vkrfnv wr WIS jurzwk udlvhv wzr lvvxhv1 Iluvw/ ehfdxvh
oderxu0dxjphqwlqj whfkqrorjlfdo surjuhvv lv vxemhfw wr shupdqhqw
67
Wdeoh F= Hvwlpdwlrq uhvxowv1 Jryhuqphqw h{shqglwxuh
#iTi?_i?|
@h@M*i
W?_iTi?_i?|
@h@M*i
jw4
jw5
Y Dw4
Y Dw5
jw
_ +5
5,+
65|@|
BL<| i Ti?_|hi
jw
feb
fD
fe
ff
D.S
ffb
ffb
B#
Y Dw
fb
f.
fHS
ffH
f
f2
f
ff
ff
f
fSb
ff
S2D
L|itG ** @h@M*it @hi ? *L}t @?_ _i@|L?t uhL4 |hi?_ 5|@?_@h_ ihhLht ? |@*Ut
5,+ t t|@?_@h_ ihhLh Lu hi}hittL?c *i 6t|@| t |i 6t|@|t|U L? |i )TL|itt
|@| @** T@h@4i|iht @hi ?t}?U@?|
Iljxuh 6= Hpslulfdo lpsxovh uhvsrqvh1 Jryhuqphqw h{shqglwxuh
68
vkrfnv/ zh qhhg wr pdnh dq dvvxpswlrq derxw wkh hqwu| dqg h{lw ri
upv> zlwkrxw wklv dvvxpswlrq/ d shupdqhqw whfkqrorj| vkrfn zrxog
lpso| srvlwlyh2qhjdwlyh surwv lq wkh qhz vwhdg| vwdwh1 Wkh vhfrqg
lvvxh lv wkdw ehfdxvh zh kdyh dvvxphg d jurvv rxwsxw surgxfwlrq
ixqfwlrq dqg lpshuihfw frpshwlwlrq/ zh odfn wkh gdwd wr frqvwuxfw d
vhulhv iru wkh whfkqrorj| vkrfnv zklfk lv frqvlvwhqw zlwk wkh
dvvxpswlrq ri rxu prgho1 Lqvwhdg zh frqvwuxfw wkuhh sur{| vhulhv iru
wkh vkrfnv wr WIS jurzwk1
Olnh Urwhpehuj dqg Zrrgirug +4<<8, zh dvvxph wkdw
oderxu0dxjphqwlqj whfkqrorjlfdo surjuhvv lv vxemhfw wr shupdqhqw
vkrfnv/ lh wkdw }w iroorzv d udqgrp zdon1+47, Wkhq wkh jurzwk udwh ri
whfkqrorjlfdo surjuhvv a }w lv dq lqghshqghqwo| glvwulexwhg yduldeoh
aw @ w > w Q +3> ,
+84,
dqg wkh rqo| sdudphwhuv zh qhhg wr fdoleudwh duh wkh vwhdg|0vwdwh ydoxh
} dqg wkh vwdqgdug ghyldwlrq ri wkh vkrfnv/ 1 Zkhq vkrfnv wr }w duh
shupdqhqw/ zh qhhg wr dgguhvv wkh lvvxh ri surwv lq wkh vwhdg| vwdwh1
Dujxdeo|/ dq| surwv zrxog eh frpshwhg dzd| lq wkh orqj uxq e|
hlwkhu hqwu| ru h{lw/ lh d fkdqjh lq wkh qxpehu ri upv ru lqgxvwulhv
dqg khqfh lq wkh {hg frvwv1 Zh gr qrw dwwhpsw wr prgho hqwu| dqg
h{lw vshflfdoo|/ exw lqvwhdg dvvxph/ lq olqh zlwk Urwhpehuj dqg
Zrrgirug +4<<8,/ wkdw hqwu| iroorzv wkh iroorzlqj huuru fruuhfwlrq
surfhvv ri wkh irup=
oq Lw @ oq +L}w Qw , . +4 , oq Lw4
+85,
uhhfwlqj wkdw hqwu| dqg h{lw wdnh wlph1 Wklv lpsolhv d uhgxfhg0irup
odz ri prwlrq=
Law @ +4 , Law4
+86,
Wkh nh| ihdwxuh ri wklv surfhvv iru wkh prgho lv wkh dvvxpswlrq wkdw
wkh qxpehu ri upv2lqgxvwulhv lv kljko| shuvlvwhqw1 Zh iroorz
Urwhpehuj dqg Zrrgirug +4<<8, dqg vhw @ 3=351 Qrwlfh wkdw wklv
dgmxvwphqw surfhvv lv frqvlvwhqw zlwk uhodwlyho| kljk hqwu| dqg h{lw
Ee At @tt4T|L? t M) ?L 4i@?t ??LULtQ?_ii_c +@? Ebb. tLt |@|
|i4TLh@h) ThL_U||) tLU!t @hi 4TLh|@?| u |i t|@?_@h_ + 4L_i* t |L @UUL?|
uLh |i Mt?itt U)U*i 4L4i?|t
69
udwhvwkh dvvxpswlrq grhv qrw suhfoxgh dq dgmxvwphqw lq Lw exw
suhyhqwv kljk0iuhtxhqf| pryhphqw lq Lw iurp ehlqj wkh pdlq exvlqhvv
f|foh ghwhuplqdqw lq wkh prgho1
Glhuhqwldwlqj wkh jurvv rxwsxw surgxfwlrq ixqfwlrq dqg uhduudqjlqj
jlyhv wkh iroorzlqj h{suhvvlrq iru wrwdo idfwru surgxfwlylw| jurzwk=
oq }w @
4
vK
+|w vN nw vK kw
+87,
vH hw vP pw + 4, Lw ,
zkhuh orzhu0fdvh ohwwhuv duh qdwxudo orjv ri yduldeohv dqg dqg vl duh
wkh vwhdg|0vwdwh pdun0xs dqg uhyhqxh vkduhv uhvshfwlyho|1 Lpsolflw lq
wklv ghulydwlrq lv wkh dvvxpswlrq wkdw doo yduldwlrqv lq rxwsxw fdq eh
dvfulehg wr fkdqjhv lq idfwru lqsxwv> wklv hvvhqwldoo| lv dq
dssur{lpdwlrq durxqg d vwhdg|0vwdwh sdwk zkhuh wkh pdun0xs dqg
idfwru vkduhv fdq uhdvrqdeo| eh dvvxphg frqvwdqw1 Wklv grhv krzhyhu
phdq wkdw zh duh dvfulelqj yduldwlrqv lq rxwsxw fdxvhg e| yduldwlrqv lq
wkh pdun0xs dqg idfwru vkduhv wr fkdqjhv lq WIS1
Uhprylqj wkh phdqv iurp +87, dqg vxevwlwxwlqj wkh h{suhvvlrq iru wkh
hqwu| ri qhz upv jlyhv wkh iroorzlqj h{suhvvlrq iru wkh vkrfnv wr jurvv
rxwsxw WIS jurzwk=
4
+|w vN nw vK kw vH hw vP pw ,
vK
+88,
zkhuh +O, lv d odj sro|qrpldo1+48, Wklv h{suhvvlrq fdqqrw/ krzhyhu/
eh xvhg wr fdofxodwh wkh vkrfnv wr jurvv rxwsxw WIS jurzwk dv wkhuh
duh qr wlph vhulhv gdwd iru jurvv rxwsxw dqg wkh wzr w|shv ri pdwhuldov
dydlodeoh iru wkh Xqlwhg Nlqjgrp1 Lq d shuihfwo| frpshwlwlyh prgho/
wklv lv qrw dq lvvxh= idfwru vkduhv duh frqvwdqw/ dqg +88, fdq eh
uhgxfhg wr lqfoxgh rqo| ydoxh dgghg/ fdslwdo dqg oderxu lqsxwv1
Dvvxplqj qr vxevwlwxwlrq ehwzhhq Y dqg T zrxog kdyh vlplodu hhfwv1
Zlwk lpshuihfw frpshwlwlrq dqg vxevwlwxwlrq ehwzhhq lqsxwv/ wkh
surgxfwlrq ixqfwlrq fdqqrw eh fdswxuhg e| d cuhgxfhg* ydoxh0dgghg
surgxfwlrq ixqfwlrq wkdw grhv qrw lqfoxgh hlwkhu pdwhuldov ru pdwhuldov
sulfhv1 Wkh uhdvrq zk| +88, fdqqrw eh uhzulwwhq lq whupv ri mxvw
ydoxh0dgghg rxwsxw/ oderxu/ fdslwdo dqg wkh sulfh ri lqwhuphgldwh
oq }w @ +O,
ED Aihi t @ tMt|@?|@* @4L?| Lu @*}iMh@ ?L*i_ ? }L?} uhL4 EDe |L EDD
#i|@*t Lu |t t|iT @?_ |i ThiUti UL4TLt|L? Lu lEu @hi @@*@M*i L? hi^it|
6:
lqsxwv lv ehfdxvh wkh frqwulexwlrq ri lqwhuphgldwh lqsxwv wr jurvv
rxwsxw h{fhhgv wkhlu uhyhqxh vkduh1
Wr looxvwudwh wklv sureohp/ zh xvh wkh ghqlwlrq ri uhdo ydoxh dgghg/
YwD =
+89,
YwD @ \w sH
w Hw Pw
Glhuhqwldwlqj dqg uhduudqjlqj zh jhw=
vY ywD @ |w vH sH
w vH hw vP pw
+8:,
zkhuh vY lv wkh vkduh ri ydoxh dgghg lq uhyhqxhv dqg vpdoo ohwwhuv
vljqli| qdwxudo orjv1 Xvlqj wklv h{suhvvlrq wr vxevwlwxwh wkh jurzwk udwh
ri jurvv rxwsxw rxw ri +87, jlyhv
oq }w
@
4 vY ywD vN nw vK kw + 4, Lw
vK
+8;,
. +4 , vH hw . +4 , vP pw . vH sH
w
Urwhpehuj dqg Zrrgirug +4<<8, frqvwuxfw wkh vkrfnv wr WIS jurzwk
xvlqj wkh iroorzlqj h{suhvvlrq=
W I S4 @
4 vY ywD vN nw vK kw + 4, Lw
vK
+8<,
Dowkrxjk wkh| dovr dvvxph d jurvv rxwsxw surgxfwlrq ixqfwlrq dqg
lpshuihfw frpshwlwlrq/ wkh| dyrlg wkh suhylrxvo| phqwlrqhg sureohp
e| dvvxplqj wkdw wkhuh lv qr vxevwlwxwdelolw| ehwzhhq ydoxh dgghg dqg
wkh lqwhuphgldwh lqsxwv1 Wklv phdqv wkdw ghvslwh wkh dvvxpswlrq ri
lpshuihfw frpshwlwlrq/ wkh frqwulexwlrq ri wkh pdwhuldov wr jurvv
rxwsxw htxdov wkhlu uhyhqxh vkduh/ dqg WIS jurzwk fdq eh hvwlpdwhg
mxvw xvlqj gdwd rq ydoxh0dgghg rxwsxw/ oderxu dqg fdslwdo1 Krzhyhu/
wkh hodvwlflw| ri vxevwlwxwlrq rewdlqhg iurp hvwlpdwlqj +6<, vxjjhvwv
wkdw ydoxh dgghg dqg wkh lqwhuphgldwh lqsxwv djjuhjdwru duh lqghhg
vxevwlwxwhv lq surgxfwlrq1 Vr phdvxulqj wkh vkrfnv xvlqj +8<,
jhqhudwhv whfkqrorj| vkrfnv zklfk duh lqfrqvlvwhqw zlwk rxu prgho
dvvxpswlrqv/ ehfdxvh lw ljqruhv fkdqjhv lq sulfhv dqg yroxphv ri
pdwhuldov1
Jlyhq wkdw wlph vhulhv gdwd rq wkh sulfh ri lpsruwhg lqwhuphgldwh
lqsxwv lv dydlodeoh/ lw lv srvvleoh wr clpsuryh* wkh hvwlpdwhv rewdlqhg
6;
iurp +8<, e| xvlqj wkh iroorzlqj h{suhvvlrq wr hvwlpdwh wkh vkrfnv wr
WIS jurzwk=
W I S5
@ oq }w @
4
++4 , vH hw . +4 , vP pw ,
vK
+93,
4 vY ywD vN nw vK kw vH sH
w + 4, Lw
vK
zkhuh oq }w lv wdnhq iurp +8;,1 Wkhvh hvwlpdwhv wdnh dffrxqw ri
fkdqjhv lq wkh sulfh ri lpsruwhg pdwhuldov1 Krzhyhu/ wkh hvwlpdwhv duh
vwloo lqfrqvlvwhqw zlwk wkh dvvxpswlrqv ri rxu prgho/ dv wkh vkrfnv
hvwlpdwhg lq +93, vwloo gr qrw wdnh dffrxqw ri fkdqjhv lq pdwhuldov1
Dq dowhuqdwlyh zd| ri hvwlpdwlqj wkh vkrfnv wr WIS jurzwk lv wr xvh
wkh gxdo dssurdfk1 Wklv lqyroyhv xvlqj wkh frvw ixqfwlrq gxdo wr wkh
dvvxphg surgxfwlrq ixqfwlrq wr ghulyh dq h{suhvvlrq iru wkh jurzwk
udwh ri pdujlqdo frvw/ vxevwlwxwlqj pdujlqdo frvw rxw xvlqj wkh rxwsxw
sulfh dqg wkh pdun0xs dqg wkhq uhduudqjlqj wklv wr jlyh dq h{suhvvlrq
iru WIS jurzwk1 Wkh wrwdo yduldeoh frvw ixqfwlrq lv=
Zw
H
P
> Uw > Sw > Sw
+94,
Fw @ Iw f
}w
Wkh pdujlqdo frvwv duh
PFw @
CFw
Zw
@f
> Uw > SwH > SwP
CIw
}w
+95,
Glhuhqwldwlqj/ zh jhw wkh pdujlqdo yduldeoh frvw ixqfwlrq dv
pfw @
}fZ
fU
f H
fS P
Z fZ
zw . uw . S sH
sP
oq }w +96,
w .
w f
f
f
f
f
Sulfhv duh vhw dv d pdun0xs rq pdujlqdo yduldeoh frvwv +lq vwhdg| vwdwh
wkh pdun0xs htxdov {hg frvwv shu xqlw ri rxwsxw,/ vr wkh jurzwk udwh
ri pdujlqdo frvw htxdov wkh jurzwk udwh ri sulfh plqxv wkh jurzwk lq
wkh pdun0xs=
+97,
pfw @ sw oq w
Vxevwlwxwlqj +97, lqwr +96, dqg xvlqj Vkhskdug*v ohppd jlyhv wkh
iroorzlqj h{suhvvlrq iru WIS jurzwk=
W I S6 @
4 P
sw . oq w vK zw vN uw vH sH
w vP sw
vK
+98,
6<
Iljxuh 7= Dowhuqdwlyh phdvxuhv ri WIS jurzwk
T FP1
TFP2
T FP3
0.0 6
0.0 4
0.0 2
0
-0.0 2
-0.0 4
19 76
1 97 9
19 82
1 98 5
19 88
19 91
1 99 4
19 97
Wkh dgydqwdjhv ri xvlqj wklv h{suhvvlrq wr hvwlpdwh wkh vkrfnv wr WIS
jurzwk duh wkdw wlph vhulhv gdwd duh dydlodeoh ru fdq eh uhodwlyho| hdvlo|
frqvwuxfwhg iru doo wkh sulfh gdwd/ dqg doo wkh lqsxwv lq wkh surgxfwlrq
ixqfwlrq duh wdnhq dffrxqw ri1 Wkh glvdgydqwdjh lv wkdw lw uhtxluhv gdwd
rq wkh pryhphqwv lq wkh pdun0xs/ zklfk lv hqgrjhqrxv wr rxu prgho1
Li zh dvvxph wkdw oq w fdq eh ljqruhg/ rxu phdvxuh ri WIS zloo dovr
uhhfw f|folfdo fkdqjhv lq wkh pdun0xs1
Jlyhq wkdw zh fdqqrw frqvwuxfw hvwlpdwhv ri wkh WIS vkrfnv zklfk duh
frqvlvwhqw zlwk rxu prgho dvvxpswlrqv xvlqj hlwkhu wkh sulpdo ru wkh
gxdo dssurdfk/ zh frqvwuxfw sur{lhv iru wkh whfkqrorj| vkrfnv xvlqj
htxdwlrqv +8<,/ +93, dqg +98,1 Iljxuh 7 frpsduhv wkh surshuwlhv ri wkh
wkuhh sur{lhv ri WIS jurzwk1 Wkh wzr sulpdo phdvxuhv ri WIS jurzwk
duh kljko| fruuhodwhg1 Wkh pdlq glhuhqfh ehwzhhq wkhp lv wkdw wkh
phdvxuh zklfk sduwldoo| doorzv iru wkh vxevwlwxwdelolw| ri ydoxh0dgghg
dqg lqwhuphgldwh pdwhuldov/ W I S5 / lv pruh yrodwloh wkdq wkh phdvxuh
wkdw dvvxphv qrq0vxevwlwxwdelolw|/ W I S4 1 Wklv uhhfwv kljk yrodwlolw| lq
sulfhv ri lpsruwhg pdwhuldov1 Wkh gxdo phdvxuh ri WIS jurzwk/ W I S6 /
lv qrw kljko| fruuhodwhg zlwk wkh wzr sulpdo phdvxuhv/ lq sduwlfxodu lw
lv ohvv yrodwloh1 Wklv frxog eh hlwkhu ehfdxvh wkh gxdo phdvxuh wdnhv
dffrxqw ri doo wkh lqsxwv lq wkh surgxfwlrq ixqfwlrq/ ru ehfdxvh upv
devrue vrph ri wkh vkrfnv wr lqsxw sulfhv lq wkh pdun0xs1
73
e #)?@4U ThLTih|it Lu |i 4L_i*
Wklv vhfwlrq h{soruhv wkh g|qdplf surshuwlhv ri wkh prgho e| dqdo|vlqj
wkh uhvsrqvh wr h{rjhqrxv vkrfnv1 Wklv dqdo|vlv kdv wzr sduwv= uvw/ zh
dqdo|vh lpsxovh uhvsrqvh ixqfwlrqv/ wudflqj wkh uhvsrqvh ri wkh
hqgrjhqrxv yduldeohv wr d rqh r0vkrfn wr rqh ri wkh h{rjhqrxv
yduldeohv1 Vhfrqg/ zh lqyhvwljdwh wkh prgho*v vwrfkdvwlf surshuwlhv
xvlqj d vlpxodwlrq dssurdfk/ fdofxodwlqj vwdqgdug vwdwlvwlfv iru d
vhtxhqfh ri hqgrjhqrxv yduldeohv jhqhudwhg dv d uhvsrqvh wr d vhtxhqfh
ri udqgrp h{rjhqrxv vkrfnv zlwk vwrfkdvwlf surshuwlhv vlplodu wr wkrvh
revhuyhg lq wkh XN hfrqrp|1 Lq wklv sduw ri wkh h{huflvh/ zh dovr uxq
wkh prgho xvlqj wkh vhtxhqfhv ri vkrfnv hvwlpdwhg suhylrxvo|wkh
txhvwlrq ri lqwhuhvw lv zkhwkhu wkh prgho lv deoh wr uhsolfdwh revhuyhg
ehkdylrxu dqg wr dvvhvv wkh uhodwlyh lpsruwdqfh ri wkh vkrfnv lq
h{sodlqlqj rxwsxw xfwxdwlrqv1
e
#i4@?_ tLU!t
Lq wkh olwhudwxuh rq wkh f|folfdolw| ri pdun0xsv/ dwwhqwlrq kdv irfxvhg
rq wkh hhfwv ri ghpdqg vkrfnvlqghhg d pdlq srlqw ri wkh uhvhdufk
surjudpph lv wr vkrz wkdw wkh lqwhudfwlrq ehwzhhq pdun0xsv dqg
ghpdqg vkrfnv lv lpsruwdqw/ dqg wkdw d frxqwhuf|folfdo pdun0xs zloo
dpsoli| wkh rxwsxw uhvsrqvh ri d ghpdqg vkrfn1 Lq wkh prgho dqdo|vhg
khuh/ d ghpdqg vkrfn lqfuhdvhv ydoxh dgghg e| d qhjdwlyh zhdowk hhfw=
dq lqfuhdvh lq jryhuqphqw h{shqglwxuh lqfuhdvhv wkh pdujlqdo xwlolw| ri
zhdowk/ wkhuhe| lqfuhdvlqj oderxu vxsso|1 Vxppdulvlqj wkh glvfxvvlrq
lq wkh lqwurgxfwlrq/ wkh suhvhqfh ri d pdun0xs dorqh dhfwv wkh
txdqwlwdwlyh uhvsrqvh= d pdun0xs juhdwhu wkdq xqlw| hhfwlyho| uhgxfhv
wkh vorsh ri wkh oderxu ghpdqg fxuyh dqg khqfh lqfuhdvhv wkh
uhvsrqvlyhqhvv ri hpsor|phqw wr d fkdqjh lq zdjhv/ dv glvfxvvhg lq wkh
lqwurgxfwlrq1 Wkh idfw wkdw wkh pdun0xs ydulhv f|folfdoo| dhfwv wkh
srvlwlrq ri wkh oderxu ghpdqg fxuyh1
Wkh nh| lvvxh lv krz wkh ghpdqg vkrfn dhfwv h{shfwhg ixwxuh surwv
uhodwlyh wr fxuuhqw rxwsxw/ lh wkh udwlr [w @\w 1 Dq lqfuhdvh lq
jryhuqphqw h{shqglwxuh lqfuhdvhv wkh pdujlqdo xwlolw| ri zhdowk w / dqg
wkh pdujlqdo xwlolw| lv rq d ghfuhdvlqj sdwk edfn wrzdugv vwhdg| vwdwh/
a w.4 1 Wklv lqfuhdvhv wkh uhdo lqwhuhvw udwh dqg orzhuv wkh
aw A lh glvfrxqw idfwru/ whqglqj wr orzhu [w iru d jlyhq rxwsxw sdwk1 Surylghg
74
d jryhuqphqw h{shqglwxuh vkrfn h{sdqgv fxuuhqw rxwsxw dqg wkdw wkh
g|qdplf uhvsrqvh ri rxwsxw iroorzv d ghfuhdvlqj sdwk/ wkh udwlr [w @\w
zloo xqdpeljxrxvo| ghfuhdvh1 Lq wkh fxvwrphu pdunhw prgho/ wkh
dvvxpswlrq lv wkdw wkh pdun0xs uhvsrqgv srvlwlyho| wr vxfk d ghfuhdvh
+ ? 3,/ ohdglqj wr d surf|folfdo pdun0xs uhvsrqvh1 Wkh idfw wkdw wkh
pdun0xs pryhv surf|folfdoo| uhgxfhv wkh srvlwlyh rxwsxw dqg
hpsor|phqw hhfwv ri vxfk d ghpdqg vkrfn= dq lqfuhdvh lq wkh
pdun0xs vkliwv wkh oderxu ghpdqg fxuyh wr wkh ohiw/ dqg wklv rvhwv wkh
hpsor|phqw uhvsrqvh exw pdjqlhv wkh zdjh uhvsrqvh1 Lq frqwudvw/ wkh
lpsolflw frooxvlrq prgho suhglfwv wkdw wkh f|folfdo pryhphqwv lq wkh
pdun0xs zloo dpsoli| wkh rxwsxw uhvsrqvh= wkh pdun0xs uhvsrqgv
frxqwhuf|folfdoo| + A 3,/ vkliwlqj wkh oderxu ghpdqg vfkhgxoh wr wkh
uljkw/ khqfh dpsoli|lqj wkh rxwsxw pxowlsolhu1 Wkh lpsxovh uhvsrqvh
ixqfwlrqv xqghu wkh wzr glhuhqw dvvxpswlrqv derxw pdun0xs g|qdplfv
duh looxvwudwhg lq Iljxuh 81 Wkh ghfuhdvh lq zdjhv lq wkh fxvwrphu
pdunhw prgho lv txdqwlwdwlyho| odujhu wkdq wkh ghfuhdvh lq wkh lpsolflw
frooxvlrq prgho/ uhhfwlqj wkh vkliw ohiwzdug ri wkh oderxu ghpdqg
fxuyh> wkh krxuv dqg rxwsxw uhvsrqvh lq wkh lpsolflw frooxvlrq prgho lv
wkxv odujhu dqg forvhu wr wkh hpslulfdo lpsxovh uhvsrqvh ixqfwlrq/
hvwlpdwhg lq Vhfwlrq 6151 Vwloo/ wkh rxwsxw pxowlsolhu lv vpdoohu wkdq wkh
hpslulfdoo| hvwlpdwhg= zlwk d jryhuqphqw h{shqglwxuh vkduh ri ydoxh
dgghg ri durxqg 55(/ wkh hpslulfdo pxowlsolhu lv forvh wr 3178/ zkloh
wkh pxowlsolhu lq wkh lpsolflw frooxvlrq prgho lv dssur{lpdwho| 3149 dqg
lq wkh fxvwrphu pdunhw prgho rqo| 31371 Dv ehiruh/ wkh sursdjdwlrq ri
wkh vkrfn odujho| uhhfwv wkh shuvlvwhqfh ri wkh h{rjhqrxv vkrfnv udwkhu
wkdq lqwhuqdo sursdjdwlrq1
Udwkhu wkdq lqwhusuhwlqj wklv dv hylghqfh lq idyrxu ri wkh lpsolflw
frooxvlrq prgho/ zh lqwhusuhw wklv dv hylghqfh wkdw wkh prgho grhv qrw
fdswxuh wkh hhfwv ri vkrfnv wr jryhuqphqw h{shqglwxuh zhoohyhq lq
wkh lpsolflw frooxvlrq prgho/ wkh hodvwlflw| ri wkh pdun0xs/ / zrxog
kdyh wr lqfuhdvh e| d idfwru ri wkuhh wr dffrxqw iru wkh vl}h ri wkh
hpslulfdo pxowlsolhu> dqg wkh sursdjdwlrq ri wklv lqlwldo hhfw zrxog
odujho| uhhfw wkdw shuvlvwhqfh lq jryhuqphqw h{shqglwxuh1
e2
tLU! |L |i ThUi Lu 4TLh|i_ 4@|ih@*t
Wklv vhfwlrq frqvlghuv wkh uhvsrqvh wr d vkrfn wr sH
w / wkh uhdo sulfh ri
lpsruwhg pdwhuldov1 Uhfdoo wkdw wkhvh vkrfnv duh lghqwlhg dv vkrfnv wr
75
Figure 5: Model impulse response. Government expenditure
(A) Customer markets
(B) Implicit collusion
43
wkh qrplqdo sulfh ri lpsruwv/ dqg wkdw wkh vwrfkdvwlf surfhvv xvhg khuh
doorzv iru lqwhudfwlrq ehwzhhq odjjhg ydoxhv ri uhdo dqg qrplqdo sulfhv1
Wklv lv lpsruwdqw/ ehfdxvh wkh h{rjhqrxv yduldeohv wkhpvhoyhv duh
shuvlvwhqwd vkrfn wr saH
w lv sursdjdwhg h{rjhqrxvo|/ dqg wklv dhfwv
wkh uhvsrqvh ri wkh hqgrjhqrxv yduldeohv1 Zlwkrxw vxfk shuvlvwhqfh/ d
vxusulvh lqfuhdvh lq saH
w pd| kdyh vxevwdqwldo lpphgldwh lpsdfw rq
rxwsxw exw lv qrw sursdjdwhg vwurqjo|/ vr wkdw wkh hhfwv ri d vkrfn glh
rxw txlfno|1
Wklv lv uhhfwhg lq wkh lpsxovh uhvsrqvh ixqfwlrqv jhqhudwhg e| wkh
prgho/ uhsruwhg lq Iljxuh 9= lq erwk wkh fxvwrphu pdunhw prgho dqg
wkh lpsolflw frooxvlrq prgho/ wkh sdwkv ri rxwsxw/ ydoxh dgghg/
hpsor|phqw dqg zdjhv doo iroorz wkh sdwk ri wkh uhdo sulfh/ ghvfulehg lq
Vhfwlrq 6141 Lqghshqghqw ri wkh fkrlfh ri pdun0xs ehkdylrxu/ d qrplqdo
sulfh vkrfn lv sursdjdwhg lq wkh uhdo yduldeohv ryhu 9 wr ; txduwhuv/
zkhuh wkh hpslulfdo uhvsrqvh lqglfdwhv d vwurqjhu sursdjdwlrq1 Wkh
shdn hhfw glhuv vkduso| dfurvv wkh wzr prghov= lq wkh lpsolflw
frooxvlrq prgho/ wkh frxqwhuf|folfdo ehkdylrxu ri wkh pdun0xs dpsolhv
wkh uhvsrqvh/ e| pdjqli|lqj wkh oderxu ghpdqg uhvsrqvh1 Lq wkh
fxvwrphu pdunhw/ zkhuh wkh pdun0xs pryhv surf|folfdoo|/ wkh rxwsxw
uhvsrqvh lv gdpshqhg/ eulqjlqj lw forvhu lqwr olqh zlwk wkh pdjqlwxgh
ri wkh uhvsrqvh revhuyhg lq wkh hpslulfdo lpsxovh uhvsrqvh ixqfwlrq1
Qrwlfh wkdw wkh fxvwrphu pdunhw lqgluhfwo| vxssruwv wkh lghd ri d
judgxdo sdvv0wkurxjk iurp lpsruw sulfhv wr grphvwlf sulfhv1+49, Hyhq
lq d prgho vxfk dv wkh fxuuhqw rqh zkhuh sulfhv duh ixoo| h{leoh/
surf|folfdo pryhv lq wkh pdun0xs fdq ohdg wr judgxdo sdvv0wkurxjk
ehfdxvh dq lqfuhdvh lq wkh sulfh ri lpsruwhg pdwhuldov hhfwlyho|
wuljjhuv d uhfhvvlrq1 Wklv uhvxow vxssruwv wkh frqfoxvlrqv uhdfkhg e|
PfFdooxp dqg Qhovrq +4<<;, zkr dovr wuhdw lqqrydwlrqv lq wkh sulfh ri
lpsruwhg pdwhuldov dv dq h{rjhqrxv cvxsso| vkrfn*1 Lq wkhlu prgho
grphvwlf lqdwlrq dqg uhdo rxwsxw duh srvlwlyho| fruuhodwhg/ vr dowkrxjk
d srvlwlyh sulfh vkrfn zloo udlvh wkh frqvxphu sulfh lqgh{ gluhfwo|/ lw
zloo orzhu lqdwlrq ehfdxvh lw lv lq hhfw d qhjdwlyh vxsso| vkrfn1 D
pdun0xs hhfw olnh wkh rqh ghvfulehg deryh zloo zrun lq dgglwlrq wr
wklv hhfw= qrw rqo| zrxog sdvv0wkurxjk eh ghod|hg gluhfwo| wkurxjk dq
lqdwlrq hhfw/ exw dovr glplqlvkhg e| d ghfuhdvh lq wkh pdun0xs1
ES W? |i i4ThU@* *|ih@|hic |i T33*i t uLh4*@|i_ @t @ *@U! Lu } ULhhi*@|L?
Mi|ii? ??L@|L?t ? |i hi@* i U@?}i h@|i @?_ ??L@|L?t ? ?@|L?Q|L|
?L4?@* @h@M*it @?_ @ 4Lhi i*@MLh@|i 4L_i* Lu |h@_ic i U@??L| @__hitt |i T33*i
_hiU|*)
77
Iljxuh 9= Prgho lpsxovh uhvsrqvh1 Sulfh ri lpsruwhg pdwhuldov1
+D, Fxvwrphu pdunhwv
+E, Lpsolflw frooxvlrq
78
e
AiU?L*L})
Uhfdoo wkdw whfkqrorj| vkrfnv duh shupdqhqwd vkrfn wr }w lv d
shupdqhqw lqfuhdvh lq wkh ohyho ri oderxu0dxjphqwlqj whfkqrorjlfdo
surjuhvv/ dqg khqfh d shupdqhqw vkliw lq rxwsxw dqg lwv frpsrqhqwv wr
d qhz wuhqg sdwk1 Lq Iljxuh 4/ d srvlwlyh vkrfn wr wkh ohyho ri
whfkqrorj| fruuhvsrqgv wr d shupdqhqw vkliw wr wkh uljkw ri wkh oderxu
ghpdqg fxuyh/ zkhuh wkh pdjqlwxgh ri wkh vkliw ghshqgv rq wkh vl}h ri
wkh vkrfn dqg wkh hodvwlflwlhv ri vxevwlwxwlrq ehwzhhq wkh idfwruv1 Dv
ehiruh/ wkh suhvhqfh ri d pdun0xs dorqh dhfwv wkh pdjqlwxgh ri wkh
vkrfn= d shupdqhqw lqfuhdvh lq wkh pdujlqdo surgxfw ri oderxu zloo/ iru
dq| jlyhq ohyho ri fdslwdo/ vkliw wkh oderxu ghpdqg fxuyh uhodwlyho| ohvv
lq wkh suhvhqfh ri d pdun0xs1 Lq dgglwlrq/ d whfkqrorj| vkrfn lqfuhdvhv
wkh suhvhqw ydoxh ri ixwxuh zhdowk/ dqg wkh suhvhqfh ri d pdun0xs whqgv
wr lqfuhdvh wklv zhdowk hhfw1 Li wkh zhdowk hhfw lv vxflhqwo| odujh
dqg grplqdwhv wkh lqwhuwhpsrudo vxevwlwxwlrq hhfw/ wkhq krxuv pd|
dfwxdoo| ghfuhdvh lq uhvsrqvh wr dq lqfuhdvh lq whfkqrorj|1
Wkh uhvsrqvh ri wkh prgho xqghu wkh wzr glhuhqw dvvxpswlrqv derxw
pdun0xs g|qdplfv duh looxvwudwhg lq Iljxuh :1 D vkrfn wr whfkqrorj|
fkdqjhv wkh vwhdg|0vwdwh sdwk/ dqg wkh g|qdplf dgmxvwphqw diwhu wkh
vkrfn lv jryhuqhg e| wkh idfw wkdw d shupdqhqw vkrfn wr whfkqrorj|
ohdyhv wkh fdslwdo vwrfn ehorz lwv ghvluhg ohyhowkh pdujlqdo xwlolw| ri
zhdowk lv deryh lwv qhz +dqg kljkhu, orqj0uxq ohyho/ dqg wklv lq hhfw
lqfuhdvhv wkh uhdo lqwhuhvw udwh deryh wkh orqj0uxq ohyho> uhvrxufhv duh
vkliwhg wrzdugv fdslwdo dffxpxodwlrq dqg lqyhvwphqw ryhuvkrrwv lwv qhz
orqj uxq0ohyho +qrw looxvwudwhg,1 Wklv dgglwlrqdo dffxpxodwlrq fdq eh
eurxjkw derxw wkurxjk vhyhudo fkdqqhov1 Iluvw/ iru d jlyhq ohyho ri
lqsxwv/ rxwsxw kdv lqfuhdvhg gxh wr wkh lqfuhdvh lq wkh ohyho ri
whfkqrorj|1 Vhfrqg/ oderxu vxsso| fdq lqfuhdvh= wkhuh lv d pryhphqw
dorqj wkh oderxu vxsso| fxuyh/ iroorzlqj wkh vkliw lq oderxu ghpdqg/
dqg srvvleo| d vkliw lq wkh oderxu vxsso| fxuyh/ li wkh pdujlqdo xwlolw|
ri zhdowk lqfuhdvhv iroorzlqj wklv vkrfn/ lh li wkh lqfhqwlyh wr
dffxpxodwh fdslwdo lv vxflhqwo| vwurqj1 Erwk wkhvh hhfwv whqg wr
lqfuhdvh wkh wrwdo dprxqw ri dydlodeoh uhvrxufhv1 Wklug/ frqvxpswlrq
fdq eh orzhuhg ehorz lwv +qhz, orqj0uxq ohyho/ lqfuhdvlqj uhvrxufhv
dydlodeoh iru lqyhvwphqw iru d jlyhq ohyho ri hpsor|phqw dqg rxwsxw1
Qrwlfh wkdw dowkrxjk rxwsxw dqg zdjhv duh ehorz wkhlu qhz
vwhdg|0vwdwh ohyhov/ erwk zloo jurz dw dq| srlqw lq wlph diwhu wkh vkrfn1
Iru wkh g|qdplfv ri wkh pdun0xs/ lw lv djdlq wkh udwlr [w @\w wkdw
79
Iljxuh := +D, Whfkqrorj| vkrfnv1 Fxvwrphu pdunhwv
+l,
+ll,
7:
Iljxuh := +E, Whfkqrorj| vkrfnv1 Lpsolflw frooxvlrq
+l,
+ll,
7;
pdwwhuv> wklv udwlr lv/ ri frxuvh/ vwdwlrqdu|1 Wkh lpsruwdqw srlqw lv wkdw
rxwsxw lv ehorz lwv qhz wuhqg ohyho dqg lv lqfuhdvlqj= iru d jlyhq
glvfrxqw idfwru dqg vhtxhqfh ri pdun0xsv/ wklv zloo uhtxluh wkdw [w
xqghuvkrrwv lwv qhz orqj0uxq ohyho pruh wkdq \w / dqg wkdw wkh udwlr ri
wkh wzr ghfuhdvhv1 Lq dgglwlrq/
wkh
pdujlqdo xwlolw| ri zhdowk h{fhhgv
a
a
lwv qhz orqj0uxq ohyho w A w.4 1 Wklv lpsolhv wkdw uhdo lqwhuhvw udwhv
duh ehorz vwhdg|0vwdwh ohyhov= wkh glvfrxqw idfwru lq fdofxodwlqj wkh
h{shfwhg suhvhqw ydoxh ri ixwxuh surwv ghfuhdvhv/ ohdglqj wr d ghfuhdvh
lq [w 1
Lw vkrxog eh hpskdvlvhg wkdw dowkrxjk rxwsxw lv ehorz lwv qhz wuhqg
ohyho/ rxwsxw lv jurzlqj deryh wkh rog wuhqg ohyho dw doo srlqwv1 Wklv lv
lpsruwdqw lq wkh vlpxodwlrq h{huflvh= li wkh wuhqg lv phdvxuhg fruuhfwo|/
lh ghwuhqghg e| wkh ohyho ri }w / wkhq wkh g|qdplf surshuwlhv ri wkh
pdun0xs pd| glhurxwsxw lv ehorz wuhqg dqg vr lv wkh udwlr [w @\w /
vr lq wkh fxvwrphu pdunhw prgho/ wkh pdun0xs zloo eh deryh vwhdg|
vwdwh/ dqg ehorz lq wkh lpsolflw frooxvlrq prgho1 Lq wkh vlpxodwlrq
h{huflvh/ krzhyhu/ wkh ghwuhqglqj phwkrg lv qhfhvvdulo| wkh vdph dv wkh
rqh dssolhg wr wkh gdwd/ lh d Krgulfn0Suhvfrww owhu1 Li wkh owhu
vprrwkhv rxwsxw vxflhqwo|/ wkhq wkh lqlwldo uhvsrqvh ri rxwsxw pd| eh
lqwhusuhwhg dv dq deryh0wuhqg lqfuhdvh> lq wkh ehqfkpdun fdvh/ wklv
pdwhuldolvhv dqg wkh pdun0xs suhvhuyhv lwv f|folfdo surshuwlhv1
Dsduw iurp wkh pdun0xs/ wkh g|qdplf uhvsrqvhv ri wkh hqgrjhqrxv
yduldeohv duh vlplodu lq whupv ri wkhlu sdwwhuqv/ exw glhu lq
pdjqlwxghwkh fxvwrphu pdunhw prgho dpsolhv wkh uhvsrqvh wr d
whfkqrorj| vkrfn zkloh wkh lpsolflw frooxvlrq prgho gdpshqv lw/ lq
frqwudvw wr wkh ghpdqg vkrfn1 Xqvxusulvlqjo|/ lq erwk fdvhv wkh rxwsxw
uhvsrqvh wr d shufhqwdjh fkdqjh lq }w lv odujhu wkdq wkh uhvsrqvh wr
ghpdqg vkrfnv1
ee
54*@|L? ThLTih|it
Wklv vhfwlrq ixuwkhu fkdudfwhulvhv wkh txdqwlwdwlyh surshuwlhv ri wkh
prgho lq d vlpxodwlrq h{huflvh1 Wkh sxusrvh ri wkh h{huflvh lv wr
hvwdeolvk krz zhoo wkh prgho shuirupv lq wkh sduwlfxodu glphqvlrqv ri
lqwhuhvw/ dqg zkhwkhu wkh prgho lv vwdeoh iru vpdoo shuwxedwlrqv lq wkh
xqghuo|lqj sdudphwhuv1 Iroorzlqj vwdqgdug g|qdplf jhqhudo
htxloleulxp phwkrgrorj|/ wkh whfkqltxh lv wr vxemhfw wkh prgho wr
7<
vhtxhqfhv ri h{rjhqrxv udqgrp vkrfnv/ fdofxodwh wkh hqgrjhqrxv
uhvsrqvh dqg fkdudfwhulvh wklv uhvsrqvh e| d vhw ri vwdqgdug vwdwlvwlfv1
Wkhvh vwdwlvwlfv duh wkhq frpsduhg zlwk d vhw ri vwdwlvwlfv fdofxodwhg rq
djjuhjdwh XN gdwd1 Lq wkh vhqvlwlylw| dqdo|vlv/ vhohfwhg sdudphwhuv duh
ydulhg wr vhh wkh h{whqw wr zklfk wklv dhfwv wkh prgho*v surshuwlhv1
Zh xvh wkuhh vwdwlvwlfdo wlph vhulhv lq wklv frpsdulvrq1 Wr frpsduh
zlwk wkh prgho*v ydoxh0dgghg/ zh xvh wkh JGS +rxwsxw, vhulhv/ dqg iru
dyhudjh zdjhv wkh uhylvhg dyhudjh hduqlqjv lqgh{1 Erwk vhulhv duh
vhdvrqdoo| dgmxvwhg dqg wdnhq gluhfwo| iurp wkh Rfh iru Qdwlrqdo
Vwdwlvwlfv1 Wkh phdvxuh ri wrwdo krxuv zrunhg lv wkh vhulhv xvhg lq wkh
Edqn ri Hqjodqg*v iruhfdvwlqj prgho +vhh Edqn ri Hqjodqg +4<<<,,1
Wklv vhulhv lv edvhg rq wkh Oderxu Irufh Vxuyh| +OIV, phdvxuh ri wrwdo
krxuv zrunhg lq doo lqgxvwulhv iurp 4<<50<:1 Sulru wr wklv/ iurp 4<;7 wr
4<<5/ wkh fruuhvsrqglqj vhulhv iurp wkh rog OIV lv xvhg1 Wklv lv
dgmxvwhg wr wdnh lqwr dffrxqw wkh idfw wkdw wkh rog OIV rqo| fryhuv
Juhdw Eulwdlq udwkhu wkdq wkh hqwluh Xqlwhg Nlqjgrp1 Sulru wr 4<;7/
qr zkroh0hfrqrp| phdvxuh h{lvwv= wkh klvwrulfdo uhodwlrq ehwzhhq
pdqxidfwxulqj dqg hfrqrp|0zlgh wrwdo krxuv srvw~4<;7 lv xvhg
wrjhwkhu zlwk wkh vhulhv iru pdqxidfwxulqj krxuv wr frqvwuxfw d orqjhu
uxq ri gdwd1 Doo vhulhv duh txduwhuo| dqg vhdvrqdoo| dgmxvwhg/ dqg wkh
vdpsoh uxqv iurp 4<96 T4 wr 4<<; T61
Wkh ehqfkpdun prgho lv fdoleudwhg dv ghvfulehg deryh1 Wkh ohqjwk ri
wkh vlpxodwlrq lv vhw dw wkh vdpsoh ohqjwk ri wkh djjuhjdwh vhulhv/ lh 473
txduwhuv1 Iru wkh vwdwlvwlfv wr eh frpsdudeoh/ wkh vhulhv jhqhudwhg e|
wkh prgho dqg wkh XN djjuhjdwhv duh vxemhfw wr wkh vdph ghwuhqglqj
phwkrg= wkh vhulhv jhqhudwhg e| wkh prgho duh frqyhuwhg iurp
ghyldwlrqv durxqg d vwrfkdvwlf wuhqg lqwr ohyhov/ dqg wkhq/ olnh wkh XN
gdwd/ owhuhg zlwk d Krgulfn0Suhvfrww owhu1 Wkh gdwd xvhg iru wklv
dqdo|vlv fruuhvsrqg wr wkh gdwd xvhg lq wkh Edqn*v pdfurhfrqrphwulf
prgho +iru ghwdlov vhh Edqn ri Hqjodqg +4<<<,,1
Wdeoh G uhsruwv vwdqgdug ghyldwlrqv uhodwlyh wr rxwsxw/ rxwsxw
fruuhodwlrq dqg uvw0rughu dxwrfruuhodwlrq ri rxwsxw/ hpsor|phqw/ dqg
dyhudjh zdjhv iurp XN gdwd dqg iurp wkh wzr prghov ri pdun0xsv1 Iru
looxvwudwlyh sxusrvhv/ wkh wdeoh dovr uhsruwv vwdwlvwlfv iru lpsruwhg dqg
grphvwlfdoo| surgxfhg pdwhuldovdowkrxjk zh gr qrw kdyh htxlydohqw
wlph vhulhv hylghqfh/ wkh surshuwlhv ri wkhvh vhulhv duh ri lqwhuhvw1
Dv h{shfwhg/ wkh prgho odujho| uhsolfdwhv wkh surshuwlhv ri wkh vwdqgdug
g|qdplf jhqhudo htxloleulxp prghov/ vxfk dv Kdqvhq +4<;8, ru
83
Wdeoh G= Vwdqgdug vwdwlvwlfv1 XN gdwd dqg prgho
XN gdwd
Yduldeoh
{
{|
{ Y D
414; 31;:
4145 31:5
4133 31;3
316<
317;
4133
{
Y D
{
Krxuv zrunhg 41;6
Dyhudjh zdjhv 41:8
JGS
4189
Lpsolflw frooxvlrq prgho
Yduldeoh
Krxuv
Lps1 pdw1
Grp1 pdw1
Dy1 zdjh
Jurvv rxwsxw
Pdun0xs
Ydoxh dgghg
{
Y D
{
415:
51;8
515<
4169
515:
313<
5167
fD
ff
f22
fe
f22
ff
f2e
3188
4156
31<;
318;
31<:
3137
4133
ffS
f
ffe
ff2
ffe
ff
fff
{|
{ Y D
{
31::
31:9
31:6
31:7
31:5
31;4
31:4
ffe
ffe
ffD
ffD
ffD
ff
ffS
31;3
31;:
31<;
31<;
31<;
03193
4133
ffe
ff
ff
ff
ff
ffH
fff
Fxvwrphu pdunhw prgho
Yduldeoh
Krxuv
Lps1 pdw1
Grp1 pdw1
Dy1 zdjh
Jurvv rxwsxw
Pdun0xs
Ydoxh dgghg
{
Y D
{
317:
41<<
4184
31<<
418<
314;
513<
ffD
f2f
fD
ff
fS
ff2
f22
3156
31<9
31:6
317;
31:9
313<
4133
ff
ffb
ff
ff
ff2
ff
fff
{|
{ Y D
{
31:9
31:8
31:4
31:5
31:4
31:6
31:4
ffe
ffe
ffS
ffD
ffS
ffD
ffS
31::
31;7
31<<
31<;
31<<
31;<
4133
Qrwhv= { lv wkh vwdqgdug ghyldwlrq ri wkh yduldeoh [/ Y{D wkh vwdqgdug ghyldwlrq
ri [ uhodwlyh wr JGS/ { wkh dxwrfruuhodwlrq frhflhqw dqg {{|Y D wkh fruuhodwlrq
frhflhqw zlwk JGS1 Vwdqgdug huuruv lq lwdolfv1
84
ffD
ffe
fff
ff
fff
ff
fff
Gdqwklqh dqg Grqdogvrq +4<<6,1 Olnh wkh vwdqgdug prghov/ wkh prgho
jhqhudwhv lqvxflhqw uhodwlyh yduldelolw| lq hpsor|phqw/ dqg dyhudjh
zdjhv wkdw duh wrr kljko| fruuhodwhg zlwk rxwsxw1 Zkloh wkhvh duh
lqghhg sureohpdwlf ihdwxuhv ri wkh prghov/ wkh| duh qrw d sduwlfxodu
ihdwxuh ri wklv lpshuihfwo| frpshwlwlyh prgho ru ri wkh g|qdplf
surshuwlhv ri wkh pdun0xs1 Xqolnh wkh vwdqgdug prghov/ wkh prghov
zlwk lpshuihfw frpshwlwlrq jhqhudwh h{fhvvlyh yduldelolw| lq rxwsxw
jlyhq wkh vwdqgdug ghyldwlrqv ri wkh vkrfnv1 Wklv lv fdxvhg e| d pxfk
kljkhu yrodwlolw| lq wkh vkrfnv wr a }w > zh zloo uhwxuq wr wklv lq wkh
iroorzlqj vhfwlrq1
Frpsdulqj wkh wzr prghov ri pdun0xsv lq whupv ri wkhlu vwdqgdug
vwdwlvwlfv/ wkh prvw qrwdeoh glhuhqfh lv wkdw iru jlyhq vwdqgdug
ghyldwlrqv ri wkh vkrfnv/ wkh lpsolflw frooxvlrq prgho jhqhudwhv odujhu
devroxwh xfwxdwlrqv lq rxwsxw dqg odujhu uhodwlyh yduldelolw| lq
hpsor|phqw dqg zdjhv1
Wkh lpsruwdqw lvvxh lv wkh g|qdplfv ri wkh pdun0xs/ dqg lq sduwlfxodu
wkh furvv0fruuhodwlrq zlwk rxwsxw1 Wdeoh H uhsruwv furvv0fruuhodwlrqv dw
ydulrxv krul}rqv1 Wkh vljqv ri wkh frqwhpsrudqhrxv fruuhodwlrqv
frqup wkh wkhruhwlfdo sulruvjlyhq wkh vwrfkdvwlf surshuwlhv ri wkh
vkrfnv dqg wkh sduwlfxodu fdoleudwlrq/ wkh fxvwrphu pdunhw prgho
jhqhudwhv surf|folfdo pdun0xsv/ zkloh wkh lpsolflw frooxvlrq prgho
jhqhudwhv frxqwhuf|folfdo pdun0xsv1 Wkh furvv0fruuhodwlrq vwuxfwxuh
suhvhuyhv wkh vljq lq erwk odjv dqg ohdgv/ exw lq erwk fdvhv wkh
frqwhpsrudqhrxv fruuhodwlrq lv wkh kljkhvw1 Wklv lv ehfdxvh wkh
pdun0xs lv d frqwhpsrudqhrxv frqwuro yduldeoh/ zlwk w ehlqj vhw diwhu
wkh vkrfnv ri shulrg w kdyh ehhq revhuyhgh{shulphqwv zlwk d prgho
zkhuh w lv phdvxuhg zlwk uhvshfw wr lqirupdwlrq dw w 4/ lh wkh
pdun0xs lv d suhghwhuplqhg yduldeoh/ vxjjhvw wkdw wklv pd| ohdg wr d
pxfk ulfkhu odj dqg ohdg vwuxfwxuh ri wkhvh furvv0fruuhodwlrqv1 Lq
jhqhudo/ wklv frpsolfdwhv wkh g|qdplfv ri wkh prgho frqvlghudeo|/
wdnlqj lw ixuwkhu dzd| iurp d vwdqgdug g|qdplf jhqhudo htxloleulxp
prgho1
Lq whupv ri wkh fdoleudwlrq h{huflvh/ wklv lv wkh lpsruwdqw uhvxowwkh
fxvwrphu pdunhw prgho dqg wkh lpsolflw frooxvlrq prgho jhqhudwh
furvv0fruuhodwlrqv ehwzhhq wkh pdun0xs dqg rxwsxw wkdw duh
xqdpeljxrxvo| vljqhg dw vkruw ohdgv dqg odjv1 Zlwk wkh hpslulfdo
hylghqfh iru wkh Xqlwhg Nlqjgrp vxjjhvwlqj d +zhdno|, surf|folfdo
pdun0xs/ wkh vlpxodwlrq h{huflvh vxjjhvwv wkdw wkh fxvwrphu pdunhw
85
Wdeoh H= Furvv0fruuhodwlrq ehwzhhq pdun0xs dqg rxwsxw
m
L_i*
W4T*U| UL**tL?
+w.m >Y Dw ,
Y D
t|@?_@h_ ihhLh
t|L4ih 4@h!i|
+w.m >Y Dw ,
Y D
t|@?_@h_ ihhLh
2
f
2
f
f2.
fe2
fSf
feD
f2b
fD
fD
fe
f
ffb
ff
f
fD
fS
fS
fSf
fHb
fSS
feD
f2.
f2
ff
ff.
ff
ff.
f
f
prgho lv wkh cfruuhfw* prgho lq wkh vhqvh wkdw lw uhsolfdwhv wkh
sduwlfxodu ihdwxuh wkdw zh duh lqwhuhvwhg lq1 Lq dgglwlrq/ wkh wdeoh
vxjjhvwv wkdw zh fdqqrw dffrpprgdwh wkh glhulqj ylhzv rq wkh
hpslulfdo f|folfdo surshuwlhv ri wkh pdun0xs zlwklq rqh wkhruhwlfdo
prghorqo| wkh fxvwrphu pdunhw prgho gholyhuv wkh surf|folfdo
ihdwxuhv zh fodlp wr revhuyh hpslulfdoo| lq wkh Xqlwhg Nlqjgrp1
Jlyhq wklv frqfoxvlrq/ lw lv lqwhuhvwlqj wr hvwdeolvk wkh h{whqw wr zklfk
wklv uhvxow lv dhfwhg e| fkdqjhv lq sdudphwhuv1 Udwkhu wkdq fduu|lqj
rxw d ixoo0vfdoh vhqvlwlylw| dqdo|vlv/ zh irfxv rq wkh sdudphwhuv iru
zklfk wkh fdoleudwlrq h{huflvh vxjjhvwhg wkh juhdwhvw xqfhuwdlqw|1
Xqolnh lq wkh lpsolflw frooxvlrq prgho/ qr sdudphwhu uhvwulfwlrqv wlh
grzq wkh hodvwlflw| ri wkh pdun0xs zlwk uhvshfw wr wkh udwlr ri h{shfwhg
ixwxuh surwv wr fxuuhqw rxwsxw1 Wkh ehqfkpdun lv vhw dw 3=6 dqg wkh
vwdqgdug vwdwlvwlfv/ lqfoxglqj wkh furvv0fruuhodwlrq ehwzhhq rxwsxw dqg
wkh pdun0xs/ duh qrw vljqlfdqwo| dhfwhg iru vpdoo fkdqjhv ri wklv
ydoxh1 Odujh pdun0xs hodvwlflwlhv fdq krzhyhu dhfw wkh g|qdplfv
vljqlfdqwo|/ dv uhyhdohg e| Wdeoh I1 Wkh wdeoh uhsruwv uhvxowv iurp
h{shulphqwv zkhuh wkh hodvwlflw| ri wkh pdun0xs lv lqfuhdvhg +lq
devroxwh ydoxh, wr 3=: dqg 41
Dq lqfuhdvh lq wkh hodvwlflw| ri wkh pdun0xs gdpshqv wkh yrodwlolw| lq
rxwsxw/ hpsor|phqw dqg zdjhv> iru d vxflhqw +devroxwh, lqfuhdvh lq
wkh hodvwlflw|/ wkh pdun0xs ehfrphv forvh wr df|folfdo/ zlwk d qhjdwlyh
fruuhodwlrq dw doo odjv1 Wr vkhg vrph oljkw dw wklv uhvxowv/ Iljxuh ; sorwv
wkh lpsxovh uhvsrqvh ri wkh h{shfwhg suhvhqw ydoxh ri ixwxuh surwv [w
dqg fxuuhqw rxwsxw \w / wkh pdlq ghwhuplqdqwv ri wkh pdun0xs/ lq
86
Wdeoh I= Vhqvlwlylw| dqdo|vlv1 Furvv0fruuhodwlrq ehwzhhq pdun0xs dqg
rxwsxw lq wkh fxvwrphu pdunhw prgho
f
f.
m
2
f
2
fS
fS
fSf
fHb
fSS
feD
f2.
t|@?_@h_ ihhLh
f2
ff
ff.
ff
ff.
f
f
+w.m >Y Dw ,
Y D
ff2
f2
fDf
fHD
fS
feS
f2
t|@?_@h_ ihhLh
ff
ff
ffb
ffD
ffH
f
f
+w.m >Y Dw ,
Y D
ffH
f
feD
fHb
fSe
feD
ff
ffH
ffb
ffH
ff
ff.
ff
f2
+w.m >Y Dw ,
Y D
t|@?_@h_ ihhLh
uhvsrqvh wr d jryhuqphqw h{shqglwxuh vkrfn iru @ i3=6> 4j zkhuh
0316 fruuhvsrqgv wr wkh ehqfkpdun ydoxh1 D odujh uhvsrqvh lq wkh
pdun0xs +zkhq @ 4, lpsolhv d uhyhuvdo lq wkh vljq ri wkh
fruuhodwlrq ehwzhhq wkh pdun0xs dqg rxwsxw1 Wkh h{sodqdwlrq iru wklv
lv vlpso| wkdw zlwk d vxflhqw pdun0xs uhvsrqvh/ rxwsxw zloo h{fhhg lwv
qhz vwhdg|0vwdwh ohyho lq uhvsrqvh wr d whfkqrorj| vkrfn> wklv zloo irufh
wkh udwlr [w @\w wr lqfuhdvh udwkhu wkdq ghfuhdvh1 Wklv lv looxvwudwhg lq
Iljxuh ;= dq lqfuhdvh lq rxwsxw sxvkhg wkh h{shfwhg suhvhqw ydoxh ri
surwv ixuwkhu xs wkdq fxuuhqw rxwsxw/ fdxvlqj d idoo lq wkh pdun0xs1 Lw
uhtxluhv d odujh hodvwlflw| wr ryhuvkrrw wkh qhz orqj0uxq ohyho/ dqg wkh
xqghuvkrrwlqj sdwk pd| dsshdu pruh sodxvleoh= li rxwsxw ryhuvkrrwv lwv
qhz orqj0uxq ohyho/ wkhq lq doo exw wkh uvw shulrg rxwsxw jurzwk zrxog
eh idoolqj1 Wklv lpsolhv d frxqwhuidfwxdo vkdus h{sdqvlrq dqg surorqjhg
frqwudfwlrq1 Qrwh dovr wkdw wkh fruuhodwlrq lv qrw olqhdu lq wkh
sdudphwhu = lq wkh lqwhuydo ehwzhhq 3168 dqg 3198/ wkh htxdwlrqv lq
wkh v|vwhp ehfrphv olqhduo| ghshqghqw dqg qr vroxwlrq fdq eh irxqg1
Wklv dovr lpsolhv wkdw wkh vsolw ehwzhhq whpsrudu| dqg shupdqhqw
whfkqrorj| vkrfnv pdwwhuv= zlwk d srvlwlyh exw whpsrudu| whfkqrorj|
vkrfn/ rxwsxw zloo h{fhhg lwv vwhdg|0vwdwh ohyho/ dqg wkh pdun0xs lq wkh
fxvwrphu pdunhw prgho pd| ehfrph frxqwhuf|folfdo1 Khuh zh kdyh
fkrvhq wr prgho whfkqrorj| vkrfnv dv shupdqhqw dqg khqfh vkxw grzq
wkdw srwhqwldo fkdqqho ri frxqwhuf|folfdolw| lq wkh pdun0xs1 Lw lv qrw
87
!"
# $
%%
reylrxv wkdw wklv lv wkh uljkw fkrlfhwkh lvvxh lv dw wkh khduw ri wkh
frqwuryhuv| ri shupdqhqw yv whpsrudu| dqg ghpdqg yv vxsso| vkrfnv>
lw grhv krzhyhu uh0hpskdvlvh wkh lpsruwdqfh ri whfkqrorj| vkrfnv lq
ghwhuplqlqj wkh surshuwlhv ri wkh prgho1+4:,
Wkh surgxfwlrq ixqfwlrq sdudphwhuv hvwlpdwhg suhylrxvo| vxjjhvw d
kljk ghjuhh ri vxevwlwxwdelolw| lq wkh wkuhh hodvwlflwlhv lq wkh surgxfwlrq
ixqfwlrq1 Uhfdoo wkdw wkh ixqfwlrqdo irup kdv ehhq uhvwulfwhg wr doorz
vxevwlwxwlrq ehwzhhq fdslwdo dqg oderxu/ NK / ehwzhhq lpsruwhg dqg
grphvwlfdoo| surgxfhg pdwhuldov/ HP / dqg ehwzhhq wkh ydoxh dgghg
dqg pdwhuldov djjuhjdwruv Yw dqg Tw / Y T 1 Wdeoh J vxjjhvwv wkdw wkh
ehqfkpdun uhvxowv zlwk wkh fxvwrphu pdunhwv prgho duh idluo| urexvw
zkhq orzhulqj wkhvh hodvwlflwlhv/ uhsruwlqj uhvxowv iurp irxu vlpxodwlrqv
zkhuh wkh hodvwlflwlhv ri vxevwlwxwlrq duh ydulhg1 Wkh uhvxowv lq wkh uvw
wzr froxpqv duh rewdlqhg e| orzhulqj wkh hodvwlflw| ri vxevwlwxwlrq
ehwzhhq oderxu dqg fdslwdo wr 319 dqg wr 315 uhvshfwlyho|1 Uhgxflqj
wklv hodvwlflw| uhgxfhv wkh devroxwh yduldelolw| lq rxwsxw dqg wkh
uhodwlyh yduldelolw| lq hpsor|phqw uhodwlyh wr wkh ehqfkpdun/ exw wkh
dxwrfruuhodwlrq sdwwhuqv uhpdlqv odujho| xqfkdqjhg1 Zkhq wkh
hodvwlflw| ri vxevwlwxwlrq ehwzhhq lpsruwhg dqg grphvwlfdoo| surgxfhg
pdwhuldov lv orzhuhg/ wkh uhodwlyh vwdqgdug ghyldwlrq ri wkh odujhu
frpsrqhqw ri Tw / Pw / lqfuhdvhv zkloh wkdw ri wkh vpdoohu/ Hw / ghfuhdvhv1
Vlploduo|/ zkhq wkh hodvwlflw| ri vxevwlwxwlrq ehwzhhq Yw dqg Tw
ghfuhdvhv/ wkh uhodwlyh yrodwlolw| lq wkh eljjhu frpsrqhqw/ Yw / lqfuhdvhv
zkloh wkdw ri Tw ghfuhdvhv1
Lq dgglwlrq wr wkhvh uhvxowv/ wkh prgho*v vhqvlwlylw| wr fkdqjhv lq wkh
suhihuhqfh sdudphwhuv + dqg KZ ,/ wkh vwhdg|0vwdwh ohyho ri uhdo
lqwhuhvw udwhv dqg wkh vwhdg|0vwdwh ohyho ri wkh pdun0xs kdv ehhq
dvvhvvhg1 Iru vpdoo fkdqjhv lq wkhvh sdudphwhuv/ wkh vlpxodwlrq
surshuwlhv duh qrw dhfwhg pdwhuldoo| dqg zh gr qrw uhsruw ixuwkhu
uhvxowv iurp wkhvh h{shulphqwv1
Dv d qdo dvvhvvphqw ri wkh prgho/ zh vlpxodwh wkh hqgrjhqrxv
uhvsrqvh wr d sduwlfxodu vhtxhqfh ri vkrfnv= zh wudfh wkh uhvsrqvh ri
rxwsxw/ hpsor|phqw/ zdjhv dqg wkh pdun0xs iru wkh vhtxhqfh ri vkrfnv
wkdw zh kdyh ghulyhg iru wkh Xqlwhg Nlqjgrp1 Wkh sxusrvh ri wkh
h{huflvh lv wr jhw dq lghd ri wkh uhodwlyh lpsruwdqfh ri wkh vkrfnv dqg
wkh h{whqw wr zklfk wkh prgho jhqhudwhv f|fohv wkdw orrn olnh wkh rqhv
wkdw kdyh dfwxdoo| ehhq revhuyhg1
E. 6Lh
@ _tUttL? Lu Tih4@?i?| t |i4TLh@h) |iU?L*L}) tLU!tc tii +@? Ebb.
89
Wdeoh J= +D, Vhqvlwlylw| dqdo|vlv1 Fkdqjhv lq hodvwlflw| ri vxevwlwxwlrq
ehwzhhq oderxu dqg fdslwdo
Hodvwlflw| ri vxevwlwxwlrq @ 3135
Yduldeoh
{
Y D
{
Krxuv
Lps1 pdw1
Grp1 pdw1
Dy1 zdjh
Jurvv rxwsxw
Pdun0xs
Ydoxh dgghg
419;
4156
3167
418:
316:
3143
4167
m
+w.m >Y Dw ,
Y D
t|@?_@h_ ihhLh
f2f
f.
ffD
fb
ffD
ff
fe
4159
31<6
3159
414:
315;
313:
4133
{|
{ Y D
{
31:5
31;3
31;6
31:4
31;5
319<
31:3
ffS
fD
ffe
ffH
ffD
ff
fff
ffS
ffe
ff
ffS
ff
ffS
ffS
031<7
3154
314;
031<4
315:
31;:
4133
ff2
f
fD
ff
fe
ff
fff
06 05 04 3
4
5
6
3149 3169 3193 31;< 3199 3178 315:
f2
ff
ff.
ff
ff.
f
f
Hodvwlflw| ri vxevwlwxwlrq @ 319
Yduldeoh
{
Y D
{
Krxuv
Lps1 pdw1
Grp1 pdw1
Dy1 zdjh
Jurvv rxwsxw
Pdun0xs
Ydoxh dgghg
3165
41:4
4148
3178
4157
3153
41<5
m
+w.m >Y Dw ,
Y D
t|@?_@h_ ihhLh
ffe
fb
f
ffD
f2
ff2
f2f
314:
31<3
3193
3157
3198
3143
4133
{|
{ Y D
{
31;4
31:9
31:5
31;4
31:5
31:4
31:4
ff
f
ff2
ff
ff2
ff
fff
ff
ffe
ffD
ffe
ffD
ffD
ffS
06 05 04
3
4
5
6
3149 3167 3189 31;: 3196 3175 3157
f2
f
ffH
8:
ff
ff.
f
f2
03157
31:9
31<;
319:
31<;
31<8
4133
fD
ffD
ff
ff.
ff
ff
fff
Wdeoh J= +E, Vhqvlwlylw| dqdo|vlv1 Fkdqjhv lq hodvwlflw| ri vxevwlwxwlrq
Hodvwlflw| ri vxevwlwxwlrq ehwzhhq
grphvwlf dqg lpsruwhg pdwhuldov @ 3134
Yduldeoh
{
Y D
{
Krxuv
Lps1 pdw1
Grp1 pdw1
Dy1 zdjh
Jurvv rxwsxw
Pdun0xs
Ydoxh dgghg
317:
418<
418;
31<<
418<
314;
513<
m
+w.m >Y Dw ,
Y D
t|@?_@h_ ihhLh
ffD
fD
fD
ff
fS
ff2
f22
3156
31:9
31:9
317;
31:9
313<
4133
{|
{ Y D
{
31:9
31:5
31:5
31:5
31:4
31:6
31:4
ff
ff
ff
ff
ff2
ff
fff
ffe
ffD
ffD
ffD
ffS
ffD
ffS
31::
31<:
31<:
31<;
31<<
31;<
4133
ffD
ff
ff
ff
fff
ff
fff
06 05 04
3
4
5
6
3149 3169 3193 31;< 3199 3178 315:
f2
ff
ff.
ff
ff.
f
f
Hodvwlflw| ri vxevwlwxwlrq
ehwzhhq Y dqg T @ 3134
Yduldeoh
{
Y D
{
Krxuv
Lps1 pdw1
Grp1 pdw1
Dy1 zdjh
Jurvv rxwsxw
Pdun0xs
Ydoxh dgghg
3187
41<4
418:
31<<
4194
314:
513;
m
+w.m >Y Dw ,
Y D
t|@?_@h_ ihhLh
ff.
fH
f.
ff
f.
ff2
f22
3159
31<5
31:9
317;
31::
313;
4133
{|
{ Y D
{
31:;
31:7
31:3
31:6
31:4
31:4
31:4
ffe
ff.
ff
ff
ff
fff
fff
ffe
ffD
ffS
ffD
ffS
ffD
ffS
06 05 04
3
4
5
6
314: 316; 3197 31<8 31:4 317; 315<
f
ffb
ffS
8;
ff
ffS
ff
f2
3197
31<5
31<<
31<;
4133
31<8
4133
ff.
ff2
fff
ff
fff
ff
fff
Figure 9: Output. Model vs data
(A) Customer markets
(B) Implicit collusion
59
Iljxuh < sorwv wkh ghyldwlrqv iurp wuhqg/ ghulyhg xvlqj d
Krgulfn0Suhvfrww owhu/ ri rxwsxw jhqhudwhg e| wkh fxvwrphu pdunhw
dqg lpsolflw frooxvlrq prgho/ dqg ri dfwxdo rxwsxw1 Dv qrwhg
suhylrxvo|/ erwk prghov jhqhudwh h{fhvvlyh f|folfdo xfwxdwlrqv lq
rxwsxw/ dowkrxjk wkh f|folfdo sdwwhuq lv eurdgo| uljkwwkh fruuhodwlrq
ehwzhhq dfwxdo dqg jhqhudwhg rxwsxw lv kljk1 Wkh wzr frpshwlqj
prghov jhqhudwh doprvw lghqwlfdo f|folfdo sdwwhuqv/ zlwk wkh
xfwxdwlrqv lq wkh fxvwrphu pdunhw prgho ehlqj vpdoohu1
Wkhvh xfwxdwlrqv duh odujho| gulyhq e| wkh whfkqrorj| vkrfnv1 Iljxuh
43 looxvwudwhv wkh rxwsxw sdwkv jhqhudwhg e| wkh fxvwrphu pdunhw
prgho zkhq vhohfwhg vkrfnv duh cwxuqhg r* e| lqvhuwlqj vhtxhqfhv zlwk
qr vkrfnv lq sodfh ri wkh rqhv ghulyhg iurp gdwd1 Lq sdqho +D,/ zkhuh
rqo| whfkqrorj| vkrfnv duh qrq0}hur/ wkh prgho plplfv wkh sdwk zlwk
doo vkrfnv qrq0}hur forvho|/ zkloh lq sdqhov +E, dqg +F,/ zlwk rqo|
vkrfnv wr jryhuqphqw h{shqglwxuh dqg wr wkh sulfh ri lpsruwhg
pdwhuldov uhvshfwlyho|/ lw jhqhudwhv lqvxflhqw yrodwlolw| lq rxwsxw1
Wklv lv d sduwlfxoduo| lpsruwdqw revhuydwlrq ehfdxvh ri wkh vhqvlwlylw|
ri wkh whfkqrorj| phdvxuh= uhfdoo wkh suhylrxv ghprqvwudwlrq wkdw d
WIS phdvxuh edvhg rq d surgxfwlrq ixqfwlrq wkdw doorzv iru
vxevwlwxwlrq ehwzhhq wkh pdwhuldov lqgh{ dqg wkh fdslwdo2oderxu lqgh{
lpsolhv kljkhu yrodwlolw| lq WIS jurzwk/ fdxvhg e| wkh yrodwlolw| lq wkh
sulfh ri lpsruwhg pdwhuldov1 Li zh xvh wkh phdvxuh vxjjhvwhg e|
Urwhpehuj dqg Zrrgirug +4<<8, dqg/ lqfrqvlvwhqw zlwk wkh prgho
dvvxpswlrqv/ dvvxph qr vxevwlwxwlrq ehwzhhq Y dqg T/ wkh xfwxdwlrqv
lq prgho rxwsxw duh gdpshqhg vwurqjo|1 Dv looxvwudwhg lq Iljxuh 44/
wklv dovr lpsolhv wkdw wkh f|folfdo sdwwhuq dffrugv ehwwhu zlwk wkh
dfwxdo rxwsxw vhtxhqfh1 Wklv phuho| xqghuolqhv wkdw zkdw lv lpsruwdqw
iru rxwsxw g|qdplfv +ru lq jhqhudo/ wkh g|qdplfv ri uhdo yduldeohv,
whqgv wr eh uhdo vkrfnv/ dqg khuh pdlqo| vkrfnv wr WIS jurzwk/ dqg
wkdw wkh lqwhuqdo sursdjdwlrq phfkdqlvp ri wkh prgho lv zhdn1+4;,
EH At @tt4T|L? L*_ @*tL L*_ u i @tt4i_ |i4TLh@h) ThL_U||) tLU!tQ
|iti L*_ @**L @ Mi||ih U)U*U@* |c M| L*_ ?L| U@?}i |i UL?U*tL? |@| A6
tLU!t _hi L|T|
93
Iljxuh 43= Rxwsxw sdwk1 Prgho yv gdwd1 Rqh vkrfn rqo|
+D, Surgxfwlylw| vkrfnv rqo|
+E, Jryhuqphqw h{shqglwxuh vkrfnv rqo|
+F, Vkrfnv wr sulfh ri lpsruwhg pdwhuldov rqo|
94
Iljxuh 44= Rxwsxw/ prgho yv gdwd1 Dowhuqdwlyh surgxfwlylw| vkrfnv
D L?U*tL?
Wklv sdshu kdv fkdudfwhulvhg wkh g|qdplfv ri pdun0xsv/ rxwsxw dqg
hpsor|phqw lq d g|qdplf jhqhudo htxloleulxp prgho/ edvhg rq
Urwhpehuj dqg Zrrgirug +4<<8,1 Zh frpsduh wzr prghov ri wkh
pdun0xs fdoleudwhg rq XN gdwd dqg qg wkdw wkh| uhvxow lq
furvv0fruuhodwlrq vwuxfwxuhv zlwk rxwsxw wkdw doorz xv wr glvwlqjxlvk
wkhp hpslulfdoo|rqo| wkh fxvwrphu pdunhw prgho surylghv d
surf|folfdo pdun0xs lq olqh zlwk wkh hylghqfh lq Vpdoo +4<<:,1 Zh vkrz/
lq wkh lpsxovh uhvsrqvh dqdo|vlv/ wkdw wkh fxvwrphu pdunhw prgho ohdgv
wr d judgxdo sdvv0wkurxjk ri vkrfnv wr wkh sulfh ri lpsruwhg pdwhuldov/
lq dffrugdqfh zlwk hpslulfdo hylghqfh rq uhdo h{fkdqjh vkrfnv1 Lq
dgglwlrq/ zh ghprqvwudwh wkdw wkh frqlfwlqj ylhzv rq wkh f|folfdolw| ri
wkh pdun0xs fdqqrw eh dffrpprgdwhg zlwklq rqh prghowkh uhvxow
wkdw wkh pdun0xs lv surf|folfdo lq wkh fxvwrphu pdunhw dqg
frxqwhuf|folfdo lq wkh lpsolflw frooxvlrq prgho vwdqg iru d zlgh udqjh ri
sdudphwhu ydoxhv dqg lv dssduhqwo| qrw dhfwhg e| wkh vshflf
vhtxhqfh ri vkrfnv1 Zh frqfoxgh wkdw iru dq djjuhjdwh ylhz ri pdun0xs
ehkdylrxu/ wkh fxvwrphu pdunhw prgho lv wkh prvw dssursuldwh1
95
Lw lv zruwk uhfdoolqj wkdw wklv prgho lqkhulwv vrph ri wkh zhdnqhvvhv ri
vwdqgdug g|qdplf jhqhudo htxloleulxp prghov1 Lq sduwlfxodu/ wkh
lqwhuqdo sursdjdwlrq phfkdqlvp lv zhdn dqg wkh g|qdplfv ri wkh
prgho duh odujho| jlyhq e| wkh vwrfkdvwlf surshuwlhv ri wkh h{rjhqrxv
vkrfnv1 Wkh g|qdplfv ri wkh pdun0xs gr qrw dhfw wklv= wkh ehkdylrxu
ri wkh pdun0xs dhfwv wkh lpphgldwh uhvsrqvh wr d vkrfn exw grhv qrw
lwvhoi surylgh d vwurqjhu sursdjdwlrq phfkdqlvp1
96
TTi?_
5L*?} |i 4L_i*
Wklv whfkqlfdo dsshqgl{ surylghv ghwdlov ri krz wkh prgho lv vroyhg1 Dv
phqwlrqhg suhylrxvo|/ lq wklv dqdo|vlv zh irfxv rq +vpdoo, exvlqhvv
f|foh xfwxdwlrqv durxqg d vwhdg|0vwdwh jurzwk sdwk/ doorzlqj iru
qrq0vwdwlrqdu|/ vwrfkdvwlf oderxu0dxjphqwlqj whfkqlfdo surjuhvv dqg iru
ghwhuplqlvwlf srsxodwlrq jurzwk1
5|@|L?@h) hiThiti?|@|L?
Wr dqdo|vh wklv hfrqrp|/ zh wudqvirup wkh qrq0vwdwlrqdu| htxloleulxp
frqglwlrqv lqwr vwdwlrqdu| rqhv/ dqg pdnh d orj~olqhdu dssur{lpdwlrq
durxqg wkh vwhdg|0vwdwh ri wklv vwdwlrqdu| v|vwhp/ xvlqj wkh phwkrg
ghvfulehg e| Nlqj/ Sorvvhu/ dqg Uhehor +4<;;e,1 Ghqh=
\w
@
w
N
@
\w
Yw
w @ Tw >
> Yw @
> T
}w Qw
}w Qw
}w Qw
Nw
K
zw
w
w @
> K
> z
w @
> w @ w }w
}w4 Qw
Qw
}w
+D4,
Zlwk wkh krprjhqhlw| dvvxpswlrq rq f+=, dqg k+=,/ wkh yduldeohv
ghqhg lq +D4, duh vwdwlrqdu| jlyhq vwdwlrqdu| surfhvv iru wudqviruphg
h{rjhqrxv yduldeohv/ ghqhg dv=
w @ Jw > Lw @ Lw > }w @ }w > Q @ Qw
J
}w Qw
}w Qw
}w4
Qw4
+D5,
Wkh htxloleulxp frqglwlrqv +5<, wr +69, duh uhzulwwhq lq whupv ri wkhvh
yduldeohv1 Lq wkhlu vwdwlrqdu| irup/ wkhvh htxloleulxp frqglwlrqv duh=
Qw f +zw > w ,
zw
w
+D6,
@f
> w }w @ f z
w > Fw @
}w Qw
}w
w @ k +zw > w ,
K
+D7,
4 > K L !> T H > Pa sH H P
\ I + }w , N
w
w
w
w
mw
w
w w
w.4 Q . +4 ,N
w
w . N
w }w 4 . J
@ f z
w > +D8,
97
;
4<
3
4
}
w.4 > K
w.4
?
@ 4
Y
N
\
Y
w
.4
N
w
.4
w.4 C
}w.4
Hw
. 4 D @
w
=
> w.4
+D9,
4 > K @ z
+D:,
\Y Yw > Tw YK + }w , N
w
w
w w
w > P
w
@ w sH
+D;,
\T Yw > Tw TH H
w
w > P
w
@ w
+D<,
\T Yw > Tw TP H
Wkh vwdwh yduldeoh dffrxqwlqj iru h{shfwhg ixwxuh surwv fdq eh zulwwhq
dv
+
,
w.4
w
4
[
\
[
w
.4
w
.4
w.4
}w.4
@ Hw .
+D43,
w
w.4
Lw
Lw.4
Lw.4
Qrwlfh wkdw lqyhvwphqw lv lpsolflw lq wkh uhvrxufh frqvwudlqw=
w + }w ,4 . "
w.4 Q @ +4 , N
w
N
2
5|i@_) t|@|i
Iru wkh sxusrvh ri fdoleudwlrq dqg vlpxodwlrq/ zh vroyh wkh prgho iru
lwv qrq0vwrfkdvwlf vwhdg| vwdwh1 Ohw qr vxevfulsw w lqglfdwh vwhdg|0vwdwh
ydoxh1 Iluvw/ zh dvvxph wkdw lq vwhdg| vwdwh/ wkhuh duh qr sxuh surwv1
Iurp wkh Hxohu wkhruhp dqg wkh dvvxpswlrq wkdw \ / T dqg Y kdyh
frqvwdqw uhwxuqv wr vfdoh/ jurvv rxwsxw +lh rxwsxw diwhu {hg frvwv kdyh
ehhq ghgxfwhg, fdq eh zulwwhq dv
\ @ \T TH H . \T TP P . \Y YK }K . \Y YN N \Y !L
+D44,
Zh ghqh wrwdo idfwru frvwv dv
WIF
@ sH H . P . zk . uN
\T TP
\ Y YK
\Y YN
\T TH
H.
P.
K.
N
@
\ . \Y !L
@
+D45,
98
Sxuh surwv duh ghqhg dv rxwsxw plqxv idfwru frvwv
4
\Y !L
@ \ WIF @ 4 \ 4
\Y !L
@
4
\ +D46,
Wkh dvvxpswlrq ri qr sxuh surwv lpsrvhv wkh frqglwlrq wkdw
\ @ W I F/ lh wkdw=
\Y !L
\Y Y !L
\ Y
@ + 4, \
@ + 4,
Wkh idfwru vkduhv fdq eh zulwwhq
vH
@
vK
@
\T TH H T
\T TP P T
sH H
P
@
> vP @
@
> +D47,
WIF
T \
WIF
T
\
\Y YK K I
\Y YN N I
zK
uN
@
> vN @
@
+D48,
WIF
I \
WIF
I \
Wklv dovr lpsolhv wkdw=
vH . vP . vK . vN @ 4
+D49,
Zh fdq wkhq zulwh wkdw
\
\Y Y
\
@ \Y Y . \T T @ \Y Y . \T TH H . \T TP P ,
@ 4 +vH . vP ,
+D4:,
Wkh nh| lv wkdw ehfdxvh wkhuh duh }hur surwv/ W I F @ \ 1 Dovr/ wkh
vkduh ri ydoxh dgghg lq wrwdo rxwsxw lv
vY
sH H . P
Y
@4
\
\
\T TP
\T TH
H
P @ 4 +vH . vP ,
@ 4
\
\
@
+D4;,
Vroylqj iru wkh vwhdg|0vwdwh fdslwdo wr lqyhvwphqw udwlr/ zh fdq wlh grzq
wkh fdslwdo lqyhvwphqw udwlr dv d ixqfwlrq ri wkh prgho sdudphwhuv rqo|=
+ } ,4 . "
Q
+4 , N
@ N
99
Q } +4 ,, @ } [
N+
}
N
@
"
Q } . + 4,
"
"
4 uN
@
u \
\
N
u }
vN
@
Q } . + 4,
+D4<,
+D53,
Vlploduo|/ wkh vwhdg|0vwdwh uhdo udwh ri uhwxuq diwhu ghsuhfldwlrq/ u /
lv wlhg grzq dv=
+ } , +u . 4, @ 4
+ } ,
4.u @
[
L
L
[@
4
L
\ @L . [@
4
\ @L
4
L
\ @L . [@
+
@
+ } ,
@
@
4.u
+ } ,
@
4
,
[
\
.
,
L
L
[
L
4.u
@
4.u
+ } ,
Dovr/ f+z> , lv krg+4, dqg k+z> , lv krg+4,=
Kz K
Fz F
@ 3
@ 4
dqg=
@
@
4z
K
F
\
4z
K@
FY
Y \
+D55,
+ } ,
4
@ 4
Fz
Kz
+D54,
@
9:
4
vK
vf ^4 +vH . vP ,`
+D56,
6*U|@|L?t @hL?_ t|i@_) t|@|i
Iru vpdoo ghyldwlrqv durxqg vwhdg| vwdwh/ zh fdq dssur{lpdwh wkh
dfwxdo ydoxh ri d yduldeoh ]w e| lwv ghyldwlrqv iurp vwhdg| vwdwh1 Li zh
xvh dq dssur{lpdwlrq wr pdnh wkh v|vwhp olqhdu/ zh fdq wkhq vroyh dqg
wudqvirup edfn lqwr ohyhov1 Wklv lv edvlfdoo| d Wd|oru dssur{lpdwlrq=
]w
] . ]w
]w
]w
]aw @ oq
oq
@ oq 4 .
]
]
]
]
Iru d ixqfwlrq ri d yduldeoh i +]w ,/ zh qg
i 3 +]w , ]
i 3 +]w ,
i 3 +]w , ]
]w
g oq i+]w ,
m] ]w @
m] @
m] ]aw
g]w
i+]w ,
i +]w ,
]
i +]w ,
Wkh whup i i++]]ww,,] fdq xvxdoo| eh zulwwhq dv dq hodvwlflw| ru d vkduh/
hydoxdwhg lq vwhdg| vwdwh1 Wklv v|vwhp ri qrq0olqhdu htxdwlrqv lv wkhq
dssur{lpdwhg durxqg wkh +qrq0vwrfkdvwlf, vwhdg| vwdwh e| zkdw lv
hhfwlyho| d uvw0rughu Wd|oru dssur{lpdwlrq1 Wkh rxwfrph lv d v|vwhp
ri olqhdu glhuhqfh htxdwlrqv/ zulwwhq lq yduldeohv ghqhg lq ghyldwlrqv
iurp vwhdg| vwdwh/ zkhuh wkh frhflhqwv duh ghulydwlyhv hydoxdwhg dw
vwhdg| vwdwh1 Ohw ckdwwhg*
ghqrwh shufhqwdjh ghyldwlrq iurp
yduldeohv
]
w
a
vwhdg| vwdwh/ hj ]w @ oq ] 1 Wkh dssur{lpdwlrqv ri +5<, wr +69, duh
wkhq
4 a
a w Hw
aw . 4
\w T
T
@ saH
aw
+D57,
w .
Y T
HP
4 a
aw P
aw . 4
aw
\w T
T
@ aw
+D58,
Y T
HP
4 a
4 a
aw
\w Yaw .
Yw K
@ z
aw . aw
+D59,
Y T
KN
4 a
4 a
aw
\w Yaw .
Yw N
@ uaw . aw
+D5:,
Y T
KN
3
\aw
Hw a w.4
aw saH . vP P
a w . vF vY faw . vJ vY J
aw
@ vH H
+D5;,
w
k
l
vL vY
a w.4 +4 , N
a w a }w
Q } Hw N
. Q }
. + 4,
u
a w.4 aw .
Hw uaw.4
@ Hw +D5<,
4.u
9;
w
a w Law
[
a w \aw
@ [
aw .
a w.4 @ Hw +D63,
a w.4 Law.4
[
+D64,
4.u
4.u
4
}
a
a
a
w.4 .
a
. \w.4 Lw.4
4.u
4 w.4
zkhuh
faw
aw
K
a w . fz z
@ f aw
a w . Kz z
@ K aw
+D65,
+D66,
Khuh/ m +m @ iY> Tj> iKNj> iHPj, duh wkh hodvwlflwlhv ri vxevwlwxwlrq
lq wkh surgxfwlrq ixqfwlrq/ l +l @ ifj> ifzj> iKj> iKzj, duh
hodvwlflwlhv ri vxevwlwxwlrq lq xwlolw| whupv/ dqg lv wkh hodvwlflw| ri wkh
ixqfwlrq zlwk uhvshfw wr lwv dujxphqw1 Wkh sdudphwhuv vm
im @ F> L> Jj duh wkh h{shqglwxuh vkduhv ri ydoxh dgghg ri
frqvxpswlrq/ lqyhvwphqw/ dqg jryhuqphqw h{shqglwxuh/ zkloh vl
il @ Y> P> Hj duh vkduhv ri wrwdo frvwv ri wkh lqgh{ Y / grphvwlfdoo|
surgxfhg pdwhuldov Pw dqg lpsruwhg pdwhuldov Hw 1 Wkh uhpdlqlqj
sdudphwhuv duh u/ wkh vwhdg|0vwdwh uhdo udwh ri lqwhuhvw ehiruh
ghsuhfdwlrq/ dqg / wkh vwhdg|0vwdwh pdun0xs1 Wr frpsohwh wkh v|vwhp/
wkh iroorzlqj wkuhh dffrxqwlqj uhodwlrqv wkdw ghqh wkh g|qdplfv ri wkh
djjuhjdwruv Yw > Tw dqg \w duh qhhghg=
a w a w . vK K
a w + 4, Law
vN N
+D67,
Yaw @
+4 +vH . vP ,,
a
a
a w @ vH Hw . vP Pw
+D68,
T
+vH . vP ,
aw
\aw @ +4 +vH . vP ,, Yaw . +vH . vP , T
+D69,
dv zhoo dv d vshflfdwlrq ri wkh vwrfkdvwlf surfhvv iru wkh h{rjhqrxv
yduldeohv1 Wkhvh duh glvfxvvhg lq ghwdlo lq Vhfwlrq 5191
e
5L*|L? 4i|L_
Wkhvh htxdwlrqv/ +D57, wr +D69,/ ghqh d v|vwhp ri olqhdu/ udwlrqdo
h{shfwdwlrqv glhuhqfh htxdwlrqv1 Lq g|qdplf surjudpplqj
9<
a w dqg
whuplqrorj|/ zh ghqh wkh frqwuro yduldeohv ri wkh v|vwhp dv faw > K
aw
a w / wkh fr0vwdwh yduldeohv dv a w / wkh hqgrjhqrxv vwdwh yduldeohv dv N
a w / dqg wkh h{rjhqrxv vwdwh yduldeohv dv a }w / saH
a
a
dqg [
/
J
dqg
L
1
Zh
w
w
w
vroyh wkh orj0olqhdulvhg sureohp iru srolf| uxohv wkdw jlyh frqwuro
yduldeohv dw w dqg hqgrjhqrxv vwdwh yduldeohv dw w . 4 dv ixqfwlrqv ri
hqgrjhqrxv dqg h{rjhqrxv vwdwh yduldeohv dw w1 Wkh wdvn lv wr vroyh d
uvw rughu vwrfkdvwlf glhuhqfh htxdwlrq ri wkh irup
Pff Fw
Pvv3 Vw.4 . Pvv4 Vw
@ Pfv Vw . Pfh Hw
+D6:,
@ Pvf3 Fw.4 . Pvf4 Fw . Pvh3 Hw.4 . Pvh4 Hw
zkhuh Fw lv d yhfwru ri frqwuro yduldeohv/ Vw lv d yhfwru ri hqgrjhqrxv
vwdwh dqg fr0vwdwh yduldeohv/ dqg Hw lv d yhfwru ri h{rjhqrxv vwdwh
yduldeohv1 Khuh zh ghqh=
Fw
HVw
Ow
Hw
@
@
@
@
a w > Haw > P
a w> z
a w > Yaw > w
faw > K
aw > \aw > T
aw
N
a
aw
w > [
a w > saH > Law =
a }w > J
w
:3
+D6;,
+iuihi?Uit
Deuhx/ G +4<;9,c , |hi4@* ,^*Mh@ Lu *}LTL*t|U 5Tih}@4it<c
aLh?@* Lu ,UL?L4U AiLh)c VL* bc T@}it b22D
Edqn ri Hqjodqg +4<<<,c ,UL?L4U L_i*t @| |i @?! Lu ,?}*@?_c
@?! Lu ,?}*@?_
Ehdq/ F dqg V|prqv/ M +4<;<,c Ai? vi@ht Lu ht A<c ,+
@UhLiUL?L4Ut ??@* c VL* ..c T@}it HHDD
Elov/ P +4<;:,c Ai )U*U@* i@Lh Lu @h}?@* Lt|t @?_ hUi<c
4ihU@? ,UL?L4U +ii c VL* ..c T@}it HHD.
Gdqwklqh/ M S dqg Grqdogvrq/ M E +4<<6,c i|L_L*L}U@* @?_
,4ThU@* Wttit ? +i@* t?itt )U*i AiLh)<c ,hLTi@? ,UL?L4U
+ii c VL* .c T@}it D
Gl{lw/ D N dqg Vwljolw}/ M H +4<::,c L?LTL*|U L4Ti||L? @?_
T|44 hL_U| #iht|)<c 4ihU@? ,UL?L4U +ii c VL* S.c
T@}it 2b.fH
Kdqvhq/ J G +4<;8,c W?_tM*i w@MLh @?_ Ai t?itt )U*i<c
aLh?@* u L?i|@h) ,UL?L4Utc VL* Sc T@}it fb2.
Kdvnho/ M/ Pduwlq/ F dqg Vpdoo/ L +4<<8,c hUic @h}?@* Lt|t
@?_ |i t?itt )U*i<c uLh_ **i|? Lu ,UL?L4Ut @?_ 5|@|t|Utc VL* D.c
T@}it 2Db
Kroodqg/ D dqg Vfrww/ D +4<<;,c Ai #i|ih4?@?|t Lu |i Nk
t?itt )U*it<c ,UL?L4U aLh?@*c VL* fHc T@}it cfS.b2
Luhodqg/ S Q +4<<;,c t|L4ih 6*Ltc L?|ihU)U*U@* @h!Ttc @?_
|i |T| ,giU|t Lu AiU?L*L}) 5LU!t<c aLh?@* Lu @UhLiUL?L4Utc
VL* fc T@}it SebS
Nlqj/ U J/ Sorvvhu/ F L dqg Uhehor/ V W +4<;;d,c hL_U|L?c
BhL| @?_ t?itt )U*it W Ai @tU iLU*@ttU@* L_i*<c aLh?@* Lu
L?i|@h) ,UL?L4Utc VL* 2c T@}it bD22
Nlqj/ U J/ Sorvvhu/ F L dqg Uhehor/ V W +4<;;e,c hL_U|L?c
BhL| @?_ t?itt )U*it WW i #hiU|L?t<c aLh?@* Lu L?i|@h)
,UL?L4Utc VL* 2c T@}it fbe
:4
PfFdooxp/ E W dqg Qhovrq/ H +4<<;,c L4?@* W?UL4i A@h}i|?}
? @? Ti?,UL?L4) T|43?} L_i*<c #tUttL? T@Tih L SS.Dc
@|L?@* hi@ Lu ,UL?L4U +iti@hU
Skhosv/ H V dqg Zlqwhu/ V J +4<:3,c T|4@* hUi L*U) ?_ih
|L4t|U L4Ti||L?<c ? i*Ttc , Ei_c UhLiUL?L4U 6L?_@|L?t Lu
,4T*L)4i?| @?_ W?@|L? AiLh)c Lh|L? M*tiht
Udyq/ P R +4<<:,c ih4@?i?| @?_ Ah@?tt|Lh) 5LU!tc @?_ |i Nk
t?itt )U*i<c aLh?@* Lu TT*i_ ,UL?L4i|hUtc VL* 2c T@}it 2.eH
Urhjhu/ Z +4<<8,c @? W4TihuiU| L4Ti||L? , T*@? |i #gihi?Ui
i|ii? h4@* @?_ #@* hL_U||) i@thitq ,t|4@|it uLh N5
@?u@U|h?}<c aLh?@* Lu L*|U@* ,UL?L4)c VL* fc T@}it Sf
Urwhpehuj/ M M dqg Zrrgirug/ P +4<<4,c @h!Tt @?_ |i
t?itt )U*i<c ? *@?U@h_c a @?_ 6tUihc 5c Ei_tc ,+
@UhLiUL?L4Ut ??@*c T@}it S2H
Urwhpehuj/ M M dqg Zrrgirug/ P +4<<5,c *}LTL*t|U hU?}
@?_ Ai ,giU|t Lu }}hi}@|i #i4@?_ L? ,UL?L4U U||)<c aLh?@* u
L*|U@* ,UL?L4)c VL* ffc T@}it cDc2f.
Urwhpehuj/ M M dqg Zrrgirug/ P +4<<8,c #)?@4U Bi?ih@*
,^*Mh4 L_i*t | W4TihuiU|*) L4Ti||i hL_U| @h!i|t<c ?
LL*i)c A 6 Ei_ 6hL?|iht Lu t?itt )U*i +iti@hUc T@}it 2ebc
h?Ui|L? N?iht|) hitt
Urwhpehuj/ M M dqg Zrrgirug/ P +4<<9,c W4TihuiU| L4Ti||L?
@?_ |i ,giU|t Lu ,?ih}) hUi W?Uhi@tit L? ,UL?L4U U||)<c aLh?@* Lu
L?i)c hi_| @?_ @?!?}c VL* 2Hc T@}it Deb..
Vpdoo/ L +4<<:,c Ai U)U*U@*|) Lu 4@h!Tt @?_ ThL| 4@h}?tG tL4i
i_i?Ui uLh 4@?u@U|h?} @?_ tihUit<c @?! Lu ,?}*@?_ `Lh!?} @Tih
L .2
:5
© Copyright 2026 Paperzz