Impatience and climate policy

(1)

CICERO Working Paper 2001:1

Impatience and climate policy

H. Asbjørn Aaheim December 2000

CICERO

Center for International Climate and Environmental Research

P.O. Box 1129 Blindern N-0318 Oslo, Norway Phone: +47 22 85 87 50

Fax: +47 22 85 87 51 E-mail: [email protected]

Web: www.cicero.uio.no

CICERO Senter for klimaforskning P.B. 1129 Blindern, 0318 Oslo

Telefon: 22 85 87 50 Faks: 22 85 87 51 E-post: [email protected]

Nett: www.cicero.uio.no

(2)

FLFHUR Zrunlqj Sdshu 5334=4 4

4 Lqwurgxfwlrq

lv vhqvlwlyh wr wkh fkrlfh ri glvfrxqw udwh ehfdxvh ri wkh orqj wlph vsdq iurp zkhq juhhqkrxvh jdvhv duh hplwwhg wr wkh dwprvskhuh wr wkh hhfwv ri wkhvh hplvvlrqv kdyh glvdsshduhg1 Iru FR5/ zklfk frqvwlwxwhv wkh exon ri juhhqkrxvh jdv hplvvlrqv/ lw wdnhv pruh wkdq 533 |hduv ehiruh wkh frqfhqwudwlrqv dgghg iurp d wrq ri hplvvlrqv kdv ehhq uhgxfhg wr 426 ri wkh lqlwldo lqfuhdvh1 Pruh wkdq ;33 |hduv pd| sdvv ehiruh wkh frqfhqwudwlrqv duh uhgxfhg wr 43 shufhqw1 Wkh pdmru sduw ri wkh ehqhwv ri dedwhphqw wkhuhiruh glplqlvk dv d uhvxow ri wkh glvfrxqwlqj1

Wklv lv zk| hfrqrplf dqdo|vhv vhogrp vxssruw vwurqj phdvxuhv wr plwljdwh folpdwh fkdqjh1 Krzhyhu/ pdq| shrsoh qg lw gl!fxow wr dffhsw wkdw lw vkrxog eh luudwlrqdo wr fxw suhvhqw hplvvlrqv li lw lv frqvlghuhg olnho| wkdw wkhvh hplvvlrqv uhsuhvhqw d vhulrxv wkuhdw wr wkh zhoiduh ri ixwxuh jhqhudwlrqv1 Lq uhfhqw |hduv/

qhz dujxphqwv iru dsso|lqj orzhu glvfrxqw udwhv iru folpdwh srolf| dqdo|vlv wkdq iru rwkhu ghflvlrqv kdyh ehhq sxw iruzdug1 Wkh uhqhzhg lqwhuhvw lq wkh glvfrxqwlqj lvvxh lv gxh wr wkh h{whqvlrq ri wkh wlph iudph lq dqdo|vhv ri folpdwh fkdqjh/ frpsduhg zlwk frqyhqwlrqdo hfrqrplf dqdo|vhv1 Dvvxpswlrqv wkdw duh dffhswdeoh zlwklq wkh qrupdo wlph iudph ri 48 0 63 |hduv pd| eh lqdssursuldwh zkhq wkh shuvshfwlyh h{whqgv wr kxqguhgv ru wkrxvdqgv ri |hduv1

Shukdsv wkh prvw ghedwhg lvvxh lv zkhwkhu ru qrw lqwhuwhpsrudo frpsdulvrqv ri vrfldo zhoiduh vkrxog uh hfw lpsdwlhqfh1 Wklv vhhpv wr eh pruh gl!fxow wr dffhsw ehfdxvh wkh fkdqjh ri wlph iudph wxuqv wkh suhvhqfh ri lpsdwlhqfh lq wkh zhoiduh ixqfwlrq lqwr d txhvwlrq ri lqwhujhqhudwlrqdo htxlw|/ zkloh lw lv xvxdoo|

h{sodlqhg zlwk uhihuhqfh wr wkh suhihuhqfhv ri vlqjoh lqglylgxdov1 Fohduo|/ lpsd0 wlhqfh ghqlhv lqwhuwhpsrudo htxlw| iurp wkh rxwvhw1 Vr zk| gr hfrqrplvwv lqvlvw rq lpsdwlhqfh li wkh| frqvlghu glvfrxqwlqj wr eh d sureohp lq dqdo|}lqj dedwh0 phqw ri orqj0whup hqylurqphqwdo sureohpvB Iluvw/ vrph hfrqrplvwv dovr glvfdug lpsdwlhqfh1 Udpvh| ^53`/ zkr lv wkh idwkhu ri prghuq hfrqrplf jurzwk wkhru|/

fodlpv wkdw lw lv xqhwklfdo/ dqg Folqh ^9` frqvlghuv glvfrxqw udwhv zlwkrxw lpsd0 wlhqfh1 Rq wkh rwkhu kdqg/ rqh fdqqrw lpsrvh lqwhujhqhudwlrqdo htxlw| zlwkrxw sureohpv1 Lq ghyhorslqj lqwhuwhpsrudo uhfxuvlyh xwlolw| ixqfwlrqv/ Nrrspdqv

^48`/ ^49` vkrzhg wkdw lpsdwlhqfh lv hpehgghg lq d vwdqgdug zhoiduh ixqfwlrq e|

ghidxow/ li h{whqghg wr dq lqqlwh qxpehu ri shulrgv1 Odwhu/ Gldprqg ^;` vkrzhg wkdw rqh fdqqrw fodlp erwk lqwhujhqhudwlrqdo htxlw| dqg h!flhqf| ri d xwlolw|

ixqfwlrq ryhu dq lqqlwh wlph krul}rq1 Iru qdwxudo uhdvrqv/ hfrqrplvwv qg lw kdug wr jlyh xs h!flhqf|1 Pdqqh ^4:` dovr vkrzv wkdw d }hur udwh ri lpsdwlhqfh ohdgv wr dq xquhdolvwlfdoo| kljk vdylqjv udwlr1

Rwkhu dujxphqwv idyrulqj dq dgmxvwphqw ri wkh glvfrxqw udwh lq dqdo|vhv ri folpdwh srolf| uhodwh wr wkh surshuwlhv ri wkh hfrqrp| wkdw lv dhfwhg e|

joredo zduplqj1 Zhlw}pdqq ^59` vkrzv zk| xqhtxdo ghyhorsphqw ri fdslwdo dqg hqylurqphqwdo vwrfnv ohdgv wr v|vwhpdwlfdoo| glhuhqw glvfrxqw udwhv iru ruglqdu| lqyhvwphqwv dqg lqyhvwphqwv lq hqylurqphqwdo suhvhuydwlrq1 Dffruglqj wr Zhlw}pdqq ^5:`/ xqfhuwdlqw| derxw hqylurqphqwdo lpsdfwv ri suhvhqw dfwlrqv lq wkh yhu| orqj whup vkrxog eh glvfrxqwhg dw wkh orzhvw srvvleoh glvfrxqw udwh/

ehfdxvh wkh suhvhqw ydoxh lv frpsohwho| grplqdwhg e| wkh rxwfrph ri wkh prvw

(3)

FLFHUR Zrunlqj Sdshu 5334=4 5 xqiruwxqdwh fdvh1 Pruh lqwxlwlyh dujxphqwv iru dsso|lqj orz glvfrxqw udwhv iru orqj0whup hqylurqphqwdo ehqhwv duh surylghg e| h1j1 Kdvvhopdq hw1 do1 ^46`

dqg Kdvvhopdqq ^45`1 Wkh| fodlp wkdw wkh luuhyhuvlelolw| ri d fdwdvwurskh/ dqg xqfhuwdlqw| dv wr zkhq wkh gdpdjh iurp folpdwh fkdqjh rffxuv mxvwlhv wkh xvh ri orz glvfrxqw udwh iru lqyhvwphqwv lq wkh plwljdwlrq ri folpdwh fkdqjh1 Qrugkdxv ^4;` dqg Khdo ^47` uhvsrqg wkdw lw lv gl!fxow wr dfklhyh frqvlvwhqf|

e| wkh xvh ri glhuhqw glvfrxqw udwhv1 Wkh fkrlfh ri glvfrxqw udwh lv wkhuhiruh ohiw wr wkh dqdo|vw/ zklfk lv dovr sureohpdwlf1

Wklv ghedwh frpprqo| fhqwhuv rq zkhwkhu ru qrw lw lv mxvwldeoh wr pdnh frqvlghudeoh lqyhvwphqwv lq wkh suhvhqw wr plwljdwh ixwxuh folpdwh fkdqjh1 L txhvwlrq wklv srlqw ri ghsduwxuh1 L dujxh wkdw wkh suhvhqw qrz ru qhyhu shu0 vshfwlyh ryhuorrnv wkh lvvxh riwlplqj1 Wkh fkrlfh ri glvfrxqw udwh qrw rqo| lq0 glfdwhv zkhwkhu d jlyhq lqyhvwphqw pljkw eh frvw0hhfwlyh dw wkh suhvhqw wlph/

exw dovr lqglfdwhv suhflvho| zkhq lw zloo ehfrph vr1 Lq rwkhu zrugv/ d glvfrxqw udwh dfwv dv dq lqfhqwlyh wr doorfdwh frqvxpswlrq dqg dedwhphqw ryhu wlph1 D fkdqjh wkh shuvshfwlyh ri wklv ghedwh lv lpsruwdqw ehfdxvh wkh fxuuhqw dssurdfk/

zlwk lwv hpskdvlv ri suhvhqw ydoxh/ whqgv wr ljqruh wkh frqvhtxhqfhv ri suhvhqw lqdfwlrq1

Wkh txhvwlrq ri krz pxfk wr gr ryhu d fhuwdlq shulrg ri wlph/ rq wkh rwkhu kdqg/ lv sulpdulo| d txhvwlrq ri vxvwdlqdelolw|1 Wkh sureohp lv fohduo| looxvwudwhg e| wkh hruwv pdgh e| wkrvh zkr idyru orz glvfrxqw udwhv lq rughu wr pdnh folpdwh srolf| ehfrph ehqhfldo *hqrxjk*1 Wkh frpprq ylhz lv wkdw li d orz glvfrxqw udwh fdq eh mxvwlhg/ vr fdq frqvlghudeoh dedwhphqw1 Exw wklv grhv qrw pdwfk wkh lqwhusuhwdwlrq ri wkh glvfrxqw udwh dv uh hfwlqj wkh uhwxuq rq lqyhvwphqwv1 D orz glvfrxqw udwh phdqv wkdw wkh uhwxuq lv orz/ zklfk lqglfdwhv wkdw wkh lqyhvwphqwv vkrxog eh uhgxfhg1

Fdq wkhvh wzr vljqdov eh mrlqhgB Wkh dqvzhu lv qr/ xqohvv wkh dprxqw ri suhvhqw lqyhvwphqw lv frpsduhg zlwk wkh dprxqw ri ixwxuh lqyhvwphqw1 Wr pdnh vxfk d frpsdulvrq/ prvw ri wkh idfwruv nqrzq iurp wkh suhylrxv ghedwhv rq wkh glvfrxqw udwh/ vxfk dv wkh udwh ri lpsdwlhqfh/ wkh suhihuhqfhv iru lqwhujhq0 hudwlrqdo htxlw|/ dqg wkh uhwxuq rq fdslwdo duh lpsruwdqw/ exw wkhlu urohv duh wr vrph h{whqw glhuhqw iurp zkdw lv xvxdoo| eholhyhg1 Wkh pdujlqdo frvw ri dedwhphqw/ iru h{dpsoh/ lv lpsruwdqw/ exw kdv kdugo| ehhq wrxfkhg xsrq lq wkh ghedwh1

Wklv sdshu dlpv wr dvvhvv wkh glvfrxqw udwh iru dqdo|vlv ri folpdwh srolf|

iurp wkh shuvshfwlyh ri wkh ohyho dqg sdwk ri dedwhphqw1 Iru vxfk dq dqdo|vlv/

zh qhhg wr dvvhvv erwk wkh glvfrxqw udwh dqg wkh vxvwdlqdelolw| fulwhulrq1 Lq wkh qh{w vhfwlrq L suhvhqw wkh wudglwlrqdo dssurdfk edvhg rq wkh Udpvh|0prgho ri hfrqrplf jurzwk/ dqg h{whqg wkh prgho wr lqfoxgh plwljdwlrq dqg lpsdfwv ri folpdwh fkdqjh1 Wkh glvfrxqw udwhv duh lpsolflw lq wkh hqgrjhqrxv sdwkv iru wkh vkdgrz sulfhv rq fdslwdo dqg frqfhqwudwlrqv ri folpdwh jdvhv1 Wkh vxvwdlqdelolw|

fulwhulrq fdq qdwxudoo| eh ghqhg e| wkh vwhdg| vwdwh vroxwlrq1 Lq wkh iroorzlqj vhfwlrqv/ sduwlfxodu dwwhqwlrq lv sdlg wr wkh zhoiduh ixqfwlrq/ dqg wr wkh uroh ri lpsdwlhqfh1 Lw lv vkrzq wkdw wkh wudglwlrqdo dgglwlyh zhoiduh ixqfwlrq/ zklfk lqfoxghv d udwh ri lpsdwlhqfh/ lv srruo| irxqghg exw gl!fxow wr uhsodfh zlwk dowhuqdwlyhv wkdw phhw pruh lqwxlwlyh dqg hwklfdoo| dffhswdeoh surshuwlhv ri vrfldo suhihuhqfhv1 Wkh phvvdjh lv wkdw dq lqwhuwhpsrudo dgglwlyh zhoiduh ixqfwlrq zlwk d jlyhq udwh ri lpsdwlhqfh idlov wr surylgh d shuihfw ghvfulswlrq ri vrfldo suhihuhqfhv/ exw lw lv gl!fxow wr frph xs zlwk jrrg dowhuqdwlyhv1 Pruhryhu/ wkh frqyhqwlrqdo dssurdfk wr glvfrxqwlqj lv qrw dv glvfulplqdwlqj wrzdugv orqj0whup

(4)

hqylurqphqwdo lpsdfwv dv lv riwhq fodlphg1

5 Wkh edvlf dssurdfk wr glvfrxqwlqj

Ryhu wkh sdvw ghfdghv d frqvhqvxv kdv hphujhg derxw wkh pdlq ghwhuplqdqwv iru wkh glvfrxqw udwh1 Wklv lv pdlqo| gxh wr Duurz ^4`/ zkrvh iudphzrun lv h{0 whqghg wr lqfoxgh hqylurqphqwdo frqvhtxhqfhv lq Gdvjxswd hw do1 ^:`1 Krzhyhu/

wkh edvlfv vwhp iurp Udpvh|*v ^53` prghoolqj ri wkh frqvxpswlrq0vdylqj ghfl0 vlrq1 Udpvh|*v srlqw ri ghsduwxuh zdv wr qg wkh rswlpdo doorfdwlrq ri suhvhqw dqg ixwxuh frqvxpswlrq/ jlyhq ghfolqlqj pdujlqdo xwlolw| ri lqvwdqwdqhrxv frq0 vxpswlrq dqg d uhwxuq rq vdylqjv1 Lq lwv vlpsohvw irup/ Udpvh|*v prgho dlpv dw pd{lplvlqj xwlolw| ryhu dq lqqlwh wlph krul}rq/ jlyhq wkdw hfrqrplf jurzwk lv frqvwudlqhg e| wkh lqfuhdvh lq wkh fdslwdo vwrfn1 Folpdwh fkdqjh fdq eh lq0 foxghg lq wkh Udpvh| prgho e| h{whqglqj wkh surgxfwlrq ixqfwlrq wr ghshqg rq wkh ohyho ri joredo zduplqj/ zklfk ohdgv wr d gdpdjh1 Joredo zduplqj lv ghwhuplqhg e| dgglqj hplvvlrqv wr wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv ryhu wlph1 Zh uvw glvfxvv wkh glvfrxqw udwh lq wkh vlpsohvw Udpvh| prgho/ odwhu h{whqglqj lw wr wkh fdvh ri folpdwh fkdqjh1

514 Wkh Udpvh| prgho

Ghqrwh e| {w frqvxpswlrq dww=Wkh surgxfwlrq ri jrrgv dqg vhuylfhv duh jhq0 hudwhg e| wkh surgxfwlrq ixqfwlrq i+nw,/ zkhuh nw lv uhdo fdslwdo1 Wkh lqvwdq0 wdqhrxv xwlolw| ixqfwlrq/ x+{w,/ kdv wkh wudglwlrqdo +zhdn, surshuwlhv +x³_{ 3> x³³_{{ 3,/ dqg frpsdulvrqv ri frqvxpswlrq ohyhov ryhu wlph duh dgmxvwhg e|

d frqvwdqw udwh ri sxuh wlph suhihuhqfh/ ru lpsdwlhqfh/31 Dv d fxulrvlw|/ lw vkrxog eh phqwlrqhg wkdw Udpvh| dujxhg wkdw @ 3 iru hwklfdo uhdvrqv1 Wkh prgho lv

pd{_{ ] ₄

3 x+{w,h^wgw>

vxemhfw wr

nb @i+nw,{w= +4,

Wkh nh| idfwru wr wkh vroxwlrq ri wklv sureohp lv wkh ghyhorsphqw ri wkh sulfh ri fdslwdo dorqj wkh rswlpdo sdwk1 Wklv sulfh h{suhvvhv wkh ydoxh ri fdslwdo dw hdfk srlqw lq wlph1 Khqfh/ wkh udwh ri fkdqjh lq wkh ydoxh ri fdslwdo surylghv wkh dssursuldwh udwh dw zklfk wkh vwrfn ri fdslwdo fdq eh frpsduhg dw glhuhqw wlphv/ wkdw lv/ wkh glvfrxqw udwh1 Wkh lqwhuwhpsrudo htxloleulxp frqglwlrq iru wklv sureohp lv +vhh Eodqfkdug dqg Ilvkhu ^7`,

@i_n³ @_{{b

{.= +5,

Wklv lv wkh vr0fdoohg Udpvh| uxoh iru rswlpdo vdylqj1 Wkh idfwru{@{x³³{{@x³{

fkdudfwhul}hv wkh fxuydwxuh ri wkh xwlolw| ixqfwlrq1 Lw lv vrphwlphv fdoohg wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq ehfdxvh lw phdvxuhv wkh xwlolw| ri prylqj frqvxpswlrq iurp rqh srlqw lq wlph wr dqrwkhu1 D kljk_{lqglfdwhv d vwurqjo|

fxuyhg xwlolw| ixqfwlrq/ dqg ylfh yhuvd1 Li_{ lv kljk/ wkh xwlolw| lv vwurqjo| di0 ihfwhg e| d fkdqjh ri frqvxpswlrq ryhu wlph1 Lq wkh frqwh{w ri lqwhuwhpsrudo

(5)

FLFHUR Zrunlqj Sdshu 5334=4 7 prghov/ d kljk_{lv xvxdoo| lqwhusuhwhg dv d uhvxow ri kljk suhihuhqfhv iru lqwhu0 jhqhudwlrqdo htxlw| ehfdxvh lw phdqv wkdw wkh pdujlqdo xwlolw| ri frqvxpswlrq jurzwk lv udslgo| ghfolqlqj1

Wkh Udpvh| uxoh lv dq htxloleulxp frqglwlrq whoolqj krz wr edodqfh suhvhqw dqg ixwxuh frqvxpswlrq1 Zkhq +5, dssolhv/ wkh fkrlfh ehwzhhq dgglqj d xqlw ri rxwsxw wr suhvhqw frqvxpswlrq yhuvxv vdylqj dqg lqyhvwlqj lq wkh fdslwdo vwrfn wr jdlq pruh lq wkh ixwxuh duh htxdoo| zhljkwhg1 Wkh pdujlqdo xwlolw| ri suhvhqw frqvxpswlrq htxdov wkh jdlq ri vxevwlwxwlqj frqvxpswlrq iurp d ixwxuh srlqw ri wlph soxv wkh jdlq wkdw iroorzv iurp lpsdwlhqfh1 Dq dvvhvvphqw ri wkh pdujlqdo xwlolw| ri dgglqj wr suhvhqw frqvxpswlrq wkhuhe| surylghv rqh zd| wr dvvhvv wkh glvfrxqw udwh1 Wkh dowhuqdwlyh lv wr hvwlpdwh wkh uhwxuq rq fdslwdo1

Vrph surshuwlhv ri wkh glvfrxqw udwh dvvhvvhg e| +5, duh zruwk qrwlqj1 Iluvw/

wkh glvfrxqw udwh lv d uhvxow ri dq htxloleulxp frqglwlrq1 Khqfh/ rqh fdqqrw fkrrvh d glvfrxqw udwh rq wkh edvlv ri/ h1j1/ vrph lqglfdwlrq ri wkh uhwxuq rq fdslwdo zlwkrxw frqvlghulqj wkh frqvhtxhqfhv iru wkh pdujlqdo xwlolw| ri suhvhqw frqvxpswlrq1 Li/ iru h{dpsoh/ wkh xwlolw| ixqfwlrq lv orjdulwkplf/{ lv frqvwdqw dqg lqghshqghqw ri wkh frqvxpswlrq ohyho1 Li d kljk glvfrxqw udwh lv fkrvhq ehfdxvh lw eholhyhg wkdw wkh uhwxuq rq fdslwdo lv kljk/ lw iroorzv wkdw frqvxpswlrq jurzwk dovr kdv wr eh kljk1 Zlwk d jlyhq lqlwldo fdslwdo vwrfn/ lw lv wkhuhiruh qhfhvvdu| wr vwduw dw d orz ohyho ri frqvxpswlrq/ wr pdqdjh wkh uhtxluhg jurzwk1 Fruuhvsrqglqjo|/ d orz udwh ri lpsdwlhqfh ohdgv wr d kljk jurzwk udwh lq frqvxpswlrq1 Djdlq/ wkh lqlwldo ohyho ri frqvxpswlrq zloo kdyh wr eh orz1 Rqh pd| whvw wkh mxvwlfdwlrq ri d }hur udwh ri lpsdwlhqfh e| dvnlqj zkhwkhu wkh uhvxowlqj rswlpdo ohyho ri lqlwldo frqvxpswlrq lv frpsdwleoh zlwk revhuydwlrqv1 Pdqqh ^4:` dujxhv wkdw lw lv qrw1 Kh wkhuhiruh frqfoxghv wkdw wr eh uhdolvwlf/ rqh qhhgv wr lqfoxgh lpsdwlhqfh lq wkh suhihuhqfh vwuxfwxuh1

D vhfrqg surshuw| iroorzlqj iurp +5, lv wkdw wkh glvfrxqw udwh lv qrw frqvwdqw ryhu wlph1 Lw fkdqjhv dffruglqj wr wkh lqfuhdvh lq wkh fdslwdo vwrfn lq d jurzwk surfhvv1 Zlwk d voljkw h{whqvlrq ri wkh prgho/ rqh frxog dovr lqfoxgh whfkqr0 orjlfdo fkdqjh/ zklfk dovr lpsolhv d fkdqjh lq wkh pdujlqdo uhwxuq rq fdslwdo ryhu wlph1 Krzhyhu/ whfkqrorjlfdo fkdqjh pd| frxqwhudfw d ghfolqlqj pdujlqdo uhwxuq rq fdslwdo dv wkh fdslwdo vwrfn lqfuhdvhv1 Ryhu d shulrg ri 43 wr 53 |hduv/

rqh pd| wkhuhiruh ghihqg d frqvwdqw glvfrxqw udwh dv dq dssur{lpdwlrq1 Zkhq dqdo|}lqj orqj0whup lvvxhv/ vxfk dv folpdwh fkdqjh/ wklv dvvxpswlrq lv jhqhudoo|

wrr vwurqj1

Wkh idfw wkdw wkh glvfrxqw udwh lv yduldeoh ryhu wlph pdnhv lw lqfrqvlvwhqw wr fodlp wkdw glvfrxqwlqj pd| ohdg wr ixwxuh glvdvwhuv1 D ixwxuh glvdvwhu zloo ohdg wr d ghfolqh lq frqvxpswlrq1 Li wkh udwh ri lpsdwlhqfh lv forvh wr }hur/

ghfolqlqj frqvxpswlrq zloo uh hfw d qhjdwlyh uhwxuq rq fdslwdo/ zklfk fdqqrw eh rswlpdo ehfdxvh lw zloo dozd|v eh ehwwhu wr uhiudlq iurp lqyhvwphqwv dqg frqvxph lpphgldwho| xqohvv wkh pdujlqdo uhwxuq rq fdslwdo lv srvlwlyh1

Wkh udwh ri lpsdwlhqfh lv/ krzhyhu/ fuxfldo1 Li lw lv kljk/ suhvhqw frqvxps0 wlrq pd| eh suhihuuhg wr vxfk dq h{whqw wkdw lw ehfrphv rswlpdo wr uhgxfh frqvxpswlrq lq wkh orqj whup1 Wr wdnh d forvhu orrn dw wklv fdvh/ frqvlghu jxuh 4/ zklfk glvsod|v d judsklfdo looxvwudwlrq ri wkh vroxwlrq ri wkh Udpvh| prgho1 Wkh gdvkhg olqh ghpdufdwhv wkh vl}h ri wkh fdslwdo vwrfn dw zklfk frqvxpswlrq vkliwv iurp dq lqfuhdvlqj sdwk wr d ghfuhdvlqj sdwk1 Wklv lv zkhuh wkh udwh ri lpsdwlhqfh lv orzhu ru kljkhu wkdq wkh uhwxuq rq fdslwdo1 Wkh fdslwdo vwrfn lq0 fuhdvhv ru ghfuhdvhv ghshqglqj rq zkhwkhu frqvxpswlrq lv orzhu ru kljkhu wkdq wkh rxwsxw/i+n,1 Wkh duurzv lqglfdwh gluhfwlrqv iru wkh jurzwk lq frqvxpswlrq

(6)

f’_k x

k k

f(k), x

k x

f(k)

δ

Iljxuh 4= Rswlpdo hfrqrplf jurzwk lq wkh Udpvh| prgho dqg vwrfn ri fdslwdo1

Zlwk dq lqqlwh wlph krul}rq/ wkh fdvh zkhuh wkh fdslwdo vwrfn dssurdfkhv }hur lv fohduo| vxerswlpdo1 Dorqj dq rswlpdo frqvxpswlrq sdwk zlwk ghfolqlqj fdslwdo vwrfn/{jurzv dw dq dffhohudwlqj udwh1 Khqfh wkh v|vwhp zloo vrrqhu ru odwhu froodsvh1 Wklv fdqqrw eh rswlpdo li wkh wlph krul}rq h{whqgv wr lqqlw|1 Vlqfh wkh udwh ri uhwxuq lq wkdw fdvh h{fhhgv wkh udwh ri lpsdwlhqfh/ wkh pdujlqdo xwlolw| ri dgglqj wr wkh fdslwdo vwrfn lq rughu wr srvwsrqh wkh froodsvh lv srvlwlyh1 Wkh fdvh ri d ghfolqlqj frqvxpswlrq udwh lv dovr vxerswlpdo zkhq wkh wlph krul}rq lv lqqlwh ehfdxvh { dssurdfkhv }hur zkloh n jurzv wr lqqlw|1 Wklv fdqqrw eh rswlpdo dv orqj dv wkh xwlolw| ghshqgv rq frqvxpswlrq rqo|1

Zh duh wkhuhiruh ohiw zlwk wkh orqj0whup vwdwlrqdu| vroxwlrq zklfk jlyhv d frqvwdqw frqvxpswlrq udwh dqg d frqvwdqw fdslwdo vwrfn1 Wklv lpsolhv wkdw wkh glvfrxqw udwh htxdov wkh udwh ri lpsdwlhqfh1 Wkdw lv/ lq wkh yhu| orqj whup/

Wkh txhvwlrq lv zkhwkhufrxog dssurdfk }hur1 Lq wkh fdvh ri dq xqerxqghg surgxfwlrq ixqfwlrq +i_n³ A3,/ wkh vwdwlrqdu| vroxwlrq zrxog wkhq vkliw lqqlwho|

idu rxw wr wkh uljkw lq wkh gldjudp1 Zh kdyh frqfoxghg douhdg| wkdw wkh rswl0 pdo frqvxpswlrq sdwk kdv wr eh qrq0ghfuhdvlqj1 Dvvxph wkdw iru d jlyhq wlph krul}rq/ W/ wkh rswlpdo lqlwldo frqvxpswlrq lv{^W₃1 H{whqglqj wkh wlph krul}rq zlwkrxw dgglqj wr wkh lqlwldo fdslwdo vwrfn/ lqlwldo rswlpdo frqvxpswlrq zrxog kdyh wr eh orzhuhg1 H{whqglqj wkh wlph krul}rq ixuwkhu wr lqqlw| phdqv wkdw lqlwldo frqvxpswlrq zrxog kdyh wr eh orzhuhg wr }hur1 Exw lqlwldo rswlpdo frq0 vxpswlrq htxdo wr }hur lv qrw ri dq| lqwhuhvw1 Wkhuh kdv wr eh vrphwklqj zurqj zlwk wkh prgho1 Wkh prvw iuhtxhqw vroxwlrq lv dovr wkh vlpsohvw/ qdpho| wr dffhsw d srvlwlyh udwh ri lpsdwlhqfh1

(7)

515 Lqfoxglqj folpdwh fkdqjh

Wkh suhylrxv vhfwlrq vxppdul}hv wkh edvlf hohphqwv wkdw jr lqwr ghwhuplqlqj wkh glvfrxqw udwh= wkh vrfldo uhwxuq rq lqyhvwphqwv/ wkh suhihuhqfhv iru lqwhujhq0 hudwlrqdo htxlw|/ dqg wkh suhihuuhg udwh ri lpsdwlhqfh1 Wkhvh lvvxhv duh vwdqgdug lq glvfxvvlrqv derxw wkh fkrlfh ri glvfrxqw udwh wr eh dssolhg rq wkh ehqhwv ri phdvxuhv wr plwljdwh folpdwh fkdqjh1 Wklv lv lqwxlwlyho| uhdvrqdeoh/ ehfdxvh lw lv hylghqw wkdw li wkh lvvxhv duh lpsruwdqw iru dq dvvhvvphqw ri wkh glvfrxqw udwh iru lqyhvwphqwv lq uhdo fdslwdo/ wkh| duh dovr lpsruwdqw iru wkh ehqhwv ri folpdwh phdvxuhv1 Lq dq hfrqrplf shuvshfwlyh/ lw lv lpsruwdqw wr dwwdlq lqwhuwhpsrudo h!flhqf|/ zklfk lv wkh dlp ri htxdwlrq +5,1 Phdvxuhv wr plwljdwh folpdwh fkdqjh vkrxog/ wkhuhiruh/ eh vxemhfw wr wkh vdph frqvwudlqwv dv vrfldo frvwv lpsrvhg wr dfklhyh rwkhu phdqv1

Rq wkh rwkhu kdqg/ wkh dvvhvvphqw deryh xvhv d yhu| vlpsoh prgho/ zklfk grhv qrw lqfoxgh folpdwh lpsdfwv ru folpdwh srolf|1 Wkh glvfrxqw udwh zdv olqnhg gluhfwo| wr uhdo fdslwdo/ zklfk lv wkh rqo| vwrfn yduldeoh1 Ehqhwv ru frvwv ri lqyhvwphqwv zhuh vxssrvhg wr eh dgghg wr surgxfwlrq lq dffrugdqfh zlwk wkh surgxfwlrq ixqfwlrq1 Wkh dlp ri folpdwh phdvxuhv lv/ krzhyhu/ wr dyrlg gdpdjh iurp wkh lqfuhdvh lq wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv1 Lq rwkhu zrugv/ zh kdyh qrw |hw wrog wkh zkroh vwru| derxw wkh glvfrxqwlqj ri folpdwh phdvxuhv1

Wr lqfoxgh folpdwh h{solflwo|/ zh dvvxph wkdw wkh ohyho ri frqfhqwudwlrqv ri juhhqkrxvh jdvhv/ vw/ dhfwv surgxfwlylw| vxfk wkdw wkh surgxfwlrq ixqfwlrq lv i+nw> vw,/ dqgi_v³ ?31 Wkh sureohp fdq qrz eh uhirupxodwhg wr

pd{_{>| ] ₄

3 x+{w,h^wgw>

vxemhfw wr

nb4w @i+n4w> vw,{w|w> +6, b

v@+vw, .i+n4w> vw,j+|w,= +7, Htxdwlrq +7, ghvfulehv wkh ghyhorsphqw ri frqfhqwudwlrqv ri juhhqkrxvh jdvhv lq wkh dwprvskhuh1 Wkh whup+vw,uhsuhvhqwv wkh fkhplvwu| ri wkh dwprvskhuh/ dqg ³_v?3lv xvxdoo| fdoohg wkh qdwxudo udwh ri ghfd| iru juhhqkrxvh jdvhv1 Wklv lv d frpsoh{ ixqfwlrq/ lqghhg/ exw zh zloo qrw h{soruh lwv surshuwlhv ixuwkhu/ h{fhsw wr dvvxph wkdw wkh ghulydwlyh lv qhjdwlyh1 Wkh frh!flhqw jlyhv hplvvlrqv shu xqlw ri surgxfwlrq dqgj+|w,uhsuhvhqwv wkh surgxfwlylw| ri dedwhphqw phdvxuhv lpsrvhg dww1 Zh dvvxph wkdwj_|³ A3dqgj_||³³ ?31

Wkh vwdwlf htxloleulxp frqglwlrqv iru wkh sureohp duh

t4@x³_{h^w> +8,

tv@t4

j_|³= +9,

Wkh yduldeohv t4 dqg tv duh wkh vkdgrz sulfhv ri uhdo fdslwdo dqg juhhqkrxvh jdv frqfhqwudwlrqv/ uhvshfwlyho|1 Dffruglqj wr +8, wkh ydoxh ri d xqlw ri fdslwdo lv htxdo wr wkh glvfrxqwhg pdujlqdo xwlolw| ri rswlpdo frqvxpswlrq/ zklfk lv wkh edvlf frqglwlrq iru wkh dvvhvvphqw ri wkh glvfrxqw udwh/ dv srlqwhg rxw e|

(8)

FLFHUR Zrunlqj Sdshu 5334=4 : Duurz ^4`1 Htxdwlrq +9, lpsolhv wkdw wkh ydoxh ri juhhqkrxvh jdv frqfhqwudwlrqv lv htxdo wr wkh pdujlqdo frvw ri dedwhphqw/ zklfk lv h{suhvvhg rq wkh uljkw kdqg vlgh1

Wkh glvfrxqw udwh lv ghqhg e| wkh udwh dw zklfk wkhvh sulfhv fkdqjh ryhu wlph/ wkdw lv ₄ @ bt4@t4 dqg _v @ btv@tv1 Wkh| duh h{suhvvhg e| phdqv ri wkh lqwhuwhpsrudo htxloleulxp frqglwlrqv=Iru uhdo fdslwdo/ zh kdyh

₄@ +4

j_|³,i_n³ @_{{b

{.> +:,

dqg iru wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv/

_v@³_v. +j³_|,i_v³ @_||b

| .₄= +;,

Wkh whup | @ |j_||³³ @j_|³ fkdudfwhul}hv wkh fxuydwxuh ri wkh surgxfwlylw| ri dedwhphqw frvwv1

Zh uhfrjql}h wkh Udpvh| uxoh lq +:,/ h{fhsw wkdw wkh uhwxuq rq fdslwdo kdv ehhq uhgxfhg e| wkh udwh @j³_|1 Wkh pdujlqdo frvw ri hplvvlrq fxw lq rswlpxp fdq eh h{suhvvhg dv4@j_|³/ dqg@j³_|fdq wkhuhiruh eh lqwhusuhwhg dv wkh dedwhphqw frvw udwh1 Wkh whup rq wkh ohiw kdqg vlgh lv wkh vrfldo uhwxuq rq fdslwdo1 Lw pd|

eh zruwk qrwlqj wkdw wkh glhuhqfh ehwzhhq wkh *sulydwh* uhwxuq/ i_n³/ dqg wkh vrfldo uhwxuq ghshqgv rq wkh surgxfwlylw| ri dedwhphqw hruwv dqg qrw gluhfwo|

rq wkh gdpdjh1 Wkh lpsruwdqfh ri wklv uhvxow/ zklfk jrhv edfn wr Duurz ^4` dqg Duurz dqg Nxuw} ^B`/ lv wkdw rqh fdqqrw/ lq sulqflsoh/ glvfrxqw wkh ehqhwv ri folpdwh phdvxuhv rqo| rq wkh edvlv ri wkh sulydwh uhwxuq rq fdslwdo dqg wkh vrfldo gdpdjh ri folpdwh fkdqjh1 Rqh qhhgv wr nqrz zkdw wkh rswlpdo dedwhphqw hruwv duh lq rughu wr qg wkh glvfrxqw udwh1

Wkh ohyho ri dedwhphqw lv qrw ghwhuplqhg e| wkh glvfrxqw udwh iru surgxf0 wlyh fdslwdo iurp +:,/ exw iurp wkh lqwhuwhpsrudo htxloleulxp frqglwlrq iru wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv/ +;,1 Wkh ohiw kdqg vlgh ri +;, frqvlvwv ri wkh udwh ri ghfd|/ zklfk lv qhjdwlyh1 Pruhryhu wkhuh duh wzr whupv uhodwhg wr wkh pdujlqdo gdpdjh frvw/i_v³1 Wkh uvw/i_v³/ lv hplvvlrq uhgxfwlrqv fdxvhg e|

kljkhu gdpdjh uhodwhg wr folpdwh fkdqjh1 Wklv whup dulvhv iurp wkh dvvxpswlrq wkdw hplvvlrqv duh ghshqghqw rq wkh rxwsxw/ dqg qrw rq wkh xvh ri lqsxw idfwruv1 Li hplvvlrqv ghshqg rq wkh xvh ri lqsxw idfwruv/ h1j1 n rqo|/ wklv whup zrxog eh }hur1 Wkh vhfrqg whup/ j³_|i_v³ lv wkh hhfw rq gdpdjh iurp dedwhphqw/ zklfk lv wkh vkdgrz sulfh ri gdpdjh lq whupv ri frqfhqwudwlrq xqlwv1 Wkh ohiw kdqg vlgh uhsuhvhqwv wkh qhw *uhwxuq* rq wkh folpdwh v|vwhp1 Lwv wzr frpsrqhqwv duh wkh qdwxudo fkdqjh lq frqfhqwudwlrqv dqg wkh fkdqjh fdxvhg e| kxpdq dfwlylwlhv1 Zkhwkhu wkh ryhudoo uhwxuq lv srvlwlyh ru qhjdwlyh lq d jlyhq vwdwh ri wkh zruog ghshqgv rq wkh ohyho ri dedwhphqw hruwv1

Zlwk wzr vwdwh yduldeohv dqg wzr frqwurov/ lw lv gl!fxow wr dqdo|}h wkh orqj0 whup ghyhorsphqw dqg vwdelolw| frqglwlrqv ri wkh v|vwhp1 Sh}}h| dqg Zlwkdjhq

^4<` vkrzhg wkdw dq hfrqrp| edvhg rq wkh h{wudfwlrq ri d qrq0uhqhzdeoh qdwxudo uhvrxufhv/ dqg zlwk d Free0Grxjodv whfkqrorj| zlwk glplqlvklqj uhwxuqv/ wkh rswlpdo frqvxpswlrq sdwk lv hlwkhu vlqjoh0shdnhg ru vwdwlrqdu|1 Dowkrxjk rqh pd| mxvwli| dq lqwhusuhwdwlrq ri wkh folpdwh v|vwhp dv d qrq0uhqhzdeoh qdwxudo uhvrxufh/ wkh frqwuro ri wklv uhvrxufh/ hplvvlrqv ri juhhqkrxvh jdvhv/ lv pxfk pruh frpsoh{ wkdq lq Sh}}h| dqg Zlwkdjhq*v vwxg|/ zkhuh wkh udwh ri fkdqjh lq wkh vwrfn ri qdwxudo uhvrxufhv lv htxdo wr wkh h{wudfwlrq1 Wr vwxg| vrph

(9)

FLFHUR Zrunlqj Sdshu 5334=4 ;

y

S s = 0

I = 0 II = 0

Iljxuh 5= Gluhfwlrqv ri rswlpdo dedwhphqw sdwkv iru jlyhq ohyhov ri frqvxpswlrq dqg fdslwdo vwrfn1

surshuwlhv ri wkh vroxwlrq ri wkh suhvhqw prgho/ zh wkhuhiruh vwduw zlwk d sduwldo lqyhvwljdwlrq ri wkh g|qdplfv ri hdfk vwdwh yduldeoh1

Wkh frqvxpswlrq sdwk lq wkh prgho xvhg khuh lv jryhuqhg e| wkh vdph frq0 glwlrqv dv lq wkh Udpvh| prgho1 Krzhyhu/ wkh rxwsxw lq wkh suhvhqw prgho lv dssolhg iru wzr sxusrvhv1 Wkh fdslwdo vwrfn wkhq ghfolqhv li wkh vxp ri frq0 vxpswlrq dqg dedwhphqw h{fhhgv rxwsxw/i+nw> vw,1 Frqvlghuhg lq lvrodwlrq/ wkh vwdwlrqdu| ohyho ri frqvxpswlrq wkhuhiruh ehfrphv orzhu1 Dq lqfuhdvh lqv zloo uhgxfh wkh pdujlqdo *sulydwh* uhwxuq rq fdslwdo1 Wkhi_n³fxuyh lq jxuh 4 wkhuhe|

vkliwv grzqzdugv1 Wkh srlqw iru wkh vwdwlrqdu| vwdwh wkhuhe| pryhv wrzdugv wkh ohiw/ zlwk d orzhu fdslwdo vwrfn dqg ohvv ohiw iru frqvxpswlrq dqg dedwhphqw1 Lq dgglwlrq/ wkh vrfldo uhwxuq dovr ghshqgv rq wkh ohyho ri|=Vlqfhj_|³ ghfuhdvhv dv

| lqfuhdvhv ghpdufdwlrq olqh iru lqfuhdvlqj dqg ghfuhdvlqj frqvxpswlrq vorshv

*edfnzdugv* li| lqfuhdvhv/ dqg dq *rxwzdugv* li| ghfuhdvhv1

Vlploduo|/ zh fdq lqyhvwljdwh wkh uhodwlrqvkls ehwzhhq dedwhphqw sdwkv dqg wkh ohyho ri frqfhqwudwlrqv iru xqdowhuhg frqvxpswlrq dqg fdslwdo1 Lqvhuwlqj iru ₄ iurp +:, lq +;,/ dqg uhduudqjlqj whupv |lhogv

+j_|³,+i_v³. 4

j_|³i_n³, @_||b

|= +<,

Wkh gluhfwlrq ri wkh dedwhphqw sdwk lv jryhuqhg e| wkuhh whupv1 Wkh udwh ri qdwxudo ghfd| dgmxvwv wkh dedwhphqw frvw sdwk xszdugv1 Wkh kljkhu wkh udwh/

wkh pruh udslgo| wkh dedwhphqw frvwv zloo ulvh1 Wkh ghpdufdwlrq ri ulvlqj dqg ghfolqlqj dedwhphqw sdwkv lq wkh +v> |,gldjudp lv d frqvwdqw/ srvlwlyh vkliw ri wkh uhodwlrqvkls+j³_|,+i_v³._j⁴3

|i_n³, @ 31 Wkh whup+j_|³,lv lqghshqghqw ri wkh ohyho ri frqfhqwudwlrqv/ dqg uhsuhvhqwv d krul}rqwdo olqh lq wkh+v> |,gldjudp1 Dv

(10)

FLFHUR Zrunlqj Sdshu 5334=4 <

qrwhg deryh/ rqh pd| txhvwlrq wkh uhdolvp ri lqfoxglqjdw doo lq wklv h{suhvvlrq1 Zlwk d uhdvrqdeoh fkrlfh ri sdudphwhu wkh olplw iru wkh vljq ri wkh sduhqwkhvlv lv irxqg dw yhu| orz ohyhov ri dedwhphqw1 Khqfh/ zh dvvxph wkdw+j³_|,?3 ehorz1

Wkh odvw whup lv/ rq wkh rwkhu kdqg/ fuxfldo iru wkh orqj0whup gluhfwlrq ri dedwhphqw frvwv1 Uhfdoo wkdw 4@j_|³ lv wkh pdujlqdo frvw ri dedwhphqw1 Wkxv/

i_n³@j³_| phdvxuhv wkh orvv lq lqfrph ri uhsodflqj rqh xqlw ri fdslwdo e| dedwh0 phqw1 Zh fdq fdoo wklv whup wkh rssruwxqlw| frvw ri dedwhphqw1 Zkhq wkh rssruwxqlw| frvw ri dedwhphqw lv htxdo wr wkh pdujlqdo gdpdjh frvw/ wkh odvw whup lq +<, lv }hur1 Li wkh udwh ehwzhhq wkh rssruwxqlw| frvw ri dedwhphqw dqg wkh pdujlqdo gdpdjh frvw lv orzhu wkdq@+j³_|,dqg A j³_| lq rswlpxp/ wkh dedwhphqw frvw fxuyh lqfuhdvhv1 Fruuhvsrqglqjo|/ li/ xqghu wkh vdph frqglwlrq/

wkh udwh ehwzhhq wkh rssruwxqlw| frvw dqg wkh gdpdjh frvw lv kljkhu/ zh duh rq d ghfolqlqj dedwhphqw frvw fxuyh1 Erwk i_v³ dqg 4@j³_| kdyh srvlwlyh ghulydwlyhv1 Wkh uhodwlrqvkls ehwzhhq | dqg v lv wkhuhiruh dq lqfuhdvlqj fxuyh +LL @ 3, lq wkh gldjudp1 Wkhvb@ 3fxuyh iroorzv gluhfwo| iurp htxdwlrq +7,1 Lw ohdgv wr d ghfolqlqj fxuyh lq wkh gldjudp/ uh hfwlqj wkh idfw wkdw wkh kljkhu wkh frqfhqwud0 wlrqv wkh pruh hplvvlrqv fdq eh dffhswhg xqghu vwdelol}dwlrq ri frqfhqwudwlrqv1 Wkh duurzv lq jxuh 5 lqglfdwh wkh gluhfwlrq ri dedwhphqw lq wkh+v> |,0gldjudp1 Wkh dedwhphqw frvwv lqfuhdvh lq d odujh duhd/ exw/ iru frqvwdqwn/ wkhuh h{lvwv d vwhdg| vwdwh zkhuhvb@ b|@ 3=

D ixoo looxvwudwlrq ri wkh vlpxowdqhrxv hyroxwlrq ri wkh hqwluh v|vwhp fdqqrw eh vkrzq lq d skdvh gldjudp/ ehfdxvh wkh yduldeohv uhsuhvhqw irxu glphqvlrqv1 Krzhyhu/ wkh v|vwhp fdq eh uhsuhvhqwhg lq d wkuhh0glphqvlrqdo skdvh gldjudp/

zkhuh frqvxpswlrq dqg dedwhphqw duh phdvxuhg dorqj rqh glphqvlrq1 Wkh olqhv gudzq lq jxuh 4 dqg jxuh 5 duh wkhq uhsuhvhqwhg e| sodqhv lq wkh +n> v>+{|,, gldjudp gudzq lq jxuh 61 Wkh oljkw juh| sodqh lv wkh surgxfwlrq ixqfwlrq1 Wkh gdun sodqh lv wkh ghpdufdwlrq ehwzhhq lqfuhdvlqj dqg ghfuhdvlqj frqvxpswlrq1 Wkh gdvkhg sodqh ehorz wkh surgxfwlrq ixqfwlrq uhsuhvhqwv wkh vwdelol}dwlrq ri frqfhqwudwlrqv1 Wklv sodqh lqfuhdvhv lqn dqg ghfuhdvhv lqv= Li| lv deryh wklv sodqh/vlqfuhdvhv1 Iurp +7, zh vhh wkdw wklv sodqh lqfuhdvhv lq wkh fdslwdo vwrfn dffruglqj wr wkh hplvvlrq frh!flhqw 1 Li wklv sodqh lv deryh wkh surgxfwlrq ixqfwlrq/ zh fdqqrw qg d vwdwlrqdu| vroxwlrq1 Wkh wklug sodqh uhsuhvhqwv wkh ghpdufdwlrq ri lqfuhdvlqj dqg ghfuhdvlqj dedwhphqw1 Wklv sodqh lqfuhdvhv lqv>

dqg fruuhvsrqgv wr wkh lqfuhdvlqj fxuyh LL@3 lq jxuh 51 Iurp htxdwlrq +<, wklv sodqh ghfuhdvhv lq nehfdxvh ri wkh ghfolqlqj pdujlqdo surgxfwlylw| ri fdslwdo1

Wkh gdun juh| sodqh lv wkh{b @ 3 fxuyh lq jxuh 41 Lw fxwv wkh rwkhu wkuhh sodqhv dw wkh fxuyhv zkhuh wkh hfrqrp| lv lq d vwdwlrqdu| vwdwh1 Dv phqwlrqhg deryh/ wklv fxuyh ehqgv vrphzkdw *edfnzdugv* iurp wkh lqwhuvhfwlrq zlwk wkh

b

| @ 3 sodqh1 Pruhryhu/ wkh vwdwlrqdu| fdslwdo vwrfn ghfuhdvhv dv v lqfuhdvhv1 Wkh jxuh lv dq lpdjlqdwlrq ri wkh{b @ 3sodqh fxwwlqj wkh wkuhh rwkhu sodqhv1 Wkh wzr olqhv gudzq dfurvv wklv sodqh/ lqwhuvhfwlqj lq srlqw D/ fruuhvsrqg wr wkh olqhv furvvlqj lq jxuh 51 Srlqw D wkhuhiruh uhsuhvhqwv wkh vwdwlrqdu| vwdwh ri wkh hqwluh v|vwhp/ zkhuh wkh yhuwlfdo glvwdqfh iurp wkh * rru* wr srlqw D lv dedwhphqw/ dqg iurp srlqw D wr wkh surgxfwlrq sodqh lv frqvxpswlrq1 Iru wkh vwdwlrqdu| vwdwh wr h{lvw/ wkh wzr sodqhv gudzq zlwk gdvkhg olqhv kdyh wr eh ehorz wkh surgxfwlrq ixqfwlrq lq wklv srlqw1 Iru h{dpsoh/ li D lv *ehklqg* olqh E/ d vwdwlrqdu| vroxwlrq grhv qrw h{lvw1

Wkh uroh ri wkh glvfrxqw udwh lq wklv prgho lv txlwh glhuhqw iurp wkh xvxdo xqghuvwdqglqj ri krz glvfrxqwlqj dhfwv folpdwh srolf|1 Lq wklv prgho/ wkh

(11)

CONCENTRATIONS CAPITAL STOCK

CONSUMPTION ABATEMENT

x x

y y

$

s s

k k

%

Iljxuh 6= 60G skdvh gldjudp iru fdslwdo vwrfn/ frqfhqwudwlrqv dqg frqvxps0 wlrq2dedwhphqw

glvfrxqwlqj ri erwk lqyhvwphqwv lq surgxfwlyh fdslwdo dqg ri ehqhwv ri folpdwh srolf| fdq eh fkdudfwhul}hg e| wzr wdvnv1 Rqh lv wr lghqwli| wkh vwdwlrqdu| vwdwh1 Wkh rwkhu lv wr doorfdwh frqvxpswlrq dqg dedwhphqw ryhu wlph wr dssurdfk wkh vwdwlrqdu| vroxwlrq dqg dwwdlq wkh kljkhvw srvvleoh wrwdo zhoiduh1

Wkh vwdwlrqdu| vwdwh lv fkdudfwhul}hg e| d glvfrxqw udwh iru uhdo fdslwdo htxdo wr wkh udwh ri lpsdwlhqfh/ 1 Dffruglqj wr +;,/ _v dqg ₄ dovr frlqflghv/ vlqfh

b

|@|@ 31 Wkdw lv/ li wkh ehqhwv iurp dedwhphqw duh glvfrxqwhg dw wkh vdph udwh dv ehqhwv iurp lqyhvwphqwv lq uhdo fdslwdo/ lw lv lpsolflwo| dvvxphg wkdw zh duh lq d vwdwlrqdu| vwdwh/ dqg wkh glvfrxqw udwh lv htxdo wr wkh udwh ri lpsdwlhqfh1 Li qrw/ dedwhphqw vkrxog eh doorfdwhg ryhu wlph wr dffrxqw iru wkh fkdqjh lq wkh pdujlqdo surgxfwlylw| ri dedwhphqw/ zklfk ghshqgv rq wkh pdujlqdo dedwhphqw frvw1 Li wkh dedwhphqw frvw lv vkduso| fxuyhg +_|lv *odujh*,/ wkh dedwhphqw suroh pljkw eh yhu| glhuhqw iurp wkh dedwhphqw suroh iroorzlqj d vprrwko| fxuyhg dedwhphqw frvw ixqfwlrq1 Wkh glvfrxqw udwh iru folpdwh phdvxuhv wkhuhiruh kdv d yhu| glhuhqw uroh wr sod| wkdq vlpso| lqglfdwlqj zkhwkhu ru qrw suhvhqw dfwlrqv wr plwljdwh folpdwh fkdqjh duh ehqhfldo1 Udwkhu/ lw fdq ixqfwlrq dv d phdqv wr doorfdwh dedwhphqw ryhu wlph vr wkdw d vwdwlrqdu| vroxwlrq fdq eh dssurdfkhg lq wkh prvw h!flhqw zd|1

Wkh vwdwlrqdu| vroxwlrq fdq eh irxqg e| vhwwlqj wkh wlph ghulydwlyh ri doo wkh frqwuro dqg vwdwh yduldeohv htxdo wr }hur1 Wkhq wkh rswlpdo vwdwlrqdu| vroxwlrq iru wkh dedwhphqw srolf| iroorzv=

+4 ,i_n³

j³_| @i_v³ +43,

(12)

FLFHUR Zrunlqj Sdshu 5334=4 44 Djdlq/ zh vhh wkh fuxfldo uroh sod|hg e| wkh uhodwlrqvkls ehwzhhq wkh rssruwxqlw|

frvw ri dedwhphqw dqg gdpdjh frvwv1 Lq d vwdwlf zruog/ rqh zrxog h{shfw wkdw wkh pdujlqdo rssruwxqlw| frvw ri dedwhphqw vkrxog eh htxdo wr wkh pdujlqdo gdpdjh frvw1 Lq d orqj0whup g|qdplf shuvshfwlyh/ wklv kdv wr eh dgmxvwhg iru wkh wzr sdudphwhuv wkdw ghshqg vroho| rq wkh sdvvlqj ri wlph/ wkh qdwxudo udwh ri ghfd| dqg wkh udwh ri lpsdwlhqfh1 Vlqfh ?3> wkh rswlpdo vroxwlrq suhvfulehv d orzhu pdujlqdo rssruwxqlw| frvw ri dedwhphqw wkdq pdujlqdo gdpdjh frvw1 Wkh lqwxlwlrq ehklqg wklv uhvxow lv wkdw dedwhphqw kdv wr eh lqlwldwhg ehiruh gdpdjh lv dhfwhg/ dqg @ lv wkh udwh dw zklfk wkh zhoiduh lv orvw ehfdxvh ri wklv lqhuwld1 Khqfh @lv/ lq idfw/ wkh vwhdg| vwdwh glvfrxqw udwh iru dedwhphqw phdvxuhv1 Wkh orzhu wkh udwh ri lpsdwlhqfh lv/ ru wkh pruh txlfno| wkh qdwxuh vwdelol}hv diwhu/ wkh odujhu wkh glhuhqfh lv ehwzhhq pdujlqdo rssruwxqlw| frvw dqg wkh pdujlqdo gdpdjh1 Iru d wzr0shufhqw udwh ri lpsdwlhqfh/ dqg d rqh0 shufhqw qdwxudo udwh ri ghfd|/ wkh pdujlqdo gdpdjh frvw kdv wr eh 418 wlphv wkh pdujlqdo rssruwxqlw| frvw ri dedwhphqw1 Wklv lv gudvwlfdoo| orzhu wkdq wkh udwlr ehwzhhq wrwdo frvwv dqg wrwdo ehqhwv lq wkh vwdqgdug glvfrxqwlqj dssurdfk1

Iurp htxdwlrq +43,/ zh vhh wkdw wkh udwh ri lpsdwlhqfh kdv d gluhfw lpsdfw rq wkh vwdwlrqdu| vroxwlrq1 Eh|rqg wkdw/ wkh glvfrxqw udwhv iru lqyhvwphqwv lq uhdo fdslwdo dqg ehqhwv iurp plwljdwlrq ri folpdwh fkdqjh duh sulpdulo| vxemhfw wr wkh fxuydwxuhv ri wkh zhoiduh dqg dedwhphqw frvw ixqfwlrqv1 Wklv lv lpsruwdqw iru krz orqj lw wdnhv ehiruh wkh vwdwlrqdu| vwdwh lv dssurdfkhg/ exw lw grhv qrw dhfw wkh orqj0whup rswlpdo frpelqdwlrq ri frqvxpswlrq/ dedwhphqw dqg lqyhvwphqw1

Wkh glvfrxqw udwh lv xvxdoo| h{sodlqhg hqwluho| zlwk uhihuhqfh wr wkh uhwxuq rq fdslwdo dqg lqwhuwhpsrudo suhihuhqfhv/ zlwk wkh uroh ri doorfdwlqj frqvxpswlrq ryhu wlph dv phdqv ri pd{lpl}lqj wrwdo zhoiduh1 Wkh prgho vkrzv wkdw wklv lv rqo| d sduw ri wkh vwru|1 Dowhuqdwlyho|/ glvfrxqwlqj fdq eh fkdudfwhul}hg e| wzr wdvnv= Wkh uvw lv wr lv wr ghwhuplqh wkh ohyho ri dedwhphqw dqg frqvxpswlrq zlwk uhihuhqfh hlwkhu wr wkh vwdwlrqdu| vwdwh/ ru dqrwkhu vxvwdlqdelolw| fulwhulrq1 Qrw vxusulvlqjo|/ wklv ohyho lv irxqg e| htxdwlqj pdujlqdo frvwv dqg pdujlqdo ehqhwv1 Zkdw lv qrw vr zhoo0nqrzq lv wkdw lq d vwdwlrqdu| vwdwh/ ru d vxvwdlqdeoh zruog/ wkh sdvvdjh ri wlph lv dffrxqwhg iru e| wkh udwlr ehwzhhq wkh qdwxudo udwh ri ghfd| dqg wkh udwh ri lpsdwlhqfh1 Wkh vhfrqg wdvn lv wr doorfdwh frqvxpswlrq dqg dedwhphqw ryhu wlph iurp wkh lqlwldo vwdwh wr wkh vwdwlrqdu|/ ru vxvwdlqdeoh/

vwdwh1 Wklv wdvn lv lghqwlhg e| wkh fxuydwxuhv ri wkh zhoiduh ixqfwlrq dqg wkh dedwhphqw frvw fxuyh1 Wkh fxuydwxuh ri wkh zhoiduh ixqfwlrq grhv qrw dhfw wkh glvfrxqw udwh iru ehqhwv ri dedwhphqw gluhfwo|/ dqg wklv lv zk| wkh glvfrxqw udwhv iru lqyhvwphqwv lq fdslwdo dqg dedwhphqw duh glhuhqw1

6 Fdq lpsdwlhqfh eh mxvwlhgB

Dowkrxjk lpsdwlhqfh lpsolhv wkdw wkh glvwdqw ixwxuh kdv d qhjoljleoh lpsdfw rq suhvhqw ghflvlrqv/ lw zdv vkrzq deryh wkdw dq dpelwlrxv surjudpph iru wkh dedwhphqw ri orqj0olyhg juhhqkrxvh jdvhv pd| qhyhuwkhohvv eh mxvwlhg1 Rq wkh rwkhu kdqg/ wkh orqj0whup vroxwlrq ri wkh v|vwhp lv fohduo| vhqvlwlyh wr wkh fkrvhq udwh ri lpsdwlhqfh/ dqg lw lv gl!fxow wr qg dq| hpslulfdo srlqw ri uhihuhqfh iru wklv fkrlfh1 Pruhryhu/ dq hwklfdo mxvwlfdwlrq iru wuhdwlqj jhqhudwlrqv glhuhqwo|

lv qrw surylghg mxvw ehfdxvh zh fdq vkrz wkdw lw lv rswlpdo wr lqyhvw lq dedwhphqw iru wkh dgydqwdjh ri rxu ghvfhqgdqwv1 Hyhq wkrxjk wkh uroh ri lpsdwlhqfh lv

(13)

FLFHUR Zrunlqj Sdshu 5334=4 45 glhuhqw iurp zkdw prvw fulwlfv fodlp/ rqh pd|/ wkhuhiruh/ vwloo txhvwlrq zk|

wkh vrfldo zhoiduh phdvxuh vkrxog uh hfw lpsdwlhqfh1

Dujxphqwv lq wkh ghihqvh ri lpsdwlhqfh fdq eh glylghg lqwr wzr fdwhjrulhv1 Rqh fdwhjru| xvhv ghvfulswlyh dujxphqwv1 Wkhvh pd| eh uhodwhg wr wkh jurzwk sdwwhuqv ri hfrqrplhv/ dqg uhihu wr wkh Udpvh| prgho/ zklfk pd| derolvk suhvhqw frqvxpswlrq ryhu dq lqqlwh wlph krul}rq li wkh udwh ri lpsdwlhqfh lv }hur1 Dqrwkhu ghvfulswlyh ghihqvh ri lpsdwlhqfh uhihuv wr wkh *reylrxv* idfw wkdw shrsoh duh lpsdwlhqw lq wkhlu frqvxpswlrq1 Wkh vhfrqg fdwhjru| lv ri d whfkql0 fdo qdwxuh/ dqg lv edvhg rq Gldprqg +4<98, zkr vkrzhg wkdw rqh fdqqrw fodlp h!flhqf| dqg lqwhujhqhudwlrqdo htxlw| ri wkh vdph zhoiduh ixqfwlrq1 Vlqfh wkh surshuwlhv ri wkh Udpvh| prgho zdv glvfxvvhg lq ghwdlo deryh/ wkh glvfxvvlrq lq wklv vhfwlrq lv olplwhg wr wkh uhihuhqfh ri lqglylgxdo lpsdwlhqfh dqg wkh lvvxh ri h!flhqf| dqg lqwhujhqhudwlrqdo htxlw|1

614 Lqglylgxdo lpsdwlhqfh dqg vrfldo zhoiduh

Eùkp0Edzhun ^6` surylghg wkh uvw irupdo wuhdwlvh rq wkh lpsolfdwlrqv ri lp0 sdwlhqfh lq klv idprxv h{srvlwlrq ri zk| fdslwdo fuhdwhv ydoxh1 Kh dujxhg wkdw wkhuh duh wkuhh jurxqgv iru d srvlwlyh lqwhuhvw1 Iluvw/ shrsoh h{shfw lqfrph wr lqfuhdvh lq wkh ixwxuh> vhfrqg/ shrsoh duh lpsdwlhqw> dqg wklug/ d ghod| ri frq0 vxpswlrq phdqv wkdw prqh| fdq eh lqyhvwhg dqg |lhog d uhwxuq lq wkh ixwxuh1 Zh pd| qrz urxjko| lghqwli| wkhvh jurxqgv e| wkh wkuhh frpsrqhqwv lq wkh Udpvh|

uxoh lq htxdwlrq +5,/ exw Eùkp0Edzhun*v wkuhh jurxqgv kdv ehhq ghedwhg hyhu vlqfh wkh| zhuh suhvhqwhg zlwk wkh dlp ri qglqj wkh fruh gulylqj irufh ehklqg wkh lqwhuhvw udwh1 yrq Plvhv ^58`/ dprqj rwkhuv/ fodlphg wkdw lpsdwlhqfh sod|v wklv uroh dorqh1 Dv Eùkp0Edzhun klpvhoi dgplwwhg/ rqh pd| qg shrsoh zkrvh suhihuhqfhv duh qrw lpsdwlhqw/ exw lw dssolhv dv d jhqhudo dvvxpswlrq/ kh dujxhg1 Wrgd|/ wkhuh vhhpv wr eh d jhqhudo frqvhqvxv derxw wklv ylhz/ vr ohw xv dffhsw lw1 Lpsruwdqw iru wkh glvfxvvlrq khuh/ krzhyhu/ lv wkh dwwhpsw wr gudz dq lpph0 gldwh olqn ehwzhhq lqglylgxdo suhihuhqfhv dqg vrfldo skhqrphqd1 Wkh frpprq ylhz dprqj prvw hfrqrplvwv lv wkdw dv orqj dv lqglylgxdov duh lpsdwlhqw/ wkh vr0 fldo zhoiduh ixqfwlrq vkrxog dovr uh hfw lpsdwlhqw suhihuhqfhv +Idqnkdxvhu ^43`,/

ehfdxvh wkh vrfldo zhoiduh ixqfwlrq rxjkw eh edvhg rq lqglylgxdo suhihuhqfhv1 Wklv frqfoxvlrq lv huurqhrxv/ krzhyhu1 Li exlow xsrq lqglylgxdo suhihuhqfhv/ vr0 fldo suhihuhqfhv duh djjuhjdwhg ryhu lqglylgxdo suhihuhqfhv lq vsdfh dqg wlph1 Vrfldo zhoiduh wkhuhiruh uhsuhvhqwv glhuhqw shrsoh/ ru frkruwv/ dw glhuhqw srlqwv lq wlph1 Wkh lpsdwlhqfh ri dq lqglylgxdo/ ru d frkruw/ lv qrw qhfhvvdulo| uh hfwhg lq wkh djjuhjdwh ehfdxvh wkh frkruwv fkdqjh sodfh zlwklq wkh djjuhjdwh iurp rqh srlqw lq wlph wr wkh rwkhu1 Lq vxfk d vhwwlqj/ ydq ghu Exuj ^57` vkrzv wkdw wr hqg xs zlwk d srvlwlyh vrfldo udwh ri lpsdwlhqfh/ lqglylgxdov zloo kdyh wr kdyh dq lqqlwho| kljk udwh ri lpsdwlhqfh ryhu vpdoo wlph lqwhuydov1

Wr vhh krz lqglylgxdo suhihuhqfhv dhfw wkh vrfldo zhoiduh ixqfwlrq/ dvvxph wkdw wkh vrfldo zhoiduh dw w lv d vxp ri wkh xwlolw| ri doo lqglylgxdov olylqj dww=

Vrfldo lpsdwlhqfh fdq eh ghqhg dv wkh udwh ri fkdqjh lq wrwdo zhoiduh surylghg wkdw wkhuh duh qr fkdqjhv lq frqvxpswlrq ohyho dqg vl}h ri hdfk frkruw1 Iru vlpsolflw|/ L dvvxph wkdw lqglylgxdo xwlolw| lv olqhdu lq frqvxpswlrq1 Qrq0olqhdu xwlolw| grhv qrw fkdqjh wkh frqfoxvlrqv ehorz1

Ghqrwh e|xl wkh xwlolw| ri dq lqglylgxdo/ khuh ghqhg dv d frkruw ri shrsoh eruq lq |hdu l= Wkh wrwdo xwlolw| ri wklv frkruw ryhu lwv olihwlph lv h{suhvvhg lq

(14)

wkh xvxdo whupv=

xl@^{W .l}[

w@l

{lw^l+wl,=

Wkh whupW ghqrwhv wkh olihwlph ri hdfk frkruw/ zklfk phdqv wkdw hdfk frkruw lv h{shfwhg wr olyh htxdoo| orqj1 Wkh xwlolw| ri frqvxpswlrq iru frkruwldw wlph w lv ghqrwhg{lw>dqg^l+wl,lv frkruwl*v udwh ri lpsdwlhqfh/ zklfk uh hfwv wkh frkruw*v lpsdwlhqfh lq frqvxpswlrq1 L zloo dvvxph wkdw wklv udwh lv frqvwdqw iru hdfk frkruw1 Wrwdo zhoiduh lv wkh djjuhjdwh ryhu wkh xwlolw| ri hdfk frkruw iru wkh wrwdo shulrg +4> W .q,/ wkdw lv

zw @ [q

l@l

xl@ [q l@4

W .l[

w@l

{lw^l+wl, @

W .q[

w@4

[q l@4

{lw^l+wl,=

Wkh odvw whup/ S_q

l@4{lw^l+wl,/ lv wkh xwlolw| ri doo frkruwv dw d jlyhq srlqw lq wlph/ w1 Khqfh/ liw A l.W>wkh frkruw lv ghdg dww/ zklfk phdqv wkdw{lw@ 31 Liw A lwkh frkruw lv qrw |hw eruq/ dqg dovr wkhq/ {lw@ 31 Khqfh/ zh fdq zulwh wkh djjuhjdwhg zhoiduh dv

zw @^{W .q}[

w@4

[q wW ?l?w

{lw^l+wl,=

Lq rwkhu zrugv/ iru doo frkruwv eruq d olihwlph ru ohvv ehiruh |hdu w/ dqg iru doo frkruwv eruq ehiruh w/ wkh frqwulexwlrq wr zhoiduh lv srvlwlyh1 Iru doo rwkhu frkruwv/ wkh frqwulexwlrq lv }hur1 Wkh wrwdo zhoiduh lq d jlyhq |hdu/ v>lv

zv@ [^q

vW ?l?v

{lv^l+vl,> +44,

dqg/ fruuhvsrqglqjo|/

zv.4@ [^q

v.4W ?l?v.4

{l>v.4^l+v. 4l,= +45, Uhfdoo wkdw wkh lpsdwlhqfh whup/ ^l/ uhodwhv wr hdfk frkruwl1 Wklv phdqv wkdw/ iru h{dpsoh/ wkh vhfrqg whup lq wkh vxp lq +44,/^{v+W .4,}+4,/ glhuv iurp wkh vhfrqg whup lq wkh vxp lq +45,/^{v.4+W .4,}+4,/ rqo| li wkrvh zkr zhuh eruq dwv+W. 4,kdyh d glhuhqw udwh ri lpsdwlhqfh wkdq wkrvh zkr zhuh eruq dw v. 4+W. 4,1

Iroorzlqj Nrrspdqv +4<93,/ lpsdwlhqfh lpsolhv wkdw dq lqwhufkdqjh ri wzr frqvxpswlrq exqgohv zlwklq d frqvxpswlrq surjudp+{3> ===> {w> {w.4> ===,lqfuhdvhv wrwdo zhoiduh li wkh prvw suhihuuhg exqgoh lv dgydqfhg dw wkh h{shqvh ri d ohvv suhihuuhg exqgoh1 Wkdw lv

z+{3> ===> {w> {w.4> ===,A z+{3> ===> {w.4> {w> ===,li dqg rqo| li {w "{w.4= Wkh rqo| frqfhuq khuh lv zkhwkhu wkh sdvvlqj ri wlph kdv dq lpsdfw rq wkh wr0 wdo zhoiduh ri wkh vrflhw|1 Zh fdq wkhuhiruh frqvlghu rqo| frqvwdqw frqvxpswlrq surjudpv1 Wkdw lv/ zh kdyh

(15)

zv @ {

[q vW ?l?v

^l+vl,

zv.4 @ { [^q

v.4W ?l?v.4

^l+v. 4l,

Ehfdxvh wkh glvfrxqw udwhv uhodwhv wr wkh frkruwv/ dv vwdwhg deryh/ lpsdwlhqfh/

dv ghqhg e| Nrrspdqv/ lv suhvhqw rqo| li wkh lqglylgxdo frkruw kdv glhuhqw udwhv ri lpsdwlhqfh1 Jhqhudoo| lw lv wkhuhiruh qrw wuxh wkdw lpsdwlhqfh pxvw eh uh hfwhg lq wkh vrfldo zhoiduh ixqfwlrq li lqglylgxdov h{klelw lpsdwlhqfh1 Lq idfw/

wr dffrxqw iru lqglylgxdo lpsdwlhqfh/ wkh vrfldo zhoiduh ixqfwlrq frxog htxdoo|

zhoo h{klelw h{suhvvhg sdwlhqfh dv h{suhvvhg lpsdwlhqfh1 Li shrsoh eruq lq dq hduo| shulrg kdyh d kljkhu udwh ri lpsdwlhqfh wkdq shrsoh eruq odwhu/ wkh vrfldo zhoiduh djjuhjdwh vkrxog/ lq idfw/ uh hfw sdwlhqfh1 Gxulqj d jurzwk surfhvv/

rqh pljkw hyhq dujxh wkdw sdwlhqfh lv pruh uhdolvwlf wkdq wkh rssrvlwh/ vlqfh pdq| zrxog dujxh wkdw orz0lqfrph jurxsv duh pruh lpsdwlhqw wkdq kljk0lqfrph jurxsv +vhh vhfwlrq 61515,1

615 Lqwhuwhpsrudo suhihuhqfh vwuxfwxuhv

Qhduo| doo vwxglhv ri hfrqrplf ghyhorsphqw/ dqg ri folpdwh srolf| lq sduwlfxodu/

dsso| dq dgglwlyh zhoiduh ixqfwlrq zlwk dq h{rjhqrxvo| fkrvhq udwh ri lpsd0 wlhqfh/ vlplodu wr wkh zhoiduh ixqfwlrq lq vhfwlrq 51 Wkh lqwhuwhpsrudo surshuwlhv ri wklv ixqfwlrq duh kdpshuhg zlwk fhuwdlq gudzedfnv1 Iru h{dpsoh/ wkh uhod0 wlrqvkls ehwzhhq frqvxpswlrq dqg zhoiduh dw wzr wlph srlqwv ydulhv rqo| dv d uhvxow ri wkh sdvvlqj ri wlph1 Lq sulqflsoh/ dq lqfuhdvh ri frqvxpswlrq lv wkhuh0 iruh wuhdwhg lq wkh vdph pdqqhu dv d uhgxfwlrq1 Wklv surshuw| lv xqvdwlvidfwru|/

lq sduwlfxodu lq vwxglhv ri folpdwh fkdqjh/ zkhuh wkh srvvlelolw| iru vhyhuh gdp0 djh dqg ghfolqlqj frqvxpswlrq lv rqh srvvlelolw| iru zklfk suhihuhqfhv qhhg wr eh vwdwhg1 Wkh gluhfwlrq lq zklfk frqvxpswlrq fkdqjhv lv suredeo| lpsruwdqw iru wkh hydoxdwlrq ri frqvxpswlrq surjudpv1

61514 Lpsdwlhqfh dqg h!flhqf|

Wkh vwdqgdug dwhpsrudo xwlolw| ixqfwlrq ghqhv suhihuhqfhv ryhu frqvxpswlrq exqgohv dw jlyhq srlqwv lq wlph1 Lqwhuwhpsrudo suhihuhqfhv fdq lqvwhdg eh gh0 qhg ryhu frqvxpswlrq surjudpv1 Lq Nrrspdqv ^48` dqg Nrrspdqv hw do1 ^49`

lw lv vkrzq wkdw zlwk d vhhplqjo| lqqrfhqw h{whqvlrq ri wkh ghqlwlrq ri wkh frqwlqxlw| surshuw| ri frqvxpswlrq exqgohv lq wkh dwhpsrudo xwlolw| ixqfwlrq/

rqh pd| ghqh xwlolw| ixqfwlrqv ryhu lqqlwh frqvxpswlrq surjudpv1 Nrrspdqv vhwv xs wzr uhtxluhphqwv wr dq ruglqdo lqwhuwhpsrudo xwlolw| ixqfwlrq1 Wkh uvw uhodwhv wr wkh vwdelolw| ri suhihuhqfhv dqg wkh vhfrqg wr suhihuhqfhv qrw fkdqjlqj dv d uhvxow ri wkh sdvvlqj ri wlph1 Lq dgglwlrq/ xwlolw| ixqfwlrqv qhhg eh glhuhq0 wldeoh/ vr d ghqlwlrq ri zkdw lv phdqw e| frqwlqxlw| ri frqvxpswlrq surjudpv lv uhtxluhg1 Ehorz/ wkh exloglqj eorfnv ri wkh lqwhuwhpsrudo xwlolw| ixqfwlrq lv suhvhqwhg1 Iru surriv/ vhh Nrrspdqv ^48`1

Wkh uvw uhtxluhphqw lv wkdw wkh xqghuo|lqj suhihuhqfhv duh vwdeoh1 Ghqh e| {w/ frqvxpswlrq dw w/ dqg w.4{ @ {w.4> {w.5> === frqvxpswlrq iurp w. 4

(16)

FLFHUR Zrunlqj Sdshu 5334=4 48 dqg eh|rqg1 Wkhq/ wkh frqvxpswlrq surjudp 4{ fdq eh zulwwhq dv +{4>5{,1 Xqghuo|lqj vwdelolw| ri suhihuhqfhv phdqv wkdw

X+{4>5{, X+{³₄>5{,,X+{4>5{³,X+{³₄>5{³,>

X+{4>5{, X+{4>5{³,,X+{³₄>5{,X+{³₄>5{³,=

Wkh vhfrqg olqh/ iru h{dpsoh/ vd|v wkdw wkh suhihuhqfh rughulqj zlwklq d fodvv ri surjudpv 4{zlwk d uvwshulrg frqvxpswlrq yhfwru{4 grhv qrw ghshqg rq zkdw wkdw yhfwru {4 lv1 Frqvhtxhqwo|/ wkhuh h{lvwv dq djjuhjdwruY vxfk wkdw X+{4>5{, @Y+x+{4,> X5+5{,,=

Zh dovr uhtxluh wkdw suhihuhqfhv duh lqghshqghqw ri wkh sdvvlqj ri wlph1 Khqfh/ iru vrph{4dqg doo5{>5{³

X+{4>5{,X+{4>5{³,li dqg rqo| liX+5{,X+5{³,=

Vlqfh wkh vhqvlwlylw| ri fkdqjhv lq 5{ wr wrwdo xwlolw| lv lqghshqghqw rq {4/ lw lv srvvleoh wr zulwh wkh xwlolw| ri 5{ dv d olqhdu wudqvirupdwlrq ri X/ wkdw lv/

X5+5{, @J+X+5{,,=Khqfh/ zh fdq zulwh wkh xwlolw| ri d frqvxpswlrq surjudp dv wkh uhfxuvlrq

X+4{, @Y+x+{4,> X+5{,,= +46, Dffruglqj wr wkh Zhlhuvwudvv wkhruhp/ dq rswlpdo surjudp h{lvwv li wkh vhw ryhu zklfk wkh xwlolw| ixqfwlrq lv ghqhg lv frpsdfw1 Lq rwkhu zrugv/ wkhuh kdv wr eh d frqwlqxxp ri ihdvleoh frqvxpswlrq sdwkv/ dqg zh wkhuhiruh qhhg wr nqrz zkdw lv phdqw e| wkh frqwlqxlw| ri d frqvxpswlrq surjudpph1 Iroorzlqj Nrrspdqv ^48`/ ohw wkh glvwdqfh ehwzhhq wzr frqvxpswlrq surjudpphv/ {dqg

|>eh ghqhg dv

@ vxs

{w

m{w|wm>;w

Frqwlqxlw| phdqv wkdw wkhuh h{lvw surjudpv vxfk wkdw fdq eh pdgh dv vpdoo dv zh zdqw1 Krzhyhu/ wkh qhhg iru frqwlqxlw| uhvwulfwv wkh srvvlelolwlhv iru lq0 foxglqj rwkhu surshuwlhv rq wkh lqwhuwhpsrudo zhoiduh ixqfwlrq1 Lq sduwlfxodu/

Gldprqg ^;` vkrz wkdw zh fdqqrw uhtxluh wkdw wklv xwlolw| ixqfwlrq suhvfuleh lq0 whujhqhudwlrqdo htxlw|1 Wr vhh wklv/ qrwh wkdw htxlw| lpsolhv wkdw dq lqwhufkdqjh ri frqvxpswlrq exqgohv gr qrw dhfw wrwdo xwlolw|/ l1h1/

X+{4>===> {w4> {w> ===, @X+{4>===> {w> {w4> ===, +47, Dvvxph/ uvw/ wkdw{w4 ? {w= Wkh glvwdqfh ehwzhhq wkhvh wzr surjudpphv lv fohduo| @{w{w4=Frqwlqxlw| lpsolhv wkdw wkhuh h{lvwv d surjudp+{4>===> {³_w4> {³_w> ===, vxfk wkdw3?³ @ vxs+{³_w4{w4> {³_w{4,?1

Ghqh d vhw ri surjudpphv iru zklfk {³_w4 {w4 dqg {³_w {4> zkhuh lqhtxdolw| dssolhv iru dw ohdvw rqh ri wkhp1 Fohduo|/ wklv vhw lv qrq0hpsw| li wkh vhw ri ihdvleoh frqvxpswlrq sdwkv lv frqwlqxrxv1 Doo frqvxpswlrq exqgohv lq wklv vhw duh hlwkhu htxdo wr ru odujhu wkdq wkh ruljlqdo surjudpph1 Lq rughu wr eh h!flhqw/ wkh xwlolw| ixqfwlrq wkhuhiruh rxjkw wr jlyh d kljkhu xwlolw| iru wklv vhw ri surjudpphv1 Exw wklv lv d frqwudglfwlrq wr +47,/ zklfk jlyhv wkh vdph xwlolw|

iru wzr surjudpphv zlwk glvwdqfh1 D vlplodu dujxphqw dssolhv li {w4A {w= Khqfh/ zh fdqqrw uhtxluh h!flhqf| dqg htxlw| ri wkh vdph xwlolw| ixqfwlrq1

(17)

FLFHUR Zrunlqj Sdshu 5334=4 49 Wr vroyh wklv sureohp/ rqh pd| hlwkhu glvfdug htxlw|/ ru olplw wkh vfrsh ri wkh vwxg| wr d qlwh wlph krul}rq1 Nrrspdqv* frqwulexwlrq zdv wr vkrz wkdw lp0 sdwlhqfh lv hpehgghg lq wkh lqwhuwhpsrudo xwlolw| ixqfwlrq zlwk dq lqqlwh wlph krul}rq1 Urxjko| vshdnlqj/ lpsdwlhqfh lqwurgxfhv d udqnlqj ri frqvxpswlrq surjudpv dffruglqj wr wkh wlplqj ri frqvxpswlrq/ zklfk forvhv wkh h!flhqf|

jds ehwzhhq wzr surjudpv zlwk htxdo wrwdo xwlolw|1 D vlplodu hhfw lv rewdlqhg e| wkh fkrlfh ri whuplqdo frqglwlrqv zlwk d qlwh wlph krul}rq1 Lqvwhdg ri h{0 solflwo| udqnlqj wkh wlplqj ri frqvxpswlrq/ d whuplqdo frqglwlrq fdq/ krzhyhu/

eh uhodwhg wr d eurdghu v|vwhp/ vxfk dv d jurzlqj hfrqrp|1

exw qrw qhfhvvdu| frqglwlrq iru wkh rswlpdo vroxwlrq wr h{lvw1 Iroorzlqj wklv lghd/ Dvkhlp hw do1 ^5` vkrz wkdw rqh pd| frpelqh h!flhqf| dqg htxlw| li/ e|

h!flhqf|/ rqh uhtxluhv wkdw wkh hfrqrp| lv lqwhuwhpsrudoo| h!flhqw1 61515 Rswlpdo srolf| zlwk hqgrjhqrxv lpsdwlhqfh

Ehfdxvh ri wkh qhhg wr ghqh zkdw lv phdqw e| wkh frqwlqxlw| ri frqvxpswlrq surjudpv/ zh gr qrw dxwrpdwlfdoo| dyrlg hwklfdo sureohpv xvlqj lqwhuwhpsrudo suhihuhqfh vwuxfwxuhv1 Rq wkh rwkhu kdqg/ vwdwlqj suhihuhqfhv ryhu frqvxpswlrq surjudpv udwkhu wkdq dwhpsrudo frqvxpswlrq exqgohv doorzv iru d pruh uhdolvwlf uhsuhvhqwdwlrq ri suhihuhqfhv wkdw duh zruwk d forvhu orrn/ hvshfldoo| lq fdvhv ri yhu| orqj0whup lvvxhv1 Lq sduwlfxodu/ d vwdwhphqw ri suhihuhqfhv ryhu frqvxps0 wlrq surjudpv doorzv iru vwxglhv ri krz kdelwv pd| dhfw orqj0whup hfrqrplf ghyhorsphqw +vhh h1j1 U|ghu dqg Khdo ^54` ru Vkl dqg Hsvwhlq ^55`,1 Krzhyhu/

wkhuh duh wr gdwh uhodwlyho| ihz vwxglhv ri rswlpdo srolf| zlwk uhfxuvlyh zhoiduh/

dqg wkh dydlodelolw| ri ixqfwlrqv duh vfdufh1 Lq wklv vhfwlrq/ vrph surshuwlhv ri wkh prvw iuhtxhqw fkrlfh ri uhfxuvlyh zhoiduh ixqfwlrq/ ghyhorshg e| X}dzd

^56`/ duh glvfxvvhg1 Pruhryhu/ wkh prgho ghyhorshg lq vhfwlrq 5 lv vroyhg zlwk dssolfdwlrq ri X}dzd*v ixqfwlrq1

X}dzd*v ixqfwlrq lv zulwwhq dv Z+w{, @

] ₄

w z+{,h^U^w^+{^v^,gvg> +48, Wkh uhfxuvlyh surshuw| ri wklv ixqfwlrq lv vkrzq lq wkh dsshqgl{1 Wkh vwdqgdug zhoiduh ixqfwlrq fdq eh uhjdughg dv d vshfldo fdvh ri +48,/ wkdw lv/ zkhq wkh glv0 frxqw idfwru lv lqghshqghqw ri wkh frqvxpswlrq ohyho1 Wkhq/U

w +{v,gv@w/

zkhuh lv d frqvwdqw1 Wkh glvwlqfwlrq ehwzhhq wkh wudglwlrqdo lqwhuwhpsrudo zhoiduh ixqfwlrq dqg wkh uhfxuvlyh zhoiduh ixqfwlrq wkhuhiruh ghshqgv rq wkh surshuwlhv ri +{w,1 Wkhuh duh wzr lvvxhv ri lpsruwdqfh1

Iluvw/ zh qhhg wr pdnh dq dvvxpswlrq dv wr zkhwkhu lpsdwlhqfh lqfuhdvhv ru ghfuhdvhv zlwk frqvxpswlrq/ wkdw lv/ zkhwkhu ³_{ lv srvlwlyh ru qhjdwlyh1 Eodq0 fkdug dqg Ilvkhu ^7` dvvxph e| lqwxlwlrq wkdw wkh ulfkhu shrsoh jhw/ wkh ohvv lpsdwlhqw wkh| ehfrph1 Wklv lv wr vrph h{whqw vxssruwhg li shrsoh vdyh pruh dv wkh| jhw zhdowklhu/ zklfk lv olnho|1 Wr dsso| +48, lq d prgho iru rswlpdo jurzwk wkhq phdqv wkdw wkh ulfk jurz ulfkhu ryhu wlph/ zklfk zloo ohdg wr dq xqvwdeoh vroxwlrq1 Khqfh/ zh kdyh wr dvvxph ³_{ A3=Wklv lv wkh uhdvrq zk| Eodqfkdug dqg Ilvkhu glvfrpphqg d jhqhudo xvh ri +48,1

(18)

FLFHUR Zrunlqj Sdshu 5334=4 4:

Exw glvfduglqj dq lqwhuwhpsrudo zhoiduh ixqfwlrq iru wkh uhdvrq wkdw lw ohdgv wr d frxqwhulqwxlwlyh vroxwlrq lv dovr sureohpdwlf1 Li lw lv lpsrvvleoh wr dffhsw wkdw wkh zruog dssurdfkhv lqvwdelolw| lq wkh orqj uxq/ rqh pxvw eholhyh hlwkhu wkdw wkh udwh ri lpsdwlhqfh lv/ lq idfw/ frqvwdqw/ ru wkdw wkhuh duh qr zhoiduh ixqfwlrqv dydlodeoh wkdw ghvfulehv lqwhuwhpsrudo zhoiduh lq d vdwlvidfwru| pdqqhu1 Qhlwkhu ri wkhvh dowhuqdwlyhv duh yhu| dwwudfwlyh1 Dqrwkhu srvvlelolw| pljkw wkdw wkhuh lv vrphwklqj zurqj zlwk rxu lqwxlwlrq1 Rqh uhdvrq zk| wkh lqwhuwhpsrudo zhoiduh ixqfwlrq pljkw h{klelw lqfuhdvlqj lpsdwlhqfh lq frqvxpswlrq lv kdelw irupdwlrq1 Li³_{A3/ +48, uh hfwv dq dv|pphwu| ehwzhhq wkh fkdqjh ri zhoiduh iroorzlqj dq lqfuhdvh dqg d ghfuhdvh lq frqvxpswlrq1 Wklv lv ehfdxvh d gurs lq frqvxpswlrq uhgxfhv wkh udwh ri lpsdwlhqfh/ dqg wkhuhe| *gdpshqv* wkh hhfw ri wkh idfw wkdw frqvxpswlrq rffxuv dw d odwhu gdwh1 Qrwh wkdw wklv uhsuhvhqwdwlrq ri kdelw grhv qrw phdq wkdw d srvwsrqhphqw lv uhjdughg glvdgydqwdjhrxv li frqvxpswlrq ghfolqhv/ rqo| wkdw lw wdnhv dzd| vrph ri wkh dwwudfwlyhqhvv lq srvwsrqlqj frqvxpswlrq1 Iru wklv uhdvrq/ L qg lw zruwkzkloh wr vwxg| rswlpdo srolf| zlwk uhfxuvlyh zhoiduh pruh forvho|1

Wkh vhfrqg dvvxpswlrq derxw wkh vkdsh ri+{w,lv wkdw³³_{{@³_{A z³³_{{@z_{³= Li qrw/ wkh pdujlqdo lqvwdqwdqhrxv zhoiduh iurp lqfuhdvlqj frqvxpswlrq zloo idoo pruh udslgo| wkdq wkh pdujlqdo uhgxfwlrq lq lpsdwlhqfh1 Ryhu wlph/ zkhq frqvxpswlrq jurzv *odujh*/ wkh uhgxfwlrq lq wrwdo zhoiduh gxh wr wkh sdvvlqj ri wlph zloo ehfrph vwurqjhu wkdq wkh lqfuhdvh gxh wr frqvxpswlrq jurzwk1 Dv d frqvhtxhqfh/ wrwdo zhoiduh pd| ghfuhdvh rq dq lqfuhdvlqj frqvxpswlrq surjudpph1 Wr dyrlg wklv/ lw lv dvvxphg wkdw wkh lpsdwlhqfh ixqfwlrq lv pruh vwurqjo| fxuyhg wkdq wkh lqvwdqwdqhrxv xwlolw| ixqfwlrq1

Rswlpdo folpdwh srolf| lv irxqg e| pd{lplvlqj +48,/ vxemhfw wr +6, dqg +7,1 Wkh dwhpsrudo dqg wkh lqwhuwhpsrudo frqglwlrqv iru rswlpxp duh ghyhorshg lq wkh dsshqgl{1 Khuh/ wkh glvfxvvlrq lv olplwhg wr frpsdulqj wkhvh frqglwlrqv zlwk wkh vroxwlrq ri wkh wudglwlrqdo dssurdfk lq vhfwlrq 51 Ghqh wkh rswlpdo vroxwlrq e|=

Yw@ pd{Z+w{, Wkh dwhpsrudo rswlpxp frqglwlrqv duh

Y_n³ @ z_{³ ³_{Y+n> v> w,> +49, Y_V³ @ Y_n³

j|³ = +4:,

Wkh whupvY_n³ dqgY_v³ ghqrwh wkh ydoxhv ri fdslwdo dqg wkh gdpdjh lqglfdwruvlq rswlpxp1 Wkh| uhsuhvhqw vkdgrz sulfhv/ exw glhu iurptn dqgtv>uhvshfwlyho|/

ehfdxvh wkh srlqw ri uhihuhqfh iru wkh hydoxdwlrq lvw1 Khqfh/ Y_l³ fruuhvsrqgv wr tlh^w=

Ghqrwh wkh hodvwlflw| ri frqvxpswlrq lq rswlpxp e|ho{Z@ ^{C+z.Y ,}_C{ _z.Y^{ dqg wkh hodvwlflw| ri lpsdwlhqfh e|ho{@ _C{^C^{1 Pruhryhu/ ohw+{,eh d phdvxuh iru wkh fxuydwxuh ri wkh zhoiduh ri lqvwdqwdqhrxv frqvxpswlrq/ vlplodu wr wkh lqwhusuhwdwlrq ri_{/ exw dssolhg wr wkh uhfxuvlyh zhoiduh ixqfwlrqZ+w{,lqvwhdg ri x+{w,1 Dv lq wudglwlrqdo zhoiduh ixqfwlrq/ +{,pd| h{suhvv wkh suhihuhqfhv iru lqwhuwhpsrudo htxlw|1 Wkhq/ wkh lqwhuwhpsrudo rswlpxp frqglwlrqv fdq eh zulwwhq dv=

₄@+{,{b

{.+{,+4 . ho{

ho{Z, +4;,

(19)

FLFHUR Zrunlqj Sdshu 5334=4 4;

.i_V³+j³_|, @ 4_||b

| +4<,

Zh qrwh/ uvw/ wkdw wkh frqglwlrqv uhodwhg wr wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv dqg dedwhphqw duh h{dfwo| wkh vdph dv lq wkh wudglwlrqdo prgho1 Khqfh/

folpdwh srolf| lv dhfwhg e| wkh lqwurgxfwlrq ri uhfxuvlyh zhoiduh rqo| lqgluhfwo|

e| wkh qhz frqglwlrqv iru wkh frqvxpswlrq0vdylqjv ghflvlrq1 Wkh vwdwlf frq0 glwlrq +49, frqwulexwhv wr orzhu wkh vkdgrz sulfh/ vlqfh ³_{ A 31 Wklv fdq eh frqvlghuhg dv d uhgxfwlrq lq wkh vxsso| ri vdylqjv/ dqg ohdgv/ lq lvrodwlrq/ wr d orzhu jurzwk udwh dqg kljkhu lqlwldo frqvxpswlrq1 Wkh uhdvrq lv vlpso| wkdw wkh pdujlqdo zhoiduh ri lqfuhdvhg frqvxpswlrq lv uhgxfhg dv d uhvxow ri d kljkhu udwh ri lpsdwlhqfh1

Wkh lqwhuwhpsrudo frqglwlrq lv fkdqjhg wr uh hfw wkh uhodwlyh hodvwlflwlhv ri lpsdwlhqfh dqg zhoiduh1 Vlqfh ³_{ A 3/ wkhuh duh wzr uhdvrqv zk| hqgrjhqrxv lpsdwlhqfh frqwulexwhv wr lqfuhdvlqj wkh glvfrxqw udwh1 Wklv lv sduwo| ehfdxvh lpsdwlhqfh lqfuhdvhv e| lqfuhdvlqj frqvxpswlrq1 Lq dgglwlrq/ zkhq lpsdwlhqfh lqfuhdvhv/ wkh lpsruwdqfh ri lpsdwlhqfh dovr lqfuhdvhv/ zklfk lv zk| wkh odvw whup lq wkh sduhqwkhvlv lv lqfoxghg1 Wklv odwwhu hhfw ghfolqhv ryhu wlph/ krzhyhu/

ehfdxvh zh kdyh dvvxphg wkdw³³_{{@³_{A z³³_{{@z_{³=

Dv glvfxvvhg e| Fkdqj ^8`/ wklv fkdqjh lq wkh udwh ri lpsdwlhqfh fdq eh looxvwudwhg e| d edfnzdug vorslqj lqwhuvhfwlrq srlqw iru i_n³ dqg +{,lq Iljxuh 41 Li zh xvh wkh suhvhqw gd| udwh ri lpsdwlhqfh dv d uhihuhqfh iru frpsdulqj orqj0whup vroxwlrqv ri wudglwlrqdo dqg uhfxuvlyh zhoiduh ixqfwlrqv/ wklv zrxog ohdg wr d vwhdg| vwdwh vroxwlrq dw d orzhu ohyho ri surgxfwlrq dqg frqvxpswlrq1 Dv qrwhg/ wkh hhfw rq folpdwh srolf| zrxog iroorz lqgluhfwo| iurp wklv qhz vlwxdwlrq1 Iru wkh vwhdg| vwdwh vroxwlrq/ zh fdq vhh wkh hhfw e| frqvlghulqj jxuh 61 Uhfxuvlyh zhoiduh phdqv wkdw wkh gdun sodqh vorshv edfnzdugv/ sxvklqj srlqw D *lqzdugv* dorqj wkh grwwhg fxuyh wkdw uhsuhvhqwv wkh lqwhuvhfwlrq ri wkh wzr sodqhv1 Zlwkrxw d irupdo surri/ zh vhh wkdw wklv ohdgv wr orzhu ohyho ri fdslwdo/ dqg orzhu frqfhqwudwlrqv1 Wkh vwhdg|0vwdwh ohyho ri dedwhphqw zloo eh gudzq xszdugv dv wkh fdslwdo vwrfn lv uhgxfhg/ exw grzqzdugv ehfdxvh wkh frqfhqwudwlrqv duh orzhu1 Zklfk hhfw lv wkh vwurqjhvw fdqqrw eh frqfoxghg zlwkrxw ixuwkhu vshflfdwlrqv ri wkh prgho1

7 Frqfoxvlrqv

Hfrqrplvwv kdyh kdg wurxeoh h{sodlqlqj wr rwkhu shrsoh zk| ehqhwv ri folpdwh phdvxuhv rxjkw wr eh glvfrxqwhg ehfdxvh lw vhhpv fohdu wr dq|rqh wkdw glvfrxqw0 lqj zloo pdnh wkh exon ri ehqhwv iurp folpdwh srolflhv glvdsshdu1 Ihz vhhp wr eh frqylqfhg e| wkh xvxdo dvvhuwlrq wkdw rqh vkrxog frpsduh wkh uhwxuq ri dq| lqyhvwphqw zlwk wkh uhwxuq ri wkh ehvw dydlodeoh dowhuqdwlyh1 Wklv lv zk| lw lv xvhixo wr jr pruh ghhso| lqwr wkh ghwhuplqdqwv ri wkh glvfrxqw udwh/ dqg dvn zkhwkhu wkhuh lv vrphwklqj zlwk folpdwh fkdqjh wkdw pd| vsxu d uhrulhqwdwlrq zlwk uhvshfw wr krz glvfrxqwlqj ehqhwv ri folpdwh phdvxuhv lv h{sodlqhg1

Wkh pdlq dqvzhu wr wklv odvw txhvwlrq lv |hv/ dqg wklv kdv ehhq nqrzq iru txlwh vrph wlph +vhh h1j1 Kddyhopr ^44`,1 Folpdwh phdvxuhv +dv zhoo dv pdq|

rwkhu hqylurqphqwdo srolf| phdvxuhv, dlp dw frqwuroolqj wkh ghyhorsphqw ri d vwrfn/ wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv1 Prvw rwkhu lqyhvwphqwv dlp dw frqwuroolqj d glhuhqw vwrfn/ wkh vwrfn ri surgxfwlyh fdslwdo1 Wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv ghyhors dw d glhuhqw vshhg wkdq wkh fdslwdo vwrfn1 Dv d

(20)

FLFHUR Zrunlqj Sdshu 5334=4 4<

uhvxow/ dovr wkh ydoxh ri wkh frqfhqwudwlrqv hyroyhv glhuhqwo| iurp wkh ydoxh ri uhdo fdslwdo1 Wklv lpsolhv/ wdnlqj lqwr frqvlghudwlrq krz glvfrxqw udwhv duh ghqhg/ wkdw wkh glvfrxqw udwh iru ehqhwv ri folpdwh phdvxuhv glhuv/ lq jhqhudo/

iurp wkh glvfrxqw udwh iru rwkhu lqyhvwphqwv1

Wkh pdmru remhfwlyh ri glvfrxqwlqj lv wr surylgh wkh uljkw lqfhqwlyhv iru vwhhu0 lqj wkh frqfhqwudwlrqv ri juhhqkrxvh jdvhv dqg wkh vwrfn ri uhdo fdslwdo lq vxfk d zd| wkdw wkh wrwdo zhoiduh lv pd{lpl}hg1 Lq vxfk d shuvshfwlyh/ wkh glhuhqw udwhv ri glvfrxqw iru lqyhvwphqwv lq uhdo fdslwdo dqg dedwhphqw ri juhhqkrxvh jdvhv fdq eh h{sodlqhg e| krz frqvxpswlrq dqg dedwhphqw dhfw wrwdo zhoiduh/

gluhfwo| ru lqgluhfwo|1 Wkh gluhfw zhoiduh jdlq iurp frqvxpswlrq jurzwk fdq eh fkdudfwhul}hg e| wkh fxuydwxuh ri wkh zhoiduh ixqfwlrq1 Wklv uh hfwv shrsohv*

hydoxdwlrq ri suhvhqw frqvxpswlrq yhuvxv ixwxuh zhdowk/ dqg lv d zhoo0hvwdeolvkhg uhvxow lq hfrqrplfv1 Vlploduo|/ wkh lqgluhfw zhoiduh jdlq iurp ulvlqj dedwhphqw lv fkdudfwhul}hg e| wkh *surgxfwlylw|* ri dedwhphqw frvwv/ wkdw lv/ krz pxfk hplvvlrqv duh fxw e| dedwhphqw dw glhuhqw vwdjhv ri wkh dedwhphqw ohyho1 Wkh glhuhqw glvfrxqw udwhv fdq wkhuhiruh eh h{sodlqhg e| wkh idfw wkdw wkh fxuyd0 wxuhv ri wkh zhoiduh ixqfwlrq dqg wkh dedwhphqw frvw ixqfwlrq duh glhuhqw1 Sxw lq d vlpsohu zd|/ dedwhphqw dqg rwkhu lqyhvwphqwv duh glvfrxqwhg glhuhqwo|

ehfdxvh lw lv dgydqwdjhrxv wr iroorz glhuhqw wlph surohv lq rughu wr uhdfk wkh ryhudoo jrdo ri pd{lpxp wrwdo zhoiduh ryhu wlph1

Glhuhqw glvfrxqw udwhv pdwwhu/ wkhuhiruh/ rqo| wr wkh h{whqw wkdw frqvxps0 wlrq dqg dedwhphqw duh dfwxdoo| lqfuhdvlqj +ru fkdqjlqj,1 Wkh ohyho ri wkh glvfrxqw udwhv/ dqg wkhuhe| wkh ohyho ri frqvxpswlrq dqg dedwhphqw ryhu wlph/

ghshqgv rq krz shrsoh hydoxdwh wkh dgydqwdjh ri kdylqj d jlyhq dprxqw frq0 vxphu jrrgv wrgd| frpsduhg zlwk kdylqj wkhq surylghg vrph wlph lq wkh ixwxuh1 Wkh dvvhvvphqw ri d ohyho iru dedwhphqw dqg frqvxpswlrq fdq eh dgguhvvhg e|

wkh vwdwlrqdu| vwdwh vroxwlrq1 Wkhq/ lw wxuqv rxw wkdw wkh lqwxlwlyh uxoh iru dedwhphqw dssolhv/ qdpho| wkdw wkh pdujlqdo frvw ri dedwhphqw vkrxog eh uh0 odwhg gluhfwo| wr wkh pdujlqdo ehqhwv lq whupv ri kljkhu surgxfwlylw|1 Krzhyhu/

wkh sdvvlqj ri wlph dhfwv wkh ydoxh ri wkhvh ehqhwv iru wzr uhdvrqv1 Iluvw/ qd0 wxuh lwvhoi uhgxfhv wkh frqfhqwudwlrqv e| d fhuwdlq udwh// dqg wkh zhoiduh ri wkh surgxfwlylw| |lhog lv uhgxfhg e| wkh udwh ri lpsdwlhqfh li zh frpsduh zlwk wkh wlph dw zklfk dedwhphqw kdv wr wdnh sodfh1 Wklv h{sodlqv wkh dgmxvwhg rswlpxp fulwhulrq iru dedwhphqw lq vwhdg| vwdwh/ vkrzq lq htxdwlrq +43,1 Wkhuhiruh/ d kljk glvfrxqw udwh grhv qrw qhfhvvdulo| phdq wkdw ixwxuh ehqhwv duh qhjohfwhg1 Lq idfw/ lq prvw fdvhv lw zloo qrw uh hfw vxfk ylhz/ exw udwkhu d frqvlghudwlrq ri zkhq pdmru dedwhphqw vkrxog eh lqlwldwhg1

Rq wkh rwkhu kdqg/ glvfrxqwlqj lv frqwuryhuvldo iurp dq hwklfdo srlqw ri ylhz ehfdxvh lw uh hfwv dq dffhswdqfh ri wkh dvvxpswlrq wkdw jhqhudwlrqv vkrxog qrw eh wuhdwhg htxdoo|1 Dv zh kdyh vhhq/ wkhuh lv qr hwklfdo mxvwlfdwlrq iru wklv/

qrw hyhq li zh lqvlvw wkdw zhoiduh ixqfwlrqv vkrxog eh edvhg rq suhihuhqfhv ri lpsdwlhqw shrsoh1 Wkh dvvxpswlrq ri lpsdwlhqfh lv pruh d txhvwlrq ri whfk0 qlfdo frqyhqlhqfh1 Rqh pd|/ krzhyhu/ txhvwlrq wkh vwdqgdug dvvxpswlrq ri d frqvwdqw udwh ri lpsdwlhqfh/ hvshfldoo| vlqfh fhuwdlq wlph0vshflf surshuwlhv ri suhihuhqfhv/ lq sduwlfxodu kdelw irupdwlrq/ duh glvuhjdughg1 Wkh lqfoxvlrq ri dq hqgrjhqrxv udwh ri lpsdwlhqfh/ uh hfwlqj kdelw irupdwlrq/ fkdqjhv rswlpdo srolf| voljkwo|1 Wkh vwdwlrqdu| ohyho wr zklfk wkh vwrfn ri uhdo fdslwdo dqg frq0 fhqwudwlrqv ri juhhqkrxvh jdvhv ulvhv jrhv grzq/ exw zh fdqqrw vd| iru vxuh zkhwkhu wkh ohyho ri dedwhphqw lqfuhdvhv ru ghfuhdvhv1

Lw vkrxog eh dgghg/ qdoo|/ wkdw wr lqfoxgh kdelw irupdwlrq pd| uh hfw suhi0

(21)

FLFHUR Zrunlqj Sdshu 5334=4 53 huhqfhv lq d pruh uhdolvwlf zd|/ exw lw grhv qrw vroyh wkh hwklfdo sureohp ri lqwhujhqhudwlrqdo htxlw|1 Udwkhu/ wkh rssrvlwh lv wkh fdvh1 Kdelw irupdwlrq lpsolhv dq dffhswdqfh ri wkh fodlp wkdw shrsoh zkr duh ulfk kdyh wkh uljkw wr uhpdlq vr/ d vr0fdoohg judqgidwkhulqj sulqflsoh1 Wklv vroxwlrq ri wkh sureohp ri exughq vkdulqj fdq dovr eh wudfhg lq wkh N|rwr Surwrfro/ zkhuh frxqwulhv duh frpplwwhg wr uhgxflqj hplvvlrqv lq sursruwlrq wr suhvhqw ohyhov1 Vwloo/ wkhuh duh uhdvrqv wr uhtxhvw ixuwkhu ghyhorsphqw lq wklv duhd1

(22)

D Rswlpdo srolf| zlwk uhfxuvlyh zhoiduh

D ixqfwlrq wkdw lv frqvlvwhqw zlwk Nrrspdq*v ^48` ghqlwlrq ri lqwhuwhpsrudo xwlolw| zdv ghyhorshg e| X}dzd ^56`/ dqg fdq eh zulwwhq lq wkh irup/

Z+w{, @] ₄

w z+{,h^U^w^+{^v^,gvg> +53, zkhuh w{olpw$3+{w>{w.w> {w.5w> ===,lv wkh frqvxpswlrq surjudpph vwduw0 lqj iurp w1 Wr vlpsoli| qrwdwlrq ehorz/ ghqh wkh glvfrxqw idfwru

G+{>w> , @h^U^w^+{^v^,gv +54, Wkh glvfrxqw idfwru fdq eh vhsdudwhg lqwr wlph lqwhuydov/

G+{>w> , @hÛ^w^w.w^+{^v^,gvÛ^w.w ^+{^v^,gv@hÛ^w^w.w^+{^v^,gvhÛ^w.w ^+{^v^,gv= Khqfh/ G+{>w> , @G+{>w> w. w,G+{>w. w> ,1 Qrwh wkdwG+{>w> w. w, @ G+{w,zkhqw$3=Zh pd| wkhuhiruh zulwh wkh zhoiduh ixqfwlrq dv wkh uhfxu0 vlrq

w$3olp Z+w{, @z+{w, .G+{w,Z+w.w{,= +55, Qrz/ frqvlghu wkh rswlpl}dwlrq sureohp lq vhfwlrq 6=

pd{_{>| Z+w{,>

v1w1

gnw @ ^i+nw> Vw,{w|w`gw>

gVw @ ^Vw.i+nw> Vw,j+|w,`gw=

Iluvw/ zh irup wkh ydoxh ixqfwlrq Y+n> V> w, @ pd{_{>| i] _W

w z+{v,G+{v>w> W,gvj>

Zh dvvxph wkdw Y+n>V> w, lv frqwlqxrxv dqg wzlfh glhuhqwldeoh ryhu dq| dg0 plvvleoh sdwk1 Ehfdxvh ri wkh vhsdudelolw| surshuwlhv dvvxphg deryh/ zh fdq ghyhors wkh ydoxh ixqfwlrqY+n> V> w,wr

pd{_{>| i] _w.w

w z+{v,G+{v>w> w. w,gv.] _W

w.wz+{v,G+{v> 3> w. w,G+{v>w. w> W,gvj pd{_{>| i

] _w.w

w z+{v,G+{v>w> w. w,gv.G+{w>w> w. w, ] _W

w.wz+{v,G+{v>w. w> W,gvj pd{{>| i] _w

3 z+{v,G+{v>w> w. w,gv.G+{w>w> w. w,Y+n. n> V. V> w. w,j> +56, zklfk lv wkh ixqgdphqwdo uhfxuuhqfh uhodwlrq1 Vxewudfwlqj Y+n> V=w, rq erwk

vlghv/ dqg lqvhuwlqj d *qhxwudo* hohphqwY+n. n> V. V> w. w,jlyhv 3 @ pd{

{>| i] _w

3 z+{v,G+{3> 3>w,gv.G+{3> 3>w,Y+n. n> V. V> w. w, Y+n. n> V. V> w. w, .Y+n. n> V. V> w. w,Y+n> V> w,j=