12 Money in representative agent models

45
Prqh| lq uhsuhvhqwdwlyh djhqw
prghov
Zkdw lv prqh|B Wklv vhhpv dq rgg txhvwlrq wr dvn1 Fohduo| lw lv wkh
sk|vlfdo lwhp groodu eloo/ ehdgv/ jrog frlqv xvhg wr sd| iru jrrgv
lq rughu wr dyrlg wkh cgrxeoh frlqflghqfh ri zdqwv* uhtxluhg lq eduwhu
h{fkdqjh1 Wkh fodvvlf wkhru| ri lwv hphujhqfh zlwkrxw vrfldo clqyhqwlrq*
lv gxh wr Phqjhu +4;<5, zkr ghyhorsv dq hyroxwlrqdu| prgho1 Lq wklv d
jrrg zlwk dssursuldwh fkdudfwhulvwlfv +wudqvsruwdelolw|/ zlgh xvh hwf1, lv
uhtxhvwhg lq h{fkdqjh iru jrrgv e| vhoohuv vhfxuh lq wkh nqrzohgjh wkdw
vrphrqh hovh zloo zdqw lw lq h{fkdqjh iru wkh jrrgv wkh| pd| zlvk wr ex|
vhh Zklwh +4<<<,1 Lq ixooolqj wklv ixqfwlrq prqh| zloo qrw sd| lqwhuhvw
ehfdxvh ri wkh lqfrqyhqlhqfh lqyroyhg> prqh| kroglqjv zrxog kdyh wr
eh gdwhg rq wkhlu idfh wr sd| lqwhuhvw/ olnh ehduhu erqgv1 Exw jlyhq
wkh iuhtxhqf| zlwk zklfk prqh| fkdqjhv kdqgv/ iru hdfk ehduhu wr jhw
lqwhuhvw gxh ri d ihz shqqlhv zrxog lqyroyh reylrxvo| eljjhu wudqvdfwlrqv
frvwv1 +Puv1 Jurfhu kdv wr jlyh |rx lqwhuhvw rq |rxu srxqg qrwh zkhq
|rx sd| iru |rxu euhdg/ diwhu doorzlqj iru zkhq |rx dftxluhg lw dq
xqkdss| odg|1,
Rwkhu wkhrulhv ri krz lw hphujhg duh ri lqwhuhvw exw dsshdu lq wkh
hqg qrw wr grplqdwh Phqjhu*v lqvljkw1 Zh glvfxvv Zdoodfh*v +4<;3,
ryhuodsslqj0jhqhudwlrq/ ROJ/ prgho ehorz> lq lw prqh| kdv d vwruh0
ri0ydoxh uroh lq wkh devhqfh ri erqgv1 Dv d vwruh ri ydoxh prqh| kdv wkh
vhulrxv sureohp wkdw lw grhv qrw sd| h{solflw lqwhuhvw1 Hyhq lq sulpl0
wlyh vrflhwlhv zlwkrxw sdshu erqgv wkhuh duh zd|v ri vwrulqj ydoxh wkdw
|lhog h{shfwhg uhwxuq fdwwoh/ lqyhqwrulhv/ furs fuhglw duudqjhphqwv
hwf11 Wkhuh lv dq dqdorjrxv sureohp zlwk prqh| dv d vwruh ri ydoxh lq
wkh Ehzoh|Wrzqvhqg wxuqslnh prgho1
Wkh vhdufk0wkhruhwlf prghov ri Nl|rwdnl dqg Zuljkw +4<;</ 4<<4/
4<<6, duh ri juhdw lqwhuhvw dqg hvvhqwldoo| d frqwulexwlrq wr Phqjhuldq
wkhru|1 Wkh| ghulyh wkh rswlpdo vhdufk vwudwhj| ri d uhsuhvhqwdwlyh djhqw
639
Prqh| lq Uhsuhvhqwdwlyh Djhqw Prghov
63:
zlwklq dq hqylurqphqw zkhuh wkhuh duh d ydulhw| ri jrrgv zklfk pd|
ixqfwlrq dv prqh|/ dw fxuuhqf| ehlqj rqh exw zlwk qr lqwulqvlf ydoxh/
wkh rwkhuv +frpprglw| prqh|v, kdylqj lqwulqvlf ydoxh1 Wkh| orrn iru
Qdvk htxloleuld zlwk glhuhqw prqh|v1 Ri frxuvh wkh sureohp zlwk wkhvh
htxloleuld lv krz rqh udwkhu wkdq dqrwkhu lv vhohfwhg1 Phqjhu*v hyrox0
wlrqdu| prgho lpsolflwo| dvvxphv lq dgglwlrq vrph vruw ri dgdswlyhqhvv>
shrsoh cjudylwdwh* wrzdugv wkh lqlwldoo| prvw surplvlqj frpprglw| prqh|
dqg odwhu dv wkh dwwudfwlrqv ri dw prqh| ehfrph hylghqw +ru wkh lqwhu0
hvwv ri d prqrsro| fxuuhqf| surylghu/ pruh olnho|/ ehfrph grplqdqw,
wkh| cmxps* wr d dw fxuuhqf| htxloleulxp1 Lq wkhlu 4<<6 sdshu Nl|rwdnl
dqg Zuljkw vkrz irupdoo| wkdw dw fxuuhqf| dgrswlrq lv rq d nqlih0hgjh>
li prvw shrsoh ghflgh wr dgrsw lw hyhu|rqh zloo/ exw li ihz wklqn ri grlqj
vr/ qrrqh zloo1 Vhojlq +4<<:, vkrzv wkdw d vsrqwdqhrxv pryh wr dw fxu0
uhqf| lv xqolnho| li wkhuh lv dgdswlyh ohduqlqj> udwkhu/ shrsoh duh olnho|
wr dgrsw frpprglw| prqh| uvw dv wkh forvhvw dowhuqdwlyh wr eduwhu dqg
wkhq vwd| zlwk lw1 Wkhvh prghov fdq eh wkrxjkw ri dv wkh ehjlqqlqjv ri
d irupdol}dwlrq ri Phqjhu*v lghdv1
Vxssrvh zh dffhsw wkh hyroxwlrqdu| wkhru| ri prqh|*v hphujhqfh1
Wkhq e| wkh vdph hyroxwlrqdu| sulqflsohv zh zrxog h{shfw prqh|*v ixqf0
wlrqv lq wlph wr eh shuiruphg e| qhz phdqv1 Phdqv ri sd|phqw hphujh
wkdw xvh prqh| dv d xqlw ri dffrxqw exw iru prvw ri wkh wlph dyrlg xvlqj
prqh| dv d phdqv ri sd|phqw fohdulqj v|vwhpv/ edqn dffrxqwv/ gh0
srvlw dffrxqwv zlwk qrqedqn lqwhuphgldulhv/ hprqh| yrxfkhuv dqg vr
rq1 Wkh ruljlqdo sk|vlfdo prqh| lv wkh cedvh* ri wklv v|vwhp ri cfuhglw
sd|phqw* +vr0fdoohg ehfdxvh |rx sd| zlwk dq lqwhuhvw0ehdulqj edodqfh,
ehvlghv ghqlqj wkh xqlw +vhh Zklwh +4<<<, djdlq iru d frpsdfw klvwru|
ri lqwhuphgldwlrq dorqj wkhvh olqhv,1
Dv wklv hyroxwlrq surfhhgv zh vkrxog revhuyh wkdw frpshwlwlrq eh0
wzhhq lqwhuphgldulhv gulyhv wkh uhzdugv wr wkhvh ixqfwlrqv ri prqh| grzq
wr wkh frvw ri surylvlrq iru hdfk1 Vr hhfwlqj sd|phqw yld d fohdulqj v|v0
whp zloo frppdqg d frpshwlwlyh ihh/ zkloh kroglqj d ghsrvlw zloo frp0
pdqg wkh udwh ri lqwhuhvw= wkhuh lv cxqexqgolqj* zkhuhe| zh fdq frqvlghu
ghsrvlwv dv khog iru wkhlu uhwxuqv dv vwruhv ri ydoxh dqg sd|phqw dv d vhs0
dudwh vhuylfh grqh iru d ihh1 Fdvk dorqh/ ehlqj surylghg e| d prqrsro|
vxssolhu +wkh jryhuqphqw w|slfdoo|,/ zloo qrw eh vxemhfw wr frpshwlwlrq
e| dvvxpswlrq zlwklq lwv cgrpdlq*> krzhyhu/ hyhq khuh hyroxwlrq pd| eh
surgxflqj frpshwlwlrq dv lw ehfrphv hdvlhu +yld wkh lqwhuqhw iru h{dpsoh,
wr xvh dowhuqdwlyh xqlwv ri dffrxqw zlwk jhqhudo dffhswdelolw|1
Khqfh wr lqwurgxfh cprqh|* lqwr uhsuhvhqwdwlyh djhqw prghov zh pxvw
dvn zkdw rxu sxusrvh lv= wr prgho d sduwlfxodu hduolhu vwdjh ri prqhwdu|
hyroxwlrq ru wr dwwhpsw wr prgho cprghuq wlphv* +ru lqghhg wkh srvvl0
eoh ixwxuh,B Lq zkdw iroorzv zh lqwurgxfh vhyhudo h{dpsohv ri prqh|
63;
H{whqglqj dqg Ghhshqlqj wkh Prghov
lq wkh hfrqrp|/ vwduwlqj zlwk prghov zklfk zh ylhz dv klvwrulfdo lq lq0
whqwlrq dqg jrlqj rq wr prghov ri wkh khuh0dqg0qrz1 Ilqdoo| zh glvfxvv
zkhuh vrph whqghqflhv lq prqhwdu| srolf| dqg lq srwhqwldo frpshwlwlrq
ehwzhhq prqhwdu| dxwkrulwlhv pljkw ohdg lq pdfurhfrqrplf ehkdylrxu1
Zh ehjlq zlwk prghov zklfk wuhdw prqh| dv d vwruh ri ydoxh +lw pd|
dovr eh d phdqv ri sd|phqw exw vr lv dq| rwkhu dvvhw ru dq| jrrg,1
Uhsuhvhqwdwlyh djhqw prghov idfh d ixqgdphqwdo gl!fxow| lq wklv fdvh=
prqh|/ ghqhg dv d qrq0lqwhuhvw ehdulqj dvvhw/ kdv qr uroh wr sod| xqohvv
rqh lv fuhdwhg iru lw e| dq duwlfldo frqvwudlqw1 Frqvlghu wkh prghov ri
fkdswhu 441 Lq prghov olnh Oxfdv*/ zlwk lqqlwho|0olyhg djhqwv/ wkhuh lv
qrwklqj wr vwrs wudghv lq iuxlw +glhuhqw vruwv suhvxpdeo|, ehlqj sdlg iru
e| dq lqwhuhvw0ehdulqj fodlp rq iuxlw0lq0jhqhudo1 Lq wkh Ehzoh|0Wrzqvhqg
prgho/ zlwk sdudooho frppxqlwlhv/ jryhuqphqw erqgv fdq eh xvhg lq
sd|phqw1 Lq wkh ROJ prghov/ djdlq jryhuqphqw erqgv fdq eh xvhg1 Vr
prqh| lv lqihulru wr lqwhuhvw0ehdulqj fodlpv dqg zloo kdyh qr ydoxh1
Dffruglqjo|/ wkhvh prghov kdyh d sureohp1 Wr uhvroyh lw/ wkh Ehzoh|0
Wrzqvhqg prgho dvvxphv dzd| jryhuqphqw erqgv/ ohdylqj prqh| dv
wkh rqo| dvvhw +srvvleo| d jryhuqphqw oldelolw|,1 Zh duh suhvxpdeo| wr
dvvxph wkdw wklv lv d yhu| sulplwlyh hfrqrp| lq zklfk wkhuh lv qr ohjdo
ru rwkhu lqiudvwuxfwxuh wr vxssruw ohqglqj dqg eruurzlqj1
ROJ prghov dwwhpsw dq rswlpl}lqj0djhqw h{sodqdwlrq lq wkh suhv0
hqfh ri d ixoo phqx ri dvvhwv1 Wkh| dffhsw wkdw prqh| pxvw rhu d |lhog
htxdo wr wkdw ri erqgv/ ru lw fdqqrw eh khog +zloo ehfrph ydoxhohvv/ zlwk
wkh csulfh ohyho* lq whupv ri lw ehfrplqj lqqlwh,1 Wkhq htxloleuld pxvw
lqyroyh d udwh ri lq dwlrq +ru htxlydohqw prqh| fuhdwlrq dqg glvwulexwlrq
wr h{lvwlqj kroghuv, htxdo wr wkh udwh ri lqwhuhvw1
Lw lv qrw reylrxv zkdw dssolfdwlrq wkhvh prghov kdyh1 Wkh ROJ
dvvxpswlrq +zlwk qr ehtxhvwv, euhdnv xs lqwhu0jhqhudwlrqdo olqnv dqg
rshqv xs d uroh iru jryhuqphqw dv dq lqwhu0jhqhudwlrqdo lqwhuphgldu|/ dv
zh vdz lq fkdswhu 44 zlwk jryhuqphqw erqgv1 Lw lv iudqno| gl!fxow wr vhh
zk| d jryhuqphqw zrxog sulqw prqh| lq vxfk d zruog zkhq erqgv fdq eh
lvvxhg1 +D frpshwlwlyh edqnlqj v|vwhp sd|lqj lqwhuhvw rq prqh| zrxog
eh d glhuhqw pdwwhu1, Krzhyhu/ d qxpehu ri wkhvh prghov kdyh judiwhg
rq wr wkh ROJ frqvwudlqw dgglwlrqdo frqvwudlqwv= wkdw rqo| prqh| +qr
erqg, h{lvwv/ dqg wkdw wkhuh duh uhjxodwru| uhtxluhphqwv wr xvh lw +h1j1 wr
sd| wd{hv,1 +Lqflghqwdoo|/ wklv uhjxodwru| wkhru| ri prqh|/ suhvxpdeo|
prwlydwhg e| wkh jryhuqphqw*v ghvluh iru uhyhqxh/ lv wkh qhduhvw dq| ri
wkhvh prghov jhwv wr dq rswlpl}lqj wkhru| ri zk| prqh| h{lvwv1, Wkhvh
frqvwudlqwv sodfh vxfk ROJ prghov dorqjvlgh wkh Ehzoh|0Wrzqvhqg rqh
dv h{dplqdwlrqv ri hfrqrplhv dw d udwkhu hduo| vwdjh ri prqhwdu| hyrox0
wlrq1
Prqh| lq Uhsuhvhqwdwlyh Djhqw Prghov
63<
LQWURGXFLQJ PRQH\ LQWR EHZOH\WRZQVH0
QG DQG ROJ PRGHOV
Lq fkdswhu 44 zh vkrzhg wkdw erwk wkh Ehzoh|0Wrzqvhqg dqg ROJ
prghov jdyh jryhuqphqw erqgv d uroh lq lqwhuphgldwlqj ehwzhhq jurxsv
zklfk zrxog qrw ohqg wr hdfk rwkhu1 Krzhyhu/ jryhuqphqw erqgv duh d
uhodwlyho| vrsklvwlfdwhg lqvwuxphqw/ frplqj odwh lq hfrqrplf hyroxwlrq1
Prqh| fdph uvw/ dv d jryhuqphqw0edfnhg phglxp ri h{fkdqjh dqg vwruh
ri ydoxh1 Lw lv lqwhuhvwlqj wr dvn krz wkhvh hfrqrplhv zrxog ehkdyh li
wkhuh zhuh rqo| prqh| wr shuirup wklv lqwhuphgldu| ixqfwlrq1
Li prqh| lv wkh rqo| dydlodeoh vwruh ri ydoxh/ wkhq wr dfklhyh wkh
rswlpdo frqvxpswlrq0vprrwklqj +wkdw lv/ lqwhuphgldwlrq, wkdw erqgv
dfklhyhg +lq fkdswhu 44, zh uhtxluh sulfhv wr eh idoolqj dw wkh udwh ri
wlph suhihuhqfh1
Frqvlghu qrz wkh Ehzoh|0Wrzqvhqg prgho1 Frqvxphuv zlwk W 0
shulrg olyhv pd{lpl}h
W
[
M@
! x+fw , kw +sw fw . pkw sw |w pkw4 ,
+4,
w@3
E
D
zlwk pD
4 @ 3> p4 @ p jlyhq> D djhqwv uhfhlyh | @ % +rgg shulrgv,/
E
| % +hyhq,> E djhqwv uhfhlyh | @ | % +rgg,/ % +hyhq,1 Wkh uvw0rughu
frqglwlrqv duh=
3@
M
@ ! x! +fw , kw sw
fw
+5,
M
@ kw . kw.4 @ 3 iru pkw A 3
pkw
3 iru pkw @ 3
3@
+6,
M
@ kw @ 3 iru pkW A 3
pkw
3 iru pkW @ 3
+7,
Wkh odvw/ whuplqdo/ frqglwlrq lpsolhv wkdw iru prqh| wr eh khog lq
E
W +dv lw pxvw eh iru pdunhw fohdulqj/ l1h1 QD pD
W . QE pW @ P,/ wkhq
W !
W !
x +fW ,@sW @ 31 Exw zlwk x +fW , A 3 +vlqfh fW ? 4/ wkhuh fdq eh
qr vdwxudwlrq ri zdqwv,/ sW pxvw eh lqqlwh= prqh| zloo doo eh vshqw lq
wkh odvw shulrg xqwlo lw lv zruwkohvv1 Zrunlqj edfn wr W 4/ vlqfh kW @ 3/
643
H{whqglqj dqg Ghhshqlqj wkh Prghov
vr zloo kW 4 @ 3/ dqg prqh| zloo ehfrph zruwkohvv wkurxjkrxw1 Vr wr
kdyh d prqhwdu| hfrqrp| zh kdyh wr ohw W $ 4 1 Djhqwv olyh iruhyhu
+d sdudeoh iru wkh krxvhkrog zklfk wuhdwv wkh lqwhuhvwv ri lwv ghvfhqgdqwv
dv lwv rzq/ vxemhfw wr wkh frqvwdqw udwh ri wlph suhihuhqfh/ ,1
Lq wklv fdvh/ wkh whuplqdo frqglwlrq ehfrphv iru pkW A 3/
olpW " +kW , @ 3
Li sulfhv duh idoolqj dw wkh udwh 4 / zh nqrz wkdw doo djhqwv zloo
vprrwk wkhlu frqvxpswlrq rswlpdoo| +dv lq fkdswhu 44,> frqvxpswlrq
zloo eh frqvwdqw1 Zklfkhyhu djhqwv duh kroglqj prqh| dw wlph w kdyh
x! +fw ,
x! +fw.4 ,
@
sw
sw.4
+8,
Exw D dqg E djhqwv krog prqh| lq dowhuqdwh shulrgv +dv zh vdz zlwk
erqgv lq fkdswhu 44,1 Khqfh +8, krogv lq dowhuqdwh shulrgv iru D dqg
E djhqwv= vr lw dozd|v krogv iru rqh vhw ri djhqwv1 Wkhuhiruh vlqfh fw
@ 1 Xqiruwxqdwho|/ wklv ylrodwhv wkh whuplqdo
dqg x! duh frqvwdqw/ sws.4
w
w
w,
frqglwlrq/ vlqfh kw @ s3 x!w dqg wkhuhiruh olpW " +kW , @ x!+f
9@ 3
s3
+xqohvv s3 @ 4,1
Zkdw wklv phdqv lv wkdw dv wlph jrhv rq wkh suhvhqw ydoxh ri prqh|
kroglqjv +glvfrxqwhg e| wkh udwh ri wlph suhihuhqfh, grhv qrw glplqlvk= vr
shrsoh qg wkh| kdyh vxusoxv prqh| dqg vshqg lw1 Wklv udlvhv sW dqg wkh
sdwk ehiruh lw/ dqg s grhv qrw idoo e| +4, hdfk shulrg1 Wklv vroxwlrq lv
wkhuhiruh qrw dq htxloleulxp1 Lw iroorzv wkdw wkh Sduhwr0rswlpdo vroxwlrq
lq wklv prgho lv lpsrvvleoh li rqo| prqh|/ qrw erqgv/ h{lvwv1
Rqo| htxloleuld zkhuh sulfhv idoo pruh vorzo| wkdq +4, duh srvvleoh1
Rqh vxfk htxloleulxp lv d frqvwdqw sulfh rqh= wklv lv reylrxvo| lqwhuhvwlqj
dv rqh ri wkh ihdwxuhv hqfrxudjlqj wkh hyroxwlrq ri prqh| zrxog eh lwv
vwdelolw| lq sxufkdvlqj srzhu1 Iljxuh 4514 looxvwudwhv krz sulfh vwdelolw|
+rhulqj d }hur udwh ri uhwxuq, dfklhyhv ohvv wkdq shuihfw frqvxpswlrq
vprrwklqj1
Ohw s eh frqvwdqw dw s @ 41 Wkhq D djhqwv iru h{dpsoh zkr jhw % lq
rgg shulrgv zloo frqvxph f lq rgg shulrgv/ f lq hyhq +f ! f ,/ vr
wkdw
x! +f ,
@ 4@
x! +f,
+9,
E djhqwv frqvxph f lq hyhq shulrgv/ f lq rgg +zlwk +9, kroglqj dv iru
D djhqwv,1 f . f @ | iru erwk vhwv ri djhqwv/ vr wkdw f @ | f 1
Vlqfh wkhuh duh wkh vdph qxpehuv +Q @ QD @ QE , ri hdfk/ wklv dovr
vdwlvhv pdunhw fohdulqj1
Prqh| lq Uhsuhvhqwdwlyh Djhqw Prghov
644
&$W
%
\ −ε
&%W
D DXWDUFK\
ε
α
\ −ε
α
E RSWLPXP
F PRQHWDU\
HTXLOLEULXP
ƒ
$
&$W
ε
&%W
$DJHQWVPRYHWRDQLQWHUQDOHTXLOLEULXPDWFJLYHQ]HUR
LQWHUHVWUDWH
OHQG
αE\DFTXLULQJPRQH\%DJHQWVIDLOWRUHDFK
DQLQWHUQDORSWLPXP
Iljxuh 4514= Prqhwdu| htxloleulxp zlwk d vwdwlf sulfh ohyho lq wkh Ehzoh|
Wrzqvhqg prgho
E djhqwv zloo krog hqrxjk prqh| dw wkh hqg ri rgg shulrgv wr ex|
f lq wkh hyhq shulrgv/ exw zloo uxq wklv prqh| edodqfhv grzq wrwdoo| lq
wkh hyhq shulrg1 Wkh uhdvrq lv wkdw lq wkh hyhq shulrg x! +f , A x! +f ,/
vr wkdw wkh| zrxog udwkhu vshqg wkhq wkdq zdlw dqg zloo h{kdxvw wkhlu
prqh| kroglqjv lq wkh dwwhpsw= lq whupv ri wkh uvw rughu frqglwlrqv/
wkh| duh dw d fruqhu vroxwlrq zlwk pkw @ 3 +w hyhq,1 D djhqwv iroorz
wkh vdph sdwwhuq lq wkh dowhuqdwh shulrg1 Wkh uhvxow lv wkdw wkh prqh|
vxsso| lv sdvvhg iurp D wr E hdfk shulrg1
Lw lv srvvleoh wr jhw wr Sduhwr0rswlpdo htxloleulxp li vrph zd| fdq
eh irxqg wr sd| lqwhuhvw rq prqh| zkloh vwloo vdwlvi|lqj wkh whuplqdo
frqglwlrq1 Iru h{dpsoh/ li sulfhv idoo dw 4 shu shulrg dqg wkh prqh|
vxsso| lv uhgxfhg dw wklv udwh dovr e| oxps vxp wd{hv/ wkhq wklv frqglwlrq
zloo eh vdwlvhg1 Exw lw lv ri frxuvh d kljko| duwlfldo vfhqdulr/ kdugo|
vxlwhg wr wkh lghd wkdw wklv lv d sulplwlyh hfrqrp|1
645
H{whqglqj dqg Ghhshqlqj wkh Prghov
PRQH\ LQ WKH ROJ PRGHO
Lq wkh ROJ h{dpsoh ri fkdswhu 44/ zh irxqg wkdw/ li wkh jryhuqphqw
eruurzhg wr qdqfh d wudqvihu wr wkh lqlwldo rog jhqhudwlrq/ dqg fkrvh wklv
wudqvihu zhoo/ lw frxog shuihfwo| vprrwk hdfk jhqhudwlrq*v frqvxpswlrq
sdwwhuq dfurvv lwv |rxwk dqg rog djh1 E| idlolqj hyhu wr sd| r wkh ordq
lw pdlqwdlqv wklv shuihfw vprrwklqj iru hyhu1 Wkh uhdo lqwhuhvw udwh lv
}hur> ehfdxvh wkh hfrqrp| lv qrw jurzlqj/ wklv lv lq wklv h{dpsoh wkh
Sduhwr0rswlpdo rxwfrph1 Lq wklv prgho li wkh hfrqrp| zhuh jurzlqj dw
wkh udwh q/ wkh Sduhwr0rswlpxp zrxog eh zkhuh u @ q1
Wklv lv looxvwudwhg lq jxuh 4515/ zklfk vkrzv wkh }hur0vdylqj htxl0
oleulxp +frqvxpswlrq lv | % lq |rxwk dqg % lq rog djh, zlwk u @ ur dqg
wkh ihdvleoh uhdoorfdwlrq dorqj wkh olqh +4 . q,/ dfklhyhg zkhq u @ q1
Wklv uhdoorfdwlrq lv ihdvleoh ehfdxvh wkh qh{w jhqhudwlrq lv +4 . q, wlphv
wkh fxuuhqw> vr li frqvxpswlrq lv wudqvihuuhg e| hdfk ri wkh |rxqj wr
wkh rog hdfk shulrg/ wkhq hdfk ri wkh rog/ ehlqj ohvv lq qxpehu/ zloo hqmr|
+4 . q,1 E| jlylqj xs frqvxpswlrq wr wkh rog wrgd| wkhq wkh |rxqj
jxdudqwhh dq rog djh lq zklfk wkh| zloo hqmr| +4 . q,1 +Wr dfklhyh wklv
htxloleulxp/ wkh jryhuqphqw pxvw lqfuhdvh lwv ghew dqg vshqglqj hdfk
shulrg dw wkh udwh q,1
F W KW
VORSHLV −3 RSWLPXP
ε
DXWDUFK\
δ3 VORSHLV −U
ƒ
δ
\ε
F W KW
Iljxuh 4515= Ihdvleoh uhdoorfdwlrq ri frqvxpswlrq iurp }hur vdylqjv +dxwdufk|,
wr rswlpdo lqwhujhqhudwlrqdo ohqglqj lq jryhuqphqw erqgv wkh ryhuodsslqj
jhqhudwlrqv prgho zlwk jurzwk/ Z1
Lq wkh h{dpsoh ri fkdswhu 44/ zkhuh q @ 3/ dqg erqgv sdlg }hur
Prqh| lq Uhsuhvhqwdwlyh Djhqw Prghov
646
lqwhuhvw/ zh frxog hdvlo| wklqn ri prqh| shuiruplqj h{dfwo| wkh vdph
ixqfwlrq dv erqgv1 Ohw wkh jryhuqphqw lvvxh fxuuhqf|/ K+w,/ lqvwhdg ri
j
erqgv dqg ohw wkh sulfh ohyho eh s+w,1 Wkhq zh fdq wuhdw K+w,
s+w, @ O +w,
dv htxlydohqw wr wkh suhylrxv erqgv/ dqg wkh htxloleulxp zrxog eh wkh
vdph1 Li fxuuhqf| lvvxh zhuh wr eh nhsw frqvwdqw dw K3 / s+w, zrxog wkhq
eh frqvwdqw dw s3 = s+w.4,
s+w, @ 4/ lpso|lqj d }hur lqwhuhvw udwh rq prqh|/
wkh Sduhwr0rswlpdo htxloleulxp dv ehiruh1
Vr lq wklv fdvh/ prqh| frxog shuirup wkh vdph frqvxpswlrq0vprrwk0
lqj ixqfwlrq dv erqgv/ mxvwli|lqj lwv uroh lq hduo| vrflhwlhv zlwk qr jurzwk
dqg dq xqvrsklvwlfdwhg qdqfldo pdunhw1 Lw zrxog dovr eh srvvleoh iru
dq| jryhuqphqw wr vzds erqgv iru prqh| zlwk qr hhfw rq dq|wklqj
+sulfhv/ frqvxpswlrq/ lqwhuhvw udwhv, lq wkh hfrqrp|= wkhuh lv qr qhhg
iru wkh qhz pl{ ri erqgv dqg prqh| K ! +w, . Oj! +w, wr fkdqjh s+w, vlqfh
K 3 +w,
K+w,
j!
j
s+w, .O +w, @ s+w, .O +w, dqg/ jlyhq wkdw jryhuqphqw oldelollhv duh wkh
vdph ryhudoo/ qrwklqj hovh fkdqjhv hlwkhu1 Wklv lv wkh cuhdo eloov* grfwulqh/
wkdw rshq pdunhw rshudwlrqv dowhulqj wkh vxsso| ri prqh| kdyh qr hhfw1
Pdwwhuv ehfrph pruh frpsolfdwhg lq jurzlqj hfrqrplhv/ zkhuh wkh
rswlpdo u @ q A 31 Khuh lw zloo riwhq eh wkh fdvh wkdw d prqhwdu| htxl0
oleulxp fdqqrw eh rswlpdo1 Hvvhqwldoo|/ wklv lv ehfdxvh lq d prqhwdu|
htxloleulxp sulfhv zrxog kdyh wr eh idoolqj dw wkh udwh q1 \hw wkh jry0
huqphqw pxvw uxq d ghflw lq rughu wr shuplw lqwhuphgldwlrq ehwzhhq
jhqhudwlrqv1 Wr qdqfh wklv lw pxvw lvvxh pruh dqg pruh fxuuhqf|/ zklfk
zloo eh lqfrqvlvwhqw zlwk idoolqj sulfhv1
Krzhyhu/ wkhuh zloo xvxdoo| h{lvw vxerswlpdo prqhwdu| htxloleuld1 D
odujh olwhudwxuh h{lvwv h{sorulqj wkh surshuwlhv ri vxfk htxloleuld= iru dq
lqwurgxfwlrq wr lw vhh Vdujhqw +4<;:,1
Wkh pdlq gl!fxow| zlwk xvlqj ROJ prghov iru wkh vwxg| ri prqhwdu|
hfrqrplhv lv wkhlu hpskdvlv rq prqh| dv d vwruh ri ydoxh= wklv dqg wkh
duelwudjh ehwzhhq prqh| dqg rwkhu qdqfldo dvvhwv surgxfhv vrphzkdw
vwudqjh uhvxowv1 Iru sulplwlyh vwdwlf hfrqrplhv zlwkrxw dowhuqdwlyhv wr
prqh|/ krzhyhu/ ROJ prqhwdu| prghov rhu lqwhuhvwlqj lqvljkwv1
WKH FDVK0LQ0DGYDQFH PRGHO
Wr uhvroyh wkh sureohp ri prqh|*v ydoxh lq d prgho ri d prghuq hfrqrp|/
lw lv vlpso| dvvxphg wkdw prqh| lv qhhghg iru wudqvdfwlrqv1 Prqh| khuh
lv wkh sk|vlfdo xqlw ri fdvk> doo rwkhu dvvhwv dqg oldelolwlhv duh erqgv/
sk|vlfdo dvvhwv/ ru fodlpv wr wkhp +h1j1 htxlwlhv,1 Lw lv dvvxphg wkdw
wr pdnh wudqvdfwlrqv fdvk lv uhtxluhg1 Lq rqh vxfk prgho/ Oxfdv*/ wkhuh
lv d cfdvk0lq0dgydqfh* frqvwudlqw +iroorzlqj Forzhu/ 4<98,/ zkhuhe| lw lv
647
H{whqglqj dqg Ghhshqlqj wkh Prghov
dvvxphg wkdw vshqglqj fdq rqo| eh fduulhg rxw zlwk prqh|/ zklfk pxvw
wkhuhiruh eh dffxpxodwhg lq dgydqfh +Oxfdv/ 4<;3,1
Vlplodu wr wklv/ dqg xvhg lq rwkhu prghov ri wkh vdph jhqhudo w|sh/
lv wkh dvvxpswlrq wkdw prqh| ckdv xwlolw|* dqg lv dq dujxphqw ri wkh
frqvxphu*v xwlolw| ixqfwlrq1 Hhfwlyho|/ wklv lv ehlqj mxvwlhg e| wkh
Forzhu frqvwudlqw/ dqg lv phuho| dq dowhuqdwlyh zd| ri h{suhvvlqj lw/
lpso|lqj d ghjuhh ri vxevwlwxwdelolw| ri prqh| lq wudqvdfwlrqv/ zkhuhdv
wkh fdvk0lq0dgydqfh prgho dvvxphv qrqh1
Li prqh| kdv qr xvh/ lw ehfrphv ydoxhohvv1 Wklv lv reylrxv hqrxjk1
Frqvlghu vrph frzulh vkhoov zklfk duh vwdwhg e| vrph yloodjh fklhiwdlq wr
eh fxuuhqf|= li qr rqh qhhgv wkhp lq rughu wr h{fkdqjh jrrgv +dqg wkh|
duh xvhohvv iru dq| rwkhu sxusrvh,/ wkhq wkhuh duh wzr srvvlelolwlhv1 Rqh lv
wkdw wkh| kdyh d ydoxh lq h{fkdqjh= exw li vr/ wkh| zloo eh vshqw lq rughu
wr rewdlq wkdw ydoxh iurp wkh jrrgv iru zklfk wkh| fdq eh h{fkdqjhg1
Vlqfh hyhu|rqh zloo jhw ulg ri wkhp lq wklv zd|/ wkh| idoo lq ydoxh xqwlo
wkh| duh zruwkohvv wkh rwkhu srvvlelolw|1
Wklv lv zruwk eulh | ghprqvwudwlqj1 Vxssrvh wkh fklhiwdlq +jryhuq0
phqw, lvvxhv fxuuhqf| ri P shu fdslwd dqg zloo vshqg P
s3 @ j3 zkhuh sw @
wkh sulfh ohyho/ j3 @ jryhuqphqw vshqglqj lq shulrg 3= wkh jryhuqphqw
vshqgv qr pruh/ jw @ 3 +w 4,1 Li wkhuh h{lvw rwkhu dvvhwv zlwk d jurvv
uhwxuq Uw / wkhq iru fxuuhqf| wr eh khog lw pxvw kdyh dq htxdo uhwxuq vr
wkdw
sw
tw.4 . gw.4
@ Uw @
+:,
sw.4
tw
zkhuh tw lv wkh sulfh ri wkh dvvhwv +wuhhv, dqg gw wkhlu glylghqg +iuxlw,1
Qrz frqvlghu wkh frqvxphu*v exgjhw frqvwudlqw zkhq wkh pdunhw htxl0
oleulxp krogv=
P
pw.4
tw v .
. gw @ fw . tw v .
+w 4,
+;,
sw
sw
zkhuh v @ wkh qxpehu ri wuhhv shu fdslwd dqg pw.4 @ wkh qxpehu ri
fxuuhqf| xqlwv khog iru shulrg w . 41
pw.4
Vlqfh fw @ gw e| jrrgv pdunhw htxloleulxp/ lw iroorzv wkdw P
sw @ sw >
wkdw lv/ P
sw lv qhyhu vshqw/ exw lw jurzv dw wkh udwh +Uw 4, hdfk shulrg/
vr wkdw lwv suhvhqw glvfrxqwhg ydoxh lv P
sw ri frxuvh1
Lw iroorzv wkdw li wkhuh lv pdunhw htxloleulxp wkh frqvxphu fdq rqo|
vshqg gw lq hyhu| shulrg> dqg |hw li P
sw A 3/ kh zloo idlo wr vshqg doo klv
zhdowk ryhu klv lqqlwh olihwlph> wkdw lv/ kh zloo vshqg doo h{fhsw P
sw +lq
suhvhqw ydoxh,1 Wklv lv vxe0rswlpdo1 Wkh frqvxphu zloo wkhuhiruh dwwhpsw
wr vshqg pruh wkdq gw / zklfk zloo gulyh wkh sulfh ohyho wr lqqlw|1 Vlqfh
4@s4 @ 3/ prqh| zloo qrw eh khog lq shulrg 3/ vr wkdw 4@s3 @ 3/ dqg wkh
jryhuqphqw zloo eh xqdeoh wr vshqg dq|wklqj +j3 @ 3,1
Prqh| lq Uhsuhvhqwdwlyh Djhqw Prghov
648
Lq rughu wr jlyh prqh| ydoxh/ lw pxvw eh jlyhq d vrxufh ri xvhixo0
qhvv1 Dv zh kdyh vhhq/ rqh zd| wr srlqw wr wklv lv wkurxjk wkh Forzhu
frqvwudlqw/ uhylyhg e| Oxfdv +4<;3,1
Lq Oxfdv* prgho hdfk krxvhkrog kdv wr dftxluh prqh| ADEHKD jrlqj
wr ex| jrrgv iru frqvxpswlrq1 Wkhq zkloh rqh phpehu lv vkrsslqj iru
frqvxpswlrq/ wkh rwkhu krxvhkroghu lv vhoolqj wkh krxvhkrog surgxfw iru
prqh|/ zklfk lv wdnhq lqwr wkh qh{w shulrg1 Wkhq wkdw sduw ri lw qrw
qhhghg iru vkrsslqj lv h{fkdqjhg iru lqfrph0|lhoglqj dvvhwv> dqg wkh
zkroh vhtxhqfh lv uhshdwhg/ vwduwlqj djdlq zlwk vkrsslqj2vhoolqj odwhu lq
wkdw shulrg1
w
Ehjlq dw wkh vwduw ri d shulrg/ zkhuh wkh krxvkrog kdv prqh|/ P
sw / dqg
dvvhwv= rqh wuhh/ vw4 @ 4 zkrvh sulfh lv uw +{w ,/ dqg jryhuqphqw qrplqdo
ok +{ ,
ghew w sw w / zkhuh {w lv wkh vwdwh ri wkh hfrqrp| +wkh yhfwru ri fxuuhqw
hqgrjhqrxv dqg h{rjhqrxv yduldeohv,1 Doo txdqwlwlhv duh phdvxuhg shu
fdslwd= hyhu|rqh qrz jrhv lqwr d vhfxulwlhv0wudglqj vhvvlrq zkhuh dvvhwv
duh dftxluhg dqg wd{hv duh sdlg1 Wkh pdunhw0fohdulqj frqglwlrqv iru wklv
duh=
psw . pjw @ Pw.4
+<,
vw @ 4
+43,
k
ow.4
+{w.4 , @ ow.4 +{w.4 ,
+44,
zkhuh Pw.4 dqg ow.4 +{w.4 , duh uhvshfwlyho| wkh prqh| vxsso| dqg jry0
s
+{w.4 , wkh sulydwh
huqphqw fodlpv lvvxhg e| wkh jryhuqphqw dqg psw / ow.4
j
vhfwru*v ghpdqg iru wkhvh> pw lv wkh jryhuqphqw*v ghpdqg iru prqh|1
Kdylqj dftxluhg prqh|/ jryhuqphqw dqg sulydwh djhqwv jr lqwr d
jrrgv wudglqj vhvvlrq lq zklfk krxvhkrog surgxfwv duh vrog iru prqh|
wr jryhuqphqw dqg krxvhkrog frqvxphuv=
sw gw @ psw . pjw
+45,
lv wkh krxvhkrog lqfrph fduulhg lqwr wkh qh{w shulrg lq wkh irup ri d
prqh| kroglqj1 gw lv wkh kduyhvw shu wuhh1
sw jw @ pjw
+46,
sw fw @ psw
+47,
duh wkh jryhuqphqw*v dqg krxvhkrog*v frqvxpswlrq uhvshfwlyho|1 +Zh
zloo dvvxph wkdw wkh qrplqdo udwh ri lqwhuhvw lv srvlwlyh vr grplqdwlqj
wkh }hur uhwxuq rq prqh| vr wkdw frqvxphuv qhyhu krog pruh prqh|
wkdq wkh| sodq wr vshqg1 Khqfh +47, lv dq htxdolw|1,
649
H{whqglqj dqg Ghhshqlqj wkh Prghov
Htxdwlrqv +45,0+47, lpso| wkdw wkh jrrgv pdunhw fohduv=
sw fw . sw jw @ sw gw
+48,
dqg prqh|0pdunhw fohdulqj +<, wkhq lpsolhv wkh Txdqwlw| Wkhru|=
Pw.4 @ sw gw
+49,
Qrz frqvlghu wkh krxvhkrog*v frqvxpswlrq ghflvlrq lq wklv iudph0
zrun1
"
S
! x+fw , vxemhfw wr
Lw pd{lpl}hv Hw@3
w@3
w +{w , psw
psw
. w . uw +{w ,vw .
sw
@ sw fw
w.4 +{w.4 , @
4
sw
]
s
+{w.4 ,q+{w.4 > {w ,g{w.4
ow.4
os +{w.4 , psw sw fw
sw gw vw
. uw.4 +{w.4 ,vw . w.4
.
sw.4
sw.4
sw.4
+4:,
Htxdwlrq +4:, vwdwhv wkdw wkh krxvhkrog*v ehjlqqlqj shulrg uhdo zhd0
owk/ w +{w ,/ pxvw eh vshqw rq prqh|/ wd{hv + w ,/ dqg jryhuqphqw fodlpv
+dw fxuuhqw sulfh q+{w. 4 > {w , iru frqwlqjhqf| {w.4 ,> wkdw frqvxpswlrq
fdq rqo| eh fduulhg rxw e| prqh|> dqg wkdw qh{w shulrg*v ehjlqqlqj
zhdowk zloo eh surgxfhg e| wklv shulrg*v lqfrph +lq wkh irup ri prqh|
@sw gw vw , gh dwhg e| qh{w shulrg*v sulfhv/ qh{w shulrg*v ydoxh ri wkh wuhhv
dqg jryhuqphqw fodlpv dftxluhg wklv shulrg/ dqg wkh ydoxh ri dq| prqh|
dftxluhg e| wkh krxvhkrog ehiruh wkh jrrgv wudglqj vhvvlrq exw qrw xvhg
iru frqvxpswlrq wkhq1 Krzhyhu rxu dvvxpswlrq wkdw wkh qrplqdo uhwxuq
rq erqgv lv srvlwlyh lpsolhv wkdw wkhuh zloo eh qr vxfk vxusoxv prqh|
edodqfh/ ehfdxvh prqh| lv rqo| xvhixo iru ex|lqj frqvxpswlrq jrrgv/
dqg wkh fdvk0lq0dgydqfh frqvwudlqw lv elqglqj= psw @ sw fw 1
Wkh frqvxphu*v Odjudqjhdq lv wkhq=
O @ Hw@3
4
.
sw
"
[
w x+fw , w i+fw . Ww . uw +{w ,vw
w@3
]
os w.4 +{w.4 ,q+{w.4 > {w ,g{w.4 u+{w ,vw4 ows +{w ,
j
sw
gw4 iw4 vw4
sw
+4;,
+4<,
+53,
Wkh uvw rughu frqglwlrqv |lhog=
q+{w.4 > {w , @
x! +gw.4 jw.4 ,Pw.4 gw.4
i+{w.4 > {4 ,
x! +gw jw ,Pw.5 gw
+54,
Prqh| lq Uhsuhvhqwdwlyh Djhqw Prghov
x! +gw.4 jw.4 ,
=
uw +{w , @
x! +gw jw ,
Pw.4 gw.4
uw.4 +{w.4 , .
i+{w.4 > {w ,g{w.4
Pw.5
64:
] +55,
+56,
Htxdwlrq +4<, lv wkh sulfh ri d qrplqdo erqg zklfk sd|v rxw zkhq
{w.4 rffxuv dqg lw lv vhw mxvw dv lw zdv hduolhu/ dv wkh h{shfwhg grphv0
wlf pdujlqdo xwlolw| ri frqvxpswlrq lq wkdw hyhqw uhodwlyh wr lwv fxuuhqw
pdujlqdo xwlolw|> h{fhsw wkdw lq wklv fdvh lw lv wkh pdujlqdo xwlolw| ri frq0
vxpswlrq ri d qrplqdo udwkhu wkdq d uhdo xqlw wkdw lv dvvhvvhg/ vr wkdw lw
lv gh dwhg e| wkh sulfh ohyho zkhq {w.4 rffxuv1 +Ri frxuvh fw @ gw jw
e| wkh pdunhw0fohdulqj frqglwlrq1,
Htxdwlrq +53, lv wkh sulfh ri d uhdo dvvhw +wkh wuhh,1 Djdlq wkh sulflqj
phwkrg lv wkh vdph dv ehiruh exw qrz wkh dvvhw*v glylghqg lv lq prqh0
wdu| irup +sw gw , dv lw fdq rqo| eh
U h{fkdqjhg iru prqh|1 Vr dsduw iurp
wkh h{shfwhg ixwxuh uhdo sulfh/ uw.4 +{w.4 ,i+{w.4 > {w ,g{w.4 / lwv ydoxh
ghshqgv rq Uwkh h{shfwhg ydoxh ri wkh glylghqg uhfhlyhg wklv shulrg exw
sw
i+{w.4 > {w ,g{w.4 1
vshqw qh{w/ sgww.4
Zkdw hhfwv grhv jryhuqphqw srolf| kdyh rq wklv hfrqrp|B
Wkh jryhuqphqw*v exgjhw frqvwudlqw lv=
]
ow +{w , Pw.4 Pw
4
jw @ w .
.
ow.4 +{w.4 ,q+{w.4 > {w ,g{w.4 +57,
sw
sw
sw
Jryhuqphqw srolf| frqvlvwv ri fkrrvlqj vhtxhqfhv iru jw / w dqg Pw /
frqglwlrqdo rq {w / dqg frqvlvwhqwo| zlwk +54,1 Qrz ehfdxvh lq wklv hfrq0
rp| wkh rxwsxw pxvw eh vkduhg ehwzhhq jryhuqphqw dqg sulydwh frq0
vxpswlrq/ lw lv lpsrvvleoh iru frqvxpswlrq wr eh dhfwhg e| wkh sdw0
whuq ri wd{dwlrq ru prqh| vxsso|/ jlyhq wkh jryhuqphqw frqvxpswlrq
vhtxhqfh1 Krzhyhu/ wkh prqh| vxsso| vhtxhqfh +dqg vr dovr wkh wd{0
dwlrq vhtxhqfh li lw lv dowhuhg dv d uhvxow, grhv dhfw wkh uhdo ydoxh ri
uhdo dvvhwv +wuhhv,1 Khqfh wkhuh duh uhdo hhfwv ri prqhwdu| srolf| dqg
wd{dwlrq/ ehfdxvh wkh uhwxuqv iurp uhdo dvvhwv fdq rqo| eh hqmr|hg e|
ehlqj h{fkdqjhg iru prqh| dqg wkhq vshqw1 Lw iroorzv wkdw li wkhuh zdv
lqyhvwphqw lq wklv hfrqrp|/ exw wkh uhwxuqv iurp lqyhvwphqw frxog rqo|
eh hqmr|hg e| h{fkdqjh iru prqh|/ wkhq sulydwh frqvxpswlrq zrxog eh
dhfwhg e| wd{dwlrq dqg prqhwdu| srolf|= wkh pl{ ehwzhhq sulydwh lq0
yhvwphqw dqg frqvxpswlrq zrxog eh fkdqjhg1 Rqo| li wkhuh zhuh dq
lqyhvwphqw yhklfoh +vxfk dv dq lqgh{hg erqg, rhulqj frqvxpswlrq srv0
vlelolwlhv txlwh lqghshqghqw ri wkh sulfh ohyho zrxog wklv fhdvh wr eh wkh
fdvh1 Exw vxfk d yhklfoh pd| eh uxohg rxw lq wklv fdvk0lq0dgydqfh zruog
+hyhq rq dq lqgh{hg erqg wkh lqgh{hg glylghqg kdv wr eh sdlg dw d
sduwlfxodu wlph wr eh h{fkdqjhg odwhu iru jrrgv lq prqhwdu| h{fkdqjh,1
64;
H{whqglqj dqg Ghhshqlqj wkh Prghov
XQSOHDVDQW PRQHWDULVW DULWKPHWLF UHYLV0
LWHG
Ohw xv uhwxuq wr wkh txhvwlrq dgguhvvhg lq fkdswhu := zkhwkhu wljkwhu
prqh| wrgd| zloo uhgxfh lq dwlrq shupdqhqwo| lq wkh devhqfh ri fkdqjhv
lq jryhuqphqw vshqglqj dqg wd{ vhtxhqfhv1 Zh fdq uhzulwh wkh jryhuq0
phqw exgjhw frqvwudlqw=
]
+Pw.m.4 Pw.m ,
P4 P3 [
.
g{w.m
N@
t+{w.m > {w ,
+58,
s3
sw.m
m@4
zkhuh N lv wkh suhvhqw ydoxh ri wkh +jw w , vhtxhqfh dqg t+{w.m > {w , @
3 x3 +fW .m ,
i+{ > { , lv wkh suhvhqw +w0shulrg, uhdo glvfrxqwhg ydoxh ri
x3 +fw ,
w.m
w
d xqlw ri frqvxpswlrq li {w.m rffxuv +wkdw lv/ lw lv d frqwlqjhqw uhdo
glvfrxqw udwh,1
Xvlqj wkh txdqwlw| wkhru|/ zh fdq uhzulwh wklv=
]
P3 [
.
t+{w.m > {w ,+gw.m sw.m4 gw.m4 @sw.m ,g{w.m
N @ g3 s3
m@4
+59,
3
Li P4 lv orzhuhg/ s3 idoov +P3 lv jlyhq,/ vr P
s3 ulvhv dqg wkhuh lv d orzhu
fxuuhqw lq dwlrq wd{ zklfk pxvw eh rvhw e| d kljkhu ixwxuh lq dwlrq wd{
+sw.m @sw.m4 ,= exw ehfdxvh t+{w.m > {w , lv ri wkh rughu m +fw.m ehlqj
udqgrp dqg vwdwlrqdu|,/ ixwxuh lq dwlrq kdv wr eh kljkhu e| wkh rughu
m 1
SRVVLEOH PRGHUQ HYROXWLRQV RI PRQH\
DQG PRQHWDU\ SROLF\= WKH UHHPHUJHQFH
RI FRPPRGLW\ PRQH\B
Lq fkdswhu 9 zh frqvlghuhg wkh hylghqfh rq rswlpdo prqhwdu| +lqwhuhvw
udwh, ihhgedfn srolf|1 Zh vdz wkdw uxohv lq zklfk prqhwdu| frqglwlrqv
uhdfwhg wr ghyldwlrqv ri lq dwlrq dqg rxwsxw iurp wkhlu uhvshfwlyh wdujhwv
+ri corz* lq dwlrq dqg wkh qdwxudo udwh, jdyh jrrg uhvxowv lq frqglwlrqv
zkhuh shrsoh vljqhg ryhuodsslqj zdjh +ru sulfh, frqwudfwv ri vrph jlyhq
vwuxfwxuh1 Krzhyhu/ wkh txhvwlrq dulvhv zkhwkhu wkhvh uxohv uhpdlq rswl0
pdo zkhq wklv vwuxfwxuh lv hqgrjhqrxv> zh vxjjhvwhg wkhuh wkdw wklv zdv
qrw vr exw udwkhu rqh irxqg wkdw uxohv rhulqj sulfh vwdelolw| +wkdw lv/
zkhuh dq| h{fhvv ryhu wkh lq dwlrq wdujhw zklfk qhhg qrw eh }hur lv fodzhg edfn lq vxevhtxhqw shulrgv, surgxfhg d jdlq lq pdfur vwdelolw|
Prqh| lq Uhsuhvhqwdwlyh Djhqw Prghov
64<
ehfdxvh shrsoh ohqjwkhqhg wkh shulrg iru zklfk qrplqdo frqwudfwv zrxog
eh vljqhg1 D irupdo uxoh ri wklv vruw zrxog eh rqh zlwk d sulfh0ohyho wdu0
jhw lq zklfk wkhuh zdv/ iru h{dpsoh/ ixoo fodzedfn ri d txduwhuo| lq dwlrq
huuru lq wkh iroorzlqj txduwhu1 Wklv lv h{dplqhg lq Plqirug hw do1 +5334,
zkhuh wkh ghwdlov fdq eh irxqg1 Wkh uxoh lv dqdorjrxv wr d frpprglw|
vwdqgdug lq zklfk li sulfhv ulvh/ lpso|lqj wkdw wkh uhodwlyh sulfh ri wkh
frpprglw|/ vd| jrog/ kdv idoohq/ wkhq ohvv jrog lv surgxfhg/ gulylqj wkh
sulfh xs djdlq dqg wkh jhqhudo sulfh ohyho grzq djdlq1 Zlwk dw prqh|
ehlqj wdujhwhg rq wkh jhqhudo sulfh ohyho lq whupv ri prqh|/ wkh vxs0
so| ri prqh| zrxog uhdfw pruh udslgo| wkdq jrog surgxfwlrq xqghu wkh
jrog vwdqgdug> iru h{dpsoh/ vxssrvh wkdw surgxfwlylw| urvh/ udlvlqj wkh
vxsso| ri jrrgv dqg vr gulylqj grzq wkh sulfh ohyho/ wkhq wkh prqh| vxs0
so| zrxog ulvh lq wkh iroorzlqj shulrg wr dffrpprgdwh wklv surgxfwlylw|
lqfuhdvh/ sxvklqj sulfhv edfn xs djdlq1
Li krxvhkrogv duh xvlqj wkhlu zdjh frqwudfw vwuxfwxuh wr khos plqlpl}h
wkh yduldqfh ri wkhlu frqvxpswlrq sdwwhuq/ wkhq wkh| vkrxog uhdfw wr
wklv juhdwhu fhuwdlqw| lq sulfhv e| uhgxflqj wkh lqgh{dwlrq hohphqw +ru
htxlydohqwo| ohqjwkhqlqj wkhlu frqwudfw shulrg,1 Wklv zloo phdq +vhh
jxuh 4516, wkdw wkh djjuhjdwh vxsso| fxuyh zloo eh dwwhu +pruh qrplqdo
uljlglw|, dqg wkh djjuhjdwh ghpdqg fxuyh zloo eh vwhhshu +ohvv uhdfwlrq
ri prqh| ghpdqg wr ulvlqj sulfhv ehfdxvh ohvv hhfw rq zdjhv> dqg vr
ohvv prqhwdu| wljkwhqlqj fdxvhg e| ulvlqj sulfhv/ vr ohvv idoo lq rxwsxw
dqg hpsor|phqw,1 Wklv lq wxuq lpsolhv wkdw uhdo vkrfnv kdyh ohvv hhfw
rq erwk rxwsxw dqg sulfhv1 Vr surylghg sxuh prqhwdu| vkrfnv duh nhsw
orz wklv lv d uhflsh iru jhqhudo pdfur vwdelolw|1
FRQFOXVLRQV
Prqh| lv d frpprglw| xvhg dv d xqlw ri dffrxqw dqg d phdqv ri sd|phqw1
Lwv hyroxwlrqdu| ruljlq prvw olnho| olhv lq wkh hphujhqfh ri d frpprglw|
zlwk wkh uljkw fkdudfwhulvwlfv wr dyrlg wkh grxeoh frlqflghqfh ri zdqwv
lpsolhg e| eduwhu1 Lwv hyroxwlrq surfhhghg wrzdugv wkh prghuq frp0
shwlwlyh edqnlqj v|vwhp lq zklfk prqh| uhpdlqv wkh xqlw ri dffrxqw/
lvvxhg e| d prqrsro| jryhuqphqw/ exw lwv sd|phqwv ixqfwlrq kdv odujho|
ehhq wdnhq ryhu e| fohdulqj v|vwhpv edvhg rq fuhglw/ lq zklfk shrsoh
fdq hduq lqwhuhvw rq wkhlu dvvhwv dv zhoo1 Lwv hyroxwlrq frqwlqxhv wrgd|/
zlwk wkh prqrsro| uroh ri jryhuqphqw lq ghflglqj wkh xqlw ri dffrxqw
srvvleo| frplqj xqghu frpshwlwlyh wkuhdw iurp rwkhu jryhuqphqwv dqg
hyhq shukdsv wkh sulydwh vhfwru1 Wkh prghov zh irfxv rq lq wklv fkds0
whu duh lqwhqghg wr xqghuvwdqg krz prqh| lqwhudfwv zlwk wkh hfrqrp|
dw glhuhqw vwdjhv ri prqhwdu| hyroxwlrq1 Wkh Ehzoh|0Wrzqvhqg prgho
653
H{whqglqj dqg Ghhshqlqj wkh Prghov
$ +LJK,QGH[DWLRQ
3
>φ W @
$ /RZ
,QGH[DWLRQ
$
%
+LJK
,QGH[DWLRQ
/RZ,QGH[DWLRQ>PW @
(PSOR\PHQW
Iljxuh 4516= Wkh hhfw ri uhgxfhg lqgh{dwlrq rq vorshv ri 7 dqg ( fxuyhv
^| surgxfwlylw| vkrfn> 6| prqhwdu| vkrfn`
h{sodlqv krz prqh| shuplwv wzr sdudooho +sulplwlyh, frppxqlwlhv zklfk
duh wrr uhprwh iurp hdfk rwkhu wr ohqg gluhfwo| wr hdfk rwkhu/ wr ohqg lq0
gluhfwo| yld prqh| kroglqj lq wkh devhqfh ri lqwhuhvw0ehdulqj lqvwuxphqwv
lvvxhg e| d jryhuqphqw ru vrph wuxvwhg lqwhuqdwlrqdo lqwhuphgldu|1 Wkh
ROJ prgho vkrzv krz prqh| fdq eh d vwruh ri ydoxh zklfk shuplwv rqh
jhqhudwlrq wr ohqg wr wkh qh{w lqgluhfwo| djdlq lq wkh devhqfh ri erqgv=
krzhyhu lq d jurzlqj hfrqrp| wkhuh duh gl!fxowlhv lq dfklhylqj d prqh0
wdu| htxloleulxp zklfk lv Sduhwr rswlpdo1 Wkh odvw prgho zh orrnhg dw/
Oxfdv* prgho ri fdvk0lq0dgydqfh/ vlpso| dvvxphv prqh| lv qhhghg iru
wudqvdfwlrqv> jlyhq wkdw/ lw prghov krz dq hfrqrp| zlwk lwv ixoo sdqrso|
ri rwkhu qdqfldo lqvwuxphqwv zrxog ehkdyh1 Ehfdxvh uhwxuqv rq rwkhu
dvvhwv kdyh wr eh wxuqhg lqwr prqh| ehiruh wkh| fdq eh hqmr|hg/ fkdqjhv
lq prqhwdu| srolf| fdq dowhu wkh h{shfwhg udwh ri uhwxuq rq wkhvh dqg
vr kdyh uhdo hhfwv1 Ilqdoo|/ zh frqvlghuhg krz prqh| dqg prqhwdu|
srolf| pljkw ghyhors> fxuuhqwo| zh revhuyh ryhuodsslqj zdjh +dqg sulfh,
frqwudfwv zklfk pdnh lw rswlpdo wr sxuvxh uxohv ri prqhwdu| uhdfwlrq wr
ghyldwlrqv iurp lq dwlrq dqg rxwsxw wdujhwv1 Krzhyhu/ li frqwudfwv duh
hqgrjhqrxv/ dv vxuho| wkh| duh ryhu vrph wlph shulrg/ wkhq lw pd| zhoo
eh ghvludeoh wr sxuvxh sulfh0ohyho vwdelolw|> wklv ohqjwkhqv frqwudfwv dqg
fuhdwhv juhdwhu dxwrpdwlf vwdelolw| lq wkh hfrqrp|1