Area Conference on Industrial Organisation

A joint Initiative of Ludwig-Maximilians-Universität and Ifo Institute for Economic Research
Area Conference on Industrial
Organisation
CESifo Conference Centre, Munich
19-20 April 2002
Reallocation of Corporate Resources and
Managerial Incentives in Internal Capital
Markets
Sandro Brusco & Fausto Panunzi
CESifo
Poschingerstr. 5, 81679 Munich, Germany
Phone: +49 (89) 9224-1410 - Fax: +49 (89) 9224-1409
E-mail: [email protected]
Internet: http://www.cesifo.de
+i@**LU@|L? Lu LhTLh@|i +itLhUit @?_
@?@}ih@* W?Ui?|it ? W?|ih?@* @T|@* @h!i|t
5@?_hL htUL=
6@t|L @??3g
#iUi4Mih 2ff
W `i |@?! #i?t BhL4Mc 6*Lh@? Oi_ihc aLt‚
i @h? @?_ i|hL AL44@t?L uLh ih)
i*Tu* UL44i?|t Ai T@Tih @t @*tL Mi?i€|i_ uhL4 UL44i?|t M) T@h|UT@?|t |L ti4
?@ht i*_ @| |i ,hLTi@? N?iht|) W?t|||ic N?iht_@_ _i *U@?|ic N?iht|
@ _
hitU@c N?iht|
@ _ L*L}?@c |i WWW 6LhL _i 6?@?3@t E@_h_c |i ,+ UL?uihi?Ui
L? LhTLh@|i BLih?@?Ui ELh4@)ihc W|@*) @?_ |i V 5,A UL?uihi?Ui EWtU@c W|@*)
`i=hi4@? hitTL?tM*i uLh @** ihhLht
#iT@h|@4i?|L _i ,UL?L4‚@ _i *@ ,4Thit@c N?iht_@_ @h*Lt WWW _i @_h_c
@**i @_h_c 2Sc 2Hbf Bi|@ui @_h_ E5T@? Ai* ne b S2e bf. ,4@*G
MhtUL9i4TU4it
g #T@h|4i?|L _ 5Ui?3i ,UL?L4Uic N?iht|
@ _ L*L}?@ @?_ ,+c @33@ 5U@h
@** 2c ef2S L*L}?@ EW|@*) Ai* nb f2 DHSD2. ,4@*G u@t|LT@??39?
MLUUL?|
4
7C-C
Glyhuvlhg upv riwhq wudgh dw d glvfrxqw zlwk uhvshfw wr wkhlu irfxvhg
frxqwhusduwv1 Wkh olwhudwxuh kdv wulhg wr h{sodlq wkh dssduhqw plvdoorfdwlrq
ri uhvrxufhv zlwk oree|lqj dfwlylwlhv ru srzhu vwuxjjohv1 Zh vkrz wkdw gl0
yhuvlfdwlrq fdq ghvwur| ydoxh hyhq zkhq uhvrxufhv duh hflhqwo| doorfdwhg
h{ srvw1 Zkhq pdqdjhuv ghulyh xwlolw| iurp wkh ixqgv xqghu wkhlu sxuylhz/
prylqj ixqgv dfurvv glylvlrqv pd| glplqlvk wkhlu lqfhqwlyhv1 Wkh h{ dqwh
uhgxfwlrq lq pdqdjhuldo lqfhqwlyhv fdq pruh wkdq rvhw wkh lqfuhdvh lq up
ydoxh gxh wr wkh h{ srvw hflhqw uhdoorfdwlrq ri ixqgv1 Wklv hhfw lv urexvw
wr wkh lqwurgxfwlrq ri prqhwdu| lqfhqwlyhv1Zh dsso| rxu prgho wr wkh dqdo0
|vlv ri wkh rswlpdo uhdoorfdwlrq srolf| dqg wr wkh hhfw ri wkh dv|pphwu|
dprqj glylvlrqv1 Lq jhqhudo lw lv rswlpdo iru khdgtxduwhuv wr frpplw qrw
wr uhdoorfdwh dw ohdvw d iudfwlrq ri ixqgv1 Dv d uhvxow/ wkh lqyhvwphqw lq
d jlyhq glylvlrq lv pruh vhqvlwlyh wr wkh glylvlrq*v fdvk rz wkdq wr rwkhu
glylvlrqv* fdvk rz1 Dv|pphwulhv lq vl}h dqg jurzwk survshfwv lqfuhdvh wkh
glyhuvlfdwlrq glvfrxqw1
5

W?|hL_U|L?
Wkh dqdo|vlv ri wkh doorfdwlrq ri ixqgv dprqj glhuhqw glylvlrqv ri d frq0
jorphudwh up lv d uhodwlyho| |rxqj wrslf1 Vwhlq +5334, surylghv d uhfhqw
vxuyh|1 Wkh jhqhudo wkhph frplqj iurp wkh hpslulfdo olwhudwxuh lv wkdw gl0
yhuvlhg upv wudgh rq dyhudjh dw d glvfrxqw uhodwlyh wr d sruwirolr ri irfxvhg
upv lq wkh vdph lqgxvwulhv/ dv uhsruwhg e| Ehujhu dqg Rihn +4<<8,/ Vhu0
ydhv +4<<9, dqg Olqv dqg Vhuydhv +4<<<,1 Pruhryhu/ wkh *;3v vdz d surfhvv ri
glvpdqwolqj ri glyhuvlhg upv/ gulyhq e| wkh lghd wkdw wkh glylvlrqv zrxog
eh pruh hflhqwo| pdqdjhg dv vwdqg0dorqhv1 Exw li wkhuh lv e| qrz d zlgh
frqvhqvxv rq wkh lghd wkdw d glyhuvlfdwlrq glvfrxqw h{lvwv/ lw lv pxfk ohvv
fohdu zk| wklv lv wkh fdvh1
Vwhlq +4<<:, kdv srlqwhg rxw wkdw lqwhuqdo fdslwdo pdunhwv fdq fuhdwh
ydoxh lq qdqfldoo| frqvwudlqhg upv1 Lq Vwhlq*v zrugv/ _Vlpso| sxw/ lqgl0
ylgxdo surmhfwv pxvw frpshwh iru vfdufh ixqgv/ dqg khdgtxduwhuv* mre lv wr
slfn zlqqhuv dqg orvhuv lq wklv frpshwlwlrq1% Vwhlq ghqrwhv wklv dfwlylw| ri
khdgtxduwhuv lq d frqjorphudwh up dv czlqqhu0slfnlqj*1 Frqwudu| wr wkh
hpslulfdo qglqjv/ Vwhlq*v prgho vxjjhvwv wkdw lqwhuqdo fdslwdo pdunhw vkrxog
fuhdwh ydoxh dqg wkxv d suhplxp iru glyhuvlhg upv1 Rqh srvvleoh zd| wr
vroyh wklv dssduhqw sdudgr{ lv wr dujxh wkdw wkh glvfrxqw ri glyhuvlhg upv
lv gxh wr plvdoorfdwlrq ri uhvrxufhv lq lqwhuqdo fdslwdo pdunhwv1 Iru lqvwdqfh/
Udmdq hw do1 +5333, qg wkdw pxowl0vhjphqw upv doorfdwh uhodwlyho| pruh
fdslwdo wr czhdn* olqhv ri exvlqhvv wkdq wkhlu vwdqg0dorqh frxqwhusduwv/ dqg
uhodwlyho| ohvv wr vhjphqwv lq cvwurqj* olqhv ri exvlqhvv1 Vfkduivwhlq +4<<;,
qgv wkdw wkh lqyhvwphqw ri frqjorphudwh glylvlrqv lv yluwxdoo| lqvhqvlwlyh wr
lqyhvwphqw rssruwxqlwlhv/ dv phdvxuhg e| wkh lqgxvwu| J*v1 Odprqw +4<<:,
vkrzv wkdw uhvrxufh doorfdwlrq lq glyhuvlhg upv lv glhuhqw iurp wkdw lq
irfxvhg upv dqg ohvv vhqvlwlyh wr lqglfdwruv ri lqyhvwphqw ydoxh vxfk dv
Wrelq*v J1
Krzhyhu/ wkh hylghqfh rq lqhflhqw doorfdwlrq ri ixqgv kdv ehhq glv0
sxwhg1 Zklwhg +5334, srlqwv rxw wkdw wkh lqhflhqf| uhvxowv dsshduhg lq
wkh olwhudwxuh pd| dfwxdoo| eh gxh wr wkh lqfruuhfw phdvxuhv dgrswhg iru wkh
lqyhvwphqw rssruwxqlwlhv ri wkh glylvlrqv1 Vkh vkrzv wkdw zkhq phdvxuh0
phqw sureohpv duh wdnhq lqwr dffrxqw/ wkh hylghqfh ri lqhflhqw doorfdwlrq
ri ixqgv glvdsshduv1 Fkhydolhu +5333, dqdo|}hv wkh lqyhvwphqw ehkdylru ri d
vdpsoh ri upv ehiruh dqg diwhu glyhuvli|lqj phujhuv/ qglqj qr hylghqfh ri
d fkdqjh lq lqyhvwphqw ehkdylru1 Wklv lpsolhv wkdw/ li wkhuh lv lqhflhqf| lq
wkh lqyhvwphqw ehkdylru ri wkh glylvlrqv ri frqjorphudwh upv/ vxfk lqh0
flhqf| grhv qrw dsshdu wr eh gxh wr wkh suhvhqfh ri lqwhuqdo fdslwdo pdunhwv1
6
Lq wklv sdshu zh dujxh wkdw lq rughu wr h{sodlq wkh glyhuvlfdwlrq glvfrxqw
zh gr qrw qhhg wr dvvxph dq| plvdoorfdwlrq ri ixqgv lq lqwhuqdo fdslwdo
pdunhwv1 Frqjorphudwhv fdq ghvwur| ydoxh hyhq li uhvrxufhv duh +h{ srvw,
hflhqwo| doorfdwhg1 Li pdqdjhuv ghulyh xwlolw| iurp wkh ixqgv xqghu wkhlu
sxuylhz/ wkh srvvlelolw| ri lpsohphqwlqj d czlqqhu0slfnlqj* srolf|/ zkloh
rswlpl}lqj uhvrxufhv doorfdwlrq h{ srvw/ uhgxfhv pdqdjhuldo lqfhqwlyhv h{
dqwh1 Wdnlqj dzd| iurp wkh pdqdjhu wkh fdvk rz jhqhudwhg e| wkh glylvlrq
kdv wkh qhjdwlyh lpsolfdwlrq ri uhgxflqj wkh h{ dqwh lqfhqwlyhv iru glylvlrq
pdqdjhuv wr vshqg hruw wr jhqhudwh wkh fdvk rz1 Wkh uhgxfhg pdqdjhuldo
lqfhqwlyhv fdq pruh wkdq rvhw wkh jdlqv ri uhdoorfdwlqj ixqgv wr wkh prvw
surwdeoh glylvlrqv1 Lq rwkhu zrugv/ czlqqhu0slfnlqj* lv erwk wkh euljkw dqg
wkh gdun vlgh ri lqwhuqdo fdslwdo pdunhwv1
Zh frqvlghu d wzr0shulrg prgho zlwk wzr glylvlrqv dqg d khdgtxduwhuv1
Glylvlrq pdqdjhuv uhfhlyh sulydwh ehqhwv lq sursruwlrq wr wkh jurvv uhwxuq
ri wkh glylvlrq wkh| uxq1 khdgtxduwhuv pd{lpl}hv wrwdo up ydoxh1 Lq wkh
uvw shulrg wkh wzr glylvlrq pdqdjhuv kdyh wr h{huw d qrq0yhuldeoh hruw
wr lqfuhdvh wkh suredelolw| ri vxffhvv ri d surmhfw douhdg| lq sodfh1 Wkh fdvk
rz jhqhudwhg e| wkh h{lvwlqj surmhfw zloo eh uhlqyhvwhg lqvlgh wkh up lq wkh
vhfrqg shulrg1 Ehiruh wkh vhfrqg shulrg/ wkh khdgtxduwhuv uhfhlyhv d vljqdo
rq wkh vhfrqg shulrg surwdelolw| ri wkh wzr glylvlrqv dqg uhdoorfdwhv ixqgv1
Zkhq glylvlrqv rshudwh dv vwdqg0dorqhv/ hdfk glylvlrq uhlqyhvwv wkh fdvk rz
jhqhudwhg e| wkh uvw shulrg surmhfw1 Rq wkh frqwudu|/ lq wkh glyhuvlhg
up wkh khdgtxduwhuv zloo uhglvwulexwh wkh fdvk rz wr wkh prvw surwdeoh
glylvlrq1 Wkh uhglvwulexwlrq kdv wzr hhfwv= rq rqh kdqg lw fuhdwhv ydoxh/
exw rq wkh rwkhu kdqg lw uhgxfhv wkh uhqw iru wkh pdqdjhu ri wkh +h{ srvw,
ohvv surwdeoh glylvlrq1 Dqwlflsdwlqj wkh srvvlelolw| ri ehlqj h{sursuldwhg/
hdfk glylvlrq pdqdjhu zloo uhgxfh klv hruw1 Frqvhtxhqwo|/ wkh wrwdo fdvk
rz wr eh uhlqyhvwhg lq shulrg wzr zloo ghfuhdvh1
Wklv revhuydwlrq kdv wkh iroorzlqj lpsolfdwlrqv1 Iluvw/ d surw0 pd{lpl}0
lqj khdgtxduwhuv zloo idfh d wlph lqfrqvlvwhqf| sureohp1 Rqfh wkh ixqgv duh
jhqhudwhg/ khdgtxduwhuv zrxog olnh wr h{huflvh czlqqhu slfnlqj* wr wkh kljkhvw
h{whqw srvvleoh1 Krzhyhu/ wklv h{ srvw pd{lpl}lqj ehkdylru e| khdgtxduwhuv
zloo uhgxfh h{ dqwh lqfhqwlyhv dw wkh glylvlrqdo ohyho/ dqg lw pd| fdxvh d
orvv ri ydoxh iru wkh frusrudwlrq1 Wkh glvfrxqw lv sduwlfxoduo| vhyhuh zkhq
glylvlrqv duh h{ srvw vlplodu lq whupv ri lqyhvwphqw rssruwxqlwlhv/ l1h1 zkhq
czlqqhu0slfnlqj* kdv d olplwhg srwhqwldo iru fuhdwlqj ydoxh1 Lq wklv fdvh wkh
qhjdwlyh hhfw rq wkh uhgxfhg pdqdjhuldo lqfhqwlyhv zloo grplqdwh1 Rq wkh
rwkhu kdqg/ zkhq h{ srvw glyhuvlw| lq lqyhvwphqw rssruwxqlwlhv lv odujh/ vr
wkdw wkh surwv jhqhudwhg iurp uhdoorfdwlrq duh dovr odujh/ wkhq wkh dgydq0
7
wdjhv ri czlqqhu0slfnlqj* grplqdwh ryhu wkh uhgxfhg pdqdjhuldo lqfhqwlyhv/
dqg wkh glyhuvlhg up wudghv dw d suhplxp1 Frqyhuvho|/ glyhuvlw| lq wkh
h{ dqwh surwdelolw| ri glylvlrqv lqfuhdvhv wkh olnholkrrg wkdw d frqjorphudwh
wudghv dw d glvfrxqw1 Li rqh glylvlrq kdv d yhu| kljk suredelolw| ri kdylqj
h{ srvw wkh ehvw lqyhvwphqw rssruwxqlwlhv/ wkh lqfhqwlyhv iru wkh pdqdjhu ri
wkh rwkhu glylvlrq duh uhgxfhg1 Wkhuhiruh wkh fdvk rz wkdw fdq eh uhdoor0
fdwhg wr wkh prvw surwdeoh glylvlrq lv dovr uhgxfhg/ olplwlqj wkh jdlqv ri
czlqqhu0slfnlqj*1
Vhfrqg/ li khdgtxduwhuv kdv vrph frpplwphqw srzhu wkhq d srolf| zklfk
ohwv glylvlrqv wr uhwdlq sduw ri wkh fdvk rz jhqhudwhg/ lqghshqghqwo| ri wkh
lqyhvwphqw rssruwxqlwlhv/ pd| zhoo pd{lpl}h h{shfwhg surw1 Olqnlqj lq0
yhvwphqw wr wkh fdvk rz surgxfhg dqg qrw rqo| wr lqyhvwphqw rssruwxqlwlhv
uhvwruhv wkh uljkw edodqfh ehwzhhq wkh jrdo ri doorfdwlqj uhvrxufhv hflhqwo|
dqg surylglqj lqfhqwlyhv wr glylvlrq pdqdjhuv12 Zh qh{w srvh wkh iroorzlqj
txhvwlrq= Li wkh h{ dqwh surwdelolwlhv ri wkh glylvlrqv duh glhuhqw/ zklfk
rqh vkrxog eh doorzhg wr uhwdlq pruh fdvkB Khuh wkhuh duh wzr hhfwv dw
zrun1 Rq rqh kdqg/ doorzlqj wkh h{ dqwh prvw surwdeoh glylvlrq wr uhwdlq
dq h{wud groodu lv ohvv frvwo|/ vlqfh wkh groodu lv pruh olnho| wr hqg xs dw
lwv ehvw xvh1 Rq wkh rwkhu kdqg/ doorzlqj wkh zhdnhu glylvlrq wr uhwdlq dq
h{wud groodu kdv d vwurqjhu lqfhqwlyh hhfw1 Wkh uhdvrq lv wkdw wkh pdqdjhu
ri wkh ohvv surwdeoh glylvlrq lv pruh olnho| wr eh ch{sursuldwhg* ri wkh fdvk
jhqhudwhg zkhqhyhu khdgtxduwhuv grhv qrw frpplw wr ohw wkh glylvlrq uhwdlq
ixqgv1 Wkhuhiruh/ ixqg uhwhqwlrq lv hvshfldoo| dssuhfldwhg e| wkh pdqdjhuv
ri zhdnhu glylvlrqv/ dqg wkh| zloo uhvsrqg pruh vwurqjo| wr wkh srvvlelolw|
ri ixqg uhwhqwlrq wkdq wkh pdqdjhuv ri vwurqj glylvlrqv1 Wkh lqwhusod| ri
wkhvh wzr irufhv jhqhudwhv wzr glhuhqw uhjlphv1 Zkhq wkh glhuhqfh lq
surwdelolw| lv vpdoo wkhq wkh jhqhudo ohyho ri ixqg uhwhqwlrq lv kljk/ vlqfh
surylglqj lqfhqwlyhv lv pruh lpsruwdqw wkdq hflhqw h{ srvw uhdoorfdwlrq ri
fdslwdo/ dqg wkh glylvlrq zklfk lv h{ dqwh pruh surwdeoh lv doorzhg wr uh0
wdlq pruh ri lwv fdvk1 Zkhq wkh glhuhqfh lq surwdelolw| lv odujh wkhq wkh
jhqhudo ohyho ri ixqg uhwhqwlrq zloo eh vpdoo1 Krzhyhu/ lq wklv fdvh lw lv wkh
ohvv surwdeoh glylvlrq zklfk lv doorzhg wr uhwdlq pruh1 Wklv sdwwhuq ri ixqg
uhwhqwlrq lv d srwhqwldoo| whvwdeoh suhglfwlrq ri wkh prgho1

W| t ?L| ?UL44L? |L LMtihi _iht€i_ €h4t |h@_?} @| @ Thi44 +@@?
i| @*
E2fff hiTLh| |@| b TihUi?| Lu _iht€i_ €h4t ? |ih t@4T*i |h@_i_ @| @ Thi44
? bbf
2
5? @?_ 5|*3 EbbH €?_ |@| |i ?it|4i?| ? @ }i? _tL? t 4Lhi ti?t|i |L
|i _tL?<t U@t L |@? |L L|ih _tL?t< U@t L OLiihc `|i_ E2ff tLt
|@| |i hit*| 4@) Mi _i |L 4i@thi4i?| ihhLht
8
Wklug/ wkh glyhuvlfdwlrq glvfrxqw lv juhdwhu zkhq d glylvlrq zlwk d
juhdwhu srwhqwldo iru lpphgldwh fdvk jhqhudwlrq lv sdluhg zlwk d glylvlrq zlwk
srru fdsdflw| ri lpphgldwh fdvk jhqhudwlrq exw jrrg jurzwk survshfwv1 Lq
wklv fdvh wkh pdqdjhu ri wkh uvw up zloo ihdu h{sursuldwlrq ri wkh fdvk rz
jhqhudwhg lq idyru ri wkh kljk0jurzwk up/ wkxv uhgxflqj wkh frqjorphudwh*v
ydoxh1
Wkh edvlf lqwxlwlrq wkdw h{ srvw lqwhuihuhqfh e| wkh sulqflsdo pd| eh
kdupixo iru lqfhqwlyhv lv ri frxuvh qrw qhz1 Iru h{dpsoh/ Djklrq dqg Wl0
uroh +4<<:, vkrz wkdw li wkh sulqflsdo lqwhuyhqhv wrr riwhq lq wkh ghflvlrqdo
surfhvv/ lw fdq vwlh wkh djhqw*v lqlwldwlyh1 Urwhpehuj dqg Vdorqhu +4<<7,
glvfxvv krz/ lq wkh suhvhqfh ri lqfrpsohwh frqwudfwv/ upv pd| ehqhw iurp
uhvwulfwlqj wkh vfrsh ri wkhlu dfwlylwlhv1 Wkhlu edvlf lghd lv wkdw glyhuvlhg
upv kdyh d zlghu udqjh ri surmhfwv wr lpsohphqw/ dqg iru vrph uhdvrq wkh|
fdqqrw lpsohphqw doo ri wkhp1 Dv d uhvxow/ wkh| duh pruh olnho| wr lpsoh0
phqw d surmhfw wkdw lw lv qrw lghdo iru surylglqj h{ dqwh lqfhqwlyhv1 Wkh| dovr
sursrvh dq dssolfdwlrq ri wkhlu prgho wr lqwhuqdo fdslwdo pdunhwv/ vkrzlqj
wkdw lw pd| eh rswlpdo wr irufh hdfk glylvlrq wr xvh rqo| ixqgv wkdw lw kdv
jhqhudwhg lwvhoi1 Zh h{whqg wkhlu dujxphqw srlqwlqj rxw wkh fdvhv zkhuh
lqwhuqdo fdslwdo pdunhwv duh ohvv olnho| wr eh ehqhfldo1 Pruhryhu zh vkrz
wkdw doorzlqj glylvlrqv wr uhwdlq d iudfwlrq ri wkhlu fdvk rz fdq eh vxshulru
wr erwk d sxuh lqwhuqdo fdslwdo pdunhw dqg wr d qduurz exvlqhvv vwudwhj|1
Rwkhu sdshuv zklfk suhvhqw dujxphqwv vlplodu wr rxuv lqfoxgh Jdxwlhu
xoohu +5334, dqg
dqg Khlghu +5333,/ Lqghuvw dqg Odx{ +5333,/ Lqghuvw dqg P‚
Gh Prwwd +4<<<,1 Jdxwlhu dqg Khlghu +5333, dqdo|}hv d prgho lq zklfk
glylvlrq pdqdjhuv pxvw uvw surgxfh fdvk rz wkdw zloo eh wkhq uhlqyhvwhg
lqvlgh wkh frpsdq|1 Krzhyhu/ lq wkhlu prgho surmhfwv kdyh d {hg vl}h +dq
h{wuhph yhuvlrq ri ghfuhdvlqj pdujlqdo uhwxuqv ri fdslwdo, dqg frqvhtxhqwo|
wkh vfrsh iru zlqqhu0slfnlqj lv uhgxfhg1 Lq Lqghuvw dqg Odx{ +5333, hruw
lv gluhfwhg wr wkh jhqhudwlrq ri lqyhvwphqw rssruwxqlwlhv udwkhu wkdq fdvk/
dqg wkhlu irfxv lv rq wkh uroh sod|hg e| oltxlglw| frqvwudlqwv1 Lqghuvw dqg
P‚
xoohu +5334, lv dovr irfxvhg rq wkh lpsdfw ri frqjorphudwhv rq qdqflqj
frqvwudlqwv1 Gh Prwwd +4<<<, vwxglhv d prgho lq zklfk glylvlrq pdqdjhuv
wu| wr lqxhqfh wkh lqwhuqdo fdslwdo pdunhw*v dvvhvvphqw ri wkhlu glylvlrq lq
rughu wr errvw wkhlu ohyho ri ixqglqj1 Lq klv prgho wkh glhuhqfh ehwzhhq
h{whuqdo dqg lqwhuqdo fdslwdo pdunhwv lv wkh lqirupdwlrqdo dgydqwdjh ri wkh
5|i? E2fff TL?|t L| |@| 4@?@}iht< ?Ui?|it 4@) Mi M*?|i_ i? |i) _L ?L|
@i *|4@|i @|Lh|)
OLiihc t 4L_i* @__hittit @ _gihi?| ttic ?@4i*) L
_iUi?|h@*3i_ @?_ ih@hUU@* €h4t _gih ? |ih4t Lu |ih @M*|) |L }i?ih@|i ?uLh4@|L?
@ML| ?it|4i?| ThLiU|t @?_ @**LU@|i U@T|@* |L |iti ThLiU|t iUi?|*)
9
odwwhu/ zkloh lq rxu prgho wkh glvwlqfwlyh ihdwxuh lv wkh khdgtxduwhuv* delolw|
wr uhdoorfdwh ixqgv dfurvv glylvlrqv dqg lqirupdwlrqdo dv|pphwulhv gr qrw
sod| dq| uroh1
Dw odvw/ d qxpehu ri sdshuv kdyh wulhg wr h{sodlq wkh cglyhuvlfdwlrq
glvfrxqw* dv wkh frqvhtxhqfh ri d qrq0udqgrp vhohfwlrq ri upv wkdw ehfrph
frqjorphudwh1 Sdshuv lq wklv olwhudwxuh lqfoxgh Fdpsd dqg Nhgld +4<<<,/
Ioxfn dqg O|qfk +4<<<, dqg Pdnvlprylf dqg Skloolsv +5334,1 Wkh jhqhudo
srlqw lv wkdw zhdnhu upv pd| kdyh d frpsdudwlyh dgydqwdjh lq phujlqj1
Wkxv/ hyhq li frqjorphudwh upv zrun hflhqwo|/ d glyhuvlfdwlrq glvfrxqw
dsshduv1 Rxu srlqw lv glhuhqw/ dqg lw lv uhodwhg wr wkh glhuhqw pdqdjhuldo
lqfhqwlyhv surylghg e| frqjorphudwh upv1
Dv d qdo uhpdun/ zh vwuhvv wkdw zh gr qrw zdqw wr dujxh wkdw uhvrxufhv
duh lqghhg doorfdwhg hflhqwo| lq lqwhuqdo fdslwdo pdunhwv1 Srzhu vwuxj0
johv/ lqxhqfh dfwlylwlhv hwf1 duh vxuho| suhvhqw lq prvw frusrudwlrqv/ dqg
vxfk lqhflhqflhv frqwulexwh wr uhgxflqj wkh ydoxh ri glyhuvlfdwlrq1 Rxu
srlqw lv wkdw glyhuvlhg upv pd| zhoo wudgh dw d glvfrxqw hyhq li lqwhuqdo
fdslwdo pdunhwv doorfdwh ixqgv hflhqwo|1 Wkh rswlpdo srolf| h{ srvw lv qrw
qhfhvvdulo| wkh rswlpdo srolf| h{ dqwh1
Wkh sdshu lv rujdql}hg dv iroorzv1 Lq vhfwlrq 5 zh ghvfuleh wkh edvlf
prgho1 Vhfwlrq 6 looxvwudwhv wkh pdlq hhfwv ri wkh czlqqhu slfnlqj* srolf|/
frpsdulqj wkh shuirupdqfh ri d glyhuvlhg frusrudwlrq lq zklfk ixqgv duh
doorfdwhg h{ srvw hflhqwo| zlwk wkh shuirupdqfh ri d cvwdqg dorqh* up1
Zh dovr vkrz wkdw wkh edvlf uhvxowv vwloo krog zkhq wkh up fdq surylgh
prqhwdu| lqfhqwlyhv wr wkh pdqdjhuv1 Vhfwlrq 7 glvfxvvhv dssolfdwlrqv ri
wkh edvlf prgho1 Zh uvw frqvlghu wkh lpsdfw ri dv|pphwu| lq vl}h ehwzhhq
wkh wzr glylvlrqv1 Qh{w/ zh dqdo|}h wkh rswlpdo fdslwdo doorfdwlrq srolf|/
dqg zh vkrz wkdw lw pd| eh rswlpdo wr pdnh wkh lqyhvwphqw ri d glylvlrq
ghshqg rq wkh fdvk rz jhqhudwhg e| wkh glylvlrq1 Vhfwlrq 8 frqwdlqv wkh
frqfoxvlrqv/ dqg dq dsshqgl{ froohfwv wkh surriv1
2
Ai L_i*
K
Rxu prgho kdv wkuhh djhqwv= khdgtxduwhuv + , dqg wzr glylvlrq pdqdjhuv/
 dqg
2 1 Hdfk glylvlrq kdv dvvhwv lq sodfh dqg qhz lqyhvwphqw rssru0
wxqlwlhv1 Glylvlrq pdqdjhuv ghulyh sulydwh ehqhwv iurp wkh dvvhwv ri wkhlu
glylvlrqv rqo|/ zkloh khdgtxduwhuv lv lqwhuhvwhg lq wrwdo uhwxuqv1 Wkh wlplqj
ri wkh prgho lv dv iroorzv1
P
P
Dw
w @ 3 pdqdjhu P
zrunv zlwk wkh dvvhwv douhdg| lq sodfh lq klv
:
glylvlrq1 Wkh h{lvwlqj surmhfw fdq hlwkhu vxffhhg ru idlo1 Li lw idlov/
lw surgxfhv d fdvk rz htxdo wr @ 31 Li wkh surmhfw vxffhhgv/ wkh
fdvk rz lv @ 41 Wkh suredelolw| ri vxffhvv lv ghwhuplqhg e| wkh
ohyho ri hruw h{huwhg e| wkh pdqdjhu1 Zh dvvxph wkdw li wkh pdqdjhu
fkrrvhv d ohyho ri hruw / wkhq wkh h{lvwlqj surmhfw lv vxffhvvixo zlwk
A2
suredelolw| 1 Wkh glvxwlolw| ri hruw lv + , @ 2
1e
F
F
h
h
w =
h #h
n
v
Dw @ 8 wkh wzr pdqdjhuv dqg khdgtxduwhuv revhuyh d vljqdo wkdw
surylghv lqirupdwlrq rq wkh surgxfwlylw| ri wkh lqyhvwphqw surmhfwv
dydlodeoh dw @ 4 lq wkh wzr glylvlrqv1
w
w
Dw @ 4 khdgtxduwhuv revhuyhv wkh fdvk rz surgxfhg e| wkh wzr
glylvlrqv/  dqg 2 dqg uhglvwulexwhv ixqgv wr wkh glylvlrqv1 Wkh rog
dvvhwv lq sodfh duh ixoo| ghsuhfldwhg1 Zh dvvxph wkdw wkh up kdv
qr dffhvv wr h{whuqdo qdqfh/D vr wkdw  . 2 lv wkh wrwdo dprxqw ri
ixqgv wkdw fdq eh uhlqyhvwhg lq shulrg 51 khdgtxduwhuv kdv wkh srzhu wr
doorfdwh ixqgv dfurvv glylvlrqv lq d glyhuvlhg up1 Zh ghqrwh zlwk wkh ixqgv dvvljqhg wr glylvlrq 1 Zh dvvxph wkdw khdgtxduwhuv doorfdwh
doo ixqgv wr wkh glylvlrqv/ vr wkdw  . 2 @  . 2 1 Wklv pd| eh
mxvwlhg hlwkhu e| dvvxplqj wkdw hdfk glylvlrq dozd|v kdv vxflhqwo|
surwdeoh lqyhvwphqw surmhfwv1 Dowhuqdwlyho|/ zh pd| dvvxph wkdw
khdgtxduwhuv kdv d suhihuhqfh iru hpsluh0exloglqj1
F
F
F F
N
l
N
N
Iru vlpsolflw| zh zloo dvvxph wkdw wkh vljqdo
v
@
v
v
H Uh v A H Uh2 v
fdq rqo| wdnh wzr ydoxhv1
/ zkloh li
rffxuv 1 Lq rughu wr vlpsoli| ixuwkhu zh dvvxph=
1
wkhq wkh h{shfwhg uhwxuq lq wkh uvw glylvlrq lv kljkhu wkdq lq
wkh vhfrqg/ wkdw lv
.
F
l |lhogv d fdvk rzS N Uh
w
Dw @ 5 wkh lqyhvwphqw lq glylvlrq
Li
F
v
@
v2
wkh rssrvlwh
H Uh v H Uh2 v2 U H Uh2 v H Uh v2 U=
e
Lt| Lu |i hit*|t ? tiU|L? L*_ uLh @?) EA ?Uhi@t?} @?_ UL?i D
At E@_4||i_*) i |hi4i @tt4T|L? @**Lt t |L uLUt L? |i U@ti ihi u?_t
@hi tU@hUi @?_ |t ??ihTU!?} @t @ }ih TL|i?|@* |L Uhi@|i @*i M) hi@**LU@|?}
hitLhUit
@UhLtt _tL?t
S
t ? 5U@hut|i? @?_ 5|i? E2fffc i @tt4i |@| ? |i tiUL?_ TihL_ |i _tL?@*
4@?@}iht
_L ?L| @i |L i ih| @?) igLh| At t LMLt*) ?L?itti?|@*
.
L|i |@| i @tt4i |@| |i ThL€|@M*|) Lu i@U _tL? ? TihL_ 2 t i L}i?Lt
W? @ 4Lhi }i?ih@* uh@4iLh!c |i hi|h? L? |i ?it|4i?| Lu i@U _tL? tL*_ Mi @
u?U|L? Lu ML| 4@?@}ih@* igLh| @?_ *U!< Ai? |ihi L*_ Mi @ UL?|ih@*?} igiU|G
UL4Ti||L? uLh u?_t 4@) MLLt| 4@?@}ih@* ?Ui?|it 6Lh @ |hi@|4i?| Lu |t U@ti tii
W?_iht| @?_ w@ E2fff
@
@ @
;
@
UU
Zh ghqh @ orvv ri jhqhudolw|=
dqg dvvxph
UA
31 Wkh dvvxpswlrq
UA
lw mxvw vd|v wkdw wkh glylvlrqdo pdqdjhu fdq dw prvw
s
vtxdqghu frpsohwho| wkh ixqgv rewdlqhg1 Dw odvw/ zh ghqh
dqg dvvxph
v v2
Su +
@
s# >
s
, @ 4
3 lv zlwkrxw
@ Su +
+3 4,1 Jlyhq wkh dvvxpswlrq rq wkh vxssruw ri
v
v v
@
,/
/ zh kdyh
1
Ilqdoo| zh qhhg wr vshfli| wkh remhfwlyh ixqfwlrqv ri wkh khdgtxduwhuv
dqg ri glylvlrq pdqdjhuv1 Khdgtxduwhuv pd{lpl}hv wrwdo uhwxuqv1 Frqfhuq0
lqj glylvlrq pdqdjhuv/ zh iroorz Vwhlq +4<<:, dqg dvvxph wkdw hdfk glylvlrq
pdqdjhu uhfhlyhv sulydwh ehqhwv ri frqwuro wkdw duh sursruwlrqdo wr wkh
jurvv rxwsxw ri lwv glylvlrq1 Pruh suhflvho|/ zh dvvxph wkdw d glylvlrq pdq0
!# >
N Uh H
+3 4, ri wkh fdvk rz
djhu uhdsv sulydwh ehqhwv htxdo wr d iudfwlrq
jhqhudwhg e| klv glylvlrq lq wkh vhfrqg shulrg
1
Wklv dvvxpswlrq lp0
solhv wkdw hdfk pdqdjhu dozd|v suhihuv pruh fdslwdo wr ohvv/ exw frqglwlrqdo
rq ehlqj jlyhq d fhuwdlq dprxqw ri fdslwdo hdfk pdqdjhu wulhv wr lqyhvw lw lq
wkh prvw surwdeoh surmhfw dydlodeoh1 Lq rwkhu zrugv/ pdqdjhuv duh hpsluh
exloghuv/ exw wkh| suhihu pruh surwdeoh hpsluhv wr ohvv surwdeoh hpsluhv1
Ixuwkhupruh/ lw lv wkh srvvlelolw| ri uhdoorfdwlqj uhvrxufhv dfurvv glylvlrqv
wkdw pd| fuhdwh d glyhujhqfh ri lqwhuhvwv ehwzhhq wkh khdgtxduwhuv dqg wkh
glylvlrq pdqdjhuv1 Zlwkrxw wkh srvvlelolw| ri czlqqhu0slfnlqj* wkhuh zrxog
eh qr frqlfw ri lqwhuhvwv ehwzhhq khdgtxduwhuv dqg glylvlrqv1
Dv lw lv frpprq lq wklv olwhudwxuh/ zh ehjlq rxu dqdo|vlv e| dvvxplqj
qrq0uhvsrqvlyhqhvv ri pdqdjhuv wr prqhwdu| lqfhqwlyhv1 Lq vxevhfwlrq 614/
zh frqvlghu wkh fdvh lq zklfk prqhwdu| lqfhqwlyhv fdq eh surylghg1
Irupdoo|/ wkh xwlolw| ixqfwlrq ri wkh pdqdjhu ri glylvlrq
X h > N > Uh
l
lv jlyhq e|=
!N Uh n h
2
@
5
Iru vlpsolflw| zh dvvxph wkdw sulydwh ehqhwv duh sv|fklf/ wkdw lv wkh| gr
qrw ghulyh iurp cvwhdolqj* ru plvxvlqj wkh frpsdq|*v dvvhwv1
sulydwh ehqhwv fdq eh wkrxjkw ri lq d qxpehu ri zd|v=
b
Dv xvxdo/
wkh xvxdo shunv/
wkh sv|fklf ehqhwv iurp hpsluh exloglqj/ hwf1 Ilqdoo|/ wkh ulvn0iuhh lqwhuhvw
udwh lv qrupdol}hg wr }hur dqg doo djhqwv duh ulvn qhxwudo1
`i UL*_ i^@**) @tt4i |@| _tL? 4@?@}iht _ihi Th@|i Mi?i€|t @*tL uhL4 |i
€ht| TihL_ U@t L At L*_ ?L| @giU| Lh hit*|t Ai U@*U*@|L?t uLh |t U@ti U@?
Mi buL?_ ? htUL @?_ @??3 E2fff
Bi? |@| i @tt4i |@| Th@|i Mi?i€|t @hi @ UL?t|@?| uh@U|L? Lu |i _tL?<t
}hLtt L|T|c |t @tt4T|L? @t ?L tihLt 4T*U@|L?t uLh |i @?@*)tt Wu Th@|i
Mi?i€|t ihi i |h@U|i_ @| |i i Ti?ti Lu ThL€|tc i tL*_ 4*|T*) |i i@_^@h|iht<
ThL€| M) @ UL?t|@?|
H
<
Ai h}| @?_ |i #@h! 5_i Lu `??ihU!?}<
Lq wklv vhfwlrq zh zdqw wr kljkoljkw wkh edvlf wudgh0r ehwzhhq lqfhqwlyh
h{ srvw hflhqf|1 Zh wkhuhiruh frqvlghu wzr dowhuqdwlyh ch{0
wuhph* rujdql}dwlrqdo irupv= Wkh cvwdqg0dorqh* rqh/ lq zklfk glylvlrqv duh
frpsohwho| vhsdudwhg dqg qr lqwhuqdo fdslwdo pdunhw h{lvwv/ dqg wkh sxuh
lqwhuqdo fdslwdo pdunhw rqh/ lq zklfk fdslwdo lv hqwluho| dvvljqhg wr wkh h{
srvw prvw hflhqw glylvlrq1
Lq wkh vwdqg0dorqh fdvh/ e| h{huflvlqj hruw h
dw w @ 3 wkh pdqdjhu ri
glylvlrq l rewdlqv dq h{shfwhg dprxqw h
4 . +4 h
, 3 @ h
ri ixqgv iru
uhlqyhvwphqw dw wlph 41 Jlyhq wkh lqirupdwlrq dw wlph 3/ wkh h{shfwhg fdvk
rz jhqhudwhg e| wkrvh ixqgv dw w @ 5 lv=
surylvlrq dqg
^s
+U . , . +4 s
, U` h
@ ^U . s
` h
zkhuh sQ @ s dqg s @ 4 s1 Wkh sureohp ri P
dw wlph }hur lv=
pd{ ! ^U . s ` h
e
n
h
5
Wkh qhfhvvdu| dqg vxflhqw frqglwlrq iru d pd{lpxp lv=
h @
! +U . s ,
n
+4,
Zh fdoo h7
wkh vroxwlrq wr wklv htxdwlrq1 Wr rewdlq lqwhulru vroxwlrqv iru h
zh lpsrvh wkh iroorzlqj
Dvvxpswlrq 4 U . ? n
Wkh vxp ri wkh h{shfwhg surw xqghu wkh vwdqg dorqh vroxwlrq lv jlyhq e|=
7
7 @ +U . s, h7
Q . +U . +4 s, , h =
Frqvlghu qrz wkh sxuh LFP fdvh1 Qrz khdgtxduwhuv revhuyhv v dw wlph
w @ 3=8 dqg wkhq doorfdwhv hqwluho| wkh ixqgv wr glylvlrq l1QB Wkhuhiruh/
glylvlrq l idfhv d suredelolw| 4 s ri kdylqj }hur ixqgv dqg d suredelolw|
At 4T*U@|L? Lu Lh 4L_i* t i |hi4i @?_ | t _i |L |i @tt4T|L? |@| |i
hi|h? |L |i ?it|4i?| ? i@U _tL? t *?i@h `| _iUhi@t?} 4@h}?@* hi|h?t Lu
?it|4i?|c ii? |i *itt ThL€|@M*i _tL? UL*_ LM|@? @ TLt|i @4L?| Lu u?_t
QB
43
s ri kdylqj doo ixqgv1 Wkh wrwdo h{shfwhg dprxqw ri ixqgv lv hQ . h 1 Wkh
sureohp iru P lv wkhuhiruh=
h
pd{ !s +U . , +h . h! , n e
5
dqg wkh qhfhvvdu| dqg vxflhqw frqglwlrq iru d pd{lpxp lv=
h @
!s +U . ,
n
+5,
Ohw xv fdoo hU
wkh vroxwlrq wr wklv htxdwlrq1 Wkh h{shfwhg surw iru
khdgtxduwhuv lv jlyhq e|=
. hU
U @ +U . , hU
Q
Zh fdq qrz frpsduh wkh h{shfwhg surwv xqghu wkh sxuh LFP dqg vwdqg
dorqh irupv1 Zh kdyh=
U
k
7 @ +4 s, hU
. shU
Q
U
+U . s, h7
. +U . +4 s, , h7 hU
Q hQ
Wklv fdq eh uhdg dv iroorzv1 Wkh whup
. shU
+4 s, hU
Q
+6,
l
uhsuhvhqwv wkh czlqqhu slfnlqj* hhfw1 Zlwk suredelolw| +4 s,/ wkh vhfrqg
glylvlrq lv wkh pruh surwdeoh rqh1 Lq wkh VD fdvh/ wklv grhv qrw ohdg wr
dq| h{wud ixqglqj iru wkh up1 Lq wkh LFP fdvh/ glylvlrq 5 rewdlqv wkh fdvk
1 H{shfwhg surw wkhuhiruh lqfuhdvhv
jhqhudwhg e| glylvlrq 4/ wkdw lv hU
Q
1
D
vlplodu
hhfw
lv
dw zrun zkhq glylvlrq 4 lv wkh pruh
e| +4 s, hU
Q
surwdeoh1 Wklv whup lv wkh euljkw vlgh ri lqwhuqdo fdslwdo pdunhwv= Uhvrxufhv
duh h{0srvw doorfdwhg wr wkh ehvw lqyhvwphqw1
Wkh nh| srlqw ri rxu sdshu lv wkh vhfrqg whup1 Qrwlfh wkdw wklv whup
@ h7
zrxog eh }hur li zh kdg hU
1 Krzhyhu/ vlqfh=
!+U . s ,
!s +U . ,
?
n
n
zh kdyh hU
? h7
1 Wklv lv wkh clqfhqwlyh hhfw* ghqrwlqj wkh uhgxfwlrq
lq h{shfwhg surwv dv d frqvhtxhqfh ri wkh uhgxfhg lqfhqwlyhv wkdw pdq0
djhuv kdyh zkhq ixqgv duh uhglvwulexwhg dfurvv glylvlrqv1 Lq idfw/ wkh whup
44
+U . s , ghqrwhv wkh jurvv h{shfwhg uhwxuq rq ixqgv lqyhvwhg lq wkh gl0
ylvlrq/ dqg wkh uhgxfwlrq lq h{shfwhg surw lq glylvlrq l lv htxdo wr wkh
U
, wlphv wklv
uhgxfwlrq lq wkh dprxqw ri ixqgv jhqhudwhg +wkdw lv/ h7
h
uhwxuq1
Wkh vljq ri U 7 ghshqgv rq wkh sdudphwhuv dv iroorzv1
Sursrvlwlrq 4 d, Iru dq| jlyhq yd
s
U
9
? 9
? 3
A 9
U
A3
s@ Q
s
Q
>4
vlplodu + lv
Sduw d, vwdwhv wkdw zkhq wkh glylvlrqv duh h{shfwhg wr eh
vpdoo, wkhuh lv qrw pxfk srlqw lq uhdoorfdwlqj ixqgv/ vr wkdw wkh suhgrplqdqw
hhfw ri lqwhuqdo fdslwdo pdunhwv lv wkh uhgxfwlrq lq lqfhqwlyhv1 Lq wklv fdvh
wkh glyhuvlhg up wudghv dw d glvfrxqw zlwk uhvshfw wr wkh vwdqg0dorqh
ehqfkpdun1QQ Dv lqfuhdvhv uhdoorfdwlrq fuhdwhv pruh ydoxh/ dqg wkh euljkw
vlgh ri lqwhuqdo fdslwdo pdunhwv hyhqwxdoo| suhydlov1
glhuhqfhv lq surwdelolw|
Sduw e, dgguhvvhv wkh lvvxh ri krz
ehwzhhq wkh wzr glylvlrqv ohdg wr d kljkhu ru orzhu glyhuvlfdwlrq glvfrxqw
+ru suhplxp,1 Wkh dgydqwdjh ri wkh LFP irup lv dw lwv pd{lpxp zkhq
wkh wzr glylvlrqv duh
lghqwlfdo1 Dv wkh glhuhqfh ehwzhhq glylvlrqv
lqfuhdvhv/ wkh VD irup ehfrphv pruh dsshdolqj1 Wr judvs wkh lqwxlwlrq/
frqvlghu wkh fdvh ri s forvh wr 41 Wkhq wkh pdqdjhu ri glylvlrq 4 lv doprvw
vxuh ri rewdlqlqj doo wkh ixqgv lq dq lqwhuqdo fdslwdo pdunhw/ zklovw wkh
pdqdjhu ri glylvlrq 5 dqwlflsdwhv wkdw lw zloo rewdlq qr ixqgv doprvw vxuho|1
Iru glylvlrq 4 lqfhqwlyhv duh dv lq wkh vwdqg dorqh fdvh1 Pruhryhu/ vlqfh s
lv forvh wr 4 zlqqhu slfnlqj grhv qrw dgg pxfk ydoxh1 Rq wkh rwkhu kdqg/
wkh lqfhqwlyhv iru wkh pdqdjhu ri glylvlrq 5 duh forvh wr 3 lq wkh fdvh ri
wkh lqwhuqdo fdslwdo pdunhw/ zkloh khu hruw lv srvlwlyh lq wkh vwdqg dorqh
fdvh1 Wkxv/ lqwhuqdo fdslwdo pdunhwv duh ohvv ghvludeoh zkhq glylvlrqv duh
yhu| glyhuvh1 Wkh qhjdwlyh lpsdfw ri dv|pphwu| rq glyhuvlhg up kdv dovr
ehhq srlqwhg rxw e| Udmdq 1 +5333,1 Wkh| qg wkdw dv glyhuvlw| lq
rssruwxqlwlhv dprqj glylvlrqv lqfuhdvhv/ lqyhvwphqw ehfrphv ohvv hflhqw
dqg upv duh ohvv ydoxdeoh1 Wkh glhuhqfh ehwzhhq wkhlu sdshu dqg rxuv
lv wkdw lq wkhlu prgho lqhflhqflhv duh fdxvhg e| ixqgv ehlqj wudqvihuuhg
oxh ri
vxfk wkdw li
7
wkhq
U
> e, iru dq| jlyhq ydoxh ri
uhdfkhv lwv pd{lpxp dw
lqwhuydo
7
dqg
wkhuh h{lvwv d ydoxh
/ zkloh li
dqg
wkhq
wkh glhuhqfh
U
/ dqg lw lv d ghfuhdvlqj ixqfwlrq ri
U
7
rq wkh
1
h{ srvw
h{ dqwh
h{ dqwh
hw do
QQ
W| t Lh| t|hitt?} @| |t TL?| |@|c @t @** |i T@Tiht ? |t *|ih@|hic i _L ?L|
@?@*)3i ) |ihi t ?L tT?Lg Lu |i |L _tL?t
45
iurp glylvlrqv zlwk jrrg lqyhvwphqw rssruwxqlwlhv wr glylvlrqv zlwk srru
rssruwxqlwlhv/ zkloh lq rxu prgho ixqgv dozd|v jr wr wkh prvw surwdeoh
glylvlrq dqg lqhflhqflhv duh fdxvhg e| d uhgxfwlrq lq lqfhqwlyhv1

L?i|@h) W?Ui?|it
Zh qrz vkrz wkdw wkh edvlf prgho lv urexvw wr wkh suhvhqfh ri prqhwdu|
lqfhqwlyhv iru glylvlrq pdqdjhuv1
Wkh pdqdjhuv* xwlolw| ixqfwlrq qrz ehfrphv
D > z @ z . !N U
D n h
X h > N > U
5
zkhuh z lv wkh zdjh sdlg wr wkh pdqdjhu ri glylvlrq l1 khdgtxduwhuv pd{l0
pl}hv wrwdo uhwxuqv qhw ri wkh zdjhv sdlg wr glylvlrq pdqdjhuv1 Zh dvvxph
wkdw pdqdjhuv duh surwhfwhg e| olplwhg oldelolw|/ l1h1 z 31 Lq sulqflsoh/
wkhuh duh ydulrxv revhuydeoh hyhqwv rq zklfk wkh zdjh fdq eh frqglwlrqhg=
Krz pxfk fdvk lv surgxfhg lq wkh glylvlrq/ krz ixqgv duh doorfdwhg dfurvv
glylvlrqv dqg wkh uhwxuq rq fdslwdo lq hdfk glylvlrq1 Krzhyhu/ vlqfh wkh
sureohp lv wr surylgh lqfhqwlyhv wr h{huw hruw/ wkh rqo| hyhqw rq zklfk lw
pdnhv vhqvh wr frqglwlrq wkh zdjh lv wkh dprxqw ri fdvk F jhqhudwhg lq wkh
glylvlrq1 Jlyhq wkh olplwhg oldelolw| frqvwudlqw dqg wkh ulvn0qhxwudo iudph0
zrun/ wkh rswlpdo zdjh frqwudfw zloo sd| }hur wr wkh pdqdjhu zkhq F @ 3
dqg d srvlwlyh dprxqw zkhq F @ 41 Zlwkrxw dpeljxlw|/ ohw z ghqrwh wkh
zdjh sdlg wr wkh pdqdjhu ri glylvlrq l zkhq wkh uvw shulrg fdvk rz ri
glylvlrq l lv 41 Lq wkh vwdqg0dorqh fdvh/ wkh sureohp ri P lv=
h
^
z . !+U . s ,` h n pd{
e
5
Wkh qhfhvvdu| dqg vxflhqw frqglwlrq iru d pd{lpxp lv=
z . !+U . s ,
h @
n
Ye%
Zh ghqrwh e| h7
+z , wkh vroxwlrq wr wklv htxdwlrq/ dqg revhuyh wkdw Y
@
Q 1 Wkh vxp ri wkh h{shfwhg surw iru wkh wzr glylvlrqv xqghu wkh vwdqg dorqh
&
vroxwlrq lv qrzQ =
7
7 @ +U . s zQ, h7
Q +zQ , . +U . +4 s, z , h +z ,=
`i @tt4i |@| @}it @hi T@_ @| |i i?_ Lu |i tiUL?_ TihL_c tL |@| @** |i U@t
L ThL_Ui_ @| |i i?_ Lu TihL_  t hi?it|i_ 5?Ui i @tt4i |@| |i ?|ihit| h@|i
t f 4@?@}iht Mi@h ?L ULt| uLh |i _i*@) ? |i T@)4i?|
Q
46
Wkh rswlpdo zdjh iru hdfk glylvlrq lv jlyhq e| wkh frqglwlrq=
7
Ch
h7
+z , . +U . s z , Cz
@3
Ye%
dqg xvlqj wkh h{suhvvlrq rewdlqhg iru h7
+z , dqg Y
zh kdyh=
z7 @
4!
+U . s ,
5
zkhuh z7 A 3 vlqfh ! ? 41 Qrwh wkdw lq wkh vwdqg0dorqh fdvh wkh pdqdjhu
ri wkh prvw surwdeoh glylvlrq +l1h1 wkh rqh zlwk kljkhu s , lv jlyhq d pruh
kljk0srzhuhg lqfhqwlyh vfkhph/ vlqfh wkh fdvk surgxfhg e| wkh dvvhwv lq
sodfh lv pruh ydoxdeoh1 Vxevwlwxwlqj wkh rswlpdo zdjh lq wkh h{suhvvlrq iru
h7
+z , zh kdyh
4.!
h7
+7,
@ 5n +U . s ,=
Ilqdoo|/ vxevwlwxwlqj lq wkh h{suhvvlrq iru wrwdo surwv zh kdyh=
4
7 @
n
+4 . !,+U . s,
5
+4 . !,+U . +4 s,,
.
5
Zh wxuq qrz wr wkh dqdo|vlv ri wkh LFP fdvh1 Dv ehiruh/ zlwk suredelolw|
4 s glylvlrq l kdv }hur ixqgv dqg zlwk suredelolw| s uhfhlyhv wkh wrwdo
h{shfwhg dprxqw ri ixqgv hQ . h 1 Wkh sureohp iru P lv wkhuhiruh=
h
pd{ !s +U . , +h . h! , . h z n e
5
dqg wkh qhfhvvdu| dqg vxflhqw frqglwlrq iru d pd{lpxp lv
h @
z . !s +U . ,
=
n
+z , wkh vroxwlrq wr wklv htxdwlrq1 Wkh h{shfwhg surw iru
Ohw xv fdoo hU
khdgtxduwhuv lv jlyhq e|=
+zQ , . hU +z ,
U @ +U . , hU
Q
+zQ , zQ hU +z , z
hU
Q
Wkh rswlpdo zdjh iru hdfk glylvlrq lv jlyhq e|
U
gh
+z ,
+z , . +U . z , @3
hU
gz
47
dqg xvlqj wkh h{suhvvlrq iru hU
+z , zh kdyh
zU @
+4 !s ,
+U . ,=
5
Qrwh wkdw lq dq lqwhuqdo fdslwdo pdunhw prqhwdu| lqfhqwlyhv duh pruh kljk
srzhuhg iru wkh ohvv surwdeoh glylvlrq +l1h1 wkh rqh zlwk orzhu s ,1 Wkh
lqwxlwlrq lv wkh iroorzlqj= iurp wkh khdgtxduwhuv ylhzsrlqw wkh pdujlqdo
uhwxuq ri d xqlw ri fdvk lv U ./ lqghshqghqwo| ri wkh glylvlrq wkdw surgxfhv
wkh fdvk rz1 Rq wkh rwkhu kdqg/ devwudfwlqj iurp prqhwdu| lqfhqwlyhv/
wkh ohvv surwdeoh glylvlrq zrxog h{huw d orzhu hruw1 Wr frpshqvdwh iru wkh
uhgxfhg lqfhqwlyhv/ d kljkhu zdjh lv sdlg wr wkh pdqdjhu ri glylvlrq 51
Dqrwkhu lqwhuhvwlqj frpsdulvrq lv ehwzhhq wkh zdjh sdlg lq wkh vwdqg0
dorqh fdvh dqg wkh LFP fdvh1 Zh kdyh=
Ohppd 4
Wkh rswlpdo zdjh sdlg wr glylvlrq pdqdjhuv lv dozd|v pruh kljk 0
srzhuhg lq dq lqwhuqdo fdslwdo pdunhw wkdq lq wkh vwdqg0dorqh fdvh/ wkdw lv
zU z7 1
Wkh lqwxlwlrq lv wkh iroorzlqj= Lq wkh LFP fdvh glylvlrq pdqdjhuv kdyh
uhgxfhg qrq0prqhwdu| lqfhqwlyhv zlwk uhvshfw wr wkh VD fdvh/ vr wkdw lw lv
qhfhvvdu| wr surylgh vwurqjhu prqhwdu| lqfhqwlyhv lq rughu wr dfklhyh wkh
vdph ohyho ri hruw1 Pruhryhu/ wkh pdujlqdo ydoxh ri hruw lv kljkhu zkhq dq
lqwhuqdo fdslwdo pdunhw lv dfwlyh/ ehfdxvh ri wkh srvvlelolw| ri zlqqhu0slfnlqj1
Erwk dujxphqwv ohdg wr wkh frqfoxvlrq wkdw zdjhv duh kljkhu lq wkh LFP
fdvh1
Zh fdq qrz frpsduh pdqdjhuv* hruw lq wkh VD fdvh dqg lq wkh LFP
fdvh1 D sulrul wkh uhvxow lv qrw reylrxv1 Rq rqh kdqg/ wkh VD irup surylghv
vwurqjhu qrq0prqhwdu| lqfhqwlyhv iru hruw1 Rq wkh rwkhu kdqg/ Ohppd 4
vkrzv wkdw prqhwdu| lqfhqwlyhv duh vwurqjhu lq wkh LFP fdvh1 Wkh dqvzhu
lv surylghg lq wkh iroorzlqj=
li dqg rqo| li !U1
h7
Ohppd 5 hU
Zkhq wkh ydoxh fuhdwhg e| zlqqhu slfnlqj lv kljk + lv kljk, khdgtxduwhuv
rhuv glylvlrq pdqdjhuv d kljk0srzhuhg frpshqvdwlrq vfkhph wkdw lqgxfhv
d kljk ohyho ri hruw1 Dovr/ zkhq sulydwh ehqhwv duh vpdoo +! vpdoo, pdqdjh0
uldo lqfhqwlyhv ghulyh prvwo| iurp prqhwdu| frpshqvdwlrq/ zklfk lv kljkhu
lq wkh LFP fdvh1
48
Wkh prvw lpsruwdqw lvvxh lv wkh uhodwlyh ydoxh ri wkh vwdqg0dorqh fdvh
nR E-n{
lq wkh
ylv0
d0ylv wkh lqwhuqdo fdslwdo pdunhw1 Vxevwlwxwlqj h @
&
surw ixqfwlrq zh kdyh=
U @ n4
2 2
4
.
!s
4
.
!+4
s,
+U . , 5
. +U . ,
5
Frpsdulqj wklv h{suhvvlrq zlwk 7 zh kdyh wkh iroorzlqj=
Sursrvlwlrq 5 d, wkhuh lv d ydoxh 9 vxfk wkdw U 7 li dqg rqo|
U
9
li A 1 e, Wkh glhuhqfh 7 ghfuhdvhv lq s ryhu wkh udqjh
 > 4 li A !U/ dqg lqfuhdvhv rwkhuzlvh1
2
Rqo| zkhq wkh h{shfwhg ydoxh ri zlqqhu0slfnlqj lv vxflhqwo| kljk dq lq0
whuqdo fdslwdo pdunhw lv vxshulru wr wkh vwdqg0dorqh fdvh1 Rwkhuzlvh wkh
dgyhuvh dhfw rq lqfhqwlyhv ri wkh uhdoorfdwlrq ri frusrudwh uhvrxufhv lv wkh
grplqdqw hhfw dqg wkh ydoxh ri wkh frqjorphudwh lv ohvv wkdq wkh vxp ri
wkh vwdqg0dorqh ydoxhv ri wkh wzr glylvlrqv1 Wkhuhiruh/ wkh edvlf uhvxow ri
wkh suhylrxv vhfwlrq vwloo krogv= Wkh LFP irup lv vxshulru zkhq lv odujh/
dqg wkh VD irup lv vxshulru zkhq lv vpdoo1
Zkhq zh orrn dw wkh lpsdfw ri dv|pphwu| rq wkh frpsdudwlyh dgydqwdjh
ri wkh LFP irup/ wkh uhvxowv duh txdolwdwlyho| wkh vdph dv lq wkh fdvh ri
qr prqhwdu| lqfhqwlyhv zkhq A !U/ wkdw lv dq lqfuhdvh lq dv|pphwu|
ghfuhdvhv wkh dgydqwdjh ri wkh LFP irup1 Krzhyhu/ wkh rssrvlwh lv wuxh
zkhq ? !U +reylrxvo|/ xqghu wkh pdlqwdlqhg dvvxpswlrq s A 2 ,1
Wr xqghuvwdqg ehwwhu zk| lw lv vr/ uvw qrwlfh wkdw lq wkh LFP fdvh zh
kdyh=
4 4 . 4 ! +U . ,
U
.
h
@
hU

2
n
5
U
zklfk lv lqghshqghqw ri s1 Wkhuhiruh/ +U . , hU
.
h
lv frqvwdqw

2
zlwk uhvshfw wr s lq wkh LFP fdvh/ dqg dq| yduldwlrq pxvw frph iurp wkh
U
U U
h{shfwhg vdodu| sd|phqw/ hU
 z . h2 z2 1 Zkhq s lqfuhdvhv/ wkh
h{shfwhg zdjh eloo ghfuhdvhv1 Wkh uhdvrq lv wkdw dq lqfuhdvh lq s lqfuhdvhv
qrq0prqhwdu| lqfhqwlyhv lq glylvlrq 4/ dqg ghfuhdvhv qrq0prqhwdu| lqfhq0
wlyhv lq glylvlrq 51 Wkh rswlpdo uhvsrqvh e| khdgtxduwhuv lv wr ghfuhdvh
zU dqg lqfuhdvh z2U / pdlqwdlqlqj wkh vxp zU . z2U frqvwdqw1
Dv d frqvhtxhqfh ri wkh fkdqjhv lq qrq0prqhwdu| dqg prqhwdu| lqfhqwlyhv/
U
U
U
hU
 lqfuhdvhv dqg h2 ghfuhdvhv/ zkloh wkh vxp h .h2 uhpdlqv frq0
U
vwdqw1 Wklv lq wxuq lpsolhv wkdw wkh orz vdodu| z lv sdlg pruh riwhq/ dqg
49
U
wkh kljk vdodu| z2U lv sdlg ohvv riwhq1 Wkh h{shfwhg
sd|phqwhU
 z .
U
U
U
uhpdlqv frq0
hU
2 z2 wkhuhiruh ghfuhdvhv1 Vlqfh +U . , h . h2
U
vwdqw/ lqfuhdvhv1 Lq idfw zh kdyh1
CU @ 4 +U . ,2 !2 +5s 4,
Cs
5n
Frqvlghu qrz wkh VD fdvh1 Zh nqrz wkdw/ glhuhqwo| iurp zU / dq lqfuhdvh
lv dovr lqfuhdvlqj lq s / dq rssrvlwh hhfw
lq s lqfuhdvhv z7 1 Vlqfh h7
uhvshfw wr wkh LFP fdvh lv dw zrun= qrz wkh kljkhu vdodu| lv sdlg pruh
riwhq/ vr wkdw wkh wrwdo zdjh eloo lqfuhdvhv1 Krzhyhu/ zh dovr kdyh d srvlwlyh
hhfw/ vlqfh fdvk hqgv xs pruh iuhtxhqwo| lq wkh glylvlrq zlwk wkh kljkhu
uhwxuq1 Wkh vxp ri wkh wzr hhfwv pxvw eh srvlwlyh/ vlqfh khdgtxduwhuv
zrxog qrw rswlpdoo| fkrrvh wr sd| d kljkhu h{shfwhg vdodu| li wklv glg qrw
surgxfh d kljkhu surw1 Lq idfw zh kdyh=
C7 @ 4 +4 . !,2 2 +5s 4,
Cs
5n
Ohw xv qrz frpsduh wkh wzr ghulydwlyhv1 Frqvlghu uvw wkh h{wuhph fdvh lq
zklfk ! @ 3/ vr wkdw wkh frqglwlrq A !U lv dozd|v vdwlvhg1 Lq wklv fdvh
wkh LFP irup lv lqvhqvlwlyh wr s1 Wkh rqo| lqfhqwlyhv duh prqhwdu|/ dqg
ixqgv duh dozd|v doorfdwhg wr wkhlu ehvw xvh1 Ixuwkhupruh/ li ! @ 3 hruw
dqg vdodu| duh wkh vdph lq erwk glylvlrqv1 Wklqjv duh glhuhqw iru wkh VD
irup1 Djdlq/ lqfhqwlyhv duh rqo| prqhwdu|/ exw qrz khdgtxduwhuv rswlpdoo|
fkrrvh d kljkhu vdodu| iru wkh glylvlrq zlwk wkh kljkhu s dqg d orzhu vdodu|
iru wkh wkh rwkhu glylvlrq1 Wkh wrwdo txdqwlw| ri hruw uhpdlqv frqvwdqw/ exw
qrz pruh hruw lv surgxfhg lq wkh kljk0uhwxuq glylvlrq1 Wklv lqfuhdvhv wkh
ydoxh ri 7 / vlqfh lw lv qrz pruh olnho| wkdw ixqgv rewdlq d kljk uhwxuq1
Wkhuhiruh/ wkh frpsdudwlyh dgydqwdjh ri wkh VD irup yv1 wkh LFP lqfuhdvhv
zkhq s lqfuhdvhv1 Zkhq ! lv vxflhqwo| vpdoo d vlplodu hhfw lv dw zrun1
Frqvlghu qh{w wkh rwkhu h{wuhph fdvh lq zklfk lv yhu| forvh wr 3/ vr
wkdw wkh frqglwlrq ? !U lv dozd|v vdwlvhg1 Lq wklv fdvh d fkdqjh lq s
kdv edvlfdoo| qr lpsdfw rq 7 / vlqfh wkh rswlpdo dprxqwv ri vdodu| dqg
hruw uhpdlq wkh vdph lq erwk glylvlrqv1 Dw wkh vdph wlph/ lw uhpdlqv wuxh
wkdw U lqfuhdvhv ehfdxvh ri wkh uhgxfwlrq lq wkh zdjh eloo1 Wkhuhiruh/ lq
wklv fdvh lw lv wkh LFP irup wkdw lqfuhdvhv lwv frpsdudwlyh dgydqwdjh zkhq
s lqfuhdvhv1
4:
e
TT*U@|L?t
Lq wklv vhfwlrq zh h{whqg wkh edvlf prgho lq wzr gluhfwlrqv1 Iluvw/ zh dqd0
o|}h wkh fdvh lq zklfk d glylvlrq lv odujhu wkdq wkh rwkhu +ru lw kdv d kljkhu
surwdelolw| dw wlph 3,1 Zh vkrz wkdw wkh sureohp ri pdqdjhuldo lqfhqwlyhv
lv pruh vhyhuh zkhq d glylvlrq zklfk lv odujh dqg pdwxuh +d fdvk frz, lv
frpelqhg zlwk d vpdoohu exw pruh udslgo| jurzlqj glylvlrq +d ulvlqj vwdu,1
Vhfrqg/ zh vwxg| wkh fdvh lq zklfk wkh khdgtxduwhuv fdq frpplw wr ohdyh
wr hdfk glylvlrq d iudfwlrq ri wkh fdvk rz surgxfhg/ wkhuhe| olplwlqj wkh
vfrsh ri czlqqhu0slfnlqj*1 Zh vkrz wkdw lw pd| eh rswlpdo wr ohw wkh gl0
ylvlrqv uhwdlq d iudfwlrq ri wkhlu fdvk rz dqg zh fkdudfwhul}h wkh rswlpdo
uhwhqwlrq uxohv1 Lq erwk fdvhv zh jr edfn wr wkh edvlf prgho dqg zh ljqruh
iru vlpsolflw| prqhwdu| lqfhqwlyhv1
e
@t Lt @?_ +t?} 5|@ht
Zh qrz lqyhvwljdwh krz dv|pphwulhv lq glylvlrqv* vl}h lpsdfw rq wkh frq0
jorphudwh glvfrxqw1 Pruh suhflvho|/ zh duh lqwhuhvwhg lq wkh iroorzlqj txhv0
wlrq1 Dvvxph wkdw glylvlrq 4 lq shulrg 4 lv pruh surwdeoh wkdq glylvlrq 5/ lq
wkh vhqvh wkdw lw surgxfhv d kljkhu h{shfwhg fdvk rz1 Grhv wklv dv|pphwu|
lq vl}h dhfw wkh frqjorphudwh glvfrxqwB Dqg lq zklfk gluhfwlrqB
Dvvxph wkdw glylvlrq 4 kdv dvvhwv lq sodfh wkdw surgxfh d fdvk rz htxdo
wr F A 4 zlwk suredelolw| h> zkhuh h ghqrwhv dv ehiruh wkh hruw h{huwhg
e| wkh pdqdjhu ri glylvlrq 4/ dqg d fdvk rz htxdo wr 3 rwkhuzlvh1 Wkh fdvh
F @ 4 lv wkh rqh zh kdyh dqdo|}hg lq vhfwlrq 61 Li s A 2 wkhq glylvlrq 4
surgxfhv d kljkhu h{shfwhg fdvk rz iru d jlyhq dprxqw ri pdqdjhuldo hruw
wkdq glylvlrq 5 dqg lw dovr kdv ehwwhu lqyhvwphqw rssruwxqlwlhv1 Lqvwhdg/ li
s ? 2 wkhq glylvlrq 4 lv d cfdvk frz* dqg glylvlrq 5 d culvlqj vwdu*1
Dv xvxdo/ frqvlghu uvw wkh vwdqg0dorqh fdvh1 Wkh sureohp ri P dw wlph
}hur lv=
h2
!
+U . s, Fh n
pd{
eQ
5
Wkh qhfhvvdu| dqg vxflhqw frqglwlrq iru d pd{lpxp lv=
h7
 @
Zh zloo frqvlghu ydoxhv ri
F
! +U . s, F
n
vxfk wkdw
dozd|v lqwhulru1
E-n{
&
? 4/
vr wkdw wkh vroxwlrq lv
Wkh sureohp iru glylvlrq 5 lv xqfkdqjhg dqg wkxv
4;
! +U . +4 s,,
n
h7
2 @
Wkh h{shfwhg fdvk rz surgxfhg lq wkh vwdqg0dorqh fdvh lv
7
h7
 F . h2 @
l
!k
2
+U . s, F . U . +4 s,
n
Wkh vxp ri wkh h{shfwhg surw iru wkh wzr glylvlrqv xqghu wkh vwdqg dorqh
vroxwlrq lv jlyhq e|=
7
@
s, , h7
2 @
q
r
!
2 2
2
^U . s` F . ^U . +4 s,`
n
7 . +U . +4
+U . s, Fh
@
Qrwlfh wkdw erwk wrwdo fdvk rz dqg wkh vxp ri surwv duh lqfuhdvlqj lq
F1
Frqvlghu qrz wkh sxuh LFP fdvh1 Glylvlrq 4 idfhv d suredelolw| 4
kdylqj }hur ixqgv dqg d suredelolw|
P
s ri kdylqj doo ixqgv1
lv wkhuhiruh=
!s +U . , +h F . h2, n
pd{
eQ
h2
5
dqg wkh qhfhvvdu| dqg vxflhqw frqglwlrq iru d pd{lpxp lv=
hU

@
!
s +U . , F
n
Wkh sureohp ri glylvlrq 5 lv xqfkdqjhg dqg wkxv=
hU
2
@
!
+4 s, +U . ,
n
Wkh wrwdo h{shfwhg fdvk rz surgxfhg lq wkh LFP fdvh lv
hU
F . hU

2
@
k
l
!
2
+U . , sF . +4 s,
n
Wkh h{shfwhg surw iru khdgtxduwhuv lv jlyhq e|=
U
@
@
+U . ,
hU
F . hU

2
k
@
l
!
2
2
+U . , sF . +4 s,
n
4<
s ri
Wkh sureohp iru
Qrwh wkdw dovr lq wkh LFP fdvh erwk wkh wrwdo h{shfwhg fdvk rz surgxfhg
dqg h{shfwhg surwv duh lqfuhdvlqj lq
F
vlqfh wkh hruw ri wkh pdqdjhu ri
glylvlrq 4 lv lqfuhdvhg zlwkrxw dhfwlqj wkh hruw ri wkh pdqdjhu ri glylvlrq
51 Pruh lq jhqhudo/ zkhq
kljkhu
F
F
lqfuhdvhv erwk U dqg 7 lqfuhdvh/ vlqfh d
lv htxlydohqw wr d ehwwhu whfkqrorj|1
Zh fdq qrz dgguhvv wkh txhvwlrq ri wkh lpsdfw ri glhuhqfhv lq
F rq wkh
frqjorphudwh glvfrxqw1 Iroorzlqj wkh dqdo|vlv ri vhfwlrq 6/ wkh glhuhqfh lq
surw ehwzhhq wkh LFP fdvh dqg wkh VD fdvh fdq eh zulwwhq dv=
U
+U . s,
7 @ U F
h7
 F h
Glhuhqwldwlqj zlwk uhvshfw wr
g+U
gF
7 ,
k
F
+4
F . shU
s, hU

2
+U . +4 s,,
l
U
h7
2 h2
zh rewdlq=
k
C hU
F

@ +4 s, CF
l
k
C h7 F hU F
+U . s,  CF 
l
Wklv h{suhvvlrq fdq eh lqwhusuhwhg dv iroorzv1 Zkhq zh orrn dw wkh glhuhqfh
ehwzhhq U dqg 7 zh kdyh wr uhphpehu wkdw wkhuh duh wzr pdlq
frpsrqhqwv= Wkh czlqqhu0slfnlqj* sduw dqg wkh clqfhqwlyh0uhgxfwlrq* sduw1
Wkh uvw frpsrqhqw lv jlyhq e| wkh uvw whup/ dqg lw idyruv wkh LFP
irup zkhq
F
lqfuhdvhv1 Zh kdyh=
k
F
C hU

CF
l
@5
!
s +U . , F A 3
n
wkdw lv/ wkh h{shfwhg fdvk rz ri wkh uvw glylvlrq lqfuhdvhv zkhq
F
lq0
fuhdvhv1 Wklv lqfuhdvhg fdvk rz lv xvhg pruh hflhqwo| zkhq glylvlrq 5
lv pruh surwdeoh/ zklfk kdsshqv zlwk suredelolw| 4
s/ dqg lq vxfk fdvhv
surw lqfuhdvhv e| dq dprxqw 1 Qrwlfh wkdw wkh pdjqlwxgh ri
ghshqgv rq
s1
Y ^eU
`
Q
Y
Lq sduwlfxodu/ wkh orzhu wkh suredelolw| ri uhfhlylqj ixqgv/
wkh zhdnhu lv wkh hhfw ri dq lqfuhdvhg
F
rq wkh h{shfwhg fdvk rz1
Wkh vhfrqg whup uhsuhvhqwv wkh lqfhqwlyh0uhgxfwlrq frpsrqhqw1 Xvlqj
wkh irupxodv iru
h7

dqg
hU

zh kdyh=
U F @
h7
 F h
vr wkdw=
k
U F
C h7
 F h
CF
!
2
+4 s, UF
n
l
53
@5
!
+4 s, UF
n
Dv
F
lqfuhdvhv/ wkh lqfhqwlyh hhfw ehfrphv vwurqjhu1 Wkh h{shfwhg fdvk
rz jurzv xqghu erwk uhjlphv/ exw lw lqfuhdvhv pruh xqghu wkh vwdqg dorqh
irup= Zkhq glylvlrq 4 ehfrphv odujhu wkh glhuhqfh ehwzhhq wkh fdvk rz
surgxfhg xqghu wkh VD irup dqg xqghu wkh LFP irup lv pruh surqrxqfhg1
F 1 Qrwlfh
U F lv kljkhu zkhq s lv vpdoohu1
F
h
h7


Lqwxlwlyho|/ dq lqfuhdvh lq F vkrxog jlyh d frpsdudwlyh dgydqwdjh wr wkh
Wklv lv d frqvhtxhqfh ri wkh idfw wkdw hruw ghshqgv olqhduo| rq
dovr wkdw wkh glhuhqfh
LFP irup zkhq wkh zlqqhu slfnlqj hhfw lv vwurqjhu/ zklfk kdsshqv zkhq
lv odujh1 Ixuwkhupruh/ zkhq
s
lv orz dq lqfuhdvh lq
F
kdv d zhdn hhfw
/ dqg d vwurqjhu hhfw rq lqfhqwlyh uhgxfwlrq/ wkdw lv wkh glhuhqfh
rq hU

h7 F hU F 1 Lq idfw zh kdyh wkh iroorzlqj uhvxow1

Sursrvlwlrq 6 Wkh glhuhqfh U 7 lv lqfuhdvlqj lq F li s2 A U2/

dqg ghfuhdvlqj rwkhuzlvh1
Wkh uhvxow lpsolhv wkdw wkh LFP irup lv pruh olnho| wr eh glvdgydqwdjhrxv
zkhq d cfdvk frz* dqg d culvlqj vwdu* duh sxw wrjhwkhu1 Lq wklv fdvh
s
lv
vpdoo/ vlqfh wkh glylvlrq zlwk wkh kljkhu fdvk rz lv wkh rqh zlwk wkh zruvw
lqyhvwphqw rssruwxqlwlhv1 Wkxv/ zh duh olnho| wr kdyh s2 ? U2 / dqg dq
lqfuhdvh lq dv|pphwu| uhgxfhv wkh frpsdudwlyh dgydqwdjh ri wkh LFP irup
yv1 wkh VD irup1 Rwkhu wklqjv htxdo/ zh vkrxog wkhuhiruh revhuyh rq dyhudjh
d kljkhu frqjorphudwh glvfrxqw zkhq glylvlrqv duh dv|pphwulf lq fxuuhqw
surwdelolw| dqg jurzwk survshfwv/ zlwk wkh glylvlrqv kdylqj pruh fdvk dovr
ehlqj wkh rqhv zlwk srruhu jurzwk survshfwv1
e2
T|4@* +i@**LU@|L? +*it
Zh srlqwhg rxw lq wkh suhylrxv vhfwlrq wkdw wkh sxuh LFP fdvh lv wkh uhvxow
ri wkh lqdelolw| wr frpplw rq wkh sduw ri khdgtxduwhuv1 Vxfk dvvxpswlrq lv
suredeo| h{wuhph1 Lq jhqhudo wkh rujdql}dwlrqdo vwuxfwxuh fdq eh kduqhvvhg
lq d zd| wkdw pdnhv +dw ohdvw sduwldo, frpplwphqw qrw wr lqwhuihuh srvvleoh1
Iru h{dpsoh/ khdgtxduwhuv pd| ghflgh wkdw rqo| lqyhvwphqw surmhfwv ri d
fhuwdlq vl}h qhhg vxshulru dssurydo/ doorzlqj glylvlrqv wr vshqg wkhlu rzq
fdvk rq surmhfwv ri vpdoohu vl}h1 Wklv zrxog uhvxow lq jlylqj uhdo dxwkrulw|
wr glylvlrqdo pdqdjhuv ryhu d sduw ri wkh fdvk rz wkh| jhqhudwh1
Lq wklv vhfwlrq zh ljqruh wkh h{dfw plfurhfrqrplf phfkdqlvp wkurxjk
zklfk frpplwphqw lv dwwdlqhg/ dqg zh vlpso| dvvxph wkdw khdgtxduwhuv
fdq ghflgh wkh iudfwlrq ri ixqgv wkdw hdfk glylvlrqdo pdqdjhu lv doorzhg wr
uhwdlq dqg dxwrpdwlfdoo| uhlqyhvw lq wkh glylvlrq1
54
Doorzlqj glylvlrqv wr uhwdlq d iudfwlrq ri wkhlu fdvk rz kdv frvwv dqg
ehqhwv iru wkh khdgtxduwhuv1 Rq wkh rqh kdqg/ wkh ixqgv uhwdlqhg e| wkh
glylvlrq pd| eh uhlqyhvwhg lq d vxerswlpdo surmhfw iurp wkh khdgtxduwhuv
srlqw ri ylhz +ohvv czlqqhu0slfnlqj*,> rq wkh rwkhu kdqg/ uhwdlqlqj ixqgv
errvwv pdqdjhuldo lqlwldwlyh1 Wkh rswlpdo ghjuhh ri lqwhuyhqwlrq rswlpdoo|
edodqfhv wkhvh wzr frqwudvwlqj hhfwv1
Iru vlpsolflw| zh devwudfw djdlq
iurp prqhwdu| lqfhqwlyhv dqg zh jr edfn wr wkh fdvh zkhuh pdqdjhuv duh

prwlydwhg rqo| e| sulydwh ehqhwv1 Zh dvvxph s A 2 / vr wkdw wkh uvw
glylvlrq lv
pruh surwdeoh1 Zh ghqrwh e| wkh vkduh ri lwv fdvk
rz F uhwdlqhg e| glylvlrq l/ vr wkdw lq hdfk shulrg d wrwdo dprxqw ri
h{ dqwh
 , F . +4 2, F2 fdq eh uhdoorfdwhg e| wkh khdgtxduwhuv wr
wkh glylvlrq zlwk wkh kljkhvw h{shfwhg uhwxuq1 Wkh pdqdjhu ri glylvlrq l
fdvk +4
pd{lpl}hv=
! h ^U . s ` . s +U . ,
2
+4
,h . +4 ! ,h9! n h5
vr wkdw wkh IRF iru glylvlrq l lv=
h
@
!
^s
+U . , . +4
n
s , U` 1
+8,
Zh ghqrwh e| h
+
, wkh xqltxh vroxwlrq wr wklv htxdwlrq1
Zh vwduw rxu dqdo|vlv iurp wkh iroorzlqj vlpsoh revhuydwlrq1
Ohppd 6 Vxssrvh wkdw K zdqwv wr lpsohphqw wkh vdph ohyho ri hruw h9 @
!
!
lq wkh wzr glylvlrqv/ dqg ohw  dqg 2 eh wkh wzr ohyhov ri ixqg
!
!
uhwhqwlrq wkdw dwwdlq wklv jrdo1 Wkhq  ? 21
h @ h2
Wkh sursrvlwlrq vwdwhv wkdw zkhqhyhu wkh vdph ohyho ri hruw lv uhtxluhg
lq wkh wzr glylvlrqv/ wkhq wkh
kljkhu
ohvv
surwdeoh glylvlrq lv doorzhg wr uhwdlq d
vkduh ri lwv rzq fdvk rz1 Wkh lqwxlwlrq lv wkh iroorzlqj= gxh wr
wkh uhdoorfdwlrq ri ixqgv rshudwhg e| khdgtxduwhuv/ lw lv kdughu wr prwlydwh
wkh pdqdjhu ri wkh ohvv surwdeoh glylvlrq/ vlqfh vkh dqwlflsdwhv d kljkhu
suredelolw| wkdw ixqgv zloo eh doorfdwhg wr wkh rwkhu glylvlrq1 Wkxv/ lq rughu
wr h{wudfw d jlyhq dprxqw ri hruw/ lw lv qhfhvvdu| wr doorz pdqdjhuv ri ohvv
surwdeoh glylvlrqv wr uhwdlq dqg kdyh glvfuhwlrq ryhu d odujhu iudfwlrq ri
wkhlu fdvk rz1
*|L} |i U@t L Lu i@U €h4 U@? Mi i|ih f Lh c Lh hi|i?|L? h*i t ?L| |i
4Lt| }i?ih@* L?ic t?Ui i _L ?L| @**L UL?_|L??} |L |i hi@*3@|L? Lu |i U@t L
Lu _tL? !
Q
55
Qrwlfh krzhyhu wkdw lq jhqhudo lw zloo eh rswlpdo iru khdgtxduwhuv wr
lpsohphqw glhuhqw ohyhov ri hruw ehwzhhq wkh wzr glylvlrqv1 Lq sduwlfxodu/
holflwlqj hruw iurp wkh vhfrqg glylvlrq lv pruh frvwo| wkdq holflwlqj hruw
iurp wkh uvw glylvlrq/ vlqfh d juhdwhu vkduh ri ixqgv kdv wr eh glyhuwhg wr
wkh ohvv surgxfwlyh glylvlrq +lq h{shfwhg ydoxh,/ dqg wklv ohdgv wr d ghfuhdvh
lq wkh dprxqw ri hruw zklfk lv rswlpdo wr uhtxluh wr wkh ohvv surwdeoh
glylvlrq1 Wklv lq wxuq uhgxfhv wkh vkduh ri ixqgv qhfhvvdu| wr lpsohphqw wkh
ghvluhg ohyho ri hruw1 Ghshqglqj rq zklfk rqh ri wkh wzr hhfwv suhydlov/
wkh vkduh ri ixqgv dvvljqhg wr wkh ohvv surwdeoh glylvlrq pd| eh kljkhu ru
orzhu wkdq wkh vkduh dvvljqhg wr wkh pruh surwdeoh glylvlrq1
Wkh sureohp fdq eh irupdoo| dgguhvvhg dv iroorzv1 khdgtxduwhuv* h{0
shfwhg surw fdq qrz eh zulwwhq dv=
+ > 2 ,
@
2, h2 +2,, +U . , . 2h2 +2, U`
. +4 s, ^ h + , U . +h2 +2 , . +4  , h + ,, +U . ,`
s ^+h + , . +4
Wkh sureohp lv wkhuhiruh=
pd{
Q Mdfcoc Mdfco
+ > , 1

2
Xvlqj wkh h{suhvvlrqv iru h + , dqg h + , jlyhq e| +8, dqg frqvlghulqj
ydoxhv ri wkh sdudphwhuv iru zklfk dq lqwhulru vroxwlrq h{lvwv/ zh kdyh wkdw
wkh xqltxh vroxwlrq lv jlyhq e|=
+U . , +U s,
W @
5 +4 s, U
+U . , +U +4 s, ,
W @
5sU
zkhqhyhu wkh sdudphwhuv duh vxfk wkdw W # +3> 4,1 Li - 3R-3R- 4 wkhq
wkh vroxwlrq lv W @ 4/ dqg li - 3R-3R- 3 wkhq wkh vroxwlrq lv W @ 31
Pruh jhqhudoo|/ revhuyh wkdw wkh rswlpdo iudfwlrq W iru d glylvlrq kdylqj
suredelolw| s ri ehlqj wkh prvw surwdeoh rqh fdq eh zulwwhq dv=
+U . , +U s,
W +s , @
5 +4 s, U
Qrwh wkdw W +s, fdq eh vwulfwo| srvlwlyh/ wkdw lv lw fdq eh rswlpdo wr doorz
glylvlrqv wr uhwdlq d iudfwlrq ri wkh fdvk rz wkh| kdyh surgxfhg1 Wklv


2
2

2
E
n{E
2E
E
n{E
2E
{
{
56
{
{
uhvxow vkhgv d glhuhqw oljkw rq wkh qglqjv ri Vklq dqg Vwxo} +4<<;,1 Wkh|
vkrz wkdw wkh doorfdwlrq ri ixqgv grhv qrw rqo| ghshqg rq wkh lqyhvwphqw
rssruwxqlwlhv/ exw dovr rq wkh fdvk rz jhqhudwhg e| hdfk glylvlrq1 Pruh
suhflvho|/ wkh ixqgv doorfdwhg wr d glylvlrq duh pruh vhqvlwlyh wr wkh glylvlrq*v
fdvk rz wkdq wr rwkhu glylvlrqv* fdvk rz1 Wklv lv frqvlvwhqw zlwk wkh
qglqj ri rxu prgho1 Vxssrvh wkh fdvk rz ri glylvlrq l lv lqfuhdvhg e| rqh
groodu1 Wkhq wkh lqyhvwphqw ri glylvlrq l zloo eh lqfuhdvhg e| W .+4 W,s1
Hdfk glylvlrq lv doorzhg wr uhwdlq d iudfwlrq W ri lwv fdvk rz1 Wkh uhpdlqlqj
iudfwlrq +4 W, zloo eh doorfdwhg wr glylvlrq l rqo| zkhq lw kdv wkh prvw
surwdeoh lqyhvwphqw rssruwxqlw|/ l1h1 zlwk suredelolw| s1 Vxssrvh qrz wkh
fdvk rz ri glylvlrq m lv lqfuhdvhg e| rqh groodu1 Wkhq wkh lqyhvwphqw ri
glylvlrq l zloo eh lqfuhdvhg e| s+4 W,1 Glylvlrq l zloo fdswxuh d iudfwlrq
+4 W, ri wkh fdvk rz ri glylvlrq m rqo| zkhq lwv lqyhvwphqw rssruwxqlw|
lv wkh prvw surwdeoh1 Lw lv hdv| wr qrwh wkdw
W . +4 W ,s A s +4 W ,
zkhqhyhu W A 3 dqg2ru W A 3 dqg s A 31 Vklq dqg Vwxo} +4<<;, lqwhusuhw
wkh kljkhu vhqvlwlylw| ri d glylvlrq*v lqyhvwphqw wr lwv fdvk rz wkdq wr rwkhu
glylvlrqv* fdvk rz dv hylghqfh wkdw ixqgv duh qrw doorfdwhg hflhqwo| lq dq
lqwhuqdo fdslwdo pdunhw1 Dffruglqj wr rxu prgho/ wklv kljkhu vhqvlwlylw| lv
frqvlvwhqw zlwk wkh xvh ri dq rswlpdo fdvk rz uhglvwulexwlrq srolf| e| wkh
khdgtxduwhuv1
Wkh jhqhudo irup ri W +s, lv wkh iroorzlqj1 Wdnh U dv jlyhq1 Zkhq wkh
h{ srvw glhuhqfh lq surwdelolw| lq wkh vhfrqg shulrg lv vpdoo + lv vpdoo,
wkh jdlq iurp uhdoorfdwlrq lv olplwhg1 Lw lv wkhuhiruh rswlpdo wr surylgh
wkh kljkhvw srvvleoh lqfhqwlyhv + W @ 4,1 Dv lqfuhdvhv wkh jdlqv iurp
uhdoorfdwlrq ehfrph odujh1 Dw vrph srlqw W wkh jdlqv iurp uhdoorfdwlrq
ehfrph wrr odujh/ vr lw lv rswlpdo wr vdfulfh lqfhqwlyhv* surylvlrq1 Zh
wkhuhiruh hqwhu dq lqwhuphgldwh udqjh lq zklfk wkhuh lv rqo| d sduwldo lqwhuqdo
fdslwdo pdunhw= d iudfwlrq ri wkh fdvk rz lv ohiw wr wkh glylvlrq wkdw kdv
surgxfhg lw/ lq rughu wr errvw pdqdjhuldo hruw1 Zkhq wkh glhuhqfh lq
h{0srvw surwdelolw| lv yhu| kljk +s U iru hdfk l, d sxuh lqwhuqdo fdslwdo
pdunhw lv rswlpdo +W @ W @ 3,1
Wdnh qrz dv jlyhq1 Zkhq U lv forvh wr }hur/ lw lv yhu| frvwo| wr
doorz d glylvlrq wr uhwdlq ixqgv1 Wkhuhiruh/ wkh rswlpdo srolf| zloo eh wr
vhw W +s, @ 31 Dv U lqfuhdvhv lw ehfrphv ohvv frvwo| wr surylgh lqfhqwlyhv
e

2
`|i_ E2ff @t U@**i?}i_ 5? @?_ 5|*3 EbbH<t hit*|t @h}?} |@| |i) 4@)
Mi U@ti_ M) 4i@thi4i?| ihhLht
QA
57
wkurxjk ixqg uhwhqwlrq/ vr wkdw hyhqwxdoo| W +s , ehfrphv srvlwlyh1 Dw odvw/
iru U vxflhqwo| kljk wkh frpsdudwlyh dgydqwdjh ri zlqqhu slfnlqj lv vr orz
wkdw d frqjorphudwh up suhihuv wr frqfhqwudwh rq surylglqj lqfhqwlyhv/ vhw0
wlqj W +s, @ 41 Zh froohfw wkh jhqhudo surshuwlhv ri W +s, lq wkh iroorzlqj
ohppd1
Ohppd 7
+ ,
Wkh iudfwlrq W s lv ghfuhdvlqj lq
dqg lqfuhdvlqj lq U1
Zkhq lqfuhdvhv wkh ehqhwv ri zlqqhu slfnlqj duh juhdwhu/ vlqfh doorfdwlqj
ixqgv wr wkh pruh surwdeoh glylvlrq |lhogv d kljkhu uhwxuq1 Wklv orzhuv wkh
rswlpdo ohyho ri W1 Wkhuh lv qr frxqwhuydlolqj hhfw/ vlqfh wkh zd| lq zklfk
wkh ohyho ri hruw ghshqgv rq W lv lqghshqghqw ri 1 Lq idfw/ wkh glhuhqfh
ehwzhhq wkh ohyho ri hruw h{huflvhg dw dqg dw lv=
!
h + , h @ +4 s , U
n
dqg lw lv wkhuhiruh lqghshqghqw ri 1 Wklv lpsolhv wkdw wkh frvw ri ghfuhdvlqj
lv qrw dhfwhg e| dq lqfuhdvh lq 1 Wkh qhw hhfw li wkhuhiruh wkdw d kljkhu
pdnhv d orzhu rswlpdo1
Dq lqfuhdvh lq U lqfuhdvhv e| wkh vdph dprxqw wkh surwdelolw| ri erwk
uhwdlqhg dqg uhdoorfdwhg ixqgv1 Lw wkhuhiruh pdnhv wkh fdslwdo jhqhudwhg
pruh surgxfwlyh/ lqghshqghqwo| ri lwv doorfdwlrq1 Wklv lq wxuq lpsolhv wkdw
lw ehfrphv frqyhqlhqw wr lqfuhdvh wkh txdqwlw| ri ixqgv jhqhudwhg1 Wkh rqo|
zd| wr dfklhyh wklv lv e| lqfuhdvlqj wkh iudfwlrq ri ixqgv uhwdlqhg1 Wklv
h{sodlqv zk| W ghshqgv srvlwlyho| rq U1
Zh qrz frph wr wkh txhvwlrq ri krz W ghshqgv rq s/ wkh suredelolw|
ri vxffhvv ri wkh glylvlrq1
Lq jhqhudo/ wkhuh lv d whqvlrq ehwzhhq wzr irufhv1 Vxssrvh khdgtxduwhuv
zdqwv wr jhqhudwh rqh +h{shfwhg, h{wud xqlw ri fdvk iurp wkh glylvlrqv1 Lq
rughu wr gr wkdw/ lw kdv wr lqfuhdvh wkh vkduh uhwdlqhg e| wkh glylvlrqv/
zklfk lq wklv prgho lv wkh rqo| zd| wr jhqhudwh pruh ixqgv lq wkh uvw
shulrg1 Zklfk glylvlrq vkrxog eh doorzhg wr uhwdlq pruh ixqgvB Lqfuhdvlqj
wkh vkduh uhwdlqhg e| wkh pruh surwdeoh glylvlrq lv ehwwhu ehfdxvh dq|
groodu uhpdlqlqj lq wkh glylvlrq rewdlqv d kljkhu uhwxuq1 Rq wkh rwkhu kdqg/
zh kdyh=
Ch !
@ n +4 s , U1
C
vr wkdw dq lqfuhdvh lq lv ohvv hhfwlyh lq jhqhudwlqj pruh ixqgv zkhq
s lv kljk1 Surylglqj lqfhqwlyhv wkurxjk ixqg uhwhqwlrq lv pruh hhfwlyh
58
lq glylvlrqv zlwk h{ dqwh orz surwdelolw|/ vlqfh wkhvh glylvlrqv duh pruh
olnho| qrw wr eh wkh zlqqhu lq wkh qh{w shulrg/ dqg wkhuhiruh kdyh d orzhu
suredelolw| ri vhhlqj djdlq dq| groodu wdnhq dzd| iru uhdoorfdwlrq1
Zh fdq qrz qdoo| dqdo|}h wkh lpsdfw ri s rq 9 +s ,1
Sursrvlwlrq 7 Wkh iudfwlrq 9 +s , lv ghfuhdvlqj lq s li A U/ dqg lw lv
lqfuhdvlqj lq s
rwkhuzlvh1
Zh fdq wkhuhiruh ghwhfw wzr glhuhqw uhjlphv1 Zkhq lv orz dqg U lv
kljk wkhq wkh iudfwlrqv 9 dqg 29 ri uhwdlqhg ixqgv zloo eh kljk/ dqg wkh
prvw surwdeoh glylvlrqv zloo eh doorzhg wr uhwdlq d kljkhu iudfwlrq ri ixqgv1
Wkhvh duh vlwxdwlrqv lq zklfk czlqqhu slfnlqj* grhv qrw dgg pxfk ydoxh wr
wkh up/ vr lqfhqwlyh surylvlrq lv wkh uhohydqw lvvxh1 Lq wklv fdvhv zh dovr
revhuyh wkdw wkh iudfwlrq ri ixqgv uhwdlqhg e| wkh pruh surwdeoh glylvlrq
lv kljkhu1 Wkh lqwxlwlrq ri zk| lw vkrxog eh vr frphv iurp ohppd 7= Dv U
lqfuhdvhv/ lw lv rswlpdo wr lqfuhdvh pruh wkh iudfwlrq ri wkh prvw surwdeoh
glylvlrq/ vlqfh hdfk groodu sxw lq wkh pruh surwdeoh glylvlrq kdv d kljkhu
uhwxuq1
Wkh vhfrqg uhjlph lv wkh rqh lq zklfk wkh frqglwlrq A U lv vdwlvhg1
Lq wklv fdvh zlqqhu slfnlqj lv surwdeoh/ dqg wkh glylvlrqv duh doorzhg wr
uhwdlq rqo| d vpdoo iudfwlrq ri ixqgv/ li dq|1 Krzhyhu/ lw lv wkh ohvv surwdeoh
glylvlrq zklfk lv doorzhg wr uhwdlq pruh1 Djdlq/ zh fdq orrn dw ohppd 7
iru lqwxlwlrq1 Dv lqfuhdvhv wkh iudfwlrq ri uhwdlqhg ixqgv ghfuhdvhv iru
doo glylvlrqv/ exw lw lv wkh prvw surwdeoh glylvlrq zklfk vxhuv wkh kljkhvw
uhgxfwlrq1 Dq lqfuhdvh lq surylghv lqfhqwlyhv iru wkh glylvlrqv/ vlqfh zh
kdyh Ye
Y { @ & s 1 Wklv rffxuv ehfdxvh glylvlrqdo pdqdjhuv uhfrjql}h wkdw dq|
groodu jhqhudwhg iru lqyhvwphqw zloo kdyh d kljkhu sd|rxw/ dqg vr h{huflvh
pruh hruw1 Iurp wkh srlqw ri ylhz ri khdgtxduwhuv/ wklv phdqv wkdw d orzhu
ohyho ri lv qhfhvvdu| wr h{wudfw d jlyhq ohyho ri hruw/ vr wkh rswlpdo ohyho
ri ixqg uhwhqwlrq ghfuhdvhv1 Wkh ghfuhdvh lv vwurqjhu lq glylvlrqv zlwk d kljk
s / vlqfh lq vxfk glylvlrqv wkh lqfuhdvh lq kdv d vwurqjhu lpsdfw rq h 1
Rqh hpslulfdo lpsolfdwlrq ri Sursrvlwlrq 7 lv wkdw wkhuh vkrxog eh d
fruuhodwlrq ehwzhhq wkh jhqhudo ohyho ri ixqg uhwhqwlrq dqg wkh +dssduhqw,
plvdoorfdwlrq ri ixqgv1 Frqjorphudwh upv wkdw doorz kljk ohyhov ri ixqg
uhwhqwlrq zloo dovr doorz pruh uhwhqwlrq e| wkh h{ dqwh pruh surwdeoh gl0
ylvlrqv1 Rq wkh rwkhu kdqg/ frqjorphudwh upv wkdw doorz iru orz ohyhov ri
ixqg uhwhqwlrq duh pruh olnho| wr shuplw d kljkhu uhwhqwlrq udwh wr glylvlrqv
zlwk srruhu survshfwv1
59
D
L?U*tL?t
Wklv sdshu kdv dujxhg wkdw rqh ri wkh glvwlqfwlyh ihdwxuhv ri lqwhuqdo fds0
lwdo pdunhwv/ wkdw lv wkh delolw| ri khdgtxduwhuv wr uhdoorfdwh ixqgv dfurvv
glylvlrqv +zlqqhu0slfnlqj, lv dvvrfldwhg erwk zlwk frvwv dqg ehqhwv1 Wkh
ehqhwv ghulyh iurp wudqvihuulqj ixqgv wr wkh prvw surwdeoh glylvlrqv> wkh
frvwv ghulyh iurp wkh zhdnhqlqj ri pdqdjhuldo lqfhqwlyhv1 Lq rwkhu zrugv/
zlqqhu0slfnlqj lv vlpxowdqhrxvo| wkh gdun dqg wkh euljkw vlgh ri lqwhuqdo
fdslwdo pdunhwv1 Rxu wkhru| fdq h{sodlq zk| frqjorphudwh upv wudgh dw
d glvfrxqw +ru dw d suhplxp, zlwk uhvshfw wr wkhlu irfxvhg frxqwhusduwv1
Pruh lpsruwdqwo|/ lw grhv vr zlwkrxw dvvxplqj dq| lqhflhqf| lq wkh door0
fdwlrq ri frusrudwh uhvrxufhv1 Zh vkrz wkdw h{ dqwh glyhuvlw| lq glylvlrqv*
surwdelolw| lqfuhdvhv wkh lqhflhqf| ri dq lqwhuqdo fdslwdo pdunhw/ frqup0
lqj wkh qglqjv ri Udmdq hw do1 +5333,1 Ixuwkhupruh/ dq lpsolfdwlrq ri rxu
prgho lv wkdw doorzlqj glylvlrqv wr uhwdlq d iudfwlrq ri wkh fdvk rz wkh|
kdyh surgxfhg luuhvshfwlyh ri wkhlu lqyhvwphqw rssruwxqlwlhv fdq eh dq rs0
wlpdo srolf| iru wkh khdgtxduwhuv vlqfh lw ohdgv wr d ehwwhu edodqfh ehwzhhq
wkh wzr frqlfwlqj jrdov ri doorfdwlqj rswlpdoo| h{ srvw frusrudwh uhvrxufhv
dqg surylglqj pdqdjhuv zlwk dghtxdwh h{ dqwh lqfhqwlyhv wr shuirup1 Do0
orzlqj glylvlrqv wr vshqg glvfuhwlrqdu| d iudfwlrq ri wkhlu fdvk rz lv d zd|
ri ghohjdwlqj uhdo dxwkrulw| wr glylvlrq pdqdjhu1 Dv zh nqrz iurp Djklrq
dqg Wluroh +4<<:,/ wkh jdlqv ri ghohjdwlrq lq whupv ri errvwhg pdqdjhuldo
lqlwldwlyh fdq pruh wkdq frpshqvdwh wkh orvv lq frqwuro ri khdgtxduwhuv1
Dq lpsruwdqw fdyhdw lv wkdw zh kdyh qrw dgguhvvhg wkh uhdvrqv zk| gl0
ylvlrqv wkdw duh yhu| glhuhqw lq whupv ri wkhlu surwdelolw| duh eurxjkw
wrjhwkhu lq wkh vdph up dqg zk| vrph glylvlrqv duh qrw vsxq0r lq wkrvh
flufxpvwdqfhv zkhuh frqjorphudwhv duh lqhflhqw1 Pruhryhu/ zh kdyh dv0
vxphg wkdw doo uhvrxufhv duh lqwhuqdoo| jhqhudwhg/ ljqrulqj wkh uroh ri h{0
whuqdo qdqflqj1 Wkhvh olplwdwlrqv qrwzlwkvwdqglqj/ zh eholhyh wkdw wkh
dqdo|vlv ri lqwhuqdo fdslwdo pdunhwv lq whupv ri doorfdwlrq ri ghohjdwlrq ri
dxwkrulw| pd| eh d surplvlqj gluhfwlrq iru ixwxuh uhvhdufk1
5:
TTi?_
Surri ri Sursrvlwlrq 41 Xvlqj wkh h{suhvvlrq iru U 7 jlyhq e|
U jlyhq e| +4, dqg +5, zh rewdlq=
+6, dqg wkh h{suhvvlrqv iru h7
/ h
U 7 @
l
!k
!
+
U . s,2 . +U . +4 s,,2 @
+
U . ,2 n
n
l
!k
5
s +4 s, 2 U2
n
Wklv lv d vwulfwo| lqfuhdvlqj ixqfwlrq ri dqg d vwulfwo| ghfuhdvlqj ixqfwlrq
ri U1 Iru d jlyhq sdlu U/ wkh ixqfwlrq uhdfkhv d pd{lpxp dw s @ 2 dqg
lv ghfuhdvlqj lq s iru s A 2 1
Surri ri Ohppd 41 Wkh frqglwlrq zU A z7 lv htxlydohqw wr=
4 !s +U . , A 4 ! +U . s ,
5
5
ru/ diwhu vlpsolfdwlrqv/
+ . !U,+4 s, 3
zklfk lv dozd|v vdwlvhg1
Surri ri Ohppd 51 Vxevwlwxwlqj iru wkh h{suhvvlrqv irxqg iru wkh rswlpdo
zdjh/ zh kdyh=
@ +4 . !s5,+nU . , =
hU
Frpsdulqj lw zlwk h{suhvvlrq ri h7
jlyhq e| +7, |lhogv wkh uhvxow diwhu vrph
hdv| pdqlsxodwlrqv1
Surri ri Sursrvlwlrq 51 Ghqh wkh udwlr=
2
2
+> s, @ +U . , n2R . +U . , nE2 3R
EnE-nR{ 2
3R{ 2
7 +> s,
. EnE-nE
2
2
Frqvlghu srlqw d,1 Gluhfw frpsxwdwlrq vkrzv wkdw/ iru hdfk s=
U +> s, A 4
U +3> s, ? 4
olp
7
{<n" 7 +> s,
+3> s,
U
5;
dqg=
U
E{cR
C 7 E{
cR
C
A3
wkxv hvwdeolvklqj wkh uhvxow1
Qh{w/ frqvlghu srlqw e,1 Xvlqj wkh h{suhvvlrqv iru U dqg 7 jlyhq
lq wkh wh{w zh kdyh=
C U 7
+ . 5! . !U, +!U ,+5s 4,
@
Cs
5n
Li s A 2 wkhq wkh vljq ri wkh ghulydwlyh lv htxdo wr wkh vljq ri +!U ,/
wkxv hvwdeolvklqj wkh uhvxow1
Surri ri Sursrvlwlrq 61 E| gluhfw frpsxwdwlrq zh kdyh=
g+U 7 ,
@ 5 ! +4 s, s2 U2 F
gF
n
vr wkdw wkh glhuhqfh U 7 lqfuhdvhv zlwk F zkhq s2 A U2 dqg
ghfuhdvhv rwkhuzlvh1
Surri ri Ohppd 61 E| htxdwlqj wkh wzr uljkw kdqg vlghv ri wkh wzr uvw
rughu frqglwlrqv/ zh kdyh=
3
3
s +U . , . +4 s,  U @ +4 s,+U . , . s 2 U
vr wkdw=
3 5s 4 +U . , +4 s, !
2@
 1
.
s
U
s
!
!
Wkh frqglwlrq 2 A  lv htxlydohqw wr=
5s 4 +U . ,
s
U
.
+4 s, ! !
 A 
s
ru=
+U . , !
A 
U
zklfk lv dozd|v vdwlvhg vlqfh zh kdyh uhvwulfwhg dwwhqwlrq wr 
Surri ri Ohppd 71
Ohw 9 eh wkh ydoxh ri vxfk wkdw=
+U . 9 , +U s
9 ,
@4
5 +4 s
, 9 U
5<
# ^3> 4`1
dqg ohw 99 eh wkh ydoxh ri vxfk wkdw=
+U . 99 , +U s
99 ,
@3
5 +4 s
, 99 U
Wkhq 9 +s
, @ 4 iru 9 dqg 9 +s
, @ 3 iru 99 / vr wkdw wkh
ixqfwlrq lv frqvwdqw1 Li # +9 > 99 , wkhq 9 +s
, lv jlyhq e|=
9 +s
, @
vr wkdw=
+U . , +U s
,
5 +4 s
, U
C 9
4 s
2 . U2
? 31
@
C
5 +4 s
, 2 U
Wr suryh wkdw 9 +s
, lv lqfuhdvlqj U/ ohw U9 eh wkh ydoxh vxfk wkdw=
+U9 . , +U9 s
,
@3
5 +4 s
, U9
dqg U99 wkh ydoxh vxfk wkdw=
+U99 . , +U99 s
,
@4
5 +4 s
, U99
Iru U U9 zh kdyh 9 +s
, @ 3> dqg iru U
9
 - n{ R
U # +U9 > U99 , wkhq= Y
Y- @ 2 E!R {- A 3=
U99
zh kdyh 9 +s
, @ 41 Li
Surri ri Sursrvlwlrq 71 Zkhq lv frqvwdqw wkhq YRY
lq zklfk +s , # +3> 4, zh kdyh=
@ 31 Lq wkh uhjlrq
C
4 +U2 2 ,
@
Cs
5 +4 s ,2 U
Y
vr wkdw wkh vljq ri lv htxdo YR
lv htxdo wr wkh vljq ri +U ,1
63
+iuihi?Uit
^4` Djklrq/ Sklolssh dqg Mhdq Wluroh +4<<:, cIrupdo dqg Uhdo Dxwkrulw|
lq Rujdql}dwlrqv*/ Mrxuqdo ri Srolwlfdo Hfrqrp|/ 438+4,= 405<1
^5` Ehujhu/ Sklols dqg Hol Rihn +4<<8, cGlyhuvlfdwlrq*v Hhfw rq Ilup
Ydoxh*/ Mrxuqdo ri Ilqdqfldo Hfrqrplfv/ 6:+4,= 6<0981
^6` Euxvfr/ Vdqgur dqg Idxvwr Sdqxq}l +5333, cUhdoorfdwlrq ri Frusrudwh
Uhvrxufhv dqg Pdqdjhuldo Lqfhqwlyhv lq Lqwhuqdo Fdslwdo Pdunhwv*/ Il0
qdqfldo Pdunhwv Jurxs G1S1 6861
^7` Fdpsd/ Mrvh dqg Vlpl Nhgld +4<<<, cH{sodlqlqj wkh Glyhuvlfdwlrq
Glvfrxqw*/ plphr/ Qhz \run Xqlyhuvlw| dqg Kduydug Xqlyhuvlw|1
^8` Fkhydolhu/ Mxglwk +5333, cZkdw gr Zh Nqrz derxw Furvv0
Vxevlgl}dwlrqB Hylghqfh iurp wkh Lqyhvwphqw Srolflhv ri Phujlqj
Ilupv*/ Zrunlqj sdshu/ Xqlyhuvlw| ri Fklfdjr1
^9` Gh Prwwd/ Dgroir +4<<<, cPdqdjhuldo Lqfhqwlyhv dqg Lqwhuqdo Fdslwdo
Pdunhwv*/ plphr/ PLW1
^:` Ioxfn/ ]vx}vdqqd dqg Dqwkrq| O|qfk +4<<<, cZk| Gr Ilupv Phujh
dqg wkhq GlyhvwB D Wkhru| ri Ilqdqfldo V|qhuj|*/ Mrxuqdo ri Exvlqhvv/
:5= 64<06791
^;` Jdxwlhu D{ho dqg Ioruldq Khlghu +5333, cZkdw Gr Lqwhuqdo Fdslwdo
Pdunhwv GrB Uhglvwulexwlrq yv1 Lqfhqwlyhv*/ plphr/ Orqgrq Vfkrro ri
Hfrqrplfv1
^<` Lqghuvw/ Urpdq dqg Fkulvwldq Odx{ +5333, cLqfhqwlyhv lq Lqwhuqdo Fdsl0
wdo Pdunhwv= Fdslwdo Frqvwudlqwv/ Frpshwlwlrq dqg Lqyhvwphqw Rssru0
wxqlwlhv*/ plphr/ Xqlyhuvlw| Froohjh Orqgrq1
^43` Lqghuvw/ Urpdq dqg Krojhu P‚
xoohu +5334, cFrusrudwh Eruurzlqj dqg
Ilqdqflqj Frqvwudlqwv*/ FHSU zrunlqj sdshu Qr 5:;71
^44` Odprqw/ Rzhq +4<<:, cFdvk Iorz dqg Lqyhvwphqw= Hylghqfh iurp Lq0
whuqdo Fdslwdo Pdunhwv*/ Mrxuqdo ri Ilqdqfh/ 85+4,= ;6043<1
^45` Olqv/ Nduo dqg Khqul Vhuydhv +4<<<, cLqwhuqdwlrqdo Hylghqfh rq wkh
Ydoxh ri Frusrudwh Glyhuvlfdwlrq*/ Mrxuqdo ri Ilqdqfh/ 87+9,= 55480
556<1
64
^46` Pdnvlprylf/ Yrmlvody dqg Jrugrq Skloolsv +5334, cGr Frqjorphudwh
Ilupv Doorfdwh Uhvrxufhv Lqhflhqwo|B*/ plphr/ Xqlyhuvlw| ri Pdu|0
odqg1
^47` Udmdq/ Udjkxudp/ Khqul Vhuydhv dqg Oxljl ]lqjdohv +5333, cWkh Frvw
ri Glyhuvlw|= Wkh Glyhuvlfdwlrq Glvfrxqw dqg Lqhflhqw Lqyhvwphqw*/
Mrxuqdo ri Ilqdqfh/ 88+4,= 680;31
^48` Urwhpehuj/ Mxolr dqg Jduwk Vdorqhu +4<<7, cEhqhwv ri Qduurz Exvl0
qhvv Vwudwhjlhv*/ Dphulfdq Hfrqrplf Uhylhz/ ;7+8,= 46630467<1
^49` Vfkduivwhlq/ Gdylg +4<<;,/ cWkh Gdun Vlgh ri Lqwhuqdo Fdslwdo Pdunhwv
LL*/ QEHU Zrunlqj Sdshu Qr1 96851
^4:` Vfkduivwhlq/ Gdylg dqg Mhuhp| Vwhlq +5333, cWkh Gdun Vlgh ri Lqwhuqdo
Fdslwdo Pdunhwv= Glylvlrqdo Uhqw0Vhhnlqj dqg Lqhflhqw Lqyhvwphqw*/
Mrxuqdo ri Ilqdqfh/ 88 +9,= 586:0971
^4;` Vhuydhv/ Khqul +4<<9, cWkh Ydoxh ri Glyhuvlfdwlrq Gxulqj wkh Frq0
jorphudwh Phujhu Zdyh*/
Mrxuqdo ri Ilqdqfh/
84
+7,= 45340581
^4<` Vklq/ K|xq Kdq dqg Uhq
h Vwxo} +4<<;, cDuh Lqwhuqdo Fdslwdo Pdunhwv
HflhqwB*/
Txduwhuo| Mrxuqdo ri Hfrqrplfv>
446+5,=
8640851
^53` Vwhlq/ Mhuhp| +4<<:, cLqwhuqdo Fdslwdo Pdunhwv dqg wkh Frpshwlwlrq
iru Frusrudwh Uhvrxufhv*/
Mrxuqdo ri Ilqdqfh/
85+4,=
4440661
^54` Vwhlq/ Mhuhp| +5333,/ cLqirupdwlrq Surgxfwlrq dqg Fdslwdo Doorfdwlrq=
Ghfhqwudol}hg yv1 Klhudufklfdo Ilupv*/ plphr/ PLW1
^55` Vwhlq/ Mhuhp| +5334,/ cDjhqf|/ Lqirupdwlrq dqg Frusrudwh Lqyhvwphqw*/
iruwkfrplqj/
Kdqgerrn ri Hfrqrplfv dqg Ilqdqfh
/
Qruwk Kroodqg1
^56` Zklwhg/ Wrql P1 +5334, cLv Lw Lqhflhqw Lqyhvwphqw wkdw Fdxvhv wkh
Glyhuvlfdwlrq GlvfrxqwB*/
Mrxuqdo ri Ilqdqfh/
65
89+8,=
499:0<41