Advertising with Spillover Effects in a Differential Oligopoly Game With Differentiated Goods

Lambertini, Luca; Cellini, Roberto Working Paper Advertising with Spillover Effects in a Differential Oligopoly Game With Differentiated Goods Quaderni - Working Paper DSE, No. 430 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca; Cellini, Roberto (2002) : Advertising with Spillover Effects in a Differential Oligopoly Game With Differentiated Goods, Quaderni - Working Paper DSE, No. 430, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4868 This Version is available at: https://hdl.handle.net/10419/159271 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Dgyhuwlvlqj zlwk Vslooryhu Hhfwv lq d Glhuhqwldo Roljrsro| Jdph zlwk Glhuhqwldwhg Jrrgv Urehuwr Fhoolql  0OxfdOdpehuwlql   Glsduwlphqwr gl Hfrqrpld h Phwrgl Txdqwlwdwlyl Xqlyhuvlwã gl Fdwdqld Fruvr Lwdold 88/ <845< Fdwdqld/ Lwdo| id{ 6<03<806:38:7 h0pdlo fhoolqlCper{1xqlfw1lw  Glsduwlphqwr gl Vflhq}h Hfrqrplfkh Xqlyhuvlwã ghjol Vwxgl gl Erorjqd Vwudgd Pdjjlruh 78/ 73458 Erorjqd/ Lwdo| id{ 6<0384053<5997 h0pdlo odpehuwlCvser1xqler1lw Mdqxdu| 47/ 5335 Devwudfw Zh frqvlghu d glhuhqwldwhg roljrsro| zkhuh upv frpshwh ãodFrxuqrw lq wkh pdunhw skdvh/ dqg hdfk upv pd| lqyhvw lq dgyhuwlvlqj dfwlylw|/ wr hqodujh lwv pdunhw vl}h1 Hdfk up*v dgyhuwlvlqj hruw kdv srvlwlyh h{whuqdo hhfwv rq wkh pdunhw vl}h ri doo ulydov1 Zh ghulyh wkh rshq0orrs +dqg wkh frlqflghqw forvhg0orrs, Qdvk htxloleulxp/ dqg wkh rswlpdo ehkdylrxu ri d fduwho lqyroylqj doo upv vhwwlqj erwk txdqwlwlhv dqg dgyhuwlvlqj hruwv vr dv wr pd{lpl}h mrlqw surwv1 Wkh frpsdudwlyh dvvhvvphqw ri wkhvh htxloleuld vkrzv wkdw d fduwho pd| surgxfh d vwhdg| vwdwh zkhuh vrfldo zhoiduh lv kljkhu wkdq wkh vrfldo zhoiduh ohyho dvvrfldwhg zlwk wkh qrq0frrshudwlyh vhwwlqj1 Wklv lv gxh wr wkh srvlwlyh h{whuqdolwlhv iurp dgyhuwlvlqj dfwlylw|1 Nh|zrugv= dgyhuwlvlqj/ glhuhqwldo jdphv/ fdslwdo dffxpxodwlrq/ forvhg0 orrs htxloleuld/ h{whuqdolw|1 MHO Fodvvlfdwlrq= G76/ G<5/ O46/ P6:1 4 Lqwurgxfwlrq Zh suhvhqw d g|qdplf roljrsro| prgho zlwk glhuhqwldwhg surgxfwv/ zkhuh upv frpshwh ãodFrxuqrw lq wkh pdunhw skdvh/ dqg hdfk up lqyhvwv lq dgyhuwlvlqj dfwlylwlhv dlphg dw hqodujlqj lwv pdunhw vl}h1 Wkh dgyhuwlvlqj lq0 yhvwphqw fduulhg rxw e| dq| up vsloov ryhu wr wkh ulydov1 Wklv k|srwkhvlv vhhpv wr eh txlwh uhdolvwlf/ exw kdv qrw |hw ixoo| lqyhvwljdwhg e| wkh dydlodeoh olwhudwxuh1 4 Rxu lghd lv wkdw dgyhuwlvlqj dfwlylw| iru/ vd|/ d preloh whohskrqh ehqhwv doo surgxfhuv ri whohskrqhv/ hyhq li surgxfwv duh glhuhqwldwhg1 Xqghu wklv uhvshfw/ dgyhuwlvlqj lv/ dw ohdvw wr vrph h{whqw/ d sxeolf jrrg1 Frqvh0 txhqwo|/ wkh pdunhw uhjxodwlrq suhvfulswlrqv/ frqfhuqlqj wkh frpshwlwlrq lq wkh pdunhw skdvh dv zhoo dv wkh frpshwlwlrq lq dgyhuwlvlqj hruwv/ duh qrw wulyldo1 Zh frqvlghu d g|qdplf iudphzrun/ zkhuh wkh dgyhuwlvlqj hruwv kdyh orqj0odvwlqj hhfwv1 Dv wr wkh pdunhw ghflvlrq/ zh frqvlghu dq roljrsro|/ zlwk d srvlwlyh ghjuhh ri vxevwlwxwdelolw| ehwzhhq jrrgv/ xqghu olqhdu pdu0 nhw ghpdqg ixqfwlrq> lw lv zhoo nqrzq wkdw wklv frpshwlwlrq prgho lqfoxghv prqrsrolvwlf frpshwlwlrq dqg krprjhqrxv roljrsro| dv sduwlfxodu fdvhv +vhh Vshqfh/ 4<:9/ Ylyhv 4<;8/ Fhoolql dqg Odpehuwlql/ 4<<;,1 Zh vkrz wkdw wklv g|qdplf v|vwhp pd| frqyhujh wr d vwhdg| vwdwh1 Wkh ghwhuplqdqwv ri wkh vwhdg| vwdwh pdunhw vl}hv dqg dgyhuwlvlqj hruwv duh uvw dqdo|}hg lq wkh qrq0frrshudwlyh jdph/ xvlqj wkh rshq0orrs dqg wkh forvhg0orrs Qdvk htxl0 oleuld dv vroxwlrq frqfhswv1 Dv d pdwwhu ri idfw/ wkhvh htxloleuld frlqflgh lq rxu iudphzrun/ vlqfh wkh sureohp dw kdqg lv vwdwh0olqhdu +vhh Grfnqhu hw do1/ 5333,1 Wkxv/ wkh vroxwlrq ri wkh glhuhqwldo roljrsro| jdph ehqhwv erwk iurp wkh hdv| dqdo|wlfdo surfhgxuh ri wkh rshq0orrs frqfhsw dqg iurp wkh wlph frqvlvwhqf| surshuw| ri wkh forvhg0orrs frqfhsw1 Wkhq/ zh dqdo|}h wkh rswlpdo ghflvlrqv e| d +v|pphwulf, fduwho pdgh xs e| doo upv lq wkh pdunhw/ dlplqj dw wkh pd{lpxp mrlqw surwv zlwk uhvshfw wr erwk rxwsxw dqg dgyhuwlvlqj hruw1 Pdunhw vl}hv/ lqyhvwphqwv 4 Wkh olwhudwxuh rq g|qdplf prghov ri dgyhuwlvlqj fdq eh eurdgo| sduwlwlrqhg lqwr wzr fodvvhv= wkh uvw hvwdeolvkhv d gluhfw uhodwlrqvkls ehwzhhq wkh udwh ri fkdqjh lq vdohv + ru pdunhw vkduhv , dqg wkh dgyhuwlvlq hruwv ri upv + vhh/ h1j1/ Ohlwpdqq dqg Vfkplwhqgrui/ 4<:;/ Ihlfkwlqjhu/ 4<;6/ dqg Hulfnvrq/4<;8 , > wkh vhfrqg frqvlghuv dgyhuwlvhphw dv dq lq0 vwuxphqw wr lqfuhdvh wkh vwrfn ri jrrgzloo ru uhsxwdwlrq dqg hvwdeolvkhv d olqn ehwzhhq wkh dgyhuwlvlqj hruwv ri d up dqg khu pdunhw ghpdqg + Vhwkl/ 4<::/ Ihuvkwpdq/ 4<;7/ dqg Mûujhqvhq dqg ]dffrxu/ 4<<</ Sljd/ 5333 , 1 Vhh Mrujhqvhq + 4<;5 , / Hulfnvrq + 4<<4 , / Ihlfkwlqjhu/ Kduwo dqg Vhwkl + 4<<7 , / Grfnqhu hw do1 + 5333/ fk1 44 , iru h{kdxvwlyh vxuyh|v1 4 dqg surgxfwlrq ghflvlrqv duh frpsduhg dfurvv wkh vwhdg| vwdwhv ri doo wkh diruhphqwlrqhg uhjlphv1 D uhohydqw uhvxow lv wkdw vrfldo zhoiduh lq vwhdg| vwdwh pd| zhoo eh odujhu lq wkh fduwho htxloleulxp/ wkdq lq wkh qrq0frrshudwlyh Frxuqrw htxloleulxp1 Wklv lv gxh wr wkh h{whuqdolw| hhfw dhfwlqj upv* dgyhuwlvlqj dfwlylwlhv/ dqg wkh dvvrfldwhg lqh!flhqf| dhfwlqj wkh Frxuqrw vhwwlqj1 Wkh vwuxfwxuh ri wkh sdshu lv dv iroorzv1 Vhfwlrq 5 looxvwudwhv wkh edvlf vhwxs1 Vhfwlrq 6 uvw vkrzv wkdw wkh rshq0orrs lqirupdwlrq vwuxfwxuh dqg wkh forvhg0orrs lqirupdwlrq vwuxfwxuh duh lppdwhuldo wr wkh uhvxow ri wklv glhuhqwldo jdph/ vlqfh wkh jdph lv d olqhdu vwdwh rqh1 Wkhq/ lw ghyhorsv wkh vroxwlrq ri wkh Qdvk htxloleulxp dqg vwxglhv wkh g|qdplf surshuwlhv ri wkh vwhdg| vwdwh1 Vhfwlrq 7 ghdov zlwk wkh rswlpdo fduwho ehkdylrxu1 Wkh frpsdudwlyh dvvhvvphqw ri doo htxloleuld lv lq Vhfwlrq 81 Vhfwlrq 9 eulh | frqfoxghv1 5 Wkh edvlf vhwxs Zh frqvlghu dq roljrsro| jdph sod|hg ryhu |5 d fc4Ghqh wkh vhw ri upv +sod|huv, dv 5ic2cc jHdfk up surgxfhv rqh jrrg/ wkdw lv glhuhqwldwhg zlwk uhvshfw wr wkh jrrg surgxfhg e| dq| rwkhu up 1 Ohw R  ghqrwh wkh sulfh ri jrrg /dqg%  wkh txdqwlw| ri jrrg 1Ilupidfhv wkh iroorzlqj ghpdqg ixqfwlrq/ eruurzhg iurp Vshqfh +4<:9, dqg hpsor|hg e| Vlqjk dqg Ylyhv +4<;7,/ Ylyhv +4<;8,/ Odpehuwlql +4<<:,/ Fhoolql dqg Odpehuwlql +4<<;/ 5335,/ lqwhu dold= R  E |'  E |%  E |([   '  %  E |+4, Yduldeoh   ghvfulehv wkh pdunhw vl}h ru wkh uhvhuydwlrq sulfh iru jrrg 1 dqg (duh sdudphwhuv/ zlwk f( Qrwlfh wkdw sdudphwhu (fdswxuhv wkh ghjuhh ri vxevwlwxwdelolw| ehwzhhq dq| sdlu ri glhuhqw jrrgv1 Lq wkh olplw fdvh ('fcjrrgv duh lqghshqghqw dqg hdfk up ehfrphv d prqrsrolvw1 Lq wkh rssrvlwh olplw fdvh ('/ wkh jrrgv surgxfhg e| glhuhqw upv duh shuihfw vxevwlwxwhv dqg wkh prgho froodsvhv lqwr wkh krprjhqhrxv roljrsro| prgho1 Zh dvvxph wkdw wkh pdunhw vl}h pd| eh lqfuhdvhg e| upv/ wkurxjk dgyhuwlvlqj dfwlylwlhv1 Pruh suhflvho|/ zh dvvxph wkdw wkh hruwv pdgh e| 5 dq| up dhfwv lwv rzq pdunhw vl}h/ dv zhoo dv wkh pdunhw vl}h ri wkh ulydov1 Lq sduwlfxodu/ wkh g|qdplfv ri wkh pdunhw vl}h ri up lv ghvfulehg e| wkh iroorzlqj htxdwlrq= _  E| _|     E|'&  E|n[   '  &  E|B  E|+5, zkhuh &  lv wkh hruw lq dgyhuwlvlqj pdgh e| up /flv d sdudphwhu fdswxulqj wkh h{whuqdo hhfw ri wkh dgyhuwlvlqj ri d up rq wkh pdunhw vl}h ri glhuhqw upv/ Bflv d ghsuhfldwlrq sdudphwhu lqglfdwlqj wkdw pdunhw vl}hvkulqnvdvwlphjrhve|1 Dgyhuwlvlqj dfwlylwlhv duh dvvxphg wr kdyh ghfuhdvlqj pdujlqdo surgxfwly0 lw|/ l1h1/ wkh| hqwdlo d txdgudwlf frvw S  E &  E | ' k 2 E &  E | 2 zlwk k:f1Pruh0 ryhu/ surgxfwlrq hqwdlov d frqvwdqw pdujlqdo surgxfwlrq frvwv= S  E %  E | ' S%  E |zlwk S:f1 Txdqwlwlhv i%  j   ' dqg dgyhuwlvlqj hruwv i&  j   ' duh wkh frqwuro yduldeohv/ zkloh pdunhw vl}hv i  j   ' duh vwdwh yduldeohv1 Hdfk sod|hu fkrrvhv wkh sdwk ri khu frqwuro yduldeohv ryhu wlph/ lq rughu wr pd{lpl}h wkh suhvhqw ydoxh ri khu surw rz/ vxemhfw wr +l, wkh prwlrq odzv uhjduglqj wkh vwdwh yduldeohv/ dqg +ll, wkh lqlwldo frqglwlrqv1 Irupdoo|/ wkh sureohp ri sod|hu pd| eh zulwwhq dv iroorzv1 Wkh remhfwlyh ixqfwlrq lv= 4@ %  E |  c&  E |  a  ] " f Z  E|e 3 4| _| +6, zkhuh wkh idfwru e 3 4| glvfrxqwv ixwxuh jdlqv/ dqg wkh glvfrxqw udwh 4lv dv0 vxphg wr eh frqvwdqw dqg frpprq wr doo sod|huv1 Lqvwdqwdqhrxv surwv duh= Z  E |'#  E |%  E |([   '  %  E |$%  E |k 2 E &  E | 2 S%  E |+7, Ixqfwlrq +6, lv vxemhfw wr wkh vhw ri g|qdplf frqvwudlqwv ri w|sh +5,/ dqg wr wkh vhw ri lqlwldo frqglwlrqv i  E f '   f j   ' Wr vroyh wkh sureohp/ zh kdyh wr lqwurgxfh vshflf k|srwkhvhv derxw wkh lqirupdwlrq vwuxfwxuh ri sod|huv1 6 6 Wkh qrq0frrshudwlyh roljrsro| Wkh dydlodeoh olwhudwxuh rq glhuhqwldo jdphv dssolhg wr upv* ehkdylrxu pdlqo| frqfhqwudwhv rq wzr nlqgv ri vroxwlrq frqfhswv= rshq0orrs dqg forvhg0 orrs htxloleuld1 Lq wkh iruphu fdvh/ upv suhfrpplw wkhlu ghflvlrqv rq wkh frqwuro yduldeohv wr d sdwk ryhu wlph/ l1h1/ wkh| ghvljq wkh rswlpdo sodq dw wkh lqlwldo gdwh dqg wkhq vwlfn wr lw iruhyhu1 Wkh uhvxowlqj rshq0orrs Qdvk htxloleulxp lv rqo| zhdno| wlph frqvlvwhqw dqg wkhuhiruh/ lq jhqhudo/ lw lv qrw vxejdph shuihfw1 Lq wkh odwwhu/ upv gr qrw suhfrpplw wr dq| sdwk dqg wkhlu vwudwhjlhv dw dq| lqvwdqw ghshqg rq wkh suhfhglqj klvwru|1 Lq wklv vlwxdwlrq/ wkh lqirupdwlrq vhw xvhg e| upv lq vhwwlqj wkhlu vwudwhjlhv dw dq| jlyhq wlph lv riwhq vlpsolhg wr eh rqo| wkh fxuuhqw ydoxh ri wkh vwrfnv ri vwdwh yduldeohv dw wkdw wlph1 Wkh uhohydqw htxloleulxp frqfhsw/ lq wklv fdvh/ lv wkh forvhg0orrs phpru|ohvv Qdvk htxloleulxp/ zklfk lv vwurqjo| wlph frqvlvwhqw dqg wkhuhiruh vxejdph shuihfw1 D uhqhphqw ri wkh forvhg0orrs Qdvk htxloleulxp/ zklfk lv nqrzq dv wkh ihhgedfn Qdvk htxloleulxp/ fdq dovr eh dgrswhg dv wkh vroxwlrq frqfhsw= zkloh lq wkh forvhg0orrs phpru|ohvv fdvh wkh lqlwldo dqg fxuuhqw ohyhov ri doo vwdwh yduldeohv duh wdnhq lqwr dffrxqw/ lq wkh ihhgedfn fdvh rqo| wkh fxuuhqw vwrfnv ri vwdwhv duh frqvlghuhg1 5 Wkh olwhudwxuh rq glhuhqwldo jdphv ghyrwhv d frqvlghudeoh dprxqw ri dwwhqwlrq wr lghqwli|lqj fodvvhv ri jdphv zkhuh wkh forvhg0orrs htxloleuld gh0 jhqhudwh lqwr rshq0orrs htxloleuld1 Wklv lqwhuhvw lv prwlydwhg e| wkh iroorzlqj uhdvrq1 Zkhqhyhu dq rshq0orrs htxloleulxp lv d ghjhqhudwh forvhg0orrs htxl0 oleulxp/ wkhq wkh iruphu lv dovr vxejdph shuihfw> wkhuhiruh rqh fdq uho| rq wkh rshq0orrs htxloleulxp zklfk/ lq jhqhudo/ lv pxfk hdvlhu wr ghulyh wkdq wkh forvhg0orrs rqh1 6 Zh vkrz odwhu wkdw/ lq wkh suhvhqw prgho/ wkh frlqflghqfh ehwzhhq forvhg0orrs dqg rshq0orrs Qdvk htxloleulxp dulvhv1 Zh vwduw e| frqvlghulqj wkh vroxwlrq ri wkh roljrsro| jdph xqghu wkh forvhg0orrs lqirupdwlrq vwuxfwxuh1 Wkh g|qdplf rswlpl}dwlrq sureohp uh0 txluhv wr frqvlghu wkh iroorzlqj Kdplowrqldq ixqfwlrq iru hdfk sod|hu= 5 Iru roljrsro| prghov zkhuh upv iroorz ihhgedfn uxohv/ vhh Vlpddq dqg Wdnd|dpd + 4<:; , / Ihuvkwpdq dqg Ndplhq + 4<;:/ 4<<3 , / Grfnqhu dqg Kdxj + 4<<3 , / lqwh u dold 1Irud fohdu h{srvlwlrq ri wkh glhuhqfh dprqj wkhvh htxloleulxp vroxwlrqv vhh Ed vdu dqg Rovghu + 4<;5/ ss1 64;065:/ dqg fkdswhu 9/ lq sduwlfxodu Sursrvlwlrq 914 , ru Grfnqhu hw do1 + 5333 , 1 6 Fodvvhv ri jdphv zkhuh wklv frlqflghqfh dulvhv duh looxvwudwhg lq Fohpkrxw dqg Zdq + 4<:7 , > Uhlqjdqxp + 4<;5 , > Phkopdqq dqg Zloolqj + 4<;6 , > Grfnqhu/ Ihlfkwlqjhu dqg Mûu0 jhqvhq + 4<;8 , > Ihuvkwpdq + 4<;: , > Ihuvkwpdq/ Ndplhq dqg Pxoohu + 4<<5 , 1 Iru dq ryhuylhz/ vhh Phkopdqq + 4<;; , / Ihuvkwpdq/ Ndplhq dqg Pxoohu + 4<<5 , / dqg Grfnqhu hw do1 + 5333 , 1 7 K  E|e 3 4| %#  E|%  E|([   '  %  E|$%  E|k 2E&  E| 2 S%  E|n+8, nb  E|#&  E|n[   '  &  E|B  E|$n n[   '  b  E|#&  E|n[   '  &  E|B  E|$& zkhuh b  E |'>  E |e 4| lv wkh frvwdwh yduldeoh +hydoxdwhg dw wlph |,dvvrfldwhg e| sod|hu zlwk vwdwh yduldeoh   Wkh uvw rughu frqglwlrqv dqg wkh dgmrlqw htxdwlrqv +iru hdfk sod|hu ,duh= YK  E c | Y%  E |'f( +9, YK  E c | Y&  E |'f( +:, YK  E  Y  E |[   '  YK  E  Y%  E | Y% W  E | Y  E |[   '  YK  E  Y&  E | Y& W  E | Y  E |'Yb  E | Y| 4b  ;'c2 +;, Wkhvh frqglwlrqv kdyh wr eh frqvlghuhg dorqj zlwk wkh lqlwldo frqglwlrqv i  Ef '   f j   ' dqg wkh wudqvyhuvdolw| frqglwlrqv/ zklfk vhw wkh qdo ydoxh ri wkh vwdwh dqg2ru fr0vwdwh yduldeohv1 Lq sureohpv ghqhg ryhu dq lqqlwh wlph krul}rq/ lw lv xvxdo wr vhw *4 | <" b  E |  E |'firu doo  +<, AIurp +9, dqg +:, zh rewdlq uhvshfwlyho|=   E |S2%  E |([   '  %  E |'f +43, k&  E |nb  E |n[   '  b  E |'f +44, 8 dqg wkhq/ xqghu wkh v|pphwu| k|srwkhvlv S   '  %  E|'E%c wkh rswlpdo uxohv= % W  E|'   E|S 2n(E +45, & W  E|'b  E|nEb  E| k+46, Lq wkh dgmrlqw htxdwlrqv +;,/ wkh whupv YK  E  Y%  E | Y% W  E | Y  E |cYK  E  Y&  E | Y& W  E | Y  E |+47, fdswxuh wkh vwudwhjlf lqwhudfwlrq dprqj upv wkurxjk wkh ihhgedfn iurp vwdwhv wr frqwurov/ zklfk lv e| ghqlwlrq devhqw xqghu wkh rshq0orrs vroxwlrq frqfhsw1 Zkhqhyhu wkh h{suhvvlrqv lq +47, duh }hur iru doo c wkhq wkh forvhg0 orrs phpru|ohvv htxloleulxp froodsvhv lqwr wkh rshq0orrs Qdvk htxloleulxp1 Wklv kdsshqv lq wkh suhvhqw prgho ehfdxvh wkh rswlpdo ydoxhv ri wkh frqwuro yduldeohv ri hdfk sod|hu duh qrw dhfwhg e| vwdwh yduldeohv glhuhqw iurp lwv rzq pdunhw vl}h1 Zlwk uhihuhqfh wr wkh odehov xvhg e| Grfnqhu hw do1 +5333/ fk1 :,/ wkh sureohp dw kdqg lv d olqhdu vwdwh glhuhqwldo jdph/ dqg wkh forvhg0orrs vroxwlrq froodsvhv lqwr dq rshq0orrs vroxwlrq/ zklfk lv vwurqjo| wlph frqvlvwhqw/ ru Pdunry shuihfw +vhh dovr/ h1j1/ Gulvnloo dqg PfFdhuw|/ 4<;</ ss1 65:05;,1 Dffruglqjo|/ frqglwlrqv +;, uhzulwh dv iroorzv= Yb  E| Y| '%  E|b  E|Bb  E|4+48, Yb  E | Y| 'b  E |Bb  E |4+49, Glhuhqwldwlqj +46, zlwk uhvshfw wr wlph/ soxjjlqj +48, dqg uhduudqjlqj/ zh rewdlq= Y& W  E | Y| ' k E 4nBk&  E |  E |S 2n( E +4:, Wkh g|qdplf htxdwlrq +4:,/ dorqjvlgh zlwk wkh g|qdplf frqvwudlqw xqghu v|pphwu| frqglwlrq/ _  E | * _| ' E n E &  E |B  E |/pd|ehuhzulw0 whq/ lq pdwul{ whup/ dv iroorzv +vxevfulsw lv rplwwhg/ dv lw lv uhgxqgdqw xqghu v|pphwu|,= 9 dqg Mdftxhplq/ 4<;;> Ndplhq/ Pxoohu dqg ]dqj/ 4<<5> Vx}xpxud/ 4<<5/ lqwhu dold/ ru 0 lq wkh vdph yhlq 0 Qrupdq dqg Wklvvh/ 4<<9,1 9Frqfoxvlrqv Lq wkh iruhjrlqj dqdo|vlv/ zh kdyh vkrzq wkdw wkh frqvlghudwlrq ri wkh srvlwlyh h{whuqdo hhfwv iurp dgyhuwlvlqj ri d up xsrq rwkhu upv* pdunhw vl}h kdv qrw d wulyldo lpsdfw rq wkh frqfoxvlrq derxw wkh vrfldo ghvludelolw| ri fduwho ehkdylrxu ri upv1 Lq sduwlfxodu/ zh kdyh wdnhq d glhuhqwldo jdph dssurdfk wr vwxg| wkh g|qdplf rswlpdo sodqv ri upv wkdw surgxfh glhuhqwldwhg jrrgv dqg pdnh dgyhuwlvlqj hruwv1 Wkh dgyhuwlvlqj hruwv ri dq| up kdyh orqj0 odvwlqj hhfwv rq khu rzq pdunhw vl}hv/ dv zhoo dv rq wkh pdunhw vl}h di doo wkh ulydo upv1 Xqghu wklv uhvshfw/ dgyhuwlvlqj lv/ dw ohdvw wr vrph h{whqw/ d sxeolf jrrg1 Zh kdyh irfxvhg rq wkh vwhdg| vwdwh vroxwlrqv ri wkh g|qdplf v|vwhp xqghu +xvxdo, v|pphwu| frqglwlrqv/ dqg zh kdyh vkrzq wkdw/ zkhq d phdqlqjixoo vwhdg| vwdwh h{lvwv/ lw lv d vdggoh srlqw1 Wkdqnv wr wkh srvlwlyh h{whuqdo hhfwv ri dgyhuwlvhphqw/ lw lv qr orqjhu wuxh wkdw wkh fduwho ehkdylrxu/ l1h1 wkh ehkdylrxu ri upv dlplqj dw wkh pd{lpxp mrlqw surw/ lv qhfhvvdulo| ghwulphqwdo wr wkh vwhdg| vwdwh vrfldo zhoiduh1 Wklv uhvxow lv frpprq wr wkh frqfoxvlrq ri rwkhu lqyhvwljdwlrqv/ zkhuh wkh uhohydqfh ri suh0pdunhw ehkdylrxu/ olnh lqyhvwphqw lq surgxfw ru surfhvv lqqrydwlrq/ fdq ohdg wr wkh frqfoxvlrq wkdw wkh fduwhol}dwlrq lv vrfldoo| ghvludeoh1 D glhuhqw lqwhuhvw lq wkh suhvhqw prgho frxog eh prwlydwhg/ lq rxu rslq0 lrq/ e| wkh idfw wkdw lwv irupdol}dwlrq lv ri olqhdu vwdwh w|sh/ dqg khqfh wkh rshq0orrs Qdvk htxloleulxp lv wlph0frqvlvwhqw/ vlqfh lw lv d ghjhqhudwh forvhg0 orrs htxloleulxp1 Frqvlghu/ xqghu wklv uhvshfw/ wkdw/ wkh dydlodeoh olwhudwxuh rq glhuhqwldo jdphv ghyrwhv frqvlghdeoh hruwv wr sursrvh prghov wkdw duh erwk wlph0frqvlvwhqw +dv xqghu wkh forvhg0orrs lqirupdwlrq vwuxfwxuh, dqg hdvlo| vroydeoh +olnh xqghu wkh rshq0orrs htxloleulxp lqirupdwlrq vwuxfwxuh,1 46 Uhihuhqfhv ^4` Edvdu/ W1 dqg J1M1 Rovghu +4<;5/ 4<<8 2 ,/ G|qdplf Qrqfrrshudwlyh Jdph Wkhru|/ Vdq Glhjr/ Dfdghplf Suhvv1 ^5` Fhoolql/ U1 dqg O1 Odpehuwlql +4<<;,/ D G|qdplf Prgho ri Glhuhqwl0 dwhg Roljrsro| zlwk Fdslwdo Dffxpxodwlrq/ Mrxuqdo ri Hfrqrplf Wkh0 ru|/;6/ 4780881 ^6` Fhoolql/ U1 dqg O1 Odpehuwlql +5333,/ Qrq0Olqhdu Pdunhw Ghpdqg dqg Fdslwdo Dffxpxodwlrq lq d Glhuhqwldo Roljrsro| Jdph/ zrunlqj sdshu qr1 6:3/ Ghsduwphqw ri Hfrqrplfv/ Xqlyhuvlw| ri Erorjqd1 ^7` Fhoolql/ U1/ O1 Odpehuwlql/ +5335,/ D Glhuhqwldo Jdph Dssurdfk wr Lqyhvwphqw lq Surgxfw Glhuhqwldwlrq/ Mrxuqdo ri hfrqrplf G|qdplfv dqg Frqwuro/iruwkfrplqj1 ^8` Fohpkrxw/ V1 dqg K1\1 Zdq/ Mu1 +4<:7,/ D Fodvv ri Wulolqhdu Glhuhqwldo Jdphv/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssolfdwlrqv/47/ 74<0571 ^9` g*Dvsuhprqw/ F1/ D1 Mdftxhplq +4<;;,/ Frrshudwlyh dqg Qrqfrrshu0 dwlyh U)G lq Gxrsro| zlwk Vslooryhuv/ Dphulfdq Hfrqrplfv Uhylhz/ :;/ 446606:1 ^:` Grfnqhu/ H1M1 dqg D1D1 Kdxj +4<<3,/ Wdulv dqg Txrwdv xqghu G|0 qdplf Gxrsrolvwlf Frpshwlwlrq/ Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv/ 5</47:08<1 ^;` Grfnqhu/ H1M1/ J1 Ihlfkwlqjhu dqg V1 Mûujhqvhq +4<;8,/ Wudfwdeoh Fodvvhv ri Qrq}hur0Vxp Rshq0Orrs Qdvk Glhuhqwldo Jdphv= Wkhru| dqg H{dpsohv/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssolfdwlrqv/78/ 4:<0<:1 ^<` Grfnqhu/ H1M1/ V1 Mûujhqvhq/ Q1 Ydq Orqj/ J1 Vrujhq +5333,/ Glhuhqwldo Jdphv lq Hfrqrplfv dqg pdqdjphqw Vflhqfh/ Fdpeulgjh Xqlyhuvlw| Suhvv/ Fdpeulgjh1 ^43` Gulvnloo/ U1D1 dqg V1 PfFdhuw| +4<;<,/ G|qdplf Gxrsro| zlwk Dg0 mxvwphqw Frvwv= D Glhuhqwldo Jdph Dssurdfk/ Mrxuqdo ri Hfrqrplf Wkhru|/7</ 65706;1 47 ^44` Hulfnvrq/ J1P1 +4<;8,/ D Prgho ri Dgyhuwlvlqj Frpshwlwlrq/ Mrxuqdo ri Pdunhwlqj Uhvhdufk/55/ 5<:06371 ^45` Hulfnvrq/ J1P1 +4<<4,/ G|qdplf Prghov ri Dgyhuwlvlqj Frpshwlwlrq/ Gruguhfkw/ Noxzhu1 ^46` Ihlfkwlqjhu/ J1 +4<;6,/ Wkh Qdvk Vroxwlrq ri dq Dgyhuwlvlqj Glhuhq0 wldo Jdph= Jhqhudol}dwlrq ri d Prgho e| Ohlwpdqq dqg Vfkplwhqgrui/ LHHH Wudqvdfwlrqv rq Dxwrpdwlf Frqwuro/5;/ 437707;1 ^47` Ihlfkwlqjhu/ J1/ U1I1 Kduwo dqg S1V1 Vhwkl +4<<7,/ G|qdplf Rswlpdo Frqwuro Prghov lq Dgyhuwlvlqj= Uhfhqw Ghyhorsphqwv/ Pdqdjhphqw Vflhqfh/73/ 4<805591 ^48` Ihuvkwpdq/ F1 +4<;7,/ Jrrgzloo dqg Pdunhw Vkduhv lq Roljrsro|/ Hfr0 qrplfd/84/ 5:40;51 ^49` Ihuvkwpdq/ F1 +4<;:,/ Lghqwlfdwlrq ri Fodvvhv ri Glhuhqwldo Jdphv iru Zklfk wkh Rshq0Orrs lv d ghjhqhuwdhg Ihhgedfn Qdvk Htxloleulxp/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssolfdwlrqv/88/ 54:0641 ^4:` Ihuvkwpdq/ F1 dqg P1L1 Ndplhq +4<;:,/ G|qdplf Gxrsrolvwlf Frpsh0 wlwlrq zlwk Vwlfn| Sulfhv/ Hfrqrphwulfd/88/ 44840971 ^4;` Ihuvkwpdq/ F1 dqg P1L1 Ndplhq +4<<3,/ Wxuqslnh Surshuwlhv lq d Ilqlwh0Krul}rq Glhuhqwldo Jdph= G|qdplf Gxrsro| zlwk Vwlfn| Sulfhv/ Lqwhuqdwlrqdo Hfrqrplf Uhylhz/64/7<0931 ^4<` Ihuvkwpdq/ F1 dqg H1 Pxoohu +4<;7,/ Fdslwdo Dffxpxodwlrq Jdphv ri Lqqlwh Gxudwlrq/ Mrxuqdo ri Hfrqrplf Wkhru|/66/ 65506<1 ^53` Ihuvkwpdq/ F1/ P1 Ndplhq dqg H1 Pxoohu +4<<5,/ Lqwhjudo Jdphv= Wkh0 ru| dqg Dssolfdwlrqv/ lq Ihlfkwlqjhu/ J1 +hg1,/ G|qdplf Hfrqrplf Prg0 hov dqg Rswlpdo Frqwuro/ Dpvwhugdp/ Qruwk0Kroodqg/ 5<:06441 ^54` Iulhgpdq/ M1Z1 +4<;6,/ Dgyhuwlvlqj dqg Roljrsrolvwlf Htxloleulxp/ Ehoo Mrxuqdo ri Hfrqrplfv/47/7970:61 ^55` Mûujhqvhq/ V1 +4<;5,/ D Vxuyh| ri Vrph Glhuhqwldo Jdphv lq Dgyhu0 wlvlqj/ Mrxuqdo ri Hfrqrplf G|qdplfv dqg Frqwuro/7/ 67409<1 48 ^56` Mûujhqvhq/ V1 dqg J1 ]dffrxu +4<<<,/ Htxloleulxp Sulflqj dqg Dg0 yhuwlvlqj Vwudwhjlhv lq d Pdunhwlqj Fkdqqho/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssolfdwlrqv/435/ 4440581 ^57` Ndplhq/ P1L1/ H1 Pxoohu dqg L1 ]dqj +4<<5,/ Frrshudwlyh Mrlqw Yhq0 wxuhv dqg U)G Fduwhov/ Dphulfdq Hfrqrplf Uhylhz/;5/ 45<6046391 ^58` Odpehuwlql/ O1 +4<<:,/ Sulvrqhuv* Glohppd lq Gxrsro| +Vxshu,Jdphv/ Mrxuqdo ri Hfrqrplf Wkhru|/::/4;40<41 ^59` Ohlwpdqq/ J1 dqg Z1H1 Vfkplwhqgrui +4<:;,/ Surw Pd{lpl}dwlrq wkurxjk Dgyhuwlvlqj= D Qrq}hur Vxp Glhuhqwldo Jdph Dssurdfk/ LHHH Wudqvdfwlrqv rq Dxwrpdwlf Frqwuro/56/ 9790831 ^5:` Phkopdqq/ D1 +4<;;,/ Dssolhg Glhuhqwldo Jdphv/Qhz\run/Sohqxp Suhvv1 ^5;` Phkopdqq/ D1 dqg U1 Zloolqj +4<;6,/ Rq Qrqxqltxh Forvhg0Orrs Qdvk Htxloleuld iru d Fodvv ri Glhuhqwldo Jdphv zlwk d Xqltxh dqg Ghjhq0 hudwh Ihhgedfn Vroxwlrq/ Mrxuqdo ri Rswlpl}dwlrq Wkhru| dqg Dssol0 fdwlrqv/74/ 7960:51 ^5<` Qhuoryh/ P1 dqg N1M1 Duurz +4<95,/ Rswlpdo Dgyhuwlvlqj Srolf| xqghu G|qdplf Frqglwlrqv/ Hfrqrplfd/5</ 45<0751 ^63` Qrupdq/ J1 dqg M1I1 Wklvvh +4<<9,/Surgxfw Ydulhw| dqg Zhoiduh xqghu Wrxjk dqg Vriw sulflqj Uhjlphv/ Hfrqrplf Mrxuqdo/439/ :90<41 ^64` Sljd/ F1 +5333,/ Frpshwlwlrq lq d Gxrsro| zlwk Vwlfn| Sulfh dqg Dg0 yhuwlvlqj/ Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq/4;/ 8<809471 ^65` Uhlqjdqxp/ M1 +4<;5,/ D Fodvv ri Glhuhqwldo Jdphv iru Zklfk wkh Forvhg Orrs dqg Rshq Orrs Qdvk Htxloleuld Frlqflgh/ Mrxuqdo ri Rs0 wlpl}dwlrq Wkhru| dqg Dssolfdwlrqv/69/ 5860951 ^66` Uh|qrogv/ V1V1 +4<;:,/ Fdsdflw| Lqyhvwphqw/ Suhhpswlrq dqg Frpplw0 phqw lq dq Lqqlwh Krul}rq Prgho/ Lqwhuqdwlrqdo Hfrqrplf Uhylhz/ 5;/9<0;;1 ^67` Vhwkl/ V1S1 +4<::,/ G|qdplf Rswlpdo Frqwuro Prghov lq Dgyhuwlvlqj= D Vxuyh|/ VLDP Uhylhz/4</9;80:581 49 ^68` Vlpddq/ P1 dqg W1 Wdnd|dpd +4<:;,/ Jdph Wkhru| Dssolhg wr G|0 qdplf Gxrsro| Sureohpv zlwk Surgxfwlrq Frqvwudlqwv/ Dxwrpdwlfd/ 47/4940991 ^69` Vlqjk/ Q1 dqg [1 Ylyhv +4<;7,/ Sulfh dqg Txdqwlw| Frpshwlwlrq lq d Glhuhqwldwhg Gxrsro|/ UDQG Mrxuqdo ri Hfrqrplfv/48/ 87908871 ^6:` Vshqfh/ D1P1 +4<:9,/ Surgxfw Glhuhqwldwlrq dqg Zhoiduh/ Dphulfdq Hfrqrplf Uhylhz/99/ 73:07471 ^6;` Vx}xpxud/ N +4<<5,/ Frrshudwlyh dqg Qrqfrrshudwlyh U)G lq Gxrsro| zlwk Vslooryhuv/ Dphulfdq Hfrqrplfv Uhylhz/;5/ 463:0531 ^6<` Ylyhv/ [1 +4<;8,/ H!flhqf| ri Ehuwudqg dqg Frxuqrw Htxloleuld zlwk Surgxfw Glhuhqwldwlrq/ Mrxuqdo ri Hfrqrplf Wkhru|/69/ 49904:81 4: