BUNDLING INFORMATION GOODS: PRICING, PROFITS AND EFFICIENCY ppt

%XQGOLQJ,QIRUPDWLRQ*RRGV 3ULFLQJ3URILWVDQG(IILFLHQF\ <DQQLV%DNRV DQG(ULN%U\QMROIVVRQ )LUVW'UDIW'HFHPEHU &XUUHQW'UDIW)HEUXDU\ )RUWKFRPLQJLQ0DQDJHPHQW6FLHQFH &RS\ULJKWE\<DQQLV%DNRVDQG(ULN%U\QMROIVVRQ   6WHUQ6FKRRORI%XVLQHVV1HZ<RUN8QLYHUVLW\HPDLOEDNRV#VWHUQQ\XHGX  6ORDQ6FKRRORI0DQDJHPHQW0DVVDFKXVHWWV,QVWLWXWHRI7HFKQRORJ\HPDLOHULNE#PLWHGX %XQGOLQJ,QIRUPDWLRQ*RRGV 3ULFLQJ3URILWVDQG(IILFLHQF\ $%675$&7 :HVWXG\WKHVWUDWHJ\RIEXQGOLQJDODUJHQXPEHURILQIRUPDWLRQJRRGVVXFKDVWKRVHLQFUHDVLQJO\DYDLODEOH RQWKH,QWHUQHWDQGVHOOLQJWKHPIRUDIL[HGSULFH:HDQDO\]HWKHRSWLPDOEXQGOLQJVWUDWHJLHVIRUD PXOWLSURGXFWPRQRSROLVWDQGZHILQGWKDWEXQGOLQJYHU\ODUJHQXPEHUVRIXQUHODWHGLQIRUPDWLRQJRRGVFDQ EHVXUSULVLQJO\SURILWDEOH7KHUHDVRQLVWKDWWKHODZRIODUJHQXPEHUVPDNHVLWPXFKHDVLHUWRSUHGLFW FRQVXPHUVYDOXDWLRQVIRUDEXQGOHRIJRRGVWKDQWKHLUYDOXDWLRQVIRUWKHLQGLYLGXDOJRRGVZKHQVROG VHSDUDWHO\$VDUHVXOWWKLVSUHGLFWLYHYDOXHRIEXQGOLQJPDNHVLWSRVVLEOHWRDFKLHYHJUHDWHUVDOHVJUHDWHU HFRQRPLFHIILFLHQF\DQGJUHDWHUSURILWVSHUJRRGIURPDEXQGOHRILQIRUPDWLRQJRRGVWKDQFDQEHDWWDLQHG ZKHQWKHVDPHJRRGVDUHVROGVHSDUDWHO\2XUPDLQUHVXOWVGRQRWH[WHQGWRPRVWSK\VLFDOJRRGVDVWKH PDUJLQDOFRVWVRISURGXFWLRQIRUJRRGVQRWXVHGE\WKHEX\HUW\SLFDOO\QHJDWHDQ\EHQHILWVIURPWKH SUHGLFWLYHYDOXHRIODUJHVFDOHEXQGOLQJ :KLOHGHWHUPLQLQJRSWLPDOEXQGOLQJVWUDWHJLHVIRUPRUHWKDQWZRJRRGVLVDQRWRULRXVO\GLIILFXOWSUREOHP ZHXVHVWDWLVWLFDOWHFKQLTXHVWRSURYLGHVWURQJDV\PSWRWLFUHVXOWVDQGERXQGVRQSURILWVIRUEXQGOHVRIDQ\ DUELWUDU\VL]H:HVKRZKRZRXUPRGHOFDQEHXVHGWRDQDO\]HWKHEXQGOLQJRIFRPSOHPHQWVDQG VXEVWLWXWHVEXQGOLQJLQWKHSUHVHQFHRIEXGJHWFRQVWUDLQWVDQGEXQGOLQJRIJRRGVZLWKYDULRXVW\SHVRI FRUUHODWLRQV:HILQGWKDWZKHQGLIIHUHQWPDUNHWVHJPHQWVRIFRQVXPHUVGLIIHUV\VWHPDWLFDOO\LQWKHLU YDOXDWLRQVIRUJRRGVVLPSOHEXQGOLQJZLOOQRORQJHUEHRSWLPDO+RZHYHUE\RIIHULQJDPHQXRIGLIIHUHQW EXQGOHVDLPHGDWHDFKPDUNHWVHJPHQWEXQGOLQJPDNHVWUDGLWLRQDOSULFHGLVFULPLQDWLRQVWUDWHJLHVPRUH SRZHUIXOE\UHGXFLQJWKHUROHRIXQSUHGLFWDEOHLGLRV\QFUDWLFFRPSRQHQWVRIYDOXDWLRQV7KHSUHGLFWLRQVRI RXUDQDO\VLVDSSHDUWREHFRQVLVWHQWZLWKHPSLULFDOREVHUYDWLRQVRIWKHPDUNHWVIRU,QWHUQHWDQGRQOLQH FRQWHQWFDEOHWHOHYLVLRQSURJUDPPLQJDQGFRS\ULJKWHGPXVLF BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB :HWKDQN7LPRWK\%UHVQDKDQ+XQJ.HQ&KLHQ)UDQN)LVKHU0LFKDHO+DUULVRQ3DXO.OHLQGRUIHU7KRPDV 0DORQH5REHUW3LQG\FN1DQF\5RVH5LFKDUG6FKPDOHQVHH-RKQ7VLWVLNOLV+DO9DULDQ$OEHUW:HQJHU %LUJHU:HUQHUIHOWIRXUDQRQ\PRXVUHYLHZHUVDQGVHPLQDUSDUWLFLSDQWVDWWKH8QLYHUVLW\RI&DOLIRUQLDDW %HUNHOH\0,71HZ<RUN8QLYHUVLW\6WDQIRUG8QLYHUVLW\8QLYHUVLW\RI5RFKHVWHUWKH:KDUWRQ6FKRROWKH :RUNVKRSRQ,QIRUPDWLRQ6\VWHPVDQG(FRQRPLFVDQGWKH:RUNVKRSRQ0DUNHWLQJ6FLHQFHDQG WKH,QWHUQHWIRUPDQ\KHOSIXOVXJJHVWLRQV 3DJH ,QWURGXFWLRQ 2YHUYLHZ 0LOOLRQVRIGLJLWDOLQIRUPDWLRQJRRGVFDQEHGLVWULEXWHGDOPRVWFRVWOHVVO\YLDGDWDQHWZRUNVVXFKDVWKH ,QWHUQHW7KHWHFKQRORJ\FRQWLQXHVWRDGYDQFHZLWKEUHDWKWDNLQJVSHHG\HWH[LVWLQJWKHRU\DQGSUDFWLFHIDLOWR SURYLGHFOHDUJXLGDQFHRQKRZGLJLWDOLQIRUPDWLRQJRRGVVKRXOGEHSDFNDJHGSULFHGDQGVROG$WRQHHQGRI WKHVSHFWUXPWHFKQRORJLHVVXFKDVPLFURSD\PHQWVLQFUHDVLQJO\HQDEOHWKHVDOHDQGGHOLYHU\RIVPDOOXQLWVRI LQIRUPDWLRQEXWLQWKLVDUWLFOHZHGUDZDWWHQWLRQWRWKHRSSRVLWHHQG:HDQDO\]HWKHVWUDWHJ\RIEXQGOLQJD ODUJHQXPEHURILQIRUPDWLRQJRRGVDQGVHOOLQJWKHEXQGOHIRUDIL[HGSULFH:HILQGWKDWLQDYDULHW\RI FLUFXPVWDQFHVDPXOWLSURGXFWPRQRSROLVWZLOOH[WUDFWVXEVWDQWLDOO\KLJKHUSURILWVE\RIIHULQJRQHRUPRUH EXQGOHVRILQIRUPDWLRQJRRGVWKDQE\RIIHULQJWKHVDPHJRRGVVHSDUDWHO\,QDGGLWLRQZHSURYLGHFULWHULDIRU WKHRSWLPDOGHVLJQDQGSULFLQJRIEXQGOHV 7KHNH\LQWXLWLRQEHKLQGWKHSRZHURIEXQGOLQJLVWKDWFRQVXPHU·VYDOXDWLRQIRUDFROOHFWLRQRIJRRGVW\SLFDOO\ KDVDSUREDELOLW\GLVWULEXWLRQZLWKDORZHUYDULDQFHSHUJRRGFRPSDUHGWRWKHYDOXDWLRQVIRUWKHLQGLYLGXDO JRRGV7KHODUJHUWKHQXPEHURIJRRGVEXQGOHGWKHJUHDWHUWKHW\SLFDOUHGXFWLRQLQWKHYDULDQFH%HFDXVH XQFHUWDLQW\DERXWFRQVXPHUYDOXDWLRQVLVWKHHQHP\RIHIIHFWLYHSULFLQJDQGHIILFLHQWWUDQVDFWLRQV0\HUVRQ DQG6DWWHUWKZDLWHWKLVSUHGLFWLYHYDOXHRIEXQGOLQJFDQEHYHU\YDOXDEOH)RULQVWDQFHFRQVXPHU YDOXDWLRQVIRUDQRQOLQHVSRUWVVFRUHERDUGDQHZVVHUYLFHRUDGDLO\KRURVFRSHZLOOYDU\$PRQRSROLVW VHOOLQJWKHVHJRRGVVHSDUDWHO\ZLOOW\SLFDOO\PD[LPL]HSURILWVE\FKDUJLQJDSULFHIRUHDFKJRRGWKDWH[FOXGHV VRPHFRQVXPHUVZLWKORZYDOXDWLRQVIRUWKDWJRRGDQGIRUJRHVVLJQLILFDQWUHYHQXHVIURPVRPHFRQVXPHUV ZLWKKLJKYDOXDWLRQV$OWHUQDWLYHO\WKHVHOOHUFRXOGRIIHUDOOWKHLQIRUPDWLRQJRRGVDVDEXQGOH8QGHUDYHU\ JHQHUDOVHWRIFRQGLWLRQVWKHODZRIODUJHQXPEHUVJXDUDQWHHVWKDWWKHGLVWULEXWLRQRIYDOXDWLRQVIRUWKHEXQGOH KDVSURSRUWLRQDWHO\IHZHUH[WUHPHYDOXHV$V6FKPDOHQVHHKDVDUJXHGVXFKDUHGXFWLRQLQ´EX\HU GLYHUVLW\µW\SLFDOO\KHOSVVHOOHUVH[WUDFWKLJKHUSURILWVIURPDOOFRQVXPHUV +LVWRULFDOO\YHU\ODUJHEXQGOHVRIJRRGVKDYHW\SLFDOO\EHHQXQSURILWDEOHDQGKHQFHXQFRPPRQLQSUDFWLFH 7KH\KDYHDOVREHHQKRSHOHVVO\FRPSOH[WRPRGHOWKHQXPEHURISRVVLEOHLQWHUDFWLRQVLVVLPSO\WRRODUJHWR GHULYHJHQHUDOUHVXOWV+DQVRQDQG0DUWLQ0F$GDPV$VDUHVXOWODUJHEXQGOHVKDYHUHFHLYHG OLWWOHDWWHQWLRQ+RZHYHUDVZHVKRZEHORZWKHDGYHQWRIGLJLWDOLQIRUPDWLRQJRRGVZLWKYHU\ORZPDUJLQDO FRVWVFDQPDNHEXQGOLQJKXQGUHGVRUHYHQWKRXVDQGVRIXQUHODWHGJRRGVDSURILWDEOHVWUDWHJ\)XUWKHUPRUH WKHPRGHOLQJIUDPHZRUNWKDWZHLQWURGXFHSURYLGHVVWURQJUHVXOWVUHJDUGLQJWKHSURILWDELOLW\RIEXQGOLQJ HYHQXQGHUUHODWLYHO\ZHDNDVVXPSWLRQV8QOLNHHDUOLHUZRUNRXUPRGHOGRHVQRWEHFRPHPRUHFRPSOH[DV WKHQXPEHURIJRRGVLQFUHDVHV,QVWHDGWKHSUHFLVLRQRIRXUDQDO\VLVLQFUHDVHVZLWKWKHQXPEHURIJRRGV FRQVLGHUHGPDNLQJRXUIUDPHZRUNVXLWDEOHIRUXQGHUVWDQGLQJWKHHFRQRPLFVRIODUJHEXQGOHV2XUPRGHOFDQ H[SODLQWKHSUHYDOHQFHRIODUJHEXQGOHVRILQIRUPDWLRQJRRGVDQGSURYLGHVJXLGHOLQHVIRUWKHXVHRIPRUH FRPSOH[VWUDWHJLHVVXFKDVPL[HGEXQGOLQJZKLFKLQYROYHVVLPXOWDQHRXVO\VHOOLQJDODUJHEXQGOHDQGRQHRU PRUHVXEVHWVRIWKHEXQGOH 3DJH 7KHEXQGOLQJOLWHUDWXUH %XQGOLQJKDVPDQ\SRWHQWLDOEHQHILWVLQFOXGLQJFRVWVDYLQJVLQSURGXFWLRQDQGWUDQVDFWLRQFRVWV FRPSOHPHQWDULWLHVDPRQJWKHEXQGOHFRPSRQHQWVDQGVRUWLQJFRQVXPHUVDFFRUGLQJWRWKHLUYDOXDWLRQV (SSHQ+DQVRQHWDO:HIRFXVRQWKLVODVWEHQHILWRIEXQGOLQJZKLFKZDVILUVWGLVFXVVHGE\6WLJOHU LQDSDSHUVKRZLQJKRZEXQGOLQJFRXOGLQFUHDVHVHOOHUV·SURILWVZKHQFRQVXPHUYDOXDWLRQVIRUWZR JRRGVZHUHQHJDWLYHO\FRUUHODWHG$GDPVDQG<HOOHQLQWURGXFHGDWZRGLPHQVLRQDOJUDSKLFDO IUDPHZRUNIRUDQDO\]LQJEXQGOLQJDVDGHYLFHIRUSULFHGLVFULPLQDWLRQ%\LQWURGXFLQJDVHWWLQJZLWKD PXOWLSURGXFWPRQRSROLVWWZRJRRGVQRUHVHOOLQJLQGHSHQGHQWDQGDGGLWLYHFRQVXPHUYDOXDWLRQVDQGOLQHDU ´XQLWGHPDQGVµLHFRQVXPHUVEX\HLWKHU]HURRURQHXQLWIRUWKHVHWZRJRRGVWKH\FRPSDUHXQEXQGOHG VDOHVWRSXUHEXQGOLQJRIIHULQJRQO\WKHFRPSOHWHEXQGOHDQGPL[HGEXQGOLQJRIIHULQJERWKWKHFRPSOHWH EXQGOHDQGVXEVHWVRIWKHEXQGOH 8VLQJVW\OL]HGH[DPSOHVWKH\LOOXVWUDWHWKDWYDULRXVW\SHVRIEXQGOLQJFDQ EHPRUHRUOHVVSURILWDEOHWKDQXQEXQGOLQJ 7KHPRUHIRUPDODQDO\VHVE\6FKPDOHQVHH0F$IHH0F0LOODQDQG:KLQVWRQDQG6DOLQJHU DOVRIRFXVHGRQEXQGOHVRIWZRJRRGV6FKPDOHQVHHDVVXPHGDELYDULDWH*DXVVLDQGLVWULEXWLRQRIUHVHUYDWLRQ SULFHVDQGWKURXJKDFRPELQDWLRQRIDQDO\WLFGHULYDWLRQDQGQXPHULFDOWHFKQLTXHVVKRZHGWKDWSXUHEXQGOLQJ W\SLFDOO\UHGXFHVWKHGLYHUVLW\RIWKHSRSXODWLRQRIFRQVXPHUVWKHUHE\HQDEOLQJVHOOHUVWRH[WUDFWPRUH FRQVXPHUV·VXUSOXV+HIRXQGWKDWEXQGOLQJFDQLQFUHDVHSURILWVLIWKHYDOXDWLRQVRIWKHWZRJRRGVDUH QHJDWLYHO\FRUUHODWHGDVVXJJHVWHGE\6WLJOHUDQG$GDPVDQG<HOOHQEXWFDQDOVREHWUXHLIWKHYDOXDWLRQVDUH LQGHSHQGHQWRUHYHQSRVLWLYHO\EXWQRWSHUIHFWO\FRUUHODWHG 0F$IHH0F0LOODQDQG:KLQVWRQDQDO\]HGDVHWWLQJZLWKDPXOWLSURGXFWPRQRSROLVWDQGDFRQWLQXXPRI FRQVXPHUYDOXDWLRQV7KH\GHULYHGDVHWRIFRQGLWLRQVXQGHUZKLFKPL[HGEXQGOLQJRIWZRJRRGVGRPLQDWHV XQEXQGOHGVDOHV6DOLQJHUGHYHORSVDJUDSKLFDOIUDPHZRUNWRDQDO\]HWKHSURILWDELOLW\DQGZHOIDUH LPSOLFDWLRQVRIEXQGOLQJWZRJRRGVSULPDULO\LQWKHFRQWH[WRILQGHSHQGHQWOLQHDUGHPDQGIXQFWLRQV+H ILQGVWKDWEXQGOLQJWZRJRRGVWHQGVWREHSURILWDEOHZKHQFRQVXPHUYDOXDWLRQVDUHQHJDWLYHO\FRUUHODWHGDQG KLJKUHODWLYHWRPDUJLQDOFRVWV 0RUHUHFHQWO\$UPVWURQJVKRZVWKDWIRUDVSHFLDOFODVVRIFDVHVWKHRSWLPDOWDULIILQWKHPXOWLSURGXFW FDVHFDQEHGHWHUPLQHGXVLQJWKHWHFKQLTXHVW\SLFDOO\XVHGLQWKHVLQJOHSURGXFWFDVH+HILQGVWKDWLQKLV VHWWLQJWKHRSWLPDOEXQGOHSULFHZLOODOPRVWDOZD\VLQHIILFLHQWO\H[FOXGHVRPHORZGHPDQGFRQVXPHUV +RZHYHUKHGRHVQRWH[SORUHWKHLPSOLFDWLRQVRILQFUHDVLQJWKHQXPEHURIJRRGV 7KHUHDUHIHZJHQHUDOUHVXOWVIRUEXQGOHVRIPRUHWKDQWZRJRRGV0F$GDPVIRXQGWKDWWKHH[LVWLQJ DQDO\WLFDOPDFKLQHU\IRUDQDO\]LQJPL[HGEXQGOLQJFRXOGQRWEHHDVLO\JHQHUDOL]HGWRHYHQWKUHHJRRGV EHFDXVHRIWKHLQWHUDFWLRQVDPRQJVXEEXQGOHV,QJHQHUDOSULFHVHWWLQJIRUPL[HGEXQGOLQJRIPDQ\JRRGVLV   )RUWKHUHPDLQGHURIWKLVDUWLFOHWKHXQPRGLILHGWHUPEXQGOLQJUHIHUVWRSXUHEXQGOLQJXQOHVVRWKHUZLVHVSHFLILHG 3DJH DQ13FRPSOHWHSUREOHPUHTXLULQJWKHVHOOHUWRGHWHUPLQHDQXPEHURISULFHVDQGTXDQWLWLHVWKDWJURZV H[SRQHQWLDOO\DVWKHVL]HRIWKHEXQGOHLQFUHDVHV+DQVRQDQG0DUWLQ $SSURDFKLQWKLVDUWLFOH 8QOLNHWKHDERYHDUWLFOHVRXUDSSURDFKLVPRVWDSSOLFDEOHWRODUJHEXQGOHVRIJRRGVVXFKDVWKHWKRXVDQGVRI LQIRUPDWLRQJRRGVDYDLODEOHYLDDW\SLFDORQOLQHVHUYLFH:HDUHDEOHWRERXQGWKHSURILWVGHULYHGIURPDQ\ EXQGOHRIQJRRGVZLWKILQLWHYDULDQFHDQGWRH[SORUHKRZWKHRSWLPDOEXQGOHSULFHFKDQJHVXQGHUYDULRXV FRQGLWLRQV,QSDUWLFXODUZHGUDZRQZHOOHVWDEOLVKHGVWDWLVWLFDOWKHRUHPVWRFKDUDFWHUL]HWKHSUREDELOLVWLF YDOXDWLRQVRIODUJHFROOHFWLRQVRIJRRGV:HILQGWKDWVRPHRIWKHUHVXOWVLQWKHOLWHUDWXUHIRUEXQGOHVRIWZR JRRGVGRQRWJHQHUDOL]HWRODUJHUEXQGOHV)RULQVWDQFH6DOLQJHUVKRZVZKHQFRQVXPHUVKDYH LQGHSHQGHQWOLQHDUGHPDQGVEXQGOLQJWZRJRRGVLQFUHDVHVFRQVXPHUV·VXUSOXVZKHQEXQGOLQJZDV SURILWDEOH,WWXUQVRXWWKDWWKLVLVQRWW\SLFDOEXQGOHVRIPRUHWKDQWZRJRRGVZLOODOZD\VUHGXFHFRQVXPHUV· VXUSOXVZKHQWKHJRRGVKDYHLQGHSHQGHQWOLQHDUGHPDQGV2WKHUUHVXOWVIURPWKHEXQGOLQJOLWHUDWXUHDUH VWUHQJWKHQHGLQRXUVHWWLQJVRPHWLPHVGUDPDWLFDOO\VREXQGOLQJLVSURILWDEOHIRUDEURDGHUVHWRIFRQGLWLRQV DQGPD\HYHQEHDEOHWRH[WUDFWQHDUO\DOOWKHYDOXHIURPDFROOHFWLRQRIJRRGV 6HFWLRQSUHVHQWVWKHEDVLFPRGHOLQJIUDPHZRUNDQGNH\UHVXOWVUHJDUGLQJWKHSUHGLFWLYHYDOXHRIEXQGOLQJ IRUWKHJHQHUDOFDVHRILQIRUPDWLRQJRRGVZLWKLQGHSHQGHQWYDOXDWLRQV7KHVHLQFOXGHWKHDV\PSWRWLF RSWLPDOLW\RIEXQGOLQJZKHQPDUJLQDOFRVWVDUH]HURWKHVXERSWLPDOLW\RIEXQGOLQJZKHQPDUJLQDOFRVWV H[FHHGDFULWLFDOWKUHVKROGDQGDVXIILFLHQWFRQGLWLRQIRUILQLWHEXQGOHVWREHPRUHSURILWDEOHWKDQXQEXQGOHG VDOHV6HFWLRQDSSOLHVWKHJHQHUDOPRGHOWRVHYHUDOVSHFLILFFDVHVLQFOXGLQJEXQGOHVRIJRRGVWKDWKDYHLLG LQGHSHQGHQWDQGLGHQWLFDOO\GLVWULEXWHGYDOXDWLRQVEXQGOHVRIFRPSOHPHQWVRUVXEVWLWXWHVDQGEXQGOLQJLQ WKHSUHVHQFHRIEXGJHWFRQVWUDLQWV,QHDFKFDVHZHGHULYHDQLQHTXDOLW\IRUEXQGOHSURILWVDVDIXQFWLRQRI EXQGOHVL]H:HDOVRFRQVLGHUVHYHUDOW\SHVRIFRUUHODWLRQLQWKHYDOXDWLRQVRIWKHLQIRUPDWLRQJRRGV:H SUHVHQWGLVFULPLQDWLQJPHFKDQLVPVWKDWVLJQLILFDQWO\LQFUHDVHWKHEHQHILWVRIEXQGOLQJIRUJRRGVZLWKRWKHU W\SHVRIFRUUHODWHGGHPDQGVSURYLGHGWKHVRXUFHRIWKHXQGHUO\LQJFRUUHODWLRQFDQEHLGHQWLILHGHLWKHU GLUHFWO\RULQGLUHFWO\WKURXJKFRQVXPHUVEHKDYLRU:HDOVRVKRZWKDWPL[HGEXQGOLQJFDQEHPRUHSURILWDEOH WKDQSXUHEXQGOLQJZKHQFRQVXPHUYDOXDWLRQVDUHQRWGUDZQIURPWKHVDPHGLVWULEXWLRQDVLWLQGXFHV FRQVXPHUVWRVHOIVHOHFW6HFWLRQFRPSDUHVWKHLPSOLFDWLRQVRIRXUDQDO\VLVZLWKVRPHHPSLULFDOHYLGHQFH DQGSURYLGHVVRPHFRQFOXGLQJUHPDUNV 7KH%DVLF0RGHOZLWK,QGHSHQGH QW9DOXDWLRQV $V\PSWRWLFUHVXOWVIRUODUJHEXQGOHV :HEHJLQE\FRQVLGHULQJDVHWWLQJZLWKDVLQJOHVHOOHUSURYLGLQJQLQIRUPDWLRQJRRGVWRDVHWRIFRQVXPHUVΩ (DFKFRQVXPHUFDQFRQVXPHHLWKHURUXQLWVRIHDFKLQIRUPDWLRQJRRGDQGUHVDOHLVQRWSHUPLWWHGRULV 3DJH XQSURILWDEOHIRUFRQVXPHUV )RUHDFKFRQVXPHU ω ∈ΩOHWv ni () ω GHQRWHWKHYDOXDWLRQRIJRRGLZKHQD WRWDORIQJRRGVDUHSXUFKDVHG:HDOORZv ni WRGHSHQGRQQVRWKDWWKHGLVWULEXWLRQVRIYDOXDWLRQVIRU LQGLYLGXDOJRRGVFDQFKDQJHDVWKHQXPEHURIJRRGVSXUFKDVHGFKDQJHV 6XFKDFROOHFWLRQRIUDQGRP YDULDEOHVvv v nn nn12 (), (), , () ω ω ω K LVVRPHWLPHVUHIHUUHGWRDVDWULDQJXODUDUUD\RIUDQGRPYDULDEOHVDQG FDQEHGHQRWHGE\V n  V n nn nn v vv vv v =  ! " $ # # # # 11 21 22 12 MO L (1-good bundle) (2-good bundle) ( -good bundle) M n /HW x n v nnk k n = = ∑ 1 1 EHWKHSHUJRRGYDOXDWLRQRIWKHEXQGOHRIQLQIRUPDWLRQJRRGV/HW p n * q n * DQG π n * GHQRWH WKHSURILWPD[LPL]LQJSULFHSHUJRRGIRUDEXQGOHRIQJRRGVWKHFRUUHVSRQGLQJVDOHVDVDIUDFWLRQRIWKH SRSXODWLRQDQGWKHVHOOHU·VUHVXOWLQJSURILWVSHUJRRG$VVXPHWKHIROORZLQJFRQGLWLRQVKROG $ 7KHPDUJLQDOFRVWIRUFRSLHVRIDOOLQIRUPDWLRQJRRGVLV]HURWRWKHVHOOHU $ )RUDOOQFRQVXPHUYDOXDWLRQVv ni DUHLQGHSHQGHQWDQGXQLIRUPO\ERXQGHG ZLWKFRQWLQXRXVGHQVLW\ IXQFWLRQVQRQQHJDWLYHVXSSRUWPHDQ µ ni DQGYDULDQFH σ ni 2  $ &RQVXPHUVKDYHIUHHGLVSRVDO,QSDUWLFXODUIRUDOO n >1 vv nk k n nk k n = − = − ∑∑ ≥ 1 1) 1 1 (  8QGHUWKHVHFRQGLWLRQVZHILQGWKDWVHOOLQJDEXQGOHRIDOOQLQIRUPDWLRQJRRGVFDQEHUHPDUNDEO\VXSHULRUWR VHOOLQJWKHQJRRGVVHSDUDWHO\)RUWKHGLVWULEXWLRQVRIYDOXDWLRQVXQGHUO\LQJPRVWFRPPRQGHPDQG IXQFWLRQVEXQGOLQJVXEVWDQWLDOO\UHGXFHVWKHDYHUDJHGHDGZHLJKWORVVDQGOHDGVWRKLJKHUDYHUDJHSURILWVIRU WKHVHOOHU$VQLQFUHDVHVWKHVHOOHUFDSWXUHVDQLQFUHDVLQJIUDFWLRQRIWKHWRWDODUHDXQGHUWKHGHPDQGFXUYH   :HDVVXPHWKDWWKHSURGXFHUVRILQIRUPDWLRQJRRGVFDQXVHWHFKQLFDOOHJDODQGVRFLDOPHDQVWRSUHYHQWXQDXWKRUL]HG GXSOLFDWLRQDQGWKXVUHPDLQPRQRSROLVWV+RZHYHU%DNRV%U\QMROIVVRQDQG/LFKWPDQHPSOR\RXU IUDPHZRUNWRVWXG\DVHWWLQJZKHUHXVHUVVKDUHWKHJRRGV$OVRZHGRQRWFRQVLGHUVWUDWHJLFLQWHUDFWLRQVZLWKRWKHU SDUWLHVDOWKRXJKRXUDQDO\VLVLVDOVRDSSOLFDEOHWRPRQRSROLVWLFFRPSHWLWLRQDPRQJSURYLGHUVRIVLPLODUSURGXFWV %DNRVDQG%U\QMROIVVRQH[WHQGRXUIUDPHZRUNWRVWXG\FRPSHWLWLRQDPRQJEXQGOHUV  )RULQVWDQFHWKHYDOXHRIDZHDWKHUUHSRUWPD\EHGLIIHUHQWZKHQSXUFKDVHGDORQHIURPLWVYDOXHZKHQSXUFKDVHG WRJHWKHUZLWKWKHPRUQLQJQHZVKHDGOLQHVDVWKH\ERWKFRPSHWHIRUWKHFRQVXPHU·VOLPLWHGWLPH6LPLODUO\RWKHU IDFWRUVVXFKDVJRRGVWKDWDUHFRPSOHPHQWVRUVXEVWLWXWHVGLPLQLVKLQJUHWXUQVDQGEXGJHWFRQVWUDLQWVPD\DIIHFW FRQVXPHUYDOXDWLRQVDVDGGLWLRQDOJRRGVDUHSXUFKDVHG  7RVLPSOLI\WKHQRWDWLRQZHZLOORPLWWKHDUJXPHQW ω ZKHQSRVVLEOH  ,QWKLVDUWLFOHZHZLOOW\SLFDOO\XVHWKHSKUDVHLQIRUPDWLRQJRRGVDVVKRUWKDQGIRUJRRGVZLWK]HURRUYHU\ORZ PDUJLQDOFRVWVRISURGXFWLRQ  ,Hsup ( ) ,,ni ni v ω ω 16 <∞IRUDOOQL()in≤ DQG ω ∈Ω  7KLVDVVXPSWLRQLPSOLHVWKDWDGGLQJDJRRGWRDEXQGOHFDQQRWUHGXFHWKHWRWDOYDOXDWLRQRIWKHEXQGOHDOWKRXJKLW PD\UHGXFHWKHPHDQYDOXDWLRQ 3DJH FRUUHVSRQGLQJO\UHGXFLQJERWKWKHGHDGZHLJKWORVVDQGFRQVXPHUV·VXUSOXVUHODWLYHWRVHOOLQJWKHJRRGV VHSDUDWHO\0RUHIRUPDOO\ 3URSRVLWLRQ$V\PSWRWLF3URILWV&RQVXPHUV6XUSOXVDQG(IILFLHQF\IRU%XQGOLQJ *LYHQDVVXPSWLRQV$$DQG$DVQLQFUHDVHVWKHGHDGZHLJKWORVVSHUJRRGDQGWKHFRQVXPHUV·VXUSOXV SHUJRRGIRUDEXQGOHRIQLQIRUPDWLRQJRRGVFRQYHUJHVWR]HURDQGWKHVHOOHU·VSURILWSHUJRRGLVPD[LPL]HG 3URRI$OOSURRIVDUHLQWKH$SSHQGL[ 7KHLQWXLWLRQEHKLQG3URSRVLWLRQLVWKDWDVWKHQXPEHURILQIRUPDWLRQJRRGVLQWKHEXQGOHLQFUHDVHVWKHODZ RIODUJHQXPEHUVDVVXUHVWKDWWKHGLVWULEXWLRQIRUWKHYDOXDWLRQRIWKHEXQGOHKDVDQLQFUHDVLQJIUDFWLRQRI FRQVXPHUVZLWK´PRGHUDWHµYDOXDWLRQVQHDUWKHPHDQRIWKHXQGHUO\LQJGLVWULEXWLRQ6LQFHWKHGHPDQGFXUYH LVGHULYHGIURPWKHFXPXODWLYHGLVWULEXWLRQIXQFWLRQIRUFRQVXPHUYDOXDWLRQVLWEHFRPHVPRUHHODVWLFQHDUWKH PHDQDQGOHVVHODVWLFDZD\IURPWKHPHDQ)LJXUH 0. 5 0. 25 0. 75 1 0 0. 2 0. 4 0. 6 0. 8 1 0. 2 0. 4 0. 6 0. 8 1 0. 2 0. 4 0. 6 0. 8 1 SHUJRRG 0. 25 0. 75 1 0 3ULFH SHUJRRG 4XDQWLW\IRUEXQGOHDVDIUDFWLRQRIWRWDOSRSXODWLRQ 0. 5 0. 25 0. 75 1 0 SHUJRRG 3ULFH 3ULFH *RRG *RRGV *RRGV 0. 5 )LJXUH'HPDQGIRUEXQGOHVRIDQGLQIRUPDWLRQJRRGVZLWKLLGYDOXDWLRQVXQLIRUPO\GLVWULEXWHGLQ >@OLQHDUGHPDQGFDVH 3URSRVLWLRQLVIDLUO\JHQHUDO:KLOHLWDVVXPHVLQGHSHQGHQFHRIWKHYDOXDWLRQVRIWKHLQGLYLGXDOJRRGVLQD EXQGOHRIDJLYHQVL]HHDFKYDOXDWLRQPD\EHGUDZQIURPDGLIIHUHQWGLVWULEXWLRQ )XUWKHUPRUHYDOXDWLRQV PD\FKDQJHDVPRUHJRRGVDUHDGGHGWRDEXQGOH$VZHVKRZODWHU3URSRVLWLRQFDQEHLQYRNHGWRVWXG\ VHYHUDOVSHFLILFVHWWLQJVVXFKDVGLPLQLVKLQJUHWXUQVIURPWKHFRQVXPSWLRQRIDGGLWLRQDOJRRGVRUJRRGVWKDW DUHVXEVWLWXWHVRUFRPSOHPHQWV   7KHDQDO\VLVLQWKLVVHFWLRQDVVXPHVLQGHSHQGHQFHWRSURYLGHDXVHIXOEDVHOLQHIRULVRODWLQJWKHHIIHFWVRIEXQGOLQJ 7KLVLVDVVXPSWLRQLVOLNHO\WREHPRVWHPSLULFDOO\UHOHYDQWZKHQFRQVXPHUVGLIIHUVLJQLILFDQWO\LQWKHLUWDVWHVDFURVV JRRGVEXWQRWLQWKHLUWRWDOH[SHQGLWXUHRQWKHHQWLUHVHWRIJRRGV7KHFDVHZKHUHFRQVXPHUYDOXDWLRQVDUH FRUUHODWHGLVDQDO\]HGLQVHFWLRQ 3DJH 7KH5ROHRI0DUJLQDO&RVWV ,QWKHEDVLFPRGHOZHDVVXPHWKDWPDUJLQDOFRVWVDUH]HUR:KLOHYHU\ODUJHEXQGOHVZLOOW\SLFDOO\FRQWLQXHWR EHSURILWDEOHHYHQLQWKHSUHVHQFHRIQRQ]HUREXWVPDOOPDUJLQDOFRVWVEXQGOLQJEHFRPHVXQSURILWDEOHIRU JRRGVZLWKVXEVWDQWLDOPDUJLQDOFRVWV3URSRVLWLRQVKRZVWKDWDVH[SHFWHGEXQGOLQJJRRGVZLWKVXIILFLHQWO\ KLJKPDUJLQDOFRVWVLVQHLWKHUSURILWDEOHQRUVRFLDOO\HIILFLHQW 7KXVRXUPRGHOSUHGLFWVWKDWVLQFHELWVDUH GUDPDWLFDOO\FKHDSHUWRUHSURGXFHWKDQDWRPVWKHRSWLPDOEXQGOLQJVWUDWHJLHVGLIIHUVXEVWDQWLDOO\IRU LQIRUPDWLRQJRRGVDVFRPSDUHGWRSK\VLFDOJRRGV 3URSRVLWLRQ0DUJLQDO&RVWV&DQ0DNH%XQGOLQJ8QSURILWDEOH 8QGHUDVVXPSWLRQV$DQG$WKHUHLVDPDUJLQDOFRVWc > 0IRUHDFKLQIRUPDWLRQJRRGWKDWPDNHVEXQGOLQJ UHVXOWLQORZHUSURILWVDQGKLJKHUGHDGZHLJKWORVVWKDQVHOOLQJWKHJRRGVVHSDUDWHO\ $VSRLQWHGRXWE\6FKPDOHQVHHEXQGOLQJFDQLQFUHDVHDVHOOHU·VSURILWVE\UHGXFLQJWKHGLVSHUVLRQRI EX\HUYDOXDWLRQV+RZHYHULIPDUJLQDOFRVWVDUHODUJHWKHVHOOHUZLOOXVXDOO\ZDQWWRLQFUHDVHUDWKHUWKDQ GHFUHDVHWKHGLVSHUVLRQRIYDOXDWLRQV)RUH[DPSOHLIWKHPDUJLQDOFRVWLVJUHDWHUWKDQWKHPHDQYDOXDWLRQ EXQGOLQJZLOOGHFUHDVHSURILWVEHFDXVHLWGHFUHDVHVWKHIUDFWLRQRIEX\HUVZLWKYDOXDWLRQVLQH[FHVVRIWKHWRWDO PDUJLQDOFRVWRIWKHEXQGOH,QJHQHUDOWKHWKUHVKROGDWZKLFKEXQGOLQJEHFRPHVOHVVSURILWDEOHWKDQ XQEXQGOHGVDOHVGHSHQGVRQWKHGLVWULEXWLRQRIYDOXDWLRQVIRUWKHLQGLYLGXDOJRRGV (YHQZLWK]HURPDUJLQDOFRVWVWKHEHQHILWVRIEXQGOLQJPD\EHHOLPLQDWHGLIWKHEXQGOHLQFOXGHVJRRGVWKDW KDYHQHJDWLYHYDOXHWRVRPHFRQVXPHUVHJSRUQRJUDSK\RUDGYHUWLVHPHQWV,QDGGLWLRQZKLOHWHFKQRORJ\LV UDSLGO\UHGXFLQJWKHPDUJLQDOFRVWVRIUHSURGXFWLRQDQGWUDQVPLVVLRQWKHWLPHDQGHQHUJ\DXVHUPXVWVSHQG WRLGHQWLI\DQLQIRUPDWLRQJRRGFDQSUHVHQWDEDUULHUWRWKHOLPLWLQJUHVXOWRI3URSRVLWLRQ)RUH[DPSOH %U\QMROIVVRQDQG.HPHUHUILQGWKDWWKHFRJQLWLYHFRVWRIOHDUQLQJWKHFRPPDQGVDFFRXQWVIRUD VXEVWDQWLDOIUDFWLRQRIWKHSULFHRIVSUHDGVKHHWVRIWZDUH 5HVXOWVIRU%XQGOHVRI)LQLWH6L]H :KLOH3URSRVLWLRQVKRZVWKDWIRUDVXIILFLHQWO\ODUJHQVHOOLQJJRRGVDVDEXQGOHFDQEHVLJQLILFDQWO\PRUH SURILWDEOHWKDQXQEXQGOHGVDOHVSXUHEXQGOLQJGRHVQRWQHFHVVDULO\LQFUHDVHSURILWVIRUVPDOOQ ,Q SDUWLFXODULIWKHVHOOHULVDEOHWRH[WUDFWDODUJHIUDFWLRQRIWKHSRWHQWLDOVXUSOXVHYHQZKHQWKHJRRGVDUHVROG   6DOLQJHUSURYHVDVLPLODUSURSRVLWLRQIRUEXQGOLQJWZRJRRGV,QJHQHUDOWKHPDUJLQDOFRVWWKUHVKROGWKDW PDNHVEXQGOLQJXQSURILWDEOHDWWKHOLPLWDVn →∞LVVXEVWDQWLDOO\KLJKHUIRUDODUJHEXQGOHWKDQIRUDEXQGOHRIWZR JRRGV)RUH[DPSOHLIYDOXDWLRQVDUHLQGHSHQGHQWDQGXQLIRUPO\GLVWULEXWHGLQ[, ] max 0 v DPDUJLQDOFRVWcv>. max 14 VXIILFHVWRPDNHEXQGOLQJWZRJRRGVXQSURILWDEOHZKLOHWKHWKUHVKROGEHFRPHVDOPRVWWKUHHWLPHVKLJKHU cv>. max 41 DVn →∞7KHWKUHVKROGLVPXFKORZHUZKHQWKHGLVWULEXWLRQRIYDOXDWLRQVKDVDORQJWDLORIORZYDOXHV HJLIRIWKHFRQVXPHUVKDYHQRYDOXHIRUHDFKJRRGDQGWKHUHPDLQLQJFRQVXPHUVKDYHXQLIRUPO\GLVWULEXWHG YDOXDWLRQVLQ[, ] max 0 v WKHQEXQGOLQJLVXQSURILWDEOHIRUcv>. max 0041  ,QFRQWUDVW0F$IHH0F0LOODQDQG:KLQVWRQILQGWKDWPL[HGEXQGOLQJRIWZRJRRGVDOZD\VGRPLQDWHV XQEXQGOHGVDOHVZKHQFRQVXPHUYDOXDWLRQVDUHLQGHSHQGHQW)RUDODUJHQXPEHURIJRRGVDQGXQGHUWKHFRQGLWLRQV IRU3URSRVLWLRQSXUHEXQGOLQJFDSWXUHVQHDUO\WKHHQWLUHYDOXHFUHDWHGE\WKHLQIRUPDWLRQJRRGVVRPL[HG EXQGOLQJFDQQRWGRVXEVWDQWLDOO\EHWWHU+RZHYHUDVVKRZQLQVHFWLRQVDQGWKHSUHVHQFHRIPDUJLQDOFRVWVRU FRUUHODWHGGHPDQGVFDQPDNHPL[HGEXQGOLQJVXEVWDQWLDOO\PRUHSURILWDEOHWKDQSXUHEXQGOLQJ 3DJH VHSDUDWHO\WKHQWKHUHPD\EHOLWWOHRUQREHQHILWIURPVPDOODPRXQWVRIEXQGOLQJ)RUH[DPSOHLIFRQVXPHU YDOXDWLRQVIRULQGLYLGXDOJRRGVDUHLLGWDNLQJYDOXHVHLWKHUv H ZLWKSUREDELOLW\U RUv L ZLWK SUREDELOLW\²U WKHSURILWPD[LPL]LQJSULFH p 1 * ZLOOVHOOWRDOOKLJKYDOXDWLRQFRQVXPHUVDQGZLOO H[WUDFWPRVWSRWHQWLDOVXUSOXVDVq 1 * $EXQGOHRIWZRJRRGVZLOOKDYHSHUJRRGYDOXDWLRQVRIv H ZLWK SUREDELOLW\r 2  1 2 ()vv HL + ZLWKSUREDELOLW\U²U DQGv L ZLWKSUREDELOLW\()1 2 − r 7KH SURILWPD[LPL]LQJSULFHIRUWKLVEXQGOHLV p 2 * DQGUHVXOWVLQVDOHVWRDIUDFWLRQq 2 * RIFRQVXPHUV \LHOGLQJERWKORZHUSURILWVDQGKLJKHUGHDGZHLJKWORVV 7RIXUWKHUVWXG\XQGHUZKDWFRQGLWLRQVEXQGOLQJZLOOGRPLQDWHVHSDUDWHVDOHVHYHQIRUVPDOOQZHDVVXPH WKHIROORZLQJFRQGLWLRQWKDWLPSOLHVD´VLQJOHFURVVLQJµSURSHUW\IRUWKHSHUJRRGGHPDQGV ppq nnn = () $ 6LQJOH&URVVLQJRI&XPXODWLYH'LVWULEXWLRQV&RQGLWLRQ6&'&7KHGLVWULEXWLRQRIFRQVXPHUYDOXDWLRQV LVVXFKWKDW Prob Probxx nn n n −<≤ − < ++ µε µ ε 11 IRUDOOQDQG ε  ,QSUDFWLFHWKH6&'&LVQRWYHU\UHVWULFWLYH,WKROGVIRUPRVWFRPPRQGHPDQGIXQFWLRQVLQFOXGLQJOLQHDU VHPLORJDQGORJORJGHPDQGDVZHOODVDQ\GHPDQGIXQFWLRQEDVHGRQD*DXVVLDQGLVWULEXWLRQRIYDOXDWLRQV *LYHQWKH6'&'LILWLVPRUHSURILWDEOHWREXQGOHDFHUWDLQQXPEHURIJRRGVVD\ $ nWKDQWRVHOOWKHPVHSDUDWHO\ DQGLIWKHRSWLPDOSULFHSHUJRRGIRUWKHEXQGOHLVOHVVWKDQWKHPHDQYDOXDWLRQ µ n WKHQEXQGOLQJDQ\QXPEHU RIJRRGVJUHDWHUWKDQ $ nZLOOIXUWKHULQFUHDVHSURILWVFRPSDUHGWRWKHFDVHZKHQWKHDGGLWLRQDOJRRGVRUDOO JRRGVDUHVROGVHSDUDWHO\0RUHIRUPDOO\ 3URSRVLWLRQ0RQRWRQLF%XQGOLQJ3URILWV *LYHQDVVXPSWLRQV$$$DQG$LI ππ ** n > 1 DQG p nn * < µ WKHQEXQGOLQJDQ\QXPEHURIJRRGVnn≥ $ ZLOOPRQRWRQLFDOO\LQFUHDVHWKHVHOOHU·VSURILWVFRPSDUHGWRVHOOLQJWKHPVHSDUDWHO\ 6LQFHWKHXQLIRUPGLVWULEXWLRQRIYDOXDWLRQVXQGHUO\LQJOLQHDUGHPDQGVDWLVILHV$VVXPSWLRQ$ZKHQWKH YDOXDWLRQVDUHLQGHSHQGHQWDQGVLQFHEXQGOLQJWZRJRRGVZLWKLQGHSHQGHQWOLQHDUGHPDQGVDQG]HUR PDUJLQDOFRVWLVSURILWPD[LPL]LQJIRUWKHVHOOHU6DOLQJHUWKHIROORZLQJFRUROODU\IROORZVIURP 3URSRVLWLRQ &RUROODU\D:LWKLQGHSHQGHQWOLQHDUGHPDQGVIRUWKHLQGLYLGXDOJRRGVEXQGOLQJDQ\QXPEHURIJRRGVZLWK ]HURPDUJLQDOFRVWLQFUHDVHVWKHVHOOHU·VSURILWV   )RUVLPSOLFLW\WKLVH[DPSOHXVHVDGLVFUHWHGLVWULEXWLRQRIFRQVXPHUYDOXDWLRQV7KHUHVXOWVZRXOGQRWEHPDWHULDOO\ HIIHFWHGLIDVLPLODUFRQWLQXRXVGLVWULEXWLRQZHUHXVHGLQVWHDGHJDELPRGDOFRQWLQXRXVGLVWULEXWLRQZLWKDYHU\ VPDOOSUREDELOLW\RIDQ\YDOXDWLRQEHWZHHQDQGDQGSHDNVZLWKDUHDVRIMXVWXQGHUDWDQGDW  7KHZHDNODZRIODUJHQXPEHUVUHTXLUHVWKLVSUREDELOLW\WRGHFUHDVHDVO σε 22 n 27 IRUDOOGLVWULEXWLRQVZLWKILQLWH PHDQDQGYDULDQFHEXWLWGRHVQRWJXDUDQWHHPRQRWRQLFLW\ 3DJH ,WLVLQWHUHVWLQJWRFRQWUDVWWKHEXQGOLQJDSSURDFKZHDQDO\]HKHUHZLWKFRQYHQWLRQDOSULFHGLVFULPLQDWLRQ 6XSSRVHWKHUHDUHPFRQVXPHUVLQWKHVHWΩ,IDVLQRXUVHWWLQJHDFKRIWKHFRQVXPHUVSRWHQWLDOO\KDVD GLIIHUHQWYDOXHIRUHDFKRIWKHQJRRGVWKHQPQSULFHVZLOOEHUHTXLUHGWRFDSWXUHWKHFRPSOHWHVXUSOXVZKHQ WKHJRRGVDUHVROGVHSDUDWHO\)XUWKHUPRUHSULFHGLVFULPLQDWLRQUHTXLUHVWKDWWKHVHOOHUFDQDFFXUDWHO\LGHQWLI\ FRQVXPHUYDOXDWLRQVDQGSUHYHQWFRQVXPHUVIURPEX\LQJJRRGVDWSULFHVPHDQWIRURWKHUV7KXVWKH FRQYHQWLRQDODSSURDFKWRSULFHGLVFULPLQDWLRQRSHUDWHVE\LQFUHDVLQJWKHQXPEHURISULFHVFKDUJHGWR DFFRPPRGDWHWKHGLYHUVLW\RIFRQVXPHUYDOXDWLRQV,QFRQWUDVWWKHEXQGOLQJDSSURDFKPLJKWEHFDOOHG 3URFUXVWHDQSULFHGLVFULPLQDWLRQVLQFHLWRSHUDWHVRQDRQHVL]HILWVDOOSULQFLSOH %XQGOLQJUHGXFHVWKH GLYHUVLW\RIFRQVXPHUYDOXDWLRQVVRWKDWLQWKHOLPLWVHOOHUVQHHGFKDUJHRQO\RQHSULFHGRQRWQHHGWR LGHQWLI\GLIIHUHQWW\SHVRIFRQVXPHUVDQGGRQRWQHHGWRHQIRUFHDQ\UHVWULFWLRQVRQZKLFKSULFHVFRQVXPHUV SD\ $VWKHQXPEHURIJRRGVLQWKHEXQGOHLQFUHDVHVWRWDOSURILWDQGSURILWSHUJRRGLQFUHDVH7KHSURILW PD[LPL]LQJSULFHSHUJRRGIRUWKHEXQGOHVWHDGLO\LQFUHDVHVJUDGXDOO\DSSURDFKLQJWKHSHUJRRGH[SHFWHG YDOXHRIWKHEXQGOHWRWKHFRQVXPHUV)LJXUHVKRZVWKLVIRUWKHFDVHRIOLQHDUGHPDQG7KHQXPEHURIJRRGV QHFHVVDU\WRPDNHEXQGOLQJGHVLUDEOHDQGWKHVSHHGDWZKLFKGHDGZHLJKWORVVDQGSURILWFRQYHUJHWRWKHLU OLPLWLQJYDOXHVGHSHQGRQWKHDFWXDOGLVWULEXWLRQRIFRQVXPHUYDOXDWLRQV n=1, 2, 3, 6, 50, 500 Profit/ good Price/ good )LJXUH3URILWDVDIXQFWLRQRISULFHSHUJRRGIRUEXQGOHVRIYDU\LQJQXPEHURIJRRGVQZLWKLGHQWLFDOOLQHDU GHPDQGVDQGYDOXDWLRQVLQ>@VWHHSHUFXUYHVUHIOHFWODUJHUQ7KHSURILWPD[LPL]LQJSULFHLVWKHSRLQWDWWKH PD[LPXPRIHDFKFXUYH,QWKHOLPLWWKHSULFHSHUJRRGDSSURDFKHVWKHPHDQYDOXDWLRQ 7KHHIILFLHQF\DQGSURILWJDLQVWKDWEXQGOLQJRIIHUVLQRXUVHWWLQJFRQWUDVWZLWKWKHPRUHOLPLWHGEHQHILWV LGHQWLILHGLQSUHYLRXVZRUNSULQFLSDOO\DVDUHVXOWRIRXUIRFXVRQEXQGOLQJODUJHQXPEHUVRIJRRGVDQGRQ LQIRUPDWLRQJRRGVZLWK]HURPDUJLQDOFRVWV$QLPSRUWDQWLPSOLFDWLRQRIRXUDQDO\VLVLVWKDWWKHEHQHILWVRI EXQGOLQJJURZDVWKHQXPEHURIJRRGVLQWKHEXQGOHLQFUHDVHV7KLVLPSOLHVDIRUPRIVXSHUDGGLWLYLW\ELJJHU EXQGOHVZLOOEHPRUHSURILWDEOHWKDQVPDOOHUEXQGOHVHYHQZKHQWKHJRRGVLQYROYHGDUHLGHQWLFDO   ,WLVVWUDLJKWIRUZDUGWRGHULYHDQDORJRXVFRUROODULHVIRURWKHUFRPPRQGHPDQGIXQFWLRQVVXFKDVVHPLORJRU FXPXODWLYHWUXQFDWHGQRUPDOWKDWVDWLVI\$VVXPSWLRQ$ . 22 1 3 2 3  ,QDJUHHPHQWZLWK3URSRVLWLRQDVQDSSURDFKHVLQILQLWWKHSHUJRRGSURILWVDSSURDFK µ ZKLFKLVWKH PD[LPXPSRVVLEOHYDOXH0LQLPXPSURILWVIRUVPDOOHUEXQGOHVDUHDOVRHDVWRFRPSXWH)LJXUHGHSLFWVWKH PLQLPXPSURILWVIRUDEXQGOHRILQIRUPDWLRQJRRGVWKDWKDYHLLGYDOXDWLRQVZLWK σ µ 22 1 3 = 7KUHHFDVHV DUHVKRZQWKHORZHUERXQGIRUDQGLVWULEXWLRQLPSOLHGE&RUROODUDWKHORZHUERXQGIRUDGLVWULEXWLRQ WKDWLVVPPHWULFDURXQGWKHPHDQDQGWKHDFWXDOSURILWVIRUDQRUPDOGLVWULEXWLRQRIYDOXDWLRQV)RUH[DPSOH LIFRQVXPHUYDOXDWLRQVDUHLLGDQGVPPHWULFDURXQGWKHPHDQZLWK σ µ 22 1 3 = WKHQWKHVHOOHUFDQUHDOL]H SURILWVRIDWOHDVWRIWKHWRWDODUHDXQGHUWKHGHPDQGFXUYHZLWKDEXQGOHRIQRPRUHWKDQJRRGV 3DJH Profits (Lower Bound) i.i.d. valuations with σ 2 / µ 2 = 1/3 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 10 15 20 25 30 3 5 40 45 50 5 5 60 1 0 0 1000 10 0 00 Number. 1/3 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 5 10 15 20 25 30 3 5 40 45 50 5 5 60 1 0 0 1000 10 0 00 Number of Goods in Bundle Profits (as share of total surplus) Normal Distribution Symmetric Distribution General