Social Culture and Economic Performance - Duke Economics

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pxvw eh wuxh= ý ü li F @ Fu > Y è 4 . Fu ý Z^ø äø rwkhuzlvh ý ü li F @ Fu > Y è 4 . Fu ý Zý^ø üäø = Y è 4 ý Züø äø li F @ FM > rwkhuzlvh

Wkh dfwlylw| dqg vnloo dftxlvlwlrq fkrlfhv lq dq D0fxowxudo htxloleulxp duh sruwud|hg lq ý ü ý ü ý ü ý ü Iljxuh +5e, zkhuh zh kdyh gh?qhg Y•^ ä ø @ 4.Fu ýZ^ø äø dqg Y•ü äø @ 4ýZüø ä ø dv wkh wkuhvkrog glvxwlolw| ydoxhv wkdw uhvshfwlyho| d vnloohg dqg dq xqvnloohg zrunhu duh ý ü ý ü zloolqj wr lqfxu wr eh d phpehu ri wkh holwhv1 Qrwh wkdw Z^ø ä ø ý Züø äø è Fu ehfdxvh 44

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Lqhtxdolw| +5, hvwdeolvkhv wkdw lq d fxowxudo htxloleulxp/ d vlqjoh0furvvlqj surshuw| +VFS, ri wkh fxowxudo dfwlylw| lv hqgrjhqrxvo| jhqhudwhg1 Pruh vshfl?fdoo|/ ohw xv ghqrwh wkh qhw ehqh?w wr xqghuwdnh dfwlylw| D iru d vnloohg dqg dq xqvnloohg zrunhu zlwk wkh vdph xwlolw| ý ü ý ü ý ü ý ü w|sh Y uhvshfwlyho| e| e h^ > Y > ä ø @ Y ý Y•^ äø dqg e hü > Y > äø @ Y ý Y•ü ä ø = Lq0 ý ü ý ü htxdolw| +5, |lhogv wkdw e h^ > Y > ä ø ? e hü > Y > ä ø iru hyhu| w|sh Y= Lq rwkhu zrugv/ lq dq| D0fxowxudo htxloleulxp/ d vnloohg zrunhu lv pruh zloolqj wkdq dq xqvnloohg rqh wr hqgxuh glvx0 wlolw| iurp dfwlylw| D wr eh dq holwh/ zklfk lq wxuq mxvwl?hv D0zrunhuv dv holwhv1 Xqghuwdnlqj dfwlylw| D ehfrphv d vljqdolqj lqvwuxphqw iru vnloohg zrunhuv gxh wr wkh hqgrjhqrxvo| jhq0 hudwhg VFS/ zklfk gl>huv iurp Plfkdho Vshqfh*v +4 d fkdudfwhulvwlf ri wkh dfwlylw| D lwvhoi1 Qrz L zloo surylgh wkh qhfhvvdu| dqg vx!flhqw frqglwlrq iru wkh h{lvwhqfh ri D0fxowxudo 45

htxloleuld1 Iru dq| ä ø 5 øu > Ohppd 5 whoov xv krz zrunhuv pdnh dfwlylw| dqg vnloo dftxl0 vlwlrq fkrlfhv1 Iru hyhu| srvwxodwhg ydoxh ri äø L fdq wkhq fdofxodwh wkh sursruwlrq ri wkh vnloohg dprqj D0zrunhuv1 Vshfl?fdoo|/ zh gh?qh d pdsslqj ô = ^3> 4` $ ^3> 4` e| ; bu +•3M +Tì^ +Zø ,,, ý øü ? li ä ø 5 øu ì T b •3M + ^ EZ ø ä,,nbM +•3M +Tìü EZ ø ä,, u+ ô ä @ = 3 rwkhuzlvh zkhuh wkh qxphudwru ri wkh iudfwlrq lv wkh wrwdo pdvv ri vnloohg D0zrunhuv +vhh wkh vkdghg duhd lq Iljxuh 5e, dqg wkh ghqrplqdwru lv wkh wrwdo pdvv ri D0zrunhuv1 Hyhu| ?{hg srlqw ri wkh pdsslqj ô zloo fruuhvsrqg wr dq D0fxowxudo htxloleulxp1 Dq looxvwudwlrq ri ô lv surylghg lq Iljxuh +4e,1 Wkh zd| ô lv gh?qhg kljkoljkwv wkh orfdo sxeolf jrrg dvshfw ri jurxs uhsxwdwlrq lq fxowxudo htxloleuld1 Wkdw lv/ D0zrunhuv* vnloo lqyhvwphqw fkrlfhv ghshqg rqo| rq wkh ?upv* shufhswlrq ri wkh sursruwlrq ri wkh vnloohg dprqj D0zrunhuv1 Lq frqwudvw wkh ?up*v shufhswlrq lv d sxeolf jrrg lq wkh edvlf prgho1 û ý ü ú Ohw ü ì pd{Z ø Máu ô ä ø ý äø eh wkh pd{lpdo gl>huhqfh ehwzhhq wkh ixqfwlrq ô dqg wkh lghqwlw| pds1 Lw vkrxog eh fohdu wkdw ü lv zhoo gh?qhg lq whupv ri wkh sulplwlyhv ri wkh prgho/ uhjdugohvv ri dq| htxloleulxp frqvlghudwlrqv1 Sursrvlwlrq 5 Wkhuh h{lvwv dw ohdvw rqh D0fxowxudo htxloleulxp l> ü è 3= Zkhq ü è 3 krogv/ Sursrvlwlrq 5 hvwdeolvkhv wkdw wkh hfrqrp| dgplwv dw ohdvw rqh D0fxowxudo htxloleulxp1 Lq idfw pxowlsoh D0fxowxudo htxloleuld pd| h{lvw1 Pruhryhu/ ehvlghv wkh qrq0fxowxudo htxloleulxp zklfk zh nqhz dozd|v h{lvwv/ dq hfrqrp| pd| vlpxowdqhrxvo| dgplw D0 dqg E0fxowxudo htxloleuld ehfdxvh dq dqdorjrxv qhfhvvdu| dqg vx!flhqw frqglwlrq iru wkh h{lvwhqfh ri E0fxowxudo htxloleuld grhv qrw qhfhvvdulo| h{foxgh ü è 3= Lq wklv vhqvh h{dfwo| zklfk jurxs ri zrunhuv zloo uhfhlyh suhihuhqwldo wuhdwphqw fdq eh txlwh duelwudu|1 Wkh iroorzlqj sursrvlwlrq hvwdeolvkhv wkdw hfrqrplhv zklfk dgplw d fxowxudo htxloleulxp fdq eh uhdglo| frqvwuxfwhg1 Sursrvlwlrq 6 Il{ i{^ > Fu > FM > èu > i^ > iü j vdwlvi|lqj D40D61 Iru dq| äø 5 lqw +øu , > wkhuh h{lvwv vrph frqwlqxrxv dqg vwulfwo| lqfuhdvlqj glvwulexwlrq ixqfwlrq K vxfk wkdw äø lv wkh sursruwlrq ri wkh vnloohg dprqj D0zrunhuv lq vrph D0fxowxudo htxloleulxp ri wkh hfrqrp| i{^ > Fu > FM > èu > i^ > iü > Kj=

46

Zhoiduh1 Lq d fxowxudo htxloleulxp/ wkh qhz whfkqrorj| lv dgrswhg e| d srvlwlyh pdvv ri zrunhuv1 Lq wkh phdq wlph/ vrph zrunhuv duh hqgxulqj wkh glvxwlolw| ri dfwlylw| D lq rughu wr eh phpehuv ri wkh holwhv1 Wkh wudgh0r> lv lq idyru ri zhoiduh lpsuryhphqw=45 Sursrvlwlrq 7 Dq| fxowxudo htxloleulxp Sduhwr grplqdwhv wkh qrq0fxowxudo htxloleulxp1 Surri= Zlwk qr orvv ri jhqhudolw| frqvlghu dq D0fxowxudo htxloleulxp1 E0zrunhuv duh h{dfwo| dv zhoo r> dv wkh| duh lq wkh qrq0fxowxudo htxloleulxp1 E| uhyhdohg suhihuhqfh D0zrunhuv duh vwulfwo| ehwwhu r> wkdq wkh| duh lq wkh qrq0fxowxudo htxloleulxp1 Uhfdoo wkdw lq Vshqfh +4 lq wkh vhsdudwlqj htxloleulxp1 Wklv lv gxh wr wkh dvvxphg qhjdwlyh uhodwlrqvkls ehwzhhq delolw| dqg lqyhvwphqw frvw/ zklfk lv qhfhvvdu| iru vljqdolqj wr dulvh lq klv prgho1 L fdq glvshqvh zlwk wklv qhjdwlyh uhodwlrqvkls ehfdxvh wkh VFS lv hqgrjhqrxvo| jhqhudwhg1 Wklv h{sodlqv wkh gl>huhqfh lq zhoiduh surshuwlhv iurp Vshqfh1

7

D Qxphulfdo H{dpsoh Lq wklv vhfwlrq L surylgh dq looxvwudwlyh qxphulfdo h{dpsoh= Wkh Edvlf Hfrqrp|1 Frqvlghu dq hfrqrp| zkhuh {^ @ 7> Fu @ 3=;> FM @ 4=5>

èu @ 6@;/ èM @ 8@;= Wkh whvwlqj whfkqrorj| lv jlyhq e| Su ^ëM mh^ ` @ Su ^ë u mhü ` @ 6@7 dqg Su ^ë M mhü ` @ Su ^ëu mh^ ` @ 4@7= Qrwh wkdw lw vdwlv?hv POUS1 Ohw ä eh wkh sursruwlrq ri vnloohg zrunhuv lq wkh srsxodwlrq1 Lw lv whglrxv exw vwudljkw0 iruzdug wr vkrz wkdw wkh htxloleulxp zdjh vfkhgxohv lghqwl?hg lq Ohppd 4 duh= ã 4 li ä é 4@43 z +ä> ë M , @ 45ä@ +4 . 5ä, li ä A 4@43 ã 4 li ä é 4@5 z +ä> ëu , @ 7ä@ +6 ý 5ä, li ä A 4@5= Wkh h{shfwhg sulydwh ehqh?w ri vnloo dftxlvlwlrq lv E +ä, @ 4@5^z +ä> ëM , ý z +ä> ëu ,` diwhu froohfwlqj whupv1 Lw lv d vlqjoh shdnhg ixqfwlrq1 Vlpsoh fdofxodwlrqv |lhog øu @ ^=6;57> 3=:8` > dqg øM @ >= Qrwh pruhryhu wkdw èu @ 6@; ? plq øu = Khqfh dvvxpswlrqv D5 dqg D6 duh vdwlv?hg1 Sursrvlwlrq 4 suhglfwv wkdw wkh qhz whfkqrorj| zloo qrw eh dgrswhg lq dq| 45

Li/ lqvwhdg ri D7/ wkh xwlolw| frvwv ri xqghuwdnlqj dfwlylw| ø duh wkh vdph iru hyhu| zrunhu/ wkhq ø0 zrunhuv zloo eh lqgl>huhqw ehwzhhq wkh ø0fxowxudo dqg wkh qrq0fxowxudo htxloleuld1 Lq wklv fdvh wkh ø0fxowxudo htxloleulxp lqyroyhv pl{hg vwudwhjlhv e| orz frvw zrunhuv1

47

htxloleulxp ri wklv edvlf hfrqrp|1 Lw lv lqwhuhvwlqj wr qrwh wkdw khuh wkh surgxfwlylw| jdlq ri wkh qhz whfkqrorj|/ {^ ý 4> lv pruh wkdq grxeoh hyhq wkh kljk vnloo dftxlvlwlrq frvw FM = Fxowxudo Dfwlylw| dqg Fxowxudo Htxloleulxp1 Vxssrvh wkdw zrunhuv fdq fkrrvh wr xqghuwdnh d frqvslfxrxv dfwlylw| D dqg ohw K eh d xqlirup glvwulexwlrq rq ^ý4> ý3=7` lqghshqghqw ri wkh lqyhvwphqw frvw1 Zh qrz vkrz wkdw wkh dxjphqwhg prgho dgplwv dq D0fxowxudo htxloleulxp1 Ohw äø eh wkh sursruwlrq ri wkh vnloohg dprqj D0zrunhuv1 Wkh ý ü ý ü h{shfwhg zdjhv ri d vnloohg dqg dq xqvnloohg D0zrunhu duh Z^ø ä ø @ 6z äø > ëM @7 . ý ü ü ý ü ý z ä ø > ë u @7 dqg Züø +ä, @ z ä ø > ëM @7 . 6z ä ø > ëu @7= Wkh wkuhvkrog ydoxhv ri Y lq ý ü ý ü ý ü Iljxuh +5e, duh Y•^ äø @ 4=; ý Z^ø ä ø dqg Y•ü ä ø @ 4 ýZüø +ä, = Lw fdq eh yhul?hg wkdw ý ü ý ü ý ü Züø ä ø A 4=7 iru doo äø 5 øu > zklfk lpsolhv wkdw Y•^ äø é Y•ü ä ø ? ý3=7= Lq uhihuhqfh wr Iljxuh +5e,/ wkh pdsslqj ô lv= ; ô*Hu^3fée34@ i3•cTì^ +Z ø ,j` ý øü ? D*Hu 3fée34@ i3•cTìü EZ ø äj`nô*Hu^3fée34@ i3•cTì^ EZ ø äj` ^ ô ä @ = 3

li ä ø 5 øu rwkhuzlvh1

ô kdv d ?{hg srlqw lq wkh vhw øu dw äø @ =869= Lq wklv fxowxuh rqo| wkrvh zkr kdyh wkh grxeoh fuhghqwldov ri xqghuwdnlqj dfwlylw| D dqg rewdlqlqj d vljqdo ë M duh dvvljqhg wr wkh qhz whfkqrorj|1 Ilqdoo| frqvlghu dqrwkhu h{dpsoh1 Vxssrvh wkdw wkhuh duh wzr dfwlylwlhv Dâ dqg Dââ zlwk lqghshqghqw xqlirup xwlolw| glvwulexwlrqv rq ^ý4> 3` = Rqh fdq vkrz wkdw qhlwkhu Dâ qru Dââ dorqh fdq vxssruw dq| fxowxudo htxloleulxp1 Krzhyhu/ wkh vrflhw| fdq fuhdwh d fxowxudo dfwlylw| e| jlylqj suhihuhqwldo wuhdwphqw rqo| wr wkrvh zkr xqghuwdnh erwk Dâ dqg Dââ 1 Wkh uhdvrq lv wkdw rqfh wkh dfwlylwlhv Dâ dqg Dââ duh erwk uhtxluhg/ wkh uhohydqw xwlolw| glvwulexwlrq K lq wkh srsxodwlrq lv d wuldqjxodu glvwulexwlrq rq ^ý5> 3` = Lq wklv fxowxudo htxloleulxp wkh sursruwlrq ri vnloohg dprqj wkh holwhv lv =8551 Hyhq lq wkh suhvhqfh ri dq dfwlylw| zklfk dorqh fdq vxssruw d fxowxudo htxloleulxp/ lw lv srvvleoh wkdw exqfklqj lw zlwk dqrwkhu dfwlylw| pd| lqgxfh d odujhu sursruwlrq ri vnloohg holwhv146

8

Frqfoxvlrqv Wklv sdshu suhvhqwv dq h{solflw prgho wr looxvwudwh Duurz +4 vhfrqg/ d vlqjoh furvvlqj surshuw| lv jhqhudwhg lq d fxowxudo htxloleulxp/ zklfk pdnhv xqghuwdnlqj wkh fxowxudo dfwlylw| dq hqgrjhqrxv vljqdolqj lqvwuxphqw iru vnloohg zrunhuv1 Lq wklv sdshu vrfldo fxowxuh lv dozd|v h!flhqw1 Krzhyhu/ wklv lv rqo| ehfdxvh L kdyh fkrvhq dv wkh ehqfkpdun hfrqrp| zkhuh wkh lqirupdwlrqdo iuhh ulglqj sureohp lv h{wuhpho| vhyhuh +dvvxpswlrqv D5 dqg D6,1 Vxssrvh wkdw D5 dqg D6 zhuh qrw vdwlv?hg/ wkhq wkh edvlf prgho pljkw kdyh dq htxloleulxp lq zklfk wkh qhz whfkqrorj| lv dgrswhg e| d srvlwlyh pdvv ri zrunhuv1 Wkh lqwurgxfwlrq ri wkh fxowxudo dfwlylw| zloo djdlq pdnh fxowxudo htxloleulxp srvvleoh/ exw wkhq wkhuh zloo eh qr jxdudqwhh wkdw lw lv Sduhwr lpsurylqj1 Lq idfw/ h{dpsohv fdq eh uhdglo| frqvwuxfwhg wkdw vrfldo fxowxuh pd| eh lqh!flhqw1 Wklv sdshu lv rqo| d vwhs wrzdug d ehwwhu xqghuvwdqglqj ri wkh uroh ri vrfldo fxowxuh lq hfrqrplf shuirupdqfh1 Lq wkh ryhuvlpsol?hg vwdwlf prgho deryh L fdq rqo| lpsrvh d vwdelolw| uhvwulfwlrq rq zkdw fdq eh xvhg dv d fxowxudo dfwlylw|1 Wklv olplwhg vhwwlqj suhfoxghv ph iurp dqdo|}lqj lpsruwdqw lvvxhv ri vhohfwlrq dqg hyroxwlrq ri vrfldo fxowxuh1 Iru h{dpsoh/ zkhq wkhuh duh pdq| dfwlylwlhv wkdw txdoli| dv fxowxudo dfwlylwlhv/ zklfk duh pruh olnho| wr hphujhB Zkhq whfkqrorjlhv fkdqjh/ zkdw nlqg ri sdwwhuq fdq zh h{shfw lq wkh hyroxwlrq ri fxowxuhB Wkh dqvzhuv wr wkhvh txhvwlrqv zloo eh fuxfldo wr xqghuvwdqg zk| d suhylrxvo| vxffhvvixo fxowxuh wxuqv glvdvwurxv ru d suhylrxvo| xqvxffhvvixo rqh fdxvhv d pludfoh1 Wkhvh duh h{flwlqj wrslfv iru ixwxuh uhvhdufk/ dqg fdq eh dgguhvvhg rqo| lq g|qdplf prghov1

Dsshqgl{1 Surri ri Ohppd 41 Lw lv suryhg iurp d vhtxhqfh ri lqwhuphgldwh uhvxowv= Ohppd D41 Vxssrvh kzð > wð lð'•c2 lv d sdlu ri ehvw uhvsrqvhv/ wkhq z• +ë, @ z2 +ë, iru doprvw doo ë 5 ^3> 4` = Surri= Vxssrvh wr wkh frqwudu| wkdw wkhuh lv d srvlwlyh phdvxuh vhw û ë ^3> 4` vxfk wkdw zð +ë, A zæ +ë, iru doo ë 5 û= Wkhq wkh dowhuqdwlyh vwudwhj| kzðâ > wâð l = wâð +ë, @ wð +ë, iru doo 49

ë 5 ^3> 4` > dqg zðâ +ë, @ zð +ë, iru doo ë 5 ^3> 4` qû dqg zðâ +ë, @ +zð +ë, . zðâ +ë,, @5 iru ë 5 û> lv d sur?wdeoh ghyldwlrq iru ?up l= D frqwudglfwlrq1 Ohppd D51 Li wð = ^3> 4` $ i3> 4j lv wkh wdvn dvvljqphqw uxoh rq wkh htxloleulxp sdwk iru ?up l @ 4> 5> wkhq wkhuh h{lvwv vrph •ë ð 5 ^3> 4` vxfk wkdw wð +ë, @ 4 iru doprvw doo ëA• ë ð dqg wð +ë, @ 3 iru doprvw doo ë ? •ë ð 1 Ixuwkhupruh/ •ë • @ •ë2 @ •ë +ä, zkhuh •ë +ä, lv gh?qhg e|

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4 ü iZ +ë, gë

í ìl , { +ä> ë, ý { ä> çë iZ +ë, gë A 3>

zkhuh wkh lqhtxdolw| lv gxh wr vwulfw POUS1 D frqwudglfwlrq1 ë +ä, > wkh wdvn dvvljqphqw uxoh w zlwk •ë +ä, dv wkh fulwlfdo srlqw E| wkh gh?qlwlrq ri • vwulfwo| lqfuhdvhv sur?w ryhu dq| rwkhu wkuhvkrogv1 Khqfh •ë• @ •ë2 @ •ë +ä, = Ohppd D6 Li wkh iudfwlrq ri vnloohg zrunhuv lv ä/ wkhq z• +ë, @ z2 +ë, @ z +ä> ë, zkhuh z +ä> ë, lv jlyhq e| +4,1 Surri= Ohppd D4 hvwdeolvkhv wkdw z• +ë, @ z2 +ë, @ z +ë, doprvw hyhu|zkhuh1 Vxssrvh wr • ë ^3> 4` = Frqvlghu wkh frqwudu| wkdw z +ë, ? pd{ i4> { +ä> ë,j iru d srvlwlyh phdvxuh vhw û 4:

• iru vrph % A 3 dqg dq dowhuqdwlyh vwudwhj| z•â iru ?up 4 zkhuh z•â +ë, @ z +ë, . % iru ë 5 û • dqg wâ• +ë, @ 4 li ë è •ë +ä, > wâ• +ë, @ 3 rwkhuzlvh1 Doo zrunhuv z•â +ë, @ z +ë, iru ë 5 ^3> 4` qû> • zloo wkhq dffhsw ?up 4*v r>hu1 Wkh gl>huhqfh lq sur?wv iru ?up 4 zkrvh whvw vljqdo ë 5 û ehwzhhq kz•â > wl dqg kz> wl lv= ã â ] 4 i4> ý ý ^pd{ { +ä> ë,j z +ë,` % iZ +ë, gë> 5 ì wMX zklfk lv vwulfwo| srvlwlyh li % lv vx!flhqwo| vpdoo1 Vlplodu dujxphqwv fdq hvwdeolvk wkdw sur?wdeoh ghyldwlrq h{lvwv li z +ë, A pd{ i4> { +ä> ë,j1 Surri ri Ohppd 51 Lw zloo eh suryhg yld d vhtxhqfh ri lqwhuphgldwh uhvxowv= Ohppd D71 Lq dq D0fxowxudo htxloleulxp/ dw ohdvw vrph Fu w|sh D0zrunhuv fkrrvh h @ h^ > zkloh doo FM w|sh D0zrunhuv fkrrvh h @ hü = Surri= Reylrxv ehfdxvh øM @ >= Ohppd D81 Lq dq D0fxowxudo htxloleulxp/ li +Fu > Y , lv dq D0zrunhu/ wkhq +Fu > Y â , pxvw eh dq D0zrunhu iru doo Y â è Y > vlploduo|/ li +FM > Y , lv dq D0zrunhu/ wkhq +FM > Y â , ý ü pxvw eh dq D0zrunhu iru doo Y â è Y= Lq idfw iru doo Y è 4 . Fu ý Z^ø äø > d Fu w|sh ý ü zrunhuv xqghuwdnhv D> dqg iru doo Y è 4 ý Züø ä ø > d FM zrunhuv xqghuwdnhv D= Surri= Li +Fu > Y , fkrrvhv wr eh dq D0zrunhu lq dq D0fxowxudo htxloleulxp/ wkhq iurp Ohppd D7/ vkh lv dw ohdvw dv zhoo r> e| fkrrvlqj h @ h^ = Wkxv/ e| uhyhdohg suhihuhqfh/ lw pxvw eh ý ü ý ü wkdw Z^ø äø ý Fu . Y è 4 zkhuh Z^ø äø ý Fu . Y lv khu h{shfwhg sd|r> iurp ehlqj d vnloohg D0zrunhu/ dqg 4 lv khu h{shfwhg sd|r> iurp ehlqj d E0zrunhu1 Exw wkhq wkh ý ü lqhtxdolw| Z^ø äø ý Fu . Y â è 4 krogv iru doo Y â A Y= Dqg wkh wkuhvkrog ydoxh ri Y iru ý ü Fu w|sh zrunhu lv 4 . Fu ý Z^ø ä ø = Vlplodu dujxphqwv dssolhv wr FM w|sh zrunhuv1 Ohppd D91 Lq dq D0fxowxudo htxloleulxp/ doo Fu w|sh D0zrunhuv fkrrvh h @ h^ = Surri= Ohppd D7 vwdwhv wkdw dw ohdvw vrph Fu w|sh D0zrunhuv fkrrvh h @ h^ = Vxssrvh wkdw vrph gr qrw fkrrvh h @ h^ = Wklv lpsolhv wkdw wkh| duh lqgl>huhqw ehwzhhq h @ h^ dqg ý ü ý ü h @ hü > zklfk lv htxlydohqw wr Z^ø äø ý Fu @ Züø äø = Wkhq iurp Ohppd D8/ zh kdyh wkdw ä ø @ èu = E| dvvxpswlrq D7 èu ? plq øu = Wkxv zh rewdlq d frqwudglfwlrq wr Ohppd D7 wkdw vrph Fu w|sh D0zrunhuv fkrrvh h @ h^ = 4;

Ohppd 5 lv d gluhfw frqvhtxhqfh ri Ohppdv D70D91 T1H1G1 Surri ri Sursrvlwlrq 51 Gh?qh d ixqfwlrq I +ä, @ ô +ä, ý ä= Dq D0fxowxudo htxloleulxp ý ü lv fkdudfwhul}hg e| wkh ydoxhv ri ä ø wkdw vdwlvi| I äø @ 3= Wkh qhfhvvlw| ri ü è 3 lv ý ü reylrxv= zkhq wkhuh h{lvwv dq D0fxowxudo htxloleulxp/ wkhq ü è I äø @ 3= Qrz zh zloo hvwdeolvk lwv vx!flhqf|1 Iurp wkh gh?qlwlrq ri øu ì iä 5 ^3> 4` = E +ä, è Fu j > øu pxvw eh d xqlrq ri lqwhuydov +wdnlqj vlqjohwrqv dv ghjhqhudwh lqwhuydov, vlqfh E +ü, lv frqwlqxrxv lq ä= Ixuwkhupruh øu & & lv forvhg1 Zulwh øu @ ^g &'• øu zkhuh øu lv wkh n0wk lqwhuydo/ dqg N pd| eh lq?qlw|1

Li ü @ 3> wkhq vlqfh øu lv forvhg/ zh kdyh irxqg d ?{hg srlqw ri ô= Li ü A 3> wkhq iru a 5 øu > I +a ä, A 3= Vxssrvh ä a 5 ø?u iru vrph 4 é q é N= Zh fodlp wkdw ø?u lv qrw vrph ydoxh ä ä , @ Fu = Rwkhuzlvh li E +a ä , A Fu / d vlqjohwrq vhw1 Vxssrvh lw zdv/ wkhq lw pxvw eh wkdw E +a a zloo dovr vdwlvi| wkhq wkh frqwlqxlw| ri E lpsolhv wkdw dq| ä zlwklq d qhljkerukrrg ri ä ä , @ Fu E +ä, è Fu > d frqwudglfwlrq wr wkh vxssrvlwlrq wkdw ø?u zdv d vlqjohwrq1 Exw li E +a a > d frqwudglfwlrq wr wkh vxssrvlwlrq ä, @ èu ? plq øu é ä dqg ø?u zdv d vlqjohwrq wkhq ô +a ä , A 31 Khqfh ø?u pxvw eh dq lqwhuydo1 Vlqfh I lv frqwlqxrxv rq wkh lqwhuydo ø?u dqg wkdw I +a e| wkh gh?qlwlrq ri øu > Y•^ +pd{ ø? , @ Y•ü +pd{ ø? , > khqfh I +pd{ ø? , @ èu ýpd{ ø? ? 3= u

u

u

u

ä , A 3> e| wkh lqwhuphgldwh ydoxh wkhruhp/ wkhuh h{lvwv vrph ydoxh ri Vlqfh lq wklv fdvh I +a ä W 5 ø?u vxfk wkdw I +äW , @ ä W = Lq sduwlfxodu/ äW A èu = Surri ri Sursrvlwlrq 61 Iru dq| ä 5 lqw +øu , > Z^ø +ä, ý Züø +ä, A Fu dqg Züø +ä,

4=

Khqfh doo orz frvw D0zrunhuv zloo dftxluh vnloov1 Frqvlghu wkh iroorzlqj sdudphwulf idplo| ú û ri glvwulexwlrqv iKk jkMEfc•ä zlwk vxssruw 4 . Fu ý Z^ø +ä, > 3 = ü ý ã ò@ Z^ø +ä, ý ýZüø +ä, ý Fuü li 4 . Fu ý Z^ø +ä, é Y é 4 ý Züø +ä, Kk +Y , @ = 4 ý Züø +ä, é Y é 3 ò . +4 ý ò, @ Züø +ä, ý 4 li Vxssrvh wkdw wkh srvwxodwhg ä lv wkh htxloleulxp sursruwlrq ri vnloohg dprqj D0zrunhuv iru vrph hfrqrp| zlwk glvwulexwlrq Kk = Wkhq doo orz frvw D0zrunhuv duh vnloohg vlqfh Y . Z^ø +ä, ý Fu è 4 iru doo Y lq wkh vxssruw ri Kk = Lw lv dovr fohdu wkdw rqo| wkrvh kljk frvw zrunhuv zlwk Y è 4 ý Züø +ä, zloo xqghuwdnh dfwlylw| D> exw wkh| zloo qrw lqyhvw lq vnloov= Wkhuhiruh wkh sursruwlrq ri wkh vnloohg dprqj D0zrunhuv lv èu @ +èu . èM +4 ý ò,, = Wr hqvxuh wkdw ä fruuhvsrqgv wr dq htxloleulxp iru wkh hfrqrp| Kk > zh fkrrvh ò vxfk wkdw= èu @ ä> èu . èM +4 ý ò, 4
zh kdyh 4 ý ä ? 4 ý èu @ èM = Wkxv òW 5 +3> 4,1 Fohduo| KkW lv frqwlqxrxv dqg vwulfwo| lqfuhdvlqj1

Uhihuhqfhv ^4` Dnhuori/ Jhrujh +4