The Impact of Youth Unemployment Policy

(1)

Socialforskningsinstituttet The Danish National Institute of Social Research

The Impact of Youth Unemployment Policy

Trine Filges & Birthe Larsen

Labour Market Policy Working Paper 2:2002

Working Paper

Working papers-skærm.indd1 28-01-2002, 16:35

(2)

The Impact of Youth Unemployment Policy

Trine Filges and Birthe Larsen

Labour Market Policy Working Paper 2:2002

The Working Paper Series of The Danish National Institute of Social Research contain interim results of research and preparatory studies. The Working Paper Series provide a basis for professional discussion as part of the research process.

(3)

Wkh Lpsdfw ri \rxwk Xqhpsor|phqw Srolf|

Wulqh Ilojhv

^|

dqg Eluwkh Oduvhq

^}

Mdqxdu| 5;/ 5335

Devwudfw

Wklv sdshu h{dplqhv wkh lpsdfw rq xqhpsor|phqw/ xqhpsor|phqw glv0 wulexwlrq/ zdjhv dqg zhoiduh ri \rxwk Xqhpsor|phqw Surjudpphv +\XSv,1 Wkh dlp ri \XS lv wr lqfuhdvh wkh qxpehu ri |rxqj shrsoh dftxlulqj vnloov1 Zh dvvxph wkdw wkh \XSv duh d frpsohwh vxffhvv dqg frqvhtxhqwo| dqdo|vh zkdw kdsshqv zkhq wkh qxpehu ri vnloohg zrunhuv lqfuhdvhv uhodwlyho| wr wkh qxpehu ri xqvnloohg zrunhuv1 Wkh uhvxowv ghshqg rq wkh surgxfwlylw|

ri wkh vnloohg zrunhuv zkhq hpsor|hg lq wkh *xqvnloohg vhfwru* uhodwlyho| wr wkh surgxfwlylw| ri wkh xqvnloohg zrunhu1

Nh|zrugv= Vnloo/ xqhpsor|phqw/ vhdufk MHO fodvvlfdwlrqv= M4;/ M6;/ M9;

41 Lqwurgxfwlrq

Uhgxfwlrq ri kljk Hxurshdq xqhpsor|phqw udwhv kdv ehhq ri juhdw frqfhuq wr wkh srolwlfldqv lq wkh Hxurshdq frxqwulhv wkurxjkrxw wkh qlqhwlhv1 Lq pdq| frxq0 wulhv/ wklv frqfhuq kdv uhvxowhg lq srolf| dwwhpswv lq rughu wr jkw wkh sureohp1 Glhuhqw srolflhv kdyh ehhq xvhg dqg d vkliw iurp sdvvlyh xqhpsor|phqw srolf|/

jhqhudoo|/ vlpso| lq wkh irup ri xqhpsor|phqw lqvxudqfh sdlg wr xqhpsor|hg zrunhuv wrzdugv dfwlyh oderxu pdunhw srolf| kdv ehhq lqlwldwhg lq pdq| frxq0 wulhv1 Iru h{dpsoh/ Qlfnhoo dqg Ydq Rxuv +5333, dqdo|vhv wkh Gxwfk dqg Eulwlvk fdvhv1 Erwk frxqwulhv kdyh h{shulhqfhg d odujh gurs lq xqhpsor|phqw gxulqj wkh qlqhwlhv1 Ghqpdun kdv h{shulhqfhg dq hyhq odujhu idoo lq xqhpsor|phqw gxulqj wkh vdph shulrg1 Zh wkhuhiruh jlyh d vkruw ghvfulswlrq ri wkh srolf| lpsohphqwhg

Frpphqwv iurp sduwlflsdqwv dw wkh Hxurshdq Hfrqrplf Dvvrfldwlrq Frqihuhqfh/ Odxvdqqh 5334/ duh judwhixoo| dfnqrzohgjhg1

|Fhqwuh iru Uhvhdufk lq Vrfldo Lqwhjudwlrq dqg Pdujlqdol}dwlrq dqg wkh Gdqlvk Qdwlrqdo Lqvwlwxwh ri Vrfldo Uhvhdufk/ Khuoxi Wuroohv Jdgh 44/ GN04385 Frshqkdjhq N/ H0pdlo dgguhvv=

wliCv1gn/ Skrqh= .78 667; 3<59/ Id{= .78 667; 3;66

} Fhqwuh iru Uhvhdufk lq Vrfldo Lqwhjudwlrq dqg Pdujlqdol}dwlrq dqg Frshqkdjhq Exvlqhvv Vfkrro/ Ghsduwphqw ri Hfrqrplfv/ Vroemhuj Sodgv 6/ GN05333 Frshqkdjhq I1 H0pdlo dgguhvv=

eo1hfrCfev1gn

(4)

lq Ghqpdun lq wkh qlqhwlhv/ hyhq wkrxjk vrph ri wkh vdph hohphqwv pd| eh irxqg lq wkh srolf| frqgxfwhg lq rwkhu frxqwulhv1

Lq Ghqpdun lq 4<<9 d uhirup gluhfwhg wrzdugv wkh xqhpsor|hg/ orz0hgxfdwhg

|rxwk zdv lpsohphqwhg/ wkh \rxwk Xqhpsor|phqw Surjudpph +\XS,1 Wkh dlp ri wklv uhirup zdv wr lpsuryh wkh hpsor|phqw srvvlelolwlhv iru xqhpsor|hg/ orz0 hgxfdwhg |rxwk e| prwlydwlqj wkhp wr xqghuwdnh dq hgxfdwlrq1⁴ Wkh prwlydwlrq zdv wkh idfw wkdw wkh xqhpsor|phqw udwh ri vnloohg zrunhuv zdv +dqg vwloo lv, ohvv wkdq wkh xqhpsor|phqw udwh ri xqvnloohg zrunhuv lq Ghqpdun/ dqg wklv rughulqj kdv ehhq suhvhqw lq pruh wkdq wzhqw| |hduv1 Jurhv dqg Krop +4<<<, dqdo|vh wkh xqhpsor|phqw udwhv lq Ghqpdun ghshqghqw xsrq hgxfdwlrqdo ohyho iru wkh shulrg 4<;304<<;1 Wkh xqhpsor|phqw udwh ri vnloohg zrunhuv lv ehorz wkh dyhudjh1 Wklv rughulqj kdv ehhq wkh vdph wkurxjkrxw wkh shulrg/ dqg wkhuh lv hylghqfh ri ixu0 wkhu srodul}dwlrq= wkh xqhpsor|phqw udwh ri qrq0vnloohg zrunhuv nhhsv lqfuhdvlqj/

zkhuhdv wkh| h{shfw erwwohqhfn sureohpv iru kljkhu hgxfdwhg zrunhuv1

Lq wklv sdshu zh dqdo|vh wkh orqj0whup hhfwv rq xqhpsor|phqw/ zdjhv dqg zhoiduh ri wkh \XS1 Ixuwkhupruh/ zh frqvlghu glvwulexwlrq hhfwv ri wkh \XS1 Zh vhw xs d vhdufk prgho glvwlqjxlvklqj ehwzhhq vnloohg dqg xqvnloohg zrunhuv1 Zh gr qrw prgho wkh ghflvlrq wr dftxluh vnloov/ udwkhu zh dvvxph wkdw dq h{rjhqrxv iudfwlrq ri wkh oderxu irufh kdv douhdg| dftxluhg vnloov1⁵ Frqvhtxhqwo|/ zrunhuv duh hlwkhu vnloohg ru xqvnloohg1 Wkhuh duh wzr vhfwruv/ d vnloohg vhfwru zkhuh wkhuh lv d plqlpxp vnloo uhtxluhphqw dqg dq xqvnloohg vhfwru zlwkrxw dq| vnloo uhtxluhphqwv1 Xqvnloohg zrunhuv fdqqrw eh kluhg lq wkh vnloohg vhfwru zkhuhdv vnloohg zrunhuv fdq eh kluhg lq erwk wkh xqvnloohg dqg wkh vnloohg vhfwru1 Wklv ghqlwlrq ri vnloov hqdeohv xv wr glvwlqjxlvk ehwzhhq vnloohg mrev dqg xqvnloohg mrev dqg vnloohg zrunhuv dqg xqvnloohg zrunhuv1 Zh dvvxph wkdw vnloohg zrunhuv vhdufk iru mrev lq erwk vhfwruv dv wkh ydoxh ri hpsor|phqw lq wkh xqvnloohg vhfwru lv kljkhu wkdq wkh ydoxh ri ehlqj xqhpsor|hg1 Wkh dvvxpswlrq wkdw zrunhuv vhdufk iru mrev lq erwk vhfwruv lv dovr irxqg lq Jdxwlhu +5334, dqg Doeuhfkw dqg Yurpdq +5333,1 Ixuwkhupruh lw lv lq dffrugdqfh zlwk uhdo olih ehkdylrxu ri vnloohg zrunhuv lq Ghqpdun1 Lq 4<<;/ 63 shufhqw ri wkh zrunhuv zkr duh hpsor|hg lq wkh xqvnloohg vhfwru duh vnloohg zrunhuv dffruglqj wr wkh Gdqlvk Plqlvwu| ri Oderxu +5333,1 Zh dvvxph wkdw zrunhuv fdq gluhfw wkhlu vhdufk/ l1h1/ wkh| nqrz zkhwkhu wkh| dsso|

iru d vnloohg mre ru dq xqvnloohg mre1 Ilupv/ krzhyhu/ gr qrw nqrz h{dqwh zklfk w|sh ri zrunhuv dssolhv iru wkh ydfdqf|1 Wkh prwlydwlrq ehklqg wkhvh dvvxpswlrqv lv wkdw lw lv qrw gl!fxow iru d up wr lqirup derxw wkh vnloo uhtxluhphqwv iru d mre/ exw wkh| fdqqrw h{dqwh h{foxgh vrph zrunhuv iurp dsso|lqj iru wkh mre1

4\rxqjlqvxuhg shuvrqv/ ohvv wkdq 58 |hduv/ zlwkrxw dq| irupdo hgxfdwlrq eh|rqg vhfrqgdu|

vfkrro/ dqg zkr kdyh ehhq xqhpsor|hg iru 9 prqwkv gxulqj wkh odvw < prqwkv/ duh jlyhq dq rhu ri 4; prqwkv vshfldoo| ghvljqhg yrfdwlrqdo hgxfdwlrq1 Wklv rhu frqwdlqv dq lqfhqwlyh wr xqghuwdnh ruglqdu| hgxfdwlrq ru wr qg d mre vlqfh xqhpsor|phqw ehqhwv duh fxw e| 83 shufhqw zkloh lq wkh vshfldo hgxfdwlrq surjudpph1 Uhixvdo wr sduwlflsdwh lq wkh vshfldo hgxfdwlrq surjudpphv lv iroorzhg e| d vdqfwlrq/ lw zloo uhvxow lq d wrwdo orvv ri xqhpsor|phqw ehqhwv1

5Vhh Ilojhv dqg Oduvhq +5334, iru dq dqdo|vlv ri wkh ghflvlrq wr dftxluh vnloov1

(5)

Frqvhtxhqwo|/ vnloohg zrunhuv vhdufk iru mrev lq erwk vhfwruv zkhuhdv xqvnloohg zrunhuv/ nqrzlqj wkh| duh xqhpsor|deoh lq wkh vnloohg vhfwru/ vhdufk iru mrev lq wkh xqvnloohg vhfwru rqo|1 Wkh zdjh sdlg lq wkh vnloohg vhfwru lv kljkhu wkdq wkh zdjh rhuhg lq wkh xqvnloohg vhfwru1 Wklv lpsolhv wkdw vnloohg zrunhuv suhihu wr zrun lq wkh vnloohg vhfwru dqg frqvhtxhqwo|/ wkh| vhdufk iru mrev lq wkh vnloohg vhfwru zkloh hpsor|hg lq wkh xqvnloohg vhfwru1⁶

Zh dvvxph wkdw wkh \XS lv d frpsohwh vxffhvv lq wkh vhqvh wkdw lw dfwxdoo| lq0 fuhdvhv wkh uhodwlyh qxpehu ri zrunhuv dftxlulqj vnloov1 Frqvhtxhqwo|/ zh dqdo|vh wkh orqj0whup hhfw ri wkh \XS e| frqvlghulqj dq h{rjhqrxv lqfuhdvh lq wkh qxpehu ri vnloohg zrunhuv uhodwlyh wr wkh qxpehu ri xqvnloohg zrunhuv1 Xvxdoo|/ lq vwdqgdug vhdufk prghov/ wkh xqhpsor|phqwudwh ri d vshflf zrunhu jurxs lv qrw dhfwhg e| wkh qxpehu ri zrunhuv lq wkh jurxs1 Lq wkh suhvhqw prgho/ krzhyhu/

d odujhu iudfwlrq ri vnloohg zrunhuv lpsolhv wkdw wkh xqhpsor|phqw udwh ri vnloohg zrunhuv dv zhoo dv wkh xqhpsor|phqw udwh ri xqvnloohg zrunhuv duh dhfwhg1 Wkh uhvxowv ghshqg rq wkh pdujlqdo surgxfwlylw| ri wkh vnloohg zrunhu zkloh hpsor|hg lq wkh xqvnloohg vhfwru uhodwlyho| wr wkh surgxfwlylw| ri wkh xqvnloohg zrunhu1

Li wkh pdujlqdo surgxfwlylw| ri d vnloohg zrunhu hpsor|hg lq wkh xqvnloohg vhf0 wru uhodwlyh wr wkh pdujlqdo surgxfwlylw| ri dq xqvnloohg zrunhu lv orzhu wkdq d wkuhvkrog/q^W/ zkhuh lq jhqhudoq^Wlv juhdwhu wkdqcwkhq wkh hhfwv duh dv iroorzv1 D odujhu iudfwlrq ri vnloohg zrunhuv lpsolhv wkdw wkh xqhpsor|phqw udwh ri vnloohg zrunhuv dv zhoo dv wkh xqhpsor|phqw udwh ri xqvnloohg zrunhuv lqfuhdvhv1 Wkh srv0 lwlyh hhfw rq wkh xqhpsor|phqw udwhv lv gxh wr wkh dvvxpswlrq wkdw xqhpsor|hg vnloohg zrunhuv vhdufk iru mrev lq erwk wkh vnloohg dqg wkh xqvnloohg vhfwru1 Wkh

\XS dovr lqgxfhv d qhjdwlyh hhfw rq wkh wrwdo xqhpsor|phqw udwh1 Wklv uhvxowv iurp wkh pryhphqw ri zrunhuv iurp wkh kljk xqhpsor|phqw jurxs/ wkh xqvnloohg zrunhuv/ wr wkh orz xqhpsor|phqw jurxs/ wkh vnloohg zrunhuv1 Khqfh/ wkh qhw hhfw ri wkh \XS rq wkh wrwdo xqhpsor|phqw udwh lv dpeljxrxv1 Vlploduo|/ wkhuh lv erwk d qhjdwlyh dqg d srvlwlyh zhoiduh hhfw1 Lqfuhdvlqj wkh uhodwlyh qxpehu ri vnloohg zrunhuv lpsolhv prylqj zrunhuv iurp wkh jurxs jlylqj ulvh wr wkh orzhvw zhoiduh wr wkh jurxs ri zrunhuv jlylqj ulvh wr wkh kljkhvw zhoiduh/ wkhuhe| lqgxf0 lqj dq lqfuhdvh lq wrwdo zhoiduh1 Krzhyhu/ wkh \XS kdv d qhjdwlyh lpsdfw rq wkh wrwdo zhoiduh udwh lq wkh hfrqrp|1 Frqvhtxhqwo|/ wkh wrwdo zhoiduh hhfw ri wkh

\XS lv dpeljxrxv1 Frqvlghulqj zdjh hhfwv zh vkrz wkdw zdjhv lq wkh xqvnloohg vhfwru ghfuhdvhv gxh wr wkh \XS1 Frqfhuqlqj glvwulexwlrq hhfwv/ zh vkrz wkdw wkh \XS lqgxfhv d pruh xqhtxdo zdjh dqg xqhpsor|phqw glvwulexwlrq1 Lqlwldoo|/

wkh zdjh udwh lq wkh vnloohg vhfwru lv kljkhu wkdq zdjhv lq wkh xqvnloohg vhfwru1 Wkh \XS zruvhqv wklv xqhtxdo zdjh glvwulexwlrq1 Olnhzlvh/ xqhpsor|phqw ri wkh xqvnloohg zrunhuv lv lqlwldoo| kljkhu wkdq xqhpsor|phqw ri wkh vnloohg zrunhuv1 Wkh \XS zruvhqv wkh xqhtxdo xqhpsor|phqw glvwulexwlrq wrr1

Zkhq wkh pdujlqdo surgxfwlylw| ri wkh vnloohg zrunhu hpsor|hg lq wkh xq0 vnloohg vhfwru furvvhv wkh wkuhvkrogq^W wkhq wkhvh qglqjv duh vrphzkdw prglhg1

6Wklv lqirupdwlrq vwuxfwxuh lv dovr irxqg lq Jdxwlhu +5334,1

6

(6)

Lq wklv fdvh/ zh pd| vhh d ghfuhdvh lq zdjh glvshuvlrq dqg wkh xqhpsor|phqw udwhv ri erwk vnloohg dqg xqvnloohg zrunhuv dv zhoo dv dq lqfuhdvh lq wkh zhoiduh udwh ri wkh hfrqrp|1

Rxu frqwulexwlrq vkrxog eh frpsduhg wr wzr uhodwhg sdshuv1 Jdxwklhu +5334, kdv ehhq zulwwhq sdudooho zlwk rxu sdshu dqg gr kdyh vrph vlplodulwlhv wr rxu sdshu1 Krzhyhu/ lq klv sdshu zdjhv duh rqo| ixqfwlrqv ri h{rjhqrxv sdudphwhuv1 Rxu sdshu kdv zdjhv dv ixqfwlrqv ri wudqvlwlrq udwhv dv wkhvh duh lpsruwdqw iru zrunhu*v idoo edfn srvlwlrq1 Zh fdq wkhuhiruh vwxg| lpsdfwv ri d fkdqjh lq wkh iudfwlrq ri vnloohg zrunhuv rq zdjhv dqg zdjh glvwulexwlrq1 Ixuwkhupruh/ zh fdq frqvlghu lpsdfwv rq zhoiduh1 Vdlqw0Sdxo +4<<9, shuirup d vlplodu h{shulphqw dv zh gr/ exw lq d pruh uhvwulfwlyh prgho/ lq wkh vhqvh wkdw wkhuh lv d {hg qxpehu ri mrev lq wkh hfrqrp| dqg zrunhuv hpsor|hg lq wkh xqvnloohg vhfwru gr qrw vhdufk rq0wkh0mre1

Wkh sdshu lv rujdql}hg dv iroorzv1 Wkh prgho lv suhvhqwhg lq Vhfwlrq 21 Lq Vhfwlrq/ wkh htxloleulxp lv ghvfulehg dqg Vhfwlrqefrqvlghuv wkh hhfwv ri \XS1 Wkh odvw vhfwlrq frqfoxghv1

51 Wkh prgho

Lq wklv vhfwlrq zh vhw xs wkh prgho wr dqdo|vh wkh orqj0whup hhfwv rq xqhp0 sor|phqw/ zdjhv dqg zhoiduh ri \XS1 Zh gr qrw prgho wkh ghflvlrq wr dftxluh vnloov/ udwkhu zh dvvxph wkdw dq h{rjhqrxv iudfwlrq ri wkh oderxu irufh kdv do0 uhdg| dftxluhg vnloov1 Frqvhtxhqwo|/ wkh oderxu irufh frqvlvwv ri zrunhuv zkr duh hlwkhu vnloohg ru xqvnloohg1 Lq rughu wr irfxv rq wkh orqj whup lpsdfw ri wkh \XS/

zh dqdo|vh wkh frqvhtxhqfhv ri dq h{rjhqrxv lqfuhdvh lq wkh qxpehu ri vnloohg zrunhuv1

Wkhuh duh wzr vhfwruv/ vhfwru lv wkh vnloohg vhfwru dqg vxssolhv mrev rqo| wr vnloohg zrunhuv1 Vhfwru2/ lv wkh xqvnloohg vhfwru dqg vxssolhv mrev wr erwk vnloohg dqg xqvnloohg zrunhuv1 Wkh pdujlqdo surgxfw lq vhfwrulv kljkhu wkdq lq vhfwru 21 Vnloohg zrunhuv/ krzhyhu/ suhihu wr zrun lq vhfwru dv wkh pdujlqdo surgxfw lv kljkhu wkdq lq vhfwru2 dqg frqvhtxhqwo| wkh zdjh sdlg lv dovr kljkhu wkdq wkh zdjh rhuhg lq vhfwru21

5141 Zrunhuv

Ohw K^. dqg K^L ghqrwh wkh h{shfwhg suhvhqw ydoxhv ri olihwlph xwlolwlhv ri eh0 lqj hpsor|hg dqg xqhpsor|hg/ uhvshfwlyho|1 Vxevfulsw ' 7c ghqrwhv vnloohg uhvshfwlyho| xqvnloohg zrunhuv dqg ' c2 ghqrwhv zklfk vhfwru wkh zrunhu lv hpsor|hg lq1 Wkh olihwlph xwlolwlhv ri d vnloohg zrunhu duh wkhq=

oK^.₇ ' _r|n^EK^L₇ K^.₇c +514, oK^L₇ ' K|nREK^.₇K^L₇ nR2

K^.₇²K^L₇

c +515,

(7)

oK^.₇² ' _r²|nR

K^.₇K^.₇² ^

K^.₇²K^L₇

c +516,

zkhuhKlv wkh xqhpsor|phqw lqvxudqfh/o lv wkh glvfrxqw udwh dqg ^ lv dq h{rjh0 qrxv iudfwlrq ri fxuuhqwo| hpsor|hg zrunhuv ohdylqj wkhlu mrev1 Wkh zdjh udwh ri d vnloohg zrunhu zrunlqj lq vhfwru lv ghqrwhg _r/ dqg ²_r lv wkh zdjh udwh ri d vnloohg zrunhu hpsor|hg lq vhfwru21 | lv d oxps vxp wd{sdlg e| doo zrunhuv1 R lv wkh wudqvlwlrq udwh ri vnloohg zrunhuv lqwr hpsor|phqw lq vhfwrudqgR2 lv wkh wudqvlwlrq udwh ri zrunhuv lqwr hpsor|phqw lq vhfwru 21 Qrwh wkdw xqhpsor|hg vnloohg zrunhuv vhdufk iru d mre lq erwk vhfwru dqg vhfwru 2/ dqg wkdw vnloohg zrunhuv hpsor|hg lq vhfwru2 vhdufk iru d mre lq vhfwru 1

Wkh olihwlph xwlolwlhv ri xqvnloohg zrunhuv duh=

oK^.² ' _?²|n^EK^L K^.²c +517, oK^L ' K|nR₂EK^.²K^Lc +518, zkhuh_?² lv wkh zdjh udwh ri dq xqvnloohg zrunhu hpsor|hg lq vhfwru21 Qrwh wkdw wkh wudqvlwlrq udwhv ri vnloohg dqg xqvnloohg zrunhuv lqwr hpsor|phqw lq vhfwru2 duh htxdo1 Khqfh/ dv dujxhg ehorz/ zh dvvxph wkdw vnloohg dqg xqvnloohg zrunhuv frpshwh iru mrev lq vhfwru2 rq htxdo whupv1

5151 Ilupv

Ilupv vxsso| mrev ghshqghqw xsrq wkh zdjh dqg wkhlu klulqj frvwv1 Ilupv vxsso|

rqh mre hdfk1 Ohw + eh wkh pdujlqdo surgxfw ri d vnloohg zrunhu lq vhfwru / @+/

@ / zkhuh @+ : K/ lv wkh pdujlqdo surgxfw ri dq xqvnloohg zrunhu lq vhfwru 2/ qdoo|/ @q+ lv wkh pdujlqdo surgxfw ri d vnloohg zrunhu zrunlqj lq vhfwru 5/

@q dqg @q+ : K1 Khqfh/ zh kdyh wkdw K 4? E@q+c @+1 Qrwh/ wkdw zh dvvxph wkdw wkh vnloov ri vnloohg zrunhuv gr qrw lpso| wkdw wkh| qhfhvvdulo| duh deoh wr surgxfh pruh wkdq xqvnloohg zrunhuv zkhq zrunlqj lq wkh xqvnloohg vhfwru1 Rqo| zkhq wkh vnloohg zrunhuv duh hpsor|hg lq wkh vnloohg vhfwru gr wkh| pdnh xvh ri wkhlu vnloov dqg frqvhtxhqwo| duh deoh wr surgxfh dw *wkhlu pd{lpxp*1 Zh gr qrw uhvwulfwq wr eh odujhu ru vpdoohu wkdq rqh/ rqo| ohvv wkdq _@1 Wkh hpslulfdo hylghqfh rq wklv lvvxh lv pl{hg +vhh E fkho 5333 iru d vxuyh|, dqg wkh lqwxlwlrq lv dv iroorzv1 Li q ' / wkdw lv/ d vnloohg zrunhu hpsor|hg lq vhfwru 2 kdv wkh vdph pdujlqdo surgxfwlylw| dv dq xqvnloohg zrunhu hpsor|hg lq vhfwru 21 Wklv dvvxpswlrq lpsolhv wkdw iru h{dpsoh/ zkhq zrunlqj dv d fohdqhu/ lw grhv qrw pdwwhu iru wkh pdujlqdo surgxfw zkhwkhu |rx duh deoh wr frqwuro d fudqh ru qrw1 Wkh qxpehu ri x uhpryhg lq d gd| lv wkh vdph1 Rq wkh rwkhu kdqg/ zkhq frqwuroolqj d fudqh lw vxuho| pdwwhuv iru wkh pdujlqdo surgxfw zkhwkhu |rx duh wudlqhg wr gr wkdw ru qrw1 Liq :cwkh vnloohg zrunhu lv pruh surgxfwlyh wkdq wkh xqvnloohg zrunhuv/ zkloh hpsor|hg lq vhfwru 21 Wklv pd| kdsshq/ iru h{dpsoh/

lq wkh wrxulvw vhfwru/ zkhuh vrph nqrzohgjh ri odqjxdjh pd| eh dq dgydqwdjh hyhq lq xqvnloohg nlqg ri mrev/ ru lw pd| kdsshq li vnloohg zrunhuv kdyh d kljkhu

8

(8)

ohyho ri mre vdwlvidfwlrq ru ehwwhu khdowk vwdwxv1 Wkh fdvh ri q / fruuhvsrqgv wr wkh vlwxdwlrq zkhuh vnloohg zrunhuv zkhq shuiruplqj xqvnloohg nlqg ri mrev duh vr eruhg dqg iuxvwudwhg wkdw lw qhjdwlyho| lq xhqfhv wkhlu ohyho ri mre vdwlvidfwlrq dqg khdowk vwdwxv dqg frqvhtxhqwo| orzhuv wkhlu surgxfwlylw|1⁷

Wkh h{shfwhg suhvhqw ydoxhv ri d oohg mre lq vhfwru / ' c2/ K^ac dqg ri d ydfdqw mre/K^T / duh ghwhuplqhg e| wkh htxdwlrqv=

oKâ ' +_rÊKâ K^Tc +519,

oK^T ' b

K^a K^T

&c +51:,

oK^a₂ ²_?

' @+_?²^EK^a₂ _?²

K^T₂c +51;,

oK^a₂ ²_r

' @+q²_r ERn^ EK^a₂ _r²

K^T₂c +51<, oK^T₂ ' b2EbK^a₂

_?²

n Eb K^a₂ ²_r

K^T₂&c +5143, zkhuh b/ ' c2/ lv wkh up*v wudqvlwlrq udwh/ dqg b lv wkh suredelolw| wkdw wkh zrunhu vhdufklqj iru d mre lq vhfwru2 lv xqvnloohg1 Wkh gluhfw frvw dvvrfldwhg zlwk mre vxsso| lv jlyhq e|&1 Iuhh hqwu| lpsolhv wkdw mrev duh vxssolhg dv orqj dv lw lv surwdeoh/ l1h1 xqwlo K^T ' f Xvlqj wklv frqglwlrq dqg frpelqlqj htxdwlrqv +519,0+51:,/ uhvshfwlyho| +51;,0+5143, jlyhv wzr htxdwlrqv wr ghwhuplqh wkh upv*

wudqvlwlrq udwhv=

&nb+_r

on^ ' fc +5144,

&nb2

b

@+_?² on^

n Eb

@+q_r² on^nR

' f +5145, 5161 Pdwfklqj dqg Xqhpsor|phqw

Wkh zrun irufh lv glylghg lqwr wzr jurxsv dffruglqj wr vnloov1 Zh qrupdol}h wkh wrwdo oderxu irufh wr rqh1 Wkh qxpehu ri zrunhuv zkr kdyh dftxluhg vnloov lv jlyhq e|\ dqg wkh qxpehu ri xqvnloohg zrunhuv lv \1 ./ ' 7c c dqg ' c2/

ghqrwhv wkh qxpehu ri zrunhuv ri w|sh hpsor|hg lq vhfwru / dqgL/ '7c / ghqrwhv wkh qxpehu ri xqhpsor|hg zrunhuv ri w|sh 1 Wkh rzv ri zrunhuv lv looxvwudwhg lq Iljxuh 41

7D qxpehu ri XV vwxglhv vkrz wkdw ryhutxdolhg zrunhuv kdyh d orzhu ohyho ri mre vdwlvidfwlrq dqg pruh khdowk sureohpv dqg frqvhtxhqwo| kdyh kljkhu udwhv ri vklunlqj ru devhqwhhlvp wkdq fruuhfwo| doorfdwhg zrunhuv/ vhh E fkho 53331

(9)

ES

1 U_S #p1 US

E_S²

E_S²#q US#p2

E_N²

U_N #p2

E_N² #q

U_N ES

1 #q

E_S²#p₁

Iljxuh 4= Oderxu Pdunhw Iorzv1

Lq vwhdg| vwdwh/ lq rzv duh htxdo wr rxw rzv1 Wkh htxdwlrqv ghwhuplqlqj htxl0 oleulxp xqhpsor|phqw iru xqvnloohg dqg vnloohg zrunhuv/ hpsor|phqw iru vnloohg zrunhuv hpsor|hg lq vhfwru dqg lq vhfwru2 dqg wrwdo qxpehuv duh=

R2L ' ^.²c ERnR2L7 ' ^

.₇ n.₇² c R

L7n.₇²

' ^.₇c

R₂L₇ ' E^nR.₇²c L n.² ' \c L7n.₇ n._r² ' \

Wkh qxpehu ri zrunhuv lq hdfk ri wkh fdwhjrulhv lv wkhq jlyhq e|=

.₇ ' R

^nR\c +5146,

.₇² ' R₂

^nRL7c +5147,

L_r ' \ ^

^nRnR₂c +5148,

.² ' E\ R2

^nR2c +5149,

L ' E\ ^

^nR2 +514:,

Wkh xqhpsor|phqw udwhv duh erwk lqfuhdvlqj lq wkh vhsdudwlrq udwh/ ^/ dqg ghfuhdvlqj lq wkh wudqvlwlrq udwhR21 Lq dgglwlrq wkh xqhpsor|phqw udwh ri vnloohg zrunhuv ghfuhdvhv zlwkR1 Wkh hpsor|phqw udwhv doo ghfuhdvh zlwk wkh vhsdudwlrq

:

(10)

udwh/ ^1 Lq dgglwlrq/ wkh hpsor|phqw udwh ri vnloohg zrunhuv lq vhfwru lqfuhdvhv zlwk wkh wudqvlwlrq udwhR dqg wkh hpsor|phqw udwh ri xqvnloohg zrunhuv lq vhfwru 2 lqfuhdvhv zlwk wkh wudqvlwlrq udwh R21 Wkh hpsor|phqw udwh ri vnloohg zrunhuv hpsor|hg lq vhfwru2lqfuhdvhv zlwk wkh wudqvlwlrq udwhR₂ dqg ghfuhdvhv zlwk wkh wudqvlwlrq udwh R1

Wkh hpsor|phqw survshfwv ri vnloohg zrunhuv duh ehwwhu wkdq iru xqvnloohg zrunhuv/ vhh htxdwlrq +5148, dqg +514:,=

L

\ : L7

\

Ixuwkhupruh/ wkh udwh ri vnloohg zrunhuv hpsor|hg lq vnloohg mrev lv kljkhu wkdq wkh udwh ri xqvnloohg zrunhuv hpsor|hg lq xqvnloohg mrev li wkh wudqvlwlrq udwh lqwr mrev lq vhfwru lv kljkhu wkdq wkh wudqvlwlrq udwh lqwr mrev lq vhfwru 2=

R : R2 , .₇

\ : .²

\ +514;,

Zh zloo rqo| frqvlghu sdudphwhu ydoxhv iru zklfk wkh hpsor|phqw udwh ri vnloohg zrunhuv lq vnloohg mrev lv kljkhu wkdq wkh hpsor|phqw udwh ri xqvnloohg zrunhuv lq xqvnloohg mrev1 Wklv lv ghqlwho| wkh prvw uhdolvwlf vlwxdwlrq1 Jurhv dqg Krop +4<<<, qg wkdw wkh xqhpsor|phqw udwh iru xqvnloohg zrunhuv lq Ghqpdun kdv ehhq deryh dyhudjh/ zkhuhdv wkh xqhpsor|phqw ri vnloohg zrunhuv kdv ehhq ehorz dyhudjh/ wkurxjkrxw wkh shulrg 4<;304<<;1 Ixuwkhupruh/ wkh sklorvrsk|

ehklqg wkh \XS lv wr lqfuhdvh wkh hpsor|phqw survshfwv ri |rxqj shrsoh e|

prwlydwlqj wkhp wr xqghuwdnh dq hgxfdwlrq1 Li wkh lqhtxdolw| lq +514;, lv uhyhuvhg wkh \XS grhv qrw pdnh dq| vhqvh1

Wkh qxpehu ri pdwfkhv iruphg lq vhfwru/ uhvshfwlyho|/ vhfwru 2/ lv jlyhq e|

wkh pdwfklqj ixqfwlrqv=

%ETc s7 's Ts

s7/ ' c2c +514<, zkhuh T lv wkh qxpehu ri ydfdqflhv lq vhfwru / dqg s7 lv wkh wrwdo qxpehu ri zrunhuv vhdufklqj iru d mre lq vhfwru phdvxuhg lq h!flhqf| whupv1 s lv d phdvxuh ri wkh zrunhuv vhdufk h!flhqf|1 Qrwh wkdw wkh qxpehu ri pdwfkhv kdv srvlwlyh uvw rughu ghulydwlyhv lq T dqg s7/ qhjdwlyh vhfrqg rughu ghulydwlyhv/

srvlwlyh furvv sduwldo ghulydwlyhv dqg lv krprjhqrxv ri ghjuhh rqh lqT dqgs71 Slvvdulghv ;9 dqg Eodqfkdug dqg Gldprqg ;< surylgh hpslulfdo mxvwlfdwlrq iru wkh Free0Grxjodv pdwfklqj ixqfwlrq zlwk htxdo h{srqhqwv1

Wkh zrunhuv* wudqvlwlrq udwh/ R/ ' c2c lv htxdo wr wkh qxpehu ri pdwfkhv glylghg e| wkh qxpehu ri vhdufkhuv1 Wkh upv* wudqvlwlrq udwh lv htxdo wr wkh qxpehu ri pdwfkhv glylghg e| wkh qxpehu ri ydfdqflhv1 Zh phdvxuh wkh wudq0 vlwlrq udwhv lq whupv ri oderxu pdunhw wljkwqhvv/ w1 Oderxu pdunhw wljkwqhvv lv d phdvxuh ri krz wljkw wkh pdunhw lv/ dqg lv jlyhq e| ydfdqflhv uhodwlyh wr wkh qxpehu ri vhdufkhuv lq h!flhqf| whupv1 Wkh wudqvlwlrq udwhv ehfrph=

(11)

R 'ss

wc b 's w₃

/ w ' T

s7c ' c2 +5153, Kdylqj ghwhuplqhg wkh htxloleulxp xqhpsor|phqw dqg hpsor|phqw udwhv dqg kdylqj irxqg h{suhvvlrqv iru wkh wudqvlwlrq udwhv/ zh fdq uhzulwh wkh htxloleulxp frqglwlrqv iru wkh wudqvlwlrq udwhv/ htxdwlrq +5144, dqg +5145,1 E| xvh ri htxdwlrq +5146,0+5149, zh fdq h{suhvv wkh suredelolw| wkdw d zrunhu dsso|lqj iru d mre lq wkh xqvnloohg vhfwru lv xqvnloohg/ bc lq whupv riRc ' c2=

b ' L

L nL7 '

n _3\^\ _R^R_nR²^{n^}₂_n^ +5154, E| xvh ri +5153, zh jhw=

b ' s

R/ ' c2 +5155,

Khqfh htxdwlrq +5144, dqg +5145,/ jlylqj wkh htxloleulxp frqglwlrqv iru wkh wudqvlwlrq udwhv R dqgR₂ uhvshfwlyho|/ fdq eh uhzulwwhq wr=

x ' &n s R

+_r on^

' fc +5156,

[ ' &n s R2

#_@+32

on^? ^@+q3_on^nR^r²

n _3\^\ _R^R_nR²^{n^}₂_n^ n @+q _r² on^nR

$

' f +5157,

5171 Zdjh ghwhuplqdwlrq

Zh dvvxph wkdw zkhq d up dqg d zrunhu phhw/ wkh| edujdlq ryhu wkh zdjh zlwk dq htxdo edujdlqlqj srzhu1 Zdjhv wkhq vsolw wkh pdwfk vxusoxv ehwzhhq wkh up dqg wkh zrunhu htxdoo|1 Wkh upv kdyh wkh rswlrq ri klulqj wkh zrunhu zlwk dq h{shfwhg uhwxuq ri uhvshfwlyho|Kâ/Kâ₂E_r² dqgKâ₂E²_?/ ru qrw klulqj wkh zrunhu dqg srvh wkh ydfdqf| djdlq zlwk h{shfwhg uhwxuqv ri K^T dqg K^T₂/ uhvshfwlyho|1 Wkh zrunhu*v h{shfwhg uhwxuq iurp dffhswdqfh lv jlyhq e| wkh h{shfwhg uhwxuqv iurp kroglqj d mre/ K^.₇/ K^.₇² dqgK^.²1 Wkh uhwxuq iurp qrw dffhswlqj wkh mre lv iru xqhpsor|hg vnloohg dqg xqvnloohg zrunhuv jlyhq e|K^L₇ uhvshfwlyho|K^L1 Vnloohg zrunhuv hpsor|hg lq vhfwru2zloo jhqhudoo| kdyh wkh rswlrq ri uhwxuqlqj wr wkhuh fxuuhqw mrev/ ehiruh djuhhlqj wr irup d qhz pdwfk1 Krzhyhu/ wklv rswlrq zloo qrw eh dydlodeoh rqfh wkh qhz mre lv dffhswhg/ khqfh wkh zrunhu*v rqo| dowhuqdwlyh diwhu wkh qhz pdwfk lv iruphg lv xqhpsor|hg vhdufk1 Li zh dvvxph wkdw zdjhv fdq eh uhqhjrwldwhg dw dq| wlph/ wkh h{shfwhg qhw vxusoxv iurp dffhswlqj wkh mre lv wkh ydoxh ri kroglqj wkh mre plqxv wkh ydoxh ri ehlqj xqhpsor|hg= K^.₇K^L₇1⁸

8Wklv dvvxpswlrq lv pdgh lq Slvvdulghv +4<<7,/ lpso|lqj wkdw wkh zdjh ri d vnloohg zrunhu grhv qrw ghshqg rq zkhwkhu kh lv xqhpsor|hg ru lv hpsor|hg lq vhfwru51

<

(12)

Khqfh/ xvlqj wkh htxloleulxp frqglwlrqvK^T ' fdqg K^T₂ ' f/ wkh zdjhv/ _r/ ²_r/ _?²/ duh ghwhuplqhg e| wkh htxdwlrqv +vhh Slvvdulghv +<3,,=

K^a

K^.₇K^L₇

' fc K^a₂E_r²

K^.₇²K^L₇

' fc K^a₂E_?²

K^.²K^L

' f

Xvlqj htxdwlrq +514,0+518, dqg htxdwlrq +519,0+5143, wkh uvw rughu frqglwlrqv fdq eh uhzulwwhq=

+_r

on^ _rKR2 ²_r3K onRnR2n^

on^nR ' fc +5158,

@+q²_r

on^nR ²_rK

onRnR2 n^ ' fc +5159,

@+²_?

on^ _?² K

on^nR2 ' f +515:,

Wkh zdjhv pd| eh zulwwhq dv=

_r ' %+n E%Kn%E@q+Kc _r² ' %f@q+n E%fK/

_?² ' %@+n E%K/

zkhuh % ' _R^{on^nR}_n2Eon^ c %f ' _R₂^{on^nR}_n2Eon^nR^nR²c % ' _R₂^{on^nR}_n2Eon^² c % ' E%_R₂_n2Eon^nR^R² 1 Lw lv vkrzq lq Dsshqgl{ wkdw wkh zdjh rughulqj lv dv iroorzv1 Lq jhqhudo zh kdyh wkdw_r : _r² dqg_r : _?²1 Ixuwkhupruh/

_?² W_r² iruq Vqc zkhuhq lv ghqhg e| wkh htxdwlrq=

@q+K

@+K ' ER₂ non^ ER₂n 2 Eon^nR

ER2 n 2 Eon^ ERnR2non^c +515;, dqgq :1

Wkh zdjh udwh lq vhfwru lv kljkhu wkdq zdjhv lq vhfwru 2 dv wkh pdujlqdo surgxfwlylw| lv wkh kljkhvw lq vhfwru 1⁹ Frqfhuqlqj wkh uhodwlyh zdjhv ri wkh vnloohg dqg xqvnloohg zrunhuv hpsor|hg lq vhfwru 2 zh kdyh wzr glhuhqw fdvhv/

ghshqghqw xsrq wkhlu uhodwlyh surgxfwlylwlhv/ l1h1 ghshqglqj rqq1 Frqvlghu uvw

(13)

wkh fdvh zkhuhq q1 Wkh zdjh ri dq xqvnloohg zrunhu lv kljkhu wkdq wkh zdjh ri d vnloohg zrunhu hpsor|hg lq vhfwru2li wkh surgxfwlylw| ri wkh vnloohg zrunhuv hpsor|hg lq vhfwru2 lv orzhu wkdq ru qrw wrr pxfk kljkhu wkdq wkh surgxfwlylw|

ri dq xqvnloohg zrunhu1 Wkh lqwxlwlrq lv dv iroorzv1 D vnloohg zrunhu*v h{shfwhg uhwxuq ri qrw dffhswlqj d mre lq vhfwru 2 lv kljkhu wkdq wkh xqvnloohg zrunhu*v h{shfwhg uhwxuq/K^L₇ :K^L/ lpso|lqj d kljkhu zdjh ghpdqg ri wkh vnloohg zrunhuv1 Rq wkh rwkhu kdqg/ dv vnloohg zrunhuv vhdufk iru d mre lq vhfwruzkloh hpsor|hg lq vhfwru2/ wkh up kdv wr eh frpshqvdwhg iru wkh kljkhu ulvn ri ehlqj vhsdudwhg iurp wkh zrunhu1 Wkh odvw hhfw grplqdwhv/ lpso|lqj xqvnloohg zrunhuv uhfhlyh d kljkhu zdjh wkdq vnloohg zrunhuv zkhq hpsor|hg lq vhfwru 21

Frqvlghu qrz wkh fdvh/ q : q/ wkdw lv/ wkh surgxfwlylw| ri d vnloohg zrunhu hpsor|hg lq vhfwru2lv pxfk odujhu wkdq wkh surgxfwlylw| ri dq xqvnloohg zrunhu1 Wklv frqglwlrq lv pruh olnho| wr eh vdwlvhg/ wkh kljkhu wkh ohyho ri xqhpsor|phqw lqvxudqfh1 Lq wklv fdvh/ wkh uhodwlyho| kljkhu surgxfwlylw| ri wkh vnloohg zrunhu pruh wkdq frpshqvdwhv iru wkh kljkhu ulvn ri vhsdudwlrq1 Khqfh/ wkh vnloohg zrunhu hpsor|hg lq vhfwru2 uhfhlyhv d kljkhu zdjh wkdq wkh xqvnloohg zrunhu1

61 Htxloleulxp

Wkh htxloleulxp lv irxqg e| lqvhuwlqj wkh htxloleulxp zdjhv/ htxdwlrq +5158,0 +515:,/ lqwr htxdwlrq +5156, dqg +5157,/ jlylqj wkh htxloleulxp wudqvlwlrq udwhv/

R dv d ixqfwlrq ri R2/ RER2/ dqg R2 dv d ixqfwlrq ri R/ R2ER/ uhvshfwlyho|1 Rqo| wkh htxloleulxp wudqvlwlrq udwh/ R2/ gluhfwo| ghshqgv xsrq wkh qxpehu ri vnloohg zrunhuv/ \1 Krz wklv wudqvlwlrq udwh lv dhfwhg ghshqgv rq wkh uhodwlyh surgxfwlylwlhv ri wkh zrunhuv hpsor|hg lq vhfwru 5 uhodwlyho| wr wkh vhsdudwlrq udwh frqfhuqlqj d vshflf vhfwru 2 zrunhu sdlu1 Lq jhqhudo/ wkh lpsdfw rq wkh wudqvlwlrq udwh idflqj zrunhuv dsso|lqj iru mrev lq vhfwru 2 ghshqgv rq wkh glhuhqfh ehwzhhq wkh up0ydoxhv dwwdfkhg wr wkh wzr w|shv ri zrunhuv1 Wkh ydoxh dwwdfkhg wr wkh vnloohg zrunhu hpsor|hg lq vhfwru 2 ghshqgv srvlwlyho|

rq wkh zrunhu*v surgxfwlylw| dqg qhjdwlyho| rq wkh zrunhu*v wudqvlwlrq udwh lqwr hpsor|phqw lq vhfwru/R=

K^a₂ ²_?

K^a₂ ²_r

' @+²_?

on^ @q+_r² on^nR

Wkh kljkhu wklv wudqvlwlrq udwh/ R/ lv/ wkh vkruwhu wkh pdwfk/ dqg khqfh/ wkh orzhu wkh h{shfwhg uhwxuq iru wkh vhfwru 2up zkhq hpsor|lqj d vnloohg zrunhu1

Wkh htxdwlrq=

@+²_?

on^ @q+²_r

on^nR ' @+K

R2n 2 Eon^ @q+K

R2n 2 Eon^nR ' fc ghqhvq^W/ zkhuh q^W :EVhh Dsshqgl{ ,1

44

(14)

Li wkh surgxfwlylw| ri wkh vnloohg zrunhu zkloh hpsor|hg lq vhfwru lv qrw wrr pxfk kljkhu wkdq wkh surgxfwlylw| ri wkh xqvnloohg zrunhu/q q^Wczh kdyh wkdw zkhq wkh qxpehu ri vnloohg zrunhuv lqfuhdvhv/\lqfuhdvhv/ wkh wudqvlwlrq udwh lqwr hpsor|phqw lq vhfwru2ghfuhdvhv/ vhh Dsshqgl{ =

YR₂

Y\m_{q q}^W 'Y[*Y\ Y[*YR2 f

Zkhq wkh uhodwlyh qxpehu ri vnloohg zrunhuv lqfuhdvhv/ wkh suredelolw| wkdw d zrunhu dsso|lqj iru d mre lq vhfwru 2 lv vnloohg/ lqfuhdvhv wrr1 Wkh ydoxh ri oolqj d mre zlwk d vnloohg zrunhu lv ohvv wkdq wkh ydoxh ri oolqj wkh mre zlwk dq xqvnloohg zrunhu lq vhfwru 2 lpso|lqj/ wkdw ihzhu ydfdqflhv duh vxssolhg lq wkh xqvnloohg vhfwru dqg frqvhtxhqwo| wkh wudqvlwlrq udwh R2 ghfuhdvhv1 Khqfh/

zkhq wkh qxpehu ri vnloohg zrunhuv lqfuhdvhv uhodwlyh wr wkh qxpehu ri xqvnloohg zrunhuv/ wkh fkdqfh ri jhwwlqj d mre lq wkh xqvnloohg vhfwru ghfuhdvhv1

Frqvlghu qrz wkh fdvh zkhuh q : q^W1 Zh vkrz lq Dsshqgl{ wkdw wkh wudqvlwlrq udwh lqwr hpsor|phqw lqwr vhfwru2c R2 lv lqfuhdvlqj zlwk \ li q^W q G

YR₂

Y\mq:q^W 'Y[*Y\ Y[*YR2 :fc

Frqfhuqlqj wkh wudqvlwlrq udwh R/ lw lv vkrzq lq Dsshqgl{ / wkdw R dv d ixqfwlrq riR₂ lv qhjdwlyho| vorshg=

YR

YR2 mRER₂ 'Yx*YR₂ Yx*YR f

Zh fdq jr rqh vwhs ixuwkhu dqg vkrz wkdw lq jhqhudo wkh vorsh olhv ehwzhhq }hur dqg plqxv rqh=

YR

YR2 mRER2 f ;q :f +614,

Wkh h{lvwhqfh ri d vwdeoh htxloleulxp uhtxluhv wkdw= ^_R_{_R}² m R₂ER : 1 Lq dsshqgl{ zh vkrz wkdw=

_R2

_R mR2ER: l q qc

zkhuhq : q^W1 H{lvwhqfh uhtxluhv wkdwq fdq qrw eh wrr odujh1 Frqvhtxhqwo|/ zh uhvwulfwq wr eh ohvv wkdq 4?

@cq

Deryh zh kdyh vkrzq wkdw zkhq wkh wudqvlwlrq udwh lqwr hpsor|phqw lq vhf0 wru2 ghfuhdvhv +lqfuhdvhv,/ wkh wudqvlwlrq lqwr hpsor|phqw lq vhfwru lqfuhdvhv +ghfuhdvhv,1 Wkh wudqvlwlrq udwh/R/ lv qrw gluhfwo| ghshqghqw xsrqR21 Wkh hhfw zrunv wkurxjk wkh zdjh ri zrunhuv hpsor|hg lq wkh vnloohg vhfwru/ _r/ vhh htxd0 wlrq +5156, dqg +5158,1 Li wkh ydoxh ri xqhpsor|phqw lqfuhdvhv gxh wr d kljkhu

(15)

wudqvlwlrq udwh lqwr hpsor|phqw lq wkh xqvnloohg vhfwru/R₂/ wklv whqgv wr lqfuhdvh wkh zdjh ri vnloohg zrunhuv hpsor|hg lq wkh vnloohg vhfwru +lw lv vkrzq ehorz wkdw

Y_r

YR2 : f,1 Wkh kljkhu zdjh lpsolhv wkdw wkh ydoxh ri kdylqj d oohg mre lq wkh vnloohg vhfwru ghfuhdvhv/ fruuhvsrqglqj wr wkdw ohvv ydfdqflhv duh vxssolhg/ dqg frqvhtxhqwo| wkh wudqvlwlrq udwhR ghfuhdvhv1

Wkh htxloleulxp wudqvlwlrq udwhv/R dqgR2ciru\htxdo wrfDuhvshfwlyho|fH dqgq q^W duh looxvwudwhg lq Iljxuh 51

p₂

p1

p₁(p₂)

p2(p1, =0,5)

p2(p1, =0,8)

45^%

Iljxuh 5= Htxloleulxp Wudqvlwlrq Udwhv1

Wkh qhjdwlyho| vorshg vwhhs olqh uhsuhvhqwv R dv d ixqfwlrq ri R2/ jlyhq lq htxdwlrq +5156,/ dqg wkh wrs olqh uhvshfwlyho| wkh erwwrp olqh/ uhsuhvhqwR₂ dv d ixqfwlrq ri Rc jlyhq lq htxdwlrq +5157,/ zkhq \ ' fD uhvshfwlyho| \ ' fH dqg q' 1

Dv vkrzq deryh/ zkhq \ lqfuhdvhv/ wkh fxuyh jlylqj R₂ dv d ixqfwlrq ri R pryhv grzqzdug iru q q^W/ lpso|lqj wkdw R2 ghfuhdvhv/ zkhuhdv R lqfuhdvhv1 Iru q ' q^W wkh fxuyh grhv qrw pryh dqg iru q : q^W wkh R2ER fxuyh pryhv xszdugv zkhqhyhu \ lqfuhdvhv1 Lq wklv fdvh/ R₂ lqfuhdvhv fruuhvsrqglqj wr d orzhu R1

71 Hhfwv ri \XS

Zh dqdo|vh wkh lpsdfw ri \XS e| frqvlghulqj dq lqfuhdvh lq wkh qxpehu ri vnloohg zrunhuv/ nhhslqj frqvwdqw wkh wrwdo qxpehu ri zrunhuv/ l1h1/ zh frqvlghu wkh hhfwv ri lqfuhdvlqj wkh h{rjhqrxv sdudphwhu \1 Lq grlqj vr zh lpsolflwo| dvvxph wkh ehvw ri doo zruogv/ qdpho| wkdw wkh \XS lv d frpsohwh vxffhvv1 Zh h{dplqh wkh lpsdfw rq zdjhv/ zdjh glvshuvlrq/ xqhpsor|phqw dqg zhoiduh1

46

(16)

7141 Zdjhv

Zkhq wkh qxpehu ri vnloohg zrunhuv lqfuhdvhv/ wkh zrunhu*v wudqvlwlrq udwhv duh dhfwhg/ dv ghvfulehg deryh lq wkh htxloleulxp vhfwlrq1 Wkh lpsdfw rq zdjhv ghshqg rq wkh uhodwlyh surgxfwlylwlhv ri zrunhu hpsor|hg lq vhfwru 21 Lq wkh fdvh zkhuh wkh surgxfwlylw| ri d vnloohg zrunhuv hpsor|hg lq vhfwru2 lv qrw wrr pxfk kljkhu wkdq wkh surgxfwlylw| ri dq xqvnloohg zrunhu/ q q^Wc zh kdyh wkh iroorzlqj1 Wkh wudqvlwlrq udwh iru zrunhuv vhdufklqj iru d mre lq vhfwru2/R2/ zloo ghfuhdvh dqg wkh wudqvlwlrq udwh iru zrunhuv vhdufklqj iru d mre lq vhfwru 4/ R/ lqfuhdvhv1 Dv vkrzq lq Dsshqgl{ wkh ghfuhdvh lq R₂ fruuhvsrqgv wr d ghfuhdvh lq doo zdjhv/ zkhuhdv wkh lqfuhdvh lq R kdv qr lpsdfw rq zdjhv uhfhlyhg e|

xqvnloohg zrunhuv/ whqg wr ghfuhdvh zdjhv iru vnloohg zrunhuv hpsor|hg lq vhfwru 5 dqg whqg wr lqfuhdvh zdjhv iru vnloohg zrunhuv hpsor|hg lq vhfwru 41 Zkhqhyhu q : q^W wkh uhvxowv duh wkh rssrvlwh1 Wkh wrwdo lpsdfw rq zdjhv uhvxowlqj iurp dq lqfuhdvh lq\ lv jlyhq e| wkh iroorzlqj sursrvlwlrq=

Sursrvlwlrq 7141 Zkhq wkh iudfwlrq ri vnloohg zrunhuv lqfuhdvhv/ \ lqfuhdvhv/

zh kdyh iru q q^W +q : q^W, wkdw vhfwru 5 zdjhv ghfuhdvh +lqfuhdvh,/ ^__{_\}²^? f +:3,/^__{_\}²^r f +:3, zkhuhdv wkh hhfw rq wkh vhfwru 4 zdjh udwh lv dpeljxrxv/

__r _\ Vf1

Surri1 Zdjhv fkdqjh zlwk \ dffruglqj wrr=

__?² _\ ' _%

_R₂ YR2

Y\ E@+Kc __r²

_\ ' Y%f

YR₂ nY%f

YR YR

YR₂ YR2

Y\ E@q+Kc __r

_\ '

E+KY% YR

YR

YR2 n E@q+K Y%

YR

YR2 n Y%

YR2

YR₂ Y\

Wkh vljqv ri ^__{_\}^?² dqg ^__{_\}^r² duh htxdo wr wkh vljq ri ^YR_Y\²/ iru zklfk zh ghulyhg frqglwlrqv lq wkh suhylrxv vhfwlrq1 Dv ^_%_{_R}₂ : fc ^_%_{_R} : f dqg ^_%_{_R} fc wkh vljq rq wkh wklug ghulydwlyh lv lq jhqhudo dpeljxrxv1 Lq Dsshqgl{ lw lv vkrzq wkdw

Y%

YR n ^Y%_YR : f1 Krzhyhu/ wkh ghfuhdvh lq R₂ kdv d gluhfw qhjdwlyh lpsdfw rq _r1 Dv ^YR_YR₂ m RER2 f/ zh fdqqrw ghwhuplqh zklfk hhfw zloo grplqdwh1

Frqvlghu wkh fdvh zkhq q q^W1 Wkh vkliw ri zrunhuv iurp wkh xqvnloohg wr wkh vnloohg oderxu irufh lv dvvrfldwhg zlwk zdjh ghfuhdvhv iru zrunhuv hpsor|hg lq vhfwru 21 Wklv lv vr dv wkh wudqvlwlrq udwh iru zrunhuv lqwr vhfwru 2 ghfuhdvhv/

ghfuhdvlqj wkhlu edujdlqlqj srzhu1 Wkh wudqvlwlrq udwh idflqj vnloohg zrunhuv frqfhuqlqj wkhlu mre vhdufk lqwr vhfwru lqfuhdvhv/ whqglqj wr lqfuhdvh vnloohg zrunhuv* edujdlqlqj srzhu1 Wkh lpsdfw rq vhfwru 4 zdjhv lv wkhuhiruh dpeljxrxv1 Frqvlghu qh{w wkh fdvh zkhuhq : q^W1 Wkh wudqvlwlrq udwh lqwr hpsor|phqw lq vhfwru 2 lqfuhdvhv fruuhvsrqglqj wr zdjh ghfuhdvhv iru wkh zrunhuv hpsor|hg lq

(17)

wklv vhfwru1 V|pphwulfdoo| zh kdyh d ghfuhdvh lq wkh wudqvlwlrq udwh lqwr vhfwru

/ djdlq lqgxflqj dq dpeljxrxv lpsdfw rq vhfwru 4 zdjhv1 7151 Zdjh glvshuvlrq

Lq wkh suhylrxv vhfwlrq zh kdyh vkrzq wkdw vhfwru 5 zdjhv ghfuhdvh +lqfuhdvh, iru q q^W +q : q^W, dqg wkh lpsdfw rq vhfwru 4 zdjhv lv dpeljxrxv/ iroorzlqj dq lqfuhdvh lq wkh iudfwlrq ri vnloohg zrunhuv1 Lq wklv vhfwlrq zh hydoxdwh zkdw kdsshqv wr zdjh glvshuvlrq/ erwk ehwzhhq xqvnloohg dqg vnloohg zrunhuv dqg lq0 ehwzhhq vhfwru 4 dqg vhfwru 5 zrunhuv1 Zdjh glvshuvlrq ehwzhhq xqvnloohg dqg vnloohg zrunhuv hpsor|hg lq vhfwru 5 lv ghqhg dv

` (2 '_?²_r² Wfiru q Vqc zkhuhq : lv ghqhg lq htxdwlrq +515;,1

Zdjh glvshuvlrq ehwzhhq vnloohg zrunhuv hpsor|hg lq vhfwru dqg xqvnloohg zrunhuv lv srvlwlyh=

` (_r?'_r²_?:f

Ilqdoo|/ zdjh glvshuvlrq ehwzhhq vnloohg zrunhuv hpsor|hg lq vhfwru dqg2 lv srvlwlyh=

` (_rr '_r_r² :f

zdjh glvshuvlrq ehwzhhq vnloohg zrunhuv lq vhfwru 4 dqg xqvnloohg zrunhuv lq0 fuhdvhv +ghfuhdvhv, iru q q^W +q^W q q,7 ^{Y` (}_Y\^r? : f _{Y` (}

Y\r? f

/ zdjh glvshuvlrq lq0ehwzhhq vnloohg zrunhuv lqfuhdvhv +ghfuhdvhv, iru q q^W +q^W q,

Y` (rr

Y\ :f _{Y` (}_rr

Y\ f

c dqg wkh vljq ri wkh fkdqjh lq zdjh glvshuvlrq ehwzhhq vhfwru 5 zrunhuv lv lqghwhuplqdwh1

Surri1 Glhuhqwldwlqj zdjh glvshuvlrq zlwk uhvshfw wr\ zh rewdlq=

Y`(₂

Y\ ' __?²

_\ _²_r _\ c Y`(r?

Y\ ' __r

_\ __?² _\ c Y`(rr

Y\ ' __r

_\ __r² _\

Vxevwlwxwlqj iru wkh ghulydwlyhv jlyhv=

Y`(2

Y\ '

_%

_R2 E@+K _²_r

_R2 _²_r _R

YR

YR2

Y\c Y`(r?

Y\ '

__r _R

YR

YR₂ n E@q+KY%

YR₂ _%

_R₂E@+K YR2

Y\c Y`(_rr

Y\ '

__r _R

YR

YR2 n E@q+KY%

YR2 __r²

_R2 _²_r _R

YR YR2

YR₂ Y\

48

(18)

Xvlqj@q c @ c dqg ^YR_YR₂ flw lv vkrzq lq Dsshqgl{ ( wkdw wkh vljq ri wkh vhfrqg ghulydwlyh lv srvlwlyh iru q q^W dqg qhjdwlyh iru q^W q qc7 zkhuh q7 lv ghwhuplqhg e| wkh htxdwlrq=

@q+K

@+K ' ERn 2 Eon^ ER2n 2 Eon^nR² 2 Eon^nR ER2n 2 Eon^² c

dqg wkh vljq ri wkh odvw ghulydwlyh lv srvlwlyh iru q q^W dqg qhjdwlyh iru q : q^W1

Iru q q^W zh wkhq kdyh wkh iroorzlqj1 Erwk zdjhv ri vhfwru 5 ghfuhdvh dqg zh fdqqrw ghwhuplqh zklfk ri wkh wzr zdjhv ghfuhdvhv wkh prvw1 Zdjh glvshuvlrq lq0ehwzhhq vnloohg zrunhuv hpsor|hg lq vhfwru 4 dqg xqvnloohg zrunhuv zloo lqfuhdvh1 Khqfh/ hyhq li wkh zdjh udwh iru vnloohg vhfwru 4 zrunhuv vkrxog ghfuhdvh/ lw ghfuhdvhv ohvv wkdq wkh zdjh udwh uhfhlyhg e| xqvnloohg zrunhuv1

Zdjh glvshuvlrq lq0ehwzhhq vnloohg zrunhuv hpsor|hg lq wkh wzr vhfwruv lq0 fuhdvhv iru q q^W1 Wkh vkliw ri zrunhuv iurp wkh xqvnloohg wr wkh vnloohg oderxu irufh lqgxfhv d pruh xqhtxdo zdjh glvwulexwlrq1 Wkh lqwxlwlrq lv khuh wkdw erwk vnloohg zrunhuv qhjrwldwlqj zdjhv lq vhfwru 4 dqg vhfwru 5 duh htxdoo| dhfwhg e|

wkh idoo lq wkh wudqvlwlrq udwh lqwr hpsor|phqw lq vhfwru 5/ zkhuhdv wkh kljkhu wudqvlwlrq udwh lqwr vhfwru 4 hpsor|phqw whqgv wr lqfuhdvh vhfwru 4 zdjhv1 Iru q : q^W wkh wudqvlwlrq udwh lqwr vhfwru 5 lqfuhdvhv zlwk \1 Lq wklv fdvh/ zdjh glvshuvlrq lq0ehwzhhq vnloohg zrunhuv hpsor|hg lq vhfwru 4 dqg xqvnloohg zrunhuv zloo ghfuhdvh li q q7 dqg zdjh glvshuvlrq lq0ehwzhhq vnloohg zrunhuv hpsor|hg lq wkh wzr vhfwruv ghfuhdvhv1

7161 Xqhpsor|phqw

Frqvlghulqj xqhpsor|phqw/ zh uvw frqvlghu wkh lpsdfw rq wkh xqhpsor|phqw udwhv iru vnloohg dqg xqvnloohg zrunhuv zkhq lqfuhdvlqj wkh qxpehu ri vnloohg zrunhuv uhodwlyh wr wkh qxpehu ri xqvnloohg zrunhuv1

xqhpsor|phqw udwhv iru vnloohg dqg xqvnloohg zrunhuv lqfuhdvh +ghfuhdvh,/ ^_EL_{_\}⁷^*\ : f+ f,c^_EL^*E3\_{_\} :f + f, iruq q^WcEq : q^W1

Surri1 Glhuhqwldwlqj wkh wzr xqhpsor|phqw udwhv zlwk uhvshfw wr\zh rewdlq=

_EL7*\

_\ ' L7

_R

_R2 mRER2 n ERnR2n^

_R2

_\c +714,

_EL*E\

_\ ' L

ER₂n^ _R2

_\c +715,

zkhuh zh vkrzhg deryh +vhh htxdwlrq +614,, wkdw=

YR

YR2 mRER2 f, YR

YR2 mRER2 n :fc ;q :f

(19)

Wkh qhw hhfw rq erwk xqhpsor|phqw udwhv ri lqfuhdvlqj wkh qxpehu ri vnloohg zrunhuv lv srvlwlyh iru q q^W1 Wkh xqhpsor|phqw udwh ri xqvnloohg zrunhuv lq0 fuhdvhv ehfdxvh wkh ydoxh ri hpsor|lqj d mre zlwk d vnloohg zrunhu lv ohvv wkdq wkh ydoxh ri oolqj wkh mre zlwk dq xqvnloohg zrunhu1 Zkhq wkh qxpehu ri vnloohg zrunhuv lqfuhdvhv uhodwlyh wr wkh qxpehu ri xqvnloohg zrunhuv wkh suredelolw| ri hpsor|lqj d vnloohg zrunhu lq vhfwru 2 lqfuhdvhv1 Wklv lpsolhv wkdw ihzhu ydfdq0 flhv duh vxssolhg lq wkh xqvnloohg vhfwru dqg wkh xqhpsor|phqw udwh frqvhtxhqwo|

Frqvlghu wkh fdvh zkhuh q q^W1 Frqfhuqlqj wkh xqhpsor|phqw udwh ri vnloohg zrunhuv/ wkhuh lv d srvlwlyh dv zhoo dv d qhjdwlyh hhfw1 Wkh srvlwlyh hhfw lv gluhfwo| gxh wr d idoo lq wkh qxpehu ri ydfdqflhv vxssolhg lq wkh xqvnloohg vhfwru1 Wkh qhjdwlyh hhfw dulvhv ehfdxvh wkh wudqvlwlrq udwh lqwr hpsor|phqw lq wkh vnloohg vhfwru/R/ lqfuhdvhv1 Wkh wudqvlwlrq udwh/R/ lv qrw gluhfwo| ghshqghqw xsrq R21 Wkh hhfw zrunv wkurxjk wkh zdjh ri vnloohg zrunhuv hpsor|hg lq wkh vnloohg vhfwru1 Dv wkh ydoxh ri xqhpsor|phqw ghfuhdvhv gxh wr wkh orzhu wudqvlwlrq udwh lqwr hpsor|phqw lq wkh xqvnloohg vhfwru/ wklv whqgv wr ghfuhdvh wkh zdjh ri vnloohg zrunhuv hpsor|hg lq wkh vnloohg vhfwru +f1l1i1 sursrvlwlrq 715,1 Wkh orzhu zdjh lpsolhv wkdw wkh ydoxh ri kdylqj d oohg mre lq wkh vnloohg vhfwru lqfuhdvhv/

lpso|lqj wkdw pruh ydfdqflhv duh vxssolhg1 Wkh qhw hhfw rq wkh xqhpsor|phqw udwh lv xqdpeljxrxvo| srvlwlyh1 Zkhq q : q^W/ wkh uhvxow lv wkh uhyhuvh= kljkhu R2 dqg orzhu R uhvxowlqj lq orzhu xqhpsor|phqw iru vnloohg zrunhuv/ L71

Frqvhtxhqwo|/ wkh xqhpsor|phqw udwhv ri erwk vnloohg dqg xqvnloohg zrunhuv lqfuhdvh iruq q^W/ duh xqdhfwhg iruq 'q^W dqg ghfuhdvh iru q : q^W dv d uhvxow ri wkh odujhu iudfwlrq ri vnloohg zrunhuv1

Ixuwkhupruh/ lw fdq eh vkrzq wkdw wkh odujhu iudfwlrq ri vnloohg zrunhuv kdupv wkh xqvnloohg zrunhuv wkh prvw zkhq q q^W/ dv wkh lqfuhdvh lq wkh xqhpsor|0 phqw udwh lv kljkhu iru xqvnloohg zrunhuv wkdq iru vnloohg zrunhuv1 Iru q : q^W/ wkh ghfuhdvh lq xqvnloohg xqhpsor|phqw lv odujhu wkdq wkh ghfuhdvh lq vnloohg xqhpsor|phqw=

_EL7*\

_\ _EL*E\ _\

' ^_R2

_\

# E^nR2²

_R

_R2 mRER2 n E^nRnR2²

$ f ' f:f sJo

;?

=

q q^W q 'q^W q : q^W Zh wkhuhiruh kdyh wkh uhvxow=

Sursrvlwlrq 7171 D odujhu iudfwlrq ri vnloohg zrunhuv lqgxfhv d pruh xqhtxdo xqhpsor|phqw glvwulexwlrq iru q q^W/ kdv qr lpsdfw iru q ' q^W dqg d pruh htxdo xqhpsor|phqw glvwulexwlrq iruq : q^W

4:

(20)

Dqrwkhu lqwhuhvwlqj wklqj wr frqvlghu lv wkh qxpehu ri vnloohg zrunhuv hp0 sor|hg lq vhfwru 51 Wkh kljkhu xqhpsor|phqw udwh ri vnloohg zrunhuv zloo whqg wr lqfuhdvh wkh udwh ri vnloohg zrunhuv hpsor|hg lq wkh xqvnloohg vhfwru dqg wkh orzhu ydfdqf| vxsso| lq vhfwru 2 zloo whqg wr ghfuhdvh wklv udwh1 Wkh qhw hhfw fdq eh vkrzq wr eh qhjdwlyh1 Zh rewdlq wkh uhvxow1

Sursrvlwlrq 7181 Wkh udwh ri vnloohg zrunhuv hpsor|hg lq vhfwru 5 ghfuhdvhv +xqdhfwhg2lqfuhdvhv,/ ^_E^.7²*\

_\ f+' f* :f, iruq q^W +q 'q^W*q : q^W,1 Surri1 Glhuhqwldwlqj .₇²*\ zlwk uhvshfw wr\ zh kdyh=

_E.₇²*\

_\ ' L7

^nR 3 CR₂

YR

YR2 mRER₂ n

YR2

_\

^nRnR₂ n

YR2

_\ E^nRR2YR

YR2 mRER2^YR_{_\}²

^nR

4 D

Uhzulwlqj wkh htxdwlrq zh jhw=

_E.₇²*\

_\ ' L7

E^nR YR2

_\

YR

YR2 mRER2

R2

^nRnR2 n R2

^nR

n ^nR

^nRnR2

c

vkrzlqj wkdw wkh qhw hhfw lv qhjdwlyh iruq q^W dqg srvlwlyh iru q : q^W1 Wkh odvw wklqj wr frqvlghu lq wklv vhfwlrq lv wkh lpsdfw rq wkh wrwdo xqhpsor|0 phqw udwh1 Wkh wrwdo xqhpsor|phqw udwh lq wkh hfrqrp| lv jlyhq e|=

L ' L7nL

L7nL n.₇² n.₇ n.² 'L₇nL Zh fdq frqfoxgh wkh iroorzlqj=

Sursrvlwlrq 7191 Wkh lpsdfw rq wrwdo xqhpsor|phqw iroorzlqj dq lqfuhdvh lq wkh qxpehu ri vnloohg zrunhuv lv dpeljxrxv iruq q^W dqg qhjdwlyh iruq q^W1 Surri1 Wkh hhfw ri lqfuhdvlqj wkh uhodwlyh qxpehu ri vnloohg zrunhuv rq wrwdo xqhpsor|phqw lv=

_L _\ '

L7

\ L

\

YR2

_\

YR

YR₂ mRER₂ n

L7

RnR₂n^ n L

R₂n^

c

zkhuh wkh uvw sduhqwkhvlv frqwdlq d qhjdwlyh ydoxh dqg wkh odvw/ vtxduhg/

sduhqwkhvlv frqwdlq d srvlwlyh ydoxh1 Dv ^YR_{_\}² f + f, iru q q^W +q q^W wkh qhw hhfw lv dpeljxrxv iruq q^W dqg qhjdwlyh iru q q^W1

Dv wkh xqhpsor|phqw udwh ri vnloohg zrunhuv lv ohvv wkdq wkh xqhpsor|phqw udwh ri xqvnloohg zrunhuv= ^L_\⁷ _3\^L f/ wkh vxp ri wkh uvw wzr whupv lv qhj0 dwlyh1 Khqfh/ prylqj zrunhuv iurp wkh kljk xqhpsor|phqw jurxs wr wkh orz xqhpsor|phqw jurxs reylrxvo| ghfuhdvhv wkh wrwdo xqhpsor|phqw udwh1 Wkh odvw

(21)

whup lv wkh vxp ri wkh hhfwv rq wkh vnloohg dqg xqvnloohg zrunhu*v xqhpsor|phqw udwhv uhvshfwlyho|1 Dv wkh| duh erwk srvlwlyh iru q q^W/ wkhvh hhfwv frqwulexwh wr dq lqfuhdvh lq wkh wrwdo xqhpsor|phqw udwh1 Lq wklv fdvh/ wkh wrwdo hhfw lv wkxv dpeljxrxv1 Iru q : q^W erwk xqhpsor|phqw udwh ghfuhdvhv/ frqvhtxhqwo|/

xqhpsor|phqw ghfuhdvhv1

Wr frqfoxgh/ zh kdyh wzr glvwlqfw fdvhv1 Zkhq q q^W/ zkdw kdsshqv zkhq wkh uhodwlyh qxpehu ri vnloohg zrunhuv lqfuhdvhv lv wkdw/ dowkrxjk wkh fkdqfh ri jhwwlqj d mre lq wkh vnloohg vhfwru lqfuhdvhv/ R lqfuhdvhv/ lw lv qrw hqrxjk wr rxw0 zhljk wkh idoo lq wkh wudqvlwlrq udwh lqwr hpsor|phqw lq wkh xqvnloohg vhfwru/ R21 Khqfh/ wkh xqhpsor|phqw udwhv ri erwk vnloohg dqg xqvnloohg zrunhuv lqfuhdvh1 Wkh rqo| srvlwlyh hhfw rq wrwdo hpsor|phqw lv wkh pryhphqw ri zrunhuv iurp wkh kljk xqhpsor|phqw jurxs wr wkh orz xqhpsor|phqw jurxs1 Ixuwkhupruh/ lql0 wldoo|/ wkh hpsor|phqw survshfwv ri dq xqvnloohg zrunhu lv zruvh wkdq iru d vnloohg zrunhu1 Dv wkh xqhpsor|phqw udwh ri xqvnloohg zrunhuv lqfuhdvhv pruh wkdq wkh xqhpsor|phqw udwh ri vnloohg zrunhuv/ wkh uhodwlyh srvlwlrq ri dq xqvnloohg zrunhu lq whupv ri hpsor|phqw survshfwv zruvhq1 Wkh odujhu iudfwlrq ri vnloohg zrunhuv lqgxfhv d pruh xqhtxdo xqhpsor|phqw glvwulexwlrq1 Zkhqq : q^W/ wkh wudqvlwlrq udwh lqwr vhfwru 2 lqfuhdvhv/ wkhuhe| erwk xqhpsor|phqw udwhv dqg wrwdo xqhp0 sor|phqw ghfuhdvhv1 Ixuwkhupruh/ wkh ghfuhdvh lq xqhpsor|phqw iru xqvnloohg zrunhuv lv odujhu wkdq wkh ghfuhdvh lq xqhpsor|phqw iru vnloohg zrunhuv/ lqgxflqj d pruh htxdo xqhpsor|phqw glvwulexwlrq1 Krzhyhu/ lq wklv fdvh wkhuh zloo eh d odujhu iudfwlrq ri vnloohg zrunhuv hpsor|hg lq vhfwru1

7171 Zhoiduh

Lq wklv vhfwlrq zh h{dplqh wkh lpsdfw rq zhoiduh iurp dq lqfuhdvh lq wkh iudfwlrq ri vnloohg zrunhuv1 Jlyhq doo zrunhuv sd| d oxps vxp wd{wkh jryhuqphqw exgjhw frqvwudlqw lv jlyhq dv=

KEL_rnL '|

.₇ n.₇² n. nL₇nL '| Wkh zhoiduh ixqfwlrq lv ghulyhg wr eh=

` '`₇n`₇²n`c zkhuh

`₇ ' .₇+T& ' R

^nR\

+&^R

s

c

`₇²n`² ' .²@+n.₇²@q+T2&

' \ ^R₂

E^nRnR2 E^nR

@q+&R₂

s E^nR n E\ R₂

^nR2

@+&^R₂ s

c

4<

(22)

xvlqj wkh jryhuqphqw exgjhw frqvwudlqw wr holplqdwh wd{hv dqg xqhpsor|phqw lqvxudqfh1 `₇ lv zhoiduh dvvrfldwhg zlwk vhfwru 4 dqg `₇²n`² lv zhoiduh dvvr0 fldwhg zlwk vhfwru 51 Zhoiduh lv lqfuhdvlqj lq hpsor|phqw dqg surgxfwlylw| dqg ghfuhdvlqj lq ydfdqf| frvwv1 Lw fdq eh vkrzq wkdw=

`₇

\ : `₇²

\ / `₇

\ : `²

\ li R : R₂ +716,

`²

\ : `₇²

\ li R : R₂ dqg q q +717, zkhuh q : q ^W lv ghqhg e| wkh htxdwlrq=

@q+K

@+K ' ER2n 2 E^nR ER₂n 2^

Rn^

^

n K

@+Kl

' f/

l ' RERnR2n 2^ ER2n 2^

^ER2n^ ERnR2n^ ERn^

Wkh zhoiduh udwh frqfhuqlqj vnloohg zrunhuv hpsor|hg lq wkh vnloohg vhfwru/

vhfwru/ lv wkh kljkhvw1 Lq jhqhudo wkh zhoiduh udwh frqfhuqlqj xqvnloohg zrunhuv lv kljkhu wkdq wkh zhoiduh udwh frqfhuqlqj vnloohg zrunhuv hpsor|hg lq vhfwru 21 Krzhyhu/ zkhq vnloohg zrunhuv hpsor|hg lq vhfwru 5 kdyh d pxfk kljkhu surgxfwlylw| wkdq dq xqvnloohg zrunhu/ wklv kljkhu surgxfwlylw| pd| frpshqvdwh iru wkh kljkhu ulvn ri vhsdudwlrq frqqhfwhg wr wklv zrunhu dqg frqvhtxhqwo| wkh zhoiduh udwh ri xqvnloohg zrunhuv lv ohvv wkdq wkh zhoiduh udwh ri vnloohg zrunhuv hpsor|hg lq vhfwru21

Frqvlghulqj dq lqfuhdvh lq wkh uhodwlyh qxpehu ri vnloohg zrunhuv/ zh fdq vkrz wkh iroorzlqj=

Sursrvlwlrq 71:1 Wkh wrwdo zhoiduh hhfw ri lqfuhdvlqj wkh qxpehu ri vnloohg zrunhuv lv dpeljxrxv iruq q^W dqg srvlwlyh iru q q^W1

Surri1 Glhuhqwldwlqj wkh zhoiduh htxdwlrq zlwk uhvshfw wr wkh iudfwlrq ri vnloohg zrunhuv/ \/ zh rewdlq=

`₇

\ n`₇²

\ `²

\n\YE`₇*\

Y\ nE\YE`²*E\

Y\ n\YE`₇²*\

Y\ c +718, zkhuh=

YE`₇*\

Y\ ' _R

_R₂ _R2

_\

+ &^R

s E^nRR&

s

^

E^nR²c +719, YE`²*E\

Y\ ' ^

E^nR2²

@+&R₂^

s E^nR₂&R₂ s

_R₂

_\c +71:, YE`₇²*\

Y\ '^

^nR3R2_R

^nRnR_R22

@q+

^nR ^&R_s² R2

@q+

E^nR²_R

_R2 n_s^&

^nRnR2

_R2

_\ +71;,