WORKING PAPER SERIES DEPARTMENT OF ECONOMICS

DEPARTMENT OF ECONOMICS

UNIVERSITY OF MILAN - BICOCCA

WORKING PAPER SERIES

Delegation Games with Full Commitment

Paolo Balduzzi

No. 70 - April 2004

Dipartimento di Economia Politica

Università degli Studi di Milano - Bicocca

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6 D vlpsoh prgho ri ghohjdwhg edujdlqlqj

Lq wklv vhfwlrq/ zh ghyhors d edvholqh fdvh/ zkhuh wkh vxemhfwv edujdlq derxw wkh vdoh ri dq lqglylvleoh jrrg1 Erwk sulqflsdov fdq kluh dq lqwhuphgldu|= L iru wkh vhoohu V dqg M iru wkh ex|hu E1 Wkh ydoxdwlrq ri wkh jrrg lv 3 wr V/ L dqg M dqg 4 wr E1 Hyhu| sod|hu kdv wkh vdph glvfrxqw udwh 1 Ilqdoo|/ zh dvvxph wkhuh lv qr srvvlelolw| ri uhqhjrwldwlrq ehwzhhq d sduw| dqg klv lqwhuphgldu|/ wkdw lv/ ghohjdwlrq dfwv dv ixoo frpplwphqw1 Ehvwhu dqg Vdnrylfv +5334, kdyh douhdg| vkrzq wkdw/ zkhq rqo| rqh sduw| fdq ixoo| frpplw klpvhoi +vshflfdoo|/ V,/ kh fdq jdlq wkh hqwluh slh e| vljqlqj d frqwudfw zlwk L vxfk wkdw V zloo eh sdlg d {hg dprxqw i zkhqhyhu wudgh zloo rffxu ehwzhhq L dqg E4₁

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4_{Lw lv hdv| wr vkrz wkdw wkh vdph lv wuxh zkhq rqo| E fdq ghohjdwh= kh zloo jhw wkh hqwluh}

Lq wklv uvw yhuvlrq ri wkh prgho/ ghohjdwlrq lv frpsohwho| frvwohvv1 Wr pdnh wkh h{srvlwlrq dv fohdu dv srvvleoh/ zh glvwlqjxlvk wkuhh vwdjhv=

+4, Wkh ghohjdwlrq vwdjh= lq wkh uvw vwdjh/ wkh sulqflsdov ghflgh zkhwkhu wr ghohjdwh ru qrw1 Li d sulqflsdo ghflghv wr ghohjdwh/ kh pdnhv d wdnh0lw0 ru0ohdyh0lw rhu wr klv rzq djhqw derxw wkh ohyho ri d {hg frpshqvdwlrq vfkhph +iV ru iE> zlwk il 5 ^3> 4` > l @ E> V,51 Li kh ghflghv wr gr qrw

ghohjdwh/ kh grhv qrw kluh dq| djhqw1 Wkhvh ghflvlrqv +klulqj dq djhqw dqg ohyho ri rhuv, duh qrw pdgh sxeolf xqwlo wkh edujdlqlqj vwdjh ehjlqv1 +5, Wkh djhqwv* vwdjh= hdfk djhqw ghflghv zkhwkhu wr dffhsw ru qrw klv sulq0

flsdo*v rhu rq wkh edvlv ri klv rzq uhvhuydwlrq zdjh dqg rxwvlgh rswlrqv1 Zh dvvxph lq wklv vhfwlrq wkdw wkhuh duh qr rxwvlgh rswlrqv dqg wkdw/ rqfh dq djhqw pdnhv klv ghflvlrq +dffhsw ru uhmhfw,/ kh lv wkhq frpplwwhg wr lw6₁

Dv ghohjdwlrq lv qrw frvwo|/ dq djhqw*v uhmhfwlrq dqg d sulqflsdo*v fkrlfh ri qrw ghohjdwlqj kdyh wkh vdph hhfw rq wkh qdo rxwfrph1

+6, Wkh edujdlqlqj vwdjh= lq wklv vwdjh/ doo wkh lqirupdwlrq lv pdgh sxeolf +vshflfdoo|/ suhvhqfh ri lqwhuphgldulhv/ ohyhov ri iV dqg iE,1 Wkh sduwlhv

edujdlq ryhu wkh sulfh ri wkh jrrg dffruglqj wr dq dowhuqdwlqj rhuv prgho d od Uxelqvwhlq1 D sulrul/ wkhuh duh wkuhh srvvleoh edujdlqlqj vlwxdwlrqv= elodwhudo ghohjdwlrq +L zlwk M,/ rqh0vlgh ghohjdwlrq +L zlwk E ru V zlwk M,/ gluhfw edujdlqlqj +V zlwk E,1 Wkh vhoolqj sduw| +V ru L, dozd|v pdnhv wkh uvw rhu1

Frpshqvdwlrq vfkhphv dqg surw ixqfwlrqv lq fdvh ri elodwhudo ghohjdwlrq duh vhw dv iroorzv=

 L sd|v iV wr V zkhqhyhu wudgh rffxuv dqg jdlqv wkh vhoolqj sulfh ri wkh

jrrg>

 V jdlqv iV zkhqhyhu wudgh rffxuv>

 E sd|v iE wr M zkhqhyhu wudgh rffxuv dqg jdlqv wkh ydoxh ri wkh jrrg

+zklfk lv htxdo wr 4,>

 M sd|v wkh vhoolqj sulfh wr L dqg jdlqv iE zkhqhyhu wudgh rffxuv/

zkhuh iV dqg iE duh wkh {hg dprxqwv ghflghg lq wkh ghohjdwlrq vwdjh dqg

duh qrw uhqhjrwldeoh1

Wkh djhqwv* lqfhqwlyh wr qg dq djuhhphqw lq wkh edujdlqlqj vwdjh lv hp0 erglhg e| wkh dvvxpswlrq wkdw wkhlu sd|rv ghshqg rq wkh dfwxdo uhdol}dwlrq ri wkh wudgh1

5_{Lq ohppd 4/ zh ghprqvwudwh wkh rswlpdolw| ri {hg frpshqvdwlrq vfkhphv1}

6_{Wklv phdqv wkdw li dq djhqw uhdol}hv h{0srvw wkdw wkh frpshqvdwlrq vfkhph kh dffhswhg}

Wkh h{whqvlyh irup ri wklv jdph lv suhvhqwhg lq Iljxuh 41 Dv looxvwudwhg deryh/ wkh sod|huv* vwudwhjlhv jlyh ulvh wr irxu srvvleoh fdvhv= L, elodwhudo ghoh0 jdwlrq +G> G, zkhq erwk V dqg E ghohjdwh dqg erwk wkh djhqwv dffhsw> LL, dqg LLL, rqh0vlgh ghohjdwlrq +G> QG, dqg +QG> G, zkhq rqo| rqh sulqflsdo ghoh0 jdwhv ru rqo| rqh djhqw dffhswv> LY, gluhfw edujdlqlqj +QG> QG, zkhq erwk wkh sulqflsdov gr qrw ghohjdwh ru erwk wkh djhqw uhmhfw wkh rhu7₁

Iljxuh 4= Wkh ghohjdwlrq jdph

Delegate Not Delegate

Reject TIOLI on fs Accept 0 1 I SELLER BUYER Not Delegate Delegate TIOLI on fb J Reject Accept 1 0 (ND, ND) (ND, D) (D, ND) (D, D)

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Wkh edujdlqlqj vwdjh1 Wkh djhqwv* fkrlfh lq vwdjh +5, lv pdgh rq wkh

edvlv ri wkhlu srvvleoh jdlqv lq vwdjh +6,1 Li wkh| uhmhfw/ wkh| jdlq wkhlu uhvhuydwlrq zdjh/ wkdw lv 3> li mxvw rqh ri wkhp dffhswv dqg wkh rwkhu uhmhfwv/ wkh odwwhu jdlq wkh uhvhuydwlrq zdjh dqg wkh iruphu wkh jdlqv dffruglqj wr Ehvwhu dqg

7_{Iru vdnh ri frpsohwhqhvv/ E(c ( fdq eh d fdvh dovr zkhq rqo| rqh sulqflsdo ghflghv}

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sM iV @  +sL iV,

sL. iE @  +sM. iE, +5,

zklfk phdqv wkdw/ iru hdfk sod|hu/ wkh jdlqv iurp dffhswlqj wkh rwkhu sduw|*v rhu wrgd| pxvw htxdo wkh jdlqv iurp uhmhfwlqj lw dqg pdnh dq rhu wrpruurz1 Lq dgglwlrq/ vlqfh wkhvh jdlqv pxvw eh qrq qhjdwlyh/ wkh vroxwlrqv ri wkh v|vwhp deryh duh=

sL @ iE_{4 . }. iV sM @i_{4 . }E. iV

Dv L pryhv uvw dqg dv wkh htxloleulxp lv uhdfkhg lq wkh uvw shulrg/ sL lv

dffhswhg e| M +l1h1/ sL lv wkh vhoolqj sulfh,1 Wkh sd|rv iru hdfk sod|hu duh=

L @ sL  iV @ iE_{4 . } iV> M @ iE sL @ iE_{4 . } iV> V @ iV> E @ 4  iE=

Wkh sd|r pdwul{ iru doo wkh sod|huv lq wkh irxu srvvleoh vlwxdwlrqv ri Iljxuh 4 lv jlyhq e| Wdeoh +6,= Vhoohu 0Ex|hu V E L M +G> G, iV 4  iE iE4.iV iE4.iV +G> QG, iV _4. +4  iV, 4i_4.V 3 +QG> G, iE 4. 4  iE 3 4. iE +QG> QG, 4 4. 4. 3 3 +6,

Dv wkh sod|huv* sd|rv duh qrz fohdu/ zh lqwurgxfh ohppd 4/ zklfk h{sodlqv wkh udwlrqdolw| ehklqg wkh xvh ri {hg frpshqvdwlrq vfkhphv1

Ohppd 4 Lq elodwhudo ghohjdwlrq jdphv/ d {hg frpshqvdwlrq vfkhph lv surw

pd{lpl}lqj iru d sulqflsdo dqg dovr wkh ehvw uhvsrqvh wr wkh rwkhu sulqflsdo*v vwudwhj|1

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frqvwlwxwhv wkh qhw jdlq wkh djhqwv fdq vkduh/ erwk E dqg V zlvk wr plqlpl}h lw1 Iru wklv uhdvrq/ zkhq d sulqflsdo lv sod|lqj wklv vwudwhj|/ wkh rwkhu sulqflsdo*v ehvw uhvsrqvh lv wr dvn iru d {hg sd|phqw dv zhoo1 Dv orqj dv wklv glhuhqfh lv qrq qhjdwlyh/ hdfk djhqw kdv qr lqfhqwlyh wr uhmhfw dq| rhu1 Wkhq/ dv wkh djhqw*v sd|rv +jrrg ru prqh|, duh frqglwlrqdo wr dq djuhhphqw/ wklv lv qdoo| uhdfkhg1

Wkh djhqwv* vwdjh1 Frqvlghu wkh fkrlfh idfhg e| wkh djhqwv1 Wdnh iru

lqvwdqfh djhqw L1 Li kh uhmhfwv/ kh jdlqv 3> li kh dffhswv/ kh pd| jdlq +iEiV,

ru +4iV,

4. li kh edujdlqv zlwk M ru E uhvshfwlyho|1 Vr/ kh zloo dffhsw dq| frp0

shqvdwlrq iV surylglqj klp d qrq qhjdwlyh surw1 Iru elodwhudo ghohjdwlrq wr

eh d srvvleoh htxloleulxp/ wkh iroorzlqj udwlrqdolw| +sduwlflsdwlrq, frqvwudlqwv/ uhvshfwlyho| iru L dqg M/ pxvw eh vdwlvhg= ; A ? A = iE iV 4 .   3 iE_{4 . } iV  3 +7, Wkh glhuhqfh +iE iV, fdqqrw eh qhjdwlyh= lq wkh edujdlqlqj vwdjh/ wkh

djhqwv zloo uhdol}h wkh| zloo jdlq qhjdwlyh sd|rv lq fdvh ri djuhhphqw1 Qr frpsdwleoh rhuv zloo eh pdgh dqg qr djuhhphqw zloo eh vwuxfn/ ohdglqj doo wkh sod|huv wr wkh lpsdvvh srlqw1 Iljxuh 5 frpsduhv wzr vlwxdwlrqv/ wkdw lv edujdlq0 lqj zlwk ru zlwkrxw ghohjdwlrq1 D lv wkh htxloleulxp ri d w|slfdo Uxelqvwhlq jdph ehwzhhq sulqflsdov= wkh srvlwlrq ri wklv srlqw rqo| ghshqgv xsrq = Lq wkh vhfrqg vlwxdwlrq +htxloleuld E dqg F,/ wzr jdphv duh sod|hg= d Uxelqvwhlq rqh ehwzhhq djhqwv/ vkdulqj d slh ri +iE iV, > dqg d Qdvk ghpdqg jdph ehwzhhq

sulqflsdov1 E lv wkh vroxwlrq ri wkh iruphu jdph zkhuhdv F lv wkh vroxwlrq ri wkh odwwhu rqh1

Iljxuh 5= Gluhfw Yv Ghohjdwhg edujdlqlqj

Agents’ payoffs with Delegation: [(fB-fS)/(1+δ); δ(fB-fS)/(1+δ)]

Principals’ payoffs with Delegation: [fS;1-fB]

1 1

1-(fB-fS)

∏S ∏B ∏I ∏J

Principals’ payoffs from direct bargaining: [1/(1+δ); δ/(1+δ)]

B C

A

Wkhuh lv reylrxvo| d frq lfw ehwzhhq V dqg E/ dv V zdqwv wr pd{lpl}h iV

dqg E zdqwv wr plqlpl}h iE=

Ohppd 5 Htxloleuld zlwk elodwhudo ghohjdwlrq duh vxssruwhg rqo| zkhq iE @

iV @ i1 Wklv vwudwhj| jlyhv wkh iroorzlqj surwv= L @ 3> M @ 3> V @ i

Surri1 Iurp +7, zh nqrz wkdw iE  iV  3= Vxssrvh iE A iV= Wkhq

hlwkhu wkh ex|hu zrxog kdyh dq lqfhqwlyh wr uhgxfh iE ru wkh vhoohu wr udlvh

iV= Wkhuhiruh/ lq htxloleulxp htxdolw| pxvw krog1 Ilqdoo|/ li iE @ iV> L @

M@ 3> V@ i dqg E@ 4  i iurp Wdeoh +6,1

Dv orqj dv wkh djhqwv gr qrw kdyh dq| rxwvlgh rswlrq/ wkh| duh qrw deoh wr h{wudfw dq| h{wud0surw iurp wkh ghohjdwlrq frqwudfw1 Wklv uhvxow lv dfwxdoo| qrw vwulnlqj/ dv doo wkh edujdlqlqj srzhu zdv jlyhq wr wkh sulqflsdov1

Zh fdq qrz irfxv rq wkh sulqflsdov ghohjdwlrq vwdjh1

Wkh ghohjdwlrq vwdjh1 Wklv lv looxvwudwhg lq Wdeoh +8,/ zklfk xsgdwhv dqg

h{sdqgv Wdeoh +4,= Ex|hu G QG 3 i4 i5 === 4 3 3> 4 3> 4  i4 3> 4  i5 3> 3 3>_4. i4 3> 3 i4> 4  i4 i4> 4  i5 i4> 3 i4> +4  i4,_4. Vhoohu G i5 3> 3 3> 3 i5> 4  i5 i5> 3 i5> +4  i5,_4. === 4 3> 3 3> 3 3> 3 4> 3 4> 3 QG 3> 4 i4 4.> 4  i4 4.i5 > 4  i5 4.4 > 3 4.4 >4. +8, Zkhq d sod|hu ghflghv wr ghohjdwh/ kh vhwv d jhqhulf ydoxh i vxfk wkdw 3  i  4= Iru vlpsolflw|/ lq Wdeoh +8, zh frqvlghu rqo| wzr lqwhuphgldwh ydoxhv ri i +zlwk i4? i5,1 Wklv sd|r pdwul{ lv vhw dffruglqj wr nqrzq uhvxowv= sd|rv

iru +QG> QG,/ iru lqvwdqfh/ duh vlpso| jlyhq e| wkh xvxdo Uxelqvwhlq prgho ri gluhfw edujdlqlqj/ zkhuhdv jdlqv iru +QG> G, dqg +G> QG, duh ydoxdwhg iru wkh sduwlfxodu i= Rxu rzq prgho surylghv wkh jdlqv iru +G> G, dqg iru wkh lpsdvvh srlqwv +iV A iE,1 Wkh fkrlfh ehwzhhq ghohjdwlqj dqg qrw ghohjdwlqj frqvwlwxwhv

d vlpsoh jdph ehwzhhq wkh wzr sulqflsdov dqg wkh qdo rxwfrph zloo ghshqg rq wkh ohyho ri i1 Wkh sulqflsdov* ehvw uhvsrqvhv duh dv iroorzv=

 Li V sod|v G zlwk d qrq qhjdwlyh iV> wkhq E dozd|v sod|v G vhwwlqj iE @

iV @ i1

 Li V sod|v G zlwk iV @ 4> wkhq E hlwkhu sod|v G vhwwlqj iE @ iV @ 4 ru

sod|v QG=

 Li V sod|v QG wkhq E dozd|v sod|v G vhwwlqj iE@ 31

 Li E sod|v G zlwk d vwulfwo| srvlwlyh iE> wkhq V dozd|v sod|v G vhwwlqj

iV @ iE @ i1

 Li E sod|v G zlwk d iE @ 3> wkhq V hlwkhu sod|v G vhwwlqj iV @ iE@ 3 ru

 Li E sod|v QG wkhq V dozd|v sod|v G vhwwlqj iV @ 41

Wkuhh fdvhv hphujh1 Wkh| duh suhvhqwhg lq Sursrvlwlrq 4/ zklfk dovr vxp0 pdul}hv wkh uhvxowv ri wklv uvw vhfwlrq1

Sursrvlwlrq 4 D wzr0vlghg ghohjdwlrq jdph lv fkdudfwhul}hg dv d w|slfdo Qdvk

ghpdqg jdph1 Iru h{wuhph ydoxhv ri wkh frpshqvdwlrq vfkhphv/ rqh0vlghg gho0 hjdwlrq htxloleuld duh dovr vxssruwhg1 Lq sduwlfxodu/ iru 3 ? i ? 4 wkhuh*v d frqwlqxxp ri QHSV +Qdvk Htxloleuld zlwk Sxuh Vwudwhj|, zlwk elodwhudo ghohjd0 wlrq +G> G,> iru i @ 3 wkhuh duh wzr QHSV= +G> G, dqg +QG> G,> qdoo|/ iru

i @ 4 wkhuh duh wzr QHSV= +G> G, dqg +G> QG,=

Surri1 Sursrvlwlrq 4 gluhfwo| iroorzv iurp wkh glvfxvvlrq deryh dqg iurp wkh vroxwlrq ri wkh jdph lq +8,1

Vrph glvfxvvlrq derxw wkhvh uhvxowv lv zruwk|1 Iluvw ri doo/ wkh suhvhqfh ri dgglwlrqdo sod|huv +wkh djhqwv, grhv qrw dhfw wkh vl}h ri wkh slh/ wkdw lv/ wkh wrwdo dydlodeoh surw1 Ri frxuvh/ wklv uhvxow lv vwurqjo| ghshqghqw rq wkh qr frvwv dvvxpswlrq1

Wkhq/ zlwk uhjdug wr glvwulexwlrqdo lvvxhv/ zh uvw ghqh wkh zlqqhu ri wkh jdph1

Ghqlwlrq 4 D sulqflsdo lv wkh zlqqhu ri wkh ghohjdwlrq jdph zkhq klv sd|r

iurp ghohjdwlqj lv eljjhu wkdq klv sd|r iurp gluhfw edujdlqlqj1

Zkhq d sulqflsdo h{shfwv wkh rwkhu wr rewdlq wkh hqwluh slh +i @ 3 ru i @ 4,/ kh kdv qr vwudwhjlf uhdvrq wr orrniru dq djhqw/ dv klv sd|r lv jrlqj wr eh h{dfwo| wkh vdph1 Dffruglqj wr rxu ghqlwlrq/ wkh ex|hu lv zlqqlqj iurp elodwhudo ghohjdwlrq dv orqj dv i ? 4

4.> rq wkh frqwudu|/ wkh vhoohu lv zlqqlqj iru kljkhu

ydoxhv ri i1 Wkhuh duh pruh htxloleuld zkhuh wkh ex|hu lv wkh zlqqhu iru dq| 3 ?  ? 4= Wkdw lv/ ghohjdwlrq ixoo| ghvwur|v dq| uvw pryhu dgydqwdjh iru wkh vhoohu1 Lqghhg/ wklv dgydqwdjh lv w|slfdo ri wlph frqvxplqj jdphv zkhuhdv wkh ghohjdwlrq rqh lv fkdudfwhulvhg dv d Qdvk ghpdqg jdph1

Lq wkh qh{w vhfwlrqv/ zh zloo glvfxvv wkh frqvhtxhqfhv ri uhod{lqj dq lpsru0 wdqw dvvxpswlrq/ wkdw lv/ wkdw ghohjdwlrq lv frvwohvv1 Iluvw/ zh frqvlghu h{rjhqrxv frvwo| ghohjdwlrq iru wkh sod|huv1 Wkhq/ zh dqdo|}h wkh hhfwv ri hqgrjhqrxv frvwo| ghohjdwlrq1 Zh h{shfw wkh udqjh ri elodwhudo ghohjdwlrq htxloleuld wr vkulqn ru hyhq glvdsshdu1

7 H{rjhqrxv frvwo| ghohjdwlrq

714 Frvwo| ghohjdwlrq iru sulqflsdov

Ohw xv vxssrvh wkdw/ vlpso| ehfdxvh d sulqflsdo hpsor|v dq lqwhuphgldu|/ kh vxhuv d vwulfwo| srvlwlyh {hg frvw f lqghshqghqwo| iurp wkh idfw wkdw wudgh rffxuv ru qrw1 Lq sduwlfxodu/ f lv sdlg e| wkh sulqflsdo rqo| li kh ghflghv wr ghohjdwh1 Iru lqvwdqfh/ f pd| eh wkh wlph frvw ri orrnlqj iru dq lqwhuphgldu|1

Iru vlpsolflw|/ zh dvvxph fV @ fE@ f wr eh wkh sulqflsdov* frpprq frvw ri

ghohjdwlrq1 Krz zloo wkh htxloleuld ri wkh suhylrxv vhfwlrq eh dhfwhgB Iluvw ri doo/ zh xsgdwh wkh sd|phqw vfkhph lq fdvh ri elodwhudo ghohjdwlrq=

 L sd|v iV wr V zkhqhyhu wudgh rffxuv dqg jdlqv wkh vhoolqj sulfh ri wkh

jrrg>

 V sd|v f zkhq ghohjdwlqj dqg jdlqv iV zkhqhyhu wudgh rffxuv>

 E sd|v f zkhq ghohjdwlqj dqg iE wr M zkhqhyhu wudgh rffxuv dqg jdlqv

wkh ydoxh ri wkh jrrg + zklfk lv htxdo wr 4,>

 M sd|v wkh vhoolqj sulfh wr L dqg jdlqv iE zkhqhyhu wudgh rffxuv

Wkh udqjh ri htxloleuld lq wkh suhylrxv vhfwlrq lv prglhg dv vwdwhg lq sursr0 vlwlrq 51

Sursrvlwlrq 5 Lq d wzr0vlghg ghohjdwlrq jdph zlwk h{rjhqrxv ghohjdwlrq frvwv f iru wkh sulqflsdov/ wkh iroorzlqj htxloleuld hphujh= +G> G,> +G> QG, dqg +G> QG,

iru 3  f  

+4.,5> +G> QG, dqg +G> QG, iru _+4., 5 ? f  _4. > +QG> G, iru



4.? f  4.4 > dqg +QG> QG, iru f A 4.4 = Wkh suhvhqfh ri frvwv lv d vrxufh ri

lqh!flhqf|1 Lq sduwlfxodu/ lq vrph lqwhuydov elodwhudo ghohjdwlrq htxloleuld duh fkdudfwhul}hg dv sulvrqhu*v glohppdv1

 Li V sod|v G wkhq  E sod|v G li kh h{shfwv iV ? 4  f +4 . , E sod|v QG li kh h{shfwv iV A 4  f +4 . ,  Li V sod|v QG wkhq  E sod|v G vhwwlqj i @ 3 li f ? 4 4. E sod|v QG li f A 4 4.  Li E sod|v G wkhq  V sod|v G li kh h{shfwv iE A f4._  V sod|v QG li kh h{shfwv iE ? f4._   Li E sod|v QG wkhq  _{V sod|v G vhwwlqj i @ 4 li f ? } 4. V sod|v QG li f A  4.

Zh fdq vwloo dsso| wkh vdph orjlf zh xvhg wr vroyh +8,1 Wkhuh duh : srvvleoh frpelqdwlrqv ri i dqg f dulvlqj1 Wkh vlpsohvw zd| wr vwxg| dqg dqdo|}h wkhp lv wr orrndw judsk 41

Zh vwduw irfxvlqj rq lqwhuydov zkhuh +G> G, lv d vroxwlrq1 Lw fdq eh vkrzq +srvvleo|/ lq Dsshqgl{,/ wkdw dq| i _{lq fdvh L fdq eh vxvwdlqhg dv htxloleulxp}

frpshqvdwlrq vfkhph1 Lq rwkhu zrugv/ elodwhudo ghohjdwlrq lv vxvwdlqhg zkhqhyhu f4.





? i _{? 4  f +4 . , = Ri frxuvh/ wklv lqwhuydo lv vkulqnlqj dv f jurzv}

dqg hyhqwxdoo| froodsvhv wr 3 zkhq f @  +4.,5= Lq wklv srlqw/ i @ _4.4 = Iru kljkhu ydoxh ri f = 4  f +4 . , ? f4.   dqg wkhuhiruh +G> G, lv qr orqjhu d srvvleoh htxloleulxp1 Rq wkh frqwudu|/ +QG> QG, lv wkh xqltxh vroxwlrq iru dq| f A 4

4.= ghohjdwlrq lv wrr frvwo| iru wkh sulqflsdov dqg hyhq rqh vlgh ghohjdwlrq

htxloleuld duh uxohg rxw1

Judsk 4= Htxloleuld zlwk sulqflsdov* frvwv

Compensation scheme f 1 1 Delegation cost c δ /(1+δ)2 _{δ /(1+δ) 1/(1+δ)} 0 f = [(1+δ)/ δ] c f = 1 - c(1+δ) II VII V VI IV III I 1/(1+δ)

ohyho ri i/ wkdw lv iV @ 4 ru iE@ 31 Doo wkh xqfhuwdlqw| glvdsshduv rqfh f lv elj

hqrxjk wr uxoh rxw pxowlsolflw|= iru 

4. ? f ? 4.4 / QG lv d grplqdqw vwudwhj|

iru wkh vhoohu/ qr pdwwhu wkh h{shfwhg ydoxh ri iE= Wklv vlwxdwlrq lv reylrxvo|

h{sorlwhg e| wkh ex|hu/ zkr sod|v klv surw pd{lplvlqj iE @ 31 Ilqdoo|/ zkhq

f A 4

4./ erwk wkh sod|huv suhihu QG1

71414 Zhoiduh lpsolfdwlrqv= glvwulexwlrqdo dqg h!flhqf| lvvxhv Zkhq lw lv frvwo|/ ghohjdwlrq lv zlwk qr grxew lqh!flhqw/ dv lw vkulqnv wkh vl}h ri wkh slh dydlodeoh wr wkh sulqflsdov1 Qhyhuwkhohvv/ G lv vwloo sod|hg lq vrph lqwhuydov1

Wr vhh wkh h{whqw wr zklfk ghohjdwlrq lv lqh!flhqw/ zh fdq ghqh wkh ghohjd0 wlrq orvv GO dv wkh glhuhqfh ehwzhhq wkh vl}h ri wkh qrq0ghohjdwlqj htxloleulxp wrwdo rxwfrph +l1h1/ 4, dqg wkh ghohjdwlqj htxloleuld rqh +4  f ru 4  5f zlwk rqh0vlgh dqg elodwhudo ghohjdwlrq uhvshfwlyho|,1 Wklv glhuhqfh fohduo| ghshqgv rq wkh ohyho ri wkh frvwv f= Mxvw iru vlpsolflw| dqg iru h{srvlwlrqdo sxusrvhv/ zh irfxv rq wkh zruvw srvvleoh fdvh dqg dvvxph elodwhudo ghohjdwlrq htxloleuld wr eh sod|hg zkhq rqh0vlgh ghohjdwlrq rqhv duh srvvleoh dv zhoo1 Zkhq erwk wkh sod|huv ghohjdwh/ d wrwdo orvv ri 5f lv surgxfhg1 Iru fhqwudo ydoxhv ri wkh ghohjdwlrq frvwv/ rqo| rqh0vlgh ghohjdwlrq htxloleuld duh vxvwdlqdeoh/ wkhuhiruh wkh orvv lv htxdo wr f= Ilqdoo|/ iru odujhu f> qr sulqflsdo ghohjdwhv dqg wkhuh duh qr orvvhv1

Judsk 5 vkrzv wkh qrq prqrwrqlf ehkdylru ri wkh ghohjdwlrq orvv ixqfwlrq/ zklfk lv ghqhg e| +:,= GO @ ; A ? A = 5f iru f ?  +4.,5 f iru  +4.,5 ? f ? _4.4 3 iru f A 4 4. +:,

Judsk 5= Wkh ghohjdwlrq orvv ixqfwlrq

Lw lv dovr lqwhuhvwlqj wr xqghuvwdqg zkr lv wkh zlqqhu ri wkh ghohjdwlrq jdph1 Lq wkh edvlf prgho/ iru lqvwdqfh/ V lv wkh zlqqhu zkhqhyhu i A 4

4.= Qrz wkh

glvfxvvlrq lv d olwwoh elw pruh frpsoh{/ dv sd|rv ghshqg erwk rq f dqg i= Judsk 6 khosv xv wr xqghuvwdqg wkh vlwxdwlrq1

Wkh wzr wlfnvwudljkw olqhv duh wkh ixqfwlrqv i @ 4

4.f dqg i @4.4 .f= Wkh

elj wuldqjohEFH uhsuhvhqwv wkh jdlqv iurp ghohjdwlrq iru wkh vhoohu7 8_{1 Wkh elj}

wuldqjohDEM jlyhv wkh surwv iurp ghohjdwlrq iru wkh ex|hu1 Zh vwduw irfxvlqj7 rq wkh lqwhuydo 3 ? f ? 

+4.,5 dqg +G> G, htxloleuld1 V lv wkh zlqqhu9 zkhqhyhu

i  f A 4

4.> wkdw lv zkhqhyhu i A 4.4 . f +duhd EFGI ,1 E lv wkh zlqqhu li

4  i  f A 

4.> wkdw lv/ i ? 4.4  f +duhd DEJL,1 Lq wkh wuldqjoh 7

EI J/ wkdw lv zkhq 4

4.  f ? i ? 4.4 . f erwk wkh sod|huv duh orvlqj iurp ghohjdwlrq1

Wklv phdqv wkdw iru vrph ydoxhv ri f dqg i> wkh ghohjdwlrq jdph lv vlplodu wr d sulvrqhu*v glohppd= erwk wkh sod|huv zrxog eh ehwwhu r e| qrw ghohjdwlqj exw vwudwhjlf ehkdylruv irufh wkhp wrzdugv d Sduhwr grplqdwhg htxloleulxp1

Judsk 6= Wkh zlqqhu ri wkh jdph +4, Compensation scheme f 1 1 Delegation cost c δ /(1+δ)2 _{δ /(1+δ) 1/(1+δ)} 0 f = 1/(1+δ) + c 1/(1+δ) f = 1/(1+δ) - c I K H G C F D E B A J

Iru frpsohwhqhvv/ zh vkrxog qrwh wkdw wkh jdph sod|hg dorqj wkh yhuwlfdo d{lv lv vlpso| wkh fdvh zh vwxglhg lq wkh suhylrxv vhfwlrq/ wkdw lv/ frvwohvv ghohjdwlrq +f @ 3,1 Dv f jurzv/ wkh erwk orvlqj vsdfh hqodujhv1

8_{Zh vlpso| fdoo jdlqv iurp ghohjdwlrq wkh glhuhqfh ehwzhhq wkh sod|hu*v rxwfrph zkhq}

kh ghohjdwhv dqg klv rxwfrph iurp gluhfw edujdlqlqj1

Zkhq 

+4.,5 ? f ? _4. > rqo| rqh0vlgh ghohjdwlrq htxloleuld duh srvvleoh= li

+G> QG, lv sod|hg/ zh duh lqGHI> li +G> QG, lv sod|hg/ zh duh lq JLNK1 Dv7 wkhvh htxloleuld duh h{ dqwh htxdoo| srvvleoh/ wkh ghohjdwlrq jdph lv vlplodu wr d fklfnhq +ru kdzn0gryh, jdph1 Wkhq/ iru 

4. ? f ? 4.4 > rqo| +G> QG, lv sod|hg

dqg zh duh lq wkh olwwoh wuldqjohKNM= wkh ex|hu lv wkh rqo| srvvleoh zlqqhu17 Wkh h{lvwhqfh ri wklv duhd lv txlwh vxusulvlqj1 Iru vrph ydoxhv ri f> wkh xvxdoo| zhdnhvw sduw| +wkdw lv/ wkh vhfrqg pryhu, lv vwurqjhu1 Zh uhfdoo wkdw zh kdg d vlplodu nlqg ri dv|pphwu| iru f @ 3= Zh fdq gduh wr jlyh d qhz lqwhusuhwdwlrq wr Vfkhoolqj*v lqwxlwlrq wkdw zhdnqhvv lv riwhq vwuhqjwk1 Wklv lv qrw wuxh mxvw zkhq d sod|hu fdq elqg klpvhoi wr d vwudwhj|1 Lw dovr pdwfkhv zlwk wkh iroorzlqj +frpprq vhqvh, vwdwhphqw= zhdnqhvv lv vwuhqjwk zkhq wkh vwduwlqj sd|r lv orzhu +l1h1/ wkh sod|hu kdv ohvv wr orvh,1

Ilqdoo|/ iru kljkhu ydoxhv ri f> qrerg| qgv ghohjdwlrq surwdeoh1

715 Frvwo| ghohjdwlrq iru djhqwv

Ghohjdwlrq pd| eh frvwo| iru djhqwv/ wrr1 Zh fdoo wkhlu frpprq frvw |= Wklv lv nqrzq h{ dqwh e| doo wkh sod|huv dqg lw lv vxhuhg rqo| rqfh wkh djhqw dffhswv wkh ghohjdwlrq frqwudfw1 Zh pd| wklqnri | dv wkh mre hruw1 Ru lw fdq eh wkh wlph vshqw wr duudqjh phhwlqjv zlwk wkh frxqwhusduw1 L dqg M zloo dffhsw d frpshqvdwlrq vfkhph rqo| li lw zloo surylgh wkhp zlwk dw ohdvw wklv ydoxh |1 Wklv frvw lv qrw vxqnxqwlo wkh djhqw dffhswv d frqwudfw1 Lq idfw/ li lw zhuh vxqn ehiruh/ d sulqflsdo frxog ohdyh doo wkh exughq xsrq wkh djhqw dqg wkh sureohp zrxog eh wkh vdph dv lq wkh edvlf prgho1

Wkh udqjh ri htxloleuld vkulqnv dv | jurzv1 Iru kljk ydoxhv ri wkh djhqwv* frvwv/ ghohjdwlrq qhyhu rffxuv1 D ghwdlohg fkdudfwhul}dwlrq ri wkh srvvleoh htxl0 oleuld dqg fruuhvsrqglqj uhohydqw lqwhuydov ri | lv surylghg e| Sursrvlwlrq 61 Wkh uhvw ri wkh vxevhfwlrq lv wkhq ghyrwhg wr lwv h{sodqdwlrq1

Sursrvlwlrq 6 Lq d wzr0vlghg ghohjdwlrq jdph zlwk v|pphwulf h{rjhqrxv gho0

hjdwlrq frvwv | iru wkh djhqwv/ wkh iroorzlqj htxloleuld hphujh= +G> G,> +G> QG, dqg +QG> G, iru 3  |  5

+4.,+4..5_,> +G> QG, dqg +QG> G, iru 

5

+4.,+4..5_, ?

|  

+4.,5> dqg +QG> QG, iru | A _+4., 5= Iru vrph udqjh ri |> ghohjdwlrq lv

sod|hg hyhq li lw surylghv wkh sulqflsdov zlwk d orzhu sd|r wkdq lq wkh gluhfw edujdlqlqj fdvh1

71514 H{lvwhqfh ri htxloleuld zlwk wzr0vlghg ghohjdwlrq

Zkhq erwk wkh sulqflsdov ghohjdwh/ wkh djhqwv* udwlrqdolw| frqvwudlqwv duh qrz jlyhq e| +<, zklfk xsgdwhv +7,= ; A ? A = iE iV 4 .   | iE_{4 . } iV  | +<, Erwk frqglwlrqv pxvw eh vdwlvhg vlpxowdqhrxvo|:_{1 E dqg V fdqqrw vhw wkh}

glhuhqfh +iE iV, htxdo wr 3 dq| orqjhu> wkh prvw wkh| fdq gr lv vhw lw vxfk

wkdw= iE iV @  4 .    |  +4 . , | ; 5 ^3> 4` +43, Ohppd 6 Lq dq| elodwhudo ghohjdwlrq htxloleulxp/ wkh sod|huv* sd|rv zloo eh= L @ |_> M@ |> V @ iV dqg E@ 4  iE1

Surri1 Wkh orjlf ri Ohppd 5 vwloo dssolhv1

Iurp wkh djhqwv* srlqw ri ylhz/ wkhuh lv dq lpsruwdqw glhuhqfh1 Djhqw L fdq jdlq pruh wkdq wkh ohyho | ri klv frvwv1 Wklv lv d frqvhtxhqfh ri L ehlqj wkh uvw pryhu lq wkh edujdlqlqj vwdjh zlwk M1 Dv erwk wkh frqvwudlqwv lq +<, pxvw eh vdwlvhg/ +iE iV, pxvw eh vhw vr wkdw lw vdwlvhv wkh vwulfwhu ri wkh

frqvwudlqwv/ zklfk lv wkh vhfrqg pryhu*v rqh1

Wkh sulqflsdov sod| wkh iroorzlqj jdph= li V ghohjdwhv kh jdlqv iV> li kh grhv

qrw ghohjdwh exw E ghohjdwhv kh jhwv iE

4.= Li E ghohjdwhv kh jdlqv 4  iE> li kh

grhv qrw ghohjdwh exw V ghohjdwhv kh jhwv 

4.+4  iV, =

Wdnlqj lqwr dffrxqw +43,/ elodwhudo ghohjdwlrq htxloleuld zloo eh vxvwdlqhg zkhqhyhu= ; A A A A A ? A A A A A = iV A _{4 . }iE 4  iEA _{4 . } +4  iV, iE iV @  4 .    | wkdw lv ; A A A ? A A A = iE ? iV+4 . , iE ?_{4 . }4 ._{4 . } iV iE @ iV.  4 .    | +44,

Wkh wzr lqhtxdolwlhv lq +44, uhvshfwlyho| uhsuhvhqw wkh udwlrqdolw| frqvwudlqwv iru wkh vhoohu dqg wkh ex|hu wr sod| elodwhudo ghohjdwlrq1 Wkh odvw ixqfwlrq +htxdo0 lw|, lv wkh djhqwv* sduwlflsdwlrq frqvwudlqw +43,1 Zh ghslfw doo wkhvh ixqfwlrqv lq judsk 7/ lq rughu wr looxvwudwh wkh udqjh ri srvvleoh elodwhudo htxloleuld lq whupv ri frpshqvdwlrq vfkhphv1

Wkh lqhtxdolwlhv ghwhuplqh d srvvleoh duhdDEF zkhuh elodwhudo ghohjdwlrq lv7 dq htxloleulxp1 Dv wkh djhqwv* sduwlflsdwlrq frqvwudlqw lv elqglqj +vhh Ohppd

:_{Vxssrvh wkh glhuhqfh Es3 s7 lv vhw vxfk wkdw M zloo qrw eh deoh wr jdlq dq| srvlwlyh}

:,/ wkh udqjh ri htxloleuld uhgxfhv wr wkh vhjphqw DE= Wkh suhvhqfh ri | lq0 xhqfhv wkh srvlwlrq ri wkh olqh iE @ iV .4._ |= Lqghhg/ zkhq | @ 3> zh

nqrz wkdw dq| iV @ iE @ i fdq eh vxvwdlqhg dv htxloleulxp +dqg lqghhg wkh

vhjphqw DE vxvwdlqv hyhu| i 5 ^3> 4`,1 Iru kljkhu ydoxhv ri |> wklv vhw ri htxl0 oleuld vkulqnv +wkh wklfn vwudljkw olqh vkliwv xszdugv, dqg hyhqwxdoo| froodsvhv wr d vlqjohwrq +srlqw F, zkhq | @ 5 +4.,+4..5_,= Lq wklv fdvh= iV @ V @ _4..4 5 dqg iE @_4..4.5 +E@  5 4..5,= Judsk 7= Htxloleuld zlwk h{rjhqrxv djhqwv* frvwv Compensation scheme fB 1 1 Compensation scheme fS 1/(1+δ) 0 fB = [1/(1+δ)] + [δ/(1+δ)] fS 1/(1+δ) fB = (1+δ) fS C _B A fB = fS + (1+δ/δ)y 1/(1+δ+δ2₎ (1+δ)/(1+δ+δ2₎

Iru lqwhuphgldwh ydoxhv ri |/ dq| frpshqvdwlrq vfkhph ehwzhhq D dqg E lv vxvwdlqdeoh dv elodwhudo ghohjdwlrq htxloleulxp1 Lq whupv ri iV> wkh udqjh

ri htxloleuld lv jlyhq e|= 4. 5  |  iV  4  +4., 5  |1 Lq whupv ri iE> wklv fruuhvsrqgv wr= 4.  5 |  iE  4  +4 . , |

Zhoiduh lpsolfdwlrqv ri elodwhudo ghohjdwlrq Zh zlvk wr rewdlq vrph lqvljkwv derxw wkh zhoiduh lpsolfdwlrqv ri wkhvh htxloleuld1 Lq judsk 8 zh suhvhqw wkh vdph lqirupdwlrq dv lq judsk 7/ exw zlwk uhvshfw wr wkh sulqflsdov* surwv;₁

Wkh suhvhqfh ri d vwulfwo| srvlwlyh frvw iru wkh djhqwv lv dozd|v lqh!flhqw iurp wkh sulqflsdov* srlqw ri ylhz1 Lqghhg/ erwk ri wkhp pxvw jlyh xs d vkduh ri wkh hqwluh slh zkhq klulqj dq djhqw1 Wkrxjk/ wkh glvwulexwlrq ri sd|rv lv vxfk wkdw/ iru vrph ydoxhv ri |> dw ohdvw rqh ri wkh wzr sduwlhv lv ehwwhu r1

7/ zh fdq vwdwh wkdw iru 3  |  5

+4.,6> wkh ex|hu lv wkh zlqqhu zkhqhyhu

klv sd|r lv eljjhu wkdq 

4. +dq| frpshqvdwlrq vfkhph rq wkh vhjphqw GH,1

Rq wkh frqwudu|/ wkh vhoohu lv wkh zlqqhu zkhqhyhu iV A _4.4 +vhjphqw HI ,=

Qhyhuwkhohvv/ htxloleuld zkhuh d sulvrqhu*v glohppd dulvhv duh srvvleoh1 Lq wkhvh htxloleuld 4  iE ? _4. dqg iV ? _4.4 = Wklv lqwhuydo lv uhsuhvhqwhg e| dq|

vhjphqw vlplodu wr DE> wkdw lv/ dq| sruwlrq ri djhqwv* sduwlflsdwlrq frqvwudlqw olqh lqDEH1 Iru7 5

+4.,6  | ? 

5

+4.,+4..5_,> erwk wkh sulqflsdov vwloo ghohjdwh

hyhq li wkh| duh erwk zruvh r1 Ilqdoo|/ iru kljkhu ydoxhv ri |> elodwhudo ghohjdwlrq qhyhu rffxuv1 Judsk 8= Wkh zlqqhu ri wkh jdph +5, Buyer’s profits ΠB 1 1 Seller’s profits ΠS δ /(1+δ) 1/(1+δ) 0 ΠB = δ/(1+δ) - δ/(1+δ) ΠS y = δ2 _{/(1+δ) (1+δ+ δ}2₎ ΠB = 1 - (1+δ) ΠS C B A ΠB = 1 - ΠS - (1+δ/δ)y D E F y = δ2 _/(1+δ)3 y = 0

Djdlq/ zlwk v|pphwulf lqwhuphgldulhv* frvwv/ wkh ghohjdwlrq jdph lv fkdudf0 whul}hg +dw ohdvw iru vrph lqwhuydov ri |, dv d w|slfdo sulvrqhu*v glohppd= E dqg V zrxog eh erwk ehwwhu r e| qrw ghohjdwlqj1 Dv lq wkh suhylrxv vxevhfwlrq/ wkhvh htxloleuld dulvh dv ghohjdwlrq lv d grplqdqw vwudwhj| lq wkh lqwhuydo1 Wkh glhuhqfh zlwk wkh fdvh ri v|pphwulf sulqflsdov* frvwv lv wkdw qrz wkh orvv lv qrw d ghdgzhljkw rqh/ dv wkh lqwhuphgldulhv duh jdlqlqj dqg vkdulqj lw1

71515 H{lvwhqfh ri htxloleuld zlwk rqh0vlghg ghohjdwlrq

Iru vrph ydoxhv ri iE> iV dqg |> htxloleuld zlwk rqh0vlghg ghohjdwlrq duh dv zhoo

srvvleoh1 Li zh jr edfnwr +44,/ lw*v fohdu wkdw/ vd|/ V zloo suhihu QG zkhqhyhu kh h{shfwv E wr sod| iEA iV+4 . , = Olnhzlvh/ E zloo suhihu QG zkhqhyhu kh

+d, Rqo| wkh ex|hu ghohjdwhv> iru wklv wkuhh sod|huv fdvh/ wkh djhqw M*v sduwlf0 lsdwlrq frqvwudlqw lv 

+4.,iE | dqg wkh ex|hu*v udwlrqdolw| frqvwudlqw lv

4  iE  _+4., = Wkhuhiruh/ zh kdyh= V @ |_> M @ |> E @ 4  iE @ 4 4.   | li |   +4.,5 dqg V @ _4.4 > M @ |> E@ _4. li | A _+4., 51 Vxssrvh V zdqwv wr ghyldwh dqg wr sod| G= Kh zrxog jdlq iE

+4., hydoxdwhg

lq iE @4._ |> wkdw lv |_= Dv lq wkh edvholqh prgho/ iru h{wuhph ydoxhv

ri iE> wkh vhoohu lv lqglhuhqw zkhwkhu wr ghohjdwh ru qrw1

+e, Rqo| wkh vhoohu ghohjdwhv> lw fdq eh hdvlo| vkrzq wkdw qrz wkh djhqw L*v sduwlflsdwlrq frqvwudlqw lv  +4.,+4  iV,  |= Wkh uhvxowlqj sd|rv duh V @ iV @ 4  +4 . , |> L @ |> E @ | li |  _+4., 5 dqg V @ _4.4 > L @ |> E @_4. li | A _+4., 51 Vxssrvh E zdqwv wr ghyldwh dqg wr sod| G= Kh zrxog jdlq  +4.,+4  iV, hydoxdwhg lq iV @ 4  +4 . , |> wkdw lv

|= Djdlq/ iru h{wuhph ydoxhv ri iV> wkh ex|hu lv lqglhuhqw zkhwkhu wr

ghohjdwh ru qrw1

Zh uhfdoo iurp wkh suhylrxv vxevhfwlrq wkdw jdlqv iurp rqh0vlghg ghohjdwlrq htxloleuld zhuh qrw hyhqo| glvwulexwhg ehwzhhq wkh sulqflsdov1 Wklv lv qrw wuxh lq wklv fdvh= rqh0vlghg ghohjdwlrq htxloleuld duh vxvwdlqhg lq wkh vdph lqwhuydo iru erwk wkh sulqflsdov1 Wkh suhylrxv dv|pphwu| lv qrz rvhw e| dq dgglwlrqdo rqh1 Wkh zhljkw ri | lv glhuhqw zkhwkhu wklv frvw lv vxvwdlqhg e| wkh uvw pryhu ru e| wkh vhfrqg rqh1 Wkh ex|hu kdv wr jlyh xs wr pruh dv kh kdv wr frpshqvdwh d vhfrqg pryhu djhqw zlwk d vwulfwo| srvlwlyh ghohjdwlrq frvw1

Judsk 9 vkrzv krz wkh ohyho ri | lq xhqfhv wkh sulqflsdov* sd|rv=

Judsk 9= Rqh 0vlgh ghohjdwlrq htxloleuld

Wkh xsshu grzqzdug vorslqj olqh uhsuhvhqwv wkh vhoohu*v sd|rv1 V zloo gho0 hjdwh dv orqj dv wkhvh duh kljkhu wkdq klv sd|rv iurp gluhfw edujdlqlqj/ wkdw lv 4

4.1 Wkh orzhu grzqzdug vorslqj olqh uhsuhvhqwv wkh ex|hu*v sd|rv1 E zloo

ghohjdwh dv orqj dv wkhvh duh kljkhu wkdq klv sd|rv iurp gluhfw edujdlqlqj/ wkdw lv 

4.1 Lw lv fohduo| vkrzq e| wkh judsk wkdw wklv kdsshqv iru wkh vdph ydoxh ri

|> wkdw lv  +4.,5=

Iru kljkhu ydoxhv ri |> dv douhdg| vwdwhg lq sursrvlwlrq 9/ ghohjdwlrq qhyhu rffxuv1

716D frpsdulvrq ehwzhhq sulqflsdov* dqg djhqwv* frvwv

Lq wkh edvlf prgho zlwkrxw frvwv/ zh zhuh frqfhuqhg e| wkh suhvhqfh ri pxowlsoh htxloleuld1 Wkhvh duh vwloo srvvleoh zlwk srvlwlyh ohyhov ri frvwv exw wkhlu udqjh lv gudpdwlfdoo| uhgxfhg hyhq iru yhu| orz ohyhov ri f dqg |= Qhyhuwkhohvv/ wkh lpsdfw ri f dqg | lv glhuhqw1 Lq +45, zh frpsduh wkh wkuhvkrog ydoxhv ri wkhvh frvwv1

Djhqwv* frvwv duh pruh uhvwulfwlyh derxw wkh srvvlelolw| ri elodwhudo ghohjdwlrq htxloleuld1 D uvw srvvleoh h{sodqdwlrq lv wkdw/ iurp wkh sulqflsdov* srlqw ri ylhz/ d wrwdo djhqwv* frvw ri 5| lv dfwxdoo| sdlg dv | .|

+A 5|, e| wkh sulqflsdov1

Qhyhuwkhohvv/ wklv lv qrw ixoo| vdwlvidfwru|1 Iru  @ 4 wkh lpsdfw ri wkh wzr frvwv lv vwloo glhuhqw +f @ 4

7 dqg | @ 49 zlwk elodwhudo ghohjdwlrq htxloleuld,/ hyhq li

wkh wrwdo frvw lv qrz h{dfwo| htxdo wr 5|= Wklv uvw h{sodqdwlrq rqo| kljkoljkwv wkh idfw wkdw wkh frvw f lv sdlg lq d Qdvk ghpdqg jdph dqg/ e| frqvwuxfwlrq/ lv lqghshqghqw ri wlph1 Rq wkh rwkhu kdqg/ | lv vxssruwhg e| lqglylgxdov sod|lqj d Uxelqvwhlq jdph/ zkhuh sd|rv +dqg frvwv, duh dhfwhg e| hohphqwv olnh wkh ghjuhh ri lpsdwlhqfh +,1 Htxloleulxp +G> G, +G> QG, +QG> G, Pd{ f  +4.,5 _4. _4.4 Pd{ | 5 +4.,+4..5_{, +4.,} 5 _+4., 5 +45, Dqrwkhu srvvlelolw| lv wkh iroorzlqj1 Iurp wkh sulqflsdov* srlqw ri ylhz/ f lv dq lqglylgxdo frvw1 Vxssrvh V kdv d srvlwlyh f dqg E kdv qrw frvwv= rqo| wkh vhoohu*v vwudwhj| zloo eh dhfwhg1 Dv uhjdugv |> lw dhfwv erwk wkh sulqflsdov1 Erwk ri wkhp kdyh wr vxssruw d vkduh ri wklv frvw wr eh vxuh wkh djhqwv zloo dffhsw wkhlu rhuv1 Lq dgglwlrq/ vxssrvh L kdv qr frvwv dqg M kdv d srvlwlyh rqh1 Dv wkh rqo| uhohydqw frqvwudlqw lv wkh prvw elqglqj rqh/ wkh sulqflsdov vwudwhj| zrxog vwloo eh wkh vdph dv lq wkh fdvh zlwk v|pphwulf djhqwv* frvw<₁

717 D jhqhudo prgho ri h{rjhqrxv frvwo| ghohjdwlrq

Lq wklv vxevhfwlrq/ zh suhvhqw d jhqhudo prgho zkhuh ghohjdwlrq lv frvwo| erwk iru wkh djhqwv dqg wkh sulqflsdov1 Zh zloo rqo| suhvhqw wkh uhvxowv dqg wkh

<_{Lq wkh rssrvlwh fdvh/ zkhq rqo| L kdv d srvlwlyh ghohjdwlrq frvw/ wkh pd{lpxp + vxssruwlqj}

elodwhudo ghohjdwlrq htxloleuld lv B

pdlq lqwxlwlrqv/ dv wkh zd| wr vroyh wkh jdph kdv ehhq douhdg| glvfxvvhg lq wkh suhylrxv vxevhfwlrqv1

71714 H{lvwhqfh ri htxloleuld zlwk elodwhudo ghohjdwlrq

Wkh frqglwlrqv iru wkh h{lvwhqfh ri elodwhudo ghohjdwlrq htxloleuld duh jlyhq e| d prglfdwlrq ri +44,= ; A A A ? A A A = iV f A _{4 . }iE 4  iE f A _{4 . } +4  iV, iE iV @4._ | +46, Judsk : vkrzv krz wkh udqjh ri pxowlsoh htxloleuld txlfno| vkulqnv1

Judsk := Elodwhudo ghohjdwlrq htxloleuld zlwk h{rjhqrxv frvwv

Compensation scheme fB 1 1 Compensation scheme fS c 0 fB = 1/(1+δ) + δ/(1+δ) fS - c 1/(1+δ) - c fB = (1+δ)fS - (1+δ)c C B A fB = fS + (1+δ/δ)y

Iurp d jhrphwulf srlqw ri ylhz/ wkh suhvhqfh ri f lq xhqfhv wkh srvlwlrq +lqwhufhsw srlqw zlwk wkh d{hv, ri wkh sulqflsdov* frqvwudlqwv1 Lq sduwlfxodu/ srv0 lwlyh frvwv lpso| d grzqzdug vkliw ri wkhvh olqhv1 Zlwk uhjdug wr |> lw pryhv wkh djhqwv* frqvwudlqw olqh xszdugv1 Iurp +46,/ zh fdq zrunrxw d uhodwlrq ehwzhhq wkh ydoxhv ri f dqg | vxfk wkdw elodwhudo ghohjdwlrq htxloleuld duh srvvleoh1 Lq sduwlfxodu/ zh kdyh wkdw=

|  5

+4 . ,4 .  . 5  f_{4 .  . } +4 . ,5 ru f _{+4 . ,} 5  |4 .  .  5

vxssruw elodwhudo ghohjdwlrq htxloleuld1 Dv zh vkrxog h{shfw iurp wkh suhylrxv vxevhfwlrq/ iru f @ 3> wkh pd{lpxp ohyho ri h{rjhqrxv frvwv vxssruwlqj elodwhudo ghohjdwlrq htxloleuld lv | @ 5 +4.,+4..5_,= Judsk ;= Frvwv/ lpsdwlhqfh dqg htxloleuld +4, Principals’ cost c Agents’ costs y δ2_/[(1+δ)(1+ δ+ δ2_)] 1/4 0 _δ/(1+δ)2 1/6 0 < δ < 1 δ = 1

Rq wkh frqwudu|/ zkhq | @ 3> wkh pd{lpxp ohyho ri h{rjhqrxv sulqflsdo frvwv wxuqv rxw wr eh f @ 

+4.,5=

8 Ghohjdwlrq zlwk hqgrjhqrxv rssruwxqlwlhv

Zh qrz lqwurgxfh d glhuhqw nlqg ri frvw/ wkdw lv dq hqgrjhqrxv txdqwlw| { zklfk fdq eh frqvxphg e| wkh djhqwv rqo| rqfh wkh ghohjdwlrq surfhvv lv ryhu1 Wklv dgglwlrqdo slh ri vl}h { lv vkulqnlqj dffruglqj wr wkh djhqw*v rzq glvfrxqw idfwru +zh vwloo dvvxph wkdw doo wkh vxemhfwv kdyh wkh vdph ,1 Wkdw lv/ lwv dfwxdo vl}h ghshqgv rq wkh sduwlfxodu wlph wkh sod|huv uhdfk dq djuhhphqw1 Zh zloo uhihu wr { erwk dv d frvw/ jlyhq wkdw lw lq xhqfhv wkh frpshqvdwlrq vfkhph dq djhqw zloo uhtxluh/ dqg dv dq rssruwxqlw|/ ehfdxvh lw lv d vrxufh ri dgglwlrqdo lqfrph1 Zh fdq lqwhusuhw { dv d zdjh iurp d glhuhqw mre= dq djhqw fdq ghflgh wr gr mxvw wklv mre dqg jdlq {> ru wr dffhsw wkh ghohjdwlrq frqwudfw dqg wkhq jr edfnwr zkdw uhpdlqv ri {=

Sursrvlwlrq 7 fkdudfwhul}hv wkh vhw ri htxloleuld ri wklv jdph1

Sursrvlwlrq 7 Lq d wzr0vlghg ghohjdwlrq jdph zlwk hqgrjhqrxv ghohjdwlrq frvwv { iru erwk djhqwv/ wkh iroorzlqj htxloleuld hphujh= elodwhudo ghohjdwlrq htxloleuld

iru 3  {  5

+4.,> +QG> G, htxloleuld iru 3 ? {  4. > +G> QG, htxloleuld iru

3 ? {  4

4.> dqg gluhfw edujdlqlqj htxloleuld iru { A4.4 =

Surri1 Wkh uhvw ri wkh vhfwlrq zloo surylgh wkh surri iru wklv sursrvlwlrq1

814 Htxloleuld zlwk elodwhudo ghohjdwlrq

sM iV. { @  +sL  iV. {,

iE sL. { @  +iE sM. {,

Dv xvxdo/ wklv v|vwhp ri vlpxowdqhrxv htxdwlrqv phdqv wkdw wkh jdlqv iurp dffhswlqj wkh rwkhu sduw|*v rhu wrgd| pxvw htxdo wkh jdlqv iurp uhmhfwlqj lw dqg pdnh dq rhu wrpruurz1 Wkh uvw htxdwlrq uhihuv wr L dqg wkh vhfrqg wr M1 Wdnlqj L/ iru lqvwdqfh/ klv surwv duh ghwhuplqhg e| wkh vhoolqj sulfh kh jdlqv iurp M/ e| wkh frpshqvdwlrq vfkhph kh pxvw sd| wr V dqg e| wkh rssruwxqlw| {= Dv dq djuhhphqw ehwzhhq L dqg M lv vwuxfnlq wkh uvw shulrg/ wkh uhohydqw vroxwlrq ri wkh v|vwhp deryh lv jlyhq e| L*v uvw rhu/ wkdw lv=

sL @iE_{4 . }. iV . {4  _{4 . }

Wkh vhoolqj sulfh sL lv qrz fruuhfwhg e| wkh suhvhqfh ri dq dgglwlrqdo whup/

wkdw lv {4

4.= Wkh suhvhqfh ri srvlwlyh hqgrjhqrxv frvwv srvlwlyho| lq xhqfhv wkh

htxloleulxp vhoolqj sulfh1 Wkh zd| wklv lq xhqfh zrunv lv pruh fohdu zkhq zh orrndw wkh sd|rv iru hdfk ri wkh sod|huv=

; A A A A A ? A A A A A = L @ sL  iV. { @ iE i_{4 . }V. 5{ M @ iE sL. { @ L @ iE i_{4 . }V. 5{ V @ iV E @ 4  iE

Zh fdq h{suhvv wkh djhqwv* sd|rv dovr lq d glhuhqw zd|/ kljkoljkwlqj wkh zhljkw ri wklv dgglwlrqdo rssruwxqlw|= ; A ? A = L @ iE_{4 . } iV . 5{_{4 . }4 M @ iE_{4 . } iV . 5{_{4 . }

Wkh wrwdo dgglwlrqdo rssruwxqlw| +5{, lv vsolw ehwzhhq wkh djhqwv dffruglqj wr wkh xvxdo Uxelqvwhlq*v uxoh1 Wkh djhqwv gr qrw vlpso| edujdlq dfurvv wkh vkduh ri slh ohiw e| wkh sulqflsdov +iE iV,1 Wkh| dovr frqvlghu wkhlu dgglwlrqdo

rssruwxqlw|/ zklfk lv h{whuqdo wr wkh edujdlqlqj sureohp1 Lq rwkhu zrugv/ dq djhqw*v sd|r lv lq xhqfhg dovr e| wkh rwkhu djhqw*v rssruwxqlw|1 L*v sd|r lv qrw rqo| dxjphqwhg e| {> lw lv dfwxdoo| dxjphqwhg e| d txdqwlw| 5{ 4

4. A {1

Wklv phdqv wkdw L/ dv uvw pryhu/ lv dovr deoh wr rewdlq vrph ri M*v hqgrjhqrxv rssruwxqlw|1 Orrnlqj edfn dw wkh vhoolqj sulfh sL> lq wkh edujdlqlqj surfhvv lw

vhhpv wkdw L jlyhv xs wr d vkduh 

4. ri klv { dqg jdlqv d vkduh 4.4 ri M*v vkduh

+wklv fdq eh pruh hdvlo| vhhq e| doorzlqj iru glhuhqw frvwv/ vd|/ {L dqg {M,1

Hyhqwxdoo|/ L kdv wzr eljjhu vkduhv +5{ 4

Zh nqrz wkdw { sod|v wkh uroh ri wkh djhqwv* uhvhuydwlrq zdjh dv zhoo> lqghhg/ wkh| pd| ghflgh wr uhixvh wkh ghohjdwlrq frqwudfw dqg wr frqvxph gluhfwo| wkh txdqwlw| {= Wkh iroorzlqj djhqwv* udwlrqdolw| frqvwudlqwv pxvw eh vlpxowdqhrxvo| vdwlvhg43₌ ; A ? A = iE iV. 5{ 4 .   { iE iV. 5{ 4 .   { wkdw lv/ iE iV  {4  _ ; 5 ^3> 4` +48,

Wkh frqglwlrq lq +48, lv glhuhqw iurp wkh rqh lq +43,/ zkhq zh lqwurgxfhg h{rjhqrxv djhqwv* frvwv1 Wklv lv gxh wr wkh idfw wkdw { lv qrw rqo| d srvlwlyh uhvhuydwlrq zdjh exw dovr d vrxufh ri dgglwlrqdo surw zlwklq wkh edujdlqlqj sur0 fhvv +l1h1/ lw dsshduv dovr rq wkh ohiw kdqg vlgh ri wkh lqhtxdolw|,1 Lq htxloleulxp/ wkh sulqflsdov { wkh frpshqvdwlrq vfkhphv vxfk wkdw wkh htxdolw| krogv= Ohppd 7 Lq dq| elodwhudo ghohjdwlrq htxloleulxp/ wkh sod|huv* sd|rv zloo eh= L @ {_> M @ {> V @ iV dqg E@ 4iE1 Wkh udqjh ri wkh sulqflsdov* sd|rv

lv dv iroorzv= 4

5 {  V  4 +4,+4.,_ {> +4  , {  E  4 +4,+4.,_5 {= Surri1 Wkh orjlf ri Ohppd 5 vwloo dssolhv wr wkh uvw sduw ri Ohppd 71 Dv uhjdugv wkh vhfrqg sduw/ zh mxvw qrwh wkdw wkh jdph lv vroyhg wkurxjk wkh v|vwhp lq +49,= ; A A A A ? A A A A = iV A _{4 . }4 iE._{4 . }4 { 4  iEA _{4 . } +4  iV. {, iE @ iV. {+4  ,_ +49,

Wkh wzr lqhtxdolwlhv ghwhuplqh wkh udqjh ri ydoxhv ri iV dqg iE vxssruwlqj el0

odwhudo ghohjdwlrq htxloleuld1 Wkh| uhtxluh wkh sulqflsdov* surwv iurp elodwhudo ghohjdwlrq +ohiw kdqg vlgh, wr eh kljkhu wkdq surwv iurp rqh0vlgh ghohjdwlrq +uljkw kdqg vlgh,1 Wkh sduwlfxodu qdwxuh ri { lv vxfk wkdw d sulqflsdo fdq hqmr| vrph ri wkh rwkhu sulqflsdo*v djhqw rssruwxqlw| hyhq li wkh iruphu lv qrw gho0 hjdwlqj +uljkw kdqg vlgh ri wkh lqhtxdolwlhv,1 Wkh qdo htxdolw| lv wkh djhqwv* sduwlflsdwlrq frqvwudlqw +zklfk lv elqglqj, dqg gluhfwo| frphv iurp +48,1 Zh

43_{Djdlq/ li wkh glhuhqfh Es3 s7 lv vhw vxfk wkdw M zloo qrw eh deoh wr jdlq dq| srvlwlyh}

rewdlq htxloleuld zlwk elodwhudo ghohjdwlrq iru {  5

+4., dqg {5  iV  4 {

+ru/ dowhuqdwlyho|/ 4.5

5 {  iE 4  {,1

Judsk < looxvwudwhv wklv lqwhuydo lq whupv ri { dqg iV= Judsk <= Htxloleuld zlwk hqgrjhqrxv rssruwxqlwlhv

Compensation scheme f_S 1 Opportunity x δ2_{/(1+δ) δ}2 _δ 0 f_S = x/δ2 f_S = 1 - (x/δ) 1/(1+δ)

Wkh olplw ydoxh ri { lv eljjhu wkdq wkh rqh zlwk h{rjhqrxv frvwv +|, iru dq| vwulfwo| srvlwlyh 1 Judsk 43 frpsduhv wkhvh wzr glhuhqw frvwv1 Lq wkh uhohydqw lqwhuydo/ wkdw lv iru  5 ^3> 4` > | lv frqfdyh zkloh { lv frqyh{1 Zkhq wkh vxemhfwv duh yhu| sdwlhqw + forvh wr 4,/ vhqvleoh ohyhov ri { fdq vwloo vxssruw elodwhudo gho0 hjdwlrq htxloleuld1 Rq wkh frqwudu|/ wkhvh duh vxssruwhg rqo| iru vpdoo ydoxhv ri |1 Wkh glhuhqfh jhwv vpdoohu wkh pruh wkh lqglylgxdov ehfrph pruh lpsdwlhqw1

Judsk 43= Hhfw ri  rq wkh djhqwv* frvwv Discount factor δ Agents’ costs 0 1 y = d2_{/[(1 + d)(1 + d + d}2_)] x = d2_{/(1 + d)} 1/2 1/6

Wklv lpsdfw lv glhuhqw dv/ zlwk dgglwlrqdo rssruwxqlwlhv ri surw +{,/ ghohjdwlrq lv ohvv gdpdjlqj wkdq zlwk h{rjhqrxv frvwv iurp wkh sulqflsdov* srlqw ri ylhz1 Lq dgglwlrq/ iru { @ 5

+4.,> wkh sd|rv ri wkh sulqflsdov duh wkh iroorzlqj= V @

4

4. dqg E @4.

5

edujdlqlqj +iru dq| srvvleoh ,1 Wkh ohyho ri { srvlwlyho| lq xhqfh wkh sulqflsdov* sd|rv dqg hyhq iru kljk ydoxhv ri wkhvh frvwv wkh sulqflsdov duh qhyhu zruvh r zlwk uhvshfw wr gluhfw edujdlqlqj1

815 Htxloleuld zlwk rqh0vlghg ghohjdwlrq

Htxloleuld zlwk rqh0vlghg ghohjdwlrq duh srvvleoh dv zhoo1 Zh kdyh wkh iroorzlqj fdvhv=

+d, Zkhq rqo| V ghohjdwhv/ E dqg L vroyh wkh iroorzlqj jdph= 

sE iV. { @  +sL iV. {,

sL. 4 @  +sE. 4,

Wkh uvw htxdwlrq uhihuv wr wkh xvxdo vhoohu*v djhqw sureohp1 Wkh vhfrqg htxdwlrq h{sodlqv wkh ex|hu*v ghflvlrq= kh fdq dffhsw wkh vhoolqj sulfh sursrvhg e| L wrgd| dqg jdlq wkh ydoxh ri wkh jrrg/ ru uhmhfw lw dqg sursrvh d glhuhqw vhoolqj sulfh wrpruurz1 Wkh uhohydqw vroxwlrq ri wklv v|vwhp lv sL @4.i_4.V{_4.

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