Cycles and Multiple Equilibria in the Market for Durable Lemons
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- Γερβάσιος Πανταζής
- 6 χρόνια πριν
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Transcript
1 Cyce Mue Equbr he Mre for Durbe Leo Mre C W Je Eru Uery Roer Vr Kryche Tberge Iue Roer Jury Abrc: We ege he ure of re fure yc ero of Aerof 97 where ec cohor of urbe goo eer he re oer e I he yc oe equbr wh quey ffere roere eerge Tycy equbr of he yc oe eer wh hgher quy w orer o e w ore h eer of ower quy Aog oher hg we how for y rbuo of quy h here ex fe uber of cycc equbr where goo re re wh cer uber of ero fer eerg he re Key Wor: Dyc Trg Ayerc Iforo Ery Durbe Goo JEL Cfco: D8 Are for Correoece: Tberge Iue Roer H6-8 Burg Ou 5 36 PA Roer The Neher; e: fx: E-: ryche@feweurnl We h Hugo Hoehy u Roy for hefu uggeo We he ge fro coe e by eber of he uece urg reeo of h er he Jue 999 coferece o "Thry Yer fer "The Mre for Leo" Roer he 999 Euroe Meeg of he Ecooerc ocey go
2 Irouco ce he oeerg wor of Aerof 97 y ecoo he ue he re fure ue o yerc foro oherwe erfecy coee re The r oe ue c re wh oc ge whoe uo ee o quy r reu h oy ow quy goo re re f ee f he buyer re wg o y ore h he reero rce of eer for ech u quy ee o Wo Th o-ce eo robe ffec rge ecru of re cug urce re I y ce cug he cc eco-h cr re he goo uer coero urbe goo Durby rouce wo cocg fcor he ue goo re: goo o re y ero c be offere for e he fuure o ew cohor of oe eer y eer he re oer e Je Roy b he ege oe of he ue h re whe urby excy e o ccou yc oe Je Roy 999 re he ue wheher ge oc of goo c be re oer e They how h y yc coee equbru goo eeuy w be re The e beh h reu h ow quy eer he e cee o w before eg core o hgh quy eer Oce cer ow que re o oy reey hgh que re he re Couer c rec h eer of ffere que w or heee o ffere e ero hece hey re wg o y hgher rce er ero The equbru hu oe whch hgher que re o er ero hgher rce Je Roy 999b re he e ue he coex of re where ec cohor of goo wh ufory rbue quy eer he re oer e I uch re he fe reeo of he c equbru uer ere eeco equbru he yc oe I fc he uque ory equbru o he oy equbru where rce erge quy re re wey oooc oer e They how h here ex e oe oher equbru howeer where goo re re wh fe e fer hey he eere he
3 re Thee equbr re cycc rce que he ee h oce goo re re rce que w f U o he oe goo re o howeer he yc roce of rce que ooocy creg I h er we exe he y of Je Roy 999b uber of wy Fr we rex he uo h eery ero cohor of ufory rbue que eer he re Ie we ow for y rbrry rbuo whch fe oe regury coo eco our reu re roger he ee h we how he exece of fe uber of equbr where goo re re wh fe e fer hey he eere he re Fy we how he exe o whch he ufor rbuo ec I ur ou h for e of ue of he oe' reer e of rbuo whch he reey e robby he eghborhoo of he c equbru obe o coruc yc equbru wh ooocy creg rce que u o he oe eeryhg o We roe exe where h he ce Hece he equbru coruco for he ufor rbuo oe o exe ury o he c of rbuo Oher exg erure o ere eeco h focue o rou rocee uch gg creeg hrough whch he ffcue of rg uer yerc foro y be reoe h ehze he roe of o-re uo h coex uch cerfco erere eg Th er cor oe by ore bc ue whch o uere he org Aerof er z he fucog of he rce ech erfecy coee re whe rer he re foro I or o uer he ure of re fure ue o ere eeco before yzg he roe of uo gg hee fure Our ecfc oe foow We coer coee re for erfecy urbe goo where oe eer re rey fore bou he quy of he goo hey ow Ech ero cohor of eer of equ ze wh ec bu rbrry rbuo of quy eer he re The e e 3
4 oee he foowg e wy Buyer re ec he u e for y ge quy buyer wge o y excee he reero rce of eer for h quy A buyer o o ow he quy her wge o y ero equ he exece uo of goo re h ero Moreoer here re ore buyer h eer ech ero o h equbru rce equ he exece uo Oce re goo re o re-o he e re The Aerof-Wo oe c be coere he c ero of our oe The ere eeco robe e h equbru oy cer rge of ow que re The fey reee ero of c equbru oucoe o equbru our yc oe Hece he ue of exece of yc equbr ey reoe I h yc equbru hgh quy goo re uo foreer We cocere o he exece of oher equbr wh ore ereg roere - where rce erge quy re fucue oer e We roe chrcerzo reu yg h uch equbr he rge of quy whch eeuy re he re excee h he ory c oucoe Moreoer eer of ffere que wh ech cohor of er ere heee ou oer e A he ue ue of ow quy goo ower h h of hgh quy goo ow quy eer e erer h hgh quy eer he ower of goo wh ower quy re erer ower of hgher quy goo w oger I orer o hghgh he wg ec of he ere eeco robe o o e cer he hr cor bewee he roere of equbr of our oe wh hoe of he c oe he r of he y eoe o rog he exece of fe uber of equbru where eery oe eer eerg he re re wh cer fe uber of ero fer eerg he re The reu obe he er roe ffere erece o he ere eeco robe I he c Aerof-Wo oe he ere eeco robe ee for ce Guh W 997 Hee Lzzer 999b Lzzer 999 W 999 Our y ber oe reebce o h by obe 99 of urbe goo oooy where ew cohor of couer eer he re oer e Ue our frewor here o correo bewee he uo of buyer eer h oe 4
5 fe ef he fc h reey hgh quy goo co be re ee he oe g fro re I he yc re for urbe goo he eo robe o o uch he oby of rg reey hgh quy goo bu rher h eer wh reey hgh quy goo ee o w oger orer o re 3 o he co of wg becoe or fcor he wefre o rg ue o yerc foro There re hree or ereor fcor he re whch eere he re yc he o-ory equbr of our oe Fr oce cer rge of quy re oy eer of hgher quy goo re ef he re whch e o roe he rbuo of quy of oey rbe goo he fuure eco he ery of ew cohor of oe eer wh goo of obe quy ue he erge quy of oey rbe goo - hey co be guhe by buyer fro hgher quy eer ef oer fro he Fy e rogree oc of ure goo ccuue fro he he ew cohor of rer eerg he re y ero becoe cregy e gfc eerg he rbuo of quy of rbe goo 4 The er orgze foow eco e ou he oe he equbru coce oe rery reu eco 3 roe chrcerzo reu The reu of he er reg o he exece of fe uber of equbr where goo re re wh fe e fer ery o he re re oue eco 4 eco 5 cocue Proof re coe he Aex 3 There re cer uo whch he fc h eer h we for og e gh ce ow rher h hgh quy Th wou be rue for exe whe he buyer c ec quy - hgh uo buyer re ore ey o ec eec he reey hgh quy houe - eg uo goo of reey ow quy for er ero Tyor 998 A er wh r r h of Ve 997 A e erer our oe ege o uer he ure of he eo robe o we o o ow for y echoogy whch c recy ofy he foro rucure 4 If here o ery of eer fer he ero or equey f buyer c guh he ero of ery of eer he re he oy he fr fcor ree I h ce h bee how erer for fry geer rbuo of quy ee Je Roy 998 h eery equbru goo re re fe e Vce 99 yze yc uco ge wh r feure 5
6 Moe Coer Wr re for erfecy urbe goo whoe quy eoe by re bewee where Te cree exe by K Ech e ero e of eer I eer he re I he e of eer U I I he ero of ery of eer I Ech eer eowe wh oe u of he urbe goo of quy Le he o Lebegue eure of eer fro he e I who ow goo of quy e h or equ o be fuco { } µ whch eee of We ue h µ I creg bouey couou wh reec o he Lebegue eure µ rcy The eure of eer who eer he re ech ero rcy oe o µ Ech eer ow he quy of he goo he eowe wh ere fow uy fro owerh of he goo u he e Therefore he eer' reero rce he coue u of gro uru ue o owerh we ue h excy equ o o he er ero gro uru Ech e ero e of buyer wh eure rger h µ eer he re A buyer re ec he u e A buyer' uo of quy equ o where Thu uer fu foro buyer' uo excee he eer' A buyer ow he ex e rbuo of he eer wh reec o que bu o o ow he quy of he goo offere by rcur eer Whe buyer buy goo he ee he re foreer A yer cou he fuure wh coo cou fcor They re r eur ro ge We w eoe exece quy of he goo fro eer coo o he fc h he beog o cer ube uch h { I } η I µ foow h I µ µ I I I I η { I } h ue efe for I I 6
7 I orer o he ere eeco robe he c oe we ue E where η { I } { I} E he ucoo exece quy of goo E η Th e h he c Aerof-Wo ero of h oe h rge equbru quy whch we w eoe by { η { I [ ] } } x : To fy our y we e he foowg wo uo Throughou h er we ue h hee uo ho Bcy uo ure h he rbuo of quy uffcey we-behe for oe efeghborhoo of Auo Le µ be rcy creg bouey couou wh reec o Lebegue eure o [ µ ] uch h for y µ for oe µ Moreoer e µ M µ : Auo The eure fuco µ f µ Ge equece of re rce { } µ µ µ µ M µ fferebe fuco ech eer chooe wheher or o o e f he chooe o e he e ero whch o e If he chooe o o e h gro uru equ o herefore h e uru equ o zero whe f he ece o e ero h gro uru herefore h e uru equ The e of e ero whch o o e for eer ge by T rgx{ } rg x{ } 7
8 If for he T Ech oe eer chooe e ero T whch o e Le { } I be e of eg eco We w eoe e of he eer who choe e ero for re J foow h J { I } Th geere cer rbuo of que oer e ero he exece quy of he goo offere for e e ero η η { } whe { } J I he eco h foow we w ue he foowg oo: µ µ µ o h for he eure of eer fro µ µ µ J I whoe goo re of quy fro he rge [ ] : µ µ { I [ ]} µ µ b η µ o h for µ f f ; µ η he exece quy of goo fro eer who beog o I whoe goo re of quy fro he rge [ ] η { [ ]} η η η I The foowg e e he couy of η : Le For µ he fuco η rcy creg couou fuco Moreoer η η Mη η η Mη uch h O he equbru h buyer' execo of quy ero where rcy oe eure of goo offere for e u equ he exece quy h e ero A buyer re ec we ue h her execo of quy ero re yerc eoe by E 8
9 A yc equbru equbru where yer roy xze her objece execo re fufe re cer eery ero Defo A yc equbru ecrbe er of: equece of rce { } e of eg eco { } I execo { E } E uch h: eer xze: re oy equece of buyer' quy T for I e eer chooe e ero o b Buyer xze re cer: If { } µ J he E e f here rcy oe ou of re e ero he ech buyer er zero e uru o h he ffere bewee buyg o buyg re cer If { } µ J he E e f zero eure of re occur e ero he ech buyer c er o zero e uru Hece o buyg o for h h ero c Execo re fufe whe re occur: If { } µ J he E η Execo re reobe ee f o re occur: For E Ge he e-u ecrbe boe coo -c re que r coo y h ee ero whch o zero eure of eer e o e buyer' beef hou be reobe Th coo ure h ury e o re y ero co be ue equbru of he yc oe Ge he coo he wge o y hece he rce y ero rerce fro beow by eer wh ow eough que refer o e g h rce rher h o e 3 Chrcerzo of equbru We r he y chrcerzg he roere of y yc equbru I he Prooo 3 beow we fr rgue h f goo of cer quy e ero he goo wh ower que h he eere he re before ero 9
10 w o e h ero Th fc ow u o efe for ech ero rg eer he eer of he hghe quy ero I o ow u o efe he uru of he rg eer ero e Th r of he Prooo 3 bcy foow fro he fc h he ue ue of ow que ower h he ue ue of hgh que o h ow que re ore rey o e The eco r of he Prooo 3 rgue h he rg eer y ero e o-ege e uru Th e h he oher eer e rcy oe uru The hr r of he Prooo 3 rgue h he rg eer ero ffere bewee eg ero eg he fr fuure ero whch quy rger h h ow quy o Prce h fuure ero w be hgher refecg hgher erge quy bu he coue uru uch h he eer ffere The r of he Prooo 3 y h f ex he hghe quy h w eer be o y yc equbru eher equ o or uch h he eer e zero uru I cer h f eer e zero e uru rce fuure ero co be hgher h eer w he cee o w e h fuure ero The Prooo 3 rgue h f he hghe quy o yc equbru e rcy oe uru he u be equ o Prooo 3 equbru chrcerzo Ay yc equbru h he foowg roere For [ ] uch h f he J { [ ] } e eery ero whch rcy oe ou of re occur he e of quy re rge [ ] where he rg quy re ero A eer who re he re by ero ow goo of quy o rger h refer o re h e ero b
11 c Le { } e he fr ero fer where The e rg eer ero ju ffere bewee eg h ero he fr ex ero where rg quy rger h h ow quy ˆ he ˆ If rg x ˆ ˆ e f ero ˆ he rg quy he fr rge oe for ubeque ero he eher h he hghe obe quy or he uru of he correog rg eer zero I ey ee h he fey reee oucoe of he c oe yc equbru of our oe Hece exece of equbru o rey ue here We w how h he yc oe here re fey y oher equbr ech oe rg fro cer eghborhoo of c equbru quy 4 Equbr Trg Goo We w ow how h for y eure fuco µ whch fe Auo - for geerc ue of he reer here ex fe uber of yc equbr coerg que u o A we rey ow h our oe h e oe equbru geer exece roof r Th why we ue coruce roof howg how o f equbru equece of rg que h uch h que u o re re The fc h here re fey y yc equbr foow fro he fc h f here ex yc equbru coerg que u o rg fro oe he here o ex yc equbru coerg que u o rg fro ' wh '
12 Before we w go o he e of he y we fr rouce or reer Auo ow u o efe reer whch ecrbe he reo bewee he rbuo of quy oer he rge [ ] he rg rbuo ef: µ µ f µ Obouy rcy oe We w ow rgue h geercy u be h To h e coer he uru of he rg eer he c oe fuco of : η µ µ µ µ Thu he uru of he rg eer c be wre o o uoe he h 5 Th wou y h oe rgh eghborhoo of Bu h corc he uo h he hghe c equbru quy I he ufor ce we he 6 A he ufor ce ere eeco e h he ufor rbuo ec ce of he ce whe I ubeco 4 we w r wh h e ce whch geerze he y Je Roy 999b We how h oe c coruc "oooc" equece of rg que h re rcy creg oer e u goo re o The reo why he ce ce c be ee by oog equo If we chooe o be guhe fro oher he he eco ero he eure of que ju boe h re o ye o wo e hgh 5 The ce where o-geerc ce
13 he org eure If he rbuo of que he eco ero uch h ew "c" equbru eerge h rger h A he eco ero we c wre o obe o chooe coe eough o uch h If y o be obe o coruc uch "oooc" equbru we how h by exe I ubeco 4 we how h yc equbr eerhee ex f where oe ecreg fuco of The of equbru we ob o oooc howeer rg que rcy ecree for oe e ero fer whch hey rcy cree u goo re o The geer heore roe ubeco 43 A he coruco here becoe que coce ubeco 4 4 re o roe for cc reo beow The coruco of equbru ue "equbru equece" whch efe Defo 4 A equbru equece U que fuco of e Θ { } T U T K T Moreoer for U : Θ fe equece of rg T he er beg efe oer oe rge U uch h equbru coo Defo ho for couou for K T ; T for T b K ; c η he rce ero couou ; for K T e he rce ero excee he rg quy h ero ; 6 Th ey foow fro he fc h µ f 3
14 e for K T where { } T e he rg eer ero ju ffere bewee eg h ero he fr ex ero where rg quy rger h h ow quy 7 The boe efo oe o y he exece of equbru equece Howeer ey o ee h here ex e oe equbru equece ey µ { } Θ uch h eoe boe coo re ry fe The roery of equbru equece we ue h f here yc equbru wh rg que { } by cer equbru equece Θ { } T fferece equo ey T T T T T T uch h for K T c be ecrbe U T he here oy oe whch ree rce rg que for T K o rce rg que for K T h foow fro b boe Iuey T urze he ree roere of he equece of rg que u o e ero T Our uroe herefore o f equbru equece uch h T for oe T 4 The ce where I h ubeco we roe he exece of creg equece T { } where T whe A he ufor rbuo ec ce he reu obe h eco how o wh exe he reu obe Je Roy 999b c be geerze o ow for oher ye of rbuo fuco The foowg heore co ee of he for reu 7 Th u wy ex e oe boe ey T 4
15 Theore 4 For y for y geerc ue of here ex fe uber of yc equbr uch h goo re o fe e fer eerg he re The roof co of hree e I Prooo 4 we roe h obe o coruc equbru equece of rbrry egh where rg que { } re rcy creg ery coe o he c equbru quy Uer hee crcuce he fferece equo e he foowg for: 3 I oher wor he rg eer ero ju ffere bewee eg h ero he ex ero We w eoe uch oooc equbru equece U Θ c yc equbru whch be o he "yc equbru of ye I" Prooo 4 If he here ex fe uber of Θ U T uch h for T for K Θ U U fferebe b for U Prooo 4 e h f uch h: ; Moreoer we c coruc equbru equece of rbrry og egh uch h ero here w be ore eer wh hgh quy goo h he uber of eer wh ow quy Th ow u o ex he equbru equece Θ for oe ore ero Nex Prooo 4 we roe h whe we re be o coruc equbru equece of rbrry egh where rg que beog o cer eghborhoo of he we c ex uch wy h he uru of he rg quy cou be e y ue bewee recey ge y equbru equece coruc oher equece Θ where Θ wh uch h More we c Θ Θ coer he 5
16 whoe er The coo uer whch he Prooo 4 ho re he e he cocuo reche Prooo 4 Th oe er ubeco we w o e ue of Prooo 4 If here ex T uch h for T Θ U uch h for U T uch h for for y b ; U T U ; c U he for y U Θ U uch h: Prooo 4 e u h f we cou re goo for y e ero herefore ccuue "hgh quy eer" he we c orgze re uch wy h he e ero of he equbru equece "o" eer who refer o e h ero w he goo of quy ery coe o Fy Prooo 43 we roe h f we re be o re goo og equbru h fro cer rge of que uch h he rce he ero of he equbru equece c be e y ue bewee he rg quy buyer' uo of he rg quy he we c ex h equbru equece uch wy h wer rge of que cou be re wh he e roere Dog o fer fe uber of ero we geercy c coruc equbru equece where e goo re re by he ero Prooo 43 If for T [ uch h for y U 8 Θ U uch h: T uch h for U ; 8 Here we o' e co bewee A we ee U o be oey oe e whe re boury o 6
17 b ; c ; he eher uch h for y T T or uch h Θ b for y ˆ ˆ ] [ ] U U Θ T uch h for U uch h: T for U ; ; Prooo 43 bcy y h f we he coruce equbru equece for uffcey rge uber of ero he we c eher e ure h fer oe ore e ero he ex rg quy c be choe reey fr fro he ree rg quy uch h erbe roere re e ce b or we c rech ce Prooo e ogeher ge u rge r of he roof of Theore 4 4 The ce of I h eco we coruc equbru equece for he ce whe We fr roe exe howg why he y of he reou ubeco oe o coue o be The exe how h whe re here oe o ex uch h uch h ffere bewee eg ero eg ero 7
18 Exe 4 Le u e 3 eure fuco µ uch h µ f µ f The c equbru quy for h ce uque equ 55 3 I y yc equbru we u he [ ] oherwe we wou he The foowg cure 4 how he grh of fuco X X X where X X X c quy X Fgure 4 I ey o ee h for y ue of we ge boe he uru he eco ero ege // We w ow roe h f reey rcury f he we re be o coruc fey y yc equbr uch h goo fro he rge [ ] re re The equbru equece o-oooc Noe h he reer cofguro yze here ry oer wh he reer cofguro yze he reou ubeco The reu we w roe fory e Theore 4 beow 8
19 Theore 4 For y for y geerc ue of here ex fe uber of yc equbr uch h goo re o fe e fer eerg he re I orer o roe h heore we oy ee o how h whe o obe o coruc equbru equece of rbrry rge egh where rg que { } re ery coe o he c equbru quy We w coruc equece h rcy ecreg for oe e oy he rg quy T excee reou oe We eoe uch equece "equbru equece of ye II" wre U Θ I h ce our fferece equo becoe he foowg ye: K Prooo 44 If Moreoer for K 4 he here ex fe uber of U T uch h for T Θ U U fferebe b for U ; Θ uch h: Noe h he cocuo reche Prooo 44 re ec o he cocuo reche Prooo 4 o h we c e ue of Prooo 4 43 o ge he roof of Theore 4 43 The Geer ce Fy we roe h for y ue of we re be o coruc fey y yc equbr uch h goo [ ] re re The rucure of he correog equbru equece becoe xure of he equbru equece of ye I ye II 9
20 Theore 43 For y geerc ue of here ex fe uber of yc equbr uch h goo re o fe e fer eerg he re Ag e ubeco 4 he oy hg we ee o roe h obe o coruc equbru equece of rbrry rge egh where rg que { } re ery coe o he c equbru quy Th he coe of Prooo 45 Prooo 45 There ex fe uber of Θ U uber T uch h for T Θ U uch h: for K b for U fferebe Moreoer here ex oe U ; The fferece wh ree Prooo 4 44 h here he equbru equece coruce rou ry cooe of creg ubequece ry cooe of ecreg ubequece Therefore we ee o ce o ee rc of he whoe equbru equece Noe h he cocuo reche Prooo 45 re ec o he cocuo reche Prooo 4 o h we c e ue of Prooo 4 43 o ge he roof of Theore 43 5 Cocuo I h er we he roe ffere erece o he wy he ere eeco robe y fe ef urbe goo re where ery e ce he e re I he c Aerof-Wo oe ere eeco reu hgh quy goo o beg be o re ee he oe g fro re The fe reeo of h c equbru o equbru he yc oe where urbe goo re coee re Our reu h er
21 howeer y h here re fey y oher equbr where goo re o wh fe e fer eerg he re I ech of hee yc equbr he rg quy h o he fr ero e eghborhoo of he c equbru Th reu ho rue for geerc ue of he reer goerg he behor of buyer eer he rbuo of que he ouo of eer Referece Aerof G 97 "The re for eo: Que ucery he re ech" Qurery Jour of Ecooc Guh R M W 997 "Leg oe he eo robe" Joho Grue choo of Mgee Core Uery eo Hee I A Lzzer 999 "Ierferg wh ecory Mre" R Jour of Ecooc 3:- Hee I A Lzzer 999b "Aere eeco Durbe Goo Mre" Aerc Ecooc Reew 895: 95-5 Je M Roy 999 "Trg Durbe Goo Wr Mre wh Ayerc Iforo" Tberge Iue Dcuo Per TI 98-5/ Je M Roy 999b "O he Nure of he eo robe Durbe Goo Mre" For Iero Uery Worg Per 99-4 Lzzer A 999 "Iforo Reeo Cerfco Ierere" R Jour of Ecooc forhcog obe J 99 "Durbe goo oooy wh ery of ew couer" Ecooerc Tyor C 999 "Te-o-he-re g of quy" Reew of Ecooc ue
22 Ve N 997 "O he Iforo Roe of Que: Durbe Goo Couer' wor-of-ouh Couco" Iero Ecooc Reew Vce D R 99 "Dyc uco" Reew of Ecooc ue 57 W M 999 "Leg Leo Mor Hzr" Joho Grue choo of Mgee Core Uery eo Wo C 979 "Equbru ere eeco" Aerc Ecooc Reew Wo C 98 "The ure of equbru re wh ere eeco" Be Jour of Ecooc 8 3 Aex Fro ow o we w ue he foowg oo b η η K ; 5 µ F ; 6 c y ; 7 z y y ; 8 e ; f g g g x ; η g g [ ] h f f ; 9 ϕ
23 Proof of Le o reque Proof of Prooo 3 We roe ee of he rooo equey Le u e y ero of oe ou of re y J rgx J o h By he efo of yc equbru we c wre: { } Th e for µ e J Now e y : o for eer wh goo of quy e he who re he re cer ero he o ye re o o re h ero Thu we c efe J { [ ] } { J } u he ey o ee h Fy f µ for oe he we e J b By he equbru efo for E E o h Thu f µ J we he If µ o for he rg eer o re ero ecery coo J o for c uoe σ The we c f eer of quy σ uch h e he he re by ero By efo of { } he w re ero Bu c be how h h o o: [ ] σ σ σ σ o o obe h h obe o he A r rgue how uoe σ We w how h h ce uoe o ˆ ˆ ˆ The u be Le u e eer of quy ˆ ˆ 3
24 { σ} ˆ uch h ˆ ˆ By efo of { } he w eer re becue for x If he howeer re { } ˆ ˆ ero ˆ he wou ge { σ} { σ} ˆ ˆ ˆ whch corco o u be he ce h ˆ ˆ ˆ Proof of Prooo 4 Ug he fc h we exre he exece quy o ero er of µ η : µ η µ µ µ η η Now we coer he fferece coo 3 wh η I c be wre for η η The r of he roof by uco A fr we w roe h f coo o be roe exce re rue for oe he uch h hoe coo re o rue for Nex we w how h here ex uch h hoe coo re fe Fy we w how h for oe T we ge T herefore for T U U T uoe h for oe uch h for U : ; U { } { } ; b re couou fuco fferebe o h we c wre: o o ; c y z ; 4
25 for K ; e for K β where β We w roe h he U U uch h for U equo eere uque ue of fuco of ; ; b re couou fuco fferebe o h we c wre: o o ; c y z ; β where β G Le u fr coer he ef-h e of he equo fuco η η I ey ee h eug G G for y uch h G ye G U o G Ao for y A G couou fuco here u ex eghborhoo U for y U re boh U uch h G Fy G rcy creg couou fuco wr Tg hee fc ogeher yg he eree o heore we c rw he foowg cocuo For U here ex uque couou fuco uch h η couou fuco o 5
26 6 To roe he re of e b of our uco e we w ow how h boh re fferebe fuco For K he fferece coo 3 c be wre η η Tg he fr ffere of h ey wr ug we ge: µ µ µ η µ η µ µ µ η µ η where Tg he ere excy o ccou ye µ µ η µ µ µ µ η µ µ f f f f f f f f Rewrg ge 3 o we c wre A by uo boh re fferebe o Ao uru efe by η o fferebe Nex we roe r c of he uco rgue To h e we c rewre 3 ug 7 8 he foowg wy: y y y z z 3 4
27 y y Ug he uco uo we he z z o foow fro 4 h y urue y z ry we c ge he foowg exreo for he fr ffere of he : y 5 z y 6 Ag by our uco uo we he foow h The oy r of he uco rgue we he o roe re β β where for U β U uch h for U ubrcg 5 for fro 5 for we c wre y y y y 7 We c wre he fferece coo 3 or he fr ffere of whch wr becoe y ubug o 7 ye Rewrg ge 8 We c exre he boe equo er of β e of where β I c be how h 7
28 8 [ ] β β By our uco uo we he β o foow h β herefore β β β L L 9 Now g e u coer he fferece equo 3 ug u fro o rewrg ge Tg he fr ffere wr ug 9 we ge: β β β β β β β β β β β β A re fferebe hey c be wre foow: o The boe equy e h U U uch h for U o or where Now we w how h uco uo re for Le u fr coer he fuco η I er of he fuco µ c
29 be wre µ µ U µ fferebe couou fuco oer Fro he efo of foow h o eghborhoo ey Hece here ex U U uch h for Obouy ef couou fferebe U The ug he efo of y z we ge y z Fy ug ye coequey uco rgue β Th e he roof of he We fh he roof by howg h for oe T we u he T hece for T U : To ee h coer he U T equece The fr er of h equece equ Moreoer he equece ecreg wh rcy ege cree y y Thu here ex fr ege er of T where T I e h for whch c be eoe by U T T 9
30 Proof of Prooo 4 I ce whe we he our fferece equo c be wre 3 or η Fuco η Rewrg ye rcy cree wr ere fuco whch eere for o here ex fuco of Th fuco efe for og η The η y y Ug we c wre: µ η µ µ µ where µ η µ y η η g µ F where g F were efe 9 6 correogy The we ge: y y η y y F K g µ g y F µ ϕ where K ϕ were efe 5 correogy Now e u e y uch h { } y { } The we e 3 uch h rge T uch h 3
31 T g µ x 4M 4 η µ η Prooo 4 for h T here ex correog U T Now e u e he ube Uˆ ˆ T 3 b ; T ; x K T ˆ 3 U T T c u ϕ T U By he uo of he Θ T T U T T uch h for UˆT : T wy obe ϕ ; T Now we w roe h f for oe T UˆT here ex we-efe fuco eere by ϕ he uch h y A fr we roe he exece of howg h f η η ϕ : η ϕ η η µ η µ µ µ η µ µ T 4 η µ µ µ µ µ µ hu here ex uch h y eere by Ug he fc h for T ϕ we c oe wr y 9 : y g ϕ 4Kµ K Fϕ 9 Aoher ouo wy ege oe' fy 3
32 I c be how h he exreo uer he qure roo boe oe herefore y uquey efe by Now we w how h x x we ge: ϕ y K µ Fϕ x 4Kµ Fϕ Ug he we-ow equy h for g µ g g T µ herefore y Now we w roe h T T ˆ T T UT Θ T uch h for U T K T for U T ϕ uoe o he for T for ce we he uco: for T ϕ T ϕ UˆT Bu h Le fx y coer he foowg equece UˆT of uber: { } { } T T The forer cree boue o The er o cree Bu f we e of ϕ we ge corco: ϕ ϕ for T o u be he ce h T T U T Θ T uch h for T K T for Θ U T Now we w roe by uco h for Θ uch h ϕ uoe h for oe ϕ T ϕ T U T U U for U T U Θ uch h for 3
33 33 T K for U ϕ ϕ I e h y y y y ϕ o ug u he fferece equo 3 for fro o we ge or { } u x ϕ T T T T K o for U Fy e u coer ϕ : ϕ ϕ y y y o ϕ O he oher h ϕ ϕ couou o U uch h ϕ h e he uco Bu f ϕ he we he w how boe e u e U U o h for U h e he roof
34 Proof of Prooo 43 I reou y we were coerg fuco of we w coer ˆ ˆ ˆ ˆ ˆ fuco of We efe he foowg fuco: Now where y ˆ ˆ ˆ ˆ ˆ ˆ ˆ I he e r before we rouce fuco ˆ ˆ ˆ ˆ ˆ yˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ϕˆ ˆ ˆ ˆ ˆ Tg he excy [ ˆ η ˆ ˆ uber Coergece ufor hece ˆ ˆ ye h he cuy ex for ˆ where rbrry couou foow h ˆ ˆ ˆ ˆ ˆ f ˆ ˆ ˆ ˆ ˆ : η ˆ ˆ ˆ ˆ f ˆ ˆ η ˆ ˆ The we efe ˆ ˆ ˆ o boury where ˆ ˆ η ˆ ˆ or ˆ by g correog of he fuco ˆ ˆ ˆ whe h ye ˆ ˆ ˆ ˆ ˆ η ˆ ˆ ˆ ˆ Fy we efe ˆ ˆ foow If for oe [ ] ˆ for ˆ [ ˆ ] for K here ex fuco ˆ ˆ ˆ ˆ ˆ ˆ h ˆ η ˆ ˆ ˆ ˆ ˆ he we e ˆ ˆ ˆ ˆ ˆ ˆ ˆ uch I c be ey ee h f η ˆ ˆ ˆ h he foowg rereeo : ˆ ˆ ˆ ˆ ˆ ˆ he Th c be oe by coerg wo ce ey ϕ ˆ ϕ ˆ The forer ye whe he er oe K ˆ y ϕ The f reu he rghforwr ˆ efe oy f ˆ ˆ The exreo ˆ ϕ e ˆ If ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ η or ˆ ˆ ˆ ˆ y ˆ for oe he fuco ˆ o fuco y ore bu fer ero he bee e hoe fuco w eer be eue uch o 34
35 ˆ ˆ K ˆ ˆ K ˆ ˆ ˆ The ue of h rc o ubue coex fuco K by her og for ery rge whe K he eure of "ow quy goo" becoe eggbe core o he eure of "hgh quy goo" L fuco ˆ ˆ K wou he bee excy he e K f here h bee o ery of ew eer 3 Now e u fx ˆ e y ˆ ˆ If for oe we he obe he fuco ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ η ˆ ˆ ˆ he e e he here ex he ex fuco ey ˆ ˆ ˆ ˆ ˆ ˆ uch h ˆ ˆ [ ˆ ] We w how h ˆ ˆ η ˆ ˆ ˆ ˆ ˆ ˆ ˆ for ˆ ˆ uoe o h e h for y y ˆ uch h η ˆ ˆ ˆ ˆ { } ge fe equece ˆ ˆ { ˆ ˆ } boue o ˆ ˆ uch h eher ˆˆ ˆ ˆ ˆ ˆ Le fx y ˆ ˆ or The forer wey creg Bu h e h he er h eher { η ˆ ˆ ˆ } η ˆ ˆ ˆ ˆ ˆ fferece equo ˆ ˆ ˆ ˆ ˆ ˆ { ˆ ˆ ˆ } ˆ ˆ ge re o corco: o oy wo obe re ef: Ce ˆ ˆ uch h for ˆ η ˆ ˆ ˆ ˆˆ whe η ˆ ˆ Tg of he K ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ; 3 No ery ce ecrbe Je Roy
36 b Ce ˆ ˆ uch h for K ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ η whe ˆˆ The ee roof of he Ce o reque We roe h he Ce uch h for y T T µ uch h I oher wor h ce here ex fe uber equbru equece uch h goo re re he ero Θ I he Ce we efe ˆ ˆˆ We how h ] The we roe h eher we he he e reu he Ce or for y T T uch h for U U T Θ U uch h for U Proof of Theore 4 Coequey yg Prooo 4 4 we ge he foowg reu: for y h for y T uch h for U U T Θ U uch Now we c ee h we re uer he coo of Prooo 43 f we e Here we guh hree ce Ce For y K here ex uch h for y T uch h for [ ] U U Θ U uch h: T for U ; ; 36
37 Bu h ce we ge fe equece { } where h corc wh o fer oe e ˆ we u ee eher he Ce or Ce 3 b Ce There ex ˆ uch h here oe o ex ˆ ˆ ] I ccorce wh Prooo 43 we c e cocuo: uch h for y T T Θ uch h I oher wor here re fe uber of equbru equece uch h goo re o he ero he rg uru rcy oe ere fro zero I h ce we c coruc fe uber of yc equbr by coceg equbru equece eg we e Θ { } e yc equbru be he foowg equece of rg eer: { } f f uch h c Ce 3 There ex ˆ uch h here ex ˆ Noe here h ˆ eere er of he reou o ˆ he eure fuco µ reer I oher wor ˆ where ˆ µ Ω ˆ µ ˆ Ω µ Therefore he ce whe Ω oe oeror o-geerc Proof of Prooo 44 We beg wh og he ye of fferece equo 4 wr : K 3 37
38 38 We w oo for uch equece of fuco { } fyg 4 h for K I h ce we he µ µ ubug h o he fr ffere of 3 we ge Hece we c wre o o The o o I foow h f he Thu here ex eghborhoo ey U uch h for U herefore Therefore here ex equece of fuco { } uch h coo o be roe re fe exce he oe we oy he o how h f he here ex U U uch h for U Ge he rucure of { } we c wre: µ µ µ µ K coequey ubug h o he fr ffere of 4 ye
39 herefore o Th e h here ex eghborhoo U U uch h for U og o he coo cuy uffce oe Noe here h o f he uffce coo The we chec wheher : Hece foow h whe o here ex [ ] T uch h for T U U Θ U uch h for U for K by coruco Fy we w roe h f e uffcey rge h uch h 39
40 Le u coer he ro o o I c be wre o Therefore T Th e h we cuy c e y uch h T T uch h for Θ U U uch h Proof of Prooo 45 uoe we he obe equbru equece U uch h for K Θ where U c be rereee o o where for U We rouce he foowg ew rbe: 4 I er of uru c be rereee o There ex e oe of uch equece ey { } where Now we w coruc ew equbru equece Θ Θ he foowg wy We w ree he whoe rucure of K for Θ e I oher wor for K we u f ce er Aoher rue h for K 4
41 4 Hg oe h we c ee h ech of he equece { } for K equbru equece Θ Now we he o f uch h for K oher wor he eer of quy u be ffere bewee eg e ero Looey eg we ry o coruc or of equbru equece of ye II ug Θ ge cooe e of We w how h f he c be oe for y f he Ayg he e roceure he roof of he Prooo 44 we ge he foowg fferece equo: 5 Tg he fr ffere of 5 ug efo of 4 we ge: Ug he fc h ye or Hece we c wre : o o o The o o
42 I foow h for y K og 4 Thu here ex eghborhoo ey U U uch h for U herefore Therefore here ex ye of fuco { } whch fe he fferece equo 5 uch h coo of Θ Θ re fe exce he oe ow we he o f uch h Ge he rucure of { } we c wre: coequey µ µ µ K µ ubug h o he fr ffere of 5 ye herefore o Th e h here ex eghborhoo U U uch h og o he coo cuy uffce oe Now we w chec wheher : 4 Noe h h coo ry fe for 4
43 43 herefore I foow h whe Now we w coruc he ere equbru equece Θ our objece o ge Θ uch h We r fro Θ Θ uoe h we co e I h ce we e oher wor we oo for I foow h hece oe ef eghborhoo of We e Θ Θ Θ where equbru equece ree he ecrbe roceure g Now we w roe h fer oe e we ge To o o we e ower of wh I h ce we he: uoe for y he
44 44 Bu h e h h corc wh Hece here ex oe uber uch h we ge { } Θ The we chec wheher or o If he we re oe I he ce we w how h obe o f uch h equbru equece Θ Θ Θ we ge We e f he Θ Θ Θ oe eghborhoo of If for oe he foow h or O he oher h we he e uo h Therefore here ex uch h j j j whe j j j we he for Θ Θ Θ : j j j j j j I oher wor we c coue wh Θ Θ Θ
45 45 Now we w how h oe ge we w ge uoe o h e h for The we c eue g o ccou h : j j j j j j j j j o for rge herefore 6 The foow h becue oherwe we wou he for rge coequey fy
46 h corc wh our uo h for o here ex ubequece uch h Tg uer og h ubequece ye: Ug he equy 6 we ge: x x x Thu we he roe he foowg ee: herefore uch h Bu h e h for y : og o here ex T uch h for T U U ΘU uch h for U 46
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