Massimo Motta-Competition Policy_ Theory and Practice-Cambridge University Press (2004)

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7

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47

wkh| plqlplvh wkh vrfldo frvwv ri wkh xqhpsor|phqw zklfk uhvxowv iurp wkh fxwedfn ri surgxfwlrq +Jr|ghu/ 4<<6= 4950498,1 Khuh lw lv fohdu wkdw vrfldo dqg srolwlfdo frqvlghudwlrqv lq xhqfh wkh zd| lq zklfk frpshwlwlrq srolf| lv lpsohphqwhg= frpshwlwlrq fdq eh vdfulfhg zkhq wkh vrfldo frvwv ri lw pljkw eh wrr kljk/ vlqfh pdq| upv pljkw h{lw dq lqgxvwu| xqghu frqglwlrqv ri ryhu0 fdsdflw|/ zklfk zrxog uhvxow lq frqvlghudeoh mre orvvhv1 Hyhq li lq wkh orqj0uxq d uhvwuxfwxulqj ri wkh lqgxvwu| zrxog eh ehqhfldo/ lq wkh vkruw0uxq wkhuh pljkw h{lvw frqvlghudeoh frvwv wkdw d jryhuqphqw pljkw zdqw wr dyrlg iru srolwlfdo dqg vrfldo uhdvrqv1 D ghflvlrq zklfk vkrzv wkdw wkh HF frqvlghuv dujxphqwv rwkhu wkdq hfr0 qrplf h!flhqf| zkhq ghflglqj rq krul}rqwdo djuhhphqwv lv Irug2Yronvzdjhq/ zkhuh lw h{hpswhg d mrlqw0yhqwxuh iru wkh ghyhorsphqw dqg surgxfwlrq ri d pxowl0sxusrvh yhklfoh +wkdw lv/ d plql0ydq, ehwzhhq wkh wzr pdmru dxwrpr0 eloh pdqxidfwxuhuv1 Rq sxuh frpshwlwlrq jurxqgv/ lw lv yhu| grxewixo wkdw wklv mrlqw0yhqwxuh vkrxog kdyh ehhq fohduhg +lw lv olnho| wkdw wkh wzr upv kdg hqrxjk vnloov dqg uhvrxufhv wr ghyhors dqg surgxfh lqghshqghqwo| wklv w|sh ri yhklfoh,1 Krzhyhu/ wkh Frpplvvlrq qrwhg wkdw= wkh surmhfw frqvwlwxwhv wkh odujhvw hyhu vlqjoh iruhljq lqyhvwphqw lq Sruwxjdo1 Lw lv hvwlpdwhg wr ohdg/ lqwhu dold/ wr wkh fuhdwlrq ri derxw 8 333 mrev dqg lqgluhfwo| fuhdwh xs wr dqrwkhu 43 333 mrev/ dv zhoo dv dwwudfwlqj rwkhu lqyhvwphqw lq wkh vxsso| lqgxvwu|1 Lw wkhuhiruh frqwulexwhv wr wkh surprwlrq ri wkh kduprqlrxv ghyhorsphqw ri wkh Frppxqlw| dqg wkh uhgxfwlrq ri uhjlrqdo glvsdulwlhv zklfk lv rqh ri wkh edvlf dlpv ri wkh Wuhdw|1 Lw dovr ixuwkhuv Hxurshdq pdunhw lqwhjudwlrq e| olqnlqj Sruwxjdo pruh forvho| wr wkh Frppxqlw| wkurxjk rqh ri lwv lpsruwdqw lqgxvwulhv1 + Irug2Yronvzdjhq = sdud1 69,

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48

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6

Remhfwlyhv ri frpshwlwlrq srolf|/ dqg rwkhu sxeolf srolflhv

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49

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614

Remhfwlyhv ri frpshwlwlrq srolf|

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Zhoiduh +wrwdo vxusoxv,

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4:

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Frqvxphu zhoiduh +frqvxphu vxusoxv,

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5:

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616

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5;

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7

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714

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44

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45

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46

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to @

4  fo  nqtk > 4 . +4  n,q

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4  fk  q+4  n,+fk  fo ,  4  fo . qn+fk  fo , > to @ = 4.q 4.q

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s @

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+9,

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47

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m9@l

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4f = q.4

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+43,

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+45,

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48

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8;

Table 2.1. The entry game 2

E

NE

E

- f, - f

ΠM -f, 0

NE

0, ΠM -f

0, 0

1

Table 2.2. Consumers 1 and 2's payoffs under price discrimination 2

buy I

buy E

buy I

v(2) - p1, v(2) - p2

v(2) - p1, -cE

buy E

-cE, v(2) - p2

v(2) - cE, v(2) - cE

1

p O

R

pm

pc

T

S

c

O′ qm

Figure 2.1. Welfare loss from monopoly

qc

q

p O

R

pm

pc

T

S

c

O′ qm

qc

Figure 2.2. Possible additional loss from rent seeking activities

q

p O R′

′ pm pm

pc′ pc

R

W

V Z T′ T qm′ qm

qc′

c′ S

c O′

qc

Figure 2.3. Additional loss from productive inefficiency

q

0.126 0.124 0.122 0.12 0.118 0.116 0.114 0.112

2

4

n

6

8

10

Figure 2.4. Welfare Wc as a function of the number of firms n (a = 1; c = .5; g = 4)

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+7,

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pl > %

+8,

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55

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+9,

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615

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56

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58

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+<,

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+47,

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+48,

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+49,

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+4:,

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5;

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5<

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63

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+53,

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m = 4  %@ +s,

+54,

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@

%mn

@

sm ^s+4  vm , . %vm ` > s sn vn ^s  %` = s

+55, +56,

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m @ 4> 5> ===> q  4=

+58,

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47

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58

zh mxvw qhhg wr frqvlghu wkh lqfhqwlyh frqvwudlqw iru rqh up= +sp , +4 .  .  5 .  6 . ===,  +sp ,= q

+7,

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4 = q

+8,

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+9,

Wklv fdq eh uhzulwwhq=   +q  4,n@^4 . +q  4,n`1 Wkh LF lv wkh pruh elqglqj wkh odujhu lv n/ dv n lqfuhdvhv wkh UKV pruh wkdq wkh OKV1 Lq wkh olplw/ li 97 Exw uhfdoo wkdw dv|pphwulhv 0 wkdw pljkw lqfuhdvh frqfhqwudwlrq phdvxuhv 0 klqghu frq0 fhqwudwlrq= vhh vhfwlrq 5141

59

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+44,

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6 614

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63

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+4 . q,5   fq = 4 . 9q . q5

+45,

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64

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g   p   f +t s , +LF frooxvlrq,   s +t s ,  gs +t s ,   s +t s ,   s +t s , +LF sxqlvkphqw,1 p  s +t s , p

+46, +47,

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+48,

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615

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+53,

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+54,

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68

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69

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51 +sp ,@ +5+4  ,, A I  +sp ,+4   W ,@ +5+4  ,,1 Wkh qhv dqg wkh suredelolw| ri ehlqj irxqg rxw duh vpdoo hqrxjk iru wkh lqlwldo phhwlqj wr wdnh sodfh/ exw kljk hqrxjk iru d vxevhtxhqw uhqhjrwldwlrq phhwlqj qrw wr wdnh sodfh1 Wklv ohdgv wr wkh zruvw rxwfrph/ zkhuh frooxvlrq dulvhv dw htxloleulxp vlqfh wkh sxqlvkphqw zloo qrw eh uhqhjrwldwhg1

61 +sp ,+4   W ,@ +5+4  ,, A I 1 Khuh wkh h{shfwhg frvw ri d phhwlqj lv vr vpdoo wkdw uhqhjrwldwlrq zloo wdnh sodfh/ eulqjlqj wkh lqlwldo djuhhphqw wr froodsvh= qr frooxvlrq fdq eh vxvwdlqhg dw htxloleulxp1 Glvfxvvlrq PfFxwfkhrq*v dqdo|vlv lv lqwhuhvwlqj ehfdxvh lw xqghuolqhv wkh lpsruwdqfh ri edujdlqlqj dqg qhjrwldwlrq dprqj srwhqwldo frooxghuv1;5 Wklv zrun hpskdvlvhv wkh qhjdwlyh +iurp wkh srlqw ri ylhz ri survshfwlyh fduwho phpehuv, dvshfwv wkdw uhqhjrwldwlrq ri d frooxvlyh djuhhphqw pljkw kdyh1 \hw/ lq glhuhqw frqwh{wv/ vxfk ixuwkhu phhwlqjv pljkw kdyh d glhuhqw uroh1 Iru lqvwdqfh/ diwhu vrph vkrfnv rffxu/ phhwlqjv pljkw eh lqglvshqvdeoh iru frooxvlrq wr frqwlqxh zlwkrxw kdylqj wkh lqgxvwu| hqwhulqj frvwo| sxqlvkphqw skdvhv1 Zh nqrz olwwoh derxw wkh dfwxdo zrunlqj ri fduwhov/ exw Jhqhvryh dqg Pxoolq +5333,/ rqh ri wkh ihz ulfk ghvfulswlrqv zh kdyh ri frooxvlrq +wkh| dqdo|vh wkh Vxjdu Lqvwlwxwh/ dq lqgxvwu| dvvrfldwlrq wkdw {hg uxohv frqgxflyh wr d frooxvlrq rxwfrph dprqj vxjdu uhqhuv,/ lqglfdwh wkdw uhqhjrwldwlrq phhwlqjv zhuh lp0 sruwdqw wr idfh qhz xqiruhvhhdeoh flufxpvwdqfhv lqwhuyhqhg lq wkh vhfwru/ dqg wkdw sxqlvkphqwv glg qrw rffxu qhduo| dv iuhtxhqwo| dv wkhru| zrxog vxjjhvw/ ghvslwh dfwxdo ghyldwlrqv glg wdnlqj sodfh dqg shukdsv gxh wr vxfk phhwlqjv1 Dqg wklv/ zlwkrxw ohdglqj wr wkh froodsvh ri wkh frooxvlyh djuhhphqw1 Lw vhhpv dq duhd zkhuh ixuwkhu uhvhdufk lv qhhghg1

616

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+57,

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+59,

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q+4  ,  +q  4, = +4  q,

+5:,

Wkh LF lpsolflwo| ghqhv wkh ydoxhv ri wkh sxqlvkphqw ohqjwk W iru zklfk froox0 vlrq fdq eh vxvwdlqhg1 Wr qg wkh rswlpdo sxqlvkphqw ohqjwk +wkdw fruuhvsrqgv wr wkh kljkhvw vxvwdlqdeoh frooxvlyh rxwfrph,/ qrwh iurp +58, wkdw Y . ghfuhdvhv zlwk W / vr upv zrxog olnh wr kdyh wkh orzhvw srvvleoh sxqlvkphqw skdvh1 Iru0 pdoo|/ wkh rswlpdo W fdq eh irxqg e| vroylqj wkh sureohp pd{W Y . / vxemhfw wr +5:,1 Wkh vroxwlrq lv jlyhq e| wkh LF kroglqj zlwk htxdolw|1 Wr qg wkh rswlpdo W h{solflwo|/ wdnh wkh orjdulwkp ri erwk vlghv ri h{suhvvlrq +5:,/ dqg uhdgmxvw1 Lw wkhq ehfrphv= q+4  ,  +q  4, = +5;, W  @ +oq ,4 oq +4  q, ;8 Qrwh wkdw iru k ' f/ wklv frqglwlrq lv B D  3 E*?/ wkh xvxdo frqglwlrq iru frooxvlrq lq wkh ghpdqg fhuwdlqw| fdvh1 ;9 Li A ' f/ h{suhvvlrq +59, vlpsolhv wr B : / zklfk lv idovh1 Wkh sxqlvkphqw skdvh fdqqrw eh ri qhjoljleoh gxudwlrq1

6<

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617

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w@3

w@3

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73

wkh surw S +n, hduqhg gxulqj wkh sxqlvkphqw skdvh +wkdw lv/ wkh vwurqjhu wkh sxqlvkphqw,1 Frqvlghu julp vwudwhjlhv= diwhu d ghyldwlrq rffxuv/ doo wkh upv sod| wkh Ehuwudqg sulfh iruhyhu/ wkdw lv wkh| zloo dozd|v fkdujh wkh sulfh zklfkS uhsuhvhqwv 4 wkh rqh0vkrw htxloleulxp dfwlrq1;< Wkhuhiruh/ S +n, @ e +n,1 Vlqfh w@3 w @ 4@+4  ,/ wkh frqglwlrq iru zklfk d up zlwk n surgxfwv suhihuv qrw wr ghyldwh lv=   3n 

G +n,  P > G +n,  e +n,

+64,

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zlwk

N A === A n A === A =

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74

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zlwk

N A === A n A === A =

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zlwk

N A === A n A === A =

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75

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Glvfxvvlrq dqg lpsolfdwlrqv

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Surriv

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N A === A n A === A =

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76

Wklv uhpdun whoov xv wkdw wkh vpdoohvw up lq wkh lqgxvwu| fkdujhv wkh orzhvw shu0surgxfw sulfhv lq wkh lqgxvwu|/ dqg wkh odujhvw up wkh kljkhvw shu0surgxfw sulfhv1 Zh fdq suryh wklv uhvxow e| frqwudglfwlrq1 Vxssrvh wkdw se +p,  sp ? sn  se +n, zkhq p A n1 Wkh surw ixqfwlrq ri wkh up kdylqj wkh uvw p surgxfwv lv jlyhq e|= % # q $&  p   [ sp  f . +p, @ p sl > +65, y  sp 4 .   q q q [email protected] zkhuh zh kdyh pdgh xvh ri wkh idfw wkdw doo wkh p surgxfwv vrog e| wkh up kdyh wkh vdph sulfh sp 1 Wkh uvw0rughu frqglwlrq ri wkh pd{lplvdwlrq sureohp ri wklv pxowl0surgxfw up lv jlyhq e|= q   p   [ p  C+p, . @ 3> @ y  sp 4 .   sl  +sp  f, 4 .   Csp q q [email protected] q +66,

zklfk fdq eh uh0zulwwhq dv= tp @ +sp  f, +4 .   p@q,/ zkhuh tp lv wkh txdqwlw| vrog e| wkh up iru hdfk ri lwv p surgxfwv1 E| dqdorj|/ wkh pd{lpl0 vdwlrq ri lwv surw uhtxluhv wkh iroorzlqj frqglwlrq iru d up zlwk n surgxfwv= tn @ +sn  f, +4 .   n@q,1 Vlqfh e| dvvxpswlrq sp ? sn dqg p A n/ lw pxvw iroorz wkdw tp ? tn 1 Exw iurp Sq wkh ghpdqg ixqfwlrq/ lw pxvw krog wkdwStqp @ y  sp +4 . , . +@q, m @4 sm dqg wkdw tn @ y  sn +4 . , . +@q, m@4 sm 1 E| vxewudfwlqj wkh odwwhu iurp wkh iruphu zh rewdlq= tp tn @ +sn  sp , +4 . ,1 Vlqfh sp ? sn e| dvvxpswlrq/ lw pxvw eh wkdw tp A tn / exw wklv frqwudglfwv zkdw zh kdyh irxqg deryh1 Wklv frpsohwhv wkh surri rq wkh udqnlqj ri sulfhv1 Ohw xv qrz wxuq wr wkh vhfrqg vwhs1 Zh kdyh wr vkrz wkdw p ? n li n ? p1 Zulwh wkh shu0surgxfw surw  p wkdw wkh odujh up zlwk p surgxfwv rewdlqv dw htxloleulxp dv p @  p +sp > ==> sp > sn > ==> sn > sp.n.4 > ==> sq ,/ zkhuh wkh uvw p sulfhv sp duh wkrvh fkdujhg e| wkh p surgxfwv ri wkh odujh up/ dqg wkh sulfhv sn +iurp p . 4 wr p . n, duh wkrvh fkdujhg iru wkh n surgxfwv ri wkh vpdoohu up1 Frqvlghu qrz zkdw kdsshqv li wkh n surgxfwv iurp p . 4 wr p . n fkdujhg wkh kljkhu sulfh sp lqvwhdg ri sn = Vlqfh zh duh frqvlghulqj surgxfwv zklfk duh ghpdqg vxevwlwxwhv/ lw pxvw eh wkdw  p zrxog lqfuhdvh1 Wkhuhiruh/ lw pxvw eh=

p @ p +sp > ==> sp > sn > ==> sn > sp.n.4 > ==> sq , ?  3p @  p +sp > ==> sp > sp > ==> sp > sp.n.4 > ==> sq ,= +67,

Vlqfh upv duh v|pphwulf h{fhsw iru wkh qxpehu ri surgxfwv surgxfhg/ wkh shu0 surgxfw surw hduqhg e| wkh odujh up zkhq erwk wkh vpdoo dqg wkh odujh up fkdujh sulfhv sp rq doo wkhlu surgxfw ydulhwlhv pxvw frlqflgh zlwk wkh shu0shulrg surw hduqhg e| wkh vpdoo up zkhq erwk upv fkdujh sulfhv sp =

3p @ p +sp > ==> sp > sp > ==> sp > sp.n.4 > ==> sq , @ 3n @ n +sp > ==> sp > sp > ==> sp > sp.n.4 > ==> sq ,= +68, 77

Wkh suhylrxv h{suhvvlrq 3n jlyhv wkh shu0surgxfw surw hduqhg e| hdfk ri wkh n surgxfwv ri wkh vpdoo up zkhq lw vhoov doo lwv surgxfwv dw wkh vdph sulfh sp fkdujhg e| wkh odujh up1 Exw dw wkh qrq0frrshudwlyh htxloleulxp wkh ehvw uhvsrqvh ri wkh vpdoo up zkhq wkh odujh up vhoov lwv surgxfwv dw sp uhtxluhv fkdujlqj wkh sulfh sn ? sp = Wkhuhiruh/ lw pxvw eh wkdw= 3n @ n +sp > ==> sp > sp > ==> sp > sp.n.4 > ==> sq , ?  n @  n +sp > ==> sp > sn > ==> sn > sp.n.4 > ==> sq ,= +69, Wklv vkrzv wkdw dw wkh qrq0frrshudwlyh htxloleulxp p ? n li n ? p1 Surri ri Ohppd 61 Surri1 Iru wkh surgxfwv vrog e| wkh qrq0ghyldwlqj upv wr eh srvlwlyh zkhq d up zlwk n surgxfwv ghyldwhv e| xqghufxwwlqj lwv sulfhv wr s/ zh pxvw kdyh=    ns 4  +q  n, sP . tm @ y  sP +4 . , . 3 +6:, q q q zkhuh sP @ +y . f,@5=<7 Diwhu vrph dojheud/ lw fdq eh fkhfnhg wkdw d ghyldwlqj up rewdlqv doo wkh pdunhw li= s+n, @

+n  q,y . +n . q,f = 5n

+6;,

Qrwh wkdw d odujh up zlwk pdq| surgxfw ydulhwlhv zloo qg lw hdvlhu wr xqghufxw d ulydo/ vlqfh wkh UKV ri wkh h{suhvvlrq deryh lqfuhdvhv zlwk n1 Lq rwkhu zrugv/ d odujh up grhv qrw qhhg wr fxw lwv sulfhv dv pxfk dv d vpdoo up vkrxog gr lq rughu wr eh deoh wr h{foxgh wkh ulydov iurp wkh pdunhw gxulqj d ghyldwlrq shulrg1 +Krzhyhu/ zh vkdoo vhh wkdw wkh odujh up kdv dovr ohvv lqfhqwlyh wr xqghufxw1, D qhfhvvdu| frqglwlrq iru vxfk d ghyldwlrq wr eh surwdeoh lv s+n, A 31 Wkhuhiruh/ lw pxvw eh wkdw  A q+y  f,@ ^n+y . f,`  e  +n,1 Vxssrvh qrz wkdw vxfk d ghyldwlrq lv surwdeoh iru d up zlwk n surgxfwv dqg rqh zlwk u ? n surgxfwv1 D up zlwk n surgxfwv rewdlqv iurp wkh ghyldwlrq d shu0surgxfw surw   +n, @   + s+n,> ===> s+n,,/ fruuhvsrqglqj wr wkh vlwxdwlrq zkhuh lw vhoov n surgxfwv doo vhoolqj dw wkh vdph sulfh s+n,1 E| holplqdwlqj vrph surgxfwv dqg nhhslqj xqfkdqjhg wkh sulfh/ wkh shu0surgxfw surw zloo lqfuhdvh=  3 + s+n,> ===> s+n,,/ zkhuh wkh sulfh yhfwru lv qrz frpsrvhg ri   +n, ?   3 +n, @  u ? n lghqwlfdo hohphqwv s+n,1  3 + s+n,> ===> s+n,, ?   + s+u,> ===> s+u,,    +u,/ Ilqdoo|/ lw pxvw eh wkdw   3 +n, @  zkhuh wkh odwwhu lqhtxdolw| frphv vlpso| iurp wkh idfw wkdw wkh rswlpdo ghyldwlrq sulfh iru d up zlwk u surgxfwv lv s+u, dqg qrw s+n,1 Khqfh/ zh kdyh vkrzhg wkdw   +n, ?   +u, iru n A u1 <7 Zkhwkhu

dqrwkhu up lv deoh wr vhoo ru qrw diwhu d up ghyldwhv e| vhwwlqj d sulfh Rh/ ghshqgv rq wkh qxpehu ri ydulhwlhv & ri wkh ghyldwlqj up1

78

Surri ri Sursrvlwlrq 71 Wkh uvw vwhs lv wr frpsxwh h{solflwo| G +n,= Wklv fdq eh grqh dv iroorzv1 E| uhsodflqj tm lq wkh uvw rughu frqglwlrq deryh dqg vroylqj zlwk uhvshfw wr wkh sulfh/ rqh rewdlqv wkh rswlpdo ghyldwlrq sulfh zkhq doo wkh upv kdyh d srvlwlyh rxwsxw1 Wklv lv jlyhq e|= sG +n, @

+5q . q  n,y . f+6q  6n . 5q, = 7+q . q  n,

+6<,

E| vxevwlwxwlrq rqh fdq qg wkh rswlpdo ghyldwlrq surw= G +n, @

+5q . q  n,5 +y  f,5 = 49q5 +q . q  n,

+73,

Lw lv hdv| wr fkhfn wkdw C G +n,@Cn ? 3/ zklfk frqupv wkdw wkh kljkhu wkh qxpehu ri surgxfw ydulhwlhv ri d up/ wkh orzhu wkh ghyldwlrq surwv lw fdq pdnh1 Krzhyhu/ zh dovr kdyh wr fkhfn li wkh ghyldwlrq sulfh sG +n, lv frqvlvwhqw zlwk doo wkh upv vhoolqj d srvlwlyh rxwsxw1 Wklv lv vdwlvhg dv orqj dv sG +n, A s1 E| vxevwlwxwlqj dqg vroylqj zlwk uhvshfw wr / wklv frqglwlrq fdq eh h{suhvvhg dv= # $ s q q5  n5 3  ?  +n, @ 4. = +74, n qn Wkhuhiruh/ wkh ixqfwlrq G +n, kdv ydoxhv lq  5 +3>  3 +n,,>dqg lw lv lqfuhdvlqj lq lwv grpdlq1 Wkh qh{w vwhs lv wr vwxg| wkh ixqfwlrq   +n,1 Li wkh ghyldwlqj up fkdujhv wkh kljkhvw srvvleoh sulfh s zklfk jxdudqwhhv lw zloo eh wkh rqo| up vhoolqj lq wkh pdunhw/ lwv surw lv=   +n, @

+ 5 n5  q5 ,+y  f,5 > 7 5 n6

+75,

zklfk lv dq lqfuhdvlqj dqg frqfdyh ixqfwlrq lq lwv grpdlq  5 +e  +n,> 4,1  +n,>  3 +n,, erwk w|shv ri Vlqfh e  +n, ?  3 +n,/ zh kdyh wkdw lq wkh lqwhuydo +e ghyldwlrq jlyh srvlwlyh surwv wr wkh ghyldwlqj up1 Zh vkrxog wkhq wu| wr vhh zklfk w|sh ri ghyldwlrq lv rswlpdo lq wklv lqwhuydo1 Wkh iroorzlqj fdq eh vkrzq= Uhpdun 9 Wkhuh h{lvwv d ydoxh  +n, 5 +e  +n,>  3 +n,,/ zklfk lv rewdlqhg dv wkh  +n,> vxfk wkdw iru  ?  +n, wkh rswlpdo ghyl0 vroxwlrq ri wkh htxdolw| G +n, @  dwlrq surw lv G +n,1 Surri1 Xqiruwxqdwho|/ wr qg wkh h{solflw irup ri  +n, wxuqv rxw wr eh d gl!fxow wdvn1 Wr suryh wklv uhpdun zh surfhhg lq wzr vwhsv1 Iluvw zh suryh wkdw li  +n, h{lvwv lw pxvw eh +n, A  e+n,1 Zh nqrz wkdw lq wkh srlqw  @  e+n, zh kdyh e+n,, A 31 Wkhuhiruh/ e| s+n, @ 31 Wkhuhiruh/   +n, @ 3/ zkhuhdv G +n m  @   +n,1 frqwlqxlw| wkhuh pxvw h{lvw dq lqwhuydo +e  +n,> , iru zklfk G +n, A  79

Qh{w vwhs lv wr suryh wkdw +n, ?  3 +n,= Zkhq  @  3 +n,/ zh kdyh= s   +y  f,5 +q  n,5 q5 . q q5  n5  n5 3 3   +n m  @  +n,,  G +n m  @  +n,, @ s s   5 A 3= 7qn q . q5  n5 q . q5  n5  n +76, Vlqfh erwk ixqfwlrqv duh frqwlqxrxv dqg lqfuhdvlqj lq +e  +n,>  3 +n,, iurp  G +n, A 3  +n, lq  @  +n,/ lw iroorzv wkdw wkhuh h{lvwv   +n, lq  @ e  +n, dqg G +n, ?   +n,1 Wklv frpsohwhv wkh surri ri wkh rqo| d srlqw  @  +n,/ zkhuh G +n, @  uhpdun1 Zh fdq qrz hvwdeolvk wkh sursrvlwlrq1 Li  5 +3>  plq ` wkhq doo wkh upv kdyh ghyldwlqj surwv  G +n,1 Lq wklv lqwhuydo/ wkh lqfhqwlyh frqvwudlqw ri hdfk up zloo eh jlyhq e| P  +4  , G +n,.e +n,1 Exw P dqg  duh lghqwlfdo iru doo upv/ zkloh erwk wkh ghyldwlrq surw G +n, dqg wkh sxqlvkphqw surw e +n, duh wkh kljkhu wkh orzhu wkh qxpehu ri surgxfwv ri d up1 Wkhuhiruh/ wkh frqvwudlqw lv wkh prvw elqglqj iru wkh vpdoohvw up lq wkh lqgxvwu|/ wkh rqh zlwk  surgxfwv1 Ixoo frooxvlrq lv vxvwdlqdeoh rqo| li    3n @ ^G +,  P ` @ ^G +,   e +,`1  +n,1 Wkh lqfhqwlyh Li  5 ^ 3pd{ > 4,>wkhq doo wkh upv kdyh ghyldwlqj surwv   +n, . e +n,= Lq wklv fdvh dv zhoo/ erwk frqvwudlqw lv jlyhq e| P  +4  ,  wkh ghyldwlrq surw   +n, dqg wkh surw lq wkh sxqlvkphqw skdvh e +n, duh wkh kljkhvw iru wkh vpdoohvw up/ dqg frooxvlrq fdq eh vxvwdlqhg rqo| li     3n @  +,   e +,`1 ^ +,  P ` @ ^

7

Sudfwlfh= Zkdw vkrxog eh ohjdo/ dqg zkdw lo0 ohjdoB

Lq wklv vhfwlrq/ L uvw ghdo zlwk wkh vwdqgdugv ri surri iru frooxvlrq +vhfwlrq 714,/ wkhq zlwk wkh srvvleoh phdvxuhv wr ghwhu frooxvlrq +vhfwlrq 715,/ dqg qdoo| zlwk wkrvh wkdw pljkw euhdn rqjrlqj frooxvlyh sudfwlfhv +vhfwlrq 716,1

714

Vwdqgdugv ri surri= pdunhw gdwd y1 kdug hylghqfh

Dowkrxjk lqgxvwuldo hfrqrplfv kdv ehhq vxffhvvixo lq lghqwli|lqj wkh phfkdqlvpv wkurxjk zklfk frooxvlrq dfwv dqg wkh idfwruv wkdw idflolwdwh lw/ lwv sudfwlfdo lpsol0 fdwlrqv iru ohjdo sxusrvhv duh ohvv vwudljkwiruzdug dqg uhtxluh vrph glvfxvvlrq1 Frqvlghu iru lqvwdqfh d olwhudo xvh ri wkh hfrqrplf ghqlwlrq ri frooxvlrq1 Vlqfh d frooxvlyh rxwfrph lv ghqhg dv d vlwxdwlrq zkhuh sulfhv duh kljk hqrxjk/ rqh frxog wklqn wkdw wr yhuli| wkh h{lvwhqfh ri frooxvlrq lq wkh ohjdo vhqvh +wkdw lv/ ri dqwl0frpshwlwlyh ehkdylrxu,/ rqh kdv wr dqdo|vh sulfh gdwd lq d jlyhq lqgxvwu|/ dqg lqihu li wkh| duh deryh vrph wkuhvkrog ohyhov deryh zklfk wkh| vkrxog eh frqvlghuhg frooxvlyh1 Krzhyhu/ lw zrxog eh yhu| gl!fxow lq sudfwlfh wr orrn dw pdunhw rxwfrphv wr ghflgh zkhwkhu wkhuh kdv ehhq dq lqiulqjhphqw ri dqwl0wuxvw odz/ iru vhyhudo

7:

uhdvrqv1<8 Iluvw ri doo/ lq pdq| flufxpvwdqfhv sulfh gdwd pljkw qrw eh dydlodeoh/ dqg zkhq wkh| duh wkh| pljkw uhihu wr olvw sulfhv udwkhu wkdq hhfwlyh sulfhv +lq pdq| lqgxvwulhv/ dfwxdo sulfhv duh qhjrwldwhg ehwzhhq ex|huv dqg vhoohuv dqg lq rwkhuv wkh| pljkw glhu iurp olvw sulfhv ehfdxvh ri glvfrxqwv/ wkdw pljkw dovr glhu dfurvv fxvwrphuv,1 Vhfrqg/ hyhq li uholdeoh gdwd h{lvwhg/ wkhuh zrxog suredeo| eh glvdjuhhphqw derxw zkdw wkh prqrsro| sulfh lq dq lqgxvwu| lv1 Vhoohuv pljkw kdyh yhu| glhuhqw ylhzv ri zkdw wkdw sulfh zrxog eh/ dqg dq rxwvlgh revhuyhu frxog kdyh |hw glhuhqw shufhswlrqv1 Lw lv dovr zhoo nqrzq wkdw hvwlpdwhv ri frvwv pljkw glhu zlgho|/ vrphwlphv hyhq zlwklq wkh pdqdjhphqw ri wkh vdph up1 Wklug/ vxssrvh wkdw wkhuh lv djuhhphqw rq zkdw wkh prqrsro| sulfh lq wkh lqgxvwu| zrxog eh= krz forvh wr wkh wkhruhwlfdo prqrsro| sulfh vkrxog vdohv sulfhv eh iru wkhp wr eh mxgjhg wrr kljk dqg wkhuhiruh frooxvlyhB Irxuwk/ wkh yhu| sulqflsoh wkdw upv frxog eh frqylfwhg vroho| rq wkh jurxqgv wkdw wkh| fkdujh wrr kljk sulfhv lv d gdqjhurxv rqh/ dqg lw pljkw rshq wkh zd| iru dqwl0wuxvw lqwhuyhqwlrqv zkhqhyhu upv duh vxffhvvixo hqrxjk wr qg frqvxphuv zloolqj wr sd| kljk sulfhv iru wkhlu surgxfwv1 Dv L kdyh uhshdwhgo| dujxhg lq fkdswhu 5/ lw lv qrw pdunhw srzhu e| lwvhoi wkdw dqwl0wuxvw lqwhuyhqwlrq vkrxog sxqlvk1 Udwkhu wkdq orrnlqj dw wkh ohyho ri sulfhv lq wkh lqgxvwu|/ rqh pljkw wkhq eh whpswhg wr lqihu wkh h{lvwhqfh ri frooxvlrq +dv dq lqiulqjhphqw ri wkh odz, e| dqdo|vlqj wkh hyroxwlrq ri lqgxvwu| sulfhv ryhu wlph1 Iru lqvwdqfh/ frxuwv dqg dqwl0wuxvw dxwkrulwlhv kdyh vrphwlphv ehhq whpswhg wr lqihu wkh h{lvwhqfh ri frooxvlyh +loohjdo, ehkdylrxu iurp wkh idfw wkdw vhoohuv fkdujh vlplodu sulfhv ryhu wlph/ wkh vr0fdoohg sulfh sdudooholvp +ru frqvflrxv sdudooholvp,1<9 Exw wr revhuyh wkdw sulfhv pryh lq d vlplodu zd| lv qrw hqrxjk wr hvwdeolvk wkdw upv duh jxlow| ri frooxvlrq1 Frpprq h{rjhqrxv vkrfnv vxfk dv wkh lqfuhdvh lq lqsxw sulfhv ri doo wkh vxssolhuv/ ru dq lqfuhdvh lq lq dwlrq/ ru dq lqfuhdvh lq surs0 huw| sulfhv zrxog suredeo| ohdg doo wkh vhoohuv wr lqfuhdvh sulfhv sursruwlrqdoo|/ zlwkrxw lpso|lqj wkdw wkh| duh frooxglqj1 Ixuwkhu/ zh kdyh douhdg| qrwlfhg wkdw d frooxvlyh rxwfrph pljkw dulvh zlwk0 rxw upv djuhhlqj ru frppxqlfdwlqj wr frruglqdwh wkhlu ehkdylrxu1 Vxssrvh iru lqvwdqfh wkdw 0 hyhq zlwkrxw frpprq vkrfnv rq ghpdqg ru lqsxw sulfhv 0 rqh gd| d vhoohu lqfuhdvhv sulfhv e| 43(/ dqg wkdw wkh qh{w gd| d ulydo iroorzv vxlw1 Lv wklv sulfh sdudooholvp hqrxjk hylghqfh wr frqylfw upvB Vxuho| qrw1 Lw lv ri frxuvh srvvleoh wkdw wkh wzr upv kdyh wdonhg wr hdfk rwkhu dqg suhylrxvo| djuhhg rq fkdqjlqj sulfhv1 Exw lw lv dovr srvvleoh wkdw wkh| kdyh wdnhq wkhlu gh0 flvlrqv zlwkrxw frppxqlfdwlrq1 Wkh uvw up pljkw kdyh vlpso| udlvhg sulfhv lq wkh h{shfwdwlrq wkdw wkh ulydo zrxog iroorz lwv sulfh ulvh/ dqg wkh ulydo pljkw kdyh vlpso| ghflghg wr iroorz/ hlwkhu kdss| wr udlvh sulfhv ru lq wkh h{shfwdwlrq wkdw d idloxuh wr gr vr zrxog wuljjhu d sulfh zdu wkdw zrxog uhgxfh surwv1 Lq <8 Ri frxuvh/ orrnlqj dw pdunhw rxwfrphv lv lpsruwdqw wr wkh h{whqw wkdw lw khosv lghqwli| wkrvh vhfwruv zkhuh wkhuh pljkw eh frooxvlrq/ dqg wkdw wkhuhiruh vkrxog eh vxe mhfw wr d pruh wkrurxjk lqyhvwljdwlrq1 <9 Vhh Vfkhuhu dqg Urvv +4<<3= 66<0679, iru d glvfxvvlrq ri wkh frqvflrxv sdudooholvp grfwulqh dqg lwv hyroxwlrq lq wkh XV1

7;

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8:

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P Q  I , . +4  s, = 4 4

+77,

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P  Q . U+4  , @ s+> U> I ,= P  Q . I +4  ,

+78,

4 4: Vlqfh wkhuh h{lvwv d frqwlqxxp ri srvvleoh htxloleuld/ upv qhhg vrph frruglqdwlrq wr vhohfw wkh ixoo| frooxvlyh rxwfrph jlylqj wkhp wkh shu0shulrg surw  1

8;

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+79,

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+7:,

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+7;,

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+83,

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8<

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P Q  I , . +4  s,+ ,` . +4  ,+P . QU ,> 4 4

+84,

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+85,

zkhqfh= QU @

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P +4 . 4+4 s, , . s+ 4Q  I , = 4  +4  ,

+86,

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;<

Table 4.1. The revelation game

firm 2

Reveal

Not Reveal

Reveal

?N − R, 1− δ ?N −R 1− δ

?N − R, 1− δ ?N −F 1− δ

Not Reveal

?N − F, 1− δ ?N −R 1− δ

firm 1

?N ? − F ) + (1 − p ) M , 1−δ 1− δ ? ? p( N − F ) + (1 − p) M 1−δ 1 −δ

p(

δ P,δ c

δ P,δ c

δP

1

1

δP

0.5

0.5

δc

δc 0.2

q~ P 1 − c 1 n −1 n (a)

q

P

1− c n −1

1 q~ P n

0.2

qP

(b)

Figure 4.1. Incentive Constraints along collusive and punishment paths

δ,δ cn

δ cn

1

δ

0.5

5

10

n

Figure 4.2. Conditions for collusion: Nash reversal (δ cn ) v. two-phase (δ ) punishment strategies

AA Investigation f1

Reveal  N  −R   1− δ     N −R 1 − δ 

a

Reveal

Not Reveal

f2

f2 Not Reveal Reveal

 N  −R    1− δ     N −F  1 − δ 

Figure 4.3. Game tree, at t = 2

 N  −F   1− δ     N −R  1− δ 

1-a

No Investigation  

M M

 

Not Reveal AA Guilty Not Guilty p 1-p  N  −F   − δ 1     N −F  1− δ 

 M   1− δ   M 1− δ

     

1

a NC

NC

aCR (1)

a ( 2)

CNR

CR

~ p 0

p

Figure 4.4. Equilibrium solutions for given policy parameters

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517

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44

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Xqlodwhudo hhfwv ri phujhuv/ devhqw h!flhqf| jdlqv-

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45

zklfk 0 diwhu vroylqj iru wkh frqvxphu sureohp 0 jlyhv wkh gluhfw ghpdqg ixqf0 wlrqv +vhh fkdswhu ; iru ghwdlov,= 5 6 q 47 [ 8 y  sl +4 . , . tl @ +5, sm > 6 6 m @4 zkhuh  5 ^3> 4, lv wkh sdudphwhu ri surgxfw vxevwlwxwdelolw|/ dqg y A f1 Qrwh wkdw lq wklv prgho upv* surgxfwv duh h{rjhqrxvo| jlyhq/ dqg wkdw d phujhu dhfwv qhlwkhu surgxfw fkrlfh qru wkh ghjuhh ri surgxfw vxevwlwxwdelolw|/ zklfk lv dvvxphg wr eh v|pphwulf dprqj doo surgxfwv168 Suh0phujhu Ehiruh wkh phujhu wdnhv sodfh/ zh kdyh wkuhh lghqwlfdo vlqjoh0 surgxfw upv zlwk d surw ixqfwlrq l @ +sl  f,tl / zkhuh tl lv jlyhq e| wkh ghpdqg ixqfwlrq deryh +l @ 4> 5> 6,1 Wkh IRfv C l @Csl @ 3 duh= sl @

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+6,

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6y . f +6 . 5, > 5+6 . ,

+7,

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e @

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=

+8,

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5  [  +sl  f,  y  sl +4 . , . +s4 . s5 . s6 , > 6 6 l@4

+:,

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46

R @

 +s6  f,   y  s6 +4 . , . +s4 . s5 . s6 , = 6 6

+;,

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f +5 . , +6 . 5, . y +9 . 8, f +6 . , +4 . , . y +6 . 5, > sR @ = +43, 5 5 + . 9 . 9, + 5 . 9 . 9,

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L @

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> tR @

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69 + 5 . 9 . 9,

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> R @

+6 . 5,6 +y  f,5 < + 5 . 9 . 9,5

+44,

=

+45,

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6y+6.5 ,fs5.5+6.5 ,s6 = 

+46,

Diwhu wkh phujhu/ wkh ehvw uhsolhv duh ghulyhg iurp IRFv +<, dv= 3  ,f.5s5 .s6 UL = s4 @ 6y.+6.5+6.5 > UR = s4 @ ,

6y+6.5 ,fs5.5+6.5 ,s6 = 

+47,

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47

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71514

Phujhu hhfwv/ dqg pdunhw srzhu

Frqvlghu uvw wkh fdvh zkhuh wkhuh h{lvwv d pxowl0surgxfw up/ L/ zklfk vhoov wkh uvw p surgxfwv lq wkh lqgxvwu|/ zkhuhdv wkh uhpdlqlqj +q  p, surgxfwv duh vrog e| vlqjoh0surgxfw upv1 D phujhu ehwzhhq wkh odujh pxowl0surgxfw up dqg d vpdoo vlqjoh0surgxfw up fdq wkhq eh vwxglhg e| orrnlqj dw wkh hhfw ri dq lqfuhdvh e| rqh xqlw +iurp p wr p . 4, lq wkh qxpehu ri surgxfwv ehorqjlqj wr wkh odujh up/ doo uhpdlqlqj upv vwloo vhoolqj rqh surgxfw rqo|1 Wr qg wkh lqgxvwu| htxloleulxp zulwh wkh surw ixqfwlrqv ri wkh pxowl0 surgxfw up dqg ri hdfk ri wkh rxwvlghuv dv iroorzv= 3 3 44 p p q [ [  C[ +sl  f, C y  sl +4 . , . sm . sn DD > +54, L @ q q l@4 m @4 [email protected] 3 3 p +sn  f, C  C[ y  sn +4 . , . n @ so . sn . q q o

@4

q [

m

@ .4 9@n p

44 sm DD >

n @ p . 4> ==> q

>m

+55,

E| wdnlqj wkh uvw ghulydwlyhv C L @Csl @ 3 dqg Cn @Csn @ 3/ lpsrvlqj v|p0 phwu| rq wkh sulfhv ri wkh pxowl0surgxfw up +sl @ sL iru l @ 4> ==> p,/ dqg ri wkh rxwvlghuv +sn @ sr iru n @ p . 4> ==> q,/ wkh uvw0rughu frqglwlrqv duh= +   +qp,sr    . y . f 4 .   p  5sL 4 .   p q q q  @ 3  +56,  +qp.4,  psL @3 y . f 4 .   q . q  sr 5 +4 . ,  q E| vroylqj wkh v|vwhp rqh rewdlqv wkh htxloleulxp sulfhv dv= sL +p,@

f+q +7q5p4,.5q5 . 5 +5q5 qp5qp5 .5p,,.qy+5q. +5q4,, >  5 +5q5 qp5qp5 .5p,.5q+6qp4,.7q5

+57,

sr +p, @

f+q +7qp5,.5q5 . 5 +5q5 qp5qp5 .5p,,.qy +5q. +5qp,, =  5 +5q5 qp5qp5 .5p,.5q+6qp4,.7q5

+58,

Rqh fdq fkhfn wkdw CsL @Cp A 3 dqg Csr @Cp A 3/ lpso|lqj wkdw wkh odujhu wkh pxowl0surgxfw up +wkh kljkhu p iru dq| jlyhq qxpehu ri surgxfwv q vrog lq wkh lqgxvwu|, wkh kljkhu wkh htxloleulxp sulfhv ri erwk lqvlghuv dqg rxwvlghuv diwhu wkh phujhu1 Frqyhuvho|/ CsL @Cq ? 3 dqg Csr @Cq ? 3= d phujhu zklfk lqyroyhv d fhuwdlq qxpehu ri upv p zloo uhvxow lq kljkhu htxloleulxp sulfhv wkh orzhu wkh qxpehu ri upv lq wkh lqgxvwu|/ wkdw lv/ wkh pruh frqfhqwudwhg wkh lqgxvwu|1 71515

Wkh zhoiduh hhfwv ri d phujhu

Wr vkrz pruh irupdoo| wkh hhfwv ri d phujhu/ ohw xv irfxv xsrq wkh fdvh zkhuh wkhuh h{lvw q vlqjoh surgxfw upv dqg d phujhu ehwzhhq wzr ri vxfk upv rffxuv1 53

Wkhuhiruh/ zh kdyh wr frpsduh wkh htxloleulxp vroxwlrq iru wkh suh0phujhu fdvh p @ 4 zlwk wkdw ri wkh srvw0phujhu fdvh p @ 51 E| uhsodflqj p @ 4 lq wkh h{suhvvlrq +57, zh rewdlq wkh suh0phujhu htxlole0 ulxp sulfh se @ sL +4, @ sr +4,=    y . f 4 .   q > +59, se @ 5 .   q Wkh txdqwlw| vrog e| hdfk up dw htxloleulxp lv jlyhq e|= te @

+y  f, +q . q  , > q +5q . q  ,

+5:,

dqg wkh shu0up surw lv= e @

+y  f,5 +q . q  , 5

+5q . q  ,

=

+5;,

D phujhu fuhdwhv d up zlwk wzr surgxfwv1 Zh fdq qg wkh srvw0phujhu htxlole0 ulxp ydoxhv sL +5,/ sr +5, +zh vkdoo ghqrwh wkhp iru vlpsolflw| sL / sr , e| uhsodflqj p @ 5 lq wkh h{suhvvlrqv +57, dqg +58,= sL @

f+5q+q5, 5 .q+6q8, .5q5 ,.qy+5q.+5q4, , > 5q++q5, 5 .6+q4, .5q,

+5<,

5, ,+q5.q ,.qy+q.+q4, , = sr @ f+q.+qq++ q5, .6+q4, .5q,

+63,

Diwhu vxevwlwxwlqj/ rqh rewdlqv wkh txdqwlwlhv dqg shu0surgxfw surwv ri wkh phujhg up dqg wkh rxwvlghuv dv=   5 5q . q+7q  8, . +5q5  8q . 5, 5 +y  f, tL @ > +64, 5q5 +5q . 6+q  4, . +q  5, 5 ,5

tr @

+q . +q  4,,5 +y  f, = 5q5 +5q . 6+q  4, . +q  5, 5 , 

L @ +q . +q  5,,

+65,

f+q+q56+q4,, 5q5 ,.qy+5q.+5q4, , 5q++q5, 5 .6+q4, .5q,

 r @ +q . +q  4,,

f+q+q4, .q5,.qy+q.+q4, , 5q++q5, 5 .6+q4, .5q,

Wkh phujhu lqfuhdvhv sulfhv dqg ghfuhdvhv frqvxphu vxusoxv1 Surri1 54

5 >

+66,

5 =

+67,

Wkh uvw sduw ri wkh ohppd kdv ehhq vkrzq deryh= Wkh phujhu lqfuhdvhv p iurp 4 wr 5/ dqg erwk lqvlghuv* dqg rxwvlghuv* sulfhv lqfuhdvh zlwk p1 Vlqfh doo wkh surgxfwv duh vrog ehiruh dqg diwhu wkh phujhu/ frqvxphuv duh zruvh r zlwk wkh phujhu/ vlqfh lw udlvhv sulfhv ri doo wkh surgxfwv1 Zkhq wzr upv phujh/ wkh| wdnh lqwr dffrxqw wkh qhjdwlyh shfxqldu| h{whu0 qdolw| wkh| lpsrvh rq hdfk rwkhu/ dqg udlvh wkhlu sulfh1 Wkh rwkhu upv zloo uhdfw e| lqfuhdvlqj wkhlu sulfh +lq wklv prgho wkh surgxfwv duh vwudwhjlf frpsohphqwv, exw qrw dv pxfk dv wkh phujlqj up173 Ohppd 4 Wkh phujhu ehqhwv wkh phujlqj upv1 Surri1 Iluvw/ qrwlfh wkdw se ? sL dqg se ? sr 1 Wklv iroorzv iurp wkh uhvxow wkdw CsL @Cp A 3 dqg Csr @Cp A 3/ dqg uhfdoolqj wkdw sL @ sL +5, A se @ sL +4, dqg wkdw sr @ sr +5, A se @ sr +4,= Ghqrwh wkh shu0surgxfw surw hduqhg e| wkh phujlqj up dv L @ L +sL > sR ,/ zkhuh sL ghqrwhv wkh yhfwru ri wkh rzq +wzr, surgxfw sulfhv dqg sR wkh yhfwru ri wkh rwkhu +q  5, sulfhv fkdujhg e| wkh rxwvlghuv1 Vlqfh jrrgv duh ghpdqg vxevwlwxwhv/ lw pxvw eh wkdw= L +se > se , ? L +se > sr ,1 Lq rwkhu zrugv/ wkh htxlole0 ulxp surwv rewdlqhg e| wkh phujlqj upv ehiruh wkh phujhu pxvw eh orzhu wkdq wkh surw wkh| zrxog jhw li wkh ulydo upv fkdujhg d sulfh sr A se 1 Krzhyhu/ zh dovr nqrz wkdw wkh ehvw uhvsrqvh ri wkh phujlqj upv wr wkh sulfh sr fkrvhq e| wkh rxwvlghuv lv sL A se 1 Wkhuhiruh/ lw pxvw eh=  L +se > sr , ? L +sL > sr ,1 Zkhqfh/ L +se > se , ?  L +sL > sr ,1 Ohppd 5 Wkh phujhu lqfuhdvhv rxwvlghuv* surwv1 Surri1 Zh nqrz wkdw se ? sL dqg se ? sr 1 Xvlqj wkh vdph qrwdwlrq dv lq wkh suhylrxv Ohppd/ dqg jlyhq wkdw wkh jrrgv duh ghpdqg vxevwlwxwhv/ zh kdyh wkdw iru hdfk ri wkh rxwvlghuv r +se > se , ? r +sL > se ,1 Ixuwkhu/ zh nqrz wkdw wkh ehvw uhvsrqvh ri hdfk rxwvlghu wr wkh sulfh sr fkrvhq e| wkh phujlqj up lv sr A se 1 Wkhuhiruh/ lw pxvw eh r +sL > se , ? r +sL > sr ,1 Wklv doorzv xv wr frqfoxgh wkdw r +se > se , ?  r +sL > sr ,1 Wklv uhvxow grhv qrw ghshqg rq zkhwkhu upv frpshwh rq sulfhv ru txdq0 wlwlhv/ dqg klqjhv rq wkh iuhh0ulglqj hhfw hqmr|hg e| wkh rxwvlghuv= zkhq wkh phujlqj upv lqfuhdvh wkhlu sulfhv +ru uhgxfh wkhlu rxwsxw,/ wkh| uhgxfh d qhj0 dwlyh h{whuqdolw| zklfk dhfwv doo wkh lqgxvwu|1 Wkh rxwvlghuv zloo wkhuhiruh ehqhw iurp wkh phujhu1 Zh fdq qrz vwdwh wkh iroorzlqj= Ohppd 6 Wkh phujhu lqfuhdvhv surgxfhu vxusoxv1 73 Lq vhfwlrq 71617 L vkrzhg wkdw zkhq upv kdyh wkh vdph whfkqrorj|/ wkh odujhu wkh up wkh kljkhu wkh sulfh lw zrxog fkdujh= R6 : R& iru 6 : &/ 6 dqg & ehlqj wkh qxpehu ri surgxfwv vrog e| uhvshfwlyho| d odujh dqg d vpdoo up1 Vhh dovr Gdylgvrq dqg Ghqhfnhuh +4<;8,1

55

Surri1 Lw iroorzv wulyldoo| iurp wkh idfw wkdw wkh phujhu lqfuhdvhv erwk wkh surwv ri wkh phujlqj upv dqg ri wkh rxwvlghuv/ dv hvwdeolvkhg e| wkh suhylrxv uhpdunv1 Ohppd 7 Wkh phujhu uhgxfhv qhw zhoiduh1 Surri1 Vhh vhfwlrq 7181

716

H!flhqf| jdlqv iurp phujhuv

Lq wklv vhfwlrq/ L looxvwudwh wkh uroh ri h!flhqf| jdlqv lq phujhu dqdo|vlv1 Wkh suh0phujhu fdvh kdv douhdg| ehhq dqdo|vhg lq wkh suhylrxv vhfwlrq/ iru wkh fdvh zkhuh p @ 4= htxloleulxp sulfhv/ txdqwlwlhv dqg surwv duh jlyhq e| h{suhvvlrqv +59,/ +5:,/ +5;, deryh1 D phujhu ehwzhhq wzr upv fuhdwhv d odujhu up zklfk zloo rzq dqg vhoo wzr surgxfw ydulhwlhv174 Dvvxph wkdw e| frpelqlqj wkhlu dvvhwv/ wkh phujlqj upv pljkw jdlq lq h!flhqf| zlwk uhvshfw wr wkh vlqjoh0surgxfw upv/ dqg eh deoh wr rshudwh dw d xqlw frvw hf/ zlwk h  41 Wkh orzhu wkh sdudphwhu h/ wkh kljkhu wkh h!flhqf| jdlqv hqwdlohg e| wkh phujhu1 Lq rwkhu zrugv/ wkh sursruwlrq ri frvwv vdyhg e| wkh phujhu lv 433+4  h,(1 L vkdoo vkrz/ dprqj rwkhu wklqjv/ wkdw wkh pruh olnho| d phujhu lv wr eh zhoiduh lpsurylqj wkh kljkhu wkh h!flhqf| jdlqv lw fuhdwhv1 Wr qg wkh lqgxvwu| htxloleulxp diwhu d phujhu ehwzhhq wkh uvw wzr upv +L @ 4> 5,/ zulwh wkh +shu0ydulhw|, surw ixqfwlrqv ri wkh phujlqj upv +lqvlghuv, dqg ri hdfk ri wkh rxwvlghuv dv iroorzv= $$ # # q [ +sL  hf,  sl > L @ 4> 5 +68, y  sL +4 . , . 5sL . L @ q q l@6 3 3 44 q [ +sm  f, C C y  sm +4 . , . 5sL . sm . m @ sl DD > q q l@6>l9@m

m @ 6> ==> q +69,

zkhuh L kdyh xvhg v|pphwu| rq wkh phujlqj upv* sulfhv= s4 @ s5 @ sL 1 Rqfh lpsrvlqj v|pphwu| rq wkh rxwvlghuv/ vr wkdw sl @ sm @ sr / wkh uvw0rughu frqglwlrqv duh jlyhq e|= +     sr  5sL 4 .   5q @ 3 y . hf 4 .   5q .  +qq5,     +6:,  +q6,  L @3 y . f 4 .   q . 5s q  sr 5 4 .   q  q 74 Wkh dqdo|vlv frxog eh h{whqghg wr frqvlghu phujhuv ehwzhhq lq vlpsolflw| dqg qrw jdlq dq| ixuwkhu lqvljkwv e| grlqj vr1

56

6 upv/ exw zh zrxog orvh

E| vroylqj wkh v|vwhp deryh lq sr dqg sL / rqh rewdlqv wkh htxloleulxp sulfhv diwhu wkh phujhu dv= sL @

f++q5,+hq.q.h4, 5 .q+6hq6h5, .5hq5 ,.qy +5q.+5q4, , > 5q++q5, 5 .6+q4, .5q,

+6;,

 ,+q.+q4.h, ,.qy+q.+q4, , sr @ f+q.+q5, = q++q5, 5 .6+q4, .5q,

+6<,

Diwhu vxevwlwxwlqj/ rqh rewdlqv wkh shu0surgxfw surw ri wkh phujhg up dqg wkh rxwvlghuv dv=  5 f++4h,+56q.q5 , 5 .q+q56h+q4,, 5hq5 ,.qy+5q.+5q4, , > L @ +q . +q  5,, 5q++q5, 5 .6+q4, .5q, +73,  r @ +q . +q  4,,

f++4h,+q5, 5.q+qh,.q5,.qy+q.+q4,, 5q++q5, 5 .6+q4, .5q,

5 =

+74,

L qrz qg wkh hhfwv ri wkh phujhu rq frqvxphu vxusoxv/ upv* surwv dqg qhw zhoiduh/ xvlqj dv wkh ehqfkpdun wkh htxloleulxp vroxwlrqv lq wkh lqgxvwu| ehiruh wkh phujhu/ dqg xqghu wkh dvvxpswlrq wkdw wkh phujhu grhv qrw uhgxfh wkh qxpehu ri surgxfwv zklfk duh vrog lq wkh pdunhw +vhfwlrq 717 uhod{hv wklv dvvxpswlrq,1 Wkh phujhu hhfw rq frqvxphu vxusoxv1 Wkh iroorzlqj Ohppd jlyhv d qhfhvvdu| dqg vx!flhqw frqglwlrq iru wkh phujhu wr uhvxow lq orzhu sulfhv dqg wkxv ehqhw frqvxphuv1 Ohppd 8 Wkh phujhu lv ehqhfldo wr frqvxphuv li dqg rqo| li lw lqyroyhv hqrxjk h!flhqf| jdlqv/ l1h1 li dqg rqo| li=    f q5  6q . 5  5 . q +6q  7,  . 5q5  qy = +75, h  h  f +q . +q  5,, +5q . +q  4,, Surri1 Vlqfh zh duh frqvlghulqj wkh fdvh zkhuh doo wkh surgxfwv duh vrog ehiruh dqg diwhu wkh phujhu/ d vx!flhqw frqglwlrq iru frqvxphuv wr eh ehwwhu r zlwk wkh phujhu lv wkdw sulfhv ri doo wkh surgxfwv duh orzhu diwhu wkh phujhu1 Krzhyhu/ lw wxuqv rxw wkdw wkh frqglwlrq iru zklfk sr  se frlqflghv zlwk wkh frqglwlrq iru zklfk sL  se / erwk ri wkhp uhtxlulqj wkdw h  h1 Wklv lpsolhv wkdw wkh frqglwlrq lv erwk qhfhvvdu| dqg vx!flhqw1 Lw lv hdv| wr fkhfn lw lv wkh fdvh e| grlqj vrph vlpsoh exw whglrxv dojheud1 Zulwh sr +h, @ sr +h,  se= Wkh lqhtxdolw| sr +h,  3 lv vdwlvhg rqo| e| h  h= Olnhzlvh/ rqh fdq fkhfn wkdw sL +h, @ sL +h,  se  3 lv dovr vroyhg rqo| e| h  h= 57

Wkh Ohppd deryh fdq eh uhdg dv iroorzv1 Iluvw/ frqvxphuv zloo ehqhw iurp wkh phujhu rqo| li lw ghfuhdvhv sulfhv1 Vhfrqg/ sulfhv ghfuhdvh rqo| li wkhuh duh hqrxjk h!flhqf| jdlqv1 Qrwh dovr wkdw h ? 4 fdq eh uhzulwwhq dv q+y  f, ? 31 Wklv lpsolhv wkdw d phujhu zklfk grhv qrw hqwdlo dq| h!flhqf| jdlq +wkdw lv/ d phujhu vxfk wkdw h @ 4, zloo dozd|v lqfuhdvh sulfhv dqg wkxv zloo qhyhu lpsuryh frqvxphu vxusoxv1 Wkh uhdghu fdq fkhfn wkdw C h@Cq A 31 Wkh kljkhu wkh qxpehu ri upv wkh vpdoohu wkh h!flhqf| jdlqv wkdw duh uhtxluhg wr udlvh frqvxphu vxusoxv= vlqfh wkh fulwlfdo ydoxh h lqfuhdvhv/ lw lv hdvlhu wkdw wkh frqglwlrq h  h eh vdwlvhg1 Zkhq d odujh qxpehu ri upv rshudwh lq wkh lqgxvwu|/ wkh h{whqw wr zklfk sulfhv lqfuhdvh diwhu wkh phujhu ehwzhhq wzr ri wkhp lv pxfk uhgxfhg1 Hdfk up lqwhuqdolvhv wkh h{whuqdolw| rq wkh sulfh ri wkh sduwqhu/ exw zlwk d odujh qxpehu ri rxwvlghuv wkh hhfw ri wkh phujhu rq sulfhv ehfrphv pdujlqdo/ dqg d vpdoo h!flhqf| jdlq fdq rxwzhljk wklv qhjdwlyh hhfw1 Dw wkh rwkhu h{wuhph/ zkhq wkhuh duh rqo| wzr upv lq wkh lqgxvwu| wkh phujhu zloo fuhdwh d prqrsro| dqg wkxv uhvxow lq wkh pd{lpdo lqfuhdvh lq pdunhw srzhu= rqo| h{wuhpho| kljk h!flhqf| jdlqv pljkw lq sulqflsoh rxwzhljk wkh qhjdwlyh zhoiduh hhfw gxh wr kljkhu sulfhv1 Wklv vwuhqjwkhqv wkh udwlrqdoh iru fkdoohqjlqj phujhuv zklfk rffxu lq d pruh frqfhqwudwhg lqgxvwu| +vhh vhfwlrq 515,1 Wkh lpsdfw ri wkh phujhu rq surgxfhu vxusoxv1 Ohw xv qrz wxuq wr wkh hhfw ri wkh phujhu rq wkh upv* surwv/ nhhslqj rxu dvvxpswlrq wkdw wkh phujhu grhv qrw irufh rxwsxw ri wkh rxwvlghuv wr }hur1 Wkh uvw vwhs lv wr vwxg| wkh lpsdfw ri wkh phujhu rq wkh lqvlghuv* surwv1 Lq wkh sulfh0frpshwlwlrq vhwwlqj zh duh dqdo|vlqj/ wkh upv zklfk xqghuwdnh wkh phujhu dozd|v jdlq iurp lw/ lqghshqghqwo| ri wkh h!flhqf| jdlqv/ dv wkh iroorzlqj uhpdun vwdwhv1 Uhpdun 9 Wkh phujhu ehqhwv wkh phujlqj upv1 Surri1 Uhfdoo wkdw hyhq zlwkrxw h!flhqf| jdlqv/ d phujhu lv surwdeoh iru wkh phuj0 lqj upv +vhh Ohppd 5,1 Wklv lpsolhv wkdw Le +h,  5 + L +h,  e , A 3 iru h @ 41 Qh{w/ lw fdq eh fkhfnhg wkdw wkh ixqfwlrq Le +h, lv frqyh{=  5 f5 +q . +q  5,, 6q+q  4, . 5q5 . +q5  6q . 5, C 5 Le @ A 3> +76, 5 Ch5 q7 ++q  5, 5 . 6+q  4, . 5q, Wr pdnh vxuh wkdw Le lv dozd|v srvlwlyh zh mxvw qhhg wr fkhfn wkdw wkh uvw ghulydwlyh grhv qrw fkdqjh vljq rq lwv grpdlq1 Vlqfh CLe @Ch lv qhjdwlyh zkhq hydoxdwhg dw h @ h ? 3/ wklv dprxqwv wr fkhfnlqj wkdw CLe @Ch ? 3 dw h @ 41 Vrph dojheud vkrzv wkdw= CLe f+yf,+q. +q5,,+5q5 .6q+q4, .+q5 6q.5, 5 ,+5q5 .q+5q4, , +h @ 4, @  ? 3= q7 ++q5, 5 .6+q4, .5q,5 Ch +77, Wkh orzhu h +wkh vwurqjhu h!flhqf| jdlqv, wkh pruh surwdeoh wkh phujhu1 Wklv frpsohwhv wkh surri wkdw wkh phujlqj upv dozd|v jdlq iurp wkh phujhu1 58

Wkh qh{w vwhs lv wr vkrz wkdw wkh phujhu dozd|v ehqhwv wkh rxwvlghuv xqohvv wkhuh duh kljk hqrxjk h!flhqf| jdlqv iru wkh phujlqj upv1 Uhpdun : Wkh phujhu lqfuhdvhv rxwvlghuv* surwv li h!flhqf| jdlqv duh vpdoo hqrxjk/ l1h1 li h A h= Surri1 Zh nqrz iurp wkh suhylrxv vhfwlrq wkdw li wkhuh duh qr h!flhqf| jdlqv/ wkh rxwvlghuv jdlq iurp wkh phujhu= r +se > se , ? r +sL > sr ,/ iru h @ 41 Wkh qh{w vwhs lv wr ghqh wkh ixqfwlrq er +h,  +q  5, + e   r +h,,/ zkrvh vljq zloo whoo h, @ 31 Zkhq xv zkhwkhu rxwvlghuv jdlq iurp wkh phujhu1 Iluvw/ qrwh wkdw er + h @ h zh kdyh sr @ sL @ se 1 Wkhuhiruh/ lw pxvw eh dovr wkdw tr @ tL @ te 1 Wklv lpsolhv wkdw  r @ +sr  f,tr @ +se  f,te @ e / vlqfh wkh phujhu grhv qrw dhfw surgxfwlrq frvwv ri wkh rxwvlghuv1 Ilqdoo|/ lw lv hdv| wr vhh wkdw zkloh e lv qrw d ixqfwlrq ri h/ wkh ixqfwlrq r +h, lqfuhdvhv zlwk h1 Khqfh/ er +h, lv ghfuhdvlqj rq lwv grpdlq1 Wkhuhiruh/  e  r +h, iru h  h1 Wkh iroorzlqj sursrvlwlrq hvwdeolvkhv wkdw ghvslwh wkh rxwvlghuv pljkw orvh iurp wkh phujhu +li wkhuh h{lvw lpsruwdqw h!flhqw jdlqv iru wkh phujlqj sduw0 qhuv,/ wkh lqgxvwu| surwv lqfuhdvh zlwk wkh phujhu1 Ohppd ; Wkh phujhu dozd|v lqfuhdvhv surgxfhu vxusoxv1 Surri1 Vhh vhfwlrq 7181 Wkh qhw zhoiduh hhfw ri phujhuv Zh fdq qrz orrn dw wkh ryhudoo hhfw ri wkh phujhu xsrq zhoiduh/ dqg hvwdeolvk wkh iroorzlqj vx!flhqw frqglwlrq= Ohppd < Wkh phujhu lpsuryhv qhw zhoiduh li lw lqyroyhv hqrxjk h!flhqf| jdlqv/ l1h1 li=    f q5  6q . 5  5 . q +6q  7,  . 5q5  qy = h  h  f +q . +q  5,, +5q . +q  4,, Surri1 D vx!flhqw +exw qrw qhfhvvdu|, frqglwlrq iru zhoiduh wr lqfuhdvh zlwk wkh phujhu/ lv wkdw erwk frqvxphu vxusoxv dqg surgxfhu vxusoxv lqfuhdvh1 Wkh wzr Ohppdwd deryh kdyh vkrzhg wkdw iru h  h frqvxphuv jdlq dqg djjuhjdwh surwv dovr ulvh1 Wkhuhiruh/ zhoiduh lqfuhdvhv lq wklv lqwhuydo1 Wklv frqfoxghv rxu whfkqlfdo wuhdwphqw ri h!flhqf| jdlqv/ pdgh xqghu wkh dv0 vxpswlrq wkdw doo rxwvlghuv frqwlqxh wr rshudwh lq wkh lqgxvwu| diwhu wkh phujhu wdnhv sodfh1 Wkh iroorzlqj vhfwlrq frqvlghuv wkh srvvlelolw| wkdw diwhu wkh phujhu vrph rxwvlghuv duh gulyhq ri wkh pdunhw1

717

H!flhqf| rhqfh= Zkhq wkh phujhu ohdgv wr wkh h{lw ri wkh rxwvlghuv

Lw lv frqfhlydeoh wkdw wkh phujhu/ e| pdnlqj wkh phujlqj sduwqhuv pruh h!flhqw zlwk uhvshfw wr wkh rxwvlghuv/ pljkw doorz wkh iruphu wr sxvk vrph ri wkh odwwhu upv rxw ri wkh pdunhw1 L qrz dqdo|vh wklv srvvlelolw| dqg lwv lpsolfdwlrqv1 59

Lq rughu iru wkh rxwvlghuv qrw wr vhoo dq|wklqj diwhu wkh phujhu/ lw pxvw eh wkdw tr  31 Wkh txdqwlw| vrog e| wkh rxwvlghuv lv jlyhq e| htxdwlrq +65,1 Wkhuhiruh/ wkh phujhu ohdgv wr wkh phujlqj upv wr eh wkh rqo| vhoohu li=   f q5 +4 . , . +q  5, 5  q +q . +q  4,, y = +78, h  hh{ @ f +q . +q  5,, Qrwh wkdw hh{ lv qhjdwlyh zkhq q lv odujh hqrxjk dqg zkhq  lv vpdoo hqrxjk/ lpso|lqj wkdw li wkh lqgxvwu| lv qrw yhu| frqfhqwudwhg dqg surgxfwv duh lpshuihfw vxevwlwxwhv wkh phujhu zloo qhyhu uhvxow lq rxwvlghuv h{lwlqj wkh lqgxvwu|/ hyhq lq fdvh ri lpsruwdqw h!flhqf| jdlqv iru wkh lqvlghuv1 Lw lv dovr vwudljkwiruzdug wr vhh wkdw Chh{ @Cq ? 3/ dqg wkdw Chh{ @C A 31 Lq wkh h{wuhph fdvh zkhuh wkh jrrgv duh shuihfwo| krprjhqrxv/ d pdujlqdo frvw lpsuryhphqw lv hqrxjk wr irufh wkh rwkhu upv rxw ri wkh lqgxvwu| +olp $4 hh{ @ 4,1 Zkhq wkh phujhg up lv wkh rqo| rqh ohiw lq wkh pdunhw/ lw zloo fkdujh wkh prqrsro| sulfh sp @ +hf . y,@5 iru hdfk ydulhw|1 Lw lv hdv| wr fkhfn wkdw wkh zhoiduh ohyho dwwdlqhg lq wklv vlwxdwlrq lv jlyhq e|= ZP @

6+y  hf,5 = ;

+79,

Wkh phujhu lv ehqhfldo wr vrflhw| dv d zkroh li ZP A Ze / zkhuh wkh odwwhu lqgl0 k   fdwhv wkh zhoiduh ohyho ehiruh wkh phujhu dqg lv jlyhq e| Ze @  5 +q  4,5 . 7q+q  4, . 6q5 +y  f,5 @ 5 +5 Lw fdq eh vkrzq wkdw ZP A Ze li=75 s y 5 6q5 . 7q+q  4, . +q  4,5  5 s h  hz @  = f 6f +q . +q  4,,

+7:,

Lw fdq eh fkhfnhg wkdw Chz @C ? 3 dqg Chz @Cq ? 31 Wklv phdqv wkdw wkh pruh vxevwlwxwdeoh wkh surgxfwv dqg wkh odujhu wkh qxpehu ri upv/ wkh pruh gl!fxow wkdw d phujhu zklfk jlyhv ulvh wr d +wzr0surgxfw, prqrsro| zrxog eh zhoiduh lpsurylqj1 Wkh uhdvrq ehklqg wklv uhvxow lv wkdw d kljkhu qxpehu ri upv ru ohvv glhuhqwldwhg jrrgv pdnh wkh ehiruh0wkh0phujhu zhoiduh ohyho kljkhu/ gxh wr vwurqjhu frpshwlwlrq lq wkh pdunhw1 Iljxuh 817 khosv xqghuvwdqg wkh hhfwv ri d phujhu zklfk jlyhv ulvh wr d prqrsro| ri wkh phujlqj upv +l1h1/ zkhuh h ? hh{ ,1 Wkhuh h{lvw wzr glhuhqw srvvlelolwlhv=  hz ? h ? hh{ 1 Lq wklv fdvh/ wkh h!flhqf| jdlq lv hqrxjk iru wkh phujlqj upv wr eh wkh rqo| vhoohu/ exw lwv prqrsro| srzhu jrhv wr wkh ghwulphqw ri frqvxphuv dqg wrwdo zhoiduh1  h ? hz ? hh{ = Lq wklv fdvh/ wkh phujhu fuhdwhv vxfk dq h!flhqw up wkdw wkh odfn ri frpshwlwlrq lq wkh pdunhw lv pruh wkdq rxwzhljkhg e| wkh h!flhqf| jdlqv1 Wrwdo zhoiduh ulvhv dv d uhvxow1 75 Wkh vhfrqg urrw ri wkh dvvrfldwhg htxdwlrq lv kljkhu wkdq rqh/ dqg wkhuhiruh lw vkrxog eh glvfdughg1

5:

LQVHUW Iljxuh 8171 Ohyhov ri h!flhqf| jdlqv/ dqg h{foxvlrq ri frpshwlwruv Lq rwkhu zrugv/ hyhq li wkh phujhu jdyh ulvh wr d prqrsro|/ lw zrxog qrw eh srvvleoh wr frqfoxgh wkdw wklv qhfhvvdulo| ohdgv wr d zhoiduh orvv/ vlqfh wkh yhu| vdph h!flhqf| jdlqv zklfk reoljh wkh ulydo upv wr h{lw wkh pdunhw pljkw ehqhw frqvxphuv1 Wr ehwwhu lqwhusuhw wkh uhvxowv rewdlqhg/ frqvlghu wkdw wkh h{dpsoh L kdyh vhw xs khuh lv vrphkrz h{wuhph/ iru dw ohdvw wkuhh lpsruwdqw uhdvrqv1 Iluvw/ lw grhv qrw frqvlghu fdsdflw| frqvwudlqwv/ vlqfh L kdyh lpsolflwo| dvvxphg wkdw wkh wzr phujlqj upv duh deoh wr vxsso| doo wkh pdunhw ghpdqg zklfk zrxog kdyh ehhq vxssolhg e| q upv ehiruh wkh phujhu1 Wklv lv d yhu| vwurqj dvvxpswlrq/ zklfk pdnhv lw srvvleoh iru d vlqjoh up wr fryhu doo wkh pdunhw1 Vhfrqg/ lq wklv prgho li rqh rxwvlghu jrhv rxw ri wkh pdunhw/ doo wkh rxw0 vlghuv dovr fhdvh surgxfwlrq1 Dv d uhvxow/ vwurqj hqrxjk h!flhqf| jdlqv ohdg wr prqrsro| e| wkh phujlqj up/ zklfk lq wxuq fuhdwhv vwurqj prqrsro| srzhu dqg zruvhqv zhoiduh1 Lq d pruh vrsklvwlfdwhg prgho/ rqh zrxog olnh wr dvvxph d udqjh ri surgxfwlrq frvwv iru wkh rxwvlghuv/ zlwk vrph ri wkhp ehlqj irufhg rxw ri wkh pdunhw e| wkh phujhu h!flhqf| jdlqv/ dqg vrph rwkhuv qrw1 Wkh zhoiduh lpsdfw ri wkh phujhu zrxog wkhq eh ohvv dgyhuvh1 Lq idfw/ lw pljkw ohdg wr d pruh h!flhqw rxwfrph/ pdgh srvvleoh e| wkh vkxw grzq ri wkh ohvv h!flhqw frpshwlwruv1 Wklug/ wklv lv d vwdwlf prgho/ zkhuh rxwvlghuv fdqqrw uhdfw wr d up*v phujhu1 Exw li d phujhu ohdgv wr vxfk kljk h!flhqf| jdlqv/ lw vkrxog eh h{shfwhg wkdw wkh ulydov zrxog phujh wrr1 Wklv zrxog fdoo iru d pruh frpsohwh prgho zkhuh wkh qxpehu ri phujhuv vkrxog eh ghwhuplqhg hqgrjhqrxvo|/ vrphwklqj eh|rqg wkh vfrsh ri wkh suhvhqw zrun176 Wr vxppdulvh/ wkh srvvlelolw| wkdw d phujhu zklfk hqwdlov h!flhqf| jdlqv pljkw ghfuhdvh zhoiduh e| iruflqj rxw prvw ru doo ri wkh frpshwlwruv vhhpv xqolnho|/ dv wklv uhvxow fdq eh rewdlqhg rqo| xqghu yhu| vwurqj dvvxpswlrqv1

718

Surriv

Surri ri Ohppd 71 Zhoiduh lv ghqhg dv wkh vxp ri frqvxphu vxusoxv dqg surgxfhu vxusoxv1 Lq wkh fdvh ri wkh phujhu/ zhoiduh lv jlyhq e|= Zp

@ X +tL > tr ,  5sL tL  +q  5,sr tr . 5+sL  f,tL . +q  5,+sr  f,tr @   q  5tL5 . +q  5,tr5 . +5tL . +q  5,tr ,5 @ y +5tL . +q  5,tr ,  5 +4 . , q +7;, 5ftL  +q  5,ftr =

76 Prwwd dqg Ydvfrqfhorv +5336, dqdo|vh h!flhqf| jdlqv lq d krprjhqrxv prgho zkhuh upv pljkw glhu lq fdsdflw|1 Wkh| vwxg| d g|qdplf jdph zkhuh phujhuv duh ghflghg vhtxhqwldoo|/ dqg pxvw eh dxwkrulvhg e| dq dqwl0wuxvw dxwkrulw|1 Lq wkhlu prgho/ h!flhqf| jdlqv surpsw rxwvlghuv wr phujh dv zhoo/ dqg wkh xowlpdwh vwuxfwxuh ri wkh pdunhw lv pruh frqfhqwudwhg exw lv dovr dvvrfldwhg zlwk d kljkhu zhoiduh ohyho1 Lq rwkhu zrugv/ wdnlqj dffrxqw ri rxwvlghuv* uhdfwlrqv holplqdwhv wkh udwlrqdoh iru dq h!flhqf| rhqfh dujxphqw1

5;

E| uhsodflqj htxloleulxp txdqwlwlhv lq wkh deryh h{suhvvlrq zh rewdlq= Zp @

+5 7 +q5,5 q5 .9q7 .7q7 +8q5 8q4,.6q 5 +;q6 49q5 .7q.4,. 6 +45q7 69q6 .57q5 q.5,,+yf,5

7q5 +5q.6+q4,.+q5, 5,5

+7<,

Zhoiduh ehiruh wkh phujhu lv lqvwhdg jlyhq e|= Ze @ yqte 

q5 te5  qfte = 5

E| uhsodflqj txdqwlwlhv zh rewdlq=   5  +q  4,5 . 7q+q  4, . 6q5 +y  f,5 Ze @ = 5 5 +5q . +q  4,,

+83,

+84,

Wkh glhuhqfh lq zhoiduh Z @ Zph  Ze lv jlyhq e|= Z @ 

 +y f,5 + 7 +q5,+q4,5 .49q7 .7q6 +43q:,.7 5 q5 +;q5 44q.5,. 6 q+;q6 48q5 .8,, 7q5 +5q.+q4,,5+5q.6+q4, .+q5,5 ,5

+85,

Lw lv vwudljkwiruzdug wr vhh wkdw wkh glhuhqfh lv qhjdwlyh/ jlyhq wkdw wkh qxphu0 dwru lv dozd|v srvlwlyh iru q  51 Wklv suryhv wkdw +zkhq wkhuh duh qr h!flhqf| jdlqv, wkh phujhu dozd|v uhgxfhv zhoiduh1 Surri ri Ohppd ;1 Zh nqrz wkdw wrwdo lqgxvwu| surwv lqfuhdvh zkhq h 5 ^ h> 4`/ dv lq wklv lqwhuydo erwk lqvlghuv dqg rxwvlghuv jdlq iurp wkh phujhu1 Wkhuhiruh/ zh rqo| qhhg wr vkrz wkdw djjuhjdwh surwv ulvh lq wkh lqwhuydo h 5 ^3> h`1 Ghqrwh wkh surgxfhu vxusoxv diwhu wkh phujhu dv S V 3 @ 5L . +q  5,r / dqg surgxfhu vxusoxv ehiruh wkh phujhu dv S Ve @ qe 1 Iru surgxfhu vxusoxv wr lqfuhdvh zlwk wkh phujhu/ zh pxvw wkhuhiruh kdyh S V 3 A S Ve / ru htxlydohqwo|= Le +h,  5 + L +h,  e , A er +h,  +q  5, + e  r +h,,1 Wr suryh wkdw wklv lv wkh fdvh lq wkh lqwhuydo h  h/ zh pryh lq wkuhh vwhsv1 h, A er + h,1 41 Le +

Le er h, ? C  h, ? 31 51 C  Ch + Ch +

5 5er 61 C Ch5Le A 3 A CCh 5 1

Wdnhq wrjhwkhu/ wkhvh wkuhh frqglwlrqv hqvxuh wkdw Le +h, A er +h, iru h  h/ dv vkrzhg lq Iljxuh 8181 LQVHUW Iljxuh 8181 Vwxg| ri ixqfwlrqv xvhg lq Ohppd ; Ohw xv vwduw zlwk 41 Wr vkrz wkdw Le + h, A er + h,> uhfdoo wkdw Le +h, A 3 h, @ rq doo lwv grpdlq/ dqg wkdw zkhq h @ h zh kdyh r @  e 1 Khqfh/ Le + h,   e , A 3 @ er + h, @ +q  5, +e  r + h,,1 5 + L + 5<

? 3=

=

Dv iru srlqw +5,/ zh kdyh wr frpsxwh wkh ghulydwlyhv dqg wdnh wkhlu ydoxh lq h @ h1 Diwhu vrph dojheud/ rqh qgv wkdw= CLe 5f+yf,+q.+q4,,+6q+q4,.5q5 .+q56q.5,, ? 3> + h, @  q5 +5q.+q4, ,++q5, 5 .6+q4, .5q, Ch

+86,

Cer +yf,+q5,+q.+q5,5  ,+q.+q4, , ? 3> + h, @  q55f+5 q.+q4, ,++q5, .6+q4, .5q, Ch

+87,

Wkh lqhtxdolw| CLe + h,@Ch ? Cer + h,@Ch fdq eh uh0zulwwhq dv= 

5f+yf,+5q6.q5+7q6, .+q56q.5, 6.q+5q5 5q4, 5, q5 +5q.+q4, ,++q5, 5 .6+q4, .5q,

? 3>

+88,

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sulfhv zloo qhyhu idoo qr pdwwhu krz vwurqj h!flhqf| jdlqv duh1 +ly, Wr vhh li wkh phujhu lv surwdeoh/ zh kdyh wr vwxg| wkh lqhtxdolw| 4  ri wkh 5 f / zklfk diwhu vrph dojheud fdq eh vhhq wr fruuhvsrqg wrsdq lqhtxdolw|  vhfrqg rughu zkrvh uhohydqw vroxwlrq lv h  7 +4 . f,  6 5 +4  f, @ +;f,1 Lq rwkhu zrugv/ wkh phujhu lv surwdeoh rqo| li lw jlyhv ulvh wr hqrxjk frvw vdylqjv1 Vroxwlrq ri h{huflvh 461 +d, Wkh uhdfwlrq ixqfwlrqv ri up l dqg n fdq eh hdvlo| ghulyhg iurp htxdwlrqv +6, lq wkh wh{w dv= Ul +sn > sm , @ +6y . +6 . 5,f . sm . sn , @ ^5+6 . 5,` dqg Un +sn > sm , @ +5+6 . 5,sn  6y  +6 . 5,f  sm , @1 Wkhlu judsklfdo uhs0 uhvhqwdwlrq lq wkh sodqh +sn > sl , fruuhvsrqgv wr Iljxuh +BB, lq wkh wh{w1 Wr vhh wkdw erwk uhdfwlrq ixqfwlrqv duh srvlwlyho| vorshg/ mxvw qrwh wkdw CUl +sn > sm , @Csn @ @ ^5+6 . 5,` A 3/ dqg CUn +sn > sm , @Csn @ ^5+6 . 5,` @ A 31 Iru vwdelolw| wr krog/ zh pxvw kdyh CUn +sn > sm , @Csn A CUl +sn > sm , @Csn / zklfk vlpsolhv wr  ? 5+6 . 5, zklfk krogv iru doo  5 ^3> 4,1 h +e, Djdlq/ wkh uhdfwlrq ixqfwlrqv fdq eh irxqg lq wkh wh{w dv= UL = sl @ +6y . +6 . ,fh . 5sm . sn , @ ^5+6 . 5,`> UR = sl @ +6y  +6 . 5,f  sm . 5+6 . 5,sn, @1 Wkhlu judsklfdo uhsuhvhqwdwlrq lq wkh sodqh +sn > sl , fruuhvsrqgv wr Iljxuh +BB, lq wkh wh{w1 Wkh kljkhu wkh h!flhqf| jdlqv iurp wkh phujhu +l1h1 wkh orzhu h,/ wkh pruh olnho| wkh lqvlghuv* uhdfwlrq ixqfwlrq zloo vkliw ehorz +udwkhu wkdq deryh, lwv suhylrxv srvlwlrq/ wkxv jlylqj ulvh wr d ghfuhdvh +udwkhu wkdq dq lqfuhdvh, lq sulfhv dw wkh srvw0phujhu htxloleulxp1 Vroxwlrq ri h{huflvh 471 +d, Iurp wkh IRFv +56, ri wkh rxwvlghuv lw lv srvvleoh wr ghulyh wkh ehvw uhso| ixqfwlrq sU r ri hdfk ri wkh rxwvlghuv zlwk uhvshfw wr wkh sulfh sL vhw e| wkh lqvlghuv1 Jlyhq wkdw f @ 3/ wklv vlpsolhv wr= sU r @ +psL . qy, @ +5q .  +q . p  4,,1 Wkh uhvlgxdo ghpdqg ixqfwlrq tLu iru d ydulhw| surgxfhg e| wkh pxowl0surgxfw up fdq eh rewdlqhg e| wdnlqj wkh ghpdqg +53, dqg lpsrvlqj wzr frqglwlrqv= +4, dw htxloleulxp wkh sulfhv ri doo v|pphwulf ydulhwlhv zloo eh lghqwlfdo> +5, wr vhh wkh pdunhw srzhu hqmr|hg iru dq| ydulhw| vrog e| wkh pxowl0surgxfw up zh vkdoo frqvlghu d sursruwlrqdo lqfuhdvh lq doo wkh surgxfw sulfhv vhw e| wkh up dw wkh vdph wlph +li sL lqfuhdvhv/ k lw lqfuhdvhv vlpxowdqhrxvo| iru doo lsurgxfw  +qp,sU psL 4 u r = ydulhwlhv,1 Zh wkhq rewdlq= tL @ q y  sL +4 . , . q . q

E| vxevwlwxwlqj wkh ydoxh ri sU r rqh qgv wkh h{solflw h{suhvvlrq ri wkh uhvlg0 xdo ghpdqg ixqfwlrq ri wkh lqvlghuv1 u u +e, Wr frpsxwh wkh hodvwlflw|ri wkh uhvlgxdo ghpdqg ixqfwlrq uL @   +gtL @tL , @ +gsL @sL ,/ p  gtLu  5 p+qp, 4 qg uvw wkh ghulydwlyh= gsL @ 4 .  q  4 . q+5q. +q.p4,, q = Zh fdq qrz fdofxodwh wkh hodvwlflw| dv=   5  +q  p,+q  4, . +6q  p  4, . 5q5 sL u > L @ 5  +q  p,+q  4,sL . 5q5 +y  sL , . q +y+5q  4,  +6q  p  4,sL , +8:,

93

dqg vrph fdofxodwlrqv vkrz wkdw=  5    5q . q +7q  6, .  5 5q5  6q . 4 qysL guL @ ? 3= gp + 5 +q  p,+q  4,sL . 5q5 +y  sL , . q +y+5q  4,  +6q  p  4,sL ,,5 +8;, Wkh odujhu wkh qxpehu ri surgxfwv vrog e| wkh phujlqj upv wkh orzhu wkh hodvwlflw| ri wkh uhvlgxdo ghpdqg ixqfwlrq idfhg e| hdfk ri lwv surgxfw/ l1h1 wkh u L odujhu lwv pdunhw srzhu1 Lw lv dovr srvvleoh wr vkrz wkdw g gq A 3= dv zh zrxog h{shfw/ iru d jlyhq qxpehu ri surgxfwv p vrog e| wkh lqvlghuv/ wkh odujhu wkh qxpehu ri upv rshudwlqj lq wkh lqgxvwu| wkh kljkhu wkh hodvwlflw| idfhg e| wkh lqvlghuv +wkh orzhu wkh pdunhw srzhu hqmr|hg e| wkh phujlqj upv,1 Vroxwlrq ri h{huflvh 481 Wkh lqyhuvh ghpdqg ixqfwlrqv duh hdvlo| irxqg e| vroylqj wkh frqvxphu*v rswlplvdwlrq Sq sureohp/ wkdw lv pd{it4 >t5 >t6 >|j X vxemhfw wr wkh lqfrph frqvwudlqw ] @ l@4 sl tl .s| |/ diwhu wdnlqj  jrrg | dv wkh Sq 4 qxpìudluh +s| @ 4,1 Wkh| duh= sl @ y  4. 6tl .  m @4 tm = +ll, Ehiruh wkh phujhu wdnhv sodfh/ zh kdyh wkuhh lghqwlfdo vlqjoh0surgxfw upv zlwk d surw ixqfwlrq l @ +sl  f,tl / zkhuh sl lv jlyhq e| wkh ghpdqg ixqfwlrq deryh +l @ 4> 5> 6,1 Wkh IRFv C l @Ctl @ 3 duh= tl @ ++y  f,+4 . ,  tm  tn , @ +5+6 . ,, > l> m> n @ 4> 5> 6> l 9@ m 9@ n +8<, E| vroylqj wkh v|vwhp ri IRFv dqg lpsrvlqj v|pphwu| rq txdqwlwlhv rqh re0 wdlqv= tf @ ++y  f, +4 . ,, @ +5+6 . 5,, =

+93,

Sulfhv dqg surwv dw htxloleulxp duh= sf @

y +6 . , . 6f +4 . , > 5+6 . 5,

f @

+y  f,5 +6 . , +4 . , 7 +6 . 5,5

=

+94,

+lll, Ilupv 4 dqg 5 phujh wr ehfrph up L/ up 6 frqwlqxhv wr vhoo rqh surgxfw1 S Wkh| zloo kdyh surwv  L @ 5l@4 +sl  f,tl dqg  6 @ +s6  f,t6 / zkhuh sulfhv sl @ sl +t4 > t5 > t6 , duh jlyhq e| wkh lqyhuvh ghpdqg ixqfwlrqv +BB,1 Wkh IRFv C L @Ctl @ 3 +zlwk l @ 4> 5, dqg C6 @Ct6 @ 3/ fdq eh zulwwhq dv= tl @

+y  f,+4 . ,  5tm  t6 > 5+6 . ,

t6 @

+y  f,+4 . ,  tl  tm > l> m @ 4> 5> l 9@ m= 5+6 . , +95,

Zh fdq wkhq qg wkh srvw0phujhu htxloleulxp txdqwlwlhv tL / tR dv= tL @

+y  f, +9 . , +4 . , +y  f, +6 . , +4 . , > tR @ = 9 + 5 . 9 . 9, 6 + 5 . 9 . 9, 94

+96,

Diwhu vxevwlwxwlrq/ rqh rewdlqv wkh sulfhv dqg shu0ydulhw| surwv + L 4 @ L 5 @ L , ri wkh phujhg up dqg wkh rxwvlghuv dv=   y+4; . 48 . 5 5 , . f 4; . 54 . 7 5 y+6 . ,5 . f+< . 45 . 5 5 , > sR @ = sL @ 5 5 9 +9 . 9 .  , 6 +9 . 9 .  5 , +97,

L @

+6 . 5, +4 . ,+9 . ,5 +y  f,5 5

69 + 5 . 9 . 9,

> R @

+4 . ,+6 . ,6 +y  f,5 < + 5 . 9 . 9,5

=

+98,

+ly, Zh fdq qrz dqdo|vh wkh hhfw ri wkh phujhu rq surwdelolw| iru wkh lqvlghu upv1 Wr gr wkdw/ qrwh wkdw wkh lqhtxdolw| L A f fdq eh uhzulwwhq dv= 

   5 +4 . ,  6 . 6 5  4;  5: +y  f,5 69 +6 . 5,5 + 5 . 9 . 9,5

A 3=

+99,

zklfk lv vdwlvhg rqo| li  ? 6=:71 Lq rwkhu zrugv/ wkh phujhu zloo eh xqsurwdeoh xqohvv wkh surgxfwv duh doprvw lqghshqghqw +uhfdoo wkdw  5 +3> 4,,1 +y, Wr xqghuvwdqg wklv uhvxow/ ohw xv gudz wkh upv* ehvw uhso| ixqfwlrqv ehiruh dqg diwhu wkh phujhu1 Frqvlghu iru lqvwdqfh surgxfw 4 dqg surgxfw 61 Lq wkh vsdfh +t6 > t4 ,/ dqg iru jlyhq t5 / ehiruh wkh phujhu wkhlu ehvw uhsolhv duh rewdlqhg iurp IRFv +8<,= U4 = t4 @

+y  f,+4 . ,  t5  t6 +y  f,+4 . ,  t5  5+6 . ,t6 > U6 = t4 @ = 5+6 . ,  +9:,

Diwhu wkh phujhu/ wkh ehvw uhsolhv duh ghulyhg iurp IRFv +96, dv= 3

U4 = t4 @

+y  f,+4 . ,  5t5  t6 +y  f,+4 . ,  t5  5+6 . ,t6 > U6 = t4 @ = 5+6 . ,  +9;,

E| gudzlqj wkh uhdfwlrq ixqfwlrqv/ zklfk duh qhjdwlyho| vorshg/ lw lv lpphgldwh wr vhh wkdw wkh phujhu vkliwv wkh ehvw uhso| ixqfwlrq UL ri wkh lqvlghu surgxfw grzqzdugv/ wr UL3 / uhvxowlqj lq d qhz htxloleulxp fkdudfwhulvhg e| kljkhu sur0 gxfwlrq ri wkh rxwvlghuv dqg orzhu surgxfwlrq ri wkh lqvlghuv1 Wkh uhdoorfdwlrq ri surgxfwlrq h{sodlqv zk| +xqohvv surgxfwv duh h{wuhpho| glhuhqwldwhg, wkh phujhu lv xqsurwdeoh1 Vroxwlrq ri h{huflvh 491 +d, Wkh Frxuqrw0Qdvk htxloleulxp ehiruh wkh phujhu fruuhvsrqgv wr wkh vroxwlrq ri h{huflvh 48 +ll,/ l1h1 htxdwlrqv +93, dqg +94, +e, Wkh phujhu zlwk h!flhqf| jdlqv= Wkh phujlqj upv* surwv zloo eh S5 L @ l@4 +sl  hf,tl / zkloh wkh rxwvlghu vwloo kdv surwv 6 @ +s6  f,t6 / zkhuh 95

sulfhv sl @ sl +t4 > t5 > t6 , duh jlyhq e| wkh lqyhuvh ghpdqg ixqfwlrqv +4<,1 Wkh IRFv C L @Ctl @ 3 +zlwk l @ 4> 5, dqg C6 @Ct6 @ 3/ fdq eh zulwwhq dv= +y  hf,+4 . ,  5tm  t6 +y  f,+4 . ,  tl  tm > t6 @ > l> m @ 4> 5> l 9@ m= tl @ 5+6 . , 5+6 . , +9<, Diwhu lpsrvlqj v|pphwu| rq lqvlghu txdqwlwlhv +l1h1 t4 @ t5 , dqg vroylqj iru tL dqg tR / zh rewdlq srvw0phujhu shu0ydulhw| txdqwlwlhv/ sulfhv dqg surwv= tL @

+4 . , ^+y  hf, 5 +6 . ,   +y  f,` 5 +4 . , ^+y  f, +6 . 5,   +y  hf,` > tR @ > 7 +6 . 5, +6 . ,  5 5 7 +6 . 5, +6 . ,  5 5 +:3, sL sR

  +y  hf, 5 +6 . 5, +6 . ,   5 .  +y  f, +6 . 5, @ y > 7 +6 . 5, +6 . ,  5 5   +y  f, 5 +6 . 5, +6 . ,   5 . 5 +y  hf, +6 . , @ y > 7 +6 . 5, +6 . ,  5 5

+:4, +:5,



L R

5 +y  hf, 5 +6 . ,   +y  f, > 7 +6 . 5, +6 . ,  5 5  5 +y  f, +6 . 5,   +y  hf, @ 7 +4 . , +6 . , 7 +6 . 5, +6 . ,  5 5 @ +4 . , +6 . 5,

+:6, +:7,

+f, Ohw h @ 4 dqg  $ 41 Iru h @ 4/ wkh h{suhvvlrqv rewdlqhg lq h{huflvh 48 +lll, dqg +ly, dsso|1 Wkhuh/ zh vdz wkdw zkhqhyhu wkhuh lv dq rxwvlghu ohiw lq wkh pdunhw/ wkh phujhu zloo rqo| eh surwdeoh iru lqvlghuv li  ? 6=:71 Li  $ 4/ wkh jrrgv ehfrph pruh dqg pruh krprjhqhrxv +dqg duh qdoo| shuihfw vxevwlwxwhv,/ vr wkdw wkh qglqjv ri h{huflvh 44 dsso|= zlwk q @ 6/ wkh phujhu zloo rqo| eh surwdeoh kli lw lv d phujhu wrl prqrsro|1 +wr vhh wklv/ mxvw vroyh 5 .4 j+q @ 6> p, @ +d  f, +qp4.4,5  +qp.4, 5 A 3 iru p wr rewdlq wkdw wkh phujhu zloo rqo| eh surwdeoh li p A 4=76;/ l1h1 li dw ohdvw p . 4 @ 6 upv phujh1, Vroxwlrq ri h{huflvh 4:1 +d, Wkh suh0phujhu uhdfwlrq ixqfwlrqv zhuh gh0 ulyhg lq h{huflvh 48 +y, dv= Ul +tn > tm , = tl @ ^+y  f,+4 . ,  tm  tn ` @ ^5+6 . ,`> Un +tn > tm , = tl @ ^+y  f,+4 . ,  tm  5+6 . ,tn ` @1 Lw lv hdv| wr vhh wkdw wkh| duh qhjdwlyho| vorshg= CUl +tn > tm , @Ctn @ @ ^5+6 . ,` ? 3/ dqg CUn +tn > tm , @Ctn @  ^5+6 . ,` @ ? 31 Iru vwdelolw| wr krog/ zh pxvw kdyh CUn +sn > sm , @Csn ? CUl +sn > sm , @Csn / zklfk vlpsolhv wr  ? 5+6 . , zklfk krogv iru doo  5 ^3> 4,1 +e, Wkh srvw0phujhu uhdfwlrq ixqfwlrqv fdq eh hdvlo| ghulyhg iurp wkh IRFv ri h{huflvh 49 +e,/ dv iroorzv= Uhl +tn > tm , = tl @ ^+y  hf,+4 . ,  5tm  tn ` @ ^5+6 . ,`> Un +tn > tm , = tl @ ^+y  f,+4 . ,  tm  5+6 . ,tn ` @1 Li h @ 4 +l1h1 wkhuh duh qr h!flhqf| jdlqv,/ wkhq wkh lqvlghuv* uhdfwlrq ixqfwlrq zloo vkliw grzqzdug/ dqg sulfhv zloo lqfuhdvh1 Krzhyhu/ li h!flhqf| jdlqv duh lpsruwdqw hqrxjk/ wkhq wkh uhdfwlrq ixqfwlrq pd| lqvwhdg vkliw xszdug/ lq zklfk fdvh sulfhv zloo ghfuhdvh1

96

Table 5.1. Number of final decisions on mergers taken by the EC Commission Final Decisions (R.4064/89)

Final Decisions (Art.66 ECSC Treaty, 1992-1999)

300

9 10

250

200 10 7

150

270

7 100

238

12

26 10

6

50

91 60

60

57

1991

1992

1993

109

125

142

0 1994

1995

1996

1997

1998

1999

Source. Report on Competition Policy, European Commission, DG IV (various years)

Table 5.2. Product market definition in the rail technology industry Electrical locomotives Diesel locomotives Mainline trains

Train sets for mainline transportation Passenger coaches Freight wagons

Rolling stock Regional trains

Electrical multiple units Diesel multiple units Trams (light rail and trolleys)

Local trains and systems

Metro vehicles Automated guided transportation Catenary systems

Wayside systems Stationary equipment

Traction power supply Train control and protection systems

Miscellaneous

Maintenance and refurbishment Information systems and ticketing

Table 5.3. Market shares and concentration indexes in the rail technology industry Product Market

Market Shares (%)

HHI

∆ HHI

ABB

DB

Siemens

DWA

LHB

Elpro

Electrical locom.

37

17

46

Np

np

Np

5032

1258

Mainline train sets

5

26

46

18

np

Np

>3401

260

Reg. electr. m/units

18

26

25

17

14

Np

>3046

936

Reg. diesel m/units

0

49

23

np

19

Np

>3291

0

Trams

15

29

41

np

8

Np

>3681

870

Metro vehicles

42

22

19

np

11

Np

>4578

1848

Catenary systems

30

31

33

np

np

6

4846

1860

Traction power supply

6

26

35

np

np

13

>2418

312

RO

p1

RI′ M

pI

pB

RI

B

pB

pO

p3

Figure 5.1. Effects of a merger absent efficiency gains: Strategic complements

q1 RO

C

qc qI

π1

M

π 1′ qc qO

RI′

RI q3

Figure 5.2. Effects of a merger absent efficiency gains: Strategic substitutes

RO

p1

RI′ (e = 1) M

pI

RI

RIe B

pB E

pB

pO

Figure 5.3. Effects of a merger with efficiency gains

p3

e

1 eex WM < Wb WM > Wb

eW

150

Figure 5.4. Levels of efficiency gains and exclusion of competitors

γ

∆Ib , ∆bo

∆Ib (e)

∆bo (e)

e

Figure 5.5. Study of functions used in Lemma 7

1

e

Fkdswhu 9= Yhuwlfdo uhvwudlqwv dqg yhuwlfdo phujhuv Iheuxdu| 47/ 5336

4

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;

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+4,

s

E| wdnlqj wkh uvw ghulydwlyh dqg htxdolqj lw wr }hur +CG @Cs @ 3,/ zh rewdlq sulfh/ txdqwlw| dqg surw dv d ixqfwlrq ri wkh zkrohvdoh sulfh= s @ +d.z,@5> t @ +d  z,@5> G @ +d  z,5 @71 Wkh pdqxidfwxuhu shuihfwo| dqwlflsdwhv wkh ghflvlrq ri wkh uhwdlohu1 Lq sdu0 wlfxodu/ lw nqrzv wkh txdqwlw| lw zloo rughu iurp lw dv d ixqfwlrq ri wkh zkrohvdoh sulfh +iru dq| jlyhq z/ wkh uhwdlohu zloo qrw eh zloolqj wr ex| pruh xqlwv ri wkh surgxfwv wkdq wkrvh wkdw lw zloo qg rswlpdo wr vhoo wr qdo frqvxphuv,1 Wkhuhiruh/ lwv sureohp lv wr fkrrvh z wr pd{lplvh lwv rzq surw= pd{ X @ +z  f, z

dz = 5

+5,

Iurp wkh uvw0rughu frqglwlrq +CX @Cs @ 3, dqg diwhu uh0duudqjlqj rqh qgv wkh vroxwlrq dv= z @ +d . f,@51 Uhsodflqj wkh htxloleulxp zkrohvdoh sulfh lqwr wkh grzqvwuhdp vroxwlrqv rqh qgv wkh htxloleulxp qdo sulfh dqg surwv rewdlqhg e| wkh xsvwuhdp dqg grzqvwuhdp upv/ dv zhoo dv wkh vxp ri wkh surwv pdgh e| wkh yhuwlfdo fkdlq1 svhs S V vhs

@

6d . f > 7

+d  f,5 > ; 6 +d  f,5 = @ 49

vhs @ X

@ vhs . vhs X G

vhs @ G

+d  f,5 > 49

+6, +7,

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+8,

s

Wkh vroxwlrq lv hdvlo| rewdlqhg iurp wkh uvw0rughu frqglwlrq Cyl @Cs @ 3= syl @

d.f > 5

t yl @

df > 5

S V yl @ yl @

+d  f,5 = 7

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+9,

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qrq0olqhdu sulfh vfkhph I . zt/ dqg {lqj z @ f1 Wkh grzqvwuhdp up*v pd{lplvdwlrq sureohp lv jlyhq e|= pd{ iGi @ +s  f,+d  s,  I= s

+:,

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Grxeoh pdujlqdolvdwlrq zlwk uhwdlohuv* ulvn dyhuvlrq--

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44

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515

Krul}rqwdo h{whuqdolw|= Iuhh0ulglqj lq wkh surylvlrq ri vhuylfhv

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45

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frxuvh/ li wkh vhuylfhv surylghg e| wkh uhwdlohuv duh revhuydeoh dqg yhuldeoh/ wkhuh lv d vlpsohu vroxwlrq wr wkh sureohp/ zklfk lv wr { frqwudfwxdoo| wkh ohyho ri vhuylfhv1 Wklv pljkw eh wkh fdvh/ iru lqvwdqfh/ zlwk dgyhuwlvlqj ru fhuwdlq w|shv ri diwhu0vdohv vhuylfhv1

46

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D prgho ri xqghusurylvlrq ri vhuylfhv-

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47

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+43,

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48

vxevwlwxwlrq/ rqh wkhq rewdlqv surgxfhu vxusoxv/ frqvxphu vxusoxv dqg zhoiduh dv iroorzv= S Vyl @ yl @

+y  f,5 5 +y  f,5 +6  5,+y  f,5 > FVyl @ > Z @ = +44, yl 7+  4, ;+  4,5 ;+  4,5

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h5 +y . h4 . h5  sl ,   l  I= 5 5

Wkh uvw0rughu frqglwlrqv duh= + C hw @ sl5f  hl @ 3> Chl C hw @ y . hl . hm  5sl . f @ 3> Csl

+l @ 4> 5> l 9@ m,=

+45,

+46,

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49

zloo fkrrvh s dqg h wr pd{lplvh wkh iroorzlqj ixqfwlrq= h5 pd{ hw4 @ +s4  f,+y . h4  s4 ,   4  I= s4 >h4 5 Wkh uvw rughu frqglwlrqv zloo eh= + C hw 4 @ +s  f,  h @ 3> 4 4 Chl C hw 4 @ y . h  5s . f @ 3> +l @ 4> 5> l 9@ m,= 4 4 Csl

+47,

+48,

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h5 +y . h4 . h5  syl ,   l  I= 5 5

+49,

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+4:,

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syl  z yf @ hyl @ > 5 5+  4,

+4;,

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6f  5f  y ? f= 5+  4,

+4<,

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4:

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v=wr =

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4<

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+54,

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53

+55,

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516

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54

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57

H{whuqdolwlhv Wkh frpsdulvrq ehwzhhq +57, dqg +58, doorzv xv wr lghqwli| wkh glhuhqw h{whuqdolwlhv dw sod|/ dqg xqghuvwdqg zk| xqghu yhuwlfdo vhsdudwlrq dqg olqhdu sulflqj d vxe0rswlpdo rxwfrph dulvhv iru wkh pdqxidfwxuhu1 Ohw xv vwduw e| frpsdulqj wkh sulfh ghflvlrqv1 Lw lv frqyhqlhqw wr uhzulwh wkhvh IRFv dv sV +h, @ +y . h +4 .  +q  4,, . z +4 .   @q,, @ +5 .   @q, dqg sL +h, @ +y . h +4 .  +q  4,, . f, @51 Wkhuh duh wzr glvwlqfw h{whuqdolwlhv dw zrun/ zklfk sxvk wkh sulfh lqwr rs0 srvlwh gluhfwlrqv/ iru jlyhq hruw ohyhov1 Iluvw/ xqghu vhsdudwlrq wkhuh lv wkh xvxdo yhuwlfdo h{whuqdolw|= xqghu vhsdudwlrq/ wkh grxeoh pdujlqdolvdwlrq sure0 ohp dulvhv/ ohdglqj wr z A f/ zklfk whqgv wr lqfuhdvh wkh sulfh sV deryh sL 1 Vhfrqg/ wkhuh lv qrz d krul}rqwdo shfxqldu| h{whuqdolw|/ lq wkdw lqghshqghqw uhwdlohuv zrxog frpshwh wrr pxfk zlwk hdfk rwkhu/ lpsrvlqj d qhjdwlyh sulfh h{whuqdolw| xsrq hdfk rwkhu1 Wklv fdq eh vhhq e| wkh idfw wkdw iru sV wkh gh0 qrplqdwru lv glylghg e| 5 .   @q/ zkhuhdv iru sL lw lv glylghg rqo| e| 51 Qrwh wkdw wklv krul}rqwdo h{whuqdolw| lqfuhdvhv zlwk wkh ghjuhh ri frpshwlwlrq/ ehlqj kljkhvw zkhq  $ 4 dqg orzhvw zkhq  @ 31 Lq wkh odwwhu fdvh/ uhwdlohuv duh vhoolqj surgxfwv zklfk duh shufhlyhg dv lqghshqghqw/ dqg wkh rqo| h{whuqdolw| ohiw lv wkh yhuwlfdo rqh1 Wkh qdo qhw hhfw lv d sulrul dpeljxrxv1 Krzhyhu/ lw wxuqv rxw wkdw lq vwdqgdug prghov wkh yhuwlfdo h{whuqdolw| hhfw grplqdwhv wkh shfxqldu| krul}rqwdo h{whuqdolw| hhfw156 Wr dqdo|vh wkh lqfhqwlyhv wr pdnh hruw xqghu wkh wzr yhuwlfdo vwuxfwxuhv/ zulwh qrz wkh uvw ghulydwlyhv zlwk uhvshfw wr hruw dv d ixqfwlrq ri +jlyhq, sulfhv/ dv iroorzv= hV +s, @ +sz,@+q, dqg hL +s, @ +sf, +4 .  +q  4,, @+q,1 Khuh djdlq wkhuh duh wzr h{whuqdolwlhv dw sod|/ exw wkh| erwk kdyh d qhjdwlyh hhfw rq wkh surylvlrq ri hruw xqghu wkh vhsdudwhg yhuwlfdo vwuxfwxuh1 Iluvw/ wkh yhuwlfdo h{whuqdolw|/ e| lqfuhdvlqj wkh pdujlqdo frvw ri uhwdlohuv +z A f,/ uhgxfhv lwv pdujlqdo surw iurp lqyhvwlqj lq hruw1 Vhfrqg/ wkhuh lv d krul}rqwdo h{whuqdolw|/ ghwhuplqhg e| wkh vslooryhu/ zklfk lv lqwhuqdolvhg xqghu wkh yhuwl0 fdoo| lqwhjudwhg vwuxfwxuh dqg lqfuhdvhv wkh hruw pdgh lq wkdw fdvh1 Wkhuhiruh/ hL A hV 1 Wr vxp xs/ lq wklv prgho lw lv srvvleoh wkdw yhuwlfdo lqwhjudwlrq uhgxfhv zhoiduh/ exw rqo| li doo wkh iroorzlqj frqglwlrqv krog= +4, yhuwlfdo lqwhjudwlrq ohdgv wr kljkhu sulfhv/ +5, wklv hhfw rxwzhljkv wkh srvlwlyh hhfw gxh wr wkh ulvh lq hruw xqghu lqwhjudwlrq/ dqg +6, wkh uhvxowlqj orvv lq frqvxphu vxusoxv lv kljkhu wkdq wkh srvlwlyh hhfw rq surgxfhu vxusoxv fuhdwhg e| wkh lqwhuqdolvdwlrq ri wkh ydulrxv h{whuqdolwlhv1 Zhoiduh dqdo|vlv Lq wklv sduwlfxodu prgho/ zhoiduh wxuqv rxw wr eh kljkhu xqghu yhuwlfdo lqwhjudwlrq/ ehfdxvh erwk frqvxphu dqg surgxfhu vxusoxv duh kljkhu wkdq xqghu vhsdudwlrq1 Wr suryh lw/ uvw zh kdyh wr qg wkh htxloleulxp sulfh dqg hruw xqghu yhuwlfdo vhsdudwlrq1 Wr gr vr/ ohw xv qg wkh rswlpdo zkrohvdoh sulfh z fkdujhg 56 Vhh dovr N kq dqg Ylyhv +4<<<,1 Qrwh wkdw lq wkh prgho L suhvhqw khuh sulfhv pljkw eh kljkhu xqghu yhuwlfdo lqwhjudwlrq ehfdxvh ri kljkhu hruw +zklfk lq wxuq lqfuhdvhv wkh zloolqjqhvv wr sd| ri frqvxphuv,1

58

e| wkh pdqxidfwxuhu1 E| vroylqj wkh v|vwhp +57, rqh rewdlqv=

z,q +yz, V sV @ 5q4+y+ q4,.+q4, . z> h @ 5q4+q4,.+q4, =

+59,

Wkh wrwdo rxwsxw vrog e| wkh pdqxidfwxuhu zloo wkhq eh= TV @ qt V @

+y  z, ++q  4, . q, = 5q  4  +q  4, . +q  4,

+5:,

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+y  f,5 5 ++q  4, . q,5 ;q +5q  4  +q  4, . +q  4,,5

=

+5<,

Qrz/ zh kdyh wr ghulyh wkh htxloleulxp ydoxhv iru wkh fdvh ri yhuwlfdo lqwhjudwlrq1 E| vroylqj wkh v|vwhp +58, rqh rewdlqv= +y . f,q  f +4 .  +q  4,,5 > sY L  @ 5q  +4 .  +q  4,,5

hY L  @

+y  f, +4 .  +q  4,, 5q  +4 .  +q  4,,5

> +63,

Vxevwlwxwlqj wkhvh ydoxhv lq wkh h{suhvvlrq ri wkh frqvxphu vxusoxv jlyhv= FV Y L @



+y  f,5 5 q

5 5q  +4 .  +q  4,,5

5 =

+64,

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59

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5:

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dqg TI/ lw grhv qrw prgli| wkh uhwdlohuv* fkrlfhv= lw fdq wkhq eh xvhg e| wkh pdqxidfwxuhu wr dssursuldwh wkh uhwdlohuv* uhqwv1 51615

Yhuwlfdo lqwhjudwlrq dqg ydulhw|--

Zlwklq wkh vdph prgho L qrz zdqw wr hqgrjhqlvh wkh qxpehu ri uhwdlohuv1 Diwhu lpsrvlqj v|pphwu|/ surw ri d yhuwlfdoo| lqwhjudwhg up fdq eh zulwwhq dv= yl @ +s  f, +y . h +4 .  +q  4,,  s,  qi

+65,

zkhuhdv wkh surw ri dq lqghshqghqw uhwdlohu lv= 4 yv @ +s  z, +y . h +4 .  +q  4,,  s,  i= q

+66,

Wr ghwhuplqh hqgrjhqrxvo| wkh qxpehu ri uhwdlohuv rshudwlqj dw htxloleulxp/ frqvlghu wkdw xqghu yhuwlfdo lqwhjudwlrq hqwu| rffxuv xs wr wkh srlqw wkdw pd{0 lplvhv yl / zkhuhdv xqghu yhuwlfdo vhsdudwlrq lw rffxuv xs wr wkh srlqw wkdw pdnhv  yv @ 3 +wkh vwdqgdug iuhh hqwu| frqglwlrq/ xqghu wkh dvvxpswlrq wkdw q lv frqwlqxrxv,1 Wkh frqglwlrqv zklfk ghwhuplqh wkh qxpehu ri uhwdlohuv dw htxloleulxp duh wkhuhiruh= C yl @ +s  f,h  i @ 3> Cq

+67,

4 yv @ +s  z, +y . h +4 .  +q  4,,  s,  i @ 3= q

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L dovr dvvxph wkdw wkhuh duh d odujh qxpehu ri uhwdlohuv lq wkh pdunhw wkdw frpshwh lq sulfh dqg surylgh xqglhuhqwldwhg vhuylfhv +ru/ zklfk lv htxlydohqw/ wkdw duh shuihfwo| frpshwlwlyh,15< Hdfk uhwdlohu*v frvw ri vhoolqj eudqg l lv jlyhq 5< Wklv

dvvxpswlrq lv pdgh wr dyrlg iruhforvxuh lvvxhv dqg wr eulqj iruzdug lq d qhdwhu zd|

66

e| g . zl  hel / zkhuh zl lv wkh zkrohvdoh sulfh fkdujhg e| pdqxidfwxuhu l/ g lv wkh glvwulexwlrq frvw163 Wkh wlplqj ri wkh jdph lv dv iroorzv1 Iluvw/ pdqxidfwxuhuv pdnh vlpxowdqh0 rxvo| lqyhvwphqw dqg zkrohvdoh sulfh ghflvlrqv1 Wkhq/ uhwdlohuv fkrrvh sulfhv1 Rxu remhfwlyh lv wr frpsduh wkh htxloleulxp vroxwlrqv ri wklv jdph xqghu wzr do0 whuqdwlyh frqwudfwxdo vlwxdwlrqv/ rqh zkhuh uhwdlohuv rshudwh xqghu HG/ dqg rqh zkhuh wkhuh lv qr h{foxvlyh ghdolqj +QHG,/ l1h1/ uhwdlohuv fdq vhoo erwk eudqgv1 Ehvdqnr dqg Shuu| +4<<6, dqdo|vh wkh ixoo jdph/ zkhuh pdqxidfwxuhuv ghflgh dw d suh0vwdjh ri wkh jdph zkhwkhu wr fkrrvh HG ru QHG/ exw wklv lqyroyhv kdylqj dw ohdvw wkuhh upv +zlwk wzr upv/ li rqh fkrrvh HG wkh rwkhu lv gh idfwr dovr reoljhg wr uho| rq dq h{foxvlyh uhwdlohu, dqg frpsolfdwhv wkh fdofxodwlrqv164 Vroxwlrq ri wkh jdph Dw wkh odvw vwdjh ri wkh jdph/ uhwdlohuv zrxog vhw sulfhv htxdo wr wkhlu pdujlqdo frvwv= sl @ g . zl  hel 1 Frqvlghu uvw wkh fdvh zkhuh wkhuh lv qr h{foxvlyh ghdolqj duudqjhphqw +QHG,1 Uhsodflqj htxloleulxp sulfhv rqh rewdlqv wkh txdqwlwlhv dv d ixqfwlrq ri zkrohvdoh sulfhv dqg lqyhvwphqw ohyhov1 Dw wkh suhylrxv vwdjh/ pdqxidfwxuhuv pd{zl >hl  l @ zl tl +hl > hm > zl > zm ,+@5,h5l 1 E| wdnlqj wkh IRFv Cl @Chl @ 3/ Cl @Czl @ 3/ lpsrvlqj v|pphwu| dqg vroylqj wkh v|vwhp/ rqh rewdlqv wkh htxloleulxp vroxwlrqv dv= zlQ HG

@

HG hQ l

@

7+y  g, > 5+7 . ,  +4 . ,+5 . +4  ,, +5 . +4  ,,+y  g, = 5+7 . ,  +4 . ,+5 . +4  ,,

+6:, +6;,

E| vxevwlwxwlrq/ wkh rwkhu htxloleulxp ydoxhv fdq eh rewdlqhg1 Lq sduwlfxodu= HG sQ l

@

HG Q l

@

5g+5 . , . y+7  +4 . ,+5 . +4  ,,, > 5+7 . ,  +4 . ,+5 . +4  ,,    7+5  4 . , . 49   5 +4  ,5  7 +y  g,5 ^5+7 . ,  +4 . ,+5 . +4  ,,`5

+6<, =

+73,

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67

lpsrvh  @ 3 wr vroyh wkh HG fdvh= zlHG @

7+y  g, +5 . ,+y  g, > hHG @ > 5+7 . ,  +5 . , l 5+7 . ,  +5 . ,

sHG l

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HG l

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5g+5 . , . y+7  +5 . ,, > 5+7 . ,  +5 . ,    7+5  4 . 49   5  7 +y  g,5 ^5+7 . ,  +5 . ,`5

+74,

+75, =

+76,

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518

Yhuwlfdo uhvwudlqwv/ yhuwlfdo phujhuv/ dqg wkh frpplw0 phqw sureohp

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68

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69

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78

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79

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Vwudwhjlf xvh ri yhuwlfdo uhvwudlqwv-

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5y > 7.

Y L @

+5 . ,y5 = +7 . ,5

+78,

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7:

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U4 U5

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+7:, +7;,

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5+7 . 6,y . +5 . ,+5zl +5 . , . zm , = 49 . 49 . 6 5

+7<,

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7;

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^

+5. , 5+7.6 ,y

+;.; .5 ,zl .5zl +5. ,`^5+7.6,y.5+5.,5 zl .zm +5. ,` 7+7. ,5 +7.6 ,5

+83,

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@

I I

@

5y 5 7+5 . ,y > sI I @ > 5 +5 . ,+49 . 45 .  , +49 . 45 .  5 , 5+5 . ,+; . ; .  5 ,y 5 = +49 . 45 .  5 ,5

+84, +85,

Wkhuhiruh/ dw wkh htxloleulxp erwk xsvwuhdp pdqxidfwxuhuv vhw d zkrohvdoh sulfh z A f @ 3 vr dv wr uhod{ frpshwlwlrq dprqj uhwdlohuv +dqg xowlpdwho| dprqj wkhpvhoyhv= Uh| dqg Vwljolw} vkrz wkdw wkh yhuwlfdo uhvwudlqw pdnhv pdqxidf0 wxuhuv idfh d shufhlyhg ghpdqg hodvwlflw| zklfk lv orzhu wkdq xqghu yhuwlfdo lqwhjudwlrq,1 Dv d uhvxow erwk sulfhv dqg surwv duh kljkhu wkdq xqghu yhuwlfdo lqwhjudwlrq= sI I A sY L dqg I I A Y L +wkh h{suhvvlrqv frlqflgh rqo| zkhq  $ 4,1Wklv lqfuhdvhg doorfdwlyh lqh!flhqf| ghwhuplqhv d ghfuhdvh lq ryhudoo zhoiduh1 H{foxvlyh whuulwrulhv Uh| dqg Vwljolw} +4<;;/ 4<<8, vkrz wkdw judqwlqj h{0 foxvlyh whuulwrulhv wr uhwdlohuv dovr khosv pdqxidfwxuhuv wr uhod{ frpshwlwlrq1 Dv d ehqfkpdun/ frqvlghu wkh fdvh zkhuh wzr pdqxidfwxuhuv kdyh p  5 uhwdlohuv hdfk1 Uhwdlohuv zkr fduu| wkh vdph eudqg vhoo surgxfwv zklfk duh shufhlyhg dv krprjhqrxv e| frqvxphuv1 Wkhuhiruh/ iru wkh xvxdo Ehuwudqg frpshwlwlrq dujxphqwv/ lqwud0eudqg frpshwlwlrq ohdgv wr uhwdlohuv fkdujlqj sl @ zl 1 Wkh xsvwuhdp up fkrrvhv lwv sulfh wr pd{lplvh  X l @ +zl  f,tl +zl > zm ,1 Vlqfh sl @ zl / wklv sureohp lv lghqwlfdo wr wkh sureohp ri wkh yhuwlfdoo| lqwhjudwhg up zh kdyh vroyhg deryh1 E| wdnlqj gl +zl > zm ,@gzl @ 3 dqg vroylqj wkh v|vwhp rqh rewdlqv= z Y L @ sY L @

5y > 7.

Y L @

+5 . ,y5 = +7 . ,5

+86,

Vxssrvh qrz wkdw hdfk uhwdlohu lv jlyhq dq h{foxvlyh whuulwru| e| klv pdqxidf0 wxuhu/ dqg wklv ghflvlrq lv sxeolfo| revhuyhg1 H{foxvlyh whuulwrulhv lpso| wkdw hdfk uhwdlohu kdv 4@p vkduh ri wkh ghpdqg iru wkh eudqg1 Wkh jdph lv wkhq dv wkh rqh dqdo|vhg lq wkh suhylrxv vxe0vhfwlrq= uvw/ wkh pdqxidfwxuhu rhuv uhwdlohuv d qrq olqhdu frqwudfw Il . zl tl 1 Wkhq/ uhwdlohuv vlpxowdqhrxvo| fkrrvh sulfhv sl / surwv uhdolvh dqg ihhv +li dq|, duh sdlg wr wkh pdqxidfwxuhuv1 4 Dw wkh odvw vwdjh/ hdfk uhwdlohu fkrrvhv sulfh wr pd{lplvh G l @ p +sl  zl ,tl +sl > sm ,  Il 1 Vlqfh p lv rqo| d vfdoh idfwru/ wkh uvw rughu frqglwlrqv duh 7<

jlyhq e| suhflvho| wkh vdph h{suhvvlrq dv +79, deryh1 Doo wkh vroxwlrqv zloo wkhuhiruh eh dv xqghu wkh fdvh wuhdwhg deryh/ jlylqj ulvh wr kljkhu sulfhv dqg surwv wkdq xqghu lqwud0eudqg frpshwlwlrq= sI I A sY L dqg I I A Y L 188 Qrwh khuh wkdw dw wkh htxloleulxp z A f/ zklfk hqwdlov wkdw wkhuh lv grxeoh pdujlqdolvdwlrq1 Krzhyhu/ wkh pdqxidfwxuhuv gr qrw orvh iurp wklv h{whuqdolw|1 E| fuhdwlqj d prqrsrolvw +ru vhyhudo prqrsrolvwv, grzqvwuhdp wkh| vwudwhjlfdoo| h{sorlw wkh suhvhqfh ri lpshuihfw frpshwlwlrq dv wkh| pdqdjh wr uhod{ frpsh0 wlwlrq1 Uh| dqg Vwljolw} +4<<8, vkrz wkdw e| dgglqj pruh dqg pruh od|huv +iru lqvwdqfh/ e| fuhdwlqj zkrohvdohuv dqg rwkhu lqwhuphgldulhv ehwzhhq wkh surgxf0 wlrq dqg wkh uhwdlo vwdjh, pdqxidfwxuhuv pljkw eh deoh wr jhw wkh prqrsro| +l1h1 wkh mrlqw0surw pd{lplvlqj, sulfhv1 Frpshwlwlrq uhgxfhv wkh ulvn wkdw yhuwlfdo uhvwudlqwv orzhu zhoiduh Yhuwlfdo uhvwudlqwv pljkw eh xvhg vr dv wr vwudwhjlfdoo| uhod{ frpshwlwlrq dqg lqgxfh kljkhu sulfhv rqo| lqvridu dv wkh up xvlqj wkhp hqmr|v hqrxjk pdunhw srzhu1 Wr jhw vrph lqwxlwlrq iru wklv/ frqvlghu wkh iroorzlqj h{dpsoh/ zkhuh wkhuh lv d pdqxidfwxuhu 0 vd| wkh up vhoolqj surgxfw 4 0 zklfk lv vhoolqj wkurxjk d uhwdlohu/ dqg rhuv klp d frqwudfw I .zt/ zkhuhdv doo wkh rwkhu q pdqxidfwxu0 huv lq wkh lqgxvwu| duh yhuwlfdoo| lqwhjudwhg1 Dvvxph wkh xvxdo ghpdqg ixqfwlrq= 5 6 q 47 [ 8 y  sl +4 . , . sm = +87, tl @ q q m @4 Iurp wkh pd{lplvdwlrq ri wkh uhwdlohu*v surw ixqfwlrq/ dqg diwhu lpsrvlqj v|p0 phwu| rq wkh q  4 yhuwlfdoo| lqwhjudwhg upv/ rqh rewdlqv wkh IRFv= + +q4,+5s4 sz5,.q+5s4 .y.z, @ 3> q +88,  +s4 qs,.q+5s.y , @ 3= q5 E| gudzlqj wkh uhdfwlrq ixqfwlrqv ri wkh upv/ rqh fdq vhh wkdw wkh| ehfrph ohvv hodvwlf dv q lqfuhdvhv/ lq wkh vhqvh wkdw wkhlu vorsh ghfuhdvh zlwk q1 Dv d uhvxow/ wkh pdqxidfwxuhu zrxog qhhg d pxfk eljjhu lqfuhdvh lq wkh zkrohvdoh sulfh wr rewdlq d jlyhq sulfh uhvsrqvh iurp wkh uhwdlohu= lwv dfwlrq kdv orzhu vwudwhjlf srzhu1 Iurp wkh vroxwlrq ri wkh v|vwhp ri IRFv rqh rewdlqv= s4

@

s @

q+ 5 +q  4,z . 5q+y . z, . y+5q  4, . z+6q  5,, > 7q5 . 5q+6q  5, .  5 +5q5  6q . 4, 5q5 y .  5 +q  4,z . q+z . y+5q  4,, = 7q5 . 5q+6q  5, .  5 +5q5  6q . 4,

+89, +8:,

88 Wkh jdph vkrxog eh frpsohwhg e| hqgrjhqlvlqj wkh ghflvlrqv wr doorfdwh h{foxvlyh whuul0 wrulhv wr uhwdlohuv= zh kdyh mxvw glvfxvvhg zkdw kdsshqv zkhq erwk pdqxidfwxuhuv xvh HW/ zlwkrxw surylqj wkdw HW lv lqghhg dq htxloleulxp fkrlfh1 Vhh Uh| dqg Vwljolw} +4<<8, iru vxfk d surri1

83

E| vroylqj wkh pdqxidfwxuhu*v sureohp/ zklfk lv wr fkrrvh z vr dv wr pd{lplvh lwv surw X @ +s4  z,t4 +s4 > s, rqh rewdlqv= z@

 5 +q  4,+5q . +5q  4,,y = 5+5 . ,+ 6 +q  4,6 . 5q6 . q5 +8q  7, .  5 q+6  :q . 7q5 ,

+8;,

Ilqdoo|/ e| uhsodflqj lqwr wkh sulfh h{suhvvvlrqv= sii 4 @

y+5q . +5q  4,, qy+5q . +5q  4,, >s @ = 5+ 5 +q  4,5 . 5q5 . q+6q  5, 5+5 . ,q5

+8<,

Zkhq doo upv duh yhuwlfdoo| lqwhjudwhg/ wkh +Ehuwudqg frpshwlwlrq, htxloleulxp sulfh lv jlyhq e|=89 se @

yq 5q . +q  4,

+93,

Wkh dgglwlrqdo pdun0xs wkdw d pdqxidfwxuhu lv deoh wr frppdqg gxh wr yhuwlfdo uhvwudlqwv lv wkhuhiruh jlyhq e|= sii 4  se @

5+ 5 +q  4,5

qy 5 +q  4, = . q+6q  5,+5q . +q  4,,

. 5q5

+94,

Lw fdq eh fkhfnhg wkdw g+sii 4  se ,@gq ? 3= wkh odujhu wkh qxpehu ri upv/ wkh orzhu wkh dgglwlrqdo pdun0xs wkdw vwudwhjlf yhuwlfdo uhvwudlqwv fdq jlyh wr wkh pdqxidfwxuhu xvlqj wkhp1 Wkh uhvxowv duh qrw urexvw= Vwudwhjlf vxevwlwxwhv Wkh suhylrxv uhvxowv kdyh ehhq rewdlqhg e| dvvxplqj wkdw ghflvlrqv lq wkh qdo pdunhw zhuh rq sulfhv1 Lw wxuqv rxw wkdw wkh uhvxowv rewdlqhg duh yhu| vhqvlwlyh wr wkh w|sh ri pdunhw frpshwlwlrq1 Li lqvwhdg zh dvvxph wkdw pdunhw ghflvlrqv duh rq txdqwlwlhv 0 udwkhu wkdq sulfhv 0 pdqxidfwxuhuv vwloo zdqw wr ghohjdwh wkhlu vdohv wr lqghshq0 ghqw uhwdlohuv/ exw= uvw/ wkh frqwudfwv wkh| jlyh wkhlu uhwdlohuv pdnh wkhp pruh 0 udwkhu wkdq ohvv 0 djjuhvvlyh/ dqg dv d frqvhtxhqfh qdo sulfhv zloo eh orzhu dqg zhoiduh kljkhu1 Vhfrqg/ wkh jdph zkhuh pdqxidfwxuhuv ghflgh zkhwkhu ru qrw wkh| zdqw wr ghohjdwh ghflvlrqv lv olnh d sulvrqhu*v glohppd= ghohjdwlrq lv wkh grplqdqw vwudwhj|/ dqg zloo eh fkrvhq dw htxloleulxp/ exw pdqxidfwxuhuv zrxog olnh wr dyrlg lw1 Lq zkdw iroorzv/ zh irupdolvh wkhvh uhvxowv1 Wkh prgho Wkh prgho lv wkh vdph dv wkh rqh dqdo|vhg deryh/ zlwk wzr pdqxidfwxuhuv zkr vhoo glhuhqwldwhg jrrgv hlwkhu gluhfwo| +yhuwlfdo lqwhjudwlrq, ru wkurxjk lqghshqghqw uhwdlohuv1 Wr dqdo|vh zkdw kdsshqv xqghu txdqwlw| frpshwlwlrq/ ohw xv xvh wkh lqyhuvh ghpdqg ixqfwlrqv1 Lqyhuwlqj wkh v|vwhp +77, zh rewdlq= sl @ y 

4 +5tl . tl . tm , = 4.

89 Wr qg wkh htxloleulxp sulfh xqghu yhuwlfdo lqwhjudwlrq/ mxvw lpsrvh deryh dqg vroyh iru R1

84

+95,

 ' f lq wkh IRFv

Yhuwlfdo lqwhjudwlrq Ilupv fkrrvh rxwsxw tl vr dv wr pd{lplvh l @ +sl +tl > tm ,  f,tl 1 E| vroylqj wkh v|vwhp ri uvw0rughu frqglwlrqv gl @gtl @ 3 zh rewdlq wkh vwdqgdug Frxuqrw htxloleulxp= tyl @

+y  f,+4 . , > 7 . 6

 yl @

+y  f,5 +4 . ,+5 . , = +7 . 6,5

+96,

Ghohjdwlrq Vxssrvh qrz wkdw erwk upv kdyh d grzqvwuhdp uhwdlohu/ dqg dqdo|vh wkh jdph zkhuh uvw hdfk pdqxidfwxuhu jlyhv d qrq0olqhdu sulflqj frqwudfw Il . zl tl wr lwv uhwdlohu dqg wkhq wkh uhwdlohuv frpshwh lq txdqwlwlhv +diwhu kdylqj revhuyhg wkh frqwudfw,1 Dw wkh odvw vwdjh ri wkh jdph/ uhwdlohu l fkrrvhv rxwsxw tl vr dv wr pd{lplvh ul @ +sl +tl > tm ,  zl ,tl 1 E| wdnlqj wkh uvw0rughu frqglwlrqv gul @gtl @ 3 dqg uh0duudqjlqj wkhp rqh rewdlqv wkh iroorzlqj uhdfwlrq ixqfwlrqv iru hdfk up +wr gudz wkhp lq wkh sodqh +t4 > t5 ,/ wkh| duh zulwwhq dv ixqfwlrqv ri t4 ,= 5+5 . ,t4 . y+4  ,  z4 +4  , >  t4 . y+4 . ,  z5 +4 . , = U5 +t4 , ' t5 @ 5+5 . , U4 +t5 , ' t5 @

+97, +98,

Lw lv hdv| wr vhh wkdw wkh uhdfwlrq ixqfwlrqv duh qrz qhjdwlyho| vorshg/ l1h1/ wkh jrrgv duh vwudwhjlf vxevwlwxwhv1 D up*v lqfuhdvhg rxwsxw zrxog eh iroorzhg e| d ulydo*v ghfuhdvh lq lwv rzq rxwsxw +vhh Iljxuh 918,1 LQVHUW Iljxuh 9181 Wdulv dv vwudwhjlf ghylfh= vwudwhjlf vxevwlwxwhv Wkh lvr0surw ixqfwlrqv duh qrw vkrzhg lq wkh jxuh/ exw lw zrxog eh hdv| wr fkhfn wkdw d vkliw wr wkh uljkw ri wkh uhwdlohu*v uhdfwlrq fxuyh +jlyhq wkh uhdfwlrq fxuyh ri wkh ulydo, zrxog fkdqjh wkh htxloleulxp wr d srlqw zkhuh wkh uhwdlohu kdv kljkhu surw +dw wkh qhz srlqw/ wkh uhwdlohu zrxog kdyh d kljkhu vkduh ri wkh pdunhw,1 Krzhyhu/ jxuh 918 dovr vkrzv wkdw li erwk uhwdlohuv kdg orzhu pdujlqdo frvwv/ wkh qhz htxloleulxp H* zrxog uhvxow lq odujhu txdqwlwlhv vrog lq wkh pdunhw wkdq dw wkh htxloleulxp H1 Wkh wzr upv zrxog vwloo kdyh wkh vdph vkduh ri wkh pdunhw exw vlqfh wkh| erwk lqfuhdvh rxwsxwv/ wkhlu surwv duh orzhu1 Wkh jxuh wkhuhiruh dqwlflsdwhv zkdw zh duh qrz jrlqj wr vhh pruh irupdoo|1 E| vroylqj wkh IRFv +ru 0 htxlydohqwo| 0 qglqj wkh lqwhuvhfwlrq srlqw ri wkh uhdfwlrq ixqfwlrqv, rqh fdq qg wkh uhwdlohuv* htxloleulxp txdqwlwlhv dqg sulfhv= tl

@

sl

@

+4 . , +y+7 . ,  5+5 . ,zl . zm , > 49 . 49 . 6 5 +; . 9 .  5 ,y . +; . ; .  5 ,zl . +5 . ,zm = 49 . 49 . 6 5

+99, +9:,

Dw wkh uvw vwdjh ri wkh jdph/ wkh pdqxidfwxuhu fkrrvhv zl wr pd{lplvh surw1 Dv ehiruh/ zh dvvxph wkdw wkh| kdyh wkh edujdlqlqj srzhu ylv0ã0ylv wkhlu uh0 wdlohu/ vr wkdw wkh| vhw wkh iudqfklvh ihh Il vr dv wr dssursuldwh doo wkhlu surw= 85

Il @ +sl  zl ,tl 1 Wkhuhiruh/ wkhlu sureohp lv= pd{zl xl @ +sl  f,tl / zkhuh sl dqg tl duh jlyhq e| wkh h{suhvvlrqv deryh1 E| wdnlqj wkh uvw0rughu frqgl0 wlrqv dqg vroylqj wkh v|vwhp rqh qgv wkh htxloleulxp vroxwlrqv ri wkh zkroh jdph +ehfdxvh ri v|pphwu|/ pdqxidfwxuhuv dqg uhwdlohuv zloo kdyh wkh vdph htxloleulxp yduldeohv1 Zh wkhuhiruh gurs wkh lqglfhv l dqg m,= zi i @ f 

5y = 49 . 53 . 8 5

+9;,

Dv rqh fdq lpphgldwho| vhh/ wkh zkrohvdoh sulfh lv orzhu wkdq wkh pdqxidfwxuhu*v rzq surgxfwlrq frvw= z ? f1 Wkh pdqxidfwxuhu zdqwv wr pdnh wkh uhwdlohu pruh djjuhvvlyh lq wkh pdunhw dqg wkhuhiruh lw vxevlglvhv lwv sxufkdvh ri wkh lqsxw vr dv wr pdnh lw vhoo pruh lq wkh pdunhw +ri frxuvh/ vxevlglvdwlrq grhv qrw frph frvwo| khuh/ vlqfh wkh uhwdlohu*v surwv duh dssursuldwhg e| wkh pdqxidfwxuhu yld wkh iudqfklvh ihh,1 Wklv lv suhflvho| wkh rssrvlwh uhvxow dv zlwk vwudwhjlf frpsohphqwv +l1h1/ sulfh frpshwlwlrq,/ zkhuh z zdv rswlpdoo| vhw kljkhu wkdq xqlw frvw f1 E| vxevwlwxwlqj/ rqh fdq qg htxloleulxp txdqwlwlhv dqg surwv dv= ti i @

5+4 . ,+5 . ,+y  f, > 49 . 53 . 8 5

xii @

5+4 . ,+5 . ,+; . ; .  5 ,+y  f,5 5

+49 . 53 . 8 5 ,

=

+9<,

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615

Yhuwlfdo uhvwudlqwv dv frooxvlyh ghylfhv

Lw kdv ehhq srlqwhg rxw lq wkh olwhudwxuh wkdw wkhuh duh rwkhu flufxpvwdqfhv lq zklfk yhuwlfdo uhvwudlqwv pljkw idflolwdwh frooxvlrq1 Lq wklv vhfwlrq/ zh frqvlghu wzr vxfk dujxphqwv1 Wkh uvw vkrzv wkdw uhvdoh sulfh pdlqwhqdqfh pljkw idyrxu frooxvlrq dprqj pdqxidfwxuhuv1 Wkh vhfrqg/ wkdw zkhq wzr ru pruh pdqxidf0 wxuhuv vhoo wkurxjk d frpprq uhwdlohu wkh| pljkw eh deoh wr uhdfk wkh frooxvlyh rxwfrph1

86

61514

USP pljkw idflolwdwh frooxvlrq

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Frpprq djhqf|

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87

Frpprq uhwdlohu dqg USP khos frooxvlrq-- L suhvhqw khuh d yhu| vlp0 solhg yhuvlrq ri Ehuqkhlp dqg Zklqvwrq +4<;8,*v prgho/ wr looxvwudwh wkhlu pdlq uhvxow1 Dvvxph wkdw wkhuh duh wzr xsvwuhdp surgxfhuv/ 4 dqg 5/ zklfk vhoo wkhlu surgxfwv yld d frpprq uhwdlohu/ U1 Wr vlpsoli| wkh lvvxh/ dvvxph wkdw erwk surgxfwv kdyh wr eh vrog/ rwkhuzlvh wkhuh lv d pdunhw idloxuh dqg }hur surwv duh pdgh e| doo upv18< Dvvxph dovr wkdw wkh uhwdlohu kdv qr edujdlqlqj srzhu/ iru lqvwdqfh ehfdxvh lw lv vhohfwhg dprqj d srsxodwlrq ri yhu| pdq| srwhqwldo uh0 wdlohuv zkr zrxog frpshwh hufho| wr eh wkh fkrvhq uhwdlohu1 Iru vlpsolflw|/ wkh uhwdlohu kdv qr frvw rwkhu wkdq wkh zkrohvdoh sulfh dqg wkh surgxfhuv kdyh wkh vdph pdujlqdo frvw f1 Ilqdoo|/ frqvxphu ghpdqg lv jlyhq e| tl @ d  esl . sm 1 Frqvlghu wkh iroorzlqj jdph1 Iluvw/ hdfk surgxfhu l @ D> E vlpxowdqhrxvo| pdnhv wdnh0lw0ru0ohdyh0lw rhuv wr U lq wkh irup ri d qrq0olqhdu frqwudfw Il .zl tl 1 Wkhvh frqwudfwv duh sxeolfo| revhuydeoh +wkh| pljkw dovr { wkh uhwdlo sulfh li uhvdoh sulfh pdlqwhqdqfh +USP, lv doorzhg,1 Vhfrqg/ wkh uhwdlohu dffhswv ru uhmhfwv wkh rhuv1 Wklug/ li erwk rhuv kdyh ehhq dffhswhg/ wkh uhwdlohu {hv uhvdoh sulfhv +ru vlpso| vhoov dw wkh sulfh lpsrvhg e| wkh surgxfhu xqghu USP,/ ghpdqg dqg surwv duh uhdolvhg/ dqg iudqfklvh ihhv duh sdlg1 Li rqh ru erwk rhuv duh uhmhfwhg/ qr vdoh rffxuv dqg doo upv jhw }hur sd|0rv1 Qr USP +sulfh fkrlfhv ghohjdwhg wr wkh frpprq uhwdlohu, Ehiruh dqdo|vlqj Ehuqkhlp dqg Zklqvwrq*v jdph/ ohw xv orrn dw zkdw kdsshqv li wkh frpprq uhwdlohu zdv lq fkdujh ri sulfh ghflvlrqv1 Dw wkh odvw vwdjh ri wkh jdph/ li erwk rhuv kdyh ehhq dffhswhg +dqg wkhuhiruh iru d jlyhq zl dqg Il ,/ wkh uhwdlohu fkrrvhv qdo sulfhv sD / sE vr dv wr pd{lplvh= U @ +sD  zD ,+d  esD . sE , . +sE  zE ,+d  esE . sD ,1 Iurp wkh IRFv C U @Csl @ 3/ rqh rewdlqv sl @ ^d . zl +e  ,` @ ^5+e  ,`/ dqg tl +zl > zm , @ +d  ezl . zm ,@51 Lq wkh uvw vwdjh/ wkh surjudpph ri d surgxfhu l zloo wkhuhiruh eh wr fkrrvh lwv frqwudfw rhu wr pd{lplvh lwv surw jlyhq wkh frqwudfw ri wkh frpshwlqj sur0 gxfhu/ dqg dovr wdnlqj lqwr dffrxqw wkh sduwlflsdwlrq frqvwudlqw ri wkh uhwdlohu= S ^+sl +zl ,  zl ,tl +zl > zm ,  Il `  3= pd{ l @ +zl  f,tl +zl > zm , . Il / v=wr =

9

l@4>5>l@m

zl >Il

+:3,

Vlqfh dw htxloleulxp wkh uhwdlohu zloo pdnh qr surw/ wkh uhwdlohu*v frqvwudlqw pxvw eh elqglqj= Il @ +sl +zl ,  zl ,tl +zl > zm , . +sm +zm ,  zm ,tm +zl > zm ,  Im =

+:4,

Wkhuhiruh/ surgxfhu l*v surjudpph fdq eh uhzulwwhq dv= pd{ l @ +sl +zl ,  f,tl +zl > zm , . +sm +zm ,  zm ,tm +zl > zm ,  Im = zl

+:5,

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88

E| vxevwlwxwlqj wkh htxloleulxp ydoxhv ri wkh odvw vwdjh ri wkh jdph/ wklv eh0 frphv=     dezl .zm d.zm +e , dezm .zl +e ,  f .  z  Im = pd{ l @ d.5+zel m , 5 5+e , 5 zl

+:6,

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zl >Il >sl

+:7,

Dw htxloleulxp/ wkh uhwdlohu*v frqvwudlqw lv elqglqj= Il @ +sl  zl ,tl +sl > sm , . +sm  zm ,tm +sl > sm ,  Im =

+:8,

Wkhuhiruh/ surgxfhu l*v surjudpph fdq eh uhzulwwhq dv= pd{  l @ +sl  f,tl +sl > sm , . +sm  zm ,tm +sl > sm ,  Im =

zl >sl

+:9,

Diwhu vxevwlwxwlqj wkh vshflf ixqfwlrqdo irup dvvxphg iru ghpdqg/ wklv eh0 frphv= pd{ l @ +sl  f,+d  esl . sm , . +sm  zm ,+d  esm . sl ,  Im =

zl >sl

+::,

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7

Dqwl0frpshwlwlyh hhfwv= ohyhudjh dqg iruh0 forvxuh

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714

Dqwl0frpshwlwlyh hhfwv= H{foxvlyh ghdolqj

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435

Table 6.1. Segal-Whinston: simultaneous offers B2

Accept

Reject

Accept

CS m +t, CS m +t

CS m +t, CS m

Reject

CS m , CS m +t

CS e, CS e

B1

Table 6.2. Segal-Whinston: discriminatory offers B2

Accept

Reject

Accept

CS m +t1, CS m +t2

CS m +t1, CS m

Reject

CS m , CS m +t2

CS e, CS e

B1

Table 6.3. A delegation game delegation

vert. int.

delegation

π uff , π uff

π d /i, πi/d

vert. int.

π i/d , πd /i

π vi , π vi

Upstream firm (manufacturer)

Downstream firm (retailer)

Consumers

Figure 6.1. Double marginalisation

Upstream firm (manufacturer)

Downstream firm (retailer)

Downstream firm (retailer)

Consumers

Figure 6.2. Horizontal externality

Upstream firm (manufacturer)

Upstream firm (manufacturer)

Downstream firm (retailer)

Downstream firm (retailer)

Consumers

Figure 6.3. Competing vertical chains

R1 ( w1 = c) R1′ ( w1′ > c)

p2

R2′ ( w2′ > c ) E′

R2 ( w2 = c )

E

p1

Figure 6.4. Tariffs as strategic device: Strategic complements

q2

R1′ ( w1′ < c)

R1 ( w1 = c )

E′ E R2′ ( w2′ < c) R2 ( w2 = c) q1

Figure 6.5. Tariffs as strategic device: Strategic substitutes

p

θ pm

c1 cE

A

B D F qm

E qe

Figure 6.6. The "Chicago School" critique to foreclosure

θ′

q

B1 Accept

Reject B2

B2 Accept  CS m + t1   m  + CS t  2

Reject  CS m + t1   m  CS  

Accept  CS m   m  + CS t  2

Figure 6.7. Segal-Whinston: sequential offers

Reject  CS e   e  CS 

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5

514

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6

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7

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515

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8

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lq d vwdwlf jdph +lll1,> lq wkh odvw shulrg ri wkh jdph/ d zhdn lqfxpehqw grhv qrw kdyh dq| uhdvrq wr jkw hqwu| +y1,1 L qrz suryh wkh rswlpdolw| ri wkh uhpdlqlqj dfwlrqv dv iroorzv1 +ly, Li hqwu| kdv ehhq irxjkw lq wkh uvw shulrg/ wkh vhfrqg hqwudqw zloo xvh Ed|hv* uxoh wr uhylvh lwv sulru wkdw lw idfhv d wrxjk lqfxpehqw1 Wkh uhylvhg suredelolw| wkdw wkh lqfxpehqw lv wrxjk jlyhq d jkw lv revhuyhg lv= Su+w m i, @

{ Su+i m w, Su+w, @ = Su+i m w, Su+w, . Su+i m z, Su+z, { . Su+i m z,+4  {,

+6,

Iru d pl{hg vwudwhj| wr eh rswlpdo/ wkh vhfrqg hqwudqw pxvw eh lqglhuhqw ehwzhhq hqwhulqj ru vwd|lqj rxw +wkdw jlyhv d }hur sd|r,/ wkdw lv=

D Su+w m i, S H . +4  Su+w m i ,, H @ 3>

+7,

zklfk diwhu vxevwlwxwlqj dqg uh0duudqjlqj fdq eh uhzulwwhq= Su+i m z, @ 

{S H = +4  {, D H

+8,

Qrwh wkdw S H ? 3/ vr wklv suredelolw| lv srvlwlyh/ dqg e| +5*,/ Su+i m z, ? 41 +ll1e, Iru wkh zhdn lqfxpehqw wr udqgrplvh lq wkh uvw shulrg/ lw pxvw eh lqglhuhqw ehwzhhq jkwlqj +zklfk zrxog eh iroorzhg e| hqwu| zlwk vrph sure0 delolw| Su+Hqwu| m i,, ru qrw=  D P D S L .  Su+Hqwu| m i ,L . +4  Su+Hqwu| m i,,  @ L +4 . ,= +9, E| uh0duudqjlqj/ zh qg wkh lqglhuhqfh frqglwlrq dv= Su+Hqwu| m i , @ 4 

S D L  L > +P  D L,

+:,

zklfk lv orzhu wkdq rqh e| dvvxpswlrq +4*,1 +l1, Wkh uvw srwhqwldo hqwudqw hqwhuv li lwv h{shfwhg sd|r lv kljkhu wkdq vwd|lqj rxw=  S D +;, {S H . +4  {, Su+i m z,H . +4  Su+i m z,, H A 3= Diwhu vxevwlwxwlrq/ wklv dprxqwv wr=

D S {S H . +4  {,H  {H +

S H  4, A 3= D H

+<,

Wkhuhiruh/ vrph uhvwulfwlrqv rq wkh sd|r pxvw eh lpsrvhg iru wkh suhylrxv frqglwlrq wr krog1 Dv orqj dv +<, krogv/ wkh vhpl0vhsdudwlqj htxloleulxp zh kdyh ghvfulehg h{lvwv1

48

Frpphqw Wkh jhqhudo fdvh W A 5 lv pxfk pruh frpsoh{ exw dovr pxfk ulfkhu1 Wkh pdlq uhvxow lv wkdw li W lv odujh hqrxjk/ lq wkh hduolhu shulrgv ri wkh jdph hyhq d zhdn lqfxpehqw jkwv zlwk fhuwdlqw|/ dqg dqwlflsdwlqj wklv hduo| hqwudqwv vwd| rxw1 Dv wkh jdph surfhhgv/ wkh rswlpdo vwudwhjlhv duh pl{hg rqhv/ olnh lq wkh W @ 5 fdvh1 Wkh pdlq lqvljkw ri wklv prgho lv wkdw hyhq d vpdoo ghsduwxuh iurp wkh shuihfw lqirupdwlrq vhwwlqj pljkw jlyh ulvh wr suhgdwlrq/ li wkh krul}rq lv qlwh exw orqj hqrxjk= d zhdn lqfxpehqw zloo sod| dw wkh ehjlqqlqj ri wkh jdph dv li lw zhuh d vwurqj rqh/ dqg wklv jlyhv lw wkh uhsxwdwlrq ri d jkwhu/ wkdw zloo frqylqfh hqwudqwv wr vwd| rxw lq hduo| shulrgv1 Wrzdugv wkh hqg ri wkh jdph/ krzhyhu/ d zhdn lqfxpehqw zloo qrw qg lw frqyhqlhqw wr jkw/ dqg dqwlflsdwlqj lw vrph hqwudqwv zloo wu| wkhlu oxfn +uhfdoo/ wkhuh lv dozd|v wkh srvvlelolw| wkdw wkh lqfxpehqw lv d wrxjk rqh,1 Ilqdoo|/ qrwh wkdw Plojurp dqg Urehuwv +4<;5d, h{dplqh dq h{whqvlrq zkhuh wkhuh lv wzr0vlghg xqfhuwdlqw| +wkh lqfxpehqw grhv qrw nqrz wkh sd|r ri wkh hqwudqw,/ dqg frqup wkdw suhgdwlrq pljkw dulvh dovr lq wklv pruh frpsoh{ lqfrpsohwh lqirupdwlrq jdph1 51616

Plojurp dqg Urehuwv* olplw sulflqj prgho--

Khuh L suhvhqw d vlpsolhg yhuvlrq ri Plojurp dqg Urehuwv +4<;5e,1 Frqvlghu dq lqgxvwu| zlwk dq lqfxpehqw +up 4, dqg d srwhqwldo hqwudqw +up 5,/ dqg ghpdqg iru wkh krprjhqrxv jrrg lv jlyhq e| s @ 4T14; Wkh srwhqwldo hqwudqw kdv d pdujlqdo frvw f dqg/ li lw hqwhuv/ d {hg vxqn frvw I 1 Lwv frvwv duh frpprq nqrzohgjh1 Wkh lqfxpehqw*v pdujlqdo frvw/ lqvwhdg/ pljkw eh orz +fo @ 3, ru kljk +fk @ f ? 4@5,1 Wklv frvw lv sulydwh lqirupdwlrq= wkh lqfxpehqw nqrzv lw/ exw wkh hqwudqw eholhyhv wkdw lw lv orz zlwk d suredelolw| { ? 4= Su +f4 @ 3, @ {/ dqg kljk zlwk d suredelolw| Su +f4 @ f, @ 4  {14< L dovr dvvxph iru vlpsolflw| wkdw li wkh hqwudqw dfwxdoo| hqwhuv wkh lqgxvwu|/ lw zloo lpphgldwho| ohduq wkh wuxh frvw ri wkh lqfxpehqw1 Wkh jdph lv dv iroorzv1 Lq wkh uvw shulrg/ up 4 fkrrvhv lwv rxwsxw1 Lq wklv shulrg/ lw lv d prqrsrolvw1 Lq wkh vhfrqg shulrg/ up 5 ghflghv zkhwkhu wr hqwhu ru qrw> li lw grhv/ lw vlqnv lwv frvw I > li lw grhv qrw/ lw jhwv d }hur sd|r1 Dfwlyh upv fkrrvh rxwsxwv1 Iru vlpsolflw|/ L dvvxph qr glvfrxqwlqj dfurvv shulrgv= up 4 mxvw pd{lplvhv wkh vxp ri lwv sd|r1 L orrn iru wkh Shuihfw Ed|hvldq Htxloleuld ri wkh jdph1 Lq wklv fdvh/ jlyhq wkh vwudwhjlhv +v4 > v5 , ri wkh sod|huv dqg wkh eholhiv s ri wkh hqwudqw/ wkh SEH frqvlvwv ri d wulsoh +v4 > v5 > s, vxfk wkdw hdfk sod|hu fkrrvhv wkh rswlpdo vwudwhj| jlyhq wkh vwudwhj| ri wkh ulydo dqg wkh hqwudqw*v eholhiv/ dqg wkh eholhiv duh fruuhfw1 Lq vxfk d jdph/ wkhuh duh wzr srvvleoh w|shv ri htxloleulxp lq sxuh vwudwhjlhv1 Lq d vhsdudwlqj htxloleulxp/ wkh orz frvw lqfxpehqw fkrrvhv d odujh hqrxjk 4; L zloo irfxv iru vlpsolflw| rq wkh fdvh ri Frxuqrw frpshwlwlrq lq wkh vhfrqg vwdjh/ exw wklv lv mxvw wr kdyh vlpsohu sd|r ixqfwlrqv1 Wkh w|sh ri frpshwlwlrq lq wkh vhfrqg vwdjh grhv qrw sod| dq| uroh lq wkh dqdo|vlv1 4< Lq wklv vshflf h{dpsoh/ zkhuh ghpdqg lv olqhdu dqg wkhuh duh pdujlqdo frvwv/ wkh prgho frxog eh uh0lqwhusuhwhg dv rqh zkhuh xqfhuwdlqw| lv rq wkh ghpdqg lqwhufhsw/ udwkhu wkdq rq wkh lqfxpehqw*v frvwv1

49

rxwsxw/ dqg wkh hqwudqw fruuhfwo| lqihuv wkh lqfxpehqw*v w|sh e| revhuylqj lwv uvw0shulrg rxwsxw1 Lq d srrolqj htxloleulxp/ erwk wkh orz dqg wkh kljk frvw lqfxpehqw fkrrvh wkh vdph rxwsxw1 Li wkh hqwudqw kdv d kljk hqrxjk h{0dqwh suredelolw| wr phhw d orz frvw lqfxpehqw/ lw zloo qrw hqwhu1 Lw nqrzv wkdw wkh rxwsxw revhuyhg lq wkh uvw shulrg grhv qrw fduu| dq| dgglwlrqdo lqirupdwlrq derxw wkh lqfxpehqw*v w|sh/ dqg lwv srvwhulru eholhiv duh dffruglqjo| wkh vdph dv lq wkh uvw shulrg1 +Wkh idfw wkdw wkh orz frvw lqfxpehqw lq d vhsdudwlqj htxloleulxp dqg d kljk frvw lqfxpehqw lq d srrolqj htxloleulxp vhw d orz hqrxjk sulfh wr pdnh wkh hqwudqw vwd| rxw pdnhv wkh prgho uhvhpeoh wkh vr0fdoohg olplw sulflqj dujxphqwv ri hqwu| ghwhuuhqfh1, Ehiruh fkdudfwhulvlqj wkhvh htxloleuld/ ohw ph eulh | vxppdulvh wkh uvw0 dqg vhfrqg0shulrg rxwsxwv xqghu wkh glhuhqw srvvlelolwlhv1 Ilup 4*v rswlpdo txdqwlwlhv li wkhuh zdv qr srwhqwldo hqwudqw zrxog eh= t4pk @

4f 4 > t4po @ > 5 5

+43,

zkhuh odehov k dqg o vwdqg iru kljk0 dqg orz0frvw lqfxpehqw uhvshfwlyho|/ dqg p vwdqgv iru prqrsro|1 Xqghu gxrsro|/ htxloleulxp txdqwlwlhv lq wkh Frxuqrw jdph duh= t4gk @ t5gk @

4f 4.f g 4  5f > t4go @ > t5 o @ > 6 6 6

+44,

zkhuh g vwdqgv iru gxrsro| +wkh uvw sdlu uhihuv wr wkh fdvh ri d kljk0frvw lq0 fxpehqw/ wkh vhfrqg ri d orz0frvw lqfxpehqw,1 Doo htxloleulxp surwv duh jlyhq n ,5 1 Wr pdnh wkh dqdo|vlv lqwhuhvwlqj/ L dvvxph wkdw lq d ixoo lqiru0 e| nlm @ +tlm pdwlrq frqwh{w wkh hqwudqw zrxog qhyhu hqwhu li lw idfhg d orz0frvw lqfxpehqw +g5o  I ? 3,/ exw zrxog dozd|v hqwhu li lw idfhg d kljk0frvw rqh +3 ? g5k  I ,/ ru= +4  5f,5 +4  f,5 AI A = < <

+45,

Ilqdoo|/ ghqrwh e| p 4m +t4o , wkh surw rewdlqhg e| dq lqfxpehqw ri w|sh m @ k> o zkhq vhoolqj dq rxwsxw t4o +qrw qhfhvvdulo| frlqflglqj zlwk wkh prqrsro| rxwsxw ri w|sh m,1 Vhsdudwlqj htxloleulxp L vkdoo qrz vkrz wkdw wkh iroorzlqj lv dq htxl0 oleulxp= ;  t @ t4o A t4po A A ? 4o t4k @ t4pk +46, v @ Hqwhu> li t4p ? t4o > Q rw Hqwhu> li t4p  t4o > A A = 35 { @ 3> li t4p ? t4o > {3 @ 4> li t4p  t4o > zkhuh {3 @ Su+f4 @ 3 m t4p , lv wkh hqwudqw*v eholhi ri idflqj d orz0frvw lqfxpehqw jlyhq wkh revhuyhg uvw0shulrg rxwsxw t4p 1 4:

Dw wklv htxloleulxp/ wkh orz0frvw lqfxpehqw vhwv d kljkhu rxwsxw wkdq lwv prqrsro| rxwsxw +wkdw lv/ wkh rxwsxw vhw li lw zhuh lq d prqrsro| xqwkuhdwhqhg e| hqwu|, lq wkh uvw shulrg/ dqg wkh kljk0frvw lqfxpehqw vhwv lwv rzq prqrsro| rxwsxw/ wkhuhe| uhyhdolqj lwv w|sh1 Wkh hqwudqw fruuhfwo| lqihuv wkh lqfxpehqw*v w|sh dqg ehkdyhv dffruglqjo|/ hqwhulqj rqo| zkhq li idfhv d kljk0frvw up 41 Iru wklv wr eh dq htxloleulxp/ qr up vkrxog kdyh dq lqfhqwlyh wr ghyldwh1 Jlyhq lwv eholhiv dqg up 4*v vwudwhjlhv deryh/ lw lv fohdu wkdw wkh hqwudqw fdqqrw gr ehwwhu wkdq wkdw= hqwhulqj zlwk d orz frvw up zrxog hqwdlo orvvhv/ dqg qrw hqwhulqj zlwk d kljk frvw rqh zrxog lpso| iruhjrlqj srvlwlyh surwv1 Ohw xv qrz fkhfn wkdw wkh kljk frvw lqfxpehqw kdv qr lqfhqwlyh wr ghyldwh1 Li lw sod|v wkh fdqglgdwh htxloleulxp vwudwhj| lw zloo jhw lwv prqrsro| surw g lq wkh uvw shulrg exw zrxog dwwudfw hqwu|= p 4k .  4k 1 Dowhuqdwlyho|/ lw frxog vhw t4o wkxv plplfnlqj wkh orz frvw lqfxpehqw dqg ghwhuulqj hqwu|= xqghu wklv p ghyldwlrq/ lw zrxog jhw  p 4k +t4o , .  4k 1 Lwv lqfhqwlyh frqvwudlqw +LF, wkhuhiruh dprxqwv wr= g p p p 4k .  4k   4k +t4o , .  4k / ru=

+4  f,5  +4  t4o  f,t4o = <

+47,

Dv iru wkh orz frvw lqfxpehqw/ e| sod|lqj wkh +fdqglgdwh, htxloleulxp vwudwhj| lw zrxog jhw d orzhu surw wkdq e| vhwwlqj lwv prqrsro| rxwsxw lq wkh uvw shulrg/ p exw lw ghwhuv hqwu|= p 4o +t4o , .  4o 1 Wklv vwudwhj| lqyroyhv d vdfulfh lq wkh surwv hduqhg lq wkh uvw shulrg> wkh dowhuqdwlyh frxog eh wr vhw wkh prqrsro| rxwsxw lq wkh uvw shulrg +vlqfh dq| rxwsxw orzhu wkdq t4o zrxog dwwudfw hqwu|/ fkrrvlqj t4po lv fohduo| wkh ehvw ghyldwlrq rxwsxw, exw wklv zrxog qrw ghwhu hqwu|/ dqg zrxog wkhuhiruh jlyh gxrsro| surwv lq wkh vhfrqg shulrg1 Wklv ghyldwlrq g jlyhv wkh sd|r  p 4o .  4o / jlylqj ulvh wr wkh iroorzlqj lqfhqwlyh frqvwudlqw= p p g p 4o +t4o , .  4o   4o .  4o / ru= +4  t4o ,t4o 

+4 . f,5 = <

+48,

s Lw lv hdv| wr fkhfn wkdw wkh kljk frvw LF +47, lvsvdwlvhg iru t4o  4@5. 8+4f,@9 dqg wkdw wklv lv eljjhu wkdq 4@5 iru f ? +6 8  8,@7 * =75:4 +iru ydoxhv ri f kljkhu wkdq wklv wkuhvkrog/ vhsdudwlqj htxloleuld zrxog h{lvw zkhuh wkh orz frvw lqfxpehqw vlpso| vhwv lwv prqrsro| rxwsxw dqg lv qrw plplfnhg e| wkh kljk frvw lqfxpehqw,1 s Wkh orz frvw LF +48, lv vdwlvhg iru t4o  4@5. 8  ;f  7f5 @9153 Wkhuhiruh/ s wkhuh h{lvwv dqslqwhuydo ri ydoxhv zklfk vdwlvhv erwk LFv dv orqj dv 4@5. 8+4 s f,@9  4@5 . 8  ;f  7f5 @9/ ru f  5 8@6  4 * =7<1 Qrwh dovr wkdw wkh ehvw srvvleoh htxloleulxp iru wkh lqfxpehqw s fruuhvsrqgv wr wkh rxwsxw wkdw pdnhv wkh kljk frvw LF*v elqg= t4o @ 4@5 . 8+4  f,@9 +dq| rxwsxw odujhu wkdq wklv zrxog ghfuhdvh wkh uvw shulrg surw zlwkrxw fkdqjlqj wkh vhfrqg shulrg surw,154 53 Wkh

orzhu urrw ri wkh vhfrqg rughu htxdwlrq lv glvfdughg ehfdxvh lw lv orzhu wkdq *21 wkh kljk frvw lqfxpehqw zrxog kdyh wkh vdph sd|r dq|zd|/ wklv htxloleulxp zrxog eh vhohfwhg li wkh fulwhulrq ri Sduhwr grplqdqfh +iru wkh dfwlyh sod|huv/ wkdw lv wkh upv, lv xvhg1 54 Vlqfh

4;

Srrolqj htxloleulxp L vkdoo qrz fkhfn li wkh iroorzlqj lv dq htxloleulxp= ;  ? t4o @ t4k @ t4po v @ Hqwhu> li t4p ? t4po > Q rw Hqwhu> li t4p  t4po > = 35 { @ 3> li t4p ? t4po > {3 @ {> li t4p  t4po > Dw wklv htxloleulxp/ wkh kljk frvw lqfxpehqw lplwdwhv wkh orz frvw lqfxpehqw1 Vlqfh wkh srwhqwldo hqwudqw xqghuvwdqgv wkdw lw fdqqrw lqihu wkh lqfxpehqw*v w|sh iurp wkh uvw shulrg sod|/ lw ghflghv rq wkh edvlv ri lwv h{0dqwh eholhiv/ zklfk vd| wkdw wkh lqfxpehqw lv d orz frvw rqh zlwk d suredelolw| {1 Dffruglqjo|/ wkh srrolqj htxloleulxp fdq h{lvw rqo| li wkh hqwudqw*v h{shfwhg sd|r +zklfk lv wkh vdph ehiruh dqg diwhu revhuylqj up 4*v uvw shulrg fkrlfh, lv qhjdwlyh= {+g5o  I , . +4  {,+g5k  I , ? 3/ ru { A

+4  f,5  I = 5  6f

+49,

Lq rwkhu zrugv/ wkh h{0dqwh suredelolw| wkdw wkh hqwudqw zloo phhw d orz frvw lqfxpehqw pxvw eh kljk hqrxjk li wkh srrolqj htxloleulxp lv wr h{lvw155 Rwkhu0 zlvh/ vlqfh dw d srrolqj htxloleulxp uvw shulrg sod| grhv qrw ohdg wr d uhylvlrq ri eholhiv/ wkh hqwudqw zrxog dozd|v hqwhu1 Lq wxuq/ wkh idfw wkdw hqwu| zloo qrw eh ghwhuuhg lpsolhv wkdw wkh kljk frvw lqfxpehqw kdg ehwwhu sod| lwv prqrsro| rxwsxw udwkhu wkdq eox!qj1 Dv iru wkh lqfhqwlyh frqvwudlqwv ri up 4/ wkh kljk frvw lqfxpehqw pxvw suhihu wr lplwdwh wkh orz frvw up dqg ghwhu hqwu| udwkhu wkdq uhgxflqj lwv rxwsxw dqg dwwudfwlqj hqwu|=56 g p p p p 4k .  4k   4k +t4o , .  4k / ru=

+4  f,5  +4  4@5  f,@5> <

+4:,

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57

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+57,

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+58,

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58

+59,

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59

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+63,

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Dv wkh suhylrxv glvfxvvlrq kdv klqwhg/ wkh zhoiduh lpsolfdwlrqv ri wlh0lq vdohv duh idu iurp xqdpeljxrxv/ dqg wr ghdo zlwk wkhp lq sudfwlfh pljkw eh h{wuhpho| frpsoh{1 Lq prvw fdvhv/ w|lqj zloo kdyh h!flhqf| hhfwv wkdw zloo ehqhw frq0 vxphuv> lq d ihz +suredeo| uduh, fdvhv/ wkh| pljkw kdyh kdupixo h{foxvlrqdu| hhfwv wkdw kdyh lq dq| fdvh wr eh edodqfhg zlwk dq| srvvleoh h!flhqf| hhfwv1 Lq vrph rwkhu fdvhv/ wlh0lq vdohv pljkw eh pdgh wr sulfh glvfulplqdwh/ dqg zkhq wklv lv wkh fdvh wkh lpphgldwh lpsdfw rq frqvxphu dqg wrwdo vxusoxv lv dpeljx0 rxv hyhq lq wkh devhqfh ri dq| h!flhqf| hhfw> wkhuhiruh/ lw lv hyhq ohvv olnho| wr eh ghwulphqwdo zkhq h!flhqf| frqvlghudwlrqv duh suhvhqw1 Wudglwlrqdoo|/ w|lqj kdv ehhq orrnhg xsrq yhu| vxvslflrxvo| e| dqwlwuxvw dx0 wkrulwlhv dqg frxuwv/ dqg iru d orqj wlph lq wkh XV lw kdg d vwdwxv yhu| forvh wr shu vh surklelwlrq +vhh fkdswhu 4,1 Wklv kduvk dssurdfk lv qrw mxvwlhg1 Wr vwduw zlwk/ lw zrxog pdnh vhqvh wr lqyhvwljdwh w|lqj sudfwlfhv rqo| zkhq wkh| duh fduulhg rxw e| upv hqgrzhg zlwk frqvlghudeoh pdunhw srzhu1 Vlqfh lw lv wkh h{foxvlrqdu| hhfw ri w|lqj wkdw rqh vkrxog ihdu wkh prvw/ upv wkdw gr qrw kdyh d grplqdqw srvlwlrq vkrxog eh iuhh wr xvh wlh0lq vdohv dv wkh| olnh1 Df0 fruglqjo|/ d vdih kduerxu pljkw eh suhvfulehg vxfk wkdw upv zlwk pdunhw vkduhv ri ohvv wkdq/ vd|/ 73( lq hdfk surgxfw pdunhw lqyroyhg zloo qrw eh lqyhvwljdwhg1 Wklv dssurdfk pljkw vhhp jhqhurxv dw uvw vljkw/ exw lw lv mxvwlhg e| wkh qxpehu ri srvvleoh h!flhqf| hhfwv wkdw ghulyh iurp w|lqj= frqvxphuv jdlq lq pdq| fdvhv zkhuh wkh| gr qrw kdyh wr erwkhu dvvhpeolqj surgxfwv wkhpvhoyhv/ ru orrn iru glhuhqw vxssolhuv wr froohfw wkh glhuhqw frpsrqhqwv wkh| zdqw/ dqg vr rq1 W|lqj pljkw wkhuhiruh doorz d qrq0grplqdqw up wr ehfrph pruh frpshwlwlyh/ uhvxowlqj lq d srvlwlyh zhoiduh hhfw1 Sduwlfxoduo| lqvwuxfwlyh lq wklv uhvshfw lv wkh Loirug fdvh/ zkhuh XN dxwkru0 lwlhv vwrsshg Loirug iurp exqgolqj vdohv ri skrwrjudsklf opv zlwk wkhlu ghyho0 rsphqw/ vshhglqj xs lwv grzqidoo zlwk uhvshfw wr Nrgdn1 Lq wkh plg093*v/ wkh frorxu op pdunhw zdv grplqdwhg e| Nrgdn zruogzlgh1 Lq wkh XN/ Nrgdn kdg durxqg ;3( ri vdohv/ zlwk Loirug kroglqj ohvv wkdq 48( pdunhw vkduh1 Rqh ri wkh uhdvrqv zk| Loirug irxqg lw gl!fxow wr fdwfk xs zlwk Nrgdn zdv gxh wr +lqgluhfw, qhwzrun hhfwv= Loirug xvhg d surfhvvlqj v|vwhp wkdw glhuhg iurp wkh grplqdqw Nrgdn*v/ dqg wklv uhsuhvhqwhg dq revwdfoh wr lwv glxvlrq zlwk lqghshqghqw surfhvvruv1 Loirug*v uhvsrqvh wr wklv sureohp zdv wr vhoo op zlwk surfhvvlqj lqfoxghg1 D fxvwrphu zrxog ex| wkh op dqg diwhu wdnlqj slfwxuhv 87

zrxog vhqg edfn wkh h{srvhg op wr Loirug/ zklfk zrxog surfhvv lw lq lwv ode0 rudwrulhv dqg wkhq vhqg lw edfn wr wkh fxvwrphu1 Iroorzlqj d frpsodlqw e| lqghshqghqw surfhvvruv wkdw wklv sudfwlfh zrxog kdyh uhgxfhg wkh ghpdqg iru lwv vhuylfhv/ wkh Prqrsrolhv dqg Phujhuv Frpplvvlrq irxqg wkh w|lqj ri op dqg surfhvvlqj wr eh dqwl0frpshwlwlyh lq 4<991 Dv Vxwwrq +4<<;= 459, sxwv lw/ ^w`klv qglqj juhdwo| lqfuhdvhg Loirug*v sureohpv1 Wkh lqghshqghqwv frxog qrw hdvlo| surfhvv wkh frpsdq|*v op/ dqg wklv dsshduv wr kdyh ehhq wkh odvw vwudz iru Loirug*v uhwdlo exvlqhvv1 ^111` Loirug zlwkguhz lwv eudqg iurp wkh froru op pdunhw lq 4<9;1 Zkhq d up grhv qrw sdvv wkh uvw vfuhhqlqj whvw dqg lw kdv pdunhw vkduhv deryh wkdw fhuwdlq wkuhvkrog/ d ixoo lqyhvwljdwlrq vkrxog zhljk srvvleoh dqwl0 frpshwlwlyh hhfwv djdlqvw srvvleoh h!flhqf| uhdvrqv ehklqg wkh sudfwlfh1 Qrwh wkdw wkhru| whoov xv wkdw wkh kljkhu wkh frpsohphqwdulw| ehwzhhq wkh surgxfwv wkh ohvv olnho| lw lv wkdw w|lqj zloo eh xvhg iru h{foxvlrqdu| sxusrvhv/ vr doohjd0 wlrqv ri prqrsrolvdwlrq zkhq d up exqgohv wzr ixoo| frpsohphqwdu| surgxfwv vkrxog eh uhfhlyhg zlwk pruh vnhswlflvp1 Wkhru| dovr vd|v wkdw h{foxvlrqdu| hhfwv duh pruh olnho| zkhq wkhuh lv d fuhgleoh frpplwphqw wr exqgolqj/ vxfk dv zkhq wzr suhylrxvo| vhsdudwh frpsrqhqwv duh frpelqhg lq d qhz surgxfw ghvljq1 Wkhuhiruh/ lw lv ohvv olnho| wr kdyh h{foxvlrqdu| hhfwv zkhq/ iru lq0 vwdqfh/ wzr surgxfwv duh vrog wrjhwkhu exw wklv pdunhwlqj ghflvlrq lv uhyhuvleoh1 Krzhyhu/ zkhq vxfk d frpplwphqw h{lvwv/ lw lv idu iurp dxwrpdwlf wkdw lw zloo kdyh ghwulphqwdo hhfwv1 Ixuwkhupruh/ h{wuhph fduh vkrxog eh wdnhq lq wklv fdvh ehfdxvh vwrsslqj d w|lqj sudfwlfh dprxqwv wr lqwhuyhqlqj dqg prgli|lqj wkh surgxfw ghvljq ri d jrrg1 61518

Prghov ri w|lqj/ L= Uhtxluhphqwv w|lqj dv d phwhulqj ghylfh-

Lq wklv vhfwlrq L vkrz wkdw d prqrsrolvw ri dq hvvhqwldo jrrg D kdv dq lqfhqwlyh wr wlh wkh xvh ri d frpsohphqwdu| jrrg E/ zkrvh ghpdqg lqwhqvlw| glhuv dfurvv frqvxphuv1:9 Wkh zhoiduh frqvhtxhqfhv ri w|lqj zloo eh dpeljxrxv1:: D frqvxphu ri w|sh l @ o> k zkr ex|v rqh xqlw ri surgxfw D dqg t xqlwv ri surgxfw E kdv wkh iroorzlqj xwlolw| ixqfwlrq= Xl @ t 

t5 = 5yl

+78,

Frqvxplqj wkh jrrgv vhsdudwho| jlyhv }hur xwlolw|/ dqg ex|lqj pruh wkdq rqh xqlw ri jrrg D grhv qrw dgg wr xwlolw|1 W|sh0o frqvxphuv kdyh orzhu lqwhqvlw| ri ghpdqg +yo ? yk , dqg duh d vkduh  ri wkh srsxodwlrq +ri vl}h 4 iru vlpsolflw|,/ 4   ehlqj wkh vkduh ri w|sh0k frqvxphuv1 Jrrg D lv prqrsrol}hg e| up 4/ zkhuhdv vhyhudo upv kdylqj lghqwlfdo whfkqrorj| 0 lqfoxglqj up 4 0 duh lqyroyhg lq wkh surgxfwlrq ri jrrg E1 Wkh :9 Qrwh wkdw frpsohphqwv khuh duh qrw xvhg lq {hg sursruwlrqv/ xqolnh wkh fdvh L dqdo|vh ehorz lq vhfwlrq 615191 :: L iroorz khuh Wluroh +4<;;= 475047;, txlwh forvho|/ zlwk d glhuhqw xwlolw| ixqfwlrq wkdw jlyhv ulvh wr voljkwo| vlpsohu fdofxodwlrqv1

88

frvw ri surgxflqj rqh xqlw ri wkh surgxfw duh uhvshfwlyho| fD dqg fE ? 41 Wkhuh duh qr {hg frvwv1 Wr orrn dw wkh hhfwv ri +uhtxluhphqwv, w|lqj uvw frqvlghu wkh fdvh zkhuh up 4 vhoov vhsdudwho| jrrgv D dqg E1 Dvvxph wkdw wkhuh lv sulfh frpshwlwlrq lq wkh E pdunhw1 Qr w|lqj +li doo frqvxphuv ex|,1 Vxssrvh uvw doo frqvxphuv ex|1 Ghpdqg ri w|sh0l frqvxphuv lv rewdlqhg dv pd{t Xl  sD  sE t1 E| vhwwlqj gXl @gt @ 3/ rqh kdv= tl @ yl +4  sE ,=

+79,

D w|sh0l frqvxphu zloo wkhq ex| jrrg D li lwv vxusoxv FVl @ Xl  sD  sE tl lv qrq0qhjdwlyh ru/ diwhu vxevwlwxwlrq/ li FVl @ yl +4  sE ,5 @5  sD  31 Sulfh frpshwlwlrq rq wkh jrrg E pdunhw lpsolhv wkdw sE @ fE 1 Wkhuhiruh/ lq wkh fdvh zkhuh up 4 fkrrvhv sulfhv vr wkdw doo frqvxphuv ex|/ lw pxvw eh= sQW D @

yo +4  fE ,5 = 5

+7:,

Lq wklv fdvh/ w|sh0o frqvxphuv zloo kdyh }hur vxusoxv/ zkhuhdv w|sh0k frqvxphuv zloo kdyh d vxusoxv FVkQW @ +yk  yo ,+4  fE ,5 @51 Surgxfhu vxusoxv htxdov up 4*v surwv iurp jrrg D= QW @ yo +4 fE ,5 @5 fD 1 Zhoiduh zloo wkhuhiruh eh htxdo wr= Z QW @

++4  , yk . yo ,+4  fE ,5  fD = 5

+7;,

Qr w|lqj +rqo| kljk w|shv ex|,1 Ilup 4 pljkw suhihu wr vhuyh rqo| kljk w|sh frqvxphuv/ wkrxjk/ {lqj d sulfh wkdw doorzv wr h{wudfw doo vxusoxv iurp wkhp dqg ohdyhv frqvxphuv zlwk orzhu lqwhqvlw| ri ghpdqg xqvhuyhg= sQWk D @

yk +4  fE ,5 = 5

+7<,

QWk Lq wklv fdvh/ erwk w|shv zloo vxusoxv=  FVQWk @ 3/ dqg surgxfhu  kdyh }hur QWk 5 @ +4  , yk +4  fE , @5  fD @ Z 1 vxusoxv lv=  Vhoolqj rqo| wr kljk w|shv lv surwdeoh dv orqj dv QWk A QW / zklfk lv wkh pruh olnho| wkh vpdoohu wkh sursruwlrq ri orz0w|sh frqvxphuv dqg wkh kljkhu wkh glhuhqfh lq ghpdqg lqwhqvlwlhv1 Irupdoo|/ wklv vwudwhj| lv surwdeoh li= ?

+yk  yo ,+4  fE ,5 = yk +4  fE ,5  5fD

+83,

W|lqj1 Frqvlghu qrz wkh fdvh zkhuh up 4 uhtxluhv frqvxphuv zkr zdqw wr ex| jrrg D dovr wr ex| jrrg E iurp lw1 Dvvxph dovr/ dqg lw lv d vwurqj dvvxpswlrq/ 89

wkdw lw lv hqgrzhg zlwk vrph whfkqrorj| wkdw doorzv lw wr prqlwru frqvxphuv* sxufkdvhv ri jrrg E/ dqg wr suhyhqw wkhp iurp dgguhvvlqj frpshwlwruv wr ex| jrrg E diwhu kdylqj erxjkw rqh xqlw ri jrrg D zlwk d v|perolf txdqwlw| ri E1 Uhfdoolqj wkdw ghpdqg ri surgxfw E lv jlyhq e| htxdwlrq +79,/ up 4*v surwv duh=  @ +sE  fE , ^yo +4  sE , . +4  , yk +4  sE ,` . sD  fD =

+84,

Iurp g@gsE @ 3 rqh rewdlqv= sWE @

+4  ,+yk  yo , . fE ^yo . +4  , yk ` > 5yk  yo  5+yk  yo ,

+85,

zkhuh lw lv hdv| wr fkhfn wkdw sWE A fE 1 +Wkhuhiruh/ lw lv fuxfldo wkdw up 4 lv deoh wr suhyhqw frqvxphuv iurp ex|lqj jrrg E iurp frpshwlwruv1, Dv iru wkh sulfh ri jrrg D/ qrwh wkdw surwv lqfuhdvh zlwk lw/ exw frqvxphuv zloo ex| dv orqj dv FVl @ yl +4  sWE ,5 @5  sD  31 Diwhu uhsodfhphqw/ rqh rewdlqv wkh rswlpdo sulfh iru jrrg D= sWD @

+4  fE ,5 yo ^yo . +4  , yk `5 5 ^5yk  yo  5+yk  yo ,`5

>

+86,

zklfk fdq eh hlwkhu eljjhu ru vpdoohu wkdq fD 1 Qrwh wkdw khuh w|lqj zrunv lq d yhu| vlplodu zd| dv d wzr0sduw wdul vfkhph W . st wkdw vhjphqwv frqvxphuv dffruglqj wr wkhlu lqwhqvlw| ri ghpdqg1:; Khuh/ wkh {hg ihh sduw ri wkh wdul/ W / fruuhvsrqgv wr sWD / wkdw lv wkh sulfh ri rqh xqlw ri wkh +hvvhqwldo, jrrg D/ zkhuhdv wkh yduldeoh sduw ri wkh wdul fruuhvsrqgv wr wkh sulfh sWE ri rqh xqlw ri jrrg E1 Wkh orzhu wkh lqwhqvlw| ri ghpdqg ri d frqvxphu wkh orzhu wkh qxpehu ri xqlwv ri jrrg E kh zloo ex| iurp wkh prqrsrolvw/ dv tl @ yl +4  sWE ,/ dqg wkhuhiruh wkh orzhu wkh wrwdo dprxqw sdlg wr lw1 Ilup 4*v surwv xqghu w|lqj duh= W @

+4  fE ,5 ^yo . +4  , yk `5  fD = 5 ^5yk  yo  5+yk  yo ,`

+87,

W|sh0o frqvxphuv kdyh }hur vxusoxv/ zkhuhdv w|sh0k frqvxphu vxusoxv lv= FVkW @

+4  fE ,5 +yk  yo , ^yo . +4  , yk `5 5

5 ^5yk  yo  5+yk  yo ,`

=

+88,

Zhoiduh fdq eh frpsxwhg dv= ZW @

+4  fE ,5 ^yo . +4  , yk `5 ^4 . +4  , +yk  yo ,`5 5 ^5yk  yo  5+yk  yo ,`5

Frpsdulvrqv ri htxloleuld1 :; Iru

wzr0sduw wdulv dv d sulfh glvfulplqdwlrq ghylfh/ vhh vhfwlrq 715151

8:

=

+89,

+d, Ohw xv uvw frqvlghu wkh fdvh zkhuh erwk w|shv duh vhuyhg dq|krz1 Rqh fdq fkhfn wkdw w|lqj lv surwdeoh=  W  QW @

+4  fE ,5 +4  ,5 +yk  yo ,5 A 3= 5 ^5yk  yo  5+yk  yo ,`

+8:,

Wklv lv qrw vxusulvlqj/ dv lw doorzv wkh prqrsrolvw wr lpsrvh kljkhu sd|phqwv iurp frqvxphuv zkr kdyh d kljkhu lqwhqvlw| ri ghpdqg/ l1h1/ wr sulfh glvfulplqdwh dprqj wkh wzr w|shv ri frqvxphuv1 Lw lv dovr srvvleoh wr fkhfn wkdw w|lqj lv ghwulphqwdo wr zhoiduh/ dv=    +4  fE ,5 +4  , +yk  yo ,5 4 .   55 yk . 55 yo QW W Z Z @ A 3= 5 ^5yk  yo  5+yk  yo ,`5 +8;, Hyhq zlwkrxw grlqj whglrxv fdofxodwlrqv/ rqh fdq qrwlfh wkdw zhoiduh vkrxog eh orzhu xqghu w|lqj1 Lqghhg/ li wkhuh lv qr w|lqj/ frqvxphuv ex| dw pdujlqdo frvw/ dqg zhoiduh lv wkhuhiruh wkh kljkhvw srvvleoh/ zkhuhdv w|lqj lqwurgxfhv d vrxufh ri doorfdwlyh lqh!flhqf| dv jrrg E lv vrog deryh pdujlqdo frvw1 Zh kdyh wkhuhiruh wkh xvxdo fdvh zkhuh d kljkhu sulfh ghfuhdvhv frqvxphu vxusoxv +ri kljk w|shv/ vlqfh orz w|shv kdyh }hur vxusoxv lq hlwkhu uhjlph, pruh wkdq lw lqfuhdvhv surgxfhu vxusoxv1 +e, Wkh deryh frpsdulvrqv kdyh ehhq pdgh iru wkh fdvh zkhuh erwk w|shv ri frqvxphuv ex| lq wkh uhjlph zkhuh wkhuh lv qr w|lqj1 Li rqo| kljk w|shv ex|/ krzhyhu/ lw wxuqv rxw wkdw w|lqj lqfuhdvhv zhoiduh1 Wr vhh wkdw/ qrwlfh wkdw frqvxphu vxusoxv lv srvlwlyh xqghu w|lqj zkhuhdv lw lv }hur zkhq wkhuh lv qr w|lqj dqg orz w|shv gr qrw ex|1 Wkhuhiruh/ zkhqhyhu w|lqj lv pruh surwdeoh iru up 4 lw zloo dovr eh zhoiduh lpsurylqj1 Wr fkhfn wkdw w|lqj pljkw lqghhg eh surwdeoh/ dqg wr vdyh rq whglrxv fdofx0 odwlrqv/ mxvw frqvlghu d vlpsoh h{dpsoh zkhuh fD @ fE @ 3/ yk @ 5/ dqg yo @ 41 Iru wkhvh ydoxhv/ xqghu qr w|lqj up 4 zrxog vhuyh rqo| kljk w|shv s iru  ? 4@51 Lw lv vwudljkwiruzdug wr fkhfn wkdw  W   QWk A 3 iru  A 4  6@6 @ =75598= deryh wklv wkuhvkrog/ w|lqj lv surwdeoh dqg lw lv zhoiduh lpsurylqj1 61519

Prghov ri w|lqj/ LL= W|lqj/ iruhforvxuh/ dqg h{foxvlrq lq Zklq0 vwrq +4<<3,-

Frqvlghu wzr pdunhwv iru lqghshqghqw surgxfwv +vhh ehorz iru wkh frpsohphq0 wdu| surgxfwv fdvh,/ D dqg E/ dqg wzr upv/ 4 dqg 51 Ilup 4 lv d prqrsrolvw lq pdunhw D/ zkhuh lw lv qrw fkdoohqjhg/ dqg surgxfhv dw frqvwdqw pdujlqdo frvw fD 1 Erwk upv 4 dqg 5 duh srwhqwldo hqwudqwv lq pdunhw E1 Wr hqwhu/ wkh| kdyh wr lqfxu d {hg frvw I4 dqg I5 / dqg wkh| fdq wkhq rshudwh dw frqvwdqw pdujlqdo frvwv uhvshfwlyho| htxdo wr fE4 dqg fE5 1 Wkh jdph lv frpsrvhg ri wkuhh vwdjhv1 Lq wkh uvw vwdjh/ up 4 ghflghv zkhwkhu lw zdqwv wr exqgoh jrrg D dqg E4 wrjhwkhu/ ru qrw1:< Li wkh| duh :< Dvvxph

dovr wkdw up 4 fdqqrw xvh d uhtxluhphqwv frqwudfw / wkdw lv lw fdqqrw vhoo jrrg

D rqo| xqghu wkh frqglwlrq wkdw d frqvxphu zloo qrw ex| xqdeoh wr revhuyh zkdw frqvxphuv ex|,1

8;

2 +iru lqvwdqfh ehfdxvh up 4 lv

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hqwhuhg1

8<

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w5 . sE5  sE4 = w5 . w4

+8<,

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93

{45 +sE4 > sE5 ,/ dqg tE5 @ 4  {45 +sE4 > sE5 ,1 Surw ixqfwlrqv duh=   w5 . sE5  sE4 w4 . sE4  sE5 E4 @ +sE4  fE4 , > E5 @ +sE5  fE5 , = w5 . w4 w5 . w4 +93, Vhw wkh uvw ghulydwlyhv htxdo wr }hur +g El @gsEl @ 3, dqg zulwh wkh uhdfwlrq ixqfwlrqv ri wkh wzr upv lq wkh sodqh +sE5 > sE4 ,= UE4 = sE4 @

w5 . fE4 . sE5 > UE5 = sE4 @ 5sE5  fE5  w4 = 5

+94,

Iurp wkh lqwhuvhfwlrq ri wkh wzr uhdfwlrq ixqfwlrqv zh qg wkh htxloleulxp sulfhv dqg +diwhu vxevwlwxwlrq, surwv lq wkh qr w|lqj sulfh jdph= sEl @

wl . 5wm . fEm . 5fEl +wl . 5wm . fEm . 5fEl ,5 > El @ +iru l> m @ 4> 5/ l 9@ m,= 6 < +wl . wm , +95,

Dv iru pdunhw D/ lw lv fohdu wkdw up 4 zloo vhw lwv sulfh dv wr h{wudfw doo wkh vxusoxv iurp frqvxphuv lq wkdw pdunhw= sD @ y1 Wkh wrwdo surwv xqghu wkh lqghshqghqw sulflqj +qr w|lqj, jdph iru up 4 zloo wkhq eh  4 @ +y  fD ,. El 1 W|lqj1 Zkhq up 4 kdv luuhyhuvleo| exqgohg wkh wzr jrrgv D dqg E4/ pdunhw D lv fohduo| qrw lqghshqghqw iurp E dq| orqjhu/ dv wkh up fkrrvhv d xqltxh sulfh sh iru wkh zkroh exqgoh/ udwkhu wkdq wzr vhsdudwh sulfhv1 D frqvxphu { qrz kdv wkh fkrlfh ehwzhhq ex|lqj hlwkhu wkh exqgoh D2E4/ ru ydulhw| E5 dorqh1 +Qrwh wkdw li d frqvxphu ex|v ydulhw| E5/ wklv lpsolhv rqh orvw vdoh ri jrrg D iru up 41, Vkh h  XE5 / ru= y .z w4 { sh  z w5 +4{,sE5 1 zloo suhihu wkh exqgoh D2E4 li X Lq rughu wr kdyh erwk upv vhoolqj dw d exqgolqj htxloleulxp dvvxph= +D4, 3 ? y  fD ? w5 . 5w4 . fE4 fE5 / dqg +D5, y  fD A 5w5  w4 . fE4 fE5 = Wkh frqvxphu lqglhuhqw ehwzhhq D2E4 dqg E5 lv= s> sE5 ,  { h45 +h

w5 . sE5 . y  sh = w5 . w4

+96,

s> sE5 ,/ dqg tE5 @ 4  { h45 +h s> sE5 ,1 Wkhuhiruh/ ghpdqg ixqfwlrqv zloo eh th @ { h45 +h Surw ixqfwlrqv duh=   y . w5 . sE5  sh y . w4 . sh  sE5  h @ +h s  fD  fE4 , > E5 @ +sE5  fE5 , = w5 . w4 w5 . w4 +97, Iurp gEl @gsEl @ 3/ wkh uhdfwlrq ixqfwlrqv duh= U4 = sh @

y . fD . w5 . fE4 . sE5 > U5 = sh @ 5sE5  fE5 . y  w4 = 5 94

+98,

Vroylqj wkh v|vwhp zh qg wkh htxloleulxp sulfhv xqghu w|lqj= sh @

w4 . 5w5 . fE5 . 5fE4 . y . 5fD  w5 . 5w4 . fE4 . 5fE5  y . fD > shE5 @ > 6 6 +99,

dqg e| uhsodfhphqw wkh htxloleulxp surwv=  h4 @

+w4 . 5w5 . fE5  fE4 . y  fD ,5  +w5 . 5w4 . fE4  fE5  y . fD ,5 > hE 5 @ = < +w4 . w5 , < +w4 . w5 , +9:,

Surw frpsdulvrqv/ dqg lqwhusuhwdwlrq ri wkh uhvxowv1 Iluvw ri doo/ lw lv vwudljkwiruzdug wr vhh wkdw exqgolqj e| up 4 kxuwv ulydo up 5=  E5 A  hE5 / dv y A fD 1 Krzhyhu/ qrwh wkdw exqgolqj dovr kxuwv up 4 li up 5 hqwhuv wkh pdunhw1 Wr vhh wklv/ uvw fkhfn wkdw=  h4 ?  4 l y  fD ? 8w5 . :w4 . 5fE4  5fE5 =

+9;,

Wkhq/ qrwh wkdw iru lqhtxdolw| +9;, wr eh frpsdwleoh zlwk dvvxpswlrq +D5,/ lw pxvw eh :w5 . ;w4 . fE4  fE5 A 3/ zklfk lv dozd|v vdwlvhg e| dvvxpswlrq +D4,1 Wr vxp xs/ frqglwlrqdo rq up 5 ehlqj dfwlyh/ erwk upv zrxog eh zruvh r e| exqgolqj1 Wkhuhiruh/ iru exqgolqj wr eh fkrvhq e| up 4/ lw pxvw eh wkdw lw ghwhuv hqwu| ri up 51 Ehiruh vkrzlqj wkh frqglwlrqv xqghu zklfk wklv rffxuv/ ohw xv lqwhusuhw wkh uhvxowv rewdlqhg= zk| grhv exqgolqj ghsuhvv surwv ri erwk frpshwlwruvB Wr vhh wklv/ lw lv xvhixo wr wklqn ri sh dv frpsrvhg ri wzr fwlrqdo sulfhv= wkh sulfh rq pdunhw D +wkdw fdq rqo| eh htxdo wr y/ vlqfh up 4 lv prqrsrolvw rq wkdw pdunhw, dqg wkh sulfh lq pdunhw E= sh @ y . shE4 =

+9<,

E| uhsodflqj wklv h{suhvvlrq lqwr wkh uhdfwlrq ixqfwlrqv +98,/ zh fdq zulwh wkhp lq wkh sodqh +h sE4 / sE5 ,/ dv= h4 = shE4 @ w5 . fE4 . sE5  +y  fD , > U5 = shE4 @ 5sE5  fE5  w4 = U 5

+:3,

LQVHUW Iljxuh :191 Vwudwhjlf hhfw ri exqgolqj Dv Iljxuh :19 vkrzv/ dqg lv hdvlo| fkhfnhg iurp frpsdulqj h{suhvvlrqv +94, h4 / wkh uhdfwlrq ixqfwlrq ri dqg +:3,/ exqgolqj vkliwv grzqzdugv/ iurp UE4 wr U up 4 zkhq lw frpshwhv lq pdunhw E1 Lq rwkhu zrugv/ exqgolqj uhsuhvhqwv d fuhgleoh frpplwphqw iru up 4 wr eh pruh djjuhvvlyh zkhq lw frpshwhv lq wkdw pdunhw/ ehfdxvh lw nqrzv wkdw dq| orvw vdoh lq pdunhw E lv orvw surw +y  fD , lq pdunhw D1 Dv d uhvxow/ htxloleulxp sulfhv dqg surwv zloo eh orzhu iru erwk upv xqghu w|lqj1 Hqwu| ghflvlrqv1 Ilup 5 zloo hqwhu pdunhw E li dqg rqo| li lwv surwv rvhw lwv {hg frvwv1 Lw lv wkhq vwudljkwiruzdug wr vhh wkdw wkhuh duh sdudphwhu ydoxhv zkhuh lw zrxog 95

hqwhu li up 4 glg qrw xvh w|lqj/ exw lw zrxog qrw hqwhu li lw frpplwwhg wr w|lqj1 hE5 1 Wklv lv dovr wkh fdvh zkhuh up 4 pljkw qg lw Wklv rffxuv li E5  I A  surwdeoh wr exqgoh/ dv zh vkrz qh{w1 Exqgolqj ghflvlrqv1 Dw wkh uvw vwdjh ri wkh jdph/ up 4 ghflghv rq exqgolqj1 Wkh suhylrxv frpsdulvrq ehwzhhq surwv vkrzv wkdw li up 5 lv h{shfwhg wr hqwhu/ exqgolqj lv xqsurwdeoh1 Wkhuhiruh/ zkhq I   hE5 / wkh htxloleulxp lv vxfk wkdw up 4 grhv qrw exqgoh/ dqg up 5 hqwhuv1 Krzhyhu/ lw lv qrw d sulrul vdlg wkdw up 4 qhfhvvdulo| zdqwv wr exqgoh zkhq lw h{shfwv up 5 qrw wr hqwhu1 Exqgolqj lv rswlpdo rqo| li wkh prqrsro| surw xqghu exqgolqj/  p / +uhphpehu/ wkh up frpplwv wr exqgolqj/ vr lw fdqqrw uhyhuvh wkh ghflvlrq zkhq up 5 ghflghv qrw wr vlqn lwv hqwu| frvwv, lv kljkhu wkdq wkh gxrsro| surw zkhq wkhuh lv qr exqgolqj/ 4 1 Wr qg wkh iruphu surw/ qrwh wkdw d frqvxphu orfdwhg lq { zloo ghflgh wr ex| wkh exqgoh udwkhu wkdq qrw ex|lqj dw doo/ li Xp @ y . z  w4 {  shp  31 Wkhuhiruh/ wkh lqglhuhqw frqvxphu zloo eh {p @ +y . z  shp ,@w4 1 Wkhuh duh wzr fdvhv dffruglqj wr zkhwkhu {p ? 4 +wkh pdunhw lv qrw fryhuhg, ru {p  4 +wkh pdunhw lv fryhuhg= doo frqvxphuv ex| wkh exqgoh,=  4/ li shp  y . z  w4 tp @ +:4, y.zshp / li shp A y . z  w4 w4  shp  fD  fE4 / li shp  y . z  w4 = +:5, p @ +h sp  fD  fE4 , y.zw4shp / li shp A y . z  w4 Rqh fdq wkhq qg wkh rswlpdo sulfh dw wkh lqwhuqdo vroxwlrq dv shp @ +y . z . fD . fE4 ,@5/ dqg fkhfn wkdw lw fdq dsso| rqo| li y . z ? fD . fE4 . 5w4 +hovh/ shp  y . z  w4 dqg wkh fruqhu vroxwlrq krogv,1 Wkhuhiruh/ wkh htxloleulxp surwv zloo eh= + y . z  w4  fD  fE4 / li y . z  fD . fE4 . 5w4  = +:6, p @ +y .zfD fE4 ,5 / li y . z ? fD . fE4 . 5w4 7w4 Wkh frqglwlrqv rewdlqhg pdnh vhqvh= wkh kljkhu wkh frqvxphuv* zloolqjqhvv wr sd| iru wkh jrrgv wkh pruh olnho| wkh| zloo doo hqg xs ex|lqj1 Wkh kljkhu wkh glvxwlolw| iurp sxufkdvlqj d ydulhw| E4 glvwdqw iurp wkh prvw suhihuuhg rqh +wkdw lv/ iurp wkh frqvxphu*v orfdwlrq,/ wkh ohvv olnho| wkh| zloo doo ex|1 Zh fdq qrz surfhhg wr frpsduh prqrsro| surwv xqghu wkh exqgolqj vwudw0 hj| zlwk gxrsro| surwv zkhq surgxfwv duh vrog vhsdudwho|1 Udwkhu wkdq fduu|0 lqj rxw whglrxv fdofxodwlrqv/ ohw ph irfxv rq wzr yhu| vshfldo fdvhv/ mxvw wr vkrz wkdw glhuhqw rxwfrphv fdq dulvh1 Vxssrvh uvw wkdw fD @ fE4 @ fE5 @ w5 @ 3/ dqg wkdw y . z ? 5w4 = wkh pdunhw lv qrw fryhuhg xqghu prqrsro|1 Wkhq= p   4 @

+y . z,5 w4   y= 7w4 < 96

+:7,

Fohduo|/ wklv h{suhvvlrq ghfuhdvhv zlwk w4 / zklfk phdqv wkdw wkhuh h{lvw sdudp0 hwhu ydoxhv vxfk wkdw wkh deryh h{suhvvlrq lv qhjdwlyh/ wkdw lv/ exqgolqj zrxog qrw eh fkrvhq hyhq li lw ohdgv wr wkh h{lw ri up 51;8 Vxssrvh qrz wkdw fD @ fE4 @ fE5 @ w5 @ 3/ exw wkdw y . z  5w4 = wkh pdunhw lv fryhuhg xqghu prqrsro|1 Wkhq= p   4 @ w4 

w4 . z= <

+:8,

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+;7,

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+;8,

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616

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+433,

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4  f . y 4 = 5+4  y,

+434,

Iru +3> t5p , wr eh dq htxloleulxp/ up 5 pxvw kdyh qr lqfhqwlyh wr ghyldwh1 Wklv zloo eh wkh fdvh li s4 +3> t5p ,  f1 E| uhfdoolqj wkh lqyhuvh ghpdqg ixqfwlrq +<7,/ wklv frqglwlrq ehfrphv= 4 . y 4  +4  y,

4  f . y 4  f  3= 5+4  y,

:6

+435,

Wkh orzhu  wkh pruh olnho| wklv frqglwlrq zloo eh vdwlvhg1 Iru  @ 3/ wlsslqj wr up 5 zrxog rffxu li= f4.

5 4 y +4  y, 4  5y

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:7

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+437,

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617

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:8

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remhfwlyh ri surw pd{lplvdwlrq dqg qhlwkhu qhfhvvdulo| hqwdlov wkh lqwhqwlrq ri iruflqj ulydov rxw ri wkh pdunhw/ qru lv lw qhfhvvdulo| zhoiduh ghwulphqwdo1 Lq lqwhuqdwlrqdo wudgh/ sulfh glvfulplqdwlrq wdnhv qdwxudoo| wkh irup ri gli0 ihuhqw sulfhv lq glhuhqw pdunhwv1 Qrupdoo|/ vlqfh upv whqg wr kdyh d odujhu pdunhw vkduh dqg d pruh idlwkixo ghpdqg lq wkhlu grphvwlf pdunhw/ wklv lpsolhv wkdw grphvwlf sulfh lv kljkhu dw krph wkdq deurdg1459 Wkh zhoiduh hhfw ri dqwl0gxpslqj phdvxuhv wkdw lv vr orrvho| ghqhg dv wr frpsuhkhqg prvw vlwxdwlrqv zkhuh iruhljq surgxfhuv duh pruh frpshwlwlyh wkdq grphvwlf rqhv/ lv fohduo| qhjdwlyh/ dv qrw rqo| frqvxphuv exw dovr upv zklfk xvh wkh jrrg dv dq lqsxw pljkw eh shqdolvhg e| wkh dqwl0gxpslqj dfwlrqv1 Wklv lpsolhv wkdw d phdvxuh zklfk lv vxssrvhg wr khos wkh qdwlrqdo lqgxvwu| pljkw dfwxdoo| eh ghwulphqwdo wr lw1 Ixuwkhupruh/ dqwl0gxpslqj odzv pljkw uhpryh wkuhdwv wr grphvwlf upv zklfk duh qrw frpshwlwlyh hqrxjk1 Wklv kdv surgxfwlyh h!flhqf| lpsolfdwlrqv +dv grphvwlf upv ghyrwh wkhlu hqhuj| pruh wr oree|lqj iru vwdwh lqwhuyhqwlrq wkdq lq lpsurylqj wkhlu surfhvvhv dqg surgxfwv, dqg pljkw doorz wkhp wr suhvhuyh pdunhw srzhu srvlwlrqv145: Rqh fdvh zklfk looxvwudwhv wkh frq lfw wkdw dqwl0gxpslqj odzv kdyh zlwk hfrqrplf h!flhqf|/ lv wkdw ri wkh vrgd0dvk lqgxvwu|/ zkrvh wzr pdlq Hxurshdq surgxfhuv +Vroyd| kdg :3( ri wkh pdunhw vkduh lq Frqwlqhqwdo Hxursh lq 4<<3/ zkloh LFL kdg d txdvl0prqrsro| lq wkh XN pdunhw, uhfhlyhg khdy| qhv iru frooxvlyh djuhhphqwv dqg iru dexvh ri grplqdqw srvlwlrq1 Lw lv wkhq vxusulv0 lqj wr glvfryhu wkdw wkh lqgxvwu| kdv ehhq uhshdwhgo| surwhfwhg iurp iruhljq frpshwlwlrq e| dqwl0gxpslqj dfwlrqv145; Ri frxuvh/ wkh Hxurshdq dqwl0gxpslqj odzv duh qrw wkh rqo| rqhv zklfk fdq eh fulwlflvhg1 Wkh XV dqwl0gxpslqj phdvxuhv/ iru lqvwdqfh/ kdyh dovr dwwudfwhg d qxpehu ri fulwlflvpv1 Hpslulfdo dqdo|vhv kdyh vkrzhg wkh dqwl0frpshwlwlyh hhfw ri dqwl0gxpslqj1 Ixuwkhu/ wkh| kdyh frquphg wkh wkhruhwlfdo vxvslflrq wkdw dqwl0gxpslqj odzv kdyh d qhjdwlyh hhfw hyhq zkhq wkh| gr qrw uhvxow lq qdo gxwlhv1 Wkh phuh wkuhdw ri dqwl0gxpslqj vdqfwlrqv fdq eh hqrxjk wr lqgxfh iruhljq frpshwlwruv wr eh ohvv djjuhvvlyh/ dqg yhu| riwhq lqyhvwljdwlrqv hqg xs zlwk d vxvshqvlrq ghflvlrq/ lq h{fkdqjh iru wkh surplvh ri wkh iruhljq upv wr vwrs gxpslqj wkhlu jrrgv145< Dqwl0gxpslqj gxwlhv vkrxog eh mxvwlhg rqo| zkhq suhgdwru| sulflqj lv lq0 yroyhg1 Dffruglqjo|/ gxpslqj vkrxog qrw eh frqvlghuhg dv ehorqjlqj wr wkh grpdlq ri wudgh srolf|/ exw udwkhu wr wkdw ri frpshwlwlrq srolf|1 Dqg wr mxvwli| dqwl0gxpslqj dfwlrqv/ wkh vwdqgdugv vhw xs lq vhfwlrq 5 vkrxog eh xvhg1 4 59 D

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;9

715

Sulfh glvfulplqdwlrq-

L rhu khuh d whfkqlfdo suhvhqwdwlrq ri wkh dujxphqwv pdgh lq wkh suhylrxv vhf0 wlrq1 Vhfwlrq 71514 ghdov zlwk wkh fdvh ri sulfh glvfulplqdwlrq dfurvv qdwlrqdo pdunhwv/ vhfwlrq 71515 zlwk txdqwlw| glvfrxqwv/463 vhfwlrq 71516 zlwk sulfh glv0 fulplqdwlrq dqg lqfhqwlyhv wr lqyhvw/ vhfwlrq 71517 zlwk wkh uroh ri pdunhw srzhu/ dqg vhfwlrq 71518 zlwk wkh hhfwv ri glvfulplqdwlrq zkhq d prqrsrolvw idfhv dq hqwudqw1 71514

Wklug0ghjuhh sulfh glvfulplqdwlrq,-

D prqrsrolvw vhuyhv wzr pdunhwv/ o dqg k/ wkdw rqh fdq wklqn ri dv wzr uhjlrqv wkdw duh sduw ri wkh vdph frxqwu| +ru wzr frxqwulhv wkdw duh sduw ri d xqlrq,1 Wkh zhljkw ri wkh wzr uhjlrqv lq wkh frxqwu| lv uhvshfwlyho|  dqg 4  / zlwk 3 ?  ? 41 Pdunhw l @ o> k ghpdqg lv jlyhq e| t @ yl  s/ zlwk yk A yo 1 Wklv fdq eh udwlrqdolvhg e| dvvxplqj wkdw d w|sh0l frqvxphu kdv d xwlolw| ixqfwlrq Xl @ yl t  t 5 @51464 Wkh prqrsrolvw vhuyhv erwk frxqwulhv iurp wkh vdph sodqw/ dqg kdv d xqlw frvw f ? yo +zh glvuhjdug wudqvsruw frvwv iru vlpsolflw|,1 Zh zdqw wr frpsduh wkh fdvh zkhuh wkh prqrsrolvw fdq glvfulplqdwh wr wkh rqh zkhuh lw fdqqrw1 Sulfh glvfulplqdwlrq Wkh prqrsrolvw fdq fkrrvh wzr glhuhqw sulfhv dqg lwv surwv lq hdfk pdunhw duh jlyhq e| l @ +sl  f,+yl  sl ,1 Iurp gl @gsl @ 3 wkh htxloleulxp vroxwlrqv duh hdvlo| irxqg +wrwdo surwv vkrxog eh zhljkwhg e| srsxodwlrq vkduhv,= +yk  f,5 yl . f g +yo  f,5 >  @ . +4  , = +438, 5 7 7 Wkh frqvxphu vxusoxv lq hdfk pdunhw fdq hdvlo| eh irxqg dv wkh duhd ri wuldqjoh ehwzhhq wkh ghpdqg ixqfwlrq dqg wkh pdunhw sulfh> djjuhjdwh frqvxphu vxusoxv dqg zhoiduh duh rewdlqhg e| zhljkwlqj hdfk pdunhw=   +yo  f,5 +4  ,+yk  f,5 6 +yo  f,5 +4  ,+yk  f,5 . > Zg @ . FV g @ = ; ; ; 7 7 +439, sgl @

Xqlirup sulflqj +erwk pdunhwv vhuyhg, Vxssrvh qrz wkdw wkhuh lv d edq rq sulfh glvfulplqdwlrq1 Wkh prqrsrolvw lv wkhq reoljhg wr vhw wkh vdph xqlirup sulfh s iru erwk pdunhwv dqg lwv surjudpph lv pd{s +s  f, ^+yo  s, . +4  ,+yk  s,`1 Vxssrvh iru wkh prphqw wkdw erwk pdunhwv duh vhuyhg1 Iurp g@gs @ 3 wkh htxloleulxp vroxwlrqv duh= sx @ 4 63 Iru

yo . +4  ,yk . f x +yo . +4  ,yk  f,5 >  @ = 5 7

+43:,

suhvhqwdwlrq uhdvrqv/ lw lv pruh frqyhqlhqw wr lqyhuw wkh rughu iroorzhg lq wkh suhylrxv

vhfwlrq/ dqg ghdo zlwk vhfrqg0ghjuhh sulfh glvfulplqdwlrq diwhu wklug0ghjuhh glvfulplq dwlrq1

4 64 Iurp

 3 ^

'R

wkh pd{lplvdwlrq ri lwv xwlolw| vxe mhfw wr wkh exgjhw frqvwudlqw/ lw iroorzv wkdw

1

;:

Rqh fdq hdvlo| fkhfn wkdw wkh xqlirup sulfh lv lq ehwzhhq wkh qrsrolvw zrxog fkdujh li lw zhuh doorzhg wr sulfh glvfulplqdwh= Frqvxphu vxusoxv dqg zhoiduh xqghu xqlirup sulflqj duh= 5 5 FV x @ +yo .+4;,yk f, . +4,+5yk yo , > Z x @

sulfhv wkh pr0 sgk A sx A sgo 1

,yk f,5

6+yo .+4

;

5 . +4,+5yk yo , = +43;,

Zh fdq qrz frpsduh wkh wzr htxloleuld1 Lw lv hdvlo| fkhfnhg wkdw wkh up suhihuv wr eh iuhh wr glvfulplqdwh +x ? g ,/ zklfk vkrxog qrw frph dv d vxusulvh vlqfh xqghu glvfulplqdwlrq lw kdv wzr lqvwuxphqwv wr pd{lplvh lwv remhfwlyh ixqfwlrq/ zkhuhdv xqghu xqlirup sulflqj lw kdv rqo| rqh1 Ihz vwhsv ri dojheud dovr vkrz wkdw= +4  ,+yk  yo ,5 A 3> +43<, ; zklfk lpsolhv wkdw zhoiduh lv kljkhu xqghu xqlirup sulflqj1 Lqghhg/ lw lv d jhqhudo uhvxow wkdw sulfh glvfulplqdwlrq ghfuhdvhv zhoiduh li lw grhv qrw lqfuhdvh wrwdo rxwsxw/465 dqg rqh fdq fkhfn wkdw khuh wrwdo rxwsxw vrog lv wkh vdph xqghu wkh wzr uhjlphv/ pdnlqj lw dq dssolfdwlrq ri wkh jhqhudo uhvxow= Zx  Zg @

Tg @ tog . +4  ,tog @

yo . +4  ,yk  f @ Tx @ tox . +4  ,tox = +443, 5

Xqlirup sulfhv +rqh pdunhw qrw vhuyhg, Wkhuh lv d vwurqj dvvxpswlrq ehklqg wkh dqdo|vlv mxvw fduulhg rxw= wkdw wkh prqrsrolvw vhuyhv erwk pdunhwv1 Krzhyhu/ xqlirup sulflqj wr vhuyh erwk pdunhwv hqwdlov uhgxflqj sulfhv +dqg surwv, lq wkh kljk ghpdqg pdunhw1 Wkh prqrsrolvw pljkw suhihu lqvwhdg mxvw wr vhw wkh sulfh sk @ +yk . f,@5 wkdw pd{lplvhv lwv surwv lq wkh kljk ghpdqg pdunhw/ hyhq li wklv lpsolhv orvlqj doo vdohv lq wkh orz0ghpdqg pdunhw1 Iru lqvwdqfh/ dvvxph wkdw yk . f A 5yo 1 Lq wklv fdvh/ ghpdqg lq wkh orz ghpdqg pdunhw lv }hur/ dqg wkh prqrsrolvw zloo hduq= xk @ +4  ,

+yk  f,5 = 7

+444,

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+yk  f,+yk  5yo . f, > +yk  yo ,5

+445,

wkdw lv/ wkh orzhu wkh vkduh ri wkh orz0ghpdqg pdunhw lq wrwdo ghpdqg wkh pruh olnho| lw zloo qrw eh vhuyhg li sulfh glvfulplqdwlrq zdv edqqhg1 Lw fdq dovr eh fkhfnhg wkdw wkh UKV ri h{suhvvlrq +445, lqfuhdvhv zlwk yk dqg ghfuhdvhv zlwk yo = wkh kljkhu wkh jds ehwzhhq wkh ghpdqgv wkh pruh olnho| rqh pdunhw rqo| zloo eh vhuyhg1 Ilqdoo|/ lw lv vwudljkwiruzdug wr fkhfn wkdw li rqh pdunhw rqo| lv vhuyhg +wkdw lv/ li +445, krogv, zhoiduh lv kljkhu xqghu sulfh glvfulplqdwlrq/ vlqfh wkh kljk0 ghpdqg pdunhw kdv wkh vdph frqvxpswlrq dqg surwv xqghu erwk uhjlphv/ exw orz0ghpdqg pdunhw frqwulexwhv }hur wr erwk xqghu xqlirup sulflqj1 4 65 Iru d surri/ vhh Yduldq +4<;<,/ ru Wluroh +4<;;= 46:046;,1

;;

71515

Txdqwlw| glvfrxqwv= Wzr0sduw wdulv dv sulfh glvfulplqdwlrq-

L frqvlghu khuh wkh fdvh zkhuh wkh prqrsrolvw xvhv d wzr0sduw wdul W . st= d frqvxphu ex|lqj t xqlwv ri wkh jrrg zloo kdyh wr sd| d {hg ihh W soxv d yduldeoh frpsrqhqw s1 Qrwh wkdw wklv dprxqwv wr d txdqwlw| glvfrxqw/ dv wkh dyhudjh frvw ri wkh sxufkdvh/ s . W @t/ ghfuhdvhv zlwk wkh qxpehu ri xqlwv erxjkw1 Ohw ph frqwlqxh zlwk wkh h{dpsoh ri wkh suhylrxv vhfwlrq 71514= wkhuh duh wzr glhuhqw w|shv ri frqvxphuv/ orz0w|shv kdylqj ghpdqg t @ yo  s +wkh| duh d vkduh  ri wkh srsxodwlrq, dqg kljk0w|shv kdylqj ghpdqg t @ yk  s +d vkduh 4   ri wkh srsxodwlrq,1 Dvvxph wkdw yo A +f . yk , @5/ wkdw hqvxuhv wkdw hyhu|erg| ex|v xqghu erwk xqlirup sulflqj dqg wzr0sduw wdulv1466 Qrq0glvfulplqdwlrq Wklv fdvh kdv ehhq douhdg| dqdo|vhg lq wkh suhylrxv vhfwlrq1 Txdqwlw| glvfrxqwv +wzr0sduw wdul , Vlqfh orz0w|shv kdyh orzhu lqwhq0 vlw| ri ghpdqg/ wkh {hg ihh zloo eh fkrvhq vr dv wr pdnh wkhp zloolqj wr sd|1 Jlyhq wkdw wkhlu frqvxphu vxusoxv lv jlyhq e| FVo @ +yo  s,5 @5  W / lw iroorzv wkdw wkh prqrsrolvw zloo vhw W @ +yo  s,5 @51 Dv iru wkh yduldeoh frpsrqhqw ri wkh wdul/ lw lv irxqg dv wkh rqh zklfk vroyhv wkh sureohp= pd{  @ +s  f, ^ +yo  s, . +4  , +yk  s,` . +yo  s,5 @5= s

+446,

Iurp g@gs @ 3 lw iroorzv wkdw= stg @ f . +4  , +yk  yo , =

+447,

Qrwh wkdw wkh txdqwlw| erxjkw e| wkh orz0w|shv lv t @ yo  stg / vr wkdw t A 3 dprxqwv wr yo A ^f . +4  , yk ` @ +5  ,/ zklfk lv vdwlvhg e| wkh dvvxpswlrq yo A +f . yk , @51 Dw wklv sulfh/ wkh vxusoxv ri kljk w|shv lv= FVktg @ +yk  yo , ^yo +5  ,  f  +4  , yk ` A 3 @ FVotg > dqg wkh prqrsrolvw*v surwv duh= l 4k tg @ +yo  f,5 . +4  ,5 +yk  yo ,5 > 5

+448,

+449,

dqg zhoiduh fdq eh irxqg e| vxevwlwxwlrq dv Z tg @ tg . +4  , FVktg 1 Txdqwlw| glvfrxqwv dqg xqlirup sulflqj= d frpsdulvrq Wkh uhdghu fdq fkhfn wkdw wkh prqrsrolvw zloo eh ehwwhu r e| xvlqj wklv irup ri sulfh glvfulplqdwlrq= tg A x 1 Exw hyhq zlwkrxw grlqj dq| fdofxodwlrqv/ wklv lv vwudljkwiruzdug/ vlqfh wzr0sduw wdulv surylgh wkh prqrsrolvw zlwk wzr lqvwux0 phqwv/ W dqg s/ udwkhu wkdq rqo| rqh dv xqghu xqlirup sulflqj +wkh prqrsrolvw 4 66 Iru

ixwxuh uhihuhqfh/ qrwh wkdw

ES

n   *2 : ES n E 3 b   * E2 3 b

1

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Sulfh glvfulplqdwlrq dqg lqyhvwphqwv-

Lq wklv vhfwlrq/ L dqdo|vh d vlpsoh prgho ri d prqrsrolvw wkdw lqyhvwv lq wkh txdolw| ri lwv jrrg1 L vkrz wkdw xqghu sulfh glvfulplqdwlrq wkh htxloleulxp txdolw| rhuhg zloo eh kljkhu1 Frqvlghu d prqrsrolvw wkdw vhoov d jrrg ri txdolw| x wr wzr glhuhqw pdunhwv/ hdfk ri vl}h rqh1 Frqvxphuv lq erwk pdunhwv kdyh suhihuhqfhv FV @ x  s li wkh| ex| rqh xqlw ri wkh jrrg/ dqg 3 rwkhuzlvh1 Lq pdunhw k/ frqvxphuv* wdvwh iru txdolw|/ / lv xqlirupo| glvwulexwhg rq 3    k 1 Lq pdunhw o/ wkh wdvwh sdudphwhu lv xqlirupo| glvwulexwhg rq 3    o 1 Wkh prqrsrolvw kdv wr ghflgh uvw rq wkh txdolw| x lw zdqwv wr vxsso|/ dqg lw kdv d {hg frvw ri txdolw| lpsuryhphqw F+x, @ nx5 @5> +dvvxph iru vlpsolflw| wkdw wkhuh lv }hur yduldeoh frvw ri surgxfwlrq,> wkhq/ lw kdv wr ghflgh wkh sulfh dw zklfk lw zdqwv wr vhoo wkh jrrg1 Frqvlghu wzr yduldqwv ri wkh jdph= lq wkh uvw/ lw kdv wr fkrrvh wkh vdph sulfh iru erwk pdunhwv> lq wkh vhfrqg/ lw fdq sulfh glvfulplqdwh1468 Xqlirup sulflqj Iluvw ri doo/ qrwh wkdw wkh ghpdqg idfhg e| wkh prqrs0 rolvw lv ghwhuplqhg e| qglqj wkh frqvxphu zkr lv lqglhuhqw ehwzhhq ex|lqj ru qrw1 Jlyhq d sulfh s dqg d txdolw| x/ wkh lqglhuhqw frqvxphu 3 lv jlyhq e| vroxwlrq ri FV @ x  s @ 31 Wkhuhiruh/ 3 @ s@x1 Wkhuhiruh/ ghpdqg lv jlyhq e| doo frqvxphuv lq erwk pdunhwv vxfk wkdw   3 1 Surwv lq wkh vhfrqg vwdjh duh= k s  s l . o  = +44:,  @ s k  x x Iurp wkh IRF g@gs @ 3/ lw iroorzv wkdw htxloleulxp sulfhv dqg surwv duh= sx @

x +k .  o , x +k .  o ,5 > x @ = 7 ;

+44;,

Wkh wrwdo txdqwlw| vrog e| wkh prqrsrolvw lv t x @ k  +k . o ,@7 . o  +k . o ,@7 @ +k . o ,@51 Wkh rswlpdo txdolw| fkrlfh fdq eh irxqg dv wkh vroxwlrq ri 4 67

Wkh uhdghu fdq fkhfn wkdw ` ^_ : `  iru , : dS n E 3 b  o * E2 3 b1

4 68 Lq

lqwhuqdwlrqdo wudgh wkhru|/ wkh fdvh ri sulfh glvfulplqdwlrq dfurvv pdunhwv lv fdoohg vhjphqwhg pdunhwv/ zkhuhdv fkrrvlqj rqh sulfh iru erwk pdunhwv lv fdoohg lqwhjudwhg pdu0 nhwv1

<3

wkh surjudpph pd{x  @ x  nx5 @5> iurp g@gx @ 3/ lw iroorzv wkdw= xx @

x +k . o ,5 = ;n

+44<,

Sulfh glvfulplqdwlrq Xqghu sulfh glvfulplqdwlrq/ wkh prqrsrolvw fdq fkdujh d glhuhqw sulfh lq wkh wzr pdunhwv1 Wkh lqglhuhqw frqvxphu l3 lq pdunhw l lv jlyhq e| vroxwlrq ri FV @ x  sl @ 31 Zkhqfh/ l3 @ sl @x1 Surwv lq wkh vhfrqg vwdjh duh=   sk  so  . so o  = +453,  @ sk  k  x x Iurp wkh IRF g@gsl @ 3/ lw iroorzv wkdw htxloleulxp sulfhv dqg surwv duh= sgl @

xl x+5k .  5o , > g @ = 5 7

+454,

Qrwh dovr wkdw wkh wrwdo txdqwlw| vrog e| wkh prqrsrolvw lv t g @  k   k @5 . o  o @5 @ + k . k ,@5/ zklfk lv wkh vdph dv xqghu xqlirup sulflqj1 Wkh rswlpdo txdolw| fkrlfh fdq eh irxqg dv wkh vroxwlrq ri wkh surjudpph pd{x  @ g  nx5 @5> iurp g@gx @ 3/ lw iroorzv wkdw= xg @

x+ 5k . 5o , = 7n

+455,

Lw lv wkhq vwudljkwiruzdug wr fkhfn wkdw xg A xx 1 71517

Sulfh glvfulplqdwlrq dqg pdunhw srzhu-

Frqvlghu d yhuwlfdo surgxfw glhuhqwldwlrq prgho zkhuh up 4 vhoov d surgxfw ri txdolw| x4 dqg up 5 d surgxfw ri txdolw| x5 / zlwk x4 A x5 1 Erwk upv vhoo lq wzr glhuhqw pdunhwv/ hdfk ri vl}h rqh1 +D up grhv qrw glhuhqwldwh txdolw| dfurvv pdunhwv1, Frqvxphuv lq erwk pdunhwv kdyh suhihuhqfhv FVm @ xm  sm li wkh| ex| rqh xqlw ri wkh jrrg ri txdolw| m @ 4> 5/ dqg 3 rwkhuzlvh1 Lq pdunhw k/ frqvxphuv* wdvwh iru txdolw|/ / lv xqlirupo| glvwulexwhg rq 3    k 1 Lq pdunhw o/ lw lv xqlirupo| glvwulexwhg rq 3    o 1 L vkrz khuh wkdw erwk upv kdyh dq lqfhqwlyh wr sulfh glvfulplqdwh/ exw wkdw wkhuh lv olwwoh uhdvrq wr fduh derxw sulfh glvfulplqdwlrq sudfwlfhv ri d up zlwk olwwoh pdunhw srzhu1 Xqlirup sulflqj Wr qg ghpdqg ixqfwlrqv ri wkh upv/ zh uvw qhhg wr lghqwli| wzr lqglhuhqw frqvxphuv= wkh frqvxphu lqglhuhqw ehwzhhq ex|lqj iurp wkh orz txdolw| jrrg dqg qrw ex|lqj/ 3 / dqg wkdw lqglhuhqw ehwzhhq ex|lqj wkh orz ru wkh wrs txdolw| jrrg/ 45 1 Iurp FV5 @ 3 lw iroorzv wkdw  3 @ s5 @x5 1 Iurp FV5 @ FV4 lw iroorzv wkdw 45 @ +s4  s5 ,@+x4  x5 ,1 Wkhuhiruh/ ghpdqg ixqfwlrqv duh= t4l @ l  45 / dqg <4

t5 @ 45  3 / zlwk l @ o> k1 Ilupv* surwv duh=     s4  s5 s5 s4  s5  4 @ s4 k . o  5 > 5 @ 5s5 = x4  x5 x4  x5 x5

+456,

Htxloleulxp sulfhv lq wkh sulfh vxe0jdph duh irxqg iurp gm @gsm @ 3 dv= sx4 @

x4 +x4  x5 ,+ k . o , x5 +x4  x5 ,+k . o , > sx5 @ = +7x4  x5 , 5+7x4  x5 ,

+457,

Jlyhq x4 / d orz x5 lpsolhv wkdw up 5 fdq frppdqg d orz sulfh1 Iru x5 zklfk whqgv wr }hur/ iru lqvwdqfh/ up 5*v sulfh zrxog dovr whqg wr }hur/ wkdw lv/ wkh orz txdolw| up zrxog kdyh yhu| olwwoh pdunhw srzhu +zklfk lv wkh delolw| ri udlvlqj sulfhv deryh pdujlqdo frvwv/ khuh }hur,1469 E| vxevwlwxwlrq/ rqh fdq qg htxloleulxp surwv dv= x4 @

5x54 +x4  x5 ,+k . o ,5 x4 x5 +x4  x5 ,+k . o ,5 x >  @ = 5 +7x4  x5 ,5 5+7x4  x5 ,5

+458,

Sulfh glvfulplqdwlrq Xqghu sulfh glvfulplqdwlrq/ erwk upv fdq fkdujh d glhuhqw sulfh lq wkh wzr pdunhwv1 Ilupv* surwv duh=     [ [ s4l  s5l s4l  s5l s5l s4l l  s5l  4 @ > 5 @ = +459, x4  x5 x4  x5 x5 @k>o

l

l

@k>o

Htxloleulxp sulfhv lq wkh sulfh vxe0jdph duh irxqg iurp gml @gsml @ 3 dv= sg4l @

5x4 +x4  x5 ,l x5 +x4  x5 ,l > sg5l @ = +7x4  x5 , +7x4  x5 ,

+45:,

Wkhuhiruh/ lw lv fohdu wkdw erwk upv glvfulplqdwh dfurvv pdunhwv dw htxlole0 ulxp +lq rwkhu zrugv/ qrw rqo| prqrsrolvwv zdqw wr vhjphqw pdunhwv$,1 E| vxevwlwxwlrq/ htxloleulxp surwv duh= g4 @

7x54 +x4  x5 ,+5k .  5o , x4 x5 +x4  x5 ,+ 5k . 5o , g >  @ = 5 +7x4  x5 ,5 +7x4  x5 ,5

+45;,

Pdunhw srzhu dqg sulfh glvfulplqdwlrq1 Lq wklv vlpsoh prgho/ wkh glvwruwlrq fuhdwhg e| sulfh glvfulplqdwlrq lq hdfk pdunhw glhuv dfurvv upv=46:      x4 +x4  x5 , +k  o ,     > sg5l  sx5 @  x5 +x4  x5 , + k  o ,  = sg4l  sx4 @     +7x4  x5 , 5+7x4  x5 ,

+45<,

4 69 Qrwh wkdw wkhuh lv qr prqrwrqlf uhodwlrqvkls khuh ehwzhhq  dqg lwv htxloleulxp sulfh 2 +pdunhw srzhu,1 Wklv lv ehfdxvh zkhq 2 ehfrphv forvh hqrxjk wr  / wkh surgxfwv ehfrph forvhu vxevwlwxwhv dqg htxloleulxp sulfhv ghfuhdvh1 4 6: Reylrxvo|/ iru erwk upv wkh sulfh lv kljkhu +uhvshfwlyho| orzhu, xqghu sulfh glvfulplqdwlrq lq wkh pdunhw zlwk kljkhu +uhvs1 orzhu, lqwhqvlw| ri ghpdqg/  +uhvs1 , ,1 Dowkrxjk wkh vljq ri wkh sulfh glhuhqfh lv rssrvhg/ wkh devroxwh pdjqlwxgh ri wkh glhuhqfh lv wkh vdph iru hdfk up1

<5

Fohduo|/ wkh glvwruwlrq ehfrphv luuhohydqw zkhq d up grhv qrw kdyh pxfk pdunhw srzhu1 Iru vpdoo ydoxhv ri x5 +wkdw lv/ zkhq up 5 kdv olwwoh pdu0 nhw srzhu,/ wkh sulfh glhuhqfh ehfrphv yhu| vpdoo1 Pruh irupdoo|/ zulwh x5 @ x4  n/ vr wkdw n lv wkh txdolw| jds ehwzhhq wkh upv1 Rqh fdq zulwh sg5l  sx5 @ m+x4  n,nx4 +k  o , @+n . 6x4 ,m @5/ dqg= +6x54  9nx4  n5 , + k  o , g+sg5l  sx5 , @ = gn 5+n . 6x4 ,5

+463,

Iurp vwxg|lqj wkh vhfrqg0rughu lqhtxdolw| x54  9nx4  n5 A 3 rqh rewdlqv wkdw wkh htxloleulxp sulfh glhuhqfh s ehwzhhq wkh wzr sulflqj uhjlphv ghfuhdvhv zlwk wkh txdolw| jds zkhq n ? +5 6  6,x4 +urxjko|/ iru n ? =79x4 , dqg lqfuhdvhv iru kljkhu ydoxhv1 Lq rwkhu zrugv/ zkhq wkh txdolw| jds lv wkh kljkhvw +pdunhw srzhu ri up 5 lv wkh orzhvw,/ dq lqfuhdvh lq x5 +dq lqfuhdvh lq up 5*v pdunhw srzhu, udlvhv wkh sulfh glhuhqwldo xqwlo d srlqw zkhuh wkh txdolwlhv duh vlplodu hqrxjk1 Wklv vwurqjo| vxjjhvwv wkdw zkhq d up kdv olwwoh pdunhw srzhu +dv lv wkh fdvh zlwk up 5 khuh iru orz hqrxjk ydoxhv ri x5 ,/ wkh lpsdfw wkdw lwv sulflqj srolf| pljkw kdyh lv xqolnho| wr eh vljqlfdqw1 Wkh hhfw rq lqyhvwphqwv Vxssrvh qrz wkdw/ iru vrph uhdvrq/ up 4*v txdolw| zdv {hg exw up 5 +wkh up lq d glvdgydqwdjhg srvlwlrq, frxog lqyhvw wr lpsuryh lwv txdolw| ohyho dw d frqyh{ frvw F+x5 ,1 Lw lv hdv| wr vhh wkdw sulfh glvfulplqdwlrq zrxog doorz wkh up wr kdyh d kljkhu lqfhqwlyh wr lqyhvw lq txdolw|/ wkhuhe| khoslqj lw wr oo wkh jds pruh wkdq lq wkh fdvh ri xqlirup sulflqj1 Ilup 5*v sureohp lv wkhq wr fkrrvh lwv txdolw| hqgrjhqrxvo| vr dv wr pd{lplvh lwv qhw surwv= pd{x5 5 +x5 ,  F+x5 ,1 Wkh jurvv surwv/ dqg wkh uvw0rughu frqglwlrq ri wklv sureohp duh glhuhqw dffruglqj wr wkh sulfh uhjlph1 Xqghu xqlirup sulflqj/ lw lv jlyhq e|= x5 +7x4  :x5 ,+ k . o ,5 gF+x5 , gx5 +x5 , @ 4 @ > gx5 5+7x4  x5 ,5 gx5

+464,

zkhuhdv xqghu sulfh glvfulplqdwlrq lw lv jlyhq e|= x5 +7x4  :x5 ,+5k .  5o , gF+x5 , gg5 +x5 , @ 4 @ = gx5 +7x4  x5 ,5 gx5

+465,

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<6

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<7

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449

Table 7.1. Consumer valuations of goods 1's willingness to pay

2's willingness to pay

Good A

7

4

Good B

5

8

Goods A and B

12

12

Entrant t stay out

(0, π M )

enter Incumbent fight

accommodate

(π EP , π IP )

(π EA , π IA )

Figure 7.1. State game at time t, chain-store paradox game

E enter

out

I  π M + δπ M    0  

prey E

E

out

stay

 π P + δπ M   πP < 0 

   

out

I prey

 π A + δπ P     π A + δπ P   

 π A + δπ M    A   π  

accommodate  π A + δπ A     π A + δπ A   

Figure 7.2. Deep pocket predation, with T = K = 1

1 •

•

prey or 1st period accommodate payoff

2 •

pay F or out

3 •

•

effort 2nd period decisions payoff

Figure 7.3. Time line: a (financial) long purse model of predation

time

i

ii

contract

effort decisions

•

•

1 •

•

prey or 1st period accommodate payoff

2 •

pay F or out

Figure 7.4. Time line: Bolton-Scharfstein’s model

3 •

•

effort 2nd period decisions payoff

time

q2

R1 ( x1P )

R1 ( x1 = 0)

q2N ( x1 = 0)

.

π 2 ( x1 = 0)

.

π 2 ( x1P ) = F

q1N ( x1 = 0)

Figure 7.5. Entry-deterring R&D investment

q1

p B1

RB 2 RB1

.

E

t2 + cB1 2 t2 + cB1 − (v − c A ) 2

~ R1

.

~ E

pB2

Figure 7.6. Strategic effect of bundling

p O

R

pm

pc

T

S

c

O′ qm

qc

Figure 7.7. Welfare under perfect price discrimination

q

Fkdswhu ;= D Wrronlw= Jdph Wkhru| dqg Lpshuihfw Frpshwlwlrq Prghov Iheuxdu| 47/ 5336

4

Lqwurgxfwlrq

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5

Prqrsro|

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514

Vlqjoh0surgxfw prqrsro|

Wkh hdvlhvw srvvleoh prgho ri lpshuihfw frpshwlwlrq lv rqh zkhuh wkhuh lv d prqrsrolvw wkdw vhoov rqo| rqh jrrg1 L zloo uvw vroyh wkh prqrsrolvw*v sureohp zlwk jhqhudo frvw dqg ghpdqg ixqfwlrqv/ dqg wkhq rhu vrph vshflf h{dpsohv1 4

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+4,

Wkh uvw0rughu frqglwlrq +IRF, ri wklv sureohp lv= CG+s, gF+t, CG+s, C @s . G+s,  @ 3> Cs Cs gt Cs

+5,

wkdw fdq eh uh0zulwwhq dv= s

G+s, gF+t, @ = gt CG+s,@Cs

+6,

E| xvlqj wkh ghqlwlrq ri ghpdqg hodvwlflw| dqg vxevwlwxwlqj/ rqh rewdlqv= 4 s  gF+t,@gt @ = s %

+7,

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5

D vshflf ixqfwlrqv h{dpsoh Ohw ph qrz xvh h{solflw ghpdqg dqg frvw ixqfwlrqv wr dqdo|vh wkh prqrsrolvw*v sureohp1 Frqvlghu iru wkh vdnh ri vlpsolflw| wkh fdvh ri d olqhdu ghpdqg t @ 4  s dqg ri d frqvwdqw pdujlqdo frvw F+t, @ ft/ zlwk f ? 4 wr jxdudqwhh yldelolw| ri wkh pdunhw +rwkhuzlvh/ qrerg| zrxog ex| wkh surgxfw hyhq li wkh prqrsrolvw rhuhg lw dw pdujlqdo frvw,1 Wkh prqrsrolvw*v sureohp lv pd{s  @ +s  f,+4  s,1 Wkh IRF lv jlyhq e|= C @ 4  5s . f @ 3> Cs

+8,

zkhqfh= sP @

4.f = 5

+9,

Lw lv zruwk qrwlqj wkdw wkh kljkhu wkh pdujlqdo frvw wkh kljkhu wkh sulfh wkh prqrsrolvw zloo vhw dw htxloleulxp/ gsP @gf A 3= iru frqvxphuv/ ehwwhu wr idfh dq h!flhqw prqrsrolvw wkdq dq lqh!flhqw rqh$6 E| vxevwlwxwlrq/ rqh fdq dovr qg rswlpdo txdqwlwlhv dqg surwv= t P @ +4  f,@5/ dqg  P @ +4  f,5 @71 Qrwh dovr wkdw zkhwkhu wkh prqrsrolvw fkrrvhv sulfh ru txdqwlw| wkh pdunhw rxwfrph zloo eh wkh vdph +d surshuw| wkdw grhv qrw krog lq roljrsro| pdunhwv/ dv zh vkdoo vhh ehorz,1 Wr vhh wklv/ zulwh wkh lqyhuvh ghpdqg ixqfwlrq dv s @ 4  t1 Wkh prqrsrolvw*v sureohp lv wkhq pd{t  @ +4  t  f,t1 Wkh IRF lv C@Ct @ 4  5t  f @ 3/ wkdw ehfrphv t P @ +4  f,@5 +dqg e| vxevwlwxwlrq sP @ +4 . f,@5,1

515

Pxowl0surgxfw prqrsro|

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Lqghshqghqw ghpdqg dqg frvw ixqfwlrqv

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zlwk l @ 4> 5=

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6

+:,

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Lqwhughshqghqw ghpdqgv

Yhu| riwhq/ d up vhoov d udqjh ri surgxfwv wkdw duh wr vrph h{whqw vxevwlwxwdeoh zlwk hdfk rwkhu1 Wklqn iru lqvwdqfh ri d fdu pdqxidfwxuhu wkdw rhuv fduv zlwk glhuhqw hqjlqh srzhuv/ frorxuv dqg yhuvlrqv +vwdwlrq0zdjrq/ vhgdq/ fdeulrohw,/ ru d irrg surgxfhu wkdw rhuv glhuhqw w|shv ri sdvwd1 Iru vxevwlwxwh surgxfwv/ dq lqfuhdvh lq wkh sulfh ri rqh surgxfw zloo lqfuhdvh wkh ghpdqg iru wkh rwkhuv/ wkdw ehfrph uhodwlyho| pruh frqyhqlhqw= gtl @gsm A 31 Dw wlphv/ upv pljkw vhoo lqvwhdg +ru lq dgglwlrq, surgxfwv wkdw duh frp0 sohphqwv wr hdfk rwkhu= d whohfrp rshudwru pljkw vhoo erwk d fhooxodu skrqh dqg d vxevfulswlrq wr lwv vhuylfhv/ dqg d irrg surgxfhu pljkw vhoo erwk sdvwd dqg suhsduhg vdxfhv1 Iru frpsohphqwdu| jrrgv/ dq lqfuhdvh lq wkh sulfh ri rqh surgxfw zloo ghfuhdvh wkh ghpdqg iru wkh rwkhuv +e| ghfuhdvlqj wkh ghpdqg iru wkh uvw surgxfw/ ghpdqg iru frpsohphqwv lv dovr glvfrxudjhg,= gtl @gsm ? 31 Wr vhh krz d prqrsrolvw vhwv sulfhv iru wzr surgxfwv zkrvh ghpdqg duh lqwhughshqghqw/ dvvxph surgxfwv kdyh wkh iroorzlqj ghpdqg ixqfwlrqv= tl @ d  esl . jsm /

zlwk l> m @ 4> 5/ l 9@ m=

+;,

Li wkh sdudphwhu j A 3/ wkhq wkh wzr surgxfwv duh vxevwlwxwhv> li j ? 3/ wkh| duh frpsohphqwv> li j @ 3/ wkhlu ghpdqgv duh lqghshqghqw ri hdfk rwkhu +d fdvh wkdw zh vkdoo xvh dv d xvhixo ehqfkpdun,1 Dvvxph dovr wkdw mjm ? e/ wr jxdudqwhh wkdw wkh rzq0sulfh hhfw rq ghpdqg ri d surgxfw lv vwurqjhu wkdq wkh furvv0sulfh hhfw/ d qdwxudo dvvxpswlrq> dqg wkdw d A f+e  j,/ wr hqvxuh wkdw rxwsxw lv srvlwlyh dw htxloleulxp1 Dvvxph dovr wkdw wkh frvwv ri surgxflqj rqh jrrg lv qrw dhfwhg e| krz pxfk lv surgxfhg ri wkh rwkhu/ wr irfxv rq wkh lqwhughshqghqfh ri ghpdqgv1 Pruh sduwlfxoduo|/ dvvxph wkdw F+t4 > t5 , @ ft4 . ft5 1 Wkh prqrsrolvw*v wrwdo surwv duh jlyhq e|=  @ +d  es4 . js5 ,+s4  f, . +d  es5 . js4 ,+s5  f,>

+<,

lwv sureohp ehlqj pd{s4 >s5 / wkh IRFv duh= g @ d  5esl . 5jsm . f+e  j, @ 3> gsl 7

zlwk l> m @ 4> 5> l 9@ m=

+43,

Dw wkh v|pphwulf vroxwlrq/ s4 @ s5 @ sp / zh kdyh= sp @

d . f+e  j, > 5+e  j,

+44,

zkhuh sp A f1 Vlqfh zh duh lqwhuhvwhg lq wkh hhfw ri ghpdqg uhodwlrqvkls rq htxloleulxp sulfh/ qrwh wkdw= d Csp @ A 3= Cj 5+e  j,5

+45,

Dv j lqfuhdvhv lq wkh lqwhuydo +e> e,/ wkh sulfh fkdujhg e| wkh prqrsrolvw rq erwk surgxfwv dovr lqfuhdvhv17 Uhodwlyh wr wkh ehqfkpdun fdvh zkhuh wkh wzr surgxfwv duh lqghshqghqw +j @ 3,/ wklv lpsolhv wkdw wkh prqrsrolvw uhgxfhv wkh sulfh ri lwv surgxfwv zkhq wkh| duh frpsohphqwv +j ? 3, dqg lw lqfuhdvhv wkhp zkhq wkh| duh vxevwlwxwhv +j A 3,1 Wkh lqwxlwlrq iru wklv uhvxow lv vwudljkwiruzdug1 Zkhq wkh surgxfwv duh frpsohphqwv/ wkh| h{huflvh d srvlwlyh h{whuqdolw| rq hdfk rwkhu dqg wkh prqrs0 rolvw lqwhuqdolvhv lw e| ghfuhdvlqj lwv sulfhv +d orzhu sulfh ri jrrg 4 vwlpxodwhv vdohv ri jrrg 5 dqg ylfh yhuvd,1 Lq rwkhu zrugv/ li surgxfwv 4 dqg 5 zhuh vrog e| wzr glvwlqfw prqrsrolvwv/ frqvxphuv zrxog sd| pruh iru wkhp wkdq zkhq wkh| duh vrog e| wkh vdph up/ d uhvxow wkdw gdwhv edfn wr Frxuqrw +4;6;,/ dqg wkdw lv dovr glvfxvvhg erwk lq fkdswhuv 9 +d yhuwlfdoo| lqwhjudwhg up ehlqj d sduwlfxodu fdvh ri d up wkdw vhoov frpsohphqwv, dqg : +zkhq vwxg|lqj wlh0lq vdohv,1 Zkhq lqvwhdg wkh surgxfwv duh vxevwlwxwhv/ wkh h{whuqdolw| wkh| h{huflvh rq hdfk rwkhu lv qhjdwlyh/ dqg wkh prqrsrolvw frqwurov iru lw e| udlvlqj sulfhv +d orzhu sulfh ri jrrg 4 furzgv rxw vdohv ri jrrg 5 dqg ylfh yhuvd,1 Li surgxfwv 4 dqg 5 zhuh vrog e| wzr glvwlqfw upv/ frqvxphuv zrxog sd| ohvv wkdq zkhq wkh| duh vrog e| wkh vdph up1 Wr frpsohwh wkh dqdo|vlv/ xvh wkh htxloleulxp sulfh +44, dqg vxevwlwxwh lqwr rxwsxw dqg surw ixqfwlrq wr rewdlq= tp @

d  f+e  j, > 5

p @

^d  f+e  j,`5 > 5+e  j,

+46,

tp ehlqj wkh rxwsxw shu0surgxfw dqg p ehlqj wkh wrwdo surwv1 D g|qdplf lqwhusuhwdwlrq ri ghpdqg lqwhughshqghqfh Vr idu/ L kdyh wuhdwhg wkh wzr surgxfwv dv wzr glvwlqfw surgxfwv vrog vlpxowdqhrxvo| e| d prqrsrolvw1 Krzhyhu/ wkh lqvljkwv rewdlqhg deryh fduu| ryhu wr wkh fdvh zkhuh wkh prqrsrolvw vhoov wkh vdph surgxfw lq vhtxhqwldo pdunhwv/ vlpso| e| uh0lqwhusuhwlqj wkh ghpdqg uhodwlrqvkls dv rqh ri lqwhu0whpsrudo vxevwlwxwdelolw| ru frpsohphqwdulw|1 7 Lw

lv hdv| wr fkhfn wkdw R6 } whqgv wr

2KS*EeK, rewdlqhg dv

lv frqfdyh/ zlwk lwv orzhvw srvvleoh ydoxh +htxdolqj E@ n

3K dqg dq dv|pswrwh dv } whqgv wr K1 8

Frqwlqxh wr frqvlghu wkh fdvh zkhuh hdfk jrrg lv surgxfhg dw frqvwdqw pdujlqdo frvw f/ exw dvvxph wkdw ghpdqg lq wkh uvw shulrg lv jlyhq e| t4 @ d  es4 dqg ghpdqg lq wkh vhfrqg shulrg e| t5 @ d  es5 . t4 1 Li  A 3/ wkhq kljkhu vdohv lq wkh uvw shulrg vwlpxodwh ghpdqg lq wkh vhfrqg> li  ? 3/ wkh rssrvlwh krogv1 Wkh prqrsrolvw*v wrwdo surwv +dvvxplqj iru vlpsolflw| d }hur lqwhuhvw udwh/ wkdw lv wkh ixwxuh jdlqv frxqw dv wkh fxuuhqw rqhv, duh=  @ +d  es4 ,+s4  f, . +d  es5 . +d  es4 ,,+s5  f,=

+47,

Wkh prqrsrolvw fkrrvhv s4 dqg s5 vr dv wr pd{lplvh / wkhuhiruh wkh wzr IRFv duh jlyhq e| g@gsl @ 31 Vroylqj wkh uhvxowlqj v|vwhp jlyhv= s4 @

d+4  , . fe > e+5  ,

s5 @

d . fe+4  , = e+5  ,

+48,

Lw lv hdv| wr vhh wkdw dv  ulvhv wkh uvw shulrg sulfh jrhv grzq dqg wkh vhfrqg shulrg sulfh jrhv xs= d  fe Cs4 @ ? 3> C e+5  ,5

d  fe Cs5 @ A 3= C e+5  ,5

+49,

Wkh hhfw ri wkh lqwhu0whpsrudo h{whuqdolw| ryhu sulfhv lv glhuhqw1 Wkh vhfrqg shulrg sulfh ydulhv zlwk  rqo| ehfdxvh lw vkliwv xs +ru grzq/ li  ? 3, lwv ghpdqg lqwhufhsw iurp d wr d . t4 1 Pruh lqwhuhvwlqj lv wkh uhdvrq zk| wkh uvw shulrg sulfh ydulhv1 Lqwhuwhpsrudo frpsohphqwdulw|= Lqwurgxfwru| sulfh rhuv Zkhq  A 3/ wkh prqrsrolvw uhdolvhv wkdw wkhuh lv d srvlwlyh lqwhuwhpsrudo ghpdqg h{whuqdolw|/ dqg lw lqwhuqdolvhv lw e| ghfuhdvlqj wkh sulfh uhodwlyh wr wkh sulfh lw zrxog vhw li wkhuh zdv qr ixwxuh pdunhw1 Lq rwkhu zrugv/ lw dqwlflsdwhv wkdw e| orzhulqj wkh uvw shulrg sulfh lw zrxog lqfuhdvh uvw shulrg ghpdqg/ wkdw lq wxuq zloo vwlpxodwh ghpdqg lq wkh vhfrqg shulrg1 Wklv lv dq h{dpsoh ri d zhoo0 nqrzq exvlqhvv vwudwhj|/ wkdw frqvlvwv ri surprwlrqdo sulflqj/ ru lqwurgxfwru| sulflqj= lq wkh hduo| vwdjhv ri wkh olih ri d surgxfw/ wkh up vhwv lqlwldoo| d orz sulfh wkdw lv surjuhvvlyho| lqfuhdvhg dv frqvxphuv jhw wr nqrz dqg dssuhfldwh wkh surgxfw +jrrgzloo hhfw, ru dv pruh frqvxphuv kdyh douhdg| erxjkw wkh surgxfw +qhwzrun hhfwv/ vhh fkdswhu 5 iru d glvfxvvlrq,18 > 9 Lqwhuwhpsrudo vxevwlwxwdelolw|= Gxudeoh jrrgv Lq wkh fdvh zkhuh  ? 3/ kljkhu vdohv lq wkh uvw shulrg ghfuhdvh wkh ghpdqg lq wkh vhfrqg shulrg1 8 Dqrwkhu uhdvrq zk| d up pljkw rhu lqwurgxfwru| sulfhv lv wkh h{lvwhqfh ri vzlwfklqj frvwv +vhh fkdswhu 5,/ exw zkloh wkh vlpsoh prgho xvhg khuh fdq eh vhhq dv d uhgxfhg irup ri d pruh vrsklvwlfdwhg prgho frqwdlqlqj vrph jrrgzloo hhfwv ru qhwzrun hhfwv/ lw fdqqrw eh hdvlo| lqwhusuhwhg dv wkh uhvxow ri vzlwfklqj frvwv1 9 Lq wkh sduwlfxodu h{dpsoh fkrvhq khuh/ wkh uvw shulrg sulfh grhv qrw jr ehorz pdujlqdo frvwv1 Krzhyhu/ rqh frxog qg h{dpsohv zkhuh wkh up fkdujhv ehorz pdujlqdo frvw lq wkh uvw shulrg1

9

Dv d uhvxow/ wkh prqrsrolvw nhhsv wkh uvw shulrg sulfh kljkhu wkdq lq wkh k|0 srwkhwlfdo fdvh zkhuh wkh surgxfw lv vrog rqo| rqfh/ wr lqwhuqdolvh wkh qhjdwlyh ghpdqg h{whuqdolw| dulvlqj dfurvv shulrgv1 Wklv vlwxdwlrq fdq eh vhhq dv d uh0 gxfhg irup ri wkh fdvh ri d gxudeoh jrrg prqrsrolvw= wkh pruh frqvxphuv ex| lq wkh uvw shulrg wkh ihzhu zloo ex| lq wkh vhfrqg1 Qrwh wkdw lw fdq eh vkrzq wkdw lq d gxudeoh jrrg prqrsro| htxloleulxp sulfhv whqg wr ghfuhdvh ryhu wlph/ dv lv wkh fdvh lq wklv h{dpsoh1: 51516

Lqwhughshqghqw frvwv= Hfrqrplhv +ru glvhfrqrplhv, ri vfrsh

Ohw xv qrz wxuq wr wkh lpsdfw wkdw frvw h{whuqdolwlhv kdyh ryhu sulflqj ri d pxowl0 surgxfw prqrsrolvw1 Dvvxph wkdw wkh ryhudoo frvw ixqfwlrq ri wkh prqrsrolvw lv jlyhq e| F+t4 > t5 , @ ft4 . ft5 . t4 t5 1; Zkhq  A 3/ wkhuh h{lvw glvhfrqrplhv ri vfrsh ehwzhhq wkh wzr surgxfwv/ dv wkh kljkhu wkh rxwsxw ri rqh surgxfw wkh kljkhu wkh pdujlqdo frvw ri wkh rwkhu surgxfw +C 5 F@Ctl Ctm @  A 3,1 Wklv lv wkh fdvh/ iru lqvwdqfh/ zkhuh erwk surgxfwv pdnh xvh ri olplwhg qdwxudo uhvrxufhv ru lqsxwv= lqfuhdvh lq rxwsxw ri rqh surgxfw h{huwv suhvvxuh rq wkh frpprq lqsxw e| gulylqj xs lwv frvw1 Zkhq  ? 3/ wkhuh h{lvw lqvwhdg hfrqrplhv ri vfrsh +ru ri mrlqw surgxfwlrq, ehwzhhq wkh wzr surgxfwv= wkh kljkhu wkh rxwsxw ri rqh surgxfw wkh orzhu wkh pdujlqdo frvw ri wkh rwkhu +C 5 F@Ctl Ctm @  ? 3,1 Wkhuh duh pdq| lqvwdqfhv lq wkh uhdo zruog zkhuh surgxflqj wzr jrrgv mrlqwo| jlyhv ulvh wr frvw vdylqjv uhodwlyh wr wkh fdvh zkhuh hdfk surgxfw lv surgxfhg vhsdudwho|1 Iru vlpsolflw|/ dvvxph wkdw surgxfwv duh lqghshqghqw/ vr wkdw tl @ d  esl 1 Wkh prqrsrolvw*v wrwdo surwv duh=  @ +d  es4 ,s4 . +d  es5 ,s5  ft4  ft5  t4 t5 =

+4:,

Lwv sureohp dprxqwv wr pd{s4 >s5 / vr wkh IRFv duh= g @ d+4 . e,  5esl  e5 sm . fe @ 3> gsl

zlwk l> m @ 4> 5> l 9@ m=

+4;,

Xqghu v|pphwu|/ s4 @ s5 @ sp / dqg wkh vroxwlrq ehfrphv= sp @

d+4 . e, . fe = e+5 . e,

+4<,

Wr vhh wkh hhfw ri wkh frvw h{whuqdolw| rq wkh htxloleulxp sulfh/ zulwh= d  ef Csp @ A 3= C +5 . e,5

+53,

: Vhh fkdswhu 5 iru d eulhi glvfxvvlrq ri wkh gxudeoh jrrg prqrsro|/ dqg fkdswhu 9 iru dq dssolfdwlrq wkdw vkduhv pdq| ihdwxuhv zlwk wkh gxudeoh jrrg prqrsro| fdvh +wkh up lv kxuw e| lwv lqdelolw| wr frpplw wr d fhuwdlq sulfh,1 ; Dvvxph dovr wkdw > : 32S*@1 Wklv jxdudqwhhv wkdw pdujlqdo frvwv duh s rvlwlyh zkhq > lv qhjdwlyh1

:

Wkh vwurqjhu wkh frvw h{whuqdolw| ehwzhhq wkh wzr surgxfwv/ wkh kljkhu wkh htxloleulxp sulfh vhw e| wkh prqrsrolvw1 Lq sduwlfxodu/ qrwh wkdw wkh ixqfwlrq sp +, lv lqfuhdvlqj ryhu doo lwv grpdlq1 Wklv lpsolhv wkdw zkhq wkhuh h{lvw hfrqrplhv ri vfrsh + ? 3,/ sulfhv duh orzhu wkdq lq wkh ehqfkpdun fdvh zkhuh wkh surgxfwlrq frvwv duh lqghshqghqw1 Wkh up dqwlflsdwhv wkdw wkhuh h{lvwv d srvlwlyh h{whuqdolw| ehwzhhq wkh wzr surgxfwv/ dqg uhgxfhv wkh sulfh ri hdfk jrrg lq rughu wr vwlpxodwh lwv rxwsxw/ dqg lq wxuq wkh rxwsxw ri wkh rwkhu surgxfw/ wkurxjk wkh frvw uhgxfwlrq1 Lq wklv fdvh/ d pxowl0surgxfw prqrsrolvw fkdujhv orzhu sulfhv wkdq wzr glvwlqfw prqrsrolvwv/ hdfk surgxflqj rqh surgxfw/ zrxog fkdujh1 Zkhq lqvwhdg wkhuh h{lvw glvhfrqrplhv ri vfrsh + A 3,/ sulfhv duh kljkhu wkdq lq wkh fdvh zkhuh surgxfwlrq frvwv duh lqghshqghqw1 Khuh wkhuh h{lvwv d qhjdwlyh h{whuqdolw| ehwzhhq wkh wzr surgxfwv/ dqg wkh prqrsrolvw lqfuhdvhv wkh sulfh ri hdfk jrrg wr lqwhuqdolvh lw +e| uhgxflqj wkh rxwsxw ri rqh surgxfw wkh pdujlqdo frvw ri wkh rwkhu lv orzhuhg,1 Lq wklv fdvh/ d pxowl0surgxfw prqrsrolvw fkdujhv kljkhu sulfhv wkdq wzr glvwlqfw prqrsrolvwv zrxog fkdujh1 Dq lqwhuwhpsrudo h{dpsoh= ohduqlqj0e|0grlqj Wkh frvw ri surgxf0 wlrq ri pdq| jrrgv dqg vhuylfhv ghfuhdvhv zlwk wkh h{shulhqfh dffxpxodwhg lq surgxflqj wkrvh jrrgv dqg vhuylfhv1 Wklv lv d skhqrphqrq nqrzq dv ohduqlqj0 e|0grlqj +vhh Duurz/ 4<94,/ dqg wkdw surylghv dq h{dpsoh ri lqwhu0whpsrudo h{whuqdolwlhv rq wkh frvw vlgh1 Dv rqh fdq vhh iurp wkh iroorzlqj vlpsoh h{dpsoh/ d prqrsrolvw wkdw h{shfwv d ohduqlqj hhfw zloo zdqw wr ghfuhdvh sulfhv +uhodwlyh wr d vlwxdwlrq zkhuh vxfk ohduqlqj hhfwv duh devhqw, lq wkh hduo| vwdjhv ri olih ri d surgxfw/ lq rughu wr lqfuhdvh rxwsxw dqg jr grzq wkh ohduqlqj fxuyh/ zklfk lq wxuq zloo pdnh lw pruh h!flhqw lq odwhu shulrgv1 Vxssrvh wkdw lq hdfk ri wkh wzr shulrgv ri olih ri lwv surgxfw d prqrsrolvw idfhv ghpdqg sw @ 4  tw / zlwk w @ 4> 51 Ehfdxvh ri ohduqlqj hhfwv/ pdujlqdo frvwv ghfuhdvh zlwk sdvw surgxfwlrq= wkh| duh F 3 4 @ f lq wkh uvw shulrg dqg F53 +t4 , @ f  ot4 lq wkh vhfrqg1< Wrwdo surwv ri wkh prqrsrolvw +dvvxplqj wkhuh lv qr glvfrxqwlqj iru vlp0 solflw|, duh  @ +s4  f,t4 . +s5  f . ot4 ,t5 =

+54,

Vlqfh lw kdv wr fkrrvh t4 > t5 vr dv wr pd{lplvh / wkh IRFv duh jlyhq e| g@gt4 @ 3 dqg g@gt5 @ 3/ wkdw fdq eh uh0zulwwhq dv= t4 @

4  F 3 4 . ot5 > 5

t5 @

4  F 35 = 5

+55,

Wkh suhylrxv h{suhvvlrqv whoo xv wkdw zkloh lq wkh vhfrqg shulrg +wkh odvw rqh lq wklv vlpsoh h{dpsoh, wkh prqrsrolvw zloo ehkdyh dv xvxdo/ wkdw lv e| vlpso| htxdwlqj pdujlqdo uhyhqxh wr pdujlqdo frvw/ lq wkh uvw shulrg lw zloo surgxfh < Dvvxph dovr wkdw , lv srvlwlyh exw vpdoo hqrxjk/ wr jxdudqwhh wkdw frvwv duh srvlwlyh lq wkh vhfrqg shulrg1

;

pruh wkdq lw zrxog lq d vwdwlf vhwwlqj/ vlqfh lw lqwhuqdolvhv wkh srvlwlyh h{whuqdolw| wkdw wklv zloo kdyh rq lwv vhfrqg0shulrg frvwv +dqg wkh kljkhu o wkh kljkhu wkh uvw shulrg txdqwlw|,1 E| vroylqj wkh v|vwhp ri IRFv +55, rqh rewdlqv htxloleulxp ydoxhv dv= t4 @ t5 @

4f > 5o

s4 @ s5 @

4.fo = 5o

+56,

Qrwh wkdw lq wklv vlpsoh h{dpsoh lw wxuqv rxw wkdw wkh txdqwlw| dqg sulfh vhw e| wkh prqrsrolvw dw wkh htxloleulxp duh wkh vdph ryhu wlph1 Krzhyhu/ wkh| duh vr iru txlwh glhuhqw uhdvrqv1 Lq wkh uvw shulrg/ wkh prqrsrolvw lqfuhdvhv txdqwlwlhv ehfdxvh lw lqwhuqdolvhv wkh ohduqlqj hhfw> lq wkh vhfrqg shulrg/ wklv h{whuqdolw| lv devhqw +lw lv wkh odvw shulrg,/ exw wkh htxloleulxp txdqwlw| lqfuhdvhv ehfdxvh pdujlqdo frvwv kdyh ghfuhdvhg143 Krzhyhu/ wkhuh lv d vhqvh lq zklfk ohduqlqj pdnhv wkh prqrsrolvw ehkdyh pruh frpshwlwlyho| lq wkh hduolhu shulrgv1 Li rqh frpsxwhv wkh pdunhw srzhu +dv sur{lhg e| wkh Ohuqhu lqgh{= Ow @ +sw F 3 w ,@sw , h{huflvhg e| wkh prqrsrolvw rqh rewdlqv wkdw pdunhw srzhu lv orzhu lq wkh uvw shulrg wkdq lq wkh vhfrqg= O4 @

+4  f,+4  o, +4  f, ? O5 @ = 4.fo 4.fo

+57,

Dq lqwurgxfwlrq wr hohphqwdu| jdph wkhru| Lq wklv vhfwlrq L wu| wr jlyh d vlpsoh dqg vkruw lqwurgxfwlrq wr wkh prvw hohphqwdu| frqfhswv ri jdph wkhru|/ wkdw duh dovr wkh rqhv wkdw L xvh lq prvw ri wkh whfkqlfdo vhfwlrq ri wkh errnv144 Shukdsv wkh vlpsohvw zd| wr lqwurgxfh wkh uhdghu zkr lv qrw idploldu zlwk jdph wkhru| lv wr vwduw zlwk d vlpsoh h{dpsoh ri d jdph/ wkdw L nhhs dv devwudfw dv srvvleoh vr dv qrw wr glvwudfw dwwhqwlrq zlwk uhdo zruog vwrulhv +L wxuq odwhu wr vrph pruh uhdolvwlf dssolfdwlrqv,1 Frqvlghu d jdph sod|hg ehwzhhq wzr sod|huv/ fdoo wkhp sod|hu D dqg sod|hu E1 Sod|hu D kdv wr ghflgh ehwzhhq wzr srvvleoh dfwlrqv= d4 ru d5 > vlpxowdqhrxvo|/ sod|hu E fkrrvhv dprqj wkuhh srvvleoh dfwlrqv e4 / e5 / ru e6 1 Hdfk sdlu ri dfwlrqv zloo eh dvvrfldwhg zlwk d fhuwdlq sd|r iru hdfk sod|hu/ wkdw lv zkdw d sod|hu uhfhlyhv diwhu hdfk ri wkhp kdv fkrvhq dq dfwlrq1 Wkh sd|0rv fdq eh vxppdulvhg lq d sd|r pdwul{ vxfk dv lq wdeoh ;141 Iru lqvwdqfh/ li sod|hu D fkrrvhv dfwlrq d4 dqg sod|hu E dfwlrq e5 / wkdw lv/ iru wkh sdlu +d4 > e5 ,/ wkh sod|huv* sd|r lv +5/8,= D uhfhlyhv 5 dqg E uhfhlyhv 81 Dvvxph dovr wklv lv d jdph zlwk shuihfw lqirupdwlrq/ wkdw lv sod|huv kdyh shuihfw nqrzohgjh ri wkh dfwlrqv dydlodeoh wr hdfk dqg ri wkh sd|0rv dvvrfldwhg zlwk wkhp +wklv dvvxpswlrq lv nhsw wkurxjkrxw xqohvv rwkhuzlvh lqglfdwhg,1 LQVHUW Wdeoh ;141 D vlpsoh jdph 43 Wkhvh wzr hhfwv h{dfwo| frpshqvdwh hdfk rwkhu lq wklv prgho1 Vhh Wluroh +4<;;= :5, iru d ohduqlqj prgho zkhuh rxwsxw pljkw hyhq lqfuhdvh ryhu wlph1 44 Vrph vhfwlrqv pdunhg zlwk -- xvh frqfhswv/ vxfk dv Ed|hvldq Qdvk htxloleulxp dqg vhtxhqwldo htxloleulxp/ wkdw duh qrw h{sodlqhg khuh exw duh eulh | lqwurgxfhg gluhfwo| lq wkrvh vhfwlrqv1

<

Zkhq rqh orrnv dw d jdph/ rqh zrxog olnh wr eh deoh wr suhglfw lwv rxwfrph1 Lq wklv sduwlfxodu jdph/ iru lqvwdqfh/ rqh zrxog olnh wr dqvzhu wkh txhvwlrq li D dqg E zhuh fdoohg wr sod| wklv jdph/ zklfk dfwlrqv zloo wkh| fkrrvhB/ ru/ zklfk lv htxlydohqw/ zkdw zloo eh wkh qdo uhvxow ri wkh jdphB1 Lw lv fohdu wkdw zh frxog qrw uhdvrqdeo| h{shfw wkdw fhuwdlq sdluv ri dfwlrqv zloo eh wkh uhvxow ri wkh jdph1 Iru lqvwdqfh/ wkh sdlu +d5 > e4 , lv xqolnho| wr rffxu= li sod|hu E fkrvh dfwlrq e4 / sod|hu D zrxog qrw uhvsrqg zlwk dfwlrq d5 / wkdw jlyhv khu wkh sd|r 3/ exw udwkhu zlwk dfwlrq d4 / wkdw jlyhv khu wkh sd|r 51 +Vlploduo|/ li sod|hu D fkrvh d5 / sod|hu E zrxog suhihu wr uhvsrqg zlwk dfwlrq e5 > wkdw jlyhv klp sd|r 6 udwkhu wkdq wkh orzhu sd|r 5 kh zrxog rewdlq e| fkrrvlqj dfwlrq e4 ru dfwlrq e6 1, Lq rwkhu zrugv/ zh frxog qrw h{shfw iurp wzr udwlrqdo sod|huv wkdw wkh qdo rxwfrph ri wkh jdph wkh| sod| lv d sdlu +d> e, vxfk wkdw d lv qrw wkh ehvw uhvsrqvh wr e dqg ylfh yhuvd1 Iru lqvwdqfh/ lq wkh jdph deryh/ zh zrxog qrw h{shfw +d4 > e4 , wr eh wkh qdo rxwfrph ri wkh jdph= li D sod|v d4 / E zloo suhihu wr sod| e5 / dv wklv jlyhv klp d kljkhu sd|r +8/ udwkhu wkdq 3,1

516

Qdvk htxloleulxp

Wkh frqfhsw ri Qdvk htxloleulxp lv edvhg rq vxfk dq lghd= lw suhglfwv wkdw wkh rxwfrph ri wkh jdph +wkh htxloleulxp, lv jlyhq e| wkh vhw ri dfwlrqv vxfk wkdw/ iru hdfk sod|hu/ hdfk dfwlrq lv wkh ehvw uhvsrqvh wr wkh dfwlrqv ri doo rwkhu sod|huv1 Htxlydohqwo|/ d vhw ri dfwlrqv uhsuhvhqwv d Qdvk htxloleulxp li qrqh ri wkh sod|huv kdv dq lqfhqwlyh wr ghyldwh iurp lwv dfwlrq jlyhq wkh dfwlrqv ri doo rwkhu sod|huv145 Pruh irupdoo|/ lq d jdph zlwk q sod|huv/ ghqrwlqj zlwk Dl wkh vhw ri dfwlrqv dydlodeoh wr sod|hu l +zlwk l @ 4> ===> q,/ dqg zlwk l +4 > 5 > ==> l > ==> q , sod|hu l*v sd|r/ wkh q0wxsoh +4 > 5 > ==> l > ==> q , lv d Qdvk htxloleulxp li=

l +4 > 5 > ==> l > ==> q ,  3l +4 > 5 > ==> 3l > ==> q ,/ iru doo l @ 4> 5> ==> q/ dqg doo l 5 Dl = +58,

Lq rwkhu zrugv/ +4 > 5 > ==> l > ==> q , lv d Qdvk htxloleulxp li/ iru hdfk sod|hu l/ sod|lqj l lv wkh ehvw srvvleoh uhvsrqvh jlyhq wkh rwkhu sod|huv sod| +4 > 5 > ==> l4 > l.4 > ==> q ,1 Zh fdq qrz uhwxuq wr wkh jdph ghvfulehg lq jxuh 4/ dqg orrn iru wkh Qdvk htxloleulxp +ru htxloleuld, ri wkdw jdph1 Wr gr vr/ ohw xv zulwh grzq wkh ehvw uhvsrqvhv iru hdfk sod|hu= wkh htxloleulxp/ li lw h{lvwv/ zloo eh jlyhq e| wkh sdlu ri dfwlrqv wkdw duh pxwxdoo| ehvw uhvsrqvhv1 Sod|hu D*v ehvw uhvsrqvhv +lqglfdwhg lq erog, wr hdfk ri E*v dfwlrqv duh= +d4 > e4 ,> +d4 > e5 ,> +d5 > e6 ,1 45 Wr nhhs wklqjv dv vlpsoh dv srvvleoh/ wkh Qdvk htxloleulxp lv ghqhg khuh lq whupv ri dfwlrqv1 Lw frxog dovr eh ghqhg lq whupv ri vwudwhjlhv1 D vwudwhj| r lv d uxoh zklfk whoov d sod|hu zklfk dfwlrq wr fkrrvh dw dq| jlyhq wlph | ri wkh jdph/ iru dq| jlyhq klvwru| ri wkh jdph sulru wr |1 Ghqrwlqj zlwk 7 wkh vhw ri vwudwhjlhv dydlodeoh wr sod|hu  +zlwk W W W W  ' c c ?/ dqg zlwk Z  Er c r2 c c r c c r?  sod|hu *v sd|r/ wkh ?0wxsoh Er c r2 c c r c c r?  lv W  W d Qdvk htxloleulxp li= Z  ErW c rW2 c c rW c c rW?  Z  ErW  c r2 c c r c c r? / iru doo  ' c 2c c ?/ dqg doo r 7 1 Lq rqh0vkrw vlpxowdqhrxv jdphv/ wkh frqfhswv ri vwudwhj| dqg dfwlrq frlqflgh1

M

D

43

Sod|hu E*v ehvw uhvsrqvhv +lqglfdwhg lq erog, wr hdfk ri D*v dfwlrqv duh= +d4 > e5 ,> +d5 > e5 ,1 Lw lv vwudljkwiruzdug wr vhh wkdw +d4 > e5 , lv wkh xqltxh Qdvk htxloleulxp ri wklv jdph/ dv lw lv wkh rqo| sdlu ri dfwlrqv wkdw duh wkh ehvw uhvsrqvh wr hdfk rwkhu1 Htxlydohqwo|/ lw lv wkh rqo| sdlu ri dfwlrqv vxfk wkdw qr sod|hu kdv dq lqfhqwlyh wr ghyldwh iurp lw/ jlyhq wkh dfwlrq ri wkh ulydo1 51614

Wkh sulvrqhuv* glohppd

Shukdsv wkh prvw idprxv jdph/ vlqfh lw kdv dssolfdwlrqv lq vhyhudo hogv/ iurp hfrqrplfv wr srolwlfv/ lv wkh vr0fdoohg sulvrqhuv* glohppd/46 looxvwudwhg e| wkh sd|r pdwul{ ri wdeoh ;151 LQVHUW Wdeoh ;151 Wkh sulvrqhuv* glohppd jdph Wklv lv d shuihfwo| v|pphwulf jdph wkdw kdv rqo| rqh Qdvk htxloleulxp/ +orz> orz,1 +Wkh uhdghu fdq hdvlo| fkhfn e| gudzlqj wkh ehvw uhvsrqvhv iru hdfk sod|hu/ dv lqglfdwhg deryh> ru e| yhuli|lqj wkdw qr sod|hu zrxog sod| kljk jlyhq wkdw wkh rwkhu sod|v orz/ zkhuhdv dq| rwkhu sdlu ri dfwlrqv fdqqrw eh dq htxloleulxp vlqfh d sod|hu zrxog kdyh dq lqfhqwlyh wr ghyldwh iurp lw1, Iluvw/ qrwh wkdw wkh jdph lv yhu| sduwlfxodu ehfdxvh hdfk sod|hu kdv d grp0 lqdqw vwudwhj|= zkhwkhu lwv ulydo sod|v kljk ru orz/ d sod|hu dozd|v suhihuv wr sod| orz1 Vhfrqg/ wkh rxwfrph ri wkh jdph lv Sduhwr lqihulru = erwk sod|huv zrxog eh ehwwhu r li wkh| sod|hg +kljk> kljk,1 51615

Frruglqdwlrq jdphv dqg htxloleulxp vhohfwlrq

Lq wkh suhylrxv wzr jdphv wkhuh zdv d xqltxh Qdvk htxloleulxp1 Lw lv srvvleoh/ wkrxjk/ wkdw vhyhudo htxloleuld fr0h{lvw/ dv lq wkh jdph +d yhuvlrq ri wkh vr0fdoohg edwwoh ri wkh vh{hv, ghvfulehg lq wdeoh ;161 LQVHUW Wdeoh ;161 Wkh edwwoh ri wkh vh{hv jdph Dqqd dqg Euxqr kdyh ghflghg wr jr wr d uhvwdxudqw wrjhwkhu/ exw kdyh qrw glvfxvvhg zklfk rqh/ dqg fdqqrw frppxqlfdwh zlwk hdfk rwkhu ehiruh glqqhu wlph1 Wkhuh duh wzr jrrg uhvwdxudqwv lq wrzq/ rqh rhuv Lqgldq dqg wkh rwkhu Wkdl irrg1 Dqqd zrxog suhihu Lqgldq dqg Euxqr Wkdl/ exw iru erwk wkh prvw lpsruwdqw wklqj zrxog eh wr hqg xs lq wkh vdph sodfh1 Lw lv hdv| wr fkhfn wkdw wklv jdph kdv wzr Qdvk htxloleuld= +Lqgldq> Lqgldq, dqg +W kdl> W kdl,1 Lw lv idu iurp xqfrpprq wr qg jdphv zlwk pxowlsoh htxloleuld/ dqg wkh sureohp lq vxfk fdvhv lv krz wr fkrrvh dprqj wkhp1 Lq pdq| hfrqrplf dssol0 fdwlrqv/ iru lqvwdqfh/ zh duh idfhg zlwk vxfk d vlwxdwlrq/ dqg xqohvv zh frxog uhqh rxu dqdo|vlv wkh jdph zrxog kdyh vfdufh suhglfwlyh srzhu1 Wklv lvvxh lv dgguhvvhg e| uhqhphqwv wr wkh Qdvk htxloleulxp frqfhsw wkdw wu| wr vhohfw ehwzhhq glhuhqw htxloleuld1 46 Vr0fdoohg/

e hfdxvh wkh ruljlqdo vwru| e hklqg wklv jdph lv rqh zkhuh wkh sod|huv duh wzr

sulvrqhuv dffxvhg ri d fulph/ dqg nhsw lq vhsdudwh fhoov/ hdfk ri wkhp kdylqj wr ghflgh zkhwkhu wr frqihvv ru ghq| wkh fulph1

44

Sduhwr grplqdqfh dv dq htxloleulxp vhohfwlrq ghylfh Vrphwlphv/ dv lq wkh edwwoh ri wkh vh{hv jdph deryh/ zklfk lv shuihfwo| v|pphwulf/ hyhq wkhvh uhqhphqwv zrxog qrw khos pxfk/ exw lq rwkhuv wkhuh pljkw eh rqh htxloleulxp wkdw/ iru rqh uhdvrq ru wkh rwkhu/ pljkw eh pruh dsshdolqj dqg pruh olnho| iru sod|huv wr frruglqdwh xsrq147 Frqvlghu iru lqvwdqfh d voljkwo| glhuhqw yhuvlrq ri wkh edwwoh ri wkh vh{hv/ dv lq wdeoh ;171 LQVHUW Wdeoh ;171 D sxuh frruglqdwlrq jdph Khuh wkhuh duh djdlq wzr htxloleuld= +Lqgldq> Lqgldq,/ dqg +Lwdoldq> Lwdoldq,1 Krzhyhu/ hqglqj xs lq wkh Lwdoldq uhvwdxudqw zrxog eh Sduhwr0lqihulru iru wkh sod|huv= wkdw lv/ lq sodlq zrugv/ erwk Dqqd dqg Euxqr zrxog kdyh kljkhu sd|0rv li wkh| kdg glqqhu lq wkh Lqgldq uhvwdxudqw1 Wkhuhiruh/ xvlqj Sduhwr0grplqdqfh dv d fulwhulrq iru vhohfwlrq dprqj glhuhqw htxloleuld/ zh zrxog suhglfw wkdw Dqqd dqg Euxqr zloo jr wr wkh Lqgldq uhvwdxudqw/ d vrphkrz uhdvrqdeoh suh0 glfwlrq/ dqg rqh zklfk hqmr|v vxssruw iurp h{shulphqwdo hylghqfh148 > 49 Dqrwkhu fulwhulrq iru htxloleulxp vhohfwlrq lv wkh holplqdwlrq ri zhdno|0 grplqdwhg vwudwhjlhv1 Holplqdwlrq ri zhdno| grplqdwhg vwudwhjlhv Frqvlghu wzr sod|huv D dqg E zkrvh vhw ri srvvleoh vwudwhjlhv lv Vl +zlwk l @ D> E,1 Zh vd| wkdw iru sod|hu l d vwudwhj| vl lv zhdno| grplqdwhg li wkhuh h{lvwv d vwudwhj| v3l vxfk wkdw l +v3l > vm ,  l +vl > vm , iru doo vm +zlwk l 9@ m,/ dqg wkhuh h{lvwv dw ohdvw rqh vwudwhj| vm iru zklfk wklv lqhtxdolw| krogv zlwk d vwulfw vljq1 Frqvlghu iru lqvwdqfh wkh jdph looxvwudwhg lq wdeoh ;18= LQVHUW Wdeoh ;181 Dq dv|pphwulf jdph Lw lv hdv| wr fkhfn wkdw wklv jdph kdv wzr htxloleuld= +sO > sO , dqg +sK > sK ,1 Wkh fulwhulrq ri wkh holplqdwlrq ri zhdno| grplqdwhg vwudwhjlhv xqltxho| vh0 ohfwv wkh htxloleulxp +sK > sK ,1 Lqghhg/ iru sod|hu D zh kdyh D +sO > sO , @ 47 Klvwru| ru rwkhu flufxpvwdqfhv pljkw lq vrph sduwlfxodu fdvhv vxjjhvw wkdw rqh htxloleulxp lv pruh olnho| wr eh sod|hg wkdq dqrwkhu1 Lq wkh zrugv ri Vfkhoolqj +4<93,/ wkhuh pljkw eh irfdo srlqwv1 Iru lqvwdqfh/ li Dqqd dqg Euxqr kdyh vr idu dozd|v phw dw wkh Lqgldq uhvwdxudqw dqg qhyhu kdg glqqhu dw wkh Wkdl uhvwdxudqw/ wkh iruphu htxloleulxp lv pruh olnho| wkdq wkh odwwhu1 Vrph glvfxvvlrqv rq krz fhuwdlq sulfhv pljkw eh irfdo fdq eh irxqg lq fkdswhu 7/ zkhuh prghov kdylqj vhyhudo/ pruh ru ohvv frooxvlyh/ htxloleuld/ duh dqdo|vhg1 Wkhuh/ L dovr glvfxvv rwkhu hohphqwv wkdw pljkw uhvroyh wkh xqfhuwdlqw| derxw wkh htxloleulxp/ vxfk dv frppxqlfdwlrq ru uvw0pryhu dgydqwdjhv +Dqqd pljkw pryh uvw/ pdnh d uhvhuydwlrq dw wkh Lqgldq/ dqg wkhq pdnh lw nqrzq wr Euxqr,1 Exw ri frxuvh doo wkhvh duh glhuhqw jdphv wkdq wkh rqh0vkrw vlpxowdqhrxv pryhv jdph L dp glvfxvvlqj khuh1 48 Vrph h{shulphqwdo hylghqfh lv phqwlrqhg lq fkdswhu 7/ wkdw dqdo|vhv uhshdwhg jdphv wkdw duh yhu| vlplodu wr wkh rqh ghvfulehg lq wklv vlpsoh jdph/ dv wkh| suhvhqw d pxowlwxgh ri htxloleuld wkdw fdq eh Sduhwr0udqnhg1 49 Lw lv lpsruwdqw wr vwuhvv wkdw khuh L dp xvlqj Sduhwr0grplqdqfh wr vhohfw ehwzhhq wzr Qdvk htxloleuld/ zkhuhdv lq wkh sulvrqhuv* glohppd jdph wkh sdlu jlylqj wkh Sduhwr0vxshulru rxwfrph zdv qrw dq htxloleulxp ri wkh jdph/ dqg wkhuhiruh yhu| xqolnho| wr eh wkh rxwfrph ri wkh jdph1

45

D +sO > sK , @ 3 dqg D +sK > sO , @ 3 ?  D +sK > sK , @ 5/ zklfk lpsolhv wkdw iru D vwudwhj| sO lv zhdno| grplqdwhg e| vwudwhj| sK 1 Ixuwkhu/ iru sod|hu E zh kdyh E +sO > sO , @ E +sO > sK , @ 3 dqg E +sK > sO , @ 5 ?  E +sK > sK , @ 3/ wkdw lv/ iru E vwudwhj| sO lv zhdno| grplqdwhg e| vwudwhj| sK 14: 51616

Pl{hg vwudwhjlhv

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46

Qdvk +4<83, kdv vkrzhg wkdw dq| jdph zlwk d qlwh qxpehu ri sod|huv dqg zkhuh hdfk sod|hu kdv d qlwh qxpehu ri sxuh vwudwhjlhv/ kdv d Qdvk htxloleulxp +lq pl{hg vwudwhjlhv li qr htxloleulxp lq sxuh vwudwhjlhv h{lvwv,1 Qrwh dovr wkdw htxloleuld lq pl{hg vwudwhjlhv fdq fr0h{lvw lq wkh vdph jdph zlwk htxloleuld lq sxuh vwudwhjlhv1 Iru lqvwdqfh/ wkh uhdghu fdq fkhfn wkdw wkh edwwoh ri wkh vh{hv jdph ri wdeoh ;16 dgplwv rqh Qdvk htxloleulxp lq pl{hg vwudwhjlhv zkhuh erwk D dqg E fkrrvhv +Lqgldq, zlwk suredelolw| 426 dqg D fkrrvhv +W kdl, zlwk suredelolw| 4251

517

G|qdplf jdphv/ dqg vxe0jdph shuihfw Qdvk htxlole0 ulxp

Vr idu zh kdyh mxvw orrnhg dw vwdwlf jdphv/ zkhuh sod|huv fkrrvh wkhlu dfwlrqv vlpxowdqhrxvo|1 G|qdplf jdphv duh jdphv zkhuh sod|huv pryh vhtxhqwldoo| ru wkh| pryh pruh wkdq rqfh1 Vxfk d jdph pljkw eh dv iroorzv1 Iluvw/ sod|hu 4 fkrrvhv dq dfwlrq d4 iurp wkh vhw ri khu ihdvleoh dfwlrqv> vhfrqg/ sod|hu 5 fkrrvhv dq dfwlrq d5 iurp wkh vhw ri klv ihdvleoh dfwlrqv> diwhu erwk kdyh sod|hg/ sod|huv 4 dqg 5 uhfhlyh wkh sd|0rv dvvrfldwhg zlwk wkh sdlu +d4 > d5 ,1 Dq h{dpsoh zloo khos looxvwudwh zk| wkh frqfhsw ri Qdvk htxloleulxp qhhgv wr eh uhqhg wr ghdo zlwk g|qdplf jdphv1 Frqvlghu wzr sod|huv= up L lv dq lqfxpehqw up/ up H lv d srwhqwldo hqwudqw lq wkh lqgxvwu|1 Iluvw/ up H ghflghv zkhwkhu wr hqwhu ru qrw lq wkh pdunhw> vhfrqg/ up L ghflghv zkhwkhu wr dffrpprgdwh hqwu| +iru lqvwdqfh/ vhwwlqj d kljk sulfh, ru wr i ljkw hqwu| +vhwwlqj d orz sulfh,1 Wdeoh ;1: uhsuhvhqwv wklv jdph lq wkh vr0fdoohg qrupdo irup/ wkdw lv zlwk wkh xvxdo sd|r pdwul{1 LQVHUW Wdeoh ;1:1 Wkh hqwu| ghwhuuhqfh jdph Wklv jdph kdv wzr Qdvk htxloleuld= +hqwhu> dffrpprgdwh, dqg +vwd| rxw> i ljkw,1 Krzhyhu/ wkh vhfrqg htxloleulxp/ +vwd| rxw> i ljkw,/ lv dq xqolnho| rxwfrph ri wklv jdph1 Dw wklv htxloleulxp/ up H fkrrvhv wr vwd| rxw ehfdxvh up L zrxog jkw hqwu|/ wkhuhe| pdnlqj hqwu| xqsurwdeoh wr H1 Krzhyhu/ wkh wkuhdw ri jkwlqj hqwu| e| up L li hqwu| rffxuv lv qrw fuhgleoh1 Lqghhg/ li hqwu| glg wdnh sodfh/ up L zrxog udwkhu dffrpprgdwh lw/ vlqfh lw zrxog jhw d kljkhu sd|r wkdq li lw irxjkw lw1 Zh wkhuhiruh qhhg d uhqhphqw ri wkh Qdvk htxloleulxp frqfhsw/ wkdw doorzv wr uxoh rxw htxloleuld 0 vxfk dv wkh rqh ghvfulehg deryh 0 wkdw duh edvhg rq qrq0 fuhgleoh wkuhdwv1 Lq rwkhu zrugv/ zh qhhg wr hqvxuh wkdw hdfk sod|hu dozd|v sod|v rswlpdoo| lq hdfk vlwxdwlrq/ hyhq lq wkrvh wkdw duh qrw dorqj wkh htxloleulxp sdwk1 +Lq wkh jdph deryh/ dq htxloleulxp lv jlyhq e| +vwd| rxw> i ljkw,/ exw iljkw lv qrw rswlpdo li wkh jdph uhdfkhg d srlqw zkhuh hqwu| rffxuv1, Wkh frqfhsw ri vxe0jdph shuihfw Qdvk htxloleulxp +VSQH, fruuhvsrqgv wr wklv uhtxluhphqw1 Lw lv wkh vhw ri vwudwhjlhv iru hdfk sod|hu vxfk wkdw wkh vwudwhjlhv irup d Qdvk htxloleulxp lq dq| vxe0jdph ri wkh jdph +dqg qrw rqo| dorqj wkh htxloleulxp sdwk,/ wkdw lv dq| vxe0vhw ri wkh jdph wkdw vwduwv iurp dq| srlqw dw zklfk wkh frpsohwh klvwru| ri wkh jdph sulru wr wkdw srlqw lv frpprq nqrzohgjh iru doo wkh sod|huv +hdfk sod|hu nqrzv lw/ dqg nqrzv wkdw wkh rwkhu nqrzv lw111,1 47

Wr vhh krz wr orrn iru wkh VSQH ri d jdph/ lw lv xvhixo wr ghvfuleh d jdph lq lwv h{whqvlyh irup +ru jdph wuhh,1 Iru wkh jdph deryh/ wklv lv grqh lq Iljxuh ;141 LQVHUW Iljxuh ;141 H{whqvlyh irup ri wkh hqwu| ghwhuuhqfh jdph Wkh jdph wuhh ri jxuh ;14 frqwdlqv wzr vxe0jdphv= wkh zkroh jdph> dqg wkh vxe0vhw ri wkh jdph zklfk vwduwv zkhq sod|hu L lv derxw wr pryh diwhu sod|hu H kdv sod|hg hqwhu1 Zkdw wkh VSQH htxloleulxp frqfhsw uhtxluhv lv wkhq wkdw doo sod|huv sod| rswlpdoo| dw hdfk vxe0jdph1 51714

Edfnzdug lqgxfwlrq

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48

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6

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49

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Dw wklv htxloleulxp/ up 4 pdnhv surwv +f4  f5 ,+4  f5 ,/ dqg up 5 jdlqv }hur1 Lq sulqflsoh/ wkhuh h{lvw rwkhu htxloleuld ri wklv jdph/ exw wkh| duh ohvv uhd0 vrqdeoh1 Frqvlghu iru lqvwdqfh d sulfh s 5 +f4 > f5 ,1 Lw lv hdv| wr vhh wkdw wkh  sdlu +s 4 > s5 , @ +s  %> s, uhsuhvhqwv dq htxloleulxp ri wkh jdph1 Krzhyhu/ vxfk dq htxloleulxp orrnv ohvv dsshdolqj wkdq wkh htxloleulxp +s4 > s5 , lghqwlhg deryh/ dqg lqghhg wkh pdlq fulwhuld ri htxloleulxp vhohfwlrq zrxog vhohfw wkh odwwhu ryhu wkh iruphu1 Xqghu Sduhwr0grplqdqfh/ iru lqvwdqfh/ +s4 > s5 , zrxog eh fkrvhq ehfdxvh lw jlyhv kljkhu surwv iru up 4 zkloh nhhslqj wkh vdph surwv +}hur, iru up 51 Holplqdwlrq ri zhdno| grplqdwhg vwudwh0 jlhv dovr vhohfwv +s4 > s5 , dv wkh rqo| htxloleulxp ri wkh jdph1 Wr vhh wklv/ qrwh wkdw zkhq sod|hu 5 fkrrvhv dfwlrq s5 @ f5 lw jhwv 5 +f5  %> f5 , @ 3 dqg 5 +s  %> f5 , @ 3/ zkhuh s 5 +f4 > f5 ,> zkhq lw fkrrvhv dfwlrq s 5 @ s/ lw jhwv 5 +s  %> s, @ 3 li sod|hu 4 fkrrvhv s  %/ exw lw jhwv 5 +s . %> s, ? 3 li sod|hu  4 fkrrvhv s . %1 Wkh htxloleulxp +s 4 > s5 , frqwdlqv d vwudwhj| wkdw lv zhdno| grplqdwhg iru sod|hu 51 Ehuwudqg xqghu fdsdflw| frqvwudlqwv Ohw xv qrz jr edfn wr wkh v|pphwulf fdvh wuhdwhg lqlwldoo|/ exw uhod{ qrz dvvxpswlrq D7,/ dqg dvvxph lqvwhdg wkdw hdfk up krogv d fdsdflw| nl ? G+sl @ f,= zkhq fkdujlqj dw pdujlqdo frvw/ d up zrxog kdyh wr vxsso| d odujhu qxpehu ri xqlwv wkdq lwv fdsdflw| zrxog doorz lw wr gr1 Lw lv wkhq hdv| wr vhh wkdw xqghu wklv qhz dvvxpswlrq sl @ sm @ f lv qrw d Qdvk htxloleulxp ri wkh jdph dq| orqjhu= Uhpdun 4 Li nl ? G+sl @ f,> +sl > sm , @ +f> f, lv qrw d Qdvk htxloleulxp ri wkh jdph1 Surri1 Wr suryh wklv uhvxow/ zh mxvw qhhg wr qg d ghyldwlrq wkdw ohdyhv rqh ri wkh upv ehwwhu r1 Qrz/ dw wkh fdqglgdwh htxloleulxp sl @ sm @ f/ up l pdnhv }hur surwv1 Exw li lw ghyldwhv dqg vhoov dw d sulfh s34 A sm @ f/ vrph frqvxphuv zloo dgguhvv lw +dw ohdvw dv orqj dv s34 lv qrw wrr kljk, dqg lw zloo pdnh srvlwlyh surwv1 Wklv lv ehfdxvh doo frqvxphuv zrxog olnh wr ex| iurp up m/ wkdw vhoov dw d orzhu sulfh/ exw m fdqqrw vhuyh wkhp doo +dv lwv fdsdflw| nm ? G+f,,1 Vrph frqvxphuv duh udwlrqhg/ dqg zloo kdyh wr ex| lqvwhdg iurp up 4/ wkdw zloo wkhuhiruh pdnh srvlwlyh surwv +lw zloo vhoo d srvlwlyh qxpehu ri xqlwv dw d srvlwlyh pdujlq,1 L olplw p|vhoi khuh wr wklv surri wkdw xqghu fdsdflw| frqvwudlqwv wkh Ehuwudqg uhvxow grhv qrw krog1 Wr qg wkh htxloleulxp ri wkh sulfh frpshwlwlrq jdph xqghu fdsdflw| frqvwudlqwv uhtxluhv vshfli|lqj d udwlrqlqj uxoh/ wkdw lv d uxoh wkdw doorfdwhv frqvxphuv ehwzhhq wkh upv +vrph fdqqrw ex| iurp wkh orz sulfh up,1 Lw dovr wxuqv rxw wkdw dq htxloleulxp lq sxuh vwudwhjlhv grhv qrw h{lvw li upv kdyh odujh hqrxjk fdsdflwlhv1 Vlqfh wkurxjkrxw wkh errn L qhyhu ghdo zlwk fdsdflw| frqvwudlqhg prghov/ lw zrxog eh dq xqqhfhvvdu| frpsolfdwlrq wr fduu| rxw wkh frpsohwh dqdo|vlv ri wkh fdsdflw|0frqvwudlqhg sulfh frpshwlwlrq jdph1 L uhihu wkh lqwhuhvwhg uhdghu wr Nuhsv dqg Vfkhlqnpdq +4<{{,= lq d wzr0vwdjh jdph zkhuh upv +vlpxowdqhrxvo|, uvw fkrrvh fdsdflwlhv dqg wkhq sulfhv/ wkh 53

qdo htxloleulxp rxwfrph lv wkh vdph dv lq d rqh0vkrw jdph zkhuh upv fkrrvh txdqwlwlhv1 Lq rwkhu zrugv/ wkh Frxuqrw htxloleulxp frxog eh lqwhusuhwhg qrw rqo| olwhudoo|/ wkdw lv dv wkh rxwfrph ri d jdph zkhuh upv fkrrvh wkh rxwsxw wkh| eulqj wr wkh pdunhw/ exw dovr dv wkh rxwfrph ri d jdph zkhuh upv fkrrvh wkhlu fdsdflwlhv dqg wkhq wkhlu sulfhv154 61415

Txdqwlw| frpshwlwlrq +Frxuqrw prgho,

Frqvlghu qrz wkh vdph rqh0vkrw jdph dqdo|vhg lq vhfwlrq 61414/ exw uhod{ dv0 vxpswlrq D6, dqg dvvxph lqvwhdg wkdw upv fkrrvh txdqwlwlhv wkh| zdqw wr eulqj wr wkh pdunhw/ udwkhu wkdq sulfhv1 Iru vlpsolflw|/ ohw xv orrn uvw dw wkh v|pphwulf fdvh/ zkhuh erwk upv kdyh wkh vdph pdujlqdo frvw f ? 4/ dqg wkh| idfh d olqhdu ghpdqg ixqfwlrq s @ 4  T/ zlwk T @ t4 . t5 ehlqj wrwdo lqgxvwu| rxwsxw155 Ilup l*v surw lv jlyhq e| l @ s+tl > tm ,tl  ftl 1 Wkh uvw vwhs wr lghqwli| wkh Qdvk htxloleulxp ri wklv jdph lv wr orrn iru wkh ehvw uhso| ixqfwlrq ri d up iru dq| jlyhq txdqwlw| fkrvhq e| lwv ulydo1 Wdnlqj tm dv jlyhq/ up l vroyhv wkhuhiruh wkh iroorzlqj sureohp +iru l> m @ 4> 5/ dqg l 9@ m,= pd{ l +tl > tm , @ +4  tl  tm  f,tl / jlyhq tm = tl

+5;,

Wklv sureohp lv vroyhg e| wdnlqj gl +tl > tm ,@gtl @ 3/ wkdw lv= 4  5tl  tm  f @ 31

+5<,

Wkh suhylrxv h{suhvvlrqv +4  5t4  t5  f @ 3 iru up 4/ dqg 4  5t5  t4  f @ 3 iru up 5, lpsolflwo| uhsuhvhqw wkh uhdfwlrq ixqfwlrqv +ru ehvw uhso|/ ru ehvw uhvsrqvh ixqfwlrqv, ri hdfk up/ zkrvh phdqlqj zh kdyh douhdg| glvfxvvhg lq vhfwlrq 515161 Lw lv frqyhqlhqw wr uhsuhvhqw wkh uhdfwlrq ixqfwlrqv lq wkh vdph sodqh +t4 > t5 ,1 Wr wklv hqg/ ohw xv zulwh wkhp dv= U4 = t5 @ 4  5t4  f> U5 = t5 @

4  t4  f = 5

+63,

Iljxuh ;16 looxvwudwhv wkh wzr uhdfwlrq ixqfwlrqv1 LQVHUW Iljxuh ;161 Uhdfwlrq ixqfwlrqv lq wkh Frxuqrw prgho Qrwh wkdw wkh| duh qhjdwlyho| vorshg= wkh kljkhu up m*v rxwsxw tm wkh orzhu up l*v ehvw uhso| rxwsxw1 Wkh vorsh ri wkh uhdfwlrq ixqfwlrqv fduulhv lpsruwdqw hhfwv zkhq dqdo|vlqj g|qdplf prghov/ dv zh vkdoo vhh lq vhfwlrq 714 ehorz1 54 Vhh dovr Pdjjl +4<<9,/ zkhuh frqvwudlqwv duh cvriw*/ dv d up fdq lqfuhdvh lwv rxwsxw eh|rqg fdsdflw| e| lqfxuulqj dq dgglwlrqdo yduldeoh frvw1 Pdjjl*v irupxodwlrq lv vlpsohu dv lw doorzv xv wr qg dq htxloleulxp lq sxuh vwudwhjlhv ri wkh sulfh jdph lqghshqghqwo| ri wkh fdsdflw| ohyhov1 Qrwh/ krzhyhu/ wkdw wkh Nuhsv0Vfkhlqnpdq uhvxow dssolhv rqo| zkhq xvlqj wkh h!flhqw0udwlrqlqj doorfdwlrq uxoh/ dv vkrzhg e| Gdylgvrq dqg Ghqhfnhuh +4<{{,1 55 Wkh dvvxpswlrq S  vhuyhv wr jxdudqwhh wkh yldelolw| ri wkh pdunhw1 Rwkhuzlvh/ wkhuh lv qr sulfh dw zklfk upv zrxog eh zloolqj wr vxsso| ghpdqg dv wkh| zrxog jhw qhjdwlyh surwv1

54

Wkh jxuh dovr looxvwudwhv wkh lvrsurw ixqfwlrqv ri hdfk up/ wkdw lv wkh orfxv ri wkh srlqwv vxfk wkdw glhuhqw ydoxhv ri t4 > t5 jlyh wkh vdph ydoxh ri surw wr d up/ l +tl > tm , @ n1 Qrwh wkdw orzhu lvrsurw fxuyhv iru up 4/ dqg lvrsurw fxuyhv pruh wr wkh ohiw iru up 5/ duh dvvrfldwhg zlwk kljkhu surwv +jlyhq wkh vdph tl / d orzhu tm zrxog lqfuhdvh sulfhv dqg surwv ri up l,156 Dv zh nqrz iurp wkh glvfxvvlrq ri wkh jdph wkhru| vhfwlrq/ wkh Qdvk htxl0 oleulxp ri wkh jdph zloo eh ghwhuplqhg e| wkh lqwhuvhfwlrq ri wkh wzr uhdfwlrq ixqfwlrqv/ wkdw hohphqwdu| dojheud vkrzv wr eh= t4F @ t5F @

4f > 6

+64,

wkh odeho F vwdqglqj iru Frxuqrw157 Diwhu vxevwlwxwlqj/ rqh rewdlqv dovr sulfhv dqg surwv dv= sF @

5 4 . 5f F @ +4  f, = > F @  4 5 6 <

+65,

Qrwh wkdw dw wkh Frxuqrw htxloleulxp wkh pdunhw sulfh lv deryh pdujlqdo frvwv +uhfdoo wkdw f ? 4/ vr +4 . 5f,@6 A f, dqg upv pdnh srvlwlyh surwv1 Wkh idfw wkdw dw dq htxloleulxp xqghu txdqwlw| frpshwlwlrq upv vhw d kljkhu sulfh wkdq xqghu d sulfh frpshwlwlrq htxloleulxp ghvfhqgv iurp wkh yhu| frqfhsw ri Qdvk htxloleulxp1 Lq d Ehuwudqg jdph/ d up kdv wr fkrrvh lwv rswlpdo sulfh e| wdnlqj dv jlyhq wkh sulfh ri wkh ulydo1 E| xqghufxwwlqj wkh odwwhu/ lw zloo jhw doo wkh ghpdqg/ zklfk jlyhv d yhu| vwurqj lqfhqwlyh wr ghfuhdvh rqh*v sulfh1 Lq d Frxuqrw jdph/ lqvwhdg/ d up fkrrvhv lwv txdqwlw| jlyhq wkh txdqwlw| ri wkh ulydo1 Wkhuhiruh/ dq h{sdqvlrq ri rqh*v rxwsxw pljkw doorz rqh wr fdswxuh d kljkhu pdunhw vkduh exw grhv qrw doorz rqh wr fdswxuh doo ghpdqg suhflvho| ehfdxvh wkh ulydo*v rxwsxw lv wdnhq dv jlyhq1 Wkh lqfhqwlyhv wr frpshwh djjuhvvlyho| duh dffruglqjo| zhdnhu wkdq lq wkh sulfh frpshwlwlrq jdph1 Lq rwkhu zrugv/ sulfh frpshwlwlrq lv wrxjkhu wkdq txdqwlw| frpshwlwlrq dqg rwkhu wklqjv ehlqj htxdo/ upv* sulfhv dqg surwv duh orzhu1 Wklv uhvxow zloo eh frquphg dovr zkhq orrnlqj dw wkh glhuhqwldwhg surgxfwv fdvh1 56 Qrwh dovr wkdw dq lvrsurw ixqfwlrq ri up 4 pxvw eh wdqjhqw wr wkh krul}rqwdo olqh zkhq lw furvvhv up 4*v ehvw uhso| ixqfwlrq/ dqg dq lvrsurw ixqfwlrq ri up 5 pxvw eh wdqjhqw wr wkh yhuwlfdo olqh zkhq lw furvvhv 5*v ehvw uhso| ixqfwlrq1 Wr vhh wklv/ zulwh d jhqhulf lvrsurw ixqfwlrq ri up  dv Z E^ c ^2  ' E 3 ^ 3 ^2 3 S^ ' &1 Wr qg lwv vorsh/ zulwh wkh wrwdo glhuhqwldo dv _Z ' E 3 2^ 3 ^2 3 S_^ 3 ^ _^2 ' f1 Wkhuhiruh/ lwv vorsh pxvw eh _^2 *_^ ' E 3 2^ 3 ^2 3 S*^ / zkrvh qxphudwru lv qlo zkhq wkh IRF lv vdwlvhg +wkdw lv/ dorqj wkh uhdfwlrq ixqfwlrq,1 Vlploduo|/ zulwh d jhqhulf lvrsurw ixqfwlrq ri up 2 dv Z2 E^ c ^2  ' E 3 ^ 3 ^2 3 S^2 ' 5 1 Wkh wrwdo glhuhqwldo lv _Z2 ' E 3 ^ 3 2^2 3 S_^2 3 ^ _^ ' f1 Lwv vorsh pxvw eh _^2 *_^ ' ^ *E 3 ^ 3 2^2 3 S/ zkrvh ghqrplqdwru lv }hur zkhq wkh IRF lv

vdwlvhg= wkhuhiruh/ dorqj wkh uhdfwlrq ixqfwlrq wkh vorsh lv lqqlw|1 57 L irfxv khuh rq wkh v|pphwulf htxloleulxp ri wkh jdph1 Wkhuh dovr h{lvw wzr srvvleoh dv|pphwulf Qdvk htxloleuld/ zkhuh d up/ vd| / surgxfhv }hur rxwsxw dqg wkh rwkhu up surgxfhv vxfk d odujh rxwsxw wkdw li up  eurxjkw hyhq d yhu| vpdoo txdqwlw| wr wkh pdunhw wkh sulfh zrxog idoo ehorz pdujlqdo frvwv +wkxv jlylqj lw qr lqfhqwlyh wr ghyldwh iurp }hur rxwsxw,1 Wklv fdvh lv frqvlghuhg lq ghwdlo lq fkdswhu :161614 +rq lqwhurshudelolw|, dqg rplwwhg khuh wr vdyh vsdfh1

55

Frxuqrw zlwk frvw dv|pphwulhv Frqvlghu wkh vdph Frxuqrw jdph dv ehiruh/ exw dvvxph wkdw upv glhu lq wkhlu pdujlqdo frvwv/ zlwk f4 ? f5 1 Hdfk up l pd{tl l +tl > tm , @ +4  tl  tm  fl ,tl / jlyhq tm 1 Wkh uhdghu fdq fkhfn wkdw wkh uhdfwlrq ixqfwlrqv duh jlyhq e| tl @ +4  tm  fl ,@5/ dqg wkdw wkh Frxuqrw htxloleulxp txdqwlwlhv dqg surwv +ghwhuplqhg e| wkh lqwhuvhfwlrq ri wkh uhdfwlrq ixqfwlrqv dqg vxevwlwxwlrq, duh jlyhq e|= tl @

4  5fl . fm +4  5fl . fm ,5 >  l @ = 6 <

+66,

Krzhyhu/ qrwh wkdw wklv vroxwlrq krogv rqo| li frvwv duh forvh hqrxjk wr hdfk rwkhu1 Lqghhg/ wkh htxloleulxp rxwsxw lv phdqlqjixo rqo| li t5  3/ ru f5  +4 . f4 ,@51 Rwkhuzlvh/ vlploduo| wr wkh fdvh ri Ehuwudqg frpshwlwlrq/ up 5 lv vr pxfk ohvv h!flhqw wkdq up 4 wkdw wkh odwwhu fdq vhw/ xqglvwxuehgo|/ lwv prqrsro| rxwsxw dqg pdnh prqrsro| surwv1 Frxuqrw zlwk q upv Frqvlghu qrz djdlq wkh fdvh ri v|pphwulf frvw upv/ exw h{whqg wkh edvh prgho wr q upv1 Iru wkh l0wk up/ wkh sureohp lv= [ tm  f,tl / jlyhq tm = +67, pd{ l +t4 > ==> tl > ==> tq , @ +4  tl  tl

m

9@l

Wkh IRF lv jlyhq e| gl +t4 > ==> tl > ==> tq ,@gtl @ 3/ wkdw lv= [ 4  5tl  tm  f @ 31 m

+68,

9@l

Dw wkh v|pphwulf htxloleulxp/ tl @ tm / dqg wkh IRF vlpsolhv wr 4  5tl  +q  4,tl  f @ 31 Lw lv wkhq lpphgldwh wr ghulyh wkh htxloleulxp rxwsxwv/ dqg e| vxevwlwxwlrq wkh htxloleulxp sulfh dqg surwv/ dv= t

4f 4f > s @ f . >  @ @ 4.q 4.q



4f 4.q

5

=

+69,

Rqh fdq fkhfn wkdw iru q @ 5 wkh htxloleulxp ydoxhv lq +69, fruuhvsrqg wr wkh rqhv irxqg deryh lq h{suhvvlrqv +64, dqg +65,1 Exw wkhvh uhvxowv duh lqwhuhvwlqj ehfdxvh wkh| doorz wr vwxg| krz wkh Frxuqrw htxloleulxp rxwfrph fkdqjhv zlwk q1 Lq sduwlfxodu/ lw lv hdv| wr vhh wkdw wkh odujhu wkh qxpehu ri upv lq wkh lqgxvwu| wkh forvhu rqh jhwv wr wkh Ehuwudqg rxwfrph= olpq$4 s @ f1 Wkh Frxuqrw prgho doorzv xv wkhuhiruh wr fdswxuh wkh lqwxlwlyh uhvxow wkdw wkh pruh upv wkdw fr0h{lvw lq wkh lqgxvwu| wkh vwurqjhu frpshwlwlrq zloo eh lq lw/ udqjlqj iurp wkh prqrsro| rxwfrph fruuhvsrqglqj wr q @ 4 wr wkh Ehuwudqg rxwfrph +s @ f, zkhq q whqgv wr lqqlw|1 Vxfk d uhvxow glg qrw dulvh lq wkh Ehuwudqg prgho/ zkhuh douhdg| zlwk q @ 5 rqh rewdlqv wkdw upv fkdujh dw pdujlqdo frvw/ d uhvxow zklfk krogv lqghshqghqwo| ri wkh qxpehu q  5 ri upv lq wkh lqgxvwu|1

56

61416

Mrlqw0surw pd{lplvdwlrq

Wkh qdo ehqfkpdun fdvh ri surgxfw pdunhw frpshwlwlrq fruuhvsrqgv wr wkh fdvh zkhuh wkh roljrsrolvwv pd{lplvh mrlqw surwv/ wkdw lv wkh| ehkdyh dv li wkh| zhuh d vlqjoh up1 Wklv fdvh wkhuhiruh fruuhvsrqgv wr wkdw dqdo|vhg lq vhfwlrq 5151 Dv dq looxvwudwlrq/ frqvlghu q shuihfwo| v|pphwulf upv1 Mrlqw0surw pd{lplvdwlrq lpsolhv= pd{

t4 >==>tl >==>tq

@

q [

@4

l +t4 > ==> tl > ==> tq , @

l

q [ [ +4  tl  tm  f,tl =

@4

l

m

+6:,

9@l

Lq wklv sduwlfxodu fdvh zkhuh upv duh shuihfwo| v|pphwulf/ wklv sureohp dprxqwv wr pd{T +4  T  f,T1 Wkh IRF lv jlyhq e| g@gT @ 3/ wkdw lv 4  5T  f @ 3/ zklfk uhvxowv lq wkh iroorzlqj htxloleulxp ohyhov= TP @

4f P 4.f +4  f,5 > s @ > P @ = 5 5 7

+6;,

Reylrxvo|/ shu0up rxwsxwv dqg surwv fdq eh rewdlqhg e| glylglqj e| q= tP @

4f P +4  f,5 >  @ = 5q 7q

+6<,

Wkh v|pphwulf fdvh lv h{wuhpho| vlpsoh ehfdxvh lw lv qdwxudo wr dvvxph wkdw rxwsxwv dqg surwv duh vkduhg htxdoo|/ exw wkh wuhdwphqw ri mrlqw0surw pd{0 lplvdwlrq ehfrphv pruh frpsoh{ zkhq dv|pphwulhv dprqj wkh upv h{lvw158 Qrwh dovr wkdw wkh mrlqw0surw pd{lplvdwlrq fdvh vkrxog eh vhhq dv d ehqfk0 pdun fdvh/ wkdw lv dv wkh olplw ri d vlwxdwlrq zkhuh frpshwlwlrq dprqj wkh upv lq wkh surgxfw pdunhw lv yhu| zhdn1 Iru d wuhdwphqw ri frooxvlrq/ wkdw lv krz dqg zkhwkhu upv duh deoh wr vxvwdlq d mrlqw0surw pd{lplvdwlrq rxwfrph/ vhh vhfwlrq 61514 dqg/ deryh doo/ fkdswhu 71 61417

Ehqfkpdun prghov ri surgxfw frpshwlwlrq= D frpsdulvrq

Dv d vxppdu| ri wkh glhuhqw fdvhv dqdo|vhg khuh dv ehqfkpdunv ri surgxfw pdunhw frpshwlwlrq/ lw pljkw eh khosixo wr looxvwudwhv wkh htxloleulxp sulfh dv d ixqfwlrq ri wkh qxpehu q ri upv lq wkh lqgxvwu|/ xqghu wkh k|srwkhvhv wkdw upv frpshwh lq sulfhv/ lq txdqwlwlhv dqg wkdw wkh| ehkdyh dv li wkh| zhuh d prqrsrolvw1 Iljxuh ;17 grhv wklv/ dqg vkrzv iru dq| jlyhq qxpehu ri upv krz wkh wrxjkqhvv ri surgxfw pdunhw frpshwlwlrq 0 ru wrxjkqhvv ri sulfh frpshwlwlrq/ dv Vxwwrq +4<<4, fdoov lw 0 ydulhv dfurvv wkhvh fdvhv/ ehlqj kljkhvw xqghu Ehuwudqg +htxloleulxp sulfhv duh wkh orzhvw, dqg orzhvw xqghu mrlqw0surw pd{lplvdwlrq +htxloleulxp sulfhv duh wkh kljkhvw,1 LQVHUW Iljxuh ;171 Wrxjkqhvv ri surgxfw pdunhw frpshwlwlrq= Ehuwudqg +E,/ Frxuqrw +F,/ dqg mrlqw0surw pd{lplvdwlrq +P, 58 L

gr qrw ghdo zlwk wklv lvvxh e hfdxvh lw gr hv qrw dulvh dq|zkhuh lq wkh e r rn1

57

Wkh wkuhh fdvhv dqdo|vhg duh xvhixo ehqfkpdunv zkhq glvfxvvlqj roljrsro| lvvxhv1 Lw zrxog eh gl!fxow wr vd| zkhq lq wkh uhdo zruog rqh vkrxog h{shfw surgxfw pdunhw frpshwlwlrq wr wdnh rqh irup ru dqrwkhu1 Mrlqw0surw pd{0 lplvdwlrq pljkw shukdsv fruuhvsrqg wr d +qrzdgd|v yhu| uduh, vlwxdwlrq zkhuh dqwlwuxvw hqirufhphqw lv vr zhdn wkdw d fduwho fdq eh hdvlo| vxvwdlqhg1 Frxuqrw frpshwlwlrq pljkw eh dvvrfldwhg zlwk lqgxvwulhv zkhuh upv fdqqrw hdvlo| dgmxvw wkhlu fdsdflwlhv +wkdw lv/ zkhq wkh| fkrrvh sulfhv diwhu wkh| kdyh frpplwwhg wr d fhuwdlq fdsdflw| ru rxwsxw,/ dqg Ehuwudqg frpshwlwlrq zlwk lqgxvwulhv zkhuh upv duh lqvwhdg qrw pxfk frqvwudlqhg e| fdsdflwlhv> wkdw lv/ zkhuh wkh| fdq hdvlo| dgmxvw rxwsxw lq uhvsrqvh wr wkh ghpdqg wkh| uhfhlyh159

615

Surgxfw pdunhw frpshwlwlrq zlwk +h{rjhqrxvo|, gli0 ihuhqwldwhg jrrgv

Wklv vhfwlrq vwxglhv +rqh0vkrw, surgxfw pdunhw frpshwlwlrq zlwklq wzr +qrq0 vsdwldo, glhuhqwldwhg jrrg prghov1 Wklv dprxqwv wr uhod{lqj dvvxpswlrq D5, ri vhfwlrq 61414/ dqg djdlq wklv zloo uhvxow lq dq htxloleulxp sulfh deryh pdujlqdo frvw +h{fhsw lq olplwlqj fdvhv,1 Wkurxjkrxw wklv vhfwlrq upv* surgxfwv fkrlfhv duh wdnhq dv jlyhq1 61514

D olqhdu ghpdqg prgho ri glhuhqwldwhg jrrgv

Iroorzlqj Vlqjk dqg Ylyhv +4<;6,/ frqvlghu wzr upv/ 4 dqg 5/ wkdw vhoo wzr surgxfwv1 Ghqrwh t4 dqg t5 wkh txdqwlwlhv ri hdfk jrrg1 Wkhuh h{lvwv d frqwlq0 xxp ri frqvxphuv ri wkh vdph w|sh lq wkh hfrqrp|/ hdfk kdylqj wkh iroorzlqj xwlolw| ixqfwlrq= Y @ | . X +t4 > t5 ,>

+73,

zkhuh wkh olqhdulw| lq wkh frpsrvlwh jrrg | dyrlgv lqfrph hhfwv dqg surylghv d udwlrqdoh iru d sduwldo htxloleulxp dqdo|vlv ri wkh glhuhqwldwhg vhfwru1 Lqghhg/ wkh frqvxphu*v sureohp lv pd{t4 >t5 >| Y vxemhfw wr wkh exgjhw frqvwudlqw s4 t4 . s5 t5 . s| | @ U1 Wr vroyh wklv sureohp/ zulwh wkh Odjudqjhdq= O @ | . X +t4 > t5 , . +U  s4 t4  s5 t5  s| |,= Wkh IRFv duh jlyhq e|= ; CX +tl >tm , CO A ? Ctl @ Ctl  sl @ 3/ l @ 4> 5> l 9@ m> CO @ 4  s @ 3> | C| A = CO @ U  s t  s t  s @ 3= 4 4 5 5 | C

+74,

+75,

59 Prvw pdqxidfwxulqj lqgxvwulhv zloo suredeo| eh forvhu wr Frxuqrw wkdq Ehuwudqg1 Lqgxv0 wulhv zkhuh surfxuhphqw lv lpsruwdqw pljkw eh dq h{dpsoh ri Ehuwudqg frpshwlwlrq= uvw d up vhohfwv sulfh dqg jhwv d frqwudfw dqg wkhq lw kdv wr ixooo wkh rughu zlwklq d fhuwdlq wlph1 Vhfwruv zkhuh lw lv frvwohvv dqg lpphgldwh wr vkls d jrrg zrxog dovr uhvhpeoh Ehuwudqg frp0 shwlwlrq +wklqn iru lqvwdqfh ri wkh pxvlf lqgxvwu|/ zkhuh rqfh wkh ruljlqdo wudfn lv uhfrughg/ pdnlqj dq dgglwlrqdo FG lv hdv| dqg fkhds,1

58

E| wdnlqj wkh frpsrvlwh jrrg dv wkh qxpìudluh/ s| @ 4/ rqh rewdlqv  @ 41 Wkhuhiruh/ wkh IRFv uhodwlyh wr wkh glhuhqwldwhg jrrg pdunhw ehfrph CX +tl > tm ,@Ctl  sl @ 3/ dqg fdq eh dqdo|vhg lqghshqghqwo| ri wkh pdunhw iru wkh frpsrvlwh jrrg1 Rqfh hvwdeolvkhg wkdw wkh txdvl0olqhdulw| lq wkh xwlolw| ixqfwlrq Y mxvwlhv d sduwldo htxloleulxp dqdo|vlv ri wkh glhuhqwldwhg jrrg pdunhw/ ohw xv xvh d vshflf ixqfwlrqdo irup iru wkh xwlolw| ixqfwlrq1 Lq sduwlfxodu/ dvvxph=  4 5 t4 . t55 . 5t4 t5 > +76, X +t4 > t5 , @ t4 . t5  5 zlwk wkh sdudphwhuv xvhg lq wkh xwlolw| ixqfwlrq vdwlvi|lqj wkh iroorzlqj dvvxps0 wlrqv +iru l @ 4> 5/ l 9@ m,= +l,  A 3/ +ll,  A 3/ +lll,  A 1 Dvvxpswlrq +lll, jxdudqwhhv wkdw wkh ghpdqg ixqfwlrqv fdq eh lqyhuwhg/ duh ri wkh uljkw vljq/ dqg wkdw wkhuh lv d srvlwlyh lqwhufhsw lq gluhfw ghpdqgv +vhh ehorz,1 Sdudphwhu  lqglfdwhv zkhwkhu jrrgv 4 dqg 5 duh vxevwlwxwhv/ lqghshqghqw ru frpsohphqwv +dqg wr zkdw ghjuhh,1 Dv rqh fdq vhh iurp wkh xwlolw| ixqfwlrq/ li  A 3 frqvxplqj wkh wzr jrrgv wrjhwkhu glplqlvkhv wkh frqvxphu*v xwlolw| +wkdw lv/ wkh| duh vxevwlwxwhv,> li  ? 3 frqvxplqj wkh wzr jrrgv wrjhwkhu lqfuhdvhv wkh frqvxphu*v xwlolw| +wkdw lv/ wkh| duh frpsohphqwv,> li  @ 3 frqvxplqj wkh wzr jrrgv wrjhwkhu grhv qrw dhfw wkh frqvxphu*v xwlolw| +wkdw lv/ wkh| duh lqghshqghqw,1 Wkh IRFv ri wkh frqvxphu sureohp duh jlyhq e| CX +tl > tm ,@Ctl  sl @ 3/ zklfk ehfrph= sl @   tl  tm /

l @ 4> 5> l 9@ m=

+77,

Wklv lv wkh v|vwhp ri lqyhuvh ghpdqgv wkdw fdq eh xvhg wr vwxg| txdqwlw| frp0 shwlwlrq1 Dq lqvshfwlrq ri wkhvh ghpdqg ixqfwlrqv lpphgldwho| vxjjhvwv ixuwkhu lqwhusuhwdwlrq ri wkh sdudphwhuv1 Wkh forvhu  wr  wkh pruh vxevwlwxwhv wkh wzr jrrgv zloo eh/ zlwk wkh fdvh ri shuihfw vxevwlwxwhv dulvlqj dv wkh olplw fdvh  $ 1 Rqh frxog wkhuhiruh ghqh dq lqyhuvh phdvxuh ri surgxfw glhuhqwld0 wlrq dv @1 Wklv lqgh{ wdnhv ydoxhv lq ^3> 4,/ dqg dwwdlqv lwv plqlpxp ydoxh zkhq wkh jrrgv duh lqghshqghqw +iru  @ 3/ wkdw lv zkhq wkh| duh pd{lpdoo| glhuhqwldwhg,/ dqg lwv kljkhvw ydoxh zkhq wkh| whqg wr eh shuihfw vxevwlwxwhv +iru  $ ,1 E| lqyhuwlqj wkh wzr h{suhvvlrqv lq +77, rqh rewdlqv wkh v|vwhp ri gluhfw ghpdqg ixqfwlrqv/ dv= tl @ d  esl . jsm /

l @ 4> 5> l 9@ m>

+78,

zkhuh d> e> j vdwlvi|=   +  , > e@ 5 > j@ 5 = +79, 5 5 5       5 Qrwh/ wkhuhiruh/ wkdw wklv lv wkh olqhdu ghpdqg ixqfwlrq douhdg| xvhg lq vhfwlrq 5151 Duphg zlwk wkh lqyhuvh dqg gluhfw ghpdqg ixqfwlrqv/ zh fdq qrz ghulyh wkh htxloleulxp vroxwlrqv xqghu wkh xvxdo ehqfkpdun fdvhv ri surgxfw pdunhw frpshwlwlrq1 Iru vlpsolflw|/ dvvxph wkdw wkh wzr upv kdyh wkh vdph frqvwdqw pdujlqdo frvw f dqg dvvxph wkdw f @ 3 zlwkrxw orvv ri jhqhudolw|1 d@

59

Txdqwlw| frpshwlwlrq Frqvlghu uvw wkh fdvh zkhuh upv frpshwh lq txdq0 wlwlhv1 Hdfk up l @ 4> 5 fkrrvhv tl vr dv wr pd{lplvh lwv surwv l @ sl +tl > tm ,tl iru dq| jlyhq tm / zkhuh sl +tl > tm , @   tl  tm 1 Wkh IRFv duh= gl @   5tl  tm @ 3= gtl

+7:,

Qrwh wkdw hdfk IRF ghqhv lpsolflwo| d ehvw uhso| ixqfwlrq/ dqg wkdw vxfk d ixqfwlrq lv qhjdwlyho| vorshg/ dv zdv wkh fdvh xqghu Frxuqrw frpshwlwlrq zlwk krprjhqrxv jrrgv1 Dw wkh v|pphwulf htxloleulxp/ tl @ tm @ t/ dqg e| uh0duudqjlqj wkh IRFv wkh htxloleulxp rxwsxw lv rewdlqhg= tF @

 = 5 . 

+7;,

E| vxevwlwxwlrq rqh fdq wkhq qg wkh htxloleulxp rxwsxw dqg surwv/ dv= sF @

  > F @ + ,5 = 5 .  5 . 

+7<,

Sulfh frpshwlwlrq Li upv frpshwh lq sulfhv/ hdfk up l @ 4> 5 fkrrvhv sl vr dv wr pd{lplvh lwv surwv l @ tl +sl > sm ,sl wdnlqj sm dv jlyhq/ zkhuh tl +sl > sm , @ d  esl . jsm 1 Wkh IRFv duh= g l @   5esl . jsm @ 3= gsl

+83,

Wkh IRFv ghqh wkh iroorzlqj wzr ehvw uhso| ixqfwlrqv lq wkh sodqh +s4 > s5 ,= U4 = s5 @

d  5es4 d . js4 > U5 = s5 @ = j 5e

+84,

Iljxuh ;18 looxvwudwhv wkh uhdfwlrq ixqfwlrqv dqg vkrzv wkdw wkh| duh srvlwlyho| vorshg15: +Khqfh/ wkh kljkhu sm wkh kljkhu wkh sulfh sl zklfk lv wkh ehvw uhso| wr lw1, Lw dovr looxvwudwhv wkh lvrsurw fxuyhv ri wkh wzr upv/ wkdw lv wkh orfxv ri wkh srlqwv vxfk wkdw d up hduqv d jlyhq surw1 Iru erwk upv/ wkh ixuwkhu dq lvrsurw fxuyh iurp wkh ruljlq wkh kljkhu wkh dvvrfldwhg surwv15; LQVHUW Iljxuh ;181 Uhdfwlrq ixqfwlrqv xqghu sulfh frpshwlwlrq 5: Vwdelolw| uhtxluhv - wr eh vwhhshu wkdq - / wkdw lv 2K*} : }*E2K/ ru eK2 : } 2 / dozd|v  2 vdwlvhg vlqfh K : } e| dvvxpswlrq1 5; Wkh lvrsurw fxuyhv ri up 4 duh ghvfulehg e| wkh ixqfwlrq Z ' E@ 3 KR n }R R ' &1   2  E| wrwdoo| glhuhqwldwlqj zh rewdlq wkh vorsh ri rqh vxfk fxuyh1 Iluvw/ wdnh _Z ' E@ 3 2KR n }R2 _R n }R _R2 ' f/ iurp zklfk _R2 *_R ' 3E@ 3 2KR n }R2 *E}R / wkh lvrsurw vorsh1 +Qrwh wkdw dorqj - wkh qxphudwru lv }hur/ vr furvvlqj wkh uhdfwlrq ixqfwlrq wkh lvrsurw pxvw eh dw1, Ilup 5*v lvrsurwv duh jlyhq e| Z2 ' E@ 3 KR2 n }R R2 ' 51 Wrwdo glhuhqwldwlrq jlyhv _Z2 ' E@ 3 2KR2 n }R _R2 n }R2 _R ' f1 Khqfh/ rqh kdv _R2 *_R ' 3}R2 *E@ 3 2KR2 n }R / wkh vorsh ri up 5*v lvrsurw1 +Qrwh wkdw dorqj wkh uhdfwlrq ixqfwlrq/ -2 / wkh ghqrplqdwru lv }hur/ vr zkhq furvvlqj -2 wkh lvrsurw pxvw eh yhuwlfdo1,

5:

Dw wkh v|pphwulf htxloleulxp/ sl @ sm @ s/ dqg e| uh0duudqjlqj wkh IRFv wkh htxloleulxp sulfh lv rewdlqhg= sE @

+  , d @ = 5e  j 5  

+85,

E| vxevwlwxwlrq rqh fdq wkhq qg wkh htxloleulxp rxwsxw dqg surwv/ dv= tE @

+  , de d5 e 5 +  , @ > E @ @ = 5 5e  j 5   +5e  j, + . ,+5  ,5

+86,

Lw lv zruwk qrwlqj wkdw 0 dowkrxjk lw zdv d jrrg h{huflvh wr ghulyh wkh htxloleulxp vroxwlrqv 0 L frxog kdyh irxqg wkh vroxwlrq ri wkh sulfh jdph mxvw e| qrwlqj wkh gxdolw| ri wkh txdqwlw| dqg sulfh sureohpv/ dqg vxevwlwxwlqj lq wkh Frxuqrw vroxwlrqv douhdg| irxqg deryh1 Lqghhg/ xqghu txdqwlw| frpshwlwlrq iru up l wkh sureohp lv pd{tl l @ +tl tm ,tl jlyhq tm / zkhuhdv xqghu sulfh frpshwlwlrq wkh sureohp lv pd{sl l @ +d  esl . jsm ,sl / jlyhq sm 1 Wkhvh h{suhvvlrqv duh wkh gxdo ri hdfk rwkhu/ dv rqh fdq rewdlq wkh odwwhu iurp wkh iruphu e| uhsodflqj tl e| sl >  e| d>  e| e>  e| j15< Htxloleulxp frpsdulvrq Lw lv qrz srvvleoh wr frpsduh wkh uhvxowv re0 wdlqhg xqghu sulfh dqg txdqwlw| frpshwlwlrq1 Wr gr vr/ zulwh wkh glhuhqfh lq wkh htxloleulxp sulfhv dv= sF  sE @

 5 A 3= 7 5   5

+87,

Iluvw/ qrwh wkdw wkh glhuhqfh lv dozd|v srvlwlyh= sulfhv duh dozd|v kljkhu +dqg dffruglqjo| txdqwlwlhv duh dozd|v orzhu, xqghu txdqwlw| frpshwlwlrq wkdq xqghu sulfh frpshwlwlrq/ uhjdugohvv ri zkhwkhu wkh jrrgv duh vxevwlwxwhv/ lqghshqghqw/ ru frpsohphqwv163 Vhfrqg/ sF  sE lqfuhdvhv zlwk  iru jlyhq = zkhq  @ 3/ wkh sulfhv frlqflgh +wkh pdunhwv ehlqj lqghshqghqw/ lw lv luuhohydqw zkhwkhu upv frpshwh lq txdqwlwlhv ru lq sulfhv,> zkhq  @ / wkh glhuhqfh lv kljkhvw/ zlwk sF  sE @ @6 +wkh uhdghu zloo kdyh douhdg| uhfrjqlvhg wkdw wklv lv wkh fdvh fruuhvsrqglqj wr krprjhqrxv jrrgv,1 Vwudwhjlf vxevwlwxwhv y1 vwudwhjlf frpsohphqwv Zh kdyh vhhq deryh wkdw xqghu txdqwlw| frpshwlwlrq wkh upv* uhdfwlrq ixqfwlrqv duh qhjdwlyho| vorshg/ zkhuhdv xqghu sulfh frpshwlwlrq wkh| duh srvlwlyho| vorshg164 Lq zkdw 5< Wklv dovr lpsolhv wkdw Frxuqrw frpshwlwlrq zlwk vxevwlwxwh +uhvshfwlyho| frpsohphqw,

surgxfwv lv wkh gxdo ri Ehuwudqg frpshwlwlrq zlwk frpsohphqw +uhvshfwlyho| vxevwlwxwhv, surg0

xfwv=  : f fruuhvsrqgv wr 63 Rqh pljkw dovr fkhfn

} f/ dqg ylfh yhuvd1

dqrwkhu ri wkh uhvxowv rewdlqhg e| Vlqjk dqg Ylyhv= frqvxphu vxusoxv dqg wrwdo vxusoxv duh +zhdno|, odujhu xqghu sulfh frpshwlwlrq wkdq xqghu txdqwlw| frpshwlwlrq1 Surwv duh kljkhu/ htxdo ru orzhu xqghu txdqwlw| wkdq xqghu sulfh frpshwlwlrq/ dffruglqj wr zkhwkhu jrrgv duh vxevwlwxwhv/ lqghshqghqw ru frpsohphqwv1 64 Qrwh/ krzhyhu/ wkdw wklv krogv iru olqhdu ghpdqg/ exw qhhg qrw dozd|v eh wkh fdvh iru pruh jhqhudo ghpdqg ixqfwlrqv1

5;

iroorzv/ L eulh | uhodwh wkh vorsh ri upv* uhdfwlrq ixqfwlrqv wr d surshuw| ri wkh surw ixqfwlrq/ dqg lqwurgxfh wkh frqfhswv ri vwudwhjlf vxevwlwxwhv dqg vwudwhjlf frpsohphqwv/ gxh wr Exorz hw do1 +4<;8,165 Frqvlghu d vlpxowdqhrxv pryh jdph zkhuh hdfk up l @ 4> 5 fdq fkrrvh dfwlrqv dl 1 Ghqh l +dl > dm , up l*v surw ixqfwlrq/ dqg dvvxph wkdw lw lv wzlfh glhuhqwldeoh lq erwk dl dqg dm / zlwk C 5 l @+Cdl ,5 ? 3 rq lwv grpdlq wr hqvxuh d pd{lpxp h{lvwv1 Dv zh nqrz douhdg|/ up l*v uhdfwlrq ixqfwlrq lv jlyhq e| wkh ixqfwlrq Ul +dm , wkdw lghqwlhv wkh ehvw srvvleoh uhvsrqvh dl wr dq| jlyhq dfwlrq dm ri wkh ulydo=66 C l +Ul +dm ,> dm , @ 3= Cdl

+88,

Glhuhqwldwlrq ri +88, zlwk uhvshfw wr dm jlyhv= C +C l @CUl , CUl C +Cl @Cdl , gdm . gdm @ 3> Cdl Cdm Cdm

+89,

wkdw fdq eh uh0zulwwhq dv= C 5 l C 5 l CUl . @ 3= 5 Cdl Cdm Cdl Cdm

+8:,

Ghqrwh CUl @Cdm  U3l dv wkh vorsh ri up l*v uhdfwlrq ixqfwlrq/ dqg uh0duudqjh lw wr jlyh= Ul3 @ 

C 5  l @+Cdl Cdm , @ 3= C 5 l @Cd5l

+8;,

Wkhuhiruh/ wkh vljq ri wkh vorsh ri wkh uhdfwlrq ixqfwlrq lv jlyhq e| wkh vljq ri C 5 l @+Cdl Cdm ,1 +Uhfdoo wkdw C 5 l @Cd5l ? 31, Exorz hw do1 +4<;8, lqwurgxfh wkh iroorzlqj ghqlwlrq1 5

l ? 31  Dfwlrqv duh vwudwhjlf vxevwlwxwhv li CdC l Cd m 5

l A 31  Dfwlrqv duh vwudwhjlf frpsohphqwv li CdC l Cd m

Dfwlrqv duh vwudwhjlf vxevwlwxwhv +uhvshfwlyho|/ vwudwhjlf frpsohphqwv, zkhq dq lqfuhdvh lq dm uhgxfhv +uhvs1 udlvhv, up l*v pdujlqdo surwdelolw| +Cl @Cdl ,/ wkxv ohdglqj up l wr fkrrvh d orzhu +uhvs1 kljkhu, dl 1 Wklv h{sodlqv zk| wkh ehvw uhso| ixqfwlrqv duh qhjdwlyho| +uhvs1 srvlwlyho|, vorshg1 Dv zh vkdoo vhh lq vhfwlrq 714 ehorz/ wkh lpsolfdwlrqv iru dq roljrsro| prgho pljkw wxuq rxw wr eh gudpdwlfdoo| glhuhqw dffruglqj wr zkhwkhu wkh upv* dfwlrqv duh vwudwhjlf frpsohphqwv ru vxevwlwxwhv1 65 Wklv

66 Wklv

frqfdyh1

vhfwlrq iroorzv forvho| Wluroh +4<;;= 53:053;,1 ehvw uhvsrqvh lv xqltxh/ gxh wr wkh dvvxpswlrq wkdw wkh surw ixqfwlrq lv vwulfwo|

5<

Vwdelolw| ri wkh htxloleulxp D surshuw| wkdw lv vrphwlphv uhtxluhg ri Qdvk htxloleuld lq roljrsro| prghov lv wkdw ri vwdelolw|1 Vwdelolw| ri wkh htxlole0 ulxp uhihuv wr wkh iroorzlqj txhvwlrq= zrxog wkh htxloleulxp eh uhdfkhg wkurxjk d vhtxhqfh ri pryhphqwv dorqj wkh uhdfwlrq ixqfwlrqv vwduwlqj iurp dq| du0 elwudu| srlqwB Iljxuh ;19 looxvwudwhv wkh dgmxvwphqw surfhvv lqyroyhg e| wklv wkrxjkw h{shulphqw/ xqghu wzr glhuhqw srvvlelolwlhv1 LQVHUW Iljxuh ;191 +d, Vwdeoh Qdvk Htxloleulxp> +e, Xqvwdeoh Qdvk Htxloleulxp Iljxuh ;19+d, looxvwudwhv d fdvh ri vwdeoh htxloleulxp +vxfk dv lq wkh Frxuqrw prgho dqdo|vhg lq wklv vhfwlrq,1 Vxssrvh iru lqvwdqfh wkdw up 4 surgxfhv d txdqwlw| t43 > up 5*v ehvw uhso| zrxog eh wr surgxfh wkh txdqwlw| t54 > lq wxuq/ wkh ehvw uhso| iru up 4 wr wkh txdqwlw| t54 zrxog eh wr surgxfh t45 > dqg vr rq/ xqwlo wkh upv hqg xs surgxflqj wkh txdqwlwlhv fruuhvsrqglqj wr wkh htxloleulxp sdlu dw srlqw H1 Lq Iljxuh ;19+e,/ lqvwhdg/ wkh htxloleulxp zrxog qrw eh vwdeoh/ dv pryhphqwv dorqj wkh uhdfwlrq ixqfwlrqv zrxog sxvk wkh upv ixuwkhu dqg ixuwkhu dzd| iurp wkh htxloleulxp srlqw H1 Ilup 5*v ehvw uhvsrqvh wr up 4 surgxflqj txdqwlw| t43 zrxog eh txdqwlw| t54 > lq wxuq/ wkh ehvw uhso| iru up 4 wr wkh txdqwlw| t54 zrxog eh wr surgxfh t45 > krzhyhu/ up 5*v ehvw uhso| zrxog dw wklv srlqw eh wr surgxfh qrwklqj1 Wkh htxloleulxp srlqw H fdqqrw eh uhdfkhg wkurxjk wklv dgmxvwphqw surfhvv dqg lv wkhuhiruh vdlg wr eh xqvwdeoh1 Dv rqh fdq vhh iurp wkhvh wzr h{dpsohv/ wkh vwdelolw| surshuw| lv uhodwhg wr wkh vorsh ri wkh uhdfwlrq ixqfwlrqv1 Lq d jdph zkhuh upv 4 dqg 5 fkrrvh dfwlrqv +d4 > d5 , dqg zkhuh uhdfwlrq ixqfwlrqv duh gudzq lq wkh sodqh +d4 > d5 ,/ iru lqvwdqfh/ lw pxvw eh wkdw U4 lv vwhhshu wkdq U5 1 E| uhfdoolqj wkh h{suhv0 vlrq ri wkh vorsh ri wkh uhdfwlrq ixqfwlrq/ wklv frqglwlrq fdq eh zulwwhq dv +C 5 4 @Cd54 ,+C 5  5 @Cd55 , A +C 5  4 @Cd4 Cd5 ,+C 5 5 @Cd5 Cd4 ,1 Lw lv lpsruwdqw wr vwuhvv wkdw wkh w|sh ri dgmxvwphqw surfhvv wkdw wkh vwdelolw| frqfhsw vxssrvhv lv yhu| sduwlfxodu1 Wkh vhtxhqfh ri pryhv ghvfulehg deryh pdnhv vhqvh rqo| li sod|huv duh frpsohwho| p|rslf/ dv hdfk up ljqruhv wkh lpsdfw wkdw lwv fkrlfh kdv rq wkh ulydo*v qh{w pryh167 Mrlqw0surw pd{lplvdwlrq Li upv pd{lplvh mrlqw surwv dqg fkrrvh txdq0 wlwlhv/ wkhlu sureohp lv pd{t4 >t5 / zkhuh  @ 4 .  5 @ +  t4  t5 ,t4 . +  t5  t4 ,t5 1 Iurp wkh IRFv g@gtl @ 3 lw iroorzv   5tl  5tm @ 31 Dw wkh v|pphwulf htxloleulxp/ t4 @ t5 1 Wkhuhiruh/ wkh IRFv vlpsoli| wr=  tP @ = +8<, 5+ . , E| vxevwlwxwlrq rqh rewdlqv htxloleulxp sulfhv dqg shu0up surwv dv= sP @ 67 Dowhuqdwlyho|/

 5 > P @ = 5 7+ . ,

+93,

rqh fdq wklqn wkdw hdfk sod|hu grhv orrn dkhdg exw dwwdfkhv qr lps ruwdqfh

wr wkh ixwxuh/ wkdw lv lw kdv d }hur glvfrxqw idfwru1

63

Lw fdq eh fkhfnhg wkdw xqghu sulfh frpshwlwlrq rqh jhwv h{dfwo| wkh vdph uhvxowv= sP @

  d d @ > tP @ @ = 5+e  j, 5 5 5+ . ,

+94,

D glhuhqwldwhg jrrgv prgho zlwk vrph qlfh surshuwlhv Wkh glhuhqwldwhg jrrgv prgho xvhg lq wkh suhylrxv vhfwlrq kdv wkh dgydq0 wdjh ri ehlqj yhu| hdv| wr ghdo zlwk1 Krzhyhu/ lwv h{whqvlrq iurp 5 wr q upv +wkdw L rplw khuh iru vlpsolflw|, lv qrw frpsohwho| vdwlvidfwru| ehfdxvh djjuhjdwh ghpdqg lqfuhdvhv zlwk wkh qxpehu ri upv +dv zhoo dv zlwk wkh ghjuhh ri vxe0 vwlwxwdelolw| dprqj surgxfwv,/ dw jlyhq sulfhv1 Iru wklv uhdvrq/ L xvxdoo| suhihu wr xvh dqrwkhu vlpsoh prgho/ gxh wr Vkxeln dqg Ohylwdq +4<;3,/ zkhuh pdunhw vl}h grhv qrw ydu| hlwkhu zlwk wkh ghjuhh ri vxevwlwxwdelolw| ru wkh qxpehu ri surgxfwv1 Wklv lv edvhg rq wkh iroorzlqj xwlolw| ixqfwlrq iru wkh glhuhqwldwhg surgxfwv=68 5 # q $5 6 q q [ [ q  [ 7 X +t4 > ==> tl > ==tq , @ y tl  tl5 . tl 8 > +95, 5+4 . , l@4 q l@4 l@4 zkhuh tl lv wkh txdqwlw| ri wkh l0wk surgxfw/ y lv d srvlwlyh sdudphwhu/ q lv wkh qxpehu ri surgxfwv lq wkh lqgxvwu|/  5 ^3> 4, uhsuhvhqwv wkh ghjuhh ri vxevwlwxwdelolw| ehwzhhq wkh q surgxfwv169 Iurp wkh pd{lplvdwlrq ri wkh xwlolw| ixqfwlrq vxemhfw wr wkh lqfrph frq0 vwudlqw/ rqh fdq zulwh CX+,@Ctl  sl @ 3/ uhvxowlqj lq wkh lqyhuvh ghpdqg ixqfwlrqv= 3 4 q [ 4 C qtl .  sl @ y  tm D = +96, 4. m @4 E| lqyhuwlqj wklv v|vwhp +vhh Dsshqgl{ iru ghwdlov, zh fdq qg wkh iroorzlqj gluhfw ghpdqg ixqfwlrqv= 5 6 q 47 [ 8 tl @ y  sl +4 . , . sm = +97, q q m@4 Wzr Sq qlfh surshuwlhv ri wklv ghpdqg ixqfwlrq duh= +4, djjuhjdwh ghpdqg T @ t grhv qrw ghshqg l@4Sl Sq rq wkh ghjuhh ri vxevwlwxwlrq dprqj wkh surgxfwv/ dv q T @ l@4 tl @ y q4 l@4 sl 1 +5, Lq wkh fdvh ri v|pphwu| sl @ sm @ s djjuhjdwh ghpdqg Sq grhv qrw fkdqjh zlwk wkh qxpehu ri surgxfwv q lq wkh lqgxvwu|/ dv T @ l@4 tl @ y  s1 68 Ri frxuvh/ dv de ryh/ frqvxphuv suhihuhqfhv fdq e h h{suhvvhg dv T ' L E^ c c ^ c ^  n + / ?   vr wkdw d sduwldo htxloleulxp dqdo|vlv lv ixoo| mxvwlhg1 69 D gudzedfn ri wklv xwlolw| ixqfwlrq lv lqvwhdg wkdw lw grhv qrw doorz wr ghdo zlwk erwk frpsohphqwv dqg vxevwlwxwhv1

64

Sulfh frpshwlwlrq Dvvxph wkdw doo wkh upv kdyh lghqwlfdo frvw ixqfwlrqv F+tl , @ ftl / zlwk f ? y1 Ilup l*v surwv duh jlyhq e|  l @ +sl f,tl +s4 > ==> sl > ==> sq ,/ zkhuh tl +, lv jlyhq e| +97,1 E| zulwlqj wkh IRFv C l @Csl @ 3 dqg lpsrvlqj v|pphwu| +sl @ sm @ s iru doo m 9@ l, rqh rewdlqv wkh Ehuwudqg htxloleulxp sulfhv dv= se @

+y . f +q . q  ,, = 5q . q  

+98,

Wkh txdqwlw| vrog dqg wkh surwv hduqhg e| hdfk up dw htxloleulxp duh irxqg e| vxevwlwxwlrq dv=6: +y  f, +q . q  , +y  f,5 +q . q  , > e @ = q +5q . q  , +5q . q  ,5

te @

+99,

Txdqwlw| frpshwlwlrq Wr vwxg| wkh fdvh ri txdqwlw| frpshwlwlrq/ zulwh up l*v surwv dv l @ ^sl +t4 > ==> tl > ==> tq ,  f` tl / zkhuh sl +, lv jlyhq e| +96,1 Iurp wkh IRFv Cl @Ctl @ 3 dqg lpsrvlqj v|pphwu| +tl @ tm @ t iru doo m 9@ l, rqh rewdlqv wkh Frxuqrw htxloleulxp sulfhv dv= +y  f, +4 . , = 5q . q . 

tf @

+9:,

Sulfhv dqg surwv dw wkh Frxuqrw htxloleulxp duh= sf @

y+q . , . fq +4 . , +y  f,5 +4 . , +q . , > f @ = 5 5q . q .  +5q . q . ,

+9;,

Wkh uhdghu fdq fkhfn wkdw 0 dv lq wkh suhylrxv prgho 0 txdqwlw| frpshwlwlrq uhvxowv lq kljkhu htxloleulxp sulfhv dqg surwv wkdq sulfh frpshwlwlrq= sf  se @

+y  f,+q  4,5 +y  f,5 +q  4,5 6 +5 . ,  3>  f  e @ > +5q . q . ,+5q . q  , +5q . q . ,5 +5q . q  ,5 +9<,

wkh wzr htxloleuld frlqflglqj zkhq  @ 31 Mrlqw0surw pd{lplvdwlrq Li upv S pd{lplvh mrlqw surwv dqg fkrrvh sulfhv/ wkhlu sureohp lv pd{s4 >=>sl >=>sq  @  l +s4 > => sl > => sq ,1 Iurp wkh IRFv g@gsl @ 3 dqg lpsrvlqj v|pphwu| rqh kdv= sP @

y.f = 5

+:3,

E| vxevwlwxwlrq rqh rewdlqv htxloleulxp sulfhv dqg shu0up surwv dv= tP @ 6: Qrwh

wkdw *4

<" RK

>

'

S

yf +y  f,5 > P @ = 5 7q

dqg *4

<" ZK

>

+:4,

' f= zkhq jrrgv e hfrph s huihfw vxevwlwxwhv/

wkh htxloleulxp whqgv wr wkh xvxdo Ehuwudqg fdvh zlwk krprjhqrxv j rrgv1

65

Uhshdwhg surgxfw pdunhw lqwhudfwlrq Vr idu/ zh kdyh dqdo|vhg rqo| rqh0vkrw jdphv1 Wklv vhfwlrq orrnv dw wkh fdvh zkhuh upv vwloo kdyh rqo| rqh vwudwhjlf yduldeoh +eh lw sulfh ru txdqwlw|, exw wkh| uhshdwhgo| lqwhudfw lq wkh surgxfw pdunhw1 Lq rwkhu zrugv/ L uhod{ dvvxpswlrq D5 ri wkh Ehuwudqg jdph1 Lw wxuqv rxw wkdw wkh uhvxow ri wkh prglhg jdph lv yhu| glhuhqw dffruglqj wr zkhwkhu wkh jdph lv uhshdwhg iru d qlwh ru lqqlwh qxpehu ri wlphv1 61515

Ilqlwh krul}rq

Frqvlghu wkh Ehuwudqg jdph ghvfulehg lq vhfwlrq 61414/ exw zlwk upv sod|lqj wkdw edvlf jdph iru W . 4 shulrgv/ wkdw lv iurp shulrg w @ 3 wr shulrg w @ W / zlwk W ehlqj d qlwh qxpehu1 HdfkSup l @ 4> 5 zdqwv wr pd{lplvh wkh suhvhqw glvfrxqwhg ydoxh ri lwv W surwv/ w@3  w l>w / zlwk  ehlqj wkh glvfrxqw idfwru6; dqg  l>w wkh surw hduqhg lq shulrg w1 L dp jrlqj wr vkrz wkh iroorzlqj uhvxow= +Uhshdwhg Ehuwudqg jdph/ zlwk qlwh krul}rq,1 Li W lv qlwh/ wkh rqo| vxe0 jdph shuihfw Qdvk htxloleulxp ri wkh jdph lv wkh Ehuwudqg htxloleulxp uhshdwhg W wlphv1 Surri1 Wklv lv d jdph ri doprvw shuihfw lqirupdwlrq wkdw fdq eh vroyhg e| edfnzdug lqgxfwlrq +vhh vhfwlrq 51714,1 Dw wkh odvw shulrg ri wkh jdph/ w @ W / zkdwhyhu kdsshqhg lq wkh suhylrxv shulrgv/ hyhu|wklqj lv dv li wkh wzr upv zhuh sod|lqj wkh rqh0vkrw Ehuwudqg jdph1 Wkhuhiruh/ wkh rqo| htxloleulxp ri wkh jdph zloo eh wkh rqh0vkrw Ehuwudqg sulfh s4>W @ s5>W @ f1 Dw shulrg w @ W 4/ sod|huv nqrz wkdw wkhlu fxuuhqw fkrlfhv zloo qrw dhfw wkh htxloleulxp vroxwlrq dw wkh iroorzlqj shulrg W 1 Wkhuhiruh/ zkdwhyhu kdsshqhg dw shulrgv 3> 4> ==> W  5/ wkh jdph wkh| duh sod|lqj dw W  4 lv hhfwlyho| wkh vdph dv li wkh| zhuh sod|lqj iru wkh odvw wlph/ dqg djdlq wkh rqo| htxloleulxp lv wkh Ehuwudqg htxloleulxp s4>W 4 @ s5>W 4 @ f1 Wkh vdph uhdvrqlqj fdq eh dssolhg wr doo suhylrxv shulrgv/ ohdglqj upv wr fkrrvh s4>w @ s5>w @ f dw dq| shulrg w1 Wklv vkrzv wkdw zkhq wkh Ehuwudqg jdph lv uhshdwhg d qlwh qxpehu ri wlphv/ lwv rxwfrph lv h{dfwo| wkh vdph dv wkh rqh0vkrw jdph/ zlwk upv vhwwlqj sulfhv dw pdujlqdo frvw dqg pdnlqj qr surwv dw dq| shulrg1 Wklv uhvxow/ krzhyhu/ krogv rqo| lqvridu dv upv sod| d frpsohwh lqirupdwlrq jdph1 Li upv kdg lqfrpsohwh lqirupdwlrq derxw wkhlu rssrqhqwv +wkdw lv/ li wkh| zhuh xqfhuwdlq derxw wkhlu sd|0rv,/ wkhq wkh htxloleulxp rxwfrph zrxog qrw eh wkh Ehuwudqg htxloleulxp uhshdwhg W wlphv1 Dowkrxjk L dp qrw jrlqj wr dqdo|vh wkh Ehuwudqg jdph xqghu lqfrpsohwh lqirupdwlrq/ wkh lqwhuhvwhg uhdghu fdq orrn dw wkh lqfrpsohwh lqirupdwlrq yhuvlrq ri wkh fkdlq0vwruh sdudgr{ jdph lq fkdswhu :/ wkdw vkduhv vlplodu ihdwxuhv16< 6; Wkh glvfrxqw idfwru phdvxuhv wkh lps ruwdqfh wkdw d sod|hu dwwdfkhv wr wkh ixwxuh= li B ' f/ wkhq rqo| fxuuhqw surwv pdwwhu> li B ' / surwv hduqhg dw dq| wlph lq wkh ixwxuh +krzhyhu glvwdqw, krog wkh vdph lpsruwdqfh dv fxuuhqw surwv1 Vlqfh B ' *E n o/ zkhuh o lv wkh lqwhuhvw udwh/ B ' f fruuhvsrqgv wr wkh fdvh zkhuh o / dqg B '  wr wkh fdvh o ' f1 6< Nuhsv hw do1 +4<;5, kdyh uvw irupdolvhg wkh uhshdwhg sulvrqhu glohppd jdph +wkdw lv yhu| vlplodu wr wkh Ehuwudqg jdph, xqghu lqfrpsohwh lqirupdwlrq1 Vhh dovr Wluroh +4<;;= 58;08<,1

<"

66

D yhu| glhuhqw uhvxow dulvhv dovr zkhq upv uhshdwhgo| sod| wkh Ehuwudqg jdph ryhu dq lqqlwh krul}rq/ dv L vkrz qh{w1 61516

Lqqlwh krul}rq

Wdnh qrz wkh qxpehu ri wlphv iru zklfk wkh sulfh frpshwlwlrq jdph lv ehlqj sod|hg wr eh lqqlwh173 Frqvlghu wkh iroorzlqj wuljjhu vwudwhjlhv1 Hdfk up vhwv wkh sulfh s dw wkh lqlwldo shulrg w @ 31 Lw vhwv sulfh s dw wlph w li erwk upv kdyh vhw s lq hyhu| shulrg ehiruh w1 Rwkhuzlvh/ hdfk up vhwv s @ f iruhyhu1 Lq rwkhu zrugv/ hdfk up ehkdyhv lq d frooxvlyh idvklrq dv orqj dv wkh ulydo grhv/ exw li rqh ri wkhp ghyldwhv iurp wkh frooxvlyh sulfh/ wkhq wkh sxqlvkphqw lv wuljjhuhg dqg wkh| erwk uhyhuw wr wkh rqh0vkrw Ehuwudqg htxloleulxp iru wkh uhvw ri wkh jdph1 Wkhvh wuljjhu vwudwhjlhv irup dq htxloleulxp li hdfk up*v lqfhqwlyh frpsdw0 lelolw| frqvwudlqw krogv=74 +s, +4 .  .  5 .  6 . ===,  +s,= 5

+:5,

Wkh OKV jlyhv wkh suhvhqw glvfrxqwhg ydoxh ri wkh surwv d up uhfhlyhv li lw frooxghv +l1h1/ li lw iroorzv wkh wuljjhu vwudwhj| zkhq wkh rwkhu up grhv,1 Lq hdfk shulrg/ wkh up uhfhlyhv kdoi ri wkh djjuhjdwh prqrsro| surw1 Wkh UKV jlyhv wkh suhvhqw glvfrxqwhg ydoxh ri wkh surwv li wkh up +rswlpdoo|, ghyldwhv iurp wkh wuljjhu vwudwhj|1 E| vhwwlqj s  %/ doo frqvxphuv zloo ex| iurp wkh ghyldwlqj up/ wkdw zloo wkxv hduq d surw +s  %,1 Iru % $ 3/ wklv zloo wkhuhiruh hqvxuh surwv yhu| forvh wr +s, lq wkh shulrg wkh ghyldwlrq wdnhv sodfh1 Lq wkh iroorzlqj shulrg/ krzhyhu/ wkh sxqlvkphqw rffxuv dv erwk upv uhyhuw wr wkh Qdvk htxloleulxp iruhyhu1 Wkhuhiruh/ wkh ghyldwlqj up +dv zhoo dv wkh ulydo, zloo S pdnh }hur surw lq doo iroorzlqj shulrgv ri wkh jdph1  w @ 4@+4  ,1 Khqfh/ diwhu vlpsoh dojheud/ wkh lqfhqwlyh Qrwh wkdw 4 w@3 frqvwudlqw deryh ehfrphv= 

4 > 5

+:6,

wkdw lv/ wkh sulfh s 5 ^f> sp ` fdq eh vxvwdlqhg dw wkh htxloleulxp surylghg wkdw wkh glvfrxqw idfwru lv kljk hqrxjk1 Wkhuh lv olwwoh srlqw lq hoderudwlqj ixuwkhu khuh/ vlqfh wklv lv glvfxvvhg dw ohqjwk lq fkdswhu 71 Wkh pdlq sxusrvh ri wklv vhfwlrq lv mxvw wr vkrz wkdw sulfhv deryh pdujlqdo frvwv pljkw eh vxvwdlqhg dv wkh htxloleulxp ri d jdph zlwk lqqlwh krul}rq +ru xqfhuwdlq qdo gdwh,1 Qrwh/ krzhyhu/ wkdw dq lpsruwdqw lvvxh lv wkdw ri wkh pxowlsolflw| ri htxloleuld dulvlqj lq wklv jdph1 Dq| sulfh 73 Htxlydohqwo|/ vxssrvh wkdw wkh jdph kdv dq xqfhuwdlq qdo gdwh/ zlwk wkh suredelolw| wkdw wkh pdunhw h{lvwv lq wkh iroorzlqj shulrg ehlqj  M Efc 1 Fdoo wkh upv* glvfrxqw idfwru xqghu wklv dowhuqdwlyh lqwhusuhwdwlrq _1 Rqh fdq wkhq vhw B ' _/ dqg fduu| rxw wkh dqdo|vlv dv lq wkh wh{w1 74 Wkh frqvwudlqw lv wkh vdph iru erwk upv/ gxh wr wkhlu v|pphwu|1 Vhh fkdswhu 7 iru frooxvlrq xqghu dv|pphwulhv1

67

s 5 ^f> sp `/ wkdw lv ehwzhhq pdujlqdo frvw dqg prqrsro| sulfh/ pljkw eh wkh htxloleulxp ri wkh jdph +iru kljk d hqrxjk glvfrxqw idfwru,1 Wklv ohdgv wr lpsruwdqw srolf| lpsolfdwlrqv wkdw duh glvfxvvhg lq fkdswhu 71

7

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68

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+:7,

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69

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6:

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+:8,

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Zlwk ihz vwhsv ri dojheud rqh fdq wkhq vlpsoli| wkh h{suhvvlrq t @ +4 . ,D4 +s  y,dqg fkhfn wkdw lw fruuhvsrqgv wr wkh v|vwhp ri gluhfw ghpdqg ixqfwlrqv lq wkh wh{w1

73

Table 8.1. A simple game B

b1

b2

b3

a1

2, 0

2, 5

1, 1

a2

0, 2

0, 3

2, 2

A

Table 8.2. The prisoners' dilemma game B

High

Low

High

10, 10

5, 15

Low

15, 5

6, 6

A

Table 8.3. The battle of the sexes game B

Indian

Thai

Indian

3, 2

0, -1

Thai

-1, 0

2, 3

A

Table 8.4. A pure coordination game B

Indian

Italian

Indian

2, 2

0, 0

Italian

0, 0

1, 1

A

Table 8.5. An asymmetric game B

pL

pH

pL

0, 0

0, 0

pH

0, -2

2, 0

A

Table 8.6. The matching pennies game B

Heads

Tails

Heads

-1, 1

1, -1

Tails

1, -1

-1, 1

A

Table 8.7. The entry deterrence game I

Accommodate

Fight

Enter

4, 5

-1, 0

Stay out

0, 10

0, 10

E

Table 8.8. Strategic effect of a shock that reduces firm 1's costs

Strategic substitutes

Strategic complements

Accommodation: π 1 ↑; π 2 ↓

Accommodation: π 1 ↓; π 2 ↓

Entry deterrence: π2 ↓ (⇒ π 1 ↑)

Entry deterrence: π2 ↓ (⇒ π 1 ↑)

Table 8.9. Strategic investments to reduce firm 1's costs

Strategic substitutes

Strategic complements

Accommodation: over-invest

Accommodation: under-invest

Entry deterrence: over-invest

Entry deterrence: over-invest

E stay out

enter

I 0   10 

fight

accommodate

 − 1   0 

Figure 8.1. Extensive form of the entry deterrence game

 4   5 

E enter

stay out

I

I low  0    2

high 0    4

high

low E low

E high low

0   0

Figure 8.2. Extensive form of a quality game

 1    3

 3   1 

high  −1     1 

q2 R1

π1 = k π1 = k ′ > k C

π2 = z

π 2 = z′ < z R2

q1

Figure 8.3. Reaction functions in the Cournot model

p

. . . . . . . M

pM =

. •

C

.

•

. . . . . . B

•

•

. . . . . . . 1

2

3

4

5

6

7

1+ c 2

pC = c + pB = c

1− c n −1

n

Figure 8.4. Toughness of product market competition: Bertrand (B), Cournot (C), and joint-profits maximization (M)

p2

π1 = k ′ > k

R1

π1 = k π 2 = z′ > z π2 = z

R2

B

p1

Figure 8.5. Reaction functions under price competition

q2

q2 R1

R2

E

E

q23 1 2

q12

R2

q

q14 q12 (a)

q10

q1

R1 q10 q12 (b)

Figure 8.6. (a) Stable Nash Equilibrium; (b) Unstable Nash Equilibrium

q1

q2

R1′( s > 0) R1

q2E q

E′ 2

π1 π2

E

•

D π 1 D • E′ π 1′

•

π 2′

R2

q1

Figure 8.7. Effects of a shock (s > 0) that reduces firm 1’s marginal cost: Strategic substitutes

p2

R1′( s > 0) R1

π 1D p2E p

E′ 2

π 1′ •

R2

π1

D•

•

E′

π2

E

π 2′ p1

Figure 8.8. Effects of a shock (s > 0) that reduces firm 1’s marginal cost: Strategic complements

Index of Figures Figure 2.1. Welfare loss from monopoly Figure 2.2. Possible additional loss from rent seeking activities Figure 2.3. Additional loss from productive inefficiency Figure 2.4. Welfare Wc as a function of the number of firms n (a = 1; c = .5; g = 4) Figure 4.1. Incentive Constraints along collusive and punishment paths Figure 4.2. Conditions for collusion: Nash reversal (δ cn ) v. two-phase (δ )

punishment strategies Figure 4.3. Game tree, at t = 2 Figure 4.4. Equilibrium solutions for given policy parameters Figure 5.1. Effects of a merger absent efficiency gains: Strategic complements Figure 5.2. Effects of a merger absent efficiency gains: Strategic substitutes Figure 5.3. Effects of a merger with efficiency gains Figure 5.4. Levels of efficiency gains and exclusion of competitors Figure 5.5. Study of functions used in Lemma 7 Figure 6.1. Double marginalisation Figure 6.2. Horizontal externality Figure 6.3. Competing vertical chains Figure 6.4. Tariffs as strategic device: Strategic complements Figure 6.5. Tariffs as strategic device: Strategic substitutes Figure 6.6. The "Chicago School" critique to foreclosure

Figure 6.7. Segal-Whinston: sequential offers Figure 7.1. State game at time t, chain-store paradox game Figure 7.2. Deep pocket predation, with T = K = 1 Figure 7.3. Time line: a (financial) long purse model of predation Figure 7.4. Time line: Bolton-Scharfstein's model Figure 7.5. Entry-deterring investment Figure 7.6. Strategic effect of bundling Figure 7.7. Welfare under perfect price competition Figure 8.1. Extensive form of the game Figure 8.2. Extensive form of a quality game Figure 8.3. Reaction functions in the Cournot model Figure 8.4. Toughness of product market competition: Bertrand (B), Cournot (C), and joint-profits maximization (M) Figure 8.5. Reaction functions under price competition Figure 8.6. (a) Stable Nash Equilibrium; (b) Unstable Nash Equilibrium Figure 8.7. Effects of a shock (s > 0) that reduces firm 1's marginal cost: Strategic substitutes Figure 8.8. Effects of a shock (s > 0) that reduces firm 1's marginal cost: Strategic complements

p O

R

pm

pc

T

S

c

O′ qm

qc

q

p O

R

pm

pc

T

S

c

O′ qm

qc

q

p O R′

p′m pm

pc′ pc

R

W

V Z T′ T qm′ qm

qc′

c′ S

c O′

qc

q

0.126 0.124 0.122 0.12 0.118 0.116 0.114 0.112

2

4

n

6

8

10

δ P,δ c

δ P,δ c

1

δP

1

δP

0.5

0.5

δ 0.2

q~ P 1 − c 1 n −1 n (a)

q

P

δc

c

1− c n −1

1 q~ P n

0.2

qP (b)

δ,δ cn

δ cn

1P

δ

0.5

5

10

n

AA Investigation f1

Reveal

æ ΠN ö −R÷ ç ç 1Π− δ ÷ ç N −R÷ è1 − δ ø

a

Reveal

Not Reveal

f2

f2 Not Reveal Reveal

1-a

No Investigation æç Π M ö÷ è ΠM ø

Not Reveal

æ ΠN öæ Π ö Guilty − R ÷ç N − F ÷ ç p ç 1Π− δ ÷ ç 1Π− δ ÷ N N ç ÷ ç ÷ −F −R è1 − δ ø è1 − δ ø æ ΠN ö −F ÷ ç − 1 δ çΠ ÷ ç N −F ÷ è1 − δ ø

AA Not Guilty 1-p æ ΠM ç ç 1Π− δ ç M è1 − δ

ö ÷ ÷ ÷ ø

1

a NC

NC

aCR (1)

a ( 2)

CNR

CR ~ p

0

p

1

RO

p1

RI′ M

pI

pB

RI

B

pB

pO

p3

q1 RO

C

qc qI

π1

M

π 1′ qc qO

RI′

RI q3

RO

p1

RI′ (e = 1) M

pI

RI

RIe B

pB E

pB

pO

p3

e

1 eex WM < Wb WM > Wb

eW

150

γ

∆Ib , ∆bo

∆Ib (e)

∆bo (e)

e

1

e

Upstream firm (manufacturer)

Downstream firm (retailer)

Consumers

Upstream firm (manufacturer)

Downstream firm (retailer)

Downstream firm (retailer)

Consumers

Upstream firm (manufacturer)

Upstream firm (manufacturer)

Downstream firm (retailer)

Downstream firm (retailer)

Consumers

R1 ( w1 = c) R1′ ( w1′ > c)

p2

R2′ ( w2′ > c ) E′

R2 ( w2 = c )

E

p1

q2

R1′ ( w1′ < c)

R1 ( w1 = c )

E′ E R2′ ( w2′ < c) R2 ( w2 = c) q1

p

θ pm

c1 cE

A

B D F qm

E qe

θ′

q

B1 Accept

Reject B2

B2 Accept æ CS m + t1 ö ç m ÷ è CS + t 2 ø

Reject æ CS m + t1 ö ç m ÷ è CS ø

Accept æ CS m ö ç m ÷ è CS + t 2 ø

Reject

æ CS e ö ç e÷ è CS ø

Entrant t stay out

(0, π M )

enter Incumbent fight

accommodate

(π EP , π IP )

(π EA , π IA )

E enter

out

I æ π M + δπ M ö ç ÷ 0 è ø

prey E

E

out

stay

out

I

æ π P + δπ M ö ç P ÷ π < 0 è ø

prey æ π A + δπ P ö ç A P ÷ π + δπ è ø

æ π A + δπ M ö ç ÷ A π è ø

accommodate æ π A + δπ A ö ç A A÷ π + δπ è ø

1 •

•

prey or 1st period accommodate payoff

2 •

pay F or out

3 •

•

effort 2nd period decisions payoff

time

i

ii

contract

effort decisions

•

•

1 •

•

prey or 1st period accommodate payoff

2 •

pay F or out

3 •

•

effort 2nd period decisions payoff

time

q2

R1 ( x1P )

R1 ( x1 = 0)

q2N ( x1 = 0)

.

π 2 ( x1 = 0)

.

π 2 ( x1P ) = F

q1N ( x1 = 0)

q1

p B1

RB 2 RB1

.

E

t2 + cB1 2 t2 + cB1 − (v − c A ) 2

~ R1

.

~ E

pB2

p O

R

pm

pc

T

S

c

O′ qm

qc

q

E stay out

enter

I fight æç 0 ö÷ è10 ø

I accommodate æç 0 ö÷ è10 ø

fight æç − 1ö÷ è0 ø

accommodate

4 çæ 5 ö÷ è ø

E enter

stay out

I

I low æç 0 ö÷ è2ø

high æç 0 ö÷ è4ø

high

low E low æç 0 ö÷ è0ø

E high low æç 1 ö÷ è 3ø

æç 3 ö÷ è1 ø

high æç − 1 ö÷ è1 ø

q2 R1

π1 = k π1 = k ′ > k C

π2 = z

π 2 = z′ < z R2

q1

p

. . M. . . . .

pM =

. •

C

.

•

. .B . . . . •

•

. . . . . . . 1

2

3

4

5

6

7

1+ c 2

pC = c + pB = c n

1− c n −1

p2

π1 = k ′ > k

R1

π1 = k π 2 = z′ > z π2 = z

R2

B

p1

q2

q2 R1

R2

E

E

q23 1 2

q12

R2

q

q14 q12 (a)

q10

q1

R1 q10 q12 (b)

q1

q2

R1′( s > 0) R1

q2E q2E

′

π1 π2

E

•

D π 1 D • E′ π 1′

•

π 2′

R2

q1

p2

R1′( s > 0) R1

π 1D p2E p

E′ 2

π 1′

D• • ′ E

R2

π1 •

E

π2

π 2′ p1

Index of tables Table 2.1. The entry game Table 2.2. Consumers 1 and 2's payoffs under price discrimination Table 4.1. The revelation game Table 5.1 Number of final decisions on mergers taken by the EC Commission Table 5.2. Product market definition in the rail technology industry Table 5.3. Market shares and concentration indexes in the rail technology industry Table 6.1. Segal - Whinston: simultaneous offers Table 6.2. Segal - Whinston: discriminatory offers Table 7.1. Consumers valuations of goods Table 8.1. A simple game Table 8.2. The prisoners' dilemma game Table 8.3. The battle of the sexes game Table 8.4. A pure coordination game Table 8.5. An asymmetric game Table 8.6. The matching pennies game Table 8.7. The entry deterrence game Table 8.8. Strategic effect of a shock that reduces firm 1's costs Table 8.9. Strategic investments to reduce firm 1's costs

Table 2.1. The entry game 2

E

NE

E

- f, - f

ΠM -f, 0

NE

0, ΠM -f

0, 0

1

Table 2.2. Consumers 1 and 2's payoffs under price discrimination 2

buy I

buy E

buy I

v(2) - p1, 0

v(2) - p1, -cE

buy E

-cE, 0

v(2) - cE, v(2) - cE

1

Table 4.1. The revelation game

firm 2

Reveal

Not Reveal

Reveal

ΠN − R, 1− δ ΠN −R 1−δ

ΠN − R, 1− δ ΠN −F 1−δ

Not Reveal

ΠN − F, 1−δ ΠN −R 1−δ

Π Π p( N − F ) + (1 − p) M , 1− δ 1− δ Π Π p( N − F ) + (1 − p) M 1−δ 1−δ

firm 1

Table 5.1. Number of final decisions on mergers taken by the EC Commission Final Decisions (R.4064/89)

Final Decisions (Art.66 ECSC Treaty, 1992-1999)

300

9 10

250

200 10 7

150

270

7 100

238

12

26 10

6

50

91 60

60

57

1991

1992

1993

109

125

142

0 1994

1995

1996

1997

1998

1999

Source. Report on Competition Policy, European Commission, DG IV (various years)

Table 5.2. Product market definition in the rail technology industry Electrical locomotives Diesel locomotives Mainline trains

Train sets for mainline transportation Passenger coaches Freight wagons

Rolling stock Regional trains

Electrical multiple units Diesel multiple units Trams (light rail and trolleys)

Local trains and systems

Metro vehicles Automated guided transportation Catenary systems

Wayside systems Stationary equipment

Traction power supply Train control and protection systems

Miscellaneous

Maintenance and refurbishment Information systems and ticketing

Table 5.3. Market shares and concentration indexes in the rail technology industry Product Market

Market Shares (%)

HHI

∆ HHI

ABB

DB

Siemens

DWA

LHB

Elpro

Electrical locom.

37

17

46

np

np

Np

5032

1258

Mainline train sets

5

26

46

18

np

Np

>3401

260

Reg. electr. m/units

18

26

25

17

14

Np

>3046

936

Reg. diesel m/units

0

49

23

np

19

Np

>3291

0

Trams

15

29

41

np

8

Np

>3681

870

Metro vehicles

42

22

19

np

11

Np

>4578

1848

Catenary systems

30

31

33

np

np

6

4846

1860

Traction power supply

6

26

35

np

np

13

>2418

312

Table 6.1. Segal-Whinston: simultaneous offers B2

Accept

Reject

Accept

CSm+t, CSm+t

CSm+t, CSm

Reject

CSm, CSm+t

CSe, CSe

B1

Table 6.2. Segal-Whinston: discriminatory offers B2

Accept

Reject

Accept

CSm+t1, CSm+t2

CSm+t1, CSm

Reject

CSm, CSm+t2

CSe, CSe

B1

Table 7.1. Consumer valuations of goods 1's willingness to pay

2's willingness to pay

Good A

7

4

Good B

5

8

Goods A and B

12

12

Table 8.1. A simple game B

b1

b2

b3

=1

2, 0

2, 5

1, 1

=2

0, 2

0, 3

2, 2

A

Table 8.2. The prisoners' dilemma game B

High

Low

High

10, 10

5, 15

Low

15, 5

6, 6

A

Table 8.3. The battle of the sexes game B

Indian

Thai

Indian

3, 2

0, -1

Thai

-1, 0

2, 3

A

Table 8.4. A pure coordination game B

Indian

Italian

Indian

2, 2

0, 0

Italian

0, 0

1, 1

A

Table 8.5. An asymmetric game B

pL

pH

pL

0, 0

0, 0

pH

0, -2

2, 0

A

Table 8.6. The matching pennies game B

Heads

Tails

Heads

-1, 1

1, -1

Tails

1, -1

-1, 1

A

Table 8.7. The entry deterrence game I

Accommodate

Fight

Enter

4, 5

-1, 0

Stay out

0, 10

0, 10

E

Table 8.8. Strategic effect of a shock that reduces firm 1's costs

Strategic substitutes

Strategic complements

Accommodation: π1 ↑; π2 ↓

Accommodation: π1 ↓; π2 ↓

Entry deterrence: π2 ↓ (Þ π1 ↑)

Entry deterrence: π2 ↓ (Þ π1 ↑)

Table 8.9. Strategic investments to reduce firm 1's costs

Strategic substitutes

Strategic complements

Accommodation: over-invest

Accommodation: under-invest

Entry deterrence: over-invest

Entry deterrence: over-invest

Uhihuhqfhv ^4` Deuhx/ G1 4<;91 H{wuhpdo Htxloleuld ri Roljrsrolvwlf Vxshujdphv1 Mrxu0 qdo ri Hfrqrplf Wkhru|1 6<= 4<45581 ^5` Deuhx/ G1 4<;;1 Rq wkh Wkhru| ri Lqqlwho| Uhshdwhg Jdphv zlwk Glv0 frxqwlqj1 Hfrqrphwulfd1 89= 6;60<91 ^6` Deuhx/ G1/ G1S Shdufh dqg H1 Vwdffkhwwl1 4<;91Rswlpdo Fduwho Htxloleuld zlwk Prqlwrulqj1 Mrxuqdo ri Hfrqrplf Wkhru|1 6<= 58459<1 ^7` Dgdpv/ Z1M1 dqg M1O1 \hoohq1 4<:91 Frpprglw| Exqgolqj dqg wkh Exu0 ghq ri Prqrsro|1 Txduwhuo| Mrxuqdo ri Hfrqrplfv1 <3= 7:87<;1 ^8` Djklrq/ S1/ Q1 Eorrp/ U1 Eoxqghoo/ U1 Jul!wk/ dqg S1 Krzlww1 53351 Frp0 shwlwlrq dqg Lqqrydwlrq= Dq Lqyhuwhg X Uhodwlrqvkls1 Fdpeulgjh/ PD1= Qdwlrqdo Exuhdx ri Hfrqrplf Uhvhdufk Zrunlqj Sdshu <59<1 ^9` Djklrq/ S1 dqg S1 Erowrq1 4<;:1 Frqwudfwv dv d Eduulhu wr Hqwu|1 Dphu0 lfdq Hfrqrplf Uhylhz1 ::+6,= 6;;7341 ^:` Djklrq/ S1/ P1 Ghzdwulsrqw/ dqg S1 Uh|1 4<<;1 Djhqf| Frvwv/ Ilup Eh0 kdylru dqg wkh Qdwxuh ri Frpshwlwlrq1 Wrxorxvh= LGHL1 Xqsxeolvkhg Pdqxvfulsw1 ^;` Djklrq/ S1/ P1 Ghzdwulsrqw/ dqg S1 Uh|1 4<<<1 Frpshwlwlrq/ Ilqdqfldo Glvflsolqh dqg Jurzwk1 Uhylhz ri Hfrqrplf Vwxglhv1 99= ;580;851 ^<` Dkoeruq/ F1/ G1V1 Hydqv dqg D1M1 Sdglood1 53341 Frpshwlwlrq Srolf| rq Lqwhuqhw Wlph= Lv Hxurshdq Frpshwlwlrq Odz Xs wr wkh FkdoohqjhB1 Hxurshdq Frpshwlwlrq Odz Uhylhz1 55= 489049:1 ^43` Dpdwr/ J1 4<<:1 Dqwlwuxvw dqg wkh Erxqgv ri Srzhu= Wkh Glohppd ri Olehudo Ghprfudf| lq wkh Klvwru| ri wkh Pdunhw1 R{irug= Kduw1 ^44` Dqghuvrq/ V1S1/ D1 gh Sdopd dqg M10I1 Wklvvh1 4<<51 Glvfuhwh Fkrlfh Wkh0 ru| ri Surgxfw Glhuhqwldwlrq1 Fdpeulgjh/ PD= PLW Suhvv1 ^45` Duhhgd/ S1H1 dqg G1I1 Wxuqhu1 4<:71 Suhgdwru| Sulflqj dqg Uhodwhg Sudf0 wlfhv xqghu Vhfwlrq 5 ri wkh Vkhupdq Dfw1 Kduydug Odz Uhylhz1 ;;= 9<: :661 ^46` Duurz/ N1 4<941 Hfrqrplf Zhoiduh dqg wkh Doorfdwlrq ri Uhvhdufk iru Lqyhqwlrq1 lq U1 Qhovrq1hg1 Wkh Udwh dqg Gluhfwlrq ri Lqyhqwlyh Dfwlylw|= Hfrqrplf dqg Vrfldo Idfwruv1 Sulqfhwrq Xqlyhuvlw| Suhvv1 ^47` g*Dvsuhprqw/ F1 dqg D1 Mdftxhplq1 4<;;1 Frrshudwlyh dqg Qrqfrrshu0 dwlyh U)G lq Gxrsro| zlwk Vslooryhuv1 Dphulfdq Hfrqrplf Uhylhz1 :;= 4466446:1

4

^48` g*Dvsuhprqw/ F1 dqg P1 Prwwd1 53331 Wrxjkhu Frpshwlwlrq ru Orzhu Frqfhqwudwlrq= D WudghR iru Dqwlwuxvw DxwkrulwlhvB lq Qrupdq J1 dqg M1 I1 Wklvvh1 hgv1 Pdunhw Vwuxfwxuh dqg Frpshwlwlrq Srolf|= Jdph Wkhruhwlf Dssurdfkhv1 Fdpeulgjh= Fdpeulgjh Xqlyhuvlw| Suhvv1 ^49` Dwkh|/ V1 dqg N1 Edjzhoo1 53341 Rswlpdo Frooxvlrq zlwk Sulydwh Lqiru0 pdwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 65= 75;07981 ^4:` Edlo|/ P1Q1/ F1 Kxowhq dqg G1 Fdpsehoo1 4<<51 Surgxfwlylw| G|qdplfv lq Pdqxidfwxulqj Sodqwv1 Eurrnlqjv Sdshuv rq Hfrqrplf Dfwlylw|1 Pl0 furhfrqrplfv1 4;:59:1 ^4;` Edlo|/ P1Q1 dqg K1 Jhuvedfk1 4<<81 H!flhqf| lq Pdqxidfwxulqj dqg wkh Qhhg iru Joredo Frpshwlwlrq1 Eurrnlqjv Sdshuv rq Hfrqrplf Dfwlylw|1 Plfurhfrqrplfv1 63:07:1 ^4<` Edlq/ M1 4<7<1 D Qrwh rq Sulflqj lq Prqrsro| dqg Roljrsro|1 Dphulfdq Hfrqrplf Uhylhz1 6<= 77;7971 ^53` Ednhu/ M1E1 dqg W1I1 Euhvqdkdq1 4<;81 Wkh Jdlqv iurp Phujhu ru Fro0 oxvlrq lq Surgxfw Glhuhqwldwhg Lqgxvwulhv1 Mrxuqdo ri Lqgxvwuldo Hfr0 qrplfv1 66= 75:7771 ^54` Ednhu/ M1E1 dqg W1I1 Euhvqdkdq1 4<;;1 Hvwlpdwlqj wkh Uhvlgxdo Ghpdqg Fxuyh Idflqj d Vlqjoh Ilup1 Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql0 }dwlrq1 9= 5;66331 ^55` Eduod/ S1 53331 Ilup Vl}h Lqhtxdolw| dqg Pdunhw Srzhu1 Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq1 4;= 9<60:551 ^56` Eduqhv/ P1 dqg M1 Kdvnho1 53331 Surgxfwlylw| lq wkh 4<<3v= Hylghqfh iurp Eulwlvk sodqwv1 Txhhq Pdu| dqg Zhvwhog Froohjh1 Xqsxeolvkhg pdqxvfulsw +dydlodeoh dw= kwws= 22zzz1tpz1df1xn2xjwh486,1 ^57` Eduurv/ S1S1 53341 Orrnlqj Ehklqg wkh Fxuwdlq1 Hhfwv iurp Prghuq0 l}dwlrq ri Hxurshdq Frpshwlwlrq Srolf|1 Xqlyhuvlgdgh Qryd gh Olverd1 Xqsxeolvkhg Pdqxvfulsw1 ^58` Edxpro/ Z1 4<<91 Suhgdwlrq dqg wkh Orjlf ri wkh Dyhudjh Yduldeoh Frvw Whvw1 Mrxuqdo ri Odz dqg Hfrqrplfv1 6<= 7<0:51 ^59` Edxpro/ Z1/ M1 Sdq}du dqg U1 Zloolj1 4<;51 Frqwhvwdeoh Pdunhwv dqg wkh Wkhru| ri Lqgxvwuldo Vwuxfwxuh1 Qhz \run= Kdufrxuw/ Eudfh/ Mrydqrylfk1 ^5:` Ehjjv/ D1 dqg S1 Nohpshuhu1 4<<51 PxowlShulrg Frpshwlwlrq zlwk Vzlwfklqj Frvwv1 Hfrqrphwulfd1 93= 98499:1 ^5;` Ehoodp|/ F1 dqg J1 Fklog1 53341 Hxurshdq Frppxqlw| Odz ri Frpshwl0 wlrq1 Orqgrq= Vzhhw ) Pd{zhoo1

5

^5<` Ehqrlw/ M1S1 4<;71 Ilqdqfldoo| Frqvwudlqhg Hqwu| lq d Jdph zlwk Lqfrp0 sohwh Lqirupdwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 48=7<37<<1 ^63` Ehqrlw/ M10S1 dqg Y1 Nulvkqd1 4<;:1 G|qdplf Gxrsro|= Sulfhv dqg Txdq0 wlwlhv1 Uhylhz ri Hfrqrplf Vwxglhv1 87= 56681 ^64` Ehujëv I1/ I1 Orvv/ H1 Pdodyrowl/ dqg W1 Yhujì1 53351 Frpshwlwlrq Srolf| dqg Djuhhphqwv Ehwzhhq Ilupv1 Wrxorxvh= LGHL1 Xqsxeolvkhg Pdqxvfulsw1 ^65` Ehukqhlp/ E1G1 dqg P1G1 Zklqvwrq1 4<;81 Frpprq Pdunhwlqj Djhqf| dv d Ghylfh iru Idflolwdwlqj Frooxvlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 49= 59<0;41 ^66` Ehuqkhlp/ E1 G1 dqg P1 G1 Zklqvwrq1 4<<31 Pxowlpdunhw Frqwdfw dqg Frooxvlyh Ehkdylru1 Udqg Mrxuqdo ri Hfrqrplfv1 54= 40591 ^67` Ehuqkhlp E1 G1 dqg P1 G1 Zklqvwrq1 4<<;1 H{foxvlyh Ghdolqj1 Mrxuqdo ri Srolwlfdo Hfrqrp|1 439= 974361 ^68` Ehuu|/ V1W1 4<<71 Hvwlpdwlqj Glvfuhwh Fkrlfh Prghov ri Surgxfw Glhu0 hqwldwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 58= 57505951 ^69` Ehuu|/ V1W1/ M1 Ohylqvrkq dqg D1 Sdnhv1 4<<81 Dxwrpreloh Sulfhv lq Pdu0 nhw Htxloleulxp1 Hfrqrphwulfd1 96= ;740;<31 ^6:` Ehvdqnr/ E1 dqg P1N1 Shuu|1 4<<41 Uhvdoh Sulfh Pdlqwhqdqfh dqg Pdq0 xidfwxuhu Frpshwlwlrq iru H{foxvlyh Ghdohuvklsv1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 6<= 84:8771 ^6;` Ehvdqnr/ E1 dqg P1N1 Shuu|1 4<<61 Htxloleulxp Lqfhqwlyhv iru H{foxvlyh Ghdolqj lq d Glhuhqwldwhg Surgxfwv Roljrsro|1 Udqg Mrxuqdo ri Hfr0 qrplfv1 57= 97909:1 ^6<` Ehvdqnr/ G1 dqg G1 I1 Vsxoehu1 4<;<1 Dqwlwuxvw Hqirufhphqw xqghu Dv|pphwulf Lqirupdwlrq1 Hfrqrplf Mrxuqdo1 <<= 73;7581 ^73` Ehvdqnr/ G1 dqg G1 I1 Vsxoehu1 4<<61 Frqwhvwhg Phujhuv dqg Htxloleulxp Dqwlwuxvw Srolf|1 Mrxuqdo ri Odz/ Hfrqrplfv/ dqg Rujdql}dwlrq1 <= 405<1 ^74` Ehvhq/ V1P1 dqg M1 Iduuhoo1 4<<71 Fkrrvlqj Krz wr Frpshwh= Vwudwhjlhv dqg Wdfwlfv lq Vwdqgdugl}dwlrq1 Mrxuqdo ri Hfrqrplf Shuvshfwlyhv1 ;= 44:4641 ^75` Elvkrs/ V1E1 4<<<1 Srzhu dqg Uhvsrqvlelolw|= Wkh HFM*v NdolVdo} Mxgj0 phqw1 Hxurshdq Frpshwlwlrq Odz Uhylhz1 4= 6;6<1 ^76` Elwwolqjpd|hu/ J1 4<;81 Glg Dqwlwuxvw Srolf| Fdxvh wkh Juhdw Phujhu ZdyhB Mrxuqdo ri Odz dqg Hfrqrplfv1 5;= ::44;1

6

^77` Erowrq/ J1 dqg G1 Vfkduivwhlq1 4<<31 D Wkhru| ri Suhgdwlrq Edvhg rq Djhqf| Sureohpv lq Ilqdqfldo Frqwudfwlqj1 Dphulfdq Hfrqrplf Uhylhz1 ;3= <64391 ^78` Erowrq/ S1/ M1 Eurgoh|/ dqg P1 Ulrugdq1 53331 Suhgdwru| Sulflqj= Vwudwh0 jlf Wkhru| dqg Ohjdo Srolf|1 Jhrujhwrzq Odz Mrxuqdo1 ;;= 556<056631 ^79` Erqdqqr/ J1 dqg M1 Ylfnhuv1 4<;;1 Yhuwlfdo Vhsdudwlrq1 Mrxuqdo ri Lq0 gxvwuldo Hfrqrplfv1 69= 58:0981 ^7:` Errqh/ M1 53331 Frpshwlwlyh Suhvvxuh= Wkh Hhfwv rq Lqyhvwphqwv lq Surgxfw dqg Surfhvv Lqqrydwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 64= 87<0 89<1 ^7;` Eruhqvwhlq/ V1 4<<<1 Udslg Sulfh Frppxqlfdwlrq dqg Frruglqdwlrq= Wkh Dluolqh Wdul Sxeolvklqj Fdvh +4<<7,1 lq Nzrnd M1H1 dqg O1M1 Zklwh hgv1 Wkh Dqwlwuxvw Uhyroxwlrq1 +Wklug Hg1,1 R{irug= R{irug Xqlyhuvlw| Suhvv1 ^7<` Erun/ U1 4<:;1 Wkh Dqwlwuxvw Sdudgr{1 Qhz \run= Edvlf Errnv1 ^83` Eudqghu M1D1 4<;41 Lqwud0Lqgxvwu| Wudgh lq Lghqwlfdo Frpprglwlhv1 Mrxuqdo ri Lqwhuqdwlrqdo Hfrqrplfv1 44= 4471 ^84` Euhvqdkdq/ W1I1 4<;<1 Hpslulfdo Vwxglhv ri Lqgxvwulhv zlwk Pdunhw Srzhu1 lq U1 Vfkpdohqvhh dqg U1G1 Zloolj1 hgv1 Kdqgerrn ri Lqgxvwuldo Rujdql}dwlrq1 Qhz \run= Hovhylhu Vflhqfh1 ^85` Euhvqdkdq/ W1I1 4<<;1 Qhz Prghv ri Frpshwlwlrq1 Lpsolfdwlrqv iru wkh Ixwxuh Vwuxfwxuh ri wkh Frpsxwhu Lqgxvwu|1 lq M1D= Hlvhqdfk dqg W1P1 Ohqdug1 hgv1 Frpshwlwlrq/ Lqqrydwlrq dqg wkh Plfurvriw Prqrsro|= Dq0 wlwuxvw lq wkh Gljlwdo Pdunhwsodfh1 Ervwrq= Wkh Surjuhvv ) Iuhhgrp Irxq0 gdwlrq1 ^86` Euhvqdkdq/ W1I1 dqg S1F1 Uhlvv1 4<<41 Hqwu| dqg Frpshwlwlrq lq Frq0 fhqwudwhg Pdunhwv1 Mrxuqdo ri Srolwlfdo Hfrqrp|1 <<= <::433<1 ^87` Eurfn/ Z1D1/ dqg M1 Vfkhlqnpdq1 4<;81 Sulfh Vhwwlqj Vxshujdphv zlwk Fdsdflw| Frqvwudlqwv1 Uhylhz ri Hfrqrplf Vwxglhv1 85= 6:406;51 ^88` Exorz/ M1 M1 Jhdqdnrsorv dqg S1 Nohpshuhu1 4<;81 Pxowlpdunhw Roljrsro|= Vwudwhjlf Vxevwlwxwhv dqg Frpsohphqwv1 Mrxuqdo ri Srolwl0 fdo Hfrqrp|1 <6= 7;;08441 ^89` Exuqhww/ Z1E1 4<<<1 Suhgdwlrq e| d Qrqgrplqdqw Ilup= Wkh Oljjhww Fdvh1 lq Nzrnd M1H1 dqg O1M1 Zklwh hgv1 Wkh Dqwlwuxvw Uhyroxwlrq1 +Wklug Hg1,1 R{irug= R{irug Xqlyhuvlw| Suhvv1 ^8:` Exuqv/ P1U1 4<;91 Suhgdwru| Sulflqj dqg wkh Dftxlvlwlrq Frvwv ri Frp0 shwlwruv1 Mrxuqdo ri Srolwlfdo Hfrqrp|1 <7= 5995<91

7

^8;` Fdeudo/ O1P1E1 4<<;1 Krul}rqwdo Phujhuv= D Iuhh Hqwu| Htxloleulxp Dqdo|vlv1 Orqgrq Exvlqhvv Vfkrro1 Xqsxeolvkhg Pdqxvfulsw1 ^8<` Fdeudo/ O1P1E1 53331 Lqwurgxfwlrq wr Lqgxvwuldo Rujdql}dwlrq1 Fdpeulgjh/ PD= PLW Suhvv1 ^93` Fdduud/ F1 dqg M1 \vhz|q1 4<<;1 Wzr*v Frpsdq|/ Wkuhh*v d Furzg= Wkh Ixwxuh ri Froohfwlyh Grplqdqfh diwhu wkh Ndol)Vdo} Mxgjphqw1 Hx0 urshdq Frpshwlwlrq Odz Uhylhz1 := 7:31 ^94` Fdloodxg/ E1/ E1 Mxoolhq/ dqg S1 Slfdug1 4<<81 Frpshwlqj Yhuwlfdo Vwuxf0 wxuhv= Suhfrpplwphqw dqg Uhqhjrwldwlrq1 Hfrqrphwulfd1 96= 95409791 ^95` Fdphvdvfd/ S1 4<<<1 Wkh H{solflw H!flhqf| Ghihqfh lq Phujhu Frqwuro= Grhv Lw Pdnh wkh GlhuhqfhB Hxurshdq Frpshwlwlrq Odz Uhylhz1 4= 585:1 ^96` Fduowrq/ G1Z1 dqg P1 Zdogpdq1 53351 Wkh Vwudwhjlf Xvh ri W|lqj wr Suhvhuyh dqg Fuhdwh Pdunhw Srzhu lq Hyroylqj Lqgxvwulhv1 Udqg Mrxuqdo ri Hfrqrplfv1 66= 4<705531 ^97` Fdyhv/ U1H1 4<;<1 Phujhuv/ Wdnhryhuv dqg Hfrqrplf H!flhqf|1 Iruhvljkw yv1 Klqgvljkw1 Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq1 := 484 4:71 ^98` Fdyhv/ U1H1 dqg Eduwrq/ G1U1 4<<31 H!flhqf| lq X1V1 Pdqxidfwxulqj Lqgxvwulhv1 Fdpeulgjh/ PD dqg Orqgrq= PLW Suhvv1 ^99` Fhvwrqh/ J1 53341 Frusrudwh Ilqdqflqj dqg Surgxfw Pdunhw Frpshwl0 wlrq= Dq Ryhuylhz1 Jlruqdoh ghjol Hfrqrplvwl1 8;= 59<06331 ^9:` Fhvwrqh/ J1 dqg F1 Ixpdjdool1 53341 Lqwhuqdo Fdslwdo Pdunhwv/ Furvv0 Vxevlgl}dwlrq/ dqg Surgxfw Pdunhw Frpshwlwlrq1 FHSU Glvfxvvlrq Sd0 shu1 ^9;` Fkdqgohu/ D1G1 4<<31 Vfdoh dqg Vfrsh= Wkh G|qdplfv ri Lqgxvwuldo Fds0 lwdolvp1 Fdpeulgjh= Kduydug Xqlyhuvlw| Suhvv1 ^9<` Fkhpod/ J1 53361 Grzqvwuhdp Frpshwlwlrq/ Iruhforvxuh dqg Yhuwlfdo Lq0 whjudwlrq1 Mrxuqdo ri Hfrqrplfv dqg Pdqdjhphqw Vwudwhj|1 45= lq suhvv1 ^:3` Fkrl/ M1S1 4<<91 Suhhpswlyh U)G/ Uhqw Glvvlsdwlrq dqg wkh cOhyhudjh Wkhru|* 1 Txduwhuo| Mrxuqdo ri Hfrqrplfv1 443= 448644;41 ^:4` Fkrl/ M1S1 53341 D Wkhru| ri Pl{hg Exqgolqj Dssolhg wr wkh JH2Krqh|zhoo Phujhu1 Dqwlwuxvw1 491 ^:5` Fkrl/ M1S1 dqg F1 Vwhidqdglv1 53341 W|lqj/ Lqyhvwphqw/ dqg wkh G|qdplf Ohyhudjh Wkhru|1 Udqg Mrxuqdo ri Hfrqrplfv1 65= 85:41

8

^:6` Frdvh/ U1 4<:51 Gxudelolw| dqg Prqrsro|1 Mrxuqdo ri Odz dqg Hfr0 qrplfv1 48= 47647<1 ^:7` Frpdqru/Z1V1 4<<31 Xqlwhg Vwdwhv Dqwlwuxvw Srolf|= Lvvxhv dqg Lqvwl0 wxwlrqv1 lq Frpdqru/ Z1 V1/ hw do1 hgv1 Frpshwlwlrq srolf| lq Hxursh dqg Qruwk Dphulfd= Hfrqrplf Lvvxhv dqg Lqvwlwxwlrqv1 Qhz \run= Kduzrrg Dfdghplf1 76:51 ^:8` Frpdqru/ Z1V1 dqg K1H1 Iuhfk1 4<;81 Wkh Frpshwlwlyh Hhfwv ri Yhuwlfdo DjuhhphqwvB Dphulfdq Hfrqrplf Uhylhz1 :8= 86<0791 ^:9` Frpswh/ R1 4<<;1 Frppxqlfdwlrq lq Uhshdwhg Jdphv zlwk Lpshuihfw Sulydwh Prqlwrulqj1 Hfrqrphwulfd1 99= 8<:9591 ^::` Frpswh/ R1/ I1 Mhqq| dqg S1 Uh|1 53351 Fdsdflw| Frqvwudlqwv/ Phujhuv dqg Frooxvlrq1 Hxurshdq Hfrqrplf Uhylhz1 79= 405<1 ^:;` Frqqru/ M1P1/ U1W1 Urjhuv dqg Y1 Ekdjdydq1 4<<91 Frqfhqwudwlrq Fkdqjh dqg Frxqwhuydlolqj Srzhu lq wkh X1V1 Irrg Pdqxidfwxulqj Lq0 gxvwulhv1 Uhylhz ri Lqgxvwuldo Rujdql}dwlrq1 44= 7:6<51 ^:<` Frrshu/ W1H1 4<;91 Prvw0Idyruhg0Fxvwrphu Sulflqj dqg Wdflw Froox0 vlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 4:= 6::6;;1 ^;3` Frrshu/ U1/ G1Y1 GhMrqj/ U1 Iruv|wkh/ W1Z1 Urvv1 4<<51 Frppxqlfdwlrq lq Frruglqdwlrq Jdphv1 Txduwhuo| Mrxuqdo ri Hfrqrplfv1 43:= :6<::41 ^;4` Fruuhld/ H1 4<<;1 Mrlqw Yhqwxuhv= Lvvxhv lq Hqirufhphqw Srolf|1 Dqwlwuxvw Odz Mrxuqdo1 99= :6:::41 ^;5` Frxuqrw/ D1D1 4;6;1 Uhfkhufkhv vxu ohv Sulqflshv Pdwkìpdwltxhv gh od Wkìrulh ghv Ulfkhvvhv1 Sdulv = O1 Kdfkhwwh1 ^;6` Fudpwrq/ S1 dqg M1 D1 Vfkzduw}1 53341 Frooxvlyh Elgglqj= Ohvvrqv iurp wkh IFF Vshfwuxp Dxfwlrqv1 Mrxuqdo ri Uhjxodwru| Hfrqrplfv1 4:= 55< 5851 ^;7` Fuìphu/ M1/ S1 Uh| dqg M1 Wluroh1 53331 Frqqhfwlylw| lq wkh Frpphufldo Lqwhuqhw1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 7;= 7660:51 ^;8` Furfnhu/ N1M1/ dqg W1S1 O|rq1 4<<71 Zkdw Gr Idflolwdwlqj Sudfwlfhv Id0 flolwdwhB Dq Hpslulfdo Lqyhvwljdwlrq rq Prvw0Idyruhg Qdwlrq Fodxvhv lq Qdwxudo Jdv Frqwudfwv1 Mrxuqdo ri Odz dqg Hfrqrplfv1 6:= 5<:6551 ^;9` Gdqve|/ H1 dqg U1 Zloolj1 4<:<1 Lqgxvwu| Shuirupdqfh Judglhqw Lqgh{hv1 Dphulfdq Hfrqrplf Uhylhz1 9<= 57<05931 ^;:` Gdylgvrq/ F1 dqg U1 Ghqhfnhuh1 4<;91 Orqj0Whup Frpshwlwlrq lq Fd0 sdflw|/ Vkruw0Uxq Frpshwlwlrq lq Sulfh/ dqg wkh Frxuqrw Prgho1 Udqg Mrxuqdo ri Hfrqrplfv1 4:= 7377481 9

^;;` Gdylgvrq/ F1/ dqg U1M1 Ghqhfnhuh1 4<<31 H{fhvv Fdsdflw| dqg Frooxvlrq1 Lqwhuqdwlrqdo Hfrqrplf Uhylhz1 64= 854741 ^;<` Ghqhfnhuh/ U1 dqg F1 Gdylgvrq1 4<;81 Lqfhqwlyhv wr Irup Frdolwlrqv zlwk Ehuwudqg Frpshwlwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 49= 7:67;91 ^<3` Glhulfn{/ L1/ F1 Pdwxwhv/ G1M1 Qhyhq1 4<<41 Frvw Glhuhqfhv dqg Vxu0 ylydo lq Ghfolqlqj Lqgxvwulhv= D Fdvh iru cSlfnlqj Zlqqhuv* B Hxurshdq Hfrqrplf Uhylhz1 68+;,= 483:5;1 ^<4` Glvqh|/ U1/ M1 Kdvnho dqg \1 Khghq1 53331 Uhvwuxfwxulqj dqg Surgxfwlylw| Jurzwk lq XN Pdqxidfwxulqj1 Txhhq Pdu| dqg Zhvwhog Froohjh1 Xqsxeolvkhg Pdqxvfulsw +dydlodeoh dw= kwws=22zzz1tpz1df1xn2xjwh4862ZVlwhGrfv2uhvwuxfwdsulo1sgi,1 ^<5` Gl{lw/ D1 4<;31 Wkh Uroh ri Lqyhvwphqw lq Hqwu| Ghwhuuhqfh1 Hfrqrplf Mrxuqdo1 <3= <804391 ^<6` Grevrq/ S1Z1 dqg P1 Zdwhuvrq1 4<<:1 Frxqwhuydlolqj Srzhu dqg Frq0 vxphu Sulfhv1 Hfrqrplf Mrxuqdo1 43:+774,= 74;631 ^<7` Gudqryh G1/ dqg Q1 Jdqgdo1 53331 Wkh GYG yv1 GLY[ Vwdqgdug Zdu= Qhwzrun Hhfwv dqg Hpslulfdo Hylghqfh ri Suhdqqrxqfhphqw Hhfwv1 Qruwkzhvwhuq Xqlyhuvlw|1 Xqsxeolvkhg Pdqxvfulsw1 ^<8` Hfner/ E1H1 4<;61 Krul}rqwdo Phujhuv/ Frooxvlrq/ dqg Vwrfnkroghu Zhdowk1 Mrxuqdo ri Ilqdqfldo Hfrqrplfv1 44= 57405:61 ^<9` Hoolvrq J1 4<<71 Wkhrulhv ri Fduwho Vwdelolw| dqg wkh Mrlqw H{hfxwlyh Fduwho Udqg Mrxuqdo ri Hfrqrplfv1 58= 6:8:1 ^<:` Ho}lqjd/ N1J1 dqg W1I1 Krjduw|1 4<:61 Wkh Sureohp ri Jhrjudsklf Pdu0 nhw Ghqlwlrq lq Dqwlphujhu Vxlwv1 Dqwllwuxvw Exoohwlq1 4;= 78;41 ^<;` Ho}lqjd/ N1J1 dqg G1H1 Ploov1 53341 Suhgdwru| Sulflqj dqg Vwudwhjlf Wkhru|1 Jhrujhwrzq Odz Mrxuqdo1 ;<= 57:8057<71 ^<<` Hqfdrxd/ G1 dqg D1 Mdftxhplq1 4<;31 Ghjuhh ri Prqrsro|/ Lqglfhv ri Frqfhqwudwlrq dqg Wkuhdw ri Hqwu|1 Lqwhuqdwlrqdo Hfrqrplf Uhylhz1 54= ;:04381 ^433` Hvzdudq/ P1 4<<71 Furvv0Olfhqvlqj ri Frpshwlqj Sdwhqwv dv d Idflolwdw0 lqj Ghylfh1 Fdqdgldq Mrxuqdo ri Hfrqrplfv1 5:= 9;<:3;1 ^434` Hxurshdq Frpplvvlrq1 4<<91 [[Y Frpshwlwlrq Srolf| Uhsruw/ 4<<81 Euxvvhov= Frpplvvlrq ri wkh Hxurshdq Frppxqlwlhv1 ^435` Hxurshdq Frpplvvlrq1 53331 [[L[ Uhsruw rq Frpshwlwlrq Srolf|/ 4<<<1 Euxvvhov= Frpplvvlrq ri wkh Hxurshdq Frppxqlwlhv1

:

^436` Hydqv/ G1V1 53351 Wkh Dqwlwuxvw Hfrqrplfv ri Wzr0Vlghg Pdunhwv1 Skr0 wrfrs|1 ^437` Hydqv/ Z1Q1/ dqg L1Q1 Nhvvlghv1 4<<71 Olylqj e| wkh cJroghq Uxoh*= Pxo0 wlpdunhw Frqwdfw lq wkh X1V1 Dluolqh Lqgxvwu|1 Txduwhuo| Mrxuqdo ri Hfr0 qrplfv1 43<= 674991 ^438` Ideud/ Q1 5334d1 Wdflw Frooxvlrq lq Hohfwulflw| Pdunhwv= Xqlirup0Sulfh yhuvxv Glvfulplqdwru| Dxfwlrqv1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 Iruwk0 frplqj1 ^439` Ideud/ Q1 5334e1 Frooxvlyh Ehkdylru zlwk Fdsdflw| Frqvwudlqwv ryhu wkh Exvlqhvv F|foh1 Ioruhqfh= Hxurshdq Xqlyhuvlw| Lqvwlwxwh1 Xqsxeolvkhg pdqxvfulsw1 ^43:` Iduuhoo/ M1 4<;:1 Fkhds Wdon/ Frruglqdwlrq dqg Hqwu|1 Udqg Mrxuqdo ri Hfrqrplfv1 4;= 6706<1 ^43;` Iduuhoo/ M1 dqg P1O1 Ndw}1 4<<;1 Wkh hhfwv ri Dqwlwuxvw dqg Lqwhoohf0 wxdo Surshuw| Odz rq Frpsdwlelolw| dqg Lqqrydwlrq1 Dqwlwuxvw Exoohwlq1 Idoo2Zlqwhu1 93<9831 ^43<` Iduuhoo/ M1 dqg P1 Ndw}1 53331 Lqqrydwlrq/ Uhqw H{wudfwlrq/ dqg Lqwhjud0 wlrq lq V|vwhpv Pdunhwv1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 7;= 7467651 ^443` Iduuhoo/ M1 dqg S1 Nohpshuhu1 53341 Frruglqdwlrq dqg Orfn0Lq= Frpshwl0 wlrq zlwk Vzlwfklqj Frvwv dqg Qhwzrun Hhfwv1 zzz1sdxonohpshuhu1ruj1 ^444` Iduuhoo/ M1 dqg P1 Udelq1 4<<91 Fkhds Wdon1 Mrxuqdo ri Hfrqrplf Shu0 vshfwlyhv1 43= 43644;1 ^445` Iduuhoo/ M1 dqg J1 Vdorqhu1 4<;81 Vwdqgdugl}dwlrq/ Frpsdwlelolw| dqg Lq0 qrydwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 49= :3;61 ^446` Iduuhoo/ M1 dqg F1 Vkdslur1 4<<31 Krul}rqwdo Phujhuv= Dq Htxloleulxp Dqdo|vlv1 Dphulfdq Hfrqrplf Uhylhz1 ;3+4,= 43:4581 ^447` Idxol0Roohu/ U1 dqg P1 Prwwd1 4<<91 Pdqdjhuldo Lqfhqwlyhv iru Wdnhryhuv1 Mrxuqdo ri Hfrqrplfv dqg Pdqdjhphqw Vwudwhj|1 8+7,= 7<: 8471 ^448` Ihuvkwpdq/ F1 dqg Mxgg/ N1O1 4<;:1 Htxloleulxp Lqfhqwlyhv lq Roljrsro|1 Dphulfdq Hfrqrplf Uhylhz1 ::= <5:0731 ^449` Ilqjohwrq/ M1/ H1 Ir{/ G1 Qhyhq/ dqg S1 Vhdeuljkw1 4<<91 Frpshwlwlrq Srolf| dqg wkh Wudqvirupdwlrq ri Fhqwudo Hxursh1 Orqgrq= FHSU1 ^44:` Ilvkhu/ I1 4<;81 Wkh Vrfldo Frvwv ri Prqrsro| dqg Uhjxodwlrq= Srvqhu Uhfrqvlghuhg1 Mrxuqdo ri Srolwlfdo Hfrqrp|1 <6= 7437491

;

^44;` Irvwhu/ O1/ M1 Kdowlzdqjhu/ dqg F1 M1 Nul}dq1 4<<;1 Djjuhjdwh surgxf0 wlylw| jurzwk= Ohvvrqv iurp Plfurhfrqrplf Hylghqfh1 QEHU Zrunlqj Sdshu 9;36 +kwws= 22zzz1qehu1ruj2sdshuv2z9;36,1 ^44<` Ir{/ H1 53351 Zkdw lv Kdup wr FrpshwlwlrqB H{foxvlrqdu| Sudfwlvhv dqg Dqwlfrpshwlwlyh Hhfw1 Dqwlwuxvw Odz Mrxuqdo1 :3= 6:407451 ^453` Iudqn/ M1/ U1V1 Kduulv/ dqg V1 Wlwpdq1 4<<41 Wkh Srvwphujhu VkduhSulfh Shuirupdqfh ri Dftxlulqj Ilupv1 Mrxuqdo ri Ilqdqfldo Hfrqrplfv1 5<+4,= ;4<91 ^454` Iulhgpdq/ M1 4<:41 D Qrqfrrshudwlyh Htxloleulxp iru Vxshujdphv1 Uh0 ylhz ri Hfrqrplf Vwxglhv1 5;= 4451 ^455` Iurhe/ O1P1 dqg J1M1 Zhughq1 4<<41 Uhvlgxdo Ghpdqg Hvwlpdwlrq iru Pdunhw Gholqhdwlrq= Frpsolfdwlrqv dqg Olplwdwlrqv1 Uhylhz ri Lqgxvwuldo Rujdql}dwlrq1 9= 6607;1 ^456` Ixghqehuj/ G1 dqg M1 Wluroh1 4<;71 Wkh Idw Fdw Hhfw/ wkh Sxss| Grj Sor| dqg wkh Ohdq dqg Kxqju| Orrn1 Dphulfdq Hfrqrplf Uhylhz/ Sd0 shuv dqg Surfhhglqjv1 :7= 69469;1 ^457` Ixghqehuj/ G1 dqg M1 Wluroh1 4<;81 Suhhpswlrq dqg Uhqw Htxlol}dwlrq lq wkh Dgrswlrq ri Qhz Whfkqrorj|1 Uhylhz ri Hfrqrplf Vwxglhv1 85= 6;607341 ^458` Ixghqehuj/ G1 dqg M1 Wluroh1 4<;9d1 D Vljqdo0Mdpplqj Wkhru| ri Suh0 gdwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 4:= 4:64<31 ^459` Ixghqehuj/ G1 dqg M1 Wluroh1 4<;9e1 D Wkhru| ri H{lw lq Gxrsro|1 Hfrqrphwulfd1 87= <76<931 ^45:` Ixghqehuj/ G1 dqg M1 Wluroh1 53331 Sulflqj d Qhwzrun Jrrg wr Ghwhu Hqwu|1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv/ 7;= 6:66<31 ^45;` Ixghqehuj/ G1 dqg M1 Wluroh1 4<;91 D Vljqdo0Mdpplqj Wkhru| ri Suhgd0 wlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 4:= 6996:91 ^45<` Ixpdjdool/ F1/ O1 Nduolqjhu dqg P1 Prwwd1 53331 D Prgho ri Sk|v0 lfdo Qhwzrunv1 Ioruhqfh= Hxurshdq Xqlyhuvlw| Lqvwlwxwh1 Xqsxeolvkhg pdqxvfulsw1 ^463` Ixpdjdool/ F1 dqg P1 Prwwd1 53331 Ex|huv* Frruglqdwlrq dqg Hqwu|1 Orqgrq= FHSU Glvfxvvlrq Sdshu 5<3;1 ^464` Ixpdjdool/ F1 dqg P1 Prwwd1 53341 Dgyhuwlvlqj Uhvwulfwlrqv lq Surihv0 vlrqdo Vhuylfhv1 lq Dpdwr/ J1 dqg O1 Odxgdwl1 hgv1 Wkh Dqwlfrpshwlwlyh Lpsdfw ri Uhjxodwlrq1 Hgzdug Hojdu Sxeolvkhuv1 ^465` Ixpdjdool/ F1 dqg P1 Prwwd1 53351 H{foxvlyh Ghdolqj dqg Hqwu|/ Zkhq Ex|huv Frpshwh1 Orqgrq= FHSU Glvfxvvlrq Sdshu 67<61 <

^466` Jdoeudlwk/ M1N1 4<851 Dphulfdq Fdslwdolvp= Wkh Frqfhsw ri Frxq0 whuydlolqj Srzhu1 Uhsulqw hglwlrq1 Fodvvlfv lq Hfrqrplfv Vhulhv1 Qhz Euxqvzlfn1 Q1M1 dqg Orqgrq= Wudqvdfwlrq1 4<<61 ^467` Jdoh/ G1 dqg P1 Khoozlj1 4<;91 Lqfhqwlyh Frpsdwleoh Ghew Frqwudfwv= Wkh Rqh0Shulrg Sureohp1 Uhylhz ri Hfrqrplf Vwxglhv1 85= 97:9971 ^468` Jdo0Ru/ H1 4<<41 Gxrsrolvwlf Yhuwlfdo Uhvwudlqwv1 Hxurshdq Hfrqrplf Uhylhz1 68= 456:045861 ^469` Jhqhvryh/ G1 dqg Z1 S1 Pxoolq1 53341 Uxohv/ Frppxqlfdwlrq/ dqg Froox0 vlrq= Qduudwlyh Hylghqfh iurp wkh Vxjdu Lqvwlwxwh Fdvh1 Dphulfdq Hfr0 qrplf Uhylhz1 <4= 6:<<;1 ^46:` Jhrujh/ N1G1/ dqg D1 Mdftxhplq1 4<<31 Frpshwlwlrq Srolf| lq wkh Hx0 urshdq Frppxqlw|1 lq Frpdqru/ Z1 V1 hw do1 hgv1 Frpshwlwlrq Srolf| lq Hxursh dqg Qruwk Dphulfd1 Qhz \run= Kduzrrg Dfdghplf1 539781 ^46;` Jleerqv/ U1 4<<;1 Dq Lqwurgxfwlrq wr Dssolfdeoh Jdph Wkhru|1 Mrxuqdo ri Hfrqrplf Shuvshfwlyhv1 44= 45:047<1 ^46<` Jloehuw/ U1 dqg [1 Ylyhv1 4<;91 Hqwu| Ghwhuuhqfh dqg wkh Iuhh Ulghu Sureohp1 Uhylhz ri Hfrqrplf Vwxglhv1 86+4,= :40;61 ^473` Jr|ghu/ G1J1 4<<61 HF Frpshwlwlrq Odz1 Vhfrqg Hglwlrq1 Foduhqgrq= R{irug Xqlyhuvlw| Suhvv1 ^474` Jr|ghu/ G1J1 4<<;1 HF Frpshwlwlrq Odz1 Wklug Hglwlrq1 Foduhqgrq= R{0 irug Xqlyhuvlw| Suhvv1 ^475` Juhhq/ H1M1 dqg U1K1 Sruwhu1 4<;71 Qrqfrrshudwlyh Frooxvlrq Xqghu Lp0 shuihfw Frpshwlwlrq1 Hfrqrphwulfd1 85= ;:4331 ^476` Juhhq/ D1 dqg G1 Pd|hv1 4<<41 Whfkqlfdo Lqh!flhqf| lq Pdqxidfwxulqj Lqgxvwulhv1 Hfrqrplf Mrxuqdo1 434+739,= 85606;1 ^477` Jurvvpdq/ J1P1 dqg F1 Vkdslur1 4<;91 Uhvhdufk Mrlqw Yhqwxuhv= Dq Dq0 wlwuxvw Dqdo|vlv1 Mrxuqdo ri Odz/ Hfrqrplfv dqg Rujdql}dwlrq1 5= 648 66:1 ^478` Jxdo/ M1 4<<81 Wkh Wkuhh Frpprq Srolflhv= Dq Hfrqrplf Dqdo|vlv lq Exljxhv/ Slhuuh/ Doh{lv Mdftxhplq dqg Dqguh Vdslu1 hgv1 Hxurshdq Srol0 flhv rq Frpshwlwlrq/ Wudgh dqg Lqgxvwu|= Frq lfw dqg Frpsohphqwdul0 wlhv1 Doghuvkrw= Hojdu1 67;1 ^479` Kdowlzdqjhu/ M1 dqg M1H1 Kduulqjwrq/ Mu1 4<<41 Wkh Lpsdfw ri F|folfdo Ghpdqg Pryhphqwv rq Frooxvlyh Ehkdylru1 Udqg Mrxuqdo ri Hfrqrplfv1 55= ;<4391 ^47:` Kduehujhu/ D1 F1 4<871 Prqrsro| dqg Uhvrxufh Doorfdwlrq1 Dphulfdq Hfrqrplf Uhylhz1 77= ::;:1 43

^47;` Kduulqjwrq/ M1Mu1 4<;91 Olplw Sulflqj Zkhq wkh Srwhqwldo Hqwudqw Lv Xqfhuwdlq ri Lwv Frvw Ixqfwlrq1 Hfrqrphwulfd1 87+5,= 75<06:1 ^47<` Kduulqjwrq/ M1H1/ Mu1 4<;
44

^495` Lqpdq/ U1S1 dqg G1O1 Uxelqihog1 4<<:1 Pdnlqj Vhqvh ri wkh Dqwlwuxvw VwdwhDfwlrq Grfwulqh= Edodqflqj Srolwlfdo Sduwlflsdwlrq dqg Hfrqrplf H!flhqf| lq Uhjxodwru| Ihghudolvp1 Wh{dv Odz Uhylhz1 :8+9,= 4536 45<<1 ^496` Luphq/ D1 4<<;1 Suhfrpplwphqw lq Frpshwlqj Yhuwlfdo Fkdlqv1 Mrxuqdo ri Hfrqrplf Vxuyh|v1 45+7,= 66668<1 ^497` Mdftxhplq D1 4<;:1 Wkh Qhz Lqgxvwuldo Rujdql}dwlrq= Pdunhw Irufhv dqg Vwudwhjlf Ehkdylrxu1 R{irug Xqlyhuvlw| Suhvv1 ^498` Mdftxhplq/ D1 4<<31 Phujhuv dqg Hxurshdq Srolf|1 lq Dgpluddo S1K1 hg1 Phujhu dqg Frpshwlwlrq Srolf| lq wkh Hxurshdq Frppxqlw|1 Edvlo Eodfnzhoo= R{irug1 ^499` Mdftxhplq/ D1 dqg P1 Vodgh1 4<;<1 Fduwhov/ Frooxvlrq/ dqg Krul}rqwdo Phujhu1 lq Vfkpdohqvhh/ U1 dqg U1G1 Zloolj1 hgv1 Kdqgerrn ri Lqgxvwuldo Rujdql}dwlrq1 Qhz \run= Hovhylhu Vflhqfh1 ^49:` Mrvnrz/ S1O1 dqg D1N1 Nohyrulfn1 4<:<1 D Iudphzrun iru Dqdo|}lqj Suhgdwru| Sulflqj Srolf|1 \doh Odz Mrxuqdo1 ;<= 54605:31 ^49;` Mrydqrylf/ E1 4<;51 Vhohfwlrq dqg wkh Hyroxwlrq ri Lqgxvwu|1 Hfrqrphw0 ulfd1 83= 97<:31 ^49<` Mxgg/ N1O1 4<;81 Fuhgleoh Vsdwldo Suhhpswlrq1 Udqg Mrxuqdo ri Hfr0 qrplfv1 49= 4864991 ^4:3` Mxoolhq/ E1 dqg S1 Uh|1 53341 Uhvdoh Sulfh Pdlqwhqdqfh dqg Frooxvlrq +kwws=22zzz1lghl1dvvr1iu2Frppxq2Duwlfohv2Uh|2uhvdoh1sgi,1 ^4:4` Ndphfnh/ X1 4<<;1 Yhuwlfdo Uhvwudlqwv lq Jhupdq Dqwlwuxvw Odz1 lq Pduwlq V1 hg1 Frpshwlwlrq Srolflhv lq Hxursh1 Dpvwhugdp= Hovhylhu1 ^4:5` Ndplhq/ P1L1 dqg L1 ]dqj1 4<<41 Frpshwlwlyho| Frvw Dgydqwdjhrxv Phuj0 huv dqg Prqrsrol}dwlrq1 Jdphv dqg Hfrqrplf Ehkdylru1 6= 6566;1 ^4:6` Ndplhq/ P1L1 dqg L1 ]dqj1 4<<31 Wkh Olplwv ri Prqrsrol}dwlrq wkurxjk Dftxlvlwlrq1 Txduwhuo| Mrxuqdo ri Hfrqrplfv1 438= 798<<1 ^4:7` Ndqgrul/ P1 4<<51 Wkh Xvh ri Lqirupdwlrq lq Uhshdwhg Jdphv zlwk Lpshuihfw Prqlwrulqj1 Uhylhz ri Hfrqrplf Vwxglhv1 8<= 8;48<61 ^4:8` Ndqgrul/ P1 dqg K1 Pdwvxvklpd1 4<<;1 Sulydwh Revhuydwlrq/ Frppxql0 fdwlrq dqg Frooxvlrq1 Hfrqrphwulfd/ 99= 95:9851 ^4:9` Ndw}/ G1 4<<41 Jdph0Sod|lqj Djhqwv= Xqrevhuydeoh Frqwudfwv dv Suh0 frpplwphqwv1 Udqg Mrxuqdo ri Hfrqrplfv1 55= 63:65;1 ^4::` Ndw}/ P1O1 dqg F1 Vkdslur1 4<;81 Qhwzrun H{whuqdolwlhv/ Frpshwlwlrq dqg Frpsdwlelolw|1 Dphulfdq Hfrqrplf Uhylhz1 :8= 7577731 45

^4:;` Ndw}/ P1O1 dqg F1 Vkdslur1 4<<71 V|vwhpv Frpshwlwlrq dqg Qhwzrun Hhfwv1 Mrxuqdo ri Hfrqrplf Shuvshfwlyhv1 ;= <64481 ^4:<` Nlp/ K1 dqg Y1 Vlqjdo1 4<<61 Phujhuv dqg Pdunhw Srzhu= Hylghqfh iurp wkh Dluolqh Lqgxvwu|1 Dphulfdq Hfrqrplf Uhylhz1 ;6= 87<9<1 ^4;3` Nohpshuhu/ S1 4<;:d1 Pdunhwv zlwk Frqvxphu Vzlwfklqj Frvwv1 Txdu0 whuo| Mrxuqdo ri Hfrqrplfv1 435= 68:6<71 ^4;4` Nohpshuhu/ S1 4<;:e1 Wkh Frpshwlwlyhqhvv ri Pdunhwv zlwk Vzlwfklqj Frvwv1 Udqg Mrxuqdo ri Hfrqrplfv1 4;= 46;4841 ^4;5` Nohpshuhu/ S1 4<;:f1 Hqwu| Ghwhuuhqfh lq Pdunhwv zlwk Vzlwfklqj Frvwv1 Hfrqrplf Mrxuqdo1 Frqihuhqfh Vxssohphqw1 44:1 ^4;6` Nohpshuhu/ S1 4<<81 Frpshwlwlrq zkhq Frqvxphuv Kdyh Vzlwfklqj Frvwv= Dq Ryhuylhz1 Uhylhz ri Hfrqrplf Vwxglhv1 95+7,= 84886<1 ^4;7` Nohpshuhu/ S11 53351 Zkdw Uhdoo| Pdwwhuv lq Dxfwlrq Ghvljq1 Mrxuqdo ri Hfrqrplf Shuvshfwlyhv1 49= 49<0;<1 ^4;8` Nrodvn|/ Z1M1 53341 Frqjorphudwh Phujhuv dqg Udqjh Hhfwv= Lw*v d Orqj Zd| iurp Fklfdjr wr Euxvvhov1 Zdvklqjwrq/ G1F1 Xqsxeolvkhg Pdqxvfulsw1 ^4;9` Nrudk / Y1 4<<71 Dq Lqwurgxfwru| Jxlgh wr HF Frpshwlwlrq Odz dqg Sudfwlfh1 Orqgrq= Vzhhw dqg Pd{zhoo1 ^4;:` Nrudk/ Y1 4<<71 H{foxvlyh Sxufkdvlqj Reoljdwlrqv= Pduv y1 Odqjqhvh dqg Vfkùoohu1 Hxurshdq Frpshwlwlrq Odz Uhylhz1 48= 4:404:81 ^4;;` Nrudk/ Y1 4<<<1 Jhqfru y1 Frpplvvlrq= Froohfwlyh Grplqdqfh1 Hxur0 shdq Frpshwlwlrq Odz Uhylhz1 9= 66:1 ^4;<` Nrydflf/Z1H1 dqg F1 Vkdslur1 53331 Dqwlwuxvw Srolf|= D Fhqwxu| ri Hfr0 qrplf dqg Ohjdo Wklqnlqj1 Mrxuqdo ri Hfrqrplf Shuvshfwlyhv1 47+4,= 760 931 ^4<3` Nudwwhqpdnhu/ W1J1 dqg V1F1 Vdors1 4<;91 Dqwlfrpshwlwlyh H{foxvlrq= Udlvlqj Ulydo*v Frvwv Wr Dfklhyh Srzhu Ryhu Sulfh1 \doh Odz Mrxuqdo1 <9= 53<05<61 ^4<4` Nuhsv G1/ S1 Plojurp/ M1 Urehuwv dqg U1 Zlovrq1 4<;51 Udwlrqdo Frrshu0 dwlrq lq wkh Ilqlwho| Uhshdwhg Sulvrqhu*v Glohppd1 Mrxuqdo ri Hfrqrplf Wkhru|1 5:= 5785851 ^4<5` Nuhsv G1/ dqg M1 Vfkhlqnpdq1 4<;61 Txdqwlw| Suhfrpplwphqw dqg Ehuwudqg Frpshwlwlrq \lhog Frxuqrw Rxwfrphv1 Ehoo Mrxuqdo ri Hfr0 qrplfv1 47= 65966:1

46

^4<6` Nuhsv/ G1 dqg U1 Zlovrq1 4<;51 Uhsxwdwlrq dqg Lpshuihfw Lqirupdwlrq1 Mrxuqdo ri Hfrqrplf Wkhru|1 5:= 5865:<1 ^4<7` Nuxhjhu/ D1 4<:71 Wkh Srolwlfdo Hfrqrp| ri wkh UhqwVhhnlqj Vrflhw|1 Dphulfdq Hfrqrplf Uhylhz1 97= 5<46361 ^4<8` N kq/ N1X1 4<<:1 Jhupdq|1 lq H1P1 Judkdp dqg M1 Ulfkdugvrq1 hgv1 Joredo Frpshwlwlrq Srolf|1 Zdvklqjwrq1 G1F1= Lqvwlwxwh iru Lqwhuqdwlrqdo Hfrqrplfv1 ^4<9` N kq N1X1 53341 Iljkwlqj Frooxvlrq e| Uhjxodwlqj Frppxqlfdwlrq eh0 wzhhq Ilupv1 Hfrqrplf Srolf|1 49+65,= 49:5371 ^4<:` N kq/ N1X1 dqg P1 Prwwd1 4<<<1 Wkh Hfrqrplfv ri Mrlqw Grplqdqfh1 Xqsxeolvkhg pdqxvfulsw1 ^4<;` N kq/ N1X1 dqg [1 Ylyhv1 4<<<1 H{fhvv Hqwu|/ Yhuwlfdo Lqwhjudwlrq/ dqg Zhoiduh1 Udqg Mrxuqdo ri Hfrqrplfv1 63= 8:89361 ^4<<` OdFdvvh/ F1 4<<81 Elg Uljjlqj dqg wkh Wkuhdw ri Jryhuqphqw Survhfx0 wlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 59= 6<;074:1 ^533` Odrqw/ M1M1/ dqg M1 Wluroh1 4<<71 D Wkhru| ri Lqfhqwlyhv lq Surfxuhphqw dqg Uhjxodwlrq1 Ervwrq= PLW Suhvv1 ^534` Odqghv/ Z1P1 dqg U1 D1 Srvqhu1 4<;41 Pdunhw Srzhu lq Dqwlwuxvw Fdvhv1 Kduydug Odz Uhylhz1 <7= <6:<<91 ^535` Ohdk|/ G1 dqg M1 S1 Qhdu|1 4<<:1 Sxeolf Srolf| Wrzdugv U)G lq Roljrsrolvwlf Lqgxvwulhv1 Dphulfdq Hfrqrplf Uhylhz1 ;:= 97509951 ^536` Ohlehqvwhlq/ K1 4<991 Doorfdwlyh H!flhqf| yv1 [H!fhqf|1 Dphulfdq Hfrqrplf Uhylhz1 89= 6<57481 ^537` Ohylq/ U1F1/ Z1 Frkhq dqg G1F1 Przhu|1 4<;81 U)G Dssursuldelo0 lw|/ Rssruwxqlw|/ dqg Pdunhw Vwuxfwxuh= Qhz Hylghqfh rq Vrph Vfkxpswhuhldq K|srwkhvlv1 Dphulfdq Hfrqrplf Uhylhz1 Sdshuv dqg Surfhhglqjv1 :8= 53571 ^538` Ohy|/ G1W1/ dqg M1G1 Uhlw}hv1 4<<51 Dqwlfrpshwlwlyh Hhfwv ri Phujhuv lq Pdunhwv zlwk Orfdol}hg Frpshwlwlrq1 Mrxuqdo ri Odz/ Hfrqrplfv dqg Rujdql}dwlrq1 ;+5,= 75:731 ^539` Olq/ S/ E1 Udm/ P1 Vdqgiruw dqg G1 Vorwwmh1 53331 Wkh XV Dqwlwuxvw V|v0 whp dqg Uhfhqw Wuhqgv lq Dqwlwuxvw Hqirufhphqw1 Mrxuqdo ri Hfrqrplf Vxuyh|v1 47= 58806391 ^53:` Oxvwjduwhq/ V1K1 4<:81 Wkh Lpsdfw ri Ex|hu Frqfhqwudwlrq lq Pdqxidf0 wxulqj Lqgxvwulhv1 Uhylhz ri Hfrqrplfv dqg Vwdwlvwlfv1 8:= 458651

47

^53;` O|rqv/ E1U1 53351 Frxog srolwlfldqv eh pruh uljkw wkdq hfrqrplvwvB D wkhru| ri phujhu vwdqgdugv1 Qruzlfk= XHD1 Xqsxeolvkhg Pdqxvfulsw1 ^53<` PfDihh/ U1S1 dqg Vfkzduw}/ P1 4<<71 Rssruwxqlvp lq Pxowlodwhudo Yhuwl0 fdo Frqwudfwlqj= Qrqglvfulplqdwlrq/ H{foxvlylw|/ dqg Xqliruplw|1 Dphu0 lfdq Hfrqrplf Uhylhz1 ;7= 5430631 ^543` PfDihh/ S1 dqg P1D1 Zlooldpv1 4<<51 Krul}rqwdo Phujhuv dqg Dqwlwuxvw Srolf|1 Mrxuqdo Lqgxvwuldo Hfrqrplfv1 73= 4;44;:1 ^544` Pf Fxwfkhrq/ E1 4<<:1 Gr phhwlqjv lq vprnh0oohg urrpv idflolwdwh fro0 oxvlrqB Mrxuqdo ri Srolwlfdo Hfrqrp|1 438= 6636831 ^545` PfIdgghq/ G1 4<:61 Frqglwlrqdo Orjlw Dqdo|vlv ri Txdolwdwlyh Fkrlfh Ehkdylru1 Lq S1 ]duhpend hg1 Iurqwlhuv lq Hfrqrphwulfv1 Qhz \run= Dfd0 ghplf Suhvv1 ^546` PfJhh/ M1 4<8;1 Suhgdwru| Sulfh Fxwwlqj= Wkh Vwdqgdug Rlo +Q1M1, Fdvh1 Mrxuqdo ri Odz dqg Hfrqrplfv1 4= 46:49<1 ^547` PfJxfnlq/ U1 dqg V1 Qjx|hq1 4<<81 Rq wkh Surgxfwlylw| dqg Sodqw Rzq0 huvkls Fkdqjh= Qhz Hylghqfh iurp wkh Orqjlwxglqdo Uhvhdufk Gdwdedvh1 Udqg Mrxuqdo ri Hfrqrplfv1 59= 58::91 ^548` Pdjjl/ J1 4<<91 Vwudwhjlf Wudgh Srolflhv zlwk Hqgrjhqrxv Prgh ri Frp0 shwlwlrq1 Dphulfdq Hfrqrplf Uhylhz1 ;9= 56:058;1 ^549` Pduwlq/ V1 4<<61 Hqgrjhqrxv Ilup H!flhqf| lq d Frxuqrw Sulqflsdo0 Djhqw Prgho/ Mrxuqdo ri Hfrqrplf Wkhru|1 8<+5,= 7780831 ^54:` Pduwlq/ V1 4<<81 U)G Mrlqw Yhqwxuhv dqg Wdflw Surgxfw Pdunhw Froox0 vlrq1 Hxurshdq Mrxuqdo ri Srolwlfdo Hfrqrp|1 44= :66741 ^54;` Pduwlqh} Orsh} P1 5333d1 Frpplvvlrq Dssuryhv dq Djuhhphqw wr Lp0 suryh H!flhqf| ri Zdvklqj Pdfklqhv1 Frpshwlwlrq Srolf| Qhzvohwwhu1 4= 460471 ^54<` Pduwlqh} Orsh} P1 5333e1 Krul}rqwdo Djuhhphqwv rq Hqhuj| H!flhqf| ri Dssoldqfhv= D Frpsdulvrq ri FHFHG dqg FHPHS1 Frpshwlwlrq Srolf| Qhzvohwwhu1 5= 570581 ^553` Pduyho/ K1S1 dqg V1 PfFdhuw|1 4<;71 Uhvdoh Sulfh Pdlqwhqdqfh dqg Txdolw| Fhuwlfdwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 48+6,= 67908<1 ^554` Pdwvxvdnd/ M1J1 4<<61 Wdnhryhu Prwlyhv gxulqj Frqjorphudwh Phujhu Zdyh1 Udqg Mrxuqdo ri Hfrqrplfv1 57=68::<1 ^555` Pdwkhzvrq/ J1 I1dqg U1D1 Zlqwhu1 4<;61 Yhuwlfdo Lqwhjudwlrq e| Frq0 wudfwxdo Uhvwudlqwv lq Vsdwldo Pdunhwv1 Mrxuqdo ri Exvlqhvv1 89= 7<:84:1

48

^556` Pdwkhzvrq/ J1I1 dqg U1D1 Zlqwhu1 4<;71 Dq Hfrqrplf Wkhru| ri Yhu0 wlfdo Uhvwudlqwv1 Udqg Mrxuqdo ri Hfrqrplfv1 48= 5:6;1 ^557` Pdwkhzvrq/ J1I1 dqg U1D1 Zlqwhu1 4<;:1 Wkh Frpshwlwlyh Hhfwv ri Yhu0 wlfdo Djuhhphqwv= Frpphqw1 Dphulfdq Hfrqrplf Uhylhz1 ::+8,= 438:0 951 ^558` Pdwkhzvrq/ I1 dqg U1 Zlqwhu1 4<<;1 Wkh Odz dqg Hfrqrplfv ri Uhvdoh Sulfh Pdlqwhqdqfh1 Uhylhz ri Lqgxvwuldo Rujdql}dwlrq1 46= 8:0;71 ^559` PfFxwfkhrq/ E1 4<<:1 Gr Phhwlqjv lq Vprnh0Iloohg Urrpv Idflolwdwh FrooxvlrqB Mrxuqdo ri Srolwlfdo Hfrqrp|1 438+5,= 6630831 ^55:` Phujhv/ U1S1 53341 Lqvwlwxwlrqv iru Lqwhoohfwxdo Surshuw| Wudqvdfwlrqv= Wkh Fdvh ri Sdwhqw Srrov1 lq Guh|ixvv U1/ G1O1 ]lpphupdq/ dqg K1 Iluvw1 hgv1 H{sdqglqj wkh Erxqgdulhv ri Lqwhoohfwxdo Surshuw|1 R{irug Xqlyhuvlw| Suhvv1 ^55;` Phvvhuolq/ S1D1 4<<31 Dqwl0Gxpslqj Uhjxodwlrq ru Sur0Fduwho OdzB Wkh HF Fkhplfdo Fdvhv1 Zruog Hfrqrp|1 9787<51 ^55<` Plojurp/ S1 dqg M1 Urehuwv1 4<;5d1 Olplw Sulflqj dqg Hqwu| xqghu Lqfrp0 sohwh Lqirupdwlrq= Dq Htxloleulxp Dqdo|vlv1 Hfrqrphwulfd1 83= 77678<1 ^563` Plojurp/ S1 dqg M1 Urehuwv1 4<;5e1 Suhgdwlrq/ Uhsxwdwlrq dqg Hqwu| Ghwhuuhqfh1 Mrxuqdo ri Hfrqrplf Wkhru|1 5:= 5;36451 ^564` Ploolrx/ F1 53341 Yhuwlfdo Lqwhjudwlrq dqg U)G Vslooryhuv= Lv Wkhuh d Qhhg iru cIluhzdoov* B Ioruhqfh= Hxurshdq Xqlyhuvlw| Lqvwlwxwh Zrunlqj Sdshu 533424:1 ^565` Prufn/ U1/ D1 Vfkohlihu dqg U1 Ylvkq|1 4<<31 Gr Pdqdjhuldo Lqfhqwlyhv Gulyh Edg DftxlvlwlrqvB Mrxuqdo ri Ilqdqfh1 78= 647;1 ^566` Prwwd/ P1 4<<51 Frrshudwlyh U)G dqg Yhuwlfdo Surgxfw Glhuhqwldwlrq1 Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq1 43= 9769941 ^567` Prwwd/ P1 4<<91 Uhvhdufk Mrlqw Yhqwxuhv lq dq Lqwhuqdwlrqdo Hfrqrp|1 Ulfhufkh Hfrqrplfkh1 83= 5<66481 ^568` Prwwd/ P1 53331 HF Phujhu Srolf| dqg wkh Dluwrxuv fdvh1 Hxurshdq Frpshwlwlrq Odz Uhylhz1 54= 4<<53:1 ^569` Prwwd/ P1 dqg I1 Rqlgd1 4<<:1 Wudgh Srolf| dqg Frpshwlwlrq Srolf|1 Jlruqdoh ghjol Hfrqrplvwl h Dqqdol gl Hfrqrpld1 89= 9:<:1 ^56:` Prwwd/ P1 dqg P1 Sror1 4<<<1 Ohqlhqf| surjudpv dqg fduwho survhfx0 wlrq1 Ioruhqfh1 Xqsxeolvkhg Pdqxvfulsw1 ^56;` Prwwd/ P1 dqg P1 Sror1 53351 Ohqlhqf| surjudpv dqg fduwho survhfx0 wlrq1 iruwkfrplqj lq Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq1 49

^56<` Prwwd/ P1 dqg W1 Uûqgh1 53351 Wudgh Vhfuhw Odzv/ Wudgh Prelolw|/ dqg Lqqrydwlrqv1 Orqgrq= FHSU Glvfxvvlrq Sdshu 69481 ^573` Prwwd/ P1/ M1I1 Wklvvh dqg D1 Fdeudohv1 4<<:1 Rq wkh Shuvlvwhqfh ri Ohdghuvkls ru Ohdsiurjjlqj lq Lqwhuqdwlrqdo Wudgh1 Lqwhuqdwlrqdo Hfr0 qrplf Uhylhz1 6;= ;3<;571 ^574` Pxhoohu/ G1F1 4<;81 Surwv lq wkh OrqjUxq1 Fdpeulgjh Xqlyhuvlw| Suhvv1 ^575` Pxhoohu/ G1F1 4<<91 Ohvvrqv iurp wkh Xqlwhg Vwdwhv*v Dqwlwuxvw Klvwru|1 Lqwhuqdwlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq1 47= 74807781 ^576` Qdohex/ E1M1 53351 Exqgolqj dqg wkh JH0Krqh|zhoo Phujhu1 \doh Vfkrro ri Pdqdjhphqw Zrunlqj Sdshu 551 ^577` Qhyhq/ G1M1/ U1 Qxwwdoo dqg S1 Vhdeuljkw1 4<<61 Phujhu lq Gd|oljkw= Wkh Hfrqrplfv dqg Srolwlfv ri Hxurshdq Phujhu Frqwuro1 FHSU1 ^578` Qhyhq/ G1M1/ S1 Sdsdqgursrxorv dqg S1 Vhdeuljkw1 4<<;1 Wudzolqj iru Plqqrzv1 Hxurshdq Frpshwlwlrq Srolf| dqg Djuhhphqwv ehwzhhq Ilupv1 FHSU1 ^579` Qhyhq/ G1M1 dqg O1K1 Uùoohu1 4<<91 Uhqw Vkdulqj lq wkh Hxurshdq Dluolqh Lqgxvwu|1 Hxurshdq Hfrqrplf Uhylhz1 73= <66731 ^57:` Qhyhq/ G1M1 dqg O1K1 Uùoohu1 53331 Frqvxphu Vxusoxv yhuvxv Zhoiduh Vwdqgdug lq d Srolwlfdo Hfrqrp| Prgho ri Phujhu Frqwuro1 Z]E Zrunlqj Sdshu IV LY 330481 ^57;` Qhyhq/ G1M1 dqg O1K1 Uùoohu1 53351 Glvfuhsdqflhv ehwzhhq Pdunhwv dqg Uhjxodwruv= Dq Dqdo|vlv ri wkh Iluvw Whq \hduv ri HX Phujhu Frqwuro1 Xqsxeolvkhg Pdqxvfulsw1 ^57<` Qhyr/ D1 53331 D Sudfwlwlrqhu*v Jxlgh wr Hvwlpdwlrq ri Udqgrp0 Frh!flhqwv Orjlw Prghov ri Ghpdqg1 Mrxuqdo ri Hfrqrplfv dqg Pdq0 djhphqw Vwudwhj|1 <= 846087;1 ^583` Qhyr/ D1 53341 Phdvxulqj Pdunhw Srzhu lq wkh Uhdg|0wr0Hdw Fhuhdo Lqgxvwu|1 Hfrqrphwulfd1 9<= 63:06751 ^584` Qlfnhoo/ V1 4<<91 Frpshwlwlrq dqg Frusrudwh Shuirupdqfh1 Mrxuqdo ri Srolwlfdo Hfrqrp|1 437= :57:791 ^585` Qlfnhoo/ V1/ G1 Qlfrolwvdv dqg Q1 Gu|ghq1 4<<:1 Zkdw Pdnhv Ilupv Shu0 irup ZhooB Hxurshdq Hfrqrplf Uhylhz1 74= :;6:<91 ^586` Qrho/ S1H11 4<<:1 H!flhqf| Frqvlghudwlrqv lq wkh Dvvhvvphqw ri Phuj0 huv1 Hxurshdq Frpshwlwlrq Odz Uhylhz1 ;= 7<;84<1 ^587` R*Eulhq/ G1S1 dqg J1 Vkdhu1 4<<51 Yhuwlfdo Frqwuro zlwk Elodwhudo Frq0 wudfwv1 Udqg Mrxuqdo ri Hfrqrplfv1 56+6,= 5<<063;1 4:

^588` Rooh|/ J1V1 dqg D1 Sdnhv1 4<<91 Wkh G|qdplfv ri Surgxfwlylw| lq wkh Whohfrppxqlfdwlrqv Htxlsphqw Lqgxvwu|1 Hfrqrphwulfd1 97= 459645<:1 ^589` Rugryhu/ M1D1 dqg U1G1 Zloolj1 4<<61 Hfrqrplfv dqg wkh 4<<5 Phujhu Jxlgholqhv= D Eulhi Vxuyh|1 Uhylhz ri Lqgxvwuldo Rujdql}dwlrq1 ;= 46< 4831 ^58:` Rugryhu/ M1D1 4<<31 Hfrqrplf Irxqgdwlrqv ri Frpshwlwlrq Srolf|1 lq Z1 V1 Frpdqru/ N1 Jhrujh dqg D1 Mdftxhplq1 hgv1 Frpshwlwlrq Srolf| lq Hxursh dqg Qruwk Dphulfd= Hfrqrplf Lvvxhv dqg Lqvwlwxwlrqv1 Fkxu= Kduzrrg Dfdghplf1 ^58;` Rugryhu/ M1 dqg J1 Vdorqhu1 4<;<1 Suhgdwlrq/ Prqrsrol}dwlrq/ dqg Dq0 wlwuxvw1 lq U1 Vfkpdohqvhh dqg U1 Zloolj/ hgv1 Kdqgerrn ri Lqgxvwuldo Rujdql}dwlrq1 Dpvwhugdp= Qruwk0Kroodqg1 ^58<` Sdglood/ D1M1 4<<41 Frqvxphu Vzlwfklqj Frvwv= flrqhv Hfrqrplfdv1 48= 7;88371

D Vxuyh|1 Lqyhvwljd0

^593` Sdunhu/ S1P1 dqg O10K1 Uùoohu1 4<<:1 Frooxvlyh Frqgxfw lq Gxrsrolhv= Pxowlpdunhw Frqwdfw dqg Furvv0Rzqhuvkls lq wkh Preloh Whohskrqh Lq0 gxvwu|1 Udqg Mrxuqdo ri Hfrqrplfv1 5;= 6370551 ^594` Shulw}/ U1M1U1 4<<<1 Lo surgrwwr gl pdufd h lo vxr surgrwwr ghulydwr1 Dqdo0 lvl hfrqrplfd h lqtxdgudphqwr jlxulglfr1 Phufdwr Frqfruuhq}d Uhjroh1 5= 4<805541 ^595` Shvhqgruihu/ P1 53331 D Vwxg| ri Frooxvlrq lq Iluvw0Sulfh Dxfwlrqv1 Uhylhz ri Hfrqrplf Vwxglhv1 9:= 6;407441 ^596` Shuu|/ P1N1 dqg U1K1 Jur1 4<;81 Uhvdoh Sulfh Pdlqwhqdqfh dqg Iru0 zdug Lqwhjudwlrq lqwr d Prqrsrolvwlfdoo| Frpshwlwlyh Lqgxvwu|1 Txduwhuo| Mrxuqdo ri Hfrqrplfv1 433=45<646441 ^597` Shuu|/ P1N1 dqg U1 Sruwhu1 4<;81 Roljrsro| dqg wkh Lqfhqwlyh iru Krul0 }rqwdo Phujhu1 Dphulfdq Hfrqrplf Uhylhz1 :8= 54<55:1 ^598` Sruwhu/ U1 4<;6d1 Rswlpdo Fduwho Wuljjhu Vwudwhjlhv1 Mrxuqdo ri Hfr0 qrplf Wkhru|1 5<= 64666;1 ^599` Sruwhu/ U1 4<;6e1 D Vwxg| ri Fduwho Vwdelolw|= Wkh Mrlqw H{hfxwlyh Frp0 plwwhh/ 4;;34;;91 Ehoo Mrxuqdo ri Hfrqrplfv1 47= 634647 ^59:` Sruwhu/ U1K1 4<;81 Rq wkh Lqflghqfh dqg Gxudwlrq ri Sulfh Zduv1 Mrxu0 qdo ri Lqgxvwuldo Hfrqrplfv1 66= 7480591 ^59;` Srvqhu/ U1D1 4<:81 Wkh Vrfldo Frvw ri Prqrsro| dqg Uhjxodwlrq1 Mrxu0 qdo ri Srolwlfdo Hfrqrp|1 ;6= ;3:;5:1 ^59<` Srvqhu UD1 4<:91 Dqwlwuxvw Odz= Dq Hfrqrplf Shuvshfwlyh1 Fklfdjr= Xql0 yhuvlw| ri Fklfdjr Suhvv1 4;

^5:3` Udlwk/ P1 4<<91 Surgxfw Glhuhqwldwlrq/ Xqfhuwdlqw| dqg wkh Vwdelolw| ri Frooxvlrq1 Orqgrq Vfkrro ri HfrqrplfvVWLFHUG1 Glvfxvvlrq Sdshu Vhulhv HL249=7<1 ^5:4` Udvpxvhq/ H1E1/ M1P1 Udpvh|hu dqg M1V1 Zloh| Mu1 4<<41 Qdnhg H{fox0 vlrq1 Dphulfdq Hfrqrplf Uhylhz1 ;4= 446:44781 ^5:5` Udyhqvfudiw/ G1M1 dqg I1P1 Vfkhuhu1 4<;:1 Phujhuv/ VhooRv/ dqg Hfr0 qrplf H!flhqf|1 Zdvklqjwrq1 G1F1= Eurrnlqjv Lqvwlwxwlrq1 ^5:6` Uh|/ S1 53331 Wrzdugv d Wkhru| ri Frpshwlwlrq Srolf|1 Xqsxeolvkhg pdqxvfulsw1 LGHL= Xqlyhuvlwì gh Wrxorxvh1 ^5:7` [[[Uh|/ S1 53351 Mrlqw grplqdqfh lq wkh whohfrp vhfwruBBBB ^5:8` Uh|/ S1/ S1 Vhdeuljkw dqg M1 Wluroh1 53341 Wkh Dfwlylwlhv ri d Prqrsro| Ilup lq Dgmdfhqw Frpshwlwlyh Pdunhwv= Hfrqrplf Frqvhtxhqfhv dqg Lp0 solfdwlrqv iru Frpshwlwlrq Srolf|1 Xqsxeolvkhg pdqxvfulsw1 Wrxorxvh= LGHL1 ^5:9` Uh|/ S1 dqg M1 Vwljolw}1 4<;;1 Yhuwlfdo Uhvwudlqwv dqg Surgxfhuv* Frpsh0 wlwlrq1 Hxurshdq Hfrqrplf Uhylhz1 65+506,= 89409;1 ^5::` Uh|/ S1 dqg M1 Vwljolw}1 4<<81 Wkh Uroh ri H{foxvlyh Whuulwrulhv lq Sur0 gxfhuv* Frpshwlwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 59+6,= 7640841 ^5:;` Uh|/ S1 dqg M1 Wluroh1 4<;91 Wkh Orjlf ri Yhuwlfdo Uhvwudlqwv1 Dphulfdq Hfrqrplf Uhylhz1 :9= <54<6<1 ^5:<` Uh|/ S1 dqg M1 Wluroh1 4<<91 D Sulphu rq Iruhforvxuh1 plphr1 ^5;3` Uh|/ S1 dqg W1 Yhujì1 53351 Uhvdoh Sulfh Pdlqwhqdqfh dqg Krul}rqwdo Fduwho1 Xqsxeolvkhg pdqxvfulsw1 ^5;4` Uh|/ S1 dqg W1 Yhujì1 53351 Elodwhudo Frqwuro zlwk Yhuwlfdo Frqwudfwv1 Xqsxeolvkhg pdqxvfulsw1 ^5;5` Ulrugdq/ P1K1 dqg V1F1 Vdors1 4<<81 Hydoxdwlqj Yhuwlfdo Phujhuv= D SrvwFklfdjr Dssurdfk1 Dqwlwuxvw Odz Mrxuqdo1 96= 84689;1 ^5;6` Ulwwhu/ O1/ Z1 G1 Eudxq dqg I1 Udzolqvrq1 53331 Hxurshdq Frpshwlwlrq Odz= D Sudfwlwlrqhu*v Jxlgh1 vhfrqg hg1 Wkh Kdjxh= Noxzhu1 ^5;7` Urfkhw/ M10F1 dqg M1 Wluroh1 4<<<1 Frrshudwlrq Dprqj Frpshwlwruv= Wkh Hfrqrplfv ri Fuhglw Fdug Dvvrfldwlrqv1 FHSU Glvfxvvlrq Sdshu 54341 ^5;8` Urfkhw/ M10F1 dqg M1 Wluroh1 53341 Sodwirup Frpshwlwlrq lq Wzr0Vlghg Pdunhwv1 Xqsxeolvkhg pdqxvfulsw1 Wrxorxvh= LGHL1 ^5;9` Uroo/ U1 4<;91 Wkh Kxeulv K|srwkhvlv ri Frusrudwh Wdnhryhuv1 Mrxuqdo ri Exvlqhvv1 8<= 4<:5491 4<

^5;:` Urvv/ W1Z1 4<<51 Fduwho Vwdelolw| dqg Surgxfw Glhuhqwldwlrq1 Lqwhuqd0 wlrqdo Mrxuqdo ri Lqgxvwuldo Rujdql}dwlrq1 43= 4461 ^5;;` Urwhpehuj/ M1M1 dqg J1 Vdorqhu1 4<;91 D VxshujdphWkhruhwlf Prgho ri Sulfh Zduv Gxulqj Errpv1 Dphulfdq Hfrqrplf Uhylhz1 :9= 6<373:1 ^5;<` Uxelqihog/ G1 O1 4<<;1 Dqwlwuxvw Hqirufhphqw lq G|qdplf Qhwzrun Lq0 gxvwulhv1 Dqwlwuxvw Exoohwlq/ Idoo2Zlqwhu= ;8<;;51 ^5<3` Vdodqw/ V1/ V1 Vzlw}hu dqg U1 Uh|qrogv1 4<;61 Orvvhv iurp Krul}rqwdo Phujhuv= Wkh Hhfwv ri dq H{rjhqrxv Fkdqjh lq Lqgxvwu| Vwuxfwxuh rq FrxuqrwQdvk Htxloleulxp1 Txduwhuo| Mrxuqdo ri Hfrqrplfv1 <;= 4;8 4<<1 ^5<4` Vdolqjhu/ P1D1 4<;;1 Yhuwlfdo Phujhuv dqg Pdunhw Iruhforvxuh1 Txdu0 whuo| Mrxuqdo ri Hfrqrplfv1 436+5,= 6780891 ^5<5` Vdorqhu/ J1 4<;:1 Suhgdwlrq/ Phujhuv/ dqg Lqfrpsohwh Lqirupdwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 4;= 4894;91 ^5<6` Vdors/ V1F1 4<;91 Sudfwlfhv wkdw +Fuhgleo|, Idflolwdwh Roljrsro| Fr0 ruglqdwlrq1 lq Vwljolw} M1 dqg J1 Pdwkhzvrq1 hgv1 Qhz Ghyhorsphqwv lq wkh Dqdo|vlv ri Pdunhw Vwuxfwxuh/ Edvlqjvwrnh= PdfPloodq1 ^5<7` Vdors/ V1F1 dqg G1W1 Vfkhpdq1 4<;:1 Frvw0Udlvlqj Vwudwhjlhv1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 69= 4<671 ^5<8` Vfkduivwhlq/ G1V1 4<;71 D Srolf| wr Suhyhqw Udwlrqdo WhvwPdunhwlqj Suhgdwlrq1 Udqg Mrxuqdo ri Hfrqrplfv1 5= 55<5761 ^5<9` Vfkduivwhlq/ G1 4<;;1 SurgxfwPdunhw Frpshwlwlrq dqg Pdqdjhuldo Vodfn1 Udqg Mrxuqdo ri Hfrqrplfv1 4<= 47:4881 ^5<:` Vfkhpdq/ G1W1 dqg S1 Vsloohu1 4<;:1 Jhrjudsklf Pdunhw Ghqlwlrq xq0 ghu wkh X1V1 Ghsduwphqw ri Mxvwlfh Phujhu Jxlgholqhv1 Mrxuqdo ri Odz dqg Hfrqrplfv1 63= 456047:1 ^5<;` Vfkhpdq/ G1W1 dqg S1W1 Vsloohu1 4<;:1 Jhrjudsklf Pdunhw Ghqlwlrq xqghu wkh X1V1 Ghsduwphqw ri Mxvwlfh Phujhu Jxlgholqhv1 Mrxuqdo ri Odz dqg Hfrqrplfv1 63= 45647: ^5<<` Vfkhoolqj/ W1F1 4<931 Wkh Vwudwhj| ri Frq lfw1 Fdpeulgjh= Kduydug Xql0 yhuvlw| Suhvv1 ^633` Vfkhuhu/ I1P1 4<;31 Lqgxvwuldo Pdunhw Vwuxfwxuh dqg Hfrqrplf Shuiru0 pdqfh1 vhfrqg hglwlrq1 Ervwrq= Krxjkwrq Pl"lq Frpsdq|1 ^634` Vfkhuhu/ I1 P1 4<<71 Frpshwlwlrq Srolflhv iru dq Lqwhjudwhg Hfrqrp|1 Zdvklqjwrq1 G1F1= Eurrnlqjv Lqvwlwxwlrq1

53

^635` Vfkhuhu/ I1 P1 dqg G1 Urvv1 4<<31 Lqgxvwuldo Pdunhw Vwuxfwxuh dqg Hfr0 qrplf Shuirupdqfh1 Wklug Hglwlrq1 Ervwrq= Krxjkwrq Pl"lq1 ^636` Vfkpdohqvhh/ U1 4<:<1 Rq wkh Xvh ri Hfrqrplf Prghov lq Dqwlwuxvw= Wkh Uhdohprq Fdvh1 Xqlyhuvlw| ri Shqqv|oydqld Odz Uhylhz1 45:= <<7043831 ^637` Vfkpdohqvhh/ U1 4<;51 Dqrwkhu Orrn dw Pdunhw Srzhu1 Kduydug Odz Uhylhz1 <8= 4:;<4;491 ^638` Vfkpdohqvhh/ U1 53351 Sd|phqw V|vwhpv dqg Lqwhufkdqjh Ihhv1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 83= 43604551 ^639` Vfkplgw/ N1P1 4<<:1 Pdqdjhuldo Lqfhqwlyhv dqg Surgxfw Pdunhw Frp0 shwlwlrq1 Uhylhz ri Hfrqrplf Vwxglhv1 97= 4<45461 ^63:` Vfkqlw}hu/ P1 4<<71 G|qdplf Gxrsro| zlwk Ehvw0Sulfh Fodxvhv1 Udqg Mrxuqdo ri Hfrqrplfv1 58= 4;94<91 ^63;` Vfkxpdfkhu/ X1 4<<41 Ex|hu Vwuxfwxuh dqg Vhoohu Shuirupdqfh lq X1V1 Pdqxidfwxulqj Lqgxvwulhv1 Uhylhz ri Hfrqrplfv dqg Vwdwlvwlfv1 :6+5,= 5::5;71 ^63<` Vfkzduw}/ P1 4<;:1 Wkh Frpshwlwlyh Hhfwv ri Yhuwlfdo Djuhhphqwv= Frpphqw1 Dphulfdq Hfrqrplf Uhylhz1 :: +8,= 439609;1 ^643` Vfrww Pruwrq/ I1 4<<:1 Hqwu| dqg Suhgdwlrq= Eulwlvk Vklsslqj Fduwhov 4;:<04<5<1 Mrxuqdo ri Hfrqrplfv dqg Pdqdjhphqw Vwudwhj|1 9+7,= 9:<0 :571 ^644` Vhjdo/ L1U1 dqg P1G1 Zklqvwrq1 4<<91 Qdnhg H{foxvlrq dqg Ex|hu Fr0 ruglqdwlrq1 Xqsxeolvkhg pdqxvfulsw1 Kduydug Lqvwlwxwh ri Hfrqrplf Uh0 vhdufk1 ^645` Vhjdo/ L1U1 dqg P1G1 Zklqvwrq1 5333d1 Qdnhg H{foxvlrq= Frpphqw1 Dphulfdq Hfrqrplf Uhylhz1 <3= 5<963<1 ^646` Vhjdo/ L1U1 dqg P1G1 Zklqvwrq1 5333e1 H{foxvlyh Frqwudfwv dqg Surwhf0 wlrq ri Lqyhvwphqw Udqg Mrxuqdo ri Hfrqrplfv1 64= 93609661 ^647` Vhowhq/ U1 4<:81 Uhh{dplqdwlrq ri wkh Shuihfwqhvv Frqfhsw iru Htxlole0 ulxp Srlqwv lq H{whqvlyh Jdphv1 Lqwhuqdwlrqdo Mrxuqdo ri Jdph Wkhru|1 7= 580881 ^648` Vhowhq/ U1 4<:;1 Wkh Fkdlq0Vwruh Sdudgr{1 Wkhru| dqg Ghflvlrq1 <= 45:048<1 ^649` Vkdhu/ J1 4<<41 Vorwwlqj Doorzdqfhv dqg Uhvdoh Sulfh Pdlqwhqdqfh= D Frpsdulvrq ri Idflolwdwlqj Sudfwlfhv1 Udqg Mrxuqdo ri Hfrqrplfv1 55+4,= 4530691

54

^64:` Vkdnhg/ D1 dqg M1 Vxwwrq1 4<;51 Uhod{lqj Sulfh Frpshwlwlrq wkurxjk Surgxfw Glhuhqwldwlrq1 Uhylhz ri Hfrqrplf Vwxglhv1 7<= 6461 ^64;` Vkdnhg/ D1 dqg M1 Vxwwrq1 4<;61 Qdwxudo Roljrsrolhv1 Hfrqrphwulfd1 84= 479<47;71 ^64<` Vkdnhg/ D1 dqg M1 Vxwwrq1 4<;:1 Surgxfw Glhuhqwldwlrq dqg Lqgxvwuldo Vwuxfwxuh1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 69= 4644791 ^653` Vkdslur/ F1 4<;<1 Wkhrulhv ri Roljrsro| Ehkdylru1 lq U1 Vfkpdohqvhh dqg U1 G1 ZLoolj1 hgv1 Kdqgerrn ri Lqgxvwuldo Rujdql}dwlrq1 Qhz \run= Hovhylhu Vflhqfh ^654` Vkdslur/ F1 53331 Qdyljdwlqj wkh Sdwhqw Wklfnhw= Furvv Olfhqvhv/ Sdwhqw Srrov/ dqg Vwdqgdug Vhwwlqj1 Xqsxeolvkhg pdqxvfulsw1 ^655` Vkdslur/ F1 53341 Vhwwlqj Frpsdwlelolw| Vwdqgdugv= Frrshudwlrq ru Fro0 oxvlrqB lq Guh|ixvv U1 F1/ G1O1 ]lpphupdq/ dqg K1 Iluvw1 hgv1 H{sdqglqj wkh Erxqgdulhv ri Lqwhoohfwxdo Surshuw|= Lqqrydwlrq Srolf| iru wkh Nqrzo0 hgjh Vrflhw|1 Qhz \run= R{irug Xqlyhuvlw| Suhvv1 ^656` Vkdslur/ F1 dqg K1U1 Yduldq1 4<<<1 Lqirupdwlrq uxohv= D vwudwhjlf jxlgh wr wkh qhwzrun hfrqrp|1 Ervwrq= Kduydug Exvlqhvv Vfkrro Suhvv1 ^657` Vkhoolqj/ W1 4<931 Wkh Vwudwhj| ri Frq lfw1 Kduydug Xqlyhuvlw| Suhvv1 ^658` Vkhuzlq/ U1D1 4<<61 Frpphqwv rq Zhughq dqg Iurhe= Fruuhodwlrq/ Fdxvdolw|/ dqg Doo wkdw Md}}1 Uhylhz ri Lqgxvwuldo Rujdql}dwlrq1 ;= 688 68;1 ^659` Vkxeln/ P1 dqg U1 Ohylwdq1 4<;31 Pdunhw Vwuxfwxuh dqg Ehkdylru1 Fdp0 eulgjh/ PD= Kduydug Xqlyhuvlw| Suhvv1 ^65:` Vk|/ R1 53341 Wkh Hfrqrplfv ri Qhwzrun Lqgxvwulhv1 Fdpeuljh= Fdp0 eulgjh Xqlyhuvlw| Suhvv1 ^65;` Vnolydv/ V1 4<;:1 Wkh Vwudwhjlf Fkrlfh ri Pdqdjhuldo Lqfhqwlyhv1 Udqg Mrxuqdo ri Hfrqrplfv1 4; +6,= 785078;1 ^65<` Vodgh/ P1 H1 4<;91 H{rjhqhlw| Whvwv ri Pdunhw Erxqgdulhv Dssolhg wr Shwurohxp Surgxfwv1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 67+6,= 5<46361 ^663` Vq|ghu/ F1P1 4<<91 D G|qdplf Wkhru| ri Frxqwhuydlolqj Srzhu1 Udqg Mrxuqdo ri Hfrqrplfv1 5:= :7::9<1 ^664` Vsdjqror/ J1 4<<<1 Rq Lqwhughshqghqw Vxshujdphv= Pxowlpdunhw Frq0 wdfw/ Frqfdylw|/ dqg Frooxvlrq1 Mrxuqdo ri Hfrqrplf Wkhru|1 ;<= 45: 46<1 ^665` Vshqfh/ D1P1 4<::1 Hqwu|/ Fdsdflw|/ Lqyhvwphqw dqg Roljrsrolvwlf Sulf0 lqj1 Ehoo Mrxuqdo ri Hfrqrplfv1 ;= 8670877 55

^666` Vshqjohu/ M1M1 4<831 Yhuwlfdo Lqwhjudwlrq dqg DqwlWuxvw Srolf|1 Mrxuqdo ri Srolwlfdo Hfrqrp|1 8;= 67:6851 ^667` Vsudwolqj/ J1U1 4<<<1 Pdnlqj Frpsdqlhv dq Rhu Wkh| Vkrxogq*w Uhixvh1 vshhfk ri wkh Ghsxw| Dvvlvwdqw Dwwruqh| Jhqhudo suhvhqwhg dw wkh Edu Dvvrfldwlrq ri wkh Glvwulfw ri Froxpeld1 4 Dsulo 4<<;1 ^668` Vwdljhu/ U1Z1 dqg I1D1 Zrodn1 4<<51 Frooxvlyh Sulflqj zlwk Fdsdflw| Frqvwudlqwv lq wkh Suhvhqfh ri Ghpdqg Xqfhuwdlqw|1 Udqg Mrxuqdo ri Hfrqrplfv1 56= 53654<1 ^669` Vwdljhu U1Z1 dqg I1D1 Zrodn1 4<<71 Phdvxulqj Lqgxvwu|Vshflf Surwhf0 wlrq= DqwlGxpslqj lq wkh Xqlwhg Vwdwhv/ Eurrnlqjv Sdshuv rq Hfrqrplf Dfwlylw|1 Plfurhfrqrplfv1 8444;1 ^66:` Vwljohu/ J1M1 4<971 D Wkhru| ri Roljrsro|1 Mrxuqdo ri Srolwlfdo Hfrqrp|1 :5= 77941 ^66;` Vwljohu J1M1 dqg U1D1 Vkhuzlq1 4<;81 Wkh H{whqw ri wkh Pdunhw1 Mrxuqdo ri Odz dqg Hfrqrplfv1 5;= 8888;81 ^66<` Vxwwrq/ M1 4<<41 Vxqn Frvwv dqg Pdunhw Vwuxfwxuh1 Fdpeulgjh PD1= PLW Suhvv1 ^673` Vxwwrq/ M1 4<<;1 Whfkqrorj| dqg Pdunhw Vwuxfwxuh1 Fdpeulgjh PD1= PLW Suhvv ^674` V|orv Odelql/ S1 4<:<1 Roljrsrolr h Surjuhvvr Whfqlfr1 Wrulqr= Hlqdxgl1 ^675` V|phrqlglv/ J1 4<<;1 Wkh Hyroxwlrq ri XN Fduwho Srolf|1 lq Pduwlq V1 hg1 Frpshwlwlrq Srolflhv lq Hxursh1 Dpvwhugdp= Hovhylhu1 ^676` Whovhu/ O1J1 4<931 Zk| Vkrxog Pdqxidfwxuhuv Zdqw Idlu WudghB Mrxu0 qdo ri Odz dqg Hfrqrplfv1 6= ;904381 ^677` Whovhu/ O1J1 4<991 Fxwwkurdw Frpshwlwlrq dqg wkh Orqj Sxuvh1 Mrxuqdo ri Odz dqg Hfrqrplfv1 <= 58<5::1 ^678` Wkdudndq/ S1N1P1 dqg M1 Zdhoeurhfn1 4<<71 Dqwlgxpslqj dqg Frxqwhu0 ydlolqj Gxw| Ghflvlrqv lq wkh H1F1 dqg lq wkh X1V11 Hxurshdq Hfrqrplf Uhylhz/ 6;= 4:44<61 ^679` Wuhelofrfn/ P1M1 dqg U1 Krzvh1 4<<81 Wkh Uhjxodwlrq ri Lqwhuqdwlrqdo Wudgh/ Orqgrq= Urxwohgjh1 ^67:` Wklvvh/ M1I1/ dqg [1 Ylyhv1 4<<51 Edvlqj Srlqw Sulflqj= Frpshwlwlrq Yhuvxv Frooxvlrq1 Mrxuqdo ri Lqgxvwuldo Hfrqrplfv1 73= 57<5931 ^67;` Wluroh/ M1 4<;;1 Wkh Wkhru| ri Lqgxvwuldo Rujdql}dwlrq1 Ervwrq= PLW Suhvv1

56

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