Copyright. William Joseph Brennan

Transcription

1 Coyrght by Wlla Joeh Brea 005

2 The Dertato Cottee for Wlla Joeh Brea ertfe that th the aroved vero of the followg dertato: Arle Prg ad Caaty Behavor Cottee: Maxwell Sthobe, Co-Suervor. Preto MAfee, Co-Suervor Dale O. Stahl Davd Sbley Peter Wloxe Vjay Mahaja

3 Arle Prg ad Caaty Behavor by Wlla Joeh Brea, B.S.; M.S. Dertato Preeted to the aulty of the Graduate Shool of the Uverty of Texa at Aut Partal ulfllet of the eureet for the Degree of Dotor of Phloohy The Uverty of Texa at Aut Augut, 005

4 Dedato: I dedate th dertato to y aret Wlla Joeh Brea, Jr. ad Kathlee Mary Brea who have uorted e every way toward y oleto of the Ph.D. They have alway eouraged e to do y bet ad th work to ther uort ad redt. I would alo lke to dedate th dertato to y ter Kaela Brea ad brother Patrk ad J Brea. They have alo bee very uortve wrtg th dertato. I d lke to thak y gradother Kathlee Dllo ad Godother Magg Dua for all of ther eourageet. I d lke to thak Ale Chakkalakal for all of her uort. I d lke to thak y fred ludg Mke ell, Mark Daege, Pat Deey, Darre ad Mhaela Cook, r. Er Shel, Phl ad Mega Tok, Bra ad Marta Coor, Chrta rak, Mary ega, Doug Maurer, a Kuazawa who eouraged e to jut la [the age] together, Mela ala, Da ad Carol Per, Paul abarre, Sa ad Kathlee auh, al ad Na Kueh, the rao faly, ad Da ad a Gayor. All of the lu ay ore uetoed fred eouraged e ad lteed to ay a overato about the roe ad about lfe geeral. I d lke to thak Dr. udo eh, Dr. lly Stoller, ad Sady Tradahl for ther uort. ally I d lke to thak y ule Gerry Dllo for h uort. At oe ot he ad I were about the ae ot dfferet Ph.D. rogra but duty to h outry ha reveted h fro reahg th tage at th ot h lfe.

5 Akowledgeet: I would lke to thak y Co-Char, Dr.. Preto MAfee, for all of h oet, uggeto, ad feedbak o th dertato. I a greatly hoored to have had h erve. I would alo lke to thak y other Co-Char, Dr. Max Sthobe, for all of h oet ad uggeto. I a alo truly hoored to have had h erve. e too ha bee very helful, eeally ear the ed of the roe. I would lke to thak Dr. Pete Wloxe for all of h oet, uggeto, ad uort o the dertato ad h hel develog the C outer rogra that allowed e to vew the arle data Chater. I would lke to thak Dr. Dale Stahl, Dr. Davd Sbley, ad Dr.Vjay Mahaja for all of ther oet ad uggeto a ottee eber. I would lke to thak Dr. Doug Day for attedg y oral defee ad rovdg trgug ueto ad uort durg the defee ad durg the Uverty of Texa Job Plaeet Sear. I would alo lke to thak Dr. Da Slek, Dr. Ve Gera, Dae Eldrdge, ad the Uverty of Texa Job Plaeet Sear for ueful oet. I would lke to thak Dr. Ele. Eheverr-Carroll, Dr. oue Woltz, Dr. Darryl Youg, uha Wu, Shae Carboeau, Abe Du, Brett Wedlg, a Dko, ad h Przao for all of ther ueful oet ad duo. ally, I would lke to thak Vva Golda-effler for all of her uort o the adtratve ad eroal level. Truly he oe great Graduate Coordator to have o your de. Ay error are trtly y ow. v

6 Arle Prg ad Caaty Behavor Publato No. Wlla Joeh Brea, Ph.D. The Uverty of Texa at Aut, 005 Suervor:. Preto MAfee Maxwell Sthobe Stadard odel of re dero Butter, Vara, Burdett & Judd gve oe exlaato of how eulbra of fr ellg the ae rodut at dfferet re our by rovdg ouer wth dered forato or loyalty but a oo reervato re. Th odel exted thee odel by havg bue traveler ayg ore tha tourt. There a otuu of re broke by a ga rght above the oooly re of the tourt. I th rego, exeted fr roft are lower, a fr do ot ake u for the drete lo of leure traveler. The odel oared to Data Bak A a te-eret rado ale of arle tket terary. Idvdual arle route are how to have u to everal eak the etated re kerel dety. Th where the theory athe the data: arle fare luter aroud re. Kre ad Sheka geerate a lted aaty odel that geerate re radozato whe two fr have ueual aate. The large aaty fr ha the ablty to et re lower tha the aller fr. Butter, Vara, Burdett & Judd develo re dero odel a derbed before. A yetr two-fr odel develoed that ha both feature, lted aaty ad lted re forato, thu ufyg two v

7 darate lterature. The lower re fr ell out to aaty, whle the hgher re fr reeve the leftover utoer fro the old out fr lu the loyal utoer who dd ot ee the lower red fr. Th lead to re radozato. ted forato ad aaty are thu detal eoo effet. There a varety of ze of loyal utoer wth the arle, hotel, ad ar retal oae. A re radozato odel reated wth oe large fr of loyal utoer ad ay aller fr havg the ae ze of loyal utoer. r radoze by hargg the oooly re for the loyal utoer ad doutg to obta a grou of hoer eeg the lowet re of all the fr. The larget fr ha a ato at the hghet re the dtrbuto. The aller fr oete for hoer at all re the dtrbuto ad have o ato. v

8 Table of Cotet 0. INTODUCTION. 0. Bue ad eure Traveler reuet ler Progra.. 0. Dfferg Sze of reuet ler Progra.5 0. The Elte reuet ler Yeld Maageet ad Caaty tato The e of ow-cot Carrer ad allg Average Tket Pre. 0.7 Pre Dero. 0.8 Suary of Develoet Se Deregulato Suary of Dertato uture eearh...5. PICE DISPESION WIT DIEING CONSUME VAUES..6. Itroduto 6. terature evew Motvato.. 6. The Model...5 Coaratve Stat Exlag the Grah: Mathg the Coaratve Stat to Arle Data...6a Dug the Couter Geerated Grah....6b Mathg the Couter Grah to the Arle Data 7.7 Aaly ad Coluo.. v

9 . CAPACITY AND ANDOM PICES...9. Itroduto..9. The Syetr Two-r Model 5. Geeralzg to the Cae Where All But Oe of the r are Sold to Caaty Geeralzg to the Ayetr Model Buldg Caaty rt ad the Settg Pre Coluo...8. EQUIIBIUM PICE ANDOMIZATION WIT ASYMMETIC CONSUME OYATY Itroduto..85. The Model wth Two r.87. Coaratve Stat wth the Two r Model The Model wth r arge ad - Slar Sall r.58.5 Coaratve Stat wth the Modfed -r Model 6.6 Coluo... 7 APPENDIX PD, VAUATION CASE, AND DATA GAPS... BIBIOGAPY..6 VITA...68 x

10 Itroduto 0-. Bue ad eure Traveler Se deregulato, the arle dutry ha bee a dya ad extg dutry roe to ay hage. Several ajor tred develoed. The frt that arle foud that ot all traveler are the ae. Soe traveled ore freuetly tha other ad were wllg to ay uh ore for ther tket. Arle foud that a ot deregulato evroet there are bue traveler ad leure traveler. Bue traveler are thoe wllg to ay to dollar for flght ad akg reeat bue. Arle deed o th aller grou of traveler to ake u a ubtatal roorto of ther reveue. There are a varety of etate of how bg the overall grou of bue traveler grou to the arle. Durg the exao of the late 990, Uted Arle had 9% of t reu bue traveler ayg to fare geeratg 6% of the etre oay reveue! Braga, Carey, MCartey The reveue rodued today by th grou of traveler uh lower today. Southwet Arle etate that thrty fve eret of t aeger ay full fare Koeg. The Natoal Bue Travel Aoato etate that oe thrd of traveler deeed to be bue aout for oe half of arle reveue Bjorhu. The Aera Traort Aoato ore lberally etate that 5% - 55% of arle aeger that travel for bue uroe ore 60 to 70 eret of a arle reveue edor. Soe traveler obe bue wth leure o th etate ould be hgh.

11 Deregulato reulted the otued re of the leure la. eure traveler geerally earh vgorouly for lower fare a they are ot a wllg to ay a hgh fare a bue traveler. They geerally do ot ake a ay reeat tr a bue traveler. There are geerally ore leure traveler tha bue traveler, eve though bue traveler are the oe akg the tr uh ore freuetly. ro the etate above there are aywhere fro oe half to two thrd leure traveler o a average flght. 0-. reuet ler Progra ro a arle eretve, bue traveler are ther bet utoer beaue they are wllg to ay hgher fare ad they travel ore freuetly. Arle wat the bue traveler ad the leure traveler f they otue to ake reeat tr to kee akg reeat traato wth ther oay. The freuet fler rogra wa frt develoed by Aera Arle 98, ad oo oed by other arle, to reward traveler that ade freuet tr wth a artular arle Baley, Graha, ad Kala,. 60. Traveler ould reeve erk uh a early hek-, earate reervato atae, ugrade to frt la, free flght to regular detato the arle erve, ad free flght to exot detato that the arle erved uh a the Carbbea, Euroe, or awa. Eah of thee erk deed o the uber of freuet fler le that a traveler eared wth a arle. Th, tur, deeded o a ba level how ofte a

12 traveler traveled wth a arle, how far wth a arle a traveler travel, what fare a traveler ay, ad today what reogzed tatu the traveler ha wth the arle. The ore le a traveler obta, the ore hoe the traveler ha wth whh award a be redeeed wth the le. The freuet fler rogra hel reate a loyal bae of traveler to dvdual arle. Bue traveler who lt ther bue betwee everal arle loe out o the arual of beeft that lag ther bue wth oe arle ould rovde. The etve the freuet fler rogra trog for traveler to be loyal to a artular arle or two. Th, tur aue traveler freuet fler rogra, eeally bue traveler, to be le re etve to other arle. Th a be ee wth the followg exale. Suoe that a bue traveler hort,000 le of the 5,000 award level for a free reu frt la tket the eak eao betwee the Uted State ad Euroe ad dere the award for a laed vaato ad urhag a roudtr tket tr betwee New York ad o Agele a dtae that would ut h or her over the award lt. ow uh heaer ut the tket be o a oetg arle that the traveler ha o te for the bue traveler to wth or eve gve oderable oderato, eeally f the bue traveler ot footg the bll? Southwet Arle ad eward Progra doe ot deed o atual le flow but o eget flow. Thu a Aut Dalla ove eget out a uh toward a ad eward redt a a Phoex Baltore eget. Mot other arle have a u floor of 500 le that a be eared by a eget o ther freuet fler rogra. Mot of the legay arrer have etablhed the lat deade rogra that reward traveler that ake heavy travel wth the at year. There are geerally three level baed o the uber of eget traveled or the uber of atual le traveled. The reward for obtag oe of thee level are aoated wth freuet fler wth very hgh leage uh a early hek-, earate reervato atae, et. Ug Aera Arle Advatage eward Progra foud at

13 eure traveler a be ofteed a lar way to a leer extet by freuet fler rogra. Wherea leure traveler ay ot travel a uh a bue traveler, they a auulate freuet fler le through rooto uh a redt ard ot, auto retal, ad hotel rogra. Aordg to ady Petere, Publher/Edtor Ide lyer agaze, 0% of all freuet fler le are auulated by ot flyg htt:// The fatet growg eget of the oulato wth freuet fler le thoe that are `leage ouer ot freuet fler Ibd. Wth freuet fler le beg auulated by a la of eole ot travelg very uh, the oaoal flght by thee ouer ore lkely to be doe o the arle that they have le, eeally f they are loe to a free tket. Wth the troduto of freuet fler rogra, a la of traveler loyal to a artular arle ha bee reated e deregulato. Mot arrer evolved to hub ad oke oerato whh oe ty the hub where ot flght orgate ad deart ad the other te the etwork feed to the hub. Paeger travelg throughout the etwork ake oeto throughout the hub thu allowg a arle to erve ay arket wth oe or fewer oeto. Arle oeted over the ze of ther etwork ad ofte the large arle had ore tha ore tha oe hub ther etwork to broade ther oe. Cobed wth freuet fler rogra, arle ould ue the breath of ther etwork to w aroval of the reeat freuet fler, who eeded to ue the erve of the etwork ofte. Th obato eeally work well hub te, where arle offer ther bet erve oto erve everal te a day to ay detato. Borete 989 foud evdee

14 that fare are hgher hub te tha o-hub te at a hgher eretle of fare. Arle ould offer the exave etwork ready for bue ue whe traveler are ayg to fare ad the offer the reah of that ae etwork to exot loato at the te freuet fler le are redeeed for free tr. 'You a fly a llo le o roter [Arle], but t wll ever get you to Par,' ad Uted [Arle] okea Jao Shehter. 'It' ale veru orage. We have a uathed global route. We have frt la' Agular. C Dfferg Sze of reuet ler Progra The heer ze of Uted Arle veru roter Arle ot oly athed aual reveue, etwork ze, but the ze of freuet fler rogra. Uted boat of aroxately llo freuet fler eber whle roter Arle reort over oe llo freuet fler eber htt:// There a varety the ze of freuet fler rogra. Table how th dfferee a t lt atual erollet of the freuet fler rogra 5. The forato oe fro htt:// 5 webfler.o alo lt hotel freuet tay eberh total ad Aera Exre Meberh eward total. 5

15 Table : Sze of reuet ler Progra US, Caada, Mexo Arle Meber Data eorted a of: AeroMexo,600,000 6/0 Ar Caada 6,000,000 /0 Alaka Arle,700,000 8/0 Aloha Arle 00,000 /0 Aera Wet,00,000 8/0 Aera Arle 8,000,000 /05 Cotetal Arle 9,000,000 /0 Delta Arle 5,000,000 9/0 roter Arle,000,000 9/0 awa Arle 880,000 8/0 Mexaa Arle 85,000 8/0 Mdwet Arle,65,000 /0 Northwet Arle 5,000,000 8/0 Uted Arle,000,000 /0 US Arway,00,000 /0 The lt above oly how erollet uber. ady Petere, Publher/Edtor Ide lyer agaze, etate that oly 7 8% of erolled freuet fler are atually atve htt:// urtherore, fler do eroll ore tha oe rogra. Thu the atual erollet uber do ot tell the etre tory of the 6

16 dfferg ze of freuet fler bae for eah arle. If forato wa avalable, the total of freuet fler ould be broke dow to etroolta area. Though Aera Arle ha a hgher freuet fler eberh total tha Northwet Arle, the total Northwet Arle WorldPerk eber Detrot, Mhga uh hgher tha Aera Arle AAdvatage eber beaue Northwet Arle ha a hub Detrot. roter Arle ay be loer ze to Uted Arle the Dever etroolta area ter of freuet fler eberh, where both arle have hub. 0-. The Elte reuet ler The freuet traveler of the freuet fler ted to brg arle a gfat orto of ther reveue. Arle reward th grou of traveler wth extra le, freuet ugrade to frt la, rorty allg to a -800 le, ad rorty boardg. Uually, arle have a ter la wth hgher ter beg for hgher travel wth the arle. Travel a be defed a wth the at year or lfete travel wth the arle. Aera Arle, for tae, ha Gold, Platu, ad Exeutve Platu for t bet utoer. Thee traveler otrbute ore to the arle reveue we tha the average freuet fler. Petero etate that Uted Arle ha aroxately 800,000 total fler eber ther three elte lae Aguar, C0. Petero etate that there are aroxately 07,000 total fler that have over a llo le wth a artular arle wth 50,000 obed wth Aera, Delta, ad Uted 7

17 Arle Stoller, 8B 6. or Aera Arle, a traveler earg oe llo le wth the arle ear a lfete Gold Meberh. Whether eaured by uulatve atvty traveled for oe arle or by uulatve atvty the at year for oe arle, th la of freuet traveler ay be aother way to eaure the tregth ad reah of arle freuet fler rogra Yeld Maageet ad Caaty tato Aother ajor tred that otued after deregulato wa the otued re of yeld aageet yte. Arle tarted to vetory ther eat a outerzed vetory yte well before deregulato frt to rovde real te forato to arle eloyee ad travel aget akg reervato for aeger. Seat were oartetalzed to varou ategore refletg the dfferet tye of rule aoaed eah tarff or fare. Durg the erod of regulato, eah tarff or fare wa otrolled by the Cvl Aeroaut Board Wahgto, DC. Sale ad rooto were reet the regulato era. owever, the reet yte of yeld aageet tll had to wat, a t wa uberoe to aly to Wahgto to have aroval for fare ode haged o route ror to deregulato. oute that had fare haged were et or eve halleged by oetg arle Baley, Graha, ad Kala. 6. After deregulato, the tool were et lae for the growth re dero. Arle ket the fare la yte. New lae of dout la tarff aeared for the 6 Aordg to Petero at webfler.o, the ero wth the hghet ulaed freuet fler leage total ha llo le! 8

18 leure traveler offerg great avg off the tadard oah fare Baley, Graha, ad Kala,. 7. Se thee heaer eat were ot a roftable a a tadard oah eat at a hgh re, arle foud way to otrol how they old dout eat. Arle ue re drato tat tarff how ay day advae the aeger urhag the tket advae or whether the aeger tayg over o a Saturday ght to detere f a dout tarff ale bd. More ubtly, arle et a lt o the uber of eat that a be old at the dout fare o a flght ba bd. Oe that dout eat aaty lt reahed o a flght aeger have to ay the ext hgher fare or hooe aother flght wth the ae dout eat. To kee trak of the exloo of ew tarff lae, rule, ad vetory o eah la o route aro ther etwork, arle eeded the erve of ohtated outerzed reervato yte. Arle ue the outerzed reervato yte to otatly readjut ther vetory of eat to axze ther oble reveue. The etre roe develoed wa alled yeld aageet yte that axzed reveue fro every arle eat gve arket odto Sloae. Thee odto luded ot to the arle, re that oettor arle are hargg, htoral tred of bue ad leure traveler, the overall eooy erforae, ad uly ad dead odto wth a gve ubla of eat, flght, ad route bd. Arle uually hage flght wth ther yte lowly o uly of overall eat o a flght ay be fxed but dead eah day o a artular flght a be ute varable Shtt ad Wlla. Arle ue everal year worth of data o flght to redt aeger traff to foreat dead o a route Shtt ad Wlla. Subategore of eat wth earate tarff are tally 9

19 dtrbuted aro eah flght a t tally etered to the yte alot a year ror to dearture Shtt. Arle leave a ay eat a redted to bue traveler, who are wllg to ay uh ore tha the leure traveler, whle ellg eat to the leure traveler bd. A dead odto hage betwee leure ad bue traveler, arle hage vetore of eat betwee lae of eat o a flght. Suoe ore bue traveler ake ore early bookg o the Aera Arle Dalla Tula 8:00 P.M. flght tha exeted. The Aera wll redue the uber of dout eat avalable for bookg ad reae the uber of hgher red bue lae of eat. The adjutet ould be week out or wth hour of dearture Shtt. Th adjutet roe geerally vble to the ouer. The et reult of yeld aageet a reae reveue for the arle ad a reae load fator bd. Every day arle fae a roble of erhg vetory, aely eat that go ety whe flght take off. A of ow, ajor arle geerally kee a fxed hedule wth or adjutet over te wth euet ad flght for a et erod of te. Wth et aaty level to arket, arle fae the roble of uerta dead to ther arket, oete ot eough aaty, oete too uh aaty. Colatg the uertaty of dead the uertaty of the x of traveler betwee leure ad bue traveler. Yeld aageet hel arle axze reveue gve the aaty they have o the route ad ze ety eat. 0

20 Due to yeld aageet, arle offered dout eat lted aaty. Traveler are odtoed to ho aroud for the lowet fare oble. If oe arle ell out dout eat o a flght or uote a hgher re beaue t doe ot offer hea eat o that flght, utoer wll ho for aother flght for the ae arrer that offer lower re. If a arrer flght eleto old out of the lowet fare, thoe ot overly loyal to a artular arle wll go to other arle to obta hea eat. Thu arle that do o ell out to aaty have a redual dead fro other arle that old out ther vetory. Th yte of offerg dout eat otatly flux. Se the tart of deregulato, arle bega exeretg wth hagg fare or fare lae otatly or offerg all out fare war. I the late 980 Uted Arle ade 0,000 fare hage daly alto,. urtherore, Uted otored advaed tket ale o 0,000 of t total flght the outer yte ad haged the fare la dtrbuto of eat o 5,000 flght daly. bd. Chage aaty were reoble for ot of Uted daly fare hage bd. If aeger total reaed or a oettor redued aaty, the fare reaed at Uted bd. Arle oete ru yte-wde ale. Soete thee ale are rooted by the arle reoe to yte-wde bookg beg lower tha aaty. Arle wll ut a erta eretage of eat o dout ther whole yte for oe te. Soete ale are arle atg retalato to aother arle for oe ato. Aera Arle, for tae, reoded to Northwet uderuttg t lfed fare

21 truture 99 wth a 50% off all dout oah eat ad oved the day advae urhae reureet to eve day advae Saluka,. D. The roe of uttg eat o dout ha beoe refed that arle are targetg ef arket or ef erod of te wthout gog to a broad re war. Aera Arle, for tae, ut eat o dout for Mother Day weeked 005 Aera Arle Pre eleae /7/05. The Mother Day Sale dd ot lead to a broad fare war. Arle have develoed weekly eal broadated by eal that ut erta route o ale. Southwet Arle ha take the roe further wth t DING! rogra, offerg eal to artatg outer uer for a few hour Southwet Arle Pre eleae: ebruary 8, 995. By ltg the erod of dout to a hort erod of te, arle are avodg all-out re war o all of ther dout eat The e of ow-cot Carrer ad allg Average Tket Pre There ha bee a re of the low ot arrer e deregulato. Southwet Arle ha grow to oe of the to doet aeger arle fro a all regoal arle at the begg of deregulato. Aera Tra Ar, Ideedee Ar, Jet Blue, roter, Aera Wet, Ar Tra, ad Alaka are all ow odered low ot arrer. All of the exet for Alaka have bee reated e deregulato. Other lowot arrer uh a Mdway Arle, Peole Exre, ad Pedot Arle ether were erged or ludated. Eve the ajor or legay arrer are oyg the low ot arle a Ted ad Sog are earate low-ot arle wth Uted ad Delta

22 reetvely. ow ot arrer arket hare jued fro 0% 000 to lghtly over 0% 00 Walh,. B. More revely, the eretage of eat arket loger tha,000 le the otguou Uted State erved by low ot arrer ha jued fro % 999 to 7% 00 Mayard, Seto, 7. The overall effet of the troduto of the yeld aageet yte ad lowot arrer wa that average arle fare fell a a reult of deregulato a arle oeted for the leure traveler. The Arle Traort Aoato reort that doet aeger yeld er aeger le fell 8.8 eret betwee 978 ad 00, adjutg for the Couer Pre Idex Ar Traort Aoato 00 Eoo eort.. The Geeral Aoutg Offe reort that betwee 979 ad 99 fare er aeger le, adjuted for flato, fell 9% for all outy arort, % for edu zed outy arort, ad 8% for large outy arort GAO /5/ The Arle Traort Aoato reort that oal total ludg teratoal travel arle reveue er aeger le fell 0.6% fro. et er aeger le 99 to.7 et er aeger le 00 Ar Traort Aoato 00 Eoo eort. 7. Th aelerato of the eroo of arfare betwee 99 ad 00 tralate to a average lo of $.90 fro $.0 to $7.0 - for a oe- way flght of 000 le. Eve wth the olated yeld aageet yte avalable, there ha bee a exloo dout eat for ar travel.

23 0-7. Pre Dero Dete the dele average tket re, there ha bee a re of re dero wth the dutry. Arle douted for the leure traveler but raed re for bue fare. Bue traveler ad evety eret hgher fare 000 tha 980 aburger 7. eure fare fell the ae erod a bue fare roe aburger 8. ecipse advor, a ubdary of Aera Exre Bue Travel, etate that the tyal bue fare refudable, three day advae o route t otor $0 oe-way 00 Aera Exre Bue Travel Motor Pre eleae //0. The ae ubdary etate that average fare ad by t utoer travelg o route otored by the ae BTM grou are at a lower $7 Ibd 9. Thee bue fare are uh hgher tha the average oe-way fare lted by the Aera Traort Aoato. Yeld aageet yte take advatage of th wde read of wllge to ay aog ouer wth ay dfferet fare. Pre dero the arle dutry a be ee by lookg at the eod uarter of 995 Data Bak a, whh a 0% rado ale of arle tket. The followg hart how adjuted oe-way fare dfferee o eleted route both 7 Soe of that ga betwee 980 ad 00 ha eroded. I a SABE tudy bue fare average roudtr fare urhaed three day fro dearture fell % betwee May 00 ad May 00 eed A. Wth the troduto of Slare by Delta Jauary 005 that ut the axu oe-way fare to $99 o doet route, the tred of lower bue fare otue well to 005 htt:// I July 005, Delta raed athed by other legay arrer the axu oe-way fare to $599 due to hgher ol re. 8 Jae gg, arle aalyt at Credt Sue rt Boto reort that leure fare have bee reeg u betwee 00 ad 00 due to dout arrer rag re Gratha.. Th tred ha otued today wth the legay arrer rag fare o a oule of te th year due to hgh ol re. 9 Thee average bue fare lude full fare, heaer orefudable fare, ad egotated ororate dout Aera Exre Bue Travel Motor Pre eleae //0.

24 dreto luded betwee the 5%le ad 75%le lu the ea, eda, ad tadard devato. Iterare are ether oe-way or loed loo roud tr. I the ae of a loed-loo roud-tr, the fare dvded by two to get the oe-way fare. Table : oute Pre Dero oute # Pax 5%le Meda Mea 75%le Dfferee Stadard Std. Both d 75%le Devato Dev./ Dreto Qtr 95 5%le Mea AXEW 7,0 $5 $9 $5.9 $5 $00 $ OMAGA 9,800 $7 $67.5 $.55 $00.50 $8.50 $ MCOSEA,990 $5.50 $96 $.0 $7 $9.50 $ ASPX 7,500 $7 $0 $50.06 $67 $0 $ SOOD 7,570 $0.50 $90.50 $78.7 $8 $50.50 $ DUSAN 8,80 $50.50 $9 $9.0 $09.5 $58.75 $ AUSBOI,0 $5 $.50 $7.5 $.50 $88.50 $ AMSP 9,00 $95.50 $.50 $. $0.50 $5 $7. 0. ANNO 70 $5 $76.50 $79.6 $0 $95 $ AXDW 0,70 $ $89.50 $59.6 $ $8 $ PDEN 50,0 $8 $6.50 $. $0.50 $66.50 $ ICGEG 60 $75 $0.50 $7.68 $0 $65 $ BOSMIA 7,00 $6.50 $5 $6.7 $77.50 $6 $ TPAGA 88,0 $ $ $5. $8.50 $68.50 $

25 DTWPIT 0,070 $.50 $86 $.6 $6.50 $05 $ PVDO 8,0 $88.50 $0 $6.70 $57 $68.50 $ PWMAX,90 $7 $ $.6 $66 $9 $ ATDCA 05,50 $90.50 $07.50 $09.7 $05.50 $5 $ AMADCA,50 $88.50 $08.50 $6. $90.50 $0 $ AXOD 97,550 $.50 $8.50 $56.80 $99.50 $58 $ PXBOS 59,60 $7.50 $85.50 $7. $57 $09.50 $ IAGA 57,0 $75.50 $87 $79.7 $ $75 $ BIAX,80 $.50 $6 $89.79 $ $86.50 $ SAVANC 0 0 $0 $09.50 $5.7 $0 $0 $ STSEA 5,80 $9 $.50 $6 $0 $9 $ SDCT 500 $9 $6 $96 $59 $0 $ MCISJC 8,990 $7.50 $ $6.69 $.50 $06 $ NOD 0,870 $0 $06 $8.9 $87.50 $87.50 $ IADSO,80 $65.50 $ $05.79 $7 $5.50 $ ANCAI,660 $69 $98 $90 $99 $0 $ Average 5,78 $.5 $79.9 $09. $7.9 $5. $ Weghted $7. $65. $0.89 $67.5 $50.0 $ Average 0 Delta the oly arrer reortg aeger betwee thee two te. reuet fler ake u alot half of the 0 traveler. If freuet fler are elated, the 50%le o fare ad ju to $0. The 5%le, eda, average, ad 75%le fare re to $9.50, $, $9., $6.50 reetvely f freuet fler ad o-reveue aeger are elated fro the ale. 6

26 Table how that there dero all route. The u tadard devato fare $0.8 o the Phoex, AZ to a Vega, NV route. owever, the average fare o that route extreely low at $ Thu the tadard devato/average o that route ot the lowet the ale. Soe route have extree re dero. or tae, the outo Iterotetal, TX New York, NY aguarda arort ha a tadard devato of $.5 ad a whog $75 dfferee betwee fare at the 5 th eretle level ad 75 th eretle level! Whe eaured by tadard devato/average re, the outo-new York route aroxately 50% hgher tha the Phoex a Vega route. A le regreo wth aeger beg the deedet varable ad eda, tadard devato, dtae, duy for low ot arrer, ad uber of oto arrer beg the deedet varable rodue the followg reult: Table A Sle egreo wth Paeger a Deedet Varable Suared 0.79 Adj. Suared ,.0 Prob > Ideedet var. Coeffet Stadard error t Prob > t Meda Stadard dev Dtae ow Cot # Noto Carrer otat

27 Obeyg the law of dead, the oeffet of eda egatve.. Iteretg that tadard devato otve ad gfat. arger arket geerally have ore varato re by the eaure of tadard devato. The uber of oto arrer extreely gfat ad date a large orto of the varato aeger. A better regreo ght lude the erfdahl dex ad adjut for the freuet fler/oreveue aeger Suary of Develoet Se Deregulato There have bee everal tred e deregulato. There ha bee the re of two dfferet tye of traveler bue ad leure traveler. Average fare ad by thee bue traveler are uh hgher tha leure traveler. There a la of bue traveler that travel uh ore tha the average ubl. The develoet of freuet fler rogra ark a way for arle to reward thoe otrbutg the ot reveue. Not all arle are the ae ter of freuet fler. A ha bee ee, Aera Arle freuet fler rogra today alot ffty te the ze of aller roter Arle. htt:// Wth the dfferee of bue traveler wllge to ay for flght oared to leure traveler, arle have develoed ohtated yeld aageet yte that lt the aaty of dout eat favor of lat ute bue traveler. Oe If ea ued tead of eda, the regreo ot a robut a eda. Addg 5le ad 75le a deedet varable lower the adjuted uared ad. Ug 75le u 5le rae the adjuted uared but lower. I th regreo of addg 75le u 5le, eda beoe gfat. Stadard devato/ea gfat. 8

28 dout eat ell out, hgher fare reult. Arle redt ug at data, urret eoo ad arket odto, what eatg aate hould be for the varou lae of traveler. Should odto hage or ot go a redted, arle ake adjutet to the yte. The re of yeld aageet yte ad low ot arrer ha reulted two tred: dereag overall average fare ad re of re dero. Average fare have falle e deregulato. The Aera Traort Aoato reort average doet reveue er aeger le ha falle alot 9% e 978, adjutg for CPI flato Aera Traort Aoato 00 Eoo eort.. The ATA reort that average fare o a oe-way 000 le flght are $ Ibd. The average fare e the ave re dero de the dutry. Tyal three day refudable fare are four te that aout 00 Aera Exre Bue Travel Motor Pre eleae //0. The average weghted tadard devato of a ale of thrty arket fro the eod uarter 995 $ The weghted tadard devato dvded by the weghted ea reult a rato of 0.7. Thu oe a olude that there re dero wth the dutry Suary of Dertato Th dertato odel oe of thee tred wth the arle dutry e deregulato. Chater oe odel re dero the dutry wth dfferg value of bue ad leure traveler. Arle are the ae ze of loyal utoer, who 9

29 ee oly oe re. Eah arle ha the ae aout of loyal traveler, who wll ay ay re at had u to ther reervato value. Arle oete for a grou of hoer, who earh all the arle for the lowet re. Arle a hooe to erve the bue arket, whh have both loyal utoer ad hoer. Arle a erve the leure arket, whh alo have a dfferet aout of loyal utoer ad hoer. I ether arket, fr radoze over a rage of re a there o ure re trategy. Uder erta odto of roft ot beg too bg the leure arket ad roft ot beg too bg ervg oly the loyal bue traveler, arle wll hooe to erve both arket. r radoze ther re over a terval of re large eough to erve both arket. I overg both arket, there a ga re edately above the reervato re of the leure traveler, where arle do ot radoze. I th rego, roft are le tha rego of re edately lower tha the oooly re of bue traveler ad oooly re of leure traveler. Thu there are two rego where fare luter. The odel a be exteded to the ae where there are ore tha two ouer valuato. A uulate dtrbuto futo ad robablty dety futo are geerated wth three ouer valuato, reultg two ga ad three rego where fare luter. The odel oared to the 995 Org ad Detato Data Bak a eod uarter. Soe larte are revealed. Ug the Eaehkov ethod of kerel dety etato, everal arle arket fare are grahed. Slar eah of thee arket fare luter aroud oe or everal re. Th luterg of re athe what the theory redt. 0

30 Chater two of the dertato odel re dero ad aaty ltato foud the arle arket. Two fr odeled ell to utoer that ee oe, oe, or both re. The utoer that ee oe re are loyal utoer. Thoe that ee both fr re are hoer. r have a aaty ltato. Eah fr aot erve the etre arket of t loyal utoer lu the hoer. Whe the re of oe fr lower tha the other, the lower red fr ell out to aaty. The hgher red fr ell to t loyal utoer lu the left-over hoer that aot urhae fro the lower red fr. Thu the hgher re fr reeve a extra bou of utoer that ould ot buy fro the lower red fr. Aug that there eough aaty the arketlae for the hgher red fr to hadle t loyal utoer lu the left-over hoer that ould ot urhae fro the lower red fr, fr radoze betwee okg ther ow loyal utoer lu the llover utoer at the oooly re ad doutg to w over the hoer ad ell out to aaty. The lowet re the dtrbuto roft at the oooly re of the hgher red fr dvded by aaty. The odel a be exteded to the ae of w fr where w- ell out to aaty. The lowet re the dtrbuto tll oooly roft of the hghet red fr dvded by aaty. Aga th ae aue that the hghet re fr ha eough aaty to hadle the left-over ouer fro the other fr. I the ae of ayetr aaty, there ot a eual lowet re the dtrbuto. If total aaty all, the aller fr ha the allet lowet re ad thu ha the ato at the hghet re. The aller fr ha ore lak to uderut the

31 hgher aaty fr. If total aaty hgh, the the ato hft to the larger fr a t ha ore lak t aaty to uderut the aller fr. I both ae, t aued that the aller fr ha eough aaty to hadle t loyal utoer lu the left-over utoer of the hgh aaty fr whe the low aaty fr ha a lower re. Whe the two fr lay a two tage gae that volve ettg aate frt ad the aoug re, the otu eulbru a yetr oe a the reato urve of both fr ro eah other at the yetr ot. Wth a ot of buldg atal ad roft beg lower wth ore aaty, the two fr wll buld the allet oble aaty. Eah fr et the re to the oooly re. Chater three odel re dero ad dfferg level of loyalty of utoer. The frt half of the aer ha two fr: oe larger the aout of loyal utoer that t erve tha the other. The eod half of the aer ha ay fr: oe fr that larger loyal utoer tha the other ae zed fr. The etu the ae eah half of the aer. The loyal utoer of eah fr ay whatever re at had. Eah fr ha the etve to harge ther grou of loyal utoer the oooly re. The larget fr ha ore of the o whe the re at the oooly re, the larget fr ear ore roft tha the aller fr. owever, there a grou of o-loyal hoer that buy fro the lowet re fr. If there are eough of thee hoer, all fr lower ther re to try to w the hoer. r re over a terval a there o ure trategy o a re that work wth thee two dfferet grou of utoer.

32 The terval of radozato of re for the larget fr ot the ae a the terval of re for the allet fr. The larget fr ha a lower u re tha the aller fr. Th le that the larget fr ha a ato t dtrbuto at the oooly re. Wth the ato, the larget fr doe ot oete a aggrevely ad thu all fr have the ae terval of re. Ireag the grou of loyal utoer for the larget fr ofte oetto for all fr e the larget fr lae ore weght o the oooly re. Ireag the grou of loyal utoer for the aller fr ha o effet o the aller fr robablty dtrbuto a they oete for hoer at all re ther dtrbuto. Proft for the aller fr, however, reae whe the loyal utoer for the aller fr reae. Ireag the loyal utoer for the aller fr lower the ato for the larget fr ad reae the doutg of the larget fr. Ireag the utoer grou of hoer aue all fr to be ore aggreve doutg. The lowet re fall ad uulatve robablty weght reae at every re below the oooly re. The ato for the larget fr dereae. If the loyal grou of utoer for the larget fr reaed at the ae te the loyal utoer for the allet fr are dereaed, the all fr dout le ad ore weght laed o the oooly re for the larget fr. If the loyal grou of utoer for the larget fr reaed at the ae te the hoer are dereaed, the the aller fr dout le. The larger fr geerally dout le, eeally f the uber of fr grow larger. If the loyal grou of utoer for the allet fr reaed at the exee of the hoer, the the aller fr dout le, beaue

33 there are le hoer. Uder th earo, the larget fr dout ore at lower re ad dout le at hgher re. The uulatve robablty of the u re reated ad aled to the odel. Ireag the loyal utoer for the larget fr dereae the uulatve robablty of the u re. Ireag the loyal utoer for the aller fr reae the uulatve robablty of the u re at all re exet at the oooly re. At the oooly re, the u re tatt zero, thu howg that the ato ha to be wth the larget fr. Ireag the hoer geerally reae the uulatve robablty of the u re. Ireag the uber of fr lower the uulatve robablty of the u re. Ireag the aout of loyal utoer for the larget fr, whle dereag the loyal utoer for the aller fr, lower the uulatve robablty of the u re. Ireag the aout of loyal utoer for the larget fr, whle dereag the hoer, geerally lower the uulatve robablty of the u re. Ireag the aout of loyal hoer for the aller fr whle dereag the grou of hoer lower the uulatve robablty of the u re a the uber of fr grow larger. Thee three hater, thu beg odelg oe of the haratert of the arle dutry e deregulato. All three hater have re dero betwee fr. A arle have olated yeld aageet tratege to axze reveue, thee three hater odel arle rg behavor a a xed trategy over a terval of re. All three hater odel loyal utoer. reuet fler are ortat to arle bue trategy. Modelg the trategy arle ue to kee freuet fler ayg hgh fare

34 take a varety of aroahe the dertato. Chater oe earate loyal utoer a bue ad leure traveler whle hater three odel dfferet ze of loyal utoer. Chater two uue that t oder aaty ltato that arle fae. oyal traveler, lu the redual utoer that are ratoed fro buyg fro the lower red fr, urhae fro the hgher red fr uture eearh uture reearh ould lude aaty ltato wth dfferg valuato of utoer. Arle ay te ell out ther lower re eat ad the leave the hghet eat left for the bue traveler. Dfferg ot of the arle dutry ould be odeled. ow would the addto of a lower ot arrer affet re dtrbuto? Extedg thee odel to ultle erod ght allow ore flexblty odelg freuet fler. Arle oete allow twety fve eret of eat o erta route to be award ad o-reveue aeger. ally, etwork ad ult-arket effet a be tuded. All three hater fou o oe artular route. What hae f there ore tha oe route? Thu lke the ever hagg arle dutry, there a everlatg uly of future odel that a be ued to further exla the dutry. 5

35 : Pre Dero wth Dfferg Couer Valuato -. Itroduto I the lat few year ueto have are o how the growg fluee of the teret ad eletro oere wll ultately at fr rg ower. Early aaly the re foued o the Bertrad eretve, derbg the very dhed rg ower. Mot redto were that old eooy fr wth etablhed arket ower e.g., brk ad ortar oae that dd ot adjut ukly to th ew edu of oere were gog to be uderut, ad that the oly fr gag arket hare would be the lowet ot fr, thoe had adjuted to the ew yberae world by ellg at razor th arg. Thee redto are o loger foud the atrea bue re. Brk ad ortar oae that were low to eter yberae are ot geerally ovg toward bakruty a ay had tally redted. May have etered the world of yberae, but tll odut a gfat orto of ther bue away fro yberae, whle alot all of the hgh flyg teret oae tok valuato have rahed. The lear-ut redto of how dutry wa gog to be haed have ether bee dredted or extreely louded by reet arket evet. What the doe the re of the teret ea for arket? The awer ay le arket that have already had a htory wth eletro oere. Thoe dutre that had already bee ubjeted to eletro re 6

36 oetto have oly otued to evolve wth the readg of yberae. ow the teret affet thee dutre ay be a revew of how other ay be affeted the future. Oe uh dutry the arle dutry, whh ha had to oted wth the eergee of teret tket ale. I Otober 999, the Wall Street Joural ra a frot age artle dug the at of the eletro oere o arle rg. I th artle, Wharto Bue Shool Profeor Er Cleo argue that the ole travel aget hel revet a "fullale re war for arle tket Cleo, 999." urther, he tate that "Eoot were offeded by th [reult]. They ad t a't be table - the Iteret ut rovde erfet oetto. [owever,] t 't [a erfetly oettve evroet]. If a rodut olex eough, eller a avod oetto by ervg dfferet utoer." The offeded eoot referred by Cleo geerally aue, the Jevo tradto, that there oe re that lear arket. Mot eoo odel are bult uder th auto. owever, there are eoot who have how how the world outde of oe learg re a ext, well before the Iteret beae atrea. I fat, three atter uggetg eulbra beyod oe re have beoe ore wderead: rooto, ouo, ad ajor re wg. or exale, Stgler 96 reort of a earler tudy about the detal ake ad odel of Cheverolet, whh have a tadard devato of $ o a average re of $6 ebruary 959, eve after aug a "average aout of 'hgglg.'" Th work out to be lu or u.% for two tadard devato. 7

37 rt, a the ortae of rooto grew, ajor retaler uh a Alberto, Wal-Mart, ad Target otue today to roote ale forato to ther ouer by the al or ewaer. lyer adverte dfferet te that are o ale eah week. Today, ajor arle, uh a Southwet Arle, eal ther utoer about the arket that have redued re that week. Seod, the dtrbuto of ouo have beoe ubutou: Suday ewaer have ther ow ouo eto or two. Booklet are dtrbuted aroud aue, tore, ad all. Pakaged rodut have ther ow ouo. Store have kok geeratg tore ouo. ally, ouer a rt ouo off the teret. Ther ue ay be advertg toward a ew rodut or to get ouer to buy ore of a erta rodut. Soete ouo try to ete ouer to hage ther ro-brad urhag habt uh a what brad of haburger bu they urhae wth the haburger eat Nazareo 00. Couo reurg two or three dfferet te urhaed before the ouo fae avg aed o to the ouer are ore oo today. Deartet tore have ouo that have ouer hooe what te reeve the eretage dout. ally, oe arket have ajor re wg. etaler ay offer a all out ale of u to 0% - 50% off tead of a otrolled weekly rooto. Deartet tore are ore lkely to rate th behavor tha the retaler uh a Wal-Mart. Arle alo egage th ort of behavor. arge fare war ofte develo betwee the large arrer that ed wth ever hagg deadle whe the ale wll ed. Soete a dvdual arle lay guerrlla tat: offerg large re dout wth arket Though the dtrbuto of ouo have reaed otat the at fve year, the redeto of ouo ha deled the lat fve ad te year. Nazareo 8 8

38 gog out of oe ty for a day or two oly, hog that the other arle wll ot ath fare or at leat ot adverte to ath fare 5. Th aer buld uo revou odel exlag delberate re radozato by fr. Prevou odel th fold uh a Vara 980, Butter 977, ad Burdett ad Judd 98 - have aued that ouer have a oo valuato. Th odel geeralze ther reult to ae where ouer have ultle valuato. Arle, for tae, have bue traveler ad leure traveler ttg ext to eah other oah. Th aer addree the ueto of whe ad how fr radoze ther re to dtguh the hoer ad o-hoer whe there are ultle ouer valuato, ad the eulbru truture of re dero ad how that dero deed o dead truture. The aer orgazed to 7 eto. Seto - derbe revou odel of re dero. Seto - rovde evdee of ultle ouer valuato ad t eral ortae. Seto - derbe the tal odel ad how what read of ouer valuato would eed to our for delberate re radozato by fr to lude all utoer. Coaratve tat are Seto -5. Seto -6 rovde a aaly the grah geerated by Matheata by luggg value fro the odel fro thoe grah geerated by arle data fro eleted route. Seto -7 gve the fal aaly of what leared fro th tye of odel. Aedx - ota grah fro Matheata ad arle data.. 5 or tae, ArTra Arway aoued a 8 hour ale to all detato out of Baltore ad Phladelha ArTra Pre eleae Augut 7, 00. 9

39 -. terature evew The law of oe re ad t oettve eulbru foudato date to Jevo 87. Startg wth Daod 97 there have bee a large uber of earh-theoret exlaato of o-oettve rg. Exlaato have dffered both auto- dfferg earh ot of ouer, dfferg ablte to re drate ouer tye, ad dfferet forato ee by ouer ad oluo overgee to oooly re, dvergee to a few re, ad dero over a re terval. Daod 97 gave the frt earh theoret exlaato of overgee to o-oettve re. Daod ehazed h aer that adjutet roee hould be baed o real fluee the eooy rather tha beg a arbtrary roe deged to yeld the oettve eulbru. e develo a dya odel that had re overge to the oooly re due to the ot of earh. Daod aue that the fr yetrally erve a arket for a drete good every erod; eah havg / frato of the erta arket dead urve X t. Daod ue the ot of earh two way. rt, he deotrate how the ouer hoke re t for the good oe erod fro ow would be greater tha the hoke re today t due to the ot of earh. Ug th dvdual ouer behavor, he how that the u re t the arket that allow everyoe to buy the rodut or the 0

40 u of all ouer hoke re a artular erod ut be odereag over te. Whe t roftable betwee the u full artato re t < *, the otal oooly re fr wll re t ad the re t, where ouer wll fd t worthwhle to wth to aother fr. Gve that t odereag over te, the re harged the arket t overge to the oooly re *. othhld 97 ehoed Daod hallegg eoot to develo odel that have edogeou aue for re dero. ro th hallege, there beae three dfferet hool of thought how re a vary. r wth rg ower a eloy re drato to dtguh betwee the varyg tye of ouer. Ug obervable haratert or hoe that ouer have ade theelve fro a lt or eu, fr dtguh betwee the varou tye of urhaer. The dero of re oe fro ouer eah ategory ayg dfferet re. Exale of re drato lude earatg ouer tye by age or ouo. Two a ue foued by the re drato lterature ha foued o how re drato affet welfare ad the role of arbtrage. Doe welfare rove or dele wth re drato oared wth ufor rg? or tae, Shaleee 98 ad Vara 985 et odto o outut where re drato wll ehae or at leat ot har welfare. Nahata, Krzyztof, ad Otazewk 990 takle the welfare ueto a dfferet agle. They how that that re ay reae or dereae wth ubarket due to thrd degree re drato but roft wll reae ulaterally.

41 Bede welfare ue, re drato ha foued o the ue of arbtrage. O the global level, wll aother fr go to bue to a arket where fr are re dratg? By buyg at the lower re ad the ellg t at a lghtly hgher re to the egeted ouer ayg hgher re the fr hel break dow the effet of re drato. The ue of arbtrage a alo be odeled o a eroal level. r offer a eu of re for ouer to hooe. Seeg th eu, a dvdual ouer ay rereet h or her tye or hage behavor to get a lower re. Takg th ouer arbtrage to aout, fr reate the rght etve for ouer. r axze roft ubjet to dvdual ratoalty otrat eurg that all tye of ouer artate - ad dvdual oatbly otrat akg ure eah offered eu oly aeal to thoe rght tye. The budle offered to the ouer are et by th ethod o that the etve for a tye to hooe a re offerg ot dreted to that tye zed. Other fator a exla re dero bede re drato o the ba of obervable haratert. I thee dero odel, fr offer a rado re ad t th radoe that the oure of re dero. ather tha offerg a gle re for ure, fr egage xed trategy behavor. Mxed trategy behavor by fr ay rodue hgher roft tha ure trategy behavor whe ouer have otve earh ot or whe ouer do ot have all the relevat forato. Mxed trategy behavor alo ould rovde the oly oluto ay of thee gae a ure trategy behavor aot ret o ay re due to the etve fag eah fr. r ould uderut eah other at a artular re to ga

42 ore arket hare oly to be further uderut at lower re. Th roe otue to argal ot where roft are the hgher at the oooly re. The other two lae of odel exlag re dero fall uder the xed trategy eulbra. The eod ategory of odel exlag re dero are fro earh ot of ouer. A tyal exale obert ad Stahl 99. They aue advertg reahe oe ouer but ot other. The other ut exed effort earhg to fd the re. I ther odel, whe earh ot reae, oe ouer ght fd t too otly to earh for the bet avalable re. r fearg the otetal of log ther bae ouer radoly offer oe dout to due eole to ho at ther tore. r adverte ore heavly thoe dout to ouer. To ake u for thee dout, fr rae the hghet re that the ufored hoer ay. Couer beoe ore fored about the otetal dout a the robablty of beg oletely ufored about ay tore offer fall. eervato re of ouer, where the exeted ga ad ot of earh are eual, re wth the reaed dero of re ad ot of earh. Whe reervato re re to the ot of ouer valuato of the objet or erve, ouer tart edg ther earh at the ot where reervato re eual valuato. Salo 977, Salo ad Stgltz 977, Carlo ad MAfee 98, ob 98, ad Stahl 989 are other exale of th la of odel. The thrd la of odel exlag re dero foue o the ex ot forato reeved by ouer rather tha the earh roe. A olated ellg trategy ouled wth the olete ablty of ouer to ee all relevat forato

43 allow ouer to vew fr rg a rado. Vara 980 ehazed th ot h odel that ouer aot lear the rg behavor of fr over te. Of th fal aroah to exlag re dero by delberate radozato the year after othhld' hallege, three odel - Butter 977, Vara 980, ad Burdett ad Judd 98 - ehaze the ex ot forato reeved by ouer. Uderlyg eah of thee three odel the olete ablty of ouer to ee the re of other oettor. Soe ouer were fored of the lowet re the arketlae ad oe ouer are le tha fully fored, hog oly at oe fr. Dero are thee odel a fr balae betwee aturg oooly roft fro ouer hog fro oe fr ad aturg the etre grou of ouer that earh for the lowet re the arketlae. The uber of ad that ouer reeve the al drve Butter' reult. Vara outrght aue the heterogeety ouer by lttg h ouer to fored ad ufored tye. Burdett ad Judd dvde the fored ad ufored by allowg the robablty of ouer eeg ore tha oe re to betwee zero ad oe. Butter 977 develoed a odel whh fr ed out ad at a fxed ot er ad.e. alg ot otfyg ouer whh tore rug the eal. A ouer ay or ay ot reeve ad fro fr. If a ouer reeve ad the al, he wll urhae fro the lowet re ad. Seller kow the lt re of ouer ad the dtrbuto of re that other eller offer but do ot kow exatly how ay ad that a ouer wll reeve. I hoog the uber of ad to ell out, eller wll wegh the exeted beeft of edg oe ore ad out agat the exeted ot of

44 rtg the ad. The re radozato reult fro two fore the odel: rt, eller wll ot re at argal ot to ature the etre arket e they wll loe oey after fatorg the fxed ot er ad. Seod, eller aot oetrate advertg above th re e aother fr a uderut the targeted re by elo ad ature the etre arket. Thu eller wll have to radoze eulbru. Vara 980 develo a oooltally oettve odel to derbe re dero by fr. Cot by fr are aued to be detal. r et ther re by the week. Cutoer are dvded betwee the fored ad the ufored tye. Thoe ouer that are fored wll kow what every fr hargg the arketlae ad ho at the fr wth the lowet re. The ufored ouer wll radoly hooe oe tore ad ay whatever the re that the tore hargg. Vara aue that the uber of ufored ouer all eough that fr wll have to alo eek the bue of fored ouer. ke Butter, Vara how that there aot be a a ot at ay other re lower tha the oooly re beaue there a be aother fr that a ature the etre arket of fored ouer by jut rg elo lower tha the oetrated re. I a ef ae that the ot futo ha a fxed ot ad o argal ot, Vara fd that dtrbuto of re wll be oetrated at the edot of the ot dtrbuto: at the argal ot ad at the oooly re. Thu, the re dety futo of the fxed ad o varable ot ae would ugget a atter of fr oaoally rug ale to get a orto of the fored ouer. 5

45 Burdett ad Judd 98 geerate eulbra wth re dero through the earh atter of ouer. r rodue a gular good or erve at the ae argal ot ad elet the re that they offer ouer. Couer have the ae reervato re, earh ot ad ae ethodology of earhg. I a euetal earh, ouer obta a uber of re uote at a te. A re dero eulbru our the euetal earh whe the robablty of eeg oly oe re le tha oe. The ae re dero reult hold f the earh beoe oy - ouer do ot kow how ay re uote they wll ee wth a gve earh. Burdett ad Judd alo fd a ae of re dero a ore otraed oeuetal earh. I th tye of all or othg tye of earh, ouer tally have to dede how ay re uote to reeve before ay a be oberved. A re dero eulbru ext whe the robablty of a ouer eeg oly oe re le tha oe but the robablty of eeg oe or two re exatly oe. -. Motvato Butter 977, Vara 980, ad Burdett ad Judd 98 th thrd la of odel have tarted how fr ater to ouer havg dfferet ex ot forato. owever, a ltg ot thee odel that ouer are aued to have the ae valuato for the good or erve. Th aer wll exltly exlore how fr wll 6

46 attet to dere re wth dfferg ouer valuato ad forato level. 6 I wll develo a gae-theoret odel fatorg both a dfferee of ouer valuato ad aout of forato reeved by ouer. A th odel develoed, the arle dutry rovde a exellet otvato develog the odel. Certa bue traveler ay be wllg to ay uward of everal te the value of the lowet ale re bought by leure traveler. Yet, both tye of ouer ould be ttg ext to eah other oah. Th wde valuato of arle erve by ouer alo aoaed by rad re wg wth the dutry. gure 8 5 are arle oe way route re data take fro a te eret rado ale of Data Bak a the eod uarter of 995. oud-tr tket are deooed to oe-way fare for eae of oaro. The fgure are draw ug a kerel dety wth the Eaehkov ethod wth the otal badwth. Note that there are ultle ode ot of the fgure. gure 0, for tae, how four ajor ode o the oe way tket re betwee Wahgto oald eaga Natoal Arort ad New York aguarda Arort. Meaol-St. Paul Iteratoal Arort to Chago O are Iteratoal Arort alo how four ode gure 9. Soete the dety aroahe zero betwee ode a gure a how o the 6 Salo 977 develoed a odel fatorg re drato ad dero for a ooolt. owever, ouer were oly dfferetated ther ot of earh ad ot of ther uderlyg valuato for the good. ob 98 develo a elaborate ult-fr re dero odel wth dfferet ouer earh ot, but aga there ot a deedee betwee the ot of earhg ad rvate valuato. Weeth 98 develo a odel loet to the objetve of th odel. e further develo uo Salo by geeralzg the ooolt odel to lude a jot dtrbutoal futo g, v ag over all ouer. I h aer the ouer ot to earh ad v ouer valuato. Weeth fd odto h odel geeratg re dero, but aga thee reult oly hold for the ooolt. 7

47 Meaol-St. Paul Iteratoal Arort to Atlata artfeld Iteratoal Arort route. The revou lterature aot exla the ultle ode or the zero dety the arle re data. Vara 980, for tae, a geerate at ot two ode, where the lowet ode the tae whe re are o ale ad a hgh ode, where re are at a oooly re. e, however, aot geerate lae h dtrbuto where the dee of re are eual to zero. Th outer what ee gure 0. e aue that all ouer have the ae valuato. Th, however, doe ot ee to ft oletely the ultle ode that are oberved o oe of the route the arle data, or the exeree that all traveler are alke. Bue traveler are wllg to ay u to everal te the fare that leure traveler ay. The odel develoed here ororate a ultle valuato of ouer. Two valuato are ued leure traveler ad bue traveler. Th lfato a ealy be exteded to ultle valuato to gve a reult of ultle ode the re dety. Iortat the odel develoed here that there a terval where re are zero, thu athg what oberved the arle re data. gure, for tae, how th ga ute learly the re dety geerated by the odel. Other exlaato, re drato ad rodut dfferetato, ould ee a the doat exlaato behd why there dero behd arle fare. Advae urhae reureet ad Saturday ght tay reureet have bee well doueted a arle ue re drato deve to get dfferet ouer to ay dfferet re. owever, a re drato tory doe ot fully exla the rg 8

48 behavor eloyed by arle. The Saturday ght reureet beg weakeed a Delta Arle aoued Jauary 005 t Slare that lowered the axu tket re ad elated Saturday ght tay reureet to obta the lower fare Delta Arle Pre eleae Ja, 005. urtherore, advae urhae reureet are eodary to arle re radozato. Soete durg ale, arle lae 7 day advae fare heaer tha the or day advae fare. 789 Th ake a trt re drato tory by advae urhae dffult to jutfy. gure a -k ad gure a -f further deooe arket fare to oah Y ad oah dout YD ad break dow the overall arket how other dagra to dvdual arrer fare dtrbuto for the New York Cty arort aguarda, Newark, ad JK to Orlado route ad Meaol-St. Paul Arort to Atlata route. or ot arrer, oah Y fare are walk-u fare ad oah dout YD fare are fare wth advae urhae. Coah Y fare ay alo be refudable ad hageable wherea oah dout YD fare ay have retrto o refud ad hage oe urhaed. Pre drato betwee the oah Y ad oah dout YD ght be the tal exlaato behd the dfferee of fare, a Y fare are geerally hgher tha YD fare. owever, further aaly of eah fare la, how re dero wth ultle hu. 7 or tae, Ideedee Ar aoued a ale o Augut, 005 for oto detato out of Wahgto, DC reurg u to a eve day advae urhae for travel betwee Augut 9, 005 ad Deeber, 005. htt:// 8 O July 5, 005 Delta Arle aoued a fall ale that had ale fare wth a te day advae urhae reureet for the travel erod betwee Augut, 005 ad Noveber 6, 005. htt://ew.delta.o/ew.dex.f 9 Southwet Arle ha teret fare ale that reure 7 or day advae urhae. O July 5, 005 Southwet aoued a day advae urhae teret fare ale for uerou rego the Uted State effetve for travel fro Augut 8 through Otober 8, 005. htt:// 9

49 or tae, gure d how that there aother eak for oe-way fare above $00 for Delta Arle for the oah dout YD tket bede the ajor eak of $0 oe-way. Kw Arle gure e ha ore of a rooued eod hu at fare jut above $00 oe way for t oah dout tket bede t ajor eak at $0 oe-way. Tra World Arle ha two eak at $50 ad $0 oe way for the oah dout tket YD. Aera Tra Ar ha two eak at $67 ad $76 oe way for the ae oah dout YD tket. gure ad d how ultle eak for Northwet ad Delta Arle reetvely for oah dout YD tket. Thu there re dero wth luterg of fare otrollg for re drato by ug oly douted oah YD tket that have advae urhae reureet ad fee baed refud ad hage retrto. Iteretgly, there oe luterg of fare to ultle eak for the ext fare la hgher tha the douted tket. Th our o the full oah Y tket la the New York Cty to Orlado route. Cotetal Arle, for tae gure, ha re dero wth ultle eak o t full oah fare Y tket. There a ajor eak at above $00 oe-way ad two hgher eak at $50 ad $00 oe-way. Cotetal Arle alo a aller eak at $5 oe way for t full oah Y fare whh about the ae re a the gle eak for oah dout YD tket for the arle gure. Delta Arle ha two loe eak for t oah Y fare at $75 ad $00 oe-way but alo ha a aller eak at $75 oe-way gure j. US Arway ha a ajor eak at $0 for t full oah Y fare ad a aller eak at $70 oe-way gure k. Thu gure k how that there re dero wth luterg of fare 0

50 eve at the ore exeve full oah Y fare. Pre drato aot exla the dero betwee the fare ategore betwee oah dout YD ad oah Y fare. I fat, the re drato tory outered by there ot beg a eat earato of varou tye of ouer ayg earate fare. Dete all of the trate uber of rule aoayg eah tye of fare, there are alway lever ouer ad travel aget workg aroud retrto to ay lower fare. Uatated ouer ato ay reult fro tarff retrto. 0 urtherore, arle, lke ay other fr, do ot alway have the luxury of lealy dfferetatg ther rodut to earate arket. Geerally, arle have lttle flexblty to dfferetate ther rodut, a t hard to do uh wth oah eat loely aed together a oat aluu tube,000 feet above the groud. Arle otatly adjutg vetory, tarff rule, fare, ad eforeet evdee agat a eat earato dvdg ouer tye. urther ooudg the dffulty of eatly earatg dfferet ouer tye, arle fae the uue roble of keeg eough of the hghet valuato ouer ayg the to re. The ajor US arle the dout arrer uh a Southwet Arle have le of th roble greatly deed o thee ouer ayg to fare. A zeable orto of ther reveue geerated fro th all grou of hgh reveue 0 or tae, to beat the hort-ote hgh fare, bue ouer travelg freuetly o a route eg MSP DW a urhae a tal oe-way tket MSP DW ad the roud tr tket o the revere terary DW MSP, ug the outboud orto of the tket to retur hoe ad hagg the retur tket of the roudtr MSP DW for the aller hage fee oe bue dtate aother tr rather tha urhag a ew roudtr MSP DW ad urrg the large fare reultg fro a urhae of a hort-ote tket. Arle have raked dow o th rate, but they aot do aythg to traveler that read th tat betwee arle uh a urhae the tal oe-way tket o Northwet Arle betwee MSP ad DW ad Aera Arle roudtr betwee DW ad MSP.

51 aeger ayg u to everal te the aout of the lowet douted fare. If eough of thee utoer a ealy fgure out how to beat the yte, the treedou avg wll be reaed at the exee of the ajor arle. Arle have to be reatve to kee a uh of th grou a oble ayg the to fare. ow arle oerate th ey evroet of ervg o ay dfferet tye of ouer a hel rovde a otvato behd develog a re dero odel wth ouer wth dttly dfferet valuato. A a reult of thee urret arle rate, ouer aot be ure what fare to exet whe they urhae a tket. There ot a great way that ouer a adjut the tg of ther urhae to reeve a artular fare. O oe had, erta arket ould have a lted uber of dout eat ell out way advae of the dearture date. Predtg whe th evet our oble; t ay be eve dffult to fd out what the lowet re the arketlae. owever, urhag a tket way advae of the travel date doe ot guaratee ouer the lowet fare. Arle ay lae vetory o ale at ay te, eve jut a hort te erod before uog deartg flght. Predtg whe the ext ale, let aloe what route wll be overed by fare ale early oble. urtherore, arle radly hage ther aaty otrolled fare la eat for oted fare very freuetly. Couer ad travel aget have very lttle forato o edate arle rg objetve o artular flght, day, te, ad/or etwork. All arle have agreed ther 999 oroe wth the US Goveret to dlay the lowet urret fare a arket. The fare telf, aaty aged to the fare, or tarff rule ould radly hage.

52 Bede the arle dutry, the re of the teret ad eletro oere ay alo add other exale where re dero ay oext wth dfferet buyer valuato. The re of oere through thee ew ea ha oly aelerated the ablty of fr to offer varyg re. Wthout eloyg dret re drato ethod, fr ay be able to re to dfferet ouer tye. Perod weekly fler akg utout ouo ad full ewaer adverteet of uog ale have gve way to ole auto, eal by oae ltg te o ale, ad outerzed earh ege urveyg ultle oay te for the lowet re oble. It o urre that US arle are aog the frt to tegrate eletro oere wth ther rg tratege. Durg the at few year ay of the ajor US arle have develoed a rogra of ealg weekly Iteret eal to ouer regtered at ther webte. Soe are eve gog further: Uted Arle aoued a aag to ublh twety dfferet "DalE-are[]" eah day that tarted o Otober 0 th, 999 Uted Arle Pre eleae - Otober 0, 999. Southwet Arle releaed th year ther DING! rogra lkg uer outer to ther yte to gve uker ae to lower fare. Soete thee fare are oly vald for a few hour Southwet Arle Pre eleae ebruary 8, 005. Th odel wll allow for ultle fr to oete ouer wth dfferg valuato ad foratoal gatherg ablte. Gve th heterogeety, wll fr radoze re ad how uh wll re flutuate? Wll a fr fd t ore worthwhle to ealze to oly oe ed of ouer? I t oble for a fr to fd a trategy of ellg the ae rodut ultaeouly to both ed of the arket? By

53 addreg thee ueto, th odel wll add the ae of heterogeeou ouer valuato to the re radozato lterature. Th wll gve aother trutured exlaato about the arle dutry ad obly offer a gle of how eletro oere oe arket ay evolve. -. The Model Aue that there are fr: through fr. r ell t rodut at re. Couer ho for oe dvble ut, uh a a arle tket. Aue that all fr the arket have the ae argal ot of rovdg erve. We wll lfy the varou tye of ouer dow to two: hgh valuato bue tye ad low valuato leure traveler. et be the roorto of gh tye, - be the roorto of low tye. Couer are lt to the fored ad ufored tye. Of the roorto of tye ouer, let be the roorto of ouer that ee the re of oly oe fr. Keeg wth our arle exale, thee ufored tye ould alo be thought a flexble or loyal traveler, oly wllg to urhae fro oe artular arle. Thu the total roorto of ufored hgh tye ouer. Aue tally that thee ufored hgh tye ouer are radoly dered betwee the fr. Thu / be the orto of hgh tye ouer wllg to buy fro oly oe fr. The roorto of ouer that ee all re -. If a fr re

54 lowet aog the fr, t wll ature thee - fored hgh tye ouer lu t hare of the hgh tye ouer /. Of - low tye ouer, let β be the roorto of ouer that ee oly oe re. Aga aue that thee β ouer are lt evely betwee eah fr. Thu - β/ ouer ee re by all fr ad the roorto β/ ee oly oe fr' re. The fr that re the lowet wll ature - - β - β/ of the low tye. The frt ter the low tye eeg both re wherea the eod ter the hare of the low tye eeg oly oe re. The followg hart uarze the ale to hgh ad low tye, gorg te whh we wll how do ot are eulbru. et ad be the re that are the axu reervato valuato of the hgh ad low tye. Table Sale to gh ad ow Tye: Sale to gh Tye Sale to ow Tye lowet of all fr re f β β f ot lowet of all fr re β If the fr rg the lowet aog the low tye, t wll alo ature the hgh tye: - fored ad / ufored ouer, or the uer left orer of the table. Prg the lowet aog all fr ature the to two boxe of the above hart. After 5

55 oe algebra, the hghet roorto T of ouer a fr a ature whe rg the lowet : T β Call the total roorto of fored tye a fr a obly ature β ke the Vara 980 odel of ale, fr radoze betwee okg ouer eeg oly ther re ad aturg all of the ouer eeg what all fr harge. owever, the trategy beoe ore olated e the fr fae ouer wth two dfferet valuato. A fr ay hooe to ell oly to the hgh tye of ouer f t harge a re above the hghet re that the low valuato ouer are wllg to ay. A fr ould alo et t trategy of extratg oooly roft o low tye that ee oly oe re, log oe reveue o the hgh tye, ad obly aturg the fored hgh tye f the other fr re hgher. ally, a fr ould ature all of the fored ouer by a low re, at the exee of oooly roft o the ufored hgh ad low tye. Wth thee four dfferet ouer tye: fored hgh, ufored hgh, fored low, ad ufored low, we wll look for yetr eulbru our odel. We wll frt larfy the rage of re a fr wll harge. rt, a fr wll ot harge hgher tha, the hghet oble ouer valuato, th doated. Se ouer are urhag oe drete ut, the oooly re a fr ould harge the hgh tye oly. kewe, the oooly re that a fr ould harge t low 6

56 tye. Should a fr re above, t wll oly ature the bue of the hgh tye. et λ be the u re that a fr would obly harge. Th u re λ greater tha or eual to the argal ot, e at ay re <, alo doated. I a yetr eulbru let be the uulatve robablty dtrbuto of [ λ ],. There are three ortat feature of o the terval betwee thee two 0 0 re. rt, f <, the <. If ot, ad, <, a fr ould rae t re to ad reae ouer roft. Th aue that roft at fro a ool of ufored hgh valued utoer are bouded below by roft at fro a ool of ufored hgh ad low valued ouer. Th alo aue roft at fro a grou of hgh valued ufored ouer are bouded above by roft at fro a larger ool of total hgh ad low valued fored ad hgh ad low valued ufored ouer. Both of thee tateet are rove a Prooto. Seod, there o a ot ad therefore o te o [ λ, ]. Suoe there a a ot of the terval λ,. The, aother fr ould ake a drete ju roft by rg ε below th a ot. The fr dog th ature the etre grou of fored ouer that were ayg the re at the a ot for oly a all lo of ε below the a ot re. Beaue the etve too great to re below the a ot re, o a ot urvve. ally, wll be trtly reag exet over a terval where fr are wthg over fro olletg oooly roft fro the low tye ad ellg oly to the hgh tye. Prooto 9h how that a u ga where flat, aug low ad hgh valuato are uffetly earated. 7

57 A fr a target a uber of ouer tye. A fr ay dede oly to ature the bue of t hare of the tye ufored ouer. At the oooly re, the exeted roft er ouer are: π M π / If a fr dede to ature ore tha ufored tye, the the fr' re wll have to be lowet aog all fr to ature all of the fored ouer wth a valuato greater tha or eual to the re. Gve that the robablty of ay fr havg the lowet re the arket le tha oe, exeted er ouer roft π e ut eual er ouer roft π M at the re. Ulke the Vara odel, the uber of fored ouer a fr ature deed o whether the re harged above or below the hghet valuato of eah ouer grou. Whe the re below, the exeted roft er ouer : π e h π [ / ] l π { / β / [ β ]} - f f > 5 The frt ter of euato are the exeted roft fro the ouer that oly ee oe re fro the fr. Wth robablty oe th gve er ouer roft of [/] -. The eod ter the exeted roft fro the hgh tye that ee all re. Wth robablty - the other fr - wll all have hgher re. I th otgey, fr wll ature the roft of the fored tye ouer. ow tye ouer do ot atter euato beaue the re above the valuato of the low tye ouer. If the re harged le tha or eual to, the low tye 8

58 ouer wll atter. I th earo, the frt ad eod ter euato 5 are the exeted roft fro the hgh ad low tye ouer that ee oly that fr' re. The eod ter euato 5, - β/, the low tye that ee oly oe re. The thrd ter euato 5 the exeted er ouer roft that a fr a exet fro the fored ouer. Wth robablty - the other fr - wll have hgher re, thu allowg fr to get the bue of both fored hgh tye - ad the fored low tye - - β. Whe the re exatly eual, the fr wll tll get the bue fro the ufored hgh tye, fored hgh tye, ufored low tye, ad fored low tye wth robablty -. The exeted er fr roft whe hargg : π l { / - β/ - [ β]} 6 et: π M π [/ - β/] - 7 be the oooly roft fro hargg, aug that there are o hgh tye. The roft euato for a fr : h π π l π f > euato f λ euato 5 I eulbru, all fr wll radoze over re o that exeted roft are otat at ay re wth otve dety: π M π h π l / - 8 The drete ju of utoer that are wllg to buy at reate a dotuty the re dtrbuto of fr. A the re ove fro to g, a fr loe a 9

59 drete uber of ouer the ufored low tye whle gag a ε reae reveue. The hage of exeted roft er ouer ovg fro to g : g g β 9 I a re dero eulbru, th exeted dfferee roft ovg fro to g ut eual zero. g β g g β 0 0 or all re reae above, euato 9 wll be egatve e the beeft gaed o reag the re by ε greatly overhadowed by the drete lo of reveue fro low tye ouer eeg oe re: - β/ -. Ay arku re over ut be large eough to outer the drete lo of tye ouer. et M be the lowet re that greater tha where exeted roft are the ae. We kow that the argal robablty f eual zero the rego betwee ad M, or the terval,m due to the lower exeted roft. Gve that f 0 th rego ad that there are o a ot o the rg dtrbuto, M. Thu, the oluto to euato 0 well defed ad le. Th aer tude the roerte of yetr xed trategy rg eulbru whh all fr radoze aordg to a uulatve dtrbuto futo deotrated gure. I the ae of eulbru extg wth th tye of re dero, there wll be three edogeou varable that wll be olved th odel: ÿ, g, ad λ. The exogeou varable are,,,, β, ad. Solvg for the uulatve re dtrbuto ÿ reure aoutg for the flat rego betwee ad 50

60 M. Solvg for ÿ eaer f we earate out the flat rego ad olve for the two rego o ether de of the flat ee of the uulatve dtrbuto. et the overall robablty dtrbuto of re ÿ be broke to two ee: ÿ ad ÿ. et ÿ be the orto of the re dtrbuto whe M ad ÿ be the orto of the re dtrbuto whe λ. g the flat rego of re betwee ÿ ad ÿ. gure g 0 ô M Pre 5

61 5 Table 5 Proerte of : Pre Value of ÿ Proerte of ÿ P l l l P œ l,] / β Ireag P œ,m f 0 Cotat g. ad P œ [M, / Ireag P ---- The ef value of the uulatve dtrbuto futo gure are gve Table 6. Table 6: Pro / β 5 Pro / 7 Pro λ β 9 Pro 5 g β β β

62 Prooto : The uulatve dtrbuto futo derbed Table 6 a yetr eulbru ff a or β greater tha 0 b or β le tha [/ - β/] - / - d / - T - e {/ - β/ [ β][ - ] - } - / - œ [l, ] gure below uarze Prooto. O the horzotal ax ad o the vertal ax. Two odto elled forth fro rooto oe are grahed the thrd odto tatg that the ufored oooly roft at ut be eual or greater tha the ufored oooly roft at the rght had dagoal le. To the left of th le Prooto atfed. The botto tragle thrd rego where Prooto doe ot hold. ere where there a eulbru for low tye oly. kewe the left-had dagoal le extedg fro the org the odto for Prooto d. Th art of Prooto ellg forth the odto that the ufored oooly roft at the oooly re ut be le tha the total ouer roft at the re atfed below the left dagoal le. Above the left dagoal le Prooto d fal. ere there a eulbru otg oly of the 5

63 hgh tye. Betwee the two dagoal le rooto oe atfed ad a re dero eulbru wth two dfferet ouer tye ext. - Codto for Pro d fal gh Tye Oly Eulbru Pre d. wth o. valuato β β Codto for Pro fal ow Tye Oly - gure Proof e: Suoe {/ - β/ [ β][ - ] - } - > / - ad the yetr eulbru derbed Table 6 hold. The fr wll fd t roftable to radoze at re hgher tha. Th volate the yetr eulbru derbed Table 6. Suoe 5

64 {/ - β/ [ β][ - ] - } - < / ad the yetr eulbru derbed Table 6 hold. The fr wll fd t ot worthwhle to radoze ad re oly at. Th volate the yetr eulbru derbed Table 6. Proof a: Suoe that ad β eual 0 ad the yetr eulbru hold Table 6. The fro Prooto e: {/ - β/ [ β][ - ] - } - / - [ - ] - } - 0 There wll be o radozato volatg the eulbru derbed Table 6. Proof b: Suoe that that ad β eual ad the yetr eulbru hold Table 6. The fro Prooto e: {/ - / [ ][ - ] - } - / - {/ - / [0][ - ] - ]} - / - [/ - / ] - / - 55

65 Thu there o radozato volatg the eulbru derbed Table 6. Proof : Prooto deal wth the ae whe t ay be roftable for fr to ell oly to the low tye. I th ae, the lo of low tye ouer too large for a fr to ake u wth hgher re. Suoe that there a yetr eulbru derbed Table 6 but [/ - β/] - > / -. Th ould hae f there a all dfferee betwee ad, or f there are uh aller uber of hgh tye ufored ouer tha low tye ufored ouer. Thu t wll ot be roftable for a fr to go after the hgh tye. I a yetr eulbru, th le that M >. elag g wth euato 9 ad reeberg gve the hghet read betwee ad M: / - -[/ - β/ - - ] - 0 Note euato that ha to be le tha oe e. If the - - ] - wll dro out e - 0. I th ae, fr wll ot radoze to the hgh tye ad thu M >. Whe M >, exeted roft at wll be lower tha exeted roft at. r wll ot fd t worthwhle to re hgher tha. Thu M < ad for a fr to go after the hgh tye: / > [/ - β/] - 56

66 57 Proof d: Th ext art of the rooto deal wth the ae whe t ay be roftable to ell oly to the hgh tye. A fr foug oly o hgh tye wll get exeted roft of / -. The ot ouer a fr ould get at the re [/ - - β/ - - β] or T. Whe fr fd t roftable oly to fou o the hgh tye, the t ut be the ae that the addto of - - β/ - - β ouer fro doutg doe ot ake u for the lot reveue fro the ufored hgh tye. Exeted roft wll fall by ellg alo to the low tye. Suoe that there a yetr eulbru derbed Table 6 but / - > T -. Thu t wll be roftable for a fr to oly go after the hgh tye. I a yetr eulbru, th le that M. Settg euato 9 to eual zero, ovg fro to g a yetr eulbru derbed Table 6 ut reult zero roft: 0 g g β 9a Suoe M ad g 0. Pluggg to euato 9a: [ ] [ ] [ ] 0 / / / β β [ ] [ ] [ ] 0 / β β [ ] [ ] 0 / β β

67 58 Thu M aot hold. ro euato 9a, there a u for whh ε a be to ake the euato hold. Th u g wll be foud a later rooto. Proof Syetr Eulbru: Suoe Prooto a e hold but there o yetr eulbru a derbed Table 6. A. Prooto e a be ued to fd : [ ] β β Aga, we relae ÿ wth ÿ, ue our defto for total roorto of fored tye ad obe ter: β β / β 5 The rato lar to ÿ exet that the low tye ow eed to be luded the aaly. The frt ter uerator tll derbe the dereae roft that a fr ear fro the ufored hgh tye whe t re far below - whe the [λ,] terval. owever, by rg low, the fr obta the low valuato ouer. The

68 eod ter the uerator the exeted reveue obtaed fro the ufored low tye. Th ter egatve beaue the fr wll ga the bue of thee ouer whe t re o the [λ,] terval. Thu the uerator derbe the et hage reveue fro rg below. The deoator the total reveue obtaed fro all fored ouer : both low ad hgh tye. B. Prooto e a be ued to fd : The β ter dro out of Prooto e e the re above : 6 elag ÿ wth ÿ. ad olatg the ÿ ter we have: [ ] [ ] [ ] / 7 The rato derbg ÿ ute traghtforward: I the eod ter the uerator derbe the reveue lot whe a fr re below the oooly re. The deoator the total arket roft that a be eared fro the fored ouer at the re. Th rato ad etre exreo wll eual zero ad oe reetvely whe, thu athg the earler tateet that the hghet re a fr wll harge. 59

69 C. l a be foud by ettg 0: Pluggg for λ: β λ λ [ β ] λ / 0 8 eeber [ β] or the total roorto of fored ouer. Pluggg to euato 8a ad lfyg: λ β λ λ λ β λ λ { [ β ] β} λ λ [ β ] β β 9a 9b λ [ β ] β λ 9 β T [ β β 0 60

70 6 ] [ ] [ β β a Euato 9a, 9b, 0, ad a are odfed vero of 9 that ake oaratve tat alulato eaer. D. g a be foud by luggg g ad for the re Prooto e ad the ubtratg the two euato: [ ][ ] g g g [ ][ ] β β [ ] [ ] 0 g g g β β Addg ad ubtratg [ ] to both de gve: [ ] [ ][ ] 0 g g g β Slfyg we get: [ ] [ ][ ] g g g β [ ] g g β β [ ] g g β β [ ] g g β β

71 6 g g β π β g g β π β π g β π β β π β π g β π β π β g g g ter ter π π β a g [ ][ ] [ ] β b M β π β π β The ga g betwee ad M deed o a rato betwee the fored low tye ad the fored hgh tye. Th rato adjuted by a otat alled g ter β. Th otat g ter ubtrated fro the uerator ad

72 added to the deoator. g ter adjut for the ae that the ufored hgh ad low tye are ot the ae. I the ae that the roorto of ufored low ad hgh tye the ae, the the otat eual zero. g ter wll alo beoe aller, the ore ueve the roorto of low ad hgh tye. The overall rato g large whe there are a large uber of ufored tye. A fr wll have to re ubtatally hgher tha to ake u exeted reveue fro thee lot low tye. Note that fro art A D the oluto ath the yetr eulbru derbed Table 6. Thu Prooto a e gve the yetr oluto derbed Table 6. QED Prooto : A eulbru wth oly hgh tye ext ff: a 0 < < b [/ - β/] - < / - / - > T - d {/ [ - ][ - ] - } - / - œ [l, ] Proof : Suoe that / - < T - ad the uvarate eulbru wth hgh tye hold. The fr fd t roftable to radoze to the low tye. Th volate the uvarate eulbru wth oly hgh tye. 6

73 Proof d: Suoe {/ [ - ][ - ] - } - > / - ad the uvarate eulbru wth hgh tye hold. The fr wll fd t roftable to radoze at re hgher tha. Th volate the uvarate eulbru wth hgh tye. Suoe {/ [ - ][ - ] - } - < / ad the uvarate eulbru wth hgh tye oly hold. The fr wll fd t ot worthwhle to radoze ad re oly at. Proof b: Suoe that [/ - β/] - > / - ad the uvarate eulbru wth hgh tye hold. The fr wll fd t ore roftable to et the oooly re at. Th volate the uvarate hgh eulbru. Suoe that [/ - β/] - / - ad the uvarate hgh eulbru hold. The fro Prooto : / - > T - [/ - β/] - > [/ - β/ β] - Th aot our. 6

74 Proof a: Suoe that 0 ad the uvarate eulbru wth hgh tye hold. The Prooto b a be wrtte a: [0 - β/] - < 0 Th volated. Suoe that ad the uvarate eulbru wth hgh tye hold. The Prooto d a be wrtte a: {/ [0][ - ] - } - / - The the uvarate hgh eulbru volated e there o. Proof Eulbru wth oly gh Tye: Suoe Prooto a d hold but there o uvarate eulbru wth hgh tye oly. A. a be olved fro Prooto d: {/ [ - ][ - ] - } - / - 65

75 66 ] [ / Note that the ae a Prooto. It alo the ae a Vara 980. B. The lowet re l a be olved by ettg 0. / 0 λ λ λ λ λ λ λ λ λ

76 Th the ae lowet re a Vara 980. QED Prooto : A eulbru wth oly low tye ext ff: a or β greater tha 0 b or β le tha [/ - β/] - > / - d / - < T - e {/ - β/ [ β][ - ] - } - [/ - β/] - œ [l, ] Proof : Suoe [/ - β/] - < / - ad the uvarate low eulbru hold. The fr wll fd t worthwhle to et the oooly re at. Proof d: 67

77 Suoe / - T - ad the uvarate low eulbru hold. The fro Prooto d fr wll ot radoze to the low tye ad oly go after the hgh tye. Proof e: Suoe {/ - β/ [ β][ - ] - } - > [/ - β/] - ad the uvarate low eulbru hold. The fr wll fd t roftable to radoze at re hgher tha. Th volate the uvarate low eulbru. Suoe {/ - β/ [ β][ - ] - } - < [/ - β/] - ad the uvarate eulbru wth low tye hold. The fr wll ot fd t worthwhle to radoze ad re at. Proof a: Suoe ad β eual 0 ad the uvarate low eulbru hold. The Prooto a be wrtte a: [0/ - 0/] - > 0/ - 68

78 69 Th volated a both de eual 0. Proof b: Suoe ad β eual ad the uvarate low eulbru hold. The Prooto e a be wrtte a: { / [ 0-0][ - ] - } - [/] - The uvarate low eulbru volated a fr wll ot radoze. Proof Eulbru wth oly ow Tye: Suoe that Prooto a e hold but there o uvarate low eulbru. A. a be foud fro Prooto e: [ ] β β β β β β / β

79 70 B. The lowet re l a be foud by ettg 0. / 0 λ λ β λ λ β [ ] λ β λ β β λ β β β λ QED gure are reated fro the odel ug Matheata a uulatve dtrbuto forat. The old le the grah of etre futo ÿ. The rght-had grah the grah of ÿ wherea the left-had grah ÿ. The dahed orto of the grah the orto of ÿ ad ÿ that are ot art of ÿ. Below are three grah gure,, ad 5 howg whe Prooto hold ad doe ot hold. gure a ae where Prooto hold ad there a eulbru a derbed Table III.

80 gure where Prooto fal ad a uvarate low eulbru. gure 5 where Prooto d fal ad a uvarate hgh eulbru. CD gure Pre gure : λ 0.8 β M 550 gure a exale of re dero eulbru wth two dfferet utoer tye ad three fr. The heavy old le the uulatve dtrbuto futo of re for eah fr. A wll be how, both rooto oe odto hold for gure. or the frt art of Prooto, odto ut hold. Proft of the 7

81 ufored tye at re ut be at leat a great a roft of the ufored tye at. eeatg odto : / [/ - β/] / [ / /] [ ] [. ] Thu Prooto hold for gure. gure wth the re dero wth two dfferet ouer tye ae Prooto d. eeatg odto : [/ - - β/ - - β] / - [ / / ] / [ ]

82 CD gure Pre gure : β I gure the old le oly at the left had uulatve dtrbuto futo. Th ot a eulbru of fr radozg to two dfferet ouer tye. Itead, th a eulbru wth low tye oly. r fd t roftable oly to radoze below. Prooto fal for gure. eeatg odto : / [/ - β/] / [ / /] [ ] 50 The above euato doe ot hold e 9 < 7

83 Thu odto fal whe the value orreodg to gure are ued. There o eed for fr to radoze to exluvely to go after the hgh tye oly. There are ore roft to be ade by gog after the lower tye oly. CD gure Pre gure 5: β I gure 5 fr radoze oly at the old le. They fd t roftable oly to erve the hgh tye; t ot roftable to go after the low tye. Aga, th ot a eulbru of fr radozg to two dfferet ouer tye. Prooto b fal for th eulbru. eeatg odto : [/ - - β/ - - β] / - [ / / ] / 7

84 [ ] The above euato doe ot hold e 60 < 85.5 Thu odto fal whe the value orreodg to gure 5 are ued. I th ae, Prooto b fal. The three fr fd t roftable oly to radoze to the hgh tye. -5. Coaratve Stat Gve thee four edogeou oluto,, l, ad g - how doe varyg the exogeou araeter -, β,,, ad affet the four edogeou oluto? Mot oaratve tat are ute tutve. or a few oaratve tat, t eaer to break ad β rato to ther raw ouer uber. et t total aout of ouer the arketlae. et U total aout of ufored hgh ouer. U The roorto of ufored hgh ouer t. et U total aout of ufored low ouer. The roorto of ufored low tye ouer U β. t Table 7 below defe the varou exogeou ad edogeou varable. Table 8 lt where oaratve tat of the varable a be foud the followg rooto. 75

85 Soe of the ore ortat oaratve tat lude /, /, /, l/, /, ad /. The oaratve tat /, /, / a be ee by oarg two route the Aedx: gure 6 o Agele to oolulu ad gure 7 Meaol to Chago O are. gure 6 aear to have ore leure traveler tha bue traveler a the dety rather large aroud low re. gure 7: Meaol to Chago O are ha ore bue traveler o the eak dety aear to be hgher re tha o Agele to oolulu. Whe further oderg the argal ot of travelg fro o Agele to oolulu uh hgher tha the argal ot of travelg fro Meaol to Chago O are due to the large dfferee ar dtae the two route, the fare adjuted for dtae are eve lower for gure 6 ad hgher for gure 7. Two route that are ore eudtat ter of ar leage are gure 8 Wahgto Natoal New York aguarda ad gure o Agele to a Vega. The fare the Wahgto New York route are hgher tha the fare the o Agele to a Vega. Sefally the axu fare the Wahgto New York route aear hgher tha the o Agele to a Vega route. There are ode well above $50 the Wahgto New York route wherea the hghet ode the o Agele a Vega route below $80. There aear to be le dety betwee $0 ad $50 the Wahgto New York route tha the o Agele - a Vega route. Thee two route gve oe redee to the oaratve tat /, /, ad l/. The dety reahe hgher eak for a lower axu re the o Agele a Vega route tha for the New York Wahgto route, a oaratve 76

86 tat redt. The u et of re aear to be lower the o Agele a Vega route tha the Wahgto New York route, jut a oaratve tat redt. Table 7: Varable Defto Varable Exogeou/ Defto Edogeou Edogeou Cuulatve re dtrbuto futo of eah fr Edogeou Uer orto of uulatve re dtrbuto for eah fr ag where fr ell to hgh tye ouer oly Edogeou ower orto of uulatve re dtrbuto for eah fr ag where fr ell to both hgh ad low tye ouer g Edogeou lat orto of uulatve re dtrbuto above the oooly low re for eah fr where t uroftable for fr to radoze to ouer l Edogeou owet re ag eah fr uulatve re β Exogeou Exogeou dtrbuto futo U Proorto of ufored hgh tye ouer t Proorto of ufored low tye ouer U β t Exogeou Proorto of hgh tye ouer Exogeou Moooly re of hgh tye 77

87 Exogeou Moooly re of low tye Exogeou Margal ot Exogeou Nuber of fr t Exogeou Total uber of ouer U Exogeou Total uber of ufored hgh tye ouer oaratve tat ot how aer U Exogeou Total uber of ufored low tye ouer oaratve tat ot how aer Varable Exogeou/ Defto Edogeou Below Table 8 that gve a ooldated lae to revew reult of th eto of oaratve tat. The etry the frt row ad frt olu read a ÿ/. The oaratve tat ÿ/ le tha zero ad the atual roof a be foud ea b after Prooto 6. 78

88 Table 8: Coaratve Stat Prevew ÿ ÿ l g ea b after Pro 7b Pro 8d Pro 9b Pro 6 < 0 > 0 > 0 f < 0 β [ ] > β β < 0 f β [ ] < β β β N/A Pro 7d > 0 or < 0 N/A Pro 7 < 0 Pro 8a > 0 Pro 8b > 0 Pro 9 > 0 f - < - < 0 f - - Pro 9a < 0 ea a after Pro 7a Pro 8 Pro 9e Pro 6 < 0 > 0 > 0 f > β < 0 < 0 f < β N/A N/A N/A Pro 9d > 0 ea after Pro 7e Pro 8e Pro 9g Pro 6 < 0 < 0 > 0 < 0 79

89 Pro 6a Pro 6b Pro 6d Pro 6 < 0 exet at < 0 < 0 0 t Pro a Pro b Pro Pro 9f > 0 > 0 < 0 > 0 f U > U < 0 f U < U Prooto : or all, f ad β aroah zero ad the odto of Prooto hold, the uulatve re dety hft ootoally toward the lowet re ad the lowet re get lower toward argal ot. I other word let ÿ, β be the eulbru a derbed Table. ÿ, β Ø d, where d a ot a o argal ot. The arket beoe ored of fr very oaoally rag re to a oderate or hgh level ad the ret of the te of fr hargg very, very loe to argal ot. Proof: Provg th rooto reure etablhg thee ot: a ÿ/t > 0 b ÿ/t > 0 λ/t < 0 d " x >, l, β 0 ö x, β 80

90 8 Proof a: / 7 earragg to ut t ad U for : / t U t U / U t U /t U t U U t U 0 > U t U U t U Wth the otve relatoh betwee t ad, reag t to fty aue to reae. Se t ad have a egatve relatoh by t U, reag t to fty whe U held otat aue to fall to ear zero. Thu ad have a egatve relatoh whe t reaed ad U held otat. Thu reag t to fty holdg U otat or lowerg to zero aue to reae for re alog t terval. Alot o ouer ay the full oooly re a all ouer reeve oe ort of re break o ther urhae.

91 8 Proof b: / β 5 earragg ÿ to ut t ad U for ad t ad U for β: / t U t U t U t U [ ] / U t U t U U /t [ ] [ ] [ ] [ ] U t U t U U frat /t [ ] [ ] 0 > U t U t U U frat Wth the otve relatoh betwee t ad, reag t to fty aue to reae. Se t ad have a egatve relatoh by t U ad t ad β have a egatve relatoh by t U β, reag t to fty whe U ad U are held otat aue ad β to fall to ear zero. Thu ad ad β jotly have a egatve relatoh whe t reaed ad U ad U are held otat. I a future rooto β ad are how ot to have a ole egatve relatoh. Thu the jot relatoh eeded wth ad β. Thu reag t to fty holdg U ad U otat keeg

92 8 rooto oe atfed or lowerg ad β to zero aue to reae for re alog t terval. r are ofte rug ale to ature the fored ouer the arketlae. Proof : λ β 9 earragg λ to ut t ad U for ad t ad U for β: λ t U t U t U λ U t U t U λ/t [ ] U t U t U λ/t 0 < U t U t U Wth a egatve relatoh betwee λ ad t, reag t to fty aue λ to fall ear argal ot. Se t ad have a egatve relatoh by t U ad t ad β have a egatve relatoh by t U β, reag t to fty whe U ad U

93 are held otat aue ad β to fall to ear zero. A wll be how a future rooto both λ/ ad λ/β are greater tha zero. Thu whe ad β to fall to ear zero, λ fall ear argal ot. Proof d: Prooto a how that a ad β aroah zero, ÿ, ÿ are ootoally reag whle l ootoally dereag to argal ot. Th art of the rooto, how that ay re x above argal ot ha x, β. We frt kow that for ay re x <, x, β 0 a fr wll ot uffer loe. or the uer orto of the dtrbuto: x x x / Whe ö 0, the - ter the deoator aroahe whle the uerator aroahe 0. Thu the frato aroahe 0 ad x ö. or the lower art of the dtrbuto: x x [ β ] β x x / Whe,β ö 0, both - ad - β aroah the deoator reultg the deoator aroahg x. The uerator aroahe 0 a both ter aroah 0. Thu the frato aroahe 0 for all x > ad thu x, β ö. 8

94 Wth x ö ad x, β ö whe,β ö 0, x, β aroahe a ot a. Th ot a d a ot a aroahg argal ot a Prooto how the lowet re aroahg argal ot ootoally. Se x, β reae ootoally ad the lowet re l dereae ootoally to argal ot a,β ö 0, x, β ootoally aroahe d. gure 6 below a exale of a eulbru of ad β aroahg zero wth rooto oe holdg. Note that ot of the weght of the uulatve dtrbuto loated ear λ 50.86, whh extreely ear argal ot 50. The uulatve dtrbuto futo doe ot really flatte out at the araeter value ad β 0.00 utl after reahe 0.8. Elo alulated at 65 ad th at heght of.98. Thu fro M 65 to 000, the uulatve dtrbuto futo ute flat. r geerally ru extree ale ot of the te a ot utoer are extreely fored. Oaoally, fr rae ther re aywhere fro oderate to hgh level to ature roft fro the few ufored utoer. A ad β get loer to zero th our le ad tead Bertrad rg our ore ofte. 85

95 CD gure Pre gure 6: λ β M QED Prooto 5: If ad β aroah oe ad the odto of Prooto hold, dero the arketlae dereae. The uulatve dtrbuto futo hft t weght toward the oooly re ad. λ otoed rght below ad M otoed jut below. The robablty dety ootoally aroahe a bodal dtrbuto at ad though t ever beoe oe. 86

96 Proof: ro Prooto :. /t > 0 le Δ / Δ < 0. /t > 0 le Δ / Δ < 0 ad Δ / Δβ < 0. λ/t < 0 le Δλ / Δ > 0 ad Δλ / Δβ > 0 The frt two ot date that ad are ootoally reag a,β ö. The thrd ot date that λ ootoally reag a,β ö. Now t eed to be how that a x, β 0 for x <., β b g,β x, β for x > M, β d, β, β e, β, β Proof 5a: or re x le tha l, x, β 0. λ [ β ] β 87

97 A,β ö, the braketed exreo the deoator beoe zero. l a be rewrtte a λ. Prooto ay the followg eualte ut hold: [/ - β/] - / - [/ - β/ - - -β] - Takg the lt,β ö, the eualty redue to: / - / - / Thu l ö whe,β ö. All re x, below have x, β 0. Proof 5b: β g [ ] [ ][ ] b The rght had ter of the uerator ad deoator beoe 0. Thu g a be rewrtte a g g 88

98 89 Euato gve the axu ga betwee ad g : / - -[/ - β/ - - ] - 0 Takg the lt a,β ö : / - / Pluggg g for - verfe that M : Proof 5: ro Prooto, the uer orto u: λ Takg the lt a ö : l λ Proof 5d:

99 90 / l / l / l / l Proof 5e: / β /, l β β /, l β β /, l β β

100 l β [ β ], β / Alyg ôtal ule by takg / b of the uerator ad deoator: l [ ], β / l, β / Se ad β are gettg larger, ÿ ad ÿ are gettg aller for ot re. The lowet re λ gettg larger a fr are ot oetg teely for the hoer. gure 7 ad 8 rovde a exale of ear lt ae. Thee are exale of eulbra where fr radoze betwee two dfferet tye of ouer. Geerally, the wdow for thee tye of eulbra ute all, wth oly the whole uber workg for the gve araeter gure 8. Wth thee tye of eulbra wth two dfferet tye of ouer, there ot uh re dero. Pre boue betwee a all terval below ad ludg the hghet re ad a terval rght below ad ludg the oooly re for the low tye. The lowet re λ jut below whle M jut below. A ad β aroah oe, the dtrbuto a two ouer eulbru aroahe a bodal dtrbuto at ad. 9

101 gure 7 CD Pre gure 7: λ 8.75 β M 99.8 CD gure Pre gure 8: λ 6.6 β 0.99 M

102 QED Prooto 6: or all ad β uh that the odto of Prooto hold, f aroahe fty, the uulatve re dety ootoally hft toward the hghet re ad the lowet re get lower a ore fr eter the arket. I other word, ÿ ö d, where d a ot a o the oooly hgh re. The arket ored of the very oaoal extree ale of ouer ayg the lowet re ad the ret of the te of everyoe ayg the oooly hgh re. Proof: Th rooto deal wth the ae whe there extree dero the arket ad what hae f ore fr eter the arket. Provg th rooto reure etablhg thee four ot: a / < 0 exet at. b / < 0. g/ 0 d λ/ < 0 e x 0 for x < f Proof 6a: 9

103 9 Calulatg / reure the trk of takg l of both de of euato 7: / 7 - / l- l Takg / o both de wll allow u to alulate /. / l l l

104 95 l / l The value of the rght had exreo deed o the value of. The u of l a wll detere whether / greater or le tha zero. The frt ter, or the atural log ter, wll be le tha zero e the frato de the exreo le tha oe. ow loe to zero th frato de the atural log exreo detere the g of the /. A, the left had ter a beoe uh le tha egatve oe beaue the ter uare braket that take the atural log of aroahe zero. The rght had ter of euato a aroahe oe but th ot eough to outerbalae a large egatve ter. Euato a the egatve ad thu euato or / < 0. The exeto that for re, ÿ beoe teeer, lookg ore lke a ke at, wth a reae. ere ot utoer are ore lkely to be ayg the oooly re. Proof 6b: To at redug the rawlg ybol, wll be abbrevated:

105 96 / β - frat /- - fratd fratn 5 ke /, alulatg / reure the trk of takg l of both de of euato 5: / ] [ β l- l β Takg / o both de wll allow u to alulate /. / [ ] / l frat [ ] [ ] l frat [ ] frat l [ ] l frat / l β 6

106 The value of the rght had exreo deed o the value of atural log futo. The aaly detal to alulatg /. The u of β l 6a wll detere whether / greater or le tha zero. A aug Prooto oe atfed, / < 0 ff the atural log ter le tha egatve oe. Whe th hae, euato 6a wll be atfed. Exag the uare braket ter the atural log futo left-had ter ore loely, β [ β ] whe get very large the deoator get very large. The etre uare braket ter aroahe zero ad the atural log ter beoe a large egatve ter. The rght had euato 6a aroahe oe, but ot able to offet uh a large egatve uber. Thu / < 0. Proof 6: β β π g β π Dvdg the uerator ad deoator by euato gve β β g β b 97

107 Takg the dervatve of g wth reet to gve a reult of zero. Gve that both ad are hftg dowward the flat rego g rea otat wth a hage the uber of fr. Proof 6d: λ/ [ β ] [ β β ] < 0 The reae the uber of fr aue fr to reagly oete for the hoer. owever, ther exoetal odd of wg the hoer fall a aroahe fty uh fater tha the fall exeted roft of reevg of ther ufored roft. r oete vgorouly for the hoer a the lowet re λ fall whe reae to fty. owever, ot uh weght laed o re aroud λ beaue the odd of wg agat the other fr ot o good. Proof 6e: l x β / x β x x / There are two oetg fator: the the deoator of the frato de the braket ad the /- the exoet. The frato de the braket aroahe 98

108 99 0 a get larger but the th root the exoet doate ad / x x x β aroahe oe. Thu x 0 l x. / l / x x x for x <. Aga there are two oetg fator: the the deoator of the frato de the braket ad the /- the exoet. Aga, the frato de the braket aroahe 0 a get larger, but the th root the exoet doate ad / x x aroahe oe. Thu x 0 l < M x Proof 6f: l / l / 0 l gure 9 a exale of a eulbru of two dfferet tye of ouer where allowed to get large. gure 9 ha the ae araeter value a gure

109 exet for the larger value of. gure 9 blow u o the vertal ax to how extreely all for all re exet for thoe rght aroud. Th a redted by the oaratve tat. A redted, λ ha oved very low rght aroud argal ot. M, ad therefore ε, have the ae value fgure ad 9, jut a the odel redted. Thu, fr otly re at but oaoally ru extree ale o the all hae they have wg the fored ouer. A get larger, oly get aller for ot re, the exeto beg re edately le tha. Alo λ he loer to argal ot. CD gure Pre gure 9: λ 5.89 β M 550 QED Th ext lea ha bee already foud revou lterature: 00

110 ea : a / < 0 b / < 0 / < 0 Proof: The edogeou araeter ÿ a oly be affeted by four araeter:,, ad. The ae of wa dealt Prooto 6. Varyg,, ad β wll have o effet o the hgh orto of the uulatve robablty dtrbuto ÿ. Proof ea a: / / Obvouly, / < 0 e / / / < 0 See Prooto 7a for grah. Proof ea b: / / / / 0

111 / / [ ] / / < 0 5 See Prooto 7b for grah. Proof ea : / 0 / / < 0 See Prooto 7e for grah. QED Prooto 7:a / < 0 b / < 0 / < 0 d: Whe low ad, /β < 0 0

112 I the ae of a hgher, f > 0 a ad /β < 0 a λ e / < 0 <, /β Proof 7a: Calulatg / farly ba: β / 5 / / β / / < 0 7 gure 0 ad rovde a exale of what hae f dereaed, holdg all other exogeou varable otat. Note that all of ÿ reae whe dereae fro 00 to 700. Both ÿ ad ÿ re wth a fall. 0

113 CD gure Pre gure 0: λ.66 β M 550 CD gure Pre gure : λ β M 550 0

114 The ext three oaratve tat alulato are ore legthy. Proof 7b: β / 5 / [ frat] [ / β / [ [ β β [ frat] [ [ β ] β [ frat] [ β ] [ β β ] / / < 0 ff the uerator o the rght-ot frato greater tha zero, or [ β ] [ β β ] > 0. Th odto a be rewrtte a: [ β ] > [ β β ] a We kow fro rooto that fr wll ell to both low ad hgh tye ouer rather tha jut to low tye f: 05

115 / > [/ - β/] - earragg the above euato to ake a eaer oaro wth odto a: > 8 β If the left de of a a be how to be greater tha the left de of euato 8 [ β ] [ β β ] > β b the we wll have how that odto a hold by takg advatage of tratvty: [ β ] [ β β ] > β > > The lat relato - > hold e the doa of. Ug a rato tet: ettg [ β ] [ β β ] a/b ad /d β ad - - β- β β-β - -β- > -β- β - b Subtratg the rght-had de o that b 0: ad - β - β- β- - β - - β - > 0 b Thu ad -β[ - β - - ] - > 0 b Thu odto hold lyg / < 0. gure ad rovde a exale of hagg ad holdg all other exogeou varable otat. The effet of dereag fro gure to gure a reae ÿ. Both ÿ ad ÿ re wth a fall. 06

116 CD gure Pre gure : λ β M 80.8 CD gure Pre gure : λ β M

117 Proof 7: / [ frat] [ frat] [ frat] [ [ β β β [ / ] [ β] ] β [ β [ β ] β[ β ] [ *fratn] [ frat] [ β β [ frat] [ β [ β β ] [ frat] β β 9 / < 0 f the uerator of the rght frato greater tha zero. earragg th, we obta odto d: β β > d Aga, we ue euato twety eght to ake eaer oaro wth d: 08

118 > 8 β Iff d hold, the the left had de of d hould be greater tha the left-had de of 8 or odto e: β β > β e the we wll have how that odto d hold by takg advatage of tratvty: [ β ] [ β β ] > β > > f β Aga, ug a rato tet: lettg β a/b ad /d β ad - β β - - β > - β b Subtratg the rght had de to allow b to eual zero: ad - β β - - β - β > 0 b - β β - β - β β β β [ - - β- ] > 0 e β - - < Thu odto d hold lyg / < 0. gure ad 5 rovde a exale of hagg ad holdg all other exogeou varable otat. The effet of dereag fro gure to gure 5 a dereae ÿ. 09

119 Se ÿ doe ot deed o, there o hage ÿ whe hage. Aother way to derbe the effet of hagg o ÿ that a hgh le that there a all dtae betwee ÿ ad ÿ a gure deotrate. Whe dereae, there le hgh ouer tye ad ore low ouer tye. r dout ore to the low tye ad ÿ hgher at low tye, or further away fro ÿ a gure 5 deotrate. CD gure Pre gure : λ 88.8 β M 7.5 0

120 Pre CD gure 5 gure 5: λ 0.88 β M 550 Proof 7d: / β Calulatg /β gve the leat traghtforward awer Prooto 7. /β [ ] ] [ [ frat β [ ] [ [ frat β

121 [ frat] [ frat] [ β β] [ ] [ frat] [ ] 0 /β < 0 f the thrd braketed ter [ - ] euato 0 greater tha zero. Wrtg th a odto g we have: > g Aga, we ue euato 8 to ake eaer oaro wth d: > 8 β Ulke the revou two dervatve, we aot ay that - greater tha -, e the deoator the eod ter of le tha oe. We β do kow that - really oooly roft of eah fr at e a a be fatored out fro the deoator of euato 0: [ frat] 0a Wth a all that - >. If - greater tha -, the /β < 0 everywhere o [λ,]. Th ottute ae where the frato of ufored hgh tye large. The oote ae whe the ufored hgh tye are ot large. If

122 - le tha - but greater tha λ -, the /β 0 for oe o the terval [λ,], where - t -. I th ae, ÿ rotate outerlokwe a β reaed. Pre ear have /β > 0 whle re ear λ have /β < 0. gure 6 through 9 rovde exale of how ÿ hage wth β. gure 6 ad 7 rovde a ae where /β < 0 everywhere o [λ,]. I thee two dagra - 5 greater tha The two edot of ÿ gure 6 ad 7 ath the ae of /β < 0 everywhere o [λ,]. λ gure 7 lower tha λ gure 6 ad lower gure 6 tha gure 7. Th orreod wth ÿ beg hgher gure 7 wth a lower β 0. tha gure 6 where β 0.6. gure 8 ad 9 deotrate the ae where re ear ad for the ot art of ÿ have /β > 0 whle re ear λ have /β < 0. I thee dagra - le tha ÿ doe a lght rotato to the outerlokwe dreto. I gure 8 wth a hgher β, λ ad are both hgher tha λ ad reetvely gure 9. Th etu ea that f the two ÿ fro gure 8 ad 9 are draw o the ae dagra, the two would ro. /β lghtly otve at re ear ad egatve at all other re. The four dagra alo deotrate that hagg β aloe doe ot rodue large hage ÿ. Vual eto a verfy th reult betwee gure 6 ad 7 ad betwee gure 8 ad 9. A a added hek, λ doe ot vary wdely, eeally betwee gure 8 ad 9. doe ot vary wdely, eeally whe oderg

123 gure 6 ad 7. Thu the deterate reult of /β ot a ortat e the atual varae ÿ fro β all. CD gure Pre gure 6: λ 8.7 β M 80.

124 CD gure Pre gure 7: λ.07 β M CD gure Pre gure 8: λ 00 β M 6. 5

125 Pre CD gure 9 gure 9: λ 9. β M Proof 7e: / β / [ ] β β β 0 < β

126 gure 0 ad rovde a exale of hagg ad holdg all other exogeou varable otat. The effet of dereag fro gure 0 to gure a reae ÿ. Both ÿ ad ÿ re wth a fall. CD gure Pre gure 0: λ 0 β M 55. g 5. 7

127 CD gure Pre gure : λ 0. β M 78.8 g 8.8 QED Prooto 8: a λ/β > 0 b λ/ > 0 λ/ > 0 d λ/ > 0 e λ/ > 0 Prooto 8 the oaratve tat dealg wth the lowet re l. All of the oaratve tat are otve. 8

128 Proof 8a: Coaratve tat alulato o λ are traghtforward: λ/β [ ] [ β β] [ β β] > 0 gure 6 9 rovde two exale that deotrate that λ/β > 0. Whe β fall fro 0.6 gure 6 to 0. gure 7, λ fall fro 8.7 gure 6 to.07 gure 7. The ae tred our gure 8 ad 9. Whe β fall fro 0. gure 8 to 0.05 gure 9, λ fall fro 00 gure 8 to 9. gure 9. Proof 8b: λ/ [ β β] β [ β β] [ β β β ] [ β β] [ β] [ β β] > 0 gure ad 5 rovde a exale that deotrate that λ/ > 0. Whe fall fro 0.8 gure to 0. gure 5, λ fall fro 88.8 gure to 0.88 gure 5. 9

129 Proof 8: λ/ [ β ] β > 0 gure 0 ad rovde a exale that deotrate that λ/ > 0. Whe fall fro 00 gure 0 to 700 gure, λ fall fro.66 gure 0 to gure. Proof 8d: λ/ [ β β] [ β β] [ β β ] [ β β] [ β β] [ β β] [ β β] [ β β] [ β ] [ β β] > 0 5 gure ad rovde a exale that deotrate that λ/ > 0. Whe fall fro 0.9 gure to 0. gure, λ fall fro gure to gure. 0

130 Proof 8e: λ T 0 λ/ T [ β ] [ β ] 6 The deoator euato 6 alway greater tha the abolute value of the uerator. Thu the left frato alway betwee 0 ad -. Thu λ/ > 0. gure 0 ad rovde a exale that deotrate that λ/ > 0. Whe fall fro 00 gure 0 to 50 gure, λ fall fro 0 gure 0 to 0. gure. Prooto 9: a g/ < 0 b g/ < 0 f g/ > 0 f : g/β < 0 f - - g/β > 0 f - < - d g/ > 0 e g/ > 0 f > β β [ ] < β β β [ ] > β β g/ < 0 f < β

131 f g/t> 0 f U > U. g/t< 0 f U < U. g g/< 0 f h M g [ ] f Proof: Whe fdg the oaratve tat for g, t ortat to kee d oe bakgroud forato. rt, g eaure the ga betwee the two uulatve dtrbuto futo ad. A eah of thee exogeou araeter hage, oe or both of the uulatve dtrbuto futo wll hage. urtherore fxed for all oaratve tat alulato exet obvouly whe fdg ε/. Thu a hage, hage a well: reag a ÿ reae ad dereag a ÿ dereae. Se M, the ga g a be thought a a le oetg M ad ldg u ad dow both uulatve dtrbuto futo. Wth o uh gog o fdg the oaratve tat for g, euato wll be rewrtte wth abbrevated ter g ter, g D, ad g N to redue the rawlg of oe ter. β β π M β π

132 β β π g β π β π g ter g π gter g N g gd a b Proof 9a: g/ [ β / β ] gd [ π β ] g N Exadg the left ter of the uerator: [ β / β ] g D g D [ β / ][ π g ] [ β ][ π g ] Exadg the rght ter arryg the egatve g: [ π β ] g D ter [ π ][ β π g ] [ β ][ β π g ] After a oule of aellato, the frt ter redue to: ter [ β / ][ π ] [ β ][ π ] The eod ter redue to: [ π ][ β π g ] ter ter ter

133 Cobg the two ter the uerator: [ β π ] [ β π ] [ β β ] / π g/ < 0 ff - β β - - < 0 β ff β > - 7 We kow fro euato 8 that β > 8 Euato 6 hold ff β β > β ff - β - - ββ > - β ff - β β - β β - β > - β ff - β β β - β > 0 ff - β - β > 0 Se th lat exreo hold, euato 6 hold ad thu g/ < 0. gure ad 5 rovde a exale that deotrate that g/ < 0. Whe fall fro 0.8 gure to 0. gure 5, g re fro 7.5 gure to 50 gure 5.

134 5 Proof 9b: g/ [ ] [ ] / / / / g g g D N D π β Aga exadg the left ter of the uerator: [ ][ ] [ ][ ] ter ter g g π π β / / Exadg the rght ter of the uerator arryg the egatve g: [ ][ ] [ ][ ] ter ter g g π β π β π / / After aellato, the left de of the uerator redue to: [ ][ ] [ ][ ] ter g π π β / / The rght de of the uerator beoe: [ ][ ] [ ][ ] ter g π β π β π / / Cobg both de of the uerator yeld: [ ] [ ] / / / g ter π β π β urther lfato yeld: [ ] [ ] [ ] [ ] β β β [ ] [ ] [ ] β β β β [ ] β β β β

135 Thu, β β β [ ] g/ β gd 8 Euato 8 otve ore ofte tha t egatve. The β ter the rght had exreo geerally drvg the ze of the exreo frot of. Whe > β or whe lghtly le tha β, euato 8 otve. β le tha oe or lghtly greater tha oe, whh eough to kee β β le tha β [ ]. Whe β > >, the exreo egatve a β large. Prooto 7b ad ea a deotrate that / ad / have the ae g: both beg egatve. g/ attet to fd whh orto of the dtrbuto flatte out fater wth a hage. A hgh alha aue both uulatve futo to be teeer eeally. owerg fro a hgh value wll flatte fater e t wa very tee to beg wth ad thu lower the ga betwee the two futo at. Proof 9: g/β [ π / ] gd [ / ] g N gd Exadg the left ter of the uerator: 6

136 [ π ][ π g ] [ / ][ π g ] ter ter Exadg the rght ter of the uerator arryg the egatve g: [ / ][ β π g ] ter After aellato ad oe rearragg, the uerator beoe: [ π ][ π g ] [ / ][ π ] ter π [ π β / / ] π [ π / ] π [ ] Puttg the uerator bak to the etre frato gve: π g/β [ ] ε D 9 ke rooto 7d, th euato wll be otve f > ad egatve otherwe. Aga, ÿ doe ot deed o β ad thu doe ot hage wth β. If - < -, the a reae β wll aue ÿ to vot outerlokwe aroud oe re o the [λ. terval. wll reae ad the ga g betwee ad M wll reae. If - > - the a reae β wll aue to dereae everywhere [λ,] ad thu g wll dereae. 7

137 Aother way to ee th reult reeberg that β the roorto of ufored low tye ouer. A fr wll have to reae t re by a greater g fro to M to outer the drete lo of thee ouer ot wllg to ay ore tha. If le tha, the the lo of the low tye ouer ot a bad for fr beaue there a large eough roorto of ufored hgh tye ouer that fr a rely uo. owever, f there ot eough of thee ufored hgh tye e < the the lo of the low tye ortat to fr ad g wll have to reae a β reae. Proof 9d: Th oaratve tat tell about the loe of the two uulatve robablty dtrbuto futo. Ulke the other art rooto 9, both uulatve robablty dtrbuto futo rea otat whe hagg. If teeer tha, the reag wll arrow g. If teeer tha the reag wll wde g. g g/ g N D [ g ] g [ g ] ter D gd ter g N g N gd gter g N g gd g N g D g ter D [ β ] g D g N g g D gd ter 0 8

138 The frt ter g N g D of the uerator otve. The ter geerally large ule extreely all. If very all, g D the left had ter beoe very all ad ε N alo beoe all. Thu the left-had ter g N g D the uerator of euato 0 very all wth beg all. owever, wth beg all, the eod ter the uerator beoe otve o the uerator rea otve. A larger ea that the g N g D ter the uerator beoe large ad a hadle the ubtrato of g ter. Thu, the uerator rea otve all ae of, β,,, ad uh that Prooto hold. Thu g/ > 0. Proof 9e: β / g g / g g/ [ ] [ ] D N D Exadg the left ter of the uerator: [ β / ][ π g ] ter Exadg the rght de of the uerator: [ / ][ β π g ] ter eobg the two de of the uerator: [ g ter ][ β / / ] [ / ][ β ] g ter β 9

139 0 Pluggg the lfed uerator bak to the orgal alulato: g/ D ε β If > β, the g/ > 0. The g ju fro to M wll be aller a the roorto of ufored hgh tye fall. e ufored hgher tye ouer wll outer the drete lo of low tye at re above. Proof 9f: g β π β π β g t U U t U t U t U U t U t U g U U U t U U U U t U g U U t U U t U U t U U t g/t [ ] [ ] [ ] U U t U U t U U t U U t U U U

140 [ ] [ ] [ ] U U t U U t U U U U U t U U t [ ] U U t U U t U U tu U t U [ ] U U t U U t U U t U U t U U [ ] U U t U U t U U U U t U U t [ ] U U t U U t U U t U U U t [ ] U U t U U t U U t U U U U tu [ ] U U t U U t U U U t U U U U t [ ] U U U U t U U t tu tu [ ] U U U U t U U t U U t U U t [ ] U U U U t U U t U U t tu [ ] U U U U t U U t U U g/t> 0 f U > U. g/t< 0 f U < U.

141 Proof 9g: g β π β π β β β β g g der ur g/ [ ] [ ] [ ] der ur ur der g g g g β β β β [ ] [ ] [ ] der ur g g β β β β [ ] [ ] [ ] β β β β [ ] [ ] [ ] / β β β β β [ ] [ ] [ ] der ur g g / β β β [ ] [ ] [ ] / β β β β [ ] [ ] [ ] / β β β β [ ] [ ] [ ] der ur g g β β β β

142 Euato egatve, evaluated at varou araeter of a,b,, ad. If the exreo lfed further, the rght had ter of der ur g g lfed rodue a ter π β the uerator that aear to doate all other ter. Thu g/ < 0. gure 0 ad rovde a exale that deotrate that g/ < 0. Whe fall fro 00 gure 0 to 50 gure, ε re fro 5. gure 0 to 8.8 gure. Proof 9h: eeatg Euato 0: I a re dero eulbru, the exeted dfferee roft ovg fro to g ut eual zero. 0 g g g β β 0 Mzg th wth reet to g gve the u oble g for a gve et of araeter value. Note that g aot be egatve beaue Euato 0 would ot hold. g Mzg euato 0 wth reet to g: [ ] [ ] 0 g g g g β β [ ] [ ] 0 : g f g g OC ε [ ] [ ] g g f g g

143 [ ] [ ] g f g g f g g [ ] [ ] f f g g β g β [ ] g β g β g β 5-6. Exlag the Grah: Mathg the Coaratve Stat to Arle Data -6a. Dug the Couter Geerated Grah ro gure -, there a be x oluo that a be draw. rt, etry by fr hage the re dtrbuto. The lowet re get lower whle ore ad ore weght get laed o the hghet re. gure 9 a exale of ay fr the

144 arket. I that dagra, the ke the uulatve dtrbuto at tart whe eual about.0. A eod oluo that a be draw that hftg the roorto of ufored ouer to the extree of zero ad oe aue the uulatve dtrbuto to ollae o erta re. Whe ad β ollae to ear zero a gure 6, the uulatve dtrbuto futo early ollae to a ot a ear argal ot. r re at th low re alot alway wth a very oaoal re arku to take advatage of whatever few ufored ouer there are. Whe ad β ove to ear oe a gure 7 ad 8, the uulatve dtrbuto futo ove to a ear bodal dtrbuto futo at the oooly hgh re ad oooly low re. r eetally radoze betwee thee two re. A thrd oluo that the two dtrbuto ÿ ad ÿ beoe loer together a get loe to oe. gure how a ae where eghty eret. g all ad oly a all orto of the weght ut o the dtrbuto below. gure 5 how the roorto of bue traveler fallg to 0.. g reae ad the dtae betwee the two futo ÿ ad ÿ reae. A a be ee fro the two dagra, the roorto of hgh tye a bg drver where ÿ draw. The odto of Prooto whh yeld a dered eulbru ay be volated for oe arle arket. gure 5 how a ae of a eulbru where there oly oe grou of utoer. I th ae t the hgh utoer oly a the old le o the ÿ orto of the grah oly. ere t doe ot ake ee for fr to dout below. Wth o loe to oe, fr do ot have to dout uh e there are o 5

145 few fored utoer. gure aother ae of a eulbru where there oly oe tye of utoer beg erved by fr. Th te, however, t the low tye of utoer. There are too few ufored hgh tye wth eual to 0. ad too few hgh tye geeral wth beg 0. for fr to lae ay weght o re above ad loe the bue of leure utoer. ere the roorto of ufored low tye hgher a β eual 0.6. The fored low tye are ortat deterg a eulbru wth two tye. I gure, β 0. ad 0.9. Dete th large, there tll a eulbru wth two dfferet tye. r ru extree ale oaoally to ature the grou of fored low tye. There tll ot uh weght o re below a the uulatve dety at rather low. Eve though the ufored hgh tye hgh, there tll a eulbru wth two dfferg tye of ouer beaue there are ore fored low tye - β ad le ufored low tye β. owever, the hae of the uulatve dtrbuto futo detered by the roorto of ufored hgh tye ad roorto of hgh tye. gure 8 ad 9 ad a leer extet gure 6 ad 7 deotrate that ÿ doe ot hage uh wth a hage the ufored low tye β, holdg all other varable otat. gure ad 5 are robablty dety futo grah of the odel. gure the df fro gure 5 uulatve dtrbuto araeter. gure 5 ha ew araeter ot eoutered gure 9. I both of thee dagra there a ke the robablty dety futo ear the lowet re l ad ear the hghet re. The ke the df at the lowet re thker tha the hghet re. The rego 6

146 above M both dagra ha oe robablty a, but ot a ggat ke. There a rego betwee ad M both dagra where robablty dety zero. Oe a olude fro thee dagra, that the fare luter aroud erta rego, eeally for the lower re. Th hold u a three valuato ae a gure 6 ad 7 date. I thee dagra, there are two rego where there o radozato by dvdual fr, a t le roftable to radoze thee rego. The uulatve dtrbuto re betwee l ad, M ad, ad M ad gure 6. The robablty dety ha three area of luterg, two below $600 ad oe above $700. The lowet area of luterg below $80 ha the hghet eak of the lower two area that have dety. -6b. Mathg the Couter Grah to the Arle Data gure 8 through 5 are kerel dety etate take fro the eod uarter 995 arle aeger Data Bak a. Data Bak a a te eret rado ale of all aeger terare wth the US ad teratoal. Aog oe of the varable luded the data et the terare that aeger take, the total fare ad for the terary, the oto eget a aeger take o the terary, the dtae of eah eget, the arle arryg o the eget, tye of eat frt la, bue, or oah o eah eget, ad the uber of aeger takg th exat ae terary ad ayg the ae fare. or tae, oe etry the data et ght be 5 aeger ayg $ for a Aut Provdee roudtr tr o Aera Arle oetg 7

147 Chago O are eah way wth eah of the four eget beg oah. or the uroe of thee arket, the data bak leaed u of teratoal flght ad terary beyod a oto flght wth the exeto of Aut Provdee where there are o arrer offerg oto erve. The arket hoe are ot deged to be a rereetatve eleto of the US arket, but rovde eough of a otrat that oe of the varyg feature of the US arle arket a be exloted for tudy. Below eah kerel etate a arket hare breakdow of the varou arrer that erve the arket. The talle bede eah arrer eed to be ultled by te to get atual aeger total e th a te eret ale. gure 8 a exale of a rarly leure arket o Agele to oolulu. There a ke of aeger that are ayg zero dollar a lot of eole ue freuet fler award o th eget. The there a large gle eak oewhere at two hudred dollar oe way ad the very all bu at extreely hgh fare. Thee fare are everal te the fare of the ot freuetly ad leure fare. Mot eole ay the douted fare ad few aeger atually ay the oted bue fare. Th arket ha eve oto arrer ervg the route ad the hare betwee eah arrer geerally fall betwee te ad eghtee eret. There are arket that have a trog bue utoer reee. gure 9 a exale of th. ere there a large roorto of bue tye or. The Meaol Chago O are arket 995 wa kow a oe of the to reveue grog yeld er le route the Uted State. The three arrer Uted, Aera, ad Northwet Arle all offered bue traveler uerou flght oto for the to hooe to fly 8

148 the route. There a arrow, but har ke aroud $00 a lot of bue traveler ad that fare oe way betwee the two te. The there are two aller ke the robablty dety futo $00 ad $70 for ore of the leure traveler. gure a aother exale of a trog bue arket. Th fgure how a ae of two large arrer oetg betwee ther hub: Meaol ad Atlata. There alo a old x of leure traveler, judgg by the two earate hu for eah tye of ouer. Oe hu, datg the leure traveler, ha t eak the $80 rage. The other hu at $0 oe way ad th hu ha the ae heght ad early the ae wdth a the oe the $80 rage. Breakg dow the Meaol to Atlata arket to douted oah YD ad full oah Y fare reveal that there tll re dero wth ultle eak at the douted oah YD level. The overall oah dout YD arket fare dtrbuto gure b how two large eak. Northwet Arle how two eak the dout oah YD tket gure. Delta Arle how four loal eak the douted oah YD oe-way fare. The larget two eak for Delta are at $75 ad $0 oe-way. Ulke the New York Cty Orlado arket, Delta full oah Y oe-way fare gure e how very lttle re dero. There oe arrow eak at $5 oe way wth a dety heght that about 0 te the heght of the Delta oah dout YD fare. Table reveal that Delta old.9% of t fare o the Meaol Atlata route a full oah tket, alot four te a uh a the New York Cty to Orlado arket. Northwet Arle oly old.5% of t tket o the Meaol 9

149 Atlata route a full oah Y tket. Mot of Northwet Y tket are red jut above $0. Thee are ot lkely free tket redeeed by freuet fler. gure 5 the oly ae of a arket where there o oto erve rovded by arle betwee the two te. The arket Aut to Provdee uh aller tha the ret of the arket that have bee tuded, but a good arket to look at beaue there are everal way that a aeger a oet betwee the two te. A aeger a ake gle lae oeto o Aera through Chago, Delta through Cat or Atlata, or Uted through Chago. Th arket how that there are two dfferet tye of utoer the arket. The ode the fare $00 oe way but there aother zeable ke at $580 oe way. Th $580 ha bee the hghet fare that ke of all the ty ar hoe gure 8 through 5. Aera Arle ha the doat arket oto the arket wth over 0% arket hare. There are arket that fall the ddle of thee two extree where there are leure traveler ad a aller reee of bue traveler. gure a how a arket of otly leure traveler betwee New York Cty ad Orlado, lorda. ere the roorto of utoer wth a reervato value of, -, ght be hgh. There a ke aeger ayg $50 oe way. Th ght be beaue there are o ay leure aeger o the route or that there the reee of low ot arrer Kw Arle or Aera Tra Ar o the route that aue other arrer o the route to ath ther lower fare. After the large ke at $50, there oly a oule of aller ke at $00 ad $00 oe way, uggetg that there tll are relatvely few bue traveler o the route that are ayg hgher fare. 0

150 owever, f the overall arket broke dow to douted oah YD fare ad oah Y fare, there tll dero wth eah fare la. Soe arle, lke Delta, Kw, TWA, ad Aera Tra Ar have ultle eak at the douted oah YD fare. Other arle, uh a Cotetal, Delta, ad US Arway have ultle eak at the full Y oah fare. gure a how the ave uber of leure tket that are old o the route. owever, Table date that the three larget arrer o the route old ay full oah Y tket. Cotetal, Delta, ad US Arway old.%, 6.8%, ad 6.% of ther total tket o the New York Cty to Orlado flght reetvely a full oah Y tket. gure aother arket that look ute lar to New York Cty to Orlado. I th ae, the arket New York Cty to Chago two arort. ke New York Cty to Orlado, there ore leure traveler ad thu a ke fare ad aroud $00. There are aller ke at the $0 ad $0 rage, datg area that bue utoer are ayg u to everal te the leure fare th arket. New York Cty to Orlado dd ot have a ke a uward a $00, thu datg that New York Cty to Chago ay have a troger bue reee tha the New York Cty to Orlado. ally, there are arket doated by low ot arrer. gure a exale of a arket doated low ot Southwet Arle. Note the everal luter of fare. There are ore eak th arket tha oe of the other arle arket. Southwet ha over a xty eret arket hare o the route wth Aera Wet ad Uted Arle havg arket hare oly the lower tee. are th arket are ubtatally le oared to the other hort hauled route Meaol Chago

151 O are. The ar leage betwee o Agele ad a Vega oly le whle the le betwee Meaol ad Chago O are 8 le. The hghet ke fare betwee o Agele ad a Vega are aroxately oe fourth a large at $75 oe way oared wth Meaol to Chago O are. The other ke fare our at $0 ad at $0 oe way, ubtatally below the oe-way fare the Meaol to Chago O are arket. There a all hu the $0 area, uggetg that f Southwet Arle wa ot the o Agele a Vega arket, other arle ght be hargg ther aeger th fare or hgher. Paeger load are ute trog th arket a there are fve te a ay aeger a the Meaol Chago O are arket. gure 8 through 5 rovde evdee that there dero fare that ouer ay ad that fare do luter aroud erta re artular arket. Th luterg heoeo a otuato fro gure, 5, ad 7, the df dagra. I oe arket, there a reeblae to the earler fgure that there are two ke the data, uh a the Meaol Atlata or Aut Provdee arket. Eah of thoe ke ould be thought of a the dfferet tye of ouer bue or leure utoer that the arle are targetg wth the d betwee the ke a the ga where fr fd t uroftable to target ouer. Eve, the ae where there are ore tha two ke the fare data, the theory ould tll aly a there ould be ore tha two tye of utoer the data. gure 7 a df exale of valuato. gure b k ad gure b - gure f otrol for re drato betwee full oah Y fare ad dout oah YD fare. Pre dero wth ultle eak

152 our the YD fare ad oete our the full oah Y fare. Thu the data ght rovde oe evdee that there heterogeety of ouer tye wth dered arle re. -7. Aaly ad Coluo Prevou re radozato odel have how that fr a delberately vary re a a ea to artto utoer wth dfferg forato. Thoe wth ore forato of re the arketlae wll o average ay a lower re tha thoe wth lted forato. Strkg the odel of Butter 977, Vara 980, ad Burdett ad Judd 98 that ouer dffer oly the aout of forato reeved the arketlae; the wllge to ay of all ouer th la of odel the ae. Gve the exeree of erta arket eeally the arle arket th auto ee ute urealt. Bede heterogeety forato, th odel allow heterogeety ouer valuato. The odel fd that uder the rght x of roorto of ouer tye ad valuato, that there wll ext a re radozato eulbru where fr ell to all ouer. The reee of the ga, g, a good dato of how well re radozato work ellg to ouer wth dfferet value. rt, f g greater tha the read betwee ay re the uort of ad, the re radozato betwee ouer wth dfferet valuato ot a feable eulbru. I th ae, fr wll erve the low ed of the arket through re dero rather tha ervg

153 both grou of ouer. Chargg the hgh tye hgher re doe ot oletely ake u for the drete lo of low tye. O the other had, too ay ufored hgh tye ay ake tee doutg uroftable for fr. I th ae g egatve. The extra reveue geerated ellg to the low tye wll ot outwegh the reveue lot by doutg to the ufored hgh tye. Oly g value the ddle wll be aoated wth eulbra wth two dfferet ouer tye. The oaratve tat eto ugget how re dero adjut to hagg ouer haratert. Soete there wll be a otet hage the etre dtrbuto of re. Ireag both ad lower the value of both ad at ay gve re ad oetrate the weght of the dtrbuto futo at the hgh ed. Wth hgher wllge to ay of the bue traveler or roorto of loyal bue traveler, fr wll dout le a they a ake greater reveue fro th aller grou of bue traveler. Chagg other ouer haratert oly affet oe art of the re dtrbuto. β ad have o effet o. Ireag dereae. ha a bg at o where laed. Ireag β ould ether reae or dereae deedg o the roorto of ufored hgh tye ad the read betwee ad. A large read betwee ad or hgh wll reult /β beg egatve. or the araeter value ued th aer, doe ot hage uh wth a hage β. Prooto 9 rovde good eretve of the how the two uulatve robablty dtrbuto futo ove whe a exogeou varable hage. A eah uulatve dtrbuto futo hage, the terval g oetg ad M

154 hage a well. Soete both futo hage a way that ake redtg g le uh a whe hage. A, the roorto of hgh tye reae, the lower robablty dtrbuto futo dereae - ovg loer to a tatoary thu redug g. Chagg - the roorto of ufored hgh tye ot a le. ere the roorto of ufored low tye β detere how the two futo ove together whe hage. A hgh β geerally le that g/ wll geerally be otve. Oe ortat reult the ae where the ufored ouer aroah oe or zero. I the ae where the ufored ad β ouer aroah zero, the uulatve dtrbuto ollae dow to the lowet re whh ear argal ot. Near Bertrad rg beoe the or wth very oaoal re ju to take aout the few ufored ouer. I the ae where the ufored ad β ouer aroah oe, there a ke the uulatve dtrbuto futo aroud the oooly hgh re for ouer ad oooly low re for ouer. The robablty dtrbuto for re beoe very loe to a bodal oe at the oooly hgh ad low re. Aother ortat reult the odel the area of reag the uber of fr ad the reultg re dtrbuto. Th reult otue oe of the teretg reult fro revou re radozato odel. Vara 980, for tae, gve a lear awer about what hae to the lowet re whe the uber of fr reae. I Vara odel, the lowet re dereae but weght aroud the hghet re 5

155 reae whe the uber of fr reae. Th odel redt the ae reult a Vara 980. The odel develoed allow oe flexblty. Two outertutve reult a our uder lted odto. rt, re radozato a our eve ae ot rereetg the uual dead ae where there a larger roorto of hgh tye tha low tye. I th tye of eulbru, the two robablty dtrbuto ad are loe together. g all a the arku above doe ot eed to be very large to ake u for the drete lo of the low tye. A all of ufored hgh tye alo hel ake uh eulbra oble. Pre radozato a alo our ae where le tha β. Norally we would exet to be larger tha β a there ould be oe orrelato betwee value ad the tedey to ho aroud. I th ae, the lower uulatve dtrbuto futo wll beoe teeer. Uder th ae, g/ hage g to be egatve. There are a few a leo leared fro the grah. rt, the fored low tye are ortat deterg eulbra wth two dfferet ouer tye. gure rovded a exale of eulbru wth two dfferet ouer tye ourrg dete the ufored hgh tye beg o loe to oe. owever, the hae of the dtrbuto futo deed o the roorto of the ufored hgh tye. gure 8 ad 9, a well to a leer extet gure 6 7 how that the hae of the ÿ urve doe ot hage draatally a β haged, holdg all other varable otat. Aother oluo that a be draw that eulbra do ext eve whe the roorto 6

156 of ufored bue ad leure ouer are ute low. Th ortat howg that th odel relevat eve whe loyalty by ouer very low or whe earh ot are low. gure 9 rovde a exale of th heoeo. A thrd reult that there are eulbra wth two dfferet tye of utoer ae whe fr would aear to ot have a etve to ru ale beaue the roorto of bue traveler who are ufored hgh. Thee eulbra eed the roorto of low tye who are fored to be hgh ad the overall roorto of bue traveler to be relatvely low. A gure how, fr otly radoze re aroud the oooly re of the bue tye, but oaoally ru extree ale to ature the fored low tye. ally, a the df grah date gure 5, fare th odel luter aroud erta re. Th hold the two valuato odel, a well a a three valuato odel deted gure 6 ad 7. ally, the te eret arle ale data how that fare luter aroud erta re, datg a lae where the theory athe the data. Soete they luter oly aroud a leure re uh a the o Agele to oolulu arket. Other te they aly luter aroud a bue re uh a the Meaol to Chago O are arket. Soete they luter ore tha oe lae oete alot yetrally a the Meaol to Atlata arket or o- yetrally a the Aut Provdee arket. Soete there are everal ode a there the Wahgto Natoal to New York aguarda arket. Th luterg of fare aroud erta re ght ugget that there are rego the dtrbuto of re that arle fd t roftable to target utoer whle rego where there are d are lae the 7

157 dtrbuto of re where t ot roftable to target ouer. Th would be a lae where the theory athe the data. Eve wth re drato rovdg a exlaato behd dfferee betwee arle fare, the re dero tory wth dfferet ouer tye tll ale the fae of re drato. Th luterg of fare oberved overall arket our eve otrollg for re drato betwee fare tye oah Y ad oah dout YD. A the Meaol - Atlata ad New York Cty - Orlado arket date, there fare luterg at the YD level for ay dvdual arle. Cluterg our for the full oah Y fare for the larget arrer ter of arket aeger hare - Cotetal, Delta, ad US Arway - o the New York - Orlado arket. Pre drato aot exla thee luterg of fare o the ublae of oah fare. Th odel beg a ortat te the dreto of ludg valuato a art of the aaly behd re radozato. uture reearh a ow lude lkg u the lterature o re drato ad re radozato e ouer both tye of odel have heterogeeou valuato. Addg ayetry the odel alo good area for future reearh. ow doe dero hage f oe fr ha ore of the ufored tye or f aother fr ha lower ot tha the extg fr? Could there be a ot to kee a erta grou of utoer loyal, or ot wllg to oare wth other fr? Doe etry by ew fr beoe deterred wth fr wth large hare of loyal hgh value utoer? Awerg thee ueto ould rovde further redto o how eletro oere wll otue to hae arket. 8

158 : Caaty ad ado Pre -. Itroduto Arle are haraterzed by radly hagg re ad oble but lted aaty. Pre are otorou for hagg freuetly ad at the lat ute. Pre dero large. r offerg heterogeeou re for the ae rodut, baed o erfet ouer forato, have bee odeled wth uh re dero odel uh a Butter 977, Vara 980, ad Burdett ad Judd 98. Couer wth the ae valuato of the good are egeted o the ba of the aout of forato that they reeve about the re fro eah of the fr the arket. The ore forato they reeve, the greater ther ablty to ho betwee offer, ad thu the lower re o average they ay for the rodut or erve. Thoe hoer wth lttle forato do ot have th luxury of oarg betwee fr o they go wth the offer that they reeve. Thee odel are forally detal to odel whh oe ouer are loked- ot by loyalty but by referee, ad other are dfferet betwee uler. Udertadg the dfferee the forato that hoer are fag, fr fae a tradeoff betwee aturg thee hoer ouer wth ore forato by rg lower veru rg hgher ad gettg ore reveue fro the ouer wth lttle forato the arketlae. The ature of th tradeoff due radozato eulbru. The reao that a ure trategy ealy exloted by rval, by a re jut below the ure trategy re. I uh a rutae, the fr better off wth a oooly re o the loyal utoer. The overall radozato of rg deed o 9

159 the dtrbuto of ouer forato, the uber of fr oetg the arketlae, ad the roft of the fr ervg utoer wth few oto. adozato work a a bet reoe th tye of tuato beaue fr a balae the rk of gog for the grou of fored ouer agat the teady reveue of the le fored hoer. Stadard odel rovde a atfatory aout of the behavor of arle wth regard to ther re radozato due a dero that oberved. Bede radozato of re, lted aaty ortat wth the arle dutry. Shedule are et ahead of te, wth ay flght deartg ether wth exe dead or wth ay exe eat that are left ety at the te of dearture. Se t otly to hft aroud hedule ore tha a few te a year, arle have to re wth the ofe of a fxed eatg allotet o route by the flght, day, week, ad oth. Th roble of aaty ot uue to the arle dutry, a hotel, tadu, heavy aufaturg whoe le aot be flexble to be haged to aother rodut ukly dutre all grale wth the ae ue of rg wth the ae aaty wth the ae erod of te. Kre ad Sheka 98 rovded a two-tage odel that allowed two fr to buld aate the frt erod ad the aoue re the eod erod. If eah fr bult aate the frt erod below the bet reoe futo of the other fr reato urve, the both fr aoued Courot re wth robablty oe the eod erod, rovded the ouer are alloated ug the effet ratog rule. 50

160 owever, f oe fr ha a larger aaty tha the other, but the u of the aate doe t lear the arket at a re of zero, re radozato wll reult. Th the xed trategy oluto to the la Bertrad Paradox; ee Baye ad Morga, 997. The aller fr wll have a hgher robablty dety at eah re tha the larger fr utl the uulatve dety futo eual oe ad wll ear a aller orto of roft due to the lower aaty ze. Yet the large fr beeft a well e art of the te t wll lae a ato o the hghet re the robablty dtrbuto ad ear ore roft due to t larger aaty ze.. The bg auto th re radozato odel that the aller fr dd ot buld eough aaty to have the oto of forg re to zero aug zero argal ot of roduto, regardle of what ato the larger fr take. Both forato ad aaty odel lead to a lar reult of re radozato. I wll how that the lar outoe of thee two dtt odel ot a adet. I the eulbru forula for the rg dtrbuto, there a ultlatve ter volvg aaty or the aout of forato that ouer are reevg. Th ter ortat the atual alulato of both the re dtrbuto ad ultately the roft that fr wll ear eah of the odel. Gve that thee tye of odel are lar how they olve for rado re, how doe a odel that eoae both kd of feature lted forato of ouer ad lted aaty of fr work? What doe rg look lke that tye of odel? Would that valdate thee two dfferet tye of odel or would oethg dfferet be ad about re radozato? Th hater awer that ueto by ythezg the two odel. 5

161 -. The Syetr Two-r Model et there be tally two fr wth a arketlae of ze otetal utoer, eah of who value the rodut at. Eah fr erve three dfferet tye of utoer. et be the roorto of fored utoer that ee the re of both fr. et be the roorto of utoer that oly ee oe fr re, alled loyal or ufored utoer. Thee ouer are relatvely ufored a they aot ee the etre arket lae. et 0 be the roorto of utoer that ee the re of ether fr ad hee do ot buy. The table below how all three lae of ouer, wth ouer lt evely betwee the two fr: Table 9: Clae of Couer for Two-r Syetr Model See r Pre: Doe Not See r Pre: See r Pre: / Doe Not See r Pre: / 0 et eah fr be edowed wth aaty to erve k ouer. et k < /. Set the argal ot of ervg ouer to zero. Suoe the re of fr le tha fr. The fr ell to 5

162 5 k k ouer. The exreo k the hare of utoer that fr, wth t lower re, a erve. The hare aled eually to both earher ad o-earher, ad ot urrgly, the total uber of utoer the aaty k. r ell to [ ] k ouer. The frt ter rereet thoe utoer oly eeg fr re. The eod ter of euato two rereet thoe utoer who aw both re that ould t buy fro fr, ad are ratoed to fr. Whe fr ha the lower re, the outoe aalogou. oyal utoer, who rereet, lay a very ortat role. Ther role aalogou lar to the role they lay the re dero odel of Butter 977 ad Vara 980. A a added feature, however, the eod ter of the hgher red fr our exale beg fr. Th ter [ ] k rereet llover fored utoer that the lower red fr ould ot erve beaue t wa aaty

163 5 otraed. Th ter ot reet the re dero odel. The effet of th ter gve the hgher red fr le of a ealty for ot wg the fored ouer ad tead add to the loyal utoer. The ore that the lower re fr otraed fro eetg the arket dead, the better advatage th to the hgher red fr. A wll be how, th etu wll gve re to re radozato leadg to a terato of aaty ad forato otrat. et k k > Ieualty ure that the hgher-red fr ot aaty otraed. Slfyg exreo : k k < < k < k k < a

164 There are o ure trategy eulbra for eah fr. Suoe that fr lay a ure trategy of a oooly re. The fr a et t re at lghtly lower ad reeve the roft of all the fored ouer. kewe, f fr lae robablty oe o ay re wth the uort the fr a uderut by a very all aout ad reeve the fored ouer ad ell all t aaty. Oe the re of fr uffetly low, fr bet reoe the oooly re, rederg fr re too low to be otal. Suoe fr j follow radozed trategy wth uulatve dtrbutve futo j. We eek a yetr xed trategy eulbru. The roft π of fr ot eual to j gve re are: [ ] k k. π j j The frt ter the earo that fr the lowet red fr, whh hae wth robablty - j. I th ae the fr ear k dollar. Wth robablty j fr wll be the hgh red fr. I that tuato fr wll ear reveue fro t loyal utoer frt ter ad reveue fro the llover utoer fro the lower red fr j lat two ter, a oted above. Pluggg the oooly re, or the hghet re the dtrbuto, lead to alulato of roft at the hgh re: 55

165 56 k π 5 Settg the two dtrbuto eual for a yetr eulbru ad olvg for the dtrbuto traghtforward ad all that reured takg the roft at the oooly re ad ettg the eual to the roft at ay other re. Dog o yeld: [ ] k k k Slfyg lead to: k k k k k k k k 6 ere a rato of the dfferee of reveue old at oe re at aaty u reveue old at the oooly re aug the fr the hgh re fr over reveue old at the ae re at aaty u reveue at aug that the fr

166 57 the hgh red fr. Se the rght had ter of the deoator wll be aller, tha the rght had ter of the uerator, wll be betwee zero ad oe. Solvg for the lowet re λ the dtrbuto alo traghtforward. Settg euato 5 to zero ad ultlyg both de by the deoator: 0 k k λ λ k k 7 λ ould have alo bee derved by olvg λ0 euato. λ ha a le terretato: oooly roft dvded by aaty. The hgher aaty, the lower λ beoe. Whe aaty at hghet, or eah fr havg eough aaty to erve the arketlae k, λ at t lowet at. Whe eah fr aaty at t lowet, whh jut able to erve the loyal utoer lu the left over hoer ratoed to the hgher red fr, k, the lowet re eual.

167 The rhetoral ueto oed the troduto a ow be awered: the aaty ad forato ltato both eter to the dtrbuto alulato a e-ultlatve faho. The odel ubue both erfet ouer forato ad aaty otrat. I fat, whe, the odel ealze to a lted aaty rg gae. I th gae, all ouer ee both re, but ouer ay ot be able to be erved by the low-red fr beaue of aaty ltato. If aaty ot too large >k, the uue yetr eulbru volve radozato. Wth <, the odel beoe rher. Now ot all ouer ee all re. r have loyal utoer thoe eeg oly ther re ad alo erve ether ther rval leftover f the fr a hgh-red fr or fae a bdg aaty otrat f t the low-red fr. Pre radozato eeary to obta eulbru. -. Geeralzg to the Cae Where All But Oe of the r are Sold to Caaty The above odel geeralze to the ae whe all but oe fr ell to aaty. Pre radozato wll tll our a a reult ad a loed for oluto a be foud for the dtrbuto of re. To ee the geeralzato, a three fr odel wll be exlored before the geeralzed odel gve. Suoe there are three fr. et be the roorto of ouer that ee all three re. kewe, let ad be the roorto of ouer that ee two ad oe re. ally, let 0 be the roorto of ouer that do ot ee ay fr re. 58

168 Thu wll be the total uber of utoer that wll ee the re of all three fr. Table 0 below how all four lae of utoer: Table 0: Cutoer Tye for Three-r Syetr Model See r Pre: all ell See r Pre: Doe Not See r Pre: See r Pre: A Doe Not See r Pre: C B D Doe Not See r Pre: all ell See r Pre: Doe Not See r Pre: See r Pre: E Doe Not See r 0 Pre: G The grou of ad ouer are broke u to thrd wth the ell beaue there are three oble obato that a ouer ould ee oe or two fr re. et eah fr be aga edowed wth aaty to erve k ouer. et the u of the uer box, or ze of the total arket, be greater tha k but le tha k or 59

169 k > > k. The ze of the total arket beaue that the uber of ouer that ee at leat oe re. Aue wthout lo of geeralty that the re of fr le tha fr ad that the re of fr le tha the re of fr three. r ell: k 8 or to k ouer. The left-had frato aaty of fr dvded by total arket ze e t otraed ultled by what the fr doe ell beg ell A, B, E, ad where fr re oted. Cell A thoe ouer that ee all three re whle ell B ad E are ae where ouer oly ee a lted art of the arketlae- two re. Cell fr loyal ouer that oly ee fr re. r ell k 9 or to k ouer. ke euato 8 the left had frato aaty of fr dvded by total arket ze whle the rght had de ter what fr atually ell. What fr atually ell broke to two art: the frt two ter fro ell C ad G are what fr ell fro ouer ot eeg the re of fr. Cell C thoe ouer that ee that fr re lower tha fr three re. Cell G fr loyal ouer that oly ee fr re. The other orto of what fr ell left over fro what fr ould ot ell due to t aaty otrat. Cell A ad E are luded what fr ell beaue ouer a ee a re of fr. 60

170 6 r three ell to k 0 ouer. The frt ter, or ell D, fr three loyal ouer. et or reader degate the reader of ouer that ould ot be erved by fr oe or two. et be the exave rght had ter. Euato 0 ow beoe:. Wth, the left had braket ter the roorto of utoer eeg lower re ot erved wth the arketlae. The rght had braket the atual ouer that ould ot be erved by the other two fr that ee fr three re. Th lude ell A, B, ad C. The rght-had braket ultled by the left had braket beaue oly a orto of ouer were ot erved by the frt two fr. Exeted roft for a fr are: π j k j j j At the hghet re the uort, exeted roft π are. By ettg π π, a yetr eulbru re dtrbuto a be foud. Dog o yeld: k k urther lfato lead to: k k k k k

171 k k k k The oluto the three fr odel wth two fr at aaty look ute lar to the two fr ae. The left ter the uerator ad deoator are aaty at a gve re, jut lke the two-fr odel. The rght ter are loyal utoer lu the reader that aot be erved by the lowet red fr. Solvg for the lowet re the dtrbuto λ the three fr odel doe by ettg 0 ad the lfyg. Settg 0 ad uarg both de yeld: 0 kλ or λ k Aga, λ ly oooly roft at the hgh re the dtrbuto dvded by aaty of a fr. The value λ ha the ae terretato th three fr ae a the two fr ae. May feature wth the two-fr odel were aed to the three fr odel wth two fr ellg to aaty. The dtrbuto ha a falar k ter the 6

172 6 uerator ad deoator ad the loyal lu reader roft are alo the uerator ad deoator. Th atter work for a hgher uber of fr otued eal ae, wth the oo feature that all fr ave oe ell to aaty. The atter a be geeralzed for a odel wth w fr ad w- ellg u to aaty. w w k w k ad k w λ, where. w w w w w w k w w w w w k w. -. Geeralzg to the Ayetr Model: The orgal two-fr odel derbed above a be exteded to the ae whe the fr are ot of eual ze. Suoe there the ae lt of utoer thoe that ee two re, thoe loyal utoer eeg oly oe fr re lt evely betwee the two fr, ad thoe utoer ot eeg ether fr re 0. Suoe that fr ha bult aaty greater tha fr or k > k. Aue that the larger fr th ae k ha aaty le tha the total arket ze / o that the aller

173 6 fr take u oe of the reag urlu utoer. Aue that the aller fr th ae k ha aaty to take u the exe urlu utoer or k k <. Suoe that fr ha t re lower tha fr re. The fr ell to k or k ouer. r ell to the utoer eeg t ow re lu thoe reag utoer that ould ot urhae fro fr, whh k. Now uoe that fr ha a lower re tha fr. r ell to t loyal utoer lu the redual utoer that ould ot be erved by fr : k. r ell to k or k utoer. Wth thee four ae, roft for the two fr a be exreed. or the larger fr, roft are [ ] k k π or the aller fr, roft are

174 65 [ ] k k π 5 Itally olvg for the dtrbuto, reveue at the oooly re et eual to reveue to reveue at ay other re wth the uort of re. or fr the roft euato are: [ ] k k k k k k k k k k k 6 Solvg for a be doe a lar faho. k k k k 7 The lowet re the dtrbuto olved by ettg 0.

175 66 k k λ k k λ Oe of the oteable thg about λ euato 6 ad 7 that they are ot eual the rooed oluto. Pre radozato wll ot our wth λ ot beg eual the fr wth the lower λ rg ueearly low. Thu, there eed to be adjutet the oluto to due the uort of re of the two fr to ode, ad the oly addate a a ot or ato the dtrbuto of re. Moreover, for a ato to ext, t ut our at, beaue otherwe, were the ato at, the rval would ever re a all terval [,ε. The fr wth the lower λ oluto ha the ato t dtrbuto. r wll have the ato t dtrbuto ff λ < λ. Th a be ee a follow. k k k k >. After lfyg ad rearragg, th odto beoe > k k k k

176 67 or 0 > k k k k k k k k whh redue to 0 > k k k k 0 > k k k k Now f k k < 8 fr two ha the ato t dtrbuto. Adjutet wll eed to be ade to robablty dtrbuto to aout for fr two havg the ato t dtrbuto. et be the leftover utoer fro fr ellg at a re lower tha fr. k. Slarly, let be the reader utoer fro fr ellg at a re lower tha fr, whh k. Se fr ha the ato at t robablty dtrbuto, the euato howg roft are the ae

177 throughout are odfed to take aout the ato. ro euato, e fr ha o ato, eualty of roft for fr gve a otat π [ ] k. Th alo gve the detal λλ. The oluto for fr, ug 5, gve λ k 9 derved fro ettg the roft of fr fro the lowet re the dtrbuto agat the roft of fr fro the hghet re the dtrbuto. The dtrbuto of fr doe ot eter the euato beaue there ot a ato fr robablty dtrbuto. At the lowet re the dtrbuto, fr reveue λk. Wth robablty oe fr doe ot have the lowet re at λ, thu fr ellg out to t aaty. At the hghet re the dtrbuto, fr reveue. Wth otve robablty, fr rele uo t loyal utoer ad the leftover utoer fro fr e t ha hgher re. The eod euato [ ] k λ k 0 t the roft fro the lowet ad hghet re fr roft euato. The dfferee that fr ha a ato t dtrbuto. The low ed rea the ae. Wth robablty oe fr ha the lowet re at re λ. eveue λk. The hgh 68

178 69 ed of the dtrbuto dfferet fro the ae above. It ot robablty oe whe fr reahe t hghet ot t dtrbuto that t ha the hghet re. That what the [ ] k ter ature. Wth robablty [ ] fr oe reeve oooly roft by rg jut below the ato of fr two. Wth robablty fr two ha the lower re at the re jut below ad thu fr oe reeve exeted roft of fro t loyal utoer ad utoer that fr two ould ot erve. λ ad - a be olved for by thee two euato. k λ [ ] k k k k k k k k k k k k k k k a

179 70 Chekg that the ato betwee zero ad oe gve two ore odto: k k k k The left odto euato reveal a relatoh betwee k ad k : k k k The g wthe beaue the deoator egatve. k k Euato the ae a what we aued before about the aller fr havg eough aaty to hadle the left-over ouer fro the hgh aaty fr. The rght odto euato the ae odto a euato 8: k k k k The g wthe beaue the deoator egatve.

180 7 k k k k k k k k k k 8 ad a be olved by luggg λ to euato ad 5 ad olvg. [ ] k k k k k k k 5 Th the ae reult a before euato 7. wll ot be the ae a euato 6 wth the ato.

181 7 [ ] k k k k k k k ` k k k k 6 If euato 8 volated, the fr oe ha the ato. Th ea that the obed aaty hgher tha. Se fr ha the ato at t robablty dtrbuto, the euato howg roft are the ae throughout are aga odfed to take aout the ato. ro euato, e fr ha o ato, eualty of roft for fr gve a otat [ ] k π. Th alo gve the detal λλ. The oluto for fr, ug, gve k λ 7 derved fro ettg the roft of fr fro the lowet re the dtrbuto agat the roft of fr fro the hghet re the dtrbuto. The dtrbuto of fr doe ot eter the euato beaue there ow ot a ato fr robablty dtrbuto. At the lowet re the dtrbuto, fr reveue λk.

182 Wth robablty oe fr doe ot have the lowet re at λ, thu fr ellg out to t aaty. At the hghet re the dtrbuto, fr reveue. Wth otve robablty, fr rele uo t loyal utoer ad the leftover utoer fro fr e t ha hgher re. The eod euato [ ] k λ k 8 t the roft fro the lowet ad hghet re fr dtrbuto. The dfferee that fr ow ha a ato t dtrbuto. The low ed rea the ae. Wth robablty oe, fr ha the lowet re at re λ. eveue λk. The hgh ed of the dtrbuto dfferet fro the ae above. It ot robablty oe whe fr reahe t hghet ot t dtrbuto that t ha the hghet re. That what the [ ] k ter ature. Wth robablty [ ] fr two reeve oooly roft by rg jut below the ato of fr oe. Wth robablty fr oe ha the lower re at the re jut below ad thu fr two reeve exeted roft of fro t loyal utoer ad utoer that fr oe ould ot erve. λ ad - a be olved for by thee two euato. λ 9 k 7

183 7 [ ] k k k k k k k k k k k 0 k k k k 0a Chekg that the ato betwee zero ad oe: 0 k k k k 0 k k k k Takg the rght odto frt: k k k k The g revere beaue the deoator egatve. k k

184 75 k k k k k k. Th the revere of odto 8 whe obed aate are large. The left odto a be redued to: k k k the deoator egatve o the g hage k k Thu the larget fr ha exe aaty. ad a be olved by luggg λ to euato ad 5 ad olvg. [ ] k k k k k k k Th the ae reult a before euato 6. wll ot be the ae a euato 7 wth the ato. [ ] k k k

185 76 k k k k k k k k -5. Buldg Caaty rt ad the Settg Pre A aother exere, both fr a be odeled to buld aaty tage oe ad the ell to utoer tage two, jut a Kre ad Sheka 98. Gve the two robablty dtrbuto, they a be lugged to euato ad 5 to detere roft a a futo of aaty ad roorto of ouer that ee oe or two re. Wth that forato, reato urve a be draw for eah fr ad a oluto to the two tage ayetr gae ght be obtaed. Pluggg - obtaed fro the ae the obed aate for fr oe ad two are aller tha - to the roft euato 5: Th the aller aaty ae where the aller fr ha the ato. Proft deed egatvely o aaty o ug the aller total aaty reult greater roft. If the larger aaty ae ued, the reato urve ro at a yetr ot ad gve the ae yetr roft euato a the low aaty ae. Se t wll be how that yetr roft j k reae wth dereag aaty, the low aaty euato evetually take hold.

186 77 k k k k k π k k k k k k 5 Pluggg, obtaed fro the aller aaty ae, to the roft euato : k k k k k k k k k π k k k k k k k k k k

187 78 k k 6 Thee relatoh a be uarzed by dexg fr wth j ad addg a otat ot of obtag atal k: > < j j j k k f k k k k j k f k π 7 Proft are aller for the lower aaty fr ad hgher for the larger fr. Proft are exatly the ae at f both fr have the ae aaty. Th feature wll be ortat deterg the oluto to the two-tage ayetr roble. Whe aaty below the other fr aaty, or k < k j, roft are j k k. Th exreo dereag k. Whe aaty above the other fr aaty, or k > k j, roft j k k k k are ether reag or dereag k, deedg o k j Thu ay ure trategy eulbru, the fr have detal aaty.

188 Catal for both fr a vary betwee o the low ed ad takg the arket o the hgh ed. Whe aaty of oe fr below the other, reag aaty reult reag roft. Thu for aaty below the other fr aaty, there a otve orrelato the two fr aate. Whe aaty above the other, dereag aaty reult reag roft. Thu for aaty above the other fr aaty, there a egatve orrelato the two fr aate ad the reato urve take o a dfferet loe. The two reato urve ro eah other whe the aate are exatly the ae or rght at the rog of the two urve. 79

189 k fr j gure k fr a.5 a.5.5 k * k j *.5 *.5 arket.75 a.5, a.5,,,.5 outde le 0.0 a.5, a.5,,,.5 de le

190 k fr j gure k fr gure oe of ay oluto to the roble of buldg aaty frt ad ettg re eod. Note that there a ot of atal that half the oooly re. Maxu roft our for eah fr at the de reato urve where atal The otrat of the ato euato ad 8 are ot draw gure o that the reato urve a be ehazed. If that otrat fro euato draw, there would be two traght le extedg fro the otal yetr oluto to the outerot ot at.5,0 ad 0,.5. All reato urve above thoe two ray wll atfy euato. Euato 8 a le oetg the ot.5,0 ad 0,.5. All reato urve below th le atfy euato 8. Thu atfyg euato ad 8 ea beg de a tragle whoe verte are the otal yetr oluto 0.5,0.5,.5,0 ad 0,.5. If the otrat that atal le tha the arket a /a oed wth the ato otrat, the reult a etago. The verte are 0.5,0.5, 0.75,0.5, 0.75,0.75, 0.75,0.75, 0.5,0.75. All reato urve de the etago atfy the three otrat. gure how the etago. 8

191 the ae for both fr: k k j.5. Th level of atal for eah fr the lowet level of atal that a be erved, f the other fr to erve the reader of the utoer eeg two re. Th level of atal. Whe atal at.5 ad aug a effet ratog rule of ervg utoer, the lower red fr wll erve.5 of utoer ad the hgher red fr erve.5 of utoer. There wll be oe re harged- the oooly re- a the lowet re λ k k eual.5/.5. Proft are There are ay uh dagra that ould be draw, but the geeral oluto bol dow to the ae dea. May reato urve rog rereetg a otuu of dtt roft level ould be draw to a dagra but for lty age oly the oluto urve are draw. Eah rog at a level of atal rereet a dfferet level of roft, ad thu the oluto are ualtatvely dtt. Due to the ot of atal, roft reae a atal uage dereae. Proft thu reah ther axu at the lowet ot of aaty due to the ot of atal. Th the oluto to the ayetr roble: a yetr oluto at the lowet addate level of atal, whh 8

192 , where both fr are eearly beoe otraed ad the oooly re harged. -6. Coluo: Pre radozato ould be exlaed the lterature by ether a lted aaty tory or a lted forato tory. A art of eah of thee lae of odel, there a ultlatve ter that eter the odel that volve aaty or the aout of forato that ouer are reevg. A two-fr yetr odel ha bee develoed that take aout both eleet aaty ltato ad forato ltato that lead to re radozato. Addg both feature together the odel ehae the ablty of the hgher red fr to ature ore utoer tha ether odel would rodue aloe the reader of the utoer that the other fr ould ot erve due to aaty ltato lu the loyal utoer that ee oly oe re. The reult of th tye of odel that there a e-ultlatve ter of aaty ad forato wth the dtrbuto alulato. There are a uber of exteo wth th odel. The odel a alo be lfed dow to a traght lted aaty odel by allowg all ouer to be fored about both fr re or ettg. The odel a be geeralzed to a loed oluto where there ore tha two fr o log a all fr ell out to aaty 8

193 exet for oe fr. ally, the two-fr odel a be arred over to ayetr tuato where aaty of oe fr greater tha the other fr. I that ae ad f total aaty of both fr ot too large, e aller tha there wll be a ato the robablty dtrbuto of the aller fr at the hghet re. If total aaty of both fr larger tha ad aller tha, the the larget fr ha the ato t rg dtrbuto. If the fr lay a Kre ad Sheka two-tage gae of buldg aaty frt before ellg to utoer, there wll be a yetr oluto to the gae at the lowet oble level of aaty. 8

194 : Eulbru Pre adozato wth Ayetr Cutoer oyalty -. Itroduto oyal or bue traveler are ortat to a arle bue. Thee traveler are geerally a all orto of the overall utoer bae yet ake a larger orto of the aual arle tr tha the average ubl. Thee utoer geerate large reveue for arle a they take ore tr ad geerally ay hgher fare tha the average fare ad by the ubl. Arle go to great legth to ourt thee traveler wth ther freuet fler rogra rewardg reeat bue wth thee traveler. Wth freuet fler rogra ad arle havg a reee ortat hub te, freuet traveler are ofte loyal wth ther bue to oe arle. The ze of the loyal utoer bae a hel detere a arle fortue. eure or ot o loyal traveler are aother grou of utoer that a arle erve. Thee utoer earh fro arle to arle for the lowet re. Thee traveler do ot ake a ay tr a the loyal traveler ad are geerally ot a ortat to a arle bue. Arle wll offer thee utoer yte-wde ale, weekly eal eal, ouo, ad lat-ute eal to ete thee traveler to book wth the. Cutoer wll the fae re that flutuate. Thee two ualtatve feature of arle travel a all grou of loyal traveler ad flutuatg re ft a great exale fro the lterature of olete forato ad rado rg. Vara 980, for tae, ue a odel of ale where there a 85

195 grou of ufored utoer o re ad fored grou of utoer o re. Vara aue h odel that fr have the ae roorto of ufored, or utoer eeg oly oe re, dvded evely betwee fr. r wll wat to harge thee utoer the oooly re. Th grou of utoer that ee oly oe re ot large eough that fr ould oetrate ellg to thee utoer. The other grou of utoer, thoe that a ee all re, ho at the fr that offer that offer the lowet re. To w thee utoer, fr wll have to offer the lowet re. r a oly offer oe re, o they have roble how do they erve both grou of utoer? r radoze ther re a terval to ature a uh exeted reveue fro the ufored tye ad exeted reveue fro the fored tye. Vara aue that the roorto of loyal utoer the ae for eah fr. Th ot a realt auto today eooy wth o ay dfferet zed fr. If th auto relaxed, how do the reult hage? What wll the robablty dtrbuto for the fr look lke? Wll there tll be ale? ow wll the ze of the fr fluee the reult? A the odel wll how fr tll radoze over a terval of re. Ulke Vara 980, the larget fr ow ha a ato of robablty at the oooly re. Ireag the larget fr loyal utoer aue all fr to dout le. Ireag the aller fr loyal utoer aue oly the larget fr to dout ore. Ireag the hoer aue the aller fr to dout ore; the larget fr ha a ore olated reato. Ireag the uber of fr aue all fr to dout 86

196 le. A tatt howg the robablty of at leat oe fr havg the lowet re a be reated. -. The Model wth Two r et there be two fr tally. et fr be the fr wth the larger roorto of loyal utoer ad fr be the fr wth a lower roorto of loyal utoer. et be the roorto of utoer loyal to fr oe. et be roorto of utoer loyal to fr two. Thu greater tha. et be the hoer that look for the bet re betwee both fr. et be reveue at re, eual to - * where uatty at re. Aue that reag. et be eual to the oooly re. or a re radozato eulbru to our, the roft at the hghet re the dtrbuto, or oooly re, ut eual the roft at the lowet re the dtrbuto or ad every re betwee the two. et, be the uulatve robablty dtrbuto futo that eah fr ha hargg re betwee the oooly re ad the lowet re the dtrbuto. The uulatve dtrbuto futo for eah fr le tha oe ad otuou at all re exet obly the oooly re for oe of the fr. I the rage of re that thee uulatve dtrbuto futo are otuou, there o rego of te or oe fr wll re elo below the te ad ature the etre arket. Other tha the ae where oe fr ha a a ot at, there are o other a ot the rego betwee ad for the ae reaog a the te. 87

197 88 The roft for eah fr a be odeled. or fr oe th : Wth robablty fr two ha the hgher re. r oe wll ear reveue fro the hoer lu t ow loyal utoer. Wth robablty fr oe ha the hgher re ad wll oly ell to t loyal utoer a. Total reveue fro radozg eual the oooly reveue fro the loyal utoer of fr oe. or fr two the roft euato Wth robablty fr oe ha the hgher re. r two wll ear reveue fro the hoer lu t ow loyal utoer. Wth robablty fr two ha the hgher re ad wll oly ell to t loyal utoer a. Total reveue fro radozg eual the oooly reveue fro the loyal utoer of fr two. Solvg for eah uulatve dtrbuto futo: [ ] Solvg for the lowet re the dtrbuto ad : 0

198 89 Note that ad are ot eual. > Deoog ad : > > > Se reag, >. Thu the u the uort ot eual. The fr wth the lower re, whh fr oe, wll have the ato. The lowet re wll eed to be reet o that t the ae for both fr ad the odel reet to take aout the ato for fr oe. The larger fr lae a otve robablty a o the uer ot the re dtrbuto that the other fr doe ot ath. Th a ot at the uer orto of the dtrbuto where the frt fr ake oooly roft fro t ufored utoer. The eod fr aot re at th oooly re but re at elo

199 below th re. I dog o, t ature wth oe u the robablty of the ato the hoer that fr oe doe ot ature at the oooly re. et - be the lt of. The roft of eah fr a be rewrtte. or fr oe th : or fr two th : r oe ha the a ot. Thu t lae otve robablty o ad harge t ufored utoer that re. Whe fr oe hargg the lowet re the dtrbuto, t ature the hoer ad tll kee t loyal utoer. owever, t ha to harge the loyal utoer a lower re of. r two re fro o the lower ed to jut below o the uer ed. ke fr oe, whe t re at, fr two reeve t loyal utoer ad the hoer. Whe fr two re at the uer ed of the re dtrbuto, t re jut below the oooly re. Th fr eetally reeve oooly reveue fro t loyal utoer ad wth robablty - - eetally oooly reveue fro hoer. Euato oe a be rearraged to olved for : Euato three tate that the lowet re the dtrbuto eual argal ot lu the frato of total exeted oooly reveue er utoer fro fr oe dvded by the 90

200 9 total exeted uatty old er utoer fro fr oe at the lowet re. Se the roorto of utoer ad uatty old at the oooly re lower, the uerator wll be lower tha the deoator thu akg le tha. Theore : The ato for fr oe at Proof: Ug euato, euato a be rearraged to olve for - - : a - betwee zero ad oe beaue frt the uerator betwee zero ad oe a larger tha. Seod the uerator - le tha the deoator. - ha a teretg terretato. If both fr are eual ze - eual oe ad

201 the ato eual zero. A the ze dfferee grow betwee the two fr holdg the hoer otat, - grow aller ad the ato - - grow larger a fr oe lae ore weght o the oooly re. If grow larger, - grow aller ad the ato - - grow larger. If grow larger, - grow larger ad the ato - - grow aller. QED The uulatve robablty futo for eah fr ad a be olved by ettg roft at ay re the dtrbuto to the roft at the hghet re the dtrbuto. or fr oe th : [ ] 5 Euato 5 lke euato that eual to ad the - ter beoe oe a eual to zero at. The roft euato for fr two at ay re : [ ] 6 Theore : The uulatve dtrbuto for fr two the two fr ae : Proof: a be olved by rearragg euato 5: 9

202 9 7 7a Note that doe ot deed o or fr two hare of loyal utoer. The hare of fr oe loyal utoer ad hoer betwee both fr are ortat deterg. The frato the dfferee betwee oooly roft for fr oe ad roft for fr oe ufored tye at a lower re dvded by the reveue by the grou of hoer at that lower re. Whe, eual oe. Whe, 0. QED Theore : The uulatve dtrbuto futo for fr oe the two fr ae Proof: a be olved larly by rearragg euato x ad luggg for - :

203 9 [ ] 8 8a deed o the loyal utoer hare of both fr oe ad two:, ad the hoer. volve the reveue of fr oe at the oooly re, reveue of fr two at re le tha the oooly re, ad the reveue of hoer at that ae re le tha the oooly re adjuted for the u of hare of fr oe ad the hoer. At. At. At 0,.

204 95 If fr two loyal utoer go away o that t oetg oly for the hoer, the ato for fr oe alo teretgly tll rea ad get larger. r two dtrbuto doe ot hage. * 0 0 * -. Coaratve Stat wth the Two r Model The tadard oaratve tat a be aked whh the hare of utoer loyal grou hoer are haged ad the effet o the uulatve dtrbuto futo of eah fr eaured. Th aroah gve everal reult. Alo, the ueto of what hae to eah fr uulatve robablty dtrbuto whe a all aout of utoer eually take away fro oe utoer grou ad added to aother utoer grou. Table lt the dfferet lea reultg fro the oaratve tat the two fr odel. The uer left-had ot etry read ÿ/ < 0, whh foud ea. The etry the frt olu ad xth row read ÿ/ - ÿ/ < 0, whh foud ea 9.

205 Table : Coaratve Stat Overvew Seto ÿ ÿ ea ea ea ea 7a ea 7 < 0 < 0 > 0 0 < 0 ea ea 5 ea 6 ea 7b ea 8 < 0 < 0 > 0 0 < 0 ea 8 ea 9 ea 0 ea ea 9 < 0 < 0 > 0 > 0 > < 0 < 0 < ea ea a ea b ea ea 0 > 0 ele 0 0 < 0 > 0 ele 0 0 ea 5 ea 6 ea 7 ea 8 ea > 0 ad < 0 > 0 < 0 < 0 > 0 ad < 0 Geerally > 0 Geerally > 0 - ea 9 ea ea ea 5 ea < 0 < 0 > 0 > 0 ad < 0 < 0 - ea 0 ea a ea a ea ea > 0 ad < 0 < 0 > 0 > 0 ad < 0 > 0 ad < 0 Geerally < 0 Geerally < 0 96

206 97 - ea > 0 hgh < 0 low ea b < 0 ea b > 0 ea 6 < 0 ea > 0 ad < 0 Geerally < 0 The frt lea deal wth hagg ot ad uatty ad ther effet o the uulatve dtrbuto. ea : 0 < Proof: Ug the ha rule: / [ ] [ ] [ ]

207 98 The braket [ ] otve e t eual 0 >. The two ter the uerator are otve. The deoator egatve e t ota the ter 0 <. Thu 0 <. QED ea : 0 < Proof: Aga ug the ha rule: / [ ] [ ] [ ] [ ] [ ]

208 99 Aga the braket [ ] otve e t eual 0 >. All art the uerator are otve. The deoator, however, egatve e 0 <. Thu 0 < QED ea : 0 > Proof: / 0 / > e 0 <. QED ea : 0 < Proof:

209 00 [ ] [ ] [ ] [ ] [ ] 0 < QED ea 5: 0 < Proof: [ ] [ ] [ ] [ ] [ ] 0 < QED

210 0 ea 6: 0 > Proof: [ ] 0 > e > QED ea 7: 7a 0 7b 0 Proof: There o uatty ad ot thu the dervatve of the ato wth reet to ot ad uatty are zero. QED

211 0 What hae to the two uulatve dtrbuto futo a the hare of loyal utoer ad hoer haged? Suoe the hare of fr oe loyal utoer reae. ow doe that affet both fr? ea 8: 0 < Proof: 0 < 9 The addto of ore ufored utoer for fr oe aue the to dout le ad try le for the grou of hoer. Proft reae for fr oe whe the loyal utoer of fr oe reae. QED The effet of a reae fr oe ufored utoer o fr two uulatve dtrbutve futo alo traghtforward: ea 9: 0 <

212 0 Proof: 0 < 0 Whe fr oe loyal utoer reae, fr two, lke fr oe, wll oetrate le o lower re a the lowet re the dtrbuto reae. Wth fr oe oetratg le o lower re ad ore o t oooly re, fr two doe ot have to radoze t re ute a aggreve to get the hoer. Thu fr two lae le weght o lower re ad ore weght o hgher re. QED ea 0: 0 > Proof: 0 > QED ea : a 0 > whe >

213 0 b 0 whe 0 < whe < Proof: QED Chagg fr two hare of loyal utoer ha a dfferet effet tha hagg fr oe hare of loyal utoer. ea : 0 whe ] 0 for re, - > Proof: At the lowet re,. Pluggg th to the above euato:

214 05 0 Thu for ay re the rage, ] 0 > e the rght de of the uerator of euato beoe larger tha the left had of the uerator. Thu ], f 0 f 0 > a QED ea : a 0 b 0 Proof:

215 There o the exreo for or o both 0 ad 0. Th reult otrat that of the revou reult whe fr oe loyal utoer were haged. r two loyal utoer have o effet fr two dtrbuto beaue fr two alway oetg for hoer at ay re t dtrbuto of re. Coetg for hoer at ay re hek fr two dere to okg t ufored utoer the oooly re. At the hghet oble re for fr two: the lt a aroahe, whh lower tha the oooly re for fr oe fr two oetg wth fr oe for the hoer eve a fr two ha ufored hoer wllg to ay that re. At re lower tha th lt re, fr two oete wth fr oe for the hoer ad t ufored ouer ay whatever re offered. r two radoze to get hoer ad e they are all oble re t uort, fr two ow ufored utoer wll have o bearg o t radozato futo. QED The ze of fr two loyal utoer doe atter fr oe dtrbuto futo: > 0. Th the oote g of the ae of fr oe loyal utoer o fr oe dtrbuto futo. Th g otve beaue the ze of ato at dereae whe reae. 06

216 07 ea : 0 < Proof: 0 < QED Calulatg ore olated: ea 5: 5a > 0 f > 5b < 0 f < 5 > 0 f <. < 0 f >. 5d > 0 f < ad >. < 0 f > ad >.. 5e > 0 f > ad <.

217 08 5f Geerally ea 5 otve at all re exet whe extreely all ad/or aroahe the ze or ad thu all Proof: [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] >0 f the uerator of euato greater tha zero. Slfyg [ ] > < a

218 Euato a wthe g beaue the deoator geerally egatve at all araeter obato of,, ad. At : >0 ff < < b Wth the ad of Matheata, the followg hart how the varou araeter obato of euato b ad the reultg g of. or eah level of, reaed whle dereaed. Mot hage are by 0.0 exet whe beoe extreely all. To ave ae, the dow arrow ad u arrow Æ are ued ext to the g of. Whe oe row ha a dow arrow followed by the ext row havg the u arrow Æ, the araeter obato of ad betwee the dow ad u row have the ae g a the two arrow row. or tae, oe a olude that the g of otve for the araeter value of 0., 0.65 ad 0.05 beaue they would ft betwee the two arrow. ow wth the oe ded arrow follow the ae log. or tae, a row wth a dow arrow wth a egatve g ea that value greater ad value le tha thoe the ae row, holdg 09

219 otat, have egatve. The araeter value 0.0, 0.0 ad 0.0 have beg egatve. The ae log work wth oe-ded row wth a u arrow. 0

220 Table : Sgg S of e b Potve Potve Potve Potve Æ Potve Negatve Potve Potve Æ Negatve Negatve Potve Potve Æ Negatve Potve Potve Negatve Negatve

221 ro the above table, oe a olude that geerally otve exet whe a relatvely low agat a. or very hgh value of a, lkely to be egatve beaue a relatvely all. or lower value of a, lkely to be otve beaue there are ay uh araeter obato where a ot all relatve to a. Whe aother araeter earh doe Matheata at the dot re value of th re beg dfferet for eah obato of, ad., the reult do ot hage exet for oe araeter value of 0.0, 0.50 ad 0.0. At : [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] The uerator of th frato otve ad therefore otve whe:

222 > Keeg ot wth a, the eualty g for allow for the oblty that the deoator to be otve revered to ould be egatve. If the deoator otve, the the eualty g greater tha for to be otve. The deoator of the rght had frato egatve whe greater tha ad otve whe greater tha : < < f > > 0 at whe hold true. > f < d > 0 at whe d hold true. > 0 our for ure whe > or whe efed euato. ug dfferet obato araeter Matheata at gve alot the detal reult at exet for a oule of araeter value 0.0, 0.50 ad 0.0 ad 0.0, 0.0 ad 0.0. Thu t afe to olude a ha

223 bee doe at for all re: relatvely low agat a. geerally otve exet whe a Aother way to ee th to reeber that a large aout of hoer reet ake t le worthwhle for fr oe to go after t ufored utoer at the oooly re ad tead dout to w over the hoer. A wll be how ea 8, the value of the ato fall a grow larger. I that ae of a large uber of hoer ad lower uber of ufored loyal fr two tye, doe ot of the fallg a reae a ot very large. The deoator of the ato rea vrtually uhaged wth fallg ad rg but the uerator of the ato fall a fallg. ere the hoer effet doate the ufored fr two tye effet. Th hage f uh larger tha. A reae reagly aue to oteably fall, the larger. If fallg at a large eough ae for a reae, the ato at reae. Th beaue the uerator reae fater tha the deoator reae. If the ato reae, the wll re at hgh re for a reae hoer a the dereae the ufored fr two tye doate the hoer effet. QED

224 5 Calulatg ore traghtforward: ea 6: 0 > Proof: 0 > r two doe ot have the luxury lke fr oe to rely uo a ato to ok t ufored utoer. r two radozg to get hoer. A reae the hoer aue fr two to dout ore to w a otetal larger rze. Thu > 0. QED ea 7: 0 < Proof: 0 < QED

225 6 ea 8: 0 < Proof: 0 < QED Aother exere a be erfored wth the three dfferet ouer tye:,, ad. oldg oe ouer tye otat, what effet wll reag ad dereag the other two ouer tye the ae roorto have o the uulatve robablty dtrbuto for eah fr, the lowet re, ad the ato for fr oe? Suoe that the hoer are held otat. What effet wll reag fr oe loyal utoer at the exee of fr two loyal utoer have o? ea 9: Ireag at the ae rate a dereaed holdg otat lower or 0 <.

226 7 Proof: By lea 8 <0 By lea >0 Thu 0 < QED Itead of the hoer beg held otat, uoe that fr two loyal or ufored utoer are held otat. What would hae to f fr oe loyal utoer are reaed at the exee of the hoer? Itutvely, oe would exet le doutg by fr oe. ea 0: 0a 0 > whe [ ] [ ] [ ] > ad the deoator otve the frato. 0b 0 < whe [ ] [ ] [ ] < ad the deoator otve the frato. If the deoator egatve, the 0 <.

227 8 0 0 > whe > 0 < whe < 0d 0 > whe > ad the deoator otve the frato. 0 < whe < ad the deoator otve the frato. If the deoator egatve, the 0. < 0e Geerally ea 0 egatve at all re exet whe extreely all ad/or aroahe the ze or ad thu all Proof: [ ] [ ] [ ] [ ] 5

228 9 or 0 > the uerator of euato 5 ut be greater tha zero. atorg out ad ubtratg o to the other de of the greater tha g: [ ] [ ] [ ] > 0 > whe [ ] [ ] [ ] > 5a At, the odto euato 5a lfe dow to [ ] > 5b A all relatve to ake t ore lkely at re ear that 0 > a the uerator of the frato euato 5a beoe aller wth volved eah ter. The revere ae hold. At, the odto euato 5a lfe dow to [ ] [ ] [ ] > [ ] [ ] [ ] > [ ] [ ] > > 5 If the deoator egatve 5 the for 0 >

229 0 < 5d Th aot our, o whe the deoator egatve euato 5, 0 <. I euato 5 t ore lkely that 0 > a the deoator geerally a egatve uber for a uffetly large eough. The followg hart how the dfferet araeter obato of euato 5 ad the reultg g of.

230 Table : Sgg S of e Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve* Potve S of e. 5

231 *Dfferet g for A detal hart a be draw for wth the ae g reultg obvouly dfferet value for euato 5b fro the varou araeter obato, exet for the oe row that ha a aterk. Oe a alo be draw for the dot betwee ad the dot re hage a the alha araeter hage. The dot hart reult for do t do t are the ae a the above hart. Gve the larty at both ed of the re terval ad at the dot, oe a olude that the behavor of the ae throughout the re terval, wth a few exeto. The hart reveal that, ad therefore, egatve for large eough, relatve to. A large ool of hoer aue the hoer effet of doutg to be doated the effet of rag re to fr oe loyal grou of utoer. If beoe relatvely all, a whe beoe large, beoe otve. Th would be the ae that the terval betwee ad ot very large. A wll be how ea a, the lower re larger a the uerator the frato of greater wth a reae ad the deoator rea uhaged wth reag

232 uhaged. Th reae of, ad therefore horteg of a all re terval, aue a reae. QED Now uoe that fr oe loyal hoer are held otat. Suoe fr two loyal utoer are reaed at the ae rate that the hoer are dereaed. What hae to? ea : a 0 > whe [ ] [ ] [ ] > b 0 < whe [ ] [ ] [ ] < 0 > whe >. 0 < whe <.

233 d Geerally 0 > hold at hgh re wth a few exeto whe all. e 0 > whe > ad > f 0 < whe < ad > or > g Geerally hgher at hgh re ad lower at low re, wth a few exeto Proof: [ ] [ ] [ ] [ ] or 0 > the uerator ut be greater tha zero. atorg out ad ubtratg o to the other de of the greater tha g: [ ] [ ] [ ] >

234 5 [ ] [ ] [ ] > 6 The greater tha g hold a t the oote of the deoator of euato a. At euato 6 lfe dow to: [ ] [ ] [ ] > > 6a The followg table how the varou araeter obato of,, ad have o. Mot value are otve exet for a few value whe all. Whe >, euato 6a alway hold a the uerator egatve ad thu 0 > at hgh re the dtrbuto.

235 Table : Sgg S of e. 6a Potve Potve Æ Negatve Negatve Æ Potve Potve Potve Æ Negatve Negatve Potve Potve Potve Potve S of e. 6a 6

236 7 At euato 6 lfe to: [ ] [ ] [ ] > [ ] [ ] [ ] > [ ] [ ] [ ] > > > > 6b Whe larger tha, for 0 >, the greater tha g euato 6b wthe to a le tha g euato 6 e there dvo by a egatve uber. < 6 Th odto 6 doe ot hold e the deoator euato 6 le tha zero, thu guarateeg 0 <. or aller ad larger, euato 6b ay ot hold thu eag 0 < for re edately above.

237 Note that the rght had de of euato 6b/6 exatly the ae a euato ad d. The dfferee that the greater tha/le tha g fl betwee the two euato thu guarateeg the oote g of ad. At, doe eter to the euato, thu oly reflet the effet of. The followg table how the varou araeter obato of,, ad have o. There are ore egatve value of oared to the revou hart. At the oooly re,a wll be how ea 6 the ato aller a the uerator ad deoator fall by the ae aout. A the revou hart dated, that lot ato weght gog to hgh re. hgher tha before ad are haged beaue the deoator aller how ea b. e weght beg laed o lower re a there are le hoer avalable to w. The followg table how that there are ore egatve outoe for, eeally whe fallg fro a relatvely hgh uber. Thee effet a fallg ato ea 6, rg lower re, ad le hoer aue fr oe to radoze ore. owever fr oe radoze ore at hgh re beaue there are le hoer avalable to 8

238 teely dout. The exeto of th behavor whe all. I th ae a few araeter obato the revou hart beoe egatve. Table 5: Sgg S of e. 6b/ Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Potve S of e. 6b/6 9

239 0 QED Suoe held otat ad reaed at the ae rate that dereaed. What the effet o? ea : a 0 < b 0 < 0 < Proof: By lea 9 0 < 0 By lea 6 0 > Thu 0 < By lea a 0 Thu 0 <

240 Thu 0 < QED ea : a 0 > b 0 > 0 > Proof a: ro ea 0: 0 > ro ea 7: 0 < 0 >

241 Proof b: ro ea b: 0 ro ea 7: 0 < Thu 0 > Proof : ro ea 0: 0 > ro ea b: 0 Thu 0 > QED ea : a 0 > f - > 0

242 b 0 < f - < 0 Proof: ro ea : ro ea 8: 0 < QED ea 5: 5a 0 > f - > 0 5b 0 < f - < 0 Proof: ro ea :

243 ro ea : 0 < ea 6: 0 < Proof: ro ea : 0 < ro ea 8: 0 < 0 < QED Shoer obta the u re offered by the two fr. The dtrbuto of the u re. I ow tur to the oaratve tat o

244 5. Eah of thee hage are reeted below. ea 7: 0 < Proof: et be rereeted by B A where A ad B A B B A < < < A ea By B ea By. QED ea 8: 0 < Proof: et be rereeted by B A where A ad B

245 6 A B B A < < < A ea By B ea By QED Suoe that the roorto of fr oe loyal utoer are haged. ow doe th affet? ea 9: 0 < Proof: By lea 8 0 < ad by lea 9 0 <.

246 7 Thu eah ter the above euato for egatve ad thu 0 <. Solvg for the exreo for : * * *

247 8 I uary, both uulatve dtrbuto futo ad ove the ae dreto whe haged. So the robablty of at leat oe fr offerg a lower re uffer whe the roorto of fr oe loyal utoer reaed. QED Calulatg the effet of hagg fr two loyal utoer o the robablty of at leat oe fr havg a lower re alo traghtforward: ea 0: 0a 0 > for [, 0b 0 whe See roof for gfae. Proof: 0 *

248 9 - the leftot ter egatve ad egatve thu otve. The loe exeto at where - eual zero thu akg the etre exreo zero. Th gfat beaue th tate that the u re ha o a ot at the oooly re. Oly the larget fr ha a ato at. The above exreo a be ultled out for the followg exreo: QED ea : a 0 > whe < b 0 < whe > 0 > whe < 0 < whe > d 0 > whe < ad a > a or a > a

249 0 e 0 > whe > ad a > a f Geerally 0 > exet whe a get large or a extreely all Proof: [ ] [ ] [ ] et M. [ ] M [ ] M [ ] M or 0 >, the ter ultled by M ut be greater tha zero e

250 M > 0. Slfyg: [ ] M [ ] M M > [ ] [ ] > [ ] > < 7 for 0 >. The g revere euato 7 beaue the deoator le tha zero. At euato 7 lfe dow to euato 7a for 0 > : < 7a The g revere beaue the deoator egatve. The dtrbuto of the u re ot very lkely to have a reae weght at hgher re whe there a reae hoer. At euato 7 lfe dow to euato 7b for 0 > :

251 < < < 7b If >, the the le tha g euato 7b revere to a greater tha g 7 for 0 > e the deoator ow otve. > 7 Se the rght had exreo egatve, 0 > wll alway our whe >. The followg hart how the dfferet araeter obato that affet the g of :

252 Table 6: Sgg S of e. 7b/ Potve Potve Æ Negatve Potve Potve Æ Negatve Potve Potve Æ Negatve Potve Potve Æ Negatve Potve Æ Negatve S of e. 7b/7

253 The g o are lar to exet for a oule ore egatve value whe beoe larger. Thu 0 > at ot value of,, ad exet whe beoe large or a beoe extreely all. QED The exere of alulatg the effet o the fr uulatve robablty dtrbuto fro holdg oe ouer grou otat whle hagg the other two oote dreto of the ae agtude a be doe wth the robablty of at leat oe fr havg the lower re. ea : a 0 > ff 7 5 > b 0 < ff 7 5 < Geerally 0 > at re ear wth oe exeto whe all

254 5 d 0 > whe > ad larger tha e 0 < whe larger tha f Geerally 0 < exet whe all Proof: [ ] [ ] [ ]

255 or >0, the uerator euato 8 ut be otve. earragg, th le: 7 5 >

256 7 7 5 > 7 5 > 8a The deoator odto 8a exatly the ae exet the oote g fro the deoator the odto foud whe the uerator of the M ter ultled to the euato. Thu the > foud here odto 8a for to be otve oote of the < g foud the odto for to be otve. 7 5 > 8a At, odto 8a lfe dow to: zero zero > 8b Th ake terretato by oral eto oble. owever, rug alulato by euato 8a ad lettg get loe to a eod bet aroah to fdg the g of. The followg hart how dfferet araeter value of oared to the rght-had de of euato 8a wth dfferg value of ad.

257 Table 7: Sgg S of e 8a Potve Æ Negatve Negatve Æ Potve Potve Æ Negatve Negatve Æ Potve Potve Æ Negatve Negatve Æ Potve Potve Potve S of e 8a 8

258 9 I ot ae, at hgh re ear, the above exreo geerally otve wth oe exeto whe all. At, there ore tae where otve: [ ] [ ] > Slfyg the uerator for the odto at : [ ] Slfyg the deoator for the odto at : [ ] > 8 or > 0 odto 8 beoe < 8d whe >. Euato 8d aot our o < 0 whe >. The followg hart how oared to the rght had de of euato 8 ad 8d whe the araeter of, ad are vared. The g for are the exat oote for the g of.

259 Table 8: Sgg S of e. 8/8d Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Æ Potve S of e. 8/8d 50

260 5 Geerally, the g of at low re the dtrbuto egatve exet whe all. QED ea : a > 0 ff 6 9 K > where 5 K b < 0 ff 6 9 K < where 5 K Geerally < 0 at hgh re ear exet whe relatvely all to d > 0 whe 5 5 > ad the deoator otve e < 0 whe 0 <

261 5 f Geerally < 0 exet whe relatvely all to Proof:

262 5 or >0 the uerator euato 9 ut be otve. Slfyg: > or >0, 6 9 K > 9a where 5 K At, euato 9a redue to: zero zero > 9b. Aga, rug alulato by euato 9b ad lettg get loe to a eod bet aroah to fdg the g of. The followg hart how dfferet araeter value of oared to the rght-had de of euato 9a wth dfferg value of ad.

263 Table 9: Sgg S of e 9a Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve Potve S of e 9a Geerally < 0 at hgh re ear exet whe relatvely all to. e weght laed o the uulatve dtrbuto futo o the u re o hgh re whe hoer are redued favor of addg loyal 5

264 55 utoer of fr oe. Th drve by both fr oe ad two doutg le teely whe hoer are take away favor of fr oe loyal utoer. At, the uerator of the frato of euato 9a redue to: At, the deoator of the frato of euato 9a redue to: > 5 5 > 9 Whe the deoator egatve the odto for 0 > hage to 5 5 < 9d

265 Se otve < 0 whe the deoator euato 9 ad 9d egatve. The followg hart how dfferet araeter value of oared to the rght-had de of euato 9 ad 9d wth dfferg value of ad : Table 0: Sgg S of e. 9/9d Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve 56

266 57 Geerally 0 < exet for all relatve to. Se th ae reult held for, 0 < exet for all relatve to. QED ea : Suoe the hoer are held otat. The robablty of havg the u re dereae f fr oe loyal utoer reaed at the ae rate a fr two loyal utoer dereaed. Proof: By lea 9 0 <. By lea 0 0. Thu 0 <. oyal utoer to fr oe atter ore to both fr dedg how uh to radoze. QED

267 -. The Model wth r large ad - Slar Sall r Now uoe that there are fr the arketlae. Suoe that there oe large fr who ha loyal utoer. Suoe that there are aller fr, eah wth loyal utoer. et tll be the roorto of hoer the arketlae. et * be the reveue eared by a fr at re. Aue aga that th reag. If the uort fro the tal roft alulato reveal dfferg lower re, the oe of the fr ha a ato. Chekg roft euato for eah tye of fr wll hel detere whether there a ato. or fr oe roft are: Se the lowet re the dtrbuto, 0 ad the above euato a be wrtte a: Solvg for : or the aller fr two through, roft are: Se the lowet re the dtrbuto, 0 ad aller fr roft euato a be rewrtte a:. Thu. Thu the 58

268 Solvg for : Note that ad are ot eual. Sefally, >. Se reag, >. Thu the uort of eah fr do ot le u. r oe wll have the ato e the lowet re t uort lower tha fr two. The roft euato wll have to be rewrtte to take to aout the ato for fr oe ad reet o that the lowet re the uort of eah fr uulatve dtrbuto futo the ae. Aga for a re radozato eulbru to our, roft at the hghet re ut eual roft at the lowet re ad all re betwee the two extree. Aga, wth the ayetr ouer grou, there wll be a ato at the hghet re that fr oe wll lae otve weght. The ret of the fr aot re at th oooly re but re at elo below th re. I dog o, they ature wth oe u the robablty of the ato the hoer that fr oe doe ot ature at the oooly re. Aga let - be the lt of. The roft of eah fr a be rewrtte. r oe roft euato : r two through eah have the ae roft euato: 59

269 60 r oe lae a a ot at t hghet ot t re dtrbuto. r oe ok t ufored ouer the oooly re. The roft for fr oe fro dog o ut eual the reveue at the lowet re fro t loyal utoer lu the hoer t k u wth robablty oe at. r two through aot ok t loyal utoer the oooly re. owever, they a re lghtly below the oooly re, okg ther loyal utoer ad wth robablty oe u the ato fr oe lae o t hghet re a k u the hoer. Euato oe a be olved for a before: The ae terretato for tll hold a before. Theore a: The ato for the odfed fr ae rea. Proof: Euato a be olved for - :

270 a - - the fr ae exatly the ae a the two fr - -. The roorto of fr oe loyal utoer, the roorto of fr two loyal utoer, ad the roorto of hoer affet the ato the ae way a they dd the two fr ae. The varable doe ot dretly affet the ze of the ato, but dretly affet the ze through. A get uh larger, the ter get aller a rereet a dvdual hare oe of the aller fr. If uhaged, the the ze of the ato wll reae whe reae. Thg get ore olated f a large reae ad dereae alo reult,, or both gettg aller. A all reae ouled wth o hage the ze of aue, or both to fall. Aga, the uulatve robablty futo for eah fr ad a be olved by ettg roft at ay re the dtrbuto to the roft at the hghet re the dtrbuto. or fr oe th : [ ] 0 Note euato 0 o the left-had de that the robablty that all other fr have hgher re ow. The roft euato for fr two at ay re : [ ] 6

271 6 Note euato o the left-had de that the robablty of all other fr havg hgher re ow the robablty of the larger fr havg a hgher re te the robablty that the lar aller fr have hgher re. Theore : I the odfed fr ae, Proof: a be olved by rearragg euato 0: the fr ae look very lar to the two fr ae exet ow the frato raed to the / - ower. ag the braketed frato to the / ower rae the overall frato ad thu lower at every re. The lato of a lower that aller - fr dout le to attet to get the hoer a ther arkethare ad thu otetal reveue fro ther loyal utoer gettg aller lower tha the hae of wg the hoer wth hgher aout of fr the

272 6 arketlae. Jut lke the two fr ae, doe ot aear the exreo for a the aller fr are radozg for hoer at every re that they ould harge. Theore 5: I the odfed fr ae Proof: a be olved by rearragg euato :

273 6 a euato the fr ae lar to euato 8a the two fr ae exet for the ter the deoator. Th ter adjut for the ae that there are aller fr oetg wth fr oe for the hoer tead of jut oe fr. A grow larger, th ter grow larger, thu akg aller a fr oe doe ot wat to oete a uh wth the other fr a reagly utthroat oetto for the hoer. The addto of th extra ter ake t ore olated to redt the effet of hage of ouer grou o exet for the ae of. -5. Coaratve Stat wth the Modfed r Model A wth the two fr odel, oe a ak how hage ouer lae ad the uber of fr affet the uulatve robablty dete of the fr the arketlae. Table how the overvew of oaratve tat for eto 5. Table lt the dfferet lea reultg fro the oaratve tat the odfed - fr

274 odel. The uer left-had ot etry read ÿ/ < 0, whh foud ea 5. The etry the frt olu ad eveth row read ÿ/ - ÿ/ < 0, whh foud ea

275 Table : Coaratve Stat Overvew Seto 5 ÿ ÿ ea 5 ea 6 ea 7 ea a ea 67 < 0 < 0 > 0 0 < 0 ea 8 ea 9 ea 0 ea b ea 68 < 0 < 0 > 0 0 < 0 ea ea ea ea 5 ea 69 < 0 < 0 > 0 > 0 > < 0 < 0 < ea 6 ea 7 ea 8 ea 9 ea 70 > 0 ele 0 0 < 0 > 0 ele 0 0 ea 50 ea 5 ea 5 ea 5 ea 7 > 0 ad < 0 > 0 < 0 < 0 > 0 ad < 0 Geerally > 0 ea 5 ea 55 ea 56 ea 57 ea 7 < 0 < 0 0 > 0 f - < 0 otat - ea 60 ea 6b ea 6 ea 65 ea 75 < 0 < 0 > 0 > 0 ad < 0 < 0 66

276 67 - ea 59 > 0 ad < 0 Geerally < 0 a reae ea 6 < 0 ea 6a > 0 ea 6 > 0 ad < 0 ea 7 > 0 ad < 0 Geerally < 0 - ea 58 > 0 hgh < 0 low ea 6a < 0 ea 6b > 0 ea 66 < 0 ea 7 > 0 ad < 0 ÿ ÿ ea 5: I the odfed fr odel 0 < Proof: et K.

277 68 et D the deoator of the frato of or. et N the uerator of the frato of or. [ ] D N K N [ ] D D [ ] D K N [ ] [ ] D N The braketed exreo [ ] otve e t eual, whh aued to be greater tha zero. The uerator otve e eah of the three ter otve. The deoator, however, egatve e. 0 < Thu the odfed -fr ae, 0 <. QED ea 6: I the odfed fr ae 0 <

278 69 Proof: [ ] Aga, the braketed exreo [ ] otve e t eual, whh aued to be greater tha zero. The left frato or K raed to the ower otve. The uerator of the rght frato otve. The deoator of the rght frato, however, ot otve e 0 < the deoator. Thu the odfed -fr ae 0 < QED ea 7: I the odfed fr odel 0 > Proof: The roof detal to ea.

279 70 / 0 / > e 0 <. QED ea 8: I the odfed fr odel 0 < Proof: [ ] D N K N [ ] D K [ ] D K [ ] D N

280 7 D K D N Se 0 <, eah uerator ter egatve. The deoator otve. Thu 0 <. QED ea 9: I the odfed fr odel 0 < Proof: Aga e 0 < the rght had frato egatve. The left had frato otve. Thu 0 <. QED ea 0: I the odfed fr odel 0 >.

281 7 Proof: The roof detal to that of ea 6. [ ] 0 > e > QED ea : I the odfed fr odel a 0 b 0 Proof: The roof exatly detal to ea 7a ad 7b. There o uatty ad ot thu the dervatve of the ato wth reet to ot ad uatty are zero. QED

282 7 Suoe that the roorto of loyal utoer to fr oe reae? ow wll that affet fr oe uulatve re dtrbuto futo? ea : I the odfed fr ae 0 < Proof: Aga, let K. et D the deoator of the frato of or. et N the uerator of the frato of or. [ ] D K N D D K N [ ] D K

283 7 [ ] D K [ ] D K 0 < D N D D D [ ] D D a Thu <0 a fr oe reod to a reae t loyal utoer wth lag le weght uo lower re. Exeted roft for fr oe reae. QED ea : I the odfed fr odel 0 <

284 75 Proof: 0 < 8 Jut lke the two fr ae, the aller fr the fr ae oete le aggrevely for hoer whe there a reae loyal utoer for fr oe. Th beaue fr oe alo oetg le aggrevely. Tye two fr tur lae le weght o lower re. QED ea : I the odfed fr odel 0 > Proof: The roof exatly detal to ea 0. 0 > QED ea 5: I the odfed fr odel 5a 0 > whe > 5b 0 whe

285 < whe < 5d A grow ze, t ore lkely that 0 >, aug that tay relatvely the ae ze Proof: The roof alot detal to ea a-. Geerally a grow larger grow aller. If tay ear the ae ze the geerally 0 >. QED ea 6: I the odfed fr ae 6a >0 for all re, ] 6b 0 whe

286 77 Proof: Chagg the loyal utoer of fr two gve a lar effet o the fr ae a the two fr ae: D 5 At : 0 > D D At : 0 D Se reag, >0 for all re the, ] terval ad eual to zero at. Th reult ot dfferet tha the two fr ae a the ze of the ato at dereae. A the ze of the ato dereae, fr oe dout ore at lower re to attrat the hoer ad ake u for lot oooly roft. QED ea 7: I the odfed fr odel 0

287 78 Proof: 0 * 0 0 * Chagg fr two loyal utoer bae ha o effet o the uulatve robablty dtrbuto the odfed fr ae. Chagg loyal utoer for the aller fr affet roft of the aller fr but ot the uulatve dtrbuto futo of the aller fr. Th the ae a the two fr ae. QED ea 8: I the odfed fr odel 0 Proof: The roof detally the ae a ea b. There o the exreo for o 0 ea 9: I the odfed fr odel 0 <

288 79 Proof: The roof detally the ae a ea. 0 < QED ea 50: I the odfed fr ae, 50a > 0 ff [ ] [ ] [ ] 6 < 50b < 0 ff [ ] [ ] [ ] 6 > 50 > 0 ff 5 < A there are ore fr or re for a gve uber of fr, ore lkely to be egatve 50d At lower re, > 0 ff <

289 80 50e Coared to, there are ore oblte for to be otve, eeally for whe fr. A re for a gve uber of fr ore lkely to be egatve Proof: [ ] D K N D D K N [ ] D K [ ] D K [ ] D K [ ] D [ ] D N

290 8 [ ] D [ ] D N [ ] [ ] D N [ ] [ ] D [ ] [ ] D D D 6 or euato 6 to be otve, the uerator ut be otve. Slfyg: >

291 8 K < where K The g reveral oe beaue the deoator egatve at all araeter obato of,,, ad exet for the ae whe very low ad very hgh ad aroahg. [ ] [ ] [ ] 6 < 6a At D D or to be otve, the uerator of the above exreo ut be otve. Slfyg: >

292 8 < < 5 < 6b The followg table how the varou araeter obato of,,, ad have o

293 Table : Sgg S of e 6b Negatve Negatve Potve Potve Æ Negatve Negatve Negatve Æ Potve Potve Æ Negatve Negatve Negatve Æ Potve Potve Æ Negatve Negatve S of e 6b 8

294 85 Geerally, th dervatve at re ear egatve a get larger. Alo f re for a gve uber of fr, < 0. The dya of hage a re fall toward : D D D D D >0 f the uerator of the above exreo greater tha zero. Slfyg: >

295 86 < < < 6 The et dfferee betwee ad that the uerator of 6 ota ore ter tha the uerator of 6b. The et reult of th there ore of a hae that otve at re ear, eeally f the roorto of hoer large ad the roorto of loyal utoer to fr oe all. Table how the varou araeter obato of,,, ad have o Note that there are ore otve value for oared to. The larget fr ore lkely to dout at low re whe the hoer are large tha at hgh re. The robablty of wg hoer at low re uh hgher. If re for a gve uber of fr, < 0.

296 Table : Sgg S of e Potve Æ Negatve Potve Negatve Potve Æ Negatve Potve Negatve Potve Æ Negatve Negatve S of e 6 QED Chagg the hoer the odfed fr ae ha the ae effet o a the two fr ae: ea 5: I the odfed fr ae > 0 87

297 88 Proof: 0 > 7 ke the two fr ae, the - aller fr dout ore whe there are ore hoer. The doutg ot a large whe the uber of fr reae. QED ea 5: I the odfed fr odel 0 < Proof: The roof detal to that of ea 7. 0 < QED ea 5: I the odfed fr odel 0 < Proof: The roof detal to that of ea 8.

298 89 0 < QED Varyg the uber of fr alo affet. ea 5: 0 < Proof: [ ] l 8 0 < Wth the reaed aller fr laorg over the hoer, fr oe fd t harder to w the hoer. Aug that the reae fr doe ot take away arket hare fro the hoer ad the loyal utoer for fr oe, the dvdual hare of tye two loyal utoer fall. Th fallg of the tye two dvdual fr hare of loyal utoer aue the ato for fr oe at to get larger. Thu fr oe lae le weght o re oetg wth the aller fr ad tead oetrate ore of t radozato at t ato at. QED

299 90 Chagg the uber of fr alo affet : ea 55: 0 < Proof: 0 l < 9 The robablty of wg the hoer dereae at every re for the aller fr whe there are ore of the. Itead the aller fr fou ore o axzg reveue fro ther loyal utoer. QED ea 56: I the odfed fr ae 0 Proof: There o thu 0. QED ea 57: If otat, 0 >

300 9 Proof: If reae whe otat, the a fallg. A fallg a le the uerator of the ato reag. Thu the ato reae. The revere ae of fallg alo hold. QED The ae exere the two fr odel a be erfored the odfed fr odel where oe ouer grou held otat ad the other two are adjuted uward ad dowward by the ae roorto to ee the effet o the fr uulatve robablty dtrbuto futo. ea 58: 58a otve ff [ ] [ ] [ ] [ ] 6 > 58b egatve ff [ ] [ ] [ ] [ ] 6 < 58 At hgh re ear, >0

301 9 58d At low re ear, <0 o log a a ot relatvely all oared to a. Proof: D D D [ ] [ ] { } D K 6 where [ ] K [ ] [ ] { } D K 6 [ ] [ ] { } D K D 6 [ ] [ ] { } D K where 5 K

302 9 otve whe the uerator of the frato of euato 0 otve. The uerator of the frato of euato 0 otve whe the - ter bgger tha K 5. Or: [ ] [ ] [ ] [ ] 6 > 0a The g ket to greater tha beaue the deoator geerally otve, exet whe low, hgh, ad aroahg. At, euato 0a redue to 6 5 > 0b or hgh re ear, euato 0b hold o atter how ay fr are reet. All alulato Matheata reult > 0. Thu at hgh re, > 0. At ad low re jut above that re euato 0a redue to: [ ] [ ] [ ] [ ] 6 > 8 5 >

303 > The tory hage at low re. There are ore egatve value of oared to. r oe radoze ore at hgh re tha at low re uder thee odto. Table how the varou. araeter obato of,,, ad have o Geerally, < 0 o log a a ot relatvely all oared to a. I the ae a relatvely all oared to a, there oly a all rage of re that are radozato. A wll be how a later rooto, reae uder thee rutae ad doate the effet o. Thu > 0. Table Sgg S of e Negatve Æ Potve Negatve Æ Potve 9

304 Negatve Æ Potve Negatve Potve Negatve Æ Potve Negatve Potve S of e 0 QED ea 59: 59a > 0 ff [ ] [ ] [ ] K > 59b < 0 ff [ ] [ ] [ ] K < where 6 K 59 Geerally < 0 exet whe very all relatve to

305 96 59d < 0 for a greater rage of araeter value a reae Proof: [ ] D D [ ] D D [ ] [ ] { } D K 6 where [ ] K [ ] [ ] { } [ ] D D 6 [ ] { } [ ] D D

306 97 [ ] { } [ ] { } [ ] D D D otve f the uerator euato otve. earragg: [ ] { } [ ] { } [ ] > [ ] [ ] [ ] K > where 6 K a At, euato a redue dow to 6 5 > b Table 5 how the varou araeter obato of,,, ad have o. May value of are egatve ad reagly o a the uber of fr re. Wth a reae t loyal utoer ad dereae hoer, fr oe a wll be how a future lea reae the ze of t ato ad dereae t radozato at hgh re. Wth a hgher uber of fr,

307 th tred < 0 reae at hgh re for ore araeter value due to t beg ore oettve to w the hoer. r oe wll reagly radoze le at thee hgher re wth ore fr beaue the odd of wg the hoer are le wth ore fr. Table 5 Sgg ad S of e b Negatve Æ Potve Negatve 0.* Potve Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve Negatve Æ 98

308 Potve S of e b * egatve for At re ear euato a be odfed to: [ ] [ ] * 6 K > where [ ] * K 5 8 > 5 8 > Whe the deoator egatve at hgh value, the greater tha g euato revere to a le tha g e there dvo by a egatve uber. 5 8 < d Table 5 above how the varou araeter obato of,,, ad have o. The g of for the araeter value gve are the ae exet for oe et of value arked by a aterk. ather tha rertg aother table, t eaer to refer to Table 5 to hek the g of

309 00. Aga ot araeter are egatve exet whe relatvely all to. A reae, there a greater rage of araeter that uort < 0. QED ea 60: < 0 Proof: By lea 0 <. By lea 6 0. Thu < 0. QED ea 6: 0 < Proof: 0 <

310 0 0 > 0 < QED ea 6: 6a 0 < 6b 0 < Proof: By ea 7 0 By ea 5 0 > Thu 0 < By ea 0 < Thu 0 < QED

311 0 ea 6: 6a 0 > 6b 0 > 6 0 > Proof 6a: ro ea : 0 > ro ea 5: 0 < 0 > Proof 6b: ro ea 8: 0

312 0 ro ea 5: 0 < Thu 0 > Proof 6: ro ea : 0 > ro ea 8: 0 Thu 0 > QED ea 6: 6a 0 > f - > 0 6b 0 < f - < 0 Proof:

313 0 ro ea 5: ro ea 5: 0 < QED ea 65: 65a 0 > f - > 0 65b 0 < f - < 0 Proof: ro ea 5: ro ea 9: 0 <

314 05 ea 66: 0 < Proof: ro ea 9: 0 < ro ea 5: 0 < 0 < QED ke the two fr odel, the robablty of the u re a be reated ad oaratve tatt a be erfored o th robablty. The robablty of the u re

315 06 The dfferee betwee the odfed fr odel ad the two fr odel the extra ter the deoator. The ae oaratve tat exere a be erfored o th ew robablty of at leat oe re beg lower. ea 67: < 0 Proof: et A et B

316 07 B A A B B A < < < A ea By B ea By. QED ea 68: < 0 Aga let A et B B A A B B A < < < A ea By B ea By QED

317 08 Suoe that fr oe loyal utoer are haged. The effet o : ea 69: <0 Proof: By lea 0 <. Se both ad are egatve, egatve. Exadg ad lfyg euato : [ ] D D D

318 09 [ ] [ ] D D D D D D D D [ ] D [ ] D D a QED Suoe that fr two loyal utoer are haged. The effet o :

319 0 ea 70: 70a > 0 for [, 70b 0 at Proof: 0 By lea > for all exet at, where 0. The aller fr uulatve robablty dtrbuto futo do ot hage whe hage. Thu fr oe uulatve robablty dtrbuto futo relatoh wth rovde the reao why th dervatve futo otve. Exadg ad lfyg euato : D D a

320 QED Suoe ow the hoer are haged. Th affet the robablty that at leat oe re lower by: ea 7: 7a > 0 ff [ ] [ ] [ ] [ ] K < 7b < 0 ff [ ] [ ] [ ] [ ] K > where [ ] K 7 Geerally > 0 exet whe very large, ad very all relatve to 7d There are ore araeter value uortg > 0 whe the uber of fr hgher 7e At ad, there are ore araeter value that uort

321 > 0 at re ear tha at Proof: D D D N D D

322 [ ] D [ ] D 8 6 [ ] D [ ] [ ] 8 6 D D [ ] 8 6 D [ ] D [ ] 9 8 D [ ] 8 6 D otve f the uerator of euato otve. earragg the uerator of euato : [ ] [ ] [ ] > [ ] [ ]

323 [ ] [ ] [ ] [ ] K < where [ ] K The eualty g wthe beaue the deoator a egatve uber. > 0 ff [ ] [ ] [ ] [ ] K < a where [ ] K At for 0 > [ ] [ ] [ ] [ ] < Ufortuately, the rght-had de of the above euato redue to zero zero. Th ake terretato by oral eto oble. owever, rug alulato by euato a ad lettg get loe to a eod bet aroah to fdg the g of at hgh re ear. Table 6 alo reveal that euato a doe ot hold or < 0 at hgh re f extreely hgh ad extreely low ad the uber of fr all. The ad 0 fr ae reveal that oly very extree value of ad aue < 0 at hgh

324 re. There a atter that there are ore otve value of for hgher value of. or tae, at 0 fr, oly otve value of ext whe eual At 0 fr, otve value ext for at all value ad whe eual a uh larger 0.00 exet for whe 0.59 ad 0.0. The 0 fr ae otrat a lar ze of at fr. At fr, all value at 0. are egatve. Table 6: Sgg # of fr S of e a Potve Æ Negatve Negatve Æ Potve Æ Negatve Negatve Potve Potve Æ 5

325 Negatve Potve Æ Potve Æ Negatve # of fr S of e a At, > 0 ff a odfed vero of euato a hold. Subttutg to euato a: [ ] [ ] [ ] [ ] K < where [ ] K [ ] [ ] [ ] 8 9 K K < where [ ] K where [ ] K 8 5 < b

326 Table 7: Sgg # of fr S of e b Potve Æ Negatve Potve Æ Negatve Potve Æ Negatve Potve Æ Negatve Potve Æ Potve Æ Negatve # of fr S of e b ro the above table ad euato b, oly gog to be egatve whe very hgh ad very low. There are alo ore outoe of 7

327 > 0 f the uber of fr larger. There are ore araeter value that uort beg otve tha, eeally at ad fr ad whe a hgh. Both the at two table reveal that geerally > 0, eeally whe the uber of fr hgh. Oly extree araeter value ake th egatve. A there are ore fr, oetto aog fr beoe ore tee over a greater uber of hoer thu reag the robablty of the u re. Whe there are ore hoer, the rze of wg the hoer larger, o aller fr lae le eha o ther loyal utoer ad ore toward radozg toward the larger grou of hoer. Wth fewer hoer, th the revere ae. ewer hoer ea le oetto betwee the ay aller fr ad the robablty of the u re dereae a fr have to dout le to w the hoer. Whe there are fewer fr, they are ore tereted the rze of hoer a uh beaue oetto le ad thu radoze ore toward the ad le toward ther loyal utoer. QED Suoe that the uber of fr haged. That affet the robablty of at leat oe re beg aller by: 8

328 9 ea 7: < 0 Proof: Se both ad are egatve, alo egatve. 0 < QED ea 7: 7a > 0 ff [ ] [ ] [ ] [ ] K K > 7b < 0 ff [ ] [ ] [ ] [ ] K K <

329 0 where [ ] 0 K where [ ] 8 6 K 7 > 0 at hgh re f all ad very large ad very all or f all ad very all ad very large 7d > 0 at low re f all ad very large ad very all 7e < 0 f large, exet very extree ae 7f The tae of < 0 reae whe there are ore fr 7g There are ore tae of < 0 at low re tha at hgh re Proof: D D

330 [ ] [ ] [ ] D D [ ] [ ] [ ] D D [ ] D [ ] 9 8 D [ ] 8 6 D [ ] D 5 [ ] [ ] D D > 0 f the uerator of euato 5 otve. earragg the uerator of euato 5: [ ] [ ] [ ] [ ] [ ] [ ] >

331 > 0 ff [ ] [ ] [ ] [ ] K K > where [ ] 0 K where [ ] 8 6 K The deoator the ae a euato exet t ha the oote g. Thu the deoator otve ad the eualty g ged otve here tead of egatve euato. [ ] [ ] [ ] [ ] K K > where [ ] 0 K 5a where [ ] 8 6 K At, > 0 ff euato 5a odfed to: [ ] [ ] [ ] [ ] > ke the oooly ae of lea 0, the above euato redue to zero zero. Thu eto a a frt oto doe ot work. A a eod bet oto, ueral alulato ug euato 5a wth value of aroahg a be ued to detere tred re ear. Below Table 8 of the alulato of dfferet araeter of,,,,, ad.

332 Table 8: Sgg S of e 5a Potve Æ Negatve Negatve Æ Potve Potve Æ Potve Negatve Negatve Æ Potve Potve Negatve Negatve Æ Potve Potve Negatve Negatve

333 Negatve Æ Potve S of e 5a The above table reveal that egatve ore tae a reae. or tae, oly the ot extree ae the 0 fr ae otve. Wherea, the three fr ae there are ay ae, eeally whe 0. e large, whe otve. Geerally a extreely large ad all or extreely large ad extreely all at hgh re aue to be otve. The rage of ad for > 0 exad a the uber of fr reae ad a dereae. At euato 5a a be lfed to: [ ] [ ] [ ] [ ] 9 8 K K >

334 5 where [ ] K where [ ] 8 6 K [ ] [ ] K K > where K where K 8 5 > > 0 ff 8 5 > 5b Table 9 below reveal oe dfferet value for the araeter,,, ad a alulated euato 5b.

335 Table 9: Sgg S of 5b Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Negatve Æ Potve 6

336 7 Aga thee value reveal that geerally egatve at lower re a the uber of fr reae. Oly extree hgh value of ad extreely low value of aue to be otve at low re. A the uber of fr dereae ad reae, the threhold at the uer rage for lower ad the threhold at the lower rage for reae for to be otve. Beaue ter arre uh le fluee tha the adjuted ter, the reult foud lea 7 volvg are early the oote g of thoe foud the ae of lea 7 where otve g. QED ea 7: 7a > 0 ff [ ] [ ] [ ] [ ] 5 K >

337 8 the deoator otve ore ae tha ot; the > g revere to < the ae where the deoator egatve whh foud ear the extree ed of the dtrbuto 7b < 0 ff [ ] [ ] [ ] [ ] 5 K < the deoator otve ore ae tha ot; the < g revere to > the ae where the deoator egatve whh foud ear the extree ed of the dtrbuto where [ ] [ ] K 7 Geerally < 0 exet whe a low oared wth a 7d < 0 exet whe extreely hgh ad extreely low 7e Geerally < 0 exet for the extree ae derbed above

338 9 Proof: [ ] D [ ] D D [ ] D [ ] D [ ] D [ ] D [ ] 9 8 D [ ] 8 6 D [ ] D [ ] 9 8 D [ ] 8 6 D

339 0 [ ] D [ ] D [ ] D [ ] 9 8 D [ ] D [ ] 8 6 D [ ] D [ ] 0 5 D [ ] D 6 [ ] D otve ff the uerator of euato 6 otve.

340 earragg: [ ] [ ] [ ] [ ] [ ] > Aug the deoator otve, whh ay ae the reult. The extree ed of the dtrbuto gve oe egatve value of the deoator. Thu the > g wthe to < the ae where the deoator egatve for to be otve. [ ] [ ] [ ] [ ] 5 K > where [ ] [ ] K [ ] [ ] [ ] [ ] 5 K > where [ ] [ ] K 6a At for to be otve at hgh re, euato 6a redue to udefed zero zero >. 6b Se euato 6b udefed by tradtoal ea, a eod bet ethod of ug varou ueral value of,,,,, ad to ee f euato 6a hold for value ear. Table 0 below uarze thee varou araeter value lu

341 the rght had de of euato 6a ad whether otve or egatve at hgh re. Geerally, egatve at hgh re ear exet whe all ze to. There are ore tae of < 0 a the uber of fr reae.

342 Table 0: Sgg S of e 6a Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve Negatve Æ Potve Negatve Negatve Æ Negatve Æ Potve S of e 6a

343 Geerally, addg ore loyal hoer at hgh re ad takg away hoer redue the weght of the robablty of the u re at hgh re. The larget fr oered about t oooly roft at t ato. The larget fr radoze le toward the hoer. The ret of the aller fr oete le teely for the hoer ad tead oetrate o t ufored hoer. At for to be otve euato 6a redue to: [ ] [ ] [ ] [ ] K > where [ ] [ ] K ad aga aug the deoator 6a otve K K > where K where K >

344 > 6 Se at the deoator a be egatve, the odto alo rewrtte whe low ad hgh: < 6d Table below lt dfferet value for,,,, ad the rght-had de of euato 6. Geerally egatve at low re wth the exeto beg f beg extreely hgh ad beg extreely low. At the dot betwee ad for the varou araeter obato of,, ad, egatve at alot all oblte for three fr, ad egatve exet for the ae exeto foud at for ad 0 fr. Thu geerally egatve wth thee few exeto.

345 Table : Sgg S of 6/6d Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Negatve Æ Potve Negatve Æ Potve Negatve Negatve Æ Negatve Æ Potve 6

346 QED ea 75: < 0. Proof: By lea 69 By lea 70 Thu QED < 0. > 0. < Coluo Vara 980 deotrated that fr eual ze radoze re betwee two grou of utoer: thoe loyal to a artular fr ad the hoer, jut loyal to the lowet oble re. The odel develoed here buld uo Vara a two fr ad a odfed fr ae where there oe large fr ad ether oe aller or eually zed aller fr oetg for the ae grou of hoer. Eah fr, lke Vara 980, ha a loyal grou of utoer wllg to ay whatever re at had. ro th etu of th odel, there a be everal oluo draw. 7

347 Jut lke Vara 980, a there o re that ha robablty oe ad all re have oe robablty greater tha zero. adozato our a fr are rg to obta the hghet oble re oble fro the ufored or loyal utoer whle o the other had doutg a uh a oble to reae the hae of obtag the grou of hoer that ee all of the re of the fr. A a eod oluo, the larget of thee fr able to lae a ato of robablty at t hghet re the dtrbuto ad ollet ore roft fro t loyal utoer. The aller fr or fr do ot have th ablty ad tead oete for the hoer at all re ther dtrbuto. The ze of the ato the dfferee of the loyal utoer bae betwee the larger ad dvdual aller fr dvded by the total uber of utoer the larger fr a ature: loyal utoer of the larger fr lu the grou of hoer. r radoze le toward obtag the hoer or the uulatve futo dereae whe the roorto of loyal or ufored fr oe utoer reae. Whe there are ore hoer tha loyal utoer of the aller fr, the larget fr lae ore weght o t ato tead a t ore roftable to ear ore oooly roft. Exeted roft for the larget fr reae. The aller fr reatg to the larget fr do ot eed to oete a teely for the hoer to w the. The larget fr radoze ore toward obtag the hoer or the uulatve futo of fr oe reae whe the roorto of loyal or ufored fr two utoer reae. Th beaue the ze of the ato dereae for the larget fr whe thee aller fr loyal utoer reae. The ze of the aller fr loyal utoer bae ha o effet o the uulatve dtrbuto futo for the 8

348 aller fr a the aller fr are oetg for hoer at every re ther dtrbuto. The roft for the aller fr, however, ay reae. Th our whe ther hare of loyal utoer reae at the exee of the hoer. Geerally the two fr ae, the larget fr tre to oete for the hoer ore or the uulatve dtrbuto futo for fr oe reae whe the hoer reae. A the uber of fr reae, t beoe le lkely that reag the uber of hoer wll reult the larget fr radozg ore to attrat the hoer at hgh re. There are ore tae of the fr oe uulatve dtrbuto futo reag whe there ore hoer at low re. Ulke the olexty of the larger fr whe hoer reae, the aller fr have a lear reult. or a aller fr, reag the ze of hoer bae reult ore doutg to attrat the hoer. All fr have a lear reult whe ore fr eter the arketlae: there le doutg to attrat the hoer. Se t beoe ore dffult to w the hoer wth ore fr the arketlae, fr rely uo ther loyal utoer ad exeted roft dereae for the aller fr, aug that the total arket ze of the aller fr otat or that the reae arket ze of the aller fr oe at the exee of hoer. Ireag oe grou of utoer at the exee of aother grou of utoer alo rovde oe teretg reult. The learet of the reult the ae of reag the larget fr loyal utoer at the ae rate that the aller fr loyal utoer are dereaed. Th aue the larget fr to dout le. The larget fr ha a reae oooly roft at t ato ad doe ot eed to dout a uh to w 9

349 over the hoer thu the doutg le tha before. Ireag the larger fr loyal utoer bae at the exee of the aller fr loyal utoer bae alo aue the aller fr uulatve dtrbuto futo to dereae. Ireag fr oe loyal utoer bae at the exee of the grou of hoer geerally aue fr oe uulatve dtrbuto futo to dereae. A the uber of fr reae, there are ore tae where th hold true. A the uber of fr reae, the ato for fr oe th ae grow larger. The aller fr uulatve dtrbuto futo dereae a fr oe ot aggrevely oetg wth the aller fr for the hoer. Ireag the aller fr loyal utoer at the exee of the hoer aue two hage fr oe uulatve dtrbuto futo. At hgher re, the uulatve dtrbuto futo reae. The ato dereae, hftg oe of that weght to the hgher re. The larget fr uulatve dtrbuto futo dereae at low re. The aller fr uulatve dtrbuto futo dereae throughout the re rage. The lowet re betwee the two fr reae. The robablty of the u re a tatt that a be reated to obe the larget ad aller fr uulatve dtrbuto futo. ro th exere, t wa foud that th robablty futo behaved very lar to the larget fr dtrbuto futo: dereag whe ore loyal utoer of the larget fr are reaed ad whe ore fr are reaed, beg otve whe ore loyal utoer of the aller fr are reaed, ad geerally otve whe the roorto of hoer are reaed. The aller fr uulatve dtrbuto futo th 0

350 u uulatve re dtrbuto hel eae the odto for whe a reae the roorto of hoer aue the robablty of the u re to reae. Ireag the uber of fr aue the u re uulatve dtrbuto to dereae. I the ae of ovg oe grou of utoer over aother, the ae g hold a fr oe exet for the ae whe the aller fr loyalty hare reaed at the exee of the hoer. I that ae, there are ore ae that the uulatve dtrbuto futo dereae. The odel reated doe exted the Vara 980 odel to the ba ayetr ae where there oe larger fr ad oe aller fr or eually zed aller zed fr. ke Vara, there tll radozato. Ulke Vara, the larget fr ow ha a ato of robablty at t oooly re. Coaratve tat geerally reveal lear reult exet for the ae of the hage of hoer o the larget fr uulatve dtrbuto futo. uture ueto ould lude hagg ot of fr ad extedg the odel beyod oe erod.

351 Aedx. PD, Valuato Cae, ad Data Grah PD gure a Pre gure a&b: λ 0.88 Pdf of g 5: β M 550 gure b PD Pre

352 PD gure 5 a Pre gure 5a&b: λ β M 689. gure 5 b PD Pre

353 CD gure Pre gure 6: 0. β λ.585 M M 700

354 PD gure 7 a Pre gure 7a&b : 0. β λ.585 M M 700 gure 7 b PD Pre 5

355 gure 8 Dety laxhlf AX - N egarr re. Peret Cu AA, CO, D,99.0. A, NW, TZ, UA, Total,

356 gure 9 Dety ordf MSP - Chago O are egarr re. Peret Cu AA, NW, UA, Total 7,

357 gure 0 Dety dalgaf Wahgto Natoal New York aguarda egarr re. Peret Cu AA D, TB 8, UA US Total,

358 gure a Dety yof NYC Arort MCO All are Both Way egarr re. Peret Cu AA CO, D 9, P JI KP, KW NW TW, TZ UA US, Total,

359 Table : Seget are Ba Code or Carrer betwee NYC ad Orlado egfareb egarr Y YD Total AA CO 5,706 0,9,859 D ,6 9,7 P JI KP 0 0 0,7,7 KW NW TW 0 0,78,70 TZ UA US 8 8,0, Total 8,6 9,987,0 gure b Dety yof NYC MCO YD All Carrer 50

360 gure ad d Dety yof NYC MCO Cotetal YD Dety yof NYC MCO Delta YD 5

361 gure e ad f Dety yof NYC MCO Kw YD Dety yof NYC MCO Tra World YD 5

362 gure g ad h Dety yof NYC MCO Aera Tra Ar YD Dety yof NYC MCO US Arway YD 5

363 gure ad j Dety yof NYC MCO Cotetal Y Dety yof NYC MCO Delta Y 5

364 gure k Dety yof NYC MCO US Arway Y 55

365 gure a Dety atlf All fare MSP AT Market Both Way Segarr re. Peret Cu B CO D, NW, UA UK Total,

366 Table : Seget are Ba Code or Carrer betwee MSP ad Atlata egfareb B CO D NW UA UK Total C Y YD,65,8, Total,99,87,85 gure b Dety atlf YD both ajor arrer 57

367 gure ad d Dety atlf YD Northwet Arle Dety YD Delta Arle atlf 58

368 gure e ad f Dety atlf Y Delta Arle Dety Y Northwet Arle atlf 59

369 gure Dety laxlaf AX - AS egarr re. Peret Cu AA B D, A P, UA, US WN, Total 5,

370 gure Dety yhf NYC three arort Chago two arort egarr re. Peret Cu AA, B, CO, D KP, NW TW, UA 5, Total 8,

371 gure 5 Dety auvdf AUS Provdee, I rearr re. Peret Cu AA CO D NW UA US Total

372 Bblograhy: Agular, ou. I roter Sght; Arle Target Uted Elte Cutoer, The Dever Pot: ebruary 7, 00, Bue,. C0. ArTra Pre eleae Augut 7, 00: ArTra Arway auhe 8-our Sale for Travel To ad ro Phladelha ArTra Arway auhe 8-our Sale for Travel To ad ro Baltore Ar Traort Aoato 00 Eoo eort. htt:// htt:// Aera Arle Pre eleae: Arl 7,005: htt:// e.jhtl Aera Exre Bue Travel Motor Pre eleae: Noveber, 00. htt://hoe.aeraexre.o/or//00/btm_rt.a Adero, Joh. Jr. Chage Arfare, Serve, ad Safety Se Deregulato Uted State Geeral Aoutg Offe tetoy before the US Seate Cottee o Coere, See, ad Traortato: Thurday, Arl 5, 996. htt:// Baley, Elzabeth E., Graha, Davd., ad Kala, Dael P. Deregulatg the Arle. Cabrdge, Maahuett: The MIT Pre, 99. Baye, Mhael. ad Joh Morga. Neeary ad Suffet Codto for Extee ad Uuee of Bertrad Paradox Outoe, Woodrow Wlo Shool of Publ ad Iteratoal Affar Eoo Workg Paer #86, Preto Uverty, 997. Bjorhu, Jefer. Bue Traveler old the Wld Card, Poeer Pre: Suday ebruary, 00. Borete, Sever. ub ad gh are: Doae ad Market Power the Arle Idutry, AND Joural of Eoo: Vol. 0, No., Autu 989,. 65. Braga, Martha, Carey, Sua, ad MCartey, Sott. ed U wth Arle, Bue Traveler Start to ght Bak, Wall Street Joural:Tueday Augut 8, 00,. A olu. 6

373 Burdett ad Judd. Eulbru Pre Dero, Eooetra: Volue 5, Nuber, July 98, Butter, Gerald. Eulbru Dtrbuto of Sale ad Advertg Pre, evew of Eoo Stude: Volue, Otober 977, Carlo, Joh A. ad. Preto MAfee. "Drete Eulbru Pre Dero," Joural of Poltal Eooy: 98, Volue 9, Nuber, Delta Arle Pre eleae Jauary, 005: Delta Slahe Everyday are U To 50 Peret A Arle Itrodue Slare Natowde 0 Ja 005 Deekere, ay ad Preto. MAfee. "Daaged Good," Joural of Eoo ad Maageet Strategy, Volue 5, No., Suer 996, Daod, Peter A. "A Model of Pre Adjutet," Joural of Eoo Theory: Jue 97, Volue, edor, z. Arle ght for Bue Traveler, Meaol-St. Paul Star Trbue: rerted The Su: Otober 9, 00. Gratha, uell. Plae are Cree gher; Swtheroo: ow-ot Carrer Able to ae Pre, Ulke Bg Couterart, The Atlata Joural-Cottuto: Thurday Marh, 00, Bue,.. aburger, To. Arle Merger May ae eavy lak; Arle Deregulato the 980 Proted a Wave of Merger, But the egulatory Clate a Chaged Se The, Star Trbue: Moday, Jue, 000, New,. A. alto, Martha M. Arle Prg: ghly Colex, otly Coettve, The Wahgto Pot: Noveber 0, 988, aal,.. eller, Jea. Arle etur Perk At a Pre, St. Peterburg Te: Marh, 00, Bue, O-e. htt:// htt:// htt://ew.delta.o/ew.dex.f htt:// htt:// 08/0/05: Ideedee Ar are Sale Offer Cool all Travel Deal 6

374 htt:// htt:// Jevo, Wlla S. The Theory of Poltal Eooy. rt Edto, odo, Eglad: Malla ad Co., 87. Koeg, Davd. Southwet ower at Mute are to ure Bue Traveler, The Aoated Pre: rerted the Seattle Pot-Itellger: rday, Augut, 00. Kre, Davd M. ad Joé A. Sheka. Quatty Preotet ad Bertrad Coetto Yeld Courot Outoe, Bell Joural of Eoo, Autu 98, Mayard, Mhele. Budget Arle Set Off O a Cro-Coutry Joy de, The New York Te: Suday Arl, 005, Seto, Suday Bue,. 7. MAfee,. Preto. Edogeou Avalablty, Cartel, ad Merger a Eulbru Pre Dero, Joural of Eoo Theory: Volue 6, Nuber, ebruary 99,. 7. MAfee,. Preto. "Multrodut Eulbru Pre Dero," Joural of Eoo Theory: Otober 995, Volue 67, Nuber, Nahata, Babu, Krzyztof Otazewk ad P.X. Sahoo. Dreto of Pre Chage Thrd Degree Pre Drato, Aera Eoo evew: Volue 80, Nuber 5, Deeber 990: Nazareo, Aala. A Seal Offer; etaler Stll Offer Couo Eve Though Ther Ue by Couer a Bee Delg, Sa Atoo Exre-New: Noveber, 00, Metro: Bue Exre,. 8. eed, Da. Cot of lyg for Bue Get Cheaer, USA Today: rday July 8, 00, New,. A. ob, afael. "Eulbru Pre Dtrbuto," evew of Eoo Stude: July 98, Volue 5, Iue : obert, Jaue ad Dale O. Stahl II. Iforatve Pre Advertg a Seuetal Searh Model, Eooetra: Volue 6, Iue, May 99,

375 othhld, Mhael. Model of Market Orgazato wth Ierfet Iforato: A Survey, Joural of Poltal Eooy: Volue 8, Nuber 6, Noveber/Deeber 97:,8 08. Salo, Steve. "The Noy Mooolt: Ierfet Iforato, Pre Dero, ad Pre Drato," evew of Eoo Stude: 977, Volue. Salo, Steve ad Joeh Stgltz. "Barga ad off: A Model of Moooltally Coettve Pre Dero," evew of Eoo Stude: Otober 977, Volue : Saluka, Ag. Aera: Dlara of the Ske, The New York Te: rday May 9, 99,. D. Shleger, Jaob M. "If E-Coere el Kll Iflato, Why Dd Pre Jut Ske?", Wall Street Joural: Otober 8, 999:. A ol 6. Shaleee, hard. Outut ad Welfare Ilato of Mooolt Thrd Degree Pre Drato, Aera Eoo evew: Volue 7, Marh 98,. 7. Shtt, Er. The Art of Devg Ar are, The New York Te: Wededay, Marh, 987, Seto D, age, Colu. Sloae, eoard. Why a $87 lght Ca Alo Cot $, The New York Te: Saturday, Arl, 988, Seto, Page 8, Colu. Southwet Arle Pre eleae: ebruary 8, 005: htt:// Southwet Arle Sed ottet are Dretly to Cutoer' Couter Dekto Southwet Arle Pre eleae: July 5, 005: htt:// Southwet Arle Offer $9 Oe-Way are ro Phoex ad Souther Calfora Cte to a Vega Southwet Arle Offer $9 - $79 Oe-Way are Selet lorda Market Southwet Arle Offer $9 Oe-Way are Selet Mdwet Market Southwet Arle Offer $9 - $69 Oe-Way are Selet Texa ad eartlad Market Southwet Arle Offer $9 - $89 Oe-Way are Selet Northwet Market Southwet Arle Offer $9 Oe-Way are Calfora Southwet Arle Offer Chea Arfare a ow a $9 Oe-Way 66

376 Stahl, Dale O. II. "Olgoolt Prg wth Seuetal Couer Searh," Aera Eoo evew: Seteber 989, Volue 79, Nuber, Stgler, George J. "The Eoo of Iforato," The Joural of Poltal Eooy: Volue 69, Nuber, Stoller, Gary. Arle Mllo Mler Get Quet Perk, USA Today: ebruary, 005, Moey,. 8B. Uted Arle Pre eleae: Otober 0, 999. Curret teret te douetg the orgal re releae: htt:// 9 Vara, al. A Model of Sale, Aera Eoo evew: Volue 70, Nuber, Seteber 980, Vara, al. Pre Drato ad Soal Welfare, Aera Eoo evew: Volue 75, Seteber 985, Walh, Chr. Arle Job Deed; But ow-cot Carrer ead rg eboud At Wage Exee, oky Mouta New: Tueday Marh 5, 005, Bue,. B. Weeth, a. "Pre Drato Baed o Ierfet Iforato: Neeary ad Suffet Codto," evew of Eoo Stude: Volue 9, 98, Wleky, Daw. Targetg: Key Strategy for Couo, BradMarketg Suleet to Suerarket New: Vol 5, No. 9, Seteber 998,. 0. Wlla, Ia. Iovato: Dark See Brg a Boot to Arle Proft, The Te odo: Suday, Noveber, 987, Iue

377 VITA Wlla Joeh Brea wa bor Oaha, Nebraka o Arl, 97, the o of Wlla Joeh Brea, Jr. ad Kathlee Mary Brea. After graduatg frt h la 990 at Creghto Prearatory gh Shool Oaha, Nebraka, he etered the Uverty of Notre Dae South Bed, Idaa. After reevg the degree of Bahelor of See Matheat ad Eoo, aga u laude, fro Notre Dae May 99, he et a year of erve outo, Texa a art of the Jeut Voluteer Cor, atg the oele Outreah Progra for Career ad eovery eoure, I. a a eloyet ealt. I Augut 995, he etered the Uverty of Texa at Aut. e reeved a Mater of See Eoo May 997. ro Augut 998 through Deeber 000, he taught fve eto of Prle of Mroeoo. I Arl 999, he ad Dr. Ele. Eheverr-Carroll ublhed the artle Are Iovato Network Bouded by Proxty Iovato, Network, ad oalte, edted by M. M. her,. Suarez- Vlla, ad M. Steer. Peraet Addre: 00 South 7 th Crle Oaha, Nebraka 685 Th dertato wa tyed by the author. 68