The Dynamics of Energy Demand of the Private Transportation Sector

Transcription

1 Th Dyamcs of Ergy Dmad of th Prvat Trasportato Sctor Rto Tar, Uvrstät Br Cofrc papr STRC 007 STRC 7 th Swss Trasport Rsarch Cofrc Mot Vrtà / Ascoa, Sptmbr

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3 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Th Dyamcs of Ergy Dmad of th Prvat Trasportato Sctor Rto Tar Uvrstät Br Schazckstrass CH-300 Br Pho: Fax: mal: rtar@gmx.ch Chrstoph Lubrgr Ecol d'géurs t d'archtcts d Frbourg Boulvard d Pérolls 8 CH-705 Frbourg Pho: Fax: mal: chrstoph.lubrgr@hfr.ch Sptmbr 007 Abstract Ths papr dscrbs th fluc of ful prcs o th dmad of car typs, car travl dmad ad ful. Th ful prc affcts th typ of car a houshold buys ad th dstac drv. I past studs, thr th short-ru or th log-ru lastcts of ful dmad wr xamd, mostly wthout cludg th stock of cars th modls. For th short-ru lastcty of ful dmad, th car stock ca b cosdrd to b costat. I th log ru, th car stock ca b cosdrd as adaptd to th w prcs ad thrfor th log ru prc lastcty should b gratr that th short ru prc lastcty. I ths modl th car stock s cosdrd. Th am of ths papr s to xam th dmad for car typs, car travl dmad ad ful th short ad log ru. W solv ths problm by stmatg a dmad fucto that dscrbs th dmad for cars ad th aual dstac drv by dvdual housholds. Ths s do by a framwork frst troducd by Dub ad McFadd (984, whr th cosumr th frst stag chooss th typ of car ad th scod stag th dstac drv. Gv th stmatd paramtrs of ths dmad fuctos, th mpact of a cras of ful prcs o th choc of th cars, th car travl dmad ad th ful dmad ca b smulatd. Th modl allows also to smulat th ffct of dmographc chags, lk th chags th spatal structur or th ag structur of th populato. Th survy s basd o data from Swtzrlad. Du to data avalablty ad th modllg framwork, so far oly housholds wth cars agd lss that 4 moth wr xamd. Kywords Ful dmad Prvat trasportato sctor Car typ dmad Travl dmad I

4 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007. Itroducto Th shar of CO mssos of th trasportato sctor o th total CO mssos s about 40% for Swtzrlad. Dspt th fact that Swss govrmt has aoucd ts dsr to rduc CO mssos of th trasportato sctor to a lvl of 8% blow th lvl of 990 by 00, th mssos wr about 9% abov th lvl of 990. O polcy for rachg th targt lvl 00 s a ful tax. I ths work th ffct of such a tax s xamd. I arlr studs, thr th short-ru or th logru lastcts of ful dmad wr xamd, mostly wthout cludg car stocks th modls. For th short-ru lastcty of ful dmad, car stocks ca b cosdrd to b costat. I th log ru, car stocks ca adapt to w prcs ad, thrfor, th log ru prc lastcty should b gratr that th short ru prc lastcty. To smulat th ful dmad for th yar 00 for dffrt lvls of ful taxs, a modl should clud th ffcts o th car spcfc ful cosumpto pr klomtr: It s assumd that f th cars cosum lss ful pr klomtr also th dmad of ful wll b lss. Furthrmor a modl should clud som dmographc mpacts o car choc ad travl dmad, sc ths mpacts ca chag ovr tm. Exampls of rlvat dmographc varabls ca b, f houshold typ s a rtrd coupl, a sgl houshold or a famly ad whthr thy lv a urba or a coutrysd ara. I prcpl th modl should also clud th scod had car markt ad th choc of a houshold o th umbr of cars. For smplcty ad du to data avalablty, th modl wll oly clud cars that ar ot oldr tha 4 moth. It s assumd that smulatos rsults for th ffct of a ful tax, wll b rprstatv for th whol st of cars. Th modl usd ths papr xplas th dmad of car travl dstac of dvdual housholds. Th ful dmad ca b calculatd multplyg th car travl dstac by th avrag ful cosumpto pr klomtr of th car of th houshold. Th modl cluds th ful prc, car attrbuts ad socodmographc attrbuts as xplaatory varabls for th car travl dmad. Th modl a s basd o th framwork frst postulatd by Dub ad McFadd. I ths framwork, th bhavour of a houshold s assumd to b as follows: Th houshold dcds to buy o car th frst stag ad a scod stag how may klomtrs to drv wth t pr yar. I th frst stag th houshold ca choos amog dffrt typs of cars. Th houshold th taks to accout th choc of a crta car ad th chooss th cosumpto lvl of all goods cludg th umbr of klomtrs t would drv by ths car. It appls ths procdur to all car modls avalabl ad th raks th cars accordg to th utlty lvl. It wll th choos th car that s o th top of th rakg. Th outcom of ths dcso procss s what s assumd to b obsrvd th data. For smplcty a frst stp oly housholds who buy a w car ar cosdrd. It wll tur out, that ths bhavour ca b capturd by th followg: For a ovrvw o th mpact of th ag ad th com o travl dmad s Budsamt für Statstk (007b, pag 8. Or at last: A smulato basd o ths subst cludg oly th ffct of a ful tax o th us of th car wll udrstmat ad a smulato cludg both th choc of th car ad th us of t wll ovrstmat th mpact of a ful tax o ful dmad. Thrfor, a uppr ad a lowr boud for th ffct of a ful tax o ful cosumpto ca b calculatd.

5 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 β p α max v ( p, y r, b, s, ε, + β ( y r + α β p + γ s + δ b + β (,,,, x x p y r s ε α p + β y r + γ s + δ b + ε, (.. whr y s th com of houshold, r s th fx costs of th car typ, ad, (.. p s th cost pr klomtr drvg that dpds strogly o th ful prc, th socodmographc varabls dotd s, ad th car attrbuts dotd b. Th socodmographc varabls s cota amog othr varabls th umbr of popl of th houshold ad th typ of ara whr th houshold lvs. Th car attrbuts cota varabls lk comfort attrbuts ad sz. Th radom trms ad ε rprst uobsrvd socodmographc varabls, uobsrvd car attrbuts ad masurmt rrors. Th radom trms ar assumd to b dpdt ad dtcally-dstrbutd radom varabls that ar corrlatd wth th radom trm ε. Both ad ε hav ma zro. Th fucto (,,,,, v p y r b s ε s a drct utlty fucto ad dcats th lvl of utlty a houshold ca rach gv ts com y ad th cost pr klomtr drv p wh choosg th car typ. Houshold wll th choos th car typ for whch hs drct utlty fucto wll yld th hghst valu. Th fucto (,,,, would drv wth car typ. x p y r s ε dscrbs th umbr of klomtrs pr yar th houshold Th crucal coomtrc problm s that th xpctd valu of ( g s ot zro ay mor: ε wh houshold chooss car typ E ε I 0. Th raso for ths dvato from zro s bcaus opto s oly chos for crta combatos of th rror trms. Sc ad ε ar corrlatd, ot all valus of ε hav th sam probablty lk th ucodtod cas ad thrfor th xpctd valu of ε gv th choc s ot zro. Dub ad McFadd show ow, that udr som assumptos o th dstrbuto of th rror trms ad ca b calculatd a smpl way. It ca b show that wh ( ( ( ε g ε th valu of E I ε s rplacd by ε E ε I g + υ th stmatd paramtrs α, β, γ ad δ ar asymptotcally cosstt wh stmatg th modl (.. by OLS. 3 I chaptr th modl of Dub ad McFadd wll b prstd ad adaptd to th problm of ths ( ε papr. It s also show how th valu of E I g ca b calculatd. I chaptr 3 th paramtrs of ths modl wll b stmatd for housholds wth cars agd lss that 4 moth usg Swss Data. Furthr thr s show, how th xpctd chag of total ful ca b calculatd for a gv scaro, lk a cras of th ful prc for xampl. I chaptr 4 cotas th coclusos of ths papr ad th futur rsarch plas o ths topc. 3 It wll b show, that ths ca b do by stmatg (.. by th maxmum lklhood mthod frst. Th th corrcto trm ca b calculatd usg th stmatd valus a th data. 3

6 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007. Th dscrt/cotuous stmato Modl. Itroducto of th Modl I ths chaptr all th lmts of th modl of Dub ad McFadd ar drvd. Th prcpal dffrc to ordary two stag modls wth slcto bas as ca b foud Maddala (983 s that th choc of th fuctoal form for th dtrmstc compot for th choc part ad th cotuous part s ot arbtrary ay mor. I th modl of Dub ad McFadd, th fuctoal form of th dtrmstc compot of th cotuous part s a Marshalla dmad fucto ad th o of th choc part s th corrspodg drct utlty fucto. Thrfor th fuctoal forms th modl of Dub ad McFadd comply to th codtos of a mcrocoomc dmad systm. I th followg, frst th modl s drvd for th most smpl fuctoal form of a Marshalla dmad fucto. Th rsult wll b slghtly dffrt of th o obtad by Dub ad McFadd, sc t wll b adaptd to th problm formulatd abov. 4 Aftr that, som assumptos for th commo stochastc trms ar mad ad out of ths, th rsultg corrcto trms for th rgrsso modl ar calculatd.. A dmad systm wth a lar Marshalla dmad fucto 5 I ths modl th dmad for drvg a aual dstac gv th choc of a crta car shall b xplad. Th dmad for othr goods s ot cosdrd. Ths task s quvalt to th dmad of th amout of a cosumr good, gv th choc of a crta budl of captal good. Th dmad for drvg a aual dstac dpds as wll o coomc as o socodmographc varabls. I th modl, t s assumd that thr xst oly two goods: Good o, th dmad for drvg a aual dstac ad good two, th umrar good that cotas all th rmag budl of goods. 6 A dmad fucto that dpds larly o th coomc varabls p, y ad r - th cost pr klomtr drvg by car typ.., th com of houshold.. ad th aual captal costs of th car - as wll as o th socodmographc varabls s ad th car attrbuts b ts most smpl fuctoal form s gv by: (,,, x x p y r s + υ α p + β y r + γ s + δ b + υ, (... 4 Th modl would b dtcal to th o of Dub ad McFadd, f t would hav b assumd that th klomtrs could hav drv by cars usg dffrt typs of ful, lk gasol or atural gas drv or v by bful cars: Dub ad McFadd xam th dmad for th housholds for lctrcty ad atural gas, gv th choc of a hous ad watr hatg systm, whch s thr a atural gas or a lctrcty basd systm or a combato of t. 5 Ths prstato follows th ma ls Hausma (98. 6 Rmd that Dub ad McFadd th dmad of two goods s xamd: Th dmad for lctrcty ad atural gas. 4

7 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Rmd that th prc p ad th com ( y r ar xprssd uts of th umrar prc. Th umrar prc s th prc dx of th budl cotag all goods apart from th dmad o klomtrs. Th prc p corrspods to th margal costs of a klomtr drvg ad dpds o th car typ ad th avrag ful prc durg th prod of usg th car. Th com y t th captal costs of th car typ, r, s usd for th com of th dmad systm. Th stochastc trm υ cotas uobsrvd socodmographc varabls s% ad car attrbuts b %, υ υ ( s, b% %. Th dtrmstc part of th choc modl s rprstd by th drct utlty fucto that corrspods to th Marshalla dmad fucto of th cotuous dmad part of th modl. Ths mas t s assumd that th houshold calculats th maxmum of utlty gv car typ, dos ths for all car typs ad th chooss th car that ylds th hghst utlty. Ths utlty calculato mpls that th houshold rcalculats all dmad goods wh havg a look at th dffrt cars ad that t s a ordary mcrocoomc utlty maxmzato calculato. Thrfor, th rsultg utlty, gv a car, ca b calculatd by computg a drct utlty fucto. Thrfor ths drct utlty fucto must corrspod to th Marshalla dmad fucto ad comply to th codtos of a mcrocoomc dmad systm. Th drct utlty fucto follows: Th startg pot s as follows: Frst a utlty lvl u 0 s dfd:, u,, 0 v p y s b y y r., Thrfor, thr must xst a combato of prcs p ad coms y houshold rmas qual 0 u v p y p s b (,,, 0. v p,,,, y r s b ca b calculatd as u. Hc, thr must xst a fucto such that th utlty of th y p, such that Th fucto y ( p ca b dtrmd by us of th followg total dffrtal: v ( p, y, s, b v ( p, y, s, b dy + 0. p y dp Trasformg ths total drvatv o gts: dy v p, y, s, b / p. dp v p, y, s, b / dy Applyg Roy's Thorm 7 ylds: 7 x ( p, y, s v p, y, s, b / p. Rmd that du to th fact, that thr s oly o good of trst ths problm, th v p, y, s, b / y dx o s lft out. Rmd also that th prc of good o, p, ad th com y ar masurd uts of p, a prc dx cotag all goods apart of good o. Th thorm of Roy s stll vald: Proof: Cosdr a dmad systm wth prcs 5

8 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 dy v p y s b p. p v p dy s b y (,,, / α p + β y + γ s + δ b + υ,,, / Solvg ths frst ordr homogous dffrtal quato ylds: 8 β p y ( p c α + γ s + δ b + υ α p β β. Choosg c 9 u0 ad solvg for 0 u o gts th drct utlty fucto: α (,,,,,, p v p y r s b υ υ β y r + + γ s + δ b + υ + α p β β. Sc drct utlty fuctos ar dfd up to a postv trasformato, th followg fucto s also fasbl. 0 β α (,,,,,, p v p y r s b υ υ + β y r + α β p + γ s + δ b + υ + υ β Trasformg ths xprsso lads to th followg:. xprssd uts of prc of th good, p%. Th thorm of Roy s th: x ( p, y, s ( ( v p, y, s, b / p v p, y, s, b / y. Wh applyg th thorm of Roy takg th trasformd prcs p p / p ad th trasformd com y y% / p%, t ca b show that th thorm rmas vald: x ( p, y, s v p, y, s, b / p v p%, y%, s, b / p% p% / p v p%, y%, s, b / p% p v p%, y%, s, b / p% v p, y, s, b / y v p%, y%, s, b / y% y% / y v p%, y%, s, b / y% p v p%, y%, s, b / y%. dy 8 α p + δ y + γ s + δ b. Th soluto of th homogous dffrtal quato dy δ y dp dp partcular soluto of th dffrtal quato y ( p P yp p δ α p + γ s p s: Ths soluto s obtad by applyg th followg gral soluto: p s: y ( p c δ α yp p + γ s α p. δ δ ( p H. Th yp yp p a + bp, b. By comparg th p coffcts th costats a ad b ca b dtrmd. 9 Applyg Roy's thorm o th drct utlty fucto that follows from ths assumpto o ca s that th Marshalla dmad fucto s rsultg. Thrfor t s fasbl to assum c u0. 0 Th thorm of Roy s stll ot volatd ths cas, sc f f ( z, f ( z > 0, z > 0 th v ( p, y f ( v ( p, y s a postv trasformato of v ( p, y, Th v ( p y f ( v ( p y x ( p y ( ( v p, y / p,,,,, v p, y / y v p, y / p f v p, y / p f v p, y / v p, y v p, y / p v p, y / p. v p, y / y f v p, y / y f v p, y / v p, y v p, y / y v p, y / y ( ( 6

9 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 β p α v ( p, y, r, s, b, υ, υ + β ( y r + α β p + γ s + δ b + υ + υ β β p Th stochastc trm υ rprsts also uobsrvd socodmographc varabls s% ad uobsrvd car attrbuts b %, but such os that ar flucg oly th choc of car typs whl as s% ad b % may cota varabls that fluc oly th dmad of drvg or both th dmad of drvg ad th choc of th car typ. Ths mas that th stochastc vctors s% ad b % may cota som compots that ar also cotad s% ad b %, but th vctors s% ad b % cota addto som varabls that oly flucs th dmad for drvg. Thrfor, th stochastc varabls υ υ ( s, b% υ ( s b% υ,. % ad % ar corrlatd. A xampl for a uobsrvd varabl that oly flucs th choc of th car s th shap of th car (stat car or lmous, f ths varabl s ot cotad th data. A xampl for a uobsrvd varabl that flucs both th choc of th dstac ad th choc of th car mght b uobsrvd attrbuts of th car, lk th tsty of oss sd th car wh drvg t..3 Th applcato of th dmad systm th modl of Dub ad McFadd Th dmad systm drvd abov s smlar to th two stag modl of Hckma (979. Sc th stochastc trms of th choc ad th dmad modl ar corrlatd, also ths modl a corrcto trm must b addd for stmatg asymptotcally cosstt paramtrs for th Marshalla dmad fucto. Th modl s dfd as follows β p α max v ( p, y, r, s, b, υ, υ max + β ( y r + α β p + γ s + δ b + υ + υ β (.3. β p (,, x x p y s + υ α p + β y r + γ s + δ b + υ. (.3. To dtrm ths corrcto trm th commo dstrbuto of th two stochastc trms plays a crucal β rol. Rwrtg th stochastc trms as (, (, p υ s b % + υ s% b % ca b wrtt as: % % ad ε υ ( s, b β p α max v ( p, y, r, s, b, υ, υ max + β ( y r + α β p + γ s + δ b + β (,, x x p y s + υ α p + β y r + γ s + δ b + ε. th modl,, 7

10 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Th stochastc trm dpds both o uobsrvd varabls that may fluc th dmad for drvg ad th choc of th cars. ε cotas varabls that fluc th dmad for drvg, but t may also cota varabls that fluc both th dmad for drvg ad for choosg a car typ. I ach cas, th stochastc trms ad ε ar ot dpdt, sc thy ar fuctos of som commo uobsrvd varabls. Th problm that th stochastc trm dpds also o p s tratd furthr blow. Frst, som assumptos ar mad ordr to smplfy th modl structur. Th commo dstrbuto of th stochastc varabls ad ε ad υ dpd o th form of th fuctos υ ( g g ad o what varabls ar cosdrd as argumts. To smplfy th modl, th followg spcal cas s of partcular trst: ( s, 0 ( s ε υ υ % υ %. I ths cas, th stochastc compot of th dmad fucto, ε, dpds oly o uobsrvd socodmographc varabls. If t s furthr assumd that th varato of th margal costs for drvg, p, btw dffrt car typs th choc st s small or at last dos oly cotrbut a small β shar of th varato of th trm (, p υ s + υ s% b% fluc of p o % %, t s rasoabl to glct th. Sc for th choc modl, th utlty fucto s oly dfd up to a postv trasformato, o could subtract th trm υ ( s ( s b% υ, p % β from. Thrfor, bcoms %. Sc th stochastc trms ad ε ar stll drv by som commo varabls, or at last som varabls that ar corrlatd, thy ar stll corrlatd. For ths spcal cas th modl s as follows: β p α max v ( p, y r, b, s, ε, max + β ( y r + α β p + γ s + δ b + β (,,, x x p y s ε α p + β y r + γ s + δ b + ε, ( wth: υ ( s, b% %, ε υ ( s %., (.3.3 It ca b assumd that th houshold do ot valuat all th cars, but oly cars that ar closd to th optmal catgory. Thrfor t s rasoabl to assum that th varato margal costs of drvg s rathr small. υ ( s, b% %, ε υ υ ( s, 0 υ ( s % %. Rmd that th stochastc vctors s% ad b % may cota som compots that ar also cotad s% ad b %. 3 ot, that oly th car travl dstac drv by th car typ chos,, ca b obsrvd. 8

11 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Sc vry smplfcato dcrass th powr of th modl, t must b dscussd, f th smplfcato s rasoabl ad what ts ffcts could b. I ths cas th assumpto that th stochastc compot of th dmad fucto dpds oly o uobsrvd socodmographc varabls dos ot sm to caus a larg dvato from th ralty, bcaus t sms ralstc that obsrvd prfrcs of th housholds fluc th dmad of th drvg dstac much mor tha uobsrvd car attrbuts lk uobsrvd comfort attrbuts. Th assumpto that th varato of margal costs amog th β dffrt cars ca b glctd th rror trm (, p υ s + υ s b ad ca oly b ustfd, wh assumg that th varato of varato of υ ( s, b% s mor problmatc υ p s% β s much bggr tha th %. Th assumpto that th stochastc compot of th utlty fucto dpds o uobsrvd socodmographc varabls sms also plausbl, sc th car choc dpds strogly o cosumr prfrcs. It sms also rasoabl to assum that thr ar uobsrvd socodmographc varabls that fluc both th dmad o dstac ad th choc of th car typ. O xampl would b that a houshold wth strog prfrcs for drvg also has strog prfrcs for a comfortabl car. 4 To sum up, th modl structur proposd abov (quatos (.3.3 ad (.3.4 sms to b rasoabl ad t wll b much asr to stmat th paramtrs tha th modl (quatos (.3. ad (.3. frst proposd..4 Th corrcto trm for th dstac dmad modl Th corrcto trm for th dstac dmad quato s cssary, bcaus of a slcto bas problm, whch mas that th rror trm of th dmad fucto dpds o th choc of th car typ. Wh glctg ths fact ad stmatg th paramtrs wthout ay corrcto trm, th stmatd paramtrs would b basd. Thrfor, a corrcto trm must b addd to th stmato ordr to gt asymptotcally cosstt stmators. I ths scto th corrcto trm for th dstac dmad modl s calculatd. Th cocpt of drvg th corrcto trm s smlar to th cass dscrbd Maddala (983 chaptr 8 ad 9. I ordr to calculat th corrcto trm for ths modl, som addtoal assumptos o th commo dstrbuto of ad ε assumptos accordg to Dub ad McFadd ar: 5 a. Th stochastc trms,.., ar dpdt ad dtcally Gumbl dstrbutd: ar cssary. Ths F ( π γ λ 3. 4 Fd bttr xampl, sc ths rathr mas that prfrcs for dstac ar corrlatd wth prfrcs for comfortabl cars. 5 Vkma (003, pag 3 ad Dub ad McFadd, pag 35. 9

12 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Th paramtr λ s a dstrbuto paramtr ad th costat γ s th Eulr-Maschro- costat. Th xpctato valu of ths dstrbuto s E ( 0 ad th varac s var b. Th codtoal xpctato valu of ε 7 gv g s: 8 σ E [ ε ] R g, R corr ( ε, λ c. Th codtoal varac of ε gv g s: var g. [ ε ] σ R d. Th corrlato btw ε ad, E [ ε ] 0 ad σ var [ ε ], R corr ( ε,., fulfls th followg proprts: λ. 6 R < ad R 0. Th stochastc trm ε ca, thrfor, b splt a compot dpdg o ad to a compot υ : [ ] ε E ε + υ g. µ ( x η γ 6 S also B Akva (985, pag 04: If x s Gumbl dstrbutd wth F ( x, th: E ( x η + ad µ var ( x π. 6µ 7 Th xprsso [ ] E ε g mas, th xpctd valu of ε, gv that th houshold has chos th car typ. Rmd also that th datast oly x - whr s th car typ chos by th houshold - ca b obsrvd. 8 Rmd that from th assumpto of larty ε α ad dpdc of ad for all, t follows that E σ [ ε g ] R ad [ ] λ cov ε, k cov α k k, α k cov k, α cov, α var k k var α corr [ ε, k ], whr var ( as dfd abov ad var [ ε ] var [ ε ] Dub ad McFadd assumd larty ε α + υ, wth υ dpdt from λ assumptos c. ad d. mpos som addtoal rstrctos o th paramtrs σ. Thrfor t sms that bhd. From ths t followd b.. Th α. Th sam assumpto, ε α + υ, wth υ dpdt from, s mad by Brhard, Bolduc ad Bélagr (996, pag 97. 0

13 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 From assumpto a. t follows that th codtoal xpctato valu of gv that th houshold has chos th opto, E I g, s qual to: ( l ( P ( θ l P falls E I ( P θ P falls g, θ 3 λ. 9 π Th paramtr λ s a arbtrary paramtr that dtrms th varac of, s also a.. By pluggg E I ( g [ ] xprsso: E ε g b., o gts aftr som rformulato th followg ( ( σ 6 P E I ( g R l ( P ( R l ( P ( π.. \ P. A quvalt rsult s: 0 ( P ( P ( σ 6 l E I ( g R ( P ( δ, δ,f, 0, f δ. π Out of ths th followg xprsso for x rsults: ( P ( P ( σ 6 l x α p + β ( y r + γ s + δ b + R ( P ( δ + υ, (.4. π wth ( P ( P ( σ 6 l E ε I R ( P ( δ g, δ,f, 0, f δ. π For th probablts P ( th stmatd probablts ( P ar substtutd: Pˆ ( Vˆ Vˆ αˆ ˆ ˆ V β y r + + γ s + δ b βˆ. (.4. ˆ, wth ˆ β p ˆ 9 For th drvato of ths xprsso s th attachmt. Th formula ca also b foud Dub ad McFadd o pag A quvalt rsult o ca gt aftr som rformulatos usg R 0 R R : ( P ( P ( π..\ σ 6 l E ε ( I g R P δ, δ,f, δ 0, f.

14 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Th paramtrs αˆ, βˆ, γˆ adδ ˆ ar th paramtr valus that ar stmatd th frst stp, wth th multomal logt modl (ML. I th scod stp, th paramtrs of th stmato quato (.4. ca b calculatd. Th corrcto trm has to b calculatd usg th stmatd probablts Pˆ ( form th frst stp. ( Pˆ ( Pˆ ( σ 6 l x α p + β ( y r + γ s + δ b + R ( Pˆ ( δ + υ. (.4.3 π Th paramtrs of th quato (.4.3 ca b stmatd usg OLS ad wll b asymptotcally cosstt. Md that th varacs of th stmatd paramtrs form OLS ar ot corrct. Ths would d to b stmatd by a procdur dscrbd Dub (98. A dscrpto of th Multomal Logt Modl s closd th attachmt. Dub (98, Two-Stag Sgl Equato Estmato Mthods: A Effccy Comparso, mmo, Massachustts Isttut of Tchology, 98.

15 Swss Trasport Rsarch Cofrc Sptmbr - 4, Data ad mprcal rsults 3. Data dscrpto Th data usd to stmat th modl coms from a survy of th Swss Fdral Statstcal Offc (FSO. Ths survy s prformd vry fv yars. About 30'000 radomly draw housholds ar trvwd by a tlpho survy. Th qustoar cotas a walth of formato o houshold travllg bhavour, rsdc charactrstcs ad a umbr of houshold charactrstcs. For ths stmato, th datast of th survy of th yar 000 was usd. For th varabl costs pr klomtr of a car typ, th avrag ful prc durg th prod th car drv was usd as a proxy. Sc all th housholds ar facd wth th sam ful prc at a crta dat, th dffrc of th avrag ful prc durg th prod th car was usd btw cars oldr tha thr yars would bcom vry small. 3 I addto to ths, t s ot ay mor crta ough, that oldr cars wr bought as w from th houshold ad thr s o formato o wh th car was bought ad about th prc. I addto, for cars bought th yar or th yar bfor th survy th moth of matrculato s avalabl. Ths allows to calculat th prod th car was usd ad, thrfor, also to calculat th avrag ful prc vry accuratly. 4 Thrfor th sampl s rstrctd to cars bought th yar or th yar bfor th survy. Th aual dstacs drv by ths cars s calculatd by dvdg th valu of th odomtr by th prod th car was usd. To dstgush car typs oly th varabl g sz catgory s avalabl for th datast of th yar Th catgors ar: Catgory o for th g sz smallr or qual to '350 ccm, th always 300 ccm stps up to 550 ccm, ad catgory sx for g sz gratr tha 550 ccm. For ths catgors, avrag aual fx costs whr calculatd by usg Swss car mport statstcs 6 ad data o car costs 7. Apart from fxd costs thr ar o car typ spcfc attrbuts avalabl th data. Thrfor, th trm δ b wll b skppd th stmato. 3 Assumg that th dat of survy for two housholds s u st 000. O houshold has a car bought 994 th othr has o bought 995. It s obvous, that th avrag ful prc durg th prod wot dffr a lot from th o of prod I cotradcto to ths, th dffrc would b much hghr, f th cars wr bought 998 ad 999 rspctvly. 4 Th day of th tlpho survy s also kow. 5 For th datast of th survy th yar 005 th xact g sz ad th 6 Always th top four car modls of th statstcs of car owrshp, Budsamt für Raumtwcklug (00, ar cosdrd. 7 For calculatg th fx costs of ths four cars, th followg data sourcs of costs wr usd: Tourg Club dr Schwz (007a ad Tourg Club dr Schwz (007b wr usd. Th valus for th avrag fx costs wr foud by a wghtd avrag of ths costs. Th wghts wr chos accordg to th umbr of cars mportd th yar

16 Swss Trasport Rsarch Cofrc Sptmbr - 4, Estmato of th dscrt cotuous Modl Th modl that wll b stmatd s as dfd th prvous chaptr. 8 Frst th choc modl wll b stmatd. β p α max v ( p, y r, b, s, ε, max + β ( y r + α β p + γ s + δ b + β.(3.. Sc th rror trms ar d Gumbl dstrbutd, th modl s a stadard Multomal Logt Modl (ML, that s solvd by th Maxmum Lklhood mthod: α, β, γ, δ ( P ( ( δ ( P ( max δ l + l, (3.. wth V α P V β y r + + γ s + δ b β β p, V ad δ,f, δ 0, f.th varabl dcats th choc of houshold. I th scod stp, th followg modl wll b stmatd by OLS mthod. ( Pˆ ( Pˆ ( σ 6 l x α p + β ( y r + γ s + δ b + R ( Pˆ ( δ + υ, (3..3 π wth ˆ ( Pˆ P bg th smulatd choc probablts from th frst stag, Vˆ Vˆ αˆ ˆ ˆ V β y r + + γ s + δ b βˆ. ˆ, ˆ β p ˆ For stmato, th followg socodmographc varabls wr cludd th modl: A dummy for lvg a dtachd hous famh, a dummy for owg a scod flat wg, th umbr of popl th houshold hhazpr ad th typ of ara agglotyp, whr typ dcats a cty ctr, lvg a agglomrato of a cty, 3 a small cty ad 4 coutrysd ara. Th varabl com s rprstd by y ad r r rprsts th fxd costs of car typ, that s assumd ot to varat btw th housholds for a gv car typ. Th varabl y r s calld _fk. For th varabl costs pr klomtr of a car typ, p, th avrag ful prc durg th prod th car was drv as a proxy, d_bp95, rspctvly for th choc modl, th ful prc two moths bfor buyg th car was usd as a proxy for what th cosumr assum of th futur ptrol prc wh thy valuat th car choc, B_bp34. Sc thr was o dummy for dsl g cars avalabl ad dsl cars ar oly a small shar of th cars, th prc for uladd ful 95 was tak as a proxy for th ful prc. Th car typs ca oly b dstgushd by th g sz catgors o to sx. 8 S quatos (.3.3, (.3.4 ad th vrso cludg th corrcto trm (.4., (

17 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 To calculat th probablts Pˆ ( th paramtrs of a larsd vrso of th choc modl (3.. was stmatd: ( max a b y r cp ds ot that th framwork has chagd du to th larsato, th paramtrs a, b, c, d ar ow calculatd dpdtly from th paramtrs α, β, γ, δ of quato ( ad ar ust usd to calculat probablts Pˆ (. Modl: Multomal Logt umbr of stmatd paramtrs: umbr of obsrvatos: 669 umbr of dvduals: 669 ull log-lklhood: It log-lklhood: Fal log-lklhood: Lklhood rato tst:85.69 Rho-squar: Adustd rho-squar: 0.07 Fal gradt orm: Varac-covarac: from aalytcal hssa Sampl fl: R:\Mkrozsus_000\MZV_Matlab\ML_Auto_tst\uwag_AzAut vr.dat Rob. Std rr Rob. t- tst Rob. p-val am Valu Std rr t-tst p-val ASC_ 0 fxd ASC_ ASC_ * * ASC_ * * ASC_ * * ASC_ * * B_azp * * B_azp * * B_bp * * B_bp * * B_ * * B_ Tabl : Estmato rsults of th choc modl Th varabls ASC_ ar altratv spcfc costats, a. ot, that o of ths costats a has to b st costat. Th oly socodmographc varabl cludd ths quato s th umbr of popl th houshold. I ths modl, th paramtr d, B_azp, was rstrctd as follows: d d, d d ad d5 d6. ot, that also for th paramtr d at last o compot has to b st to zro. 3 4 I ths cas, t was d d 0. Paramtr b, B_, s ow varatg btw th altratvs, sc th fx costs do ot varat much btw th car typ catgors ad t s assumd that th 9 Thr wr o socodmographc varabls cludd th stmato. Thrfor th trm γ s was skppd. Ths has to b chagd. Sc thr ar o car typ spcfc attrbuts avalabl, also th trm δ b was skppd. 30 It should b chckd, f ths s fasbl. It sms that Vkma (003 dd t th sam way. 5

18 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 com s a crucal varabl, wh choosg th car typ. 3 For paramtr b, th sam rstrctos hold lk for paramtr d. Th rsults show that th paramtrs show th xpctd sg: Th umbr of prsos th houshold has a postv fluc o th probablty of choosg a car wth a largr g sz, sc th lattr s postvly corrlatd wth th car sz. Th sam holds for th com. Th ptrol prc has a gatv fluc o choosg bg g szs. Th paramtrs ar ot all sgfcat. Th raso could b, that th car catgors that ar xplad ar a bad crtro to dstgush cars. A scod xplaato s that du to th lack of avalablty of car attrbuts, th varato of th rror trms s vry hgh compard to th varato of th dtrmstc part varato of th stmatd paramtrs s hgh. a + b y r + cp + d s. Thrfor, th For stmatg th paramtrs of th dmad fucto of drvg, for ach houshold th choc probablts for ach opto, Pˆ (, has to b calculatd ordr to comput th corrcto trm l ( Pˆ ( Pˆ ( ( Pˆ ( δ. Th modl to stmat s th: ( Pˆ ( Pˆ ( σ 6 l x α p + β ( y r + γ s + δ b + R ( Pˆ ( δ + υ, (3..4 π whr 6 sc sx car typs ar dstgushd th datast. Estmato of th paramtrs by OLS ylds: Sourc SS df MS umbr of obs F( 3, Modl Prob > F Rsdual R-squard Ad R-squard Total Root MSE d_km_p_a_~ Cof. Std. Err. t P> t [95% Cof. Itrval] _fk d_bp agglotyp agglotyp agglotyp hhazpr famh wg c c c c c _cos Tabl : Estmato rsults of th travl dstac dmad modl 3 It should also b chckd, f ths s fasbl. It sms that Vkma (003 dd t th sam way. 6

19 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 σ 6 ot that π R s ukow ad has to b stmatd. Du to th rstrcto R 0 oly th paramtrs R... R 5 ca b stmatd. 3 As a proxy for th margal cost of drvg, th avrag ful prc durg h prod th car was drv s. Apart form ths, th sam varabls wr usd lk th choc modl. Th rsults show that most stmatd paramtrs hav th xpctd sg: Th com of th houshold t th fx cost of th car has a postv fluc o car drvg dmad. Th plac of lvg has also a sgfcat fluc o drvg dmad: Houshold that lv agglomratos ad housholds that lv coutrysd aras hav a sgfcat hghr dmad for car drvg tha houshold lvg cts. Th dffrc btw houshold lvg small tows ad popl lvg cts s ot sgfcat. Th owrshp of a dtachd hous has a sgfcat gatv mpact o drvg dmad. It sms that popl lvg a dtachd hous mor oft stay at hom stad of vstg placs thr spar tm. Th owrshp of a scod flat dos ot hav a sgfcat mpact. Th sgs of th corrcto trm show that th hghr th probablty of choosg a car wth a largr g, th hghr th dmad for drvg th car. Ths sms rathr plausbl sc popl wth hghr prfrcs for drvg th cars may also hav a hghr prfrc for largr cars, sc ths cars ar mostly mor comfortabl. Th mpact of th avrag ful prc o car drvg dmad s gatv, but ufortuatly s ot sgfcat. Th raso for t could b th lack of varato th avrag ful prcs btw th housholds or th low sstvty to ful prcs of th housholds th short ru. For smulato, th valus of varabls should frst b pluggd th choc modl for calculatg ( ˆ ( ( ( V aˆ + b ( y r + cp ˆ + dˆ ( ( ( ( ( s wth th valus y, r, p, s rprst th put valus of th smulato ad aˆ, bˆ, cˆ, d ˆ rprst th paramtrs that wr stmatd usg th valus of th datast. ( P ( Usg ( V ( V ad l ( ( ( th smulatd choc probablts ad th corrcto ( P ( P ( ( P ( δ trm for th dmad modl ca b calculatd for ach houshold. By pluggg th put valus of th smulato ad th corrcto trm 3 σ 6 σ 6 σ 6 σ 6 σ 6 R R R R R R π π π π π ( ( ( ( l ( ˆ σ 6 P ( l R ( ˆ ( Pˆ ( P ˆ ( δ ( Pˆ. δ π P ( ˆ P ( ( ˆ ˆ ˆ σ 6 l P 6 l 6 l ˆ σ P ˆ σ P R ( ˆ P δ ˆ R ˆ ˆ P δ R P δ π P π P π P Thrfor th varabls paramtrs R... R 5. ( ˆ ˆ ( ˆ ˆ l P l P ( Pˆ ( δ P P P 7 ( ˆ δ ( C hav to b usd to stmat th

20 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 ^ ( ( ( ( l ( l ˆ ( ( ( ( ( ˆ σ 6 P ( P ( x ˆ ˆ α p + β y r + γ s + δ b + R ( P ( δ ( P δ π ( ( P ( P ( wth 6 th dmad for drvg car for ach car that ca b chos has to b stmatd. ( ( ( P xk By calculatg D, whr k dots th ful cosumpto pr klomtr of car typ, th xpctd ful dmad for th smulatd valus, D (, ca b calculatd ad compard to th ful dmad calculatd from th data: D x ( ( ( ( k. Rmark: Th formula D P ( x k ylds th xpctd log ru ffct of a polcy, sc t cluds th chag of th car stocks. Th formula ( ( short ru xk D would yld th short ru ffct of a polcy, sc t s assumd, that th car stock rmas costat. Thrfor, th log ru ffct of a cras of th ful prcs should b gratr that th short ru ffct. Bcaus th paramtrs for th ful prcs of th modl stmat wr ot sgfcat, o smulato so do so far. 8

21 Swss Trasport Rsarch Cofrc Sptmbr - 4, Coclusos, op Qustos ad futur rsarch plas Whr as othr studs lk Vkma (003 showd a gatv rlato btw ful prcs a us of cars, th stmato rsults for th data usd ths papr could ot show a sgfcat rlato btw ful prcs a car us. Th raso sms to b th data that s avalabl. Sc th dffrcs of th avrag ful prcs th houshold ar facd wth com oly from a dffrc of th prod of us of cars, ths dffrcs ar small. Thr would b also dffrcs ful prcs th dffrt rgos of Swtzrlad, but ths prcs ar ot rcordd. Morovr sc Swtzrlad s a small coutry popl could asly buy thr ful at a plac a rgo whr th ful prcs ar lowr. Thrfor o data o th actual ful prcs th housholds actually pad ar avalabl ad th data calculatd s lk mtod small. Aothr raso for th sgfcat rlatoshp btw btw ful prcs a us of cars could b, that th aual klomtrs drv s calculatd from rportd data, th odomtr valu, that ca b rathr accurat. O possblty to rduc ths problm would b to fd a rul to lmat at last th outlrs. For stac th rportd dstac drv th last yar could b usd for such a rul. Aothr possblty would b, to clud mor socodmographc formato o th housholds ordr to rduc th varac of th rror trm of th stmato problm. O th othr had, th calculatd stadard rrors of th stmator ar basd ad thrfor th calculatd t-valus could b wrog. A procdur to calculat ths stadard rror as dscrb Dub (98 should b mplmtd. Th datast form th survy th yar 005 wll cota mor formato o socodmographc varabls. Furthr thr wll b th car brad ad car modl b avalabl for most th cars. Ths wll allow for dstgushg btw mor car typs, usg mor car attrbuts ad havg mor accurat data o th fx ad varabl cost of th cars. Ths should yld mor accurat corrcto trms for th dmad of car travl quato. Ths, togthr wth th cluso of mor socodmographc varabls, would lad to a lowr th varac of th rror trm of th dmad of car travl quato. Thrfor, th stadard rror of th stmatd paramtrs, for stac also th paramtr of th ful prc, would dcras. Wh stmatg th modl wth ths data, t wll bcom clar, f th rsults ar mor satsfyg. Cosdrg th thortcal modl usd t s uclar, how much rror th larzato causs. Ths should b xamd. Furthr th othr two ways of stmatg th modl prstd Dub ad McFadd should mplmtd ad th th rsults should b compard. Aothr problm s, that ths papr t was assumd that ach houshold ows o car ad ust chooss th typ of cars. Ths s oly tru for about 50% of th housholds. Th problm of dcdg whthr to buy o or svrals cars car or ot to ow a car at all was ot cosdrd. Ths problm should b cludd th xt modl. Th rol of masurmt rrors has also to b xamd. 9

22 Swss Trasport Rsarch Cofrc Sptmbr - 4, Rfrcs Ammya, Taksh, 985, Advacd Ecoomtrcs, Oxford, Basl Blackwll, 986. B-Akva, Mosh ad Stv Lrma, 985, Dscrt Choc Aalyss: Thory ad Applcato to Travl Dmad, Th MIT Prss, 985. Brard, a-thomas; Ds Bolduc; Doald Bélagr, 996, Qubc Rsdtal Elctrcty Dmad: A Mcrocoomtrc Approach, Th Caada oural of Ecoomcs / Rvu caad d'ecoomqu, Vol. 9, o.. (Fb.,996, pp Budsamt für Statstk, 00a, Mkrozsus 00 zum Vrkhrsvrhalt, Budsamt für Statstk, uburg (Schwz, 00. Budsamt für Statstk, 00b, Mobltät dr Schwz Ergbss ds Mkrozsus 005, Budsamt für Statstk, uburg (Schwz, 00. Budsamt für Raumtwcklug, 00, Fahrlstug dr Schwzr Fahrzug, Ergbss dr prodsch Erhbug Fahrlstug (PEFA, Budsamt für Raumtwcklug, (Schwz, 00, Camro, Adra Col ud Prav K.Trvd, 006, Mcrocoomtrcs : mthods ad applcatos, Cambrdg : Cambrdg Uvrsty Prss, 006. Dub, ffry A., 98, Two-Stag Sgl Equato Estmato Mthods: A Effccy Comparso, mmo, Massachustts Isttut of Tchology, 98. Dub, ffry A. ud Dal L. McFadd, 984, A Ecoomtrc Aalyss of Rsdtal Elctrc Applac Holdgs ad Cosumpto, Ecoomtrca, Vol. 5, o., St Gr, Wllam H., 003, Ecoomtrc Aalyss, Uppr Saddl Rvr, : Prtc Hall, 003, 5th Edto. Hausma, rry A., 98, Exact Cosumr's Surplus ad Dadwght Loss, Amrca Ecoomc Rvw, Vol. 7, o. 4. (Sp., 98, S Hckma, ams, 976, "Th Commo Structur of Statstcal Modls of Trucato, Sampl Slcto ad Lmtd Dpdt Varabls ad a Smpl Estmator for Such Modls," Th Aals of Ecoomc ad Socal Masurmt, 5 (976, Hckma, ams, 977, Sampl Slcto Bas as a Spcfcato Error wth a Applcato to th Estmato of Labor Supply Fuctos, BER Workg Papr # 7, March, 977 (rvsd. Hckma, ams, 979, Sampl Slcto Bas as a Spcfcato Error, Ecoomtrca, Vol. 47, o.. (a., 979, pp Hogg ud Grag (995, Itroducto to mathmatcal statstcs,uppr Saddl Rvr (.., Prtc Hall, 995. hl, Goffry A., Advacd mcrocoomc thory, Eglwood Clffs, Prtc Hall, 99. ohso,., ud S. Kotz, 97, Dstrbuto Statstcs: Cotuous Multvarat Dstrbutos, w York: oh Wly & Sos. 0

23 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Mas-Colll, Adru, Adru Mas-Colll, Mchal D. Whsto, ad ffry R. Gr, 995, Mcrocoomc Thory, w York, Oxford Uvrsty Prss, 995. Maddala, G.S., 983, Lmtd-dpdt ad Qualtatv Varabls, Cambrdg: Cambrdg Uvrsty Prss, 983. Molchaov Ilya, 005, Wahrschlchktsthor, Vorlsugsskrpt Uvrstät Br, Wtrsmstr , lya/wk.html Molchaov Ilya, 007, Kombatork ud Wahrschlchktsrchug,Vorlsugsskrpt Uvrstät Br, Sommrsmstr 007, lya/wk.html Ols, Radall., 980, A Last Squars Corrcto for Slctvty Bas, Ecoomtrca, Vol. 48, o. 7. (ov., 980, S Roy, A. D., 95, Som Thoughts o th Dstrbuto of Eargs, Oxford Ecoomc Paprs, w Srs, Vol. 3, o.. (u., 95, S Tourg Club dr Schwz (TCS, 007a, Klomtrkost, CD ROM, Tourg Club dr Schwz (TCS, 007. Tourg Club dr Schwz (TCS, 007b, TCS Autokatalog, Vkma, Fracs, 003, Chox d véhculs t dmad d klométrag : u approch mcroécoométrqu, Mémor, Uvrsty Laval.

24 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Appdx

25 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 A Calculato of th xpctato valu of th rror trm of th dmad quato For calculatg th xpctato valu of th rror trm of th dmad quato ε gv th choc s, E ε I s, th xpctato valu of th rror trm of th choc quato E I s has to b calculatd frst. 33 As wll b show by th followg calculatos, two cass has to b dstgushd: Th cas s ad th cas s. Frst, th modl s prstd aga. A. Th Choc Modl Th choc modl s dfd as follows: U V +, * * * I s, f U > U, s. s Th radom varabls ar dpdt ad dtcally Gumbl dstrbutd. Th dstrbuto fucto ad th dsty fuctos ar: F α + β α + β (, f α α + β. ow, th xpctato valu I ( wll b calculatd. E I s shall b calculatd. Wthout loss of gralty th cas Th codtoal xpctato valu ts gral form s dfd as follows: [ A] E X [ ] P ( A E X I A, 34 I A f ω A 0 f ω A Rmd: E σ σ ε R E ε I ( s R E ( I ( s. λ λ S also chaptr.4 Th corrcto trm for dstac dmad modl or Dub ad McFadd pag S Molchaov (007, pag 5. Rmark: E [ X A] ( ω ( ω ( ( ω [ ] P ( A E X I A may also b wrtt as (!!! E X I A E X ω A ω, whr ω s a lmt of th probablty spac Ω, ω Ω. P A 35 S Molchaov (005, pag. 3

26 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 A. Probablty for I ( Startg from ths dfto frst th probablty for I (, P I has to b dfd. Ths probablty wll b usd th followg calculatos. Th varabl dcats th umbr of mutually xclusv choc optos. Th fucto I ( ca altratvly b dfd as: ( I, f < V V +,. It follows that ( P I ca b dfd as follows: V V + V V + ( (,..., P I f d d d. 36 Sc th radom varabls,..., b wrtt as follows: f (,..., f ( xprsso f,..., ca ar dpdt th commo dsty fucto. Thrfor, th tgral abov ca b smplfd to th ( ( + P I f F V V d. Isrtg for th th dsty fucto f ( ad th dstrbuto fucto F ( ylds ( α ( V V l α + β + + α + β. 37 P I α d Ths tgral ca b trasformd so that th argumt s aga a Gumbl dsty fucto 36 Rmarks: a. Th varabls,.. ar usd as radom varabls ad as valus. Sorry... b. f ( x f ( x,, f ( s wrtt as f f. Th sam holds for th dstrbuto fucto. 37 ( α ( V V + + β α + β α ( V V + + β α + β α + β α + β P I α d α d α ( ( β V α β V + + α ( V V α + β + α + β + l + α + β α + β α β + α d α d. α d 4

27 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 V V α l + β + l + α + β + + α ( V V P ( I ( + α d. α ( V V + ( F ( F (. + + A.3 Cas s : Th codtoal xpctato valu E I Th codtoal xpctato valu E I ca b calculatd as follows: ( ( ( ( (, whl ( I ( E I E I P I P ( I ( has alrady b drvd, th xprsso E ( I ( ow. 38 ( V V + V V, { } (, f I 0,ls. Sc V V + has to b calculatd E I f,..., d f,..., d d d < + Ths tgral ca b smplfd th sam way as th calculato of ( V V + K V V, { } V V + V V + ( K ( V V + f,..., d f,..., d Kd d < + K f d Kd d V V + V V + f f d Kd d V V + α β α ( V V + f ( xp d α ( V V V V + α + β + l f ( xp d ( V α + β + l + α + β d α. Trasformg ths tgral ylds aga a dsty fucto as tgrad: P I abov: 38 Th xprsso E ( I ( mas th xpctato valu of gv that all valus of ar th st whr th opto o s chos. 5

28 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 V l V α + β + l + α + β + + α ( V V E ( I ( + α d. Substtutg ( V V z a β l α + + ylds39 z α ( V V ( z V z E ( I ( + z + β + l + dz α z α ( V V ( z V z + β + l + + z d α α ( V V + ( V V β l α γ. α Rplacg P ( I + ( V V α ad pluggg for th paramtrs th valus accordg to th assumptos of Dub ad McFadd (984, α π ad β γ, ylds: λ 3 λ ( 3 E ( I ( P ( I γ l ( P ( I + γ P I l P I π π λ 3 γ Wth ths rsult E [ I ] ca b dtrmd λ 3 E s I P I λ 3 E ( I ( π l ( P ( I. P I π ( l ( P ( I P ( I, 4 39 ( V V ( V V z l α α α β + + z + β + l + d dz. α α z z z z z z E ( I ( a ( z + l ( a dz a l ( a a z dz z z 40 a l a a E Z a l a a E Z a l a a a l a a γ [ ] [ ] γ l ( γ P I P I P I 4 Ths rsult s dtcally to th rsult of Dub ad McFadd, pag 35. 6

29 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 A.4 Cas s : Th codtoal xpctato valu E I ow, th codtoal xpctato valu of, gv th choc of altratv two, E I, shall b calculatd. Sc th xprsso for P ( I s kow, oly th xprsso ( ( E I ds to b calculatd. Th dcator fucto ladg to th tgral boudars s ow ( I, f < V V +,. Hc, th followg xprsso E ( ( I V V + 3 V V3 + 4 V V4 + V V + K V V, 3 4 { } f,..., d f,..., d Kd d d d. 4 3 < + Du to th dpdc of th radom varabls,..., ths tgral ca b smplfd as follows: E ( ( I V V + 3 V V3 + 4 V V4 + V V + K V V, 3 4 { } V V + 3 V V3 + 4 V V4 + V V + ( f,..., d f,..., d Kd d d d 4 3 < + K 3 4 V V + ( f f F V V + d d. 3 Sc th tgral f chagd: 4 f d Kd d d d 4 3 V V + d caot solvd xplctly, th tgral boudars hav to b ( ( 3 V V + E I f f F V V + d d ( f f F V V + d d. 3 V V + 4 Th chag of th boudars of th tgral ar basd o th followg trasformato: > + V V + > + V < V V +. 7

30 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 For solvg E ( I ( V V + ( f F V V + d 3 α ( V V + + β f ( xp ( d V V + α ( V V + + β f ( xp d V V + 3 V V + 3 α + β α + β α ( V + + β α xp ( xp d 3 th followg xprsso ca b calculatd frst: α + β + l α β α β α xp d V V + + l α + β α β + 3 α xp + d V V + α + β α + β α xp + d 3 V V + α + β α + β α xp + d 3 V V + α β l α + β 3 α xp d V V + l + l V α α + β + + α β l α xp d V V + l V α β α α β l α xp d V V α V α V V xp d α + β + l + V α α + β + l α xp V V + l V α + β + + α α + β + l d 8

31 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Th tgrad s aga a Gumbl dsty fucto ad thrfor th dstrbuto fucto s kow. Du to ths, th tgral boudars ca b srtd to th dstrbuto fucto F ( xp xp α + β + l +. 3 F ( F ( V V + xp xp α ( V V + + β + l +. 3 Th xprsso V V + α ( V V + + β f ( xp ( d 3 s ow dtrmd: l V V α ( V V + + β 3 f ( xp ( d xp. 3 α + + β + + V V + Isrtg th rsult to th xprsso E ( I ( ylds ( ( 3 ( 3 V V + α ( V V + + β + l + 3 f ( xp V V + E I f f F V V + d d f f F V V + d d d V β l f ( xp d. Th xprsso f ( d s th ucodtod xpctato valu E [ ] valus Dub ad McFadd ths xpctato valu s [ ] α ( V V + + β + l + 3 f ( xp. For th paramtr E 0. I that cas th tgral abov ca b smplfd as follows: V β l E ( I ( f ( xp d d 9

32 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 α β V l α + β α + β 3 xp d + α ( V V + l + α + β α + β 3 xp d α + β α V + β α α ( V V xp d α + β α + β α ( V V xp d V α + α 3 + β α + β xp d α 3 + β α + β xp α + β α + β xp d P ( I α ( ( + β α + β l P I xp ( d d l( P( I ( α + β l( P( I ( α + β l( P( I ( xp d α + β l( P( I ( α + β l( P( I ( P ( I ( xp d. 30

33 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 Usg th substtuto ( P ( z α + β l, tgrad ca b trasformd to a Gumbl dsty fucto z z ( γ ( z β l ( P ( + α P ( z l ( P ( xp dz + β α α z z P ( β l ( P ( z xp + ( dz α α P ( β l ( P ( +. α ad d dz α th Pluggg th paramtr valus from Dub ad McFadd, α π ad β λ 3 γ, th xprsso abov bcoms 43 z z ( P ( I ( z l P ( I xp dz α + β α α z z P ( I ( β l ( P ( I ( z xp + ( dz α α ( P ( I ( γ l P ( I ( + γ λ 3 π λ 3 ( P ( I ( l P ( I ( E I. π Thrfor E I E bcoms ( I ( ( ( E I ( P I ( ( λ 3 P ( I ( l P ( I ( π P I ( ( P ( I λ 3 λ 3 P I l ( P ( I ( l ( P ( I (. π π 43 S Dub ad McFadd, pag 35. 3

34 Swss Trasport Rsarch Cofrc Sptmbr - 4, 007 It rmas ow to calculat E I : ( ( ( ( ( ( E I P I λ 3 E I ( l ( P ( I ( P I P I π λ ( P ( I ( P ( I ( 3 λ 3 l l π π ( ( P ( I λ 3 λ 3 P I l ( P ( I ( l ( P ( I (. π π 44 A.5 Th gral soluto of th codtoal xpctato valu: By chagg th dxs wh dog th calculatos prstd abov, th followg gral soluto ( E I s ca b drvd: λ 3 l ( P ( I ( s for s π E I ( s. 3 P ( I ( λ l ( P ( I ( s for s π P ( I ( 44 Ths rsult s qual th rsult Dub ad McFadd, pag 35. Rmark: Th rsult satsfs E P I E I + P I E I 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( P ( I P ( I P ( I. Rmark: λ 3 λ 3 P I P ( I ( l ( P ( I P ( I ( l ( P ( I π + π λ 3 λ 3 P I P ( I ( l ( P ( I l ( P ( I P ( I ( π + π λ 3 λ 3 P I P ( I ( l ( P ( I + π π P I E I + P I E I l ( P ( I ( P ( I. 3