The Multiple Discrete-Continuous Extreme Value (MDCEV) Model: Formulation and Applications

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1 Th ultpl Dscrt-Contnuous Extrm alu (DCE) odl: Formulaton and Applcatons Chandra R. Bhat Th Unvrsty of Txas at Austn Dpartmnt of Cvl Archtctural & Envronmntal Engnrng Unvrsty Staton C76 Austn Txas Tl: Fax: Emal: and Navn Eluru Th Unvrsty of Txas at Austn Dpartmnt of Cvl Archtctural & Envronmntal Engnrng Unvrsty Staton C76 Austn Txas Tl: Fax: Emal:

2 ABSTRACT any consumr choc stuatons ar charactrzd by th smultanous dmand for multpl altrnatvs that ar mprfct substtuts for on anothr. A smpl and parsmonous ultpl Dscrt-Contnuous Extrm alu (DCE) conomtrc approach to handl such multpl dscrtnss was formulatd by Bhat (5) wthn th broadr uhn-tucr (T) multpl dscrt-contnuous conomc consumr dmand modl of Wals and Woodland (983). In ths chaptr th focus s on prsntng th basc DCE modl structur dscussng ts stmaton and us n prdcton formulatng xtnsons of th basc DCE structur and prsntng applcatons of th modl. Th papr xamns svral ssus assocatd wth th DCE modl and othr xtant T multpl dscrt-contnuous modls. Spcfcally th papr dscusss th utlty functon form that nabls clarty n th rol of ach paramtr n th utlty spcfcaton prsnts dntfcaton consdratons assocatd wth both th utlty functonal form as wll as th stochastc natur of th utlty spcfcaton xtnds th DCE modl to th cas of prc varaton across goods and to gnral rror covaranc structurs dscusss th rlatonshp btwn arlr T-basd multpl dscrt-contnuous modls and llustrats th many tchncal nuancs and dntfcaton consdratons of th multpl dscrt-contnuous modl structur. Fnally w dscuss th many applcatons of DCE modl and ts xtnsons n varous flds. ywords: Dscrt-contnuous systm ultpl dscrtnss uhn-tucr dmand systms xd dscrt choc Random Utlty axmzaton.

3 . INTRODUCTION Svral consumr dmand chocs rlatd to travl and othr dcsons ar charactrzd by th choc of multpl altrnatvs smultanously along wth a contnuous quantty dmnson assocatd wth th consumd altrnatvs. Exampls of such choc stuatons nclud vhcl typ holdngs and usag and actvty typ choc and duraton of tm nvstmnt of partcpaton. In th formr cas a houshold may hold a mx of dffrnt nds of vhcl typs (for xampl a sdan a mnvan and a pc-up) and us th vhcls n dffrnt ways basd on th prfrncs of ndvdual mmbrs consdratons of mantnanc/runnng costs and th nd to satsfy dffrnt functonal nds (such as bng abl to travl on wnd gtaways as a famly or to transport goods). In th cas of actvty typ choc and duraton an ndvdual may dcd to partcpat n multpl nds of rcratonal and socal actvts wthn a gvn tm prod (such as a day) to satsfy varty sng dsrs. Of cours thr ar svral othr travlrlatd and othr consumr dmand stuatons charactrzd by th choc of multpl altrnatvs ncludng arln flt mx and usag carrr choc and transacton lvl brand choc and purchas quantty for frquntly purchasd grocry tms (such as coos rady-toat crals soft drns yoghurt tc.) and stoc slcton and nvstmnt amounts. Thr ar many ways that multpl dscrt stuatons such as thos dscussd abov may b modld. On approach s to us th tradtonal random utlty-basd (RU) sngl dscrt choc modls by dntfyng all combnatons or bundls of th lmntal altrnatvs and tratng ach bundl as a compost altrnatv (th trm sngl dscrt choc s usd to rfr to th cas whr a dcson-mar chooss only on altrnatv from a st of altrnatvs). A problm wth ths approach howvr s that th numbr of compost altrnatvs xplods wth th numbr of lmntal altrnatvs. Spcfcally f J s th numbr J of lmntal altrnatvs th total numbr of compost altrnatvs s ( ). A scond approach to analyz multpl dscrt stuatons s to us th multvarat probt (logt) mthods of anchanda t al. (999) Baltas (4) Edwards and Allnby (3) and Bhat and Srnvasan (5). In ths multvarat mthods th multpl dscrtnss s handld through statstcal mthods that gnrat corrlaton btwn unvarat utlty maxmzng modls for sngl dscrtnss. Whl ntrstng ths scond approach s mor of a statstcal sttchng of unvarat modls rathr than bng fundamntally drvd from a rgorous undrlyng utlty maxmzaton modl for multpl dscrtnss. Th rsultng multvarat modls also do not

4 collaps to th standard dscrt choc modls whn all ndvduals choos on and only on altrnatv at ach choc occason. A thrd approach s th on proposd by Hndl (999) and Dub (4). Ths rsarchrs consdr th cas of multpl dscrtnss n th purchas of multpl varts wthn a partcular product catgory as th rsult of a stram of xpctd (but unobsrvd to th analyst) futur consumpton dcsons btwn succssv shoppng purchas occasons (s also Walsh 995). Durng ach consumpton occason th standard dscrt choc framwor of prfctly substtutabl altrnatvs s nvod so that only on product s consumd. Du to varyng tasts across ndvdual consumpton occasons btwn th currnt shoppng purchas and th nxt consumrs ar obsrvd to purchas a varty of goods at th currnt shoppng occason. In all th thr approachs dscussd abov to handl multpl dscrtnss thr s no rcognton that ndvduals choos multpl altrnatvs to satsfy dffrnt functonal or varty sng nds (such as wantng to rlax at hom as wll as partcpat n out-of-hom rcraton). Thus th approachs fal to ncorporat th dmnshng margnal rturns (.. sataton) n partcpatng n a sngl typ of actvty whch may b th fundamntal drvng forc for ndvduals choosng to partcpat n multpl actvty typs. Fnally n th approachs abov t s vry cumbrsom vn f concptually fasbl to nclud a contnuous choc nto th modl (for xampl modlng th dffrnt actvty purposs of partcpaton as wll as th duraton of partcpaton n ach actvty purpos). Wals and Woodland (983) proposd two altrnatv ways to handl stuatons of multpl dscrtnss basd on sataton bhavor wthn a bhavorally-consstnt utlty maxmzng framwor. Both approachs assum a drct utlty functon U(x) that s assumd to b quas-concav ncrasng and contnuously dffrntabl wth rspct to th consumpton quantty vctor x. Consumrs maxmz th utlty functon subjct to a lnar budgt constrant whch s bndng n that all th avalabl budgt s nvstd n th consumpton of th goods; that Th approach of Hndl and Dub can b vwd as a vrtcal varty-sng modl that may b approprat for frquntly consumd grocry tms such as carbonatd soft drns crals and coos. Howvr n many othr choc occasons such as tm allocaton to dffrnt typs of dscrtonary actvts th tru dcson procss may b bttr charactrzd as horzontal varty-sng whr th consumr slcts an assortmnt of altrnatvs du to dmnshng margnal rturns for ach altrnatv. That s th altrnatvs rprsnt nhrntly mprfct substtuts at th choc occason. Th assumpton of a quas-concav utlty functon s smply a manfstaton of rqurng th ndffrnc curvs to b convx to th orgn (s Daton and ullbaur 98 pag 3 for a rgorous dfnton of quas-concavty). Th assumpton of an ncrasng utlty functon mpls that U(x ) > U(x ) f x > x.

5 s th budgt constrant has an qualty sgn rathr than a sgn. Ths bndng natur of th budgt constrant s th rsult of assumng an ncrasng utlty functon and also mpls that at last on good wll b consumd. Th dffrnc n th two altrnatv approachs proposd by Wals and Woodland (983) s n how stochastcty non-ngatvty of consumpton and cornr solutons (.. zro consumpton of som goods) ar accommodatd as brfly dscussd blow (s Wals and Woodland 983 and Phanuf t al. for addtonal dtals). Th frst approach whch Wals and Woodland labl as th Ammya-Tobn approach s an xtnson of th classc mcroconomc approach of addng normally dstrbutd stochastc trms to th budgt-constrand utlty-maxmzng shar quatons. In ths approach th drct utlty functon U(x) tslf s assumd to b dtrmnstc by th analyst and stochastcty s ntroducd post-utlty maxmzaton. Th justfcaton for th addton of such normally dstrbutd stochastc trms to th dtrmnstc utlty-maxmzng allocatons s basd on th noton that consumrs ma rrors n th utlty-maxmzng procss or that thr ar masurmnt rrors n th collcton of shar data or that thr ar unnown factors (from th analyst s prspctv) nfluncng actual consumd shars. Howvr th addton of normally dstrbutd rror trms to th shar quatons n no way rstrcts th shars to b postv and lss than. Th contrbuton of Wals and Woodland was to dvs a stochastc formulaton basd on th arlr wor of Tobn (958) and Ammya (974) that (a) rspcts th unt smplx rang constrant for th shars (b) accommodats th rstrcton that th shars sum to on and (c) allows cornr solutons n whch on or mor altrnatvs ar not consumd. Thy achv ths by assumng that th obsrvd shars for th (-) of th altrnatvs follow a truncatd multvarat normal dstrbuton (not that snc th shars across altrnatvs hav to sum to on thr s a sngularty gnratd n th -varat covaranc matrx of th shars whch can b accommodatd by droppng on altrnatv). Howvr an mportant lmtaton of th Ammya-Tobn approach of Wals and Woodland s that t dos not account for cornr solutons n ts undrlyng bhavor structur. Rathr th constrant that th shars hav to l wthn th unt smplx s mposd by ad hoc statstcal procdurs of mappng th dnsty outsd th unt smplx to th boundary ponts of th unt smplx. Th scond approach suggstd by Wals and Woodland whch thy labl as th uhn- Tucr approach s basd on th uhn Tucr or T (95) frst-ordr condtons for constrand random utlty maxmzaton (s Hanmann 978 who uss such an approach vn 3

6 bfor Wals and Woodland). Unl th Ammya-Tobn approach th T approach mploys a mor drct stochastc spcfcaton by assumng th utlty functon U(x) to b random (from th analyst s prspctv) ovr th populaton and thn drvs th consumpton vctor for th random utlty spcfcaton subjct to th lnar budgt constrant by usng th T condtons for constrand optmzaton. Thus th stochastc natur of th consumpton vctor n th T approach s basd fundamntally on th stochastc natur of th utlty functon. Consquntly th T approach mmdatly satsfs all th rstrctons of utlty thory and th stochastc T frst-ordr condtons provd th bass for drvng th probablts for ach possbl combnaton of cornr solutons (zro consumpton) for som goods and ntror solutons (strctly postv consumpton) for othr goods. Th sngularty mposd by th addng-up constrant s accommodatd n th T approach by mployng th usual dffrncng approach wth rspct to on of th goods so that thr ar only (-) ntrdpndnt stochastc frst-ordr condtons. Among th two approachs dscussd abov th T approach consttuts a mor thortcally unfd and bhavorally consstnt framwor for dalng wth multpl dscrtnss consumpton pattrns. Howvr th T approach dd not rcv much attnton untl rlatvly rcntly bcaus th random utlty dstrbuton assumptons usd by Wals and Woodland ld to a complcatd llhood functon that ntals mult-dmnsonal ntgraton. m t al. () addrssd ths ssu by usng th Gw-Hajvasslou-an (or GH) smulator to valuat th multvarat normal ntgral apparng n th llhood functon n th T approach. Also dffrnt from Wals and Woodland m t al. usd a gnralzd varant of th wll-nown translatd constant lastcty of substtuton (CES) drct utlty functon (s Polla and Wals 99; pag 8) rathr than th quadratc drct utlty functon usd by Wals and Woodland. In any cas th m t al. approach l th Wals and Woodland approach s unncssarly complcatd bcaus of th nd to valuat truncatd multvarat normal ntgrals n th llhood functon. In contrast Bhat (5) ntroducd a smpl and parsmonous conomtrc approach to handl multpl dscrtnss also basd on th gnralzd varant of th translatd CES utlty functon but wth a multplcatv log-xtrm valu rror trm. Bhat s modl labld th multpl dscrt-contnuous xtrm valu (DCE) modl s analytcally tractabl n th probablty xprssons and s practcal vn for stuatons wth a larg numbr of dscrt consumpton altrnatvs. In fact th DCE modl 4

7 rprsnts th multnomal logt (NL) form-quvalnt for multpl dscrt-contnuous choc analyss and collapss xactly to th NL n th cas that ach (and vry) dcson-mar chooss only on altrnatv. Indpndnt of th abov wors of m t al. and Bhat thr has bn a stram of rsarch n th nvronmntal conomcs fld (s Phanuf t al. ; von Hafn t al. 4; von Hafn 3; von Hafn 4; von Hafn and Phanuf 5; Phanuf and Smth 5) that has also usd th T approach to multpl dscrtnss. Ths studs us varants of th lnar xpndtur systm (LES) as proposd by Hanmann (978) and th translatd CES for th utlty functons and us multplcatv log-xtrm valu rrors. Howvr th rror spcfcaton n th utlty functon s dffrnt from that n Bhat s DCE modl rsultng n a dffrnt form for th llhood functon. In ths chaptr th focus s on prsntng th basc DCE modl structur dscussng ts stmaton and us n prdcton formulatng xtnsons of th basc DCE structur and prsntng applcatons of th modl. Accordngly th rst of th chaptr s structurd as follows. Th nxt scton formulats a functonal form for th utlty spcfcaton that nabls th solaton of th rol of dffrnt paramtrs n th spcfcaton. Ths scton also dntfs mprcal dntfcaton consdratons n stmatng th paramtrs n th utlty spcfcaton. Scton 3 dscusss th stochastc form of th utlty spcfcaton th rsultng gnral structur for th probablty xprssons and assocatd dntfcaton consdratons. Scton 4 drvs th DCE structur for th utlty functonal form usd n th currnt papr and xtnds ths structur to mor gnral rror structur spcfcatons. For prsntaton as Sctons through 4 consdr th cas of th absnc of an outsd good. In Scton 5 w xtnd th dscussons of th arlr sctons to th cas whn an outsd good s prsnt. Scton 6 provds an ovrvw of mprcal applcatons usng th modl. Th fnal scton concluds th papr.. FUNCTIONAL FOR OF UTILITY SPECIFICATION W consdr th followng functonal form for utlty n ths papr basd on a gnralzd varant of th translatd CES utlty functon: x U ( x ) ψ () 5

8 whr U(x) s a quas-concav ncrasng and contnuously dffrntabl functon wth rspct to th consumpton quantty (x)-vctor x (x for all ) and ψ and ar paramtrs assocatd wth good. Th functon n Equaton () s a vald utlty functon f ψ > and for all. Furthr for prsntaton as w assum tmporarly that thr s no outsd good so that cornr solutons (.. zro consumptons) ar allowd for all th goods (ths assumpton s bng mad only to stramln th prsntaton and should not b construd as lmtng n any way; th assumpton s rlaxd n a straghtforward mannr as dscussd n Scton 5). Th possblty of cornr solutons mpls that th trm whch s a translaton paramtr should b gratr than zro for all. 3 Th radr wll not that thr s an assumpton of addtv sparablty of prfrncs n th utlty form of Equaton () whch mmdatly mpls that non of th goods ar a pror nfror and all th goods ar strctly Hcsan substtuts (s Daton and ullbaur 98; pag 39). Addtonally addtv sparablty mpls that th margnal utlty wth rspct to any good s ndpndnt of th lvls of all othr goods. 4 paramtrs Th form of th utlty functon n Equaton () hghlghts th rol of th varous ψ and and xplctly ndcats th ntr-rlatonshps btwn ths paramtrs that rlat to thortcal and mprcal dntfcaton ssus. Th form also assums wa complmntarty (s älr 974) whch mpls that th consumr rcvs no utlty from a non-ssntal good s attrbuts f s/h dos not consum t (.. a good and ts qualty attrbuts ar wa complmnts or U f x whr U s th sub-utlty functon for th th good). Th radr wll also not that th functonal form proposd by Bhat (8) n Equaton () gnralzs arlr forms usd by Hanmann (978) von Hafn t al. (4) Hrrgs t al. (4) Phanuf t al. () and ohn and Hanmann (5). Spcfcally t should b notd that th utlty form of Equaton () collapss to th followng lnar xpndtur systm (LES) form whn : 3 As llustratd n m t al. () and Bhat (5) th prsnc of th translaton paramtrs mas th ndffrnc curvs str th consumpton axs at an angl (rathr than bng asymptotc to th consumpton axs) thus allowng cornr solutons. 4 Som othr studs assum th ovrall utlty to b drvd from th charactrstcs mbodd n th goods rathr than usng th goods as sparat ntts n th utlty functon. Th radr s rfrrd to Chan (6) for an xampl of such a charactrstcs approach to utlty. Also as w dscuss latr rcnt wor by asquz and Hanmann (8) rlaxs th assumpton of addtv sparablty but at a computatonal and ntrprtaton cost. 6

9 x U ( x ) ln ψ (). Rol of Paramtrs n Utlty Spcfcaton.. Rol of ψ Th rol of good whch s: ψ can b nfrrd by computng th margnal utlty of consumpton wth rspct to U ( x) x x ψ (3) It s obvous from abov that ψ rprsnts th basln margnal utlty or th margnal utlty at th pont of zro consumpton. Altrnatvly th margnal rat of substtuton btwn any ψ two goods and l at th pont of zro consumpton of both goods s. Ths s th cas ψ rgardlss of th valus of and. For two goods and j wth sam unt prcs a hghr basln margnal utlty for good rlatv to good j mpls that an ndvdual wll ncras ovrall utlty mor by consumng good rathr than j at th pont of no consumpton of any goods. That s th consumr wll b mor lly to consum good than good j. Thus a hghr basln ψ mpls lss llhood of a cornr soluton for good. l.. Rol of An mportant rol of th trms s to shft th poston of th pont at whch th ndffrnc curvs ar asymptotc to th axs from ( ) to... ) so that th ( 3 ndffrnc curvs str th postv orthant wth a fnt slop. Ths combnd wth th consumpton pont corrspondng to th locaton whr th budgt ln s tangntal to th ndffrnc curv rsults n th possblty of zro consumpton of good. To s ths consdr two goods and wth ψ ψ.5 and. Fgur prsnts th profls of th ndffrnc curvs n ths two-dmnsonal spac for varous valus of ( > ). To compar th profls th ndffrnc curvs ar all drawn to go through th pont (8). Th 7

10 radr wll also not that all th ndffrnc curv profls str th y-axs wth th sam slop. As can b obsrvd from th fgur th postv valus of and lad to ndffrnc curvs that cross th axs of th postv orthant allowng for cornr solutons. Th ndffrnc curv profls ar asymptotc to th x-axs at y (corrspondng to th constant valu of ) whl thy ar asymptotc to th y-axs at x. Fgur also ponts to anothr rol of th trm as a sataton paramtr. Spcfcally th ndffrnc curvs gt stpr n th postv orthant as th valu of ncrass whch mpls a strongr prfrnc (or lowr sataton) for good as ncrass (wth stpr ndffrnc curv slops th consumr s wllng to gv up mor of good to obtan unt of good ). Ths pont s partcularly clar f w xamn th profl of th sub-utlty functon for altrnatv. Fgur plots th functon for altrnatv for and ψ and for dffrnt valus of. All of th curvs hav th sam slop ψ at th orgn pont bcaus of th functonal form usd n ths papr. Howvr th margnal utlts vary for th dffrnt curvs at x >. Spcfcally th hghr th valu of th lss s th sataton ffct n th consumpton of x...3 Rol of Th xprss rol of s to rduc th margnal utlty wth ncrasng consumpton of good ; that s t rprsnts a sataton paramtr. Whn for all ths rprsnts th cas of absnc of sataton ffcts or quvalntly th cas of constant margnal utlty. Th utlty functon n Equaton () n such a stuaton collapss to ψ x whch rprsnts th prfct substtuts cas as proposd by Daton and ullbaur (98) and appld n Hanmann (984) Chang (99) Chntagunta (993) and Arora t al. (998) among othrs. Intutvly whn thr s no sataton and th unt good prcs ar all th sam th consumr wll nvst all xpndtur on th sngl good wth th hghst basln (and constant) margnal utlty (.. th 8

11 hghst ψ valu). Ths s th cas of sngl dscrtnss. 5 As movs downward from th valu of th sataton ffct for good ncrass. Whn th utlty functon collapss to th form n Equaton () as dscussd arlr. can also ta ngatv valus and whn ths mpls mmdat and full sataton. Fgur 3 plots th utlty functon for altrnatv for and ψ and for dffrnt valus of. Agan all of th curvs hav th sam slop ψ at th orgn pont and accommodat dffrnt lvls of sataton through dffrnt valus of for any gvn valu.. Emprcal Idntfcaton Issus Assocatd wth Utlty Form Th dscusson n th prvous scton ndcats that ψ rflcts th basln margnal utlty whch controls whthr or not a good s slctd for postv consumpton (or th xtnsv margn of choc). Th rol of s to nabl cornr solutons though t also govrns th lvl of sataton. Th purpos of s solly to allow sataton. Thus for a gvn xtnsv margn of choc of good and nflunc th quantty of good consumd (or th ntnsv margn of choc) through thr mpact on sataton ffcts. Th prcs functonal mchansm through whch and mpact sataton ar howvr dffrnt; controls sataton by translatng consumpton quantty whl controls sataton by xponntatng consumpton quantty. Clarly both ths ffcts oprat n dffrnt ways and dffrnt combnatons of thr valus lad to dffrnt sataton profls. Howvr mprcally spang t s vry dffcult to dsntangl th two ffcts sparatly whch lads to srous mprcal dntfcaton problms and stmaton bradowns whn on attmpts to stmat both and paramtrs for ach good. In fact for a gvn ψ valu t s possbl to closly approxmat a sub-utlty functon profl basd on a combnaton of and valus wth a sub-utlty functon basd solly on or valus. In actual applcaton t would bhoov th analyst to stmat modls basd on 5 If thr s prc varaton across goods on nds to ta th drvatv of th utlty functon wth rspct to xpndturs ( ) on th goods. In th cas that for all U Σ ψ ( / p ) whr ψ s th unt prc of good. Thn U / ψ / p. In ths stuaton th consumr wll nvst all xpndturs on th sngl good wth th hghst prc-normalzd margnal (and constant) utlty ψ / p. 9

12 both th statstcal ft. 6 -profl and th -profl and choos a spcfcaton that provds a bttr 3. STOCHASTIC FOR OF UTILITY FUNCTION Th T approach mploys a drct stochastc spcfcaton by assumng th utlty functon U(x) to b random ovr th populaton. In all rcnt applcatons of th T approach for multpl dscrtnss a multplcatv random lmnt s ntroducd to th basln margnal utlty of ach good as follows: ψ ( z ) ψ ( z ) (4) whr z s a st of attrbuts charactrzng altrnatv and th dcson mar and capturs dosyncratc (unobsrvd) charactrstcs that mpact th basln utlty for good j. Th xponntal form for th ntroducton of th random trm guarants th postvty of th basln utlty as long as ψ ( ) >. To nsur ths lattr condton ψ z ) s furthr z paramtrzd as xp( β z ) whch thn lads to th followng form for th basln random utlty assocatd wth good : ψ z ) xp( β z ). (5) ( Th z vctor n th abov quaton ncluds a constant trm. Th ovrall random utlty functon of Equaton () thn tas th followng form: x U( x ) [ xp( β z )] (6) From th analyst s prspctv th ndvdual s maxmzng random utlty subjct to th ( bndng lnar budgt constrant that E whr E s total xpndtur or ncom (or som othr appropratly dfnd total budgt quantty) p x and p s th unt prc of good. 6 Altrnatvly th analyst can stc wth on functonal form a pror but xprmnt wth varous fxd valus of for th -profl and for th -profl.

13 3. Optmal Expndtur Allocatons Th analyst can solv for th optmal xpndtur allocatons by formng th Lagrangan and applyng th uhn-tucr (T) condtons. 7 Th Lagrangan functon for th problm s: p L [ xp( β z ] ) λ E (7) whr λ s th Lagrangan multplr assocatd wth th xpndtur constrant (that s t can b vwd as th margnal utlty of total xpndtur or ncom). Th T frst-ordr condtons for th optmal xpndtur allocatons (th valus) ar gvn by: xp(β z p xp(β z p ) p ) p λ f > (8) λ < f Th optmal dmand satsfs th condtons n Equaton (8) plus th budgt constrant E. Th budgt constrant mpls that only - of th valus nd to b stmatd snc th quantty consumd of any on good s automatcally dtrmnd from th quantty consumd of all th othr goods. To accommodat ths constrant dsgnat actvty purpos as a purpos to whch th ndvdual allocats som non-zro amount of consumpton (not that th ndvdual should partcpat n at last on of th purposs gvn that E > ). For th frst good th T condton may thn b wrttn as: xp( β z λ p ) p (9) Substtutng for λ from abov nto Equaton (8) for th othr actvty purposs ( ) and tang logarthms w can rwrt th T condtons as: 7 For rasons that wll bcom clar latr w solv for th optmal xpndtur allocatons for ach good not th consumpton amounts x of ach good. Ths s dffrnt from arlr studs that focus on th consumpton of goods.

14 f > < f ( 3 ) ( 3 ) whr () β z ( )ln ln p p ( 3 ). Also not that n Equaton () a constant cannot b dntfd n th β z trm for on of th altrnatvs (bcaus only th dffrnc n th from mattrs). Smlarly ndvdualspcfc varabls ar ntroducd n th s for (-) altrnatvs wth th rmanng altrnatv srvng as th bas Gnral Economtrc odl Structur and Idntfcaton To complt th modl structur th analyst nds to spcfy th rror structur. In th gnral cas lt th jont probablty dnsty functon of th trms b f( ). Thn th probablty that th ndvdual allocats xpndtur to th frst of th goods s: P( f ( d d d... ) d 3... d J... ) () whr J s th Jacoban whos lmnts ar gvn by (s Bhat 5): J h [ h ] [ h ] ; h. () Th probablty xprsson n Equaton () s a (-)-dmnsonal ntgral. Th xprsson for th probablty of all goods bng consumd s on-dmnsonal whl th xprsson for th probablty of only th frst good bng consumd s -dmnsonal. Th dmnsonalty of th 8 Ths dntfcaton condtons ar smlar to thos n th standard dscrt choc modl though th orgn of th condtons s dffrnt btwn standard dscrt choc modls and th multpl dscrt-contnuous modls. In standard dscrt choc modls ndvduals choos th altrnatv wth hghst utlty so that all that mattrs s rlatv utlty. In multpl dscrt-contnuous modls th orgn of ths condtons s th addng up (or budgt) constrant assocatd wth th quantty of consumpton of ach good that lads to th T frst ordr condtons of Equaton ().

15 ntgral can b rducd by on by notcng that th T condtons can also b wrttn n a dffrncd form. To do so dfn ~ and lt th mpld multvarat dstrbuton of th rror dffrncs b g ~ ~... ~ ). Thn Equaton () may b wrttn n th quvalnt ( 3 (-)-ntgral form shown blow: P( g( )... J ~ ~ ~ ~... ~ ~ ) d ~ d ~ ~... d ~ (3) Th quaton abov ndcats that th probablty xprsson for th obsrvd optmal xpndtur pattrn of goods s compltly charactrzd by th (-) rror trms n dffrnc form. Thus all that s stmabl s th (-)x(-) covaranc matrx of th rror dffrncs. In othr words t s not possbl to stmat a full covaranc matrx for th orgnal rror trms (... ) bcaus thr ar nfnt possbl dnsts for f(.) that can map nto th sam g(.) dnsty for th rror dffrncs (s Tran 3 pag 7 for a smlar stuaton n th contxt of standard dscrt choc modls). Thr ar many possbl ways to normalz f(.) to account for ths stuaton. For xampl on can assum an dntty covaranc matrx for f(.) whch automatcally accommodats th normalzaton that s ndd. Altrnatvly on can stmat g(.) wthout rfrnc to f(.). In th gnral cas whn th unt prcs p vary across goods t s possbl to stmat ( ) / paramtrs of th full covaranc matrx of th rror dffrncs as just dscussd (though th analyst mght want to mpos constrants on ths full covaranc matrx for as n ntrprtaton and stablty n stmaton). Howvr whn th unt prcs ar not dffrnt among th goods an addtonal scalng rstrcton nds to b mposd. To s ths consdr th cas of ndpndnt and dntcally dstrbutd rror trms for th trms whch lads to a (-)x(- ) covaranc matrx for ~ ( 3 ) wth dagonal lmnts qual to twc th valu of scal paramtr of th trms and off-dagonal lmnts qual to th scal paramtr of th trms. Lt th unt prcs of all goods b th sam (s Bhat 5; Bhat and Sn 6; Bhat t al. 6 and Bhat t al. 9 for xampls whr th wghts or prcs on th goods n th budgt constrant ar qual). Consdr th utlty functon n Equaton (6) and anothr utlty functon as gvn blow: 3

16 ~ x U [ xp{ σ ( β z )}] (4) Th scal of th rror trms n th utlty functon n th abov xprsson s σ tms th scal of th rror trms n Equaton (6). Lt σ ( ) whr s th sataton paramtr n th orgnal Equaton (6). 9 Th T condtons for optmal xpndtur for ths modfd utlty functon can b shown to b: σ σ f > ( 3 ) σ σ < f ( 3 ) whr (5) σβ z σβ z ( )ln p σ ( )ln p ln p ln p ( 3... ) ( 3... ). If th unt prcs ar not all th sam (.. th unt prcs of at last two of th goods ar dffrnt) th T condtons abov ar dffrnt from th T condtons n Equaton (). 4. SPECIFIC ODEL STRUCTURES 4. Th DCE odl Structur Followng Bhat (5 8) consdr an xtrm valu dstrbuton for and assum that s ndpndnt of z ( ). Th s ar also assumd to b ndpndntly dstrbutd across altrnatvs wth a scal paramtr of σ (σ can b normalzd to on f thr s no varaton n unt prcs across goods). Lt b dfnd as follows: β z β z ( )ln ln p ( p ln ln p ( 3... p 3... ) ) whn th - profl s usd and whn th - profl s usd. (6) 9 Not that s lss than or qual to by dfnton bcaus s lss than or qual to and th scal σ should b non-ngatv. 4

17 5 As dscussd arlr t s gnrally not possbl to stmat th form n Equaton () bcaus th trms and trms srv a smlar sataton rol. From Equaton () th probablty that th ndvdual allocats xpndtur to th frst of th goods ( ) s: ( ) σ λ σ σ σ λ σ d J P s s Λ (7) whr λ s th standard xtrm valu dnsty functon and Λ s th standard xtrm valu cumulatv dstrbuton functon. Th xprsson n Equaton (7) smplfs to a rmarably smpl and lgant closd-form xprsson. Bhat drvd th form of th Jacoban for th cas of qual unt prcs across goods whch howvr can b xtndd n a smpl fashon to accommodat th mor gnral cas of dffrnt unt prcs. Th rsultng form for th dtrmnant of th Jacoban has a compact structur gvn by: p c c c J whr. (8) Th ntgraton n Equaton (7) also collapss to a closd form xprsson provdng th followng ovrall xprsson: ( ) )! ( / / 3 c c P σ σ σ (9) In th cas whn (.. only on altrnatv s chosn) thr ar no sataton ffcts ( for all ) and th Jacoban trm drops out (that s th contnuous componnt drops out bcaus all xpndtur s allocatd to good ). Thn th modl n Equaton (9) collapss to th standard It s mportant to not that ths compact Jacoban form s ndpndnt of th assumptons rgardng th dnsty and corrlaton structur of th rror trms.

18 NL modl. Thus th DCE modl s a multpl dscrt-contnuous xtnson of th standard NL modl. Th xprsson for th probablty of th consumpton pattrn of th goods (rathr than th xpndtur pattrn) can b drvd to b: P ( x x x... x... ) p σ 3 f p f / σ / σ ( )! whr s as dfnd arlr (s Equaton 6) and f (). Th xprsson n Equaton x () s howvr not ndpndnt of th good that s usd as th frst on (s th /p trm n front). In partcular dffrnt probablts of th sam consumpton pattrn ars dpndng on th good that s labld as th frst good (not that any good that s consumd may b dsgnatd as th frst good). In trms of th llhood functon th /p trm can b gnord snc t s smply a constant n ach ndvdual s llhood functon. Thus th sam paramtr stmats wll rsult ndpndnt of th good dsgnatd as th frst good for ach ndvdual but t s stll awward to hav dffrnt probablty valus for th sam consumpton pattrn. Ths s partcularly th cas bcaus dffrnt log-llhood valus at convrgnc wll b obtand for dffrnt dsgnatons of th frst good. Thus th prfrrd approach s to us th probablty xprsson for xpndtur allocatons whch wll provd th sam probablty for a gvn xpndtur pattrn rgardlss of th good labld as th frst good. Howvr n th cas that th frst good s an outsd numrar good that s always consumd (s Scton 5) thn p and on can us th consumpton pattrn probablty xprsson or th xpndtur allocaton probablty xprsson. Not that whn for all β'z ln p. Evn f whn Equaton (9) collapss to th NL form th scal σ s stmabl as long as th utlty tas th functonal form β'z ln p and thr s prc varaton across goods. Ths s bcaus th scal s th nvrs of th coffcnt on th ln p trm (s Hanmann 984). 6

19 4. Th ultpl Dscrt-Contnuous Gnralzd Extrm-alu (DCGE) odl Structur Thus far w hav assumd that th trms ar ndpndntly and dntcally xtrm valu dstrbutd across altrnatvs. Th analyst can xtnd th modl to allow corrlaton across altrnatvs usng a gnralzd xtrm valu (GE) rror structur. Th rmarabl advantag of th GE structur s that t contnus to rsult n closd-form probablty xprssons for any and all xpndtur pattrns. Howvr th drvaton s tdous and th xprssons gt unwldy. Pnjar and Bhat (8) formulat a spcal two-lvl nstd cas of th DCGE modl wth a nstd xtrm valu dstrbutd structur that has th followng jont cumulatv dstrbuton: F S (.. ) xp xp th nst θ s s s θ s () In th abov xprsson s (... S ) s th ndx to rprsnt a nst of altrnatvs S s th s s s S s th total numbr of nsts th altrnatvs blong to and θ ( < θ ;... ) (ds)smlarty paramtr ntroducd to nduc corrlatons among th stochastc componnts of th th utlts of altrnatvs blongng to th s nst. Wthout loss of gnralty lt... S b th nsts th chosn altrnatvs blong to and lt q q... q b th numbr of chosn altrnatvs n ach of th S nsts (thus S q q... q ). Usng th nstd xtrm valu rror dstrbuton assumpton spcfd n S Equaton () (and th abov-dntfd notaton) Pnjar and Bhat (8) drvd th followng xprsson for th multpl dscrt-contnuous nstd xtrm valu (DCNE) modl: Ths rror structur assums that th nsts ar mutually xclusv and xhaustv (.. ach altrnatv can blong to only on nst and all altrnatvs ar allocatd to on of th S nsts). 7

20 Pt ( t... t...) θ θ θ q q S s s qs th {chosn alts} nst S S J s q θ S s r r s rs s r s S S θs s s θ s s s s s th nst th nst s s s q r sum( X ) ( qs rs )! () In th abov xprsson sum( X rs ) s th sum of lmnts of a row matrx for a dscrpton of th form of th matrx X rs ). X rs (s Appndx A As ndcatd n Pnjar and Bhat (8) th gnral xprsson abov rprsnts th DCNE consumpton probablty for any consumpton pattrn wth a two-lvl nstd xtrm valu rror structur. It may b vrfd that th DCNE probablty xprsson n Equaton () smplfs to Bhat s (8) DCE probablty xprsson whn ach of th utlty s s s ). functons ar ndpndnt of on anothr (.. whn θ and q and S 4.3 Th xd DCE odl Th DCGE structur s abl to accommodat flxbl corrlaton pattrns. Howvr t s unabl to accommodat random tast varaton and t mposs th rstrcton of qual scal of th rror trms. Incorporatng a mor gnral rror structur s straghtforward through th us of a mxng dstrbuton whch lads to th xd DCE (or DCE) modl. Spcfcally th rror trm may b parttond nto two componnts ζ and η. Th frst componnt ζ can b assumd to b ndpndntly and dntcally Gumbl dstrbutd across altrnatvs wth a scal paramtr of σ. Th scond componnt η can b allowd to b corrlatd across altrnatvs and to hav a htroscdastc scal. Lt η η η... η ) and assum that η s dstrbutd multvarat normal η ~ N ( Ω ). 3 ( 3 Othr dstrbutons may also b usd for η. Not that th dstrbuton of η can ars from an rror componnts structur or a random coffcnts structur or a combnaton of th two smlar to th cas of th usual mxd logt modl (s Bhat 7). 8

21 9 For gvn valus of th vctor η on can follow th dscusson of th arlr scton and obtan th usual DCE probablty that th frst of th goods ar consumd. Th uncondtonal probablty can thn b computd as: ( ) ). ( )! ( / ) ( / ) ( 3 η σ σ η σ η η df c c P (3) whr F s th multvarat cumulatv normal dstrbuton (s Bhat 5; Bhat and Sn 6; and Bhat t al. 6). Th modl n Equaton (3) can b xtndd n a concptually straghtforward mannr to also nclud random coffcnts on th ndpndnt varabls z and random-ffcts (or vn random coffcnts) n th sataton paramtrs (f th profl s usd) or th paramtrs (f th profl s usd) Htroscdastc structur wthn th DCE framwor Consdr th cas whr thr s prc varaton across th altrnatvs and th ovrall rrors ar htroscdastc but not corrlatd. Assumng a 4-altrnatv cas for as n prsntaton th htroscdastc structur may b spcfd n th form of th followng covaranc matrx for ) ( 4 3 : ) ( ω ω ω ω σ π Cov (4) whr th frst componnt on th rght sd corrsponds to th IID covaranc matrx of ) ( 4 3 ζ ζ ζ ζ ζ and th scond componnt s th htroscdastc covaranc matrx of η. Th covaranc of rror dffrncs wth rspct to th frst altrnatv s:

22 ω ω ω ω ~ ~ ~ ~ π σ Cov ( 3 4) ω ω3 ω 6 (5) ω ω4 An nspcton of th matrx abov shows only four ndpndnt quatons (th ran condton) mplyng that at most four paramtrs ar stmabl 4. Thr ar two ways to procd wth a normalzaton as dscussd blow. Th frst approach s to normalz σ and stmat th htroscdastc covaranc matrx of η (.. ω ω ω 3 and ω 4 ). Assum that σ s normalzd to ~ σ and lt th corrspondng valus of ω b ~ ω ( 3 4). Thn th followng qualts should hold basd on Equaton (5) for any normalzaton of σ to σ ~ (q π / 6 blow): ~ ~ ω qσ ω qσ ~ ~ ~ ω ω qσ ω ω qσ ( 3 4) (6) Th abov qualts can b rwrttn as: ~ ω ω qσ q ~ σ ( 3 4) (7) ~ Th normalzd varanc trms ω must b gratr than or qual to zro whch mpls that th followng condtons should hold: ~ ω qσ qσ ( 3 4) (8) Intutvly th abov condton mpls that th normalzaton on ~ σ must b st low nough so that th ovrall tru varanc of ach rror trm ( ω qσ ) s largr than q ~σ. For xampl sttng σ to would b napproprat f th tru varanc of on or mor altrnatvs s lss than π / 6. Snc th tru varanc s unnown th bst th analyst can do s to normalz σ to progrssvly smallr valus and statstcally xamn th rsults. Th scond approach s to normalz on of th ω trms nstad of th σ trm. In ths cas from Equaton (5) w can wrt: 4 Strctly spang on can stmat all th fv paramtrs (σ ω ω ω 3 and ω 4 ) bcaus of th dffrnc n th xtrm valu dstrbutons usd for ζ and th normal dstrbutons usd for η (s Walr ). Howvr th modl wll b nar sngular and t s mportant to plac th ordr/ran constrant.

23 [ ~ ω ω qσ ω ~ ω ]; 3 4. ~ ~ qσ ω qσ ω (9) Aftr som manpulatons th abov quaton may b rwrttn as: ~ ~ ω ω ω ω ; 3 4. (3) ~ Nxt mposng th condton that th normalzd trms ω must b gratr than or qual to zro mpls th followng: ~ ω ω ω ( 3 4). (3) Th abov condton s automatcally satsfd as long as th frst altrnatv s th mnmum ~ varanc altrnatv. An assocatd convnnt normalzaton s ω snc th rsultng modl nsts th DCE modl. Th mnmum varanc altrnatv can b dtrmnd by stmatng an undntfd modl wth all th ω trms and dntfyng th altrnatv wth th mnmum varanc (s Walr t al. 4 for an quvalnt procdur for a htroscdastc spcfcaton wthn th mxd multnomal logt modl). Th abov dscusson assums thr s prc varaton across goods. In th cas of no prc varaton th scal σ s not dntfabl. In ths cas th asst procdur s to normalz σ to and th ω valu for th mnmum varanc altrnatv to zro Th gnral rror covaranc structur wthn th DCE framwor Approprat dntfcaton normalzatons wll hav to placd on σ and th covaranc matrx of η whn th analyst s stmatng an rror-componnts structur to allow corrlaton n unobsrvd factors nfluncng th basln utlty of altrnatvs snc only a (-)x(-) covaranc of rror dffrncs s dntfd. Ths can b accomplshd by mposng a structur basd on a pror blfs or ntutv consdratons. Howvr th analyst must nsur that th lmnts of th assumd rstrctd covaranc structur can b rcovrd from th (-)x(-) covaranc of rror dffrncs that s actually stmabl. In th most gnral rror covaranc structur and whn thr s prc varaton on way to achv dntfcaton s th followng: () Normalz th scal paramtr σ to b a small valu such that th varanc of th mnmum varanc altrnatv xcds π σ / 6 (snc ths

24 varanc s not nown th analyst wll hav to xprmnt wth altrnatv fxd σ valus) () Normalz ω for th mnmum varanc altrnatv to zro and (3) Normalz all corrlatons of ths mnmum varanc altrnatv wth othr altrnatvs to zro. Togthr ths normalzatons lav only ( ) / paramtrs to b stmatd and ar adquat for dntfcaton. In th cas of no prc varaton an addtonal rstrcton wll hav to b mposd. On approach would b to st ω to st th scal n th covaranc matrx of η. 4.4 Th Jont DCE-Sngl Dscrt Choc odl Th DCE modl and ts xtnsons dscussd thus far ar sutd for th cas whn th altrnatvs ar mprfct substtuts as rcognzd by th us of a non-lnar utlty that accommodats a dmnshng margnal utlty as th consumpton of any altrnatv ncrass. Howvr thr ar many nstancs whr th ral choc stuaton s charactrzd by a combnaton of mprfct and prfct substtuts (prfct substtuts corrspond to th cas whr consumrs prfr to slct only on dscrt altrnatv at any choc occason; s Hanmann 984). Th DCE modl nds to b modfd to handl such a combnaton of a multpl dscrt-contnuous choc among altrnatvs as wll as a sngl choc of on subaltrnatv wthn on or mor of th altrnatvs. W do not dscuss ths cas hr du to spac constrants but th radr s rfrrd to Bhat t al. (9) and Bhat t al. (6) Th Non-Addtv DCE odl Structur asquz and Hanmann (8) hav rcntly proposd an xtnson of Bhat s addtvly sparabl lnar Box-Cox utlty functonal form (Equaton ) to ncorporat a non-addtvly sparabl quadratc Box-Cox functonal form. Usng mor flxbl non-addtv utlty structurs allows th analyst to handl both complmntarty as wll as substtuton among goods. To wrt ths gnral non-addtv form dfn μ as: x μ. Thn a non-addtvly sparabl functonal form may b wrttn as:

25 U( x ) μ ψ m θ m μ μ. m Ths s vry gnral and collapss to Bhat s addtvly sparabl form whn θ for all and m. It collapss to th translog functonal form whn for all and to Wals and Woodland s quadratc form whn for all. Th ntrprtaton of th paramtrs s not as straghtforward as n Bhat s DCE and th probablty xprssons for th consumpton of th goods and th Jacoban do not hav smpl forms. But th gan s that th margnal utlty of consumpton of a good s not only dpndnt on th amount of that good consumd but also th amount of othr goods consumd. m 5. THE ODEL WITH AN OUTSIDE GOOD Thus far th dscusson has assumd that thr s no outsd numrar good (.. no ssntal Hcsan compost good). If an outsd good s prsnt labl t as th frst good whch now has a unt prc of on. Also for dntfcaton lt ψ x ). Thn th utlty functonal form nds to b modfd as follows: ( x U( x ) xp( { } ) ( x ) xp( β z ) (3) In th abov formula w nd whl > for >. Also w nd x >. Th magntud of may b ntrprtd as th rqurd lowr bound (or a subsstnc valu ) for consumpton of th outsd good. As n th no-outsd good cas th analyst wll gnrally not b abl to stmat both and for th outsd and nsd goods. Th analyst can stmat on of th followng fv utlty forms: {( x ) } U ( x ) xp( ) x xp( β z ) x U ( x ) xp( ) xp( )ln x β z (33) 3

26 4 ) xp( ) xp( ) ( β x z x U x { } ( ) { } ) xp( )ln xp( ) ( x z x U β x { } )ln xp( )ln xp( ) ( x z x U β x Th thrd functonal form abov s stmabl bcaus th constant paramtr s obtanng a pnnng ffct from th sataton paramtr for th outsd good. Th analyst can stmat all th fv possbl functonal forms and slct th on that fts th data bst basd on statstcal and ntutv consdratons. Th dntfcaton consdratons dscussd for th no-outsd good cas carrs ovr to th wth outsd good cas. Th probablty xprsson for th xpndtur allocaton on th varous goods (wth th frst good bng th outsd good) s dntcal to Equaton (9) whl th probablty xprsson for consumpton of th goods (wth th frst good bng th outsd good) s ( ) )! ( / / 3 f p f x x x x P σ σ σ (34) whr x f. Th xprssons for n Equaton (9) and Equaton (34) ar as follows for ach of th fv utlty forms n Equaton (33): Frst form - p x z ln ) ) ln( ( β ( ); ( ) ) ln( x Scond form - p x z ln ) ln( β ( ); ( ) ) ln( x (35) Thrd form - p x z ln ) ) ln( ( β ( ); ( ) ) ln( x

27 Fourth form - Ffth form - β z ( ) ln( x ) ln p ( ); ln( x ) x β z ln( ) ln p ( ); ln( x ) 6. APPLICATIONS Th DCE modl framwor has bn mployd n modlng a numbr of choc stuatons that ar charactrzd by multpl-dscrtnss. Ths can b broadly catgorzd nto th followng rsarch aras: () actvty tm-us analyss (adults and chldrn) () houshold vhcl ownrshp (3) houshold xpndturs and (4) Anglr s st choc Actvty Tm-Us Analyss Th DCE modl that assums dmnshng margnal utlty of consumpton provds an dal platform for modlng actvty tm-us dcsons. Th dffrnt studs on actvty tm-us ar dscrbd chronologcally blow. Bhat (5) dmonstratd an applcaton of th DCE modl to ndvdual tm us n dffrnt typs of dscrtonary actvty pursuts on wnd days. Th modlng xrcs ncludd dffrnt nds of varabls ncludng houshold dmographcs houshold locaton varabls ndvdual dmographcs and mploymnt charactrstcs and day of w and sason of yar. Bhat t al. (6) formulat a unfd utlty-maxmzng framwor for th analyss of a jont mprfct-prfct substtut goods cas. Ths s achvd by usng a sataton-basd utlty structur (DCE) across th mprfct substtuts but a smpl standard dscrt choc-basd lnar utlty structur (NL) wthn prfct substtuts. Th jont modl s appld to analyz ndvdual tm-us n both mantnanc and lsur actvts usng wnd day tm-us. apur and Bhat (7) spcfcally modld th socal contxt of actvty partcpaton by xamnng th accompanmnt arrangmnt (.. company typ) n actvty partcpaton. Snr and Bhat (7) also xamnd partcpaton and tm nvstmnt n n-hom lsur as wll as out-of-hom dscrtonary actvts wth a spcfc mphass on th accompanyng ndvduals n 5 Th summary of all th studs dscussd n ths chaptr ar compld n th form of a tabl wth nformaton on th applcaton focus th data sourc usd for th mprcal analyss th numbr and labls of dscrt altrnatvs th contnuous componnt n th mprcal contxt and th DCE modl typ mployd. Th tabl s avalabl to th radrs at: 5

28 chldrn s actvty ngagmnt. Copprman and Bhat (7) formulatd a comprhnsv framwor to consdr partcpaton and lvls of partcpaton n physcally passv and physcally actv psods among chldrn on wnd days. Laonda t al. (8) focusd thr attnton on vacaton travl n USA. Spcfcally th papr xamnd how housholds dcd what vacaton travl actvts to partcpat n on an annual bass and to what xtnt gvn th total annual vacaton travl tm that s avalabl at thr dsposal. Th modls prsntd n Snr t al. (8) offr a rch framwor for catgorzng and rprsntng th actvty-travl pattrns of chldrn wthn largr travl dmand modl systms. Th papr provds a taxonomy of chld actvts that xplctly consdrs th spatal and tmporal constrants that may b assocatd wth dffrnt typs of actvts. Pnjar t al. (9) prsntd a jont modl systm of rsdntal locaton and actvty tm-us chocs. Th modl systm tas th form of a jont mxd ultnomal Logt ultpl Dscrt-Contnuous Extrm alu (NL DCE) structur that (a) accommodats dffrntal snstvty to th actvty-travl nvronmnt attrbuts du to both obsrvd and unobsrvd ndvdual-rlatd attrbuts and (b) controls for th slf slcton of ndvduals nto nghborhoods du to both obsrvd and unobsrvd ndvdual-rlatd factors. Spssu t al. (9) formulatd a panl vrson of th xd ultpl Dscrt Contnuous Extrm alu (DCE) modl that s capabl of smultanously accountng for rpatd obsrvatons from th sam ndvduals (panl) partcpaton n multpl actvts n a w duratons of actvty ngagmnt n varous actvty catgors and unobsrvd ndvdualspcfc factors affctng dscrtonary actvty ngagmnt ncludng thos common across pars of actvty catgory utlts. Pnjar and Bhat (8) proposd th DCNE modl that capturs ntr-altrnatv corrlatons among altrnatvs n mutually xclusv substs (or nsts) of th choc st whl mantanng th closd-form of probablty xprssons for any (and all) consumpton pattrn(s). Th modl stmaton rsults provd svral nsghts nto th dtrmnants of non-worrs actvty tm-us and tmng dcsons. Rajagopalan t al. (9) prdctd worrs actvty partcpaton and tm allocaton pattrns n svn typs of out-of-hom non-wor actvts at varous tm prods of th day. Th nowldg of th actvts (and th corrspondng tm allocatons and tmng dcsons) 6

29 prdctd by ths modl can b usd for subsqunt dtald schdulng and squncng of actvts and rlatd travl n an actvty-basd mcrosmulaton framwor. 6. Houshold hcl Ownrshp Th DCE framwor wth ts capablty to handl multpl-dscrtnss lnds tslf vry wll to modl houshold vhcl ownrshp by typ. Bhat and Sn (6) modld th smultanous holdngs of multpl vhcl typs (passngr car SU pcup truc mnvan and van) as wll as dtrmnd th contnuous mls of usag of ach vhcl typ. Th modl can b usd to dtrmn th chang n vhcl typ holdngs and usag du to changs n ndpndnt varabls ovr tm. As a dmonstraton th mpact of an ncras n vhcl opratng costs on vhcl typ ownrshp and usag s assssd. Ahn t al. (8) mployd conjont analyss and th DCE framwor to undrstand consumr prfrncs for altrnatv ful vhcls. Th rsults ndcat a clar prfrnc of gasoln-powrd cars among consumrs but altrnatv ful vhcls offr a promsng substtut to consumrs. Bhat t al. (9) formulatd and stmatd a nstd modl structur that ncluds a multpl dscrt-contnuous xtrm valu (DCE) componnt to analyz th choc of vhcl typ/vntag and usag n th uppr lvl and a multnomal logt (NL) componnt to analyz th choc of vhcl ma/modl n th lowr nst. 6.3 Houshold Expndturs Th DCE framwor provds a fasbl framwor to analyz consumpton pattrns. Frdous t al. (8) mployd a DCNE structur to xplctly rcognz that popl choos to consum multpl goods and commodts. odl rsults show that a rang of houshold soco-conomc and dmographc charactrstcs affct th prcnt of ncom or budgt allocatd to varous consumpton catgors and savngs. Rajagopalan and Srnvasan (8) xplctly nvstgatd transportaton rlatd houshold xpndturs by mod. Spcfcally thy xamnd th mod choc and modal xpndturs at th houshold lvl. Th modl rsults ndcat that mod choc and frquncy dcsons ar nfluncd by pror mod choc dcsons and th usr s prcpton of safty and congston. 7

30 6.4 Anglr s St Choc asquz and Hanmann (8) formulat th non-addtv DCE modl structur to study anglr st choc. In ths study thy mploy ndvdual lvl varabls such as sll lsur tm avalabl and ownrshp status (of cabn boat or R). Furthr thy undrta th computaton of wlfar masurs usng a squntal quadratc programmng mthod. 7. CONCLUSIONS Classcal dscrt and dscrt-contnuous modls dal wth stuatons whr only on altrnatv s chosn from a st of mutually xclusv altrnatvs. Such modls assum that th altrnatvs ar prfctly substtutabl for ach othr. On th othr hand many consumr choc stuatons ar charactrzd by th smultanous dmand for multpl altrnatvs that ar mprfct substtuts or vn complmnts for on anothr. Ths boo chaptr dscusss th multpl dscrt-contnuous xtrm valu (DCE) modl and ts many varants. Rcnt applcatons of th DCE typ of modls ar prsntd and brfly dscussd. Ths ovrvw of applcatons ndcats that th DCE modl has bn mployd n many dffrnt mprcal contxts n th transportaton fld and also hghlghts th potntal for applcaton of th modl n svral othr flds. Th ovrvw also srvs to hghlght th fact that th fld s at an xctng and rp stag for furthr applcatons of th multpl dscrt-contnuous modls. At th sam tm svral challngs l ahad ncludng () Accommodatng mor than on constrant n th utlty maxmzaton problm (for xampl rcognzng both tm and mony constrants n actvty typ choc and duraton modls; s Anas 6 for a rcnt thortcal ffort to accommodat such multpl constrants) () Incorporatng latnt consdraton sts n a thortcally approprat way wthn th DCE structur (th authors ar currntly addrssng ths ssu n ongong rsarch) (3) Usng mor flxbl utlty structurs that can handl both complmntarty as wll as substtuton among goods and that do not mpos th constrants of addtv sparablty (asquz and Hanmann 8 provd som possbl ways to accommodat ths) and (4) Dvlopng asy-to-apply tchnqus to us th modl n forcastng mod. 8

31 REFERENCES Ahn J. Jong G. m Y. 8 A forcast of houshold ownrshp and us of altrnatv ful vhcls: A multpl dscrt-contnuous choc approach. Enrgy Economcs 3 (5) 9-4. Ammya T ultvarat rgrsson and smultanous quaton modls whn th dpndnt varabls ar truncatd normal. Economtrca Anas A. 6. A unfd thory of consumpton and travl. Prsntd at th Confrnc Honorng nnth A. Small Dpartmnt of Economcs Unvrsty of Calforna Irvn Fbruary 3-4. Arora N. Allnby G.. and Gntr J.L A hrarchcal Bays modl of prmary and scondary dmand. artng Scnc Baltas G. 4. A modl for multpl brand choc. Europan Journal of Opratonal Rsarch Bhat C.R. 5. A multpl dscrt-contnuous xtrm valu modl: formulaton and applcaton to dscrtonary tm-us dcsons. Transportaton Rsarch Part B 39(8) Bhat C.R. 7. Economtrc choc formulatons: altrnatv modl structurs stmaton tchnqus and mrgng drctons. In ovng Through Nts: Th Physcal and Socal Dmnsons of Travl - Slctd paprs from th th Intrnatonal Confrnc on Travl Bhavour Rsarch. W. Axhausn (Ed.) Elsvr pp Bhat C.R. 8. Th ultpl Dscrt-Contnuous Extrm alu (DCE) odl: Rol of Utlty Functon Paramtrs Idntfcaton Consdratons and odl Extnsons. Transportaton Rsarch Part B 4(3) Bhat C.R. and Sn S. 6. Houshold vhcl typ holdngs and usag: an applcaton of th multpl dscrt-contnuous xtrm valu (DCE) modl. Transportaton Rsarch Part B 4() Bhat C.R. and Srnvasan S. 5. A multdmnsonal mxd ordrd-rspons modl for analyzng wnd actvty partcpaton. Transportaton Rsarch Part B 39(3) Bhat C.R. Srnvasan S. and Sn S. 6. A jont modl for th prfct and mprfct substtut goods cas: applcaton to actvty tm-us dcsons. Transportaton Rsarch Part B 4()

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