I T L S WORKING PAPER ITLS-WP (MDCEV) model: Role of utility function parameters, identification considerations, and model extensions

Transcription

1 I T L S WORING PPER ITLS-WP-07-0 Th multpl dscrtcontnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons By Chandra R. Unvrsty of Tas at ustn Novmbr 007 ISSN 8-570X INSTITUTE of TRNSPORT and LOGISTICS STUDIES Th ustralan y Cntr n Transport and Logstcs anagmnt Th Unvrsty of Sydny Establshd undr th ustralan Rsarch Councl s y Cntr Program.

2 NUBER: Worng Papr ITLS-WP-07-0 TITLE: Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons BSTRCT: any consumr choc stuatons ar charactrzd by th smultanous dmand for multpl altrnatvs that ar mprfct substtuts for on anothr. smpl and parsmonous ultpl Dscrt-Contnuous Etrm alu DCE conomtrc approach to handl such multpl dscrtnss was formulatd by 005 wthn th broadr uhn-tucr T multpl dscrt-contnuous conomc consumr dmand modl of Wals and Woodland 98. Ths papr amns svral ssus assocatd wth th DCE modl and othr tant T multpl dscrt-contnuous modls. Spcfcally th papr proposs a nw 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 tnds 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 dscrtcontnuous modls and llustrats th many tchncal nuancs and dntfcaton consdratons of th multpl dscrtcontnuous modl structur through mprcal ampls. Th papr also hghlghts th tchncal problms assocatd wth th stochastc spcfcaton usd n th T-basd multpl dscrtcontnuous modls formulatd n rcnt Envronmntal Economcs paprs. EY WORDS: UTHORS: CONTCT: Dscrt-contnuous systm multpl dscrtnss uhn- Tucr dmand systms md dscrt choc random utlty mamzaton Chandra R. Insttut of Transport and Logstcs Studs C7 Th ustralan y Cntr n Transport anagmnt Th Unvrsty of Sydny NSW 006 ustrala Tlphon: Facsml: E-mal: tlsnfo@tls.usyd.du.au Intrnt: DTE: Novmbr 007

3 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons. Introducton ultpl dscrtnss.. th choc of multpl but not ncssarly all altrnatvs smultanously s a rathr ubqutous charactrstc of consumr dcson-mang. Eampls of multpl dscrtnss nclud stuatons whr an ndvdual may dcd to partcpat n multpl nds of mantnanc and lsur actvts wthn a gvn tm prod 005 or a houshold may own a m of dffrnt nds of vhcls such as a sdan and a pc-up truc or a sdan and a mnvan; s and Sn 006. Such multpl dscrt stuatons may b modld usng 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. problm wth ths approach howvr s that th numbr of compost altrnatvs plods wth th numbr of lmntal altrnatvs. nothr approach s to us th multvarat probt logt mthods of anchanda t al. 999 Baltas 004 Edwards and llnby 00 and and Srnvasan 005. But ths approach s not basd on a rgorous undrlyng utltymamzng framwor of multpl dscrtnss; rathr t rprsnts a statstcal sttchng of unvarat utlty mamzng modls. In both th approachs dscussd abov to handl multpl dscrtnss thr s also no plct way to accommodat th dmnshng margnal rturns.. sataton n th consumpton of an altrnatv. ddtonally and rlatd to th abov pont t s vry cumbrsom vn f concptually fasbl to nclud a contnuous dmnson of choc for ampl modlng th duratons of partcpaton n th chosn actvty purposs n addton to th choc of actvty purpos. Wals and Woodland 98 proposd two altrnatv ways to handl stuatons of multpl dscrtnss wthn a bhavorally-consstnt utlty mamzng framwor. Both approachs assum a drct utlty functon U that s assumd to b quasconcav ncrasng and contnuously dffrntabl wth rspct to th consumpton quantty vctor. Consumrs mamz th utlty functon subct to a lnar budgt constrant whch s bndng n that all th avalabl budgt s nvstd n th consumpton of th goods; that 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 brf hstory of th trm multpl dscrtnss s n ordr hr. Tradtonal dscrt choc modls focus on th slcton of a sngl altrnatv from th st of avalabl altrnatvs on a purchas occason. That s thy consdr th trm cornr soluton problm. Hanmann n hs 978 dssrtaton usd th trm gnralzd cornr soluton problm to rfr to th stuaton whr multpl altrnatvs may b chosn smultanously. Hndl 999 appars to hav bn th frst to con th trm multpl dscrtnss to rfr to th choc of multpl altrnatvs. Ths trm s also usd by Dub 004. nothr approach for multpl dscrtnss s th on proposd by Hndl 999 and Dub 004. 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 pctd 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 nt consumrs ar obsrvd to purchas a varty of goods at th currnt shoppng occason. Posson dstrbuton s assumd for th numbr of consumpton occasons and a normal dstrbuton s assumd rgardng varyng tasts to complt th modl spcfcaton. Such a vrtcal varty-sng modl of cours s dffrnt from th horzontal varty sng modl consdrd n ths papr whr th choc s consdrd to b among nhrntly mprfct substtuts at th choc occason s m t al. 00 and 005. Th assumpton of a quas-concav utlty functon s smply a manfstaton of rqurng th ndffrnc curvs to b conv to th orgn s Daton and ullbaur 980 pag 0 for a rgorous dfnton of quas-concavty. Th assumpton of an ncrasng utlty functon mpls that U > U 0 f > 0.

4 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons 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 98 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 98 and Phanuf t al. 000 for addtonal dtals. Th frst approach whch Wals and Woodland labl as th mmya-tobn approach s an tnson of th classc mcroconomc approach of addng normally dstrbutd stochastc trms to th budgt-constrand utlty-mamzng shar quatons. In ths approach th drct utlty functon U tslf s assumd to b dtrmnstc by th analyst and stochastcty s ntroducd post-utlty mamzaton. Th ustfcaton for th addton of such normally dstrbutd stochastc trms to th dtrmnstc utltymamzng allocatons s basd on th noton that consumrs ma rrors n th utltymamzng 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 mmya 974 that a rspcts th unt smpl 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 matr of th shars whch can b accommodatd by droppng on altrnatv. Howvr an mportant lmtaton of th mmya-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 smpl s mposd by ad hoc statstcal procdurs of mappng th dnsty outsd th unt smpl to th boundary ponts of th unt smpl. 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 mamzaton s Hanmann 978 who uss such an approach vn bfor Wals and Woodland. Unl th mmya-tobn approach th T approach mploys a mor drct stochastc spcfcaton by assumng th utlty functon U to b random from th analyst s prspctv ovr th populaton and thn drvs th consumpton vctor for th random utlty spcfcaton subct 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.

5 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons mong 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 lad to a complcatd llhood functon that ntals mult-dmnsonal ntgraton. m t al. 00 addrssd ths ssu by usng th Gw-Havasslou-an or GH smulator to valuat th multvarat normal ntgral apparng n th llhood functon n th T approach. lso 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 005 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-trm valu rror trm. s modl labld th multpl dscrt-contnuous trm valu DCE modl s analytcally tractabl n th probablty prssons and s practcal vn for stuatons wth a larg numbr of dscrt consumpton altrnatvs. In fact th DCE modl rprsnts th multnomal logt NL form-quvalnt for multpl dscrt-contnuous choc analyss and collapss actly to th NL n th cas that ach and vry dcsonmar chooss only on altrnatv. Indpndnt of th abov wors of m t al. and thr has bn a stram of rsarch n th nvronmntal conomcs fld s Phanuf t al. 000; von Hafn t al. 004; von Hafn 00a; von Hafn 004; von Hafn and Phanuf 005; Phanuf and Smth 005 that has also usd th T approach to multpl dscrtnss. Ths studs us varants of th lnar pndtur systm LES as proposd by Hanmann 978 and th translatd CES for th utlty functons and us multplcatv log-trm valu rrors. Howvr th rror spcfcaton n th utlty functon s dffrnt from that n s DCE modl rsultng n a dffrnt form for th llhood functon mor on ths n Scton 6. Wthn th contt of th T approach to handlng multpl dscrtnss th purpos of ths rsarch s fv-fold. Th frst obctv s to rformulat th utlty spcfcaton usd n arlr studs n a way that plctly clarfs th rol of ach paramtr n th utlty spcfcaton. Th scond obctv s to prsnt dntfcaton consdratons rlatd to both th functonal form as wll as th stochastc natur of th utlty spcfcaton. Th thrd obctv s to drv th DCE modl prsson for th cas whn thr s prc varaton across goods and to tnd th DCE modl to accommodat gnralzd trm valu GE-basd and othr corrlaton structurs. Th fourth obctv s to dscuss th rlatonshp btwn th modls of m t al. 00 th T formulatons usd n Envronmntal Economcs and th DCE formulaton. Th ffth obctv s to llustrat th tchncal ssus rlatd to th proprts and dntfcaton of th DCE modl through mprcal llustratons. Th rst of th papr s structurd as follows. Th nt 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

6 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons consdratons n stmatng th paramtrs n th utlty spcfcaton. Scton dscusss th stochastc form of th utlty spcfcaton th rsultng gnral structur for th probablty prssons and assocatd dntfcaton consdratons. Scton 4 drvs th DCE structur for th nw utlty functonal form usd n th currnt papr and tnds 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 tnd th dscussons of th arlr sctons to th cas whn an outsd good s prsnt. Scton 6 compars th arlr multpl dscrt-contnuous modls usd n th ltratur wth th on formulatd n th currnt papr. Scton 7 provds mprcal llustratons to rnforc th thortcal ssus dscussd n arlr sctons. Th fnal scton concluds th papr.. Functonal form of utlty spcfcaton W consdr th followng functonal form for utlty n ths papr basd on a gnralzd varant of th translatd CES utlty functon: U ψ whr U s a quas-concav ncrasng and contnuously dffrntabl functon wth rspct to th consumpton quantty -vctor 0 for all and ψ and ar paramtrs assocatd wth good. Th functon n Equaton s a vald utlty functon f ψ > 0 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 rlad 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. 4 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 980; pag 9. ddtonally addtv sparablty mpls that th margnal utlty wth rspct to any good s ndpndnt of th lvls of all othr goods. 5 Th form of th utlty functon n Equaton s dffrnt from that usd n arlr studs. Th rason for th spcfc functonal form adoptd hr s to hghlght th rol of th varous paramtrs ψ and and plctly ndcat th ntr-rlatonshps 4 s llustratd n m t al. 00 and 005 th prsnc of th translaton paramtrs mas th ndffrnc curvs str th consumpton as at an angl rathr than bng asymptotc to th consumpton as thus allowng cornr solutons. 5 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 006 for an ampl of such a charactrstcs approach to utlty. 4

7 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons btwn ths paramtrs that rlat to thortcal and mprcal dntfcaton ssus. 6 Fnally t should b notd that th utlty form of Equaton collapss to th followng lnar pndtur systm LES form whn 0 s ppnd ; th LES form of th typ blow appars to hav bn frst usd by Hanmann 978. U ln ψ. Rol of paramtrs n utlty spcfcaton Rol of ψ Th rol of ψ can b nfrrd by computng th margnal utlty of consumpton wth rspct to good whch s: U ψ It s obvous from abov that ψ rprsnts th basln margnal utlty or th margnal utlty at th pont of zro consumpton. ltrnatvly th margnal rat of substtuton btwn any two goods and l at th pont of zro consumpton of both ψ goods s. Ths s th cas rgardlss of th valus of and unl n arlr ψ l studs whr th basln margnal utlty and th margnal rat of substtuton n gnral ar functons of multpl paramtrs. For two goods and wth sam unt prcs a hghr basln margnal utlty for good rlatv to good mpls that an ndvdual wll ncras ovrall utlty mor by consumng good rathr than at th pont of no consumpton of any goods. That s th consumr wll b mor lly to consum good than good. Thus a hghr basln ψ mpls lss llhood of a cornr soluton for good. 6 s w wll show latr howvr th utlty form w adopt s bhavorally and obsrvatonally ndstngushabl from thos usd n 005 and m t al. 00 f s normalzd to for all and 0 < <. It s also obsrvatonally ndstngushabl from th utlty form usd n nvronmntal conomcs undr th condton that 0. Spcfcally all ths utlty forms mply an dntcal st of uhn-tucr frst ordr condtons and dmand. Howvr th varous utlty forms may not yld dntcal wlfar masurs. In our formulaton of utlty w mpos th untstabl but ntutv condton of wa complmntarty s älr 974 whch mpls that th consumr rcvs no utlty from a non-ssntal good s attrbuts f sh dos not consum t.. a good and ts qualty attrbuts ar wa complmnts or U 0 f 0 whr U s th sub-utlty functon for th th good. Th radr s rfrrd to Hanmann 984 von Hafn 004 and Hrrgs t al. 004 for a dtald dscusson of th advantags of usng th wa complmntarty assumpton. Th us of th wa complmntarty condton ssntally amounts to a cardnal normalzaton rstrcton on utlts. But as Hrrgs t al. 004 ndcat th analyst wll hav to plac som nd of a cardnal rstrcton on prfrncs anyway for wlfar masurmnt and wa complmntarty s a natural choc n many crcumstancs. W wll mantan th wa complmntary cardnal normalzaton n th rst of ths papr to smplfy th algbra though th ordnalty of utlts should always b pt n mnd. 5

8 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons Rol of n mportant rol of th trms s to shft th poston of th pont at whch th ndffrnc curvs ar asymptotc to th as from to... so that th 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 0.5 and. Fgur prsnts th profls of th ndffrnc curvs n ths two-dmnsonal spac for varous valus of > 0. To compar th profls th ndffrnc curvs ar all drawn to go through th pont 08. Th radr wll also not that all th ndffrnc curv profls str th y-as wth th sam slop. s can b obsrvd from th fgur th postv valus of and lad to ndffrnc curvs that cross th as of th postv orthant allowng for cornr solutons. Th ndffrnc curv profls ar asymptotc to th -as at y corrspondng to th constant valu of whl thy ar asymptotc to th y-as at. 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 amn th profl of th sub-utlty functon for altrnatv. Fgur plots th functon for altrnatv for 0 and ψ and for dffrnt valus of. ll 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 > 0. Spcfcally th hghr th valu of th lss s th sataton ffct n th consumpton of. It s mportant to not that th ntr rang of sataton ffcts from mmdat and full sataton flat ln to lnar sataton constant margnal utlty can b accommodatd by dffrnt valus of for any gvn valu. Rol of Th prss 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 whch rprsnts th prfct substtuts cas as proposd by Daton and ullbaur 980 and appld n Hanmann 984 Chang 99 Chntagunta 99 and rora 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 pndtur on th sngl good wth th hghst basln and constant margnal utlty.. th hghst ψ valu. Ths s 6

9 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons th cas of sngl dscrtnss. 7 s movs downward from th valu of th sataton ffct for good ncrass. Whn 0 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 plots th utlty functon for altrnatv for and ψ and for dffrnt valus of. gan 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 dntfcaton ssus 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 tnsv 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 tnsv 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 ponntatng 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 appromat a subutlty functon profl basd on a combnaton of and valus wth a sub-utlty functon basd solly on or valus. Ths s llustratd n Fgurs 4a through 4d for ψ and for dffrnt sataton lvls. In ths fgurs th subutlty functons basd solly on assum 0.. ths functons ta th form of Equaton whl thos basd solly on for ncorporatng sataton assum not that vn f fd has to b postv to allow cornr solutons. In all th fgurs th profl basd only on th -profl or th -profl tracs th profl basd on th combnaton of valus th combnaton profl rasonably wll. For modrat satatons Fgurs 4a and 4b on of th two profls dos bttr than th othr basd on how clos th and valus n th combnaton profl ar to th assumd valu of 0 for th -profl and for th -profl. For vry low and vry hgh satatons both th -profl and th -profl trac th combnaton profl vry closly. In actual applcaton t would bhoov th analyst to stmat modls basd on 7 If thr s prc varaton across goods on nds to ta th drvatv of th utlty functon wth rspct to pndturs on th goods. In th cas that for all U Σψ p whr ψ s th unt prc of good. Thn U ψ p. In ths stuaton th consumr wll nvst all pndturs on th sngl good wth th hghst prc-normalzd margnal and constant utlty ψ p. 7

10 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons both th -profl and th -profl and choos a spcfcaton that provds a bttr statstcal ft. 8 In cass whr valus ar stmatd ths valus nd to b boundd from abov at th valu of. To nforc ths condtons can b paramtrzd as [ p δ ] wth δ bng th paramtr that s stmatd. Furthr to allow th sataton paramtrs.. th valus to vary across ndvduals 005 wrts δ y whr y s a vctor of ndvdual charactrstcs mpactng sataton for th th altrnatv and s a corrspondng vctor of paramtrs. In cass whr valus ar stmatd ths valus nd to b gratr than zro whch can b mantand by rparamtrzng as p μ. ddtonally th translaton paramtrs can b allowd to vary across ndvduals by wrtng μ ϕ w whr w s a vctor of ndvdual charactrstcs for th th altrnatv and ϕ s a corrspondng vctor of paramtrs.. Stochastc form of utlty functon Th T approach mploys a drct stochastc spcfcaton by assumng th utlty functon U 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. Th ponntal form for th ntroducton of th random trm guarants th postvty of th basln utlty as long as ψ z > 0. To nsur ths lattr condton ψ z s furthr paramtrzd as p β z whch thn lads to th followng form for th basln random utlty assocatd wth good : ψ z p 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: U [ p β z ] 6 8 ltrnatvly th analyst can stc wth on functonal form a pror but prmnt wth varous fd valus of for th profl and for th -profl. 8

11 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons From th analyst s prspctv th ndvdual s mamzng random utlty subct to th bndng lnar budgt constrant that ncom or som othr appropratly dfnd total budgt quantty s th unt prc of good. E whr E s total pndtur or p and p. Optmal Epndtur llocatons Th analyst can solv for th optmal pndtur allocatons by formng th Lagrangan and applyng th uhn-tucr T condtons. 9 Th Lagrangan functon for th problm s: p L [ p β z ] λ E 7 whr λ s th Lagrangan multplr assocatd wth th pndtur constrant that s t can b vwd as th margnal utlty of total pndtur or ncom. Th T frstordr condtons for th optmal pndtur allocatons th valus ar gvn by: pβ z p p λ 0 f > 0 8 pβ z p p λ < 0 f 0 Th optmal dmand satsfs th condtons n Equaton 8 plus th budgt constrant E. 9 For rasons that wll bcom clar latr w solv for th optmal pndtur allocatons for ach good not th consumpton amounts of ach good. Ths s dffrnt from arlr studs that focus on th consumpton of goods. 9

12 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons 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 > 0. For th frst good th T condton may thn b wrttn as: p 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: f > 0 < f 0 whr 0 β z ln ln p p. lso not that n Equaton 0 a constant cannot b dntfd n th β z trm for on of th altrnatvs bcaus only th dffrnc n th from mattrs. Smlarly ndvdual-spcfc varabls ar ntroducd n th s for - altrnatvs wth th rmanng altrnatv srvng as th bas. 0. 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 ont probablty dnsty functon of th trms b f. Thn th probablty that th ndvdual allocats pndtur to th frst of th goods s: P f... d d... d d d J L... 0 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 dscrtcontnuous 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 0. 0

13 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons whr J s th Jacoban whos lmnts ar gvn by s 005: J h [ h ] [ h ] ; h. Th probablty prsson n Equaton s a --dmnsonal ntgral. Th prsson for th probablty of all goods bng consumd s on-dmnsonal whl th prsson for th probablty of only th frst good bng consumd s - dmnsonal. Th dmnsonalty of th 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 --ntgral form shown blow: P g J L... d d... d Th quaton abov ndcats that th probablty prsson for th obsrvd optmal pndtur pattrn of goods s compltly charactrzd by th - rror trms n dffrnc form. Thus all that s stmabl s th -- covaranc matr of th rror dffrncs. In othr words t s not possbl to stmat a full covaranc matr 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 00 pag 7 for a smlar stuaton n th contt of standard dscrt choc modls. Thr ar many possbl ways to normalz f. to account for ths stuaton. For ampl on can assum an dntty covaranc matr for f. whch automatcally accommodats th normalzaton that s ndd. ltrnatvly 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 matr of th rror dffrncs as ust dscussd though th analyst mght want to mpos constrants on ths full covaranc matr 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 -- covaranc matr for 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 005; and Sn 006; t al. 006 for ampls 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:

14 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons U [ p{ β z }] 4 Th scal of th rror trms n th utlty functon n th abov prsson s tms th scal of th rror trms n Equaton 6. Lt whr s th sataton paramtr n th orgnal Equaton 6. Th T condtons for optmal pndtur for ths modfd utlty functon can b shown to b: f > 0 < f 0 whr 5 β z β z ln p ln p ln p ln p 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 0. That s th utlty functon n Equaton 4 s unqu and dffrnt from th utlty functon n Equaton 6 whch mpls that th scal s dntfd. Howvr f th unt prcs ar all th sam p p t s straghtforward to not that th T condtons abov collaps actly to th T condtons n Equaton 0. In ths cas th utlty functon n Equaton 4 cannot b unquly dntfd from th utlty functon n Equaton 6 whch mpls that th scal s not dntfd thortcally. For convnnc th analyst can st th scal to. In th cas that th analyst uss a htroscdastc spcfcaton wth no varaton n unt prcs across altrnatvs th scal of on of th altrnatvs has to b st to unty smlar to th cas of th htroscdastc trm valu or HE modl of 995. Wth a gnral rror structur and no varaton n unt prcs th dntfcaton consdratons assocatd wth a standard dscrt choc modl wth corrlatd rrors apply s Tran 00; Chaptr. Not that s lss than or qual to by dfnton bcaus s lss than or qual to and th scal should b nonngatv.

15 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons 4. Spcfc modl structurs 4. Th DCE odl Structur Followng 005 w spcfy an trm 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 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 p ln p ln p ln p whn th - profl s usd and whn th - profl s usd. s dscussd arlr t s gnrally not possbl to stmat th form n Equaton 0 bcaus th trms and trms srv a smlar sataton rol. From Equaton th probablty that th ndvdual allocats pndtur to th frst of th goods s: P J λ s s Λ λ d whr λ s th standard trm valu dnsty functon and Λ s th standard trm valu cumulatv dstrbuton functon. Th prsson n Equaton 7 smplfs to a rmarably smpl and lgant closd-form prsson. drvd th form of th Jacoban for th cas of qual unt prcs across goods whch howvr can b tndd 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: J c whr c. 8 c p 6 7 Th ntgraton n Equaton 7 also collapss to a closd form prsson s ppnd B provdng th followng ovrall prsson: 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. On can also drv th prsson blow from th dffrnc form of Equaton usng th proprts of th multvarat logstc dstrbuton s ppnd C.

16 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons 4! c c P 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 pndtur s allocatd to good. Thn th modl n Equaton 9 collapss to th standard NL modl. Thus th DCE modl s a multpl dscrt-contnuous tnson of th standard NL modl. 4 Th prsson for th probablty of th consumpton pattrn of th goods rathr than th pndtur pattrn can b drvd to b:! f p f p P 0 whr s as dfnd arlr s Equaton 6 and f. Th prsson n Equaton 0 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 prsson for pndtur allocatons whch wll provd th sam probablty for a gvn pndtur 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 prsson or th pndtur allocaton probablty prsson. 4 Not that whn for all β'z ln p. Evn f whn Equaton 9 collapss to th NL form th scal 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.

17 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons 5 4. Th multpl dscrt-contnuous gnralzd trm-valu DCGE modl structur Thus far w hav assumd that th trms ar ndpndntly and dntcally trm valu dstrbutd across altrnatvs. Th analyst can tnd th modl to allow corrlaton across altrnatvs usng a gnralzd trm valu GE rror structur. Th rmarabl advantag of th GE structur s that t contnus to rsult n closdform probablty prssons for any and all pndtur pattrns. Howvr th drvaton s tdous and th prssons gt unwldy. In ths papr w provd th prssons for a spcfc nstd logt structur wth 4 altrnatvs two altrnatvs labld and n nst and th othr two altrnatvs labld and 4 n nst B th drvaton s avalabl on rqust from th author. Th cumulatv dstrbuton functon for th rror trms n th utlty prssons ta th followng form: [ ] B B B s s s s s s s s p < < < < Λ Dfn th followng: 4 4 B B B B B H 4. Thn H P 000 H H J P H B J P B B H H B J P B B 0 4 B B B B H H B H B H B J P B B B B B

18 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons Th probablts for all othr pndtur pattrns for th 4 goods can b obtand by ntrchangng th labls and Th d DCE odl Th DCGE structur s abl to accommodat flbl 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 mng dstrbuton whch lads to th d 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. 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 0 Ω. 6 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: P η c c η η! df η. whr F s th multvarat cumulatv normal dstrbuton s 005; and Sn 006; and t al Th modl n Equaton can b tndd 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. 4.. 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. ssumng a 4-altrnatv cas for as n prsntaton th htroscdastc structur may b spcfd n th form of th followng covaranc matr for : 4 5 In all th prssons corrspondng to th nstd structur abov s dntfd only whn thr s prc varaton across altrnatvs s Scton.. 6 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 md logt modl s

19 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons ω π 0 0 ω 0 0 Cov 6 0 ω 0 ω4 whr th frst componnt on th rght sd corrsponds to th IID covaranc matr of ζ ζ ζ ζ ζ 4 and th scond componnt s th htroscdastc covaranc matr of η. Th covaranc of rror dffrncs wth rspct to th frst altrnatv s: ω ω ω ω π Cov 4 ω ω ω 6 4 ω ω4 n nspcton of th matr abov shows only four ndpndnt quatons th ran condton mplyng that at most four paramtrs ar stmabl. 7 Thr ar two ways to procd wth a normalzaton as dscussd blow. Th frst approach s to normalz and stmat th htroscdastc covaranc matr of η.. ω ω ω and ω 4. ssum that s normalzd to and lt th corrspondng valus of ω b ω 4. Thn th followng qualts should hold basd on Equaton 4 for any normalzaton of to q π 6 blow: ω q ω q ω ω q ω ω q 4 5 Th abov qualts can b rwrttn as: ω ω q q 4 6 Th normalzd varanc trms ω must b gratr than or qual to zro whch mpls that th followng condtons should hold: ω q q 4 7 Intutvly th abov condton mpls that th normalzaton on must b st low nough so that th ovrall tru varanc of ach rror trm ω q s largr than q. For ampl sttng 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 7 Strctly spang on can stmat all th fv paramtrs ω ω ω and ω4 bcaus of th dffrnc n th trm valu dstrbutons usd for ζ and th normal dstrbutons usd for η s Walr 00. Howvr th modl wll b nar sngular and t s mportant to plac th ordrran constrant. 7

20 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons th analyst can do s to normalz to progrssvly smallr valus and statstcally amn th rsults. Th scond approach s to normalz on of th cas from Equaton 4 w can wrt: [ ω ω q ω ω ]; 4. ω trms nstad of th trm. In ths q ω q ω 8 ftr som manpulatons th abov quaton may b rwrttn as: ω ω ω ω ; 4. 9 Nt mposng th condton that th normalzd trms ω must b gratr than or qual to zro mpls th followng: ω ω ω 4. 0 Th abov condton s automatcally satsfd as long as th frst altrnatv s th mnmum varanc altrnatv. n assocatd convnnt normalzaton s ω 0 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. 004 for an quvalnt procdur for a Htroscdastc spcfcaton wthn th md multnomal logt modl. Th abov dscusson assums thr s prc varaton across goods. In th cas of no prc varaton th scal s not dntfabl. In ths cas th asst procdur s to normalz to and th ω valu for th mnmum varanc altrnatv to zro. 4.. Th gnral rror covaranc structur wthn th DCE framwor pproprat dntfcaton normalzatons wll hav to placd on and th covaranc matr 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 -- 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 -- 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 to b a small valu such that th varanc of th mnmum varanc altrnatv cds 8

21 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons π 6 snc ths varanc s not nown th analyst wll hav to prmnt wth altrnatv fd valus Normalz ω for th mnmum varanc altrnatv to zro and 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 matr of η. 5. Th modl wth an outsd 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. lso for dntfcaton lt ψ. Thn th utlty functonal form nds to b modfd as follows: U p p β z Not that thr s no translaton paramtr for th frst good bcaus th frst good s always consumd. s n th no-outsd good cas th analyst wll gnrally not b abl to stmat both and for th nsd goods. Th analyst can stmat on of th followng thr utlty forms: { } U p p β z U p p ln β z U p p β z 9

22 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons Th last functonal form abov s stmabl now bcaus th constant paramtr s obtanng a pnnng ffct from th sataton paramtr for th outsd good. Th analyst can stmat all th thr 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 prsson for th pndtur allocaton on th varous goods wth th frst good bng th outsd good s dntcal to Equaton 9 whl th probablty prsson for consumpton of th goods wth th frst good bng th outsd good s P whr f f p f. 8! Th prssons for n Equaton 9 and Equaton ar as follows for ach of th thr utlty forms n Equaton : Frst form - β z ln ln p ; ln Scond form - β z ln ln p ; ln 4 Thrd form - β z ln ln p ; ln 6. Comparson wth arlr multpl dscrt-contnuous modls In ths scton w dscuss how th modl dvlopd n ths papr dffrs from th modl of m t al. 00 thos n Envronmntal Economcs and th arlr modls by and collagus. Th dscusson s n th contt of th basc structur wth dntcally and ndpndntly dstrbutd rror trms across altrnatvs. 8 Th Gauss cod documntaton and tst data sts for stmatng th DCE modl wth and wthout an outsd good ar avalabl at: 0

23 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons 6. m t al. s odl m t al. 00 us th followng translatd constant lastcty of substtuton CES drct utlty form: U ψ 5 whr th suprscrpt s to dstngush th paramtrs n ths functonal form from thos n th form of Equaton. In ths scton for as n comparson w wll consdr th cas whn thr s no outsd good. In th utlty form abov ψ > 0 > 0 and 0. To mprcally dntfy th utlty form m t al. mpos th rstrcton that for all.. thy stmat th -profl. Th radr wll not that m t al. s form dos not ncorporat th wa complmntarty assumpton. Howvr ths can b asly rmdd by rvsng th utlty form abov to an mprcally ndstngushabl altrnatv form provdd blow: U ψ { } 6 Th T condtons and optmal consumptons for both Equatons 5 and 6 ar dntcal. But th lattr form assgns zro utlty to good whn t s not consumd whl stll allowng cornr solutons. Howvr n thr of th two forms th ntrprtaton of ψ s not straghtforward. Spcfcally th basln margnal utlty of a good or margnal utlty whn no quantty of th good s consumd s ψ whch dpnds on both ψ and for fd to for all. Of cours th sataton rat for good wth rspct to th basln margnal utlty s stll dtrmnd by as n th -profl basd utlty form adoptd n ths papr.. Equaton wth fd to for all. In fact th stmaton rsults and optmal consumptons from usng Equaton and Equaton 5 wth all s fd to wll b dntcal as long as 0 for all n Equaton. Th only cosmtc dffrnc wll b a shft n th constant trms btwn th ψ and ψ trms. Spcfcally f ψ p τ β ' z and ψ p τ β ' z whr th constant trm s rmovd out from th β ' z trm th followng rlatonshp wll hold btwn th two modls as long as th sam rror dstrbutons ar assumd and assumng that th frst altrnatv s consdrd as th bas: τ τ ln 7 β β.

24 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons n mportant tchncal nuanc s n ordr hr. Th form of Equaton 5 s rstrctv compard to th form of Equaton adoptd n ths papr. Spcfcally Equaton covrs th ovrall basln margnal utlty sataton spac mor compltly than Equaton 5 bcaus n Equaton whl 0 n Equaton 5. But th analyst at tms may hav to mpos th constrant 0 n th functonal form of Equaton to provd stablty n stmaton spcally whn a scal paramtr s bng stmatd wth lmtd prc varaton. nothr mportant dffrnc btwn m t al. s modl and th modl hr s th dstrbuton of th rror trms. m t al. assum that th rror trms ar ndpndnt and dntcally dstrbutd normal. Thy thn us th dffrncng form of Equaton to dvlop th probablts wth th g. functon bng a multvarat normal dnsty. Ths form rqurs an approprat dcomposton of th dnsty functon for th contnuous and dscrt componnts and multvarat normal ntgraton. Th approach s not practcal for most ralstc applcatons. s 005 notd f on consdrs th rror trms to b IID gumbl nstad of IID normal th modl structur collapss to th closd-form DCE form usd hr. 6. odls n Envronmntal Economcs Th studs n Envronmntal Economcs unl m t al. us th utlty functon corrspondng to th -profl. Ths studs always consdr th prsnc of an outsd good and so w wll consdr th cas whn thr s an outsd good n ths scton. Ths outsd good may b arbtrarly dsgnatd as th frst good n our notatonal framwor wth p as n Scton 5. Th utlty functon n th Envronmntal Economcs studs tas th LES form s von Hafn and Phanuf 005 von Hafn 00b Phanuf t al. 000 Phanuf and Hrrgs 000 and Hrrgs t al. 004: U ψ ln. 8 Th suprscrpt abov s to dstngush th paramtrs from thos n Equaton and should not b confusd wth th squar powr functon. In th functon abov th utlty accrud from zro consumpton of a good s postv snc ψ > 0 and > 0. Howvr ths can b accommodatd by r-wrtng th utlty form n th mprcally ndstngushabl altrnatv form shown blow: 9 U ψ ln 9 In partcular t can b radly sn that th T frst ordr condtons and th optmal consumptons ar dntcal for th utlty forms n Equatons 8 and 9 as also 9 von Hafn t al. 004 and von Hafn and Phanuf 00 also rcognz ths wa complmntarty problm n th functonal form of Equaton 8 whr th qualty attrbuts of good contrbut to utlty vn f th good s not consumd. Thy addrss t by ntractng th qualty attrbuts wth rathr than ncludng th qualty attrbuts n ψ.

25 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons obsrvd by Hrrgs t al Howvr n both forms th ntrprtaton of ψ s not straghtforward snc th basln margnal utlty s ψ. But for a gvn basln margnal utlty srvs as a sataton paramtr n addton to allowng cornr solutons. In fact th stmaton rsults and optmal consumptons from usng Equaton and Equaton 9 wll b dntcal cpt for a shft n th constant trms btwn th ψ and ψ trms. Spcfcally f ψ p τ β ' z and ψ p τ β ' z th followng rlatonshp wll hold btwn th two modls as long as th sam rror dstrbutons ar assumd and assumng that th frst altrnatv s consdrd as th bas: τ τ ln 40 β β n mportant pont to not about th Envronmntal Economcs studs s that thy consdr th utlty of th outsd good as bng dtrmnstc.. 0 and thn consdr th rror trms of th utlts of th nsd goods to b ndpndnt and typcally dntcally trm valu dstrbutd. To s ths consdr Equaton 7 whch s th approprat probablty prsson for th pndtur pattrn f thr ar ndpndnt and dntcally dstrbutd rror trms n all utlts wth th frst altrnatv bng th outsd good. Th quvalnt probablty prsson for th consumpton pattrn s: P J λ s s Λ λ d 4 whr p J f f p and f. Th valus of and n f wll dpnd on th utlty functon form usd n th prsnc of an outsd good. For th frst functonal form n Equaton 0 and for all. For th scond functonal form n Equaton 0 and 0 for all. For th fnal functonal form 0 and all valus ar qual across altrnatvs. Th prsson n Equaton 4 collapss to th closd form prsson provdd n Equaton yldng th DCE modl. But now assum 0 n Equaton 4 as n th Envronmntal Economcs studs. Th ntgral n Equaton 4 thn drops out and th quaton bcoms:

26 Th multpl dscrt-contnuous trm valu DCE modl: Rol of utlty functon paramtrs dntfcaton consdratons and modl tnsons Λ λ s s J P 4 Substtutng w w w λ and w w Λ th prsson may b wrttn as: p p p g g J P 4 whr g. Th form abov s th sam as th llhood functon n Equaton 0 of on Hafn and Phanuf 00. Thus th modls n Envronmntal Economcs ar obtand by assumng away stochastcty n th utlty of th outsd good. Bascally th Envronmntal Economcs studs rcognz th sngularty mposd by th budgt constrant by drctly assumng 0 so that thr ar only - rror trms n th - T condtons s Equaton 0. Th DCE modl on th othr hand rcognzs th sngularty mposd by th budgt constrant by consdrng all utlts as random and thn plctly acnowldgng th sngularty among th rror trms n th - T condtons s also m t al. 00 and Wals and Woodland 98 who us th lattr approach. Th lattr approach s concptually consstnt n consdrng th utlts of all altrnatvs as bng random strct random utlty mamzaton whl th formr approach assums that th analyst nows all consumr-rlatd and mart-rlatd factors gong nto th valuaton of th outsd good but not for th nsd goods partal random utlty mamzaton. Furthr n th Envronmntal Economcs approach f nstad of th outsd good s utlty th utlty of som othr nsd good s consdrd dtrmnstc to accommodat th sngularty w obtan dffrnt probablty prssons and probablty valus for th sam consumpton pattrn. Spcfcally f th rror trm of altrnatv l s fd to zro whr l th probablty prsson for th consumpton pattrn corrspondng to Equaton 4 s: { } p p p p p g l l l l l g g g J p P 44 whr l p s th prc of th l th good. On th othr hand f th rror trm of altrnatv l s fd to zro whr l > th probablty prsson for th consumpton pattrn s: