Uncertain lifetime, prospect theory and temporal consistency

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1 Code JEL: D81 D91 Uncerain lifeime, propec heory and emporal coniency Nicola Drouhin ver July 2006 Summary: here i an old radiion in economic of eparaing ime dicouning from uncerainy. A i ile ugge, hi paper challenge hi radiion according o Occam' razor principle. I propoe a coninuou ime life cycle model of conumpion wih uncerain lifeime and no "pure ime preference". I ue a non-parameric pecificaion of rank dependen uiliy / cumulaive propec heory o characerize he preference of he agen. I hen dicue uncerain lifeime a he unique primiive of ime preference, aking he inananeou probabiliy of dying a he "objecive par" of ime preference, and he ranformaion of he probabiliy diribuion a "ubjecive par". From a normaive poin of view he paper dicu he implicaion of adding an axiom of emporal coniency o he former model. I i proven ha ime coniency hold for a much wider cla of probabiliy weighing funcion han he ideniy one, ha characerize he expeced uiliy model. hi pecial cla of probabiliy weighing funcion provide foundaion for a conan rae of ubjecive ime dicouning which inerac muliplicaively wih he inananeou condiional probabiliy of dying (inead of addiively a in he radiional dicouned expeced uiliy approach). From a poiive poin of view, variou pecificaion of he model can accoun for many o-called anomalie in ineremporal choice, uch a decreaing rae of dicouning and ime inconien behavior. Key word : ineremporal choice; life cycle heory of conumpion and aving; uncerain lifeime; propec heory; rank dependen uiliy; emporal coniency; ime dicouning. Déparemen économie e geion Ecole normale upérieure de Cachan 61 avenue du Préiden Wilon Cachan drouhin@ecoge.en-cachan.fr Groupe de Recherche ur le Rique l'informaion e la Déciion Maion de la recherche de l'esp 30 avenue du Pd Wilon Cachan el he auhor wih o hank Mohammed Abdellaoui, Anoine d'auume, David de la Croix, Marie-Laure Cabon-Dherin, Jean-Séphane Dherin, Nahalie Echar, Berrand Munier, Jean-Marc allon and Peer Wakker for here commen on a previou verion of hi paper. 1

2 Uncerain lifeime, propec heory and emporal coniency Pluralia non e ponenda ine neceiae. William of Occam Occam' razor principle, which ae ha, for explaining phenomena, eniie hould no be muliplied wihou neceiy, i cerainly a powerful heuriic for doing "good cience". In paricular hi principle fi well wih he reducioni mehodology of economic. However, he producion of cienific knowledge i a decenralized proce. he growh of hi knowledge i he reul of many conribuion produced equenially. Some of hem anwer oher, ome of hem addre new empirical or heoreical problem, ome of hem are ucceful, ome of hem remain unnoiced for a long ime, ome of hem are imple, ome of hem require highly echnical kill and can be underand by very few, ec.. hi proce a a whole may cerainly explain he riing complexiy of cienific heory. hi can be a good hing when underlying phenomena require complexiy and ubley. Bu here are ime i become neceary o eparae he whea from he chaff and for ha o go back o Occam' razor. hi paper i preciely a ry o do ha concerning he heory of ineremporal choice when aking ino he uncerainy abou individual longeviy. here i a rong parallel beween choice facing rik and choice facing ime. Boh domain rely on normaive model buil upon rong axiomaic foundaion, von Neuman Morgenern (1947)' expeced uiliy model a regard heory of rik and Koopman (1960)' dicouned 2

3 uiliy model a regard heory of choice over ime. Boh model have been challenged on he bai of experimenal reul demonraing ha hey were no able o predic he real behavior of agen. For he domain of rik, afer Allai (1953)' eminal work, an acive reearch program ha been developed, rying o provide foundaion for a more general model of choice, able o explain, wihin an unified framework, mo of he oberved "anomalie" bu preerving ome raionaliy crieria. he idea ha agen ranform probabiliie ha been inroduced, among oher, by Kahneman and verky (1979). he main drawback of heir approach i ha agen may violae fir order ochaic dominance, which i a baic requiremen of raionaliy under rik. Quiggin (1982) olved hi problem, by auming ha agen do no ranform probabiliie of he fixed oucome bu raher he cumulaed probabiliy of he propec when hee are ranked according o he conequence. Much empirical and heoreical reearch ha been devoed o hi model, which can be conidered a generalizaion of he von Neumann Morgenern expeced uiliy heory, where he independence axiom i replaced wih ochaic dominance and a probabiliy radeoff coniency condiion (Abdellaoui (2002)). hi approach i now called currenly he "Rank Dependen Uiliy" model (RDU). here ha arien a conenu ha cumulaive propec heory (verky & Khaneman [1992]), which add he noion of reference poin o Quiggin' model, provide a good rade-off beween normaive and decripive properie of choice under rik. In he domain of choice over ime, he challenge of he dicouned uiliy model are more recen. Sroz [1956] wa he fir o inroduce explicily non-exponenial dicouning, bu i i wih he book edied by Loewenein and Eler [1992] ha he inere of economi ha been araced o he problem of anomalie in ineremporal choice (Loewenein & Prellec [1992]) or hyperbolic dicouning (Anlie (1991), Laibon (1997)). Even hough here have been many experimenal udie and ome heoreical work ince hen, no conenu ha been 3

4 reached concerning a model of choice over ime ha could be an equivalen of cumulaive propec heory for choice under rik. A already poined by Prelec & Loewenein (1991), he parallel beween choice over ime and choice under rik i no only a general analogy. In fac, he underlying mahemaical rucure of hoe wo problem i fundamenally he ame becaue he weigh aociaed wih he uiliy in each ae of naure (he probabiliy) play he ame role a he dicoun facor aociaed wih each ime period (pure ime preference). When economi croover hoe wo problem of choice over ime and choice under rik, boh weigh ac imulaneouly. he goal of hi paper i preciely o reconider he noion of pure ime preference in a conex in which here i a rik concerning life duraion a in Yaari (1965)' eminal paper. If we ake a very baic wo period dicouned uiliy (DU) model of choice over ime, he ineremporal uiliy funcion wih cerain life duraion i: (, ) δ DU c c = u c + u c (1) wih δ, he dicoun facor (i.e) pure facor of ime preference. If we inroduce lifeime uncerainy in he model, he dicouned expeced uiliy of he agen in hi andard approach will be: ( 0, 1) = (1 ) ( 0) + ( 0) + δ ( 1) = u( c ) + pδ u( c ) DEU c c p u c p u c u c 0 1 (2) wih p he probabiliy of living wo period. If we uppoe ha agen have free acce o he marke of loanable fund a he rae of inere, r, if we denoe and w, he non-financial reource of he agen for each ime period, hen an agen living he econd period i facing he conrain c w + ( w c )( + r). he model' predicion are ummarized by he fir order condiion of he opimizaion problem: w0 1 4

5 u u ( c0 ) ( c ) 1 p ( 1 r) = δ + (3) he main implicaion of hi equaion i ha he lope of he ineremporal conumpion profile depend on he difference beween ( 1+ r) and δ p. ime preference and probabiliy of urvival ac exacly he ame way. Wihin hi model, i i nearly impoible o eimae on an empirical bai wha reul from lifeime uncerainy and wha i due o pure ime preference" 1. here are wo parameer, where only one i needed. Occam' razor principle, which ae ha eniie hould no be muliplied unnecearily, hould lead u o dicard one of hoe wo parameer. A a provocaive aemen, we will claim in hi paper ha here are much more evidence ha lifeime i uncerain han ha omehing like "pure ime preference" exi. So, conrary o he andard approach in which uncerain lifeime i mere inroduced a a refinemen of he andard DU approach, we will poulae ha lifeime uncerainy i he only primiive of ime preference. We hu provide a reamen of deciion over ime a a pecial cae of deciion over rik. We will build a model ha will aify hree prerequiie. Fir, becaue we are inereed in emporal coniency, he model hould have ricly more han wo period. For echnical convenience we will work wih a coninuou ime model of ineremporal choice (i. e. life-cycle i divided ino an infiniy of "very mall" period). Second, our model of choice facing lifeime rik will be from he Cumulaive Propec heory / Rank Dependen Uiliy heory family (CP/RDU), admiing Expeced Uiliy heory a a pecial cae. hird, our model will have he mo general form poible, wih no pecial 1 Barro and Friedman (1977) and Kaz (1979) alo dicu he problem of he correpondence beween he urvival probabiliy and dicoun facor bu wihin a model auming ha agen have acce o acuarially fair 1+ r p and p diappear from he annuiie and life inurance. In hi conex, he rae of reurn on aving i Euler equaion (3). We believe ha inroducing uch coningen ae wihou iniially conidering he cae in which hey do no exi i likely o hide he mo inereing par of he problem. See Yaari (1965) for an exenive dicuion of he lifecycle model of ineremporal choice wheher agen have or have no acce o uch ae. Drouhin (2001) dicued exenively he comparaive aic and econd order properie of he wo period model of ineremporal choice wih uncerain lifeime in boh cae. 5

6 parameric pecificaion for uiliy, probabiliy diribuion of he age of deah or probabiliy weighing funcion. hi paper i organized a follow. Secion 1 will dicu he uiliy funcional repreening preference over ineremporal conumpion ream. Secion 2 will dicu he properie of he opimal conumpion ream choen a a given dae. From a normaive poin of view, ecion 3 will derive neceary and ufficien condiion for he agen o be emporally conien. We will how ha hoe condiion give foundaion for muliplicaive ime preference. Secion 4 will dicu he cae of non emporally conien preference, and derive explicily he properie of he effecive conumpion profile. he uiliy funcional 1.1. he preference of he agen when life duraion i cerain We conider an agen' choice of her conumpion profile. A conumpion profile i a funcion of ime defined on he inerval [0, ], wih 0 he age of birh and an arbirary conan, inerpreed a he maximum poible life duraion for he agen. Becaue we are inereed in underanding he way he iming of deciion influence he choice of he conumpion profile, we will denoe by [ 0, ), he dae of he deciion. Le u fir aume ha he agen, alive a dae, know for ure her age of deah. H1 We uppoe ha if an agen know wih cerainy her dae of deah, her ineremporal preference can be repreened by an ineremporal uiliy funcion aumed o be addiive, and aionary, wih no "pure ime preference" : (, ) ( ( τ )) V c = u c dτ wih 0 0 u c( τ ) > and u c( τ ) < H2 (monooniciy according o lifepan). 6

7 c: > V c, > V c, H1 and H2 implie ha u i poiive. H2 mean ha for a given conumpion profile, oucome will be alway ranked according o lifepan. When inroducing uncerainy our model will be a naural candidae for uing rank dependen uiliy Lifeime uncerainy he agen acually doe no know wih cerainy her age of deah. We aume ha for a living agen, a each age, he age of deah i an aboluely coninuou random variable defined on he inerval [, ]. We denoe by ( ) 0 π > he probabiliy deniy funcion of he random variable, aumed o be differeniable a lea once and ( ) Π he cumulaive diribuion funcion. We hu have: ( ) ( ) Π = π τ dτ (4) 1 and π τ dτ = (5) Π can be inerpreed a "he probabiliy of being dead a age, knowing you are alive a dae ", and, ( 1 ( ) ) ". Π "he probabiliy of being alive a dae, knowing you are alive a dae We can derive from Baye formula ha for : ( 1 ( ) )( 1 ( )) ( 1 ) Π Π = Π (6) and ( ) π π = 1 Π ( ) ( ) (7) In he pecial cae where =, we ge: 7

8 ( ) ( ) π π ( ) = (8) 1 Π 1.3. he choice crieria We are hu facing a pecial problem of choice in uncerainy. If we aume ha he agen i an expeced uiliy maximizer a in Yaari [1965], we have: = ( ( τ) ) τ Π EV c u c d d (9) If we now reain a more general model of choice under uncerainy, in which agen ranform oucome and probabiliy diribuion a in Quiggin [1982,1992] or verky & Kahneman [1992], we have : = ( ( τ) ) τ ( Π) (10) RDU c u c d dh h) wih ( a probabiliy weighing funcion aumed o be coninuou an wice differeniable, and uch ha: 0 0, h = h ( 1) = 1 and h ( ( ) ) 0 Π. Le u noice ha (9) i a pecial cae of (10) when h( ( ) ) Inegraing (10) by par, we obain: ( ) ( ) = 1 ( Π ) Π =Π. RDU c h u c d (11) For he agen, he expeced preen value of he uiliy ream a ime i he produc of he uiliy of conumpion a ime wih he ubjecive weigh given by he agen o he even "being alive a dae ". Equaion (11) make explici our iniial inuiion wihin he coninuou ( ) ime framework. he facor f ( ) 1 h( ( ) ) = Π i he dicoun facor applied o uiliy of he conumpion a dae viewed from dae. I depend only on he probabiliy diribuion of he age of deah and he ubjecive ranformaion of hi probabiliy diribuion. I i coninuou, derivable and ricly decreaing from one o zero on he inerval [, ]. 8

9 Uing andard acualiaion formula we can alo define he rae of dicoun of uiliy a dae viewed from dae : ( ) ( ( ) ) π ( ) f 1 h Π θ( ) = f( ) = 1 h Π ( ) (12) hu he ineremporal uiliy funcional (11) can be rewrien: θ ( ) = exp θ ( τ) τ ( ) (13) RDU c d u c d, can alo be named he rae of ime preference. A in Drouhin (2001), i can be decompoed in wo facor. he fir one, π ( ), he probabiliy deniy aociaed wih he even "dying a dae, knowing ha your alive a dae ", can be inerpreed a he objecive par of ime preference. he econd one, h ( ( ) ) 1 h( ( ) ) Π Π, depend on he way he agen ranform probabiliy diribuion. I can be inerpreed a he ubjecive par of ime preference. A hi age we ju wan o noice ha he mahemaical rucure of hoe dicoun facor and rae provide a very inereing cae. On he one hand hi mahemaical rucure i much more general han he one ha prevail in radiional exponenial or hyperbolic model of ineremporal choice. In he re of he paper we will dicu in ome deail how hoe paricular model can be found o be pecial cae of our more general model. On he oher hand he mahemaical rucure i alo much more precie han he mo general cae udied by Yaari (1964) where he dicoun facor i only aumed o be poiive and differeniable. 2. he opimal conumpion profile We are now going o inveigae he properie of he opimal choice of conumpion pah made by an agen a dae. For ha, we have fir o define he feaible e of conumpion profile. 9

10 2.1. he feaible e of conumpion profile We aume ha a each ime he living agen receive a flow of non-financial income ( w ) aumed o be coninuou and differeniable and a flow of financial income proporional o her ae ( a ). hoe income are eiher ued for curren conumpion or aved for fuure conumpion. hu, a each ime, he andard ineremporal budgeary conrain hold: a w r a c [ 0, ], = + (14) If we um he conrain a each dae weighed by he economical dicoun facor exp r, auming for purpoe of impliciy ha ( 0) a = a = 0, we obain afer ome imple manipulaion ha, for an agen living he maximum poible life-duraion : r = r e w d e c d 0 0 (15) hi i he very andard life cycle budgeary conrain. In he re of he paper, we will ake r a he rae of inere for bond, aumed, for impliciy, o be conan. For an agen deciding a a dae >0, he conrain will be: () r r a + e w d = e c d (16). For he ame reaon we can expre he oal ock of ae a dae : () r r 0 0 a = e w d e c d (17) he opimal conumpion profile planned a dae We now conider an agen a dae who ha o decide her opimal conumpion pah beween and. We denoe c ( ) he opimal conumpion pah decided a dae for he ime inerval [, ]. hu c i he oluion of he following program: 10

11 ( ) = ( Π) max RDU c 1 h u c d c a w r a c u.c. = + Becaue of he coninuiy of w( ) and he coninuiy and ric concaviy of he uiliy funcion hi program can be hown o admi a oluion ha will be coninuou and derivable (cf Yaari (1964)). he reoluion of uch problem implie o reolve a yem of differenial equaion. If, for no looing generaliy of he reul, we refue o pecify pecial "eay o ue" form for he uiliy, earning and probabiliy diribuion of he age of deah funcion, i i impoible o derive explicily he funcional form of he conumpion funcion. Wha we can only do i o derive he rae of growh of he opimal conumpion pah planned a dae. Propoiion 1: he rae of growh of he opimal conumpion pah planned a dae i: (, ) G = θ ( ) c ( ) ( ) c r c u c u c (18) Proof : he Hamilonian of he agen' program i: Fir order condiion give ( h( Π ( ) )) u( c( ) ) + λ ( ) ( w( ) + ra( ) c( ) ) H= 1- (19) H c H a ( ) ( ) = 0 λ = 1 h Π u c (20) dλ λ = ( ) = λ = r (21) d λ aking he logarihm of (20) and differeniaing according o e we ge: ( Π ) π Π ( ) λ h u c dc = + λ 1 h u c d (22) 11

12 Comparing (21) and (22), and uing definiion (12) we deduce propoiion 1. Commen: Propoiion 1 i he mo general predicion one can make wihin he life cycle heory of conumpion and aving. he rae of growh of he conumpion pah i he difference beween he rae of inere (economic dicoun rae) and he rae of ime preference, boh divided by an index of he curvaure of he uiliy funcion uually referred a he coefficien of relaive rik averion or more properly, according o Gollier (2001) a he reiance o ineremporal ubiuion. he imporan poin i ha, a in Yaari (1965) he rae of ime dicouning i no more conan and can give a wide variey of poible dynamic for conumpion. Bu conrarily o Yaari (1965) i i no only he properie of he probabiliy diribuion of he age of deah ha maer. he way agen ranform ubjecively hi probabiliy diribuion will alo maer. If we wan o go furher, we have o pecify ome more rericion o he model. We will do ha following ucceively one of he wo poin of view in deciion heory: he normaive and he poiive one. 3. he normaive poin of view. he idea ha agen "behave raionally" i he corner one of normaive deciion heory. he problem i o dicu he meaning of "behave raionally" wihin a paricular conex. hi i uually done by pecifying ome axiom of raionaliy. In he cae of our model wo axiom can be pu forward: fir order ochaic dominance; and emporal coniency. Becaue i ue a rank dependen uiliy of choice our model fulfil necearily and by conrucion he axiom of fir order ochaic dominance. he cae of emporal coniency i more inereing. 12

13 3.1. emporal coniency. An agen i aid o be emporally conien if he behave in he fuure a he ha planned in he pa. Wihin he life cycle model of conumpion i mean ha: [ ] [ ] [ ] 0,,,,, : c = c (23). Alhough he problem of emporal coniency ha been alluded in ome way by Samuleon (1937) and Allai (1947), he formal noion ha been inroduce in economic by Sroz (1956) giving birh o wo imporan line of lieraure: 1) he fir line diinguih beween myopic/naïve conumer who are no necearily conien and are no aware ha in he fuure hey will no behave a hey had planned, and ophiicaed one ha will ry o commi hemelve. In hi line, ineremporal choice i formalized by a game beween muliple emporally daed elve of he ame peron. 2) he econd line of lieraure wa o derive neceary and ufficien condiion on he dicoun funcion for a naïve agen o be ime conien. he iniial propoiion of Sroz (1956) wa ha agen i emporally conien if and only if he dicoun rae i conan i.e. he dicoun funcion i exponenial. In fac hi aemen wa no rue, becaue Sroz (1956) mi-pecified he problem. o give an anwer o hi queion you have o olve a differenial equaion wih he dicoun facor f a unknown. Sroz (1956) reolve he problem aring from a pecial form f ( ) f ( ) fac we have o ar from more general formulaion (, ) =, bu in f = f. In hi more general cae i can be hown (Burne (1976)) ha a neceary and ufficien condiion for a naïve deciion maker o be emporally conien i ha he dicoun facor i muliplicaively eparable in conumpion dae and he planning dae. Puing i in oher word i mean ha if he dicoun facor i ime dian dependen (Sroz 13

14 (1956)), emporal coniency implie exponenial dicouning; bu if he dicoun facor i no ime diance dependen hen any hape of he dicoun funcion i compaible wih emporal coniency. Le' go back o our model. he diribuion probabiliy of deah and he rank-dependen uiliy give a pecial mahemaical rucure o he dicoun rae and facor. We can noice ha θ in he mo general cae i ime-diance dependen becaue i depend on Π. I i a rong preumpion for ime inconiency. Bu in he pecial cae of expeced uiliy, θ ( ) = π ( ), whaever he form of he probabiliy diribuion, i depend no more on he planning deciion dae, i i hu "ime diance independen", emporal coniency hold. he queion i: I here ome oher cae where emporal coniency hold? Propoiion 2: Agen i emporally conien if and only if her probabiliy diribuion ranformaion funcion i of he form h( x) = 1 ( 1 ) wih ( α > 0 ) α [ ] [ ] [ ] () h x = 1 1 x 0,,,,, : c = c = c (24) Proof : (ufficiency) 1 ( 1 ) h x = θ = απ (25) x α he rae of dicoun i independen of he planning dae o he choice of conumpion i ime conien. (neceiy) If he agen i emporally conien, he fulfill equaion (23): x α A c ( ) i ricly poiive and differeniable, i implie ha: c( ) c ( ) [ 0, ], [, ], [, ] : = (26) c c 14

15 aking ino accoun (18) i implie ha: [ ] [ ] [ ] 0,,,,, : θ ( ) ( Π ) π ( ) ( Π ) π h h = = 1 h Π 1 h Π Becaue of (7) i alo implie ha: [ ] [ ] [ ] 0,,,,, : ( ) ( ) = θ (27) ( Π) ( Π ) h h = 1 h Π 1 h 1 ( ( Π ))( Π ) (28) hi equaion hould hold in he paricular cae where =. Conidering hi cae and noing ha () 0 Π =, equaion (28) alo implie : [ ] [ ] 0,,, : ( ) ( ) ( 0) h Π h = 1 h Π 1 Π ( ) (29) hi i a fir order differenial equaion wih a e of oluion fully decribed by 1 ( 1 ) h x =, wih α = h 0. x α Commen: he diribuion of he probabiliy of dying and i reamen wihin he Cumulaive Propec / Rank Dependen Uiliy heory of choice added wih an axiom of emporal coniency give behavioral foundaion o a model of ineremporal choice ha i raher imple and no le inuiive han he andard dicouned uiliy model. In paricular, equaion (25) i very inereing becaue i provide u wih a new concep of "ubjecive preference for preen conumpion", which we can name "muliplicaive ubjecive ime preference". hu he uiliy funcional can be rewrien: α d RDU ( c) e πτ τ τ u ( c( ) ) d = (30) 15

16 If α > 1, hi mean ha he agen give a pychological weigh o preen conumpion more imporan han he inananeou probabiliy of dying. he behaviour of he agen can in hi cae be called "peimiic" in he ene ha hey end o overweigh heir probabiliy of dying. A he oppoie, if α < 1, he agen will demonrae a kind of "preference for fuure conumpion", hey underweigh heir probabiliy of dying ("opimiic" behaviour). he inere of our approach wihin he "normaive poin of view" i ha ime preference i nomore poulaed a an ad-hoc parameer. I i derived wihin a normaive inerpreaion of our model auming only ineremporal choice wih uncerain lifeime when agen have RDU preference and ha agen are emporally conien. I i alo imporan o noice ha any probabiliy diribuion of he age of deah i compaible wih ime conien behaviour, a in he "pecial cae of expeced uiliy. For more clariy we can ummarize he hree exiing heory of ineremporaly conien choice ino a ynheic able (Appendix A). o go furher i i neceary o pecify ome more properie of "age of deah" probabiliy diribuion funcion. he mo imple cae (and no o unrealiic) i he one of a conan condiional probabiliy of dying a each dae. For ha o be poible we have o allow he maximum duraion of life,, o go o infiniy Infinie horizon. Le u now urn our aenion o pecial cae where i infinie and hu, conidered from birh dae he age of deah i an aboluely coninuou random variable defined on + 2. Le' in a fir age uppoe ha: +, π ( ) = π = c 2 In fac all previou finie horizon problem can be rewrien a ome degeneraed infinie horizon problem, wih he probabiliy deniy funcion going o zero afer a definie dae. 16

17 In hi cae dying i ju a andard Poion proce wih π he conan probabiliy of dying a each dae. he ineremporal uiliy funcional (30) (emporally conien agen) can be rivially rewrien: + απ( ) ( ) RDU c = e u c d (31) In hi pecial cae he andard Dicouned Uiliy model i formally equivalen o our (θ = απ). hi mean ha DU can be conidered a an approximaion of our more general model. During he wenieh cenury, he DU model ha been a very ucceful ool for underanding ineremporal choice, and all economic problem in which hoe choice are involved. Becaue i embodie he imporan idea ha agen dicoun fuure uiliy wihin he mo imple analyical framework poible (he rae of dicoun i conan), DU model ha become one of he backbone of economic heory. Bu, a our economic knowledge increae, our ool have o be more and more precie and he generaliaion of he DU model ha cerainly become a ake in he recen year. Our opinion i ha he lack of foundaion for he raher ad hoc concep of "pure ime preference" ha cerainly been an obacle for hi effor o be ucceful. hu we ake ome aifacion o ee ha, building on oher foundaion, we have been able o provide a more general model ha admi dicouned uiliy a a pecial cae. A well known juificaion for a poiive ubjecive dicoun facor (ubjecive preference for preen conumpion in he claical ene) i ha, in he cae of infinie lifeime, for he inegral (31) o be definie when he conumpion ream i bounded, θ ha o be poiive. Here, hi condiion i neceary fulfilled becaue α and π are boh poiive. Bu in our model, only α i he ubjecive par of ime preference. So if α > 1, we have "preference for fuure conumpion" in he ene we defined above and he inegral i ill definie. Preference for preen conumpion i no more a mahemaical requiie. If all agen have a poiive rae of dicouning, i i olely becaue heir life duraion i uncerain (he objecive par of ime 17

18 preference π, he inananeou condiional probabiliy of dying i alway poiive). hi reul can be generalized for much more general probabiliy diribuion. Propoiion 3: if he conumpion ream i bounded and if here exi an age ˆ uch ha for τ ˆ, π τ τ ( τ) 0 hen he inegral (30) i alway definie, even when =+. Proof: obviou. Commen: hi cae i very realiic. If he condiional probabiliie of dying a each dae were conan, hen life expecancy would be he ame a all age. he fac ha life expecancy decreae wih age implie ha he condiional probabiliie of dying increae wih age. Preference for preen conumpion i no more a requiie. 4. he decripive poin of view: ime inconien preference Occam' razor principle i no he only requiie for doing "good cience". he abiliy o "fi he daa" i cerainly even more imporan. For he la fifeen year a lo empirical evidence have been pu forward uggeing ha agen may be emporally inconien (ee Frederick & allii, for a urvey) 3. hu, from a more decripive poin view, i can be inereing o dicu hi queion wihin our model. A we have demonraed, only probabiliy weighing funcion of he cla h( x) 1 ( 1 ) = i compaible wih ime coniency. We have o dicu on an x α empirical bai if uch a probabiliy weighing funcion fi he effecive behaviour of he agen. hi dicuion i far beyond he purpoe of hi paper and a lo of empirical (experimenal) work i ill o be done. Bu a a aring poin we can make ome remark. 1. he power ranformaion funcion aociaed wih emporal conien behaviour i no andard in experimenal lieraure concerning CP/RDU. Mo of he ime, he ranformaion funcion i found o be -haped. 3 In fac i i no very clear in he lieraure if he emporal coniency i eed, many auhor believe ha in rejecing exponenial dicouning hey prove ha agen are inconien. A we have hown hing are much more uble. 18

19 h( Π ) 1 h( Π ) 1 Power ranformaion funcion 1 Π S-haped ranformaion funcion 1 Π 2. Bu we have o ak if he probabiliy weighing funcion i rik pecific. In hi cae, rik on life duraion i very pecial and may be reaed differenly by agen han rik involving money. hi poin i dicued in Bleichrod and Pino [2000] wihin a model where life duraion i he only argumen of he uiliy funcion. S-hape weighing funcion eem o fi he daa in hi cae oo. 3. If dicuing on an empirical bai i may be imporan o underand wha he model predic exacly. Propoiion 1 characerize he lope of he conumpion profile planned a a given dae. Bu fundamenally, hi planned conumpion profile i "in he mind" of he agen. Wha we have o characerize now, i he effecive conumpion profile, i.e. he equence of agen' conumpion ha can effecively be oberved he effecive conumpion pah. When an agen how ime inconien behaviour her pa planned conumpion for a given dae can acually differ from her effecive choice a hi precie dae, he only one ha i obervable. hi can be a problem, becaue wha really maer for an economic model i' hi predicive power 19

20 We will now follow an agen all over her life cycle from dae 0 o (he agen ha o be alive o have a conumpion). A each dae he agen plan i conumpion for he re of her life and effecively conume c agen, we have: c. Defining c a he "effecive conumpion pah" of a living [ 0, ], c c = (32). i he aring poin of he conumpion profile planed a dae. We have already calculaed he lope of hi funcion (18). hu we can wrie: ( τ) ( τ) r θ c( ) = c exp dτ γ ( τ) ( τ) r θ = c() exp dτ γ (33) he conumpion profile planned a dae alo aify he budgeary conrain: () r( ) r( ) (34). a + e w d = e c d he oal ock of ae a dae depend on he pa effecive conumpion pah a dae, hu () r( ) r( ) (35). a = e w d e c d 0 0 Collaping hoe la hree equaion in one, we ge: 0 0 ( τ) r r r θ τ e w( ) d e c( ) d= c() exp dτ r d γ (36) Differeniaing hi expreion give (calculaion in he appendix): Propoiion 4: he rae of growh of he effecive conumpion pah a dae i: () () c (, ) = G ( τ) r θ τ d d dτ c ( ) e γ c r c e d r d (37) Proof: Calculaion are in he appendix 2. 20

21 Commen: he econd erm of equaion (37) i ju he difference a dae beween he rae of growh of he conumpion planned a dae and he effecive rae of growh of conumpion. I i eay o verify ha when agen are emporally conien hi erm vanihe, becaue in hi cae: ( τ) r θ τ [ 0, ), τ (, ), d d = 0 γ (38) 5. Concluion. Our paper can be read from boh a decripive and a normaive poin of view. From he decripive poin of view, i emphaize he key role of lifeime uncerainy for explaining ime preference. In paricular, he ranformaion of he probabiliy of urvival offer u a "new view" concerning he debae on ubjecive ime preference. In he "old view", agen were aumed o exhibi "pure ime preference", a peculaive and ad hoc concep. In our model, ime preference, i founded on an objecive facor, derived from he probabiliy diribuion of he age of deah. Bu hi objecive facor i compleed by a ubjecive facor derived from he way he agen ranform hi objecive probabiliy diribuion wihin he CP/RDU model. Becaue of i grea generaliy our model can accoun for a lo of he ocalled anomalie in ineremporal choice. From a more normaive poin of view, which i generally he one adoped in economic heory, a model of ineremporal choice wih uncerainy have o fulfil wo main crieria of raionaliy: fir order ochaic dominance and emporal coniency. he fir one i, by conrucion, neceary fulfilled wihin he CP/RDU conex. he mo imporan reul of hi paper i o how ha emporal coniency hold for a cla of probabiliy weighing funcion much wider han he andard expeced uiliy cae (probabiliy weighing funcion i ideniy). hu our model provide foundaion for a muliplicaive rae of ime preference ha hared he 21

22 ame normaive properie han he andard DU approach bu i in fac much more general and inighful. I i much more general becaue i allow he rae of dicouning o vary from period o period. I i more inighful becaue i pu back in economic heory, a a cenral place, omehing everyone know even if we don like o know i. We are all moral bu uually we don' know when we are going o die. 6. REFERENCES Abdellaoui, M., (2002), A Genuine Rank-dependen Generalizaion of he von Neumann- Morgenern Expeced Uiliy heorem, Economerica,, 70, 2, Allai, M., [1953], Le comporemen de l'homme raionnel devan le rique : criique de poula e axiome de l'ecole Américaine., Economerica, vol. 21, n 4, pp Anlie (1991), Derivaion from raional economic behaviour from hyperbolic dicoun curve, American Economic Review, 81, 2, Barro, R. (1999), Ramey mee Laibon in he Neoclaical Growh Model, Quarerly Journal of Economic, 114, 3, Bleichrod & Pino (2000), A Parameer-Free Eliciaion of he probabiliy weighing funcion in Medical Deciion Analyi, Managemen cience, 46, 11, Burne, H. Suar (1976), A noe on conien Naïve Ineremporal deciion Making and an Applicaion o he Cae of Uncerain Lifeime; Review of Economic Sudie, 43, 3, Drouhin, N., (2001), Lifeime uncerainy and ime preference. heory and Deciion 51, Fiher, I., (1907), he rae of Inere: I Naure, Deerminaion and relaion o Economic Phenomena., Macmillan, New-York. Frederick S., G. Loewenein, ed O'Donoghue, (2002), ime Dicouning and ime Preference: A Criical Review, Journal of Economic Lieraure, 40, 2, Gollier, C. (2001), he Economic of Rik and ime, MI Pre. Kahneman, D., verky, A., (1979), Propec heory: An Analyi of Deciion under Rik, Economerica, 47, Kaz, E., (1979), A noe on uncerain lifeime, Journal of Poliical Economy, 87, 1, Koopman,. C., (1960), Saionariy Ordinal Uiliy and Impaience, Economerica, vol. 28, n 2, pp Koopman, C., (1972), Repreenaion of preference ordering over ime, in McGUIRE & RADNER (ed), Deciion and Organizaion, a Volume in Honor of Jacob Marchak., pp

23 Laibon, D. (1997), Golden egg and hyperbolic dicouning, Quarerly Journal of Economic,112, Loewenein & Eler (ed) (1992), Choice over ime, Ruel Sage Foundaion. Loewenein & alii (ed) (2003), ime and Deciion, economic and pychological perpecive on ineremporal choice, Ruel Sage Foundaion. Prelec, D. and G. Loewenein, (1991), Deciion Making over ime and Uncerainy: A Common Approach, Managemen Science, 37, 7, Quiggin, J., (1982), "A heory of Anicipaed uiliy", Journal of Economic Behavior and Organizaion, 3( 4), Quiggin, J., (1993), Generalized expeced uiliy heory, he rank dependen model, Kluwer Academic publiher. Sroz, Rober H., [1956], "Myopia and Inconiency in Dynamic Uiliy Maximiaion.", Review of Economic Sudie, vol. 23, pp verki, A., Kahneman, D.(1992), "Advance in propec heory, Cumulaive repreenaion of Uncerainy", Journal of Rik and Uncerainy. Yaari, M. E., (1964), "On he Conumer Lifeime Allocaion Proce", Inernaional Economic Review, 5, 3, Yaari, M. E., (1965), Uncerain Lifeime, Life Inurance, and he heory of he Conumer. Review of Economic Sudie, 32,

24 Appendix 1 - Comparion of he heorie of ineremporal choice when agen are emporally conien. Sandard DU model wih "pure ime preference" Reference Samuelon (1937), Ineremporal uiliy funcional Ineremporal rae of dicoun (i. e. rae of ime preference) Agen how "ubjecive preference for preen conumpion" if Koopman (1972) DU model wih "pure ime preference" + Uncerain life ime (EU Verion) Yaari (1965) θdτ τ d = ( ) EV ( c) = e ρ+ π τ τ u( c( ) ) d DU c e u c d No "pure ime preference"+ Uncerain life ime (RDU Verion)+emporal coniency Drouhin (2006) (hi paper) τ ( ) α π ( τ) dτ RDU c = e u c d = θ = θ ( ) = ρ + π ( ) θ ( ) = απ ( ) θ c θ > 0 ρ > 0 α > 1 24

25 0 0 Appendix 2 ( τ) r r r θ τ e w( ) d e c( ) d= c() exp dτ r d γ Noaion 1 r () I = e w d 2 0 r () I = e c d 0 di di 1 d = 0 = e 2 r d () c () ( τ) r θ τ I3 () = exp dτ r d γ I 4 () = exp I4 r d = ( τ) r θ τ dτ γ () = () () Z c I 3 θ ( τ) exp d γ ( τ) di4 () exp () di3 r = τ r d () θ() () + I 4 r d d = exp di [ r] () () 4 + exp 4 d I r d di4 r r θ τ = + d / d dτ d γ γ τ G = G (,) + (, τ) dτ () dz dc di3 = I3 () + c() d d d dc () = I3 c r d di () + () () d () () exp[ ] 4 c exp I 4 r d (, τ ) G dc r θ τ = = dτ γ ( τ) () () () I I Z ( τ ) c ( τ ) () () di di dz = = d d d 25

26 (, ) di4 dc () exp I4 () r d () d c = d exp I4 () r d = G G (, τ) d τ exp I 4 ( ) r d () exp I4 r d () c Uing equaion (33) we alo have I4 () = ha prove (38). c 26