Uncertin Lifetime nd Intertemporl Choice: Risk Aversion s Rtionle for Time Discounting Antoine Bommier CNRS nd University of Toulouse (GREMAQ) June 2005 Abstrct This pper mkes explicit the links between preferences over lotteries on length of life nd intertemporl choice. It shows tht the pproch used by trditionl life cycle models to ccount for uncertin survivl corresponds to strong ssumption of risk neutrlity with respect to length of life. Relxing such n ssumption leds us to develop more generl formultion of lifetime utility in which time discounting is directly relted to preferences over length of life. In prticulr, it provides n explntion for exponentil nd hyperbolic discounting which re found to result, in first pproximtion, from constnt nd hyperbolic risk version with respect to length of life. JEL clssifiction : D91, D81, J17. Keywords : intertemporl choice, life cycle model, time discounting. Ibenefited from converstions with Jerôme Bourdieu, Pierre Dubois, Sylvie Lmbert, Ronld Lee, Guy Stecklov nd prticipnts of the 16th Annul Conference of the Europen Society for Popultion Economics, the séminire d Economie Théorique of the University of Toulouse, the Journées Jourdn. I m lso grteful to n editor, José-Victor Ríos-Rull, nd three nonymous referees for their comments nd suggestions. All correspondence should be sent to Antoine Bommier, GREMAQ, Université des Sciences Sociles, Mnufcture des Tbcs - bt F, 21 Allée de Brienne, 31000, Toulouse, Frnce. E-mil: Antoine.Bommier@univtlse1.fr 1
1 Introduction It hs long been recognized by economists, nd in prticulr by Fisher (1930), tht uncertinty bout the length of life hs to ply key role in determining the trde-off between present nd future consumption. However, there re simple theoreticl rguments relting mortlity to time discounting tht hve remined unexplored. Most life cycle models tht ccount for uncertin survivl follow the pth of Yri s (1965) seminl pper. 1 According to Yri, survivl uncertinty cn be simply incorported into life cycle models by weighting the utility derived from future consumption by survivl probbilities. In Yri s model, t ny ge, remining lifetime expected utility cn then be written s: + s (t)α(t )u(c(t))dt (1) where u(c(t)) is the instnt utility of consumption t ge t nd s (t) the probbility of being live t ge t, conditionl on being live t ge. The function α(.), usully clled the subjective discount function, is presented in Yri (1965) nd in most ppers on intertemporl choice s n exogenous fctor ccounting for imptience, inherent in humn behvior nd unrelted to survivl uncertinty. This discount function is often ssumed to be exponentil, s it is the only shpe to led to time-consistent preferences (Strotz 1956), lthough more complex forms (e.g., hyperbolic) re now incresingly considered (Loewenstein nd Thler, 1989, Loewenstein nd Prelec, 1992, Libson, 1997 nd 1998, Hrris nd Libson, 2001, Angeletos & l, 2001). In this pper, I rgue tht using eqution (1) s formultion of expected lifetime utility involves mking strong implicit ssumptions on preferences over lotteries on length of life, including risk neutrlity ssumption. I explore how survivl uncertinty my ffect individul behvior nd, in prticulr, consumption smoothing, when we relx this ssumption of risk neutrlity. To do so, I propose to proceed with the following exercise: consider individuls who, in the bsence of uncertinty, would hve simple nd nice preferences (tht 1 Leung (1994) or Butler (2001) provide interesting recent exmples. 2
is dditive nd time-consistent preferences) nd try to nswer the following question: how would such individuls behve in world where length of life is uncertin? Obviously, there re multitude of possible nswers to this question, s nswers hve to depend on individuls ttitude towrd risk. However, we shll see tht ssuming preferences to be history independent nd time-consistent implies tht individuls re either risk neutrl with respect to length of life (in sense tht will be defined lter) nd hve dditive preferences or hve constnt non-zero risk version with respect to life yers nd non-dditive preferences. The former cse corresponds to Yri s model. The ltter cn be seen s n extension of this model obtined by relxing the ssumption of risk neutrlity with respect to life durtion. Although it is more complicted, especilly since it is no longer dditive, it is fr from being intrctble nd is worthy of explortion. In prticulr, I show tht in the cse of n exogenous mortlity pttern, preferences in this more generl model re represented by recursive utility functions tht cn be interpreted s extensions of the clssicl Yri utility function (eqution 1) where the discount function is endogenous. I lso show tht when considering the limit cse where length of life is much more vlued thn consumption, preferences cn be pproximted by n dditive utility function with specific discount function relted to mortlity nd preferences over length of life. Thus, even if we were only interested in the simple dditive cse, this detour through the more generl cse would be worthwhile s it provides rtionle for the subjective discount fctor. The interest of developing generl frmework which includes preferences tht do not exhibit risk neutrlity with respect to length of life cn be further emphsized by considering situtions where preferences over lotteries on length of life re not necessrily time-consistent. In prticulr, I explore the cse where risk version with respect to the end of life increses s deth becomes closer in the future. Such cse cn be esily formlized by ssuming decresing bsolute risk version with respect to length of life, insted of the constnt risk version property tht chrcterized the time-consistent cse. Anlysis of intertemporl choice under exogenous mortlity cn then be developed in similr wy nd provides interesting insights. In prticulr, I show tht hyperbolic discounting cn be seen, in first 3
pproximtion, s consequence of hyperbolic bsolute risk version with respect to length of life. The pper is structured s follows. In Section 2, I present the min ssumptions. In the third section, I introduce the notion of risk version with respect to length of life. The fourth section covers in detil the time-consistent cse, while time-inconsistent preferences will be discussed in the fifth section. The min findings of the pper, s well s pths for future reserch, re discussed in the concluding section. 2 Min ssumptions For n individul of ge wht hppened before ge belongs to history nd cnnot be mtter of choice. Individuls cre bout their lives in the future. We ssume tht, for n individul of ge, life in the future is bounded nd continuous consumption profile c C B ([, + [, R + ) nd n ge t deth t [, + [. The set of possible lives in the future of n individul of ge is denoted L nd equls the product spce: L = C B ([, + [, R + ) [, + [ Such formultion does not constrin the consumption fter deth to equl zero. Tht my seem odd t first sight, but tht is of no consequence since I will ssume tht individuls do not cre for consumption fter deth. An element of L will be typiclly denoted (c, T ). The set C B ([, + [, R + ) is endowed with the sup norm nd [, + [ with the Eucliden norm. The set L, which is the product of two normed spces, is endowed with the sup norm. The discussion will ber on choice under uncertinty. Tht mens tht I will consider preferences over M L, the set of probbility mesures on the Borel σ lgebr of L. The set of degenerted mesures will be denoted ML d.anelementofm L d will be typiclly denoted δ (c,t ). For ny two mesures M CB ([,+ [,R + ) nd b M [,+ [ theproductmesureinm L will be denoted b. In order to stress the role of uncertinty bout the length of life, I will limit the study 4
to individuls who would hve very stndrd preferences in the bsence of uncertinty, so tht identifying the consequences of uncertin lifetime will be strightforwrd. In short, I will only consider well-behved individuls defined s follows: Definition 1 Throughout the pper, the terms well-behved individuls will refer to individuls for whom: 1. Ageing is smooth process tht does not engender ny kind of discontinuity in terms of preferences. 2. At ny ge, preferences over M L (denoted by  ) re independent of pst consumption nd dmit n expected utility representtion with smooth Bernoulli utility function U. Tht mens tht there exists smooth function ½ L R U : (c, T ) U (c, T ) such tht for ny two m 1, m 2 M L m 1  m 2 U (c, T )dm 1 > U (c, T )dm 2 3. At ny ge, preferences over the set of degenerte lotteries, ML d, cn be represented by the utility function V : V (c, T )= T u(c(t))e α(t ) dt (2) where u(.) is smooth positive incresing function nd α, the rte of discount, is constnt. Such definition implies tht, in the bsence of uncertinty, well-behved individul would hve dditive preferences. He would lso hve time-consistent preferences with respect to both the choice of consumption pth nd the trde-off betweenlengthoflifend 5
consumption. The positivity of the function u lso implies tht whtever consumption level is proposed, longer life increses individul utility. 3 Risk version with respect to length of life For ny ge the Bernoulli utility function U must led to the sme preferences s the function V when we consider degenerte lotteries. This implies tht the Bernoulli utility function U must be obtined by n incresing trnsformtion from V, or from e α V,since the functions V nd e α V obviously represent the sme preferences. 2 The trnsformtion tht leds from e α V to U my depend on ll tht chrcterizes n individul of ge : his ge nd his pst consumption. However, s we ssumed tht preferences t ge re independent of pst consumption (point 2 in Definition 1), we cn ssume without loss of generlity tht this trnsformtion only depends on. Denoting this trnsformtion by x f(x, ) we obtin: T U (c, T )=f( u(c(t))e αt dt,) (3) where f(x, ) is n incresing function in x for every. The regulrity ssumptions included in the first two points of Definition 1 llow one to ssume tht f is smooth function. In Yri (1965), the function f ws implicitly ssumed to be liner in x, sinceyri considered eqution (2) s the most nturl form of Bernoulli lifetime utility functions (Yri, 1965, p 137). In this pper, I will llow the function f to tke more complex forms. The problem then involves deducing wht the resonble cndidtes for the function f(x, ) re from economic rguments. Some intuition cn be provided by considering the imginry cse of n individul who hs n exogenous consumption profile but cn choose between severl mortlity ptterns. It is cler tht such simple choices re not observed in relity since, in generl, individuls choose t the sme time between different mortlity ptterns nd different consumption profiles. However, the cse of n exogenous consumption profile cn be viewed s thought 2 For given, the fctor e α is just positive constnt tht does not ffect preferences. 6
experiment tht will provide some insights on the mening of the function f (preferences llowing choices involving different consumption profiles to be modeled will be considered in the remining sections). The consumption profile being fixed, preferences simply involve compring different lotteries on the length of life, one-dimensionl vrible. This is similr to preferences over monetry lotteries, for which the theory is well developed. In prticulr, ll the informtion cn be summrized by locl mesures of risk version: Definition 2 For ny consumption profile, c, nd ny ge,, we define the risk version with respect to the ge t deth, T,s: RAL (c, T )= 2 U (c,t ) T 2 U (c,t ) T This mesure of risk version is clerly nlogous to the Arrow-Prtt coefficient of bsolute risk version tht is found in ny economic textbook tht dels with the cse of preferences over monetry lotteries. The only difference is tht the vrible of interest is no longer welth, but life durtion. As with the Arrow-Prtt coefficient, this mesure of risk version cn be used to compute the certinty equivlents (mesured in life yers) of lotteries on life durtion. Without mking further ssumptions, we cn discuss some properties of this risk version with respect to ge t deth. Noting: I = T simple clcultions from eqution (3) give: u(c(t))e αt dt nd v(t )=e αt u(c(t )) RAL (c, T )= v0 (T ) v(t ) v(t )f xx(i,) f x (I,) (4) where f x nd f xx denote the first nd the second derivtives of the function f(x, ) with respect to x. We cn see, tht whtever the properties of the function f(x, ), well-behved individuls 7
will hve risk version with respect to ge t deth tht will depend on the consumption profile considered. This mkes sense, since rtionl individuls vlue the risks on ge t deth ccording to the plesures to be enjoyed in the event of survivl. In prticulr, there is no function f which ensures some positive risk version for ny consumption profile nd, more generlly, it would be difficult to find interesting properties for risk version with respect to ge t deth tht hold for ny consumption profile. We could however esily define some generl properties if we restrict ourselves to consumption profiles such tht v(t) =e αt u(c(t)) is constnt. Definition 3 We cll constnt flow of stisfction consumption profile ny consumption profile such tht e αt u(c(t)) is independent of t. Note tht, lthough the bove definition mkes use of the function u nd the sclr α, the concept of constnt flow of stisfction consumption profile is well-defined ordinl notion, relted to individuls preferences on ML d, but independent of the prticulr representtion of preferences which ws chosen. In fct, one my define the mrginl rte of substitution between length of life nd consumption t time zero by 1 ² 1 times the mount of instnt consumption tht n individul is willing to give up between time 0 nd 0+² 2 in order to live T +² 1 ² 2 yers, insted of T yers, when ² 1 nd ² 2 re infinitesimlly smll. This mrginl rte of substitution is unmbiguously n spect of individuls preferences on ML d,unrelted to the choice of prticulr representtion of preferences. It is then strightforwrd to check tht the constnt flow of stisfction consumption profiles re the consumption profiles such tht the mrginl rte of substitution between length of life nd consumption t time zero is independent of the length of life. Stted otherwise, constnt flow of stisfction consumption profiles re consumption profiles for which ny dditionl lpse of life of is seen s bringing the sme quntity of plesures. The notion of constnt flow of stisfction consumption profile llows us to circumvent the consequences of vritions in the qulity of life nd mkes it possible to define concept of risk version with respect to life yers s follows: 8
Definition 4 A well-behved individul will be sid to exhibit risk neutrlity (respectively: positive risk version /constnt bsolute risk version / decresing bsolute risk version / hyperbolic bsolute risk version ) with respect to length of life if, for ny constnt flow of consumption profile, nd ny current ge, he exhibits risk neutrlity (respectively: positive risk version /constnt bsolute risk version / decresing bsolute risk version / hyperbolic bsolute risk version ) with respect to ge t deth. It is pproprite to stress tht these re qulittive properties which describe wht kind of risk version would chrcterize preferences over lotteries on length of life if the qulity of life ws flt. If such properties cn be independent of the level of qulity of life, the degree of risk version will depend on this level, s shown in eqution (4). Thus, in generl, it is impossible to define degree of risk version with respect to life durtion which is independent of the qulity of life. To void potentil confusion between qulittive nd quntittive properties for risk version, I will systemticlly use the terms risk version with respect to ge t deth to refer to quntittive properties defined conditionlly on consumption profile (Definition 2) nd risk version with respect to length of life to refer to qulittive properties tht hold for ny constnt flow of stisfction consumption profile (Definition 4). From eqution (4) the link between the function f nd the notion of risk version with respect to length of life is immedite: Proposition 1 An individul of ge is risk neutrl (respectively: risk verse/risk prone) with respect to length of life if nd only if f is liner (respectively: concve/convex) in x. The concvity of f with respect to its first rgument cn therefore be interpreted in terms of risk version with respect to length of life. It is then cler tht Yri s formultion, which ssumes tht f is liner, involves ssuming risk neutrlity with respect to length of life. The reminder of the pper explins how the theory cn be extended when such n ssumption is relxed. 9
4 The cse of time-consistent preferences In this section I first show tht ssuming individuls to be time-consistent leds to restricting the study to prticulr specifictions for f. Stillf does not need to be liner nd we obtin more generl representtion of preferences thn the one used by Yri. I then discuss the consequences for intertemporl choices under uncertin lifetime. 4.1 Time-consistent gents The notion of time-consistency tht I use in this pper is stndrd. An gent is timeconsistent if he does not exhibit preference reversls, which could mke him chnge his plns s he gets older, in the bsence of ny new informtion. In order to formlly stte this non-reversl preference property we shll need few technicl definitions. For ny <bnd mesure µ M L, I will sy tht µ is degenerte till b, with intermedite consumption c 0 C([, b], R + ) if µ ({(c, T ) T b nd c(t) =c 0 (t) for ll t [, b]}) =1 According to such probbility mesure there is no chnce of dying before ge b nd consumption between ge nd b equls c 0 with probbility one. For ny µ M L tht is degenerte till ge b with intermedite consumption c 0 we define corresponding mesure µ b M Lb in the following wy. For ny mesurble subset B of L b we first define A(B) L by: A(B) ={(c, T ) L (c, T ) B nd c(t) =c 0 (t) for ll t [, b]} ndthendefine µ b (B) by µ b (B) =µ (A(B)). The mesure µ b describes wht my hppen fter ge b to n individul whose future is described by the probbility mesure µ. Definition 5 An gent is time-consistent if there do not exist two ges <b,consumption profile c 0 C([, b], R + ) nd two mesures µ 1,µ 2 M L such tht: 10
1. µ 1 nd µ 2 re degenerte till ge b with intermedite consumption c 0, 2. µ 1  µ 2, 3. The corresponding mesures µ 1 b nd µ2 b do not stisfy µ1 b  b µ 2 b. The following result shows tht preferences of time-consistent gents hve simple representtion: Theorem 1 Well-behved individuls (Definition 1) hve time-consistent preferences if nd only if there exists constnt k such tht t ny ge,, the reltion of preferences  dmits n expected utility representtion with Bernoulli utility function U given by: U (c, T )= eα k h ³ 1 exp k R i T u(c(t))e αt dt if k 6= 0 U (c, T )= R T u(c(t))e α(t ) dt if k =0 (5) Proof. See Appendix A. An immedite consequence of Theorem 1 is: Corollry 1 Well-behved individuls tht hve time-consistent preferences exhibit constnt bsolute risk version with respect to length of life. Proof. For ny constnt flow of stisfction consumption profile let us note ν(c) = e αt u(c(t)) which, by definition, is independent of t. For ny nd T we hve: U (c, T )= eα k e k(t )ν(c) ) if k 6= 0 U (c, T )=(T )ν(c) if k =0 In both cses this implies tht: RAL (c, T )= 2 U (c,t ) T 2 U (c,t ) T = kν(c) (6) so tht the risk version with respect to ge t deth is independent of the ge t deth. 11
Intuitively this lst result comes from the simple fct tht s one gets older nd closer to deth the quntity of remining life decreses. Thus ny risk version which depends on the quntity of remining life yers will led to inconsistent behvior. The constnt k in the utility function (5) is simply relted to risk version with respect to length of life. If k =0, then individuls re risk neutrl with respect to length of life. If k>0 then preferences exhibit positive risk version, while if k<0 preferences exhibit risk proneness. For given constnt flow of stisfction consumption profile, the lrger k, the stronger will be risk version with respect to ge t deth. Also, we see from eqution (6) tht for given positive k, the higher the qulity of life, the greter risk version with respect to ge t deth. This less intuitive result expresses the fct tht life with little plesure is closer to deth, in terms of utility, thn life with high level of plesure. However such result should not be overly stressed since it is likely tht differences between high nd low levels of consumption re, in terms of welfre, reltively insignificnt s compred with the difference between life nd deth. Hence, it my be the cse tht, in relity, risk version with respect to ge t deth is only wekly relted to the consumption level. 4.2 Intertemporl choice under exogenous mortlity The constnt k tht ppers in the utility function (5) hs been relted to the notion of risk version with respect to length of life. Thus, it clerly mtters for mortlity choices. In this section, we will see tht even when mortlity is exogenous (so tht individuls cn only choose between lotteries on consumption profiles) this constnt k is still key prmeter for modeling intertemporl choice. Throughout this section, s in Section 4.3, mortlity is ssumed to be exogenous. The mortlity pttern, µ, is described by n ge-specific hzrd rte of deth µ(t). It is ssumed tht µ(t) + when t +. For ny ge the survivl function t ge t (the probbility of being live t ge t 12
conditionl on being live t ge ) isgivenby: s (t) =exp( t µ(τ)dτ)dt We denote by m µ, the probbility mesure on [, + [ defined by m µ, ([t 1,t 2 ]) = s (t 1 ) s (t 2 ) for t 1 t 2 m µ, ([t 1,t 2 ]) gives the probbility tht deth occurs between ge t 1 nd t 2, knowing tht the individul ws live t ge. Theorem 2 Consider time-consistent well-behved individul (Definition 1) who fces n exogenous mortlity pttern µ. Define  µ, the reltion of preferences on M CB ([,+ [,R + ) by: ν  µ, λ (ν m µ, )  (λ m µ, ) The reltion of preferences  µ, dmits n expected utility representtion with Bernoulli utility function U µ, given by: + µ U µ, (c) = s (t)u(c(t))e α(t ) exp k Proof. See Appendix B. t u(c(τ))e ατ dτ dt (7) The utility function (7) cn be seen to be time-consistent generliztion of Yri s utility function (1), which is obtined when k =0. Although gents cnnot choose their mortlity (mortlity being exogenous), the presence of the constnt k in eqution (7) implies tht risk version with respect to length of life hs n impct on preferences over consumption profiles. The intuition is tht even if there is no control on the lottery on length of life, intertemporl relloction of consumption cn be used to modify the distribution of lifetime utility outcomes. Further nlysis of the role of the constnt k in eqution (7) provides dditionl insights. 13
Compred to the cse k =0, considering non-zero k leds us to introduce n endogenous discount function in the lifetime utility function. When k > 0, the speed t which this discount function decreses is positively ssocited with both k nd the level of consumption. Also, s noted in Section 4.1, when k is positive, risk version with respect to ge t deth increses with both k nd the level of consumption. Intuitively, wht eqution (7) sys is tht instnt utility should be discounted ccording to some fctor relted to risk version with respect to ge of deth. The mechnisms t ply then become pretty cler: 1. Risk verse individuls tend to consume erlier in the life cycle to void the prticulrly bd outcomes which would result from hving short life with low level of consumption. 3 2. Risk version with respect to ge t deth increses with the qulity of life. One interesting finding is tht, when individuls re risk verse (tht is when k>0), the rte of discount increses with consumption, so tht rich people would pper to be more imptient thn poor people. One my find this rther surprising. In prticulr, it is in contrdiction with the ssumption mde in Becker nd Mullign (1997). However, note tht the results we provide hold for given exogenous mortlity pttern. But in relity, rich people tend to hve lower mortlity rtes thn poor people nd, if we included this fct in our frmework, we might obtin results consistent with the Becker nd Mullign ssumption. 4 In fct, the effect of consumption level on the rte of discount is likely to be wek, nd therefore dominted by the effects of differentil mortlity. Indeed, s discussed t the end of the Section 4.1, if the difference between life nd deth is much greter thn the difference between high nd low levels of consumption, consumption my hve only 3 When the lottery on length of life is exogenous, the timing of consumption still llows one to mnipulte the distribution of lifetime utility outcomes. Short lives tend to correspond to low levels of lifetime utility. However privileging erly consumption is wy to mke this effect smller nd to mke the distribution of lifetime utility outcomes less unequl. 4 See Bommier (2005) for discussion on the non-trivil reltion between differentil mortlity nd time discounting. 14
minor impct on instnt utility 5 nd therefore on risk version with respect to ge t deth. Precisely, such limit cse is developed in the following subsection. 4.3 Additive preferences s the limit of priceless life Compred with Yri s formultion, the utility function (7) hs n undisputble drwbck: it is not liner. In prctice, this is source of technicl complictions nd we my wonder whether there exist liner pproximtions tht cn be used. An obvious possibility, we know, is to ssume tht k ' 0, which provides Yri s stndrd formultion. But there is nother possibility, tht does not ssume k to be smll nd provides different insights on the origin of time discounting. This involves considering the limit cse where life is much more vlued thn ny conceivble gins in consumption but where, for technologicl resons, people cnnot trde consumption for longer life. To formlize this point I definethe pricelesslifecontext sfollows: Definition 6 The priceless life context corresponds to sitution where: 1. mortlity is exogenous, 2. the instnt utility u(c(t)) cn be written: u(c(t)) = 1 + λw(c(t)) where w is bounded (t lest over the set of fesible consumption pths) nd λ is (smll) sclr, 3. well-behved individuls hve no pure time preference (α =0) Remrk tht the lst ssumption is not bsolutely necessry for wht follows. It is mde to emphsize the fct tht, even if individuls hve no pure time preferences, risk version 5 The formultion of lifetime utility chosen in Definition 1 implicitly ssumes tht instnt utility hs been normlized in such wy tht the instnt utility in the deth stte is zero. 15
with respect to length of life leds them to discount future consumption in non-trivil wy, s soon s the length of life is uncertin. The gme now involves nlyzing cses where λ is very smll. The smller λ, themore people would prefer to hve longer life even if it results in lower levels of consumption. When λ tends to zero, rtionl individuls would be redy to give up most of their consumption to live longer. However, I ssume tht they cnnot do so in the priceless life context becuse the technology to trnsform welth into longer life is unvilble (mortlity being exogenous). Considering the cses where λ is smll, n expnsion of the exponentil function provides n pproximtion of the lifetime utility given by eqution (7). Keeping only the terms of order zero nd one in λ, weget: U (c) ' R + s (t)e k(t ) dt +λ R + λ R + s (t) s (t)w(c(t))e k(t ) dt h ke k(t ) R t w(c(τ))dτ i dt For ny, thefirst term, R + s (t)e k(t ) dt, is constnt tht hs no consequences for preferences over consumption profiles. Switching the order of integrtion in the third term, we get (up to tht constnt): + + U (c) ' λ s (t)w(c(t))e 1 k(t ) s (τ) k t s (t) e k(τ t) dτ dt Integrting by prts the term in brckets leds to the following result: Theorem 3 In the limit λ 0 of the priceless life context, t ny ge, preferences over M CB ([,+ [,R + ) of time-consistent well-behved individuls dmits n expected utility representtion with Bernoulli utility function: U (c) = + s (t)α (t )w(c(t))dt (8) 16
where the discount function, α (.) (which is indexed by, sitisnotthesmetllges), is given by: α (τ) = + τ s 0 (θ + ) s (τ + ) e kθ dθ (9) Henceforth, I refer to this function α (.) s the CARA version for erly deth (CARA- AED) discount function 6. The Bernoulli utility function (8) is very much similr to the one suggested by Yri (eqution (1)). The min difference is tht the discount function is now relted to mortlity risks nd risk version with respect to length of life. Theorem 3 therefore provides rtionle for introducing discount function, even in the bsence of pure time preferences. The properties of the CARA-AED discount function cn be redily relted to mortlity nd to preferences with respect to length of life. First, if people re risk neutrl with respect to length of life (k =0), then this discount function is constnt nd cn be ignored. Second, if people re risk verse with respect to length of life (k >0), then this discount function is decresing nd leds the individul to plce reltively less weight on consumption which is further in the future, s it is most often ssumed. However, s explined in Section 4.2, rther thn pure imptience, this function expresses the fct tht risk verse individuls should consume erly in the life cycle to void the prticulrly low levels of lifetime utility tht would result from hving short life with little instntneous plesures. More generlly, the CARA-AED discount function cn be shown to stisfy the differentil eqution: τ α (τ) =µ(τ + ) α (τ) e kτ where µ(τ + ) is the hzrd rte of deth t ge τ +. In prticulr, if the risk of mortlity were nil during certin period of life, this discount function would be constnt over the sme period. This result should be rther intuitive since risk version with respect to ge t deth should not ply role in period of zero mortlity. Also, if the hzrd rte of deth 6 The term CARA is introduced here becuse time-consistent individuls hve constnt bsolute risk version with respect to length of life (Corollry 1). 17
were constnt, then this discount function would be exponentil, with rte of discount equling k, thecoefficient of bsolute risk version with respect to length of life. 7 However, demogrphic studies indicte tht the hzrd rte of deth is not constnt. After ge 40, it increses pproximtely exponentilly, t rte which lies between 8 nd 9 percent yer. But, if the coefficient of bsolute risk version with respect to length of life is somewht greter thn 0.09 per yer, then the pproximtion which involves ssuming tht the hzrd rte of deth is constnt my be resonble. In such cse, risk version with respect to length of life cn justify the use of exponentil discounting, s first pproximtion. Out of this pproximtion, it is strightforwrd to numericlly compute nd grph the CARA-AED discount function from empiricl mortlity dt (Figure 1). In the exmple drwn, which corresponds to constnt bsolute risk version of 0.10 per yer, we find tht the CARA-AED discount function is well-pproximted by n exponentil function lthough the plot of the rte of discount, τ α(τ), shows tht the fit isnotperfect(figure1c). α (τ) Lstly, we note tht the fct tht the CARA-AED discount function is not exponentil is not necessrily source of time-inconsistency since the discount function chnges with ge. In fct, time-inconsistencies only rise if the rte of discount t given ge chnges with the individul s current ge. When n individul is yers old, the rte of discount t future ge 1,whichis 1 yersinthefuture,isgivenby τ α (τ) τ=1. Simple clcultions led to: τ α (τ) α (τ) α (τ) µ( 1 ) τ=1 = µ( 1 )+ ³ R + R τ 1 (10) µ(τ 1 µ(τ 1 )e 2 )dτ 1 2 e k(τ 1 1) dτ 1 The right hnd side term is clerly independent of, implying tht the CARA-AED discount function does indeed led to time-consistent behvior. 7 In the limit λ 0, thecoefficient of bsolute risk version with respect to ge t deth does not depend on consumption nd equls k. It mkes sense, therefore, to spek of level of risk version with respect to life durtion, without specifying consumption level. 18
5 Time-inconsistent preferences From Theorem 1, we know tht well-behved individuls re time-inconsistent when the ³ function f in (3) is not exponentil in x (i.e., when fxx x f x 6= 0for some x nd ) orwhen ³ the degree of risk version ssumed by f chnges with current ge (i.e.: when f xx f x 6= 0 for some x nd ). The ltter cse is less interesting from n economic point of view. It occurs when, for psychologicl resons, individul preferences chnge with ge. It cn clerly led to time-inconsistency but it does not provide convincing insights on the fundmentl underlying ³ ³ mechnism. The more interesting cses re when fxx f x =0but fxx x f x 6= 0.There rethennopuregeeffects on individul preferences but risk version with respect to length oflifeisnotconstnt. Let us therefore ssume tht f is independent of : f(x, ) =φ(x) with, for the moment, no other ssumption thn φ 0 > 0. 8 Working long the lines of Section 4.2 we find, tht t ge, preferences over consumption pths conditionl on mortlity pttern cn be represented by the utility function: U φ µ,(c) = + s (t)u(c(t))e α(t ) φ 0 µ t u(c(τ))e ατ dτ dt where s (t) is the probbility of being live t ge t conditionl on being live t ge. The limit cse where life is much more vlued tht ny fesible gin in consumption cn be developed s in Section 4.3. In the limit λ 0 of the priceless life context (Definition 6) we find tht the bove utility function is equivlent to Yri kind utility function: + s (t)α φ (t )w(c(t))dt ³ 8 Note tht ssuming fxx f x =0nd f x > 0 implies tht f(x, ) is of the form g 1 ()φ(x) +g 2 () with g 1 ()φ 0 (x) > 0. Since positive trnsformtion of utility does not ffect preferences, it cn be ssumed without loss of generlity tht g 1 () =1, g 2 () =0nd φ 0 > 0. 19
with discount function α φ (.) given by: α φ (τ) = + τ s 0 (θ + ) s (τ + ) φ0 (θ)dθ (11) Agin, such discount function stisfies simple differentil eqution: τ αφ (τ) =µ (τ + ) α φ (τ) φ 0 (τ) which shows tht it would be constnt during period of zero mortlity. Similr to (10) we cn give the expression of the rte of discount t future ge 1 of n individul who is yers old: τ αφ (τ) α φ (τ) µ( 1 ) τ=1 = µ( 1 )+ ³ R + R τ 1 1 µ(τ 1 )e ³ µ(τ 2 )dτ 1 2 φ 0 (τ 1 ) φ 0 ( 1 ) dτ 1 Now, remrk tht if φ00 (x) φ 0 (x) is decresing function then the frction φ0 (τ 1 ) φ 0 ( 1 ) is decresing in (for < 1 < τ 1 ). Thus, it ppers tht decresing risk version with respect to length of life implies tht the rte of discount t ge 1 increses with the current ge. This is firly intuitive: s explined fter Theorem 2, when lifetime is uncertin, risk version with respect to ge t deth cretes n incentive to consume erlier in the life cycle. Under the ssumption of decresing risk version with respect to length of life, risk version with respect to ge t deth 1 increses when getting closer to ge 1 (the stock of remining life decreses). Thus the incentive to consume erlier intensifies when pproching ge 1 nd the rte of discount t ge 1 increses when getting closer to tht ge. The symmetric pttern is obtined if we ssume incresing risk version with respect to length of life. Interestingly, decresing risk version with respect to length of life genertes pttern of time discounting tht shres some similrity with the pttern ssumed by stndrd hy- 20
perbolic discounting models. 9 Therteoftimediscountingtgivengeincreseswhen getting closer to tht ge. To explore this similrity further, I considered the prticulr cse where the function φ exhibits hyperbolic bsolute risk version (HARA) 10 : φ(x) = γ 1 γ (η + x γ )1 γ with η 0 nd γ > 0. I then numericlly computed the discount function α φ (clled therefter the HARA-AED discount function) of 40 yer old mn when using relistic mortlity ptterns. The prmeters γ =2.5 nd η =0.04 were chosen so tht the rte of discount is bout 13% in the short term nd 3% fter 15 yers. Figures 1b nd 1d show tht the HARA-AED discount function is well-pproximted by generlized hyperbols nd tht the rte of discount is resonbly close to wht is predicted by hyperbolic discounting. We find therefore tht hyperbolic discounting cn ctully rise from two different kinds of ssumptions. A first possibility, followed by the stndrd literture on hyperbolic discounting, is to ssume tht individuls re risk neutrl with respect to length of life nd hve n exogenous discount function which is generlized hyperbol. The other possibility involves ssuming tht individuls hve no pure time preferences but hyperbolic risk version with respect to length of life. 11 Although these two pproches led to very similr ptterns of time discounting, they re of different nture. The stndrd dditive model with hyperbolic discounting ssumes tht ordinl preferences re time-inconsistent. In the pproch I followed, ordinl preferences re time-consistent. Inconsistencies exclusively rise from the wy individuls ccount for uncertinty, nd in prticulr for uncertin lifetime. The two pproches therefore correspond 9 Models tht ssume preferences to be dditive nd the subjective discount function to be generlized hyperbol: φ(t) =(1+αt) β with α, β > 0. Such discount functions were introduced in Hrvey (1986) nd xiomtized by Prelec (1989). See lso the discussion in Loewestein nd Prelec (1992). 10 Well-behved individuls then hve hyperbolic bsolute risk version with respect to length of life (Definition 4). 11 Hlevy (2004) suggests nother wy to obtin time-inconsistent preferences with diminishing imptience. The ide is to pply Yri s dul theory (Yri, 1987) to the cse of intertemporl choice under uncertin lifetime. 21
to fundmentlly different ssumptions on humn rtionlity. Moreover, the contrst is not purely theoreticl. Even though both pproches my hve similr predictions for intertemporl choice under n exogenous mortlity pttern, they shrply diverge when considering mortlity chnges. 12 6 Conclusion This pper explored the consequences of uncertin lifetime on individuls behvior nd in prticulr on intertemporl choice theory. I showed tht Yri s dditive utility function, which is the stndrd wy to incorporte uncertin lifetime into life cycle models, corresponds to strong ssumption on risk neutrlity with respect to length of life. I explored how one could relx this ssumption of risk neutrlity without introducing time-inconsistencies in individul preferences. This leds us to consider non-dditive preferences tht exhibit constnt bsolute risk version with respect to length of life. Relxing the risk neutrlity ssumption is costly in the sense tht it forces us to consider non-dditive utility functions. Nonetheless, dditivity cn be recovered, s first pproximtion, when considering the limit in which individuls would be willing to trde most of their consumption for longer life but cnnot do so for technologicl resons. Considering dditivity s resulting from such limit, rther thn resulting from risk neutrlity ssumption, gives interesting insights on the origin nd the structure of the subjective discount functions considered in life cycle models. I explined how such discount functions re relted to mortlity nd preferences over lotteries on length of life nd I discussed why they my be quite close to exponentil discounting when individuls hve constnt bsolute risk version with respect to length of life. Lstly, this pper briefly discussed the cse of time-inconsistent preferences. I explined why decresing risk version with respect to length of life genertes inconsistent ptterns of time discounting, with rtes of discount t given ge tht increse when pproching tht 12 See the discussion in Bommier (2005). 22
ge. In prticulr, hyperbolic risk version with respect to length of life ws shown to led, in first pproximtion, to hyperbolic discounting. Wht hs sometimes been presented s n nomly in intertemporl choice theory my therefore result from common ssumption on risk version pplied to the less usul domin of studies tht re preferences over lotteries on length of life. This pper suggests vrious extensions. First, by mking the link between time discounting nd risk version, it opens the route for life cycle theory where imptience exclusively results from the combintion of risk version nd uncertinty. Such theory would not be subject to the severe criticisms of Pigou (1920) nd Rmsey (1928) tht respectively considered time preferences s wholly irrtionl or rising from the wekness of imgintion. Humn imptience my be driven by risk version, less controversil concept. Second, it suggests tht the reltion between mortlity nd imptience my be much stronger thn ssumed in Yri s model. This is of crucil importnce when discussing the economic impct of mortlity chnges. Lst, it clls for n economic nlysis of endogenous mortlity choices tht ccounts for risk version with respect to length of life. These different points re developed in Bommier (2005, 2005b) nd Bommier nd Villeneuve (2004). References [1] Angeletos, G.M., D. Libson, A. Repetto, J. Tobcmn nd S. Weinberg, 2001, The Hyperbolic Consumption Model: Clibrtion, Simultion, nd Empiricl Evlution. Journl of Economic Perspectives 15(3):47-68. [2] Becker, G. S. nd C. B. Mullign, 1997, The Endogenous Determintion of Time Preference. Qurter Journl of Economics 112(3): 729-58. [3] Berkeley Mortlity Dtbse, http://demog.berkeley.edu/wilmoth/mortlity. 23
[4] Bommier, A., 2005, Mortlity, Time Preference nd Life Cycle Models. Working Pper GREMAQ. Downlodble from http://www-gremq.univ-tlse1.fr/working.php. [5] Bommier, A., 2005b, Life-Cycle Theory for Humn Beings. Working Pper GREMAQ. Downlodble from http://www-gremq.univ-tlse1.fr/working.php. [6] Bommier, A. nd B. Villeneuve, 2004, Risk Aversion nd the Vlue of Risk to Life. CE- Sifo Working Pper 1267. Downlodble t http://www.cesifo.de/~doccidl/1267.pdf [7] Butler, M., 2001, Neoclssicl Life-Cycle Consumption: A Textbook Exmple. Economic Theory, 17(1):209-221 [8] Fisher, I., 1930, The theory of interest. New York: Mcmilln. [9] Hlevy, Y., 2004, Diminishing Imptience nd Non-Expected Utility: A Unified Frmework. Downlodble from http://www.econ.ubc.c/hlevy. [10] Hrris, C. nd D. Libson, 2001, Dynmic Choices of Hyperbolic Consumers. Econometric 69(4): 935-957 [11] Hrvey, C., 1986, Vlue Functions for Infinite-Period Plnning. Mngement Science, 32(9):1123-1139. [12] Libson, D., 1997, Golden Eggs nd Hyperbolic Discounting. Qurterly Journl of Economics, 112(2):443-477 [13] Libson, D., 1998, Life-cycle consumption nd hyperbolic discounting. Europen Economic Review, Ppers nd Proceedings, 42: 861-871 [14] Leung, S. F., 1994, Uncertin Lifetime, the Theory of the Consumer, nd the Life Cycle Hypothesis. Econometric, 62:1233-1239 [15] Loewenstein, G. nd D. Prelec, 1992, Anomlies in Intertemporl Choice: Evidence ndninterprettion. Qurterly Journl of Economics, 107(2):573-597. 24
[16] Loewenstein, G. nd R. H. Thler, 1989, Anomlies: Intertemporl Choice. The Journl of Economic Perspectives, 3(4):181-193. [17] Pigou, A. C., 1920, The Economics of Welfre. London, Mcmilln. [18] Prelec, D, 1989, Decresing Imptience: Definition nd Consequences. Hrvrd Business School working pper. [19] Rmsey, F. P., 1928, A Mthemticl Theory of Sving. TheEconomicJournl, 38(152):543-559. [20] Strotz, R. H., 1956, Myopi nd Inconsistency in Dynmic Utility Mximiztion. The Review of Economic Studies, Vol. 23, No. 3., pp. 165-180. [21] Yri, M. E., 1965, Uncertin Lifetime, Life Insurnce, nd the Theory of the Consumer. The Review of Economic Studies, 32(2):137-150. [22] Yri, M. E., 1987, The Dul Theory of Choice Under Risk. Econometric, 55(1):95-115. 25
APPENDIX A Proof of Theorem 1 Let us first show the following result: Lemm 1 Assume time-consistency (Definition 5). Then for ny consumption profile, risk version with respect to ge t deth is independent of current ge. Proof. Consider given consumption profile c C B ([0, + [, R + ), nd three ges < b< T e. Withnobviousbuseofnottiontherestrictionofc on [, + [, [b, + [ or on [, b] will be denoted by the sme letter c. Let us show tht the ssumption of time-consistency implies tht: RAL (c, e T )=RAL b (c, e T ) For ny (smll) positive ε, definethreesimplemesuresµ, m 1 (ε) nd m 2 (ε) in M L by: µ = δ (c, T e h ) i m i 1 (ε) = 2 +( 1)i ε 3 2 + εral 4 (c, et ) h i 1 + 2 ( 1)i ε 3 2 εral 4 (c, et ) δ (c, e T +ε) δ (c, et ε) (12) for i =1, 2. Using Tylor expnsions, one hs: U (c, e T ± ε) =U (c f,t) ± ε U T (c, e T )+ 1 2 ε2 2 U T 2 (c, e T )+o(ε 5 2 ) 26
where ε 5 2 o(ε 5 2 ) 0 when ε 0. Thus U (c, T )dm i U (c, T )dµ = 1 2 ε2 RAL (c, e T ) U T (c, e T ) +2( 1) i ε 5 U 2 T (c, et ) + 1 2 U 2 ε2 T (c, T e )+o(ε 5 2 2 ) Since, by definition, RAL (c, et )= 2 U side cncel out. Therefore T 2 (c, T e ) U T (c, T e ), the first nd third terms of the right hnd U (c, T )dm i U (c, T )dµ =2( 1) i ε 5 2 U T (c, et )+o(ε 5 2 ) which implies tht for ε smll enough: m 2 (ε)  µ  m 1 (ε) The mesures µ, m 1 (ε) nd m 2 (ε) re degenerte till ge b with intermedite consumption c. The corresponding mesures µ b, m 1 b (ε) nd m2 b (ε) in M L b hve exctly the sme expression s µ, m 1 (ε) nd m 2 (ε) (see eqution (12)) with the difference tht the letter c then denotes the restriction of c on [b, + [. The ssumption of time-consistency implies tht for ε smll enough we must lso hve m 2 b(ε)  b µ b  b m 1 b(ε) (13) Now, using Tylor expnsion: U b (c, T )dm i b U b (c, T )dµ b = ε2 2 ³ Ub RAL (c, et ) RAL b (c, et ) T (c, et ) U b T (c, T e )+o(ε 5 2 ) +2( 1) i ε 5 2 27
which implies tht (13) cn hold only if RAL (c, e T )=RAL b (c, e T ) We cn now prove Theorem 1. From Lemm 1 if individuls re time-consistent it must bethecsetht: RAL (c, T )=0 for ny consumption profile nd for ny <T. In view of eqution (4) such result holds if nd only if: which is true if nd only if: where I = R T pth if nd only if: v() x h R T f xx ( h R T f x ( i u(c(t))e αt dt i u(c(t))e αt dt µ fxx (I,) + f x (I,),) =0,) µ fxx (I,) =0 f x (I,) u(c(t))e αt dt nd v() =u(c())e α. This clerly holds for ny consumption x µ fxx f x = µ fxx f x =0 or, equivlently, if nd only if: x f(x, ) correspond to constnt bsolute risk version (CARA) utility functions, with degree of risk version independent of. Since positive liner trnsformtion of Bernoulli utility function leves preferences unchnged, we cn ssume without loss of generlity tht: f(x, ) = e α ( 1 e kx ) for some k 6= 0 k or f(x, ) = e α x 28
We then obtin the Bernoulli utility function (5). Now it remins to be shown tht preferences represented by (5) re time-consistent. Consider two mesures µ 1,µ 2 M L tht re degenerte till b, with intermedite consumption c 0.Denoteµ 1 b nd µ2 b their corresponding mesures in M L b. One hs U (c, T )dµ i = θ 1 + θ 2 U b (c, T )dµ i b for i =1, 2 with θ 1 = ³ e (1 exp k R b u(c 0(t))e αt dt k nd µ θ 2 = e α( b) exp k b u(c 0 (t))e αt dt > 0 Thus U (c, T )dµ 1 > U (c, T )dµ 2 = U b (c, T )dµ 1 b > U b (c, T )dµ 2 b nd preference reversls cnnot occur. B Proof of Theorem 2 By definition of the expected utility representtion: (ν m µ, ) Â (λ m µ, ) U (c, T )d (ν m µ, ) > U (c, T )d (ν m µ, ) Write U (c, T )d (λ m µ, )= + d( s (T )) U (c, T )dλ 29
Integrting by prts: U (c, T )d (λ m µ, ) = s (T ) + + U (c, T )dλ s (T )dt + (14) T U (c, T )dλ (15) The first term equls zero since lim T + s (T ) R U (c, T )dλ =0nd U (c, ) =0. second term cn be reordered to obtin: The U (c, T )d (λ m µ, )= U µ, (c)dλ with U µ, (c) = = + + s (T ) T U (c, T )dt s (T )u(c(t ))e α(t ) exp µ T k u(c(τ))e ατ dτ dt Thus (ν m µ, )  (λ m µ, ) (nd, by definition, ν  µ, λ)ifndonlyif: U µ, (c)dν > U µ, (c)dλ Therefore, the reltion of preferences  µ, dmits n expected utility representtion with Bernoulli utility function U µ,. 30
Figure 1: Properties of the AED discount functions for 40 yer old mn Figure 1: CARA AED discount function Figure 1b: HARA AED discount function discount function 0.0 0.2 0.4 0.6 0.8 1.0 CARA AED discount function Fitted exponentil discount function 0.0 0.2 0.4 0.6 0.8 1.0 HARA AED discount function Fitted generlized hyperbol 0 5 10 15 20 0 5 10 15 20 Time from present (yers) Time from present (yers) Figure 1c: CARA AED rte of discount Figure 1d: HARA AED rte of discount Rte of discount (percent) 0 5 10 15 CARA AED rte of discount Exponentil discounting Rte of discount (percent) 0 5 10 15 HARA AED rte of discount Hyperbolic discounting 0 5 10 15 20 0 5 10 15 20 Time from present (yers) Time from present (yers) Notes : 1. Figure 1: CARA-AED discount function computed from eqution (8) with constnt bsolute risk version equlling 0.1 per yer (k =0.1). Fitted with n exponentil function (y = e 0.041 x ). 2. Figure 1b: HARA-AED discount function computed from eqution (10) with φ(x) = (0.4+ x 2.5 ) 1.5. Fitted with generlized hyperbol (y =(1+0.24 x) 0.13 0.24 ). dτ 3. Figure 1c: CARA-AED rte of discount, d α 40(τ), computed with the sme prmeters s for Figure 1. The dotted line eqution is α 40 (τ) y =0.041. dτ 4. Figure 1d: HARA-AED rte of discount, d αφ 40 (τ), computed with the sme prmeters s for Figure 1b. The dotted line eqution is α φ 40 (τ) y = 0.13 1+0.24 x. 5. Computtions re bsed on 1995 US mortlity dt, obtined from the Berkeley Mortlity Dtbse nd originlly issued by the Office of the Actury of the Socil Security Administrtion.