Dynamic Analysis of Addicion: Impaience and Heerogenous Habis Sefano Corradin Federico Perali Luca Piccoli April 25 Absrac This sudy develops a dynamic analysis of raional addicion suggesing a heoreical model ha places special emphasis on he effecs of impaience and heerogenous habis. The model describes how heerogenous habis affec he consumpion pahs viaasubjeciveraeofimepreferencevarying he rae of habi adjusmen and he paience-dependence rade off. The ineremporal srucure of preferences incorporaing impaience and heerogenous habis explains how an increase in he rae of reurn o savings implies a decrease in he raeofimepreferenceandanincreasein he elasiciy of ineremporal subsiuion. This makes a forward-looking agen more paien han a myopic one. DRAFT Acknowledgemen 1 We would like o hank he parecipans o he "Models of Household Economics for he Design and he Evaluaion of Fiscal and Social Reforms" CHILD Conference (Turin 24) for heir aenion and suppor, and in paricular prof. Ugo Colombino for his useful comens and suggesions. We also hank he parecipans o "The Paradoxes of Happiness in Economics" held in Milan (23), prof. Giampiero Cipriani, Marina Menon and Sara Borelli from Universiy of Verona. The usual assumpion in economics is ha discoun raes on fuure uiliies are consan and fixed o each person, alhough hey may differ beween persons. This assumpion is a good iniial simplificaion, bu i canno explain why discoun raes differ by age, income, educaion and oher personal characerisics or why hey change over ime for he same individual, as when a person maures from being a child o being an adul. (Becker G., Accouning for Tases, 1996) Haas School of Business, Universiy of California Berkeley, corradin@haas.berkeley.edu Deparmen of Economics, Universiy of Verona, federico.perali@univr.i Deparmen of Economics, Universiy of Verona, luca.piccoli@economia.univr.i 1
1 Inroducion The habi formaion process is affeced by economic variables and oher exogenous facors such as demographic characerisics and he psychological sae of consumers. A habi is formed when pas and curren consumpion are linked by a posiive relaion. The higher is previous consumpion, he larger he habi, and he higher should he curren consumpion level be o deliver he same uiliy. I follows ha he derived uiliy depends on he difference beween curren consumpion and a weighed sum of he quaniies consumed in he pas. Uiliy reaches a peak afer consumpion rose o a permanenly higher level. Then i declines over ime as he person becomes accusomed o ha level. Similarly, uiliy reaches a minimum jus afer consumpion fell o a permanenly lower level. Comparisons wih pas consumpion can be so effecive ha pas consumpion can be weighed more heavily han presen consumpion. When he habi formaion process is susained, hen a consumer may urn a habi ino a sae of addicion. A habi may evolve ino addicion by being exposed o habi iself. Becker and Murphy [4] define a person addiced o some goods when an increase in curren consumpion increases fuure consumpion. Consumpion of he addicive good is no equally harmful o all individuals. For example, many people can drink regularly wihou becoming alcoholiss. Addicion involves an ineracion beween people and goods. Each individual possesses a subjecive belief srucure concerning his poenial o become addiced. People of comparable wealh and educaion bu wih differen pas experiences do no share he same risk o become addiced. In essence, people have differen raes of ime preference. Par of his heerogeneiy may be explained by personal differences in pas experiences, demographic characerisics, geneic parimony, and oher exogenous facors. Theraeofimepreferenceisasubjec- ive indicaor of impaience represening he desire of an agen o anicipae and enjoy he benefis semming from higher curren consumpion. Ahighraeofimepreferencelow- ers he propensiy owards fuure uiliy in deermining curren consumpion choices. The main objecive of he presen paper is o rea he rae of ime preference as endogenous and dependen upon demographic characerisics in order o capure heerogeneiy in he addicion formaion process. We believe ha by inroducing heerogeneiy in our model, we can develop a useful ool o evidence which policies can effecively reduce alcoholism wihou srongly penalizing people who enjoy a moderae consumpion of alcoholic beverages. To mos people, curren consumpion of alcohol in moderaion provides enjoymen wihou serious side effecs. To ohers he same paern of consumpion may lead o a sae of dependence and evenually addicion 1. This developemen pah is criically affeced by personal characerisics. Analysing he causes for he low marriage rae, Akerlof [1] observes ha here are noiceable differences in he lifesyle of married and unmarried men. Married men say longer in he labour force, are less inclined o subsance abuse, commi less crime and are less likely o be vicims of crime. Moreover, hey have beer healh and are less acciden prone. A simple explanaion is advanced: low marriage raes, or, in general, soliude, will lead o increases in some social pahologies such as crime, drugs and alcohol addicion. As shown in Perali e al. [16] and [17], he gender of household head is a crucial characerisic in deermining alcohol consumpion, wih female headed households which 1 In general, addicion creaes physical absinence or wihdrawal sympoms, when he use of he subsance is disconinued, and generaes olerance, which is a physiological phenomenon requiring he individualousemoreandmoreofhesubsance([11,kennedy, 1987] and [21, Sein e al., 1988]). Tolerance for a subsance may be independen of he drug abiliy o produce physical dependence which manifess iself by he sympoms of absinence when he drug is wihdrawn. 2
consume signifinicanly less han male headed ones. Oher variables (like marial saus, educaion, he presence of children, ec.) affec alcohol consumpion and have o be aken in consideraion, bu o keep hings simple we focus on male and female singles o bring evidence of how heerogeneiy is imporan for an effecive welfare policy. Our idea is ha i can be possible o hink abou a gender-specific policy, aking ino accoun ha in general women are considered more forward looking hen men, and, possibly, o develop gender-specific insrumens for he paricular siuaions in wich addicion o alcohol seems o be more likely o rise. In fac, even if alcohol abuse is commonly considered a social bad, moderae cunsumpion, expecially of wine, is seen as par of he Ialian culure and so auspicable. Given ha he daa seem o suppor our working hypohesis, we proceed by assuming an endogenous discoun rae depending on pas consumpion as in Shi and Epsein [2], and by parameerizing he rae of ime preference o incorporae heerogenous habis nesing as special cases boh he Ramsey model, characerized by a consan rae of ime preference, and he Uzawa [22] or Obsfeld [15] models, ha assume an endogenous discoun rae depending on curren consumpion. The resuls derived from he dynamic comparaive analysis developed in his sudy are in line wih hose formulaed by he heory of raional addicion. Wih respec o he analysis of Becker and Murphy [4] and Becker and Mulligan [2], we exend he model in order o sudy he impac of habis on ineremporal consumpion pahs by he rae of ime preference and elasiciy of subsiuion varying he rae of habi adjusmen. This variaion describes he heerogeneiy of consumers who differ for a se of demographic characerisics. The sudy simulaes he behaviour of wo ypes of agens, a myopic and a forward-looking, where he myopic wih respec o he forward-looking reveals a poenial habi o alcohol for a se of characerisics ha reveal a predisposiion of he myopic o alcohol. This approach may be relevan o undersand he poliical economy dimension of addicion. An effecive policy would have he shor run objecive o find he se of acions ha makes every person discoun he fuure more highly and he long run objecive o increase he frequency of forward-looking people wihin a populaion. The sudy is srucured as follows. Secion 2 develops a rewriing of he basic dynamic opimizaion problem proposed by Shi and Epsein [2] where he consumer maximizes an isananeous uiliy funcion discouned by an endogenous rae defined wih respec o an index of pas consumpion, rying o pu in evidence he possible exensions. Secion 3 exends he basic model and develops he analyical properies of he exended models. The las secion presens he conclusions. 2 The Basic Model wih an Endogenous Discoun Rae: a Unified Noaion The seminal works of Blanchard and Fisher [5], Deaon [7] and Romer [18] have criicized he assumpion of a consan rae of ime preference as suggesed more by convenience han economic raionales. Mos of he economic lieraure represens he preference srucure in a dynamic conex using he Ramsey model (Table 1) hrough funcionals in which an addiive uiliy funcion is discouned by a consan rae. Addiiviy implies ha he marginal rae of subsiuion beween he consumpion a ime and +1 is independen of consumpion for each differen from and +1. Siuaions such as habi formaion in alcohol consumpion (bu also in drug use or sigaree smoking), or he exisence of goods as holidays and works of ar whose benefis coninue over he consumpion ac, canno be 3
described properly by an addiive preference srucure 2. A formulaion ha involves non separabiliy of preferences is suggesed by Ryder and Heal [19] who inroduce he noion of adjacen complemenariy. An increase or decrease in consumpion a 1 can induce a variaion of he marginal rae of subsiuion of curren and fuure consumpion a +1. The complemenariy is represened by a uiliy funcion ha depends on boh curren consumpion, c, and an index of pas consumpion Z z = σ c τ e σ(τ ) dτ wih σ, (2.1) which is a weighed average of pas consumpion levels. The weighs decline exponenially in he pas a exogenous adjusmen rae σ which is a measure of permanence of physical and menal effecs of pas consumpion in presen consumpion c (Table 1). As σ ges larger, less weigh is given o pas consumpion in deermining z. Therefore, he degree of addicion is more inense for a lower σ. Non separable preferences can be adoped assuming ha he consumer discouns feliciy by an endogenous discoun rae depending on 1. curren consumpion, ρ (c ), according o a model wih impaience developed by Obsfeld [15] (Table 1); 2. he index of pas consumpion z, ρ (z ), according o a model wih impaience and habis developed by Shi and Epsein [2] (Table 1). In he economic lieraure here is an open discussion if he endogenous discoun rae mus be considered increasing or decreasing wih respec o curren consumpion c.koopmans [12] suggess a decreasing rae of impaience, while Lucas and Sokey [14] observe ha an increasing rae of impaience is necessary o obain a single, sable, non degenerable equilibrium poin ino wealh disribuion in a deerminisic horizon wih a finie number of agens. According o Blanchard and Fisher [5], he assumpion of an increasing rae of impaience is difficul o defend ex ane. On he oher side Epsein [8],[9] argues ha he more a person consumes, he more discouns he fuure. In line wih Epsein, we assume ha he endogenous discoun rae, ρ (z ), is sricly increasing wih respec o curren consumpion c. This condiion is necessary for ensuring he sabiliy of he long-run opimal consumpion plan, because i guaranees ha consumpions in differen daes are subsiues. In his case as wealh and consumpion rise, he marginal privae reurn o furher savings, which depends on he marginal uiliy of fuure consumpion, falls. If ρ (z ) <, consumpion in differen raes are complemens, and a rise in presen consumpion rises he marginal uiliy of fuure consumpion. Such an assumpion is plausible in a model wih habi formaion, bu i does no seem much coheren when we consider consumpion in general. This is a furher argumen in favor of he assumpion ha he subjecive discouning of fuure uiliy rises wih consumpion. The implicaion of a discoun rae ρ (z ) sricly increasing wih respec o presen consumpion c, is ha a higher consumpion level a ime increases he discoun rae applied o uiliy a and afer. An increase in curren consumpion in induces an increase in he rae of ime preference: he consumer s desire o anicipae effecs of fuure consumpion is picked up by more curren consumpion a +1. Anincreaseincurren 2 In general, addicion creaes physical absinence or wihdrawal sympoms, when he use of he drug is disconinued, and generaes olerance, which is physiological phenomenon requiring he individual o use more and more of he subsance. Tolerance for a drug may be independen of he drug abiliy o produce physical dependence which manifess iself by he sympoms of absinence when he drug is wihdrawn. 4
consumpion a +1rises he sock of habis a +2inducing a furher increase in he discoun rae: he higher is previous consumpion, he larger he habi, and he higher mus be he curren level of consumpion o deliver he same effec. An increase in he discoun rae rises he degree of adjacen complemenariy and hence srenghens he commimen o all habis. 2.1 The Basic Model Consider an agen who can have access o a poenially harmful good a each insan of an infinie horizon. The consumpion level of he h period, corresponding o he life cycle pah C, is denoed c, while he ineremporal uiliy a ime, U (C ), is delivered from he weighed sum of all fuure flows of uiliy, u (c ). The feliciy funcion, u (c ),saisfies he Inada condiions and, in line wih Shi and Epsein [2], we assume ha he discoun rae is linear (ρ (z )=), posiive (ρ (z ) > ) and increasing (ρ (z ) > ). Over he relevan ime inerval from =o 1 =, he acual level of welfare, U, derived from he consumpion rajecory {c }, is obained inegraing all fuure flows of uiliy u (c ) discouned by he discoun facor e Θ where U (C )= Θ = Z Z u (c ) e Θ e r d (2.1.1) [ρ (z s ) r] ds, (2.1.2) subjec o he following se of equaions of moion ȧ = ra c Θ = ρ (z ) r ż = σ (c z ). (2.1.3.a) (2.1.3.b) (2.1.3.c) Expression (2.1.2), denoed as he cumulaive discoun rae, is an indicaor of accumulaed impaience obained by he difference beween he discoun rae ρ(z ), depending on consumer preferences wih respec o he ype of good and varying from agen o agen, and he rae of reurn o savings r, an opporuniy variable equal for everyone and a every. The inroducion of he cumulaive discoun rae allows us o obain a significan simplificaion in he problem solving, wihou any effec on he opimal consumpion pah. The single conrol variable is he per-capia consumpion c and he real asses per person, a, he cumulaive subjecive discoun rae, Θ, and he sock of habis, z,arehe sae variables. We assume a consan rae of habis adjusmen (σ) as well no depending on he characerisics of he individual in his secion. The conrol problem (2.1.1),(2.1.3.a),(2.1.3.b) and (2.1.3.c) is solved according o he Maximum Principle. The curren value hamilonian funcion, H d = e r H is H d n c,a, Θ,z ; eq, eϕ, f Ψ o = u (c ) e Θ + eq [ra c ] eϕ [ρ (z ) r]+ f Ψ [σ (c z )] (2.1.4) where eq = e r bq, eϕ = e r bϕ and f Ψ = e r c Ψ are he discouned cosae variables. The necessary firs-order condiions of he curren value Hamilonian funcion (2.1.4) for an inerior maximum are H d c = eq = u (c ) e Θ + f Ψ σ (2.1.5) 5
and H d = r eq eq eq = r eq r eq = a H d = r eϕ Θ eϕ eϕ = r eϕ u (c ) e Θ H d z = r f Ψ fψ fψ =(r + σ) f Ψ + eϕ ρ (z ). (2.1.6.a) (2.1.6.b) (2.1.6.c) I is convenien o rescale he cosae variables in order o eliminae Θ.Leq = eq e Θ, ϕ = eϕ e Θ and Ψ = f Ψ e Θ. Then, he firs-order necessary condiions ake he form q = u (c )+Ψ σ. (2.1.7) and, given ha q = eq e Θ, q = eq e Θ + eq e Θ Θ =+q Θ, (2.1.8) he oher firs order condiions are q = q (ρ (z ) r) ϕ = ϕ ρ (z ) u (c ) Ψ =(ρ (z )+σ) Ψ + ϕ ρ (z ), (2.1.9.a) (2.1.9.b) (2.1.9.c) and differeniaing equaion (2.1.7) wih respec o ime we obain q = u (c )ċ + Ψ σ. (2.1.1) The differenial equaion 2.1.9.b gives a coninuos-ime specificaion of he recursive srucure of consumer preferences for every feasible consumpion pah C. Ifwesolvehe differenial equaion 3 (2.1.9.b) we obain ϕ = Z u (c v ) e v ρ(zs)ds dv, (2.1.11) which is he presen value of fuure uiliies a ime and corresponds o he shadow price of he accumulaed impaience rae Θ. By equaing he wo equaions we have for q (2.1.9.a and 2.1.1), we can solve for ċ and find he Euler Equaion u (c )ċ = q (ρ (z ) r) Ψ σ = ċ c = (u (c )+Ψ σ) u (c )c ρ (z ) σ (ρ (z )+σ) Ψ + ϕ ρ (z ) (u r (c )+Ψ σ). (2.1.12) Rewriing expression (2.1.12) in erms of rae of ime preference and elasiciy of ineremporal subsiuion, he Euler Equaion becomes 1 ċ r = θ(c,z,ϕ, Ψ ), (2.1.13) η(c, Ψ ) c 3 Recall ha he soluion for a differenial equaion wih no consan coefficiens as y + P y = Q is y = e Pd Q e Pd d + ce Pd. The value ha he soluion approaches is reffered o as he seady sae so he limi for of he soluion is y = Q e Pd d. 6
where θ(c,z,ϕ, Ψ )=ρ (z ) σ (ρ (z )+σ) Ψ + ϕ ρ (z ) (u (c )+Ψ σ) (2.1.14) isheraeofimepreference,and η(c, Ψ )= (u (c )+Ψ σ) u (c )c (2.1.15) is he elasiciy of ineremporal subsiuion. 3 Exension o he Basic Model 3.1 Impaience and Heerogeneous Habis This secion exend he basic model aking advanage of he hypoesis of lineariy of he discoun rae proposed by Shi and Epsein [2] in order o obain a relaively simpler Euler Equaion where he role of he rae of habis adjusmen (σ) is widened by allowing for differences among consumers, due o he heerogeneiy of preferences. The parameer σ srongly influences consumer behavior. A proper modeling of he role of heerogeneiy in he process of habi formaion and in disinguishing differenraeofimepreferencewill also be crucial for a correc specificaion of economeric models. The endogenous rae of ime preference represens a subjecive indicaor of impaience (i.e. he desire o anicipae fuure consumpion) and can depend upon demographic variables, no only on pas consumpion pah. In his secion we propose an exension o he basic model in order o ake ino accoun he subjecive degree of impaience and, indirecly capure an imporan componen of heerogeneiy. We can obain a reformulaion of he firs order condiion of he hamilonian funcion (2.1.7) as he firs derivaive of generaing funcion wih respec o curren consumpion c. Consider he firs-order condiion of he Hamilonian funcion (2.1.7) as where q = u (c )+Ψ σ. The cosae variable, Ψ, is he shadow price of he sock of habis, z, and is defined Ψ = U (C ) z U (C )= Z Z = u (c v ) e v ρ(zs)ds dv u (c v ) e µz v v ρ(z s)ds ρ (z )e σ( s) ds dv > (3.1.1) saes he presen value of fuure uiliies valued a 4 and corresponds o ϕ,asshownin he previous secion. The preceding expression can be rewrien as Ψ = ϕ ρ (z ) Z v e σ( s) ds (3.1.2) 4 Expression (3.1.1) is derived inegraing by pars expression (2.1.11) for < v z = σe σv eσs c sds = c e σ( v) and considering ha ċ =along a locally consan pah consumpion. 7
given he condiion of lineariy of he discoun rae 5,whereϕ is he shadow price of he rae of he accumulaed impaience. Expression (3.1.2) is analyically differen from he one formulaed in he Shi and Epsein model [2]. According o our model, he firs-order condiion (2.1.7) is reformulaed as Z v q = u (c ) ϕ ρ (z ) e σ( s) ds σ = u (c ) ϕ ρ (z )ξ (σ) σ (3.1.3) wih ξ (σ) = Z v e σ( s) ds = 1 σ eσ( v) σ where, o simplify he mahemaical reamen, we assume ξ (σ) as a consan whose value is calculaed numerically by varying he rae of habis adjusmen, σ, and assuming wo values for he lower,, and he upper, v, exreme of inegraion. This reformulaion is useful o characerize he behavioural properies of he model as i will be explained in he nex secions. The Euler equaion is derived from he firs-order condiion (3.1.3). By differeniaing expression (3.1.3) wih respec o ime and considering ha ρ (z )=, q = u (c ) ċ ρ (z ) ϕ ξ (σ) σ (3.1.4) Equaing equaions (3.1.4) and (2.1.9.a) and replacing expression (2.1.9.c), a differenial equaion giving in every ime he ime rae of change of he conrol variable c is obained u (c ) ϕ ċ = ρ (z )ξ (σ) σ u (c ) ½ ¾ ϕ 1+ u (c ) /ρ (z ) u (c ) ϕ ρ ρ (z )ξ (σ) σ r. (3.1.5) (z )ξ (σ) σ Divide boh sides by curren consumpion c and define as he endogenous rae of ime preference ϕ θ (c,z,ϕ,σ)=1+ u (c ) /ρ (z ) u (c ) ϕ ρ ρ (z )ξ (σ) σ (3.1.6) (z )ξ (σ) σ as he endogenous elasiciy of ineremporal subsiuion η (c,ϕ,σ)= u (c ) ϕ ρ (z )ξ (σ) σ u (c ) c, (3.1.7) andasheeulerequaionwihrespecoheraeofreurnr ċ 1 r = θ (c,z,ϕ,σ). (3.1.8) η (c,ϕ,σ) c The model dynamics are described by equaions (2.1.9.b), (3.1.5), (2.1.3.a), (2.1.3.b) and (2.1.3.c). Convergence o he seady sae is derived equaing he sysem of equaions 5 A linear funcion is a homogeous funcion of firs degree so f (kx) =kf (x) where k is consan and x is he independen variable. 8
o zero and he ineracion of he differenial equaions defines he unique opimum ż = σ (c z )= z = c (3.1.9.a) ϕ = ϕ ρ (z ) u (c )= ϕ = u (c ) (3.1.9.b) r u (c ) ϕ ċ = ρ (z )ξ (σ) σ u [θ (c,z,ϕ (c ),σ) r] = θ (c,z )=r (3.1.9.c) ȧ = ra c = a = c r Θ = ρ (z ) r = ρ (z )=r. (3.1.9.d) (3.1.9.e) Sysem (3.1.9.a), (3.1.9.b), (3.1.9.c), (3.1.9.d) and (3.1.9.e) presens dynamic properies similar o he Shi and Epsein model [2] and similar roos of he characerisic equaion of he marix whose coefficiens are delivered by a firs-order Taylor expansion of he sysem. In his analysis, he characerisic roos depend on he rae of habi adjusmen, σ, and he ineres rae, r, assumed consan for convenience. Differen raes of habi adjusmen allow us o examine he local sabiliy of he sysem: 1. σ>σ 1 he roos are wo real unsable and wo real sable. The equilibrium poin is a saddle poin; 2. σ<σ 1 he roos are wo real unsable and wo complex wih negaive real pars and so sable. The convergence o he seady sae is cyclical. According o Shi and Epsein [2]... he cycles are local and hey dampen owards he seady sae. Cycles are more likely if habis adjus slowly or he seady sae rae of ime preference is more sensiive o he level of consumpion or he desire o smooh consumpion is weaker asiishecaseofa myopic agen in response of a rise in ineres raes. On he oher hand, cyclical behaviour is impossible, when he rae of habis adjusmen approaches σ, and as expeced, when he model reduces o he Ramsey model for σ. 3.2 Behavioral Analysis The definiion of an endogenous rae of ime preference permis o idenify a crucial feaure of heerogeneiy, separae from a generic habi formaion process. In his secion we describe and analyze he main properies and feaures of our model, aking ino accoun differences wih models presened in he lieraure. Enering a bi ino deails, we firs analize he behaviour induced by our specificaion of he endogenous rae of ineremporal subsiuion, hen we describe some characerisics of he endogenous elasiciy of ineremporal subsiuion, and finally we pu he elemens ogheer o explain our specificaion of he Euler equaion. In line wih he evidence of Ialian household budge daa, we consider he case of a myopic agen, a middle-aged woman who becomes jobless in a cerain momen of his life and has dependen children. Since being jobless, she spends her ime home and when children go o school, she drinks wine in small doses. As ime passes, he myopic become used o alcohol consumpion revealing habis wih respec o alcohol. Becoming addiced requires he accumulaion of a sock of pas consumpion beyond some criical level. Once his level is reached, consumpion follows an unsable accumulaion pah and addicion resuls. Then we analyze behaviour of he myopic agen joined wih a forward-looking agen, a fory-year-old single, saisfied wih his employmen, who does no disdain half a 9
lire of wine per meal bu he is a healh friend and a spor-loving. The likelihood ha he myopic agen reveals addicion wih respec o wine since being jobless is picked up by a rae of habis adjusmen σ (d m ) higher han he rae of he forward-looking agen, σ (d f ).Thelargerisσ ( ), he more weigh is given o pas consumpion in deermining z. Therefore, he degree of addicion is more inense for an increasing σ ( ). Heerogeneiy is described by he wo raes of habis adjusmen σ (d m ) and σ (d f ) whose values depend on demographic characerisics of he wo agens 6. In doing so, we perform a simulaion analysis imposing some arbirary coheren values o parameers. In figure 3 we presen he phase diagran 7 along wih he simulaed policy funcion for he model 8.Thisfigure serves only o confirm ha he model is well-behaves around he equilibrium poin and ha i is sable. Figure 4 shows ime pahs of consumpion for he forward looking and myopic agens. This graph represen he sysem s speed of convergence o he seady sae. The comparison of he wo pas pu in evidence a significanly differen behavior: he myopic agen have a higher seady sae level of consumpion and reach i much faser han a forward looking. Moreover, for he myopic individual fuure uiliy is discouned more heavily since he habis sock adjus more rapidly and oward a higher level, and hence also he discoun rae, This leads o a higher degree of impaience which explain he differen behavior of he wo curves. 3.2.1 The Endogenous Rae of Time Preference In he lieraure he rae of ime preference is associaed wih he slope of he indifference curves along he 45 line in he plane [c,c +1 ], where c is curren consumpion and c +1 fuure consumpion. This rae can be obained differeniaing wih respec o ime he naural logarihm of he firs-order condiion of he conrol problem considered wih negaive sign. An analyical expression of he rae of ime preference can be delivered for he considered problem oo. The firs-order condiion of he Hamilonian funcion measures a variaion of life-cycle uiliy U (C) wih respec o an infiniesimal small incremen of curren consumpion along a consan pah consumpion, as well as he rae of decrease of marginal uiliy, and a imes near q = u (c ) ϕ ρ (z )ξ (σ) σ. The rae of change, denoed U (C), is discouned by a rae Θ r U (C) =q e ( Θ r) = u (c ) ϕ ρ (z )ξ (σ) σ e ( Θ r). (3.2.1) AfeaureofU isisimpliciraeofimepreferenceθ ( ), a real valued funcion, ha is calculaed along a locally consan consumpion pah by expression (3.2.1) replacing expression (2.1.9.b) (Table 2-D3 ) 6 Noe ha σ(d m ) is referred o a myopic agen, while σ(d f ) is referred o a forward looking one. 7 Noe ha his is he phase diagram of a reduced sysem. Around he seady sae we plausibly assume ha z = c and hence he sysem is reduced by one variable. 8 In all simulaion we use a logarihmic uiliy funcion (u(c )=α log c ), wih consumpion bounded beween and 1, he discoun rae akes he linear form ρ(z )=γ + κz, and parameers are chosen o be: α =.2, γ =.2, κ =.5, ξ(σ) =2, r =.5. The rae of habis adjusmen akes he values of.2 for a forward looking agen (σ(d f ))andof.4 for a myopic one (σ(d m)). 1
θ (c,z,ϕ,σ)= h log q e ( Θ r) i ċ = = log q + (Θ + r) = 1 u (c ) ċ ρ (z ) ϕ q ξ (σ) σ + ρ (z ) ϕ u (c ) / (ρ (z )) =1+ u (c ) ϕ ρ ρ (z )ξ (σ) σ. (3.2.2) (z )ξ (σ) σ The same expression of he rae of ime preference (3.1.6) in he Euler equaion is derived. Our rae of ime preference describes a subjecive preference srucure ha links he pas, presen and fuure consumpion. The rae of ime preference in (3.2.2) incorporaes he following behavioral assumpions: ċ = 1. he memory of pas evens by he rae of habis adjusmen, σ; 2. he percepion of presen evens by he curren consumpion level, c ; ċ = 3. he anicipaion of fuure evens by he presen-value of fuure uiliies, Ψ. Consumer behaviour is non separable along ime, revealing complemenariy. Though, he imporance of he individual is cleares in he rae of ime preference in deermining wheher here is adjacen complemenariy. Presen consumpion, c, and fuure consumpion, by he presen value of fuure uiliies ϕ, depend on pas consumpion, by he rae of habis adjusmen, σ, and need no be valued equally along a locally consan consumpion pah. The rae of ime preference expresses he propensiy ha a person reveals owards fuure uiliy in deermining curren choices. An agen is more or less oriened o he fuure wih respec o he presen value of fuure uiliies. This depends on he capabiliy of anicipaing benefis of fuure consumpion and so physical and menal fuure consequences of presen and pas consumpion effecs. In line wih Shi and Epsein [2], he rae of ime preference is characerized by he same analyical properies. The rae of ime preference is sricly increasing wih respec o he presen value of fuure uiliies ϕ.anincreasein indicaes an increase in fuure consumpion and he response is o give more weigh o he presen, discouning more he feliciy u (c ). Proposiion 2 Theraeofimepreferenceθ (c,z,ϕ,σ) is sricly increasing wih respec o he presen value of fuure sream of uiliies ϕ, holding curren consumpion and he rae of habis adjusmen consan. θ (c, z,ϕ, σ) > (3.2.3) ϕ The rae of ime preference in (3.2.2) is decreasing wih respec o curren consumpion c and indicaes ha he more an agen consumes, he less is concerned wih omorrow raher han oday. In such cases here may be no need o o save agains a rainy day. The rae of ime preference approaches he greaes values when curren consumpion and he presen value of he fuure uiliy, herefore fuure consumpion, are greaes. Proposiion 3 Theraeofimepreferenceθ (c,z,ϕ,σ) is sricly decreasing wih respec o curren consumpion c, holding he rae of habis adjusmen and presen value of fuure sream of uiliies consan in a region around he seady sae. θ (c, z, ϕ, σ) < (3.2.4) c 11
As σ measures he declining marginal uiliy wih respec o ime, U (C), anincreasein he rae of habis adjusme should imply ha he marginal uiliy declines more rapidly. An increase in dangerous subsances as drugs, alcohol and smoking ends o give more weigh o curren feliciy a he expense of fuure feliciy ha is more discouned. As a resul drug addics and alcoholics end o be presen oriened. A decline of fuure feliciy reduces he benefis delivered from a low discoun rae and induces an increasingly higher rae of ime preference. Proposiion 4 Theraeofimepreferenceθ (c,z,ϕ,σ) is increasing wih respec o he rae of habis adjusmen σ, holding curren consumpion and he presen value of fuure sreamof uiliies consan in a region around he seady sae. θ (c, z, ϕ,σ) > (3.2.5) σ The analyical and behavioural properies of he rae of ime preference (Proposiions 3 and 4) allow us o describe he dynamic evoluion of an agen from a condiion of poenial habi o a sae of addicion 9. Reconsider he case of he myopic and forward-looking agen inroduced in Secion 2. From fig.1 noe he degree of impaience of he myopic agen is higher han he degree of he forward-looking because higher is he degree of habis. The propensiy o exchange curren for fuure consumpion becomes less and less considerable. The myopic agen reveals an increasing impaience since his sock of habis wih respec o alcohol is higher. The subjecive rae of ime preference of he myopic agen encloses reinforcemen and olerance, wo behavioural facors ha are closely relaed o he concep of adjacen complemenariy. Reinforcemen means ha greaer curren consumpion of a good rises is fuure consumpion in accordance while olerance means ha given levels of consumpion are less saisfying when pas consumpion has been greaer. On he oher hand, he forward-looking agen is paien, since has greaer capabiliy o anicipae he fuure consequences of presen and pas consumpion. The analysis clearly reveals a paience-dependence radeoff. A paien person has a lower sock of habis han an impaien person, since he desire o anicipae fuure consumpion is lower. I is no surprising ha addicion is more likely for people who discoun he fuure heavily since hey pay less aenion o he adverse consequences. Becker, Grossman and Murphy [3] suggesed ha poorer and younger persons discoun he fuure more heavily while Chaloupka [6] found ha less educaed persons may have higher raes of ime preference. Capabiliy of anicipaing he consequences of presen and pas consumpion depends on income, educaion, rank and degree of awareness of dangers. According o Becker and Mulligan[2]... he analysis of endogenous discoun raes implies ha even fully raional uiliy-maximizing individuals who become addiced o drugs and oher harmful subsances or behaviour are induced o place less weigh on he fuure, even if he addicion iself does no affec he discoun rae. In he Becker and Mulligan s analysis [2], addicion affecs he discoun rae hrough he rae of habis adjusmen. The degree of impaience is higher for lower values of he discoun rae and so he likelihood ha he consumer reveals addicion o a good is greaer. 3.2.2 The Endogenous Elasiciy of Ineremporal Subsiuion If he rae of ime preference is associaed o he slope of he indifference curve along a 45 line in he plane [c,c +1 ], he elasiciy of ineremporal subsiuion gives he 9 For proofs of hese proposiions see appendix A1, for a graphical evidence, see Figures 5 and 6. 12
proporionae change in he magniude of he slope in response o a proporionae change in he raio c /c +1,wherec is curren consumpion and c +1 fuure consumpion.(table 2-D4) u (c ) ϕ η (c,ϕ,σ)= ρ (z )ξ (σ) σ 1 u. (3.2.6) (c ) c The expression allows us o predic more ineremporal subsiuion of consumpion relaive o a consan elasiciy. The elasiciy η assumes higher values han he consan elasiciy and approaches o he las one a he highes presen value of fuure uiliies. An increase in he presen value of fuure uiliies, ϕ, induces an increase in fuure consumpion and he response is o give more weigh o he presen. An increase in fuure consumpion makes agen less available for absaining from curren consumpion a in favour of fuure a +1, making ineremporal subsiuion less considerable. The elasiciy declines wih respec o curren consumpion c, holding he presen value of fuure uiliies consan. This propery of he curve could have an empirical correspondence wih a consumpion analysis of alcohol. The more an agen consumes, he leas he agen is willing o sacrifice curren for fuure consumpion. The elasiciy assumes differen curvaures varying he rae of habis adjusmen σ (Figure 2). This allows us o analyse he effecs induced by habis in ineremporal subsiuion and consider again he myopic and forward-looking agens. The forward-looking agen is more available for changing his pah consumpion han he myopic one o pick he ineremporal incenives, given an equal consumpion and fuure uiliy level: he elasiciy of he forward-looking approaches higher values han he elasiciy of he myopic ha incorporaes a greaer sock of habis wih respec o alcohol han he forward-looking ype for σ (d m ) >σ(d f ). The curve of he elasiciy of he myopic is fla because she/he does no like o exchange curren for fuure consumpion: he elasiciy assumes he same values a each ϕ. The myopic does no anicipae dangers of an excessive alcohol consumpion and is no willing o exchange alcohol consumpion oday, a, for more consumpion omorrow, a +1, +2and so on. 3.2.3 The Euler Equaion Afer having analysed he analyical properies and behavioural conens of he rae of ime preference and he elasiciy of ineremporal subsiuion, we consider now he Euler equaion ċ 1 r = θ (c,ϕ,σ), (3.2.7) η (c,ϕ,σ) c where θ (c,z,ϕ,σ) is he endogenous rae of ime preference (equaion 2.2.6) and η (c,ϕ,σ) is he endogenous elasiciy of ineremporal subsiuion. The Euler equaion is differen from he canonical expression (Table 2 - A2), because i comprehends he complemenariy beween pas consumpion, σ, curren consumpion, c, and fuure consumpion by he presen value of fuure uiliies, ϕ, by he endogenous rae of ime preference and he endogenous elasiciy of ineremporal subsiuion. The complemenariy allows us o explain why an increased rae of reurn o savings, r, ends o induce more paience in consumers. Firs consider he simple case where an increased rae of reurn is compensaed holding he marginal uiliy q consan. All fuure consumpion rises since he rae of reurn is higher and curren consumpion is unchanged by marginal uiliy assumpion holding he growh rae of consumpion consan. The effec is picked 13
up by an increase in he elasiciy of ineremporal subsiuion: he agen is more in favour o he ineremporal subsiuion beween fuure and curren consumpion, given he increased rae of reurn o savings. The problem can be more complicaed. We do no consider a consan marginal uiliy and so an increased rae of reurn can lower he rae of ime preference inducing more paience on he agen. The growh rae of consumpion declines hus increasing savings. Therefore fuure consumpion and he simulaneous effec on he growh rae of consumpion is picked up by an increase in elasiciy o allow he model o approach o an anoher consumpion level equilibrium poin. The impac of an increased rae of reurn o savings on he rae of ime preference and he elasiciy of ineremporal subsiuion and indeed he growh rae of consumpion changes from person o person because people are no equally paien because of he heerogenous srucure of preferences. The analysis of he effecs induced by habis on consumpion pahs reveals how habis can influence he reacion of an agen wih respec o an increased rae of reurn. Does an increase in he rae of reurn lower he rae of ime preference of he myopic and forward-looking agens making hem more paien?an increase in he rae of reurn should have a more considerable impac on he rae of ime preference of he forward-looking han he myopic agen. This depends on he sock of habis held by he wo agens. The myopic agen is addiced o alcohol and his rae of habi adjusmen, σ (d m ), approaches o zero. This implies a fla elasiciy of ineremporal subsiuion for he myopic agen: a all he values of fuure uiliy he elasiciy assumes he same values given a curren consumpion level. The myopic s capabiliy o absain from curren consumpion in favour of savings is reduced and he agen does no reac o an increase of he ineres reurn. 4 Conclusions Tradiionally, he economic lieraure represens he srucure of preferences in a dynamic conex hrough funcionals where a uiliy funcion is discouned by a consan discoun rae. This choice, ofen adoped for he sake of of mahemaical racabiliy, does no allow o explain why he discoun rae differs by income, educaion, occupaional sanding and sex or changes over ime for he same individual. Assuming an endogenous discoun rae depending on pas consumpion as adoped in he Shi and Epsein [2], he sudy develops analyically a new formulaion of he rae of ime preference ha can be reduced wih respec o he exreme values of he rae of habis adjusmen ( and ) o he consan rae of ime preference according o he Ramsey model or o he rae of ime preference obained by an endogenous discoun rae wih respec o curren consumpion according o he Obsfeld model [15]. The rae of ime preference suppors a subjecive srucure of preferences ha comprehends he memory of pas evens, he percepion of presen evens and he anicipaion of fuure evens revealing adjacen complemenariy. The behavioural conens delivered by he dynamic comparaive analysis are in line wih he resuls of he heory of raional addicion. As regards Becker and Murphy [4] and Becker and Mulligan [2], he exension of he model allows us o verify he impac of habis produces on ineremporal consumpion pahs by he rae of ime preference varying he rae of habis adjusmen ha describes he heerogeneiy among agens. The dynamic analysis of addicion proposed by he model is characerized by he following behavioural properies: 1. an increase in he sock of habis induces an increase in he degree of impaience. The higher is previous consumpion, he larger he habi, and he higher mus be he 14
curren level of consumpion o deliver he same effec. The behavioural dynamics of an agen who evolves from habis o a sae of addicion are delivered, deriving a paience-dependence radeoff. A paien agen reveals himself forward-looking valuing he fuure more han a myopic agen, whose level of habis is higher han he firs one, because he is less worried abou he consequences of an excessive curren consumpion. 2. he higher is he incidence of pas consumpion on curren consumpion choices, he lower are he values assumed by he endogenous elasiciy of ineremporal subsiuion as well he lower is he agen s propensiy o exchange curren for fuure consumpion. 3. he heerogenous srucure of habis allows us o explain how an increase in he rae of reurn o savings ends o induce more paience in he forward-looking han myopic consumer. This means ha his model is poenially useful in applicaions wih micro-daa, specially if one can use panel-daa. Observing individual behaviour over ime, ogeher wih demographic informaions, may help o idenify he parameers in his highly nonlinear model. We believe ha an endogenous habi formaion process could play an imporan role in explaining par of unobserved heerogeneiy in individual daa. The ask of a well specified and idenified economeric model will be objecive of a furher work. 15
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A Appendices A.1 Proofs of proposiions Proposiion 2. Theraeofimepreferenceθ (c,z,ϕ,σ) is sricly increasing wih respec o he presen value of fuure sream of uiliies ϕ, holding curren consumpion and he rae of habis adjusmen consan. θ (c, z,ϕ, σ) > : ϕ Assuming ha he discoun rae is linear and increasing in z, ha he uiliy funcion akes he logarihmic form (α log c ) and ha consumpion lies beween and 1, he derivaive of he rae of ime preference θ( ) wih respec o he index of impaience ϕ is ³ σ 2 ξ(σ) 2 ρ (z ) 2 ϕ u(c ) ρ(z ) σξ(σ)ρ (z ) (u (c ) σξ(σ)ρ (z )ϕ ) 2 + (u (c ) σξ(σ)ρ (A1) (z )ϕ ) which, afer summing up and collecing σξ(σ)ρ (z ) become σξ(σ)ρ (z )[ρ(z )u (c ) σξ(σ)ρ (z )u(c )] ρ(z )(u (c ) σξ(σ)ρ (z )ϕ ) 2. (A2) We know ha ξ(σ) is posiive along wih parameer σ. We also know by assumpion ha u (c ) >, ρ (z ) >, ρ(z ) >. Because of consumpion bounds and of assumpion abou preferences form, insananeous uiliy is always negaive u(c ). The Denominaor is for sure posiive and so is he numeraor, since u(c ) is negaive he erm (ρ(z )u (c ) σξ(σ)ρ (z )u(c )) is posiive, so he whole expression is always posiive 1. Proposiion 3. Theraeofimepreferenceθ (c,z,ϕ,σ) is sricly decreasing wih respec o curren consumpion c, holding he rae of habis adjusmen and presen value of fuure sream of uiliies consan in a region around he seady sae. θ (c, z, ϕ, σ) < : c Under he same assumpion of he previous proof, he derivaive of he rae of ime preference θ( ) wih respec o curren consumpion c is ³ σξ(σ)ρ (z )u (c ) (z )u (c ) ϕ u(c) ρ(z )(u (c ) σξ(σ)ρ (z )ϕ ) ρ(z ) (u (c ) σξ(σ)ρ (z )ϕ ) 2. (A3) Summing up and expandig erms we obaing σξ(σ)ρ (z )u (c ) 2 σ 2 ξ(σ) 2 ρ (z ) 2 ϕ u (c ) ρ(z )(u (c ) σξ(σ)ρ (z )ϕ ) 2 ρ(z )σξ(σ)ρ (z )u (c )ϕ + σξ(σ)ρ (z )u (c )u(c ) ρ(z )(u (c ) σξ(σ)ρ (z )ϕ ) 2, (A4) and finally, collecing σε(σ)ρ (z ) σξ(σ)ρ (z) u (c ) 2 σξ(σ)ρ (z )u (c )ϕ + u (c )(ρ(z )ϕ u(c )) ρ(z )(u (c ) σξ(σ)ρ (z )ϕ ) 2. (A5) 1 I is possible o proove ha his proposiion is rue also if uiliy funcion akes he form of a power funcion u(c )= c1 α σξ(σ),providedha < 1, wihou any upper bound for consumpion. 1 α 1 α 18
The denominaor and he erm ouside he square brackes are for sure posiive under curren assumpions. As regards he square bracke, we analyze each single componen. u (c ) is posiive by assumpion, σξ(σ)ρ (z )ϕ is posiive since ϕ is negaive and all oher erms are posiive. The las erm u (c )(ρ(z )ϕ u(c )) is deerminan for he sign. If (ρ(z )ϕ u(c )) is negaive i implies ha cerainly he whole expression is negaive. If no one mus look o he enire square brackeed erm. In our simulaions, we find ha his proposiion is always rue, excep for a quie narrow region which correspond o very low level of consumpion and high level of fuure sreams of uiliy (see figure 5). Proposiion 4. Theraeofimepreferenceθ (c,z,ϕ,σ) is increasing wih respec o he rae of habis adjusmen σ, holding curren consumpion and he presen value of fuure sreamof uiliies consan in a region around he seady sae. θ (c, z, ϕ,σ) > : σ Under he same assumpions of previous proofs, he derivaive of he rae of ime preference θ( ) wih respe o he rae of habis adjusmen σ is ³ ρ (z ) ϕ u(c ) ξ(σ)+σξ ρ(z ) (σ) (u (c ) σξ(σ)ρ (z )ϕ ) ³ σξ(σ) ϕ u(c ) ρ(z ) ρ (z ) ϕ ξ(σ)ρ (z ) σϕ ξ (σ)ρ (z ) (u (c ) σξ(σ)ρ (z )ϕ ) 2. (A6) ³ Summing up and collecing ρ (z ) ϕ u(c) ρ(z ) we obain ³ ρ (z ) ϕ u(c ) ρ(z ) (u (c ) σξ(σ)ρ (z )ϕ ) 2 (A7) ξ(σ)+σξ (σ) (u (c ) σξ(σ)ρ (z )ϕ )+σξ(σ) ϕ ξ(σ)ρ (z )+σϕ ξ (σ)ρ (z ) (u (c ) σξ(σ)ρ (z )ϕ ) 2. Expanding and recollecing he square brackeed erms leads o ³ ρ (z ) ϕ u(c ) u ρ(z ) (c ) ξ(σ)+σξ (σ) (u (c ) σξ(σ)ρ (z )ϕ ) 2, (A8) which, in urn can be wrien as u (c )ρ (z ) ξ(σ)+σξ (σ) (ρ(z )ϕ u(c )) ρ(z )(u (c ) σϕ ξ(σ)ρ (z )) 2. (A9) The denominaor is posiive, and so are erms u (c ) and ρ (z ). ξ(σ) is an always posiive and growing funcion wih respec o σ andhenalso ξ(σ)+σξ (σ) is posiive. Again he key erm is (ρ(z )ϕ u(c )). Figure 6 evidences ha he derivaive of he rae of ime preference wih respec o he rae of habis adjusmen is slighly negaive in a quie wide region for high values of consumpion and low values of fuure sream of uiliy, bu moving owards he opposie siuaion i assumes relevan posiive values. Around he seady sae he derivaive is slighly posiive. 19
A.2 Tables and Figures Table 1: Ineremporal Uiliy Funcions Ineremporal uiliy U (C) Consan discoun rae Endogenous discoun rae R Preference independen of u (c ) e θ R d Ramsey u (c ) e θ(c τ )dτ d Obsfeld Model (199) pas consumpion Model R Preference dependen on pas u (c,z ) e θ R d Ryder u (c ) e θ(z τ )dτ d Shi consumpion and Heal Model and Epsein Model (1993) Along consan pahs Table2: Euler Equaion, Rae of Time Preference and Elasiciy of Ineremporal Subsiuion Ramsey Model u c e d Obsfeld Model (199) u c e Shi and Epsein (1993) u c e Presen paper u c e c d d z d d z d d Ineremporal uiliy U C Euler Equaion Ramsey Model r 1 c c Obsfeld Model (199) r c, 1 c c c, Shi and Epsein (1993) r c, z,, 1 c c c, Presen paper r c, z,, 1 c c,, c Ramsey Model Obsfeld Model (199) c, c 1 Rae of Time Preference u c / c u c c c Shi and Epsein (1993) c, z,, z z z u c Presen paper c, z,, 1 u c / z u c z z Elasiciy of Ineremporal Subsiuion Ramsey Model u c u c c Obsfeld Model (199) c, u c c u c c c Shi and Epsein (1993) c, u c z u c z c Presen paper c,, u c z u c c 2
Figure 1 - Rae of Time Preference of a Forward-looking and Myopic Agen RTP 4 3 2 1 Myopic -3-4.8-2 Forward-looking fuure uili.6 consumpion.4-1 Figure 2 - Ineremporal Elasiciy of a Forward-looking and Myopic Agen 1 75 ISE 5 25 Forward-looking Myopic.8-2.6-4 fuure uiliy.4 consumpion -6 21
c Figure 3: Phase Diagram and Policy Funcion.8.75.7.65.6.55 1.8 1.6 1.4 1.2 1 ϕ Consumpion Figure 4: Consumpion Time Pah for a Myopic and a Forward Looking Agen.8 Myopic.6 Forward Looking.4.2 2 4 6 8 1 12 Time 22
Figure 5: Derrivaive of θ( ) wih respec o c 2 c θ 22 2.8 4 ϕhl.6 chl.4.2 6 Figure 6: Derivaive of θ( ) wih respec o σ σ θ 1 75 5 25 2.8 4 ϕhl.6 chl.4.2 6 23