Consumption and Saving: Models of Intertemporal Allocation and Their Implications for Public Policy

Transcription

1 Consumpion and Saving: Models of Ineremporal Allocaion and Their Implicaions for Public Policy Orazio P. Aanasio and Guglielmo Weber November 11 h, 2009 Revised: February 4 h, 2010 We are graeful o a very large number of people for a number of differen reasons. Our hinking abou he issues discussed in his paper has been paricularly influenced by a se of people, several of whom have been co-auhors in several projecs. They include: Rob Alessie, James Banks, Richard Blundell, Marin Browning, Angus Deaon, Hamish Low, Tom MaCurdy, Cosas Meghir and Luigi Pisaferri. We have discussed many of he issues covered in his paper (and someimes disagreed) wih hem. We cerainly learned a lo from hem. We are very graeful o hree referees for commens and suggesions, and o he edior, for commens, suggesions, and incredible paience! Finally, he second auhor hanks ESRI, Cabine Office, Tokyo, for hospialiy, and many useful commens. The firs auhor s research was parly financed by he ESRC Professorial Fellowship RES UCL, IFS, NBER and CEPR Universiá di Padova, IFS and CEPR

2 Srucure of he paper. 1. Inroducion and moivaion 2. Facs 2.1 Average Individual behaviour 2.2 The evoluion of he cross secional evoluion of consumpion 3. The life-cycle model 3.1. Preferences 3.2. Esimaing preference parameers 3.3. Liquidiy consrains as an explanaion of excess sensiiviy 3.4. Explaining income racking and he reiremen consumpion drop 3.5. Evidence from he levels of consumpion 3.6. Simulaion resuls 4. Budge consrains and Markes: heory and evidence 4.1. Complee Markes 4.2. Exogenously incomplee markes 4.3. Endogenously incomplee markes Imperfec informaion Imperfec enforceabiliy of conracs 5. Alernaive models 5.1. Some puzzles 5.2. Modifying he basic model Relaxing geomeric discouning Relaxing ineremporal separabiliy Financial lieracy and informaion 6. Conclusions 1

3 1. Inroducion. In he early 1950s, he prevailing model of consumpion behaviour used by macro-economiss was inspired by he fundamenal psychological law menioned by Keynes (1936) in he General Theory. A ha ime, he heoreical and empirical limiaions of ha model became increasingly clear. From a heoreical perspecive, i is difficul o consruc coheren models based on ineremporal opimizing behaviour ha are consisen wih Keynes descripion of he fundamenal psychological law. From an empirical poin of view, i seemed ha Keynes view was inconsisen wih a number of facs, boh a he macro and he micro level. A he aggregae level, for insance, i was observed ha he marginal propensiy o consume ou of disposable income was lower in he shor run han in he long run. In cross secions, on he oher hand, saving raes seemed o change sysemaically wih he level of income. Moreover, i was observed ha groups of individuals wih, on average, lower level of income (such as blacks) had higher saving raes han oher groups wih higher levels of average income (such as whies) a any income level. Finally, i was observed ha saving raes are sysemaically relaed o changes in income, being higher for individuals experiencing income increases and lower for individuals experiencing income decreases (see Kaona, 1949). All hese observaions clearly conradiced he implicaions of he Keynesian model and led o he formulaion of he life cycle and permanen income models (Modigliani and Brumberg, 1954, 1980 and Friedman, 1957). These models combined heoreical consisency, in ha ineremporal consumpion and saving choices were se wihin a coheren opimizaion problem, wih he abiliy of fiing mos of he facs menioned in he previous paragraph. The saving raes of blacks was (and is) higher han ha of whies, a any income level, because he permanen income of blacks is lower and herefore, condiioning on a common income level, one selecs he blacks wih higher level of emporary shocks ha should, according o he model, be saved. Similarly, individuals wih income increases are more likely o be affeced by posiive ransiory shocks. A he macro level, shor run flucuaions in disposable income are more likely o be dominaed by he variance of emporary shocks ha would be averaged ou in he long run. Some of hese facs sill hold in modern daa, as we documen in Secion 2. The developmen of he ideas in he seminal conribuions of Modigliani and Blumberg and Friedman also led o he realizaion of oher implicaions. In a simple version of he life cycle model, if income is hump shaped and declines a reiremen, consumers will save when hey are 2

4 young o suppor consumpion in he las par of life and dis-save when hey are old. Modigliani and Blumberg hen showed ha his fac can explain he correlaion beween aggregae growh and aggregae saving: growh implies ha, in a given year, younger cohors, who are saving, are richer in life-ime erms, han older ones, who are dis-saving. The higher he rae of growh is, he larger he difference in resources beween savers and dis-savers and, herefore, he higher he aggregae rae of saving. Afer is iniial developmen, he oher imporan sep in he developmen of he life-cycle/ permanen income model, which is currenly used as he sandard workhorse of modern macroeconomics, was a rigorous reamen of uncerainy. In he lae 1970s, he conribuions of Hall (1978) (and MaCurdy (1981, 1999) in he conex of labour supply) exploied he idea of using he firs order condiions of he ineremporal opimizaion problem faced by he consumer o derive esable implicaions of he model. This approach, known as he Euler equaion approach, makes possible he empirical analysis of a problem ha is analyically inracable by circumvening he need o derive closed form soluions. This is achieved by focusing on he economic essence of he model: consumers, a he opimum, will ac o keep he marginal uiliy of wealh consan over ime. The marginal uiliy of wealh is a he same ime a sufficien saisic for consumer choices and, given is dynamic properies, can be differenced ou in a way which is analogous o he reamen of fixed effecs in economerics. The Euler equaion approach became he sandard approach as i allowed o boh es he validiy of he model and o esimae some of he srucural parameers of he uiliy funcion. A hypohesis ha received much aenion, since Hall (1978), is ha lagged values of income, or predicable changes in income, do no predic fuure consumpion once curren consumpion is accouned for. Perhaps as a consequence of his focus on esing, when i came o policy analysis and debaes he model and in paricular he empirical evidence ha has been accumulaed on i have been rarely used. One of he reasons for his divorce beween he lieraure on he life cycle model and wha should have been is pracical use in he design and evaluaion of public policy sems from he fac ha he Euler equaion does no deliver a consumpion funcion. While i can be used o es he model and esimae some of is parameers, i canno be used o deermine he effecs of specific policy changes on consumpion or saving. A he same ime, much of he evidence ha came o be perceived as he acceped view, poined o rejecions of he life cycle model ha ook he form of excess sensiiviy of consumpion o income. Indeed, in he nex secion, we ake his evidence as one of he 3

5 saring poins of our discussion of he life cycle model, of is empirical plausibiliy and of is uiliy for policy analysis. We have wo main goals: o ake a sand on where he lieraure is and wha he main issues are and o discuss he public policy implicaions of he life cycle/ permanen income models. The life cycle model can be loosely defined as a framework where individuals maximize uiliy over ime given a se of ineremporal rading opporuniies. Even a his level of generaliy, he model is of some usefulness. I esablishes a concepual framework ha reas he ineremporal allocaion of resources in a way which is similar o he allocaion of resources among differen commodiies. Decisions will hen depend on he oal amoun of resources (in he ineremporal conex: curren and fuure income as well as curren wealh), on preferences over he differen commodiies (in he ineremporal conex: presen and fuure consumpion, and possibly bequess) and on relaive prices (ineres raes and ineremporal rade opporuniies). Wihou being more specific, however, i is no possible o say much more han wha saed in he previous paragraph. Or, saying i differenly, his level of generaliy encompasses many differen ypes of behaviour and has almos no esable implicaions. In wha follows, herefore, we consruc a specific model and analyze is componens. This exercise forces us o make a number of srong assumpions and modelling choices ha we discusse below. We choose o work wih a version of he model ha is flexible enough o be brough in a serious way o he daa and ha allows us o derive specific implicaions on a number of policy-relevan quesions. We sar our approach by discussing, in Secion 2, a number of empirical findings. We refer o boh ime series and cross secional findings and we focus especially on resuls ha migh poin o empirical rejecions of he model. We organize our discussion of he empirical evidence in wo pars. We firs discuss evidence ha refers o individual consumpion behaviour. We hen move on o look a evidence derived from movemens in he disribuion of consumpion, which allows researchers o look a he funcioning of markes and he smoohing of various ypes of shocks. Afer reviewing his empirical evidence, in Secion 3 we discuss how a relaively sandard bu sufficienly rich version of he life cycle model can be made consisen wih i. Moreover, we discuss he evidence on he size of he relevan srucural parameers. Having esablished ha he model is no wildly a variance wih he daa and some of he evidence ha was presened as a rejecion of he life cycle model can be reconciled wih i if one specifies a version ha is 4

6 flexible enough, we go ahead and use he model o quanify, by using simulaions, is main properies. In paricular, we show how consumpion changes wih changes in income and ineres raes for differen values of he srucural parameers. The use of simulaions is necessary in his conex because i is no possible o obain closed form soluions. Simulaions are also useful o sudy aspecs of life cycle behaviour ha canno be sudied wih he Euler equaion approach (such as durables, housing ec.), because ransacion coss lead o infrequen adjusmens. Besides preferences and income processes, he oher imporan componen of he life cycle model is he ineremporal budge consrain. A specific hypohesis abou he naure of he ineremporal budge consrain implicily assumes a cerain marke srucure and he insrumens consumer have o move resources over ime (and across saes of he world). Secion 4, herefore, is devoed o he discussion of alernaive marke srucures, saring from he benchmark of complee markes o move on o various models of incomplee markes. One of he hemes of he paper and in paricular of Secion 3 is ha one can consruc rich versions of he life cycle of he model ha are no inconsisen wih some aspecs of he micro daa and can be useful in he conduc of policy analysis. Having said ha, i is clear ha he simples versions of he model are inconsisen wih various aspecs of he daa and ha he empirical lieraure on consumpion has accumulaed a number of puzzles. In Secion 5, we discuss some of hese puzzles and possible exensions and modificaions of he basic model. Secion 6 concludes he paper. 2. Facs. In his secion, we presen some well known facs abou consumpion behaviour boh a he aggregae and a he micro level. Our aim is o presen empirical evidence ha is or migh be relevan o judge he validiy of he life cycle model. Indeed, many of he facs ha we lis below were presened as explici ess of he life cycle/ permanen income model and someimes inerpreed as rejecions of he model. In addiion o hese facs, however, we will also repor some new evidence on old findings ha moivaed he developmen of he life cycle model. We divide he empirical evidence we presen in wo pars. We firs discuss findings ha refer o individual behaviour. In his firs subsecion, we consider how individual consumpion moves, 5

7 on average 1. We hen move on o facs abou he cross secional dispersion of consumpion and inerpre movemens in ime of hese momens as informaive abou risk sharing and insurance markes available o individuals Average Individual behaviour. As was menioned in he inroducion, he life-cycle/ permanen income model was developed o explain some facs abou consumpion. Some of hese facs were noiced in aggregae saisics: (non-durable) consumpion expendiure is less volaile han income and he marginal propensiy o consume seems o be smaller in he shor run han in he long run. These macro facs sill hold and some can also be found in micro daa (such as he relaive variabiliy of non durable consumpion and income - see Aanasio, 2000, and Aanasio and Borella, 2006). Oher facs explicily menioned by he seminal conribuions ha originaed he life cycle/ permanen income model emerged from cross secional sudies and in paricular from observaions of how saving raes vary in he cross secion wih income. As wih he macro facs, hese empirical regulariies sill hold in recen daa. If one looks a US Consumer Expendiure Survey (CEX) daa, one finds ha he saving rae of blacks is higher han ha of whies a any income level, as noed by Friedman (1957). Similar evidence can be obained in US and UK if one looks a he saving raes by curren income level of oher groups ha differ by he level of permanen income, such as households headed by individuals wih differen levels of educaion. Analogously, if one considers separaely individuals whose income has increased and individuals whose income has decreased, he saving rae of he laer is smaller han ha of he former, as noed fify years ago by Modigliani and Brumberg (1954), ciing work by Margare G. Reid. The fac ha hese empirical regulariies sill hold is imporan and we come back o hem when discussing he empirical validiy of he life cycle model. A his sage, we simply sress ha he life cycle/ permanen income model offers a coheren explanaion for hem. The main ideas behind he use of he life cycle model o explain hese facs, is ha consumers have concave uiliy funcions and herefore prefer smooh pahs of consumpion (over ime and across saes of he world) over variable ones. Therefore, only unanicipaed changes in income ha are perceived as permanen will induce subsanive changes in consumpion. Expeced and emporary changes o income should no induce a srong change in consumpion. The 1 Which momen is considered o represen he measure of locaion of he disribuion of individual consumpion is an ineresing issue which we discuss in wha follows. 6

8 explanaion of he facs menioned above boils down o he observaion ha a large fracion of he changes in income considered in hese sylized facs are emporary. For insance, if one classifies individuals wih differen levels of permanen income by he level of curren income, one will find ha, for each curren income level, individuals from he group wih a lower level of permanen income will have a higher level of emporary income, which, he model suggess, should be saved. Ineresingly, he empirical criicisms of he life cycle model ha have been accumulaing since have mainly poined ou deviaions from he predicion ha expeced changes in income should no be incorporaed ino consumpion. These deviaions can be classified ino hree groups: hose ha idenify correlaions beween expeced changes in income and consumpion a low frequencies, hose ha consider shor-run flucuaions linked o changes in earnings and income and hose ha refer o shor-run flucuaions ha are linked o ad-hoc paymens no necessarily relaed o labour supply behaviour. Low frequency, life-cycle paerns. Carroll and Summers (1991), in an influenial paper, show ha life cycle profiles of income and consumpion rack each oher. For many counries boh income and consumpion life cycle profiles are hump shaped, in ha hey increase during he firs par of he life cycle o reach a peak a few years before reiremen and decline aferwards. Groups and counries ha exhibi relaively seep income profiles also exhibi relaively seep consumpion profiles. Carroll and Summers, herefore, conclude ha income and consumpion rack each oher over he life cycle, herefore conradicing one of he main predicions of he life cycle model. We reproduce his ype of graph in Figure 1, where we repor life cycle profiles for disposable income and non durable consumpion for wo educaion groups in he UK (he FES daa used here cover he sample period). We hus adop he same mehodology as Carroll and Summers (1991). The message ha comes ou of hese picures is very similar o heirs: a life cycle frequencies, consumpion profiles do follow income profiles. (This is even more srikingly rue if oal expendiure replaces non-durable consumpion). A drawback wih his ype of graphs is ha hey average over individuals by age, irrespecive of heir year of birh. If differen generaions have access o differen life-cycle resources (as assumed in he life-cycle model) his is no he righ hing o do. In figure 2, we show wha happens when he daa are grouped in year of birh cohors and averages are hen aken by age. (In he figure, cohors are 10-year wide). There is sill evidence of income racking, even hough his is now less clear cu. 7

9 Do hese picures consiue a fundamenal rejecion of he life cycle model? In he nex secion, we will be arguing formally ha he answer is no, boh in heory and in pracice. Here we simply poin ou ha if one wans o be serious abou bringing he life cycle model o he daa, one canno ake he simples version, which is used for pedagogical reasons, bu has o ake ino accoun ha, in all likelihood, consumpion needs evolve over ime as family composiion changes. This argumen is made by Aanasio and Browning (1995). The simples way o sar considering his ype of issues is o look a life cycle profiles for consumpion ha ake ino accoun changes in needs, by considering consumpion per capia or consumpion per adul equivalen, raher han oal household consumpion. Figure 3 reproduces Figure 2, bu using consumpion per adul equivalen 2. As can be noiced he profiles for consumpion are now much flaer. We come back o hese picures and o he inerpreaion of his evidence in wha follows. Levels, By Educ Compulsory Levels, By Educ Pos-compulsory Income and Consumpion Age of Head Income Consumpion Graphs by educ Figure 1 Average income and (non-durable) consumpion by educaion 2 We are graeful o Cormac O Dea for his help wih he FES daa. 8

10 Cohor profiles Compulsory Cohor profiles Pos-compulsory Age of Head Income Consumpion Graphs by educ Figure 2 Average income and consumpion by cohor and educaion 9

11 Per-capia, by cohor Compulsory Per-capia, by cohor Pos-compulsory Age of Head Income Consumpion Graphs by educ Figure 3 Average per-capia income and consumpion by cohor and educaion Arguably he larges predicable change in income is he one ha occurs a reiremen: earnings decline considerably as individuals exi he labour force and such decline should be anicipaed. An obvious predicion of he life-cycle model of Modigliani and Brumberg (1954) is ha individuals, who should have accumulaed wealh (eiher in privae asses or in enilemens o pension benefis) should sar de-cumulaing i o keep a level of consumpion consisen o he one afforded before reiremen. Hamermesh (1984) was he firs o argue ha consumers apparenly do no save enough o achieve his aim. If households ener reiremen wih inadequae savings, hey mus cu heir consumpion level, conrary o he life-cycle model predicions. The recen lieraure has focused on esimaing how consumpion levels change around reiremen. The exisence of a consumpion fall around reiremen is documened for he UK (Banks, Blundell and Tanner, 1998), for he US (Bernheim, Skinner and Weinberg, 2001), and for Ialy (Baisin, Brugiavini, Reore and Weber, 2009) and has come o be known as he reiremen consumpion puzzle (or reiremen savings puzzle). Banks, Blundell and Tanner (1998) find ha for ages beween 60 and 67, he level of consumpion is lower han ha 10

12 prediced by a version of he life cycle model by as much as 1.5% on an annual basis. The cumulaed consumpion shorfall over his age band, where mos people reire, is around 10%. For he US, Bernheim, Skinner and Weinberg (2001) esimae a median drop of 14%, bu higher drops for low wealh, low income replacemen households. They conclude ha "31% of he sample reduce heir consumpion by a leas 35 percenage poins". Baisin, Brugiavini, Reore and Weber (2009), who use Ialian daa, esimae a 9.8 percen he par of he nondurable consumpion drop ha is associaed wih reiremen (food expendiure falls insead by 14%). Business Cycle Frequency. The evidence menioned so far refers o a relaionship beween predicable changes in income and consumpion a he life cycle frequency. Many papers have also looked a he relaionship a higher frequencies. This work is ypically based on he Euler equaions ha we will be discussing in he nex secion, bu basically ess he hypohesis ha, condiional on curren consumpion, fuure consumpion is no affeced by prediced changes in income, or curren level of income. This predicion is obviously relaed o he observaions made by he early proponens of he life cycle/ permanen income hypohesis beween he lack of srong correlaion beween changes in consumpion and income boh in cross secions and in he ime series. Many sudies in he 1980s, insead, found srong rejecions of his predicion. Campbell and Mankiw (1989), in one of he bes known and cied papers, found ha regressing changes in aggregae US log consumpion on ineres raes and changes in log disposable income, he laer variable araced a coefficien of 0.4, saisically differen from zero, even afer insrumening curren variables wih lagged ones o avoid picking up he effecs of innovaions o he level of permanen income. Cambpell and Mankiw (1991) replicae he evidence for he US for a variey of oher counries and aribue such a resul o he presence of a large number of consumers who follow a rule of humb and se heir consumpion equal or proporional o heir income. Hall and Mishkin (1982), perform a similar exercise, bu using micro daa from he US. Using daa on food consumpion from he PSID, hey find a significan correlaion beween changes in food consumpion and lagged changes in income. They inerpre his evidence as indicaing ha abou 20% of households se consumpion on he basis of curren income, raher han following he life cycle model. Anoher sudy ha has uses micro daa is by Zeldes (1989). He uses he same daa as Hall and Mishkin (1982), bu disinguishes beween consumers wih low level of asses and high level of asses and finds ha he consumpion for he former group is more linked o income han he consumpion of he laer. Zeldes (1989) explicily refers o he 11

13 possibiliy ha some consumers are affeced by liquidiy consrains and resricions o borrowing ha do no allow hem o se curren consumpion a he desired level. We come back o he issue of liquidiy consrains in he nex secion. The evidence menioned so far is relevan for he life cycle model as i explois he implicaions of he heoreical framework for changes in consumpion. In he nex secion we map direcly his evidence on he heoreical framework of he life cycle model. However, i is also possible, albei more complicaed, o derive implicaions of some version of he model for he level of consumpion. Inuiively, he heoreical framework implies ha innovaions o permanen income should be fully incorporaed in consumpion 3, while innovaions o ransiory consumpion of income should no. Therefore, if one specifies a ime series model of consumpion and income and idenifies he permanen innovaions o he laer variable, he model predics ha hese innovaions should be ranslaed one o one ino consumpion. This implies cross equaion parameric resricions on he VAR represenaion ha can be esimaed. Campbell and Deaon (1989) poined ou hese resricions and, using aggregae ime series daa, found ha consumpion seems o be oo smooh in ha i does no reac sufficienly o innovaions o he permanen componen of income. Similar findings were obained by Wes (1988), Galì (1991) and Hansen Roberds and Sargen (1991). Perhaps surprisingly, no similar es on micro daa was performed unil he recen paper by Aanasio and Pavoni (2009), who also find excess smoohness. 4 Prediced changes in income. The changes in income ha we have considered so far are large predicable changes ha occur over he life cycle and/or changes ha are likely o be relaed o changes in labour supply. In recen years, a small lieraure has developed ha sudies how consumpion varies in relaion o changes in income ha are no only predicable, bu also driven by evens ha do no have any implicaions for hours worked or labour force paricipaion. In paricular, a large number of papers have looked a he effecs of ax refunds or oher changes linked o adminisraive issues. Papers in his lieraure include Souleles (1999), Parker (1999), Hsieh (2003), Browning and Collado (2001) and Sephens (2007). Souleles, Parker, Sephens and, in par, Hsieh find ha consumpion reacs o changes in he level of resources available o consumers ha are fully predicable. Browning and Collado, on he oher hand, as well as he second par of Hsieh s paper, find ha consumers do no 3 We are absracing here from he possibiliy of insuring permanen shocks and implicily considering a consumer who has access o a fairly limied porfolio of asses o move resources over ime and across saes of he world. 4 An excepion is Deaon (1992b). 12

14 respond o such predicable changes in resources. We come back o he inerpreaion of hese resuls laer The evoluion of he cross secional evoluion of consumpion In he previous subsecion, we have lised a number of facs ha have been discussed in he lieraure on he empirical implicaions of he life cycle model. All of he evidence here referred o he properies of consumpion levels and consumpion changes, on average (eiher by looking a aggregae daa or, in he case of individual daa o regressions aimed a idenifying he behaviour of he average consumer). The evoluion of he cross secion disribuion of consumpion and income-, however, can also be very informaive abou he relevan model ha describes he daa. One of he firs papers o noice he implicaions of a simple version of he life cycle model for he evoluion of consumpion inequaliy was Deaon and Paxson (1994). These auhors noice ha if income has a uni roo, in a basic life cycle model, he cross secional secion of consumpion increases over ime. One can hen consider how he cross secional variance of consumpion for a cohor of individuals born in he same year should increase over ime as hese individuals age. Tesing his forecas for he UK, he US and Taiwan, Deaon and Paxson (1994) show ha his is effecively he case. As innovaions accumulae, he cross secional disribuion of consumpion fans ou wih age. 5 Baisin, Blundell and Lewbell (2009) use a similar argumen o explain a remarkable empirical regulariy: he cross secional disribuion of consumpion seems o be exremely well approximaed by a log normal. This is rue across a wide variey of counries. Under a sandard version of he life cycle model, a any age, (log) consumpion is given by pas (log) consumpion plus a erm ha reflecs an innovaion o permanen income. Therefore, by recursive subsiuion, one ges ha log consumpion is given by he sum of innovaions from he beginning of life o he curren age. By he cenral limi heorem, he sum of independen 5 Using repeaed cross secional daa or longiudinal daa, one can follow he evoluion of consumpion inequaliy for any given cohor and esimae how i evolves wih age and ime. The idenificaion of an average age profile for he variance of consumpion ha is common for differen ime periods and differen cohors, is complicaed by he fac ha age., ime and cohor are obviously linked and, wihou addiional resricions or srucure, i is no possible o idenify separaely age, cohor and ime effecs. Deaon and Paxson (1994) assume some resricions on ime effecs. A forhcoming issue of he Review of Economic Dynamics conains a collecion of papers from differen counries (including he US and he UK) ha underake similar exercises. The shape of he age profile in he US seems o depend crucially on wheher one considers oal household consumpion or consumpion per adul equivalen and which adul equivalence schemes are used. 13

15 innovaion converges o a normal disribuion under some regulariy assumpions, even if he individual innovaions are no normally disribued. The facs abou he evoluion of he cross secional inequaliy of consumpion and income are also used in anoher sudy by Blundell and Preson (1998). Under a specific marke assumpion, hey show ha he relaive evoluion of consumpion and income inequaliy can be used o idenify permanen and ransiory income variances. The idea is relaively simple: if consumers face a simple asse marke srucure, changes in he variance of he permanen componen of income will induce an equal increase in he cross secional variance of consumpion. Therefore he difference beween he increase in he cross secional variance of income and ha of consumpion will idenify he changes in he cross secional variance of ransiory income. The cavea abou he marke srucure in he las paragraph makes i clear ha here is a sringen relaionship beween he ype of insurance markes agens have access o and he evoluion of consumpion inequaliy. Given an iniial disribuion of consumpion (however deermined) in he presence of perfec risk sharing, ha disribuion should say consan (wih some echnical caveas we will discuss in Secion 4). Deaon and Paxson (1994) noiced ha in a foonoe and presened evidence on he evoluion of he cross secional variance of consumpion as a rejecion of he complee marke model. In an ingenious paper, Jappelli and Pisaferri (2006) exploi ha idea by looking explicily a movemens in he relaive ranking in he consumpion disribuion in an Ialian survey. As wih oher papers, hey rejec srongly he assumpion of perfec risk sharing. Similarly, Aanasio and Davis (1996) by looking a he evoluion of relaive consumpion across differen educaion groups, and relaing ha o changes in relaive wage changes inerpre he evidence of a srong correlaion a low frequencies beween hese wo variables as evidence agains he complee marke hypohesis. Ineresingly, Aanasio and Davis (1996) canno rejec he hypohesis ha a relaively high frequencies (like one year) here is no relaionship beween consumpion and relaive wage changes. This seems o indicae ha, somehow, a high frequencies wage shocks are absorbed and no refleced in consumpion. Unil he early 1990s, as repored also by Blundell, Pisaferri and Preson (2008), consumpion inequaliy has increased subsanially, mirroring he increases in inequaliy in wages and earnings. Afer he early 1990s, however, he picure is less clear. Krueger and Perri (2009) repor ha he overall cross secional variance of consumpion in he US has no increased much. Aanasio, Baisin and Ichimura (2007), insead, find ha he cross secional inequaliy of consumpion does increase even in he more recen period. Even hough boh papers use 14

16 he US Consumer Expendiure Survey (CEX), i urns ou ha he main difference in he resuls of hese wo papers sems from he daa used. The CEX is made of wo independen samples: one, called he inerview survey, in which households are asked rerospecive quesions abou heir consumpion in he quarer preceding he inerview, while he oher, he diary survey, in which households are asked o keep a diary for wo weeks. I urns ou ha fac ha Krueger and Perri use daa from he inerview survey, while Aanasio, Baisin and Ichimura inegrae daa from he wo surveys, following he pracice of he Bureau of Labour Saisics, which uses he diary survey for some commodiies and he inerview survey for ohers. The differen evidence abou he evoluion of consumpion inequaliy in he US emerging from wo differen componens of he same survey, which is also he main sources of informaion on consumpion a he micro level in he larges indusrialized counry in he world, jusifies a small digression abou he qualiy of consumpion daa. Informaion abou expendiure and even more so abou consumpion is nooriously difficul o collec in developed counries. A he same ime, he imporance of his informaion canno be undersaed. Reliable informaion on consumpion is key for a hos of issues, ranging from he consrucion of price indexes, which are used o index a variey of paymens, o he assessmen of living condiions and he measuremen of povery, o he esimaion of differen models of individual behaviour and, ulimaely, o he design of public policy. And ye, he resources spen in he collecion of reliable consumpion daa are remarkably small. The CEX is a relaively small survey whose qualiy is perceived o have been deerioraing over he years. 6 While here are signs ha daa collecion in developed counries has become harder as people seem less willing o respond o survey quesions, a redesign and improvemen of consumpion surveys is, in our opinion, very imporan. 3. The Life-Cycle Model. In he firs par of he previous secion, we menioned a number of facs, relaing o boh individual and aggregae consumpion. Afer a brief menion of he facs ha moivaed he developmen of he life cycle model (and ha sill hold in recen daases), we discussed several 6 If one aggregaes he CEX using he appropriae weighs, one obains only a fracion of aggregae Personal Consumpion expendiure as measured in he Naional Accouns. Moreover, his fracion has been declining considerably. 15

17 facs ha could be cas as criicisms of he model, in ha hey conradic some simple implicaions of he heory. To summarize, some of hese facs are: 1) The age profile of consumpion is hump shaped, apparenly racking he age profile of income for each educaion group; moreover, groups of individuals ha have seep income age profiles, seem o have seep consumpion age profiles; 2) Consumpion drops a reiremen; 3) The growh rae of consumpion seems oo sensiive o predicable changes in income; 4) Consumpion seems o reac o changes in available resources ha are fully predicable and ransiory, such as ax refunds. In his secion, we presen he life cycle model in is modern form, and discuss o wha exen i provides an explanaion for he facs lised above. Facs ha go under he firs hree headings, will be explained by he consideraion ha he model does no predic ha individuals smooh heir consumpion bu heir marginal uiliy from consumpion. We leave o he end of his secion our inerpreaion of he facs under he fourh heading. The main idea of he life cycle model is a very general one: i can be saed by saying ha consumers are supposed o allocae resources over ime in order o maximize life ime uiliy subjec o a resource consrain. A his level of generaliy, he model does no have much empirical conen and is no paricularly useful. To bring i o bear on daa and make i poenially falsifiable, we need o pu a bi more srucure on is various componens. In paricular, we have o specify he individual preferences ha inform he maximizaion problem, he naure of he processes generaing he resources available o consumers and he ype of markes hey have access o. In his secion, we specify a basic life cycle model wih an eye o he feaures ha would help us o explain some of he facs we menion above. In addiion, we also discuss how a version of he model ha does fi he available daa can be characerized and used in a variey of conexs. In Secion 4, we discuss he implicaions for he model and is applicaions of he facs abou he disribuion of consumpion discussed in he second par of Secion Preferences The version of he model we consider is one in which a consumer uni maximizes expeced uiliy over a finie inerval subjec o a se of consrains 16

18 Max E s.. T j= 0 β U( C, z, υ + j + j + j + j * W + j+ 1 = W+ j (1 + R + j ) + y + j C + j ) (1) (2) N i W + j = A + i = 1 j (3) R * + j = N i ω i = 1 + i j R + j (4) W T 0 (5) where C sands for consumpion, z for a poenially large vecor of observable variables ha affec uiliy (ha may be chosen by he consumer, or given o her his will normally include household composiion variables), and v for unobservable facors also affecing uiliy. As we shall see demographics play a key role in explaining he way consumpion varies wih age, paricularly in pre-reiremen years. We le he discoun facor β be ime varying o ake ino accoun moraliy risk (ha helps explain why consumpion falls in old age - he survival probabiliy falls wih age, and his makes he consumer progressively more impaien). Throughou he paper we neglec he issue of how decisions are aken wihin he household, and simply assume he household behaves as a uni 7. The firs consrain is a generic budge consrain where ne worh appears ogeher wih is reurn, income and consumpion. Some or all componens of income can be simulaneously deermined wih consumpion. For insance, i is possible ha income is given by he wage rae imes he number of hours worked, where he number of hours is one of he componens of z. Equaions (3) and (4) define ne worh, W, and is reurn - are he porfolio shares (or weighs). The reurn on ne worh is given by he weighed average of he individual reurns, i R + j. We assume hese reurns do nor depend on he ne posiion aken by he consumer on each of hese asses, i. A + j i ω + j 7 In he collecive model of decision making, households are normally assumed o selec efficien allocaions, as suggesed in Chiappori (1988) see Vermeulen (2002) for a survey of his in a saic seing. Browning (2000) is he firs paper o look a he implicaions of relaxing he uniary model assumpions on ineremporal decisions. Mazzocco (2007) ackles he more general problem of household decision making in a T-period uncerain world, by deriving he Euler equaions for individual and household consumpion. He looks a he case where individuals can commi o fuure allocaions of resources, and where commimen is insead no possible because separaion and divorce are a possible way ou 17

19 Equaion (5) gives he limi for oal ne worh a period T. The consumer has o die wihou deb, ha is, she has o pay back her deb wih probabiliy one. This simple resricion imposes quaniaively imporan limiaions o he abiliy o smooh consumpion. Suppose, for insance ha he income process is no bounded away from zero and can acually ake he value zero wih some posiive (small) probabiliy. If we furher assume ha he marginal uiliy of consumpion ends o infiniy a very low levels of consumpion, hen he consumer will never wan o borrow in such a siuaion. This is because he presence of deb ogeher wih he nonbankrupcy consrain and he possibiliy ha income akes he value of zero would imply assigning posiive probabiliy o zero or even negaive consumpion, which he consumer deeply dislikes. The consumer will hen never wan o borrow, even small amouns. One can generalize his o siuaions where he income process is bounded away from zero. In his case, he consumer will no wan o borrow more han he presen value of he lowes level of income. Similar consideraions apply whenever he survival probabiliy is less han one, if longeviy risks canno be fully insured. A number of imporan resricions are assumed in his formulaion. Firs, he consumer is assumed o maximize expeced uiliy. This is a srong assumpion which is ofen used in he lieraure. Someimes he Von Neumann- Morgensern framework is replaced wih differen axiomaic srucures, such as he Kreps-Poreus axiomaizaion as paramerized by Epsein and Zin (1989, 1991) 8. Second, we are assuming ha preferences are addiively separable over ime. This precludes he consideraion of various ypes of non-separabiliy, ranging from durables o habi formaion. We reurn o his issue below. Third, we are implicily assuming ha i is possible o wrie down uiliy as a funcion of a single commodiy. This pracice presupposes an aggregaion heorem of he ype sudied by Gorman (1959). The problem formulaed above is able o encompass differen versions of he model ha have been considered in he lieraure. In paricular, we rea as special cases he sandard permanen income/ life cycle model wih quadraic preferences, he so-called buffer sock saving as well as flexible versions of he model (wih an imporan role for demographics and labour supply) ha have been fied o he daa. 8 Expeced uiliy forces a negaive relaion beween risk aversion and ineremporal subsiuion, bu hese are wo disinc conceps. This promped Epsein and Zin (1989) o propose an alernaive model ha is based on Kreps and Poreus (1978) preferences. Unlike expeced uiliy opimizers, Kreps and Poreus consumers care abou he ime when uncerainy is resolved, even if hey canno ake any acion as a resul. Epsein and Zin (1989) derive a full se of firs order condiions and show ha he Euler equaion involves no only consumpion growh and he ineres rae, bu also he reurn on he marke porfolio. Epsein and Zin (1991) and Aanasio and Weber (1989) presen esimaes of he Euler equaion for his ype of preferences. 18

20 We shall show ha he flexible versions of he model can indeed explain he firs hree sylised facs presened a he beginning of he secion. In paricular, we shall show ha he hump in he age profile of consumpion is due o he inerplay of demographics and prudence, he excess sensiiviy of consumpion growh o income growh is due o he dependence of he marginal uiliy of consumpion on leisure, while he reiremen consumpion drop is due parly o adverse shocks inducing reiremen, parly o more efficien shopping ha is made convenien by he increased leisure ime. In order o prove all his, we need o work ou he soluion o he opimizaion problem. Some feaures of he soluion can be undersood by looking a he firs order condiions, ohers require he derivaion of he consumpion funcion, eiher analyically (in some special cases) or numerically. Le us sar wih a case where he consumpion funcion can be derived analyically. Le uiliy be quadraic in consumpion (and addiively separable in is oher argumens z), and assume ha a leas one financial asse is freely raded and yields a fixed real reurn, equal o he consan ime preference parameer 1 β. The firs order condiion wih respec o consumpion, or Euler equaion, implies ha consumpion is a random walk: β E ( C 1 = C (6) + I ) where I denoes informaion available a ime (Hall, 1978). If consumers have raional expecaions, hen: C + 1 = C + 1 ( + 1 = + ε E ε W ) 0 (7) for all variables W known a ime. Equaion (7) can be used o derive a consumpion funcion, in he case where no oher asse is available o he consumer (as in Bewley, 1977) and he only sochasic variable is labour income. Subsiuing (7) ino he budge consrains, Flavin (1991) shows ha consumpion is se equal o permanen income, defined as he ineres rae imes he presen value of curren and expeced fuure incomes: r r = A + E ( y k I ) (8) + r 1 + r C + 1 k = 0 Equaion (8) is derived for he special case of infinie life, bu an exension o finie life can be derived. 19

21 In his model, he firs difference in consumpion, or he error erm in (7), equals he presen value of income revisions, due o he accrual of new informaion beween periods and (+1): r 1 ΔC (9) [ E( y I ) E( y I ] + 1 = ) k k k 1+ r k = 0 (1 + r) Equaion (7) highlighs he consumpion smoohing properies of he soluion, emphasized in he seminal paper by Modigliani and Brumberg (1954). Equaion (8) makes clear he oher main implicaion of he model, ha was firs sressed in Friedman (1957): consumpion depends on he presen discouned value of fuure expeced income. The ineres rae plays he imporan role of convering fuure resources o presen ones and herefore consiues an imporan deerminan of consumpion. Equaion (8) imposes cross equaion resricions on he join ime series process for income and consumpion, as noed in Sargen (1978). Equaion (9) implies ha in appraising he effecs of a given policy, for insance a ax reform ha affecs disposable income, a disincion mus be drawn beween permanen and emporary changes (Blinder and Deaon, 1985, Poerba, 1988). Anoher implicaion of (9) is ha saving predics fuure changes in income, he so-called saving for a rainy day moive (Campbell, 1987). Quadraic uiliy implies cerainy equivalence: he consumpion funcion (8) is he same as under cerainy, once expecaions are replaced by realizaions. This is convenien for analyical purposes, bu clearly resricive, for insance in is reamen of financial decisions: quadraic preferences imply increasing absolue risk aversion in consumpion (or wealh), somehing ha is unappealing on heoreical grounds and srongly counerfacual (riskier porfolios are normally held by wealhier households). Quadraic preferences also imply ha he willingness o subsiue over ime is a decreasing funcion of consumpion: poor consumers should reac much more o ineres rae changes han rich consumers, afer allowance has been made for he wealh/income effec. The alernaive adoped in much of he lieraure has been o assume power uiliy and o allow for he exisence of a number of risky financial asses. Power uiliy, also known as iso-elasic, 1 γ C 1 or CRRA uiliy, is defined as U ( c) = ; i converges o ln(c) for γ = 1. 1 γ 20

22 Once one deviaes from quadraic uiliy, however, and/or allows for sochasic ineres raes, one loses he abiliy o obain a closed form soluion for consumpion. Many of he sudies ha made his choice, herefore, have focussed on he Euler equaions derived from he maximizaion problem faced by he consumer. The basic firs order condiions used in his lieraure are: U c = λ (10) k [ λ 1 r ) I ] + 1 ( λ = E β (11) where equaion (11) is valid as long as he k-h asse can be freely raded by consumers. Equaion (10) says ha, a each poin in ime, he marginal uiliy of consumpion equals he Lagrange muliplier associaed wih he budge consrain relevan for ha period, which is someimes referred o as he marginal uiliy of wealh. The second condiion, equaion (11), ha is derived from ineremporal opimaliy, dicaes he evoluion of he marginal uiliy of wealh. An equaion of his ype has o hold for each asse k for which he consumer is no a a corner. This is because he consumer is exploiing ha paricular ineremporal margin. The araciveness of Euler equaions is ha one can be agnosic abou he sochasic environmen faced by he consumer, he ime horizon, he possible presence of a beques moive, he presence of imperfecions in financial markes (as long as here is a leas one asse ha he consumer can freely rade), and he presence of fricions in oher variables affecing uiliy, z. All relevan informaion is summarized in he level of he marginal uiliy of wealh. The approach is concepually similar o he use of an (unobservable) fixed effec in economerics. By aking firs differences, one eliminaes he unobservable marginal uiliy of wealh and is lef only wih he innovaions o equaion (11). This approach has played an imporan role in he empirical analysis of he life-cycle model and we will come back o i. The derivaion of a closed form soluion for consumpion when cerainy equivalence does no hold is possible in he case where he uiliy funcion exhibis consan absolue risk aversion. Caballero (1991) shows ha in a modified Flavin model (wih cerain finie life and CARA preferences) he opimal consumpion age profile is fla wih no uncerainy, bu increasing wih income uncerainy. This change in he slope of he consumpion profile is labelled as 21

23 precauionary saving, because early in life consumers save more if labour income is more uncerain. Laer work by Gollier (1995) and Carroll and Kimball (1996) esablished ha a similar resul holds whenever he hird derivaive of he uiliy funcion is posiive, and his feaure of preferences is labelled prudence. Boh CARA and power uiliy exhibi prudence. The presence and size of precauionary savings is a maer of grea relevance for public policy, in so far as public insurance schemes covering such risks as unemploymen, healh and longeviy should reduce he need for consumers o accumulae asses. The grea meri of even his simple model wih prudence is ha i highlighs he need o save for rainy days even if sunny days are equally imporan. An increased variance in he shocks o income reduces consumpion even if expeced income does no change. In he case of discree variables, such as unemploymen or illness, changes in firs and second momens occur simulaneously, bu his is no he case for coninuous variables. The abiliy o disinguish beween firs and second momens effecs is of crucial imporance in he analysis of public policy, because public policy can be used o provide social insurance, by reducing he variance while keeping he mean consan. For insance, a revenue-neural ax reform ha cus axes for he rich may depress consumpion because i induces more precauionary saving (Varian, 1980, sresses he insurance role of a progressive income ax) Esimaing preference parameers The Euler equaion is paricularly useful from an empirical poin of view because i can be cas as a se of orhogonaliy condiions ha should hold in a variey of siuaions and allows esimaing preference parameers and esing he validiy of he model wihou being explici abou all he deails of he sochasic environmen faced by he consumer and wihou having o solve explicily he dynamic opimizaion problem for consumpion or oher variables joinly deermined wih consumpion. As sressed by Chamberlain (1984), esimaion of he Euler equaion requires observaions covering a long period of ime, as he orhogonaliy condiions hold in expecaion, and (bu for he special case of complee markes) sample expecaions converge o populaion expecaions only over ime (see also Hayashi, 1987). A version of he Euler equaion holds even if he consumer chooses labour supply, durable consumpion and many oher variables ha are subjec o differen ypes of adjusmen coss and fricions. I holds under a wide variey of assumpions abou he informaion se used by 22

24 he consumer and, by he law of ieraed expecaions, i holds whenever he informaion se used by he economerician is no larger han ha available o he consumer. To use i, one does no need o specify assumpions abou pension sysems, fuure wage processes, bequess moives and so on and so forh. Moreover, i reflecs he main essence of he life cycle model: he fac ha consumpion is chosen so o keep (discouned, expeced) marginal uiliy consan over ime. The Euler equaion can be used for wo purposes: esing for he validiy of some of he model assumpions, noably he abiliy of consumers o save in response o changes in ineremporal prices, and esimaing preference parameers. The firs paper o esimae a consumpion Euler equaion (Hall, 1978) was enirely devoed o esing he model, bu much of he lieraure since has done boh. Hall ook he case of quadraic uiliy and a fixed ineres rae such ha (1+r)β =1. Under hese condiions, equaion (6) obains and preference parameers are no idenified. Anoher noable feaure of Hall s version of he Euler equaion for consumpion is ha i aggregaes perfecly, because i involves linear ransformaions of he daa, and can herefore be empirically implemened in micro and aggregae daa alike. The Euler equaion (6) implies ha no variable known o he consumer a ime should help predic he change in consumpion beween and (+1) an imporan and easy o es implicaion of he ineremporal opimizaion model ha has been rejeced a number of imes on aggregae and micro daa alike (Jappelli and Pagano, 1989, Hall and Mishkin, 1982). The special feaures of he Hall s model may explain hese rejecions for his reason in he lieraure Euler equaions have been esimaed and esed for more general preference specificaions. As menioned earlier, a popular preference specificaion is he power uiliy funcion, given by 1 γ C 1 U ( c) =, ha has been used in he consumpion lieraure since he 1 γ papers by Hansen and Singleon (1982 and 1983). Is main advanage is analyic convenience, as i yields firs order condiions ha are log-linear in consumpion. However, such a specificaion also imposes srong resricions on preferences. The elasiciy of ineremporal subsiuion of consumpion is, in his conex, consan and equal o 1/γ. This implies ha he degree of ineremporal subsiuabiliy of consumpion is independen of he level of consumpion, even a very low levels of consumpion. Moreover, he same parameer governs boh he elasiciy of ineremporal subsiuion and he degree of risk aversion. This is he 23