Attanasion Weber JEL 2010

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Journal of Economic Literature 48 (September 2010): 693751http:www.aeaweb.org/articles.php?doi=10.1257/jel.48.3.693

693

1. Introduction

In the early 1950s, the prevailing modelof consumption behavior used by mac-roeconomists was inspired by the funda-mental psychological law mentioned byJohn Maynard Keynes (1936) in the GeneralTheory. At that time, the theoretical andempirical limitations of that model becameincreasingly clear. From a theoretical

perspective, it is difcult to construct coher-ent models based on intertemporal opti-mizing behavior that are consistent withKeyness description of the fundamentalpsychological law. From an empirical pointof view, it seemed that Keyness view wasinconsistent with a number of facts, both atthe macro and the micro level. At the aggre-gate level, for instance, it was observed thatthe marginal propensity to consume out ofdisposable income was lower in the short runthan in the long run. In cross sections, on theother hand, saving rates seemed to changesystematically with the level of income.Moreover, it was observed that groups of

individuals with, on average, lower levelsof income (such as blacks) had higher sav-ing rates than other groups with higher lev-els of average income (such as whites) atany income level. Finally, it was observedthat saving rates are systematically relatedto changes in income, being higher for indi-viduals experiencing income increases andlower for individuals experiencing incomedecreases (see George Katona 1949).

Consumption and Saving: Models ofIntertemporal Allocation and TheirImplications for Public Policy

O P. A G W*

This paper provides a critical survey of the large literature on the life cycle model ofconsumption, both from an empirical and a theoretical point of view. It discusses severalapproaches that have been taken in the literature to bring the model to the data, theirempirical successes, and their failures. Finally, the paper reviews a number of changes

to the standard life cycle model that could help solve the remaining empirical puzzles.

*Attanasio: UCL, IFS, NBER, and CEPR. Weber:Universit di Padova, IFS, and CEPR. We are gratefulto a very large number of people for a number of differ-ent reasons. Our thinking about the issues discussed in thispaper has been particularly inuenced by a set of people,

several of whom have been coauthors in several projects.They include: Rob Alessie, James Banks, Richard Blundell,Martin Browning, Angus Deaton, Hamish Low, Tom MaC-urdy, Costas Meghir, and Luigi Pistaferri. We have discussedmany of the issues covered in this paper (and sometimes dis-agreed) with them. We certainly learned a lot from them.We are very grateful to three referees and Erik Hurst foruseful comments and suggestions, and to the Editor forcomments, suggestions, and incredible patience! Attanasiosresearch was partially nanced by ESRC grant RES-051-27-0125. Webers research was partially nanced by MIURgrant 2007AC54X5. Weber is also grateful to ESRI, CabinetOfce, Tokyo, for hospitality and many useful discussions.


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Journal of Economic Literature, Vol. XLVIII (September 2010)694

All these observations clearly contra-dicted the implications of the Keynesianmodel and led to the formulation of thelife cycle and permanent income models(Franco Modigliani and Richard Brumberg1954, 1980; Milton Friedman 1957). Thesemodels combined theoretical consistencyin that intertemporal consumption and sav-ing choices were set within a coherent opti-mization problem with the ability of ttingmost of the facts mentioned in the previousparagraph. The saving rates of blacks was(and is) higher than that of whites at any

income level because the permanent incomeof blacks is lower and, therefore, condition-

ing on a common income level, one selectsthe blacks with a higher level oftemporaryshocks that should, according to the model,be saved. Similarly, individuals with incomeincreases are more likely to be affected bypositive transitory shocks. At the macro level,short-run uctuations in disposable incomeare more likely to be dominated by the vari-ance of temporary shocks that would beaveraged out in the long run. Some of thesefacts still hold in modern data, as we docu-ment in section 2.

The development of the ideas in theseminal contributions of Modigliani andBrumberg and Friedman also led to therealization of other implications. In a simpleversion of the life cycle model, if income ishump shaped and declines at retirement,consumers will save when they are young tosupport consumption in the last part of lifeand dissave when they are old. Modigliani

and Brumberg then showed that this fact canexplain the correlation between aggregategrowth and aggregate saving: growth impliesthat, in a given year, younger cohorts, whoare saving, are richer in lifetime termsthan older ones, who are dissaving. Thehigher the rate of growth is, the larger thedifference in resources between savers anddissavers and, therefore, the higher theaggregate rate of saving.

After its initial development, the otherimportant step in the development of thelife cycle/permanent income model, whichis currently used as the standard workhorseof modern macroeconomics, was a rigoroustreatment of uncertainty. In the late 1970s,the contributions of Robert E. Hall (1978)(and Thomas E. MaCurdy 1981, 1999 in thecontext of labor supply) exploited the idea ofusing the rst-order conditions of the inter-temporal optimization problem faced by theconsumer to derive testable implicationsof the model. This approach, known as theEuler equation approach, makes possiblethe empirical analysis of a problem that is

analytically intractable by circumventing theneed to derive closed-form solutions. Thisis achieved by focusing on the economicessence of the model: consumers, at the opti-mum, will act to keep the marginal utility ofwealth constant over time. The marginal util-ity of wealth is at the same time a sufcientstatistic for consumer choices and, given itsdynamic properties, can be differenced outin a way which is analogous to the treatmentof xed effects in econometrics.

The Euler equation approach became thestandard approach as it allowed to both testthe validity of the model and to estimatesome of the structural parameters of the util-ity function. A hypothesis that received muchattention, since Hall (1978), is that laggedvalues of income, or predictable changes inincome, do not predict future consumptiononce current consumption is accounted for.Perhaps as a consequence of this focus on

testing, when it came to policy analysis anddebates, the model and in particular theempirical evidence that has been accumu-lated on it have been rarely used. One of thereasons for this divorce between the litera-ture on the life cycle model and what shouldhave been its practical use in the design andevaluation of public policy stems from thefact that the Euler equation does not delivera consumption function. While it can be used


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695Attanasio and Weber: Consumption and Saving

to test the model and estimate some of itsparameters, it cannot be used to determinethe effects of specic policy changes on con-sumption or saving.

At the same time, much of the evidencethat came to be perceived as the accepted

view pointed to rejections of the life cyclemodel that took the form of excess sensitiv-ity of consumption to income. Indeed, inthe next section, we take this evidence as oneof the starting points of our discussion of thelife cycle model, of its empirical plausibility,and of its utility for policy analysis. We havetwo main goals: to take a stand on where theliterature is and what the main issues are and

to discuss the public policy implications ofthe life cycle/permanent income models.The life cycle model can be loosely dened

as a framework where individuals maximizeutility over time given a set of intertempo-ral trading opportunities. Even at this levelof generality, the model is of some useful-ness. It establishes a conceptual frameworkthat treats the intertemporal allocation ofresources in a way which is similar to theallocation of resources among different com-modities. Decisions will then depend on thetotal amount of resources (in the intertem-poral context: current and future income aswell as current wealth), on preferences overthe different commodities (in the intertem-poral context: present and future consump-tion, and possibly bequests), and on relativeprices (interest rates and intertemporal tradeopportunities).

Without being more specic, however, it is

not possible to say much more than what isstated in the previous paragraph. Or, sayingit differently, this level of generality encom-passes many different types of behavior andhas almost no testable implications. In whatfollows, therefore, we construct a specicmodel and analyze its components. This exer-cise forces us to make a number of strongassumptions and modeling choices that wediscuss below. We choose to work with a ver-

sion of the model that is exible enough to bebrought in a serious way to the data and thatallows us to derive specic implications on anumber of policy-relevant questions.

We start our approach by discussing a num-ber of empirical ndings in section 2. We referto both time series and cross sectional ndingsand we focus especially on results that mightpoint to empirical rejections of the model. Weorganize our discussion of the empirical evi-dence in two parts. We rst discuss evidencethat refers to individual consumption behav-ior. We then move on to look at evidencederived from movements in the distributionof consumption, which allows researchers to

look at the functioning of markets and thesmoothing of various types of shocks.After reviewing this empirical evidence,

we discuss how a relatively standard but suf-ciently rich version of the life cycle modelcan be made consistent with it in section 3.Moreover, we discuss the evidence on the sizeof the relevant structural parameters. Havingestablished that the model is not wildly atvariance with the data and some of the evi-dence that was presented as a rejection of thelife cycle model can be reconciled with it ifone species a version that is exible enough,we go ahead and use the model to quantify, byusing simulations, its main properties. In par-ticular, we show how consumption changeswith changes in income and interest ratesfor different values of the structural param-eters. The use of simulations is necessary inthis context because it is not possible to obtainclosed form solutions.

Simulations are also useful to study aspectsof life cycle behavior that cannot be studiedwith the Euler equation approach (such asdurables, housing, etc.) because transactioncosts lead to infrequent adjustments.

Besides preferences and income pro-cesses, the other important component of thelife cycle model is the intertemporal budgetconstraint. A specic hypothesis about thenature of the intertemporal budget constraint


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implicitly assumes a certain market structureand the instruments consumers have to moveresources over time (and across states of theworld). Section 4, therefore, is devoted to thediscussion of alternative market structures,starting from the benchmark of completemarkets to move on to various models ofincomplete markets.

One of the themes of the paper, and in par-ticular of section 3, is that one can constructrich versions of the life cycle of the model thatare not inconsistent with some aspects of themicro data and can be useful in the conduct ofpolicy analysis. Having said that, it is clear thatthe simplest versions of the model are incon-

sistent with various aspects of the data andthat the empirical literature on consumptionhas accumulated a number of puzzles. In sec-tion 5, we discuss some of these puzzles andpossible extensions and modications of thebasic model. Section 6 concludes the paper.

2. Facts

In this section, we present some wellknown facts about consumption behaviorboth at the aggregate and at the micro level.Our aim is to present empirical evidence thatis or might be relevant to judge the valid-ity of the life cycle model. Indeed, many ofthe facts that we list below were presentedas explicit tests of the life cycle/permanentincome model and sometimes interpreted asrejections of the model. In addition to thesefacts, however, we will also report some newevidence on old ndings that motivated the

development of the life cycle model.We divide the empirical evidence wepresent in two parts. We rst discuss nd-ings that refer to individual behavior. In thisrst subsection, we consider how individualconsumption moves, on average.1 We then

1Which moment is considered to represent the measureof location of the distribution of individual consumption isan interesting issue which we discuss in what follows.

move on to facts about the cross-sectionaldispersion of consumption and interpretmovements in time of these moments asinformative about risk sharing and insurancemarkets available to individuals.

2.1 Average Individual Behavior

As was mentioned in the introduction,the life cycle/permanent income model wasdeveloped to explain some facts about con-sumption. Some of these facts were noticedin aggregate statistics: (nondurable) con-sumption expenditure is less volatile thanincome and the marginal propensity to con-

sume seems to be smaller in the short runthan in the long run. These macro factsstill hold and some can also be found inmicro data (such as the relative variabilityof nondurable consumption and incomesee Orazio P. Attanasio 2000 and Attanasioand Margherita Borella 2006). Other factsexplicitly mentioned by the seminal contribu-tions that originated the life cycle/permanentincome model emerged from cross-sectionalstudies and, in particular, from observationsof how saving rates vary in the cross sectionwith income. As with the macro facts, theseempirical regularities still hold in recent data.If one looks at U.S. Consumer ExpenditureSurvey (CEX) data, one nds that the savingrate of blacks is higher than that of whitesat anyincome level, as noted by Friedman(1957). Similar evidence can be obtained inthe United States and the United Kingdomif one looks at the saving rates by current

income level of other groups that differ by thelevel of permanent income, such as house-holds headed by individuals with differentlevels of education. Analogously, if one con-siders separately individuals whose incomehas increased and individuals whose incomehas decreased, the saving rate of the latteris smaller than that of the former, as notedfty years ago by Modigliani and Brumberg(1954), citing work by Margaret G. Reid.


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The fact that these empirical regularitiesstill hold is important and we come back tothem when discussing the empirical valid-ity of the life cycle model. At this stage, wesimply stress that the life cycle/permanentincome model offers a coherent explana-tion for them. The main ideas behind theuse of the life cycle model to explain thesefacts is that consumers have concave util-ity functions and, therefore, prefer smoothpaths of consumption (over time and acrossstates of the world) over variable ones.Therefore, only unanticipated changes inincome that are perceived as permanent will

induce substantive changes in consumption.Expected and temporary changes to incomeshould not induce a strong change in con-sumption. The explanation of the facts men-tioned above boils down to the observationthat a large fraction of the changes in incomeconsidered in these stylized facts are tem-porary. For instance, if one classies indi-viduals with different levels of permanentincome by the level of current income, one

will nd that, for each current income level,individuals from the group with a lower levelof permanent income will have a higher levelof temporary income, which, the model sug-gests, should be saved.

Interestingly, the empirical criticisms ofthe life cycle model that have been accumu-lating since have mainly pointed out devia-tions from the prediction that expectedchanges in income should not be incorpo-rated into consumption. These deviationscan be classied into three groups: thosethat identify correlations between expectedchanges in income and consumption at low

frequencies, those that consider short-runuctuations linked to changes in earningsand income, and those that refer to short-run uctuations that are linked to ad hocpayments not necessarily related to laborsupply behavior.

2.1.1 Low Frequency, Life Cycle Patterns

Christopher D. Carroll and Lawrence H.Summers (1991), in an inuential paper,

Incomeandconsumption

800

600

400

200

Levels, by educationCompulsory

Levels, by educationPostcompulsory

Age of head

25 35 45 55 65 75 25 35 45 55 65 75

Income Consumption

Figure 1. Average Income and (Nondurable) Consumption by Education

Source: U.K. Family Expenditure Survey, 19782007.


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show that life cycle proles of income andconsumption track each other. For manycountries both income and consumptionlife cycle proles are hump shaped, in thatthey increase during the rst part of thelife cycle to reach a peak a few years beforeretirement and decline afterwards. Groupsand countries that exhibit relatively steepincome proles also exhibit relatively steepconsumption proles. Carroll and Summers,therefore, conclude that income and con-sumption track each other over the life cycle,therefore contradicting one of the main pre-

dictions of the life cycle model.We reproduce this type of graph in g-ure 1 where we report life cycle prolesfor disposable income and nondurable con-sumption for two education groups in theUnited Kingdom (the Family ExpenditureSurvey data used here cover the 19782007sample period). We thus adopt the samemethodology as Carroll and Summers(1991). The message that comes out of

these pictures is very similar to theirsatlife cycle frequencies, consumption pro-les do follow income proles. (This is evenmore strikingly true if total expenditurereplaces nondurable consumption).

A drawback with this type of graph isthat they average over individuals by age,irrespective of their year of birth. If differ-ent generations have access to different lifecycle resources (as assumed in the life cyclemodel) this is not the right thing to do. In g-ure 2, we show what happens when the dataare grouped in year of birth cohortsand

averages are then taken by age. (In the g-ure, cohorts are ten-year wide). There is stillevidence of income tracking, even thoughthis is now less clear cut.

Do these pictures constitute a fundamen-tal rejection of the life cycle model? In thenext section, we will be arguing formally thatthe answer is no, both in theory and in prac-tice. Here we simply point out that, if onewants to be serious about bringing the life

Figure 2. Average Income and Consumption by Cohort and Education


Cohort prolesCompulsory

Cohort prolesPostcompulsory

Age of head

20 40 60 80 20 40 60 80

Income Consumption

1000

500

0


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cycle model to the data, one cannot take thesimplest version, which is used for pedagogi-cal reasons, but has to take into account that,in all likelihood, consumption needs evolveover time as family composition changes.This argument is made by Attanasio andMartin Browning (1995).

The simplest way to start considering thistype of issues is to look at life cycle pro-les for consumption that take into accountchanges in needs, by considering consump-tion per capita or consumption per adult

equivalent, rather than total household con-sumption. Figure 3 reproduces gure 2 butusing consumption per adult equivalent.2 Ascan be noticed, the proles for consumptionare now much atter. We come back to thesepictures and to the interpretation of this evi-dence in what follows.

2We are grateful to Cormac ODea for his help with theFamily Expenditure Survey data.

Arguably the largest predictable changein income is the one that occurs at retire-ment: earnings decline considerably asindividuals exit the labor force and suchdecline should be anticipated. An obvi-ous prediction of the life cycle model ofModigliani and Brumberg (1954) is thatindividuals, who should have accumulatedwealth (either in private assets or in enti-tlements to pension benets), should startdecumulating it to keep a level of consump-tion consistent to the one afforded before

retirement. Daniel S. Hamermesh (1984)was the rst to argue that consumers appar-ently do not save enough to achieve thisaim. If households enter retirement withinadequate savings, they must cut theirconsumption level, contrary to the life cyclemodel predictions.

The recent literature has focused on esti-mating how consumption levels changearound retirement. The existence of a

Per capita, by cohortCompulsory

Per capita, by cohortPostcompulsory

Age of head

20 40 60 80 20 40 60 80

Income Consumption

500

400

300

200

100

Figure 3. Average Per Capita Income and Consumption by Cohort and Education



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consumption fall around retirement is doc-umented for the United Kingdom (JamesBanks, Richard Blundell, and Sarah Tanner1998), for the United States (B. DouglasBernheim, Jonathan Skinner, and StevenWeinberg 2001), and for Italy (Erich Battistinet al. 2009) and has come to be known as theretirement consumption puzzle (or retire-ment savings puzzle). Banks, Blundell, andTanner (1998) nd that, for ages between60 and 67, the level of consumption is lowerthan that predicted by a version of the lifecycle model by as much as 1.5 percent onan annual basis. The cumulated consump-tion shortfall over this age band, where most

people retire, is around 10 percent. Forthe United States, Bernheim, Skinner, andWeinberg (2001) estimate a median drop of14 percent but higher drops for low wealth,low income replacement households. Theyconclude that 31 percent of the samplereduce their consumption by at least 35 per-centage points. Battistin et al. (2009), whouse Italian data, estimate at 9.8 percent thepart of the nondurable consumption dropthat is associated with retirement (foodexpenditure falls instead by 14 percent).

2.1.2 Business Cycle Frequency

The evidence mentioned so far refers to arelationship between predictable changes inincome and consumption at the life cycle fre-quency. Many papers have also looked at therelationship at higher frequencies. This workis typically based on the Euler equations thatwe will be discussing in the next section, but

basically tests the hypothesis that, conditionalon current consumption, future consumptionis not affected by predicted changes in income,or current level of income. This prediction isobviously related to the observations made bythe early proponents of the life cycle/perma-nent income hypothesis between the lack ofstrong correlation between changes in con-sumption and income both in cross sectionsand in the time series. Many studies in the

1980s, instead, found strong rejections of thisprediction. John Y. Campbell and N. GregoryMankiw (1990a), in one of the best known andcited papers, found that regressing changes inaggregate U.S. log consumption on interestrates and changes in log disposable income,the latter variable attracted a coefcient of0.4, statistically different from zero, evenafter instrumenting current variables withlagged ones to avoid picking up the effects ofinnovations to the level of permanent income.Campbell and Mankiw (1991) replicate theevidence for the United States for a variety ofother countries and attribute such a result tothe presence of a large number of consum-

ers who follow a rule of thumb and set theirconsumption equal or proportional to theirincome.

Hall and Frederic S. Mishkin (1982)perform a similar exercise but using microdata from the United States. Using data onfood consumption from the Panel Study ofIncome Dynamics (PSID), they nd a sig-nicant correlation between changes in foodconsumption and lagged changes in income.They interpret this evidence as indicat-ing that about 20 percent of households setconsumption on the basis of current incomerather than following the life cycle model.Another study that uses micro data is byStephen P. Zeldes (1989). He uses the samedata as Hall and Mishkin (1982) but distin-guishes between consumers with a low levelof assets and a high level of assets and ndsthat the consumption for the former group ismore linked to income than the consumption

of the latter. Zeldes (1989) explicitly refersto the possibility that some consumers areaffected by liquidity constraints and restric-tions to borrowing that do not allow them toset current consumption at the desired level.We come back to the issue of liquidity con-straints in the next section.

The evidence mentioned so far is relevantfor the life cycle model as it exploits theimplications of the theoretical framework


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for changes in consumption. In the nextsection, we map directly this evidence onthe theoretical framework of the life cyclemodel. However, it is also possible, albeitmore complicated, to derive implicationsof some version of the model for the levelof consumption. Intuitively, the theoreti-cal framework implies that innovations to

permanent income should be fully incorpo-rated in consumption, while innovations to

transitory consumption of income shouldnot.3 Therefore, if one species a time seriesmodel of consumption and income and iden-ties the permanent innovations to the lat-ter variable, the model predicts that these

innovations should be translated one to oneinto consumption. This implies cross equa-tion parametric restrictions on the VAR rep-resentation that can be estimated. Campbelland Angus Deaton (1989) pointed out theserestrictions and, using aggregate time seriesdata, found that consumption seems tobe too smooth in that it does not react suf-ciently to innovations to the permanentcomponent of income. Similar ndings wereobtained by Kenneth D. West (1988), JordiGal (1991) and Lars Peter Hansen, WilliamT. Roberds, and Thomas J. Sargent (1991).Perhaps surprisingly, no similar test on microdata was performed until the recent paper byAttanasio and Nicola Pavoni (2007), who alsond excess smoothness.4

2.1.3 Predicted Changes in Income

The changes in income that we have con-sidered so far are large predictable changes

that occur over the life cycle and/or changesthat are likely to be related to changes in laborsupply. In recent years, a small literature hasdeveloped that studies how consumption

3We are abstracting here from the possibility of insur-ing permanent shocks and implicitly considering a con-sumer who has access to a fairly limited portfolio of assetsto move resources over time and across states of the world.

4 An exception is Deaton (1992a).

varies in relation to changes in income that arenot only predictable, but also driven by eventsthat do not have any implications for hoursworked or labour force participation. In par-ticular, a large number of papers have lookedat the effects of tax refunds or other changeslinked to administrative issues. Papers inthis literature include Nicholas S. Souleles(1999), Jonathan A. Parker (1999), Chang-Tai Hsieh (2003), Browning and M. DoloresCollado (2001), and Melvin Stephens (2008).Souleles, Parker, Stephens and, in part, Hsiehnd that consumption reacts to changes inthe level of resources available to consum-ers that are fully predictable. Browning and

Collado, on the other hand, as well as the sec-ond part of Hsiehs paper, nd that consumersdo not respond to such predictable changesin resources. We come back to the interpreta-tion of these results later.

2.2 The Evolution of the Cross-SectionalEvolution of Consumption

In the previous subsection, we havelisted a number of facts that have beendiscussed in the literature on the empiricalimplications of the life cycle model. All ofthe evidence there referred to the proper-ties of consumption levels and consumptionchanges, on average (either by looking ataggregate data or, in the case of individualdata, to regressions aimed at identifying thebehavior of the average consumer). The evo-lution of the cross-sectional distribution ofconsumptionand incomehowever, canalso be very informative about the relevant

model that describes the data.One of the rst papers to notice theimplications of a simple version of the lifecycle model for the evolution of consump-tion inequality was Deaton and ChristinaPaxson (1994). These authors notice that, ifincome has a unit root, in a basic life cyclemodel, the cross-sectional section of con-sumption increases over time. One can thenconsider how the cross-sectional variance of


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consumption for a cohort of individuals bornin the same year should increase over timeas these individuals age. Testing this forecastfor the United Kingdom, the United States,and Taiwan, Deaton and Paxson (1994) showthat this is effectively the case. As innova-tions accumulate, the cross-sectional distri-bution of consumption fans out with age.5

Battistin, Blundell, and Arthur Lewbel(2009) use a similar argument to explain aremarkable empirical regularitythe cross-sectional distribution of consumption seemsto be extremely well approximated by a lognormal. This is true across a wide variety ofcountries. Under a standard version of the

life cycle model, at any age, (log) consump-tion is given by past (log) consumption plus aterm that reects an innovation to permanentincome. Therefore, by recursive substitution,one gets that log consumption is given by thesum of innovations from the beginning of lifeto the current age. By the central limit theo-rem, the sum of independent innovations con-verges to a normal distribution under someregularity assumptions, even if the individualinnovations are not normally distributed.

The facts about the evolution of the cross-sectional inequality of consumption andincome are also used in another study byBlundell and Ian Preston (1998). Under a spe-cic market assumption, they show that therelative evolution of consumption and incomeinequality can be used to identify permanentand transitory income variances. The idea isrelatively simple: if consumers face a simpleasset market structure, changes in the vari-

ance of the permanent component of income

5Using repeated cross-sectional data or longitudinaldata, one can follow the evolution of consumption inequal-ity for any given cohort and estimate how it evolves withage and time. The identication of an average age pro-le for the variance of consumption that is common fordifferent time periods and different cohorts, is compli-cated by the fact that age, time, and cohort are obviouslylinked and, without additional restrictions or structure, itis not possible to identify separately age, cohort, and time

will induce an equal increase in the cross-sectional variance of consumption. Therefore,the difference between the increase in thecross-sectional variance of income and that ofconsumption will identify the changes in thecross-sectional variance of transitory income.

The caveat about the market structure inthe last paragraph makes it clear that there isa stringent relationship between the type ofinsurance markets agents have access to andthe evolution of consumption inequality. Givenan initial distribution of consumption (howeverdetermined) in the presence of perfect risksharing, that distribution should stay constant(with some technical caveats we will discuss in

section 4). Deaton and Paxson (1994) noticedthat in a footnote and presented evidence onthe evolution of the cross-sectional varianceof consumption as a rejection of the completemarket model. In an ingenious paper, TullioJappelli and Luigi Pistaferri (2006) exploit thatidea by looking explicitly at movements in therelative ranking in the consumption distribu-tion in an Italian survey. As with other papers,they reject strongly the assumption of perfectrisk sharing.

Similarly, Attanasio and Steven J. Davis(1996), by looking at the evolution of rela-

tive consumption across different educationgroups and relating that to changes in rela-

tive wage changes, interpret the evidence of astrong correlation at low frequencies betweenthese two variables as evidence against thecomplete market hypothesis. Interestingly,Attanasio and Davis (1996) cannot reject thehypothesis that, at relatively high frequen-

cies (like one year), there is no relationship

effects. Deaton and Paxson (1994) assume some restric-tions on time effects. A forthcoming issue of the Review ofEconomic Dynamics contains a collection of papers fromdifferent countries (including the United States and theUnited Kingdom) that undertake similar exercises. Theshape of the age prole in the United States seems todepend crucially on whether one considers total householdconsumption or consumption per adult equivalent andwhich adult equivalence schemes are used.


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between consumption and relative wagechanges. This seems to indicate that, some-how, at high frequencies wage shocks areabsorbed and not reected in consumption.

Until the early 1990s, as reported alsoby Blundell, Pistaferri, and Preston (2008),consumption inequality has increased sub-stantially, mirroring the increases in inequal-ity in wages and earnings. After the early1990s, however, the picture is less clear. DirkKrueger and Fabrizio Perri (2009) reportthat the overall cross-sectional variance ofconsumption in the United States has notincreased much. Attanasio, Battistin, andHidehiko Ichimura (2007), instead, nd that

the cross-sectional inequality of consump-tion does increase even in the more recentperiod. Even though both papers use theCEX, it turns out that the main differencein the results of these two papers stems fromthe data used. The CEX is made of two inde-pendent samples: one, called the interviewsurvey, in which households are asked retro-spective questions about their consumptionin the quarter preceding the interview, whilethe other, the diary survey, in which house-holds are asked to keep a diary for two weeks.It turns out that, in fact, Krueger and Perriuse data from the interview survey whileAttanasio, Battistin, and Ichimura integratedata from the two surveys, following thepractice of the Bureau of Labor Statistics,which uses the diary survey for some com-modities and the interview survey for others.

The different evidence about the evolutionof consumption inequality in the United States

emerging from two different components ofthe same survey, which is also the main sourceof information on consumption at the microlevel in the largest industrialized country inthe world, justies a small digression aboutthe quality of consumption data. Informationabout expenditure and even more so aboutconsumption is notoriously difcult to col-lect in developed countries. At the same time,the importance of this information cannot be

understated. Reliable information on con-sumption is key for a host of issues, rangingfrom the construction of price indexes, whichare used to index a variety of payments, tothe assessment of living conditions and themeasurement of poverty, to the estimation ofdifferent models of individual behavior and,ultimately, to the design of public policy. Andyet, the resources spent in the collection ofreliable consumption data are remarkablysmall. The CEX is a relatively small surveywhose quality is perceived to have been dete-riorating over the years.6 While there are signsthat data collection in developed countrieshas become harder as people seem less will-

ing to respond to survey questions, a redesignand improvement of consumption surveys is,in our opinion, very important.

3. The Life Cycle Model

In the rst part of the previous section,we mentioned a number of facts, relatingto both individual and aggregate consump-tion. After a brief mention of the facts thatmotivated the development of the life cyclemodel (and that still hold in recent datasets),

we discussed several facts that could be castas criticisms of the model, in that they con-tradict some simple implications of the the-ory. To summarize, some of these facts are:

1. The age prole of consumption is humpshaped, apparently tracking the ageprole of income for each educationgroup; moreover, groups of individuals

that have steep income age proles,seem to have steep consumption ageproles;

2. Consumption drops at retirement;

6 If one aggregates the CEX using the appropriateweights, one obtains only a fraction of aggregate PersonalConsumption expenditure as measured in the NationalAccounts. Moreover, this fraction has been decliningconsiderably.


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3. The growth rate of consumption seemstoo sensitive to predictable changesin income;

4. Consumption seems to react to changesin available resources that are fullypredictable and transitory, such as taxrefunds.

In this section, we present the life cyclemodel in its modern form and discuss to whatextent it provides an explanation for the factslisted above. Facts that go under the rstthree headings will be explained by the con-sideration that the model does not predictthat individuals smooth their consumption

but their marginal utility from consump-tion. We leave to the end of this section ourinterpretation of the facts under the fourthheading.

The main idea of the life cycle model isa very general one: it can be stated by say-ing that consumers are supposed to allocateresources over time in order to maximizelife time utility subject to a resource con-straint. At this level of generality, the modeldoes not have much empirical content andis not particularly useful. To bring it to bearon data and make it potentially falsiable,we need to put a bit more structure onits various components. In particular, wehave to specify the individual preferencesthat inform the maximization problem,the nature of the processes generating theresources available to consumers, and thetype of markets they have access to. In thissection, we specify a basic life cycle model

with an eye to the features that would helpus to explain some of the facts we mentionabove. In addition, we also discuss how aversion of the model that does t the avail-able data can be characterized and used in avariety of contexts. In section 4, we discussthe implications for the model and its appli-cations of the facts about the distribution ofconsumption discussed in the second partof section 2.

3.1 Preferences

The version of the model we consider isone in which a consumer unit maximizesexpected utility over a nite interval subjectto a set of constraints

(1) max Etj=0

Tt

t+jU(Ct+j, zt+j,vt+j),

such that

(2) Wt+j+1 = Wt+j(1 + Rt+j* )

+ yt+j Ct+j ,

(3) Wt+j =i=1

N

At+ji ,

(4) Rt+ j* = i=1

N

t+ ji Rt+ji ,

and

(5) WT 0,

where C stands for consumption, z for apotentially large vector of observable vari-ables that affect utility (that may be chosenby the consumer, or given to herthis willnormally include household compositionvariables), and v for unobservable factorsalso affecting utility. As we shall see, demo-

graphics play a key role in explaining the wayconsumption varies with age, particularlyin preretirement years. We let the discountfactor be time varying to take into accountmortality risk (that helps explain why con-sumption falls in old agethe survivalprobability falls with age, and this makesthe consumer progressively more impa-tient). Throughout the paper, we neglect theissue of how decisions are taken within the


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household, and simply assume the house-hold behaves as a unit.7

The rst constraint is a generic budgetconstraint where net worth appears togetherwith its return, income, and consumption.Some or all components of income can besimultaneously determined with consump-tion. For instance, it is possible that incomeis given by the wage rate times the numberof hours worked, where the number of hoursis one of the components ofz. Equations (3)and (4) dene net worth, W, and its return t+j

i are the portfolio shares (or weights). Thereturn on net worth is given by the weightedaverage of the individual returns, Rt+ j

i . Weassume these returns do not depend on thenet position taken by the consumer on eachof these assets,At+j

i .Equation (5) gives the limit for total net

worth at period T. The consumer has to diewithout debt, that is, she has to pay backher debt with probability one. This simplerestriction imposes quantitatively importantlimitations to the ability to smooth consump-tion. Suppose, for instance, that the incomeprocess isnot bounded away from zero andcan actually take the value zero with somepositive (small) probability. If we furtherassume that the marginal utility of consump-tion tends to innity at very low levels of con-sumption, then the consumer will never wantto borrow in such a situation. This is becausethe presence of debt together with the non-bankruptcy constraint and the possibility that

7 In the collective model of decision making, house-

holds are normally assumed to select efcient allocationsas suggested in Pierre-Andre Chiappori (1988)seeFrederic Vermeulen (2002) for a survey of this in a staticsetting. Browning (2000) is the rst paper to look at theimplications of relaxing the unitary model assumptions onintertemporal decisions. Maurizio Mazzocco (2007) tacklesthe more general problem of household decision makingin a T-period uncertain world, by deriving the Euler equa-tions for individual and household consumption. He looksat the case where individuals can commit to future alloca-tions of resources, and where commitment is instead notpossiblebecause separation and divorce are a possible

way out.

income takes the value of zero would implyassigning positive probability to zero or evennegative consumption, which the consumerdeeply dislikes. The consumer will thennever want to borrow even small amounts.One can generalize this to situations wherethe income process is bounded away fromzero. In this case, the consumer will not wantto borrow more than the present value of thelowest level of income. Similar consider-ations apply whenever the survival probabil-ity is less than one if longevity risks cannot befully insured.

A number of important restrictions areassumed in this formulation. First, the con-

sumer is assumed to maximize expectedutility. This is a strong assumption that isoften used in the literature. Sometimes theVon NeumannMorgenstern framework isreplaced with different axiomatic structures,such as the KrepsPorteus axiomatizationas parametrized by Larry G. Epstein andStanley E. Zin (1989, 1991).8 Second, weare assuming that preferences are additivelyseparable over time. This precludes theconsideration of various types of nonsepara-bility, ranging from durables to habit forma-tion. We return to this issue below. Third, weare implicitly assuming that it is possible towrite down utility as a function of a singlecommodity. This practice presupposes anaggregation theorem of the type studied byWilliam M. Gorman (1959).

8 Expected utility forces a negative relation between risk

aversion and intertemporal substitution, but these are twodistinct concepts. This prompted Epstein and Zin (1989)to propose an alternative model that is based on Kreps andPorteus (1978) preferences. Unlike expected utility opti-mizers, Kreps and Porteus consumers care about the time

when uncertainty is resolved, even if they cannot take anyaction as a result. Epstein and Zin (1989) derive a full setof rst order conditionsand show that the Euler equa-tion involves not only consumption growth and the interestrate but also the return on the market portfolio. Epsteinand Zin (1991) and Attanasio and Guglielmo Weber (1989)present estimates of the Euler equation for this type ofpreferences.


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The problem formulated above is able toencompass different versions of the modelthat have been considered in the literature.In particular, we treat as special cases thestandard permanent income/life cycle modelwith quadratic preferences, the so-calledbuffer stock saving as well as exible ver-sions of the model (with an important rolefor demographics and labor supply) thathave been tted to the data.

We shall show that the exible versionsof the model can indeed explain the rstthree stylized facts presented at the begin-ning of the section. In particular, we shallshow that the hump in the age prole of

consumption is due to the interplay ofdemographics and prudence, the excesssensitivity of consumption growth toincome growth is due to the dependenceof the marginal utility of consumption onleisure, while the retirement consumptiondrop is due partly to adverse shocks induc-ing retirement and partly to more efcientshopping that is made convenient by theincreased leisure time.

In order to prove all this, we need towork out the solution to the optimizationproblem. Some features of the solutioncan be understood by looking at the rstorder conditions, others require thederivation of the consumption function,either analytically (in some special cases) ornumerically.

Let us start with a case where the con-sumption function can be derived analyti-cally. Let utility be quadratic in consumption

(and additively separable in its other argu-mentsz) and assume that at least one nan-cial asset is freely traded and yields a xedreal return equal to the constant time pref-erence parameter (1 )/. The rst ordercondition with respect to consumption, orEuler equation, implies that consumption isa random walk:

(6) E(Ct+1|It) = Ct,

where It denotes information available attimet (Hall 1978). If consumers have ratio-nal expectations, then:

(7) Ct

+1

= Ct

+ t

+1 E(

t+

1| W

t

) = 0

for all variables Wknown at timet. Equation(7) can be used to derive a consumptionfunction in the case where no other asset isavailable to the consumer (as in Truman F.Bewley 1977) and the only stochastic vari-able is labor income. Substituting (7) intothe budget constraints, Marjorie Flavin(1981) shows that consumption is set equalto permanent income, dened as the interest

rate times the present value of current andexpected future incomes:

(8) Ct =r_

1 + rAt

+ r_1 + r

k=0

E(yt+k| It).

Equation (8) is derived for the special caseof innite life but an extension to nite lifecan be derived.

In this model, the rst difference in con-sumption, or the error term in (7), equals thepresent value of income revisions due to theaccrual of new information between periodst and (t+ 1):

(9) Ct+1 =r_

1 + rk=0

1_(1 + r)k

[E(yt+k+1| It+1) E(yt+k+1| It)].

Equation (7) highlights the consumptionsmoothing properties of the solution empha-sized in the seminal paper by Modigliani andBrumberg (1954). Equation (8) makes clearthe other main implication of the model thatwas rst stressed in Friedman (1957): con-sumption depends on the present discountedvalue of future expected income. The inter-est rate plays the important role of converting


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future resources to present ones and there-fore constitutes an important determinantof consumption. Equation (8) imposes crossequation restrictions on the joint time seriesprocess for income and consumption as notedin Sargent (1978). Equation (9) implies that,in appraising the effects of a given policy, forinstance a tax reform that affects disposableincome, a distinction must be drawn betweenpermanent and temporary changes (Alan S.Blinder and Deaton 1985; James M. Poterba1988). Another implication of (9) is that sav-ing predicts future changes in incometheso-called saving for a rainy day motive(Campbell 1987).

Quadratic utility implies certainty equiva-lence: the consumption function (8) is thesame as under certainty once expectationsare replaced by realizations. This is conve-nient for analytical purposes, but clearlyrestrictive, for instance in its treatment ofnancial decisions: quadratic preferencesimply increasing absolute risk aversion inconsumption (or wealth), something thatis unappealing on theoretical grounds andstrongly counterfactual (riskier portfoliosare normally held by wealthier households).Quadratic preferences also imply that thewillingness to substitute over time is adecreasing function of consumptionpoorconsumers should react much more to inter-est rate changes than rich consumers afterallowance has been made for the wealth/income effect.

The alternative adopted in much of the lit-erature has been to assume power utility and

to allow for the existence of a number of riskynancial assets. Power utility, also known asisoelastic, or constant relative risk aversionutility, is dened as U(c) = (C1 1)/(1 ); it converges to ln(C) for = 1.

Once one deviates from quadratic utility,however, and/or allows for stochastic interestrates, one loses the ability to obtain a closedform solution for consumption. Many of thestudies that made this choice, therefore,

have focused on the Euler equations derivedfrom the maximization problem faced by theconsumer. The basic rst order conditionsused in this literature are:

(10) Uc t = t

and

(11) t = E[t+1(1 + rt+1k ) | It],

where equation (11) is valid as long as thekth asset can be freely traded by consumers.

Equation (10) says that, at each point intime, the marginal utility of consumption

equals the Lagrange multiplier associatedwith the budget constraint relevant for thatperiod, which is sometimes referred to as themarginal utility of wealth. The second condi-tion, equation (11), that is derived from inter-temporal optimality, dictates the evolution ofthe marginal utility of wealth. An equationof this type has to hold for each asset k forwhich the consumer is not at a corner. This isbecause the consumer is exploiting that par-ticular intertemporal margin.

The attractiveness of Euler equations isthat one can be agnostic about the stochas-tic environment faced by the consumer, thetime horizon, the possible presence of abequest motive, the presence of imperfec-tions in nancial markets (as long as thereis at least one asset that the consumer canfreely trade), and the presence of frictionsin other variables affecting utility, z. Allrelevant information is summarized in the

level of the marginal utility of wealth. Theapproach is conceptually similar to the use ofan (unobservable) xed effect in economet-rics. By taking rst differences, one elimi-nates the unobservable marginal utility ofwealth and is left only with the innovationsto equation (11). This approach has playedan important role in the empirical analysis ofthe life cycle model and we will come backto it.


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The derivation of a closed-form solutionfor consumption when certainty equivalencedoes not hold is possible in the case where theutility function exhibits constant absolute riskaversion. Ricardo J. Caballero (1991) showsthat, in a modied Flavin model (with certainnite life and constant absolute risk aversionpreferences), the optimal consumptionageprole is at with no uncertainty but increas-ing with income uncertainty. This change inthe slope of the consumption prole is labeledas precautionary saving because, early in life,consumers save more if labor income is moreuncertain. Later work by Christian Gollier(1995) and Carroll and Miles S. Kimball (1996)

established that a similar result holds when-ever the third derivative of the utility functionis positive, and this feature of preferences islabeled prudence. Both constant absolute riskaversion and power utility exhibit prudence.The presence and size of precautionary sav-ings is a matter of great relevance for publicpolicy in so far as public insurance schemescovering such risks as unemployment, health,and longevity should reduce the need for con-sumers to accumulate assets.

The great merit of even this simple modelwith prudence is that it highlights the needto save for rainy days even if sunny days areequally important. An increased variance inthe shocks to income reduces consumptioneven if expected income does not change. Inthe case of discrete variables, such as unem-ployment or illness, changes in rst and sec-ond moments occur simultaneously, but thisis not the case for continuous variables. The

ability to distinguish between rst and secondmoments effects is of crucial importance in theanalysis of public policy because public policycan be used to provide social insurance, byreducing the variance while keeping the meanconstant. For instance, a revenue-neutral taxreform that cuts taxes for the rich may depressconsumption because it induces more precau-tionary saving (Hal R. Varian 1980 stresses theinsurance role of a progressive income tax).

3.2 Estimating Preference Parameters

The Euler equation is particularly usefulfrom an empirical point of view because itcan be cast as a set of orthogonality condi-tions that should hold in a variety of situ-ations and allows estimating preferenceparameters and testing the validity of themodel without being explicit about all thedetails of the stochastic environment facedby the consumer and without having to solveexplicitly the dynamic optimization problemfor consumption or other variables jointlydetermined with consumption. As stressedby Gary Chamberlain (1984), estimation of

the Euler equation requires observationscovering a long period of time, as the orthog-onality conditions hold in expectation, and(but for the special case of complete mar-kets) sample expectations converge to popu-lation expectations only over time (see alsoFumio Hayashi 1987).

A version of the Euler equation holds evenif the consumer chooses labor supply, dura-ble consumption, and many other variablesthat are subject to different types of adjust-ment costs and frictions. It holds under awide variety of assumptions about the infor-mation set used by the consumer and, by thelaw of iterated expectations, it holds when-ever the information set used by the econo-metrician is no larger than that available tothe consumer. To use it, one does not need tospecify assumptions about pension systems,future wage processes, bequests motives,and so on and so forth. Moreover, it reects

the main essence of the life cycle model: thefact that consumption is chosen so to keep(discounted, expected) marginal utility con-stant over time.

The Euler equation can be used for twopurposes: testing for the validity of some ofthe model assumptions, notably the ability ofconsumers to save in response to changes inintertemporal prices, and estimating prefer-ence parameters. The rst paper to estimate


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a consumption Euler equation (Hall 1978)was entirely devoted to testing the modelbut much of the literature since has doneboth.

Hall took the case of quadratic utility anda xed interest rate such that (1 +r)= 1.Under these conditions, equation (6) obtainsand preference parameters are not identied.Another notable feature of Halls version ofthe Euler equation for consumption is thatit aggregates perfectly because it involveslinear transformations of the data and can,therefore, be empirically implemented inmicro and aggregate data alike. The Eulerequation (6) implies that no variable known

to the consumer at time t should help pre-dict the change in consumption between tand (t+ 1)an important and easy to testimplication of the intertemporal optimiza-tion model that has been rejected a numberof times on aggregate and micro data alike(Jappelli and Marco Pagano 1989; Hall andMishkin 1982).

The special features of Halls model mayexplain these rejectionsfor this reason, inthe literature, Euler equations have beenestimated and tested for more generalpreference specications. As mentionedearlier, a popular preference specica-tion is the power utility function, given byU(c) = (C1 1)/(1 ), which has beenused in the consumption literature since thepapers by Hansen and Kenneth J. Singleton(1982 and 1983). Its main advantage is ana-lytic convenience, as it yields rst order con-ditions that are log-linear in consumption.

However, such a specication also imposesstrong restrictions on preferences. The elas-ticity of intertemporal substitution of con-sumption is, in this context, constant andequal to 1/. This implies that the degree ofintertemporal substitutability of consump-tion is independent of the level of consump-tion, even at very low levels of consumption.Moreover, the same parameter governs boththe elasticity of intertemporal substitution

and the degree of risk aversion. This is theconsequence of the assumption of intertem-poral separability and separability acrossstates of the world.

Substituting equation (10) into (11) andusing the properties of the power utilityfunction the Euler equations for consump-tion corresponding to each asset (k) are:

(12) Et ca Ct+1_Ct b (1 + rt+1k )d = 1,where is a curvature parameter (equalto the relative risk aversion parameter and

to the reciprocal of the elasticity of inter-temporal substitution) and , the subjec-tive discount factor, measures patience.Equation (12) is an orthogonality conditionstating that a particular transformation ofthe data is orthogonal to the information setused by the agent. Such a condition suggestsnaturally the use of some GMM method toestimate the unknown parameters and, tothe extent one considers a vector of vari-ables whose dimension is greater than thatof the parameter to be estimated, to test thevalidity of the model. In essence, Hall (1978)was the rst test, in a specic context, of thisorthogonality condition.

An equation such as (12) can be log-linearized to obtain (see Hansen andSingleton 1983):

(13) ln Ct+1 = t+1

+ 1_

ln (1 + rt+1k ) + t+1k ,where t+1 is a time-varying term thatdepends on the preference parameters and as well as on the conditional secondmoment of the argument of the expectedutility operator in equation (12).

Estimating equation (12) seems prefer-able because no assumption has to be madeabout the conditional variance term but will


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produce inconsistent estimates wheneverthere is measurement error in consumption.Equation (13), instead, can be consistentlyestimated if there is serially uncorrelatedmeasurement error as long as one can ndinstruments that are orthogonal to both theerror term and the time varying intercept.Attanasio and Hamish Low (2004) discussconditions under which equation (13), esti-mated under the assumption of a constantt+1, yields consistent estimates for thecurvature parameter, . Notice that the otherpreference parameter, the discount factor, isnot identied in this framework, as it getsburied into the constant.9

Of particular importance for policy analy-sis is 1/, or elasticity of intertemporal substi-tution, that tells us how the marginal rate ofsubstitution between today and tomorrowsconsumption reacts to changes in the inter-est rate, keeping lifetime utility constant.The increase in the interest rate representsa decrease in the price of future consump-tion relative to current consumption and thisinduces a substitution effect of a decreasein current consumption and a commensu-rate increase in current saving. This wouldbe counteracted by an income effect since,with a higher interest rate, a given targetlevel of future consumption is achieved withless saving. As noted by Summers (1981),wealth effects, concerning the amount thatexpected future incomes are discounted,reinforce substitution effects and also leadto a decrease in consumption or increase insaving when the interest rate goes up. These

wealth effects tend to be stronger when thetime period that the individual cares aboutis longer. Ultimately, which of these forces

9 Equations (12) and its log-linearized version (13) referto an individual asset. If the consumer has access to severalassets for which she is not at a corner, one can consider anEuler equation for each of these assets. These equationshave been used extensively to study the implications of themodel we are considering for asset pricing since Robert E.Lucas (1978) and Douglas T. Breeden (1979).

dominates depends on preference param-eters and is, therefore, an empirical issue,that depends on the size of the elasticity ofintertemporal substitution.

An inuential paper by Hall (1988)claimed that this parameter is close to zero.This nding has been challenged on variousgrounds. A low response of consumptiongrowth to the real interest rate could obtainif some consumers are liquidity constrainedor if the error term correlates with that partof the real interest rate that is explained bythe instruments. Attanasio and Weber (1993,1995) point out that aggregation bias couldbe responsible for such a low estimate: the

aggregate consumption growth rate is com-puted by taking logs of the mean of individualconsumption, whereas equation (13) impliesthat means of the logs should be takeninstead. Attanasio and Weber (1993) provideevidence that the difference between thesetwo terms is highly serially correlated, thusinvalidating lagged consumption growth as aninstrument. When they correct for this, theynd higher estimates of the elasticity of inter-temporal substitution. Attanasio and Weberuse cohort data (that is: data from repeatedcross sections that is consistently aggregatedover individuals born in the same years):when they focus on cohorts of individualswho are least likely to be liquidity constrainedand control for changes in taste shifters, theyestimate a much higher elasticity (around 0.8)using U.K. (1993) and also U.S. cohort data(1995). Recently, John Karl Scholz, AnanthSeshadri, and Surachai Khitartrakun (2006)

address the issue of how well the life cyclemodel predicts wealth holdings, and take asbenchmark case 1/= 0.33, but they alsoshow that the model ts best when they take1/= 0.67. In a recent, very ingenious paper,Gary V. Engelhardt and Anil Kumar (2007)use differences in employers matching ratesin 401(k)s and its effect on participation toidentify the elasticity of intertemporal substi-tution and obtain a point estimate of 0.74.


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In much of the macro literature, theiso-elastic specication has played apredominant role. Little attention has beenpaid to the possibility that the elasticity ofintertemporal substitution may differ acrossconsumers, particularly as a function of theirconsumption. A simple way to capture thenotion that poor consumers may be less ableto smooth consumption across periods andstates of nature is to assume that the utilityfunction does not depend on total (nondura-ble) consumption, rather on the differencebetween consumption and needs. Thus wecould have retained the analytical attrac-tion of power utility, but have (CC*) as

its argument, where C*

is an absolute mini-mum that the consumer must reach in eachand every period. This functional form isknown as StoneGeary utility in demandanalysis (see Deaton and John Muellbauer1980, chapter 3, for example), and is thesimplest way to introduce nonhomothetic-ity in a demand system.10 Attanasio andBrowning (1995) take a different route andextend the isoelastic specication by model-ing marginal utility as a quadratic functionin the logarithm of consumption. Blundell,Browning, and Costas Meghir (1991),Atkeson and Masao Ogaki (1996), and FatihGuvenen (2006) are among the few otherexamples of papers that explicitly allow forwealth-dependent elasticity of intertempo-ral substitution (see also Thomas F. Crossleyand Low 2005).

However, a recent paper by Battistin,Blundell, and Lewbel (2009) suggests that

nondurable consumption is log-normally dis-tributed, and this is consistent with the stan-dard isoelastic utility specication.

10 One could interpret external habits (Andrew B.Abel 1990; Campbell and John H. Cochrane 1999) as aspecial way to parameterize C* (by making it a fraction ofpast consumption).

3.3 Liquidity Constraints as an Explanationof Excess Sensitivity

The Euler equations (12) and (13) havebeen estimated mostly on aggregate data.In several cases, some of the model impli-cations have been rejectedgenerallyspeaking, the error term has been found tocorrelate with information available at time

t (rejection of the overidentifying restric-tions) and, in particular, with that part ofincome growth that could be explained bysuch information (excess sensitivity). A goodexample of this type of results are those inthe inuential Campbell and Mankiw (1990b

and 1991) papers, which report resultsfrom a regression like (13) where changesin aggregate consumption were related tochanges in (expectedas instrumented)disposable income. The signicance of theexpected income coefcient is interpretedin that paper as a fundamental violation ofthe basic model, caused either by rule ofthumb consumers, consuming a xed pro-portion of their disposable income, or bybinding liquidity constraints. In fact, a rea-son why excess sensitivity or violations of theoveridentifying restrictions may be detectedis because some consumers are not able toborrow and lend at the same interest rate.Binding liquidity constraints may causeexcess sensitivity if constrained individualsexperience temporary income changes: theywill change consumption by more than theintertemporal optimization problem implies.However, excess sensitivity may also have

other explanations, as we shall see later.Liquidity constraints can take severalformsin the next section we shall considermarket structures in which such constraintsare the optimal response to informationasymmetries or enforceability problems.However, much of the literature imposessuch constraints exogenously. If, in additionto the nonbankruptcy constraint consideredin the previous section, one imposes some


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exogenous and more stringent limits on theamount people can borrow, it is possible thatconsumers will be constrained in a givenperiod and the Euler equation (12) will nothold. In this case, assuming a variable rate ofinterest Rt one would have:

(14) Et[(1 + Rt+1)u(ct+1)/u(ct)] < 1.

The consumer would like to increase cur-rent consumption and, therefore, her cur-rent marginal utility is higher than in thecase in which the borrowing restriction is notbinding. The presence of a binding liquidityconstraint means that the consumer is at a

kink of the intertemporal budget constraint,so that the tangency requirement betweenthe ratio of marginal utilities and intertem-poral prices holds as a slack condition.

The presence of a binding liquidity con-straint represents an important issue for theempirical application of the Euler equa-tion. Of course, the borrowing restriction

will not be binding in every period and,when not binding, the Euler equation willhold. However, even in periods in whichthe liquidity constraint does not bind andthe Euler equation holds, the level of con-sumption will be affected as the consumertakes into account the possibility that theconstraint will bind in future periods.As pointed out, for instance, by Hayashi(1987), the presence of a borrowing restric-tion is equivalent to a shortening of thetime horizona consumer who expects toface a binding liquidity constraint n peri-

ods ahead will plan to have zero wealth inthat period, therefore behaving as if theplanning horizon was n periods.11 Notice,however, that the relationship between con-sumption atn 1 andn 2 is not affectedand the Euler equation between those two

11 Deaton (1991), simulating a stationary economy withimpatient consumers and precautionary saving, shows thatliquidity constraints are rarely binding.

periods holds as if the liquidity constraintis not operative. The liquidity constrainthas an effect on the level of consumptioneven when it is not binding. In addition tothe extreme case of an exogenously givenborrowing limit, one can consider alterna-tive borrowing restrictions. For instance,it is possible to consider the case of a dif-ference between borrowing and lendinginterest rates, or more generally, the case inwhich the interest rate varies with the posi-tion of the consumer in a given asset, typi-cally increasing with higher levels of debts.These cases have been studied, for instance,by Christopher A. Pissarides (1978) and F.

Thomas Juster and Robert P. Shay (1964).A direct way to detect binding constraintsis to ask consumers whether they appliedfor and were denied credit. Jappelli, (1990)reports that 12.5 percent of the 1982 waveof the SCF respondents answered they weredenied credit, and models the probabilityof credit denial as a function of observablecharacteristics. The problem with this typeof question is that consumers may have beendenied credit for good reasons (likely viola-tion of the no-bankruptcy condition), or mayhave decided not to apply for credit on theassumption that this would be refused tothem (a discouraged borrower effect).

Less direct tests for liquidity constraintsthat meet these criticisms are based onthe idea that the Euler equation should beviolated for groups of consumers who arelikely to be constrained, such as the youngand those whose liquid assets are particu-

larly low. This strategy was implementedby Zeldes (1989) using the ratio of liquidassets to income at time t as an indica-tor of potential constraints. Zeldes reportsevidence for liquidity constraints amonghouseholds with very low liquid assetsforthis group, consumption growth would riseby 4 percent if the constraint were relaxed.However, any sample split based on choicevariables may induce endogenous selection,


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particularly if the error term reects prefer-ence heterogeneity.

For this reason, other papers have fol-lowed a different route that works when-ever the amount borrowed depends onother variables, such as earnings (see RobAlessie, Bertrand Melenberg, and Weber1988 and Weber 1993) or collateral (such asin the case of durables, Agar Brugiavini andWeber 1994, Alessie, Michael P. Devereux,and Weber 1997, and Eun Young Chah,Valerie A. Ramey, and Ross M. Starr 1995).In this case, the presence of a binding bor-rowing restriction distorts not only theintertemporal margin but also the allocation

of resources between different commodi-ties (or leisure and consumption) within aperiod. Alessie, Devereux, and Weber notethat identication of liquidity constraints isgreatly enhanced if the relationship betweenthe borrowing limit and the choice variableis exogenously changed within the sampleperiod (this is true in their analysis of carsand nondurable expenditure, because thehire-purchase terms were heavily regulatedby the U.K. government over the rst partof the sample period, completely unregu-lated later). They do nd evidence of bind-ing liquidity constraints in some of the yearsprior to nancial liberalization but only foryoung consumers. Finally, a test for liquidityconstraints that compares the rst order con-ditions across periods to the rst order con-ditions across goods is proposed by Meghirand Weber (1996)their results suggestliquidity constraints may be binding only for

young consumers.12

12 A problem with all Euler-equations-based tests, aswell as with the direct question, is that, as Hayashi (1987)explains, the presence of an operative, albeit not bindingliquidity constraint is equivalent to a shortening of theplanning horizon. This may be the relevant informationthat is needed for policy purposes. Evidence on this can beobtained by noting that consumers that are liquidity con-straints will not be sensitive to changes in the level of theinterest rate. As they will be at a kink of an intertemporalbudget constraint, the demand for loans will be inelastic

Despite all these different approaches, themost widely cited piece of evidence for theoperation of liquidity constraints is excesssensitivity. But excess sensitivity of con-sumption to income (both at low and highfrequency) may be due to incorrect prefer-ence specication, as we argue in the nextsection.

3.4 Explaining Income Tracking and theRetirement Consumption Drop

An inuential paper by Carroll andSummers (1991) uses micro data to docu-ment excess sensitivity of consumption to

income. The authors notice not only that thelife cycle proles of income and consumptiontrack each other but that the shape of thetwo proles covary across different groupsin the population. For instance, householdsheaded by an individual with low educationhave a relatively at prole for both incomeand consumption, while households headedby better educated individuals present moreof a hump shape. This evidence has beenused to argue that consumers are impatientbut prudent to the point of holding liquidassets to buffer shocksthis has come to beknown as the buffer-stock model of savings.

The different results mentioned aboveare reminiscent of the early debate betweenLester C. Thurow (1969) and James J.Heckman (1974). The former pointed tothe covariance over the life cycle betweenincome and consumption as a rejection ofthe life cycle model, while the latter replied

that a version of the life cycle model whereconsumption and leisure were not separable

to changes in the slope of such an intertemporal budgetconstraint: the interest rate. This approach has been fol-lowed by Juster and Shay (1964) and Attanasio, PinelopiKoujianou Goldberg, and Ekaterini Kyriazidou (2008).Interest rate elasticities of credit demand have been esti-mated by David B. Gross and Souleles (2002) using U.S.credit card data and by Alessie, Stefan Hochguertel, and

Weber (2005) using Italian installment credit data.


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could well explain such a covariance. Themicro papers cited above show that con-sistent with Heckmans (1974) argument,excess sensitivity can be reconciled with theintertemporal optimization model if moregeneral, and sensible, utility functions areused. In particular, if one assumes that lei-sure affects utility in a nonadditive way, con-sumption changes respond to predictablelabor income changes, whether or not leisureis a freely chosen variable.

Micro data equations typically show theneed to take account of the effects of sometime-varying characteristics on prefer-encesdemographics and leisure. A way

to introduce this dependence is to specifyperiodt utility as:

(15) ut =(eZtCt)

1

_

1 ,

where Z contains hours of work and othertaste shifters (Attanasio and Weber 1993,1995), some of which might be unobserv-able.13 If one takes the model to micro data,one has to allow for the effect that demo-graphic variables have on utility. The fact thatwhen you have a wife and a baby a one pennybun costs three pence (Gorman) has to betaken into account if one estimates the modelon micro data. Demographics might explainconsumption changes as well as the shapeof the consumptionage prole, as arguedby Browning and Mette Ejrns (2002). Theincrease in household size early in life, anddecrease past age fty, can explain why con-sumption age proles are hump-shaped in

apparent contradiction of the consumptionsmoothing implications of the life cycle the-ory. The interaction between demographicsand prudence explains instead why the peakin consumption occurs later in life than the

13 Leisure has also been introduced in the utilityfunction in some papers that use aggregate data, such asMankiw, Julio J. Rotemberg, and Summers (1985) andCharles R. Bean (1986).

peak in household size and can generate con-sumptionincome tracking for four differ-ent education groups when labor income isuncertain as shown in Attanasio et al. (1999).

More general preference structures thatallow the elasticity of intertemporal substitu-tion to depend also on current consumptionhave been considered in the empirical litera-ture (Attanasio and Browning 1995; Blundell,Browning, and Meghir 1994; Attanasio andWeber 1995; Meghir and Weber 1996). Asin the standard case, the parameters of thesespecications can be estimated using theEuler equations and other rst order condi-tions of the optimization problem faced by

the consumer.The results obtained in the papers thatuse the Euler equation to estimate prefer-ence parameters and test the model could besummarized by saying that a exible versionof the life cycle model is not rejected by indi-vidual level data, especially if one focuses onhouseholds headed by prime aged individu-als, that is, excluding very young householdsand households on the verge of or passed theretirement age. Typically, there is no excesssensitivity of consumption growth to incomegrowth once changes in leisure and demo-graphics are taken into account.

These results show that it is possible tond a specication of preferences that isnot inconsistent with the available microdata. However, leisure and demograph-ics variables could capture the essence ofthe predictability of income and make theestimates of the excess sensitivity param-

eter imprecise. Such variables, according tothis interpretation, therefore should not beinterpreted as taste shifters. There are twopossible answers to this objection. First, ahorse race between expected income andthese other variables seems to indicate thatthe introduction of the latter does not justinate the standard error but also reducesthe size of the income coefcient. Second,once one has estimated the life cycle model


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augmented with these additional variables,one should ask whether the implied prefer-ences are sensible and predict features of thedata that were not used to estimate them.We will come back to this issue below.

Finally, aggregation issues have beenproven to be important. As pointed out inAttanasio and Weber (1993), the differencebetween the consistently aggregated equa-tion (based on the means of the logarithm ofconsumption) and what is available in macrodata (the logarithm of the mean) is a highlypersistent process that correlates with laggedinformation. Attanasio and Weber (1993)also show that results obtained with improp-

erly aggregated micro data are consistentwith results obtained with aggregate dataand indicate rejections of the model thatinstead disappear with properly aggregateddata and rich enough preference structures.

In models where utility depends on theconsumption of several goods and leisure,one can explicitly allow for home productionof goods and services (Gary S. Becker 1965,1981; Gilbert R. Ghez and Becker 1975).This has been recently emphasized in con-nection with changes in spending behavioraround retirement (Mark Aguiar and ErikHurst 2005). The availability of time-usedata allows testing for the implications ofthe model in terms of changes of the com-position of consumption over the life cycle(Aguiar and Hurst 2007, 2009).

A number of recent papers have estimatedthe effects on changes in consumption ofwell-dened predictable tax changes (such

as tax rebates, social security withholdingtax), often nding these effects to be differ-ent from zero (Parker 1999; Souleles 1999;Matthew D. Shapiro and Joel Slemrod 2003;David S. Johnson, Parker, and Souleles2006). This violation of the model predic-tions is surprising because consumptiondoes not appear to react to other anticipatedincome changes (Browning and Collado2001; Hsieh 2003).

The evidence of these natural experi-ment papers suggests that consumptionreacts to predicted changes in disposableincome only to the extent that these changesare relatively small, as noted by Browningand Crossley (2009), because small optimi-zation errors might have trivial utility costs(Cochrane 1989).

A number of recent papers report evidencein favor of a liquidity constraints interpreta-tion. Stephens (2008) shows that consump-tion reacts to the repayment of vehicle loans,and this is particularly true for young indi-viduals, who are more likely in principle tobe liquidity constrained. Sumit Agarwal,

Chunlin Liu, and Souleles (2007) investi-gate credit cardholders response to the 2001tax rebates and nd that most people rstincreased repayments but then the youngand those who were initially close to theircredit card limit started spending more (andbuilding up debt faster). The eventual risein spending could then be attributed to theoperation of liquidity constraints. Similarly,Hsieh, Shimizutani, and Hori (2010) ndthat Japanese consumers response to aspending coupon program tailored to fami-lies with children and the elderly was highestamong those with low wealth.

Another piece of evidence that apparentlycontradicts the life cycle model is the retire-ment consumption puzzle. The simple lifecycle model of Modigliani and Brumberg(1954) predicts that individuals save duringtheir working lives to keep their consumptionlevel constant once they retire. Hamermesh

(1984) was the rst paper to argue that con-sumers apparently do not save enough toachieve this aim. If households enter retire-ment with inadequate savings, they must cuttheir consumption level, contrary to the lifecycle model predictions.

The recent literature has focused on esti-mating how consumption levels changearound retirement. The existence of aconsumption fall around retirement is


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documented for the United Kingdom (Banks,Blundell, and Tanner 1998), for the UnitedStates (Bernheim, Skinner, and Weinberg2001), and for Italy (Battistin et al. 2009)and is known as the retirement consumptionpuzzle (or retirement savings puzzle).

Banks, Blundell, and Tanner use Britishcohort data and show that the standard Eulerequation, in which consumption growth is afunction of intertemporal prices and changesin demographics, overpredicts the level ofconsumption by as much as 1.5 percent onan annual basis for ages between 60 and 67.The cumulated consumption shortfall overthis age band, where most people retire, is

around 10 percent. They argue that onlya fraction of this drop can be attributed tothe increased leisure time that accompa-nies retirement. Later work by Sarah Smith(2006) uses information on food for U.K.households who retired over the sampleperiod and stresses the importance of dis-tinguishing between voluntary and involun-tary retirementa signicant drop for foodconsumption is observed only for those whoretire early because of poor health or job loss.Indeed, David M. Blau (2008) stresses thatconsumption drops at retirement can be rec-onciled with life time optimization if thereis uncertainty over layoffs, job offers, health,and mortality and retirement is a discreteevent that is freely chosen by the household.However, in Blaus model, the causal effectof retirement on consumption is zero.

Bernheim, Skinner, and Weinberg usePSID data to estimate Euler equations for

food consumption. The retirement statusis instrumented by taking age-specic pre-dicted probabilities conditional on demo-graphics. The sample is split in groups: lowwealth-to-income households drop theirconsumption most. Bernheim, Skinner, andWeinberg estimate a median drop of 14 per-cent, but higher drops for low wealth ratio,low income replacement households. Theyconclude that 31 percent of the sample

reduce their consumption by at least 35 per-centage points. The evidence they provideis consistent with the notion that consumersdo indeed enter retirement with inadequatesavings. A number of papers have furtherinvestigated the issue on U.S. dataStevenJ. Haider and Stephens (2007), who estimatea smaller consumption drop for those whoretire at the expected time; Jonathan Fisheret al. (2005), who use CEX data, deateexpenditure by the squared root of householdsize and estimate a smaller drop (around 2.5percent) for total expenditure than for foodconsumption (around 5.7 percent).

Recent papers by Aguiar and Hurst

(2005 and 2007) and Michael D. Hurd andSusann Rohwedder (2006) stress that thedrop in expenditure at retirement does notnecessarily imply an increase in the mar-ginal utility of consumption. For instance,worker-related expenditure (transport toand from work, canteen meals, and businessclothing) is no longer neededwhetherthey account for a large enough part of pre-retirement consumption is an open issue.Also, home production of services (laundry,gardening, housecleaning, cooking) maybecome advantageous, and the extra leisuretime may allow consumers to shop moreefciently. This last channel has recentlybeen stressed by Aguiar and Hurst (2005and 2007) in their careful analysis of foodconsumption around retirement, while theincrease in home production of services byrecent retirees has been documented byHurd and Rohwedder (2006), who exploit

time-use data. The literature has investi-gated as further reasons for this drop unex-pectedly low pensions or liquidity problemsas well as time-inconsistent behavior(George-Marios Angeletos et al. 2001).Another recent paper by Emma Aguila,Attanasio, and Meghir (2010), which looksat changes in consumption around retire-ment (using the longitudinal dimension ofthe CEX in the United States), nds that


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the decline in food expenditure is compen-sated by increases in nonfood items, so thatthe total is roughly constant.

Battistin et al. (2009) use Italian data andinstrument retirement with public pensioneligibility. To be more precise, they take aregression discontinuity approach and makethe identifying assumption that spend-ing behavior would be smooth around thethreshold for pension eligibility if individu-als did not retire. They estimate at 9.8 per-cent the part of the nondurable consumptiondrop that is associated with retirementinduced by eligibility (food expenditurefalls instead by 14 percent). They show that

this fall is not driven by liquidity problemsfor the less well off in the population andcan be accounted for by drops in expensesthat are work related or leisure substitutes.However, they also show that retirementinduces a signicant drop in the number ofgrown children living with their parents andthis can account for most of the retirementconsumption drop.14

As Hurst (2008) recently put it, we shouldno longer t

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