short-term planning and the life-cycle consumption puzzle

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Journal of Economic Dynamics & Control 31 (2007) 1392–1415

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Short-term planning and the life-cycleconsumption puzzle

Frank Caliendoa,�, David Aadlandb

aDepartment of Economics, Colorado State University, Fort Collins, CO 80523-1771, USAbDepartment of Economics and Finance, University of Wyoming, Wyoming, USA

Received 9 September 2004; received in revised form 27 April 2006; accepted 9 May 2006

Available online 14 July 2006

Abstract

We develop a new life-cycle consumption model where the individual has a short planning

horizon. Standard life-cycle consumers with perfect foresight and ‘hand-to-mouth’ consumers

are both special (limiting) cases of our model, where the length of the planning horizon is set to

the entire lifetime and zero, respectively. We derive an analytical solution to the (time

inconsistent) short-term planning problem, which reveals the inner workings of the model and

facilitates its use in other settings. We also show that the short-term planning mechanism is

powerful enough to generate a consumption hump with the right size and location.

r 2006 Elsevier B.V. All rights reserved.

JEL classification: C61; D91

Keywords: Short-term planning; Life-cycle consumption

1. Introduction

There is a large body of evidence from the Health and Retirement Study, theRetirement Confidence Survey, and the Survey of Consumer Finances, suggestingthat many individuals do not plan for retirement when they are young. Instead, many

see front matter r 2006 Elsevier B.V. All rights reserved.

.jedc.2006.05.002

nding author. Tel.: +970 491 0821; fax: +970 491 2925.

dress: [email protected] (F. Caliendo).

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dx.doi.org/10.1016/j.jedc.2006.05.002

mailto:[email protected]

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F. Caliendo, D. Aadland / Journal of Economic Dynamics & Control 31 (2007) 1392–1415 1393

people seem to wait until they are older, and some even wait until they are very closeto retirement to begin planning. Our primary goal in this paper is to develop a newlife-cycle consumption model that is consistent with some of the salient features ofthe survey evidence on poor planning, is mathematically rigorous, and is foundedwithin the neoclassical paradigm wherein agents choose current consumption bysolving some type of optimization problem.

The defining characteristic of our model is that the individual is forward looking,but only to a degree. In our baseline setup, we assume that the planning horizon isshort enough so that initially the agent does not plan for retirement. Although theretirement phase is ignored when young, as time passes and retirement ‘comes intoview’ the individual will actively begin to save for retirement. As time continues topass and more of the retirement period falls within the planning horizon, theindividual will continue to revise her saving plan and allocate additional income toretirement accounts.

We view short-term planning as a compromise between the standard life-cyclemodel with perfect foresight and Campbell and Mankiw’s (1989) rule-of-thumbconsumers who live hand-to-mouth and set current consumption equal to disposableincome in each period. By design, the standard model and the Campbell–Mankiwmodel are special (limiting) cases of our model, where the planning horizon is set tothe entire lifetime and zero, respectively. Our framework is general enough to allowfor both of these extreme possibilities, and all other planning horizons in between.

The short-term planning problem is time inconsistent since the planning endpointadvances continually with age. The consumption profile of such an agent will be theenvelope of initial values from a continuum of control problems as the shortplanning horizon continually slides forward along the time scale. One of the keycontributions of the paper is the derivation of an analytical solution to this time-inconsistent problem, which reveals the inner workings of the model and facilitatesits use in other settings.

The second key contribution of the paper relates to the consumption hump. Thetextbook life-cycle model predicts that consumption will grow smoothly for patientindividuals and decay smoothly for impatient individuals. Household data, however,indicate that life-cycle consumption is hump-shaped, with a peak around 50 years ofage.1 This inconsistency is a prominent ‘puzzle’ in consumption theory and hasrecently received considerable attention from macroeconomists.2 In solving ourshort-term planning problem, we find that hump-shaped consumption is a predictionof the model, and the consumption peak has the right size and location for areasonable range of parameter values.

1See Attanasio and Browning (1995), Attanasio (1999), Attanasio et al. (1999), Browning and Crossley

(2001), Carroll and Summers (1991), Gourinchas and Parker (2002), and Fernandez-Villaverde and

Krueger (2002).2The puzzle dates back to Thurow (1969), who states that ‘individuals can easily redistribute

consumption into the future by saving, but they cannot easily borrow for present consumptiony .

Consequently, the current flow of income rather than the total lifetime flow of income may dominate

current consumption expenditures.’ See Deaton (1992), Browning and Crossley (2001), and Butler (2001)

for more recent overviews.

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F. Caliendo, D. Aadland / Journal of Economic Dynamics & Control 31 (2007) 1392–14151394

Other solutions to the puzzle involve augmenting the basic (Modigliani andBrumberg, 1954; Friedman, 1957) models to include family-size effects (Attanasioet al., 1999; Attanasio and Browning, 1995; Browning et al., 1985), consumption-leisure trade-offs (Heckman, 1974; Bullard and Feigenbaum, 2004), wage incomeuncertainty and the precautionary saving motive (Nagatani, 1972; Hubbardet al., 1994; Carroll, 1994, 1997; Gourinchas and Parker, 2002), mortalityrisk (Feigenbaum, 2005; Hansen and Imrohoroglu, 2005), consumer durables(Fernandez-Villaverde and Krueger, 2001), and hump-shaped wages with rule-of-thumb consumers in the economy (Campbell and Mankiw, 1989). While thesesolutions often produce a hump with the right size and location, our solution has theadvantage of being consistent with a large body of survey evidence suggesting thatmany people do not plan for retirement until later in life. In addition, the short-termplanning mechanism is powerful enough to produce an empirically reasonable humpwithout appealing to family-size effects, consumption-leisure trade-offs, wageuncertainty, mortality risk, or durables. And, unlike the rule-of-thumb explanation,our mechanism is capable of generating a hump even for flat income profiles. Hence,our solution to the puzzle is a new and unique possibility.

The remainder of the paper is organized as follows. In Section 2, we summarize therelated evidence on retirement planning. In Section 3, we present a control modelwith short and moving planning horizons as one interpretation of this evidence.Section 4 presents simulated profiles based on various calibrations, and Section 5concludes with a discussion of some possibilities for future research.

2. Evidence and motivation

Nearly all modern theories of consumption and savings behavior share the basictenet that people look to the future when making current decisions. We do notquestion this principle. We do, however, question the standard assumption thateveryone looks all the way to death. There is mounting evidence that some peoplechoose to consider a shorter planning period when making economic decisions.Accordingly, we define a short-term planner as someone who is forward looking butwhen making economic plans for the future, sets the planning endpoint to be earlierthan his or her expected death. It is important to note that we do not view short-termplanners as ignorant of the latter stages of their life, rather they choose to ignore thisstage for various reasons: lack of self-control, financial illiteracy, distaste forcontemplating old age, or to avoid financial planning costs in an uncertainenvironment, among others. Before turning to some empirical evidence in support ofour setup, we offer a word of caution. Some of the evidence we present in favor ofshort planning horizons can be interpreted in more than one way, so in fairness wediscuss alternative interpretations as we go.

In the short-term planning model, young households make no attempt to calculatehow much they need to save for retirement. But with age, retirement eventuallycomes within their planning horizon and they begin to save. Hence, in an economywith short-term planners, age should be directly related to whether a person has

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made a saving plan and has calculated her financial needs during retirement.Yakoboski and Dickemper (1997) examine the 1997 Retirement Confidence Survey(RCS) and find that only 36% of households have performed such calculations, andimportantly, younger individuals are less likely to have done so. The Survey alsoindicates that proximity to retirement is a major factor that motivates households tosave, and that retirement planning often takes place only when households arerelatively close to retirement.3 All of this evidence mirrors the predictions of ourshort-term planning model.

An alternative interpretation of the RCS data is that the young do not have shortplanning horizons, but rather are long-term planners and choose optimally to delayretirement savings. For example, households may intentionally delay the accumula-tion of financial assets for retirement due to the initial accumulation of durablesinstead (Fernandez-Villaverde and Krueger, 2001), or due to a high discount rate.However, it is important to emphasize that ‘planning for retirement’ and ‘saving forretirement’ are not necessarily the same thing. For example, if the discount rate ishigh enough relative to the interest rate, households in the standard model willborrow while young and delay saving for retirement. This pattern is part of a time-consistent plan first conceived while young (i.e., the agent plans to start saving laterin the life cycle as an expression of her preferences). At the same time, if highdiscounting is the cause of delayed saving, then people with high discount ratesshould have calculated how much they need to save for retirement, even if the datethat saving takes place occurs later in the work life. The short-term planning modelappears to fit the above survey data better than the standard model with highdiscounting.

Similar evidence comes from the 1992 wave of the Health and Retirement Study(HRS), where respondents between the ages 51 and 61 were asked the followingquestion:

�

3

you

for

edu

wh

reg

bee

In deciding how much of their (family) income to spend or save, people are likelyto think about different financial planning periods. In planning your (family’s)saving and spending, which of the time periods listed in the booklet is mostimportant to you [and your husband/wife/partner)]?� next few months;� next year;� next few years;� next 5–10 years;� more than 10 years.

Sim

p

mul

cat

o an

ress

n in

ilarly, Ameriks et al. (2003) ask a survey of TIAA-CREF participants the following question: ‘Have

ersonally gathered together your household’s financial information, reviewed it in detail, and

ated a specific financial plan for your household’s long-term future?’ Even in this survey of highly

ed individuals, who by definition save for retirement, 27% answer ‘no’ to this question. For those

swer ‘yes’, the authors follow by asking ‘the age at which this activity was first undertaken.’ Their

ion analysis shows that having formulated such a plan, as well as the length of time such a plan has

place, both affect net worth in the expected direction.

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The standard life-cycle model assumes that agents would choose the option ‘morethan 10 years.’ 4 Surprisingly, of the 11,626 respondents that answered this question,only nine percent reported a planning horizon of more than 10 years, while almostequal numbers reported a planning horizon of one year or less (28%), a horizon of afew years (33%), and a horizon of 5–10 years (30%). With retirement just around thecorner, these are exactly the people we would expect to be concerned aboutretirement planning, yet many appear to give it little thought.5 The same questionwas also asked in the 1989–2001 waves of the Survey of Consumer Finances (SCF),and similar to the HRS findings, the ‘more than 10 years’ option was choseninfrequently (less than 20% of the time). Maybe even more compelling is the factthat in a follow-up question (‘Which of these [planning horizons] is the leastimportant to you?’’) asked in the 1989 and 1992 waves, over one third of respondentschose the ‘more than 10 years’ category. These data stand in stark contrast to thelong-term planner that conventionally inhabits life-cycle models, but are exactlywhat we would expect from short-term planners.

Similarly, in the Loewenstein et al. (1999) sample of individuals near and atretirement, respondents frequently report that they ‘should have started savingearlier.’ Moreover, ‘financial regret’ and ‘anxiety about money’ among the elderlyare strongly tried to the age at which they started saving – the later they started, thegreater the regret. Is this inconsistent with the standard model? Not necessarily. It isconceivable that those who wish they had started saving earlier are the same oneswho chose at a young age to delay retirement saving and only later regret thedecision after the onset of adverse financial news. While we recognize this possibility,in light of another HRS finding that one third of the respondents have ‘hardlythought about retirement,’ we feel it is more likely that the self-reported notion of‘financial regret’ is instead caused by agents simply not planning far enough into thefuture.

Another possible contribution of the short-term planning model is to provide arigorous theoretical foundation for the recent econometric evidence linkingretirement wealth to planning. Using the HRS data, Lusardi (1999) reports thatwealth at retirement is positively correlated with the length of the planning horizon,while controlling for a number of the standard determinants of wealth accumulationsuch as risk aversion, discounting, and bequest motives. This result is confirmed byAmeriks et al. (2003), who also verify that the causality runs from planning to wealthaccumulation, not the other way around. Indeed, Venti and Wise (1998) estimatethat ‘the primary determinant of the dispersion of wealth at retirement is evidentlythe choice to save or spend while young.’ While the work of Venti and Wise does not

4This is true under the assumption that they expect to live for at least 10 more years. The same

households in the 1992 HRS report 2001 as the average expected year of retirement, which is nearly nine

years after the completion of the survey. This is a strong indication that respondents generally expected to

live longer than 10 years.5It is possible that this survey question is read by households in a way that solicits information about the

discount rate, rather than the planning horizon. Yet, if this were true we would expect the vast majority to

answer ‘next few months’ to reveal their preference for immediate consumption. Instead, very few chose

this survey response, suggesting that the question is probably picking up the intended information.

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necessarily contradict the standard life-cycle model, since it may well be the case thathouseholds in their data set reach different levels of wealth at retirement due todifferences in preference parameters like the discount rate, Ameriks et al. (2003) andLusardi (2003) find that variation in the discount rate among households does notundo the connection between planning and wealth accumulation. In addition,Browning and Crossley (2001) suggest that heterogeneity in discount rates may notexplain heterogeneity in wealth since pre-retirement consumption growth rates arenot correlated with wealth at retirement. In our short-term planning model, smalldifferences in planning horizons can produce very large differences in wealth atretirement even for individuals who are identical in every other respect (includingdiscount rates).

Finally, we address the question of why some people fail to plan for the long run.Although this question is complex and far from well-understood, we briefly offerfour possibilities. First, a short horizon could simply come from distaste forcontemplating old age. For example, Akerlof and Dickens (1982) note that theremay be persons who ‘would simply prefer not to contemplate a time when theirearning power is diminished, andy the very fact of saving for old age (may) forcepersons into such contemplations.’

Second, a short planning horizon may reduce planning costs. When planning forthe long-term, people encounter time-intensive costs such as properly understandingpension rules and social security distributions (Lusardi, 2003). Similarly, Reis(forthcoming) notes that individuals may optimally choose to ignore implications forretirement given that there is a ‘cost in money and time of obtaining information,processing and interpreting ity or paying a financial advisor to interpret theinformation and compute the optimal financial plan.’

Third, people may choose a short horizon simply because it leads to high levels ofconsumption today with future penalties that may not be evident until late in the lifecycle. If this is true, then we would expect that retirement planning may be linked towhether the individual has a chance to learn from the successes and mistakes ofothers (Thaler, 1994). Lusardi (2003) offers some evidence which suggests that thelife experiences of older siblings and of old parents affect the planning horizon of theindividual.

Fourth, since short-term planning requires smaller current sacrifices, it mayrequire less willpower than planning for the long run. Lack of willpower has a longhistory as a possible problem with the standard life-cycle model (see Thaler, 1994;Thaler and Shefrin, 1981; Thaler and Benartzi, 2004).6

We openly admit that short-term planning is not the only way to interpret all theevidence against perfect foresight in retirement planning, but we feel that the data are

6Another interesting explanation for the lack of saving by young individuals is given by McKenzie

(1997). He argues that not all individuals perceive a unit of time in the same manner (e.g., individuals

almost invariably report that time appears to speed up as they grow older). He states: ‘it may be that

people save more as they age simply because the age at which they expect to retirey comes within the

person’s relevant time horizony. [This] inducement to save ‘is compounded by the actual or perceived

shrinking units of time extending toward retirement, which means that the rewards from saving become

ever more immediate.’ ’

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sufficiently consistent with the short-term planning model to warrant furtherexploration. In addition, since we can obtain an analytical solution to the short-termplanning problem, and since the model easily generates a consumption hump for awide range of parameters, our setup is a particularly attractive interpretation. Andperhaps most important, the short-term planning model is a natural middle pointbetween Campbell and Mankiw’s consumers with no planning horizon and thestandard consumers with a lifetime planning horizon. Both of these polar cases havebeen discussed extensively in the literature (and are limiting cases of our moregeneral model), so it seems natural to explore planning horizons between theextremes. We turn now to the control model.

3. A control model with short planning horizons

The agent enters the workforce at t ¼ 0, retires at t ¼ T , and passes away at t ¼ T .During the working period, ½0;T �, the agent receives wage income at a constant rate,w, and during the retirement period, ½T ; T �, he depletes his savings account. The dateof retirement and the date of death are exogenous and certain. The model isintentionally simple to ensure that short-term planning is the only possibleexplanation for the consumption hump.

The defining characteristic of our model is that the individual is forward looking,but only to a degree. When the individual enters the workforce he does not plan allthe way out to the date of death as in the standard model; instead, the length of theplanning horizon, x, is shorter than the entire lifetime. In our baseline setup, we notonly assume that x is less than T but we also assume it is less than T, so that initiallythe baseline agent does not plan for retirement because his planning horizon isshorter than the length of the working period.

We begin by assuming that x is constant over time. The fixed and short planninghorizon simply slides along the time scale as the individual ages. This produces adynamically inconsistent problem since the planning endpoint constantly advancesand the agent must continually solve a fresh control problem at each and everyinstant in time.7 Although we allow for learning in an extension to the model, ourbaseline model assumes the individual does not learn to be a better planner as shegets older. A short and moving planning horizon of fixed length is a simple andtransparent method for capturing the limited foresight apparent in the behavior ofmany households, while at the same time admitting analytical solutions for easyapplication and extension in future work.

For convenience we separate the agent’s life cycle into three phases. Phases 1 and 2occur during the working period of the life cycle, and phase 3 is the retirementperiod. Phase 1 is ½0;T � x�, which is the interval of time for which retirement is not

7Later, we consider the possibility that age affects the length of the planning horizon. McKenzie (1997)

suggests that older individuals are likely to have longer horizons. However, even with a fixed planning

horizon, our model naturally captures the realistic idea that individuals think more about their retirement

needs when they are older.

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in the planning horizon and therefore the retirement phase has no bearing on currentconsumption decisions. That is, for any time t0 2 ½0;T � x�, the agent’s planninghorizon, t0 þ x, is never greater than T. Thus, during the interval ½0;T � x� she never‘sees’ into retirement. On the other hand, phase 2 is the portion of the workingperiod for which retirement is in view ½T � x;T �, since for any t0 2 ½T � x;T �, theagent’s planning horizon, t0 þ x, is always greater or equal to T. During phase 2, theindividual is still working, but is aware of the need to save for retirement. Early inphase 2 she appreciates only a small portion of her true retirement needs because herplanning horizon stretches only slightly beyond the retirement threshold, and lateron in phase 2 she more fully appreciates her retirement needs since her planninghorizon now includes a larger share of the retirement period.

The control problems for phases 1 and 2 both involve dynamically inconsistentprogramming since the agent continues to revise her saving plan as her planninghorizon advances, but the problem for phase 2 is more complicated. During phase 2the agent solves a two-stage control problem at each instant in time since the stateequation switches at retirement (due to the absence of wage income).

Finally, we do not focus much on phase 3 since the key aspect of consumption thatwe are interested in (the hump) occurs well before this time. The retirement phase iscertainly important since retirement planning impacts consumption decisions duringthe working period, but consumption itself during retirement is not the focus of thepaper.

3.1. Phase 1

At any instant in time during the first phase of the working period whereretirement is not yet in view, t0 2 ½0;T � x�, the agent solves a short-horizon controlproblem:

max

Z t0þx

t0

e�rðt�t0ÞcðtÞ1�f

1� fdt, (1)

subject to

dkðtÞ

dt¼ rkðtÞ þ w� cðtÞ (2)

with kðt0Þ given and kðt0 þ xÞ ¼ 0. Here kðtÞ is the savings account balance, r is thereal rate of return, r is the discount rate, and f is the curvature parameter in theCRRA period utility function.8 The assumption kðt0 þ xÞ ¼ 0 reinforces the ideathat the individual is concerned only with the next x years of the life cycle. However,it does not imply that the capital stock runs dry before the end of the life cycle,because the agent’s planning horizon continually slides forward along the time scale

8Following empirical work, our formulation includes both the standard notion of impatience and a

short planning horizon. Ameriks et al. (2003) estimate that a household’s propensity to plan is unrelated to

the subjective discount rate. Also, Lusardi (2003) estimates that the degree of planning is a significant

predictor of wealth accumulation, even after controlling for the discount rate.

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as he grows older. That is, although the agent plans to exhaust the savings accountwithin x years, he never really exhausts the account until the end of the life cyclebecause the horizon keeps moving and he keeps re-planning. If the agent were tooptimally choose kðt0 þ xÞ, our model would of course collapse to the standardmodel with a lifetime planning horizon.

Standard application of the Maximum Principle to problem (1)–(2) gives theagent’s planned consumption path from the perspective of time t0

cðtÞ ¼½kðt0Þe

rx � wð1� erxÞ=r�ðg� rÞ

egðt0þxÞ � egt0þrx

� �egt for t 2 ½t0; t0 þ x�, (3)

where g � ðr� rÞ=f. From the perspective of time t0, Eq. (3) is the plan that theindividual perceives to be optimal, and hence the plan the individual intends tofollow. However, one of the defining features of our model is that the individual isfree to re-optimize at every instant in time, and since the planning horizoncontinually slides forward as the individual ages, the consumption plan in (3) can nolonger be optimal from the perspective of any point in time beyond t0 since theoriginal plan at time t0 obviously does not extend far enough into the future. If theagent never learns to become a better planner and continues using a short planninghorizon, then his planned and actual consumption profiles overlap only at time t0since he behaves according to his initial plan only at the initial instant in time. Henceactual consumption at time t0 is found by replacing t in (3) with t0. Yet t0 wasarbitrarily chosen to represent any point in time on ½0;T � x�. This suggests thatactual consumption during phase 1 of the working period is found by replacing all t0in (3) with t

cðtÞ ¼½kðtÞerx � wð1� erxÞ=r�ðg� rÞ

egx � erxfor t 2 ½0;T � x�. (4)

Note that cðtÞ in (4) is a function of kðtÞ and is therefore not an explicit solution;thus, it remains to find the actual time path of kðtÞ. For convenience, we rewrite (4) as

cðtÞ ¼ kðtÞz1 � z2 for t 2 ½0;T � x�, (5)

where

z1 �g� r

eðg�rÞx � 1; z2 �

½wð1� erxÞ=r�ðg� rÞ

egx � erx.

Substitute (5) into (2), and use the initial condition kð0Þ ¼ 0 to obtain

kðtÞ ¼ ðwþ z2Þf1� eðr�z1Þtg=ðz1 � rÞ for t 2 ½0;T � x�. (6)

Thus, (5) and (6) give the actual time-inconsistent program of consumption andcapital in analytical form on the interval ½0;T � x� – the program is expressed purely interm of the model’s parameters and numerical approximation is not necessary. Thisprogram is the envelope of infinitely many initial values from a continuum of plannedtime paths. If we were to plot each and every planned consumption path (i.e., plot aconsumption profile for every possible starting time using Eq. (3)), then actualconsumption could be found by connecting all the initial points from the many plannedpaths.

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3.2. Phase 2

Phase 2 is the interval of time during the working period for which the retirementperiod is in view, ½T � x;T �. For reasons apparent below, the technique that we useto solve the control problem in this section works only if the length of the planninghorizon, x, is shorter than the length of the retirement period of the life cycle. Thus,we restrict the planning horizon in this manner.

Suppose the agent is currently standing at some point during the second phase ofthe working period, t0 2 ½T � x;T �. The agent solves the following two-stage controlproblem:

max

Z t0þx

t0

e�rðt�t0ÞcðtÞ1�f

1� fdt, (7)

subject to

dkðtÞ

dt¼ rkðtÞ þ w� cðtÞ for t 2 ½t0;T �, (8)

dkðtÞ

dt¼ rkðtÞ � cðtÞ for t 2 ½T ; t0 þ x� (9)

with kðt0Þ given and kðt0 þ xÞ ¼ 0.Using the Maximum Principle for two-stage optimal control we obtain the planned

consumption profile from the perspective of time t0 (see Appendix for details)

cðtÞ ¼½kðt0Þ � wðerðt0�TÞ � 1Þ=r�ðg� rÞ

egðt0þxÞ�rx � egt0

� �egt for t 2 ½t0; t0 þ x�. (10)

To obtain actual consumption on ½T � x;T �, replace all t0 in (10) with t as explainedin the previous section

cðtÞ ¼½kðtÞ � wðerðt�TÞ � 1Þ=r�ðg� rÞ

eðg�rÞx � 1for t 2 ½T � x;T �, (11)

which can be written as

cðtÞ ¼ kðtÞz1 þ z1wð1� erðt�TÞÞ=r for t 2 ½T � x;T �. (12)

Finally, substitute (12) into the law of motion in (8) and solve to obtain the actualtime path of kðtÞ on ½T � x;T �, as a function of kðT � xÞ

kðtÞ ¼ kðT � xÞeðr�z1Þðt�TþxÞ �w

rf1� eðr�z1Þðt�TþxÞg þ

w

r

1� e�z1ðt�TþxÞ

erðT�tÞ

� �(13)

for t 2 ½T � x;T �, and with kðT � xÞ predetermined in phase 1 of the workingperiod. Thus, (12) and (13) represent the analytical solution program to thetime-inconsistent problem for phase 2. This program is the envelope of infinitelymany initial values from a continuum of two-stage planned time paths.

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3.3. Limiting cases

The standard life-cycle model and the Campbell–Mankiw (1989) model are specialcases of our more general model, where the planning horizon is set to the entirelifetime and zero, respectively. Our framework allows for both of these extremepossibilities, and all other planning horizons in between.

As x! T , phase 1 is irrelevant and we can focus on phase 2. In this case, theconsumer’s time-consistent consumption plan is obtained by setting t0 ¼ 0 and x ¼

T in (10). The result is the standard life-cycle consumption profile.At the other extreme, as x! 0; phase 1 goes to ½0;T � and phase 2 goes to ½T ;T �

and vanishes, and the consumer never does plan for retirement. Hence, we can ignorephase 2 of the working period since it does not exist. It is straightforward to showusing l’Hopital’s Rule that if kðtÞ ¼ 0 in (4), then

limx!0

cðtÞ ¼ w for t 2 ½0;T �. (14)

Hence, as the length of the planning horizon tends to zero, an individual that startsout with no retirement savings (i.e., kð0Þ ¼ 0) will simply consume his current wageincome in every period and end the working period with an empty private savingsaccount. (If we were to build social security into the model, the agent would consumeall of his disposable wage income during the working years and rely exclusively onsocial security benefits during retirement.)

3.4. A remark

The behavioral assumption implied by this model is open to a certain amount ofcriticism. It is hard to imagine that individuals would behave exactly according tothis model without ever learning or trying to accommodate their own time-inconsistent behavior.

We recognize that this behavioral implication is extreme. But on the other hand, itshares the advantage of capturing, in an analytically tractable manner, two veryrealistic ideas: first, some people do not initially take any thought for retirement;and, second, as people age and retirement approaches, they take more and morethought for their expected retirement needs. The standard model cannot account forthese ideas, and although it is unreasonable to assume people behave strictlyaccording to our model, it is clear that real-world behavior can be consistent with thegeneral qualitative features of our model.

4. Simulations and discussion

4.1. Baseline calibration

Our baseline parameters are shown in Table 1. It is common in this line of researchto assume T ¼ 40 and T ¼ 55 in order to model an agent who starts work at age 25,

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retires at age 65, and passes away at age 80. We follow the recent life-cycle literatureand set the real rate of return, r, to 3.5% (e.g., see McGrattan and Prescott, 2000;Gourinchas and Parker, 2002; Bullard and Feigenbaum, 2004; Feigenbaum, 2005).We begin by setting the discount rate equal to the real rate of return so that theshort-term planning mechanism is the only possible source of a non-flatconsumption profile in our model (the standard model will produce a flatconsumption profile when r ¼ r). With respect to the curvature parameter, we setf ¼ 1 following Attanasio (1999, p. 791) who states: ‘the evidence that emerges fromthe micro studies which use an isoelastic specification of preferences is thaty theelasticity of intertemporal substitution of consumption (EIS) is just below 1’.9

However, f does not affect the shape of the consumption profile until later in thepaper when we relax the assumption r ¼ r. Finally, we start by setting the planninghorizon, x, to 15 years, but we examine other planning horizons as well. In additionto producing a hump in the right location, the choice of a 15-year horizon isconvenient because this is also equal to the length of the retirement period, so that assoon as the individual hits retirement, her planning horizon reaches to the date ofdeath and consumption during retirement can then be obtained by solving astandard, time-consistent problem of optimal consumption. This in turn allows us toperform welfare analysis without making any major (non-standard) assumptionsabout the behavior of the agent during retirement.

Panel A of Fig. 1 shows the consumption and wage profiles over the life cycle forthe standard model where the planning horizon is the entire lifetime. Panel B showsthe corresponding time path of capital. Consumption is flat and is below wageincome due to the life-cycle motive for saving, while capital accumulates steadilyover the work life and reaches a peak at the date of retirement.

Fig. 2 corresponds to the baseline short-term planning model. Consumption is flatfrom age 25 to 50 and it equals wage income. During this phase of the work life(‘phase 1’ from Section 3.1 above), the agent’s planning horizon does not pass acrossthe retirement threshold so there is no life-cycle motive for saving. As a consequenceof the simple nature of the model (no uncertainty, etc.), the individual simplyconsumes all of her current wage income similar to a (Campbell and Mankiw, 1989)hand-to-mouth consumer. However, as the planning horizon slides along the timescale, eventually the date of retirement will come into view. For the baselineparameter values, this occurs at age 50. The agent responds by scaling backconsumption, and since the short horizon continually advances, the agent continuesto scale back in accordance with projected retirement needs. Her concern for theretirement period of the life cycle continues to increase as retirement draws nearbecause the planning horizon covers a progressively larger number of retirementyears.

Comparing capital profiles from Figs. 1 and 2 we see that the standard modelpredicts relatively steady increases across the work life, whereas the short-termmodel predicts no accumulation during the early years, and rapid accumulation

9Using the period utility function in (1), EIS ¼ 1=f.

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Capital

0

100,000

200,000

300,000

400,000

500,000

600,000

25 30 35 40 45 50 55 60 65 70 75 80

Age(B)

Consumption and wage income

0

10,000

20,000

30,000

40,000

50,000

25 30 35 40 45 50 55 60 65 70 75 80

Age

consumption wage income

(A)

Fig. 1. Textbook life-cycle model. Note: The planning horizon is the entire life cycle.

Capital

0

100,000

200,000

300,000

400,000

500,000

600,000

25 30 35 40 45 50 55 60 65 70 75 80

Age(B)

Consumption and wage income

0

10,000

20,000

30,000

40,000

50,000

25 30 35 40 45 50 55 60 65 70 75 80

Age

consumption wage income

(A)

Fig. 2. Baseline short-term planning model. Note: The planning horizon is set to 15 years.

Table 1

Baseline calibration

Parameter name Symbol Value

Date of retirement T 40

Date of death T 55

Real rate of return r 0.035

Discount rate r 0.035

Length of planning horizon x 15

Curvature parameter f 1

Initial capital stock kð0Þ 0

Wage rate w $40; 000


immediately before retirement.10 Under the baseline calibration, the short-termplanner has about half as much capital at retirement as the standard life-cycleplanner. Because short-term planners do not save early in the life cycle, our model

10In the next section we allow for the possibility that r4r, which implies that capital will accumulate

logarithmically during the first phase of the work life due only to the discrepancy between r and r. During

the second phase of the work life capital grows exponentially as the individual begins to account for

retirement consumption needs. This switch from logarithmic to exponential growth will occur precisely at

the point where retirement first comes into view.

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fits the survey evidence suggesting that many individuals do not formulate retirementplans when young. However, we note that enriched life-cycle theories can alsoproduce flat capital profiles early in life with rapid accumulation toward the end ofthe work life (e.g., Gourinchas and Parker, 2002; Fernandez-Villaverde and Krueger,2001).

One issue that has received considerable attention in recent years is the largedispersion of retirement wealth. Venti and Wise (1998) find that wealth at retirement,holding lifetime earnings constant, is largely due to preferences (the choice to save orspend) rather than resource shocks (bad health, inheritances) or investment choices.In the standard life-cycle model, there are two preference parameters that can beadjusted to produce different levels of wealth at retirement: the discount rate and thecurvature parameter on the period utility function. One advantage of the short-termplanning model is that it offers a third preference parameter that has already beenempirically tested, the length of the planning horizon, to help explain the vastdispersion of wealth at retirement.11 In moving from a 5-year to a 10-year planninghorizon, wealth at retirement roughly doubles in our model. But even small changesin planning horizons can greatly affect wealth at retirement. An increase in thehorizon from 5 to 6 years leads to a 20% increase in wealth at retirement. The short-term planning mechanism is powerful enough to account for wide variations inwealth, even for households with identical lifetime earning profiles, identicaldiscount rates, and identical attitudes toward risk.

In Panels A and B of Fig. 3, we examine planning horizons of length 10 and 5years, respectively. Note that the kink in the consumption function occurs at a laterdate as the length of the horizon decreases. This is intuitive since the agent onlybegins to save for retirement when the end of the planning horizon crosses theretirement threshold.

The short-term planning model has intuitive appeal and is consistent with casualobservation. Lusardi (1999) states the matter plainly: ‘people without a [savings] planfor retirementy may be surprised as they approach retirement at how little theyhave accumulated and suddenly realize they must accept a sharp drop in livingstandards.’ In other words, they must scale back their consumption in order toprovide for retirement as in Figs. 2 and 3. Because the planning horizon is the entirelifetime in the standard life-cycle model, the agent never wants to switch to a newsaving plan, she simply calculates a plan at the moment she enters the workforce andthen she sticks to that plan all through the life cycle. Although some people mayindeed behave this way, the data discussed earlier indicate that other people waituntil they are closer to retirement to think about a retirement plan. In ourestimation, one of the primary strengths of the short-term planning model is to allowthe individual to change her mind as she approaches retirement and to take moreand more thought for the retirement phase of the life cycle as retirement draws near.

Now, what about the consumption hump? Of course, one can easily obtain anincreasing consumption profile in the standard model with r4r. Thus, even the

11Lusardi (1999) finds that the length of the planning horizon is a strong predictor of wealth

accumulation.

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Consumption

0

10,000

20,000

30,000

40,000

50,000

25 30 35 40 45 50 55 60 65

Age

Consumption

0

10,000

20,000

30,000

40,000

50,000

25 30 35 40 45 50 55 60 65

Age(B)(A)

Fig. 3. Consumption profiles for alternative planning horizons. Notes: Short-term planning model with a

planning horizon of 10 years (Panel A) and 5 years (Panel B).


standard model has little trouble explaining the increasing portion of the hump; it isthe switch from increasing to decreasing rates of change that presents the scientificchallenge. From Figs. 2 and 3 we see that the short-term planning model does a goodjob of predicting a switch to negative consumption growth toward the latter part ofthe work life. Under the baseline calibration, however, the model does not generatethe increasing side of the hump. But in the next section we show that if r4r, as isassumed in many consumption studies, the short-term planning model can produceboth the increasing and decreasing sides of the hump.12

Finally, to assess the welfare loss from short-term planning we make a comparisonof discounted life-cycle utility: we calculate the dollar amount that must be deductedfrom the annual consumption of a long-term planner in order to force her discountedlife-cycle utility to be the same as the utility of a short-term planner. This calculationhas the interpretation of the utility cost associated with short-term planning,measured in terms of annual consumption.13 Under our baseline calibration, the lossis less than 4% of annual consumption. However, the loss increases with thecurvature parameter f and can become very large if we consider large values (e.g., iff ¼ 10, the loss is close to one-third of annual consumption). One way of using thebaseline utility loss (4%) is to build the case that short-term planning is ‘near’rational behavior. An agent may be willing to use a short planning horizon andaccept a small loss in lifetime utility in order to avoid the psychological and planningcosts (discussed in Section 2) associated with long-term financial planning thatstretches out to the date of death.

12As in our paper, other theories also require r4r to match the upside of the consumption hump

(e.g., see the work on uncertain lifetime by Hansen and Imrohoroglu, 2005; Feigenbaum, 2005; Butler,

2001); however, other studies make the opposite assumption (see the work on buffer stock models

summarized in Browning and Lusardi, 1996). We focus on the former in the next section, but we discuss

the implications of both calibrations.13This method is also used in Browning and Crossley (2001) to calculate the utility loss from following

sub-optimal consumption rules, except they calculate the compensation that a non-optimal individual

would require, rather than the deduction that an optimal individual would pay. We have chosen the latter

simply as a matter of preference.

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4.2. Alternative calibrations

It is common in the life-cycle literature to use a discount rate that is less than therate of return. For example, Bullard and Feigenbaum (2004) and Feigenbaum (2005)set the discount rate about 1 percentage point below the real rate of return. In thissection we follow suit and set r ¼ 0:025. Recall, in the baseline case with r ¼ r, thechoice of f was inconsequential. Now, however, with r4r, the curvature parameteris important since it affects willingness to capitalize on discrepancies between the tworates. Researchers have often considered values of f anywhere between 0 and 10(e.g., Bullard and Feigenbaum, 2004; Feigenbaum, 2005; Mehra and Prescott, 1985;Siegel and Thaler, 1997). Two recent calibrations of this parameter in life-cyclemodels are f ¼ 0:5 (Feigenbaum, 2005) and f ¼ 6 (Bullard and Feigenbaum, 2004).We plot consumption profiles in Fig. 4 under both calibrations.

In Panel A of Fig. 4, we set f ¼ 0:5, and in Panel B, f ¼ 6. The first thing to noteis that consumption is now increasing during phase 1 of the working period, eventhough the agent has no life-cycle motive for saving. Consumption increases duringphase 1 because the agent chooses to save more initially, taking advantage of therelatively high market rate of return. Consumption grows less steeply in Panel B thanin Panel A because the agent is less willing to make these intertemporal consumptionsubstitutions when f ¼ 6 ðEIS ¼ 1

6Þ than when f ¼ 0:5 ðEIS ¼ 2Þ. We also note thatthe location of the consumption peak is not a function of the curvature parameter(peak occurs at age 50 in both cases), and instead is purely a function of the planninghorizon. As in the baseline case, consumption will switch to a negative rate of changeas soon as the end of the agent’s planning horizon slides across the retirementthreshold. This brings us to the issue of whether the short horizon model can notonly predict a hump, but also the correct location and magnitude.

Studies typically find that the consumption peak occurs between 45 and 55 yearsof age (Attanasio, 1999; Gourinchas and Parker, 2002; Fernandez-Villaverde andKrueger, 2002, and others). In the short-term model, the agent’s consumption will

Consumption

0

10,000

20,000

30,000

40,000

50,000

25 30 35 40 45 50 55 60 65

Age

Consumption

0

10,000

20,000

30,000

40,000

50,000

25 30 35 40 45 50 55 60 65

Age(A) (B)

Fig. 4. Consumption profiles for alternative curvature parameters. Notes: Short-term planning model

with the discount rate set to 2.5%. In Panel A, f ¼ 0:5 as in Feigenbaum (2005), and in Panel B, f ¼ 6 as

in Bullard and Feigenbaum (2004). All other parameters held fixed at baseline values.

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peak in this target range as long as the planning horizon is between 10 and 20 yearsin length. Panel A of Fig. 5 reports this result. It is difficult to find empirical evidenceon the precise length of the planning horizon, although Lusardi (1999) provides someguidance based on survey data. If we use Lusardi’s estimates of the planningcomposition of the population to construct an aggregate consumption profile, wecan also obtain a pronounced hump that occurs in the late 50s.

Studies also report different peak sizes (e.g., see Gourinchas and Parker, 2002,versus Fernandez-Villaverde and Krueger, 2002), but most estimates of the ratio ofpeak consumption to initial consumption fall between 1.1 and 1.5 (Feigenbaum,2005). We treat this as a target range, and examine the parameter space that canproduce a hump of this size. In Panel B of Fig. 5, we fix the length of the planninghorizon to the baseline case of 15 years (since we know this produces a peak in theright location), and we calculate the size of the peak under various assumptionsabout the curvature parameter. If f is between 0.2 and 0.9, then the relative sizematches the data. This is a small range given that macroeconomists typicallyconsider values between 0 and 10, although the upper end of the range is close to ourbaseline value f ¼ 1, which produces a relative hump size of 1.09. Also, the range off that can produce a hump with the right size is sensitive to the spread between theinterest rate and the discount rate. If we keep the interest rate at 3.5%, but decreasethe discount rate to 1.5% so that the spread is now 2 percentage points, then theconsumption peak becomes more pronounced for all values of f. In this case, if f isbetween 0.4 and 1.8, then the peak size will match the data. Not only does the rangeof necessary f values shift to the right, but the range also more than doubles inwidth.

Of course, the theoretical profiles do not perfectly match the empirical profiles,since the theoretical ones are not as smooth at the peaks as the empirical ones. At anaggregate level, it is straightforward to smooth-out the inverted ‘V’ by averagingover a set of individuals with various planning horizons. For example, Fig. 6represents the average age-consumption profile for an economy populated with an

Age at peak consumption

35.00

40.00

45.00

50.00

55.00

60.00

65.00

5 10 15 20 25

Planning horizon

Relative size of consumption peak

1.00

1.20

1.40

1.60

1.80

2.00

0.1 0.4 0.7 1.0 1.3 1.6 1.9

Phi(A) (B)

Fig. 5. Sensitivity to planning horizon and curvature parameter. Notes: Short-term planning model, with

the discount rate set to 2.5%.

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equal mix of long-term planners and individuals with planning horizons of length 5,10, and 15 years. All individuals share the same value of f ¼ 0:5 in Panel A, and inPanel B we set f ¼ 6. Here we see that the sharp consumption peak has beensmoothed out to match the hump shape in the data, and as with our micro-levelfindings, the ratio of peak to initial consumption is sensitive to the choice of f. Withf ¼ 0:5, the peak size is 1.21, and with f ¼ 6, it is only 1.02. Both aggregate profilesreach a peak at age 50, as expected since this is precisely the age at which one-quarterof the model population (i.e., those with 15-year planning horizons) begin to scaleback their consumption in order to save for retirement. Thus, to obtain a humparound 50 years, only a portion of the population must have a planning horizon of15 years.

We interpret our findings in this section of the paper as evidence that the short-term planning mechanism is powerful enough to produce an empirically reasonableconsumption hump. However, we note that the results are sensitive to the length ofthe planning horizon, the curvature parameter for utility, and the gap between therate of return and the discount rate. If some portion of the population plan for thenext 10–20 years, if f is close to 1.0, and if the interest rate is greater than thediscount rate, then the short-term planning model is able to match the hump inempirical consumption profiles. While the values for these key parameters aresomewhat controversial, they are within ranges documented and used in otherstudies of consumer behavior.

Finally, we note that short-term planning, as a stand-alone explanation for thehump, breaks down if we set the discount rate above the interest rate. In this case,consumption would decline all across the life cycle with a kink at the date in whichretirement comes into view. That is, consumption decreases at a relatively slow rateduring the first phase of the work life, and then switches to a fast rate of decay duringthe second phase of the work life. This is important because some empirical evid-ence indicates that the discount rate is in fact larger than the interest rate(e.g., Gourinchas and Parker, 2002).

Aggregate consumption

0

10,000

20,000

30,000

40,000

25 30 35 40 45 50 55 60 65

Age

Aggregate consumption

0

10,000

20,000

30,000

40,000

25 30 35 40 45 50 55 60 65

Age(A) (B)

Fig. 6. Aggregate consumption for alternative curvature parameters. Notes: Aggregate consumption in an

economy with a mix of long- and short-term planners and with the discount rate set to 2.5%. In Panel A,

f ¼ 0:5 as in Feigenbaum (2005), and in Panel B, f ¼ 6 as in Bullard and Feigenbaum (2004). All other

parameters held fixed at baseline values.

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4.3. Robustness to learning

In the preceding analysis the length of the agent’s planning horizon remains fixedacross the life cycle. While this approach can produce a realistic consumption hump,and is instrumental in the derivation of an analytical solution to the time-inconsistent problem, it is also important to study the possibility that people learn tobecome better planners.

Learning in economic models often means that with time, the individual correctshis mistaken beliefs and becomes more rational. And it is logical to consider learningif the agent is receiving some sort of signal or feedback that reveals her errors. In thecase of short planning horizons, it is more difficult to explain exactly what signalindicates that such behavior is suboptimal. Of course, during retirement the agentwill regret his over-consumption when young, but during most of the work life heenjoys a higher level of consumption than if he followed the optimal decision ruleand planned for the long term. Thus, it is conceivable that short-term plannersreceive positive feedback during the early years as they note that their standard ofliving exceeds that of their more frugal peers, and in turn receive negative feedbackonly in the later years when the reverse is true.

Nevertheless, people may learn to become better planners with age as they observethe retirement experiences of their parents and older siblings (Lusardi, 2003) and asthey become more disciplined in other areas of life. We augment our baseline modelto include the case where the length of the planning horizon is positively related tothe age of the individual, so that the individual learns to plan further into the futureas he ages. Our main finding is that such learning acts to flatten out the upside of theconsumption hump, and it causes the peak to occur earlier than otherwise. Weconclude that if learning is slow to moderate, the consumption hump will be similarto the hump with no learning; if learning is fast so that the length of the planninghorizon quickly increases as the individual ages, then the upside of the hump isflattened out and the downside remains. Thus, the short-term planning modelaugmented with learning can still produce a hump, but the model appears to be moreof a theory of why consumption falls during middle age. Because these findings arefairly intuitive we omit the mathematical details to save space. The full details areavailable from the authors upon request.

5. Concluding remarks

We develop a model of short-term planning with a moving horizon thatcontinually slides along the time scale. The standard life-cycle model with perfectforesight and the hand-to-mouth model where current consumption equals currentincome (Campbell and Mankiw, 1989) are both special limiting cases of our moregeneral model. We then solve the model analytically and show that the actualconsumption program is the envelope of infinitely many initial values from acontinuum of planned time paths. Hump-shaped consumption is a feature of themodel. This is taken as evidence that short-term planning may be an important partof the solution to the life-cycle consumption puzzle.

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Although we have focused on the consumption hump, short-term planning maycontribute to other debates in the consumption literature. First, consider the excesssensitivity to unexpected shocks to current income. Carroll (2001) states that themarginal propensity to consume from unexpected changes in current income is muchlarger than predicted by the standard life-cycle model. Because unexpectedtemporary income shocks are smoothed over a shorter planning horizon in ourmodel, the effect on current consumption will be larger than in the standard model.This idea dates back to Friedman’s original efforts on consumption (Friedman,1963, and see Carroll, 2001, for a modern interpretation). Second, Banks et al. (1998)report a drop in consumption at retirement, which in their estimation cannot be fullyexplained within the forward-looking life-cycle framework. They conclude that ‘theonly way to reconcile fully the fall in consumption with the life-cycle hypothesis iswith the systematic arrival of unexpected adverse information,’ and according toBrowning and Crossley (2001), ‘it appears that households at retirement recognizethat they made a mistake in consumption smoothing and that they need to scalebacky’ Our model is consistent with the spirit of this explanation in that retirementtakes the agent by surprise. We note, however, that the sharp dip in retirementconsumption would require a very short planning horizon; longer planning horizonsresult in smooth declines.

Finally, we think there are a number of important ways this paper could beextended in future work. First, it would be interesting to build a model with anendogenous planning horizon. For example, augmenting the current model withplanning costs that are proportional to the length of the planning horizon may createa trade-off between short and long-term planning that leads naturally to an optimalplanning horizon.

Second, the current setup treats the date of retirement as an exogenous parameter,yet we suspect that it may be linked to retirement planning in a systematic way –those who delay planning for retirement (have a short planning horizon) may end uppostponing retirement out of necessity. A richer model could examine this potentiallinkage.

Third, in order to help us derive an analytical solution to our time-inconsistentproblem, we made the assumption that the agent plans to exhaust the capital stockby the end of the planning horizon. This assumption is a good starting point forstudying the behavior of short-term planners because it literally implies that theindividual does not see beyond the short-horizon. In reality, however, many peoplewho carefully plan their consumption for only the next few years may intend to leavethemselves some target level of assets at the end of the short horizon.

Fourth, the current model abstracts from social security taxation and from socialsecurity income during retirement. Of course, if the net present value of the socialsecurity program is negative, then a long-term planner will simply smooth this lossacross the life cycle and will reduce her consumption in each period accordingly, buta short-term planner, who does not initially consider the retirement phase of the lifecycle, will not react the same way. A related issue is whether a pay-as-you-go socialsecurity program (with a negative net present value) can improve the life-cyclewelfare of a short-term planner. This seems like an interesting area for future work.

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Fifth, the observed consumption hump corresponds to aggregate data, so it wouldbe important to see whether the consumption hump in our model can survive in amore complete, general equilibrium macroeconomic model, like the one studied byBullard and Feigenbaum (2004).

Acknowledgments

The authors have benefited from helpful conversations with Annamaria Lusardi,Kevin Huang, and from seminars at Northern Arizona University, Colorado StateUniversity, University of Wyoming, the Central European Program in EconomicTheory (CEPET), and the Midwest Economic Theory Meetings. The detailedrecommendations of two anonymous referees and Wouter DenHaan have beenparticularly helpful.

Appendix. Derivation of Eq. (10)

To solve problem (7)–(9) we use the Maximum Principle for two-stage problems.14

We start by defining the following multiplier functions: l1ðtÞ for t 2 ½t0;T � and l2ðtÞfor t 2 ½T ; t0 þ x�. The multipliers obey

dl1ðtÞdt¼ �rl1ðtÞ for t 2 ½t0;T �, (A.1)

dl2ðtÞdt¼ �rl2ðtÞ for t 2 ½T ; t0 þ x�, (A.2)

l1ðTÞ ¼ l2ðTÞ. (A.3)

Eqs. (A.1) and (A.2) describe the movement of the multipliers, and (A.3) is thetraditional continuity or matching condition which links the multipliers at the switchpoint. Given (A.1)–(A.3), the optimal control for problem (7)–(9) must satisfy

e�rðt�t0ÞcðtÞ�f ¼ l1ðtÞ for t 2 ½t0;T �, (A.4)

e�rðt�t0ÞcðtÞ�f ¼ l2ðtÞ for t 2 ½T ; t0 þ x�. (A.5)

Solve (A.1) and (A.2) to obtain l1ðtÞ ¼ a1e�rt for t 2 ½t0;T � and l2ðtÞ ¼ a2e

�rt fort 2 ½T ; t0 þ x�, where a1 and a2 are constants. Using (A.3), we note that a1 ¼ a2, sowe drop the subscript on the multiplier and write lðtÞ ¼ ae�rt for t 2 ½t0; t0 þ x�. Thus(A.4) and (A.5) can be written compactly as

e�rðt�t0ÞcðtÞ�f ¼ ae�rt for t 2 ½t0; t0 þ x�. (A.6)

14For technical details on two-stage optimal control see Kemp and Long (1977), Tomiyama (1985),

Kamien and Schwartz (1991), and Caliendo (2005).

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After solving (A.6) for cðtÞ we have

cðtÞ ¼ yegtþrt0=f for t 2 ½t0; t0 þ x�, (A.7)

where y � a�1=f and g � ðr� rÞ=f.All that remains is to pin down the constant y. This is done by substituting (A.7)

into (8) and (9) and using the boundary conditions to solve the differential equations

kðtÞ ¼ kðt0Þerðt�t0Þ þ w

Z t

t0

erðt�sÞ ds� y

Z t

t0

egsþrt0=fþrðt�sÞ ds for t 2 ½t0;T �,

(A.8)

kðtÞ ¼ y

Z t0þx

t

egsþrt0=fþrðt�sÞ ds for t 2 ½T ; t0 þ x�. (A.9)

Evaluate (A.8) and (A.9) at t ¼ T , equate (A.8) to (A.9), and then solve for y

y ¼½kðt0Þ � wðerðt0�TÞ � 1Þ=r�ðg� rÞ

egðt0þxÞþrt0=f�rx � egt0þrt0=f. (A.10)

Finally, insert (A.10) into (A.7) to obtain (10).

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short-term planning and the life-cycle consumption puzzle

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