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Working Papers Series
The Effects Of Health, Wealth, And Wages On
Labor Supply And Retirement Behavior
By Eric French
Working Papers Series
Research Department
WP 2000-02
The Effects Of Health, Wealth, And Wages On
Labor Supply And Retirement Behavior
Eric French
Federal Reserve Bank of Chicago
November 19, 2003
Comments
welcome at efrench@frbchi.org. I thank Joe Altonji, John Jones, John Kennan, Rody
Manuelli, Jonathan Parker, Dan Sullivan, and Jim Walker for detailed comments and encouragement.
Jonathan Parker also provided me with data and computer code. I also thank Dan Aaronson, Peter Arcidiacono, Hugo Benitez-Silva, Marco Cagetti, Glen Cain, Chris Carroll, Meredith Crowley, Morris Davis, Nelson
Gra , Ed Green, Alan Gustman, Lars Hansen, Yuichi Kitamura, Grigory Kosenok, Spencer Krane, Tecla
Loup, Derek Neal, Victor Rios-Rull, Marc Rysman, Karl Scholz, three very helpful referees, the editors (Orazio
Attanasio and Bernard Salanie), and numerous seminar participants. Jonathan Hao, Kate Godwin, and especially Tina Lam provided excellent research assistance. Financial support provided by the National Institute
on Mental Health. The views of the author do not necessarily re ect those of the Federal Reserve System.
Recent versions of the paper can be obtained at http://www.chicagofed.org/economists/EricFrench.cfm/.
1
Abstract
This paper estimates a life cycle model of labor supply, retirement and savings behavior
in which future health status and wages are uncertain. Individuals face a xed cost of work
and cannot borrow against future labor, pension, or Social Security income. The method
of simulated moments is used to match the life cycle pro les of labor force participation,
hours worked, and assets that are estimated from the data to those that are generated
by the model. The model establishes that the tax structure of the Social Security system
and pensions are the key determinants of the high observed job exit rates at ages 62 and
65. Removing the tax wedge embedded in the Social Security earnings test for individuals
aged 65 and older would delay job exit by almost one year. By contrast, Social Security
bene t levels, health, and borrowing constraints are less important determinants of job
exit at older ages. For example, reducing Social Security bene ts by 20% would cause
workers to delay exit from the labor force by only three months.
JEL Classi cation: C51, J22, J26
Keywords: Social Security, retirement behavior
2
1
Introduction
Why do individuals retire when they do? This paper provides an empirical analysis of the
e ects of the Social Security system and liquidity constraints on life cycle labor supply. It is
the rst structural model of labor supply and retirement behavior where individuals can save
to insure themselves against health and wage shocks as well as for retirement, but cannot
borrow against future labor, Social Security, and pension income to smooth consumption
in the face of an adverse shock. Previous structural analyses of labor supply and retirement
behavior have made diametrically opposed assumptions about a household's ability to borrow
and save. At one extreme, Gustman and Steinmeier (1986) and Burtless (1986) assume that
households can perfectly smooth consumption by borrowing and lending without limit. At
the opposite extreme, Rust and Phelan (1997) and Stock and Wise (1990) assume that
households cannot borrow or save, thus allowing no intertemporal consumption smoothing.
Clearly, neither of these extreme assumptions is correct.
Understanding the importance of borrowing constraints is critical when considering the
e ects of the Social Security rules on lifetime labor supply. For example, suppose that Social
Security were to become less generous for members of a particular cohort. This would reduce
lifetime wealth and diminish the importance of the Social Security work disincentives for
members of that cohort. The loss of wealth would cause individuals to work more hours in
order to earn more income at some stage of their lifetime. However, it is not clear when they
would do so. If individuals are liquidity constrained at the early Social Security retirement
age (62), those younger than 62 will not react to the bene t reduction. All consumption and
labor supply responses will be after age 62. On the other hand, if individuals are not liquidity
constrained, then there may be signi cant responses by those younger than 62.
The model in this paper allows for a wide range of individual behavior. The model
also captures the fact that the structure of Social Security and pensions causes declining
work incentives after age 62. Because individuals in the model can save and decumulate
3
assets, they may leave the labor force and begin dissaving shortly after age 62, as Gustman
and Steinmeier (1986) argue. However, they cannot borrow against future Social Security,
pension or labor income. Therefore, individuals may have to remain in the labor market until
they are eligible for Social Security and pension bene ts, as Rust and Phelan (1997) argue.
This paper uses the method of simulated moments to match life cycle pro les estimated
using data from the Panel Study of Income Dynamics (PSID) to life cycle pro les generated
by a dynamic programming model. I match labor force participation, hours worked, and
asset pro les. Assuming that preferences are not a ected by age (after conditioning on
health status and family size), matching these pro les allows me to identify key structural
parameters such as the intertemporal elasticity of substitution of labor supply and the time
discount factor. This allows me to consider whether the inability to borrow against future
Social Security bene ts signi cantly a ects life cycle labor supply. Moreover, I consider the
decisions of men ages 30-95, allowing me to consider when in a worker's lifetime we should
expect to see labor supply responses to changes in the Social Security rules.
The PSID data are consistent with a low level of labor supply substitutability for young
men, but a high degree of labor supply substitutability for older men. Consistent with
previous research, I nd very little life cycle variation in hours worked for men between ages
30 and 55. I also nd that work hours and labor force participation decline sharply after age
55, and especially sharply at ages 62 and 65.1 These are exactly the ages at which Social
Security, pensions, and declining wages provide strong incentives to leave the labor force.
The dynamic programming model produces reasonable preference parameter estimates. It
also captures many features of the data, including the sharp decline in labor force participation
rates between ages 55 and 70 and the especially large drops at ages 62 and 65. In order to
t both the participation and hours worked pro les, the model estimates a large xed cost
Ghez and Becker (1975) and Browning et al. (1985) estimate similar labor supply pro les, and Blau (1994)
estimates similar participation pro les. Estimates using the PSID data show that labor force participation
rates drop 71% between ages 55 and 70, reaching 13% at age 70. However, declining health can explain only a
7% decline in participation between these ages. Hours also decline during these ages, although average hours
never fall below 1200 hours per year (24 hours per week for 50 weeks per year). I argue that xed costs of
work are necessary to explain why work hours do not fall below 1200 hours per year, even though labor force
participation rates do fall to near zero at age 70.
1
4
of work. The xed cost generates a high level of labor supply substitutability at the labor
force participation margin.2 Because of the Social Security and pension incentives to leave
the labor force, those in their 60s are near the labor force participation margin. As a result,
labor supply elasticities rise from .3 at age 40 to 1.1 at age 60.
I use the model to conduct three simulations. First, I consider shifting the early retirement
age from 62 to 63. I nd that this has almost no e ect on labor supply, because forward looking
agents almost always have suÆcient nancial resources at age 62 to nance an additional year
out of the labor force. Because they have positive assets near retirement, liquidity constraints
never bind at retirement age. Second, I consider a 20% reduction in Social Security bene ts.
I nd this would cause individuals to delay job exit from the labor market by three months
in order to develop suÆcient nancial assets to o set lost retirement income. Because older
individuals are the ones most willing to substitute their labor supply, most of the labor supply
response would be after age 62. Third, I consider eliminating the tax wedge caused by the
Social Security earnings test. I nd that this would cause individuals to work an additional
one year.3 Together, these three simulations suggest that the Social Security earnings test
is the most signi cant labor supply incentive of the Social Security system. Interactions
between the Social Security system and liquidity constraints are relatively unimportant.
The rest of the paper is as follows. Section 2 develops a model of optimal lifetime decisionmaking. Section 3 describes the estimation scheme: the Method of Simulated Moments.
Section 4 describes the data. Section 5 presents parameter estimates. Section 6 describes the
policy experiments. Section 7 concludes.
See Cogan (1981) for similar ndings.
Neither making the Social Security system's budget balance nor allowing people to borrow against future
Social Security bene ts changes these results signi cantly.
2
3
5
2
The Model
2.1 The Set Up
This section describes the model of lifetime decision-making. Individuals choose consumption, work-hours, (including the labor force participation decision), and whether or not to
apply for Social Security bene ts. They are allowed to save but not borrow. When making
these decisions, they are faced with several forms of uncertainty: survival uncertainty, health
uncertainty, and wage uncertainty.
Consider a household head seeking to maximize his expected lifetime utility at age (or
equivalently, year) t, t = 1; 2; :::; T + 1. Each period that he lives, the individual receives
utility, Ut , from consumption, Ct , hours worked, Ht ; and health (or medical) status, Mt , so
that Ut = U (Ct ; Ht ; Mt ). When he dies, he values bequests of assets, At , according to a
bequest function b(At ): Let st denote the probability of being alive at age t conditional on
being alive at age t
1, and let S (j; t) = (1=st )
Qj
= s denote the probability of living to
age j , conditional on being alive at age t. Since age T + 1 is the terminal period, sT +1 = 0:
k t k
We assume that preferences take the form:
"
U (Ct ; Ht ; Mt ) + Et
+1
TX
= +1
!#
j
S (j
1; t) sj U (Cj ; Hj ; Mj ) + (1 sj )b(Aj )
;
(1)
j t
where is the time discount factor. In addition to choosing hours and consumption, eligible
individuals can choose whether to apply for Social Security bene ts; let the indicator variable
2 f0; 1g equal one if the individual has applied for bene ts. The individual maximizes
equation (1) by choosing the contingency plans fCj ; Hj ; Bj gTj =+1t , subject to the following
Bt
equations, described below: a mortality determination equation (4), a health determination
equation (5), wage determination equations (6)-(7), a spousal income determination equation
(8), and an asset accumulation equation (9).
6
The within-period utility function is of the form
U (Ct ; Ht ; Mt ) =
1
1
Ct (L Ht P Pt I fM = badg)1
!1
:
(2)
where the per period time endowment is L and the quantity of leisure consumed is L Ht
P Pt
I fM = badg: The 0-1 indicator I fM = badg is equal to one when health is bad
and 0 when health is good. Participation in the labor force is denoted by Pt , a 0-1 indicator
equal to 0 when hours worked, Ht ; equals zero. The xed cost of work, P , is measured in
hours-worked per year.4 Retirement is assumed to be a form of the participation decision.
Workers can reenter the labor force.
The parameter is between 0 and 1 and the parameter is greater than zero. The parameter has two functions. First, it controls the intertemporal substitutability of consumption
and leisure. As increases, individuals are less willing to intertemporally substitute. Second, measures the non-separability between consumption and leisure. Assuming certainty
and positive consumption and work-hours, > 1 implies that leisure and consumption are
substitutes.
The bequest function is of the form
b(At ) = B
(At + K )(1 )
:
1
(3)
where K determines the curvature of the bequest function. If K = 0 there is in nite disutility
of leaving non-positive bequests. If K > 0; the utility of a zero bequest is nite.
Given the objective function, individuals face several constraints. Mortality rates depend
Annual hours of work is clustered around both 2000 hours and 0 hours of work, a regularity in the data
that standard utility functions have a diÆcult time replicating. Fixed costs of work are a common way of
explaining this regularity in the data (Cogan (1981)). Fixed costs of work generate a reservation wage for
a given marginal utility of wealth. Below the reservation wage, hours worked is zero. Slightly above the
reservation wage, hours worked may be large. Individual level labor supply is highly responsive around this
reservation wage level, although wage increases above the reservation wage result in a smaller labor supply
response.
4
7
upon age5 and previous health status:
st+1 = s(Mt ; aget+1 ):
(4)
Next year's health status, prob(Mt+1 jMt ; aget+1 ); depends on current health status and age.
Health status follows a two-state transition matrix at each age with a typical element6
good;bad;t+1 = prob(Mt+1 = goodjMt = bad; aget+1 ):
(5)
The logarithm of wages7 at time t; ln Wt , is a function of hours-worked, age and health
status, plus an autoregressive component of wages ARt :
ln Wt = ln Ht + W (Mt ; aget ) + ARt :
(6)
The function W (Mt ; aget ) is described in detail in Section 3.2. The autoregressive component
of wages has a correlation coeÆcient and a normally distributed innovation t :
ARt = ARt 1 + t ; t N (0; 2 ):
(7)
By assumption, at time t 1 the worker knows the autoregressive component of wages (ARt 1 )
but only knows the distribution of the innovation in next period's wage (t ):
Spousal income, described in detail in Section 4.2, depends upon the individual's wage
and age:
yst = ys(Wt ; aget ):
(8)
The notation aget is redundant as both aget and t are measured in years, but I make the distinction for
the sake of clarity.
I ignore the possibility that wealth may a ect health, as the Grossman (1972) model implies.
By \wage," I am referring to the observed wage of labor market participants as well as the potential wage
of non-participants. Given this de nition of wage, another interpretation for the wage would be \productivity."
5
6
7
8
The nal constraint is the asset accumulation equation:
At+1 = At + Y (rAt + Wt Ht + yst + pbt + "t ; ) + (Bt sst) Ct ; At+1 0;
(9)
where Y (rAt + Wt Ht + yst + pbt + "t ; ) is the level of post tax income, r is the interest rate,
is the tax structure (described in Appendix A), pbt denotes pension bene ts (described in
Section 2.3 and Appendix C), "t denotes a pension accrual residual (described in Section 2.3
and Appendix C), and sst denotes Social Security bene ts net of the earnings test (described
in Section 2.2 and Appendix B).
Individuals cannot draw Social Security bene ts until age 62. By assumption, the date of
pension bene t receipt is 62. Because it is illegal to borrow against Social Security bene ts
and diÆcult to borrow against most forms of pension wealth, individuals with low asset levels
potentially must wait until age 62 to nance exit from the labor market.
2.2 Social Security
There are three major labor supply incentives provided by the Social Security system.8
All three incentives tend to induce exit from the labor market by age 65.
First, increased labor income leads to increased Social Security bene ts, but only for the
rst 35 years in the labor market. Social Security bene ts depend upon Average Indexed
Monthly Earnings, or AIMEt ; which is average earnings in the 35 highest earnings years.
However, after the rst 35 years in the labor market, AIME is only recomputed upwards if
current earnings are greater than earnings in a previous year of work. Appendix B describes
computation of AIMEt :
Second, there are incentives to begin drawing Social Security bene ts by age 65. Individuals are ineligible for Social Security bene ts before age 62. Upon application for bene ts
I use tax and bene t formulas from the Social Security Handbook Annual Statistical Supplement for the
year 1987 for several reasons. First, 1987 is relatively close to the middle year of the data. Second, there were
signi cant changes to the tax code enacted in 1986 that simplify the dynamic programming problem. Lastly,
bene t formulas have not become signi cantly more or less generous between 1987 and the end of the sample
period (although there have been reductions in the Social Security work disincentives).
8
9
the individual receives them until death. Once the individual has applied for Social Security
bene ts, bene ts depend on a progressive function of AIME and the year the individual starts
drawing bene ts. For every year before age 65 the individual applies for bene ts, bene ts
are reduced by 6.7%. This is roughly actuarially fair. For every year between ages 65 and 70
that bene t application is delayed, bene ts rise by 3%. This is actuarially unfair and thus
generates an incentive to draw bene ts by age 65.
Third, the Social Security earnings test taxes labor income for Social Security bene ciaries
at a very high rate. If a bene ciary younger than age 70 earns more labor income than a
\test" threshold level of $6,000, bene ts are taxed at a 50% rate until all bene ts have been
taxed away. Moreover, the earnings test tax on bene ts is in addition to Federal and state
income and payroll taxes. Therefore, the marginal tax rate an individual faces is the sum of
Federal, state, and payroll marginal tax rates, plus 50%. The incentive to draw bene ts by
age 65 in combination with the Social Security earnings test for Social Security bene ciaries
is a major disincentive for work after age 65.
A common misconception is that the recomputation formulas fully replace bene ts lost
through the earnings test. Although this is roughly true between ages 62 and 65, a loss of
one year's bene ts results in only a small upward revision in future bene ts after age 65. If
a year's worth of bene ts are taxed away between 62 and 65, bene ts in the future will be
raised by 6.7%. If a year's worth of bene ts are taxed away between 65 and 70, bene ts in
the future will be raised by 3%.
The formula for Social Security bene ts in the asset accumulation equation (9) captures
all of these incentives.
2.3 Pensions
Pensions are like Social Security in two important respects. First, pension wealth is illiquid
until the early retirement age, which is usually 55, 60, or 62 depending on the pension plan.9
Although it is often possible to \cash out" of pension plans, there are often penalties for doing so. For
example, there are tax penalties for drawing de ned contribution wealth before age 59 ; except in certain
hardship cases.
9
1
2
10
Therefore, I assume that pension wealth is illiquid until age 62. Second, pension bene ts
depend on the individual's work history. Because of this, pension bene ts are assumed to be
a function of AIMEt ; just like Social Security bene ts.
However, pensions are di erent than Social Security in their age-speci c incentives to
leave the labor force.10 De ned bene t pension plans are typically structured in a way that
encourages a worker to remain at a rm until the early retirement age and to leave the rm
no later than the normal retirement age (usually 62 or 65).11 These incentives, as well as my
approach to modeling these incentives, are described below.
The formula that determines de ned bene t pension plan bene ts varies greatly from rm
to rm, making it diÆcult to generalize the incentives that workers face. However, pension
bene ts typically depend on years of service at the rm, the highest annual earnings at the
rm (usually the average of the ve highest earnings years), and a formula that depends on
age and years of service at the rm.
Pension plans often provide incentives to stay at a rm until the early retirement age.
One reason for this is that a worker who leaves a rm before the early retirement age must
wait until the early retirement age-and sometimes later{to draw bene ts.12 Furthermore,
bene ts depend in part upon the nominal wage when the individual left the rm. Because
the wage at the rm is usually not adjusted for in ation, the value of the wage in real terms
will fall until he begins receiving bene ts and thus his real bene ts will fall also.13
Although pensions only provide incentives to leave the current employer, individuals who switch employers
often receive lower wages from their new employers. Therefore, it seems reasonable to assume that the declining
pension accrual on a job still results in higher compensation than on any other job that an individual could
obtain.
Health and Retirement Survey data indicate that of men aged 51-55 with a de ned bene t pension plan,
31% have an early retirement age of 55 and 21% have an early retirement age of 62. 25% have a normal
retirement age of 62 and 24% have a normal retirement age of 65. Data from employers tends to show even
more heaping of the normal retirement age at 65 (Ippolito, 1997). Note, however, that de ned contribution
pension plans do not provide strong incentives to leave the labor force. Of working male Health and Retirement
Survey respondents between ages 50 and 60, 22% only have a de ned bene t plan, 20% only have a de ned
contribution pension plan, and 11% have both.
There are other incentives to remain with some employers until the early retirement age. For example,
Federal workers with 30 years service can claim full bene ts at age 55. However, if the Federal worker leaves
his job at 54, he must wait until age 62 to draw bene ts. Therefore, leaving at age 54 instead of 55 leads to
the loss of bene ts between the ages of 55 and 61. Moreover, bene ts are not adjusted for in ation until the
individual is drawing bene ts, leading to further losses in the value of bene ts. Ippolito (1997) computes that
the loss of pension bene ts of exiting at 54 instead of 55 is equal to seven times annual earnings.
It is only after the worker receives pension bene ts that pension bene ts are adjusted for in ation.
10
11
12
13
11
The part of the pension formula that depends on age and years of services generates
the incentive to leave the rm by the normal retirement age. Up to the normal retirement
age, this pension formula component increases with age. After the normal retirement age,
it does not. Therefore, delaying exit from the rm after the normal retirement age results
in a reduction in the present value of pension bene ts. Although delaying bene t receipt
causes slightly higher annual bene ts (because years of service at the rm have increased),
the individual will receive bene ts for fewer years.
In order to account for the high pension accrual for those in their 50s and the lower
pension accrual at other ages, I take estimates of age-speci c accrual rates from Gustman et
al. (1998). Because I assume that bene ts, pbt ; depend only on AIMEt ; and the formula for
AIMEt does not account for the high accrual rates for individuals in their 50s, the formula
for AIMEt overstates pension accrual at younger ages and understates pension accrual at
older ages. To account for this problem, the variable "t represents the di erence between two
di erent methods of accounting for pension accrual. Thus "t is negative at younger ages and
is positive at older ages. Nevertheless, the average of "t is less than $1,000 at almost every
age. Construction of "t is described in Appendix C. It is treated as labor income in the asset
accumulation equation (9).14
One nal aspect of pensions is worth noting. Accrual rates tend to be higher for high
wage workers than for low wage workers. There are two reasons for this. First, the formulas
for many de ned bene t plans explicitly have higher accrual rates for high wage workers.
Second, a higher share of high wage workers tend to have pension plans. I account for this
by modeling pbt as a regressive function of AIMEt :
Note that this method of accounting for pension accrual only leads to model missepeci cation if liquidity
constraints and variable marginal tax rates a ect behavior.
14
12
2.4 Heterogeneity and Model Solution
Optimal decisions depend on the state variables, denoted Xt = (At ; Wt ; Bt ; Mt ; AIMEt )15 ,
preferences
denoted
= ( ; ; P ; B ; ; L; );
and
the
parameters that determine the data generating process for the state variables denoted =
(r; 2 ; ; ; h(Mt ; aget+1 ); fprob(Mt+1 jMt ; aget )gTt=1 ; fSt gTt=1 ; Y (:; :); fyst gTt=1 ; fpbt gTt=1 ; fsst gTt=1 ):
I solve the model backwards using value function iteration.
The model solution procedure allows for heterogeneity in the state variables, Xit ; where
i indexes individuals. However, the requirement of computational simplicity does not allow
for heterogeneity in preferences or in the data generating process for the state variables :
I assume that individual i responds only to (Xit ; ; ). Di erent realizations of the stochastic
shocks means that wages and health status will di er across individuals, so there may be
di erences in consumption, labor supply, and bene t application decisions across individuals.
However, given the same age, wage, health status, asset level, Social Security application
status, and AIME, di erent individuals will make the same decisions. See Appendix D for
details.
3
Estimation
This section describes the method of simulated moments (MSM) estimation strategy. The
goal is to estimate the preferences given the data generating process for the exogenous state
variables : Because it would be too computationally burdensome to estimate all parameters
simultaneously, I use a two-step strategy. In the rst step, I estimate some elements of and
calibrate others. I assume rational expectations, meaning that individuals know their own
state variables Xt at time t; the Markov process that determines their state variables, which
is parameterized by ; and optimize accordingly. In the second step, I use the numerical
methods described in Appendix D and the estimated data generating process for the state
Pension wealth and spousal income depend on the other state variables and are thus not state variables
themselves.
15
13
variables to simulate life cycle pro les for a large number of hypothetical individuals. The
goal is to nd preference parameters that generate simulated pro les that match the pro les
estimated from data.
The next subsection describes the MSM technique in more detail. The following subsections describe construction of the sample pro les that I match to the simulated pro les as
well as estimation of some of the elements of :
3.1 Estimation of Preferences: The Method of Simulated Moments
The MSM estimation strategy matches mean assets, hours of work, participation and
also median assets in the PSID to the corresponding moments of the same variables in a
simulated sample. The \matching" of moments is done using standard GMM techniques.
Because of problems with measurement error, I do not match high order moments.16 Using
means, however, averages out measurement error, as shown below.
The objective is to nd a vector of preferences 2 that simulates pro les that \look
like" (as measured by a GMM criterion function) the pro les from the data. I assume R7
where is a compact set. I assume the PSID data are generated by the model in Section 2,
plus measurement error in hours:
Ait = At (Xit 1 ; ; );
(10)
ln Hit = ln Ht (Xit ; ; ) + iHt if Pit > 0;
(11)
Pit = Pt (Xit ; ; )
(12)
where Ait ; ln Hit and Pit represent individual i's measured assets, log of hours-worked, and
participation decision at time t; and iHt represents measurement error in hours. I assume zero
mean measurement error in hours, E [iHt jMit ; t] = 0:17 Computation of At (Xit 1 ; ; ); ln Ht (Xit ; ; )
See Altonji (1986) and Abowd and Card (1989) for attempts to overcome the measurement error problems
that plague high frequency analyses of labor supply.
I also allow for zero mean measurement error in participation, conditional on age and health status. The
mean asset condition in equation (14) also holds if there is zero mean measurement error. However, the median
condition (13) will not typically hold in the presence of measurement error. Nevertheless, dropping the median
16
17
14
and Pt (Xit ; ; ) is by value function iteration, described in Appendix D.
Assuming that the distribution of the state variables is the same in both the simulations
and the data, it is possible to generate moment conditions for median and mean assets as
well as mean participation and hours-worked conditional upon health status, resulting in the
following 6T moment conditions:
E I fAit median(At (X; ; ))g
E [Ait jt]
E [ln HiMt jM; t]
Z
Z
1
jt = 0; for all t 2 f1; :::; T g;
2
(13)
At (X; ; )dFt 1 (X jt) = 0; for all t 2 f1; :::; T g;
(14)
ln Ht (X; ; )dFMt (X jM; t) = 0; for all t 2 f1; :::; T g; M
2 fgood; badg;
(15)
E [PiMt jM; t]
Z
Pt (X; ; )dFMt (X jM; t) = 0; for all t 2 f1; :::; T g; M
2 fgood; badg;
(16)
where median(At (X; ; )) is the median18 of the distribution of simulated assets At (X; ; ),
I f:g is the indicator function, equal to 1 if true, Ft (X ) is the cdf of the state variables at time
t; and FMt (X jM ) is the cdf of the state variables at time t given health status M: Integrals
in equations (13)-(16) are computed using Monte-Carlo integration. When evaluated at the
true preference parameters and the true distribution of the state variables, conditional on age
and, in the case of hours and participation, health status, the di erence between the data
moment and the simulated moment has an expected value of zero.
In summary, the MSM procedure I use can be described as follows. First, I estimate the
condition (13) and re-estimating the model does not have a large e ect on parameter estimates.
See French and Jones (2002) for more on using quantile conditions in a GMM framework.
18
15
life cycle pro les for hours-worked, labor force participation, and assets from the PSID data.
Second, using the same data I use to estimate pro les, I estimate the data generating processes
for health status and wages following the estimation techniques described in Sections 3.4 and
4.2. Third, I use the estimated data generating processes to simulate matrices for random
health and wage shocks as well as an initial distribution for health, wages, assets and AIME.
These are sequences of lifetime shocks for 5,000 simulated individuals, so there is a 5; 000 T
matrix of health shocks and a 5; 000 T matrix of wage shocks. Fourth, I pick an arbitrary
vector of preference parameters and compute the decision rules given those parameters and
the numerical methods described in Appendix D. The fth step is to use the decision rules
and the health and wage shocks to simulate hypothetical life cycle pro les for the decision
variables. Sixth, the simulated data and the true data are aggregated by age (and in the case
of hours and participation, by health status). Seventh, the di erence between the simulated
and true pro les is computed and the di erences are weighted up to form a distance measure.
Finally, a new vector of preference parameters is picked and the whole process is repeated.19
The preference parameters that minimize the distance between the data moments and the
^ I discuss
simulated moments described in equations (14)-(16) are the estimated parameters, :
the distribution of the parameter estimates, the weighting matrix and the overidenti cation
tests in Appendix E.
3.2 Estimation of Pro les
This section describes the life cycle pro les for assets, hours, and participation rates to be
fed into equations (13) - (16) as well as the life cycle wage pro le. When constructing pro les
I use simplex methods to search over : Because the local minimum of the GMM criterion function need
not be the global minimum, I try many di erent starting values. I check to see whether the algorithm will
nd the global minimum by simulating individuals at assumed parameter values and treating these simulated
individuals as data. I then simulate another set of individuals with di erent wage and health shocks and
with di erent initial utility function parameters. I then use the MSM algorithm to match the second set of
simulated individuals to the rst set of \data". I nd that preference parameters estimated for the second
set of individuals come very close to the \true" preference parameters of the rst set. Nevertheless, estimated
preference parameters usually do not come within two standard errors of the true parameters. This shows
that standard errors are underestimated. Footnote 48 provides further evidence that the standard errors are
underestimated.
19
16
that account for age and health e ects, I am concerned about the presence of individualspeci c e ects, year e ects, and family size e ects. To generate pro les, I estimate equation
(17), where Zit represents an observation for either assets Ait , hours ln Hit , participation Pit ,
or wages (net of the tied wage-hours e ect) ln Wit
ln Hit 20 for individual i at age t :
Zit =fi +
T
X
k
T
X
k
=1
=1
gk I fageit = kg prob(Mit = goodjMit ) +
F
X
bk I fageit = kg prob(Mit = badjMit ) + f famsizeit + U Ut + uit
f
=1
(17)
where fi is an individual-speci c e ect, famsizeit is family size, Ut is the unemployment rate,
fgk gTk=1 fbk gTk=1; ff gFf=1 ; and U are parameters, prob(Mit = badjMit ) is the probability
that health is bad given a noisy health measure Mit and prob(Mit = goodjMit ) = 1
prob(Mit = badjM ): French (2001) describes construction of prob(Mit = badjM ): If M
it
it
it
were perfectly measured, then prob(Mit = badjMit ) would collapse to a dummy variable. I
estimate equation (17) using xed-e ects to control for the individual-speci c e ect, fi : I use
a full set of age dummy variables when estimating the hours, participation, and asset pro les;
however, the wage pro le is estimated using a fourth order polynomial in age.21 For the asset
pro les I assume gk = bk for all k; that is, I do not condition on health status when
generating the asset pro le. I use a full set of dummy variables for family size famsizeit :
I use the age e ects and health e ects from equation (17) to generate the data pro les that
I will match to the simulated pro les. I set family size equal to three and the unemployment
rate to 6.5%,22 and use the mean individual-speci c e ect for individuals who were born in
1940, who are age 50, and have the average level of health for 50 year olds (see Appendix
E). Note that this approach controls for cohort e ects. The cohort e ect is just the average
Note that this identi es W (Mit; ageit):
When creating pro les with the polynomials, I estimate the polynomial using data on individuals ve
years younger and 10 years older than my sample of interest. This overcomes some of the endpoint problems
associated with polynomial smoothing.
It seems unlikely that households can properly forecast future unemployment rates. Therefore, I add in
the variance in wages coming from the unemployment rate to the variance of the innovation in the wage. See
footnote 32.
20
21
22
17
xed-e ect of all individuals in a single cohort.23
3.3 Accounting for Selection in the Wage Pro les
Unfortunately, the xed-e ects estimator does not overcome an important selection problem in the wage equation. Fixed-e ects estimators use wage observations for workers but do
not use the potential wages of non-workers. Because the xed-e ects estimator is identi ed
using growth rates for wages and not levels of wages, composition bias problems{the question
of whether high wage or low wage individuals drop out of the labor market{is not a problem
if wage growth rates for workers and non-workers are the same.24 However, if individuals
leave the market because of a sudden wage drop, such as from job loss, then wage growth
rates for workers will be greater than wage growth for non-workers. This problem will bias
wage growth upward.
In order to account for the selection problem, it is important to distinguish between three
separate objects. The rst is the unobserved average wage pro le for all individuals that
is the object of interest.25 The second is the xed-e ects wage pro le estimated using the
actual data on workers-this pro le is biased for reasons discussed above. The third is the
xed-e ects pro le using simulated workers.26 This pro le is biased for the same reason that
the pro le using actual data on workers is biased.
To correct the bias, I assume that the bias in the xed-e ects wage pro les of workers
will be the same in both the actual PSID and simulated data.27 First, I feed the estimated
(and biased) xed-e ects wage pro le into the model. Second, I solve and simulate the model
Cohort dummies would be unidenti ed if added to equation (17).
This is an important advantage of panel data over cross-sectional data. Blundell et al. (2003), who
use cross-sectional data, identify the role of selection using participation equations that rely on exclusion
restrictions. Panel data allows the econometrician to observe an individual's wages immediately before leaving
the labor market. In order to infer the wage innovation for those who leave the labor market, however, I must
use the functional forms and exclusion restrictions embedded in the model, whereas Blundell et al. provide
methods to test for the appropriateness of functional forms.
This object is W (Mt ; aget) = E [ln Wt ln HtjM; aget]:
This object converges to W (Mt; aget) + E [ARt jM; aget; Ht > 0] as the number of simulations becomes
arbitrarily large.
This is true if the simulated individuals have the same wage generating process, the same distribution of
state variables, and have the same preferences as the individuals in the data.
23
24
25
26
27
18
and estimate the xed-e ects wage pro les for both simulated workers and all simulated
individuals. Third, I compute the di erence between the two pro les so that I can estimate
the extent to which growth rates in wages are overestimated by using only simulated workers
instead of all simulated individuals. I then use this estimate of the selection bias in the
simulated wage pro le to infer the extent of selection bias in the PSID data wage pro le.
If, for example, the xed-e ects wage pro les overstate average wages at age 60 by 10% in
the simulated sample, then it is likely that wages have been overestimated at age 60 by 10%
in the PSID data. Therefore, the candidate for the unobserved average wage at age 60 is the
xed-e ects estimate from the PSID data, less 10%. This new candidate wage pro le is fed
into the model and the procedure is repeated. If, for example, the xed-e ects pro le using
simulated data still indicates a 1% upward bias, the candidate true wage pro le is reduced
by an additional 1%. This iterative process is continued until a xed point is found.28
Once the process converges, the estimated wage pro le for all individuals is fed into the
model and preference parameters are estimated using the method of simulated moments.
Upon re-estimation of the model parameters, the selection bias is recomputed and the wage
pro les are updated. The model parameters are then estimated again.
3.4 Estimation of the Health Transition Matrix
When estimating the health transition matrix in equation (5), I am concerned with both
the presence of measurement error in health status and the presence of individual heterogeneity. In order to address both of these concerns I estimate the linear probability model:
P rob(Mit = goodj i ; Mit 1 ; ageit ) = i +
K
X
k
=1
K
X
k
=0
k
k
agekit prob(Mit 1 = goodjMit 1 ) +
agekit prob(Mit 1 = badjMit 1 ) + it
(18)
If the value function were concave, it would be possible to prove that this iterative mapping was a
contraction. This cannot be proven analytically and in general cannot be proven numerically. However, based
upon carefully conducted computations it seems that a unique solution exists.
28
19
where
i
represents individual heterogeneity in capacity for good health. OLS estimates will
be inconsistent for two reasons. First,
i
is correlated with previous health status. In order
to circumvent this problem, I rst di erence equation (18), then use lags of health status
and health status interacted with age to instrument for last year's health status change and
last year's health status change interacted with age. Second, health status is measured with
non-zero mean error. As in equation (17), I take estimates of prob(Mit 1 = goodjMit 1 ) from
French (2001). When constructing the health status transition matrix, I set i equal to its
average level for individuals born in 1940.
4
Data and Calibrations
4.1 Data
I use the Panel Study of Income Dynamics (PSID) for the years 1968-1997. I drop the
Survey of Economic Opportunity (SEO) subsample to make the data more representative of
the US population.29 Because I model the behavior of a head of household, I use labor supply
variables for the male head of household and household-level asset data.
When estimating the hours-worked and labor force participation rate pro les, I use individuals born between 1922 and 1940, resulting in 18,690 person-year observations for labor
force participation rates and 15,766 person-year observations for hours-worked. For the asset pro le, I use individuals born between 1902 and 1965 to increase sample size, resulting
in 8,265 person year observations. For the wage and health pro les, I use the full sample,
resulting in 60,714 and 69,347 person-year wage and health observations.
I estimate the asset pro le using 1984, 1989, and 1994 PSID wealth surveys. Because I do
not wish my estimate of assets to be a ected by the extremely wealthy, many of whom inherit
their wealth, I exclude observations with over $1,000,000 in assets. Households in which an
entering family member brought assets into the household or an exiting family member took
assets out of the household are dropped. The PSID asset measure is fairly comprehensive.
29
The SEO subsample is a subsample of poor and minorities.
20
It includes real estate, the value of a farm or business, vehicles, stocks, mutual funds, IRAs,
Keoghs, liquid assets, bonds, other assets and investment trusts less mortgages and other
debts. It does not include pension or Social Security wealth.
Wages are computed as annual earnings divided by hours and are dropped if wages are
less than $3 per hour or greater than $100 per hour, 1987 dollars. Hours are counted as zero
if measured hours are below 300 hours-worked per year.
The PSID has only one measure of health that is asked during all years of the panel. It
is the self-reported response to \Do you have any physical or nervous condition that limits
the type of work or the amount of work that you can do?" A criticism of self-reported
health measures is that respondents often report \bad health" in order to justify being out
of the labor force. This will lead me to overestimate the e ect of health upon work-hours.
Alternatively, the coarse discretization of health status into good and bad when true health
status is likely a continuous variable potentially causes measurement error, biasing the e ect
of health status on di erent variables to zero e ect.
The PSID has poor information on mortality statistics. Therefore, I combine PSID data
with mortality statistics from the National Center for Health Statistics (NCHS).30 These
statistics use the entire US population as their sample.
4.2 Remaining Calibrations
In order to estimate preference parameters, I calibrate some of the parameters that determine the data generating process for the state variables : These are the parameters that
determine the stochastic component of wages (2 ; ); the e ect of work-hours on wages ;
the interest rate r; and spousal income.
The parameters from the wage equation (2 ; ); shown in Table 1, were estimated using
I compute mortality rates given last year's health status using Bayes' rule:
prob(Mt = goodjdeatht )
prob(deatht jMt = good) =
prob(deatht)
(19)
prob(Mt = good)
I compute prob(Mt = goodjdeatht) and prob(Mt = good) using PSID data, and prob(deatht) from the
NCHS data. When using PSID data, the estimate of prob(deatht) is about 25% lower than when using NCHS
data, indicating that the PSID underestimates mortality rates by 25%.
30
1
1
1
1
1
21
Parameter Variable
Estimate S.E.
2
variance of the innovation in wages .0141
.0014
autoregressive coeÆcient of wages .977
.017
Table 1:
The Variance and Persistence of Innovations to the Wage
equations (6) and (7) and minimum distance techniques. The model of wages allows for a
MA(1) measurement error component.31 The results indicate that = :977; wages are almost
a random walk. The estimate of 2 is .0141; one standard deviation of an innovation in the
wage is 12% of wages.32 These estimates imply that long run forecast errors may be large.
The coeÆcient ; which parameterizes the part-time wage penalty, is set at .415 and is
similar to the ndings of Aaronson and French (2001) and Gustman and Steinmeier (1986).
This implies that part-time workers (who work 1000 hours per year) earn 25% less per hour
than full-time (2000 hours per year) workers. Controlling for the fact that part-time workers
make less per hour than full-time workers eliminates most of the wage declines after age 60
that are shown in Figure 1.
The remaining calibrations are as follows. I set the pre-tax interest rate at r = :04 and
the age at which individuals receive pension bene ts at age 62. Following DeNardi (2000),
In order to obtain these estimates, I use wage residuals from the regression in equation (17) with the xed
e ect added back. I used a balanced panel for the years 1977-1986. Given equation (7), rede ne ln Wit to be
the wage residual, which has the following form:
ln Wit ln Ht = ARit + it + it
(20)
31
1
where it is assumed to be measurement error with MA(1) coeÆcient : The AR(1) component, ARit is
potentially non-stationary with autocorrelation coeÆcient and innovation it :
ARit = (t
1)
ARi1 +
t
X
j =2
j ij
(21)
French (2002) nds that most of the variance of the MA(1) component of wages is measurement error, so
assuming all of the variance of the MA(1) component of wages is measurement error seems reasonable. All
objects on the right hand side of equation (21) are assumed to be mutually orthogonal. Given that there are
10 years of data, there are 10 variances and 45 unique covariances, implying 55 moment conditions to match to
the model in equation (21). The estimates in Table 1 are similar to other estimates in the literature, although
= :977 is likely at the high end of the range. Card (1994) also nds that wages follow a highly persistent
AR(1) process.
I scaled up the variance of the innovations to re ect the additional uncertainty due to the aggregate
unemployment rate using estimates of the variance of the unemployment rate and U from equation (17).
However, the amount of volatility in wages associated with the unemployment rate is tiny, and thus this
procedure had only a small e ect on :
32
2
22
the object that determines the curvature of the bequest function, K; is set equal to $500,000.
Spousal income is assumed to follow a polynomial in age and the log of the wage.33 Because
the PSID has poor information on pensions and (until the most recent waves) Social Security,
I use spousal income when young to predict spousal pension and Social Security bene ts when
old.
5
Results
The estimated inputs into the MSM algorithm can be divided into data on the exogenous
state variables and data on decision variables. The data generating process for the exogenous
state variables, parameterized by the vector ; includes growth rates for wages conditional on
health status, health transition matrices, and mortality probabilities. The decision variables
are the pro les for hours-worked per year (by those who worked), assets, and labor force
participation. In order to identify the role of health in explaining the decline of hours near
the end of the life-cycle, pro les for hours and labor force participation rates are shown for
individuals in both good and bad health.
5.1 Pro les for the Exogenous State Variables
This section describes the pro les for wages, health transition matrices, and the survivor
probabilities. I use smoothed versions of the pro les when estimating preferences. However,
I display unsmoothed pro les to show that the pro les are precisely measured, as displayed
by their smooth appearance. On average, pro les for healthy individuals are smoother than
for unhealthy individuals. This is because there are more observations on healthy individuals
than on unhealthy individuals.
Using the methodology from Section 3.2, the top left panel of Figure 1 displays wage
pro les for males by age and health status. Most striking is the hump shape of the wage
I regress spouse's income on the husband's log wage (instrumented using education), an age polynomial,
and a set of cohort dummy variables. When I construct the spousal income pro le, I set the cohort e ect
equal to those born in 1940.
33
23
pro les for both health groups, with wages peaking near age 55. Fixed-e ects estimates show
a more rapid decline in wages after age 55 than do OLS estimates (see Ghez and Becker
(1975), Heckman (1976), and Browning et al. (1985) for pro les constructed using OLS).
The reason for this is that high wage individuals tend to remain in the labor force until older
ages than do low wage workers. Therefore, OLS estimates su er from \composition bias"
problems, where wage observations will be for all workers at age 55 but only for high wage
workers at age 65. Also striking is the small e ect of health on wages. Fixed-e ects estimates
show a smaller role for health than OLS. There are three alternative explanations for the
di erence between OLS estimates and xed-e ect estimates. First, it may be that some
other factor (e.g. childhood poverty) causes both poor health and low wages. Second, the
Grossman model (1972) predicts that individuals who have higher expected lifetime wages
invest more in health human capital when young. Therefore, the Grossman model implies
that high wages cause good health, and not vice versa as most interpretations of an OLS
regression of wages on health assume. The third explanation for the small estimated e ect
of health upon wages could be related to a selection problem. It may be only the individuals
who get lucky in the labor market who remain in the labor market after a bad health shock.
Section 5.5 discusses this third point in greater detail.
24
Probobility Dying This lear
0.04
0.00
0.12
0.08
Hourly Woge
0.16
b
0.20
Figure 1:
8[;>
8
10
12
14
16
if
~
c
ru
0
0
~
[~
0
25
Life Cycle Profiles for Exogenous State Variables
~
:<
~
g
"
if
~
g;
~
•
·- ....
----~""'·
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
I
~
-~
m
~
~
[
·.-.-.-
·:.-.- . ...
~
~
il'
:::
--~. ~~~'''"·~=-=.:·
~
~
~
~
s:
~
[ill
l
-0
~·
~
~
I
~
~
0
:;-
~
~
~
O.B
0.9
Pr-obability
1.0
0.0
0.1
O.Z
of Being
0.3
0.4
in
Bad Health This Year
0.5
~
~.
!.
~
-Q'
~
g
g
"'"'
if
~
s:
~
~
~
~
"
~
ro
~~~~~~~~~~~~~
\\
~~~~
';
';._i
~~00
00~~
_;_;]~
HH
~
,-,_
rn
~ ~
0.6
0.7
O.B
0.9
,\\_..
i",....,-.._
~
.:0
-~
.}
.'•I
}
'(•
·······--
s:
-0'
~
"~
:<
~
i-.,--.
"
-..
"·.:;..
ro
~,~
,~
I
::§:
1.0
( __ )
'=1
~
~
"
. ·::.·.·-···
~ ~:::::::::====~.:._____.__~~
·..
---.:..
•••• ~-
The two right hand side panels of Figure 1 show how health dynamics change over the lifecycle. Until age 55, very few individuals experience a change from good health to bad health.
This begins to change after age 55, with individuals becoming more and more likely to move
from good health to bad health. Note, however, there is no rapid shift in population health
that takes place only between ages 55 and 70, the ages at which labor force participation
declines most rapidly. Instead, much of the decline in population health takes place after age
70.
Lastly, the lower left panel shows mortality rates over the life cycle. Unsurprisingly,
individuals in bad health have higher mortality rates than individuals in good health.
5.2 Decision Pro les
This section describes the pro les for the decision variables. To recover preference parameters, I make two fundamental identifying assumptions. First, changes in work-hours and
consumption a ect neither health nor wages (other than through the tied wage-hours e ect).
Second, preferences depend only upon health and family size. Preferences change with age,
but only as a result of changes in health and family size. Therefore, age can be thought of
as an \exclusion restriction" which causes changes in the incentives for work and savings but
does not change preferences.
The top panel of Figure 2 shows the life-cycle pro les for hours-worked for men in good
and bad health. At any point in the life cycle, the e ect of health on hours-worked is sizeable,
but health only explains a small amount of the variation in work-hours over the life cycle.
Hours-worked begins to decline rapidly after age 55.34 This is true even when conditioning
on health status, so it appears that health status alone must have a small causal role in the
decline in the number of hours-worked near retirement.
This is similar to Browning et al. (1985), although Ghez and Becker (1975) nd that the drop-o in hours
is later in life. Ghez and Becker's result may be di erent because they use data from 1960, when the average
retirement age was later.
34
26
Annual
Hours
Worked
By
Health
Status,
··- ...
. ...
Workers
Only
·····- ...
0
=
healthy
unhealthy
34
38
42
46
50
54
58
62
66
70
Age
Labor Force
Participation
Rate
By
Health
Status
···- ....
~
0
·~
~
~
0
'l
"'
"
"'
"
healthy
unhealthy
0
34
38
42
46
50
54
58
70
Age
Non-Pension
Assets
Over the
Life
Cycle.
1987
Dollars
I_\
1 \
1
/
1
~
'....-
'I
1\ /
'
\I
./'-,,,: \./
.
I
I\/~
F-- ~.,1'·-...
. . . . -·····~-..j ......
.·....
./
34
38
42
\1
A1
11(""\\1'
/\
1
,\ , /11.
1
-....._
I
\'-JI
"
i
v
50
v '··
.r\1
_._~i\11
'
J
\I
V
mean assets
median assets
46
1
1
\/ 1
\
'i
r -{~\It-·~
1 t,, . . .
\,'[\ ''J\ \I \\j
~
f \
,, \ ,
'
A
1
1\
\/1.\ I
54
58
62
I
66
70
Age
Figure 2:
Life Cycle Profiles for Decision Variables
27
The middle panel of Figure 2 shows the life-cycle pro les for labor force participation.
Health appears to a ect labor force participation rates more than hours worked. However,
the e ect is still modest. The fraction of all individuals at age 55 who report bad health is
20% and this rises to only 37% by age 70. Therefore, the change in labor force participation
rates attributable to changes in health between ages 55 and 70 is small. The e ect can be
quanti ed using the equation
70
X
=56
t
Pt =
70 P
X
=56 M
t
t
Mt
(22)
P
where is the rst di erence operator, M
is the estimated e ect of health on participat
tion (estimated by the vertical di erence between the upper and lower pro les in the middle
panel in Figure 2) at age t; and Mt is the change in population health status between age
t 1 and t (estimated using the bottom right panel of Figure 1). This technique suggests that
declining health between ages 55 and 70 can explain a 7% drop in labor force participation
rates. Thus, of the drop in labor force participation rates from 87% to 13% between ages 55
and 70, only 10% can be attributed to declining health. Moreover, the ages at which hours
and labor force participation rates decline most rapidly coincides with those ages at which
wages decline and at which there are large pension and Social Security work disincentives.
For example, labor force participation drops 9 percentage points (or 13 percent) at age 62
and 7 percentage points (or 18 percent) at age 65.35 Therefore, it seems that wages, pensions,
and Social Security potentially play a strong role in determining the age of retirement.
The estimated e ect of health on wages, hours-worked, and labor force participation
rates is average for the literature (see Currie and Madrian (1999)). As with most studies,
I nd a statistically signi cant e ect of health. However, this is the rst study to predict
what fraction of the life cycle variation in wages, hours-worked, and labor force participation
rates is explained by health status. The above analysis shows that the amount of explained
Blau (1994) nds an even larger decline in labor force participation at age 65 using data from the Retirement History Survey.
35
28
variation is small.
Finally, the bottom panel of Figure 2 shows both mean and median assets over the life
cycle. 36 Note that young people do save. A certainty life cycle model with typical parameters
predicts that people dissave when young since wage levels are very low when young. Therefore,
the life cycle asset pro le is evidence against the standard certainty-equivalent life cycle model.
However, it is consistent with a model in which young people save in order to generate a bu er
stock of assets for insurance against bad wage shocks when old (Gourinchas and Parker, 2002;
Cagetti, 2002).
5.3 Initial Distributions
To generate the simulated initial joint distribution of assets, wages, AIME37 , and health
status, I take random draws from the empirical joint distribution of assets, wages, AIME,
and health status for individuals aged 29-31.38 I adjust the mean of log wages for healthy
and unhealthy individuals to match the estimated life cycle xed e ects pro le for wages.
Average assets at age 30 are equal to $ 42,100 and are highly correlated with wages.
5.4 Preference Parameter Estimates
Table 2 presents estimates of the parameters in the utility function for males, ages 30 95. Because relatively little is known about the extent to which tied wage-hours o ers and
selection in the wage equation may a ect parameter estimates, Table 2 presents parameter
estimates given di erent assumptions about tied wage-hours o ers selection. Because of a
Note that median assets are not much lower than mean assets. In Cagetti (2002), median assets are much
lower than mean assets. His median asset pro les are similar to mine, but his mean asset pro le implies a
higher level of assets than mine. The di erence arises because I topcode assets at $1,000,000, whereas he does
not. Topcoding has a much larger e ect on mean assets than median assets. Gourinchas and Parker's (2002)
simulated asset pro le is very di erent from those of Cagetti or myself. Their simulated asset pro le implies
that assets are almost zero until age 45. This seems to be the result of diÆculties measuring savings using
income and consumption data.
I assume all individuals enter the labor force at age 25 and work 2000 hours per year at the age 30 wage
to impute initial AIME.
This initial variance in wages will partially control for educational status. Since wages are highly persistent,
those who have higher wages in the rst period will, on average, have higher wages in the nal period than
individuals who had low wages in the initial period. Given this, the model captures the fact that college
graduates have higher wages than non-graduates.
36
37
38
29
lack of data on older individuals, I assume that individuals do not work after age 70 and
match moments only up to age 70.
Speci
Parameter and De nition
(1)
(2)
consumption weight
.578 (.003) .602 (.003)
coeÆcient of relative risk aversion, utility 3.34 (.07) 3.78 (.07)
time discount factor
.992 (.002) .985 (.002)
L leisure endowment
4466 (30) 4889 (32)
hours of leisure lost, bad health
318 (9)
191 (7)
P xed cost of work, in hours
1313 (14) 1292 (15)
B bequest weight
1.69 (.05) 2.58 (.07)
2 statistic: (233 degrees of freedom)
856
880
h;W (40) Labor supply elasticity, age 40
.37
.37
h;W (60) Labor supply elasticity, age 60
1.24
1.33
Reservation hours level, age 62
885
916
CoeÆcient of relative risk aversion
2.35
2.68
Standard errors in parentheses
Speci cations described below:
(1) Does not account for selection or tied wage-hours o ers
(2) Accounts for selection but not tied wage-hours o ers
(3) Accounts for tied wage-hours o ers but not selection
(4) Accounts for selection and tied wage-hours o ers
Table 2:
cation
(3)
.533 (.003)
3.19 (.05)
.981 (.001)
3900 (24)
196 (8)
335 (7)
1.70 (.04)
830
.35
1.10
1072
2.17
(4)
.615 (.004)
7.69 (.15)
1.04 (.004)
3399 (28)
202 (6)
240 (6)
.037 (.001)
1036
.19
1.04
1051
5.11
preference parameter estimates
One of the objects of interest in this paper is an individual's willingness to intertemporally
substitute his work-hours. At age 40, the elasticity of simulated average hours-worked given
an anticipated transitory change in the wage is .19-.37, depending upon the speci cation.39
This labor supply elasticity increases with age. At age 60, the elasticity of simulated hoursworked given an anticipated transitory change in the wage is 1.04-1.33. This increase is
due to the xed cost of work generating volatility on the participation margin. This xed
cost, P ; varies between 240 and 1313 hours, depending on the speci cation. Wage changes
cause relatively small hours changes for workers at both age 40 and age 60. However, the
substitutability of labor supply at the participation margin rises with age. By age 60, many
This calculation was made by changing the wage by 20% for all simulated individuals of a given age,
then computing the di erence in total hours-worked at that age. This causes a wealth e ect, making the
elasticity calculated herein smaller than the Frisch labor supply elasticity. Assuming certainty and Hinterior
conditions, the Frisch elasticity of leisure is
and the Frisch elasticity of labor supply is L Htt P
: However, one of the advantages of the dynamic programming approach is that it is not necessary
to assume certainty or interior conditions.
39
(1
1
)(1
(1
)
1
)(1
)
1
1
30
workers are close to indi erent between working and not working. Small changes in the wage
cause large changes in the participation rate.
The xed cost of work generates a reservation number of work-hours. Individuals will
either work more than this many hours or will not work at all. The reservation number
of work-hours depends on assets, wages, health status and AIME. At age 62, for example,
individuals never choose to work fewer than 885 hours per year in the baseline speci cation.
This is similar to Cogan's (1981) estimate of 1,000 hours per year. The xed cost of work
is identi ed by the life cycle pro le of hours-worked by workers. Note that the hours of
work pro les, presented in Figure 2, do not drop below 1,000 hours per year (or 20 hours
per week) even though labor force participation rates decline to near zero. In the absence
of a xed cost of work, we should expect hours-worked to parallel the decline in labor force
participation. When estimating the model without xed costs of work or tied wage-hours
o ers, hours-worked tends to decline to about 500 hours per year for individuals ages 65-70.
Moreover, the simulated labor supply elasticity rises very little over the life cycle.40
Most of the variation in the wage and labor supply pro les is from individuals ages 55-65.
Figure 2 shows hours-worked and labor force participation rates declining rapidly after age
60, even after the e ect of health on hours has been addressed. This decline in hours coincides
closely with the decline in wages, pension accrual, and the Social Security incentives to retire.
Therefore, the evidence from older individuals indicates that labor supply is responsive to
changes in economic incentives.
Many of the studies that estimate the intertemporal elasticity of substitution (Ghez and
Becker (1975), MaCurdy (1981), and Browning et al. (1985)) obtain identi cation from a
problematic source: the covariation of work-hours and wages of continuously employed young
workers. Young workers work many hours although on average their wage is lower than the
wage of older workers. This indicates that the intertemporal elasticity of substitution is small
For example, in the speci cation that accounts for neither selection nor tied wage-hours o ers, the elasticity
rises from .45 at age 40 to .72 at age 60. In the speci cation that accounts for tied wage wage-hours o ers but
not selection, the elasticity rises from .32 to .77.
40
31
within a certainty-equivalent environment, 41 as hours change very little but wages change a
lot over the life cycle. However, as Benitez-Silva (2000) and Low (2002) point out, younger
workers may work many hours in order to develop enough assets to bu er themselves against
the possibility of low wages when old.42 Therefore, omission of uncertainty will potentially
bias the estimated intertemporal elasticity of substitution downwards (see Domeij and Floden
(2002) for more on this point). Note that this problem is in addition to the problem of
omitting the labor force participation margin. For these two reasons, previous studies have
understated the substitutability of male labor supply.
The coeÆcient of relative risk aversion (or the inverse of the intertemporal elasticity) for
consumption is 2.2 to 5.1 (depending on the speci cation),43 which is similar to previous estimates that rely on di erent methodologies (see Auerbach and Kotliko (1987) and Attanasio
and Weber (1995) for reviews of the estimates). Identi cation of this parameter is similar to
Cagetti (2002) who estimates a bu er stock model of consumption over the life cycle using
asset data. Within this framework, a small estimate of the coeÆcient of relative risk aversion
means that individuals save little given their level of assets and their level of uncertainty.
If they were more risk averse, they would save more in order to bu er themselves against
the risk of bad income shocks in the future. I also obtain identi cation from labor supply,
as precautionary motives can explain why wages co-vary little with hours when young but a
great deal when old. More risk averse individuals work more hours when young in order to
accumulate a bu er stock of assets.
My estimate of the time discount factor, ; is larger than most estimates for three reasons.
The rst two reasons are clear upon inspection of the Euler Equation:
@Ut
@Ct
st+1(1 + r(1
For example, Ghez and Becker (1975) and Browning et al. (1985) estimate the Frisch labor supply elasticity
to be around .3 at all ages.
Both of these papers are similar to mine in that they solve dynamic programming models of labor supply
and consumption decisions under wage uncertainty. However, neither paper considers the importance of xed
costs of work. As a result, neither paper seems to be totally successful in matching the decline in work hours
after age 60.
Ut =@Ct2 Ct
= ( (1 ) 1): Note that this variable is measured
This is measured using the formula @2@U
=@Ct
holding labor supply xed. The coeÆcient of trelative
risk aversion for consumption is poorly de ned when
labor supply is exible.
41
42
43
(
)
32
@Ut+1
t ))Et @C
; where t is the marginal tax rate.44 This equation identi es st+1 (1 + r(1 t ));
t+1
although not the elements of this equation separately. Therefore, a lower value of st+1 or
(1 + r(1
t )) results in a higher value of : The rst reason for my high estimate of
is
that most studies do not include mortality risk. In my model, individuals discount the future
not by the discount rate ; but by the discount factor multiplied by the survivor function
st+1 : Since the survivor function is necessarily less than one, omitting mortality risk will
bias downwards.45 Second, the post-tax rate of return is smaller than the pre-tax rate
of return. Therefore, omission of taxes should also bias
downwards. Third, the life cycle
pro le of hours shows that young individuals work many hours even though their wage, on
average, is low. This is equivalent to stating that young people buy relatively little leisure,
even though the price of leisure (their wage) is low. Between ages 35 and 60, people buy
more leisure (i.e., work fewer hours) as they age even though their price of leisure (or wage)
increases. Therefore, life cycle labor supply pro les provide evidence that individuals are
patient. Ghez and Becker (1975), Heckman and MaCurdy (1980) and MaCurdy (1981) also
nd that (1 + r) > 1 when using life cycle labor supply data.46
The bequest parameter B varies a great deal across speci cations. However, the marginal
propensity to consume out of wealth in the nal period{which is a nonlinear function of
B ; ; ; ; and K {is very stable across speci cations. For low income individuals, the marginal
propensity to consume is 1. For high income individuals, the marginal propensity to consume
is between .025 to .045, depending on speci cation.
Note that this is not exactly correct when individuals value bequests. Also note that the Euler Equation
holds with equality when assets are positive.
Quantitatively, this e ect is small. When I estimated parameters assuming that all individuals survive to
age 65, the estimated value of fell by less than .3 percentage points.
All of these papers ignore taxes and mortality.
44
45
46
33
Annual Hours Worked, Simulations Versus Data
-- · -...
- · -...
--
healthy,
healthy,
healthy,
healthy,
Annual Hours Worked, Simulations Versus Data
data
data plus 2 s.e.s
data minus 2 s.e.s
simulations
Labor Force Participation Rote, Simulations Versus Data
Labor Force Participation Rote, Simulations Versus Dote
-:: -: :-.: : . .: : : =--- -:- . ---
e'
'
""
"
"''
e'
-healthy, data
- · -... healthy, data plus 2 s.e.s
healthy, data minus 2 s.e.s
- - healthy, simulations
'
Mean Assets, Data Versus Simulations,
1967 Dollars
""
"
""
Median Assets, Data Versus Simulations,
1967 Dollars
plus 2
- .. -
..
--
~30
data
.-ninus
si.-nuloHons
s.e.s
2
s.e.s
o\,~.--7c---oc--ce---Oc--~~=ec=cco=c=~=c~=c~.
c__________________________'c'c"______________________~
Sirnulotsd
Consumption,
''"
1987
Dollors
-----
'
'
simulations
Ags
Figure 3:
34
Simulated Profiles
Versus True Profiles
Figure 3 displays both the PSID pro les and the simulated life cycle pro les of the decision variables. It also displays 95% con dence intervals. Simulated pro les appear generally
consistent with the data, although a 2 overidenti cation test rejects the model because the
simulated pro les frequently lie outside of the con dence intervals. There are some di erences between the simulations and data that are worthy of mention. The model substantially
overpredicts labor force participation rates for unhealthy individuals. This is potential evidence of one of two things. First, that the coarse discretization of health into good and bad
is inadequate. It could be that bad health has only a small e ect on the labor supply of
some people and makes others completely unable to work. Alternatively, disability insurance
provides bene ts to those in bad health, but only if those people earn very little over the
course of the year.47 Therefore, disability insurance provides income to those who drop out
of the labor market but not to those who work part-time.
There are two reasons for the small standard errors in Table 2. First, the standard error
formulae rely on the assumption that the GMM criterion function is quadratic near the
minimum of the function. This is true in the case of a linear model, but may be a poor
approximation in the case of a non-linear model.48 For example, Gustman and Steinmeier
(1986), Palumbo (1999), Cagetti (2002), and Gourinchas and Parker (2002) all nd extremely
small standard errors using smaller data sets than the one I use. Moreover, those studies have
less variation in their data. For example, Gustman and Steinmeier (1986) only match labor
supply paths, Cagetti (2002) only matches asset pro les, whereas this study matches both
labor supply and asset pro les. Second, small standard errors may result from the assumption
that the rst stage parameter estimates of are measured with no error. Gourinchas and
Parker (2002) suggest a method that allows one to incorporate the variance of the rst stage
Individuals are only eligible for disability bene ts if their income is very low.
To address this problem, I tried an alternative technique to obtain a measure of the precision of the
estimates. I adjusted upwards by 5% and re-estimated the other parameters. A test of the unrestricted
model versus the restricted model resulted in only a narrow rejection of the restricted model (the di erence
in the statistics was 5.1, with a critical 5% value of 3.8. Also note the di erences in overidenti cation test
statistics between column 1 of Table 2 and column 1 of Table 5. When B = 0; the test statistic rises from
856 to 968. This shows that the hypothesis of B = 0 can be rejected at almost any level. This tends to show
that while the standard errors are being underestimated, the model is sharply identi ed.
47
48
2
2
35
parameter estimates into the second stage standard errors.
Note, however, that the pro les for both the decisions and beliefs are precisely estimated
as shown by their smooth appearance. This arises from the extremely large sample size used
in the analysis. Moreover, using data on labor force participation rates greatly increases
the variation in the data. For example, Heckman and MaCurdy (1980) also obtain small
standard errors in their xed-e ects Tobit speci cation for female labor supply. Therefore,
it is unsurprising that standard errors are smaller than other analyses using PSID data on
continuously employed male workers (e.g. MaCurdy (1981)).
5.5 The E ects of Selection and Tied Wage-Hours O ers on Wages
Table 2 shows preference parameter estimates with and without controls for the e ects
of selection and tied wage-hours o ers. This section describes how accounting for selection
and tied wage-hours o ers a ects parameter estimates, as well as the interpretation of the
estimates. Figure 1 shows that when using xed-e ects, the life cycle pro le for wages declines
rapidly after age 60. However, there are two reasons to suspect that the xed e ects estimates
do not represent the true productivity decline after age 60. First, as discussed in Section 3.3,
I am likely overestimating wage growth because I am using data only on the wage growth of
workers. In other words, the average wage decline for all individuals age 60+ is even sharper
than the decline shown in Figure 1. Using the methodology described in Section 3.3, I nd
that true wages are 7% lower at age 62 and are 11% lower at age 65 than what is shown in
Figure 1. Moreover, failure to account for selection leads to a 2% underestimate of the e ect
of health on wages.
In contrast, failure to account for tied wage-hours o ers may lead to a downward bias in
productivity growth after age 60. Aaronson and French (2002) and Gustman and Steinmeier
(1986) present evidence that the drop in wages after age 60 may result from the drop in
work-hours after age 60. The estimates presented herein assume that part-time (1000 hours
per year) workers are paid 25% less per hour than full-time (2000 hours per year) workers,49
49
In other words, I set = :415; which is at the high end of the estimates in the literature.
36
resulting in a productivity pro le displaying almost no decline after age 60.
Table 2 shows that the estimated xed cost of work is very sensitive to whether the wage
depends on hours-worked. Both xed costs of work and tied wage-hours o ers are potential
explanations for why hours-worked by workers do not drop below 1,000 hours. Tied wagehours o ers imply that individuals will not work a small number of hours per year, even
if the xed cost of work is small. Because of the low wages paid to part-time workers, a
small number of hours-worked results in very little labor income, making part-time work
undesirable. However, if the wage does not depend on hours-worked, a large xed cost of
work is necessary to explain why average hours-worked does not decline below 1,000 hours
per year. When estimating preferences, the model ts the data about equally well with and
without tied wage-hours o ers; that is, the data cannot distinguish whether it is tied wage
hours o ers or large xed costs of work that result in hours-worked not declining below 1,000
hours per year.
5.6 What Causes the High Job Exit Rates at Age 62?
Figure 3 shows that the model is able to replicate the high job exit rates at age 62 that
are seen in the data. There are several potential reasons for the high job exit rates at age
62: the rapid decline in pension accrual at age 62, actuarial unfairness of the Social Security
system, and liquidity constraints. This section discusses the relative importance of these
three reasons.
One important modeling decision is when to set the pension eligibility age. Because
pension income is taxed and taxation is progressive, there is a jump in an individual's marginal
tax rate if he continues to work and begins receiving pensions. This is an important labor
supply disincentive. As discussed in Section 2.3, age 62 is the median normal retirement age
for pensions. As a result, I assume that all individuals begin drawing pension bene ts at age
62. The fact that many people would be pushed into higher tax brackets if they continued
to work seems to cause about half the decline in labor supply at age 62. When I either
make taxes proportional or change the pension eligibility age, about half the decline in labor
37
supply at age 62 disappears. This result should be taken with a great deal of caution because
pension eligibility should be a choice variable and there is a great deal of heterogeneity in the
normal pension retirement age.50 Nevertheless, the correlation between labor supply and the
pension eligibility age shows the importance of considering the tax implications of pensions.
The model of pension accrual allows for discontinuous jumps in pension accrual at ages
61,62,63,64, and 65. As Gustman and Steinmeier (1986, 1999) and Stock and Wise (1990)
point out, discontinuities in pension accrual are potential explanations for the high job exit
rates at ages 62 and 65. When I force pension accrual to be smooth and re-simulate the
model, the age 62 downturn in labor force participation rates is 25% smaller. Therefore,
together the tax and accrual aspects of pensions explain most of the decline in labor supply
at age 62.
Next consider the actuarial unfairness of the Social Security system. Whether or not to
apply for bene ts at age 62 and face the Social Security earnings test depends largely upon
the assumed rate of interest. Given a 4% pre-tax interest rate, Social Security bene t accrual
is slightly negative (and is thus actuarially unfair) for individuals who face low marginal taxes
at age 62.51 However, even after using a 3% pre-tax rate of return, making Social Security
bene t accrual positive for everyone between ages 62 and 65, results in a small change in
labor force participation rates. In other words, actuarial unfairness explains only a small
part of the decline in participation rates at age 62.
Liquidity constraints are another potential explanation for the high exit rates at age 62
(Kahn (1988), Rust and Phelan (1997)). Many individuals potentially wish to borrow against
pension and Social Security bene ts in order to nance retirement before age 62, but because
Social Security wealth is illiquid, they are unable to do so. In order for liquidity constraints to
a ect consumption and labor supply decisions, future illiquid income must be high vis a vis
current income (Deaton (1991)), so that consumption (and thus leisure) will rise when income
When I set the pension eligibility age to either 55 or 65 and re-estimate the model, the model underpredicts
job exit rates at age 62. Nevertheless, the model still matches the overall decline in job exit rates rather well.
Moreover, preference parameter estimates are relatively unchanged.
Given the survivor probabilities I am using, the expected present value of bene ts should be equal at ages
62 and 63 when the post-tax rate of return is 3.0%
50
51
38
eventually rises. In order to investigate the importance of liquidity constraints, I construct
62 +ss62
the following measure of the replacement rate: pb2000
W62 : The numerator of this expression is
pension and Social Security income after age 62 if the individual applies for bene ts at age 62.
The denominator of this expression is a measure of labor income when working 2000 hours per
year. Therefore, this ratio measures the fraction of current labor income that Social Security
and pensions replace. If this ratio is equal to one at age 62, an individual can leave the
labor market at age 62 with no assets and have no decline in consumption upon retirement.
Alternatively, if this ratio is close to one, then an individual may optimally choose to have
zero assets at retirement age.
By Replacement Rate Quintile
0%-20%
Quintile
20%-40% 40%-60% 60%-80% 80%-100%
Mean Replacement Rate (Simulated) 45.3 %
61.9 %
75.1 %
92.0 %
129. %
Mean Assets (simulated)
$ 260,548 $ 200,669 $ 184,226 $ 164,863 $ 141,984
Mean Hours Decline (Simulated)*
5.72 %
10.3 %
28.8 %
53.4 %
66.1 %
By Asset Quintile
Mean Assets (Simulated)
$ 31,294 $ 88,973 $ 154,631 $ 246,243 $ 431,148
Mean Hours Decline (Simulated)*
14.5 %
13.7 %
25.4 %
37.0 %
30.2 %
Mean Assets (Data)
$15,706
$61,193
$121,096 $242,493 $521,420
*Mean hours decline is the mean hours decline between ages 61 and 62
Table 3:
The Distribution of Replacement Rates and Assets, Age 62
The top panel of Table 3 shows quintiles of the replacement rate. Note that even though
all individuals in the model face the same Social Security and pension bene t rules,52 heterogeneity in wage and labor supply histories creates a large amount of heterogeneity in the
replacement rate. Even though the median replacement rate is less than one, the average
replacement rate for those in the top quintile of the replacement rate distribution is 129%.
As a result, many simulated individuals choose to have close to zero wealth at age 62. Note
that those with high replacement rates have lower assets, on average. The evidence in Table
3 shows that having low asset levels near retirement is an optimal response to having a high
replacement rate. The lower panel of Table 3 also shows quintiles of the asset distribution
Coile and Gruber (2000) nd a large amount of heterogeneity in pension accrual rates. Therefore, I am
most likely underestimating the amount of heterogeneity in my illiquidity measure.
52
39
by both the model and by the PSID data. Note that even though I am not matching the
distribution of assets, the distribution of assets implied by the model ts the data quite well.
Nevertheless, I nd only a very small e ect of liquidity constraints on labor supply. The
mean hours decline at age 62 varies much more by replacement rate quintile than by asset
quintile.53 Moreover, those with the lowest assets have smaller hours declines at age 62 than
those with high assets. Therefore, it seems that the individuals who leave the labor force at
age 62 are those with high replacement rates and high assets. In other words, those who face
the largest jump in marginal tax rates at age 62 are the individuals who drop out of the labor
force at age 62. Moreover, almost no individuals have assets equal to zero in either the data
or the simulations at age 62. Thus, most individuals would be able to a ord one year out of
the labor market at age 61.
Figure 4:
The importance of Borrowing Constraints
Another way of testing the importance of borrowing constraints is to allow people to
borrow against their Social Security wealth (i.e. the present value of future Social Security
income).54 Figure 4 shows average hours-worked and asset levels both when the individual
can borrow against future Social Security bene ts and when the individual cannot borrow
This is measured as the change in the total number of work-hours between ages 61 and 62 for all individuals
in that quintile.
Appendix C gives formulas to compute Social Security wealth from the Social Security annuity stream.
53
54
40
against future Social Security bene ts.55 Note that borrowing constraints a ect asset growth
a great deal. It also has some e ect on hours-worked, although most of the e ect is when
young.
In sum, liquidity constraints and the actuarial unfairness of the Social Security system
explain little of the decline in work hours at age 62. Taxes and pension accrual appear to be
the driving factors.
5.7 Consumption Over the Life Cycle
Although I do not match the life cycle consumption pro le to the data, the model generates an implied life cycle consumption pro le. The bottom panel of Figure 3 shows the
geometric mean of consumption at each age. The consumption pro le displays a pronounced
hump shape. There are two reasons for the hump shape to the consumption pro le. First,
the combination of uncertainty and borrowing constraints implies that young consumers
will save more than what they would have done in the absence of borrowing constraints.56
Because their wages and thus incomes are low, their consumption must be low. Second, nonseparabilities between consumption and leisure imply that consumption tracks work hours
over the life cycle.57
This pro le can be compared to the life cycle consumption pro les of Attanasio et al.
(1999) or Gourinchas and Parker (2002). Both papers account explicitly for cohort e ects and
family size, attempt to account for time e ects, and use Consumer Expenditure Survey data.
Unsurprisingly, both papers yield similar life cycle pro les. In both studies, consumption
grows approximately 15% between age 30 and age 40, peaks around age 40, then declines 30%
by age 65.58 In contrast, my consumption pro le shows consumption rising 44% between ages
However, in both situations the individual faces uncertainty. Low (2002) shows that uncertainty may have
important e ects on life cycle labor supply.
Recall from Figure 4 that relaxing borrowing constraints has a large e ect on asset accumulation.
Identi cation of non-separabilities between consumption and leisure is tenuous in this model because
consumption data is not used. Recall that leisure and consumption are substitutes if > 1: Identi cation
of this parameter has already been discussed. Therefore, the results presented herein should not be taken as
strong evidence of non-separabilities between consumption and leisure. Instead, the results should be taken
as evidence that this preference speci cation is consistent with the evidence from other studies.
One omission of my model is family size. Although I attempt to account for family size in the rst
55
56
57
58
41
30 and 55, then declining 22% between 55 and 65. Therefore, my simulated pro le peaks at
a later age than pro les estimated using Consumer Expenditure Survey data. However, both
Attanasio et al. (1999) or Gourinchas and Parker (2002) likely underpredict consumption
of older individuals because both papers omit data on medical expenses and housing. Both
goods tend to be consumed in greater quantities later in life.59
Nevertheless, my consumption pro le provides some evidence that my baseline speci cation implies that households are \too patient". This may cause me to nd no evidence of
liquidity constraints even though they are important in the data.60 Simulated individuals
wish to save for higher consumption in the future, leading to asset levels at age 62 that are
higher than those seen in the data. Section 6.1 investigates preference speci cations with
lower discount factors.
One interesting feature of the model is the sharp decline in consumption near retirement
age. Below I show the relative importance of non-separabilities versus shocks in explaining
the consumption decline at retirement. To do this, I estimate the following model using
simulated data:
ln Cit = st + Pit + uit
(23)
where uit is a residual. Previous studies using OLS have found that is positive, indicating
that consumption falls upon exit from the labor force. The OLS estimate may be a biased
estimate of the consumption response to an anticipated exit from the labor force because
a negative wage shock can reduce both labor supply and consumption. In other words,
stage estimating equations, my approach is not fully consistent with the formal model. Attanasio and Weber
(1995) and Attanasio, Banks, Meghir and Weber (1999) emphasize the importance of family size on life cycle
consumption. However, previous versions of Gourinchas and Parker (2002) show that accounting for family
size only slightly changes the shape of the life cycle consumption pro le. For example, their estimates imply
that consumption rises 25% between 30 and 45 when not accounting for family size and 18% when accounting
for family size. Attanasio, Banks, Meghir and Weber (1999) seem to nd slightly larger e ects of family size,
but still nd a hump shaped pro le for life cycle consumption.
See Gohkale et al. (1996) for life cycle pro les that include housing and medical consumption. Properly
accounting for medical expenses and housing would therefore produce a peak in life cycle consumption later
than 40, although perhaps before 55.
Table 3 shows that I overpredict assets for those in the bottom quantile of the asset distribution.
59
60
42
the OLS estimate may just be capturing the fact that the expected present value of future
resources tends to decline when people exit the labor force. To check on the size of the bias,
I estimate equation (23) using both OLS and IVs. Similar to the approach used by Banks
et al. (1998), I use age-average values of all variables in equation (23), and instrument for
age-average values of Pit using age-average values of Pit 2 : Using this approach I obtain
an estimate of = :34; much larger than the Banks et al. (1998) estimate of .26. I also use
dummy variables equal to one if the individual is older than 62 and 65 as instruments. This
approach produces an estimate of = :36: The OLS estimate of is .37. Therefore, both
OLS and instrumental variables produce similar results. This shows that OLS estimates are
not severely biased and that shocks do not likely explain the consumption fall at retirement.61
6
Experiments
Policy makers are interested in how Social Security generosity, the early and normal Social
Security retirement age, and the Social Security earnings test a ect labor supply. To answer
these questions, I conduct four experiments. Table 4 gives accounting statistics for each of
these experiments.
The top row of Table 4 displays results from simulations under the 1987 policy environment. The second row displays results where Social Security bene ts are reduced by 20%.
Reducing bene ts has two labor supply e ects, both of which should increase hours-worked
after age 62. First, the loss of Social Security bene ts causes a loss of lifetime wealth. This
results in individuals working more hours throughout their lives, as individuals consume less
leisure given the loss of wealth. Second, reducing Social Security bene ts also e ectively
reduces the Social Security earnings test and the high marginal tax rates of the earnings
test.62 Therefore, the substitution e ect associated with a bene t cut causes individuals to
Moreover, they are similar to what the model would predict assuming certainty and interior conditions.
Assuming certainty
and interior conditions, a 1 percent change in leisure brought about by a wage change will
percent change in consumption. Given the parameter estimates in column 1 of table 2,
result in a
cutting work hours from 1,500 hours per year to 0 hours leads to consumption dropping 29%. This is again
similar to the OLS and IV estimates.
Note that if an individual receives no Social Security bene ts, there are no Social Security bene ts to be
61
(1
(1
)(
)
1)
1
62
43
work more hours when eligible for Social Security bene ts and fewer hours at younger ages.
The second row of Table 4 shows that reducing bene ts causes individuals to work more
hours throughout their lives and thus increase their assets in order to o set reduced bene ts.
To understand the magnitude of these e ects, note that the present value of Social Security
bene ts at age 62 is equal is about $132,000, on average. Cutting bene ts 20% reduces the
present value of Social Security wealth by $26,000. Due to both reduced consumption and
increased work-hours when younger than 62, age 62 asset levels are $9,800 greater when
bene ts are reduced. About 32 of this e ect is through reduced consumption, the other 31
from increased labor supply. This highlights the importance of forward-looking behavior
when considering e ects of changing the Social Security rules. Nevertheless, most of the
e ects are seen after age 62. Increased years in the labor market after age 62 replace $5,500
of the lost income. The reason for this is that most individuals are still working at age 62,
and most of the life cycle variability in hours is at the participation margin. Therefore, the
substitutability of labor supply is high after age 62. Reduced consumption when old and
reduced bequests account for the remaining lost bene ts. These labor supply e ects are
similar to those of Burtless (1986) and Krueger and Pischke (1992) and are fairly average for
the literature. A more extreme experiment of eliminating Social Security bene ts results in
an increase of average years in the labor force between age 30 and 70 to 33.71 years.
In order to check whether substitution e ects or wealth e ects drive my results, I reduce
the present discounted value of taxes paid over the life cycle by an amount equal to the
present discounted value of reduced bene ts.63 Note that this is roughly similar to eliminating
reduced by the earnings test.
This was done assuming that population growth is gp = 1%, each birthyear cohort has annual income that
is gw = 2% above the previous birthyear cohort, and mortality rates are the same for all cohorts. In this case
the net cost of the Social Security system is
63
T X
N
X
(1 + gp + gw ) T
t=30 n=1
(
t)
S30;t ssn;t
[(1
ss ) minfWn;t Hn;t ; 43; 000g] ;
(24)
where N is the number of simulations and ss is the Social Security tax. The net cost of the Social Security
system is zero when ss = :065 and ssn;t comes from the 1987 bene t formulas. This is greater than the
employee OASDI tax of .052 but less than the employer and employee contribution of .104. The net cost
is also zero when ss = 0 and ssn;t = 0: Therefore, I reduce taxes by :065 20% = :013 up to the OASDI
maximum of $43,000 when reducing bene ts 20%.
44
wealth e ects. Nevertheless, the substitution e ects of reducing bene ts remains. Results are
displayed on the third row of Table 4. Upon reducing taxes, hours-worked after age 62 are
still very high. This experiment highlights the importance of the substitution e ect generated
by the Social Security work disincentives for individuals age 62 and older.
One potential reform to the Social Security system is to shift the early retirement age
from 62 to 63. Recall that bene t recomputation formulas almost fully replace bene ts lost
through the earnings test at age 62. Therefore, if borrowing constraints do not bind, there
should be little if any work disincentive imposed by Social Security at age 62 and thus there
should be little if any e ect of shifting the Social Security early retirement age to 63. Recall
that borrowing constraints bind for very few individuals at age 62. As a result, the fourth
row of Table 4 shows that any e ect of this policy would be minor. Simulations from the
model indicate that shifting the early Social Security retirement age to 63 would leave years
in the labor force unchanged.
years
hours
PDV of PDV of
assets
worked worked labor
consumption at age
per year income
62
With borrowing constraints
1987 policies
32.60
2097
$ 1781 $ 1583
reduce bene ts
32.83
2099
$ 1789 $ 1569
reduce bene ts, reduce taxes
33.00
2115
$ 1803 $ 1586
shift early retirement age to 63 32.62
2096
$ 1781 $ 1584
eliminate earnings test, age 65+ 33.62
2085
$ 1799 $ 1594
Without borrowing constraints
1987 policies
32.39
2067
$ 1764 $ 1603
reduce bene ts
32.58
2063
$ 1770 $ 1587
reduce bene ts, reduce taxes
32.68
2078
$ 1781 $ 1602
shift early retirement age to 63 32.39
2067
$ 1764 $ 1603
eliminate earnings test, age 65+ 33.46
2063
$ 1784 $ 1616
PDV stands for present discounted value
Consumption, labor income, and assets are measured in thousands
Table 4:
$ 190
$ 200
$ 203
$ 190
$ 188
$ 158
$ 168
$ 170
$ 158
$ 154
Policy Experiments
Finally, I eliminate the Social Security earnings test for individuals ages 65 and older.64
Note, however, that all other program parameters are being held at the 1987 values. Over the past 15
years there have been important changes to the earnings test for individuals younger than 65 and to bene t
recomputation formulas for individuals aged 65 and older.
64
45
Figure 5:
The Effect of Removing the Earnings Test, Age 65+
This has large e ects. As shown in the fth row of Table 4 and also Figure 5, hours-worked
after age 65 jumps. Years in the labor force rises from 32.60 to 33.62 although average
hours-worked by workers is largely unchanged. This increase in work-hours is completely
a substitution e ect, given that eliminating the earnings test will increase lifetime wealth.
The wealth e ect from increased post-tax wages will lead individuals to consume more of
everything, including leisure. Therefore, eliminating the wealth e ects from this experiment
would lead to an even greater labor supply response.
This nal experiment provides the model with a strong out of sample test. The earnings
test was in fact abolished for individuals older than 64 in 2000. Therefore, the model predicts
that labor force participation rates for individuals should rise sharply over the coming years.65
The results in Table 4 highlights the importance of considering labor force participation
when conducting policy experiments. Most models (Auerbach and Kotliko (1987), for example) focus on hours-worked by workers and ignore the labor force participation decision.
This model suggests, however, that the dominant margin of labor supply substitutability for
men is at the labor force participation decision.
The bottom rows of Table 4 repeat the top rows, but assume that individuals can borrow
against future Social Security bene ts. Note that relaxing borrowing constraints does have
Of course, other incentives have been changing over time (Anderson et al. (1999)), so the prediction is
somewhat ambiguous.
65
46
important e ects on labor supply and asset accumulation. Asset levels are much lower at age
62 when borrowing constraints are relaxed. Hours worked per year are also lower. However,
note from Figure 4 that this re ects changes in labor supply early in life, but not near
retirement age. Moreover, note that the e ects of changing the Social Security rules on labor
supply is similar to the e ects when borrowing constraints are enforced. Elimination of the
earnings test for those older than 65 has a very large e ect on life cycle labor supply, while
shifting the early retirement age to 63 has a very small e ect. Therefore, the presence of
borrowing constraints does not a ect the predicted response of labor supply to changes in
the Social Security rules.
6.1 Sensitivity of Results to Changes in Preference Parameters and Speci cation
In this section I present evidence on the sensitivity of results to changes in the preference
parameters. Of the estimated parameters, the values of ; B , and are potentially the
most controversial, as is the source of identi cation of these parameters. Moreover, it seems
worthwhile to test the robustness of the results to alternative preference speci cations. In
this section, I evaluate whether the results in Table 4 are sensitive to changes in preference
parameters and the utility function.66 Preference parameter estimates are shown in Table 5.
One cause for concern when constructing asset pro les is that the year e ects are not well
proxied by the unemployment rate. Given that the asset data are from 1984-1994, during
which there was a rapid run up in the stock market, I may be overstating asset growth.
Therefore, I may be overestimating B and :
In order to understand the importance of this problem, I reduced asset growth 1% during
each year of the sample period and re-estimated preference parameters. Appendix F shows
that this technique likely understates asset growth over the life cycle relative to what we
would have anticipated in the absence of a run-up in the stock market. When using this
asset pro le, the estimate of B falls from 1.69 to .85. None of the other parameters changes
66
However, the importance of unobserved heterogeneity in preferences and pension accrual is not considered.
47
noticeably. Moreover, the new estimates have only a tiny e ect on the labor supply and
savings responses to changes in the Social Security rules.
However, there are other reasons to suspect that I may be overestimating B and : The
parameter B is identi ed largely o of the shape of the asset pro le, but only for individuals
younger than 70. However, Hurd (1990) nds signi cant declines in assets near the end of
the life cycle whereas the simulated pro les presented herein do not fall for older individuals.
Moreover, I omit medical expense uncertainty. Palumbo (1999) shows that uncertain medical
expenses can partly explain why the elderly run down their wealth slowly. In order to address
these concerns I tried a more extreme set of experiments. In column 1 of Table 5, I set B = 0:
Note that
rises to 1.04 when B = 0: This gives some evidence that a high value of B and
a high value of
are alternative explanations for why assets are high near age 70.
Speci cation
Parameter and De nition
(1)
(2)
(3)
(4)
consumption weight
.589 (.004) .556 (.005) .539 (.001)
coeÆcient of relative risk aversion, utility 5.68 (.07) 9.98 (.16) 6.36 (.08)
time discount factor
1.04 (.002) .95
.95
.987 (.001)
L leisure endowment
5159.0 (31) 5073 (44) 3937 (27) 5280
hours of leisure lost, bad health
559 (8)
429 (7)
175 (12)
153 (4)
P xed cost of work, in hours
1378 (15)
772 (9)
273 (3)
553 (7)
B bequest weight
0
0
0
5.62 (.26)
2 statistic: (193 degrees of freedom)
968
1093
1158
1107
h;W (40) Labor supply elasticity, age 40
.35
.41
.49
.06
h;W (60) Labor supply elasticity, age 60
2.17
.99
.50
.99
Reservation hours level, age 62
885
1226
1059
900
CoeÆcient of relative risk aversion
3.76
6.00
2.08
.566
Standard errors in parentheses
Speci cations described below:
(1) B = 0
(2) B = 0; = :95
(3) B = 0; = :95; intertemporal elasticity of substitution for consumption = (1 1) 1 = :48
(4) Separable preferences: C = :566(:0003); H = 9:82(:01); H = 3:94 1032 (2:28 1030 );
L = 5280 by assumption, and not in speci cation
Table 5:
preference parameter estimates
= 1:04 is much higher than most estimates in the literature. In order to
assess the sensitivity of my results, I set =.95 and B = 0 in column 2 of Table 5.67 Note
A value of
67
As I pointed out earlier, is identi ed partly by the life cycle labor supply pro les. Estimated pro les
48
that when B = 0 and = :95; the value of rises: because they are less patient, consumers
need to be more risk averse to generate the observed asset pro le.68 This high value of also
generates a low intertemporal elasticity of substitution for labor supply and consumption.69
Because the intertemporal elasticity of substitution for consumption in column 2 is lower
than most estimates,70 I also consider an intertemporal elasticity of substitution for consumption of .48, estimated by Attanasio et al. (1999). Results are in column 3 of Table 5. The
estimates in column 3 produce the lowest asset levels and are thus the most likely to produce
a large e ect of liquidity constraints on life cycle labor supply. Average assets are $10,000
at age 62. In order to assess the importance of liquidity constraints, Table 6 shows the same
experiments as in Table 4, but with the preferences in column 3 of Table 5. The largest
change between the results in Table 6 and 4 is the e ect of changing generosity. Reducing
bene ts now has an even larger e ect on life cycle labor supply. Because asset levels are low
at all ages, reducing bene ts has only small e ects on savings before age 62 and bequests.
Most of the response to the bene t cut is in reduced consumption and leisure after age 62.
Although lower patience factors a ect the labor supply response to Social Security generosity,
they do not a ect the labor supply response to shifting the early retirement age to 63.
One nal speci cation test is to change the utility function so that it is separable in
consumption and leisure. Consider the following utility and bequest functions.
U (Ct ; Ht ; Mt ) =
1
1
C
Ct1
C
+
1
H
H
(L Ht
P Pt I fM = badg)1
H
;
(25)
indicate that individuals cut their work hours (or equivalently, increase their leisure consumption) between
ages 50 and 60, even though the wage and pension incentives (and thus the price of leisure) are at their greatest
near age 60. These facts can only be reconciled by a high value of : However, one could argue that individuals
reduce work hours at these ages because of declining health, and that this decline is not fully captured by the
one simple health measure that I use. Individuals may be impatient, but the disutility of work rises sharply
at these ages.
Figure 4 of Cagetti (2002) shows the relationship between risk aversion and impatience more explicitly.
Note, however, that is also partly identi ed by the labor supply pro les.
Assuming certainty and interior conditions, the consumption Euler Equation is ln Ct = ln( (1+
r)) +
ln(L Ht P Pt I fM = badg):
Moreover, a wide range of values of all seem to give relatively similar criterion functions if other parameters are re-estimated. From a statistical standpoint, one can easily distinguish between di erent values of :
Nevertheless, the pro les they generate look relatively similar.
68
69
(1
70
(1
)(
)
(1
1)
1
49
1
)
1
years
hours
PDV of PDV of
assets
worked worked labor
consumption at age
per year income
62
With borrowing constraints
current policies
36.77
2003
$ 1788
reduce bene ts 20 percent
37.42
2000
$ 1805
reduce bene ts 20 percent, reduce taxes 37.66
2011
$ 1819
shift early retirement age to 63
36.77
2003
$ 1788
eliminate earnings test, age 65+
37.91
2005
$ 1811
PDV is present discounted value
Consumption, labor income, and assets are measured in thousands
Table 6: Policy Experiments, B = 0; = :95; (1 1) 1
b(At ) = B
(At + K )1
1 C
C
:
$ 1840
$ 1828
$ 1840
$ 1840
$ 1858
$ 10
$ 13
$ 14
$ 11
$8
= :48
(26)
Results from this speci cation are in column 4 of Table 5. This utility function does not t
the data as well as the non-separable preference speci cation, although there are no striking
di erences between this preference speci cation and the non-separable one. Nevertheless,
when repeating the experiments in Table 4, shifting forward the early retirement age has
almost no e ect on lifetime labor supply, whereas eliminating the Social Security earnings
test after age 65 increases years in the labor force by 1.4 years.
7
Conclusion
In this paper I estimate a dynamic structural model of labor supply, retirement, and
savings behavior where assets must be non-negative in all periods. When augmented to
include uncertainty over future wages and health status, the model ts the life cycle pro le
of assets rather well. It also does a good job of tting the life cycle pro les of hours worked
and labor force participation rates.
This allows me to assess how the Social Security system a ects life cycle labor supply. Of
central importance is whether Social Security a ects labor supply because (i) Social Security
wealth is illiquid until age 62 and/or (ii) because of the taxation and actuarial unfairness of
the system. I nd that allowing individuals to borrow against future Social Security bene ts
50
would reduce work hours when younger than 40. However, the fact that bene ts are illiquid
until 62 cannot explain the high job exit rates at 62 or 65. Instead, it seems that the taxation
and actuarial unfairness of pensions and Social Security explains the sharp decline in labor
supply at these ages.
The value of this model lies in its ability to predict how labor supply and retirement
patterns of individuals might change in response to changes in the Social Security rules.
Simulations suggest that a 20% drop in Social Security bene ts results in an increase in
labor supply throughout the life cycle. However, the e ect is rather small; individuals would
spend an additional three months in the labor force. In contrast, simulations suggest that the
elimination of the Social Security earnings test for those older than 65 will cause individuals
to delay exit from the labor force by one year, showing the important work disincentives of
the earnings test.
51
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Annual Statistical
Appendix A: Taxes
Individuals pay federal, state, and payroll taxes on income. I compute federal taxes on
income net of state income taxes using the Federal Income Tax tables for \Head of Household"
in 1987 with the standard deduction. I also use income taxes for the fairly representative
state of Rhode Island (22.96% of the Federal Income Tax level). Payroll taxes are 7.15% up
to a maximum of $43,800. Adding up the three taxes generates the following level of post
tax income as a function of labor and asset income:
Pre-tax Income (Y)
0-4440
4440-6940
6940-27440
27440-42440
42440-43800
43800-84440
84440+
Post-Tax Income
.9285Y
4123 + .796(Y-4440)
6113 + .749(Y-6940)
21468 + .6021(Y-27440)
30500 + .5262(Y-42440)
31216 + .5977(Y-43800)
55506 + .5605(Y-84440)
Table 7:
Marginal Tax Rate
.072
.204
.251
.398
.474
.403
.440
After Tax Income
Appendix B: Computation of AIME
54
The Social Security system uses the bene ciary's 35 highest earnings years when computing bene ts. The average monthly earnings over the 35 highest earnings years are called
Average Indexed Monthly Earnings, or AIME. I annualize AIME and compute it using the
following formula for individuals 30-59.
AIMEt+1 = AIMEt + (Wt Ht )=35:
(27)
I assume the individual enters the labor force at age 25. Since AIME is computed using the
35 highest earnings years, AIME increases unambiguously if the individual is younger than 60
and works. If age is 60 or greater AIME can still increase, but only if the individual earns a
great deal that year. The high earnings year will replace a low earnings year when computing
Social Security bene ts.71 Therefore, the formula for individuals 60 and older becomes
AIMEt+1 = AIMEt + maxf0; (Wt Ht AIMEt )=35g:
(28)
Lastly, AIME is capped. In 1987, the base year for the analysis, the maximum AIME level
was $43,800 in 1987 dollars.
AIME is converted into a Primary Insurance Amount (PIA) using the formula
P IAt =
8
>
>
>
>
<
>
>
>
>
:
:9 AIMEt
if AIMEt < $3; 720
$3; 348 + :32 AIMEt if $3; 720 AIMEt < $22; 392
(29)
$9; 695 + :15 AIMEt if AIMEt $22; 392
Social Security bene ts sst depend both upon the age at which the individual rst receives
Social Security bene ts and the Primary Insurance Amount. For example, pre-earnings test
bene ts for a Social Security bene ciary will be equal to PIA if the individual rst receives
bene ts at age 65. For every year before age 65 the individual rst draws bene ts, bene ts
Unfortunately, I assume that the high earnings year replaces an average earnings year, as described in
equation (28).
71
55
are reduced by 6.7% and for every year (up until age 70) that bene t receipt is delayed,
bene ts increase by 3%.72
Appendix C: Pensions
There are two important aspects of pensions for the purpose of this paper. First, pension
wealth is illiquid until a certain age (I assume until age 62). Second, pension accrual rates
are higher for individuals in their 50s than at other ages.
Consider the liquidity aspect rst. Because both Social Security bene ts and pensions
are annuities, I load pension wealth onto PIA. If the individual is age-eligible for pension
bene ts, then pension bene ts, pbt = pb(P IAt ) are:
pbt = 0 1 + 2 P IAt + 3 maxf0; P IAt
5000g
(30)
The parameters 0 ; 1 ; 2 ; 3 are taken from Gustman and Steinmeier (1999). A spline function is used to estimate 1 ; 2 ; 3 : Table 6 of Gustman and Steinmeier (1999)73 shows that the
ratio of pension wealth to Social Security wealth rises rapidly with Social Security bene ts.
Lastly, I pick the scale parameter 0 so that mean pension wealth, described in equation (34)
below, is $78,108 in 1987 pre-tax dollars at age 60, which is meant to coincide with estimates
for the male head of household in Table 5 of Gustman and Steinmeier (1999).
One problem arises from the fact that I treat the decline in Social Security bene ts that
arises from early recipiency of Social Security bene ts, described in Appendix B, as equivalent
to a decline in P IA: The problem is that P IA a ects pension bene ts. However, the model
assumes that Social Security recipiency should not a ect pension bene ts. In order to account
for this, I adjust P IA downwards in response to early Social Security bene t application as
follows. Consider next period's pension and Social Security bene ts (the later object equal to
AIME can be reduced instead of PIA for individuals who rst receive bene ts before age 65. For example,
if an individual begins drawing bene ts at age 62 we can adjust AIME to account for early retirement. We
know that adjusted AIME must result in a PIA that is only 80% of what it would have been had the individual
rst received bene ts at age 65. Using equation (29) it is straightforward to compute adjusted AIME. Age at
application, then, need not be treated as a state variable.
They provide estimates of pension wealth and Social Security wealth as a function lifetime labor income.
I convert these measures into annual bene ts.
72
73
56
P IA) after a reduction in bene ts because of early bene t application, but before the Social
Security earnings test:
pbt+1 + P IAt+1 = pbt + remt P IAt :
(31)
where remt is the fraction of remaining Social Security bene ts (for example, rem64 = :933)
and pbt+1 = pbt : Using equations (30) and (31) the new value of P IAt+1 is
P IAt+1 =
8
>
<
>
:
( 2 +remt)P IAt + 3 maxf0;P IAt 5000g
if P IAt+1 > 5000
1+ 2
5000 3 +( 2 +remt)P IAt + 3 maxf0;P IAt 5000g otherwise:
1+ 2 + 3
(32)
Next consider the fact that pension accrual is greater for individuals in their 50s than
for individuals at other ages. However, the pension accrual bene t formula in (30) does not
imply that pension accrual for individuals in their 50s is much higher than individuals in their
30s. To see the relationship between pension accrual and pension bene ts note that pension
wealth, pwt ; grows according to
pwt+1 =
8
>
>
>
>
<
>
>
>
>
:
(1=st+1 )[(1 + r)pwt + pacct ]
if living at t + 1 and aget < 61
(1=st+1 )[(1 + r)pwt + pacct
pbt ] if living at t + 1 and aget 62
0
(33)
otherwise
where pacct is pension accrual. Since neither pension accrual nor pension interest are taxed,
the appropriate rate of return on pension wealth is the pre-tax one.
I calculate the present value of current pension wealth by assuming that a worker receives no bene ts until age 62. Assuming no further pension accrual, recursively substituting
equation (33) backwards and imposing pwT +1 = 0 reveals that
pwt =
t
pbt ; where
T
1 X
S (k; t)
I fagek 62g;
1 + r k=t (1 + r)k t
t
57
(34)
(35)
and I fagek 62g is equal to one if agek 62 and is equal to zero otherwise.
Pension accrual is the increase in the present value in future bene ts caused by a rise in
future annual bene ts:
pacct = t (pb(P IAt ) pb(P IAt 1 ))
(36)
where pb(P IAt ) is the bene t level given this year's P IA and pb(P IAt 1 ) is the bene t level
given last year's P IA:
Equation (36) overstates pension accrual for individuals in their 30s and understates
pension accrual for individuals for individuals in their 50s. To overcome this problem, I take
a second pension accrual measure, where pension accrual is a function of age and labor income
pacct = pacc (Wt Ht ; aget ) :
pacc (Wt Ht ; aget ) = 0 ( 1 + 2 Wt Ht + 3 max(0; Wt Ht
15; 300)) 4 (aget ) Wt Ht :
(37)
I use a spline function with a kink at $15,300: ( 1 + 2 Wt Ht + 3 max(0; Wt Ht
15; 300))
to estimate the dependence of pension accrual on annual labor income. Table 6 of Gustman
and Steinmeier (1999)74 shows that pension accrual rates roughly triple between individuals
with extremely small incomes and individuals with incomes around $15,300. Above this
level, however, accrual rates are fairly constant. I model the age dependence of accrual rates
4 (aget ) using a weighted average of the de ned bene t, de ned contribution and combined
de ned bene t and de ned contribution pro les in Figure 2 of Gustman et al. (1998).75
Lastly, I pick the scale parameter 0 so that mean pension wealth, described below, is $53,894
in 1987 dollars at age 57, which is meant to coincide with the equation (34) measure of $78,108,
They provide estimates of pension accrual as a function lifetime household labor income. I divide lifetime
labor income by 35 to get an estimate of average annual labor income. Table 5 shows that the male in the
household accumulates, on average, 78% of the household pension wealth.
I adjust their pension accrual pro le by their assumed rate of wage growth so that pension accrual is
measured in rates then smooth their pension accrual pro le using a 20th order polynomial with dummy
variables for age greater than 61, 62, 63,64 and 65. Predicted accrual rates that are negative are set to zero.
74
75
58
but after being taxed at a 31% tax rate. This pension wealth measure is found assuming no
bene ts have been taken so far and solving equation (33) backwards
!
t
X
(1 + r)k
pwt =
pacct k :
S
(
k;
t
)
k=1
(38)
Equation (36) implies a di erent level of pension accrual than equation (37). In order to
overcome this problem the di erence between the two accrual measures is treated as income
added to the asset equation. The asset accumulation equation is then
At+1 = At + Y (rAt + Wt Ht + yst + pbt + "t ; ) + sst Ct
where " = (pacct
(39)
pacct ):
Appendix D: Numerical Methods
This section outlines the methods for computing the decision rules. Speci cally, it outlines
the methods for computing the value function, the methods for integrating the value function
with respect to uncertainty over wages, and the method to nd the optimal consumption and
hours decisions.
The value function is the solution to
(
Vt (Xt ) = max
Ct ;Ht ;Bt
st+1
X
M 2fgood;badg
Z
1
1
1fM = badg)1
Ct (L Ht P Pt
!1
+
)
Vt+1 (Xt+1 )dF (Wt+1 jMt+1 ; Wt ; t)prob(Mt+1 jMt ; t) + (1 st+1 )b(At+1 ) ;
(40)
where dF (:j:; :; :) is the conditional cdf of next period's wages. The individual is uncertain of
future wage and health shocks. The values of Ct ; Ht and Bt that solve (40) are considered
the optimal consumption and hours decisions.
Although the consumption, hours, and participation and bene t application rules have no
closed-form solutions, the rules fully characterize the decisions of the individual. The solution
59
to the worker's problem then consists of a set of consumption fCt (Xt ; ; )g1tT ; work
fHt (Xt ; ; )g1tT and bene t application fBt (Xt ; ; )g1tT rules which solve the value
function (40). A labor force participation rule Pt (Xt ; ; ) is equal to zero if Ht (Xt ; ; ) = 0
and equals one otherwise. Using these decision rules and the asset accumulation equation it
is also possible to solve for next period's asset level fAt+1 (Xt ; ; )g1tT :
The decision rules are solved for recursively, starting at time T and working backwards to
time 1. I compute the value function using value function iteration. At time T; consumption
and hours decisions will be made by maximizing equation (40), where VT +1 = b(AT +1 ):
Consumption and hours decisions are next solved for time T 1; T 2; T 3; :::; 1 by backwards
induction. Using this technique the individual decision rules at time t can be found as
functions of only the state variables at time t:
Since there is no closed form solution to the problem, the state variables are discretized
into a nite number of points on a grid and the value function is evaluated at those points.76
Because variation in assets, AIME and wages is likely to cause larger behavioral responses
at low levels of assets, AIME and wages, the grid is more nely discretized at low levels of
assets, AIME and wages. Since the value function is computed at a nite number of points, I
use linear interpolation within the grid and extrapolation outside of the grid to evaluate the
value function points that were not directly computed.
I integrate the value function with respect to the innovation in the wage using GaussHermite quadrature. Although assets at time t + 1 will be known at time t; wages at time
t + 1 will be a random variable. In practice, I use quadrature of order 5 (Judd, 1998).
I also discretize the consumption and labor supply decisions and use a grid search technique to nd the optimal consumption and hours rules. Because the xed cost of work and
the bene t application decision mean that the value function need not be globally concave,
I cannot use relatively fast hill climbing algorithms. I experimented with the neness of the
In practice, I chose 30 asset states, 10 wage states, and 10 bene t states. The grid for assets and wages
is A 2 [$0; $700;000]; W 2 [$3; $60]: There are two application states, and two health states. This requires
solving the value function at 30 10 10 2 2 = 12;000 di erent points after age 62 when the individual is
eligible to apply for bene ts and 6;000 points when younger than 62.
76
60
grids. The grids described herein seemed to produce reasonable approximations.77 Increasing
the number of grid points seemed to have a small e ect on the computed decision rules.
Figure 6 shows policy functions for both consumption and work hours. These functions are
plotted as a function of assets and wages. These functions are plotted for healthy individuals
who are age 35 and have an AIME of $7,240. For these individuals, 95% have assets between
$25,000 and $217,000 and 95% of these individuals have wages between $8 and $22. Higher
consumption and labor supply functions refer to higher wage levels.
There are a few things worth noting in the gure. First, note that labor supply drops to
zero when assets exceed a certain level. This sharp drop is caused by the xed cost of work.
Second, consumption functions are not concave or even monotonically increasing in assets.
The consumption function is convex in assets above a certain asset level. This is a result
of progressive taxation. Higher asset levels cause higher marginal tax rates. Individuals can
move to lower tax rates by consuming their assets. Third, consumption is sometimes declining in wealth for those with low wage levels. This is caused by non-seperabilities between
preferences for consumption and leisure. Recall that the given the preference speci cation,
consumption and leisure are Frisch substitutes if > 1: Note that the points where consumption falls is the same set of points where labor supply falls. Fourth, note that consumption
is never zero. At all positive wage levels, individuals will work positive hours if their wage
is zero and they have no other form of income. Figure 7 shows policy functions when the
number of asset points is set equal to 100 and all parameters are re-estimated. Note that the
Currently, there are 90 possible values for consumption. I use next period's optimal consumption rule as
an initial guess for this period's optimal consumption rule at each value of X: For most years, I search over
a space that is between 70% and 150% of next period's consumption rule. For years where there will likely
be large changes in the decision rules for a given set of state variables, such as between ages 61 and 62, I
increase the search area to 30%-300% of next period's optimal decision rule. If the new consumption rule is
near the boundary of the search space, the search space is shifted and the consumption rules are re-computed.
To nd the optimal hours decision, I use the marginal rate of substitution between consumption and leisure.
There is a diÆculty in that the marginal rate of transformation between consumption and leisure is not the
wage. Instead, taxes, pensions, and the e ect of current work-hours on Social Security bene ts distort the
relationship. Therefore, I make an initial guess by setting the marginal rate of substitution equal to the wage.
I then try 10 di erent hours choices in the neighborhood of the initial hours guess. Because the xed cost
of work may cause large discontinuous changes in optimal hours-worked (from zero hours-worked and a large
number of hours-worked), I also evaluate the value function at Ht = 0 where the space of consumption choices
is determined by next period's optimal consumption choice when Ht = 0:
77
+1
61
Figure 6:
Policy Functions, 30 asset points
policy functions are not much changed.78
Appendix E: Moment Conditions
In this appendix I describe the GMM minimization procedure where I account for the
three data problems discussed in the text. The rst data problem is that I wish to match
pro les that are uncontaminated by cohort and family size e ects. The second problem is a
When setting the number of asset states to 100, I also re-estimated all parameters. The parameter
estimates were virtually unchanged from the case where there were 30 parameters. The criterion function was
also virtually unchanged.
78
62
Figure 7:
Policy Functions, 100 asset points
selection problem. If individuals who are healthier have a greater preference for work than
unhealthy individuals, then selection of individuals into the healthy hours and participation
moment conditions will not be random. Failure to overcome this problem will lead to an
overestimate of the e ect of health on preferences for work. Healthy workers will work more
hours than unhealthy workers, not because of health but unobserved di erences in preferences
for leisure between healthy and unhealthy individuals. The third data problem is that I use
an unbalanced panel of data. Since not all individuals are seen in all moment conditions
during all time periods, some of the individual level contributions to the moment conditions
63
are \missing". Moreover, because individuals are healthy only with a certain probability,
many of the individual level contributions are \missing" with a certain probability.
I now discuss the rst two problems and their solution in greater detail. For concreteness, consider the moment condition for hours-worked for individuals in good health. Upon
estimating the xed-e ects pro le for hours-worked, I use the estimated parameters for age
and the person-speci c residual from estimation of (17). I use these estimates to generate a
predicted life cycle pro le for hours-worked. This life cycle pro le helps me generate my set
of moment conditions. I wish to set the following moment condition to zero:
E [ln Hit;M =good jbirthyear = 1940; M = good; famsize = 3] ln H~ t;M =good = 0
(41)
where ln H~ M;t is the simulated geometric mean of log hours-worked. In order to generate this
moment condition, I use parameter estimates from equation (17) and make three modi cations to hours-worked, shown in equation (42):
ln Hit;M =good = fi + E [fi jbirthyear = 1940; prob(M = good) = prob(M = goodjage = 50); ageit = 50]
E [fi jbirthyeari ; prob(Mit = good); ageit ] + g ageit + f (famsize = 3) + uit
(42)
Note the three modi cations to the data. First, there is no probability of being in good
or bad health. Instead, individuals are in either good or bad health for certain. Second, it
is not the size of the family that is used but a family size of three. In this way life cycle
family size e ects will not contaminate pro les. Third, I adjust the person speci c e ects
fi : There are two things that I wish to adjust in the person speci c e ects. First, I wish to
control for cohort e ects. Note that an individual's cohort e ect is one component of their
person speci c e ect, as a cohort e ect is an average of the xed e ects of everyone born in
that cohort. I adjust the person speci c e ect so that everyone has the same cohort e ect,
set to birthyear = 1940; which means pro les will be uncontaminated by cohort e ects. The
second aspect of the person speci c e ect that I adjust for is the possible correlation between
64
the person speci c e ect and the health status of the individual. This solves the problem of
selection into a moment condition, as the adjusted hours data in (42) should be uncorrelated
with health.
In order to generate the adjusted hours data in (42) it is necessary to predict the person
speci c e ect fi for a given age, birthyear, and health status. To predict the person speci c
xed e ect I estimate the conditional expectation of fi given birthyear and age interacted
with health status using OLS:
fi = 1 birthyear + 2 prob(Mit = good) ageit + 3 (1 prob(Mit = good)) ageit + it
(43)
where 1 ; 2 ; 3 are parameters to be estimated and birthyear is a full set of birthyear dummies, and ageit denotes a full set of age dummies.
Next I address the problem of having an unbalanced panel and the problem of not knowing
an individual's health status with certainty. If there are I separate individuals in the data
there will be a total of I possible contributions to both the healthy and unhealthy moment
conditions for hours at age t: However, not all individuals are observed working for all possible
time periods. Assume instead that there are It
I individuals observed working at age t:
The idea is to treat a moment contribution as equal to zero if it is missing.
This means that the moment condition for individuals of age t and health state M = good
is generated by
(
It
1X
ln Hit;M =good
I i=1
ln H~ t;M =good
)
prob(Mit = good)
(44)
where ln Hit;M =good is adjusted work-hours described in equation (42). The relative weight
of this moment condition rises as It ; the number of observed workers rises and as the probability that these workers are healthy rises. Note that prob(Mit = good); which determines
selection into the moment condition, might be correlated with the person speci c xed e ect
fi but will not be correlated with its adjusted value fi + E [fi jbirthyear = 1930; prob(M =
65
good) = prob(M = goodjage = 50); ageit = 50] E [fi jbirthyeari ; prob(Mit = good); ageit ] by
construction.
The value of that minimizes the (weighted) distance between the simulated pro les and
estimated pro les for assets, hours, and participation is considered to be the true value of
. De ne the vector of the 5T moment conditions as g~(; ). Assuming WT is an optimal
weighting matrix, the minimized GMM criterion function
I
g~(; )0 WT g~(; )
1+
is distributed asymptotically as Chi-squared with 5T
(45)
7 degrees of freedom if the model is
correctly speci ed. is the ratio of the number of observations to the number of simulated
observations, which tends to zero as the number of simulated observations becomes large. My
estimate of WT is the inverse of the 5T 5T variance covariance matrix of the (adjusted) data.
Pt
That is, WT 1 has a typical element along the diagonal of a variance I1 Ii=1
[ln Hit;M =good
E [ln Hit;M =good ]] prob(Mit = good) 2 and a typical element of a covariance on the o diagonal. When computing the chi-square statistic and the standard errors, the estimated
value of E [ln Hit;M =good ] is replaced with its simulated counterpart.
Under the regularity conditions stated in Pakes and Pollard (1989) and DuÆe and Singleton (1993), the MSM estimator ^ is both consistent and asymptotically normally distributed.
^ converges in distribuDenoting 0 as the true parameter vector, the estimated value of 0 ; ;
tion to
p ^
I ( 0 )
66
N (0; V );
(46)
where V is the variance-covariance matrix of ^ which is estimated by:
^ D^ ) 1
V^ = (1 + )(D^ 0 W
(47)
@ g~
D^ = j=^ :
@
(48)
Appendix F: Adjustments to the Asset Pro le
This appendix describes the adjustments to the asset pro le made in Section 6.1 and shows
that the adjustments likely lead me to understate a household's anticipated asset growth at
each age.
The procedure is as follows. First, I estimate the \excess" rate of return from the run
up in the stock market and housing wealth using procedures described in French and Jones
(2002). Second, I re-estimate the life cycle asset pro le, setting the \excess" rate of return
to zero. Therefore, the pro les can be interpreted as the likely asset pro le that would of
occurred had asset returns been equal to their historical average, and savings rates had been
unchanged. Third, I re-estimate preference parameters in the utility function.
Brie y stated, the French and Jones procedure estimates historical growth rates in asset
prices, then compares the historical rates of growth to those in the sample period. Stock price
growth (net of in ation) was 1.7% higher per year from December 1984-December 1994 than
it was over the 1950-1994 period. However, annual housing price growth was .6% lower from
December 1984-December 1994 than it was over the 1976-1994. I assume growth in all other
assets was as anticipated. Multiplying the excess rates of return by the shares of wealth in
di erent assets, I nd that rates of return during my sample period were perhaps .31% higher
than would have been anticipated. This has a tiny e ect on the results.
Next I conduct a more extreme experiment. I assume that annual rates of return were 1%
greater than anticipated. Given that I have asset data for 1984, 1989, and 1994, I reduce 1989
assets by 5% and 1994 assets by 10%. I then re-estimate the pro les using the xed-e ects
67
estimator. This asset pro le looks very similar to the cross-sectional asset pro le. Using this
procedure, I nd that the bottom panel of Figure 2 likely overstates average assets at age 70
by $46,000.
This approach likely understates the life cycle asset pro le that would have been observed
in the absence of a run-up in the stock market for two reasons. First, in the absence of a
run up in the stock market, savings rates would have been higher. If rates of return are
uncorrelated across time, then a positive rate of return shock causes only a wealth e ect and
not a substitution e ect. The wealth e ect will cause individuals to increase consumption and
leisure and thus reduce savings. Therefore, if rates of return were lower during 1984-1994,
savings rates and thus asset growth would have been higher at each age during this time
period.
The second reason that the procedure described above likely understates the \anticipated"
life cycle asset pro le is that the procedure over-corrects for the wealth gains on savings over
the sample period. For example, wealth is reduced by 10% in 1994, which is reasonable if there
is no saving or dissaving between 1984 and 1994. However, if all household wealth is from
savings in 1994, then the household has received no wealth shock. My 10% reduction in wealth
would be completely erroneous, and would lead to asset levels in 1994 being understated. This
argument is formalized below.
For simplicity, assume that is the common marginal tax rate and r = r(1 ): Consider
an individual who anticipates receiving a rate of return r on his assets but instead faces a
rate of return r + et : During most years of my sample period, et > 0: Assuming that rates
of return do not a ect savings behavior (this assumption also leads me to understate asset
growth when et > 0 for reasons described above), he saves St in period t; where St =
) + (Bt sst ) Ct : De ne observed assets at time j given the
(Wt Ht + yst + pbt + "t )(1
observed interest rates fr + et gjt=0 ; Aj ; and anticipated assets given that the rate of return
is r in every period are A : Therefore, the asset accumulation equations for observed and
j
68
anticipated assets are:
At+1 = (1 + r + et+1 )(At + St )
(49)
At+1 = (1 + r )(At + St )
(50)
and
where A0 = A0 : Therefore,
!
j
Y
Aj =
=1
t
j
X
(1 + r + et ) A0 +
t
Y
(1 + r + ek ) Sj
t=1 k=1
(51)
t
and
j
Y
A =
j
=1
t
!
j
1
X
(1 + r ) A0 +
t
Y
(1 + r ) Sj
t=1 k=1
(52)
t
Now consider another measure of assets, which is the alternative measure used in the text:
1 + r
1 + r + et
t=1
j
Y
A =
j
!!
Aj :
(53)
Note that this measure is equal to
A
j =
j
Y
=1
t
(1 + r ) A0 +
(1 + r )
t
+ ej k 1 (1 + r ) Sj
1
+
r
t=1 k=t 1
j
X
j
Y
t
(54)
P
Q
r )
Note that equation (52) is greater than equation (54) so long as jt=1 jk=t 1 1+r(1+
+ej k 1 (1+
r )t 1 Sj t < 0 Making the approximation ln(1 + r + ek ) (r + ek ) means that the term
is less than 0 if
j
X
=1
t
exp
j
X
= 1
ej
k
k t
69
1
1 (1 + r )t Sj
t
(55)
is less than 0. Note that this condition holds if average growth rates after the initial time
period are positive. My sample period is 1984-1994. Because there was a run up in stock
prices early in my sample period, this appears to be true. Although stock returns were
negative in a few years in my sample, average returns between 1984 and any time period
between 1985 and 1994 were positive.
70
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