Labor Force Attachment Beyond Normal Retirement Age
Berk Yavuzoglu∗
March 19, 2011
Abstract
Strikingly high labor force participation rates of people beyond normal retirement in the U.S.
compared to numerous European countries deserve a special attention. This paper analyzes the
labor supply, consumption and Social Security bene�ts application decision of older people
jointly, using a DP formulation. There is not any behavioral model in the literature using post-
2000 data and looking at labor supply decisions of people beyond normal retirement age. As
a counter-factual, Social Security rules in England, one of the European countries with very
high LFPR, and Spain, one of the European countries with very low LFPR, will be imposed for
Americans to see how they would behave under these rules. Lastly, a counter-factual analysis for
the case of England will be provided by solving the same model using the English Longitudinal
Study of Aging (ELSA).
1 Motivation
The �scal cost of the Social Security is becoming more important with time in the developed
countries such as USA due to an aging population. The easiest remedy for the challenge of decreasing
this cost is to increase the normal retirement age. This is not an easy decision for a government
because of political concerns. People who continue working beyond normal retirement age as well
as employers hiring them have to pay Federal Insurance Contributions Act (FICA) tax, which has
two components: Social Security tax and Medicare tax. The Social Security tax rate is 6.2 percent
of an employee's wages with a threshold of earnings equal to $106, 800, and the Medicare tax is 1.45
∗Department of Economics, University of Wisconsin-Madison, Madison, WI, 53705. E-mail: [email protected] am grateful to John Kennan, Rasmus Lentz, Karl Scholz, Salvador Navarro, Christopher Taber, Jim Walker, InsanTunali, Sera�n Grundl and Francisco Franchetti for their helpful comments.
1
percent of an employee's wages without any cap. Both employers and employees pay these FICA
tax amounts. The total amount of FICA tax that the government collects for a worker with an
annual earnings of $40, 000 is equal to $6, 200.
Social Security bene�ts are calculated using Average Indexed Monthly Earnings (AIME), which
is the average of 35 highest earnings years. If an employee works less than 35 years, zeros are
thrown into the average for the number of years less than 35. Therefore, if a person at retirement
age has 35 years of work history, delaying retirement one more year does not have any e�ect on his
retirement bene�ts as long as the amount of earnings that year does not exceed the amount of 35th
highest earnings year (except delayed retirement credit which is actuarially fair and discussed in
more detail in Section 2). Delaying retirement one more year may have some e�ect on retirement
bene�ts for workers with less than 35 years of work history. The current average and maximum
monthly Social Security bene�t levels are $1, 164 and $2, 364, respectively. The salary workers
beyond normal retirement earn is the driving force of their participation decision. It is easy to see
that the return most workers obtain on their retirement bene�ts by working one more year after
normal retirement age is either zero or small compared to the FICA tax that the government collects
from these workers. This is why �nding a way to increase labor force participation rate (LFPR) of
older people without changing the normal retirement age can be a cure to decrease the �scal cost
of the Social Security up to some degree.
There is a widespread political argument that the LFPR of older workers should be decreased
to increase employment opportunities for young. Burtless and Quinn (2002) report that when the
baby boom generation entered into the U.S. labor market, unemployment rate increased only by 0.2
percent. This result shows that the U.S. labor market is capable of absorbing new workers into the
labor force. In other words, this widespread argument is not well-founded in the U.S. case; however,
it is still a political concern.
Another interesting pattern is the increase in LFPR of people beyond normal retirement age in
the U.S. since 1995 as seen in Figure 1. This is in contrary to Burtless' (1986) �nding that as people
become more wealthy, they will be more likely to retire earlier. Blau and Goodstein (2010) discusses
that this increase is caused by the changes in the Social Security rules, increases in education levels
and the spouse labor force participation rate. However, Figures 3− 6 in the Appendix provides the
LFPR trends for elderly in the US broken down by age, gender, marital status and education. We
2
Figure 1: LFPR Trends in the U.S. for Elderly by Gender and Age
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����
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��� �� ������� �� �����������������Source: Bureau of Labor Statistics
observe that even LFPR of singles with high school or college diploma shows an increase after 1995.
An important reason behind this should be the increase overall health status of the economy.1 A
cross country comparison of labor LFPRs of di�erent age groups is provided in Table 1. Note that
this table is still not perfectly reliable since the 2006 OECD database we use include agricultural
workers2. To update this, we need to use Study of Health, Aging and Retirement (SHARE) dataset
for Europe. We cannot provide more reliable statistics before getting an access to SHARE for
which we need to submit written documents. This will be updated in future versions of this study.
Interestingly, LFPRs beyond normal retirement age, i.e. at the age groups 65 − 69, 70 − 74 and
75+ are very high in the U.S. compared to numerous developed European countries. The second
highest statistic belongs to Norway, which is caused by a higher normal retirement age. This claim
is supported by the statistics for the age group 70− 74. The next two countries with high LFPRs
beyond normal retirement age are Ireland and United Kingdom. However, LFPRs of these countries
are nearly three-�fth and one-half of the U.S. at the age group 65 − 69, two-�fth and one-third at
the age group 70− 74, and one-half and one-third at the age group 75+. Note that even though we
1Life expectancy at age 65 increased by nearly 1 year since 1995.2Labor force participation rates of elderly people in countries with high agricultural production can be naturally
high since de�nition of agricultural work is vague and scope of it is very broad.
3
Table 1: LFPRs of Di�erent Age Groups along with Retirement Ages in Di�erent CountriesCountry Early
RetirementAge
NormalRetirement
Age
LFPR,50-54
LFPR,55-59
LFPR,60-64
LFPR,65-69
LFPR,70-74
LFPR,75+
Austria 62 (57) 65 (60) 81.2% 55.2% 15.8% 7.1% 3.0% 1.3%
Belgium 60 65 (64) 71.3% 44.8% 16.0% 4.5% n/a n/a
Denmark 60 65 87.3% 83.2% 42.1% 13.1% n/a n/a
Finland 62 65 86.2% 72.9% 38.7% 7.6% 3.9% n/a
France none 60 84.1% 58.1% 15.1% 2.8% 1.2% 0.3%
Germany 63 65 85.0% 73.9% 33.3% 6.7% 3.0% 1.0%
Greece 60 (55) 62 (57) 70.3% 53.5% 32.7% 9.8% n/a n/a
Ireland none 65 73.9% 62.7% 44.8% 17.2% 7.8% 3.4%
Italy 57 65 (60) 71.2% 45.1% 19.2% 7.5% 2.9% 0.9%
Netherlands none 65 79.5% 63.9% 26.9% 8.2% n/a n/a
Norway none 67 84.6% 77.4% 57.3% 20.6% 6.0% n/a
Spain 60 65 71.3% 57.5% 34.6% 5.3% 1.6% 0.4%
Sweden 61 65 88.0% 83.0% 62.5% 13.2% 6.8% n/a
UK none 65 (60) 82.6% 71.2% 44.3% 16.3% 6.0% 1.6%
USA 62 65.5 78.3% 69.9% 48.4% 29.5% 17.8% 6.1%Notes: Parentheses indicate the eligibility age for women when di�erent. Columns 2-3 are obtainedfrom �Social Security Programs throughout the World: Europe, 2006� by U.S. Social SecurityAdministration. Columns 4-9 are obtained from 2006 Health and Retirement Survey for the U.S.and 2006 OECD database for the rest of the countries.
consider LFPR of the U.S. for the age group66− 69 by accounting the normal retirement age 65.5,
which is 27.2 percent, the discrepancy is still huge. In other words, the U.S. is doing a pretty good
job in terms of decreasing the �scal cost of the Social Security on the economy despite political
concerns. This table also provides some more interesting patterns below normal retirement age,
which may be the subject of future research.
In this paper I analyze the labor supply, consumption and Social Security bene�ts application
decision of older people jointly, using a DP formulation. Then, I will impose Social Security rules
in England, one of the European countries with very high LFPR, and Spain, one of the European
countries with very low LFPR, for Americans to see how they would behave under these rules.
Lastly, I will provide a counter-factual analysis for the case of England using English Longitudinal
Study of Aging (ELSA).
4
2 Background
Before presenting the preliminary examination and the model in detail, I should start with a dis-
cussion about the incentives that might possibly a�ect the decision to stay in the labor force after
normal retirement age. The important changes that may a�ect incentives over the time are the
2000 amendments in Social Security rule abolishing the "earnings test" after normal retirement age
and the tax rate schedule.
Since 2000, even if you work beyond normal retirement age, you may start collecting retirement
bene�ts without any exemption. This is completely in contrast with pre-2000 case, when people
lost most or all of their retirement bene�ts depending on how much they earn due to the "earnings
test" by working beyond normal retirement age. The "earning test" still applies for early retirees
and the current early retirement age is 62. The cost of choosing to work after normal retirement age
regarding Medicare is a small premium change in Part B insurance if you earn more than $85, 000.
Besides, delaying retirement bene�ts by 1 year provides an 8 percent increase in monthly retirement
bene�ts starting from the normal retirement age until you reach age 70 since 2008. This increase
in bene�ts was 5.5 percent in 1998, and it was raised by 0.5 percentage point every 2 years since
then until 2008. In 2006, CDC reported that the average life expectancy of 65 years old people in
the U.S. is 84.9 years for females and 82.2 years for males. This implies that there are around 17
years on average for males in the U.S. at the normal retirement age to collect retirement bene�ts.
Given this information, one can conclude that Social Security is actuarially fair right now3 unlike
the pre-2000 period as documented by Rust and Phelan (1997).
Blau and Goodstein (2010), by specifying an econometric model approximating the decision rule
for employment, �nd that amendments in Social Security rules account for a sizable portion of the
recent increase in labor force participation rates. Moreover, French's (2005) policy analysis for the
abolishment of earnings test with pre-2000 data is also along this line. The current Social Security
rules make workers more likely to start collecting retirement bene�ts and stay in the labor force.
Another concern in deciding to collect retirement bene�ts right away may be the tax rate sched-
3Assume that the yearly retirement bene�ts of a male is equal to $10, 000. The Social Security makes the yearlycost-of-living adjustment on the retirement bene�ts, so that we assume the real value of the bene�ts stays the same.Moreover, the current in�ation rate is around 2 percent. If this male worker delays retirement for a year, he gets10, 800/1.02 in today's value for 17 years, and if he does not delay the retirement, he gets $10, 000 for 18 years onaverage. Observe that (10, 800/1.02) ∗ 17 = 10, 000 ∗ 18.
5
ule. Up to 50% of the Social Security bene�ts are subject to taxation according to the federal laws
for a person who currently gets Social Security and �lls federal tax return as an "individual" if
his/her combined income, the sum of adjusted gross income plus nontaxable interest plus one-half
of Social Security bene�ts, is between $25, 000 and $34, 000. If his/her combined income is more
than $34, 000, up to 85 percent of his/her retirement bene�ts are taxable. Moreover, married people
can choose to �le a joint return instead of �ling a separate return if it is pro�table to do so. In that
case, if a married couple have a combined income between $32, 000 and $44, 000, up to 50 percent,
and if their combined income is above $44, 000, up to 85 percent of their retirement bene�ts are tax-
able. The precise taxable income amount for each person can be calculated using IRS Publication
Number 915.
This tax schedule may have an e�ect in deciding to collect retirement bene�ts for older workers;
however, it is important to discuss the degree of this e�ect on the labor force participation decisions.
Employed older low income workers pay no income tax out of their retirement bene�ts. As a
result, their labor force decisions will not be a�ected by this tax rate schedule. If longevity is not
signi�cantly correlated with income, these workers will be indi�erent between collecting retirement
bene�ts right away and delaying it since Social Security is actuarially fair; otherwise, they will prefer
to get their retirement bene�ts right away. 85 percent of retirement bene�ts of a middle income
employed older worker, assuming that he/she earns an amount equal to GDP per capita ($46, 000
currently) and gets average yearly Social Security bene�ts (12× $1, 164 = $14, 000), is taxable. In
other words, this employed older low income worker pays $3, 000 of his/her retirement bene�ts as
federal tax. This reduction in retirement bene�ts is considerably less than the pre-2000 case because
of the �earnings test� and unfairness of delayed retirement credit; though, it is still true that average
or rich Americans are negatively a�ected by this tax schedule. Considering that Social Security is
actuarially fair for average Americans, and actuarially fair or better for rich Americans depending
on whether the longevity is signi�cantly correlated with income, these people may easily choose to
delay getting retirement bene�ts. The current tax schedule makes elderly people more likely to stay
in the labor force compared to the past.
According to the loose argument presented here, I claim that the change in the Social Security
rules in 2000 and tax rate schedule do not cause big disincentives for older people to exit from the
labor force. The model presented in this project will �rst focus on post-2000 period and then will
6
be extended to a longer period starting from 1992, the year Health and Retirement Study (HRS)
was launched, paying special attention to the budget constraint with the hope of �nding a �rmer
evidence for the e�ect of 2000 amendments on older people's labor force decisions.
One of the reasons for high LFPRs of older people in the U.S. compared to European countries
can be the fact that the Social Security rules and the tax rate schedule do not induce workers to
stay out of the labor force, and this naturally increases participation rates. Another explanation of
these high LFPRs can be higher economic opportunities in the U.S. economy for older workers, so
they may tend to remain in the labor force. Moreover, if Social Security payments are inadequate,
poor Americans may tend to stay in the labor force. We control for these factors in our dynamic
programming model.
It is useful to discuss the contribution of this study looking at the recent work in the literature.
Our model is in between Rust and Phelan (1997) and French (2005), and improves upon them.
Di�erently from Rust and Phelan (1997), we use consumption as a decision variable, consider all
the individual level data instead of �nancially constrained people, consider the e�ect of the 2000
amendments, drop perfect control assumption over future employment status so having unemploy-
ment and out of the labor force as 2 di�erent labor force categories, include 5 di�erent health status
categories, females and supplemental security income (SSI) in our analysis and have a broader state
space. Since Rust and Phelan (1997) consider �nancially constrained people, their assumption is to
set consumption equal to income. With our extension, we have consumption as a control variable
and specify a budget constraint for an asset accumulation equation. We also allow borrowing.
Like Rust and Phelan (1997), French (2005) uses only males and the pre-2000 period and
mostly focuses on retirement behavior rather than the labor force decision beyond normal retirement
age. He has di�culty in matching labor force participation of unhealthy individuals due to coarse
discretization of health into good and bad categories. French (2005) does not allow borrowing and
focuses only on household heads. Moreover, he does not consider health expenses, health insurance
and SSI unlike our model. We employ 4 di�erent labor force states including unemployed in our
study rather than treating hours of work as a continuous control variable. Rust and Phelan (1997)
discuss that treating hours of work as a continuous variable is not reasonable since the decline in
hours of work later in life does not occur gradually.
Blau and Gilleskie (2008) investigate the e�ect of health insurance on retirement behavior and
7
again with pre-2000 data using a dynamic programming model. French and Jones (2007) has a
similar context to Blau and Gilleskie and considers data from 1992 until 2004. However, instead of
dealing with the abolishment of the "earnings test" in 2000, they assume that the "earnings test"
was abolished at the age of 67 for everyone. It seems that they miss the point of the e�ect of
this amendment. Moreover, their focus is on retirement behavior like the other studies mentioned
above. Blau and Goodstein (2010) looks at labor force participation trends of older men from
1966 to 2005 using an econometric model which is a linear approximation to the decision rule for
employment. They do not consider heterogeneity in their study, but rather use averages. Moreover,
their econometric model omits the e�ect of the "earnings test" and tax schedule.
To recapitulate, there is not any behavioral model using recent data and looking at labor supply
decisions of people beyond normal retirement age. Moreover, none of these recent studies did a
counter-factual cross country analysis.
3 Data
We use HRS data in this project, which is a is a nationally representative panel data of adults in
the U.S. aged 51+, conducted biannually and �rst �elded in 1992. It contains information on labor
force participation, health, �nancial variables, family characteristics and a host of other topics.
The results are obtained using a subsample of this data, non-disabled individuals aged 58 − 95
in between 2002 and 2008. After dropping some of the observations for the reasons given in the
following section, we are left with 18, 018 individuals with a total of 53, 293 observations. We omit
attrition problems. For the counter-factual analysis, ELSA, �rst �elded in 2002, will be used.
4 Preliminary Examination
This section provides a reduced form analysis of the labor force participation decision of people
beyond normal retirement age using 2006 cross section of HRS. This also helps determine the state
variables and the data generating process in the dynamic programming model. I also explain how
I cleaned the 2006 data in this section. The same procedure is followed for any other year.
The normal retirement age is increasing very slowly in the U.S.: It is currently 66, and it was
65 years and 6 months in 2006. Since the age data at hand has 1 year increments, I consider the
8
cuto� value for age as 66 years instead of 65 years and 6 months. A multinomial logit model of
labor force status on possible determinants for people aged 66 − 69 and 70+ in 2006 cross section
is provided.
HRS includes some con�rmation questions for the health insurance section. While generating
the health insurance data, I exploit these con�rmation questions. I also use the tracker �le released
by HRS which accounts for misspeci�ed cases of age and marital status. I de�ne marital status as
a dummy variable indicating if a person is married. Here the non-married class includes separated,
divorced, widowed, never married and other categories. Health expenses are obtained by summing
up out of pocket expenses for hospital, nursing home, outpatient surgery, doctor visit, dental,
prescription drugs, in-home health care and special facility and other health service costs in the last
2 years. There are some missing asset value observations in the data since people were not sure
about the value of their assets. Some of them reported minimum and maximum values for their
assets, and some refused to answer this question. RAND Corporation imputed these values and
provided as a separate dataset consistent with HRS. I obtain assets by summing values (or imputed
values of RAND if I do not observe a reported value) of �rst home, second home, mobile home,
business/farm, individual retirement amounts, stocks, bonds, checking/saving accounts, certi�cate
of deposits, government saving bills, treasury bills, transportation net of debts on them, value of
assets put in trusts, assets of other family members and other assets like jewelries and collections
then by subtracting mortgages, main loans, other loans and debts. There are 5 di�erent health
status categories; excellent, very good, good, fair and bad. We de�ne a dummy variable for each
category. I also have dummy variables for blacks, Social Security retirees and Medicare. Number
of other health insurance, which includes private insurance, employment insurance and government
insurance other than medicare, is also considered in the analysis. I do not include a dummy for
receiving SSI in this preliminary analysis since there is no cost of getting SSI. This is why people
get it whenever possible, which makes it an endogenous variable.
In de�ning labor force participation status, we �rst impute the hours worked and the weeks
worked observations for 3 percent of the workers who report at least one of the hours worked or
weeks worked. Then, using hours and weeks worked information, we assign workers as full-time
employed if they work more than or equal to 1, 600 hours and part-time employed otherwise. We
assign people who are listed as temporarily laid o� with blank usual hours and weeks worked
9
observations as non-participant.
The sample I use here is the respondent sample of HRS. Originally, it has 18, 469 observations.
I drop 1, 703 disabled people as well as 62 observations who report their labor force status other
than employed, unemployed and out of the labor force, 5 observations who do not know their labor
force status, 2 observations who refuse to report it and 57 observations who take a partial interview
where this question is skipped. I drop 30 respondents who work in one job and refuse to report
or do not know both how many hours in a week and weeks in a year he/she works as well as 5
respondents working in 2 jobs who do not know or refuse to report either hours worked or weeks
worked in each job. The removal of these 35 respondents does not induce an important bias since
they correspond to the 0.27 percent of the �nal sample. When I limit ages to 58 and above, I lose
3, 287 observations. I drop 6 people from the data since they have gender inconsistencies over time.
I exclude 17 respondents who do not know about their health status as well a respondent who does
not know his marital status. I exclude 10 respondents who do not know if they are receiving Social
Security, 10 respondents who refused to answer this question and a respondent with blank Social
Security information. I exclude 15 more people who do not know if they are covered by Medicare
and another respondent with blank Medicare information. I drop 97 observations with blank years
of education. I also drop 64 observations who do not know if they get medicare, 2 observations
who refuses to answer this question and 2 respondents with blank Medicaid information. Moreover,
I drop 16 observations who do not know if they get Champus, Champ�Va, Tri-Care or any other
military health plan and 2 respondents who refuse to answer this question. Finally, I drop 38
respondents who do not know the number of private health insurance they have and 7 respondents
who refuses to answer this question. In the end, I am left with a sample size of 13, 033.
I cannot include experience unfortunately as a covariate since there is only a small number of
observations for it in the data. I observe wages for less than half of the employed, and I use them in
my model to impute wages for everyone as described in Section 6.1. It is excluded in the preliminary
multinomial logit analysis. I also cannot include spouse's employment status and employment char-
acteristics in this preliminary analysis since they are observed only for selected subsamples, namely
for married and employed, respectively. I do not have any variable in HRS showing tax amount. It
will be constructed later on. This variable is also omitted from the preliminary multinomial logit
analysis. It is included in the dynamic programming model.
10
Table 2: Sample Means (Standard Deviations) of Variables by Labor Force Participation Status forAge Group 66-69Variable Full
sampleFull-TimeWorkers
Part-TimeWorkers
Unemployed Out ofLaborForce
Age 67.492(1.115)
67.356(1.101)
67.415(1.114)
67.364(1.027)
67.541(1.116)
Years of Education 12.395(3.112)
12.818(3.306)
13.249(2.798)
12.182(2.316)
12.086(3.102)
Female 0.549 0.402 0.514 0.455 0.589
Black 0.141 0.123 0.140 0.364 0.144
Married 0.681 0.656 0.727 0.455 0.676
Poor Health 0.050 0.009 0.017 0.000 0.068
Fair Health 0.187 0.114 0.128 0.091 0.219
Good Health (reference) 0.344 0.345 0.336 0.364 0.346
Very Good Health 0.309 0.376 0.372 0.364 0.278
Excellent Health 0.109 0.157 0.147 0.182 0.089
Medicare 0.945 0.843 0.941 0.818 0.969
# of Other HealthInsurance
0.737(0.621)
0.818(0.514)
0.711(0.549)
0.364(0.505)
0.730(0.657)
Health Expenses 1056.962(2162.741)
1097.823(1907.093)
1027.246(1808.048)
1429.455(2368.913)
1053.359(2294.248)
Assets (in $1,000) 762.143(2960.301)
873.029(3590.01)
847.214(3087.593)
112.636(222.792)
720.801(2782.645)
# of Children 3.387(2.056)
3.299(2.045)
3.227(1.784)
4.000(3.464)
3.443(2.110)
Receiving Social Security 0.958 0.943 0.979 1.000 0.955
Sample size 2421 351 422 11 1637
LFPR of people aged 66 to 69 is 32.4 percent, aged 70 to 74 is 21.5 percent whereas the same
statistic for people aged 75+ is 7.8 percent in the data we are using. These statistics may seem
high at a �rst glance compared to Table 1; however, it is caused by our speci�c subsample which
particularly does not include disabled people.
Moreover, unemployment rate in our sample is 1.31 percent for people aged 58−64, 0.45 percent
for people aged 66− 69, and 0.35 percent for people aged 70−74. These statistics are small compared
to the 4.8 percent overall unemployment rate in U.S.A. in 2006. Including unemployment in the
model may help predict future labor force state since many of the elderly unemployed drop out of
the labor force with time. This is why I drop the perfect control assumption over future employment
status of Rust and Phelan (1997) and treat out of the labor force and unemployment as 2 distinct
categories.
11
Table 3: Sample Means (Standard Deviations) of Variables by Labor Force Participation Status forAge Group 70+Variable Full
sampleFull-TimeWorkers
Part-TimeWorkers
Unemployed Out ofLaborForce
Age 78.267(6.439)
73.673(3.862)
74.723(4.339)
74.500(5.080)
78.825(6.512)
Years of Education 12.009(3.325)
12.742(2.958)
13.122(2.865)
13.143(2.878)
11.863(3.359)
Female 0.582 0.308 0.474 0.643 0.604
Black 0.116 0.138 0.104 0.214 0.116
Married 0.532 0.669 0.628 0.500 0.516
Poor Health 0.093 0.027 0.020 0.071 0.103
Fair Health 0.238 0.150 0.139 0.143 0.252
Good Health (reference) 0.325 0.335 0.308 0.429 0.326
Very Good Health 0.265 0.331 0.388 0.357 0.249
Excellent Health 0.080 0.158 0.145 0.000 0.070
Medicare 0.979 0.904 0.977 0.929 0.982
# of Other HealthInsurance
0.760(0.568)
0.819(0.536)
0.769(0.567)
0.857(0.770)
0.756(0.569)
Health Expenses 1745.943(7499.984)
1077.246(2088.207)
1001.720(1968.397)
742.071(885.355)
1851.477(7984.77)
Assets (in $1,000) 521.253(1569.658)
1140.232(5349.660)
718.395(1806.026)
196.815(245.99)
476.396(1137.011)
# of Children 3.287(2.252)
3.731(2.265)
3.343(1.979)
3.000(1.840)
3.264(2.277)
Receiving Social Security 0.972 0.981 0.977 1.000 0.971
Sample size 7224 260 642 14 6308
Tables 2 and 3 provide the summary statistics for the variables used in our multinomial logit
analysis for the age groups 66 − 69 and 70+. As seen from these tables, people in the labor force
have a smaller age and higher years of education on average. The proportion of males is highest
among full-time workers. Moreover, labor force participation decision is positively correlated with
the health status. We observe that 94 percent of full-time workers and 98 percent of part-time
workers in the age group 66 − 69 are receiving Social Security bene�ts. This can be seen as an
evidence for the positive e�ect of 2000 policy changes on the decision of elderly workers to start
collecting retirement bene�ts. It seems that the low saving levels of unemployed make these people
willing to work.
Now, we want to run a multinomial logit model of labor force status on the variables given
12
above. Let
y∗ij = θ′ijz + ηij for j = 1, 2, 3, 4. (1)
where i denotes individuals, y∗ij 's denote the unobserved utilities obtains from the choice of labor
force participation status j, z is the vector of explanatory variables given in Tables 2 and 3, θij 's
are the corresponding vectors of unknown coe�cients and ηij 's are the random disturbances.
Let r = max (y∗1, y∗2, y
∗3, y
∗4). Then, the labor status is given by
lfp =
1 = full-time, if r = y∗1,
2 = part-time, if r = y∗2,
3 = out of labor force, if r = y∗3,
4 = unemployed, if r = y∗4.
(2)
We assume that ηj 's satisfy the Independence of Irrelevant Alternatives (IIA) hypothesis, so they
have type I extreme value distribution. McFadden (1973) proves that this speci�cation corresponds
to the Multinomial Logit model. The choice probabilities are given by
πj = Pr(lfp = j | z) =exp(θ
′jz)
3∑k=0
exp(θ′kz)
, j = 1, 2, 3, 4. (3)
Since3∑l=0
πl = 1, we choose people who are out of the labor force as the reference group and set
θ3 = 0. Then, we obtain consistent estimates of θj 's by maximizing the following likelihood function
L =∏lfp=1
π1∏lfp=2
π2∏lfp=3
π3∏lfp=4
π4. (4)
The results of this estimation can be found in Tables 4 and 5 for age groups 66− 69 and 70+,
respectively. An irregularity in these tables are some very high coe�cient estimates in the case of
unemployment. This is due to the lack of observations or variation in these cells. For example,
among the unemployed aged 66− 69, nobody has poor health and everyone receives Social Security
bene�ts. We do not want to give a meaning to such estimates.
Log odds of staying in the labor force decreases with age except unemployed aged 66−69. Higher
13
Table 4: Multinomial Logit Estimates of Labor Force Status on Some Possible Determinants forAge Group 66-69Variable Full-Time Part-Time Unemployed
Coef. Std. Err. Coef. Std. Err. Coef. Std. Err.
Age -0.126** 0.056 -0.097* 0.05 -0.118 0.246
Years of Education 0.036 0.024 0.118*** 0.023 0.095 0.07
Female -0.808*** 0.123 -0.226** 0.114 -0.834 0.615
Black -0.08 0.195 0.237 0.169 0.882 0.556
Married -0.342** 0.136 0.202 0.131 -0.689 0.596
Poor Health -2.028*** 0.612 -1.244*** 0.401 -13.501*** 0.575
Fair Health -0.679*** 0.203 -0.380** 0.177 -1.16 1.152
Very Good Health 0.299** 0.145 0.266* 0.137 0.554 0.716
Excellent Health 0.403** 0.199 0.399** 0.184 0.985 0.817
Medicare -1.706*** 0.225 -0.861*** 0.267 -2.321*** 0.888
# of Other HealthInsurance
0.119 0.092 -0.182* 0.098 -1.098* 0.588
Health Expenses (in$1000)
0.005 0.027 -0.023 0.027 0.13 0.08
Assets (in $100000) 0.0001 0.002 0.001 0.002 -0.299** 0.142
# of Children 0.018 0.031 -0.012 0.028 0.104 0.159
Receiving Social Security 0.244 0.297 1.072*** 0.372 13.335*** 0.777
Constant 8.357** 3.774 3.656 3.421 -8.085 16.545
No. of observations 2421
Log-likelihood w/ocovariates
-2114.923
Log-likelihood withcovariates
-1966.911
Robust standard errors are in parentheses.* signi�cant at 10%; ** signi�cant at 5%; *** signi�cant at 1%.The reference group is people who are out of the labor force.
education increases full-time employment probability for people aged 70+ and part-time employment
probability for anyone above 65 compared to staying out of the labor force. Being female decreases
employment probability at any age, and being married decreases full-time employment probability
for the age group 66 − 69, and full-time and part-time employment probabilities for people aged
70+. The marriage dummy itself captures only the average e�ect on the society. In fact, the e�ect
of marriage on the labor market can be more substantial depending on spouse's labor force status.
We dig into this issue in our dynamic programming model. An interesting result is that being black
does not a�ect participation probability. The full-time and part-time employment rates are very
similar for blacks and whites in the raw data.
14
Table 5: Multinomial Logit Estimates of Labor Force Status on Some Possible Determinants forAge Group 70+Variable Full-Time Part-Time Unemployed
Coef. Std. Err. Coef. Std. Err. Coef. Std. Err.
Age -0.186*** 0.017 -0.125*** 0.01 -0.136** 0.061
Years of Education 0.049** 0.021 0.093*** 0.018 0.210* 0.116
Female -1.220*** 0.142 -0.488*** 0.101 0.118 0.572
Black 0.275 0.201 0.086 0.164 0.417 0.734
Married -0.285* 0.149 -0.179*** 0.109 -0.005 0.59
Poor Health -1.207*** 0.416 -1.307*** 0.371 -0.522 1.128
Fair Health -0.434** 0.206 -0.387** 0.155 -0.724 0.831
Very Good Health 0.146 0.163 0.383*** 0.117 0.109 0.586
Excellent Health 0.621*** 0.207 0.614*** 0.159 -12.784*** 0.369
Medicare -1.944*** 0.269 -0.491 0.36 -1.714 1.079
# of Other HealthInsurance
0.145 0.105 0.007 0.086 0.274 0.59
Health Expenses (in$1000)
-0.02 0.022 -0.036** 0.02 -0.087 0.101
Assets (in $100000) 0.008*** 0.002 0.004 0.003 -0.216* 0.131
# of Children 0.069*** 0.026 0.003 0.022 -0.08 0.123
Receiving Social Security 1.136*** 0.473 0.486** 0.383 12.946*** 0.635
Constant 11.388*** 1.369 6.443*** 0.938 -8.78 5.207
No. of observations 7224
Log-likelihood w/ocovariates
-3361.123
Log-likelihood withcovariates
-2934.116
Robust standard errors are in parentheses.* signi�cant at 10%; ** signi�cant at 5%; *** signi�cant at 1%.The reference group is people who are out of the labor force.
As people get healthier, they are more likely to participate.4 Having Medicare coverage decreases
log odds of staying in the labor force compared to staying out of the labor force for people aged
66 − 69 and full-time employment probability for the age group 70+. This is reasonable since
Medicare is one of the important determinants behind labor force decisions as discussed by Rust
and Phelan (1997).
As the number of other health insurance increases, people aged 66− 69 become less likely to be
part-time employed and unemployed compared to being a non-participant, and it is not signi�cant
for the rest of the cases. There is a question in HRS asking the primary health insurance plan to a
subset of the sample. Among our subsample, while 17.7 percent of people in the age group 66− 69
4Note that having good health is the reference for health dummies in Tables 4 and 5.
15
who responded this question identi�ed their primary insurance as di�erent than Medicare, the same
statistic is only 7.6 percent for the age group 70+. Looking at di�erent labor force status groups, we
see that while 42.0 percent of full-time workers, 15.2 percent of part-time workers, 27.2 percent of
unemployed and 7.3 percent of non-participants have a primary insurance di�erent than Medicare.
Since we control for Medicare in our multinomial logit model, the insigni�cance of the number of
other health insurance is not surprising. Health expenses in the last 2 years are insigni�cant except
for part-time workers in the age group 66−69. Since we control for health, it is reasonable to expect
that an increase in health expenses do not change log odds of being employment over staying out
of the labor force.
Assets do not seem to be an important factor determining labor force status. This may be caused
by the fact that only 2.5 percent of our sample have negative assets and 82.1 percent of our sample
are getting Social Security bene�ts. In other words, most people have enough money to survive
without working. Receiving Social Security bene�ts increases the participation probability except
full-time employment probability over staying out of the labor force for the age group 66 − 69.
This may be a sign of insu�cient Social Security bene�ts as well as an incentive caused by the
absence of a considerable monetary punishment for working while receiving Social Security bene�ts.
Number of children increases the full-time employment probability of the age group 70+, but this
is in contrast with expectations. We do not have any explanation for this increase.
We use all of the variables given in this section except race and number of children in our
dynamic programming formulation.
5 Model
The speci�cation of the dynamic programming model in this paper is close to French (2005), and
improves upon them. Rust and Phelan (1997) �nds that health care expenses and Medicare as well
as Social Security rules are the important determinants of the retirement decision. French (2005)
shows that the "earnings test" is the main reason behind the labor force participation decision of
older men, and solves the early retirement puzzle by incorporating pension bene�ts into his model.
A more recent work by Blau and Goodstein (2010) �gures out that 25 to 50 percent of the recent
increase in LFPR of older men is attributable to the Social Security rules, 16 to 18 percent to
16
increase in education and another 15 to 18 percent to increase in LFPR of married women. My
model is powerful in terms of capturing all these e�ects even though the main emphasis is on labor
supply behavior of people beyond normal retirement age.
I have a 4 dimensional vector of control variables: consumption, weeks worked in a year, hours
worked in a week and a dummy variable indicating whether the individual applied for Social Security
bene�ts. I denote consumption with ct and discretize it in my solution. There is no �xed discretiza-
tion for consumption. At each step, discretization depends on previous consumption in a way that
it clusters mostly around the previous consumption level following French (2005). Hours worked in
a week is denoted by hwt and discretized using 0, 10, 20,40, 60. Weeks worked in a year is denoted
by wwt and discretized using 0, 10, 25, 40, 50 and 99 where 99 denotes the case of unemployed. I
may use splines in the future to make hours and weeks worked continuous later on. However, there
is a possibility that an individual ends up being unemployed even though he/she looks for a job. In
other words, there is an uncertainty in �nding a job. I model this as an uncertainty in Equations
(6) and (7). bt denotes the dummy variable indicating whether the individual applied for Social
Security bene�t.
Moreover, I have a 7 dimensional vector of state variables: assets, wages, health status, medicare,
health expenses and spouse weeks worked in a year and hours worked. I use 10 asset states denoted
by At, and 5 wage states denoted by wt. There are 6 health status categories: death, poor, fair,
good, very good and excellent, which are denoted by ht taking values 0, 1, 2, 3, 4 and 5, respectively.
mt is the dummy variable for Medicare insurance. I have 4 health expenses states denoted by het.
Spouse hours worked in a year is denotes as shwt and spouse weeks worked in a year is denoted
as swwt. Altogether, I have 72, 000 di�erent state points for married and 2, 400 state points for
singles.
I have a 3 dimensional vector of type variables: female, education and number of health expenses.
There is a dummy variable indicating whether the respondent is a female denoted by ft. I have 4
education groups denoted with edt: high school dropouts, high school graduates, university dropouts
and university graduates. I assume that number of other health insurances is �xed at the initial
value and denote it with noht taking values 0, 1, 2 and 3. I will solve the dynamic programming
model for these types separately.
Social Security bene�ts (sst), pension bene�ts (pbt), spousal income (yst), health insurance
17
premiums (hipt), SSI (ssit) and an indication variable for getting it (Dt) are the variables in the
data generating process. I do not include a dummy variable for getting SSI as a state variable since
there is no cost of getting SSI. Individuals get this income whenever it is possible. I do not observe
Social Security bene�ts and pension bene�ts for every recipient. For those recipients, predicted
values are used in the analysis. I assume pension bene�ts are illiquid until age 62 following French
(2005). In data, I see spikes regarding pension bene�t receipt at ages 62 and 65. I assume that each
respondent knows his/her spouse's labor force participation decision in advance where the spouse
makes an optimal decision. I observe the dummy variable for getting SSI for most of the respondents
in the data and impute it for the rest by looking at their income and asset levels. Then, I impute
the SSI amount for recipients considering the federal rate and rules.
I model the problem as a discrete control process. Denote the control variables by d, state
variables by x, and preference parameters by θ. I am using a more detailed �ow utility function
compared to French (2005):
U(xt, dt, θt) = I(ht 6= 0)1
1− v
c 5∑i=1
I{ht=i}θCHi
t L
5
1−∑
i=1I{ht=i}θCHi
1−v
(5)
where
L = L− I(wwt 6= 99) (hwtwwt − θPwwt)− I(wwt 6= 99)θU +
θS,f (L− I(swwt 6= 99)shwtswwt) + I(swwt = 99)θSU,f . (6)
The coe�cient of relative risk aversion is given by v. θCHi measures the consumption weight.
I expect θCHi to decrease as health worsens. First, unhealthy people will need time to rest which
increases the value of leisure. Second, they will take more time to consume goods compared to
the healthy individuals which requires more leisure time. An opposite argument may be that when
people get unhealthier, they may need to make some speci�c consumptions (there should not be
many such goods) like buying a car since they cannot ride a bus any more which may increase θCHi.
I expect this e�ect to be much smaller compared to the e�ects of the �rst two arguments. θP is the
18
�xed cost of work per week, measured in hours worked per year. It can be seen as a commuting
cost. θU is the �xed cost of unemployment. Unemployed people lose some of their leisure time
due to the job search process. I expect θU < θP . θS,f measures the proportion of the additional
leisure time obtained from the spousal leisure time due to the complementarity. Multiplied with the
spousal leisure time, I get the additional leisure coming from complementarity in terms of hours of
worked. It is measured for females and males di�erently. Finally, θSU,f gives the additional leisure
time obtained from the leisure time of an unemployed spouse measured di�erently for wives and
husbands. I expect θSU,f < θS,fL .
The constraints individuals are facing in the model are part-time and full-time job �nding
determination equations, health determination equation, health expenses determination equation,
wage determination equation and asset accumulation equation.
Individuals make their optimal hours worked and weeks worked decisions. However, there is a
chance that they may end up unemployed even if they want to work. The probability of �nding a
part-time and a full-time job (depending on hours and weeks worked) next year depend on current
health status, gender, education and age. I de�ne di�erent probability functions for unemployed
and non-participants.
πppart time,good,1,high,t+1 = Pr(lfpt+1 = part time|ht = good, ft = 1, edt = high, t+ 1), (7)
πpfull timebad,0,high,t+1 = Pr(lfpt+1 = full time|ht = bad, ft = 0, edt = high, t+ 1), (8)
where p ∈{unemployed, non-participant}. I use transitions from unemployment and out of the labor
force states into full-time employment, part-time employment and unemployment to identify these
probabilities.
Health status next period depends on the current health status, education, gender, medicare,
number of other health insurance and age.
µpoor,good,high,1,1,0,t+1 = Pr(ht+1 = poor|ht = good, edt = high, ft = 1,mt = 1, noht = 0, t+ 1). (9)
Out of pocket health expenses depend on the labor force participation status, medicare, number
19
of other health insurance, age and health status last period similar to French and Jones (2007).
log(het) = ψmt +2∑j=1
γjnohjt +
2∑k=1
ϕktk +
5∑l=1
θlI(ht = l) + ςt + ξt, (10)
where
ςt = ρςςt−1 + εt, εt ∝ N(0, σ2ε ), and ξt ∝ N(0, σ2ξ ). (11)
Here ς gives the persistent component and ξ gives the transitory component of health cost uncer-
tainty.
I do not observe wages for more than half of the employed workers and use predicted wages
for them. I obtain these estimates for each cross section using Yavuzoglu and Tunali's (2011)
solution for double selection problems, which relaxes the trivariate normality assumption among the
error terms of the two selection equations and the regression equation by following the Edgeworth
expansion approach of Lee (1982). I use two Mincer-type wage equations for full-time and part-time
employment as the regression equations. In imputing wages, I exclude the probability of job-to-
job transitions which may result in higher earnings. I assume that job separation arises from a
joint decision of an employer and his/her employee. The transition rate from employment into
unemployment is just 0.1 percent over years.
Logarithm of wages in the current period depends on the employment status, health status,
education, gender and age:
log(wt) = %+ τ1t+ +τ2t2 +
5∑p=2
µpI(ht = p) +ARt, (12)
where
ARt = ρARARt−1 + ηt, ηt ∝ N(0, σ2η). (13)
Asset accumulation equation is given by:
At+1 = At+Y1(rAt+hwtwwtI{wwt 6= 99}+yst+Dt×ssit+pbt, τ1)+bt×Y2(sst, τ2)−het−ct−hipt,
(14)
where Y1(rAt+hwtwwtI{wwt 6= 99}+yst+Dt×ssit+pbt, τ1) is the level of post-tax income except
20
Social Security bene�ts, r is the interest rate, It is the indicator function, τ1 is the tax structure
including state, federal and FICA taxes, Y2(sst, τ2) is the level of post-tax Social Security bene�ts
and τ2 is the tax structure regarding Social Security bene�ts.
I assume
At > −100000 + (t− 58)100000
37∀t ∈ [58, 95] . (15)
I imposed the lower bound in an ad hoc fashion.
The Bellman equation I want to solve is given by
Vt(xt) = maxct,bt,hwt,lwt
[ut(xt, dt, θ) + β
6∑j=1
ˆV (xt+1)dF (wt+1|wt, hwt, wwt, ht, edt, ft, t)
× Pr(ht+1 = j|ht, edt, ft,mt, noht)], (16)
where β is the intertemporal discount factor and F (.|., ., ., ., ., .) is the conditional distribution of
wages next period. Note that each individual makes a choice regarding how many hours in a week
and weeks in a year to work. If an individual were unemployed or out of the labor force last period
and chooses one of full-time job or part-time job options, there is still a probability that he/she
ends up being unemployed coming from Eqn.s (6) and (7). This has bite in calculating �ow utility
via expected utility theorem and is re�ected in future periods as well. I assume that terminal age is
95 and solve the problem recursively. This assumption does not mean that everyone dies at 95, but
people die with probability 1 at age 95. In other words, the model does not account for increases
in age above 95. This is an innocuous assumption since mortality rate is very high beyond 95 and
simpli�es the problem computationally. The optimal decision rule will be given by δ = (δ0, δ1, ..., δT )
where dt = δt(xt) speci�es optimal decision dt as a function of the state variables xt for an age t
individual.
The model will be estimated in 2 steps. In the �rst step, I estimate some elements in the
data generating process and calibrate some others. The elements I want to estimate or calibrate in
the data generating process are given by ϕ = {πppart time,good,1,high,t+1and πpfull timebad,0,high,t+1 for
p ∈{unemployed, non-participant}, Pr(ht+1|ht, edt, ft,mt, noht, t+ 1), ψ, γj 's, ϕk's, θl's, ρς , σ2ε , σ
2ξ ,
r, sst, pbt, yst, hipt, Dt, ssit}. I assume rational expectations. Given the data generating process I
want to estimate the following parameters in the model φ = {v, θCHi's θP , θU , θS,f 's,θSU,f 's in the
21
�ow utility function, %, µp's, τq's, ρAR, σ2η in the wage determination equation, β} using simulated
method of moments.
England also has a very similar retirement system. An average person who continues working
after collecting retirement bene�ts will get around 400 pound on average; whereas, if they stop
working they will get around 700 pound. Moreover, delaying to collect retirement bene�ts by 1
year gives an 10 percent increase in monthly retirement bene�ts. In the end, a counter-factual
analysis for the case of England, where I solve the same model using ELSA, will be provided. By
that way, I can see if my model can be applied for countries with similar set of rules. Moreover, I
will impose Social Security rules and tax rate schedule in England and Spain to see how Americans
would behave with these set of rules. It is very interesting to see the degree of importance of these
rules among societies with di�erent labor force characteristics.
6 Estimation of the Data Generating Process
In this section, I describe the estimation procedures for the data generating process. For this
purpose, I start with imputing wages, Social Security bene�ts, pension bene�ts and health insurance
premiums since I do not observe them for everyone in the relevant subsamples. These estimates go
into the budget constraint, help determine the SSI amounts of individuals and the unknown SSI
status of 140 person-year observations.
6.1 Wages
Wage estimates are obtained for each cross section separately using Yavuzoglu and Tunali's (2011)
solution for double selection problems, which relaxes the trivariate normality assumption among the
disturbances of the two selection equations and the regression equation by following the Edgeworth
expansion approach of Lee (1982). Instead, Yavuzoglu and Tunali (2011) do not impose any condi-
tion on the form of the distribution of the random disturbance in the regression (partially observed
outcome) equation, but conveniently assume bivariate normality between the random disturbances
of the two selection equations.
22
Home− work utility :U∗0 = θ′0z + ν0, (17)
Part− time work utility :U∗1 = θ′1z + ν1, (18)
Full − time work utility :U∗2 = θ′2z + ν2. (19)
Assume that home-work (or non-participation), part�time employment and full-time employ-
ment utilities can be expressed as follows where z is a vector of observed variables, θj 's are the
corresponding vectors of unknown coe�cients and υj 's are the random disturbances. Assuming
that individuals choose the state with highest utility, their decisions can be captured using the
utility di�erences:
y∗1 = U∗1 − U∗0 = (θ′1 − θ
′0)z + (υ1 − υ0) = β
′1z + σ1u1, (20)
y∗2 = U∗2 − U∗1 = (θ′2 − θ
′1)z + (υ2 − υ1) = β
′2z + σ2u2. (21)
Note that y∗1 can be expressed as the propensity to be part-time employed rather than being a
non-participant and y∗2 as the incremental propensity to engage in full-time employment rather than
part�time employment. Then, y∗1 + y∗2 gives the propensity to engage in full-time employment over
home-work. Since the preferences of unemployed over employment options is not known, I de�ne
the unemployed as people obtaining higher utility either from part-time or full-time employment
relative to home-work following Magnac (1991). Under this assumption, the four way classi�cation
observed in the sample arises as follows:
lfp =
1 = full-time employment, if y∗1 > 0 and y∗1 + y∗2 > 0,
2 = part-time employment, if y∗1 > 0 and y∗2 < 0,
3 = home-work, if y∗1 < 0 and y∗1 + y∗2 < 0,
4 = unemployed, if y∗1 > 0 or y∗1 + y∗2 > 0.
(22)
In this case the support of (y∗1, y∗2) is broken down into three mutually exclusive regions, which
respectively correspond to lfp = 1, 2, and 3. The region for lfp = 4 is the union of those for
lfp = 1 and lfp = 2. The classi�cation in the sample is obtained via a pair from the triplet {y∗1,
23
y∗2, y∗1 + y∗2}. Normalizing the variances of y∗1 and y∗1 + y∗2 to 1 has an implication for the variance
of y∗2 (σ22 = −2ρ12 where ρ12 is the correlation between u1 and u2). This is why I may apply the
normalization to one of σ11 = σ21 and σ22 = σ22, but must leave the other variance free to take on
any positive value. In the analysis, I take σ11 = 1 and let σ22 be free. In the �rst step, I rely on
maximum likelihood estimation and obtain consistent estimates of β1, β2, ρ12 and σ2 subject to
σ1 = 1. The likelihood function is given by
L =∏lfp=1
P1
∏lfp=2
P2
∏lfp=3
P3
∏lfp=4
P4, (23)
where Pj = Pr(lfp = j) for j = 1, 2, 3, 4.
The regression equation for this problem is a Mincer-type wage equation given below where X3
is the explanatory variables given in Tables 4 and 5:
log(wage) = β′3X3 + σ3u3. (24)
The aim is to estimate β3 for lfp = 1, 2. After forming the estimates of selectivity correction
terms via �rst step estimates, I run a linear regression equation with 9 selectivity correction terms
coming from Edgeworth expansion in the second step. Note that robust correction obtained via
Edgeworth expansion nests the conventional trivariate normality correction, and therefore both the
conventional trivariate normality speci�cation and the presence of the selectivity bias can be tested
via this estimation. Details can be found in Yavuzoglu and Tunali (2011).
I present only the 2006 cross section results here to demonstrate the employed methodology. Ta-
ble 6 provides the results of the �rst step. Very low ρ12 value implies that unobserved characteristics
a�ecting the decision of part-time employment over non-participation do not a�ect the decision of
full-time employment compared to part-time employment. As expected, females have a lower partic-
ipation probability and are less likely to be full-time employed compared to part-time employment.
Blacks are more likely to be part-time employed compared to being a non-participant. This may be
caused by blacks having low assets. Along with the line of my expectations, participation pro�le is
concave with respect to age.
Moreover, as years of education increases, people are more likely to work part-time rather than
24
Table 6: Maximum Likelihood Estimates of Reduced Form Participation Equations (NormalizedVersion)
Variable First Selection Second Selection
Coef. Std. Err. Coef. Std. Err.
Black 0.090** 0.045 -0.039 0.055
Married -0.113*** 0.036 -0.221* 0.124
Age 0.099** 0.047 0.116 0.096
Age squared/100 -0.106*** 0.032 -0.111 0.084
Female -0.272*** 0.048 -0.470* 0.273
Years of Education 0.043*** 0.005 -0.025*** 0.008
Receive SS -0.250** 0.108 -0.881** 0.445
Medicare -0.276*** 0.055 -0.087 0.125
# of Other Health Insurance -0.019 0.027 0.119** 0.058
Poor Health -0.671*** 0.084 -0.226 0.28
Fair Health -0.202*** 0.044 -0.151 0.123
Very Good Health 0.138*** 0.038 -0.053 0.050
Excellent Health 0.258*** 0.051 0.023 0.076
Constant -2.361 1.784 -1.694 2.721
σ11 1 [normalized]
σ22 0.877 (0.885)
ρ12 -0.116 (0.276)
No. of observations 13077
Log-likelihood without covariates -13480.875
Log-likelihood with covariates -7923.194Robust standard errors are reported.* signi�cant at 10%; ** signi�cant at 5%; *** signi�cant at 1%.
being out of the labor force or working full-time. Since people want to realize some return on their
educational investments, they are more likely to be a participant. However, these people should have
enough savings making them unlikely to work full-time. Receiving Social Security bene�ts decreases
the full-time employment and part-time employment probabilities. Having Medicare decreases the
participation probability which is reasonable since one of the main concerns in the labor force
participation decision is health insurance as documented by Rust and Phelan (1997). I do not have
a good explanation regarding why number of other types of health insurance increases full-time
employment probability over part-time employment. With good health as the reference category,
there is positive correlation between health and participation probability. While being married
decreases participation probability, being black increases the participation probability.
Using the estimates of the �rst step, I provide least squares estimates of the log(wage) equation
for full-time and part-time employed separately in Table 7. λ's denote the selectivity correction
25
Table 7: Least Squares Estimates of the Wage EquationsVariable Full-Time Employed Part-Time Employed
Coef. Std. Err. Coef. Std. Err.
Black 0.004 0.046 -0.02 0.052
Age 0.214*** 0.077 0.233** 0.109
Age squared/100 -0.165*** 0.058 -0.181** 0.079
Female -0.113*** 0.044 -0.063 0.059
Years of Education 0.081*** 0.013 0.095*** 0.015
Poor Health -0.522*** 0.156 -0.467** 0.224
Fair Health -0.132*** 0.051 -0.078 0.066
Very Good Health 0.142*** 0.05 0.119** 0.052
Excellent Health 0.149* 0.077 0.299*** 0.076
λ1 -35.367** 17.3 -0.013 0.513
λ2 47.250** 21.027 -0.401 1.568
λ3 -10.515*** 3.759 0.044 0.335
λ4 14.609*** 5.346 -0.685 0.931
λ5 13.424*** 4.68 -0.096 0.582
λ6 -2.237 1.793 -0.019 0.066
λ7 -0.342 1.381 -0.618 0.677
λ8 -5.718* 3.13 -0.545** 0.234
λ9 -2.292** 1.039 1.200* 0.726
Constant -4.360* 2.643 -6.230* 3.747
No. of observations 795 740
R2 0.207 0.186Robust standard errors are presented.* signi�cant at 10%; ** signi�cant at 5%; *** signi�cant at 1%.
terms. The presence of selectivity bias can be tested by looking at the joint signi�cance of all the
selectivity correction terms. For both full-time and part time employment, the evidence is in favor
of the non-random selection (p− value ' 0.000 for both cases).
Conventional trivariate normality speci�cation uses only λ1 and λ2. The test for the joint
signi�cance of the remaining λ's provides evidence in favor of the robust selectivity correction for
part-time employed (p− value = 0.001) and full time employed (p− value = 0.035). The evidence
is in favor of the non-random selection for both full-time and part-time employed in 2002, and for
full-time employed in 2004 and 2008, but against non-random selection for part-time employed in
2004 and 2008 at 5 percent level of signi�cance. Moreover, the evidence is in favor of the robust
selectivity correction for part-time employed in 2002 and full-time employed in 2004 and 2008, but
in favor of the conventional trivariate normality speci�cation for full-time employed in 2002 at 5
26
percent level of signi�cance.
An interesting �nding is that the wage gap between blacks and whites disappears for elderly
workers. Wages are concave with respect to age and increases with education as expected. While
being female decreases full-time wages by 11.0%, gender is insigni�cant for part-time wages. With
good health as the reference category, it can be concluded that wages are positively correlated with
health.
Using these estimates, predicted wage values for workers without an observed wage rate are
obtained.
6.2 Social Security and Pension Bene�ts
The amount of Social Security bene�ts is missing for 6, 689 cases out of 42, 909 person-year ob-
servations for retirees and pension bene�ts for 3, 007 cases out of 19, 916 person-year observations
for pensioners. This section explains how the bene�t amounts for the retirees and pensioners are
imputed. It is possible to back out Average Indexed Monthly Earnings (AIME) for some people in
our panel data after making some assumptions on the number of years they worked up to that point.
After that, it will be possible to impute AIME values to get a better estimate of Social Security
earnings. I will do this in the future. Currently, I use the regression given below.
In the dataset, Social Security amounts deposited directly into the recipients' accounts is pro-
vided. This amount is obtained after taking copays and deductions for health expenses into account.
This is why I include variables regarding health status and health expenses in the regression equation
below. Since I am concerned about individual and year e�ects, I estimate Eqn. (24). In estimation,
Hausman and Taylor's (1981) method is employed. I choose this methodology rather than �xed
e�ects or random e�ects models because I want to exploit the variation both within and across
individuals. Moreover, I cannot apply Amemiya and MaCurdy's (1986) method since it requires a
balanced panel.5
5There are some substitutes in the data who were seen for the �rst time after 2002.
27
ssit = αi +101∑k=58
ΠkI{ageit = k}+ Πmmit + Πnohnohit + Πheheit + ΠCOLAGCOLAGt
+Πaassetsit +
5∑p=1
ΠhpI{ht = p}I{p 6= 3}+ Πffi + Πededi + uit, (25)
where αi is the individual e�ect and COLAGt is generated in a CPI like way taking cost-of-living
adjustment (COLA) amounts provided by Social Security into account. 2002 is assigned as the base
year.6
To apply this method, time-varying and time-invarying variables as well as endogenous and
exogenous variables needs to be speci�ed. Let Xit and Zi denote time-varying and time-invarying
variables, respectively. Let
Xit = [X1it,X2it] and (26)
Zi = [Z1i,Z2i], (27)
where X1it ={mt, COLAGt, I{ageit = k}107k=58, I{ht = p}5p=1,6=3
}and Zi = {femalei} denote ex-
ogenous variables, which are assumed to be uncorrelated with fi and uit, andX2 = {assetsit, nohit, heit}
and Z2 = {edi} denote endogenous variables, which are assumed to be correlated with αi but uncor-
related with uit. More assets and more education may mean higher wage earnings in the past, which
should partly be captured by αi. Moreover, richer people tend to have private health insurance and
higher out-of pocket health expenditures since they want to get the best treatment. In the �rst step,
the coe�cients of Xit are computed. Then, in the second step, the coe�cients of Zi are obtained by
running a regression of Zi on yi− X ′β where y is the independent variable and β is the estimate of
{Π′ks,Πm,Πhe,Πhe,ΠCOLAG,Πa} using X1it as an instrument for Z2i. After getting the estimates, I
predict Social Security bene�t amount for 6, 677 unobserved cases via linear prediction, so ignoring
fi.
Pension bene�ts show much more variation compared to Social Security bene�ts since individuals
have some kind of control over the money they draw monthly. This is why I still want to keep
6For example, COLA adjustment was 1.4 and 2.1 in 2003 and 2004, respectively. Therefore, I assign 100×1.014×1.021 = 103.5294 to COLAG in 2004.
28
health related variables and assets in the estimation and use Eqn. (24) substituting pbit instead of
ssit. However, instead of many di�erent age dummies, I only use ageit and age2it/100. I estimate
this equation by omitting outliers with pension bene�ts more than $10, 000/month via Hausman
and Taylor's (1981) method. Both the list of the endogenous variables and the intuition for this
endogeneity assumption is the same as the above discussion except health expenses. Health expenses
may be positively correlated with the individual e�ects. Therefore, increased health expenses may
cause pensioner to draw more bene�ts. 8.50 percent of 2, 964 imputed values generated by this
methodology are negative. The respondent with the observed pension bene�ts corresponding to
8.50 percentile draw $120 per month. I choose to replace negative imputed person-year observations
by $60/month.
6.3 Supplemental Security Income
The data does not provide information on whether 260 person-year observations receive SSI and
the amount of it for 187 observations out of 1, 446 person-year recipients. People who are below 65,
blind or disabled, single individuals having resources below $2, 000 and married individuals having
joint resources below $3, 000 cannot be entitled to supplemental security income bene�ts. The
amount of resources is found by deducting the values of �rst home and car as well as some other
seemingly unimportant resources such as wedding rings, household goods and grants for educational
attainment from the assets. Looking at the values of assets except �rst home and car, I conclude
that 139 person-year observations have resources above the limit, so I assign them as not receiving
supplemental security income. Since supplemental security income is like an extra money, any
rational individual should get this bene�t whenever it is possible. This is how I determine if the
remaining 121 person-year observations are getting supplemental security income and the amount of
it for 187 recipients.7 A crude way of evaluating the performance of this methodology is to compare
the estimates for the dummy variable receiving supplemental security income with the correct ones
coming from data.8 The results are provided in Table 8.
This methodology predicts SSI dummy correctly 88 percent of the time for people not receiving
7Married people cannot apply for SSI as singles. This rule was originated from the fact that if you are married,you can share costs like rent, which results in decreased living costs.
8I only consider people with observed SSI status who are aged 65 or above and have resources below 2, 000 or3, 000 if married. Note that his subgroup of respondents are the ones who may get SSI in theory if they do not makeenough money.
29
Table 8: Performance of the Imputation MethodologyPrediction TOTAL
Not receiving SSI Receiving SSI
Correct SSI information Not receiving SSI 7,514 1,021 8,535
Receiving SSI 249 874 1,123
TOTAL 7,763 1,895 9,658
SSI and 78 percent of the time for people receiving SSI. This provides a strong ground for using
this methodology to determine if 121 person-year observations receive SSI.
Even though some states supplement the federal SSI level, I cannot take into account these
supplement levels in imputation since I do not observe the state of residence in the data. SSI is
obtained by subtracting countable income, which I obtain by summing wage earnings, Social Security
and pension bene�ts where the �rst $20 of the highest component of income received in a month
and $65 of earnings and one-half of earnings over $65 received in a month are not counted, from
federal rates.9 Using this methodology, I predict that only 66 observations are getting SSI out of 187
respondents who report that they are getting SSI but amount of it. This may seem contradictory to
Table 8 at �rst glance; however, this is caused by the fact that 107 of the remaining 121 observations
with unknown SSI amount report resources more than $2, 000 if single and $3, 000 if married. To
see how well this methodology predicts the correct SSI amount, I plot the correct SSI amount with
the predicted ones in Figure 2. Removing the resource barrier, I get the estimates of SSI amount
for 19 more observations. In the end, I am left with 102 person-year observations for which I cannot
guess the SSI amount. I assume an SSI amount equal to $100 for them.
6.4 Health Insurance Premiums
Total premiums paid for Medicare and Medicaid as well as the premiums of 3 other health insurance
are observed for a subsample of the data. I impute total premiums paid for Medicare and Medicaid
premiums and premiums of other health insurance separately; then, sum them up to get the amount
of health insurance premiums. For total premiums paid for Medicare and Medicaid, I estimate Eqn.
9If the highest component of income is the wage earnings, then $85 of earnings and one-half of earnings over $85is not counted.
30
Figure 2: SSI vs. Predicted SSI0
200
400
600
800
1000
ssi_
amou
nt
0 200 400 600 800 1000ssi_amount_hat
(27) using Hausman and Taylor's (1981) method.
premium1it = αi + ψmmit + ψmemeit + ψa1age+ ψa2age2/100 + ϕheheit + ϕnohnohit
+ϕCOLAGCOLAGt +5∑p=1
ϕhpI{ht = p}I{p 6= 3}+ ϕffi + ϕededi + uit, (28)
where premium1it denotes the total premiums paid for Medicare and Medicaid, and meit denotes
the dummy variable for Medicare. I assume that Medicaid is the only time-varying and education
is the only time-invarying endogenous variable. Poor people may be eligible for Medicare without
paying any premium, which should be correlated with the individual e�ect determining willingness
to pay for this insurance. Also, if you are more educated, the number of other health insurance
you have should be higher. This may make you less likely to pay a high premium for Medicare. In
the estimation, I ignore the outliers de�ned as the top percentile. In the end, I get 908 negative
estimates out of 29, 428 for premium1it level which I replace with 0. This is reasonable given that
57.4 percent of the people with observed premium levels in the data do not pay any premium10.
10i.e., their premium amounts are entered as $0.
31
This should be a by-product of retirement rules.
Then, I impute the premiums of other health insurance using Eqn. (28) via Hausman and
Taylor's (1981) method.
premium2it = αi + ψmmit + ψmemeit + ψa1age+ ψa2age2/100 + ϕheheit + ϕnphnphit
+ϕCOLAGCOLAGt +
5∑p=1
ϕhpI{ht = p}I{p 6= 3}+ ϕffi + ϕededi + uit, (29)
where premium2it denotes the premiums for other health insurance, and nphit denotes the number
of private health insurance. Since I observe the premiums of only 3 other health insurance, I cannot
include anyone with more than 3 types of health insurance in the estimation. Moreover, some
people do not report premium amounts of all the other health insurance even though they have 3
or less. I get the estimation results using the rest of the sample. I assume that age and squared
age divided by 100 are the time-varying endogenous variables since they may be correlated with
the individual e�ects a�ecting premium rates. As before, I assume that the only time invarying
endogenous variable is education since more educated people are more likely to be exposed to a
healthy workplace. This should be correlated with individual e�ects determining premiums. 40 of
11, 544 imputed values are negative, and I replace them with 0 in the analysis. This is not surprising
given that 1.9 percent of the individuals with observed premium levels for other health insurance
do not pay any premium.
7 Results
In this section, I de�ne a simpler model to start solving the problem, and then provide the solution
to it. I plan to extend this model to my more comprehensive model.
I do not distinguish between unemployed and non-participants in the simpler model, and employ
a relatively simple �ow utility function, which is given by
ut(xt, dt, θ) =cθC1t
θC1+
3∑i=1
3∑j=1
θiI{lfpt = j}+ θijI{lfpt−1 = i, lfpt = j}I{lfpt−1 6= lfpt}. (30)
I have 2 constraints for the simpler model, which are wage determination and asset accumulation
32
equations. The wage determination equation employed is a simpler version of Eqn.s (11) and (12)
which is obtained by dropping health status from Eqn. (11), i.e.
log(wt) = λI(lfpt = 1) + %+ τt+ARt, (31)
where ARt denotes the autoregressive component of wages.
ARt = ρARARt−1 + ηt, ηt ∝ N(0, σ2η). (32)
Moreover, the asset accumulation equation is a simpler version of Eqn. (13) given as:
At+1 = (1 + r)At + 1000wtQt − ct, (33)
where I ignore taxes, Social Security bene�ts, SSI, spousal income, health expenses and health
insurance premiums. The reason for omitting all of these variables except taxes is that I will use
the equations given in Section 6 dropping individual speci�c e�ects and error terms to produce
expected values in the dynamic programming model. I omitted taxes at this step since tax rates
depends on the overall earnings. These variables will be added in the future versions. I use the
budget constraint given in Eqn. (14), i.e.
At > −100000 + (t− 58)100000
37∀t ∈ [58, 95] . (34)
The Bellman equation I want to solve is given by
Vt(At, wt−1, lfpt−1) = maxct,lfpt
[ut(xt, dt, θ) + βEtVt+1(At+1, wt, lfpt)]. (35)
I employ the simulated method of moments strategy where I match part-time and full-time
employment rates in the simulated data with the true data by age for people aged 59− 85. First, I
calibrate the parameters λ and r, and estimate %, τ , ρAR and σ2η. I discretize the state space using
At ∈ [−80, 000, 1, 360, 000] with increments of 80, 000 and wt ∈ [1, 57] with increments of 8. In total,
I have 456 di�erent state points. Then, I solve the dynamic programming model backwards starting
33
from the terminal age 95. I assume that at age 95, everyone is non-participant and consumes all of
their assets, i.e. c95 = A95. The next step is to randomly draw 2, 000 observations from the data.
Then, I simulate the decision rules of these people looking at the predictions of the simpler model.
In this simulation, I need to calculate the expectation EtVt+1(At+1, wt, lfpt). I do this using Gauss-
Hermite quadrature of order 5 where the uncertainty comes from wage determination equation. I
also produce sequences of lifetime wage shocks for 2, 000 simulated individuals. Subsequently, the
distance between the simulated and the true data moments are computed. This process is repeated
with di�erent choices of preference parameter vectors. The solution is given by the preference
parameters minimizing the distance between the simulated and the true data moments.
The preference parameters I want to estimate are β, θC1, θi's and θij 's. Note that there is nothing
in the simpler model about retirement. That will normally make people more likely to work than
observed data. However, I match observed participation rates with simulated ones which rules out
this possibility from age 59 to 85. Therefore, the lower participation rates in higher ages should
be re�ected in the simpler model with huge disutility parameters from participation compared to
being non-participant, i.e. θ3 � θ2 � θ1. Moreover, I expect θ21 > θ31, θ12 > θ32 and θ13 > θ23
where θij should be positive when i < j and negative otherwise.
7.1 Estimation
I set λ equal to 0.20 to ensure that full-time workers earn 20 percent more than the part-time
workers as seen in the data. This is called as part-time penalty by French (2005). Moreover, I set r
equal to 0.04. I estimate τ and % by running a �xed e�ects estimator of log(wt)−λI(lfpt = 1) on t
where τ and % are given by 0.021 and 1.029 respectively11. Then, I predict the combined residual,
i.e. �xed e�ects plus the disturbance term12. Finally, I estimate ρAR = 0.809 and σ2η = 0.065 by
running combined residual on the lagged combined residual using pooled OLS13. This estimation
implies that wages are close to random walk.
11and their standard errors are given by 0.001 and 0.065.12If we consider Eqn. (28) or any other equation estimated by Hausman and Taylor's (1981) method in Section 6,
this is given by αi + uit.13I do not include any constant in that regression. Standard deviation of ρAR is given by 0.027.
34
7.2 Solution of the DP Model
Now, I am ready to solve the DP model for the simpler model. I discretize ct ∈ [10, 000, 290, 000]
with 20, 000 increments. Since the value function depends on 2 continuous variables, namely assets
and wage, I use a weighted average of the 4 grid points closest to the intermediary point to interpolate
the value function if the intermediary point is in the set covered by the grid points. Otherwise, I
use the closest point on this set and interpolate the value function by assuming linearity. This is
not a perfect method, but it should not a�ect solutions much.
If the respondent is non-participant in the previous period, I generate his current wage by
ignoring the autoregressive part of the wage equation since I do not observe any wage in the previous
period, but include uncertainty. This is reasonable since autoregressive part of the wage is the
portion unexplained by observables with a zero average. In solving the model, if there is a 2 percent
or more chance that a person cannot satisfy borrowing constraint, i.e. Eqn. (33) does not hold, I
force him/her to work as full-time and consume nothing. Note that this constraint is not satis�ed
for only 0.12 percent of the person-year observations in the data. I simulate 2, 000 person-year
observations from the data using Mersenne Twister random number generator methodology. I use
these observations as initial points to simulate histories using the simpler model. Subsequently, I
obtain simulated moments using these histories. I use the simulated method of moments to estimate
the preference parameters. In doing this, I use the the inverse of the variance covariance matrix
of the data moments I employ as the weight matrix to calculate the distance. This methodology
provides consistent estimates. The variance covariance matrix is estimated via bootstrap.14
I interpolate consumption and labor force participation decisions using weighted average of the
relevant points in the grid to generate moments for simulated individuals, as I did in solving the
simpler DP model. Then, I generate wages and assets using wage determination and asset accu-
mulation equations in the simpler model where I include randomly generated normally distributed
wage shocks.
I obtain the estimates of the preference parameters minimizing the weighted distance between
simulated and true data moments employing Nelder Meade algorithm. These estimates are provided
14Since husband-wife decisions are interrelated, one should expect them to see in the sample together. This is whyI decided to sample households rather than individuals for bootstrap. For each bootstrap, I draw 12, 512 households,which is the total number of households in the dataset, with replacement. The estimate of the variance-covariancematrix is obtained using 1, 000 bootstrap replications.
35
Table 9: The Estimates of the Preference ParametersParameter Coef.
β 0.85
θC1 0.32
θ1 189.22
θ2 416.63
θ3 776.11
θ21 −7.24
θ31 −48.70
θ12 412.30
θ32 −78.39
θ13 277.01
θ23 250.99
in Table 9. The resulting distance is pretty small, which is only one-fourth of the smallest diagonal
element of the inverse of the estimated variance-covariance matrix. These �ndings seem very good
since they satisfy my expectations that θ3 � θ2 � θ1, θ21 > θ31, θ12 > θ32 and θ13 > θ23 where
θij should be positive when i < j and negative otherwise. The contribution of a consumption level
of $50, 000 to the utility function is just around 100 utils. Parameter estimates shows that that
disutility from participation compared to non-participation is huge.
8 Conclusion
Strikingly high labor force participation rates of people beyond normal retirement in the U.S.
compared to numerous European countries decrease the �scal cost of the Social Security on the
U.S. economy and deserve a special attention. Due to the aging population in developed countries,
it is important to understand the labor force participation decisions of older people. For this
purpose, I estimate the dynamic programming model provided in Section 5. I conduct a preliminary
multinomial logit analysis to get a feeling about the data generating process and the state variables
in the dynamic programming model. In this way, I determine to use assets, wages, health status,
medicare, number of other health insurance, health expenses and spouse employment status in the
last period as the state variables. I also include Social Security bene�ts, pension bene�ts, spousal
income, out-of-pocket health expenses, health insurance premiums and supplemental security income
in the data generating process. I provide the solution of a simpler dynamic programming model.
The 2000 amendments in the Social Security rules and the tax rate schedule are some of the
36
important determinants of older people's decision to stay in the labor force. I show that the current
law make elderly workers beyond normal retirement age more likely to start collecting retirement
bene�ts. In the future work, I may extend this model to a longer period starting from 1992 paying
special attention to budget constraint to �nd evidence for the e�ect of the 2000 amendments on
older people's labor force decisions. Another extension will be to conduct a counter-factual analysis
on England, which has a very similar retirement system as the U.S. Before doing this, I should work
on ELSA to see if there are similarities among a subset U.S. workers and British workers. By this
way, I can conclude that my model may work for the case of England. Moreover, I will impose Social
Security rules and the tax rate schedule in England and Spain to the model to see how Americans
would adapt their behavior under these rules.
The estimation of the whole model will take some more time unfortunately due to the complexity
of the model. One good thing is that the simpler model covers all the steps that will come up again
in the more comprehensive model. In other words, there is not much di�culty left in terms of coding
the program. The only di�culty will arise from the computing time which I plan to solve by using
parallel programming. I already learned how to use Message Passing Interface (MPI).
37
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APPENDIX
Here we provide the LFPR trends for elderly in the US broken down by age, gender, marital status
and education.
39
Figure 3: LFPR Trends in the U.S. for Married Elderly High School Graduates by Gender and Age
����������������������������������������������
���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��������
������� �� ������� �� �����������������
Source: Obtained using March CPS Data
Figure 4: LFPR Trends in the U.S. for Married Elderly College Graduates by Gender and Age
����������������������������������������������
���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ���� ���� ���� ���� ��� ��� ���� ���� ���� ���� ��������
������� �� ������� �� �����������������
Source: Obtained using March CPS Data
40
Figure 5: LFPR Trends in the U.S. for Single Elderly High School Graduates by Gender and Age
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Source: Obtained using March CPS Data
Figure 6: LFPR Trends in the U.S. for Single Elderly College Graduates by Gender and Age
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Source: Obtained using March CPS Data
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