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: yavuzoglu@wisc.eduI 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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��� �� ������� �� �����������������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

9 References

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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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