THE EFFECTS OF THE SOCIAL SECURITY SYSTEM
ON THE LABOR FORCE EFFORT OF OLDER MALES
JAMES LEE SWOFFORD
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
I would like to thank G.S. Maddala for directing me to
this topic, J.R. Davis, J.C. Ilenretta, and L.W. Kenny for
their many useful comments, S.B. Caudill, R.P.H. Fishe, and
R.P. Trost for their aid and support and Katherine Williams
TABLE OF CONTENTS
LIST OF TABLES . . .
LIST OF FIGURES . . .
ABSTRACT . . .
CHAPTER I INTRODUCTION AND PROBLEM STATEMENT ..
CHAPTER II THE LITERATURE CONCERNING THE EFFECTS OF
SOCIAL SECURITY ON INDIVIDUAL BEHAVIOR.
CHAPTER III A CONCEPTUAL FRAMEWORK ANALYZING THE
EFFECTS OF THE SOCIAL SECURITY SYSTEM
ON TIE INDIVIDUAL'S LABOR FORCE
DECISION . .
CHAPTER IV ECONOMETRIC LITERATURE CONCERNING
THRESHOLD MODELS . .
CHAPTER V AN ESTIMABLE MODEL OF TIE EFFECTS OF THE
SOCIAL SECURITY SYSTEM ON LABOR FORCE
OF THE ELDERLY.. . .
CHAPTER VI THE RESULTS FROM ESTIMATING THE MODEL .
CHAPTER VII CONCLUSIONS DRAWN FROM THE EMPIRICAL
RESULTS . ... 74
APPENDIX 1 A DESCRIPTION OF THE DATA AND ITS
SOURCES . .
BIBLIOGRAPHY . . .
BIOGRAPHICAL SKETCH . . .
LIST OF TABLES
1 LABOR FORCE PARTICIPATION RATE (MALES) 3
2 PERCENTAGE EMPLOYED DURING THE SURVEY YEAR 8
3 DETERMINANTS OF LABOR FORCE PARTICIPATION FOR
AGED TOTAL POPULATION AND FOR AGED MALE POPU-
LATION 1960 .. . 13
4 RESULTS FROM THE BOSKIN PAPER . .. .18
5 OLS RESULTS OF THE ENTIRE DATA SET . 61
6 OLS RESULTS INCLUDING EARNING TAX VARIABLE 63
7 OLS RESULTS OF THE NON-LIMIT INDIVIDUALS ... .65
8 TOBIT RESULTS INCLUDING THE EARNINGS TAX VARIABLE 67
9 TOBIT RESULTS WITHOUT THE ENDOWMENT VARIABLES 71
10 TIE CHANGE IN HOURS FOR A UNIT CHANGE IN X AT THE
MEAN OF THE SAMPLE . .. .73
11 DESCRIPTIVE STATISTICS OF THE DATA SET .. .80
LIST OF FIGURES
3-1. Future Oriented Person .. .. 30
3-2. Present Oriented Person . .. .32
3-3. Most Frequently Occurring Time Path of Labor 34
3-4. Effects of the Social Security Payroll Tax 38
3-5. Effects of Social Security Benefits 40
3-6. Effects of the Social Security Retirement Test .41
4-1. Simple Tobit Model . ... 43
4-2. Tobit Friction Model . 46
4-3. Two Limit Tobit Model . 49
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE EFFECTS OF THE SOCIAL SECURITY SYSTEM
ON THE LABOR FORCE EFFORT OF OLDER MALES
James Lee Swofford
Chairman: G.S. Maddala
Major Department: Economics
This research examines the relationship between the
United States social security system and the labor force
effort of elderly white males. This relationship is modeled
over the individual's life-cycle. The model is used to pre-
dict effects and these predictions are compared to empirical
results. Particular emphasis is placed on the effects of
the social security earnings tax. The empirical results
confirm the theoretical notion that the earnings tax reduces
labor force effort later in life. However, the predictions
and results imply that changing no single variable can
overcome the projected shortfall in revenue for the social
security trust fund.
INTRODUCTION AND PROBLEM STATEMENT
The decision of how much and when to participate in
the labor force is among the most important decisions an
individual economic agent ever makes. In importance it
ranks with an individual's choice of education, occupation,
and location. These are the decisions which shape the qual-
ity of life a person will have.
The choices to retire fully or in part and at what age
are also a complex maximization problem with multiple con-
straints. These choices are dependent on such factors as
the individual's expected future health, income from both
private and public sources, and consumption needs. Two of
the major constraints concerning these choices are the social
security system and the individual's time path of wages in
The social security program has three significant im-
pacts on the individual's situation. All throughout the
work-life the individual is subjected to forced savings due
to the social security tax on earnings up to the current
maximum level. Eligibility for benefits may be chosen at
age sixty-two with a reduced benefit, at age sixty-five
with a full benefit or at any age between sixty-two and
sixty-five with a prorated benefit. For up to ten years
after becoming eligible for benefits, there is a reduction
in benefits of $0.50 per $1.00 earned for all earnings above
the minimum amount all beneficiaries are allowed to earn.
The amount of benefits depends on an individual's lifetime
earnings, omitting the five lowest years. Also eligibility
can now be delayed past age sixty-five,-and benefits are
adjusted upward to compensate in part for the benefits for-
feited due to additional work.
In the last 40 years, labor force participation rates
for older white males have been declining (see Table 1).
This observation has been the catalyst for a controversy
concerning the impact of social security on the labor force
effort of the elderly.
The trend of declining labor force participation by
elderly males began immediately after World War II. The
reasons it did not become noticeable until then are that
during the war labor markets were tight which implies high
wage levels along with the social pressure to work, it
took a period of time for individuals to adjust their life
plans to the fact of the social security system, and the
fact that coverage by the system broadened over time.
Since this trend has been noticed, research on the
effects of the social security constraint has concentrated
on white males. The reasons for this are that white males
have a better continuous work history in employment covered
LABOR FORCE PARTICIPATION RATE (MALES)
(Campbell and Campbell, 1976)
Age 1940 1950 1960 1970
55 89.5% 87.8% 89.9% 88.9%
56 89.1 87.8 89.0 88.3
57 87.8 86.7 87.8 86.7
58 86.9 86.1 86.7 85.8
59 85.6 85.1 85.1 83.7
60 81.9 82.1 83.2 81.3
61 81.4 81.4 80.7 79.2
62 79.7 80.0 78.6 72.7
63 76.9 77.6 75.7 67.5
64 74.4 75.2 70.0 63.1
65 66.9 67.7 53.6 47.1
66 62.0 62.9 45.9 41.9
67 57.8 58.2 41.9 38.6
68 54.9 54.2 39.5 35.4
69 51.4 51.2 36.6 31.5
70 44.0 44.5 33.2 26.9
71 40.8 42.0 29.0 24.6
72 37.4 39.0 27.8 22.1
by social security and the changes in status which women
and blacks have undergone during this period have tended to
counteract the effects social security might have had on
them. Therefore, white males make a better group for study-
ing the effects of social security.
This area of research is of current interest because
the role and financing of the social security system are
currently being scrutinized. A large and growing short-fall
of revenue, which has been projected as the population
matures and the labor force participation of elderly men
falls, has been an important reason for study of the social
security system. This short-fall has also led to the ques-
tioning of the incentives provided by social security.
Therefore, it is important that the effects of the system on
individual behavior be fully understood so that decisions
concerning its future can be based on realistic alternatives.
It is important to know whether people are retiring
earlier because of a change in relative preference for
leisure or of incentives provided by the social security
constraint. If it is the former case, the solvency of the
system can only be realistically assured via increased
revenue. If it is the latter case, encouraging a longer
average work life can promote solvency from the out-lay side.
Only certain types of employment are covered by the
social security system. Covered employment has grown
from few types of employment to virtually all types of
Following this tradition of concern about the indi-
vidual incentives of the social security system, this
research will study micro data. Also this paper will
attempt to empirically measure the substitution effect of
the retirement test reduction of benefits. No other micro
data estimate of the magnitude of this test has been found.
The next section of this paper will review the liter-
ature concerning the effects of the social security system
on individual behavior. These effects will be modeled for
individual behavior in the third section of this paper.
Then the fourth section will review the literature on esti-
mation of threshold models which will be used for this re-
search. The empirical model and estimation techniques will
be the subject of the fifth section. The results of the
estimation will be set forth in the sixth section. The
final section contains conclusions and policy implications
from the model and the results.
THE LITERATURE CONCERNING THE EFFECTS OF
SOCIAL SECURITY ON INDIVIDUAL BEHAVIOR
As this trend of declining labor force participation
among elderly men became apparent, a literature evolved to
explain this phenomenon. This literature can be divided
into many categories. One scheme of looking at this liter-
ature divides it according to the sponsor of each paper
(Campbell and Campbell, 1976). Papers on the topic of the
behavioral effects of the Old-Age and Survivors Insurance
(OASI) program can be divided into those sponsored by the
Social Security Administration and those written under other
auspices. This division also tends to divide this literature
according to the author's conclusions.
Many of the papers by the Social Security Administra-
tion staff have been purely descriptive papers setting forth
the data. These papers have generally come to the conclusion
that OASI has caused no discernible change in the behavior
of the recipients. However, this conclusion has not been
asserted as vigorously in more recent research.
A study by Wentworth was one of the earliest research
attempts to ascertain why individuals decide to become
eligible for OASI (Wentworth, 1945a). She found that only
about five percent of the 2,380 men who were interviewed in
1941 and 1942 said that they were retiring because they
wished to do so and while they still were in good health.
However, 55.6% of the beneficiaries in this study reported
the loss of their job for involuntary reasons (termination
or lay-off), indicating that labor market conditions were
not yet fully reflecting the tightness that existed during
World War II. In a second paper that year, Wentworth
investigates why 37% of the beneficiaries interviewed
returned to work of some description during the survey
year (Wentworth, 1945b). This paper foreshadows the debate
that arose later by showing that the higher an individual's
unearned income the less likely he returned to work. Also
she presents a series of synopses from interviews with indi-
viduals who had returned to work. These interviews tend to
reinforce the idea that individuals return to work because
of monetary considerations, not because their health improved
(see Table 2).
Unfortunately the import of this idea was overlooked by
Wentworth and later Social Security staff writer. The fact
that the level of unearned income was related to the deci-
sion to re-enter the labor market indicates that OASI was
not unrelated to labor force effort.
These data came from a set of four surveys which were
completed by mid-year 1942. See footnote 2 in Wentworth
PERCENTAGE EMPLOYED DURING THE SURVEY YEAR
Unearned and and
Income Baltimore St. Louis Atlanta Los Angeles
Total 25.4 37.6 45.6 38.1
$300 33.6 51.4 59.9 40.2
$300-$599 33.2 40.2 41.5 48.1
$600-$899 20.0 24.2 34.7 33.6
$900 7.1 29.8 26.8 26.6
Includes: 12 month OASI insurance benefits, income from
assets, private annuities, retirement pay, veterans'
pensions, and imputed rent on owner-occupied dwellings.
In two articles in the 1950's, Stecker followed the lead
of the Wentworth article concluding "Voluntary quitting work
to enjoy a life of leisure is rare among old-age insurance
beneficiaries" (Stecker, 1955, p. 12). She ignored the
relationship between the level of total unearned income and
retirement. A table on this relationship, like the one in
the second Wentworth paper, was conspicuously missing in
this paper (Stecker, 1951). Rather, Stecker's conclusion
was based on the answer the beneficiary gave to a survey
question concerning the reasons for retirement. These two
papers were written to report the results of the first
national survey of over 17,000 beneficiaries, which was con-
ducted in 1951.
The next national survey was in 1963 and it was the
first survey to sample both beneficiaries and nonbenefici-
aries. Although his main point was that few (19%) of the
men interviewed said they voluntarily retired, Palmore did
point out that persons with low wages were more likely to
retire than higher paid individuals (Palmore, 1964). How-
ever, rather than seeing this as showing the effect of rela-
tive replacement ratios, the ratio of the benefit to former
earned income, on work effort, this was seen as evidence of
a benign OASI since those likely to have higher assets were
not as likely to retire.
Again, a chance to perceive an effect of OASI on indi-
vidual behavior was missed by the Social Security
Administration staff. Also, again there was no mention of
the relationship between unearned income and the decision
to return to work.
Following this tradition of finding health the most
important factor in the decision, Reno noted that this was
the main reason given by early retirees interviewed in the
"Survey of Newly Entitled Beneficiaries-' (Reno, 1971). This
survey was not as complete as the 1963 survey since it in-
cluded only OASI recipients. Also, even though this survey
took place during a period of low unemployment, job elimina-
tion and compulsory retirement were listed as important
reasons for retirement.
Finally, in a 1976 article, Bixby reviewed many of the
articles which had examined either the "Survey of Newly
Entitled Beneficiaries" or the "Retirement History Study"
(Bixby, 1976). This article somewhat modified the standard
conclusion of articles by Social Security researchers. It
allowed that the existence of pensions including OASI made
the individual regard retirement more favorably. Still,
there was no table on unearned income and return to the
labor force, and the fact that lower income individuals
retired more frequently was not seen as an effect of the
As many authors have noted, a probable cause for these
papers underestimating the influence of OASI is the use of
the interview technique (Campbell and Campbell, 1976).
There are many ways that this technique could cause
individuals to bias their response to a subjective question.
Individuals do not tend to think of themselves in a negative
way, and they do not like for others to think of them in
such a manner. Therefore, they are not likely to say or
believe they are reducing work for other than socially
acceptable reasons, such as ill health. Another point is
that all of these surveys have been conducted by represen-
tatives of the government that issues the benefits. This
means some interviewees might bias their answers out of
fear or caution, despite any reassurances they were given.
In conclusion, it is interesting to note that at vari-
ous times the analysts of the Social Security Administration
revealed that as unearned income increased, so did the
likelihood of the individual remaining retired and that the
lower an elderly person's preretirement income, the more
likely he retired. However, this information was neglected
in favor of the subjective survey answers mentioned above.
When one looks at the literature by academic and other
non-Social Security Administration staff writers, it is
clear that this literature is not as homogeneous. There is,
however, a unifying method in these papers. Academic writers
tend to analyze their data via regression analysis as
opposed to the descriptive nature of papers by the Social
Security Administration writers. When considering academic
papers it is advantageous to look at them according to the
type of data analyzed in each study. Some of the earliest
studies using regression techniques analyzed international
cross-sections to demonstrate that labor force participa-
tion rates among elderly men are affected by the existence
of retirement benefits.
In their book on Social Security, Pechman, Aaron, and
Taussig devoted a chapter to Social Security and retirement
(Pechman et al., 1968, Ch. VI). In this chapter they pre-
sented the verbal arguments concerning the impact of OASI
and the earnings test on the retirement decision. They dis-
cussed the income effect of benefits increasing the demand
for leisure if it is a normal good and the idea that the
earnings tax decreases the price of leisure by reducing the
return from work, thus increasing the quantity of leisure
demanded. Based on aggregate international data, they pre-
sented regression results linking labor force participation
rates to replacement ratios in a negative relationship. (see
Feldstein provided more results from international
cross-sections, along with developing a model of the Social
Security program's effect on savings (Feldstein, 1977).
He also found that labor force participation rates among
the elderly are negatively affected by replacement ratios.
In addition, he found that if a system had a retirement test,
it too was negatively related to labor force participation.
Further, Feldstein hypothesized that the growth of
coverage of a social security program would tend to reduce
DETERMINANTS OF LABOR FORCE PARTICIPATION FOR AGED
TOTAL POPULATION AND FOR AGED MALE POPULATION 1960
(Pechman et al., 1968)
% of Popula- Per Capita Ratio of Per
tion over National Capita Income
R Constant Retirement Agea Incomea to Wagesa
Total Population Age 65 and Over:
.4860 .43136 .89261 .06698 .25326
(2.2231) (2.5722) (2.7105)
.3593 .34483 .06649 .27496
Male Population Age 65 and Over:
.5589 .79385 -1.86350 .13378 .41291
(2.7907) (3.0893) (2.6573)
.3718 .61321 .13275 .45821
participation and that the aging of the system had two
effects. As a system ages, he suggested, more individuals
correctly anticipate the change in their situation over the
life-cycle, and, therefore, the rate of retirement would tend
to increase. However, this would be moderated by the un-
anticipated gain during the early years of the system that
allowed individuals who were saving for retirement to retire
earlier than they had planned. Also Feldstein pointed out,
that as life expectancy at the age of retirement increases,
it is likely that both more goods and more leisure will be
demanded. Unfortunately the estimated coefficients of these
factors were either ambiguous or too imprecise for infer-
ences to be drawn concerning the validity of these ideas.
Of course, two of these three effects are likely to be
small compared to the effects of benefits and the earning
tax. Also the effect of an aging system is likely to be rel-
atively small since it contains two counteracting effects.
These two papers clearly showed that social security
systems are related to the historical decline in labor force
participation rates among the elderly. Of course, these
studies have been subjected to the usual criticisms of
international cross-sections such as aggregation of unlike
earning tests, but they did bring the idea that social
IThis aging effect, if it occurs, should only take place
during the first twenty to forty years after the program
has been enacted.
security systems act only in a passive manner under scru-
tiny, a very important point. As a result of these papers,
the literature has turned to the question of the variables
that enter into the individual's decision to alter his labor
force status. To investigate the individual's decision,
researchers have had to turn to longitudinal data and away
from international cross-sections.
Soon after the Pechman, Aaron, and Taussig study,
Bowen and Finegan wrote an exhaustive study covering all
facets of labor force participation (Bowen and Finegan,
1969). This study used a microdata set from a one-thousandth
sample of 1960 census data and multiple regression analysis
to determine the factors contributing to the secular decline
of the labor force participation rate of men sixty-five
years old and older. After adjusting for other factors,
such as the aging of the population over sixty-five, they
found that other income accounted for 52% of this decline in
participation. Since social security transfers were 54% of
the other income variable, it accounted for 28% of the de-
cline in participation. This figure can be contrasted with
the 9.5% figure they estimated as the role of health to show
that Bowen and Finegan found OASI to have three times the
impact on individual behavior that health had. Also, in
this study, an attempt was made to separate the income effect
of benefits from the substitution effect of the earnings tax
in the change in participation in the labor force as age
increased from sixty-four to sixty-seven. They found only
one-half the decline in the participation rate due to other
income and health, leaving the other half to be attributed
to the substitution effect as a residual. Of course, from
an econometric point of view, this is not a valid way to
estimate the effect of a variable.
Since the decision to retire is a qualitative one and
longitudinal data are analyzed, multiple regression analysis
will not suffice. Boskin recognized this and used a logit
model to explain the decision to withdraw from the labor
force (Boskin, 1977). He used data from the "Panel Study
of Income Dynamics" for his study. Boskin's results implied
that if social security benefits were increased from $3,000
to $4,000 annually, the probability of an individual being
retired increased from 7.5% to 16%. He also used a net
earnings variable (earnings net of retirement test reduc-
tions in benefits) to estimate the effect of the retirement
test on behavior. He found that the retirement test also
tends to increase the chances of retirement. Also, Boskin,
recognizing that retirement is not totally an all or none
decision, estimated a model for quasi-retirement which he
defined as earning less than half but more than a quarter of
the individual's former salary. Current earning under a
quarter of former earnings was Boskin's definition of re--
The most controversial result in the Boskin paper was
his finding that health had a positive but insignificant
effect on the decision to participate in the labor force (see
Table 4). This result could have been due to his small
sample size (he only had 131 observations) along with his
use of estimation procedures having known asymptotic but un-
known small sample properties, or it coOld have been due to
the proxy he used for health, hours ill, capturing unex-
pected health problems among his sample. If the latter
were the case, it would imply that the life-cycling indivi-
dual will work more to meet unanticipated health problems,
rather than deplete his estate.
With its controversial finding, the Boskin result
prompted more studies, including one by Quinn in which he
analyzed the first wave of the "Retirement History Study"
data which had been collected by the Social Security Admin-
istration (Quinn, 1977). He found that social security
eligibility and health status played the most important
roles in the decision to retire. Further, he found that the
social security effect was greater on those individuals in
poor health than those individuals not in poor health.
Unfortunately, Quinn used the results of a subjective
question concerning the interviewee's health rather than
following Boskin's lead and attempting to find a less sub-
jective proxy. Also, even though the "Retirement History
Study" is a large survey, it has been a-continuing one and
RESULTS FROM THE BOSKIN PAPER
Variable Coefficient (X 10)a
Social Security Benefits plus
Income from assets 0.160
Net earnings -1.203
Age = 65 dummy 1248
Spouce's earnings -0.178
Hours ill -1.595
Log Likelihood = -144.742
x2 (L) = 7.23
Asymptotic standard errors in parentheses.
thus was constructed so that few of the respondents had
actually retired at the time of the first wave of inter-
views. Thus, Quinn shared the same problem Boskin had of
using a relatively small sample in conjunction with esti-
mation procedures with known asymptotic but unknown small
To this point, the regression analysis papers have pro-
gressed from analyzing aggregate data to analyzing micro-
data. The latest article moved back to aggregate time-
series data to discuss an idea that would need a sample of
people over their entire work life to fully investigate.
Burkhauser and Turner made the point that the labor
force decision is not totally a one time-period decision
and thus must be viewed in a life-cycle context (Burkhauser
and Turner, 1978). They pointed out that, besides the
acceleration of the decline in labor force participation
rate of elderly men, there has been another change in the
aggregate behavior of men. This change, too, can be traced
to about the time of the enactment of OASI. It is the halt
of the historic decline of the average work week of men.
Burkhauser and Turner presented aggregate regression re-
sults that suggested that labor force participation rates
among men are positively related to the social security
wealth variable used by Barro.1
This variable was developed in the debate between Barro
and Feldstein concerning the effect of OASI on savings.
This positive relationship suggests that individuals
are substituting work during the years before they are
eligible for OASI for leisure after they have become eligi-
ble for OASI. The reason this substitution would take place
is the retirement test reducing the real wage for up to ten
years after the individual becomes eligible for OASI. Since
wages are relatively lower, and thus leisure relatively
cheaper after eligibility for OASI, this substitution over
time is congruent with predictions of economic theory.
As a group, these papers by academic writers have es-
tablished, by regression analysis, that the social security
system is related to the changes in labor force participa-
tion since its enactment. These papers have also estab-
lished that the social security system is an important
variable in the individual's labor force participation de-
cision. They also have made attempts to quantify a health
variable in a less subjective manner than just asking the
individual how he feels his health is.
There is really no absolute debate between the two
groups of writers any longer (Clark et al., 1978). Rather,
it is a question of the degree of the effect of OASI
versus the effect of health. This is an important question
now, since society must now decide what kind of social secur-
ity program it wants in the future, in light of the pro-
jected shortfalls in revenue and its effect on individual
Before these effects can be measured, a model must be
developed to explain how an individual will react to a
social security program. This is done within the context
of a life-cycle model in the following section.
A CONCEPTUAL FRAMEWORK ANALYZING THE EFFECTS OF THE SOCIAL
SECURITY SYSTEM ON THE INDIVIDUAL'S LABOR FORCE DECISION
The manner in which the social security system affects
individual behavior will be examined here within the context
of a utility maximization problem.1 The individual is pic-
tured as maximizing lifetime utility. Thus, the individual
attempts to maximize:
t e [U i(Cit) + Vi(Lit) dt] (1)
in the absence of any constraints. Under this specification,
)i is the individual's internal rate of time preference, t is
continuously changing time, Ui is a functional relationship
expressing the individual's desire for consumption of goods,
Cit is the consumption of goods by the individual in time
period t, Vi is a functional expression of the individual's
desire for consumption of leisure and L. is the consumption
o leisure by the individual in time period t. Further, the
limits of integration are to, which is the start of the indi-
vidual's economic life, to T which is that t in which the
This model is an application of one set forth in Blinder
individual expires. Also the functional relationships are
assumed to be strictly convex, twice differentiable and
invariant with t.
Since consumption of goods is related to purchasing
power at any one instant t,
cit = f(rKit wthit, sit). (2)
Here cit is the proportional rate of ch-ange of goods consump-
tion, r is an aggregate rate of return on non-human assets,
K. is the individual's assets in time t, w. is the market
wage rate of the individual in time t, h. is the number of
hours the individual works during time t and sit is the indi-
vidual's savings during time t. Further, the savings of the
individual are defined as the rate of change of the capital
it = Kit, (3)
and consumption plus savings must equal earnings plus the
return on non-human assets:
Cit + st = Withit + rKit (4)
Of course the amount of leisure consumed by the individual
in any time period t depends on the number of hours the indi-
vidual works during the time period:
Lt = Mit hit (5)
Economic life begins when the individual is allowed to
choose between alternative uses of his time. So, t
would fall approximately between ages 16 to 21 in our
where Mit is the amount of usable time the individual has
to allocate. Equation 2 through 5 act as constraints on
the level of utility the individual can achieve during his
These constraints can be substituted into the objective
function, equation 1, to construct an optimal control problem
with K.t as a state variable and sit and h. as control vari-
ables. As such, the Hamiltonian is:
H(K,s,h) = e [Ui (rKi + Wi hi sit)
+ V (Mit hit) + it sit (6)
The first order conditions for a maximum are:
p t ,
IHI/aK = re Ui (Cit) + it = 0 (6.1)
-i t IU
H/D)s = -e Ui (Cit) + t = 0 (6.2)
-p.t -p.t ,
1IH/3h = -e V. (L ) + e U. (C ) W. < 0. (6.3)
i it 1 it 1 -
Usable time is twenty-four hours per day minus time re-
quired for sleeping, eating, etc.
Bequeaths can be considered a consumption good and thus the
transverse condition equals zero.
Condition 6.3 implies that if iH/Jh is less than zero then
h. will be equal to zero. This point is an important one
to notice especially in connection with estimation of the
Further, when the individual spends some time working
and therefore condition 6.3 holds as an equality, the static
utility maximization condition results:"
V. (L t) / U (Ci) = Wit. (7)
This result means that to maximize utility over his life-
cycle the economic agent must equate his marginal rate of
substitution of goods consumption for leisure consumption to
the rate at which he can substitute leisure for goods in the
market place. This is the standard neoclassical one time
The time path for the working life of the individual can
be found by differentiating equation 7 logarithmically:
T o i f
Lit V (L ) Lit C Ui (Ci) C it W (8)
Vi (Li) Lit Vi (C i) Cit W.
i 1 it i 1 it it
From condition 6.1 and 6.2 the consumption term in equation 8
can be found. First condition 6.2 must be differentiated
with respect to time, giving:
-I t ,, -U Ct .
Pi Ui (Cit) e U (C) e Cit "it = 0. (9)
Next, when equation 9 is set equal to condition 6.1, the con-
sumption term in equation 8 is found to be:
Cit Ui (Cit) Cit
S= r -pi. (10)
Ui (Cit) Cit
This result indicates that if the market rate of time prefer-
ence for assets, r, is always greater than the individual's
subjective rate of time preference, p., then the goods con-
sumption path of the individual will always be increasing.
This will happen because at such a market rate of interest
this type of individual can always gain utility by foregoing
current goods consumption, lending at the rate of interest,
and consuming goods in the future. This future goods con-
sumption will have present utility greater than the utility
from foregone consumption of goods because assets growth was
greater than the internal discount factor. Of course, by
similar reasoning, if r were always less than p. the indi-
vidual can increase his lifetime utility by borrowing to
consume more goods in the early part of his life. Thus,
this case will lead to continuously declining consumption of
From equations 7 and 9 we find that the equation of
motion of leisure for the individual is:
Lit r pi Wit/Wit (11)
t it V (Lit i ( it
So the time path of leisure and by inference, the life-cycle
pattern of work, depends on the market rate of time prefer-
ence for assets, the individual's internal rate of time pref-
erence, the time path of wages for the individual, and the
denominator term on the right hand side of equation 11.
This term will only affect the slope of the time path, not
The denominator of equation must have a positive value.
Lit is some number of hours and thus makes sense only if it
is positive. V'(Li ) is likely negative. The reasons for
this are the assumed convexity of the functions stated
earlier and that we observe all individuals consuming both
goods and leisure. If the utility from leisure were increas-
ing at an increasing rate, then the utility maximizing indi-
vidual would consume all leisure. V'(Li ) is positive since
this function is assumed strictly convex and since the indi-
vidual can not be compelled to take an extra hour of leisure
and only will take it if utility is increased. Thus, we
have the negative of a negative times a positive, divided by
a positive and the denominator term must be positive.
The types of paths of work possible from equation 11
must be ascertained before the effects of the social security
system on these paths can be modeled. For understanding
these paths it is useful to notice what must be the case if
the individual is out of the labor force. For the maximum
to occur in a corner where hit is equal to zero condition,
Wit U (Cit) < Vi (L t) (12)
Since this is not the partial derivative of just any given
amount of leisure, rather this is the partial derivative
of the utility of leisure when all usable time, Mit, is
allocated to leisure, this partial derivative can be assigned
V. (Mi) = E.. (13)
1 1 1
Further it is now clear that for the individual to be re-
tired, the following condition must hold:
Wit Ui (Ci ) < ei. (14)
This basically says that the individual gets less additional
goods from working one more hour than the individual requires
to work one more hour.
Equation 11 determines the time path of leisure and by
implication the time path of work for the individual. It can
be seen that one of three situations must exist and determine
the slope of the time path since the denominator has been
shown to be a positive value. The three situations that can
exist are the interest rate net of wage trends, r Wi/Wi,
can be equal to, greater than, or less than the individual's
internal rate of time preference, p..
If there were an individual whose r W./Wi always
equaled p., then the individual's desired amount of work
would be a constant. Such a person would either always
work or never work. Whether such an individual works or not
depends on the individual's initial endowment of assets, K
If the individual can finance the level of consumption con-
sistent with allocating all time to leisure from K then
the individual does not work. Thus, the individual's
shadow wage is related to the initial endowment. The
higher Ko is the higher the individual's shadow wage is
and the less likely the individual works for any given mar-
ket wage. If K is not large enough, the individual works
some constant amount of hours. Such individuals have no
incentive to trade hours overtime. There are probably few
individuals always in this situation but this is also the
case anytime the desired work function reaches a maximum of
The second situation that can occur is for r W./W. to
always be greater than p.. If this were the case, then the
individual's desired amount of work would continuously
decline. This implies that if the initial endowment can not
finance the amount of consumption consistent with all time
allocated to leisure, the individual works early in life
with the desired amount of time allocated to work always
declining. Such a person might be called a future-oriented
person (see Figure 3-1). The reasoning for such a path is
that if the rate at which the price of consumption falls
Figure 3-1. Future Oriented Person
overtime net of wage trends, r W./Wi, is larger than the
rate at which individual's value of consumption falls over-
time, i the individual desires to work now to acquire
assets for future goods consumption. Again, there are not
thought to be a great number of individuals always in this
situation but this is the situation anytime the desired
number of hours worked is declining.
The last situation is the opposite of the previous one.
That is r W./Wi could always be less than p.. In this
case the desired number of hours worked grows throughout
life. Thus, if the individual is working, the individual
desires to work an increasing amount. Such a person might
be called a present-oriented person (see Figure 3-2). In
this case the rate at which the price of consumption falls
in the future net of wage trends is less than the rate that
the individual's value of consumption falls overtime. There-
fore, the individual gains in lifetime utility by working
later in life. Again, this path is unlikely to occur for
many people throughout their life-cycle but it is the situa-
tion any time the desired amount of work is rising.
Since the individual's growth rate of wages is likely
to change, the relationship between r W./Wi and pi during
the life-cycle, it is unlikely that any one of the situations
previously discussed will hold over a person's entire life-
cycle. While the three paths may not occur with more than
antidotal regularity, they do represent all the parts of the
most frequently occurring path.
- F: i
Figure 3-2. Present Oriented Person
The most frequently occurring path is one where the
individual starts economic life with the market wage lower
than the individual's shadow wage. This is the case since
the typical individual begins economic life with few skills.
Thus, the individual chooses not to work for a time. How-
ever, there are economic incentives for the individual to
use some of this time investing in human capital. This
investment causes the market wage to rise to the shadow
wage and then the individual begins to work. The wage rate
typically grows at a rapid rate early in life as the returns
to human capital accumulation are at their greatest.1 How-
ever, at some point in time, wage rates peak implying that
labor force effort will also peak. Then labor force effort
declines as the market wage falls due to human capital wear-
ing out (see Figure 3-3). Human capital is not renewed
since the economic incentives for this type of investment
are reduced by the shorter time horizon. Thus, eventually,
the individual's market wage falls below the individual's
shadow wage and the person retires. This indicates at some
point, S, in time WiU.(C it) will become greater than V'(Lit)
and the individual will work. At some point in time it
is likely that the rate of growth of wages will cause
r Wi/Wi to just equal pi and there, work effort will peak.
After work effort peaks, the rate of change of wage will
This human capital development comes from learning by doing
even if no explicit human capital investment is done.
I \ ^
I \ ^
_______________I ___ L f
likely become negative as the individual's job skills become
obsolete. This implies that at some point, R, W. V. (Cit)
will again fall below E. and the individual will retire. Of
course neither retirement period has to occur for all
individuals, as there may be some who value leisure very
To finish up this discussion of paths, two additional
points should be made again. First, any number of paths
with retirement in the middle or multiple periods of retire-
ment can be formulated. For the purpose here, only the
general paths needed to be shown, and additionally, the most
commonly occurring path was described. The second point is
that while the denominator term has not been fully discussed,
what it affects is the relative slope of the paths but not
the direction. So it does not affect the conclusions drawn
concerning the time paths of labor. Now that the paths have
been set forth, the question of initial conditions must be
discussed. As long as WiU.(C it) is greater than V (Lit),
the amount of time allocated to work increases as W./W.
No one enters economic life with nothing, even if they
only have the guarantee of society's assistance. The ques-
tion of the initial endowment of the individual is important
An interesting case would be a present oriented person with
professional athletic or modeling ability. This person
might have a period out of the labor force in the middle of
because where one starts can have a large effect on how long
various retirement periods last, how long the periods of work
last and even if the individual works at all. As stated
previously, the initial endowment of capital assets, K will
affect the usual path already developed.
If the initial endowment of the individual is changed,
then the desired amount of work would be altered. If K
were increased then desired work would be less and if K were
decreased desired work would be more.
In a static sense both goods consumption and leisure
consumption are normal good. As the increase in K allows
the individual to reach a higher level of total utility
more of both are chosen. The only way more leisure can be
taken is if work is reduced, so the time path of work is
So, for the most frequently occurring path developed
previously if the endowment has increased, the desired work
path will shift down. This implies both periods of retire-
ment altered and total work effort will change. If K
increased, then the individual starts work later, retires
earlier, and works less each period. The reverse will occur
if the endowment has declined.
Now the question of how the social security system
affects the time path of labor can be addressed. As stated
previously, the system has three separate ways it affects
individuals. They are: taxes paid to the system by wage
earners of all ages, benefits paid to older individuals, and
the earnings tax on any income earned over a specified amount
while receiving benefits. Each of these will affect the
lifetime labor-leisure trade-off in its own way.
The taxes paid to the system throughout life tend to
lower wages particularly for younger and older workers whose
entire income is under the taxing limit, and thus all subject
to the tax. This tax then would tend tg delay the start of
work and encourage retirement. Also it would tend to encour-
age more work effort during the middle years of the life-
cycle when the individual is more likely to be earning above
the taxing maximum (see Figure 3-4). This would occur if
an individual makes over the limit during the middle year
of work life. The fact that the social security taxes are
not applicable over the limit means the individual's mar-
ginal wage is increased by more than the time path of wages
when earnings pass the taxing maximum. Since the initial
endowment of assets is not changed by this tax, there will
not necessarily be an hour for hour trade-off over the
The effects of benefits paid to older individuals is
less straight forward. Even though they increase income
later in life, benefits may have little effect if they are
For such an individual the wage net of the income tax rate
equals the marginal benefit rather than the wage net of the
income and social security tax rate.
However this does not imply the number of hours worked must
rd -Hl ra
fully realized in advance and their effect spread out over
the life-cycle. They may still have an important effect on
behavior, however, since benefits may be partly an inter-
generational transfer from the young of society to the old.
As such, they are an increase in K initial assets, and
will cause people to start work later, retire earlier, and
work less over their lifecycles (see Figure 3-5).
Finally, the earning tax or retirement test reduces
real wages or real potential wages for individuals ages 62
to 72 who are eligible for benefits. This reduction in real
wages means that inequality 12 will hold earlier in the
life-cycle than without the tax. Thus, individuals will be
induced to retire earlier and since this tax does not change
Ko, time working will be substituted to an earlier time
period (see Figure 3-6).
Thus, there are two substitution effects and an income
effect of the social security system. The two taxes which
reduce the marginal wage rate at certain points during the
life-cycle encourage substitution of hours overtime. The
system itself and its benefits may have an income effect if
they alter the individual's endowment.
So three effects have been attributed to the social
security system. Each of these effects tend to reduce the
labor force effort of the elderly and encourage an earlier
retirement date. Next, we turn to the problem of estimating
a model such as this one in which some observations are
likely to have a corner solution as their optimum.
j rd 4
i/ u )a
) U)n En
I 0 -P
U 0 $
U 0 4-1
ECONOMETRIC LITERATURE CONCERNING THRESHOLD MODELS
The first application of a limited.dependent variable
model in economics was by Tobin (Tobin, 1958). He pointed
out that in many of the relationships which economists study,
there is a threshold or limiting value.
When this is the case, the independent variable cannot
be correctly modeled using the classical linear regression
model because the assumption of normality of the residuals is
violated. The residuals will be truncated since there can-
not be negative residuals at a threshold or positive resid-
uals around a limiting value. If one were only interested
in whether an individual was at the limit or not, the probit
analysis would be appropriate; however, such an analysis
would ignore information in the sample concerning the re-
lationship between the dependent and independent variables
and thus would be inefficient.
Graphically with a zero threshold, the actual relation-
ship between y and x is line segments OA and OB (see figure
4-1). If the threshold value is different for each deci-
sion unit, then the relationship between y and x is curve.
OB. Clearly this shows the nonlinearity which the threshold
Figure 4-1. Simple Tobit Model
introduces. To reconcile the problems in estimation dis-
cussed above, Tobin suggested a hybrid of probit and multiple
regression analysis, which has come to be known as the Tobit
The basic Tobit model with Y. as the observed level of
the dependent variable, Y. as the desired level of the depen-
dent variable and L. as the threshold is modeled as follows:
Y. = Y. if Y. > L (1)
1 1 1
Y. = L otherwise.
Then Y. obviously depends on a set of exogenous variables
which also affect Y.:
Yi = Xli 8 + E. (2)
The likelihood function for such a model is:
L = (1/o) f (X l 6/o) T F (- Xli /o) (3)
Y. > L. Y. = L.
1 1 1 1
with f (*) the standard normal density function and F (-)
the cumulative density function of the standard normal den-
sity. As long as there are some observations not at the
threshold or limit, then both 3 and a can be estimated.
Amemiya has shown that this model will give consistent
estimates (Amemiya, 1973). Olsen proved that if f and o are
reparametrized to 3/o and 1/a, then the likelihood function
is globally concave (Olsen, 1976). Therefore, ordinary
least squares estimates can be used as initial estimates even
though they are not consistent. But as Amemiya pointed out,
if one starts with initial estimates which are consistent,
then Newton's method of one-step iteration of the maximum
likelihood equation can be used.
The first extention of the Tobit model put "friction"
into the model. What this model does is allow the dependent
variable to reach a temporary limiting value, then remain
the same over a range as the explanatory variables change
and then continue to change in value after the explanatory
values have changed enough (see figure 4-2). This model can
be represented by line segments AB and CD in figure 4-2.
In this representation, B or C is the temporary limiting
value. If the friction value is between B and C, but the
friction value changes for each individual, then the rela-
tionship of the curve AD shows the relationship.
Such a model could be formulated:
Yli l + X i i + C (4)
and Y2 = +2 + Xi 3. + Ci
2i 2 i i i
Figure 4-2. Tobit Friction Model
where L is the temporary limiting value. The likelihood
function for this specification is:
Yli + a Xi
L = i l/o f () (6)
2 'i 1 _
(a2 Xi i a X. B
S[F ( ) -F )]
1 Y2i + 2 X. e
X n f ( )
This model allows the intercept to change for the two
sets of continuous observations. The curves estimated would
be curve AD in figure 4-2.
The next extension of the Tobit model was to allow for
the existence of both a threshold and limiting value in the
same relationship (Rosett and Nelson, 1975). This "two-limit
Tobit" model is formulated:
Yi = Li if Y. Lli (7)
Yi = Yi ci if Lli < Yi f < L2i
Y. =L ifY. > L
1 2i if Yi L2i
where Lli and L2i are the upper and lower limiting value re-
spectively. Y. is again the desired level of the observed
variable Yi and both depend on a set of explanatory variables:
Y.i Xli + Ei (8)
with c. assumed to be a truncated normally distributed error
with a constant standard deviation, o. With these assump-
tions concerning the residuals, E., the likelihood function
for the "two-limit" Tobit formulation is:
L i Xi 1 Y X. p
L = r F( ) f ( ) (9)
L2i X. (
i [1 F ( )].
Graphically, this model can be represented by line
segments OA, AB, and BC when all individuals have the same
threshold (see figure 4-3). If, as is likely the case, each
individual has a different threshold, then curve OC is the
Earlier, Heckman pointed out that in labor supply Tobit
models, it is likely that at least one of the explanatory
variables is not observed all of the time (Heckman, 1974).
lie solves this censored sample problem by estimating the
general Tobit model along with the censored variable by
maximum likelihood techniques. A generalization of this
model would be:
S> L (10)
Y. = Y. if Y. > L (10)
1 1 1
Y. = L. otherwise
O A X
Figure 4-3. Two Limit Tobit Model
and Y. = X. l + E.
1 1 1
with Xli = Zid + Pi.
The likelihood function for such a specification is:
L = n g (Xi f/o) ir F (-X b/o-. (1 )
This is the same as the Tobit likelihood specification
except that g (.) is a multivariate normal density function
rather than a univariate normal density function and i does
not equal j since X i is estimated by the second term in
Nelson also deals with censoring of variables in a
Tobit framework (Nelson, 1977). However, he dealt with
censoring of the dependent variable. Further, he suggested
that censoring can be a problem when studying the labor-
With this background the problem at hand can now be
dealt with. Next, the econometric model will be developed
from the information in this and the previous section.
AN ESTIMABLE MODEL OF THE EFFECTS OF THE SOCIAL
SECURITY SYSTEM ON LABOR FORCE EFFORT OF THE ELDERLY
Since a theoretical model hypothesizing the effects of
the social security system and the problems of estimating a
model with a limiting value have been discussed, an estima-
ble model of the effects of the social security system on
the labor force effort of elderly males can be developed.
This estimable model will take into consideration the ideas
developed in the two previously mentioned sections.
For the estimable model, the individual's hours worked
per week (HRS.) were selected for the dependent variable.
This variable comes directly from the theoretical model,
which suggests that whether and how many hours each indivi-
dual chooses to work at any point in time is affected by the
social security system. The theoretical model shows partic-
ularly that all three effects of the system may be acting
on the individual's choice of weekly labor force effort
after age sixty-two.
Hours worked were selected as the dependent variable for
two reasons. Hours worked as a dependent variable are less
ambiguous than other possible choices. Often in the litera-
ture, a zero-one variable for retired or working has been
used. With this type of model, a decision must be made
whether to call individuals who work a few hours a week
retired or employed. Even though this type of dependent
variable can be expanded to include one or more semiretired
categories, such a hierarchy does not really solve the prob-
lem of the subjective decisions the researcher must make.
Rather the subjective decisions of how -many and what type of
categories to use is substituted for the part-time worker
decision. Also, more categories imply more borderline cases,
thus more decisions on the part of the researcher.
Additionally, choosing hours worked as the dependent
variable makes it possible to examine both of the effects
of the social security system which fall specifically on the
older worker. The effect of benefits can be measured as in
other studies. Further, since those individuals who fall
in the area of the earnings tax can be distinguished from
all other individuals in the sample, it is possible to
examine the effect of the earnings tax. Finally, since one
can not work less than zero hours, this dependent variable
is one with a limiting value as implied by constraint 6.3 of
the theoretical model.
It could be argued that hours worked is not truly a
continuous variable above zero. While within a given job
such as that of an assembly-line worker there is little
room for flexibility in hours worked per week, by moving
from job to job a worker can select almost any work schedule
desired. Also, with many jobs such as that of a sales-person,
hours worked are completely continuous. Overall it is felt
that it is easy enough to vary hours for most workers,
especially in the types of jobs older people tend to hold,
that using hours worked as the dependent variable is
Now that the dependent variable has been established
the question of the appropriate explanatory variables must
be addressed. From the theoretical model it can be seen
that these variables must directly or indirectly reflect,
the wage rate of the individual, W.; the individual's
initial endowment of asset, K ; the individual's internal
rate of time preference, pi; or the rate of interest, r. Of
course the interest rate these individuals face should not
vary greatly through the cross-section since each individual
has the same opportunity costs for funds. Thus, no vari-
ables were chosen to reflect it.
The wage rate the individual commands in the market
place clearly enters the empirical model from the theoreti-
cal model. The wage is afterall the rate at which the
individual can substitute income for leisure. Thus the
market wage rate, either implicitly or explicitly, is
compared to the individual's preference for use of time and
the appropriate number of hours are selected. As wages
increase the number of individuals in the labor force will
The sign of the wage coefficient can not be predicted
with certainty, however. This is due to the fact that for
those who are working there is both an income and substitu-
tion effect in connection with changes in the individual's
wage rate. Most obviously, when one's wage is increased,
there is a tendency to allocate more hours toward labor and
fewer hours toward leisure, since leisure has become rela-
tively more expensive when compared to all other goods.
However, this is not the only effect of an increase in an
individual's wage rate. The individual, after the wage
increase, has a larger income. This increase in income
means the individual will consume more of each good with a
positive income elasticity. If the income elasticity of
leisure is positive and sufficiently large, then an increase
in the wage rate may prompt the individual to buy enough
additional leisure so that labor force effort declines.
On balance, the income elasticity at current income
levels is not known. Evidence from the literature is
ambiguous. Thus, no relationship can be hypothesized
between the wage rate and hours worked.
Social security benefits or potential social security
benefits as an explanatory variable also reflects part of
the theoretical model. Benefits or potential benefits are
an attempt to measure part of Ko, the individual's endowment
at the beginning time period. The higher this endowment is,
the less likely the individual is to work. Furthermore,
the larger this endowment the fewer hours the individual
desires to work. Thus, if this variable is capturing an
intergenerational transfer and if social security increases
the individual's initial endowment, it should have a nega-
The amount of asset income, including imputed rent for
homeowners, is also an attempt to measure the person's
endowment. A person who has acquired assets which produce
a large income stream is more likely to have had a larger
initial endowment and to choose less work at any given time
period regardless of the work path. Again, the income
effects of assets income imply a negative relationship with
The relationship between hours worked and the category
of other unearned income is theoretically similar to the
two previous variables. Other unearned income including
disability payments, all non social security benefits and
all other transfers the person's nuclear family receives is
an endowment from society and to receive most if not all of
these grants, one must reduce hours worked. Also if one
is disabled or has a disabled spouse to care for, one will
not choose to work as many hours as otherwise. So for this
variable, theory predicts a negative coefficient too.
The older the person is the fewer hours of work per
week he is likely to choose. If the individual has
Other pensions such as civil service are an important
exception to this.
substituted work earlier in life for leisure later in life,
the older the person is the more likely this substitution is
completed. Also, the older the person is the more likely
the work-life will be completed whatever the original pat-
tern was planned to look like. However, if the inter-
generational transfer aspects of social security have grown
with time then the older, less endowed,- individuals may work
more. Also, this type of positive relationship may exist
if each succeeding cohort is better endowed due to rising
income or economic growth. Finally, since the relationship
between r and pi is unknown no relationship can be hypothe-
sized between hours worked and age.
The more dependents an individual has the more hours
the person is likely to desire to work each week. The
existence of dependents indicates a desire for more income,
other factors the same, which implies a greater work effort.
Also, the existence of dependents, usually a spouse, implies
more division of labor within the household, thus less
household work effort and more market work effort on the
part of the individual. Again, all of the effects are in
the same direction and a positive relationship is hypothe-
sized between hours worked and number of dependents.
Exceptions to this would be people who work a constant
amount their entire life and present oriented people.
These two groups are not thought to be large.
The amount of health expenditure by or on behalf of the
individual is an attempt to measure the health of the indi-
vidual in a less subjective manner than other studies. The
more unhealthy the individual is, the less likely and fewer
hours work effort should be. This is due to less healthy
people having a preference for leisure compared to healthy
people. This preference is due to the -extra effort that it
takes for an unhealthy person to accomplish work compared to
a healthy person. Again, all factors point to the same re-
sult, a negative coefficient for this health proxy.
The final variable selects out of the sample those
people who fall in the area of the social security earnings
tax. For these people there is a second wage effect due to
the earnings tax cutting the person's realized wage rate in
half. Such a person should work fewer hours per week than
similar persons who do not fall in the region of the earn-
ings tax. Therefore the second wage variable should be
negatively related to hours worked.
Thus, the estimable model is
HRS. B + B W. + B2 BEN. + B3 ASSETS + B4 UNY.
I o 11 2i i 3 4
+ B AGE + B DEP. + B7 HEALTH + B8DIWi + C.
B5 6GE 7 B6 Dil l
or hours worked each week, HRS, depend on the individual's
wage rate, Wi; social security benefits or potential bene-
fits, BENi; assets income, ASSETS; unearned income, UNEY;
age, AGEi; number of dependents, DEPi; health expenditure,
IIEALTI ; and the second wage effect, DW i. As discussed in
the estimating chapter the limit on the dependent variables
implies that iF, the error term, is truncated and estimation
must take this into account.
Some important caveats must be considered before turn-
ing to the actual results of the estimation process. First
of all, the wage, assets income, and potential social secur-
ity benefits variables will partly reflect certain individ-
ual's preference for or enjoyment of their work. One who
derives some utility from work will tend to work more, thus
accruing more income earning assets and a higher social
security potential, due to working more during the life-
cycle. Also, one that enjoys work is probably more produc-
tive and is likely to command a greater wage even when
that individual reduces hours worked as the work-life winds
down. Alone, this problem is not likely to be strong but it
An additional warning must be issued about the wage
coefficient. Some the the wage observations were derived
from reported salaries and the time period of the payment.
If there were errors in these variables and they were cor-
related to hours worked, then the wage coefficient is biased
toward negative one (Borjas, 1980). There is evidence that
such correlated errors exist in this data set.
A third caveat is in order concerning the assets earn-
ings variable. Some amount of the assets leading to these
earnings is likely to have been acquired in order that a
bequest can be left at the terminal time.1 Therefore high
assets earnings, implying large assets for the purpose of
bequests, may require more work effort than otherwise ex-
pected, so that the assets and thus the bequests are not
used up before the terminal time.
The final comment concerns the health variable. Most
of the individuals in the sample either-had some kind of
health insurance or qualified for medicare. Thus, the health
variable is a better proxy for health than health expendi-
ture would be for other age groups. However, expenditure
might fail to pick up the bad health of invalid persons who
nevertheless spends little on health since nothing can
be done to improve their condition. Still it is felt health
expenditure is an improvement as a proxy over the use of the
individual's answer to a question of whether their health
affects the amount of work they can do.
In summary, the coefficients of social security bene-
fits, assets income, other unearned income, health and the
social security earning tax wage variable should have
negative signs, the coefficient for the number of dependants
should be positive, and the sign of the coefficients for age
and wage can not be predicted. In the following section,
the results of the estimation of the model will be discussed.
The theoretical model terminated at zero as bequests were
modeled as a consumption good.
THE RESULTS FROM ESTIMATING THE MODEL
The empirical results from estimating the model which
was developed in the previous section will be presented in
this section. These results will be contrasted with the pre-
dictions of the estimable and theoretical models.
The first several sets of results to be discussed are of
ordinary least squares (OLS) estimation technique. This type
of result is presented for comparative purposes. Ordinary
least squares estimation was accomplished by using the
Statistical Analysis System software package.
Table 5 gives the results of ordinary least squares
estimation using the entire data set after winnowing out
incomplete observations. This process leaves a data set of
2099 observations which is a reasonably large cross-section.
Other than the benefits or potential benefits (Ben) and the
assets income (Assets) variable, all the variables in this
model have the expected signs. Since this estimation pro-
cedure is not a theoretically correct one, the coefficients
will not be discussed in detail. However, it should be
noted that the variables are all significant influences on
labor force effort (HRS) as both the theoretical and empiri-
cal models predict.
OLS RESULTS OF THE ENTIRE DATA SET
F = 75.76
N = 2096
Though these results only explain twenty percent of the
variation in the data, the significance of each of the vari-
ables suggests that the model is on the right track. Also,
given the sample size and that the sample is a cross-section,
the fit is not terrible.
However, these results do not represent the entire
estimable model as developed in the preceding section. The
variable designed to measure the effect of the social se-
curity earnings test has'been omitted. This variable,
Dl wage, is a second wage effect for individuals who fall
in the region of this earnings tax.
While this second wage variable has only a marginal
effect on the results, when included in the model (see
Table 6), it does seem to be measuring the desired effect.
The coefficient of this variable is statistically, signifi-
cantly different from zero and has the expected sign. The
fit of the regression improves with the inclusion of the
variable, which is what one would expect since the fit of
any model can generally be improved by cluttering it up with
more exogenous variables. Also as one might expect, the
F-statistic declines, since adding a marginally significant
variable decreases the statistical validity of the regres-
As discussed in the section on estimating a model with
a limited dependent variable, simple ordinary least squares
is not the correct estimation technique. Also, as discussed
OLS RESULTS INCLUDING EARNING TAX VARIABLE
- Standard Error
F = 66.87
R2 = .21
N = 2096
ill that section and illustrated by Figure 4-1, ordinary least
squares estimation of the non-limit observations somewhat
approaches the Tobit model. The Tobit model is the theoret-
ically correct estimation procedure, of course. Thus, again,
for the purpose of comparison, ordinary least squares esti-
mates of the non-limit observations are presented in
These results are not very good at all. The coeffi-
cient of determination was reduced in half. The wage
coefficient changed signs. Also the F-statistic declined
indicating a decline in the regression's validity. Thus,
the overall decline in regression quality, the wage coeffi-
cient switching signs, the economic theory and the econo-
metric theory all indicate a limited dependent variable
approach should be investigated.
Two other points concerning the partitioned sample re-
gression are as follows. The number of observations indicate
there is a reasonable limit-non-limit division in the sample,
fifteen and eighty-five percent respectively, which bodes
well for the Tobit analysis. Also, the earnings tax vari-
able is significant. This is really not surprising, since
all of the observations eliminated were limit or zero hours
worked observations, thus a special sample was selected
which had to include all those individuals who were affected
by the earnings tax.
Now the maximum likelihood results can be discussed.
As mentioned before, the complete explanation of theoretical
OLS RESULTS OF THE NON-LIMIT INDIVIDUALS
R2 = .10
N = 1764
F = 24.22
considerations of estimating models with a limit to the de-
pendent variable is in an earlier section of this paper, but
briefly, the limiting value means the residuals are trun-
cated and thus the ordinary least squares assumption of
normally distributed residuals with zero means is violated.
For these reasons, the correct estimation of the model is
with a Tobit procedure.
The Tobit results are presented in Table 8. Overall
the change in these results are what would be expected in
going from ordinary least squares to Tobit estimation. The
stacking of the limit observations implies the least squares
coefficients will be too large. This is confirmed by the
Tobit results in which all the coefficients except the one
for wages are smaller than the corresponding least squares
The wage coefficient is positive and significant.
This indicates for this sample the positive substitution
effect and labor force participation dominate the negative
income effort. This variable is one of the most precise
The coefficient for the individual's number of depend-
ents (Dep) is the largest. This coefficient is also pre-
cisely estimated. The sign and significance are congruent
with the theory suggesting that households divide labor
among members. Thus, the more dependents the member of the
household who specializes in labor force effort has, the
TOBIT RESULTS INCLUDING THE EARNINGS TAX
Assymptotic Standard Errors
Log of the Likelihood Function = -8190.86
N = 2092
greater the income needs are, implying greater labor force
The health (Health) variable is an interesting one.
The fact that it has a negative coefficient indicates that a
good quantifiable proxy for health was found. Also, the
variable passed the test of being significantly different
from zero at the ninety-five percent confidence level. Also,
of note, is that it was negative and significant in all the
ordinary least squares results. Taken together, all of
these results suggest that bad health reduces labor force
effort as expected.
The benefits variable, while significant as expected,
seems to have a counter intuitive positive sign. There are
several possible explanations for this sign. Benefits are
tied to lifetime earnings, so high benefits may be related
to a preference for work to consume goods or income over
leisure. Also, this benefits variable includes a potential
benefit for those working and someone with a high potential
benefit must have had high earnings, and thus probably has a
high wage now. Thus, the benefits variable may be picking
up some of the positive wage effect. Also, if the individ-
ual is a true life-cycler, benefits may not have the
suggested negative coefficient if there is no intergenera-
tional transfer. The sign of the benefits variable is
probably due to a combination of the reasons just presented
and is not as counter-intuitive as first suggested.
The coefficient of age, has a positive and highly sig-
nificant sign. Also, this sign is the reverse of the least
squares results. An explanation for the positive sign is
that there has been growth of the initial endowment overtime
due to economic growth. This means that younger members of
the sample have larger Ko than older members of the sample.
Accordingly the older individuals work longer hours.
The unearned income variable (UNEY) has the expected
sign and is precisely measured. Higher other pensions and
government payments leading to less labor force effort is
what is theoretically and intuitively expected. This is
partially due to the work restrictions that are conditional
on many of the sources of unearned income. This is particu-
larly true of government sources of unearned income.
The assets income variable, like the benefits variable,
seems to have the wrong sign. This was somewhat anticipated
in the estimable model section. The positive sign of the
assets income variable is probably due to preferences for
income over leisure among people with assets. Also, desires
for bequeaths may lead to this relationship. Thus, it is
not counter intuitive as it might seem. The coefficient
is significant as predicted.
Finally, the earnings tax variable is negative and
significant. This is as theory would predict. It says that
individuals who fall in the area of the earning tax have a
smaller wage effect than those outside this area. This is
because their real wages are lower, by fifty percent, than
their money wages. An estimate of this effect has pre-
viously been lacking.
As has been seen, there is some endogeneity in the
variables attempting to capture the effect social security
has on the initial endowment and attempting to measure the
effect of the endowment. Moreover, two of these three vari-
ables have counter intuitive signs. Fo- these reasons the
model has been estimated with the assets, benefits, and
unearned income variables deleted. These results in Table 9
show that the results were only slightly changed by the
deletion of the endowment variables. This and particularly
the fact that the log of the likelihood function changed
only slightly indicates the endowment variables added very
little to the model and were not good proxies.
Since these results are from a nonlinear estimation
procedure, the coefficients are not the first derivatives
and the predicted value of y, y, is not the predicted hours
worked. As Poirier and Melino have demonstrated, the change
in hours for a unit change in 'an independent variable is the
coefficient multiplied by ratio of the truncated variance
of y to untruncated variance (Poirier and Melino, 1978).
Since this ratio is less than one, the true effects for the
mean value of this sample are smaller than the coefficients
(see Table 10). Since the limit is zero, the expected hours
TOBIT RESULTS WITHOUT THE ENDOWMENT VARIABLES
Assymptotic Standard Errors
Log of the Likelihood Function -8302.37
N = 2096
THE CHANGE IN HOURS FOR A UNIT CHANGE IN X
AT THE MEAN OF THE SAMPLE
Truncated s = 411.68
S = 142.81
E(y) = [yF(y) + of(y)]
which takes the probability working into account. In
Table 10, the elasticities for the mean values of the labor
supply equation are also presented. The elasticities are
calculated as (oHRS/9X) (X/HRS).
Table 10 indicates no one variable has a large effect
on the labor force effort of older males. For example,
the health expenditure elasticity implies that an illness
which costs the individual $4786 will induce an eight hour
reduction in labor force effort per week.
The wage coefficient directly measures the effect of an
increase in the wage rate on the labor supply of those who
are not subject to the social security earnings tax. The
coefficient in Table 10 indicates that a one dollar rise in
the wage rate leads to a 0.27 hour rise in weekly labor
supply. On the other hand, the sum of the wage and D1 wage
coefficients gives the effect of a one dollars increase in
the person's gross wage rate if the individual is subject
to the social security earnings tax. The coefficients in
Table 10 imply that a one dollar increase in the person's
gross wage rate results in a .26 hour increase in hours
worked. Since this income is taxed at a fifty percent rate,
a one dollar increase in the individual's net wage is esti-
mated to bring about a .52 hour rise in weekly labor supply.
As before, f(-) is the standard normal density function and
F (-) is the cumulative density function of the standard
These estimates suggest that individuals subject to the
earnings tax are more responsive to changes in net wages
than are individuals not subject to the earnings tax. This
may be because the income effect associated with wage
changes is smaller for the former group.
The wage effects in Table 9 could be similarly analyzed.
The very large negative coefficient of 91 wage, however,
suggests that this variable may be correlated with the
In general it can be seen that neither of the policy
variables, wages nor benefits, offer much hope for solving
the social security financing problem. The elasticities of
neither indicate the kind of responsiveness in hours worked
that would provide a feasible solution.
The results of estimation have been presented and com-
pared to the predictions of theory. Next, these results
will be discussed in light of political reality, as conclu-
sions are drawn and policy implications discussed.
CONCLUSIONS DRAWN FROM THE EMPIRICAL RESULTS
Two sets of conclusions can be drawn from the results
just presented. These are conclusions concerning the model
or academic conclusions and policy implications.
The most important conclusion concerning the model is
that a non-subjective health variable can work. Boskin had
the right idea in his 1977 study, but a bad proxy for the
individual's state of health. Health, as measured by health
expenditure by or on behalf of the individual, had the ex-
pected negative effect on labor force effort. Also, when
measured by the correct technique, this effect was found to
be significant. This result is counter to that found by
Boskin. However, his explanation that the sixty to sixty-
five cohort has had improving health over time while eligi-
bility age has been rigid is still an important idea to keep
in mind. This health proxy is also very appealing consider-
ing the alternative of using the individual's answer to a
subjective question concerning health.
Also, modeling labor force effort as a Tobit model
worked out well. It does not seem that hours worked as a
dependent variable are discontinuous, particularly for an age
group in which many individuals take part-time employment.
A'lain, the Tobit model is conceptually appealing because
people do work different numbers of hours. Also, the alter-
natives specifications are not appealing. Ordinary least
squares is definitely incorrect and any scheme of work-
-retired or work-semiretired causes the researcher to make
too many decisions that can influence the results.
Since the sign of the earnings ta.x coefficient was
correct and it was significant, the earnings tax variable
likely measured the intended effect. This is an important
aspect of the model, since measures of the effect of the
earnings tax have previously been lacking.
The final academic conclusion is that the economic
variables, such as wage rate, have a more important effect
than social security variables. This is important to keep
in mind when considering the policy implications of the
social security variables. The social security system does
not dictate retirement patterns, but it clearly influences
This influence leads directly to policy implications.
Policy implications are germane because of the projected
fiscal shortfall of the system. All implications will be
discussed in light of alleviating this projected shortfall.
Despite its significance, the earnings tax coefficient
implies two things. One, the size of the earnings tax co-
efficient indicates that the earnings tax probably does not
reduce the taxes paid to the system significantly. Second,
there is probably no short-run way to change the earnings
tax and help the system. This is because the social security
system provides another incentive to shift hours worked to
the middle of the life-cycle. The other incentive is the
ceiling on taxable income.
The benefits coefficient does not suggest a solution
to the revenue problem either. It is not feasible to reduce
benefits and increasing them will only exacerbate the prob-
However, Boskin's point concerning the eligibility age
does offer a possible longer run solution. If health among
the sixty to sixty-five cohort has improved, one solution to
future revenue problems could be to index the eligibility
age to life expectancy. This would, of course, have to be
done with a lag for political reasons. But such a move
would reduce out-flows and may increase revenue.
The bottom line is that this model offers no short-run
solutions to projected social security revenue shortfalls.
Indexing the eligibility age and/or eliminating the earnings
Taxes will reduce more significantly if all work hours
are shifted into periods where earnings are above the tax-
Thlis idea does not actually imply that the health of indi-
viduals within the cohort will not have a negative effect
on their labor force effort.
tax are the only possible long-run solutions. Probably
neither of these will solve the whole problem, so society
is left with raising taxes and/or erroding real benefits
via inflation and/or transfers from general revenues to
meet at least part of the projected social security revenue
A DESCRIPTION OF THE DATA AND ITS SOURCES
The data used in this research come from the "Retire-
ment History Study" of the Social Security Administration.
Specifically, the 1973 interview and the "Summary of Social
Security Earnings" of the "Retirement History Study" are
the main data sources.
The "Retirement History Study" is a ten-year study of
the elderly population which was conducted by the Social
Security Administration. The study began in 1969 with a
national sample of 11,153 persons between ages 58 and 63.
Every two years, for a total of ten years, this sample was
reinterviewed as completely as possible. Since the 1973
wave of interview was the data base for this research, the
individuals under study were between ages 62 and 67.1 As
a supplement to the "Retirement History Study," the Social
Security Administration compiled the earnings covered by
social security of the 11,153 individuals into the "Summary
of Social Security Earnings" data set. Also, as stated
Two variables, education and race, which were expected not
to have changed over time were taken from the 1969 wave of
earlier, white males were selected from these data sets
since they have a more continuous history in social security
Selecting only white males and the normal attrition due
to death, incomplete response, and lost individuals, winnowed
the sample to 2,099 observations. This is a reasonable
number of observations for the estimation techniques used.
Further, as expected with this age range, there is a reason-
able division between those at the limit, 16% and those
working some hours above the limit, 84%.
For each individual age, hours worked, market wage
rate, health expenditure by or on behalf of, assets income,
social security benefits or potential benefits, other un-
earned income, and number of dependents was either reported,
calculated, or estimated. Further, two zero-one variables
were constructed; one, D2 for working and not working and
the other, Dl separating out the individuals earning income
above the exempt amount and receiving some benefits. This
second zero-one variable identified those individuals operat-
ing in the region of the earnings tax. The descriptive
measures of the data are reasonable suggesting that there is
not too much "noise" in these data (see Table 11).
The age variable was simply calculated from the indi-
vidual's date of birth. Each observation in the data fell
between ages 62 and 67, which indicates all cases, where a
wife was reporting for a deceased spouse, were deleted.
DESCRIPTIVE STATISTICS OF THE
Hours worked were assigned the value zero if the person
rc(ported he was out of the labor force. Otherwise, hours
worked per week were the reported number of hours. Individ-
uals who reported more than sixty hours per week were
recorded as working sixty hours, since more than sixty hours
were thought to be an exageration.
The individual's wage rate came from a number of
sources. For the individual who worked, the wage rate was
either their reported hourly wage or their reported salary
divided by the appropriate number of hours for the time
period the individual claimed for the salary. For the
individuals who did not work, the wage rate came from their
last reported wage divided by the appropriate number of
hours. This figure was than inflated by an aggregate wage
inflator to arrive at a 1973 money wage. Also, these pro-
cedures led to some very large rates which were truncated
by letting any wage over twenty dollars per hour equal
twenty dollars plus ten percent of the reported wage.
Health expenditure came from the person's reported
spending. Since most of these individuals were eligible for
medicare, spending on health is likely invariant with in-
come and thus a good quantifiable measure of health.
The number of dependents is based on a relatively
simple calculation. It is the sum of the number of spouses,
living parents, fully supported children, and one-half times
partially supported children.
Assets income also is a simple calculation. It is
respondent's total income from interest, dividends, and
rent in 1972, plus spouse's total income from interest,
dividends, and rent in 1972, plus income from interest,
dividends and rent in 1972 of children under age 18, plus
ten percent of the value of the family owned home, if there
is one. Of course, this is a family assets income variable.
The unearned income variable is also a family unit
variable and includes a sum of eleven different possible
pensions or public assistance payments. From this, social
security benefits are subtracted since they enter the model
in a separate manner.
Potential benefits from the social security system
were calculated using the formula in effect in 1973, the
'Summary of Social Security Earnings' data. The potential
benefit is the relevant variable, since benefits are not
observed for working individuals.
The two zero-one variables described previously com-
plete the data set. The first one was used to allow for the
second wage effect for those who were affected by the
earnings tax. The second dummy variable was used to separ-
ate the sample in working and retired groups for the non-
limit ordinary least squares estimation.
These are available upon request.
This program is available upon request and was written by
Steve Caudill of the University of Florida.
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James L. Swofford was born in Waco, Texas, in 1953.
He attended public school in Orlando, Florida. He
received an undergraduate degree from the University of
Richmond, Richmond, Virginia, in 1974. He is presently a
visiting instructor at the University of Central Florida.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
G.S. Maddala, Graduate
Research Professor of
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
J.R. avis, Professor or
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
( .W. Kenny, Assistant
Professor of Economis'.
I certify that I have read. this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
J.C. \enretta, Assistant
Professor of Sociology.
This dissertation was submitted to the Graduate Faculty
of the Department of Economics in the College of Business
Administration and the Graduate Council, and was accepted
as partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
Dean for Graduate Studies and
UNIVERSITY OF FLORIDA
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