The effects of the social security system on the labor force effort of older males


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The effects of the social security system on the labor force effort of older males
Physical Description:
vi, 87 leaves : ill. ; 28 cm.
Swofford, James Lee, 1953-
Publication Date:


Subjects / Keywords:
Social security -- United States   ( lcsh )
Labor supply -- United States   ( lcsh )
Men -- United States   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1981.
Includes bibliographical references (leaves 84-86).
Statement of Responsibility:
by James Lee Swofford.
General Note:
General Note:

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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aleph - 000296355
notis - ABS2717
oclc - 08078856
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Full Text








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

my typist.














S vi


S 22

S 42

S 51


RESULTS . ... 74




. 78

. 82



Table Page



LATION 1960 .. . 13










Figure Page

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



James Lee Swofford

June, 1981

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.



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

later life.

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


(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
work now.

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.



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


(Wentworth, 1945b)

Philadelphia Memphis
Unearned and and
Income Baltimore St. Louis Atlanta Los Angeles

Total 25.4 37.6 45.6 38.1

Less than
$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

Greater than
$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

Table 3).

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


(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
(2.2870) (2.6504)

Male Population Age 65 and Over:

.5589 .79385 -1.86350 .13378 .41291
(2.7907) (3.0893) (2.6573)

.3718 .61321 .13275 .45821
(2.5687) (2.4845)

aStandard errors.

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


(Boskin, 1977)

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

sample properties.

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.



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 t
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
-p~t ,
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

empirical model.

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

period result.

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
It i
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

goods overtime.

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 direction.

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,

6.3 implies:

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

a value:

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
1 1
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 e
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
i i
discussed. As long as WiU.(C it) is greater than V (Lit),

the amount of time allocated to work increases as W./W.
1 i

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
their life-cycle.

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

4-) 03

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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.

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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 _
1 F

(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)
1 2

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

correct model.

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
1 1


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 )
1 2

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-

leisure trade-off.

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.



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

probably increase.

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-

tive coefficient.

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

hours worked.

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

does exist.

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 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.




Standard Error

























R2 .20
R .20

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




- Standard Error









Dl wage



















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

Table 7.

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












D1 wage

R2 = .10

Standard Error










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





Assymptotic Standard Errors








D1 wage

















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

work are:



Variable B.

Assymptotic Standard Errors
















Log of the Likelihood Function -8302.37

N = 2096




Variable B.








D1 Wage









F' (X)


















Truncated s = 411.68

S = 142.81

- 2
- '.35


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
normal density.

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

omitted variables.

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.



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-
able limit.

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




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

covered employment.

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.













DL wage









































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.

June 1981.

Dean for Graduate Studies and


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