The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Social Security Reform
with Heterogeneous Mortality
WP 20-09
John Bailey Jones
Federal Reserve Bank of Richmond
Yue Li
University at Albany, SUNY
Social Security Reform with Heterogeneous Mortality
John Bailey Jones
Yue Li∗
July 17, 2020
Abstract
Using a heterogeneous-agent, life-cycle model of Social Security claiming, labor
supply and saving, we consider the implications of lifespan inequality for Social
Security reform. Quantitative experiments show that welfare is maximized when
baseline benefits are independent of lifetime earnings, the payroll tax cap is kept
roughly unchanged, and claiming adjustments are reduced. Eliminating the earnings
test and the income taxation of Social Security benefits provides additional gains.
The Social Security system that would maximize welfare in a “2050 demographics”
scenario, characterized by longer lifespans and an increased education-mortality
gradient, is similar to the one that would maximize welfare today.
JEL: E21, H24, H55, I38, J11
Keywords: Social Security, Mortality, Labor Supply, Welfare
∗
John Bailey Jones: Federal Reserve Bank of Richmond, john.jones@rich.frb.org. Yue Li: University
at Albany, SUNY, yli49@albany.edu. For helpful comments, we thank seminar participants at American
University, University of Houston, William and Mary, the ESRC Conference on Inequality and the Insurance Value of Transfers over the Life Cycle and the U.S.C FBE Workshop on the Economics of Health and
Aging. The research reported herein was performed pursuant to grant RDR18000003 from the U.S. Social
Security Administration (SSA) funded as part of the Retirement and Disability Research Consortium.
The opinions and conclusions expressed are solely those of the author(s) and do not represent the opinions or policy of SSA, any agency of the Federal Government, NBER, Federal Reserve Bank of Richmond
or the Federal Reserve System. Neither the U.S. Government nor any agency thereof, nor any of their
employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for
the accuracy, completeness, or usefulness of the contents of this report. Reference herein to any specific
commercial product, process or service by trade name, trademark, manufacturer, or otherwise does not
necessarily constitute or imply endorsement, recommendation or favoring by the U.S. Government or any
agency thereof.
1
Introduction
One of the most common proposals for stabilizing the Social Security system is to
increase the normal retirement age (NRA). An appealing feature of this proposal is that it
counters the secular trend toward longer lives, a major cause of Social Security’s financial
difficulties, by effectively delaying the age at which full benefits start. It is also wellknown, however, that the longevity gap between rich and poor is large and growing over
time (National Academies of Sciences, Engineering, and Medicine 2015; Chetty et al.
2016). This has generated concerns that raising the NRA would cut disproportionately
the benefits of the poor (e.g., Krugman 2015). While these concerns can be mitigated by
changing the progressivity of the benefit formula (Cremer et al. 2010), the more general
question remains. If high-income workers live longer than low-income workers, should
they work longer as well? Should they receive higher annual benefits?
When considering such questions, several conflicting principles come into play. (Cremer et al. 2010 provide a nice encapsulation. See also Ndiaye 2018.) Mirrlees’s (1971)
canonical framework highlights the tension between equalizing consumption and encouraging work by the most productive. In our setting, this implies that high-productivity
workers should retire at older ages. With heterogeneous mortality, “equal consumption”
must be defined as well. Solutions to social planners’ problems often involve equating the
weighted marginal utility of consumption across all surviving agents. This can imply equal
per-period consumption flows, but larger lifetime transfers to the long-lived. Inevitably,
optimal pension design is a quantitative exercise.
To examine these issues, we develop a heterogeneous-agent, life-cycle model of Social
Security claiming, labor supply and saving. In the model, individuals face uncertain
health, wages, medical spending and survival, the distributions of which vary by education
level. The government collects income, payroll and consumption taxes, and provides Social
Security, Medicare, Disability Insurance and means-tested social insurance. Individuals
1
can continue working while receiving Social Security benefits, but they may face financial
disincentives to do so. After calibrating the model to the current U.S. economy and Social
Security rules, we evaluate potential reforms.
In particular, we solve the following planning problem. Consider a stationary crosssectional distribution of individuals that differ along a variety of demographic and economic dimensions. Holding fixed both aggregate Social Security expenditures and revenues, and maintaining general government budget balance, what are the Social Security
rules that maximize the ex-ante utility of a newborn? We consider three sets of parametric
reforms:
1. Changes to the payroll tax earnings cap and tax rate.
2. Changes to the formula converting an individual’s earnings history to his baseline
benefit (the Primary Insurance Amount, or PIA).
3. Changes to the trade-off between the age at which an individual first receives her
Social Security benefits – and thus the length of the benefit stream – and the size
of the annual benefit. Increases in the NRA can be interpreted as a special case.
This trade-off is formally embodied in the early retirement penalties and delayed
retirement credits, which increase or decrease an individual’s annual benefits proportionally to his PIA.
All of the reforms that we study have appeared before as proposals, enacted changes
or both. While the first two sets of reforms affect how Social Security benefits depend on
wage realizations and work decisions, the third affects how benefits depend on claiming
decisions. Because individuals can simultaneously work and receive Social Security benefits, their work and claiming decisions may appear to be disconnected. This is not the
case, however, because the rate at which earnings translate into Social Security benefits
is a function of the age at which the benefits are claimed and because benefit receipt itself
generates work disincentives. These disincentives include benefit reductions through the
Social Security earnings test and the way in which the income taxation of Social Security
benefits increases the income taxation of earnings (Jones and Li 2018).
2
Each set of reforms embodies the canonical trade-off between redistribution and productive efficiency. Raising the payroll tax cap lowers the tax rate but applies it to a
broader range of earnings. This reduces taxes for most workers but raises marginal tax
rates for the most productive. Linking Social Security benefits to lifetime earnings increases the returns to work but reduces transfers from high to low earners. Raising the
returns to delayed claiming rewards longer careers but punishes those with low longevity.
Our general finding is that, relative to the Social Security policies currently in place,
the policies that would maximize welfare would reduce work incentives in order to redistribute resources from high to low earners. Under these policies, the PIA would be
independent of lifetime earnings, and claiming adjustments would be smaller, while the
upper bound on taxable earnings would remain at more or less its current value. Collectively the reforms cause both earnings and employment to fall by 1-2%. We show,
however, that eliminating the earnings test and the income taxation of Social Security
benefits, as recommended by Jones and Li (2018), reverses more than two-thirds of the
fall in earnings and more than half of the fall in employment. Combining the two sets of
reforms also results in larger welfare gains. Because the earnings test and benefit taxation apply at older ages, when the elasticity of labor supply is especially high (French and
Jones 2012; Karabarbounis 2016; Ndiaye 2018), eliminating them is an especially effective way to encourage work. Removing the provisions uncouples claiming decisions from
retirement decisions; under the joint reforms, almost everyone claims benefits at age 62.
We then consider how heterogeneous mortality affects Social Security reform in the
face of population aging. We construct a hypothetical “2050 demographics” scenario
characterized by longer lifespans, lower population growth and an increased educationmortality gradient. We find that the Social Security system that would maximize welfare
in the 2050 demographic environment is quite similar to the one that would maximize
welfare today. In both cases, the PIA would be independent of individual earnings; the
payroll tax cap would be higher in 2050 than at present; but the claiming adjustments
3
would not. Although increased longevity suggests that larger claiming adjustments are
needed to promote longer careers, increased longevity also implies that the adjustments
needed to induce claiming delays (and longer careers) are smaller.
The literature on Social Security reforms is immense (see, e.g., Feldstein and Liebman
2002). In their review, Börsch-Supan et al. (2016) contrast “parametric” reforms, where
the basic design of the system is unchanged, with more fundamental “systemic” reforms,
such as the switch from a pay-as-you-go to a fully funded system. We restrict ourselves to
parametric reforms, holding fixed Social Security’s “aggregate footprint.”1 Because the
absolute size of a country’s Social Security system can have first-order effects on its aggregate capital stock, our approach allows us to focus on distributional concerns. Within
the literature on parametric reforms, our contribution is to consider all the reforms simultaneously, quantitatively, and while accounting for heterogeneity in income and health.
Our exercise also stands out in its breadth: among other possibilities, we consider flat
benefits and what is effectively a single claiming age.
While it has been long recognized that heterogeneous mortality affects the lifetime
progressivity of Social Security (Aaron 1977), and multiple studies have sought to quantify this effect (recent analyses include Goda et al. 2011; Bosworth and Burke 2014;
National Academies of Sciences, Engineering, and Medicine 2015; and Sheshinski and
Caliendo 2018), there has been relatively little progress in quantifying its implications for
optimal policies. Laun et al. (2019) and Sánchez-Romero et al. (2019) consider how to
maintain fiscal sustainability in the presence of heterogeneous demographic change. (Also
see Conesa et al. 2019.) Although set in Norway, the structure of Laun et al.’s (2019)
model is quite similar to ours. However, they focus on a handful of reforms, including
some that change the pension system’s aggregate footprint, while we seek optimal policies
1
An alternative approach used in the literature is to fix tax rates and require that benefit changes
maintain Social Security budget balance (e.g., Huggett and Ventura 1999; Nishiyama and Smetters 2008;
Bagchi 2019). When operating within a stationary demographic environment, this constraint is much the
same as ours. The two constraints differ if one considers responses to major demographic changes.
4
with the footprint held fixed. Sánchez-Romero et al. (2019) consider systemic reforms in
a general equilibrium framework with no health or wage uncertainty. Using a life-cycle
model with heterogeneous mortality rates, Bagchi (2019) examines reforms to the Social
Security benefit formula, but he does not consider claiming decisions. As we show below,
claiming decisions and retirement decisions are at times closely linked, and changing the
claiming adjustments is one of our principal reforms. Huggett and Ventura (1999) and
Nishiyama and Smetters (2008), who also consider reforms to the benefit formula, do not
allow for claiming decisions or heterogeneous mortality risk.
Huggett and Parra (2010) find the system of life-cycle taxes that implements the social
planner’s solution for a cohort of individuals, holding fixed the net resources extracted
from that cohort. In a separate exercise, they find that the optimal Social Security
benefit function, if considered in isolation, would decrease modestly in lifetime earnings.
Ndiaye (2018) expands Huggett and Parra’s (2010) framework to include a retirement
choice. He finds that the optimal Social Security system would tie benefits more closely
to earnings and strengthen the claiming adjustments. Huggett and Parra (2010) and
Ndiaye (2018) show that Social Security reforms in isolation recover only a portion of the
utility gains achievable under an optimal tax system. On the other hand, to make their
exercise tractable, they impose a number of simplifications. Among the most important of
these are the assumptions that all agents share a common, fixed lifespan and that agents
receiving Social Security benefits are unable to work. In our robustness exercises, we
show that these restrictions increase the sensitivity of retirement to claiming incentives
and thus favor large claiming adjustments.
The remainder of the paper is organized as follows. In Section 2, we describe our
model, while in Section 3, we describe its calibration. In Section 4, we present the policies
that would maximize welfare in the current demographic environment. In Section 5, we
discuss the 2050 demographic environment and the policies that would maximize welfare
therein. In Section 6, we perform a number of robustness exercises. We conclude in
5
Section 7.
2
Model
Our behavioral model is similar to those of Imrohoroğlu and Kitao (2012) and Jones
and Li (2018) but contains a considerably richer treatment of health, mortality and wages,
including heterogeneity related to education. Our description borrows heavily from Jones
and Li (2018).
2.1
Demographics
The population consists of overlapping generations. Each period represents two years.
Let j ∈ {1, 2, . . . , J} denote age, where J represents the maximum lifespan. The population grows at the constant rate χ. Among other dimensions, new individuals differ in
terms of education level (e). We assume the distribution of education is exogenous and
constant over time.
Let sj (hj , e) denote the survival rate between periods, which depends on each individual’s age, health status (hj ), and education level. Health can take on five potential
values: good (hj = 1), bad (hj = 2), work limitation (hj = 3), in a nursing home (hj = 4),
and disabled (hj = 5). Health status affects individuals through five channels: the time
endowment, the survival probability, health transitions, medical expenditures, and access
to Disability Insurance (DI). We assume that all individuals in the disabled health state
receive DI – our empirical definition of the disability state is DI receipt – and thus abstract away from the DI application decision and uncertainty over DI receipt. Because
the reforms we consider are not intended for the disabled, we view this as a reasonable
simplification.
6
2.2
Preferences
Each period surviving individuals receive utility from consumption (c) and leisure (l)
according to the function u(c, l). Leisure, in turn, depends on hours of work, nj , health,
and labor market participation in the previous period (nj−1 ):
lj = 1 −
X
φhke I{hj =k} − φne I{nj >0} − φre I{nj−1 =0 and nj >0} − nj ,
(1)
k
where IA is the 0-1 indicator function that takes the value of 1 when event A occurs.
The term φhke I{h=k} reflects the time cost of being in health status k for a person with
education level e. This cost is normalized to 0 for individuals in good health and is
intended to reproduce the empirical observation that unhealthy people work less. The
term φne I{n>0} captures the fixed time costs of work for education level e. This term is
intended to reproduce the observation that most people work full time or not at all (see,
e.g., Cogan 1981 and French and Jones 2012). The term φre I{nj−1 =0 and nj >0} captures the
time cost of reentering the labor market. Similar specifications for leisure have been used
in, among other studies, French (2005) and French and Jones (2011). We assume that
disabled individuals and nursing home residents cannot work.
When they die, individuals receive warm-glow utility from bequests according to the
function v(a), where a denotes the amount of assets bequeathed. Future utility is discounted using the factor β.
2.3
Earnings
Individuals that work at age j receive the wage wj ,
nj ζ
.
wj = ω(e, j, hj ) · ηj · min 1,
n̄
7
(2)
Wages depend on age, education and health through ω(e, j, hj ). ηj is an idiosyncratic
productivity shock following a Markov process with transitions Πηj (e, ηj , ηj+1 ). The final
term, min{1, [nj /n]ζ }, imposes a penalty for working less than the full-time workload of n̄.
Following Aaronson and French (2004), we set ζ to 0.415, implying that half-time workers
are paid 25% less than full-time workers. Imposing the part-time earnings adjustment
leads total earnings, Wj = wj nj , to have increasing returns to scale in hours of work.
This feature combines with the fixed time cost of work to encourage full-time work.
2.4
Medical Expenditures and Health Insurance
Each individual’s health status (hj ) changes stochastically over the life cycle, following
a Markov process with the age-dependent transition probability πjh (hj+1 | e, hj ). Total
medical expenditures, denoted by mj = mj (hj , j ), depend on age, current health and
the shock j , an i.i.d. process with the stationary distribution Π(). Health insurance
coverage is universal: Medicare covers DI recipients (after a two-year waiting period) and
all individuals 65 and older (j ≥ J M ), and private health insurance covers the rest of the
population.2 The medical expenses paid by the individuals themselves can be split into
two parts: insurance premia, pj , which are paid at the beginning of each period before the
medical spending shocks for that period are revealed; and copayments, Qj (hj , j ), which
are paid at the end of each period after the shocks are revealed. Premia and copayments
follow
pj =
ppriv
if j < J M and hj−1 6= 5
pmcr + psupp otherwise,
(1 − κpriv )mj (hj , j )
if j < J M and hj−1 6= 5
Qj (hj−1 , hj , j ) =
(1 − κmcr − κsupp )mj (hj , j ) otherwise,
2
The model abstracts from heterogeneity in private insurance access. Dynamic models with variable
insurance eligibility and insurance take-up include Jeske and Kitao (2009) and Pashchenko and Porapakkarm (2013).
8
where the superscripts “priv,” “mcr,” and “supp” denote, respectively, private insurance,
Medicare, and Medicare supplement insurance plans. κi , i ∈ {priv, mcr, supp} is the
insurer payment rate for insurance type i.
2.5
Government
The government collects taxes and provides social insurance. The difference between
the government’s revenues and its transfer spending is absorbed by direct spending (G).
We assume further that the government collects all bequests, which it then distributes
equally among surviving individuals, giving each the transfer B. This commonly used,3
if clearly counterfactual, assumption simplifies the model greatly.
Old-Age Insurance (OAI). Let J E denote the Early Retirement Age (ERA), J N
denote the Normal Retirement Age (NRA), and J L denote the Late Retirement Age
(LRA). Individuals not receiving DI benefits (hj 6= 5) can choose any age from the ERA
of 62 to the LRA of 70 to claim their Social Security benefits. OAI recipients receive
benefits according to Ak ss(Ej ), where: Ak is an adjustment factor based on the benefit
claiming age k, designed to capture early retirement penalties and delayed retirement
credits; Ej is an index of the individual’s 35 highest years of earnings, commonly known
as Average Indexed Monthly Earnings (AIME); and ss(·) is a function that converts the
earnings index into the Primary Insurance Amount (PIA).
Beneficiaries who are below the NRA and have labor income in excess of the earning
limit yjet have their benefits withheld at a rate of τjet : for each additional dollar earned,
Social Security benefits are reduced by τjet until all benefits are withheld. Let Tjet denote
benefits lost through the earnings test, and let ss∗j denote the remaining benefits. We
have
3
Papers that employ this assumption include Kitao (2014), Conesa et al. (2019) and Laun et al. (2019).
9
Tjet (ssj , Wj ) = min ssj , τjet max{0, Wj − yjet } ,
(3)
ss∗j (ssj , Wj ) = ssj − Tjet (ssj , Wj ).
(4)
Any such reductions in current benefits, however, are offset by permanent increases in
future benefits, implemented in our framework through increases in Ej+1 .4 The net incentive generated by the earnings test depends on whether the increases in future benefits
are actuarially fair; because the current crediting formula is considered actuarially fair for
the average person, for most workers the net tax rate associated with the earnings test is
small.
Social Security benefits are also subject to income taxes. An important feature of these
taxes is that the amount of Social Security benefits subject to taxation is increasing in the
beneficiaries’ total income. At certain income levels, each additional dollar of earnings,
in addition to being taxable itself, adds 50-85 cents of Social Security benefits to taxable
income, increasing the effective marginal income tax rate on these earnings by 50% to
85%. Jones and Li (2018) show that this provision is potentially quite important.
Disability Insurance (DI). Each period before J N , non-disabled individuals face the
risk, given by πjh (hj+1 = 5 | e, hj ), of moving onto the DI rolls. We assume that individuals
entering the DI system remain in the system until they reach J N , consistent with the
extremely low rate (less than 1% per year) at which DI beneficiaries are terminated due
to medical recovery. DI beneficiaries receive a benefit of dij = di(Ej ), which receives the
same tax treatment as OAI benefits. Upon reaching J N , DI recipients are transferred
to OAI. We capture this in the model by having DI recipients transit to the other four
health states at age J N with probability πJhN −1 (hJ N | e, hJ N −1 = 5). Former DI recipients
also receive a productivity value (ηJ N ) drawn from the stationary productivity distribution
4
This is a simplification, as in actual practice benefits are adjusted only after a person reaches J N .
10
for their education levels.
Taxes. The government collects income taxes, consumption taxes, payroll taxes and
Medicare premia. Payroll taxes consist of two parts: a Medicare tax imposed on all
earned income at the flat rate τ mcr and a Social Security tax imposed on earned income
up to the taxable threshold y ss at the flat rate τ ss . Consumption taxes are imposed on
all consumption goods at the flat rate τ c . Income taxes are progressive and are based
on taxable income yj according to the tax function T (yj ). Taxable income itself is the
sum of asset income (raj ), earnings (Wj ) and the taxable portion of OAI or DI benefits,
SS(ss∗j + dij , raj , Wj ).
Means-Tested Social Insurance. Means-tested social insurance can be thought
of as a combination of TANF, SNAP, SSI, uncompensated medical care and Medicaid.
Following Hubbard et al. (1995), we assume that this program provides a consumption
floor of c. At the beginning of each period, means-tested transfers are given by
trj = max 0, (1 + τ c )c − yjd ,
(5)
where yjd denotes total financial resources – the sum of assets, after-tax income and distributed bequests, less insurance premiums – prior to receiving means-tested transfers:
yjd = (1 + r)aj + Wj + ss∗j (ssj , Wj ) + dij − T raj + Wj + SS(ss∗j + dij , raj , Wj )
− τ mcr Wj − τ ss min{Wj , y ss } + B − pj .
2.6
(6)
The Individual’s Problem
Individuals can be characterized by their age j and the seven-element state vector
xj = {e, bj−1 , nj−1 , aj , ηj , hj , Ej }, where e records the education level, bj−1 is an indicator
function for OAI receipt in the previous period, nj−1 records labor force participation in
the previous period, aj records assets carried over from period j −1, ηj is the idiosyncratic
11
productivity shock, hj is health status, and Ej is the earnings index. Note that bj−1 , nj−1 ,
and ηj are not active state variables for DI beneficiaries.5
At the beginning of each period, non-DI beneficiaries choose labor hours and whether
to file an OAI claim (if they are age-eligible and have not already claimed). Claiming
allows them to receive OAI benefits from the current period forward and is not reversible.
At this point, individuals’ financial resources consist of their labor income, assets and
asset income, DI or OAI benefits, and lump-sum bequest transfers, net of taxes and
health insurance premiums. If this amount is below the consumption floor, government
transfers via means-tested insurance bridge the gap. Individuals then choose how much
to consume out of their (post-transfer) financial resources. They can save, but borrowing
constraints prevent them from consuming more than their current resources.
At the end of each period, the medical expenditure shock j is realized, which determines future assets, aj+1 . The survival shock comes next. Individuals who die receive
warm-glow utility from bequests, while surviving individuals realize their new productivity
(ηj+1 ) and health status (hj+1 ) and enter the next period with state vector xj+1 .
In recursive form, the individual’s problem is
Vj (xj ) = max
cj ,nj ,bj
u(cj , lj ) + βEj [sj (e, hj )Vj+1 (xj+1 ) + (1 − sj (e, hj ))v(aj+1 )] ,
subject to equations (1)-(6) and:
(1 + τ c )cj ≤ yjd + trj ,
(7)
aj+1 = yjd + trj − (1 + τ c )cj − Qj (hj−1 , hj , j ),
(8)
Ej+1 = fj (Ej , Wj , bj−1 , bj , hj ),
(9)
nj = 0, if hj ≥ 4.
(10)
5
On the other hand, to capture the delay in Medicare eligibility, when considering DI recipients we
add lagged health to the state vector. This allows us to distinguish new DI recipients from those who
have received DI for at least two years.
12
Equation (7) prohibits borrowing to fund current goods consumption. Equation (8) describes the law of motion for assets. Note that individuals are allowed to take on medical
expense debt. Equation (9) describes the law of motion of the earnings index Ej , which
depends on age, the index’s current value, current earnings, and DI and OAI recipiency
status. Equation (10) prevents nursing home residents and DI beneficiaries from working.
2.7
Stationary Equilibrium
Our approach will be to take wages and the interest rate from the data and to find
the private insurance premium and government policies that produce budget balance for
a stationary distribution of individuals. Our equilibrium concept is identical to the one
used in Kitao (2014). Appendix A provides a formal definition.
3
Calibration
3.1
Data
We have two main data sources: the Medical Expenditure Panel Survey (MEPS)
and the Health and Retirement Study (HRS). The MEPS is a two-year rotating panel,
with each individual interviewed five times over the two-year period. In each interview,
individuals provide information on demographics, employment, income, health insurance
coverage, health conditions and medical spending. We use all 18 panels of MEPS, spanning
1996-2014, to estimate statistics for individuals under age 65. While the MEPS has data
on older individuals, it does not track individuals who enter nursing homes.6 We thus
turn to the HRS, a panel survey of individuals aged 50 and above. The initial cohort
of the HRS was first interviewed in 1992, with follow-up interviews approximately every
other year since then. Younger cohorts enter in later years. We use wave 3 (1996) to
6
The share of individuals in nursing homes is almost zero for those under age 65, making MEPS a
suitable dataset for this group.
13
wave 12 (2014) of HRS, the same period covered by the MEPS, to estimate statistics
for individuals aged 65 and above. We also utilize the Panel Study of Income Dynamics
(PSID) for wage and hours-worked data at younger ages, and the Survey of Consumer
Finances (SCF) for asset data.
Because our model is one of singles, and thus does not account for child-rearing and
secondary earners, our data for employment, wages and OAI claiming are only for men.
For all other uses, we include both genders.7 Unless otherwise noted, the data are expressed at a two-year frequency, the same as the model period; we aggregate the MEPS
data to this frequency. Because we calibrate the benchmark economy to match the 2012
U.S. economy, all nominal values are denominated in 2012 dollars.
3.2
Demographics and Health Status
Individuals enter the economy at age 24 with two possible levels of education: with
a bachelor’s degree (e = 1) and without a bachelor’s degree (e = 0). The fraction of
24-year-olds with a bachelor’s degree is set to 0.309, the share of college graduates among
those aged 25 and older in 2012 reported by the Census (U.S. Census Bureau 2019b).
For each education level, we set the initial distribution of health to match that found
in the MEPS at age 24. Individuals face uncertain mortality and die with probability 1
after age 106. The population grows at a constant annual rate of 1.1 percent, the average
growth rate for the United States.
We measure health status, hj ∈ {1, 2, 3, 4, 5}, as follows. Individuals who are currently
receiving DI benefits are classified as disabled (hj = 5); this state is eliminated at age 66,
when DI recipients are transferred to OAI. Individuals over 65 who are currently in a
nursing home with a stay of 60 days or longer are classified as being in a nursing home
(hj = 4). Among the remainder, individuals are classified as having work limitations
7
Because the SCF measures assets at the household level, assets for married couples are split evenly.
14
(hj = 3) if they report having health problems that limit their work capacity, as having
bad health (hj = 2) if their self-reported health status is fair or poor, and as having
good health (hj = 1) if their self-reported health status is excellent, very good, or good.
Appendix B provides more detail.
3.3
Survival Rates
Given that our data cover 18 years, we estimate period (for 2012) rather than cohort
survival rates. Following Attanasio et al. (2010), we proceed in three steps. First, we
estimate from the MEPS and HRS a logit model expressing the two-year survival rate
as a function of age, education, health and calendar year. Second, we use the results of
the logit estimation, ŝj (e, hj ), to calculate survival premia, ∆j (e, hj ), and approximate
survival rates, s̃j (e, hj ):
∆j (e, hj ) = ŝj (e, hj ) − ŝj (0, 1),
s̃j (e, hj ) = s̃j (0, 1) + ∆j (e, hj ).
(11)
(12)
The final step is to calibrate s̃j (e, 1) so that the model matches aggregate survival rates
for calendar year 2010 (from Bell and Miller 2005). A more detailed description of our
procedure can be found in Appendix B. Our estimates imply that at age 24, people with
a college degree expect to live an additional 56.7 years; the average remaining lifespan for
those without a degree is 52.7 years.
3.4
Health Transitions
We estimate health-transition probabilities using a multinomial logit model. Model
predictors include age, education, health status, interactions of education and health with
age, and a linear year trend; once again our base year is 2012. Given the differences in
15
health categories (see Section 3.2), we estimate separate transition models for the MEPS
and the HRS. The details of our procedure, including a description of how we handle the
switch from the MEPS health measures at age 64 to the HRS health measures at age 66,
can be found in Appendix B.
Our estimates show that at any given age, individuals without a bachelor’s degree on
average have worse health than individuals with a bachelor’s degree. Because bad health
leads to lower survival rates, the mortality disadvantages of the less-educated are mediated
in part through poorer health. Of particular note is the probability of DI receipt. At age
64, over 20% of those without a bachelor’s degree are receiving DI. The corresponding
fraction for those with a bachelor’s degree is half as large.
3.5
Medical Expenditures and Health Insurance
For individuals 65 and younger, we use medical spending data from the MEPS. The
MEPS contains measures of both total and out-of-pocket medical expenditures. We set
the coinsurance rate for private insurance to 27.2%, the ratio of out-of-pocket medical
expenditures to total expenditures for those covered by private insurance.
In contrast to the MEPS, the HRS contains quality data only for out-of-pocket medical
expenses: the way in which we impute total medical expenses for the HRS can be found
in Appendix B. We assume that for those receiving Medicare, Medicare and Medicare
supplement insurance cover 54.7% and 12.5%, respectively, of total expenditures (?)Table
8]de2016medical. The Medicare premium is set to $3,310 to match the sum of part B and
part D premia over a two-year span. The premia for private and Medicare supplement
insurance are set in equilibrium, and take the values of $4,640 and $3,210, respectively,
for the benchmark economy.
We allow the medical spending shock to take on three values. As in Kitao (2014),
we capture the long tail in the distribution of medical expenses by using: a small shock
16
with a 60% probability, a medium shock with a 35% probability, and a large shock with
a 5% probability. We report the values of these shocks in Appendix B.
3.6
Wages
We estimate the wage profile ω(e, j, hj ) and the process for the wage shock ηj from
our data. Because each MEPS panel spans only two years, we use wage data from the
PSID for ages 24-64. We use HRS data for older ages.
We begin by estimating a fixed-effects wage profile. The coefficients from this regression do not provide an unbiased estimate of ω(e, j, hj ), however, because they are
estimated only on the wages of those who work. To control for this bias, we apply the
correction developed in French (2005). Appendix C describes the procedure and presents
(in Figure 14) both the uncorrected and corrected wage profiles. With the corrections
imposed, wages fall steadily after age 55, and at any age, individuals in worse health
receive lower wages.
We use the residuals from the first-step fixed-effects regression to estimate the AR(1)
productivity shock ηj , again following the procedure used by French (2005). Appendix C
describes the procedure and shows (in Table 5) the resulting parameter estimates. Individuals with a bachelor’s degree experience larger but less persistent shocks than individuals
without a bachelor’s degree. The initial distribution of productivity shocks by education
and by health is set to match the distribution of the estimated fixed effects.
3.7
Government
We calibrate government policy parameters to those in effect in 2012.
OAI. Social Security benefits depend on the index Ej , the beneficiary’s average earnings over his 35 highest earnings years. Because keeping track of 35 years of earnings is
infeasible, we use an approximation for Ej similar to that used by Imrohoroğlu and Kitao
17
(2012). We assume that before the ERA, Ej equals the individual’s average earnings.
Between the ERA and the benefit claiming age, Ej is updated whenever current earnings
are greater than the earnings index. Once OAI benefits have been claimed, Ej is updated
only to reflect the benefits that are withheld due to the earnings test.8 We also impose
the statutory restriction that in each period the earnings applied to Ej is bounded above
by the payroll tax cap y ss .
OAI benefits are given by Ak ss(Ej ), where ss(·) is a piecewise linear function, and Ak
is an adjustment factor based on the benefit claiming age k.9 The rules for ss(·), {Ak }k ,
and the earnings test are based on the rules facing the 1943-54 birth cohorts. To facilitate
our search for optimal policies, we do not use the rules themselves but use parsimonious
approximations; this allows us to search over a small set of parameters when finding the
optimum. The approximations are
ss(E)
b
= ss1 + ss2 (E + ss4 )
bk =
A
k
JN
ss3
, where ss4 =
max{−ss1 , 0}
ss2
1/ss3
,
(13)
A
.
(14)
In the approximation of ss(·), equation (13), ss1 (when positive) is the benefit level for
those with no earnings, while ss2 and ss3 control how benefits rise with average earnings.
ss4 is an adjustment factor that ensures the benefit is non-negative. Variants of this
approximation appear in Bagchi and Jung (2019) and Cottle-Hunt and Caliendo (2019).
In the approximation of Ak , equation (14), A controls the sensitivity of the adjustment
to the claiming age k. When benefits are claimed at the normal retirement age, J N
(= 66 for the benchmark economy), there is no adjustment. As shown in Figure 1, the
approximations work very well.
In actual practice, Ej continues to be updated for high current earnings even after benefit claiming.
This is a relatively rare event, however, and in the interest of simplicity, we rule it out.
9
Following Jones and Li (2018), we capture the effect of Ak and the earnings test through adjustments
to Ej ; this allows us to find the consumer’s decision rules without having to keep track of k.
8
18
7
10 4
1.4
6
Ratio of Benefit to PIA
1.3
5
PIA
4
3
2
1
Statutory
Fitted
0
0
0.5
1
1.5
2
AIME
2.5
10 5
1.2
1.1
1
0.9
0.8
0.7
62
Statutory
Fitted
63
64
65
66
67
68
69
70
Age
(a) PIA formula, ss(Ej )
(b) Claiming adjustment, Ak
Figure 1: Social Security Rules: Statutory and Approximated
DI. DI benefits are based on average earnings at the time the benefits are claimed.
There is no adjustment for early claiming: dij = ss(Ej ). At age 66, DI beneficiaries are
transferred to OAI.
Taxes. The Social Security payroll tax rate τ ss is 6.2 percent, and the taxable earnings
limit y ss is $220,200. The Medicare payroll tax rate τ mcr is 1.45 percent. (Recall that τ ss
and τ mcr are calibrated to only the employee share (one-half) of payroll taxes.) Following
Gouveia and Strauss (1994), we use the following tax function:
Te(y) = λ0 [y − (y −λ1 + λ2 )−1/λ1 ].
We use the values of λ0 and λ1 estimated by Gouveia and Strauss (1994), and we set λ2
so that in a balanced-budget equilibrium, direct government spending (G) equals about
23% of total earned income.10
As described in Section 2.5, the portion of Social Security benefits subject to income
10
Direct spending is about 15% of GDP (The World Bank 2016), and labor income comprises about
two-thirds of GDP, consistent with the baseline specification. The consumption tax is set to 6%, the
value found by Mendoza et al. (1994).
19
taxes depends on the net benefits themselves, ss∗j (ssj , Wj ), and “combined income,” YjCI =
raj + Wj + 0.5ss∗j . The exact formula is
0
if YjCI < 25000
SS(ss∗j , raj , Wj ) =
min{0.5ss∗j , 0.5(YjCI − 25000)}
if 25000 ≤ YjCI < 34000
min{0.85ss∗ , 4500 + 0.85(Y CI − 34000)} otherwise.
j
j
Means-Tested Insurance. The consumption floor is set to $3,500 per year, which
is roughly the value estimated by De Nardi et al. (2010) ($3,800 in 2012 dollars).
3.8
Preferences
The flow utility function is specialized as:
u(cj , lj ) =
1−σ
1
cγj l1j −γ
,
1−σ
with leisure, l, defined in equation (1). γ determines the weight on consumption relative
to leisure. We set σ to 7.5, the (approximate) value estimated by French (2005) and
French and Jones (2011) and used by Jones and Li (2018), who employ utility functions
and wage processes very similar to ours. Deceased individuals derive utility from bequests
according to
v(a) = ψ1
(ψ2 + max{a, 0})γ(1−σ)
,
1−σ
(15)
where ψ2 is set to $500,000, as in De Nardi (2004) and French (2005). The maximum
operator ensures that debt due to medical expenditure shocks is waived upon death.
The remaining preference parameters are calibrated by fitting the model to a set of
labor supply – employment and hours – and asset targets. The employment targets come
20
from our principal data sets, the MEPS and the HRS,11 while our asset targets come from
the 2013 SCF. To control for business cycle fluctuations, we regress employment and the
inverse hyperbolic sine of assets (for multiple SCF waves) on aggregate unemployment
and GDP growth rates (along with age and cohort dummies) and replace the realized
business cycle effects with the effects predicted at the average rates of unemployment and
GDP growth. We assume that the average worker spends 33% of his time endowment at
work.12
Table 1 lists the parameters and associated targets, along with the model’s fit. The set
of parameters that determine the fixed cost of working and the time cost of not being in
good health (φne , φhje , e ∈ {0, 1}, j ∈ {2, 3}) are calibrated to match average employment
by health and education for individuals aged 66-75.13 The parameter φre , the cost of
reentering the labor market, is calibrated to match the reentry rate of individuals aged
66-75. We target those aged 66-75 because we want the model to match the employment
incentives facing older individuals. The discount factor β is set to match the median
assets of individuals aged 46-54, an age group accumulating wealth for retirement. (The
pre-tax interest rate is set to 0.05 per year, the value suggested in Cooley (1995).) ψ1 is
calibrated to match median assets for individuals aged 76-85, an older group closer to
death.
3.9
Model Fit
Although the model is calibrated to match several broad targets, it can be assessed
along dimensions that are not targeted.
Employment and Hours: Figure 2(a) shows the model’s fit of aggregate employ11
A person is considered employed if he worked more than 500 hours in the HRS or PSID, or earned
more than 500×$7.25 in the MEPS.
12
If an individual has 16 waking hours each day, and work absorbs exactly one-third of his annual time
endowment (365 × 16 = 5, 840), he will work 1,947 hours per year.
13
The time cost for those on DI or in a nursing home is set to that for people with work limitations.
21
Table 1: Calibrated Parameters
Para.
Interpretation
Value
Target
Data
Model
γ
β
ψ1
φn0
φn1
φh20
φh21
φh30
φh31
φre
Consumption weight
Discount
Bequest intensity
Participation cost, NC
Participation cost, C
Bad health cost, NC
Bad health cost, C
Limitation cost, NC
Limitation cost, C
Reentry cost
0.323
0.948
1,605
0.036
0.014
0.029
0.010
0.063
0.056
0.013
Average hours of workers
Median assets (000s), 46-55
Median assets (000s), 76-85
Empl., 66-75, good health, NC
Empl., 66-75, good health, C
Empl., 66-75, bad health, NC
Empl., 66-75, bad health, C
Empl., 66-75, limitation, NC
Empl., 66-75, limitation, C
Reentry, 66-75
0.333
86.1
168.2
0.335
0.452
0.225
0.376
0.103
0.224
0.066
0.334
86.5
167
0.334
0.456
0.223
0.372
0.103
0.223
0.066
Note: NC and C denote, respectively, individuals without and with a four-year college degree.
ment. The model overstates employment at younger ages, especially prior to age 30. At
these ages, only the disabled fail to work. In the model, individuals in their 20s want to
accumulate precautionary savings (Low 2005); there is no possibility of parental support.
The model also rules out the possibility of continuing education. On the other hand,
the model captures quite well the employment decline from age 50 forward, as well as
the persistence of employment into fairly old ages. Figure 2(b) compares the model’s
predictions for aggregate working hours to those found in the PSID and HRS. The close
fit of total hours indicates that at younger ages, the model simultaneously over-predicts
employment and under-predicts hours conditional on employment.
Figure 3 shows the model’s ability to reproduce life-cycle trends in employment across
education and health groups. Consistent with data, in the model, individuals without a
bachelor’s degree retire earlier than individuals with a bachelor’s degree, and individuals
with bad health or work limitations retire earlier than individuals with good health.
Assets and OAI Claims: Figure 4 shows median assets by age, revealing a good
fit. Despite low earnings at young ages, individuals start saving very early in their careers
in order to accumulate precautionary savings (Low 2005). In the model, assets increase
22
0.4
1
0.9
0.35
0.8
0.6
Total Hour
Rate of employment
0.3
0.7
0.5
0.4
0.25
0.2
0.15
0.3
0.1
0.2
Model
Data
0.1
0
20
30
Model
Data
0.05
40
50
60
70
80
0
20
90
30
40
50
Age
60
70
80
90
Age
(a) Employment
(b) Hours
Figure 2: Employment Rates and Total Hours by Age
1.2
1.2
1
1
Rate of employment, college
Rate of employment, non-college
Source: Employment data from the MEPS and HRS. Hours data from the PSID and
HRS.
0.8
0.6
Model, good
Model, bad
Model, limitation
Data, good
Data, bad
Data, limitation
0.4
0.2
0
20
30
40
0.8
0.6
Model, good
Model, bad
Model, limitation
Data, good
Data, bad
Data, limitation
0.4
0.2
50
60
70
80
90
0
20
Age
30
40
50
60
70
80
90
Age
(a) No college
(b) College
Figure 3: Employment Rates by Age, Health and Education
Source: Employment data from the MEPS and HRS.
until people reach their early 60s; as people start to retire, they spend down their assets
and wait for the optimal time to claim OAI benefits. Asset levels increase between age
23
70 and 84, and are almost unchanged after age 85. Three mechanisms account for the
slow asset decumulation after retirement: the bequest motive, precautionary savings for
longevity risks and the need to pay for medical expenses, especially nursing home costs.
3.5
10 5
Model
Data
3
Median assets
2.5
2
1.5
1
0.5
0
20
30
40
50
60
70
80
90
100
Age
Figure 4: Median Assets by Age
Note: Asset data from the 2013 SCF.
Figure 5 displays OAI recipiency rates.14 Although the model does not target these
rates, Figure 5(a) shows that it matches the general claiming patterns. The model understates age-62 claiming in 2012 by about 17 percentage points (pp). This may be due
to the absence within the model of illiquid housing wealth.15 If individuals who wish to
retire early have few liquid assets, they may need to claim their Social Security benefits at
younger ages (Kahn 1988; Kaplan and Violante 2014). Illiquid housing may also explain
why assets in the data do not dip when people are in their early 60s. Figure 5(b) shows
that within the model, people without a college degree are more likely to claim early,
consistent with their shorter lifespans. It bears noting that Figure 5 does not account
for DI. By age 62, about 16% of the population is on DI, most of them without a college
degree.
14
In the model, as in actual practice, DI beneficiaries are transferred to OAI at age 66.
Pashchenko and Porapakkarm (2018), who study claiming in some detail, find it to be sensitive to
discount and interest rates.
15
24
In recent years, there has been an ongoing trend toward later claiming, even though
people born between 1943 and 1954 – the cohorts making retirement decisions in our analysis – face the same NRA and claiming adjustments. In addition to the claiming decisions
observed in 2012, Figure 5(a) shows claiming in 2016, which more closely resembles the
1
1
0.9
0.9
0.8
0.8
Rate of OAI Recipiency
Rate of OAI Recipiency
model’s predictions.
0.7
0.6
0.5
0.4
0.3
0.2
0
62
63
64
65
66
67
0.6
0.5
0.4
0.3
0.2
Model
Data, 2012
Data, 2016
0.1
0.7
Non-college
College
0.1
68
0
62
Age
63
64
65
66
67
68
Age
(a) Aggregated, data and model
(b) Disaggregated, model
Figure 5: OAI Recipiency Rates by Age, Education and Health
Note: The claiming rate for the data is computed as the age-by-age ratio of male retired worker
beneficiaries for 2012 (year-end) from the SSA Statistical Supplement (Social Security Administration
2014, Table 5.A1.1) to the estimated male population (U.S. Census Bureau 2019a, average of July 2012
and July 2013 and average of July 2016 and July 2017). In panel (b), solid series are for individuals
without a bachelor’s degree, and dashed series are for individuals with a bachelor’s degree.
Income and Lifespan Heterogeneity: Table 2 shows, for three model specifications, mean lifetime earnings (discounted at an annual rate of 5%) and life expectancy
for each lifetime-earnings quintile. The first two columns show results for the baseline
model. Individuals in the top earnings quintile earn nearly four times as much as those
in the bottom quintile, and they live 11 years longer. By way of comparison, (Chetty et
al., 2016, Figure 3) found that in 2015, the age-40 life expectancy of men in the bottom
household income quartile was about 10 years lower than that of men in the top quartile.
25
The difference for women was six years. Noting that we condition on lifetime earnings
rather than current earnings, our mortality gradient is arguably similar to Chetty et al.’s
(2016).
Table 2: Earnings and Expected Lifespan by Lifetime-Earnings Quintile
Earnings Quintile
Bottom
Fourth
Third
Second
First
Ratio Q5/Q1
4
Baseline
Lifetime Expected
Earnings Lifespan
($000s)
(Years)
(1)
(2)
441
694
869
1,109
1,703
3.86
No Educational
Differences
in Demographics
Lifetime Expected
Earnings Lifespan
($000s)
(Years)
(3)
(4)
70.8
77.9
79.0
80.1
82.0
1.16
445
695
862
1,092
1,672
3.75
71.4
78.3
79.0
79.4
80.0
1.12
No Health or
Educational
Differences
Lifetime Expected
Earnings Lifespan
($000s)
(Years)
(5)
(6)
577
778
927
1,105
1,439
2.49
71.9
78.6
79.4
79.6
80.1
1.11
Welfare-Maximizing Policies
We turn to finding the policies that maximize aggregate welfare, which we define as
the ex-ante lifetime utility of a newborn. We search over five parameters: the payroll
tax cap y ss ; the parameters ss1 , ss2 and ss3 of the benefit function ss(E)
b
(see equabk (see
tion (13)); and the curvature parameter A of the claiming adjustment function A
equation (14)). However, we require the welfare-maximizing policy to generate the same
aggregate expenditures – to do this, we adjust all benefits proportionally – as the baseline
policies, eliminating a degree of freedom. Because our goal is to maximize welfare while
holding Social Security’s aggregate footprint fixed, we also adjust payroll taxes so that
the reformed system collects the same revenues as the baseline policies. We assume that
changes in the payroll tax rate (2·∆τ ss ) and the payroll tax cap (y ss ) apply only to workers
26
and report them accordingly.
We maintain general government budget balance by changing the tax parameter λ0 ,
holding public goods (G) fixed. We fix DI benefits at their benchmark levels as well.
Although interactions between OAI and DI are undoubtedly important, in our model, DI
claiming is treated as exogenous. We include DI recipiency to ensure that our analysis of
OAI reforms is not driven by disability-related outcomes; we do not analyze DI itself.
4.1
Benchmark Environment
Column (1) of Table 3 summarizes the optimal policies for our benchmark environment.
The optimal policies have three important features: (1) the cap on taxable earnings
changes only modestly; (2) the PIA does not depend on average lifetime earnings; (3) the
adjustments for early or late claiming are smaller than the current adjustments.
Cap on taxable earnings (y ss ). The first line of Table 3 shows that the optimal
cap on taxable earnings equals 197% of median AIME, roughly the current level (see
column (0)). Raising the upper bound on taxable earnings has two main effects. The
first is that a larger tax base leads to a lower payroll tax rate (Figure 6(a)), increasing
employment among both the college- and non-college-educated (Figure 6(b)). The second
is that it raises the marginal tax rate for workers with earnings near the previous cap.
These are among the most skilled workers in the economy. Figure 6(c) show that average
earnings (over all individuals) generally fall as the tax cap rises. Even as employment
increases, the decrease in labor supply among the most skilled is enough to reduce average
earnings. The effect is most pronounced among the college-educated.
Figure 6(a) shows that increasing the cap leads to lower bequests. The increased
employment associated with the lower payroll tax rate leads workers to claim their Social
Security benefits at older ages. Retirees receive higher annual benefits, reducing the wealth
they carry into very old ages. Given that Social Security expenditures are held fixed, it
27
Table 3: Current and Optimal Policies in Stationary Equilibrium
Current Demographics
Current Optimal 1 Optimal 2
Benefit Taxation and
Earnings Test
2050 Demographics
Current Optimal 1 Optimal 2
Yes
Yes
No
Yes
Yes
No
(0)
(1)
(2)
(3)
(4)
(5)
2.03
2.18
3.09
3.10
42.8
43.2
43.7
21.0
28.9
37.8
33.9
33.9
33.9
34.4
34.8
35.1
0.80
1.23
0.74
1.33
0.81
1.22
0.80
1.23
81.5%
100.0%
1.8%
36.8%
26.8%
99.8%
85.4%
100.0%
66.6%
25.6%
74.4%
38.4%
69.4%
37.8%
72.7%
40.7%
75.9%
0.33
87.4
133.8
6.8
74.1
6.8%
25.5%
73.4%
0.33
97.1
200.5
10.5
87.1
5.0%
25.8%
72.3%
0.33
95.9
205.2
10.7
86.3
4.8%
26.1%
73.1%
0.33
96.5
208.8
11.2
87.3
5.5%
25.8%
23.6
3.3
20.3
28.5
4.2
24.3
28.5
4.2
24.3
28.5
4.2
24.3
1.15%
1.91%
-0.06%
0.48%
0.94%
0.25%
0.48%
1.81%
-0.31%
0.48%
1.69%
-0.25%
1.50%
1.98%
0.37%
2.22%
2.76%
0.92%
Tax cap / median
2.18
1.97
age-62 AIME
Age-66 benefit (PIA) percentiles (000s)
10th percentile
30.7
43.6
median
42.1
43.7
90th percentile
55.1
43.9
Benefits relative to age-66 benefit
age 62
0.74
0.78
age 70
1.32
1.26
Benefit claiming
age 62
6.0%
30.7%
age 66
66.0%
100.0%
Employment
age 62
69.6%
64.1%
age 70
24.8%
21.5%
Aggregate behavior
employed
76.4%
75.1%
hours if employed
0.33
0.33
earnings ($000s)
87.7
86.7
assets ($000s)
123.3
129.3
bequests ($000s)
6.1
6.4
consumption ($000s)
73.5
72.9
τss
6.2%
6.5%
λ0
25.8%
25.9%
Residual government budget (000s)
revenues
23.6
23.6
expenses
3.3
3.3
balance
20.3
20.3
Internal (annual) rate of return on OAI and DI
all households
1.15%
1.15%
no college degree
1.61%
1.96%
college degree
0.54%
-0.12%
Consumption equivalent variation
all households
1.16%
no college degree
1.38%
college degree
-0.06%
28
2.31%
2.57%
0.88%
is not surprising that the median benefit available to age-66 claimers – the median PIA
– changes little.16 The small differences that do appear are due to changes in claiming
patterns and changes in the maximum benefit, which is tied to the cap: to maintain the
expenditure target, these necessitate small changes in PIA levels.
The final panel of Figure 6 shows welfare effects, which we measure as the ex-ante consumption equivalent variation (CEV), the proportional increase in lifetime consumption
needed to make a newborn individual in the benchmark economy as well off as a newborn
in the counterfactual economy.17 Because we construct Figure 6 using the optimal values
bk , the CEV, which is relative to the current Social Security policies, is
of ss(E)
b
and A
generally positive. Although expanding the cap from its benchmark level reduces aggregate earnings, most households benefit from the lower payroll tax rate. Low values of the
taxable limit thus lead to significant welfare losses. This dynamic reverses once the cap
reaches its optimal value, although the losses from setting the cap above its optimal value
are small.
PIA Formula (ss(E)).
b
Returning to Table 3, column (1) shows that the optimal PIA
function is flat in lifetime earnings. This is a radical departure from current U.S. rules,
where individuals at the 90th percentile of the PIA distribution receive 70% more than
those at the 10th percentile, but not unprecedented internationally. As Börsch-Supan
et al. (2016, Table 1) document, although most countries include an earnings-related
component in their public pensions, a non-trivial number provide only a basic benefit.
As with any form of worker compensation, changing the PIA formula generates both
income and substitution effects. Figure 7 provides more detail. Although the PIA depends
on three parameters (subject to an expenditure constraint), to simplify the discussion,
Figure 7 characterizes policies with a single variable, the 90-10 benefit ratio. Higher values
16
Because we account for earnings test credits in our simulations by modifying AIME (Ej ), we calculate
this hypothetical benefit using age-62 AIME.
1/[γ(1−σ)]
lifetime utility
−utility from bequestsbenchmark
17
We calculate this as lifetime utility reform −utility from bequests
− 1, with all utility
benchmark
benchmark
measured ex-ante.
29
0.78
Bequest transfers
Median PIA
Employee tax rate
Income tax rate
1.3
1.2
1.1
0.77
0.76
0.75
0.74
1
0.9
0
OPT
Benchmark
1.4
0.79
Employment rate
1.5
Ratio to benchmark
0.8
OPT
Benchmark
1.6
All
NC
C
0.73
1
2
3
4
0.72
0
5
1
(a) Equilibrium variables
1
0.995
5
0.5
0.99
0
0.985
-0.5
0.98
-1
0.975
0
4
OPT
Benchmark
1.5
All
NC
C
CEV %
Earnings/Benchmark
1
3
(b) Employment rate
OPT
Benchmark
1.005
2
Cap/median age-62 AIME
Cap/median age-62 AIME
1
2
3
4
-1.5
0
5
Cap/median age-62 AIME
All
NC
C
1
2
3
4
5
Cap/median age-62 AIME
(c) Earnings
(d) CEV %
Figure 6: Effects of Changing the Limit on Taxable Earnings
of this dispersion ratio correspond to a stronger positive relationship between earnings and
benefits. Figure 7(c) shows that tying the PIA more tightly to AIME increases aggregate
earnings. Holding total benefits fixed, a steeper earnings gradient causes benefits for
non-college workers to fall. This income effect, which encourages work, is reinforced by
the substitution effect from tying benefits to earnings. Employment and earnings both
30
1.01
1
0.99
0.98
0.96
0.95
1
1.5
0.77
All
NC
C
0.76
0.75
Bequest transfers
Median PIA
Employee tax rate
Income tax rate
0.97
OPT
0.78
Employment rate
1.02
0.79
Benchmark
1.03
Ratio to benchmark
0.8
Benchmark
1.04
OPT
1.05
0.74
2
2.5
0.73
1
3
1.5
90-10 benefit ratio
OPT
1
1.2
Benchmark
1.4
Benchmark
1
3
All
NC
C
0.8
0.995
CEV %
Earnings/Benchmark
1.005
2.5
(b) Employment rate
OPT
(a) Equilibrium variables
1.01
2
90-10 benefit ratio
0.99
0.6
0.4
0.985
0.2
All
NC
C
0.98
0.975
1
1.5
2
2.5
0
-0.2
1
3
1.5
2
90-10 benefit ratio
90-10 benefit ratio
(c) Earnings
(d) CEV %
2.5
3
Figure 7: Effects of Changing the PIA Formula
rise monotonically in the 90-10 ratio. For college-educated workers, the income effect of
linking benefits to earnings is negative, at least for high earners, and both employment
and income fall (Figure 7(b)). The increase in non-college earnings is steeper, however,
and these workers constitute the majority of the labor force. Aggregate earnings rise.
Figure 7(a) shows that the relationship between benefit dispersion and bequests is
U-shaped. Tying benefits more closely to earnings reduces the need of richer households
31
to save, reducing bequests, which by construction are luxury goods. On the other hand,
higher values of the 90-10 ratio also result in higher benefits for the rich and thus higher
wealth. Around the benchmark value of the 90-10 ratio, the second effect starts to dominate the first, and bequests begin to rise.
Figure 7(d) shows that, labor supply notwithstanding, the welfare effects of changing
the PIA formula are driven by redistribution. As the 90-10 ratio increases, benefits are
moved from less-educated to more-educated workers, lowering the welfare of the first group
and raising the welfare of the second. Within education groups, there is also redistribution
between individuals with different wage and health histories. Overall, moving from the
current PIA function to a flat one raises the CEV by 0.7%.
Figure 7(d) also shows that welfare is increasing even as the benefit formula becomes
completely flat. Allowing the benefits to decrease in lifetime earnings would increase
welfare even further. This is because the tax and transfer system as a whole is not
sufficiently redistributive. Given that we examine only part of this system, we view a flat
Social Security benefit formula as a natural lower bound for our exercise.
1.4
Benchmark
OPT
AF
AF, NC
AF, C
1.3
Adjustment
1.2
1.1
1
0.9
0.8
0.7
62
63
64
65
66
67
68
69
Claiming age
Figure 8: Claiming age adjustments
32
70
bk ). Figure 8 shows both the current (dashed black line)
Claiming Adjustment (A
and optimal (solid blue line) benefit adjustments for early or late claiming relative to
benefits at the NRA. The figure also shows the actuarially fair adjustments based on the
model’s mortality rates and a post-tax interest rate of 3% per year. Comparing slopes
reveals that under the current rules, the Social Security adjustments are more or less
actuarially fair prior to the NRA but less than fair post-NRA.
In Figure 9(a), median age-66 benefits are plotted against the size of the claiming
adjustment, measured as the ratio of age-70 to age-62 benefits. Two mechanisms are in
play. First, larger claiming adjustments will induce delayed claiming. This can raise or
lower aggregate expenditures, depending on the sizes of the adjustments. Second, holding
claiming dates fixed, larger adjustments will reduce expenditures on early claimers and
raise expenditures on those who claim late. Figure 9(a) shows that when the claiming
adjustments are small, and most individuals claim early, increasing the claiming adjustment generates cost savings and thus higher benchmark PIA levels. In this region of the
policy space, larger claiming adjustments raise annual benefits for late claimers not only
directly but also indirectly through higher values of the PIA. As adjustments continue to
grow, the expenditure effects reverse, and base benefits fall.
Figure 9 also shows that as claiming adjustments increase, and more workers delay
claiming, employment and earnings both rise. Figure 9(a) shows that payroll taxes fall
until the 70-62 benefit ratio reaches 1.6. After this point, delayed claiming reduces Social
Security-related income taxes to such an extent that payroll taxes have to be increased.
Higher earnings initially lead bequests to rise, but as the adjustments grow and workers
claim at increasingly later dates, they increasingly rely on their Social Security benefits.
The welfare effects of the claiming adjustments are driven both by their direct effects on
benefits and on their induced effects on labor supply. Larger claiming adjustments initially
lead to higher statutory benefits and significantly higher earnings, increasing welfare. As
adjustments continue to rise, their effect on the PIA reverses, earnings growth diminishes,
33
Employment rate
0.95
0.9
0.8
1
Bequest transfers
Median PIA
Employee tax rate
Income tax rate
1.2
1.4
1.6
1.8
2
0.77
All
NC
C
0.76
0.75
0.74
0.73
0.72
1
2.2
1.2
1.4
(a) Equilibrium variables
OPT
OPT
1.5
1
0.99
2
2.2
2
2.2
CEV %
0.5
0.98
0.97
All
NC
C
0
-0.5
All
NC
C
0.96
0.95
1
1.8
(b) Employment rate
Benchmark
Earnings/Benchmark
1
1.6
70 to 62 benefit ratio
70 to 62 benefit ratio
1.01
Benchmark
0.78
1
0.85
OPT
0.79
Benchmark
1.05
Ratio to benchmark
0.8
Benchmark
OPT
1.1
1.2
1.4
1.6
1.8
2
-1
-1.5
1
2.2
70 to 62 benefit ratio
1.2
1.4
1.6
1.8
70 to 62 benefit ratio
(c) Earnings
(d) CEV %
Figure 9: Effects of Changing the Claiming Age Adjustments
and benefits shift from the short-lived (low earners) to the long-lived. The CEV begins to
fall. Moving from the optimal claiming adjustment to the one currently in place reduces
CEV by about 0.5%.
Discussion. Relative to the policies currently in place, the optimal Social Security
policies reduce work incentives in order to redistribute resources from high to low earners.
Benefits are disconnected from earnings; and the rewards to delayed claiming (and longer
34
careers) are reduced; only the payroll tax is relatively unchanged. Comparing the first
two columns of Table 3 shows that under the optimal policies, the number of people
claiming benefits at age 62 rises by 25 pp. The aggregate employment rate declines by
1.7%, and average earnings decline by 1.1%. Receiving smaller benefits, high-earnings
workers save and bequeath more, raising aggregate bequests. Consumption declines by
0.8%. Nonetheless, there is a CEV gain of 1.16%. The gain goes entirely to those without
college degrees; the college-educated experience a loss of about -0.06%.
To quantify the extent to which Social Security redistributes resources, we calculate
internal rates of return on Social Security benefits and taxes for the college- and noncollege-educated. Column (0) of Table 3 shows that the existing system is already fairly
redistributive, with annual returns of 1.61% and 0.54%, a difference of 1.1 pp, for those
without and with college degrees, respectively. Our estimates contrast with Goda et al.
(2011), who calculate returns using Social Security earnings histories. They find rates of
return to be moderately progressive (i.e., decreasing in lifetime earnings) for women and
slightly regressive for men. (Our mortality rates apply to both sexes.) A major reason
for the difference is that our model accounts for benefit taxation and DI. Removing these
two elements causes our estimated rate of return for the non-college-educated to fall to
0.87% and our estimated rate for the college-educated to rise to 0.62%. Our results differ
more from Sánchez-Romero et al.’s (2019) estimates for men born in 1960, which are fairly
regressive, with rates of return for the 2nd and 4th income quintiles of 1.1 pp and 2.4 pp,
respectively (Table 4, case DB-II). It seems unlikely that the differences are due solely
to the modelling of Social Security benefits, as the ratio between the top and bottom
quintiles of lifetime Social Security benefits reported by Sánchez-Romero et al. (2019,
Figure 2), 1.8, is similar to our 90-10 ratio for the PIA, 1.79. DI and benefit taxation
again account for some of the difference.
In any event, the optimal Social Security rules described in column (1) of Table 3 are far
more redistributive than the current rules. The rate of return for those without a college
35
degree is 2.1 pp higher than the rate for those with a degree, which is in fact negative. It
is notable that under our optimal policies, the less-educated earn a higher rate of return
despite our utilitarian welfare criterion. A utilitarian social planner, seeking to equate
marginal utilities among the living, will naturally place more weight on the long-lived.
A handful of papers have examined the effects of changing the progressivity of the
Social Security benefit formula, employing a range of modeling assumptions and reaching a variety of conclusions. Using a model with no earnings-mortality links whatsoever,
Nishiyama and Smetters (2008) conclude that benefits should be increasing more rapidly
in earnings than under current rules, arguing that the distortions associated with a progressive benefit formula outweigh the gains from risk-sharing. Huggett and Parra (2010),
in contrast, find that benefits should be decreasing in lifetime earnings. While they also
treat earnings and mortality as independent, their analysis differs from Nishiyama and
Smetters (2008) in terms of equilibrium concept and household composition, in both cases
being more similar to ours. On the other hand, Ndiaye (2018), whose model differs from
Huggett and Parra’s (2010) primarily in having a retirement decision, reaches the same
general conclusion as Nishiyama and Smetters (2008). Ndiaye (2018) also finds that the
claiming adjustments should be amplified; we discuss this finding in the robustness section
below.
Bagchi (2019) develops a model in which mortality depends on earnings. He also finds
that the optimal benefit formula is one that is less redistributive, i.e., has benefits more
closely tied to earnings, than the current rules. One potential reason why Bagchi’s (2019)
finding differs from ours is the assumption of endogenous mortality. In his framework,
labor supply decisions affect length of life, while in our framework, exogenous differences
in health and education, which affect mortality, also affect earnings. As Hall and Jones
(2007) show, spending on longevity affects lifetime utility in ways fundamentally different
from that of spending on consumption. Bagchi and Jung (2019) find that when mortality
is exogenous but tied to health, the optimal benefit is independent of earnings; in an
36
earlier version of Bagchi (2019), where mortality depends on wealth, Bagchi (2016) finds
a flat benefit to be optimal. Bagchi’s papers differ from our model in other ways: they
all impose closed-economy equilibrium, while we hold wages and interest rates fixed; we
consider claiming decisions, the earnings test, and benefit taxation, while they do not.
4.2
Benchmark Environment, no Earnings Test or Benefit Taxation
Because the amount of Social Security benefits subject to income taxation increases
in total income, the income taxation of these benefits can significantly raise the marginal
tax rate on earnings, discouraging individuals from simultaneously working and receiving
benefits. This dynamic, along with the earnings test, implies that claiming decisions are
in part retirement decisions. Jones and Li (2018) showed that, under the existing benefit
formula, eliminating both the earnings test and benefit taxation increases labor supply
and welfare. We now introduce these reforms, increasing the payroll tax to maintain
budget balance, and solve again for optimal policies. Column (2) of Table 3 shows the
results. Column (2) shows that the optimal taxation limit, benefit formula and claiming adjustments are more or less independent of the earnings test or benefit taxation.
Eliminating the latter two provisions, however, generates significant increases in employment, earnings, consumption and welfare. Over half of the employment loss generated
by the reforms in column (1) is offset by the reforms in column (2), as is 70% of the
earnings loss. Consumption actually rises relative to its baseline value. The combined
reforms considered in column (2) allow Social Security to be more redistributive than at
present, while imposing fewer distortions than in column (1). Notably, college- as well as
non-college-educated workers benefit from the joint reforms.
Removing the earnings test and the income taxation of benefits uncouples the claiming
and labor supply decisions. As we move from column (0) to column (1), the increase in
37
age-62 claiming, from 6.0% to 30.7%, is accompanied by a fall in age-62 employment,
from 69.6% to 64.1%. In contrast, moving from column (1) to column (2) shows that the
large increase in age-62 claiming, from 30.7% to 81.5%, is accompanied by an increase in
age-62 employment, from 64.1% to 66.6%. Most of the people not claiming OAI benefits
at age 62 are those on DI.18 Claiming becomes an actuarial decision, and almost everyone
finds it worthwhile to claim at the ERA of 62.19
Because we scale benefits to keep total expenditures constant, the optimal adjustment
is not unique. When everyone claims at age 62, one could simultaneously reduce the early
claiming penalty, which would raise benefits, and reduce the benefits available at the
normal retirement age, which would lower them. Benefits at age 62 would be unchanged
and everyone would continue to claim at that age. The results shown in column (2) are
found with the curvature parameter A set to 3.5.20
5
Heterogeneous Mortality and Fiscal Sustainability
Recent history suggests that the longevity gap between rich and poor will continue
to grow (National Academies of Sciences, Engineering, and Medicine 2015; Chetty et al.
2016), even as ongoing trends in aggregate longevity and fertility leave Social Security increasingly unsustainable. As we evaluate proposals to restore fiscal balance, distributional
concerns may play a role.
To consider these interactions, we introduce what we call “2050 demographics.” First,
we reduce the annual population growth rate n to 0.43%, to match the growth rate between
18
In calculating claiming rates, we treat individuals who claim OAI at age 62 but transition to the DI
state at age 64 as not receiving OAI at age 62. Because our measure of disability is positive only when a
person receives DI benefits, we find it plausible to assume these people applied simultaneously for OAI
and DI at age 62 and received approval for DI by age 64.
19
About 2.5% of population delays claiming without receiving DI. This group consists of individuals
with high medical debt, who would be at the consumption floor even if they received Social Security
benefits. By delaying, they raise their future benefits without affecting their current income. As long as
delaying raises their benefits, they will continue to do so.
20
This restriction also applies to the results for the 2050 demographics shown in column (5).
38
2050 and 2055 projected by the World Bank (median variant). Second, following Conesa
et al. (2019), we linearly extrapolate the fraction of the population with a four-year degree
to 2050, using Census data for the period 1970-2018 (U.S. Census Bureau 2019b). This
suggests that in 2050, roughly half (49.6%) of the population will have a four-year degree.
The next step is to use the SSA projections (Bell and Miller 2005) to estimate the
increase in life expectancy between 2010 and 2050. We assume that this increase accrues
only to those with college degrees. Using our baseline and 2050 estimates of educational attainment, we find that in order to match the SSA lifespan predictions, the life
expectancy of college graduates must increase from 80.75 to 85.79 years.
It is almost surely the case that these changes to mortality and education will be
accompanied by changes in health and wages. If improved mortality implies improved
health, it is likely that both work capacity (e.g., Coile et al. 2016) and wages at older ages
will rise. On the other hand, as more people earn college degrees, the composition of the
two education groups will change (Bound et al. 2015). For this exercise, we assume that
improved mortality “stretches” the wage profile for college graduates, replacing ω(C, j, hj )
, hj . We stretch their health transition probabilities
with max ω(C, j, hj ), ω C, j 80.75
85.79
in a similar way. Finally, we decrease proportionally the mortality rates for the collegeeducated to achieve our life expectancy target of 85.79 years. The wage, health and
mortality profiles for those without a college degree remain at their baseline values.
We begin by imposing the current Social Security rules, using the claiming adjustments
for someone born in 1960. To achieve budget balance, we proportionally decrease the
current benefit formula until we match aggregate benefits for our benchmark economy.
Li (2018) and Laun et al. (2019) show that such an adjustment, as opposed to shifting
the benefit schedule rightward in the claiming age, is more efficient.21 In any event, a
proportional adjustment is in keeping with our practice of fixing Social Security’s total
21
In particular, these papers find that raising the early retirement age leads to a significant increase in
DI use, offsetting any savings in retirement benefits.
39
expenditures. Column (3) of Table 6 shows the results for this “current-rules” scenario.
Introducing the 2050 demographics lowers the aggregate employment rate from 76.4%
(column (0) of Table 3) to 74.4%. This reduction is due entirely to an older population;
at every age, employment rates are higher in 2050 than at present. An older population
also implies a reduction in Social Security benefits: the median PIA falls from $42,100
at present to $28,900. PIA values fall further because of delayed claiming. In 2050, only
1.8% of the population applies for OAI at age 62. With longer lives and lower benefits,
individuals increase their saving, and assets rise by over 60%. The rise in consumption is
much smaller, around 20%.
Column (4) shows the policies that are optimal when the earnings test and benefit
taxation remain in effect. These are quite similar to the optimal policies for the current demographic environment shown in column (1). Once again, benefits are flat in
AIME. The claiming adjustments are also similar across demographic regimes. Increased
longevity affects the optimal adjustment in opposing ways. With longer lifespans, there is
a greater need to promote work at older ages. This argues for larger adjustments. On the
other hand, with longer lives, the same credits are more likely to induce claiming delays;
this can be seen by comparing claiming in column (1) with its 2012 equivalent. Smaller
credits may therefore be more cost-effective. In the case at hand, the net effect is that
the adjustments are smaller than those in Table 3 but only slightly so. The one exception
is the optimal payroll tax limit, which rises about 40% under the 2050 demographics.
Recalling from Figure 6(d) that the welfare effects of changing the payroll tax cap vary
little across a wide range of values, the reason for the higher optimal cap is unclear.
Column (5) repeats the exercise in column (2) of Table 3 – finding the optimal policies when the earnings test and income taxation of benefits are removed – for the 2050
environment. Once again, the change in demographics has little effect on the optimal
policies.
There are, of course, many ways to bring Social Security into balance. In Appendix D,
40
we repeat the exercises of columns (3)-(5) under the assumption that the Social Security
benefit formula remains at its absolute 2012 values and payroll taxes are raised to balance
the budget. Comparing the two sets of results shows that our findings are robust to how
the Social Security budget is balanced.
6
Alternative Specifications
6.1
Demographics Do Not Depend on Education
In our model, health and income interact in two general ways. First, individuals with
a college degree enjoy better health, lower mortality and higher wages. Second, within
either education group, individuals with worse health have lower wages and a smaller time
endowment, leading to fewer hours of work and shorter lifespans. The first mechanism
reflects an endowment realized upon birth, while the second reflects shocks realized over
the entire life cycle.22 To disentangle these mechanisms, we estimate a single set of health
and mortality processes for all education levels, re-calibrate the model, and search again
for optimal policies. The third and fourth columns of Table 2 show the revised mortality
gradient. Removing education-related differences in health reduces the mortality gradient
somewhat – the lifespan difference between the bottom and top quintiles falls from 11.2 to
8.6 years – but has almost no effect on the earnings quintiles. The first three columns of
Table 7 (in Appendix E) present the policies that would be optimal in this environment.
Comparing these policies to those in the first three columns of Table 3 shows that the
education-related demographic effects have very little effect on the optimal policies. Even
the welfare effects are similar to those shown in Table 3; college-educated individuals, who
still receive higher wages, still benefit less from the reforms.
22
The health shocks also include endowment shocks, namely shocks to initial health.
41
6.2
No Health- or Education-Related Differences
Next, we simplify the model further, by removing health and education entirely, and
re-calibrate it again. Individuals now share a common wage and mortality process and
differ only in the realizations of these processes. The final two columns of Table 2 show
the mortality gradient for this specification. Even without health shocks, the mortality
gradient is significant, with people in the top lifetime-earnings quintile living 8.2 years
longer than those in the bottom. This reflects the small but significant risk of dying at
a very young age. On the other hand, removing health- and education-related effects
compresses the lifetime-earnings distribution: the ratio of the top earnings quintile to the
bottom falls from 3.86 to 2.49. Nonetheless, columns (4)-(6) of Table 7, which present
the policies that would be optimal in this environment, show that these policies remain
similar to the optimal policies for the baseline environment.
6.3
Claiming and Retirement are Linked
In the baseline model, Social Security claiming and labor supply are distinct decisions.
Although the earnings test and benefit taxation may discourage Social Security recipients
from working, in both the baseline model and real life many do. A common simplification,
however, is to assume that Social Security recipients cannot work (e.g., Ndiaye 2018).
Because claiming is usually assumed to be irreversible (as we do), in this environment
retirement is irreversible as well.
The first three columns of Table 4 show results for the specification in which claiming
Social Security leads to permanent retirement; the model is recalibrated to match the
data. In such a framework, the principal way to increase labor supply at older ages is to
encourage delayed claiming. The optimal claiming adjustments shown in columns (2) and
(3) are thus much larger than those currently in place. Under current rules, the benefit
for someone claiming at age 70 is 77% larger than that of someone claiming at 62. In
42
column (2), the difference is 497%. This differs significantly from the baseline model, in
which the optimal claiming adjustments are smaller than the ones currently in place. On
the other hand, the optimal benefit formula is still flat, as in the baseline model, and the
optimal payroll tax cap is modestly smaller than the current one, which is also similar to
our findings for the baseline model. It bears noting that, relative to the baseline model,
eliminating the earnings test and benefit taxation has a smaller effect on labor supply and
welfare. With labor supply tied to Social Security, the earnings test is irrelevant; benefit
taxation continues to generate wealth effects and alter after-tax rates of return.
6.4
Fixed Lifespans
The final alternative we consider is one where, in addition to linking claiming and
retirement, we remove all health- and education- related effects and assume that everyone
shares a common, certain lifespan. This specification is very similar to the one considered
by Ndiaye (2018), who extends Huggett and Parra (2010) to allow for a flexible retirement
date. Huggett and Parra (2010) and Ndiaye (2018) need a tractable framework because
they derive optimal systems of taxes and transfers, finding the solution to social planning
problems. However, they also consider piecemeal reforms to the current Social Security
system that can be compared to our results.
Columns (4)-(6) of Table 4 show that the optimal policies in this environment involve
significant claiming adjustments. The optimal policies also involve a positive relationship between lifetime earnings and Social Security benefits. This contrasts with most of
our specifications, where the optimal benefit differs only by claiming date. By way of
comparison, Ndiaye (2018) finds that the optimal policies also involve a large claiming
adjustment and benefits that increase in lifetime earnings (to a greater extent than under the current rules). Our results suggest that these findings may be driven in part by
modeling assumptions.
43
Table 4: Current and Optimal Policies: Alternative Specifications
Claiming = Retirement
& Retirement Irreversible
Current Optimal 1 Optimal 2
Benefit Taxation and
Earnings Test
Columns (1)-(3)
& Fixed Lifespans
Current Optimal 1 Optimal 2
Yes
Yes
No
Yes
Yes
No
(1)
(2)
(3)
(4)
(5)
(6)
1.64
2.18
2.61
2.62
29.6
29.6
29.6
30.7
42.1
55.1
23.8
28.6
34.2
31.8
33.9
36.2
0.457
2.091
0.745
1.320
0.460
2.077
0.536
1.800
0.0%
17.5%
9.4%
51.9%
0.0%
0.0%
0.0%
0.4%
80.2%
42.4%
90.5%
27.8%
96.5%
53.8%
96.8%
58.2%
78.7%
0.335
90.7
130.8
6.8
76.7
6.94%
25.26%
83.6%
0.336
94.2
119.8
7.7
80.3
6.20%
25.80%
88.2%
0.331
96.6
119.3
7.7
82.3
5.77%
24.84%
88.6%
0.332
97.2
119.5
7.7
82.7
5.99%
24.63%
24.1
3.3
20.8
20.3
1.9
18.4
20.3
1.9
18.4
20.3
1.9
18.4
1.12%
1.58%
0.45%
-0.19%
-0.14%
-0.12%
0.79%
0.77%
Tax cap / median
2.18
1.65
age-62 AIME
Age-66 benefit (PIA) percentiles (000s)
10th percentile
30.7
26.3
median
42.1
26.3
90th percentile
55.1
26.4
Benefits relative to age-66 benefit
age 62
0.745
0.398
age 70
1.320
2.378
Benefit claiming
age 62
0.7%
0.0%
age 66
46.8%
17.3%
Employment
age 62
79.3%
80.0%
age 70
25.7%
34.7%
Aggregate behavior
employed
77.2%
78.1%
hours if employed
0.339
0.336
earnings ($000s)
90.1
90.1
assets ($000s)
123.3
131
bequests ($000s)
6.2
6.6
consumption ($000s)
75.5
76.2
τ ss
6.20%
6.65%
λ0
25.80%
25.45%
Residual government budget (000s)
revenues
24.1
24.1
expenses
3.3
3.3
balance
20.8
20.8
Internal (annual) rate of return on OAI and DI
all households
1.13%
1.13%
no college degree
1.54%
1.64%
college degree
0.57%
0.35%
Consumption equivalent variation
all households
0.92%
no college degree
1.05%
college degree
0.31%
44
1.32%
1.49%
0.54%
7
Conclusion
In this paper, we consider the implications for Social Security reform of the large
and growing income gradient in mortality. We construct a heterogeneous-agent, lifecycle model of Social Security claiming, labor supply and saving and calibrate it to the
economic and demographic environment for 2012. Under our calibration, agents in the
top lifetime-earnings quintile live on average 11 years longer than those at the bottom.
We then examine changes to: the formula converting an individual’s earnings history to
his baseline benefit; the payroll tax earnings cap and tax rate; and the trade-off between
the age at which an individual first receives her Social Security benefits and the size of the
annual benefit. Our analysis suggests that welfare is maximized when baseline benefits are
independent of lifetime earnings, the payroll tax cap is set at roughly its current value,
and claiming adjustments are reduced. Eliminating the earnings test and the income
taxation of Social Security benefits would yield additional welfare gains.
We then examine a hypothetical “2050 demographics” scenario characterized by longer
lifespans, lower population growth and a steeper education-mortality gradient. The Social
Security system that would maximize welfare in the 2050 demographic environment is
quite similar to the one that would maximize welfare today. While longer careers would
be beneficial in the 2050 environment, agents with longer lifespans would be willing to
work longer even if the Social Security rules did not change.
There are a number of promising avenues along which to extend our framework. The
first would be to introduce marriage and spousal benefits, as in Sánchez-Marcos and
Bethencourt (2018), Nishiyama (2019) and Borella et al. (2019). A second is to consider
the entire tax and transfer system, rather than just Social Security. It is not immediately
obvious how to make such an analysis tractable. Our results suggest that commonly
used demographic simplifications, such as a fixed lifespan, may have important policy
implications.
45
References
Aaron, Henry J., “Demographic Effects on the Equity of Social Security Benefits,” in
“The Economics of Public Services,” Palgrave Macmillan, 1977, pp. 151–173.
Aaronson, Daniel and Eric B. French, “The Effect of Part-Time Work on Wages:
Evidence from the Social Security Rules,” Journal of Labor Economics, 2004, 22 (2),
329–352.
Attanasio, Orazio, Sagiri Kitao, and Giovanni L. Violante, “Financing Medicare:
A General Equilibrium Analysis,” in “Demography and the Economy,” University of
Chicago Press, 2010, pp. 333–366.
Bagchi, Shantanu, “Differential Mortality and the Progressivity of Social Security,”
Working Paper 16-263, W.E. Upjohn Institute for Employment Research 2016.
, “Differential Mortality and the Progressivity of Social Security,” Journal of Public
Economics, 2019, 177, 104044.
and Juergen Jung, “Health Risk and the Optimality of Social Security,” 2019.
Manuscript.
Bell, Felicitie C. and Michael L. Miller, “Life Tables for the United States Social
Security Area 1900-2100,” 2005.
Borella, Margherita, Mariacristina De Nardi, and Fang Yang, “Are Marriagerelated Taxes and Social Security Benefits Holding Back Female Labor Supply?,” Technical Report, National Bureau of Economic Research 2019.
Börsch-Supan, Axel, Klaus Härtl, and Duarte Nuno Leite, “Social Security and
Public Insurance,” in John Piggott and Alan Woodland, eds., Handbook of the Economics of Population Aging, Vol. 1, Elsevier, 2016, pp. 781–863.
Bosworth, Barry and Kathleen Burke, “Differential Mortality and Retirement Benefits in the Health and Retirement Study,” CRR WP 2014-4, Center for Retirement
Research at Boston College 2014.
Bound, John, Arline T. Geronimus, Javier M. Rodriguez, and Timothy A.
Waidmann, “Measuring Recent Apparent Declines in Longevity: The Role of Increasing Educational Attainment,” Health Affairs, 2015, 34 (12), 2167–2173.
46
Chetty, Raj, Michael Stepner, Sarah Abraham, Shelby Lin, Benjamin Scuderi, Nicholas Turner, Augustin Bergeron, and David Cutler, “The Association
between Income and Life Expectancy in the United States,” JAMA, 2016, 316 (6),
1750–1766.
Cogan, John, “Fixed Costs and Labor Supply,” Econometrica, 1981, 49 (4), 945–963.
Coile, Courtney C., Kevin Milligan, and David A. Wise, “The Capacity to Work
at Older Ages: Introduction and Summary,” in David A. Wise, ed., Social Security
Programs and Retirement Around The World: The Capacity to Work at Older Ages,
Chicago: University of Chicago Press, 2016, pp. 1–33.
Conesa, Juan Carlos, Timothy J. Kehoe, Vegard M. Nygaard, and Gajendran
Raveendranathan, “Implications of Increasing College Attainment for Aging in General Equilibrium,” NBER Working Paper No. 26000, National Bureau of Economic
Research 2019.
Cooley, Thomas F., Frontiers of Business Cycle Research, Princeton University Press,
1995.
Cottle-Hunt, Erin and Frank Caliendo, “Social Security Reform: Three Rawlsian
Options,” 2019. Manuscript.
Cremer, Helmuth, Jean-Marie Lozachmeur, and Pierre Pestieau, “Collective
Annuities and Redistribution,” Journal of Public Economic Theory, 2010, 12 (1), 23–
41.
De Nardi, Mariacristina, “Wealth Inequality and Intergenerational Links,” The Review
of Economic Studies, 2004, 71 (3), 743–768.
, Eric B. French, and John Bailey Jones, “Why Do the Elderly Save? The Role
of Medical Expenses,” Journal of Political Economy, 2010, 118 (1), 39–75.
,
,
, and Jeremy McCauley, “Medical Spending of the US Elderly,” Fiscal
Studies, 2016, 37 (3-4), 717–747.
De Nardi, Mariacristina, Eric B. French, John Bailey Jones, and Rory McGee,
“Couples’ and Singles’ Savings after Retirement,” 2019. Work in Progress.
Feldstein, Martin and Jeffrey B. Liebman, “Social Security,” Handbook of Public
Economics, 2002, 4, 2245–2324.
47
French, Eric B., “The Effects of Health, Wealth, and Wages on Labour Supply and
Retirement Behaviour,” The Review of Economic Studies, 2005, 72 (2), 395–427.
and John Bailey Jones, “The Effects of Health Insurance and Self-Insurance on
Retirement Behavior,” Econometrica, 2011, 79 (3), 693–732.
and , “Public Pensions and Labor Supply over the Life Cycle,” International Tax
and Public Finance, 2012, 19 (2), 268–287.
Goda, Gopi Shah, John B. Shoven, and Sita Nataraj Slavov, “Differential Mortality by Income and Social Security Progressivity,” in “Explorations in the Economics
of Aging,” University of Chicago Press, 2011, pp. 189–204.
Gouveia, Miguel and Robert P Strauss, “Effective Federal Individual Income Tax
Functions: An Exploratory Empirical Analysis,” National Tax Journal, 1994, pp. 317–
339.
Hall, Robert E. and Charles I. Jones, “The Value of Life and the Rise in Health
Spending,” The Quarterly Journal of Economics, 2007, 122 (1), 39–72.
Hubbard, R. Glenn, Jonathan Skinner, and Stephen P. Zeldes, “Precautionary
Saving and Social Insurance,” Journal of Political Economy, 1995, 103 (2), 360–399.
Huggett, Mark and Gustavo Ventura, “On the Distributional Effects of Social Security Reform,” Review of Economic Dynamics, 1999, 2 (3), 498–531.
and Juan Carlos Parra, “How Well Does the US Social Insurance System Provide
Social Insurance?,” Journal of Political Economy, 2010, 118 (1), 76–112.
Imrohoroğlu, Selahattin and Sagiri Kitao, “Social Security Reforms: Benefit Claiming, Labor Force Participation, and Long-Run Sustainability,” American Economic
Journal: Macroeconomics, 2012, 4 (3), 96–127.
Jeske, Karsten and Sagiri Kitao, “U.S. Tax Policy and Health Insurance Demand:
Can a Regressive Policy Improve Welfare?,” Journal of Monetary Economics, 2009, 56
(2), 210–221.
Jones, John Bailey and Yue Li, “The Effects of Collecting Income Taxes on Social
Security Benefits,” Journal of Public Economics, 2018, 159, 128–145.
Kahn, James, “Social Security, Liquidity, and Early Retirement,” Journal of Public
Economics, 1988, 35, 97–117.
48
Kaplan, Greg and Giovanni L. Violante, “A model of the consumption response to
fiscal stimulus payments,” Econometrica, 2014, 82 (4), 1199–1239.
Karabarbounis, Marios, “A Road Map for Efficiently Taxing Heterogeneous Agents,”
American Economic Journal: Macroeconomics, 2016, 8 (2), 182–214.
Kitao, Sagiri, “A Life-Cycle Model of Unemployment and Disability Insurance,” Journal
of Monetary Economics, 2014, 68 (0), 1 – 18.
Kopecky, Karen A. and Tatyana Koreshkova, “The Impact of Medical and Nursing
Home Expenses on Savings,” American Economic Journal: Macroeconomics, 2014, 6
(3), 29–72.
Krugman, Paul, “Republicans Against Retirement,” The New York Times, August 17
2015, p. A19.
Laun, Tobias, Simen Markussen, Trond Christian Vigtel, and Johanna Wallenius, “Health, Longevity and Retirement Reform,” Journal of Economic Dynamics
and Control, 2019, 103, 123–157.
Li, Yue, “Paradoxical Effects of Increasing the Normal Retirement Age: A Prospective
Evaluation,” European Economic Review, 2018, 101, 512–527.
Low, Hamish W., “Self-insurance in a Life-cycle Model of Labour Supply and Savings,”
Review of Economic Dynamics, 2005, 8 (4), 945–975.
Mendoza, Enrique G., Assaf Razin, and Linda L Tesar, “Effective Tax Rates
in Macroeconomics: Cross-Country Estimates of Tax Rates on Factor Incomes and
Consumption,” Journal of Monetary Economics, 1994, 34 (3), 297–323.
Mirrlees, James A., “An Exploration in the Theory of Optimum Income Taxation,”
The Review of Economic Studies, 1971, 38 (2), 175–208.
National Academies of Sciences, Engineering, and Medicine, The Growing Gap
in Life Expectancy by Income: Implications for Federal Programs and Policy Responses,
Washington, DC: The National Academies Press, 2015.
Ndiaye, Abdoulaye, “Flexible Retirement and Optimal Taxation,” Working Paper No.
2018-18, Federal Reserve Bank of Chicago 2018.
Nishiyama, Shinichi, “The Joint Labor Supply Decision of Married Couples and the US
Social Security Pension System,” Review of Economic Dynamics, 2019, 31, 277–304.
49
and Kent A. Smetters, “The Optimal Design of Social Security benefits,” Research
Paper 2008-197, Michigan Retirement Research Center 2008.
Pashchenko, Svetlana and Ponpoje Porapakkarm, “Quantitative Analysis of Health
Insurance Reform: Separating Regulation from Redistribution,” Review of Economic
Dynamics, 2013, 16 (3), 383 – 404.
and , “Accounting for Social Security Claiming Behavior,” WP 2018-8, Center for
Retirement Research 2018.
Sánchez-Marcos, Virginia and Carlos Bethencourt, “The Effect of Public Pensions on Women’s Labor Market Participation over a Full Life Cycle,” Quantitative
Economics, 2018, 9 (2), 707–733.
Sánchez-Romero, Miguel, Ronald D. Lee, and Alexia Prskawetz, “Redistributive
Effects of Different Pension Systems When Longevity Varies by Socioeconomic Status,”
NBER Working Paper No. 25944, National Bureau of Economic Research 2019.
Sheshinski, Eytan and Frank N. Caliendo, “Social Security and the Increasing
Longevity Gap,” Working Paper 2018.001, The Center for Growth and Opportunity
2018.
Social Security Administration, “Annual Statistical Supplement to the Social Security
Bulletin, 2013,” SSA Publication No. 13-11700 2014.
The World Bank, “Global Development Indicators,” 2016. http://data.worldbank.
org/indicator/NE.CON.GOVT.ZS.
U.S. Census Bureau, “Annual Estimates of the Resident Population by Single Year
of Age and Sex: April 1, 2010 to July 1, 2017,” 2019. https://factfinder.census.
gov/faces/tableservices/jsf/pages/productview.xhtml?src=bkmk.
, “Table A-1. Years of School Completed by People 25 Years and Over, by Age and
Sex: Selected Years 1940 to 2018,” 2019. https://www.census.gov/data/tables/
time-series/demo/educational-attainment/cps-historical-time-series.
html.
50
For Online Publication: Appendices
Appendix A: Definition of equilibrium
Our approach will be to take wages and the interest rate from the data and to find
the private insurance premium and government policies that produce budget balance for
a stationary distribution of individuals.
Definition 1. A stationary equilibrium is a collection of government policies, a lump-sum
transfer, a private health insurance premium, individual decision rules, and a distribution
µ(x̃), x̃ = [x0j , h− , j , j]0 (h− is lagged health) of individuals, such that the following conditions hold.
1. Given government policies, the lump-sum transfer and the private health insurance
premium, the decision rules solve the individual problem described in subsection 2.6;
2. The government budget is balanced:
Z h
τ c c + T ra + W + SS(ss∗ + di, ra, W ) + 2τ mcr W + 2τ ss min{W, y ss }
i
+ Tjet (ss, W ) + pmcr I{j≥J M or h− =5} µ(x̃) dx̃
Z
= G+
ss + di + tr + κmcr mI{j≥J M or h− =5} µ(x̃) dx̃;
3. Health insurers earn zero profits:
p
priv
p
Z
supp
Z
I{j<J M
and h− 6=5}
µ(x̃) dx̃ =
Z
Z
I{j≥J M
or h− =5}
µ(x̃) dx̃ =
51
κpriv mI{j<J M
and h− 6=5}
κsupp mI{j≥J M
or h− =5}
µ(x̃) dx̃;
µ(x̃) dx̃;
4. Total lump-sum transfers equal the non-negative assets of deceased individuals, with
negative values of assets interpreted as uncompensated care:
Z
(1 + r)
(1 − s) max{0, a0 } µ(x̃) dx̃ = (1 + χ)B;
5. The distribution of individuals is stationary.
Appendix B: Survival rates, health transition probabilities and
medical expenditures
Health Measurement:
We allow health, hj , to take on 5 potential values: good (hj = 1), bad (hj = 2),
work limitation (hj = 3), in a nursing home (hj = 4), and disabled (hj = 5). Individuals
who are currently receiving DI benefits are classified as disabled (hj = 5); this state is
eliminated at age 66, when DI recipients are transferred to OAI. In the MEPS, we classify
a respondent as a DI beneficiary if he/she receives Social Security or SSI income. In the
HRS, we classify a respondent as a DI beneficiary if he/she ever receives Social Security
or SSI income for his/her disabilities. Individuals over 65 who are currently in a nursing
home with a stay of 60 days or longer are classified as being in a nursing home (hj = 4).
Among the remainder, individuals are classified as having work limitations (hj = 3) if
they report having health problems that limit their work capacity. In the MEPS, the
question on work limitations is “is anyone in the family limited in any way in the ability
to work at a job, do housework, or go to school because of an impairment or a physical
or mental health problem?” The corresponding question in the HRS is “do you have any
impairment or health problem that limits the kind or amount of paid work you can do?”
Individuals without work limitations are classified as having bad health (hj = 2) if their
self-reported health status is fair or poor, and as having good health (hj = 1) if their
self-reported health status is excellent, very good or good.
52
Survival Rates:
Following Attanasio et al. (2010), we estimate the survival function in three steps.
First, we estimate a logit model expressing the two-year survival rate as a function of age,
education, health and calendar year:
P
α0 + α1 j + α2 ei + m α3m Ihijt =m
P
+ m α4m Ihijt =m × j + α5 (t − 2012)
,
Pr(alivei,j+1,t+2 | aliveijt ) =
P
α + α j + α2 ei + m α3m Ihijt =m
1 + exp 0P 1
+ m α4m Ihijt =m × j + α5 (t − 2012)
exp
where ei is individual i’s education level and hijt is her current health status. Our estimation sample appends HRS data for older individuals to MEPS data for younger ones.
To improve precision, individuals with work limitations or on DI are grouped together,
as the two groups have similar survival rates. Given that our data cover 18 years, we
estimate period (for t = 2012) rather than cohort survival rates.
Second, we use the results of the logit estimation, ŝj (e, hj ), to calculate survival premia,
∆j (e, hj ), and approximate survival rates, s̃j (e, hj ):
∆j (e, hj ) = ŝj (e, hj ) − ŝj (0, 1),
s̃j (e, hj ) = s̃j (0, h) + ∆j (e, hj ).
Figure 10(a) shows the resulting survival premia.
The final step is to calibrate s̃j (0, 1) to match aggregate survival rates:
X
e,hj
ρ(e, hj )s̃j (e, hj ) =
X
ρ(e, hj ) s̃j (0, 1) + ∆j (e, hj ) = sj ,
e,hj
where sj is the gender-weighted survival rate for age group j in year 2010 (from Bell and
Miller 2005), and ρj (e, hj ) is the share of age-j individuals in our data with education e
53
and health hj . To reduce the effects of sampling error, we fit ρj (e, hj ) to a fourth-order
polynomial, separately for two segments of age groups, and use the predicted values in
the calculations.
0.1
1
0
0.8
0.7
-0.1
-0.2
h=1
h=2
h=3
h=4
h=5
Survival rate
Health Premium on survival rates
0.9
-0.3
0.6
0.5
0.4
h=1
h=2
h=3
h=4
h=5
0.3
0.2
-0.4
0.1
-0.5
20
40
60
80
100
0
20
120
Age
40
60
80
100
120
Age
(a) Premia
(b) Rates
Figure 10: Estimated Survival Premia and Rates by Age, Health and Education
Note: Solid series are for individuals without a bachelor’s degree, and dashed series are for individuals
with a bachelor’s degree. Individuals receiving DI (h = 5) have the same survival rates as those with
work limitations (h = 3).
Figure 10(b) shows our estimated survival rates. Survival rates decrease with age.
Relative to those in good health, those in bad health are more likely to die. Mortality
rates are even higher for those with work limitations (who are combined with the disabled)
and highest for those in a nursing home. Conditional on age and health, individuals with
a college degree have higher survival rates than those without. At age 24, people with a
college degree expect to live an additional 56.7 years; the average remaining lifespan for
those without a degree is 52.7 years.
Health Transition Probabilities:
We estimate health transition probabilities using a multinomial logit model. Model
predictors include age, education, health status, interactions of education and health with
54
age, and a linear year trend.
exp
Pr(hi,t+2,j+1 = k) =
ζ0k
k
+ ζ1k j + ζ2k ei + ζ3k ei × j + m ζ4m
Ihijt =m
P k
+ m ζ5m Ihijt =m × j + ζ6k (t − 2012)
X
exp
l
P
ζ0l
+
+
P
ζ1l j
+
ζ2l ei
+
l
m ζ5m Ihijt =m
ζ3l ei
×j+
×j+
ζ6l (t
l
m ζ4m Ihijt =m
,
(16)
P
− 2012)
where Pr(hi,t+2,j+1 = k) is the probability that individual i of age j in calendar year t
transits to health state k in the next model period. Consistent with the assumption that
DI beneficiaries do not transit out of that state before reaching the NRA, the estimating
sample is restricted to non-DI beneficiaries. We experimented with richer specifications,
which included non-linear terms in age and the interactions of these non-linear terms with
other controls. Probably because of the limited number of observations, the coefficients
for these additional terms were not statistically significant.
Given the differences in health categories (see section 3.2 in the main text), we estimate
separate transition models for the MEPS and the HRS. For the MEPS, we estimate
transitions at an annual frequency, to utilize more observations, and calculate two-year
transition probabilities using
Pr(hj+1 | e, hj ) =
X
Pr(hj+1 | e, hj+0.5 = m) × Pr(hj+0.5 = m | e, hj ).
(17)
m
The model switches from the MEPS categories to the HRS categories between ages 64
and 66. We estimate the probabilities for this transition as follows. First, we estimate
transition probabilities from hj ∈ {1, 2, 3, 5} to hj+1 ∈ {1, 2, 3, 4} for individuals aged 6268 in the HRS sample. Second, we adjust the coefficients {ζ0k }k , k ∈ {1, 2, 3, 5} so that the
model-generated distribution of health by education at age 66 matches the distribution
found in the HRS. For DI beneficiaries (h = 5) at age 64, the likelihood of transitioning to
the good health, bad health, work limitation or nursing home state is, respectively, 0.0627,
55
0.0618, 0.8674 and 0.0081 for those without a bachelor’s degree; and is, respectively,
0.0857, 0.0303, 0.8840 and 0.0001 for those with a bachelor’s degree.
1
1
0.9
0.9
0.8
0.8
0.6
0.5
0.4
0.7
h
h
h
h
h
=1
=2
=3
=4
=5
Probabilities
Probabilities
0.7
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
20
40
60
80
100
h
h
h
h
h
0.6
0
20
120
=1
=2
=3
=4
=5
40
60
Age
(a) Good health
120
1
0.8
0.7
=1
=2
=3
=4
=5
0.9
0.8
0.7
Probabilities
h
h
h
h
h
0.9
Probabilities
100
(b) Bad health
1
0.6
0.5
0.4
0.6
0.4
0.3
0.2
0.2
0.1
0.1
40
60
80
100
h
h
h
h
0.5
0.3
0
20
80
Age
0
65
120
Age
70
=1
=2
=3
=4
75
80
85
90
95
100
105
Age
(c) Work limitation
(d) Nusing home
Figure 11: Health Transition Probabilities by Age
Note: Solid series are for individuals without a bachelor’s degree, and dashed series are for individuals
with a bachelor’s degree.
Figure 11 shows two-year transition probabilities for h ∈ {1, 2, 3, 4}. (Recall that the
DI state is absorbing until age 66.) Each panel shows transition probabilities associated
with an initial health status (h rather than h0 ). At any current state, people with a college
56
degree are more likely to transition to good health. People are more likely to exit good
health as they age. Between ages 64 and 66, individuals transition out of DI and begin
to transition into nursing homes.
Figure 12 compares the model-generated distribution of health at each age to that in
the data and shows that the constructed transition probabilities work well.
1
1
0.9
0.9
0.8
0.8
0.7
0.7
h=1
h=2
h=3
h=4
h=5
0.5
0.4
0.6
Share
Share
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
20
40
60
80
100
120
0
20
Age
h=1
h=2
h=3
h=4
h=5
40
60
80
100
120
Age
(a) Without a bachelor’s degree
(b) With a bachelor’s degree
Figure 12: Distribution of Health Status by Age and by Education: Data vs. Model
Note: Data series are represented by points, and model series are represented by solid lines.
Medical Spending in the HRS:
While the MEPS contains measures of both total and out-of-pocket medical expenditures, the HRS contains quality data only for out-of-pocket spending.23 We approximate
total medical spending in the HRS by multiplying out-of-pocket medical spending by the
ratio of total to out-of-pocket spending found in the Medicare Current Beneficiary Survey
(MCBS).
An additional problem with the HRS data is that our model explicitly accounts for
Medicaid (through the consumption floor), but the HRS measure of out-of-pocket spend23
This amount includes the cost of end-of-life care. We thank Eric French for sharing the code for
processing the HRS exit files.
57
ing excludes Medicaid payments. To address this issue, we follow Kopecky and Koreshkova
(2014) and restrict the HRS medical spending sample to those who are unlikely to receive
Medicaid. In our case, we restrict the sample to those in the top three quintiles of the
permanent income (PI) distribution, the construction of which we describe in the next
paragraph. De Nardi et al. (2016, Table 8) show that households in the top three current
income quintiles of the MCBS have similar medical expenditures, including low levels of
Medicaid spending. Using the top three PI quintiles of the HRS, our measure of total
medical expenses at older ages equals the HRS out-of-pocket spending measure multiplied
by 4.5, the ratio of total to out-of-pocket spending for the top three income quintiles in
the MCBS (?)Table 8]de2016medical.
To estimate PI, we follow De Nardi et al. (2019). We first calculate the sum of household pension income and Social Security income. We then regress this sum on a quadratic
function of age, interacted with employment status and household structure (dummies for
single male, single female, and couples), using fixed effects. The percentile rank of each
person’s fixed effect is our measure of PI.
Estimated Medical Spending:
We allow the medical spending shock to take on three values. Following Kitao (2014),
we use: a small shock with a 60% probability, a medium shock with a 35% probability,
and a large shock with a 5% probability. To increase the cell size, individuals are pooled
into 10-year age bins (a 15-year bin for the oldest). Figure 13 shows our estimated
expenditures.24
24
Not shown in Figure 13 are the values for the nursing home state. Due to the small number of
observations, we estimate one set of values for all ages. These are, from small to large, $21,520, $319,330
and $896,650.
58
6000
7
h=1
h=2
h=3
h=5
5000
10 4
6
5
Dollars
Dollars
4000
3000
4
3
2000
2
1000
1
0
20
40
60
80
100
120
0
20
Age
40
60
80
100
120
Age
(a) Small
4.5
h=1
h=2
h=3
h=5
(b) Medium
10 5
4
3.5
Dollars
3
2.5
2
1.5
h=1
h=2
h=3
h=5
1
0.5
0
20
40
60
80
100
120
Age
(c) Large
Figure 13: Total Medical Expenditures by Age, Health and Shock Realization
Note: Expenditures measured at a two-year frequency. Data for health statuses 3 and 5 are combined
over ages 24-65, and data for health statuses 2 and 3 are combined over ages 66+.
59
Appendix C: Wages
Recall that wages are given by
nj ζ
wj = ω(e, j, hj ) · ηj · min 1,
.
n̄
(18)
We measure worker productivity, ω(e, j, hj )ηj , as wages divided by the part-time wage
penalty. Because each MEPS panel spans only two years, we use wage data from the
Panel Study of Income Dynamics (PSID) for ages 24-64. We use HRS data for older ages.
We begin by estimating a standard fixed-effects wage profile:
ln
wij
n
min 1,
o =
nij ζ
n̄
4
X
υ
θ0υ j +
υ=1
4
X
θ1υ ei j υ + γi + ςij ,
(19)
υ=1
where wij is the hourly wage rate for individual i of age j, nij is working hours, γi is
an individual fixed effect and ςij is a time-varying residual. We allow wages to differ by
education and age but not health, because wages differ little across health statuses after
they are adjusted for the part-time penalty and because the disaggregated estimates are
less precise. Figure 14(a) shows the fixed-effects estimates.
The fixed-effects profiles show wages continuing to rise even at very old ages. These
profiles are not an unbiased estimate of ω(e, j, hj ), however, because they are estimated
only on the wages of those who work. If individuals with unusually large wage decreases
are more likely to retire, the fixed-effects estimate of ω(e, j, hj ) will overstate wage growth
at older ages. We account for this bias with the correction developed in French (2005).
In brief, using our model to simulate wage and employment histories and estimate fixedeffects wage profiles for workers, we set ω(e, j, hj ) so that the simulated estimates of
{θ0υ , θ1υ }υ match those found in the data. This involves an interactive process where
the model is solved and simulated, and the estimate of ω(e, j, hj ) is updated to reflect
differences between the PSID and simulated fixedeffects profiles. Figure 14(b) shows the
60
2.4
2.4
No college
College
2
2
1.8
1.8
1.6
1.4
1.2
1.6
1.4
1.2
1
1
0.8
0.8
0.6
20
30
40
Good, NC
Bad, NC
Lim., NC
Good, C
Bad, C
Lim., C
2.2
Efficiency units
Efficiency units
2.2
50
60
70
80
90
0.6
20
30
Age
40
50
60
70
80
Age
(a) PSID data (fixed effects)
(b) Model
Figure 14: Age Efficiency Profiles
corrected profiles, which are allowed to vary by health. Wages fall steadily after age 55,
and the wages for those with work limitations are noticeably lower. The initial level of
the wage rate is set to be the average fixed effect for all individuals of age 24, and it is
$15.8 per hour for those without a bachelor’s degree and $19.4 for those with a bachelor’s
degree.
To estimate the process for the productivity shock ηj , we again follow French (2005)
and assume that the fixed-effects residual, γi + ςij , follows
γi + ςij = ηij + εij + κεij−1 ,
(20)
where εij is white noise measurement error. Assuming that ηij follows an AR(1) process,
we can find its parameters, and the parameters of the MA(1) measurement-error process,
by matching the autocovariances of γi + ςij , using standard error components calculations.
Table 5 reports the parameters for ηj . The initial distribution of productivity shocks by
education and by health is set to match the data distribution of the fixed effects.
61
90
Table 5: Persistence and Variance of the Productivity Shocks
Autocorrelation Coefficient
Innovation Variance
Non-college
College
0.933
0.021
0.921
0.032
Appendix D: 2050 demographics under different budget balancing approaches
In the “2050 demographics” scenario examined in the main text, we balanced the
Social Security budget by reducing benefits. In Table 6, we compare our findings for this
exercise (shown in columns (1)-(3)) to one in which the Social Security budget is balanced
through higher taxes. In particular, columns (4)-(6) of Table 6 repeat the exercises of
columns (1)-(3) under the assumptions that the Social Security benefit formula remains
at its absolute 2012 values and payroll taxes are raised to balance the budget. Comparing
the two sets of columns shows that the optimal policies are robust to how the Social
Security budget is balanced.
62
Table 6: Current and Optimal Policies: 2050 Demographics
2012 Aggregate SS Spending
Current Optimal 1 Optimal 2
Benefit Taxation and
Earnings Test
Current
2012 SS Rules
Optimal 1 Optimal 2
Yes
Yes
No
Yes
Yes
No
(1)
(2)
(3)
(4)
(5)
(6)
3.10
2.18
3.09
3.09
34.4
34.8
35.1
28.6
39.3
51.5
45.9
45.9
45.9
46.7
47.0
47.4
0.80
1.23
0.74
1.33
0.80
1.23
0.80
1.23
85.4%
100.0%
3.8%
42.2%
32.7%
99.8%
85.4%
100.0%
72.7%
40.7%
73.3%
33.2%
65.3%
31.4%
69.0%
35.7%
73.1%
0.33
96.5
208.8
11.2
87.3
5.5%
25.8%
72.4%
0.33
96.2
168.8
9.5
84.0
9.0%
25.8%
70.9%
0.33
94.5
173.5
9.6
82.6
8.7%
26.3%
71.9%
0.33
95.3
179.5
10.3
84.1
9.4%
25.8%
28.5
4.2
24.3
27.6
4.2
23.3
27.6
4.2
23.3
27.5
4.2
23.3
0.48%
1.69%
-0.25%
0.43%
0.70%
0.30%
0.43%
1.64%
-0.28%
0.43%
1.55%
-0.25%
1.52%
2.04%
0.29%
2.72%
3.33%
1.27%
Tax cap / median
2.18
3.09
age-62 AIME
Age-66 benefit (PIA) percentiles (000s)
10th percentile
21.0
33.9
median
28.9
33.9
90th percentile
37.8
33.9
Benefits relative to age-66 benefit
age 62
0.74
0.81
age 70
1.33
1.22
Benefit claiming
age 62
1.8%
26.8%
age 66
36.8%
99.8%
Employment
age 62
74.4%
69.4%
age 70
38.4%
37.8%
Aggregate behavior
employed
73.4%
72.3%
hours if employed
0.33
0.33
earnings ($000s)
97.1
95.9
assets ($000s)
200.5
205.2
bequests ($000s)
10.5
10.7
consumption ($000s)
87.1
86.3
τ ss
5.0%
4.8%
λ0
25.8%
26.1%
Residual government budget (000s)
revenues
28.5
28.5
expenses
4.2
4.2
balance
24.3
24.3
Internal (annual) rate of return on OAI and DI
all households
0.48%
0.48%
no college degree
0.94%
1.81%
college degree
0.25%
-0.31%
Consumption equivalent variation
all households
1.50%
no college degree
1.98%
college degree
0.37%
63
2.22%
2.76%
0.92%
Appendix E: Robustness to demographic assumptions
In Table 7, we compare economic outcomes under the current Social Security system
to those that occur when the payroll tax cap, PIA formula and claiming adjustments are
chosen optimally. We consider two model environments. The first environment, shown in
columns (1)-(3), differs from the baseline environment in that the processes for health and
mortality do not differ by education level. In the second environment, shown in columns
(4)-(6), health- and education-related differences are removed entirely. All agents face
the same mortality and wage risks and differ only in terms of their wage and mortality
realizations. It bears noting that there is no DI in columns (4)-(6), as no one is disabled.
Table 7 shows that the optimal policies in these revised environments differ little from
their baseline counterparts.
64
Table 7: Current and Optimal Policies: Alternative Demographic Assumptions
No Education Differences
Current Optimal 1 Optimal 2
Benefit Taxation and
Earnings Test
No Health or
Educational Differences
Current Optimal 1 Optimal 2
Yes
Yes
No
Yes
Yes
No
(1)
(2)
(3)
(4)
(5)
(6)
3.03
2.18
1.89
1.88
41.6
42.1
42.5
30.7
42.1
55.1
42.9
43.9
44.8
42.3
42.9
43.5
0.803
1.229
0.745
1.320
0.776
1.269
0.803
1.229
81.2%
100.0%
5.0%
67.0%
26.4%
100.0%
100.0%
100.0%
67.5%
26.0%
82.3%
25.1%
79.7%
22.0%
80.4%
27.0%
76.4%
0.332
86.5
130
7
73.1
6.44%
25.70%
80.8%
0.330
88
108.2
5.7
72.8
6.20%
25.80%
79.8%
0.330
87.3
112.9
5.9
72.4
6.34%
25.90%
80.5%
0.330
88.1
119.5
6.5
73.8
6.53%
25.29%
23.2
3.3
19.9
21.2
2.8
18.5
21.2
2.8
18.5
21.2
2.7
18.5
1.08%
1.78%
-0.20%
0.52%
0.51%
0.48%
0.65%
2.15%
Tax cap / median
2.18
2.81
age-62 AIME
Age-66 benefit (PIA) percentiles (000s)
10th percentile
30.7
42.0
median
42.1
42.8
90th percentile
55.1
43.6
Benefits relative to age-66 benefit
age 62
0.745
0.796
age 70
1.320
1.240
Benefit claiming
age 62
6.5%
17.8%
age 66
73.9%
100.0%
Employment
age 62
69.7%
68.3%
age 70
24.6%
21.8%
Aggregate behavior
employed
76.7%
75.9%
hourse if employed
0.333
0.332
earnings ($000s)
86.9
86
assets ($000s)
123
125.4
bequests ($000s)
6.5
6.5
consumption ($000s)
72.8
72.1
τ ss
6.20%
6.13%
λ0
25.80%
26.08%
Residual government budget (000s)
revenues
23.2
23.2
expenses
3.3
3.3
balance
19.9
19.9
Internal (annual) rate of return on OAI and DI
all households
1.08%
1.08%
no college degree
1.50%
1.80%
college degree
0.38%
-0.24%
Consumption equivalent variation
all households
0.86%
no college degree
1.05%
college degree
0.05%
65
2.03%
2.21%
1.21%
@FedFRASER