The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Why Do Couples and Singles
Save During Retirement?
WP 21-09
Mariacristina De Nardi
University of Minnesota, Federal
Reserve Bank of Minneapolis,
CEPR, and NBER
Eric French
University of Cambridge, CEPR,
and IFS
John Bailey Jones
Federal Reserve Bank of Richmond
Rory McGee
University of Western Ontario and
IFS
Why Do Couples and Singles Save During
Retirement?
Mariacristina De Nardi, Eric French, John Bailey Jones
and Rory McGee*
May 27, 2021
Abstract
While the savings of retired singles tend to fall with age, those of retired
couples tend to rise. We estimate a rich model of retired singles and couples
with bequest motives and uncertain longevity and medical expenses. Our
estimates imply that while medical expenses are an important driver of the
savings of middle-income singles, bequest motives matter for couples and highincome singles and generate transfers to nonspousal heirs whenever a household
member dies. The interaction of medical expenses and bequest motives is a
crucial determinant of savings for all retirees. Hence, to understand savings,
it is important to model household structure, medical expenses, and bequest
motives.
* De
Nardi: University of Minnesota, Federal Reserve Bank of Minneapolis, CEPR, and NBER.
French: University of Cambridge, CEPR, and IFS. Jones: Federal Reserve Bank of Richmond.
McGee: UWO and IFS. We thank Wilbert van der Klaauw, Monica Costa Dias, Tom Crossley,
Ami Ko, Melanie Lührmann, Tatyana Koreshkova, Victor Rı́os-Rull and many seminar participants for helpful comments, and Karolos Arapakis, Justin Kirschner, Jeremy McCauley, Ali Merali
and Rachel Rodgers for research assistance. We gratefully acknowledge support from NSF grant
SES-2044748, the Social Security Administration through the Michigan Retirement Research Center (MRRC grant UM17-15), the Economic and Social Research Council (Centre for Microeconomic
Analysis of Public Policy at the Institute for Fiscal Studies (RES-544-28-50001), and Grant Inequality and the insurance value of transfers across the life cycle (ES/P001831/1)). The views expressed
in this paper are those of the authors and not necessarily those of the National Bureau of Economic
Research, the Federal Reserve Bank of Minneapolis, the Federal Reserve Bank of Richmond, the
IFS, or the CEPR.
1
1
Introduction
The savings of couples and singles behave very differently over the retirement
period. While the savings of singles tend to fall with age, those of retired couples
tend to increase or stay constant until one of the spouses dies. Why is this the case?
Do couples face different medical spending and longevity risks? How important are
risks to the surviving spouse? Do couples have different attitudes toward bequests?
Answering these questions helps us understand the savings of all retirees. Understanding why households save, in turn, is essential to predict how households would
react to policy reforms and other changes.
Previous work has mainly focused on singles. These studies find that medical
expenses are an important reason why single retirees hold wealth, but that the desire
to leave bequests might also be important. Much less is known about the reasons
why retired couples save.
Couples differ from singles in several notable ways. Being in a couple allows its
members to pool their longevity and medical expense risks and potentially reduces
medical spending due to spousal caregiving, but it also exposes each member to
their spouse’s risks, including the income loss and high medical expenses that often
accompany a spouse’s death. A couple also cares about leaving resources to the
surviving spouse and potentially about leaving bequests to other heirs. Disentangling
these motivations is important because couples account for about 50% of people age
70 and older and tend to be richer.
We start our analysis by documenting couples’ and singles’ savings, medical expenditures, and longevity. We then construct a rich model of retiree saving that
incorporates heterogeneity in life expectancy and medical expenses and includes the
large jump in the medical expenses that is typically observed before a death. We
model means-tested social insurance and bequest motives, both upon death of the
last survivor and upon the death of one of the people in a couple, in which case net
worth is optimally split between the surviving spouse and other heirs. We estimate
our model using the Method of Simulated Moments (MSM) and Assets and Health
Dynamics of the Oldest Old (AHEAD) data. Finally, we use our estimated model to
evaluate the extent to which retirees’ savings are driven by medical expenses, bequest
motives, and their interaction.
Our paper provides multiple contributions. We establish new facts on the savings
of singles and couples during retirement: While the savings of retired singles stay
roughly constant or fall during retirement, those of couples increase as long as both
spouses are alive. Net worth drops sharply (by $160,000 on average) when the first
spouse dies. A large share of this drop is explained by high medical expenses before
2
death, but not the majority, which is accounted for by transfers to nonspousal heirs.
By the time the second spouse dies, much of the couples’ wealth has vanished.
We use our estimated model to quantify the effects of various saving motives
during retirement for couples and singles. Our results for singles reinforce earlier
findings: Most singles save mainly to self-insure against future medical expenses, but
at the top of the permanent income (PI) distribution, bequests are also important,
especially when interacting with medical expense risk.
Our results for couples reveal several new findings. First, bequest motives have
large effects on the savings of older couples in the top two PI terciles. That is, at age
84, couples in the highest PI tercile would hold 26% less wealth (that is, $325,000
instead of $437,000) if they did not have bequest motives. Couples in the middle PI
tercile would hold about 14% less ($172,000 instead of almost $200,000). In contrast,
couples at the bottom, just like singles at the bottom and median PI level, would
barely change their wealth. Thus, more couples than singles save to leave bequests,
and bequest motives affect their savings by a much larger extent. This is for at least
two reasons. First, couples are wealthier, and we estimate bequests to be a luxury
good. Second, couples not only receive direct utility from leaving bequests at the
time of one spouse’s death, but also indirect utility from the expectation that the
last surviving spouse will leave bequests at his or her death.
Second, in the presence of bequest motives, medical expenses have a smaller effect
on the savings of couples at all PI levels. For instance, at age 84, the median wealth of
couples in the top PI tercile would fall by less than 10% (from $437,000 to $417,000)
if they did not face any medical expenses. This suggests that medical expenses are
only one of the reasons why they continue to save at very advanced ages.
Third, while the savings of couples are driven more by bequests than by medical
expenses, the interaction of medical expenses (including for the surviving spouse)
and bequest motives has large effects. At age 84, the median wealth of couples in
the top PI tercile would fall by more than 65% (from $437,000 to $150,000) if they
did not face any medical expenses and had no bequest motives.
Because wealth saved for bequests also helps insure against medical spending, the
incremental effects of bequest motives or medical spending, even when summed, can
be much smaller than their joint effect. This is due to the way bequest and precautionary saving motives interact. When medical expenses are uncertain, households
that reduce their consumption to finance future medical expenses may die with unspent wealth. By giving these unspent funds value, bequest motives reduce the
opportunity cost of self-insuring through saving. Moreover, medical expenses introduce uncertainty over whether a household will actually leave a bequest. This might
generate precautionary saving aimed at increasing the likelihood that a bequest is
3
made. A third way in which these motives interact is that medical expenses have a
negative wealth effect, which tends to decrease both savings and bequests.
Fourth, couples save to provide resources to the surviving spouse after the death
of the first spouse. This is important because there are considerable differences in
the life expectancy of each spouse. At age 70, married women and men expect to live
15.8 years and 11.5 years, respectively. In addition, while the average life expectancy
of the couple is 13.7 years, the oldest survivor expects to live for 17.9 years. During
their remaining years alone, survivors face higher medical spending (including from
an elevated risk of entering a nursing home) and receive lower income. As a result,
couples wish to leave resources to the surviving spouse upon a partner’s death. This
has a large effect on savings at all PI levels. If couples instead placed no weight on the
survivor, at age 84 their median wealth in the middle PI tercile would fall by more
than 17% (from $199,000 to $165,000). Studies omitting this risk will understate
precautionary saving motives.
Fifth, when one member of a high-PI couple dies, even though most of the estate
is usually left to the surviving spouse, large bequests to nonspousal heirs are often
distributed. Changes to saving motives thus affect both the saving of couples and
singles and the division of wealth between the surviving spouse and nonspousal heirs.
For example, when bequest motives are removed, couples save less, but when one
spouse dies all of the estate is left to the survivor. In fact, we find that removing
bequest motives often increases the amount of wealth held by widows and widowers.
Taken together, our results show that bequest motives play a larger role for the
savings of couples than those of singles and for the savings of high-PI households
than those of low-PI households, and that the interaction of precautionary savings
and bequests are most important for couples and high-PI households. In contrast,
providing for the surviving spouse is important for all couples.
We reach similar conclusions when we look at all households, couples and singles, together. For median wealth, and at lower percentiles, the effects of removing
medical spending exceed the effects of eliminating bequest motives. At higher asset
percentiles, and for the mean, bequest motives have larger effects. The interaction of
bequest and precautionary saving motives is large across the entire asset distribution.
The rest of the paper is organized as follows. Section 2 summarizes the findings
of key related papers and motivates our main modeling choices. Section 3 briefly
introduces our data and displays the key facts that we seek to understand. Section 4
introduces our model, and section 5 details our estimation procedure. Section 6
displays important features of the data affecting households’ saving decisions and
evaluates our model fit. To build additional trust in the predictions of our estimated
model, Section 7 evaluates the performance of our model in terms of important
4
additional aspects of the data that we do match at the estimation stage. Section 8
evaluates the impact of medical expenses and bequest motives for singles and couples,
and Section 9 concludes.
2
Related literature and modeling choices
The goal of this paper is to document and explain the differences in the saving behavior of couples and singles and their contribution to aggregate retirement
savings. Although much of the structural work on retiree saving has focused on
singles,1 a few papers, including Braun, Kopecky and Koreshkova [9] and Nakajima
and Telyukova [43], include couples.
Our paper builds on these contributions by modeling what happens when a spouse
dies. As in Poterba, Venti, and Wise [49], we find that net worth falls when households lose a spouse and that this fall is too large to be attributable to medical and
other end-of-life expenses. A novel aspect of our framework is that we allow households to leave bequests to nonspousal heirs when only one of the spouses dies. This
feature proves important in matching the fall in wealth observed at the time of the
first spouse’s death.
Bequests to heirs also occur when the final member of a household dies. The desire
to leave such bequests has received considerable attention as a potential explanation
why households retain high levels of wealth at very old ages, as in Ameriks, Caplin,
Laufer, and Van Nieuwerburg [4], Ameriks, Briggs, Caplin, Shapiro, and Tonetti [3],
Dynan, Skinner, and Zeldes [23], and Lockwood [41]. In addition, De Nardi, French,
and Jones [19] show that this kind of bequest motive helps fit the asset and Medicaid
recipiency profiles of singles.
Besides modeling household demographic transitions and bequest motives toward
the surviving spouse and other heirs, we take into account health and medical expense
risk, social insurance for couples and singles, and heterogeneity in life expectancy by
PI and marital status. These features are important because households face potentially large out-of-pocket medical and nursing home expenses (Braun, Kopecky, and
Koreshkova [9], French and Jones [27, 28], Palumbo [47], Feenberg and Skinner [25],
and Marshall, McGarry, and Skinner [42]), which generate precautionary savings (as
in Kopecky and Koreshkova [38], and Laitner, Silverman, and Stolyarov [39]). But
these risks are partially insured by means-tested programs such as Medicaid and
Supplemental Security Income, which provide strong saving disincentives (Hubbard,
Skinner, and Zeldes [32], and De Nardi, French, and Jones [18]). Finally, previous
1
De Nardi, French, and Jones [20] provide a detailed literature review.
5
work has shown that high-income individuals live longer than low-income individuals (Attanasio and Emmerson [6], and Rı́os-Rull and Pijoan Mas) and that married
people live longer than single people.
3
3.1
Savings, medical spending, and Medicaid
The AHEAD dataset
We use the AHEAD dataset, which begins with households of non-institutionalized
individuals age 70 or older in 1993/94 and subsequently surveys their survivors every two years. Appendix A describes the details of our sample selection and data
work. Three important decisions are that (i) to abstract from labor supply decisions, we only consider retired households; (ii) to be consistent with the transitions
in the model we drop household who either get married or divorced during the sample period; and (iii) due to the well-known under-reporting of assets and medical
spending in the initial wave, we drop the 1993/94 data. Our 1996 data include 4,634
households, of whom 1,388 are couples and 3,246 are singles. This represents 24,274
household-year observations for which at least one household member was alive.
We distinguish three types of household: single man, single woman, and couple.
Single people include those who are divorced, never married, or widowed when first
observed, and those who become widowed over the sample period.
3.2
Permanent Income
At retirement time, permanent income (PI) captures substantial household exante heterogeneity: Households with different PI ranks receive different flows of
retirement income and face different processes for health, mortality and medical
expenses. A contribution of this paper is distinguishing heterogeneity from risks.
To estimate a household’s PI, we first sum all of its annuitized income sources
(Social Security benefits, defined benefit pension benefits, veterans benefits and annuities).2 Because there is a roughly monotonic relationship between lifetime earnings and our annuity income measure, this measure is also a good indicator of its
income during the working period. We then construct a PI measure comparable
across households of different ages and sizes. To do so, we regress annuity income
on a household fixed effect, dummies for household structure, a polynomial in age,
2
Since we model means-tested social insurance from SSI and Medicaid explicitly in our model
(through a consumption floor), we do not include SSI transfers.
6
and interactions between these variables. The rank order of each household’s estimated fixed effect provides our measure of its PI. This is a time-invariant measure
that follows the household even after one of its members dies. See Appendix B for
details.
3.3
Savings, medical expenses, and bequest
Our measure of wealth (or net worth, or assets) is the sum of all assets less
mortgages and other debts.3 Figure 1 displays median net worth, conditional on
birth cohort and PI tercile, for different configurations of couples and singles. We
break the data into 4 cohorts containing people who in 1996 were ages 72-77, 7883, 84-89, and 90-102. For clarity we display data for two cohorts in this picture
(Appendix I displays all of our data, as well as model fit). To construct these
profiles, we calculate medians for each cohort for those alive in each calendar year
(an unbalanced panel). The line for each cohort starts at the cohort’s average age in
1996.
Panel (a) displays net worth for current singles, that is all people who are single
at a given age and in a given cohort. It highlights that the savings of elderly singles
depend on their income (which is predetermined at retirement). Individuals with the
lowest lifetime incomes reach retirement with little wealth and run them down as
they age. After that point, they rely on annuitized income and government insurance
to fund their consumption and medical expenses. Elderly singles in the middle and
top of the income distribution also run down their wealth as they age, but very
slowly, and carry some wealth into very old ages.
Panel (b) shows the net worth profiles that would arise if we were to limit the
sample to only people who were initially single, without including those who become
single upon being widowed. Because the newly single tend to have more wealth, the
trajectories in Panel (a) slope down less than those in Panel (b); the two sets of
trajectories are nonetheless quite similar.
Panel (c) of Figure 1 plots the net worth of current couples, including couples as
long as they remain intact but dropping them when they become single. Panel (c)
reveals several interesting results. First, relative to singles in the same PI tercile,
couples in the lowest PI tercile reach retirement with more wealth. Second, the asset
profiles of the lower- and middle-income surviving couples display no decumulation.
Third, couples in the top PI tercile increase their net worth until almost age 90.
3
The AHEAD has information on the value of housing and real estate, autos, liquid assets
(which include money market accounts, savings accounts, Treasury bills, etc.), individual retirement
accounts, Keoghs, stocks, the value of a farm or business, mutual funds, bonds, and other assets.
7
Median Net Worth by Cohort and Income: Data, Initial Singles
270
240
240
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
Median Net Worth by Cohort and Income: Data, Singles
270
210
180
150
120
90
60
30
0
72
210
180
150
120
90
60
30
76
80
84
88
92
96
0
72
100
76
80
84
Age
96
(a)
(b)
Median Net Worth by Cohort and Income: Data, Initial Couples
100
400
550
350
500
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
92
Median Net Worth by Cohort and Income: Data, Couples
600
450
400
350
300
250
200
150
100
300
250
200
150
100
50
50
0
72
88
Age
76
80
84
88
92
96
0
72
100
Age
76
80
84
88
92
96
100
Age
(c)
(d)
Figure 1: Median wealth, AHEAD data. Panel (a): current singles. Panel (b):
initial singles. Panel (c): current couples. Panel (d): initial couples. Each line shows
medians for a cohort-PI cell over the period 1996-2014, plotted against average cohort
ages. Thicker lines denote higher PI groups.
8
Panel (d) of Figure 1 reports median wealth for the population of those who
are initially in a couple but might have subsequently lost a spouse, in which case
we report the wealth of the surviving spouse. Relative to intact couples, initial
couples in the higher PI terciles decumulate their wealth more quickly. Some of this
decumulation may reflect the higher decumulation rates of singles, but some is also
due to the loss of wealth that occurs when one of the spouses dies.
To formalize and quantify this observation, we study the difference in net worth
between two sets of couples: those in which one spouse dies and those that are similar
in multiple dimensions 6 years prior to death, except for the fact that they do not
experience a spousal death during a certain time period.4 This matching analysis
accounts for observable initial heterogeneity across couples and thus identifies the
impact of death and the declining health that precedes it.
We use these data to estimate the following event study specification:
ai,t = fi +
4
X
gj + dj Di × 1{t − Ti = j} + ei,t
(1)
j=−4
where ai,t denotes the wealth of household i in calendar year t, and Di indicates
whether i belongs to the group that does not lose a spouse (Di = 0) or to the group
that does (Di = 1). We normalize the date of death, Ti , to occur at year 0 and follow
the households for three waves before and two waves after the death. The coefficients
of interest are the parameters {dj }, which show the extent to which wealth rise or
fall for households who experience the death of a spouse, relative to households in
the matched sample who experience only a placebo death at date Ti .5
Panel (a) of Figure 2 reports our estimated death-related asset decline (the values
of dj ). On average, the wealth of households experiencing a death declines by an
additional $160,000 compared to those not experiencing it. While the decline begins
up to 6 years in advance of death, a substantial share of the decline occurs in the final
two years. Part of this decline reflects a gradual worsening of health and a related
increase in medical expenses. The rest reflects bequests made at the time of death
of the first spouse and changes in consumption. The gap between the treatment
and control groups continues to widen during the post-death years. This may reflect
additional death-related effects, for instance the loss of a spouse’s income, or other
differences between intact couples and singles.
4
We match on PI, initial wealth, and age. See Appendix C for details and Jones, De Nardi,
French, McGee and Rodgers [35] for additional outcomes.
5
The coefficients gj and dj are defined relative to their value 6 years before death (the omitted
indicator category). In interpreting the summation, recall that AHEAD interviews occur every
other year.
9
-5
-200
1000s of 2014 Dollars
-100
0
1000s of 2014 Dollars
0
5
10
15
100
Some of these asset declines are explained by high medical expenses prior to
death. The AHEAD contains high-quality data on what the household spends outof-pocket on private insurance premia, drug costs, hospital stays, nursing home care,
doctor visits, dental visits, and outpatient care, including those incurred during the
last year of life. To construct our measure of out-of-pocket medical spending, we sum
all of these components, along with funeral, other end-of-life expenses, and imputed
Medicare insurance contributions.
Panel (b) of Figure 2 shows that when a member of the household dies, there is
a sharp increase in medical expenses: they are $8,000 larger during each of the two
years preceding the death of a spouse, compared with similar couples experiencing
no death. Over the 6 years preceding death, the total difference in medical expenses
between these two households surpasses $27,000. Thus high end-of-life medical ex= 17% of the average fall in wealth. The effect for singles
penses can explain 27,000
160,00
is even larger, totalling almost $40,000.
-6
-4
-2
0
Years from Death
Data
2
4
-6
95% CI
-4
-2
0
Years from Death
Data
(a)
2
4
95% CI
(b)
Figure 2: Average changes around the death of a spouse, initial couples,
AHEAD/MCBS data. Panel (a): Mean asset decline around the death of a spouse.
Panel (b): Mean increase in the sum of out-of-pocket medical and death related expenses. Solid lines: estimates, shaded region: 95% confidence interval. Death dates
are centered at year 0.
If not medical and other end-of-life expenses, what explains the drop in wealth
at the time of death of a spouse? When one spouse dies, a family member of the
deceased, usually the spouse, is asked how the estate is split between the surviving
spouse and other heirs, such as children. Comparing the wealth of the newly single
with nonspousal transfers shows that most of the wealth is left to the surviving
10
spouse, but that, on average $79,000 are transferred to other heirs around the time
of death of the first spouse.
Transfers to other heirs at the time of death of a spouse are large and support
the view that bequest motives are important. Although we do not require our model
to match the decline in wealth around the death of a spouse or transfers to other
heirs, our estimated structural model matches these aspects of the data very well,
and thus helps validate its predictions.
3.4
Medicaid recipiency
Medicaid Recipiency by Cohort and Income: Data, All
0.6
0.55
0.5
Medicaid Recipiency
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
72
76
80
84
88
92
96
100
Age
Figure 3: Medicaid recipiency by age, income, and cohort.
Because previous work has shown that means-tested social insurance programs
affects saving, it is important to match the extent to which retirees use such programs.
Figure 3 plots Medicaid recipiency rates conditional on age, PI tercile, and cohort
for all households in our sample (the AHEAD has reliable measures of Medicaid
recipiency, but not spending). It shows that the lowest-income retirees are most
likely to end up on Medicaid and that, and consistently with the asset-tested nature
of these programs (high-asset households are not eligible) and the asset profiles shown
in Figure 1, Medicaid recipiency increases with age.
4
The model
A retired household maximizes utility by choosing savings, current and future
consumption, and, upon death of his or her spouse, the split of wealth between
11
himself or herself and other heirs. These choices also determine final bequests upon
the death of the last household member.
Households make these choices at household head age t, t = tr , tr + 1..., T + 1,
where tr is the initial period that we consider and T is the maximum potential
lifespan. For tractability, we assume that wives are always two years younger than
their husbands, so that one age is sufficient to characterize a household. Consistent
with the AHEAD data frequency, our time period is two years long.
The household begins the period in one of the following possible marital statuses:
a couple, a newly widowed man or woman, or a single man or single woman. Single
men, single women, and couples use the cash on hand they have at the beginning
of a period to consume and save. After that, mortality, health and medical expense
shocks occur, income net of taxes is received, and government transfers take place.
At this point, the amount of cash-on-hand that is carried into the next period is
known, and the household enters next period.
People who are newly widowed have an additional decision, which is how to
divide their estate between themselves and bequests to others. Once that decision is
made, they become single men or women, with wealth equal to the remainder of their
estates. When the final member of a household dies, all remaining net worth goes
to its heirs. Consistent with reality, our timing implies that medical costs associated
with death are collected before any bequests can be made and that Medicaid pays
bills that are incurred even when a patient dies.
4.1
Preferences
The per-period utility functions for singles and couples are given by
uS (c) =
c1−ν
,
1−ν
uC (c) = 2
(c/η)1−ν
,
1−ν
1 < η ≤ 2,
ν ≥ 0,
(2)
respectively, where c is total consumption and the parameter η determines the extent
to which couples enjoy economies of scale in the transformation of consumption goods
to consumption services. The household weighs future utility with the factor β.
The household derives utility θj (b) from leaving bequest b, where θ0 (b) is the
utility from bequests when there are no surviving members in the household, while
θ1 (b) is the utility from bequests when there is a surviving spouse. It takes the form
θj (b) = φj
(b + κj )1−ν
,
1−ν
(3)
where κj determines the curvature of the bequest function, and φj determines its intensity. De Nardi [16] used this functional form to help explain savings over the
12
life cycle. It supports several interpretations of the “bequest motive:” dynastic
or “warm glow” altruism (as in Becker and Tomes [7], Abel and Warshawsky [1],
or Andreoni [5]); strategic motives (as in Bernheim, Schleifer and Summers [8]
or Brown [10]); or some form of utility from wealth itself, as in (Carroll [12] and
Hurd [33]).
4.2
4.2.1
Sources of uncertainty and budget constraints
Income
Because there is much less income uncertainty during retirement than during the
working period, we simplify the model by assuming that the household’s non-asset
income at time t, yt (·), is a deterministic function of the household’s permanent
income, I, age, and family structure ft (single man, single woman, or couple).
yt (·) = y(I, t, ft ).
(4)
We do not include received bequests as a source of income, because very few households aged 70 or older receive them.
4.2.2
Health and survival uncertainty:
Health and survival are individual, rather than household-level, variables and we
use gender, g ∈ {h, w}, to differentiate between men and women. A person’s health
status, hsg , indicates whether he or she is in a nursing home, in bad health, or in
good health. The transition probabilities for a person’s future health status depend
on that person’s current health status, permanent income, age, gender and marital
status
πt (·) = Pr(hst+1 |hst , I, t, g, ft ).
(5)
Survival depends on the same variables. Let st (I, g, hst , ft ) denote the probability
that an individual alive at age t survives to age t + 1.
4.2.3
Medical expense uncertainty
We use mt+1 to denote the sum of medical spending that is either paid out-ofpocket by the household or covered by Medicaid between periods t and t + 1. While
we treat this total as exogenous, the division of expenses between the household and
Medicaid depends on the household’s financial resources, the total expenses that
must be covered, and the level of the consumption floor. In other words, mt+1 gives
13
the household’s maximum possible medical spending obligation, but because of social
insurance the amount faced by poorer households may be much smaller.
We allow mt+1 to depend on the health status of each family member at both
the beginning and end of the period, permanent income, age, household’s family
structure (also differentiating single men and single women) at the beginning and
end of each period, and an idiosyncratic component, ψt+1 .
h
w
ln mt+1 = m(hsht , hsw
t , hst+1 , hst+1 , I, t + 1, ft , ft+1 )
(6)
h
w
+ σ(hsht , hsw
t , hst+1 , hst+1 , I, t + 1, ft , ft+1 ) × ψt+1 .
We normalize the variance of ψt+1 to 1.
Allowing medical expenses to depend on the household’s composition and health
status at both the beginning of a period (which was realized at the very end of the
previous period) and the period’s end (which will be carried over into the subsequent
period) allows us to capture the jump in medical spending that occurs when a family
member dies – that is ft changes – and to incorporate the impact of two subsequent
health realizations on medical spending. The latter better helps us account for the
cost of prolonged periods of bad health or nursing home stays. Our timing implies
that the value of mt+1 is not known to the household when it decides how much to
consume between periods t and t + 1.
Following Feenberg and Skinner [25] and French and Jones [27], we assume that
ψt+1 can be decomposed as
ψt+1 = ζt+1 + ξt+1 ,
ξt+1 ∼ N (0, σξ2 ),
(7)
ζt+1 = ρm ζt + t+1 ,
N (0, σ2 ),
(8)
t+1 ∼
where ξt+1 and t+1 are serially and mutually independent. We discretize ξ and ζ,
using the methods described in Tauchen [52].
4.2.4
Budget constraints
Let at denote net worth at the beginning of period t and r denote their constant
pre-tax rate of return. Total post-tax income is given by Υ(r at + yt (·), τft ), with the
vector τft summarizing the tax code, which depends on family structure. Define the
resources available before government transfers as
x
et = at + Υ r at + yt (·), τft − mt ,
(9)
To capture Medicaid and SSI, we assume that government transfers bridge the gap
between a minimum consumption floor and the household’s financial resources,
n
o
trt (e
xt , ft ) = max 0, cmin (ft ) − x
et ,
(10)
14
where we allow the guaranteed consumption level cmin to vary with family structure.
To save on state variables we follow Deaton [17] and sum the household’s financial
resources after government transfers into cash-on-hand:
xt = at + Υ(r at + yt (·), τft ) − mt + trt (e
xt , ft ).
(11)
Households divide their cash-on-hand between consumption and savings:
at+1 = xt − ct ,
ct ∈ [cmin (ft ), xt ],
∀t.
(12)
(13)
Equation (13) ensures that consumption is at least as high as the consumption floor
and that savings are non-negative.
Next period’s resources before transfers and next period’s cash-on-hand can then
be expressed in terms of this period’s cash-on-hand and are, respectively,
x
et+1 = (xt − ct ) + Υ r (xt − ct ) + yt+1 (·), τft+1 − mt+1 ,
(14)
xt+1 = xt − ct + Υ r (xt − ct ) + yt+1 (·), τft+1 − mt+1 + trt+1 (e
xt+1 , ft+1 ).
(15)
4.3
Recursive formulation
Let ft = S indicate a single-person household. The value function for a single
person of age t and gender g is
g
Vt (xt , hst , I, ζt ) = max uS (ct ) + βst (I, g, hst , S)
ct
g
× Et Vt+1 (xt+1 , hst+1 , I, ζt+1 )
+ β[1 − st (I, g, hst , S)]Et θ0 (xt+1 ) ,
(16)
subject to Equations (4)-(8), and (13)-(15).
A newly-single person – one who was part of a couple in the previous period
and single now – distributes bequests towards other heirs before making savings and
consumption decisions as a single person:
n
o
ng
g
Vt (xt , hst , I, ζt ) = max θ1 (bt ) + Vt (xt − bt , hst , I, ζt ) ,
(17)
bt
subject to equation
bt ∈ [0, xt − cmin (ft )],
15
(18)
which prohibits the surviving spouse from using bequests to become eligible for
government transfers.
The value function for couples (ft = C) can be written as
w
h
C
Vt (xt ,hst , hst , I, ζt ) = max uC (ct )
ct
h
C
h
w
+ βst (I, w, hsw
,
C)s
(I,
h,
hs
,
C)E
V
(x
,
hs
,
hs
,
I,
ζ
)
t
t
t+1
t
t
t+1 t+1
t+1
t+1
w
h
nw w
w
+ βst (I, w, hst , C)[1 − st (I, h, hst , C)]Et Vt (xt+1 , hst+1 , I, ζt+1 )
h
nh
h
h
+ β[1 − st (I, w, hsw
,
C)]s
(I,
h,
hs
,
C)E
V
(x
,
hs
,
I,
ζ
)
t
t
t+1
t
t
t+1 t+1
t+1
h
+ β[1 − st (I, w, hsw
(19)
t , C)][1 − st (I, h, hst , C)]Et θ0 (xt+1 ) ,
subject to Equations (4)-(8) and (13)-(14). The dating of the continuation value for
new widows, Vtnw (·), reflects that wives are one model period (two years) younger
than their husbands. We solve our model numerically: see Appendix E for more
details.
5
Estimation
We adopt a two-step strategy to estimate the model. In the first step, we estimate
or calibrate those parameters that, given our assumptions, can be cleanly identified
outside our model. These include health transitions, out-of-pocket medical expenses,
and mortality rates from raw demographic data. In addition, we fix the discount
factor β at an annual value of 0.97, and we set the consumption floor for couples to
be a multiple of the consumption floor for singles according to statutory rules for
Medicaid and Supplemental Social Insurance.
In the second step, we estimate the rest of the model’s parameters, which include risk aversion, the consumption equivalence scale, bequest parameters, and the
consumption floor for singles,
∆ = ν, η, φ0 , φ1 , κ0 , κ1 , cmin (ft = S) ,
with the method of simulated moments (MSM), taking as given the parameters that
were estimated in the first step. In particular, we find the parameter values that
allow the simulated life-cycle decision profiles to “best match” (as measured by a
GMM criterion function) those from the data. Appendix E provides more details on
the mechanics of our MSM procedure.
16
Because our goal is to explain why retirees save so much and do so at rates that
differ by income, we match moments of the distribution of net worth by cohort,
age, and PI tercile. Because we wish to study differences in savings patterns of
couples and singles, we match asset profiles for the singles and couples separately.
Finally, because Medicaid is an important program insuring the medical expenses
and consumption of the poor we also match Medicaid recipiency. More specifically,
the moment conditions that comprise our estimator are given by
1. The 25th percentile, median, and 75th percentile of asset holdings by PI tercilecohort-year for current singles (including bequests).
2. The 25th percentile, median, and 75th percentile of asset holdings by PI tercilecohort-year for those who are currently couples with both members currently
alive.6
3. Medicaid recipiency by PI tercile-cohort-year for all households currently alive.
Appendix F contains a detailed description of these moment conditions, the weighting matrix in our GMM criterion function, and the asymptotic distribution of our
parameter estimates.
When there is a death in a couple, we drop the household from the current
couples’ moments and add the surviving spouse to the appropriate moment condition
for current singles. This is consistent with our assumption that singles differ only
in their state variables, regardless of marital history. Such an assumption also holds
well in the data, meaning that, conditional on our state variables the savings of
recent singles are very similar to those of the people we were single in the past. (See
Figure 1.) This is perhaps not surprising, given that the vast majority of single
people in our sample were married at some point in the past.
Our estimation targets do not include the falls in wealth that follow the first
spouse’s death because we use them to validate our estimated model by applying the
same estimation procedure to the AHEAD data and our model-simulated data.
When estimating our model, we face two well-known problems. First, in a crosssection, older households were born in earlier years than younger households and,
due to secular income growth, have lower lifetime incomes. Because of this, the
asset levels of households in older cohorts will likely be lower as well. As a result,
comparing older households born in earlier years to younger households born in
later years leads to understated asset growth. We address this problem by starting
6
We also include bequests for the small number of couples where both spouses die between
adjacent waves.
17
our simulations with initial conditions that come from the data (including wealth,
income, age, and so on). This allow us to capture much of the heterogeneity across
cohorts. Second, lower-income households and singles tend to die at younger ages
than higher-income households and couples. The average survivor in a cohort thus
has higher lifetime income, and thus more wealth, than the average deceased member
of the same cohort. This “mortality bias” is more severe at older ages, when a greater
share of the cohort members are dead. As a result, not accounting for mortality bias
leads to overstated asset growth. We address this problem by modeling mortality
as a function of key observables that include gender, permanent income and marital
status. Our simulated profiles thus incorporate the same mortality bias as the data.
6
Estimation results
In section 3 we report some of the key facts about savings and Medicaid recipiency
that we require our estimated model to match. In this section we report some of
the most relevant features of our first-step model estimates, and then discuss our
second-step estimates.
6.1
First-step estimation results
As they are key elements affecting saving behavior, the most important features of
our first-step estimates pertain to life expectancy and nursing home risk, income and
its drop when one of the spouses dies, and medical spending. Appendix G describes
our first-step estimation procedures in detail.
6.1.1
Life expectancy and nursing home risk
We estimate health transitions and mortality rates simultaneously by fitting the
transitions observed in the AHEAD to a multinomial logit model. We allow the
transition probabilities to depend on age, sex, marital status, current health status,
PI, and interactions of these variables.
Table 1 shows life expectancies at age 70 for singles and couples, respectively,
that we obtain when we use our estimated transition probabilities to simulate demographic histories, beginning at age 70, for different gender-PI-health-family structure
combinations. It shows that rich people, women, married people, and healthy people
live much longer than their poor, male, single, and sick counterparts. For instance, a
single man at the 10th permanent income percentile and in a nursing home expects
to live only 3.0 more years, while a single woman at the 90th percentile and in good
18
Income
Percentile
Men
Nursing
Bad
Home Health
Good Nursing
Health Home
Women
Bad
Health
Good
Health
All
Singles
10
50
90
3.0
3.0
2.9
6.9
7.8
8.1
8.7
10.3
10.9
4.1
4.1
3.8
11.3
12.3
12.5
13.2
14.9
15.4
10.2
11.5
12.0
Couples
10
50
90
2.7
2.8
2.7
7.8
9.4
10.4
9.8
12.2
13.5
4.0
4.0
3.9
12.1
13.7
14.6
14.1
16.3
17.3
11.3
13.4
14.5
Single Men
Married Men
Single Women
Married Women
9.0
11.5
13.9
15.8
Oldest Survivor∗
Probability that Oldest Survivor is Woman
Notes:
∗
17.9
63.7%
Oldest survivor among households that were a couple at age 70.
Table 1: Life expectancy in years, conditional on reaching age 70
health expects to live 15.4 more years. The far right column of the top two panels
shows average life expectancy conditional on permanent income, averaging over both
genders and health states. Singles at the 10th percentile of the permanent income
distribution live on average 10.2 years, while singles at the 90th percentile live on
average 12.0 years.
People in couples live about 2 years longer than singles: single women live on
average 13.9 years versus 15.8 for married women but, conditional on PI and health,
the differences in longevity are much smaller. Thus, married people live longer than
singles, but a significant part of the difference is explained by the fact that married
people tend to have higher PI and to be in better health. The bottom part of Table 1
shows the expected years of remaining life for the oldest survivor in a household when
both the man and the woman are 70. On average the last survivor lives 17.9 more
19
years. The woman is the oldest survivor 63.7% of the time.
Income
Percentile
Men
Bad
Good
Health Health
Women
Bad
Good
Health Health
All
Singles
10
50
90
23.6
22.8
20.3
25.3
24.8
22.8
35.8
35.5
32.2
37.9
38.2
35.8
32.8
32.5
30.1
Couples
10
50
90
17.3
16.6
14.6
19.2
18.8
16.8
34.4
34.1
31.4
37.0
37.3
34.5
28.7
28.7
26.3
Single Men
Married Men
Single Women
Married Women
26.4
19.5
37.2
36.3
Table 2: Probability of ever entering a nursing home, conditional on being alive at
age 70
Table 2 shows that single men and women face on average a 26% and 37% chance
of being in a nursing home for an extended stay (at least 60 days in a year), respectively, while married men and women face on average a 20% and 36% chance of
being in a nursing home for an extended stay. Married people are much less likely
to transition into a nursing home at any age, but married people, especially women,
often become single as their partner dies. Furthermore, married people tend to live
longer than singles, so they have more years of life to potentially enter a nursing
home. Compared to gender or marital status, permanent income and age 70 health
have a smaller effect on ever being in a nursing home. This is because those with
high permanent income, or in good health, are less likely to be in a nursing home at
each age but they tend to live longer.
20
6.1.2
Income
As noted above, we model non-asset income as a function of PI, age, and family
structure. Figure 4 presents predicted income profiles for those at the 20th and 80th
percentiles of the PI distribution. The average income of couples ranges from about
$14,000 at the 20th percentile to over $30,000 at the 80th . As a point of comparison,
median wealth holdings for these two groups at age 74 are $70,000 and $330,000,
respectively.
To evaluate how household income drops when one of the spouses dies, Figure 4
displays three scenarios for each PI level, all commencing from the income of a couple.
Under the first scenario, the household remains a couple until age 100. Under the
second one, the man dies at age 80. Under the third one, the woman dies at age 80.
Our estimates imply that couples in which the husband dies at age 80 suffer a 40%
decline in income, while couples in which the wife dies at 80 suffer a 30% decline in
income.
20000
10000
0
2005 dollars
30000
Income, by Permanent Income Percentile, Initially Couples
70
80
90
100
age
20th percentile couple
20th percentile to female
20th percentile to male
80th percentile couple
80th percentile to female
80th percentile to male
Figure 4: Income, conditional on permanent income and family structure. All households begin as couples, then either stay in a couple or switch to being to a single
man or single woman at age 80
The income losses that occur at the death of a spouse reflect the fact that even
though Social Security and private defined benefit pensions have survivors’ benefits,
these benefits replace only a fraction of the deceased spouse’s income. By way of
example, a wife can base her Social Security benefits either on her own Social Security
contributions, or on those of her husband (assuming the husband is retired). Should
the husband die, his benefits would cease. If the wife’s benefits were based on her
husband’s contributions, after his death her individual benefit would double, from
21
50% to 100% of the husband’s benefit. In spite of this, the spouse would receive only
(100%/[100% + 50%]) = 67% of the household’s original Social Security benefits.7 If
the wife’s benefits were based on her own contributions, her individual benefit would
increase only if her husband’s contributions were higher than hers. The relative
decrease in household benefits would then depend on each spouse’s contributions.
Our fixed effects regression estimates, which capture the average drop in income at
the time of a spouse’s death, average over all these cases.
6.1.3
Medical spending
Because our model explicitly accounts for Medicaid payments, which depend on
a household’s savings, the measure of medical expenses we need is the sum of out-ofpocket spending by the household and Medicaid payments.8 Although the AHEAD
contains detailed data on out-of-pocket medical spending, it does not contain Medicaid payments. A key empirical contribution of this paper is to construct the sum
of these two components by combining out-of-pocket medical spending information
in the AHEAD with Medicaid payment data from the Medicare Current Beneficiary
Survey (MCBS).
The MCBS contains extremely high quality administrative and survey information on both Medicaid payments and out-of-pocket medical spending (De Nardi,
French, Jones, and McCauley [21]). Its main limitation for our purposes is that
the data are collected at the individual, rather than the household level: although
the MCBS contains marital status, it lacks information on the medical spending or
health of the spouse. To exploit the strengths of each dataset, we use the conditional
mean matching procedure described in Appendix D to impute a Medicaid payment
for each individual in each period in the AHEAD. The procedure preserves both the
mean and the distribution of combined medical spending, conditional on Medicaid
recipiency, age, income, out-of-pocket spending and other health and medical utilization variables. The regression of Medicaid payments on these variables has an R2
statistic of 0.67, suggesting that our predictions are accurate.
We model medical spending as a function of a polynomial in age, marital status,
health of each spouse (at the beginning and the end of the period), interactions of
these variables, and persistent and transitory spending shocks (see Equation (6)).
7
These calculations also operate in reverse, with “husband” and “wife” exchanged. To perform
the calculations, we make several assumptions, including that both spouses begin receiving benefits
at the normal retirement age. See https://socialsecurity.gov/planners/retire/yourspouse.
html for more details of calculation of spousal benefits.
8
A large share of the elderly’s medical spending is covered by Medicare co-pays, which are not
means-tested. Our measure is net of those co-pays.
22
To estimate these profiles, we use fixed effects rather than ordinary least squares for
two reasons. First, differential mortality causes the composition of our sample to
vary with age, whereas we are interested in how medical expenses vary for the same
individuals as they grow older. Second, cohort effects are ubiquitous and using fixed
effects we construct the profiles for our model using the mean fixed effect for the
cohort with an average age of 74 in 1996.
We estimate the medical spending persistence parameter ρm and the variance of
the transitory and persistent medical spending shocks using a standard error components model. Our estimates imply that approximately 40% of the cross-sectional
variation in log medical spending is explained by observables. Of the remaining
cross-sectional variation, 40% comes from the persistent shock and 60% from the
transitory shock. Our estimated value of ρm is 0.85. See Appendix G.4 for details.
Mean OOP + Medicaid by Income Tercile: Model, Singles
Mean OOP + Medicaid by Income Tercile: Model, Couples
28
55
50
top
24
OOP + Medicaid (000s of 2014 dollars)
OOP + Medicaid (000s of 2014 dollars)
26
second
third
22
20
18
16
14
12
10
8
top
second
third
40
35
30
25
20
15
10
6
4
72
45
76
80
84
88
92
96
100
Age
5
72
76
80
84
88
92
96
100
Age
(a)
(b)
Figure 5: Mean medical spending. Panel (a): current singles. Panel (b): current
couples.
Figure 5 shows model-predicted mean medical spending for the AHEAD cohort
aged 72-77 in 1996, where we simulate the households through age 100.9 Three
general trends are apparent. First, medical expenses rise rapidly with age, in part
because older individuals are more likely to reside nursing homes or die.10 Second,
spending rises only modestly with PI, if at all. Although our underlying coefficients
show that medical expenses rise with PI when health is held constant, lower-income
households are often in worse health. Finally, the spending of singles (left panel) is
roughly half that of couples (right panel).
9
Because of the structure of our medical spending model, which requires two periods of health
realizations, the simulation results start at age 76.
10
Our simulations include end-of-life spending.
23
Figure 6 shows out-of-pocket medical spending. We find out-of-pocket expenses
by simulating our estimated structural model and calculating Medicaid payments.
Subtracting these payments from total medical spending (shown in Figure 5) yields
out-of-pocket spending. Because Medicaid covers a larger share of medical expenses
in poorer households, out-of pocket spending has a strong income gradient. Likewise,
as medical expenses rise with age, the share covered by Medicaid rises as well. This
leads out-of-pocket medical spending to rise more slowly with age than total spending, especially among the low-income. For instance, among singles in the bottom
PI tercile, out-of-pocket medical spending stays between $3,700 and $5,400; among
those at the top, spending rises by a factor of 3, from $5,600 to $18,000.
Mean OOP Medex by Income Tercile: Model, Singles
Mean OOP Medex by Income Tercile: Model, Couples
19
30
top
second
OOP Medex (000s of 2014 dollars)
OOP Medex (000s of 2014 dollars)
17
32.5
third
15
13
11
9
7
5
3
72
27.5
top
second
third
25
22.5
20
17.5
15
12.5
10
7.5
76
80
84
88
92
96
100
5
72
76
Age
80
84
88
92
96
100
Age
(a)
(b)
Figure 6: Mean out-of-pocket medical spending.
Panel (b): current couples.
Panel (a): current singles.
To find the implications of our estimated spending process for lifetime medical spending, we update the calculations in Jones, De Nardi, French, McGee and
Kirschner [34] to use our latest estimates. In particular, we simulate a large number of demographic and medical spending histories and calculate discounted sums at
each age. Figure 7 shows that, as of age 70, single households will on average incur
$63,000 of medical expenses over the remainders of their lives, and that nearly 10%
of them will incur medical expenses in excess of $126,000. The corresponding values
for initial couples are $129,000 for the mean and $234,000 for the 90th percentile.
Some of this total may be picked up by Medicaid, but it is nonetheless significant.
The spending statistics for singles rise with age until age 90. This is not merely
an artifact of mortality bias. Although older people have less time to live, Figure
5 shows that medical expenses rise with age. In other words, at any point in time,
most survivors have yet to incur the bulk of their lifetime expenses. As De Nardi,
24
French, and Jones [18] and others have pointed out, this “backloading” of medical
expenses gives it the potential to be an important driver of saving. In contrast, as
couples become singles their medical expenses fall.
Remaining Lifetime Medical Spending of Surviving Households
Medical Spending (000s of 2014 dollars)
250
225
200
175
90th %tile, Initial Couples
90th %tile, Initial Singles
Means, Initial Couples
Means, Initial Singles
150
125
100
75
50
25
0
70
74
78
82
86
Age
90
94
98
102
Figure 7: Mean and 90th percentile of remaining lifetime medical spending for surviving households, initial singles and initial couples
6.2
Second-step estimation results
Table 3 reports our estimated preference parameters. Our estimate of ν, the
coefficient of relative risk aversion, is 3.7, a value similar to that estimated for retired
singles in De Nardi, French, and Jones [18], and to those typically used in the life
cycle literature.
Our estimate of consumption equivalence scale η is almost identical to the “modified” OECD scale. It also lies within the confidence interval estimated by Hong and
Rı́os-Rull [31], who estimate it using data on life insurance holdings and a structural
life-cycle model of consumption and saving decisions. After reviewing a variety of
estimates, Fernandez-Villaverde and Krueger [26] argue in favor of similar economies
of scale. In combination with our estimated value of ν, our estimated η implies
that the consumption of a couple must be 1.63 times that of a single to equate the
marginal utility of consumption across the two groups.11 Intuitively, η is identified
11
To see this note that the marginal utilities of singles and couples are
25
∂uS (cS )
∂c
= (cS )−ν and
ν: coefficient of RRA
3.698
(0.09073)
η: consumption equivalence scale
1.514
(0.2036)
φ0 : bequest intensity, single (in 000,000s)
133.3
(25.5)
κ0 : bequest curvature, single (in 000s)
9,175
(1,020)
φ1 : bequest intensity, surviving spouse
7,581
(1,411)
κ1 : bequest curvature, surviving spouse (in 000s)
244.7
(25.77)
cmin (f = 1): annual consumption floor, singles
4,108
(113.8)
Notes: Standard errors in parentheses. We set cmin (f = 2) = 1.5 · cmin (f = 1).
Table 3: Estimated second-step parameters
from the extent to which couples save, relative to singles, to self-insure future risks.
Our estimates for θ0 and κ0 govern the strength of terminal bequest motives.
Panel (a) of Figure 8 plots their implications in terms of the share of resources that
would be left as bequests by single households who know that they will die next
period and by married couples who know that both members of their couple will
die next period for sure. In that case, single households with less than $29,600 in
resources will leave no bequests,12 while those having $100,000 will leave over 75% of
their resources in bequests. In contrast, couples require almost $50,000 in resources
before they choose to leave bequests to other heirs. These differences in behavior
for singles and couples do not come from bequest parameters, which we constrain
to be the same when there are no survivors. Rather, they are due to the difference
in household size between couples and singles and to the presence of equivalence
scales. Together, these parameters imply that bequests are luxury goods for singles
(i.e., the threshold level of wealth above which households leave bequests is higher)
and that they are even more of a luxury good, but are weaker for couples. Panel
∂uC (cC )
∂c
12
C −ν
1/ν
C
2
2
= η1−ν
(c ) . Equating these marginal utilities yields ccS = η1−ν
.
This is an annual value. In our two-year framework, the threshold will (approximately) double.
See Appendix H.
26
Share Allocated to Bequests
Share Allocated to Bequests
1
.8
.6
.4
.2
0
0
50
100
150
200
1
.8
.6
.4
.2
0
0
50
Cash on Hand (1000s USD)
Singles
100
150
200
Cash on Hand (1000s USD)
Couples
Singles
(a)
DFJ
Lockwood
ABCST
L&T
(b)
Figure 8: Estimated Bequest Motives. Panel (a): expenditure share allocated to
bequests for couples (solid) and singles (dashed) facing certain death in the next
period. Panel (b): comparing our results for singles (dashed red), with those in De
Nardi, French and Jones[18] (DFJ), Lockwood [41], Ameriks, Briggs, Caplin, Shapiro
and Tonetti [3] (ABCST), and Lee and Tan [40] (L&T).
(b) of Figure 8 compares our estimated bequest motives for singles with those from
several previous papers and displays that our estimated bequest motives for singles
are of similar strength and slightly more of a luxury good than those estimated by
Lockwood [41] and Lee and Tan [40].
Our estimated consumption floor implies that the consumption of a single household is bounded below at $4,108 per year. The floor for couples is set to 150% of this
value. Our estimated floor is best interpreted as an “effective” consumption floor
that accounts for transactions costs, stigma, and other determinants of Medicaid
usage.
6.3
Model fit
We require our model to match the observed heterogeneity in savings and Medicaid recipiency by age, cohort, income, and wealth levels. More specifically, we target
the 25th percentile, the median, and the 75th percentile of wealth, conditional on
PI tercile, by cohort and age, for both couples and singles. We also target average
Medicaid recipiency by cohort, age, and PI tercile.
These targets help separately identify household precautionary savings, bequest
motives, and the degree to which bequest motives are luxury goods.
Figure 9 plots median net worth by age, PI, and birth cohort for current couples
27
(Panel (a)) and current singles (Panel (b)), in the data (solid line) and from our
model (dashed line). For clarity, we only show targeted moments for two of our birth
cohorts in each panel of these set of graphs, but the model fit is similarly good across
cohorts (See Appendix I).
Figure 9 highlights several important features of saving behavior during retirement. First, higher-PI households dissave more slowly than lower-PI households
throughout retirement and, second, that this force is stronger for couples, as long
both members of the couple stay alive. More specifically, singles in the lowest PI
tercile have almost no wealth, while couples in the same PI tercile hold onto their
retirement savings until age 90. Singles in the middle PI tercile start decumulating
their wealth from the time that we start tracking them (age 75), while couples of
the same age and PI tercile display flat asset trajectories over much of the period
that we observe them. Finally, households in the highest PI tercile, and especially
current couples, have highest savings during their retirement years. Our estimated
model matches all of these aspects of the data.
Median Net Worth: Data (Solid) vs. Model (Dashed), Singles
Median Net Worth: Data (Solid) vs. Model (Dashed), Couples
270
600
550
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
240
210
180
150
120
90
60
30
0
72
500
450
400
350
300
250
200
150
100
50
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
(a)
(b)
Figure 9: Median wealth. Panel (a): current singles. Panel (b): current couples.
Solid lines: cohorts aged 72-77 and 84-89 in 1996. Dashed lines: model simulations.
Figure 10 compares the Medicaid recipiency profiles generated by our model
(dashed line) to those in the data (solid line) and shows that our model also matches
the patterns of Medicaid usage across age and household PI. Matching these features
of Medicaid recipiency means that our model is able to capture household reliance
on social insurance, in addition to matching the risk of the catastrophic medical
spending that leads higher-income households to utilize Medicaid. Matching Medicaid recipiency rates is important for identifying the consumption floor, which in
28
Medicaid Recipiency: Data (Solid) vs. Model (Dashed), All
0.8
0.7
Medicaid Recipiency
0.6
0.5
0.4
0.3
0.2
0.1
0
72
76
80
84
88
92
96
100
Age
Figure 10: Medicaid Recipiency. Solid lines: cohorts aged 72-77 and 84-89 in 1996.
Dashed lines: model simulations.
turn is important to evaluate the role of precautionary savings because a low consumption floor implies high consumption risk, and thus a high role of precautionary
savings. The role of precautionary savings is in turn important for understanding
bequest motives, since if precautionary savings are high, the role for bequest motives
is weaker.
Appendix I shows all the moments we match and our model’s fit. Matching
multiple asset percentiles across multiple PI terciles requires the model to reproduce
saving behavior across a wide variety of income flows, household structures and initial
asset holdings. High-PI households and couples save at higher rates than their lowPI and single counterparts, and this is especially visible at the 25th and 75th asset
percentiles as shown in Figures 18 and 19. Among couples at the top of the PI
distribution, wealth rises with age at the 75th percentile of wealth, but are constant
at the 25th percentile. Among singles at the bottom of the PI distribution, wealth
is constant at the 75th percentile, while at the 25th percentile it falls rapidly, or is
0 throughout. This variation disentangles the saving motives operating within our
model.
7
Model validation
To build additional trust in the predictions of our estimated model, we turn to
comparing its implications with some aspects of the data that are important given
our question but are not targeted by our estimation procedure. These aspects of the
data are medical spending and net worth dynamics around the time of death of one
29
-5
1000s of 2014 Dollars
0
5
10
15
of the spouses in a couple.
-6
-4
-2
0
Years from Death
Data
95% CI
2
4
Simulations
Figure 11: Average change in out-of-pocket medical expenses around death of a
spouse, initial couples, relative to similar households with no death. Estimates (solid
blue line) and their 95% confidence interval (shaded region) from the AHEAD data
and differences-in-differences estimates (dashed red line) from model-simulated data.
Death dates are centered at year 0.
Our first validation exercise focuses on the rise in out-of-pocket medical spending
around the death of a spouse and is shown in Figure 11. In this graph, we use the
same matched-differences-in-differences estimation as described in Figure 2 for both
actual and model-generated data. Unlike Figure 2, this figure excludes Medicaid
payments and focuses on the out-of-pocket component because it is this part that
captures the extent to which asset declines at death are driven by medical expenses.
Figure 11 shows that our model generates a trajectory of out-of-pocket medical expenses around the time of death that matches almost exactly the one observed in the
data. This is notable for two reasons. First, because we did not require the model
to match it by construction. Second, because although the sum of out-of-pocket and
Medicaid payments is exogenous in our model, a household’s out-of-pocket spending
component is partly determined by their wealth, which determines Medicaid eligibility and thus the share of expenses paid by Medicaid. Consequently, the fact that our
model closely replicates the dynamics of out-of-pocket medical spending through the
dynamics of saving and means-tested transfers, demonstrates that we endogenously
generate the correct exposure of wealth to medical expense risk.
Our second validation exercise returns to the asset dynamics of couples when
one of their spouses dies. Here, we display results not only for our full sample of
matched initial couples (Panel (a)) but also for those in the top half of the 1996
30
100
50
1000s of 2014 Dollars
-200
-100
0
0
-300
1000s of 2014 Dollars
-150
-100
-50
-400
-200
-6
-4
-2
0
Years from Death
Data
95% CI
(a) Initial Couples
2
4
-6
Simulations
-4
-2
0
Years from Death
Data
95% CI
2
4
Simulations
(b) Initial Couples, Above Median Wealth
Figure 12: Average change in wealth around death of a spouse for initial couples
(Panel (a)) and initial couples in the top half of the 1996 wealth distribution (Panel
(b)). Differences-in-differences estimates (solid blue line) and their 95% confidence
interval (shaded region) from the AHEAD data and differences-in-differences estimates (dashed red line) from model-simulated data. Death dates are centered at
year 0.
initial wealth distribution (Panel (b)). We add this panel because matching well the
asset dynamics of wealthier households around death implies that our luxury good
specification of bequest motives matches households’ saving behavior of wealthier
households. Figure 12 shows that the net worth dynamics around a spousal death
generated by our model lie well within the 95% confidence interval of their AHEAD
data counterpart, even when we condition by household wealth. For the full sample
of initial couples both the data and the simulations predict a decline of $160,000
while for the sample of wealthier households we replicate the decline of $255,000
around death.
Our model thus captures well both medical spending and net worth dynamics
around the time of death of a spouse across the wealth distribution, which implies
that it also matches the extent to the decline in wealth at death which is due to
changes in out-of-pocket medical expenses as opposed to bequests to other heirs than
the spouse. These features of the data, in turn, have important implications about
the relative importance of precautionary savings and bequest motives, forces that
help identify risk aversion and bequest motive parameters (given the consumption
floor, which is pinned down by Medicaid recipiency).
31
8
What drives savings?
Median Net Worth: Experiment (Solid) vs. Baseline (Dashed), Initial Singles
250
500
Median Net Worth: Experiment (Solid) vs. Baseline (Dashed), Couples
225
450
200
400
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
To quantify the savings determinants of retired households, we perform several
decomposition exercises, calculating the changes in total retiree wealth that occur as
we switch off various saving motives. In doing so, we assume that household wealth
accumulation prior to retirement is unchanged. Because of this, our decompositions
provide a lower bound on the savings changes, compared to an environment in which
people can re-optimize their savings at younger ages. Our approach allows us to focus
on the drivers of post-retirement savings and to better understand policy reforms that
are not announced far in advance.
More specifically, to perform our decomposition, we fix our model parameters at
their estimated values and change one feature of the model at a time, recompute the
optimal saving decisions, and compare the resulting asset accumulation profiles to
those generated by our baseline model. To make the differences clearer, we display
the simulated median asset profiles for our youngest cohort, which has an average
age of 74 in 1996. When couples lose a spouse and become singles, we drop them
from the analysis. We do the same when singles die. Therefore, the profiles embed
the same mortality bias that is in the data.
175
150
125
100
75
50
25
0
72
350
300
250
200
150
100
50
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
(a) Singles
(b) Couples
Figure 13: Median wealth by permanent income tercile: baseline model (dashed
lines) and model with no medical expenses (solid lines), initial singles (Panel (a))
and current couples (Panel (b)).
We start by asking whether medical expenses are important drivers of saving
behavior. To answer this question, we zero out both expected and realized medical
expenses and report the corresponding asset trajectories.
32
Panel (a) of Figure 13 shows that the poorest singles, in the bottom PI tercile,
save almost nothing to cover medical expenses: with little wealth, they choose to
rely on Medicaid to pay for their medical needs. It is singles in the middle PI tercile
who are most sensitive to medical spending: at age 84, removing medical expenses
would cause their wealth to fall by nearly 40%, from $87,000 to $54,000. These
individuals have enough wealth for medical expenses to pose a concern, but not
enough for bequest motives to dominate. At the top PI tercile, eliminating medical
spending has a smaller effect on savings. At age 84, their wealth would fall by 7%,
from $238,000 to $221,000. The wealth they hold to leave bequests allows them to
self-insure against medical expenses (as noted by Dynan, Skinner and Zeldes [23]).
Interestingly, after age 90 the savings of richer singles are higher without medical
expenses. This is because eliminating medical spending leaves households with more
resources to consume or save.
Panel (b) shows that the effect of medical spending on couples’ wealth is relatively
small at every PI tercile, suggesting that medical expenses are only one of the reasons
why they continue to hold wealth at very advanced ages. The largest effect occurs for
couples in the top PI tercile, for whom net worth at age 84 declines from $437,000 to
$417,000, by 5%. Comparing Panel (a) and Panel (b) shows that medical expenses
have a larger effect on the savings of singles than of couples.
Median Assets: Experiment (Solid) vs. Baseline (Dashed), Couples
500
225
450
200
400
Assets (000s of 2014 dollars)
Assets (000s of 2014 dollars)
Median Assets: Experiment (Solid) vs. Baseline (Dashed), Initial Singles
250
175
150
125
100
75
50
25
0
72
350
300
250
200
150
100
50
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
(a) Singles
(b) Couples
Figure 14: Median wealth by permanent income tercile: baseline model (dashed
lines) and model with no bequest motives (solid lines). Singles (Panel (a)) and
couples (Panel (b)).
We next consider the role of our estimated bequest motives by shutting down the
utility from leaving any bequests (that is, we set φ0 = φ1 = 0) while holding all other
33
Median Net Worth: Experiment (Solid) vs. Baseline (Dashed), Initial Singles
250
500
Median Net Worth: Experiment (Solid) vs. Baseline (Dashed), Couples
225
450
200
400
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
preference parameters constant. Figure 14 reports the resulting effects on savings.
For singles, eliminating bequest motives has little effect on the median savings of the
bottom two PI terciles, although they do reduce median savings at the top tercile
from $238,000 to $197,000 (17%) at age 84.
For couples, bequest motives are more important and have a large effect on savings
for the top two PI terciles. Panel (b) of Figure 14 shows while savings barely change
for couples in the bottom third of the PI distribution, eliminating bequest motives
reduces savings at age 84 from $199,000 to $172,000 (14%) in the middle tercile and
from $437,000 to $325,000 (26%) in the top tercile.
Thus, more couples than singles save to leave bequests, and bequest motives
affect their savings by a much larger extent. This is for at least two reasons. First,
couples are wealthier, and bequests are a luxury good. Second, couples not only
receive direct utility from leaving bequests at the time of one spouse’s death, but
also indirect utility from the expectation that the last surviving spouse will leave
bequests at her or his death.
175
150
125
100
75
50
25
0
72
350
300
250
200
150
100
50
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
(a) Singles
(b) Couples
Figure 15: Median wealth by permanent income tercile: baseline model (dashed
lines) and model with no bequest motives and no medical expenses (solid lines).
Singles (Panel (a)) and couples (Panel (b)).
While medical expenses and bequest motives taken each in isolation have relatively modest effects on the saving behavior of most households, their interaction
could have a much larger impact. To quantify the importance of these interactions,
we eliminate both medical expenses and bequest motives.
The resulting savings profiles are displayed in Figure 15. Because singles in the
bottom and middle PI tercile are too poor for bequest motives to actively influ34
ence their saving behavior (as seen in Figure 14), the effects on their savings from
eliminating both bequest motives and medical expenses are very similar to those in
the case in which we only eliminate medical expenses (Figure 13). In contrast, for
those in the top PI tercile, for whom the bequest motive is active, savings falls from
$238,000 to $82,000 (by 66%). This drop is almost three times larger than the sum
of the drops in the previous two experiments (24%).
This is due to the way bequest and precautionary saving motives interact. When
medical expenses are uncertain, households that reduce their consumption to finance
future medical expenses may end up dying with unspent wealth. In the absence of
bequest motives, these accidental bequests have no value. Bequest motives reduce
the opportunity cost of self-insuring through saving.13 At the same time, medical
expenses introduce uncertainty over the size of the household’s bequest. This might
generate precautionary saving aimed at increasing the size of the bequest.
Panel (b) of Figure 15 displays the corresponding results for couples. The savings
of couples in the bottom PI tercile fall from $62,000 to $41,000 (by 34%) at age 84.
For couples in the middle PI tercile, savings falls from $199,000 to $99,000 (by
50%) at age 84. Because bequest motives are active for this group, unlike for the
corresponding singles, the interaction of bequest and precautionary saving motives is
much larger for middle PI couples than singles. This interaction is also very strong
for couples in the top PI tercile. Rather than accumulating wealth in retirement,
couples at the top tercile would run down their wealth holdings from $437,000 to
$150,000 (66%) by age 84.
Finally, couples save to provide resources to the surviving spouse after the death
of the first spouse. This is important because while at age 70 married men live on
average 11.5 years and women live 15.8 years (or on average 13.7 years), the oldest
survivor lives on average 17.9 years. During these remaining years the survivor
faces lower income and high medical spending, including medical spending from an
elevated risk of being in a nursing home when single. Studies omitting this risk will
understate this precautionary savings motive.
To quantify the effect of this force, Figure 16 compares our baseline savings with
those implied by our model when we shut down the weight on the utility of the
surviving spouse. It shows that the desire to leave resources to the surviving spouse
upon a partner’s death has a large effect on savings at all PI levels. For instance,
among couples in the lowest PI tercile, median wealth at age 84 would fall from
$62,000 in the benchmark to $46,000 when there is no concern for the welfare of the
surviving spouse, a 26% drop. Median-asset couples in the second PI tercile at age
13
A similar argument applies to saving against longevity risk and for the decision to self-insure
instead of purchasing formal Long Term Care insurance (See Lockwood [41] for an example).
35
Median Net Worth: Experiment (Solid) vs. Baseline (Dashed), Couples
500
Net Worth (000s of 2014 dollars)
450
400
350
300
250
200
150
100
50
0
72
76
80
84
88
92
96
100
Age
Figure 16: Median wealth for couples by permanent income tercile: baseline model
(dashed lines) and model with no weight on the surviving spouse (solid lines).
84 hold $199,000 in the benchmark economy and $165,000 in this counterfactual, a
17% drop. At the highest PI tercile, median wealth falls from $437,000 to $368,000
at the same age, a 16% drop.
Taken together, Figures 13-16 show that medical spending is a more important
determinant of savings for singles than for couples. In contrast, bequest motives affect
the savings of couples and high-PI households more than those of singles and lowPI households. The figures also show that the interaction of precautionary savings
and bequests are most important for couples and high-PI households. Moreover,
providing for the surviving spouse is important for all couples.
For clarity, Figures 13-16 only display profiles for households whose composition
does not change. Thus, couples do not experience a decline in net worth upon death
of the first spouse and singles do not include the newly widowed, who tend to be
richer. To assess the saving motives across all retired households simultaneously,
Table 8 aggregates the preceding simulations.14
The first column of Table 8 shows summary statistics for our baseline specification. The distribution of wealth is right skewed, with the mean ($372,300) more
than twice as large as the median ($150,100). The second column shows the effects
of eliminating medical expenses. Consistent with Figure 13, medical expenses have
little effect on the saving of the wealthiest: the 75th percentile falls by 2.7% and the
14
Formally, we take the same joint distribution of household initial conditions as in the data
and simulate a panel accounting for the realisations of individual mortality, health and medical
expenditure shocks. We use all retired households alive at each age in the simulated panel, except
for their wealth when first observed which we take as given – and calculates summary statistics for
wealth.
36
Percentage Change from Baseline
25th Percentile
Median
75th Percentile
Mean
Baseline
Wealth
No Medical
Expenses
No Bequest
Motives
No Bequests
or Medical
Expenses
No Weight
on Surviving
Spouse
50.0
150.1
391.9
372.3
-61.4%
-20.9%
-2.7%
-3.0%
8.3%
6.9%
-8.4%
-17.4%
-61.9%
-45.6%
-42.8%
-44.2%
-83.3%
-36.33%
-25.3%
-27.4%
Notes: Wealth in baseline simulations are expressed in thousands of 2014 dollars.
Table 4: Aggregate Implications of Retiree Saving Motives
mean, which is dominated by the richest households, falls only 3.0%. The effects are
much larger in the lower half of the wealth distribution: the median falls by 20.9%.
When medical expenses are eliminated, middle-wealth singles reduce their wealth
and middle-wealth couples transfer a larger fraction of their estates from surviving
spouses to nonspousal heirs.
The effects reverse in the third column, where we eliminate bequest motives.
Bequests are luxury goods and their elimination causes mean wealth to fall by 17.4%.
In contrast, eliminating bequest motives leads median wealth to increase by 6.9%.
Couples save less when bequest motives are removed, but when one spouse dies all
of the estate is left to the surviving spouse. For middle-wealth households, the latter
effect dominates. Taken together, the second and third column reinforce our earlier
findings that medical expenses are more important for poorer households and singles,
while bequest motives are more important for the rich.
The fourth column also reinforces Figures 13-16, in that the effect of eliminating
medical expenses and bequest motives together is large and exceeds the sum of
the individual changes. This is a key finding because the previous literature has
focused on the role of precautionary saving motives due to medical spending without
a bequest motive. Given that our results present robust evidence of bequest motives,
and that their interaction with medical expenses has a very large impact on savings,
our results show that it is essential to model both bequest motives and medical
spending.
The final column of Table 8 shows that setting to zero the weight that couples
37
place on the surviving spouse also has large effects. This is not surprising because
this experiment leads couples to bequeath all their wealth to nonspousal heirs upon
the death of the first spouse, but highlights that it is important to model couples
and spousal deaths to fully capture life cycle saving motives.
9
Conclusions
This paper documents new facts about the retirement savings and risks of couples
and singles. It then develops and estimates a rich structural model that contains these
risks and explains their savings.
In terms of new facts, we compare the saving behavior of couples and singles.
Households with high PI, and especially couples, tend to accumulate wealth, whereas
those with low PI, and especially singles, tend to decumulate it. Furthermore, upon
the death of a spouse, there are large drops in wealth, and bequests are often distributed.
To understand the relative roles of risks versus bequest motives as drivers of
these savings decisions, it is crucial to properly measure the risks faced by couples
and singles. While medical expenses for couples and singles are not very different
on a per capita basis until someone dies, medical spending jumps at the time of a
spouse’s death. At the same time, household income falls, exposing households to a
substantial loss in financial resources upon a spouse’s death.
We embed these estimated risks within a rich structural model of savings that incorporates heterogeneity in financial resources, life expectancy, and medical expense
risks, including the large jump in medical expenses that is typically observed before
a death. Our model also accounts for means-tested social insurance and bequest
motives. It allows households to leave bequests upon death of the last survivor. In
addition, and very importantly, households can choose to distribute bequests when
the first member of the couple dies (net worth is split between the surviving spouse
and other heirs).
We estimate our model using MSM, targeting the savings of both couples and
singles and Medicaid recipiency rates by cohort, PI, and wealth levels. Our model
fits all of these important features of the data well. In addition, although we do
not require our model to match the dynamics of out-of-pocket medical spending and
wealth around the time of a spouse’s death, our model also matches those well. This
validates key predictions of our model.
Finally, we use our estimated model to measure the extent to which the savings
of retirees are driven by medical expenses, bequest motives, and their interaction.
We uncover several new findings. Medical expenses are more important for singles,
38
whereas bequest motives are more important for couples. This is due to two main
reasons. First, couples not only receive direct utility from leaving bequests at the
time of one spouse’s death, but also indirect utility from the expectation that the
last surviving spouse will leave bequests upon their own death. Second, bequests
are a luxury good, and couples are wealthier. We also find that introducing bequest motives reduces the effects of medical expenses on savings. This is because
savings intended for medical spending that go unspent due to early death are still
valued when bequest motives are present. This finding shows that we need to model
both medical spending and bequests. Lastly, although medical spending is important, the desire to transfer resources to one’s heirs other than their spouse is key to
understanding the wealth drops at the death of a first spouse.
Hence, to understand the drivers of savings, it is essential to model the bequest
motives and medical spending of both couples and singles. Why people save has
important policy implications. Our results suggest that couples and high-PI singles
can easily self-insure against medical spending risk because they save to leave bequests. Low-PI singles are well insured through Medicaid and do not need to save
for medical expenses. In contrast, middle-PI singles set aside significant amounts of
wealth for precautionary purposes because they are not rich enough to want to leave
bequests but are too rich to qualify for Medicaid. It is thus the middle-PI singles
who would respond most to changes in public health insurance.
Medicaid also includes rules determining its relative generosity for couples and
singles. Evaluating whether these rules are well designed requires carefully modeling
and quantifying the saving motives of all retirees.
39
References
[1] Andrew B Abel and Mark Warshawsky. Specification of the joy of giving: Insights from altruism. The Review of Economics and Statistics, 70(1):145, 1988.
[2] Joseph G. Altonji and Lewis M. Segal. Small sample bias in gmm estimation of
covariance structures. Journal of Business and Economic Statistics, 14(3):353–
366, 1996.
[3] John Ameriks, Joseph Briggs, Andrew Caplin, Matthew D. Shapiro, and Chris
Tonetti. Long-term care utility and late in life saving. Journal of Political
Economy, 128(6):2375–2451, 2020.
[4] John Ameriks, Andrew Caplin, Steven Laufer, and Stijn Van Nieuwerburgh.
The joy of giving or assisted living? using strategic surveys to separate bequest
and precautionary motives. Journal of Finance, 66(2):519–561, 2011.
[5] James Andreoni. Giving with impure altruism: Applications to charity and
ricardian equivalence. Journal of Political Economy, 97:1447–1458, 1989.
[6] Orazio P. Attanasio and Carl Emmerson. Mortality, health status, and wealth.
Journal of the European Economic Association, 1(4):821–850, 2003.
[7] Gary S. Becker and Nigel Tomes. Human capital and the rise and fall of families.
Journal of Labor Economics, 4(3):1–39, 1986.
[8] B. Douglas Bernheim, Andrei Shleifer, and Lawrence H. Summers. The strategic
bequest motive. Journal of Political Economy, 93(5):1045–1076, 1985.
[9] R. Anton Braun, Karen A. Kopecky, and Tatyana Koreshkova. Old, sick, alone
and poor: A welfare analysis of old-age social insurance programs. The Review
of Economic Studies, 84(2):580–612, 2017.
[10] Meta Brown. End-of-life transfers and the decision to care for a parent. mimeo,
2003.
[11] Moshe Buchinsky. Recent advances in quantile regression models: A practical
guideline for empirical research. Journal of Human Resources, 33:88–126, 1998.
[12] Christopher D. Carroll. Why do the rich save so much? NBER Working Paper
6549, 1998.
40
[13] Gary Chamberlain. Comment: Sequential moment restrictions in panel data.
Journal of Business & Economic Statistics,, 10(1):20–26, 1992.
[14] Martin David, Roderick J. A. Little, Michael E. Samuhel, and Robert K. Triest. Alternative methods for cps income imputation. Journal of the American
Statistical Association, 81(393):29–41, 1986.
[15] James B. Davies and Antony F. Shorrocks. The distribution of wealth. In
Anthony B. Atkinson and François Bourguignon, editors, Handbook of Income
Distribution, pages 605–675. Handbooks in Economics, vol. 16. Amsterdam;
New York and Oxford: Elsevier Science, North-Holland, 2000.
[16] Mariacristina De Nardi. Wealth inequality and intergenerational links. Review
of Economic Studies, 71(3):743–768, 2004.
[17] Angus Deaton. Saving and liquidity constraints. Econometrica, 59(5):1221–
1248, 1991.
[18] Mariacristina De Nardi, Eric French, and John Bailey Jones. Why do the elderly
save? the role of medical expenses. Journal of Political Economy, 118(1):39–75,
2010.
[19] Mariacristina De Nardi, Eric French, and John Bailey Jones. Medicaid insurance
in old age. American Economic Review, 106(11):3480–3520, 2016.
[20] Mariacristina De Nardi, Eric French, and John Bailey Jones. Savings after
retirement: A survey. Annual Review of Economics, 8(1):177–204, 2016.
[21] Mariacristina De Nardi, Eric French, John Bailey Jones, and Jeremy McCauley.
Medical spending of the u.s. elderly. Fiscal Studies, 37(3-4):717–747, 2016.
[22] Darrell Duffie and Kenneth J. Singleton. Simulated moments estimation of
markov models of asset prices. Econometrica, 61(4):929–952, 1993.
[23] Karen E. Dynan, Jonathan Skinner, and Stephen P. Zeldes. The importance of
bequests and life-cycle saving in capital accumulation: A new answer. American
Economic Review, 92(2):274–278, 2002.
[24] Itzik Fadlon and Torben Heien Nielsen. Family health behaviors. American
Economic Review, 109(9):3162–91, 2019.
[25] Daniel Feenberg and Jonathan Skinner. The risk and duration of catastrophic
health care expenditures. Review of Economics and Statistics, 76:633–647, 1994.
41
[26] Jesús Fernández-Villaverde and Dirk Krueger. Consumption over the Life Cycle:
Facts from Consumer Expenditure Survey Data. The Review of Economics and
Statistics, 89(3):552–565, 2007.
[27] Eric French and John Bailey Jones. On the distribution and dynamics of health
care costs. Journal of Applied Econometrics, 19(4):705–721, 2004.
[28] Eric French and John Bailey Jones. The effects of health insurance and selfinsurance on retirement behavior. Econometrica, 79(3):693–732, 2011.
[29] Miguel Gouveia and Robert P Strauss. Effective federal individual income tax
functions: An exploratory empirical analysis. National Tax Journal, pages 317–
339, 1994.
[30] Nezih Guner, Remzi Kaygusuz, and Gustavo Ventura. Income taxation of us
households: Facts and parametric estimates. Review of Economic Dynamics,
17(4):559–581, 2014.
[31] Jay H. Hong and José-Vı́ctor Rı́os-Rull. Life Insurance and Household Consumption. The American Economic Review, 102(7):3701–3730, 2012.
[32] R. Glenn Hubbard, Jonathan Skinner, and Stephen P. Zeldes. Precautionary
saving and social insurance. Journal of Political Economy, 103(2):360–399, 1995.
[33] Michael D. Hurd. Mortality risk and bequests. Econometrica, 57(4):779–813,
1989.
[34] John Bailey Jones, Mariacristina De Nardi, Eric French, Rory McGee, and
Justin Kirschner. The lifetime medical spending of retirees. Economic QuarterlyFederal Reserve Bank of Richmond, 104(3):103, 2018.
[35] John Bailey Jones, Mariacristina De Nardi, Eric French, Rory McGee, and
Rachel Rodgers. Medical spending, bequests, and asset dynamics around the
time of death. Economic Quarterly, Fourth Quarter:135–157, 2020.
[36] Thomas F. Juster, James P. Smith, and Frank Stafford. The measurement and
structure of household wealth. Mimeo, 1998.
[37] Ruud H. Koning.
http://www.xs4all.nl/
Kernel:
A gauss library for kernel estimation.
42
[38] Karen Kopecky and Tatyana Koreshkova. The impact of medical and nursing
home expenses and social insurance policies on savings and inequality. American
Economic Journal: Macroeconomics, 6:29–72, 2014.
[39] John Laitner, Dan Silverman, and Dmitriy Stolyarov. The role of annuitized
wealth in post-retirement behavior. American Economic Journal: Macroeconomics, 10(3):71–117, 2018.
[40] Siha Lee and Kegon T. K. Tan. Bequest Motives and the Social Security Notch.
October 2019.
[41] Lee M. Lockwood. Incidental bequests and the choice to self-insure late-life
risks. American Economic Review, 108(9):2513–50, 2018.
[42] Samuel Marshall, Kathleen M. McGarry, and Jonathan S. Skinner. The risk of
out-of-pocket health care expenditure at the end of life. Working Paper 16170,
National Bureau of Economic Research, 2010.
[43] Makoto Nakajima and Irina A Telyukova. Home equity in retirement. International Economic Review, 61(2):573–616, 2020.
[44] Whitney K. Newey. Generalized method of moments specification testing. Journal of Econometrics, 29(3):229–256, 1985.
[45] Whitney K. Newey and Daniel L. McFadden. Large sample estimation and
hypothesis testing. In Robert Engle and Daniel L. McFadden, editors, Handbook
of Econometrics, Volume 4. Elsevier, Amsterdam, 1994.
[46] Ariel Pakes and David Pollard. Simulation and the aysmptotics of optimization
estimators. Econometrica, 57(5):1027–1057, 1989.
[47] Michael G. Palumbo. Uncertain medical expenses and precautionary saving near
the end of the life cycle. Review of Economic Studies, 66:395–421, 1999.
[48] Jorn-Steffen Pischke. Measurement error and earnings dynamics: Some estimates from the PSID validation study. Journal of Business & Economics
Statistics, 13(3):305–314, 1995.
[49] James M. Poterba, Steven F. Venti, and David A. Wise. The composition and
drawdown of wealth in retirement. Journal of Economic Perspectives, 25:95–117,
2011.
43
[50] James Powell. Estimation of semiparametric models. In Robert Engle and
Daniel L McFadden, editors, Handbook of Econometrics, Volume 4. Elsevier,
Amsterdam, 1994.
[51] Susann Rohwedder, Steven J. Haider, and Michael Hurd. Increases in wealth
among the elderly in the early 1990s: How much is due to survey design? Review
of Income and Wealth, 52(4):509–524, 2006.
[52] George Tauchen. Finite state markov-chain approximations to univariate and
vector autoregressions. Economics Letters, 20(2):177–181, 1986.
44
Appendix A: Sample selection and data handling
Our main dataset is the AHEAD data, which is a cohort within the Health and
Retirement Study (HRS).
To keep the dynamic programming problem manageable, we assume a fixed difference in age between spouses, and we take the average age difference from our data.
In our sample, husbands are on average 3 years older than their wives; in the model
we assume the difference is one period, or 2 years. To keep the data consistent with
this assumption, we drop all households where the wife is more than 4 years older or
10 years younger than her husband.
We do not use 1993/94 assets, nor medical expenses, due to underreporting issues
(Rohwedder, Haider, and Hurd [51]) and thus begin with 6,047 households. Because
we only allow for household composition changes through death, we drop households
where an individual enters a household or an individual leaves the household for
reasons other than death. Fortunately, attrition for reasons other than death is a
minor concern in our data. We drop 401 households who get married, divorced,
were same sex couples, or who report making other transitions not consistent with
the model, 753 households who report earning at least $3,000 in any period, 171
households with a large difference in the age of husband and wife, and 87 households
with no information on the spouse in a household. As a result, we are left with 4,634
households, of whom 1,388 are couples and 3,246 are singles. This represents 24,274
household-year observations where at least one household member was alive.
An advantage of the AHEAD relative to other datasets is that it provides panel
data on health status and nursing home stays. We assign individuals a health status
of “good” if self-reported health is excellent, very good or good, and a health status
of “bad” if self-reported health is fair or poor. We assign individuals to the nursing
home state if they were in a nursing home at least 120 days since the last interview
(on average 60 days per year) or if they spent at least 60 days in a nursing home
before the next scheduled interview and died before that scheduled interview.
The AHEAD has information on the value of housing and real estate, autos,
liquid assets (which include money market accounts, savings accounts, T-bills, etc.),
IRAs, Keoghs, stocks, the value of a farm or business, mutual funds, bonds, “other”
assets and investment trusts less mortgages and other debts. One problem with asset
data is that the wealthy tend to underreport their wealth in virtually all household
surveys (Davies and Shorrocks [15]). This could lead us to understate asset levels at
all ages. However, Juster, Smith, and Stafford [36] show that the wealth distribution
of the AHEAD matches up well with aggregate values for all but the richest 1 % of
households. Given that we match the quantiles (conditional on permanent income)
45
rather than means, underreporting at the very top of the wealth distribution should
not affect our results.
Social Security and pension benefits are included in our measure of annual income, hence differences in Social Security and pension wealth appear in our model as
differences in the permanent income measure we use to predict annual income (and
other state variables).
Appendix B: Computing permanent income
We assume that log income evolves as following
ln yit = κ(t, fit ) + h(Ii ) + ωit ,
(20)
where κ(t, fit ) is a flexible function of age t and family structure fit (i.e., couple,
single man or single woman) and ωit represents measurement error. The variable Ii
is the household’s percentile rank in the permanent income (PI) distribution. Since
it is a summary measure of lifetime income at retirement, it should not change during
retirement and is thus a fixed effect over our sample period. However, income could
change as households age and potentially lose a family member.
To estimate Equation (20) we first estimate the fixed effects model
ln yit = κ(t, fit ) + αi + ωit ,
(21)
which allows us to obtain a consistent estimate of the function κ(t, fit ). Next, note
that as the number of time periods over which individual i is observed (denoted Ti )
becomes large,
Ti
Ti
1 X
1 X
ln yit − κ(t, fit ) − ωit =
ln yit − κ(t, fit ) = αi = h(Ii ). (22)
plim
Ti t=1
Ti →∞ Ti t=1
We thus calculate the PIPranking Ii for every household in our sample by taking the
i
percentile ranking of T1i Tt=1
[ln yit − κ
b(t, fit )], where κ
b(t, fit ) is the estimated value
of κ(t, fit ) from Equation (21). Put differently, we take the mean residual per person
from the fixed effects regression (where the residual includes the estimated fixed
effect), then take the percentile rank of the mean residual per person to construct Ii .
However, we also need to estimate the function h(Ii ), which converts the estimated index Ii back to a predicted level of income that can be used in the dynamic
programming model. To do this we estimate the function
[ln yit − κ(t, fit )] = h(Ii ) + ωit
46
(23)
where the function h(Ii ) is a flexible functional form. In practice we model κ(t, fit )
as a third order polynomial in age, dummies for family structure, and family structure interacted with an age trend. When estimating Equation (23), we replace the
function κ(t, fit ) with its estimated value. We model h(Ii ) as a fifth order polynomial
in our measure of permanent income percentile.
Given that we have, for every member of our sample, t, fit , and estimates of Ii and
the functions κ(., .), h(.), we can calculate the predicted value ln ybit = κ
b(t, fit )+ b
h(Ibi ).
It is ln ybit that we use when simulating the model for each household. A regression of
ln yit on ln ybit yields a R2 statistic of .74, suggesting that our predictions are accurate.
Appendix C: Matching methodology
We match each household that suffers a death to a household that did not experience a death at the same date or within the window shown in our graphs, but did
experience a death within 6-10 years. (Fadlon and Nielson [24] impose a similar restriction in their analysis of health shocks.) More specifically, for each household we
take a matched control household from the set of households who have the same initial (1996) household composition, have a PI percentile within 15 percentage points
and where their 1996 wealth differs by no more than 2.5% (or $5,000 in levels). We
then use a random number generator to select a single control household from this
set giving each household an equal probability of being selected. Our event study
specification then uses a sample of 476 households who experience the death of a
spouse (2,383 household-year observations) matched to 476 households who do not
experience a death in the sample window (2,624 household-year observations). Jones,
De Nardi, French, McGee and Rodgers [35] present more results on this.
Appendix D: Imputing Medicaid payments and outof-pocket medical expenses
Our goal is to estimate the data generating process for the sum of Medicaid payments and out-of-pocket expenses: this is the variable ln(m) in Equation (6) of the
main text. The AHEAD data contain information on out-of-pocket medical spending, but not on Medicaid payments. Fortunately, the Medicare Current Beneficiary
Survey (MCBS) has extremely high quality information on both Medicaid payments
and out-of-pocket medical spending. One drawback of the MCBS, however, is that
although it has information on marital status and household income, it does not
include information on the medical spending or health of the spouse.
47
We use a two-step imputation procedure to exploit the strenghts of both datasets.
First step of imputation procedure
We use the MCBS to infer Medicaid payments for recipients, conditional on
observable variables that exist in both the MCBS and the AHEAD datasets. Let
oopit denote out-of-pocket medical expenses for AHEAD person i at time t, and let
M cdit denote the dollar value of Medicaid payments.
To impute M cdit , we follow David, Little, Samuhel and Triest [14] and French
and Jones [28] and use the following predictive mean matching regression approach.
First, using every member of the MCBS sample with a positive Medicaid indicator
(i.e., a Medicaid recipient), we regress the variable of interest M cd on the vector of
observable variables z, yielding M cd = zβ + ε. Second, for each individual j in the
[
MCBS we calculate the predicted value M
cdjt = zjt β̂, and for each member of the
[
sample we calculate the residual ε̂jt = M cdjt − M
cdjt . Third, we sort the predicted
[
value M cdjt into deciles and keep track of all values of ε̂jt within each decile.
The variables zjt include nursing home status, number of nights spent in a nursing home, an age polynomial, total household income, marital status, self-reported
health, race, visiting a medical practitioner (doctor, hospital or dentist), out-ofpocket medical spending, education and death of an individual. Because the measure of medical spending in the AHEAD is medical spending over two years, we
take two-year averages of the MCBS data to be consistent with the structure of the
AHEAD. The regression of M cdjt on zjt yields a R2 statistic of .67, suggesting that
our predictions are accurate.
Second step of imputation procedure:
Next, for every individual i in the AHEAD sample with a positive Medicaid
[
indicator, we impute M
cdit = zit β̂ using the value of β̂ estimated using the MCBS.
Then we impute εic for each member of the AHEAD sample by finding a random
[
[
individual j in the MCBS with a value of M
cdjt in the same decile as M
cdit in the
]
[
AHEAD, and set εit = ε̂jt . The imputed value of M cdit is M cdit = M cdit + εit , and
]
the imputed value of the sum mit is m
e it = oopit + M
cdit .
As David, Little, Samuhel and Triest [14] point out, our imputation approach is
equivalent to hot-decking when the “z” variables are discretized and include a full
set of interactions. The advantages of our approach over hot-decking are two-fold.
First, many of the “z” variables are continuous, and it seems unwise to discretize
them. Second, we use a large number of observable variables “z” because we find
that adding extra variables greatly improving goodness of fit when imputing Medicaid
payments. Even a small number of variables generates a large number of hot-decking
cells because hot-decking uses a full set of interactions. Thus, in this context, hot
48
decking is too data intensive.
We predict Medicaid payments for 3,756 household-wave observations with Medicaid recipiency in the AHEAD and report the results of this imputation exercise
in Table 5. Households where the last surviving member died between the previous
and current waves of the sample have the largest imputed Medicaid payments. In
contrast, couples have the smallest Medicaid payments per individual, but Medicaid
payments at the household level are larger than for either single men or women. The
imputed Medicaid payments for each spouse in a couple are approximately equal
and the results for couples are not driven by the expenditures of only the husband
or wife. In Table 5, new widows and widowers (those whose spouse has died between
the current and preceding waves of the AHEAD sample) are included in the rows for
single men and single women – on average the Medicaid payments of dead spouses
are less than 20% of the single households’ Medicaid payments.
The distribution of imputed Medicaid expenditures in the AHEAD is close to
that in the MCBS. The mean imputed Medicaid expenditure, M cdit , for Medicaid
recipients in the AHEAD is $14,050. This is lower than the corresponding value
of $16,000 in the MCBS. This difference is due to the distribution of observable
variables, z, in both the MCBS and AHEAD. In particular, the number of nights
in a nursing home is 20% lower in the AHEAD than in the MCBS, which is in part
due to the fact that AHEAD respondents were not in a nursing home when they
answered the survey for the first time.
Family
Structure
Dead
Couples
Single Men
Single Women
Number of Medicaid
Eligible Households
Mean
Standard
Deviation
1,040
287
351
2,078
21,800
15,300
11,600
12,300
31,500
25,700
21,500
21,800
Table 5: Imputed Medicaid payments for Medicaid beneficiaries in AHEAD
49
Appendix E: Computing optimal decision rules and
MSM estimates
We compute the value functions by backward induction. We start with singles. At
time-T , we find the value function and decision rules by maximizing Equation (16),
subject to the relevant constraints, using VTg+1 = θ0 (xt − ct ), g = h, w. This yields
the value function VTg . We then find the value function and decision rules at time
T − 1 by solving Equation (16) with VTg . Continuing this backward induction yields
decision rules for periods T − 2, T − 3, ..., tr .
We find the value function for new widows and widowers, Vtng , and the split of
the estate between spousal and nonspousal heirs, by solving Equation (17), subject
to the relevant constraints. These calculations utilize the value function for singles,
Vtg , described immediately above.
We find the time-T value function for couples, VTc by maximizing Equation (19),
subject to the relevant constraints and the value function for the singles, and setting
the time-T + 1 continuation value to θ0 (xt − ct ). This yields the value function VTc
and the decision rules for time T. We then find the decision rules at time T − 1
by solving Equation (19) using VTc and VTng , g = h, w. Continuing this backward
induction yields decision rules for time T − 2, T − 3, ..., 1.
More specifically, to solve our value functions, we discretize the persistent component and the transitory components of the health shock, ξ and ζ, using the methods
described in Tauchen [52]. We assume a finite number of permanent income categories and we break the space of cash-on-hand into a finite number of grid points.
We use linear interpolation within the grid and linear extrapolation outside of the
grid to evaluate the value function at cash-on-hand values that we do not directly
compute. The end result is a set of decision rules for each possible combination of
cash-on-hand, PI, health status, and persistent health shock (ζ).
The mechanics of our MSM approach are as follows. We compute life-cycle histories for a large number of artificial households. Each of these households is endowed
with an initial value of the state vector (t, ft , xt , I, hsht , hsw
t ) that is drawn from the
data for 1996. In addition, each household is assigned the entire health and mortality history that is recorded for this same household in the AHEAD data. This
way we generate attrition in our simulations that exactly follows the attrition in the
data (including by initial wealth and mortality). The simulated medical expenditure
shocks ζ and ξ are Monte Carlo draws from discretized versions of our estimated
shock processes. Our decision rules, combined with the initial conditions and simulated shocks, allows us to simulate a household’s wealth, medical expenses, health,
and mortality.
50
We construct life-cycle profiles from the artificial histories in the same way that we
compute them from the real data. We use these profiles to construct moment conditions and evaluate the match using our GMM criterion. As done when constructing
the figures from the AHEAD data, we drop cells with fewer than 10 observations
from the moment conditions. We search over the parameter space for the values
that minimize the criterion. Appendix F details our moment conditions, the weighting matrix in our GMM criterion function, and the asymptotic distribution of our
parameter estimates.
Appendix F: Moment conditions and the asymptotic distribution of parameter estimates
We estimate the parameters of our model in two steps. In the first step, we
estimate the vector χ, the set of parameters than can be estimated without explicitly
using our model. In the second step, we use the method of simulated moments
(MSM) to estimate the remaining parameters,
which are contained in the M × 1
vector ∆ = ν, η, φ0 , φ1 , κ0 , κ1 , cmin (ft = S) .
ˆ of the “true” preference vector ∆0 is the value of ∆ that minOur estimate, ∆,
imizes the (weighted) distance between the estimated life cycle profiles for wealth
and Medicaid recipiency in the data and those generated by the model.
For each calendar year t ∈ {t0 , ..., tT } = {1996, 1998, ..., 2014},15 we match the
25th, 50th and 75th percentile of wealth for 3 permanent income terciles in 4 birth
year cohorts, both for singles and couples. Because the 1996 (period-t0 ) distribution
of simulated wealth is bootstrapped from the 1996 data distribution, we match wealth
for the period 1998 to 2014, hence 9 waves. In addition, we require each cohortincome-age cell have at least 15 observations to be included in the GMM criterion.
As a result, we end up with 414 asset targets.
We construct the moment conditions for wealth by building on French and Jones [28]
(useful references include Buchinsky [11] and Powell [50]). Suppose that household
i of family type f belongs to birth cohort c and permanent income tercile p. Let
aqf cqt (∆, χ) denote the model-predicted qth asset quantile for households in households i’s group at time t.16 Assuming that observed wealth have a continuous conditional density, af cqt will satisfy
q
Pr ait ≤ af cpt (∆0 , χ0 ) | f, c, p, t, household i observed at t = πq ,
15
Because we do not allow for macro shocks, in any given cohort, t is used only to identify the
individual’s age.
16
The elements of χ include the permanent income boundaries.
51
where πq is the probability value associated with quantile q – when q denotes a
median, πq would equal 1/2. The preceding equation can be rewritten as a moment
condition. In particular, applying the indicator function produces
E 1{ait ≤ aqf cpt (∆0 , χ0 )} − πq | f, c, p, t, household i observed at t = 0.
Letting Ip denote the values contained in the pth permanent income tercile, we can
convert this conditional moment equation into an unconditional one (e.g., Chamberlain [13]):
E [1{ait ≤ aqf cpt (∆0 , χ0 )} − πq ] × 1{fit = f } × 1{ci = c} × 1{Ii ∈ Ip }
× 1{household i observed at t} t = 0,
(24)
for q ∈ {.25, .5, .75}, f ∈ {single, couple}, c ∈ {1, 2, 3, 4}, p ∈ {1, 2, 3},and t ∈
{t1 , t2 ..., tT }.
We also match Medicaid recipiency rates. We divide individuals into 4 cohorts,
match data from 9 waves, and stratify the data by permanent income, but combine
couples and singles. Let ucpt (∆, χ) denote the model-predicted recipiency rate for
households in cohort c and permanent income tercile p at time t. Let uit be the
{0, 1} indicator that equals 1 when household i receives Medicaid. The associated
moment condition is
E uit − ucpt (∆0 , χ0 ) × 1{ci = c} × 1{Ii ∈ Ip }
× 1{individual i observed at t} t = 0
(25)
for c ∈ {1, 2, 3, 4}, p ∈ {1, 2, 3},and t ∈ {t1 , t2 ..., tT }.
To summarize, the moment conditions used to estimate our model consist of: the
moments for asset quantiles described by Equation (24) and the moments for the
Medicaid recipiency rates described by Equation (25). We have a total of J = 505
moment conditions.
Suppose we have a dataset of I independent individuals that are each observed
at up to T separate calendar years. Let ϕ(∆; χ0 ) denote the J-element vector of
moment conditions described immediately above, and let ϕ̂I (.) denote its sample
c I denote a J × J weighting matrix, the MSM estimator ∆
ˆ is given
analog. Letting W
by
I
c I ϕ̂I (∆; χ0 ),
argmin
ϕ̂I (∆; χ0 )0 W
1+τ
∆
52
where τ is the ratio of the number of observations to the number of simulated observations.
In practice, we estimate χ0 using the approach described in the main text. Computational concerns, however, compel us to treat χ0 as known in the analysis that
follows.
Under the regularity conditions stated in Pakes and Pollard [46] and Duffie and
ˆ is both consistent and asymptotically normally
Singleton [22], the MSM estimator ∆
distributed:
√
ˆ − ∆0
I ∆
N (0, V),
with the variance-covariance matrix V given by
V = (1 + τ )(D0 WD)−1 D0 WSWD(D0 WD)−1 ,
where S is the variance-covariance matrix of the data;
D=
∂ϕ(∆; χ0 )
∂∆0
(26)
∆=∆0
c I }.
is the J×M gradient matrix of the population moment vector; and W = plimI→∞ {W
Moreover, Newey [44] shows that if the model is properly specified,
I
ˆ χ0 )0 R−1 ϕ̂I (∆;
ˆ χ0 )
ϕ̂I (∆;
1+τ
χ2J−M ,
where R−1 is the generalized inverse of
R = PSP,
P = I − D(D0 WD)−1 D0 W.
c I converges to S−1 ,
The asymptotically efficient weighting matrix arises when W
the inverse of the variance-covariance matrix of the data. When W = S−1 , V
simplifies to (1 + τ )(D0 S−1 D)−1 , and R is replaced with S.
Even though the optimal weighting matrix is asymptotically efficient, it can be
biased in small samples. (See, for example, Altonji and Segal [2].) Hence, we use a
“diagonal” weighting matrix, as suggested by Pischke [48]. This diagonal weighting
scheme uses the inverse of the matrix that is the same as S along the diagonal and
has zeros off the diagonal of the matrix.
We estimate D, S, and W with their sample analogs. For example, our estimate
of S is the J × J estimated variance-covariance matrix of the sample data. When
53
estimating this matrix, we use sample statistics, so that apqt (∆, χ) is replaced with
the sample median for group pqt.
One complication in estimating the gradient matrix D is that the functions inside the moment condition ϕ(∆; χ) are non-differentiable at certain data points; see
equation (24). This means that we cannot consistently estimate D as the numerical
derivative of ϕ̂I (.). Our asymptotic results therefore do not follow from the standard GMM approach, but rather the approach for non-smooth functions described
in Pakes and Pollard [46], Newey and McFadden [45] (section 7), and Powell [50].
To find D, it is helpful to rewrite equation (24), for the case of medians, as:
Pr fit = f & ci = c & Ii ∈ Ip & individual i observed at t ×
#
"Z median
af cpt (∆0 ,χ0 )
1
= 0.
(27)
f ait f, c, Ii ∈ Ip , t dait −
2
−∞
It follows that the rows of D for the median asset moments are given by
Pr fit = f & ci = c & Ii ∈ Iq & individual i observed at t ×
∂amedian (∆0 ; χ0 )
f cpt
median
f af cpt
f, c, Ii ∈ Ip , t ×
.
(28)
∂∆0
In practice, we find f amedian
f cpt |f, c, p, t , the conditional p.d.f. of wealth evaluated
at the median amedian
f cpt , with a kernel density estimator written by Koning [37]. The
derivatives for the 25th and 75th percentiles are found in analogous fashion.
Appendix G: First stage estimates
In the first-step estimation procedure, we estimate or calibrate those parameters
that, given our assumptions, can be cleanly identified outside our model. In particular, we estimate health transitions and mortality rates from raw demographic data,
estimate income profiles from the AHEAD, and calibrate the returns to saving and
income taxes using external sources. We use our measure of total medical expenses
(which combines data from the AHEAD and the MCBS) to estimate the process for
medical expenses as a function of exogenous household-level state variables which
are observed in the AHEAD.
We outline the estimation and calibration of these objects next.
54
G.1
Health and mortality
Each person’s health status, hsg , has four possible values: dead; in a nursing
home; in bad health; or in good health. We allow transition probabilities for an
individual’s health depend on his or her current health, age, household composition
f , permanent income I, and gender g.17 It follows that the elements of the health
transition matrix are given by
πi,j,k (t, fit , Ii , g = Pr hsgi,t+2 = k hsgi,t = j; t, fi,t , Ii , g ,
(29)
with the transitions covering a two-year interval, as the AHEAD interviews every
other year.18 We estimate health/mortality transition probabilities by fitting the
transitions observed in the AHEAD to a multinomial logit model.19
We enumerate each possible health status as follows
0
Dead
1
Nursing Home
(30)
hs =
2
Bad Health
3
Good Health.
Our multinomial logit assumption gives the following expression for health and mortality transitions
exit βk
πi,j,k (t, fit , Ii , g = P3
,
(31)
xit βh
h=0 e
with βk denoting the coefficient vector for future outcome k. The coefficient for
death, β0 , is normalized to zero. We jointly estimate health transitions and survival
17
We do not allow health transitions to depend on medical spending. The empirical evidence on
whether medical spending improves health, especially at older ages, is surprisingly mixed; see, for
example, the discussion in De Nardi, French and Jones [19]. Likely culprits include reverse causality
– sick people have higher expenditures – and a lack of insurance variation – almost every retiree
gets Medicare.
18
As discussed in De Nardi, French and Jones [19], one can fit annual models of health and
medical spending to the AHEAD data. The process becomes significantly more involved, however,
especially when accounting for the dynamics of two-person households.
19
We do not control for cohort effects. Instead, our estimates are a combination of period (crosssectional) and cohort probabilities. While our AHEAD sample covers 18 years, it is still too short
to track a single cohort over its entire post-retirement lifespan. This may lead us to underestimate
the lifespans expected by younger cohorts as they age. Nevertheless, lifespans have increased only
modestly over the sample period. Accounting for cohort effects would have at most a modest effect
on our estimates.
55
probabilities at the individual level using a maximum likelihood estimator. We allow
the transition probabilities to depend on xit which includes age, sex, current health
status, marital status, and permanent income. In particular we include a third order age polynomial, indicators for gender and marital status (interacted with an age
quadratic), single woman (interacted with a first order age polynomial), contemporaneous indicators for health (interacted with age), and a quadratic in permanent
income (interacted with a first order polynomial in age and marital status).
Using the estimated transition probabilities, we simulate demographic histories,
beginning at age 70, for different gender-PI-health combinations. We report moments
of these histories in Section 6.1.1.
G.2
Income
Our procedure for estimating the income profiles described in section 6.1.2 is part
of the procedure for inferring permanent income described in Appendix B. We set
the annual rate of return on saving to 4%.
G.3
The tax system
We parameterize the tax system using the functional form developed by Gouveia
and Strauss [29]. Average tax rates for total pre-tax income, y, are given by
−1
t(y, τft ) = bft [1 − (sft y pft + 1) pft ],
(32)
where bft , sft and pft are parameters of the tax system that can differ between
couples and singles, and τft = bft , sft , pft is their union. We use the estimates of
τ for married and singles households with no children provided by Guner et al. [30].
After-tax income Υ(y, τft ) is then
Υ(y, ft ) = (1 − t(y, τft )) × y.
(33)
To reflect differences in the generosity of Medicaid, and other means tested social
insurance programs available to the elderly, between couples and singles we impose
that the consumption floor for couples is 150% of the value we estimate for singles.
G.4
Medical spending
Our medical spending measure is the sum of expenditures paid out-of-pocket
plus those paid by Medicaid (see Appendix D for its construction). The AHEAD’s
56
measure of medical spending is backward-looking: in each wave, the household is
asked about the expenses it incurred since the previous interview.
Let mi,t denote the expenses incurred by household i between ages t − 1 and t.
We observe household’s health at the beginning and the end of this interval, that is,
at the time of the interview conducted at age t − 1 and at the time of the interview
conducted at age t. We assume that medical expenses depend upon a household’s
PI, its family structure at both t − 1 and t, the health of its members at both dates,
and an idiosyncratic component ψi,t :
h
w
ln mi,t = m(hshi,t−1 , hsw
i,t−1 , hsi,t , hsi,t , fi,t−1 , fi,t , Ii , t) + υi,t ,
(34)
h
w
υi,t = σ(hshi,t−1 , hsw
i,t−1 , hsi,t , hsi,t , fi,t−1 , fi,t , Ii , t) × ψi,t .
(35)
We normalize the variance of ψ to 1, hence σ 2 (·) gives the conditional variance of υ.
Including current and lagged family structure indicators allows us to account for
the jump in medical spending that occurs in the period a family member dies. Likewise, including health indicators for both periods allows us to distinguish persistent
health episodes from transitory ones.
To show how we estimate m(·) and σ(·), write Equation (6) as
ln mit = x1i β1 + x2it β2 + ϑi + ςit ,
(36)
where x1i denotes a vector of time-invariant variables, x2it denotes a vector of timevarying variables, ϑi is an unobserved person-specific term, and ςit captures any
remaining variation. We assume that E(ςit | ϑi ) = 0.
We estimate Equation (36) in three steps. First, we regress log medical spending on the time-varying factors in Equation (34), namely age, household structure,
and health, and interaction terms (such as gender and PI interacted with the time
varying variables) using a fixed effects estimator. In particular we regress log medical spending on a fourth order age polynomial, indicators for single man (interacted
with an age quadratic), single woman (interacted with an age polynomial), the contemporaneous and lagged values of indicators for {man in bad health, married man
in a nursing home, single man in a nursing home, woman in bad health, married
woman in a nursing home, single woman in a nursing home}, whether the man died
(interacted with age and permanent income), whether the woman died (interacted
with age, and permanent income).
Because fixed effects regression cannot identify the effects of time-invariant factors, which are combined into the estimated fixed effects, in the second step we
collect the residuals from the first regression, inclusive of the estimated fixed effects,
and regress them on the time-invariant factors, namely a quadratic in permanent
57
income and a set of cohort dummies. The level of m(·) is set to be consistent with
the outcomes of the cohort aged 72-76 in 1996.
A key feature of our spending model is that the conditional variance and the
conditional mean of medical spending depends on demographic and socioeconomic
factors, through the function σ(·) shown in Equation (35). In the third step of our
estimation procedure, we use the coefficients for m(·) in hand, found in the previous
2 (·), we square
two steps, to back out the residual υ from Equation (6). To find σ[
2
the residuals and regress υ on the demographic and socioeconomic variables in
Equation (35).
We assume that ψi,t can be decomposed as
ψi,t = ζi,t + ξi,t ,
ξi,t ∼ N (0, σξ2 ),
ζi,t = ρm ζi,t−2 + i,t ,
i,t ∼
N (0, σ2 ),
(37)
(38)
where ξi,t and i,t are serially and mutually independent. With the variance of ψi,t
normalized to 1, σξ2 can be interpreted as the fraction of idiosyncratic variance due
to transitory shocks.
We estimate the parameters of Equations (7) and (8) using a standard error
components method. Although the estimation procedure makes no assumptions on
the distribution of the error terms ψi,t , we assume normality in the simulations. [27]
shows that if the data are carefully constructed, normality captures well the far right
tail of the medical spending distribution.20
Approximately 40% of the cross-sectional variation in log medical spending is
explained by observables, which are quite persistent. Of the remaining cross-sectional
variation, 40% comes from the persistent shock ζ and 60% from the transitory shock
ξ. In keeping with the results in Feenberg and Skinner [25], French and Jones [27]
and De Nardi, French and Jones [18], we estimate substantial persistence in the
persistent component, with ρm = 0.85.
20
To help us match the distribution of medical spending, we bottom code medical spending at
10% of average medical spending. French and Jones [27] also bottom code the data to match the far
right tail of medical spending. Because we include Medicare B payments in our medical spending
measure, which most elderly households pay, for the vast majority of households these bottom
coding decisions are not important.
58
Appendix H: The marginal propensity to consume
Consider an individual with no income and no medical expenses. In the last
period of life, this individual solves
1−ν
1
1 1−ν
c
+ βφ0
(1 + r)(a − c) + κ0
,
max
c
1−ν
1−ν
where c denotes consumption and a denotes beginning-of-period-wealth. The firstorder condition for an interior solution is
−ν
c−ν = βφ0 (1 + r) (1 + r)(a − c) + κ0 ,
Define
ϕ = [βφ0 (1 + r)]1/ν .
(39)
We can then solve for bequests, b = (1 + r)(a − c) :
1+r
b
(ϕa − κ0 ) .
= b + κ0 ⇒ b =
ϕ a−
1+r
1+r+ϕ
(40)
It immediately follows that for
intentional bequests, the marginal propensity to
b
∂
bequeath out of wealth, ∂a
, is
1+r
MP B =
ϕ
.
1+r+ϕ
(41)
With non-negative bequests, however, it follows from Equation (40) that the propensity to bequeath is identically zero unless
κ0
a>a= .
(42)
ϕ
It is widely accepted that M P B (or M P C = 1 − M P B) and a provide a better
characterization of the bequest motive than φ0 and κ0 (see, e.g., De Nardi et al.,
2010 or Lockwood, 2018). When one expects to die at a future date rather than next
period, the effective threshold where the bequest motive becomes operative will be
higher than a.
There is a one-to-one mapping between the vectors (M P B, a) and (φ0 , κ0 ), and
thus if one of these objects can be identified, the other one can be as well. Using
equations (42), we know κ0 = ϕa. It then follows from equations (39) and (41) that:
ϕ = (1 + r)
MP B
1−M
PB
ϕν
MP B
⇒ φ0 =
= (1 + r)
β(1 + r)
1 − MP B
59
ν
1
.
β(1 + r)
We also need to map parameters between models with one-year and two-year
period lengths. An important intermediate step in this process is to consider the
sequence of 1-year problems where the individual lives two periods and then dies.
We will use hats to designate parameters for this one-year model, and unmarked
b +
parameters for the two-year framework. To simplify the algebra, suppose that β(1
r̂) = 1, so that b
c is the same in the two periods of life. The present value budget
constraint is
b̂
2 + r̂
+
,
a=b
c
1 + r̂
(1 + r̂)2
which can be rewritten as
b
c=
1 + r̂
2 + r̂
b̂
a−
.
(1 + r̂)2
(43)
∂
b̂
\
If the marginal propensity to bequeath out of starting wealth is M
P B = ∂a
,
(1+r̂)2
the marginal propensity to consume on a annual basis, found by differentiating
equation (43), is:
i
1 + r̂ h
∂ĉ
\
\
=
1−M
PB .
(44)
M
PC =
∂a
2 + r̂
Next, consider the asset threshold b
a. Recall that in the final period of life, wealth
below b
a are consumed. Let e
a denote the asset threshold two periods prior to death.
By definition, when wealth two periods prior to death equal e
a, b̂ = 0. Equation (43)
then implies that the asset threshold one period prior to death is
1 + r̂
1 + r̂
e
a =
e
a.
b
a = (1 + r̂) e
a−
(45)
2 + r̂
2 + r̂
\
To complete the mapping, suppose that 1 + r = (1 + r̂)2 , M P B = M
P B, and
a=e
a. Equations (44) and (45) then imply that
1 + r̂
\
M
PC =
M P C,
2 + r̂
1 + r̂
b
a=
a.
2 + r̂
(46)
(47)
In other words, when we operate at a one-year frequency, the marginal propensity to
consume and the asset threshold are (more or less) half their two-year counterparts.
60
Appendix I: Additional model fits
Figure 17 compares the Median asset profiles generated by the model (dashed
line) to those in the data (solid line) for all of the birth cohorts and permanent
income terciles that we match in estimation. We reproduce profiles for the cohorts
reported in the main text, but also show profiles for the cohorts that were omitted.
In all cohorts, the model matches differences in the level and trajectory of wealth by
age and household permanent income.
Median Net Worth: Data (Solid) vs. Model (Dashed), Couples
550
550
500
500
450
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
Median Net Worth: Data (Solid) vs. Model (Dashed), Couples
600
450
400
350
300
250
200
150
100
350
300
250
200
150
100
50
50
0
72
400
76
80
84
88
92
96
0
72
100
76
80
84
Age
92
96
100
Median Net Worth: Data (Solid) vs. Model (Dashed), Singles
200
240
180
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
Median Net Worth: Data (Solid) vs. Model (Dashed), Singles
270
210
180
150
120
90
60
30
0
72
88
Age
160
140
120
100
80
60
40
20
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
Figure 17: Median wealth. Top panel: current couples. Bottom panel current singles:
Solid lines: cohorts aged 72-77, 84-89 in 1996 in the left-hand panel and ages 78-83
and 90+ in 1996 in the right-hand side panel. Dashed lines: model simulations.
Figure 18 plots the corresponding data for the 75th percentile of the asset distribution. Comparing the 75th wealth percentiles to the median ones towards the
beginning of the retirement period, when we start tracking our sample, shows that
61
75th %tile Net Worth: Data (Solid) vs. Model (Dashed), Couples
75th %tile Net Worth: Data (Solid) vs. Model (Dashed), Couples
1400
1100
1000
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
1200
1000
800
600
400
200
900
800
700
600
500
400
300
200
100
0
72
76
80
84
88
92
96
0
72
100
76
80
84
Age
88
92
96
100
Age
75th %tile Net Worth: Data (Solid) vs. Model (Dashed), Singles
75th %tile Net Worth: Data (Solid) vs. Model (Dashed), Singles
700
600
550
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
600
500
400
300
200
100
500
450
400
350
300
250
200
150
100
50
0
72
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
Figure 18: 75th wealth percentile. Top panel: current couples. Bottom panel current
singles: Solid lines: cohorts aged 72-77, 84-89 in 1996 in the left-hand panel and ages
78-83 and 90+ in 1996 in the right-hand side panel. Dashed lines: model simulations.
the 75th asset percentiles are about twice the medians for the two highest PI groups
and much larger than twice the median for the lowest PI tercile. In addition, the 75th
asset percentiles show even more evidence of wealth accumulation during retirement
than do median wealth, both for couples and singles. Finally, the 75th percentiles
suggest that current couples tend to save more during their retirement than singles,
even given similar initial wealth. As with the medians, the model does a good job
of matching the 75th percentile of asset holdings across PI groups and ages, both
for singles and couples. For example, it reproduces the flatter profiles of lower PI
groups, while still generating eventual wealth holdings of more than $1,000,000 for
the top terciles.
Figure 19 plots the corresponding data for the 25th percentile of the asset distribution. Comparing the 25th wealth percentiles to the medians towards the beginning
62
25th %tile Net Worth: Data (Solid) vs. Model (Dashed), Couples
25th %tile Net Worth: Data (Solid) vs. Model (Dashed), Couples
300
270
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
240
250
200
150
100
50
210
180
150
120
90
60
30
0
72
76
80
84
88
92
96
0
72
100
76
80
84
Age
25th %tile Net Worth: Data (Solid) vs. Model (Dashed), Singles
96
100
90
100
80
90
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
92
25th %tile Net Worth: Data (Solid) vs. Model (Dashed), Singles
110
80
70
60
50
40
30
20
70
60
50
40
30
20
10
10
0
72
88
Age
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
Figure 19: 25th wealth percentile. Top panel: current couples. Bottom panel current
singles: Solid lines: cohorts aged 72-77, 84-89 in 1996 in the left-hand panel and ages
78-83 and 90+ in 1996 in the right-hand side panel. Dashed lines: model simulations.
of the retirement period, when we start tracking our sample, shows that the 25th
asset percentiles are about half the median ones for the two highest PI groups, with
the exception of the lowest PI tercile for singles, where it is almost always 0. In addition, the 25th asset percentiles show even more evidence that couples and singles’
wealth evolves differently as the wealth holdings of singles decline. As with the 75th
percentile, it seems that current couples tend to save more during their retirement
than singles, even given similar initial wealth. Our model, in addition to matching
the median and the 75th percentile of asset holdings by age, PI, and cohort, also does
a good job of generating the differential deccumulation we see for the 25th percentile.
Finally, Figure 20 compares the Medicaid recipiency profiles generated by the
model (dashed line) to those in the data (solid line) for the same birth cohorts and
permanent income terciles and shows that our model matches important patterns of
63
Medicaid Recipiency: Data (Solid) vs. Model (Dashed), All
Medicaid Recipiency: Data (Solid) vs. Model (Dashed), All
0.8
0.65
0.6
0.7
0.55
0.5
Medicaid Recipiency
Medicaid Recipiency
0.6
0.5
0.4
0.3
0.2
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.1
0.05
0
72
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
(a)
(b)
Figure 20: Medicaid Recipiency. Solid lines: cohorts aged 72-77, 84-89 in 1996
in panel (a) and ages 78-83 and 90+ in 1996 in panel (b). Dashed lines: model
simulations.
Medicaid usage. We reproduce the cohorts reported in the main text and add the
cohorts not shown there. The model matches difference in the level of usage by age
and household permanent income for all cohorts. It does especially well in capturing
the run-up in Medicaid use by wealthier households at very old ages.
Because individuals who are single, poor or sick die at younger ages, in an unbalanced panel the sample composition changes over time, introducing mortality bias.
Figure 21 compares the mortality bias found in the AHEAD data (left panel) with
that predicted by the model (right panel). The solid lines show median wealth for all
households (both singles and couples) observed at a given point in time, even if all
members died in a subsequent wave, that is, the unbalanced panel. The dashed lines
show median wealth for the subsample of households with at least one member still
alive in the final wave, that is, the balanced panel. They show that the asset profiles
for households who survived to the final wave (the balanced panel) have much more
of a downward slope. The difference between the two sets of profiles confirms that
people who died during our sample period tended to have lower wealth than the
survivors.
Our model matches the extent of the mortality bias found in the data, without
being required to do so by construction. As in the data, restricting the profiles to
long-term survivors reveals much more asset decumulation.
64
Median Net Worth: Everyone in Sims (Solid) vs. Survivors (Dashed), All
400
400
350
Net Worth (000s of 2014 dollars)
Net Worth (000s of 2014 dollars)
Median Net Worth: Everyone in Data (Solid) vs. Survivors (Dashed), All
450
350
300
250
200
150
100
250
200
150
100
50
50
0
72
300
76
80
84
88
92
96
100
Age
0
72
76
80
84
88
92
96
100
Age
(a)
(b)
Figure 21: Mortality Bias. Left panel: data. Right panel: model simulations. Solid
lines: all household in the data at a given point in time. Dashed lines: households
with at least one member surviving to 2014.
65
@FedFRASER