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Economic Quarterly— Volume 104, Number 3— Third Quarter 2018— Pages 103–135
The Lifetime Medical
Spending of Retirees
John Bailey Jones, Mariacristina De Nardi, Eric French,
Rory McGee, and Justin Kirschner
D
espite nearly universal enrollment in the Medicare program,
most elderly Americans still face the risk of catastrophic health
care expenses. There are many gaps in Medicare coverage:
for example, Medicare does not pay for long hospital and nursing
home stays and requires copayments for many medical goods and services. Medical spending is thus a major …nancial concern among elderly households. In a recent survey, a- uent individuals were more
worried about rising health care costs than about any other …nancial
issue (Merrill Lynch Wealth Management 2012).
Several papers (De Nardi et al. 2010; Kopecky and Koreshkova
2014; Ameriks et al. 2015) show that health care costs that rise with
age and income explain much of the US elderly’s saving behavior.1
Di¤erences in medical spending risk are also important in explaining
cross-country di¤erences in the consumption (Banks et al. 2016) and
John Bailey Jones: Federal Reserve Bank of Richmond, jbjones.albany@gmail.com.
Mariacristina De Nardi: Federal Reserve Bank of Chicago, UCL, CEPR, and
NBER, e-mail: denardim@nber.org. Eric French: UCL, CEPR, and IFS, e-mail:
eric.french.econ@gmail.com. Justin Kirschner: Federal Reserve Bank of Richmond,
justin.kirschner@rich.frb.org. Rory McGee: UCL and IFS, rory.mcgee.13@ucl.ac.uk.
For helpful comments we thank Sara Ho, Marios Karabarbounis, Christian Matthes,
Kerry Pechter, and John Weinberg. De Nardi and French gratefully acknowledge
support from Norface Grant (TRISP 462-16-120). French gratefully acknowledges
support from the Economic and Social Research Council (Centre for Microeconomic
Analysis of Public Policy at the Institute for Fiscal Studies (RES-544-28-50001)
and from 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 Federal Reserve Bank of Chicago, the Federal Reserve
Bank of Richmond, the IFS, the NBER, or the CEPR.
1
Additional mechanisms proposed to explain the “elderly savings puzzle,” or the
slow decumulation of assets in old age, include bequests (De Nardi 2004; and Lockwood
2012) and the desire of older individuals to remain in their current homes (Nakajima
and Telyukova 2012). See De Nardi et al. (2016b) for a review.
DOI: https://doi.org/10.21144/eq1040301
104
Federal Reserve Bank of Richmond Economic Quarterly
savings decisions (Nakajima and Telyukova 2018) of elderly households.
More generally, the literature on the macroeconomic implications of
health and medical spending is growing rapidly. Recent studies have
considered important questions such as: bankruptcy (Livshits et al.
2007); the adequacy of savings at retirement (Skinner 2007; Scholz et al.
2006); annuitization (Pashchenko 2013; Lockwood 2012; Reichling and
Smetters 2015); portfolio choice (Love 2010; Hugonnier et al. 2013);
optimal taxation of health (Boerma and McGrattan 2018); and health
insurance reform (Pashchenko and Porapakkarm 2013; Jung and Tran
2016; Conesa et al. 2018).
All of the aforementioned studies rely on accurate measures of medical risk and medical spending. But even though there is a large literature documenting annual medical spending at older ages, there has
been relatively little work documenting the distribution of cumulative
lifetime spending. Yet it is in many ways lifetime totals, rather than
spending in any given year, that are most important for saving decisions
and household …nancial well-being. The canonical permanent income
hypothesis posits that forward-looking agents base their consumption
not on their current income but on the average income they expect to
receive over their lifetimes. The same logic applies to medical expenses.
Households care not only about the risk of catastrophic expenses in a
single year, but also about the risk of moderate but persistent expenses
that accumulate into catastrophic lifetime costs.
In this paper, we estimate the distribution of lifetime medical spending for retired households whose heads are 70 or older. Our focus is
out-of-pocket spending, the payments made by households themselves.
High out-of-pocket expenses, however, can leave households …nancially
indigent and reliant on Medicaid, the means-tested public insurance
program. Medicaid eligibility depends on …nancial as well as healthrelated factors (De Nardi et al. 2012). Our benchmark spending estimates therefore include payments made by Medicaid to capture all of
the medical spending risk that households potentially face. In accounting terms, our benchmark estimates measure the medical spending not
covered by Medicare or supplemental private insurance, although they
do include Medicare and supplemental private insurance premia. In
economic terms, our estimates measure the medical spending risk that
wealthier households would face and the medical spending risk that less
wealthy households would face were Medicaid not available (absent any
other changes in their insurance). We also consider an alternative measure of out-of-pocket spending that excludes Medicaid payments.
Our main dataset is the Health and Retirement Study (HRS), which
has high-quality information on out-of-pocket medical spending over
the period 1995 to 2014. Because the HRS does not have Medicaid
Jones et al.: The Lifetime Medical Spending of Retirees
105
payment data, we impute Medicaid payments using the the Medicare
Current Bene…ciary Survey (MCBS). Ideally, these data would allow us
to estimate medical spending directly by calculating discounted sums
of household spending histories. Unfortunately, even the HRS, which
has a very long panel dimension for a survey of its type, is not long
enough to track all 70-year-olds through the ends of their lives. We thus
resort to models.2 Our data allow us to estimate dynamic models of
health, mortality, and out-of-pocket medical spending. Medical spending depends on age, household composition, health, and idiosyncratic
shocks. Simulating our estimated models allows us to construct household histories, calculate discounted sums, and ultimately compute the
distribution of lifetime medical spending.
This paper uses the estimated model of De Nardi et al. (2018),
which builds on earlier analyses of the HRS data by French and Jones
(2004) and De Nardi et al. (2010, 2016a). French and Jones (2004)
show that medical spending shocks are well described by the sum of
a persistent AR(1) process and a white noise shock.3 They also …nd
that the innovations to this process can be modeled with a normal
distribution that has been adjusted to capture the risk of catastrophic
health care costs. Simulating this model, they …nd that in any given
year 0.1 percent of households receive a health cost shock with a present
value of at least $125,000 (in 1998 dollars). That paper abstracts away
from much of the variability in costs coming from demographics or
observable measures of health. De Nardi et al. (2010, 2016a) extend
the spending model to account for health and lifetime earnings, but
they consider only singles and do not control for end-of-life events (see
French et al. [2006] and Poterba et al. [2017] on the importance of
these events). The model used here addresses both shortcomings by
including couples and singles and accounting for the additional medical
expenditures incurred at the end of life.
Closely related papers include Fahle et al. (2016), who document
the HRS medical spending data in some detail, and Hurd et al. (2017),
who use the HRS to calculate the lifetime incidence and costs of nursing home services. Alemayehu and Warner (2004) construct a measure
of lifetime spending by combining data from the MCBS and the Medical Expenditure Panel Survey with detailed data for Blue Cross Blue
Shield members in Michigan. However, their estimates only distinguish gender, current age, and age of death— abstracting from health
and marital status, among other factors— and for each of these groups
only mean expenditures are estimated.
2
3
Using a parametric model also improves our ability to measure tail risks.
Feenberg and Skinner (1994) …nd a similar result. See also, Hirth et al. (2015).
106
Federal Reserve Bank of Richmond Economic Quarterly
Of particular note is Webb and Zhivan (2010), who use the HRS
to estimate the distribution of lifetime expenses at ages 65 and above.
Our paper is complementary to theirs but di¤ers along two dimensions. The …rst is methodology. While both papers rely on simulation,
our approach is to combine a three-state model of health (good, bad,
or nursing home) with a two-component idiosyncratic shock and to
control for socioeconomic status with a measure of permanent income
(PI). In contrast, Webb and Zhivan (2010) estimate a rich model of
stochastic morbidity and mortality with multiple health indicators and
assume that medical expenditures are a function of these health conditions, along with a collection of socioeconomic indicators.4 In their
framework, all of the variation in medical spending is due to variation in these controls; there are no residual shocks. In contrast, in our
framework, the idiosyncratic shocks capture any spending variation not
attributable to age, PI, health, or household composition. The second
major di¤erence between our exercise and Webb and Zhivan’s (2010)
is the spending measure. As discussed above, the HRS data exclude
expenses covered by Medicaid, which otherwise might have been paid
out of pocket. Webb and Zhivan (2010) address this issue by excluding households that receive Medicaid. But all else equal, Medicaid
bene…ciaries tend to have higher medical expenses, in part because
households that face overwhelming medical expenses are more likely to
qualify for Medicaid (De Nardi et al. 2016a,c). Our approach is to impute the missing Medicaid expenditures, using data from the MCBS,
and work with the sum of out-of-pocket and Medicaid expenditures.
To put our results in context, we also analyze the HRS out-of-pocket
spending measure. Comparing the two measures reveals the extent to
which Medicaid reduces out-of-pocket expenditures.
We …nd that lifetime medical spending during retirement is high
and uncertain. Households who turned 70 in 1992 will on average incur
$122,000 in medical spending, including Medicaid payments, over their
remaining lives. At the top tail, 5 percent of households will incur more
than $300,000 and 1 percent of households will incur over $600,000 in
medical spending inclusive of Medicaid. The level and the dispersion
of remaining lifetime spending diminishes only slowly with age. The
reason for this is that as they age, surviving individuals on average
have fewer remaining years of life, but they are also more likely to live
to extremely old age when medical spending is very high. Although
PI, initial health, and initial marital status have large e¤ects on this
4
Using data from Catalonia, Carreras et al. (2013) perform a similar analysis.
Jones et al.: The Lifetime Medical Spending of Retirees
107
spending, much of the dispersion in lifetime spending is due to events
realized in later years.
We …nd that Medicaid lowers average lifetime expenditures by 20
percent. It covers the majority of the medical costs of the poorest
households and signi…cantly reduces their risk. Medicaid also reduces
the level and volatility of medical spending for high-income households,
but to a much smaller extent.
The rest of the paper is organized as follows. In Section 1, we
discuss some key features of the datasets that we use in our analysis,
the HRS and the MCBS, and describe how we construct our measure
of medical spending. In Section 2, we introduce our model and describe
our simulation methodology. We discuss our results in Section 3 and
conclude in Section 4.
1.
DATA
The medical spending models used here were developed and estimated
as inputs for the structural savings model used in De Nardi et al.
(2018). Our description of these models and the underlying data thus
borrows heavily from the text of that project.
The HRS
We use data from the Asset and Health Dynamics Among the Oldest
Old (AHEAD) cohorts of the HRS. The AHEAD is a sample of noninstitutionalized individuals aged 70 or older in 1993. These individuals
were interviewed in late 1993/early 1994 and again in 1995/96, 1998,
2000, 2002, 2004, 2006, 2008, 2010, 2012, 2014, and 2016. We use
data for ten waves, from 1995/96 to 2014. We exclude data from the
1994 wave because medical expenses are underreported (Rohwedder et
al. 2006), and we exclude data from the 2016 wave because they are
preliminary.
We only consider retired households, de…ned as those earning less
than $3,000 in every wave. Because our demographic model allows for
household composition changes only through death, we drop households
that get married or divorced or report other marital transitions not
consistent with the model. Consistency with the demographic model
also leads us to drop households that: have large di¤erences in ages; are
same-sex couples; or have no information on the spouse. This leaves us
with 4,324 households, of whom 1,249 are initially couples and 3,075
are singles.
Households are followed until both members die; attrition for other
reasons is low. When the respondent for a household dies, in the next
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Federal Reserve Bank of Richmond Economic Quarterly
wave an “exit”interview with a knowledgeable party— usually another
family member— is conducted. This allows the HRS to collect data on
end-of-life medical conditions and expenditures (including burial costs).
Fahle et al. (2016) compare the medical spending data from the “core”
and exit interviews in some detail.
The HRS has a variety of health indicators. We assign individuals
to the nursing home state if they were in a nursing home at least 120
days since the last interview or if they spent at least sixty days in
a nursing home before the next scheduled interview and died before
that scheduled interview. We assign the remaining individuals a health
status of “good”if their self-reported health is excellent, very good, or
good and a health status of “bad”if their self-reported health is fair or
poor.
The HRS collects data on all out-of-pocket medical expenses, including private insurance premia and nursing home care. The HRS
medical spending measure is backward-looking: medical spending in
any wave is measured as total out-of-pocket expenditures over the preceding two years. It is thus not immediately obvious whether medical
spending reported in any given wave should be expressed as a function of medical conditions reported in that wave or those reported in
the prior wave. Our empirical spending model includes indicators from
both sets of dates. French et al. (2017) compare out-of-pocket medical
spending data from the HRS, MCBS, and MEPS. They …nd that the
HRS data match up well with data from the MCBS. They also …nd that
the HRS matches up well with the MEPS for items that MEPS covers
but that the HRS is more comprehensive than the MEPS in terms of
the items covered.
To control for socioeconomic status, we construct a measure of lifetime earnings or “permanent income” (PI). We …rst …nd each household’s “nonasset” income, a pension measure that includes Social Security bene…ts, de…ned bene…t pension bene…ts, veterans bene…ts, and
annuities. Because there is a roughly monotonic relationship between
lifetime earnings and these pension variables, postretirement nonasset
income is a good measure of lifetime permanent income. We then use
…xed e¤ects regression to convert nonasset income, which depends on
age and household composition as well as lifetime earnings, to a scalar
measure comparable across all households. In particular, we assume
that the log of nonasset income for household i at age t follows
ln yit =
i
+ (t; fit ) + ! it ;
(1)
where: i is a household-speci…c e¤ect; (t; fit ) is a ‡exible function of
age and family structure fit (i.e., couple, single man, or single woman);
and ! it represents measurement error. The percentile ranks of the
estimated …xed e¤ects, b i , form our measure of permanent income,
Jones et al.: The Lifetime Medical Spending of Retirees
109
Ibi . Because we study retirees, in our simulations we treat Ibi as timeinvariant.
The MCBS
While the HRS contains reasonably accurate measures of out-of-pocket
medical spending, it does not contain Medicaid payments. To circumvent this issue, we use data from the 1996-2010 waves of the MCBS.
The MCBS is a nationally representative survey of Medicare bene…ciaries. Survey responses are matched to Medicare records, and medical
expenditure data are created through a reconciliation process that combines survey information with Medicare administrative …les. MCBS
respondents are interviewed up to twelve times over a four-year period,
resulting in medical spending panels that last up to three years. We
use the same sample selection rules for the MCBS that we use for the
HRS data.
The MCBS data include information on marital status, health,
health care spending, and household income. One drawback of the
MCBS is that it does not have information on the medical spending or
health of the spouse.
Our Medical Spending Measure
Because the HRS medical spending data exclude expenses covered by
Medicaid, which otherwise might have been paid out of pocket, they
are censored. If the incidence of Medicaid were random, we could simply drop Medicaid recipients from our sample. However, this is not the
case, because Medicaid bene…ciaries tend to have higher medical expenses, in part because households that face overwhelming medical expenses are more likely to qualify for Medicaid (De Nardi et al. 2016a,c).
Our approach is to use MCBS data to impute the missing Medicaid expenditures in the HRS and to then sum observed out-of-pocket and
imputed Medicaid expenditures into a single cost measure. In addition
to removing the censoring, our measure allows us to assess the spending risk that older households would face in the absence of Medicaid.
Knowing this risk is key to assessing the e¤ects of Medicaid itself.
We proceed in two steps. First, we use the MCBS data to regress
Medicaid payments for Medicaid recipients on a set of observable variables found in both datasets. This regression has an R2 statistic of
0.67, suggesting that our predictions are fairly accurate. Second, we
impute Medicaid payments in the AHEAD data using a conditional
mean-matching procedure, a procedure very similar to hot-decking. We
combine the regression coe¢ cients with the observables in the HRS to
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Federal Reserve Bank of Richmond Economic Quarterly
predict Medicaid payments, then add to each predicted value a residual drawn from an MCBS household with a similar value of predicted
medical spending. We describe our approach in more detail in the
Appendix.
Although our principal spending measure is the sum of out-ofpocket and Medicaid payments, we also analyze out-of-pocket spending
by itself. The extent to which out-of-pocket spending is lower and/or
less volatile than combined spending directly re‡ects the extent to
which Medicaid shields households from medical expenses.
2.
THE MODEL
Our model of lifetime medical spending consists of two parts. The …rst
is a Markov Chain model of health and mortality. The second part
is the model of medical expenditure ‡ows, where medical spending
over any given interval depends on health, family structure, and the
realizations of two idiosyncratic shocks.
Health and Mortality
Let hshit and hsw
it denote the health of, respectively, the husband h and
the wife w in household i at age t. Each person’s health status, hsg ,
has four possible values: dead; in a nursing home; in bad health; or
in good health. We assume that the transition probabilities for an
individual’s health depend on his or her current health, age, household
composition, permanent income I, and gender g 2 fh; wg.5 It follows
that the elements of the health transition matrix are given by
i;j;k (t; fit ; Ii ; g
= Pr hsgi;t+2 = k hsgi;t = j; t; fi;t ; Ii ; g ;
(2)
with the transitions covering a two-year interval, as the HRS interviews
every other year.6 We estimate health/mortality transition probabilities by …tting the transitions observed in the HRS to a multinomial
logit model.7
5
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 (De Nardi et al. 2016a). Likely culprits include reverse causality—
sick people have higher expenditures— and a lack of insurance variation— almost every
retiree gets Medicare.
6
As discussed in De Nardi et al. (2016a), one can …t annual models of health and
medical spending to the HRS data. The process becomes signi…cantly more involved,
however, especially when accounting for the dynamics of two-person households.
7
We do not control for cohort e¤ects. Instead, our estimates are a combination of
period (cross-sectional) and cohort probabilities. While our HRS sample covers eighteen
years, it is still too short to track a single cohort over its entire postretirement lifespan. This may lead us to underestimate the lifespans expected by younger cohorts as
Jones et al.: The Lifetime Medical Spending of Retirees
111
Table 1 Life Expectency in Years, Conditional on Reaching
Age 70
Income
Percentile
Nursing
Home
Single Individuals
10
3.03
50
3.02
90
2.91
Married Individuals
10
2.73
50
2.77
90
2.74
Men
Bad
Health
Good
Health
Nursing
Home
Women
Bad
Health
6.92
7.78
8.11
8.68
10.29
10.94
4.07
4.05
3.80
11.29
12.29
12.51
13.18
14.86
15.37
7.83
9.39
10.39
9.82
12.18
13.50
3.95
3.99
3.88
12.10
13.74
14.59
14.05
16.27
17.28
Good
Health
Table 1 shows the life expectancies implied by our demographic
model for those still alive at age 70. The …rst panel of Table 1 shows
the life expectancies for singles under di¤erent con…gurations of gender,
PI percentile, and age-70 health. The healthy live longer than the sick,
the rich (higher PI) live longer than the poor, and women live longer
than men. For example, a single man at the 10th PI percentile in a
nursing home expects to live only 3.0 more years, while a single woman
at the 90th percentile in good health expects to live 15.4 more years.
The second panel of Table 1 shows the same results for married men
and women. Married people live longer than singles of the same health
and PI unless they are in a nursing home, in which case the di¤erences
are small.8
Table 2 shows life expectancies for married households, that is, the
average length of time that at least one member of the household is
still alive or, equivalently, the life expectancies for the oldest survivors.
While wives generally outlive husbands, a nontrivial fraction of the oldest survivors are men, and the life expectancy for a married household
is roughly two years longer than that of a married woman.
Another key statistic for our analysis is the probability that a 70year-old will spend signi…cant time (more than 120 days) in a nursing
home before he or she dies. Nursing home incidence di¤ers relatively
they age. Nevertheless, lifespans have increased only modestly over the sample period.
Accounting for cohort e¤ects would have at most a modest e¤ect on our estimates.
8
The results for couples reported in Tables 1 and 2 are based on the assumption
that the two spouses have the same health at age 70. While our model allows an individual’s health transition probabilities to depend on his or her marital status, they do
not depend on the spouse’s health. Spousal health a¤ects the life expectancy calculations only in that healthy spouses live longer.
112
Federal Reserve Bank of Richmond Economic Quarterly
Table 2 Life Expectancy of a Couple (Oldest Survivor) in
Years, Conditional on Reaching Age 70
Income
Percentile
Nursing
Home
Bad
Health
Good
Health
10
50
90
4.51
4.61
4.50
13.94
15.91
16.85
15.93
18.40
19.41
modestly across the PI distribution. Although high-income people are
less likely to be in a nursing home at any given age, they live longer,
and older individuals are much more likely to be in a nursing home.
In contrast, the e¤ects of gender are pronounced, as are the e¤ects
of marital status for men. While 37 percent of single women and 36
percent of married women alive at age 70 will enter a nursing home
before they die, the corresponding quantities for single and married men
are 26 percent and 19 percent, respectively.9 The di¤erences between
men and women are largely driven by the di¤erences in life expectancy
and marital status. Because women tend to live longer than men, they
are more likely to live long enough to enter a nursing home. Moreover,
although being married reduces the probability of entering a nursing
home, wives tend to outlive their husbands. Women who are married
at age 70 tend to be widows for several years, at which point they
face the higher probability of entering a nursing home faced by women
who are single at age 70. It is not surprising that the two groups face
similar nursing home risk. In contrast, husbands usually die before
their wives, so that men married at age 70 rarely face the high risk of
transitioning into a nursing home faced by their single counterparts.
Individuals initially in good health are 2 to 3 percentage points more
likely to spend time in a nursing home than those initially in bad health,
as nursing home risk is higher at older ages, and those initially in good
health live longer.
Because all households in the HRS are initially noninstitutionalized,
our estimates understate the fraction of individuals in nursing homes
at any age. Our simulations begin with the second wave of the AHEAD
cohort, at which point roughly 3 percent of men and 1 percent of women
in the simulations had entered nursing homes. However, the HRS does a
good job of tracking individuals as they enter in nursing homes. French
9
These …gures depend on the distribution of PI and initial health across men and
women. We construct these distributions with bootstrap draws from the second wave of
the AHEAD, using households whose heads were between 70 and 72 in the …rst wave.
Jones et al.: The Lifetime Medical Spending of Retirees
113
and Jones (2004) show that by 2000 the HRS sample matches very well
the aggregate statistics on the share of the elderly population in a
nursing home. We also understate the number of nursing home visits
because we exclude short-term visits: as Friedberg et al. (2014) and
Hurd et al. (2017) document, many nursing home stays last only a few
weeks and are associated with lower expenses. We focus only on the
longer and more expensive stays faced by households.
Medical Spending
Our preferred medical spending measure is the sum of expenditures
paid out-of-pocket plus those paid by Medicaid. Let mit denote the
expenses incurred between ages t and t + 2. We observe the household’s health at the beginning and the end of this interval, that is, at
the time of the interview conducted at age t and at the time of the
interview conducted at age t + 2. Accordingly, we assume that medical
expenses depend upon a household’s PI, its family structure at both t
and t+2, the health of its members at both dates, and the idiosyncratic
component i;t+2:
h
w
ln mit = m(Ii; t + 2; hshi;t; hsw
i;t; hsi;t+2; hsi;t+2; fi;t; fi;t+2) + @i;t+2;(3)
@i;t+2 =
w
(Ii; t + 2; hshi;t; hsi;t
; hshi;t+2; hsw
i;t+2; fi;t; fi;t+2)
i;t+2:(4)
The variance of is normalized to 1, so that 2 ( ) gives the conditional
variance of @. Including both family structure indicators allows us to
account for the jump in medical spending that occurs in the period
when a family member dies. Likewise, including health indicators from
both periods allows us to distinguish persistent health episodes from
transitory ones. Finally, we include cohort dummies in the regression.10
While we allow medical spending to depend on PI, we otherwise
treat it as exogenous. This is a common assumption, and it allows us
to calculate lifetime expenditures without solving a formal behavioral
model. It bears noting that if households expect to make large medical expenditures near the ends of their lives, they will save for these
expenses even when they are discretionary. On the other hand, the
e¤ects of policy reforms depend on the extent to which households can
10
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), both the contemporaneous and lagged values of indicators for
{man in bad health, man in a nursing home, woman in bad health, 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), a quadratic in permanent
income, and cohort dummies.
114
Federal Reserve Bank of Richmond Economic Quarterly
control their medical spending. De Nardi et al. (2010, 2016b) discuss
this issue in some detail.
We estimate m( ) in two steps. In the …rst step, we regress log medical spending on the time-varying factors in equation (3), namely age,
household structure, and health, using a …xed-e¤ects estimator. Fixed
e¤ects regression cannot identify the e¤ects of time-invariant factors,
however, as they are not identi…ed separately from the estimated …xed
e¤ects. To address this problem, we take the residuals from the …rst
regression, inclusive of the estimated …xed e¤ects, and regress them on
the time-invariant factors, namely permanent income and a set of cohort dummies. In the simulations, we use the dummy coe¢ cient for the
cohort aged 72-76 in 1996. The level of m( ) is thus set to be consistent
with the outcomes of this youngest cohort.
With the coe¢ cients for m( ) in hand, we can back out the residual
@. A key feature of our spending model is that the conditional variance
as well as the conditional mean of medical spending depends on demographic and socioeconomic factors, through the function ( ) shown in
2 ( ), we square the residuals (the @s) from equaequation (4). To …nd d
2
tion (3) and regress @ on the demographic and socioeconomic variables
in equation (4).
An accurate estimate of the lifetime medical expenditure distribution requires an accurate model of the intertemporal correlation of the
idiosyncratic shock i;t . Following Feenberg and Skinner (1994) and
French and Jones (2004), we assume that i;t can be decomposed as
i;t
i;t
=
=
i;t
+
i;t ;
m i;t 2
+
i;t
i;t ;
N (0;
i;t
2
);
N (0;
(5)
2
);
(6)
where i;t and i;t are serially and mutually independent. With the
variance of i;t normalized to 1, 2 can be interpreted as the fraction
of idiosyncratic variance due to transitory shocks. We estimate the
parameters of equations (5) and (6) using a standard error components
method. Although the estimation procedure makes no assumptions on
the the distribution of the error terms i;t , we assume normality in
the simulations. French and Jones (2004) show that if the data are
carefully constructed, normality captures well the far right tail of the
medical spending distribution.11
11
To help us match the distribution of medical spending, we bottom code medical
spending at 10 percent of average medical spending. French and Jones (2004) 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.
Jones et al.: The Lifetime Medical Spending of Retirees
115
Approximately 40 percent of the cross-sectional variation in log
medical spending is explained by the observables, which are quite persistent. Of the remaining cross-sectional variation, 40 percent comes
from the persistent shock and 60 percent from the transitory shock
. In keeping with the results in Feenberg and Skinner (1994), French
and Jones (2004), and De Nardi et al. (2010), we estimate substantial
persistence in the persistent component, with m = 0:85.
Quantitative Approach
After estimating our model, we assess its implications through a series
of Monte Carlo exercises. The simulations begin at age 72 (re‡ecting
medical spending between 70 and 72) and end at age 102.12 Each
simulated household receives a bootstrap draw of PI, initial health, and
initial marital status from the HRS data used to estimate the health
and spending models.13 The household also receives initial values of
and drawn from their unconditional distributions. We then use our
Markov Chain model of health and mortality (equation (2)) to simulate
demographic histories for each household and give each household a
sequence of idiosyncratic shocks consistent with equations (5) and (6).
Combining these inputs through equation (3) yields medical spending
histories. We generate 1 million such histories and calculate summary
statistics at each age.
3.
RESULTS
Unconditional Spending Distributions
Figure 1 shows our model’s implications for the cross-sectional distribution of our preferred medical spending measure, the sum of costs
paid either out of pocket or by Medicaid. Costs are expressed in 2014
dollars.
Figure 1a, in the upper left corner, summarizes the health care
expenditures of surviving households. Mean and median expenditures
are shown, along with the 90th, 95th, and 99th percentiles. The results
are dated by the beginning of the spending interval: the numbers for
age 72 describe the medical expenses incurred between ages 72 and 74
12
In couples, wives are assumed to be three years younger than husbands— the data
average— and are thus initially 69. Single women are assumed to be 72.
13
Although the simulations begin at age 72, we take bootstrap draws from the set
of people aged 72 to 74 in 1996. This gives us a larger pool of households to draw from,
which should in turn improve the accuracy of our exercise. We set the initial values of
lagged (age 70) health and marital status equal to their age-72 values, allowing us to
calculate medical expenditures made between the ages of 70 and 72.
116
Federal Reserve Bank of Richmond Economic Quarterly
Figure 1 Unconditional Distribution of Annual and Lifetime
Medical Expenditures. Figures Show Mean, 50th,
90th, 95th, and 99th Percentiles of the Distribution
by people alive at both dates. Expenditures are expressed in annual
terms. The medical expenses of surviving households rise rapidly with
age. For example, mean medical spending rises from $5,100 per year at
Jones et al.: The Lifetime Medical Spending of Retirees
117
age 70 to $29,700 at age 100. The upper tail rises even more rapidly,
with the 95th percentile increasing from $13,400 to $111,200.14
Figure 1b shows end-of-life costs, which include burial expenses;
the results for age 72 describe the expenses incurred by households
who die between ages 72 and 74.15 On average, end-of-life medical
expenses exceed those of survivors. Mean end-of-life expenses range
from $11,000 at age 72 to $34,000 at age 100.
Figure 1c plots our main variable of interest, lifetime expenditures.
At each age, we calculate the present discounted value of remaining
medical expenditures from that age forward, using an annual real discount rate of 3 percent. These lifetime totals are considerable. At age
70, households will, on average, incur over $122,000 of medical expenditures over the remainder of their lives. The top 5 and 1 percent of
spenders will incur spending in excess of $330,000 and $640,000, respectively. One might expect the lifetime totals to fall rapidly as households
age and near the ends of their lives. This is not the case. A household
alive at age 90 will on average spend more than $113,000 before they
die. The 95th percentile of remaining lifetime spending is higher at age
90 than at age 70. The slow decline of lifetime costs is due mostly to
the tendency of medical costs to rise with age. Households that live to
older ages have shorter remaining lives but higher annual expenditure
rates.
A number of papers have considered whether medical expenses rise
with age generically or mostly because older people are more likely to
incur end-of-life expenses: see the discussion in De Nardi et al. (2016c).
In our spending model, both forces are present. The top row of Figure 1
shows that there is considerable age growth in the medical expenses of
both survivors and the newly deceased. Nonetheless, except for the
99th percentile at the oldest ages, the end-of-life expenses shown in
Figure 1b are larger than the expenses faced by survivors of the same
age (Figure 1a).
Figure 1d presents the annuitized spending associated with these
lifetime totals. For each household, we convert lifetime expenses into
the constant spending ‡ow that would, over that household’s realized
lifespan, have the same present value. The average annuity payment
rises from $9,100 at age 70 to $31,800 at age 100. Comparing Figures
14
In general, our estimated model matches well the distribution of medical spending found in the raw data. However, the model overstates the 99th percentile of the
medical spending distribution after age 90. Given the low probability of having medical spending in the 99th percentile, along with the low probability of living much past
age 90, this discrepancy should have only a modest impact on our estimated lifetime
spending distribution.
15
We de…ne a dead household as one that has no members. Couples who become
singles are classi…ed as survivors.
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Federal Reserve Bank of Richmond Economic Quarterly
1a and 1d shows that mean annuitized spending at age 70 is almost
double mean current spending. This re‡ects the rapid growth in medical spending that occurs as households age. The 95th and 99th percentiles of the annuitized medical spending distribution are also higher
than the corresponding percentiles of the current spending distribution. One might think that those who have high medical spending
in the present will usually have lower medical spending in the future,
leading annuitized spending, which is essentially an average, to be less
dispersed than current spending. The wide variation in annuitized
spending found in Figure 1d thus shows that medical spending is persistent and that those with high spending in the present are likely to
have high spending in the future.
Lifetime Medical Spending Determinants and
Medical Spending Risk
The graphs presented in Figure 1 show that the medical costs of older
households are high, rising with age, and widely dispersed. A signi…cant
portion of this variation, however, is due to factors that are known
to the household (PI, health, marital status, the persistent shock ).
The spending distributions that individual households actually face,
conditional on what they know at any point in time, can be quite
di¤erent.
Figures 2 and 3 compare the mean, 90th, 95th, and 99th percentiles
of lifetime medical spending at age 70 for di¤erent values of PI and initial health and marital status. Figure 2 shows results for households at
the very bottom of the income distribution (PI = 0). Lifetime spending varies greatly across the distribution of initial health and marital
status. Some trends are apparent:
1. Women have higher lifetime medical expenditures than men.
2. People initially in good health have higher lifetime expenditures
than those initially in bad health. This re‡ects their longer life
expectancies, combined with the tendency of medical costs to
rise with age.
3. Households initially in nursing homes have the highest lifetime
expenditures in spite of their high mortality. Nursing home care
is expensive, and most people— more than 70 percent of men and
60 percent of women— outside a nursing home at age 70 never
have an extended nursing home visit.
Figure 3 shows results for households at the very top of the income distribution (PI = 1). Households at the top of the income
Jones et al.: The Lifetime Medical Spending of Retirees
119
Figure 2 Distributions of Lifetime Medical Expenditures by
Initial Health and Household Structure, PI = 0
distribution spend considerably more than those at the bottom, often
well in excess of 50 percent more. By way of example, consider 70-yearold couples where both members are initially in good health. With a PI
rank of 0, these couples would, on average, spend $104,000 over their
120
Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Distributions of Lifetime Medical Expenditures by
Initial Health and Household Structure, PI = 1
remaining lives. With a PI rank of 1, they would spend over $165,000.
Households with higher income may have higher lifetime expenditures
because they live longer or because they have higher expenses at any
given age. Figure 4, which compares the same two groups in more
Jones et al.: The Lifetime Medical Spending of Retirees
121
Figure 4 Annual and Lifetime Medical Expenses of Couples
in Initial Good Health, with PI Ranks of 0 (Left
Column) and 1 (Right Column)
detail, shows that both e¤ects are present. The top two panels of this
…gure compare annual expenditures for surviving individuals. While
high-income households typically spend more each year, at earlier ages
and higher percentiles the opposite is often true.
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Federal Reserve Bank of Richmond Economic Quarterly
Figures 2-3 show that a signi…cant part of the dispersion in retiree
medical spending can be attributed to health and demographic factors
known at the very beginning of retirement. On the other hand, Figure
4 shows that spending remains dispersed even after conditioning on
these factors. For example, the gaps between the conditional means
and 99th percentiles of lifetime spending shown in Figures 4c and 4d are
of roughly the same size as the unconditional gap shown in Figure 1c.
Another potentially predictable source of spending variation is the
persistent idiosyncratic component of medical spending, . The importance of the initial idiosyncratic shocks can be seen in Figure 5. The
two panels in the left-hand column of this …gure are directly comparable to the corresponding columns in Figure 4; the only di¤erence is that
the results in the new graphs are generated using a permanent income
rank of 0.5. The two panels in the right-hand column di¤er from those
on the left in that all the simulated histories begin with = = 0. This
can be seen in Figure 5b, where the distribution of annual expenses is
initially degenerate. Comparing the panels in the top row shows that
the e¤ects of the shocks last for several years. On the other hand,
the bottom panels show that eliminating the initial spending shocks
has a relatively small e¤ect on the dispersion of lifetime expenditures.
Knowing the initial idiosyncratic shocks removes little risk.
There are three reasons why the e¤ects of the initial shocks wear o¤.
First, as we document above, a signi…cant portion of the variation in
annual medical spending is driven by the health status of the household.
Second, the transitory component accounts for a large fraction of the
residual variation, and imposing an initial condition has no e¤ect on
future transitory shocks. Finally, the e¤ect of the initial realization
of the persistent component declines with age, as the persistence
parameter m is less than 1.
The previous results notwithstanding, a large part of the elderly’s
medical spending uncertainty is due to the idiosyncratic shocks. Recall
that only about 40 percent of the cross-sectional variation in log medical spending is explained by the observables. Likewise, if we remove all
of the idiosyncratic shocks in our simulations, so that the only uncertainty is health and household structure, the unconditional variation of
lifetime medical spending is only a fraction of its original value.
Out-of-Pocket Medical Spending
Our baseline measure of medical spending is the sum of payments
made out of pocket and Medicaid. A number of recent papers have
argued that Medicaid signi…cantly reduces the out-of-pocket spending
risk faced by older households. Brown and Finkelstein (2008) con-
Jones et al.: The Lifetime Medical Spending of Retirees
123
Figure 5 Annual and Lifetime Medical Expenses of Couples
in Initial Good Health, with (Left Column) and
without (Right Column) Initial Idiosyncratic
Spending Shocks
clude that Medicaid crowds out private long-term care insurance for
about two-thirds of the wealth distribution. De Nardi et al. (2016a)
…nd that most single retirees, including those at the top of the income
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Federal Reserve Bank of Richmond Economic Quarterly
distribution, value Medicaid at more than its actuarial cost. While
both of these papers model Medicaid formally, as part of the budget
set in a dynamic structural model, it is also useful to assess the program in a less structured way. In particular, repeating our Monte Carlo
exercises with the HRS out-of-pocket measure, which excludes Medicaid, allows us to compare the pre- and post-Medicaid distribution of
medical spending.
Figure 6 compares unconditional distributions. The two panels in
the left-hand column of this …gure show results for our baseline spending measure; the panels in the right-hand column show results for outof-pocket spending alone. The …rst row of Figure 6 compares annual
expenditures for survivors. At age 70, mean out-of-pocket expenditures
($4,200) are about 18 percent less than mean combined expenditures
($5,100). In other words, Medicaid covers about 18 percent of the
total for 70-year-olds. However, at older ages and higher spending percentiles, out-of-pocket expenditures are considerably lower. The second
row of Figure 6 shows lifetime expenditures. At age 70, mean lifetime
out-of-pocket expenses are about 20 percent lower than mean combined
expenditures. This di¤erence may seem small given the di¤erences in
the …rst row, but end-of-life expenditures (not shown) are fairly similar
across the two spending measures.
Because Medicaid is means-tested, it is most prevalent at the bottom of the income distribution. To show this more clearly, Figure 7
compares the annual spending of surviving households at di¤erent points
of the PI distribution. Consistent with Figures 4 and 5, we look at couples where both spouses were initially in good health. The top row of
Figure 7, which compares the two spending measures for households at
the bottom of the PI distribution, shows that Medicaid picks up a large
share of these households’medical expenditures. At age 70, mean outof-pocket expenditures are about 45 percent lower than mean combined
expenditures, meaning that Medicaid constitutes about 45 percent of
the total. The share of costs covered by Medicaid rises rapidly with age,
however, to around 85 percent. The bottom row of Figure 7 repeats the
comparison for the top of the PI distribution. Not surprisingly, Medicaid covers a much smaller fraction of these households’expenditures.
Figure 8 compares lifetime spending totals. The top row of this
…gure shows that at the bottom of the income distribution, Medicaid
covers 57 percent of lifetime costs as of age 70. At older ages and
higher percentiles, it covers even more. The bottom row shows results
for households at the top of the income distribution. Medicaid covers
21 percent of lifetime costs at age 70, with the fraction rising to nearly
30 percent at age 100. While most high-income households do not
receive Medicaid, those that do receive it qualify under the Medically
Jones et al.: The Lifetime Medical Spending of Retirees
125
Figure 6 Unconditional Distribution of Annual and Lifetime
Medical Expenditures, with (Left Panels) and
without (Right Panels) Medicaid Payments
Needy provision, which assists households whose …nancial resources
have been exhausted by medical expenses. Such households tend to
have high medical expenses and tend to receive large Medicaid bene…ts
(De Nardi et al. 2016a).
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 7 Annual Medical Expenses of Couples in Initial
Good Health, with (Left Panels) and without
(Right Panels) Medicaid Payments
4.
DISCUSSION AND CONCLUSIONS
In this paper, we use the health and spending models developed in
De Nardi et al. (2018) to simulate the distribution of lifetime medical
expenditures as of age 70, adding to the handful of studies on this topic.
Jones et al.: The Lifetime Medical Spending of Retirees
127
Figure 8 Lifetime Medical Expenses of Couples in Initial
Good Health, with and without Medicaid Payments
We also assess the importance of Medicaid in reducing the lifetime
medical spending risk. The simulations show that lifetime medical
spending is high and uncertain and that the level and the dispersion
of this spending diminish only slowly with age. Although PI, initial
health, and initial marital status have large and predictable e¤ects,
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Federal Reserve Bank of Richmond Economic Quarterly
much of the dispersion in lifetime spending is due to events realized at
older ages. The poorest households have the majority of their medical
costs covered by Medicaid, which signi…cantly reduces their spending
volatility as well. Medicaid also reduces the level and volatility of
medical spending for high-income households, albeit to a much smaller
degree.
The paper closest to ours is Webb and Zhivan (2010), which we
discussed in our introduction. Webb and Zhivan (2010) also …nd that
lifetime out-of-pocket medical spending is high and widely dispersed
and that the level and conditional dispersion of this spending diminish
only slowly as households age. The levels of their estimated expenses,
however, are even higher than ours, even though their spending measure excludes Medicaid. For instance, they …nd that 65-year-old couples
with high school degrees and no chronic diseases will on average spend
about $300,000 over their remaining lives, and 5 percent of these households will spend well over $600,000.16 We …nd that 70-year-old couples
with a PI rank of 0.5 and good initial health will on average spend
about $150,000 over their remaining lives and that 5 percent of these
households will spend in excess of $380,000.
One likely reason why Webb and Zhivan (2010) …nd higher medical
spending is that they estimate their model using a pooled cross-section
regression. They then correct for cohort bias— the fact that, for instance, the medical spending of a 90-year-old observed in 1996 is likely
to be lower than the medical spending a 70-year-old observed in 1996
would face when she turned 90 in 2016— by allowing their simulated
medical expenses to grow over time, independent of age, at long-term
historical rates. In contrast, we estimate our spending model using
a …xed e¤ects regression with no time controls, so that our age e¤ects
measure the year-to-year spending growth that households realized over
the sample period. Because medical spending growth has been fairly
slow in recent years— and out-of-pocket spending was reduced by the
introduction of Medicare Part D in 2006— Webb and Zhivan’s (2010)
assumed growth rates likely exceed recent experience.17 A second, related, reason is that Webb and Zhivan’s (2010) estimates are for the
cohort turning 65 in 2009, while our results are for the cohort that
16
We in‡ate their results from 2009 to 2014 dollars— roughly 10 percent— using the
CPI.
17
Webb and Zhivan (2010) assume that the real growth rate of per capita health
costs exclusive of long-term care follows a stochastic process with a mean of 4.2 percent
per year, consistent with the data for 1960-2007. They assume that long-term care costs
grow by 1.1 percent per year. According to the National Health Expenditure Accounts
(Center for Medicare and Medicaid Services 2018), between 2002 and 2012, per capita
personal health care spending for those 65 and older grew at a real (CPI-de‡ated) rate
of 0.94 percent per year.
Jones et al.: The Lifetime Medical Spending of Retirees
129
turned 70 in 1992. This represents a more than twenty-year gap in
birth dates, during which time medical spending rose at every age.
We conclude by pointing out some caveats to our analysis. We
assume, as do many other empirical papers, that medical spending is
exogenous, while in reality it is a choice variable. Although the demand for some medical goods and services is extremely inelastic, the
demand for others might be elastic. Nursing home care, for example,
is a bundle of medical and nonmedical commodities, and the latter can
vary greatly in quality, with the choice between a single and a shared
room being just one example. It is also worth noting that our analysis
excludes payments made by Medicare and private insurers. Medicare
substantially reduces out-of-pocket medical expenses throughout the
retiree population (Barcellos and Jacobson 2015). While the combination of out-of-pocket and Medicaid expenditures considered here may
be su¢ cient for some analyses, such as studies of household saving,
other analyses require that all health costs be accounted for. Extending our exercise to include all medical expenditures would be useful,
and we leave it to future research.
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Federal Reserve Bank of Richmond Economic Quarterly
APPENDIX:
IMPUTING MEDICAID EXPENDITURES
Let i index individuals in the HRS. De…ne oopit as out-of-pocket medical expenses, M edit as Medicaid payments, and mit as the sum of outof-pocket and Medicaid payments that we wish to plug in the model.
To impute M edit , which is missing in the HRS, we follow David et
al. (1986) and French and Jones (2011) and use a predictive meanmatching regression approach. There are two steps to our procedure.
First, we use the MCBS data to regress Medicaid payments (for Medicaid recipients) on observable variables that exist in both datasets. Second, we impute Medicaid payments in the HRS data using a conditional
mean-matching procedure, a procedure very similar to hot-decking.
First Step Estimation Procedure
Let j index individuals in the MCBS. For the subsample of the MCBS
with a positive Medicaid indicator (i.e., a Medicaid recipient), we regress
the variable of interest, M edjt , on the vector of observable variables zjt ,
yielding M edjt = zjt + "jt . We include in zjt nursing home status,
the number of nights spent in a nursing home, a fourth-order age polynomial, total household income, marital status, self-reported health,
race, visiting a medical practitioner (doctor, hospital, or dentist), outof-pocket medical spending, education, and death of an individual. Because the measure of medical spending in the HRS is medical spending
over two years, we take two-year averages of the MCBS data to be
consistent with the structure of the HRS. The regression of M edjt on
zjt yields a R2 statistic of 0.67, suggesting that our predictions are
accurate.
Next, for every observation in the MCBS subsample we calculate
d
the predicted value M
edjt = zjt b and the residual ^"jt = M edjt
d
M
edjt . We then sort the observations into deciles by predicted values,
d
fM
edjt gj;t , keeping track of the residuals, f^"jt gj;t , as well.
Second Step Estimation Procedure
For every observation in the HRS sample with a positive Medicaid
d
indicator, we impute M
edit = zit b , using the values of b estimated from
the MCBS. Then, we impute "it for each observation of this subsample
d
by …nding a random observation in the MCBS with a value of M
edjt
Jones et al.: The Lifetime Medical Spending of Retirees
131
d
in the same decile as M
edit , and setting ^"it = ^"jt . The imputed value
d
of M edit is M edit + ^"it .
As David et al. (1986) 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 hotdecking are twofold. First, many of the “z” variables are continuous.
Second, to improve goodness of …t we use a large number of “z” variables. Because hot-decking uses a full set of interactions, this would
result in a large number of hot-decking cells relative to our sample size.
132
Federal Reserve Bank of Richmond Economic Quarterly
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Rohwedder, Susann, Steven J. Haider, and Michael D. Hurd. 2006.
“Increases in Wealth Among the Elderly in the Early 1990s: How
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“Are Americans Saving ‘Optimally’for Retirement?” Journal of
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Economic Quarterly— Volume 104, Number 3— Third Quarter 2018— Pages 137–151
Ine ciency in a Simple
Model of Production and
Bilateral Trade
Zachary Bethune, Bruno Sultanum, and Nicholas Trachter
M
any assets, both real and …nancial, are traded in secondary
over-the-counter (OTC) markets after their initial issuance
(e.g., real estate, municipal bonds, Treasuries, asset-backed
securities, etc.). The apparent importance of OTC markets has prompted
an explosion of research on the topic. There is a large literature studying OTC markets with no ex-ante investment (see, Du¢ e et al. 2005,
Du¢ e et al. 2007, Lagos and Rocheteau 2009, and Hugonnier et al.
2014, among others). A new and growing literature studies how the
market structure a¤ects ex-ante decisions. In Farboodi et al. (2017),
the market structure interacts with the investors’ incentives to invest
in negotiation skills (which a¤ects bargaining power). In Farboodi et
al. (2016), the market structure interacts with the incentives to invest
on the speed at which an investor will meet others in the secondary
market. In Bethune et al. (2017), the market structure a¤ects the incentives to produce new assets and to introduce them into the market.
In these last three papers, the equilibrium is constrained-ine¢ cient.
Our goal in this paper is to study a source of ine¢ ciency that stems
from these models in a much simpler setup. To that end, we will work
with a simpli…ed version of Bethune et al. (2017).
We study a simple model of production and trade in OTC markets. The model is meant to capture some important features of these
markets. First, as in the seminal paper Du¢ e et al. (2005), trade is
resolved in decentralized markets, where the buyer and seller in a trade
The views expressed herein are those of the authors and do not necessarily re‡ect those of the Federal Reserve Bank of Richmond or the Federal Reserve System. We thank the editor and the three referees for useful comments. Email:
nicholas.trachter@rich.frb.org.
138
Federal Reserve Bank of Richmond Economic Quarterly
meeting bargain over the terms of trade. Second, agents face uncertainty over whether their potential trade counterparty wants to or can
trade with her. Third, as in Bethune et al. (2017), we endogenize the
asset level by allowing agents to produce assets at some cost. Unlike
these papers, and a large part of the literature on the topic, we do not
include search frictions in the model. Instead, as in Babus and Kondor
(2013) or Gofman (2014), we model the market with no search frictions.
The model is static. The economy is populated by two agents who
produce and trade assets. The two agents di¤er in the level at which
they value assets: one agent’s asset valuation is l , while the other
agent’s asset valuation is h , with h > l . At the beginning of the only
period in the model, agents are faced with an opportunity to produce an
asset, but the production cost is random. Then, agents meet and trade.
When trade happens, the trade surplus is divided among the trade
participants using Nash bargaining, where denotes the bargaining
power of the buyer. In the economy, the low-valuation agent acts as an
intermediary: she produces the asset with the intention of o- oading
it during the meeting with the high-valuation agent. Thus, from now
on, we refer to this agent as the intermediary. The high-valuation
agent intends to obtain an asset. She can do so by either producing
it herself or by enjoying the intermediation services provided by the
low-valuation agent. We refer to this agent as the investor. We solve
for the unique equilibrium in this economy, and we study its e¢ ciency
properties. The model is described and analyzed in Section 1.
We …nd that the equilibrium is never constrained-e¢ cient in the
sense that there is no way to split the surplus between the intermediary
and the investor, i.e., no choice of 2 [0; 1], that makes the equilibrium e¢ cient. This result is consistent with Bethune et al. (2017),
but it is di¤erent from what is found in many search models (for example, in the seminal work of Lagos and Wright [2005]) where there
is always a such that the equilibrium is constrained-e¢ cient. The
fundamental di¤erence lies in the fact that in these other environments
there is a one-sided hold-up problem, while in ours— as in most of
the recent literature on OTC markets— there is a double-sided holdup problem.1 The double-sided hold-up problem arises because both
parties in a trade meeting have to make a costly investment. In order
to have trade between the intermediary and investor, the intermediary
has to produce the asset— pay a cost and gain the option of selling the
asset, and thus she will be held up by the investor in the trade meeting.
Likewise, for trade to happen, the investor has to refrain herself from
1
Recent examples are Farboodi et al. (2016) and Farboodi et al. (2017).
Bethune, Sultanum, Trachter: Ine ciency in Simple Model of Production & Bilateral Trade139
producing the asset— to arrive at the trade meeting empty handed, she
must give up the option of producing an asset, and thus she will also
be held up by the intermediary in the trade meeting. The …rst hold-up
problem could be solved by assigning all the bargaining power to the
intermediary, i.e., = 0 (given that the intermediary acts as the seller
in the trade meeting). The second hold-up problem could be solved by
assigning all the bargaining power to the investor, i.e., = 1 (given
that the investor acts as the buyer in the trade meeting). Both cannot
be handled together. Thus, the equilibrium is ine¢ cient. We study
e¢ ciency in Section 2.
Interestingly, the ine¢ ciency implies that there can be over- or
underinvestment in asset production, which translates into an over- or
undersupply of assets in the economy. When is low, the ine¢ ciency
implies that there is overinvestment and thus an excess supply of assets.
When is high, the ine¢ ciency implies that there is underinvestment
and thus an ine¢ ciently low asset supply. We study the e¤ect of the
ine¢ ciency on the asset supply in detail in Section 3.
Our result in this paper is the same result found in Bethune et
al. (2017). In that paper, we study a dynamic model of OTC trade
with production, where meetings among potential traders are subject
to search frictions, as in Du¢ e et al. (2005) or Hugonnier et al. (2014).
In this paper, we show that the ine¢ ciency result does not stem from
search frictions and its potential ine¢ ciency related to the Hosios condition (Hosios 1990). In fact, there is no search in our model here. We
also show that the ine¢ ciency result vanishes when both intermediary
and investor can perfectly forecast the actions of their trade counterparties through the equilibrium. We show this by studying the case
where the upper bound of the production cost equals h , which can
be interpreted as the limit as the upper bound of the cost distribution
approaches h . It happens that, in this limit, the high-valuation agent
invests for any and for any draw of production cost. This is perfectly
forecastable by the intermediary, who therefore does not produce. We
study this in Section 4. Finally, Section 5 concludes.
1.
MODEL
We study a simple two-agent, one-period model of asset trading. Agents
di¤er in their valuation of the asset. One agent values the asset at level
l , while the other agent values the asset at level h , where h > l .
Agents can hold either zero or one unit of the asset. At the beginning
of the period, both agents are confronted with an opportunity to invest
in producing an asset. The investment cost c is random, drawn from
the cumulative distribution G(c). This distribution is assumed to be
140
Federal Reserve Bank of Richmond Economic Quarterly
uniform in the support [c; c] with g(c) @G(c)=@c. To keep things as
easy as possible, we restrict attention to the case where 0 = c = l <
h < c = 1. The restriction implies that (i) agents with a low valuation
of the asset only produce with the intention to sell and thus play the
role of intermediaries, and (ii) because h < c, we can abstract from
potential kinks and discontinuities when analyzing optimal behavior.2
Given (i), we refer to agents with low valuation as intermediaries and
to agents with high valuation as investors. Finally, once production
decisions are made, agents meet and trade. We restrict them to using
a Nash bargaining protocol to resolve trade. We let 2 [0; 1] denote
the bargaining power of the buyer. We later argue that the investor
plays the role of the buyer in the trade meeting, while the intermediary
plays the role of the seller in the trade meeting. Thus, we can think of
as the bargaining power of the investor.
We can solve this model using a backward-induction argument.
First, conditional on an asset distribution among agents, we solve the
problem where the two agents meet and trade. Then, with this solution
at hand, we solve the investment stage of the model.
Trade. When agents meet and trade, the outcomes depend on
who enters this stage holding an asset. When both investors are either
holding or not holding an asset, there is trivially no trade. If the
investor enters the trade stage holding an asset, there is no trade. This
follows because the maximum willingness to pay for the asset by a
intermediary is l , which is lower than the cost of giving up the asset for
the investor, which is h . As a result, there is no positive surplus to be
split by the two parties involved in trade. The only case where there will
be trade is when the intermediary is holding an asset and the investor is
not. Because h > l , the total surplus of trading is positive, and thus
trade will happen. However, the way the surplus is split among the
two traders depends on the way bargaining power is assigned to them.
Let p denote the transfer, or price, that the seller— the intermediary—
requires to transfer the asset to the buyer— the investor. The optimal
price p solves
max(p
p
1
l)
(
h
p) :
(1)
Notice that when agents trade: (i) the intermediary receives p in exchange for giving up the asset, which she values at l , and (ii) the
investor receives the asset, which she values at h , and pays p. Operat2
Because c > h , there is always a draw for the production cost for which neither
agent wants to produce. On the ‡ip side, because c = 0 < l , there is aways a draw
for the production cost for which both agents want to produce. These two observations,
coupled with full support for the production cost in [c; c], guarantees that the problem
is smooth.
Bethune, Sultanum, Trachter: Ine ciency in Simple Model of Production & Bilateral Trade141
ing with the …rst-order condition of the problem presented in equation
(1) provides an expression for the price,
p=
l
+ (1
)
h
;
(2)
which is simply a weighted average of the asset valuation of the seller
and buyer. When the full bargaining power is assigned to the intermediary, i.e. = 0, we obtain that p = h . That is, the intermediary
obtains all the surplus of the trade. Likewise, when the full bargaining
power is assigned to the investor, i.e.
= 1, we obtain that p = l .
That is, the investor obtains all the surplus of the trade. In fact, given
that @p=@ = ( h
l ) < 0, the price decreases as the bargaining
power of the investor increases, implying that a higher fraction of the
total surplus of trade is captured by the investor.
Investment. Given the solution to the trade stage, we can now
solve the problem of both the intermediary and investor being faced
with a production opportunity. They both have to decide for which
set of production costs c they are willing to produce. Notice that once
investors reach the trade stage, the investment cost is a sunk cost,
thus it does not directly a¤ect the way the surplus is divided by the
trade meeting participants. It is also straightforward to argue that the
value of producing an asset for either the intermediary or the investor
is higher the lower the cost actually paid to produce. This implies that
both agents should produce the asset when they draw low costs and
potentially neither produces the asset for high costs. Let cl denote the
threshold such that the intermediary produces if she draws a cost c
below cl . Likewise, let ch denote the threshold such that the investor
produces if she draws a cost c below ch . Characterizing these two
thresholds involves striking a balance between the bene…ts of investing
and not investing.
Let Ul (ijch ) denote the utility of producing the asset for the intermediary when the investor’s production threshold is given by ch .
Likewise, let Ul (nijch ) denote the utility of not producing the asset for
the intermediary when the investor’s production threshold is given by
ch . A similar logic applies for describing Uh (ijcl ) and Uh (nijcl ).
When the intermediary does not produce, we trivially obtain Ul (nijch ) =
0. When the intermediary produces at cost c, we get that
Ul (ijch ) = G(ch ) l + [1 G(ch )]p c =
[1 G(ch )] ( h
c;
l)
h
G(ch )(
h
l)
where the last expression follows by using equation (2). With probability G(ch ), the investor produces an asset, and thus the intermediary
keeps the asset for herself, providing her l utiles. With probability
[1 G(ch )], the investor does not produce, which implies that trade
will occur at the trading stage, and the intermediary obtains p.
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Federal Reserve Bank of Richmond Economic Quarterly
The threshold cl is such that if c = cl , we get Ul (nijch ) = Ul (ijch ).
Using the expressions above and solving for cl provides
cl =
G(ch )(
h
l)
h
[1
G(ch )] (
l)
h
:
(3)
Now we solve the problem of an investor. If the investor produces
the asset, she gains the asset and does not trade later on. Thus,
Uh (ijcl ) = h c. If she does not produce the asset, her utility is
given by
Uh (nijcl ) = G(cl )[
h
p] = G(cl ) [
l]
h
;
where the last expression follows by using equation (2). The investor
only gets utility from not investing when the intermediary invests,
which occurs with probability G(cl ). In this event, her gains from
trade are h p, or a fraction of the total trade surplus h
l . As
before, the threshold ch makes the investor indi¤erent. Then, we have
that
ch =
G(cl ) [
h
l]
h
:
(4)
From equations (3) and (4), we can compute the following expressions,
@cl
=
@ch
g(ch )(1
)(
h
l)
<0 ;
@ch
=
@cl
g(cl ) (
h
l)
<0:
The …rst expression implies that if the intermediary expects the investor
to be more likely to produce, then she will be less willing to produce
herself. This occurs because the only reason the intermediary produces
an asset is to attempt to sell the asset to the investor. If the probability
of selling the asset falls (because the investor is more likely to hold
an asset), there are lower incentives for the intermediary to produce.
Likewise, because of the same logic, the second expression implies that
the investor is less likely to produce if the intermediary is more likely to
do it. That is, for both agents, their actions are strategic substitutes.
Equilibrium. An equilibrium is a pair of thresholds cl and ch that
solve equations (3) and (4). Solving these two thresholds provides
cl =
[1
1
h ] h (1
(1
)
)
2
h
2 [0;
h)
; ch =
[1
1
(1
(1
)
)
h] h
2
h
2 [0;
h]
:
Obviously, the fact that we were able to obtain closed-form expressions
for both thresholds implies that the equilibrium exists and that it is
unique. Moreover, the fact that the thresholds are interior and below
< 1. The thresholds are
h follows from the fact that h < 1 and
not above h , as there is always a pro…table deviation that increases
pro…ts. For example, if ch > h , the investor could trivially increase
her expected utility by setting cl = h . The fact that the threshold cl
Bethune, Sultanum, Trachter: Ine ciency in Simple Model of Production & Bilateral Trade143
is strictly below h follows from the fact that cl < ch . This follows, as
there is a positive probability that the intermediary cannot sell the asset
to the investor; because the intermediary values the asset at l < h ,
it follows that cl < ch .
2.
EFFICIENCY
The constrained-e¢ cient allocation is a set of thresholds cel and ceh that
solves the following problem,
Z ce Z ce
l
h
[ l + h c~l c~h ]dG(~
ch )dG(~
cl )
max
e e
cl ;ch
c
c
+ [1
G(cel )]
Z
ceh
[
G(ceh )]
c~h ]dG(~
ch ) + [1
h
c
Z
cel
[
h
c~l ]dG(~
cl ) ;
c
where the problem already assumes that if the intermediary produces
an asset and the investor does not, the e¢ cient outcome is for the
intermediary to transfer the asset to the investor. There are three terms
in the welfare expression. The …rst term accounts for the case where
both agents draw a cost below their respective production thresholds
and produce. Because both produce, both derive utility at the end of
the period from holding the asset and both pay the production cost.
The second term accounts for the case where the intermediary does
not produce and the investor does produce. In this case, there is no
trade and the investor keeps the asset she produces, obtaining utility
h minus the production cost incurred. The third term accounts for
the case where only the intermediary produces. In this case, she pays
the production cost and transfers the asset to the investor, who values
the asset at h .
The …rst-order condition with respect to cel is given by
Z ce
Z ce
h
h
e
e
[ l + h cl c~h ]dG(~
ch )
[ h c~h ]dG(~
ch ) + [1 G(ceh )](
g(cl )
c
c
Operating with this expression provides
cel =
h
G(ceh )(
l)
h
:
(5)
Likewise, we can obtain the …rst-order condition with respect to ceh and
operate to obtain
ceh =
h
G(cel )(
l)
h
:
(6)
Then, it is straightforward to obtain expressions for the thresholds,
cel = ceh =
h
1+
:
h
(7)
h
cel )
=0:
144
Federal Reserve Bank of Richmond Economic Quarterly
Having solved for the constrained-e¢ cient allocation, we are now ready
to discuss the e¢ ciency of the equilibrium. We do so in the next proposition.
Proposition 1 For any 2 [0,1], the equilibrium, characterized by
{cl ,ch }, is not constrained-e¢ cient.
This powerful result follows from the fact that there is no way of
choosing the bargaining power parameter so as to satisfy cl = cel and
ch = ceh . One way to proof this result is the following. Notice that in
the constrained-e¢ cient allocation we have that the thresholds satisfy
cel = ceh . Then, if the equilibrium were to be e¢ cient, a necessary
condition is that cl = ch . However,
cl
1
=
ch
1
h (1
(1
)
)
h
=1
h (1
1
) + (1
(1
(1
) h
)
h)
<1;
so that cl < ch .
There is a key di¤erence between the social values presented in
equations (5) and (6) and private values presented in equations (3) and
(4). When the planner evaluates the social value of allocating an asset
to the intermediary, the planner takes into account that the intermediary will, with some probability, transfer the asset to the investor. In
doing so, the planner takes into account the entire surplus generated by
transferring an asset from the intermediary to the investor, or the entire
surplus from trade. When the intermediary evaluates her reservation
value, however, she takes into account only a fraction of the surplus.
The reason is because she faces a hold-up problem— her decision to
invest in producing an asset occurs before meeting with the buyer for
the asset, i.e., the investor. The intermediary gives up c utiles (the cost
of production) and gains the option of selling the asset to the investor.
As usual in hold-up problems, the only way that the equations for the
social value and reservation value of the intermediary coincide is if the
intermediary has all the bargaining power when selling the asset to the
investor. That is, = 0. This is clear when we compare the equations
describing the social and private values for the intermediary.
In an analogous way to what happens with the intermediary, when
the planner evaluates the social value of the production by the investor,
the planner takes into account that the investor will, with some probability, obtain the asset from the intermediary. In doing so, the planner
takes into account the entire loss of surplus generated by passing an
asset from the intermediary to the investor. When the investor evaluates her reservation value, she takes into account only a fraction ,
her bargaining power, of this surplus. The investor also faces a hold-up
problem— her decision not to invest in producing an asset occurs before
Bethune, Sultanum, Trachter: Ine ciency in Simple Model of Production & Bilateral Trade145
Figure 1 Welfare in the Equilibrium Allocation and in the
Constrained-E cient Allocation
meeting with the intermediary. That is, in order to gain the option of
buying the asset from the intermediary, she gives up the option of producing the asset herself, and this option is gone once she meets with
the intermediary. The only way that the equations for the social value
and reservation value of an investor coincide is if the investor has all
the bargaining power when buying an asset. That is, = 1.
Fixing the two sides of the hold-up problem is not possible, as this
would require that both intermediary and investor have all the surplus
generated by a trade. Both agents will never value the gains from trade
in the same way the planner does, and as a result, the outcome of a
decentralized equilibrium cannot replicate the planner’s solution— no
matter how investors bargain over gains from trade. Figure 1 presents
the welfare of both equilibrium and constrained-e¢ cient allocations for
two di¤erent values of h as we vary . As discussed, because there is
no for which the constrained-e¢ cient allocation can be decentralized,
the equilibrium welfare lies below the constrained-e¢ cient welfare for
all 2 [0; 1].
3.
ASSET DISTRIBUTION
Through altering production incentives, the ine¢ ciency manifests in
the asset distribution of the economy. In this section, we aim to provide
146
Federal Reserve Bank of Richmond Economic Quarterly
some insights into how this happens. To this end, let s denote the asset
level in the economy. There are three possible asset levels: 0, 1, or 2.
There will be zero assets if both agents do not produce, there will be
one asset if only one of them produces, and there will be two assets if
both produce. To compute the expected asset level, we …rst need to
compute the probability of each of the three potential outcomes. We
have that
Pr(s = 1) = (1 G(cl ))G(ch )+(1 G(ch ))G(cl ) = cl +ch 2cl ch ; Pr(s = 2) = G(cl )G(ch ) = cl ch ;
and where Pr(s = 0) = 1 Pr(s = 1) Pr(s = 2). Then, the expected
asset level is given by E(s) = Pr(s = 1) + 2 Pr(s = 2) = cl + ch , which
provides the following expressions for the equilibrium and e¢ cient allocation,
E(s ) =
2
h
h (1
1
)(1 + )
h
> 0; E(se ) = 2
2
1
+
)vh
(1
h
2 (0; 1):
We can combine these two expressions to obtain a relationship between the asset level in both cases:
E(s ) = E(se )
[2
h (1
2(1
)(1 + ](1 +
(1
)vh2 )
h)
;
(8)
so that the equilibrium has a higher (lower) asset level than the e¢ cient
allocation if the second term on the right-hand side of the expression
is larger (smaller) than one. Interestingly, this term can be above or
below one, depending on the value for . In particular, when = 0, we
get that it is larger than one, and when = 1, we get that it is smaller
than one. This implies the following result.
Proposition 2 The equilibrium allocation exhibits a lower asset level
than in the constrained-e¢ cient allocation when = 1, and it exhibits
a higher asset level than in the constrained-e¢ cient allocation when
= 0.
In equilibrium, when = 1, all the gains from trade go to the
investor. Thus, the intermediary has no incentives to produce the asset.
However, the planner would like the intermediary to produce the asset
with some probability, as this increases the chances that the investor
ends up holding the asset. As a result, the asset supply is ine¢ ciently
low when is high. When = 0, the intermediary has maximum
incentive to produce the asset (as she gets all the gains from trade)
but so does the investor, as she would get no gains from trading with
the intermediary. As a result, the asset supply is ine¢ ciently high
when is low. Figure 2 presents the asset level in the equilibrium
Bethune, Sultanum, Trachter: Ine ciency in Simple Model of Production & Bilateral Trade147
and constrained-e¢ cient allocations for two levels of h . As seen in
the …gure, consistent with the discussion about
= 0 and
= 1,
for low values of the bargaining parameter , the equilibrium exhibits
overproduction, while for high values of , it exhibits underproduction.
The …gure also shows that there is a value for the bargaining power
such that the equilibrium asset supply equates the asset supply in the
constrained-e¢ cient allocation. However, given our impossibility result
in Proposition 1, we know that the allocation is ine¢ cient even when
the equilibrium asset supply matches the e¢ cient asset supply. What
happens is the distribution of assets across the two agents is ine¢ cient,
in the sense that even though we may …nd a such that E(s ) = E(se ),
at this it must be the case that the equalities required for e¢ ciency,
i.e., cl = cel and ch = ceh , are not satis…ed. In fact, given that cl <
ch and cel =ceh = 1, it is the case that, for a given asset level, the way
assets are distributed across the two agents is never the same as in the
e¢ cient allocation. This observation allows us to provide an alternative
interpretation of the ine¢ ciency result. The double hold-up problem
a¤ects e¢ ciency by distorting both the level of the asset supply and
the way assets are distributed across agents. There is always a way of
choosing in order to …x the …rst distortion: from equation (8), and
by continuity argument, it is clear that there always exists a way to
choose such that E(s ) = E(se ). However, given cl =ch 6= cel =ceh for
all , there is no way of correcting the second distortion.
4.
AN EXAMPLE WHERE THE EQUILIBRIUM IS
CONSTRAINED-EFFICIENT
Although the model we developed here has no search frictions, in the
sense that at the production stage agents know who they will meet at
the trading stage, there is a fundamental friction in the model that
stems from the fact that both parties in trade, when faced with the
opportunity to produce, are unaware if their trade counterparty will
be holding an asset or not when they meet at the trade stage. This
friction is important for understanding the ine¢ ciency result presented
in Proposition 1. To study this in some detail, we allow the upper
bound of the production cost distribution c to be in the set [ h ; 1].
Recall that, before, we had assumed c = 1. Notice that the ine¢ ciency
results discussed above hold for c > h . Thus, think of the case c = h
as the limiting case when c # h .
When ch 2 [ h ; 1], the equilibrium thresholds, equations(3) and (4),
reduce to the following expressions,
cl =
h (1
) 1
ch
c
; ch =
h
1
cl
c
:
148
Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Asset Level in the Equilibrium Allocation and in
the Constrained-E cient Allocation
Likewise, the thresholds in the constrained-e¢ cient allocation, equations (5) and (6), reduce to
cel =
h
1
ceh
c
; ceh =
1
h
cel
c
:
Suppose c = h . In the constrained-e¢ cient allocation, we obtain
cel + ceh = h . Because the production cost distribution is uniform, the
density is the same at every production cost. As a result, the e¢ cient
allocation cares about maximizing the production probability, which
equals h , but it is silent on how production is divided between the
intermediary and the investor. Notice that the e¢ cient allocation can
be attained with trade, for example, if only the intermediary produces
(cel = h and ceh = 0), with cases where trade sometimes happens (both
agents having interior production thresholds), or cases with no trade
(which occurs when only the investor produces).
In particular, the planner can attain the constrained-e¢ cient allocation by setting cel = 0 and ceh = h : the intermediary does not
produce, and all production is carried over by the investor. Now we
turn to the equilibrium thresholds. Combining cl and ch provides an
expression for ch ,
ch =
1
(1
1
h
)
(1
)
h
=
h
;
Bethune, Sultanum, Trachter: Ine ciency in Simple Model of Production & Bilateral Trade149
Figure 3 Welfare and the Upper Bound of Production Cost
Distribution h = 1=3
where we already used that c = h . Also, we immediately obtain that
cl = 0. In other words, when c = h , the equilibrium is constrainede¢ cient for any 2 [0; 1]. This occurs because by reducing c up to h
we are taking to zero the probability that the investor does not produce. This implies that the intermediary can forecast perfectly that the
investor will arrive to the trade meeting holding an asset. Likewise, because cl = 0, the investor can forecast perfectly that the intermediary
will arrive to the trade meeting not holding an asset. Thus, the intermediary does not produce as she knows she cannot o- oad the asset
to the investor, and the investor produces as she knows she cannot
acquire the asset from the intermediary. Figure 3 presents this result
for this simple example. It shows that as c approaches h = 1=3, the
equilibrium welfare approaches the welfare of the constrained-e¢ cient
allocation. It shows this in the left panel for the = 0 case and in the
right panel for the = 1 case.
5.
CONCLUSION
In this paper, we show that there is a generic source of ine¢ ciency in
OTC markets stemming from the fact that agents must make production choices (or, in general, trade decisions) ex ante without knowing
150
Federal Reserve Bank of Richmond Economic Quarterly
the choices of their trade counterparties ex post. Due to the lack of
a large mass of agents, individual actions have strong e¤ects on other
agents’decisions. This implies that the equilibrium is not constrainede¢ cient, independent of the way agents split the surplus among them.
We then show that the ine¢ ciency result vanishes in an example
where the equilibrium provides more information to market participants at the production stage, thus improving their forecasting power
regarding the actions of other players in the market. In the example
we study, when the investor always invests, the intermediary never invests, so agents know with certainty if their trade counterparties will
be holding an asset or not when they meet. We show that this attains
e¢ ciency.
Certainly, the model is a simple one. However, the ine¢ ciency result seems robust. Given the growing interest in understanding trading
in OTC markets, we believe that understanding the welfare properties
of models of OTC trading is of paramount importance and calls for
further research.
Bethune, Sultanum, Trachter: Ine ciency in Simple Model of Production & Bilateral Trade151
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