Optimal Progressive Income Taxation in a Bewley-Grossman FrameworkWe appreciate comments

53 Pages·2017·1.32 MB·English

Optimal Progressive Income Taxation in a
Bewley-Grossman Framework∗
Juergen Jung† Chung Tran‡
Towson University Australian National University
November 21, 2017
Abstract
We study the optimal degree of progressivity of income taxation in an environment
where individuals are exposed to both income and health risks over the lifecycle. We ar-
gue that the optimal degree of tax progressivity is a(cid:27)ected by health factors and insurance
arrangements. We formulate a Bewley-Grossman model that matches U.S. data, including
the U.S. tax and transfer system and the pre-2010 health insurance system. Our quanti-
tative analysis indicates that the progressive income tax system is an important channel
to provide social insurance for low income Americans who have limited access to private
health insurance. The optimal progressive income tax system includes a tax break for in-
come below $38,000 and high marginal tax rates of over 50 percent for income earners
above $200,000. The tax progressivity (Suits) index(cid:22)a Gini coe(cid:30)cient for the income tax
contribution by income(cid:22)of the optimized income tax system is around 0.53, compared to
0.17 in the benchmark tax system. Welfare gains from switching to the optimal tax system
amount to over 5 percent of compensating consumption. Indeed, the optimal income tax
system is more progressive than the current U.S. tax code and also more progressive than
the optimized systems reported in previous studies, including Conesa and Krueger (2006)
and Heathcote, Storesletten and Violante (2017). This (cid:28)nding challenges the view that the
US personal income tax code should be less progressive. Finally, we demonstrate that the
optimal degree of tax progressivity is in(cid:29)uenced by the design of public transfer programs
and the parametric speci(cid:28)cations of the income tax function.
JEL: E62, H24, I13, D52
Keywords: Health and income risks; Inequality; Social insurance; Tax progressivity;
Suits index; Optimal taxation; General equilibrium.
∗We appreciate comments from Dirk Krueger, Mariacristina De Nardi, Greg Kaplan and participants of the
UNSW/Treasury Joint Workshop on Fiscal Policy Modeling, the 2017 Midwest Macroeconomics Meetings, and
theeconomicseminarsattheUniversityofAuckland,theAustralianNationalUniversityandTowsonUniversity.
†Department of Economics, Towson University, U.S.A. Tel.: [phone], E-mail: [email protected]
‡ResearchSchoolofEconomics,TheAustralianNationalUniversity,ACT2601,AUS.Tel.: [phone],
E-mail: [email protected]
1
1 Introduction
Theliteraturehasdocumentedahighdegreeofinequalityinincome,wealthandconsumptionof
U.S.households. Thesocialinsuranceliteratureviewsincomeriskasoneimportantcontributor
tothisheterogeneity(e.g., Heathcote, StoreslettenandViolante(2008)andKaplan(2012)). On
the other hand, the health economics literature has focused on health as an additional source
of heterogeneity (e.g., Deaton and Paxson (1998) and Kippersluis et al. (2009)). In order to
equalize opportunities, almost all advanced economies have instituted tax systems where the
marginal tax rate increases with income and public transfers are targeted to disadvantaged
groups including low income households, the sick and the unemployed (compare Table 1). As
a result, tax systems and social insurance systems tend to be highly progressive as a whole and
play a key role in shaping the income distribution across households and over time.
The optimal progressive tax literature has focused on frameworks where income risk is the
only source of inequality (e.g., see Conesa and Krueger (2006) and Heathcote, Storesletten
and Violante (2017)). In this paper, we aim to better understand the optimal progressivity
of the U.S. income tax system in a lifecycle framework where both income risk and health
risk are sources of lifetime inequality. We (cid:28)rst document some stylized facts on health status,
income and health expenditures over the lifecycle, using the U.S. medical expenditure panel
surveydata(MEPS)anddemonstratetheimportantroleofhealthfactorsassourcesoflifecycle
inequality. We then construct a simple partial equilibrium two-period model with income and
health risks and illustrate how the interaction between income and health risks a(cid:27)ects the
optimal degree of tax progressivity. Finally, we formulate a dynamic general equilibrium model
with an endogenous distribution of income, consumption and health expenditure and conduct
quantitative analyses of various progressive income tax systems. Our main goal is to identify
quantitativelytheroleofhealthfactorsintheoptimaldesignofaprogressiveincometaxsystem
in a framework with health risk and health insurance.
Our quantitative model is based on two workhorse models in the literature. The workhorse
model in the optimal tax literature is an incomplete-markets heterogenous agents model ini-
tially developed by Bewley (1986) and extended by Huggett (1993) and Aiyagari (1994). In the
Bewley-Huggett-Aiyagari environment with idiosyncratic income risk and incomplete markets,
progressive income taxation essentially can improve welfare through two channels. First, pro-
gressive taxes lead to a more equal distribution of income and wealth, and therefore to more
equitable distributions of consumption. Second, in the absence of private insurance markets,
progressive taxes provide a partial insurance substitute and can generate more stable house-
hold consumption paths over time through income redistribution from (cid:16)lucky(cid:17) individuals to
(cid:16)unlucky(cid:17) ones who experience large negative shocks to income. However, health risk, medical
expenditures and health insurance arrangements are largely absent in this literature.
In order to incorporate health factors, we combine the Bewley model with the Grossman
model of health capital accumulation (Grossman (1972)). In the Grossman environment, indi-
viduals value their health in addition to a consumption goods basket and have a strong motive
to smooth both health and the consumption bundle over the lifecycle. Health a(cid:27)ects house-
hold consumption through direct and indirect channels. First, the utility of consumption itself
is a(cid:27)ected by the health status of an individual. Second, health is a co-determinant of labor
earnings and therefore a(cid:27)ects the household’s ability to purchase (cid:28)nal consumption goods. In
addition, smoothing health over the lifecycle requires healthcare spending, which subsequently
reduces funds available for purchasing (cid:28)nal consumption goods. The simultaneous presence of
both income and health risks and the institutional insurance arrangements that lower a house-
hold’s exposure to such risk shape the distributions of income, wealth and consumption. In
our framework, progressive income taxes and public health insurance serve as policy tools to
2
provide social insurance against income and health risks.
Our modeling framework allows us to model the lifecycle structure of health risk in con-
junction with income risk as observed in the data. In the model, health care spending and
health insurance take-up rates over the lifecycle are endogenous and jointly determined with
consumption, savings and labor supply. The benchmark model is calibrated to U.S. data and
incorporates the lifecycle patterns of shocks to income and health. The model subsequently
matches labor supply, asset holdings, consumption and health expenditures over the lifecycle.
Health expenditures are low early in life because of high initial health capital and low health
risk. Health expenditures then rise exponentially later in life because individuals are exposed
to more frequent and larger health shocks. The benchmark model also reproduces the hump-
shaped lifecycle pro(cid:28)le of private health insurance take-up rates, the income distribution from
the Panel Study of Income Dynamics (PSID) as well as macroeconomic aggregates from the
National Income and Product Accounts (NIPA).
We use the calibrated model to quantitatively explore the shape of the optimal progressive
income tax system. Our main results are summarized as follows: First, the optimal income tax
system is highly progressive and imposes a tax break for income below $38,000, followed by a
jump in the marginal tax rate to 25 percent. The marginal tax rate then increases further to
over 40 percent for income above $100,000 and to over 50 percent for income above $200,000.
The large zero-tax bracket at the low end of the income distribution is mainly driven by the
high demand for social insurance by the poor unhealthy population who has limited access
to private health insurance. The high tax rates at the upper end of the income distribution
are required to meet the (cid:28)nancing needs of government spending and transfer programs. The
optimal tax system in our model is more progressive than the optimal tax systems reported in
prior literature that largely abstracts from health risk and health insurance institutions (e.g.,
Conesa and Krueger (2006) and Heathcote, Storesletten and Violante (2017)).
In our setting, the progressive income tax system is an important channel to supplement
missingcomponentsofsocialhealthinsurance. Lowincomeindividualsfaceahigherprobability
to end up in poor health states than higher income individuals. Many of them do not have
adequate access to health insurance through the mixed public/private U.S. health insurance
system. These individuals are not poor enough to qualify for Medicaid and not rich enough
to buy private health insurance. Gruber (2008) refers to this group as the poor working class
whose income is below the median income but above the eligibility thresholds for public health
insurance and income transfers. In the model the poor working class bene(cid:28)ts most from the
optimized tax system. The zero-tax at the lower end of the income distribution allows these
individuals to invest more in health and increase their non-medical consumption. We therefore
observe an increase in medical spending of the uninsured in the optimized tax system. This
mechanismismissingfromtheframeworksusedinpriorstudies. Ourresultsimplythatignoring
health risk and institutional features of the U.S. health insurance system can lead to misleading
conclusions about the optimal progressive tax policy in the U.S.
In order to measure the progressivity level of the U.S. income tax system, we compute the
Suits index(cid:22)a Gini coe(cid:30)cient for income tax contributions by income. According to Suits
(1977), the Suits index varies from +1 (most progressive) where the entire tax burden is borne
by households of the highest income bracket, through 0 for a proportional tax, to −1 (most
regressive) where the entire tax burden falls on households of the lowest income bracket. Our
calculations indicate that the U.S. income tax system in the benchmark model is not very
progressive with a Suits index of 0.17. The optimal tax system is much more progressive with a
Suits index of 0.54. Under the optimal tax system income inequality decreases signi(cid:28)cantly. We
observethattheafter-tax-incomeGinicoe(cid:30)cientdecreasesfrom0.38inthebenchmarkeconomy
to 0.31 in the economy with optimized progressivity. We also observe large welfare gains of 5.5
3
percent of compensating lifetime consumption at the aggregate level when switching from the
benchmark to the optimal tax system. This outcome is mainly driven by large welfare gains
of low income individuals that dominate the welfare losses of the higher income groups. These
(cid:28)ndings imply that replacing the existing progressive income tax code with a less progressive
system such as the one currently discussed by the Trump administration can cause welfare
losses.
Wenextanalyzehowdi(cid:27)erentdesignsoftheU.S.healthinsurancesystema(cid:27)ecttheoptimal
level of tax progressivity. We consider three alternative health insurance systems: (i) the U.S.
health insurance system after the introduction of the A(cid:27)ordable Care Act (ACA), (ii) a system
with universal Medicare for all, and (iii) a system without health insurance contracts where
individuals can only self-insurance.
We (cid:28)nd that the ACA provides channels that redistribute resources from healthy, high
income types to sicker, low income types through premium subsidies and an expansion of Medi-
caid(cid:22)apublichealthinsuranceprogramforlowincomeindividuals. Ourresultsshowthatwith
the ACA in place, the optimal tax progressivity decreases slightly compared to the pre-ACA
scenario. The Suits index decreases to 0.52.
The optimal tax schedule changes signi(cid:28)cantly when alternative insurance designs are con-
sidered. When Medicare is extended to everyone in the model economy, the optimal income
tax system becomes much less progressive. The tax break at the lower end of the income
distribution is smaller and the marginal tax rates imposed on top earners are much lower at
approximately 31 percent for incomes over $200,000. This result implies that the universal
expansion of Medicare signi(cid:28)cantly reduces the residual demand for social insurance provided
through the progressive income tax system.
Finally, we examine how the parametric speci(cid:28)cation of the income tax function a(cid:27)ects the
optimaltaxprogressivity. Inourbenchmarkmodel,weusethetwoparameterspeci(cid:28)cationfrom
Benabou (2002) with a non-negative tax restriction to remove transfer components embedded
in the tax function. As an extension, we (cid:28)rst remove this restriction and use the income tax
function as in Heathcote, Storesletten and Violante (2017). Our results shown that the optimal
progressiveincometaxfunctionshiftsdownandmarginaltaxratesarenegativeatthelowendof
the income distribution. These negative taxes are conditional cash transfers that induce poorer
individuals to work and save in order to receive these transfer payments. Next, we consider
the three parameter speci(cid:28)cation from Gouveia and Strauss (1994) which was used in Conesa
and Krueger (2006). We (cid:28)nd that the shapes of the optimal tax function di(cid:27)er signi(cid:28)cantly
and welfare gains from optimizing progressivity using the three parameter speci(cid:28)cation are
smaller at 1.08 percent of compensating lifetime consumption compared to 5.5 percent with
the 2-parameter polynomial. Indeed, the transfer policy to the low income households at the
bottom of the distribution strongly a(cid:27)ects the shape of the optimal marginal tax function. Our
(cid:28)ndings imply that the parametric speci(cid:28)cation of the tax function as well as the design of
transfer policies are important for evaluating the optimal level of progressivity of the income
tax code.
Related literature. Our work is connected to di(cid:27)erent branches of the quantitative
macroeconomics and health economics literature. First, our paper is closely related to the
optimal progressive income taxation literature. In a seminal paper, Varian (1980) shows an-
alytically how social insurance can be provided via a progressive tax system. More recently,
Conesa and Krueger (2006) quantify the optimal progressivity of the income tax code in a
dynamic general equilibrium model with household heterogeneity due to uninsurable labor pro-
ductivityrisk. Theyshowthataprogressivetaxsystemservesasapartialsubstituteformissing
income-insurance markets and results in a more equal distribution of income. Erosa and Kore-
4
shkova (2007) analyze the insurance role of the U.S. progressive income tax code in a dynastic
model with human capital accumulation. Chambers, Garriga and Schlagenhauf (2009) quantify
interactions between progressive income taxes and housing policies to promote home ownership
in an overlapping generations model with housing and rental markets. Krueger and Ludwig
(2016) compute optimal tax- and education policies in an economy where progressive taxes
provide social insurance against idiosyncratic wage risk but distort the education and human
capital decision of households. Stantcheva (2015) characterizes the optimal income tax and
human capital policies in a dynamic lifecycle model of labor supply with risky human capital
formation. McKay and Reis (2016) study the optimal generosity of unemployment bene(cid:28)ts and
progressivity of income taxes in a model with macroeconomic aggregate shocks and individual
unemployment risk. These studies abstract from the implications of health risk and the e(cid:27)ects
of the health insurance system on the optimal progressivity of the income tax system. Our
study includes these components.
Heathcote, Storesletten and Violante (2017) develop a tractable general equilibrium model
to study the optimal degree of progressivity of a tax and transfer system. They focus on the
trade-o(cid:27)s between risk sharing and the incentives to work and invest in skills. They show
how preferences, technology and the market structure in(cid:29)uence the optimal degree of tax and
transfer progressivity. Note that, in order to obtain analytical results they abstract from inter-
temporal consumption and savings decisions and more realistic government transfer programs.
We analyze a similar problem, but we focus on the quantitative aspects which allows for more
realistic features in the model. In our model health is an important source of heterogeneity
across individuals and over the lifecycle. In addition, we explicitly model the main components
of the U.S. social insurance system including Food Stamp programs, Social Security, Medicaid
and Medicare. As a direct result, the optimal income tax system in our setting is more pro-
gressive than the one in Heathcote, Storesletten and Violante (2017). Our results illustrate
the quantitative importance of accounting for health, health risk and the design of the health
insurance system in determining the optimal degree of progressivity of the income tax system.
Our paper is related to the literature on incomplete markets macroeconomic models with
heterogeneousagentsaspioneeredbyBewley(1986)andextendedbyHuggett(1993)andAiya-
gari (1994). This Bewley model has been applied widely to quantify the welfare cost of public
insurance for idiosyncratic income risk (e.g., Hansen and Imrohoroglu (1992), (cid:157)mrohoro§lu,
(cid:157)mrohoro§lu and Joines (1995), Golosov and Tsyvinski (2006), Heathcote, Storesletten and Vi-
olante(2008),Conesa,KitaoandKrueger(2009)andHuggettandParra(2010)). Thisliterature
focuses on the welfare cost triggered by income/labor productivity risk in combination with a
lack of insurance for non-medical consumption. Recently, Capatina (2015) demonstrates that
health shocks are another important source of idiosyncratic risk faced by individuals over the
lifecycle. In our study we add to this literature by incorporating idiosyncratic health risk into
a Bewley framework. We incorporate the micro-foundations of a health capital accumulation
mechanism based on the Grossman model which endogenizes medical spending. Our research
mergestheworkhorsemodelsfromthehealtheconomicsandthemacro/public(cid:28)nanceliterature
to analyze the optimal income tax progressivity in the presence of income and health risk in
combination with a realistic depiction of the U.S. health insurance system.
Our work contributes to a growing macro-public (cid:28)nance literature that extends the Gross-
man model of health capital accumulation (Grossman (1972)). This literature incorporates
health shocks, insurance markets and general equilibrium channels using a more realistic in-
stitutional setting (e.g., Jung and Tran (2007), Yogo (2016), Fonseca et al. (2013), Scholz and
Seshadri (2013a) and Jung and Tran (2016)). Jung and Tran (2015) explore the welfare im-
plications of optimal health insurance policies. Di(cid:27)erent from this paper, we quantitatively
characterize the optimal progressivity of the income tax system while taking the redistribution
5
e(cid:27)ects of the health insurance system into account. We then demonstrate how changes to the
health insurance system a(cid:27)ect the optimal level of income tax progressivity.
Our paper is connected to the literature on high marginal tax rates for top income earners.
DiamondandSaez(2011)advocatesfortaxinglaborearningsatthehighendofthedistribution
at very high marginal rates in excess of 75 percent. Badel and Huggett (2015) assess the conse-
quences of increasing the marginal tax rate on U.S. top income earners using a human capital
model. Guner, Lopez-Daneri and Ventura (2016) analyze how e(cid:27)ective a progressive income
tax system is in raising tax revenues. Kindermann and Krueger (2015) (cid:28)nd that high marginal
labor income tax rates are an e(cid:27)ective tool for social insurance in a large-scale stochastic over-
lapping generations model with optimal marginal tax rates for top 1 percent earners of close
to 90 percent. Di(cid:27)erent from these studies, we focus on the optimal marginal tax rates across
the entire income distribution. Moreover, we base our analysis on a health capital model where
health risk is an additional source of heterogeneity in addition to labor market risk. We also
(cid:28)nd that very high tax rates at the top are an essential component of the optimal progressive
tax system. More importantly, we highlight that such high optimal marginal tax rates at the
topareinter-dependentwiththemarginaltaxratessetatthebottomoftheincomedistribution
and the government transfer policies already in place.
The paper is structured as follows. Section 2 describes the insurance and incentive trade-o(cid:27)
in a two-period model. Section 3 presents the full dynamic model. Section 4 describes our
calibration strategy. Section 5 describes our experiments and quantitative results. Section 6
is devoted to sensitivity analysis. Section 7 concludes. The Appendix presents all calibration
tables and (cid:28)gures.
2 Facts and analytical analysis
2.1 Stylized facts
The Medical Expenditure Panel Survey (MEPS) is a longitudinal survey for the U.S. that pays
particular attention to medical expenditures and (cid:28)nancing. We use data from 1999(cid:21)2009 to
track health, health expenditures, the sources of health (cid:28)nancing and income over the lifecycle.
Health status. Due to the biological process health status is highly correlated with age.
There are various proxy measures of health conditions in empirical health studies. The MEPS
provides two measures: the Short-Form 12 Version 2 (SF-12v2) and a self-reported health
status.1 Panel 1 of Figure 1 displays the physical and mental components of the SF-12v2 index
over the lifecycle. Young individuals report relatively high levels of both physical and mental
health(capital). Mentalhealthexhibitsaslight"M"shapeoverthelifecycle. Youngindividuals
(around age 20) and very old individuals (around age 75 and higher) report the lowest mental
health status. Individuals in the age range between 40 and 55 have lower mental health status
than younger cohorts in their thirties and older cohorts in their sixties. The average physical
health consistently decreases as an individual ages. In addition, we construct a "healthy" index
using self-reported health status measures (1. excellent, 2. very good, 3. good, 4. fair, and 5.
poor health). An individual is considered to be healthy if the health status measure is either
excellent, very good, or good and unhealthy otherwise. This classi(cid:28)cation is standard in the
1The SF-12v2 includes twelve di(cid:27)erent health measures about physical and mental health. We use physical
and mental measures. Both measures use the same variables to construct the index but the physical health
index puts more weight on variables measuring physical health components and the mental health index puts
moreweightonvariablesmeasuringmentalhealthcomponents. Ware,KosinskiandKeller(1996)providesmore
details on construction of the SF-12v2 index. The self-reported health status measure is reported as either: 1.
excellent, 2. very good, 3. good, 4. fair, or 5. poor.
6
literature. This lifecycle pattern of health is similar to the one using the physical component of
the SF-12v2 index.
Health expenditure. Panel 2 of Figure 1 reports the lifecycle pattern of health expendi-
tures, expressed in 2009 US dollars. Individual health expenditures are relatively low at young
ages but increase signi(cid:28)cantly at later ages. This is mainly driven by depreciation of health
over the lifecycle. On average, individuals in their twenties spend about $1,500 per year on
healthcare whereas older individuals in their (cid:28)fties spend about $4,000 per year. After the
age (cid:28)fty, health expenditures rise very fast. The highest expenditures are incurred by the very
old towards the end of their life and amount to approximately $12,000 on average per year in
2009. This indicates that the accumulation of bad health shocks over the lifecycle drives up
the health expenditures of an individual. In order to get a sense for the distribution of health
expenditures we report the Gini coe(cid:30)cient of health expenditure per age group. Inequality in
health expenditures decreases by age. The Gini coe(cid:30)cient of health expenditures is very high
at around 0.8 when individuals are younger than 40 and then sharply drops as individuals get
older. Higher Gini coe(cid:30)cients at younger ages indicate that health expenditures among the
young are much more concentrated than health expenditures of the old. This is probably due
to relatively rare, but catastrophic health events among the young. Lower Gini coe(cid:30)cients at
older ages imply that the higher incidence of health problems at higher ages "equalizes" health
spending across individuals. Moreover, the availability of public health insurance programs
plays a role in reducing uneven access to health care services and therefore evens out health
expenditure di(cid:27)erences across di(cid:27)erent income groups as well.
Health (cid:28)nancing. TheU.S.healthinsurancesystemisamixedsystemwherepublichealth
insurance programs target the retired population (Medicare) and the poor (Medicaid) while the
majority of working individuals obtain private health insurance via their employers. Panels 3
and4ofFigure1displaythe(cid:28)nancingsourcesofhealthexpendituresandtheinsurancetake-up
rates over the lifecycle. Despite the many di(cid:27)erent types of insurances, there is a large number
of Americans lacking health insurance. The U.S. health insurance system failed to provide
insurance for about 50 million people in 2010. The employer-based group health insurance
policies (GHI) cover only around 60 percent of the working-age population while individual-
based health insurance policies (IHI) cover less than 6 percent. A large number of healthy
and young individuals do not have health insurance, either by choice or by circumstance. The
fraction of the uninsured is highest among young workers below 35. Medicaid picks up less
than 10 percent of workers by covering low income individuals. Consequently, the system leaves
about 25 percent of the working population without health insurance which contributes to high
insurance premiums and poor risk pooling. Gruber (2008) points out the modal uninsured
person is a member of the working poor class, whose income is below median income but above
thefederalpovertylevelandthereforenoteligibleforMedicaidandothergovernmentprograms.
Private insurance reimbursements and out-of-pocket payments are the two major funding
sources for medical spending of the working age population. The fraction of health expenditure
(cid:28)nanced by private insurance and Medicaid decreases with age, whereas the fraction of health
expenditures (cid:28)nanced by out-of-pocket funds increases moderately. Around the retirement age
of 65 there is a big shift in the magnitude of (cid:28)nancing from private insurance toward public
insurance including Medicare, Veteran’s bene(cid:28)ts, and other state run insurance plans.
Health and income. Figure 2 presents the age-pro(cid:28)les of wage income for unhealthy and
healthy individuals. There is a signi(cid:28)cant gap between the wage income pro(cid:28)le between the two
groups. Unhealthy individuals exhibit a lower income path over the lifecycle. This con(cid:28)rms
that health risk is an important source of lifetime inequality.
To better understand income and health risks we compute the coe(cid:30)cients of variation for
income and health expenditures over the lifecycle in Figure 3. For comparison, we also include
7
data from the Health and Retirement Survey (HRS). Average wage income varies over the
lifecycle and follows a hump shape. However, the coe(cid:30)cient of variation for wage income is
fairly stable around 0.9 before age sixty, and then rises sharply after retirement age. The
coe(cid:30)cient of variation for household income tracks the coe(cid:30)cient of variation for wage income
very well for individuals of working age, and stays stable until the end of life. On the other
hand, the coe(cid:30)cient of variation for health expenditure is four to (cid:28)ve times larger and also
varies sharply over the lifecycle. The coe(cid:30)cient of variation for health expenditure is largest
between age 20(cid:21)30. Note that health expenditures are relatively low for young individuals but
soisincome. Thisindicatesthathealthexpenditureriskcanbesigni(cid:28)cantforyoungindividuals
given their low income and their lack of access to credit. This is partly due to the structure
of the U.S. health insurance system that lets many young individuals opt out of the health
insurance system or does not provide enough (cid:28)nancial aid to help cover insurance.
Overall, health expenditures make up a signi(cid:28)cant fraction of income over the lifecycle.
When comparing health expenditures as fraction of household income, we observe an increase
over the lifecycle. Figure 4 presents average health expenditures over average income and
average out-of-pocket health expenditure over average income for each age group. At the end of
their life individuals spend on average 60 percent of their income on health care. Medicare and
Medicaid are the main sources of health (cid:28)nancing for the elderly. As a consequence, the share
of out-of-pocket health expenditures as fraction of income is contained at less than 8 percent
despite the high levels of overall health spending by this age group.
2.2 Analytical framework
We next construct a simple model to highlight the connection between a tax- and transfer
system and an individual’s income and health risks.
Endowments and preferences. Individuals live for two periods with certainty. They
supply labor elastically in period 1 and do not work in period 2. The individual is born with
skill (cid:15) at the beginning of period 1 and faces health shocks (cid:15) at the beginning of period 2.
I H
These shocks are drawn from a joint distribution
(cid:20) (cid:15) (cid:21) (cid:18)(cid:20) µ (cid:21) (cid:20) σ2 σ (cid:21)(cid:19)
I ∼ f I , I IH ,
(cid:15) µ σ σ2
H H IH H
with density f((cid:15) ,(cid:15) ) where the average labor shock is normalized as µ = 0 and µ > 0 and
I H I H
σ2 > 0, σ2 > 0 and σ > 0. Health shocks will result in compulsory health care expenditures.
I H IH
Individuals derive utility from consumption in both periods and disutility from work in period
1. Their lifetime utility is: u(c )−φv(n)+βE[u(c )], where u(.) and v(.) are utility functions
1 2
with the usual properties, c and c are consumption in periods 1 and 2, respectively, n is
1 2
labor supply, φ scales the level of disutility of labor, β is the time discount factor and E is the
expectations operator.
Government. The government runs two separate tax and transfer programs. The (cid:28)rst
program is a redistribution program where the government imposes a payroll tax τ to (cid:28)nance
I
a lump-sum payment of d in the (cid:28)rst period. The second program is a public health insurance
program (Medicare) that covers a fraction (1 − ρ) of the health care expenditures (cid:15) in the
H
second period. This program is (cid:28)nanced by Medicare tax τ .
H
8
Household problem. The household maximization problem can be summarized as
max {u(c )−φv(n)+βE[u(c )]} s.t.
1 2
{c1,c2,n,s}
c +s = (1−τ −τ )wn(cid:15) +d,
1 I H I
c = Rs−ρ(cid:15) ,
2 H
where w is the wage rate, R is the interest rate, s is savings in period 1, d is the lump-sum
transfer and ρ is the coinsurance rate. The (cid:28)rst order conditions are
∂n :u(cid:48)(c )(1−τ −τ )w = φv(cid:48)(n), (1)
1 I H
∂s :u(cid:48)(c ) = βRE(cid:2)u(cid:48)(c )(cid:3). (2)
1 2
Government problem. The government clears the two transfer programs separately so
that
(cid:90) (cid:90) (cid:90)
τ wn(cid:15) = (1−ρ)(cid:15) f((cid:15) ,(cid:15) )d(cid:15) d(cid:15) = (1−ρ)µ ,
H I H I H I H H
(cid:15)I (cid:15)H (cid:15)I
(cid:90) (cid:90)
τ wn(cid:15) f((cid:15) )d(cid:15) = d×f((cid:15) )d(cid:15) := D,
I I I I I I
(cid:15)I (cid:15)I
The required tax rates are given by
(1−ρ)µ
H
τH = (cid:82) , (3)
wn(cid:15) f((cid:15) )d(cid:15)
(cid:15)I I I I
D
τI = (cid:82) . (4)
wn(cid:15) f((cid:15) )d(cid:15)
(cid:15)I I I I
The government maximizes the following problem
(cid:26)(cid:90) (cid:27)
V (τ ,ρ) = max u(wn(1−τ −τ )−s)−φv(n)+βE[u(Rs+d+(cid:15) −ρ(cid:15) )]
I I H I H
{τI,ρ} (cid:15)I
s.t.(1), (2), (3) and (4).
Whenthegovernmentchoosesτ andρviaexpressions(3) and (4) respectively,itautomatically
I
(cid:28)xes D and τ . Since the population in each period is normalized to one we also have that
H
d = D.
Optimal taxation and insurance policy. The government (cid:28)rst order conditions are
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:20) (cid:21)(cid:21)
∂V ∂n ∂τ ∂s ∂n ∂s ∂d
= u(cid:48)(c ) w (1−τ −τ )−n H − −φv(cid:48)(n) +βE u(cid:48)(c ) R + −(cid:15) ,
1 I H 2 H
∂ρ ∂ρ ∂ρ ∂ρ ∂ρ ∂ρ ∂ρ
(5)
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:20) (cid:21)(cid:21)
∂V ∂n ∂s ∂n ∂s ∂d
= u(cid:48)(c ) w (1−τ −τ )−n − −φv(cid:48)(n) +βE u(cid:48)(c ) R + ,
1 I H 2
∂τ ∂τ ∂τ ∂τ ∂τ ∂τ
I I I I I I
(6)
where we know from expressions (3) and (4) that
∂τ µ (cid:32)n+(1−ρ) ∂n(cid:33)
H H ∂ρ
= − ,
∂ρ wn n
(cid:18) (cid:19)
∂D ∂n
= w n+τ ,
I
∂τ ∂τ
I I
∂D ∂n
= τ w .
I
∂ρ ∂ρ
9
Substituting these expressions and the (cid:28)rm FOCs into the government FOCs (5) and (6) we
get two government Euler equations
∂V (cid:32)n+(1−ρ) ∂n(cid:33) (cid:20) (cid:18) ∂n (cid:19)(cid:21)
= u(cid:48)(c )µ ∂ρ +βE u(cid:48)(c ) τ w −(cid:15) = 0,
1 H 2 I H
∂ρ n ∂ρ
(cid:20) (cid:18) (cid:19)(cid:21)
∂V ∂n
= −u(cid:48)(c )wn+βE u(cid:48)(c )w n+τ = 0.
1 2 I
∂τ ∂τ
I I
This system describes the trade-o(cid:27) between current cost and future bene(cid:28)ts of changes in the
replacement rate ρ and the labor tax rate τ which (cid:28)nances lump-sum transfers in the second
n
period.
Thus, the optimal level of social insurance provided by an income tax system depends on
economy-based fundamentals including preferences, the evolution of income and health risks
over the lifecycle, and the structure of a health insurance system. In particular, the optimal
income tax rate depends on the structure of health and income shocks and level of social
insurance provided by the health insurance program. In the next section, we formulate a more
realistic model of the U.S. economy and quantify the optimal degree of progressivity of the US
income tax system. In addition, we explore how di(cid:27)erently designs of a health insurance system
a(cid:27)ect the optimal tax progressivity.
3 The model
3.1 Technologies and (cid:28)rms
There are two production sectors in the economy, which are assumed to grow at a constant
rate g. Sector one is populated by a continuum of identical (cid:28)rms that use physical capital K
and e(cid:27)ective labor services N to produce a non-medical consumption good c with a normalized
price of one. Firms in the non-medical sector are perfectly competitive and solve the following
maximization problem
max F (K,N)−qK −wN, (7)
{K, N}
taking the rental rate of capital q and the wage rate w as given. Capital depreciates at rate δ
in each period. Sector two, the medical sector, is also populated by a continuum of identical
(cid:28)rms that use capital K and labor N to produce medical services m at a price of p . Firms
m m m
in the medical sector maximize
max p F (K ,N )−qK −wN . (8)
m m m m m m
{Km, Nm}
3.2 Demographics, preferences and endowments
Theeconomyispopulatedwithoverlappinggenerationsofindividualswholiveuptoamaximum
of J periods. Individuals work for J periods and are retired thereafter. Individuals survive
1
each period with age dependent survival probability π . Deceased agents leave an accidental
j
bequest that is taxed and redistributed equally to the working age population. The population
grows exogenously at an annual rate n. We assume stable demographic patterns, so that age j
agentsmakeupaconstantfractionµ oftheentirepopulationatanypointintime. Therelative
j
sizes of the cohorts alive µ and the mass of individuals dying in each period µ˜ (conditional
j j
10