Optimal Progressive Income Taxation in aBewley-Grossman Framework∗

Juergen Jung†

Towson University

Chung Tran‡

Australian National University

November 21, 2017

Abstract

We study the optimal degree of progressivity of income taxation in an environmentwhere individuals are exposed to both income and health risks over the lifecycle. We ar-gue that the optimal degree of tax progressivity is a�ected by health factors and insurancearrangements. We formulate a Bewley-Grossman model that matches U.S. data, includingthe 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 channelto provide social insurance for low income Americans who have limited access to privatehealth 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 earnersabove $200, 000. The tax progressivity (Suits) index�a Gini coe�cient for the income taxcontribution by income�of the optimized income tax system is around 0.53, compared to0.17 in the benchmark tax system. Welfare gains from switching to the optimal tax systemamount to over 5 percent of compensating consumption. Indeed, the optimal income taxsystem is more progressive than the current U.S. tax code and also more progressive thanthe optimized systems reported in previous studies, including Conesa and Krueger (2006)and Heathcote, Storesletten and Violante (2017). This �nding challenges the view that theUS personal income tax code should be less progressive. Finally, we demonstrate that theoptimal degree of tax progressivity is in�uenced by the design of public transfer programsand the parametric speci�cations of the income tax function.

JEL: E62, H24, I13, D52Keywords: 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 theUNSW/Treasury Joint Workshop on Fiscal Policy Modeling, the 2017 Midwest Macroeconomics Meetings, andthe economic seminars at the University of Auckland, the Australian National University and Towson University.†Department of Economics, Towson University, U.S.A. Tel.: 1 (812) 345-9182, E-mail: jjung@towson.edu‡Research School of Economics, The Australian National University, ACT 2601, AUS. Tel.: +61 2 6125 5638,

E-mail: chung.tran@anu.edu.au

1

1 Introduction

The literature has documented a high degree of inequality in income, wealth and consumption ofU.S. households. The social insurance literature views income risk as one important contributorto this heterogeneity (e.g., Heathcote, Storesletten and Violante (2008) and Kaplan (2012)). Onthe other hand, the health economics literature has focused on health as an additional sourceof heterogeneity (e.g., Deaton and Paxson (1998) and Kippersluis et al. (2009)). In order toequalize opportunities, almost all advanced economies have instituted tax systems where themarginal tax rate increases with income and public transfers are targeted to disadvantagedgroups including low income households, the sick and the unemployed (compare Table 1). Asa result, tax systems and social insurance systems tend to be highly progressive as a whole andplay 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 theonly source of inequality (e.g., see Conesa and Krueger (2006) and Heathcote, Storeslettenand Violante (2017)). In this paper, we aim to better understand the optimal progressivityof the U.S. income tax system in a lifecycle framework where both income risk and healthrisk are sources of lifetime inequality. We �rst document some stylized facts on health status,income and health expenditures over the lifecycle, using the U.S. medical expenditure panelsurvey data (MEPS) and demonstrate the important role of health factors as sources of lifecycleinequality. We then construct a simple partial equilibrium two-period model with income andhealth risks and illustrate how the interaction between income and health risks a�ects theoptimal degree of tax progressivity. Finally, we formulate a dynamic general equilibrium modelwith an endogenous distribution of income, consumption and health expenditure and conductquantitative analyses of various progressive income tax systems. Our main goal is to identifyquantitatively the role of health factors in the optimal design of a progressive income tax systemin a framework with health risk and health insurance.

Our quantitative model is based on two workhorse models in the literature. The workhorsemodel 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 theBewley-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 moreequitable 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 �lucky� individuals to�unlucky� ones who experience large negative shocks to income. However, health risk, medicalexpenditures and health insurance arrangements are largely absent in this literature.

In order to incorporate health factors, we combine the Bewley model with the Grossmanmodel 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 motiveto smooth both health and the consumption bundle over the lifecycle. Health a�ects house-hold consumption through direct and indirect channels. First, the utility of consumption itselfis a�ected by the health status of an individual. Second, health is a co-determinant of laborearnings and therefore a�ects the household's ability to purchase �nal consumption goods. Inaddition, smoothing health over the lifecycle requires healthcare spending, which subsequentlyreduces funds available for purchasing �nal consumption goods. The simultaneous presence ofboth 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. Inour framework, progressive income taxes and public health insurance serve as policy tools to

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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 andhealth insurance take-up rates over the lifecycle are endogenous and jointly determined withconsumption, savings and labor supply. The benchmark model is calibrated to U.S. data andincorporates the lifecycle patterns of shocks to income and health. The model subsequentlymatches 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 healthrisk. Health expenditures then rise exponentially later in life because individuals are exposedto more frequent and larger health shocks. The benchmark model also reproduces the hump-shaped lifecycle pro�le of private health insurance take-up rates, the income distribution fromthe Panel Study of Income Dynamics (PSID) as well as macroeconomic aggregates from theNational Income and Product Accounts (NIPA).

We use the calibrated model to quantitatively explore the shape of the optimal progressiveincome tax system. Our main results are summarized as follows: First, the optimal income taxsystem is highly progressive and imposes a tax break for income below $38, 000, followed by ajump in the marginal tax rate to 25 percent. The marginal tax rate then increases further toover 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 thehigh demand for social insurance by the poor unhealthy population who has limited accessto private health insurance. The high tax rates at the upper end of the income distributionare required to meet the �nancing needs of government spending and transfer programs. Theoptimal tax system in our model is more progressive than the optimal tax systems reported inprior 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 supplementmissing components of social health insurance. Low income individuals face a higher probabilityto end up in poor health states than higher income individuals. Many of them do not haveadequate access to health insurance through the mixed public/private U.S. health insurancesystem. These individuals are not poor enough to qualify for Medicaid and not rich enoughto buy private health insurance. Gruber (2008) refers to this group as the poor working classwhose income is below the median income but above the eligibility thresholds for public healthinsurance and income transfers. In the model the poor working class bene�ts most from theoptimized tax system. The zero-tax at the lower end of the income distribution allows theseindividuals to invest more in health and increase their non-medical consumption. We thereforeobserve an increase in medical spending of the uninsured in the optimized tax system. Thismechanism is missing from the frameworks used in prior studies. Our results imply that ignoringhealth risk and institutional features of the U.S. health insurance system can lead to misleadingconclusions 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 theSuits index�a Gini coe�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 borneby households of the highest income bracket, through 0 for a proportional tax, to −1 (mostregressive) where the entire tax burden falls on households of the lowest income bracket. Ourcalculations indicate that the U.S. income tax system in the benchmark model is not veryprogressive with a Suits index of 0.17. The optimal tax system is much more progressive with aSuits index of 0.54. Under the optimal tax system income inequality decreases signi�cantly. Weobserve that the after-tax-income Gini coe�cient decreases from 0.38 in the benchmark economyto 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 thebenchmark to the optimal tax system. This outcome is mainly driven by large welfare gainsof low income individuals that dominate the welfare losses of the higher income groups. These�ndings imply that replacing the existing progressive income tax code with a less progressivesystem such as the one currently discussed by the Trump administration can cause welfarelosses.

We next analyze how di�erent designs of the U.S. health insurance system a�ect the optimallevel of tax progressivity. We consider three alternative health insurance systems: (i) the U.S.health insurance system after the introduction of the A�ordable Care Act (ACA), (ii) a systemwith universal Medicare for all, and (iii) a system without health insurance contracts whereindividuals can only self-insurance.

We �nd that the ACA provides channels that redistribute resources from healthy, highincome types to sicker, low income types through premium subsidies and an expansion of Medi-caid�a public health insurance program for low income individuals. Our results show that withthe ACA in place, the optimal tax progressivity decreases slightly compared to the pre-ACAscenario. The Suits index decreases to 0.52.

The optimal tax schedule changes signi�cantly when alternative insurance designs are con-sidered. When Medicare is extended to everyone in the model economy, the optimal incometax system becomes much less progressive. The tax break at the lower end of the incomedistribution is smaller and the marginal tax rates imposed on top earners are much lower atapproximately 31 percent for incomes over $200, 000. This result implies that the universalexpansion of Medicare signi�cantly reduces the residual demand for social insurance providedthrough the progressive income tax system.

Finally, we examine how the parametric speci�cation of the income tax function a�ects theoptimal tax progressivity. In our benchmark model, we use the two parameter speci�cation fromBenabou (2002) with a non-negative tax restriction to remove transfer components embeddedin the tax function. As an extension, we �rst remove this restriction and use the income taxfunction as in Heathcote, Storesletten and Violante (2017). Our results shown that the optimalprogressive income tax function shifts down and marginal tax rates are negative at the low end ofthe income distribution. These negative taxes are conditional cash transfers that induce poorerindividuals to work and save in order to receive these transfer payments. Next, we considerthe three parameter speci�cation from Gouveia and Strauss (1994) which was used in Conesaand Krueger (2006). We �nd that the shapes of the optimal tax function di�er signi�cantlyand welfare gains from optimizing progressivity using the three parameter speci�cation aresmaller at 1.08 percent of compensating lifetime consumption compared to 5.5 percent withthe 2-parameter polynomial. Indeed, the transfer policy to the low income households at thebottom of the distribution strongly a�ects the shape of the optimal marginal tax function. Our�ndings imply that the parametric speci�cation of the tax function as well as the design oftransfer policies are important for evaluating the optimal level of progressivity of the incometax code.

Related literature. Our work is connected to di�erent branches of the quantitativemacroeconomics and health economics literature. First, our paper is closely related to theoptimal 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 adynamic general equilibrium model with household heterogeneity due to uninsurable labor pro-ductivity risk. They show that a progressive tax system serves as a partial substitute for missingincome-insurance markets and results in a more equal distribution of income. Erosa and Kore-

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shkova (2007) analyze the insurance role of the U.S. progressive income tax code in a dynasticmodel with human capital accumulation. Chambers, Garriga and Schlagenhauf (2009) quantifyinteractions between progressive income taxes and housing policies to promote home ownershipin an overlapping generations model with housing and rental markets. Krueger and Ludwig(2016) compute optimal tax- and education policies in an economy where progressive taxesprovide social insurance against idiosyncratic wage risk but distort the education and humancapital decision of households. Stantcheva (2015) characterizes the optimal income tax andhuman capital policies in a dynamic lifecycle model of labor supply with risky human capitalformation. McKay and Reis (2016) study the optimal generosity of unemployment bene�ts andprogressivity of income taxes in a model with macroeconomic aggregate shocks and individualunemployment risk. These studies abstract from the implications of health risk and the e�ectsof the health insurance system on the optimal progressivity of the income tax system. Ourstudy includes these components.

Heathcote, Storesletten and Violante (2017) develop a tractable general equilibrium modelto study the optimal degree of progressivity of a tax and transfer system. They focus on thetrade-o�s between risk sharing and the incentives to work and invest in skills. They showhow preferences, technology and the market structure in�uence the optimal degree of tax andtransfer 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 morerealistic features in the model. In our model health is an important source of heterogeneityacross individuals and over the lifecycle. In addition, we explicitly model the main componentsof the U.S. social insurance system including Food Stamp programs, Social Security, Medicaidand 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 illustratethe quantitative importance of accounting for health, health risk and the design of the healthinsurance 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 withheterogeneous agents as pioneered by Bewley (1986) and extended by Huggett (1993) and Aiya-gari (1994). This Bewley model has been applied widely to quantify the welfare cost of publicinsurance for idiosyncratic income risk (e.g., Hansen and Imrohoroglu (1992), �mrohoro§lu,�mrohoro§lu and Joines (1995), Golosov and Tsyvinski (2006), Heathcote, Storesletten and Vi-olante (2008), Conesa, Kitao and Krueger (2009) and Huggett and Parra (2010)). This literaturefocuses on the welfare cost triggered by income/labor productivity risk in combination with alack of insurance for non-medical consumption. Recently, Capatina (2015) demonstrates thathealth shocks are another important source of idiosyncratic risk faced by individuals over thelifecycle. In our study we add to this literature by incorporating idiosyncratic health risk intoa Bewley framework. We incorporate the micro-foundations of a health capital accumulationmechanism based on the Grossman model which endogenizes medical spending. Our researchmerges the workhorse models from the health economics and the macro/public �nance literatureto analyze the optimal income tax progressivity in the presence of income and health risk incombination with a realistic depiction of the U.S. health insurance system.

Our work contributes to a growing macro-public �nance literature that extends the Gross-man model of health capital accumulation (Grossman (1972)). This literature incorporateshealth 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 andSeshadri (2013a) and Jung and Tran (2016)). Jung and Tran (2015) explore the welfare im-plications of optimal health insurance policies. Di�erent from this paper, we quantitativelycharacterize the optimal progressivity of the income tax system while taking the redistribution

5

e�ects of the health insurance system into account. We then demonstrate how changes to thehealth insurance system a�ect the optimal level of income tax progressivity.

Our paper is connected to the literature on high marginal tax rates for top income earners.Diamond and Saez (2011) advocates for taxing labor earnings at the high end of the distributionat 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 capitalmodel. Guner, Lopez-Daneri and Ventura (2016) analyze how e�ective a progressive incometax system is in raising tax revenues. Kindermann and Krueger (2015) �nd that high marginallabor income tax rates are an e�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 closeto 90 percent. Di�erent from these studies, we focus on the optimal marginal tax rates acrossthe entire income distribution. Moreover, we base our analysis on a health capital model wherehealth risk is an additional source of heterogeneity in addition to labor market risk. We also�nd that very high tax rates at the top are an essential component of the optimal progressivetax system. More importantly, we highlight that such high optimal marginal tax rates at thetop are inter-dependent with the marginal tax rates set at the bottom of the income distributionand the government transfer policies already in place.

The paper is structured as follows. Section 2 describes the insurance and incentive trade-o�in a two-period model. Section 3 presents the full dynamic model. Section 4 describes ourcalibration strategy. Section 5 describes our experiments and quantitative results. Section 6is devoted to sensitivity analysis. Section 7 concludes. The Appendix presents all calibrationtables and �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 paysparticular attention to medical expenditures and �nancing. We use data from 1999�2009 totrack health, health expenditures, the sources of health �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 MEPSprovides two measures: the Short-Form 12 Version 2 (SF-12v2) and a self-reported healthstatus.1 Panel 1 of Figure 1 displays the physical and mental components of the SF-12v2 indexover the lifecycle. Young individuals report relatively high levels of both physical and mentalhealth (capital). Mental health exhibits a slight "M" shape over the lifecycle. Young individuals(around age 20) and very old individuals (around age 75 and higher) report the lowest mentalhealth status. Individuals in the age range between 40 and 55 have lower mental health statusthan younger cohorts in their thirties and older cohorts in their sixties. The average physicalhealth consistently decreases as an individual ages. In addition, we construct a "healthy" indexusing 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 eitherexcellent, very good, or good and unhealthy otherwise. This classi�cation is standard in the

1The SF-12v2 includes twelve di�erent health measures about physical and mental health. We use physicaland mental measures. Both measures use the same variables to construct the index but the physical healthindex puts more weight on variables measuring physical health components and the mental health index putsmore weight on variables measuring mental health components. Ware, Kosinski and Keller (1996) provides moredetails 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.

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literature. This lifecycle pattern of health is similar to the one using the physical component ofthe 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 youngages but increase signi�cantly at later ages. This is mainly driven by depreciation of healthover the lifecycle. On average, individuals in their twenties spend about $1, 500 per year onhealthcare whereas older individuals in their �fties spend about $4, 000 per year. After theage �fty, health expenditures rise very fast. The highest expenditures are incurred by the veryold towards the end of their life and amount to approximately $12, 000 on average per year in2009. This indicates that the accumulation of bad health shocks over the lifecycle drives upthe health expenditures of an individual. In order to get a sense for the distribution of healthexpenditures we report the Gini coe�cient of health expenditure per age group. Inequality inhealth expenditures decreases by age. The Gini coe�cient of health expenditures is very highat around 0.8 when individuals are younger than 40 and then sharply drops as individuals getolder. Higher Gini coe�cients at younger ages indicate that health expenditures among theyoung are much more concentrated than health expenditures of the old. This is probably dueto relatively rare, but catastrophic health events among the young. Lower Gini coe�cients atolder ages imply that the higher incidence of health problems at higher ages "equalizes" healthspending across individuals. Moreover, the availability of public health insurance programsplays a role in reducing uneven access to health care services and therefore evens out healthexpenditure di�erences across di�erent income groups as well.

Health �nancing. The U.S. health insurance system is a mixed system where public healthinsurance programs target the retired population (Medicare) and the poor (Medicaid) while themajority of working individuals obtain private health insurance via their employers. Panels 3and 4 of Figure 1 display the �nancing sources of health expenditures and the insurance take-uprates over the lifecycle. Despite the many di�erent types of insurances, there is a large numberof Americans lacking health insurance. The U.S. health insurance system failed to provideinsurance for about 50 million people in 2010. The employer-based group health insurancepolicies (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 healthyand young individuals do not have health insurance, either by choice or by circumstance. Thefraction of the uninsured is highest among young workers below 35. Medicaid picks up lessthan 10 percent of workers by covering low income individuals. Consequently, the system leavesabout 25 percent of the working population without health insurance which contributes to highinsurance premiums and poor risk pooling. Gruber (2008) points out the modal uninsuredperson is a member of the working poor class, whose income is below median income but abovethe federal poverty level and therefore not eligible for Medicaid and other government programs.

Private insurance reimbursements and out-of-pocket payments are the two major fundingsources for medical spending of the working age population. The fraction of health expenditure�nanced by private insurance and Medicaid decreases with age, whereas the fraction of healthexpenditures �nanced by out-of-pocket funds increases moderately. Around the retirement ageof 65 there is a big shift in the magnitude of �nancing from private insurance toward publicinsurance including Medicare, Veteran's bene�ts, and other state run insurance plans.

Health and income. Figure 2 presents the age-pro�les of wage income for unhealthy andhealthy individuals. There is a signi�cant gap between the wage income pro�le between the twogroups. Unhealthy individuals exhibit a lower income path over the lifecycle. This con�rmsthat health risk is an important source of lifetime inequality.

To better understand income and health risks we compute the coe�cients of variation forincome and health expenditures over the lifecycle in Figure 3. For comparison, we also include

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data from the Health and Retirement Survey (HRS). Average wage income varies over thelifecycle and follows a hump shape. However, the coe�cient of variation for wage income isfairly stable around 0.9 before age sixty, and then rises sharply after retirement age. Thecoe�cient of variation for household income tracks the coe�cient of variation for wage incomevery well for individuals of working age, and stays stable until the end of life. On the otherhand, the coe�cient of variation for health expenditure is four to �ve times larger and alsovaries sharply over the lifecycle. The coe�cient of variation for health expenditure is largestbetween age 20�30. Note that health expenditures are relatively low for young individuals butso is income. This indicates that health expenditure risk can be signi�cant for young individualsgiven their low income and their lack of access to credit. This is partly due to the structureof the U.S. health insurance system that lets many young individuals opt out of the healthinsurance system or does not provide enough �nancial aid to help cover insurance.

Overall, health expenditures make up a signi�cant fraction of income over the lifecycle.When comparing health expenditures as fraction of household income, we observe an increaseover the lifecycle. Figure 4 presents average health expenditures over average income andaverage out-of-pocket health expenditure over average income for each age group. At the end oftheir life individuals spend on average 60 percent of their income on health care. Medicare andMedicaid are the main sources of health �nancing for the elderly. As a consequence, the shareof out-of-pocket health expenditures as fraction of income is contained at less than 8 percentdespite 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 transfersystem and an individual's income and health risks.

Endowments and preferences. Individuals live for two periods with certainty. Theysupply labor elastically in period 1 and do not work in period 2. The individual is born withskill εI at the beginning of period 1 and faces health shocks εH at the beginning of period 2.These shocks are drawn from a joint distribution[

εIεH

]∼ f

([µIµH

],

[σ2I σIH

σIH σ2H

]),

with density f (εI , εH) where the average labor shock is normalized as µI = 0 and µH > 0 andσ2I > 0, σ2

H> 0 and σIH > 0. Health shocks will result in compulsory health care expenditures.Individuals derive utility from consumption in both periods and disutility from work in period1. Their lifetime utility is: u (c1)−φv (n)+βE [u(c2)] , where u (.) and v (.) are utility functionswith the usual properties, c1 and c2 are consumption in periods 1 and 2, respectively, n islabor supply, φ scales the level of disutility of labor, β is the time discount factor and E is theexpectations operator.

Government. The government runs two separate tax and transfer programs. The �rstprogram is a redistribution program where the government imposes a payroll tax τI to �nancea lump-sum payment of d in the �rst period. The second program is a public health insuranceprogram (Medicare) that covers a fraction (1 − ρ) of the health care expenditures εH in thesecond period. This program is �nanced by Medicare tax τH .

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Household problem. The household maximization problem can be summarized as

max{c1,c2,n,s}

{u (c1)− φv (n) + βE [u(c2)]} s.t.

c1 + s = (1− τI − τH)wnεI + d ,

c2 = Rs− ρεH ,where w is the wage rate, R is the interest rate, s is savings in period 1, d is the lump-sumtransfer and ρ is the coinsurance rate. The �rst order conditions are

∂n :u′ (c1) (1− τI − τH)w = φv′ (n) , (1)

∂s :u′ (c1) = βRE[u′ (c2)

]. (2)

Government problem. The government clears the two transfer programs separately sothat

τH

∫εI

wnεI =

∫εH

∫εI

(1− ρ) εHf (εI , εH) dεIdεH = (1− ρ)µH ,

τI

∫εI

wnεIf (εI) dεI =

∫εI

d× f (εI) dεI := D,

The required tax rates are given by

τH =(1− ρ)µH∫

εIwnεIf (εI) dεI

, (3)

τI =D∫

εIwnεIf (εI) dεI

. (4)

The government maximizes the following problem

V (τI , ρ) = max{τI ,ρ}

{∫εI

u (wn (1− τI − τH)− s)− φv (n) + βE [u (Rs+ d+ εI − ρεH)]

}s.t. (1) , (2) , (3) and (4).

When the government chooses τI and ρ via expressions (3) and (4) respectively, it automatically�xes D and τH . Since the population in each period is normalized to one we also have thatd = D.

Optimal taxation and insurance policy. The government �rst order conditions are

∂V

∂ρ= u′ (c1)

[w

(∂n

∂ρ(1− τI − τH)− n∂τH

∂ρ

)− ∂s

∂ρ

]− φv′ (n)

∂n

∂ρ+ βE

[u′ (c2)

[R∂s

∂ρ+∂d

∂ρ− εH

]],

(5)

∂V

∂τI= u′ (c1)

[w

(∂n

∂τI(1− τI − τH)− n

)− ∂s

∂τI

]− φv′ (n)

∂n

∂τI+ βE

[u′ (c2)

[R∂s

∂τI+∂d

∂τI

]],

(6)

where we know from expressions (3) and (4) that

∂τH∂ρ

= −µHwn

(n+ (1− ρ) ∂n∂ρ

n

),

∂D

∂τI= w

(n+ τI

∂n

∂τI

),

∂D

∂ρ= τIw

∂n

∂ρ.

9

Substituting these expressions and the �rm FOCs into the government FOCs (5) and (6) weget two government Euler equations

∂V

∂ρ= u′ (c1)µH

(n+ (1− ρ) ∂n∂ρ

n

)+ βE

[u′ (c2)

(τIw

∂n

∂ρ− εH

)]= 0,

∂V

∂τI= −u′ (c1)wn+ βE

[u′ (c2)w

(n+ τI

∂n

∂τI

)]= 0.

This system describes the trade-o� between current cost and future bene�ts of changes in thereplacement rate ρ and the labor tax rate τn which �nances lump-sum transfers in the secondperiod.

Thus, the optimal level of social insurance provided by an income tax system depends oneconomy-based fundamentals including preferences, the evolution of income and health risksover the lifecycle, and the structure of a health insurance system. In particular, the optimalincome tax rate depends on the structure of health and income shocks and level of socialinsurance provided by the health insurance program. In the next section, we formulate a morerealistic model of the U.S. economy and quantify the optimal degree of progressivity of the USincome tax system. In addition, we explore how di�erently designs of a health insurance systema�ect the optimal tax progressivity.

3 The model

3.1 Technologies and �rms

There are two production sectors in the economy, which are assumed to grow at a constantrate g. Sector one is populated by a continuum of identical �rms that use physical capital Kand e�ective labor services N to produce a non-medical consumption good c with a normalizedprice of one. Firms in the non-medical sector are perfectly competitive and solve the followingmaximization problem

max{K, N}

F (K,N)− qK − wN, (7)

taking the rental rate of capital q and the wage rate w as given. Capital depreciates at rate δin each period. Sector two, the medical sector, is also populated by a continuum of identical�rms that use capital Km and labor Nm to produce medical services m at a price of pm. Firmsin the medical sector maximize

max{Km, Nm}

pmFm (Km, Nm)− qKm − wNm. (8)

3.2 Demographics, preferences and endowments

The economy is populated with overlapping generations of individuals who live up to a maximumof J periods. Individuals work for J1 periods and are retired thereafter. Individuals surviveeach period with age dependent survival probability πj . Deceased agents leave an accidentalbequest that is taxed and redistributed equally to the working age population. The populationgrows exogenously at an annual rate n. We assume stable demographic patterns, so that age jagents make up a constant fraction µj of the entire population at any point in time. The relativesizes of the cohorts alive µj and the mass of individuals dying in each period µj (conditional

10

on survival up to the previous period) can be recursively de�ned as µj =πj

(1+n)yearsµj−1 and

µj =1−πj

(1+n)yearsµj−1, where years denotes the number of years per model period.

In each period individuals are endowed with one unit of time that can be used for work n orleisure. Individual utility is denoted by function u (c, n, h) where u : R3

++ → R is C2, increasesin consumption c and health h, and decreases in labor n. We assume a Cobb-Douglas typeutility function of the form

u (c, n, h) =

((cη ×

(1− n− 1[n>0]nj

)1−η)κ × h1−κ)1−σ

1− σ,

where nj is an age dependent �xed cost of working as in French (2005), η is the intensityparameter of consumption relative to leisure, κ is the intensity parameter of health servicesrelative to consumption and leisure, and σ is the inverse of the inter-temporal rate of substitution(or relative risk aversion parameter).

Individuals are born with a speci�c skill type ϑ that cannot be changed and that togetherwith their health capital hj and an idiosyncratic labor productivity shock εnj determines their

age-speci�c labor e�ciency e(ϑ, hj , ε

nj

). The transition probabilities for the idiosyncratic pro-

ductivity shock εnj follow an age-dependent Markov process with transition probability matrix

Πnj . An element of this transition matrix is de�ned as the conditional probability Pr

(εni,j+1|εni,j

),

where the probability of next period's labor productivity εni,j+1 depends on today's productivityshock εni,j .

2

3.3 Health capital

Health capital evolves according to hj = H(mj , hj−1, δ

hj , ε

hj

), where hj denotes current health

capital, hj−1 denotes health capital of the previous period, δhj is the depreciation rate of healthcapital and εhj is an idiosyncratic health shock. The exogenous health shock εhj follows a Markovprocess with age dependent transition probability matrix Πh

j . Transition probabilities to nextperiod's health shock εhj+1 depend on the current health shock ε

hj so that an element of transition

matrix Πhj is de�ned as the conditional probability Pr

(εhj+1|εhj

). Individuals can buy medical

services mj at price pm to improve their health capital. Speci�cally, the law of motion of healthcapital follows

hj =

Investment︷ ︸︸ ︷φjm

ξj +

Trend︷ ︸︸ ︷(1− δhj

)hj−1 +

Disturbance︷︸︸︷εhj . (9)

This law of motion is an extension of the deterministic framework in Grossman (1972). The�rst two components are indeed similar to the original deterministic form in Grossman (1972);meanwhile, the third component can be thought of as a random depreciation rate as in Grossman(2000).

2Heathcote, Storesletten and Violante (2017) model government spending in terms of public goods that enterthe preferences directly. In their setting, the utility from the consumption of public goods and social insuranceare two main channels of welfare gains. In our analysis, we abstract from the former and focus on risk sharingthrough social insurance programs. We model all public spending programs explicitly. Our modeling approachis similar to the one in Conesa and Krueger (2006). However, we take into account both labor productivity andhealth risks as well as the institutional setup of the U.S. healthcare sector.

11

3.4 Health insurance

In the benchmark economy we introduce the main features of the U.S. health insurance systembefore the implementation of the A�ordable Care Act in 2010. The health insurance marketconsists of private health insurance companies that o�er two types of health insurance policies:(i) an individual health insurance plan (IHI) and (ii) a tax deductible group health insuranceplan (GHI). Individuals are required to buy insurance one period prior to the realization of theirhealth shock in order to be insured in the following period. The insurance policy needs to berenewed each period. By construction, agents in their �rst period are thus not covered by anyinsurance. The government provides public health insurance with Medicaid for the poor andMedicare for retirees. To be eligible for Medicaid, individuals are required to pass an incomeand asset test. The health insurance state inj for workers can therefore take on the followingvalues:

inj =

0 not insured,1 Individual health insurance (IHI),2 Group health insurance (GHI),3 Medicaid.

After retirement (j > J1) all agents are covered by public health insurance which is a combina-tion of Medicare and Medicaid for which they pay a premium, premR.

An agent's total health expenditure in any given period is pinjm × mj , where the price of

medical services pinjm depends on insurance state inj . The out-of-pocket health expenditure of

a working-age agent is given by

o (mj) =

{pinjm ×mj , if inj = 0,

ρinj ×(pinjm ×mj

), if inj > 0

(10)

where 0 ≤ ρinj ≤ 1 are the insurance state speci�c coinsurance rates. The coinsurance ratedenotes the fraction of the medical bill that the patient has to pay out-of-pocket.3 A retiree'sout-of-pocket expenditure is o (mj) = ρR ×

(pRm ×mj

), where ρR is the coinsurance rate of

Medicare and pRm is the price that a Medicare patient pays for medical services.Insurance companies. Insurance companies o�er IHI to any working individual and

charge an age and health dependent premium, premIHI (j, h). In addition, workers are randomlyassigned to employers who o�er GHI which is indicated by random variable εGHI = 1. The GHIpremium, premGHI, is tax deductible and group rated so that insurance companies are notallowed to screen workers by health or age. If a worker is not o�ered group insurance fromher employer, i.e., εGHI = 0, the worker can still buy IHI. However, the worker is subjected toscreening and the insurance premium is not tax deductible. There is a Markov process thatgoverns the group insurance o�er probability. It is a function of the individual's permanent skill

type ϑ. Let Pr(εGHIj+1 |εGHIj , ϑ

)be the conditional probability that an agent has group insurance

status εGHIj+1 at age j + 1 given she had group insurance status εGHIj at age j. All conditionalprobabilities for group insurance status are collected in a 2 × 2 transition probability matrixΠGHIj,ϑ .For simplicity we abstain from modeling insurance companies as pro�t maximizing �rms

and simply allow for a premium markup ω. Since insurance companies in the individual marketscreen customers by age and health, we impose separate clearing conditions for each age-health

3For simplicity we include deductibles and co-pays into the coinsurance rate.

12

type, so that premiums, premIHI (j, h) , adjust to balance

(1 + ωIHIj,h

)µj

∫ [1[inj(xj ,h)=1]

(1− ρIHI

)pIHIm mj,h (xj,h)

]dΛ (xj,−h) (11)

= Rµj−1

∫ (1[inj−1,h(xj−1,h)=1]prem

IHI (j − 1, h))dΛ (xj−1,−h) ,

where xj,−h is the state vector for cohort age j not containing h since we do not want toaggregate over the health state vector h in this case. The clearing condition for the grouphealth insurances is simpler as only one price, premGHI, adjusts to balance

(1 + ωGHI

) J1∑j=2

µj

∫ [1[inj(xj)=2]

(1− ρGHI

)pGHIm mj (xj)

]dΛ (xj) (12)

= R

J1−1∑j=1

µj

∫ (1[inj(xj)=2]prem

GHI)dΛ (xj) ,

where ωIHIj,h and ωGHI are markup factors that determine loading costs (�xed costs or pro�ts).Variables ρIHI and ρGHI are the coinsurance rates, and pIHIm and pGHIm are the prices for healthcare services of the two insurance types. The respective left-hand-sides in the above expressionssummarize aggregate payments made by insurance companies whereas the right-hand-sides ag-gregate the premium collections one period prior. Since premiums are invested for one period,they enter the capital stock and we therefore multiply the term with the after tax gross interestrate R. The premium markups generate pro�ts which are redistributed in equal (per-capita)amounts of πpro�ts to all surviving agents.4

3.5 Fiscal policy

The government administers various government programs that are �nanced by a combinationof taxes.

Progressive income taxes. The government imposes a progressive income tax code onhousehold incomes. The tax schedule is given by a parametric function τ (y) = y − λy(1−τ),where τ (y) denotes net tax revenues as a function of pre-tax income y , τ is the progressivityparameter, and λ is a scaling factor to match the U.S. income tax revenue. This tax functionis fairly general and captures the common cases:

(1) Full redistribution: τ (y) = y − λ and τ ′ (y) = 1 if τ = 1,

(2) Progressive: τ ′ (y) = 1−<1︷ ︸︸ ︷

(1− τ)λy(−a1) and τ ′ (y) > τ(y)y if 0 < τ < 1,

(3) No redistribution (proportional): τ (y) = y − λy and τ ′ (y) = 1− λ if τ = 0,

(4) Regressive: τ (y) = 1−>1︷ ︸︸ ︷

(1− τ)λy(−τ) and τ ′ (y) < τ(y)y if τ < 0.

4Notice that ex-post moral hazard and adverse selection issues arise naturally in the model due to informationasymmetry. Insurance companies cannot directly observe the idiosyncratic health shocks and have to reimburseagents based on the actual observed levels of health care spending. Adverse selection arises because insurancecompanies cannot observe the risk type of agents and therefore cannot price insurance premiums accordingly.They instead have to charge an average premium that clears the insurance companies' pro�t conditions. Indi-vidual insurance contracts do distinguish agents by age and health status but not by their health shock.

13

This tax function has a long tradition in public �nance (see Musgrave (1959), Kakwani (1977))and was implemented into a dynamic setting by Benabou (2002) and more recently used inHeathcote, Storesletten and Violante (2017). This tax function is �exible and can be usedto model the transfer programs. Heathcote, Storesletten and Violante (2017) abstract frommodeling government transfer programs and allow negative income taxes to act as govern-ment transfers. We instead model government spending explicitly including Medicaid and FoodStamp programs for low income individuals. To avoid double counting we impose a non-negativetax payment restriction, τ (y) ≥ 0. This non-negative tax restriction eliminates all governmenttransfers embedded in the progressive tax function.5 In our benchmark setting, the progressiveincome tax function has the following speci�cation

τ (y) = max[0, y − λy(1−τ)

],

Spending programs. The government has the following spending programs: social secu-rity, social transfers to low income earners, public health insurance, and general governmentconsumption. The social security program operates on a Pay-As-You-Go (PAYG) principle inwhich the government collects a payroll tax τSS from the working population to �nance socialsecurity bene�ts tSS for retired individuals. The PAYG program is self-�nanced

J∑j=J1+1

µj

∫tSSj (xj) dΛ (xj) =

J1∑j=1

µj

∫taxSSj (xj) dΛ (xj) .

In addition, the government provides social insurance through a social transfer program(T SI) that guarantees a minimum consumption level, i.e., Food Stamp program. Public healthinsurance consists of Medicare and Medicaid. Medicare is �nanced by a Medicare tax (taxMed

j )

and premium payments (premR) and Medicaid is �nanced by general tax revenues. In addi-tion, the government needs to �nance government consumption (CG) which is exogenous andunproductive. Finally, the government collects accidental bequests from deceased individualsand redistributes them as lump-sum payments tBeqj to all surviving working-age households.

Balanced budget. The government collects consumption tax revenue at a �at rate, τC ,and income tax revenue at a progressive rate to balance its budget every period. The governmentbudget constraint is given by

CG + TSI +

J1∑j=2

µj

∫1[inj(xj)=3]

(1− ρMAid

)pMAidm mj (xj) dΛ (xj) (13)

+

J∑j=J1+1

µj

∫ (1− ρR

)pRmmj (xj) dΛ (xj)

=

J∑j=1

µj

∫ [τCc (xj) + taxj (xj)

]dΛ (xj) +

J∑j=J1+1

µj

∫premR (xj) dΛ (xj) +

J1∑j=1

µj

∫taxMed

j dΛ (xj) ,

where ρMAid and ρR are the coinsurance rate of Medicaid and of the combined Medicare/Medicaidprogram for the old, respectively.

The government collects and redistributes accidental bequests in a lump-sum fashion to allworking-age households

J1∑j=1

µj

∫tBeqj (xj) dΛ (xj) =

J∑j=1

µj

∫aj (xj) dΛ (xj) , (14)

5We will relax this restriction in our sensitivity analysis.

14

where µj and µj are measures of the surviving and deceased agents at age j in time t, respec-tively.

3.6 Household problem

Individuals at age j ≤ J1 are workers and thus exposed to labor shocks. Old individuals,j > J1, are retired (nj = 0) and receive pension payments. They do not face labor marketshocks anymore. The agent state vector at age j is given by

xj ∈ Dj ≡

(aj , hj−1, ϑ, ε

nj , ε

hj , inj

)∈ R+ ×R+ ×R+ ×R+ ×R− × Iw if j ≤ J1,(

aj , hj−1, ϑ, εhj , inj

)∈ R+ ×R+ ×R+ ×R− × IR if j > J1,

(15)

where aj is the capital stock at the beginning of the period, hj−1 is the health state at thebeginning of the period, ϑ is the skill type, εnj is the positive labor productivity shock, εhj is anegative health shock, inj is the insurance state and Iw = {0, 1, 2, 3} denotes the dimensionof the insurance state of workers and IR = {1} is the sole insurance state for retirees as everyretiree is on a combined Medicare/Medicaid program. After the realization of the state variables,agents simultaneously chose from their choice set

Cj ≡

{(cj , nj ,mj,aj+1, inj+1) ∈ R+ × [0, 1]×R+ ×R+ × Iw if j ≤ J1,

(cj ,mj,aj+1) ∈ R+ ×R+ ×R+ if j > J1,

where cj is consumption, nj is labor supply, mj are health care services, aj+1 are asset holdingsfor the next period and inj+1 is the insurance state for the next period in order to maximize theirlifetime utility. All choice variables in the household problem depend on state vector xj . Wesuppress this dependence in the notation to improve readability. The household optimizationproblem is

V (xj) = max{Cj}{u (cj , nj , hj) + βπjE [V (xj+1) | xj ]} s.t. (16)

(1 + τC

)cj + (1 + g) aj+1 + o (mj)

+ 1[j≤J1∧inj+1=1]premIHI (j, h) + 1[j≤J1∧inj+1=2]prem

GHI + 1[j>J1]premR

= yj − taxj + tSIj ,

0 ≤ aj+1, 0 ≤ nj ≤ 1 and (9) .

Variable τC is a consumption tax rate, o (mj) is out-of-pocket medical spending dependingon insurance type, yWj is the sum of all income including labor, assets, bequests, and pro�ts.Household income and tax payments are de�ned as

yj = e(ϑ, hj , ε

nj

)njw + 1[j > JW ]t

Socj (ϑ) +R

(aj + tBeq

)+ πpro�ts,

taxj = τ (yj) + taxSSj + taxMcarej ,

yj = yj − aj − tBeq − 1[inj+1=2]premGHI − 0.5

(taxSSj + taxMed

j

),

taxSSj = τSoc ×min(yss, e

(ϑ, hj , ε

nj

)njw − 1[inj+1=2]prem

GHI),

taxMcarej = τMcare ×

(e(ϑ, hj , ε

nj

)njw − 1[inj+1=2]prem

GHI),

tSIj = max [0, c+ o (mj) + taxj − yj ] .

Variable w is the market wage rate and R is the gross interest rate and πpro�ts denotes pro�tsfrom insurance companies. Variable yj is taxable income, τ (yj) is the progressive income

15

tax payment and taxSSj is the social security tax with marginal rate τSS that �nances the

social security payments tSSj . The maximum contribution to social security is yss. The socialinsurance payment tSIj guarantees a minimum consumption level c. If social insurance is paidout, then automatically aj+1 = 0, so that social insurance cannot be used to �nance savings.For each xj ∈ Dj let Λ (xj) denote the measure of age j agents with xj ∈ Dj . Then expressionµjΛ (xj) becomes the population measure of age-j agents with state vector xj ∈ Dj that is usedfor aggregation.

4 Calibration

For the calibration we distinguish between two sets of parameters: (i) externally selected param-eters and (ii) internally calibrated parameters. Externally selected parameters are estimatedindependently from our model and are either based on our own estimates using data fromthe Medical Expenditure Panel Survey (MEPS) or estimates provided by other studies. Wesummarize these external parameters in Table 2. Internal parameters are calibrated so thatmodel-generated data match a given set of targets from U.S. data. These parameters are pre-sented in Table 3. Model generated data moments and target moments from U.S. data arejuxtaposed in Table 4.6

4.1 Technologies and �rms

We impose a Cobb-Douglas production technology using physical capital and labor as inputs forthe �nal goods and the medical sector respectively: F (K,N) = AKαN1−α and Fm (Km, Nm) =AmK

αmm N1−αm

m . We set the capital share α = 0.33 and the annual capital depreciation rate atδ = 0.1. They are both similar to standard values in the calibration literature (e.g., Kydlandand Prescott (1982)). The capital share in production in the health care sector is lower atαm = 0.26 which is based on Donahoe (2000) and our own calculations. We abstract fromchanges in production technologies or other possible causes of excess cost growth in the U.S.health sector.

4.2 Demographics, preferences and endowments

One model period is de�ned as 5 years. We model households from age 20 to age 95 whichresults in J = 15 periods. The annual conditional survival probabilities, supplied by CMS, areadjusted for period length. The population growth rate for the U.S. was 1.2 percent on averagefrom 1950 to 1997 according to the Council of Economic Advisors (1998). In the model thetotal population over the age of 65 is 17.7 percent which is very close to the 17.4 percent in thecensus.

We choose �xed cost of working, nj , to match labor hours per age group. Parameter σ = 3.0and the time preference parameter β = 1.001 to match the capital output ratio and the interestrate. The intensity parameter η is 0.43 to match the aggregate labor supply and κ is 0.89 tomatch the ratio between �nal goods consumption and medical consumption. In conjunctionwith the health productivity parameters φj and ξ from expression (9) these preference weightsalso ensure that the model matches total health spending and the fraction of individuals withhealth insurance per age group.

We allow for 4 permanent skill types ϑ is skill type. The permanent skill types are de�ned asaverage individual wages per wage quartile. The e�ciency unit of labor, i.e., labor productivity,

6More details of our calibration strategy and the solution algorithm can be found in Jung and Tran (2016).

16

evolves over the lifecycle according to

ej (ϑ, hj , εn) = (ej,ϑ)χ ×

(exp

(hj − hj,ϑhj,ϑ

))1−χ

× εn for j = {1, ..., J1} , (17)

where ej,ϑ is the average productivity of labor of the (j, ϑ) types. We estimate ej,ϑ fromMEPS data using average wages which results in hump-shaped lifecycle earnings pro�les. Inaddition, labor productivity can be in�uenced by health. The idiosyncratic health e�ect on laborproductivity is measured as deviation from the average health hj,ϑ per skill and age group. Inorder to avoid negative numbers we use the exponent function. Parameter χ = 0.85 measuresthe relative weight of the average productivity vs. the individual health e�ect. Finally, theidiosyncratic labor productivity shock εn is based on Storesletten, Telmer and Yaron (2004).We discretize this process into a �ve state Markov process following Tauchen (1986).

4.3 Health capital

We use the health index Short-Form 12 Version 2 (SF −12v2) in MEPS data to measure healthcapital.7 The SF − 12v2 includes twelve health measures of physical and mental health. It iswidely used to assess health improvements after medical treatments in hospitals. The SF−12v2has continuous value and varies between between 0 (worst) and 100 (best).

We �rst de�ne a space for health capital in the model with a minimum health capital levelof hminm and a maximum health capital level of hmaxm . We set the maximum health capital levelhmaxm and map the health index from MEPS data to the health capital space in the model. Notethat, we normalize health capital and health production parameters according to the maximumhealth level. The lower bound of the health grid hmin

m is calibrated.We classify individual health status into four groups by age-cohort and health capital quartile

(i.e., group 1 has health capital in the 25th percentile whereas group 4 has health capital inthe top quartile). We assume that individuals in group 1 are in the best health status, sothat there is negligibly small or no health shock. Meanwhile, individuals in the other healthgroups experience negative health shocks. Group 2 experiences a �small� health shock, group3 experiences a �moderate� health shock, and group 4 su�ers from a �large� health shock. Thetransition probability matrix of health shocks Πh is calculated by counting how many individualsmove across health groups between two consecutive years in MEPS data where we also adjustfor period length.

In order to measure the magnitudes of health shocks, we compute the average healthcapital of group, hij,d with i = {1, 2, 3, 4}. The average health capital per age group is de-

noted{h1j,d > h2

j,d > h3j,d > h4

j,d

}. We measure the shock magnitude in terms of relative dis-

tance from an average health state of each group to the average health state of group 1,(h{i}j,d − h

1j,d

). The vector of shock magnitude in percentage deviation is de�ned as εh%

j ={0,

h2j,d−h1j,d

h1j,d,h3j,d−h

1j,d

h1j,d,h4j,d−h

1j,d

h1j,d

}. This vector is scaled by the maximum health capital level in

the model hmaxm and used as the shock levels in the model.

The natural rate of health depreciation δhj per age group is calculated by focusing on indi-viduals with group insurance and zero health spending in any given year. We then postulatethat such individuals did not incur a negative health shock in this period as they could easilya�ord to buy medical services m to replenish their health due to their insurance status. Thisallows us to back out the depreciation rate from expression (9).

7See Ware, Kosinski and Keller (1996) for further details about this health index.

17

To the best of our knowledge, there are no suitable estimates for health production processesin equation (9), especially within macro modeling frameworks. A recent empirical contributionby Galama et al. (2012) �nds weak evidence for decreasing returns to scale which implies ξ < 0.In our paper we calibrate ξ and φj together to match aggregate health expenditures and themedical expenditure pro�le over age (see Figure 5). We assume a grid of 15 health states forour calibration in order to reduce the computational burden.

4.4 Health insurance

Group Insurance O�ers. MEPS data contain information about whether individuals havereceived a group health insurance o�er from their employer i.e., o�er shock εGHI = {0, 1}.8 The

transition matrix Πh with elements Pr(εGHIj+1 |εGHIj , ϑ

)depends on the permanent skill type ϑ.

We then count how many individuals with a GHI o�er in year j are still o�ered group insurancein j + 1. We smooth the transition probabilities and adjust for the �ve-year period length.

Insurance premiums and coinsurance Rates. Insurance companies set premiums ac-cording to a person's age and health status. Premiums premIHI (j, h) will adjust to clear ex-pression (11) . Age and health dependent markup pro�ts ωIHIj,h are calibrated to match the IHItake-up rate by age group. Similarly, premGHI adjusts to clear expression (12) and the markuppro�t ωGHI is calibrated to match the insurance take-up rate of GHI. The coinsurance rateis de�ned as the fraction of out-of-pocket health expenditures over total health expenditures.Coinsurance rates therefore include deductibles and copayments. We use MEPS data to es-timate coinsurance rates of γIHI, γGHI, γMAid and γMcare for individual, group, Medicaid andMedicare insurance, respectively.

Price of medical services. The base price of medical services pm is endogenous as wemodel the production of medical services via expression (8). According to Shatto and Clemens(2011) we know that prices paid by Medicare and Medicaid are close to 70 percent of the pricespaid by private health insurance who themselves pay lower prices than the uninsured due totheir market power vis-a-vis health care providers (see Phelps (2003)). Various studies havefound that uninsured individuals pay an average markup of 60 percent or more for prescriptiondrugs as well as hospital services (see Playing Fair, State Action to Lower Prescription DrugPrices (2000), Brown (2006), Anderson (2007), Gruber and Rodriguez (2007)). Based on thisinformation we pick the following markup factors for the �ve insurance types in the model:[

pnoInsm , pIHIm , pGHIm , pMaidm , pMcare

m

]= [0.70, 0.25, 0.10, 0.0, −0.10]× pm.

When the experiments are run, this relative pricing structure is held constant so that Medicaidand Medicare remain the programs that pay the lowest prices for medical services. Thus,providers are assumed to not being able to renegotiate reimbursement rates.

4.5 Fiscal policy

Taxes. The consumption tax rate, τC , is set to 5 percent. We follow Guner, Lopez-Daneri andVentura (2016) and choose a1 = 0.053 as the progressivity level in the benchmark model. Wecalibrate the tax level parameter to match the relative size of the government budget so thata0 = 1.095.

Social security. In the model, Social Security bene�t payments are de�ned as a functionof average labor income by skill type: tSoc (ϑ) = Ψ (ϑ) × w × L (ϑ), where Ψ (ϑ) is a scaling

8We use OFFER31X, OFFER42X, and OFFER53X where the numbers 31, 42, and 53 refer to the interviewround within the year (individuals are interviewed �ve times in two years). We assume that an individual waso�ered GHI when either one of the three variables indicates so.

18

vector that determines the total size of pension payments as a function of the average wageincome by skill type. Total pension payments amount to 4.1 percent of GDP, similar to thenumber reported in the budget tables of the O�ce of Management and Budget (OMB) for 2008.The Social Security system is self-�nanced via a payroll tax of τSS = 9.4 percent similar toJeske and Kitao (2009). The Social Security payroll tax is collected on labor income up to amaximum of $97, 500.

Medicare and Medicaid. According to data from CMS (Keehan et al. (2011)) the share oftotal Medicaid spending on individuals older than 65 is about 36 percent. Adding this amount tothe total size of Medicare results in public health insurance payments to the old of 4.16 percentof GDP. Given a coinsurance rate of γR = 0.20, the size of the combined Medicare/Medicaidprogram in the model is 3.1 percent of GDP.9 The premium for Medicare is 2.11 percent of percapita GDP as in Jeske and Kitao (2009). The Medicare tax τMcare is 2.9 percent and is notrestricted by an upper limit (see Social Security Update 2007 (2007)).

According to MEPS data, 9.2 percent of working age individuals are on some form of publichealth insurance. We therefore set the Medicaid eligibility level in the model to 70 percent ofthe FPL (i.e., FPLMaid = 0.7×FPL), which is the average state eligibility level (Kaiser (2013))and calibrate the asset test level, aMaid, to match the Medicaid take-up rate.10 Setting the agedependent coinsurance rate for Medicaid γMaid

j to MEPS levels, Medicaid for workers is 0.5

percent of GDP in the model which underestimates Medicaid spending of workers in MEPS.11

Overall, the model results in total tax revenue of 21.8% of GDP and residual (unproductive)government consumption of 12 percent. The latter adjusts to clear the government budgetconstraint (13) .

4.6 The benchmark model

Figures 5, 6 and 8 and Table 4 in the Appendix show that the benchmark model matchesthe relevant elements of the MEPS data quite well. The model closely tracks average medicalexpenditures by age group (Figure 5, Panel [1]) and reproduces the extremely right skeweddistribution of health expenditures shown (Figure 5, Panel [2]). Overall, the model generatestotal health expenditures of 12 percent of GDP. In addition, the model matches the insurancetake-up percentages of IHI, GHI and Medicaid by age group as shown in Figure 5, Panels [3],[4] and [5] respectively.

The model reproduces the hump-shaped patterns of asset holdings (Figure 6, Panel [1]).However, the lack of a formal bequest motive in the model generates a shift in asset holdingsfrom retirees to the working age population. On the other hand, the model provides a close�t to the average household income over the lifecycle (Figure 6, Panel [2]). Retired individualsdecrease their consumption faster than in the data which is a result of the low asset holdingsof the elderly (Figure 6, Panel [3]). The model provides a close �t for the lifecycle pattern oflabor supply (Figure 6, Panel [4]).

Figure 7 displays health accumulation and wage income by health status. The model repro-duces the fact that unhealthy individuals follow a lower income path over the life cycle. Healthrisk is important source of lifetime inequality in our model. Figure 8 compares the model in-

9Our model cannot match the NIPA number because it is calibrated to MEPS data which only accounts forabout 65-70 percent of health care spending in the national accounts (see Sing et al. (2006) and Bernard et al.(2012)).

10All model experiments that expand the Medicaid program are percentage expansions based on the modelthreshold, FPLMaid. Compare Remler and Glied (2001) and Aizer (2003) for additional discussions of Medicaidtake-up rates.

11Overall Medicaid spending in MEPS, workers and retirees, accounts for about 0.95 to 1.02 percent of GDPaccording to Sing et al. (2006), Keehan et al. (2011) and Bernard et al. (2012).

19

come and wage distribution to data from MEPS. The model matches the lower and upper tailsof the income distribution with around 12 percent individuals with income below 133 percentof the Medicaid eligibility level (MaidFPL). Finally, Table 4 compares �rst moments from themodel to data moments from MEPS, CMS, and NIPA.

5 Quantitative analysis

We �rst characterize the optimal income tax system for the benchmark economy using a 2-parameter income tax function following Benabou (2002). We then analyze how the optimallevel of progressivity changes after the introduction of the A�ordable Care Act of 2010 whichis a large scale reform that redistributes income from high income and healthy individuals tolow income and sicker individuals. Finally, we analyze to what extent the particular parametricassumption about the income tax polynomial a�ects our results by using di�erent parameteri-zations from the literature.

5.1 The optimal degree of tax progressivity

The government problem. In order to characterize the optimal level of progressivity inthe tax function in this large-scale OLG model with uninsurable idiosyncratic risk and en-dogenous health spending we follow the Ramsey tradition and restrict the choice dimensionof the government similar to Conesa, Kitao and Krueger (2009). We assume that the socialwelfare function�de�ned as the ex-ante lifetime utility of an individual born into the sta-tionary equilibrium�depends on the two parameters of the income tax polynomial so thatWF (λ, τ) =

∫V (xj=1|λ, τ) dΛ (xj=1). The government's objective is to choose the tax pa-

rameter values {λ, τ} in order to maximize the social welfare function taking the decision rulesof consumers and �rms as well as competitive equilibrium conditions into account. All otherpolicy variables are kept unchanged.

We implement the maximization by searching over a grid of values for progressivity parame-ter τ while letting the scaling parameter λ adjust to keep the government budget (23) balanced.The level of exogenous government consumption CG remains identical to the benchmark gov-ernment consumption level CG. The tax maximization problem can be written as

WF ∗ = max{λ, τ}

∫WF (λ, τ) (18)

s.t.J∑j=1

µj

∫taxj (λ, τ, xj) dΛ (xj) = CG + TSI (λ, τ) + Medicaid(λ, τ) + Medicare(λ, τ)

− τCC (λ, τ)−Medicare Prem(λ, τ) - Medicare Tax(λ, τ),

where the terms on the right hand side of the constraint are aggregates that depend on the taxparameters due to tax distortions and general equilibrium price e�ects.

Macroeconomic aggregates. Macroeconomic and welfare e�ects are summarized in Table5. In the optimal tax system the government cuts taxes for households with incomes below US$60, 000 per year and imposes higher taxes on households with incomes higher than that. Thetax increases distort individuals' incentives to save and work. Capital in the non-medical andmedical sectors decreases. Weekly hours worked decrease from 29.4 hours to 29.0 hours. Thesedistortions subsequently lead to e�ciency losses and lower GDP by about 6 percent.

20

The optimal progressivity of income taxes. Figure 9 presents the results of the taxoptimization. The optimal income tax system includes a tax break for income earners belowUS$ 38, 000 followed by a jump in the marginal tax rate to 25 percent and a steep increasethereafter to top marginal rates of over 40 percent and over 50 percent for income earners aboveUS$ 100, 000 and above US$ 200, 000, respectively. The optimal progressivity level isτ∗ = 0.247, which is higher than the U.S. benchmark case of τ = 0.053. In Panel [1] of Figure 9 wecompare the tax burdens of the U.S. benchmark to the optimal tax system. Under the optimaltax system (red dotted line) poor and low income households pay almost zero taxes, whereashigher-middle and high income households pay signi�cantly more than under the benchmarkcase. Panel [2] presents the average tax rates per income group and Panel [3] displays themarginal tax rates. The working poor class, who are not at the bottom but below medianincome level, are bene�t most from this long tax break. The tax burden is shifted to individualsat the upper end of the income distribution. The optimal income tax system is indeed moreprogressive than the benchmark system.

The optimal tax system in our model is more progressive than the optimal tax systems inprevious studies that abstract from health and health risk. For comparison, we plot the optimaltax systems in Conesa and Krueger (2006) and Heathcote, Storesletten and Violante (2017).Heathcote, Storesletten and Violante (2017) use a similar progressive income tax function withno non-negative tax restriction. They �nd an optimal progressivity parameter of τ∗ = 0.084 anda scaling parameter of λ = 0.233. The optimal income tax system is less progressive than onein the benchmark model, where the estimated progressive parameter is τ = 0.161. Note that,they do not explicitly track Social Security, Medicare and Medicaid. In fact, they include allgovernment spending programs by adding public goods consumption (CG) into household pref-erences. Conesa and Krueger (2006) use a di�erent parametric speci�cation of the tax functionbased on Gouveia and Strauss (1994). Their optimal tax is a proportional tax of 17.2 percentwith a �xed deduction of about US$ 9, 400. Health risk and institutional details of the U.S.health insurance system are abstracted. Labor productivity risk is the only source of householdheterogeneity. Retirees are not exposed to income shocks after retirement. Di�erently, individ-uals are exposed to both risks through life course in our framework, especially more to incomerisk during working ages and more to health risk during retirement ages.

Our result is in line with the more recent literature on income taxation which also �nds muchhigher marginal tax rates in the range of 75− 90 percent for top income earners than previousstudies (e.g., Diamond and Saez (2011), Badel and Huggett (2015) and Kindermann and Krueger(2015)). In particular, Kindermann and Krueger (2015) show that very high marginal tax ratesfor the top one percent are primarily driven by the social insurance bene�ts that these hightaxes imply. In their model�in order to match the very high concentration of labor earningsand wealth in the data�they require that households have the opportunity to work for veryhigh wages with very low probability. Then, as a result of precautionary motives, the laborsupply of these households is not strongly a�ected by high marginal taxes. The intuition isthat during the periods of high labor productivity households work hard and earn the majorityof their lifetime income. There is a similar mechanism in our framework where households areexposed to health risks throughout the lifecycle. Precautionary motives are relatively strongin our setting because households face severe health shocks at the end of their life as shown inDe Nardi, French and Jones (2010). Labor supply and savings of high skill households are notstrongly a�ected by high marginal tax rates. From the social welfare perspective it is optimalto impose very high rates on high income households so that the government can provide socialinsurance against idiosyncratic earnings- and health risk through progressive taxes.

In panel [4], we present the income distribution of the benchmark case and the optimaltax case. The fraction of the poor households with income below US$ 20, 000 decreases in

21

the optimal tax system. Income inequality decreases after introducing the optimal tax system.The Gini coe�cient for after tax income decreases from 0.38 to 0.31. The increase in incomeequality is due to the more progressive income tax system as well as general equilibrium e�ectsthat reduce the number of top income earners in the population.

Suits index. In order to obtain a better measure of how the level of tax progressiv-ity changes under di�erent tax systems we follow Suits (1977) and construct Lorenz-type taxcontribution curves and the Suits index. The Suits index is an aggregate measure of tax progres-sivity which is widely used in the empirical public �nance and tax policy literature. However,it is rarely used in the macro/public �nance literature. Intuitively, the Suits index measuresconcentration of aggregate tax contributions by income group. Figure 11 illustrates the tax-income Lorenz curve and its relationship to the Suits index. These Lorenz-type curves for taxcontributions of the lowest to highest income group provide an aggregate measure of tax pro-gressivity and the relative contributions by income group. The Suits index is in essence a Ginicoe�cient for tax contribution inequality. It varies from +1 (most progressive) where the entiretax burden is allocated to members of the highest income bracket, through 0 for a proportionaltax, and to −1 (most regressive) where the entire tax burden is allocated to members of thelowest income bracket.

Panel [1] in Figure 12 presents the standard Lorenz curve for gross income. Panel [2] displaysthe Suits curve for income tax contributions by income group. Panel [3] presents the Suits curvefor the total tax contribution (progressive income tax and other taxes) by income group.

As seen in panel [2], the Suits curve for the income tax contribution in the optimal taxsystem is �atter at the bottom and steeper at the top compared to the benchmark tax system.This indicates that there are signi�cant changes in the allocation of tax burdens across incomegroups. The Suits index is 0.17 in the benchmark economy. The optimal tax system is veryprogressive with a Suits index around 0.53.

Welfare. The optimal tax system improves risk sharing across agents and redistributesincome toward low income households which can result in welfare gains if distortions of thetax system remain small enough. In order to assess the variation of welfare e�ects across theincome distribution, we compute compensating consumption by permanent income (skill) types.As expected, we �nd that the welfare e�ects vary signi�cantly across the four permanent skilltypes. Workers with medium and high skill levels experience welfare losses, while low skillworkers experience welfare gains in the new steady state. The welfare gains are mainly due toimprovement in health conditions for the unhealthy low income type individuals. Under thebenchmark income tax and health insurance systems, many low skill individuals do not investenough in their health because of limited access to the health insurance system. Under theoptimal income tax system, with the tax break at the lower end of income distribution, theseunhealthy individuals are in better position to improve their health conditions. The medicalspending of the uninsured indeed increases by 15 percent. The tax break at the lower endof the income distribution is important as it provides subsidies to the unhealthy low incomeindividuals who left out of the health insurance system. That is, the progressive income taxsystem supplements the incomplete social insurance of the health insurance system. The welfaregain for the lowest skill type can be large at up to 21 percent of lifetime consumption whereaswelfare losses can amount to 33 percent for the highest income types. Overall, there is a netwelfare gain of 5.5 percent at the aggregate level when switching to the optimal income taxsystem.

The positive welfare outcome indicates that the welfare gains resulting from better risksharing and redistribution dominate the welfare losses caused by tax distortions. Thus, theprogressive income tax system is an important channel to provide social insurance for unhealthylow income individuals who are left out of the pre-2010 US health insurance system.

22

5.2 Health insurance and tax progressivity

The health insurance system provides a mechanism to insure against health risk and incomerisk as it redistributes income to low income and relatively sicker households. The design of thehealth insurance system has implications for the optimal degree of tax progressivity. In thisesection we analyze how the design of a health insurance system changes the optimal degree ofprogressivity of the income tax system. We consider three alternative health insurance systems:(i) The US health insurance system with the introduction of the A�ordable Care Act (hereafter,ACA); (ii) Universal health insurance system: the government provides health insurance for allindividuals.

5.2.1 The A�ordable Care Act (ACA)

We study how the ACA a�ects the optimal progressive income tax system.The ACA represents the most signi�cant reform of the U.S. health care system since the

introduction of Medicare in 1965. The key policy instruments embedded in the ACA are: (i)an insurance mandate enforced by penalties, (ii) the introduction of insurance exchanges withpremium subsidies, (iii) a Medicaid expansion and (iv) new taxes on high income earners. Assuch the ACA provides a large redistribution program from healthy high income types to sickerlow income types as shown in Jung and Tran (2016). In this section we analyze how the ACAchanges the optimal progressivity level of the U.S. income tax system.

The following features of the ACA are added to the model. First, we introduce a penaltyof 2.5 percent of taxable income on workers without health insurance which enters the budgetconstraint as

penaltyj = 1[insj+1=0] × 0.025× yj ,

where 1[insj=0] is an indicator variable equal to one if the household has no health insurance.12

Furthermore, we do not allow IHI companies to screen anymore. The price setting in GHIand IHI markets is now similar, except for the fact that IHI premiums are not tax deductible.Second, workers who are not o�ered insurance from their employers are eligible to buy healthinsurance through insurance exchanges at subsidized rates according to

subsidyj =

max(

0, premIHIj − 0.03yj

)if 1.33 FPLMaid ≤ yj < 1.5 FPLMaid,

max(

0, premIHIj − 0.04yj

)if 1.5 FPLMaid ≤ yj < 2.0 FPLMaid,

max(

0, premIHIj − 0.06yj

)if 2.0 FPLMaid ≤ yj < 2.5 FPLMaid,

max(

0, premIHIj − 0.08yj

)if 2.5 FPLMaid ≤ yj < 3.0 FPLMaid,

max(

0, premIHIj − 0.095yj

)if 3.0 FPLMaid ≤ yj < 4.0 FPLMaid.

(19)

The subsidies ensure that the premiums that an individual pays at the health insurance exchangefor IHI will not exceed a certain percentage of her taxable income yj at age j. Third, the ACAexpands the Medicaid eligibility threshold to 133 percent of the FPL and removes the assettest. After the reform is implemented all individuals with incomes lower than 133 percentof the FPLMaid will be enrolled in Medicaid. Finally, the reform is �nanced by increases incapital gains taxes for individuals with incomes higher than $200, 000 per year (or $250, 000for families). In the model we use a �at income tax on individuals with incomes higher than

12 We do not model employer penalties.

23

$200, 000. Summarizing, we can write the new household budget constraint with the ACA as(1 + τC

)cj + (1 + g) aj+1 + oW (mj) + 1[inj+1=1]prem

IHI + 1[inj+1=2]premGHI

= yj + tSIj − taxj − 1[inj+1=0]penaltyj + 1[inj+1=1]subsidyj − taxACAj .

The optimal tax system. We solve for a new steady state with the described featuresof the ACA as our new benchmark. We then use the new level of unproductive governmentconsumption CG and solve the government maximization problem (18) with the ACA in placefor the optimal tax progressivity rate τ∗. We report the results of this exercise in Table 6,column [2] and Figure 10.

The optimal progressivity level is τ∗ = 0.24 in the economy with the ACA insurance system.It is slightly lower than τ∗ = 0.247 in the benchmark case prior to the ACA. Figure 10 comparesthe two optimal tax systems: one before ACA and one after ACA. As seen in panel [3], theoptimal marginal tax schedule shifts left after introducing the ACA. This indicates that the�xed deduction is smaller and the marginal tax rates are larger for each income group whichimplies less progressivity overall as measured by the Suits index. Households with annual incomearound $35,000 would have to pay taxes in the ACA-case whereas before they were not taxedat all. The ACA provides a signi�cant redistribution of wealth from healthy high income typesto sicker low income types through subsidies and the expansion of Medicaid. The governmentfactors in the redistribution that is already in place through the ACA when it optimizes theincome tax code and the result is that the new optimal tax system is slightly less progressivethan before with a slightly lower Suits index of 0.52.

Welfare. The welfare e�ects are similar to the baseline analysis. The welfare gain for thelowest skill type is relatively larger, up to 22 percent of lifetime consumption and the welfareloss the highest income types is also larger, up to 36 percent. Overall, there is still a netwelfare gain of 3.14 percent at the aggregate level when switching to the optimal income taxand ACA health insurance systems. Yet, the welfare gains resulting from better risk sharingand redistribution still dominate the welfare losses caused by tax distortions.

5.2.2 Universal public health insurance

In this section we consider an experiment in which the government extend public health insur-ance for all individuals. The out-of-pocket health expenditure is given by

o (mj) = ρ× (pm ×mj) (20)

Speci�cally, we assume that the coinsurance rate ρ = 0.2 which is identical to the calibratedvalue of Medicare in the US version of the model. We refer to this universal public healthinsurance system as Medicare for all. We keep all the structural parameter values for preferences,technologies, labor productivity and health shocks from the US-calibration, but all privateinsurance programs are eliminated. We also maintain social security, foodstamp programs,government consumption and taxes as in the benchmark model. The universal medicare systemis self-�nanced by a payroll (Medicare) tax.

We follow a similar procedure to search for the optimal progressive income tax system.We display the optional income tax schedule in Figure 10. The e�ects of switching from thebenchmark model to the optimal tax system with the universal health insurance system arereported in column [4] of Table 6.

Macroeconomic aggregates. The tax increases to �nance the universal public healthinsurance largely distort individuals' incentives to save and work. Weekly hours worked decreasefrom 29.4 hours in the benchmark model to 27.06 hours. The change in the health insurance

24

sector a�ects the sectoral production structure and the allocation of capital and labor acrosssectors. Capital in the non-medical sector decreases by 14 percent; meanwhile, capital in themedical sector is almost unchanged. Output in the non-medical sector drops by 10 percent;meanwhile, output in the medical sector increases by 2.5 percent. These distortions lead tosigni�cant e�ciency losses and lower GDP by about 10 percent.

The optimal tax system. Universal Medicare provides (social) health insurance for alltypes of individuals in the economy. The poor working class, who are left out in the pre-2010healthcare system, are now coverd by in the universal health insurance system. As a result,the demand for social insurance prodivided through the progressive income tax system is nowlesser. Our results show that when Medicare is extended to everyone the optimal degree ofprogressivity decreases signi�cantly. The optimal tax schedule shifts down toward zero as seenin Figure 10. The tax break at the lower end of the income distribution is reduced to around US$22, 000, which is much smaller than US$ 38, 000 in the benchmark model. The marginal incometax rates are relatively lower. The rates imposed on incomes above US$ 200, 000 are reducedto 32 percent, compared to over 50 percent in the bechmark model. This reduction is mainlydriven by a relatively smaller need for tax revenue from the top end of income distribution.Thus, the expansion of public (social) health insurance signi�cantly reduces the demand forsocial insurance provided through the progressive income tax system. Interestingly, this newoptimal income tax code looks relatively closer to the one in Conesa and Krueger (2006).

Welfare. The welfare e�ects of switching to the optimal progressive income tax systemare di�erent when the universal public health insurance system is already in place. There arestill large welfare gains for the lowest skill type group, but large welfare losses for all otherskill groups when comparing to the steady state welfare in the benchmark model. The overalle�ect is signi�cantly negative. The net welfare loss is -3.4 percent in terms of compensatingconsumption at the aggregate level. Welfare losses caused by the distortions on incentivesgains dominate the welfare gains resulting from improving risk sharing and income distribution.This welfare result indicates that the social insurance role of a progressive income tax systeminteracts with the social insurance role of the public health insurance system.

6 Sensitivity analysis

In this section we conduct sensitivity analysis for three cases: (i) No health insurance, (ii) Nohealth risk, and (iii)Alternative progressive income tax functions.

6.1 Self insurance only

We consider a hypothetical economy in which the health insurance system is removed. In thissetting, individuals rely their own income to cover health expenditures. The out-of-pockethealth expenditure is given by

o (mj) = pm ×mj , (21)

We refer to this version of the model as self insurance only from here onward. The modelstill inherits the structural parameter values for preferences, technologies, labor productivitiesand health shocks from the US calibration. However, the health insurance system is eliminated.All �scal policies including social security, food stamp, general government consumption andtaxes are kept unchanged. We solve the government maximization problem (18) with no healthinsurance and �nd the optimal tax progressivity rate τ∗ .

We display the optimal progressive tax schedule in Figure 10 and report the macroeconomicand welfare e�ects in column [5] of Table 7. For comparison we �rst repeat steady state valuesin the benchmark in Column [1].

25

In the environment where there is no health insurance, individuals are exposed more tohealth risks, especially at the end of life cycle. They rely on private savings and labor incomesto cover their healthcare costs. They therefore work longer and save more in order to insure themself against health shocks. These precautionary motives raise capital accumulation and laborsupply at aggregate level, which subsequently lead to a large expansion of the economy. Morespeci�cally, output increases by 7 percent, compared to the benchmark economy. Consumptionof non-medical goods increases by 4.5 percent whereas consumption of medical goods decreases7 percent.

6.2 No health risk

We study how health risk a�ect the optimal degree of tax progressivity. We consider a case inwhich health capital evolves over the lifecycle as

hj =

Investment︷ ︸︸ ︷φjm

ξj +

Trend︷ ︸︸ ︷(1− δhj

)hj−1. (22)

This law of motion is an extension of the deterministic framework in Grossman (1972). Wesolve the government maximization problem (18) with this law of motion for health capital. Wesummarize the results in column [6] of Table 7.

6.3 Alternative tax functions

We analyze how robust our results are with respect to a di�erent tax function speci�cation.Two parameter speci�cation. We �rst relax the assumption that the tax payment has

to be non-negative and use the original speci�cation as in Benabou (2002)

τ (y) =[y − λy(1−τ)

].

This speci�cation allows for additional government transfers for poor households. We re-calibrate the model and solve again for the optimal tax progressivity where λ∗ = 1.687 andτ∗ = 0.1666 . We again report marginal tax rates, average tax rates and tax payments byincome group in Figure 14. The optimal marginal tax rates are negative for households withvery low income. This implies that the poor households receive transfers from the governmentvia the tax system itself. The tax payments become positive only when household incomes riseabove US$ 5, 000. The marginal tax rates are around 20 percent when household incomes reachUS$ 30, 000. Households with incomes above US$ 100, 000 face marginal tax rates of over 35percent.

We �nd that the marginal tax rates are substantially lower when removing the restrictionof non-negative tax contribution as the entire tax function shifts downward. The households atthe bottom of the income distribution now receive transfers through negative tax rates, whilethe households at the top pay less in taxes in the new optimal tax system. It is important tonote that taxable incomes includes labor and capital incomes. The negative marginal tax ratesinduces low income individuals to work and save more as the amount of transfer bene�ts dependson their labor and capital incomes. In our model, there are two social insurance programs forthe poor, which are means-tested. The negative tax rates partially mitigate the adverse e�ectsof the asset and income tests. This subsequently has implications for aggregate e�ciency andincome distribution in our general equilibrium setting. Our results indicate that the governmenttransfers to the low income a�ect the shape of the optimal marginal tax function, especially taxrates for top income individuals.

26

The optimal progressive income tax system is more progressive than the one in Heathcote,Storesletten and Violante (2017). The main reason is that we have a di�erent modelling ap-proach. We model health is an important source of heterogeneity across individuals and overlife cycle. In adddition, we explicitly model the main social insurance programs including FoodStamp, Social Security, Medicaid and Medicare. The di�erent results illustrate the quantitativeimportance of accounting for health, health risk and health insurance system in determiningthe optimal degree of progressivity of the income tax system.

Three parameter speci�cation. We now consider a commonly used polynomial forprogressive income taxation based on Gouveia and Strauss (1994)

τ (y) = τ0

[y −

(y−τ1 + τ2

)−1/τ1],

where y is taxable income. This speci�cation has three parameters and covers several specialcases. For τ1 = −1, it is a constant tax independent of income τ (y) = −λτ2. For τ1 → 0, it isa purely proportional system. For τ1 > 0 it is a progressive tax system. Guner, Kaygusuz andVentura (2014) estimate this three parameter speci�cation using U.S. public use tax data and�nd it performs better than the two parameter speci�cation.

In our benchmark calibration we use τ0 = 0.258, τ1 = 0.768 and τ2 = 0.031. The �rst twoparameter values are from Gouveia and Strauss (1994), while the last parameter is calibrated tobalance the government budget in the benchmark economy. We solve for an optimal taxationproblem in which the government problem is to choose the tax parameter values {τ0, τ1, τ2}in order to maximize the social welfare function accounting for the optimal decisions of �rmsand households and competitive equilibrium conditions. Our results show that the parametervalues for the optimal tax system are τ0 = 0.517, τ∗1 = 2.167. and τ∗2 = 0.000108. Figure 15presents the results for the three parameter speci�cation of the tax polynomial.

As seen in Panel [3] the optimal tax system starts from very low marginal tax rates on lowincome agents, but marginal tax rates increase sharply as household income rises. Householdswith income around US$ 40, 000 face a similar marginal tax rate of around 17 percent. However,middle income households with income higher than US$ 50, 000 face much higher marginal taxrates in our setting. Households with income over US$ 90, 000 pay marginal tax rates above40 percent. The optimal tax system is di�erent from the ones found in previous studies. Inparticular, Conesa and Krueger (2006) use the same tax polynomial and search for the optimalprogressivity of the U.S. income tax code in a heterogeneous household model. However, they donot model health risk and a detailed representation of U.S. health insurance system. They �ndthat households with income higher than US$ 9, 400 pay the equivalent of a proportional taxof 17.2 percent. We �nd that the optimal progressive income tax system is more progressive inour model. In our model, individuals at the lower end of income distribution pay relatively lesstaxes; meanwhile, the ones at the upper end pay relatively more taxes. That is, the lower taxesare an indirect mechanism to provide social health insurance for the low income who are morelikely in bad health conditions, but do not have access to Medicaid or private health insurance.Yet, the social insurance role of the progressive income tax system is more pronounced ourmodel, so that it supplements the incomplete role of the social health insurance system.

Interestingly, the new optimal tax system under a three parameter speci�cation is quitedi�erent from the optimal tax system with a two parameter speci�cation in the previous section.In particular, the marginal tax rates for low and middle income households are di�erent betweethe two optimal systems, compared Figure 15 and Figure 14. The marginal tax rates forthose households are relatively higher in the three parameter speci�cation. Interestingly, themarginal tax rates for households at the top of the income distribution are quite similar undertwo di�erent speci�cations. Guner, Kaygusuz and Ventura (2014) provide estimates of di�erent

27

tax functions using micro data from the U.S. Internal Revenue Service. They argue that theirestimates are ready to use for applied work in macroeconomics and public �nance. Our resultimplies that the optimal progressive income tax system is sensitive to the speci�cation of thetax polynomial.

In addition, we compare the e�ects of the ACA reform on the optimal income tax systemwith the three parameter speci�cation. We present the e�ects in Figure 16. The introduction ofthe ACA results in similar e�ects as under the tax regime model with a 2−parameter polynomial.The optimal tax function is shifts to the left and the welfare maximizing progressivity level ofthe income tax code decreases. This �nding is similar to our earlier results with the exceptionthat the welfare gains measured in terms of compensating consumption are somewhat smaller.

7 Conclusion

In this paper we investigate the optimal level of progressivity in income taxation in a Bewley-Grossman model of health capital where individuals face uninsurable income and health risksover the lifecycle. In our setting health a�ects the labor market productivity of workers sothat health serves as a consumption and as an investment good. Individuals subsequentlysmooth their consumption in the presence of idiosyncratic earnings shocks using a limited setof instruments. In addition to earnings shocks, individuals are also exposed to idiosyncratichealth shocks. Individuals choose to invest in their health via purchases of medical services.The income and wealth distribution is a function of both stochastic, persistent, and exogenousearning risk and health risk and endogenous health capital accumulation.

We calibrate the benchmark model to match the U.S. economy in 2010. We then quantita-tively characterize the optimal progressivity of the U.S. income tax system. We show that theoptimal system is very progressive with much higher levels of progressivity than the U.S. bench-mark income tax system. Our �ndings highlight that the progressive income tax system playsa key role in shaping the income distribution across households. Income inequality decreasesunder the optimal system as measured by after-tax income Gini coe�cients. A fundamental taxreform that switches the current U.S. income tax system to a more progressive welfare maxi-mizing system results in large welfare gains. However, this switch generates winners and losersin welfare terms. The poor and lower-middle income households gain because of lower taxeswhereas high income earners su�er from higher taxes. Implementing such a reform is thereforepolitically problematic.

In addition, we quantify the optimal progressivity of the income tax code after the imple-mentation of the ACA, a large scale health insurance reform that redistributes income from highincome healthy types to low income sicker types. Our results indicate that the optimal level ofprogressivity is a�ected by the interaction between income and health risks over the lifecycleas well as on the design of the health insurance system. Finally, we �nd that the parametricspeci�cation of the progressive tax function matters quantitatively for the design of the optimalprogressive income tax system.

In this paper we only focus on the progressive income tax system. Broader tax instrumentscan be investigated in our framework. For example, a switch from a progressive income tax to aprogressive consumption tax is an interesting case. Analyzing the optimal design of the wholetax and transfer system in this environment is another step forward. We leave these issues forfuture work. In addition, we leave bequest motives, health state dependence of survival andtransition dynamics for future extensions.

28

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32

8 Appendix

8.1 Appendix A: Recursive equilibrium

Given transition probability matrices{

Πnj

}J1j=1

and{

Πhj

}Jj=1

, survival probabilities {πj}Jj=1

and exogenous government policies{tax (xj) , τ

C , τSS , c, yss}Jj=1

, a competitive equilibrium is

a collection of sequences of distributions {µj ,Λj (xj)}Jj=1 of individual household decisions

{cj (xj) , nj (xj) , aj+1 (xj) ,mj (xj) , inj+1 (xj)}Jj=1 , aggregate stocks of physical capital and ef-fective labor services {K,N,Km, Nm} , and factor prices {w, q,R, pm} such that

(a) {cj (xj) , nj (xj) , aj+1 (xj) ,mj (xj) , inj+1 (xj)}Jj=1 solves the consumer problem (16),

(b) the �rm �rst order conditions hold in both sectors

w = FN (K,N) = pmFm,Nm (Km, Nm) ,

q = FK (K,N) = pmFm,Km (Km, Nm) ,

R = q + 1− δ,

(c) markets clear

K +Km =

J∑j=1

µj

∫(a (xj)) dΛ (xj) +

J∑j=1

∫µjaj (xj) dΛ (xj) ,

N +Nm =

J1∑j=1

µj

∫ej(xj)nj (xj) dΛ (xj) ,

(d) the aggregate resource constraint holds

G+ (1 + g)S +

J∑j=1

µj

∫(c (xj) + pmm (xj)) dΛ (xj) = Y + pmYm + (1− δ)K,

(e) the government programs clear

J∑j=J1+1

µj

∫tSSj (xj) dΛ (xj) =

J1∑j=1

µj

∫taxSS (xj) dΛ (xj) ,

MG +

TSI︷ ︸︸ ︷J∑j=1

µj

∫tSIj (xj) dΛ (xj) +G =

J∑j=1

µj

∫ [τCc (xj) + taxj (xj)

]dΛ (xj) , (23)

(f) the accidental bequest redistribution program clears

J1∑j=1

µj

∫tBeqj (xj) dΛ (xj) =

J∑j=1

∫µjaj (xj) dΛ (xj) ,

(g) the insurance system is self-�nanced so that insurance payouts over all participants equalpremium contributions and/or ear marked tax collections and

(h) the distribution is stationary µj+1,Λ (xj+1) = Tµ,Λ (µj ,Λ (xj)) where Tµ,Λ is a one periodtransition operator on the distribution.

33

8.2 Appendix B: Tables

Marginal Tax Rate Single Taxable Income Married Filing Jointly

10% $0 - $9,325 $0 - $18,65015% $9,326 - $37,950 $18,651 - $75,90025% $37,951 - $91,900 $75,901 - 153,10028% $91,901 - $191,650 $153,101 - $233,35033% $191,651 - $416,700 $233,350 - $416,70035% $416,7001 - $418,400 $416,701 - $470,70039.6% $418,401+ $470,700+

Table 1: 2017 Marginal tax rates in the US

34

Parameters: Explanation/Source:

- Periods working J1 = 9- Periods retired J2 = 6- Population growth rate n = 1.2% CMS 2010

- Years modeled years = 75 from age 20 to 95

- Total factor productivity A = 1 Normalization

- Growth rate g = 2% NIPA

- Capital share in production α = 0.33 Kydland and Prescott (1982)

- Capital in medical services prod. αm = 0.26 Donahoe (2000)

- Capital depreciation δ = 10% Kydland and Prescott (1982)

- Health depreciation δh,j = [0.6%− 2.13%] MEPS 1999/2009

- Survival probabilities πj CMS 2010

- Health Shocks Technical Appendix MEPS 1999/2009

- Health transition prob. Technical Appendix MEPS 1999/2009

- Productivity shocks see Section 3 MEPS 1999/2009

- Productivity transition prob. Technical Appendix MEPS 1999/2009

- Group ins. transition prob. Technical Appendix MEPS 1999/2009

- Price for medical care

for uninsuredνnoIns = 0.7 MEPS 1999/2009

- M price markup for

IHI insuredνIHI = 0.25 Shatto and Clemens (2011)

- M price markup for

GHI insuredνGHI = 0.1 Shatto and Clemens (2011)

- M price markup for

MedicaidνMaid = 0.0 Shatto and Clemens (2011)

- M price markup for

MedicareνMcare = −0.1 Shatto and Clemens (2011)

- Coinsurance rate: IHI in % γIHIj ∈ [22, 46, 48, 49, 50, 53, 52, 50] MEPS 1999/2009

- Coinsurance rate: GHI in % γGHIj ∈ [33, 33, 33, 34, 36, 36, 45, 50] MEPS 1999/2009

- Medicare premiums/GDP 2.11% Jeske and Kitao (2010)

- Medicaid coinsurance rate in % γMaidj ∈ [11, 14, 17, 16, 17, 18, 20, 22]

Center for Medicare and

Medicaid Services (2005)

- Public coinsurance rate retired in % γR = 20Center for Medicare and

Medicaid Services (2005)

- Payroll tax Social Security: τSoc = 9.4% IRS

- Consumption tax: τC=5.0% Mendoza et al. (1994)

- Payroll tax Medicare: τMed = 2.9% Social Security Update (2007)

Table 2: External parametersThese parameters are based on our own estimates from MEPS and CMS data as well as other studies.

35

Parameters: Explanation/Source: Nr.M.

- Relative risk aversion σ = 3.0 to match KY and R 1

- Preference on consumption

vs. leisure:η = 0.43 to match labor supply and p×M

Y1

- Preference on c and lvs. health

κ = 0.75 to match labor supply and p×MY

1

- Discount factor β = 1.0 to match KY and R 1

- GHI markup pro�ts ωGHI = 0 to match GHI take-up 1

- IHI markup pro�ts ωj,h ∈ [0.6− 1.5] to match spending pro�le 8

- Health production productivity φj ∈ [0.2− 0.45] to match spending pro�le 15

- TFP in medical production Am = 0.4 to match p×MY

1

- Production parameter of health ξ = 0.26 to match p×MY

1

- E�ective labor services production χ = 0.85 to match labor supply 1

- Health productivity θ = 1.0 used for sensitivity analysis 1

- Pension replacement rate Ψ = 40% to match τ soc 1

- Fixed time cost of labor lj ∈ [0.0− 0.7] to match average work hours 9

- Minimum health state hmin = 0.01 to match health spending 1

- Asset test level aMaid = $150, 000 to match Medicaid take-up 1

-Total number of

internal parameters:44

Table 3: Internal parametersWe choose these parameters in order to match a set of target moments in the data.

Moments Model Data Source Nr.M.

- Medical expenses HH income 17.6% 17.07% CMS communication 1- Workers IHI 5.6% 7.2% MEPS 1999/2009 1- Workers GHI 61.1% 62.2% MEPS 1999/2009 1- Workers Medicaid 9.6% 9.2% MEPS 1999/2009 1- Capital output ratio: K/Y 2.7 2.6− 3 NIPA 1- Interest rate: R 4.2% 4% NIPA 1- Size of Social Security/Y 5.9% 5% OMB 2008 1- Size of Medicare/Y 3.1% 2.5− 3.1% Dept. of Health (2007) 1- Medical spend. pro�le Figure 5 Figure 5 MEPS 1999/2009 15- IHI insurance take-up pro�le Figure 5 Figure 5 MEPS 1999/2009 7- Medicaid ins. take-up pro�le Figure 5 Figure 5 MEPS 1999/2009 7- Average labor hours Figure 6 Figure 6 PSID1984-2007 7Total number of moments 44

Table 4: Matched data momentsWe choose internal parameters so that model generated data matches data from MEPS, CMS, and

NIPA.

36

[1] Benchmark [2] Optimal TaxOutput (GDP ) 100 94.06Capital (Kc) 100 93.26Capital (Km) 100 99.30

Weekly hours worked 29.40 29.00Non- Med. Consumption (C) 100 92.75Med. consumption (M) 100 99.41Med. spending (pmM) 100 100.46

Workers insured ( 78.59 75.51Medicaid (%) 9.56 6.18

Interest rate (r in %) 5.07 5.08Wage rate (w) 100.00 99.95

Gini (Total income) 0.44 0.41Gini (Net income) 0.38 0.31Progressivity parameter (τ) 0.053 0.0247Suits index (Income tax) 0.17 0.53

Welfare (CEV): 0 +5.54• Income Group 1 (Low) 0 +21.10• Income Group 2 0 +12.10• Income Group 3 0 −9.71• Income Group 4 (High) 0 −33.10

Table 5: Macroeconomic and welfare e�ects when switching from the benchmarksystem to the optimal tax systemThis table presents steady state results comparing the benchmark economy to the equilibrium outcome

with the optimal progressive income tax system (column [2]). Data in rows marked with the % symbol

are either fractions in percent or tax rates in percent. The other rows are normalized with values of

the benchmark case. Each column presents steady-state results. CEV values are reported as percentage

changes in terms of lifetime consumption of a newborn individual with respect to consumption levels in

the benchmark.

37

[1] Bench. [3] ACA - Opt [4] Medicare - OptOutput (GDP ) 100 92.41 89.98Capital (Kc) 100 91.19 85.93Capital (Km) 100 101.17 99.76

Weekly hours worked 29.40 28.20 27.06Non- Med. Consumption (C) 100 89.78 85.46Med. consumption (M) 100 101.23 102.82Med. spending (pmM) 100 96.79 92.84

Workers insured (%) 78.59 99.61 100Medicaid (%) 9.56 10.01 100

Interest rate (r in %) 5.07 5.08 5.35Wage rate (w) 100.00 99.97 98.66

Gini (Total income) 0.44 0.42 0.44Gini (Net income) 0.38 0.30 0.33Suits index (Income tax) 0.17 0.52 −Progressivity parameter (τ) 0.053 0.24 0.114

Welfare (CEV): 0 +3.14 −3.40• Income Group 1 (Low) 0 +22.37 +20.01• Income Group 2 0 +8.87 −2.25• Income Group 3 0 −13.92 −20.16• Income Group 4 (High) 0 −36.54 −29.07

Table 6: The e�ects of switching to the optimal tax systems with di�erent healthinsurance arrangements.This table presents steady state results of the optimal progressive income tax systems with di�erent

health insurance systems: [1] Benchmark - the US health insurance system before the ACA reform,

[2] ACA: the US health insurance system after the ACA, [3] Universal Medicare for all: Public health

insurance (Medicare) for all individuals, and [4] Self Insurance Only - No health insurance system. Datain rows marked with the % symbol are either fractions in percent or tax rates in percent. The other rows

are normalized with values of the benchmark case. Each column presents steady-state results. CEV

values are reported as percentage changes in terms of lifetime consumption of a newborn individual with

respect to consumption levels in the benchmark.

38

[1] Bench. [5] Self Ins. - Opt [6] No Health Risk - OptOutput (GDP ) 100 107.13 94.06Capital (Kc) 100 115.28 93.26Capital (Km) 100 99.05 99.30

Weekly hours worked 29.40 31.14 29.00Non- Med. Consumption (C) 100 104.53 92.75Med. consumption (M) 100 93.03 99.41Med. spending (pmM) 100 87.72 100.46

Workers insured ( 78.59 0 75.51Medicaid (%) 9.56 0 6.18

Interest rate (r in %) 5.07 4.50 5.08Wage rate (w) 100.00 102.82 99.95

Gini (Total income) 0.44 0.41 0.41Gini (Net income) 0.38 0.39 0.31Suits index (Income tax) 0.17 0.53 −Progressivity parameter (τ) 0.053 0.112 0.102

Welfare (CEV): 0 −0.44 −• Income Group 1 (Low) 0 −1.04 −• Income Group 2 0 −1.98 −• Income Group 3 0 −1.47 −• Income Group 4 (High) 0 −0.99 −

Table 7: The e�ects of the optimal tax system when elimilating health risksThis table presents steady state results comparing the benchmark economy to the equilibrium outcome

with the optimal progressive income tax system (column [2]). Data in rows marked with the % symbol

are either fractions in percent or tax rates in percent. The other rows are normalized with values of

the benchmark case. Each column presents steady-state results. CEV values are reported as percentage

changes in terms of lifetime consumption of a newborn individual with respect to consumption levels in

the benchmark.

39

8.3 Appendix C: Figures

(a) Health Status

0.00

5.00

10.00

15.00

20.00

in $

1,0

00

20 40 60 80Age

Average Median

Source: MEPS 1996−2007

A. Health expenditure profile

0.60

0.65

0.70

0.75

0.80

Gin

i

20 40 60 80Age

Source: MEPS 1996−2007

B. Gini coefficient of health expenditures

(b) Health Expenditure

0.00

5.00

10.00

15.00

20.00

in $

1,0

00

20 40 60 80Age

Other

Worker’s compensation

State insurance

Federal insurance

Tricare

CHAMPUS

Veteran’s benefits

Private insurance

Medicaid

Medicare

Out−of−pocket

Source: MEPS 1996−2007

Health expenditure by financing source

(c) Health Financing (d) Insurance Take Up Rate

Figure 1: Health status, spending and �nancing over the life cycle

Figure 2: Health status and income over the life cycle from MEPS 2000-2009

40

0

10

20

30

40

50

in $

1,0

00

20 40 60 80 100

Age

MEPS

HRS

Source: MEPS-HRS 2000-2012

Average Wage-Income

(a) Wage Incomes

.5

1

1.5

2

2.5

C.V

.

20 30 40 50 60 70

Age

wage-Inc

HH-Inc

Source: MEPS 2000-2012

Coefficient of Variation: Wage-Income and HH-Income

(b) Coe�cient of Variation

0

5

10

15

in $

1,0

00

20 40 60 80 100

Age

Avge-MEPS

Avge-HRS

Med-MEPS

Med-HRS

Source: MEPS-HRS 2000-2012

Health Expenditures

(c) Health Expenditures

1

2

3

4

5

6

C.V

.

20 30 40 50 60 70

Age

Health Exp

OOP Exp

HH-Inc

Source: MEPS-HRS 2000-2012

Coefficient of Variation: Health-Exp. and HH-Income

(d) Coe�cient of Variation

Figure 3: Health expenditure and income risks over the life cycle

41

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îðòðð

ìðòðð

êðòðð

îð ìð êð èð ïðð

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Åëà ػ¿´¬¸ »¨°»²¼·¬«®» ¿­ û ±º ·²½±³»

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Åêà ÑÑÐ Ø»¿´¬¸ »¨°»²¼·¬«®» ¿­ û ±º ·²½±³»

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Figure 4: Health expenditures as fraction of incomeOOP is out of pocket.

42

20 30 40 50 60 70 80 90Age

0

20

40

60

80

100

120

140M

ed-S

pend

. % o

f Inc

ome

Ret

ired:

65

[1] Medical Spending in % of Income

Data MEPSData CMSModel

0 25 50 75 100Med. spending ($1,000)

0

10

20

30

40

50

60

70

%

[2] Med-Spend. Distr.

DataModel

30 40 50 60Age

0

20

40

60

80

100

%

[3] IHI %

DataModel

30 40 50 60Age

0

20

40

60

80

100

%

[4] GHI %

DataModel

30 40 50 60Age

0

20

40

60

80

100

%

[5] Medicaid %

DataModel

Figure 5: Health expenditure and insurance take-upModel vs. data from MEPS 2000-2009

43

20 40 60 80Age

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Norm

alize

d Lo

g As

sets

[1] AssetsDataModel

20 40 60 80Age

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Norm

alize

d Lo

g In

com

e

[2] IncomeDataModel

20 40 60 80Age

0.0

0.2

0.4

0.6

0.8

1.0

Norm

alize

d Lo

g Co

nsum

ption

[3] Consumption

DataModel

20 30 40 50 60Age

20

22

24

26

28

30

32

34

Hour

s

[4] Avge Labor Hours per WeekDataModel

Figure 6: Moment matching using PSID 1984-2007 and CPS 1999-2009Blue lines are model generated data moments and black dotted lines are PSID data in Panel 1 and 2

and CPS data in Panel 3.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5Health Capital

0.0

0.2

0.4

0.6

0.8

1.0

Mea

n W

age

Inco

me

(Nor

mal

ized

)

MEPSModel

20 30 40 50 60Age

0.0

0.2

0.4

0.6

0.8

1.0

Mea

n W

age

Inco

me

(Nor

mal

ized

)

MEPS-HealthyModel-HealthyMEPS-SickModel-Sick

Figure 7: Moment matching using MEPS 2000-2009Blue dots are model generated data moments and green dots lines are from PSID data.

44

0 20 40 60 80 100 120 140Income in $1,000

0

1

2

3

4

5

Freq

uenc

y in %

[1] Income Distribution

DataModel

0 20 40 60 80 100 120 140Wage in $1,000

0

1

2

3

4

5

6

7

Freq

uenc

y in \

%

[2] Wage Distribution

DataModel

Figure 8: Moment matching using MEPS 2000-2009Blue dots are model generated data moments and green dots lines are from PSID data.

45

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

60

70

Taxe

s in

$1,

000

[1] Total TaxBenchmarkConesa & Krueger (2006)-OptimalBenchmark-Optimal

0 50 100 150 200Income in $1,000

5

0

5

10

15

20

25

30

35

40

%

[2] Average Tax RateBenchmarkConesa & Krueger (2006)-OptimalBenchmark-Optimal

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

%

[3] Marginal Tax Rate

BenchmarkConesa & Krueger (2006)-OptimalBenchmark-Optimal

0 50 100 150 200Disposable Income in $1,000

0

1

2

3

4

5

Freq

uenc

y in

%

[4] After-Tax Income DistributionBenchmarkBenchmark-Optimal

Figure 9: The optimal income taxes in the benchmark pre-ACA economyProgressive income taxes of the pre-ACA Benchmark case are based on Guner, Lopez-Daneri and Ventura

(2016) and use the tax polynomial introduced in Benabou (2002). The Conesa and Krueger (2006) case

is based on a model without health shocks and health insurance and uses a tax polynomial based on

Gouveia and Strauss (1994).

46

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

60

70

Taxe

s in

$1,

000

[1] Total TaxBenchmark-OptimalACA-Optimal

0 50 100 150 200Income in $1,000

5

0

5

10

15

20

25

30

35

40

%

[2] Average Tax RateBenchmark-OptimalACA-Optimal

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

%

[3] Marginal Tax RateBenchmark-OptimalACA-Optimal

0 50 100 150 200Disposable Income in $1,000

0

1

2

3

4

5

Freq

uenc

y in

%

[4] After-Tax Income DistributionBenchmark-OptimalACA-Optimal

Figure 10: The ACA and optimal income tax systemProgressive income taxes of the pre-ACA Benchmark case are based on Guner, Lopez-Daneri and Ventura

(2016) and use the tax polynomial introduced in Benabou (2002). The red-circled line is the optimal

tax without the ACA and the green-triangle line is the optimal tax with the ACA. The Conesa and

Krueger (2006) case is based on a model without health shocks and health insurance and uses a tax

polynomial based on Gouveia and Strauss (1994).

47

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Suits Index = A/(A+B)

A

B

Cumulative percent of income (%)

Cum

ulat

ive

perc

ent o

f tax

es (

%)

Tax−Lorenz Curve and Suits Index

Figure 11: Lorenz curves and Suits Index for the income taxesThe Tax Lorenz-type curve and Suits index measure the degree of disproportionality betweenpretax income and tax contributions by means of a relative concentration curve. The Suitsindex is essentially a Gini coe�cient for tax contributions by income group. It varies from +1(most progressive) where the entire tax burden is allocated to households of the highest incomebracket, through 0 for a proportional tax, and to −1 (most regressive) where the entire taxburden is allocated to households of the lowest income bracket.

48

0.0 0.2 0.4 0.6 0.8 1.0Population (cumulative)

0.0

0.2

0.4

0.6

0.8

1.0

Inco

me

(cum

ulat

ive)

[1] Lorenz-Curves: Income45Bench: IncomeBench: After-Tax Inc.Bench-Optimal: After-Tax Inc.

0.0 0.2 0.4 0.6 0.8 1.0Income (cumulative)

0.0

0.2

0.4

0.6

0.8

1.0

Inco

me-

Tax

(cum

ulat

ive)

[2] Suits-Curves: Income-Tax45BenchmarkBenchmark-Optimal

0.0 0.2 0.4 0.6 0.8 1.0Income (cumulative)

0.0

0.2

0.4

0.6

0.8

1.0

Tota

l-Tax

(cum

ulat

ive)

[3] Suits-Curves: Total-Tax45BenchmarkBenchmark-Optimal

0.0 0.2 0.4 0.6 0.8 1.0Population (cumulative)

0.0

0.2

0.4

0.6

0.8

1.0

Afte

r-Tax

Inco

me

(cum

ulat

ive)

[4] Lorenz: After-Tax Inc. & ACA45ACAACA-Optimal

0.0 0.2 0.4 0.6 0.8 1.0Income (cumulative)

0.0

0.2

0.4

0.6

0.8

1.0

Inco

me-

Tax

(cum

ulat

ive)

[5] Suits: Income-Tax & ACA45ACAACA-Optimal

0.0 0.2 0.4 0.6 0.8 1.0Income (cumulative)

0.0

0.2

0.4

0.6

0.8

1.0

Tota

l-Tax

(cum

ulat

ive)

[6] Suits: Total-Tax & ACA45ACAACA-Optimal

Figure 12: Income and Income Tax Lorenz CurvesPanel [1] presents Lorenz curves of taxable income (pre-ACA Benchmark), Panel [2] presents Suits curves

of progressive income taxes based on Suits (1977) and Panel [3] presents Suits curves of total taxes,

that is consumption taxes, progressive income taxes, and payroll taxes for social security and Medicare.

For the ACA case total taxes also include a new tax on investment income of high income earners and

penalties for being uninsured. Panel [4-6] present the cases with the ACA.

Progressive income taxes of the pre-ACA Benchmark case are based on Guner, Lopez-Daneri and Ventura

(2016) and use the tax polynomial introduced in Benabou (2002). The ACA case uses the same tax

structure as the pre-ACA Benchmark case.

49

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

60

70

Taxe

s in

$1,

000

[1] Total TaxUS-OptACA-OptNo-Ins-OptNo-H-Shock-OptUPHI

0 50 100 150 200Income in $1,000

5

0

5

10

15

20

25

30

35

40

%

[2] Average Tax Rate

US-OptACA-OptNo-Ins-OptNo-H-Shock-OptUPHI

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

%

[3] Marginal Tax Rate

US-OptACA-OptNo-Ins-OptNo-H-Shock-OptUPHI

Figure 13: Sensitivity analysis of optimal taxesProgressive income taxes of the pre-ACA Benchmark case are based on Guner, Lopez-Daneri and Ventura

(2016) and use the tax polynomial introduced in Benabou (2002). The red-circle line is the optimal tax

without the ACA, the green-triangle line is the optimal tax with the ACA, the orange-diamond line is

the optimal tax without any health insurance contracts and the brown-square line is the optimal tax

without any idiosyncratic health shocks.

50

Figure 14: The optimal Income tax system based on the two parameter functionfrom Benabou (2002)

Progressive income taxes of the Benchmark case are based on the original form in Benabou (2002) and

Guner, Lopez-Daneri and Ventura (2016). We remove the restriction of non-negative income taxes.

That, the government now can transfer to the low income households via negative taxes. The Conesa

and Krueger (2006) case is based on a model without health shocks and health insurance and uses a tax

polynomial based on Gouveia and Strauss (1994).

51

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

60

70

80

Taxe

s in

$1,

000

[1] Total TaxBenchmarkConesa & Krueger (2006)-OptimalBenchmark-Optimal

0 50 100 150 200Income in $1,000

0

5

10

15

20

25

30

35

40

%

[2] Average Tax RateBenchmarkConesa & Krueger (2006)-OptimalBenchmark-Optimal

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

%

[3] Marginal Tax Rate

BenchmarkConesa & Krueger (2006)-OptimalBenchmark-Optimal

0 50 100 150 200Disposable Income in $1,000

0

1

2

3

4

Freq

uenc

y in

%

[4] After-Tax Income DistributionBenchmarkBenchmark-Optimal

Figure 15: The optimal income taxes based on the three parameter function fromGouveia and Strauss (1994)Progressive income taxes of the benchmark case are based on the tax polynomial introduced in Gouveia

and Strauss (1994). The Conesa and Krueger (2006) case is based on a model without health shocks

and health insurance and uses a tax polynomial based on Gouveia and Strauss (1994). The red dotted

line is the optimal taxes.

52

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

60

70

80

Taxe

s in

$1,

000

[1] Total TaxBenchmark-OptimalACA-Optimal

0 50 100 150 200Income in $1,000

0

5

10

15

20

25

30

35

40

%

[2] Average Tax RateBenchmark-OptimalACA-Optimal

0 50 100 150 200Income in $1,000

0

10

20

30

40

50

%

[3] Marginal Tax RateBenchmark-OptimalACA-Optimal

0 50 100 150 200Disposable Income in $1,000

0

1

2

3

4

Freq

uenc

y in

%

[4] After-Tax Income DistributionBenchmark-OptimalACA-Optimal

Figure 16: The ACA and the optimal income tax system based on Gouveia andStrauss (1994)The progressive income taxes are based on Gouveia and Strauss (1994). The red dotted line is the

optimal tax without the ACA and the green dotted line is the optimal tax with the ACA. The Conesa

and Krueger (2006) case is based on a model without health shocks and health insurance and uses a tax

polynomial based on Gouveia and Strauss (1994)

53