Optimal Progressive Income Taxation in a Bewley-Grossman Framework

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1 Optimal Progressive Income Taxation in a Bewley-Grossman Framework Juergen Jung Towson University Chung Tran Australian National University November 21, 217 Abstract We study the optimal degree of progressivity of income taxation in an environment where individuals are exposed to both income and health risks over the lifecycle. We argue that the optimal degree of tax progressivity is aected by health factors and insurance arrangements. We formulate a Bewley-Grossman model that matches U.S. data, including the U.S. tax and transfer system and the pre-21 health insurance system. Our quantitative analysis indicates that the progressive income tax system is an important channel to provide social insurance for low income Americans who have limited access to private health insurance. The optimal progressive income tax system includes a tax break for income below $38, and high marginal tax rates of over 5 percent for income earners above $2,. The tax progressivity (Suits) indexa Gini coecient for the income tax contribution by incomeof the optimized income tax system is around.53, compared to.17 in the benchmark tax system. Welfare gains from switching to the optimal tax system amount to over 5 percent of compensating consumption. Indeed, the optimal income tax system is more progressive than the current U.S. tax code and also more progressive than the optimized systems reported in previous studies, including Conesa and Krueger (26) and Heathcote, Storesletten and Violante (217). This nding challenges the view that the US personal income tax code should be less progressive. Finally, we demonstrate that the optimal degree of tax progressivity is inuenced by the design of public transfer programs and the parametric specications of the income tax function. JEL: E62, H24, I13, D52 Keywords: Health and income risks; Inequality; Social insurance; Tax progressivity; Suits index; Optimal taxation; General equilibrium. We appreciate comments from Dirk Krueger, Mariacristina De Nardi, Greg Kaplan and participants of the UNSW/Treasury Joint Workshop on Fiscal Policy Modeling, the 217 Midwest Macroeconomics Meetings, and the 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) , jjung@towson.edu Research School of Economics, The Australian National University, ACT 261, AUS. Tel.: , chung.tran@anu.edu.au 1

2 1 Introduction The literature has documented a high degree of inequality in income, wealth and consumption of U.S. households. The social insurance literature views income risk as one important contributor to this heterogeneity (e.g., Heathcote, Storesletten and Violante (28) and Kaplan (212)). On the other hand, the health economics literature has focused on health as an additional source of heterogeneity (e.g., Deaton and Paxson (1998) and Kippersluis et al. (29)). In order to equalize opportunities, almost all advanced economies have instituted tax systems where the marginal tax rate increases with income and public transfers are targeted to disadvantaged groups including low income households, the sick and the unemployed (compare Table 1). As a result, tax systems and social insurance systems tend to be highly progressive as a whole and play a key role in shaping the income distribution across households and over time. The optimal progressive tax literature has focused on frameworks where income risk is the only source of inequality (e.g., see Conesa and Krueger (26) and Heathcote, Storesletten and Violante (217)). In this paper, we aim to better understand the optimal progressivity of the U.S. income tax system in a lifecycle framework where both income risk and health risk are sources of lifetime inequality. We rst document some stylized facts on health status, income and health expenditures over the lifecycle, using the U.S. medical expenditure panel survey data (MEPS) and demonstrate the important role of health factors as sources of lifecycle inequality. We then construct a simple partial equilibrium two-period model with income and health risks and illustrate how the interaction between income and health risks aects the optimal degree of tax progressivity. Finally, we formulate a dynamic general equilibrium model with an endogenous distribution of income, consumption and health expenditure and conduct quantitative analyses of various progressive income tax systems. Our main goal is to identify quantitatively the role of health factors in the optimal design of a progressive income tax system in a framework with health risk and health insurance. Our quantitative model is based on two workhorse models in the literature. The workhorse model in the optimal tax literature is an incomplete-markets heterogenous agents model initially developed by Bewley (1986) and extended by Huggett (1993) and Aiyagari (1994). In the Bewley-Huggett-Aiyagari environment with idiosyncratic income risk and incomplete markets, progressive income taxation essentially can improve welfare through two channels. First, progressive taxes lead to a more equal distribution of income and wealth, and therefore to more equitable distributions of consumption. Second, in the absence of private insurance markets, progressive taxes provide a partial insurance substitute and can generate more stable household consumption paths over time through income redistribution from lucky individuals to unlucky ones who experience large negative shocks to income. However, health risk, medical expenditures and health insurance arrangements are largely absent in this literature. In order to incorporate health factors, we combine the Bewley model with the Grossman model of health capital accumulation (Grossman (1972)). In the Grossman environment, individuals value their health in addition to a consumption goods basket and have a strong motive to smooth both health and the consumption bundle over the lifecycle. Health aects household consumption through direct and indirect channels. First, the utility of consumption itself is aected by the health status of an individual. Second, health is a co-determinant of labor earnings and therefore aects the household's ability to purchase nal consumption goods. In addition, smoothing health over the lifecycle requires healthcare spending, which subsequently reduces funds available for purchasing nal consumption goods. The simultaneous presence of both income and health risks and the institutional insurance arrangements that lower a household's exposure to such risk shape the distributions of income, wealth and consumption. In our framework, progressive income taxes and public health insurance serve as policy tools to 2

3 provide social insurance against income and health risks. Our modeling framework allows us to model the lifecycle structure of health risk in conjunction with income risk as observed in the data. In the model, health care spending and health insurance take-up rates over the lifecycle are endogenous and jointly determined with consumption, savings and labor supply. The benchmark model is calibrated to U.S. data and incorporates the lifecycle patterns of shocks to income and health. The model subsequently matches labor supply, asset holdings, consumption and health expenditures over the lifecycle. Health expenditures are low early in life because of high initial health capital and low health risk. Health expenditures then rise exponentially later in life because individuals are exposed to more frequent and larger health shocks. The benchmark model also reproduces the humpshaped lifecycle prole of private health insurance take-up rates, the income distribution from the Panel Study of Income Dynamics (PSID) as well as macroeconomic aggregates from the National Income and Product Accounts (NIPA). We use the calibrated model to quantitatively explore the shape of the optimal progressive income tax system. Our main results are summarized as follows: First, the optimal income tax system is highly progressive and imposes a tax break for income below $38,, followed by a jump in the marginal tax rate to 25 percent. The marginal tax rate then increases further to over 4 percent for income above $1, and to over 5 percent for income above $2,. The large zero-tax bracket at the low end of the income distribution is mainly driven by the high demand for social insurance by the poor unhealthy population who has limited access to private health insurance. The high tax rates at the upper end of the income distribution are required to meet the nancing needs of government spending and transfer programs. The optimal tax system in our model is more progressive than the optimal tax systems reported in prior literature that largely abstracts from health risk and health insurance institutions (e.g., Conesa and Krueger (26) and Heathcote, Storesletten and Violante (217)). In our setting, the progressive income tax system is an important channel to supplement missing components of social health insurance. Low income individuals face a higher probability to end up in poor health states than higher income individuals. Many of them do not have adequate access to health insurance through the mixed public/private U.S. health insurance system. These individuals are not poor enough to qualify for Medicaid and not rich enough to buy private health insurance. Gruber (28) refers to this group as the poor working class whose income is below the median income but above the eligibility thresholds for public health insurance and income transfers. In the model the poor working class benets most from the optimized tax system. The zero-tax at the lower end of the income distribution allows these individuals to invest more in health and increase their non-medical consumption. We therefore observe an increase in medical spending of the uninsured in the optimized tax system. This mechanism is missing from the frameworks used in prior studies. Our results imply that ignoring health risk and institutional features of the U.S. health insurance system can lead to misleading conclusions about the optimal progressive tax policy in the U.S. In order to measure the progressivity level of the U.S. income tax system, we compute the Suits indexa Gini coecient for income tax contributions by income. According to Suits (1977), the Suits index varies from +1 (most progressive) where the entire tax burden is borne by households of the highest income bracket, through for a proportional tax, to 1 (most regressive) where the entire tax burden falls on households of the lowest income bracket. Our calculations indicate that the U.S. income tax system in the benchmark model is not very progressive with a Suits index of.17. The optimal tax system is much more progressive with a Suits index of.54. Under the optimal tax system income inequality decreases signicantly. We observe that the after-tax-income Gini coecient decreases from.38 in the benchmark economy to.31 in the economy with optimized progressivity. We also observe large welfare gains of 5.5 3

4 percent of compensating lifetime consumption at the aggregate level when switching from the benchmark to the optimal tax system. This outcome is mainly driven by large welfare gains of low income individuals that dominate the welfare losses of the higher income groups. These ndings imply that replacing the existing progressive income tax code with a less progressive system such as the one currently discussed by the Trump administration can cause welfare losses. We next analyze how dierent designs of the U.S. health insurance system aect the optimal level of tax progressivity. We consider three alternative health insurance systems: (i) the U.S. health insurance system after the introduction of the Aordable Care Act (ACA), (ii) a system with universal Medicare for all, and (iii) a system without health insurance contracts where individuals can only self-insurance. We nd that the ACA provides channels that redistribute resources from healthy, high income types to sicker, low income types through premium subsidies and an expansion of Medicaida public health insurance program for low income individuals. Our results show that with the ACA in place, the optimal tax progressivity decreases slightly compared to the pre-aca scenario. The Suits index decreases to.52. The optimal tax schedule changes signicantly when alternative insurance designs are considered. When Medicare is extended to everyone in the model economy, the optimal income tax system becomes much less progressive. The tax break at the lower end of the income distribution is smaller and the marginal tax rates imposed on top earners are much lower at approximately 31 percent for incomes over $2,. This result implies that the universal expansion of Medicare signicantly reduces the residual demand for social insurance provided through the progressive income tax system. Finally, we examine how the parametric specication of the income tax function aects the optimal tax progressivity. In our benchmark model, we use the two parameter specication from Benabou (22) with a non-negative tax restriction to remove transfer components embedded in the tax function. As an extension, we rst remove this restriction and use the income tax function as in Heathcote, Storesletten and Violante (217). Our results shown that the optimal progressive income tax function shifts down and marginal tax rates are negative at the low end of the income distribution. These negative taxes are conditional cash transfers that induce poorer individuals to work and save in order to receive these transfer payments. Next, we consider the three parameter specication from Gouveia and Strauss (1994) which was used in Conesa and Krueger (26). We nd that the shapes of the optimal tax function dier signicantly and welfare gains from optimizing progressivity using the three parameter specication are smaller at 1.8 percent of compensating lifetime consumption compared to 5.5 percent with the 2-parameter polynomial. Indeed, the transfer policy to the low income households at the bottom of the distribution strongly aects the shape of the optimal marginal tax function. Our ndings imply that the parametric specication of the tax function as well as the design of transfer policies are important for evaluating the optimal level of progressivity of the income tax code. Related literature. Our work is connected to dierent branches of the quantitative macroeconomics and health economics literature. First, our paper is closely related to the optimal progressive income taxation literature. In a seminal paper, Varian (198) shows analytically how social insurance can be provided via a progressive tax system. More recently, Conesa and Krueger (26) quantify the optimal progressivity of the income tax code in a dynamic general equilibrium model with household heterogeneity due to uninsurable labor productivity risk. They show that a progressive tax system serves as a partial substitute for missing income-insurance markets and results in a more equal distribution of income. Erosa and Kore- 4

5 shkova (27) analyze the insurance role of the U.S. progressive income tax code in a dynastic model with human capital accumulation. Chambers, Garriga and Schlagenhauf (29) quantify interactions between progressive income taxes and housing policies to promote home ownership in an overlapping generations model with housing and rental markets. Krueger and Ludwig (216) compute optimal tax- and education policies in an economy where progressive taxes provide social insurance against idiosyncratic wage risk but distort the education and human capital decision of households. Stantcheva (215) characterizes the optimal income tax and human capital policies in a dynamic lifecycle model of labor supply with risky human capital formation. McKay and Reis (216) study the optimal generosity of unemployment benets and progressivity of income taxes in a model with macroeconomic aggregate shocks and individual unemployment risk. These studies abstract from the implications of health risk and the eects of the health insurance system on the optimal progressivity of the income tax system. Our study includes these components. Heathcote, Storesletten and Violante (217) develop a tractable general equilibrium model to study the optimal degree of progressivity of a tax and transfer system. They focus on the trade-os between risk sharing and the incentives to work and invest in skills. They show how preferences, technology and the market structure inuence the optimal degree of tax and transfer progressivity. Note that, in order to obtain analytical results they abstract from intertemporal consumption and savings decisions and more realistic government transfer programs. We analyze a similar problem, but we focus on the quantitative aspects which allows for more realistic features in the model. In our model health is an important source of heterogeneity across individuals and over the lifecycle. In addition, we explicitly model the main components of the U.S. social insurance system including Food Stamp programs, Social Security, Medicaid and Medicare. As a direct result, the optimal income tax system in our setting is more progressive than the one in Heathcote, Storesletten and Violante (217). Our results illustrate the quantitative importance of accounting for health, health risk and the design of the health insurance system in determining the optimal degree of progressivity of the income tax system. Our paper is related to the literature on incomplete markets macroeconomic models with heterogeneous agents as pioneered by Bewley (1986) and extended by Huggett (1993) and Aiyagari (1994). This Bewley model has been applied widely to quantify the welfare cost of public insurance for idiosyncratic income risk (e.g., Hansen and Imrohoroglu (1992), mrohoro lu, mrohoro lu and Joines (1995), Golosov and Tsyvinski (26), Heathcote, Storesletten and Violante (28), Conesa, Kitao and Krueger (29) and Huggett and Parra (21)). This literature focuses on the welfare cost triggered by income/labor productivity risk in combination with a lack of insurance for non-medical consumption. Recently, Capatina (215) demonstrates that health shocks are another important source of idiosyncratic risk faced by individuals over the lifecycle. In our study we add to this literature by incorporating idiosyncratic health risk into a Bewley framework. We incorporate the micro-foundations of a health capital accumulation mechanism based on the Grossman model which endogenizes medical spending. Our research merges the workhorse models from the health economics and the macro/public nance literature to analyze the optimal income tax progressivity in the presence of income and health risk in combination with a realistic depiction of the U.S. health insurance system. Our work contributes to a growing macro-public nance literature that extends the Grossman model of health capital accumulation (Grossman (1972)). This literature incorporates health shocks, insurance markets and general equilibrium channels using a more realistic institutional setting (e.g., Jung and Tran (27), Yogo (216), Fonseca et al. (213), Scholz and Seshadri (213a) and Jung and Tran (216)). Jung and Tran (215) explore the welfare implications of optimal health insurance policies. Dierent from this paper, we quantitatively characterize the optimal progressivity of the income tax system while taking the redistribution 5

6 eects of the health insurance system into account. We then demonstrate how changes to the health insurance system aect 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 (211) advocates for taxing labor earnings at the high end of the distribution at very high marginal rates in excess of 75 percent. Badel and Huggett (215) assess the consequences of increasing the marginal tax rate on U.S. top income earners using a human capital model. Guner, Lopez-Daneri and Ventura (216) analyze how eective a progressive income tax system is in raising tax revenues. Kindermann and Krueger (215) nd that high marginal labor income tax rates are an eective tool for social insurance in a large-scale stochastic overlapping generations model with optimal marginal tax rates for top 1 percent earners of close to 9 percent. Dierent from these studies, we focus on the optimal marginal tax rates across the entire income distribution. Moreover, we base our analysis on a health capital model where health risk is an additional source of heterogeneity in addition to labor market risk. We also nd that very high tax rates at the top are an essential component of the optimal progressive tax system. More importantly, we highlight that such high optimal marginal tax rates at the top are inter-dependent with the marginal tax rates set at the bottom of the income distribution and 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 our calibration strategy. Section 5 describes our experiments and quantitative results. Section 6 is devoted to sensitivity analysis. Section 7 concludes. The Appendix presents all calibration tables and gures. 2 Facts and analytical analysis 2.1 Stylized facts The Medical Expenditure Panel Survey (MEPS) is a longitudinal survey for the U.S. that pays particular attention to medical expenditures and nancing. We use data from to track 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 MEPS provides two measures: the Short-Form 12 Version 2 (SF-12v2) and a self-reported health status. 1 Panel 1 of Figure 1 displays the physical and mental components of the SF-12v2 index over the lifecycle. Young individuals report relatively high levels of both physical and mental health (capital). Mental health exhibits a slight "M" shape over the lifecycle. Young individuals (around age 2) and very old individuals (around age 75 and higher) report the lowest mental health status. Individuals in the age range between 4 and 55 have lower mental health status than younger cohorts in their thirties and older cohorts in their sixties. The average physical health consistently decreases as an individual ages. In addition, we construct a "healthy" index using self-reported health status measures (1. excellent, 2. very good, 3. good, 4. fair, and 5. poor health). An individual is considered to be healthy if the health status measure is either excellent, very good, or good and unhealthy otherwise. This classication is standard in the 1 The SF-12v2 includes twelve dierent health measures about physical and mental health. We use physical and mental measures. Both measures use the same variables to construct the index but the physical health index puts more weight on variables measuring physical health components and the mental health index puts more weight on variables measuring mental health components. Ware, Kosinski and Keller (1996) provides more details on construction of the SF-12v2 index. The self-reported health status measure is reported as either: 1. excellent, 2. very good, 3. good, 4. fair, or 5. poor. 6

7 literature. This lifecycle pattern of health is similar to the one using the physical component of the SF-12v2 index. Health expenditure. Panel 2 of Figure 1 reports the lifecycle pattern of health expenditures, expressed in 29 US dollars. Individual health expenditures are relatively low at young ages but increase signicantly at later ages. This is mainly driven by depreciation of health over the lifecycle. On average, individuals in their twenties spend about $1, 5 per year on healthcare whereas older individuals in their fties spend about $4, per year. After the age fty, health expenditures rise very fast. The highest expenditures are incurred by the very old towards the end of their life and amount to approximately $12, on average per year in 29. This indicates that the accumulation of bad health shocks over the lifecycle drives up the health expenditures of an individual. In order to get a sense for the distribution of health expenditures we report the Gini coecient of health expenditure per age group. Inequality in health expenditures decreases by age. The Gini coecient of health expenditures is very high at around.8 when individuals are younger than 4 and then sharply drops as individuals get older. Higher Gini coecients at younger ages indicate that health expenditures among the young are much more concentrated than health expenditures of the old. This is probably due to relatively rare, but catastrophic health events among the young. Lower Gini coecients at older ages imply that the higher incidence of health problems at higher ages "equalizes" health spending across individuals. Moreover, the availability of public health insurance programs plays a role in reducing uneven access to health care services and therefore evens out health expenditure dierences across dierent income groups as well. Health nancing. The U.S. health insurance system is a mixed system where public health insurance programs target the retired population (Medicare) and the poor (Medicaid) while the majority of working individuals obtain private health insurance via their employers. Panels 3 and 4 of Figure 1 display the nancing sources of health expenditures and the insurance take-up rates over the lifecycle. Despite the many dierent types of insurances, there is a large number of Americans lacking health insurance. The U.S. health insurance system failed to provide insurance for about 5 million people in 21. The employer-based group health insurance policies (GHI) cover only around 6 percent of the working-age population while individualbased health insurance policies (IHI) cover less than 6 percent. A large number of healthy and young individuals do not have health insurance, either by choice or by circumstance. The fraction of the uninsured is highest among young workers below 35. Medicaid picks up less than 1 percent of workers by covering low income individuals. Consequently, the system leaves about 25 percent of the working population without health insurance which contributes to high insurance premiums and poor risk pooling. Gruber (28) points out the modal uninsured person is a member of the working poor class, whose income is below median income but above the 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 funding sources 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 health expenditures nanced by out-of-pocket funds increases moderately. Around the retirement age of 65 there is a big shift in the magnitude of nancing from private insurance toward public insurance including Medicare, Veteran's benets, and other state run insurance plans. Health and income. Figure 2 presents the age-proles of wage income for unhealthy and healthy individuals. There is a signicant gap between the wage income prole between the two groups. Unhealthy individuals exhibit a lower income path over the lifecycle. This conrms that health risk is an important source of lifetime inequality. To better understand income and health risks we compute the coecients of variation for income and health expenditures over the lifecycle in Figure 3. For comparison, we also include 7

8 data from the Health and Retirement Survey (HRS). Average wage income varies over the lifecycle and follows a hump shape. However, the coecient of variation for wage income is fairly stable around.9 before age sixty, and then rises sharply after retirement age. The coecient of variation for household income tracks the coecient of variation for wage income very well for individuals of working age, and stays stable until the end of life. On the other hand, the coecient of variation for health expenditure is four to ve times larger and also varies sharply over the lifecycle. The coecient of variation for health expenditure is largest between age 23. Note that health expenditures are relatively low for young individuals but so is income. This indicates that health expenditure risk can be signicant for young individuals given their low income and their lack of access to credit. This is partly due to the structure of the U.S. health insurance system that lets many young individuals opt out of the health insurance system or does not provide enough nancial aid to help cover insurance. Overall, health expenditures make up a signicant fraction of income over the lifecycle. When comparing health expenditures as fraction of household income, we observe an increase over the lifecycle. Figure 4 presents average health expenditures over average income and average out-of-pocket health expenditure over average income for each age group. At the end of their life individuals spend on average 6 percent of their income on health care. Medicare and Medicaid are the main sources of health nancing for the elderly. As a consequence, the share of out-of-pocket health expenditures as fraction of income is contained at less than 8 percent despite the high levels of overall health spending by this age group. 2.2 Analytical framework We next construct a simple model to highlight the connection between a tax- and transfer system and an individual's income and health risks. Endowments and preferences. Individuals live for two periods with certainty. They supply labor elastically in period 1 and do not work in period 2. The individual is born with skill ɛ I at the beginning of period 1 and faces health shocks ɛ H at the beginning of period 2. These shocks are drawn from a joint distribution [ ɛi ɛ H ] f ([ µi µ H ] [ σ 2, I σ IH with density f (ɛ I, ɛ H ) where the average labor shock is normalized as µ I = and µ H > and σ 2 I >, σ2 H > and σ IH >. Health shocks will result in compulsory health care expenditures. Individuals derive utility from consumption in both periods and disutility from work in period 1. Their lifetime utility is: u (c 1 ) φv (n)+βe [u(c 2 )], where u (.) and v (.) are utility functions with the usual properties, c 1 and c 2 are consumption in periods 1 and 2, respectively, n is labor supply, φ scales the level of disutility of labor, β is the time discount factor and E is the expectations operator. Government. The government runs two separate tax and transfer programs. The rst program is a redistribution program where the government imposes a payroll tax τ I to nance a lump-sum payment of d in the rst period. The second program is a public health insurance program (Medicare) that covers a fraction (1 ρ) of the health care expenditures ɛ H in the second period. This program is nanced by Medicare tax τ H. σ IH σ 2 H ]), 8

9 Household problem. The household maximization problem can be summarized as max {u (c 1) φv (n) + βe [u(c 2 )]} s.t. {c 1,c 2,n,s} c 1 + s = (1 τ I τ H ) wnɛ I + d, c 2 = Rs ρɛ H, where w is the wage rate, R is the interest rate, s is savings in period 1, d is the lump-sum transfer and ρ is the coinsurance rate. The rst order conditions are n : u (c 1 ) (1 τ I τ H ) w = φv (n), (1) s : u (c 1 ) = βre [ u (c 2 ) ]. (2) Government problem. The government clears the two transfer programs separately so that τ H wnɛ I = (1 ρ) ɛ H f (ɛ I, ɛ H ) dɛ I dɛ H = (1 ρ) µ H, ɛ I ɛ H ɛ I τ I wnɛ I f (ɛ I ) dɛ I = d f (ɛ I ) dɛ I := D, ɛ I ɛ I The required tax rates are given by τ H = τ I = (1 ρ) µ H ɛ I wnɛ I f (ɛ I ) dɛ I, (3) D ɛ I wnɛ I f (ɛ I ) dɛ I. (4) The government maximizes the following problem { } V (τ I, ρ) = max u (wn (1 τ I τ H ) s) φv (n) + βe [u (Rs + d + ɛ I ρɛ H )] {τ I,ρ} ɛ I s.t. (1), (2), (3) and (4). When the government chooses τ I and ρ via expressions (3) and (4) respectively, it automatically xes D and τ H. Since the population in each period is normalized to one we also have that d = D. Optimal taxation and insurance policy. The government rst order conditions are V ρ = u (c 1 ) [ w ( n ρ (1 τ I τ H ) n τ H ρ ) s ρ ] φv (n) n ρ + βe [ u (c 2 ) [ R s ρ + d ρ ɛ H [ ( ) V n = u (c 1 ) w (1 τ I τ H ) n s ] φv (n) n [ [ + βe u (c 2 ) R s + d τ I τ I τ I τ I τ I τ I where we know from expressions (3) and (4) that τ H ρ = µ H wn ( n + (1 ρ) n ρ n ( n + τ I n τ I D = w τ I D ρ = τ Iw n ρ. ), ), (5) ]], (6) ]], 9

10 Substituting these expressions and the rm FOCs into the government FOCs (5) and (6) we get two government Euler equations V ρ = u (c 1 ) µ H ( ) n + (1 ρ) n [ ( ρ + βe u (c 2 ) τ I w n )] n ρ ɛ H =, [ ( )] V = u (c 1 ) wn + βe u n (c 2 ) w n + τ I =. τ I τ I This system describes the trade-o between current cost and future benets of changes in the replacement rate ρ and the labor tax rate τ n which nances lump-sum transfers in the second period. Thus, the optimal level of social insurance provided by an income tax system depends on economy-based fundamentals including preferences, the evolution of income and health risks over the lifecycle, and the structure of a health insurance system. In particular, the optimal income tax rate depends on the structure of health and income shocks and level of social insurance provided by the health insurance program. In the next section, we formulate a more realistic model of the U.S. economy and quantify the optimal degree of progressivity of the US income tax system. In addition, we explore how dierently designs of a health insurance system aect 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 constant rate g. Sector one is populated by a continuum of identical rms that use physical capital K and eective labor services N to produce a non-medical consumption good c with a normalized price of one. Firms in the non-medical sector are perfectly competitive and solve the following maximization problem max F (K, N) qk wn, (7) {K, N} taking the rental rate of capital q and the wage rate w as given. Capital depreciates at rate δ in each period. Sector two, the medical sector, is also populated by a continuum of identical rms that use capital K m and labor N m to produce medical services m at a price of p m. Firms in the medical sector maximize max p mf m (K m, N m ) qk m wn m. (8) {K m, N m} 3.2 Demographics, preferences and endowments The economy is populated with overlapping generations of individuals who live up to a maximum of J periods. Individuals work for J 1 periods and are retired thereafter. Individuals survive each period with age dependent survival probability π j. Deceased agents leave an accidental bequest that is taxed and redistributed equally to the working age population. The population grows exogenously at an annual rate n. We assume stable demographic patterns, so that age j agents make up a constant fraction µ j of the entire population at any point in time. The relative sizes of the cohorts alive µ j and the mass of individuals dying in each period µ j (conditional 1

11 π on survival up to the previous period) can be recursively dened as µ j = j (1+n) years µ j 1 and µ j = 1 π j (1+n) years µ j 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 or leisure. Individual utility is denoted by function u (c, n, h) where u : R++ 3 R is C 2, increases in consumption c and health h, and decreases in labor n. We assume a Cobb-Douglas type utility function of the form u (c, n, h) = (( c η ( ) ) 1 η κ ) 1 σ 1 n 1 [n>] n j h 1 κ 1 σ where n j is an age dependent xed cost of working as in French (25), η is the intensity parameter of consumption relative to leisure, κ is the intensity parameter of health services relative 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 specic skill type ϑ that cannot be changed and that together with their health capital h j and an idiosyncratic labor productivity shock ɛ n j determines their ( ) age-specic labor eciency e ϑ, h j, ɛ n j. The transition probabilities for the idiosyncratic productivity shock ɛ n j follow an age-dependent Markov process with transition probability matrix ) Π n j (ɛ. An element of this transition matrix is dened as the conditional probability Pr n i,j+1 ɛn i,j, where the probability of next period's labor productivity ɛ n i,j+1 depends on today's productivity shock ɛ n i,j.2, 3.3 Health capital ( ) Health capital evolves according to h j = H m j, h j 1, δj h, εh j, where h j denotes current health capital, h j 1 denotes health capital of the previous period, δj h is the depreciation rate of health capital and ɛ h j is an idiosyncratic health shock. The exogenous health shock ɛh j follows a Markov process with age dependent transition probability matrix Π h j. Transition probabilities to next period's health shock ɛ h j+1 depend on the current health shock ɛh j so that an element of transition ) matrix Π h j (ɛ is dened as the conditional probability Pr h j+1 ɛh j. Individuals can buy medical services m j at price p m to improve their health capital. Specically, the law of motion of health capital follows Investment Trend {}}{{}}{ Disturbance ( ) {}}{ h j = φ j m ξ j + 1 δj h h j 1 + ε h j. (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 (2). 2 Heathcote, Storesletten and Violante (217) model government spending in terms of public goods that enter the preferences directly. In their setting, the utility from the consumption of public goods and social insurance are two main channels of welfare gains. In our analysis, we abstract from the former and focus on risk sharing through social insurance programs. We model all public spending programs explicitly. Our modeling approach is similar to the one in Conesa and Krueger (26). However, we take into account both labor productivity and health risks as well as the institutional setup of the U.S. healthcare sector. 11

12 3.4 Health insurance In the benchmark economy we introduce the main features of the U.S. health insurance system before the implementation of the Aordable Care Act in 21. The health insurance market consists of private health insurance companies that oer two types of health insurance policies: (i) an individual health insurance plan (IHI) and (ii) a tax deductible group health insurance plan (GHI). Individuals are required to buy insurance one period prior to the realization of their health shock in order to be insured in the following period. The insurance policy needs to be renewed each period. By construction, agents in their rst period are thus not covered by any insurance. The government provides public health insurance with Medicaid for the poor and Medicare for retirees. To be eligible for Medicaid, individuals are required to pass an income and asset test. The health insurance state in j for workers can therefore take on the following values: not insured, 1 Individual health insurance (IHI), in j = 2 Group health insurance (GHI), 3 Medicaid. After retirement (j > J 1 ) all agents are covered by public health insurance which is a combination of Medicare and Medicaid for which they pay a premium, prem R. An agent's total health expenditure in any given period is p in j m m j, where the price of medical services p in j m depends on insurance state in j. The out-of-pocket health expenditure of a working-age agent is given by { in p j m m j, if in j =, o (m j ) = ( ) ρ in j p in j m m j, if in j > (1) where ρ in j 1 are the insurance state specic coinsurance rates. The coinsurance rate denotes the fraction of the medical bill that the patient has to pay out-of-pocket. 3 A retiree's out-of-pocket expenditure is o (m j ) = ρ R ( p R ) m m j, where ρ R is the coinsurance rate of Medicare and p R m is the price that a Medicare patient pays for medical services. Insurance companies. Insurance companies oer IHI to any working individual and charge an age and health dependent premium, prem IHI (j, h). In addition, workers are randomly assigned to employers who oer GHI which is indicated by random variable ɛ GHI = 1. The GHI premium, prem GHI, is tax deductible and group rated so that insurance companies are not allowed to screen workers by health or age. If a worker is not oered group insurance from her employer, i.e., ɛ GHI =, the worker can still buy IHI. However, the worker is subjected to screening and the insurance premium is not tax deductible. There is a Markov process that governs the group ( insurance ) oer probability. It is a function of the individual's permanent skill type ϑ. Let Pr ɛ GHI, ϑ be the conditional probability that an agent has group insurance j+1 ɛghi j status ɛ GHI j+1 at age j + 1 given she had group insurance status ɛghi j at age j. All conditional probabilities for group insurance status are collected in a 2 2 transition probability matrix Π GHI j,ϑ. For simplicity we abstain from modeling insurance companies as prot maximizing rms and simply allow for a premium markup ω. Since insurance companies in the individual market screen customers by age and health, we impose separate clearing conditions for each age-health 3 For simplicity we include deductibles and co-pays into the coinsurance rate. 12

13 type, so that premiums, prem IHI (j, h), adjust to balance ( ) [ ( 1 + ω IHI j,h µj 1 [inj (x j,h)=1] 1 ρ IHI ) ] p IHI m m j,h (x j,h ) dλ (x j, h ) (11) ( ) = Rµ j 1 1 [inj 1,h(x j 1,h)=1] premihi (j 1, h) dλ (x j 1, h ), where x j, h is the state vector for cohort age j not containing h since we do not want to aggregate over the health state vector h in this case. The clearing condition for the group health insurances is simpler as only one price, prem GHI, adjusts to balance ( 1 + ω GHI ) J 1 [ ( µ j 1 [inj (x j )=2] 1 ρ GHI ) ] p GHI m m j (x j ) dλ (x j ) (12) j=2 J 1 1 ( = R µ j 1 [inj (x j )=2]prem GHI) dλ (x j ), j=1 where ωj,h IHI and ωghi are markup factors that determine loading costs (xed costs or prots). Variables ρ IHI and ρ GHI are the coinsurance rates, and p IHI m and p GHI m are the prices for health care services of the two insurance types. The respective left-hand-sides in the above expressions summarize aggregate payments made by insurance companies whereas the right-hand-sides aggregate 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 interest rate R. The premium markups generate prots which are redistributed in equal (per-capita) amounts of π prots to all surviving agents Fiscal policy The government administers various government programs that are nanced by a combination of taxes. Progressive income taxes. The government imposes a progressive income tax code on household incomes. The tax schedule is given by a parametric function τ (ỹ) = ỹ λỹ (1 τ), where τ (ỹ) denotes net tax revenues as a function of pre-tax income ỹ, τ is the progressivity parameter, and λ is a scaling factor to match the U.S. income tax revenue. This tax function is fairly general and captures the common cases: (1) Full redistribution: τ (ỹ) = ỹ λ and τ (ỹ) = 1 if τ = 1, <1 {}}{ (2) Progressive: τ (ỹ) = 1 (1 τ)λỹ ( a1) and τ (ỹ) > τ(ỹ) ỹ if < τ < 1, (3) No redistribution (proportional): τ (ỹ) = ỹ λỹ and τ (ỹ) = 1 λ if τ =, >1 {}}{ (4) Regressive: τ (ỹ) = 1 (1 τ)λỹ ( τ) and τ (ỹ) < τ(ỹ) ỹ if τ <. 4 Notice that ex-post moral hazard and adverse selection issues arise naturally in the model due to information asymmetry. Insurance companies cannot directly observe the idiosyncratic health shocks and have to reimburse agents based on the actual observed levels of health care spending. Adverse selection arises because insurance companies 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' prot conditions. Individual insurance contracts do distinguish agents by age and health status but not by their health shock. 13

14 This tax function has a long tradition in public nance (see Musgrave (1959), Kakwani (1977)) and was implemented into a dynamic setting by Benabou (22) and more recently used in Heathcote, Storesletten and Violante (217). This tax function is exible and can be used to model the transfer programs. Heathcote, Storesletten and Violante (217) abstract from modeling government transfer programs and allow negative income taxes to act as government transfers. We instead model government spending explicitly including Medicaid and Food Stamp programs for low income individuals. To avoid double counting we impose a non-negative tax payment restriction, τ (ỹ). This non-negative tax restriction eliminates all government transfers embedded in the progressive tax function. 5 In our benchmark setting, the progressive income tax function has the following specication [ τ (ỹ) = max, ỹ λỹ (1 τ)], Spending programs. The government has the following spending programs: social security, social transfers to low income earners, public health insurance, and general government consumption. The social security program operates on a Pay-As-You-Go (PAYG) principle in which the government collects a payroll tax τ SS from the working population to nance social security benets t SS for retired individuals. The PAYG program is self-nanced J J 1 µ j µ j j=j 1 +1 t SS j (x j ) dλ (x j ) = j=1 tax SS j (x j ) dλ (x j ). 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 health insurance consists of Medicare and Medicaid. Medicare is nanced by a Medicare tax (tax Med j ) and premium payments (prem R ) and Medicaid is nanced by general tax revenues. In addition, the government needs to nance government consumption (C G ) which is exogenous and unproductive. Finally, the government collects accidental bequests from deceased individuals and redistributes them as lump-sum payments t Beq j to all surviving working-age households. Balanced budget. The government collects consumption tax revenue at a at rate, τ C, and income tax revenue at a progressive rate to balance its budget every period. The government budget constraint is given by J 1 C G + T SI ( + µ j 1 [inj (x j )=3] 1 ρ MAid ) p MAid m m j (x j ) dλ (x j ) (13) = + J j=j 1 +1 j=2 µ j (1 ρ R ) p R mm j (x j ) dλ (x j ) J [τ µ C j c (x j ) + tax j (x j ) ] dλ (x j ) + j=1 J j=j 1 +1 µ j J 1 prem R (x j ) dλ (x j ) + µ j where ρ MAid and ρ R are the coinsurance rate of Medicaid and of the combined Medicare/Medicaid program for the old, respectively. The government collects and redistributes accidental bequests in a lump-sum fashion to all working-age households J 1 J µ j t Beq j (x j ) dλ (x j ) = µ j a j (x j ) dλ (x j ), (14) j=1 5 We will relax this restriction in our sensitivity analysis. j=1 j=1 tax Med j dλ (x j ), 14

15 where µ j and µ j are measures of the surviving and deceased agents at age j in time t, respectively. 3.6 Household problem Individuals at age j J 1 are workers and thus exposed to labor shocks. Old individuals, j > J 1, are retired (n j = ) and receive pension payments. They do not face labor market shocks anymore. The agent state vector at age j is given by ( ) a j, h j 1, ϑ, ɛ n j x j D j, ɛh j, in j R + R + R + R + R I w if j J 1, ( ) a j, h j 1, ϑ, ɛ h j, in (15) j R + R + R + R I R if j > J 1, where a j is the capital stock at the beginning of the period, h j 1 is the health state at the beginning of the period, ϑ is the skill type, ɛ n j is the positive labor productivity shock, ɛh j is a negative health shock, in j is the insurance state and I w = {, 1, 2, 3} denotes the dimension of the insurance state of workers and I R = {1} is the sole insurance state for retirees as every retiree is on a combined Medicare/Medicaid program. After the realization of the state variables, agents simultaneously chose from their choice set { (c j, n j, m j, a j+1, in j+1 ) R + [, 1] R + R + I w if j J 1, C j (c j, m j, a j+1 ) R + R + R + if j > J 1, where c j is consumption, n j is labor supply, m j are health care services, a j+1 are asset holdings for the next period and in j+1 is the insurance state for the next period in order to maximize their lifetime utility. All choice variables in the household problem depend on state vector x j. We suppress this dependence in the notation to improve readability. The household optimization problem is V (x j ) = max {u (c j, n j, h j ) + βπ j E [V (x j+1 ) x j ]} s.t. (16) {C j } ( 1 + τ C ) c j + (1 + g) a j+1 + o (m j ) + 1 [j J1 in j+1 =1]prem IHI (j, h) + 1 [j J1 in j+1 =2]prem GHI + 1 [j>j1 ]prem R = y j tax j + t SI j, a j+1, n j 1 and (9). Variable τ C is a consumption tax rate, o (m j ) is out-of-pocket medical spending depending on insurance type, yj W is the sum of all income including labor, assets, bequests, and prots. Household income and tax payments are dened as y j = e ( ϑ, h j, ε n ) j nj w + 1 [j > JW ]t Soc j (ϑ) + R ( a j + t Beq) + π prots, tax j = τ (ỹ j ) + tax SS j + tax Mcare j, ỹ j = y j a j t Beq 1 [inj+1 =2]prem GHI.5 ( tax SS tax SS j = τ Soc min tax Mcare j = τ Mcare j + tax Med ) j, ( ȳ ss, e ( ϑ, h j, ε n j ) nj w 1 [inj+1 =2]prem GHI), ( e ( ϑ, h j, ε n j ) nj w 1 [inj+1 =2]prem GHI), t SI j = max [, c + o (m j ) + tax j y j ]. Variable w is the market wage rate and R is the gross interest rate and π prots denotes prots from insurance companies. Variable ỹ j is taxable income, τ (ỹ j ) is the progressive income 15