Retirement Financing: An Optimal Reform Approach

1.1 Related Literature

Our paper contributes to various strands in the literature on policy reform. We contribute to the large and growing literature on retirement financing, most of which studies the implications of a specific set of policy proposals. For example, Nishiyama and Smetters (2007) studied the effect of privatization of Social Security. Kitao (2014) compared different combinations of tax increase and benefit cuts within the current Social Security system. McGrattan and Prescott (2017) proposed phasing out Social Security and Medicare benefits and removing payroll taxes. Blandin (2018) studied the effect of eliminating the Social Security maximum earnings cap.

We depart from the existing literature in two important aspects. First, we do not restrict the set of policies at the outset. Therefore, our results can inform us about which policy instrument is an essential part of a reform. As a result, we find that changing the marginal tax rates on labor earnings is not a major contributor to an optimal policy reform. Second, we focus explicitly on Pareto optimal policies and derive the condition that can inform us about the feasibility of Pareto improving policy reforms. In that regard, our paper is close to Conesa and Garriga (2008), who characterized a Pareto optimal reform in an economy without heterogeneity within each cohort and found Pareto optimal linear taxes (a Ramsey exercise).

Our paper is also related to a large literature on optimal policy design. The common approach in this literature is to take a stand on specific social welfare criteria and find optimal policies that maximize social welfare. For example, Conesa and Krueger (2006) and Heathcote, Storesletten, and Violante (2017) studied the optimal progressivity of a tax formula for a parametric set of tax functions, while Huggett and Parra (2010) and Heathcote and Tsujiyama (2015) did the same using a Mirrleesian approach that does not impose a parametric restriction on policy instruments (similar to our paper). One drawback of this approach is that it relies on the choice of the social welfare function. Consequently, the resulting policy proposals can improve efficiency while at the same time provide redistribution across individuals.6 Moreover, the resulting policies are conditional on a particular welfare function which might or might not be conforming to the political institutions that are determinants of government policies in a certain country. The benefit of our approach is that it does not rely on an arbitrary welfare function by providing nonnegative gains to all individuals. To the best of our knowledge, this is the first paper that proposes this approach to optimal policy reform in a dynamic quantitative setting.7

Our paper also contributes to the literature on dynamic optimal taxation over the life cycle. Similarly to Weinzierl (2011), Golosov, Troshkin, and Tsyvinski (2016), and Farhi and Werning (2013b), we provide analytical expressions for distortions and summarize insights from those expressions. However, unlike these cited works, which focus on labor distortions over the life cycle, we focus on intertemporal distortions. Furthermore, we emphasize the role of policy during the retirement period, thus relating our work to Golosov and Tsyvinski (2006), who studied the optimal design of the disability insurance system, and Shourideh and Troshkin (2017) and Ndiaye (2018), who focused on an optimal tax system that provides incentive for an efficient retirement age.

Another strand of literature our paper is related to studies the role of Social Security in providing longevity insurance. Hubbard and Judd (1987), İmrohoroǧlu, İmrohoroǧlu, and Joines (1995), Hong and Ríos-Rull (2007), and Hosseini (2015) (among many others) have examined the welfare-enhancing role of providing an annuity income through Social Security when the private annuity insurance market has imperfections. Caliendo, Guo, and Hosseini (2014) pointed out that the welfare-enhancing role of Social Security in providing annuitization is limited because Social Security does not affect individuals' intertemporal trade-offs. In this paper, we pinpoint the optimal distortions and policies that address this shortcoming in the system by emphasizing that any optimal retirement system (whether public, private, or mixed) must include features that affect individuals' intertemporal decisions on the margin. In our proposed implementation, those features take the form of a nonlinear subsidy on assets.

Finally, our paper is related to the literature on the observed lack of annuitization in the United States. Friedman and Warshawsky (1990) showed that if one is to consider the high fees (what they referred to as “load factor”) on annuities provided in the market together with adverse selection, the standard model without bequest motives can go a long way in explaining the lack of annuitization. Diamond (2004) and Mitchell, Poterba, Warshawsky, and Brown (1999) pointed to taxes on insurance companies as well as high overhead costs (marketing and administrative costs as well as other corporate overhead) behind the high transaction costs. In particular, observing that the government cost of handling Social Security is much lower, Diamond (2004) suggested government-provided annuities—a task that our saving subsidies achieve. Our paper can be thought of as a quantitative evaluation of this idea in reforming the retirement benefit system in the United States.

The rest of the paper is organized as follows: Section 2 lays out a two-period OLG framework where we provide intuition for our results; in Section 3, we describe the benchmark model used in our quantitative exercise; in Section 4, we calibrate the model; in Section 5, we discuss our quantitative results in steady state; in Section 6, we discuss reforms in an aging economy; in Section 7, we study various robustness exercises; and in Section 8, we present our conclusions.

3.1 Demographics, Preferences, and Technology

Time is discrete, and the economy is populated by overlapping generations. A cohort of individuals is born in each period . The number of newborns grows at rate . Upon birth, each individual draws a type from a continuous distribution that has density . This parameter determines three main characteristics of an individual: life-cycle labor productivity profile, survival rate profile, and discount factor. In particular, an individual of type θ has a labor productivity of at age j. We assume that and thus refer to individuals with a higher value of θ as more productive. Everyone retires at age R, and for .

Moreover, an individual of type θ and of age j who is born in period t has a survival rate (this is the probability of being alive at age , conditional on being alive at age j).17 Nobody survives beyond age J (with for all θ and t). As a result, the survival probability at age j for those who are born in period t is

Additionally, an individual of type θ has a discount factor given by . Thus, that individual's preferences over streams of consumption and hours worked are given by

(6)

Here, and are consumption and hours worked for an individual of θ at j who is born in period t.

We assume that the economy-wide production function uses capital and labor and is given by . In this formulation, is aggregate per capita stock of capital, and is the aggregate effective units of labor per capita. Effective labor is defined as labor productivity, , multiplied by hours, . Its aggregate value is the sum of the units of effective labor across all individuals alive in each period. In other words,

where is the share of type θ of age j in the population in period t. Finally, capital depreciates at rate δ. Therefore, the return on capital net of depreciation is .

3.2 Markets and Government

We assume that individuals supply labor in the labor market and earn a wage per unit of effective labor. In addition, individuals have access to a risk-free asset and cannot borrow. The assets of the deceased in each period t convert to bequests and are distributed equally among the living population in period t.18 Our main assumption here is that annuity markets do not exist. As discussed in Section 2, this assumption is in line with the observed low volume of trade in annuity markets in the United States and other countries.19

The government uses nonlinear taxes on earnings from supplying labor, including the Social Security tax, while we assume that there is a linear tax on capital income and consumption. The revenue from taxation is then used to finance transfers to workers and Social Security payments to retirees. While transfers are assumed to be equal for all individuals, Social Security benefits are not and depend on individuals' lifetime income.

Given the above market structure and government policies, each individual born in period t faces a sequence of budget constraints of the following form:20

(7)

Here, is the rate of return on assets ; and are the earnings tax and asset tax functions, respectively; are transfers to working individuals; is the retirement benefit from the government; and is the income earned from bequests. The dependence of retirement benefits on lifetime earnings is captured by , which is given by

All tax functions and transfers can potentially depend on age and birth cohort (e.g., along a demographic transition).

There is a corporate tax rate paid by producers. Therefore, the return on assets, , is equal to .21 We assume that the government taxes households' holding of government debt at an equal rate and, therefore, the interest paid on government debt is also .

Given the above assumptions, the government budget constraint is given by

(8)

where is per capita government purchases, is per capita government debt, and is population growth rate at t, which can be calculated as a function of mortality rates and . Finally, goods and asset market clearing implies

(9)

(10)

(11)

3.3 Equilibrium

The equilibrium of this economy is defined as allocations where individuals maximize (6) subject to (7), while the government budget constraint (8), market clearings (9), (10), and (11) must hold. The equilibrium is stationary (or in steady state) when all policy functions, demographics parameters, allocations, and prices are independent of calendar period t.

This sums up our description of the economy. In the next section, we describe our approach to analyzing an optimal reform within the framework specified above. Note that we have not specified any details of the status quo policies yet. We will do that in Section 4 where we impose detailed parametric specifications of the U.S. tax and Social Security policies and calibrate this model to the U.S. data. We can then apply our optimal reform approach to the calibrated model and conduct our optimal reform exercise.

When the tax function and Social Security benefits are calibrated to those for the United States, we refer to the resulting equilibrium allocations and welfare as status quo allocations and welfare. We refer to the status quo welfare of an individual of type θ who is born in period t by .

3.4 Remark on Annuity Markets

Throughout the analysis in this paper, we assume that there are no markets for annuities. This is in line with the observed lack of annuitization in the United States. As Poterba (2001), Benartzi, Previtero, and Thaler (2011), and many others have mentioned, the annuity market in the United States is very small. According to Hosseini (2015)'s calculation based on HRS, only 5 percent of the elderly hold private annuities in their portfolio.22 Moreover, the offered annuities have very high transaction costs and low yields (see Friedman and Warshawsky (1990) and Mitchell et al. (1999)), and are not effectively used by individuals (see Brown and Poterba (2006)).23

Various reasons have been proposed as leading to lack of annuitization in the United States: the presence of Social Security as an imperfect substitute, adverse selection in the annuity market, low yields on offered annuities due to overhead and other costs, bequest motives, and complexity of choice faced by individuals (see Benartzi, Previtero, and Thaler (2011) and Diamond (2004)). All of these reasons warrant government intervention in annuity markets. In our paper, we have focused on the extreme case where the government fully takes over the annuity market. This role for the government was also discussed in detail by Diamond (2004). It would be interesting to study the case where annuity markets are present and government intervention crowds out the private market. This, however, is beyond the scope of our paper.

3.5 Optimal Policy Reform in the Quantitative Framework

Our optimal policy-reform exercise is very similar to the one in the two-period model provided in Section 2. It builds on the positive description of the economy in Section 3. In particular, we use the distribution of welfare implied by the model in Section 3 and consider a planning problem that chooses policies in order to minimize the cost of delivering this distribution of welfare, the status quo utility profile , to a particular representative cohort of individuals. We show how the efficiency tests discussed in Section 2 extend to the dynamic environment. For simplicity, we assume steady state and do not consider the changes in prices resulting from the reforms. Later, in our quantitative exercise, we allow for both transitions and changes in prices.

3.5.1 A Planning Problem

The set of policies that we allow for in our optimal reform are very similar to those described in Section 3. In particular, we allow for nonlinear and age-dependent taxation of assets. Moreover, we allow for nonlinear and age-dependent taxation of earnings together with flat Social Security benefits (i.e., Social Security benefits are independent of lifetime earnings). Therefore, given any tax and benefit structure, each individual maximizes utility (6) subject to the budget constraints (7).

The planning problem associated with the optimal reform finds the policies described above to maximize the net revenue for the government (i.e., present value of receipts net of expenses). In this maximization, the government is constrained by the optimizing behavior by individuals—as described above, the feasibility of allocations and the requirement that each individual's utility must be above . We also focus on the steady-state problem for the government and ignore issues related to transition.

Using standard techniques, in the 8 we show that the problem of finding Pareto optimal reforms for each generation24 can be written as a planning problem and in terms of allocations. This planning problem maximizes the revenue from delivering an allocation of consumption and labor supply over the life of a generation subject to an implementability constraint and a minimum utility requirement given by

(P1)

subject to

(12)

(13)

(14)

The objective in the above optimization problem is equal to the present discounted value of government tax receipts net of outlays from a given cohort of individuals.25

3.5.2 Test of Pareto Optimality

Given our environment and the optimal reform problem described above, we can provide tests of Pareto optimality. Note that as before, the labor and savings wedge is defined as

(15)

where w and r are steady-state wage and interest rate, respectively.

The following proposition presents tests of Pareto optimality for the quantitative framework developed above:

Proposition 4.The wedges induced by any Pareto optimal tax schedule must satisfy the following equality constraints:

(16)

(17)

as well as the following inequality:

(18)

where is the intertemporal elasticity of substitution at age 0.

Moreover, when the first-order conditions are sufficient for describing the behavior of consumers, the above conditions are also sufficient for any Pareto optimal tax schedule.26

The above formulas are the equivalents of the optimality formulas in Section 2. In particular, as it might be apparent, equations (16) and (17) are the equivalents of the tax-smoothing relationship (3).

Equation (16) is identical to (3) and has a similar intuition. It states that, at the optimum, the behavioral response of the government revenue to a perturbation of marginal tax rate of earnings should be equal to that of asset taxes at every age and for every type. Note that in the dynamic model, a perturbation of any tax rate affects all margins—assets and earnings over the life cycle—through an income effect. As a result, writing the precise equation that describes the behavioral increase in government revenue in response to a tax increase is rather cumbersome. Nevertheless, (16) is identical to its equivalent in the two-period model and, therefore, the intuition behind it is the same. Similarly to the discussion in Section 2, (16) can be written as

which again identifies the main forces that lead to distortion of savings, that is, market incompleteness and higher saving demand by more productive individuals.

Similarly, equation (17) states that the behavioral increase in government revenue must be equated for an increase in taxes at age j and age . Note that since these tax perturbations are done in two different ages, in order for them to be welfare neutral, the magnitude of the perturbations must be adjusted. The last term in (17) performs this age adjustment; by definition of the savings wedge, . In other words, the age adjustment is equal to the ratio of marginal utilities of consumption adjusted by the interest rate. Finally, as before, inequality (18) is equivalent to the nonnegativity of the implied Pareto weights.

Equation (17) is informative about the behavior of savings wedges—marginal tax rates—over time. Specifically, it highlights the role of the changes in the gradient of the survival over the life cycle, that is, . An increase in this gradient with age leads to a decline in subsidies or increase in taxes on savings. This goes back to a mechanism that we have already discussed in Section 2: a higher gradient of survival leads to a higher value of consumption late in life and higher demand for saving. Taxation of savings is then a way to prevent productive individuals from earning less and saving less and, as a result, reducing the deadweight loss of taxation of earnings. From an alternative perspective, when the gradient of survival is high, saving increases quickly with productivity, which in turn reduces the density of number of workers at a certain asset level. Hence, the tax-smoothing intuition in Section 2 would imply that taxes (subsidies) must be higher (lower). As we show in our quantitative exercise, the gradient of survival is positive and increases with age. This would imply that subsidies become more progressive as individuals age.

Another way of getting an understanding about Pareto optimality tests is to use (16) and rewrite (17) as

(19)

This relationship can be thought of as the tax-smoothing relationship between earnings taxes across two ages. In our quantitative exercise, we use this relationship to shed light on Pareto optimality of earnings taxes.

Finally, note that whether the above tests are satisfied determines the Pareto optimality of a tax system. Away from the optimum, that is, when these conditions are violated, in general it is difficult to determine what margins must be adjusted. In our numerical simulations, however, often the magnitude of the violation is indicative of the significance of the reform. In other words, when the above conditions are violated significantly, the gains from a reform in the associated policies are significant. In what follows, we show that the main source of violation is the equations associated with savings wedges and, thus, they are the main source of gains.

4.1 Earning Ability Profiles

We assume that individual productivity at age j can be written as

where θ is an individual fixed effect, while is an age-dependent productivity shifter given by

To estimate the productivity parameters, we follow a large part of the literature (e.g., Altig, Auerbach, Kotlikoff, Smetters, and Walliser (2001), Nishiyama and Smetters (2007), and Shourideh and Troshkin (2017)) and use the effective reported labor earnings per hour in Panel Study of Income Dynamics as a proxy for . We calculate this as the ratio of all reported labor earnings to total reported hours. For labor earnings, we use the sum over a list of variables on salaries and wages, separate bonuses, the labor portion of business income, overtime pay, tips, commissions, professional practice or trade payments, and other miscellaneous labor income converted to constant 2000 dollars. In order to avoid well-known issues in the raw data, we use Heathcote, Perri, and Violante's (2010) version of the PSID data. The resulting estimated parameters are , , , and .

Moreover, we assume the type-dependent fixed effect θ has a Pareto-lognormal distribution with parameters . This distribution approximates a lognormal distribution with parameters and at low incomes and a Pareto distribution with parameter at high values. It therefore allows for a heavy right tail at the top of the ability and earnings distribution. For this reason, it is commonly used in the literature (see Golosov, Troshkin, and Tsyvinski (2016), Badel and Huggett (2014), and Heathcote and Tsujiyama (2015)).28 We choose the tail parameter and variance parameter to be and , respectively. The location parameter is set to so that has mean 0. With these parameters, the cross-section variance of log hourly wages in the model is 0.36. Also, the ratio of median hourly wages to the bottom decile of hourly wages is 2.3. These statistics are consistent with the reported facts on the cross-section distribution of hourly wages in Heathcote, Perri, and Violante (2010).

4.2 Demographics and Mortality Profiles

Population growth is constant and is equal to 1 percent. The model period is 1 year. Individuals start earning income at age 25, they all retire at age 65, and nobody survives beyond 100 years of age. Each individual has a Gompertz force of mortality

(20)

The Gompertz distribution is widely used in the actuarial literature (see, e.g., Horiuchi and Coale (1982)) and economics (see, for example, Einav, Finkelstein, and Schrimpf (2010)). The second term in equation (20) determines the changes in mortality by age and is common across all types. The first term is decreasing in θ and determines the gradient of mortality in the cross section. Therefore, a higher-ability person has a lower mortality at all ages. The key parameter is , which determines how mortality varies with ability. To choose this parameter, we use data on the male mortality rate across lifetime earnings deciles reported in Waldron (2013). She used Social Security Administration data to estimate mortality differentials at ages 67–71 by lifetime earnings deciles. Table I shows the estimated annual mortality rates for 67- to 71-year-old males born in 1940.29 This piece of evidence points to large differences in death rates across different income groups, with the poorest deciles almost four times more likely to die than the richest decile. We use these data to calibrate parameter .

Parameter is chosen to match the average survival probability from cohort life tables for the Social Security area by year of birth and sex for males of the 1940 birth cohort (Table 7 in Bell and Miller (2005)). Finally, is chosen so that mortality at age 25 is 0. The parameters that give the best fit to the mortality data in Table I and average mortality data are , , and . Figure 1 shows the fit of the model in terms of matching mortality across the lifetime earnings deciles in Waldron (2013). Once we have the mortality hazard , we can find the survival probability .

Using this parameterization, we find there are 4 workers per each retiree in the steady state. This is consistent with U.S. Census Bureau estimates.30

4.3 Preferences and Technology

We assume a constant relative risk aversion over consumption, , and constant Frisch elasticity for disutility over hours worked, . The risk aversion parameter is , and the elasticity of labor supply is . The weight of leisure in utility ψ is chosen so that, in the model, the average number of annual hours worked is 2000.

To capture the heterogeneity in the discount factor across different ability types, we assume

We choose to match a capital to output ratio of 4.31 The other parameter, , determines the degree of heterogeneity in the discount factor. The larger is , the larger is the dispersion in the discount factor across ability types. We choose this parameter to match the wealth Gini index of 0.78 based on the 2007 Survey of Consumer Finances (SCF).

The aggregate production function is Cobb–Douglas with a capital share parameter , and the depreciation rate is . These are chosen to match the average ratio of capital income and investment relative to GDP in the United States over the period 2000–2010.

4.4 Social Security

Social Security taxes are levied on labor earnings, up to a maximum taxable, as in the actual U.S. system. Benefits are paid as a nonlinear function of the average taxable earnings over lifetime.32 Let e be labor earnings and be maximum taxable earnings. We set equal to 2.47 times the average earnings in the economy, . The Social Security tax rate is .33 There is also a Medicare tax rate, , which applies to the entire earnings.

Each individual's benefits are a function of that individual's average lifetime earnings (up to ). We use the same benefit formula that the U.S. Social Security Administration uses to determine the primary insurance amount (PIA) for retirees:

To account for Medicare benefits, we assume each individual in retirement will receive an additional transfer independent of that individual's earnings history. We choose this value so that the aggregate Medicare benefits are 3 percent of GDP.34

4.5 Income Taxes and Government Purchases

In addition to Social Security, the government has an exogenous spending G, which we assume to be 8 percent of GDP.35 For the income tax function, we use

where y is taxable income. During the working age, the taxable income for each individual is , in which is labor earnings and is the Social Security and Medicare payroll taxes that the worker pays. The second term reflects the effective tax credit individuals get for the portion of Social Security tax paid by their employers. We assume retirement benefits are not taxed.

The tax function of this form is extensively used to approximate the effective income taxes in the United States. The parameter τ determines the progressivity of the tax function, while λ determines the level (the lower λ is, the higher are the total tax revenues for a given τ). Heathcote, Storesletten, and Violante (2017) estimated a value of 0.151 for τ, based on PSID income data and income tax calculations using NBER's TAXSIM program. We use their estimated value for τ and choose λ. We refer to this tax function as HSV tax function. The left panel in Figure 2 illustrates the resulting marginal and average taxes as functions of annual earnings in constant 2000 dollars.

Finally, we assume the government debt is 47 percent of GDP.36 The transfers are chosen such that the government budget constraint (equation (8)) is satisfied in stationary equilibrium.

To summarize, individuals pay three different types of taxes on their earnings: HSV nonlinear tax, Social Security payroll tax (subject to a maximum taxable cap), and Medicare tax. In addition, they receive the transfer prior to retirement. The right panel in Figure 2 shows the resulting marginal and average tax on the sum of all these taxes and transfers. The discontinuity in the marginal tax is due to Social Security's maximum taxable earnings cap. In addition to earnings taxes, we assume that there is a proportional tax on consumption. This tax allows us to match the government's balance sheet. In particular, part of the government's revenue comes from consumption tax, which is not captured by the earnings tax and transfers, as estimated by Heathcote, Storesletten, and Violante (2017). In our steady-state analysis, the value of this consumption tax, represented by , is fixed and is set to 5.5 percent, as calculated in Mendoza, Razin, and Tesar (1994). In our analysis of the economy under transition, we assume that this consumption tax is increased to finance the increase in the retirement benefits paid out by the government.

Finally, we assume that there is a (corporate) capital income tax of 33 percent, which is paid by the firms. We assume that this rate is fixed and remains unchanged under the reform. As a result, the implied after-tax return on all assets is and is the same for everyone.37 This is also the interest rate that the government pays on its debt. We assume that households do not pay any tax on their savings—beyond the corporate income tax. In general, measurement of savings taxes in the cross section is very difficult.38 This is because of the vast differences in the tax code in the treatment of different types of savings. In reality, a significant fraction of savings are held in tax-deferred retirement accounts39 (which are tax deductible and are treated as income during retirement), whose tax treatment is possibly progressive. On the other hand, richer individuals who hold stocks and bonds can have more sophisticated strategies to minimize their tax burden. These facts motivate us to use no savings taxes in our benchmark calibration. In order to check the robustness of this assumption, we provide two robustness exercises in the Supplemental Material: an exercise with a flat and positive savings tax and one with a progressive savings tax.40

4.6 Calibration Results

Table II lists the parameters that are either taken from other studies, or estimated or calculated independent of the model structure. Their sources and estimation or calculation procedures are outlined in the previous paragraphs. Table III lists the parameters that are calibrated using the model by matching some moments in the U.S. data. The top panel lists the parameter values. The bottom panel shows the targeted moments in the data and the resulting values in the model.

As a check of the model's ability to capture the extent of inequality in the data, we compute the concentration of earnings and wealth in the model and compare them with the data. The results are presented in Figure 3. The left panel shows the concentration of earnings. The dashed line indicates the commutative share of earnings at each commutative population share for individuals age 25 to 60 in the CPS (1994). The solid line shows the same measure in the model. Overall, the model does a good job at capturing the extent of earnings inequality in the data. The Gini index of earnings is 0.43 in the model and 0.46 in the data. Moreover, the model is able to capture the concentration of earnings at the top. The share of earnings of the top 1 percent is 8 percent in the model and 6 percent in the data. This is achieved through the use of a Pareto-lognormal distribution for ability distribution (even though we did not directly target this moment).

Finally, the right panel in Figure 3 shows the concentration of wealth. The dashed line is the cumulative share of wealth owned by each cumulative population share in the SCF (2007). The model matches the Gini index of wealth by construction (see Table III). Heterogeneity in the discount factor allows us to generate a high concentration of wealth in the model. The share of wealth owned by the top 1 percent is 23 percent in the model and 29 percent in the data. In the Supplemental Material, we plot consumption and earnings profiles in the model and discuss their relationship to the data.

5.1 Test of Pareto Optimality

We start our analysis by testing the Pareto optimality of the status quo allocations. We do this by computing the intertemporal and intratemporal distortions for the status quo allocations and checking how much the formulas (16), (17), and (18) are violated.

In Figures 4 and 5, we plot the implications of Pareto optimality tests for the status quo economy. Figure 4 plots the performance of the tests for labor wedges.41 The left panel depicts the inequality (18); the dashed red line is the left-hand side, and the black solid line is the right-hand side. As it illustrates, the inequality only fails to hold over a small range of earnings. This is where effective earnings taxes are regressive, due to the Social Security maximum taxable earnings cap (see Figure 2). In this range, the term is a large negative number. This pushes the left-hand side of inequality (18) up. The right panel depicts the change required in the labor wedge so that the tax smoothing relationship in (19) holds. As we see, the percentage change in the labor wedges to restore (19) is around ±0.5 percent. In other words, it suggests that given the earnings taxes at , that is, age 25, and the redistributive motives they represent, earnings taxes are roughly optimal and should not change by much. This observation is suggestive of one of our main findings: that earnings taxes are not a large source of inefficiency of the tax code.

Moving to the efficiency properties of savings taxes, Figure 5 shows the efficiency properties of savings wedges. In the left panel, we focus on working life and examine (16). The figure depicts the change required in savings taxes so that (16) holds at ages 30, 40, and 50. Interestingly, for young people, savings wedges must increase. This is because these young individuals face a borrowing constraint (they cannot borrow) and thus face a negative wedge on their intertemporal savings margin.42 For individuals with mid-values of lifetime earnings, the required change is minimal, close to 0. This is mainly because their mortality risk is small and not very sensitive to their lifetime income. Finally, for working-age rich individuals, the savings wedge must increase. This is because of the discount rate differentials for productive workers. As mentioned in Section 2, since more productive individuals value consumption more in the future, a tax on savings of an individual incentivizes everyone with a higher productivity to work harder and save more. Nevertheless, this figure illustrates that a reform must significantly change the tax treatment of savings.

The right panel of Figure 5 depicts the change required in savings taxes so that (17) holds with equality—holding the values on the RHS fixed. As it can be seen, savings wedges must decline significantly for the majority of individuals older than 65. This is mainly capturing the fact that markets are incomplete, and a subsidy to savings completes the market, that is, provides annuity insurance to workers. Note that the required change in savings subsidies is small at the extremes. At the bottom of the distribution, individuals face a binding borrowing constraint and thus face a negative wedge. Therefore, the required decline in their savings wedge is not high. For individuals at the top of the income distribution, the mortality risk is very small and, as a result, the required decline in their savings wedge is small.

In summary, the results of our tests suggest the following. First, earnings taxes pass the Pareto optimality tests to a great extent, except around the Social Security earnings cap. Second, savings taxes strongly fail the Pareto optimality tests. This result suggests that in a reform, the focus must be on asset taxes as opposed to earnings taxes. Our numerical results below confirm this intuition.

5.2 Optimal Policies

We solve for optimal policies using the planning problem (P1) outlined in Section 3.5. These are (1) nonlinear, age-dependent taxes on assets upon survival, ; (2) nonlinear, age-dependent taxes on labor income, ; (3) transfers to workers before retirement, ; and (4) transfers to workers after retirement, . Note that transfers are independent of individual choices, but they do depend on age. Note also that the level of transfers and assets of households is not uniquely determined, due to the presence of lump-sum transfers. As a result, we choose transfers such that the lowest-ability type opts not to hold any asset. Moreover, we assume that individuals face linear consumption taxes. We fix the consumption tax rate at the calibrated level for the status quo economy. This assumption eases the comparison of labor income taxes across economies.43 Finally, we fix the corporate income tax rate at the calibrated status quo level. This implies that the pre-tax return on assets is the same in the status quo economy and in the optimal reform.

Figure 6 shows the optimal marginal and average labor income tax functions for ages (solid lines). We also plot the status quo tax functions for comparison (dashed lines). Notice that except for the region where there is a sharp drop in the status quo tax rates (due to Social Security maximum taxable earnings), the optimal taxes are very close to those in the status quo. Furthermore, there is little age dependence in the optimal labor income taxes.44 These results imply that there is little room for improvement in efficiency by reforming labor income taxes. In essence, our exercise confirms the insight from the Pareto optimality tests performed in Section 5.1 regarding earnings taxes.

The left panel of Figure 7 shows the optimal marginal taxes (subsidies) on assets for ages . Since mortality is larger for asset-poor individuals, the rates are larger for these individuals at all ages. In contrast, asset-rich individuals have higher ability, and hence lower mortality. The inefficiency due to the absence of an annuity market is smaller for these individuals; therefore, asset subsidies are smaller (taxes are higher). In this sense, optimal asset taxes (subsidies) are progressive. Figure 7 also illustrates that subsidies are large, around 5 percent, and thus can play an important role in the provision of retirement benefits by the government.

The right panel shows the average marginal rates at each age from 65 to 85 years in comparison to the average mortality of the population. The difference between the two implies the following: first, progressivity of the subsidies is significant and cannot be ignored; second, policies are above and beyond completing the annuity market, as would be the case in a world where mortality were observed by the government (or mortality were uniform in the population).

As before, the implied magnitudes of asset subsidies and their progressivity confirm the results of our optimality tests. In other words, asset subsidies are an important part of our Pareto optimal reform.

5.3 Sources of Retirement Income

It is useful to compare the sources of retirement income in the status quo economy and that of the optimal reform. This comparison would shed light on the burden of the reform for the government and on changes in individual budgets.

Table IV compares the share of government transfers out of the total income for retired individuals (asset income plus government transfers). In our calculation for the status quo economy, the government transfers consist of Social Security (and Medicare) benefits. For comparison, we also include the share of government transfers in retirement income as measured in the CPS data (reported in Poterba (2014)).45

The numbers in our status quo economy are close to the CPS data, particularly for the lower half of the income distribution.

An important feature of the reform economy is the significant reduction in the share of government transfers in retirement income for all income groups except the top quartile. This is mainly a result of the presence of asset subsidies. In particular, asset subsidies imply that individuals will save more. As a result, asset income constitutes a higher fraction of retirement income and, therefore, the share of government transfers in income declines.

5.4 Aggregate Effects of Reforms

Table V shows the summary statistics of the aggregate variables for our economy. In the first column, we report the aggregate quantities in the calibrated benchmark with the status quo U.S. policies. The second column shows the aggregate variables under Pareto optimal reform policies, holding factor prices fixed. In this case, the stock of capital in the economy is 12.29 percent higher relative to the status quo. This is due to higher incentives to save provided by optimal asset subsidies. As a result, GDP is higher by 4.33 percent and consumption by 1.66 percent relative to the status quo. However, consumption as share of GDP falls slightly from 0.69 to 0.67. This is, again, due to a higher desire for savings under optimal reform policies. Overall, the present discounted value of consumption, net of labor income, for each cohort falls by 11.08 percent in the optimal reform relative to the status quo. In terms of flow of consumption, this is equivalent to a 0.82 percent fall in consumption for all types in all ages. That is the amount of decline in status quo consumption needed to equalize the present discount value of consumption, net of labor income, equalized across status quo and optimal reform allocations.

The third column in Table V shows aggregate quantities under Pareto optimal policies with endogenous factor prices (but with benchmark demographics, i.e., current U.S. demographics). In this case, the capital stock is higher by only 5.14 percent. This is due to the general equilibrium effect of the lower real return (3.81 percent relative to 4.05 percent). GDP is higher by 1.72 percent and consumption by 0.58 percent relative to the status quo. The cost savings in this case are significantly larger relative to the case with fixed factor prices. In other words, the present discounted value of consumption, net of labor income, for each generation falls by 29.43 percent (this is equivalent to a 2.18 percent fall in the flow of consumption for all types at all ages). This large difference in cost savings can be accounted for entirely by the fall in the interest rate.46

5.5 Distributional and Budgetary Effects of the Reform

While our exercise keeps the distribution of welfare the same, an optimal reform can affect the allocation of resources across individuals. In this section, we describe the effect of our optimal reform exercise on the distribution of allocations.

Figure 8 shows the Lorenz curve for earnings and wealth distribution for status quo and efficient allocations. As we see, the optimal reform policies do not have a significant effect on the distribution of earnings, which is in line with the fact that earnings taxes exhibit very little change. On the other hand, the efficient distribution of assets is less concentrated than in the status quo. In particular, the wealth Gini under reform policies is 0.64, which is significantly lower than the wealth Gini of 0.78 under the status quo. This is mainly because the consumption of low-productivity individuals increases late in life, due to subsidies on assets and, as a result, the asset distribution becomes less skewed.

Table V shows how the optimal reform affects the government's tax revenue and transfers. There is little difference in total tax revenue and total transfers as a fraction of GDP. However, the nature of transfers changes significantly in an optimal reform. Pure transfers before and after retirement fall as a percentage of GDP; instead, asset subsidies, which amount to 8 percent of GDP, are introduced. Optimal reform policies can achieve the same welfare as status quo policies by collecting more taxes and transferring fewer resources. This is possible because optimal reform policies remove inefficiencies due to a lack of annuitization and inefficiencies in the status quo income tax.

6.1 An Aging Economy

We assume that the status quo economy is initially in a steady state determined by the calibrated parameters, as described in Section 4. The economy then experiences a demographic transition which starts at and ends in 50 years. At the conclusion of the demographic transition, the population growth is 0.5 percent (down from 1 percent), consistent with U.S. Census Bureau's projections (see Colby and Ortman (2015)). In addition, the new population mortality rates match the mortality rates of 2040 birth cohort males (Table 7 in Bell and Miller (2005)). We calibrate equation (20) to match the differences in mortality rates among lifetime earnings deciles reported in Waldron (2013), as well as the new population mortality rates.47 All parameters change gradually according to a linear trend over the 50-year transition period. These assumptions imply that the ratio of workers to retirees falls from 4 (its current value) to 2.4 (its projected value). This is consistent with U.S. Census Bureau's projections (see Colby and Ortman (2015)).

6.2 Transition in the Status quo Economy

In order to solve for optimal policies, we need to know the distribution of lifetime welfare for each birth cohort along the transition path for the status quo economy. Since, under the status quo and in an aging economy, the Social Security program is not sustainable, we have to take a stance on what status quo policies will be implemented in order to make the Social Security system sustainable. To this end, we make the following assumptions. First, we assume that the income tax schedules and Social Security benefit formula do not change. Second, the debt to GDP ratio is held constant at its initial calibrated value of 47 percent. Third, and most importantly, we assume that the consumption tax adjusts in each period to balance the government budget constraint and hence finance the transition. It is important to note that, due to political uncertainties, it is impossible to know how status quo policies evolve in response to demographic changes. Here, we use the simplest benchmark to conduct our analysis. However, our methodology can be applied to any alternative assumption for the future path of status quo policies.

The second column in Table VI shows how the demographic change and continuation of status quo policies affect the aggregates. Since mortality is lower, individuals live longer and, therefore, have a higher demand for savings. This, in turn, increases the stock of capital by 7.96 percent. However, due to the lower number of workers as share of population, the labor input falls by 9.26 percent, resulting in a 2.13 percent decline in GDP.

While continuation of the status quo policies does not change the tax revenue as percentage of GDP, there is a significant increase in old-age transfers as percentage of GDP. This is because there are more retirees in the economy. On the other hand, to offset the effect of a rise in old-age transfers on government budget, the consumption tax rate must rise to 10 percent (from the original value of 5.5 percent). This increase in the consumption tax rate increases the share of government revenue from consumption tax and contributes to a decline in inequality. As a result, inequality overall does not change very much. The cross-sectional distribution of earnings and wealth in the new steady state are depicted in Figure 9. There is no change in the distribution of earnings, while the distribution of wealth becomes slightly more unequal (the wealth Gini index rises from 0.78 to 0.79).

6.3 Reform Exercise

Using the time path of the distribution of welfare for each generation, we solve the problem of minimizing the resource cost of delivering the status quo welfare to each individual in each birth cohort. We do this while keeping the corporate income tax rate and consumption tax rate at their status quo level.48

A complication that arises when performing an optimal policy reform in an economy in transition is the treatment of existing generations: generations that are alive at the time of the reform. The complication arises from an information problem. At the time of the reform, households who have worked and saved previously have revealed their types. Thus, if the government has a flexible enough tax function (e.g., generation-specific taxes on their assets at the time of the reform), it can achieve first best and fully bypass the incentive problem. We think this ability of the government to completely bypass the incentive problem is unrealistic. It also creates a discontinuity on allocations for people who are alive at the time of the reform relative to future generations, which makes it harder to accept it as a reasonable reform.

In order to solve this problem, we make the following assumptions: any person who is alive at the beginning of the reform () will face the status quo policies together with an additional one-time lump-sum transfer. All other individuals will face optimal reform policies. Note that this means that the generations that are alive at the start of the reform receive all the gains from the reform.

6.4 Optimal Reforms

Our quantitative exercise for the transition mainly confirms our previous findings in our steady-state analysis: asset subsidies play a key role in the reform, while earnings taxes do not change it by much. Figure 10 shows the changes in the earnings taxes over time. Since in the course of transition to the new steady state, inequality remains somewhat constant, earnings taxes should not change by much. Furthermore, asset subsidies are still significant, although slightly lower, due to the decline in the mortality rate (Figure 11).

The last column of Table VI shows the impact of these policies on aggregate allocations and on government budget. Capital stock rises more relative to the status quo economy. This leads to a smaller decline in GDP and aggregate consumption.49 Figure 12 shows the path of the aggregate variables over the transition. The jump in the primary surplus as share of GDP is due to the initial lump-sum distribution.

Importantly, reform policies reduce the cost of delivering the status quo welfare to each birth cohort. Under optimal reform policies, the present discounted value of consumption net of labor income for a newborn is 4.24 percent lower relative to what it would be under the continuation of the status quo policies in the steady state (this is equivalent to 0.98 percent lower consumption for all types and all ages). As we discuss above, we distribute these resources to those who are alive at the start of the reform in a lump-sum fashion. This transfer is equivalent to 10.5 percent of GDP in the initial steady state.

Overall, we view the results of our quantitative exercises, one for the aging economy and one in the steady state, as pointing towards the importance of asset subsidies to all individuals as an integral part of any fiscal policy reform. This is in contrast with much of the discussion in policy circles on earnings tax reform (reform of the payroll taxes, etc.).

7.1 Optimal Privatization Reform

As we discussed in Section 5, savings subsidies play an important role in our Pareto optimal reforms. In particular, the optimality tests and the optimal reform exercise indicate a reform of the earnings taxes does not seem to play an important role. One might, however, think that this is due to the generality and flexibility of the asset taxes. Here, we briefly describe an exercise that further highlights the role of asset subsidies and their progressivity. The details of the analysis are in the 8.

In this exercise, we assume that there are no asset taxes or subsidies. This exercise is similar to a particular proposal that has received considerable attention in the literature: privatization of retirement financing. More precisely, this is the proposal to eliminate Social Security retirement benefits and reduce payroll taxes and move towards a save-for-retirement system.50 These privatization policies differ from our optimal reform policy in two very important ways. First, our optimal reform policy does not involve a major adjustment of labor income taxes. Second, our optimal reform policy relies crucially on asset subsidies.

We solve for the best reform policies that feature no old-age transfers and no asset taxes or subsidies. In this regard, the efficiency gains from these policies can be viewed as an upper bound on what can be gained through privatization policies. The additional constraint that we impose on the planning problem relative to that in Section 5 is that the savings wedge as defined in (15) must be 0. This implies that earnings taxes are chosen without constraint, and savings subsidies are 0 for everyone.

Figure 13 (left panel) shows the optimal marginal taxes under privatization policies. Note that marginal rates are lower than the status quo, especially at the lower income levels. Moreover, the drop in marginal taxes matches the level of payroll taxes. In this regard, our optimal policies mimic a key feature of the privatization proposals. However, there is also a crucial difference that our optimal labor tax rates are negative for the poorest individuals. The no-subsidy restriction tilts the optimal profiles of consumption towards younger ages. To accommodate this higher consumption, low-income individuals must work more. The negative marginal income tax provides the incentive needed for these low-ability individuals to increase their work effort.

Under privatization policies, the present discounted value of consumption, net of labor income, rises relative to the status quo under all scenarios regarding prices and demographics. In other words, imposing zero taxes on savings—as opposed to subsidizing them—is stringent enough on allocations that it raises the costs of delivering utilities in the steady state. This highlights the importance of subsidies in any reform. Details of the calculations and aggregate effects of privatization policies are provided in the Supplemental Material.51

While this exercise highlights the importance of savings subsidies, one can also question how important the progressivity of the subsidy system is in a reform. In the Supplemental Material, we perform an optimal reform exercise while imposing that savings taxes must be linear. Our calculations establish that two-thirds of the gains can be achieved by linear subsidies and optimal earnings taxes. Thus, progressivity of subsidies is an integral part of an optimal reform.

7.2 Alternative Status quo Tax Function

One of our key findings in Section 5 is that the status quo earnings tax function in the United States fails the Pareto efficiency test only at the maximum Social Security taxable income. This is partly because, for the most part, we used a smooth function to approximate the U.S. earnings taxes. A concern is that the actual earnings tax in the United States contains many thresholds which lead to a non-smooth tax function and could potentially lead to inefficiencies. To address this concern, we repeat our exercise using the calculations of effective marginal tax rates on labor income provided by the Congressional Budget Office; see Harris (2005). In particular, for the status quo earnings taxes, we use the effective marginal federal (and state and local) tax rate for a head of household with one child in 2005.52 This tax approximation includes many intricate features of the tax code including EITC phase-in and phase-out, AMT, CTC, and itemized deductions.

Figure 14 depicts the earnings tax test. Despite a non-smooth status quo tax function, the earnings tax function fails the inequality test only at the Social Security maximum taxable income. However, it comes close to being violated at a lower income level as well. This level of earnings is the large drop in the effective marginal tax rate due to transition from the EITC phase-out (which implies an effective marginal rate of 31 percent) to the 15 percent bracket. Furthermore, as the right panel in Figure 14 depicts, the deviations from the tax smoothing equation (19) are higher than before and of a magnitude of up to 3 percent. Nevertheless, as depicted in Figure 15, the optimal labor income taxes are not very far from optimal. Intuitively, despite having many ups and downs, there is not much variation in the marginal income taxes relative to a smooth approximation to this schedule. As a result, the earnings taxes do not vary by much in an optimal reform.53

7.3 Additional Motives for Saving

In our analysis so far, we have assumed that the drop in income during retirement and demand for insurance against mortality risk are the only motives for saving. In other words, a large source of inefficiency comes from households' desire to finance old-age consumption and self-insure against outliving their assets. Absent any other motive for saving, the model may over-emphasize the role of life-cycle saving and, hence, exaggerates inefficiencies caused by the annuity market incompleteness. To check the robustness of our findings, in this section we consider other motives for saving commonly considered in the literature: out-of-pocket medical expenditures and bequest motives.

7.3.1 Out-of-Pocket Medical Expenditures

As De Nardi, French, and Jones (2010) documented, out-of-pocket medical expenditures rise rapidly with age and income. As people get older, this increase in their medical needs provides a strong motive to save. This motive is even stronger for those with a higher lifetime income. To examine how this additional saving motive affects our results, we introduce exogenous out-of-pocket medical expenditure to the model.54 We assume these medical expenditures increase with age and ability type θ. More specifically, let be the medical expenditure of a person of type θ at age j. In other words, we assume for all θ. Moreover, to focus on saving in old age, we assume there are no out-of-pocket medical expenditures prior to retirement, that is, for all . Finally, we assume monotonicity with respect to age. For , we assume if for all θ.

Let be the total consumption expenditure. Individual preferences are given by

In other words, individuals have preferences over non-medical expenditure and hours worked.55 The rest of the model is identical to what we described in Section 3. The formulation of the optimal policy problem is also very similar. Since medical expenditures vary with type, the implementability constraint is different from (13) and is given by

The above constraint captures the fact that, due to the presence of type-dependent medical expenditure, individuals with different types value consumption differently, and this can be used to change the incentive to work. In particular, the presence of medical expenditure that increases with types leads to forces towards the optimality of positive taxes on savings. Due to higher medical expenditure, productive individuals have a stronger incentive to save, which creates an inelastic source of income for the government and can be taxed in order to lower the deadweight loss of earnings taxation.

In order to calibrate the out-of-pocket medical expenditure profiles, we closely follow De Nardi, French, and Jones (2010) and use data from the AHEAD survey between 1996 and 2006. We allow medical expenditure to depend on age and permanent income ranking (the individual's average income quantile, which can be thought of as associated with θ). This is depicted in the left panel of Figure 16. Furthermore, in order to better match the patterns of asset decumulation, we assume that σ, the coefficient of absolute risk aversion, takes a value of 2. As before, we calibrate the average discount factor in order to match the capital output ratio. We leave the variation in the discount factor, represented by parameter , the same as in the benchmark model.

To show how well the model captures the pattern of dissaving in retirement, we plot the median assets by permanent income quintile in the model as well as the medial assets by permanent income quintile in the AHEAD data in Figure 16 (right panel). The data are based on De Nardi, French, and Jones (2010) calculations for AHEAD cohorts who were 74 and 84 years old in 1996. As we see, the model (solid line) captures the pattern of dissaving very well except for the assets of the top income quintile.

Using the calibrated model, we compute the optimal earnings tax and asset subsidies. These are presented in Figures 17 and 18. As these figures demonstrate, there are no significant differences in optimal policies derived from the model with medical expenditures relative to the ones presented in the previous sections. In other words, the presence of medical expenditure does not change the prescription of our model about policy reforms. As we have mentioned before, the presence of medical expenditure that increases with earnings leads to forces towards taxation of savings. Yet, our analysis shows that such forces are not strong enough to overcome the forces to subsidize savings. This is mainly because the gradient of medical expenditure is not large enough to generate a strong motive for savings taxes.

Finally, Table VII shows the effect of optimal policies on aggregate quantities. The last row presents the efficiency gains, measured in decline in the present discounted value of lifetime consumption, next to labor income for each cohort. The magnitude of the cost savings is not very different from the ones in the main exercise. This is mainly because of the way optimal reforms affect consumption profiles over the lifetime. In particular, due to annuitization, consumption does not fall as people age and, thus, the same level of utilities can be delivered with a lower level of consumption. As a result, when we calculate the present values, the drop in consumption early in life is more pronounced because of discounting. Because of this, the cost saving measures do not drop significantly.

Table VII. Aggregate Effects of Optimal Policies With out-of-Pocket Medical Expendituresa

	Current U.S. (1)	Continue (2)	Optimal reform
			(3)	(4)	(5)
Factor prices
Interest rate (%)	4.05	3.14	4.05	3.84	3.06
Wage	1	1.11	1	1.02	1.12
Change (%) (relative to status quo)
GDP	–	1.70	2.18	0.9	1.92
Consumption	–	0.19	0.34	−0.31	−0.06
Capital	–	16.21	7.63	3.95	17.97
Labor input	–	−8.24	−2.02	−1.39	−8.93
PDV of net resources	–	–	−9.67	−28.76	−7.94
Consumption equivalence			0.66	1.97	0.99

• a Column (1) is the benchmark calibration to the current U.S. economy. Column (2) is the continuation of the U.S. status quo policies (with the consumption tax adjusted to balance government's budget constraint). Column (3) is the optimal reform policies with prices and demographics fixed at the current U.S. values. Column (4) is the optimal reform policies with equilibrium prices but fixed demographics (at the current U.S. levels). Column (5) is the optimal reform policies with equilibrium prices and future demographics. In columns (3) and (4), the percentage change in the PDV is calculated relative to column (1). In column (5), the percentage change in the PDV is calculated relative to column (2).

In summary, the inclusion of out-of-pocket medical expenditure results in a richer model that is able to capture more details in the patterns of asset accumulation or decumulation. However, the model's implication for an optimal policy does not change. Moreover, the efficiency gains from implementing optimal policies, although lower, are still significant and imply that the reform is effective even in the presence of out-of-pocket medical expenditure.

7.3.2 Bequest Motive

Another potentially important reason for saving is the bequest motive. When individuals want to leave assets behind, either for their descendants (altruism) or for the society (joy-of-giving), they save more. In order to investigate the robustness of our results to the addition of this motive, we extend the model in Section 7.3.1 to allow for bequest motives. We assume that individuals have joy-of-giving bequest motives (see De Nardi, French, and Jones (2010), Lockwood (2012), and De Nardi and Yang (2016), among many others) given by

(21)

where is the amount of bequest left by type θ if they do not survive to age . We use a utility from bequest that implies that bequests are luxury goods. Moreover, we allow the government to differentially tax savings conditional on survival and bequests. In order to calibrate this model, we match the moments related to the distribution of bequests in the United States: the fraction of the deceased individuals that leave no bequests and the ratio of bequests to wealth in the aggregate.

Using this calibrated model, we perform an optimal policy reform exercise on the calibrated model that includes the medical expenditure profiles estimated in Section 7.3.1. Figure 19 depicts the optimal asset subsidies compared to those in the benchmark model. Especially for lower values of assets, optimal subsidies are as large as those in the benchmark models. This is mainly because it is optimal for these individuals not to leave bequests. For higher values of assets, subsidies fall relative to the benchmark model due to the demand for bequests by richer individuals. Nevertheless, our benchmark implications for optimal policies remain roughly unchanged. An integral part of this policy is bequest taxation. In particular, for many individuals, bequests must be fully taxed away in order to solve the market incompleteness problem faced by these households. Interestingly, in an optimal policy reform, most of the cost savings come from a reduction in bequests. This is because the only way for individuals to save is a risk-free asset and, as a result, bequests are too high for the status quo economy. The detailed theoretical and quantitative analysis of this model is in the Supplemental Material.