Constrained Inefficiency over the Life-cycle Dongya Koh∗ University of Arkansas June 13, 2018

Abstract We quantitatively explore the extent to which the savings and human capital investments of U.S. households depart from efficient levels—measured by the constrained optimum— through an individual’s life-cycle. This question is readily applicable to a number of current issues of interest (e.g., the student debt crisis). Using a life-cycle human capital model with uninsurable idiosyncratic income shocks, we show that the median household saves an average of 18.5% less and invests an average of 11% more time in human capital than the constrained efficient levels. As the income compositions of consumption-poor younger households rely more on labor income than on asset income, an income transfer from old age to youth with the higher market wages in the constrained optimum makes individuals better off, whereas a wealth transfer from youth to old age provides more self-insurance for income risks in old age. We demonstrate that the redistribution of income and wealth not only across individuals but also within an individual’s life plays a key role in improving welfare over the life-cycle of an individual. We further extend this framework to explore the life-cycle effects and welfare implications of student debt burdens after completing higher education in the U.S. Keywords: Constrained efficiency, life-cycle, uninsurable idiosyncratic shocks, saving, human capital investment JEL Codes: D15, D52, D62, H23, J24

∗

[email protected]; https://sites.google.com/site/dongyakoh/

1

Introduction

To what extent do saving and investment in human capital over an individual’s course of life depart from efficient levels? In this study, we quantitatively characterize constrained efficient profiles in a life-cycle human capital model with uninsurable idiosyncratic income shocks, and we compare them to realistically calibrated competitive equilibrium age profiles of U.S. households. We adopt the notion of constrained efficiency, which was recently studied by D´avila et al. (2012) in a standard incomplete market model with uninsurable idiosyncratic income shocks, as the baseline efficiency level of a profile. In this efficient allocation, the planner improves market allocations by internalizing the effect on market prices without changing market structures and household budget constraints. In other words, households incorporate the effects on market prices into their decisions, thereby acting differently from self-interested optimization.1 In an incomplete market with idiosyncratic income shocks, constrained inefficiency, which is sometimes known as the “pecuniary externality,” arises from two sources: i) incomplete insurance and ii) wealth and income inequality. However, in a life-cycle setting, such inefficiency varies by age for at least two reasons.2 First, because the future income risks that individuals face vary by age, incomplete insurance induces age-varying precautionary saving. For example, older cohorts nearing retirement have to insure against only a few years of future income risks. Therefore, their precautionary saving could be less than that of younger cohorts. Second, increasing income and consumption profiles over the life-cycle imply that larger fractions of younger cohorts tend to be classified as consumption-poor within a cross-sectional distribution. Therefore, a unit increase in the consumption of a younger cohort contributes to welfare improvement to a larger extent than the same unit increase in the consumption of an older cohort does. To quantify the inefficiency of age profiles of U.S. households, we first calibrate an overlappinggenerations model with endogenous human capital accumulation in which households face idiosyncratic human capital shocks, a borrowing constraint, a progressive tax on labor income, proportional taxes on asset returns and consumption, and social security payments. The realisticallycalibrated competitive equilibrium profiles of household’s income, consumption, saving, labor supply, and human capital investment are taken as our benchmark profiles. To examine the inefficiency, we compute constrained efficient profiles and juxtapose them with the benchmark profiles. In a life-cycle setting, the consumption-poor are generally younger households. Further, the overor under-accumulation of capital in the benchmark economy hinges critically on the factor income composition of those younger households. Data from U.S. households clearly shows that younger households are both wealth-poor and income-poor, but they tend to have income compositions 1

The constrained optimum is an allocation in which the planner maximizes a utilitarian objective, assigning equal weights to all households. However, the constrained optimum is not necessarily equivalent to Pareto efficiency, as the planner’s hands are tied by existing incomplete market structures and household budgets. 2 Conesa et al. (2009) and Peterman (2016) also argue that the elasticity of labor supply may vary by age. As such, the labor supply and human capital investment of young cohorts reacts less elastically to wage changes than those of older cohorts do. This property also differentiates welfare effects of price changes by age.

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that rely more on labor income than on asset income. Therefore, based on D´avila et al. (2012), it is clear that the consumption-poor younger households become better off when the return on assets (the market wage) decreases (increases) in the constrained optimum. Our main findings regarding the life-cycle profiles show that the median U.S. household is currently saving too little and investing too much in human capital (or work) over its life-cycle. More specifically, the median U.S. household’s saving is, on average, 18.5% lower than the constrained efficient levels of saving. In aggregate, the capital-labor ratio is 10% lower in the competitive equilibrium than in the constrained optimum. Due to the under-accumulation of capital in the competitive equilibrium, the interest rate is 22.7% higher and the effective wage is 3.3% lower in the benchmark economy than in the constrained optimum. On the other hand, U.S. households currently spend, on average, 11% more time on human capital investment than the constrained efficient level. Essentially, the over-investment in human capital is mostly concentrated in the early part of the life-cycle. The hours of work that U.S. households spend is also lower in the constrained optimum than the actual hours worked in the benchmark economy. The joint effect of less time investment in human capital and fewer working hours in the constrained optimum therefore imply that the effective working hours should be on average 2.2% higher in the early part of the life-cycle. The marginal benefit of less time investment in human capital today yields higher labor income today at the given wage and level of human capital, whereas the marginal cost is a lower future labor income due to lower human capital accumulation. As the income composition of the consumption-poor younger cohorts is relatively stronger for labor income than for asset income, an income transfer from old age to youth with a higher wage in the constrained optimum makes younger cohorts better off, whereas a wealth transfer from youth to old age provides more self-insurance against the income risks that older cohorts are exposed to in old age. In a life-cycle model, the constrained inefficient allocations can be improved by the redistribution of income and wealth not only across an individual’s own life but also across individuals of the same age cohort. To observe distributional effects within each age cohort, we also examine the constrained efficient profiles of top and bottom 10% of income earners. The top 10% of income earners save up to 26% more and spend 17.3% less time on human capital investment in the constrained optimum. On the contrary, the asset levels of the bottom 10% of income earners throughout their lives are almost at the efficient level. The human capital investment of the bottom 10% would be 5% lower at the efficient level at the age of approximately 34, but it is almost at its efficient level thereafter. In summary, the constrained inefficiency present in the benchmark economy is largely corrected by the top income earners, whereas the bottom income earners reap benefits through the changes in market prices and, hence, through the changes in the factor income composition. We further extend this analysis to the current issue of the student debt burden as a consequence of higher education attainment. Although student loans (either private or public) allow younger cohorts to attain higher education, the debt burden financially constrains young debt holders for several years after graduation (i.e., the student debt crisis). For instance, while they repay their debt, young households have to minimize the accumulation of self-insurance against future income risks and have to refrain from purchasing their own houses. In other words, student loans allow 2

young cohorts to improve their human capital through higher education at the cost of having a student debt burden after graduation, which affects these cohorts’ investment decisions over their life-cycles. Using our life-cycle framework, we explicitly characterize the life-cycle profiles of debt holders in the benchmark economy and provide the welfare implications of student debt over the life-cycle. Essentially, accumulated student debt causes the current U.S. debt holders to save too little and invest too much time in human capital (and hours of work). This result is mainly due to the fact that the market wage is set below its efficient level, and, therefore, the debt holders allocate their time toward raising their future labor income at the cost of saving and consuming less in the early part of the life-cycle. This study sheds light on constrained efficiency in a competitive equilibrium with uninsurable idiosyncratic shocks in which the planner can directly affect the household’s decisions without changing market structures and household budget constraints. The possibility of constrained inefficiency in competitive equilibrium that Diamond (1967) first posed in a one-period, one-good stock market economy under technological uncertianty was followed by Hart (1975), Diamond (1980), Stiglitz (1982), Loong and Zeckhauser (1982), Newbery and Stiglitz (1984), Greenwald and Stiglitz (1986), and Geanakoplos et al. (1990). Two closely related recent studies are the steady state analyses of constrained efficiency by D´avila et al. (2012) and Park (2017). In a steady state with a calibrated Aiyagari model, D´avila et al. (2012) show that increasing aggregate capital improves welfare and eventually makes the consumption-poor better off while making the consumption-rich worse off. Park (2017) extends this analysis to endogenous human capital model in an incomplete market with exogenous labor income shocks. She examines the two sources of constrained inefficiency arising from monetary investment in human capital. In the steadystate analysis, the constrained inefficient allocations can be improved by the redistribution of income and wealth across households (consumption-rich to consumption-poor) and across states. This study differs from those steady-state analyses in that we apply the notion of constrained efficiency to a life-cycle framework. The main contribution of this study to the literature is that the constrained inefficiency can vary by age over the life-cycle and can be further improved if the planner redistributes income and wealth not only within cross-sectional distribution but also within an individual’s life. Another contribution is that the life-cycle analysis practically enables us to apply the notion of constrained efficiency to current issues of interest (e.g., the student debt crisis). The rest of the paper is organized as follows. In section 2, we first analytically investigate life-cycle properties of constrained inefficiency. In turn, we introduce a benchmark overlappinggenerations model with endogenous human capital to characterize the competitive equilibrium profiles. In section 3, the model is numerically solved and is mapped onto the age profiles of U.S. households. Section 4 examines the constrained inefficiency inherent in the competitive equilibrium levels of saving and human capital investment by comparing the competitive equilibrium profiles to the constrained efficient profiles. In section 5, we discuss the implementation of constrained efficient profiles with a modified tax code. We incorporate student debt into the life-cycle framework to understand the welfare implications in section 6. In section 7, we show that our results are robust to different parameterizations. Finally, section 8 concludes.

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2

Life-cycle Analysis

2.1

Life-Cycle Properties of Constrained Inefficiency

To illustrate the life-cycle properties of constrained inefficiency, we extend the simple two-period model in D´avila et al. (2012) to a longer finite horizon. We assume that households live until age J > 1. At each age j < J, labor income is the source of uncertainty, εj , and households decide how much to save, aj+1 , given an initial wealth heterogeneity, a0 > 0, with a time discount, β. Output is produced by a technology, f , taking aggregate capital K and aggregate labor L at each point in time and the market is perfectly competitive. For simplicity, there is no population growth. The distribution of households over age-j assets and labor income, (aj , εj ) ∈ χ, is denoted as ψj , and let B ∈ χ be a Borel set on χ. For the constrained optimum, the planner solves the following problem: max E aj+1

J Z X j=0 χ

β j u(fl (K, L)εj + fk (K, L)aj − aj+1 )dψj

subject to K =

J Z X j=0 χ

ψj+1 =

Z χ

aj dψj ,

L=

J Z X j=0 χ

(1)

(2)

εj dψj ,

Qj ((aj , εj ), B)dψj .

(3)

where Qj is a transition function at age j that an age-j households with (aj , εj ) transits to the set B at age j + 1. Further, equation (3) characterizes the law of motion for the distribution of age-j households. Note that the planner maximizes the present discounted lifetime utility of an age cohort, whereas the aggregation of capital and labor is taken across all age cohorts at a point in time. Assuming interior solutions, the first-order condition of the planner’s problem with respect to aj+1 yields a typical Euler equation plus an additional marginal benefit/cost from the change in market prices, as follows: u0 (cj ) = βfk (K 0 , L)Eu0 (cj+1 ) + ∆k ,

(4)

where ∆k =

J X

βfkk (K 0 , L)K 0

j=0

Z χ

Eu0 (cj+1 )



aj+1 εj+1 − dψj+1 K0 L 





=

J X j=0

βfkk (K 0 , L)K 0

Z χ

  a j+1  − 1 0  | K {z }

Eu0 (cj+1 ) 

wealth heterogeneity

4



+ |

εj+1 {z L }

1−

    dψj+1 . 

incomplete insurance

(5)

It is clearly shown that the constrained inefficiency arises from incomplete insurance and wealth heterogeneity. Note that in D´avila et al. (2012)’s two-period model in which J = 1, the wedge becomes 0

∆k = βfkk (K , L)K

0

Z χ



0

Eu (c1 )

a1 ε1 −1 + 1− 0 K L 





dψ1 ,

(6)

which reproduces their conclusion that the sign of the wedge, the marginal benefits or costs through the change in market prices, is mostly determined by the proportion of the consumptionpoor households in the cross-sectional distribution of households. This result is mainly due to the factor composition of income is weighted by the marginal utility. As such, the wedge amounts to placing a higher weight on the consumption-poor households. A simple comparison of equations (5) and (6) implies that the constrained inefficiency varies by age, as none of life-cycle trajectories of consumption, assets, and labor income remain constant across different ages. Even assuming that income shocks are independent and identically distributed (iid), the degree of wealth heterogeneity and the marginal-utility weights differ by age due to changes in asset positions and consumption over the life-cycle. In fact, empirical evidence shows that there is a greater concentration of wealth-poor and consumption-poor households among younger households than among older households. In Figure 1, we plot the fraction of households whose income, nondurable consumption, and net worth are below 10th percentile of all households using U.S. household data.3 The figure shows that 29% of 23-year-old age cohorts belong to the bottom 10% income group, 21% belong to the bottom 10% consumption group, and 31% belong to the bottom 10% net worth group. These rates monotonically decline by age, and these proportions become 6.7%, 6.6%, and 5.8%, respectively, at age 50. There is clearly a concentration of income-, consumption-, and wealth-poor households in the early stage of life. Therefore, whether the U.S. economy over- or under-accumulates the aggregate capital is predominantly determined by the fraction of the consumption-poor households among younger cohorts. This result implies that there should clearly be redistribution not only across households but also between earlier and later stages of life within an individual’s lifetime. We explicitly examine these two redistribution channels—the life-cycle redistribution channel and the cross-sectional redistribution channel—in the next section.

2.2

The Benchmark Model

In this benchmark model, we apply the tax structures of Conesa et al. (2009) to an overlapping generations model with endogenous human capital accumulation. Demographics The economy is populated by J + 1 overlapping generations. In each period, a continuum of a new age cohort is born. The population grows at a constant rate n. Defining a fraction of age j 3

This evidence on inequality is also documented by Diaz-Gimenez et al. (1997) using the 1992 SCF and the 1984-85 and 1989-1990 PSID samples and is further updated with the 1998 SCF and the 1994-95 PSID by Rodriguez et al. (2002).

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households in the population as µj , the demographics satisfy µj = µj−1 /(1+n) for all j = 1, ..., J P and Jj=0 µj = 1. Preferences and per-period decisions Households live until age J < ∞ and retires at age JR ≤ J.4 Households are ex ante heterogeneous in their initial human capital (h0 ), initial wealth (a0 ), and learning ability (θ0 ). Households are employed throughout their lives until retirement. Throughout the course of life at each age j < J, a household chooses how much to consume (cj ) and save (aj+1 ). Also, at each working age j ≤ JR , one unit of time endowment is allocated into labor and leisure (nj ∈ [0, 1]), and a fraction of working hours are allocated to human capital investment (sj ∈ [0, 1]). There is a borrowing limit a ≥ 0. A household maximizes its time-separable expected lifetime utility over P consumption and leisure as E Jj=0 β j u(cj , 1 − nj ). Labor income risks Households encounter idiosyncratic labor income shocks (εj ) through human capital at each working age. The human capital shocks affect the stock of human capital accumulated in the previous period. These shocks are iid and normally distributed across individuals with mean zero and variance of σε2 , εj ∼ N (0, σε2 ), and the shock is uninsurable due to the absence of full insurance contracts. Human capital accumulation Human capital technology incorporates both learning-by-doing (LBD) and Ben-Porath (1967)’s on-the-job training (OJT) technology.5 The human capital accumulation is governed by the human capital depreciation (δh ), a learning ability (θ0 ), the current stock of human capital (hj ), hours of work (nj ), and time invested in training (sj ):6 hj+1 = exp(εj+1 )H(hj , nj , sj ; θ0 ), where the accumulation of human capital before the shock arrival is defined as H(hj , nj , sj ; θ0 ) = (1 − δh )hj + θ0 (hj nj sj )γ . We assume that the skills represented by the stock of human capital are general and labor market is competitive. 4 In this life-cycle model, we assume that fertility choice and family structures are not endogenous. Therefore, an individual represents a household ,and the two terms are interchangeably used. 5 One caveat about a life-cycle human capital model with OJT technology (e.g Huggett et al. (2011)) is that consumption growth tends to be three- to four-times higher than income growth. This difference is due to insufficient consumption insurance associated with uninsurable income risks provided in the model. Therefore, in order to quantitatively match the consumption growth, the model requires one of these features: 1) an unrealistically high consumption or capital income tax, 2) a positive initial asset that provides an extra insurance, or 3) utility from leisure. As the unrealistically high taxes distort constrained efficient allocations, we incorporate an elastic labor supply in the form of learning-by-doing into the benchmark model with a natural borrowing limit to match consumption growth. 6 The implementation of OJT or LBD human capital technology is justified in the human capital literature despite their known inherent pros and cons. For example, OJT can endogenously generate a hump-shaped income growth due to the decrease in time spent on human capital together with the depreciation of human capital. On the other hand, the LBD technology is more widely used in the literature due to its realistic economic interpretation despite the fact that the decline in the labor supply is not as dramatic as time investment in human capital and, hence, labor income is increasing but is not as hump-shaped as in the case of OJT.

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Taxes and transfers The government receives tax income by levying a flat-rate consumption tax (τc ), a capital income tax (τk ), a social security tax (τss ), and a potentially progressive labor income tax (T (y)). After retirement, households receive social security payments (yss ). Household budget constraint Given the features of the model, the household budget constraint for age cohort j at each point in time is: (1 + τc )cj + aj+1 = y˜j − T (˜ yj ) + (1 + r(1 − τk ))aj , (1 + τc )cj + aj+1 = yss + (1 + r(1 − τk ))aj ,

for j ≤ JR for j > JR .

where r is the rate of return on saving and y˜j is taxable labor income. The taxable labor income is the pre-tax labor income minus the employee’s portion of the social security tax upto a wage base limit, y¯. (

y˜j =

whj nj (1 − sj ) − 0.5τss min(whj nj (1 − sj ), y¯) 0

if j ≤ JR if j > JR .

where w is the real wage on effective labor. Production technology Production in this economy follows a constant returns to scale (CRS) technology, Y = AK α L1−α

(7)

where K is an aggregate capital stock, L is an aggregate effective labor, and A is Hick-neutral technical change. We assume that there is no aggregate shock for simplicity and, hence, A is constant over time. Initial Distribution An individual’s initial assets, human capital, and learning ability are trivariate log-normally distributed, where the log of the random variable follows a three-dimensional mean vector and 3 × 3 covariance matrix, 



ma  M=  mh  , mθ

σa2 λah σh σa λaθ σθ σa σh2 λhθ σθ σh  Σ=  λah σa σh . 2 λaθ σa σθ λhθ σh σθ σθ 



Aggregation At each point in time, a household at age j is characterized by an individual state xj = (aj , hj ; θ0 ) where aj ∈ A is the amount of saving/borrowing, hj ∈ H is the stock of human capital, and θ0 ∈ Θ0 is the innate (learning) ability, where the sets of control variables are convex.7 Define a 7

Note that the current stock of human capital h ∈ H encompasses the current realization of the human capital shock ε ∈ E; therefore, the current human capital stock and physical assets are enough to characterize the current state of a household.

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state space as X = A×H×Θ0 . Then, for age-j households at a particular point in time, we define a joint probability measure Ψj over the probability space (X , B(X ), Ψj ), where B(X ) is the Borel σ-algebra on X . The distribution of age-j households at each point in time is denoted as Ψj (xj ) for xj ∈ B(X ). The transition function of individual states of each age j given the current state xj is characterized as Qj (xj , B) = P r(xj+1 ∈ B|xj ) for all B ∈ B(X ). Further, the transition R of the household distribution of each state is characterized as Ψj+1 (B) = X Qj (xj , B)dΨj (xj ) for all B ∈ B(X ). In equilibrium, the goods, labor, and capital markets clear at the competitive prices at each point in time: K=

J X

µj

Z X

j=0

L=

JR X

µj

j=0 J X j=0

µj

Z X

Z X

aj dΨj (xj )

(8)

hj nj (xj )(1 − sj (xj ))dΨj (xj )

(9)

cj (xj )dΨj (xj ) + (1 + n)K 0 + G = Y + (1 − δ)K

(10)

where sj (xj ), nj (xj ), and cj (xj ) are the policy functions of human capital investment, labor supply, and consumption, respectively, given the current state xj of each age j. Government budget balance The government budget balances when G=

J Z X j=0 X

(τc cj (xj ) + rτk aj + T (˜ yj (xj )))dΨj (xj ),

(11)

and the social security policies satisfy JR Z X j=0 X

τss min{whj nj (1 − sj (xj )), y¯}dΨj (xj ) =

J X

Z

j=JR +1 X

yss dΨj (xj ).

(12)

Then we define the benchmark competitive equilibrium. Definition 2.1. A competitive equilibrium with tax distortions is a collection of policy funcR and {Wj (aj )}Jj=JR +1 , tions {aj+1 (xj ), hj+1 (xj ), cj (xj ), nj (xj ), sj (xj )}Jj=0 , value functions {Vj (xj )}Jj=0 factor prices {r, w}, aggregate input factors {K, L}, and distributions of individual states by age {Ψj (xj )}Jj=1 , given the social security payments, yss , government spending G, a set of taxes (τc , τk , τss , {T : R+ → R+ }), and an initial distribution N (M, Σ), such that 1. Given factor prices, the household policy functions solve the recursive household prob-

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lem, for j ≤ JR , Vj (aj , hj ; θ0 ) =

max

cj ,nj ,sj ,aj+1 ,hj+1

u(cj , 1 − nj ) + βEVj+1 (aj+1 , hj+1 ; θ0 )

s.t. (1 + τc )cj + aj+1 = y˜j − T (˜ yj ) + (1 + r(1 − τk ))aj hj+1 = exp(εj+1 )H(hj , nj , sj ; θ0 ) y˜j = whj nj (1 − sj ) − 0.5τss min(whj nj (1 − sj ), y¯) aj ≥ −a

(13) (14) (15) (16) (17)

2. And for j > JR Wj (aj ) = cmax u(cj ) + βWj+1 (aj+1 ) ,a j

(18)

j+1

s.t. (1 + τc )cj + aj+1 = yss + (1 + r(1 − τk ))aj aj ≥ −a

(19) (20)

3. Competitive factor prices equal the marginal products of the input factors in the proY duction technology: w = (1 − α) YL and r = α K − δk , 4. Markets clear, as in equations (8)-(10) 5. Government budgets balance, as in equations (11)-(12) 6. The distributions of households in each state and at each age Ψj+1 (B) = X Qj (xj , B)dΨj (xj ) for all B ∈ B(X ) and j < J are consistent with the household’s policy functions. R

3

Quantitative Analysis

3.1

Functional Forms and Calibration

In this section, we discuss the calibration of model parameters. The calibration of tax parameters mostly follows Conesa et al. (2009). Table 1 shows the overview of the predetermined and calibrated parameters. Demographics We set J = 58 and JR = 40 by assuming that individuals enter the labor market at age 23, retire at age 62, and live until age 80.8 Population growth n is 0.011 based on the BLS estimate. 8

There are two peak retirement ages, 62 and 65, and we choose the former. The choice is solely due to the sample size. We eliminate individuals with their annual working hours below 520 hours and above 5200 hours and those with hourly nominal wages less than half of that year’s minimum wage. Hence, the data samples do not explicitly exclude individuals who switch from full-time to part-time employment or to being unemployed. Also retirees, students, housewives, and permanently disabled individuals are dropped.

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Production technology The average capital income share, α, is set to be 0.322 from NIPA data. As we disregard aggregate effects in the model,  total α factor productivity (A) is set to K = 1, given that the capital-output normalize the return to human capital w = (1 − α)A L ratio K/Y is 2.947. The benchmark elasticity of human capital production, γ, is set to be 0.7 as estimated in the literature. This parameter requires a sensitivity check. Depreciation By targeting the annual average rate of return to capital, r, as 4.2% and the capital-output ratio K/Y as 2.947, the depreciation of capital δ is set at 0.067 from the equilibrium condition r + δ = FK (K, N ).

Preferences We assume an isoelastic utility function as follows: u(cj , 1 − nj ) =

cj1−ψ 1−ψ

+

1−φ j) χ (1−n . 1−φ

The coefficient of relative risk aversion, ψ, is unidentified from the model. We set χ so that the average working hours are approximately 0.3. We calibrate (ψ, φ) by matching the age-profiles of consumption and hours of work, as in Kaplan (2012). The discount factor β is calibrated to match 4.2% of the annual rate of return on physical capital. Initial heterogeneity The parameters for initial heterogeneity, (ma , mh , mθ , σh2 , σθ2 , λhθ ), are identified by targeting data moments of the log wage profile. We assume no initial assets from the fact that younger cohorts tend to be wealth-poor. As we observe that the variance of initial assets has a negligible effect on life-cycle profiles, we also assume no initial heterogeneity in assets (σa2 = 0), which in turn implies no correlation with initial human capital and learning ability (λah = λaθ = 0). According to Huggett et al. (2011), a heterogeneity in learning ability captures the increasing variance of log incomes over the life-cycle, and, hence, in the absence of heterogeneity in learning ability, they show that the variance becomes flat across different ages. Therefore, we target the initial mean log wage to identify the average initial level of human capital (mh ), the variance of the initial wage to identify the variance of initial human capital (σh2 ), the average wage growth in the first ten years to identify the initial distribution of learning ability (mθ ), and the first ten years of wage variance for the variance of initial learning ability (σθ2 ). Further, the wage decline in old age in the data identifies the depreciation of human capital (δh ), as the model implies that sj ≈ 0 in old age. Finally, the correlation between initial human capital and learning ability, (λhθ ), can be identified from the variance of wages, {var(w˜j )}JR j=0 . Government Policies As of 2017, the social security taxes in the U.S. were 12.4% in total and the wage base limit was $127,200. Therefore, we set the payroll tax rate τss to be 12.4% and the maximum labor income y¯ to be 2.5 times the average income, as in Conesa et al. (2009). The social security benefits (yss ) are determined by the social security budget balance. We use 6% for the consumption tax rate (τc ) based on Mendoza et al. (1994) and 40% for the capital income tax rate (τk ) based on Domeij and Heathcote (2004). For labor income tax, we implement the

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nonlinear tax function estimated by Gouveia and Strauss (1994), as follows: T (y) = τ0 (y − (y −τ1 + τ2 )−1/τ1 ) This functional form allows for progressivity in the tax system and becomes a flat rate tax as τ1 → 0. We set the tax parameters at τ0 =0.258 and τ1 =0.768 and determine τ2 by the government budget balance. Human Capital Shocks Labor income risk in the model stems from human capital shocks, where the iid shocks have persistent effects on labor income through human capital accumulation. Note that there are several different ways to calibrate the variance of persistent income shocks employed in the literature. For example, Storesletten et al. (2004) associate the growth of income and consumption dispersion with the distribution of the income process. More specifically, the variance of individual-specific fixed effects and the variance of transitory shocks can be identified from an intercept of age-profile income inequality at age 25, and the variance of permanent shocks can be identified from the growth of income inequality. We cannot utilize this calibration method, as our individual-specific learning ability σθ2 spans income profiles across ages and, in turn, generates growth in the income dispersion over the life-cycle. Another calibration strategy is introduced by Huggett et al. (2011). The distribution of human capital shocks in their life-cycle human capital model is identified from the income processes of older cohorts. This calibration method is justified from the typical feature of a life-cycle human capital model, similar to ours, that the time investment in human capital becomes negligible as an individual approaches retirement. Therefore, almost all income changes in old age should be attributed to income shocks rather than the endogenous choices of individuals. One concern with this methodology is that the sizes of the permanent income shocks could be smaller in old age than at younger ages, and, hence, the identification of permanent income shocks using the income processes of older cohorts could underestimate the size of income risk. Essentially, this concern should be inconsequential, as Karahan and Ozkan (2012) show that the persistence and size of persistent shocks increase as an individual approaches retirement. Therefore, in our benchmark parameterization, we use the estimate of the variance of human capital shocks of Huggett et al. (2011), σε =0.111.9

3.2

A Computational Algorithm to Solve for Household Problem

To solve for the household life-cycle problem characterized in equations (13)–(20), we apply a hybrid of endogenous grid methods (ENGM) and exogenous grid methods (EXGM) (see Carroll (2005), Barillas and Fern´andez-Villaverde (2007), Ludwig and Sch¨on (2016)). First, the first 9

We need not restrict the estimation of this parameter to the older cohort’s income profiles. Instead, we can directly estimate variances of human capital shocks σε2 by matching the auto-covariance of the wage estimated from PSID data covering 1968-1997. This alternative estimate of the income process is checked in the online appendix.

11

order conditions (FOCs) and envelope conditions of the household problem for j ≤ JR are: uc (cj , 1 − nj ) = λcj 1 + τc

(21)

un (cj , 1 − nj ) = λcj whj (1 − sj )(1 − 0.5τss Iy˜j ≤¯y )(1 − T 0 (˜ yj )) + λhj exp(εj+1 ) λcj whj nj (1 − 0.5τss Iy˜j ≤¯y )(1 − T 0 (˜ yj )) = λhj exp(εj+1 )

∂H ∂nj

∂H ∂sj

(22) (23)

(1 + τc )cj + aj+1 = y˜j − T (˜ yj ) + (1 + r(1 − τk ))aj hj+1 = exp(εj+1 )H(hj , nj , sj ; θ0 ) a βEVj+1 = λcj

(24) (25) (26)

h = λhj βEVj+1 ∂H = γθ0 (hj nj sj )γ−1 hj sj ∂nj ∂H = γθ0 (hj nj sj )γ−1 hj nj ∂sj ∂H = (1 − δh ) + γθ0 (hj nj sj )γ−1 nj sj ∂hj Vja = λcj (1 + r(1 − τk ))

(27) (28) (29) (30) (31)

Vjh = λcj wnj (1 − sj )(1 − 0.5τss Iy˜j ≤¯y )(1 − T 0 (˜ yj )) + λhj exp(εj+1 )

∂H ∂hj

(32)

where Vja and Vjh indicate partial derivatives of the value function with respect to aj and hj , and λcj and λhj are Lagrange multipliers associated with the household budget constraint in equation (16) and human capital accumulation in equation (15). Combining equations (22) and (23) yields an intra-temporal optimality condition: un (cj , 1 − nj ) = λcj whj (1 − 0.5τss Iy˜j ≤¯y )(1 − T 0 (˜ yj )).

(22’)

Equation (23) indicates the intertemporal optimality condition for investing an additional unit of time in human capital. Combining equations (23) and (32), the envelope condition for hj becomes "

Vjh

=

λcj w(1

1 − δh − 0.5τss Iyj ≤¯y )(1 − T (˜ yj )) nj + γθ0 (hj nj sj )γ−1

#

0

(32’)

Similarly, the FOCs and envelope conditions of the household problem for j > JR are: u0 (cj ) = (1 + τc )λcj

(33)

a βWj+1 = λcj

(34)

(1 + τc )cj + aj+1 = yss + (1 + r(1 − τk ))aj Wja = λcj (1 + r(1 − τk ))

(35) (36)

12

The computational algorithm for a working-age household proceeds in the following manner: ˜ 0 for ˜⊗Θ Step 1. For the endogenous grid methods, define grid points on (aj+1 , hj , θ0 ) ∈ A˜ ⊗ H ˜ denotes a set of Nx discretized points j ≤ JR and on aj+1 ∈ A˜ for j > JR , where a set X ˜ of a convex set X, X = {xmin = x0 , x1 , ..., xNx −1 , xNx = xmax }.10 For the exogenous grid ˜ 0 for j ≤ JR and on aj ∈ A˜ for ˜⊗Θ methods, define grid points on (aj , hj , θ0 ) ∈ A˜ ⊗ H j > JR ˜ we Step 2. [EXGM] At j = J, aJ+1 = 0 for any given aJ . Therefore, for each grid point aJ ∈ A, obtain c∗J = (yss + (1 + r(1 − τk ))aJ )/(1 + τc ) WJa (aJ ) = u0 (c∗J )(1 + r(1 − τk ))/(1 + τc ) ˜ Step 3. [ENGM&EXGM] At JR + 1 ≤ j < J, we find (ˆ cj , a ˆj ) for each grid point aj+1 ∈ A: a u0 (ˆ cj ) = (1 + τc )βWj+1 (aj+1 )

a ˆj = ((1 + τc )ˆ cj + aj+1 − yss )/R(1 − τk ). and using a derived policy function gj (ˆ aj ) = aj+1 , interpolate a policy function gj (aj ) = ∗ ˜ Finally, if a∗ ≥ −a, proceed aj+1 with respect to each exogenous grid point, aj ∈ A. j+1 to the j − 1 problem. Otherwise, find a∗j+1 and c∗j for the particular exogenous states ˜ A˜{a :a∗ ≥−a} using a nonlinear solution method.11 aj ∈ A\ j j+1 Step 4. At j = JR , it is optimal to invest no time in human capital, sj = 0.12 Otherwise, the computational algorithm follows as in step 5. 10

Note that there are two reasons that our endogenous grid methods cannot take exogenous grid points on ˜ 0 . First, we cannot identify an optimal pair of (aj+1 , hj+1 ) that maps into the ˜⊗Θ (aj+1 , hj+1 , θ0 ) ∈ A˜ ⊗ H ˜ 0 , which ˜⊗Θ model state variables (aj , hj , θ0 ). Instead, we map aj+1 into aj ∈ A conditional on (hj , θ0 ) ∈ H allows us to skip one nonlinear equation associated with aj+1 . Second, as the human capital accumulation involves iid shocks to current human capital, it is not straightforward to map hj+1 into a specific (aj , hj , θ0 ). 11 The hybrid use of endogenous and exogenous grid methods are computationally more efficient than the use of only exogenous grid methods as long as very few states are binding at the borrowing constraint. 12 From j = JR to j = JR + 1, there is a transformation of the state space from three dimensions to one dimension. That is, the household problem at j = JR becomes Vj (aj , hj ; θ0 ) = s.t.

max

cj ,aj+1 ,sj

u(cj , 1 − nj ) + βWj+1 (aj+1 )

(1 + τc )cj + aj+1 = y˜j − T (˜ yj ) + (1 + r(1 − τk ))aj y˜j = whj nj (1 − sj ) − 0.5τss min(whj nj (1 − sj ), y¯) aj ≥ −a.

As the stock of human capital accumulated up until JR and initial productivity becomes irrelevant to the benefits after retirement, it is trivial that sJR = 0. Therefore, the problem is identical to that in Step 3 except for the envelope condition for human capital Vjh .

13

Step 5. [ENGM&EXGM] At j < JR , for each grid point (aj+1 , hj , θ0 ) ∈ X˜ , we first numerically solve a nonlinear equation (22’) and (23) for n ˆ j and sˆj . An optimal pair of (ˆ nj , sˆj ) with h c h a ˆ ˆ Vj+1 and Vj+1 determines Lagrange multipliers λj and λj from equations (26) and (27).13 Then, we find cˆj from equation (21) and a ˆj from equation (24). With three policy functions gjn (ˆ aj , hj , θ0 ) = n ˆ j , gjs (ˆ aj , hj , θ0 ) = sˆj , and gja (ˆ aj , hj , θ0 ) = aj+1 , interpolate new policy ∗ s ∗ n functions gj (aj , hj , θ0 ) = nj , gj (aj , hj , θ0 ) = sj , and gja (aj , hj , θ0 ) = a∗j+1 for each exogenous grid point, (aj , hj , θ0 ) ∈ X˜ . Finally, if a∗j+1 ≥ −a, proceed to the j−1 problem. Otherwise, find n∗j and s∗j for the particular exogenous states (aj , hj , θ0 ) ∈ X˜ \X˜{(aj ,hj ,θ0 ):a∗j+1 ≥−a} using a nonlinear solution method. Given the solutions (n∗j , s∗j , a∗j+1 ), we compute envelope conditions Vja and Vjh from equations (31) and (32’) for each (aj , hj , θ0 ) ∈ X˜ .

3.3

A Computational Algorithm for Competitive Equilibrium Profiles

ˆ yˆss , τˆ2 ). Step 1. Guess (β, Step 2. Solve for life-cycle profiles using the algorithm in Section 3.2. Then simulate (cij , sij , aij+1 )profiles of 10000 individuals given the initial conditions (ai0 , hi0 , θ0i ) randomly drawn from a distribution of N (M, Σ). Step 3. Aggregate capital, effective labor, government spending, and outlays as in equations (8), (9), (11), and (12), respectively. Step 4. If the government budget and social security budgets balance, and the simulated capitalˆ K labor ratio K ˆ equals the targeted capital-labor ratio L =4.347, then stop. Otherwise, redo L the process with a new guess.

3.4

A Computational Algorithm for Constrained Efficient Profiles

The constrained optimum can be found when the planner improves on market allocations by internalizing the effect on prices without completing the market and without changing households’ budget constraints. In other words, households depart from their self-interested optimization by internalizing the effect on market prices. To solve for the efficient allocations, the planner maximizes a utilitarian objective assigning equal weights to all households with initial assets (a0 , h0 ; θ): max

JR X

µj

j=0

Z χ

Vj (xj )dΨj +

J X j=JR +1

s.t. Ψj+1 = Gj (Ψj ) 13

µj

Z χ

Wj (aj )dΨj (37)

A two-dimensional nonlinear solver is not needed if there is no social security tax and a progressive labor income tax.

14

The FOCs of the planner’s problem with respect to aj+1 and the envelope condition with respect to aj are uc (cj , 1 − nj ) a = βEVj+1 = λcj 1 + τc Vja = λcj (1 + FK (1 − τk )) + ∆k ∆k = =

J X i=0 J X

Z

µi

χ

µi

χ

i=0

(39)

λci [FKK (1 − τk )ai + FLK hi ni (1 − si )(1 − 0.5τss Iy˜i ≤¯y )(1 − T 0 (˜ yi ))] dΨi "

Z

(38)

λci FKK K

#

(1 − τk )ai hi ni (1 − si ) − (1 − 0.5τss Iy˜i ≤¯y )(1 − T 0 (˜ yi )) dΨi K L

(40)

As individuals after retirement do not face any idiosyncratic income risks, their savings must be optimal at the given prices. The FOCs of the planner’s problem with respect to sj and the envelope condition with respect to hj are h λcj FL hj nj (1 − T 0 (˜ yj ))(1 − 0.5τss Iyj ≤¯y ) + hj nj ∆h = βEVj+1 exp(εj+1 )

∆h = =

JR X i=0 JR X

µi µi

Z χ

Z

i=0

χ

∂Hj ∂sj

(41)

λci [FKL (1 − τk )ai + FLL hi ni (1 − si )(1 − 0.5τss Iy˜i ≤¯y )(1 − T 0 (˜ yi ))] dΨi "

λci FKL K

#

(1 − τk )ai hi ni (1 − si ) − (1 − 0.5τss Iy˜i ≤¯y )(1 − T 0 (˜ yi )) dΨi K L

h Vjh = λcj FL nj (1 − sj )(1 − T 0 (˜ yj ))(1 − 0.5τss Iy˜j ≤¯y ) + nj (1 − sj )∆h + βEVj+1 exp(εj+1 )

(42) ∂Hj ∂hj (43)

As we combine the FOCs and the envelope condition, the new envelope condition becomes Vjh

=

n

λcj FL (1

0

− T (˜ yj ))(1 − 0.5τss Iyj ≤¯y ) + ∆h

o

"

#

1 − δh nj + . γθ0 (hj nj sj )γ−1

(44)

The FOCs of the planner’s problem with respect to nj is un (cj , 1 − nj ) = λcj FL hj (1 − sj )(1 − 0.5τss Iy˜j ≤¯y )(1 − T 0 (˜ yj )) + λhj exp(εj+1 ) + hj (1 − sj )∆h

∂H ∂nj (45)

Then, the following algorithm computes the constrained efficiency. ˆ = {Ψ ˆ 1, Ψ ˆ 2, . . . , Ψ ˆ J }. A good initial guess could be a comStep 1. Guess an initial distribution Ψ petitive equilibrium distribution.14 14

Considering the benchmark distribution as an initial guess, the computational algorithm mimics the transition from the competitive equilibrium distribution to the constrained efficient distribution.

15

Step 2. Compute the value functions and policy functions as in the competitive equilibrium and ˜ simulate the distribution Ψ. ˆ − Ψ| ˜ < δΨ , then stop. Otherwise, repeat the process with the new distribution Ψ. ˜ Step 3. If |Ψ

4

Quantitative Results

In this section, we first discuss the competitive equilibrium profiles of average households as benchmark profiles of U.S. households. Then, we compare the aggregate moments of the competitive equilibrium and constrained efficiency. Further, a comparison of the life-cycle profiles of the median household in the competitive equilibrium and under constrained efficiency shows a redistribution of income and wealth within an individual’s course of life to achieve an efficient profile. Finally, we examine the redistribution of income and wealth across households by comparing the constrained efficient profiles of the top and bottom 10% of income earners. Competitive Equilibrium Profiles As we discussed in detail in section 3.1, the set of model parameters in the life-cycle human capital model are calibrated to match the mean lifecycle profiles of hourly wages, nondurable consumption, and hours of work estimated from U.S. household data. Figure 2 shows the data match of the model-simulated mean life-cycle profiles. Both the lifetime hourly wage and consumption are hump-shaped in the data, as the wage profile reaches its peak around age 45âĂŞ50 and the consumption profile peaks at age 55.15 Although the growth of the model-simulated effective wage (measured as the real wage times current human capital, wh) falls somewhat short of that of the wage in the data, the model-simulated wage and consumption profiles demonstrate almost identical hump shapes (panel (a) and (b)). Hours of work also exhibit a mild hump shape that peaks at age 30âĂŞ50 over the life-cycle, whereas the model-simulated hours of work show a similar hump shape but with an earlier peak at age 30 followed by a substantial decline afterward (panel (c)). The prediction of higher working hours at young ages found in the model is a typical feature of a model with a limited ability to borrow (see Kaplan (2012) and Peterman (2016)). Even with the natural borrowing limit assumed in the model, model households are predicted to have less access to financial wealth than U.S. households have. Thus, model households require more hours of work due to precautionary motives at young ages. The model nonetheless matches the average number of hours of work at around -0.95 (in log points). The time investment in human capital accumulation starts at 35% of total working hours at age 23, and it exhibits the typical decline of Ben-Porath human capital technology until it reaches zero at retirement. Asset accumulation is monotonically increasing until retirement and drops thereafter. The dispersion of each variable over the life-cycle matches the data profiles as well. The 15

Due to the presence of unearned income in the model and due to the fact that only nondurable consumption is included in the consumption data, we do not expect to match the level of the consumption profile. Therefore, we normalize the data and model consumption profiles at the level of age 23 to match only the growth of consumption.

16

hourly wage dispersion in the data increases from 0.168 at age 23 to 0.34 at age 62, while the model-simulated wage variance increases from 0.09 at age 23 to 0.39 at age 62. The variance of consumption remains rather flat, at approximately 0.15, with wide confidence intervals in the data, whereas the model-predicted variance rises from 0.05 at age 23 to 0.16 at age 47 and then flattens out until retirement.16 The variance of asset holdings drastically increases after age 50 and then plummets after retirement. The variance of time investment sharply increases until age 46 and then drops afterward. The variance of labor supply starts substantially low but marginally increases over time in the model. Incorporating the initial heterogeneity in households’ distaste for working allows the model to match the initial dispersion. However, as shown in equation (5), the size of constrained inefficiency hinges critically on the average labor supply weighted by the marginal utility of consumption, and, therefore, the contribution from the dispersion of labor supply is of second order. Now, we take these profiles as the benchmark, and any deviations of the benchmark profiles from the constrained efficient levels are considered to be constrained inefficiency. Next, we characterize the degree of inefficiency embedded in these profiles by studying constrained efficient profiles. Aggregate Variables We first compare the aggregate moments in the competitive equilibrium and constrained optimum calculated from the general equilibrium allocations of all age cohorts at a particular point in time.17 As stated earlier, the inefficiency in the competitive equilibrium arises from incomplete insurance and wealth dispersion. To present the effect from incomplete insurance, we also demonstrate the aggregate moments of the full-insurance economy in which all income risks are insured by a provision of Arrow-Debreu securities.18 Table 2 compares the aggregate variables of three economies with tax distortions: competitive equilibrium, constrained optimum, and full insurance. Consistent with the results of D´avila et al. (2012), aggregate capital in the competitive equilibrium is under-accumulated compared to capital in the constrained optimum. More specifically, the capital-output ratio is 6.9% lower in the benchmark economy than in the constrained optimum, whereas the capital-labor ratio is 10% lower. This result is due to the 7.8% under-accumulation of capital together with the 2.6% over-supply of effective labor. Due to the under-accumulation of the capital-labor ratio in the competitive equilibrium, the interest rate is 22.7% higher and the effective wage is 3.3% lower in the benchmark economy than in the constrained optimum. These effects on the aggregate variables can also be inferred from the 16

Kaplan (2012) shows that the variance of consumption profiles is sensitive to the choice of household equivalence scales. We use a modified-OECD equivalence scale where we assign 1 to the first adult, 0.5 to each additional adult, and 0.3 to each additional child. 17 Due to the absence of aggregate shock in the model, general equilibrium allocations that maximize one’s lifetime welfare and clear markets are stationary equilibrium. In computing the constrained efficient allocations, the algorithm in fact mimics the transition from the stationary benchmark distribution to the stationary constrained optimum distribution. 18 A full-insurance economy is computed by taking σε → 0 in the competitive equilibrium economy rather than by adding full-insurance Arrow-Debreu securities. Taking the values of other model parameters in the competitive equilibrium as given, we re-compute the market clearing prices in the full-insurance economy. All taxes in the competitive equilibrium are kept unchanged. Note that removing idiosyncratic income shocks removes up to 80% of the wealth dispersion, mainly after age 55. It is trivial that no heterogeneity in initial wealth together with no income shocks make the model a deterministic representative agent model.

17

calculated capital and labor wedges in the constrained optimum. The sign of the capital wedge that internalizes the effect on prices in saving is positive (∆k > 0), whereas the sign of labor wedge is negative (∆h < 0). Therefore, there is an under-accumulation of capital as well as an over-supply of effective labor in the competitive equilibrium. Also note that the aggregate effects of constrained inefficiency are smaller than those in D´avila et al. (2012) or Park (2017). This result is mainly due to the tax distortions present in the model that inhibit the constrained efficient accumulation of capital. In a full-insurance economy in which we shut down the channel of inefficiency from incomplete insurance, the over-accumulation of assets induced by precautionary saving vanishes. Therefore, in the competitive equilibrium, capital is over-accumulated by 20.2% in terms of the capital-labor ratio, resulting in a 25.7% higher interest rate. Life-Cycle Redistribution Channel The primary question of this study is to what extent savings and investment in human capital over the life-cycle are shifted away from their efficient levels. We now answer this question by comparing the competitive equilibrium life-cycle profiles to the constrained efficient profiles. Figure 4 plots a median household’s life-cycle profiles of labor income, consumption, hours of work, asset accumulation, and time investment in human capital. The competitive equilibrium profiles of the median household are not too different from those of the mean household in Figure 2. The life-cycle asset profile in panel (a) of Figure 4 shows that the median U.S. household’s saving is, on average, 18.5% lower than the constrained efficient level of saving. More specifically, the actual saving is 23% lower at age 30, 30% lower at age 40, and 26% lower at age 50. At retirement, saving is 8.9% below its efficient level. The median household would save more over the course of its life, thereby increasing aggregate capital and, in turn, lowering the rate of return on assets. The suggested redistribution of wealth from youth to old age may improve an individual’s lifetime welfare through the effects on market prices, as households are typically wealth-poor (as well as consumption-poor) in the early part of the lifecycle, and, hence, younger households become better off by shifting the factor compositions of their incomes more toward labor income than toward capital income. The constrained efficient profiles of time investment in human capital in panel (b) of Figure 4 show that U.S. households currently spend on average 11% more time on human capital investment than the constrained efficient level. The over-investment in human capital is concentrated mostly in the early stage of the life-cycle. At age 40, for instance, the median household should have spent 24.2% of its working hours on human capital investment rather than the actual 28.3% of time that they currently spend (5.6% lower). On the other hand, the number of hours that U.S. households spend working should also be lower in the constrained optimum than the number of hours that they actually do spend working (see panel (e)). U.S. households currently spend on average 0.39 of their unit time (-0.94 in logs) working, whereas a fraction of those working hours are used to invest in human capital. However, the constrained efficient hours of work are 0.38 on average. At age 40, the households should spend 0.40 of their unit time working instead of the current 0.41 (2.5% lower). The joint effect of less time investment in human capital and less working hours in the constrained optimum imply that effective working hours (excluding hours spent on human capital investment, nj (1 − sj )) should be, on average, 2.2% higher in the early stage of life (say, up to age 40), or 3.1% higher (5.6-2.5=3.1) at age 40. 18

An increase in the market wage of 3.4% due to a higher capital-labor ratio in the constrained optimum attracts more income-poor young households to work by lowering their time spent on human capital investment, which eventually yields 1.3% higher labor income and 1.7% higher consumption at age 23 (panels (c) and (d)). The marginal cost of lowering human capital accumulation in the constrained optimum, however, is labor income loss as well as consumption loss in the old age. Therefore, the constrained efficient allocation implies that there should clearly be an income transfer from old age to youth over the life-cycle. For comparison, we also plot full-insurance profiles along with the other profiles. Considering full insurance, human capital is no longer a risky asset, and, hence, the precautionary motive vanishes. This change yields approximately 43% lower asset accumulation over the life-cycle in panel (a). Surprisingly, however, the time investment in human capital in the full-insurance economy is also 27% lower at age 23 (panel (b)) even though human capital is no longer a risky asset. Simultaneously, working hours are also lower by 10.6% under full insurance. The joint effect is a 2.7% higher effective labor supply. This result is mostly due to the income effect of labor supply. A decrease in capital accumulation lowers the wage by 5.8%. As the wage declines across all ages and is considered to be a decline in the permanent wage, the income effect of labor supply dominates the substitution effect. Therefore, the median household provides more effective labor supply as well as more leisure time to improve lifetime utility. Under full insurance, the lower accumulation of human capital results in lower income and consumption levels, and households achieve almost perfect consumption smoothing over the life-cycle. The comparison of the median household’s profiles clearly shows a redistribution channel through the course of the life-cycle. As younger cohorts are relatively more consumption-poor than older cohorts are, an income transfer from old age to youth supports more consumption in the early part of life, whereas a wealth transfer from youth to old age provides more self-insurance for income shocks to which elderly people are exposed. To explicitly answer the primary questions, U.S. households currently save too little and invest too much in human capital (or work) over their life-cycle. Thus, they have to save more and invest (or work) less to improve their lifetime welfare. Cross-Sectional Redistribution Channel In a life-cycle model, a newly introduced redistribution channel across an individual’s life plays a crucial role in improving social welfare, whereas the conventional redistribution channel across households is still in effect. To examine cross-sectional distribution effects within each age cohort, we study the constrained efficient profiles of the top and bottom 10% of income earners. First, we classify the top and bottom 10% of income earners by their initial human capital. Therefore, they may not necessarily remain in the top or bottom throughout their lives. In Figure 5, the top 10% of income earners save up to 26% (panel (a)) in the constrained optimum, whereas their human capital investment in the constrained optimum is up to 17.3% lower than what they actually invest (panel (b)). In other words, the current asset holdings of the top 10% are too small in comparison to the efficient level, whereas they spend too much time on human capital investment. The top 10% would also work 11% less as they approach retirement. By contrast, the asset levels of the bottom 10% 19

income earners throughout their life-cycles are almost at the efficient level. Their human capital investment is 5% lower in the constrained optimum at approximately 34 years of age, but it is almost at its efficient level thereafter. The bottom 10% of households work 2.6% fewer hours in the constrained optimum, mostly in the early part of life. The changes in factor income composition through market prices explain the results. The top 10% of earners are born with high human capital. Therefore, their labor incomes are relatively high even in the early part of life, whereas they are all initially wealth-poor with no assets. Therefore, a small drop in the relative return on assets has a minuscule effect on capital income, while a rise in the market wage undoubtedly makes them better off. Therefore, they would save more than they actually do to raise the market wage. However, the bottom 10% of earners are born with low human capital and no wealth. Hence, these income- and wealth-poor households work more to raise their labor income composition when the market wage is higher. Nevertheless, they cannot have a significant effect on market prices, as they are financially constrained. Instead, they can raise their hours of work in the early stage of life by lowering human capital investment so that they can obtain higher labor incomes. In summary, the constrained inefficiency present in the benchmark economy is largely corrected by top income earners, whereas lower income earners reap benefits through the change in market prices and, hence, through the change in the factor income composition.

5

Implementation of Constrained Efficient Profiles

The characterized constrained efficient profiles can also be implemented by a modified current tax system. The envelope conditions (39), (44), and (45) for the constrained optimum indicate that the constrained inefficiency (∆k , ∆h ) should be incorporated into the capital income tax and progressive labor income tax with a lump-sum transfer so that the net transfer to each household remains unchanged, as follows: τbk,j (xj ) = τk −

∆k rλcj

Tbj0 (˜ yj , xj ) = T 0 (˜ yj ) −

wλcj (1

∆h . − 0.5τss Iyj ≤¯y )

To obtain the constrained efficiency, the tax wedges should take into account the households’ marginal utility of consumption. Thus, the tax wedges should be conditioned not only on household states (x) but also on their ages, j. The previous quantitative results show that the sign of the saving wedge is positive (∆k > 0). Hence, the modified capital income tax should be lower than the current tax rate, τk , which incentivizes households to save more. However, younger households are relatively consumption-poor, so their marginal utilities of consumption are relatively higher. Therefore, the capital income tax for younger households would be closer to the current tax rate than that for the older households would be. The marginal labor income tax, on the other hand, would become higher than the current rate because of the negative labor wedge (∆h < 0). 20

To incentivize younger households to invest less time in human capital, the marginal tax would be lower for the (consumption-poor) younger households than for the older households. This marginal utility effect on the modified tax code implies progressivity in both capital and labor income. As consumption is highly correlated with labor income, the income-poor would have to pay a higher capital income tax and a lower labor income tax in the constrained optimum. Figure 6 shows the computed average tax incentives for each age cohort that implement the constrained efficient profiles. While the current proportional capital income tax rate is 40%, the implemented tax rate would be 25% at age 23 and the rate declines by age, so that after age 45, households receive tax credits (panel (a)). On the contrary, the marginal labor income tax would be 4.2% instead of actual 0.02%, and this rate would increase as households age so that the tax inhibits the over-investment in human capital and the over-supply of labor (panel (b)). Despite the conceptual difference between constrained efficiency and optimal taxation, the agedependent labor income tax code for the constrained optimum echoes the results in the optimal tax literature. In a life-cycle setting, the optimal capital income tax has to be positive, the labor income tax has to be age-dependent, or the labor income tax has to be progressive (see Erosa and Gervais (2002), Gervais (2012), Conesa et al. (2009), Peterman (2016)), which resemble the tax implementation of constrained efficiency. However, there is a sharp contrast between the constrained optimum and optimal taxation in that the constrained efficiency directly affects market allocations, which indirectly change the factor income composition through market prices, whereas optimal taxation directly affects market prices for the provision of insurance against idiosyncratic shocks and redistribution across households with different abilities.

6

Life-Cycle Effects of Student Debt

The constrained efficiency analysis in a life-cycle model allows us to extend it to a variety of current issues. In this section, we provide the welfare implications of the student debt burden over the course of life. As financing for higher education through college student loans has become ubiquitous, the total student debt and its default rate after starting the repayment have been dramatically rising in the U.S. About two thirds of college graduates have a certain amount of debt, which accounts for the second-largest source of household debt after mortgage debt. Admittedly, the provision of student loans (either private or public) allows more students to attain higher education and, hence, attain (supposedly-)higher productivity. However, after completing higher education with debt, young households become financially constrained while they repay the debt and, hence, the debt undoubtedly affects their investment decisions. For instance, with the student debt to be paid off, young households have to minimize the accumulation of selfinsurance against future income risks and have to refrain from purchasing their own houses. A number of studies have examined the effect of student debt on human capital (Fos et al., 2017) and labor market outcomes (Chapman (2015), Justin Weidner (2016), Gervais and Ziebarth (2017)). Nevertheless, little is known about the student debt effects, not only on labor income, but also on consumption, saving, labor supply, and human capital investment, as well as its 21

welfare implications over the life-cycle. Therefore, we apply our life-cycle framework to explore the life-cycle profiles of student debt holders and their constrained efficient levels. The quantitative exercise in the previous section assumed that all households start their labor market experiences with no assets and that their initial human capital (and hence initial labor income) is independent of initial debt. We can relax this assumption to allow a fraction of households to start their life-cycle profiles with debts (σa > 0). In addition, we assume that the initial debt correlates with initial human capital (λah 6= 0). Assuming that approximately half of households enter the labor market with debt, we maintain our original assumption that the average asset holdings of initial age households is initially zero (ma = 0). We also retain our assumption that the initial debt has no effect on households’ learning ability (λaθ = 0). To calibrate these parameters, we use two pieces of empirical evidence: the student-debt-to-income ratio and the correlation between debt and initial labor income. Using NLSY97 data, we obtain a student-debt-to-income ratio for student debt holders of 0.94. Furthermore, the correlation between initial assets and initial labor income to be corr(a0 , h0 ) =0.35, implying that the more debt an individual owes, the less labor income s/he earns. Knowing that NLSY97 data has major sample size limitations, several issues need to be addressed regarding the latter data moment. First, the correlation between the debt amounts and labor market outcomes is mixed in the empirical literature. For example, some authors demonstrate a negative (positive) correlation between asset holdings (student debt) and post-graduation earnings (Minicozzi (2005), Chapman (2015)). On the contrary, some other studies show the opposite result (Justin Weidner (2016), Gervais and Ziebarth (2017)). In terms of the effects on human capital or graduate school enrollment, Fos et al. (2017) argue that there is a negative relationship between debt and the enrollment rate, whereas Monks (2001) claims no effect. Given the mixed results in the literature, our choice of 0.35 is rather arbitrary, and instead of supporting or refuting any empirical evidence, we redo our exercise with a negative correlation of assets and initial income, corr(a0 , h0 ) = -0.35. In fact, the welfare implications of student debt do not qualitatively change with either a positive or negative correlation between initial assets and initial human capital. Figure 7 shows the model-simulated life-cycle profiles of student debt holders in the benchmark economy and in the constrained optimum. As the welfare implications of student debt do not qualitatively change with either a positive (green line) or negative (orange line) correlation between initial assets and initial human capital, we describe the life-cycle profiles of student debt holders in the positive case. It is clear that debt holders are initially wealth-poor and that their main source of income is labor income. Therefore, our previous analysis indicates that debt holders are better off when the market wage is high. In the constrained optimum, in which the equilibrium wage would be 3.4% higher, the debt holders would finish paying off their debt by age 31, as compared to age 32 in the benchmark case, or they would save 22% more on average (panel (a)). They would also invest less time in human capital. The joint effects of 3.9% less time investment in human capital (panel (b)) and 1.9% fewer hours of work (panel (e)) would yield, on average, a 2.1% greater effective labor supply until age 53 and a 0.013% lower effective labor supply after age 54. As a consequence, debt holders would earn, on average, 1.6% more until age 46 and 0.18% less in the later stage before retirement (panel (c)). Similarly, they would consume 1.25% more until 22

age 42 and consume 1.05% less in old age (panel (d)). Again, the life-cycle profiles of student debt holders convey the same message as before. The constrained efficient allocations through the equilibrium prices induce a wealth transfer from youth to old age on the one hand and income and consumption transfers from old age to youth on the other hand. Considered differently, the student debt causes current U.S. debt holders to save too little and invest too much time in human capital accumulation. This result is mainly due to the fact that the equilibrium wage is set below its efficient level, and, therefore, debt holders would allocate their time to raise their future labor income at the cost of saving and consuming less in the early stage of life.

7

Sensitivity Analysis

Finally, we check whether our results are sensitive to different values of model parameters. To save space, we only describe the results, and all of the figures are shown in the online appendix. Borrowing Limits — In the benchmark model, we assume a natural borrowing limit (a > 0) such that households can borrow up to the amount needed to pay off their debt before the end of life. In fact, the lower-income households tend to borrow in the early stage of life. Now, we assume a tighter borrowing capacity, that is, a zero borrowing limit (a = 0). Holding other parameters unchanged, we re-calibrate the model to match the data profiles and re-compute the constrained efficient profiles. The tighter borrowing capacity intensifies the precautionary motives, which in turn generate a higher consumption growth over the life-cycle as well as higher labor supply and human capital investment in the early part of life. Thus, the tighter borrowing constraint has an indirect effect on the market prices and, hence, on the capital and labor wedges (∆k , ∆h ). Figure 1 in the online appendix shows the median household’s benchmark profiles versus constrained efficient profiles. The direction of inefficiency over the life-cycle is qualitatively identical to the results with natural borrowing limits. That is, the U.S. households currently save too little and invest too much in human capital (or work) over the life-cycle. The median household’s profiles clearly show a redistribution channel through the course of life. As younger cohorts are relatively more consumption-poor than older cohorts are, an income transfer from old age to youth supports more consumption in the early part of life, whereas a wealth transfer from youth to old age provides more self-insurance to the income shocks that they are exposed to in old age. Thus, a tighter borrowing capacity does not alter the direction of redistribution over the life-cycle. No Labor-Leisure Choice (Huggett et al., 2011) — For this sensitivity check, we apply Conesa et al. (2009)’s tax structures to Huggett et al. (2011)’s endogenous life-cycle human capital model. Huggett et al. (2011)’s model assumes no labor-leisure choice, which is equivalent to setting nj = 1 for all j with no disutility from the labor supply in our benchmark model. One caveat about Huggett et al. (2011)’s time investment model is that the consumption growth tends to be three- to four-times higher than the income growth because of insufficient consumption insurance provided in the model. Therefore, in order to match the consumption growth, the model requires one of these features: 1) unrealistically high consumption or capital income tax, 2) a positive initial asset that provides an extra insurance, or 3) an elastic labor supply. Our benchmark model 23

indeed incorporates the elastic labor supply nj ∈ [0, 1]. As the unrealistically high taxes distort constrained efficient allocations, we instead use a positive initial assets for this exercise to provide a large degree of self-insurance to the young cohorts, which successfully flattens out the lifetime consumption growth. The results are qualitatively no different from the benchmark results. The capital-output ratio is 19% higher and the capital-labor ratio is 29.5% higher in the constrained optimum. This result is due to an 8.6% under-accumulation of capital together with an 18.4% over-supply of effective labor. Due to the under-accumulation of capital-labor ratio in the competitive equilibrium, the interest rate is 41.9% lower and the wage is 8.7% higher in the constrained optimum. The lifecycle asset profile shows that the constrained efficient saving until age 56 is up to 43% higher than the competitive equilibrium saving. More specifically, the constrained efficient saving is 22.1% higher at age 30, 42.4% higher at age 40, and 24.8% higher at age 50. Prior to the retirement, the constrained efficient asset accumulation slows down and its peak at the retirement is 15% lower than the peak of the competitive equilibrium. The median households would save more in the early stage of life, thereby increasing its aggregate capital and lowering the rate of return on assets. This result indicates the redistribution of wealth from youth to old age for an efficient allocation. On the contrary, time investment in human capital drops by up to 64% before the retirement. In the constrained efficient allocations, at age 23, the median households spends 16.2% of its time on investment, whereas this value is 34.8% in the competitive equilibrium allocations. Since human capital is a risky asset, increasing risk-free assets and reducing risky assets provides more insurance to the households. The decline of time investment in human capital implies an increase in the current labor supply, which eventually yields 3.2% more labor income and 1.4% more consumption at age 23. However, the lower human capital accumulation, which slows down the income growth over the life-cycle, yields 7.5% less labor income in the constrained optimum at retirement. Despite the fact that an increase in the aggregate wage by 8.7% due to the accumulation of capital partially offsets labor income loss of older cohorts in the constrained optimum, there is clearly an income transfer from old age to youth over the life-cycle.

8

Conclusion

In this study, we quantify the degree of inefficiency in saving and human capital investment across different ages over the life-cycle. We analyze to what extent saving and human capital investment are driven by the age-varying inefficiency and determine the efficient levels of saving and human capital investment. For this analysis, we adopt a notion of constrained efficiency in which the planner can internalize the effect on market prices (interest rate and wage) without controlling household budgets and without providing extra insurance contracts. We find that the constrained inefficiency over the life-cycle can differ by age and can be further improved by the redistribution of income and wealth not only across individuals and states but also within an individual’s life-cycle. The main contribution of this study is that we explicitly demonstrate a redistribution channel 24

arising from the life-cycle property to achieve a constrained efficient allocation. Our analysis shows two channels of redistribution for the constrained optimum. The first is a lifecycle redistribution channel. Our quantitative life-cycle analysis shows that the median household’s wealth is under-accumulated at all ages until retirement by the constrained inefficiency, whereas there is an over-investment in human capital mostly in the early part of life. Thus, there must be a wealth transfer from youth to old age and income and consumption transfers from old age to youth for a constrained optimum. The second redistribution channel is cross-sectional. When we examine the constrained optimum profiles by income group, the top 10% of income earners save substantially more and invest less in human capital in the constrained optimum. On the contrary, the bottom 10% of income earners’ savings are almost at the efficient levels, and their human capital investment would be lower only at around age 34. Thus, the constrained inefficiency would be corrected mostly by the incomerich; and, as a consequence, the income-poor would become better off by receiving more labor income through an increase in market wage. The life-cycle analysis of constrained efficiency with endogenous human capital accumulation can be applied to a welfare analysis of college student debt. Although college student loans allow young cohorts to obtain higher education, accumulated student debt causes current U.S. student debt holders to save too little and invest too much time in human capital accumulation. Essentially, in a constrained optimum with a 3.4% higher wage, the debt holders would be better off by saving more and invest less in human capital. Thus, the constrained efficient allocations again require a wealth transfer from youth to old age and income and consumption transfers from old age to youth. Our future research will include additional applications to current issues.

9

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Appendix A

Data and Age Profiles

Data Sources To construct age profiles of the hourly wage and labor supply, we use familyindividual longitudinal data from 1980âĂŞ2002 waves of the Panel Study of Income Dynamics (PSID). We use cleaned PSID data provided by Heathcote et al. (2010). The PSID interview has been conducted biannually since 1997. The estimates of the income process and our results do not change significantly by extending or shortening the sample period. For the estimation of age profiles, we use the hourly wage calculated as annual labor income divided by annual hours of work. To calculate the labor supply as a fraction of the total available time normalized to one in the model, we divide annual hours of work by 5200 hours. For consumption profiles, we use the 1980âĂŞ2002 waves of the Consumer Expenditure Survey (CEX). The CEX is a repeated panel of a quarterly interview survey, although estimating consumption profiles across age groups does not require a panel structure. We use cleaned CEX data provided by Krueger and Perri (2006). Even though the CEX provides household consumption expenditure by item, we use nondurable expenditures to estimate our consumption profiles. The definition of nondurable expenditures is the same definition used in Krueger and Perri (2006). As the survey on consumption expenditures is conducted on a household level, we apply the modified-OECD equivalence scale, which assigns a value of 1 to the first adult, a value of 0.5 to each additional adult, and a value of 0.3 to each additional child, to convert household consumption expenditure to adult-equivalent units.19 Sample selection Our sample selection criteria closely follows Kaplan (2012), as we require consistency between the PSID and CEX samples. For the PSID, we drop the SEO and Latino samples that were added to the survey in 1990. For both the PSID and the CEX, we retain male heads of the household between 23-62 years of age. We drop self-employed individuals and retain individuals (household) with at least four years of consecutive income observations. We also eliminate individuals with annual working hours below 520 and above 5,200 hours and those with an hourly nominal wage less than half of that year’s minimum wage. Retirees, students, housewives, and permanently disabled individuals are dropped. We deflate the income and consumption variables by Consumer Price Index (2005=100 based) to convert them from nominal to real values. These sampling criteria apply to both the PSID and CEX data sets. In addition, we choose CEX samples with four consecutive observations. Estimation of Age Profiles To identify the age effect of a variable x, it is common practice to run a linear regression with a full set of age dummies, year dummies, cohort dummies, and 19

The sensitivity of consumption profiles to the choice of equivalent scales is reported in Kaplan (2012).

28

other observable individual characteristics, such as race and education: i xia,c,t = βa Da + βt Dt + βc Dc + θa,c,t Xa,c,t + a,c,t ,

(46)

where xia,c,t is a variable (e.g., labor income or consumption) of household i at age a belonging i is a set of observable to cohort c in year t; Dk is a full set of dummies for k ∈ {a, c, t}; Xa,c,t individuals characteristics; and a,c,t is the residual. Then, the estimated coefficients on the age dummies βba characterize the age profile of the variable. One well-known problem with this regression is that age, cohort, and year effects are not linearly independent. Therefore, more restrictions on this regression are required. The most common practice used in the literature is to control for either year or cohort effects. In other words, only two of the three dummy controls can be operative. The age profiles shown in Figure 2 and 3 control for year effects, but controlling for cohort effects instead does not substantially change our results. To compute the variance of a variable over the life-cycle, we take the estimated residuals from the previous regression, ba,c,t , square them, and run a regression again with a full set of age dummies and either year or cohort dummies. We assume that the residuals are now stationary, so observable characteristics do not need to be included in this regression. The confidence intervals for each age profile are constructed by bootstrap.

29

Table 1: Calibrated and Predetermined Parameters Parameters Value Demographics n 0.011 40 JR J 58 Preferences and Technology r 0.042 β 0.966* α 0.322 0.067 δk γ [0.7,0.9] ψ 2.603* φ 4.387* 0.052* χ δh 0.023* Initial conditions mh 2.734* -1.021* mθ σh 0.291* σθ 0.285* λhθ 0.003* Shock process σε 0.111 Tax system τss 0.124 y¯ 2.5×E(y) yss 1.002* τc 0.06 τk 0.40 0.258 τ0 τ1 0.768

Target from BLS data working age from 23 to 62 age up to 80 annual net interest rate at 4.2% average capital-output ratio K/Y =2.947 average capital income share Y αK −r 0.7 as a benchmark from Huggett et al. (2011) consumption age profile labor supply age profile average hours of work labor income decline in old age average initial wage at age 23 average wage growth initial variance of wage profile variance growth of wage profile the covariance of initial wage and initial wage growth Huggett et al. (2011) Data Data social security budget balance Mendoza et al. (1994) Domeij and Heathcote (2004) Gouveia and Strauss (1994) Gouveia and Strauss (1994)

Notes: * on the second column indicates a calibrated parameter value. Otherwise, parameter values are predetermined.

30

Figure 1: Fraction of the Bottom Ten Percent of Households in Income, Consumption, and Net Worth

Notes: The figure shows the fractions of the bottom 10% of households in terms of income, consumption, and wealth for each age cohort. Income is measured by labor income using PSID data for 1980-2002. Consumption is measured by nondurable expenditures using CEX data for 1980-2002. Wealth is measured by net worth using the Survey of Consumer Finances (SCF) for 1992-2016. As the tenth percentile of each variable changes by year, we first calculate the number of age cohorts below the benchmark threshold by year and then add the numbers over the years. Further, we divide the number of age cohorts below the benchmark threshold by the total number of age cohorts to obtain of each age cohort below the benchmark threshold. More precisely, we plot the P the P fraction i i measure Qj = N1j t i 1{qj,t ≤ q¯x,t } for j ∈ {23, 24, ..., 62}, where Nj is the total number of age-j cohorts, qj,t is an individual i’s variable q ∈ {income, consumption, net worth} at age j in year t, and q¯x,t is the x-percentile threshold in year t.

31

Figure 2: Data vs. Model: Mean Profiles (a) Hourly Wage

(b) Consumption

(c) Hours of Work

(d) Human Capital Investment

(e) Assets

Notes: We plot mean profiles of the hourly wage, consumption, asset accumulation, time investment in human capital, and hours of work. The data sources and the estimation of profiles are described in Appendix A. Modelsimulated profiles are plotted on top of data profiles with 95% confidence intervals for hourly wage (panel (a)), consumption (panel (b)), and hours of work (panel (c)).32 Since asset accumulation (panel (e)) and time investment in human capital (panel (d)) are not targeting any data-counterpart mean profiles, only model-simulated profiles are plotted. Hourly wage, consumption, and hours of work are logged.

Figure 3: Data vs. Model: Variance Profiles (a) Hourly Wage

(b) Consumption

(c) Hours of Work

(d) Human Capital Investment

(e) Assets

Notes: We plot the variance of the hourly wage, consumption, asset accumulation, time investment in human capital, and hours of work over the life-cycle. Data sources and the estimation of profiles are described in Appendix A. Model-simulated profiles are plotted on top of data profiles with 95% confidence intervals for the hourly wage (panel (a)), consumption (panel (b)), and hours of work 33 (panel (c)). Since asset accumulation (panel (e)) and time investment in human capital (panel (d)) are not targeting any data-counterpart variances, only model-simulated variances are plotted. Hourly wage, consumption, and hours of work are logged.

Table 2: Comparison of Resource Allocations and Market Prices

Interest rate (r) Wage (w) Aggregate output (Y ) Aggregate capital (K) Aggregate labor (L) Aggregate consumption (C) Capital-output ratio (K/Y ) Capital-labor ratio (K/L) Capital wedge (∆k ) Labor wedge (∆h )

Competitive Equilibrium

Constrained Optimum

Full Insurance

4.22% 0.999 7.182 21.126 4.874 4.826 2.942 4.335 -

3.44% 1.034 7.249 22.907 4.751 4.764 3.161 4.817 0.000145 -0.001048

5.68% 0.942 5.723 14.856 4.121 3.990 2.596 3.605 -

Notes: For a comparison of aggregate variables in the competitive equilibrium, the constrained optimum, and the full-insurance economy, we re-compute the market-clearing prices for each economy and calculate model-simulated aggregate variables.

34

Figure 4: Constrained Efficient Profiles: Median Households (a) Assets

(b) Human Capital Investment

(c) Hourly Wage

(d) Consumption

(e) Hours of Work

Notes: Panels (a)–(e) plot median model-simulated profiles of labor income, consumption, asset accumulation, time investment in human capital, and hours of work in the competitive equilibrium (green solid line), the constrained optimum (blue dashed line), and the full-insurance economy (black dotted line).

35

Figure 5: Distributional Effects: Top 10% versus Bottom 10% of Income Earners (a) Assets

(b) Human Capital Investment

(c) Hourly Wage

(d) Consumption

(e) Hours of Work

Notes: We plot model-simulated profiles of labor income, consumption, asset accumulation, time investment in human capital, and hours of work in the competitive equilibrium (“Eq,” solid line) and constrained optimum (“Eff,” dashed line). To observe the distributional effect within each age cohort, we plot the profiles separately for the top 10% (orange line) and bottom 10% (green line) of36 income earners. The top and bottom 10% of households are classified by their initial human capital.

Figure 6: Implementation of Constrained Efficiency (a) Capital Income Tax

(b) Marginal Labor Income Tax

Notes: We plot the capital income tax and marginal labor income tax for each age cohort that implements the constrained efficient profiles (blue line). For comparison, we also plot the current tax rates that we employ in our benchmark analysis (black dashed line). Note that even though the tax codes under the constrained optimum are conditioned on household characteristics and age, we only plot the average tax rate for each age cohort.

37

Figure 7: Student Debt Effects on Constrained Inefficiency (a) Assets

(b) Human Capital Investment

(c) Hourly Wage

(d) Consumption

(e) Hours of Work

Notes: We plot model-simulated profiles of labor income, consumption, asset accumulation, time investment in human capital, and hours of work in the competitive equilibrium (“Eq,” solid line) and constrained optimum (“Eff,” dashed line). Due to mixed evidence on the correlation between initial assets and initial human capital in the empirical literature, we plot the profiles separately 38 for the cases in which corr(a0 , h0 ) < 0 (orange line) and corr(a0 , h0 ) > 0 (green line).