Economic Analysis of Social Security Survivors Benefits For Dependent Children

Transcription

1 Economic Analysis of Social Security Survivors Benefits For Dependent Children Yue Li June 30, 2015 Abstract This paper develops a lifecycle model with a warm glow bequest motive to analyze social security survivors benefits for depend children (SBDC). The model features differential mortality by income group and price discrimination in the private life insurance market. The model parametrizes the bequest motive to match actual life insurance holdings, and identifies several preference attributes that lead male agents to leave insufficient bequests to support survivors. The model finds that removing SBDC harms both agents and surviving widows with dependent children, and that either allocating more benefits to young agents or expanding the size of SBDC is welfare improving. JEL: H55, E21, D64 Keywords: Social Security, Life Insurance Demand, Differential Mortality, Bequest Motive Department of Economics, SUNY Albany. yli49@albany.edu. I am especially grateful to Daniele Coen-Pirani and Marla Ripoll for their invaluable advice, guidance and support. I also thank Laurence Ales, John Duffy, Sewon Hur, Alejandro Badel, Satyajit Chatterjee, Pablo D Erasmo, Jeremy Greenwood, Samuel Myers, and Kim Ruhl for helpful discussions; the participants in University of Pittsburgh Macroeconomic Seminars and the 88th Annual Meetings of Western Economic Association International for helpful comments. All remaining errors are mine.

2 1 Introduction In 2014, about 1.9 million surviving children received social security monthly paychecks, and the average benefit amount was $814. These payments are called survivors benefits for dependent children (SBDC). The program of SBDC provides each surviving child with 75 percent of the deceased parent s primary insurance amount (PIA) until he/she reaches the age of 18. In 2014, the total benefit outlays of SBDC was $19 billion (Social Security Annual Statistical Supplement). In spite of the importance of SBDC, no previous papers have formally studied the benefit and the cost of the program. To fill the gap in the literature, this paper considers that SBDC are substitutes to private life insurance, and develops a life-cycle model with a warm glow bequest motive to capture the demand for life insurance and study the influence of SBDC. The model features differential mortality by income group: conditional on age, low-income individuals are more likely to die than high-income individuals are. 1 In an economy where the price of life insurance contracts are actuarially fair for each person, low-income individuals not only expect to live for a shorter period, but also need to purchase deceased contingent claims at a more expensive price than high-income individuals do. The provision of SBDC compensates for both mortality inequalities and price discrimination. An alternative model without differential mortality by income group understates the welfare loss from removing SBDC by 80 percent. 1 For instance, Cristia (2009) finds that among males aged 35-49, the mortality rate of individuals in the bottom quintile of the life earning distribution is 6.4 times as large as that of individuals in the top quintile. 1

3 Previous literature mostly focuses the bequest motive of old individuals, and little is known about this motive among young and middle-age individuals. 2 This paper makes a first step to characterize the bequest motive for male agents of all age groups using actual life insurance holdings. The paper identifies three preference attributes. First, to match the lifecycle profile of life insurance holdings, the bequest motive of young agents needs to be weaker than that of old agents. Second, to match the difference in life insurance holdings across household types, agents need to have an additional bequest motive to support under-age-18 consumption of children and post-retirement consumption of wives. Last, to match the difference in life insurance holdings between high and low income groups, bequests need to be luxury goods, as argued in De Nardi (2004). The model consists of heterogeneous male agents who face uncertainties of survival, income, and household types. Agents allocate resources among consumption, risk-free assets, and life insurance to maximize expected lifetime utility from consumption and bequests. Since life insurance holdings are determined by agents preferences, its amount may not be financially sufficient to protect survivors from consumption drops upon the decease of agents. In this regard, SBDC force agents with dependent children to hold certain among of deceased contingent claims and alleviate consumption drops among surviving widows with dependent children. On the agent s side, SBDC improve equality by redistributing resources from high-income agents who have small marginal utility of consumption to low-income agents who have large marginal utility of consumption. However, SBDC create in- 2 For example, see De Nardi (2004), Ameriks et al. (2007), Laitner and Sonnega (2012), and De Nardi and Yang (2014). 2

4 efficiency by distorting the allocation between consumption and bequests. Given the two opposing channels, whether SBDC improve welfare is an open question that needs a model to answer. 3 The model is able to match both lifecycle patterns of life insurance holdings and the difference in holdings across income groups and household types. Since the amount of bequests is chosen to maximize agents lifetime utility, about 9 percent of wives would suffer from a significant (>20%) consumption drop if the agent dies. Although the degree of mismatch between life insurance holdings and financial vulnerabilities in the model is smaller than that found in Bernheim et al. (2003a,b), the paper makes an important first step to explain the source of this mismatch. This paper further calculates the progressivity of social security using information about simulated agents. Results suggest that the traditional approach of calculating the internal rate of return based on a sample of individuals who accumulate enough earning history produces biased estimates, because low-income individuals are more likely to die early and be excluded from the sample. Correcting the survival selection makes social security less progressive, and adding SBDC makes it more progressive. The paper implements several counterfactual policy experiments to study welfare implications of SBDC. The first experiment removes SBDC, and causes welfare losses that are equivalent to a 0.2 percent drop in all-period consumption for agents and a 5.0 percent drop in all-period consumption for surviving widows with dependent children. The second experiment keeps the total SBDC benefits unchanged, but adjusts the allocation of benefits across age groups. Agents welfare is maximized 3 The model abstracts from tax distortions, because the labor tax rate that is needed to fund SBDC is only 0.15 percent. 3

5 under a policy that allows the ratio of per period benefit for each surviving child to PIA to start at a high level of 192 percent for agents aged 22-24, drop by 20 percentage points at each subsequent three years of age, and reach the low level of zero for agents aged 49 and older. The last experiment keeps the allocation across age groups unchanged, but adjusts the size of SBDC. The model finds that a policy that allows each surviving child to receive 245 percent of the agent s PIA per period (until reaching the age of 18) maximizes agents welfare. Note that both policies that maximize agents welfare also improve the welfare of surviving widows with dependent children. This paper is related to three strands of the literature. The first strand of literature develops models to predict the demand for life insurance. This literature has two different approaches. The first approach specifies the bequest preference independent of life insurance holdings (Chambers et al., 2003, 2011; Love, 2010; Hubener et al., 2013a,b). This approach finds that it is hard to match the lifecycle profile of life insurance holdings and it is puzzling why households do not purchase enough life insurance to smooth family consumption in the event of member s death. To address the puzzle, this paper adopts the second approach (see Hong and Ríos-Rull (2007, 2012)) and uses actual life insurance holdings to define preferences. Different from Hong and Ríos-Rull (2007, 2012), this paper introduces differential mortality by income group, and focuses on characterizing the bequest motive and discussing welfare implications of SBDC. The second strand of literature empirically examines the effect of death on survivors (Auerbach and Kotlikoff, 1987, 1991; Hurd and Wise, 1996; Bernheim et al., 4

6 2003a,b). These papers document that a significant number of households do not purchase adequate amount of life insurance to smooth consumption in the event of adult member s death. This paper provides a theoretical explanation for the insufficient holdings. The last strand of literature uses models with heterogeneous agents to examine social security programs (Imorohoroǧlu et al., 1995; Conesa and Krueger, 1999; De Nardi et al., 1999; Imorohoroǧlu and Kitao, 2012; Kitao, 2014; Hosseini, Forthcoming; Li, 2015). This paper extends the literature by studying the SBDC program, an important component of the social security system that receives little attention previously. The remainder of the paper is organized as follows: section 2 describes the model; section 3 presents the calibration; section 4 shows the benchmark economy; section 5 conducts policy experiments; section 6 discusses model assumptions; and section 7 concludes. 2 Model This section presents the model. Agents in the model can be thought as males who are either singles or household heads. 2.1 Demographics The economy is populated by a constant size of overlapping generations. Each generation lives up to J periods. Agents supply one unit of labor in the first Jr 1 periods and retire from the period of Jr. Let s(e, j) ( s(e,j) e 0, s(e,j) j 0) denote 5

7 the probability of survival, where j is an age index, and e is an index recording average earnings up to the current period. f denotes a household type, the process of which follows a finite-state Markov chain with transitions P (f f, j). Household types determine the presence of spouses and children in the agent problem. Household types combined with ages further determines the number of children C n (f, j), and the age of children C a (f, j). 2.2 Government and Insurance Firms Government: The government collects taxes from labor at a rate of τ l, from consumption at a rate of τ c, and from interests at a rate of τ k. The government operates OAI and survivors benefits, which include both SBDC and survivors benefits for aged spouses. Let S r (e, j, f) denote household OAI benefits. Survivors benefits S s (e, j, f) are a lump-sum payment that equals the present value of future additional social security benefits that survivors can receive if an agent deceases in the next period. The amount of tax revenue that is in excess of social security expenses is used for direct spending G. Insurance Firm: A representative insurance firm operates in a competitive market. This firm perfectly observes individual mortality rates and sells life insurance contracts at a unit price of p(e, j) = (1 s(e, j))/(1+r), where r denotes the interest rate of one-period risk-free assets. 4 This market setting indicates that agents cannot sell deceased contingent claims (or purchase survival contingent claims), and that agents with high mortality rates need to purchase deceased contingent claims at a 4 This paper assumes that there is no asymmetric information in the life insurance market, which is consistent with the evidence in Cawley and Philipson (1999). 6

8 more expensive price than agents with low mortality rates. 2.3 Agent Problem Agents are characterized by a state vector φ = (a, e, η, ι, j, f), where a denotes holdings of risk-free assets, η represents a permanent productivity type, and ι represents a transitory productivity shock that follows a finite state Markov chain with stationary transitions Q(ι ι). wε(j)ηι represents agents earnings, where w is the wage rate, and ε(j) is the age-efficient profile. Household earnings are denoted by W (wε(j)ηι, f, j), which is a reduced form representation that captures wives earnings. Average earning index e is a key state variable that determines individual survival rates, life insurance prices, and social security benefits, and has the following form: (1) e(j) = wε(j)ηι j = 1 ((j 1)e(j 1) + wε(j)ηι)/j 1 < j < Jr. e(j 1) j Jr Given prices and taxes, agents choose consumption c, risk-free assets k, life insurance x, and bequests b to maximize their expected lifetime utility. Formally, the dynamic problem solved by an agent of age group j = 1,..., J can be written as follows: V (φ) = max {u(c, j, f) + β(1 s(e, c 0,x 0,k 0,b 0 j))v(b, j, f) + βs(e, j) V (φ )Q(ι ι)p (f f, j)} f ι 7

9 subject to: (2) (1 + τ c )c + k + p(e, j)x a(1 + r(1 τ k )) + (1 τ l )W (wε(j)ηι, f, j) + S r (e, j, f) (3) (4) b k (1 + r) + x + S s (e, j, f), a =A(k, f, f, j) where V ( ) is the value function, and β is a discount factor. u(c, j, f) represents utility flows from consumption. v(b, j, f) represents utility flows from bequests. Note that the model restricts both holdings of risk-free assets and life insurance to be non-negative. Constraint (2) is a budget constraint. Constraint (3) shows that bequests consist of three components: life insurance, risk-free assets, and survivors benefits. 5 Equation (4) demonstrates that the amount of risk-free assets may change respect to the dynamics of household types. 2.4 First Order Conditions The following two first order conditions characterize the solution to the agent s problem. 5 The model abstracts from estate taxes, since bequests left for spouses are not taxable and the majority of single agents in the model have a bequest amount smaller than the exclusion cap, which is $1 million in

10 Trade-off between consumption and bequests: (5) u c(c, j, f) }{{} β(1 + r)(1 + τ c )v b(b, j, f) }{{} benefit of additional c cost of exchanging deceased contingent claims for additional c where u c(c, j, f) denotes the marginal utility of consumption, and v b (b, j, f) denotes the marginal utility of bequests. The allocation between consumption and bequests is optimal if holdings of life insurance are positive. Since SBDC provide deceased contingent claims that cannot be resold in the private market, this program may distort the allocation between consumption and bequests by mandating individuals to hold more deceased contingent claims than they desire. This distortion constitutes the cost of providing or expanding SBDC. Trade-off between current consumption and future consumption: (6) u c(c, j, f) }{{} benefit of additional c β(1 + r(1 τ k )[(1 s(e, j))v b(b, j, f)(1 + τ c ) + s(e, j)e f,ι u c(c, j, f ) a k ], }{{ } cost of reducing risk-free assets for additional c where E is an expectation operator. The allocation between current consumption and future consumption is optimal if holdings of risk-free assets are positive. The effect of SBDC on the intertemporal allocation is ambiguous, and depends on actual taxation and benefit schedules. 9

11 3 Calibration The section discusses the calibration. 3.1 Data The main data source is the Survey of Income and Program Participation (SIPP) 2001 Panel. This paper uses the 2001 Panel because panels that are more recent do not ask life insurance face values. The SIPP is a three-year panel survey that contains detailed information about demographics, income, public program participation, and household portfolios. The model uses demographic information in waves 1 and 9 to characterize the process of household types and the change in assets due to marriage and divorce, uses income information in waves 7 to 9 to specify the earning process, and uses life insurance face value and demographics reported in wave 9 to identify the bequest motive. In addition, the model collects information about assets from the 2004 Survey of Consumer Finances (SCF), and information about government programs from various government reports. 6 The model specifies the survival process to simultaneously match average mortality rates in the US period life table and mortality ratios by income group reported in Cristia (2009). 3.2 Demographics One period in the model is defined as three years. Agents enter the economy at the age of 22, retire at the age of 67, and definitely die at the age of 102. This 6 The model uses the SCF instead of the SIPP to obtain household net worth, because the SIPP systematically understands this variable (Czajka et al., 2003). 10

12 age structure corresponds to set Jr to 16 and J to 27. The size of population is normalized to Household Types The model incorporates four household types: married with dependent children, denoted by f = 1, married without dependent children, denoted by f = 2, single with dependent children, denoted by f = 3, and single without dependent children, denoted by f = 4. Household types change stochastically over time and affect earned income, social security benefits, asset stocks, and preferences for consumption and bequests. Figure 1 displays the distribution of household types by age group. Transition matrices are provided in Appendix A Married w/ children Married w/o chidren Single w/ children Single w/o children Percent Age Figure 1: Distribution of Household Types by Age Group Figure 2 presents the life cycle profile of the number and the age of children for 11

13 Average Number Age (a) Average Number of Children Average Age Age (b) Average Age of Children Figure 2: Child Information those who have dependent children. Because the SIPP does not contain good information about children who live outside of households (with their divorced mothers), the model assumes that conditional on age and dependent children status, the number of children C n (f, j) does not differ by marriage status. The number of children jointly with the age of children determine the number of stacked years that all children need to spend before reaching the age of 18, which is a key parameter that affects the bequest motive and the amount of SBDC. Since the number of stacked years is invariant to the heterogeneity of child ages within a household, the model assumes that all children in a household are of the same age C a (f, j). Note that the model abstracts from the heterogeneity of child numbers within age groups, because as discussed in subsection 6.3, it is not clear how the number of children affects the bequest motive. 12

14 3.4 Labor Endowments The permanent productivity type η is assumed to be drawn from a log-normal distribution, which is characterized by a mean of zero and a standard deviation of σ d. The logarithm of the transitory productivity shock follows the below AR(1) process with persistence ρ and a conditional variance σs: 2 ln ι = ρ ln ι + o, o N(0, σ 2 s). Following Conesa et al. (2009), σ 2 d, ρ, and σ2 s are jointly calibrated to match the increasing variance of log earnings over the life cycle, and take the value of 0.43, 0.91, and 0.26, respectively. 7 The age-efficient productivity profile ε(j) is normalized to 1 for the first age group and changes over the life cycle according to the profile reported in Hansen (1993). The annual wage rate w is set to $ to match the average wage of individuals aged Household earnings W (wε(j)ηι, f, j) have the following form: α f,j wε(j)ηι if f 2 W (wε(j)ηι, f, j) = wε(j)ηι otherwise. where α f,j is a parameter that controls wives earnings (See Figure 3). 7 η is discretized using two equal probability states, and ι is discretized by seven grids using the Rouwenhorst s method (Cooley, 1995). Further details are reported in Appendix B. 13

15 With children Without children α f,j Age Figure 3: Family Income Multipliers 3.5 Survival Function Table 1: Mortality Ratios Data (Cristia, 2009) Model Earning quintile (1) (2) (3) (4) (5) (6) Bottom Second Third Fourth Top Notes: Mortality ratios are computed as the ratio of one income group s mortality rate to the average mortality rate of one age category. As shown in the first three columns of Table 1, low-income individuals are more likely to die than high-income individuals are. In order to formalize differential mortality 14

16 th 75th Probability of Being Alive Age Figure 4: Probability of Being Alive by Income Group by income group, the survival function is parameterized in the following form: (7) [ ] 1 (1 s j ) 1 + min{g(j 1), 0}( e e j e s(e, j) = j ) 2 + s adj (j) if e e j [ ] 1 (1 s j ) 1 + max{h(j 1), 0}( e e j e j ) 2 + s adj (j) otherwise where s j is the average survival rate reported in the US period life table. e j is the mean of earning index e over all alive agents of age group j. g( ) and h( ) are fourth order polynomial functions that governs the process of transforming income differences into mortality differences. Parameters of g( ) and h( ) are set to match the mortality ratios reported in Table 1. s adj (j) is an adjustment factor that equalizes average survival rates in the model to those in the data. Further details about these parameters are provided in Appendix C. The comparison between the first three and the last three columns in Table 1 15

17 shows that model generated mortality ratios match well with the data. To better illustrate the mortality difference, Figure 4 displays the probability of being alive for two representative agents, one at the 25 percentile of the distribution of current average earnings (e) and the other at the 75 percentile. This figure demonstrates that there are large inequalities in survival rates: the agent at the 25 percentile has a 25 percent chance of dying before reaching the retirement age, which is 13 percentage points greater than that of the agent at the 75 percentile. The figure stops at the age of 76, because after that mortality ratios collapse to one, and survival rates no longer depend on previous earnings. 3.6 Government OAI and Survivors Insurance OAI: OAI benefits include benefits for agents and spouses, and have the below form: S r (e, j, f) = 0 if j < Jr P IA(e) if j Jr, f 3 zp IA(e) otherwise where the multiplier z is set to 1.57 to match the average ratio of couples benefits to husband s benefits. P IA(e) is a piece-wise linear function of average earnings that matches the 2003 social security benefit formula. SBDC: Each surviving child is entitled to receive 75 percent of the deceased father s PIA as long as he/she is under the age of 18. The present value of all benefit 16

18 flows is: S s c(e, j, f) =0.75P IA(e)C n (f, j)i Ca(f,j)+1<6[ 5 (C a(f,j)+1) a j =0 (1 + r) a j (8) + (1 + r) (6 (Ca(f,j)+1)) (C a (f, j) C a (f, j))], where C a (f, j) is the smallest integer that is greater than or equal to C a (f, j). Survivors Benefits for Aged Spouses: Each period, a retied widow (Aged 67 and older) is entitled to receive the maximum of the deceased husband old-age benefits (P IA(e)) and herself old-age benefits ((z 1)P IA(e)). The additional (per period) benefits received upon the decease of husbands are (2 z)p IA(e). The present value of all additional benefits is: (9) 0 if j < Jr Ss(e, s j, f) = j f J j j f j (2 z)p IA(e) f i j s f (i 1) otherwise (1+r) j f j Note that to match the average age difference between couples, in the model, wives are three years (one period) younger than husbands are. Moreover, a husband does not derive utility from survivors benefits for aged spouses, unless his wife reaches the retirement age and is entitled to receive the full benefits immediately upon his death. Total Survivors Benefits: S s (e, j, f) = Sc(e, s j, f) + Ss(e, s j, f). 17

19 3.6.2 Taxes and Government Expenditure According to the NBER Taxsim, the capital tax rate τ k is set to percent, and the labor tax rate τ l is set to percent. The consumption tax rate is set to 6.34 percent. Government expenditure G is a residual that balances the government budget, and is kept unchanged in all experiments. 3.7 Risk-Free Assets Interest Rate: Following Cooley (1995), the pre-tax interest rate r is set to 5 percent per annum. Transitions: The model assumes that people who maintain the same marital status carry the same amount of risk-free assets (a = k ) across periods, and estimates the change in assets due to divorce and marriage using the below equation: ln a ijt+1 = ln a ijt + γ f1 I fijt =1,f ijt+1 >2 + γ f2 I fijt =2,f ijt+1 >2 + γ f3 I fij >2,f ijt γ j + ɛ ijt where i denotes the individual number, j denotes the individual age group, t denotes the time of observation. γ j is an age group fixed effect. ɛ ijt are error terms. The omitted category is those who maintain the same marital status. Based on estimated coefficients, percentage changes in risk-free assets across periods are constructed and are reported in Table This transition matrix only applies to working-age agents, and all retired agents have a = k, because the model assumes that transitions of household types after retirement are due to the spouse s death. 18

20 Table 2: Percentage Change in Risk-Free Assets across Periods Married w/ children Married w/o children Single Married w/ children Married w/o children Single Preferences Utility flow from consumption is: (10) u(c, j, f) = ((1 + I f 2 + λ 1 C n (j, f)) λ 2 c) 1 σ /(1 σ), where σ is the relative risk aversion parameter, which is set to 1.5 as the common value used in the literature. I is an indicator function that takes the value of one if the subscript condition is true. λ 1 converts child consumption into adult equivalents. λ 2 [ 1, 0) measures the economics of scale in household consumption. Following Greenwood et al. (2000), λ 1 is set to 0.4 and λ 2 is set to Hereinafter, (1 + I f 2 + λ 1 C n (j, f)) λ 2 c is referred to as the actual consumption per adult, which is a comparable measure of consumption for agents of different household types. Utility flow from bequests is: (11) v(b, j, f) = (λ 3 j λ 4 )(1 + λ 5 (min{ j Jr, (b + κ)1 σ 1})λ6 Rs(f, j) + λ 7 Dc(f, j)), 1 σ where Rs(f, j) is the number of years a surviving spouse is expected to live after retirement if the husband dies in the next period. Dc(f, j) is the number of stacked years for all children to reach the age of 18 if the father dies in the next period. 19

21 Rs(f, j) and Dc(f, j) have the following forms: Rs(f, j) = jd f J jd f j jd f 1 ( s f (i))(1 s f (jd f ))(jd f j + 1) i j 1 } {{ } life expectancy max{jr j, 0} }{{} working periods if f 2 0 otherwise Dc(f, j) = C n (f, j) max{6 (C a (f, j) + 1)), 0}, where jd f denotes the wife s age of death. s f (j) is a one period survival rate of an age j female in the US period life table. λ 3 reflects the basic bequest motive that does not change over the life cycle, and is set to match the average life insurance face value of singles aged (age group 1). λ 4 represents the change in the bequest motive as people age, and is set to match the average life insurance face value of working-age (ages 22-66) singles. λ 5 and λ 6 jointly control the additional bequest motive to support surviving spouse s post-retirement consumption. Specifically, λ 5 captures the motive to support a wife who is already retired, and is set to match the average life insurance face value of married men aged without dependent children, whose wives do not work in the next period. λ 6 captures the extent to which the bequest motive is reduced by the number of years that a wife can work, and is set to match the average life insurance face value of working-age (22-66) married men without dependent children, whose wives still work in the next period. λ 7 controls the additional bequest motive to support under-age-18 consumption of children, and is calibrated to match the average life insurance face value of working-age (22-66) married men with dependent 20

22 children. κ > 0 measures the extent to which bequests are a luxury good, and is calibrated to match the difference in life insurance face value between working-age men with above median household income and working-age men with below median household income. The discount factor β represents the general incentive to save for the future, and is calibrated to match the average net worth of households with a male head aged (the last age group before retirement). Table 3: Calibrated Parameters Para. Value Targets Model(k) Data (k) κ 7.5k x of men aged above median income x of men aged below median income λ x of single men aged λ x of single men aged λ x of married men aged w./o. children λ x of married men aged w./o. children λ x of married men aged w. children β 0.92 a of households with a male head aged Notes: All targets are average values. The information about life insurance x is collected from the SIPP. The information about assets a is collected from the SCF. The SCF sample excludes observations in the top 10 percent of the wealth distribution. To better understand the bequest motive, Figure 5 displays the consumption equivalence of the first dollar bequest, which is the amount of annual actual consumption per adult that equalizes the marginal utility of consumption to the marginal utility of the first dollar bequest. As illustrated by the figure, the bequest motive has three attributes: 1. Bequest motive becomes stronger as people age (λ 4 > 0). The solid line marked with circles illustrates the life-cycle change in the bequest 21

23 Annual Consumption 1.8 x Married w/ children Married w/o children Single w/ children Single w/o children Age Figure 5: Consumption Equivalence of the First Dollar Bequest motive for a single man without dependent children. For this household type, the marginal utility of the first dollar bequest for a years old is 4.7 times greater than that for a years old. However, the growth in bequest motive becomes smaller as people age: the marginal utility of the first dollar bequest for a years old is only 1.3 times as that for a years old. This indicates that the bequest motive grows quickly during working periods, but it is quite stable after retirement. 2. Bequest motive varies by the characteristics of survivors: husbands have an additional bequest motive to support post-retirement consumption of wives (λ 5 > 0), but this motive is reduced by the number of years that a wife can work (λ 6 > 0); fathers have an additional bequest motive to support under-age-18 consumption of children 22

24 (λ 7 > 0). The difference between the solid line marked with circles and the solid line marked with crosses illustrates the additional bequest motive towards spouses. Of all age groups, married men aged have the greatest incentive to support their spouses retirement: their marginal utility of leaving the first dollar bequest is 2.1 times as that of a single man in the same age group. The difference in marginal utility becomes smaller for younger men, because their wives can work for a longer period. The difference also becomes smaller for older men, because their wives are expected to live for a shorter period. The difference between the solid line marked with circles and the solid line marked with diamonds shows the additional bequest motive towards children. For men aged 22-36, the marginal utility of leaving the first dollar bequest for single men with dependent children is about 2.5 to 2.7 times as large as that for single men without dependent children. The difference in marginal utility drops as agents age, and shrinks to almost zero for agents aged 52 and older. 3. Bequests are luxury goods (κ > 0). The annual consumption equivalence of the first dollar bequest ranges from $3.5k for married men aged with dependent children to $16.1k for single men aged without dependent children. 4 Benchmark Economy The section presents the benchmark economy. 23

25 4.1 Life Insurance Profiles Life Insurance Face Value 2.5 x Model Data Life Insurance Face Value 2.5 x Below Median, Model Below Median, Data Above Median, Model Above Median, Data Age (a) Average Age (b) By Income Group 2.5 x Model Data 2.5 x Model Data Target Life Insurance Face Value Life Insurance Face Value Age Age (c) Married with Children (d) Married without Children Figure 6: Life Insurance Holdings by Age Group 24

26 Figure 6(a) compares the life cycle profile of average life insurance holdings generated by the model with that in the data. The model correctly captures the inverse U-shape of life insurance face value: it starts at a low level of $24k, increases gradually as people age, reaches the peak level of $155k at age group 40-42, and declines afterwards. Note that the model peak is very close to the data peak of $153k reached at age group 43-45, although this is not a targeted moment. The initial increase is due to the rise in the bequest motive, and the later drop is attributed to the accumulation of risk-free assets. Survivors benefits largely affect life insurance demand, specifically, SBDC slow down the increase in life insurance holdings early in life, and survivors benefits for aged spouses accelerate the decline in life insurance holdings after retirement. Figure 6(b) displays life insurance face value separately for agents with above median household income and agents with below median household income. The model reasonably well matches the difference across income groups. In particular, it captures that the peak of life insurance holdings is reached later in life for low-income agents than for high-income agents. This is because the model correctly specifies the impact of SBDC on life insurance demand, and indeed both income groups have their greatest holdings of deceased contingent claims at the same age group of Figures 6(c) and 6(d), respectively, show the insurance profile for married men with dependent children and married men without dependent children. The model matches well both profiles, because it incorporates the additional bequest motive to support wives and children. Although the model is capable of matching average life insurance holdings, it overstates the percentage of people who purchase life insurance. 25

27 4.2 Consumption and Asset Profiles Annual Consumption x Age (a) Consumption Risk Free Assets x Age (b) Risk-Free Assets Figure 7: Life Cycle Profiles for Men Who Never Married Figure 7 presents the profiles of consumption and risk-free assets for men who have never married. This group of agents comprise about 13 percent of the population, and are selected to isolate the effect from changes in household types on consumption and assets. To control mortality differences, the presented profiles are constructed under a counterfactual assumption that all individuals live to 102 years old. The model is capable of producing a hump-shaped consumption profile: due to the binding borrowing constraint, consumption increases quickly among young agents; after accumulating positive assets, consumption continues to rise because agents are patient; consumption reaches its maximum around retirement, and drops slightly afterwards due to the absence of annuity markets. The profile of risk-free assets 26

28 is also hump-shaped: assets increase quickly during working periods, but decrease slowly after retirement due to the strong bequest motive among old individuals. 4.3 Bequests Left to Surviving Children 12 x All Social Security Payments Life Insurance Risk Free Assets 12 x All Social Security Payments Life Insurance Risk Free Assets Received Bequests Received Bequests Death Age of the Father (a) Low PP Type Death Age of the Father (b) High PP Type Figure 8: Received Bequests by Income Group Figure 8 plots the amount and the composition of bequests left by deceased fathers, separately for fathers from the high permanent productivity (PP) type and fathers from the low PP type. On average, bequests left by a high PP type father are three times as large as that left by a low PP type father. The difference in bequests across income groups is greater than the difference in earnings, because high-income individuals want to allocate a larger share of their resources into bequeathable wealth than do low-income individuals. SBDC constitute the primary source of bequests 27

29 for young and low-income fathers, and a secondary source of bequests for old and high-income fathers. 4.4 Mismatch Between Life Insurance Holdings and Financial Vulnerabilities Following Bernheim et al. (2003a,b), the degree of financial vulnerabilities is determined by comparing the next period actual consumption per adult when both spouses are alive with that when the husband deceases. The model defines that a wife is not financially vulnerable to the decease of husband if the event does not cause a drop in actual consumption per adult, and that a wife is significantly vulnerable to the decease of husband if the event causes a drop that is greater than 20 percent (hereinafter referred to as a significant drop in consumption ). The method for simulating financial vulnerabilities is described in Appendix D. Table 4: Life Insurance Holdings of Working-Age Husbands and Financial Vulnerabilities of Their Wives Life ins. ($k) Needed life ins. ($k) Significant drop in con. (%) (1) (2) (3) All Low PP High PP In Table 4, column (1) reports the amount of insurance generated by the model; column (2) reports the amount of insurance that is needed to eliminate financial 28

30 vulnerabilities; and column (3) reports the percentage of wives who are significantly vulnerable to the death of husbands. As illustrated by the table, the model is capable of generating a mismatch between life insurance holdings and financial vulnerabilities: average life insurance holdings are 20 percent less than the needed amount to eliminate financial vulnerabilities, and about 9 percent of wives would experience a significant drop in consumption if the husband deceases. The degree of financial vulnerabilities varies a lot according to the husband s characteristics: young and low-income husbands are more likely to purchase insufficient life insurance than old and high-income husbands are. The above patterns about financial vulnerabilities are consistent with the data patterns documented in Bernheim et al. (2003a). However, the degree of financial vulnerabilities produced by the model is smaller than that found in the data, which means there may be other behavioral factors that are not incorporated in the model lead individuals to under-insure against the future risk of death. 4.5 Allocation between Consumption and Bequests Figure 9(a) displays the average marginal benefit of additional consumption and the average marginal cost of exchanging deceased contingent claims for additional consumption (refer to Equation (5)). As demonstrated by the figure, the average marginal benefit of consumption is always greater than the average marginal cost, which indicates some agents are forced to hold more deceased contingent claims than they desire. To control the difference in marginal utility of consumption across income groups 29

31 14 x u c β(1+r)(1+τ c )v b All Low PP High PP Marginal Utility Age (a) Benefits and Cost of Additional Consumption (u c β(1+r)(1+τc )v b )/u c Age (b) Normalized Deviations Figure 9: Deviations from the Optimal Allocation between Consumption and Bequests and over life cycle, Figure 9(b) presents a normalized measure of this deviation ( u c(c,j,f) β(1+r)(1+τ c )v b (b,j,f) ), separately for low PP and high PP types. As shown by u c (c,j,f) this figure, in the early stage of life, the low PP type deviates more from the optimal allocation than does the high PP type. This is because low-income young fathers are very likely to demand zero life insurance given the high ratio of SBDC to average earnings. While in the late stage of life, the result is reversed, because the high PP type is more likely to demand annuities (have zero life insurance) given the low ratio of OAI benefit to previous earnings. 30

32 Table 5: Internal Rates of Return (%) OAI OAI OAI+SBDC (1) (2) (3) (4) (5) (6) 25th th th At least 67 years old Yes Yes No No No No Differential mortality by income Yes No Yes No Yes No 4.6 Social Security Progressivity Due to the difficulty of calculating lifetime earnings for those who decease at an early age, past studies on social security progressivity (for example, see Goda et al. (2011) and Bosworth and Burke (2014)) use a sample who survive to a relatively old age. Due to this sample limitation, previous studies do not consider the value of SBDC. To overcome these two issues, this paper calculates the measure of progressivity using information on simulated agents. Table 5 reports the internal rate of returns for individuals at the 25, 50, and 75 percentile in the distribution of lifetime earnings. Lifetime earnings are the average of (conterfactual) earnings from ages 22 to 66 regardless whether a person actually survive. Internal rates of returns are calculated to equalize the present value of a 10.6 percent labor tax (the portion used to fund the OAI and survivors insurance trust fund) to the present value of benefits. The first two columns report the internal rate of return for OAI benefits among a sub-sample of individuals who survive to the age of 67. These results are comparable to the traditional approach that excludes 31

33 individuals who do not accumulate enough earning history. Similar to previous studies, this paper finds that the OAI program is progressive and adding differential mortality by income group reduces the degree of progressivity. Columns (3) and (4) repeat the same exercise for the whole population. This sample change substantially reduces the internal rate of return, because a non-trivial number of individuals do not survive to the retirement age and receive zero OAI benefits. Moreover, due to differential mortality by income group, this change has a greater effect on people at the 25 percentile of the lifetime earning distribution than people at the 75 percentile, which indicates the traditional approach of restricting the analysis to a sub-sample who survive to a relatively old age overstates the progressivity of OAI. Columns (5) and (6) add the SBDC and recalculate the internal rate of return. Incorporating SBDC raises the internal rate of return, and the increase is greater at the 25 percentile than at the 75 percentile, which indicates adding SBDC raises the social security s progressivity. 9 5 Policy experiments This section implements policy experiments to study the long-term consequences of reforming SBDC. 9 Note that all calculations are based on the model s structure, and the results should be interpreted in a qualitative way. 32

34 5.1 Removing SBDC The first experiment removes SBDC. To maintain a balanced budget, this experiment also reduces the labor tax rate by 0.15 percentage point. Column (2) of Table 6 reports the changes relative to the benchmark economy. On the agents side, removing SBDC does not affect much average consumption or average holdings of risk-free assets, but it raises average life insurance face value by $28.8k. As mandatory holdings of deceased contingent claims decline, the allocation between consumption and bequests is improved. This policy reduces the residual of Condition (5) by 20.5 percent. Nevertheless, this policy slightly increases the marginal utility of current-period consumption among those who have a binding borrowing constraint in the benchmark economy, and worsens the intertemporal allocation between current consumption and future consumption. On the survivors side, removing SBDC reduces the average bequests left to widows with dependent children by 1.0 percent. The degree of financial vulnerabilities among widows with dependent children is different from that among wives with dependent children for two reasons. First, due to differential mortality by income group, a surviving widow is more likely to have a low-income husband and be financially vulnerable towards the husband s death. Second, due to rising mortality rates with age, a surviving widow is more likely to have an old husband and be financially secured towards the husband s death. Overall, the second difference dominates. In the benchmark economy, only about 1.8 percent of surviving widows with dependent children suffer from a significant drop in consumption. Removing SBDC substantially raises this number to 6.6 percent. 33

35 Table 6: Comparison between Economies Benchmark Removal Optimal allocation Optimal size (1) (2) (3) (4) Agents Annual consumption 32.76k 0.04% -0.01% -0.27% Life insurance 85.99k 33.47% -6.08% % Risk-free assets k 0.08% 0.05% -0.62% u (c) β(1 + r)(1 + τ c )v (b) % 24.23% % u (c) (k = 0) % -0.12% 0.74% Surviving widows w/ children Received bequests k -1.00% 2.19% 17.73% Significant drop in con. 1.76% % % % CEV Agents -0.20% 0.04% 0.18% Low PP type -0.38% 0.06% 0.45% High PP type 0.07% 0.01% -0.20% Widows w/ children -4.87% 3.73% 23.30% Notes: Column (1) describes the benchmark economy. Columns (2)-(4) report changes relative to the benchmark economy. To estimate the welfare influence, consumption equivalence variations (CEV) are calculated by asking the percentage change in all-period consumption that is needed for a new individual of one group in the benchmark economy to be indifferent between living in the benchmark economy and in the new economy. A positive number indicates welfare gains. Results suggest that the removal of SBDC induces welfare losses that are equivalent to a 0.2 percent drop in all-period consumption for agents. A newborn individual prefers to live in an economy with SBDC: the gain from redistribution and better intertemporal allocation dominates the loss from distorting the consumption/bequest allocation. The welfare loss for widows with dependent children is equivalent to a 4.9 percent drop in all-period consumption The life-cycle problem solved by a surviving widow with dependent children is defined in ap- 34

36 The loss for widows is much greater than that for agents, because SBDC significantly alleviates the mismatch between life insurance holdings and financial vulnerabilities. Last, examining heterogeneity by income indicates that the removal of SBDC benefit the high PP type agents at the cost of the low PP type agents. 5.2 Optimal Allocation In order to find an optimal allocation that maximizes agents welfare, the second experiment studies a set of policies that keep the total spending of SBDC unchanged, but adjust the allocation of SBDC across age groups. Specifically, this experiment replaces Sc(e, s j, f), described in Equation (8), with the following one: Sc(e, s j, f) = max{a 1 + a 2 (j 1), 0}P IA(e)C n (f, j)i Ca(f,j)+1<6[ + (1 + r) (6 (Ca(f,j)+1)) (C a (f, j) C a (f, j))], 5 (C a(f,j)+1) a j =0 (1 + r) a j where a 1 represents the starting level of the ratio of per period benefit for each surviving child to PIA (hereinafter referred to as the benefit to PIA ratio ). a 2 represents the change of the benefit to PIA ratio at each subsequent age group. Under the current rules, the benefit to PIA ratio is kept unchanged at As shown in Figure 10(a), on average, each 100 percentage point increase in a 1 can be funded by a 17.5 percentage point decrease in a 2. Figure 10(b) displays the relationship between the benefit to PIA ratio for age group 1 (a 1 ) and the ex-ante utility of agents. Compared to the benchmark economy, pendix D. 35

37 a newborn agent prefers to live in an economy that provides more generous SBDC to young fathers. Under the optimal policy that maximizes ex-ante utility, the benefit to PIA ratio starts at a high level of 192 percent for fathers aged 22-24, drops by 20 percentage points at each subsequent three years of age, and reaches the low level of zero for fathers aged 49 and older. Since on average the PIA replaces 40 percent worker s earnings, this optimal allocation would allow each surviving child to receive about 70 percent (instead of the 30 percent in the benchmark economy) of the deceased father s average earnings if the father dies before the age of 30. The amount of SBDC drops quickly as the father becomes older, and is smaller than that in the benchmark economy if the father deceases after the age of 40. % Change at Each Subsequent Period (a 2 ) Policy Experiments Current Level Optimal Level Benefits to PIA Ratio for Age Group 1 (a 1 ) Ex Ante Utility Policy Experiments Current Level Optimal Level Benefits to PIA Ratio for Age Group 1 (a 1 ) (a) Policy Candidates (b) Ex-Ante Utility Figure 10: Adjust Allocation Across Age Groups As reported in Column (3) of Table 6, compared to the benchmark economy, the policy that achieves the optimal allocation of SBDC has almost zero effects on av- 36