Bequests and Heterogeneity in Retirement Wealth

Transcription

1 Bequests and Heterogeneity in Retirement Wealth Fang Yang University at Albany - SUNY June Abstract The data show large dispersion in households wealth holdings at retirement. In addition, the empirical correlation between household lifetime earnings and retirement wealth is much lower in the data than in many quantitative models. This paper quantifies and analyzes the implications of a life cycle model with intergenerational links (in the form of voluntary bequest motives and intergenerational transmission of earnings ability) that also explicitly allows for defined benefit pensions, history-dependent social security, and a government-provided minimum consumption floor. The key findings are that this model matches well the observed wealth differences at retirement and their correlation with lifetime incomes, and that intentional bequests play a key role in generating those results. Keywords: Wealth Inequality; Retirement; Earnings Shocks; Bequest; Minimum Consumption; Social Security. JEL Classification: E21; J14 I am grateful to my thesis advisors, Michele Boldrin and Mariacristina De Nardi, for numerous suggestions and continuous encouragement. I would like to thank John Boyd, V. V. Chari, Betty Daniel, Zvi Eckstein, John B. Jones, Larry Jones, and seminar participants at various seminars and conferences for helpful comments and suggestions. The research reported herein was pursuant to a 2007 Sandell grant from the U.S. Social Security Administration (SSA) funded as part of the Retirement Research Consortium (RRC). The findings and conclusions expressed are solely those of the author and do not represent the views of SSA, any agency of the Federal Government, or the RRC. Mailing Address: Department of Economics, BA 109, University at Albany, State University of New York, Albany, NY fyang@albany.edu. URL: fy

2 1 Introduction Many papers document that households with similar characteristics, such as lifetime income, age, and family structure, hold very different amounts of wealth at retirement (see, among others, Hurst et al. (1998), and Grafova et al. (2006)). In particular, Hendricks (2007a) documents that the correlation coefficient between lifetime earnings and wealth at retirement (0.61) is much less than unity. He shows that substantial wealth differences remain after controlling for lifetime earnings and age: The average of the Gini coefficients in wealth at retirement across lifetime earnings deciles is 0.54, compared with 0.62 in the full sample. Various economists (see, for example, Bernheim et al. (2001), and Hendricks (2007a)) argue that these features of the data are inconsistent with most life-cycle models of consumption-saving behavior, and thus constitute a challenge to such theories and their policy implications. In this paper I analyze the relationship between lifetime earnings and wealth at retirement in a model with the following features: voluntary and accidental bequests and intergenerational productivity transmission, governmentprovided minimum consumption, defined benefit pensions and a more realistic modeling of Social Security rules. Each of those features has been shown to be important to understanding the saving behavior and wealth distribution in the US. 1 However, little is known about how those features affect the wealth at retirement. I find that a model with those features matches well the observed data. The correlation coefficient between lifetime earnings and retirement wealth (0.75) is much lower than unity. Substantial wealth differences remain after controlling for lifetime earnings and age: The average of the Gini coefficients (0.53) is very high. In this model, households are ex-ante identical in wealth. 2 Wealth dif- 1 See, for example, the discussion of bequests in De Nardi (2004), consumption floor in Hubbard et al. (1995), and history dependence of Social Security in Scholz et al. (2006). 2 I assume limited sources of heterogeneity, and abstract from ex-ante heterogeneity in wealth holdings by race (see for example Smith (1995), and Altonji and Doraszelski (2005)), by education (see for example Hubbard et al. (1995), and Cagetti (2003)), by preferences (see for example Krusell and Smith (1998), Samwick (1998), and Hendricks (2007b)), and by selfcontrol (see for example Ameriks et al. (2007)). I also abstract from ex-post heterogeneity 2

3 ferences at retirement remain, after controlling for lifetime earnings, for the following two reasons. First, when borrowing constraints prevent households from smoothing consumption intertemporally, households that differ in the timing of earnings over the life cycle will have levels of retirement wealth that differ substantially. Secondly, inheritance adds another source of wealth heterogeneity among households with similar lifetime earnings. The bequest motive in the benchmark model increases the wealth of some households. In addition, since the calibrated bequest motive implies that bequests are a luxury good, it also increases the desire to leave bequests for the households receiving bequests, thus increasing their saving rate, which leads for more wealth inequality and less correlation of wealth to lifetime earnings. I compare the benchmark model with three models without bequest motives that differ in distributing inheritances: evenly distributed accidental bequests at age 50; random inheritances at age 50 as in Hendricks (2007a); random inheritances at ages 35 to 55. Those models without bequest motives generate a very tight relationship between lifetime earnings and retirement wealth, although adding random inheritances reduces it: The correlation coefficients are 0.86, 0.81, and 0.80, respectively, and the average of the Gini coefficients are 0.40, 0.46, and 0.46, respectively. This comparison shows that bequest motives are important in explaining the discrepancies between model and data. The model with bequest motives and without intergenerational transfer of productivity reduces the correlation coefficient to 0.73 from 0.75 in the benchmark model. The main reason is that, the distribution of inheritance received is independent of households earnings, which strength wealth holdings from inheritance among lower earnings deciles, thus reducing the correlation coefficient between lifetime earnings and retirement wealth. I then perform policy experiments to investigate the distributional effects of means-tested minimum consumption programs, Social Security, and pensions. in wealth holdings by marital status (see for example Cubeddu and Rios-Rull (2003), and Guner and Knowles (2007)), by children (see for example Scholz and Seshadri (2007)), by rate of return (see for example Guvenen (2006)), by health expenditure (see for example De Nardi et al. (2010)). 3

4 Means-tested minimum consumption programs reduce the buffer stock needs among income-poor households and increase wealth inequality. In a model with no consumption floor, income-poor households have much more incentive to save to insure against bad income shocks. This tends to decrease wealth inequality. Due to the persistence of earnings, those low-income households are more concentrated in low lifetime earnings deciles. As a result, removal of the government-provided minimum consumption level reduces the average of the Gini coefficients considerably to Pensions and Social Security have very different effects on the joint distribution of retirement wealth and lifetime earnings. Removing private pensions raises the correlation coefficient because it raises average wealth for households in the high earnings deciles. The Gini coefficients decrease a lot at the higher earnings deciles, since those people now increase saving for retirement. I then further remove history-dependent Social Security. The Gini coefficients increase at the lower earnings deciles since wealthy households increase saving while poor households, due to government transfers, do not increase saving. The Gini coefficients at the higher earnings deciles decrease since poor households increase saving relatively more than rich households. Adding equal Social Security decreases the Gini coefficients at the lower earnings deciles since poor households, having become disqualified for government transfers, increase saving. The Gini coefficients at the higher earnings deciles increase since poor households decrease saving relatively more than rich households. This paper builds on a large literature that studies the economy-wide wealth inequality (see, among others, Huggett (1996), Quadrini (2000), Casteneda et al. (2003), De Nardi (2004), and Cagetti and De Nardi (2006)). In particular, this paper is related to several other studies on retirement saving. Engen et al. (1999), Engen et al. (2004), and Scholz et al. (2006) study the adequacy of household retirement saving. Those papers abstract from the intergenerational links of bequests and earnings ability. Gokhale et al. (2001) abstract from voluntary bequest motives. Hendricks (2007a) assumes that households receive bequests at the same age and there is neither a bequest motive, intergenerational transmission of earnings ability, nor 4

5 a government-provided consumption floor. In my model, households expect and receive different amounts of inheritances at different times and hold on more wealth due to intentional bequest motives, which generates more heterogeneity in retirement wealth among households with similar lifetime earnings. A consumption floor further discourages wealth accumulation for low-income households. The paper is organized as follows. Section 2 presents some empirical results documenting the relationship between wealth at retirement and lifetime earnings. Section 3 presents the model and the calibration of the model. Section 4 shows the quantitative results of the benchmark model and investigates the role of bequests and inheritance heterogeneity. Section 5 conducts sensitivity analysis. Section 6 performs policy experiments. Section 7 concludes. 2 Empirical Findings This section presents some empirical results from Hendricks (2007a) showing the relationship between wealth at retirement and lifetime earnings. Hendricks (2007a) uses households data from the Panel Study of Income Dynamics (PSID) on wealth reported at age 65. Earnings consist of labor income (net of income tax payments and Social Security contributions) received by both the household head and the spouse, which include wages, salaries, bonuses, overtime payments, and the business part of labor income. Lifetime earnings are the present value of earnings between the ages of 18 and 65, discounted to age 65. Wealth at retirement includes financial wealth, real estate, business wealth, vehicles, life-insurance policies, private annuities, IRAs, trusts, and other miscellaneous assets minus debts. Wealth at retirement is positively but not perfectly correlated with lifetime earnings: The correlation coefficient between lifetime earnings and wealth at retirement is Substantial wealth differences remain after controlling for lifetime earnings and age: Households with similar lifetime earnings hold diverse amounts of wealth at retirement age. Table 1 displays the Gini coefficient of retirement wealth for each lifetime 5

6 earnings decile. We observe that controlling for age and lifetime earnings reduces wealth inequality: The average of the Gini coefficients in retirement wealth across lifetime earnings deciles is 0.54, compared with 0.62 in the full sample. However, at all levels of lifetime earnings there is large dispersion in wealth at retirement: The Gini coefficients are all above 0.4. The degree of wealth inequality declines with lifetime earnings decile until the last two deciles. 3, 4 Earnings decile mean Table 1: Gini coefficient of retirement wealth by lifetime earnings decile in PSID (Hendricks (2007a)) The findings that retirement wealth and lifetime earnings are not perfectly correlated and that households with similar lifetime earnings hold diverse amounts of wealth at retirement age constitute a challenge to our theories of saving behavior. 3 The Benchmark Model The model is a discrete-time overlapping generations economy with an infinitelylived government. 5 I focus on a stationary equilibrium in which the interest rate is constant at r and the wage rate is at w. Households face uninsurable 3 Similar findings arise from Health and Retirement Study (Venti and Wise (2000)). To check the robustness of those findings, Hendricks (2007a) restricts samples to contain only households with similar characteristics. The main findings are not affected by household characteristics such as numbers of children, marital breakups, self-employment, or stock holdings. 4 Survey of Consumer Finances has a better coverage of high-earner and high-wealth households than PSID and Health and Retirement Study. However, as is shown in Juster, Smith and Stafford (1999), and Cagetti (2003), for lower quartiles, those data sets give similar information. 5 The major difference between my model and De Nardi (2004) is that I add a governmentprovided minimum consumption level, defined benefit pensions, and a more realistic modeling of Social Security rules. Another difference is that, in my model children have full information about their parents while in her model children only observe parent s income shock once. 6

7 idiosyncratic earnings shocks and the only financial instrument is a one-period bond. Net assets must be non-negative. There are no private annuity markets to insure against mortality risk. Parents and their children are linked by bequests and intergenerational transmission of earnings ability. Each agent enters the economy at age 20 and starts consuming, working, and paying taxes. After retirement, the agent no longer works but receives benefits from defined benefit plan provided by the firm and from Social Security provided by the government Government and Defined Benefit Plan Government taxes capital and labor income at rates τ a and τ l, respectively, to finance government spending and government transfers, and taxes earnings at the rate τ ss to finance Social Security. The Social Security benefits that agents receive, P (ỹ), are linked to their average annual earnings ỹ. There is one representative firm producing goods according to the aggregate production function F (K; L), where K is the aggregate capital stock and L is the aggregate labor input. The firm maintains a defined benefit plan, which is financed by contributions on each worker s behalf at the rate τ DB, such that wage rate w equals F 2 (K; L)/(1 + τ DB ). Pension benefit, DB(ỹ), is linked to each individual s average annual earnings Demographics, Preferences, and Labor Productivity Each period in the model is assumed to be 5 years. Agents start their economic life at the age of 20 (age t = 1). At the beginning of period 4, the agent s children are born. The agents retire at t = 10 (when they are 65 years old). From t = 10, each person faces a positive probability of dying, given by (1 p t ), which is exogenous and independent of other household characteristics. 8 6 In the model, defined contribution pension wealth is treated as wealth accumulated through household saving. 7 Scholz et al. (2006) propose an alternative model of defined benefit plan, in which pension benefits are linked to individuals average earnings in the past 5 year before retirement. 8 I make the assumption that people do not die before age 65 to reduce computational time. This assumption does not affect the results since in the US the number of adults dying 7 The

8 maximum life span is T = 14 (before turning 90 years old). The population grows at a constant rate n. I assume that preferences are time separable with a constant discount factor β. Agents derive utility from consumption and from bequests transferred to their children upon death. 9 The period utility function from consumption is given by U(c) = (c 1 γ 1)/(1 γ). Following De Nardi (2004), the utility from bequests b is denoted by φ(b) = φ 1 [(b + φ 2 ) 1 γ 1], in which the term φ 1 measures the strength of bequest motives, and φ 2 reflects the extent to which bequests are luxury goods. 10 The total productivity of worker i at age t is given by y i t = e zi t +ɛ t, in which ɛ t is the deterministic age-efficiency profile. The process for the stochastic productivity shock z i t is: z i t = ρ z z i t 1 + µ i t, µ i t N(0, σ 2 µ). To capture the intergenerational correlation of earnings, I assume that productivity of worker i at age 55 is transmitted to children j at age 20 as follows: z j 1 = ρ h z i 8+ν j, ν j N(0, σ 2 h ). 3.3 The Household s Recursive Problem I assume that children have full information about their parents state and infer the size of the bequests they are likely to receive based on this information. The household s state variables are given by x = (t, a, z, ỹ, S p ), where a denotes the agent s financial assets carried from the previous period. The last term S p denotes the parent s state variables. From t = 1 to t = 9 (from 20 to 60 years of age), the agent does not face any survival uncertainty. Let V I (t, a, z, ỹ) and V (t, a, z, ỹ, S p ) denote the value function of an agent whose parent is dead and is alive, respectively. (1) V (t, a, z, ỹ, S p ) = max U(c) + βp t+7 E [ V (t + 1, a, z, ỹ, S p) ] c,a +β(1 p t+7 )E [ V I (t + 1, a + a p/n, z, ỹ ) ] before age 65 is small. 9 Note that this form of impure bequest motive implies that an individual cares about the total bequests left to his/her children, but not about the consumption of his/her children. 10 If φ 2 > 0, the marginal utility of small bequests is bounded, while the marginal utility of large bequests declines more slowly than the marginal utility of consumption. 8

9 subject to (2) (3) (4) (5) (6) c + a = (1 τ l τ ss )wy + [1 + r(1 τ a )]a, a 0, ỹ = [ (t 1)ỹ + wy/5 ] /t, { [ (t + 6)ỹp + wy p /5 ] /(t + 7) if t < 3 ỹ p = ỹ p a p = a p(s p ). otherwise } where N is the average number of kids determined by the growth rate of the population. The expected value of the value function is taken with respect to (z, z p), conditional on (z, z p ). The agent s resources depend on labor endowment y, and asset holdings a. Average earnings for children and parents evolve according to equations (4) and (5), respectively. Equation (6) shows the law of motion of assets for the parents from their decision rule. The value function of an agent whose parent is dead is defined as (7) V I (t, a, z, ỹ) = max U(c) + βe [ V I (t + 1, a, z, ỹ ) ] c,a subject to (2), (3), and (4). From t = 10 to t = 14 (from 65 to 85 years of age), the agent does not work any more. The agent faces a positive probability of dying, in which case the agent derives utility from bequeathing his/her assets. The agent receives Social Security benefits and pensions from defined benefit plan. Following Hubbard et al. (1995), I assume that government transfers T r provide a consumption floor c, as is specified in equation (10). (8) V (t, a, ỹ) = max c,a U(c) + βp t V (t + 1, a, ỹ) + (1 p t )φ(a ) subject to (3) and (9) (10) c + a = [1 + r(1 τ a )]a + (1 τ l )[P (ỹ) + DB(ỹ)] + T r, T r = max{0, c [1 + r(1 τ a )]a (1 τ l )[P (ỹ) + DB(ỹ)]}. 9

10 3.4 Calibration Parameters Value Demographics n annual population growth 1.2% p t survival probability see text Preference γ risk aversion coefficient 1.5 Labor productivity ɛ t age-efficiency profile see text ρ z AR(1) coef. of 5-year prod. process 0.85 σµ 2 innovation of 5-year prod. process 0.30 ρ h AR(1) coef. of prod. inheritance process 0.40 σh 2 innovation of prod. inheritance process 0.37 Government policy τ a tax on capital income 20% c minimum consumption 0.21 τ l tax on labor income 21.7% P (ỹ) Social Security benefit see text τ ss Social Security tax 10.0% Firm DB(ỹ) pension from defined benefit plan see text τ DB contribution rate to defined benefit plan 3.9% r interest rate 5% Table 2: Parameters used in the benchmark model Table 2 lists the parameters taken from other studies. 11 I set the population growth rate, n, to the average value of population growth from 1950 to 1997 from the Council of Economic Advisors (1998). The p t s are the vectors of conditional survival probabilities for people older than 65 and is set to the survival probabilities for people born in 1965 (Bell et al. (1992)). I take the risk aversion coefficient, γ, to be 1.5. The deterministic age-profile of labor productivity ɛ t is taken from Hansen (1993). 12 The persistence ρ z and variance σ 2 µ of the stochastic productivity process are estimated from PSID data (Altonji and Villanueva (2002)). 13 The persistence is low and variance is high because this refers to income in a 5- year period. I take the persistence ρ h of the productivity inheritance process 11 I report parameters at annual frequency, unless stated otherwise. 12 Since I impose mandatory retirement at the age of 65, I set ɛ t = 0 for t > De Nardi (2004) provides a detailed discussion of the estimation process. 10

11 from Zimmerman (1992) and Solon (1992), and the variance σh 2 from De Nardi (2004). The capital income tax τ a is set to be 20% (Kotlikoff et al. (1999)). The minimum consumption floor c is 0.21 of average household income. 14 Tax rate on labor τ l is set at 21.7% so that the ratio of government spending to output is 0.18 (Council of Economic Advisors (1998)). The Social Security benefit P (ỹ) resembles the Old Age and Survivor Insurance component of Social Security system and is set as follows: P (ỹ) = 0.9min(ỹ, 0.2)+0.32max(0, min(ỹ, 1.24) 0.2)+0.15max(0, min(ỹ, 2.47), 1.24). In this formula, the bend points, expressed as average earnings, and marginal rates are taken from Huggett and Ventura (2000). Tax on labor income τ ss is set at 10.0% to balance Social Security budget. The defined benefit is equal to DB(ỹ) = 0.49max[0, min(ỹ, 0.72) 0.58] max[0, min(ỹ, 0.88) 0.72] +0.28max[0, min(ỹ, 1.06) 0.88] max[0, min(ỹ, 1.28) 1.06] +0.16max[0, min(ỹ, 1.59) 1.28] max[0, min(ỹ, 2.12) 1.59] +0.02max[0, min(ỹ, 3.23) 2.12]. The bend points, expressed as average earnings, and marginal rates are chosen to match the holding of defined benefit wealth relative to Social Security wealth by lifetime earnings from Health and Retirement Study reported in Scholz et al. (2006). Contribution rate τ DB to defined benefit plan is 3.9% to balance pension budget. The annual interest rate is set at 5%. I choose parameters β, φ 1, φ 2 to match the ratio of wealth to after-tax earnings (Hendricks (2007a)), bequest-wealth ratio (Gale and Scholz (1994)), and 90th percentile of bequest distribution normalized by income (Hurd and 15, 16 Smith (2002)). 14 In 1992, the consumption floor was $8, 159 (Scholz et al. (2006)) and the average income was $38, 840 (Census Bureau). Hubbard et al. (1995) use a slightly higher value. 15 I use the distribution for single decedents instead of the one for all decedents. As is argued in De Nardi (2004), typically a surviving spouse inherits a large share of the estate, consumes part of it, and only leaves the remainder to the couple s children. 16 Discount factor affects saving and average wealth in the economy. The term φ 1 measures 11

12 Moment Data Model Wealth-earnings ratio Bequest-wealth ratio th percentile of bequest distribution Parameters Value β discount factor 0.96 φ 1 weight of bequests in utility function φ 2 shifter of bequests in utility function 16.5 Table 3: Parameters calibrated from the model To better gauge the quantitative implications of the model, I begin evaluating the exogenous earnings process that I feed into the model. Table 4 compares values for the lifetime earnings distribution in the model with those in the PSID. The earnings process used in the model generates a skewed distribution of lifetime earnings comparable with the data. 17 Percentile (%) Gini PSID a All models a From Hendricks (2007a) Table 4: Lorenz curve of lifetime earnings 4 Numerical Results I will study the benchmark model and the quantitative role of bequests motives and the distribution of inheritance. I first present the benchmark model. I then investigate the role of bequests motives by comparing three models without bequest motives that differ in distributing inheritances. 18 First, to see how much wealth inequality can be generated by the life-cycle structure when the strength of bequest motives, thus I choose the aggregate bequest as a moment. The term φ 2 reflects the extent to which bequests are luxury goods, affecting the bequest distribution, especially the high end of it. 17 The earnings process used in the model also produces lifetime earnings at each decile similar as those reported in the PSID. Results are available from the author upon request. 18 Details about computing the benchmark model are provided in the Appendix. 12

13 only earnings uncertainty is activated, I turn off bequest motives and assume accidental bequests are equally redistributed among 50-year-old people. The second experiment adds random inheritances uncorrelated with earnings at age 50 as in Hendricks (2007a). Third, I further make the time of inheritances random. Finally, I keep bequest motives but change intergenerational persistence in earnings which endogenously change the distribution of inheritance. In particular, if there is no intergenerational link of earnings, inheritances are evenly distributed by lifetime earnings decile. In each experiment, I solve for the equilibrium and then simulate 100,000 households starting from age 20 drawn from the initial distribution. I define retirement wealth to be the wealth at age 65, and lifetime earnings to be the total earnings from ages 20 to 60, discounted to age Wealth Distribution Table 5, row one, shows values for the wealth distribution at retirement. In the PSID, wealth at age 65 is highly unevenly distributed. The benchmark model with intergenerational links and bequest motives generates a skewed retirement wealth distribution that is comparable with the data, except for the top 1% of the wealth holding. The three models without intergenerational links and bequest motives fail to replicate a skewed retirement wealth distribution. Compared with the benchmark, removing of intergenerational persistence in earnings (ρ h = 0) reduces wealth inequality slightly. Table 5, row two, reports values for the wealth distribution for the whole economy. Wealth for the whole economy is more unevenly distributed than wealth at retirement both in the data and in all models. This shows that a large amount of wealth dispersion in the economy is due to differences in age. Those three models without intergenerational links and bequest motives fail to generate a skewed wealth distribution. The benchmark model with intergenerational links and bequest motives generates a skewed wealth distribution that is comparable with the data. Higher intergenerational persistence of earnings ability leads to more wealth accumulation across generations and increases 13

14 Percentile (%) Age Gini PSID (Hendricks (2007a)) Benchmark model Change bequest motives and inheritance Equal inher. at age Random inher. at age Random inher. at ages Change intergenerational persistence in earnings ρ h = ρ h = All PSID (Hendricks (2007a)) Benchmark model Change bequest motives and inheritance Equal inher. at age Random inher. at age Random inher. at ages Change intergenerational persistence in earnings ρ h = ρ h = Table 5: Lorenz curve of wealth wealth inequality Wealth Inequality and Lifetime Earnings I now look at the relationship between retirement wealth and lifetime earnings Benchmark Model The benchmark model generates a relationship between lifetime earnings and retirement wealth close to the data. The correlation coefficient between retirement wealth and lifetime earnings of 0.75, although a bit higher than that in the data (0.61), is quite low. Substantial wealth difference remains after 19 De Nardi (2004) provides a detailed discussion of the effects of intergenerational links on wealth distribution. 14

15 controlling for lifetime earnings and age: The average of the Gini coefficients (0.53) is almost as high as the data counterpart. Earnings decile mean PSID a Model a From Hendricks (2007a) Table 6: Gini coefficient of retirement wealth by lifetime earnings decile Table 6 illustrates the Gini coefficients of retirement wealth for each lifetime earnings decile. We notice two important features. First, after controlling for age and lifetime earnings, there is still large wealth inequality in the benchmark model: All the Gini coefficients are above Second, the degree of wealth inequality declines as lifetime earnings increases, as is observed in the data. To better gauge the amount of wealth dispersion at retirement generated by the benchmark model, Figure 1 compares the retirement wealth distributions for the 2nd, 5th, and 9th lifetime earnings deciles in the model with those in PSID (Hendricks (2007a)). The model successfully replicates the fact that households with similar lifetime earnings hold diverse amounts of wealth. At each lifetime earnings decile, households in the lower wealth deciles hold very little wealth while households in the higher wealth deciles hold much more wealth. Compared with the data, the model overstates the high wealth percentiles at the 2nd earnings decile but overstates low wealth percentiles at the 9th earnings decile. In the benchmark model, retirement wealth inequality arises among household with similar lifetime earnings because households differ in the timing of earnings over the life cycle and in the amount and timing of inheritance received. Let us now try to understand the role of inheritance heterogeneity in the heterogeneity in retirement wealth. The model with intergenerational links of bequests and earnings ability endogenously generates differences in the timing and amount of inheritance. To see how large the variation in inheritance is, Table 7 reports values for 15

16 Wealth at retirement th earning decile (benchmark) 5th earning decile (PSID) 2nd earning decile (benchmark) 2nd earning decile (PSID) Wealth at retirement Benchmark PSID Wealth percentile (a) 2nd and 5th lifetime earnings deciles Wealth percentile (b) 9th lifetime earnings decile Figure 1: Cumulative distribution of wealth (normalized by average household aftertax earnings) the discounted lifetime inheritance distribution. In the PSID, inheritances are highly unevenly distributed with a Gini coefficient of % of the households receive very little or no inheritance. The top 1% of the households receive 35% of all the inheritances. The model generates a skewed inheritance distribution that is comparable with the data. Treating bequests as luxury goods and modeling the transmission of earnings ability across generations are essential to match the observed skewness in the inheritance distribution. First, the marginal utility from bequeathing is finite at zero bequests, which helps to generate a large fraction of households without any inheritance. Secondly, some large inheritances are transmitted across generations because of the voluntary bequests. Because the marginal utility of bequests declines more slowly than the marginal utility of consumption, the richest households have strong bequest motives to save some assets for their children even when very old. When there is a positive correlation between parents and children in earnings, their offspring are more likely to be earnings-rich and tend to leave more wealth to their offspring, thus generating a skewed inheritance distribution. Percentile (%) Gini PSID (Hendricks (2007a)) Benchmark model Table 7: Lorenz curve of inheritance distribution 16

17 Table 8 shows the fraction of lifetime inheritance received by households in each lifetime earnings decile. In the PSID, there is a positive correlation between lifetime inheritance and lifetime earnings. The benchmark model generates an increasing relation between inheritance and lifetime earnings. Modeling the transmission of earnings ability across generations and a highly correlated lifetime earnings process is key in generating this pattern. The monotonicity relation is weaker in the data. This might due to the fact that, in reality individuals differ by numbers of siblings and marital status and each married couple might receive inheritances multiple times. More inheritances at the lowest earnings decile might help the model to generate less correlation between retirement wealth and earnings. Earnings decile PSID (Hendricks (2001)) Benchmark model ρ h = Table 8: Fraction of inheritance received by lifetime earnings decile (%) Now I show the effect of inheritance heterogeneity on retirement wealth. The upper panel in Table 9 shows extra wealth holding at selected percentile among those who did inherit, compared with those who did not, in selected lifetime earnings deciles. At each earnings decile, those who never inherited hold less wealth than those who have inherited, and the difference increases as the wealth percentile increases. The reason is, with operative bequest motives, those who have inherited hold a large part of the inheritances at retirement The Role of Bequests I now show the quantitative role of bequests and inheritance distribution. Table 10 shows some statistics summarizing the relationship between retirement wealth and lifetime earnings in the PSID and those in each model. In the model in which accidental bequests are equally redistributed, the only source of heterogeneity is the timing of earnings shocks. Table 10 shows 17

18 Model Earnings Percentile Decile Benchmark 2nd th th th All Random inher. at age 50 All Table 9: Extra wealth holding at selected percentiles among those who have inherited (normalized by average household after-tax earnings) that this model implies a very tight relationship between lifetime earnings and retirement wealth. The correlation coefficient of 0.86 is much stronger and the mean Gini of 0.40 is much lower than the data counterparts. The comparison indicates that the timing of earnings shocks and the existence of borrowing constraints in the bonds market alone are not enough to generate the observed relationship between lifetime earnings and retirement wealth. Corr(W, E) a Mean Gini b PSID (Hendricks (2007a)) Benchmark 0.75 (0.14) 0.53 (-0.01) Change bequest motives and inheritance Equal inher. at age (0.25) 0.40 (-0.14) Random inher. at age (0.20) 0.46 (-0.08) Random inher. at ages (0.19) 0.46 (-0.08) Change intergenerational persistence in earnings ρ h = (0.11) 0.53 (0.00) ρ h = (0.16) 0.54 (0.00) a The correlation coefficient between lifetime earnings and retirement wealth b The average of the Gini coefficients of retirement wealth within lifetime earnings deciles Table 10: Relationship between retirement wealth and lifetime earnings (The numbers in the parentheses are deviations from the data) I then look at the random inheritance models without bequest motives. The probabilities of inheritance are (0.50, 0.20, 0.10, 0.10, 0.05, 0.04, 0.01) and the corresponding amounts are (0.0, 0.3, 1.2, 3.0, 6.8, 18.5, 84.6) of av- 18

19 erage after-tax earnings. 20 Adding random bequests uncorrelated with earnings at age 50 in an otherwise standard life-cycle model weakens the relationship between lifetime earnings and retirement wealth. The distribution of inheritance received is independent of households earnings. This mechanism strengths wealth holdings from inheritance among lower earnings deciles, thus reducing the correlation coefficient between lifetime earnings and retirement wealth. However, the correlation coefficient (0.81) is still higher than that in the data and the mean Gini coefficient (0.46) is still lower than that in the data. This comparison shows that random bequests are not sufficient to generate the observed relationship between lifetime earnings and retirement wealth. The reason is, without an operative bequest motive, those who have received large inheritances will consume a large part of them before retirement. In the benchmark model households hold on more wealth due to operative bequest motives, which generates more heterogeneity in retirement wealth for given lifetime earnings. As is shown in Table 9, extra wealth holdings among those who have inherited are much smaller in the random bequest model than in the benchmark model with bequest motives. I now make the time of receiving inheritance random at ages 35 to 55. The probabilities of receiving inheritances at ages 35 to 55 coincide with parents death probabilities. The amounts of inheritances at each age are adjusted by discounting. This modification further weakens the relationship between lifetime earnings and retirement wealth: A household who is borrowing constrained and receives an inheritance earlier consumes more of it and holds on less wealth at retirement than an otherwise identical household who receives an inheritance later. However, the quantitative effect is small. I now look at the model where parents care about leaving bequests to their children but there is no intergenerational transfer of productivity (ρ h = 0). In this case the correlation between retirement wealth and lifetime earnings is reduced to 0.73, from 0.75 in the benchmark economy. The first reason is that, without intergenerational link of productivity, inheritances are evenly distributed by lifetime earnings decile, which is against what is observed in the 20 Those are taken from Table 8 in Hendricks (2007a). 19

20 data (Table 8). Some households at the low lifetime earnings deciles receive unrealistically large amount of inheritances, weakening the correlation between lifetime earnings and retirement wealth. The second reason is that, as in the benchmark model, with an operative bequest motive, those who inherited large estates from their parents will consume only a small part of their inheritances by the age of 65. Thus the heterogeneity of inheritance adds a lot to the heterogeneity of retirement wealth. On the contrary, the modification that parents and children s productivity are perfectly correlated (ρ h = 1) brings the correlation coefficient further away from the data. The reason is that, higher intergenerational persistence in productivity generates stronger positive correlation between lifetime inheritance and lifetime earnings. As is shown in Table 8, the amount of inheritance received by the lowest two earnings decile is only 4.7%. 5 Sensitivity Analysis I now conduct sensitivity analysis and the results are shown in Table 11. Both earnings and wealth in the data may subject to measurement error. To address this issue, I add random noise into the sample simulated from the benchmark model. I assume that the log of observed earnings for household i at age t, êi t, follows êi t = e i t + v i t, in which e i t is the log of true earnings and v i t is the random error that follows v i t = ρ v e i t + ι i t, ι i t N(0, σ 2 ι ). The observed log wealth at age 65 follows Ŵ i = W i + η i, and η i N(0, ση). 2 η i and ι i t are uncorrelated with each other. I use ρ v = 0.104, and σ 2 ι = (Bound et al. (1989)). 21 The measurement error of wealth may be substantial as well. I choose a small variance σ 2 η = 0.01 as a lower bound. Adding measurement error substantially reduces the correlation coefficient. The Gini coefficient of retirement wealth is slightly higher than that in the benchmark. The reason is that, by grouping households into deciles, rather than looking at each household, some random noise averages out. 21 Using 1982 PSID earnings, Bound et al. (1989) reported that σ 2 v/(σ 2 v + σ 2 e) = 0.15, and ρ v = In the model σ 2 e is 0.831, which gives σ 2 v = and σ 2 ι =

21 In the next experiment I assume that, as in De Nardi (2004), children only observe their parent s productivity at age 50. Without perfect information, children of poor parents will save less whereas children of richer parents will save more, compared with the benchmark model. This enhances the effect of inheritance heterogeneity on generating heterogeneity in retirement wealth for given lifetime earnings. However, the quantitative effect is small, indicating that children can infer other relevant information of their parents, thus the amounts of future inheritance, very well based on their parents productivity. Corr(W, E) a Mean Gini b PSID (Hendricks (2007a)) Benchmark measurement error limited information a The correlation coefficient between lifetime earnings and retirement wealth b The average of the Gini coefficients of retirement wealth within lifetime earnings deciles Table 11: Relationship between retirement wealth and lifetime earnings 6 Policy Experiments The benchmark model with bequest motives, consumption floor, Social Security, and pensions captures the fundamental determinants of households saving and wealth accumulation. I keep parameters as in the benchmark model and conduct policy experiments to study the distributional effects of policies that affect welfare of retirees: minimum consumption, Social Security, and pensions. Among those, minimum consumption programs are linked tightly with assets, while Social Security and pensions are linked closely with lifetime earnings. 6.1 The Role of Government-provided Minimum Consumption Shutting down government-provided minimum consumption reduces the mean Gini by 0.04 to Table 13 illustrates the Gini coefficients of retirement wealth for each lifetime earnings decile. As in Hubbard et al. (1995) and 21

22 Corr(W, E) a Mean Gini b Benchmark No consumption floor No pension No pension, no Social Security No pension, equal Social Security a The correlation coefficient between lifetime earnings and retirement wealth b The average of the Gini coefficients of retirement wealth within lifetime earnings deciles Table 12: Relationship between retirement wealth and lifetime earnings Scholz et al. (2006), a means-tested minimum consumption guarantee provides strong incentives for low-income individuals not to save. Due to the persistence of earnings process, those low-income households are more concentrated in low lifetime earnings deciles. Without a consumption floor, the Gini coefficients decrease a lot at the lowest 3 earnings deciles, since there are fewer poor households in those deciles. Gini coefficients at the higher deciles barely change, since households in those deciles receive high pensions and Social Security income and do not qualify for a transfer. Earnings decile Benchmark no c no DB no DB, no SS no DB, equal SS Table 13: Gini coefficient of retirement wealth by lifetime earnings decile 6.2 The Role of Pensions and Social Security I first deactivate defined benefit pensions. This modification raises the correlation coefficient substantially because it raises average wealth for households in the high earnings deciles. Compared with the benchmark model with pensions, the Gini coefficients decrease a lot at the highest 5 earnings deciles, since they now increase saving for retirement. 22

23 I then further remove history-dependent Social Security. The Gini coefficients increase at the lower earnings deciles since wealthy households increase saving while poor households, due to government transfers, do not increase saving. The Gini coefficients at the higher earnings deciles decrease since poor households increase saving relatively more than rich households. I then add the same amounts of Social Security for all retirees. The amount of Social Security, determined from government budget, is large enough that no retiree qualifies for a government transfer. Adding equal Social Security decreases the Gini coefficients at the lower earnings deciles since poor households, having become disqualified for government transfers, increase saving. The Gini coefficients at the higher earnings deciles increase since poor households decrease saving relatively more than rich households. 7 Conclusions Empirical studies using micro data find that there is large heterogeneity in retirement wealth among households with similar lifetime earnings, and raise doubts about the ability of a standard life-cycle model of saving behavior to reproduce the observed facts. I use an incomplete-market life-cycle model with intergenerational links of bequests and earnings ability, government-provided minimum consumption, a history-dependent Social Security system, and a defined benefit pension. I show that this model with earnings heterogeneity and inheritance heterogeneity generates a substantial amount of heterogeneity in retirement wealth for given lifetime earnings. This suggests that a properly specified life-cycle model with bequest motives, consumption floor, and Social Security pensions captures the fundamental determinants of households saving and wealth accumulation. This framework might shed light on understanding the effects of policy reforms that affect saving. I show that government-provided minimum consumption, pensions, and Social Security have very different distributional effects. In a separate paper, Yang (2013) studies the consequences of eliminating Social Security in a similar environment to the one constructed in this 23

24 paper, and find that the presence of bequest motives reduces life-cycle saving and thus reduces the gains from Social Security reform. 8 Appendix: Computation of the Benchmark Model For a given set of parameters, I solve for the steady state equilibrium as follows: 1. Solve the optimal consumption and saving plans recursively. 2. Guess an initial joint distribution of parents and children at the beginning of the life cycle, compute the associated stationary distribution of households using the policy functions. 3. Check whether the implied joint distribution of parents and children at the beginning of the life cycle is consistent with the initial guess. If so, an equilibrium is found. Otherwise, go back to step 2 with an updated initial guess. References [1] J.G. Altonji, and U. Doraszelski, The Role of Permanent Income and Demographics in Black/White Differences in Wealth, Journal of Human Resources, 40(1), [2] J.G. Altonji, and E. Villanueva, The Effect of Parental Income on Wealth and Bequests, NBER Working Paper [3] J. Ameriks, A. Caplin, J. Leahy, and T. Tyler, Measuring Self-Control Problems, American Economic Review, 97, [4] F.C. Bell, A.H. Wade, and S.C. Goss, Life Tables for the United States Social Security Area , Actuarial Study No. 107, Social Security Administration, Office of the Actuary. [5] D. Bernheim, J. Skinner, and S. Weinberg, What Accounts for the Variation in Retirement Wealth Among U.S. Households? American Economic Review, 91, [6] J. Bound, C. Brown, G. Duncan, and W. Rodgers, Measurement Error in Cross-sectional and Longitudinal Labor Market Surveys: Validation Study Evidence, NBER working paper

25 [7] M. Cagetti, Wealth Accumulation over the Life Cycle and Precautionary Savings, Journal of Business and Economic Statistics, 21(3), [8] M. Cagetti, and M. De Nardi, Entrepreneurship, Frictions, and Wealth, Journal of Political Economy, 114(5), [9] A. Castaneda, J. Diaz-Gimenez, and J. Rios-Rull, Accounting for the U.S. Earnings and Wealth Inequality, Journal of Political Economy, 111(4), [10] T.F. Cooley, and E. Prescott, Economic Growth and Business Cycles, In Thomas F. Cooley eds., Frontiers of Business Cycle Research (Princeton: Princeton University Press, 1995), [11] Council of Economic Advisors, Economic Report of the President. United States Government Printing Office, Washington. [12] L.M. Cubeddu, and J. Rios-Rull, Families as Shocks, Journal of the European Economic Association, 1, [13] M. De Nardi, 2004, Wealth Inequality and Intergenerational Links, Review of Economic Studies, 71, [14] M. De Nardi, E. French, and J. Jones, 2010, Why do the Elderly Save? The Role of Medical Expenses, Journal of Political Economy, 118, [15] E.M. Engen, W.G. Gale, C. Carroll, and D.I. Laibson, The Adequacy of Retirement Saving, Brookings Papers on Economic Activity, 1999, [16] E.M. Engen, W.G. Gale, and C.E. Uccello, Lifetime Earnings, Social Security Benefits, and the Adequacy of Retirement Wealth Accumulation, Social Security Bulletin, 66(1), [17] W. Gale, and J.K. Scholz, Intergenerational Transfers and the Accumulation of Wealth, Journal of Economic Perspectives, 8, [18] J. Gokhale, L.J. Kotlikoff, J. Sefton, and M. Weale, Simulating the Transmission of Wealth Inequality via Bequests, The Journal of Public Economics, 79, [19] I. Grafova, K. McGonagle, and F. Stafford, Functioning and Well-Being in the Third Age, in Jacquelyn B. James and Paul Wink, ed., The Crown of Life: Dynamics of the Early Post Retirement Years (Springer Publishing Company, New York, 2006). 25