How Family Status and Social Security Claiming Options Shape Optimal Life Cycle Portfolios

Abstract

We show how optimal household decisions regarding work, retirement, saving, portfolio allocations, and life insurance are shaped by the complex financial options embedded in U.S. Social Security rules and uncertain family transitions. Our life cycle model predicts sharp consumption drops on retirement, an age-62 peak in claiming rates, and earlier claiming by wives versus husbands and single women. Moreover, life insurance is mainly purchased on men’s lives. Our model, which takes Social Security rules seriously, generates wealth and retirement outcomes that are more consistent with the data, in contrast to earlier and less realistic models.

Two underexplored but important factors drive household optimal life cycle saving and investment decisions: labor market activity and family status. These are influential because decisions about work hours, as well as retirement, affect labor earnings, which in turn shape how people spend, save, invest, and build up retirement benefits. Not only are wages uncertain, but so too is family status because of marriage or divorce, the arrival or departure of children, and spousal death. Each of these poses important risks to the household financial position. For instance, the arrival of children changes both household spending and saving patterns because of higher consumption needs, college costs, and child support in the case of marital dissolution (Love 2010). Additionally, children influence finances directly, as well as indirectly, in that they alter the amount of time that household members, especially mothers, can work to earn income essential to build up financial assets (Kimmel and Connelly 2007). Premature death of one spouse can likewise devastate the survivor’s wellbeing (Sevak, Weir, and Willis 2003/04).

Life cycle financial decisions are also powerfully shaped by mandatory national retirement system rules. In the United States, the Social Security program pays retirees a deferred inflation-indexed life annuity involving complex claiming options and cash-flow patterns that depend on age, work history, and family status. These benefits represent a substantial component of total household assets. For example, the median baby boomer household on the verge of retirement has accumulated $600,000, of which 40% is Social Security wealth.¹ The risk/return profile of the Social Security asset should therefore have important consequences for how households manage their financial wealth, both during work lives and in retirement. Moreover, when to exercise the option to claim Social Security benefits is one of the most crucial and complex financial decisions facing U.S. workers (Shoven and Slavov 2014).

This study is the first to incorporate all of these rich and complex elements influencing the household over its life cycle in a calibrated portfolio and consumption choice model. We do so by accounting for the fact that U.S. Social Security rules give an eligible worker the option of claiming retirement benefits at any point between age 62 and 70. The retiree’s monthly lifelong benefit depends on his earnings history and claiming age, with a reduction of 5%–6.67% per year if he claimed before the Full Retirement Age (FRA), which is currently age 66, and an increment of 8% per year for deferring claiming after that age. For example, someone born between 1943 and 1954 would increase lifetime monthly benefits by 76% if he claimed benefits as late as age 70 instead of as early as age 62.² Moreover, the financial decision as to when to claim Social Security benefits is different from, although related to, the retirement decision about when to leave the labor force (Coile et al. 2002). For example, a worker can leave work at age 62, delay claiming Social Security benefits until age 70 to accrue higher deferred benefits, and live on financial assets for consumption purposes in the interim period. Alternatively, he could claim at the earliest possible age (62) by accepting lower benefits and continue to work, so he could receive income from both employment and Social Security. The Social Security system also has detailed rules for spouse and widow(er) benefits that may shape optimal financial wealth and claiming patterns in important ways.

Our model thus considers that the decision to delay claiming is equivalent to purchasing a deferred joint and survivor life annuity with fixed real benefits: individuals may give up near-term benefits in exchange for higher lifelong benefits starting later. Prior theoretical studies (Yaari 1965; Davidoff, Brown, and Diamond 2005) show that life annuities are highly beneficial for most households. This is because they provide protection against individual longevity risk, and they pay survivors a “mortality credit” above what can be achieved through capital market investments. Yet empirical research also shows that many households are reluctant to voluntarily convert accumulated assets into a life annuity, a phenomenon referred to as the “annuity puzzle” (e.g., Inkmann, Lopez, and Michaelides 2011).

Interestingly, most Americans—49% of men and 55% of women—claim Social Security retirement benefits at the earliest age 62 (Social Security Administration 2012) so they do not exercise the option to “purchase” higher benefits by delaying claiming. This is surprising because the higher benefits from delaying are quite attractive compared to buying an annuity in the private market. The system’s adjustment factors for delayed claiming are based on an actuarial calculation, but they are independent of sex and type of annuity (single or joint and survivor), and they have remained essentially unchanged over the past 3 decades despite the long-term rise in life expectancy (Myers 1985; Shoven and Slavov 2014). Moreover, the public system does not charge administrative fees (unlike private insurance companies) nor does it hold risk capital to cover the possibility of adverse selection because delayed claiming. Even more striking is the fact that many Americans have accumulated substantial wealth in defined contribution pensions, which they can use to finance consumption during retirement and delay claiming to receive the higher Social Security benefits. Overall, the high take-up rates at early claiming age would appear to be rather puzzling.

Previous theoretical analyses have not taken into account how Social Security benefit rules and claiming dynamics influence household portfolio behavior in the context of a rich household optimization framework over the complete life cycle. This study fills the gap, allowing for risky asset returns, as well as uncertainty in family status, mortality, labor income, and retirement income, and jointly modeling work, saving, investment, insurance, and claiming decisions. Previous studies have also assumed retirement benefits are a fixed-proportion final labor income, which makes retirement benefits extremely risky. By contrast, our more realistic formulation shows that households have an alternative to altering their financial portfolios and consumption patterns when shocks arrive: they can delay claiming benefits. This is important because our results imply lower wealth accumulations and earlier retirement, both consistent with observed behavior.

Using data from the Panel Study of Income Dynamics (PSID), we calibrate wage rate dynamics by age, sex, education, and family status. In addition, we calibrate the effect of childcare needs on household time using the American Time Use Survey (ATUS). We track individual work histories for each person separately, and we realistically model Social Security benefits with spousal and survivor benefits, as well as delayed claiming options. In this environment, people make decisions about saving, investment (stocks/bonds/life insurance products), work hours, and benefit claiming.

Our simulation results show that incorporating family status, endogenous work effort, and realistic Social Security claiming rules has a powerful effect on household decisions. In our model and in line with evidence, consumption follows a hump-shaped pattern, rising over the work life and falling markedly around retirement. In addition, married women optimally claim their own Social Security benefits 2 years earlier than single women do, whereas married men claim more than a year later. This strategy effectively hedges longevity risk. We also find that, compared with childless couples, couples with children invest about 10% less of their financial wealth in risky assets and buy more life insurance before retirement.

After developing some key normative implications from our model, we then show that model predictions align well with three stylized facts not predicted by existing life cycle portfolio-choice models (e.g., Cocco, Gomes, and Maenhout 2005). First, because of nonseparable preferences over consumption and leisure, our model generates a sharp drop in consumption around retirement, which comports with evidence. Second, our model predicts a large peak in Social Security claiming at the earliest age of 62—49% of women and 34% of men—which again is what we observe in reality. Third, we show that the model’s predicted claiming ages for persons in different family states align well with patterns in the Health and Retirement Study (HRS).

To illustrate the importance of our model innovations, we also compare findings from our baseline case with those from two simpler approaches used in previous studies. One abstracts from uncertainty regarding fertility, whereas the other calculates Social Security benefits as a prespecified percentage of only last year’s labor income rather than average lifetime earnings as per the actual rules. We show that eliminating fertility risk reduces precautionary saving and hence financial wealth, with little effect on work and benefit-claiming patterns. Having Social Security benefits depend on the worker’s earnings history instead of just his final salary makes retirement benefits less volatile, which in turn induces households to save less and to claim benefits earlier. In both cases, children prove to have large effect on work patterns, but a relatively small effect on claiming decisions.

Our research builds on and extends the literature initiated by Merton (1969) on life cycle consumption and portfolio-choice models. Research has sought to make those models more realistic by introducing new sources of risk (Campbell and Viceira 2001; Cocco, Gomes, and Maenhout 2005), important nonfinancial assets such as housing (Cocco 2005) or life annuities (Horneff, Maurer, and Stamos 2008; Inkmann, Lopez, and Michaelides 2011); and endogeneity of labor supply (Bodie, Merton, and Samuelson 1992; Gomes, Kotlikoff, and Viceira 2008) or the retirement age (Farhi and Panageas, 2007; Chai et al. 2011). Nevertheless, most prior studies take the perspective of an individual representative agent, rather than examining the possibly differing perspectives of households of varying sizes and compositions. Love’s (2010) work is an important and invaluable exception, as his was the first model to incorporate the effect of family and marital status risk on portfolio and saving choice. ³ Drawing on PSID data and the Urban Institute’s Model of Income in the Near Term (MINT), he fitted family transition probabilities, housing cost processes, and labor income paths that depended on age, sex, marital status, and children. His main results were that children living at home lead to less accumulation of financial assets, and households with children have substantially higher demand for term life insurance than singles do. Nevertheless, that important study did not examine the effect of endogenous labor supply and claiming ages on optimal household patterns, considering Social Security rules. By contrast, our more realistic formulation of Social Security benefit options produces results that differ materially from prior research, which assumed that retirement benefits are simply a fixed fraction of final labor earnings.

Other work related to ours in the household finance literature includes Shoven and Slavov (2014) and Coile et al. (2002), both of which explored benefit-claiming options under U.S. Social Security system rules. Yet neither of these studies integrated the portfolio-choice problem within a household lifetime optimization framework. Hubener, Maurer, and Rogalla (2014) developed a multiperson portfolio-choice model, allowing for investments in risky stocks, annuities, and life insurance purchases. However, that paper focused only on retired couples, so it did not examine work patterns. It also included a simple Social Security benefit rule rather than the more realistic one we describe next.

1. The Life Cycle Optimization Model

In our model, agents face the risk of exogenous family transitions throughout their working lives and into retirement. In the following, x(y) denotes a woman (man). Time t = (0,…,T), is measured in years. At time t = 0 (assuming age 20 for women and 24 for men), the individual starts working life as either single or married; we assume that the 4-year age difference between spouses is fixed over the life cycle. Each individual has an uncertain life span and may live for a maximum of T = 80 years.

1.1 Family dynamics

The state variable family s_t is modeled at each point in time as a Markov chain with 35 discrete states. Before retirement, the possible family states are never married, married couple, divorced, and widowed.⁴ We further differentiate each of these states for the woman and man. In addition, a household can have between zero and three children. Possible retirement states for couples include only the wife being retired, only the husband being retired, and both spouses being retired. When modeling spousal benefits, it is also necessary to differentiate these states with respect to the age when the husband claimed his own retirement benefits. A complete list of all possible family states is given in Online Appendix A.

The time-dependent transition matrix Π_i,j,t = Prob (s_t+1 = i|s_t = j) for this Markov chain is influenced by five factors: mortality, marriage, divorce, fertility, and children leaving the household. We abstract from multiple births and divorces during retirement. We only allow married couples to receive children, and we treat three or more children as the same family state.⁵ In the case of a divorce, children are assumed to stay with their mothers.⁶ At the end of our projection horizon T (age 100 for women and 104 for men), we set the survival probability to zero. In the following, we describe the model for couples and refer to the single case only when it is not a straightforward simplification of ignoring the absent partner.

1.2 Financial products

Individuals may select from three different financial products to manage their liquid wealth: riskless bonds, risky stocks, and term life insurance. Bonds are characterized by a constant annual real gross rate of return R_­0. The distribution of the stock return R_­t is assumed lognormal and serially independent.

Each period the individual i ε {x,y} may purchase a 1-year term life insurance contract. If the insured person dies within the period [t,t + 1), any surviving spouse or children receive the face value $L_t^i$ at time t+1. If the insured person survives, no payments are distributed because no cash value is built up by the term life insurance contract. According to the actuarial principle of equivalence, the premium ${LP}_t^i$ charged by the insurer equals the present value of the expected payout plus a percentage expense loading ${\delta }_t^i$, which is given by

$${LP}_t^i=(1+{\delta }_t^i)·(1-p_t^i)·\frac{L_t^i}{R_0}.$$ (1)

Here $p_t^i$ denotes the probability from a mortality table that individual i conditional on being alive at time t survives to time t + 1. The (age-dependent) loading factor ${\delta }_t^i$ reflects expenses covered by the insurance company for administration and to control for adverse selection.

1.3 Time budget and labor income

Each individual has an (exogenously given) total time budget of Ω waking hours per day. Depending on family status, sex, and age, a certain amount of time must be spent on childcare ${\theta }_{s,t}^i$. Before retirement, the worker can decide how much of the remaining time he will spend in the paid labor market ${\tau }_t^i$ to generate labor income or utility-increasing leisure $l_t^i$. Working for pay inflicts (unpaid) commuting time (e.g., for travel) of ${\tau }_{\mathrm{t},\text{trav}}^i$. Accordingly, the time budget equation for each individual is given as follows:

$$\mathrm{\Omega }={\theta }_{s,t}^i+{\tau }_t^i+{\tau }_{\mathrm{t},\text{trav}}^i+l_t^i$$ (2)

Depending on the time devoted to paid work ${\tau }_t^i$, each agent earns uncertain labor income specified as follows:

$$Y_{t+1}^i={\tau }_t^i·w_{s,t}^i·P_t^i·{\epsilon }_{t+1}^i.$$ (3)

Here $w_{s,t}^i$ denotes the deterministic wage rate component, which depends on sex, age, education, and family status. The variable ${\epsilon }_t^i$ is an independent identically lognormal distributed transitory income shock, and $P_t^i$ is the permanent component of the wage rate with independent identical lognormal shock ${\eta }_t^i$ evolving according to

$$P_{t+1}^i=P_t^i·{\eta }_{t+1}^i.$$ (4)

Both, the transitory shock and the permanent wage component $P_t^i$ (and its shock ${\eta }_{t+1}^i$) depend on education and sex. For couples, the transitory shocks ${\epsilon }_{t+1}^x$ and ${\epsilon }_{t+1}^y$ of both partners are uncorrelated such that transitory income risk is diversified across the partners. Yet we assume the permanent shocks ${\eta }_{t+1}^x$ and ${\eta }_{t+1}^y$ to be perfectly correlated, which allows us to keep the model tractable by using one permanent factor P_t for the couple’s income (or else it would be necessary to track each spouse’s permanent wage component separately, resulting in an additional state variable).

1.4 Social Security retirement income

Anytime from age 62 to 70, each adult has the possibility of claiming Social Security retirement benefits. Retirement benefits payable to an individual are equal to his Primary Insurance Amount (PIA), which is based on a progressive function of the average lifetime earnings.⁷ The Social Security retirement benefit is given by:

$$Y_{t+1}^{i,\text{ret}}={\mathit{PIA}}_t^i·{\lambda }^i·{\epsilon }_{t+1}^{i,\text{ret}},$$ (5)

with λⁱ being the adjustment factor (relative to the FRA) for early claiming (a reduction) or delayed retirement (a credit), and ${\epsilon }_t^{i,\text{ret}}$ is a lognormal transitory shock with a mean of one which reflects out-of-pocket medical and other expenditure shocks (as in Love 2010). We use the PIAs for each spouse as state variables. After claiming retirement benefits, individuals still have the opportunity to continue working until age 70. If they do, each dollar earned above the retirement-earnings exempt amount is taxed at 50 cents, consistent with the U.S. Social Security rules.⁸

After both partners have claimed their retirement benefits, the partner with the lower retirement income may elect to receive spousal benefits instead of own benefits. These amount to 50% of the partner’s benefits, unless the spousal benefits are claimed before reaching FRA, whereupon a permanent reduction of up to 30% applies. In contrast to own retirement benefits, claiming spousal benefits after the FRA is not incentivized with an increase of lifelong payments. Because tracking the claiming age for spousal benefits would imply an additional state variable, our framework only allows claiming spousal benefits at the FRA.⁹ After that, a partner receives spousal benefits if these exceed the own already-claimed old-age retirement benefits. Another rule is that if one partner claims after his FRA, the delayed retirement credit only increases his own benefits, but not his partner’s spousal benefits. To exclude the delayed retirement credit for spousal benefits, we use separate retirement states for different claiming ages of the husband.¹⁰

When a spouse passes away, the surviving spouse may switch from (own or spousal benefits) to widow(er) benefits. These are equal to 100% of the deceased spouse’s benefits. In our model, this is not an active decision; instead, these benefits are automatically paid if the widow(er) benefits are higher. If retirement benefits have not yet been claimed, the PIA of the surviving spouse is substituted in place of the PIA of the deceased spouse. Accordingly, we do not track if the widow(er)’s PIA results from own work history or that of a deceased spouse.

This quite realistic formulation of Social Security benefit options differs from and extends substantially the typical approach taken in prior portfolio-choice life cycle studies. That is, the usual approach until now has been to assume that the worker’s retirement benefit is given by a fixed proportion of his last labor income as of a prespecified date.¹¹ Although such an approach reduces the computational burden of the model substantially (it saves at least one state variable), it has the drawback that the individual’s work history is not fully reflected in his Social Security benefits. Prior studies have also not modeled spousal or survivor payments, ignoring the reality that one spouse can claim first on her own account and later switch to alternative benefit payment streams. As we will see, this realism plays a crucial role in understanding the interaction between Social Security and family transitions.

1.5 Wealth dynamics

Besides determining how much time to spend working, in each period, the household must also decide how much of its liquid wealth (W_t) to spend on consumption (C_t), life insurance premiums for ( ${LP}_t^x,{LP}_t^y$) for the wife (husband) x(y), and how to allocate savings to bonds B_t and stocks S_t. The household is liquidity-constrained so it cannot borrow to finance consumption and life insurance purchases:

$$W_t=C_t+{LP}_t^x+{LP}_t^y+B_t+S_t$$ (6)

$${LP}_t^x\ge 0{LP}_t^y\ge 0S_t\ge 0B_t\ge 0.$$ (7)

Next period’s liquid wealth is given by any remaining wealth including capital market returns, labor income ( $Y_t^i$), and Social Security benefits ( $Y_t^{i,\text{ret}}$), less income taxes according to proportional rates ϑ_labor and ϑ_ret and housing expenses h_s,t:

$$w_{t+1}=S_t·R_{t+1}+B_t·R_0+(1-{\mathbb{I}}_{t+1}^x)L_t^x+\left(1-{\mathbb{I}}_{t+1}^y\right)L_t^y+\left(\left(y_t^x+Y_t^y\right)·(1-{\vartheta }_{\text{labor}})+\left(y_t^{x,\text{ret}}+Y_t^{y,\text{ret}}\right)·(1-{\vartheta }_{\text{ret}})\right)·\left(1-h_{s,t}\right).$$ (8)

The indicator variables ${\mathbb{I}}_t^x$ and ${\mathbb{I}}_t^y$ are equal to one if the corresponding spouse is alive at time t and zero otherwise. If one of the spouses i dies, the remaining spouse j receives the payment from the life insurance contract $L_t^i$. There are additional cash flows caused by family transitions that are not represented in Equation (8). If a child leaves the household, we assume the parents must pay for the child’s college education (here designated as a lump sum). Further, a divorced woman with children receives child support payments, whereas a divorced husband with children must devote a certain fraction of his income for child support.

1.6 Preferences and optimization

We posit that the household has a time-additive utility function with constant relative risk aversion γ, and that utility derives from a composite good consisting of consumption C_t and effective leisure l_t. Depending on the number of adults and children present in the household, total consumption is normalized by a scaling factor ϕ_s (Love 2010; Hubener, Maurer, and Rogalla 2014). For a single adult, leisure is identical to time devoted to leisure, whereas for a couple, “effective” leisure is given by the geometric mean of both spouses’ leisure times:¹²

$$l_t=\sqrt{l_t^x·l_t^y}.$$ (9)

The relative importance between consumption and leisure is given by a modified Cobb- Douglas function $u_t(C_t,l_t)=\frac{1}{1-\gamma }{\left(\frac{C_t}{{\varphi }_s}·l_t^{\alpha }\right)}^{1-\gamma }$, whereby leisure preferences are governed by the parameter α. The higher α is, the less the family is willing to increase work hours and reduce leisure time to raise consumption.¹³ The household’s expected lifetime utility can be expressed by the recursive Bellman equation:

$$\begin{gathered} J_t\left(W_t,P_t,{\mathit{PIA}}_t^x,{\mathit{PIA}}_t^y,s_t\right) \\ =\underset{C_t,{\tau }_t^x,{\tau }_t^y,S_t,B_t,{LP}_t^x,{LP}_t^y}{max}\left\{\frac{1}{1-\gamma }{\left(\frac{C_t}{{\varphi }_s}·l_t^{\alpha }\right)}^{1-\gamma }+\beta \\ {E}_t\left[J_{t+1}\left(W_{t+1},P_{t+1},{\mathit{PIA}}_{t+1}^x,{\mathit{PIA}}_{t+1}^y,s_{t+1}\right)\right]\right\}, \end{gathered}$$ (10)

where β represents the time preference factor. The value function is governed by the state variables financial wealth W_t, the permanent wage component P_t, the PIA, ${\mathit{PIA}}_t^x$ and ${\mathit{PIA}}_t^y$, and the family state s_t. The controls are consumption C_t, working time τ_t, investments in stocks S_t or bonds B_t, and premiums for life insurance purchases ${LP}_t^x$ and ${LP}_t^y$.

The expectation of the household’s future value function is the sum over all possible family states weighted using the transition probability Π_{St, St+1}.

$$E_t\left[J_{t+1}\left(W_{t+1},P_{t+1},{\mathit{PIA}}_{t+1}^x,{\mathit{PIA}}_{t+1}^y,s_{t+1}\right)\right]=\sum _s{\mathrm{\Pi }}_{s_t,s}{E}_t\left[J_{t+1}\left(W_{t+1},P_{t+1},{\mathit{PIA}}_{t+1}^x,{\mathit{PIA}}_{t+1}^y,s_{t+1}=s\right)\right]$$ (11)

An exception is the case of divorce, the only instance in which a household is split into two separate units, each with a different utility function. In this case, we assume equal bargaining weights between the spouses and weight the two utility functions equally¹⁴:

$$E_t[J_{t+1}]=\frac{1}{2}{E}_t[J_{t+1}(s_{t+1}=\text{divorced}\text{woman})]=\frac{1}{2}{E}_t[J_{t+1}(s_{t+1}=\text{divorced}\text{man})].$$ (12)

If one spouse dies, the desire to provide for the surviving partner is reflected in the corresponding value function of the surviving spouse. If the last or both spouses die, they may wish to provide for their children or leave a bequest. The strength of this bequest motive is given by the parameter b_s,t, which is zero in the states without underage children. If children are present in the household, the strength depends both on the number of children (i.e., the family state) as on the age of the mother (as a proxy for the age of the children). The corresponding utility is given by remaining wealth normalized by the bequest parameter and multiplied by the available time budget¹⁵:

$$\begin{array}{cc}{J}_t=\frac{1}{1-\gamma }{\left(\frac{W_t}{b_{s,t}}·{\mathrm{\Omega }}^{\alpha }\right)}^{1-\gamma }& \text{for}{b}_{s,t}>0\\ J_t=0& \text{for}{b}_{s,t}=0\end{array}\}\text{if}\text{both}\text{spouses}\text{have}\text{died.}$$ (13)

Between ages 62 and 69, each spouse has the opportunity to claim his Social Security benefits. At age 70, no further delayed retirement credit can be earned and claiming is mandatory. If the utility of a retirement state exceeds the utility of the current state calculated from Equation (10), the utility of the current state is replaced by the higher value, and the couple switches to the retirement state.¹⁶ Online Appendix B provides details on solving the model.

2. Model Calibration and Parameterization

2.1 Family process calibration

The drivers of family state transitions are the hazards of marriage, divorce, fertility, children leaving the household, and mortality. We calibrate marriage and divorce probabilities using the Urban Institute’s MINT model (Smith et al. 2010). There, current age and sex are related to marriage and divorce hazard rates, the number of previous marriages, and the duration of the current marriage time since last marriage. To parameterize the transition probability matrix, we simulate a population of 1,000,000 people with an initial marriage rate of 20%, for which we track the number and duration of marriages.¹⁷ These then evolve according to the MINT hazard rates. We derive the transition probability Π_i,j,t by dividing the number of transitions in our simulated population from state i to state j at age t by the number of paths in state i. In the MINT model, the number of children does not affect hazard rates for marriage and divorce, so these transitions are independent of the number of children. Fertility-driven transitions probabilities are determined in a subsequent step.

For the transitions between family states with different numbers of children, we use 2009 values of the all-race fertility rates from the National Vital Statistics Reports (Martin et al. 2011). Reported fertility rates are adjusted for the fact that in our model only married couples get children.¹⁸ We assume that children leave the household when they turn age 18. Because our state variables track only the numbers of children but not their ages, we again simulate a population with the already-calibrated fertility, marriage, and divorce transitions, and we track the ages of the children and have them leave the household after 18 years. The transition probability to states with one fewer child Π_i,j,t is given by the number of paths at age t with a child turning 18 in state i, divided by the total number of paths in state i.

Mortality transitions to widow or widower states are given by sex and age-dependent 1-year survival probabilities, for which we use the U.S. 2009 population life table in the National Vital Statistics Report (Arias 2014). We assume survival probabilities are independent of family status.

2.2 Time budget and childcare time

Each spouse has a (fixed) total time budget available for leisure, child support, and paid work (see Equation 2) of Ω= 16 hours per day, and the possible workweek consists of 5 days (relevant for distinguishing between full-, part-time, and overtime work). We further assume a year to have 52 weeks (relevant for transformation to annual values) and a month to be 1/12^th of a year (relevant for determining the PIA).

To calibrate state and age-dependent childcare time ${\theta }_{s,t}^i$ we use data from 2003–2011 waves of the ATUS. This study provides detailed information on how much time households spend on several activities. We divide the ATUS respondent sample into four subgroups: married women, married men, single women, and single men, and we drop observations older than age 66. Then we regress time spent on the childcare-related activities on a set of dummies for the number of children (with one dummy representing three or more children), and a second-/third-order polynomial in the number of years until the youngest child turns age 18.¹⁹ A graphical representation of the results is provided in Figure 1 (details on the ATUS sample and coefficient estimates appear in Online Appendix C).

Figure 1. Time spent on home production by age of youngest child by sex and marital status.

Figure 1 is a graphical representation of results from our home-production time regression using the American Time Use Sample. Time spent on home production is shown for the case of one child in the household. For each of the subsamples, the final data point reports home-production time when no child is present. In the life cycle model, the difference in these levels is used for the parameterization of childcare time ${\theta }_{s,t}^i$. Additional details are provided in Online Appendix C (Table C1).

In general, women allocate more time than men do to home production. Moreover, children boost women’s time devoted to nonmarket activities more than men’s. Single women spend less time on these activities than married women do, but the effect of (at least the first two) children is about the same for both female groups. For someone with no children, the set of child dummies and the age of the youngest child are set to zero, so the regression constant term reflects time spent on home production when no children are present. The parameter ${\theta }_{s,t}^i$ captures only the marginal effect of children on home-production time, so rather than setting ${\theta }_{s,t}^i$ to the estimated home-production time for each family state, s, we set ${\theta }_{s,t}^i$ to the difference in home production time with reference to someone having a similar marital status but with no children (e.g., married couple with two children versus a married couple with no children). As noted above, our state variables do not directly track the ages of children at home. Instead, for the calibration of transition probabilities, we simulate a population keeping track of the children’s ages. For each path, childcare time is calculated according to the regression results,²⁰ and the value ${\theta }_{s,t}^i$ is derived by computing the mean over all paths for corresponding family state s at (parent’s) age t.

We also use the ATUS for calibrating the time needed to commute to work. The sample mean for those who worked at least an hour for pay and traveled to work less than 4 hours on the diary day is ${\tau }_{\mathrm{t},\text{trav}}^x=0.64\text{hours}$ for women and ${\tau }_{\mathrm{t},\text{trav}}^y=0.79\text{hours}$ for men.

2.3 Wage rate calibration

We estimate the deterministic component of the wage rate process $w_{s,t}^i$ and the variances of the permanent and transitory wage shocks ${\eta }_t^i$ and ${\epsilon }_t^i$ using the 1975–2011 waves of the PSID; dollar values are in $2011. Besides age, sex, and education, we are especially interested in the effect of the family status and work hours on the hourly wage. In our dataset, some respondents directly report a wage in terms of dollars per hour; for the remainder of the observations, we infer hourly wage rates by dividing annual income by annual work hours. Annual work hours are given by the hours worked per week multiplied by 52. To capture the effects of children, the wage rate equation includes a vector of dummy variables for the number of children under 18 in the household, a polynomial in the respondent’s age, an indicator of whether a spouse is present in the household, and a set of dummies representing work levels: full time for pay (between 20 and 40 hours per week), part time (more than zero but less than 20 hours per week), or overtime (over 40 hours per week). In doing so, we capture the effect of children on parents’ earnings when they cannot work full time.

For couple households, we treat spouses as separate observations. By using the wage rate as the dependent variable, we limit the sample to the working population for which we compute this quantity. We also eliminate all observations with hourly wage rates below $5 (below minimum wage laws) and extreme observations (above the 99^th percentile of each wave). Further, we divide the sample into four subgroups by sex and education: men with a high school education, women with a high school education, men with a college education, and women with a college education.

Table 1 shows ordinary-least-squares regression results of the factors associated with (the natural logarithm of) hourly wages.²¹ For both education groups, men receive higher wages than women do and the gap widens with age. ²² For all subgroups, living with a spouse is significantly associated with higher wages, of more than 10% for men and from 3.5%–7.0% for women.²³ Having children only slightly decreases men’s wages (significant only for those with a college education), whereas it significantly decreases women’s wages. For women with a high school (college) education, having one, two, or three+ children decreases wage rates by 6.4%, 9.5%, and 17.0%, respectively (9.1%, 11.0%, and 18.0%). For all four groups, there are large wage reductions for part-time work (up to 12.5%), whereas working overtime yields a significant bonus.

Table 1. Wage rate regression coefficient estimates.

Table 1 shows estimated coefficients from multivariate regressions of the natural logarithm of the wage rate, using Panel Study of Income Dynamics data for respondents age 20–69 from waves 1975–2011 and the corresponding estimates of variances of permanent and transitory shocks to the log wage rate. Control variables include a third-order polynomial in age, dummies for the number of children under 18 in the household, presence of a spouse in the household, and dummies for working part time (under 20 hours per week) or overtime (over 40 hours a week). Shock variances are estimated by regressing the squared difference between waves in unexplained log wage on the time lag between waves and a constant vector. Coefficients on wave dummies are not shown. Standard errors are given in parentheses, and the number of stars indicates levels of significance (10%, 5%, and 1%, respectively).

	Men, high school		Women, high school		Men, college		Women, college
Coefficient estimates
Constant	1.274 (0.075)	***	0.706 (0.066)	***	0.430 (0.142)	***	−0.156 (0.119)
Age/100	6.407 (0.598)	***	12.741 (0.532)	***	13.610 (1.100)	***	19.793 (0.934)	***
Age2/10000	−6.429 (1.525)	***	−26.420 (1.341)	***	−20.326 (2.727)	***	−38.556 (2.340)	***
Age3/1000000	−0.203 (1.240)		17.438 (1.071)	***	8.990 (2.174)	***	24.077 (1.875)	***
1 child	−0.024 (0.005)	***	−0.064 (0.005)	***	−0.020 (0.008)	**	−0.091 (0.007)	***
2 children	−0.002 (0.005)		−0.095 (0.005)	***	0.016 (0.008)	**	−0.110 (0.008)	***
3+ children	−0.038 (0.006)	***	−0.170 (0.006)	***	−0.012 (0.010)		−0.180 (0.011)	***
Spouse present	0.118 (0.006)	***	0.035 (0.004)	***	0.104 (0.009)	***	0.070 (0.006)	***
Part-time work	−0.088 (0.023)	***	−0.125 (0.008)	***	−0.096 (0.030)	***	−0.083 (0.012)	***
Overtime work	0.049 −0.004	***	0.080 −0.005	***	0.064 −0.005	***	0.058 −0.006	***
Shock variances
Permanent	0.0106 (0.00055)	***	0.0104 (0.00055)	***	0.0153 (0.00099)	***	0.0147 (0.00084)	***
Transitory	0.0324 (0.00149)	***	0.0269 (0.00139)	***	0.0455 (0.00261)	***	0.0357 (0.00220)	***

To estimate the variances ${\sigma }_{\eta }^2$ and ${\sigma }_{\epsilon }^2$ of shocks ${\eta }_t^i$ to permanent wages and ${\epsilon }_t^i$ to transitory income, we follow the well-established procedure introduced by Carroll and Samwick (1997), also used by Cocco, Gomes, and Maenhout (2005) and Love (2010). The idea is that the residual of the observed log wage in the PSID and the predicted log wage from our regression results can be attributed to permanent wage and transitory shocks P_t + ln ε_t. Let r_i,d = (ln P_t+d + ln ε_t+d) − (ln P_t + ln ε_t) be the difference of these residuals of waves for individual i, d years apart. Under the assumption of serially uncorrelated and independent shocks, this difference has a variance of $d{\sigma }_{\eta }^2+2{\sigma }_{\epsilon }^2$. Regressing the squared differences $r_{i,d}^2$ on the time span d between waves, and a constant vector of 2’s yields the estimate for these variances.

The results of our calibration appear at the bottom of Table 1. As was the case for the deterministic wage component, we split the sample by education and sex. Compared with Love (2010), who based his empirical analysis on a broader definition of household income (including public transfers and unemployment compensation, as well as labor income), our estimate of the variance of permanent shocks ${\sigma }_{\eta }^2$ is about the same for the less educated, and slightly higher for the college educated. In addition, the variances of permanent income shocks are very similar for men and women, whereas the difference is more pronounced for transitory shocks. Because in the case of couples we assume the same permanent wage shock, the variance is set equal to the average of the man’s and the woman’s values. Our variance of transitory shocks ${\sigma }_{\epsilon }^2$ is considerably lower for both educational groups. As retirement income is purely a public transfer in our model, we set the variance of transitory retirement income shocks to ${\sigma }_{{\epsilon }^{\mathit{ret}}}^2=0.0784$ (as in Love 2010).

2.4 Preferences, financial products, and other parameters

Following several other studies in the life cycle literature, we adopt the household consumption-scaling factors proposed by Citro and Michael (1995), ϕ_s = (A + 0.7 · K)^0.7, with A being the number of adults and K being the number of children in the household. Our calibration of bequest strength b_s,t is motivated by a provisional motive, that is, to provide for children’s consumption and education costs. We set b_s,t to zero for any family states without children present, which applies to all retirement states, among others. Otherwise, we assume that an annuity must be purchased to finance consumption for each left-behind child until his 18^th birthday, plus a lump sum sufficient to cover the cost of 4 college years.²⁴ As children’s ages are not explicitly tracked in our model, we again use the same simulation technique as before for the family transition probabilities and childcare times to derive mean values of b_s,t for family state s at each age t.

For our base case, we follow prior studies (Gomes, Kotlikoff, and Viceira 2008; Cocco and Gomes 2012) in using a relative risk-aversion parameter of γ = 5 and a time discount rate to β = 0.96 (the latter is around the average of the risk-free return and the expected return of the risky asset). The leisure-preference parameter is of key importance for our model, and we use α = 0.9 because for this value, the optimal life cycle profiles for hours worked per week roughly match the average work hours in the PSID data used for the calibration of the wage rate (see Online Appendix D). The risk-free rate is set to R­₀ = 1.02, and we assume an equity premium for stock returns of E[R_t] = 4% with a standard deviation of stock returns of 20%. Life insurance contracts are priced according to the 2001 Commissioners Standard Ordinary Mortality Table (American Academy of Actuaries 2002). As in Gomes, Kotlikoff, and Viceira (2008), labor earnings are taxed at a rate of ϑ_labor = 30%, and retirement benefits at ϑ_ret = 15%.

We follow Love (2010) in calibrating several other parameters. For instance, we use his estimation of housing costs h_s,t from PSID data; for child support, divorced men are assumed to pay 17%/25%/30% of their labor income for 1/2/3+ dependent children; divorced women with children receive the corresponding fraction of a single man’s income as if he works for 40 hours per week; when a child turns age 18, the household pays 40% of its permanent (full-time) income for college costs upon his departure; in the case of divorce, wealth is split evenly between spouses after deducting 10% of assets for divorce costs. ²⁵

2.5 Retirement income parameters

When a single individual marries, we must make some assumptions about the new partner. First, we posit that the new partner has the same permanent wage rate component P_t as the single individual had before marriage. Second, the PIA of the new husband is an age-dependent multiple of the wife’s PIA, ranging from 1.06 in their early 20’s to 1.09 just before retirement. Third, the financial wealth brought by the husband into the couple’s wealth is an age-dependent multiple of the wife’s wealth, ranging from 1.08 early on, and 1.12 late in life.²⁶

When a couple divorces, the partner with lower retirement-benefit claims is entitled to spousal benefits, and after the former partner’s death, to widow(er) benefits. Our model does not track the PIA of former partners, so we increase the PIA of a divorced woman (man) to 70.85% (58.23%) of the former partner if her (his) own PIA is smaller. This is motivated by the following consideration: an annuity paying $50 per year to a woman as soon as her former husband reaches FRA, and $100 after his death, as long as the woman lives, has the same actuarially fair present value as an annuity paying the woman $70.85 per year (because of the age difference and the asymmetry in the mortality rates, the value for men is only $58.23).

For the piecewise linear function converting average lifetime earnings into the PIA, we use the Social Security bend point specification of 2011. The deduction (bonus) for claiming early (late) old-age retirement benefit is calculated according to Social Security claiming rules as of the FRA equal to 66. Lifelong retirement or spousal benefits are reduced by 5%–6.67% per year if the individual claims prior the FRA. Retirement benefits increase by 8% for each year claiming after that age (see Online Appendix B).

The adjustment factors for claiming after age 62 are attractive compared with annuities offered in the private market. To illustrate this, we calculate the assumed interest rate of an actuarially, fairly priced deferred annuity offering the same delayed benefits as under Social Security, and the U.S. Annuity 2000 Basic Mortality Table provided by the Society of Actuaries used by insurance companies to evaluate life annuities offered in the private market (assuming no additional loadings). Results are in the 2.8%–4.8% range, depending on age and sex (see Online Appendix B), and the rates are around one percentage higher for females versus males.

Even though these yield calculations illustrate the attractiveness of the delaying Social Security, they are of course insufficient to derive optimal claiming strategies for singles or for couples. Rather, a realistically calibrated dynamic portfolio-choice model is needed incorporating the key factors driving optimal decisions: the opportunity cost of leisure (uncertain wages), the household’s financial wealth, each partner’s PIA, asset risk and return profiles, and the options to work/claim own benefits/switch from own benefits to spousal or widow benefits. Next, we turn to the results from our richer, and more realistic, life cycle model.

3. Optimal Decisions Regarding Saving, Work, Claiming, Life Insurance, and Investments

3.1 Simulations

We use the optimal controls of the baseline parameterization of our life cycle model to generate 100,000 simulated life cycles reflecting realizations of stock returns, wage rates, and marital status. We assume that 59.3% (40.7%) of the simulated households have a wage rate profile corresponding to those with a high school (college) degree (as per the 2011 wave of the PSID). We divide the sample of simulated life cycles equally into female and male paths. At the start of the simulations, 80% are singles and 20% are already married, whereas later in life, each individual randomly moves between the 35 family states. Each household is endowed with a starting financial wealth equivalent paid by a 40-hour per week job. Results are presented using individual simulated paths generated by our model conditional on survival. To do so, we modify the transition matrix Π_i,j,t for the simulation by setting the mortality of women in female paths and men in male paths to zero and rescale the other probabilities such that they sum up to one.²⁷ This procedure keeps the same number of paths even at older ages. If a single agent marries, we make the same assumptions about the new spouse as in the optimization regarding age difference, permanent income, wealth, and PIA. In the case of divorce, we follow only the ex-wife (ex-husband) in a female (male) path and ignore the other spouse.

For the reporting of aggregate quantities over all paths, such as for example average wealth, each path is weighted with its survival probability to the age in question. This gives female paths a slightly higher weight compared with male paths, especially at older ages. When sex-dependent quantities such as hours worked by women (men) are considered, we only report the average over female (male) paths. We also report results for subsamples (e.g., single or couple households). In this case, we use averages over paths in that family state at the reported age; accordingly, the samples are not the same at each age. For example, an individual who is initially modeled as single will drop out of the single sample when she marries; she can also reenter the single subsample at a later age, when a divorce occurs.

3.2 Optimal life cycle profiles

3.2.1 Population

Figure 2 reports the average life cycle profiles for the entire population of singles and couples with either a high school or college education. Panel A shows average consumption, life insurance demand, wealth level, and investment in equities. Panel B reports average work hours for men and women, and Panel C indicates the frequency of claiming Social Security benefits. In what follows, we discuss these life cycle patterns and show how the results differ from previous life cycle studies that disallowed endogenous work hours and Social Security claiming patterns, most importantly the work by Love (2010) because he also incorporated uncertain family transitions.

Figure 2. Expected life cycle profiles: Entire Population.

These three figures illustrate simulated life cycle profiles for the complete population (singles and couples with high school or college education) at various ages. Panel A shows average levels of wealth, consumption, stock holdings, and the face value of life insurance holdings for men and women. Panel B shows average work hours for men and women. Panel C shows the percentage of men and women claiming Social Security benefits at each age from 62 to 70. Averages are generated from 100,000 independent simulations based on optimal feedback controls from the baseline specification of the life cycle model. Averages for wealth, consumption, and stock holdings in Panel A aggregate across men and women weighted with survival probabilities. Model parameters include the following: risk aversion γ =5; time discount rate β= 0.96; leisure preference α=0.9; (uncertain) consumption-scaling factor ϕ depends on family size; equity risk premium 4%; initial fraction of couples 20%; and fraction with college education 40.3%.

Figure 2A shows that liquid wealth builds up gradually until age 52 when it amounts to about $173,000 on average; thereafter, people begin to draw down their assets. This profile accords with the well-known lifetime-wealth hump shape reported in other studies, but here the withdrawal of wealth starts earlier compared with models using fixed work hours for which the wealth is accumulated until attaining the fixed retirement age (Cocco, Gomes, and Maenhout 2005; Love 2010).²⁸ The reason for the wealth decumulation before retirement is that workers increasingly reduce work hours around age 50, resulting in lower labor income. To support consumption, which is rather flat until the mid-60s, households start to draw down accumulated assets. We also see that liquid wealth (W_t) and people’s investment in risky stocks (S_t) are highly correlated. The amount invested in stocks divided by the level of liquid wealth generates a relatively low and stable fraction invested in the stock market. For instance, in the first decade, the household’s stock share rises from about 20% at age 20, to 61% at age 35. Subsequently, the average allocation to stocks is quite stable, in a range of 43%–61%. After age 62, when households start to claim Social Security benefits and receive their riskless benefits, the fraction invested in stocks increases slightly, to 53%. This is of interest because U.S. households who do hold stocks have about 40%–60% of their portfolios in equity, and the shares do not vary much by age.²⁹ Compared with other life cycle studies, these levels of stock participation are relatively low.³⁰ This is mainly due to our definition of the stock fraction as the amount held in stocks divided by cash on hand, which means that the portion of cash on hand dedicated to this period’s consumption is assumed to be held in nonrisky assets.³¹

The demand for life insurance is also hump shaped with a peak in the mid-40s. Especially for couples with children, the financial risk of losing an income in the event of a spouse’s unexpected death is high. Accordingly, both men and women purchase life insurance to protect children against a parent’s death. This contrasts with Love (2010) who restricted married women’s life insurance to zero. More details on life insurance demand are provided in the next section.

Figure 2B shows that men begin their careers working an average of 43 hours per week, which they gradually reduce to around 35 hours right before the earliest possible Social Security claiming age. At age 62, their work hours drop sharply to around 28 per week; thereafter, they increasingly retire from full-time work. In their early 20s, women also work for pay more than full time (over 40 hours a week), but they pull back to about 30 weekly hours in their late 30s. Around 15% of women work part time (i.e., less than 20 hours per week) before age 62, and at age 62, average female weekly work hours drop from 27 to 22. After age 62, men’s and women’s work hours become more similar.³²

Our model also generates substantial variation across Social Security benefit-claiming ages as illustrated in Figure 2C. Men are predicted to claim benefits significantly later than women are, and for both, the model generates two peaks at the earliest and latest claiming age. Around 49% (34%) of women (men) claim at age 62. This pattern is in contrast to previous life cycle studies, which typically assumed that people must cease work and take retirement benefits at a mandatory prespecified age (Love 2010). By contrast, our model predicts that only a small minority of people will claim benefits at the FRA, whereas many claim early. This is compatible with information from the Social Security Administration (2013) indicating a substantial claiming peak at age 62: around 55% (49%) of women (men) claimed benefits at age 62, and only 12.6% (18.4%) of men (women) claimed at the FRA of 66. Thus, our model does a credible job in replicating both the variation in claiming frequencies across ages and the peak at age 62.³³

Our model also predicts that consumption follows a hump-shaped pattern, rising over the work life and falling substantially between ages 62 and 70, when many households retire. This differs from prior life cycle models that shut down endogenous work effort, claiming, and retirement (e.g., Cocco, Gomes, and Maenhout 2005; Love 2010), which typically generate a smooth average consumption profile before and after retirement. Before retirement, the consumption path reflects the pattern of children and wage rates over the life cycle. The drop in consumption around retirement arises because a composite good defined over consumption and leisure enters the households’ utility function $U_t=C_t{l}_t^{\alpha }$ (see Equation 10). Accordingly, households can substitute consumption expenditures for more leisure time when they claim Social Security benefits from age 62 onward. The public pension provides retirees with a lifetime income stream, when they sharply increase their leisure time and reduce consumption accordingly.³⁴

Interestingly, the consumption drop around retirement is exactly what real-world evidence shows. For instance, in the PSID dataset, Bernheim, Skinner, and Weinberg (2001) found a change in spending at retirement that averaged −14%. Other studies by Aguiar and Hurst (2005) and Battistin et al. (2009) confirmed a statistically significant decline in spending at retirement. These authors suggested that retirees were willingly increasing home production and curtailing consumption expenditures, a result fully compatible with our model’s predictions.

Table 2 provides more distributional details on claiming rates, retirement rates, and consumption patterns based on our 100,000 simulated life cycle paths. We measure the size of the consumption change as the difference in log-average spending 2 years before and after claiming (see also Bernheim, Skinner, and Weinberg 2001; Chai et al. 2011). This generates a negative drop in consumption for all age groups with an average of −19.7% for women and − 20.9% for men. Table 2 also shows that the optimal point to claim Social Security benefits is for most (but not all) households the same as the age of retirement; here we denote individuals as retired if they work fewer than 20 hours per week. About half of all women claim benefits and retire at the same time, and only 37% of them work after claiming. In addition, the majority of men (68%) both claims and retires at the same time. In other words, most households use their Social Security benefits to reduce work hours and substitute consumption for more leisure time. This is in line with Battistin et al. (2009), who concluded that the main reason for the retirement consumption drop was eligibility for retirement benefits, rather than liquidity constraints among the less well off. Evidently, our model with flexible retirement and work hours shows that this is not a puzzle at all.

Table 2. Retirement rates, claiming patterns, and changes in consumption at retirement.

Table 2 shows frequencies (in percent) by age groups of the simulated population claiming Social Security benefits, retiring from full-time jobs (work hours < 20h/week), and the mean change in average log consumption 2 years after and before claiming, and based on 100,000 simulated life cycle profiles.

Age	Claiming rates (%)		Retirement rate (%)		Mean cons. drop (%)
	Female	Male	Female	Male	Female	Male
< 62	-	-	14.0	3.8	-	-
62	48.4	34.0	13.3	22.6	−7.1	−10.4
63–66	12.2	21.1	10.3	10.9	−16.7	−19.9
67–70	39.4	44.9	62.4	62.7	−36.1	−29.4
Mean	65.2	65.9	66.4	66.9	−19.7	−20.9
P(claiming age = ret. age)			49.0%	67.6%
P(claiming age < ret. age)			37.4%	25.7%

Additional summary statistics providing insights about factors driving the claiming decision appear in Table 3. We report average claiming ages for males (Panel A) and females (Panel B) for 16 different wealth/income subgroups based on our simulated life cycle profiles. These subgroups are constructed using the joint distribution at age 62 with respect to quartiles on liquid wealth and quartiles of retirement income (PIA). The numbers below the average claiming age are the percentages of individuals in the different subgroups.

Table 3. Effect of wealth and accumulated Social Security benefits on claiming age.

Table 3 shows average Social Security claiming ages for various subgroups. The life cycle simulation method is identical to that used in Section 3.2 (also see notes to Figure 3). The subgroups vary by levels of wealth and PIA (at age 62). Subgroup Wealth-Q1/PIA-Q1 represents those with the 25% lowest wealth and 25% lowest PIA. Numbers in brackets show the relative frequencies of the 100,000 simulated individuals in this group.

A. Average claiming age female (percentage in subgroup)
	PIA Q1	PIA Q2	PIA Q3	PIA Q4
Wealth Q1	62.8 (13.2)	63.6 (7.5)	64.4 (3.7)	64.4 (0.6)
Wealth Q2	63.6 (7.2)	64.0 (8.3)	65.2 (6.6)	66.9 (2.9)
Wealth Q3	64.4 (3.9)	65.2 (6.7)	66.1 (8.4)	67.4 (5.9)
Wealth Q4	64.2 (0.7)	65.1 (2.5)	66.1 (6.3)	68.1 (15.6)
B. Average claiming age male (percentage in subgroup)
	PIA Q1	PIA Q2	PIA Q3	PIA Q4
Wealth Q1	62.8 (13.5)	63.5 (7.6)	63.6 (3.4)	64.0 (0.5)
Wealth Q2	64.3 (6.8)	65.7 (8.2)	66.3 (7.2)	66.9 (2.8)
Wealth Q3	64.6 (4.0)	66.0 (6.4)	67.2 (8.0)	68.2 (6.6)
Wealth Q4	66.1 (0.7)	66.5 (2.8)	67.2 (6.4)	68.8 (15.1)

One observation is that claiming ages rise with higher levels of liquid wealth (ceteris paribus), which is reasonable in that having more liquid wealth allows people to finance consumption after leaving the work force and still delay claiming to receive the higher lifetime benefits. Moreover, a higher retirement income results in later claiming, because a higher PIA is typically generated by a higher permanent wage rate, giving individuals an incentive to keep working.³⁵ Conversely, low-wage workers receive a relatively high replacement rate from Social Security benefits, giving them an incentive to claim. For instance, women having low wealth and retirement income at age 62 claim, on average, at age 62.8; this is because if they work longer they would earn little, and they lack sufficient wealth to self-finance delayed claiming. By contrast, wealthier women claim at age 68, on average, or more than 5 years later. They do so as they can earn high wages and have wealth sufficient to finance consumption during the period they delay claiming. Results for men are similar, but most claim later than women do.

It is worth noting that the high rates of early Social Security claiming cannot be fully explained by low levels of liquid wealth and retirement income, since only 13% of the simulated population is in the lowest wealth/income quartile and many more than that claim early. In addition, the deferred annuity provided by delaying the claiming age should be highly attractive from a purely financial perspective, especially for women. Nevertheless, most women still claim benefits before men do. To gain more insight into what drives these results, we turn next to a separate analyses of single versus couple households.

3.2.2 Singles

Figure 3 presents expected life cycle profiles by age for simulated single sample members. Panel A shows that, for them, wealth builds up gradually until age 56, when it amounts to about four times average consumption. Thereafter, singles begin to draw down assets to compensate for fewer hours of work. Their wealth levels are relatively flat between ages 65 and 80, for two reasons. First, the singles gradually claim their Social Security benefits between age 62 and 70, but they do not completely withdraw from the labor force. Instead, they work part time up to the earnings test exempt amount (this corresponds to about 19 hours per week for high school graduates and 14 hours for college graduates), which explains their relatively flat wealth levels up to age 70. Second, though the singles do begin drawing down their assets after age 70, mortality is also rising. For this reason, the pool of singles increasingly receives an influx of widows and widowers with higher wealth levels coming from their previously coupled state. Accordingly, the transition from couple to single tends to neutralize the aggregate effect of dissaving, which accounts for the relatively flat wealth levels to age 80.

Figure 3. Expected life cycle profiles: Single men and women.

These three panels show simulated life cycle profiles for singles at various ages. Panel A shows average levels of wealth, consumption, stock holdings, and the face value of life insurance holdings for men and women. Panel B shows work hours for men versus women. Panel C shows the percentage of households claiming Social Security benefits by sex at each age from 62 to 70. Averages are generated from 100,000 independent simulations based on optimal feedback controls. At each age, we extract the subgroup of male and female singles, and reported values are calculated as (conditional) means from the subgroup of singles. Averages for wealth, consumption, and stock holdings (Panel A) aggregate across men and women. See also notes to Figure 2

For singles, the share of financial wealth held in stocks is relatively constant over the life cycle (at 40%–60%), similar to the overall population. Singles’ average consumption is lower, but the same overall pattern prevails as taken together. We also see that singles have virtually no demand for term life insurance; they have no provisional and bequest motives as generally they have no children or partners to provide for after death (Hubener, Maurer, and Rogalla 2014) and they gain no (altruistic) utility from the transfer of wealth to the next generation. Only for single women age 30–40 is there a small positive demand for life insurance; this is generated by divorced women who must cover their children’s consumption and college education costs should they die young. There is very little life insurance demand among single men because the only case for which single men must take care of children is when they are widowers. Because young women’s mortality is very low, the few such cases do not change average life insurance demand overall.

Turning to labor supply patterns, Panel B of Figure 3 shows that single men work overtime (41 hours per week) early in their careers, and they gradually reduce time on the job to 30 hours just before retirement. From age 62 onward, single men start claiming Social Security benefits that provide them with a safe income stream for life. In conjunction with the possibility of receiving Social Security benefits and working without tax penalty up to the earnings test exempt amount, most single men reduce their average work hours sharply; after claiming, they work only part time (15–25 hours per week). Single women’s work hours are lower than men’s are because women receive lower wage rates on average. Accordingly, they are less willing to curtail their leisure time for higher consumption afforded by more work. Moreover, the single sample includes divorced women with children who are financially supported by their ex-husbands and have lower time budgets because of childcare responsibilities. Consequently, these women work less for pay compared with childless single women. This explains the slightly increasing gap of paid work hours between men and women age 35–45. In this age group, about 30% of single women have children. From age 45 to 55, the gender gap decreases because the children grow older and require less time (or leave home). After age 60, men and women exhibit very similar work patterns when children are out of the house.

Panel C of Figure 3 displays Social Security claiming patterns by age. Single women claim significantly later than their male counterparts do: the mean claiming age is 65.1 for men and 66.3 for women. Because of women’s higher life expectancy, the present value of the deferred annuity from delaying claiming is higher for women than for men. In addition, about 38% (26%) of single men (women) claim Social Security benefits at the earliest possible age of 62. These households do not find it appealing to take advantage of the additional life annuity benefits generated by delayed claiming. After the claiming peak at age 62, 6% of additional singles claim their benefits each subsequent year to 69, on average. Our detailed analysis shows that early claiming households build up relatively low wealth during their working lives and have low permanent wage rates in their 60s. Because Social Security replacement rates are progressive, the lifetime poor with low PIAs receive a higher replacement rate, enhancing their incentive to claim Social Security benefits early and work part time up to the earnings test exempt amount to augment overall income. About 16% (30%) of single men (women) delay claiming to age 70, the oldest claiming age in the model. Households with a higher permanent wage rate and consequently more financial wealth claim later because they can take advantage of the increased real annuity income from the delayed retirement credits. Moreover, they take advantage of still high wages and work longer; claiming later avoids the penalty from the earnings test.

3.2.3 Couples

Life cycle profiles for couples appear in Panels A–C of Figure 4, where we highlight several important differences compared with singles. Most importantly, wealth and consumption levels are much higher for couples because couple households have multiple members. The hump-shaped consumption path for couples is more pronounced compared with that of singles, which reflects children’s arrivals and departures over the life cycle. Interestingly, younger couples build up wealth more quickly than singles do: between ages 20 and 30, the average level of wealth for couples increases by about 14% per year, but by only by 8.5% annually for singles. Moreover, wealth relative to family size is higher for younger couples: for instance, at age 30, the ratio of average wealth to consumption for couples is 3.5, but only 2.0 for singles. This is because of couples’ higher precautionary saving motives resulting from uncertainty in family status (i.e., the births of additional children or going through a divorce), as well as having to save for their children’s college education. After about age 55, household wealth peaks and children start to leave the home, so differences between singles and couples shrink.

Figure 4. Expected life cycle profiles: Couples.

These three panels show simulated life cycle profiles for couples at various ages. Panel A shows average levels of wealth, consumption, stock holdings, and the face value of life insurance holdings for married men and women. Panel B shows work hours by sex. Panel C shows the percentage of households claiming Social Security benefits by sex, at each age from 62 to 70. Averages are generated from 100,000 independent simulations based on optimal feedback controls. At each age, we extract the subgroup of couples and reported values are calculated as (conditional) means in the subgroup of couples. Averages for wealth, consumption, and stock holdings (Panel A) aggregate across couples. See also notes to Figure 2.

Couples’ demand for life insurance is hump shaped, with insurance purchased mainly on the husband’s life: average face values peak at around $135,000 at age 37, when most couples have children and many women reduce their paid work hours substantially because of childcare demands. Demand for life insurance on the wife’s life is positive though lower than on the husband’s, topping out at $45,000. One explanation for the lower level is that female mortality is substantially lower than men’s; another is that men earn more than women, so a widower can more easily provide for the family than can a widow. In addition, the remarriage rate of widowers with children is more than twice as high as for widows, so widowers are much more likely to find a new partner to help with childcare and provide a second income. The demand for life insurance on women age 30–50 is mainly driven by couples with more than two children. In this instance, the wife’s death would impose a substantial burden on the husband, because he would need to curtail his work hours to care for the young children. Life insurance purchases of both partners combined with accumulated liquid savings cover the risk that both parents might die at once.

Interestingly, the demand for life insurance during retirement is zero for both partners. Because of generous Social Security widow benefits, retirement income proves to be rather symmetrically distributed between both partners, so only a minor portion of retirement income is lost when one spouse dies. If the husband dies first, his surviving widow receives 100% of his Social Security benefit as his widow, an amount typically higher than her own (and her spousal) benefit. In addition, the surviving spouse retains the household’s remaining liquid wealth, and when single, the widow requires lower consumption. Therefore, the death of one partner need not cause a large consumption shortfall that would need to be hedged by life insurance purchases.³⁶

The work-hour pattern for couples also differs importantly from that of singles. In their early 20s, both husbands and wives work for pay up to 50 hours per week. In contrast to single men, husbands reduce work to 43 hours around age 40, and they maintain this level until retirement, effectively working each week about 5 hours more than single men do. Wives, however, reduce their paid work hours in their late 30s to about 26 per week. Between age 40 and 55, women gradually boost their paid work to 30 hours per week, when children are older and require less home time. Despite their high work hours at younger ages, wives work for pay about 3 fewer hours per week over the life cycle, compared with single women. This specialization of work hours within the family occurs because women’s wage rates are lower than men’s, on average, and they fall further on the arrival of children. Thus, the wife shoulders most of the unpaid burden of childcare and home-production time, and she works less for pay than the husband does. Similar to the situation for singles, both husband and wife start to reduce their market work substantially in their 60s.

It is worth noting that couples’ Social Security claiming patterns in our model also differ markedly from those of singles. About 64% of married women optimally claim their own old-age Social Security benefits at the earliest possible age of 62; also, their mean claiming age is 64.3, about 2 years earlier than that of single women. By contrast, 40% of married men delay claiming up to age 70, and their mean claiming age of 66.3 is 1.2 years higher than that of single men. There are several reasons for these differences. First, married women’s PIAs are considerably lower than their husband’s. In addition, married women are eligible for spousal benefits and later for relatively generous widow benefits (100% of their husbands’ benefits). The Social Security claiming rules also permit the wife to switch from her own old-age retirement benefits to spousal benefits and/or to widow benefits when the husband passes away. Spousal benefits increase for every year of delaying after age 62 by about 8% (up to the normal retirement age 66). Because of these switching possibilities and particularly because of the generous widow benefits, early claiming for married women only reduces their retirement benefits up to the point of the husband’s death.

As a result, for most couples, the optimal lifetime strategy is for the wife to claim her own relatively lower benefits early, and to claim spousal benefits later if they are higher. In addition, the husband will claim his own old-age benefits relatively late in life. This increases his own benefits and his potential widow’s benefits after his death. Because of the high probability that the wife outlives her husband, better widow benefits are important to maximize the couple’s joint lifetime utility.

Such a strategy also effectively hedges longevity risk. If one partner dies, the surviving spouse receives the high benefits of the husband (either directly or as widow benefits) for the rest of his life. If both spouses survive for a long time, they continue to receive both incomes, (i.e., the own benefit of the husband and the spousal or own benefits of the wife). Even though the benefits for the wife are smaller, the couple profits from the consumption scaling of not having to consume twice as much as a single person.

Coincident with the results for single men, married men’s higher permanent wage rates generate later claiming patterns, on average. The few households (24%) in which wives delay claiming to age 70 also have very high wage rates. These high-earning women seek to remain in the workforce to generate high labor income and take advantage of the delayed retirement credit by claiming later.

3.3. Effect of widowhood and divorce on women’s life cycle patterns

In this section, we analyze married mothers’ optimal behavior in the event of an (unexpected) divorce or partner’s death, according to key financial outcomes (stock holding, wealth, and Social Security claiming) and hours of paid work. To do so, we present average results from 100,000 simulated life cycles based on optimal feedback controls, and focus on those paths of particular interest. Beginning with a 20-year-old single woman who marries at age 25 and has her first (second) child at age 30 (33), two cases are of special interest. In the first, the husband dies when she is age 36, whereas in the second, she becomes divorced at age 36: either way, she never remarries, or the children leave home when their mother is age 48 and 51, respectively. Such a comparison allows us to study the short- and long-term implications of an unexpected loss of a partner (ceteris paribus).

The first consequence of the husband’s exit is the loss of his earnings. Prior to the shock, the husband contributed an average of 70% of the couple’s total income (about $45,000 per year), resulting from his higher wage rate and the family’s specialization of labor. The husband worked an average of 42 weekly hours, whereas the wife worked only 21 hours for pay and devoted substantial time to childcare. Accordingly, the loss of the husband has an important effect on income flows. A second consequence is that, without the husband, the family now consists of only three members, so its consumption needs decline by 21% (the scaling factor falls from 2.34 to 1.85). Nevertheless, the family still requires 79% of previous expenditures to maintain the same per-member consumption. A third result is that the mother is now solely responsible for the children’s educational costs.

To explore the differences in financial consequences for widowhood or divorce, Figure 5 outlines life cycle profiles for the two cases. Panel A compares their average financial wealth and allocations to risky stocks, Panel B depicts women’s work hours, and Panel C presents the distribution of claiming ages.

Figure 5. Life cycle profiles of widows and divorced women.

These three panels show simulated life cycle profiles for two family state patterns for 36-year old mothers of two children: either they experience the (unexpected) death of the husband or they are (unexpectedly) divorced. Until age 36, both the widow and the divorcee have the same family states, namely being single from ages 20 to 24 and married from ages 25 to 35; followed by a first child at age 30 and a second at age 33. For both the divorcee and the widow, children leave home when the mother is age 48 and 51, respectively. Panel A shows average levels of wealth and stock holdings; Panel B displays average work hours; and Panel C shows the percentage of widowed and divorced women claiming Social Security benefits at each age from 62 to 70. Profiles are based on averages over 100,000 independent simulations (both high school and college education) of stock returns and income shocks. Simulation paths are based on optimal feedback controls from the baseline specification of the life cycle model. See also notes to Figure 2.

In their early 20s when both women are single, they work full time and build up only little savings. Upon marriage, the family starts to accumulate financial wealth and the mothers reduce their work hours substantially after children arrive. After the husband dies, in Panel A, the widow receives a large life insurance payout of about four times the couple’s annual income before the husband’s death ($153,000). This allows her to reduce her work hours and finance consumption by depleting her wealth over the next 15 years until the children leave home. Interestingly, though her financial wealth is about twice as high as before the husband’s death, she reduces her risky stock share substantially in absolute, as well as relative, terms (from 65 to 30%). This de-risking occurs because the mother uses the life insurance payment as a withdrawal plan over the next 15 years to help finance her own and her children’s consumption.

By contrast, the woman who undergoes divorce experiences about a 50% reduction in her financial wealth, though she does receive child support payments from her former spouse. After the divorce, the mother does reduce her labor market hours to devote more time to childcare, but she reduces her work time much less than the widow. In addition to child support from the ex-husband and labor earnings, she too depletes her financial wealth as long as the children are at home. Yet her financial wealth drawdown finances only a small portion of consumption, compared with the widow, and her equity exposure remains higher, at about 60%.

It is also instructive to compare the long-term consequences of widowhood or divorce on other outcomes, as well. Claiming age patterns are reported in Panel C of Figure 5, where we see that the divorcee claims Social Security benefits much earlier than the widow does, by about 1.5 years on average. This occurs for two reasons. First, because the divorced woman works much more for pay, her PIA and, hence, accrued Social Security benefits are higher than the widow’s, which encourages earlier claiming. Second, the widow’s financial wealth is much higher, so she can delay claiming to get the higher lifelong annuity. The situation for the two women is quite different in other regards, as well. The widow is better off than the divorcee because of the life insurance payout. This is because there is no divorce insurance (at least in our model) and child support payments amount to much less than the life insurance.

3.4 Effects of education and children on key financial outcomes

Next, we explore how differences in education and children can influence optimal claiming patterns and allocations of stocks, bonds, and life insurance. Using our simulation results, we differentiate between lesser-versus better-educated households, and couples with no children versus those with at least two children. Results appear in Tables 4–6, which illustrate findings for claiming ages, stock fractions, and life insurance demand.

Table 4. Effect of education and children on Social Security claiming decisions.

Table 4 shows the frequency of Social Security claiming ages by age and sex. The life cycle simulation method is identical to that used in Section 3.2 (see also notes to Figure 3). Results are shown for two education subgroups (high school and college), as well as childless couples, versus those with 2+ children.

Claiming age	High school education		College education		Couples without children		Couples with 2+ children
	Women	Men	Women	Men	Women	Men	Women	Men
62	57%	32%	36%	38%	64%	31%	64%	33%
63	3%	4%	3%	4%	3%	3%	2%	4%
64	4%	7%	4%	4%	3%	3%	3%	5%
65	4%	6%	3%	3%	2%	3%	2%	4%
66	2%	9%	2%	3%	1%	9%	1%	5%
67	3%	5%	2%	3%	1%	1%	1%	3%
68	4%	5%	4%	4%	1%	2%	1%	4%
69	4%	5%	7%	4%	2%	4%	1%	5%
70	20%	27%	39%	37%	24%	44%	26%	37%
Avg. claiming age	64.6	65.8	66.2	66.0	64.3	66.5	64.4	66.1

Table 6. Effect of education and children on normalized life insurance values.

Table 6 shows average life insurance face values by sex and age. Values are multiples of the spouse’s income assuming full-time work (40 hours per week). The life cycle simulation method is identical to that used in Section 3.2 (see also notes to Figure 3). Results are shown for two education groups (high school and college), as well as childless couples, versus those with 2+ children.

Age	High school education		College education		Couples without children		Couples with 2+ children
	Women	Men	Women	Men	Women	Men	Women	Men
25–34	0.32	1.15	0.32	1.12	0.00	1.35	0.85	2.76
35–44	0.80	1.79	0.65	1.64	0.15	2.05	1.72	3.17
45–54	0.57	1.65	0.41	1.52	0.25	2.13	1.17	2.57
55–64	0.04	1.32	0.03	1.23	0.02	1.81	0.09	1.99
65–74	0.00	0.20	0.00	0.18	0.00	0.29	0.01	0.31

Turning first to the claiming decision, Table 4 indicates the fraction of persons taking Social Security benefits between ages 62 and 70, arrayed by education and number of children. Here we see that the model predicts that less-educated women claim much earlier, with an expected claiming age of 64.6 versus 66.2 for the college educated. By contrast, men’s patterns are more similar, with nearly identical average claiming ages (65.8 and 66.0, respectively). The Social Security replacement rate is particularly generous for low-wage women, whereas higher-earning men and college-educated women have more of an incentive to remain employed. Again, it is worth noting that men’s average optimal claiming age is much higher than for women, driven by the availability of spousal and survivor benefits for married women.

When we compare childless couples and those with children, we see that claiming patterns are remarkably similar: 64% of the women claim as early as possible in both groups, and women’s expected claiming ages are also nearly identical with mothers of two or more children who claim only 0.1 years later than childless wives do. For men, having two or more children has a small effect, reducing the average claiming age by 0.4 years. Overall, the model implies that adults’ educational levels and, hence, their earnings, have a stronger effect than do children, in terms of when people exercise their retirement-benefit claiming options.

The results for the share of financial wealth held in equity appear in Table 5, which also reports differences by education. Here we see that both education groups hold nearly the same portion of their portfolios in equities during their work lives. Before retirement, the less educated dissave faster than the college educated do because Social Security offers the former such a high replacement that they need little liquid savings. As a result, the less educated hold relatively more of their overall financial wealth in nonrisky transaction accounts to finance their current consumption, reducing the funds available to invest in stocks. Turning to couples, we see that the young and the old hold similar stock fractions. Nevertheless, middle-aged couples with children are much less invested in equity: specifically, couples aged 45–64 with children hold 9 to 11 percentage points less in equities. This can be explained by the fact that, compared with childless couples, they hold more nonrisky assets in their transaction accounts to finance higher consumption expenditures. In addition, they must use part of their saving to pay for the children’s college educations, further reducing the relative amount of overall financial wealth available for stock investments.³⁷ Overall, the equity share does vary moderately with education and family status, through the profile is rather smooth by age in contrast to other studies of optimal portfolio choice that typically generate decreasing equity profiles over the life cycle (Cocco, Gomes, and Maenhout 2005; Love 2010; Gomes, Kotlikoff, and Viceira 2008).

Table 5. Effect of education and children on stock holdings as a fraction of financial wealth.

Table 5 shows average stock holdings as a fraction of financial wealth by age group. The life cycle simulation method is identical to that used in Section 3.2 (see also notes to Figure 3). Results are shown for two education subgroups (high school and college), as well as childless couples, versus those with 2+ more children.

Age	High school education	College education	Couples without children	Couples with 2+ children
25–34	51%	50%	60%	62%
35–44	59%	61%	64%	59%
45–54	53%	55%	58%	49%
55–64	43%	52%	53%	41%
65–74	45%	51%	51%	46%
75–84	44%	47%	47%	44%

Life insurance holdings also vary over the life cycle and by household type (see Table 6). The peak age for purchase is clearly when the children are young; after age 65, there is effectively no demand for further insurance because Social Security benefits provide a generous replacement rate to those losing their spouses. Those with lower wages also purchase relatively more life insurance as a multiple of their full-time labor income, compared with the college educated. This is because high school-educated couples with children seek to insure against the loss of the husband’s income in the event of his death. Although wives could return to the labor market, their low wages would be less than required to smooth consumption, compared to the college-educated women. Couples without children buy little insurance on wives, but they do carry an insurance face value of up to 213% of the full-time labor earnings on the husbands. This is because the wife’s low wage rates induce her to spend more time at home; the loss of her husband may induce more work on her part, but his demise would still impose substantial financial risk on the couple, inducing her to seek more insurance on his life. By contrast, couples with 2+ children demand much more insurance while the children are young, particularly on the husbands whose face values exceed 300% of full-time labor earnings, inasmuch as the wives are devoting much of their time to home production and not earning much. In contrast to childless couples, a substantial amount of life insurance is also bought on mothers’ lives because their possible death is a major risk to husbands with dependent children.

3.5 Effects of neglecting fertility risk and individual work histories for retirement benefits

In Section 3.3, the effect of two demographic shocks—death and divorce—are analyzed by focusing only on mothers with children. Next, we analyze how uncertainty about fertility affects optimal life cycle patterns in a simulation where we shut off uncertainty about whether and how many children will arrive, and we compare these new results with our base case to evaluate the effect on optimal saving, investment, and insurance outcomes. In this section, we also report the effect of simplifying the Social Security benefit rules in line with prior studies, to highlight the effect of integrating earnings histories that are more realistic.

We simplify fertility uncertainty by re-characterizing each family state as an age-dependent prespecified frequency of having zero, one, two, or three children, where these frequencies are taken from our base case simulations. For example, a married couple at age 40 is assumed to have zero children with a probability of 31.2%, one child with 30.1%, two children with 23.4%, and three children with 14.8%. We then assume that households know the frequencies (which are continuous in age) in advance, so this eliminates fertility risk.³⁸ In the base case model, several quantities depended on the ex post number of children, including the wage rate, childcare time, and the consumption-scaling factor. Without fertility risk, we use expected values for these based on the age-dependent child frequencies.

To evaluate the effect of eliminating fertility risk on key financial decisions, we construct subsamples from simulating our optimal life cycle models with and without fertility risk, and we compare summary statistics for the key outcomes. For financial wealth, stock fraction, life insurance face values, and work hours, we construct a cross-sectional sample by selecting all married couples at age 25, 35, 45, and 55. When households are selected at several ages, we treat these as independent observations. For the claiming age statistics, we construct a longitudinal sample by selecting paths of couples married throughout the retirement transition, or from the husband’s age 62 to the wife’s age 70. Table 7 presents sample averages, standard deviations, and median values of these quantities.

Table 7. Effect of fertility risk, flexible claiming, and PIA formulation.

Table 7 shows sample statistics (mean, standard deviation, and median) of key financial decisions for couples in the base case model (upper panel), for the case with no fertility risk (middle panel), and for the case with a fixed Social Security replacement rate. For the statistics on financial wealth, stock fractions, life insurance face values, and weekly work hours, cross-sectional samples are constructed from life cycle simulations based on optimal feedback controls (see also Figure 3) by selecting married couples at ages 25, 35, 45, and 55. For the average claiming age, a longitudinal sample is constructed by selecting married couples from the time that the husband turns 62 (earliest possible retirement age) to when the wife turns 70 (last possible retirement age).

A. Base case
	Financial wealth	Stock fraction (%)	Life insurance		Work hours		Claiming age
			Husband	Wife	Husband	Wife	Husband	Wife
Mean	176	56.9	109	24	42.5	30.1	66.3	64.3
Std. deviation	150	14.9	92	39	4.4	10.2	3.5	3.5
Median	132	60.6	86	3	43.2	30.7	67.0	62.0
B. No fertility risk
	Financial wealth	Stock fraction (%)	Life insurance		Work hours		Claiming age
			Husband	Wife	Husband	Wife	Husband	Wife
Mean	146	56.1	126	32	42.1	30.3	66.3	64.4
Std. deviation	119	14.1	90	35	3.7	5.1	3.5	3.5
Median	111	59.4	106	26	43.1	30.9	67.0	62.0
C. Fixed replacement rate
	Financial wealth	Stock fraction (%)	Life insurance		Work hours		Claiming age
			Husband	Wife	Husband	Wife	Husband	Wife
Mean	210	58.0	101	22	42.6	31.0	69.8	62.6
Std. deviation	177	14.4	92	39	4.1	9.1	0.6	1.9
Median	158	61.2	79	1	43.1	31.1	70.0	62.0

The evidence shows that children constrain the household’s time budget and require higher consumption expenditures, so if a couple could have many children, this exposes them to an important amount of risk. In fact we predict that eliminating fertility risk can have a substantial effect on financial wealth: the average falls by 17% (from $176,000 to $146,000), compared with the base case model with fertility risk (Panels A and B, Table 7). In other words, precautionary savings will rise when unexpectedly high fertility is possible, a phenomenon not present in models lacking fertility risk. As consequence of lower financial wealth in the fertility-certainty case, the household’s stock fraction is also slightly lower. This is because the household keeps a larger fraction of financial wealth in risk-free assets to finance next year’s consumption. Moreover, since financial wealth is lower, the demand for life insurance is higher, to endow a widowed spouse with sufficient liquid funds. The average face value for the husband (wife) rises to $126,000 ($32,000) versus $109,000 ($24,000) in the base case.

The effect of fertility risk on work hours is almost negligible, as is evident in Table 7. When fertility risk is eliminated, average work hours are basically unchanged for women and men only reduce their weekly workload by 0.4 hours, as there is no need to build up precautionary savings for additional children. Yet the standard deviation of work hours across the sample is greatly reduced, by 16% for husbands and 50% for wives. This is because all households have the same number of children when fertility shocks are ruled out. Without fertility risk, the average work hours profile is similar as in the base case but there is no variation across households.

Social Security benefit-claiming patterns are also mostly unaffected by fertility risk. Both the average claiming age and the variability across the sample are similar for both husbands and wives. This should come as no surprise, since the retirement decision is made well beyond the age when household fertility risk has been realized. In sum, eliminating fertility risk mainly affects saving behavior, asset allocation, and life insurance demand. Both the work hours and claiming decisions are little affected, on average.

We illustrate the importance of tracking individual work histories for computing the Social Security benefits (PIA) in Panel C of Table 7. Here we report results for a simplified model with fixed benefit-replacement rates: more precisely, this exercise follows prior research in modeling Social Security benefits as a prespecified percentage of the last permanent earnings assuming a work effort of 40 hours per week. To compare results with our base case, we calibrate the replacement rates such that average benefits for claiming at the FRA in the simplified setting are the same as those in the base case. This produces replacement rates of 52.6% (48%) for high school-educated men (women), and 41.6% (40%) for college-educated men (women). We allow for early and delayed claiming using the same adjustment factors as in the base case. The key difference between the two versions is that the level of retirement benefits for households is much riskier in the new simpler version because the retirement income is now highly sensitive to income shocks at the time of claiming.

Compared with our more realistic base case, we see two key differences: households accumulate 20% more wealth in the simpler version ($210,000 versus $176,000), and men’s claiming ages are much later (age 70 versus 66). Both are a direct consequence of the much higher risk to which the household is exposed. In other words, this experiment demonstrates that a realistic modeling of Social Security benefits has powerful consequences for household financial and retirement decisions.

3.6 Effects of Social Security family benefits

Finally, we use our calibrated theoretical model to examine the effect of family-related Social Security rules—namely spousal and widow benefits—on couples’ claiming behavior and other key financial outcomes. To illustrate this, we repeat the optimization procedure but eliminate spousal benefits, and, in a second case, we eliminate widow benefits. Based on these new optimal feedback controls, we generate 100,000 simulations for couples. Table 8 reports results. Compared with the base case, the first row (second column) indicates that wives’ claiming ages are unaffected by eliminating spousal benefits. The reason is that most women have acquired substantial retirement benefits themselves and do not depend on spousal benefits. By contrast, if widow benefits were eliminated (Column 3), this would have a substantial impact on claiming patterns: married women would on average delay claiming by 3.6 years (to age 68.6 instead of 65.0 in row 1) and work 14% more hours (in row 5). This is because wives would be exposed to a substantial risk of uninsured widowhood if they only had their single annuity on which to rely over their relatively long old age.

Table 8. Simulated behavioral effects of Social Security family benefits.

Table 8 shows average claiming ages, life insurance face values, paid work hours, and asset allocations for three cases: the base case, one without spousal benefits, and one without survivor benefits. The same simulation method is used as in Section 3.2 (see also notes to Figure 3). Simulations for the latter two cases are based on new optimal feedback controls that consider the absence of the indicated benefits. The last column reports results for the base case for never-divorced couples. Averages are shown for the subgroup of couples in the 50–69 age bracket.

	Base case	No spousal benefits	No widow benefits	No family transitions
Wife’s avg. claiming age	64.3	64.4	68.3	64.4
Husband’s avg. claiming age	66.3	66.3	65.0	66.3
Average over ages 50–69
Wifes life insurance ($000)	4.5	4.5	4.6	7.5
Husband’s life insurance ($000)	63.2	63.2	80.7	70.8
Wife’s work hours	25.7	25.8	29.9	26.4
Husband’s work hours	34.7	34.6	33.1	35.0
Stock allocation (%)	48	48	48	45

Table 8 also shows that men’s claiming patterns and work hours would be relatively unaffected by cutting spouse benefits, because few men receive spousal benefits in any event. By contrast, if widow benefits were eliminated, men would claim 1-year earlier (row 2) and work 4% fewer hours (row 6). This is because the husband’s additional work and deferred claiming would no longer contribute to enhanced widow benefits after his death. Moreover, the household would optimally buy 28% more life insurance on the husband (row 4); this would continue until the wife claimed her Social Security benefit because her own benefit thereafter would be sufficient to support her in old age. Interestingly, neither policy simulation has any measurable effect on the household’s equity share (last row). This is intuitive because the household can alter its Social Security claiming patterns and in effect “purchase” a higher annuity benefit over the remaining lifetime. In sum, the structure of Social Security options interacting with claiming and work patterns provides the household an alternative to saving more and changing its stock allocation.

Having shown that family-related Social Security benefits (especially for widows) have an important effect on key financial outcomes, we can also illustrate the relative importance of family-status uncertainty. To address the effect of family risk, we repeat the simulation for the base case but now focus only on paths of couples that never divorce; here, the only change in family status that matters is the transition from being married to widowed. The final column of Table 8 shows that average claiming ages are close to the base case: husbands claim on average at age 66.3, and wives at 64.4. Other outcomes are not much different from the base case, as well. Accordingly, we conclude that, for couples, the family-related Social Security benefits have a potent effect on claiming decisions, whereas the risk of uncertain family transition is less consequential.

4. Does Our Model Imply Reasonable Claiming Patterns?

As noted previously, four key results flow from our normative model regarding Social Security claiming patterns. First, married women will optimally claim much earlier than single women do. Second, married women optimally claim much earlier than married men do. Third, more-educated women claim later than less-educated women do. Fourth, children have little effect on men and women’s claiming patterns. Estimating a full structural model is beyond the purview of the present study, but the reader may find informative a brief analysis of actual Social Security claiming patterns in the HRS, a nationally representative longitudinal survey of Americans over the age of 50 followed over the period 1992–2010.³⁹ To this end, we define the Social Security claiming age as the number of months elapsed between turning age 62 and benefit receipt. We then regress this measure on a vector of explanatory variables that our model indicates should be importantly associated with claiming patterns: these include sex, marital status, number of living children (0, 1, 2, 3+), and educational attainment (some college versus none). To test whether claiming patterns differ by sex, all variables are interacted with the male (0/1) dummy. Tobit coefficient estimates and standard errors are reported in Table 9, where the reference category is female.

Table 9. Empirical analysis of Social Security claiming behavior.

Table 9 reports Tobit regression results of a multivariate model where the dependent variable is the number of months after age 62 that the respondent claimed Social Security benefits. Control variables are measured at age 62 and include Married, self-reported being married (versus single); College, at least some college (versus none); Children, number of living children (versus 0); Male (versus female); other terms are interactions as indicated. The mean of the dependent variable is 16.9 months, for an average claiming age of 63.4. The analysis uses the Health and Retirement Study (HRS) sample constructed by Shoven and Slavov (2014) and extended to include the 2010 wave. Standard errors are given in brackets below the coefficients, and the number of stars indicates levels of significance (10%, 5%, and 1%). See the text for further discussion.

Married	−6.915 (1.183)	***
College	4.527 (0.944)	***
1 child	3.239 (2.698)
2 children	3.159 (2.208)
3+ children	2.100 (2.07)
Male	−0.525 (3.183)
Male × married	6.888 (1.925)	***
Male × college	1.077 (1.343)
Male × 1 child	−1.862 (3.969)
Male × 2 children	−2.982 (3.304)
Male × 3+ children	−0.860 (3.136)
Constant	15.235 (2.023)	***
Number of observations	3,542

The average claiming age in the HRS dataset is 63.4,⁴⁰ yet married women claim substantially earlier (−6.915 months) than do single women, and the result is statistically significant. This agrees with our theoretical predictions, though the magnitude is a bit smaller than the simulations. Additionally, better-educated women claim later than less-educated women do (+4.527 months), again a statistically significant finding consistent with our hypotheses. Married men claim later than married women do (+6.888 months), and the result is again statistically significant and compatible with our predictions. Finally, we find no significant effects of children on women’s (or men’s) claiming patterns, a result that again conforms to the model predictions. In sum, the key variables in our model are also empirically important.

Our model also predicts a substantial peak in claiming behavior at the earliest age of 62, with the highest peak for married women and lowest for single women. When we compare claiming frequencies from the HRS for married versus single males and married versus single females, the results are consistent: over half (55%) of married females claim at age 62, but only 44.7% of married men do. The claim rate of single females age 62 is 40.6% or 3% less than for single males (see Online Appendix E). We have noted that this early claiming peak is optimal for people with low income and low wealth, and for married women whose husbands have substantially higher lifetime incomes. Therefore, early claiming by a substantial portion of the older population proves not to be a puzzle after all.⁴¹

5. Conclusions

This study shows that Social Security rules and family risk have important effects on optimal life cycle household saving and stock/bond allocation patterns, work/retirement decisions, and life insurance purchases. Our rich and complex normative model builds on previous research that included stochastic equity returns and labor income, as well as mortality risk. We extend the literature by incorporating the effect of demographic shocks on household budgets, such that the costs of children influence peoples’ direct and indirect time and money constraints. Additionally, we track men and women before, during, and after marriage, and we model the effect of having children, as well as college costs. Most importantly, our formulation of Social Security benefit options is far more realistic than in previous studies, which assumed retirement benefits were a fixed proportion of the last labor income as of some prespecified date. Not only do we model own benefits as a function of own lifetime earnings histories and benefit-claiming ages, but we also model spousal and survivor payments. These factors interact in rich and complex ways with Social Security benefit options, which in turn shape optimal saving, portfolio, and work decisions over the life cycle.

We calibrate our model with realistic evidence on time use, demographics, and wage rates, and we show that it generates reasonable saving and wealth profiles along with low and stable equity fractions, compatible with evidence. Moreover, consumption follows a hump-shaped pattern, rising over the work life and falling around retirement, consistent with the data. In contrast, previous life cycle models that shut down endogenous work effort, claiming, and retirement had much flatter consumption paths. We also show that having children reduces household equity holdings, and married women optimally work for pay much less compared with their single counterparts. Demand for life insurance is hump shaped and mainly purchased for men. Current Social Security rules induce married women to claim retirement benefits much earlier than single women and married men, consistent with our model’s predictions. Finally, we show that Social Security rules matter, in that incorporating fertility risk and individual work histories in the benefits formula is essential for better understanding of household finance and retirement patterns.

The key lesson from our analysis is that adjusting work and benefit-claiming patterns offers households an alternative to altering their financial portfolios and consumption patterns when shocks arrive. For this reason, analysts seeking to better understand why people work, save, invest, and consume will do well to acknowledge family status and Social Security as powerful behavioral influences of life cycle portfolios. In future work, it would be interesting to extend our model to incorporate health risk and housing shocks, which could influence not only claiming but also other household financial decisions. ⁴⁶