How do subjective mortality beliefs affect the value of social security and the optimal claiming ages?

2.1 The US Social Security program

Single individuals can claim their retired worker benefits earned through their own contributions from age 62 to 70, subject to an earnings test.2 Those who claim benefits at their full retirement age (FRA) receive the primary insurance amount (PIA).3 Benefits are adjusted for those who claim before or after their FRA. Benefits are reduced by 5/9 of 1% for each month retired workers claim before their FRA up to 36 months, 5/12 of 1% thereafter, and are increased by 2/3 of 1% for each month retired workers claim after their FRA until age 70.4 At a 3% real interest rate, the increase in monthly benefits from age 62 to 70 is approximately actuarially fair for retired workers with population average mortality born after 1943 (Sass et al., 2007).

Married couples can claim two additional categories of Social Security benefits: spousal benefits and survivors' benefits. The spousal benefits can be claimed by the spouse of the retired worker after age 62, provided that the spousal benefits exceed the retired worker benefits based on the spouse's own earnings record, and given that the retired worker has already claimed. The spousal benefits are 50% of the retired worker's PIA if the spouse claimed at or after the FRA. The spousal benefits are subject to a reduction of 25/36 of 1% for each month claimed before the spouse's FRA, up to 36 months, 5/12 of 1% thereafter. The survivors' benefits equal 100% of the deceased spouse's retired worker benefits if claimed at or after the FRA of the living spouse, which can be greater or less than the PIA, depending on the initial claiming age of the deceased spouse. The benefits can be claimed as early as age 60 and the reductions vary with the birth cohort of the living spouse. The benefits are bounded by a floor of 71.5% of the deceased spouse's PIA if claimed before the FRA of the living spouse and 82.5% of the deceased spouse's PIA if claimed at or after the FRA of the living spouse. The reductions in spousal and survivors' benefits are also approximately actuarially fair for older workers with population average mortality (Sass et al., 2007).

2.2 Research on subjective mortality beliefs

All previous studies of subjective mortality beliefs focus on whether individuals could form unbiased estimates about their life expectancy and have not considered the degree of uncertainty surrounding their age of death. Hurd and McGarry (1995) analyze subjective survival probability estimates provided by participants in the 1992 wave of the HRS. They show these estimates aggregate to life table probabilities and covary with risk factors. Hurd and McGarry (2002) further show that the above beliefs are predictive of subsequent mortality over a 2-year time horizon and respond in the panel to the receipt of new information. Sondergeld et al. (2002) find that, on average, individuals approaching retirement mis-state life expectancy by only small amounts.5 In contrast, Elder (2007) finds that HRS participants are pessimistic about their chances of surviving to age 75 but optimistic about their chances of surviving to age 85 relative to the predictions of life tables. He finds that higher subjective survival probabilities are associated with households holding a larger proportion of their wealth in stocks. But this relationship does not hold for individuals who assess their survival probability at 100%, suggesting that these individuals may differ in some way from those whose responses are close to, but less than, 100%. Problems arise in the interpretation of focal point responses.6 Gan et al. (2005) propose a Bayesian updating methodology for recovering the individual's true underlying beliefs. Although their approach produces life tables that vary with individuals' responses, it is not self-evident that these tables better reflect individuals' true underlying beliefs than the raw responses.

2.3 Research on Social Security claiming decisions

As discussed above, the Social Security program is an actuarially fair system, but the expected present value calculations do not incorporate the value of longevity insurance provided by Social Security benefits. An extensive literature, for example, Mitchell et al. (1999) and Brown and Poterba (2000), shows that the value of an annuity substantially exceeds its money's worth, defined as the expected present value of the income stream, discounted by an interest rate and annual survival probabilities, divided by the premium paid. This is because households face an uncertain age of death. If they choose not to annuitize, they run the risk of outliving their wealth if they decumulate too rapidly or forgoing valuable consumption opportunities if they decumulate too slowly. Given plausible preference parameters, Coile et al. (2002) and Sun and Webb (2011) find that it is optimal for retired workers to postpone claiming for a number of years. Coile et al. (2002) show that the optimal claiming age for a hypothetical single male born in 1930 could be as late as age 65, depending on his wealth level. Sun and Webb (2011) calculate the optimal claiming ages for typical nonbudget-constrained married couples born in 1946 are between age 67 and 70. The above papers assume that individuals' annual mortality risk equals that predicted by annuitant or SSA cohort life tables. Brown (2003) and Sun and Webb (2011) show that although mortality rates vary significantly by socioeconomic status, they have little effect on annuity valuation. Although the average household in high mortality groups faces a smaller chance of surviving to advanced old age, it will still want to restrict current consumption to safeguard against that possibility, should it choose not to annuitize.

Several researchers have undertaken empirical analyses of Social Security claiming decisions. Hurd et al. (2004) find a small but significant relationship between the subjective mortality beliefs and the age of claiming, as do Coile et al. (2002). They show individuals with low probability of surviving to age 75 are more likely to make the joint decision of retiring and claiming Social Security benefits early. But the average delay is much less than theoretical calculations suggest is optimal. Recently, the structural framework has made significant advancements in explaining the early claiming behaviors. Gustman and Steinmeier (2005) estimate a structural model of Social Security claiming decisions and attribute the prevalence of early claiming behaviors to heterogeneity in rates of time preference. Benitez-Silva et al. (2007) attribute the prevalence of early claiming behaviors to uncertainty as to whether scheduled benefits will be paid. Gustman and Steinmeier (2015) show that claiming decisions are influenced by retired workers' beliefs regarding the sustainability of the Social Security program, the expected returns of financial assets, and the utility aggregation of married couples. Pashchenko and Porapakkarm (2023) show that the low demand for public annuity can be explained by the combination of bequest motives and impatience. Bairoliya and McKiernan (2022) find that claiming frictions, including Social Security marital rules, labor income and medical expenditure risks, misbeliefs in mortality rates and in institutional details, and bequest motives, together account for 78% of early claiming behaviors in the data.

3.1 Focal point responses

We investigate individuals' subjective mortality beliefs using the HRS data. The HRS is a national representative panel of older individuals first interviewed in 1992. Individuals have been reinterviewed every 2 years. Starting from the first wave, individuals have been asked to self-assess their probabilities of surviving to age 75 and 85. However, the responses suffer from focal point biases, with considerable bunching of responses at 0%, 50%, 100%, and so on. It is unclear how to understand these responses. Do individuals literally mean that they rate their survival probability at 0% or 100%, or somewhere close to these numbers? In 2008, individuals who gave focal point responses were asked follow-up questions designed to elicit information regarding their true underlying beliefs. We use the answers to investigate the nature and extent of focal point biases.

In 2008, individuals aged less than 65 in the HRS were again asked to self-assess the probabilities that they would live to age 75 and 85 ( and hereafter). As in previous waves, a substantial proportion of individuals provided focal point estimates of 0%, 50%, 100%, and so on. In contrast to previous waves, individuals were asked follow-up questions that were designed to test the degree of precision of their estimate. The series of questions are listed in Appendix A and Figure A1 of Appendix A summarizes the response patterns. Of the 5553 individuals who were asked the question, only 213 individuals (3.8%) either refused to answer or said that they did not know, 410 individuals (7.4%) gave a nonrounding probability, whereas 4930 individuals (88.8%) gave forecasts that were multiples of 10% or 25%, and were therefore potential focal point responses.

A total of 227 individuals answered that they had 0% chance of living to age 75 and 861 individuals said they were 100% sure they can survive to age 75. A total of 153 (67.4%) and 500 (58.1%) of them, respectively, stated that it was a precise estimate. A total of 1154 individuals answered they had a 50% chance of living to age 75. 448 (38.8%) of them stated that it meant equally likely that they would survive to age 75. Among those 2689 individuals who answered multiples of 10% or 25% but other than 0%, 50%, or 100%, 851 individuals (31.7%) stated it was an exact probability. So, individuals assessing 0% or 100% appear more confident of their forecasting abilities than those assessing other probabilities.

Individuals who answered 0% and 100% and said their forecast was an approximation were generally able to provide a more accurate estimate. The averages of the new estimates are 25.5% and 85.6%, respectively, considerably different than the original answers, suggesting that their original responses may not accurately reflect their true beliefs. Individuals who answered other multiples of 10% or 25% and said that their forecast was an approximation were asked to provide a range for their estimate. On average, the range length is 17.9% and the median of the range is 64.4%, exactly the same as the average of the original focal point forecasts of 64.4%. We interpret this as indicating that individuals attach a plausible degree of imprecision to their subjective mortality estimate.

The differences between the original estimates and the answers from the follow-up questions, especially for those who initially answered 0% or 100%, suggest the necessity of correcting for focal point biases before performing further calculations. Therefore, we update the answers of the question to the more precise estimate if provided and to the median of the range if a range is provided in the follow-up questions.

3.2 Recovering subjective mortality tables

Next, we estimate subjective mortality tables for each individual in the HRS sample, and calculate their self-assessed life expectancy and the variance of age of death. From the 5553 individuals, we drop 213 and 147 individuals who did not provide original forecasts for the and questions, respectively. We further drop 616 individuals who cannot provide a more precise estimate in the follow-up questions. We finally drop 358 individuals for whom . This leaves a sample of 4219, of which 1653 are men and 2566 are women.7

We apply the methodology in Thatcher et al. (2010) to estimate individual-level subjective mortality tables. Thatcher et al. (2010) estimate a logistic mortality model of the following form:

()

where is individual i's subjective mortality rate at age . Parameters and control how mortality rates change as individual age. Using method of simulated moments, we estimate and for each individual, so that the simulated probabilities of surviving to age 75 and 85 best match their self-reported probabilities in the data. We then simulate every individual's subjective mortality rates at all ages and calculate their life expectancy, , and variance of age of death, , conditional on attaining age 62, using the following formulas:

()

()

The patterns of subjective mortality beliefs of the HRS sample are presented in Tables 1 and 2, which compare the average and median life expectancy and variance of age of death calculated using the estimated subjective mortality rates and the objective mortality rates from the SSA cohort life tables, respectively.8 Conditional on being alive at age 62, we calculate the average subjective life expectancy is 21.7 years, slightly higher than the 21.6 years predicted by the cohort life tables, showing individuals could in general accurately predict their life expectancy. This is consistent with previous literature (Hurd & McGarry, 1995, 2002; Siegel et al., 2003). However, we show that the average variance of age of death calculated using the subjective mortality tables is 88.3 years², 6.2% lower than the 94.2 years² calculated using the objective mortality tables, indicating individuals predict their age of death with greater precision. The results hold for both men and women. The average subjective variances of age of death are 86.0 and 89.8 years² for men and women, respectively, compared with the 90.5 and 96.5 years² calculated using the cohort life tables. Additionally, the median subjective variance of age of death of the HRS sample is lower, at 76.1 years², 20.9% lower than the median variance of age of death of 96.1 years² calculated using the cohort life tables.

Because there is no follow-up question, the focal point responses of the question may affect our results. We thus consider a clean sample which only consists of individuals who answered nonrounding probabilities and who answered multiples of 10% or 25% for the question and said it was an exact probability. These individuals provided a precise estimate for the question and most likely also provided a precise estimate of the question.9 We show that individuals continue to predict their life expectancy accurately. The average and median subjective life expectancies are 21.8 and 22.5 years, respectively, compared with the 21.6 and 22.6 years calculated using objective mortality tables. However, the deviations of the subjective variance of age of death in the clean sample are larger than the full sample. We calculate the average and median subjective variance of the age of death of the clean sample are 80.6 and 57.6 years² respectively, 14.4% and 40.1% lower than the 94.2 and 96.1 years² calculated using the cohort life tables. The patterns again hold for both men and women. Thus, the results are sensitive to how to handle focal point responses.

4.1 The life-cycle model

Following Coile et al. (2002) and Sun and Webb (2011), we construct an intertemporal optimization model to investigate the impact of plausible variations of subjective mortality beliefs on the value of delaying claiming Social Security benefits and the optimal claiming ages of retired workers. The model considers the Social Security claiming decision as a separate financial decision from the retirement decision and concentrates on calculating the theoretical optimal behaviors of rational, well-informed representative households.10 We consider both single households and married couples. The Single's case is straightforward. The representative single household enters the life-cycle model at age 62, retired. The terminal age is set at 100. The individual maximizes its expected discounted lifetime utility as follows:

()

where is the time discount factor. is the probability of the individual being alive at time . is the consumption. The utility function takes the common Constant Relative Risk Aversion (CRRA) form:

()

where is the coefficient of risk aversion.

The individual faces the following budget constraint:

()

where is the individual's nonannuitized financial wealth. represents the Social Security benefits payable to the individual at time , conditional on claimed at age . is the real interest rate. There is the usual nonnegative asset constraint, so that at all .

In each period, the individual decides how much to consume and whether to claim Social Security benefits. The dynamic optimization problem is solved by backward induction. The baseline calculations assume the representative individual has population average mortality rates for the 1950 birth cohort from the SSA cohort life tables.11 The individual has a PIA of $2000.12 And as in Sun and Webb (2011), we endow the household with an amount of financial wealth equal to the EPV of Social Security benefits they would receive if claimed as soon as possible. This amount of financial wealth is sufficient to ensure that the household has sufficient financial assets to fund consumption from age 62 to 70, thereby mitigating the effect of liquidity constraint. We further assume the coefficient of risk aversion equals 3 which is in range as reported in the literature (Chetty, 2006). Finally, as is conventional in the annuity literature, both the real interest rate and the time discount rate are set at 3%.

We compare the value of delaying claiming Social Security benefits using Social Security Equivalent Income (SSEI), the factor by which the benefits payable at suboptimal claiming ages must be multiplied so that the household is indifferent between claiming at the optimal claiming age and claiming at other ages in expected present value (EPV) terms or in expected utility (EU) terms. By design, SSEI is never smaller than one. It equals one at the optimal claiming age. At suboptimal claiming ages, SSEI is greater than one. Note that when the individual is risk-neutral, SSEI in EU terms equals SSEI in EPV terms which is the ratio of the EPVs claimed at the optimal age and other ages. The EPV of an individual's Social Security benefits claimed at age can be calculated as follows:

()

We next introduce the model for married couples. The representative married couple maximizes their expected lifetime utility as follows:

()

where is the survival status. It equals 1 when both spouses are alive, 2 if only the husband is alive, and 3 if only the wife is alive. is the probability of the couple being in survival status at time . denote the consumption of the husband and the wife at time , respectively. We follow Brown and Poterba (2000) and assume that the married couple has an additive utility function of the following form:

()

where measures the jointness of the couple's consumption. When equals one, all consumption is joint. When equals zero, none of the household's consumption is joint.

The couple's budget constraint can be written as follows:

()

where is the couple's joint financial wealth at time . is the household's total Social Security benefits payable in survival state at time , conditional on the combination of ages at which the husband and the wife claimed benefits ( and ). Again, is the real return of a risk-free asset and there is the nonnegative asset constraint.

In each period, the couple decide how much to consume, whether the husband should claim his Social Security retired worker benefits, and whether, conditional on the husband has claimed, the wife should claim her spousal benefits. Upon the death of the husband, spousal benefits are immediately replaced by survivor benefits. The optimal choices depend on their joint financial wealth and whether both or only one of the spouses is alive. We perform benchmark calculations for a single-earner couple and assume equals 0.5 as in Sun and Webb (2011). Both spouses are the same age with the population average mortality for the 1950 birth cohort in the baseline calculations. The husband has a PIA of $2000. We again endow the married couple with an amount of financial wealth equal to the EPV of Social Security benefits they would receive if claimed as soon as possible to rule out the effect of liquidity constraint. The rest of the parameters are the same as in the single's case.

4.2 Recovering subjective mortality tables for the representative households

Using the same approach described in Section 3.2, we calculate the representative households' annual subjective mortality rates based on different numerical exercise assumptions. We first extract the objective mortality rates, given the representative household's gender and birth cohort, from the SSA cohort life tables, and calculate the population average life expectancy and variance of age of death, conditional on attaining age 62, based on Equations (2) and (3). We then estimate the representative household's parameters and in Equation (1) using methods of simulated moments, so that the life expectancy and the variance of age of death calculated by simulated mortality rates best match the chosen life expectancy and variance of age of death, produced by population average mortality rates and adjusted as appropriate, that is, and , where and are adjustment factors based on proposed numerical experiments. We finally recover the representative individual's subjective mortality probabilities at each age using estimated parameters and .

To illustrate, Figure 1a,b compare the cumulative survival probabilities of selected levels of life expectancy and variance of age of death, conditional on alive at age 62, for men and women, respectively. The solid curves are plotted using the population average mortality rates of men and women born in 1950. The long-dashed curves are calculated using the Thatcher et al. (2010) model simulated mortality rates by matching the life expectancy and the variance of age of death produced by above cohort life tables. Although there are only two parameters in the Thatcher et al. (2010) model, the two cumulative survival probabilities are matched fairly well. Additionally, in Section 3.2, we calculate from the HRS clean sample that, on average, individuals can correctly predict their life expectancy, but the average and median variance of age of death calculated using their subjective mortality tables are 14.4% and 40.1% lower than that calculated by the cohort life tables, respectively. Thus, the dashed and dash-dotted curves compare the cumulative survival probabilities predicted by the Thatcher et al. (2010) model with the same life expectancy but 14.4% and 40.1% lower variance of age of death. The curves are steeper, indicating individuals are more certain of their age of death, relative to individuals whose mortality risks equal that predicted by the cohort life tables. Finally, the short-dashed and dotted curves plot the cumulative survival probabilities for individuals whose subjective life expectancy and subjective variance of age of death are at the 25th and 10th percentiles of the respective distribution. Except for being steeper, the curves further demonstrate a noticeable leftward shift, suggesting that the subjective life expectancies are also lower than the estimates calculated using the cohort life tables.

4.3 Results

Tables 3 and 4 report SSEIs in expected present value terms and in expected utility terms for the representative single households, respectively. The upper panel of each table reports SSEIs for single male households and the lower panel presents the calculations for single female households. Our benchmark calculations reported in the first column assume individuals have population average mortality for the 1950 birth cohort from the SSA cohort life tables. Single men with population average mortality maximize their EPV of benefits by claiming at age 64 (the SSEI at age 64 is 1.000 indicating his Social Security benefits stream has the highest EPV if claimed at age 64). As women live longer than men, they place a higher value on delay and their optimal claiming age is 67. Claiming earlier than the ages does not result in substantial EPV losses because the benefit adjustment for early claiming is approximately actuarially fair. For example, if a single man claims at age 62, the earliest possible age, the SSEI is 1.001, indicating he would only require a 0.1% increase in his age 62 benefits to achieve the same EPV of benefits as that obtained by claiming at the optimal age of 64.

However, at a coefficient of risk aversion of three, a single man maximizes his expected lifetime utility if he claims his Social Security benefits at age 69 (the SSEI at age 69 is 1.000 indicating he gets the highest expected discounted lifetime utility by claiming at age 69). Those who claim their Social Security benefits early receive lower expected discounted lifetime utility and run the risk of low consumption in the event they live an unexpectedly long time. They require their Social Security benefits to be increased each month to receive the same level of expected discounted lifetime utility as claiming at the optimal claiming age 69, reflecting the high value placed on the additional longevity insurance acquired as a result of delaying claiming. For example, a single man who claims at age 62 would require a 14.1% increase in his benefits to be as well off as he would be were he to delay until age 69. The same conclusion applies to single women. Because women usually live longer than men, they will place even higher value on delaying claiming. A single woman maximizes her expected lifetime utility by delaying claiming until age 70. If a single woman is forced to claim at age 62, she would require a 17.7% increase in her age 62 benefits to be as well off in EU terms.

We next consider how plausible variations of subjective mortality beliefs affect the value of postponing claiming Social Security benefits and the optimal claiming ages of retired workers, compared with the benchmark case. In Section 3.2, we find that the average subjective variance of age of death of the HRS participants is 6.2%–14.4% lower than that calculated from the SSA cohort life tables. Therefore, we start our experiments by decreasing the variance of age of death by 14.4%, while holding the life expectancy constant (the adjustment factors are: and ). Columns 2 in Tables 3 and 4 show SSEIs of this numerical exercise. A 14.4% decrease in variance of age of death has almost no impact on the EPV of Social Security benefits. This is not surprising, because the primary concern of the EPV calculations is on life expectancy, which stays constant in the exercise. However, the exercise reduces the interquartile survival range by 1 and 2 years for men and women, respectively. It also decreases SSEIs in EU terms at all ages, indicating individuals place a lower value on delaying claiming as they are more certain about their age of death. The effect is small in magnitude. Single men and women claiming at age 62 would require 13.0% and 16.7% increase in benefits to be as well off in EU terms, a change of 1.1% and 1.0%, respectively, from the benchmark case. In addition, the optimal claiming ages stay the same at age 69 and 70 for single men and women, respectively.

In Section 3.2, we also demonstrate that the deviation between the median variances of age of death calculated using the subjective and objective mortality tables is larger, ranging from 20.9% to 40.1%. Therefore, we conducted a second numerical exercise by decreasing the variance of age of death of the representative households by 40.1% while keeping their life expectancy the same as the benchmark case (the adjustment factors are: and ). In this exercise, the interquartile survival range decreases by 3 years for men and 5 years for women compared to the benchmark case. Columns 3 of Tables 3 and 4 report the SSEIs in EPV terms and in EU terms, respectively. Again, the SSEIs in EPV terms barely change, indicating the variation in the variance of age of death has minimal impact on the EPV of benefits. The exercise has a larger effect on the SSEIs in EU terms. Decreasing the variance of age of death by 40.1% further reduces SSEIs at age 62 for men and women by 3.3 and 3.0 percentage points to 1.108 and 1.147, respectively. Additionally, it is now optimal for single men and women to claim their benefits 1 year early than the benchmark case at age 68 and 69, respectively.

To provide a more comprehensive understanding of how subjective mortality beliefs may theoretically shape older individuals' optimal claiming decisions, we conduct two additional numerical experiments, assuming that the representative households experience relatively extreme, yet plausible, subjective mortality beliefs. More specifically, we calculate the SSEIs and optimal claiming ages for individuals whose life expectancy and variance of age of death are both at the 25th or 10th percentile of the respective subjective mortality belief distributions.13 These individuals are pessimistic and fairly confident, but not certain, that they will not survive to advanced old ages. As discussed in the introduction, we expect that individuals who have extreme confidence on predicting their age of death are likely to attribute a limited value on postponing claiming Social Security. This is because they benefit much less from the additional longevity insurance that would otherwise protect them from the risk of outliving their wealth.

Conditional on attaining age 62, the subjective life expectancy for men at the 25th percentile of the subjective mortality belief distribution is 14.6 years. It is 28.4% lower than the average subjective life expectancy and 25.5% lower than the average life expectancy calculated using the cohort mortality tables. For women at the 25th percentile, their subjective life expectancy is 17.0 years, which is 25.2% lower than the average subjective life expectancy and 25.5% lower than the average objective life expectancy. The subjective variance of age of death for men and women at the 25th percentile is 26.3 and 27.3 years², respectively. These values are 66.2% and 66.8% lower than the corresponding average subjective variance of age of death, and 70.9% and 71.7% lower than the corresponding average variance of age of death calculated using the cohort life tables, respectively. At the 10th percentile, the subjective life expectancy of men and women further reduces to 7.3 and 8.8 years, respectively, and the subjective variance of age of death further reduces to 16.4 and 16.7 years², respectively.

We report the results in Columns 4 and 5 in Tables 3 and 4, respectively. Both shorter life expectancy and lower variance of age of death reduce incentives for individuals to delay claiming. At the 25th percentile, claiming immediately at age 62 becomes optimal for both men and women in EPV calculations. The costs of early claiming in EU terms are also lower. Single men maximize their expected discounted lifetime utility at age 65 and single women maximize their expected discounted lifetime utility at age 67, 4 and 3 years earlier than the benchmark case, respectively. Those who claimed at age 62 would require 1.9 % and 4.2% increase in benefits, respectively, to be as well off as he or she would be if claimed at the optimal ages, dropping from 14.1% and 17.7% in the benchmark calculations. At the 10th percentile, it further becomes optimal for both men and women to claim their Social Security benefits at age 62, the youngest plausible age, in both EPV and EU terms.

Next, we show how subjective mortality beliefs affect Social Security claiming decisions of married couples. We concentrate on explaining the SSEIs in expected utility terms.14 Our base case is a one-earner couple. Both the husband and the wife are at the same age with population average mortality for the 1950 birth cohort. The SSEIs are reported in the first panel in Table 5. We do not report values for infeasible combinations of claiming ages—those in which the wife's claiming age is less than that of her husband. For married couples with population average mortality, their expected discounted lifetime utility is maximized when both husband and wife claim at age 67. A husband and wife claiming at age 62 would require an 8.9% increase in benefits to be as well off in EU terms.

The second and third panels, respectively, assume that both the husband and wife have 14.4% and 40.1% lower variance of age of death, holding life expectancy constant. When decreasing the variance of age of death by 14.4%, the combination of claiming ages that maximizes the households’ expected discounted lifetime utility is identical to that of the benchmark case, but the required benefits increase if both spouses claim at age 62 reduces to 8.4%. When the variance of age of death is 40.1% lower than the benchmark case, the value of delaying claiming is further reduced and the optimal combination of claiming ages is 1 year early than the benchmark case.

The fourth and fifth panels show SSEIs of husband and wife that have both life expectancy and variance of age of death that lie at the 25th or 10th percentiles of the respective distributions. Again, both lower life expectancy and lower variance of age of death reduce the value of delaying claiming Social Security. At the 25th percentile, both spouses should claim at age 65 to maximize their expected discounted lifetime utility, 2 years earlier than the benchmark scenario. A married couple that both spouses claiming at age 62 would require a 1.1% increase in their benefits to be as well off as they would be if claimed at the optimal combination of claiming ages. At the 10th percentile, it further becomes optimal for the couple to claim as soon as becoming eligible.

4.4 Sensitivity analysis

A substantial literature, including Ameriks et al. (2011), Lockwood (2012), Pashchenko (2013), Lockwood (2018), Pashchenko and Porapakkarm (2023), and Bairoliya and McKiernan (2022), have emphasized the significant influence of bequest motives on the demand for annuities. To examine the impact of bequest motives on our results, we perform robustness checks by incorporating bequest motives into the life-cycle optimization model. We employ two approaches to model bequests, drawing from Lockwood (2018) and De Nardi et al. (2016). In both studies, bequests are modeled as luxury goods. In Lockwood (2018), the utility that individuals derive from their bequests is:

()

where represents the amount of bequests left to heirs. The parameter measures the strength of the individuals' bequest motives and the parameter is a consumption floor below which households do not leave bequests. The two-parameter utility function of bequests in De Nardi et al. (2016) takes the following form:

()

where represents the estates. The parameter represents the bequest intensity and the parameter determines the curvature of the function. We take the parameter values directly from the two studies, so is 0.95 and is 24.8 thousand dollars based on Lockwood (2018) and the two parameters and in De Nardi et al. (2016) are 39.7 and 13.0 thousand dollars, respectively.15

We focus on explaining the results of single households which are reported in Table 6. Columns 1–3 assume individuals experience objective mortality rates. The variance of age of death is reduced by 14.4% (holding the life expectancy constant) in Columns 4–6. Under each mortality assumption, we compare SSEIs of three scenarios: no bequest motives, Lockwood (2018) bequest motives and De Nardi et al. (2016) bequest motives. We show that bequest motives significantly reduce retired workers' optimal claiming ages. Modeling bequest motives as in Lockwood (2018) would decrease the optimal claim ages of single men and women from ages 69 and 70 in the no bequest motives scenario to ages 65 and 67, respectively. The effect of bequest motives is smaller when using the parameter values in De Nardi et al. (2016). The optimal claiming age is age 67 for single male households and age 68 for single female households. The subjective uncertainty regarding individuals' age of death, however, again has little effect on the optimal claiming ages. In scenarios that accommodate both bequest motives and a 14.4% lower variance of age of death, the optimal claiming ages remain the same as in the scenarios that only incorporate the corresponding bequest motives.16