Subjective life expectancies, time preference heterogeneity, and wealth inequality

Foltyn, Richard; Olsson, Jonna Article Subjective life expectancies, time preference heterogeneity, and wealth inequality Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Foltyn, Richard; Olsson, Jonna (2024) : Subjective life expectancies, time preference heterogeneity, and wealth inequality, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 15, Iss. 3, pp. 699-736, https://doi.org/10.3982/QE2016 This Version is available at: https://hdl.handle.net/10419/320316 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 15 (2024), 699–736 1759-7331/20240699 Subjective life expectancies, time preference heterogeneity, and wealth inequality Richard Foltyn Department of Economics, Norwegian School of Economics and Adam Smith Business School, University of Glasgow Jonna Olsson Department of Economics, Norwegian School of Economics This paper examines how objective and subjective heterogeneity in life expectancy affects savings behavior of healthy and unhealthy people. Using data from the Health and Retirement Study, we first document systematic biases in survival beliefs across self-reported health: those in poor health not only have a shorter actual lifespan but also underestimate their remaining life time. To gauge the effect on savings behavior and wealth accumulation, we use an overlappinggenerations model where survival probabilities and beliefs evolve according to a health and survival process estimated from data. We conclude that differences in life expectancy are important to understand savings behavior, and that the belief biases, especially among the unhealthy, can explain up to a fifth of the observed health-wealth gap. Keywords. Life expectancy, preference heterogeneity, subjective beliefs, life cycle. JEL classification. D15, E21, G41, I14. 1. Introduction The determinants of the wealth distribution are of fundamental interest to economists. Standard consumption/savings theory predicts that people who place a larger weight on future states will be wealthier than people who are more impatient, all else equal. This paper explores one reason to put a higher weight on the future: the higher probability to survive to old age. However, an individual’s consumption/savings decision is not necessarily guided by the objective (statistical) survival probability but rather the individual’s beliefs about survival. The first contribution of this paper is to document new facts about a withinRichard Foltyn: [email protected] Jonna Olsson: [email protected] We are grateful to James Banks, Timo Boppart, Per Krusell, Alexander Ludwig, Hannes Malmberg, Ay¸segül ¸Sahin, Paolo Sodini, Christian Stoltenberg, Magnus Åhl, Erik Öberg, and participants in numerous conferences and seminars for helpful discussions and comments. The collection of data used in this study was partly supported by the National Institutes of Health under grants R01 HD069609 and R01 AG040213, and the National Science Foundation under awards SES 1157698 and 1623684. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE2016 700 Foltyn and Olsson Quantitative Economics 15 (2024) cohort steepness bias in survival beliefs: people overestimate the health gradient of survival. It has previously been shown (e.g., Hamermesh (1985), Elder (2013), Ludwig and Zimper (2013), Heimer,Myrseth,andSchoenle(2019)) that there is a systematic flatness bias over age: younger people tend to underestimate their survival probabilities, while older people overestimate their chances of a long life. We show that within a cohort, individuals in bad health not only have a shorter expected life span but are also relatively more downward biased about their survival chances, while individuals in good health, and thus with higher survival probability display an upward bias. These systematic biases exacerbate the life expectancy heterogeneity in the population. The differences in beliefs about survival translate into time preference heterogeneity in the population. Our second contribution is to quantify this heterogeneity and its implications for savings and wealth accumulation in an overlapping-generations model. With a stochastic health and survival process, the effective discount rate varies depending on age, health, and the forecast horizon. Over a 1-year horizon, the effective discount rate for 50-year-olds ranges from 2% for an individual in best health to around 20% for an individual in worst health. At a 10-year horizon, this gap shrinks somewhat but still amounts to 8 percentage points between best and worst health. For 70-yearolds in worst versus best health state, the difference at the 10-year horizon is close to 10 percentage points. This resulting time preference heterogeneity is in line with the dispersion (Calvet, Campbell, Gomes, and Sodini (2021)) and the age gradient (Kureishi, Paule-Paludkiewicz, Tsujiyama, and Wakabayashi (2021)) of the time preference distribution found in other empirical studies. To gauge the quantitative effect of survival heterogeneity on savings behavior and wealth accumulation, we use an overlapping-generations general-equilibrium model with uninsurable idiosyncratic shocks. Agents face heterogeneous survival risk that depends on their age and current health state, and are subject to health shocks that follow a process estimated from data. The current health state also affects labor earnings and medical expenditure risk. Besides this uncertainty, we additionally include standard persistent and transitory shocks to labor productivity during working age. After agents reach a fixed retirement age, they are entitled to retirement benefits mimicking the US social security system. Finally, our model includes probabilistic bequests that feature intergenerational persistence of income and wealth. We purposely use an otherwise standard model of consumption/savings to establish a benchmark and focus on the survival heterogeneity savings channel. We compare three scenarios. The first scenario is a standard model in which there is no health risk: all agents face the same labor earnings risk, the same medical expenditure risk, and the same survival risk, and thus have the same effective discount factor, conditional on age. In the second scenario, we introduce health risk that affects labor earnings and medical expenditures. In terms of survival risk, individuals are perfectly informed about their true survival probability conditional on health and age. In the third scenario, agents believe and act according to their subjective survival beliefs. Thus, our analysis is designed to answer the question: what if we turned off the discount factor heterogeneity implied by the biases in survival beliefs we uncovered in the empirical part? Would savings patterns look quantitatively different? Quantitative Economics 15 (2024) Subjective life expectancies 701 The simulations show that the survival expectation channel is important for understanding wealth accumulation. Not surprisingly, agents in bad health, and thus with a shorter expected life span save less than their healthy counterparts, and the differences in savings rates are large. For example, for 60-year-olds in the middle of the wealth distribution, the total savings rates of an agent in the best, and an agent in the worst health state differ by 5 percentage points when they are endowed with correct objective beliefs about survival. When we let them act according to the estimated subjective beliefs instead, the difference doubles to 10 percentage points. These differences in savings behavior translate into large differences in accumulated wealth: in the model with subjective survival beliefs, median wealth differs by 193% between those in the worst and best health states at ages 55–59. This health-wealth gradient is very close to the magnitude we observe in the data. A fifth of this difference is driven by the erroneous survival beliefs, especially by individuals in poor health underestimating their remaining life span. Thus, the biases in survival beliefs are important to understand the health-wealth gradient in older ages. This paper speaks to three broad strands of literature. The first is concerned with subjective survival expectations (Hamermesh (1985), Smith, Taylor, and Sloan (2001), Hurd and McGarry (2002), Ludwig and Zimper (2013), Elder (2013), Gan, Gong, Hurd, and McFadden (2015), Groneck, Ludwig, and Zimper (2016), Heimer, Myrseth, and Schoenle (2019), de Bresser (2023)). Many studies have documented the existence of an age bias in subjective life expectancies, and a few of the papers within this group are concerned with the implications for the consumption/savings behavior. Some predict individual survival probabilities and contrast them with elicited beliefs (Gan, Hurd, and McFadden (2005), Bissonnette, Hurd, and Michaud (2017), Grevenbrock, Groneck, Ludwig, and Zimper (2021)), but none of these look at the implications for within-cohort savings behavior in a structural model where beliefs change in the event of health shocks, or analyze the implications for wealth inequality. The second strand are macroeconomic studies pointing out the importance of heterogeneity in time preferences to explain wealth inequality (e.g., Krusell and Smith (1998), Hendricks (2007), Quadrini and Ríos-Rull (2015), Krueger, Mitman, and Perri (2016)) and studies documenting time preference heterogeneity in the population (Epper et al. (2020), Calvet et al. (2021)). Compared to these papers, we provide a micro-foundation for one source of time preference heterogeneity—differences in life expectancy—and evaluate its importance. The third is the literature about the general impact of health (including life expectancy) on wealth (Smith (1999), Lee and Kim (2008), Coile and Milligan (2009), De Nardi, French, and Jones (2009), Kopecky and Koreshkova (2014), Capatina (2015), De Nardi, Pashchenko, and Porapakkarm (2017), Poterba, Venti, and Wise (2017), Margaris and Wallenius (2023), to name a few). In contrast to these papers, we include heterogeneity in subjective life expectancy and examine its impact on savings and consumption behavior. In the next section, we describe how we estimate the health and survival process and give details about the systematic bias in survival expectations. Section 3describes 702 Foltyn and Olsson Quantitative Economics 15 (2024) the model we use to quantify the importance of the heterogeneity in survival expectations. After that, we discuss the parametrization and then we present our results. The last section concludes. The Supplemental Appendix to the working paper (Foltyn and Olsson (2024)) contains additional results and derivations. 2. Empirical evidence 2.1 Data We use the Health and Retirement Study (HRS), a representative panel of elderly US households, to investigate the evolution of health and longevity in the later stages of life. The survey includes questions about self-reported health and expectations about survival, and records the date of death, if applicable. Our analysis is based on the survey years 1992–2014 taken from the HRS data compiled by RAND, version 2018 (V2) (Health and Retirement Study (2023)).1The first cohort included in the survey was between 51 and 61 years old in 1992, and thereafter new (older and younger) cohorts were added. Many of the respondents died over the sample period, making it an ideal data set for studying survival. In this section, we first document the relationship between health and wealth, and between beliefs about survival and wealth. We then briefly describe how we estimate the objective survival probabilities. In Section 2.4, we show how average elicited beliefs about survival are biased, and in Section 2.5, we estimate a subjective life expectancy process that replicates this bias. 2.2 The health-wealth gradient and the life expectancy/savings channel The HRS asks participants to assess their health using one of the five categories excellent,very good,good,fair,orpoor. Figure 1shows net total wealth over the life cycle by self-reported health state computed for the pooled sample of all respondents.2,3The health-wealth gradient is well documented, but the underlying causal relationship is debated (Attanasio and Hoynes (2000), Deaton (2002), Duncan, Daly, McDonough, and Williams (2002), Attanasio and Emmerson (2003), Hajat, Kaufman, Rose, Siddiqi, and Thomas (2010)). One line of argument is that low economic status leads to poor health. There could be many reasons: poor people have access to less or lower-quality medical care, do not invest enough in preventive health measures, and/or have more healthdeteriorating habits. However, there are also many arguments for the reversed causality: poor health has economic consequences in itself. First, poor health may restrict the individual’s earnings potential by making it more costly to work and/or by lowering the 1The HRS (Health and Retirement Study) is sponsored by the National Institute on Aging (grant NIA U01AG009740) and is conducted by the University of Michigan. 2Net total wealth is defined as sum of housing, other real estate, vehicles, businesses, IRA and Keogh accounts, stocks, checkings, and all other savings, net of mortgages and other debts. 3In the Appendix to the working paper (Foltyn and Olsson (2024)) (henceforth referred to as the Supplemental Appendix), we disaggregate these wealth profiles by race, sex, household size, and education (see Section A.4). The overall picture remains unchanged. Quantitative Economics 15 (2024) Subjective life expectancies 703 Figure 1. Median net total household wealth by self-reported health state. Pooled sample from HRS 1992–2014. Assets are adjusted for outliers, time, and cohort fixed effects. Colors indicate the health state: dark green is excellent while red is poor health. Error bars indicate 95% confidence intervals. wage. Second, poor health may lead to large medical expenditures. Third, poor health may lower the savings incentives due to a lower survival expectancy. This last channel is the focus of this paper. If individuals adjust their savings behavior based on their survival prospects, this could be either on the basis of objective (statistical) survival probabilities or subjective survival beliefs, which are also surveyed by the HRS. To assess how wealth correlates with survival beliefs, we regress net total wealth on an indicator of whether an individual believes to have above-median survival chances compared to other respondents of the same age, race, and sex. Table 1shows a positive correlation between having abovemedian beliefs and being wealthier.4The positive relationship also holds when additionally controlling for education and couple status.5Other empirical studies corroborate the existence of the life expectancy/savings channel and suggest a causal link. For instance, Heimer,Myrseth,andSchoenle(2019) administer a novel survey and estimate that greater survival optimism correlates with higher savings rates, not only after controlling for standard demographic characteristics such as education, marital status, and income, but also financial literacy and risk tolerance. Another prediction of the life expectancy/savings channel is that individuals who receive a bad health shock, that is, a plausible decrease in life expectancy, should exhibitlowerassetgrowth.Table2reports the results from regressing the 2-year change in net total wealth (again using an inverse hyperbolic sine transformation) on a negative 4All empirical results in this paper are reported with standard errors and confidence intervals that take into account the stratification and clustering of the HRS; see the Supplemental Appendix, Section A.3. 5We apply an inverse hyperbolic sine transformation since assets are heavily skewed and contain zeros and negative values. All coefficients of interest are positive and significant at the 1% level when alternatively using assets in levels. The Supplemental Appendix, Section A.5.1 contains further information and robustness checks. 704 Foltyn and Olsson Quantitative Economics 15 (2024) Table 1. Net total wealth and above-median subjective survival beliefs. Dep. Variable: Net Total Wealth (Inverse Hyperbolic Sine Transformation) Men Women (1) (2) (3) (4) (5) (6) Above median SSB 0.418 0.230 0.221 0.575 0.374 0.353 (0.045) (0.040) (0.039) (0.032) (0.029) (0.029) Age FE Yes Yes Yes Yes Yes Yes Education FE Yes Yes Yes Yes Couple FE Yes Yes Observations 54,212 54,212 54,212 70,537 70,537 70,537 Note: The table shows the results of regressing net total wealth (after an inverse hyperbolic sign transformation) on an indicator of above-median survival beliefs compared to individuals of the same age, race, and sex. The regression includes fully interacted fixed effects as indicated. Nonblack population. Clustered standard errors in parentheses. health shock defined as an indicator for a deterioration in self-reported health between survey waves. As column 1 shows, men who experience a negative health shock decumulate their assets more compared to men of the same race, age, and initial health who do not. Column 2 additionally controls for education, while columns 3–4 show the corresponding results for women. A negative health shock is associated with a faster decumulation (or slower accumulation) of assets in all specifications. More details are given in the Supplemental Appendix to the working paper (Foltyn and Olsson (2024)) (henceforth referred to as the Supplemental Appendix); see Section A.5.2.6 While these results are indicative, a recent study by Kvaerner (2022) using the plausibly exogenous timing of cancer diagnoses shows that news about a bad health shock inTable 2. Health shocks and changes in wealth. Dep. Variable: Relative Change in Net Total Wealth Men Women (1) (2) (3) (4) Negative health shock −0.149 −0.130 −0.134 −0.115 (0.023) (0.023) (0.018) (0.018) Age FE Yes Yes Yes Yes Health FE Yes Yes Yes Yes Education FE Yes Yes Observations 61,528 61,528 80,615 80,615 Note: The table shows the results of regressing changes in net total wealth (after an inverse hyperbolic sign transformation) on an indicator for a deterioration in self-reported health between two consecutive survey waves. The regression includes fully interacted fixed effects as indicated. Nonblack population. Clustered standard errors in parentheses. 6It is possible that the decumulation of assets associated with a health deterioration is driven by lower labor income or large medical expenditures. In the Supplemental Appendix, Section A.5.2, we show that the results also hold in the subsample aged 65 and older (who are likely to be retired and on Medicare) even after accounting for out-of-pocket medical expenditures. Quantitative Economics 15 (2024) Subjective life expectancies 705 creases the probability of an immediate inter vivos transfer, suggesting a causal link between survival prospects and wealth decumulation. Thus, if a decrease in life expectancy increases own consumption and/or increases the probability of inter vivos transfers is an open question, and one interpretation of inter vivos transfers from the giver’s perspective is to view them as “consumption of gift-giving.” While both inter vivos transfers and own consumption show up as a decumulation of assets and thereby affect the health-wealth gradient in older ages in the same way, they have different implications for wealth among the younger receiving generation, and thus for the wealth distribution. Our structural model does not allow for inter vivos transfers, and consequently, leaves this question open for future research. 2.3 Objective health and survival probabilities In this paper, we examine the effect of heterogeneity in survival expectancy on savings behavior and its implications for wealth inequality through the lens of a structural model. Therefore, we need to formulate heterogeneity in survival expectations, both objective and subjective, in a way that can be used in such a model. For our quantitative model, we use a Markov process for health transitions and survival at an annual frequency. We estimate this Markov process as described in Foltyn and Olsson (2021). Conceptually, the method is a straightforward maximum likelihood estimator where the probability of observing the transitions in the data is maximized. We briefly summarize the method and estimation sample in the next few paragraphs. To put structure on the Markov process, we follow Pijoan-Mas and Ríos-Rull (2014) and use a logit model, where survival and health transitions conditional on survival are modeled as functions of the current health state and age. The probability of survival follows the usual binary logit model while, conditional on survival, health transitions are modeled using multinomial logit. For example, the one-period-ahead survival probability is given by ps t+1=1 1+e−g(xt|γ),(1) where g(•)is a function of the covariate vector xtwhich contains race, sex, age, health, and potentially other observables such as education. Survival probabilities are governed by the parameter vector γto be estimated. Transition probabilities for health conditional on survival are defined in an analogous manner.7 Estimation sample We exclude all observations with missing age, race, sex, or selfreported health, as well as individuals with only a single observation (since then we do not have any transition probability to estimate). We only consider individuals aged 50 or older.8Furthermore, we restrict the sample to maximum age of 99 years at transition start (even though individuals can be older when we observe them in the end of 7In Foltyn and Olsson (2021), we provide details about the estimation and also perform an extensive evaluation of the results. The estimated Markov process is shown to predict actual mortality very well, both short and long term. See the Supplemental Appendix, Section B for a brief overview. 8Each incoming HRS cohort is aged 51 or above, but the survey contains younger individuals who are spouses of age-eligible respondents. 706 Foltyn and Olsson Quantitative Economics 15 (2024) a transition). This leaves us with 34,196 individuals and 219,539 observations in total. We estimate the health and objective (statistical) survival process separately for the subsamples of men/women and the black/nonblack population, since it is well known that the life expectancies for these groups follow very different trajectories.9Table B.1 in the Supplemental Appendix shows descriptive statistics and the number of individuals and observations by subgroup. Results From these estimates, we construct a first-order Markov process defined on five health states and the absorbing state of death, which governs the objective health and survival probabilities. This process can be used to calculate objective life expectancies conditional on age, health, race, and sex. Not surprisingly, there is a substantial health gradient in life expectancy. For example, for a 70-year-old nonblack man in excellent health, the predicted probability of surviving an additional 10 years is approximately 75%, while the probability is just around 35% if instead starting out in poor health. For a brief overview of the results, see the Supplemental Appendix, Section B.2. It is important to note that even though the health and survival process is based on self-reported health—a subjective measure of how respondents perceive their health state—the result from the estimation is an objective statistical life expectancy for each combination of race, sex, age, and health. Self-reported health can be thought of as letting the respondents themselves aggregate the multidimensional information about their health (that is potentially unobservable to the econometrician) into a single categorical variable, and the variable can also capture subjective perceptions of the respondent. The estimated Markov process maximizes the probability of observing the health transitions and survival in the data conditional on self-reported health, irrespective of why a particular health state was reported. 2.4 Expectationerrorsinsurvivalprobabilities In the expectations survey module of the HRS, respondents are asked about the probability they assign to certain events. One of these questions is about the probability of surviving to a certain age, for example: “Using a number from 0 to 100, what do you think are the chances that you will live to be at least 100 years?”10 The exact target age depends on the respondent’s age and survey wave. For instance, in 1995, respondents below the age of 70 were asked about the probability of living until the age of 80, while respondents above the age of 85 were asked about the target age of 100. In later surveys, individuals were asked about survival beliefs for up to two target ages. 9For the remainder of the paper, the “black” sample consists of respondents who identify as black or African-American, while “nonblack” is the complementary group, which also includes Hispanics. The HRS is not large enough to disaggregate the nonblack group further, since the (unweighted) sample of personyear observations is approximately 72.7% white, 15.7% black/African-American, 9.4% Hispanic, with other ethnicities together contributing the remaining 2.3%. 10Before the respondent answers the questions about expectations, the interviewer discusses probabilities and verifies that the respondent understands the concept. Quantitative Economics 15 (2024) Subjective life expectancies 713 In what follows, we partition the sample into groups indexed by k, such that each unique combination of (x,T)forms a separate group. Denote by kall individual/target age/year observations that satisfy k=(i,j,t)|xit =xk,Tijt =Tk, that is, all observations where the individuals are asked about their survival beliefs over the same horizon, are of the same age, report the same health state, and share any other covariates in xit.Denotebyps kthe (weighted) sample average of reported survival beliefs conditional on (xk,Tk),thatis, ps k= (i,j,t)∈k wit ·φTk(xk,zit )  (i,k,t)∈k wit ,(5) where wit are the respondent-level sampling weights. Now consider the logit counterpart of (5), which we denote by  ps k=Pr(alive at agek+Tk|xk,ν) that is, the predicted probability of being alive for group k, taking into account all possible health transitions to Tk. We assume exactly the same functional form as for the objective health and survival process, but allow the parameter vector νgoverning survival to differ. The observed sample moment for each group can then be written as ps k= ps k+uk, where ukis the deviation from the group mean not explained by our model. Our aim is to minimize these group-specific residuals using the least-squares objective function J(ν)=1 W k Wkps k− ps(xk,Tk|ν)2,(6) where Wk=(i,j,t)∈kwit is the sum of weights in group k. The estimated vector νis hence the arg min of J(ν). Estimation sample We use all target ages from Table 3for the estimation of the subjective life expectancy process. In the main paper, we present the results for nonblack men, as these are later incorporated into our quantitative model. Results The estimated subjective survival beliefs for nonblack men are shown in pink in Figure 6, juxtaposing the objective survival probabilities estimated in Foltyn and Olsson (2021) in blue. As can be seen, the subjective belief about survival in health state excellent or very good is almost 100% for all ages. This does not mean that individuals in those health states believe that they will live forever. Rather they believe that death is necessarily preceded by a deterioration in health. 714 Foltyn and Olsson Quantitative Economics 15 (2024) Figure 6. One-year objective and subjective survival probabilities by health state for nonblack men (model estimates). Shaded areas indicate 95% confidence intervals. For each bootstrapped sample, we reestimate the objective health process. Figure 7summarizes the results, showing the life expectancy by age and health state using the objective and the subjective survival process. At all ages, the difference in life expectancy between the best and the worst health state is larger when using subjective life expectancies. The divergence between objective and subjective life expectancies is particularly large for men in bad health, who substantially underestimate survival at Figure 7. Life expectancy by age and health for nonblack men. Colors indicate the health state: dark green is excellent while red is poor health. In panel 7a, the black line depicts the weighted population average. Shaded areas indicate 95% confidence intervals. Quantitative Economics 15 (2024) Subjective life expectancies 715 younger ages. Conversely, individuals in all health states overestimate their chances of survival late in life. Figure C.8 in the Supplemental Appendix plots the objective and subjective life expectancy for the remaining demographic groups, which exhibit very similar patterns. Figure C.9 in the Supplemental Appendix plots the model-predicted subjective survival probabilities against their data counterparts, showing that both sets of moments are well aligned. 3. Economic model In this section, we describe the overlapping-generations model used to quantify the implications of survival heterogeneity. Time is discrete and each period corresponds to one year. Agents derive utility from consumption and face idiosyncratic risk in the form of shocks to their labor income, medical expenditures, health and survival, as well as stochastic bequests from their parents. Markets are incomplete as agents can only save in a riskless asset while borrowing is not permitted. 3.1 The agent’s problem There is a unit mass of individuals distributed across Ntcohorts according to the ergodic distribution implied by the transition matrix of survival probabilities. An individual of age t∈{1, ,Nt}and health h∈{1, ,5 }has a one-period survival probability to age t+1givenbyπs ht,withπs ht =0 in the terminal period.13 Individuals are assumed to be working for the first Tr−1 years of their life and exogenously retire in the period in which they attain age Tr. While working, they are hit by persistent and transitory labor productivity shocks. During retirement, individuals receive social security retirement benefits, which depend on their last persistent labor productivity in working age. Bequests are modeled via probabilistic intergenerational links along the lines of Straub (2019) so that children with higher lifetime income are more likely to have income-rich parents, and thus expect to receive higher bequests. Retirement Let x=(a,p,h,η,1b,t)be a retired individual’s state vector, where ais cash-at-hand, prepresents pre-retirement labor productivity, his the current health state, ηis the persistent component of medical expenditures, 1bis a bequest indicator, and tis age. In each period, an individual chooses consumption cand savings kto be invested in risk-free productive capital. Individuals earn a gross return Ron their savings and receive gross retirement income w·yr, which is proportional to the economywide wage rate w,andwhere yr=ωr·Rss(p) is a function of the average earnings profile at the time of retirement, ωr,andRss(•), which mimics the regressive replacement rate of the US social security system applied 13Since the model does not use calendar time, we from now on use tto denote age. 716 Foltyn and Olsson Quantitative Economics 15 (2024) to the last pre-retirement labor productivity p. Retirement income is taxed using the nonlinear tax schedule Ty(•)so that after-tax retirement income amounts to ι=wy r−Ty(wy r).(7) Agents receive an inheritance b∗≥0 at most once in their life, which we track using the indicator 1b.Theyareborninstate1b=1 and transition to 1b=0 when their parents die. As long as 1b=1, the tuple (b∗,1b)evolves according to b ∗,1 b∼⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ b∗(p,h,t),0 with prob. 1−πs t∗πb pht, (0, 0)with prob. 1−πs t∗1−πb pht, (0, 1)with prob. πs t∗. (8) Once 1b=0, no additional bequests are expected and (b ∗,1 b)=(0, 0)obtains with certainty. It is not possible to directly model intergenerational links between a parent and a child as this would double the number of state variables. We therefore assume that parents are exactly 30 years older (with age t∗=t+30) and survive with the age-specific population-average probability πs t∗to the next period. Conditional on parental death, children receive bequests with probability πb pht ∈(0, 1)to reflect that many parents do not leave sizeable estates. To capture the intergenerational persistence of income and wealth, we map agents into income quintiles and use the intergenerational income quintile transition matrix from Chetty, Hendren, Kline, and Saez (2014) to stochastically connect children to potential parents. This creates a positive sorting between children’s and parents’ income and wealth so that richer children are more likely to receive higher bequests. Since our mapping to income quintiles relies on the states (p,h,t), both the probability to receive a bequest πb pht and the amount received b∗are functions of (p,h,t). We describe the technical details of these linkages in Section D.1 in the Supplemental Appendix. The next period cash-at-hand is given by a=Rk +b ∗+ι−mh,η,ν,t+1+ξ,(9) where mare out-of-pocket medical expenditures that accrue between ages tand t+1, which are allowed to depend on health h, a persistent component η, and a transitory shock ν, similar to the approach taken in De Nardi, French, and Jones (2010). Because these expenditure shocks can be quite large, we assume that the government guarantees a minimum consumption level cby making a transfer ξwhenever agents do not have the resources to cover the medical expenditures themselves. The required transfer is therefore defined as ξ=max0, c+mh,η,ν,t+1−Rk −b ∗−ι. (10) In the process, whenever agents receive a positive transfer ξ>0, they are not permitted to save for the next period and, therefore, choose k=0andc=c. Nonsurvivors leave their asset holdings as bequests to their offspring. Any out-ofpocket medical bills m(η,ν,t+1)incurred in the last period of life are deducted, and Quantitative Economics 15 (2024) Subjective life expectancies 717 bequests are additionally subject to the estate tax Tb(•). Thus, after-tax bequests are given by b=max0, Rk +b ∗−mη,ν,t+1 −Tbmax0, Rk +b ∗−mη,ν,t+1, (11) where we assume that descendants are not liable for any medical bills exceeding a deceased individual’s assets.14 Finally, we impose a warm-glow bequest motive as in De Nardi (2004), Vb(b)=φ1(b+φ2)1−σ−1 1−σ, where φ1governs the weight individuals assign to leaving bequest and φ2is a parameter controlling to what extent bequests are a luxury good. To summarize, a retired individual’s maximization problem is defined by the value function Vr(a,p,h,η,1b,t)=max c≥0,k≥0c1−σ−1 1−σ+βπs htEVrx|p,h,η,1b,t +β1−πs htEVbb|h,η,1b,t subject to c+k≤aand the laws of motion (9)and(11), where x=(a,p,h,η,1 b,t+ 1)is the continuation state conditional on survival. Health hevolves according to the transitions estimated from the HRS, while the survival probabilities πs ht follow either the objective or subjective survival beliefs discussed in the previous section. Lastly, the persistent component ηof medical expenditures follows an AR(1) process. Working age Individuals of working age solve a problem almost identical to that of retirees, except that they additionally face both persistent and transitory labor earnings risk. The persistent risk component is captured by the state variable pand is assumed to follow a first-order Markov process, while the transitory shock is i.i.d. over time. Together with the age-health earnings profile ωht , they pin down an individual’s labor productivity y, which is allowed to depend on health h, y=ωhtp. (12) Moreover, workers are subject to payroll taxes Tss(•), which we model as a function of productivity, so that their after-tax labor income is ι=y−Tss(y)w−Tyy−Tss(y)w. (13) The remaining problem is the same as for retirees, including the medical expenditure shocks, the consumption floor, government transfers, and the intergenerational linkages and bequests. 14As (11) suggests, it is possible that children die in the same period as their parents so that an inheritance immediately becomes part of a child’s estate. 718 Foltyn and Olsson Quantitative Economics 15 (2024) 3.2 Technology The production side of the model is standard. Competitive firms employ labor and capital hired from households to produce a homogeneous final good, which is used for both consumption and investment. The aggregate production function is assumed to be Cobb–Douglas, F(A,K,L)=AKαkL1−αk. Capital depreciates at the rate δk. 3.3 Government We assume that the government runs a PAYGO social security system that has to balance in each period, and that transfers as well as any remaining (wasteful) government expenditures have to be fully financed by income and inheritance taxes. We first describe the social security system and thereafter the general government budget. Social security system We use a stylized version of the actual retirement income formula used in the US social security system. It captures the main features, such as a regressive replacement rate based on pre-retirement income and a cap for maximum benefits. In the model, we define retirement benefits to be a product of the economywide wage w, the average life-cycle profile component from the last year before retiring ωr, and a function that mimics the regressive replacement rate ι(p)=w·yr(p)=w·ωr·Rss(p). The replacement function Rss(•)is given by Rss(p)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ρ1pif p≤p∗ 1, ρ1p∗ 1+ρ2p−p∗ 1if p∗ 1<p≤p∗ 2, ρ1p∗ 1+ρ2p∗ 2−p∗ 1+ρ3minp∗ max,p−p∗ 2else, where p∗ 1and p∗ 2are bend points and p∗ max is the contribution and benefit base (CBB) in the social security income formula, expressed in terms of the individual’s last preretirement persistent labor state p, which becomes permanent once retired. The Supplemental Appendix, Section D.2.1 describes in detail how we map the dollar quantities taken from social security regulations to their model counterparts. The government expenditures on retirement are financed by a payroll tax. The payroll tax function is defined as Tss(y)=τss ·min{ymax,y}, where ymax represents maximum taxable earnings. The derivation of total payroll taxes raised in each period can be found in the Supplemental Appendix, Section D.2.2. To balance the social security system, we find τss such that total expenditures on social security benefits equal total payroll taxes. Quantitative Economics 15 (2024) Subjective life expectancies 719 Government budget The government needs to finance lump-sum transfers ξto households defined in (10). We denote the aggregate transfers by and provide details on how these are computed in the Supplemental Appendix, Section D.3.1. Additionally, the government finances nondiscretionary expenditures that amount to a constant fraction g of output, G=gY. We adopt the same income tax function as in Heathcote, Storesletten, and Violante (2017), which is defined as Ty(ι)=ι−λι1−τ, (14) where ιis either earnings (net of payroll taxes) or retirement income, and we denote total income tax revenue by Tinc. Additionally, the government collects Testate in estate taxes. We provide details on how to compute Testate and Tinc in the Supplemental Appendix, Sections D.3.2 and D.3.3. We assume that the progressivity parameter τin (14) is fixed, and we pin down λsuch that the government budget is balanced in each period, that is, +G=Testate +Tinc(λ). 3.4 Equilibrium The equilibrium definition is mostly standard and can be found in the Supplemental Appendix, Section D.4. The one noteworthy addition is that we require that for each cohort, after-tax estates left by parents are consistent with the bequests expected and received by children given the stochastic intergenerational links. This introduces computational complications, which we discuss in the Supplemental Appendix, Section H. 4. Calibration 4.1 Preferences We assume log preferences, that is, σ=1, and thus u(c)=logc. The common discount factor β=0.979 is set to obtain a capital-to-output ratio of 3.0 in the scenario with subjective survival beliefs. In our benchmark calibration, we shut down the warm-glow bequest motive by setting φ1=0, and hence all bequests are accidental. We discuss alternative scenarios in Section 5.4. 4.2 Externally calibrated parameters Demographics Agents are assumed to enter the economy at age 20, which corresponds to model age 1, and retire at the age of 65, implying that Tr=46. The maximum attainable age is 109, and hence we let Nt=90.15 15The reason for imposing such a high maximum age is that otherwise the scope for upward bias in beliefs about survival in old age is limited: if agents know for sure that they are going to die at the age of 100, say, any gap between objective and subjective beliefs shrinks by construction, even at younger ages. However, setting a high maximum age has no effect on the age distribution: as shown in Figure E.2, the mass of individuals aged 100+in the economy is only 0.06%. 720 Foltyn and Olsson Quantitative Economics 15 (2024) Earnings We assume that the logarithm of labor earnings follows a process with transitory and persistent shocks, logyht =logωht +log pt+logt,t∈{1, ,Tr−1}, where ωht is the age-health profile, ptis the persistent component, and tis the transitory component of earnings. The persistent component is assumed to follow an AR(1) process, logpt=ρplogpt−1+υt, with autocorrelation ρpand innovation υt iid ∼N(0, σ2 υ). The transitory shock is lognormally distributed with log t iid ∼N(0, σ2 ). The stochastic part of the wage process is therefore characterized by the parameters (ρp,σ2 υ,σ2 ),whichwesetto(0.9695, 0.0384, 0.0522), following Krueger, Mitman, and Perri (2016). We use the Rouwenhorst procedure to discretize the persistent part of the process into an five-state Markov chain, and we discretize the transitory shock into three states. The age-health profile for labor earnings is estimated for nonblack men aged 20 to 65 using PSID data (PSID (2023)). Not surprisingly, there is a strong health gradient. This is partly driven by lower wages conditional on working, and partly by a larger fraction of individuals in bad health not working at all. Since we abstract from the labor supply decision, our estimates of the age-health earnings profile captures both margins. More details can be found in the Supplemental Appendix, Section E.3. Medical expenditures Following French and Jones (2004), De Nardi, French, and Jones (2010), we estimate the medical expenditure shocks from the out-of-pocket medical expenses reported at biennial frequency in the HRS for the sample of nonblack men aged 50 and above. Since the HRS includes hardly any individuals below the age of 50, we assume that agents do not face any out-of-pocket medical costs at these ages. We impose that both the mean and variance of log medical expenditures are statedependent and given by the following process: logmit =αi+x itβ+z itγ+σ(xit )(ηit +νit ), ηit =ρmηit−1+ζit, (15) ζit iid ∼N0, σ2 ζ, νit iid ∼N0, σ2 ν. (16) The vector xit contains a third-order polynomial in age, health, as well as health interacted with age. Additionally, zit includes controls not present in the economic model such as marital status, education level, 5-year cohort dummies, and time fixed effects, as well as interactions of these terms. We run a fixed effects estimator on the level of log medical expenditures and recover the parameters governing the variances and covariances from the residuals using GMM. Once we have identified the parameters for Quantitative Economics 15 (2024) Subjective life expectancies 721 Table 4. Calibrated parameters. Parameter Description Value Source Production technology parameters αkCapital share 36% Krueger, Mitman, and Perri (2016) δkDepreciation rate 9.6% Krueger, Mitman, and Perri (2016) ATotal factor productivity 0.896 Fixes equilibrium wages at unity Social security ρ1Replacement rate bracket 1 90% 2000 SS rules ρ2Replacement rate bracket 2 32% 2000 SS rules ρ3Replacement rate bracket 3 15% 2000 SS rules b$ 1Bendpoint 1 $6384 2000 SS rules b$ 2Bendpoint 2 $38,424 2000 SS rules e$ max Contribution and benefit base (CBB) $76,200 2000 SS rules cConsumption floor $2325 5% of average annual earnings Government budget gGov. spending (share of GDP) 6% Brinca Holter, Krusell, and Malafry (2016) τTax progressivity 0.137 Brinca et al. (2016) τbMarginal tax on estates 30% Authors’ approximation medical expenditures over a 2-year period, we use a simulated method of moments procedure to recover the implied parameters at annual frequency, which yields ρm=0.920, σ2 ζ=0.084, and σ2 ν=0.457. More details can be found in the Supplemental Appendix, Section E.4. For the purpose of including medical expenditure shocks in the OLG model, we discretize the persistent component (15) using the Rouwenhorst procedure with seven states, and we discretize the transitory component (16) into five possible realizations. Bequests We assume that estates are tax exempt up to the amount χband subject to a proportional tax τbthereafter. We set χb=19.75 so that in equilibrium 2% of estates are subject to estate taxes, while τbis set to 30%.16 The intergenerational income quintile transition matrix used to link parents to children is taken from Chetty et al. (2014, Table II) and reproduced in Table D.1 in the Supplemental Appendix. Lastly, we allow the probability to receive a bequest conditional on parental death to differ by income quintile. To this end, we use the Survey of Consumer Finances (SCF) waves 1998–2007 and compute the fraction of respondents aged 60–70 who report having ever received an inheritance by each income quintile, which gives the probabilities 20.5%, 25.2%, 27.4%, 33.3%, and 40.4% for the lowest to highest quintile. Remaining externally calibrated parameters The remaining parameters are listed in Table 4. The bend points and the contribution and benefit base are reported in US dollars to facilitate the interpretation. The value for the consumption floor is similar to the levels used by De Nardi, French, and Jones (2010)orPalumbo (1999). 16The top marginal tax rate in 2023 was 40% (see https://www.irs.gov/pub/irs-pdf/i706.pdf); however, not all taxable estates fall into the top category. We choose 30% as an approximation. 722 Foltyn and Olsson Quantitative Economics 15 (2024) 4.3 Health and survival process We use the processes for health transitions and survival probabilities described in Section 2.3 (health transitions and objective survival probabilities) and Section 2.5 (subjective survival probabilities) for nonblack men. Agents enter the model at the age of 20, but the health and survival processes we estimate based on the HRS data starts at the age of 50. We therefore estimate a health process for the ages 20 to 50 using PSID data. We use data from the years 1984 to 2019 and the subsample of nonblack male household heads, and assume that survival is certain during this age span (further details can be found in the Supplemental Appendix, Section E.1). From the age of 50, we use our estimated process based on the HRS data, and agents start facing a positive probability of death. The resulting cohort sizes and distribution of health states are shown in Figure E.2 in the Supplemental Appendix. While the model is solved with five health states, in what follows we report results only for the best, middle, and worst health states to reduce visual clutter. 5. Results We solve the model under three distinct assumptions about health heterogeneity and survival expectations: (i) No health heterogeneity (NHH): In this scenario, all agents of the same age face the same earnings profile, the same medical expenditure risk, and the same survival risk. We eliminate health heterogeneity and use the average survival rates (depicted by the black line in Figure 7a), an average earnings profile, and the average medical expenditure process. The probability of receiving an inheritance and the amount received are also averaged across health.17 (ii) Objective survival heterogeneity (OSH): In the second scenario, we use the objective process for health transitions and survival probabilities described in Section 2.3. In this case, individuals are perfectly informed about their true survival probability conditional on health and age. Medical expenditures and labor earnings are allowed to differ by age and health. (iii) Subjective survival heterogeneity (SSH): In the third scenario, agents conversely form beliefs and act accordingto the subjective survival process estimated in Section 2.5. However, this subjective process does not correspond to the true survival process, which we use when simulating the model. In the remainder of this section, we first contrast the effective discount rates that arise in the objective versus subjective survival belief scenarios. These are important drivers of savings behavior, which we discuss next. We then turn to the implications for wealth accumulation across health and also briefly discuss general equilibrium effects, which are mostly unchanged across all three scenarios. In our benchmark calibration, 17These averages are computed using the age-specific health distribution implied by our estimated health transition probabilities and the initial distribution over health at age 20 observed in the PSID. Quantitative Economics 15 (2024) Subjective life expectancies 729 The last row in Table 5shows the general equilibrium results from the model with no health heterogeneity. The Gini coefficient for wealth is slightly higher than in any of the models with health heterogeneity. The driver of this is an increased number of large accidental bequests. The largest amounts bequethed stem from deaths in relatively young ages (between ages 50 and 65). In a model with no health heterogeneity, these deaths are equally likely to happen to agents in the top of the asset distribution as to agents in the bottom. With health-dependent survival on the other hand, it is more likely that deaths in relatively young ages happen to agents in poor health, who are on average poorer. Thus, the lack of health heterogeneity gives rise to larger bequests, which in turn gives rise to slightly larger wealth inequality. 5.4 A model with an active bequest motive There are many drivers of savings that could vary across health but are not included in our model: the existence of (employer-tied) health or life insurance, human capital investment, endogenous retirement decisions, portfolio composition, private pensions, and permanent characteristics such as patience, to name a few (see, for instance, De Nardi, French, and Jones (2010), Capatina (2015), or De Nardi, Pashchenko, and Porapakkarm (2017) for studies taking a broader perspective including several channels). These mechanisms could all add realism to the model and make the life-cycle profiles more in line with the data. One mechanism often introduced to capture the slow decumulation of asset in older ages is a warm-glow bequest motive. In this section, we therefore use a model calibration with an active bequest motive (φ1>0) to show how it interacts with (subjective) survival heterogeneity. 5.4.1 Calibration Where applicable, we use the same calibration as in the main text. We determine the discount factor β, the preference parameters governing the bequest motive (φ1and φ2) and the estate tax exemption threshold χbusing the method of simulated moments, that is, we minimize the weighted sum of squared distances between targeted and simulated moments from the model with subjective beliefs about survival. We again target a capital-to-output ratio of 3 and that 2% of estates should be subject to estate taxes. Additionally, we try to match the old-age, life-cycle profile of assets. To this end, we use the median wealth levels at ages 60, 65, 70, 75, 80, and 85 observed in the HRS, relative to median wealth at age 55. We choose this approach as it is quantitatively not possible to match wealth in levels and at the same time impose a capital-outputratio of 3 in a model with productive capital as the only asset (after all, most of the wealth in the data is held in residential real estate). The capital-to-output ratio and the fraction of estates subject to estate tax are perfectly matched, while the asset holdings by age and their data counterparts are shown in Table G.1 in the Supplemental Appendix. Some aspects of our model are too simplistic to match the data moments exactly. For example, because we impose an exogenous retirement age of 65, the life-cycle profile of assets peaks exactly at this age, whereas this is not the case in the data. The estimated parameters are listed in Table 6. As the table shows, the bequest luxury shifter is small. The reason is that we try to match the median asset holdings late in life. If bequests were a luxury good, the median asset level would fall quickly toward zero. 730 Foltyn and Olsson Quantitative Economics 15 (2024) Table 6. Parameters for the model with a bequest motive. Parameter Description Value Model with bequest βDiscount factor 0.942 φ1Bequest weight 11.157 φ2Bequest shifter 0.001 χbEstate tax exemption 18.333 5.4.2 Results Cross-section and life cycle Figure 11 shows the life-cycle profiles for the scenarios with objective survival heterogeneity (OSH) and subjective survival heterogeneity (SSH). Overall, due to the bequest motive, older agents do not decumulate their wealth, and the resulting median asset profile is more in line with the data in both scenarios. Before retirement, agents in excellent health have more wealth than agents in worse health. The main reason for this is the higher labor income of the former group. However, the health-wealth gradient is substantially smaller prior to retirement than in the baseline calibration without a bequest motive (compare to Figure 10), and is even reversed after the age of 75, with agents in poor health being richer. This shows that the effect of combining survival heterogeneity with a bequest motive of this type is not entirely straightforward. The expected utility from leaving a bequest is not only a function oftheamountexpectedtobehandedovertothedescendants,butalsoofthesurvival probability: agents with low (objective or subjective) survival prospects put more weight on the bequest motive. Hence, there are two effects from lower life expectancy that work in opposite directions: a shorter expected life span makes agents want to save less for their own consumption in old age, but a stronger bequest motive induces them to save more. The net effect varies depending on the calibration of bequest parameters, but the Figure 11. Median life-cycle profiles for wealth, model with an active bequest motive. Quantitative Economics 15 (2024) Subjective life expectancies 731 second mechanism is always present with a warm-glow bequest motive of this type: a shorter life span makes agents want to save more to leave bequests.22 Thus, in both the OSH and the SSH scenario, the model misses the cross-sectional correlation between wealth and health in older ages that is present in the data. The slightly stronger reversal of the health-wealth gradient in the SSH model follows directly from the biases that amplify the health-survival belief gradient. Note also that in very high ages, agents keep less assets in the SSH model than in the OHS model. Agents in the subjective belief model overestimate the probability of a long life and, therefore, put a lower weight on warm-glow bequests, resulting in lower savings. Dynamic responses to health shocks Next, we compare changes in wealth following negative health shocks in the data to their model counterparts. To this end, we regress the change in net total wealth (using an inverse hyperbolic sine transformation) on a negative health shock defined as an indicator for a deterioration in self-reported health between survey waves. We restrict the sample to ages 65 and above in order to focus on the part of the life cycle where life expectancy and bequest considerations are important drivers of savings. We simulate a panel of 100,000 agents and collapse the data to two-year frequency to replicate the biennial HRS.23 The changes in assets and health are defined analogously to the data, taking differences over 2-year periods. The results in Table 7show that the model with a bequest motive produces dynamic savings responses that are difficult to square with the data. The estimated savings response to a negative health shock in the model with an active bequest motive is effectively zero, whereas the model without an active bequest motive produces savings responses that are well in line with the data. Table 7. Health shocks and changes in wealth. Dep. Variable: Relative Change in Net Total Wealth Data Model (1) (2) No bequest Bequest Negative health shock −0.116 −0.105 −0.1081 0.0001 (0.026) (0.026) (0.001) (0.000) Age FE Yes Yes Yes Yes Health FE Yes Yes Yes Yes Education FE Yes Observations 35,821 35,821 1,297,804 1,297,804 Note: The table shows the results of regressing changes in net total wealth (after an inverse hyperbolic sign transformation) on an indicator for a deterioration in self-reported health between two consecutive survey waves. The regression includes fully interacted fixed effects as indicated. Columns 1 and 2 are the same as the first two columns in Table A.5 in the Supplemental Appendix. For the model columns, we use the SSH scenario where agents act according to their subjective beliefs. Sample restricted to nonblack males age 65 and above. Clustered standard errors in parentheses. 22To aid intuition, in Section G.1 in the Supplemental Appendix we show the mechanisms at play in a simple two-period model example. 23We start the simulation with 100,000 agents at the age of 20, of which 86,809 are still alive at the age of 65. 732 Foltyn and Olsson Quantitative Economics 15 (2024) In sum, in a model with health heterogeneity, both the cross-sectional implications and the dynamic responses to health shocks make the model with a bequest motive calibrated to match median asset holdings in old age difficult to align with data. The relative strength of the different effects we point to in this section of course varies depending on calibration, but should be kept in mind when combining a warm-glow bequest motive with dynamically evolving expected longevity. 6. Conclusions This paper explores how variation in objective and subjectively perceived life expectancy affects savings behavior of healthy and unhealthy people. Using HRS data, we show that there exists a within-cohort steepness bias in survival beliefs: individuals in bad health not only have a shorter expected life span, but are also relatively more downward biased about their survival chances, while individuals in good health, and thus with higher survival probability display a more upward bias. These systematic biases exacerbate the survival expectancy heterogeneity in the population. The differences in beliefs about survival translate into time preference heterogeneity, and consequently, savings behavior. We show that biases in beliefs about survival can explain approximately one-fifth of the differences in accumulated wealth between those in excellent versus poor health, mostly because the latter group underestimates their life expectancy. This paper ties into a strandof current researchinvestigating preference heterogeneity and its importance for individual choices and aggregate outcomes. We provide an intuitively plausible and micro-founded source of heterogeneity: the perceived probability of surviving to future states of the world. Our quantification of this channel shows that life expectancy heterogeneity is important and should be included in the list of potential sources of heterogeneity that we need to consider in our analyses. Investigating the importance of the steepness bias for within-cohort differences in terms of portfolio allocations, demand for financial products, or retirement behavior is left for future research. References Attanasio, Orazio P. and Carl Emmerson (2003), “Mortality, health status, and wealth.” Journal of the European Economic Association, 1 (4), 821–850. [702] Attanasio, Orazio P. and Hilary Williamson Hoynes (2000), “Differential mortality and wealth accumulation.” Journal of Human Resources, 35 (1), 1–29. [702] Bissonnette, Luc, Michael D. Hurd, and Pierre-Carl Michaud (2017), “Individual survival curves comparing subjective and observed mortality risks.” Health Economics, 26 (12), e285–e303. [701,710] Quantitative Economics 15 (2024) Subjective life expectancies 733 Brinca, Pedro, Hans A. Holter, Per Krusell, and Laurence Malafry (2016), “Fiscal multipliers in the 21st century.” Journal of Monetary Economics, 77, 53–69. [721] Calvet, Laurent E., John Y. Campbell, Francisco Gomes, and Paolo Sodini (2021), “The cross-section of household preferences.” Technical report, National Bureau of Economic Research. [700,701,723] Capatina, Elena (2015), “Life-cycle effects of health risk.” Journal of Monetary Economics, 74, 67–88. [701,729] Chetty, Raj, Nathaniel Hendren, Patrick Kline, and Emmanuel Saez (2014), “Where is the land of opportunity? The geography of intergenerational mobility in the United States.” The Quarterly Journal of Economics, 129 (4), 1553–1623. [716,721] Coile, Courtney and Kevin Milligan (2009), “How household portfolios evolve after retirement: The effect of aging and health shocks.” Review of Income and Wealth,55(2), 226–248. [701] de Bresser (2023), “Evaluating the accuracy of counterfactuals: Heterogeneous survival expectations in a life cycle model.” Review of Economic Studies, rdad088. [701] De Nardi, Mariacristina (2004), “Wealth inequality and intergenerational links.” The Review of Economic Studies, 71 (3), 743–768. [717] De Nardi, Mariacristina, Eric French, and John B. Jones (2009), “Life expectancy and old age savings.” The American Economic Review Papers and Proceedings, 99 (2), 110–115. [701] De Nardi, Mariacristina, Eric French, and John B. Jones (2010), “Why do the elderly save? The role of medical expenses.” Journal of Political Economy, 118 (1), 39–75. [716,720, 721,729] De Nardi, Mariacristina, Svetlana Pashchenko, and Ponpoje Porapakkarm (2017), “The lifetime costs of bad health.” Technical report, National Bureau of Economic Research. [701,723,729] Deaton, Angus (2002), “Policy implications of the gradient of health and wealth.” Health Affairs, 21 (2), 13–30. [702] Duncan, Greg J., Mary C. Daly, Peggy McDonough, and David R. Williams (2002), “Optimal indicators of socioeconomic status for health research.” American Journal of Public Health, 92 (7), 1151–1157. [702] Elder, Todd E. (2013), “The predictive validity of subjective mortality expectations: Evidence from the Health and Retirement Study.” Demography, 50 (2), 569–589. [700,701] Epper, Thomas, Ernst Fehr, Helga Fehr-Duda, Claus Thustrup Kreiner, David Dreyer Lassen, Søren Leth-Petersen, and Gregers Nytoft Rasmussen (2020), “Time discounting and wealth inequality.” American Economic Review, 110 (4), 1177–1205. [701] 734 Foltyn and Olsson Quantitative Economics 15 (2024) Foltyn, Richard and Jonna Olsson (2021), “Health dynamics and heterogeneous life expectancies.” Working Papers 2021-17, Business School—Economics, University of Glasgow, https://ideas.repec.org/p/gla/glaewp/2021_17.html.[705,712,713] Foltyn, Richard and Jonna Olsson (2024), “Subjective life expectancies, time preference heterogeneity, and wealth inequality.” Technical report, Econstor, https://hdl.handle. net/10419/294009.[702,704] French, Eric and John Bailey Jones (2004), “On the distribution and dynamics of health care costs.” Journal of Applied Econometrics, 19 (6), 705–721. [720] Gan, Li, Guan Gong, Michael D. Hurd, and Daniel McFadden (2015), “Subjective mortality risk and bequests.” Journal of Econometrics, 188 (2), 514–525. [701] Gan, Li, Michael D. Hurd, and Daniel McFadden (2005), “Individual subjective survival curves.” In Analyses in the Economics of Aging (David A. Wise, ed.), Chapter 12, 377–412, University of Chicago Press. [701,708] Grevenbrock, Nils, Max Groneck, Alexander Ludwig, and Alexander Zimper (2021), “Cognition, optimism, and the formation of age-dependent survival beliefs.” International Economic Review, 62 (2), 887–918. [701,711,712] Groneck, Max, Alexander Ludwig, and Alexander Zimper (2016), “A life-cycle model with ambiguous survival beliefs.” Journal of Economic Theory, 162, 137–180. [701,707,712] Hajat, Anjum, Jay S. Kaufman, Kathryn M. Rose, Arjumand Siddiqi, and James C. Thomas (2010), “Long-term effects of wealth on mortality and self-rated health status.” American Journal of Epidemiology, 173 (2), 192–200. [702] Hamermesh, Daniel S. (1985), “Expectations, life expectancy, and economic behavior.” The Quarterly Journal of Economics, 100 (2), 389–408. [700,701] Health and Retirement Study (2023), “RAND HRS longitudinal file 2018 (V2) public use dataset.” [702] Heathcote, Jonathan, Kjetil Storesletten, and Giovanni L. Violante (2017), “Optimal tax progressivity: An analytical framework.” The Quarterly Journal of Economics, 132 (4), 1693–1754. [719] Heimer, Rawley Z., Kristian Ove R. Myrseth, and Raphael S. Schoenle (2019), “Yolo: Mortality beliefs and household finance puzzles.” Journal of Finance, 74 (6), 2957–2996. [700,701,703,707,712] Hendricks, Lutz (2007), “How important is discount rate heterogeneity for wealth inequality?” Journal of Economic Dynamics and Control, 31 (9), 3042–3068. [701] Hurd, Michael D. and Kathleen McGarry (2002), “The predictive validity of subjective probabilities of survival.” Economic Journal, 112 (482), 966–985. [701,708] Kopecky, Karen A. and Tatyana Koreshkova (2014), “The impact of medical and nursing home expenses on savings.” American Economic Journal: Macroeconomics, 6 (3), 29–72. [701] Quantitative Economics 15 (2024) Subjective life expectancies 735 Krueger, Dirk, Kurt Mitman, and Fabrizio Perri (2016), “Chapter 11—Macroeconomics and household heterogeneity.” In Handbook of Macroeconomics, Vol. 2, 843–921, Elsevier. [701,720,721] Krusell, Per and Anthony A. Jr. Smith (1998), “Income and wealth heterogeneity in the macroeconomy.” Journal of Political Economy, 106 (5), 867–896. [701] Kureishi, Wataru, Hannah Paule-Paludkiewicz, Hitoshi Tsujiyama, and Midori Wakabayashi (2021), “Time preferences over the life cycle and household saving puzzles.” Journal of Monetary Economics, 124, 123–139. [700,724] Kvaerner, Jens Soerlie (2022), “How large are bequest motives? Estimates based on health shocks.” The Review of Financial Studies, 36 (8), 3382–3422. [704] Lee, Jinkook and Hyungsoo Kim (2008), “A longitudinal analysis of the impact of health shocks on the wealth of elders.” Journal of Population Economics, 21 (1), 217–230. [701] Ludwig, Alexander and Alexander Zimper (2013), “A parsimonious model of subjective life expectancy.” Theory and Decision, 75 (4), 519–541. [700,701,707,712] Margaris, Panos and Johanna Wallenius (2023), “Can wealth buy health? A model of pecuniary and non-pecuniary investments in health.” Journal of the European Economic Association, jvad044. [701] Palumbo, Michael G. (1999), “Uncertain medical expenses and precautionary saving neartheendofthelifecycle.”The Review of Economic Studies, 66 (2), 395–421. [721] Pijoan-Mas, Josep and José-Víctor Ríos-Rull (2014), “Heterogeneity in expected longevities.” Demography, 51 (6), 2075–2102. [705] Poterba, James M., Steven F. Venti, and David A. Wise (2017), “The asset cost of poor health.” TheJournaloftheEconomicsofAgeing, 9, 172–184. [701] PSID (2023), “Panel Study of Income Dynamics, public use dataset.” [720] Quadrini, Vincenzo and José-Víctor Ríos-Rull (2015), “Inequality in macroeconomics.” In Handbook of Income Distribution, Vol. 2, 1229–1302, Elsevier. [701] Rozsypal, Filip and Kathrin Schlafmann (2023), “Overpersistence bias in individual income expectations and its aggregate implications.” American Economic Journal: Macroeconomics, 15 (4), 331–371. [711] Smith, James P. (1999), “Healthy bodies and thick wallets: The dual relation between health and economic status.” Journal of Economic Perspectives, 13 (2), 145–166. [701] Smith, V. Kerry, Donald H. Taylor, and Frank A. Sloan (2001), “Longevity expectations and death: Can people predict their own demise?” The American Economic Review,91 (4), 1126–1134. [701,708] Straub, Ludwig (2019), “Consumption, savings, and the distribution of permanent income.” Unpublished manuscript, Harvard University. [715] 736 Foltyn and Olsson Quantitative Economics 15 (2024) Co-editor Morten Ravn handled this manuscript. Manuscript received 31 October, 2021; final version accepted 1 May, 2024; available online 24 May, 2024. The replication package for this paper is available at https://doi.org/10.5281/zenodo.11092578. The Journal checked the data and codes included in the package for their ability to reproduce the results in the paper and approved online appendices.