2.1. Agents

Agents are distinguished between their work preference (high and low disutility from working) and education status (non-college and college). Subscript $i$ denotes those agent types. Lifetime utility of type $i$ agent at the age of 25 ( $t=0$ ) is given as

(1) \begin{align} \sum ^{55}_{t=0} \left (\Pi _{j=0}^t \gamma _{h,j}\right ) \beta ^t [\!\ln c_{i,t} - \rho (h,i,t)l_{i,t} + h_{i,t}]. \end{align}

where $\beta$ is a discount factor, $\gamma _{h,j}$ is health- and age-dependent survival rate, $c_{i,t}$ is consumption, and $l_{i,t}$ is a binary variable that equals one if the agent is working and zero if retired. Thus, labor hour is indivisible. Retirement is assumed to be an absorbing state; thus, agents cannot return to work once they retire. $h_{i,t}$ is the health status of the agent. $\rho (h,i,t)$ determines the level of utility loss from work, and it is affected by the agent’s health condition $h_{i, t}$ , the agent type $i$ , and age $t$ . The preference is chosen to jointly explain the health and retirement distribution across ages. Distinguishing labor preferences between high and low disutility types enables the model to match age patterns of retirement given income and health conditions, which is difficult with homogenous labor preference.

Health status is a random process that has five discrete values (5–1), (5) being the best grade, and (1) being the worst. All agents start with the best health condition when entering the economy. Negative idiosyncratic health shock hits each individual at the end of the period, affecting the next period’s health status. The probability of receiving an idiosyncratic health shock differs across age and health status. Health status evolves as

(2) \begin{align} h_{i,t+1} = h_{i,t} + I^{h}_{i,t} i^h_{i,t} - \epsilon ^h_{i,t}, \end{align}

where $i^h_{i,t}$ is an indicator variable that has value one if the agent conducts health investment and zero otherwise, and $I^{h}_{i,t}$ is a random variable that has value one if the investment is effective and zero otherwise. Therefore, there is a health gain of one grade only if $I^{h}_{i,t} i^h_{i,t} =1$ ; that is, health investment is performed and turns out to be effective.

$\epsilon ^{h}_{i,t}$ is an exogenous idiosyncratic health shock which takes discrete values of $\{\bar{\epsilon }_h, \bar{\epsilon }_l, 0\}$ where $\bar{\epsilon }_h \gt \bar{\epsilon }_l$ . This specification means that $\epsilon ^h_{i,t}$ harms one’s health seriously (high shock), mildly (low shock), or does not affect at all. The size of the health shock is set such that the high shock lowers the health status by three levels, and the low shock lowers it by one level. That is, $\{\bar{\epsilon }_h, \bar{\epsilon }_l, 0\}$ = $\{3, 1, 0\}$ .

Period budget constraint of the agent is given by

(3) \begin{align} (1+\tau _c) c_{i,t} + k_{i,t+1} - (1+r)k_{i,t} + p^h_{i,t} i^h_{i,t} = [1-\tau _l(y_{i,t})] w_{i,t} l_{i,t} + DB_{i,t} - c^{DB}_{i,t} + RB_{i,t}. \end{align}

where $k_{i,t}$ is capital which is the only saving instrument, $r$ is the real interest rate, $p^h_{i,t}$ is the cost of health investment, and $w_{i,t}$ is wage. $\tau _c$ is the consumption tax rate, $\tau _l(y_{i,t})$ is the income tax rate which progressively depends on the level of labor income ( $y_{i,t} = w_{i,t} l_{i,t}$ ). $DB_{i,t}$ is the disability insurance benefits and $c^{DB}_{i,t}$ is the cost of applying the disability insurance benefits. $RB_{i,t}$ is the social security retired benefits. Borrowing is not allowed; thus, $k_{i,t} \geq 0 \; \forall i, t$ .

The above agent problem can be reformulated as a recursive form,

(4) \begin{align} V(x) = \max _{c, k^{\prime}, i^h, l, a^{DB}, a^{RB}} u(c,l,h) + \beta E V(x^{\prime}) \end{align}

subject to

(5) \begin{align} (1+\tau _c) c + k^{\prime} - (1+r)k + p^h(x) i^h = [1-\tau _l(x)] w(x) l + DB(x) - c^{DB}(x) + RB(x), \; k^{\prime} \geq 0.\nonumber\\ \end{align}

where $x$ represents the set of individual states, which is given by $x=\{i, t, h, k, t^{R}, t^{DB}, t^{RB}, AIME\}$ . $i$ is the agent type, $t$ is the age, $h$ is health status, $k$ is capital, $t^{R}$ is the age of retirement if the agent is retired, $t^{DB}$ , $t^{RB}$ are the ages that he claimed disability benefit and retirement benefit if receiving, respectively. $AIME$ stands for Average Indexed Monthly Income. $AIME$ is the reference income for social security benefits and is accumulated according to the agent’s labor income during their working years. Details about $AIME$ will be discussed in the following subsection. Given these individual states, agents make optimal decisions about consumption $c$ , next period’s saving $k^{\prime}$ , health investment $i^h$ , working $l$ , applying for disability benefit $a^{DB}$ , applying for retirement benefit $a^{RB}$ .

Calculating the next period’s expected value, $E V(x^{\prime})$ in equation (4) requires the evaluation of the value function under two layers of uncertainty: individual health uncertainty and aggregate health uncertainty. Individual health uncertainty arises because of the idiosyncratic health shock $\epsilon ^h= \{\bar{\epsilon }_h, \bar{\epsilon }_l, 0\}$ , of which the individual state-dependent probability $\mathbb{P}(x) = \{p_h(x), p_l(x),1-p_h(x)-p_l(x)\}$ is given. Once the aggregate health shock hits the economy, the nature of the economy changes to the pandemic state, and the probability of the health shock is reset to $\mathbb{P}^{\prime}(x) = \{p^{\prime}_h(x), p^{\prime}_l(x),1-p^{\prime}_h(x)-p^{\prime}_l(x)\}$ . Agents know that they are in the pandemic state, and they also place a positive probability ( $p^A$ ) on the case that the next period is in the pandemic state. That is, they expect the next period to be the normal state with the probability $1-p^A$ and the pandemic state with the probability $p^A$ .

2.2. Social security and tax system

The social security system consists of retirement benefits and disability insurance benefits. The retirement benefit is indexed to $AIME$ . The calculation of $AIME$ is based on the individual’s 35 highest years of earnings. Before the agent’s working span reaches 35 years, $AIME$ accumulates as

(6) \begin{align} AIME_{i, t+1} = AIME_{i, t} + \frac{w_{i,t}l_{i,t}}{35} \quad \textrm{if} \quad 0\leq t\lt 35. \end{align}

If the agent works more than 35 years, $AIME$ increases only when the labor income of the additional year exceeds the current $AIME$ .

(7) \begin{align} AIME_{i, t+1} = AIME_{i, t} + \max \left [ 0, \frac{w_{i,t}l_{i,t}-AIME_{i,t}}{35} \right ] \quad \textrm{if} \quad t\geq 35. \end{align}

The retirement benefit is determined based on a piecewise linear function of $AIME$ , the Primary Insurance Amount ( $PIA$ ).

(8) \begin{align} PIA_{i,t} = \begin{cases} 0.9 AIME & \textrm{if} \quad AIME \in [0,b_1),\\[4pt] 0.9 b_1 + 0.32(AIME-b_1) & \textrm{if} \quad AIME \in [b_1,b_2),\\[4pt] 0.9 b_1 + 0.32(b_2-b_1) + 0.15(AIME-b_2) & \textrm{if} \quad AIME \in [b_2,\infty ). \end{cases} \end{align}

Agents are eligible to claim retirement benefits once they are 62 years old. Agents can collect the benefit even though they are still working. Also, they can choose to delay the benefit even if they are retired. There is a penalty if they claim it before the full retirement age of 65, but additional credit if they delay claiming after 65. This specification implies that the actual retired benefit is defined as $RB_{i,t} = \delta _{i,t} PIA_{i,t}$ , where the value of $\delta _{i,t}$ is less than one before age 65, one at 65, greater than one after 65.

Disability benefit is also a function of the agent’s previous labor income. It is transferred to retirement benefits after age 65. Agents cannot return to work status once they start receiving disability benefits. When agents apply for disability insurance, the probability of actually receiving the benefit is less than one and is a function of age and health status. There is a cost in applying for the benefit regardless of the application’s success.

The government collects two types of tax from agents: consumption tax ( $\tau _c$ ) and income tax ( $\tau _l$ ). Consumption tax is a flat-rate tax, and income tax is a progressive tax based on labor income. Those tax revenues are used to finance retirement and disability benefits, and the remaining revenues are used for government purchases of public goods.

3.1. Model parameters

This section discusses the parameter calibration of the model. Parameters for agent preference, individual health process, and disability insurance system are calibrated similarly to the US calibration of Laun and Wallenius (Reference Laun and Wallenius2016).Footnote ¹ Regarding stochastic mortality design, the baseline death rate for each age is taken from Murphy et al. (Reference Murphy, Kochanek, Xu and Arias2021). As the actual death rate is near zero under age 50, we assign a zero death rate for agents under age 50. For agents whose age is 50 or higher, we assume that the mortality rate is higher if they are in bad health status, as their death rate is assigned to be 1.5 times the baseline death rate. If agents are in good health, the death rate is 0.875 times the baseline rate. These values will yield a weighted average death rate identical to the baseline rate when the population consists of 80 percent good and 20 percent bad health agents. The annual real interest rate is set at 3%. Discount factor $\beta$ is set at 0.996. Using the baseline survival rate ( $\gamma _t$ =1-death rate), the average value of the mortality-adjusted discount factor ( $\gamma _t \beta$ ) over age 50 is 0.979.

As mentioned in Section 2.1, there are five discrete health grades. Here we classify the top three grades as “good health” and the lower two as “bad health.” $\rho (h,i,t)$ , the relative degree of disutility from work, differs across health status(good and bad), type, and age. Disutility from working is greater when agents are in bad health conditions than they are in good status. Agent type is distinguished between work preference (high and low disutility) and education status (non-college and college). Moreover, we assume that disutility from labor increases with age. Therefore, agents receive higher disutility from working if they are high disutility type, older, and in bad health status. Combining these factors yields various values for $\rho (h,i,t)$ . The specific values of $\rho (h,i,t)$ are calibrated to match the sample retirement rate across age and education groups. Specifically, we set different parameter values for each education, health, and labor preference type and multiply those values by the age-specific labor preference scale, which varies between 0.6 (age 25) and 1 (age 80). Additionally, to match the retirement rate, we assume that college and non-college workers have 45 percent high disutility type and 55 percent low utility type.

College and non-college agents also differ in their lifetime income profiles. The lifetime labor income profile is derived by a regression model that uses wage as the dependent variable and age and its square term as regressors. The regression uses data from Luxembourg Income Study for US full-time male workers between 25 and 62 and is conducted separately for college and non-college agents.

The consumption tax rate is set at 0.075 following McDaniel (Reference McDaniel2007). The income tax rate is set progressively following Guvenen et al. (Reference Guvenen, Kuruscu and Ozkan2013) process. This process uses a regression model that assigns the tax rate as the dependent variable and different proportions of average wage earnings as regressors. US labor income tax data from the OECD tax database are used for estimation. In the model, half of the social security benefits are counted toward total taxable income, but social security benefits are exempt from tax if total taxable income is below $\$25,000$ .

$p^h_{i,t}$ the cost of health investment is calibrated to match health expenditure by health insurance type and work status. For the agents with no health insurance, their health investment cost is the same regardless of their work status, and they spend more money on health than those covered by health insurance. However, agents aged 65 or higher are covered by Medicare; thus, their payment is reduced. Agents covered with tied health insurance face increased health costs when they retire before age 65, but the worker/non-worker health cost gap decreases for those over 65 because of Medicare. Agents covered with retiree health insurance pay the lowest health cost in general. Retirement increases their health cost, but by a minor degree compared to those with tied health insurance. The proportion of workers covered by no, tied, retiree health insurance in each education group is set to follow Laun and Wallenius (Reference Laun and Wallenius2016).

The probability of idiosyncratic health shock, the success rate of health investment, that of disability insurance application, and the cost of disability insurance application are jointly calibrated to match the health distribution, the application rate for disability insurance, the incidence of disability benefit claimers, and the incidence of type 1 and 2 errors in granting disability benefits by age and education.

The probability of idiosyncratic health shock varies by age, health status, and education status. It is lower if the agent is young, in good health status, and has a college education. The difference of probability by a college education is to capture empirical findings that non-college workers are in worse health and have high disability incidences. The agents with good health conditions have a 2.5 percent chance of receiving the high shock and 13–54 percent of the low shock, depending on age and education. The agents with bad health conditions have 10 percent of receiving the high shock and have the same probability of the low shock as the 80 years old agents with good health. The probability of success in health investment similarly depends on age and health status, as it is higher when the agent is young and in good health condition.

Disability benefit is awarded conditionally on the application if the agent is in bad health, with a probability of 15 percent. There is a cost in applying for a disability benefit if the agent is currently non-retired to capture the earning loss arising from the requirement to stop working to apply for disability insurance. This cost is set at 40 percent of the agent’s labor income. Table 1 summarizes the parameter calibration of the model.

Table 1. Calibrated parameters

Table 2 compares the retirement rate, disability benefit claimer proportion, and the health status distribution of the model and data before the COVID-19 pandemic. The retirement rate is defined as the proportion of retirees in the total population. Those numbers are calculated separately for college and non-college workers and classified by age groups. The model generally matches the retirement rate well in data for each education and age group, especially for ages over 60, which we mainly analyze. In line with the literature, educational attainment matters in the retirement decision: non-college agents tend to retire earlier than college agents. The model also predicts a similar proportion of disability benefit claimers and similar health distribution shown in the data.Footnote ² The health conditions of non-college agents are generally worse than those of college agents, which leads to a higher proportion of disability benefit claimers.

Table 2. Calibrated economy and sample data (percent)

Note: In (a), the retirement rate is the proportion of retirees in the total population. In (c), good health status stands for the best three states.

Data Source: Current Population Survey (2019) for retirement rate and health distribution, Health and Retirement Study (2004) for disability benefit claimer proportion.

3.2. Aggregate health shock

We define the aggregate health shock to mimic health deterioration in the aggregate level during the COVID-19 pandemic. It increases the probability of receiving negative idiosyncratic health shock at the end of the period to capture a higher likelihood of contagion during the pandemic. Also, this probability is even higher if agents are working because they are likely to engage in more physical contact and thus are more exposed to infection risk. We assume that the probability of infection for retirees, controlling for health status and health insurance as in our model, would be lower than workers.

Among the researches that study occupational infection risk of COVID-19, Chen et al. (Reference Chen, Glymour, Riley, Balmes, Duchowny, Harrison, Matthay and Bibbins-Domingo2021), using California Department of Public Health data, report that during March through November 2020, the excessive mortality ratio compared to expected mortality is 1.22 for the total population, 1.12 for non-essential workers, and 1.25 for unemployed/missing information. This result clearly suggests that the overall health risk elevated during the pandemic. However, because unemployed are likely to be in worse health conditions, also financially more constrained to afford medical care, and as Chen et al. (Reference Chen, Glymour, Riley, Balmes, Duchowny, Harrison, Matthay and Bibbins-Domingo2021) do not control for such factors, we cannot directly interpret this result as unemployed having a higher risk of adverse health shock. Another study, Mutambudzi et al. (Reference Mutambudzi, Niedwiedz, Macdonald, Leyland, Mair, Anderson, Celis-Morales, Cleland, Forbes, Gill, Hastie, Ho, Jani, Mackay, Nicholl, O’donnell, Sattar, Welsh, Pell, Katikireddi and Demou2021), using UK Biobank data and controlling for individual characteristics, reports that essential workers, classified by healthcare workers, social and education workers, and other essential workers, have a higher risk of severe COVID-19 relative to non-essential workers by 7.43, 1.88, 1.15 times, respectively. If we assume that the infection risk of retirees is similar to that of non-essential workers, and apply the population share of essential and non-essential workers in the total workforce, workers on average have a higher risk than retirees by about 1.7 times.

We take a conservative approach and calibrate the risk ratio, defined by the ratio of the pandemic state probability of the idiosyncratic health shock to the normal state probability, to be 1.3 for workers and 1.1 for non-workers. In other words, the risk of adverse health shock, whether the shock is high or low, increases by 1.3 times for workers and 1.1 times for non-workers during the pandemic compared to the normal state.

3.3. Model solution and simulation

Given the individual set of states $x = \left\{i, t, h, k, t^{R}, t^{DB}, t^{RB}, AIME\right\}$ , the policy space for the next period’s saving $k^{\prime}$ is discretized, to evenly spaced grid points between $[0,\bar{k}_{max}]$ where $\bar{k}_{max}$ is set differently for each individual type. The distance between grid points is set as 850. Agents know their state at the beginning of the period and optimally choose their policy $\left\{c, k^{\prime}, i^h, l, a^{DB}, a^{RB}\right\}$ . Among choice variables, consumption is determined as a residual.

The value function at age 81 is known, and the value and policy functions under age 81 are determined by the backward induction method. We assume that retirement, disability benefit claim, and retirement benefit claim are absorbing states. Depending on age, agents face different disability insurance and pension application options. For example, agents have the opportunity to retire at any time, but retirement benefit application is only available to agents at least 62 years old. Also, one cannot apply for the disability benefit after age 65. Consumption choice is calculated as a residual arising from each value combination of the other choice variables. Given individual states, we calculate the lifetime utility for each combination of choices and choose the optimizing one as a policy function. The resulting value function is used to calculate the value function of the previous age.

After solving the value and policy functions, we simulate the model economy using baseline calibration, where agents make decisions given that the probability of health shock will remain normal until they die. There are 56 age cohorts from age 25 to 80. Each age cohort has 5,000 individual agents. Each cohort enters the economy as 25 years old; thus, agents at age 26 are those who entered the economy one year ago, and agents at age 80 are those who entered the economy 55 years ago. An individual health shock hits each agent at the end of each period. This provides idiosyncratic uncertainty in the model.

We also simulate a counterfactual economy where unanticipated aggregate health shock hits the economy. Agents in age cohort $j$ , $j$ = 26, $\ldots$ ,80 behave the same way as the baseline economy until age $j-1$ , and agents in cohort $j$ = 25 enter the economy. They receive the aggregate health shock at age $j$ , choose their policy, then we calculate the retirement rate of each age cohort. Their decision at age $j$ is affected by two changes induced by the shock. First, in the pandemic state, the probability of current idiosyncratic health shock changes to $\mathbb{P}^{\prime}(x) = \{p^{\prime}_h(x), p^{\prime}_l(x),1-p^{\prime}_h(x)-p^{\prime}_l(x)\}$ , $\{p^{\prime}_h(x), p^{\prime}_l(x)\} = rr(x) \times \{p_h(x), p_l(x)\}$ , where $rr(x)$ is the risk ratio defined in the section 3.2. Second, once in the pandemic state, there exists a positive probability ( $p^A$ ) that the next period is again the pandemic state where the probability of health shock remains at $\mathbb{P}^{\prime}(x)$ . This assumption provides persistency to the aggregate health shock. However, agents believe that once the economy returns to the normal state with the probability 1- $p^A$ , the pandemic does not recur. We assign a small but meaningful probability to the continuation of the pandemic by setting $p^A =0.2$ . Later, we also analyze the case in which the aggregate health shock is expected to be temporary or more persistent, as setting $p^A=0$ or $p^A=0.4$ , respectively.