3.1. Demographics

Time is discrete. The economy is populated by overlapping generations of individuals. Their age is indexed by $j=1,..,J$ . Lifetime is uncertain. Individuals reach the age $j$ with the probability $Q_{j}$ and survive to the next period with probability $q_{j}$ at that age. The size of each new cohort grows at the rate $\nu$ . The population of age $j$ is noted $L_{j}$ and obeys the following recursion: $L_{j}=\frac{q_{j-1}L_{j-1}}{1+\nu }$ .

3.2. Preferences

Individuals derive utility from consumption and leisure, which is the amount of time not spent on working. More precisely, they make decisions to maximize the following lifetime expected utility:Footnote ⁸

(1) \begin{equation} \mathbb{E}_{1}\biggl [\sum _{j=1}^{J}Q_{j-1}\beta ^{j-1}\Bigl (q_{j}\bigl (u(c_{j},l_{j})+\sigma _{S}\epsilon _{j,l_{j},\Upsilon '_{j}}\unicode{x1D7D9}_{j\leq J_{W}}\bigr ) + (1-q_{j})v(a_{j+1}) \Bigr ) \biggr ]\\[5pt] \end{equation}

Where $\beta$ is the discount factor, $c_{j}$ represents the amount of consumption, and $l_{j}$ represents the number of hours worked at age $j$ . $\Upsilon '_{j}\in \mathbb{C}_{j}(\Upsilon _{j})$ is a binary variable indicating whether an individual claims her SS benefits at age $j$ or not. The choice set $\mathbb{C}_{j}(\Upsilon{j})$ is described below. As in Keane and Wasi (Reference Keane and Wasi2016), the choice of hours worked is restricted to a discrete set $\mathbb{D}_{j}$ . Individuals can participate in the labor market up to age $J{W}$ , meaning that $\mathbb{D}_{j}=\{0\}$ if $j\gt J{W}$ . Otherwise, $\mathbb{D}_{j}$ consists of five different levels, including the possibility of not working.

$u(c,l)$ is the utility derived from consuming $c$ units of goods and working $l$ hours. It has the following functional form:

(2) \begin{equation} u(c,l)=\frac{c^{\alpha _{L}(1-\sigma )}(1-l)^{(1-\alpha _{L})(1-\sigma )}}{1-\sigma }-\chi _{\eta } \unicode{x1D7D9}_{l\gt 0} \end{equation}

Where $\sigma$ is the coefficient of relative risk aversion for total utility. $\alpha _{L}$ determines the weight on consumption relative to leisure. $\chi _{\eta }$ is a fixed cost associated to labor market participation. It depends on the health status $\eta$ of the individual. This health status variable is described below.

$((\epsilon _{j,l,\Upsilon '})_{l\in \mathbb{D}_{j},\Upsilon ' \in \mathbb{C}_{j}(\Upsilon )})_{j\in [\![1,J_{W}]\!]})$ are taste shocks to the discrete decisions of the individual. They are independent and identically distributed according to a type-I distribution. As shown by Iskhakov et al. (Reference Iskhakov, Jørgensen, Rust and Schjerning2017), these taste shocks are useful to smooth the value function, which would otherwise present a kink at the points of indifference between two levels of hours worked. We interpret them as error terms capturing omitted factors influencing the labor supply decision of individuals. $\sigma _{S}$ controls the scale of these taste shocks.

Finally, the last term of (1) is a warm-glow bequests motive. We assume the following functional form for $v(:)$ : $v(a) =B_{eq}\frac{a^{\alpha _{L}(1-\sigma )}}{1-\sigma }$ (see for example Hosseini (Reference Hosseini2015)). This bequest motive is useful to avoid that individuals decumulate their assets too quickly during the retirement period (Nardi et al. (Reference De Nardi, İmrohoroğlu and Sargent1999)).

3.3. Human capital accumulation

Human capital evolves according to a Learning-By-Doing (LBD) mechanism: the more individuals work today, the more productive they are tomorrow. The effectiveness of this mechanism depends on the ability $A$ of individuals, which is constant over the life-cycle. Hence the marginal effect of labor supply on next period human capital is positive and positively depends on the ability of the individual. Human capital is also subject to a depreciation rate $\delta _{H}$ and to idiosyncratic shocks $(\Psi _{j})_{j \in [\![1,J_{W}]\!]}$ . These shocks are independent and follow a log-normal distribution, $\log (\Psi _{j}) \sim \mathcal{N}(0,\sigma _{\Psi }^{2})$ . The evolution of the human capital $h_{j}$ of an individual working $l_{j}$ hours is then given by:

(3) \begin{equation} h_{j+1}=\Psi _{j}(h_{j}(1-\delta _{H})+A(h_{j}l_{j})^{\theta }) \end{equation}

$\theta$ is a parameter belonging to $(0,1)$ , so much that the LBD mechanism has decreasing returns-to-scale. Blandin (Reference Blandin2018) and Blandin and Peterman (Reference Blandin and Peterman2019) employ a specification very similar to (3).Footnote ⁹ As shown by Huggett et al. (Reference Huggett, Ventura and Yaron2006), heterogeneity across the endowment space $(A,h_{1})$ is necessary to reproduce the dispersion of labor earnings over the lifecycle. Thus we allow individuals to differ both in their ability $A$ and in their initial human capital $h_{1}$ . Following Blandin (Reference Blandin2018), we assume that the distribution $\Lambda$ of this endowment is log-normal, $(\log (h_{1}),\log (A)) \sim \mathcal{N}(\mu _{a},\mu _{h_{1}},\sigma _{a}^{2},\sigma _{h_{1}}^{2},\rho )$ .

3.4. Earnings

An individual with human capital level $h$ working $l$ hours earns the following labor income:

(4) \begin{equation} W(h,l)=whl\max (1,(\frac{l}{\overline{l}})^{\psi }) \end{equation}

Where $w$ is the wage rate. The last term of (4) is an adjustment factor penalizing part-time work.Footnote ¹⁰ $\overline{l}$ is the amount of hours worked corresponding to full-time work. Note that the productivity of individuals confounds with their human capital. Thus, we use the two terms interchangeably throughout the remainder of the paper. We define effective labor supply at the individual level as the product of human capital/productivity, labor supply and the part-time penalty ( $hl\max (1,(\frac{l}{\overline{l}})^{\psi })$ ).

3.5. Health

A non-negligible share of old individuals is in bad health condition, which limits their possibility to supply labor. Thus, abstracting from health in the model would lead to overestimate the labor supply response of old individuals in our experiment. Therefore, we include a health status variable $\eta$ . Individuals are either in bad health, $\eta _{j}=0$ , or in good health, $\eta _{j}=1$ . The evolution of $\eta _{j}$ is governed by age-dependent transition probabilities. Bad health lowers labor supply via a greater disutility of work.Footnote ¹¹ Hence we assume that $\chi _{A,0}\gt \chi _{A,1}$ .

3.6. Retirement System

The SS benefits of an individual depends on two factors: her Average Indexed Monthly Earnings (AIME) and the age at which she claims her SS benefits $j_{claim}$ . Individuals claim their SS benefits between the ages $J_{1}$ and $J_{2}$ . They are allowed to continue to work while receiving their SS benefits and the claiming decision is irreversible. Thus, the choice set $\mathbb{C}_{j}(\Upsilon _{j})$ of the claiming decision $\Upsilon '_{j}$ at age $j$ depends on the current claiming status $\Upsilon _{j}$ as follows:

(5) \begin{align} \begin{split} \mathbb{C}_{j}(\Upsilon _{j})&= \left \{ \begin{array}{l@{\quad}l} \{0\}& \mbox{if } j\lt J_{1}\\[5pt] \{0;\;1\}& \mbox{if } J_{1} \leq j \leq J_{2} \mbox{ and } \Upsilon _{j}=0\\[5pt] \{1\} & \mbox{otherwise} \end{array} \right . \end{split} \end{align}

We approximate the AIME by the value at age $j_{claim}$ of the variable $\overline{e}_{j}$ , which is defined by the following equation:

(6) \begin{align} \begin{split} \overline{e}_{j+1}&= \left \{ \begin{array}{l@{\quad}l} \frac{j\overline{e}_{j}+\min (W(h,l),\hat{e})}{j+1}& \mbox{if } j\lt J_{1}\\[5pt] \max (\overline{e}_{j}, \frac{j\overline{e}_{j}+\min (W(h,l),\hat{e})}{j+1})& \mbox{if } J_{1} \leq j \leq J_{2} \\[5pt] \overline{e}_{j} & \mbox{otherwise} \end{array} \right . \end{split} \end{align}

And $\overline{e}_{1}=0$ . This approximation follows French (Reference French2005) among others. It means that before age $J_{1}$ , the AIME is computed as the average of labor earnings up to the SS tax cap $\hat{e}$ . From ages $J_{1}$ to $J_{2}$ , the AIME is updated only if labor earnings allow an increase of the AIME.

The SS benefits $p$ of an individual then write: $p=\gamma (j_{claim})P(\overline{e}_{j_{claim}})$ . Where $P(.)$ is the benefit-earnings rule and $\gamma (.)$ adjusts SS benefits according to the claiming age:

(7) \begin{align} \begin{split} \gamma (j)& \left \{ \begin{array}{l@{\quad}l} \lt 1 & \mbox{if } j\lt NR\\[5pt] =1 & \mbox{if } j=NR\\[5pt] \gt 1 & \mbox{if } j\gt NR \end{array} \right . \end{split} \end{align}

Where $NR$ is the normal retirement age. Finally, before age NR, individuals who have already claimed their SS benefits are subject to the Earning Test. Hence their SS benefits $p$ are reduced by the quantity $ET(p)$ , which is given by the following expression:

(8) \begin{equation} ET(p)=\min (p,\tau _{ET}\max (0,W(h,l)-y_{ET})) \end{equation}

Where $y_{ET}$ is the earning test threshold. In the same time, their AIME is increased by $ET(p)$ . In practice, to simplify the exposition, note that the state variables of the indiviudals with respect to her SS status are $(\overline{e}_{j},p,\Upsilon _{j})$ , where $\overline{e}_{j}$ is relevant if and only if $\Upsilon _{j}=0$ , while $p$ is relevant if and only if $\Upsilon _{j}=1$ . And the next period values of these two state variables can be written as: $\overline{e}_{j+1}=f(\overline{e}_{j},W(h,l), \Upsilon _{j},j)$ and $p'=g(p,W(h,l),\Upsilon _{j},j)$ .

3.7. Government

In addition to providing SS benefits, the government spends an amount of resources equal to $G$ and realizes transfers between individuals. The government levies resources through the confiscation of accidental bequests, a corporate tax, a consumption tax, the taxation of income and the payroll tax. Corporate and consumption taxes are linear taxes respectively on positive capital income and consumption purchases. Their respective rates are $\tau _{K}$ and $\tau _{C}$ . The payroll tax, $T_{S}(.)$ , is composed of the SS tax, a linear tax at rate $\tau _{S}$ on labor earnings below the SS tax cap, and the Medicare tax, a linear tax at rate $\tau _{med}$ on all labor earnings W, $T_{S}(W)=\tau _{med}W+\tau _{S}\min (W,\hat{e})$ .

We now detail the tax and transfer schedule that incorporates both the taxation of income and the transfers to individuals. Taxable income is the sum of capital income net of corporate tax, labor earnings net of half of the SS tax and SS benefits. The post-tax and transfer income $T(a,W,p)$ associated to a level of assets $a$ , labor earnings $W$ and SS benefits $p$ (net of the earning test) is given by:Footnote ¹²

(9) \begin{equation} T(a,y,p)=\min (ra,0)+\max (T_{0}((1-\tau _{K})\max (ra,0)+y-0.5T_{S}(y)+p)^{1-\tau _{1}}-0.5T_{S}(y),\underline{y}) \end{equation}

The first term inside the max operator of (9) is the tax and transfer functional form highlighted by Heathcote et al. (Reference Heathcote, Storesletten and Violante2017). These authors show that this function provides a good fit to the US tax and transfer schedule. A limitation of this function, that they acknowledge, is that the government does not provide any transfer to individuals without any resources. To remedy this, we inspire from Hubbard et al. (Reference Hubbard, Skinner and Zeldes1995) and augment our tax and transfer schedule with an income floor $\underline{y}$ , the second term of (9). Yum (Reference Yum2018) and del Rio (Reference del Rio2015) both stress the importance of including such transfers to low-income individuals to capture well their labor supply behavior. This ingredient also appears of first importance for our main experiment. Poor-wealth and old individuals mainly rely on their SS benefit to finance their consumption. Consequently, they are particularly hit by their removal. By abstracting from transfers to low-income individuals, the model would thus overestimate their labor supply reaction. Finally, we note that our formulation of the tax and transfer schedule (9) excludes the possibility for individuals to borrow against the income floor.

3.8. Markets

Markets are incomplete. Individuals cannot purchase insurance contracts against the various shocks they face. However, they can hold assets that pay a risk-free interest rate $r$ .Footnote ¹³ Individuals aged below $J_{1}$ can borrow at the risk free rate up to an exogenous limit $\underline{a}_{j}=\underline{a}$ . Individuals older than $J_{1}$ face a tight budget constraint, $\underline{a}_{j}=0$ .

3.9. Production

The final good is produced by a representative firm with Cobb-Douglas technology:

(10) \begin{equation} Y=BK^{\alpha }H^{1-\alpha } \end{equation}

Where $Y$ is output and $K$ is the physical capital input. $H$ is the aggregate effective labor supply, hence the sum across individuals of effective labor supply. It is the quantity of interest of the paper. We assume that the total factor productivity $B$ is constant over time. $\alpha \in (0,1)$ is the capital share. Physical capital depreciates at rate $\delta$ . Production factors are paid at their marginal productivity. Noting $k=\frac{K}{H}$ , the physical to human capital ratio, we obtain:

(11) \begin{equation} w=(1-\alpha )k^{\alpha } \end{equation}

(12) \begin{equation} r=\alpha k^{\alpha -1}-\delta \end{equation}

3.10. Equilibrium

At each age $j$ , an individual is characterized by a state vector $\Omega =(a,h,\overline{e},p,\eta, A,\Upsilon, ((\epsilon _{j,l,\Upsilon '})_{l\in \mathbb{D}_{j},\Upsilon ' \in \mathbb{C}_{j}(\Upsilon )})))$ where $a$ is the amount of assets held, $h$ is the human capital stock, $\overline{e}$ is the average of past labor earnings, $p$ is the SS benefits, $\eta$ is the health status of the individual, $\Upsilon$ is a binary variable indicating whether SS benefits have been claimed or not and $A$ denotes the ability of the individual. Initially, $a$ and $\overline{e}$ are both equal to $0$ , while $h$ and $A$ are drawn from their joint distribution $\Lambda$ . $\eta$ is initially drawn from its empirical distribution and then evolves according to age-dependent transition probabilities. $\Upsilon$ is worth 0 before age $J_{1}$ , and 1 after age $J_{2}$ . $((\epsilon _{j,l,\Upsilon '})_{l\in \mathbb{D}_{j},\Upsilon ' \in \mathbb{C}_{j}(\Upsilon )})_{j\in [\![1,J_{W}]\!]})$ are independently drawn from a Type-I value distribution. At age $j$ the distribution over $\Omega$ is written $\mu _{j}(\Omega )$ . The value of being in the state $\Omega$ at age $j$ is $V_{j}(\Omega )=\max _{l\in \mathbb{D}_{j}, \Upsilon '\in \mathbb{C}_{j}(\Upsilon )}(\tilde{V}_{j}(a,h,\overline{e},p,\eta, A,\Upsilon, l,\Upsilon ')+\sigma _{S}\epsilon _{j,l,\Upsilon '})$ , where $\tilde{V}_{j}(a,h,\overline{e},p,\eta, A,\Upsilon, l,\Upsilon ')$ satisfies the following Bellman equation:

(13) \begin{align} \begin{split} \tilde{V}_{j}(a,h,\overline{e},p,\eta, A,\Upsilon, l,\Upsilon ')&=\max _{a', c}(u(c,l)+\beta q_{j}\mathbb{E}_{j}[V_{j+1}(\Omega ')]+\beta (1-q_{j})v(a'))\\[5pt] s.t.&\\[5pt] a'&\geq \underline{a}_{j}, c\gt 0\\[5pt] a'&=a+T(a,W(h,l),\Upsilon (p-\unicode{x1D7D9}_{j\lt NR}ET(p,h,l))) -c(1+\tau _{c})\\[5pt] h'&=\Psi (h(1-\delta _{H})+A(hl)^{\theta })\\[5pt] \overline{e}'&=f(\overline{e},W(h,l), \Upsilon, j)\\[5pt] p'&=g(p,W(h,l),\Upsilon, j) \end{split} \end{align}

For a variable $z$ of the problem (13), we note $z_{j}(\Omega )$ the optimal value of $z$ at the state $\Omega$ and at the age $j$ . We define a stationary competitive equilibrium as follows:

• • Individuals solve the problem (13) at each age.

• • The representative firm chooses capital stock $K$ and human capital input $H$ to maximize profits given the production function (10).

• • Capital and labor markets clear:

  (14) \begin{equation} K=\sum _{j=1}^{J}L_{j}\int a_{j}(\Omega )\mu _{j}(d\Omega ) \end{equation}
  (15) \begin{equation} H=\sum _{j=1}^{J}L_{j}\int h_{j}(\Omega )l_{j}(\Omega )\max (1,(\frac{l_{j}(\Omega )}{\overline{l}})^{\psi })\mu _{j}(d\Omega ) \end{equation}

• • The budget of the government is balanced:

  (16) \begin{align} G&=\tau _{C}\sum _{j=1}^{J}L_{j}\int c_{j}(\Omega )\mu _{j}(d\Omega )+wH+rK-\nonumber \\[5pt] &\quad \sum _{j=1}^{J}L_{j}\int T(a_{j}(\Omega ),W(h_{j}(\Omega ),l_{j}(\Omega ),\unicode{x1D7D9}_{j\geq J_{R}}P(\overline{e}_{J_{R}}(\Omega )))\mu _{j}(d\Omega ) \end{align}

• • $(\mu _{j})_{1\leq j\leq T}$ is consistent.

3.11. Resolution

Solving the problem of the individuals is challenging for two main reasons. The first issue relates to the dimension of the state space: in addition to the age, ability and health variables, the problem features three continuous state variables. The second issue is due to the non-concavity of the problem because of the discrete labor supply decision and the LBD technology. We follow Iskhakov and Keane (Reference Iskhakov and Keane2021) and we note that conditional on the choice of hours worked, the problem of individuals only features one endogenous state variable. Thus we can implement a very fast solution algorithm by using the extension of the endogenous grid method to problems with discrete and continous choices (Iskhakov et al. (Reference Iskhakov, Jørgensen, Rust and Schjerning2017)). Appendix A provides details on the resolution.

4.1. Parameters Set Externally

4.1.4. Health

We adopt a logit specification for the health transition probabilities:

(17) \begin{equation} \log \left (\frac{\mathbb{P}(\eta '=1|\eta, j,time)}{\mathbb{P}(\eta '=0|\eta, j,time)}\right )=\sum _{age=24-25}^{age=78-79}(\varkappa _{0,age}1_{age=j}+\varkappa _{1,age,\eta }1_{age=j})+\varkappa (time) \end{equation}

Where $\varkappa (time)$ is a full set of time dummies. We estimate (17) using the PSID dataset, considering as unhealthy individuals self-reporting a poor or a bad health condition. Figure 1 reports the estimated health transition probabilities.

Figure 1.

Health data.

4.1.5. Social security

The SS tax cap is set to 2.47 times the average of labor earnings as in Huggett and Ventura (Reference Huggett and Ventura1999), $\hat{e}=2.47\overline{y}$ . Concerning the function $P(.)$ , we use the US benefit-earnings rule. The latter is a concave and piecewise linear function with three bend points:

(18) \begin{align} P(\overline{e}_{R})&=(0.9-0.32)\min (\overline{e}_{R}-0.2\overline{y},0)+(0.32-0.15)\min (\overline{e}_{R}-1.24\overline{y},0)\nonumber\\[5pt] &\quad +0.15\min (\overline{e}_{R}-2.47\overline{y},0) \end{align}

Where $(0.9;\; 0.32;\; 0.15)$ are the replacement rates, and $0.2\overline{y},1.24\overline{y}$ and $2.47\overline{y}$ are the estimates of the bend points of Huggett and Ventura (Reference Huggett and Ventura1999).

We use the value of Imrohoroglu and Kitao (Reference Imrohoroglu and Kitao2012) for the Earning Test tax rate, $\tau _{ET}=0.5$ . According to the values reported by Jones and Li (Reference Jones and Li2018), the Earning Test threshold, $y_{ET}$ is linked to the SS tax cap $\hat{e}$ by the relationship $y_{ET}=\frac{\hat{e}}{7.52}$ . Individuals are allowed to claim their SS benefits from period $J_{1}=20$ to period $J_{2}=23$ . The normal retirement period NR is set to 22, this corresponds to an actual age equal to 66-67. The adjustment coefficients $\gamma (.)$ are computed as the two-year average of the annual values reported by Bagchi (Reference Bagchi2015). Finally, the SS tax rate $\tau _{S}$ is set to $10.6\%$ , while the Medicare tax rate $\tau _{med}$ is set to $2.9\%$ .

4.2. Second-Stage Estimation

The remaining parameters, $(\beta, \tau _{0},B_{eq},(\chi _{\eta })_{\eta \in \{0;1\}},\alpha _{L},\sigma, \delta _{H},\theta, \sigma _{S},\sigma _{\Psi })$ and the distributional parameters $(\mu _{a},\sigma _{a}^{2},\sigma _{h_{1}}^{2},\rho )$ , are determined internally. We constrain the model to match an annual capital-to-output ratio equal to $3$ by iterating over the discount factor $\beta$ . We balance the budget of the government via the parameter $T_{0}$ .

We compute the other parameters for our model to reproduce several life-cycle profiles of the US economy. The identification of the parameters governing human capital accumulation follows Huggett et al. (Reference Huggett, Ventura and Yaron2006) and Blandin (Reference Blandin2018). The variance of the initial log human capital, $\sigma _{h_{1}}^{2}$ , is identified by the variance of the log wage at early ages. The higher the variance of the log ability, $\sigma _{a}^{2}$ , the greater the increase in wage dispersion over the lifecycle. Thus, $\sigma _{a}^{2}$ is mostly identified by the variance of the log wage at old ages. A higher value of $\sigma _{\Psi }$ shifts the variance of the log wage upwards at all ages. Thus, this parameter is identified by the level of this profile. A higher average of the log ability $\mu _{a}$ implies a greater growth in wages over the lifecycle. Thus, $\mu _{a}$ is mainly identified by the peak of the log wage profile. The correlation between ability and initial human capital, $\rho$ , influences the shape of the variance of the log wage profile: low values produce a U-shaped profile, while values close to 1 produce an increasing profile. Thus, $\rho$ is identified by the shape of the variance of the log wage profile. The parameter $\theta$ tends to rotate the log wage profile: a lower value of $\theta$ implies a higher growth rate of wages at the beginning of the lifecycle. Thus, $\theta$ is identified by the shape of the log wage profile. At old ages the evolution of human capital tends to be driven by its depreciation since older individuals work fewer hours. Thus, $\delta _{H}$ is identified by the shape of the log wage profile at old ages.

The parameters $(\chi _{0}, \chi _{1})$ influence the labor market participation decision. Thus, they are identified by the lifecycle profile of the employment rate conditional on health. As $\sigma _{S}$ tends to infinity, the employment rate converges toward $0.5$ at all ages. Thus, a higher value for $\sigma _{S}$ rotates the employment profile counterclockwise, and this parameter is identified by the employment rate profile. The parameter $\alpha _{L}$ is identified by the lifecycle profile of hours worked conditional on working. The parameter $\underline{a}$ influences the shape of the consumption profile at young ages when the mass of poor-wealth individuals is large. The parameter $\sigma$ affects both the slope of the consumption and the assets profile. Finally, the parameter $B_{eq}$ is identified by the assets profile at old ages.

The wage, hours, employment and assets profiles are computed from the PSID, while the consumption profile is computed from the CEX. Details on the computation of the moments are relegated in Appendix C. The objective we minimize is the squared distance between the moments computed from the data and their model counterparts. Formally, let $\gamma =(B_{eq},(\chi _{\eta })_{\eta \in \{0;1\}},\alpha _{L},\sigma, \delta _{H},\theta, \sigma _{S},\sigma _{\Psi },\mu _{a},\sigma _{a}^{2},\sigma _{h_{1}}^{2},\rho )$ the set of estimated parameters, $m=(m_{i})_{i\in I}$ the set of targeted moments and $M(\gamma )=(M_{i}(\gamma ))_{i\in I}$ their model counterparts. The indirect inference estimate of $\gamma$ is :

(19) \begin{equation} \hat{\gamma }=\textrm{argmin}_{\tilde{\gamma }}(G(\tilde{\gamma }))=\textrm{argmin}_{\tilde{\gamma }}((m-M(\tilde{\gamma }))'W^{-1}(m-M(\tilde{\gamma }))) \end{equation}

The weighting matrix $W$ is diagonal. Appendix B details how we perform the minimization of the function $G(.)$ . Table 2 reports the obtained values. Appendix C details how we compute the standard errors of the estimated parameters.

4.2.1. Backfitting

From Figure 2 and 3, we see that the LBD model reproduces quite well the targeted profiles. This happens without using a large number of parameters. Table 3 also reveals that the LBD model reproduces well the various macroeconomic targets.

Figure 2.

Life-cycle profiles in the data, in the LBD model and in the exogenous model.

Figure 3.

Life-cycle profiles in the data, in the LBD model and in the exogenous model.

Notes: the figures 2 and 3 display the life-cycle profile of various variables in the LBD model, in the data and in the exogenous model. For figure (a) the data source is the CEX. For the remaining figures, the data source is the PSID. Wages and hours worked are conditional on labor market participation. The series of figure (a), (d), (f) and (h) are normalized by their initial value.

5.1. A variant with exogenous human capital

We build the variant as closely as possible to the LBD model for the differences in results between the two models to only reflect the different mechanism of human capital accumulation. For this, we assume that the variant is identical to the LBD model except that the law of accumulation (3) is now given by the following equation:

(20) \begin{equation} h_{j+1}=e^{\Psi _{j}}h_{j}\gamma _{j}(A) \end{equation}

Where $\Psi _{j} \sim \mathcal{N}(0,\,\sigma _{\Psi, j,A}^{2})$ and $\gamma _{j}(A)$ are coefficients that depend on age and ability. Importantly, (20) does not depend on any choice variable.

To calibrate the parameters of the exogenous model, we target the same moments used to estimate the LBD model. However, the set of targeted moments does not include any moments from the unconditional distribution of wages because the latter is unobserved. This is problematic here because the levels of human capital need to be similar between the two models to isolate the role played by the endogenous accumulation of human capital. Indeed, the larger the human capital at old ages, the greater the labor supply reaction of individuals in the experiment, independently of the way human capital accumulates. To avoid this, we impose restrictions on the parameters of the variant to ensure that the unconditional distribution of wages is similar between the models. First, we assume that the initial endowment distribution is the same as in the LBD model. Second, we choose $\gamma _{j}(A)$ for the average of the log growth rate of human capital conditional on ability and age to be equal in the two models, $\gamma _{j}(A)=\mathbb{E}[\log (\frac{h_{j+1}}{h_{j}})|A,j]$ . Third, we choose $\sigma _{\Psi, j,A}$ for the variance of the log growth rate of human capital conditional on ability and age to be equal in the two models, $\sigma _{\Psi, j,A}^{2}=$ $[\log (\frac{h_{j+1}}{h_{j}})|A,j]$ . These restrictions, in particular, imply that the wage profiles conditional on ability are identical in the two models.Footnote ¹⁵ A drawback of this approach is that it is more challenging for the exogenous model to replicate the moments of the wage distribution conditional on working. This is because most parameters governing the wage process have been fixed by our previous restrictions. This is evident in the fit of the exogenous model (Figure 3). We observe that the variance of the log wage is higher in the exogenous model compared to the data. However, the exogenous model replicates the other moments quite well. Alternatively, we could specify functional forms for $\gamma _{j}(A)=f_{1}(\tilde{\gamma _{1}},j,A)$ and $\sigma _{\Psi, j,A}=f_{2}(\tilde{\gamma _{2}},j,A)$ , where $f_{1}$ and $f_{2}$ are given functions, and $\tilde{\gamma _{1}},\tilde{\gamma _{2}}$ are additional parameters to be estimated. Then we would calibrate the exogenous model to reproduce the previous moments as well as moments from the unconditional distribution of the wage of the LBD model. However, this approach is more computationally demanding because there are more parameters to calibrate. Moreover, it is unclear which moments of the unconditional distribution of the wage we should include. For these reasons, we choose to pursue our first approach.

Finally, we also note that Fan et al. (Reference Fan, Seshadri and Taber2024) follows a different approach as they similarly estimate their exogenous model and their model with endogenous human capital. The reason is that their goal is to compare the ability of both models to fit the data, not to highlight the role of endogenous human capital accumulation in the difference of results between their two models.

Figure 2 and 3 report the fit of the exogenous model. Figure 4 reports the value of the Frisch elasticity by age. The aggregate Frisch elasticity of labor supply and labor market participation are respectively 0.96 and 0.87. Hence, the Frisch elasticity in the LBD model is lower than in the exogenous model. This is particularly true at young ages where the dynamic payoff of labor supply, its effects on future labor earnings, dominates its contemporaneous payoff.

Table 3.

Model performance

Notes: the table compares the values of certain moments in the Learning-By-Doing model to their target.

Figure 4.

Labor supply elasticities.

Notes: the left figure displays the elasticity of labor supply to a transitory and anticipated increase of the wage rate in the LBD model and in the exogenous model. The right figure displays the elasticity of labor supply at the extensive margin to a transitory and anticipated increase of the wage rate.

5.2. Results

In this subsection, we assess the effect of removing the SS on aggregate effective labor supply in the LBD model and determine the role of human capital in our result by performing the same experiment in the exogenous model. To implement this experiment, we set the SS tax rate $\tau _{S}$ and the level of SS benefits to 0. The level of government spending, $G$ , and all other parameters, except $T_{0}$ , remain unchanged. The parameter $T_{0}$ adjusts to balance the government’s budget. Table 4 reports the impact on the main aggregate variables in the three models. Figures 5 and 6 report the behavioral reactions of individuals. We observe that removing the SS system has a strong positive effect on aggregate effective labor supply in the LBD model ( $10.31\%$ ). For comparison, Imrohoroglu and Kitao (Reference Imrohoroglu and Kitao2009) find an effect ranging from $-0.4\%$ to $0.35\%$ , while Nishiyama and Smetters (Reference Nishiyama and Smetters2007) find an effect equal to $3.5\%$ . This is more than twice the impact observed in the exogenous model ( $5.10\%$ ). This contrasts with the findings of Fan et al. (Reference Fan, Seshadri and Taber2024). They present three models with endogenous human capital in which the effect of removing the SS on the lifetime output of individuals ranges from $-0.043\%$ to $+1.298\%$ , while in their exogenous model, the effect is $-0.308\%$ (Table D3 in the Appendix D of Fan et al. [Reference Fan, Seshadri and Taber2024]). As previously mentioned, their strategy to calibrate their exogenous model is different from ours. It implies that the human capital profiles of their main model and their exogenous model might diverge at old ages, which could confound the role of human capital accumulation in the difference of results of the experiment.

Table 4.

Macroeconomic effects

Notes: the table reports the relative change of the main macroeconomic variables when we eliminate the social security. For the interest rate, we report its annual value in the counterfactual economy. In the baseline economy of both models, the value of the annual interest rate is $4\%$ . The CEV is computed as $(\frac{\mbox{Welfare(counterfactual)}-\mbox{Utility bequests(baseline)}}{\mbox{Welfare(baseline)}-\mbox{Utility bequests(baseline)}})^{\frac{1}{\alpha _{L}(1-\sigma )}}-1$ .

Figure 5.

Life-cycle profiles in the baseline and in the counterfactual economies: LBD model.

Figure 6.

Life-cycle profiles in the baseline and in the counterfactual economies: exogenous model.

We can see from Figure 5 and Figure 6 that the result stems from a larger increase in old-age effective labor supply in the LBD model compared to the exogenous model. This is the result of both higher employment rates and greater levels of human capital at those ages. To better understand the difference in results between the two models, we decompose the effect of the SS on aggregate effective labor supply into the following five channels:

1. (i) the removal of SS benefits

2. (ii) the removal of the SS tax

3. (iii) the decrease in the level of the income tax

4. (iv) the increase in the wage rate

5. (v) the decrease in the interest rate

In Table 5, we report how each of these effects changes aggregate effective labor supply in each model. We observe that the removal of SS benefits drives the increase in effective labor supply in both models. Additionally, we can see that this effect is much larger in the LBD model than in the exogenous model. Since this effect encourages old-age labor supply through an income effect, it motivates individuals to accumulate more human capital to be productive at these ages. Conversely, the incentive to work more at old ages is weakened in the exogenous model since older individuals have a lower level of human capital at those ages.Footnote ¹⁶

We briefly comment on the other channels. The removal of the SS tax decreases labor supply in both models through a wealth effect, and this effect is greater in the LBD model. The difference in results with respect to income taxation is due to the difference in the change in the level of the income tax between the two models. In the LBD model, the government budget is almost unchanged after the removal of the SS, which explains the negligible impact of the change in income taxation on $H$ . The loss of the surplus of the SS is compensated by the increase in labor income and a small increase in capital income due to the increase in the capital stock. In the exogenous model, the increase in labor income is smaller, and capital income decreases. Thus, the government needs to increase the level of the income tax to balance its budget, which increases labor supply through a wealth effect. This wealth effect also explains the sign of the labor supply change caused by the increase in the wage rate and the decrease in the interest rate.

It is instructive to look at the effective labor response across initial types. Let us first focus on the LBD model. We compute the lifetime effective labor supply change across the endowment space. We first observe from Table 6 that there is a substantial effective labor supply response by all individuals. The response is monotonically decreasing with respect to initial human capital. Indeed, individuals at the bottom of the initial human capital distribution tend to be the poorest ones. Thus, they are particularly affected by the removal of SS benefits. The effective labor supply response with respect to ability is not monotonic. This reflects two opposite effects. On the one hand, the lower $A$ the poorer the individual, the higher the income effect due to the removal of SS benefits. On the other hand, the lower $A$ the less effective the individual is at augmenting their human capital at old ages, and the lower the effective labor supply responses at those ages.

We repeat this exercise for the exogenous model. We observe from Table 6 that the decline in the effective labor supply response is the largest for individuals with either low ability or a low initial human capital level. The reason is that individuals with a low endowment have a particularly low level of human capital at old ages in the baseline economy of the LBD model and also in the exogenous model This is a direct consequence of this endowment. Individuals with lower ability accumulate less human capital over the life-cycle, and are thus less productive at old ages. It is also due to the fact that individuals with a low endowment leave the labor force earlier. This happens because the low level of human capital is an incentive to leave the labor force, and because the negative income effect of SS benefits is stronger for individuals with low assets. Thus, the potential wage of older individuals with a bad endowment is low in the exogenous model. Because of the transfers, the reservation wage of such individuals remains greater than their potential wage for any value of their SS benefits.

5.2.1. Welfare

We also examine the welfare effects due to the removal of the SS. From Table 4, we note that the CEV is greater in the LBD model than in the exogenous model. Without the SS system, individuals finance their old-age consumption via greater savings and greater labor earnings at old ages. If human capital is fixed, then the second channel is limited because individuals have a low level of human capital at those ages. Conversely, individuals equipped with the LBD technology can more easily obtain resources on the labor market at old ages. This shows that endogenous human capital is also important for computing the welfare effects of SS systems.

Table 5.

Decomposition

Notes: the table reports the change of aggregate effective labor supply, $H$ , due to each effect. For example, the last column reports the change of $H$ when we set the interest rate to its value of the counterfactual economy.

Table 6.

Lifetime effective labor supply in the endowment space

Notes: the table reports the lifetime effective labor supply change according to the endowment of the individual. Low (respectively high) ability individuals are in the bottom (respectively top) $27\%$ of the ability distribution. Individuals with a low (respectively high) level of initial human capital are in the bottom (respectively top) $ 25\%$ of the initial human capital distribution.

6.1. Two variants to isolate the role of the selection bias

In this section, we investigate a second issue that researchers face when they assess the effect of SS reforms from a structural model. The typical approach is to specify a structural model similar to our LBD model where labor earnings are still given by equation (4), $W(h,l)=whl\max (1,(\frac{l}{\overline{l}})^{\psi })$ , except that the productivity $h_{i, a}$ of an individual $i$ of age $a$ is an exogenous process specified by the following equation:

(21) \begin{equation} \log (h_{i, a})=\gamma _{a} +\eta _{i}+\epsilon _{i,a} \end{equation}

(22) \begin{equation} \epsilon _{i,a+1}=\rho \epsilon _{i,a}+u_{i,a} \end{equation}

Where $\eta _{i}$ represents an individual fixed effect, and $\epsilon _{i,a}$ and $u_{i,a}$ are error terms. We refer to $(\gamma _{a})_{a}$ as the log productivity profile. Similar specifications are used for example in Alonso-Ortiz (Reference Alonso-Ortiz2014), Imrohoroglu and Kitao (Reference Imrohoroglu and Kitao2012), Kopecky and Koreshkova (Reference Kopecky and Koreshkova2014). In these papers, the parameters of the process defined by (21) and (22) are estimated using wage data and then incorporated into the model to conduct the experiment (or alternatively the estimates are taken from other papers). However, a problem arises because the wages of individuals are only observed when they are employed. Consequently, the estimated parameters are likely affected by selection bias, which in turn contaminates the results of the experiment. We refer to this issue as the selection bias issue. We quantify this selection bias problem as follows. We assume that the LBD model is the DGP, and we draw two samples from this DGP. The first sample is a random sample of the DGP that includes both workers and non-workers. Thus, it is a sample in which the estimation of the parameters of the process (21) and (22) is not subject to selection bias. The second sample is a random sample of the DGP that includes only observations of workers, mimicking real-world datasets. For each of these two samples, we estimate the parameters of the process (21) and (22). We then use these estimates in our structural model and calibrate the remaining parameters. Finally, we compare the results of our experiment using both samples, which allows us to quantify the extent of the selection bias problem (Figure 7).

Figure 7.

Productivity profiles.

Notes: the figure displays the productivity profile $(e^{\gamma _{a}})_{a}$ in the selection bias model and in the No selection bias model.

Figure 8.

Life-cycle profiles in the selection bias model and in the No selection bias model.

Figure 9.

Life-cycle profiles in the selection bias model and in the No selection bias model.

Notes: the figures display the life-cycle profile of various variables in the selection bias model and in the No selection bias model.

We refer to selection bias model(respectively No selection bias model) as the model calibrated with the productivity process that does (respectively does not) suffer from the selection bias issue. There are two issues related to the calibration of these two variants that are worth discussing. First, the selection bias concerns the log productivity by age $(\gamma _{a})_{a}$ , the fixed effects distribution (which is discretized with the same number of ability levels as in the LBD model), and the parameters governing the AR(1) process ( $\rho$ and $\sigma _{u}$ ). We found that the estimates of the parameters $\rho$ and $\sigma _{u}$ were quite close in both samples. Moreover, the estimate of $\rho$ is a bit larger than the values reported in other studies. Thus, we choose to set $\rho$ and $\sigma _{u}$ to standard values (the values used by Kitao (Reference Kitao2014)) for both models. Second, the set of targeted moments used to calibrate the remaining parameters is restricted compared to the estimation of the LBD model. More precisely, we do not target any moments of the wages process: the log wage profile (with and without fixed effects) and the variance of the log wage profile. Indeed the parameters of this process (equations (21) and (22)) are already estimated on our two samples without having to use the model. This is consistent with the two-stage approach employed in similar models such as Kitao (Reference Kitao2014).Footnote ¹⁷ Figure 8 and Figure 9 show the fit of the two models with respect to the targeted moments. Table 7 reports the values of the parameters of the two models. Figure 10 reports the Frisch elasticity of labor supply and labor market participation by age. Figure 7 compares the productivity profile in the two variants. It reveals that the selection bias is particularly important at old ages: the selection bias leads to an overestimate of the wage by $15\%$ at age 75.

Table 7.

Model parameters

Notes: the table displays the values of parameters set jointly to reproduce the targeted moments in the selection bias model and in the No selection bias model.

Figure 10.

Labor supply elasticities.

Notes: the left figure displays the elasticity of labor supply to a transitory and anticipated increase of the wage rate in the selection bias model and in the No selection bias model. The right figure displays the elasticity of labor supply at the extensive margin to a transitory and anticipated increase of the wage rate.

6.2. Results

In this subsection, we replicate our previous experiment in the selection bias model and in the No selection bias model. The impact on the main macroeconomic variables is reported in Table 4. We observe that the impact on $H$ in the No selection bias model is very similar to what was found inthe exogenous model. This is not surprising because the productivity process in both the exogenous model and in the No selection bias model is constructed from the unconditional distribution of the human capital process in the LBD model. Consequently, older individuals exhibit relatively low productivity, and as a result, they do not significantly increase their labor supply after the removal of the SS. In contrast, the selection bias model features older individuals with higher productivity due to the contamination of their productivity estimates by the selection bias issue. This explains why the increase in $H$ is greater in the selection bias model compared to the No selection bias model. However, this difference is smaller than the one observed between the LBD model and the exogenous model. Part of the reason for this lies in the higher disutility of labor in the selection bias model compared to the No selection bias model, as evidenced by the larger value of the parameter $\alpha _{L}$ . Since individuals are more productive in the selection bias model, a greater disutility of labor is necessary for the model to align with the observed data.

In summary, we find that: $(i)$ the endogeneity issue, assessed by comparing the results of the experiment between the exogenous model and the LBD model, leads to an underestimation of the impact on $H$ by $50.53\%$ . $(ii)$ the selection bias issue, assessed by comparing the results of the experiment between the selection bias model and the No selection bias model, leads to an overestimation of the impact on $H$ by $34.6\%$ . Consequently, the endogeneity of human capital appears to be the more significant issue. Moreover, some of the results of the experiment in the selection bias model are closer to those of the LBD model than those of the No selection bias model. Given that the LBD model is not subject to the selection bias issue and the endogeneity issue, its results are likely closer to the true results than those of our different variants. Thus, when predicting the impact of SS reforms on $H$ or GDP, the best model between the selection bias model and the No selection bias model is the selection bias model. Hence, it may be more judicious not to correct for the selection bias. However, this does not hold true for assessing the impact of SS reforms on welfare. Indeed, we observe that the CEV of the No selection bias model is closer to the CEV of the LBD model. The reason for this is that the disutility of labor is greater in the selection bias model. Consequently, the increase in labor supply due to the removal of the SS has a more negative impact on the welfare of individuals.