2.1. Estimating differences in wage risk

We estimate the stochastic process for public and private sector wages using the Brazilian matched employer–employee microdata (RAIS). We follow the estimation procedure presented in Low et al. (Reference Low, Meghir and Pistaferri2010). We first estimate separate wage regressions for each sector and then use the method of moments to identify the persistence and the variance from the residuals.

To deal with potential selection bias, we take into account the participation decision in the estimation process. To this end, we first estimate quarterly probits for participation controlling for age, gender, education, and year.Footnote ^5, Footnote ⁶ We then estimate wage equation including the selection terms obtained from the probit estimates.

We assume wages are given by:

(1) \begin{equation} \ln w_{it} = d_i+x^{\prime}_{it}\psi +u_{it} \end{equation}

where $w_{it}$ is the (real) wage, $d_i$ is the human capital, $x_{it}$ is the vector of controls, including age, sector, and year dummies, and $u_{it}$ is the persistent component of wages, which we assume to follow an AR(1) process:

\begin{equation*}u_{it}=\rho u_{it-1}+\varsigma _{it}, \quad \rho \lt 1 \text { and } \varsigma \sim N\!\left(0,\sigma _{\varsigma }\right)\end{equation*}

In order to control for selection, we have two latent indices associated with participation:

(2) \begin{equation} P_{it}^*=z^{\prime}_{it} \gamma + \pi _{it} \end{equation}

(3) \begin{equation} P_{it-1}^*=z^{\prime}_{it-1} \gamma + \pi _{it-1} \end{equation}

Taking expectations for employed workers (conditional on participation):

(4) \begin{equation} \mathbf{E} (\!\ln w_{it} | P_{it} = 1,P_{it-1} = 1) = d_t+ x^{\prime}_{it}\psi +\rho _{\varsigma \pi } \sigma _{\varsigma } \bar{\lambda }_{it} + \rho _{\varsigma \pi -1} \sigma _{\varsigma } \bar{\lambda }_{it-1} \end{equation}

where $\rho _{\varsigma \pi }$ is the correlation between the persistent component of wages and the participation equation, and $\sigma _{\varsigma }$ and $\bar{\lambda }$ are the variance and Mills ratio associated with this persistent component.

Let the residual wage growth be $g_{it}=\hat{w}_{it} - w_{it}$ , where $\hat{w}_{it}$ is the wage growth predicted by the model. In order to identify the variance of the shocks, we use the first two moments and the first-order auto-covariance of this residual, which are defined below.

\begin{align*} E(g_{it}|P_{it}=1,P_{it-1}=1) =\, & \sigma _{u} \!\left (\rho _{\varsigma \pi } \bar{\lambda }_{it} + \rho _{\varsigma \pi -1}\bar{\lambda }_{it-1} \right ) \\ E(g_{it}^2|P_{it}=1,P_{it-1}=1) =\, & \sigma _{u}^2 \!\left ( 1-\rho _{\varsigma \pi }^2 z^{\prime}_{it} \gamma \bar{\lambda }_{it} - \rho _{\varsigma \pi -1}^2 z^{\prime}_{it-1} \gamma \bar{\lambda }_{it-1} + 2 \rho _{\varsigma \pi }\rho _{\varsigma \pi -1}\bar{\lambda }_{it}\bar{\lambda }_{it-1}\right ) \\ E(g_{it}g_{it-1}) =\, & \rho \sigma _u^2 \end{align*}

with $\sigma _u^2 = \frac{\sigma _{\varsigma }^2}{1-\rho ^2}$

Given the residuals from the wage equation, we use the method of moments in order to estimate the parameters governing the productivity for each sector. More specifically, we use the first two moments and the first-level autocorrelation from the residuals in order to estimate the persistence and the variance of this process.

We first report the results for the process estimated without separating private and public sector, and then for each sector separately. Moreover, for each of these groups, we obtain the results with and without considering the participation adjustment. We are able to do this by simply imposing $\rho _{\varsigma \pi }=0$ in the case without participation. Moreover, in order to be able to identify all the parameters properly in the case with participation we assumed that: $\rho _{\varsigma \pi }=\rho _{\varsigma \pi -1}$ . Table 1 shows our results.

Table 1. Stochastic process estimation

Standard errors in parentheses. ^* $p\lt 0.05$ , ^** $p\lt 0.01$ , ^*** $p\lt 0.001$ .

We can observe that considering both sectors together the variance of the process is higher and the persistence is lower when we do not take into account the participation decision. This results still holds considering only the private sector workers. The results from including the participation estimation only for civil servants are the opposite but for them $\rho _{\varsigma \pi }$ is not statistically significant. Moreover, comparing the estimates for each sector separately we can see that the persistent is higher and the variance is lower in the public sector compared to the private sector, consistent with the fact that public sector jobs are more stable and less riskier than jobs in the private sector.

2.2. Unemployment shocks

In order to estimate the unemployment risk, we also rely on data from Rais since it allows to identify the cause of separation. For the period between 2010 and 2014, we find that the separation rate in the private sector is more than four times larger than in the public sector. Despite being small, we do observe some separation in the public sector, most of which being involuntary such as dismissal by the firm and compulsory retirement. Fig. 3 plots the exogenous separation for each sector by age for 2010.

Figure 3. Unemployment risk for private and public sector. Source: Rais (2010).

3.1. Demography

Time is discrete and at each period $t$ a new generation is born. Individuals may live for at most $T$ periods. Uncertainty regarding the time of death is captured by the fact that each individual faces a probability $\psi _{t+1}$ of surviving to the age $t+1$ conditional on being alive at age $t$ . We map the survival probability into the time invariant age profile of the population denoted by $\left \{ \mu _{t}\right \} _{t=1}^{T}$ . Let $g_{n}$ denote the population growth rate, the fraction of agents $t$ years old in the population is found using the following law of motion $\mu _{t}=\frac{\psi _{t}}{1+g_{n}}\mu _{t-1}$ , with $\mu _t \geq 0$ and $\sum _{t=1}^{T}\mu _{t}=1$ . For most of our analysis, we will focus on the steady-state allocations. Since it greatly simplifies the presentation, we shall drop all time indices, $j$ , from aggregate variables and use $t$ to represent age.

3.2. Private sector technology

Firms in the private sector produce the consumption good through a standard constant returns to scale production function:

(5) \begin{equation} Y_{p} = G^\chi K_{p}^\alpha N_{p}^{1-\alpha }, \ \alpha \in (0,1). \end{equation}

where $G$ is the public good produced by the government, which is taken as given.Footnote ⁷ Private sector firms also take prices as given and choose $K_p$ and $N_p$ to maximize profits.Footnote ⁸ The first-order conditions of a representative firm are given by:

(6) \begin{align} r = \alpha G^\chi \!\left ( \frac{K_{p}}{N_{p}} \right )^{\alpha -1} - \delta, \end{align}

(7) \begin{align} w = (1-\alpha ) G^\chi \!\left ( \frac{K_{p}}{N_{p}} \right )^{\alpha }. \end{align}

where $r$ denotes the interest rate, $w$ is the wage rate, and $\delta$ is the depreciation rate.

3.3. Government sector

We assume that the public good is produced by the government. The public good, $G$ , is produced using efficient labor units $N_{g}$ and capital $K_{g}$ according to the following technology:

(8) \begin{equation} G = A_{g} K_{g}^\alpha N_{g}^{1-\alpha }, \ A_g\gt 0. \end{equation}

This is an aggregation of labor and capital to produce public goods and services such as paved roads and the rule of law. It is important to highlight that $K_{g}$ in our model is capital, such as machines and equipment, employed in the public sector to produce public infrastructure. Capital employed in the public sector evolves according to the following law of motion:

(9) \begin{equation} K_{g,t+1} = I_{g} + (1-\delta _{g})K_{g,t}, \end{equation}

where $I_{g}$ is financed through taxes.

There are two different social security regimes: a scheme for private sector workers and a scheme for public workers. The replacement rate for civil servants is different from the one faced by private sector workers. Consequently, in the model we have two types of retirees: individuals who retired from the public sector and individuals who retired from the private sector.

In addition, we assume that the government carries out an exogenous flow of expenditure, $C_{g}$ , which includes other parts of government consumption such as military expenditure that is deemed to be unproductive in our model. $C_{g}$ is just useful to allow the model to match the actual aggregate share of government spending in the economy and it is kept constant in the quantitative exercises. In order to finance its expenditures, the government levies proportional taxes on consumption, $\tau _{c}$ , and on capital income, $\tau _{k}$ . Labor income taxes are allowed to be nonlinear. In particular, we use the specification $T(y)=y-(1-\tau _w) y^{1-\varrho }$ .

3.4. Households

Labor supply and earnings. In each period of life, agents can choose to search for a job in the private or in the public sector. Conditional on the career choice, individuals make decisions about asset accumulation and labor supply. They can also choose not to work and stay nonemployed, in which case they supply zero hours in the labor market.

An individual of age $t$ who works for $n$ hours in the private sector earns $y_p = w n_t h_{t}e^{(u+z_{p,t})}$ , where $w$ is the rental rate. The variable $u\sim \mathcal{N}\!\left(0,\sigma _{u}^{2}\right)$ is a permanent component of an individual’s skills. It is realized at birth and retained throughout one’s life. On the other hand, $z_{p,t}$ evolves stochastically according to an AR(1) process, $z_{p,t}=\varphi _{p} z_{p,t-1}+\varepsilon _{p,t}$ , with innovations $\varepsilon _{p,t}\sim N\!\left(0,\sigma _{\varepsilon _{p}}^{2}\right)$ . Analogously, an individual of age $t$ who works for $n$ hours in the public sector earns $y_{cs} = (1+\zeta )wh_{t}e^{(u+z_{g,t})}$ , where $z_{g,t}$ follows an AR(1) process given by $z_{g,t}=\varphi _{g} z_{g,t-1}+\varepsilon _{g,t}$ , with innovations $\varepsilon _{g,t}\sim N\!\left(0,\sigma _{\varepsilon _{g}}^{2}\right)$ . We allow the stochastic process for $z$ to differ to capture differences in wage risk between sectors. This to be consistent with the fact that labor legislation and job characteristics in the public sector may be different from the ones in the private sector.

In addition, agents face at each period an exogenous sector-specific age-dependent probability of job separation, $\varsigma _{m,t}$ with $m=g,p$ . As shown in Section 2, we have that $\varsigma _{p,t}\gt \varsigma _{g,t}$ in the data. In fact, low risk of unemployment should be one of the main reasons for agents to search for a job in the public sector.

The age-efficiency profile, $h_t$ , is endogenous and evolves according to a learning-by-doing technology in which individuals accumulate human capital by working. We introduce human capital accumulation to account for differences in the age-efficiency profile arising from differences in hours worked and job tenure between the two sectors. The law of motion for $h_t$ is given by

(10) \begin{equation} h_{t+1} = q_m h_t^{\xi _{m}} n_t^{1-\xi _{m}} + (1-\delta _h) h_t \; \; \text{with} \; m={g,p} \end{equation}

where $q_m$ is a scale parameter and $\xi _{m}$ are parameters that govern both the persistence of the age-efficiency profile and the impact of hours worked on its evolution.

Preferences and choices. Each person has a unit time endowment which can be directly consumed in the form of leisure, $l$ , or used in market-related activities. An agent’s period-by-period time constraint is $l_{t}+n_{t}+s_{g,t}\phi _t=1$ , where $\phi _t$ is a time cost of searching for a public job and $s_{g,t} \in [0,1)$ is the search intensity. In addition, we assume that households are subject to a non-convexity constraint due to setup costs for work, which is intended to generate an extensive margin of labor supply adjustments. In particular, we specify $n_t = \max\! \left \{ n_t - \bar{n},0 \right \}$ . Preferences are such that agents derive utility from random paths for consumption, $c$ , and leisure, $l$ , according to

\begin{equation*} U = \underset { \left\{c_t,n_t,s_{g,t}\right\}_{t=1}^{T} }{\max :} \mathbb {E} \sum _{t=1}^{T} \beta ^{t} \prod _{j=1}^{t} \psi _j u\!\left(c_t,n_t,s_{g,t}\right) \end{equation*}

where

\begin{equation*}u\!\left(c_t,n_t,s_{g,t}\right) = \frac {c_t^{1-\gamma _c}}{1-\gamma _c} + \rho _t \frac {\left(1-n_t-s_g\phi _t\right)^{\left (1+\frac {1}{\gamma _n}\right )}}{1+\frac {1}{\gamma _n}} \end{equation*}

where $\beta,\gamma _c,\gamma _n$ are, respectively, the discount factor, the risk aversion parameter, and the Frisch labor supply elasticity.

There is no on-the-job search within sector. At each period, nonemployed agents and private sector workers can search for a job in the public sector. Given the search intensity, $s_g$ , the probability of finding a public sector job next period is $\pi _g(s_g,u)$ , which also depends on the ex ante ability to be consistent with the fact that the public sector hires proportionally more skilled workers. The cost of job search in the public sector is specified as $\phi _t=\phi _0 t^{\phi _1}$ , where $\phi _0$ is a scale parameter and $\phi _1$ governs the elasticity of the application cost to age. This age dependency of $\phi _t$ is important for the model to be able to replicate the share of civil servants over the life cycle. In addition, nonemployed agents can also search for a job in the private sector and we denote the job finding rate by $\pi _p$ . In line with the empirical evidence, which suggests that the public sector is an absorbing state, we assume the civil servants cannot search.

Besides making decisions about job search and hours worked, agents can trade a risk-free asset which holdings we denote by $a_{t}$ . Asset holdings are subject to an exogenous lower bound. More precisely, for our main exercise, we assume that agents are not allowed to contract debt at any age, so that the amount of assets carried over from age $t$ to $t+1$ is such that $a_{t+1}\geq 0$ . Because no agent can hold a negative position in assets at any time, we assume without loss that asset takes the form of capital, $a_t = k_t$ , as in Aiyagari (Reference Aiyagari1994).

In addition, in order for the model to be able to reproduce the distribution of savings rate observed, we assume that individuals derive utility from leaving a bequest to their children: $\nu (a)=\eta _0 \left(1+\frac{a}{\eta _1} \right)^{(1-\gamma _c)}$ . This type of bequest motive has been called warm glow, was first introduced by Andreoni (Reference Andreoni1989), and has been used in other papers as a way to generate a wealth distribution closer to the data.Footnote ⁹ Considering a more sophisticated form of altruism would increase the number of state variables and, in some cases, would generate strategic parent–child interaction. The term $\eta _0$ reflects the individual’s concern about leaving bequests, while $\eta _1$ measures the extent to which bequests are a luxury good. In particular, when $\eta _1$ is relatively large, bequests are treated as a luxury good. Thus, richer agents choose to save in order to leave bequests, while poorer agents do not, or less frequently so.

Budget constraints. The flow budget constraint that individuals in working age face in our model is

(11) \begin{equation} (1+\tau _c)c +a^{\prime } = \left [1+(1-\tau _k)r\right ]a + \hat{y} - T(\hat{y}) \end{equation}

where $a^{\prime }$ denotes next period assets.

At age $T_{r}$ , agents retire and start collecting social security payments at an exogenously specified replacement rate of the last period earnings. There are two main differences in the calculation of retirement benefits in each sector. First, the replacement rate, $\theta _{j}$ , in the public sector is higher than in the private sector. Second, benefits in the private sector are capped by a limit denoted by $\bar{b}$ , while there is no benefit cap in the public sector. Thus, the budget constraint for retirees can be written as follows:

(12) \begin{equation} (1+\tau _c)c + a^{\prime }= \left [1+(1-\tau _k)r\right ]a + b_{j} \end{equation}

Recursive formulation of individuals’ problems. Let $V_{m,t}(\omega _{t})$ denote the value function of an individual aged $t\lt T_R$ with employment status $m=g,p,o$ , where $\omega _{t}=(a_{t},u,z_{p,t},z_{g,t},h_t)$ is the individual state space. Thus, the choice problem of individuals aged $t$ who work in the private sector can be recursively represented as follows:Footnote ¹⁰

(13) \begin{equation} \begin{aligned} V_{p}(\omega ) & = \max _{c,n,a^{\prime },s_g} u\!\left(c,n,s_g\right) \\[3pt] & \quad + \beta \Big [ \pi _g(s_g,u) E_{z^{\prime}} \max \!\left\{V_{p}(\omega ^{\prime }),V_{g}(\omega ^{\prime }),V_{o}(\omega ^{\prime })\right\} + \varsigma _{p,t} E_{z^{\prime}} V_{o}(\omega ^{\prime }) \\ & \quad + (1-\varsigma _{p,t}-\pi _g(s_g,u))E_{z^{\prime}}\max \!\left\{V_{p}(\omega ^{\prime }),V_{o}(\omega ^{\prime })\right\} \Big ] \end{aligned} \end{equation}

subject to (7). Similarly, for public sector and nonemployed workers:

(14) \begin{align} V_{g}(\omega ) = \max _{c,n,a^{\prime }} u(c,n,0) + \beta \Big [ (1-\varsigma _{g,t}) E_{z^{\prime}} \max \!\left\{V_{g}(\omega ^{\prime }),V_{o}(\omega ^{\prime })\right\} + \varsigma _{g,t} E_{z^{\prime}}V_{o}(\omega ^{\prime }) \Big ] \end{align}

(15) \begin{align} V_{o}(\omega ) &= \max _{c,a^{\prime },s_g} u(c,0,s_g) \nonumber \\[3pt] & \quad + \beta \Big[ \pi _g(u,s_g) E_{z^{\prime}} \max \!\left\{V_{g}(\omega ^{\prime }),V_{o}(\omega ^{\prime })\right\} + \pi _{p,t} E_{z^{\prime}} \max \!\left\{V_{p}(\omega ^{\prime }),V_{o}(\omega ^{\prime })\right\} \nonumber \\[3pt] & \quad + ((1-\pi _g(u,s_g)- \pi _{p,t} )E_{z^{\prime}}V_{o}(\omega ^{\prime }) \Big ] \end{align}

In addition, since agents live up to age $T$ , they start facing a mortality risk after the retirement age, $T_R$ . Hence, with probability $(1-\psi _{t+1})$ agents might die leaving a bequest.

(16) \begin{equation} V_{r,j,t}(a,u) = \max _{c,a^{\prime }} u(c,0,0,0) + \beta \Big [ \psi _{t+1} V_{r,j,t+1}(a^{\prime },u) + (1-\psi _{t+1}) \nu (a^{\prime }) \Big ] \end{equation}

Recursive competitive equilibrium. At each point in time, agents differ from one another with respect to age $t$ and to state $\omega _{t}=(a_{t},u,z_{p,t},z_{g,t},h_t)\in \Omega$ . Agents of age $t$ identified by their individual states, $\omega$ , are distributed according to a probability measure $\lambda _{t}$ defined on $\Omega$ as follows. Let ( $\Omega,\digamma (\Omega ),\lambda _{t})$ be a space of probability, where $\digamma (\Omega )$ is the Borel $\sigma$ -algebra on $\Omega$ : for each $\eta \subset \digamma (\Omega )$ , $\lambda _{t}(\eta )$ denotes the fraction of agents aged $t$ that are in $\eta$ . Given the asset $t$ distribution, $\lambda _{t}$ , $Q_{t}(\omega,\eta )$ induces the asset $t+1$ distribution $\lambda _{t+1}$ as follows. The function $Q_{t}(\omega,\eta )$ determines the probability of an agent at age $t$ and state $\omega$ transiting to the set $\eta$ at age $t+1$ . $Q_{t}(\omega,\eta )$ , in turn, depends on the agents’ policy functions and on the exogenous stochastic process for $z_m$ . Now, we have all the tools to characterize the stationary recursive competitive equilibrium. Households’ optimal behavior was previously described in detail above as well as the firms maximization problem and the government sector. It remains, therefore, to characterize the market equilibrium conditions, the aggregate law of motion, and the government budget constraint. In each period, there are two prices in this economy $(w, r)$ . The equilibrium in the labor and capital markets is defined by:

\begin{align*} K & = \sum _{t=1}^{T}{\mu _t}\displaystyle \int \limits _{\Omega } d_{a,t}(\omega )d\lambda _{t} \\ N_p & = \sum _{t=1}^{T}{\mu _t}\displaystyle \int \limits _{\Omega } d_{p,t} d_{h,t}(\omega )d\lambda _{t}\\ N_g & = \sum _{t=1}^{T}{\mu _t}\displaystyle \int \limits _{\Omega } d_{cs,t} d_{h,t}(\omega )d\lambda _{t} \end{align*}

where $d_{a,t}(\omega )$ and $d_{h,t}(\omega )$ denote the policy functions for asset holding and hours worked, respectively, while $d_{p,t}(\omega )$ and $d_{cs,t}(\omega )$ are indicator functions describing the sectoral choice.

The consumption tax rate, $\tau _{c}$ , is such that it balances the government’s budget,

\begin{equation*} C_g+I_g+(1+\zeta )wN_g+B=\tau\! \left ( wN_p+\left (1+\zeta \right ) N_g \right ) +\tau _k r K_p + \tau _c C,\end{equation*}

where $C$ denotes the aggregate consumption and $B$ denotes the total benefits. The distribution of accidental bequests is given by:

\begin{equation*}\epsilon =\sum _{t=1}^{T}{\mu _{t}\displaystyle \int \limits _{\Omega }(1-\psi _{t+1}) d_{a,t} (\omega ) d\lambda _{t}}.\end{equation*}

Finally, given the decision rules of households, $\lambda _{t}(\omega )$ satisfies the following law of motion:

\begin{equation*} \lambda _{t+1}(\eta )={\displaystyle \int \limits _{\Omega }}Q_{t}(\omega,\eta )d\lambda _{t} \forall \eta \subset \digamma (\Omega ). \end{equation*}

4.1. Calibration and estimation

Table 2 lists the value of each parameter for the Brazilian economy and includes a comment on how each was selected.

Model period and age distribution. The model period is 1 year. We assume that individuals start their lives at the age of 25 and live until the age of 80. Therefore, the extension of their lifetimes in the model is 56 periods (T = 56). The age population distribution, $\left \{ \mu _{t}\right \} _{t=1}^{T}$ , and the mortality risk are obtained from the life tables for the Brazilian population constructed by IBGE (National Central Statistical Agency) based on the 2010 census data.

Preference parameters. First, we choose the discount factor $\beta$ in such a way that the equilibrium of our benchmark economy implies a capital-output ratio around of $2.8$ , which is the value observed in the data.Footnote ¹¹ We fix the coefficient of relative risk aversion, $\gamma$ , to $2.0$ , in line with the bulk of the literature on consumption surveyed by Attanasio (Reference Attanasio1999). This value is also consistent with the literature that estimates $\gamma$ using Brazilian data, which suggests a range from 1 to 3 (see, e.g., Gandelman and Hernández-Murillo (Reference Gandelman and Hernández-Murillo2014) and Fajardo et al. (Reference Fajardo, Ornelas and Farias2012)). Then, we choose the share of leisure in the utility function, $\rho _{t}$ to match average hours for different age groups and the average labor force participation. In particular, we assume that $\rho _{t}=\rho _{0}+\rho _{1}t$ . To calibrate $\rho _{0}$ , we use the average working hours for ages 25–40 and for $\rho _{1}$ the average between 41 and 60. The first group works on average $37.86$ while the second $40.37$ of their time endowment. For the last 5 years, we specify a new profile $\rho _{t}=\rho _{60}+\rho _{2}t$ . We calibrate $\rho _{2}$ to match the average hours during those last 5 years equal to $35.16$ .Footnote ¹² Finally, we set the parameter related to the cost of work, $\bar{n}$ , so that the model matches a labor force participation rate of 83%, which is calculated using PNAD 2014 for males older than 25 in 2014.

The remainder preferences parameters are related to the utility function of bequests. The parameter $\eta _0$ represents the weight on the utility from bequeathing. Since it measures the strength of bequest motives, we calibrate $\eta _1$ so that the model matches the aggregate savings rate in the data. The term $\eta _1$ is the shifter of bequests in utility function. It reflects the extent to which bequests are luxury goods, affecting the bequest distribution, especially the high end of it. Thus, we set $\eta _1$ to match the savings rate of the richest 20% individuals. In Fig. 4, we show the simulated distribution of savings by income quintile. It can be seen that the model matches the data very well, specially among the top 20% who are responsible for almost all of the household savings.

Table 2. Parameters calibrated externally

Figure 4. Distribution of savings by income: data vs. model.

Table 3. Parameters calibrated internally

Production technologies. We set the capital share $\alpha$ at 0.36. This number is consistent with the one reported by Gomes et al. (Reference Gomes, Bugarin and Ellery-Jr2005), when the correction suggested by Gollin (Reference Gollin2002) and Young (Reference Young1995) about the self-employed income is taken into account. In addition, we assume that the capital stock depreciates at a rate of 6% per year, which is consistent to the figures used in the growth and development literature (cf., Parente and Prescott (Reference Parente and Prescott2000)). We also set $\delta _{g}=0.06$ . According to the Brazilian Institute of Geography and Statistics (IBGE), the ratio of public goods to output is roughly 16%—using information on production costs. Then, in order to match this ratio we set $A_{g}=0.80$ . To calibrate parameter $\chi$ , we rely on estimates provided by Hulten (Reference Hulten1996) who uses a cross-section of low-income countries including Latin American countries and obtains a point estimate of 0.15 for $\chi$ , which is the value we use.

Labor income shocks. We calibrate the parameters of the stochastic process for wages, $(\varsigma _p,\varsigma _g,\sigma _{\epsilon _p},\sigma _{\epsilon _g})$ , using the estimates reported in Section 2. The variance of the permanent component, $\sigma _u^2$ , is chosen to match the income Gini index at age 25. To solve the model, we use the algorithm described in Tauchen (Reference Tauchen1986) to discretize the two shocks, using 7 states to represent the permanent shock and 11 states for the persistent shock. We also use the data reported in Section 2 to calibrate the unemployment shocks. In particular, we use a second-order polynomial to interpolate the flows reported in Table 3 to obtain the age profile for $(\varsigma _p,\varsigma _g)$ by exact age.

Job-offer probability. For the transition from unemployment to private sector employment, we specify an age-dependent function $\pi _t = \pi _0 \exp{\pi _1 t}$ , where $\pi _0$ and $\pi _1$ are parameters to be chosen in such a way that the model matches key moments regarding unemployment in the data. The first moment we target is an unemployment rate of $7\%$ . To discipline the employment profile over the life cycle by the data, they also use the shape of the age profile of employment rate.

Human capital functions. The parameters of the human capital functions are calibrated as follows. First, given that the evidence for the human capital depreciation rate ranges from 0.0016 to 0.089, with most of the estimates concentrated around 0.04 (Browning et al. (Reference Browning, Hansen and Heckman1999)), we set $\delta _{h}=0.04$ . As it is usual in the macro-labor literature, the other parameters are then chosen in order to approximate the simulated earnings profiles to their counterparts estimated from the data.Footnote ¹³ This procedure is carried out for civil servants and workers and we obtain $(\xi _{g},\xi _{p})=(0.16,0.15)$ .Footnote ¹⁴ The resulting profiles are presented in Fig. 5. In order to measure the goodness of the fit, we calculate the average (percentage) deviation, in absolute terms, between the model implied earnings profiles and the data. By this measure, on average, the model implied earnings profiles differed from the data by $2.93\%$ in the case of civil servants and $1.98\%$ in the case of private workers.

Figure 5. Earnings profile and income distribution. Source: PNAD.

Public sector parameters. Based on Paes and Bugarin (Reference Paes and Bugarin2006), we assume a labor income tax rate of 18% and a capital tax of 15%. The consumption tax is determined in such a way that the government budget balances in equilibrium, which implies a tax rate equal to 30.7% in the benchmark economy. The replacement rate in the private sector, $\eta _{rp}$ , is taken from Afonso (Reference Afonso2016) who, based on microdata from the private sector social security system, provides a value of $0.82$ . In the public sector, the replacement rate, $\eta _{rg}$ , is equal to one, which is consistent with the fact that civil servants receive their last salary as benefits. We assume that agents can get the public pension system only if they start to work in the public sector 5 years before the retirement age. We also impose a ceiling on private pensions, $\bar{b}$ , such that the maximum pension in the private sector is 20% of the maximum pension in the public sector.Footnote ¹⁵ We set the public wage premium for workers to be equal to $\zeta =0.19$ , which is consistent with the estimate of the conditional public wage premium provided in regression (5), Table A.1 in Appendix A.

We specify the probability of finding a job in the public sector as $\pi _g(s_g,u)= 1-\left (\frac{\bar{u}-s_gu}{\bar{u}-s_g\underline{u}}\right )^{\iota }$ , where $(\bar{u},\underline{u})$ are the upper and the lower bound of the ex ante productivity shock, respectively. The advantage of this specification is that it has only one parameter, $\iota$ , which governs the curvature of the job finding rate in the public sector as a function of the search intensity, $s_g$ . We calibrate $\iota$ and the parameters of the cost of job search function $(\phi _0,\phi _1)$ in order to minimize the distance between a set of simulated moments and their counterparts in the data. In particular, we use as targets the total share of civil servants, 13.4%, the share of civil servants at ages 30–35 and 45–50, and the share of civil servants among the richest 20%. These data moments were taken from the 2014 Continuous National Household Sample Survey (PNAD Continua) compiled by the Brazilian Institute of Geography and Statistics (IBGE). In Fig. 6, we present the simulated and the actual flow of individuals to public jobs by age. It can be seen that this flow increases with age in the model as well as in the data. This pattern could be explained by the fact that the retirement benefits in the public sector are much more generous than in the private sector, which makes the public sector jobs more attractive as individuals approach retirement.

Figure 6. Share of civil servants by age and income. Source: PNAD.

Figure 7. Benchmark economy.