3.1 The households

Households in our baseline economy are heterogeneous and differ in their age, j ∈ J; in their education, h ∈ H; in their labor market status, $e\in {\cal E}$; in their pension rights, b ∈ B; in their pension, p ∈ P; and in their assets, a ∈ A. Sets J, H, ${\cal E}$, B, P, and A are all finite sets and we use μ _j,h,e,b,p,a to denote the measure of households of type (j, h, e, b, p, a). We think of a household in our model as a single individual, even though we use the two terms interchangeably. To calibrate the model, we use individual data of persons older than 20 in the Spanish economy.

Age. Individuals enter the economy at age 20, the duration of their lifetimes is random, and they exit the economy at age 100 at the latest. Therefore J = {20, 21, …, 100}. The parameter ψ _jh denotes the conditional probability of surviving from age j to age j + 1, for those households with educational level h.

Education. Households can either be high school dropouts with h = 1, high school graduates who have not completed college h = 2, or college graduates denoted h = 3. Therefore H = {1, 2, 3}. A household's education level is exogenous and determined forever at the age of 20.

Labor market status. Households in our economy are either employed, unemployed eligible for benefits, unemployed non-eligible, or retired. We denote workers by ω, eligible unemployed by $\varpi$, non-eligible unemployed by υ, and retirees by ρ. Consequently, ${\cal E} = \{ {\omega , \;\varpi , \;\upsilon , \;\rho } \}$. Upon entering the economy, individuals draw a job opportunity. In subsequent years, the labor market status evolves according to exogenous job separation and job finding rates, and also to the optimal retirement decision.

Workers. A worker provides labor services and receives a salary that depends on his endowment of efficiency labor units and his hours worked. This endowment has two components: a deterministic component, which we denote by ε _jh, and a stochastic component, which we denote by s.

The deterministic component depends on the household age and education, and we use it to characterize the life-cycle profiles of earnings. We model these profiles using the following quadratic functions:Footnote ⁶

(1)$$\epsilon _{\,jh} = a_{1h} + a_{2h}j + a_{3h}j^2.$$

We choose this functional form because it allows us to represent the life-cycle profiles of the productivity of workers in a very parsimonious way.

The stochastic component is independently and identically distributed across the households, and we calibrate it to match moments of the Spanish earnings and wealth distribution, following Castañeda et al. (Reference Castañeda, Díaz-Giménez and Rios-Rull2003). This component does not depend on the age or the education of the households, and we assume that it follows a first order, finite state, Markov chain, with invariant distribution given by π(s), and with conditional transition probabilities given by Γ:

(2)$$\Gamma [ {{s}^{\prime}\vert s} ] = {\rm Pr}\{ {s_{t + 1} = {s}^{\prime}\vert s_t = s} \} , \;{\rm with} s, \;{s}^{\prime}\in S.$$

We assume that the process on s takes three values and, consequently, that s ∈ S = s ₁, s ₂, s ₃. We make this assumption because it turns our that three states are sufficient to account for the Lorenz curves of the Spanish distributions of income and labor earnings in enough detail, and because we want to keep this process as simple as possible.

Every period agents receive a new realization of s. His labor productivity is then given by ε _jhs. A worker with education h and age j who supplies l hours of labor has gross labor earnings y ^l given by:

(3)$$y^l = w\epsilon _{\,jh}sl, \;$$

where the economy-wide wage rate w.

Workers face a probability of losing their job at the end of the period, denoted φ _jh. This probability is education and age dependent, and we use it to generate the observed labor market flows between employment and non-employment states within age cohorts. We model these profiles using the following functions:

(4)$$\varphi _{\,jh} = a_{4h} + a_{5h}j + a_{6h}j^2 + a_{7h}j^3.$$

Unemployed. Eligibility for unemployment benefits is conditional on having lost a job during the previous 2 years and not having started a new job yet. Eligibility expires when one of the conditions is not met. An eligible agent with education h receives unemployment benefits given $y^u = \vartheta y_h^{\bar{l}}$, where $y_h^{\bar{l}}$ is the average labor earnings of those workers with education h, and where $\vartheta < 1$ is a replacement rate.

At the end of each period, an unemployed receives a job offer with probability ξ _jh. This probability is also education and age dependent, and we use it to generate the observed labor market flows between unemployment and employment. The offer is the productivity shocks. Therefore, its amount is either s ₁, s ₂, or s ₃. Conditional on receiving an offer, the probability of receiving each one of the productivity shocks is the unconditional probability of each realization of that shock. Once a household is re-employed, the future values of s are determined by the process on s.

We model the probabilities to receive a job offer as:

(5)$$\xi _{\,jh} = a_{8h} + a_{9h}j + a_{10h}j^2 + a_{11h}j^3.$$

Retirees. Workers who are R ₀ years old or older decide whether to retire and collect the retirement pension. They take this decision after observing their current labor productivity. If they decide to retire, they lose the endowment of labor efficiency units forever and exit the labor market. Unemployed households who are R ₀ years or older are forced to retire.

Pension rights. Workers and unemployed also differ in the pension rights. These rights are used to determine the value of their pensions when they retire. The rules of the pension system, which we describe below, include the rules that govern the accumulation of pension rights, and the rules that determine the mapping from pension rights into pensions. In our model economy households take this mapping into account when they decide how much to work and when to retire. We assume that pension rights belong to the discrete set B  =  {b ₀, b ₁, …, b _m}, that m = 9, and that the spacing between points in set B is increasing. We also assume that b ₀ = 0, and that b _m = a ₁₂y, where a ₁₂ > 1, y is the model economy per capita output, measured at market prices, and a ₁₂y is the maximum covered earnings, following the Spanish public pension system.

Pensions. Retirees differ in their retirement pensions. We assume that retirement pensions belong to the set P  =  {p ₀, p ₁, …, p _m}. Since this mapping is single valued, and the cardinality of the set of pension rights, B, is 10, we let m = 9 also for P. We also assume that p ₀ = a ₁₃y, and that p _m = a ₁₄y, where p ₀ and p _m are the minimum and maximum retirement pensions, in accordance with the Spanish public pension system. Finally, we also assume that the distances between any two consecutive points in P are increasing. We make this assumption because minimum pensions play a large role in the Spanish system and this suggests that we should have a tight grid in the low end of P.

Assets. Households in our model economy differ in their asset holdings, which are constrained to being non-negative. The absence of insurance markets gives the households a precautionary motive to save. They do so by accumulating real assets which take the form of productive capital, denoted a ∈ A.Footnote ⁷

Preferences. Households derive utility from consumption, c, and disutility from labor effort, l, where labor is decided both at the extensive and intensive margins. The period utility is described by a utility flow from consumption and leisure, u(c, 1 − l). Unemployed and retired agents dedicate all the time endowment to leisure consumption. Accordingly, lifetime utility is given by

(6)$${\rm {\opf E}}\sum\limits_{\,j = 20}^{100} {\beta ^{\,j-20}} \psi _{\,jh}[ {\rm log} ( c) + \chi \displaystyle{{{( 1-l) }^{1-\gamma }} \over {1-\gamma }}] , \;$$

where β is a time discount factor, c is consumption, χ is the relative utility weight on leisure, and γ is the labor elasticity.

3.2 The representative firm

In our model economy there is a representative firm. Aggregate output, Y, is obtained combining aggregate capital, K, with the aggregate labor input, L, through a Cobb–Douglas, aggregate production function which we denote by $Y = K^\theta L_t^{1-\theta }$. We assume that factor and product markets are perfectly competitive and that the capital stock depreciates geometrically at a constant rate, which we denote by δ.

3.3 The government

The government in our model economy taxes capital income, household income, and consumption, and it confiscates unintentional bequests. It uses its revenues to consume, and to make transfers to households other than pensions. In addition, the government runs a pay-as-you-go pension system. The consolidated government and pension system budget constraint is

(7)$$G + Z + P + U = T_k + T_y + T_c + T_s + E.$$

On the expenditure side, G denotes government consumption, Z denotes government transfers other than pensions, P denotes pensions, and U denotes unemployment benefits, and, in the revenue side, T _k, T _y, and T _c, denote the revenues collected by the capital income tax, the household income tax, and the consumption tax, T _s denotes the revenues collected by the payroll tax, and E denotes unintentional bequests. Finally, we assume that the government uses the consumption tax rate to clear the government budget.Footnote ⁸

3.3.1 The fiscal policy

Expenditures. We assume that the amount of government consumption is given by G = a ₁₅Y*, where Y* is the model economy output at market prices. Transfers other than pensions are delivered to those households whose income is below a minimum income level, y = a ₁₆y. In this case, these households receive a transfer from the government, denoted by t _r = y. We already defined unemployment benefits, and we describe pension expenditures in the next section.

Revenues. We assume that the proportional capital income and consumption tax rates are given by τ _k, and τ _c. Moreover, we assume that the assets that belong to the households that exit the economy are confiscated by the government. To model the household income tax, we use the following function:

(8)$$\tau _y( {y_t^b } ) = a_{17}\{ {y_t^b -{[ {a_{18} + {( {y_t^b } ) }^{{-}a_{19}}} ] }^{{-}1{\rm /}a_{19}}} \} , \;$$

where $y_t^b$ is the income tax base. This expression, where a ₁₇, a ₁₈, and a ₁₉ are parameters, is the function chosen by Gouveia and Strauss (Reference Gouveia and Strauss1994) to model effective personal income taxes in the United States, and it is also the functional form chosen by Calonge and Conesa (Reference Calonge and Conesa2003) to model effective personal income taxes in Spain.Footnote ⁹ Finally, we describe payroll taxes in the next section.

3.3.2 The pension system

In our benchmark model economy we choose the payroll tax and the pension system rules so that they replicate as closely as possible the Régimen General de la Seguridad Social of the Spanish pay-as-you-go pension system in 2018, which is our calibration target year. See Díaz-Saavedra (Reference Díaz-Saavedra2020) for a description of the Spanish public pension system.

Payroll taxes. In our model economy, as in Spain, the payroll tax is capped and workers older than the full entitlement retirement age, which we denote by R ₁, are exempt from paying payroll taxes. Specifically, the payroll tax function is the following:

(9)$$t_s( {y^l} ) = \left\{{\matrix{ 0 & {{\rm if\;}j > R_1} \cr {{\rm otherwise}} & {\left\{{\matrix{ {\tau_{ss}y^l} & {{\rm if\;}y^l < {\bar{y}}^l} \cr {\tau_{ss}{\bar{y}}^l} & {{\rm otherwise}} \cr {} & {} \cr } } \right.} \cr {} & {} \cr } , \;} \right.$$

where parameter τ _ss is the payroll tax rate and $\bar{y}^l = b_m = a_{12}y$ is the maximum covered earnings. Finally, we also assume that eligible unemployed also pay social security contributions, so that the payroll tax function becomes t _s(y ^u) = τ _ssy ^u.

Retirement ages. In our model economy the early retirement age is R ₀. Workers who choose to retire early pay a penalty, λ _j, which is determined by the following function

(10)$$ {\it \lambda} _j = \left\{{\matrix{ {a_{20}-a_{21}( {\,j-R_0} ) } & {{\rm if\;}j < R_1} \cr 0 & {{\rm if\;}j \ge R_1} \cr {} & {} \cr } } \right., \;$$

where a ₂₀ and a ₂₁ are parameters which we choose to replicate the Spanish early retirement penalties.

Retirement pensions. A household of age j ≥ R ₀, that chooses to retire, receives a retirement pension, p(b), which we compute following the Spanish pension system rules. The main component of the retirement pension is its regulatory base, RB, which averages labor earnings up to the maximum covered earnings, during the last N _b = 21 years prior retirement. If a household has not reached the full entitlement retirement age, its pension is subject to an early retirement penalty. If the household is older than R ₁, its pension claims are increased by 3% for each year worked after this age. The regulatory base is multiplied by a pension replacement rate, ϕ, which we use to replicate the pension expenditures to output ratio. Finally, retirement pensions are bounded by a minimum and a maximum pension.

Note that the regulatory base takes into account a long period of time. Consequently, it can be relatively frequent that contribution gaps occur; that is, periods to be taken into account to determine the amount of the pension in which the household does not credit any contribution. This is the case, for instance, of non-eligible unemployed. In order to mitigate the negative effects of these gaps, the Spanish pension rules established that these unlisted periods will be integrated with fictitious quotes. In our model economy, we assume that these fictitious quotes are y ^fq = a ₂₂y.

In our benchmark model economy we calculate the retirement pensions using the following formula:

(11)$$p( b ) = \phi ( {1.03} ) ^v( {1- {\lambda}_j} ) RB, \;$$

where ϕ denotes the replacement rate, and v denotes the number of years that the worker remains in the labor force after reaching the full entitlement retirement age. The regulatory base, RB, is exactly equal to the pension rights at the time of retirement. Consequently, it is defined as:

(12)$$RB = \displaystyle{1 \over {N_b}}\mathop {\mathop \sum }\limits_{s = j-N_b}^{j-1} {\rm min}\{ {y_s^l , \;{\bar{y}}^l} \} .$$

Note that labor earnings, $y_s^l$, is replaced by $y_s^u$ or $y_s^{fq}$ in the case of eligible or non-eligible unemployed households (see below). Expressions (11) and (12) replicate most of the features of Spanish retirement pensions. The main difference is that in our model economy the pension replacement rate is independent of the number of years of contributions. We abstract from this feature of Spanish pensions because it requires an additional state variable. Finally, we require that p ₀ ≤ p(b) ≤ p _m.

3.4 The households’ decision problem

Individuals with education h are heterogeneous in five dimensions x = {j, e, b, p, a}, where j is age, e is employment status, b is pension rights, p is pensions, and a is private savings. The households' problem is described recursively. Let V _h(x) be the value function of an individual with education h in state x.Footnote ¹⁰

Workers. We start with employed individuals that are younger than the minimum retirement age, specifically j < R ₀. In this way we can abstract, for now, from the retirement decision. An individual of education level h, with age j, stochastic productivity s, pension rights b, and private savings a, faces the following optimization problem:

(13)$$\matrix{ {V_h( {\,j, \;s, \;b, \;a} ) = } & {\mathop {\max }\limits_{( {c, l, {a}^{\prime}} ) } \{ {} u( {c, \;1-l} ) + \beta E[ {} ( {1-\varphi_{\,jh}} ) \sum\nolimits_{{s}^{\prime}\in S} {\Gamma ( {{s}^{\prime}\vert s} ) V_h( {\,j + 1, \;{s}^{\prime}, \;{b}^{\prime}, \;{a}^{\prime}} ) } } \cr {} & { + \varphi _{\,jh}V_h( {\,j + 1, \;\varpi , \;{b}^{\prime}, \;{a}^{\prime}} ) {\rm ] \} \ }} \cr {} & {} \cr } $$

subject to

$$( {1 + \tau_c} ) c + {a}^{\prime} = y^l + ( {1 + r( {1-\tau_k} ) } ) a-t_s( {y^l} ) -\tau _yy^b + I_{t_r}, \;$$

where y ^b = (1 − τ _k)ra + y ^l − t _s(y ^l) is the income base of the personal income tax, and $I_{t_r}$ is an indicator function that takes value 1 if households are eligible for public transfers other than pensions. In addition, the law of motion of pension rights is:

(14)$${b}^{\prime} = \left\{{\matrix{ 0 & {{\rm \;\;\;if\;\;\;\;}j < R_0-N_b} \cr {b + ( {\min \{ {y^l, \;{\bar{y}}^l} \} {\rm /}N_b} ) } & {{\rm \;\;\;if\;\;\;\;}R_0-N_b \le j < R_0{\rm , \;}} \cr {[ {b( {N_b-1} ) + \min \{ {y^l, \;y^{\bar{l}}} \} } ] {\rm /}N_b} & {{\rm \;\;\;if\;\;\;\;}j \ge R_0.} \cr {} & {} \cr } } \right.$$

Eligible unemployed. A household currently unemployed and eligible for unemployment benefits, aged j < R ₀, solves the following problem:

(15)$$\eqalign{V_h( j, \;\varpi , \;b, \;a) & = \mathop {\max }\limits_{( c, {a}^{\prime}) } \{ u( c, \;1) \beta E[ \xi _{\,jh}\sum\nolimits_{s\in S} \pi ( s) V_h( j + 1, \;s, \;{b}^{\prime}, \;{a}^{\prime}) \cr& \quad + ( 1-\xi _{\,jh}) V_h( j + 1, \;{u}^{\prime}, \;{b}^{\prime}, \;{a}^{\prime}) ] \}} $$

subject to

$$( {1 + \tau_c} ) c + {a}^{\prime} = y^u + ( {1 + r( {1-\tau_k} ) } ) a-\tau _s( {y^u} ) -\tau _yy^b, \;$$

where y ^b = (1 − τ _k)ra and u′ is $\varpi$ if the current period is the first period that the unemployed collects unemployment benefits, and u′ is υ if it is the second period. Note that eligible unemployed households do not receive public transfers other than pensions, since we assume that unemployment benefits are well above the minimum income level y, which entitles families to receive these public transfers.

The law of motion for pension rights is in this case:

(16)$${b}^{\prime} = \left\{{\matrix{ 0 & {{\rm \;\;\;if\;\;\;\;}j < R_0-N_b} \cr {b + ( {y^u{\rm /}N_b} ) } & {{\rm \;\;\;if\;\;\;\;}R_0-N_b \le j < R_0 .} \cr {} & {} \cr } } \right.$$

Non-eligible unemployed. A household currently unemployed and non-eligible for unemployment benefits, aged j < R ₀, solves the following problem:

(17)$$V_h( j, \;\varpi , \;b, \;a) = \max _{( c, {a}^{\prime}) }\{ u( c, \;1) \beta E[ \xi _{\,jh}\sum\limits_{s\in S} \pi ( s) V_h( j + 1, \;s, \;{b}^{\prime}, \;{a}^{\prime}) + ( 1-\xi _{\,jh}) V_h( j + 1, \;{u}^{\prime}, \;{b}^{\prime}, \;{a}^{\prime}) ] \} $$

subject to

$$( {1 + \tau_c} ) c + {a}^{\prime} = ( {1 + r( {1-\tau_k} ) } ) a-\tau _yy^b + I_{t_r}, \;$$

where y ^b = (1 − τ _k)ra and the law of motion for pension rights is:

(18)$${b}^{\prime} = \left\{{\matrix{ 0 & {{\rm \;\;\;if\;\;\;\;}j < R_0-N_b} \cr {b + ( {y^{\,fq}{\rm /}N_b} ) } & {{\rm \;\;\;if\;\;\;\;}R_0-N_b \le j < R_0 .} \cr {} & {} \cr } } \right.$$

Retired. Retired individuals do not receive labor income. They finance consumption with past private savings and pension payments. The problem is a standard consumption-savings decision, with survival risk and a certain maximum attainable age, assumed to be J = 100. At age j = 99, the continuation value is zero because the agent exists the economy next period with probability one. Before that, the retired household solves:

(19)$$V_h( j, \;\rho , \;p, \;a) = \mathop {\max }\limits_{( c, {a}^{\prime}) } \{ u( c, \;1) + \beta \psi _j[ V_h( j + 1, \;\rho , \;p, \;{a}^{\prime}) ] \} $$

subject to

$$( {1 + \tau_c} ) c + {a}^{\prime} = p + ( {1 + r( {1-\tau_k} ) } ) a-\tau _yy^b, \;$$

where y ^b = p + (1 − τ _k)ra. Retired households, similarly to eligible unemployed, are not eligible in any case to receive public transfers other than pensions, since we assume that the minimum retirement pension is well above the minimum income level.

Retirement decision. Recall that we assume that unemployed households who are R ₀ years or older are forced to retire. On the other hand, a worker aged j ≥ R ₀ must decide if to retire or not from the labor market. In this case, she chooses the optimal plan after solving problems 13 and 19.

3.5. Equilibrium

A detailed description of the equilibrium process of this model economy can be found in Appendix 1.

4.1. The initial distribution of households

Individuals are assumed to be born at the age of 20 and they can live a maximum of J = 100 years. After age 100, death is certain. The sequence of conditional average survival probabilities $\{ {\psi_j} \} _{j = 1}^J$ is taken directly from the Spanish Institute of Statistics (INE) at 2018. The initial share of age groups in the population, μ _j, is then calculated from the relation:

(20)$$\mu _{\,j + 1} = \displaystyle{{\psi _j} \over {( {1 + n} ) }}\mu _j, \;$$

where n is the growth rate of the population, which has averaged 0.67% per year in Spain over the last 50 years. Finally, we also take from INE the distribution by education across cohorts, μ _jh, at 2018.Footnote ¹¹

4.2. Parameters and targets

To characterize our model economy fully, we must choose the values of a total of 70 parameters. Of these 70 parameters, 20 describe the government policy, 21 describe the endowment of efficiency labor units profiles, 24 describe the employment and unemployment risk functions, two describe the production technology, and the remaining three describe the household preferences. To choose the values of these 70 parameters, we need 70 equations which formalize our calibration targets.

4.3. Equations

To determine the values of the 70 parameters that identify our model economy, we do the following. First, we assign values to 51 parameters that can be estimated directly using equations that involve either one parameter only, or one parameter and our guesses for (K, L). These include, for instance, the deterministic productivity profiles and the probabilities governing employment transitions. Second, we use the model and a system of 19 non-linear equations to calibrate the 19 remaining parameters. Most of these equations require various statistics in our model economy to replicate the values of the corresponding Spanish statistics in 2018. We describe the determination of both sets of parameters in the subsections below.

4.3.1 First stage of calibration

The life-cycle profile of earnings. We measure the deterministic component of the process on the endowment of efficiency labor units independently of the rest of the model. We estimate the values of the parameters of the three quadratic functions that we describe in expression (1), using the age and educational distributions of hourly wages reported by the Instituto Nacional de Estadística (INE) in the Encuesta de Estructura Salarial (2010) for Spain.Footnote ¹² This procedure allows us to identify the values of nine parameters directly.

The employment risk. We also measure the exogenous probability to find/lose a job independently of the rest of the model. We estimate the values of the parameters of the functions that we describe in expressions (4) and (15), using the age and educational distributions of employment flows reported by the INE in the Encuesta de Población Activa (2018) for Spain, and the implied exogenous probabilities to fin/lose a job are shown in Figure 3.Footnote ¹³ This procedure allows us to identify the values of 24 parameters directly.

Figure 3. The probabilities for find/lose a job (%).

The pension system. In 2018 in Spain, the payroll tax rate paid by households was 28.3% and it was levied only on the first 45,014 euros of annual gross labor income, which corresponds to 141.06% of per GDP per person aged 20+. Consequently, we set a ₁₂ = 1.4106, so that the maximum covered earnings is b _m = 1.4106y.Footnote ¹⁴ We also assume that the payroll tax rate is τ _ss = 0.235. The rationale for our choice is because the part of the payroll tax that would finance the expenditure of retirement and widowhood pensions is limited to 23.5% of earnings.

Our choice for the number of years used to compute the retirement pensions in our benchmark model economy is N _b = 21. This is because in 2018 the Spanish Régimen General de la Seguridad Social took into account the last 21 years of contributions prior to retirement to compute the pension. Our choice for the first and normal retirement ages are R ₀ = 62 and R ₁ = 66, so that to identify the early retirement penalty function, we choose a ₂₀ = 0.28 and a ₂₁ = 0.07. This is because we have chosen R ₀ = 62.

We assume that the minimum and the maximum pension are also directly proportional to per capita income. Our targets for the proportionality coefficients are a ₁₃ = 0.2362 and a ₁₄ = 1.1390. These numbers correspond to their values in 2018.Footnote ¹⁵ Finally, we assume that the fictitious quotes are also a proportion a ₂₂ = 0.24 of per capita output, so that we set y ^fq = 0.24y.

Government policy. To specify part of the government policy, we must choose the values of government consumption, G, of the parameters of the income tax function, of the tax rate on capital income, τ _k, and of the tax rate on consumption, τ _c. We target the output shares of G and T _k, so that they replicate the GDP shares of government consumption and capital income taxes. According to the INE, in 2018, government consumption was 236,342 million euros, and the corporate profit tax collected 29,711 million euros. Consequently, we set a ₁₆ = 0.1961 and τ _k = 0.0964.

To identify the income tax function, we must choose the values of parameters a ₁₇, a ₁₈, and a ₁₉. Since a ₁₇ and a ₁₉ are unit-independent, we use the values reported by Calonge and Conesa (Reference Calonge and Conesa2003) for these parameters, namely, a ₁₇ = 0.45 and a ₁₉ = 1.071. Finally, the government budget is an additional equation that allows us to obtain residually the consumption tax rate.

Preferences. Of the three parameters in the utility function, we choose the value of γ directly. Specifically, we choose σ = 4.0.

Technology. According to the Spanish National Institute of Statistics (INE) data, the capital income share in Spanish GDP was 0.4846 in 2018. Consequently, we choose θ = 0.4846.

Adding up. So far we have determined the values of 51 parameters either directly or as functions of our guesses for (K, L) only. We report their values in Tables 2 and 3.

Table 2. First stage of calibration (life-cycle profiles)

Table 3. First stage of calibration (single parameters)

4.3.2 Second stage of calibration

We still have to determine the values of 19 parameters. To find the values of those 19 parameters we need 19 equations. Of those equations, 15 require that model economy statistics replicate the value of the corresponding statistics for the Spanish economy in 2018, and four are normalization conditions (Table 4).

Table 4. Macroeconomic aggregates and ratios in 2018 (%)^a

^a Variable Y* is GDP at market prices.

^b Variable h denotes the average share of disposable time allocated to the market.

^c Variable DSS denotes the social security deficit.

Aggregate targets. According to the BBVA database, in 2016 the value of the Spanish capital stock was 3,281,631 million euros.Footnote ¹⁶ According to the INE in 2016 the Spanish gross domestic product at market prices was 1,113,840 million euros. Dividing these two numbers, we obtain K/Y = 2.94, which is our target value for the model economy capital to output ratio.

According to the INE, private consumption plus indirect taxes was 654,574 million euros in 2018, and unemployment benefits amounted 17,469 million. That same year, and according to the Spanish Instituto Nacional de la Seguridad Social, pension deficit was 21,038 million euros. Consequently, the ratios of these variables to GDP at market prices are 54.35%, 1.32%, and 1.75%. Finally, and according to the Spanish Ministerio de Inclusión, Seguridad Social y Migraciones, the sum of minimum income transfers delivered by the central government amounted 0.13% of Spanish GDP in 2019.

Finally, and according to the Encuesta de Población Activa (INE), in Spain in 2018 an average worker devoted 33.9 hours per week to labor market activities. If we assume that the total endowment per week is 98 hours (=14 × 7), it means that an average worker devoted 34.59% of his disposable time to labor market activities.

Distributional targets. We target the three Gini indexes and five points of the Lorenz curves of the Spanish distributions of earnings, income, and wealth. We have taken these statistics from the INE, the OECD, and Budría and Díaz-Giménez (Reference Budria and Díaz-Giménez2007), and we report them in bold face in Table 5. As we said before, Castañeda et al. (Reference Castañeda, Díaz-Giménez and Rios-Rull2003) argue in favor of this calibration procedure to replicate the inequality reported in the data. These targets give us a total of eight additional equations.

Table 5. The distributions of earnings, income, and wealth^a

^a The source for the Spanish data of earnings and income are the Spanish National Institute of Statistics (INE) and the OECD. The source for the Spanish data of wealth is the 2004 Encuesta Financiera de las Familias Españolas as reported in Budría and Díaz-Giménez (Reference Budría and Díaz-Giménez2006).

Normalization conditions. In our model economy there are four normalization conditions. The transition probability matrix on the stochastic component of the endowment of efficiency labor units process is a Markov matrix and therefore its rows must add up to one. This gives us three normalization conditions, and finally, we also normalize the first realization of this process to be s(1)  =  1 (Tables 6 and 7).

Table 6. Second stage of calibration (the stochastic component of the endowment process)

^a π*(s)% denotes the invariant distribution of s.

Table 7. Second stage of calibration (preferences, technology, and public policies)

In the Technical Appendix of Díaz-Saavedra (Reference Díaz-Saavedra2020), we describe in detail how we solve the system of 15 non-linear equations needed to find the value of these parameters.

4.4 Calibration results

In this section we show that our calibrated, benchmark model economy replicates reasonably well both most of the Spanish statistics that we target, and also untargeted moments in our calibration procedure. To differentiate targeted from untargeted moments, we present the first group of Spanish statistics in bold face.

Macroeconomic aggregates and ratios. In Table 8 we report the macroeconomic aggregates and ratios in Spain and in the benchmark model economy for 2018. We find that the benchmark model economy does a good job in replicating most of the main Spanish macroeconomic aggregates and ratios. The only exceptions are the inheritance tax revenues and the consumption tax collections. This is not surprising because the assets that belong to the households that exit our model economy are confiscated by the government, and because the consumption tax rate is determined residually to satisfy the government budget.

Table 8. Macroeconomic aggregates and ratios in 2018 (%)^a

^a In this table, variable Y* is GDP at market prices.

^b Variable h denotes the average share of disposable time allocated to the market.

^c Variable DSS denotes the social security deficit.

The pension system. Some of the more important features of the Spanish economy that our model economy should approximate are those related to public pension system, e.g., the shares of retirees collecting minimum and maximum pensions, and the internal rate of return of the Spanish public pension system. This is so since the Spanish pension system redistributes pension income mainly through a minimum pension, and also a cap of the payroll tax that is well above the maximum pension. We find that the 35.9% of all retirees in our model economy receive the minimum pension, and only 4.1% of retirees receive the maximum pension. In Spain, these same numbers are 34.8% and 3.2% (see Table 9).

Table 9. The Spanish pension system in 2018^a

^a The source of Spanish data for the shares of minimum and maximum pensions is the Spanish Social Security Statistics and the Ministry of Inclusion, Social Security and Migration.

^b Share of minimum pensions.

^c Share of maximum pensions.

^d Average internal rate of return of pensions.

There are several studies that analyze the internal rate of return of the Spanish pension system. For instance, Barea (Reference Barea1996) find that the average internal rate of return of the Régimen General de la Seguridad Social is 2.1%. On the other hand, Jimeno and Licandro (Reference Jimeno and Licandro1999) find that this same number may reach 5.4%. Finally, and in a recent study, Moraga and Ramos (Reference Moraga and Ramos2020) find that this number is 3.5%. In our case, the model economy predicts an average internal rate of return of 2.1%. The main reason why Jimeno and Licandro (Reference Jimeno and Licandro1999) and Moraga and Ramos (Reference Moraga and Ramos2020) obtain higher internal rates of return is due to the fact that these authors assume that the payroll tax rate is only 15% and 18% of gross earnings respectively, while our model economy assumes that this same rate is 23.5%. The rationale for our choice is because the part of the payroll tax that would finance the expenditure of retirement and widowhood pensions is limited to 23.5% of earnings.Footnote ¹⁷ Overall, we find these results very encouraging since we did not target explicitly any of these statistics in our calibration procedure.

Retirement behavior. Another salient feature of the Spanish economy that our model economy should approximate, if we are to take its results seriously, is the retirement behavior of Spanish households. In Figure 4, we report the probabilities of exiting the labor force due to retirement.Footnote ¹⁸ Two features stand out from this comparison. Qualitatively, our model economy does a good job in replicating the general shape of the retirement hazards observed in Spain, including the peaks observed at ages 62 and 65.Footnote ¹⁹

Figure 4. The retirement hazards in Spain and the model economy (%)*.

*The Spanish data are reported by the Spanish Instituto de la Seguridad Social and they correspond to 2018.

Quantitatively, the peak at age 62 is higher in our model economy. For instance, at that same age the hazard in our model economy is 40.9% while in Spain it is only 34.8%. One of the main reasons for this is that our replacement rate is independent of the number of years of contributions to the pension system, while in Spain the pension replacement rate is an increasing function of this number.

We also analyze the composition of acquired wealth between the legal retirement ages. We find that those households that choose to retire between the ages of 62 and 66, and are located in the 30th percentile of the wealth distribution, have a social-security-wealth-to-liquid-assets ratio of 2.1. We also find that this number is 1.0 for those located in the 90th percentile. Unfortunately, we could not find these data for Spain, but we think that this decreasing relationship is very plausible. For instance, Poterba et al. (Reference Poterba, Venti and Wise2011) find that these figures are 2.9 and 0.9 in the United States for individuals aged 65–69 years old. Finally, and once again, these results can be interpreted as an overidentification condition, since we did not use them as a calibration target.

Life-cycle profiles. Figure 5 shows life-cycle profiles of average hours worked as a percentage of disposable time, average consumption, and average assets. We find that hours are mainly in the range of 30–40%, which decline gradually as individuals age. The overall patterns of the hour profiles are consistent with the data, for example, as reported by Díaz-Saavedra (Reference Díaz-Saavedra2017) for Spain (Figure 5), and French (Reference French2005) for the case of the United States.

Figure 5. Life-cycle profiles of hours worked, consumption, and assets in the model economy.

Figure 5 also displays the usual patterns of average asset holdings over the life cycle. That is, individuals accumulate wealth during their working lifetime for two main reasons. First, in order to accumulate stock savings against uncertainty about earnings and longevity, and second, to build the stock of savings for old-age consumption. However, since households are not altruistic in our model economy, consumption grows continuously until age 80, as they deplete their assets after leaving the labor market at a higher rate than they would if they were to leave inheritances. For instance, at the age range of 45–54 years, the median net wealth in Spain in 2017 was 3.6 times GDP per person aged 20+, and in our model economy it is 3.4. However, at the age range of 65–74 years, these figures are 5.6 in Spain, but only 4.6 in our model economy.Footnote ²⁰

Inequality. In Table 10 we report the Gini indexes and selected points of the Lorenz curves for earnings, income, and wealth in Spain and in our benchmark model economy. The statistics reported in bold face are our eight calibration targets. The source for the Spanish data of earnings and income are the INE and the OECD. The source for the Spanish data of wealth is the 2004 Encuesta Financiera de las Familias Españolas as reported in Budría and Díaz-Giménez (Reference Budría and Díaz-Giménez2006).The model economy statistics correspond to 2018.

Table 10. The distributions of earnings, income, and wealth^a

^a The source for the Spanish data of earnings and income are the Spanish National Institute of Statistics (INE) and the OECD. The source for the Spanish data of wealth is the 2004 Encuesta Financiera de las Familias Españolas as reported in Budría and Díaz-Giménez (Reference Budria and Díaz-Giménez2007).

We find that our model economy replicates the Spanish Gini indexes of earnings, income, and wealth reasonably well – the largest difference is only 0.01. When we look at the top tail of the distributions we find that the share of wealth owned by the top 10% of the wealth distribution is 8.2 percentage points higher in Spain. This disparity was to be expected, because it is a well-known result that overlapping generation model economies that abstract from bequests fail to account for the large shares of wealth owned by the very richest households in the data.Footnote ²¹

Moreover, our model economy also comes close to replicating the Gini index of pensions. According to Conde-Ruiz and Profeta (Reference Conde-Ruiz and Profeta2007), in 2000 this number was 0.32 in Spain and in our model economy it is 0.29 in our calibration year.

The labor market. Figure 6 shows the profiles of employee and unemployed households by age and education. When carrying out this comparison we must keep in mind that there are some fundamental differences between Spain and our model economy. In Spain, working-age people fall into one of five categories: employed, unemployed, retired, inactive, and other non-participants. In our model economy we only have three of these categories: employed, unemployed, and retired. Since these differences necessarily would distort our comparisons, we opted for excluding inactive and other non-participants people from the Spanish data.

Figure 6. The share of workers and unemployed in Spain and the model economy (%)*.

*The shares of workers are the shares of workers in the sum of workers and unemployed. We compute this share for Spain from the Encuesta de Población Activa (2018), reported by the INE.

We find that the model economy replicates these profiles well. If we look at the fine print, we find that the largest difference is observed in the case of dropouts households between 30 and 32 years old, since the model predicts that their employment rates are around 5 percentage points lower than those observed in Spain.

5.1 The experiments

We compare a steady-state benchmark allocation with different steady-state allocations of economies that differ in the mortality risk and in their pension rules. We describe below all simulations in turn (Table 11).

Table 11. The model economies: mortality risk and pension reform

The benchmark economy. Our benchmark economy is the economy that we modelled and calibrated to approximate the Spanish economy in 2018, but with the counterfactual assumption that survival risk only differs across ages, and it is given by the average age-dependent survival probabilities estimated by the INE in 2018. The rationale for this choice is because it allows us to quantify the degree of progressiveness embodied in the Spanish pension system. Additionally, we report the results of this same economy but assuming that these probabilities also differ by educational type, in order to analyze the loss of progressivity due to the current gap in life expectancy. In other words, what we expect from these exercises is to understand how much lifetime benefits and taxes change when we assume that less-educated workers have less longevity, whereas more-educated individuals live longer.

Alternative economies. The next simulations assume that mortality risk depends on both age and education, but with the projected long-term demographic and educational distributions. First, we simulated an economy called experiment 0. Essentially it is a do-nothing policy, where the pension rules follow the 2011 and 2013 pension reforms in Spain. Consequently, in this economy the first retirement age is 63 and the full entitlement retirement age is 67. The number of years of labor income used to compute the pension is the last 25 before retirement.Footnote ²² We simulate this economy, because from its comparison with our previous results, we can obtain a measure of how much progressiveness of the pension system is lost in the long-run because of the expected increase in lifetime gaps (Figure 7).

Figure 7. The unconditional survival probabilities in 2060 (%).

The next simulations are counterfactual experiments that allow us to evaluate how parametric pension reforms that do not differentiate between socioeconomic groups may restore pension income redistribution in the long run. Thus, at this point it is crucial first to identify which are the elements of this system that generate an income redistribution from the earnings poor to the earnings rich. Conde-Ruiz and González (Reference Conde-Ruiz and González2018) show that in many cases this is mainly achieved through the existence of a minimum pension. Consequently, in experiment 1 we assume that the minimum pension, as a share of per capita GDP, is increased by 50%.Footnote ²³

Additionally, one can explore the consequences of changes in the regulatory base, which is the main component of the Spanish retirement pension, for the progressiveness of the pension system. At this point, two main potential changes emerge. First, in Spain, from 2022 the regulatory base will be computed using labor earnings from the last 25 years before retirement, but contributions are broadly on the basis of the working lifetime. Furthermore, as Jimeno (Reference Jimeno2003) points out, because the distribution of labor income is much more unequal in the later periods of working life, the resulting pension distribution is more unequal under the current system than it would be under a system that takes into account the contribution bases or the contributions actually made during longer periods of working life, as occurs in other European countries with defined benefit pension systems.Footnote ²⁴ Consequently, if a person's salary doubles in their final years, their pension will double. Thus, there is a subsidy from people whose earnings grow more slowly to those whose earnings grow rapidly later in their working lifetime. The former group tends to be low earners, and the latter the high earners. Thus, on average, final-earnings schemes redistribute from low to high earners. Therefore, extending the averaging period of the regulatory base should reduce retirement pensions especially for the more educated workers, thus increasing the progressiveness of the pension system. In experiment 2 we explore this hypothesis by assuming that the regulatory base computes the average labor earnings for the entire working lifetime.

Lastly, we carry out a final simulation, experiment 3, in which we assume that there is both an increase in the minimum pension and in the number of years used to calculate the pension. In other words, we are applying the reforms of experiments 1 and 2 jointly. The rationale for doing this is given in the next section.

5.2 Welfare effects

We follow Conesa and Krueger (Reference Conesa and Krueger1999) and compute the consumption equivalent variation measure (CEV) for newborns before and after the reforms in 2060. Specifically, we compute the welfare change of a reform for a newborn with education h and labor status e, by asking by how much this newborn's consumption has to be increased, holding leisure constant, in the old steady state, so that her expected utility equals that under the specific reform. Consequently, and given the form of the utility function, we compute

(21)$$CEV_h( e ) = e^{( {V_h^\ast ( e ) -V_h( e ) } ) {\rm /}\kappa }-1, \;$$

where V _h(e) and $V_h^\ast ( e )$ are the value functions of a newborn with education h and labor status e under the current and the reformed pension system, respectively, and $\kappa = \mathop \sum \nolimits _{j=20} ^{J-1} \beta ^{j-20}\psi _{jh}$. For example, a CEV _h(e) of 0.01 implies that a newborn with education h and labor status e will enjoy an increase in welfare equivalent to receive 1% higher consumption under the current pension system.Footnote ²⁵

Finally, we also compute a weighted average consumption equivalent variation. That is, the model implicitly defines some social welfare weights under which a distributional outcome arises. Thus, the weighted average consumption equivalent variation, ($\overline {CEV}$), is given by

(22)$$\overline {CEV} = \mathop \int \! \int \nolimits \mathop \nolimits CEV_h( e ) d\mu _ed\mu _h.$$

In fact, we use a weighted social welfare function where the weights are given by the projected long-term measure of newborns by education and labor status.

5.3 The environments

All the model economies described in experiments 0–3 share the demographic, educational, employment, and government policy scenarios that we now describe.

The demographic scenario. The long-term demographic scenario replicates the demographic projection for Spain in the year 2060 estimated by the INE in 2018. In panel A of Figure 8 we plot the projected distribution by age of Spanish population in 2060. According to this projection, the old-age dependency ratio is 52.76 (=[34.54/65.45] × 100) percent that same year.Footnote ²⁶

Figure 8. The age and educational distribution in 2060.

However, a problem we have faced when carrying out our simulations is the lack of data regarding the mortality risk across the different educational groups. Therefore, to solve this lack of data, we have had to estimate it, for which we have carried out a process that consisted of several stages. First, from the data reported by Permanyer et al. (Reference Permanyer, Spijker, Blanes and Renteria2018) we extrapolate current tendencies on life expectancies at age 35 by educational type to the year 2060. Second, as input we use the average probabilities calculated by the INE in 2018 and 2060. Third, we assume that the differences in these probabilities across educational groups appear after age 40, and that this difference reaches a maximum at age 70, and then decreases. Finally, using iterative methods we generate profiles of survival probabilities by age and educational type that produce a life expectancy at the age of 35 consistent with the estimates for the period 2012–2015 provided by Permanyer et al. (Reference Permanyer, Spijker, Blanes and Renteria2018) or our extrapolation for 2060 of each educational group.

The educational scenario. In panel B of Figure 8 we plot the assumed distribution of education in 2060, and this panel shows that high school households is the most numerous educational group within the population. See Díaz-Saavedra (Reference Díaz-Saavedra2020) for an explanation of how we obtain this educational distribution for the Spanish population in 2060.

The employment scenario. We assume that employment risk continues to be characterized by the age-dependent probabilities in/out of employment that we estimated for Spain in 2018.

The government policy scenario. Recall that the consolidated government and pension system budget constraint in our model economy is

(23)$$G + P + U + Z = T_k + T_s + T_y + T_c + E.$$

In this expression G is exogenous and the remaining variables are endogenous. In all model economies the income, capital income, and payroll tax rates are identical and they remain unchanged at their calibrated values. The consumption tax rates differ across economies because we change it to clear the government budget. Every other variable in expression (23) varies and differs across both economies because they are all endogenous.

Finally, all of pension parameters that characterize the pension policy are already set because both, our assumption that all model economies introduce the main pension rules following the 2011 and 2013 pension reforms in Spain, and also because the aforesaid described pension policy experiments.