This section presents the model economy. We study an overlapping generations economy with heterogeneous households, a representative firm, and a government. We use the same model in related work in Brogueira de Sousa et al. (2021). Although we calibrate the model economy to Spanish data, here we describe an economy that features a backpack system, in addition to the PAYG pension system. For the calibration procedure, we set one parameter in the model to zero—the Backpack contribution rate—and match the model to Spanish data. In the policy reform exercise, we compare the calibrated model to alternative Backpack economies.
Time is discrete and runs forever, and each time period represents one calendar year. All model quantities depend on calendar time t, but we omit this dependence since we focus on steady-state equilibria. In a steady state, as defined below, all aggregate variables and individual policy functions are constant with respect to calendar time. We begin with a description of household heterogeneity.
The households
Households in our economy are heterogeneous and differ in their age, \(j \in J\); in their education, \(h \in H\); in their productivity level \(z \in {\mathcal {Z}}\); in their labor market status \(s \in S\); in their private assets, \(a \in A\); and in their backpack savings, \(b \in B\). Sets J, H, \({\mathcal {Z}}\), S, A, and B are all finite sets and we use \(\mu _{j,h,z,s,a,b}\) to denote the measure of households of type (j, h, z, s, a, b). They also differ in their claims to different social insurance systems: unemployment benefits UB, retirement PAYG pensions P, and government transfers TR. 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.
Age
Individuals enter the economy at age 20, the duration of their lifetimes is random, and they exit the economy at age \(T=100\) at the latest. Therefore, \(J=\{20,21,...,100\}\). The parameter \(\psi _j\) denotes the conditional probability of surviving from age j to age \(j+1\). The notation makes explicit that the exogenous probabilities depend on age j, but not on education or other factors.
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 productivity
Individuals receive an endowment of efficiency labor units every period. This endowment has two components: a deterministic component, denoted \(\epsilon _{h,j}\) and a stochastic component, denoted by z. The deterministic component depends on the household’s age and education, and we use it to characterize the life cycle profiles of earnings. The stochastic component is independently and identically distributed across households, and we use it to generate earnings and income dispersion in the economy. 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 conditional transition probabilities given by \(\Gamma \):
$$\begin{aligned} \Gamma \left[ z'|z\right] = \text {Pr}\left\{ z_{j+1}=z'|z_j=z\right\} ,\;\text {with}\;z,z'\in Z. \end{aligned}$$
(1)
Every period agents receive a new realization of z. Total labor productivity is then given by \(\epsilon _{h,j}z\). A worker who supplies l hours of labor has gross labor earnings y given by:
$$\begin{aligned} y = \omega \epsilon _{h,j} z l, \end{aligned}$$
(2)
where the economy-wide wage rate \(\omega \).
Labor market status
In the model, an agent is either employed, unemployed, non-active, or retired. Among the unemployed, there are individuals who are eligible to receive unemployment benefits and access their backpack savings (workers who have recently been laid off), and others who are not eligible (either because eligibility expired, or because they quit work). Worker decide when to retire (with a minimum retirement age), leaving the labor force permanently once they do. Upon entering the economy, individuals randomly draw a job opportunity and then decide to work or not during the first period. Similarly, in subsequent years the labor market status evolves according to both optimal work and job search decisions (described below), and exogenous job separation and job finding probabilities.
Employed
An individual with a job in the beginning of the period, and who decides to work, is employed in that period and his labor market status is denoted by \(s=e\). An employed worker provides labor services and receives a salary that depends on his efficiency labor units and hours worked. He faces a probability of losing the job at the end of the period, denoted \(\sigma _{j}\). This probability is age dependent, and we use it to generate the observed labor market flows between employment and non-employment states within age cohorts.
Unemployed
An agent may not have a job opportunity at the beginning of a period, because he lost his job last period, because he quit his job, or because he was unemployed last period and did not find (or did not accept) a new job offer. Without a job, households may actively search for a job offer next period. If they do actively search we label them as unemployed. Unemployed agents who have lost a job are eligible for unemployment benefits and to use accumulated backpack savings (we refer to them as unemployed eligible, with \(s=ue\)). A formal description of eligibility criteria is given below. Agents who have quit work are not eligible for unemployment compensation (we often refer to this group as unemployed non-eligible, \(s=un\)). Active job searchers receive a job offer at the end of the period with probability \(\lambda ^{\mathbf{u }}_{j}\). The probabilities are again age dependent, and we use it to generate the observed labor market flows between unemployment and employment.
Non-active
Agents without a job and who do not actively search for a new one are labeled non-active, with \(s=n\). Those agents are not eligible for unemployment benefits nor to collect backpack savings, and receive a job offer for next period with a lower probability than an unemployed agent, \(\lambda ^{\mathbf{n }}_{j}<\lambda ^{\mathbf{u }}_{j}\). This probability is also age dependent, and we use it to generate the observed labor market flows between non-activity and employment.
Retirees
In our model, workers optimally decide whether to retire and leave the labor force (with a minimum retirement age). They take this decision after observing their current labor productivity. If they decide to retire, \(s=r\), they lose the endowment of labor efficiency units for ever and exit the labor market. Depending on the retirement savings systems in place, they receive PAYG retirement pension payments and may receive additionally a backpack annuity payment.
Private assets
Households in our model economy endogenously differ in their asset holdings, which are constrained to being nonnegative. The absence of insurance markets give the households a precautionary motive to save. They do so by accumulating real assets which take the form of productive capital, denoted \(a\in A\). Different retirement pension systems affect, among others, the agents’ private savings decisions.Footnote 3
Backpack assets
Workers accumulate backpack savings while they work. These savings result from a mandatory contribution out of workers’ salaries, and are invested in productive capital and earn the real rate of return in the international capital market. When workers lose a job, they can access their accumulated savings and decide how much to keep in their individual accounts or how much to use, while out of work, to finance consumption. A formal description of the decision problem is given below. At retirement, backpack assets are converted into retirement pension payments (an actuarially fair life annuity).
Households derive utility from consumption, and disutility from labor and the search effort. Labor is decided both at the extensive and intensive margins, while search is a discrete choice. The period utility is described by a utility flow from consumption and the utility cost of time allocated to market work and to job search. Non-active and retired agents dedicate all the time endowment to leisure consumption. Accordingly, lifetime utility is given by
$$\begin{aligned} {\mathbb {E}} \sum _{j=20}^{100} \beta ^{j-20}\psi _{j} \Big [ u(c_j, l_j) - \gamma e_j \Big ], \end{aligned}$$
(3)
where \(\beta \) is a time discount factor, u satisfies standard assumptions, \(c_j\) is consumption and \(l_j\) is labor supply, and \(\gamma \) represents a job search utility cost. \(l_j\) can take values between 0 and 1, while \(e_j\) equals 1 in periods of active job search and is zero otherwise. Survival probabilities \(\psi _{j}\) determine average life expectancy in the economy, a central object in our analysis.Footnote 4
At the beginning of each period, z, households’ stochastic productivity component, is realized.Footnote 5 When entering the economy (at age 20) agents additionally learn their education level and draw a job opportunity, that they can either accept of reject. For older households, if they start a period with a job opportunity, they decide whether to work and if so, by how much. If they lost their job or decided not to work in the previous period, they choose whether to search for a new job or not. Depending on these decisions, individuals then spend the period working, unemployed or inactive. Wages and unemployment benefits are received, and decisions on consumption and savings are taken. At the end of the period, workers observe the job separation shock, and unemployed or inactive learn if they found a job for next period. Households can choose to retire at the beginning of the period, and once they do they leave the labor market permanently.
Technology
The firm. In our model economy, there is a representative firm. Aggregate output depends on aggregate capital, K, and on the aggregate labor input, L, through a constant returns to scale, Cobb–Douglas, aggregate production function of the form
$$\begin{aligned} Y = K^\theta L^{1-\theta }. \end{aligned}$$
(4)
Factor and product markets are perfectly competitive and the capital stock depreciates geometrically at a constant rate, \(\delta \). The firm rents capital in the international capital market at an interest rate r and hires workers in the domestic market at a wage rate \(\omega \) per efficiency unit of labor.
Insurance markets. An important feature of the model is that there are no insurance markets for the stochastic component of the endowment shock, for unemployment risk, or survival risk.
Backpack system
The BP economy features a fully funded employment fund, financed by individual worker contributions. Workers may choose to use all or a fraction of the BP savings during periods of involuntary unemployment. Every individual enters the economy without backpack claims. For every period of employment, a worker sees a fraction \(\tau _b\) of his gross labor earnings deducted and invested into a personal savings account, which is remunerated at the capital market rate of return, r. If \(b_t\) is the level of backpack assets at the beginning of an employment period, then next period’s backpack evolves according to:
$$\begin{aligned} b_{t+1} = \tau _b y + (1+r(1-\tau _k))b_t, \end{aligned}$$
(5)
with \(\tau _k\) being the capital income tax rate. When a worker loses his job, his backpack assets can be allocated to finance consumption (present or future, as he can choose to save the backpack assets). Next period’s backpack assets become a choice variable for the involuntary unemployed. In contrast, if a worker chooses to quit his job while still in the labor force, he keeps the backpack but cannot withdraw. In that period, the backpack evolves according to
$$\begin{aligned} b_{t+1}= (1+r(1-\tau _k))b_t. \end{aligned}$$
(6)
Upon retirement, backpack assets can be used to buy a lifetime annuity or added to private savings. If the worker decides retire at age R and allocate b amount of BP savings to the purchase of the annuity contract, he receives in return:
$$\begin{aligned} p^B(b) = b\left[ 1 + \sum _{t=1}^{T-R}\frac{\prod _{i=0}^t\psi _{R+i}}{(1+r)^t}\right] ^{-1}. \end{aligned}$$
(7)
The aggregate amount of backpack assets is invested in the capital market and adds to the stock of productive capital available in the economy. Since this is an individual, fully funded system, the aggregate amount of BP assets used to purchase annuity contracts equals the total amount of annuity payments received by retirees. Hence, we do not include it in the social security budget equation, shown below.
Pay-as-you-go system
The PAYG system is an unfunded defined contribution pension system, where pension payments mostly depend on individual workers’ history of salaries, among other factors. In the model, pension payments depend on average earnings during the \(N_b\) years prior to retirement. In Spain, as in many other countries where a PAYG system exists, there is a minimum retirement age after which worker can decide to retire. We denote it by \(R_0\). In order to capture the heterogeneity in pension payments that arises from different lifetime earnings histories, but at the same time reduce the dimensionality of the problem, we model pension payments that differ for each educational group (instead of each individual). Specifically, pension payments for retirees of educational group h are:
$$\begin{aligned} p^S_h=p_r{\bar{y}}^S_{h}, \end{aligned}$$
(8)
where \({\bar{y}}^S_{h}\) is the average earnings of households in educational group h during the last \(N_{b}\) years before the retirement age, \(R_0\), and \(p_r\) is a replacement rate. \({\bar{y}}^S_{h}\) is computed as:
$$\begin{aligned} {\bar{y}}^S_{h} = \frac{1}{N_{b}}\sum _{j=R_0-N_{b}}^{R_0-1} {\bar{y}}_{j,h} \end{aligned}$$
(9)
where \({\bar{y}}_{j,h}\) is the average gross labor earnings of workers aged j and with education h. We assume that there are no early retirement penalties, nor minimum or maximum pensions.
The government
Here we describe the government programs other than retirement pensions, discussed above, and the government and social security budgets that we model separately.
Unemployment benefits. The government taxes workers and provides unemployment benefits to the unemployed. Eligibility for unemployment benefits—denoted \(\mathbbm {1}_{UB}=1\), below—is conditional on: (i) having lost a job (i.e., a job separation) and not having started a new job yet, (ii) on actively searching for a job, and (iii) having been unemployed for less than a given number of periods, \({\bar{d}}\). Eligibility expires when one of the conditions is not met, and non-eligibility is an absorbing state. Eligible agents receive unemployment benefits given by \(u_b = b_{0} {\bar{y}}_{j,h}\), where \(b_0\in (0,1)\) is a replacement rate and \({\bar{y}}_{j,h}\) is the average labor earnings of workers in age group j and with education h. Unemployment benefits are financed with payroll taxes, described below.
Other transfers Households below an income level \(y<\overline{t_{r}}\) receive a transfer from the government, denoted TR. Eligibility for transfers is conditional on income only and denoted by \(\mathbbm {1}_{TR}=1\). Eligible households receive an amount \(t_r=b_{1}\overline{t_{r}}\).
We model the government budget restriction with two separate identities. Unemployment benefits and PAYG pensions are financed with payroll taxes and form the social security budget. Other government expenditures and revenues form the government budget.Footnote 6
The government taxes capital income, household income and consumption, and it confiscates (part of the) unintentional bequests. It uses its revenues to finance an exogenous flow of public consumption and debt, and to make transfers to low-income households. In addition, the government provides unemployment benefits and runs a PAYG pension system.
The government budget constraint is then:
$$\begin{aligned} G_t+T_{r,t}+D_{t+1}&= T_{k,t}+T_{y,t}+T_{c,t}+E_t+(1+r)D_t, \end{aligned}$$
(10)
$$\begin{aligned} U_{b,t} + P_t&= T_{p,t}, \end{aligned}$$
(11)
where \(G_t\) denotes government consumption, \(T_{r,t}\) denotes government transfers, \(T_{k,t}\), \(T_{y,t}\), and \(T_{c,t}\), denote the revenues collected with the capital income tax, the household income tax, and the consumption tax, and \(E_t\) denotes unintentional bequests taxed by the government. \(U_{b,t}\) denotes unemployment benefits, \(P_t\) denotes pension payments in period t and \(T_{p,t}\) denotes revenues collected with the payroll tax. In the remaining of the paper, we assume that the level of public debt is fixed at the baseline calibration year level, \(D_{t+1}=D_t\).
Capital income taxes Capital income taxes are given by \(\tau _k y_k\), where \(\tau _k\) is the tax rate on gross capital income \(y_k = r a\). a denotes households’ capital holdings and r the economy rate of return on capital.
Payroll taxes Payroll taxes are proportional to before-tax labor earnings: \(\tau _p y\).
Backpack taxes Similarly, contributions to accumulate assets in the individual Backpack account are given by: \(\tau _b y\).
Consumption taxes Similarly, consumption taxes are simply \(\tau _{c}c\), where \(\tau _c\) is the consumption tax rate and c is consumption.
Income taxes We assume a simplified income tax formula according to which the income tax is proportional to the income level: \(\tau _y {\hat{y}}\), where \(\tau _y\) is a tax rate parameter and \({\hat{y}}\) is the tax base. The income tax base depends on the employment status. If a household is employed:
$$\begin{aligned} {\hat{y}}= (1-(\tau _p +\tau _b))y + r(1-\tau _k)a. \end{aligned}$$
(12)
For the unemployed and non-active agents,
$$\begin{aligned} {\hat{y}}= r(1-\tau _k)a, \end{aligned}$$
(13)
and for a retired household:
$$\begin{aligned} {\hat{y}}= r(1-\tau _k)a + p^S_h. \end{aligned}$$
(14)
Individual decision problem
We describe the problem in the BP economy, i.e., a steady-state economy with a Backpack system and a PAYG pension system. The households’ problem is described recursively. To simplify the notation, we omit the dependence of the value functions on the state variables age, education, private savings, backpack savings, and unemployment spell duration.
We first state the decision problem of a worker at the beginning of the period after the job acceptance decision was taken. Given the value functions, we define below the job acceptance and retirement decisions. An individual who is currently employed decides how much to consume c, save \(a'\), and work \(l\in \left[ 0,1\right] \), according to the following optimization problem:
$$\begin{aligned} W = \max _{c,l,a'} \Bigg \{ u(c, l) + \beta {\mathbb {E}} \Big [ (1-\sigma _j) J + \sigma _j U \Big ] \Bigg \} \end{aligned}$$
(15)
subject to:
$$\begin{aligned} (1+\tau _c)c + a' + \tau _y{\hat{y}} + (\tau _p+\tau _b)y \le (1+r(1-\tau _k))a + y + TR(y) , \end{aligned}$$
(16)
the backpack law of motion,
$$\begin{aligned} b' = \tau _b y + (1+r(1-\tau _k))b, \end{aligned}$$
(17)
and a non-borrowing constraint:
$$\begin{aligned} a' \ge 0. \end{aligned}$$
(18)
Gross labor income is \(y = \omega \epsilon z l\), income tax base \({\hat{y}} = (1-\tau _p-\tau _b)y + r(1-\tau _k)a\) and government transfers for low-income households are denoted by \(TR(y) = t_r \mathbbm {1}_{TR}(y)\), where \(\mathbbm {1}_{TR}(y)=1\) if \(y<\bar{t_r}\) and zero otherwise, as explained above. While working, backpack asset \(b'\) accumulate in the worker’s individual BP account, according to (17).
Equation (15) reads in the following way: the first term account the utility flow from consumption and labor. In the discounted continuation value, the expectation operator accounts for survival risk, all possible continuation histories of the realization of the stochastic component \(z'\in {\mathcal {Z}}\), and two distinct labor market outcomes: With probability \(1-\sigma _j\), the worker keeps the job in the next period (and therefore is not eligible to claim unemployment benefits), with value denoted J that depends on next period’s private and backpack assets, respectively \(a'\) and \(b'\), and the new realization of idiosyncratic productivity \(z'\); alternatively, with probability \(\sigma _j\), the job is destroyed and the worker starts next period without a job, with value U. This value depends on the number of periods after an involuntary job separation (relevant to determine eligibility for unemployment benefits), d. In the first period after a layoff, \(d=0\). \(z'\) follows the Markov chain described in (1).
Workers can start the period without a job. A job searcher who faced a job separation shock and has yet to start a new job has access to his backpack savings and, depending on how long he has been without working, may be eligible to receive unemployment benefits. He solves a consumption-savings problem, a job search problem, and a portfolio problem for the allocation of his private and backpack savings. At the beginning of the period, the individual state vector is given by private asset holdings a, backpack savings b, stochastic productivity z, and layoff duration d. Given the current state, the agent chooses consumption, future asset holdings and the search effort \(e \in \{0,1\} \) according to:
$$\begin{aligned} U = \max _{c,a',b',e} \Bigg \{&u(c) - \gamma e + \beta {\mathbb {E}} \Big [ e \Big (\lambda ^{\mathbf{u }}_j J + (1-\lambda ^{\mathbf{u }}_j) U\Big ) + (1-e) \Big (\lambda ^{\mathbf{n }}_j J + (1-\lambda ^{\mathbf{n }}_j) N\Big ) \Big ] \Bigg \} \end{aligned}$$
(19)
subject to
$$\begin{aligned} (1+\tau _c) c + a' + b'(e) + \tau _y {\hat{y}} \le (1+r(1-\tau _k))(a + b) + UB(d,e) + TR(y),\qquad \end{aligned}$$
(20)
and
$$\begin{aligned} a', b'(e)&\ge 0. \end{aligned}$$
(21)
The first term in Eq. (19) inside the curly brackets is the flow utility from consumption and the utility cost of search, \(\gamma e\). The expected continuation value takes into account the survival probability and the evolution of the stochastic productivity component, z. Higher search effort (\(e=1\)) translates into higher probability of finding a job: \(\lambda ^{\mathbf{u }}_j>\lambda ^{\mathbf{n }}_j\). The trade-off in the job search problem is inside the expectation operator: With high search effort in the current period, with utility cost \(\gamma \), the agent finds a job next period with probability \(\lambda ^{\mathbf{u }}_j\); with low search effort (\(e=0\)), a job arrives with lower probability \(\lambda ^{\mathbf{n }}_j\). If the worker finds a job, he decides in the beginning of next period whether to work or not at that job, with an option value J which depends on beginning of period assets and labor productivity. If search is not successful the worker continues unemployed next period with probability \((1-\lambda ^{\mathbf{u }}_j)\), with value U which depends on assets, productivity and unemployment duration \(d'=d+1\). If the unemployed worker decides not to search (\(e=0\)) and does not find a job, he becomes non-eligible for unemployment insurance benefits and may again search for a job next period, with associated value N.
Equation (20) is the budget constraint. Total income is used to finance consumption expenditures, next period assets and income taxes, with the income tax base given by \({\hat{y}} = r(1-\tau _k)a\). The right-hand side is the sum of beginning of period private and backpack assets, plus after-tax return, unemployment benefits UB(d, e) and government transfers for low-income households, TR(y). The laid-off worker may be entitled to unemployment benefits: \(UB(d,e) = u_b \mathbbm {1}_{UB}(d,e)\), with \(\mathbbm {1}_{UB}(d,e)=1\) indicating eligibility for unemployment benefits. Formally:
$$\begin{aligned} \mathbbm {1}_{UB}(d,e) = {\left\{ \begin{array}{ll} 1 &{}\text {if } e = 1 \text { and } d \le {\bar{d}}, \\ 0 &{}\text {otherwise}. \end{array}\right. } \end{aligned}$$
(22)
The state variable d evolves deterministically according to \(d'=d+1\) if the worker continues unemployed in the following period, and \(d=0\) in the period immediately after a separation shock. We make two important simplifying assumptions here. The search effort is dichotumous: Either can the agent actively search for a job (\(e=1\)), or he does not search (\(e=0\)). Additionally, the possibility of using backpack assets while unemployed and searching for a job is represented by \(b'(e)\) in the constraint (20): The laid-off worker can use his backpack savings to finance present (or future) consumption if he searches for a new job, but cannot increase backpack holdings other than through wage contributions (i.e., while working). Formally:
$$\begin{aligned} b'(e) {\left\{ \begin{array}{ll} \le &{} \tau _b y + (1+r(1-\tau _k))b,\text {if } e = 1 , \\ =&{} \tau _b y + (1+r(1-\tau _k))b,\text {if } e = 0. \end{array}\right. } \end{aligned}$$
(23)
The fact that unemployed eligible (\(e=1\)) workers can use some or all of accumulated BP savings is reflected in the \(\le \) sign, above. There is a non-borrowing constraint given by (21).
Finally, an individual can start the period without a job because he has decided not to work or not to search in previous periods, not having found a new job yet. In this scenario, he solves the following problem:
$$\begin{aligned} N = \max _{c,a',e} \Bigg \{&u(c) - \gamma e + \beta {\mathbb {E}} \Big [ e \Big (\lambda ^{\mathbf{u }}_j J + (1-\lambda ^{\mathbf{u }}_j) N\Big ) + (1-e) \Big (\lambda ^{\mathbf{n }}_j J + (1-\lambda ^{\mathbf{n }}_j) N\Big ) \Big ] \Bigg \}, \end{aligned}$$
(24)
subject to
$$\begin{aligned} (1+\tau _c) c + a' + \tau _y {\hat{y}} \le (1+r(1-\tau _k))a + TR(y), \end{aligned}$$
(25)
and
$$\begin{aligned} a'&\ge 0, \end{aligned}$$
(26)
$$\begin{aligned} b'&= (1+r(1-\tau _k))b . \end{aligned}$$
(27)
As above, \({\hat{y}} = r(1-\tau _k)a\). The decision problem is similar to (19). But in this case, the unemployed worker is not eligible for unemployment benefits, and he also cannot use backpack assets. Accordingly, the evolution of BP assets is given by (27).
Retired individuals are not in the labor market and have no endowment of efficiency units of labor. They finance consumption with past private savings, backpack annuity payments, and PAYG 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 exits the economy next period with probability one. During retirement, the retired household solves a standard consumption-savings problem taking into account survival probabilities and total income:
$$\begin{aligned} V (a) = \max _{c,a'} \Bigg \{ u(c) + \beta {\mathbb {E}} \Big [ V (a') \Big ] \Bigg \} , \end{aligned}$$
(28)
subject to
$$\begin{aligned} (1+\tau _c) c + a' + \tau _y {\hat{y}} \le (1+r(1-\tau _k))a + p^S_h + p^B(b) + TR(y). \end{aligned}$$
(29)
Pension payments and backpack annuities are part of the income side of the budget constraint. In this case, \({\hat{y}} = r(1-\tau _k)a + p^S_h + p^B(b)\). After retirement, labor market productivity is always zero, and hence, expectations take into account only the survival risk.
To close the description of the household’s problem, we define the job acceptance and retirement decisions. These jointly pin down the value of having a job offer at the beginning of a period. For a household older or at the minimum retirement age (as defined by the PAYG pension system rules), \(j\ge R_0\):
$$\begin{aligned} J&= \max \Big \{ V, \max \{ W, N \}\Big \}. \end{aligned}$$
(30)
The outermost \(\max \) operator represents the retirement decision, while the inner operator is the job acceptance decision. Younger households, \(j<R_0\), make only the job acceptance decision.
Stationary equilibrium
The formal definition of a stationary equilibrium in the open economy, as well as in the closed economy, is postponed to Appendix G.