3.1 Demographics

The economy is populated by J overlapping generations with a model period that equals a calendar year. At the beginning of each period, a new generation is born. The population grows at a constant rate n and the cohort size of newborns is normalized to unity. Individuals enter the economy at (model) age j = 1 (i.e., age 20) with perfect health and may live up to a maximum of J years after which they pass away with certainty. After working in the labor market they may be hit by health shocks which may induce them to apply for DP. Alternatively, if eligible, they may claim (and immediately receive) an OAP pension starting at the ERA j _B. Throughout their entire life, agents face idiosyncratic survival risk, which is determined by age and individual health status h. The conditional survival probability of an agent to survive from age j − 1 to age j is denoted by ψ _j(h) with ψ _J+1(h) = 0.

3.2 Preferences and endowments

Households have preferences over stochastic streams of consumption $c_j {\unicode{x2A7E}} 0$, work and non-work l _j ∈ {0, 0.2, 0.5} and the application for disability benefits d _j ∈ {0, 1}. They maximize discounted expected utility

$$U = {\rm {\mathbb E}}\left[{\sum\limits_J^{\,j = 1} {\beta^{\,j-1}u( c_j, \;l_j, \;d_j) } } \right]\quad {\rm with}\;u( c_j, \;l_j, \;d_j) = \displaystyle{{c_j^{1-1/\gamma } } \over {1-1/\gamma }}-\chi _{\,j, s}I_{l_j > 0}-\xi _{\,j, s}d_j.$$

Expectations are formed concerning survival, productivity, health, and employment risk and future utility is discounted with the constant time discount factor β. The intertemporal elasticity of substitution (IES) in consumption is denoted by γ. Agents face participation cost χ _j,s when employed depending on age and education. The indicator variable $I_{l_j > 0}$ is zero when retired and one when working. Finally, following Laun et al. (Reference Laun, Markussen, Vigtel and Wallenius2019) and Galaasen (Reference Galaasen2021), we assume that applying for DP induces stigma cost ξ _j,s, meant to capture psychological factors of accepting work limitations but also time cost and hassle to write a DP application. Stigma cost also depends on age and education, they are iid across households with a log-normal density function, so thatFootnote ⁹

$$\xi _{\,j, s}\sim LN( \mu _{\,j, s}, \;\sigma _\xi ^2 ) , \;$$

where ${\rm \Psi }_\xi ( {\cdot} )$ denotes the cumulative distribution function (at a specific age).

Labor productivity: The modeling of productivity risk is taken from Kindermann and Püschel (Reference Kindermann and Püschel2021) who have studied administrative data from the German pension insurance system to investigate the properties of individual labor earnings dynamics over the life cycle. They found that workers are exposed to significant earnings risk which could not be captured by the standard AR(1) process for log-earnings. On the one hand, individual contribution records show periods where employees do not pay contributions because they are out of official employment due to informal work, illness, etc. On the other, some employees typically spend a significant fraction of working years in low-income episodes where they only make about 10 percent of average annual labor earnings. Due to these ‘mini-jobs’, the distribution of labor earnings becomes bimodal with quite distinct dynamics across educational groups.

Kindermann and Püschel (Reference Kindermann and Püschel2021) therefore distinguish between households with high-school education (s = 0) and college education (s = 1), where the initial fraction of college-educated households is denoted by ω _s. All workers of education level s share a common deterministic age-specific labor productivity profile e _j,s. Knowing their educational level, workers are again divided into two permanent subgroups m, which indicate whether they face a stable (m = 0) or an unstable career path (m = 1). The probability to draw the state m = 1 is independent of education and denoted by ω _m. Throughout their working life, individuals' labor productivity is due to idiosyncratic shocks η. Productivity of individuals with a stable career (m = 0) follows a standard AR(1) process

(1)$$\eta ^ + { = } \rho \eta + \epsilon ^ + \quad {\rm with}\;\epsilon ^ + {\sim} N( 0, \;\sigma _{\epsilon , s}^2 ) , \;$$

where innovations $\epsilon ^ +$ are iid across households with education level s. In the following, we omit all state indices and only indicate the next period variables with ‘ + ’ to simplify notation. To capture low-income episodes for individuals with unstable careers (m = 1), this standard shock process (1) is augmented by a persistent (but not permanent) low productivity shock η ₀. Productivity in this low-productivity state is independent of age and education. For households with unstable careers, the transition into and out of low earnings is modeled as a first-order discrete Markov process. Those who are currently normal earners (i.e., where η ≠ η ₀) face the education-specific probability $\pi _{low, 0}^s$ to transit into the low-earning state in the next period (i.e., where η ⁺ = η ₀), while currently low earner households face a probability $\pi _{low, 1}^s$ to remain in the low-earning state. In the initial year, a fraction $\omega _{low}^s$ of individuals with education s in the unstable career subgroup starts as a low-earning individual.

Health and employment risk: In our model households are also exposed to health risks that affect their work capacity. We distinguish a good health state (h = 0), where households can work without impairment, a medium state (h = 1), where they face some minor injuries and suffer from mental problems that reduce work capacity slightly and a bad health state (h = 2), where work capacity is reduced severely.Footnote ¹⁰ These shocks again follow a first-order discrete Markov process where households face probabilities of the future health state π _h(h ⁺|j, s, h) that depend on age, education, and the current health state:

(2)$$h^ + { = } f( j, \;s, \;h) .$$

To model the impact of these health shocks on productivity, we further augment the productivity process described above by assuming that a fraction $\pi _h^{nw}$ of employees in the health state h stay at home (i.e., do not receive income and pay no pension contributions), while the productivity of the remaining fraction is reduced by θ _h,s percent. Bad health therefore not only reduces labor income and pensions, it may also reduce the eligibility for early OAP receipt if the number of contributory years is not reached.Footnote ¹¹

Consequently, labor productivity z(j, s, m, η, h) depends on age, education, career stability, the idiosyncratic productivity shock, and the health state. Individual labor income y is then given by

$$y = w \times z( j, \;s, \;m, \;\eta , \;h) \times l, \;$$

where the wage per efficiency unit w is multiplied by labor productivity and the working hours.

Individuals enter the labor market without assets (a ₁ = 0), during their working years they accumulate assets to finance retirement and to self-insure against uncertainty. Since our model abstracts from annuity markets, individuals who die before the maximum age of J may leave accidental bequest b that will be distributed equally in a lump-sum fashion among all individuals below age 60.

3.3 Accumulation of pension wealth and benefit calculation

As already explained above, the statutory German pension system is based on so-called ‘earning points’, which reflect the relative income position (and therefore contribution level) during employment years. Agents who receive an average income $\bar{y}$ in a specific year accumulate one earning point in their retirement account. In the case of higher income, earning points are increased proportionally (up to the contribution ceiling which is roughly $2\bar{y}$) and vice versa in the case of lower income. Consequently, earning points ep in the model are accumulated as

(3)$$ep^ + { = }\; ep + \min [ y/\bar{y}; \;2.0] $$

until agents decide to stop working at age j _R and receive retirement benefits. We abstract from temporary retirement and from separating the decisions of benefit claiming and labor market exit as in İmrohoroğlu and Kitao (Reference İmrohoroğlu and Kitao2012) or Jones and Li (Reference Jones and Li2023). As already discussed above, OAP claiming typically happens during employment, and although there are only a few restrictions there is hardly any full-time work after retirement. Consequently, retirement is an absorbing state and no earning points are accumulated in the model after retirement.

Figure 2 shows the retirement windows for OAP and DP in the model. Early retirement with an OAP is possible either with or without deductions in and after the ERA j _B if the contribution record is either more than 35 years or more than 45 years, respectively. All other employees retire with an OAP at the NRA j _N or delayed up to the maximum retirement age j _E in which all remaining employees are forced to retire. Since those with health problems might end up without labor income in some periods, not all employees qualify for early retirement. To check eligibility for early retirement, we keep track of the number of non-contributory years before retirementFootnote ¹²:

(4)$$nc^ + { = } \left\{{\matrix{ {nc} \hfill & {{\rm if\ }y > 0} \hfill \cr {nc + 1} \hfill & {{\rm otherwise} .} \hfill \cr } } \right.$$

Figure 2. Overlapping retirement windows for OAP and DP.

In contrast to OAP, retirement with DP depends on the individual health status. Agents enter the labor market in full health (h = 0), but after one year their health may deteriorate which reduces or even temporarily eliminates their work capacity. Employees may then either stay in the labor market and hope for future health improvements or restrict labor market participation from l = 0.5 to l = 0.2 and submit an application for DP.Footnote ¹³ After one year, the pension insurance accepts an application with an (exogenous) probability of 1 − q and the agent starts receiving DP benefits.Footnote ¹⁴ If the application is rejected with probability q, the agent either works again full time or files another application depending on the actual health condition or decides to retire with OAP. The acceptance/rejection of the application depends on the health status or other characteristics such as age, etc. As shown in Figure 2, DP retirement is possible before reaching j _N, so that the two retirement windows may overlap in some periods for specific individuals.

If agents are eligible for OAP benefits they may retire without any delay. At the retirement age j _R, the accumulated earning points $ep_{j_R}$ are multiplied with an adjustment factor ν(j _R) which takes into account early or late retirement, and the annual point value (APV), which reflects the worth of one earning point in pension income. For simplicity, we derive the APV from the replacement rate of a so-called ‘standard pensioner’, who has worked for 45 years and always received an average income:

(5)$$p = \nu ( j_R) \times ep_{\,j_R} \times \underbrace{{\kappa \times \bar{y}/45}}_{{{\rm APV}}}.$$

Since all agents in the model enter the labor market at age 20 (j = 1) and the NRA for OAP in the initial equilibrium is 65 years (j _N = 46), the standard pensioner has worked for 45 years and accumulated ep ₄₆ = 45 earning points at retirement. The adjustment factor at j _N is unity, so that OAP benefits of the standard pensioner are simply computed by $p = \kappa \times \bar{y}$ where κ denotes the replacement rate. For each year of early retirement before j _N, OAP benefits are reduced by 3.6 percent, and for each year of delayed retirement they are increased by 6 percentage points, i.e.:

$$\nu \left( {\,j_R} \right) = \left\{ {\matrix{ {1-\left( {\,j_N-j_R} \right) \times 0.036,} & {\,j_R < j_N} \cr {1 + \left( {\,j_R-j_N} \right) \times 0.060,} & {\,j_R{\unicode{x2A7E}}j_N} \cr } } \right..$$

In principle, the calculation of DP benefits is based on the same formula (5). However, earning points and the adjustment factor are computed slightly differently. Since DP retirement is typically much earlier than OAP retirement (see Table 1), accumulated earning points are upgraded to the MAA j _MA:

$$ep_{\,j_R} = ep_{\,j_R} \times \max \left[{1.0; \;\displaystyle{{\,j_{MA}-1} \over {\,j_R-1}}} \right].$$

In the initial equilibrium, the MAA is set at 60 (j _MA = 41). Someone who receives DP benefits starting at age 50 (j _R = 31) therefore receives a 33 percent increase in accumulated earning points. In addition, the DP adjustment factor ν(j _R) is defined relative to the DRA j _D, reductions of DP benefits are limited to 10.8 percent, and no increase in DP benefits is granted for delayed retirement, i.e.:

$$\matrix{ {\nu ( j_R) } \hfill & = \hfill & {\left\{{\matrix{ {1.0-\min [ 0.108; \;( j_D-j_R) \times 0.036] , \;} \hfill & {\,j_R < j_D} \hfill \cr {1.0} \hfill & {\,j_R {\unicode{x2A7E}} j_D.} \hfill \cr } } \right.} \hfill \cr } $$

3.4 The dynamic optimization problem

To describe the individual decision process we need to distinguish three retirement states rs for each household, which specify the labor market situation in the previous year. If the agent has worked before (rs = 0) he/she needs to decide about retirement (with DP or OAP), consumption, and savings. If the agent has either filed a successful DP application or has already received a DP benefit before (rs = 1), he/she only needs to decide about consumption and savings. Of course, the same applies when the agent was already retired with OAP benefits before (rs = 2), but the benefit calculation is different. Therefore, the current state of a household is described by a vector

$$x = ( j, \;s, \;m, \;rs, \;a, \;ep, \;nc, \;\eta , \;h) $$

where a ∈ [0, ∞] defines the financial assets at the beginning of the period and the remaining variables summarize the household's current age, education, career stability, retirement state, earning points, non-contributory periods, labor productivity, and health. In each period the age-j cohort is fragmented into subgroups, according to the initial distribution at age j = 1 as well as idiosyncratic shocks to productivity and health and optimal household decisions. Let Φ(x) be the corresponding cumulated measure so thatFootnote ¹⁵

$$\smallint {\rm d}\Phi ( x) = 1\quad {\rm with}\;x = ( 1, \;s, \;m, \;0, \;0, \;0, \;0, \;\eta , \;0) $$

must hold since we have normalized the initial cohort size to be unity and endowed initial agents with productivity η. In the following, we will omit the state index x for every variable whenever possible.

Optimal household decisions regarding consumption, savings, and retirement with DP or OAP will be formulated recursively starting in the last phase of life (see Figure 2). Since we abstract from bequest motives, households who have survived until the final age of J simply consume their resources:

$$V( x) = u( c, \;0, \;0) \quad {\rm with}\;c = [ ( 1 + r) a + p-T( p, \;ra) ] /( 1 + \tau ^c) , \;$$

where p either denotes DP (rs = 1) or OAP (rs = 2) benefits, T(p, ra) defines the (progressive) income tax levied on pension income and asset income, and τ ^c is the consumption tax rate (see below).

Younger households who are beyond the maximum retirement age j _E must be also retired (i.e., rs ∈ {1, 2}), but they may still face health shocks that affect their life expectancy. Optimal consumption c(x) and savings a ⁺ (x) are then derived according to

(6)$$V( x) = V_2( x) = \mathop {\max }\limits_{c, a^ + } u( c, \;0, \;0) + \beta \psi _{\,j + 1}( h) {\rm {\mathbb E}}[ {V( x^ + ) \vert h} ] $$

subject to (2) and the budget constraint

$$a^ + { = } ( 1 + r) a + p-T( p, \;ra) -( 1 + \tau ^c) c.$$

Of course, the same also applies to all retired households who are below the maximum retirement age j _E and in one of the absorbing states rs ∈ {1, 2}. Figure 3 illustrates the decision problem of the household. The lower part describes someone who does not apply for DP so that the value function V ₂(x) applies to OAP recipients. The upper part of Figure 3 shows someone who applies for DP, so that the value functions V ₂(x ⁺) refer to DP recipients.

Figure 3. Decision problem of household.

Those who are in or beyond j _N and were still working full time in the previous year (i.e., rs = 0) have to decide whether to continue working or to exit the labor market and receive OAP benefits. In the case of working, they need to account for employment risk which is captured by the (expected) value function

$$V_0( x) = \pi _h^{nw} V_0^{nw} ( x) + ( 1-\pi _h^{nw} ) V_0^w ( x) , \;$$

where the exogenous employment probability $1-\pi _h^{nw}$ is independent of age and increasing with better health. If the household is employed, optimal consumption and saving is derived from

(7)$$V_0^w ( x) = \mathop {\max }\limits_{c, a^ + } u( c, \;0.5, \;0) + \beta \psi _{\,j + 1}( h) {\rm {\mathbb E}}[ {V( x^ + ) \vert \eta , \;h} ] $$

subject to (1)–(4) and the budget constraint

(8)$$a^ + { = } ( 1 + r) a + y-T_p( y) -T( y-T_p( y) , \;ra) -( 1 + \tau ^c) c, \;$$

where T _p(y) defines the contribution to the pension system.Footnote ¹⁶ If the household is not working the same optimization problem (7) is solved to derive $V_0^{nw} ( x)$, but in this case y = l = 0. Since at this age DP application is not possible anymore, the expected value function V ₀(x) is shown in the lower part of Figure 3.

In the case of retirement, the optimization problem (6) applies. The participation decision l(x) is then derived from maximizing

(9)$$V( x) = \hat{V}( x) = \mathop {\max }\limits_l {\mkern 1mu} [ V_0( x) , \;V_2( x) ] , \;$$

shown again in the lower part of Figure 3. If the household decides to retire, the future retirement status changes to the absorbing state rs = 2. Otherwise, he/she remains in rs = 0. The same decision applies to working agents in good health who are in the early retirement window for OAP.

Agents younger than j _N, who are in the medium or bad health state (i.e., h ∈ {1, 2}) and who have worked in the previous year may also consider applying for DP. In the latter case (shown in the upper part of Figure 3) they also face employment risk as in the previous situation. The expected value function is therefore defined by

$$\tilde{V}_1( x) = \pi _h^{nw} \tilde{V}_1^{nw} ( x) + ( 1-\pi _h^{nw} ) \tilde{V}_1^w ( x) .$$

When computing the value functions in the two situations they need to take into account that on the one hand they can work only part-time and face stigma cost during the application period and on the other their application may be rejected with probability q. In the case of employment, optimal consumption and savings are derived from

(10)$$\tilde{V}_1^w ( x) = \mathop {\max }\limits_{c, a^ + } \displaystyle{{c^{1-1/\gamma }} \over {1-1/\gamma }}-\chi + \beta \psi _{\,j + 1}( h) \{ {q{\rm {\mathbb E}}[ {V( x^ + ) \vert \eta , \;h} ] + ( 1-q) {\rm {\mathbb E}}[ {V_2( x^ + ) \vert h} ] } \} $$

subject to (1)–(4) and the budget constraint (8), which now may also include unintended bequest b received at ages younger than 60 (i.e., j < 41). As before, the situation without employment is solved quite similarly but then y = l = 0. Note that $\tilde{V}_1( x)$ denotes the expected value function in case of an application net of stigma cost. If the application is accepted with probability 1 − q, agents receive DP benefits in the next period, they are assigned to the retirement state rs = 1 and face no further productivity uncertainty.Footnote ¹⁷ In case of denial, expected future utility depends on future productivity and health conditions. In case of good health, they may still decide (if they are eligible) to retire early with OAP benefits, or they have to return to the labor market. If they end up again in the medium or bad health state, they may apply again for DP benefits.Footnote ¹⁸

The application decision d(x) depends on individual stigma cost ξ(x). Since the distribution of this cost is given by the age- and education-dependent cumulative function ${\rm \Psi }_\xi$, the probability that an individual in state x decides to apply is

$$P( d = 1\vert x) = {\rm \Psi }_\xi ( \tilde{V}_1( x) -\hat{V}( x) ) .$$

Ignoring stigma cost, the difference $\tilde{V}_1( x) -\hat{V}( x)$ reflects the expected utility gain from a DP application. The term ${\rm \Psi }_\xi ( {\cdot} )$ on the right side then denotes the fraction of households with state x with stigma cost below this utility gain, who will apply for DP. The utility in case of not applying for DP

$$\matrix{ {\hat{V}( x) } \hfill & = \hfill & {\left\{{\matrix{ {\max [ V_0( x) ; \;V_2( x) ] } \hfill & {\,j_B {\rm \leqslant } j < j_N} \hfill \cr {V_0( x) } \hfill & {\,j < j_B.} \hfill \cr } } \right.} \hfill \cr } $$

reflects the options for those in and above the ERA who may either continue to work or receive OAP and the younger ones who can only continue to work. The value function at this age (shown on the left side of Figure 3) before DP application is therefore a weighted average of the value functions with and without disability application:

$$V( x) = P( d = 1\vert x) V_1( x) + ( 1-P( d = 1\vert x) ) \hat{V}( x) , \;$$

where now V ₁(x) (in contrast to $\tilde{V}_1( x)$) also includes the expected stigma cost of applicants.

Finally, the value function of healthy young employees at age j < j _B is given by (7) except that the budget constraints (8) now include unintended bequest b. Table 3 summarizes the final decision problem of an agent in retirement state rs = 0 depending on age and health status.

Table 3. Decision problem for working individuals

When entering the labor market everybody is healthy and working, so there is only the decision problem of how much to consume and save. Those who are younger than j _B and not healthy have to decide whether to file a DP application or work. Of course, healthy agents will always work or may stay at home. The period between the early and the NRA is most interesting. Healthy agents (i.e., where h = 0) with a long contribution record have to choose whether to work or to retire early with OAP. Those with work limitations (i.e., h ∈ {1, 2}) may in principle file a DP application, work or retire (if qualified) with OAP. These are exactly the groups where the government may induce even more instead of less retirement. Finally, at ages between the NRA j _N and the end of the retirement window j _E, all households still working have to decide whether to continue working or receive an OAP.

3.5 The government sector

The government in our model splits into a general budget and a pension system. Both budgets are closed separately. We abstract from public debt and corporate taxes so that the general government expenditure G, which is fixed in absolute terms, is financed by income and consumption taxes, i.e.:

(11)$$T_y + \tau ^cC = G, \;$$

where C defines aggregate consumption and T _y the revenues of income taxation. The latter is computed from

$$T_y = \smallint T( y, \;p, \;ra) {\rm d}\Phi ( x) \quad {\rm with}\;T( y, \;p, \;ra) = 2T_{16}( \tilde{y}/2) + \tau _rra.$$

Due to deferred taxation of pensions, taxable income $\tilde{y}$ is either computed as the difference between gross labor income net of pension contributions $T_p( y) = \tau ^p\min [ y, \;2\bar{y}]$ or – after retirement – as public pensions:

$$\tilde{y} = y-T_p( y) + p.$$

Given taxable income, we apply the German progressive tax code of 2016 T ₁₆( ⋅ ) to labor income and assume that all households are married couples (i.e., full income splitting). Concerning taxable interest income we apply a constant rate τ ^r which reflects the flat capital income tax in Germany.

The pension system pays in every period disability and old-age benefits p(x) to retired households and collects payroll contributions from labor income below the contribution ceiling. The budget of the pension system must be balanced in every period by adjusting the contribution rate τ ^p:

(12)$$\int p ( x) {\rm d}\Phi ( x) = \tau ^p\int {\min } [ y( x) ; \;2\bar{y}] {\rm d}\Phi ( x) .$$

3.6 The production sector

The production sector is populated by firms that employ capital K and effective labor L from perfectly competitive factor markets to produce a single good according to the Cobb–Douglas production technology

$$Y = AK^\alpha L^{1-\alpha }, \;$$

with α being the capital share in production and A the technology parameter. Capital is rented from households through an intermediary at the riskless rate r and depreciates over time with rate δ. Labor inputs are paid the competitive wage w. Factor prices are then determined by marginal productivity conditions, i.e.:

(13)$$w = ( 1-\alpha ) A\left({\displaystyle{K \over L}} \right)^\alpha $$

(14)$$r = \alpha A\left({\displaystyle{K \over L}} \right)^{\alpha -1}-\delta .$$

3.7 Equilibrium conditions

Given a specific fiscal policy, an equilibrium path of the economy has to solve the household decision problems (6), (7), (9), and (10) reflect competitive factor prices (13) and (14) and balance aggregate inheritances with unintended bequests

$$\smallint b( x) {\rm d}\Phi ( x) = \int {\displaystyle{{1-\psi _{\,j + 1}( h) } \over {1 + n}}( 1 + r) a^ + ( x) {\rm d}\Phi ( x) .} $$

Furthermore, in the closed economy aggregation holds,

$$\eqalign{& L = \smallint z( x) l( x) {\mkern 1mu} {\rm d}\Phi ( x) , \;\cr & C = \smallint c( x) {\rm d}\Phi ( x) , \;\cr & K = \smallint a^ + ( x) {\rm d}\Phi ( x) , \;} $$

the budgets of the general government (11) and the pension system (12) are balanced and the goods market clears in every period, i.e.:

$$Y = C + G + ( n + \delta ) K.$$

The computational method to solve the model numerically follows the Gauss–Seidel procedure of Auerbach and Kotlikoff (Reference Auerbach and Kotlikoff1987). We start with a guess for aggregate variables, bequest distribution, and policy parameters. Then we compute factor prices, individual decision rules, and value functions which involve discretization of the state space and interpolation, see Fehr and Kindermann (Reference Fehr and Kindermann2018). Next, we obtain the distribution of households and aggregate assets, labor supply and consumption as well as payroll and consumption taxes to update the initial guesses. The procedure is repeated until the initial guesses and the resulting values of macro variables and policy parameters have sufficiently converged.

4.1 Demographics, health, and productivity

In our model, one period covers one year. Agents therefore start their economic life at age 20 (j = 1) and may receive health shocks after the initial work year. They may start to retire early with OAP at age 63 (j _B = 44) or work until age 70 (j _E = 51) and face a maximum possible life span of 99 years (J = 80). The specification of the retirement window reduces the state space significantly, but this restriction is not very binding. In 2019 only 0.5 percent of retirees retired after age 69, see Deutsche Rentenversicherung (2021). The growth rate of the population is set at n = 0.0065 which reflects the average population growth from 2012 until 2017 and generates a fairly realistic old-age dependency ratio. The ratio of cohorts aged 65+ relative to cohorts at ages 20–64 is 31.6 percent in the model while it was at roughly 34 percent around 2012, see DRV (2021, 288). Finally, the share of college-educated workers ω _s = 0.2373 as well as the share of workers with a stable career ω _m = 0.5 is taken from Kindermann and Püschel (Reference Kindermann and Püschel2021, 24) who base their estimates on administrative data from the German pension insurance (Versichertenkontenstichprobe, 2017).

4.1.1 Health transitions and survival probabilities

Since labor productivity is affected by the health state h, we need to describe the health process over the life cycle first. The model distinguishes three life-cycle phases for health transitions. Appendix 1A shows that disability applications only increase slightly between ages 20 and 44. Here we assume that agents may receive only a bad health shock with probability $\pi _{j, s}^h$ from which they potentially recover, but not until they become 45 and face a new health transition process. Consequently

$$\pi _h( h^ + {\rm \mid }j, \;s, \;h) = \left\{{\matrix{ {\pi_{\,j, s}^h } \hfill & {h = 0, \;h^ + { = }\; 2} \hfill \cr {1-\pi_{\,j, s}^h } \hfill & {h = 0, \;h^ + { = } \;0} \hfill \cr {1.0} \hfill & {h = 2, \;h^ + { = } \;2} \hfill \cr {0.0} \hfill & {{\rm otherwise}, \;} \hfill \cr } } \right.$$

where the probabilities $\pi _{j, s}^h$ are derived as in Fehr et al. (Reference Fehr, Kallweit and Kindermann2013) from an exponential function

$$\pi _{\,j, s}^h = {\rm \varrho }_s \times \exp ( \upsilon _s \times j) $$

with ϱ_s = (0.0018, 0.0005) and υ_s = (0.062, 0.048). This procedure generates bad health probabilities between 0.2 percent and 0.9 percent which in turn leads to realistic disability applications and disability rates close to those reported in Hagen et al. (Reference Hagen, Himmelreicher, Kemptner and Lampert2010) for these two groups.

Appendix 1A documents that in 2019 more than 80 percent of disability applications were submitted by individuals older than age 44. Following Jürges et al. (Reference Jürges, Thiel, Bucher-Koenen, Rausch, Schuth, Börsch-Supan and Wise2015), we applied a principal component analysis to estimate two health transition matrices for each age group from the German sub-sample of the Survey of Health, Ageing and Retirement in Europe. The data selection and more details on our estimation approach are explained in Appendix 1C. The resulting health transition probabilities for the both age and education groups are reported in Tables 4 and 5.

Table 4. Health transition matrix for age group 45–64 (j = 26, …, 45)

Table 5. Health transition matrix for age group 65+ (j = 46, …, 80)

Table 6 in the upper part compares the model results for the two educational groups in three phases of their life cycle. Note that health deteriorates with rising age for both education groups. In each age group considered, college graduates are significantly healthier than high-school graduates. This will determine the difference in life expectancy discussed below. In the lower part of Table 6 the two educational groups are aggregated in each age and health cell considered and the resulting distribution of health states generated by the transition matrices is compared with the respective self-assessed health data for Germany during the period 2008–19 from Eurostat (2021). The model seems to slightly overstate bad health for the oldest group, but overall the calibrated health transition seems to be quite realistic.Footnote ¹⁹

Table 6. Distribution of health status in the data and the model (in %)

* Source: Eurostat (2021), average between 2008 and 2019.

Given the health transition matrices, we can determine the age- and health-dependent survival probabilities ψ _j(h). The calibration starts with the average male survival probabilities $\bar{\psi }_j$ taken from the Human Mortality Database (HMD, 2020). We follow the approach of Kindermann and Püschel (Reference Kindermann and Püschel2021) and compute the health-dependent probabilities according to

$$\psi _j( h) = \displaystyle{1 \over {1 + \exp ( -\iota _h \times {\bar{x}}_j) }}\quad {\rm with}\;\bar{x}_j = {-}\log \left({\displaystyle{1 \over {{\bar{\psi }}_j}}-1} \right).$$

The three parameters $\iota _h$ are calibrated to match exactly the average life expectancy of newborn and 65-year-old men in Germany as well as the difference in life expectancy at age 65 for the two education groups. Table 7 reports these targeted data moments and the resulting education-specific life expectancies.

Table 7. Education-specific and targeted life expectancies

The HMD (2020) reports an average life expectancy of newborn men in Germany of 79.5 years and 83 years for those who have reached age 65. In addition, Luy et al. (Reference Luy, Wegner-Siegmundt, Wiedemann and Spijker2015) report that 65-year-old college graduates in Germany face a 2.5-year higher life expectancy than high-school graduates. We determine the three parameters $\iota _h$ by solving a non-linear equation system in life expectancies. As Table 8 shows, the resulting education-specific difference in life expectancy even amounts to 4.5 years at birth, but then decreases to 2.5 years for those who have reached retirement age. Table 8 reports the values of the exogenously specified and internally calibrated parameters.

Table 8. Parameter values: demographics, health, and productivity

4.1.2 Productivity and employment over the life cycle

Given health transitions and survival probabilities, the productivity process over the life cycle can be calibrated with our simulation model to match empirical targets taken from Kindermann and Püschel (Reference Kindermann and Püschel2021).Footnote ²⁰ As explained before, individual labor income is defined by y = w × z(j, s, m, η, h) × l. We normalize the wage rate to unity in the initial equilibrium and specify the productivity term

$$z( j, \;s, \;m, \;\eta , \;h) = \left\{{\matrix{ {\exp ( e_{\,j, s} + \eta ) \times \theta_{h, s}} \hfill & {{\rm employed\ normal\ earners}\;( 1-\pi_h^{nw} ) } \hfill \cr {\exp ( \eta_0) \times \theta_{h, s}} \hfill & {{\rm employed\ low\ earners}\;( 1-\pi_h^{nw} ) } \hfill \cr 0 \hfill & {{\rm unemployed}\;( \pi_h^{nw} ) , \;} \hfill \cr } } \right.$$

which is similar to the one in Kindermann and Püschel (Reference Kindermann and Püschel2021), but allows for health-related productivity shocks θ _h,s and temporary non-working $\pi _h^{nw}$.

The calibration starts with a guess for the innovation variance $\sigma _{\epsilon , s}^2$ and then proceeds in two steps. In the first step, given the guess for the innovation variance, we calibrate the age-productivity profile e _j,s. Following Kindermann and Püschel (Reference Kindermann and Püschel2021) we assume the following functional form

(15)$$e_{\,j, s} = b_{0, s} + b_{1, s}\displaystyle{{\min ( j, \;j_{M, s}) } \over {10}} + b_{2, s}\left[{\displaystyle{{\min ( j, \;j_{M, s}) } \over {10}}} \right]^2 + b_{3, s}\left[{\displaystyle{{\min ( j, \;j_{M, s}) } \over {10}}} \right]^3, \;$$

which captures both a hump-shaped (j _M,s = ∞) and a stagnating (j _M,s < 60) life-cycle labor productivity profile where productivity is constant from age j _M,s onward. The parameters b _i,s and j _M,s of the polynomials are selected to match specific incomes y _j,s for the normal earner group. More specifically, the profile is combined with health-related productivity effects from Capatina (Reference Capatina2015) and a productivity shock η _s that follows an education-specific AR(1) process with autocorrelation parameters $\hat{\rho }_s$ taken from Kindermann and Püschel (Reference Kindermann and Püschel2021) and innovation variances $\sigma _{\epsilon , s}^2$ specified before. The processes for the two education levels are discretized using a Rouwenhorst method as described in Kopecky and Suen (Reference Kopecky and Suen2010) with seven approximation points.Footnote ²¹ The resulting incomes are then fit to the fixed effects derived by Kindermann and Püschel (Reference Kindermann and Püschel2021).Footnote ²² In the second step, we use the parametrization for the low-earner group provided by Kindermann and Püschel (Reference Kindermann and Püschel2021) in combination with the computed incomes of normal earners to simulate the earnings processes of the complete model and generate an unconditional variance of earnings. If the latter does not match with the targeted values, the iteration starts again with a new guess for the innovation variances. The final unconditional variances exactly match the values reported in Table 8. Table 8 summarizes the resulting parameter values and Appendix 1D shows the match of the age-productivity profiles. Table 8 also reports the exogenous transition probabilities for stable and unstable careers of the two education groups. A fairly high fraction of college graduates start in the low-earner group, but afterward, the chance to transition into a low-earnings episode is very small (<1% for both education groups). Being in a low-income state, however, has quite some persistence. The average duration of a low-earning episode is 6.24 and 3.7 years for high school and for college-educated, respectively.

Finally, the non-working probabilities $\pi _h^{nw}$ are specified as follows: assuming a non-working probability of healthy individuals of 4 percent, the respective probabilities of individuals with medium and bad health are selected to match realistic fractions of long-term (i.e., those with at least 35 years of contributions) and very long-term insured (i.e., those with at least 45 years of contributions) in the data. Appendix 2A shows that the resulting fractions of non-working households increase up to 40 percent for elderly cohorts.

4.2 Preference technology and government parameters

To calibrate the parameters of the utility function we first set the IES γ at 0.5, which is in the range of commonly used parameters in these types of models, see Conesa et al. (Reference Conesa, Kitao and Krueger2009, 33). The time preference rate β is set to 0.98 to calibrate a realistic capital-output share of 330 percent. Labor disutility

$$\chi _{\,j, s} = \zeta _{1, s}j + \zeta _2\max [ 0, \;j-( j_N-1) ] $$

increases with age, but differently in the two age groups. There is an additional utility cost of ζ ₂ if households would still work after the NRA. Consequently, as in Seibold (Reference Seibold2021) or Dolls and Krolage (Reference Dolls and Krolage2023), the NRA acts as an RP that changes with the reforms and affects retirement decisions directly. The parameter ζ ₂ reduces employment after j _N to a minimum and ζ _1,s are calibrated to match the retirement pattern of the two education types for OAP. As Figure 4 shows, the specification of labor market disutility helps to generate with the model the two peaks in retirement observed in the data (when we abstract from retirement before j _B).

Figure 4. Old-age retirement inflow pattern in model and data.

Source: Computed from https://statistik-rente.de for 2014.

Similarly, j _N also acts as an RP for the stigma cost $\xi _{j, s}\sim LN( \mu _{j, s}, \;\sigma _\xi ^2 )$ that are associated with DP claiming. The latter now decreases when the claiming age approaches j _N. Assuming a polynomial form for the expected value

$$\mu _{\,j, s} = \varphi _{0, s}( j_N-j) + \varphi _{1, s}( j_N-j) ^2\quad {\rm with}\;\varphi _{0, s}, \;\varphi _{1, s} {\unicode{x2A7E}} 0, \;$$

φ_0,s are calibrated to match the respective retirement ages for DP. Concerning the two remaining values we set φ_1,1 to zero and calibrate the value φ_1,0 to match the shares of the two skill classes in DP inflows. High-skilled then have lower stigma costs when applying for DP, which can be motivated by the fact that they have lower stress to apply. Finally, the variance $\sigma _\xi ^2$ of the stigma cost is independent of age and education and specified to match the relative share of DP retirees in the data. The resulting age pattern of the DP inflow is discussed in Figure 5.

Figure 5. DP inflows in the model and the data 2014.

Source: Computed from https://statistik-rente.de for 2014.

The capital share in production is set at α = 0.30 to match the capital income share. In addition, we choose a value for the technology parameter A to normalize the wage rate for effective labor to unity. The depreciation rate δ on capital is set at 5 percent which guarantees together with the population growth rate a realistic investment-to-GDP ratio of 21.8 percent for Germany.

Government tax policy in our model reflects the main features of the German tax system. We neglect public debt and corporate taxes and fix the public goods expenditures to 20 percent of output. The German income tax code of 2016 is applied to labor and pension income, that is, the marginal tax rate schedule rises after a basic allowance from 15.8 to 44.3 percent. We assume that the transition toward deferred pension taxation is already finished. Consequently, pension contributions are fully deductible from the tax base while pension benefits are fully taxable.Footnote ²³ Concerning the tax base we apply the German income splitting method to compute individual labor tax revenue. Returns from savings are taxed linearly at the rate of 15 percent. This reflects the quasi-dual income taxation in Germany. The resulting income tax revenue in the initial equilibrium is 10.5 percent of GDP. The budget is balanced by the consumption tax, where the equilibrium tax rate of 16 percent generates a revenue of 9.5 percent of GDP.

Concerning the pension system we consider an initial equilibrium which reflects the situation in Germany before the implementation of the retirement reform 2007, see Table 2. Consequently, j _B and j _N for OAP benefits are set to 44 and 46, respectively and the j _D for receiving DP benefits without deductions is set at 44. Finally, j _MA is set at 41. We assume that all DP applicants in good health are rejected if they file a DP application, while the rejection rate for individuals in medium or bad health shock is set to the average rejection rate, see Deutsche Rentenversicherung (2021). Finally, replacement rates are identical for both pension types and are set at κ = 0.56 which yields a realistic contribution rate. Table 9 summarizes the parameter values and the respective calibration targets or data sources.

Table 9. Parameter values: preferences, technology, and government

4.3 Initial equilibrium

Given the dynamics of health and productivity over the life cycle, our chosen preference, technology, and policy parameters generate an initial equilibrium that reflects some key indicators of the German pension system in the years when the NRA increase was phased in. Table 10 compares some model results with respective microdata in 2016 provided by the research center of the German pension insurance, see FDZ-RV (2019) and Appendix 1E.Footnote ²⁴ More than 75 percent of cohorts aged 60–65 were long-term insured, while only 1.4 percent qualify for very long-term insured. Table 10 also shows that our model captures important empirical facts concerning retirement behavior. On the one hand and quite surprisingly, the average retirement age for OAP is (more or less) independent of skill level in our data. On the other, the retirement age for DP benefits rises significantly with the skill class. As in the data, the average retirement age in the model rises significantly with the educational background, since high-skilled retire later with DP, and high-skilled have a lower share of DP. As already shown in Figure 4, our modeling of labor disutility allows us to match not only the average OAP retirement ages but also to capture the twin peaks of retirement at the ERA and NRA.

Table 10. Long-term insured (in %) and average retirement ages in model and data

* Source: FDZ-RV(2019): SUFVVL2016.

The mapping of the DP retirement is much more complicated since the decision to file a DP application depends on many different factors and parameters, such as current health and employment conditions, retirement incentives, and the stigma cost associated with the application. Nevertheless, our model replicates DP inflow and DP retirement ages quite well. Figure 5 compares the age pattern of DP inflows in the model benchmark and the year 2014. For younger ages, the model generates a lower DP inflow as in the data. This is mostly due to two reasons: first, bad health is very unlikely in that period, and (more importantly) stigma costs at young ages are prohibitively high. However, the shape and the peak of the inflow are captured quite well.

As shown in Table 1, the effective retirement age has increased significantly since 2011. Consequently, the shift in the profile can be easily explained and even improves the calibration. Besides that, the two curves are very similar, especially with the steep increase after age 45.

The first column in Table 11 reports again the average retirement ages from Table 10 as well as some key aspects of the initial equilibrium. Total outlays of the pension system amount to 12.6 percent of GDP, about 7 percent of total pension expenditure is spent for DP benefits and DP pensioners receive on average about 70 percent of average OAP benefits. These results from the model match the budget figure reported in Table 1 quite well.Footnote ²⁵ The numbers describe quite well the situation of the German pension system in the period 2012–13 when Reform 2007 was implemented. Note that only 1.9 and 7.8 percent of 50–65-year-old individuals with a medium and bad health state respectively apply for DP benefits. These rates seem quite low but they reflect economic and psychological factors in the model such as expectations about future recovery, benefits, existing wealth, and the stigma cost of applications.Footnote ²⁶ In combination with the exogenously set rejection rates they determine the pension structure.

Table 11. Economic effects of recent pension reforms

Concerning some key macroeconomic statistics, the capital-output ratio of 330 percent is close to the target value of 320 percent which we derived from official data. Of course, the consumption and investment fractions of GDP are somewhat higher than in the official data, reflecting the fact that Germany is a net exporter and we model a closed economy. The endogenously determined interest rate is at 3.1 percent and the unintended bequests which are distributed lump sum to working cohorts amount to 3.2 percent of GDP. Finally, the budget of the general government is financed by income and consumption taxes and closed by the consumption tax rate which is at 16.2 percent. This rate is not unrealistic, given the very stylized modeling of the income tax system.

5.1 Impact of pension Reforms 2007 and 2018

In this section, we implement successively the reform packages of 2007 and 2018 in the partial (PE) and the GE. Columns 2–4 of Table 11 report the consequences of an increase in the NRAs for OAP and DP benefits by two years. To isolate the impact of the RPs and the tax and price adjustment, the second column shows the effects when the RPs for labor disutility and stigma cost as well as prices and tax rates remain unchanged. As in Li (Reference Li2018) this results in a strong shift toward DP retirement especially at older ages, so that DP applications, budget share, and DP retirement age increase significantly. While relative contributions remain almost constant, increased deductions reduce the aggregate benefits resulting in a surplus of the pension system. Due to lower benefits in old age, people save more, but higher savings are not balanced by higher capital demand. In the third column the RPs for labor disutility and stigma cost are adjusted, so that households now retire later and shift back to OAP retirement. Employment increases significantly compared to the initial equilibrium, but the pension budget as well as the capital market are still not balanced. This highlights the fact that in our model RPs are important for retirement behavior as was already pointed out by Seibold (Reference Seibold2021) or Dolls and Krolage (Reference Dolls and Krolage2023). Finally, column ‘GE effect’ shows the consequences of the reform, when all budgets and markets are balanced by tax and price adjustments. The higher capital stock increases the wage rate and the pension budget is balanced by a reduction in the contribution rate by 1.7 percentage points. Note that the increase in the effective retirement age for OAP benefits only by 0.7 years up to 64.7 years is mainly due to the adjusted RPs.Footnote ²⁷ Similarly, the DP retirement age increases by 1.7 years up to age 54.6, again mainly due to the adjusted labor disutility and stigma cost.Footnote ²⁸ Later retirement of DP pensioners (relative to OAP pensioners who retire now with higher deductions) also increases the relative DP benefit level and the DP budget share slightly. Mainly due to later retirement and higher deductions, the total pension budget decreases significantly to 11.5 percent of GDP and the contribution rate is reduced by 1.7 percentage points. At the macro level, the delay of retirement (and the fall in contributions) increases labor supply, employment, and savings significantly. Although households retire – but less than two years – later, they receive lower pensions compared to before, and hence they save more. Higher wages and employment increase income tax revenues which induces a fall in the consumption tax rate by 1.1 percentage points. Not surprisingly, the reform increases the long-run welfare of all households by about 3.6 percent of initial consumption.

The remaining columns of Table 11 consider Reform 2018 which adds to the previous one the increase in the MAA from 60 to 67 (i.e., j _MA increases from 41 to 48). To better understand the economic effects, we first simulate the resulting increase in benefits while fixing the retirement behavior of DP pensioners at the Reform 2007 level. Consequently, the non-behavioral ‘mechanical effect’ of the reform keeps DP and OAP retirement ages as in the previous simulation but increases the DP benefit ratio as well as the DP share in the pension budget significantly.Footnote ²⁹ Due to expected higher DP benefits the capital accumulation is dampened, the contribution rate increases slightly and factor prices remain roughly constant relative to the previous simulation.

Allowing individuals to adjust their retirement behavior in the column ‘GE effect’ induces a strong shift toward DP retirement. Households now retire earlier so that the OAP retirement age falls slightly and the DP retirement age falls significantly by one year. Not surprisingly, the DP share of the pension budget rises further up to 11.6 percent, which in turn raises the total budget to 12.5 percent of GDP and the contribution rate by one percentage point (relative to the Reform 2007). At the macro level, labor supply now decreases as well as capital accumulation, further reducing the wage by 0.5 percent. The higher DP generosity, therefore, induces a massive increase in DP applications, especially in the medium health state, and (with constant rejection rates) an inflow of DP pensioners at all ages, so that the whole economy suffers from the policy reform. As it turns out, higher contribution and tax rates as well as the lower long-run wage reduce now long-run welfare by 3.3 percent of initial consumption (relative to Reform 2007). At first sight, these consequences of Reform 2018 in the model seem to be very strong. However, they are in line with observed reactions from the empirical studies mentioned above. Closer inspection shows that the reform increases average DP benefits by about 17 percent and applications increase by 36 percent. Consequently, the resulting elasticity of 2.1 is higher than the application elasticity computed by Mullen and Staubli (Reference Mullen and Staubli2016) for Austria, but it is not completely out of range.

5.2 Sensitivity analysis

The results of Table 11 suggest that without further adjustments Reform 2018 will most likely induce a dramatic shift toward disability retirement which reduces the average retirement age, raises the pension budget, and hurts the whole economy significantly. Note, however, that the simulation assumed unaltered rejection rates for DP applicants after the full implementation of Reform 2018. This seems quite unrealistic since on the one hand households in better health will start to apply for DP benefits and on the other one can expect a stricter screening process after the full implementation of the reform. Consequently, the second column of Table 12 shows the consequences when the rejection rate increases from 49 to 60 percent for all applicants (in bad and medium health). Comparing this simulation with the last column of Table 11 (replicated in the first column of Table 12) shows that households now apply later when their health has deteriorated. As a consequence, average DP retirement age increases significantly, DP benefits in the pension budget as well as pension expenditure as a share of GDP decrease. Higher labor supply and lower contribution rates have a positive effect on the overall economy so long-run welfare increases compared to the previous simulation.

Table 12. Sensitivity analysis: alternative rejection rates for DP applications

Changes are reported in % of initial equilibrium if not stated otherwise.

* Changes relative to ‘No reform equilibrium’.

Of course, one might also argue that our benchmark assumption that rejection rates are the same in the bad and the medium health states is somewhat unrealistic. In practice, it is more likely that rejection rates increase with the better health condition of the applicants in all scenarios considered. To analyze the impact of this conjecture, the right part of Table 12 assumes differentiated rejection rates of 55 and 45 percent for the medium and bad health states, respectively. Of course, now fewer (more) households with medium (bad) health apply in the ‘No reform equilibrium’ compared to the initial equilibrium reported in Table 11, but the average rate remains at 49 percent. The average DP retirement age is now a little bit lower, but overall the equilibrium without reforms has hardly changed. Differentiated DP rejection rates also have only minor consequences for the effects of Reforms 2007 and 2018. Comparing the first and the last columns of Table 12 only shows a slightly lower average DP retirement age and DP benefit level, but at the aggregate level there is hardly any difference.

The simulations reported so far did not take into account population aging which might affect the results significantly. According to the main variant of the most recent population projection (Statistisches Bundesamt, 2022), average life expectancy at birth will increase in Germany from currently 79.5 to 84.6 years by 2070. In combination with a modest fertility and migration scenario, this implies an increase in the dependency ratio from currently 31 percent up to more than 46 percent in 2070. To analyze the impact of the older population structure in future years, the left part of Table 13 simulates the economy without and with the two pension reforms with the projected future life expectancy and dependency ratio. The first column (‘No reform equilibrium’) illustrates an economy in 2070 with initial NRAs and unchanged MAA. The contribution rate would then rise to 29 percent and the pension budget would increase to almost 20 percent of GDP.Footnote ³⁰ The two reforms now induce a stronger increase in DP applications and higher inflows. Consequently, the average DP retirement age is lower compared to the benchmark and the DP share in the pension budget is higher. Overall the two reforms have now a stronger impact than in the previous simulations. The contribution rates and the consumption tax rates fall stronger (in Reform 2007) and rise further (in Reform 2018) than in Table 11. Consequently, the welfare effects of the two reforms are also more significant.

Table 13. Sensitivity analysis: aging and fixed DP inflows

Changes are reported in % of initial equilibrium if not stated otherwise.

* Changes relative to ‘No reform equilibrium’.

The right part of Table 13 shows a scenario without aging but with a reduced application elasticity. More specifically, it is now assumed that all households in bad health below age 45 (i.e., j < 26) file a DP application, and 10 percent of them are accepted. Individual decisions on whether to apply or not to apply for DP are restricted to ages 46 and older. This somewhat improves the replication of the data in Figure 5 but due to the higher DP inflow at younger ages, the average DP retirement age decreases to 50 years in the equilibrium without the reforms. Earlier DP retirement increases the share of DP in the pension budget – despite the slightly lower relative DP benefit level – and increases the contribution rate. The two pension reforms considered now only affect DP retirement after age 45. Reform 2018 now increases the average DP retirement age, because there is no reaction of households younger than age 45. Nevertheless, the impact of both reforms on the pension budget as a share of GDP as well as the share of DP in the pension budget are quite similar as in the respective benchmark simulations of Table 11.

As already discussed, the demographic pressure on the pension budget will continuously rise, so further reforms that increase the retirement age will be required.Footnote ³¹ The final sensitivity analysis reported in Table 14 therefore analyzes two modest and two more radical reforms of the OAP system intended to induce households to retire later in the future. The two modest reforms adjust the eligibility criteria for early OAP retirement and are shown in the first and second columns of Table 14. The two remaining policy reforms in Table 14 implement an isolated and a combined increase of the adjustment factors for early retirement from 3.6 to 6.3 percent, which reflects the average actuarial value in Europe and the United States, see Börsch-Supan et al. (Reference Börsch-Supan, Härtl and Leite2018).

Table 14. Sensitivity analysis: alternative OAP reforms

Changes are reported in % of initial equilibrium if not stated otherwise.

Quite surprisingly, Reform 2007 has not adjusted the ERA and/or the required number of contribution years for early OAP benefits. Households who have contributed 35 years could still retire at age 63 when they accept a benefit reduction of 14.4 percent, which partly explains the widespread early retirement in Germany and the very modest increase in the effective OAP retirement age shown in Table 11. The first column of Table 14 increases, therefore, increases the ERA from currently 63 to 65 (i.e., j _B from 44 to 46), while the second column keeps the ERA at 63, but increases the required number of contribution years for early retirement from 35 to 37.

As they are supplements to Reforms 2007 and 2018, the considered scenarios need to be compared with the last column of Table 11. The increase in j _B has a significant effect on retirement behavior. On the one hand there is a shift toward DP applications and DP retirement, while on the other households now retire significantly later with OAP benefits. As a consequence, both the OAP and DP retirement ages increase significantly by 1.1 and 0.4 years, respectively. The total pension budget (and the contribution rate) is hardly affected, but the share of DP benefits in the pension budget rises. Later retirement increases employment and the revenues from income taxes, so that the consumption tax rate could be reduced. The slight reduction in long-run welfare also reflects the lower wage rate implied by the reform.

Keeping the current ERA, but increasing the required number of contribution years for early retirement even induces a stronger shift toward DP retirement. As the second column shows, the OAP retirement age rises in this case only slightly and employment is hardly affected. As a consequence, the pension budget increases slightly as well as the contribution rate. Slightly lower wages and higher consumption taxes now result in a reduction of long-run welfare by 0.4 percent.

The phased increase of the adjustment factor from currently 3.6 up to 6.3 percent in the two remaining columns of Table 14 has a very strong effect. When the increase only applies to OAP benefits, it induces again a strong shift toward DP applications/retirement and later OAP retirement. Consequently, the effective retirement age increases for both groups significantly, while at the same time, the pension budget and contribution rate are reduced. People now work longer and save more, which in turn also increases income tax revenues and lowers the consumption tax rate, so that long-run welfare increases now by 2 percent.

Increasing the adjustment rate for both pension types has the strongest effects on long-run retirement, the pension budget, and the aggregate economy. The effective retirement ages for OAP and DP benefits increase by 1.0 and 1.4 years, respectively. Lower benefits increase aggregate savings and the capital stock and reduce the contribution rate by 2 percentage points. Higher employment and wages boost income tax revenues so that the consumption tax rate decreases further by 1.3 percentage points. Consequently, long-run welfare now increases by even 4.1 percent.

Summing up, three major conclusions emerge from the sensitivity analysis of this subsection: first, the qualitative (and major quantitative) results for the reforms in 2007 and 2018 are quite robust concerning differentiated rejection rates, aging, and lower flexibility concerning DP claiming.Footnote ³² Second, future reforms of OAP eligibility rules and/or benefit calculation may improve the fiscal situation and dampen the long-run pressure on the whole economy, but these OAP reforms typically induce unintended substitution toward DP pensions. Third, a better future response to dampen the negative consequences of Reform 2018 might be an improved ‘activation policy’ for sick elderly with enhanced medical rehabilitation options. In the end, this would affect retirement behavior quite similar to an increase in DP rejection rates.