2.1 Basic setup

We assume an open economy with an exogenous but time-varying world interest rate. Physical capital moves freely across borders. Human capital and labour, however, are assumed internationally immobile. Time-varying exogenous fertility and survival rates drive demographic change. Twenty-eight generations of individuals coexist. Individuals enter the model at the age of 18. They live at most for 28 periods of 3 years. Within each generation one fraction of the individuals is assumed to have low innate ability, others have medium ability, a third group has high innate ability. Depending on their ability, individuals will enter the model with a different initial human capital endowment and with a different productivity of schooling. Young individuals with high or medium ability will continue education when they enter the model at 18. Individuals with low ability will not. Next to education and endogenous human capital, our model also has endogenous employment. Besides studying (for high and medium ability individuals) everyone optimally allocates time to labour and leisure. As to output, domestic firms are modelled to employ physical capital and effective labour under constant returns to scale. Technology is assumed to have exogenous growth.

A central part of our model is the public pension system, the specification of which allows us to simulate a great variety of pension reforms. Finally, the government is an important actor in our model also from the side of fiscal policy. It sets tax rates on labour (both on employees and employers), consumption and capital income. It allocates resources to goods and services and pensions (to finance possible deficits in the public PAYG system). It may also borrow.

Concerning notation, superscript t denotes the time an individual or group of individuals (a generation) enter the model. Subscript j refers to the j-th period of life or, in other terms, the age. It goes from 1 to 28Footnote ⁴. When a subscript s is used, it denotes one of three levels of innate ability: low (L), medium (M) or high (H). Last but not least, time subscripts t added to aggregate variables indicate historical time.

2.2 Demography

Demographic change in our model is captured by time-varying fertility and survival rates, with the latter determining individuals’ expected length of life. Equation (1) expresses the size of the youngest generation at time t relative to the size of the youngest generation at t–1. f _t( > 0) is the time-dependent ‘fertility’ rate in the model.

(1)$$N_1^t = f_tN_1^{t-1}.$$

Equation (2) describes the evolution of the size of a specific generation over time. We denote by $sr_{j-1}^t ( { < 1} )$ the probability for each individual of generation t to survive until model age j conditional on reaching age j − 1. This survival rate is both generation and age-dependent.

(2)$$N_j^t = sr_{\,j-1}^t N_{\,j-1}^t , \;\;\;\;{\rm for\ }j = 2-28.$$

The trajectories of both f _t and $sr_j^t$ are taken as exogenous in our model. Finally, the population consists of low, medium and high ability individuals:

(3)$$N_j^t = N_{\,j, L}^t + N_{\,j, M}^t + N_{\,j, H}^t . $$

Given our assumption in this paper that the fertility and survival rates are equal across ability types, the share of each group will be constant. We assume it equal to one-thirdFootnote ⁵.

2.3 Individuals

2.3.1 Preferences

An individual with ability s (s = L, M, H) reaching age 18 and entering the model at time t maximizes expected lifetime utility described by equation (4) subject to his budget and time constraints (cf. infra). In this equation β is the discount factor and $\pi _j^t$ the unconditional probability to survive until age j.

(4)$$U_s^t = \mathop {\mathop \sum \nolimits }\limits_{\,j = 1}^{28} \beta ^{\,j-1}\pi _j^t u( {c_{\,j, s}^t , \;l_{\,j, s}^t } ) $$

with $0 < \beta < 1, \;\;\pi _1^t = 1, \;\;0 < \pi _j^t = \mathop \prod \nolimits _{i = 1}^{j-1} sr_i^t < 1$ for 1 < j < 29, and $\pi _{29}^t = 0$.

Utility depends positively on consumption $c_{j, s}^t$ and leisure time $l_{j, s}^t$ as shown in equation (5). The intertemporal elasticity of substitution in leisure is 1/θ. The relative utility value of leisure versus consumption is γ _j. It may differ by age.

(5)$$u( {c_{\,j, s}^t , \;l_{\,j, s}^t } ) = lnc_{\,j, s}^t + \gamma _j\displaystyle{{{( {l_{\,j, s}^t } ) }^{1-\theta }} \over {1-\theta }}$$

with γ _j > 0 and θ > 0 (θ ≠ 1).

2.3.2 Time constraints

Every period, an individual is endowed with one unit of time that can be split into hours worked while employed (n), education (e) and leisure (l) depending on age and innate ability. Equations (6)–(8) describe the age-dependent time constraints for medium and high ability individuals (s = M, H). Only in the first four periods an individual can spend time in post-secondary education next to working and enjoying leisure. From period 5 until 15, time can be allocated only to labour and leisure. From period and age 16 on an individual is eligible for public old-age pensions.

(6)$${\rm for}\;{\mkern 1mu} j = 1-4 \ {\mkern 1mu} ( age \,{\mkern 1mu} 18-29) \ \ \ \colon \ \ \ l_{\,j, s}^t = 1-n_{\,j, s}^t -e_{\,j, s}^t , \;$$

(7)$${\rm for}\;{\mkern 1mu} j = 5-15\ {\mkern 1mu} ( age\, {\mkern 1mu} 30-62) \ \ \ \colon \ \ \ l_{\,j, s}^t = 1-n_{\,j, s}^t , \;$$

(8)$${\rm for}\;{\mkern 1mu} j = 16-28 \ {\mkern 1mu} ( age\, {\mkern 1mu} 63-101) \ \ \ \colon \ \ \ l_{\,j, s}^t = 1.$$

Equations (9) and (10) relate to low ability individuals. Since these individuals start working earlier than individuals of medium or high ability, they can also leave the labour market earlier. They receive a public pension from period and age 15 onFootnote ⁶.

(9)$$ {\rm for}\;{\mkern 1mu} j = 1-14 \ {\mkern 1mu} ( age\, {\mkern 1mu} 18-59) \ \ \ \colon \ \ \ l_{\,j, L}^t = 1-n_{\,j, L}^t , \;$$

(10)$${\rm for}\;{\mkern 1mu} j = 15-28 \ {\mkern 1mu} ( age\, {\mkern 1mu} 60-101) \ \ \ \colon \ \ \ l_{\,j, L}^t = 1.$$

2.3.3 Budget constraints

Individuals have varying budget constraints over their life cycle depending on age and innate ability. Equation (11) describes the budget constraint faced by individuals during active life, i.e., at age j = 1 − 15 for individuals of high and medium ability, and age j = 1 − 14 for individuals of low ability.

(11)$$\eqalign{( 1 + \tau _c) c_{\,j, s}^t + a_{\,j, s}^t & = ( 1 + r_{t + j-1}) ( a_{\,j-1, s}^t + tr_{t + j-1}) \cr & \quad + w_{t + j-1}^s \varepsilon _jh_{\,j, s}^t n_{\,j, s}^t ( 1-\tau _{w, j, s}) + z_{t + j-1}.} $$

Disposable income is used to consume $c_{j, s}^t$ and accumulate non-human wealth. We denote by $a_{j, s}^t$ the stock of wealth held by a type s individual at the end of the j-th period of his life. τ _c is the tax rate applied by the government on consumption goods. When individuals assign a fraction $n_{j, s}^t$ of their time to work, with productive efficiency $\varepsilon _jh_{j, s}^t$, they earn a net labour income of $w_{t + j-1}^s \varepsilon _jh_{j, s}^t n_{j, s}^t ( {1-\tau_{w, j, s}} )$. Underlying factors are the real gross wage per unit of effective labour of ability type $s\;( w_{t + j-1}^s )$, an exogenous parameter linking productivity to age (ɛ_j), human capital $( h_{j, s}^t )$, and the average labour income tax rate (τ _w,j,s). The contribution rate cr ₁ of workers to the public pension system is included in τ _w,j,sFootnote ⁷.

Next to labour income, disposable income consists of interest income earned on assets, $r_{t + j-1}a_{j-1, s}^t$ with r _t+j−1 the exogenous world real interest rate, and lump-sum transfers received from the government z _t+j−1. A final source of income are transfers from accidental bequests tr _t+j−1 (plus interest). There are no annuity markets in our model. Transfers are uniformly distributed among the population.

From the eligible age individuals receive public pension benefits $ppt_{j, s}^t$. Equation (12) presents the budget constraint of retirees.

(12)$$\eqalign{ ( {1 + \tau_c} ) c_{\,j, s}^t + &a_{\,j, s}^t = ( {1 + r_{t + j-1}} ) ( {a_{\,j-1, s}^t + tr_{t + j-1}} ) + ppt_{\,j, s}^t + z_{t + j-1} \cr &{\rm for\;}s = M, \;H{\rm , \;\;\;} j = 16-28, \cr &{\rm for\;}s = L{\rm , \;\;\;}j = 15-28.} $$

All individuals in our model are born without assets. They also plan to consume all accumulated assets by the end of their life. A final assumption is that retired individuals cannot have negative assets. Algebraically, $a_{0, s}^t = a_{28, s}^t = 0$ and $a_{j, s}^t \ge 0$ for j > 15 (for s = H, M) or 14 (for s = L).

2.4 Human capital production

Individuals enter the model at the age of 18 with a predetermined ability-specific endowment of human capital. In equation (13), h ₀ stands for the initial time-invariant human capital endowment of a high ability individual. Low and medium ability individuals are respectively endowed with lower human capital stocks ω _Lh ₀ and ω _Mh ₀ with 0 < ω _L < ω _M < ω _H = 1.

(13)$$h_{1, s}^t = \omega _sh_0.$$

High and medium ability individuals can engage in higher education to accumulate additional human capital in the first four periods (equation (14a)). ϕ _s is a positive ability-related efficiency parameter reflecting the productivity of schooling, and σ the elasticity of human capital growth with respect to time input. After the first four periods, their human capital remains constant (equation (14b)). Since individuals with low innate ability do not study, their human capital remains constant from model age 1. Human capital does not depreciate. We have in mind that learning by doing while at work may counteract depreciation.

(14a)$$h_{\,j + 1, s}^t = h_{\,j, s}^t \left( {1 + \phi_s{( e_{\,j, s}^t ) }^\sigma } \right) \;\;\;\;\; {\rm for}\;{\mkern 1mu} j = 1-4, \;\;s = H, \;M$$

(14b)$$\eqalign{\hskip -56pt {= h_{\,j, s}^t &\hskip 1pc{\rm for}\ {\mkern 1mu} j \ge 5, \;\;s = H, \;M} \cr &\hskip -6.9pc{\rm for}\ {\mkern 1mu} j \ge 1, \;\;s = L} $$

with: 0 < σ < 1, ϕ _H, ϕ _M > 0.

2.5 The pension system

Our model includes a public PAYG pension scheme of the defined benefit type that makes pension payments to retirees out of contributions (taxes) paid by current workers and firms. Individuals receive a pension benefit from model age j = 16 (for s = H, M) or j = 15 (for s = L) on, i.e., respectively actual age 63 or 60. The amount $ppt_{j, s}^t$ they receive at the time of retirement is

(15)$$ppt_{J, s}^t = rr_s\left\{{\sum\limits_{\,j = 1}^{J-1} {\,p_j} w_{t + j-1}^s \varepsilon_jh_{\,j, s}^t n_{\,j, s}^t ( 1-\tau_{w, j, s}) \prod\limits_{l = j}^{J-1} w g_{t + l}} \right\}$$

with: J = 15 for s = L and J = 16 for s = M, H and $\mathop \sum \nolimits p_j = 1.$

The pension benefit is related to one's own contributions during active life. More precisely, the pensioner receives a fraction of the weighted average of revalued earlier net labour income. In equation (15), p _j determines the weight of net labour income earned at age j, rr _s is the net replacement rate, which can differ by ability (income), and wg is the period-wise revaluation factor applied to net labour income earned in the past. The pension will rise in the earned wage, the individual's hours of work and his productive efficiency with the latter also increasing in human capital. For retired low ability individuals the pension amount is very similar, except for the lower eligibility age of 60 (model age 15).

After the initial pension payment, the pension benefit may be revalued to adjust for a changed living standard, so $ppt_{j, s}^t$ then becomes

(16)$$ppt_{\,j, s}^t = ppt_{J, s}^t \prod\limits_{l = J}^{\,j-1} p g_{t + l}, \;{\rm for\ j} > {\rm J}$$

with pg _k the coefficient that revalues the pension benefit of period k − 1 to k.

The public pension system's budget identity is as follows:

(17)$$\eqalign{& \sum\limits_{s = M, H} {\sum\limits_{\,j = 16}^{28} {N_{\,j, s}^{t + 1-j} } } ppt_{\,j, s}^{t + 1-j} + \sum\limits_{\,j = 15}^{28} {N_{\,j, L}^{t + 1-j} } ppt_{\,j, L}^{t + 1-j} = \cr & cr\sum\limits_{s = M, H} {\sum\limits_{\,j = 1}^{15} {N_{\,j, s}^{t + 1-j} } } n_{\,j, s}^{t + 1-j} w_t^s \varepsilon _jh_{\,j, s}^{t + 1-j} + cr\sum\limits_{\,j = 1}^{14} {N_{\,j, L}^{t + 1-j} } n_{\,j, L}^{t + 1-j} w_t^L \varepsilon _jh_{\,j, L}^{t + 1-j} + GPP_t} $$

with: cr = cr ₁ + cr ₂.

The left side of equation (17) indicates total pension expenditures at time t. As public pensions are organized on a PAYG basis, this amount is financed by (a) the working population from taxes on their gross labour income applying contribution rate cr ₁ and by (b) the firms applying cr ₂. As we have mentioned before, cr ₁ is part of the labour tax rate τ _w,j,s in equation (11) while cr ₂ is a component of the employers’ social contribution rate (cf. infra). Tailored to institutional reality in Belgium, GPP _t denotes the total resources assigned to pension payments by the government to ensure that equation (17) holds.

2.6 Individual optimization and the role of the pension system

Low ability individuals will choose consumption and labour supply to maximize equation (4), taking into account their instantaneous utility function in equation (5), their time and budget constraints in equations (9)–(12), and the human capital process in equations (13) and (14b). Individuals of medium and high ability will in addition choose the fraction of time they spend in education when young. They optimize equation (4), subject to equations (5)–(8) and (11)–(14b). For details on the optimality conditions, see supplementary online Appendix B.

Building on our discussion of the pension system in the previous section, and equation (15) in particular, our focus here is on the effects on behaviour in earlier periods of life that the specific organization of public pension benefits may have. We first discuss these effects for a given way of financing. Both income and substitution effects occur:

• • For given contribution rates, a higher replacement rate rr raises the return to working (for all ability groups) and to building human capital (for high and medium ability individuals) in earlier periods. It will encourage individuals to work and to invest in education.

• • Changes in the particular weights of the periods that constitute the pension assessment base p _j may modify these incentive effects. The return to working in a particular period rises in the weight attached to that period. A shift in weight from labour income earned when young to labour income earned when older brings strong incentives to work less when young, and to work more and longer when old. This shift also includes an incentive to invest in human capital, due to the reduced opportunity cost of education and the increased return to employed human capital (for given hours worked) at higher age.

• • Pension systems that encourage individuals to work more when middle aged or older also stimulate them to study when young (at least when they have medium or high innate ability). The reason is that an increase in labour supplied during these periods raises the return to education. Following the same logic, an increase in the normal retirement age will also make it more interesting for young individuals to study. Conversely, individuals who invest more in human capital when young will also prefer to work more and longer at higher age. The reason here is that a higher level of human capital raises wages and the return to working.

• • Higher replacement rates do not only bring about substitution effects, however. Raising individuals’ lifetime consumption possibilities, they also cause adverse income effects on labour supply.

Obviously, for a proper assessment of the effects of pension systems and reforms, it is good also to consider the issue of financing. Maybe tax or contribution rates do not changeFootnote ⁸, but of course they can. In this respect, it has been shown in the literature that if an increase of the replacement rate and the future pension benefit is associated with an increase in the tax or contribution rate on labour, the positive effect on labour supply disappears. In most cases, i.e., when the present discounted value of benefits is lower than the value of the contributions, the effect may turn negative (see e.g., Cigno, Reference Cigno2008; Fisher and Keuschnigg, Reference Fisher and Keuschnigg2010). The positive effect on education will not disappear, however. A PAYG pension system with earnings-related benefits will always encourage individuals to invest in education when youngFootnote ⁹. The reason is that when the present value of future benefits is lower than the value of the contributions, an implicit tax structure results that has high tax rates on labour income in the first period of active life and lower tax rates towards the end. This subsidizes human capital formation (see also Kindermann, Reference Kindermann2015). Raising individuals’ future wages, a higher level of human capital will then recreate positive incentive effects for individuals to work when middle aged and older. All these interactions between endogenous labour and endogenous human capital, supplied by individuals of different generations and ability, clearly highlight the need for a larger scale numerical analysis of pension reform.

2.7 Firms, output and factor prices

Identical firms act competitively on output and factor markets. The constant returns to scale production function to produce a homogeneous good is given by

(18)$$Y_t = K_t^\alpha ( {A_tH_t} ) ^{1-\alpha }\quad {\rm with}\quad 0 < \alpha < 1$$

(19)$$A_t = ( {1 + g_{a, t}} ) A_{t-1}.$$

In equation (18), K _t is the stock of physical capital at time t, while A _tH _t indicates employed labour in efficiency units at that time. Technical progress is labour augmenting and occurs at an exogenous rate g _a,t. Total effective labour H _t is defined in equation (20) as a CES-aggregate of effective labour performed by the three ability groups. H _s,t indicates effective labour supply by ability, as specified in equation (21), and λ is the elasticity of substitution between the different ability types. We will impose that the input share parameters η _L, η _M and η _H sum to 1.

(20)$$H_t = \left( {\eta_HH_{H, t}^{1-( {1/\lambda } ) } + \eta_MH_{M, t}^{1-( {1/\lambda } ) } + \eta_LH_{L, t}^{1-( {1/\lambda } ) } } \right) ^{\lambda /( {\lambda -1} ) }$$

with:

(21)$$H_{s, t} = \sum\limits_{\,j = 1}^{15} {N_{\,j, s}^{t-j + 1} } n_{\,j, s}^{t-j + 1} \varepsilon _jh_{\,j, s}^{t-j + 1} , \;\;\;{\rm for}\;s = H, \;M, \;L \;{\rm and\ where\ }n_{ 15, L}^{t- 14} {\rm} = 0.$$

Competitive behaviour implies in equation (22) that firms will hire labour of each ability type up to the point where its marginal productivity is equal to the total wage cost per unit of effective labour. τ _p is the employer's social contribution rate. It includes the contribution cr ₂ to the public pension system.

(22)$$( 1-\alpha ) A_t\left({\displaystyle{{K_t} \over {A_tH_t}}} \right)^\alpha \eta _s\left({\displaystyle{{H_t} \over {H_{s, t}}}} \right)^{1/\lambda } = w_t^s ( 1 + \tau _p) , \;\quad {\rm for}\;s = H, \;M, \;L.$$

Furthermore, firms install physical capital up to the point where the after-tax marginal product of capital net of depreciation equals the exogenous world interest rate r _t:

(23)$$\left[{\alpha {\left({\displaystyle{{A_tH_t} \over {K_t}}} \right)}^{1-\alpha }-\delta } \right]( {1-\tau_k} ) = r_t, \;$$

with δ the depreciation rate of physical capital, and τ _k a tax paid by firms on capital returns. For a given interest rate, firms will install more capital when the amount of effective labour increases or the capital tax rate falls. Pension reform may affect the former.

2.8 Fiscal government

Equation (24) describes the government's budget constraint. Its revenues consist of taxes on labour income paid by workers T _nt, employer taxes on labour income T _pt, taxes on capital T _kt and consumption taxes T _ct. They are allocated to interest payments on outstanding debt r _tB _t, government purchases of goods and services G _t, pension payments GPP _t and lump-sum transfers Z _t. Fiscal deficits explain the issuance of new government bonds (B _t+1 − B _t).

(24)$$B_{t + 1}-B_t = r_tB_t + G_t + GPP_t + Z_t-T_{nt}-T_{pt}-T_{kt}-T_{ct}.$$

Except for G _t and Z _t all revenues and expenditures are determined in a straightforward way within the model (for details, see Devriendt and Heylen, Reference Devriendt and Heylen2020). For G _t, we assume that the government spends a constant amount g per capita, adjusted for technical change, as in equation (25). Lastly, lump-sum transfers Z _t can be adjusted by the government if it sets specific targets on the evolution of public debt.

(25)$$G_t = gN_tA_t.$$

2.9 Aggregate equilibrium and the current account

Equation (26) describes aggregate equilibrium defined for all generations living at time t. The LHS of this equation represents national income. It is the sum of domestic output Y _t and net factor income from abroad r _tF _t, where F _t stands for net foreign assets at the beginning of t. Accumulated foreign assets are part of aggregate private wealth Ω_t, which can also be allocated to physical capital K _t and domestic government bonds B _t (equation (27)). The RHS of equation (26) includes aggregate demand from individuals, firms and the government, while CA _t stands for the current account in period t. Equation (28) denotes that a surplus on the current account translates into more foreign assets. Equation (29) is the well-known identity relating investment to the evolution of the physical capital stock.

(26)$$Y_t + r_tF_t = C_t + I_t + G_t + CA_t$$

with:

(27)$$\hskip1pc{F_t = \Omega _t-K_t-B_t, \;$$

(28)$$CA_t = F_{t + 1}-F_t = \Delta \Omega _{t + 1}-\Delta K_{t + 1}-\Delta B_{t + 1}, \;$$

(29)$$\hskip1pc{I_t = \Delta K_{t+1} + \delta K_t.$$

4.1 Main endogenous variables

We focus our attention on variables related to macroeconomic performance (employment, education, per capita output), financial viability (pension expenditures), welfare and welfare inequality. We report aggregate dynamic effects as well as dynamic effects by ability group and by generation. To measure the welfare effects of policy changes for individuals, we compute the (constant) percentage change in baseline consumption in each period of remaining life that the individual should get to attain the same lifetime utility in the baseline as after the policy shock (following e.g., Buyse et al., Reference Buyse, Heylen and Van de Kerckhove2017; Fried et al., Reference Fried, Novan and Peterman2018)Footnote ¹⁵. Next to welfare effects for individuals of exemplary cohorts, we also report cumulative welfare measures for all current and/or all future cohorts of different abilities. Our most comprehensive aggregate welfare measure reflects the net utility gain from policy reform after the winners have hypothetically compensated all those who lose. We report the consumption volume that is equivalent to this net utility gain as a percentage of GDP in the last period before the policy reform.

All reforms considered are announced in 2014, but implemented in 2029Footnote ¹⁶. From 2014 onwards, individuals and firms can alter their behaviour. For the welfare analysis we consider individuals of eight exemplary cohorts, being the high and the low ability individuals from four different generations. More precisely, the generations observed are individuals of age 18–20 in 2011–13, individuals of age 42–44 in 2011–13, individuals of age 66–68 in 2011–13 and individuals of age 18–20 in 2035–37. Note that individuals of age 18–20 in 2011–13 and especially individuals of age 42–44 in 2011–13 can only partially adjust their behaviour. Individuals of age 66–68 in 2011–13 are not directly affected by the pension reforms. Individuals of age 18–20 in 2035–37 will be living under the new policy regime during their whole life. The reported cumulative welfare measures obviously include all individuals of current and/or future generations of specific abilities, as will be specified below (see also the note below Table 2).

Table 2. Aggregate welfare effects of alternative pension system reforms^a

^a To compute aggregate welfare effects we take three steps. First, we compute for each cohort the present discounted value of the total change in consumption (volume) over life that is required in the baseline to make the cohort equally well off in the baseline as under the policy reform. (The basis of our computation are the data for individuals that we report in Figures 3 and 6. Cohort data follow from taking into account the size (and its expected evolution) of the cohort that individuals belong to. For young individuals the data in Figures 3 and 6 apply to many periods, whereas for the oldest individuals they only apply to one remaining period). Discounting is done to 2011–13, the last period before the announcement of the policy change. For future cohorts, the present value of their required total consumption change is also discounted to 2011–13. Second, we impose that all cohorts within a considered group of cohorts (e.g., all future generations of low ability) who lose under the new policy, are compensated by the winners in that group of cohorts. Third, the present discounted value of the net aggregate consumption gain of all winners after having compensated the losers is expressed in per cent of GDP in 2011–13.

^b The first three data columns include all cohorts of low, medium and high ability; the last four columns only consider cohorts of low or high ability. The results for the medium ability cohorts are fairly close to those for the high ability cohorts (available upon request).

4.2 Increase of the retirement age and reduction of pension benefits

Figures 1–3 and the upper part of Table 2 report the effects of the two most frequently imposed reforms to make the public pension system financially viable: macroeconomic effects (Figure 1), financial effects (Figure 2), welfare effects for eight individuals (Figure 3) and aggregate welfare effects (Table 2). The first reform is an increase of the retirement age. From 2029 onwards, the normal retirement age is extended with one model period of three years. Individuals of medium and high ability will consequently face an exogenous retirement age of 66. Those of low ability will be eligible for a public pension at age 63. The second reform is a reduction in the pension benefits that individuals receive by lowering the public pension replacement rates. We introduce a permanent cut in these replacement rates for new retirees (from 2029 onwards) such that the present value of total savings in public pension expenditures is equal to the value obtained from extending the retirement age. This comes down to a 12.6%-points cut in rr _L, rr _M and rr _H in equation (15).

Figure 1. Macroeconomic effects of pension reform – part 1 (most frequent reforms).

Figure 2. Financial effects of pension reform – part 1 (most frequent reforms).

Figure 3. Welfare effects of pension reform – part 1 (most frequent reforms).

Note: Our welfare measure is the (constant) percentage change in baseline consumption in each remaining period of life that individuals should get to attain the same lifetime utility in the baseline as after the policy shock.

From a macroeconomic perspective, the long-run effects of raising the retirement age are clearly better than those of a reduction in the benefit replacement rate. Compared to the baseline, the former reform implies in the long run higher per capita output (Figure 1a), higher per capita hours worked (Figure 1b), mainly among individuals with medium or high ability (Figure 1d) and among older individuals (Figure 1f), and higher investment in education (Figure 1g). Unsurprisingly, these positive effects only manifest themselves from 2029 onwards. The main factors driving these results are the sudden increase in the active population and the decline in the number of dependent retirees at the macro level, and the perspective of working longer at the individual level. Encouraged by this perspective, young higher (and medium) ability individuals will expand their participation in higher education. The reason is that as their career length increases with three years, the return to education rises (Cervellati and Sunde, Reference Cervellati and Sunde2013). Accumulating more human capital, they will later also earn higher wages, which raises the gain from work and stimulates hours worked. Moreover, given the earnings-related linkage in the pension system, both the higher wage and the increase of hours worked will further bring a higher pension benefit. Last but not least, the drop in consumption taxes in Figure 2b, resulting from a reduction of public pension expenditures to lower levels than in 2014, reinforces the increase in purchasing power even more. The stronger accumulation of human capital and higher labour supply of older workers improve the productivity of physical capital and consequently private investment. We observe in Figure 1h an investment boom with long-lasting effects on the stock of physical capital in 2029.

Before the actual implementation of the higher retirement age, however, some of the effects go in the opposite direction. From 2014 onwards, individuals anticipate that they will have to work one period of three years longer and therefore will supply slightly less labour in earlier periods (intertemporal substitution of labour) with negative effects on private investment, GDP per capita and pension expenditures in per cent of GDP.

Our description of the effects of an extension of the retirement age directly also implies that this reform will be much less applauded by low ability individuals. They are no longer eligible to pension benefits at age 60–62 and should work longer (at an age when their taste for leisure is the highest), but they cannot enjoy the important gains on wages and future pension benefits from having a higher human capital. It then comes as no surprise in the left panel of Figure 3 that (especially middle aged) low ability individuals experience significant negative welfare effects caused by the reduction of leisure time. For high ability individuals, this negative effect is more than compensated by the increased consumption possibilities from higher wages, higher future pension benefits and lower consumption taxes. Although the latter also holds for lower ability individuals, this effect isn't strong enough. If we consider all current and all future generations, net aggregate welfare effects are clearly positive. We observe in Table 2 an increase equivalent to a consumption volume of 6.15% of initial GDP. Consistent with the left panel of Figure 3, Table 2 also reveals the rising welfare inequality that we explained. Considered as a group, current generations of low ability individuals lose (−0.19%), while current generations of high ability individuals win (+0.90%). As to future generations, the difference in welfare effects is less extreme: both high and low ability individuals experience welfare gains. However, high ability individuals again gain the most. They can optimally allocate more time to education to improve their productivity and future wage path. The latter translates (through the earnings-linkage in the public pension formula) into an increasing average pension of high ability retirees relative to low ability retirees (Figure 2d) and consequently leads to rising welfare inequality.

Summarizing, an extension of the retirement age improves long-run macroeconomic performance, the financial viability of the pension system and the welfare of all future generations. However, current generations of low ability individuals experience significant welfare losses, especially those of middle and older age. The less they can optimally adjust labour supply over their life cycle, the more negative these effects are. In addition, future low ability generations experience smaller welfare gains than their high ability counterparts. Rising welfare inequality is the resultFootnote ¹⁷.

The aggregate effects of a permanent reduction by 12.6%-points in the public pension replacement rates for all new retirees (from 2029 onwards) are not positive, quite on the contrary. Both from the perspective of macroeconomic performance and aggregate welfare, reducing everyone's own-earnings-related pension replacement rate is a bad idea. The lower replacement rates affect the utility gain from working and studying, and cause a fall in hours worked by most individuals and a reduction of investment in human capital (Figures 1b and 1g). The induced slow decrease of consumption taxes in Figure 2b (which has positive effects on the gain from work) cannot compensate this. Furthermore, undermining the marginal product of physical capital, lower effective labour will also bring about a (slight) fall in the private investment rate in physical capital (Figure 1h). All these effects lead in Figure 1a to a lower GDP per capita than in our baseline simulation. All current generations experience a loss of welfare in Figure 3 and Table 2. Future generations do not experience significant welfare effects compared to the baselineFootnote ¹⁸. The only aspect in which a reduction in the replacement rates ‘dominates’ an increase in the retirement age relates to inequality. Now both high and low ability individuals are more or less treated equally. Relative to individuals with high ability, the pension level of individuals with low ability even increases (Figure 2d). A major reason is that the former will gain less from their ability to study.

The negative effects that we find when benefit replacements are reduced to ensure the financial viability of the public pension system, contrast with the more positive observations of e.g., Fehr (Reference Fehr2000), Kotlikoff et al. (Reference Kotlikoff, Smetters and Walliser2007), Kitao (Reference Kitao2014) and Laun et al. (Reference Laun, Markussen, Vigtel and Wallenius2019). There are reasons for this difference. First, in our simulations contribution rates to the pension system are unchanged. They cannot fall in parallel with benefits, since it is an important objective to restore or maintain the financial balance of the system in times of ageing. For the marginal gain from work and education this is bad news. Second, in our open economy model the strong increase in savings induced by the reduction of the benefit replacement will largely flow out of the country, rather than raise domestic investment and the physical capital stock. Again, this is bad news for wages and the marginal gain from work. So it is for output.

4.3 Broader parametric pension reform: efficiency and equity

Our simulations in the previous section revealed clearly positive long-run effects on macro-economic performance and aggregate welfare when the retirement age is increased. Welfare inequality rises, however: current generations of low ability individuals experience welfare losses, while those of high ability gain. Also future generations experience an increase in welfare, but again individuals of high ability gain more. In Figures 4–6 we therefore investigate alternative and broader parametric reforms which try not only to improve macroeconomic performance (productive efficiency), but also equity. Our point of reference in these figures is the simulation with increased retirement age.

Figure 4. Macroeconomic effects of pension reform – part 2: productive efficiency and equity?

Notes: (1) increase in the retirement age by 3 years (see also Figures 1–3). (2) In all simulations in Figures 1–3 and in the first simulation in this figure, the weights in equation (15) have constant values of 0.067 (for s = M, H) or 0.071 (for s = L). Here, they are put at zero until age 29, and increased to 0.12 (for s = M, H) or 0.143 (for s = L) from age 48 onwards. The retirement age is not increased in this simulation. (1) + (2) + unconditional minimum pension: this simulation extends (1) + (2) by the introduction of a minimum pension of 40% of average net labour income in the economy. (1) + (2) + change repl. rates: this simulation extends (1) + (2) by increasing the replacement rate rr in equation (15) from 64% to 72% for individuals of low ability and reducing it from 54% to 49% for high ability individuals.

Figure 5. Financial effects of pension reform – part 2: productive efficiency and equity?

Figure 6. Welfare effects of pension reform – part 2: productive efficiency and equity?

The first alternative, indicated as ‘rising accrual rates’ in Figures 4–6 and Table 2, has been inspired by Buyse et al. (Reference Buyse, Heylen and Van de Kerckhove2013, Reference Buyse, Heylen and Van de Kerckhove2017). They argue in favour of a change in the weights p _j attached to past labour income in the calculation of the pension benefit (equation (15)). Labour income earned at older age should generate more pension benefits. Labour income earned at young age should generate fewer pension benefits, reducing the opportunity cost to participation in education and to building human capital. The implication is an accrual rate that rises with age (for a given replacement rate, rr _s). In line with their argument, our third pension reform reduces the weights p _j attached to net labour income earned at age 18–29 for all individuals to zero, and raises these weights for net labour income earned from age 48 onwardsFootnote ¹⁹. In the periods between (age 30–47), the weights remain unchangedFootnote ²⁰. The normal retirement age also remains unchanged in this simulation. Despite that, in the long run aggregate per capita hours worked are not much lower when only accrual rates are increased (compare (1) and (2) in Figure 4b). In the short run, hours worked are even higher. Immediately after announcement of this reform, older workers increase their hours of work, since these will now yield higher pension benefits. The same (but opposite) rationale explains why, on average, younger workers start to work fewer hours. Encouraged by the reduced opportunity cost of education when young, and the higher reward to accumulated human capital when old, individuals of medium and high ability will prefer to study. As a result, the anticipated implementation of increasing accrual rates will raise participation in higher education in Figure 4e much more than an extension of the retirement age. The change in hours worked and education, and their positive effects on the marginal productivity of physical capital, feed through in private investment in physical capital (Figure 4h) and in per capita output (Figure 4a). In the long run, per capita output is as high as when the retirement age is extended. In the short and medium-long run, however, annual per capita output is 1–2% higher.

If we then consider welfare, a comparison of Figure 6a with the left panel of Figure 3 and a comparison of the third row in Table 2 with the first row show much better effects for the current generations (in particular those of middle age) from the reform with accrual rates rising with age than from an extension of the retirement age. Moreover, the former avoids the strong increase in welfare inequality for generations that are directly affected by the reform. The downside, however, are much smaller welfare gains for future generations. We observe in Table 2 an aggregate gain for all future generations equivalent to a present discounted consumption volume of only 1.73% of GDP (compared to 4.80% of GDP after an increase of the retirement age). The main reason is that these future generations will have to finance very high pension expenditures to currently active generations, causing future consumption tax rates to rise. Figure 5b reveals a consumption tax rate that is 3–4%-points higher than when the normal retirement age is lifted. Underlying the significantly higher pension expenditures is, first, the higher number of pensioners (when the retirement age is not increased) and, second, the typical life cycle profile of labour income with individuals earning more when they are older. Attaching higher weights to labour income earned when older will consequently increase average pension benefits. As to welfare effects for current generations and welfare inequality, our results are thus fully in line with the findings of Buyse et al. (Reference Buyse, Heylen and Van de Kerckhove2017). For future generations, however, the results here are much less optimistic. The reason is that Buyse et al. (Reference Buyse, Heylen and Van de Kerckhove2017) did not account for demographic change and ageing (they assumed a constant population and population structure), nor for the increasing life cycle profile of labour income.

A fourth parametric pension reform combines the above-mentioned policies: the extension of the retirement age and pension weights rising with age. This unique pension policy mix exploits the complementarity of both reforms. While both improve macroeconomic performance, the former reform is financially viable, but strongly disadvantages the current generations of low ability. The latter reform has much better welfare effects for current generations of low ability and reduces welfare inequality somewhat, but it is too expensive and impairs the consumption possibilities of future generations (and thus their welfare). Figures 4a and 4b show among the best macroeconomic effects from this combined policy for per capita output and labour. This also holds for education and human capital accumulation (not shown)Footnote ²¹. At the same time, it reduces public pension expenditures in Figure 5a (although less so than when only the retirement age is increased). In Table 2 this policy also brings almost the strongest net aggregate welfare gain if we include all current and future generations. We observe a welfare increase equivalent to a consumption volume of 8.78% of initial GDP. One disadvantage remains, however. Although welfare effects for both current and future generations are better than when only the retirement age is adjusted, high ability individuals still win (much) more than low ability individuals. An additional correction to reduce welfare inequality is therefore needed.

To this end, we simulate two more parametric pension reforms. Starting from pension reform 4, pension reform 5 additionally imposes a minimum pension. Individuals are sure of a pension equal to at least 40% of the average net labour income per worker in the economyFootnote ²². In practice the latter implies an increase in the pension level for the low ability group (see also Figure 5c and 5d), but no ex-ante change for the other two groups. The minimum pension considered here is of the unconditional type (see also Buyse et al., Reference Buyse, Heylen and Van de Kerckhove2017). In a robustness check in Section 4.4, we also consider a minimum pension conditional on hours worked. Pension reform 6 adds to reform 4 an increase in the pension replacement rate of low ability individuals by 8%-points and reduces the pension replacement rate of high ability individuals by 5%-points. An increase in the own-earnings-related replacement rate for the individuals of low ability was another key element in the pension reform preferred by Buyse et al. (Reference Buyse, Heylen and Van de Kerckhove2017).

Compared to reform 4, the introduction of a minimum pension and an increase of the own-earnings-related replacement rate for lower ability individuals are effective in raising these individuals’ welfare and in reducing inequality between low and high ability individuals. Our simulation results reveal significant progress for low ability individuals of working age in Figures 6c and 6d in comparison with 6b. Considering macroeconomic performance (productive efficiency) and the welfare of high ability individuals, however, policy reforms 5 and 6 are very different. The introduction of an unconditional minimum pension strongly undermines the incentive to work for all low ability individuals in Figure 4c. They now receive a public pension benefit unrelated to their own past labour supply and earnings. As a result, aggregate hours worked in Figure 4b are seriously reduced, and so is GDP per capita (Figure 4a). Pension reform 6 retains all the advantages of reform 4, but reduces welfare inequality by strengthening the earnings-related link in the calculation of the public pension benefit for low ability individuals. This reform brings the best results when it comes to hours worked by low ability individuals and older individuals, without affecting hours worked by individuals of high ability. Also the effects on investment in human and physical capital are very positive. Unsurprisingly, together with aggregate hours worked, per capita GDP rises strongly in the medium to long run. In Table 2, policy 6 achieves the best aggregate welfare increase, equivalent to a consumption volume of 8.82% of initial GDP.

4.4 Robustness

In reality, minimum pensions do not have to be of the unconditional type. As an alternative, we considered a minimum pension that is conditioned on the history of individuals’ hours worked. More precisely, the government decides to raise the pension benefit of individuals who under the normal pension system (equation (15)) would not reach a certain threshold. Conditionality is such that the adjusted benefit depends on each individual's average life-cycle hours worked relative to the average life-cycle hours worked by all individuals retiring at that same moment. We define the conditional minimum pension at time t as

(30)$$ppt_{min^c, 16, L}^{t-15} = \displaystyle{{( 1/15) ( {n_{1, L}^{t-15} + \cdots + n_{15, L}^{t-15} } ) } \over {{\tilde{n}}_{lc, t}}}rr_{min}\tilde{y}_t^n , \;$$

$\tilde{y}_t^n$ denotes average net labour income per worker in the economy at time $t$, $\tilde{n}_{lc,t}$ the average life-cycle hours worked by all individuals of all abilities who retire at time $t$ and $rr_{min}$ the minimum pension replacement rate determined by the government. It is set to 0.46 so that the conditional minimum pension is comparable with the unconditional minimum pension of 40% of $\tilde{y}_t^n$ that we assumed above. Note that we use subscript $L$ on the right side of equation (30) as in practice the minimum pension only applies to the low ability group. The more an individual works over the course of his life relative to those who retire at the same moment, the higher will be his pensionFootnote ²³.

The pension reform that adds to reform 4 a conditional minimum pension performs almost as well as reform 6 (see Table 3), although the introduction of this type of pension proves to be somewhat more expensive and is a little less successful in reducing welfare inequality between current generations of low and high ability individuals. This comes as no surprise as in this reform the replacement rate of the high ability group remains untouched. In comparison with the unconditional minimum pension, the conditional minimum pension provides far fewer disincentives to work. Details are available upon request.

Table 3. Aggregate welfare effects of implementing a minimum pension conditioned on hours worked^a

^a Details on the computation of the aggregate welfare effects are provided below Table 2.

^b The first three data columns include all cohorts of low, medium and high ability; the last four columns only consider cohorts of low or high ability.

As a second robustness check, we evaluated whether our main results and conclusions depend on the imposed taste for leisure in model period 16 (γ ₁₆). The alternative values we consider are 0.54 (lower than the γ ₁₆ previously used), and 0.7 (higher). We find that they do not. Based on aggregate welfare effects and on the same criteria as used before, we find that the policy that combines an extension of the retirement age with rising accrual rates and a change in the replacement rates remains our preferred one, irrespective of the value of γ ₁₆. Here also, underlying simulation details are available upon simple request.