3.1 Demography

Demographic change in our model is captured by the evolution over time of fertility and survival rates, with the latter determining individuals' expected length of life. Equation (1) expresses the size of the youngest or “newborn” generation at time t relative to the size of the youngest generation at t−1, following among others de la Croix et al. (Reference de la Croix, Pierrard and Sneessens2013). 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)$$N_j^t = sr_{\,j-1}^t N_{\,j-1}^t, \;{\rm for}\;{\rm} j = 2-28$$

The trajectories of both f _t and $sr_j^t$ are taken as exogenous in our modelFootnote ⁶. Figures 5a and 5b show the data. Finally, the population consists of low, medium, and high ability agents:

(3)$$\eqalign{& N_j^t = N_{\,j,L}^t + N_{\,j,M}^t + N_{\,j,H}^t \cr& \quad = (v_L + v_M + v_H)N_j^t} $$

with v _s denoting their respective shares and $\sum_{s = L,M,H}v_s = 1$. Assuming the fertility and survival rates to be equal across ability types, these shares will be constantFootnote ⁷.

Figure 5.

Demographic change in Belgium.

Data source: see Figures 1 and 2. Computational details are provided in our supplementary online Appendix C.1. Note: the year indicated on the horizontal axis of Figure 5a is the first of the 3 years constituting a period in our model.

The fertility rate f _t in Figure 5a is computed as the ratio of the size of the population of actual age 18–20 in a particular 3-year period to its size 3 years earlier. We observe a peak in 1963–65 and 1966–68, when the after-war baby boomers became 18. Both periods reveal a growth rate of the youngest generation by almost 20% compared to the previous period. After a few more periods with f _t > 1 in the 1970s, we observe a decline of the youngest generation (f _t < 1) in the 1980s and 1990s. Later decades show a pattern of dampened oscillation. Figure 5b shows the evolution over time of conditional survival rates, $sr_j^t$, for individuals born in 1905, 1925, 1950, 1975, and 2014. We observe an overall rise, with higher probabilities to survive at higher age, for people born later. The increase is particularly strong at ages 75 and 90. A child born in Belgium can now (unconditionally) expect to reach age 90 with a probability of more than 60%. For a child born in 1950 that probability was less than 30%.

In line with our assumption that only physical capital is internationally mobile, we do not model migration in this paper. Our data for the exogenous fertility rate do, however, capture the impact of migration of individuals not older than 20. The data also count the children of migrants when these children reach age 18–20 (model age 1)Footnote ⁸.

3.2 Households

Individuals of the same generation and ability are grouped in a representative household of unitary mass [Merz (Reference Merz1995), Andolfatto (Reference Andolfatto1996), Boone and Heylen (Reference Boone and Heylen2019)]. As low ability individuals can be involuntarily unemployed, their household will consist both of a fraction of unemployed (u) and a fraction of employed members (1 − u). They pool their income, so consumption is equalized across household members. Medium and high ability households have only employed members.

3.2.1 Instantaneous household utility

The instantaneous utility function of a representative high or medium ability household of age j born at time t depends positively on consumption $c_{j,s}^t$ and leisure time $l_{j,s}^t$ (equation 4). Preferences are logarithmic in consumption and iso-elastic in leisure. The intertemporal elasticity of substitution in leisure is 1/θ. Furthermore, γ _j indicates the relative utility value of leisure versus consumption. It may differ by age. The instantaneous utility function of a low ability household of age j is the same except that only the leisure time of the employed fraction of the household (1 − u _t+j−1) is taken into account (equation 5). Implicitly, leisure of the unemployed fraction is assumed to be neutral for household utility.

(4)$$u\lpar {c_{\,j,s}^t, l_{\,j,s}^t} \rpar = {\rm lnc}_{\,j,s}^t + \gamma _j\displaystyle{{{\lpar {l_{\,j,s}^t} \rpar }^{1-\theta}} \over {1-\theta}}, \;{\rm for}\;s = H,M$$

(5)$$u\lpar {c_{\,j,L}^t, l_{\,j,L}^t} \rpar = {\rm lnc}_{\,j,L}^t + \gamma _j\lpar {1-u_{t + j-1}} \rpar \displaystyle{{{\lpar {l_{\,j,L}^t} \rpar }^{1-\theta}} \over {1-\theta}} $$

with γ _j > 0 and θ > 0 (θ ≠ 1).

3.2.2 Expected lifetime utility

A household that enters the model at time t maximizes expected lifetime utility described by equation (6) subject to its 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.

(6)$$U^t = {\rm }\sum\limits_{j=1}^{28} \beta ^{\,j-1}\pi _j^t u\left( {c_{\,j,s}^t ,l_{\,j,s}^t } \right)$$

with 0 < β < 1, $\pi _1^t = 1$, $0 \lt \pi _j^t = \prod \nolimits_{i = 1}^{j-1} sr_i^t \lt1$ for 1 < j < 29, and $\pi _{29}^t = 0$.

3.2.3 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 (7) to (9) describe the age-dependent time constraints for medium and high ability individuals (s = M, H). Only in the first 4 periods an individual can spend time in post-secondary education next to working and enjoying leisure. From period 5 until 15, time can exclusively be allocated to labor and leisure. From period and age 16 onwards an agent is eligible for public old-age pensions.

(7)$${\rm for}\,j = 1-4\,({\rm age}\,18-29):l_{\,j,s}^t = 1-n_{\,j,s}^t -e_{\,j,s}^t $$

(8)$$\hskip-1.5pc {\rm for}\,j = 5-15\,({\rm age}\,30-62):l_{\,j,s}^t = 1-n_{\,j,s}^t $$

(9)$$\hskip-3pc {\rm for}\,j = 16-28\,({\rm age}\,63-101):l_{\,j,s}^t = 1$$

Equations (10) and (11) relate to low ability individuals. Since these individuals start working earlier than individuals of medium or high ability, they can also leave the labor market earlier. They receive a public pension from period and age 15 onwardsFootnote ⁹. Unemployed low ability individuals cannot choose hours worked or leisure (for them $n_{j,L}^t = 0$). They only have “leisure”. As mentioned before, this does not carry positive utility.

(10)$$\hskip-1.5pc {\rm for}\,j = 1-14\,({\rm age}\,18-59):l_{\,j,L}^t = 1-n_{\,j,L}^t $$

(11)$$\hskip-3pc {\rm for}\,j = 15-28\,({\rm age}\,60-101):l_{\,j,L}^t = 1$$

3.2.4 Budget constraints

Households have varying budget constraints over their life cycle depending on age and innate ability. Equation (12) describes the budget constraint faced by households of high and medium ability during active life (periods 1–15).

(12)$$\eqalign{(1 + \tau _c)c_{\,j,s}^t + a_{\,j,s}^t = & \; (1 + r_{t + j-1})(a_{\,j-1,s}^t + tr_{t + j-1}) + w_{t + j-1}^s \varepsilon _jh_{\,j,s}^t n_{\,j,s}^t (1-\tau _{w,j,s}) \cr & -T_{\,j,s}n_{\,j,s}^t + z_{t + j-1},\;{\rm for}\;s = H,M{\rm,} \;j = 1-15} $$

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 labor income of $w_{t + j-1}^s \varepsilon _jh_{j,s}^t n_{j,s}^t \lpar {1-\tau_{w,j,s}} \rpar$. Underlying factors are the real gross wage per unit of effective labor 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 labor 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,s. Engaging in work, however, also induces costs related to child care and transportation T _j,s. Moreover, if the individuals in a household work more days a week and more weeks per year, implying higher n, these costs will rise. This explains why we have $T_{j,s}n_{j,s}^t$ in the budget constraint. One reason for T to depend on ability is for example the use of company cars, which make transportation cheaper typically for higher ability individuals. A reason for T to depend on age is the need (to pay for) for child care at low j, when households have children.

Next to labor 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 (N _k):

$$tr_k = \displaystyle{1 \over {N_k}}\left[ {\sum\limits_s \sum\limits_{j=1}^{28} (1-sr_j^{k-j} )a_{\,j,s}^{k-j} N_{\,j,s}^{k-j} } \right]$$

From the eligible pension age j = 16 onwards individuals of high and medium ability receive public pension benefits $ppt_{j,s}^t$ (see also section 3.3). Equation (13) presents these individuals' budget constraint.

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

The budget constraint of low ability households at working age (equation 14) looks slightly different. Here, only the employed fraction of the representative household (1 − u _t+j−1) earns a labor income while the unemployed part receives an unemployment benefit related to what they would have earned when employed. The policy parameter b indicates the gross benefit replacement rate.

(14)$$\eqalign{(1 + \tau _c)c_{\,j,L}^t + a_{\,j,L}^t = & \; (1 + r_{t + j-1})(a_{\,j-1,L}^t + tr_{t + j-1}) + bw_{t + j-1}^L \varepsilon _jh_{\,j,L}^t n_{\,j,L}^t u_{t + j-1} \cr & + w_{t + j-1}^L \varepsilon _jh_{\,j,L}^t n_{\,j,L}^t (1-\tau _{w,j,L})\lpar {1-u_{t + j-1}} \rpar \cr & -T_{\,j,L}n_{\,j,L}^t \lpar {1-u_{t + j-1}} \rpar + z_{t + j-1} \cr & {\rm for}\,j = 1-14.} $$

After retirement (from age j = 15 onwards) the budget constraint of low ability households is the same as the one of high or medium ability households.

(15)$$\eqalign{(1 + \tau _c)c_{\,j,L}^t + a_{\,j,L}^t = & \; (1 + r_{t + j-1})(a_{\,j-1,L}^t + tr_{t + j-1}) + ppt_{\,j,L}^t + z_{t + j-1}{\rm,} \cr &{\rm for}\,j = 15-28.} $$

All households 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).

3.3 The pension system

Our model includes a public pay-as-you-go (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) onwards, i.e., respectively actual age 63 or 60. The amount $ppt_{j,s}^t$ they receive at the time of retirement is

(16a)$$ppt_{16,s}^t = rr_s\left\{ {\displaystyle{1 \over {15}}\sum\limits_{\,j = 1}^{15} {w_{t + j-1}^s} \varepsilon_jh_{\,j,s}^t n_{\,j,s}^t (1-\tau_{w,j,s})\prod\limits_{l = j}^{15} w g_{t + l}} \right\},\,{\rm for}\,s = H,M$$

or

(16b)$$ppt_{15,L}^t = rr_L\left\{ {\displaystyle{1 \over {14}}\sum\limits_{\,j = 1}^{14} {w_{t + j-1}^L} \varepsilon_jh_{\,j,L}^t n_{\,j,L}^t (1-\tau_{w,j,L})\prod\limits_{l = j}^{14} w g_{t + l}} \right\}$$

The pension benefit is related to one's own contributions during active life. More precisely, the pensioner receives a fraction of the average of revalued earlier net labor income. In equation (16), rr _s is the net replacement rate, which can differ by ability (income), and wg is the period-wise revaluation factor applied to net labor 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 households the pension amount looks very similar, except for the lower eligibility age of 60 (model age 15). Underlying this, is our assumption that periods of unemployment, as experienced by some household members, are considered equivalent to periods of workFootnote ¹⁰.

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

(17a)$$ppt_{\,j,s}^t = ppt_{16,s}^t \prod\limits_{l = 17}^j p g_{t + l-1},\,{\rm for}\,j \gt 16\,{\rm and}\,s = H,M$$

(17b)$$ppt_{\,j,L}^t = ppt_{15,L}^t \prod\limits_{l = 16}^j p g_{t + l-1},\,{\rm for}\,j \gt 15$$

with pg _k the coefficient that revalues the pension benefit of period k − 1 to k Footnote ¹¹.

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

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

with cr = cr ₁ + cr ₂.

The left side of equation (18) 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 labor income applying contribution rate cr ₁ and by (b) the firms applying cr ₂. In equations (12) and (14), cr ₁ is thus part of the labor tax rate τ _w,j,s. Tailored to institutional reality in Belgium, GPP _t denotes the total resources assigned to pension payments by the government to ensure that equation (18) holds.

3.4 Human capital production

Individuals enter the model at the age of 18 with a predetermined ability-specific endowment of human capital. In equation (19), 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.

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

High and medium ability individuals can engage in higher education to accumulate additional human capital in the first four periods (equation 20a). 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, human capital remains constant (equation 20b). We assume that learning-by-doing offsets depreciation. Constant human capital, however, does not imply constant productive efficiency $h_j^t \varepsilon _j$, as there is still variation in the exogenous age-productivity link ɛ _j.

(20a)$$h_{\,j + 1,s}^t = h_{\,j,s}^t \lpar {1 + \phi_s{(e_{\,j,s}^t )}^\sigma} \rpar \quad {\rm for}\ j = 1-4,s = H,M$$

(20b)$$ \hskip7.55pc = h_{\,j,s}^t \quad \hskip4.77pc {\rm for}\ j \ge 5,s = H,M\fleqno $$

with: 0 < σ < 1, ϕ _H, ϕ _M > 0.

Individuals with low innate ability do not study at the tertiary level. Their human capital remains constant at the initial level:

(21)$$h_{\,j + 1,L}^t {\rm} = h_{1,L}^t. \quad \forall j$$

3.5 Household optimization

Low ability individuals will choose consumption and labor supply to maximize equation (6), taking into account their instantaneous utility function (equation 5), their time and budget constraints (equations 10, 11, 14 and 15) and the human capital process (equations 19 and 21). Individuals of medium and high ability will in addition choose the fraction of time they spend in education when young. They optimize equation (6), taking into account equation (4), and their time and budget constraints and the human capital formation process described by equations (7)–(9), (12), (13), (19) and (20). For details on the optimality conditions, we refer to our supplementary online Appendix A.

3.6 Firms, output and factor prices

Firms act competitively on the output market. The constant returns to scale production function to produce a homogeneous good is given by

(22)$$Y_t = K_t^\alpha \lpar {A_tH_t} \rpar ^{1-\alpha} \quad {\rm with}\quad 0 \lt \alpha \lt 1$$

(23)$$A_t = \lpar {1 + g_{a,t}} \rpar A_{t-1}$$

In equation (22), K _t is the stock of physical capital at time t, while A _tH _t indicates employed labor in efficiency units at that time. Technical progress is labor augmenting and occurs at an exogenous rate g _a,t. Total effective labor H _t is defined in equation (24) as a CES-aggregate of effective labor performed by the three ability groups. In this equation λ is the elasticity of substitution between the different ability types of labor, and η _L, η _M and η _H are the input share parameters. We will impose that they sum to 1.

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

Effective labor supply by the high and medium ability group is specified as

(25)$$H_{s,t} = \sum\limits_{j=1}^{15} N_{\,j,s}^{t-j + 1} n_{\,j,s}^{t-j + 1} h_{\,j,s}^{t-j + 1} \varepsilon _j,\quad {\rm for}\quad s = H,M$$

Our assumption of a competitive labor market for high and medium ability individuals implies that the total supply of effective labor will equal demand and employment for these individuals ($H_{s,t} = H_{s,t}^d$ for s = M, H). For low ability households, however, effective employment is lower than supply ($H_{L,t}^d \lt H_{L,t}$, equation 26). There is unemployment.

(26)$$H_{L,t}^d = (1-u_t)\sum\limits_{\,j = 1}^{14} {N_{\,j,L}^{t-j + 1}} n_{\,j,L}^{t-j + 1} h_{\,j,L}^{t-j + 1} \varepsilon _j$$

(27)$$\hskip-5.62pc {\rm}={\rm } (1-u_t)H_{L,t} $$

This brings us to wage formation and union involvement. For medium and high ability labor, the labor market and wage formation are assumed to be competitive. The total wage cost per unit of effective labor will be equal to the market-clearing marginal labor productivity (equation 28). τ _p is the employer social contribution rate. It includes the contribution cr ₂ to the public pension system.

(28)$$\lpar {1-\alpha} \rpar A_t\left( {\displaystyle{{K_t} \over {A_tH_t}}} \right)^\alpha \eta _s\left( {\displaystyle{{H_t} \over {H_{s,t}}}} \right)^{1/\lambda} = w_t^s \lpar {1 + \tau_p} \rpar {\rm,} \,{\rm for}\,s = H,M$$

For low ability labor, however, wages will be above the competitive level. The existence of minimum wages or union influence are obvious possible explanations. As in Sommacal (Reference Sommacal2006) and Fanti and Gori (Reference Fanti and Gori2011) firms will, therefore, choose the optimizing unemployment rate among low ability individuals (equation 29).

(29)$$\lpar {1-\alpha} \rpar A_t\left( {\displaystyle{{K_t} \over {A_tH_t}}} \right)^\alpha \eta _L\left( {\displaystyle{{H_t} \over {\lpar {1-u_t} \rpar H_{L,t}}}} \right)^{1/\lambda} = w_t^L \lpar {1 + \tau_p} \rpar $$

In the spirit of Boone and Heylen (Reference Boone and Heylen2019), we assume a union which sets the union wage $w_t^L$ in equation (30) as a markup on top of a reference wage. This reference wage is a weighted average of the competitive wage $w_t^{L,c}$, the average wage of the medium and high ability group $\lpar {w_t^M + w_t^H} \rpar /2$ and the unemployment benefit. The competitive wage is the hypothetical wage that would occur if there were no unemployment among low ability households. The weights q ₁, q ₂ and q ₃ sum to 1. We take their values from Boone and Heylen (Reference Boone and Heylen2019). μ is the wage premium which we calibrate.

(30)$$w_t^L = \left( {q_1w_t^{L,c} + q_2\displaystyle{{w_t^M + w_t^H} \over 2} + q_3bw_t^L} \right)\lpar {1 + \mu} \rpar $$

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:

(31)$$\left[ {\alpha {\left( {\displaystyle{{A_tH_t} \over {K_t}}} \right)}^{1-\alpha} -\delta} \right]\lpar {1-\tau_k} \rpar = r_t$$

with δ the depreciation rate of physical capital, and τ _k a tax paid by firms on capital returns. If the net marginal product of capital exceeds the world interest rate, capital will flow in until equality is restored. For a given interest rate, firms will install more capital when employed effective labor increases or the capital tax rate falls.

3.7 Fiscal government

Equation (32) describes the government's budget constraint. Its revenues consist of taxes on labor income paid by workers T _nt, taxes on capital T _kt, employer taxes on labor income T _pt and consumption taxes T _ct. They are allocated to interest payments on outstanding debt r _tB _t, spending on goods and services G _t, pension payments GPP _t, unemployment benefits UB _t and lump-sum transfers Z _t. Note in equation (33) our assumption that the government claims a fraction g of GDP for its expenditures on goods and services. Fiscal deficits explain the issuance of new government bonds (B _t+1 − B _t).

(32)$$B_{t + 1}-B_t = r_tB_t + G_t + GPP_t + UB_t + Z_t-T_{nt}-T_{kt}-T_{ct}-T_{\,pt}$$

with:

(33)$$G_t = gY_t$$

(34)$$UB_t = b\sum\limits_{\,j = 1}^{14} {w_t^L} \varepsilon _jh_{\,j,L}^{t + 1-j} n_{\,j,L}^{t + 1-j} u_tN_{\,j,L}^{t + 1-j} $$

(35)$$GPP_t = {\rm see}\;{\rm equation}\;({\rm 18})$$

(36)$$\eqalign{T_{nt} = & \sum\limits_{s = M,H} {\sum\limits_{\,j = 1}^{15} {\lpar {\tau_{w,j,s}-cr_1} \rpar }} w_t^s \varepsilon _jh_{\,j,s}^{t + 1-j} n_{\,j,s}^{t + 1-j} N_{\,j,s}^{t + 1-j} \cr & + \lpar {1-u_t} \rpar \sum\limits_{\,j = 1}^{14} {\lpar {\tau_{w,j,L}-cr_1} \rpar } w_t^L \varepsilon _jh_{\,j,L}^{t + 1-j} n_{\,j,L}^{t + 1-j} N_{\,j,L}^{t + 1-j}} $$

(37)$$T_{kt} = \tau _kK_t\left[ {\alpha \displaystyle{{Y_t} \over {K_t}}-\delta} \right] = \tau _k\lsqb {\alpha Y_t-\delta K_t} \rsqb $$

(38)$$T_{ct} = \tau _c\sum\limits_s {\sum\limits_{\,j = 1}^{28} {N_{\,j,s}^{t + 1-j}}} c_{\,j,s}^{t + 1-j} $$

(39)$$\eqalign{T_{\,pt} & = (\tau _p-cr_2)\sum\limits_s {w_t^s} H_{s,t}^d {\rm,} \;{\rm with}\;H_{s,t}^d = H_{s,t} \cr & \quad {\rm for}\;{\rm} s = H,M\;{\rm and}\,H_{L,t}^d = \lpar {1-u_t} \rpar H_{L,t}} $$

(40)$$Z_t = z_t\sum\limits_{\,j = 1}^{28} {N_j^{t + 1-j}} $$

Labor income taxes paid by workers τ _w,j,s are progressive. We model this by using a tax function

(41)$$\tau _{w,j,s} = \Gamma \left( {\displaystyle{{y_{\,j,s}} \over {\tilde{y}}}} \right)^\xi + cr_1\;{\rm with}\;s = \lpar {L,M,H} \rpar ,\xi \ge 0,0 \lt \Gamma \le 1$$

where y _j,s is gross labor income of a household of ability s and age j, and $\tilde{y}$ is average gross labor income. As we have mentioned before, the pension contribution rate cr ₁ is a (non-progressive) part of the labor tax rate. As in Guo and Lansing (Reference Guo and Lansing1998) and Koyuncu (Reference Koyuncu2011), ξ and Γ govern the level and slope of the tax schedule. The marginal tax rate $\tau _{w^m,j,s}$ is the rate applied to the last euro earned:

(42)$$\tau _{w^m,j,s} = \displaystyle{{ \partial \lpar {\tau_{w,j,s}y_{\,j,s}} \rpar } \over {\partial y_{\,j,s}}} = \lpar {1 + \xi} \rpar \Gamma \left( {\displaystyle{{y_{\,j,s}} \over {\tilde{y}}}} \right)^\xi + cr_1$$

This means that the marginal tax rate is higher than the average tax rate when ξ>0, i.e., the tax schedule is said to be progressive. Households are aware of the progressive structure of the tax system when making decisions. Note that, consequently, in the budget constraints average tax rates are used, while in the first-order conditions (cf. supplementary online Appendix A) marginal tax rates are used.

3.8 Aggregate equilibrium and the current account

Optimal behavior by firms and households and government spending underlie aggregate domestic demand for goods and services in the economy. Our assumption that the economy is open implies that aggregate domestic demand may differ from supply and income, which generates international capital flows and imbalance on the current account. Equation (43) 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. Aggregate accumulated private wealth is denoted by Ω_t. It can be allocated to physical capital K _t, domestic government bonds B _t and foreign assets F _t (equation 44). At the RHS of equation (43) CA _t stands for the current account in period t. Equation (45) denotes that a surplus on the current account translates into more foreign assets. Equation (46) is the well-known identity relating investment to the evolution of the physical capital stock.

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

with:

(44)$$F_t = \Omega _t-K_t-B_t$$

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

(46)$$I_t = \Delta K_{t + 1} + \delta K_t$$

5.1 Exogenous variables

The exogenous variables in our model concern demography (fertility and survival rates), the world real interest rate (Figure 6), the rate of technical progress (Figure 7), and a set of fiscal and pension policy parameters (Figure 8). We already showed and discussed the demographic variables in section 3.1. Here, we focus on the other ones.

Figure 6.

Annual world interest rate.

Figure 7.

Average annual rate of technical progress.

Figure 8.

Fiscal and pension policy variables.

Data sources and computation: see our supplementary online Appendix C.2. Notes: (1) Earlier data are assumed to be equal to their level in the earliest available year. (2) For details on policy variables not shown here (the revaluation factors wg and pg, and the contribution rates cr1 and cr2), see supplementary online Appendix C.

The assumption of an open economy with perfect capital mobility implies that the net after-tax rate of return on physical capital will always be equal to the (exogenous) world real interest rate r _t. It requires us to introduce data for this interest rate over a very long period of time. To the best of our knowledge, however, this is not readily available. Krueger and Ludwig (Reference Krueger and Ludwig2007) and—more recently—Marchiori et al. (Reference Marchiori, Pierrard and Sneessens2017) have computed highly relevant series using an OLG model and taking into account projections for future demography at the world or OECD level. Their results are fairly similar, but their data do not cover the whole period since 1950. To get data also for the earliest decades, we relied on the US stock market data from Shiller (Reference Shiller2015). Figure 6 includes his cyclically adjusted earnings/price ratio in %. We take it as a proxy for the return to physical capital in the world in the 20th century. Combining this proxy with the simulated real interest rate series for 2000–2050 from Marchiori et al. (Reference Marchiori, Pierrard and Sneessens2017), and smoothing using a third degree polynomial, gives us our world real interest rate.

Figure 7 displays the exogenous rate of labor augmenting technical change ga _t since 1951. Our main source is Feenstra et al. (Reference Feenstra, Inklaar and Timmer2015, Penn World Table 8.1). We used their data for TFP growth until 2011, after a double adjustment. First, a correction was necessary for the different treatment of hours workedFootnote ¹³. Second, we HP-filtered the corrected data to obtain the trend rate of technical change and to exclude cyclical effects. For the years until 2021, we approximate ga _t by the growth rate of productivity per hour worked as projected by the Federal Planning Bureau (2016). As of 2022, we use productivity per worker as advanced by the Belgian Studiecommissie voor de Vergrijzing (2016) as a proxy. The projected 1.5% annual growth rate after 2034 also corresponds to the projection for the rate of technical progress of the 2015 European Commission's Working Group on Ageing.

The evolution over time of the fiscal and pension policy variables in our model is shown in Figure 8. In supplementary online Appendix C.2, more detail is provided on the underlying data sources and computations. We also include the evolution of the public debt to GDP ratio in Figure 8f. We determined the lump sum transfer Z _t in equation (32) such that our simulated model exactly replicates this evolution.

5.2 Backfitting

Our model is calibrated to match as closely as possible the data in 1996–2007. Can it also, after introducing the time-varying values of the exogenous variables, replicate the data in earlier and later periods?

Figure 9 reports the results for six endogenous macro variables since the 1950s or 1960s. The different panels reveal that our model performs very well for the evolution of per capita hours worked (panel b), the capital-output ratio (panel d), and the old-age dependency ratio (panel f). For average annual per capita GDP growth (panel a) the model series is more volatile than the data, mainly due to the assumption of perfect capital mobility, but it captures well the trend observed over a very long time period. In panel c the use of a different scale makes a direct comparison of the data for investment in tertiary education with the simulated model education rate among high ability individuals (e _H) impossible. Indirectly, one can see, however, the same upward trend over time, and even the slight acceleration in the major part of the 1990s. Last but not least, panel e compares the values of the pre-tax Gini coefficient generated by the model with the data. Pre-tax income includes labor income, unemployment benefits, interest income, pensions, and excludes lump-sum transfers. We consider the unemployed members of a low ability household as a separate group. The most important thing to note is the fairly identical stability over time in the data and in our simulated Gini.

Figure 9.

Backfitting.

Data sources: (a) EU Commission, AMECO, series OVGDP and population. Data points before 1960 have been computed from the Penn World Table [8.1, series rgdpna, Feenstra et al., (2015)]. (b) OECD, Historical population data; The Conference Board Total Economy Database, 2018. (c) Enrollment rate in tertiary education: World Bank, World Development Indicators; Barro and Lee (2013). (d) Feenstra et al. (2015): series rgdpna and rkna. (e) Gini coefficient of market income: Solt (2016), (f) the population of 65 or more divided by the population of 18–64: Population forecasts 2015–2060 of the Belgian Federal Planning Bureau and Statistics Belgium.Notes: (1) Reported growth rates are averages over consecutive periods of 3 years, starting in the year indicated on the horizontal axis. (2) The reported model prediction follows from the simulated fractions of time that individuals work (n) and our assumption that potential annual hours are 2080 (see also supplementary online Appendix B). (3) Enrollment rates are multiplied by 5/12 to account for the different age groups used in our model (18–29) versus the data (18–22). (4) We computed the model Gini using Cowell's (2011) lower-bound index GL.

Figure 10 shows the variation in per capita hours by age and ability in 2005–2007. Panel a reports aggregated data by age. Panels b, c, and d break this variable down in its constituting parts by ability. Although not perfect, the match between simulated and true data is very good.

Figure 10.

Backfitting: annual per capita hours worked by age and ability in 2005–2007.

Data sources: Eurostat, Employment rates by sex, age, and education (lfsa_ergaed) and Average usual weekly hours of work by professional status (lfsq_ewhuis); OECD, Labor Force Statistics, Average usual weekly hours worked on the main job. For more details on the construction of the data, see supplementary online Appendix B.

6.1 Baseline simulation and counterfactual scenarios

Our considered demographic changes allow to study four scenarios: a baseline scenario and three counterfactuals. In each of them fiscal and pension policy parameters are kept constant (mostly at their level of 2014). Lump-sum transfers adjust to maintain a constant public debt to GDP ratio from 2014 onwards. Furthermore, the rate of technical progress is in each scenario assumed to manifest itself as projected in Figure 7.

Our “baseline simulation” assumes that the projections for fertility and survival rates in Figure 5, and the projection for the future world real interest rate in Figure 6, become true. The full black lines in the different panels of Figure 11 reveal the implied future size and structure of the population. They confirm for example the strong increase in the old-age dependency ratio in panel b, and the drop in the population at working age in panel e, most dramatically so in 2010–2040, that we also observed in Figures 1 and 2. The alternative “no demographic change” scenario counterfactually assumes that (i) the fertility rate (i.e., the growth rate of the generation of age 18–20) remains constant at its 1948–50 level, and (ii) the unconditional probability to survive is constant at the value that holds for the youngest generation of 1948–50. As such, the “no demographic change” scenario imposes a constant instead of an increasing life expectancy for generations entering our model in 1948 or later. Another assumption (iii) concerns the projected evolution of the exogenous world interest rate. Existing literature is fairly unanimous in its expectation that demographic change induces a lower world real interest rate. The majority of existing studies advances a reduction in the interest rate due to demographic change of about 0.5 to 1%-point by 2025 [see e.g., Krueger and Ludwig (Reference Krueger and Ludwig2007), Ludwig et al. (Reference Ludwig, Schelkle and Vogel2012), Attanasio et al. (Reference Attanasio, Bonfatti, Kitao, Weber, Piggott and Woodland2016), Marchiori et al. (Reference Marchiori, Pierrard and Sneessens2017)]. Among these studies, Ludwig et al. (Reference Ludwig, Schelkle and Vogel2012) project the smallest decline. The reason is their assumption of endogenous human capital and the positive effect of demographic change on human capital accumulation. Only Attanasio et al. (Reference Attanasio, Bonfatti, Kitao, Weber, Piggott and Woodland2016) predict a decline in the interest rate that exceeds 1.5%-points. Building on these existing studies, and taking into account that also in our model human capital is endogenous, we impose in our “no demographic change” counterfactual an interest rate that exceeds the baseline level by 0.5%-points from 2020 onwards. The higher interest rate gap arises gradually, starting from 1993 onwards. Our supplementary online Appendix D.1 shows this imposed counterfactual interest rate scenarioFootnote ^14, Footnote ¹⁵.

Figure 11.

Demographic change: effects on the population and its distribution.

Notes: *The population of 65 or older divided by the population of 18–64. **The population of 18–64 divided by the total adult population (18–101).

The implications of the “no demographic change” assumption for various indicators of the size and the distribution of the population are clear from Figure 11. As one of the main (expected) counterfactual results, we observe in panel c a constant ratio of the population at working age to total population, from about 2010 onwards. Unsurprisingly, the same holds for the old-age dependency ratio in panel b.

Last but not least, Figure 11 also reveals the projected size and distribution of the population in our third and fourth counterfactual simulation. These keep one of the demographic components constant at the level of 1948–50, while baseline projections are imposed for the other. In these two counterfactual simulations, we also adjust the assumed world interest rate (see supplementary online Appendix D.1)Footnote ¹⁶.

6.2 Macroeconomic effects

Arithmetically, i.e. disregarding behavioral adjustments, we found in section 2 that projected demographic change may cut off about 0.4%-points of the annual per capita growth rate in Belgium in the next 25 years. Accounting now for behavioral adjustment, Figure 12a reveals an average per capita growth gap due to demographic change of only 0.29%-points in 2017–2040. This reduced gap clearly points to significant favorable behavioral effects of demographic change. At the same time, however, these effects are not strong enough. Accumulated over 25 years, a remaining 0.29%-points decline of the annual per capita growth rate would cost about 7% of per capita income. The different panels of Figures 13 and 14 provide more insight. Anticipating, we find that falling fertility and (especially) increasing life expectancy stimulate individual labor supply, private savings, and human capital accumulation. Despite these favorable adjustments, future growth remains weak for at least three reasons. First, due to reduced fertility and the implied fall in the population at working age, investment in physical capital declines. Second, in an open economy increased household savings may leave the country (capital outflow), which reinforces the fall in physical capital investment. Third, many of the favorable adjustments have already taken place in previous decades. A comparison of the “fertility constant” counterfactual with the baseline simulation in Figure 12b highlights the key role of reduced fertility as the main driver of lower per capita growth. It explains almost the whole loss. Only a minor part is due to increasing life spans. The positive behavioral effects of higher life expectancy related to hours worked, education, and investment in Figure 14 almost compensate its negative effect resulting from the rapidly increasing number of retirees.

Figure 12.

Per capita growth effects of demographic change.

Figure 13.

Components of per capita growth in the baseline simulation and under constant demography.

Figure 14.

Demographic change: macroeconomic effects.

Figure 13 returns to the decomposition of per capita output growth that we presented at the beginning of section 2. Behavioral adjustment should show up either in changes in labor productivity growth (Figure 13a), for example, due to changes in investment in physical or human capital, or in changes in the growth rate of hours worked per person at working age (Figure 13b). Comparing our baseline simulation for both growth rates with their counterfactual under the assumption of no demographic change, we see that counterfactual growth has almost always been lower since the 1980s. Demographic change thus implied higher productivity growth and increasing hours worked. Since the 2000s, however, the difference has become smaller.

Figure 14 focuses on the behavioral responses themselves. We show our baseline simulation and three counterfactuals for hours worked, education (by those of high ability), gross investment in physical capital and savings (as captured by the accumulated stock of non-human wealth). We also include our results for the current account and for public pension expenditures in % of GDP.

If we first compare the level and evolution of the full black line in the different panels of Figure 14 (baseline simulation) with the interrupted black line (no demographic change), our main findings are the following. Demographic change induces an increase in labor supply and per capita hours worked, an increase in schooling, an increase in savings and the stock of non-human wealth, but lower gross investment, and much higher future pension expenditures. Related to the changes in savings and investment, the current account balance increases (and capital flows out).

Figure 14a shows that the increased participation in higher education is mainly due to longer life expectancy. An overall increase in survival rates encourages individuals to study since it allows individuals to enjoy the returns on their investment during a (much) longer period. Increased returns will include higher labor income before the normal retirement age (which individuals will be more likely to reach) as well as higher pensions after it (which individuals will enjoy for a longer period of time). For the latter effect to materialize, however, it is important that pensions are related to the individual's own labor income and education (see the first-order condition in supplementary online Appendix A).

The increase in human capital and its positive effect on the wage is one element making it attractive for individuals to work more. Another one is the perspective of longer life (including longer life in retirement for a given pension age), which implies the need for more resources at old age to maintain consumption standards. These effects will ultimately lead to increased average annual hours worked per person at working age in Figure 14b, with most of the action after 2010 coming from increased life expectancy. These positive effects will—again—be stronger if increased hours of work also feed through into higher future pensions (see supplementary online Appendix A).

Next to making individuals work more, the need for more resources will also make them save more and accumulate more financial assets (and consume less when young). The dominant positive impact of increasing life expectancy on savings is most clear in Figure 14c. This result supports the hypothesis that its positive effect on savings (due to the fact that people at active age and young retirees will save more) dominates its negative effect (arising from the growing number of dissavers).

The increase in individuals' labor supply and the accumulation of more human capital are two elements that could encourage firms to invest in physical capital. Both elements raise the marginal productivity of physical capital. Figure 14d confirms this. Investment would be the highest (until about 2032) if the fertility rate was kept constant at its level of 1948–50, but life expectancy was free to vary as it did in reality. The reason for the drastic reduction of investment in the baseline simulation is clear then. It follows from the drop in the population at working age (see also Figure 11e). Although individuals optimally raise their labor supply, aggregate labor input will fall due to the lower number of people. The negative effect of this fall on the need for equipment and the productivity of physical capital is an important cause of capital outflow (an increase of the current account and a reduction of net capital inflow, see Figure 11f) and disinvestment.

Another determining factor of aggregate hours worked is the unemployment rate. Our simulations (not shown) reveal no noticeable effect from demographic change (or the size of labor supply) on the unemployment rate. On the one hand, a falling number of workers could provide jobs for those who are initially unemployed. On the other hand, however, the decrease in labor supply will raise competitive wages, which translates into higher wages set by the union in equation (30). Labor demand for low ability workers will then decline also.

The strong increase in the old-age dependency ratio (Figure 11b) due to subsequent relatively smaller youngest cohorts and increasing life expectancy translates in fast-growing public pension expenditures from 2014 onwards. Over 2014–61 demographic change induces public pension expenditures to be on average 1.6% of GDP higher compared with the constant fertility and life expectancy scenario. The maximum increase is about 2% of GDP in the early and mid-2030s. Figure 14e reveals that the main cause up until 2032 is the path of fertility experienced in the past. Later, increased life expectancy is becoming more important.

The results that we showed in Figure 14 are macroeconomic results. To further substantiate our conclusions on the behavioral effects of demographic change, we include in supplementary online Appendix E the results of counterfactual simulations at the level of separate cohorts (generations) of individuals over time. We distinguish cohorts that entered the model at age 18 in 1966, 1981, 1996, and 2011. We observe for most cohorts indeed higher hours worked, increased savings, and increased time allocated to education when young.

6.3 Robustness

We tested the robustness of our results in Figures 12, 13, and 14 to variation in the projected future world real interest rate and future technical change. We simulated and compared new baselines and new “no demographic change” counterfactuals, respectively, assuming in both scenarios a higher world real interest rate (0.5%-points higher by 2030) and a lower rate of technical change (0.5%-points lower by 2035) than shown in Figures 6 and 7. The levels of all endogenous variables obviously change, but also under these alternative projections the net per capita growth effect of demographic change in 2017–2040 is very close to −0.3%-points, while the increase in pension expenditures remains about 1.6% of GDP. Also under these alternative projections, individuals increase their participation in higher education, work more and save more due to demographic change. Capital will again flow out and private investment will be lower.

In the second series of additional simulations, we investigated the robustness of our results for changes in the assumed effect of demographic change on the world real interest rate. Our “no demographic change” counterfactual simulations in Figures 12, 13 and 14 assumed a world interest rate that exceeded the baseline level by 0.5%-points from 2020 onwards. Most existing studies, however, suggest a stronger effect (see our discussion in section 6.1). As an additional robustness simulation we therefore, assumed a 1%-point higher world interest in the “no demographic change” counterfactual. Supplementary online Appendix D.2 demonstrates that our main findings in Figures 12 and 14 did not change.

As a third robustness check, we used the consumption tax rate as a fiscal instrument to close the public budget after 2014 in our simulations (rather than lump-sum transfers). Again, this did not change our results and conclusions.