2.1 Production

Production occurs within a period according to a neoclassical, constant returns to scale technology. Assuming a Cobb–Douglas production function, output produced at time $t$ , $Y_t$ is given by:

(1) \begin{equation} Y_t=A K_t^{\psi } H_t^{1-\psi } \end{equation}

where $K_t$ and $H_t$ are physical capital and the aggregate quantity of efficiency units of labor employed in production in period $t$ , respectively, $A \gt 0$ is a technological parameter, and $\psi \in (0,1)$ .

The effective labor supply in each period $t$ is given by the number of effective labor units supplied by young workers, that is, $h_tN_t$ and the number of effective labor units from elderly workers at time $t$ , that is, $\lambda h_{t-1} l_t N_{t-1}$ where $h_t$ is the human capital level for each young worker at time $t$ , $l_t$ is the labor supply during the old age, and the parameter $\lambda \in (0,1)$ captures the fraction of youth productivity maintained in old age (see Aísa et al. (Reference Aísa, Pueyo and Sanso2012), Nishimura et al. (Reference Nishimura, Pestieau and Ponthiere2018), and Hirazawa and Yakita (Reference Hirazawa and Yakita2017)). Thus, $\lambda$ close to $0$ implies a significant decline in human capital. Given $N_t=n_{t-1}N_{t-1}$ , the total amount of effective labor at time $t$ is given by:

(2) \begin{equation} H_t=N_t\!\left (h_t+\frac{\lambda h_{t-1} l_t}{n_{t-1}}\right ) \end{equation}

With perfectly mobile capital, the interest rate is equal to the world interest rate $r$ , which is assumed to be exogenous and constant over time. Thus, given a constant-returns-to-scale technology, both capital per unit of effective labor and the wage $w$ per unit of human capital are fixed and constant over time.

In accordance with existing literature, we assume that the level of human capital of children in period $t+1$ , is an increasing, strictly concave function of parental investment in education (or the share of income that parents devote to education).Footnote ¹ In particular, for the sake of simplicity, we specify the following reduced form for human capital production function (Rehme (Reference Rehme2007)):Footnote ²

(3) \begin{equation} h_{t+1}=e_t^{\epsilon } \end{equation}

where $\epsilon \in (0,1)$ .

2.2 Households

Adults (born in period $t-1$ ) derive utility from consumption in adult age, that is, $c^y_t$ , the number of children $n_t$ , the human capital of their offspring $h_{t+1}$ , from consumption $c^o_{t+1}$ , and leisure time $(1-l_{t+1})$ after retirement. Thus, the utility function is given by:

(4) \begin{equation} U^t=\ln c_t^y+\theta \ln n_t h_{t+1}+\beta [\ln c^o_{t+1}+\delta\! \ln (1-l_{t+1})] \end{equation}

where $\beta \in (0,1)$ is the overall weight attached to utility in old age, $\delta \in (0,1)$ is the weight of retirement, and $\theta$ reflects the preferences given to children (quality as well as quantity).Footnote ³

In adulthood, agents apportion their wage between consumption $c^y_t$ , saving $s_t$ , raising their children $n_t$ , investing in their children’s education $e_t$ , and paying a tax $\tau$ in order to contribute to the PAYG pension scheme. Thus, the budget constraint in adulthood is given by:

(5) \begin{equation} c^y_t=wh_t(1-\tau -qn_t-e_tn_t)-s_t \end{equation}

where $q\gt 0$ is the cost of raising each child. In old age, agents consume their savings and if they work they pay a tax to finance the PAYG scheme, whereas they receive a pension benefit $b_{t+1}$ for the period during which they do not work. Thus, the budget constraint in old age is as follows:

(6) \begin{equation} c^o_{t+1}=Rs_t+\lambda w h_{t} l_{t+1}(1-\tau )+b_{t+1}(1-l_{t+1}) \end{equation}

where $R=1+r$ is the rate of return on savings.

The government has to observe a balanced budget and a PAYG social security scheme. Thus, the revenue from taxing both the young, that is, $\tau w h_{t+1}n_tN_t$ , and the elderly, that is, $\tau w\lambda h_t l_{t+1}N_t$ is used to finance retirement pensions. Thus, pension benefits are given by:

(7) \begin{equation} b_{t+1}=\frac{\tau w (h_{t+1} n_t+\lambda h_t l_{t+1})}{1-l_{t+1}} \end{equation}

Specifically, households maximization of the the utility function (4) subject to (3), (5), (6) and $l_t\geq 0$ taking the wage, the interest rate and the pension benefit as given, leads to the following first-order conditions:

(8) \begin{equation} c^o_{t+1}=\beta R c^y_t \end{equation}

(9) \begin{equation} \frac{\theta }{n_t}=\frac{w h_t (e_t+q)}{c^y_t} \end{equation}

(10) \begin{equation} \frac{w h_t n_t}{c^y_t}=\frac{\theta \epsilon }{e_t} \end{equation}

(11) \begin{equation} \frac{w\lambda h_t(1-\tau )-b_{t+1}}{c^o_{t+1}} \leq \frac{\delta }{1-l_{t+1}} (= \; \text{if} \; l_{t+1}\gt 0) \end{equation}

Equation (8) is the standard life cycle saving first-order condition. Equation (9) equalizes the marginal gain in utility of having a child to the marginal loss in utility due to the upbringing and education cost. Equation (10) balances the marginal loss in parental utility from investing in their children’s education against the marginal gain deriving from a higher future human capital. Finally in equation (11), agents choose to work in old age if the marginal utility of leisure in old age is sufficiently low.

After substituting $b_{t+1}$ from equation (7) optimal education, fertility, retirement, and consumption at an interior solution are given by:

(12) \begin{equation} e_t=\frac{q \epsilon }{1-\epsilon } \end{equation}

(13) \begin{equation} n_t=\frac{\theta (1-\epsilon )(1-\tau )(1+\lambda/R)}{q[1+\theta +\beta (1+\delta )]} \end{equation}

(14) \begin{equation} l_{t+1}=\frac{(1-\tau )\{Rqh_t[\lambda (1+\theta +\beta )-\beta R \delta ]-\tau \theta (\lambda +R)[w h_t q \epsilon/(1-\epsilon )]^\epsilon \}}{R\lambda q h_t[1+\theta +\beta (1+\delta )]} \end{equation}

(15) \begin{equation} c^y_t=\frac{w h_t(1-\tau )(R+\lambda )}{R[1+\theta +\beta (1+\delta )]} \end{equation}

Equation (13) shows that a lower depreciation of human capital in old age, that is, a higher value of $\lambda$ , positively affects fertility via a positive income effect (Mizuno and Yakita (Reference Mizuno and Yakita2013)). In other words, a lower human capital decay increases lifetime labor income and therefore enables agents to increase consumption as well as the number of children when young. Moreover, we can observe that, in accordance with the quantity–quality trade-off literature as the return to the investment in education increases, that is, as $\epsilon$ increases for any $\epsilon \in (0,1)$ , the investment in education increases and fertility decreases. From equation (14), it is easy to note that, in accordance with Aísa et al. (Reference Aísa, Pueyo and Sanso2012) and Hirazawa and Yakita (Reference Hirazawa and Yakita2017), in the absence of pension benefits, agents choose to supply labor in old age if $\lambda$ is sufficiently high. On the other hand, the presence of a pension benefit leads to an ambiguous effect of $\lambda$ on $l_{t+1}$ . In fact if, on the one hand, a higher $\lambda$ is a stimulus to remain in the labor market, on the other hand, it increases pension benefits and therefore generates a greater incentive for retirement. Finally, a higher human capital level leads agents to retire later. In fact, if, on the one hand, agents with a higher human capital level earn a higher income in the first period of life, which could lead to an early retirement, they can also earn a higher income in old age, and this acts as an incentive to retire later (see Aísa et al. (Reference Aísa, Pueyo and Sanso2012)).

When the agents’ weight on the preference over leisure when old is sufficiently high, that is, $\delta \gt \underline{\delta }$ , with $\underline{\delta }=[Rq\lambda h_t(1+\theta +\beta )-(w h_t q \epsilon )^{\epsilon }(1-\epsilon )^{1-\epsilon }\tau (1+\lambda )]/R^2 \beta q h_t$ , then a corner solution for elderly labor supply arises, where $l_{t+1}=0$ and:

(16) \begin{equation} e_t=\frac{q \epsilon }{1-\epsilon } \end{equation}

(17) \begin{equation} n_t=\frac{(1-\epsilon )R\theta h_t(1-\tau )}{qRh_t(1+\beta +\theta )-\theta \tau (q \epsilon )^{\epsilon }(1-\epsilon )^{(1-\epsilon )}} \end{equation}

(18) \begin{equation} c^y_t=\frac{w h_t^2(1-\tau )qR}{qRh_t(1+\beta +\theta )-\theta \tau (q \epsilon )^{\epsilon }(1-\epsilon )^{(1-\epsilon )}} \end{equation}

2.3 The planning problem

In this section, we extend the social planner problem developed by van Groezen et al. (Reference van Groezen, Leers and Meijdam2003) by adding endogenous retirement. Thus, the social planner chooses consumption, education, fertility, and retirement age in order to maximize the utility of a dynastic family, subject to the resource constraint of the economy.

DEFINITION 2. The social welfare function is defined as the discounted sum of average lifetime utility of all current and future generations:

(19) \begin{equation} W_t=\sum _{i=t}^\infty \alpha ^{i-t}[\ln c^y_{i-1}+\theta \ln n_{i-1}h_{i}+\beta [\ln c^o_{i}+\delta \ln (1-l_{i})] ] \end{equation}

where $0\lt \alpha \lt 1$ is the social discount factor. Footnote ⁴

Assuming that capital does not depreciate, the total resources of the economy, comprising production and debt creation minus interest payments, are allocated to the consumption of the young and old, the cost of raising children $q$ , the expenditure for children’s education $e_t$ , and domestic investments:

(20) \begin{equation} c_t^y=Rk+w\!\left (h_t+\frac{\lambda h_{t-1}l_t}{n_{t-1}}\right )-kn_t+n_t d_{t+1}-d_t R-n_t w h_t(q+e_t)-\frac{c^o_t}{n_{t-1}} \end{equation}

where $k=K/N$ is the capital per young worker and $d_t$ is the per capita foreign debt of the country at time $t$ .Footnote ⁵

The first-order conditions of the social planning problem are therefore given by:

(21) \begin{equation} \frac{c^y_{t+1}}{c^y_{t}}=\frac{\alpha R}{n_t} \end{equation}

(22) \begin{equation} \frac{c^o_{t+1}}{c^y_{t}}=\beta R \end{equation}

(23) \begin{equation} \frac{c^y_t (\theta +\beta )}{n_t}=(q+e_t)wh_t+ k-d_{t+1}+\frac{\alpha \lambda w h_t l_{t+1}}{n^2_t} \end{equation}

(24) \begin{equation} \frac{\theta \epsilon c^y_t}{e_{t}}=n_t w h_t-\alpha w h'_{\!\!t+1}[1-(q+e_t)n_t]-\alpha ^2\frac{w \lambda h'_{\!\!t+1}l_{t+1} }{n_t} \end{equation}

(25) \begin{equation} \frac{w \lambda h_t}{c^o_{t+1}}\leq \frac{\delta }{1-l_{t+1}} (= \; \text{if} \; l_{t+1}\gt 0) \end{equation}

where $h'_{\!\!t+1}=\partial h_{t+1}/\partial e_t$ . Equations (21) and (22) show, respectively, the optimal intergenerational allocation, that is, the equality of the marginal utility of the young generation in two different periods, and the optimal intragenerational allocation, that is, the equal marginal utility of consumption of the young and old generations. Equation (23) states that the marginal benefit of having a child should be equal to the marginal loss in utility, including the wage, the time, and the education cost. Equation (24) equates the marginal cost of education with the marginal utility gain from higher human capital for children, which includes not only the conventional gains from the marginal product of human capital but also labor earnings in adulthood and old age for a given retirement period. Note that in old age, this effect is mitigated by the decay of human capital (Nishimura et al. (Reference Nishimura, Pestieau and Ponthiere2018)). Equation (25) indicates that a corner solution for labor supply in old age arises if the marginal cost of leisure in old age is lower than its marginal utility.

For the solution to be a social optimum, we assume (for technical details see Appendix A):

Thus, we can now characterize the social planner’s steady state:

PROPOSITION 1. If the preference for leisure time in old age is sufficiently low, that is, $\delta \leq \hat{\delta }$ , with $\hat{\delta }=\lambda [\alpha (1+\theta +\beta )(1+\alpha \epsilon )-\theta (1-\epsilon +\alpha \epsilon )]/\alpha \beta R (1-\epsilon +\alpha \epsilon -Rq)$ , then there exist an interior steady-state first-best optimum given by:

(26) \begin{equation} n^{**}=\alpha R \end{equation}

(27) \begin{equation} c^{y**}=\frac{\alpha w h^{**}\{(\lambda +R)[1-\epsilon (1-\alpha )]-R^2q\}}{R \{\epsilon \alpha ^2[1+\theta +\beta (1+\delta )]+\alpha [1+\theta (1-\epsilon )+\beta (1+\delta -\delta \epsilon )]-\theta (1-\epsilon )\}} \end{equation}

(28) \begin{equation} l^{**}=1- \frac{\beta \delta \alpha }{\lambda }\frac{(\lambda +R)[1-\epsilon (1-\alpha )]-R^2q}{\{\epsilon \alpha ^2[1+\theta +\beta (1+\delta )]+\alpha [1+\theta (1-\epsilon )+\beta (1+\delta -\delta \epsilon )]-\theta (1-\epsilon )\}} \end{equation}

(29) \begin{equation} e^{**}=\epsilon \frac{\alpha [(\lambda +R)(1+\theta +\beta )+R^2q(\beta \delta +\theta )]-R^2q\alpha ^2[1+\theta +\beta (1+\delta )]-R^2q\theta }{R^2 \{\epsilon \alpha ^2[1+\theta +\beta (1+\delta )]+\alpha [1+\theta (1-\epsilon )+\beta (1+\delta -\delta \epsilon )]-\theta (1-\epsilon )\}} \end{equation}

By contrast if $\delta \geq \hat{\delta }$ then a corner solution with full retirement arises:

(30) \begin{equation} n^{**}=\alpha R \end{equation}

(31) \begin{equation} c^{y**}=\frac{\alpha w h^{**}[1-qR-\epsilon (1-\alpha )]}{\epsilon \alpha ^2(1+\theta +\beta )+\alpha [1+\beta +\theta (1-\epsilon )]-\theta (1-\epsilon )} \end{equation}

(32) \begin{equation} e^{**}=\frac{\epsilon [\alpha (1+\theta +\beta +Rq\theta )-Rq\alpha ^2(1+\theta +\beta )-Rq\theta ]}{R[\epsilon \alpha ^2(1+\theta +\beta )+\alpha [1+\beta +\theta (1-\epsilon )]-\theta (1-\epsilon )]} \end{equation}

where $h^{**}={e^{**}}^{\epsilon }$ .

Proof. The proof follows immediately from the first-order condition of the social planning problem given by equations (21)–(25).

3.1 Corner solution

Let us first analyze potential government policies to implement the first-best allocation in a decentralized economy in the case of a corner solution.

Government runs a balanced budget. For simplicity, we assume that it imposes two payroll taxes, $\tau$ and $\eta$ , the first to finance policies for the elderly and the second to finance policies for the children. The latter consists of a subsidy $s_N$ per child in order to finance a fraction of the child rearing costs $q$ and a subsidy $s_H$ in order to cover a share of the educational cost. Therefore, agent’s budget constraint in adulthood becomes (see Stauvermann and Kumar (Reference Stauvermann and Kumar2018)):

(33) \begin{equation} c^y_t= wh_t[1-\tau -\eta -q(1-s_N)n_t-e_t(1-s_H)n_t]-s_t \end{equation}

Assuming a balanced budget regime in each period, the following equation in per capita terms must hold

(34) \begin{equation} (s_N q+s_H e_t)n_t=\eta \end{equation}

From equations (4), (6), (7), (33), and (34), the optimal investment in education, fertility, and consumption at the steady state are as follows:

(35) \begin{equation} e^*=\frac{(1-s_N)q\epsilon }{(1-s_H)(1-\epsilon )} \end{equation}

(36) \begin{equation} n^*=\frac{(1-\tau )R(1-\epsilon )(1-s_H)\theta }{qR\left \{(1+\beta )(1-s_H)(1-s_N)+\theta [1-\epsilon s_N-s_H(1-\epsilon )]\right \}-\theta \tau (1-s_H)(1-\epsilon )} \end{equation}

and:

(37) \begin{equation} c^{y*}=\frac{wh^*(1-\tau )(1-s_N)(1-s_H)Rq}{\left \{(1+\beta )(1-s_H)(1-s_N)+\theta [1-\epsilon s_N-s_H(1-\epsilon )]\right \}-\theta \tau (1-s_H)(1-\epsilon )} \end{equation}

Note that as a result of the quantity–quality trade-off between the number of children and their level of education, the introduction of child benefit increases the number of children but negatively affects the optimal investment in education (Stauvermann and Kumar (Reference Stauvermann and Kumar2018)). In fact, if, on the one hand, the introduction of a child allowance reduces the direct cost of child rearing, on the other hand, it increases the relative cost of education, leading parents to decrease the investments in their children’s education. For the same reasons, the introduction of an educational subsidy leads to a decline in the number of children and increases the investment in human capital. This suggests that neither education subsidization nor child benefit can achieve the social optimum on their own. The following proposition characterizes the optimal policy combination which allows to realize the first-best solution in a market economy.

PROPOSITION 2. Assuming $\alpha \geq \alpha _0$ , the optimal government policies, such that the market allocation is the same as the socially optimal allocation, are $\tau =\tau _c^*$ , $s_N=s_{N_c}^*$ , and $s_H=s_{H_c}^*$ :

(38) \begin{equation} \tau _c^*=\dfrac{\alpha [(1+\beta )\epsilon +Rq(1+\beta +\theta )]-(1-\epsilon )\theta }{\alpha ^2\epsilon (1+\beta +\theta )+\alpha [1+\beta +\theta (1-\epsilon )]-\theta (1-\epsilon )} \end{equation}

(39) \begin{equation} s_{N_c}^*=\dfrac{Rq\alpha ^2\epsilon (1+\theta +\beta )+\alpha \{Rq[1+\beta +\theta (1-\epsilon )]-\theta \epsilon (1-\epsilon )\}-\theta (1-\epsilon )^2}{Rq\{\alpha ^2\epsilon (1+\theta +\beta )+\alpha [1+\beta +\theta (1-\epsilon )]-\theta (1-\epsilon )]\}} \end{equation}

and:

(40) \begin{equation} s_{H_c}^*=\dfrac{\alpha [1+\beta +Rq\theta +\theta (1-\epsilon )]-Rq\alpha ^2(1+\theta +\beta )-\theta (1-\epsilon )}{\alpha (1+\theta +\beta +Rq\theta )-Rq\alpha ^2(1+\theta +\beta )-Rq\theta } \end{equation}

where under Assumption 1 $s_{N_c}^*\lt 1$ and $s_{H_c}^*\lt 1$ .

Proof. The first-best solution given by equations (30), (31), and (32) coincides with the market solution given by equations (35), (36), and (37) if $\tau =\tau _c^*$ , $s_N=s_{N_c}^*$ , and $s_H=s_{H_c}^*$ .

Proof. Under Assumption 1, from equations (39) and (40) $s_{N_c}^*\gt 0$ if $\alpha \gt \hat{\alpha _c}$ with $\hat{\alpha _c}\gt \alpha _0$ and $s_{H_c}^*\gt 0$ if $\alpha \gt \bar{\alpha _c}$ with $\bar{\alpha _c}\lt \alpha _0$ . See Appendix B for more technical details.

3.2 Interior solution

We now derive socially optimal government policies which allow us to replicate the first-best allocation in a competitive equilibrium when agents choose to supply labor in old age.

At the interior solution, education subsidization and child benefit are not enough to ensure that the elderly labor supply reaches the first-best level; thus, the government needs to introduce a subsidy on elderly labor supply, that is, $\tau ^o\gt 0$ . This implies that agents’ budget constraints in adulthood are given by equation (33) and, in old age, by:

(41) \begin{equation} c^o_{t+1}=Rs_t+w \lambda h_t l_{t+1}(1+\tau ^o)+b_{t+1}(1-l_{t+1}) \end{equation}

Under the assumption that the government uses a fraction $0\lt a\lt 1$ of social security tax revenues for pension benefits and a fraction $1-a$ for elderly labor supply subsidy, the internal allocation of government expenditure is, therefore, given by:

(42) \begin{equation} b_{t+1}=\frac{a \tau w n_t h_{t+1}}{1-l_{t+1}} \end{equation}

and:

(43) \begin{equation} \tau ^o=\frac{(1-a)\tau n_t h_{t+1}}{\lambda h_tl_{t+1}} \end{equation}

From equations (4), (33), (34), (41), (42), and (43), the optimal investment in education, fertility, and consumption are as follows:

(44) \begin{equation} e^*=\frac{(1-s_N)q\epsilon }{(1-s_H)(1-\epsilon )} \end{equation}

(45) \begin{equation} n^*=\frac{\theta (1-s_H)(1-\epsilon )[R(1-\tau )+\lambda (1+\tau ^o)]}{qR\{(1-s_N)(1-s_H)[1+\beta (1+\delta )]+\Psi \}} \end{equation}

(46) \begin{equation} c^{y*}=\frac{wh_t(1-s_N)(1-s_H)[R(1-\tau )+\lambda (1+\tau ^o)]}{R\{(1-s_N)(1-s_H)[1+\beta (1+\delta )]+\Psi \}} \end{equation}

(47) \begin{equation} l^*=\frac{(1+\tau ^o)\lambda \{(1-s_H)[(1+\beta )(1-s_N)qR-a\tau \theta (1-\epsilon )]+ qR\Psi \}-R(1-\tau )(1-s_H)[\delta \beta q(1-s_N)R+a\tau \theta (1-\epsilon )]}{R\lambda (1+\tau ^o)q\{(1-s_N)(1-s_H)[1+(1+\delta )\beta ]+\Psi \}} \end{equation}

where $\Psi =\theta [1-\epsilon s_N-(1-\epsilon )s_H]$ .

We therefore derive socially optimal government policies such that the market equilibrium allocation in Definition 1 coincides with that preferred by the social planner in Proposition 1.

PROPOSITION 3. Assuming $\alpha \geq \tilde{\alpha ^0}$ , in equilibrium, the socially optimal government policy is characterized by $\tau =\tau ^*$ , $a=a^*$ , $s_H=s_H^*$ , and $s_N=s_N^*$ . Human capital decay unambiguously increases $s_N^*$ and $a^*$ , while its effect on $s^*_H$ is ambiguous:

(48) \begin{equation} \tau ^*=\frac{\alpha \{(1+\beta )(\lambda +R)\epsilon +R^2q[(1+\delta )\beta +1+\theta ]\}-\theta (1-\epsilon )(\lambda +R)}{R T} \end{equation}

(49) \begin{equation} a^*=\frac{\alpha \beta \delta \{(\lambda +R)[1-\epsilon (1-\alpha )]-R^2 q\}}{\lambda T} \end{equation}

(50) \begin{equation} s_N^*=\frac{qR^2T-(1-\epsilon )\theta \{ (\lambda +R)[1-\epsilon (1-\alpha )]-R^2q\}}{qR^2T} \end{equation}

(51) \begin{equation} s_H^*=\frac{\alpha \{(\lambda +R)[1+\beta +\theta (1-\epsilon )]+R^2q(\beta \delta +\theta )\}-R^2q\alpha ^2[1+\theta +\beta (1+\delta )]-(\lambda +R)(1-\epsilon )\theta }{\alpha [(\lambda +R)(1+\theta +\beta )+R^2q(\beta \delta +\theta )]-R^2q\alpha ^2[1+\theta +\beta (1+\delta )]-R^2q\theta } \end{equation}

where Assumption 1 implies that $\alpha [R(1+\theta +\beta +Rq\theta )+\lambda (1+\theta +\beta )+R^2q\beta \delta (1-\alpha )]-R^2q\alpha ^2(1+\theta +\beta )\gt R^2q\theta$ , $T=\epsilon \alpha ^2[1+\theta +\beta (1+\delta )]+\alpha [1+\beta +\theta (1-\epsilon )(1+\delta )]-\theta (1-\epsilon )\gt 0$ , $s_{N}^*\lt 1$ and $s_{H}^*\lt 1$ .

Proof. The first-best solution given by equations (26)–(29) coincides with the market solution given by equations (44)–(47) if $\tau =\tau ^*$ , $a=a^*$ , $s_H=s_H^*$ , and $s_N=s_N^*$ .

Proof. Under Assumption 1, from equations (39) and (40), $s_{N}^*\gt 0$ if $\alpha \gt \hat{\alpha }$ with $\hat{\alpha }\gt \alpha _0$ and $s_{H}^*\gt 0$ if $\alpha \gt \bar{\alpha }$ with $\bar{\alpha }\lt \alpha _0$ . See Appendix B for technical details.

Human capital decay is a key parameter in this model, and its variation crucially affects policy combinations at the interior solution. In fact, if, ceteris paribus, human capital decay falls, child benefits decrease. In particular, from equation (50), it is easy to note that child benefit becomes a tax when human capital decay is sufficiently low, that is, $\lambda \gt \lambda _{SN}$ .Footnote ⁸ The intuition behind this result is that high human capital productivity maintained in the second period of life increases fertility through the positive income effect, as can be seen from equation (13). Thus, a sufficiently low human capital decay leads to a decentralized equilibrium with too many children, and therefore a tax on children is required in order to reach the first-best solution. As for education, the central planner, differently from the decentralized economy, considers the effect of children’s education on their labor income. Hence, when there is less decay, since the marginal productivity of education increases for the elderly, optimal education should be higher. As can be seen in equation (35), this can be achieved by changes in child or education benefit (where $s_N\lt s_H$ ). Given that $s_N$ decreases, $s_H$ can, therefore, increase or decrease. Moreover, from equation (49), it is evident that that a lower human capital decay reduces the fraction of social security tax revenues devoted to pensions, that is, $\partial{a^*}/\partial{\lambda }\lt 0$ . In Section 4, we carry out a sensitivity analysis to study the overall effect of a change in human capital decay.