6.1 Corner solution

After replacing b _t+1 from Equation (18) in Equations (25) and (26), optimal fertility and young consumption in a steady state and a market economy are given by:

(27)$$n^\ast{ = } \displaystyle{{wR\theta ( {1-\tau } ) } \over {Rq( {1 + \theta + \beta } ) -\theta w\tau }}, \;$$

and

(28)$$c^{y\ast } = \displaystyle{{wRq( {1-\tau } ) } \over {Rq( {1 + \theta + \beta } ) -\theta w\tau }}, \;$$

where we assume τ < Rq(1 + θ + β)/θw.

Comparing Equations (27) and (28) with (14) and (13), it can be seen that the optimal choices of households differ from those of a central planner. In particular, in accordance with the existing literature (see, e.g., van Groezen et al., Reference van Groezen, Leers and Meijdam2003; Cremer et al., Reference Cremer, Gahvari and Pestieau2011; Cipriani, Reference Cipriani2014), optimal choices in a market economy differ from the social optimum due to two effects. The first, known as the intergenerational transfer effect (or dependency ratio effect), means that in a PAYGO social security system, individuals do not take into account the effect each person's fertility decision has on population growth and therefore on everybody's pension benefits. In other words, they do not take into account that a higher number of children means that, in the future, there will be a higher number of workers able to support pensioners. This leads to a decentralized equilibrium with too few children. The second effect is the capital dilution effect: parents do not consider that a higher fertility rate, given the aggregate saving, lowers the capital-labor ratio. In a small open economy, the capital-labor ratio is constant and thus, if the number of children increases, savings and/or foreign debt should increase in order to maintain the capital-labor ratio at the same level. This leads to a decentralized equilibrium with too many children. Depending on which of these two effects prevails, the number of children in a market economy is either too low or too high compared to the first-best solution.

Thus, the government needs to redistribute resources across generations in order to achieve the first-best allocation in a decentralized economy. Suppose the government provides a subsidy ε _t per child in order to lower the cost of raising a child. The budget constraint in adulthood, therefore, becomes:

(29)$$c_t^y = w( {1-\tau } ) -s_t-( {q-\epsilon_t} ) n_t, \;$$

and assuming that the government devotes a fraction 0 < λ < 1 of tax revenues to child benefits and a fraction 1 − λ to social security benefits, the internal allocation of government expenditure is given by:

(30)$$b_{t + 1} = ( {1-\lambda } ) \tau wn_t, \;$$

and

(31)$$\epsilon _{t + 1} = \displaystyle{{\lambda \tau w} \over {n_{t + 1}}}.$$

Thus, the optimum policy combination is τ = τ* and λ = λ*, where:

(32)$$\tau ^\ast{ = } \displaystyle{{( {1 + \alpha } ) [ {Rq( {1 + \theta + \beta } ) \alpha -\theta w} ] } \over {w( {\alpha ( {1 + \theta + \beta } ) -\theta } ) }}, \;$$

and

(33)$$\lambda ^\ast{ = } \displaystyle{\alpha \over {1 + \alpha }}.$$

Note that for a given value of the social discount factor, i.e., α ⁰ = θw/Rq(1 + θ + β), τ* = 0, where α ⁰ > α. Thus, for this level of the social discount factor, the market outcome coincides with the first-best solution and therefore government intervention is unnecessary. On the other hand, if the social discount factor is sufficiently high, i.e., α > α ⁰, then in a market economy, the dependency ratio effect (intergenerational transfer effect) dominates the capital dilution effect, i.e., parents have too few children (n* < n**) and this in turn means higher consumption in the decentralized equilibrium c* > c**. Thus, the government should provide a positive child allowance in order to achieve the first-best outcome (see van Groezen et al., Reference van Groezen, Leers and Meijdam2003). By contrast, if α < α ⁰, children should be taxed in order to achieve the first-best solution.

From Equations (31) and (33), it follows that the first-best allocation is achieved in a market setting if the child allowance equals the present value of the pension benefit, i.e., ε = w(1 − λ*)τ*/R (see van Groezen et al., Reference van Groezen, Leers and Meijdam2003).Footnote ⁶ This coincides with the level of child allowance that, for a given value of the PAYG-tax τ, maximizes lifelong utility for the young in period t.

6.2 Interior solution

We now turn to the equilibrium where agents choose to supply labor in their old age. Thus, after replacing b _t+1 from Equation (18) in Equations (22)–(24), optimal fertility, consumption, and the labor supply in old age, in a steady state, are given by:

(34)$$n^\ast{ = } \displaystyle{{\theta w( {1-\tau } ) ( {1 + R} ) } \over {R[ {1 + \theta + \beta ( {1 + \delta } ) } ] q}}, \;$$

(35)$$c^{y\ast } = \displaystyle{{w( {1-\tau } ) ( {1 + R} ) } \over {[ {1 + \theta + \beta ( {1 + \delta } ) } ] R}}, \;$$

(36)$$l^\ast{ = } \displaystyle{{( {1-\tau } ) [ {( {1 + \beta + \theta } ) qR-\theta w\tau ( {1 + R} ) -R^2q\beta \delta } ] } \over {R[ {1 + \theta + ( {1 + \delta } ) \beta } ] q}}.$$

As in the case of the corner solution, the optimum choices of households deviate from the social optimum. However, in addition to the intergenerational transfer effect and the capital dilution effect discussed above, the impact of pensions on the labor supply in old age and the interaction between the labor supply in old age and fertility have to be considered. In particular, from Equations (21) and (9), the provision of pension benefits reduces the marginal cost of leisure in old age leading to early retirement. In addition, agents do not take into consideration the fact that a higher labor supply in old age, all else being equal, increases pension benefits and thus, from this point of view, the labor supply in old age is too low. On the other hand, agents also fail to consider that a higher labor supply in old age, for given aggregate savings and fertility, lowers the capital-labor ratio. In a small open economy, the capital-labor ratio is fixed and thus, if the labor supply in old age increases, savings and/or foreign debt should increase in order to maintain the capital-labor ratio at the same level. This leads to a decentralized equilibrium with an excessive labor supply in old age. Thus, depending on which of these effects prevails, the levels of labor supply and fertility in a competitive equilibrium will either be lower or higher than those in the first-best economy.

Therefore, as in the corner solution, to replicate the first-best allocation, the government needs to introduce certain public policies. In particular, the government is able to increase fertility and to decrease the retirement period by a mix of policies that redistribute across generations. Assuming that the government introduces a subsidy or a tax on elderly labor supply, i.e., $\tau ^o \ge0$, agents' budget constraints in adulthood and in old age become:

(37)$$c_t^y = w( {1-\tau } ) -s_t-( {q-\epsilon_t} ) n_t, \;$$

(38)$$c_{t + 1}^o = Rs_t + wl_{t + 1}( {1-\tau^o} ) + b_{t + 1}( {1-l_{t + 1}} ) , \;$$

and the government budget constraint for pensions and a child allowance scheme is:

(39)$$b_{t + 1} = \displaystyle{{( {1-\lambda } ) w} \over {1-l_{t + 1}}}( {\tau n_t + \tau^ol_{t + 1}} ) , \;$$

and

(40)$$\epsilon _{t + 1} = \lambda w\left({\displaystyle{{\tau n_t + \tau^ol_{t + 1}} \over {n_tn_{t + 1}}}} \right).$$

Thus, in equilibrium, the government can ensure that market allocations coincide with the social optimum by setting τ = τ*, τ ^o = τ ^o*, and λ = λ*, where:

(41)$$\tau ^\ast{ = } \displaystyle{{( {1 + \alpha } ) [ {qR^2\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -( {1 + R} ) \theta w} ] } \over {Rw[ {\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -\theta } ] }}, \;$$

(42)$$\tau^{o^{\ast}}{ = }{-}\displaystyle{{\alpha [ {qR^2\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -( {1 + R} ) \theta w} ] } \over {w[ {\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -\theta } ] }} = {-}\displaystyle{{\alpha R\tau ^\ast } \over {1 + \alpha }}, \;$$

and

(43)$$\lambda ^\ast{ = } \displaystyle{{[ {\alpha ( {1 + \theta + ( {1 + \delta } ) \beta } ) -\theta } ] w} \over {w[ {\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -\theta } ] + \beta \delta [ {w( {1 + R} ) -R^2q} ] }}.$$

Notice that when the social discount factor is equal to $\tilde{\alpha }^0 = \theta w( {1 + R} ) {\rm /}qR^2[ {1 + \theta + \beta ( {1 + \delta } ) } ]$, with $\tilde{\alpha }^0 > \underline{\alpha }$, government intervention is not required. With this discount factor, in fact, the intergenerational transfer effect is compensated by the capital-dilution effect and fertility and retirement are at the first-best level. For other levels of α, the command optimum does not coincide with the market solution. In particular, if $\alpha > \tilde{\alpha }^0$, parents have too few children and retire too early, i.e., n* < n**, c* > c**, and l* < l**, and therefore the government should provide a child allowance and a subsidy for the labor supply of older people in order to achieve the first-best allocation. On the other hand, if $\underline{\alpha } < \alpha < \tilde{\alpha }^0$, fertility and elderly labor supply are too high, a tax on children and elderly labor supply would be required so that agents choose to have a lower number of children and retire early. The relationship between the social discount factor and the sign of the Pigouvian tax goes in the same direction as van Groezen et al. (Reference van Groezen, Leers and Meijdam2003): in general, the market outcome differs from the command optimum depending on whether the intergenerational transfer effect prevails or is dominated by the capital-dilution effect. In our model, in addition to the standard capital-dilution effect of fertility, we have another effect on the capital-labor ratio from the labor supply of old agents. On the other hand, labor supply of older agents could be lower than optimal given its effect on pension benefits.

From Equations (41)–(43), if τ ^o = τ, it is not possible to achieve the first-best allocation in a decentralized economy. In this case, in fact, the government needs an additional instrument for the market outcome to coincide with the social optimum, particularly direct control of the retirement age. In this case, workers cannot choose the length of their retirement period but are induced by the government to work for a fixed amount of time $\bar{l}$. Thus, given that τ ^o = τ, the optimum fertility and consumption chosen by households in the steady stare are implicitly given by:

(44)$$n^\ast{ = } \displaystyle{{[ {R( {1-\tau } ) + ( {1-\lambda \tau } ) \bar{l}} ] \theta w} \over {( {1 + \theta + \beta } ) ( {q-\epsilon ( {n^\ast } ) } ) R-\theta ( {1-\lambda } ) w\tau }}, \;$$

and

(45)$$c^{y\ast } = \displaystyle{{( {R( {1-\tau } ) + ( {1-\lambda } ) \tau } ) \bar{l}) w( {q-\epsilon ( {n^\ast } ) } ) } \over {( {1 + \theta + \beta } ) ( {q-\epsilon ( {n^\ast } ) } ) R-\theta ( {1-\lambda } ) w\tau }}, \;$$

where $\bar{l}$ is given by Equation (12) and $\epsilon ( n ) = \lambda w\tau ( {n + \bar{l}} ) {\rm /}n^2$. In this case, therefore, the government can achieve the social optimum in a market economy by setting $\bar{l} = l^{{\ast}{\ast} }$, $\tau = \bar{\tau }^\ast$, and $\lambda = \overline {\lambda ^\ast }$:

(46)$$\bar{\tau }^\ast{ = } \displaystyle{{( {1 + \alpha } ) [ {qR^2\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -\theta w( {R + 1} ) } ] } \over {Rw[ {\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -\theta } ] }}, \;$$

and

(47)$$\overline {\lambda ^\ast } = \displaystyle{{R[ {\alpha ( {1 + \theta + \beta ( {1 + \delta } ) } ) -\theta } ] w\alpha ^2} \over {( {1 + \alpha } ) \{ R[ {1 + \theta + \beta ( {1 + \delta } ) } ] w\alpha ^2 + [ {( {1 + \theta + \beta -R( {\theta + \beta \delta } ) } ) w + \beta \delta qR^2} ] \alpha -\theta w) \} }}.$$

6.3 Can reforms yield a Pareto improvement? A simulation

In this section, we consider an economy initially characterized by a PAYG-pension system, no child allowance (λ = 0), and no subsidy for elderly labor supply (τ _o = 0). This could be a typical developed country, which is facing an aging population with the usual pension sustainability problem. We wish to investigate the possibility of implementing a policy to achieve the first-best and in particular, to check if such a policy could be Pareto-improving.

We consider two types of pension systems: one is characterized by a fixed retirement age, and the other by a flexible retirement regime. In the first case, agents retire at the onset of old age, whereas in the second case, agents decide when to retire, provided that they work for a minimum number of years (equivalent to the first period).

In the first case, as we have seen in Section 6, a government can achieve the first-best allocation either through a policy that consists of setting the fixed retirement age at the optimal level (l**) and implementing a child subsidy (policy 1) or through a policy which consists of a subsidy on both the elderly labor supply and fertility (policy 2). Under a flexible retirement regime, the first-best allocation can be achieved by a policy mix that includes both a child allowance and a subsidy for the labor supply of the elderly. In the simulation, we refer to this case as policy 3.

We focus on these pension reforms because, in general, they recall the main policies implemented in recent years by various governments for increasing the financial sustainability of pension systems. In fact, as discussed in Section 2, raising the retirement age has been a common reform over the last two decades in almost all OECD countries. Additionally, various countries have boosted incentives for working longer. For instance, Belgium, Canada, and Denmark have recently been enhancing work incentives for elderly workers (OECD, 2019). Finally, many countries provide various incentives for childcare, often in the form of credits for periods of maternity leave. However, as found by the OECD (2019), on average, these incentives are not enough to compensate women for lost pension income.

Of course, in many countries, the distinction between the policies that we are analyzing is not so clearly defined and often we find more complex scenarios. Thus, our aim is not to provide a comprehensive analysis of complex recent pension reforms, but to shed some light on their potential welfare impact across the generations.

We assume that one period lasts 30 years. The parameter η, that is the capital share in added value, is set at 0.3 as usually assumed. The annual interest rate is set at 1.2% per year and therefore we obtain R = 1.4 over a generation (30 years). The productivity level A is a scale parameter and is set at 1.5 in order to satisfy Assumption 1. Given R, η, and A, the capital-labor ratio is k = 0.2 and this means that w = 0.6. Following Blackburn and Cipriani (Reference Blackburn and Cipriani2002), the annual discount factor is set at 0.99 so β = (0.99)³⁰ = 0.74 over a generation. The weight of leisure relating to consumption in old age δ is set at 0.8. The child raising cost, q, is set at 0.3, in line with the empirical literature on this resource share, which estimates that children account for between 20% and 30% of the households' budget (see, e.g., Letablier et al., Reference Letablier, Luci, Math and Thévenon2009; Apps and Rees, Reference Apps and Rees2001). The initial social security rate is equal to0.15. The parameter θ is calibrated to produce a total fertility rate of 1.2 in the steady state before policy implementation. However, since the model includes single sex individuals in the model, θ is calibrated so that fertility is equal to 0.6 per individual. This gives θ equal to0.85. This parameter set yields$\widetilde{{\alpha ^0}} = 0.65$, thus we set α = 0.7.

Ten periods (i.e., generations) are considered and the economy is assumed to be in the steady state until period 3, and the new policy is announced and implemented at the beginning of period 4. Thus, until period 3, agents' choices, in the case of policy 1 and 2, are given by Equations (27) and (28), while in the case of policy 3, they are given by Equations (34)–(36).

Figure 5 shows the impact of the three policies on the utility of the currently young and all future generations.

Figure 5. Welfare analysis. Policy 1 consists of setting the fixed retirement age at the first-best level (l**) and implementing a child subsidy; policy 2 consists of a subsidy for both the elderly labor supply and fertility in a fixed retirement regime; policy 3 consists of a child allowance and a subsidy for elderly labor supply in a flexible retirement regime.

The black curve refers to lifetime utility in the case of policy 1. Here, the government sets $\bar{l} = l^{{\ast}{\ast} }$ and imposes a tax rate $\tau = \bar{\tau }^\ast$ and $\lambda = \overline {\lambda ^\ast }$ (see Equations (46) and (47)). With our parameters set, we obtain $\bar{l} = 0.57$, $\bar{\tau }^\ast{ = } 0.13$, and $\overline {\lambda ^\ast } = 0.25$.Footnote ⁷ From period 4, agents change their fertility and consumption choices and this implies an upward spike followed by a small decrease in periods 5–6 which then stabilizes at the first-best level from period 6 onwards. In fact, both fertility and consumption do not immediately reach the first-best level because they negatively depend on the fertility choices of the previous period.Footnote ⁸ Overall, lifetime utility increases compared to the initial steady state. The introduction of a child allowance actually lowers the cost of raising children and thus, all else being equal, increases the fertility and consumption of young agents (see Figure 6 in Appendix B). Moreover, forcing agents to work in old age has both a direct positive income effect and an indirect positive effect, through child allowance, on fertility (see Equation 44). Finally, the reduction in the retirement period reduces the necessity of savings, thus increasing consumption in adulthood. To sum up, utility increase from higher fertility and consumption exceeds the utility loss from the reduction of leisure time in old age. In order to give a measure of welfare change, we calculate the wealth equivalent. Specifically, this is a measure of the percentage increase in full lifetime resources needed in the initial scenario, without the policy, to achieve the level of utility reached after the introduction of the policy (Auerbach et al., Reference Auerbach, Auerbach and Kotlikoff1987; Fehr et al., Reference Fehr, Kallweit and Kindermann2017). Overall, we find that welfare gains from the initial steady state to the final one, measured in terms of wealth equivalent, amount to 24%.

However, what is most interesting to study is the effect of this policy on the welfare of the current elderly. Clearly, the effect is a priori ambiguous, since on the one hand, it increases their retirement age, which has a direct negative effect on utility, and on the other hand, it increases their labor income and pension benefits. Overall, in our example, for currently elderly agents, the gain in welfare is equivalent to a variation in wealth of 22%.

In policy 2, the government starting from a mandatory retirement system allows agents to choose how much they wish to work in old age, providing them with a subsidy in order to raise the elderly labor supply to the first-best level. Thus, from Equations (41)–(43), with our parameters set, we obtain τ* = 0.13 and λ* = 0.62, τ ^o* = −0.07. Lifetime utility follows a very similar path to policy 1, and therefore as in policy 1, there is a marked increase in lifetime utility in period 4 due to the positive impact of child benefit and the possibility of working in old age, on fertility and consumption. Regarding the currently elderly, as in the case of policy 1, there are two opposite effects on utility, that is, a positive one due to a higher consumption and a negative one due to the reduction of leisure time in old age. The total effect on the currently elderly is positive and is equivalent to a variation in wealth of 33%.

Thus, comparing policy 1 and 2, we can observe that although both lead to the same welfare gain for the currently young and future generations, because the total value of lifetime resources in the initial scenario is the same, policy 2 implies a higher welfare gain for the currently elderly. This is due to the fact that, while in policy 1 labor supply is immediately taken by the government to the first-best level, in policy 2 although the subsidy to the elderly labor supply induces agents to delay their retirement period, this is, nevertheless, above the first-best level. This, therefore, implies a minor utility loss.

Finally, regarding policy 3, there is an increase in lifetime utility, albeit smoother and lower compared to the previous policies. Indeed, in this case, agents already have the opportunity to work in old age prior to the introduction of the policy, and therefore, when compared to a scenario initially characterized by mandatory early retirement, fertility is higher and although the policy has a positive effect, the increase is lower (see Figure 6 in Appendix B). In fact, while in policies 1 and 2, the positive income effect of the labor supply in old age adds to child benefit, in policy 3, this additional effect is missing. Thus, the welfare gain from the initial steady state to the final one, measured in terms of wealth equivalent is 6%.

Regarding the currently elderly, policy 3 leads to a reduction of their welfare, i.e., −2.7%. This is due to the relatively low increase in consumption because, although the subsidy to elderly labor supply, on the one hand, has a direct positive effect on consumption, on the other hand, it has an indirect negative effect through a lower pension benefit. In particular, the latter decreases because total tax revenues, i.e., $\tau + \tau ^ol_t^\ast {\rm /}n_{t-1}^\ast$, decrease and this in turn implies an increase in the fraction of tax revenues devoted to child benefit, i.e., λ, necessary for taking fertility to the first-best level.

To conclude, on the basis of our analysis, while policies devoted to increasing retirement age and child benefits may have a positive welfare effect across all the generations in those countries that are initially characterized by an early mandatory retirement regime, they may, however, generate a worsening of welfare for some generations in countries initially characterized by a flexible retirement regime.