3.1 Households

Our framework is a simple overlapping-generations model with two generations, endogenous fertility and exogenous mortality. The representative agent maximises an expected utility:

(1)$$U = {\rm ln\;} c_{y,t} + \gamma \ln n_t + \beta \pi \ln c_{o,t + 1} $$

In the first period of life, the agent is young and chooses the level of consumption c _y,t and the optimal fertility rate n _t – the number of children per individual. In the second period, the agent chooses old age consumption c _o,t+1, facing a conditional probability π of living till the end of period t + 1, given that she survived to the first period with full probability. β ∈ (0, 1) is the usual time-preference discount factor. The parameter γ measures the relative preference with respect to the number of children: in this set-up, children generate utility just because they are born, that is, they are implicitly considered as consumption goods when parents are young. At time t, the agent has to satisfy the following budget constraint:

(2)$$c_{y,t} = w_t (1 - \tau ) - s_t - n_t qw_t $$

where w _t is the wage income, τ ∈ (0, 1) is the contribution rate decided by the social security system, s _t > 0 denotes the resources saved and invested in the capital market, and q > 0 is the fixed proportion of income spent per child – that is, the child-rearing cost. In the second period, following Miyazaki (Reference Miyazaki2014), consumption is financed by gross savings, labour income from the fraction of time spent working, and pension benefits:

(3)$$c_{o,t + 1} = \theta w_{t + 1}^e (1 - \tau )x + \displaystyle{{R_{t + 1}^e} \over \pi} s_t + p_{t + 1}^e (1 - x)$$

Here the agent will supply x ∈ [0, 1] units of labour in old age, where x is the official retirement age set by the social security system. It can also be interpreted as the fraction of the second period of life in which the agent works and gets the net wage income $\theta w_{t + 1}^e (1 - \tau )$ where θ ∈ (0, 1) is an index of productivity of old age workers compared with the younger ones; for the remaining life fraction 1 − x they receive a pension $p_{t + 1}^e $ and consume their savings. We assume that intermediaries in financial markets operate in a perfectly competitive market, hence the rate of return on savings is $R_{t + 1}^e /\pi $.

Solving the optimisation problem for n _t and s _t, we derive the optimal fertility rate and savings:

(4)$$n_t = \displaystyle{{\gamma (1 - \tau )w_t R_{t + 1}^e + \gamma \pi [\theta w_{t + 1}^e (1 - \tau )x + p_{t + 1}^e (1 - x)]} \over {qw_t R_{t + 1}^e (1 + \beta \pi + \gamma )}}$$

(5)$$s_t = \displaystyle{{\beta \pi w_t R_{t + 1}^e (1 - \tau ) - (1 + \gamma )\pi [\theta w_{t + 1}^e (1 - \tau )x + p_{t + 1}^e (1 - x)]} \over {R_{t + 1}^e (1 + \beta \pi + \gamma )}}$$

From here, we can rewrite the fertility rate in terms of the present value of lifetime income (in the square brackets):

(6)$$n_t = \displaystyle{1 \over {qw_t}} \left( {\displaystyle{\gamma \over {1 + \beta \pi + \gamma}}} \right)\left[ {w_t (1 - \tau ) + \displaystyle{\pi \over {R_{t + 1}^e}} (\theta w_{t + 1}^e (1 - \tau )x + p_{t + 1}^e (1 - x))} \right]$$

The lifetime income is distributed among consumption and child-rearing expenditure according to the parameters of the utility function and longevity. In particular, the optimal fraction of lifetime income devoted to child rearing is (γ/1 + βπ + γ), and by dividing this income for the child-rearing cost, qw _t, we derive the optimal number of children.

3.2 Firms

Firms maximise profits producing a homogeneous good that can be used both for consumption and investment. Markets are assumed to be perfectly competitive. The price of output is normalised to one, and output is defined by a Cobb–Douglas production function, $Y_t = AK_t^\alpha L_t^{1 - \alpha} $. Workers can either be young or old. In particular, there is only a fraction of old people in period t that is both alive and working. This is weighted by an index of productivity, thus labour supply is defined as:

(7)$$L_t = N_{y,t} + N_{o,t} \pi \theta x$$

where N _y,t and N _o,t are young and old age individuals at time t, with each person supplying inelastically one unit of labour when young, or an effective unit θ when old. The older working (and still living) generation is only a fraction πx of the whole old population.

As usual, we rewrite the production function in intensive form $y_t = Ak_t^\alpha $. The problem of the firm, then, is to maximise its profit function, by choosing the optimal capital–labour ratio, k. Capital is paid at the market interest rate; hence, assuming full depreciation, $R_t = \alpha Ak_t^{\alpha - 1} $. Labour is paid at the market wage, $w_t = (1 - \alpha )Ak_t^\alpha $.

The capital market is in equilibrium when firms’ demand for capital equals households’ supply of savings. The amount of capital available today is equal to the amount of resources saved in the previous period; hence if those who saved previously are the elderly of today, N _y,t−1 = N _o,t, then we have the following market-clearing condition:

(8)$$K_t = s_{t - 1} N_{o,t} $$

To express it in terms of capital–labour ratio, we have to divide both sides of the equation by L _t. Considering that the number of young individuals today is equal to the offsprings per current old person at time t − 1, that is, N _y,t = n _t−1 N _o,t, the equilibrium condition becomes:

(9)$$k_t = \displaystyle{{s_{t - 1}} \over {n_{t - 1} + \pi \theta x}}$$

3.3 Social security system

The social security system adopts a pay-as-you-go defined-contribution plan and runs a balanced budget:

(10)$$w_t \tau \; L_t = p_t \pi N_{o,t} (1 - x)$$

On the left-hand side, resources are gathered from the entire working population, using wage income as the tax base; on the right-hand side, pensions, p _t, are paid only to the living and retired old population. Substituting the labour supply and rearranging in terms of the fertility rate, pension benefits are defined as follows:

(11)$$p_t = \displaystyle{{\tau w_t} \over {\pi (1 - x)}}(n_{t - 1} + \pi \theta x)$$

3.4 General equilibrium

Given expectations of future prices $w_{t + 1}^e $ and $R_{t + 1}^e $ and substituting pension benefits in equations (4) and (5), the temporary equilibrium satisfies the conditions:

(12)$$n_t = \displaystyle{{\gamma (1 - \tau )w_t R_{t + 1}^e + \gamma \pi \theta xw_{t + 1}^e} \over {qw_t R_{t + 1}^e (1 + \beta \pi + \gamma ) - \gamma \tau w_{t + 1}^e}} $$

(13)$$s_t = \displaystyle{{\beta \pi w_t R_{t + 1}^e (1 - \tau ) - (1 + \gamma )\pi \theta xw_{t + 1}^e - \tau (1 + \gamma )w_{t + 1}^e n_t} \over {R_{t + 1}^e (1 + \beta \pi + \gamma )}}$$

Then, the stationary equilibrium with perfect foresight is:

(14)$$n^{^\ast} = \displaystyle{{\gamma [(1 - \tau )R^{^\ast} + \pi \theta x]} \over {[qR^{^\ast} (1 + \beta \pi + \gamma ) - \gamma \tau ]}}$$

(15)$$s^{^\ast} = \displaystyle{{w^{^\ast} [\pi q(\beta R^{^\ast} (1 - \tau ) - \theta x(1 + \gamma )) - \gamma \tau (1 - \tau )]} \over {[qR^{^\ast} (1 + \beta \pi + \gamma ) - \gamma \tau ]}}$$

(16)$$k^{^\ast} = \left[ {\displaystyle{A \over {2\pi x\gamma \theta (1 - \tau )}}\left( {\sqrt {4\pi ^2 qx(1 - \alpha )\alpha \beta \gamma \theta (1 - \tau )^2 + z^2} - z} \right)} \right]^{1/(1 - \alpha )} $$

where z ≡ πqx(1 + γ)θ + α(π ²qxβθ + γ(1 − τ)²) + γτ(1 − τ) > 0. Apart from the uninteresting steady state where capital is zero, k* is the unique positive solution to the dynamic equation k _t+1 = g(k _t), obtained from equation (9). Moreover, since the function g(.) is non-decreasing and concave, k* is also globally stable as shown in Appendix A.

The corresponding steady-state level of the pension benefit is:

(17)$$p^{^\ast} = \displaystyle{{\tau w^{^\ast}} \over {\pi (1 - x)}}\left( {\displaystyle{{\gamma [(1 - \tau )R^{^\ast} + \pi \theta x]} \over {[qR^{^\ast} (1 + \beta \pi + \gamma ) - \gamma \tau ]}} + \pi \theta x} \right)$$

This model encompasses Cipriani (Reference Cipriani2014). In fact, if we assume that individuals cannot work in the second period (i.e., x = 0), then the above solutions are exactly equal to those found in that article. Our extension allows us to study explicitly not only the impact of a change in the contribution rate, but also of the effects of a change in the official retirement age.

4.1 Longevity shock

In the first place, capital-per-worker in the steady-state changes due to a change in longevity as follows:

(18)$$\displaystyle{{dk^{^\ast}} \over {d\pi}} = \displaystyle{{k^{^\ast} [ - \alpha ( - \pi ^2 qx\beta \theta + \gamma (1 - \tau )^2 ) - \gamma \tau (1 - \tau )]} \over { - \pi \left( {1 - \alpha} \right)\sqrt {4\pi ^2 qx(1 - \alpha )\alpha \beta \gamma \theta (1 - \tau )^2 + z^2}}} $$

The sign of the partial derivative is in general ambiguous. However, defining this threshold value of π:

(19)$$\bar \pi \equiv \sqrt {\displaystyle{{\alpha \gamma (1 - \tau )^2 + \gamma \tau (1 - \tau )} \over {\alpha qx\beta \theta}}} $$

we have that if $0 \lt \pi \lt \bar \pi $, then (dk*/dπ) > 0 and if $\bar \pi \lt \; \pi \; \lt 1$, then (dk*/dπ) < 0. Hence if $\bar \pi \gt 1$, the derivative will always be positive. Otherwise, if the threshold value is sufficiently low, a higher survival probability reduces capital-per-worker in the long run. In general, the condition $\pi \lt \bar \pi $ is satisfied for a sufficiently high value of child preferences, γ, or a sufficiently low level of child-rearing costs, q.

The effects on the fertility rate and pensions are:

(20)$$ \displaystyle{{dn^{\ast}} \over {d\pi }} = \displaystyle{\matrix{\gamma [(1 - \tau )(dR^{\ast}{\rm /}d\pi ) + \theta x][(1 + \beta \pi + \gamma )qR^{\ast} - \gamma \tau ] - \hfill \cr [\beta R^{\ast} + (1 + \beta \pi + \gamma )(dR^{\ast}{\rm /}d\pi )]q\gamma [(1 - \tau )R^{\ast} + \pi \theta x] \hfill} \over {{[(1 + \beta \pi + \gamma )qR^{\ast} - \gamma \tau ]}^2}} $$

(21)$$\displaystyle{{dp^{^\ast}} \over {d\pi}} = \displaystyle{{\tau w^{^\ast} (\varepsilon _{w^{^\ast}, \pi} - 1)} \over {\pi ^2 (1 - x)}}(n^{^\ast} + \pi \theta x) + \left( {\displaystyle{{dn^{^\ast}} \over {d\pi}} + \theta x} \right)\displaystyle{{\tau w^{^\ast}} \over {\pi (1 - x)}}$$

where, from now on, ε_a,b is the elasticity of a with respect to b. The sign of the derivative of n is ambiguous and therefore the effect of life expectancy on pensions is also ambiguous. Obviously, if we considered only the direct effect of longevity on pensions, the effect would be negative, that is, (∂p*/∂π) < 0. On the other hand, if we ignored the general equilibrium effect of ageing on wages, for instance by assuming a small open economy, equation (21) would simplify to:

(22)$$\displaystyle{{dp^{^\ast}} \over {d\pi}} = \displaystyle{{n^{^\ast} \tau w^{^\ast} (\varepsilon _{n^{^\ast}, \pi} - 1)} \over {\pi ^2 (1 - x)}}$$

Now the derivative depends on the elasticity of fertility with respect to longevity. If fertility was positively elastic with respect to longevity (i.e., ε _n*,π > 1), then pensions would rise: the effect of ageing from above would be overwhelmed by a reduction of ageing from below. These results highlight the limits of partial analyses, which inevitably conclude that demographic consequences on the sustainability of the pension systems are unavoidable.

4.2 Social security system shocks

In order to maintain a balanced budget in the face of an increasing number of pensioners without reducing pension benefits, the typical policies adopted by governments are the increase in the payroll tax or in the retirement age. However, we find that capital-per-worker always declines following an increase in the contribution rate or an increase in the official retirement age. In fact:

(23)$$\displaystyle{{dk^{^\ast}} \over {d\tau}} = \displaystyle{{ - k^{^\ast} [\pi qx(1 + \pi \alpha \beta )\theta + \gamma (\pi qx\theta + (1 - \alpha )(1 - \tau )^2 )/[(1 - \alpha )(1 - \tau )]]} \over {\sqrt {4\pi ^2 qx(1 - \alpha )\alpha \beta \gamma \theta (1 - \tau )^2 + z^2}}} \lt 0$$

(24)$$\displaystyle{{dk^{^\ast}} \over {dx}} = \displaystyle{{Ak^{^\ast} \phi} \over {\pi x^2 (1 - \alpha )\gamma \theta (1 - \tau )}} \lt 0$$

where ϕ < 0 for any value of the parameters.Footnote ¹

The general equilibrium effect on the optimal number of children can be deduced from the following decomposition:

(25)$$\displaystyle{{dn^{^\ast}} \over {d\tau}} = \displaystyle{{\partial n^{^\ast}} \over {\partial k^{^\ast}}} \displaystyle{{dk^{^\ast}} \over {d\tau}} + \displaystyle{{\partial n^{^\ast}} \over {\partial \tau}} $$

where (∂n*/∂k*) > 0, (dk*/dτ) < 0 and the last component is given by:

(26)$$\displaystyle{{\partial n^{^\ast}} \over {\partial \tau}} = - \displaystyle{{\gamma (R^{^\ast} - n^{^\ast} )} \over {[qR^{^\ast} (1 + \beta \pi + \gamma ) - \gamma \tau ]}}$$

If R* > n*, then the partial derivative above is negative and, consequently, also the total derivative. This is in general the case, since R* is defined as the gross return of an investment lasting, say, 30 years, whereas n* is the number of children per individual. For example, where the fertility rate per couple equal to 3, which is much above the average in the developed world, the corresponding number of children per individual n* would be 1.5, implying an average yearly rate of return of 1.36%, which is clearly too low. In particular, if we consider the reasonable case in which the household fertility rate is about 2, that is, about one child per individual, we will find that for any value of R* the above condition is satisfied. Therefore, we can safely argue that in any plausible scenario the optimal number of children declines as the contribution rate increases.

The general equilibrium effect on pensions is, however, ambiguous:

(27)$$\displaystyle{{dp^{^\ast}} \over {d\tau}} = \displaystyle{{w^{^\ast}} \over {\pi (1 - x)}}[(\varepsilon _{w^{^\ast}, \tau} + 1)(n^{^\ast} + \pi \theta x) + n^{^\ast} \varepsilon _{n^{^\ast}, \tau} ]$$

In fact, if wages are inelastic with respect to τ, and the fertility rate declines as the contribution rate rises, then nothing can be said about the sign of the derivative. In the next section, we provide a numerical simulation to study this further.

The alternative policy is to increase the official retirement age, but its effect on capital-per-worker is also negative, as shown in equation (24). The increase of old-age labour supply has two implications on the production side: first, there are more workers in the economy, and second, households need to save less, thus reducing the aggregate capital stock. Also, as the labour supply increases, the wage rate falls, mitigating the effect on the saving decision.

As far as the steady-state level of fertility rate and pension benefits are concerned, the derivatives are the following:

(28)$$\displaystyle{{dn^{\ast}} \over {dx}} = \gamma \displaystyle{\matrix{[(dR^{\ast}{\rm /}dx)(1 - \tau ) + \pi \theta ][(1 + \beta \pi + \gamma )qR^{\ast} - \gamma \tau ] \hfill \cr - (dR^{\ast}{\rm /}dx)(1 + \beta \pi + \gamma )q[(1 - \tau )R^{\ast} + \pi \theta x] \hfill} \over {{[(1 + \beta \pi + \gamma )qR^{\ast} - \gamma \tau ]}^2}} $$

(29)$$\displaystyle{{dp^{^\ast}} \over {dx}} = \displaystyle{{\tau w^{^\ast} \pi} \over x}\left[ {\displaystyle{{\varepsilon _{w^{^\ast}, x} (1 - x)x - 1} \over x}} \right]\displaystyle{{(n^{^\ast} + \pi \theta x)} \over {\pi ^2 (1 - x)^2}} + \left( {\displaystyle{{\partial n^{^\ast}} \over {\partial x}} + \pi \theta} \right)\displaystyle{{\tau w^{^\ast}} \over {\pi (1 - x)}}$$

As before, it is not possible to analytically determine the sign of the derivatives. Recall, however, that from equation (14), the partial effect of an increase in the official retirement age is positive. Similarly, the direct effect on pensions is also positive.Footnote ² The general equilibrium effects are much more complicated and must take into account the effects of factors’ prices.

4.3 Case with x = 0

To simplify the analysis of policy variations, we can study the model in the case in which x = 0. As previously said, the model collapses into that of Cipriani (Reference Cipriani2014). Here we show that it is possible to study an exogenous variation in retirement also in this simpler model, although the general model we have outlined in Section 3 allows for an explicit analysis. If x = 0, individuals work only when young, while during retirement they consume their accumulated savings and the pension. Therefore, in order to study the variation in retirement age, we could focus on the length of the working life. That is, we can directly manipulate the time scale by changing the value of the time-varying parameters β and π. In particular, the discount factor can be interpreted as $\beta = \beta _y^{\Delta t} $, where β _y is the constant yearly discount factor, and Δt is the duration of the first period. Since β _y ≤ 1, β is non-increasing in Δt. Longevity, instead, is a survival probability; therefore, it is also non-increasing over time. Thus, given the monotonicity of β and π with respect to Δt, we can study the effects of an increase in the retirement age by changing these parameters. In general, labour productivity θ also changes over time; however, if we set x = 0, the parameter disappears from the model. To retain the parameter θ, we should modify both the consumer's budget constraint and the constraint of the social security system in equations (2) and (10), by multiplying w _t by θ. However, labour productivity may change non-monotonically over time: for example, θ may increase when the worker is young and is accumulating experience, while it may decrease when the mental and physical capabilities start declining. Hence, here we consider only the simple case with no change in labour productivity.

The steady-state levels of capital, fertility, and pensions are:

(30)$$k^{^\ast} = \left[ {\displaystyle{{Aq(1 - \alpha )\alpha v} \over {\gamma (\alpha + \tau (1 - \alpha ))}}} \right]^{1/(1 - \alpha )} $$

(31)$$n^{^\ast} = \displaystyle{{\gamma (1 - \tau )R^{^\ast}} \over {qR^{^\ast} (1 + v + \gamma ) - \gamma \tau}} $$

(32)$$p^{^\ast} = \displaystyle{{\tau w^{^\ast} n^{^\ast}} \over \pi} $$

where v ≡ βπ is the combined discount factor in the utility function. This formulation of the model allows us to study the impact of an extension of the working life by looking at changes of v. It is easy to see that, as expected, k* declines as the working life gets longer, that is, when a smaller discount factor v is applied. Also, the fertility rate increases as v shrinks, as shown by differentiating with respect to v:

(33)$$\displaystyle{{dn^{^\ast}} \over {dv}} = \displaystyle{{ - \gamma \alpha \; n^{^\ast}} \over {v(1 - \alpha )[qR^{^\ast} (1 + v + \gamma ) - \gamma \tau ]}} \lt 0$$

In this scenario, it is also possible to show that the relationship between contribution rate and pensions is non-monotonic. In fact by computing the derivative in the neighbourhood of τ = 0 and τ = 1, we find different signs:

(34)$$\left. {\displaystyle{{dp^{^\ast}} \over {d\tau}}} \right \vert _{\tau = 0} = \displaystyle{{\gamma (1 - \alpha )A} \over {\pi q(1 + \pi \beta + \gamma )}}\left[ {\displaystyle{{q(1 - \alpha )A\pi \beta} \over \gamma}} \right]^{\alpha /(1 - \alpha )} \gt 0$$

(35)$$\left. {\displaystyle{{dp^{^\ast}} \over {d\tau}}} \right \vert _{\tau = 1} = - \displaystyle{{\gamma (1 - \alpha )A} \over {\pi q(1 + \alpha \pi \beta + \gamma )}}\left[ {\displaystyle{{q\alpha (1 - \alpha )A\pi \beta} \over \gamma}} \right]^{\alpha /(1 - \alpha )} \lt 0$$

For sufficiently low values of τ, raising the contribution rate produces new inflows into the social security system, at the expenses of a marginal decrease in capital and fertility. However, the positive impact of a higher contribution rate outweighs the loss in the tax base and, consequently, pensions increase. The opposite argument applies when the level of taxation is already high enough. However, the overall effects of the contribution rate and retirement age on pension benefits are still ambiguous. The following section, therefore, complements the analytical results with a numerical example.

5.1 Rise in longevity

As stated in Section 4, the impact of a longer life on capital accumulation depends on the threshold value $\bar \pi $: if the probability is below the threshold, then capital will always increase in the steady state as response to the rise in longevity. Indeed, with the values assigned to the parameters, the threshold is 3.75; therefore, capital is always increasing with longevity. Fertility rate drops when longevity increases: there is, therefore, a reinforcement of ageing. The impact on pension benefits is also negative due to the prevailing effect of the larger number of pensioners per worker. Figure 1 plots fertility and pensions.

Figure 1.

Change in longevity

With these parameters, the fertility rate seems poorly sensitive to a variation of the survival probability. Consequently, the direct effect on pensions, as shown in equation (22), is negative; furthermore, also the general equilibrium effect on pensions is negative and the relationship is convex.

5.2 Pension policies

We found that ageing generates additional capital per worker in the steady state, with a consequent increase in output per worker. Nonetheless, the impact on pensions is the same as we might expect from a partial equilibrium analysis: more pensioners and fewer taxpayers imply lower pensions and may require a change in the social security system in order to avoid a decrease of the pension benefits. Thus, decision-makers typically choose either to increase the contribution rate or the retirement age. Both policies have an unambiguously negative effect on capital per worker. However, as discussed in the previous section, the final effect on pensions is in general ambiguous: it might be the case that both do not provide any improvement to equilibrium pensions.

As can be seen in Figure 2, with our chosen parameters, fertility decreases: the reduced capital accumulation generates a negative income effect that cuts fertility, thus aggravating the ageing problem. Pension benefits follow an inverted-U shape, with a maximum when the contribution rate reaches 52.3%. However, since it is highly unlikely that the contribution rate for social security would reach such a high level, we can focus only on the first half of the graph, and assume that, for any plausible level of the contribution rate, pension benefits are increasing.

Figure 2.

Change in contribution rate.

The analysis is replicated also for a change in the official retirement age (Figure 3). Like for the increase of the contribution rate, capital per worker is decreasing with the retirement age. However, in this case, an increasing fertility path does not reinforce the ageing problem as before. Pensions are always increasing in retirement age, and the size of the increase is higher than before.

Figure 3.

Change in the retirement age