Fertility, child care outside the home, and pay-as-you-go social security

Abstract

We examine the long-run effects of the pay-as-you-go (PAYG) social security scheme on fertility and welfare of individuals in an overlapping generations model, assuming that child-care services are available in the market. We show that the impact of a tax increase on fertility depends on the relative magnitudes of the standard intergenerational redistribution effect through the social security system, the (implicit) subsidy effect through tax-exemption of child rearing at home, and the price effect through changes in the relative price of market child care, and that if parental child-rearing time is inelastic, a tax cut could bring about a Pareto-improving allocation.

Appendices

Appendix A

1.1 Market child care produced with labor and goods

Assume that the production function of market child care is written as

$$ x_{t} = m^{\varepsilon }_{t} l^{{1 - \varepsilon }}_{{xt}} $$

(32)

where m _t is market goods purchased and l _xt is labor employed, and that individuals decide how much market child care is produced by employing goods and labor. The budget constraint becomes

$$ {\left( {1 - \tau } \right)}w{\left( {1 - z_{t} } \right)} = c_{t} + s_{t} + {\left( {m_{t} + wl_{{xt}} } \right)} $$

(33)

where m _t  + wl _xt is the spending for the market child care. Labor of individuals, 1 − z _t , is employed either in production of market child care, l _xt , or in goods production, 1 − z _t  − l _xt . The wage rates in both production sectors are the same by arbitrage. Assuming that Eq. 2 holds as in the text, the optimal demand plans are obtained in a similar way as in the text:

$$ c_{t} = \frac{1} {{1 + \gamma + \rho }}{\left[ {{\left( {1 - \tau } \right)}w + \frac{{P_{{t + 1}} }} {r}} \right]} $$

(34a)

$$ m_{t} = \frac{{\gamma \varepsilon \beta }} {{1 + \gamma + \rho }}{\left[ {{\left( {1 - \tau } \right)}w + \frac{{P_{{t + 1}} }} {r}} \right]} $$

(34b)

$$ l_{{xt}} = \frac{1} {w}\frac{{\gamma \beta {\left( {1 - \varepsilon } \right)}}} {{1 + \gamma + \rho }}{\left[ {{\left( {1 - \tau } \right)}w + \frac{{P_{{t + 1}} }} {r}} \right]} $$

(34c)

$$ z_{t} = \frac{1} {{{\left( {1 - \tau } \right)}w}}\frac{{\gamma {\left( {1 - \beta } \right)}}} {{1 + \gamma + \rho }}{\left[ {{\left( {1 - \tau } \right)}w + \frac{{P_{{t + 1}} }} {r}} \right]} $$

(34d)

Making use of the budget equation, we have

$$ s_{t} = \frac{\rho } {{1 + \gamma + \rho }}{\left( {1 - \tau } \right)}w - \frac{{1 + \gamma }} {{1 + \gamma + \rho }}\frac{{P_{{t + 1}} }} {r} $$

(35)

In this case, defining the cost of child rearing as

$$ C_{t} = {\left( {1 - \tau } \right)}wz_{t} + m_{t} + wl_{{xt}} $$

we have

$$ p = \frac{1} {{\theta \varepsilon \beta }}{\left( {\frac{{{\left( {1 - \tau } \right)}w\varepsilon \beta }} {{1 - \beta }}} \right)}^{{1 - \varepsilon \beta }} {\left( {\frac{{\beta {\left( {1 - \varepsilon } \right)}{\left( {1 - \tau } \right)}}} {{1 - \beta }}} \right)}^{{\beta {\left( {\varepsilon - 1} \right)}}} $$

(36)

from which we obtain ${dp} \mathord{\left/ {\vphantom {{dp} {d\tau }}} \right. \kern-\nulldelimiterspace} {d\tau } = { - p{\left( {1 - \beta } \right)}} \mathord{\left/ {\vphantom {{ - p{\left( {1 - \beta } \right)}} {{\left( {1 - \tau } \right)} < 0}}} \right. \kern-\nulldelimiterspace} {{\left( {1 - \tau } \right)} < 0}$. Therefore, assuming labor as well as goods as inputs in the market-child-care production does not alter our result essentially.

Appendix B

1.1 Derivation of dynamic equation (Eq. 16)

Combining Eqs. 14 and 15 (or Eqs. 10 and 12), we have

$$ n_{t} = \theta {\left[ {\frac{\beta } {{1 - \beta }}{\left( {1 - \tau } \right)}w} \right]}^{\beta } z_{t} $$

(37)

Substituting Eq. 37 into Eq. 14, we obtain

$$ z_{t} = \frac{{{\left( {1 - \beta } \right)}\gamma }} {{1 + \gamma + \rho }}\frac{1} {{{\left( {1 - \tau } \right)}w}}{\left[ {{\left( {1 - \tau } \right)}w + \frac{{{\left( {1 - z_{{t + 1}} } \right)}w\tau \theta }} {r}{\left( {\frac{\beta } {{1 - \beta }}{\left( {1 - \tau } \right)}w} \right)}^{\beta } z_{t} } \right]} $$

(38)

Dividing both sides of Eq. 38 by z _t and rearranging them, we have the dynamic Eq. 16.

Appendix C

1.1 Numerical example

We show a numerical example, assuming parameter values of the model as follows: $\beta = 0.3;\quad \theta = 30;\quad \gamma = 0.1;\quad \rho = 0.3{\left[ { \approx {\left( {1 + 0.05} \right)}^{{ - 24}} } \right]};\quad r = 2.033{\left[ { \approx {\left( {1 + 0.03} \right)}^{{24}} } \right]};$ and w = 1.5.

One period is considered to consist of 25 years. These parameters are set so as to have the annual growth rate of population equal to about 1.5%, i.e., about 43% per period, at the tax rate of 20%. The steady state values of variables at various tax rates are presented in Table 2.

Table 2 Numerical example

The results can be summarized as follows: (1) As Proposition 1 predicts, the child-rearing time is increasing with the tax rate; (2) at the tax rates from 0.05 to 0.35, as the condition of (2) of Proposition 2 holds, the fertility rate increases with the tax rate, but the purchases of the market child care decrease because of declining disposable income; (3) at even higher tax rates, as the tax-rate elasticity of child-rearing time becomes greater and the condition of (1) of Proposition 2 comes to be satisfied, the purchases of the market child care come to be increasing with the tax rate. In this situation, the market child care is purchased with the lowered disposable income. This reflects the fact that parents try to have many children to raise the social security benefits when retired; (4) as the weight on the utility from having children is relatively low in this example, the fertility rate rises with the tax increase.

The last column of Table 2 indicates the steady state welfare change at each tax rate. At the tax rates from 0.05 to 0.3, the tax increase diminishes the steady state welfare of individuals, while at the tax rates above 0.35 (up to 0.8, although this is not shown in the table), the tax increase improves the steady state welfare. It should be noted that the situation implied in the third column of Table 1 holds at tax rate 0.35.

Finally, to explain the mechanism of a Pareto improvement, we consider a tax cut of 0.01 from 82.5%. Suppose that the economy is initially at the steady state, then that the government cuts the tax rate by 0.01% in period 0. The time paths of the child-rearing time of individuals and the utility level of each generation are depicted in Fig. 2a,b, respectively. Although the initial tax rate is not plausible, we can see the mechanism of the Pareto improvement. As explained in the text, generation −1, retired in period 0, obtains a windfall of extra benefits because of the increased labor supply of generation 0.

Fig. 2

a Time path of parental child-rearing time. b Lifetime welfare of generations

The simulation analysis of this simple model does not generate plausible values of endogenous variables for the economy. It is partly because some other factors than those taken into account in this study affect the dynamics of the model. However, the conclusion of our study holds at least qualitatively.