Preference for Having Children, Fertility and Social Security

Abstract

In this chapter, incorporating heterogeneity in preference to having children in a small open economy model, we examine the effects of changes in the size of PAYG social security on fertility choices of individuals and population growth of the economy. In this chapter we focus on the differences in fertility rates under two benefits schemes of proportional-to-contribution or uniform benefits.

This chapter is the revised and expanded version of Hirazawa et al. (2014).

Appendix 1

1.1 1.1 Derivation of (6.34)

Inserting (6.33) into (6.32), we have

$$ \begin{array}{l}\frac{d\beta }{d\tau}\left\{1+\beta \left[\frac{z}{{\displaystyle {\int}_0^{\varepsilon_v*}dF\left(\varepsilon \right)}+{\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\left(1-zn\right)dF\left(\varepsilon \right)}}-\frac{1}{\upsilon}\right]\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \beta }dF\left(\varepsilon \right)}\right)\right\}\\ {}\kern1em =\beta \left\{\frac{1}{\tau }-\left[\frac{z}{{\displaystyle {\int}_0^{\varepsilon_v*}dF\left(\varepsilon \right)}+{\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\left(1-zn\right)dF\left(\varepsilon \right)}}-\frac{1}{\upsilon}\right]\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \tau }dF\left(\varepsilon \right)}\right)\right\}\end{array} $$

(6.39)

where, assuming $1-2z\upsilon >0$, we have $\frac{z}{{\displaystyle {\int}_0^{\varepsilon_v*}dF\left(\varepsilon \right)}+{\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\left(1-zn\right)dF\left(\varepsilon \right)}}-\frac{1}{\upsilon}\equiv V<0$.²¹

Differentiating ε _v  * in (6.30) with respect to τ, we have

$$ {\left[1+\frac{\beta }{Rw\left(1-\tau \right)}\right]}^2\frac{d{\varepsilon}_v*}{d\tau }+\frac{\left(1+\rho \right)z}{Rw\left(1-\tau \right)}\frac{d\beta }{d\tau }=-\frac{\left(1+\rho \right)z\beta }{Rw{\left(1-\tau \right)}^2}. $$

(6.40)

From (6.39) and (6.40) we have

$$ \left(\begin{array}{cc}\hfill 1+\beta V\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \beta }dF}\right)\hfill & \hfill 0\hfill \\ {}\hfill \frac{\left(1+\rho \right)z}{Rw\left(1-\tau \right)}\hfill & \hfill {\left[1+\frac{\beta }{Rw\left(1-\tau \right)}\right]}^2\hfill \end{array}\right)\left(\begin{array}{c}\hfill \frac{d\beta }{d\tau}\hfill \\ {}\hfill \frac{d{\varepsilon}_v*}{d\tau}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \frac{\beta }{\tau }-\beta V\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \tau }dF}\right)\hfill \\ {}\hfill -\frac{\left(1+\rho \right)z\beta }{Rw{\left(1-\tau \right)}^2}\hfill \end{array}\right). $$

(6.41)

From (6.41) we obtain

$$ \frac{d\beta }{d\tau }={G}^{-1}\left[\frac{\beta }{\tau }-\beta V\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \tau }dF}\right)\right]{\left[1+\frac{\beta }{Rw\left(1-\tau \right)}\right]}^2 $$

(6.42)

$$ \frac{d{\varepsilon}_v*}{d\tau }={G}^{-1}\frac{-\left(1+\rho \right)z\beta }{Rw\left(1-\tau \right)}\left\{\left[1+\beta V\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \beta }dF}\right)\right]\frac{1}{1-\tau }+\frac{1}{\tau }-V\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \tau }dF}\right)\right\} $$

(6.43)

where $G=\left[1+\beta V\left({\displaystyle {\int}_{\varepsilon_v*}^{\overline{\varepsilon}}\frac{\partial n}{\partial \beta }dF}\right)\right]{\left[1+\frac{\beta }{Rw\left(1-\tau \right)}\right]}^2$. We assume here that $G>0$.²² Therefore, it follows that:

$$ d{\varepsilon}_v*/d\tau <0 $$

(6.34)

and $d\beta /d\tau >0$.