Regional differences in childcare environment, urbanization, and fertility decline

Abstract

We theoretically demonstrate that a decline in an economy’s total fertility rate occurs with urbanization. If there are no regional differences in fertility rates, the decline in the economy’s total fertility rate should be explained by a decrease in the number of children per woman. However, if there are regional differences in fertility rates, urbanization can depress the economy’s total fertility rate even if the number of children per woman does not decline due to urbanization. We construct an overlapping generations model with two regions (u and r). Region u has a lower fertility rate, but its higher wage keeps attracting more workers who will be parents. Since the economy’s total fertility rate is the weighted sum of the low fertility rate in region u and the high fertility rate in region r, urbanization and a decline in the economy’s total fertility rate are observed simultaneously, even if urbanization does not change the regional fertility rates. An implication for population policies aimed at increasing the economy’s total fertility rate is provided. Such policies must not only focus on increasing the fertility rate in each region, but also consider migration to regions with lower (higher) fertility rates.

Appendices

Derivation of (16)

Substituting (11) and (12) into (15), we obtain

$$\begin{aligned} \frac{\phi _{t}}{k_{t}}=\left\{ \frac{b}{(1-\tau )(1-\alpha )(1-\sigma )^{-\alpha }(1-\gamma )^{-\alpha }(1-\sigma )^{\frac{1}{1-\gamma }}}\left( \frac{1}{\mu }\right) ^{\frac{\gamma }{1-\gamma }}\left[ \mu +(1-\mu )p_{t} \right] ^{\frac{\gamma }{1-\gamma }}\right\} ^{\frac{-1}{\alpha }} \end{aligned}$$

(A.1)

Considering that $p_t$ in (A.1) is a function of $\frac{\phi _t}{k_t}$ as in (14), the derivative of $\phi _t$ with respect to $k_t$ is obtained as

$$\begin{aligned} \frac{d\phi _{t}}{d k_{t}}=\frac{\phi _{t}}{k_{t}} \end{aligned}$$

(A.2)

Using the implicit function theorem for the case $\frac{\phi _t}{k_t}>0$, $\phi _t$ can be expressed as a function of $k_t$:

$$\begin{aligned} \phi _t=\Phi (k_t;\tau ). \end{aligned}$$

Then, the integration of (A.2) shows that $\phi _t$ is a linear function of $k_t$ such that

$$\begin{aligned} \phi _t=\Omega (\tau ) k_t. \end{aligned}$$

where $\Omega (\tau )$ is characterized by the parameters in (A.1).

Proof of convexity of $\Psi _1$

Considering that $s^u_t$ and $s^r$ are constant, as in (28) and (30), and differentiating (26) with respect to $k_t$, we derive

$$\begin{aligned} \frac{dk_{t+1}}{dk_t}=\frac{1}{m_t^2}\left\{ \frac{\partial \phi _t}{\partial k_t} (s_t^u-s^r) m_t -\left[ \phi _ts_t^u+(1-\phi _t)s^r \right] \frac{\partial m_t}{\partial k_t} \right\} >0. \end{aligned}$$

(B.1)

As $(s_t^u-s^r)$ is positive and $\frac{\partial m_t}{\partial k_t}$ is negative, the sign of (B.1) is strictly positive. Moreover, differentiating (B.1) with respect to $k_t$ yields the following derivative:

$$\begin{aligned} \frac{d^2k_{t+1}}{dk_t^2}=-2\frac{1}{m_t^3} \frac{\partial m_t}{\partial k_t} \left\{ \frac{\partial \phi _t}{\partial k_t} (s_t^u-s^r) m_t -\left[ \phi _ts_t^u+(1-\phi _t)s^r \right] \frac{\partial m_t}{\partial k_t} \right\} >0. \end{aligned}$$

(B.2)