Fertility, Inequality and Income Growth


This paper sets an endogenous fertility model with a two-sector model: one for the final goods sector and the other for child care service sector. Results of theoretical analysis indicate that the subsidy for children raises the labor share of the child care service sector and that it can increase fertility. An aging population reduces fertility and the labor share of the child care service sector. In addition to these results, we consider monetary policy effects on fertility. Results show that monetary policy can raise fertility and the labor share of the child care service sector by virtue of an increase in the pension benefit if a pay-as-you-go pension exists.

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Fig. 1


  1. 1.

    An aging population reduces fertility. Therefore, demand for the child care service sector decreases. However, some papers consider the grandparents’ child care in endogenous fertility, as demonstrated by Yasuoka et al. (2009).

  2. 2.

    Based on data from the Cabinet Office, Japan, the monetary stock increases considerably. In 2015, the monetary stock was 906.4 trillion JPY; it reached 974.0 trillion JPY. 7.4% rises. The inflation rate increased by 0.4% during 2015–2017. In some OECD countries, the policy of an increase in monetary stock is provided to avoid the bad economic status.

  3. 3.

    As shown by Ashraf et al. (2013), economic growth can reduce fertility because of an increase in the opportunity cost of having children, as shown by Galor and Weil (1996). Apps and Rees (2004) derive positive correlation between fertility and income. However, if the child care service cost depends on the income level, as shown by Yasuoka and Miyake (2010), then the fertility can not change fertility with income. Our paper depends positively on the income in the model with pension benefit because income growth raises the pension benefit. Then, an increase in lifetime income raises fertility.

  4. 4.

    This is a standard utility function in endogenous fertility model, as assumed by van Groezen et al. (2003) and others. The utility is pulled up directly by fertility.

  5. 5.

    This production function, which includes only labor input, is assumed by Yasuoka and Miyake (2010), Hashimoto and Tabata (2010), and others for studies in which child care or elderly care is examined. Child care and elderly care are regarded as labor-intensive services.

  6. 6.

    \(\left({\sigma }_{t}^{*}-\frac{{\sigma }_{t}^{*2}}{2}\right)+{\left(1-{\sigma }_{t}^{*}\right)}^{2}\) can be shown simply by \(\frac{{\sigma }_{t}^{*2}}{2}-{\sigma }_{t}^{*}+1\). However, our paper remains in the form of \(\left({\sigma }_{t}^{*}-\frac{{\sigma }_{t}^{*2}}{2}\right)+{\left(1-{\sigma }_{t}^{*}\right)}^{2}\). Also, \(\left({\sigma }_{t}^{*}-\frac{{\sigma }_{t}^{*2}}{2}\right){w}_{t}\) and \({\left(1-{\sigma }_{t}^{*}\right)}^{2}{w}_{t}\) respectively show the labor income share of the final goods sector and the child care service sector. We retain the form of \(\left({\sigma }_{t}^{*}-\frac{{\sigma }_{t}^{*2}}{2}\right)+{\left(1-{\sigma }_{t}^{*}\right)}^{2}\) because we can not understand the economic meanings of \(\frac{{\sigma }_{t}^{*2}}{2}-{\sigma }_{t}^{*}+1\).

  7. 7.

    The tax revenue of the final goods sector and child care service sector are shown respectively as \(\varepsilon \left({\sigma }_{t}^{*}-\frac{{\sigma }_{t}^{*2}}{2}\right){w}_{t+1}\) and \(\varepsilon {\left(1-{\sigma }_{t}^{*}\right)}^{2}{w}_{t+1}\). Because of the intergenerational population ratio \({n}_{t}\), the pension benefit per capita is given as (19).

  8. 8.

    For simplicity, \(\tau =0\mathrm{ \,and }\,\varepsilon =0\) are assumed in this section. There exist child allowance and pension benefits in this model if \(\tau \mathrm{and }\varepsilon \) are not zero. The child allowance raises demand for child care service because of a decrease in net child care cost. The pension benefit raises the demand for child care services because of an increase in lifetime income. Therefore, \({\sigma }^{*}\) in the case of \(\tau >0\, \mathrm {and} \,\varepsilon >0\) is less than \({\sigma }^{*}\) in the case of \(\tau =0\mathrm{ and }\varepsilon =0\). However, the strong effects of the aging population on the labor share of child care sector and fertility do not change.

  9. 9.

    The fertility is given by (20) if we consider the pension benefit. Then, the fertility depends on income growth. Income growth raises the pension benefit and then lifetime income increases. An increase in lifetime increase income fertility. Income growth pulls up the demand for child care service because of an increase in the fertility. Then the labor share of child care service sector rises.

  10. 10.

    (33) and (34) can be reduced to \(\frac{dn}{dq}>0\). This equation can be generally obtained in an endogenous fertility model. As reported by Fanti and Gori (2009, 2012), a child allowance can not always raise fertility because of a decrease in capital stock per capita. However, our paper can always raise fertility because child care service costs depend on the wage rate. Then, by virtue of increased fertility, the demand for child care service rises and the labor share of child care service can be pulled up.

  11. 11.

    Both the fertility and the labor share of child care service sector increase. This outcome is consistent with Fig. 1. However, the detail check must be examined to ascertain whether the model fits the data, as shown by Fig. 1.

  12. 12.

    The Appendix presents a detailed proof. The model with a pension benefit brings about the dynamics. The Appendix presents dynamics of the model with a pension.

  13. 13.

    As shown by (5), the wage rate of the child care service sector depends on the wage rate of the final goods sector. In addition to an increase in wage rate of final goods sector, an increase in the demand for the child care service sector raises the wage rate of child care service sector, which leaves a decrease in \({\sigma }^{*}\).


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We would like to thank the anonymous reviewers for the beneficial comments. Research for this paper was supported financially by JSPS KAKENHI Grant Numbers 17K03746 and 17K03791. Nevertheless, any remaining errors are the authors’ responsibility.


JSPS KAKENHI Grant numbers 17K03746 and 17K03791.

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Correspondence to Masaya Yasuoka.

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The Sign of \(\frac{{\varvec{d}}{{\varvec{\sigma}}}^{\boldsymbol{*}}}{{\varvec{d}}{\varvec{g}}}\) (37)

Defining \(L= \alpha \left(\left(1-\varepsilon \right)\left(\left({\sigma }^{*}-\frac{{\sigma }^{*2}}{2}\right)+{\left(1-{\sigma }^{*}\right)}^{2}\right)+\frac{\varepsilon n\left(1+g\right)}{1+r}\right)\) and \(R={\left(1-{\sigma }^{*}\right)}^{2}\) as shown by (21), respectively, we can depict the following figure. (Fig. 

Fig. 2

Increase in \(g\)


We can obtain the intersection of L and R and demonstrate that an increase in \(g\) reduces \({\sigma }^{*}\).

Dynamics in the Model with a Pay-as-you-Go Pension

With a pay-as-you-go pension, the dynamics occurs in the model. The dynamics in this model is shown as

$${n}_{t}=\frac{\alpha \rho \left(\left(1-\varepsilon \right)\left({\sigma }_{t}-\frac{{\sigma }_{t}^{2}}{2}+{\left(1-{\sigma }_{t}\right)}^{2}\right)+\frac{\varepsilon {n}_{t}\left(1+{g}_{t}\right)\left({\sigma }_{t+1}-\frac{{\sigma }_{t+1}^{2}}{2}+{\left(1-{\sigma }_{t+1}\right)}^{2}\right)}{1+r}\right)}{1-{\sigma }_{t}}.$$
$${n}_{t}=\rho \left(1-{\sigma }_{t}\right).$$
$$1+{g}_{t}=\frac{1}{{\sigma }_{t+1}{n}_{t}}\times \left(\left(1-\alpha -\frac{\beta }{1-\frac{1}{1+\pi }\frac{1}{1+r}}-\gamma \right)\left(1-\varepsilon \right)\left(1-\theta \right){a}^{1-\theta }\right.\left({\sigma }_{t}-\frac{{\sigma }_{t}^{2}}{2}+{\left(1-{\sigma }_{t}\right)}^{2}\right)$$
$$-\frac{\left(\alpha +\frac{\beta }{1-\frac{1}{1+\pi }\frac{1}{1+r}}+\gamma \right)\varepsilon {n}_{t}\left(1+{g}_{t}\right)\left({\sigma }_{t+1}-\frac{{\sigma }_{t+1}^{2}}{2}+{\left(1-{\sigma }_{t+1}\right)}^{2}\right)}{1+r}$$
$$1+{\pi }_{t}=\frac{1+\mu }{\left(1+{g}_{t}\right){n}_{t}}.$$

We replace (41), (43), and (44) with the following equations as

$${n}_{t}=n\left({\sigma }_{t},{\sigma }_{t+1},{g}_{t} \right),$$
$${g}_{t}=g\left({n}_{t},{\pi }_{t},{\sigma }_{t},{\sigma }_{t+1} \right),$$
$${\pi }_{t}=\pi \left({n}_{t}, {g}_{t}\right).$$

The values of \(\frac{d{\sigma }_{t+1}}{d{\sigma }_{t}}\) are shown as

$$\frac{d{\sigma }_{t+1}}{d{\sigma }_{t}}=\frac{-\rho +\frac{\partial n}{\partial {\sigma }_{t}}-\frac{\partial n}{\partial {g}_{t}}\frac{\frac{\partial g}{\partial {\sigma }_{t}}-\rho \left(\frac{\partial g}{\partial {n}_{t}}+\frac{\partial g}{\partial \pi }\frac{\partial \pi }{\partial {n}_{t}}\right)}{1-\frac{\partial g}{\partial \pi }\frac{\partial \pi }{\partial {g}_{t}}}}{\frac{\partial n}{\partial {\sigma }_{t+1}}+\frac{\frac{\partial n}{\partial {g}_{t}}\frac{\partial {g}_{t}}{\partial {\sigma }_{t+1}}}{1-\frac{\partial g}{\partial \pi }\frac{\partial \pi }{\partial {g}_{t}}}}.$$

The local stability condition is \(-1<\frac{d{\sigma }_{t+1}}{d{\sigma }_{t}}<1\) in the steady state.

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Shintani, M., Yasuoka, M. Fertility, Inequality and Income Growth. Ital Econ J (2021). https://doi.org/10.1007/s40797-021-00143-6

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  • Aging population
  • Fertility
  • Income growth
  • Monetary policy
  • Subsidy

JEL Classification

  • J11
  • J14
  • E31
  • H22