## Abstract

Introducing precautionary public health policy into an overlapping generations model with migration of households between regions, we discuss the causality between ageing and urban agglomeration. We analyze the effect of public policy to expand longevity on an equilibrium population distribution between an urban region and a rural region in a steady state. As the result of our analysis, it is possible that the promotion of precautionary public health policy leads to enlarge wage differentials and enlarge regional disparity.

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## Notes

Now we have not analyzed the causality between prevention expenditure and average life span age with econometric method in this paper. However, it is important to analyze these issues with econometric approach. Therefore, the analysis of this causality remains the issue of our future analysis.

Yakita (2011) extends a simple overlapping generation model by incorporating migration of households between regions and shows that the increase of spillover effects of regional public goods affects the population distribution in equilibrium. Introducing trans-boundary pollution caused by the manufactured goods sector in an urban area into Yakita (2011), Naito and Omori (2014) examine the effect of environmental policy on the equilibrium population distribution. Furthermore, introducing Marshallian externalities into Yakita (2011), Ikazaki (2014) shows the effect of social security system on equilibrium.

Following the previous literature such as Yakita (2011) and Naito and Omori (2017) which assume the similar utility function, for the simplicity, we assume the utility function (1). However, when we assume the utility function including consumption at both periods, we can derive the similar results.

Here we assume the competitive insurance market in this model. The saving of households is invested to the capital of production sector in region \(u\). Households who survive in the retirement generation receive interests and capital at the top of retirement generation.

As for the interpretation of urban cost indexed by \(\sigma\), we follow Yakita (2011) and Naito and Omori (2014). We consider any congestion and traffic jam as urban cost. Urban cost may include not only congestion, traffic jams but also child-care costs. The child-care costs are not identical in both regions. We also assume that urban cost is constant for simplification though it is natural to assume that this urban cost depends on the population in region \(u\).

See Appendix 2.

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## Acknowledgements

The previous version of this paper was presented in 54th annual meeting in Japan Section of RSAI and 7th Asian Regional Science Seminar. We are grateful to Osamu Keida, Hikaru Ogawa, Daisuke Nakamura, and anonymous referees for useful comments and suggestions. All errors remained in the paper are our responsibility.

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Funding was provided by Japan Society for the Promotion of Science (Grant nos. 15K03464, 16K12374, 16K03719).

## Appendices

### Appendix 1

Here, we will focus on the equilibrium value of \(G\).

From (21), we obtain

Here, let us assume that \(P/G_{t} > P^{{\prime }} (G_t)\). 1% increase in \(G_t\) produces increment of \(p\) less than 1%. In this case, the left-hand side of this equation increases with \(G_t\).

On the other hand, the right-hand side of the equation decreases with \(G_t\) because \(\theta^{{\prime }} (G_t) < 0\).

In this case, \(G_{t}\) is determined uniquely in equilibrium. It is easy to show that \(G_{t}\) is an increasing function of \(\tau\).

### Appendix 2: The effect of \(\varvec{\tau}\) on steady state under agglomeration

First of all, we define \(\varOmega\) as follows.

Thus, we rewrite \(\Delta\) as follows:

Differentiating \(\Delta\) with respect to \(\tau\), \(\frac{\partial \Delta }{\partial \tau }\) is given by

Moreover, we differentiate \(\Delta\) with respect to \(k^{*}\)

Taking account of the signs of (35) and (36), the sign of \(\frac{{{\text{d}}k^{*} }}{{{\text{d}}\tau }}\) is not determined uniquely.

Since we define \(k^{**}\) as capital accumulation at steady state where 45° line and (28) intersect, we define \(\varGamma\) as the L.H.S of (33). Differentiating \(\varGamma\) with respect to \(\tau\), \(\frac{\partial \varGamma }{\partial \tau }\) is given by

where \(\Delta\) denotes \(\frac{Abz(1 - \gamma )}{\gamma }\). Differentiating \(\varGamma\) with respect to \(k^{**}\), \(\frac{\partial \varGamma }{{\partial k^{**} }}\) is given by

Taking account of (37) and (38), the sign of \(\frac{{{\text{d}}k^{**} }}{{{\text{d}}\tau }}\) is also not determined uniquely.

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Naito, T., Ikazaki, D. & Omori, T. Precautionary public health, ageing and urban agglomeration.
*Asia-Pac J Reg Sci* **1**, 655–669 (2017). https://doi.org/10.1007/s41685-017-0056-y

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DOI: https://doi.org/10.1007/s41685-017-0056-y