Precautionary public health, ageing and urban agglomeration

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.

Appendices

Appendix 1

Here, we will focus on the equilibrium value of $G$.

From (21), we obtain

$$G_{t} \frac{{[\gamma + (1 - \gamma )p(G_{t} )]}}{{(1 - \gamma )p(G_{t} )}} = \tau Ab\left[ {\left( {(1 - \sigma )^{{\frac{ - \gamma }{{(1 - \gamma )p(G_{t} )}}}} - 1} \right)\theta k_{t} + 1} \right]$$

(34)

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.

$$\varOmega \equiv \left( {\frac{Az(1 - \alpha )}{{\gamma (1 - \sigma )^{\alpha } }}} \right)$$

Thus, we rewrite $\Delta$ as follows:

$$\Delta \equiv \varOmega (1 - \tau )[\gamma + (1 - \gamma )p]^{\alpha } \cdot [(1 - \gamma p)]^{1 - \alpha } k^{*} - k^{*}$$

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

$$\begin{aligned} & \frac{\partial \Delta }{\partial \tau } = \varOmega \left\{ { - [\gamma + (1 - \gamma )p]^{\alpha } [(1 - \gamma )p]^{1 - \alpha } k^{*} } \right. \\ & + (1 - \tau )\alpha \left[ {(1 - \alpha )\frac{{{\text{d}}p}}{{{\text{d}}G_{t} }}\frac{{\partial G_{t} }}{\partial \tau }} \right][\gamma + (1 - \gamma )p]^{\alpha } [(1 - \gamma )p]^{1 - \alpha } k^{*} \\ & + \left. {(1 - \tau )[\gamma + (1 - \gamma )p]^{\alpha } (1 - \alpha )\left[ {(1 - \gamma )\frac{{{\text{d}}p}}{{{\text{d}}G_{t} }}\frac{{\partial G_{t} }}{\partial \tau }} \right]k^{*} } \right\}\frac{ > }{ < }0. \\ \end{aligned}$$

(35)

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

$$\begin{aligned} & \frac{\partial \Delta }{{\partial k^{*} }} = \varOmega (1 - \tau )\left\{ {\alpha \left[ {(1 - \gamma )\left( {\frac{{{\text{d}}p}}{{{\text{d}}G_{t} }}\frac{{{\text{d}}G_{t} }}{{{\text{d}}k^{*} }}} \right)} \right][\gamma + (1 - \gamma )p]^{\alpha - 1} [(1 - \gamma )p]^{1 - \alpha } k^{*} } \right. \\ & + \left. {[\gamma + (1 - \gamma )p]^{\alpha } (1 - \alpha )\left[ {(1 - \gamma )\frac{{{\text{d}}p}}{{{\text{d}}G_{t} }}\frac{{\partial G_{t} }}{{\partial k^{*} }}} \right][(1 - \gamma )p]^{ - \alpha } k^{*} ] - 1} \right\}\frac{ > }{ < }0 \\ \end{aligned}$$

(36)

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

$$ \begin{aligned} & \frac{\partial \varGamma }{\partial \tau } = \frac{\varLambda }{{\{ \gamma (1 - \sigma \theta k^{**} )\}^{2} }}\left[ {\gamma (1 - \sigma \theta k^{**} )\left\{ { - \tau p\left[ {1 + \theta k^{**} (1 - \sigma )^{{\frac{ - \gamma }{(1 - \gamma )p}}} - 1} \right]} \right.} \right] \\ & + (1 - \tau )\left[ {1 + \theta k^{**} (1 - \sigma )^{{\frac{ - \gamma }{(1 - \gamma )p}}} - 1} \right]\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{\partial \tau } \\ & + (1 - \tau )p\left[ {\left( {\frac{\partial \theta }{\partial \eta }\frac{\partial \eta }{\partial p}\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{\partial \tau } + \frac{\partial \theta }{\partial p}\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{\partial \tau }} \right)(1 - \sigma )^{{\frac{ - \gamma }{(1 - \gamma )p}}} } \right. \\ & + \theta k^{**} \frac{1}{{\left. {\left. {(1 - \gamma )p^{2} } \right\}{ \ln } (1 - \sigma )^{{1 - \frac{\gamma }{(1 - \gamma )p}}} } \right]}} \\ & + (1 - \tau )p\left[ {1 + \theta k^{**} (1 - \sigma )^{{\frac{ - \gamma }{(1 - \gamma )p}}} - 1} \right] \\ & \times \gamma \sigma k^{**} \left. {\left( {\left( {\frac{\partial \theta }{\partial \eta }\frac{\partial \eta }{\partial p}\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{\partial \tau } + \frac{\partial \theta }{\partial p}\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{\partial \tau }} \right)} \right)} \right]\frac{ > }{ < }0, \\ \end{aligned} $$

(37)

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

$$\begin{aligned} \frac{\partial \varGamma }{{\partial k^{**} }} = \frac{\varLambda (1 - \tau )}{{\{ \gamma (1 - \sigma \theta k^{**} )\}^{2} }}\left[ {\gamma (1 - \sigma \theta k^{**} )\left\{ {\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{{\partial k^{**} }}} \right.\left[ {1 + \theta k^{**} (1 - \sigma )^{{\frac{ - \gamma }{(1 - \gamma )p}}} - 1} \right]} \right. \hfill \\ \quad + p\left[ {k^{**} (1 - \sigma )^{{\frac{ - \gamma }{(1 - \gamma )p}}} \left( {\frac{\partial \theta }{\partial \eta }\frac{\partial \eta }{\partial p}\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{{\partial k^{**} }} + \frac{\partial \theta }{\partial p}\frac{\partial p}{{\partial G_{t} }}\frac{{\partial G_{t} }}{{\partial k^{**} }}} \right)} \right. \hfill \\ \left.\left. {\quad + \theta (1 - \sigma )^{{^{{\frac{ - \gamma }{(1 - \gamma )p}}} }} + \theta k^{**} \frac{1}{{(1 - \gamma )p^{2} }}\ln (1 - \sigma )^{{1 - \frac{\gamma }{(1 - \gamma )p}}} } \right]\right\} \hfill \\ \quad \left. { + p\left[ {1 + \theta k^{**} (1 - \sigma )^{{\frac{ - \gamma }{(1 - \gamma )p}}} - 1} \right]\left( {\gamma \sigma \left( {\frac{\partial \theta }{{\partial k^{**} }}k^{**} + \sigma \theta } \right)} \right) } \right] - 1\frac{ > }{ < }0 \hfill \\ \end{aligned}$$

(38)

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