Rural-to-urban migration in developing economies: characterizing the role of the rural labor supply in the process of urban agglomeration and city growth

Abstract

This paper aims to contribute to the literature on rural-to-urban migration in developing economies by shedding more light on the hitherto little-investigated linkage between the rural sector and the process of urban development. To this end, we combine existing spatial rural-to-urban migration frameworks and extend them by including some key assumptions of the urban development theory. Specifically, we construct a dynamic rural-to-urban migration model in which the links between the urban and rural areas are modeled using the spatial variables of rural land rents and an urban productivity spillover, in order to investigate the contribution of the rural labor supply to urban agglomeration and city growth. The results of our model indicate that rural land rents affect the optimal level of urban agglomeration in scenarios of both unplanned and planned migration, and that the city planner’s solution of optimal agglomeration may not necessarily be inferior to the market solution. In addition, we find that due to their impact on rural workers’ incomes, the spatial variables exert an influence on the steady-state level of growth to which the urban economy can converge. Our study bears policy implications, including the need for city planners to consider spatial variables in order to achieve the desired trade-off between city growth and urban agglomeration.

Appendices

Appendix 1

Dynamics of $\pi _{t+1}^p$ and $\pi _{t+1}^f$ functions

The dynamic evolution of the technological level of the economy will depend on the shape of the function $\pi _{t+1}$. Specifically:

$$\begin{aligned} \pi _{t+1} = {\left\{ \begin{array}{ll} \pi ^0 &{} \leftrightarrow \; \delta _t = 0 \\ \pi _{t+1}^p &{} \leftrightarrow \; 0< \delta _t < 1 \; (with\, \pi _{t+1}^p> 0\wedge \,(\delta _t s_t)^\rho> \pi _t)\\ \pi _{t+1}^f &{} \leftrightarrow \; \delta _t = 1 \; (with\,\pi _{t+1}^f> 0\,\wedge \,(\delta _t s_t)^\rho > \pi _t)\\ \end{array}\right. } \end{aligned}$$

where:

$$\begin{aligned} \pi _{t+1}^p = \left[ \varTheta ^{2/(1-\gamma )} \left( \frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\right) + \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}\right] ^\rho \end{aligned}$$

and:

$$\begin{aligned} \pi _{t+1}^f = \varTheta ^{\rho /(1-\gamma )} \end{aligned}$$

From this, $\pi _{t+1}^p < 0$ when the following conditions are met: $\pi _t< \frac{\phi }{\kappa ^{u} \varTheta ^{(\gamma -1)/(1-\gamma )}} \qquad \wedge \qquad \pi _t < \frac{2\hat{r} + \bar{\varPsi } \omega }{2 \kappa ^r}\nonumber$, or $\pi _t < \frac{\phi }{\kappa ^{u} \varTheta ^{(\gamma -1)/(1-\gamma )}} \qquad \wedge \qquad \pi _t > \frac{2\hat{r} + \bar{\varPsi } \omega }{2 \kappa ^r}$, with $\varTheta ^{2/(1-\gamma )} \left( \frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\right) > \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}$. The same conditions are always satisfied when $\pi _t > \frac{\phi }{\kappa ^{u} \varTheta ^{(\gamma -1)/(1-\gamma )}} \qquad \wedge \qquad \pi _t < \frac{2\hat{r} + \bar{\varPsi } \omega }{2 \kappa ^r}\nonumber$, with $\varTheta ^{2/(1-\gamma )} \left( \frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\right) < \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi }\omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}$. Otherwise, the function $\pi _{t+1}^p$ is strictly positive.

In differentiating $\pi _{t+1}^p$ with respect to $\pi _t$, we obtain:

$$\begin{aligned} \begin{aligned} \frac{\partial {\pi _{t+1}^p}}{\partial {\pi _t}}&= \rho \left[ \varTheta ^{2/(1-\gamma )} \left( \frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\right) \right. \\&\quad \left. + \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}\right] ^{\rho -1} \\&\quad \times \left[ \frac{2 \varTheta ^{2/(1-\gamma )} \kappa ^{u} \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )} - \frac{2 \varTheta ^{1/(1-\gamma )} \kappa ^r}{\bar{\varPsi }(2 q + \omega )}\right] \end{aligned} \end{aligned}$$

In the relevant range where $\pi _{t+1}^p$ and $\delta _t$ are both positive, the first expression in brackets of $\frac{\partial {\pi _{t+1}^p}}{\partial {\pi _t}}$ is positive. The second expression in brackets of $\frac{\partial {\pi _{t+1}^p}}{\partial {\pi _t}}$ is positive as long as $\varTheta > (\frac{\kappa ^r}{\kappa ^{u}})^{\frac{1-\gamma }{\gamma }}$, which can be rewritten as $\pi _t > \frac{\phi }{\gamma } {\kappa ^r}^\frac{1-\gamma }{\gamma } (\frac{1}{\kappa ^{u}})^\frac{1}{\gamma }$, which is always the case in the relevant range in question. The function reaches its minimum at $\pi _t = \frac{\phi }{\gamma } {\kappa ^r}^\frac{1-\gamma }{\gamma } (\frac{1}{\kappa ^{u}})^\frac{1}{\gamma }$ and is monotonically increasing thereafter.

The evolution of the economy will eventually depend on the curvature of the $\pi _{t+1}$ function. In particular, the number of times that this function intersects the $45^\circ$ line (where $\pi _{t+1}^p = \pi _t$) will be a crucial factor for the dynamic path of the economy. Subsequently, in taking the second derivative of $\pi _{t+1}^p$, we get:

$$\begin{aligned} \begin{aligned} \frac{\partial ^2{\pi _{t+1}^p}}{\partial {\pi _t}^2}&= \rho (\rho -1) \bigg [\varTheta ^{2/(1-\gamma )} \bigg (\frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\bigg ) + \varTheta ^{1/(1-\gamma )} \\&\quad \times \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}\bigg ]^{\rho -2} \bigg [\frac{2 \varTheta ^{2/(1-\gamma )} \kappa ^{u} \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )} - \frac{2 \varTheta ^{1/(1-\gamma )} \kappa ^r}{\bar{\varPsi }(2 q + \omega )}\bigg ]^2 \end{aligned} \end{aligned}$$

To simplify the calculations, label $\varTheta ^{2/(1-\gamma )} \left( \frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\right) = \varPhi$ and $\varTheta ^{1/(1-\gamma )} \times \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )} = \varOmega$.

The function $\pi _{t+1}^p$ will be convex when $\frac{\partial ^2{\pi _{t+1}^p}}{\partial {\pi _t}^2}$ is positive, and this happens when both $\varPhi <0$, $\varOmega <0$. Or when $\varPhi >0$ and $\varOmega <0$, with $|\varPhi |< |\varOmega |$. Or $\varPhi <0$ and $\varOmega >0$, with $|\varPhi |> |\varOmega |$. In these three cases, the slope of the monotonically increasing function of motion will be increasing for $\forall {\pi _t}$ and therefore there will always be a single intersection point where $\pi _{t+1}^p = \pi _t$.

Conversely, $\pi _{t+1}^p$ will be concave when $\frac{\partial ^2{\pi _{t+1}^p}}{\partial {\pi _t}^2}$ is negative, and this happens when both $\varPhi >0$, $\varOmega >0$. Or when $\varPhi >0$ and $\varOmega <0$, with $|\varPhi |> |\varOmega |$. Or $\varPhi <0$ and $\varOmega >0$, with $|\varPhi |< |\varOmega |$. In these cases, the slope of the monotonically increasing function of motion will be decreasing and therefore there could be zero, one or two values of $\pi _t$ to allow tangency points (i.e., when $\pi _{t+1}^p = \pi _t$); depending on the values of the parameters, it can hence be the case that for $\forall {\pi _t}$, $\pi _{t+1}^p < \pi _t$. In this case, there will be no tangency points. On the other hand, if the condition $\pi _{t+1}^p = \pi _t$ is verified, we have one single crossing point. Finally, if $\pi _{t+1}^p > \pi _t$, there will be two tangency points.

For the study of the $\pi _{t+1}^f$ function, the calculations are relatively easier. In differentiating $\pi _{t+1}^f$, we get:

$$\begin{aligned} \frac{\partial {\pi _{t+1}^f}}{\partial {\pi _t}} = \frac{\rho }{1-\gamma } \varTheta ^{\frac{\rho }{1-\gamma }-1} \end{aligned}$$

The function is hence monotonically increasing. Taking the second derivative, we obtain:

$$\begin{aligned} \frac{\partial ^2{\pi _{t+1}^f}}{\partial {\pi _t}^2} = \frac{\rho }{1-\gamma } \left( {\frac{\rho -1 + \gamma }{1-\gamma }}\right) \varTheta ^{\frac{\rho }{1-\gamma }-2} \end{aligned}$$

Therefore, $\pi _{t+1}^f$ will be monotonically increasing and globally concave if $\gamma + \rho < 1$, and monotonically increasing but globally convex if $\gamma + \rho > 1$.

Appendix 2

$\partial {\pi _{t+1}^p}/\partial {\pi _t} > \partial {\pi _{t+1}^f}/\partial {\pi _t}$.

Proof

Consider the point where $\delta _t = 1$, and $\pi _{t+1}^p = \pi _{t+1}^f$. Recalling that:

$$\begin{aligned} \begin{aligned} \frac{\partial {\pi _{t+1}^p}}{\partial {\pi _t}}&= \rho \left[ \varTheta ^{2/(1-\gamma )} \left( \frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\right) \right. \\&\quad \left. + \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}\right] ^{\rho -1} \\&\quad \times \left[ \frac{2 \varTheta ^{2/(1-\gamma )} \kappa ^{u} \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )} - \frac{2 \varTheta ^{1/(1-\gamma )} \kappa ^r}{\bar{\varPsi }(2 q + \omega )}\right] \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial {\pi _{t+1}^f}}{\partial {\pi _t}} = \frac{\rho }{1-\gamma } \varTheta ^{\frac{\rho }{1-\gamma }-1} \end{aligned}$$

at the intersection of the two functions we have that:

$$\begin{aligned} \frac{\partial {\pi _{t+1}^f}}{\partial {\pi _t}} = \frac{\rho }{1-\gamma } {\pi _{t+1}^f} {\pi _t}^{-1} = \frac{\rho }{1-\gamma } {\pi _{t+1}^p} {\pi _t}^{-1} \end{aligned}$$

Assume now that $\frac{\partial {\pi _{t+1}^p}}{\partial {\pi _t}} > \frac{\partial {\pi _{t+1}^f}}{\partial {\pi _t}}$. This entails that:

$$\begin{aligned} \begin{aligned}&\rho \bigg [\varTheta ^{2/(1-\gamma )} \bigg (\frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )}\bigg )\\&\quad + \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}\bigg ]^{\rho -1} \\&\quad \times \bigg [\frac{2 \varTheta ^{2/(1-\gamma )} \kappa ^{u} \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + \omega )} - \frac{2 \varTheta ^{1/(1-\gamma )} \kappa ^r}{\bar{\varPsi }(2 q + \omega )}\bigg ] > \frac{\rho }{1-\gamma } {\pi _t}^{-1} \bigg [\varTheta ^{2/(1-\gamma )} \\&\quad \bigg (\frac{-2\phi }{\bar{\varPsi }(2 q + \omega )} + \frac{2 \kappa ^{u} \pi _t \varTheta ^{(\gamma -1)/(1-\gamma )}}{\bar{\varPsi }(2 q + t)}\bigg ) + \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi } \omega - 2 \kappa ^r \pi _t}{\bar{\varPsi }(2 q + \omega )}\bigg ]^\rho \end{aligned} \end{aligned}$$

Simplifying, we obtain:

$$\begin{aligned} \begin{aligned}&\left( \frac{1}{1-\gamma }-1\right) \frac{2\varTheta ^{(\gamma +1)/(1-\gamma )} \kappa ^{u}}{\bar{\varPsi }(2 q + \omega )} \\&\quad + \bigg (1 - \frac{1}{1-\gamma }\bigg ) \frac{2\varTheta ^{1/(1-\gamma )} \kappa ^{r}}{\bar{\varPsi }(2 q + \omega )} + \frac{1}{1-\gamma } \bigg [\frac{-2\varTheta ^{2/(1-\gamma )} \phi }{\pi _t \bar{\varPsi }(2 q + \omega )} \\&\quad + \varTheta ^{1/(1-\gamma )} \frac{2\hat{r} + \bar{\varPsi } \omega }{\pi _t \bar{\varPsi }(2 q + \omega )}\bigg ] < 0 \end{aligned} \end{aligned}$$

which leads to:

$$\begin{aligned} \begin{aligned}&(\pi _t \kappa ^{u})^{1/(1-\gamma )} (\gamma \phi ^{-1})^{\gamma /(1-\gamma )} - \gamma ^{\gamma /(1-\gamma )} (\pi _t \kappa ^{u})^{1/(1-\gamma )} \phi ^{\gamma /(\gamma -1)}\\&\quad + \gamma ^{-1} \left[ \hat{r} + \frac{\bar{\varPsi } \omega }{2}\right] < \kappa ^r \pi _t \end{aligned} \end{aligned}$$

or:

$$\begin{aligned} \frac{\hat{r} + \frac{\bar{\varPsi } \omega }{2}}{\gamma } < \kappa ^r \pi _t \end{aligned}$$

Since $y_{t}^{r} \ge 0$ and $0 < \delta _t \le 1$, given expression (2), the above inequality is always true. $\square$