Agglomeration, search frictions and growth of cities in developing economies

Abstract

We study agglomeration economies from a dynamic search-matching perspective to account for the growth of cities in developing economies. Agglomeration economies are not only the cause but also the consequence of migratory inflows. Without frictions, however, agglomeration economies as a labor-pull factor will attract workers instantaneously to a region, which is inconsistent with the data. We argue that agglomeration benefits combined with search and additional frictions (e.g., fixed costs and imperfect recognition of the benefits) can generate a transitory endogenous growth path that is consistent with the set of empirical regularities observed over the economic development process: gradual rural-to-urban migration and the resulting urban concentration, productivity improvement, urban unemployment, and wage and unemployment gaps across regions, which diminish with economic development. We also characterize optimal urbanization and draw some policy implications including the necessity of regulating urban concentration.

Appendix: Proof of Proposition 1

We prove Proposition 1 using the model with $\delta =FC=0$ for analytical tractability and leave the proof for the extended model to the numerical calibration section. To present the existence of equilibrium at a particular period t, we determine $\theta _t$ first for a given $p_{t}$, which is set externally by Eq. (1): ${p}'=p+g\left( {l_1 +u_1 } \right)$ and updates in each period. A closed-form solution for $\theta _t$ is not possible because of the nonlinearity of the matching probability function, but Brouwer’s fixed-point theorem can be used to prove the existence of $\theta _t$ from (18).³⁷ $q_{t}$ can be determined immediately [i.e., $q_t =q(\theta _t )]$ and then $w_{t}$ is obtained from (19). Next, with the equilibrium values of $\theta _t$, $q_{t}$ and $w_{t}$ already determined, we can solve for $u_1$ and $u_2$ using the Beveridge equation (14-2) and (20). The uniqueness of $\theta _t$ can also be shown by $\;d\left[ {\frac{1}{\gamma }\left( {\frac{p_t (1-\gamma )}{c\;}-\frac{r+\lambda }{q(\theta _t )}} \right) } \right] /\mathrm{d}\theta _t =\underbrace{\frac{(r+\lambda ){q}'(\theta _t )}{\gamma q(\theta _t )^{2}}}_{(-)}<1$ for all $\theta _t$. Other variables are recursively determined, and we can verify the existence and uniqueness of equilibrium.

To show the endogenous growth path, we make use of urban productivity updating by Eq. (1): $p^{{\prime }}=p+g(l_1 +u_1 )$. We can define the next period’s equilibrium by repeating the aforementioned process at a given value of ${p}'$. When the agglomeration externality remains positive with $g\left( {l_1 +u_1 } \right) >0$, we need to keep updating urban productivity and repeat the above process of finding equilibrium, which constitutes an endogenous transitional growth path.

Now, we prove how a greater ${p}'$ leads to a decrease in the non-urban population, suggesting a higher urban concentration. First, using Eq. (18) we obtain the following derivative:

$$\begin{aligned} \frac{\mathrm{d}\theta }{\mathrm{d}p}\;=\frac{1-\gamma }{\gamma c}/A>0,\quad \hbox {where}\quad A\;\equiv 1-\frac{(r+\lambda ){q}'(\theta )}{\gamma q(\theta )^{2}}>0. \end{aligned}$$

(28-1)

This property in turn leads to an increase in the job arrival rate $\theta q(\theta )$ and a decrease in the worker-arrival rate $q(\theta )$ in response to an increase in p.³⁸ Using these results, we can verify that w rises with p in Eq. (16) [see (28-2) below]. Finally, we can find from Eq. (22) that the non-urban population declines as p rises. To prove this in (28-5), we derive the partial derivatives (28-3) and (28-4) from equation (22) and use (28-1) and (28-2) as well.³⁹

$$\begin{aligned}&\frac{\mathrm{d}w}{\mathrm{d}p}=\gamma \left( {1+c\cdot \underbrace{\frac{\mathrm{d}\theta }{\mathrm{d}p}}_{(+)}} \right) >0, \end{aligned}$$

(28-2)

$$\begin{aligned}&\frac{\partial u_2 }{\partial \theta }=-\frac{1}{1-\beta }u_2 ^{\beta }\left( {\frac{\beta w\left( {r+\lambda } \right) \overbrace{\mathrm{d}\theta q(\theta )/\mathrm{d}\theta }^{(+)}}{[w\theta q(\theta )]^{2}}} \right) <0.\end{aligned}$$

(28-3)

$$\begin{aligned}&\frac{\partial u_2 }{\partial w}=-\frac{1}{1-\beta }u_2 ^{\beta }\left( {\frac{\beta (r+\lambda +\theta q(\theta ))}{w^{2}\theta q(\theta )}} \right) <0, \end{aligned}$$

(28-4)

$$\begin{aligned}&\frac{\mathrm{d}u_2 }{\mathrm{d}p}=\underbrace{\frac{\partial u_2 }{\partial \theta }}_{(-)}\underbrace{\frac{\mathrm{d}\theta }{\mathrm{d}p}}_{(+)}+\underbrace{\frac{\partial u_2 }{\partial w}}_{(-)}\underbrace{\frac{\mathrm{d}w}{\mathrm{d}p}}_{(+)}<0 \end{aligned}$$

(28-5)

$$\begin{aligned}&\frac{\mathrm{d}(1-u_2 )}{\mathrm{d}p}=-\underbrace{\frac{\mathrm{d}u_2 }{\mathrm{d}p}}_{(+)}>0. \end{aligned}$$

(28-6)

The last line proves that urbanization advances with urban productivity.

Finally, the wage gap arises between urban and rural sectors because Eq. (21) implies $\frac{w_{R,t} }{w_t }=\frac{\theta _t q(\theta _t )}{\left( {r+\lambda +\theta _t q(\theta _t )} \right) }<1$ in equilibrium. The unemployment gap occurs because of urban job search frictions. $\square$