Urban agglomeration and heterogeneous firms: a synthesis of Helpman and Melitz

Abstract

This study revisits the seminal findings by Helpman in 1998 in a more comprehensive setup with Melitz-type firm heterogeneity. Our model features how a change in firm heterogeneity affects agglomeration in an urban-space economy. We contribute to the literature by analytically giving explicit solutions for the threshold values of the housing preference and transport costs that are crucial in determining the equilibrium. We also obtain new findings in the case of asymmetric housing stocks. Our results provide alternative explanations to stylized facts such as “ghost cities” in countries undertaking high-speed urbanization and real estate development.

Appendices

Appendices

1.1 Appendix 1: Proof of Proposition 1

If $\beta >1/\sigma$, we have ${\mathcal {C}}>0$. Multiplying the two roots which satisfy ${\mathcal {F}}(\phi )=0$, the product is ${\mathcal {C}}/{\mathcal {A}}>0$, which implies that the two roots have the same sign. Also, it is easy to check ${\mathcal {B}}>0$ when $\beta >1/\sigma$, which implies that the slop of ${\mathcal {F}}(\phi )$ at $\phi =0$ is positive. Since ${\mathcal {F}}(\phi )$ is a convex function, we know that the two roots are both negative and ${\mathcal {F}}(\phi )>0$ for $\phi \in (0,1)$. By Eq. (12), it implies that the symmetric equilibrium is always stable.

If $\beta<\beta ^{\sharp }<1/\sigma$, we have ${\mathcal {F}}(0)={\mathcal {C}}<0$. Also, as shown by Lemma 1, if $\beta <\beta ^{\sharp }$, we have ${\mathcal {F}}(1)<0$. Note that ${\mathcal {A}}>0$ and ${\mathcal {F}}(\phi )$ is a convex function. Due to the continuities, ${\mathcal {F}}(0)<0$ and ${\mathcal {F}}(1)<0$ imply ${\mathcal {F}}(\phi )<0$ for $\phi \in (0,1)$. By Eq. (12), it implies that the symmetric equilibrium is always unstable.

If $\beta ^{\sharp }<\beta <1/\sigma$, we have ${\mathcal {F}}(0)={\mathcal {C}}<0$ and ${\mathcal {F}}(1)>0$ by Lemma 1. Because ${\mathcal {F}}(\phi )$ is a convex function, there exists a unique $\phi _b$ at which ${\mathcal {F}}(\phi _b)=0$ and we have ${\mathcal {F}}(\phi )>0$ when $\phi >\phi _b$. By the definition of $\phi _b$, we solve the $\tau _b$. $\square$

1.2 Appendix 2: Proof of Proposition 2

If $\lambda$ is very close to 1, $(1-\lambda )$ is very close to zero. As shown in Eq. (14), if $\beta >\frac{k-\sigma +1}{k\sigma -\sigma +1}$, the power of $(1-\lambda )$ is positive and, therefore, $v_1/v_2$ is very close to zero. As a result, the full agglomeration is unstable. On the other hand, if $\beta <\frac{k-\sigma +1}{k\sigma -\sigma +1}$, the power of $(1-\lambda )$ is negative and $v_1/v_2$ is close to infinity, as as result, the full agglomeration is stable. If $\beta =\frac{k-\sigma +1}{k\sigma -\sigma +1}$, the term of $(1-\lambda )^{\frac{\beta (k\sigma -\sigma +1)-(k-\sigma +1)}{k(\sigma -1)}}$ equates 1. The full agglomeration is sustainable if and only if ${\varPsi }(\tau )>1$. Also, it is easy to check that the $\varPsi (\tau )$ is an increasing function in terms of $\tau$ and, therefore, the sustain point $\tau_s$ is uniquely defined by $\varPsi (\tau )=1$, beyond which the full agglomeration is sustainable. Moreover, we have

$$\begin{aligned} \partial \left( \frac{k-\sigma +1}{k\sigma -\sigma +1}\right) /\partial k =\frac{(\sigma -1)^2}{(k\sigma -\sigma +1)^2}>0, \partial \left( \frac{k-\sigma +1}{k\sigma -\sigma +1}\right) /\partial \sigma =-\frac{k^2}{(k\sigma -\sigma +1)^2}<0, \end{aligned}$$

implying that a smaller k enlarges the range of parameters in which the full agglomeration is unstable while a smaller $\sigma$ fosters the equilibrium of full agglomeration. $\square$