Small business and the self-organization of a marketplace

Abstract

In many developing countries such as China, the typical marketplace is a cluster of small shops or booths. We investigate an economic model in which circular causality, including search and matching between buyers and sellers, forms agglomeration forces. We find that an authoritative third party that reduces search costs is important in sustaining a large marketplace. However, it is unnecessary to reduce search costs to zero. Finally, the low capital requirement of setting up a firm helps to sustain a large marketplace owing to its increased product heterogeneity.

Appendices

Appendix 1: Stability of corner solutions

In this appendix, we start from the full agglomeration case. Full agglomeration implies $\lambda =1$. By substituting $\lambda =1$ into (12) and solving $U(1,\theta )=0$, we obtain $\theta =\Psi _{U}(\lambda =1)$. Since $\theta \gtreqqless \Psi _{U}(\lambda =1)$, which implies $\overset{.}{\theta }\lesseqqgtr 0$, the economy is stable at $\lambda =1$ and $\theta =\Psi _{U}(\lambda =1)$ if and only if $\Psi _{U}(\lambda =1)>\Psi _{r}(\lambda =1)$, which implies $\overset{.}{\lambda }>0$ around $\left( \lambda ,\theta \right) =\left( 1,\Psi _{U}(\lambda =1)\right)$.

Next, we examine the no-marketplace case. By setting $\lambda =0$ in (12), we obtain $U(0,\theta )<0$. Likewise, by setting $\theta =0$ in (10), we obtain $r(\lambda ,\theta =0)<\bar{r}$.

Appendix 2: Stability of the interior outcome

In this appendix, we examine the stability of the interior outcome $(\widetilde{\lambda },\widetilde{\theta })$. Linearizing (3) around ($\widetilde{\lambda }$, $\widetilde{\theta }$) with (10) and (12) yields

$$\begin{aligned} \left( {\begin{array}{c}\overset{.}{\lambda }\\ \overset{.}{\theta }\end{array}}\right) =A\left( {\begin{array}{c}\lambda -\widetilde{\lambda }\\ \theta -\widetilde{\theta }\end{array}}\right) , \end{aligned}$$

where

$$\begin{aligned} A\equiv \left( \begin{array} [c]{cc} \partial r(\widetilde{\lambda },\widetilde{\theta })/\partial \lambda &{}\quad \partial r(\widetilde{\lambda },\widetilde{\theta })/\partial \theta \\ \partial U(\widetilde{\lambda },\widetilde{\theta })/\partial \lambda &{}\quad \partial U(\widetilde{\lambda },\widetilde{\theta })/\partial \theta \end{array} \right) =\left( \begin{array} [c]{cc} -\widetilde{\theta }\sqrt{c\mu /2\widetilde{\lambda }^{3}\phi {}^{\phantom {0^0}}} &{}\quad \sqrt{2c\mu /\widetilde{\lambda }\phi {}^{\phantom {0^0}}} \\ \mu /\phi -\sqrt{2c\mu /\widetilde{\lambda }\phi {}^{\phantom {0^0}}} &{}\quad -t \end{array} \right) . \end{aligned}$$

Thus, we obtain a characteristic equation $a^2+ [c\bar{r}\mu ^3/\phi (\phi \bar{r}t+4c\mu )^2+t]a-c\mu /\phi ^2(\phi \bar{r}t+4c\mu )$ and find two eigenvalues, $a_1$ and $a_2$; $a_{1}<0$ and $a_{2}>0$. Hence, the interior outcome ($\widetilde{\lambda }$, $\widetilde{\theta }$) is a saddle point. The simple calculation yields an eigenvector for each of $a_1$ and $a_2$:

$$\begin{aligned} \left( \begin{array} [c]{c} \eta _{1}^{(1)}\\ \eta _{2}^{(1)} \end{array} \right) =\left( \begin{array} [c]{c} -A/\left[ t-B-\sqrt{\left( B+t \right) ^2+A \mu /\phi } \right] \\ 1 \end{array} \right) \end{aligned}$$

which is derived by using $a_{1}<0$;

$$\begin{aligned} \left( \begin{array} [c]{c} \eta _{1}^{(2)}\\ \eta _{2}^{(2)} \end{array} \right) =\left( \begin{array} [c]{c} -A/\left[ t-B+\sqrt{\left( B+t \right) ^2+A \mu /\phi } \right] \\ 1 \end{array} \right) \end{aligned}$$

which is derived by using $a_{2}>0$, where $A\equiv 4c\mu ^2/\phi (\phi \bar{r}t+4c\mu )$ and $B\equiv c\bar{r}\mu ^3/\phi (\phi \bar{r}t+4c\mu )^2$. Since $\eta _{1}^{(1)}<0$ and $\eta _{1}^{(2)}>0$, the slope of a stable saddle path is negative and the slope of an unstable saddle path is positive around ($\widetilde{\lambda }$, $\widetilde{\theta }$).

Appendix 3: The equilibrium condition for full agglomeration

In this appendix, we derive the equilibrium condition for full agglomeration. First, we focus on the amount of capital required for a firm. By rewriting (17), which is equivalent to $\Psi _{U}(\lambda =1)>\Psi _{r}(\lambda =1)$, we obtain $h(\phi ^\frac{1}{2})\equiv -t\bar{r}\phi ^\frac{3}{2}-4\mu c\phi ^\frac{1}{2}+\sqrt{2 c}\mu ^\frac{3}{2}> 0$, which is a cubic function of $\phi ^\frac{1}{2}$ and has a negative discriminant and $h(0)>0$. Thus, we find that $h(\phi ^\frac{1}{2})> 0$ iff $\phi$ is between zero and only one real root of $h(\phi ^\frac{1}{2})=0$, which can be solved explicitly by using Vieta’s substitution. Next, we examine the impact of lowering the search costs for sellers. Furthermore, by solving $h(\phi ^\frac{1}{2})$ on $c^\frac{1}{2}$, we find that a marketplace is sustained when $\frac{\mu -\sqrt{\mu ^2 - 8 \bar{r} t \phi ^2}}{4 \sqrt{2\mu \phi }}< \sqrt{c}< \frac{\mu +\sqrt{\mu ^2 - 8 \bar{r} t \phi ^2}}{4 \sqrt{2\mu \phi }}$ if $\mu >\phi \sqrt{t \bar{r}}$, which satisfies condition (5). Finally, we find the impact of the transport costs and/or outside options of capital in (17). By rewriting condition (17), we obtain $\bar{r}t < (\sqrt{2 \mu }-4\sqrt{c \phi }) \mu \sqrt{c}/\phi \sqrt{\phi }$.