Information technology, market congestion, and economic geography

Abstract

I investigate the impact of economic integration, in terms of trade costs and information technology, on the interregional location patterns of retail firms by introducing three features into a new economic geography model: individuals consume only a restricted number of varieties; firms compete for customer acquisition; and the advancement of information technology enables consumers to easily perceive and buy products sold by firms in distant regions. I consider two possibilities regarding competition for attention: relative competition, in which a firm can attract more consumers by advertising more than the average level, and absolute competition, in which each firm competes with another to enhance the share of its advertisement in the market. I show that consumption variety is richer in the agglomerated region, but spatial consumption inequality shrinks as information technology develops. A reduction in trade costs may cause the redispersion of economic activities under sufficiently low trade costs only in the case of absolute competition. This is more likely when competition for attention is fierce and information technology is not fully developed. The results not only support the complementarity between information technology and cities but also provide a new insight that competition among firms is an important factor governing the impact of information technology on economic geography.

1 Introduction

One of the most critical differences between the twentieth and twenty-first centuries is the existence of the Internet. The impact of information technology on cities has been a topic of great interest since the late twentieth century in the field of urban economics. Gaspar and Glaeser (1998) concluded that telecommunications and face-to-face contacts were complements rather than substitutes, and that the growth of information technologies increased the relevance of cities.¹

Currently, most people can easily access the Internet and gain various types of information. The growth of the Internet has greatly changed our daily lives, and shopping behavior is no exception. We can buy a variety of goods anytime using our smartphones, increasing our opportunities to access a wider range and variety of goods. Dolfen et al. (2023) showed that, by using the data of all credit and debit card transaction in the USA from the Visa network, E-commerce resulted in nearly a 1% welfare gain from variety, compared with the counterfactual situation where the online shopping was completely unavailable. Fan et al. (2018) also concluded that the welfare gains from E-commerce were approximately 1.6% on average and tended to be larger in small cites in China. Suppliers have benefited from these advancements. For example, a small shop can have its website and sell its products online to consumers in distant regions. Indeed, such benefits arise not only from the progress of the Internet, but also from improvements in transportation technologies. Thus, economic integration has made interregional shopping easier. Despite these significant changes, the impact of economic integration on the agglomeration of retail firms in each region has not been examined.

By examining the real world, we can observe that some regions are rich in stores, while others are not. Several studies have focused on retail agglomeration, although most were related to intracity status. This is partly because the competition among retail firms is more likely to occur within a city. However, extant studies have found evidence of the significance of competition between brick-and-mortar and online retailers (Prince 2007; Brynjolfsson et al. 2009), implying that competition is no longer limited to urban areas. In this study, I focus on interregional differences in retail agglomeration. For example, it is possible for retail firms to concentrate in large cities to attract many consumers or to be located in less concentrated regions to avoid fierce competition.² However, it is ambiguous how economic integration regarding the progress of both transportation and information technologies has affected such location patterns. This study aims to clarify this point.

When considering the interregional differences in retail agglomeration, another important point related to economic geography should be taken into account: people can migrate across regions in pursuit of better variety availability.³ Hottman (2021) empirically showed that retail store variety significantly impacts the cost of living and become an important consumption-based agglomeration force. Owing to this type of consumer-related agglomeration force, the interregional location patterns of retail firms become complex.

To account for the interaction of these agglomeration and dispersion forces in a tractable way, I propose the approach followed in new economic geography (NEG), which originated from the seminal work of Krugman (1991), and has been widely used to examine changes in industrial locations in response to declines in trade costs.

However, because NEG models have been developed to primarily explain the location patterns of manufacturing firms, simply applying NEG models to retail firms is inadequate. Most NEG models are based on Chamberlinian monopolistic competition in which all consumers are implicitly assumed to have access to all varieties in the market. Because of this assumption, the residents in each region consume an identical number of varieties in most NEG models as long as trade costs are not prohibitive. This implicit assumption is not ideal when considering retail agglomeration because it is an important feature of the retail agglomeration that consumers can enjoy richer varieties of goods in a concentrated area. Such spatial consumption inequality has been empirically shown to be significant (Handbury and Weinstein 2015). Therefore, it is desirable to consider this finding.

To introduce this perspective of economic geography into the model, I assume that individuals have opportunities to consume a limited number of varieties. This assumption differs from standard Chamberlinian monopolistic competition models. In my model, firms are assumed to advertise their products to customers. Otherwise, consumers will not perceive their products.⁴ If information technology is sufficiently developed, advertisements are more likely to be perceived by consumers in distant regions, making interregional shopping easier.

When considering the effects of advertisements, the interaction of each firm’s advertisements should be taken into account. According to Friedman (1983a), advertising can be cooperative and predatory. Their definitions mirror those of Friedman (1983b, Ch.6), who states that advertising is cooperative if the advertisement by one firm increases the sales of all firms in the market, while advertising is predatory if an increase in advertising by one firm causes a decrease in the sales of all of its competitors.⁵ Thus, competition among firms to acquire customers is expected. To account for these effects, I assume that the number of consumers who buy the product of a certain firm is determined not only by the firm’s own advertisement but also by the aggregate level of advertisements made by all firms in the market.⁶

In addition, I consider two possible forms of advertising competition. In the first case, a firm can stand out in the market and attract more consumers by advertising more than the average level. In the second case, each firm must compete with one another to enhance the share of its advertisement in the aggregate level of advertisements made in the market. In other words, the relative and absolute levels of advertisements are important in the former and latter cases, respectively. Thus, I refer to the former case as relative competition, and the latter as absolute competition.

In summary, this study aims to capture how economic integration in terms of transportation and information technologies affects retail firms’ interregional location patterns. To accomplish this, I extend the NEG model presented by Pflüger (2004) by introducing consumers’ restricted availability of varieties, firms’ advertisements for drawing attention, and informational differences across regions. Then, some noteworthy results are obtained.

First, the number of varieties consumed by individuals in the agglomerated region is always larger than that in the less populated region. Moreover, the difference in consumption variety between the two regions decreases as the information technology develops. These results can be analytically shown in the present framework, and are in line with empirical studies, including those of Handbury and Weinstein (2015) and Fan et al. (2018).

Second, a decline in trade costs can have different impacts on spatial configuration in the cases of relative and absolute competition. When firms engage in relative competition, a reduction in trade costs leads the spatial configuration from dispersion to agglomeration, as long as competition for attention is not quite fierce and the information technology is not significantly underdeveloped. Meanwhile, in the case of absolute competition, it is possible that economic activities disperse under both high and low trade costs, and agglomerate when trade costs are at intermediate levels. The key idea behind this observation is that absolute competition causes market congestion, meaning that a firm becomes less likely to acquire customers because of the large number of advertisements made by all of its competitors. When trade costs are at intermediate levels, firms will rather be in a larger market despite fiercer competition for acquiring customers. This is because a firm’s relocation to a peripheral region significantly decreases the total demand from the market with a larger population owing to both trade frictions and informational differences. If trade costs become sufficiently low, a reduction in individual demand from the larger market is not significant because a firm’s relocation does not cause a considerable change in the price of its product. Furthermore, informational differences imply that competition for attention is milder at the periphery than at the core. Thus, some firms may be better off by relocating from the core with many competitors to the periphery with fewer competitors when trade costs are quite low. Numerical simulations indicate that economic activities are more likely to disperse when competition for attention is fierce, and the information technology is not fully developed. This is because fierce competition strengthens the effects of market congestion and the imperfect development of information technology localizes competition for attention. Therefore, these results are not only in line with the early studies that found complementarity between information technology and cities, but also reveal that the fierceness of competition among firms is an important factor that governs the impacts of information technology on economic geography.

This study is related to the literature on NEG.⁷ One of the most traditional outcomes found in NEG models is that economic activities disperse when trade costs are high, and agglomerate in a core region as trade costs drop (Krugman 1991; Fujita et al. 1999; Ottaviano et al. 2002; Forslid and Ottaviano 2003; Pflüger 2004).⁸ The mechanism of concentration is explained by the cumulative interaction between two effects: consumers prefer a region where they can obtain more varieties for lower prices (i.e., forward linkage effect), and firms tend to be located in larger market areas (i.e., backward linkage effect). However, other studies have suggested the possibility of redispersion, meaning that economic activities can disperse again under sufficiently low trade costs. Redispersion has been shown to occur because of, for example, an increase in urban costs (e.g., Tabuchi 1998; Pflüger and Südekum 2008), an increase in costs of production (e.g., Fujita et al. 1999, Ch.7; Picard and Zeng 2005), and taste heterogeneity across individuals (e.g., Tabuchi and Thisse 2002; Murata 2003). In the present study, the redispersion of economic activities can be observed because of competition among firms for acquiring customers. Thus, this study contributes to the literature by providing a new source of redispersion.

The present study is also related to the debate over whether information technology is a complement or substitute for cities. Early studies include the seminal paper by Gaspar and Glaeser (1998) and those following them, such as Kolko (2000), Leamer and Storper (2001), Sinai and Waldfogel (2004), Forman et al. (2005), and Ioannides et al. (2008). Although many of the studies in this literature agree with the complementarity between information technology and cities, some argue that this relationship is not necessarily conclusive and suggest that they are substitutes. Therefore, this issue remains controversial. The result of the present study is in favor of the complementarity. In addition, it provides new insights from the viewpoint of fierceness of competition among firms.

The remainder of this paper is organized as follows. Section 2 provides the model. Section 3 characterizes the short-run equilibrium in which no migration occurs and interregional differences in the availability of varieties emerge. Section 4 examines the long-run equilibrium in which migration occurs. I derive the conditions for a dispersed location pattern under low trade costs. In addition, I focus on what drives retail firms and consumers to agglomerate or disperse. Finally, Sect. 5 concludes the paper.

2 The model

The economy consists of two regions, indexed by 1 and 2, and there are two types of workers: skilled and unskilled. The total number of skilled workers is H. I assume that they are freely mobile across regions in the long run. The number of unskilled workers in each region is L and they are assumed to be immobile across regions. Thus, the total population of the economy is $H+2L$. Let the share of skilled workers in region 1 be $\lambda$, then the populations in regions 1 and 2 are $\lambda H+L$ and $(1-\lambda )H+L$, respectively.

Each worker derives utility from consuming two types of goods: differentiated and homogeneous goods. Differentiated goods are produced in a monopolistically competitive market and its trade between the two regions is subject to trade costs. The homogeneous good is produced in a perfectly competitive market and can be traded without any costs, which is a usual assumption in the NEG literature.

In Chamberlinian models of monopolistic competition, it is implicitly assumed that each consumer can access all varieties in the market. Thus, consumers are perfectly informed, and firms are not required to make efforts to gain customers.

In contrast with these extant models, the present model assumes that consumers do not necessarily have access to all varieties, although there are numerous varieties in the market. Imagine, for instance, a very large store that displays an enormous number of products for sale. Browsing all products is not possible; therefore, one must decide which products to buy using limited information. In this setting, firms must inform consumers of their products through advertising. To simply represent such an interaction, I assume that it is stochastically determined whether a worker can obtain an opportunity to consume a certain variety, and the probability of matching between a consumer and a variety is endogenously determined by firms’ advertisements. Thus, the greater the effort made by a firm to attract consumers, the more likely it is that a product will be perceived and bought.

Additionally, because the economy has two regions that are geographically distant from one another, the information technology is important when a consumer in a region wants to buy goods from another region. If information technology is sufficiently developed, people are more likely to gain information, even from remote regions, and interregional shopping becomes easier.

2.1 Consumers

Following Pflüger (2004), I assume that the utility function of individual i living in region $r \in \{1,2\}$ is given by

$$\begin{aligned} U_{ri}=\alpha \ln {X_{ri}}+A_{ri} . \end{aligned}$$

In the utility function, $A_{ri}$ is the amount of the homogeneous good consumed by individual i in region r, and $X_{ri}$ is the bundle of differentiated goods, defined as

$$\begin{aligned} X_{ri}=\left( \sum _{s=1}^2 \int _{\Omega _{sr}^i} x_{sr}^i(\omega )^{\frac{\sigma -1}{\sigma }} \textrm{d}\omega \right) ^{\frac{\sigma }{\sigma -1}} , \end{aligned}$$

where $\Omega _{sr}^i$ is the set of varieties produced in region $s \in \{1,2\}$ and available to individual i living in region r; $x_{sr}^i(\omega )$ is the amount of variety $\omega \in \Omega _{sr}^i$ originating in region s and consumed by individual i in region r; and $\sigma >1$ is the elasticity of substitution among varieties. Thus, $\alpha >0$ represents the intensity of preference for differentiated goods. Let $\Omega _{s}$ be the set of all varieties produced in region s, then $\Omega _{sr}^i$ is a subset of $\Omega _s$.

The homogeneous good is produced using constant returns to scale technology, and its market is perfectly competitive. Producing a unit amount of the homogeneous good requires an unskilled worker. Because the homogeneous good is freely traded, the wage of unskilled workers in both regions is one by normalizing the price of the homogeneous good to one.⁹ Let $p_{sr}(\omega )$ be the price of a differentiated good produced in region s and purchased in region r. Then, the budget constraint of individual i residing in region r is

$$\begin{aligned} \sum _{s=1}^2 \int _{\Omega _{sr}^i} p_{sr}(\omega )x_{sr}^i(\omega ) \textrm{d}\omega +A_{ri}=y_{ri}+\bar{A} , \end{aligned}$$

where $y_{ri}$ denotes the wage income of individual i. If i is unskilled, $y_{ri}=1$; and if i is skilled, $y_{ri}$ equals the wage of a skilled worker in region r, denoted by $w_{r}$. $\bar{A}$ is the initial endowment of the homogeneous good, which is common across all workers in both regions and is assumed sufficiently large to ensure that $A_{ri}$ is positive.

Utility maximization yields the demand function $x_{sr}^i(\omega )=\alpha P_{ri}^{\sigma -1}p_{sr}(\omega )^{-\sigma }$, where $P_{ri}$ is the price index faced by individual i living in region r, given by

$$\begin{aligned} P_{ri}=\left( \sum _{s=1}^2 \int _{\Omega _{sr}^i} p_{sr}(\omega )^{1-\sigma } \textrm{d}\omega \right) ^{\frac{1}{1-\sigma }} . \end{aligned}$$

(1)

Unlike Dixit and Stiglitz (1977), each worker in this scenario can face a varying price index, because the set of varieties one can consume differs across all workers.

2.2 Production and advertisements

Firms’ decision making comprises three stages. First, each firm decides whether to enter the market. Second, each firm chooses the level of advertisement. Third, each firm sets prices to maximize expected profit, because matching consumers and varieties is assumed to be stochastically done after these stages.

Differentiated goods are produced using a technology of increasing returns to scale, and each variety is produced by one firm, implying that the number of firms equals that of varieties. Then, letting $N_s$ be the number of firms located in region s, I have $|\Omega _s|=N_s$.

Each firm requires m units of unskilled workers per unit of output and f units of skilled workers as fixed inputs.¹⁰ Additionally, I let $k_s(\omega ) \in [0,\infty )$ represent the level of advertisement made by firm $\omega$ located in region s, and assume that c units of skilled workers are required per advertisement unit. For simplicity, technological parameters m, c, and f are assumed identical across all firms. Thus, the firms are symmetric.

To consider the effect of the information technology, I assume that the effectiveness of advertising is diminished when a firm wants to attract consumers living in a foreign region. This assumption is embodied in the following way: if the amount of advertisement made by firm $\omega$ in region s is $k_s(\omega )$ units, the effect of the advertisement is fully attained in the home region, whereas only $\delta k_s(\omega )$ $(0<\delta <1)$ out of $k_s(\omega )$ units of advertisement reaches region $r(\ne s)$, and the effectiveness of the advertisement is restricted to the level at which $\delta k_s(\omega )$ units of advertisement correspond. I refer to $\delta$ as the spatial discount factor for the information. Then, the total amount of advertisement reaching region r is expressed as follows:

$$\begin{aligned} \mathbb {K}_{r}=\sum _{s=1}^2 \int _{\Omega _s} \delta _{sr}k_s(\omega ) \textrm{d}\omega , \quad \text{ where } \quad \delta _{sr}=\left\{ \begin{array}{ll} \delta &{} (r \ne s) \\ 1 &{} (r=s) \end{array} \right. . \end{aligned}$$

When $\delta =0$, information is not conveyed across the regions. When $\delta =1$, the two regions share exactly the same information, and there is no difference in the set of information. Allowing $\delta$ to take some values in (0, 1), the model can examine cases lying between these two extremes, which are considered to reflect real-world situations. The development of information technology can be represented by the change in the value of $\delta$: the closer $\delta$ is to 1, the more developed is the information technology.

As it is stochastically determined whether a worker can obtain an opportunity to consume a certain variety, I let $\theta _{sr}(\omega )$ represent the probability of matching between a consumer living in region r and variety $\omega$ originating in region s. The matching probability $\theta _{sr}(\omega )$ is determined not only by the level of advertisement made by firm $\omega$ but also by the aggregate level of advertisement prevalent in region r, $\mathbb {K}_{r}$. Naturally, the more a firm $\omega$ advertises its product, the higher is the probability of matching. On the other hand, I assume $\mathbb {K}_{r}$ affects $\theta _{sr}(\omega )$ both positively and negatively, reflecting that advertising can be both cooperative and predatory (Friedman 1983a). Additionally, I consider two types of advertising competition: “relative” and “absolute.”

First, it is assumed that $\theta _{sr}(\omega )$ has the following form:

$$\begin{aligned} \theta _{sr}(\omega )=\theta _{sr}^R(\omega ) \equiv \left( \frac{\delta _{sr}k_{s}(\omega )}{\tilde{k}_r} \right) ^\gamma \left( \psi (\mathbb {K}_{r}) \right) ^{1-\gamma } , \end{aligned}$$

(2)

where $\gamma \in (0,1)$ is an exogenous parameter and $\tilde{k}_{r}$ is the weighted average of advertisements reaching region r, defined as $\tilde{k}_{r} \equiv \mathbb {K}_{r}/\sum _{s=1}^2 \delta _{sr}N_s$.¹¹ In this setting, the relative level of the advertisement is important. If a firm advertises more than the average level, it can stand out in the market and attract more consumers, consistent with the nature of advertising. However, $\mathbb {K}_{r}$ negatively affects $\theta _{sr}(\omega )$ because an increase in $\mathbb {K}_{r}$ also increases $\tilde{k}_{r}$. This can be interpreted as the predatory effect of advertising. In contrast, $\psi (\mathbb {K}_{r})$ represents the cooperative effect of advertising. I do not specify $\psi (\cdot )$ here, but impose some assumptions¹²:

$$\begin{aligned} \begin{array}{c} \displaystyle \psi (0)=0 \,,\,\, \psi '(x)>0 \,,\,\, \psi ''(x)<0 \,,\,\, \psi '''(x)>0 , \\ \displaystyle \lim _{x \rightarrow \infty } \psi (x)=1 \,,\,\, \lim _{x \rightarrow \infty } \psi '(x)=0 . \end{array} \end{aligned}$$

(3)

These assumptions imply that $\psi (\mathbb {K}_{r})$ is strictly increasing in $\mathbb {K}_{r}$ and satisfies $0 \le \psi (\mathbb {K}_{r}) <1$ and that the larger $\psi (\mathbb {K}_{r})$, the higher $\theta _{sr}(\omega )$.

Second, I assume that $\theta _{sr}(\omega )$ has the following form:

$$\begin{aligned} \theta _{sr}(\omega )=\theta _{sr}^A(\omega ) \equiv \left( \min \left\{ \frac{\delta _{sr}k_{s}(\omega )}{\mathbb {K}_{r}}, \delta _{sr} \right\} \right) ^\gamma \left( \psi (\mathbb {K}_{r}) \right) ^{1-\gamma } . \end{aligned}$$

(4)

The assumption of $\psi (\cdot )$ is the same as that in the previous case. This formula implies that the predatory effect is stronger when the number of firms is large. Thus, the more firms enter the market, the more difficult it is to be perceived by and attract consumers. Each firm must compete with one another to enhance the absolute share of advertisements in the market. I use $\min \{\,\cdot ,\,\cdot \,\}$ because $\delta _{sr}k_s(\omega )/\mathbb {K}_{r}$ may be larger than one when the number of firms is too small. Although this formula is adopted to avoid mathematical troubles, it can be justified intuitively. When the number of firms is too small, the difficulty of being perceived by consumers in the market is unaffected by the total number of firms.¹³

For convenience, I define the name of each case as follows: regarding competition for acquiring customers, firms are engaged in

• (i) “Relative competition,” if $\theta _{sr}(\omega )=\theta _{sr}^R(\omega )$;

• (ii) “Absolute competition,” if $\theta _{sr}(\omega )=\theta _{sr}^A(\omega )$.

The role of $\gamma$ should be examined, because it is one of the key parameters in this study. It is useful to consider the elasticity of $\theta _{sr}(\omega )$ with respect to $\mathbb {K}_{r}$. Focusing on the case of $0<\theta _{sr}(\omega )<1$, a simple calculation yields

$$\begin{aligned} \dfrac{\partial \theta _{sr}(\omega )}{\partial \mathbb {K}_{r}}\frac{\mathbb {K}_{r}}{\theta _{sr}(\omega )}=(1-\gamma )\mathcal {E}(\mathbb {K}_{r})-\gamma , \end{aligned}$$

(5)

for both relative and absolute competition, where $\mathcal {E}(\mathbb {K}_{r}) \equiv \mathbb {K}_{r}\psi '(\mathbb {K}_{r})/\psi (\mathbb {K}_{r})$ is the elasticity of the cooperative effect, $\psi (\mathbb {K}_{r})$, with respect to $\mathbb {K}_{r}$. Note that $\mathcal {E}(\mathbb {K}_{r})>0$ holds for all $\mathbb {K}_{r}>0$ based on assumptions (3). For a fixed $\mathbb {K}_{r}$, when $\gamma$ is close to one, (5) is likely to be negative. However, when $\gamma$ is close to zero, it is likely to be positive. Stated differently, a marginal increase in the aggregate level of advertisements, $\mathbb {K}_{r}$, causes a decrease (an increase) in $\theta _{sr}(\omega )$ when $\gamma$ is large (small). Therefore, $\gamma$ can be viewed as a measure of the dominance of predatory effects. This can also be regarded as a measure of the competitiveness of the advertising market.

Moreover, it can be easily verified that ${\partial \theta _{sr}(\omega )}/{\partial k_s(\omega )}$ decreases in $k_s(\omega )$. Thus, the marginal effect of a firm’s advertisement, $k_s(\omega )$, on the probability of matching is smaller when $k_s(\omega )$ is large, which is consistent with Arkolakis (2010).

2.3 Profit maximization

Not all consumers demand a variety $\omega$. Let $I_{sr}^\omega$ be the set of consumers living in region r who have access to variety $\omega$ originating in region s. Then, the expected total demand for variety $\omega$ produced in region s is

$$\begin{aligned} E[Q_{sr}(\omega )]=E\left[ \int _{I_{sr}^\omega } x_{sr}^i(\omega ) \textrm{d}i \right] =\alpha p_{sr}(\omega )^{-\sigma }E\left[ \int _{I_{sr}^\omega } {P_{ri}}^{\sigma -1} \textrm{d}i \right] , \end{aligned}$$

where $E[\cdot ]$ denotes the expectation operator. I assume that trade costs are represented by the iceberg type promoted by Samuelson (1952). In other words, a firm needs to produce $\tau >1$ units to export one unit to a foreign region. Thus, the expected profit of firm $\omega$ in region s is

$$\begin{aligned} E[\Pi _s(\omega )]=\sum _{r=1}^2(p_{sr}(\omega )-\tau _{sr}m)E[Q_{sr}(\omega )]-cw_{s}k_s(\omega )-fw_{s} , \end{aligned}$$

(6)

where $w_{s}$ is the wage of a skilled worker in region s, and $\tau _{sr}$ is expressed as follows:

$$\begin{aligned} \tau _{sr}=\left\{ \begin{array}{ll} \tau &{} (r \ne s) \\ 1 &{} (r=s) \end{array} \right. . \end{aligned}$$

Because there are three stages of decision making, I solve the problem backward. In the third stage, each firm chooses $p_{sr}(\omega )$ that maximizes (6), resulting in

$$\begin{aligned} p_{sr}=\frac{\tau _{sr} \sigma m}{\sigma -1} . \end{aligned}$$

(7)

I omit the notation $\omega$ because the equilibrium price is independent of $\omega$.

In the second stage, each firm decides on the level of $k_s(\omega )$, taking (7) as given. The objective function in this stage can be derived by substituting (7) into (6):

$$\begin{aligned} E[\Pi _s(\omega )]=\frac{\alpha }{\sigma }\sum _{r=1}^2 \tau _{sr}^{1-\sigma }\theta _{sr}(\omega )(H_{r}+L)\Xi _{r}-cw_sk_s(\omega )-fw_s , \end{aligned}$$

(8)

where $H_{r}$ is the number of skilled workers residing in region r and $\Xi _{i}$ is given by

$$\begin{aligned} \Xi _{r} \equiv E\left[ \left( \sum _{r'=1}^2 \tau _{r'r}^{1-\sigma }|\Omega _{r'r}^i| \right) ^{-1} \right] . \end{aligned}$$

Note that $|I_{sr}^\omega |=\theta _{sr}(\omega )(H_{r}+L)$ holds by the law of large numbers and has no uncertainty, for it is out of the expectation operator in (8). Because (8) is concave in $k_s(\omega )$, the first order condition gives the profit-maximizing level of advertisement in both relative and absolute competition as follows¹⁴

$$\begin{aligned} k_s(\omega )=\frac{\alpha \gamma }{\sigma cw_{s}}\sum _{r=1}^2 \tau _{sr}^{1-\sigma }\theta _{sr}(\omega )(H_{r}+L)\Xi _r . \end{aligned}$$

(9)

In the first stage of decision making, firms decide whether to enter the market. The assumption of free entry yields the zero profit condition. By substituting (9) into (8) and applying the zero profit condition, I obtain

$$\begin{aligned} \sum _{r=1}^2 \tau _{sr}^{1-\sigma }\theta _{sr}(\omega )(H_{r}+L)\Xi _r=\frac{\sigma fw_{s}}{\alpha (1-\gamma )} . \end{aligned}$$

(10)

Consequently, combining (9) and (10) yields

$$\begin{aligned} k_s=\frac{f\gamma }{c(1-\gamma )} . \end{aligned}$$

(11)

This implies that the amount of advertisements made by a firm is identical across the regions.

Substituting (11) into (2) and (4), it can be verified that the equilibrium probabilities of matching satisfy

$$\begin{aligned} \theta _{21}=\delta ^\gamma \theta _{11} \,\,,\,\,\, \theta _{12}=\delta ^\gamma \theta _{22} . \end{aligned}$$

(12)

These expressions do not depend on the form of advertising competition.

3 Short-run equilibrium

I consider the short-run equilibrium in which the distribution of skilled workers is fixed. From (11) and the labor market clearing conditions, $(ck_s+f)N_s=H_s$ for $s=1,2$, the number of firms in each region is derived as follows:

$$\begin{aligned} N_1=\frac{(1-\gamma )\lambda H}{f} \,,\,\, N_2=\frac{(1-\gamma )(1-\lambda )H}{f} . \end{aligned}$$

(13)

Finally, using (10), (11), and (13), the equilibrium wages are expressed as follows:

$$\begin{aligned} \begin{array}{c} \displaystyle w_1=\frac{\alpha }{\sigma H}\left\{ \frac{\lambda H+L}{\lambda +\delta ^\gamma \phi (1-\lambda )}+\frac{\delta ^\gamma \phi [(1-\lambda )H+L]}{\delta ^\gamma \phi \lambda +(1-\lambda )} \right\} , \\ \\ \displaystyle w_2=\frac{\alpha }{\sigma H}\left[ \frac{\delta ^\gamma \phi (\lambda H+L)}{\lambda +\delta ^\gamma \phi (1-\lambda )}+\frac{(1-\lambda )H+L}{\delta ^\gamma \phi \lambda +(1-\lambda )} \right] , \end{array} \end{aligned}$$

(14)

where $\phi \equiv \tau ^{1-\sigma }$ represents freeness of trade.

Equilibrium wages (14) are affected by $\gamma$ through the spatial discount factor of information, $\delta$. Here, I focus on $w_1$. Because of the zero profit condition, the profitability of firms is reflected in the wages of skilled workers. In (14), the first term in the brackets represents profitability from the home market, whereas the second term is from the foreign market. An increase in $\delta$ negatively affects the first term, because advertisements from the foreign market are more likely to be conveyed. However, it positively affects the second term, because the number of customers in the foreign region increases. As $\gamma$ measures the competitiveness of the advertising market, an increase in $\gamma$ mitigates the impact of $\delta$ on $w_1$. Thus, the inflow of foreign advertisements is less important for the home market and it becomes more difficult to acquire foreign customers.

3.1 Spatial consumption inequality

Extant NEG studies have assumed that consumers can access all varieties in the market regardless of the region in which they live. By this assumption, the number of varieties that individuals in both regions can consume is equalized. However, it has been empirically revealed that households in larger cities tend to enjoy more variety in consumption than those in peripheral regions (Handbury and Weinstein 2015).

In the present model, the availability of variety differs when the population is distributed asymmetrically because the number of firms located in each region and the probability of matching between a consumer and a variety differ across regions. These differences are sources of spatial consumption inequality.

Proposition 1

The number of varieties consumed by residents in the core region is larger than that in the peripheral region.

Proof

See Online Appendix A. $\square$

Note that Proposition 1 holds regardless of whether firms engage in relative or absolute competition. This result is consistent with that of Handbury and Weinstein (2015) and supports the validity of the present model.

Figure 1 shows the numerical examples. Without loss of generality, I focus only on the case in which $\lambda \ge 0.5$. The two curves intersect at $\lambda =0.5$, which is the case of symmetric dispersion. When the population distribution is asymmetric (i.e., $\lambda >0.5$), the number of varieties available to an individual in region 1 (the core region) is larger than that in region 2 (the peripheral region). Interestingly, at least in this example, the number of available varieties in the peripheral region first decreases and then increases, although it does not catch up with the core region.¹⁵ This increase is caused by a decrease in the number of firms located in the peripheral region. That is, the probability of matching between a variety and a resident in the peripheral region increases. This example shows that firms’ concentration in the core region may benefit the peripheral region in terms of the number of varieties available to each consumer.

Fig. 1

Number of varieties available to a consumer living in each region. The left-hand picture shows the case of relative competition, and the right-hand one shows absolute competition. The solid and dashed curves are those of regions 1 (core) and 2 (periphery), respectively. In both pictures, the vertical axis represents the number of varieties available to each consumer. $\psi (\mathbb {K})$ is specified by $\psi (\mathbb {K})=\mathbb {K}/(1+\mathbb {K})$. I use specific values of parameters: $f=1$, $c=0.5$, $H=30$, $\gamma =0.5$, and $\delta =0.5$

It is also interesting to note how spatial consumption inequality is affected by the development of information technology (i.e., an increase in $\delta$). One can conjecture that inequality decreases, which can be formally proven in my framework.

Proposition 2

The difference in consumption variety between the core and peripheral regions is reduced by an increase in $\delta$.

Proof

See Online Appendix B. $\square$

Although it is easier for the residents of both regions to access a wider range of varieties, the difference in consumption diversity decreases, implying that the variety expansion caused by an increase in $\delta$ is greater for residents in the peripheral region. This result is consistent with Fan et al. (2018), who showed through empirical and quantitative analyses that the welfare gains from E-commerce were larger in less populated regions.

Figure 2 shows how spatial consumption inequality is affected by changes in $\delta$. The population distribution is fixed at $\lambda =0.7$. The solid and dashed curves represent cases of relative and absolute competition, respectively. The vertical axis is the ratio of the number of varieties available to a resident in region 1 to that in region 2, $(\theta _{11}N_1+\theta _{21}N_2)/(\theta _{12}N_1+\theta _{22}N_2)$. For example, in the case of relative competition, if interregional shopping is almost unavailable ($\delta \rightarrow 0$), the residents in region 1 consume approximately 2.4 times more varieties than those in region 2. The difference shrinks as $\delta$ increases in both cases, as stated in Proposition 2.

Fig. 2

Difference in the consumption variety. The solid and dashed curves represent the cases of relative and absolute competition, respectively. The vertical axis represents the ratio of the number of varieties available to a resident in region 1 (core) to that in region 2 (periphery). $\psi (\mathbb {K})$ is specified by $\psi (\mathbb {K})=\mathbb {K}/(1+\mathbb {K})$. I use specific values of parameters: $f=1$, $c=0.5$, $H=30$, $\gamma =0.5$, and $\lambda =0.7$

4 Long-run equilibrium

In the previous section, I fixed the population distribution and examined the short-run equilibrium. In this section, I investigate the long-run equilibrium, in which skilled workers are mobile across regions. They migrate in pursuit of higher utility and continue to migrate until a stable spatial equilibrium is attained.

The indirect utility of skilled workers residing in region r is expressed as $V_{r}=-\alpha \ln {P_{r}}+w_{r}+\bar{A}+\alpha \ln {\alpha }-\alpha$, where I omit the notation i because all skilled workers are symmetric. Combining (1) with (7) and (12), the price indices faced by individuals in regions 1 and 2 become $P_1=p(\theta _{11}N_1+\phi \theta _{21}N_2)^{1/(1-\sigma )}=p\theta _{11}^{1/(1-\sigma )}(N_1+\phi \delta ^\gamma N_2)^{1/(1-\sigma )}$ and $P_2=p(\phi \theta _{12}N_1+\theta _{22}N_2)^{1/(1-\sigma )}=p\theta _{22}^{1/(1-\sigma )}(\phi \delta ^\gamma N_1+N_2)^{1/(1-\sigma )}$, respectively, where p is a positive constant defined as $p \equiv \sigma m/(\sigma -1)$. Then, the utility differential between regions 1 and 2 can be written as

$$\begin{aligned} \Delta V \equiv V_1-V_2=\frac{\alpha }{\sigma -1}\ln {\frac{\theta _{11}}{\theta _{22}}}+\frac{\alpha }{\sigma -1}\ln {\frac{N_1+\delta ^\gamma \phi N_2}{\delta ^\gamma \phi N_1+N_2}}+w_1-w_2 . \end{aligned}$$

(15)

Note that (15) may not be a smooth function in the case of absolute competition because $\theta _{11}^A$ and $\theta _{22}^A$ contain the minimum function, $\min \{\,\cdot ,\,\cdot \,\}$. To simplify the analysis, I impose the following assumption only in the case of absolute competition:

$$\begin{aligned} (1-\gamma )\delta >\frac{f}{H}. \end{aligned}$$

(16)

This assumption is more likely to be satisfied when H is sufficiently larger than f and excludes the case in which (15) is not smooth.¹⁶

By (11) and (13), $\mathbb {K}_1$ and $\mathbb {K}_2$ can be written as $\mathbb {K}_1=\gamma H[\lambda +\delta (1-\lambda )]/c$ and $\mathbb {K}_2=\gamma H(\delta \lambda +1-\lambda )/c$, respectively. Then, under assumption (16), $\theta _{11}/\theta _{22}$ becomes

$$\begin{aligned} \frac{\theta _{11}}{\theta _{22}}=\left\{ \begin{array}{ll} \left( \dfrac{\psi (\mathbb {K}_1)}{\psi (\mathbb {K}_2)} \right) ^{1-\gamma } &{} \text{(for } \text{ relative } \text{ competition) } \\ \left[ \dfrac{\lambda +\delta (1-\lambda )}{\delta \lambda +1-\lambda } \right] ^{-\gamma } \left( \dfrac{\psi (\mathbb {K}_1)}{\psi (\mathbb {K}_2)} \right) ^{1-\gamma } &{} \text{(for } \text{ absolute } \text{ competition) } \end{array} \right. . \end{aligned}$$

(17)

Substituting (13), (14) and (17) into (15), the utility differential can be expressed as a function of $\lambda$ as follows:

$$\begin{aligned} \begin{aligned} \Delta V(\lambda )&= \underbrace{ \frac{\alpha (1-\gamma )}{\sigma -1}\ln {\frac{\psi (\mathbb {K}_1)}{\psi (\mathbb {K}_2)}}-\frac{\mathbbm {1}_{\textrm{abs}}\alpha \gamma }{\sigma -1}\ln {\frac{\lambda +\delta (1-\lambda )}{\delta \lambda +1-\lambda }}}_{\text{ accessibility } \text{ of } \text{ varieties }}\\&\quad + \underbrace{ \frac{\alpha }{\sigma -1}\ln {\frac{\lambda +\delta ^\gamma \phi (1-\lambda )}{\delta ^\gamma \phi \lambda +1-\lambda }} }_{\text{ cost } \text{ of } \text{ living }} + \underbrace{ \frac{\alpha (1-\delta ^\gamma \phi )[(H+L)\delta ^\gamma \phi -L](2\lambda -1)}{\sigma H[\lambda +\delta ^\gamma \phi (1-\lambda )](\delta ^\gamma \phi \lambda +1-\lambda )} }_{\text{ wage } \text{ differential }} \end{aligned} \end{aligned}$$

(18)

where $\mathbbm {1}_{\textrm{abs}}$ is the indicator function that takes the value one if firms are engaged in absolute competition and zero otherwise.

As in Pflüger (2004), the third term of (18), which increases in $\lambda$ for any trade cost, reflects the effect of cost of living. The larger $\lambda$, the more firms are located in region 1, and thus, the lower the price index. The last term in (18) represents the wage differential, which reflects the difference in firm profitability. Thus, skilled workers are attracted to the agglomerated region if $w_1>w_2$ holds.¹⁷ In addition to these two forces, another force is reflected in the first and second terms of (18), which capture the effect of the accessibility of varieties, where the second term appears only in the case of absolute competition. When firms are engaged in relative competition, the first term of (18) always increases in $\lambda$, implying that it is an agglomeration force. Firms acquire more customers by operating in an agglomerated region, which also attracts skilled workers. However, when firms are engaged in absolute competition, the first and second terms are not necessarily an agglomeration force, because it becomes more difficult for firms to attract consumers as the number of firms located in the same region increases. I refer to this situation as market congestion. Thus, the effect of the accessibility of varieties may work not only as an agglomeration force but also as a dispersion force.

Using (18), the dynamics of migration are assumed to be characterized by replicator dynamics, $\dot{\lambda }=\lambda (1-\lambda )\Delta V(\lambda )$, where $\dot{\lambda }$ is the time derivative of $\lambda$. That is, when $\dot{\lambda }$ is positive, migration occurs from region 2 toward region 1, and vice versa. Then, $\lambda ^*$ is a spatial equilibrium if and only if $\dot{\lambda }=0$ holds and is stable if the differential coefficient of $\dot{\lambda }$ with respect to $\lambda$ is negative at $\lambda =\lambda ^*$. Clearly, $\lambda =1/2$ leads to $\dot{\lambda }=0$, implying that symmetric dispersion (i.e., $\lambda ^*=1/2$) is a spatial equilibrium. Therefore, the stability of the symmetric dispersion is ensured if

$$\begin{aligned} \frac{\textrm{d}\Delta V(1/2)}{\textrm{d}\lambda }<0 . \end{aligned}$$

(19)

When $\lambda =1/2$, it holds that $\mathbb {K}_1=\mathbb {K}_2=\mathbb {K}_{\textrm{sym}}$, where $\mathbb {K}_{\textrm{sym}}=\gamma (1+\delta )H/(2c)$. After some calculations, (19) can be reduced to the following inequality¹⁸:

$$\begin{aligned} Z(\phi ,\delta ,\gamma ) \equiv \chi _0(\delta ,\gamma )-2\phi \delta ^\gamma \chi _1(\delta ,\gamma )+\phi ^2\delta ^{2\gamma }\chi _2(\delta ,\gamma )>0 . \end{aligned}$$

(20)

$\chi _0(\delta ,\gamma )$, $\chi _1(\delta ,\gamma )$, and $\chi _2(\delta ,\gamma )$ are defined as follows:

$$\begin{aligned} & \chi _0(\delta ,\gamma ) \equiv 1-\frac{(2\sigma -1)H}{(\sigma -1)(H+2L)}-\frac{\sigma (1-\gamma )(1-\delta )H}{(\sigma -1)(1+\delta )(H+2L)}\left( \mathcal {E}_{\textrm{sym}}-\frac{\mathbbm {1}_{\textrm{abs}}\gamma }{1-\gamma } \right) , \\ & \chi _1(\delta ,\gamma ) \equiv 1+\frac{\sigma (1-\gamma )(1-\delta )H}{(\sigma -1)(1+\delta )(H+2L)}\left( \mathcal {E}_{\textrm{sym}}-\frac{\mathbbm {1}_{\textrm{abs}}\gamma }{1-\gamma } \right) , \\ & \chi _2(\delta ,\gamma ) \equiv 1+\frac{(2\sigma -1)H}{(\sigma -1)(H+2L)}-\frac{\sigma (1-\gamma )(1-\delta )H}{(\sigma -1)(1+\delta )(H+2L)}\left( \mathcal {E}_{\textrm{sym}}-\frac{\mathbbm {1}_{\textrm{abs}}\gamma }{1-\gamma } \right) , \end{aligned}$$

where $\mathcal {E}_{\textrm{sym}} \equiv \mathbb {K}_{\textrm{sym}} \psi ^\prime (\mathbb {K}_{\textrm{sym}})/\psi (\mathbb {K}_{\textrm{sym}})$ is the elasticity of the cooperative effect evaluated at $\mathbb {K}_{\textrm{sym}}$. The threshold values of $\phi$ that satisfy $Z(\phi ,\delta ,\gamma )=0$ are called break points (Fujita et al. 1999). As $Z(\phi ,\delta ,\gamma )$ is quadratic in $\phi$, there are at most two break points.

For a symmetric dispersion to be stable under low values of $\phi$, I assume that

$$\begin{aligned} \chi _0(\delta ,\gamma )>0 \end{aligned}$$

(21)

is satisfied.¹⁹ In the NEG literature, this inequality is called “no black-hole condition," which is required to rule out that the agglomeration forces are quite strong that the symmetric dispersion is unstable even at infinitely high trade costs (Fujita et al. 1999). Note that under assumption (21), $\chi _2(\delta ,\gamma )>0$ also holds because $\chi _2(\delta ,\gamma )>\chi _0(\delta ,\gamma )$.

4.1 An extreme case

Before examining general cases, I consider an extreme case in which there is no friction in the information (i.e., $\delta \rightarrow 1$). In this case, it can be shown that there exists only one break point,

$$\begin{aligned} \phi _{\textrm{break}}^{\textrm{ex}}=\frac{2(\sigma -1)L-\sigma H}{2(\sigma -1)L+(3\sigma -2)H}, \end{aligned}$$

regardless of the form of competition for customer acquisition. When $\delta \rightarrow 1$, assumption (21) is reduced to $2L/H>\sigma /(\sigma -1)$, which ensures that this break point is positive. Thus, the symmetric dispersion is stable as long as $\phi <\phi _{\textrm{break}}^{\textrm{ex}}$ holds and becomes unstable when $\phi$ exceeds $\phi _{\textrm{break}}^{\textrm{ex}}$. Note that this break point is exactly the same as that derived by Pflüger (2004).

The reason for obtaining the same break point as in Pflüger (2004) is simple. When $\delta \rightarrow 1$, there is no difference in consumption variety, and the first and second terms in (18) vanish. The spatial configuration is then determined by the two forces captured in Pflüger (2004), resulting in the same break point.

4.2 The case of relative competition

Although equation $Z(\phi ,\delta ,\gamma )=0$ can have at most two solutions with respect to $\phi$, it can be shown that, in the case of relative competition, this equation has at most one solution that satisfies $0<\phi <1$. If $Z(\phi ,\delta ,\gamma )$ is evaluated at $\phi =\delta ^{-\gamma }$, I obtain

$$\begin{aligned} Z(\delta ^{-\gamma },\delta ,\gamma )=-\frac{4\sigma (1-\gamma )(1-\delta )H\mathcal {E}_{\textrm{sym}}}{(\sigma -1)(1+\delta )(H+2L)}<0 , \end{aligned}$$

implying that $Z(\phi ,\delta ,\gamma )=0$ has two solutions with respect to $\phi$ but one solution is larger than $\delta ^{-\gamma }$ ($>1$). Thus, in the range $0<\phi <1$, there exists at most one solution.

These results indicate that in the case of relative competition, the break point is uniquely determined under certain conditions. It is straightforward to verify that the condition for the existence of the break point is given by $Z(1,\delta ,\gamma )<0$. Then, I obtain the following proposition.

Proposition 3

Suppose that firms are engaged in relative competition, and that assumption (21) is satisfied.

• (i) If $Z(1,\delta ,\gamma )<0$ holds, then there exists a break point, $\phi _{\textrm{break}}^{\textrm{rel}}$, such that symmetric dispersion is stable under $\phi <\phi _{\textrm{break}}^{\textrm{rel}}$ and unstable under $\phi >\phi _{\textrm{break}}^{\textrm{rel}}$, where the break point is given by

  $$\begin{aligned} \phi _{\textrm{break}}^{\textrm{rel}}=\frac{\chi _1(\delta ,\gamma )-\sqrt{\chi _1(\delta ,\gamma )^2-\chi _0(\delta ,\gamma )\chi _2(\delta ,\gamma )}}{\delta ^\gamma \chi _2(\delta ,\gamma )} . \end{aligned}$$

• (ii) Otherwise, symmetric dispersion is stable for any $\phi$.

Consider that the trade costs gradually decrease. Part (i) of Proposition 3 implies that, in the case of relative competition, the bifurcation is similar to that of Pflüger (2004). This is because the first term in (15) functions only as an agglomeration force. Economic activities first disperse, but as trade costs drop, the configuration of economic activities becomes asymmetric and a core-periphery structure emerges.

Fig. 3

Bifurcation diagram in the case of relative competition. The solid line represents stable equilibria, and the dashed line is unstable. $\psi (\mathbb {K}_{r})$ is specified by $\psi (\mathbb {K}_{r})=\mathbb {K}_{r}/(1+\mathbb {K}_{r})$. I use specific values of parameters: $\alpha =1$, $\sigma =2$, $f=1$, $c=0.5$, $H=30$, $L=40$, $\gamma =0.5$, and $\delta =0.6$

Figure 3 shows an example of a bifurcation diagram, which is similar to that in Pflüger (2004). Because $\phi$ represents the freeness of trade (an inverse measure of trade costs), $\phi$ gradually increases from zero to one. When $\phi$ exceeds $\phi =\phi _{\textrm{break}}^{\textrm{rel}} \simeq 0.13$, the symmetry breaks and economic activities begin to agglomerate in one region, which is the emergence of the core. $\phi \simeq 0.27$ is the point at which the full agglomeration equilibria become sustainable.

Fig. 4

Conditions of $\gamma$and $\delta$in the case of relative competition. $\psi (\mathbb {K}_{r})$ is specified by $\psi (\mathbb {K}_{r})=\mathbb {K}_{r}/(1+\mathbb {K}_{r})$, and I use specific values of parameters: $\sigma =2$, $f=1$, $c=0.5$, $H=30$, and $L=40$. The area colored with gray is the irrelevant one, violating (21)

Figure 4 shows the area in which this type of bifurcation is observed in the $\gamma$–$\delta$ plane. In the present model, $\gamma$ and $\delta$ affect spatial configuration. A small $\delta$ motivates firms to disperse. When $\delta$ is small, information is not conveyed enough across regions, making competition for attention more local. A large $\gamma$ also encourages firms to disperse. When $\gamma$ is large, a firm benefits less from the cooperative effect. Therefore, the advantage of agglomeration in a region is reduced.

The impacts of $\gamma$ and $\delta$ on the spatial configuration imply that there may be cases in which economic activities always disperse, irrespective of the level of trade costs. As shown in Fig. 4, the symmetric dispersion is stable for any $\phi$ when $\gamma$ is large and $\delta$ is small because the dispersion force is quite strong in this area.

Although the pattern of bifurcation is quite similar to that of Pflüger (2004), the forces of agglomeration working in the model are different. As already mentioned, the first and second terms in the utility differential (18) represent the effect of accessibility of varieties. In the case of relative competition, the second term vanishes and this effect works only as an agglomeration force, because $\psi (\mathbb {K}_1)/\psi (\mathbb {K}_2)$ increases in $\lambda$. However, the presence of information friction weakens the traditional forces of agglomeration, reflected in the third and fourth terms in (18) as the cost of living and the wage differential.²⁰ It can be examined by comparing $\phi _{\textrm{break}}^{\textrm{rel}}$ with the break point in the extreme case ($\phi _{\textrm{break}}^{\textrm{ex}}$) whether the net effects of introducing information friction and competition for attention strengthen or weaken the agglomeration force.

Proposition 4

Suppose that $\delta$ is close to one. If $\gamma$ is close to zero, then $\phi _{\textrm{break}}^{\textrm{rel}}<\phi _{\textrm{break}}^{\textrm{ex}}$ holds, whereas if $\gamma$ is close to one, then $\phi _{\textrm{break}}^{\textrm{rel}}>\phi _{\textrm{break}}^{\textrm{ex}}$ holds.

Proof

See Online Appendix D. $\square$

Proposition 4 considers a case in which information technology is well developed. When the competition for attention is mild enough (i.e., $\gamma$ is close to zero), the symmetry breaks at the higher level of trade costs than in the case of no friction in the information (i.e., $\phi _{\textrm{break}}^{\textrm{rel}}<\phi _{\textrm{break}}^{\textrm{ex}}$), implying that the agglomeration force is stronger in the present model, compared with the traditional models. Conversely, when the competition for attention is fierce enough (i.e., $\gamma$ is close to one), the symmetry breaks at the lower level of trade costs than in the traditional case (i.e., $\phi _{\textrm{break}}^{\textrm{rel}}>\phi _{\textrm{break}}^{\textrm{ex}}$), indicating that the agglomeration force becomes weaker. These results are consistent with the observation that a large $\gamma$ discourages firms to agglomerate.

Fig. 5

Comparison of $\phi _{\textrm{break}}^{\textrm{rel}}$ and $\phi _{\textrm{break}}^{\textrm{ex}}$. $\psi (\mathbb {K}_\text{r})$ is specified by $\psi (\mathbb {K}_\text{r})=\mathbb {K}_\text{r}/(1+\mathbb {K}_\text{r})$, and I use specific values of parameters: $\sigma =2$, $f=1$, $c=0.5$, $H=30$, and $L=40$. The areas colored with gray are the irrelevant ones, violating either (21) or $Z(1,\delta ,\gamma )<0$

Unfortunately, it is difficult to obtain an analytical result on the relationship between $\phi _{\textrm{break}}^{\textrm{rel}}$ and $\phi _{\textrm{break}}^{\textrm{ex}}$ for more general values of $\delta$ and $\gamma$. However, the numerical example shown in Fig. 5 implies that a similar tendency can be observed even with wider ranges of $\delta$ and $\gamma$.

4.3 The case of absolute competition

Recall that in the case of absolute competition, (16) is assumed in addition to (21) for analytical tractability. Unlike the case of relative competition, it is possible that equation $Z(\phi ,\delta ,\gamma )=0$ has two solutions with respect to $\phi$ in the case of absolute competition.

Proposition 5

Suppose that firms are engaged in absolute competition, and that assumptions (16) and (21) are satisfied.

• (i) If $Z(1,\delta ,\gamma )>0$ and $\sqrt{\chi _0(\delta ,\gamma )\chi _2(\delta ,\gamma )}<\chi _1(\delta ,\gamma )<\delta ^\gamma \chi _2(\delta ,\gamma )$ hold, then there exist two break points, $\underline{\phi }_{\textrm{break}}^{\textrm{abs}}$ and $\overline{\phi }_{\textrm{break}}^{\textrm{abs}}$, such that symmetric dispersion is stable under $\phi <\underline{\phi }_{\textrm{break}}^{\textrm{abs}}$ or $\phi >\overline{\phi }_{\textrm{break}}^{\textrm{abs}}$ and unstable under $\underline{\phi }_{\textrm{break}}^{\textrm{abs}}<\phi <\overline{\phi }_{\textrm{break}}^{\textrm{abs}}$, where the break points are given by

  $$\begin{aligned} \underline{\phi }_{\textrm{break}}^{\textrm{abs}}=\frac{\chi _1(\delta ,\gamma )-\sqrt{\chi _1(\delta ,\gamma )^2-\chi _0(\delta ,\gamma )\chi _2(\delta ,\gamma )}}{\delta ^\gamma \chi _2(\delta ,\gamma )} , \\ \overline{\phi }_{\textrm{break}}^{\textrm{abs}}=\frac{\chi _1(\delta ,\gamma )+\sqrt{\chi _1(\delta ,\gamma )^2-\chi _0(\delta ,\gamma )\chi _2(\delta ,\gamma )}}{\delta ^\gamma \chi _2(\delta ,\gamma )} . \end{aligned}$$

• (ii) If $Z(1,\delta ,\gamma )<0$ holds or $Z(1,\delta ,\gamma )=0$ and $\chi _1(\delta ,\gamma )<\delta ^\gamma \chi _2(\delta ,\gamma )$ hold, then there exists a break point, $\underline{\phi }_{\textrm{break}}^{\textrm{abs}}$, such that symmetric dispersion is stable under $\phi <\underline{\phi }_{\textrm{break}}^{\textrm{abs}}$ and unstable under $\phi >\underline{\phi }_{\textrm{break}}^{\textrm{abs}}$, where $\underline{\phi }_{\textrm{break}}^{\textrm{abs}}$ is the same as the one given in part (i).

• (iii) Otherwise, symmetric dispersion is stable for any $\phi$.

Proof

See Online Appendix E. $\square$

As implied by case (i) of Proposition 5, the spatial configuration may change from dispersion to agglomeration, and finally to dispersion again, as trade costs drop; this is called redispersion. Although the possibility of the redispersion of economic activities is widely observed in extant models of NEG, the mechanism is different.

In contrast to the case of relative competition, the first term in (15) can be a dispersion force. For firms, it becomes more difficult to gain attention from consumers as the number of rivals in the home region increases. When a core-periphery structure exists in the economy, firms located in the core region suffer from the difficulty of gaining customers. However, in the peripheral region, firms face milder competition for attention because the number of firms is smaller. If trade costs are sufficiently low, the location of a firm is not important. Thus, in an economy with quite low trade costs, firms in the core region have an incentive to relocate to the peripheral region.

Fig. 6

Bifurcation diagrams in the case of absolute competition. The solid line represents stable equilibria, and the dashed line is unstable. $\psi (\mathbb {K}_{r})$ is specified by $\psi (\mathbb {K}_{r})=\mathbb {K}_{r}/(1+\mathbb {K}_{r})$. In the left- and right-hand pictures, I let $(\gamma ,\delta )=(0.8,0.45)$ and $(\gamma ,\delta )=(0.6,0.35)$, respectively. Other parameters are set as follows: $\alpha =1$, $\sigma =2$, $f=1$, $c=0.5$, $H=30$, and $L=40$

In the present model, the redispersion process occurs when, simply put, $\gamma$ is large and $\delta$ is not very large. This represents a situation in which competition for attention is fierce, and information is not sufficiently conveyed to the other region. In this situation, when trade costs are at intermediate levels, firms would rather be in a larger market despite fiercer advertising competition because a firm’s relocation to a less populated region would significantly decrease the total demand from the larger market. This reduction is caused by two spatial frictions: trade costs and informational differences. The former increases the price of its product in the core region and the latter reduces the number of consumers residing in the agglomerated region who have access to its product. However, when trade costs fall further, the former effect is weakened, which increases the motivation for a firm to relocate to a less populated region to avoid fierce competition. Recall that a small $\delta$ also motivates firms to relocate from the core to the periphery because the competition for attention is localized. Thus, the disadvantage of relocation is dominated by the advantage, and some firms begin to leave the core region.

The left-hand picture in Fig. 6 shows an example of a bifurcation diagram. Economic activities are symmetrically dispersed when $\phi$ is less than $\phi =\underline{\phi }_{\textrm{break}}^{\textrm{abs}} \simeq 0.69$. After $\phi$ exceeds this level, economic activities begin to concentrate in one region (i.e., the core). However, further declines in trade costs (or increases in freeness of trade) decrease the motivation for firms to stay in the core region. When $\phi$ becomes sufficiently large, the dispersion force becomes dominant and some firms begin to avoid fiercer competition for attention in the core region. This is because absolute competition generates a situation where a firm becomes less likely to acquire customers because of the large number of advertisements made by all of its competitors, which I refer to as market congestion. Finally, economic activities disperse symmetrically again after $\phi$ exceeds $\phi =\overline{\phi }_{\textrm{break}}^{\textrm{abs}} \simeq 0.9$. Although full agglomeration equilibria are never stable in this example, it does not mean that full agglomeration never occurs in the case of absolute competition. Note that this is only a specific numerical simulation, and changing some variables yields full agglomeration equilibria.

Fig. 7

Conditions of $\gamma$ and $\delta$ in the case of absolute competition. $\psi (\mathbb {K}_{r})$ is specified by $\psi (\mathbb {K}_{r})=\mathbb {K}_{r}/(1+\mathbb {K}_{r})$, and I use specific values of parameters: $\sigma =2$, $f=1$, $c=0.5$, $H=30$, and $L=40$. The area colored with gray is the irrelevant one, violating (16) and (21)

The way in which $\gamma$ and $\delta$ affect the spatial configuration is shown in Fig. 7. In this numerical example, when the pair of $\gamma$ and $\delta$ is in the area labeled with “two break points,” a bifurcation pattern similar to the left-hand picture of Fig. 6 is observed. In the area labeled with “always dispersion,” the dispersion force is so strong that symmetric dispersion is always a stable equilibrium. In the area labeled with “one break point,” a bifurcation pattern is similar to that in the case of relative competition. However, even in this area, a process similar to redispersion can be observed. The right-hand picture in Fig. 6 provides an example. Economic activities begin to agglomerate in one region when $\phi$ exceeds $\phi =\underline{\phi }_{\textrm{break}}^{\textrm{abs}} \simeq 0.58$, and full agglomeration equilibria become sustainable at $\phi \simeq 0.73$. After $\phi$ goes beyond $\phi \simeq 0.94$, economic activities start to disperse, although a symmetrically dispersed configuration never emerges, even if $\phi$ is almost one.

From Fig. 7, it can be observed that in the $\gamma$−$\delta$ plane, the dispersion force is strong in the southeast area, whereas the agglomeration force is strong in the northwest area. This result implies that the importance of the marketplace is strengthened if information technology is well developed. This finding is similar to that of Gaspar and Glaeser (1998), who showed that progress in information technologies increased the relevance of cities. The results are not only in line with the idea that information technology is complementary to cities but also provide new insight that the fierceness of competition among firms is an important factor that governs the impact of information technology on economic geography.

Although the present model relies on a quasi-linear utility for tractability, the key implications derived from this study are expected to be valid even with income effects. This is because the definition of the probability of matching between consumers and varieties is independent of consumers’ income, and each firm’s profit-maximizing level of advertisements, (11), is not affected by consumers’ demand parameters. A possible outcome from introducing income effects is that the agglomeration forces are strengthened and a catastrophic bifurcation from the symmetric dispersion to the full agglomeration may become more likely.²¹ Nevertheless, the effects of the information technology and competition for attention are expected to be unchanged, leading to the same implications as in the present study.

5 Concluding remarks

This study investigated the impact of economic integration, in terms of trade costs and information technologies, on the agglomeration of retail firms. For this purpose, I assumed consumers’ limited access to varieties in monopolistic competition. Because firms must inform consumers of their products, there is another kind of competition among firms in addition to price competition: competition for acquiring customers. The fierceness of this competition is a key parameter in the proposed model.

Another key parameter is the degree of progress of information technology. I introduced the informational difference across regions, which decreased as information technology developed. This difference is the source of the spatial consumption inequality, as empirically observed by Handbury and Weinstein (2015). Because the advancement of information technology makes interregional shopping easier, spatial consumption inequality shrinks, as shown by Fan et al. (2018).

In the long-run equilibrium with migration, I show that firms basically tend to agglomerate in a core region when trade costs are low. However, the opposite result may be observed under a certain condition in the case of absolute competition: if the competition for attention is fierce enough and information technology is not fully developed, the redispersion of economic activities may occur for sufficiently low trade costs. An important mechanism behind this observation is that absolute competition leads to a situation in which a firm becomes less likely to acquire customers because of a large number of advertisements made by all of its competitors, which I refer to as market congestion in this context. When information technology is not sufficiently developed, competition for attention is more localized, and some firms may be better off by moving from the core to the periphery. This relocation is more likely to occur under sufficiently low trade costs because it does not cause a significant change in the price of its product and does not lead to a considerable decrease in individual demand from the larger market.

The results of this study support the idea that information technology is a complement to cities, which was proposed by, for example, Gaspar and Glaeser (1998). Furthermore, a new insight can be derived from the present study: the fierceness of competition among firms should also be taken into account when considering the impact of information technology on economic geography.