Theory and Evidence from Japanese Firm-to-Firm Trade

Matching and Agglomeration: Theory and Evidence from Japanese Firm-to-Firm Trade Abstract This paper shows that matching frictions and thick market externality in firm-tofirm trade shape the agglomeration of economic activity. Using panel data of firm-tofirm trade in Japan, I demonstrate that firms gradually match with alternative suppliers following an unanticipated supplier bankruptcy, and that the rate of rematching increases in the geographic density of alternative suppliers. Motivated by these empirical findings, I develop a general equilibrium model of firm-to-firm matching in input trade across space. The model reveals that the thick market externality gives rise to agglomeration externalities affecting regional production and welfare. Using the calibrated model to the reduced-form patterns of firm-to-firm matching, I estimate that the elasticity of a region’s real wage with respect to population density due to thick market externalities is approximately 0.02. This finding highlights the substantial impact of thick market externality on the overall agglomeration benefit. 1 “When an industry has thus chosen a locality for itself, it is likely to stay there long: so great are the advantages which people following the same skilled trade get from near neighbourhood to one another... And presently subsidiary trades grow up in the neighbourhood, supplying it with implements and materials, organizing its traffic, and in many ways conducing to the economy of its material.” – Alfred Marshall (1890) “Principles of Economics” 1 Introduction Firms are more productive in more densely populated areas. Scholars widely agree that a substantial portion of this productivity pattern can be attributed to agglomeration externalities. In an influential book, Marshall (1890) argues that one important reason behind the agglomeration externality is the presence of matching frictions and thick market externality in firm-to-firm trade; i.e., suppliers and buyers in densely-populated areas have more opportunities to interact and engage in trade. However, despite the recognition of these mechanisms, our understanding of how these forces shape the spatial distribution of economic activity remains limited due to insufficient data and theoretical frameworks pertaining to firm-to-firm trade within and across regions. This paper investigates how matching frictions and thick market externality in firm-to-firm trade shape the agglomeration of economic activity. Leveraging a unique panel dataset of firm-to-firm trade in Japan in conjunction with unanticipated supplier bankruptcies as a natural experiment, I empirically examine the dynamics of firm-to-firm matching following supplier bankruptcies. I find that firms gradually match with an alternative supplier after the suppliers’ bankruptcy, and that the rate of rematching increases in the geographic density of alternative suppliers. Building upon these empirical findings, I develop a general equilibrium model that captures the dynamics of firm-to-firm matching in input trade across space. I demonstrate that thick market externalities give rise to agglomeration externalities, impacting aggregate regional production and welfare. By calibrating the model to the reduced-form patterns of firm-to-firm matching, I estimate that the elasticity of a region’s real wage to population density due to thick market externalities is approximately 0.02. By comparing this value to existing estimates of overall agglomeration spillovers, I conclude that thick market externalities substantially contribute to the overall agglomeration benefits experienced in the economy. This paper is structured into three main parts. In the first part, I provide reduced-form evidence of matching frictions and thick market externality in firm-to-firm trade. Iden- 2 tifying matching frictions in this context poses a significant challenge, as observational data alone does not reveal which input buyer is in need of a supplier, and if so, what type of suppliers. Therefore, a simple comparison of observed matching patterns across locations is insufficient to establish the presence of matching frictions and thick market externality. To overcome this challenge, I leverage unanticipated supplier bankruptcies as a natural experiment. After these shocks, firms are plausibly in need of alternative suppliers. The rate at which firms match with new suppliers provides insight into the severity of matching frictions. Moreover, to what extent the matching rates increase in the density of suppliers provides information about the thick market externality. To implement this idea, I leverage a rich panel dataset of firm-to-firm trade in Japan, complemented by a comprehensive list of bankruptcies. A crucial aspect of this dataset is that it provides information on the primary causes leading to bankruptcy. Within this dataset, I focus specifically on supplier bankruptcies categorized as “unanticipated reasons,” which encompass events like the death of company representatives or natural disasters. I uncover the following patterns of firms’ responses to unanticipated supplier bankruptcies. First, I demonstrate that firms only gradually match with alternative suppliers following unanticipated supplier bankruptcies. At the same time, these firms experience a significant decline in production. These facts suggest the presence of matching friction in firm-to-firm trade. Second, I find a robust positive correlation between matching rates with new suppliers and the geographic density of alternative suppliers. This evidence provides support for the thick market externality. To address the potential concern about selective firm entry, such as firms adept at finding alternative suppliers tending to locate in areas with higher supplier densities, I show that this correlation is robust to controlling for location fixed effects and various firm characteristics, using the birth location of the CEO as exogenous variation for a firm’s location, and focusing specifically on the bankruptcies of non-primary suppliers. Third, I do not find a significant correlation between matching rates and the density of other buyers. Therefore, there is no strong evidence that the congestion externality stemming from other input buyers plays a role in the market of firm-to-firm matching. Drawing upon these empirical patterns, the second part of the paper develops a quantitative theoretical framework that captures the dynamics of firm-to-firm matching in input trade across space. The primary objective of this model is to establish a clear link between the thick market externality and the resulting agglomeration externality. Firms undertake 3 production activities utilizing intermediate goods, which can be sourced either through direct matches with suppliers or through indirect sourcing from costly intermediaries. Due to the presence of matching frictions, matching with suppliers occurs gradually over time. At the same time, the thick market externality enhances the rate at which firms establish supplier linkages in locations and sectors served by a larger number of suppliers. I show that this thick market externality gives rise to agglomeration externalities that impact regional aggregate productivity and welfare. Moreover, when accounting for population mobility, this agglomeration externality generates a force driving population concentration. In the third part of the paper, I use the developed theoretical model in conjunction with the reduced-form evidence to quantitatively assess the magnitude of the agglomeration benefit resulting from the thick market externality in firm-to-firm trade. To achieve this goal, I utilize the exact-hat algebra approach proposed by Dekle, Eaton, and Kortum (2008) to calibrate the model to the data encompassing multiple heterogeneous locations and cross-regional firm-to-firm trade. An essential aspect of this calibration process involves determining the values of key structural parameters: the elasticity of matching functions and the iceberg cost associated with indirect sourcing. I calibrate these parameters using the reduced-form effects of unanticipated supplier bankruptcies on new supplier matching rates and sales reduction. Using the calibrated model, I conduct two sets of counterfactual simulations to examine the magnitude of the agglomeration force resulting from the thick market externality. In the first set of simulations, I analyze the impacts of an increase in the population size of the Tokyo prefecture. In the baseline model, I observe that Tokyo’s real wage increases in its population size, with an elasticity of 0.137. When I abstract the thick market externality, this elasticity decreases to 0.118. Therefore, the elasticity of agglomeration benefit attributed to the thick market externality is approximately 0.02 (≈ 0.137 − 0.118). To provide a benchmark for this result, I also undertake the same counterfactual simulation under an alternate value of productivity spillovers from local population density (modeled as a separate agglomeration externality). I find that abstracting the thick market externality yields a reduction in agglomeration benefit comparable to decreasing the elasticity of local productivity spillovers by 0.03. Although on the lower end, this value is within the range of existing estimates for agglomeration productivity spillovers (from 0.02 to 0.1; i.e., Melo, Graham, and Noland 2009). These findings suggest that a significant proportion of the overall agglomeration benefit can be attributed to the thick market externality in 4 input trade. In the second set of counterfactual simulations, I investigate how these agglomeration externalities amplify the effects of exogenous productivity increases in the Tokyo prefecture. In the baseline model, I observe that Tokyo’s real wage responds to productivity growth with an elasticity of 2.5. This high elasticity is primarily driven by cost propagation effects through sectoral input-output linkages. When I abstract the thick market externality in firm-to-firm matching, this elasticity decreases to 0.241. Therefore, the presence of the thick market externality amplifies the impacts of productivity shocks on local real wages. Furthermore, I observe that abstracting the thick market externality leads to a comparable decrease in the amplification of productivity shocks as reducing the elasticity of local productivity spillovers by 0.03-0.04. Therefore, the thick market externality plays a crucial role in amplifying regional productivity shocks, and its significance is comparable to typical estimates of agglomeration productivity spillovers. The primary contribution of this paper is to provide an empirical and theoretical analysis of thick market externality in firm-to-firm trade as a source of agglomeration externality. This mechanism aligns with one of the core agglomeration theories put forth by Marshall (1890). More recently, Duranton and Puga (2004) underscores the significance of matching between different agents as one important mechanism behind agglomeration externalities. However, the empirical and theoretical exploration of this specific agglomeration mechanism has been limited due to a lack of data and theoretical framework. The closest empirical evidence to this paper is provided by Holmes (1999), who finds that firms located in denser areas tend to rely more on external suppliers for their input purchases. In contrast, this paper goes beyond cross-sectional evidence by examining the dynamics of supplier matching using detailed dynamic firm-to-firm trade data. Theoretically, Krugman and Venables (1995) and Venables (1996) propose models of agglomeration based on the love of varieties in intermediate inputs. My focus is instead on exploring matching frictions and thick market externality in firm-to-firm trade. Outside the context of firm-to-firm trade, the concept of thick market externality has been extensively studied in the context of the labor market. Drawing on the theoretical framework of search and matching frictions in the labor market (Diamond 1982, Pissarides 1985, Mortensen 1986), this body of literature has sought to estimate matching frictions and the thick market externality using aggregate data (see Petrongolo and Pissarides (2001) for a comprehensive survey) or by analyzing the reemployment patterns of unemployed individuals using micro data (Petrongolo 2001). More recent papers in this 5 literature, such as Bleakley and Lin (2012), Jäger and Heining (2022), and Macaluso (2023), have utilized exogenous separations between workers and firms, such as worker displacement or mortality, to provide empirical evidence for matching frictions and the presence of thick market externality. In this paper, I employ similar empirical and theoretical approaches in the context of firm-to-firm matching in input trade. Besides the agglomeration literature, this paper contributes to the broader body of research on production networks, trade, and economic geography. First, this paper contributes to the literature on the endogenous formation of production networks across regions and countries (see Bernard and Moxnes (2018) and Antràs and Chor (2021) for recent surveys). Closest to this paper is Eaton, Kortum, and Kramarz (2022), who develop a theoretical framework where firms in different regions endogenously establish production linkages within a general equilibrium framework.1 Building upon Eaton et al. (2022), I extend their theoretical framework in two important ways. First, I incorporate dynamic matching and separation. This feature allows me to explicitly connect reduced-form evidence on dynamic supplier matching after unanticpated supplier bankruptcy to my theoretical framework. Second, I incorporate worker mobility, following the recent literature on quantitative spatial models (Allen and Arkolakis 2014, Redding and Rossi-Hansberg 2017). This extension enables me to establish a link between thick market externality in firm-to-firm trade and agglomeration phenomena.2 This paper also contributes to the literature on the disruption of production networks. In particular, Barrot and Sauvagnat (2016), Boehm, Flaaen, and Pandalai-Nayar (2019) and Carvalho, Nirei, Saito, and Tahbaz-Salehi (2021) document how natural disaster shocks to firms or regions propagate through firm-to-firm trade networks and impact connected firms. Jacobson and Von Schedvin (2015) shows that the impact of firm bankruptcy propagates through trade credit relationships. This paper contributes to this literature by examining the process of production network recovery following an unanticipated supplier bankruptcy. 1 See Oberfield (2018) for a model with a similar aggregation property as Eaton et al. (2022) without the spatial dimension of trade, and Panigrahi (2022) for an extension of Eaton et al. (2022) to incorporate multiple dimensions of firm heterogeneity. 2 Other recent papers on endogenous firm-to-firm trade networks focus on the role of domestic and international sourcing decisions (Furusawa, Inui, Ito, and Tang 2017), transportation infrastructure (Bernard, Moxnes, and Saito 2019), propagation of shocks across production networks (Dhyne, Kikkawa, Mogstad, and Tintelnot 2020, Lim 2018 and Huneeus 2018), firm size distribution (Bernard, Dhyne, Magerman, Manova, and Moxnes 2022), misallocation of production resources (Boehm and Oberfield 2020), market power distortions (Dhyne, Kikkawa, and Magerman 2022), effects of supplying to multinational companies (AlfaroUrena, Manelici, and Vasquez 2022), and firms’ quality choices (Demir, Fieler, Xu, and Yang 2023). 6 Finally, this paper contributes to the literature on search and matching frictions in domestic and international trade. In particular, Allen (2014) and Krolikowski and McCallum (2021) develop dynamic frameworks that incorporate search and matching frictions in the formation of trade relationships. In contrast to these studies, this paper empirically and theoretically studies matching frictions in the context of production networks, where firms and industries are interconnected through input-output linkages.3 The rest of the paper is organized as follows. Section 2 describes the main dataset. Section 3 presents reduced-form evidence of matching frictions and thick market externality using unanticipated supplier bankruptcies as a natural experiment. Section 4 develops a general equilibrium model of firm-to-firm matching across space. Section 5 calibrates my model to reduced-form patterns and quantifies the agglomeration externality arising from thick market externality. Section 6 concludes. 2 Data Data Sources. This paper utilizes a primary dataset obtained from Tokyo Shoko Research (TSR), a major credit reporting company in Japan. TSR collects comprehensive panel data of firms through personal interviews or phone surveys, supplemented by public resources such as financial statements, corporate registrations, and public relations documents. Most importantly for my purpose, this dataset reports up to 24 main suppliers and buyers.4 Firms also report the ranking of their suppliers and buyers based on the transaction amount. The dataset constitutes a yearly panel spanning from 2008 to 2016, covering nearly 70% of all firms in Japan.5 I use this dataset to track the dynamic evolution of supplier-to-buyer (or firm-to-firm) trade linkages. In my analysis, I exclude supplier linkages if the firms have a major ownership linkage, which corresponds to 1.5% of all supplier linkages. In the baseline analysis, I define supplier linkages based on reports from the buyer-side firms. I demonstrate that my empirical results remain robust when including linkages reported by the supplier-side firms. The dataset provides the headquarters’ address for each firm and the addresses of 3 Other papers that embed search frictions in international trade include Chaney (2014), Brancaccio, Kalouptsidi, and Papageorgiou (2020), Dasgupta and Mondria (2018), Eaton, Jinkins, Tybout, and Xu (2022), Startz (2021), Lenoir, Martin, and Mejean (2023). 4 The censoring at 24 is practically not binding; fewer than 0.1% of firms report exactly 24 suppliers. 5 Appendix Table A.1 provides additional details of the coverage of the dataset. Appendix Figure A.1 shows that the TSR dataset exhibits similar coverage rates across different municipalities in Japan. 7 all its establishments, including their prefectures and municipalities.6 At the same time, supplier-to-buyer linkages are reported at the firm level rather than the establishment level. In my main analysis below, I proxy firms’ location using the address of their headquarters. I show that the main reduced-form empirical results hold when restricting the analysis to firms with establishments concentrated in a single prefecture to address potential mismeasurement of the transaction location. The dataset also includes information on the CEO’s name and characteristics, including birth prefecture, graduation school, age, and gender. I merge this firm-to-firm dataset with the list of bankruptcies and their main documented causes. TSR compiles this data through their investigation of the parties involved. About 2 percent of bankruptcies are categorized as “unanticipated reasons,” which TSR’s internal document describes as “bankruptcies due to unanticipated accidental problems such as the death of representatives, flood disaster, fire, earthquake, traffic accident, fraud, theft, embezzlement, etc.” In the next section, I leverage these unanticipated supplier bankruptcies as a natural experiment to offer evidence of matching frictions.7 I use this firm-to-firm trade data to provide reduced-form evidence for matching frictions and thick-market externality in Section 3 and to calibrate the quantitative general equilibrium model in Section 5. For the latter, I augment the dataset with official government statistics. First, I use population size across 47 prefectures in Japan from the Population Census, conducted by the Ministry of Internal Affairs of Japan in 2010. Second, I use input-output table at the two-digit sector level, which was created by Japan’s Ministry of International Affairs and Communications in Japan in 2011. Descriptive Patterns of Firm-to-firm Trade and Geography. Before delving into the main empirical analysis using unanticipated supplier bankruptcy, I present suggestive evidence for the thick market externality in firm-to-firm matching. First, firm-to-firm trade linkages are geographically concentrated. Panel (A) of Figure 1 illustrates the cumulative distribution function of the geographic distance between the headquarters of a supplier and a buyer. The median distance between them is 37 kilometers, significantly shorter than the median distance for all potential pairs in Japan (172 kilometers; Bernard et al. 6 There are 47 prefectures and 1719 municipalities in Japan in 2014. Table A.2 reports the list of all reasons and their proportion among reported bankruptcies. Appendix Figure A.2 shows that these unanticipated bankruptcies occur uniformly across space, suggesting that these bankruptcies are not driven by a single regional shock such as the Great Tohoku Earthquake. I also confirm the robustness of my results by excluding firms in the Tohoku and Hokkaido regions. 7 Appendix 8 2019). These findings indicate a strong tendency for firms to engage with local suppliers, although a notable fraction of trade occurs across regions. If firms tend to source from local suppliers, do firms in denser areas tend to have a higher number of suppliers? Panel (B) of Figure 1 reveals a clear positive correlation between the population density and the number of suppliers per firm at the municipality level. This finding aligns with Holmes (1999), who documents the positive correlation between the fraction of externally purchased inputs per firm and firm density in the United States. These pieces of evidence suggest the presence of matching frictions and thick market externality in firm-to-firm trade. However, they are also consistent with an alternative hypothesis that firms in different locations have varying demands for external suppliers. In the next section, I provide more direct evidence of matching frictions and thick-market externality using unanticipated supplier bankruptcy as a natural experiment. Figure 1: Spatial Patterns of Firm-to-Firm Trade (A) Distances between Suppliers & Buyers (B) Number of Suppliers & Population Density Note: Panel (A) shows the cumulative distribution functions of the geodesic headquarter distances to reported suppliers in 2008 and 2016 from TSR data. Panel (B) shows the relationship between the population density and the average number of reported suppliers per firm at the municipality level in 2008 (weighted by the buyer-side firms’ sales). 3 3.1 Reduced-Form Evidence Empirical Strategy The primary approach for examining matching frictions and thick market externality is to analyze firms’ reactions to unanticipated supplier bankruptcies. When faced with these shocks, firms are plausibly in need of alternative suppliers. The rate at which firms establish new matches with suppliers indicates the level of matching frictions. Moreover, the 9 extent to which matching rates increase in relation to supplier density provides insights into the presence of thick market externality. In this section, I formalize these ideas using a simple model of supplier-to-buyer matching to derive a specification for empirical analysis. 3.1.1 A Model of Firm-to-Firm Matching Consider a firm in region j and sector m that is looking for a supplier in sector k. Due to matching frictions, it takes some time for the firm to find a suitable supplier and establish a transactional relationship. The rate at which successful matches are formed may depend on the number of potential suppliers and the number of buyers in search of suppliers. Building on the labor search and matching literature (Diamond 1982, Pissarides 1985, Mortensen 1986), I adopt a Cobb-Douglas matching function to capture these relationships. In particular, I assume that the Poisson rate at which a link between suppliers in sector k and buyers in sector m is created per unit of geographic area in location j follows: M  S∗j,k , B∗j,m  =η  S∗j,k  λS  B∗j,m λ B , S∗j,k = ∑ ∗ Snj,k (1) n∈N ∗ represents the geographic density of suppliers producing in location n who where Snj,k can supply intermediate inputs to buyers in location j (the asterisk denotes geographic density); S∗j,k sums these up across all supplier locations n (to account for inter-region trade); B∗j,m is the geographic density of input buyers in location j and sector m; η is a parameter governing the efficiency of the matching process; and λS and λ B represent the elasticities of the matching function. The Poisson rate at which a buyer matches with a     λS   λ B −1 ∗ ∗ ∗ ∗ ∗ supplier is given by M S j,k , Bj,m /Bj,m = η S j,k Bj,m . The elasticities of the matching function, λS and λ B , capture the key externalities in the matching process. A positive value of λS indicates that a buyer’s matching rate with a supplier increases with the geographic density of suppliers, capturing the “thick market externality” (Petrongolo and Pissarides 2001). Similarly, a negative value of λ B − 1 implies that a buyer’s matching rate with a supplier decreases with the geographic density of buyers, capturing the “congestion externality.” In Section 4, I show that these externalities give rise to agglomeration externality for regional production and welfare in general equilibrium. 10 3.1.2 Empirical Specification using Unanticipated Supplier Bankruptcies I now derive the empirical specification to test matching frictions as well as thick market and congestion externality. Consider a buyer f in sector m and location j who is searching for a supplier in sector k. Based on the matching function (1), I can approximate the probability that the buyer matches with a supplier within ∆ years as follows:8  NewSuppliers f jkm∆ ≈ η∆ × 1 + λ S log S∗j,k  B + λ −1  log B∗j,m  . (2) If NewSuppliers f jkm∆ is directly observed, one can estimate equation (2) to test for thick market and congestion externality. In fact, previous studies in the labor search and matching literature estimate a version of this equation using the observed data on employment status (e.g., Petrongolo 2001). However, in the context of supplier-to-buyer matching, researchers do not directly observe NewSuppliers f jkm∆ . This is because observational data does not reveal which buyer is in need of a supplier, and if so, what type of suppliers. To overcome this challenge, I focus on the event where a buyer loses a supplier due to an unanticipated bankruptcy. Following such a loss, buyers are likely to require an alternative supplier. Therefore, the speed at which firms match with a new supplier after an unanticipated supplier bankruptcy provides information about the magnitude of matching frictions. In other words, the causal effect of supplier bankruptcy on new supplier matching reveals the presence of matching frictions in firm-to-firm trade. I implement this idea empirically using a stacked-by-event difference-in-difference method. For each “event” of an unanticipated supplier bankruptcy, I define their input buyers as “treatment firms” (i.e., firms that report the bankrupt firm as a supplier one year prior to its bankruptcy). I also select “control firms” that are comparable to the treatment firms but whose suppliers do not go bankrupt. In particular, I select control firms such that their headquarters are located in the same municipality as the treatment firms and they have a supplier in the same four-digit industry as the treatment firms’ bankrupt supplier prior to the bankruptcy. Finally, I stack observations from all bankruptcy events. Adapting equation (2) in the difference-in-difference framework, the regression specification is given as follows:   ∗λS B∗λ B −1 ∆ . Taking = 1 − exp − ηS f jkm∆ j,m j,k the first-order approximation around ∆ ≈ 0 and then around log S j,k ≈ log Bj,m ≈ 0 gives equation (2). 8 Using the properties of a Poisson process, NewSuppliers 11 NewSuppliers f jkmgt = ∑ ∆=···−2,0,···  Trt f × 1[t − BankruptYearg = ∆]×  η̃∆ + λ̃S∆ log S∗j,k + λ̃∆B log B∗j,m + θ∆ X f jkmg + ξ f g + ζ gt + e f jkmgt , (3) where f denotes the firm, t denotes the year, Trt f denotes a dummy variable for the treatment firm, g represents the bankruptcy event (i.e., the set of control and treatment firms associated with the same bankruptcy event), BankruptYearg is the year of supplier bankruptcy in event g, and η̃∆ , and λ̃S∆ , λ̃∆B are regression coefficients ∆ years after the supplier bankruptcy shock. The event-year fixed effects ζ gt ensure that the treatment effect is identified by comparing the treatment and control firms within the same event, and the firm-event fixed effects ξ f g account for time-invariant firm heterogeneity.9 X f jkmg includes a series of fixed effects and firm characteristics that control for other dimensions of treatment heterogeneity, as I further discuss in Section 3.1.3. Notice that I normalize the event-study coefficients by one year prior to the bankruptcy, i.e, η̃∆ = λ̃S∆ = λ̃∆B = 0 for ∆ = −1. I cluster the standard errors at the firm f level.10 A key advantage of the stacked-by-event difference-in-difference method is that the treatment effects {η̃∆ , λ̃S∆ , λ̃∆B } are identified solely through the comparison between treatment firms and control firms for each period within each bankruptcy event. This specification avoids the potential bias in the conventional two-way fixed-effects (TWFE) difference-in-difference framework, which includes post-treatment observations of treatment units as a part of the control group.11 The stacked-by-event difference-in-difference design is a recommended approach to address this issue (Roth et al. 2023). The regression equation (3) identifies η̃∆ as the average effect of unanticipated supplier bankruptcy (when log S∗j,k , log B∗j,m , and X f jkmg are excluded from the regression), and λ̃S∆ , λ̃∆B as the heterogeneous effects of supplier bankruptcy with respect to log S∗j,k and log B∗j,m , respectively. Under the first-order approximation of the matching function 9 Note that firm f may appear multiple times as control firms in different groups g. 10 If the treatment assignment (i.e., unanticipated supplier bankruptcy) is independent between treatment and control firms, clustering standard errors at the firm level yields asymptotically consistent standard errors (Abadie, Athey, Imbens, and Wooldridge 2023). 11 The conventional TWFE regression in this context corresponds to a specification where the regression unit is a firm-and-year combination with firm and year fixed effects, without including for event-year fixed effects. de Chaisemartin and D’Haultfoeuille (2022) and Roth, Sant’Anna, Bilinski, and Poe (2023) provide recent surveys for the biases in TWFE specifications and alternative difference-in-difference specifications to overcome them. 12 (equation 2), these regression coefficients correspond to η̃∆ ≈ η∆, λ̃S∆ ≈ η∆λS , and λ̃∆B ≈ η∆(λ B − 1). Therefore, one can infer the elasticity of matching function with supplier density (i.e., thick market externality) from λS ≈ λ̃S∆ /η̃∆ and the elasticity of matching function with supplier density (i.e., congestion externality) from λ B ≈ λ̃∆B /η̃∆ + 1. I use these relationships to calibrate λS and λ B in the quantitative analysis in Section 5. The empirical implementation of regression (3) requires a proxy for the supplier den∗ ), and the buyer density, B∗ . In my baseline analysis, I use the number sity, S∗j,k (= ∑i Sij,k j,m of firms in the bankrupt supplier’s four-digit industry that have at least one buyer in firm f ’s headquarter prefecture j in 2008 (beginning of the sample) divided by the geographic area of j as a proxy for S∗j,k . This definition of supplier density accounts for the possibility that suppliers are located outside the treatment firms’ prefecture, as often observed in the data (Figure 1). This choice of S∗j,k aligns with the definition in my structural model in Section 4, where suppliers make entry decisions to each destination market j subject to variable and fixed costs. At the same time, I demonstrate the robustness of my reducedform results by using alternative proxies, such as counting only local suppliers or taking the distance-weighted sum of the number of suppliers across different locations. I proxy B∗j,m with the geographic density of firms that belong to the same industry as firm f and whose headquarters are located in prefecture j. Similar to S∗j,k , this definition of B∗j,m aligns with my structural model in Section 4. At the same time, I present the robustness of the results by employing alternative proxies, such as the number of firms that experienced supplier bankruptcy or separation in the same input industry as the treatment firms. 3.1.3 Endogeneity Concerns There are two types of endogeneity concerns: the endogeneity of supplier bankruptcy (Trt f ) and the endogeneity of supplier and buyer densities (S∗j,k and B∗j,m ). Below I discuss potential threats to these identification assumptions and my strategy to address them. Endogeneity of Supplier Bankruptcies. A key identification assumption for studying the effects of supplier bankruptcies is that supplier bankruptcy occurs exogenously to the unobserved trends in outcome variables for buyers (new supplier matching rates). An obvious violation of this identification assumption would occur if supplier bankruptcy is caused by a decline in buyers’ input demand. To rule out this possibility, I focus solely on supplier bankruptcies that are classified as “unanticipated reasons” as discussed in Sec- 13 tion 2. In particular, I exclude supplier bankruptcies classified as “sales decline” or “debt accumulation,” as they are unlikely to meet these criteria. To further validate that “unanticipated bankruptcies” indeed occur independently of buyers’ characteristics, I show below that the characteristics of treatment and control firms are broadly similar prior to the supplier bankruptcy events. Even if supplier bankruptcy is exogenous to the treatment firms, identification may still be compromised if the supplier bankruptcy is associated with potential outcomes of control firms. One possible scenario is that supplier bankruptcies are induced by aggregate regional shocks. In particular, the Great Tohoku Earthquake in 2012 might have directly affected both treatment and control firms in the region (Carvalho et al. 2021). To address this concern, I conduct robustness checks by excluding firms headquartered in the Tohoku and Hokkaido regions (the most severely affected regions) and firms that had direct trading relationships with these regions before 2011.12 Another possible scenario that could lead to identification failure is if supplier bankruptcy directly affects the outcome variables of control firms. For one thing, control firms may be indirectly connected to bankrupt firms through firm-to-firm trade networks. I address this concern by conducting robustness checks by excluding control firms that are within second-degree proximity in the firm-to-firm trade network (e.g., supplier’s buyer or buyer’s buyer) to firms experiencing unanticipated bankruptcies. For another, control firms may be direct competitors of the treatment firms. I address this concern by conducting robustness checks by excluding control firms belonging to the same two-digit industry as the treatment firms. Endogeneity of Supplier and Buyer Densities. Even if unanticipated supplier bankruptcies (Trt f ) are exogenous so that their average treatment effects are credibly identified, the heterogeneous effects with respect to supplier and buyer densities (S∗j,k and B∗j,m ) may be biased if these proxies are correlated with other dimensions of treatment effect heterogeneity. For instance, it is possible that firms with a higher ability to find an alternative supplier tend to enter regions where the geographic density of alternative suppliers is higher. Similarly, firms that heavily rely on a particular input may preemptively enter those regions. In both scenarios, the heterogeneous effects with respect to supplier and buyer densities are confounded by the unobserved differences in firms’ characteristics. 12 In Appendix Table B.3, I provide further robustness checks by excluding all supplier bankruptcies in 2009 (the year subsequent to the Great Financial Crisis) and excluding firms headquartered in Tokyo prefecture, the most densely populated prefecture in Japan. 14 To address these concerns, I first demonstrate the robustness of my results by controlling for observed dimensions of treatment heterogeneity through a series of location fixed effects and firm-level controls (X f jkmg in regression equation 3). In particular, my results are robust to controlling for prefecture fixed effects in X f jkmg . This specification ensures that the heterogeneous effects with supplier densities are identified as a comparison of treatment effects between firms within the same prefecture but facing supplier bankruptcies in different input sectors. I also show that controlling for firm sizes and relationships with bankrupt suppliers (e.g., relationship duration, geographic distance) does not affect my results. However, these fixed effects and firm controls do not completely eliminate the identification concern. In particular, location fixed effects do not rule out the possibility that firms that are more reliant on a particular input (and therefore more responsive to supplier bankruptcies in those inputs) may proactively locate in regions where the relative density of suppliers for those inputs is higher. To address this concern, I implement two additional empirical specifications. First, I demonstrate that my results remain robust when using the supplier and buyer density evaluated at the birth prefecture of CEOs as an instrumental variable (IV) for the supplier and buyer density evaluated at the firms’ actual prefecture. The idea is that firms’ location choices are influenced not only by supplier access but also by CEOs’ preferences to be close to their birthplaces. Therefore, the birthplaces of CEOs serve as a plausibly exogenous variation for the firms’ exposure to supplier density. To ensure that this IV does not capture systematic differences in the CEOs’ ability that depend on their birthplaces (e.g., schooling, probability of entrepreneurship), I further control for the fixed effects of the CEOs’ birth prefectures and the CEOs’ observed characteristics (education level, age, and gender) interacted with treatment dummies.13 Second, I demonstrate the robustness of my results by focusing solely on cases where bankruptcy occurs for treatment firms’ non-primary suppliers. While the availability of alternative suppliers in the event of supplier disturbance may influence firms’ location choices, this consideration is arguably less significant for non-primary suppliers. Therefore, if the results remain robust when solely focusing on non-primary supplier bankruptcy, the evidence supports the interpretation that the heterogeneous effects are not substantially driven by selection. To implement this idea, I use the reported ranking 13 Following a similar idea, Bleakley and Lin (2012) use a worker’s birth location as an exogenous variation to study thick market externality in the labor market. 15 of suppliers and exclude the supplier bankruptcy events that occur for the top 1 and 2 suppliers. 3.2 Empirical Results 3.2.1 Samples After excluding firms whose accounting years are outdated for more than a year, the final sample consists of 433 treatment firms connected to 181 bankrupt suppliers, with 11,889 assigned control firms in total.14 Table 1 presents the characteristics of both treatment and control firms prior to the supplier bankruptcy events. For each statistic in the first column, the table reports the mean and the standard error within the treatment group (second column) and the control group (third column). The last column reports the p-value of the null hypothesis that the means of each statistic is identical between the treatment and control groups. No statistically significant differences are observed in the number of suppliers, sales, employment, and their growth prior to the supplier bankruptcy event. In particular, the average number of suppliers is 5.09 for treatment firms and 5.18 for control firms, with no statistically significant difference between the two. These findings support the assertion that unanticipated bankruptcies occur independently of the characteristics of the buyers. Table 1: Baseline Characteristics of Treatment and Control Firms Variable Number of Suppliers (Baseline) Number of Suppliers (Pre-Period Yearly Growth) log Sales (Baseline) log Sales (Pre-Period Yearly Growth) log Employment (Baseline) log Employment (Pre-Period Yearly Growth) Sample Size Treatment 5.09 (3.51) 0.24 (1.09) 5.44 (0.69) -0.00 (0.14) 1.05 (0.57) -1.38 (0.76) Control 5.18 (3.62) 0.23 (1.12) 5.48 (0.77) -0.01 (0.13) 1.08 (0.61) -1.42 (0.80) 433 11,889 P-value (Treatment = Control) 0.65 0.77 0.29 0.95 0.35 0.34 Note: This table shows the characteristics of treatment and control firms prior to the supplier bankruptcy events. “Baseline” indicates one year prior to the supplier bankruptcy events and “Pre-Period Yearly Growth” indicates the growth rates from two to one year prior to supplier bankruptcy events. The parentheses in Columns “Treatment” and “Control” indicate the standard error of each variable. The unit of sales is 1000 Japanese Yen, and the bases of the logs for sales and employment are 10. 14 This number of bankrupt suppliers (181) is smaller than the total number of suppliers with unanticipated bankruptcies (269; Table A.2) after removing cases where there are no control firms satisfying the criteria outlined in Section 3.1.2. 16 3.2.2 Average Effect of Unanticipated Supplier Bankruptcies I start by documenting the average effects of unanticipated supplier bankruptcy on new supplier matching rates. Figure 2 displays the regression coefficients η̃∆ from regression specification (3), excluding the terms that capture treatment heterogeneity (log S j,k , log Bj,m , X f jkmg ). “All sector” indicates the impacts on the number of new suppliers across all input sectors (i.e., the number of connected suppliers in each period that are not connected prior to supplier bankruptcy events). “Within 2-digit sector” indicates the impacts on the number of new suppliers belonging to the same 2-digit industry as the bankrupt suppliers. I present the results for both total suppliers and suppliers within the two-digit sector of bankrupt suppliers, considering the potential mismeasurement of the set of alternative suppliers using reported two-digit sector classification. In cases where a firm exits during the sample period, the last observed value of the outcome variable is used as a substitute. Figure 2: Impacts of Unanticipated Supplier Bankruptcy on New Supplier Matching Note: This figure plots the regression coefficients η̃∆ and their 95 percent confidence intervals following specification (3) against the years ∆ since the supplier bankruptcy event, where the regression omits the terms corresponding to treatment heterogeneity (log S j,k , log Bj,m , X f jkmg ). “All sector” indicates the impacts on the number of new suppliers across all input sectors (i.e., the number of connected suppliers in each period that are not connected prior to supplier bankruptcy events). “Within 2-digit sector” indicates the impacts on the number of new suppliers that belong to the same 2-digit industry as the bankrupt suppliers. In cases where a firm exits during the sample period, the last observed value of the outcome variable is used as a substitute. The figure shows that firms gradually match with a new supplier after an unanticipated supplier bankruptcy. The impacts on the number of new suppliers are approximately 0.1 in the year of the supplier bankruptcy and approximately 0.2 after 3 years from the supplier bankruptcy event. These impacts are statistically significant and above 17 zero, indicating that firms are indeed in need of an alternative supplier. At the same time, the impacts are significantly below one and gradually increasing, indicating that the process of matching with new suppliers is imperfect and takes time. Furthermore, the impact on the number of new suppliers within the same two-digit industry as the bankrupt suppliers (“within 2-digit sector”) is almost half of the impact on suppliers across all sectors (“all sector”), even though there are a total of 98 distinct two-digit sectors. This finding indicates that the newly matched suppliers are predominantly likely to belong to the same industry as bankrupt suppliers, likely substituting bankrupt suppliers. Table 2 demonstrate the robustness of these findings to alternative specifications. Column (1) reports the results of the same baseline specification. Column (2) confirms the robustness of the results when excluding firms that exit during the sample period (instead of using the last observed value prior to the firm’s exit). Table 2: Average Impacts of Unanticipated Supplier Bankruptcy on Supplier Matching Dependent Variable: Number of New Suppliers (1) (2) (3) (4) (5) (6) (7) (8) (9) −0.02 (0.03) −0.01 (0.03) −0.06 (0.04) −0.02 (0.03) 0.04 (0.04) −0.01 (0.04) −0.06 (0.05) 0.02 (0.03) −0.01 (0.02) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) Trt x 1[t - BankruptYear = 0 or 1] 0.14∗∗∗ 0.15∗∗∗ 0.09∗∗ 0.15∗∗∗ 0.23∗∗∗ 0.12∗∗ 0.15∗∗ 0.18∗∗∗ (0.04) 0.13∗∗∗ (0.04) Trt x 1[t - BankruptYear = 2 or 3] 0.19∗∗∗ (0.06) 0.21∗∗∗ (0.06) 0.19∗∗∗ (0.06) 0.20∗∗∗ (0.06) 0.37∗∗∗ (0.06) 0.18∗∗∗ (0.07) 0.21∗∗ (0.09) 0.25∗∗∗ (0.06) 0.18∗∗∗ (0.05) Baseline Exclude Exit Firms Exclude Tohoku & Hokkaido Exclude Same-Industry Control Firms Exclude Indirect Connection to Bankrupt Firms Exclude Top 1 Supplier Exclude Top 2 Suppliers Exclude Ever-Treated Control Firms Impute Pre-Period Fixed Effects 433 181 11,889 85,951 433 181 11,889 83,548 342 142 9,277 67,542 433 181 11,177 80,934 433 181 7,880 55,441 348 163 9,012 64,909 261 136 6,669 47,837 433 181 10,889 79,052 433 181 11,889 83,944 Trt x 1[t - BankruptYear = -2 or -3] Trt x 1[t - BankruptYear = -1] Specification Number of Treatment Firms Number of Bankrupt Suppliers Number of Control Firms Observations (0.04) (0.04) (0.04) (0.04) (0.04) (0.05) (0.06) Note: Results of the stacked-by-event difference-in-difference regression specification (3), omitting the terms corresponding to treatment heterogeneity (log S j,k , log Bj,m , X f jkmg ). The outcome variable of the regression is the number of newly matched suppliers relative to the baseline period (one year before the shock). Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. See the main text for additional details on each specification. Columns (3)-(7) address endogeneity concerns of supplier bankruptcy as discussed in Section 3.1.3. Column (3) presents the robustness results by excluding firms headquartered in the Tohoku and Hokkaido regions and firms that had direct trading relationships with these regions before 2011. Column (4) addresses concerns about direct competition by excluding control firms located in the same prefecture and belonging to the same two-digit 18 industry as the treatment firms.15 Column (5) demonstrates the robustness of the results by excluding control firms that are within second-degree proximity in the firm-to-firm trade network to firms experiencing unanticipated bankruptcies. The somewhat larger treatment effects observed in this specification may be attributed to the omission of larger firms from the sample, which may have the capacity to switch to in-house production. Columns (6) and (7) show the robustness of the results when excluding supplier bankruptcies that occur to a firm’s top 1 and top 2 suppliers, respectively. Columns (8) and (9) present additional specification tests for the difference-in-difference framework. Recall that my baseline specification adopts the stacked-by-event difference-in-difference design, which circumvents the potential bias in the two-way fixed-effects specification as discussed in Section 3.1.2. On top of this treatment, Column (7) shows the robustness by explicitly excluding control firms if they receive supplier bankruptcy shocks (regardless of the reasons) in the post periods, as suggested by Sun and Abraham (2021). In Column (8), I show robustness using the imputation estimator proposed by Borusyak, Jaravel, and Spiess (2023), which involves estimating the firm-and-event fixed effects solely using data before supplier bankruptcy events and then plugging these fixed effects into the regression equation. I now turn to the impacts on firm sales and exit. In Column (1) of Table 3, I report the impacts on firm sales growth defined by arc-elasticity as (Sales f ∆ − 
 Sales f 0 )/ 12 Sales f ∆ + Sales f 0 following Davis and Haltiwanger (1992), where I define Sales f ∆ = 0 if firm f exits the market. This measure ranges between -2 and 2 and approximates log sales growth when the sales growth is infinitesimally small. In Column (1), I find that supplier bankruptcy leads to a decrease in this measure of sales growth by 0.042 (after 2 or 3 years from supplier bankruptcy). This finding supports the interpretation that firms experience disruptions in production until they find a suitable alternative supplier, providing further evidence of the presence of matching frictions. Column (2) shows that there is a positive effect on the exit probability of 0.025 (after 2 or 3 years from supplier bankruptcy). This is a large effect compared to the control mean of 0.044. Column (3) shows that the effects on sales growth are not statistically significant if I exclude firms that exit the markets. Therefore, the reduction in sales observed in Column (1) is primarily driven by the exit of firms. 15 Columns (5) and (6) of Appendix Table B.1 further alleviates the concerns about direct competition by demonstrating that supplier bankruptcy has no statistically significant effects on the new supplier matching rates and sales growth of other firms within the same prefecture and industry. 19 Table 3: Average Impacts of Unanticipated Supplier Bankruptcy on Sales and Exit Dependent Variable: Sales Growth (Arc-Elasticity) Exit Sales Growth (Arc-Elasticity) (1) Trt x 1[t - BankruptYear = -2 or -3] Trt x 1[t - BankruptYear = -1] (2) −0.014 (0.013) (3) −0.005 (0.011) (0.000) (0.000) (0.000) Trt x 1[t - BankruptYear = 0 or 1] −0.034∗∗ (0.015) 0.010∗ (0.006) −0.005 (0.010) Trt x 1[t - BankruptYear = 2 or 3] −0.042∗ (0.023) 0.025∗∗∗ (0.009) 0.020 (0.013) All All Cond. on Survival -0.145 433 181 11,889 84,113 0.044 433 181 11,889 85,951 -0.059 433 181 11,889 81,989 Samples Control Mean Number of Treatment Firms Number of Bankrupt Suppliers Number of Control Firms Observations Note: Results of the stacked-by-event difference-in-difference regression specification (3) by omitting the terms corresponding to treatment heterogeneity (log S j,k , log Bj,m , X f jkmg ) with alternate outcome variables. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. For Column (2), the interaction term of the pre-period dummy and the treatment dummy is omitted because the outcome variables in the pre-period are all zero. In Appendix Table B.1, I examine the impacts of supplier bankruptcy on other firmlevel outcomes. I find that there are no statistically significant effects on firms’ employment growth or profit-to-sales ratio. I also find a positive effect on the probability of retaining existing suppliers. Although the magnitude is smaller than the response of new supplier matching, this evidence suggests that retaining a supplier is an additional mechanism through which firms cope with supplier bankruptcy shocks. In Appendix Table B.2, I explore the heterogeneous effects of new supplier matching rates and sales growth with respect to reported supplier ranking, firm size, and supplier size. When supplier bankruptcy occurs for a primary (top 1) supplier, the effects on new supplier matching rates tend to be higher (although statistically insignificant), and the negative effects on sales growth are significantly larger, suggesting that bankruptcies of primary suppliers may have a more significant impact compared to those of secondary suppliers. I also find that the effects on new supplier matching rates are significantly lower for larger treatment firms (in terms of employment size), suggesting that larger firms may have the capacity to substitute bankrupt suppliers by relying on in-house production capabilities. Lastly, I do not find statistically significant heterogeneous effects with respect to the size of the bankrupt suppliers. 20 3.2.3 Heterogeneous Effects with respect to Supplier and Buyer Density In this section, I explore the heterogeneous effects of supplier bankruptcies with respect to supplier and buyer density, thereby investigating the evidence for thick market and congestion externality. Table 4 presents the results of regression specification (3). Each column in the table corresponds to a separate regression, with Panel A and B displaying the regression coefficients for the pre-period and post-period, respectively. In all the specifications, I include prefecture fixed effects in X f jkmg (interacted with treatment dummies). This specification ensures that the heterogeneous effects with respect to supplier densities are identified by comparing the treatment effects among firms located in the same prefecture but facing supplier bankruptcies in different input sectors. Since these fixed effects saturate the coefficients of treatment status dummies (Trt f × 1[t − BankruptYearg = ∆]), I omit the average effects and solely report the heterogeneous effects. Column (1) reveals statistically significant and positive heterogeneous effects in relation to supplier density. In particular, a one log point increase in supplier density is associated with an increase in the treatment effects by 0.13 in 0-1 years (first row in Panel B) and 0.17 in 2-3 years (second row in Panel B). These effects are sizable relative to the average treatment effects (0.14 and 0.19 in Table 2). These findings support the existence of thick market externality in firm-to-firm trade. On the other hand, there are no statistically significant heterogeneous effects observed for buyer density; the estimated coefficients are small in magnitude and close to zero (third and fourth rows in Panel B). It should be noted that the lack of statistical significance is not due to imprecise estimates, as the standard errors of these coefficients are similar in magnitude to those for supplier density. Overall, these results suggest limited evidence for congestion externality. While these patterns of heterogeneous effects suggest the presence of thick market externality and the absence of congestion externality, there is a concern that they may capture other dimensions of treatment heterogeneity through the selective firm entry. To address this concern, I provide additional robustness tests as discussed in Section 3.1.3. In Column (2), I demonstrate the robustness of the results by including additional controls for heterogeneous effects related to the employment size of the firm and the bankrupt supplier, as well as the strength of the relationship with the bankrupt suppliers (reported supplier rankings, geographic distance, and relationship duration). In Column (3), I demonstrate the robustness of the results by using the supplier and buyer density evaluated at the birth prefecture of a CEO as an IV for the supplier and buyer density evaluated at the 21 Table 4: Heterogeneous Impacts of Unanticipated Supplier Bankruptcies Dependent Variable: Number of New Suppliers (1) (2) (3) (4) (5) (6) (7) (8) Trt x 1[t - BankruptYear = -3 or -2] x log Supplier Density 0.04 (0.03) 0.03 (0.03) −0.03 (0.04) 0.01 (0.04) 0.01 (0.05) 0.004 (0.03) 0.03 (0.03) 0.03 (0.03) Trt x 1[t - BankruptYear = -3 or -2] x log Buyer Density 0.02 (0.04) 0.05 (0.04) 0.09∗ (0.06) 0.06 (0.05) 0.08 (0.06) 0.02 (0.04) 0.05 (0.04) 0.03 (0.04) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.13∗∗∗ (0.04) 0.14∗∗∗ (0.04) 0.16∗∗∗ (0.06) 0.11∗∗ (0.04) 0.08∗ (0.05) 0.10∗∗∗ (0.03) 0.11∗∗∗ (0.04) 0.11∗∗∗ (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.17∗∗∗ (0.06) 0.16∗∗∗ (0.06) 0.17∗∗ (0.08) 0.17∗∗∗ (0.06) 0.23∗∗∗ (0.08) 0.10∗ (0.06) 0.13∗∗ (0.06) 0.09 (0.06) Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density −0.01 (0.04) −0.03 (0.04) −0.01 (0.06) −0.03 (0.05) −0.07 (0.06) −0.06 (0.04) −0.003 (0.04) −0.03 (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density −0.07 (0.06) −0.08 (0.06) 0.02 (0.08) −0.09 (0.08) −0.05 (0.10) −0.06 (0.07) −0.06 (0.06) −0.09 (0.06) IV: Firm Density in CEO’s Birth Prefecture Exclude Top 1 Supplier Exclude Top 2 Suppliers Exclude Tohoku & Hokkaido Exclude Same-Industry Control Firms Exclude Indirect Connection to Bankrupt Firms Panel A: Pre-period Panel B: Post-period Specification Trt FE x Year FE x Prefecture FE Trt FE x Year FE x Buyer and Supplier Size Trt FE x Year FE x Firm-Relationship Controls Trt FE x Year FE x CEO Birth Prefecture FE and CEO Characteristics Number of Treatment Firms Number of Bankrupt Suppliers Number of Control Firms Observations X X X X X X X X X X X X X X X X X X X X X X X 433 181 11,889 85,939 433 181 11,889 85,395 405 174 10,933 60,767 348 163 9,012 64,522 261 136 6,669 47,516 342 142 9,277 67,079 433 181 11,177 80,440 433 181 7,880 56,783 Note: Results of the stacked-by-event difference-in-difference regression specification (3). Each column corresponds to a separate regression, and Panel A and B report the regression coefficients for the pre-period and post-period, respectively. Supplier density is defined as the number of firms in the bankrupt suppliers’ four-digit industry and that have at least one buyer in firm f ’s headquarter prefecture j in 2008 (beginning of the sample) divided by j’s geographic area. Buyer density is defined by the geographic density of firms that belong to the same two-digit industry to firm f and whose headquarters are located in prefecture j. The standard deviation of log supplier and buyer densities among treatment firms is 1.73 and 1.78, respectively. Column (2) controls for the employment size of the firm and the bankrupt supplier, the reported supplier rankings, geographic distance to the bankrupt suppliers, and the relationship duration to the bankrupt supplier in X f jkmg . Column (3) controls for the prefecture FE for the CEO’s birthplaces and the CEO’s age, gender, and education levels. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. See the main text for additional details on each specification. firm’s actual prefecture. In Columns (4) and (5), I show robustness to my results by excluding supplier bankruptcies that occur to a firm’s top 1 and top 2 suppliers, respectively. Columns (6), (7), and (8) repeat the same robustness tests in Columns (3), (4), and (5) of Table 2 to excluding firms in and connected with Tohoku and Hokkaido regions, excluding firms within second-degree proximity to firms experiencing unanticipated bankruptcies, and excluding control firms in the same prefecture and industry of the treatment firms. Appendix Table B.3 shows that these empirical results are further robust to various alternative specifications, including by omitting exited firms from samples, excluding bankruptcies in 2009 (the year of the Great Financial Crisis) and firms in Tokyo prefecture (the densest prefecture in Japan), excluding firms that have establishments outside their headquarter locations, and excluding firms with potentially outdated accounting information. The results are further robust to alternative definitions of supplier density, such 22 as counting only local suppliers or taking the distance-weighted sum of the number of suppliers (Appendix Table B.4). The results are also robust to an alternative definition of buyer density, such as using firms that faced supplier bankruptcy or separation during the same periods (Appendix Table B.5). I also find a similar (but noisier) pattern of results by including supplier-reported supplier-to-buyer-linkages when constructing the outcome variables (Appendix Table B.6). An additional concern for the above analysis is the external validity, given its focus on a limited number of supplier bankruptcies that occur due to unanticipated reasons. To address this concern, I provide additional analyses in Appendix Tables B.7, B.8 and B.9, where I examined the impacts of supplier bankruptcies resulting from “management failures” among the list of bankruptcy reasons in Appendix Table A.2. Compared to supplier bankruptcy that occurs for “unanticipated reasons”, this type of supplier bankruptcy has a potential endogeneity concern. For example, firms that source from suppliers with problematic management practices may preemptively switch suppliers in advance. Despite this concern, I find a similar pattern of the average and heterogeneous effects on the number of new suppliers, except that there are significant pretrends in the outcome variables. These results suggest that the analysis presented above has wider applicability beyond a narrow focus on unanticipated supplier bankruptcies. 4 Theoretical Framework In this section, I develop a general equilibrium model of matching frictions in firm-to-firm trade across space. In Section 4.1, I show how firm-to-firm matching shapes aggregate productivity across regions. In Section 4.2, I embed this model into a general equilibrium with endogenous wages and population mobility. In Section 4.3, I examine how the thick market externality gives rise to agglomeration externalities in regional production and welfare. 4.1 Production and Firm-to-Firm Matching Space is partitioned into discrete locations, denoted by i, j ∈ N . A unit mass of the population decides their residential locations. I denote Li as the population size in location i. Time is continuous and denoted by t. I focus on a steady-state equilibrium in which aggregate variables are constant (e.g., wages, output). Only firm-level variables, such as 23 supplier matching status, vary by t. Without risk of confusion, the subscript t is therefore omitted from the aggregate variables. 4.1.1 Technology and Preferences There is a continuum of firms in each location. Each firm belongs to a sector k, m ∈ K and has different productivity ϕ. All firms produce a product that can be used for final consumption or intermediate input for other firms. The unit cost of production for a firm ω in location i and sector m is given by cωt = 1 γ w L,m pωt,k γkm , ϕω Ai,m i k∏ ∈K (4) where ϕω is the productivity of firm ω; Ai,m is the productivity of location i and sector m; wi is the wage in firm ω’s production location i; pωt,k is the unit cost of intermediate inputs that firm ω has access to in period t in input sector k; γL,m is the labor share in production for sector m; and γkm is the share of sector-k intermediate inputs for sector m’s production. I assume that the production function exhibits constant returns to scale such that γL,m + ∑k γkm = 1 for all m ∈ K. The dynamic matching with suppliers determines the stochastic process of pωt,k , as further described below. There are measure Ni,k of entrepreneurs who jointly own firms in location i and sector k. Each entrepreneur owns ϕ−θ measure of firms with productivity above ϕ (for any value of ϕ). Therefore, in location i and sector k, there exists a measure µi,k ( ϕ) = Ni,k ϕ−θ of firms with productivity above ϕ. Following Eaton et al. (2022), this assumption implies that the distribution of firms’ unit costs also follows a power-law distribution. In particular, I conjecture that the measure of firms in location i and sector k with unit cost below c is given by Γi,k cθ , where Γi,k represents the endogenously determined inverse cost shifter in the equilibrium. I verify this conjecture by explicitly deriving Γi,k in Section 4.1.4. Final goods are consumed by workers. Workers’ utility in location j from goods consumption is defined by: Uj = ∏ k∈K Z ω ∈Ψ j,k qk (ω ) σ −1 σ ! dω σ σ −1 α k , (5) where qk (ω ) is the consumption of goods produced by firm ω, αk is the consumption share of final goods from sector k, Ψ j,k is the set of varieties available to consumers in 24 location j, and σ > 1 is the elasticity of substitution. 4.1.2 Cross-Location Trade Both final goods and intermediate goods can be tradable across locations. When firms want to sell their final or intermediate goods in location j, they must incur a fixed cost f j,k in each period, regardless of the firms’ production location. Once this fixed cost is paid, firms can deliver their products to location j subject to an iceberg trade cost τij,k (> 1), which represents the amount of goods required to deliver one unit of goods from location i to j. After entering location j, firms determine prices for both their final and intermediate goods. For final goods, the combination of CES utility and monopolistic competition implies that firms charge a constant markup of σ/ (σ − 1) over their marginal cost, net of the iceberg trade costs. Regarding intermediate goods, I assume that all bargaining power is held by the buying-side firms, resulting in prices equal to their marginal costs.16 By following the framework of Melitz (2003) and Chaney (2008), it can be shown that there exists a cutoff value for the marginal cost (inclusive of iceberg trade costs) that determines firms’ entry into each location. Employing standard algebra (see Appendix C.1), the threshold marginal cost for entering location j and sector k, denoted as c j,k , and the measure of suppliers selling in location j, denoted as S j,k , are given by: " c j,k = F θ − σ + 1 Yj,k θσ w j f j,k Ω j,k #1/θ , S j,k = Ω j,k cθj,k , (6) F represents the demand for final goods in location j and sector k, and Ω where Yj,k j,k ≡ −θ captures the shifter for the marginal cost distribution of suppliers entering ∑i Γi,k τij,k location j (inclusive of iceberg trade costs). 4.1.3 Matching between Suppliers and Buyers Conditional on entering location j as a seller, firms match with input buyers producing in the location. To model this matching process, I adopt the framework outlined in Section 3.1.1. More specifically, the Poisson rate at which connections are formed between suppliers in sector k and buyers in sector m per unit of geographic area in location j is 16 In Appendix E.1, I examine an alternative setup in which firms apply the same markup ratio for intermediate input buyers as they do for final consumers. 25 λ B    λS   . Here, S∗j,k = S j,k /Zj is the density B∗j,m represented by M S∗j,k , B∗j,m = η S∗j,k of suppliers, where Zj is the geographic area of location j. I proxy the buyer density by the geographic density of entrepreneurs, given by Bj,m = Bj,m /Zj = Nj,m /Zj .17 Once a match occurs, two firms establish a long-term relationship if and only if the buyer does not already have a supplier in sector k.18 Once formed, the relationship persists until it is exogenously terminated at the Poisson rate ρ j,km . During the duration of the relationship, the buyer is unable to form new matches with other suppliers. In cases where a firm lacks an existing supplier relationship in input sector k, it can source inputs through intermediaries. Intermediaries, in turn, randomly acquire intermediate goods from a pool of suppliers who have entered the market to sell in location j. However, this indirect sourcing comes with costs, as intermediaries face an additional iceberg cost denoted by χ > 1. This cost reflects the drawbacks associated with indirect procurement, such as transaction costs with intermediaries or a lack of customization. The steady-state probability that a firm in location j and sector m has a direct relationship with a supplier in sector k is given by: Λ j,km = v j,km , v j,km + ρ j,km (7)  λS   λ B −1 where v j,km = η S∗j,k B∗j,m is the Poisson rate at which a buyer in location j and sector m matches with a supplier in sector k. 4.1.4 Aggregate Production and Gravity Equations Recall my earlier conjecture that the measure of firms producing in location i and sector m with unit costs below c is given by Γi,m cθ . Given the process of matching with suppliers, I can now explicitly derive the inverse cost shifter, denoted as Γi,m , as follows: −θγL,m θ Γi,m = $m Ni,m Ai,m wi 17 Alternatively, ∏ ci,k −θγkm    1 + Λi,km χθγkm − 1 , (8) k∈K one can define the relevant set of buyers as those with a productivity level above ϕ. My B baseline specification is isomorphic to this specification by multiplying η by ϕ−θλ . 18 In Appendix E.2, I explore an extension where buyer-side firms make forward-looking decisions regarding accepting matches. 26 where $m is a constant that solely depends on exogenous parameters (refer to Appendix C.2 for the derivation). This equation summarizes the main channels in which firm-to-firm matching affects aggregate regional productivity. Similarly to standard trade models with sectoral input-output linkages (e.g., Caliendo and Parro 2014), Γi,m depends on the measure of entrepreneurs (Ni,m ), location and sector productivity (Ai,m ), wages (wi ), and the cost threshold of suppliers to enter market i (ci,k ). In addition to these conventional factors, if the location and sector has a high steady-state probability of matching with a supplier (a higher Λi,km ), firms can produce at a lower cost on average (a higher Γi,m ). This effect is more pronounced if the indirect sourcing is more costly (a higher χ). Following Melitz (2003) and Chaney (2008), it is also straightforward to show that aggregate bilateral trade flows follow gravity equations. In particular, the share of expenditure in location j for sector m goods that are produced in location i, both for intermediate goods and final goods, can be expressed as follows: πij,m
−θ Γi,m τij,m =  −θ . ∑i0 ∈N Γi0 ,m τi0 j,m (9) Furthermore, following the usual property in this class of model with power law distribution of production cost, conditional on a firm in location j sourcing from a supplier in location i, the expected transaction volume does not depend on the origin i. Therefore, this expenditure share coincides with the probability that firms in location j source from a supplier in location i for sector m input. 4.2 General Equilibrium I now embed the production and firm-to-firm trade in a general equilibrium framework, which incorporates the entry of entrepreneurs into production in each location and the mobility of workers across locations. Free Entry of Entrepreneurs. In each location and sector, there are a continuum of potential entrepreneurs. For entrepreneurs to initiate firm operations, they are required to make a fixed cost payment denoted as Fj,m in the unit of local labor (separately from the fixed cost associated with making sales as described in Section 4.1.2). Under free entry of entrepreneurs, the fixed cost payment prcisely offset the post-entry profit. Therefore, 27 Nj,m F σ − 1 X j,m = , σθ w j Fj,m (10) F is the aggregate final sales generated by firms in location j F ≡ where X j,m ∑n π jn,m Yn,m −1 and sector m. The term σσθ corresponds to the share of post-entry profit (excluding sales fixed cost) as a fraction of aggregate final sales (see Appendix C.1 for details). Productivity Spillovers. The productivity of location i and sector m, Ai,m , is given by:  Ai,m = Ãi,m Li Zi ε (11) where Ãi,m is the exogenous productivity of the location and sector, and ε is the elasticity of total factor productivity (TFP) with respect to local population density. This term summarizes all other types of agglomeration spillovers that operate through TFP (e.g., knowledge spillovers, labor market pooling). Market Clearing. The final goods market, intermediate goods market, and labor market clear in equilibrium. Final goods market clearing in location i and sector k is given by: F Yi,k = αk wi Li ψi , (12) where ψi is the (exogenous) efficiency unit of labor input per population, and αk is the expenditure share for final goods in sector k. Intermediate goods market clearing condition in location i for sales from sector k to sector m is given by: I Yi,km = γkm X̃i,m , X̃i,m = ∑ j∈N σ−1 F I Yj,m0 Yj,mm 0 + ∑ σ m0 ∈K ! πij,m0 , (13) where X̃i,m denotes the total expenditure on inputs (including both labor and intermediate inputs) required by firms in location i and sector m to meet demand from all downstream sectors m0 .19 19 In the expression for X̃ i,m , recall that 1/σ fraction of final goods sales goes to firm profit, and remaining revenue (including those from intermediate goods sales) are compensated for input expenditure. 28 Finally, labor market clearing condition in location i is given by: wi Li ψi = ∑ k σ−1 γL,k X̃i,k + σθ θ−σ+1 ∑ Yj,kF πij,k + σθ Yi,kF j∈N ! . (14) In this expression, the right-hand side shows the labor demand. The first term stems from variable labor cost, the second term stems from the fixed labor cost for the entrepreneurs’ entry, and the third term stems from sales fixed cost payment (see Appendix C.1 for the derivation of each term). Steady-State Equilibrium with Exogenous Population. I first define the equilibrium without population mobility. Given population size { Li }, the steady-state equilibrium is given by the steady-state matching probability {Λi,km }, inverse cost shifter {Γi,k }, entry cut-off {ci,k } sourcing share {πij,k }, wage {wi }, measure of entrepreneurs { Ni }, and location and sector productivity { Ai,k } that satisfy equations (6), (7), (8), (9), (10), (11), and market clearing conditions as stated above. Steady-State Equilibrium with Population Mobility. I next define the equilibrium with population mobility. Workers are freely mobile across locations. The utility of workers 1/υ who reside in location j is given by U j = K j Uj L− , where K j is the exogenous resij dential amenity, L j is the population size j, Uj is the utility from goods consumption as defined by equation (5). The parameter υ governs the dispersion force, which includes housing costs, negative residential spillovers, and idiosyncratic preference heterogeneity (Allen and Arkolakis 2014). Note that, with a CES utility function given by equation (5), the welfare of residents in location j is given by the real wage such that Uj = w j /Pj , α where Pj = ∏k Pj,kk is the ideal price index for final goods given by: 1− σ Pj,k = Z c j,k 0 c1−σ Ω j,k θcθ −1 dc =
θ − σ +1 θ Ω j,k c j,k . θ−σ+1 (15) Free mobility implies that utility is equalized across locations, i.e., U j = U for all locations j. Therefore, the population size of location j is given by: Lj = K υj (w j /Pj )υ ∑` K`υ (w` /P` )υ , (16) 29
1/υ . and the aggregate welfare is given by U = ∑` K`υ (w` /P` )υ The steady-state equilibrium is defined by the steady-state matching probability {Λi,km }, inverse cost shifter {Γi,k }, entry cut-off {ci,k }, sourcing share {πij,k }, wage {wi }, measure of entrepreneurs { Ni }, location and sector productivity { Ai,k }, and population size { Li } that satisfy equations (6), (7), (8), (9), (10), (11), (16) and market clearing conditions as stated above. Appendix D.2 shows that an equilibrium always exists. The same appendix further shows that, in the single sector special case, equilibrium without population mobility is unique (up to scale) regardless of the values of λS and λ B , and provides sufficient conditions for equilibrium uniqueness with population mobility. 4.3 Discussion: Agglomeration Forces I now analyze the agglomeration effects from thick market externality. To do so, I consider the impacts of an increase in the population size L j in the steady-state equilibrium (with exogenous population). I adopt the conventional notation to denote the marginal percentage change of equilibrium variable x by d log x ≈ x 0 /x − 1. Take location j’s wage as a numeraire such that w j = 1. Using equations (6), (12), and (15), the change in
the real wage in location j, d log Uj = d log w j /Pj = −d log Pj is expressed as: d log Uj = ∑ αm m " # 1 θ−σ+1 d log L j + ∑ πij,m d log Γi,m . θ ( σ − 1) θ i (17) The first term of this expression captures the love-of-variety effects for final consumption from market size, i.e., a larger market attracts a greater number of sellers, leading to a greater variety of available goods. The second term represents the cumulative effect of changes in the inverse cost shifter weighted by the expenditure share πij,m . To further decompose the second term, I specifically focus on the change in the inverse cost shifter 30 within location j, which usually occupies the largest expenditure share in location j: d log Γ j,m = d log Nj,m + | {z } firm entry effect θεd log L j | {z } productivity spillover effect        θγkm  + ∑ d log 1 + Λ j,km χ − 1 − θγkm d log c j,k  . {z } {z } | k∈K | thick market and congestion effect (18) input cost effect The first term of this expression, referred to as “firm entry effect,” captures the cost reduction benefit that arises from a larger number of entrepreneurs entering the market. This term represents a standard market size effect observed in models with firm entry and productivity heterogeneity (Melitz 2003, Chaney 2008). The second term, referred to as “productivity spillover effect,” represents the agglomeration productivity spillovers as introduced by equation (11). The third term is the new agglomeration channel in this model through thick market externality in firm-to-firm matching. This term can be further rewritten as (see Appendix C.3 for derivation):   d log 1 + Λ j,km χθγkm − 1  = Λ j,km χθγkm − 1
1 − Λ j,km
1 + Λ j,km (χθγkm − 1)   S B × λ d log S j,k + (λ − 1)d log Bj,m , d log S j,k = d log L j , d log Bj,m = d log Nj,m . (19) (20) Equation (19) summarizes how thick market externality and congestion externality shape the agglomeration externality. The presence of thick market externality (λS > 0) implies that an increase in the measure of suppliers (d log S j,k > 0) raises the steady-state supplier matching probability. This, in turn, leads to a decrease in aggregate production costs depending on the cost of indirect sourcing (χ). At the same time, in the presence of congestion externality (λ B − 1 < 0), an increase in the measure of buyers (d log Bj,m > 0) reduces the probability of matching with suppliers and consequently increases production costs. The agglomeration effect is determined by the balance between these two 31 externalities.20 The fourth term of equation (19), refered to as “input cost effect,” captures the changes in intermediate goods costs. Similar to standard models of sectoral input-output linkages (e.g., Caliendo and Parro 2014), d log c j,k is influenced by Γn,k in all other locations n, creating a positive feedback loop of input costs that spans across locations and sectors. At the same time, d log c j,k can also respond positively to a population shock d log L j . This occurs because a larger market has the capacity to accommodate suppliers with higher costs, as indicated by equation (6). This force is analogous to the pooling externality analyzed in the labor search and matching context (e.g., Acemoglu 2001, Bilal 2023), and it operates as negative externality.21 The presence of agglomeration externality also influences the effects of exogenous productivity shocks on the economy. Consider an increase in the exogenous component of total factor productivity (TFP) in location i and sector k, d log Ãi,k > 0. Regardless of the presence of thick market or congestion externalities, this shock increases real wages in location i as a direct shock to the location and sector, and through the cost propagation as captured by the “input cost effect” in equation (18). Furthermore, if workers are mobile, the increased real wage attracts more workers to location i. If there are agglomeration externalities, this population movement amplifies the impact of the productivity shock on local welfare. 5 Quantitative Analysis In this section, I combine the reduced-form results in Section 3 and the model in Section 4 to quantitatively assess the magnitude of the agglomeration benefit arising from thick market externality in firm-to-firm trade. Section 5.1 calibrates the model and the key structural parameters. Section 5.2 quantifies the importance of agglomeration externality through a set of counterfactual simulations. 20 In a special case of a single sector, population size L j and entrepreneurs entry Nj,m are proportional to each other (Appendix D.1). In this case, equation (19) is zero under any specification of constant returns to scale (CRS) matching technology (λS + λ B = 1). 21 In Appendix D.3, I discuss how the externalities discussed above lead to inefficiency in equilibrium entry by analyzing a planning problem in a special case of a single location and sector. 32 5.1 Calibration To take the model to data, I first characterize the minimal set of sufficient statistics needed for conducting counterfactual simulations. Following the exact-hat algebra approach proposed by Dekle et al. (2008), I reformulate the counterfactual equilibrium conditions in terms of the changes in the endogenous variables between the initial and counterfactual equilibria as fully described in Appendix F. Denoting the relative change in equilibrium x by x̂ = x 0 /x (with a hat), I show that, given the values of the production and preference parameters {αk ,γL,m ,γkm , θ, σ, ε}, parameters for the matching process {λS , λ B , χ}, and a F,Λ subset of baseline equilibrium variables {Li , πij,k , Yi,k i,km }, counterfactual equilibrium can be computed in terms of the changes in the endogenous variables {Λ̂i,km , Γ̂i,k , ĉi,k , ŵ j , N̂j,m , L̂ j , Â j,m }. I map the locations in the model to the 47 prefectures in Japan. Sectors in the model correspond to 98 two-digit sectors. Table 5 summarizes the calibrated parameters and observed equilibrium variables necessary for the counterfactuals. I describe the details of these calibrated parameters in turn. Table 5: Baseline Calibration Parameters Description (1) Parameters for Production and Preference γkm intermediate goods share for production γL,m labor share for production αm final goods consumption share σ elasticity of substitution of final goods θ dispersion of productivity distribution υ elasticity of migration with respect to real wage ε elasticity of productivity spillover with population density Values input-output table (2011) input-output table (2011) input-output table (2011) 3 5 3 0.05 (2) Parameters for Firm-to-Firm Matching λS elasticity of matching function with supplier density λB elasticity of matching function with buyer density χ iceberg cost for sourcing from intermediaries 0.9 1.0 1.37 (3) Baseline Variables Li population size πij,k sourcing share Λi,km steady-state probability of matching with suppliers F Yi,k aggregate final goods consumption Population census (2010) TSR firm-to-firm trade data (2008) TSR firm-to-firm trade data (2008) market clearing condition 33 5.1.1 Production and Preference Parameters I start by calibrating a subset of production and preference parameters {αk ,γL,m ,γL,m , θ, σ, υ, ε}. Given that these parameters commonly appear in many trade and spatial equilibrium models (e.g., Caliendo and Parro 2014, Redding and Rossi-Hansberg 2017), I calibrate these parameters following the standard procedure or using central values from the existing empirical literature. Following Caliendo and Parro (2014), I calibrate the intermediate input share {γL,m }, labor share of production {γkm }, and the final goods consumption share {αm } from the input-output table of overall Japan in 2011. I set the elasticity of substitution σ to 3 and productivity dispersion parameter θ to 5, as estimated by Eaton, Kortum, and Kramarz (2011) based on firm-level export data in France. The elasticity of migration with respect to real wage υ is set to 3 from Allen and Arkolakis (2014). Lastly, I set the productivity spillovers from population density at ε = 0.05, which is the mean value of the metaanalysis conducted by Melo et al. (2009). In the subsequent quantitative analysis, I explore different values of ε to provide a benchmark for the strength of agglomeration externality arising from thick market externality. 5.1.2 Matching Parameters The key parameters that capture the thick market externality are the matching function elasticities, {λS , λ B }, and the iceberg cost for indirect sourcing, χ. As discussed in Section 4, these parameters play a pivotal role in determining the agglomeration externality resulting from thick market and congestion externalities in firm-to-firm trade. To calibrate these parameters, I utilize the impacts of unanticipated supplier bankruptcies as documented in Section 3 of this paper. I begin by calibrating the elasticities of the matching function, λS and λ B , based on the effects of unanticipated supplier bankruptcies on new supplier matching rates. By employing a first-order approximation of the model’s implied matching rates as discussed in Section 3.1.2, the average effects on new supplier matching rates after ∆ years from an unanticipated supplier bankruptcy can be approximated as η̃∆ ≈ η∆. Similarly, the heterogeneous effects with respect to the log of supplier and buyer density can be approximated as λ̃S∆ ≈ η∆λS and λ̃∆B ≈ η∆(λ B − 1). Using these relationships, I can calibrate the magnitude of thick market externality as λS ≈ λ̃S∆ /η̃∆ and that of congestion externality as λ B ≈ λ̃∆B /η̃∆ + 1. Using the estimates from Section 3 for the average effect 34 on new supplier matching within 0 to 1 years as η̃∆ = 0.14 (Column (1) of Table 2) and the heterogeneous effects with respect to supplier density as λ̃S∆ = 0.13 (Column (1) of Table 4), I calibrate the elasticity of the matching function with respect to supplier density as λS = 0.9 ≈ 0.13/0.14. On the other hand, the heterogeneous effect with respect to buyer density is λ̃∆B = −0.01, which is statistically insignificant (Column (1) of Table 4). Therefore, I calibrate λ B = 1, indicating no congestion externality. I calibrate the iceberg cost for indirect sourcing, χ, using the average impact of unanticipated supplier bankruptcy on firm sales. From equation (4), the marginal production cost of a firm in sector m increases by a factor of χγkm if it loses a supplier in sector k. Given the power law distribution of firm productivity with dispersion parameter θ, it then translates to a decrease in expected sales by a factor of χθγkm .22 Therefore, the log changes in the expected sales after supplier bankruptcy are given by ∆ log E[Sales f jkm∆ ] = −(1 − NewSuppliers f jkm∆ ) × log χθγkm , where (1 − NewSuppliers f jkm∆ ) represents the probability that the treatment firm has not yet recovered a supplier and remains unmatched with a supplier in sector k. Using the reduced-form estimates of the average impacts on the arc-elasticity of sales growth after 0-1 years since supplier bankruptcy at −0.034 (Table 3), the impacts on the number of new suppliers at NewSuppliers f jkm∆ = 0.14 (Table 2), the average value of γkm at 0.025 among my treatment firms, and a baseline calibration of θ = 5, I calibrate χ = 1.37. This implies that the cost of sourcing inputs through intermediaries is, on average, 37 percent higher than sourcing from a directly matched supplier. In Table 6, I show that the calibrated values of λS , λ B , and χ indeed closely replicate the reduced-form results in Section 3. In Columns (1), (3), and (5), I reproduce the reduced-form estimates of the average and heterogeneous effects of unanticipated supplier bankruptcies from Section 3. In Columns (2), (4), and (6), I present the modelpredicted coefficients based on responses to exogenous separation from a supplier. To obtain these coefficients, I additionally need to calibrate η, which is not required for counterfactual simulation. I do so by minimizing the sum of squared differences between the two coefficients in Columns (1) and (2). I find that the calibrated parameters closely replicate the reduced-form estimates in the remaining columns. Columns (3) and (4) show that, given my choice of λS and 22 Because of the power law distribution of marginal cost, the drop of sales is driven entirely by decreasing the entry probability to make sales in each destination market, and not by the decrease of sales conditional on this entry decision, following the same logic as Melitz (2003) and Chaney (2008). Therefore, the elasticity of substitution σ does not play a role for this value. 35 λ B , the model replicates the patterns of heterogeneous effects on new supplier matching rates with supplier and buyer densities. The model-predicted regression coefficients for the supplier density are somewhat smaller than those using actual data, arising from the first-order approximation error. Nonetheless, the model-predicted regression coefficients remain within the 95% confidence intervals of data regression. Finally, Columns (5) and (6) demonstrate that my chosen χ closely reproduces the negative effects of supplier bankruptcy on sales growth. Although my baseline selection of λS , λ B , and χ reproduces the point estimates of the reduced-form regressions, it is important to consider the associated statistical uncertainties. In Section 5.2, I address this point by conducting a sensitivity analysis using alternative values for λS , λ B , and χ, taking into account the statistical uncertainty in these parameter estimates.23 Table 6: Model Fit of Matching Parameters to Targeted Regression Coefficients Dependent variable: Data Trt x 1[t - BankruptYear = 0 or 1] Trt x 1[t - BankruptYear = 2 or 3] Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density Observations Adjusted R2 Number of New Suppliers Model Data (1) (2) 0.139 (0.040) 0.191 (0.057) 0.110 0.227 85,951 0.604 (3) 0.134 (0.041) 0.168 (0.056) −0.008 (0.042) −0.071 (0.059) 85,939 0.617 Model (4) 0.078 0.130 −0.007 −0.007 Sales Growth (Arc-Elasticity) Data Model (5) (6) −0.034 (0.015) −0.042 (0.023) −0.038 −0.036 84,113 0.530 Note: Columns (1) and (3) reproduce the reduced-form estimates of the average and heterogeneous effects of unanticipated supplier bankruptcies on new supplier matching rates (Column 1 of Table 2 and 4, respectively), and Column (5) reproduces the average effects on sales growth (Column 1 of Table 3). Columns (2), (4), and (6) are the same regression coefficients using the model-predicted responses to exogenous supplier separation. To construct the model prediction, for each treatment and control firm in my sample, I simulate the probability that firms are matched with a new supplier after ∆ years since the supplier bankruptcy using the Poisson match rate S B v j,km = η (S∗j,k )λ ( B∗j,m )λ −1 and separation rates ρ j,km , where I choose the value of η to minimize the sum of squared differences between the two coefficients in Columns (1) and (2), and I obtain ρ j,km from TSR data. I also compute the expected sales changes using the expression ∆ log E[Sales f jkm∆ ] = −(1 − NewSuppliers f jkm∆ ) × log χθγkm for Column (6). 5.1.3 Baseline Variables I calibrate population share, Li , using data from the Population Census in 2010. For the construction of the input sourcing share, πij,k , I rely on the cross-sectional patterns observed in the TSR firm-to-firm trade data from 2008. Specifically, I count the number of 23 In Appendix G, I provide additional evidence of model fit using untargeted moments, including the supplier matching rates unconditional on supplier bankruptcy and aggregate firm sales by sector and location. 36 supplier-buyer linkages from suppliers located in prefecture i and sector k to buyers located in prefecture j, and normalize this count by the total number of supplier linkages in prefecture j for input sector k. Note that, as discussed in Section 4.1.4, my model predicts that the sourcing share probability coincides with the expenditure share for intermediate and final goods. I also calibrate the steady-state probability of matching with a supplier for buyers in location j and sector m for input sector k, Λ j,km , using the TSR firm-to-firm F , by solving for the trade data in 2008. Finally, I calibrate the final goods consumption, Yj,k market clearing conditions (12), (13), and (14) given parameter values of {αk ,γL,m ,γkm , θ, σ} and observed trade shares, {πij,k }.24 5.2 Counterfactual Simulations In this section, I use the calibrated model to quantify the agglomeration externality resulting from thick market externality through two counterfactual simulations: increasing local population size and increasing local productivity. 5.2.1 Impacts of Population Size In my first counterfactual simulation, I examine the effects of increasing local population size on local production and welfare. In particular, I conduct this counterfactual simulation for Tokyo prefecture, which is the most densely populated prefecture in Japan with more than 10 percent of the overall population. To highlight the significance of thick market externality, I compare the results of this counterfactual simulation with two alternative scenarios: one without thick market externality (λS = 0 instead of λS = 0.9) and one with lower levels of productivity spillovers (ε ∈ [0, 0.05) instead of ε = 0.05). Figure 3 reports the results. In Panel (A), I plot the log changes in real wages in Tokyo prefecture (wTokyo /PTokyo ) against the log change in Tokyo prefecture’s population size. Each line indicates alternate model specifications: The red solid line (“Baseline”) reports the results under my baseline calibration of thick market externality and productivity spillovers (λS = 0.9, ε = 0.05); the green dotted line (“Shut Down Thick Market Externality”) indicates the specification where I shut down thick market externality (λS = 0); the blue dashed line (“Shut Down Productivity Spillovers”) indicates the specification where I completely shut down productivity spillovers (ε = 0). 24 The baseline values of wages, wi , and the efficiency unit of labor, ψi , are not required for the counterfactual simulation. 37 Figure 3: Effects of Increasing Population Size in Tokyo (A) Real Wage in Tokyo Prefecture (B) Elasticities under Alternative Productivity Spillovers (ε) Note: Results of the counterfactual simulation to increase population size in Tokyo prefecture. Panel (A) plots the log changes in real wages in Tokyo prefecture (wTokyo /PTokyo ) against the log change in Tokyo prefecture’s population size. Panel (B) plots the elasticity of log wage with respect to population size in Tokyo (evaluated at its 50 percent increase) against alternative value for ε. Panel (A) reveals that, across all specifications, an increase in population size in Tokyo prefecture has a positive impact on real wages in Tokyo. The relationship between population size and real wages appears to be approximately log-linear. In my baseline specification, I find that the elasticity of Tokyo’s real wage with respect to population size is 0.137. When I exclude thick market externality in firm-to-firm trade, this elasticity decreases to 0.118. Thus, the elasticity of regional welfare with respect to population density resulting from thick market externality is approximately 0.02 (≈ 0.137 − 0.118). When I instead shut down productivity spillovers, this elasticity decreases to 0.107. Therefore, the elasticity of regional welfare with population density resulting from local productivity spillovers (with ε = 0.05) is approximately 0.03 (≈ 0.137 − 0.107).25 The remaining margin of agglomeration benefit mainly comes from the increased firm entry (i.e., Section 4.3). 25 Note that the effect of population size on the real wage is smaller than the assumed value of elasticity on local productivity (ε = 0.05). This is due to the decline in the ratio of producer price to consumption price (terms of trade) as population size increases. For related discussions, refer to Ramondo, Rodríguez-Clare, and Saborío-Rodríguez (2016). 38 In Panel (B), I further examine this elasticity under different levels of productivity spillovers ε ∈ [0, 0.05] indicated in the horizontal axis. The results show that excluding thick market externality has a similar magnitude of effects compared to reducing the elasticity of local productivity spillovers by 0.03 (= 0.05 − 0.02). Although on the lower end, this value is within the range of existing estimates for agglomeration productivity spillovers (from 0.02 to 0.1; i.e., Melo et al. 2009). This evidence suggests that agglomeration benefit from thick market externality is sizable relative to overall agglomeration productivity spillovers. Table 7 presents the results of the same counterfactual simulation with additional model specifications. For each model specification indicated in Column (1), I report the elasticity of real wages in Tokyo with respect to its population size (in Column 2) and the elasticity of average real wages across all prefectures in Japan with respect to Tokyo’s population size (in Column 3). Rows (a) to (c) in Column (2) present the same results explained in Figure 3. In Column (3) of these rows, I find that the average real wage across all of Japan increases at a faster rate compared to that in Tokyo prefecture in response to the population increase in Tokyo. This result arises because other regions benefit from increased productivity in Tokyo through trade linkages, while they do not directly suffer from pooling externality resulting from the increased entry cost threshold c j,k in Tokyo prefecture.26 Table 7: Effects of Increasing Population Size in Tokyo 1) Specification (a) Baseline (λS = 0.9, λ B = 1, ε = 0.05) (b) No Thick Market Externality (λS = 0, λ B = 1) (c) No Productivity Spillover (ε = 0) (d) Alternative CRS Matching Technology (λS = 0.5, λ B = 0.5) (e) Alternative CRS Matching Technology (λS = 1, λ B = 0) 2) Elasticity of Real Wage in Tokyo 3) Elasticity of Average Real Wage (Diff. from Baseline in Parenthesis) (Diff. from Baseline in Parenthesis) 0.137 (0.000) 0.170 (0.000) 0.118 (-0.019) 0.162 (-0.007) 0.107 (-0.030) 0.162 (-0.008) 0.119 (-0.018) 0.161 (-0.009) 0.120 (-0.016) 0.160 (-0.010) Note: Results of the counterfactual simulation to increase population size in Tokyo prefecture under alternative model specifications. The figures presented are the elasticities of the effects of the counterfactual simulation on the corresponding variable with respect to population size in Tokyo, evaluated at its 50 percent increase. In Rows (d) and (e), I report the results under alternative configurations of the thick market and congestion externality that satisfy constant returns to scale (CRS) matching technology (λS = 0.5 and λS = 0.5 in Row d and λS = 1 and λ B = 0 in Row e), instead of simply eliminating thick market externality (λS = 0 and λ B = 1 in Row b). The 26 In Appendix Figure H.1, I show that the increase in real wages is positively correlated with the strength of trade linkages with Tokyo prefecture. 39 results across these three specifications are broadly similar, indicating that thick market externality generates agglomeration benefits primarily through increasing returns to scale (IRS) in matching technology. In Appendix Table H.1, I provide a sensitivity analysis of the results under alternative calibrations and model specifications. As expected, the extent to which thick market externality contributes to agglomeration benefit varies significantly depending on the values of λS , λ B , and χ. The parameters σ and θ have an impact on the overall agglomeration benefit, although the contribution of thick market externality remains relatively stable. Furthermore, I explore alternative model specifications, such as incorporating forwardlooking acceptance decisions and accounting for firm profits from intermediate goods sales. I find that they yield similar conclusions regarding the contribution of thick market externality to agglomeration benefit. 5.2.2 Impacts of Local Productivity I next investigate how these agglomeration externalities amplify the impacts of an exogenous increase in local productivity. In particular, I conduct a counterfactual simulation using the calibrated model by raising the total factor productivity (TFP) in Tokyo prefecture ( ÃTokyo,k for all k ∈ K) while allowing for population mobility. Figure 4 reports my results. In Panel (A), I plot the log changes in real wages in Tokyo prefecture against the log change in Tokyo prefecture’s TFP. Similar to Figure 3, each line represents different model specifications: baseline specification (λS = 0.9, ε = 0.05), shutting down thick market externality (λS = 0), and shutting down productivity spillovers (ε = 0). Panel (A) reveals that in all specifications, a productivity shock generates an increase in the local real wage with approximate log-linear relationships. In the baseline model, the elasticity of local real wage with respect to the productivity increase is estimated to be 2.5. This high elasticity (which is above one) is primarily driven by the input cost propagation effects through input-output linkages, as discussed in Section 4.3. When I abstract thick market externality in firm-to-firm trade, this elasticity decreases to 2.41. Therefore, the thick market externality in firm-to-firm trade amplifies the productivity shock. Hence, the presence of thick market externality amplifies the impact of the productivity shock that operates through increased population size in Tokyo prefecture. When I abstract productivity spillovers by decreasing ε from 0.05 to 0, the elasticity decreases to 2.38. Consequently, the contribution of thick market externality to amplifying productiv- 40 Figure 4: Effects of Increasing Productivity in Tokyo (A) Real Wage in Tokyo Prefecture (B) Elasticities under Alternative Productivity Spillovers (ε) ∗ Note: Results of the counterfactual simulation to increase total factor productivity (TFP) in Tokyo prefecture (ATokyo,k for all k ∈ K). Panel (A) plots the log changes in real wages in Tokyo prefecture (wTokyo /PTokyo ) against the log change in Tokyo prefecture’s TFP. Panel (B) plots the elasticity of log wage with respect to the TFP in Tokyo (evaluated at its 20 percent increase) against the alternative value for ε. ity shocks is smaller but still comparable to the effect of local productivity spillovers with ε = 0.05. It is important to note that while these agglomeration effects result in significant amplification, the direct and cost propagation effects through input-output linkages play a quantitatively dominant role. In Panel (B), I present the same elasticity under alternative values for productivity spillovers ε ∈ [0, 0.05] indicated in the horizontal axis. I find that abstracting thick market externality has a similar magnitude of effects as reducing the elasticity of local productivity spillovers by 0.03-0.04. As emphasized in Figure 3, this value is smaller but comparable in magnitude to the range of existing estimates of agglomeration productivity spillovers found in the previous literature. Table 8 provides the results of the same counterfactual simulation, including additional model specifications. Similar to Table 7, each model specification is listed in Column (1), followed by the elasticity of real wages in Tokyo (Column 2) and the impact on aggregate welfare (U ) across all prefectures in Japan (Column 3). Column (2) of Rows (a) to (c) report the same results explained in Figure 4. In Column (3) of these rows, I observe a substantial 41 increase in aggregate welfare in this counterfactual, consistent with the interpretation that other regions also benefit from the increased productivity in Tokyo through trade linkages.27 In Rows (d) and (e), I report the results under alternative configurations of the thick market and congestion externality that satisfy constant returns to scale (CRS) matching technology. Similar to the previous counterfactual, the results across these three specifications are broadly similar, indicating that thick market externality generates agglomeration benefits primarily through increasing returns to scale (IRS) in matching technology.28 Table 8: Effects of Increasing Productivity in Tokyo 1) Specification (a) Baseline (λS = 0.9, λ B = 1, ε = 0.05) (b) No Thick-Market Externality (λS = 0, λ B = 1) (c) No Productivity Spillover (ε = 0) (d) Alternative CRS Matching Technology (λS = 0.5, λ B = 0.5) (e) Alternative CRS Matching Technology (λS = 1, λ B = 0) 2) Elasticity of log Real Wage in Tokyo 3) Elasticity of log Aggregate Welfare (Diff. from Baseline in Parenthesis) (Diff. from Baseline in Parenthesis) 2.50 (0.00) 1.31 (0.00) 2.41 (-0.09) 1.25 (-0.05) 2.38 (-0.13) 1.24 (-0.06) 2.42 (-0.08) 1.25 (-0.05) 2.44 (-0.06) 1.26 (-0.05) Note: Results of the counterfactual simulation to increase total factor productivity (TFP) in Tokyo prefecture under alternative model specifications. The figures presented are the elasticities of the effects of the counterfactual simulation on the corresponding variable with respect to TFP increase in Tokyo, evaluated at its 20 percent increase. 6 Conclusion This paper shows that matching frictions and thick market externality in firm-to-firm trade shape the agglomeration of economic activity. Using yearly panel data of firm-tofirm trade in Japan, I document that firms gradually match with an alternative supplier after an unanticipated supplier bankruptcy, and that these rematching rates increase in the geographic density of alternative suppliers. I develop a general equilibrium model of firmto-firm matching in input trade across space and show that the thick market externality leads to agglomeration externality for regional production and welfare. By fitting the model to the reduced-form patterns of firm-to-firm matching, I show that the substantial impact of thick market externality on the overall agglomeration benefit. 27 From equation (16), the change in aggregate welfare is given by Û = (∑i ∑LiL (ŵi / P̂i )υ )1/υ , which ` ` is the geometric average of the changes in real wages across locations. In Appendix Figure H.2, I show that the increase in real wages is positively correlated with the strength of trade linkages with Tokyo prefecture. 28 In Appendix Table H.2, I present the sensitivity analysis of calibrated parameters. The patterns of the results are broadly similar to the observations discussed in the sensitivity analysis in Section 5.2.1 (Appendix Table H.1). In addition, a larger value of υ leads to a greater contribution of thick market externality, which is driven by a greater population movement in the counterfactual simulation. 42 This paper highlights a particular agglomeration mechanism: matching friction and thick market externality of firm-to-firm trade. This is, of course, not the only relevant agglomeration mechanism. Other agglomeration mechanisms, such as labor market pooling or knowledge spillover, are also important and provide different policy implications. Therefore, an important direction of future work is to explore various agglomeration mechanisms using spatially-granular microdata and study their equilibrium implications.