Matching and Agglomeration - Theory and Evidence from Japanese Firm-to-Firm Trade

Matching and Agglomeration: Theory and Evidence from Japanese Firm-to-Firm Trade Abstract This paper proposes a new microfoundation for the agglomeration of economic activity: increasing returns to scale in supplier-to-buyer matching. Using a yearly panel of firm-to-firm trade in Japan, I document that firms gradually match with an alternative supplier after an unanticipated supplier bankruptcy; these rematching rates increase in the geographic density of alternative suppliers, and they do not decrease in the geographic density of other buyers. Motivated by these findings, I develop a quantitative spatial equilibrium model of search and matching frictions in firm-to-firm input trade. By fitting the model to the reduced-form facts, I show that the increasing returns to scale in matching is as important as other agglomeration forces in explaining the geographic concentration of economic activity. In the counterfactuals for a reduction in cross-regional trade cost, misattributing these agglomeration mechanisms substantially biases the estimates of welfare gains. 1 1 Introduction Firms are more productive in more densely populated areas. There is a broad consensus that a non-trivial fraction of this pattern is driven by agglomeration spillovers.1 However, there is limited consensus on what precise mechanisms of agglomeration spillovers contribute to this phenomena.2 One important mechanism originating from Marshall (1890) is that suppliers and buyers in urban areas have more opportunities to meet and initiate transactions. Although intuitive, this agglomeration mechanism has been relatively unexplored both empirically and theoretically. Empirically, evidence of this type of agglomeration force is scarce due to a lack of micro-data on supplier-to-buyer linkages. Theoretically, the complexity of modeling matching frictions in production linkages that connect different regions and industries has been a challenge for building a microfounded model of this type of agglomeration. In this paper, I study how increasing returns to scale in supplier-to-buyer matching shapes the agglomeration of economic activity. I open this paper by providing a set of reduced-form facts suggestive of this agglomeration mechanism using firm-to-firm trade data in Japan. Based on the reduced-form evidence, I develop a quantitative spatial equilibrium model of search and matching frictions in firm-to-firm input trade. The model accommodates rich heterogeneity in location fundamentals and in trade frictions across regions yet provides tractable aggregate equilibrium conditions for wages, population, bilateral trade flows, and welfare. I estimate the model’s key structural parameters in a sequential procedure to fit the reduced-form evidence. Finally, through a sequence of counterfactual simulations, I show that this agglomeration mechanism has a sizable contribution to the spatial distribution of economic activity and to the welfare effects of cross-region trade cost reduction. I first provide a set of new reduced-form facts suggestive of the matching frictions and increasing returns to scale in supplier-to-buyer matching. Inspired by the labor market literature on firm-to-worker matching, I document how firms respond to unanticipated 1 See, for example, Ciccone and Hall (1996), Greenstone, Hornbeck, and Moretti (2010), Combes, Duranton, Gobillon, Puga, and Roux (2012), Gaubert (2018) for evidence of agglomeration on firm productivity. 2 Marshall (1890) identified knowledge spillovers, labor market pooling, and industry linkages as three important agglomeration mechanisms, and Duranton and Puga (2004) classified them into sharing, matching, and learning mechanisms. Empirically, Ellison, Glaeser, and Kerr (2010) studied how various proxies of different agglomeration mechanisms are correlated with observed industry coagglomeration patterns. 2 supplier bankruptcies by using a panel of firm-to-firm trade data in Japan.3 After these shocks, firms are in need of alternative suppliers. I document three facts about firms’ responses to these shocks. First, firms match with alternative suppliers only gradually and their sales decline. Second, these matching rates increase in the geographic density of alternative suppliers. Third, these matching rates are not correlated with the density of other buyers. How do these patterns of firm-to-firm trade matter for the spatial distribution of economic activity? To answer this question, I develop a spatial equilibrium model of search and matching frictions in firm-to-firm trade. The model embeds search and matching in input trade (Diamond 1982, Pissarides 1985, Mortensen 1986) in a static framework of firm-to-firm trade in space (Eaton, Kortum, and Kramarz 2016) and in a spatial equilibrium framework with population mobility (Allen and Arkolakis 2014, Redding and RossiHansberg 2017). The model accommodates two distinct agglomeration spillovers. One type of agglomeration spillover is the increasing returns to scale (IRS) in firm-to-firm matching technology. The other type of agglomeration spillover is the direct effect of local population density on productivity (summarizing all other agglomeration spillovers). I estimate these two agglomeration spillovers using the model’s equilibrium conditions and the firm-tofirm trade data. In particular, I estimate the elasticities of firm-to-firm matching technology to fit the reduced-form effects of unanticipated supplier bankruptcies. I also estimate the elasticity of production spillover from local population density by using the spatial productivity distribution revealed from the gravity equations of cross-region trade flows. My estimates suggest that both types of agglomeration spillovers are quantitatively relevant in explaining the spatial productivity distribution. Equipped with the estimated model, I undertake a sequence of counterfactual simulations to quantify these two agglomeration spillovers in general equilibrium. I show how to use the model structure and the initial values of endogenous variables to undertake these counterfactual simulations, extending the “exact-hat algebra” approach by Dekle, Eaton, and Kortum (2008). In my first counterfactual simulation, I hypothetically shut down each of these two agglomeration spillovers. When I shut down the increasing returns to scale in 3 See Petrongolo and Pissarides (2001) for a survey of estimating matching frictions in labor market. In particular, Bleakley and Lin (2012) and Jäger and Heining (2019) use exogenous separation between workers and firms (displacement of workers and the death of a worker, respectively) to identify matching frictions, similar in spirit to the use of unanticipated supplier bankruptcy in this paper. Berliant, Reed, and Wang (2006) and Bilal (2020) connect these matching frictions in labor markets to agglomeration. 3 matching, the population-density premium in wages (regression slope of the logarithms of wages on the logarithm of population density) declines by 41 percent compared to the actual relationship. In comparison, when I further shut down the spillovers from local population density (which summarizes all other agglomeration forces), the populationdensity premium becomes negative and declines by 117 percent compared to the actual relationship. Therefore, the contribution of the IRS in matching technology on observed population-wage premium is nonnegligible to that of the overall agglomeration spillovers. In my second counterfactual simulation, I study how a reduction of regional trade cost affects aggregate welfare in the presence of these two types of agglomeration spillovers. Calibrating the reduction of these trade costs from the travel time reduction from Japan’s existing highway networks, I find a 6.19 percentage point increase in welfare from this trade cost reduction. When I undertake the same counterfactual simulation by omitting the IRS in matching technology, the predicted welfare gain is 4.53 percentage points, which is 27 percent smaller. On the other hand, omitting the production spillovers from local population density does not meaningfully affect the estimates of welfare gains. This difference arises because of the differences of the margins on which agglomeration spillover arises. The trade cost reduction directly increases the pool of capable suppliers in each location when trade occurs across regions. This effect increases local productivity in the presence of IRS in matching. On the other hand, population spillovers have a limited effect on aggregate welfare because productivity gains in locations with a population increase are largely offset by the productivity losses in locations with a population decrease. This result indicates that misattributing the agglomeration mechanisms substantially biases the estimates of welfare gains of the trade cost reduction. This paper contributes to different strands of literature. First, this paper contributes to the literature on agglomeration mechanisms through matching frictions and increasing returns to scale in supplier-to-buyer matching. As noted earlier, previous work is limited due to a lack of micro-data on supplier-to-buyer linkages and the complexity of modeling matching frictions in production linkages across regions and industries. The closest empirical evidence is provided by Holmes (1999), who documents that firms in denser areas tend to have higher shares of input purchase from external suppliers. Krugman and Venables (1995) and Venables (1996) propose a theoretical model of agglomeration through the love of varieties in intermediate inputs. This paper instead focuses on matching frictions and IRS in firm-to-firm matching technology. Second, this paper contributes to the growing literature on the endogenous formation 4 of firm-to-firm production networks, and it is particularly related to general equilibrium models of firm-to-firm trade with stochastic matching between suppliers and buyers.4 My theoretical model is closest to that of Eaton, Kortum, and Kramarz (2016), who build a static model of firm-to-firm trade across space under search frictions. I extend their model to incorporate dynamic search and matching, which is consistent with the reduced-form evidence from unanticipated supplier bankruptcy.5 My model is also related to Oberfield (2018), who characterizes general equilibrium of stochastic supplier-to-buyer matching, and Lim (2018) and Huneeus (2018), who model dynamic evolution of firm-to-firm trade in a similar framework. Unlike these papers, I explicitly incorporate geographic structure in which firm-to-firm trade occurs and derive tractable equilibrium conditions for wages, population, aggregate bilateral trade flows, and welfare. This feature of my model allows me to microfound the agglomeration of economic activity through firm-to-firm trade.6 Third, this paper contributes to the growing literature on quantitative spatial equilibrium models (Allen and Arkolakis 2014, Redding and Rossi-Hansberg 2017). So far, this literature has been agnostic about the nature of agglomeration mechanisms, and it typically introduces a reduced-form assumption such that the agglomeration spillover is summarized by local population density. In contrast, this paper builds a micro-founded model of agglomeration through supplier-to-buyer matching and shows that the correct specification of agglomeration mechanisms matters for the counterfactuals for trade cost reduction.7 Fourth, this paper contributes to the literature highlighting the implication of search and matching frictions in international and domestic trade. This paper is closest to Allen (2014) and Krolikowski and McCallum (2021), who adopt a dynamic search and match4 See Bernard, Moxnes, and Saito (2019) and Dhyne, Kikkawa, Mogstad, and Tintelnot (2020) for the deterministic models of firm-to-firm trade network formation, and Bernard and Moxnes (2018) and Antràs and Chor (2021) for the broader survey of endogenous production network formation. 5 See Panigrahi (2021) for an extension of Eaton, Kortum, and Kramarz (2016) to incorporate multiple dimensions of firm heterogeneity. 6 Other papers on endogenous firm-to-firm trade networks focus on the role of domestic and international sourcing decisions (Furusawa, Inui, Ito, and Tang 2017), firm size distribution (Bernard, Dhyne, Magerman, Manova, and Moxnes 2020), misallocation of production resources (Boehm and Oberfield 2020), market power distortions (Kikkawa, Magerman, and Dhyne 2020), effects of supplying to multinational companies (Alfaro-Ureña, Manelici, and Vasquez 2020), and firms’ quality choices (Demir, Fieler, Xu, and Yang 2021). 7 Other recent applications of quantitative spatial models include Diamond (2016), Fajgelbaum and Gaubert (2020), Nagy (2020). Papers using quantitative spatial equilibrium framework to study the effects of transport infrastructure include Donaldson and Hornbeck (2016), Allen and Arkolakis (2019), Fajgelbaum and Schaal (2019), Cosar, Demir, Ghose, and Young (2020), as surveyed by Redding and Turner (2015). 5 ing framework for the relationship formation between suppliers and buyers in multiple heterogeneous locations.8 Unlike these papers, I model production networks where firms and industries are interconnected through input-output linkages, and how the increasing returns to scale in matching technology in these production networks connect to the agglomeration phenomena. The rest of the paper is organized as follows. Section 2 describes the main data set. Section 3 presents reduced-form evidence of firms’ responses to unanticipated supplier bankruptcies. Section 4 develops a spatial equilibrium model of firm-to-firm matching. Section 5 estimates the model by combining the model’s equilibrium conditions and firmto-firm trade data. Section 6 presents a sequence of counterfactual simulations. Section 7 concludes. 2 Data This paper’s main data set of firm-to-firm trade comes from Tokyo Shoko Research (TSR), a major credit reporting company in Japan. TSR collects data through face-to-face or phone interviews, and this data is complemented by public resources (financial statements, corporate registrations, and public relations documents). The data is constructed as a yearly panel from 2008 to 2016, covering nearly 70% of all firms in Japan.9 Each year, TSR’s field surveyors ask each firm to report up to 24 main suppliers and buyers.10 This paper uses a snapshot of this database at the end of each year. In my analysis, I exclude supplier linkages if there is a major ownership linkage between the firms (corresponding to 1.5% of all supplier linkages). In the baseline analysis, I define the supplier linkages as those reported by the buyer-side firms. I show that my empirical results are robust to including the linkages reported by supplier-side firms. The data set reports the addressees of each firm’s headquarters and all of its establishments. At the same time, supplier-to-buyer linkages are reported at the firm level but 8 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 (2016), Startz (2021), Lenoir, Martin, and Mejean (2020). 9 See Appendix Table A.1. Appendix Figure A.1 shows that the coverage rates of the firms in TSR data set are similar across different municipalities in Japan. Nakajima, Saito, and Uesugi (2013), Bernard, Moxnes, and Saito (2019) and Furusawa, Inui, Ito, and Tang (2017) have previously used this data set to study the spatial extent of firm-to-firm trade, and Carvalho, Nirei, Saito, and Tahbaz-Salehi (2020) used it to study the propagation of shocks from Tohoku Earthquake in Japan through production networks. 10 The censoring at 24 is practically not binding; fewer than 0.1% of firms report exactly 24 suppliers. 6 not at the establishment level. In my main analysis below, I proxy firms’ location by the location of their headquarters. I show that the main empirical results are robust to restricting firms whose establishments are concentrated in a single prefecture to deal with the mismeasurement of the location where transactions actually occur. I merge this firm-to-firm data set with the list of bankruptcies that documents their main reasons. This data is also constructed by TSR through their investigation of the involved parties. About 2 percent of bankruptcies are classified as “unanticipated reasons,” which is described as “bankruptcies due to unanticipated accidental problems such as the death of representatives, flood disaster, fire, earthquake, traffic accident, fraud, theft, embezzlement, etc,” in a TSR’s internal document. In the next section, I use these unanticipated bankruptcies of suppliers as a natural experiment to provide evidence for matching frictions.11 Before proceeding to my main empirical results using unanticipated supplier bankruptcy, I first provide suggestive evidence for the agglomeration benefit through supplier-tobuyer matching. First, supplier linkages are geographically concentrated. Panel (A) of Figure 1 shows the cumulative distribution function of the geographic distance between the headquarters of a supplier and a buyer. The median distance between a supplier and a buyer is 37 kilometers, which is substantially smaller than the median distance of all possible pairs in Japan (172 kilometers; Bernard, Moxnes, and Saito 2019). These patterns of results indicate that there is a strong tendency for firms to source from local suppliers, though a nonnegligible fraction of trade occurs across regions. If firms have a tendency to source from local suppliers, do firms in denser areas tend to match with more suppliers? Panel (B) of Figure 1 shows that there is a stark positive correlation between the population density and the number of suppliers per firm at the municipality level. This finding is in line 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 are suggestive of the link between agglomeration and matching frictions in firm-to-firm trade. However, they are also consistent with an alternative hypothesis that firms in different locations have different demand for external suppliers. In the next section, I provide more direct evidence of matching frictions 11 Appendix Table A.2 reports the list of all reasons and its share out of all reported bankruptcies. Appendix Figure A.2 shows that these unanticipated bankruptcies occur equally across space and time, suggesting that these bankruptcies are not driven by a single regional shock such as the Great Tohoku Earthquake. I also show that my results are robust to excluding firms in the Tohoku area after 2011, the year of the Great Tohoku Earthquake (Appendix Table C.4). 7 in firm-to-firm trade using unanticipated supplier bankruptcies as a natural experiment. Figure 1: Cross-Sectional Evidence of Supplier-to-Buyer Matching and Agglomeration (A) Distances between Suppliers and Buyers (B) Number of Suppliers and 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. In 2008, median and 75th percentiles of the distributions are 37 and 252 kilometers, respectively. Panel (B) shows the relationship between the population density and the average number of suppliers per firm at the municipality level in 2008, where the latter is weighted by the (buyer-side) firms’ sales. 3 Evidence from Unanticipated Supplier Bankruptcies In this section, I document how firms respond to unanticipated supplier bankruptcies by using a panel of firm-to-firm trade data in Japan. These documented patterns are jointly suggestive of the presence of matching frictions and the increasing returns to scale in matching technology as modelled in Section 4. I also use these reduced-form estimates to estimate the key model parameters in Section 5. 3.1 Average Effects of Unanticipated Supplier Bankruptcies To understand the nature of matching frictions in firm-to-firm trade, I focus on firms’ responses to unanticipated supplier bankruptcies. After these shocks, firms are in need of alternative suppliers. How these firms respond to these supplier bankruptcies is informative about the nature of matching frictions, similar in spirit to the labor search and matching literature that studies the consequences of firm-to-worker separation (Petrongolo 2001, Bleakley and Lin 2012, Jäger and Heining 2019). In order to identify matching frictions, it is important that these supplier bankruptcies are exogenous to the (buyer-side) firm. For example, if supplier bankruptcy is due to the 8 discontinuation of a product line by the buyer-side firm, the imperfect recovery simply reflects the lack of demand. Therefore, I focus on cases where the reported reason for the bankruptcy is due to “unanticipated reasons” as discussed in Section 2. To identify the impacts of unanticipated supplier bankruptcies, I implement the differencein-difference method. For each treatment firm (i.e., firms experiencing unanticipated supplier bankruptcy), I select control firms whose headquarters are in the same municipality and have a supplier in the same four-digit industry as the bankrupt supplier (one year prior to its bankruptcy). Denoting the group of treatment and control firms by g, I estimate the following regression equation: Yf gt = βPost gt × Trt f + ηgt + ξ f g + e f gt , (1) where f is the firm, t is the year, Trt f denotes the dummy variable that takes one if f is the treatment firm, Post gt is the dummy that takes one if t is after the bankruptcy, and Yf gt is the outcome variable (e.g., number of new suppliers, sales). This regression controls for the group and year fixed effects ηgt (so that β is identified off of the withingroup comparison) and firm fixed effects ξ f g (so that β does not capture time-invariant firm heterogeneity).12 Standard errors are clustered at the firm f level. In order to give an equal weight on each group g, each control firm is weighted by the inverse of the number of control firms of group g.13 To ensure that control firms are not indirectly affected through supply-chain networks, I exclude control firms that are within second-degree proximity (in firm-to-firm trade network) to firms experiencing unanticipated bankruptcies at some point in the sample period. I further exclude firms whose reported accounting year is not updated for more than one year (at the point of the baseline period). The final sample consists of 421 treatment firms that are connected to 161 bankrupt suppliers, with 10,842 assigned control firms in total. The identifying assumption of the regression equation (1) is that unobserved trend in outcome variables are uncorrelated with the unanticipated supplier bankruptcy. This assumption is plausibly satisfied if these bankruptcies are indeed unanticipated. In fact, Appendix Table ?? shows that the characteristics of treatment firms and the control firms that firm f may appear multiple times as control firms in a different group g. 80 percent of suppliers immediately exit after their unanticipated bankruptcies. Since these supplier bankruptcies likely disrupt buyers even without a formal exit, I present reduced-form effects of supplier bankruptcies rather than the instrumental variable (IV) effects of supplier exit. 12 Note 13 About 9 are broadly similar, supporting the interpretation that unanticipated supplier bankruptcies indeed happen orthogonal to observable firm characteristics. Furthermore, in Appendix Table ??, I show that there are no differential pretrends between treatment firms and control firms in the new supplier matching rates and sales growth. Fact 1. After unanticipated supplier bankruptcies, firms match with alternative suppliers only gradually, and their sales decline. Table 1 presents the impacts of unanticipated supplier bankruptcy on the number of newly matched suppliers and sales using regression specification (1). Column (1) reports the impacts on the number of new suppliers relative to the baseline period (one year before the shock). Treatment firms are significantly more likely to match with new suppliers; they match with 0.17 more suppliers within 1 year, and 0.26 more suppliers within 3 years, compared to control firms. Therefore, treatment firms are indeed in need of an alternative supplier. At the same time, these effects are significantly less than one, yet gradually increasing over time, supporting the evidence for the presence of matching frictions. Column (2) shows the impacts of supplier bankruptcy on firm sales. To accommodate firm exit during the sample period, I take the inverse hyperbolic sine (IHS) transformation of sales as an outcome variable.14 I find that the treatment firms’ sales significantly decline by 21 percent after 0 or 1 years of the shock, and the magnitude persists (insignificantly) after 2 or 3 years. These patterns of results are consistent with the interpretation that firms face disruption in production until they find a suitable alternative supplier, providing additional support for the presence of matching frictions.15 As additional robustness and supportive evidence, Appendix Table C.1 shows that these supplier bankruptcies do not significantly affect the sales of the treatment firms’ other existing suppliers, indicating that treatment firms primarily respond by rematching with new suppliers instead of simply substituting from existing suppliers. In Appendix Table ??, I show that the newly matched suppliers are significantly more likely to belong to the same industry as bankrupt suppliers, consistent with the interpretation that treatment firms are in search of an alternative supplier. Appendix Table C.2 shows that there are no heterogeneous effects by firm size, suggesting that these matching frictions are relevant x is defined by log( x + ( x2 + 1)(1/2) ). When x is large, this is approximately log( x ) + log(2), hence the regression coefficients are interpreted as percentage effect. 15 These results are also in line with the findings of Carvalho, Nirei, Saito, and Tahbaz-Salehi (2020) and Boehm, Flaaen, and Pandalai-Nayar (2019), who document that the Great Tohoku Earthquake has negatively affected the sales of the domestic and foreign buyers of directly affected firms, respectively. 14 Inverse hyperbolic sine (IHS) transformation of variable 10 Table 1: Impacts of Supplier Bankruptcy on Supplier Matching and Sales Dependent Variable: Number of New Suppliers Trt x 1[t - BankruptYear = -2 or -3] (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) Trt x 1[t - BankruptYear = -1] (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∗∗∗ (0.04) 0.15∗∗∗ (0.04) 0.09∗∗ (0.04) 0.15∗∗∗ (0.04) 0.23∗∗∗ (0.04) 0.12∗∗ (0.05) 0.15∗∗ (0.06) 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) Specification 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 Number of Treatment Firms Number of Bankrupt Suppliers Number of Control Firms Observations Note: This table reports the result of the difference-in-difference regression (equation 1). The outcome variable of the regression is the number of newly matched suppliers relative to the baseline period (one year before the shock) for Column 1 and the inverse-hyperbolic sine (IHS) transformation of sales for Column 2. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. for both small and large firms. Appendix Table C.3 presents the effects on other firmlevel outcome variables. In particular, I find no effects on profit per sales, indicating that supplier bankruptcy primarily affects firm production through its scale but not through its profitability. 3.2 Heterogeneous Effects of Unanticipated Supplier Bankruptcies In this section, I document how the matching rates with new suppliers after unanticipated supplier bankruptcies relate to the geographic density of alternative suppliers and other buyers. First, I examine how these matching rates depend on the geographic density of alternative suppliers. To do so, I estimate the heterogeneous effects of unanticipated supplier bankruptcy using the following regression: NewSuppliers f gt = Post gt × Trt f × β + γ log SupplierDensity g + θZ f g + ηgt + ξ f g + e f gt ,
(2) where SupplierDensity g is the proxy of the geographic density of alternative suppliers. In the baseline specification, I define SupplierDensity g as the number of firms in the bankrupt suppliers’ four-digit industry that have at least one buyer in firm f ’s headquarter prefecture in 2008 divided by its geographic area. These “alternative suppliers" can be located outside firm f ’s prefecture, reflecting the fact that supplier linkages happen across regions (Figure 1). I show that my results are robust to alternative measures of supplier density, in particular using the number of firms with headquarters in firm f ’s lo- 11 cation, constructed without using the information of realized supplier-to-buyer linkages (Appendix Table C.5). To control for other dimensions of firm heterogeneity that affect the matching rates with new suppliers, I include location fixed effects, industry fixed effects, and other firm and supplier characteristics as an interaction with treatment and post dummy (Z f g in the above regression) as further discussed below. Fact 2. The matching rates with suppliers after unanticipated supplier bankruptcies increase in the geographic density of alternative suppliers. Table 2 presents the results of regression (2). Column (1) shows that firms in areas with higher density of suppliers are significantly more likely to match with new suppliers after unanticipated supplier bankruptcies. To facilitate the interpretation of the coefficients, I standardize the log of the supplier density to be mean zero and standard deviation one after residualized by Zg . The magnitude of this impact heterogeneity is sizable; a one-standard-deviation increase in the seller density proxy increases the effect by 0.13 after 2 or 3 years, relative to the average effects of 0.26. This implies that the treatment effect at the 95th percentile of the supplier density distribution is almost twice the average treatment effect, while it is almost zero at the 5th percentile. Column (2) shows that these results are robust to controlling for prefecture fixed effects and the bankrupt supplier’s two-digit industry (as interaction with treatment dummy and the post dummy). With these sets of fixed effects, this regression effectively compares two treatment firms that are in the same location but faced a supplier bankruptcy in different industries with different (relative) supplier density. The results are further robust to excluding firms that have multiple (non-headquarter) establishments in different prefectures (Column 3). Therefore, although I proxy firms’ locations by their headquarter, a mismeasurement of firms’ locations unlikely affects these results. While this evidence is consistent with the interpretation that a higher density of suppliers improves the matching rates, it is also consistent with alternative stories of firm selection. In particular, firms with higher ability in finding an alternative supplier or unobservably higher demand for external suppliers may selectively enter in locations with higher supplier density. To rule out this concern as much as possible, I employ an instrumental variable (IV) approach to exploit an exogenous component of firms’ location decision unrelated to supplier matching. To satisfy the exclusion restriction, the IV has to be uncorrelated with unobserved supplier matching rates. A candidate for this IV is a factor affecting a firm’s location decision other than through access to suppliers. Here, I 12 Table 2: Matching Rates and Supplier Density 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) 0.005 (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) −0.02 (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 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: The regression specification follows equation (2). Seller density is defined as the geographic density of suppliers in the bankrupting suppliers’ four-digit industry that have at least one buyer in firm f ’s prefecture in 2008. The seller density measure is normalized to be mean 0 with standard deviation 1 after residualized by other controls Zg . Column (1) only controls for log of the geographic area of treatment firm f ’s prefecture as an interaction with treatment and post dummy. Average effects are omitted from Columns (2) to (6), as they are saturated by the fixed effects. Columns (4)-(6) present results of the IV regressions by instrumenting log supplier density by that evaluated at the firm’s ideal location for buyer access (interacted with treatment dummy and post-period dummies), and Panel B shows the first stage of this IV regression. In Column (4)-(6), I omit samples without treatment firms without any buyers throughout the period. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. construct the IV based on the idea that the access to buyers is an important determinant for a firm’s location choice, on top of the access to suppliers.16 More concretely, I use the supplier density evaluated at the firm’s ideal location purely from a buyer access perspective to instrument for the actual supplier density. For each treatment firm f , I collect all buyers of the firm f throughout my sample period. From this list of buyers, I pick the most common four-digit industry (buyerind f ). I then identify the prefecture (n∗f ) where the firm density of industry buyerind f is the highest. I then construct my IV as the supplier density of the bankrupt supplier industry at this location (n∗f ) instead of the firm’s actual location. 16 In the context of international trade, Redding and Venables (2004) show that firm entry and country welfare is affected by both supplier and buyer access. 13 Intuitively, this IV captures the following variation. Even if two firms source from the same industry, they produce different products and services and sell to different industries. These two firms tend to enter into different locations depending on the access to their buyer industry. Consequently, they face different supplier densities of alternative suppliers at the point when an unanticipated supplier bankruptcy occurs.17 Column (4) presents the results of this IV regression. Panel (B) of the same column shows that there is a strongly significant first stage, confirming that firms’ location choice is indeed influenced by buyer access (conditional on supplier access). Using these instruments, I find significantly positive heterogeneous IV effects by supplier density. These results are further robust to controlling for firms’ and bankrupt suppliers’ observable characteristics. In Column (5), I additionally control for the employment size of the (buyer-side) firms, the employment size of the bankrupt suppliers, and the number of existing suppliers of the (buyer-side) firms in baseline (interacted with treatment times post dummy). In Column (6), I show robustness to controlling for the proxies for the strength of the relationships to bankrupt suppliers, including the geographic distance to the bankrupt suppliers, the duration of the relationship to the bankrupt supplier, and the input coefficient between buyer and supplier industry (obtained from a separate inputoutput table). Appendix Table C.4 shows that these empirical results are further robust to various specification tests, including by omitting exit firms from samples (instead of inserting zero for outcome variables in baseline above), excluding bankrupt suppliers in the Tohoku area after 2011 (the year of the Great Tohoku Earthquake) and those in 2009 (the year followed by the Great Financial Crisis in 2007-08), and by employing another IV strategy using the birthplace of firm CEOs as exogenous variation of firms’ location choice. The results are further robust to alternative definitions of supplier density (Appendix Table C.5), splitting samples to manufacturing and non-manufacturing supplier bankruptcies (Appendix Table C.6), and including supplier-reported supplier-to-buyer-linkages when constructing the outcome variables (Appendix Table C.7). Lastly, Appendix Table C.9 investigates the heterogeneous effects on sales. I find 17 One concern for this IV is that firms’ main supplier industry correlates with their main buyer industry. To rule out this possibility, in the IV regression, I additionally control for the supplier density at the firm’s ideal location purely from a supplier access perspective. My strong first stage after this control suggests that there is an independent variation of main supplier and buyer industries that I exploit in the IV strategy. Relatedly, only 7 percent of treatment firms have a main supplier and buyer industries coinciding with each other at the four-digit industry. 14 a significant average increase in sales per matched supplier (instrumented by supplier bankruptcy), yet these effects are not significantly different by supplier density. These patterns of the results indicate that, in this context, firms benefit primarily by faster supplier matching but not by better supplier matching from supplier density.18 Fact 3. The matching rates with suppliers after unanticipated supplier bankruptcies are not related to the geographic density of other buyers. I now study how the matching rates are related to the density of other buyers. In urban areas, there are not only many suppliers but also many buyers. If buyers crowd out each other to match with a supplier (for example, due to suppliers’ limited production capacity), the presence of more buyers may offset the benefit of the higher supplier density.19 The regression specification is given by: NewSuppliersigt = Post gt × Trt f × ( β + γ log SupplierDensity g + δ log BuyerDensity g + θZ f g ) + ηgt + ξ f g + eigt . (3) Relative to supplier density, it is less straightforward to define the density of relevant buyers, because researchers do not directly observe which firms are looking for a supplier. Here, I take three alternative measures for buyer density. The first measure uses the number of firms in the treatment firm’s prefecture that faced an unanticipated supplier bankruptcy in the same four-digit industry in the same year (divided by the geographic area). The second measure uses the number of firms facing any types of supplier separation in the same four-digit supplier industry in the same year. The third measure uses the number of firms that belong to the same two-digit industry and prefecture as the treatment firm. Table 3 presents the results. The first two rows report the heterogeneous effects by supplier density, and the next two rows report the heterogeneous effects by buyer density with three different proxies in each column. In all three measures, I find no statistically significant evidence that matching rates relate to buyer density; the estimated coefficients are small in magnitude and close to zero. This lack of statistical significance is not the result 18 See Helsley and Strange (1990) for a model where the latter force drives the agglomeration benefits and Duranton and Puga (2004) for conceptual distinctions about these two types of channels. 19 The literature of labor search and matching often find that the presence of more unemployed workers decreases other unemployed workers’ reemployment rate (Petrongolo and Pissarides 2001). My finding of no significant association between supplier matching rates and the density of other buyers highlights the differences between labor market and supplier-to-buyer matching markets. 15 of imprecise estimates; the standard errors of these coefficients have similar magnitudes as those for supplier density, which are robustly significant and positive. To further facilitate the interpretation of these coefficients, in the second to the bottom row of Table 3, I report the P-value of the statistical hypothesis that the coefficients on supplier density and buyer density sum up to zero (γ + δ = 0). I find that this null hypothesis is rejected at the 5 percent significance level in all specifications. Therefore, an increase of the density of suppliers and buyers by the same proportion is associated with a higher matching rates. In the next section, I build a model that connects these findings to the scale effects of matching technology. Table 3: Matching Rates and Other Buyer Density Dependent Variable: Number of New Suppliers (1) (2) (3) 0.11∗∗∗ (0.04) 0.14∗∗ (0.06) 0.09 (0.08) 0.02 (0.04) −0.02 (0.07) 0.04 (0.08) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.19∗∗∗ (0.06) 0.28∗∗∗ (0.09) 0.32∗∗ (0.14) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density −0.01 (0.06) −0.12 (0.10) −0.13 (0.13) Buyers in Same Two-Digit Industry Buyers facing Unanticipated Supplier Bankruptcies Buyers facing Supplier Separation Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density Definition of Buyers Trt x Post x Prefecture FE P-Value (Null: Sum of Coeffcieints on Supplier Density + Buyer Density = 0) X X X 0.026 0.028 0.002 Observations 76,069 76,069 76,069 Note: The regression specification follows equation (3) with three alternative definitions of buyer density. Buyer density is defined using the number of firms in the treatment firm’s prefecture that faced an unanticipated bankruptcy of suppliers in the same four-digit industry in the same year divided by the geographic area (in Column 1); those facing any types of supplier separation in the same four-digit supplier industry in the same year (in Column 2); and those that belong to the same two-digit industry and prefecture as the treatment firm (in Column 3). Both supplier density and buyer density are normalized to be mean zero and standard deviation one after residualized by prefecture fixed effects and supplier industry fixed effects. The second to bottom row reports the P-value of the statistical hypothesis that the coefficients on supplier density and buyer density sum up to zero (γ + δ = 0). Standard errors are clustered at the supplier level. *p<0.1; **p<0.05; ***p<0.01. 4 Model In this section, I develop a model of spatial firm-to-firm trade under search and matching frictions. I model dynamic firm-to-firm matching following a canonical search and matching framework popularized to study labor market frictions (Diamond 1982, Pissarides 16 1985, Mortensen 1986). A firm can source intermediate inputs either from a long-term supplier or by producing in-house. When firms do not have an ongoing supplier for a particular input, they meet with a potential supplier and decide to form a long-term relationship. I embed this firm-to-firm matching in the static firm-to-firm trade framework of Eaton, Kortum, and Kramarz (2016) and in a spatial equilibrium framework with population mobility (Allen and Arkolakis 2014, Redding and Rossi-Hansberg 2017). I also incorporate multiple heterogeneous industries connected through input-output linkages following Johnson and Noguera (2012) and Caliendo and Parro (2014). 4.1 Basic Set-up Space is partitioned into a discrete number of 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. I later describe how Li is determined by the free population mobility. Time is continuous and denoted by t. In this paper, I mainly focus on a steady-state equilibrium in which aggregate variables are constant (e.g., wages, output). Only firmlevel variables such as supplier matching status vary by t. Without risk of confusion, the subscript t is therefore omitted from the aggregate variables. In each location, there is a continuum of potential producers in each sector. Sectors are denoted by k, m ∈ K. All firms produce both final goods, consumed by final goods consumers, and intermediate inputs, used for production by other firms. In this sense, each firm can be simultaneously a buyer and a supplier in intermediate input trade. Each of these firms is owned by foreigners, and hence all firm profit goes outside the economy. Each firm produces a differentiated final good. Intermediate inputs are homogeneous within each input sector. Intermediate input trade is possible only when two firms form a long-term relationship as a supplier and a buyer. I assume that each buyer-side firm can form a relationship with at most one supplier in each input sector at a time. On the other hand, suppliers can form a relationship with multiple buyers simultaneously. 17 4.2 Technology Each firm produces both final goods and intermediate inputs with the same Cobb-Douglas production technology. The unit cost by firm ω in location i in sector m is cωt = 1 γL,m w pωt,k γkm , ϕω i k∏ ∈K (4) where wi is the wage in 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, ϕω is the exogenous productivity of firm ω, γL,m is the labor share in production for sector m, and γkm is the sector-k intermediate inputs share for sector m’s production. I assume that production function exhibits constant returns to scale, i.e., γL,m + ∑k γkm = 1 for all m ∈ K.20 Both final goods and intermediate goods are tradable across locations and subject to iceberg cost. For firms in location i to sell one unit of final goods to location j, it has to ship τij,k (> 1) unit of goods. The measure of firms whose productivity is above ϕ is given by:  µi,m ( ϕ) = ϕ Ai,m −θ . (5) Here, Ai,m governs the productivity of the location and sector. Following the tradition of spatial equilibrium models, I assume that this productivity is affected both by exogenous location fundamentals and agglomeration spillovers, depending on the local population density, such that:  ι Li ∗ Ai,m = Ai,m (6) Zi ∗ is the exogenous productivity, Z is the geographic area, L /Z is the populawhere Ai,m i i i tion density in location i, and ι is the elasticity of productivity with respect to local population density. The second term captures agglomeration spillovers arising from higher population density. In my model, this term captures all other agglomeration spillovers other than the increasing returns to scale in firm-to-firm matching modeled below, such as knowledge spillover or labor market pooling. 20 For simplicity, I assume that each firm produces at a single location. When I take the model to data, I assume that all production occurs in the prefecture of the firms’ headquarters. My estimates of the agglomeration spillovers are robust to excluding firms that have establishments outside their headquarter prefecture in Section 5 (Appendix Table ??). 18 There are two possible ways to source input goods: forming a long-term relationship with a supplier or producing them in-house using local labor. This process defines the stochastic process of pωt,k . As I describe in detail below, the steady-state distribution of the unit cost follows power-law distribution given the power-law distribution of productivity ϕω stated above. I denote the measure of firms in location i and sector m whose unit cost is below c by: Hi,m (c) = Γi,m cθ , (7) where Γi,m is the “supply capacity” term, which depends on wages and the scope of supplier matching as analytically characterized below. 4.3 Final Goods Market The final goods market follows the same structure as Melitz (2003). In each period, all firms produce and sell a differentiated variety of final goods. To sell in location j, each firm has to pay a fixed cost f j,k in each period by local labor. I assume that workers do not have access to saving technology and therefore use all of their labor income for consumption in each period. The representative final goods consumer (workers) has a CES utility function: U= ∏ k∈K Z ω ∈Ω j,k qk (ω ) σ −1 σ ! dω σ σ −1 α k , (8) where qk (ω ) is the consumption of the goods produced by firm ω, αk is the consumption share of sector k final goods, Ω j,k is the set of varieties available to final goods consumers in location j, and σ > 1 is the elasticity of substitution. Because of the power law distribution of the unit cost, the model predicts a gravity equations of trade flows (Melitz 2003 and Chaney 2008). More specifically, the measure of firms that produce in location i whose unit cost net of trade cost is below c in location j is
−θ given by Hi,m (c/τij,m ) = Γi,m τij,m (c)θ . Hence, the share of final goods consumed in location j that are produced in location i is given by: πij,m
−θ
−θ Γi,m τij,m Γi,m τij,m , =  −θ = Ω j,m 0 0 τ Γ 0 ∑i ∈ N i ,m i j,m (9) 19
−θ θ where Ω j,m cθ = ∑i0 ∈ N Γi,m τij,m c is the measure of firms whose unit cost net of trade cost is below c in sales location i. Furthermore, Appendix E.3 shows that the consumer price index is given by: Pj = ς L j
− θ − σ +1 θ ( σ −1) ∏ f j,k
θ − σ +1 α k θ ( σ −1) Ω j,k
− αk θ , (10) k where ς = 4.4  θ − σ +1 θσ 1 θ 1 σ σ −1 ∏ k ( α k ) αk θθ (−σσ−+11) . Search and Matching in Intermediate Goods Market As mentioned above, there are two possible ways to source input goods: producing them in-house using local labor or forming a direct relationship with a supplier. For the first option, I assume that all firms can produce one unit of sector-k intermediate goods using ζ k unit of local labor. Furthermore, in order to produce in-house intermediate goods, firms have to pay χ − 1(≥ 0) additional unit of transaction cost per unit of output. Below, I first discuss the matching process and its technology. I then derive and solve for Bellman equations for the decision of forming a relationship conditional on a match. Lastly, I derive the steady-state match probabilities. Matching Technology. I first assume that 0 ≤ δ ≤ 1 fraction of firms ever matches with a supplier. When I fit the model to data, this parameter rationalizes the relative magnitudes of the matching rates, separation rates, and the probability of matching in a steady-state. If a potential input buyer (i.e., δ fraction of firms producing in location j) does not currently have a supplier, it stochastically draws a potential supplier. Following the approach of labor search and matching literature (Diamond 1982, Pissarides 1985, Mortensen 1986), I assume that the matching between suppliers and buyers occurs through matching technology summarized by a matching function. The total matches created in one unit of geographic area by buyers producing in location j with suppliers (that can be producing anywhere) is given by

λ
ν M S j,k , Bj,km = η S j,k Bj,km (11) where S j,k is the density of suppliers in location j whose unit cost (net of iceberg trade 20 cost) is below ζ k w j , and Bj,km is the density of buyers in location i whose unit cost is below some sufficiently large threshold ϑ and who is not currently matched with a supplier in sector k. This matching function implies that the Poisson rate for buyers to match with a
supplier is M S j,k , Bj,km /Bj,km . Similarly, the Poisson rate for suppliers to match with a
buyer is M S j,k , Bj,km /S j,k . I allow the matching technology to have a scale effect; i.e., λ + ν can be different from one. If λ + ν = 1, the matching function exhibits constant returns to scale. If λ + ν > 1, the matching function exhibits increasing returns to scale. The measure of suppliers with unit cost below ζ k w j (net of trade cost) in location j  −θ
θ
θ is given by Ω j,k ζ k w j = ∑i0 ∈ N Γi0 ,k τi0 j,k ζ k w j . The measure of buyers whose unit cost for production is below ϑ and who is looking for a supplier is Φ j,km Γ j,m ϑθ , where Φ j,km is the steady-state probability that a firm in location j and sector m has a supplier in sector k.21 Together, the Poisson rate of matching with a supplier v j,km is given by: v j,km = 1 Bj,km
M S j,k , Bj,km = η Ω j,k ζ k w j Zj
θ !λ Φ j,km Γ j,m ϑθ Zj ! ν −1 , (12) where Zj is again the geographic area of location j. Conditional on a match, the two firms jointly determine if they form a long-term relationship. The buyer cannot match with another supplier until the relationship is exogenously destroyed at the Poisson rate ρkm . This creates an option value for buyers to forego the supplier that they draw and wait for a better supplier. On the other hand, suppliers can match with other buyers simultaneously. Therefore, suppliers do not have an option value for waiting, and they always form a relationship whenever the profit is positive. Nash Bargaining and Bellman Equations for Firm-to-Firm Matching. Following the standard search and matching framework, the joint surplus of a match is split between buyers and suppliers according to Nash bargaining. Since Nash bargaining is efficient, intermediate goods are traded at supplier’s unit cost to avoid double marginalization. In addition, there is a lump-sum transfer from the buyer to the supplier in each period based on Nash bargaining. To avoid that this bargaining involves infinitely many third firms indirectly connected through the dynamic firm-to-firm trade network, I assume that the two firms bargain     Appendix ??, I analytically derive that Φ j,km = δ − Λ j,km / 1 + Λ j,km (φj,km − 1) , where and φj,km are given by equations (D.11) and (D.12) below. 21 In Λ j,km 21 only over the profit from the buyer’s final goods sales. Furthermore, I assume that the relationship-specific unit cost of the supplier is fixed at the beginning of the relationship (by assuming that the supplier can keep receiving the intermediate goods for this specific relationship from its own suppliers until the relationship ends). Under these two assumptions, the bargaining problem can be solved for each pair of suppliers and buyers, independent of the actions of other firms in the economy. I now define the Bellman equations. I first consider the value functions of the buyer. Denote the value of the buyer ω that has an ongoing relationship with a supplier in sector B ( c ), and the value of the firm that does not k at unit cost c (net of iceberg cost) by Vωt,k B . The Bellman equation for the have an ongoing supplier relationship in sector k by Uωt,k buyers when the firm is matched with a supplier is given by B ξVωt,k (c) = F Πωt,k (c) − δωt,k (c) − ρkm  B B Vωt,k (c) − Uωt,k  B + V̇ωt,k ( c ), (13) F ( c ) is ω’s final goods profit when the unit where ξ is the discount rate of the firm; Πωt,k cost of intermediate goods in sector k is c; δωt,k (c) is the lump-sum transfer to the supplier in sector k; ρkm is again the Poisson rate at which the relationship is destroyed; and B ( c ) indicates the time derivative of the value function V B ( c ). As discussed above, V̇ωt,k ωt,k F ( c )), and hence I omit Nash bargaining involves only the buyer’s final goods profit (Πωt,k the profit from intermediate goods sales from the value function.22 The Bellman equation for the buyers in location i when they are not matched with a supplier in sector k is given by B ξUωt,k =χ 1− σ F Πωt,k (ζ k wi ) + vi,km aω,k Z c  ω,k 0 B B Vωt,k (c) − Uωt,k  B dGω,k (c) + U̇ωt,k , (14) F ( ζ w ) is the where χ is the iceberg cost of producing intermediate goods in-house; Πωt,k k i profit from final goods sales when the unit cost of intermediate goods in sector k is ζ k wi ; vi,km is the Poisson rate of matching with a supplier; aω,k is the probability that the match is accepted (derived below); cω,k is the threshold of the supplier’s unit cost below which the buyer decides to form a relationship; Gω,k (·) is the cumulative distribution function of the suppliers’ unit cost conditional on match acceptance (such that Gω,k (0) = 0 and B ( c ) and U B normalize Vωt,k ωt,k by omitting the payment to suppliers in other sectors (δωt,k̃ ( c ) for B ( c ) and U B . k̃ 6= k), which equally affects both Vωt,k ωt,k 22 I 22 B B .23 Gω,k (cω,k ) = 1); and U̇ωt,k indicates time derivatives of Uωt,k Turning to the supplier’s side, the value of the supplier with unit cost c from the relationship with buyer ω is given by S ξVωt,k (c) = δωt,k (c) − ρkm  S S Vωt,k (c) − Uωt,k  S + V̇ωt,k ( c ), (15) where δωt,k (c) is again the lump-sum transfer from the buyer to the supplier, and ρkm is the Poisson rate at which the relationship is destroyed. Since the supplier does not have an option value of waiting (because it can simultaneously form a relationship with other buyers), I normalize the value of the supplier when it is not matched with ω by zero, i.e.,24 S Uωt,k = 0. (16)   S (c) − U S Because of the Nash bargaining, the joint surplus Jωt,k (c) = Vωt,k ωt,k +   B (c) − U B Vωt,k ωt,k is split between the supplier and buyer with a weight of 1 − β and β, where β denotes the bargaining power of the buyer. Therefore: S S Vωt,k (c) − Uωt,k = (1 − β) Jωt,k (c), (17) B B Vωt,k (c) − Uωt,k = βJωt,k (c). I now solve these Bellman equations to derive the decision of whether to form a relationship conditional on a match. Because suppliers have no option value, this problem comes down to the buyer’s decision of accepting the match. Buyers decide to form a relationship if the supplier’s unit cost is sufficiently low. I assume that buyers set cω,k to maximize the expected value of unmatched state. This implies that cω,k is determined so that firms are in expectation indifferent between accepting or rejecting a match when the supplier’s unit cost is cω,k : B B E[Vωt,k (cω,k )] = E[Uωt,k ], (18) where the expectation is taken with respect to intermediate goods cost other than input 23 It is straightforward to show that there exists such a threshold. The per-period profit of the buyer is strictly decreasing in c. Therefore, the joint surplus is also strictly decreasing in c, which guarantees the existence of cω,k . 24 Similarly to the normalization of the buyer’s value functions, I normalize V S ( c ) and U S ωt,k ωt,k by omitS S ting the profits from other buyers and final consumers, which equally affects Vωt,k (c) and Uωt,k . 23 sector k. In Appendix D.1, I show that cω,k and aω,k depend only on firms’ location i, sector m, and the supplier sector k, such that cω,k = ci,km and aω,k = ai,km when fixed cost for final goods sales is negligible ( f j,k → 0), and they are analytically solved to give the following expressions: ( ai,km = min 1, χ θ γkm  vi,km ai,km (σ − 1) γkm 1−β ξ + ρkm θ − (σ − 1) γkm  θ (σ−1)γkm ) , (19) and 1 θ . ci,km = ζ k wi ai,km (20) Note that the expression of ai,km in equation (D.9) is implicit because ai,km also appears in both sides of the equations. Because the right-hand side is decreasing in ai,km , equation (D.9) uniquely determines ai,km . These equations are intuitive. In equation (D.9), ai,km is increasing in vi,km because faster matching rates imply that the option value of waiting is higher. ai,km is decreasing in ρkm and ξ because an anticipation of faster separation ρkm and a higher discount rate ξ discourage firms to wait longer. ai,km is decreasing in β because a higher value of β implies a greater option value of buyers to wait for a better supplier. Lastly, ai,km is decreasing in χ because a higher transaction cost for in-house production increases the opportunity cost of waiting. In equation (D.10), ci,km is increasing in ai,km because accepting at higher probability (higher ai,km ) means that firms accept more costly suppliers (higher ci,km ). When ai,km = 1 (always accept the supplier), ci,km = ζ k wi , i.e., buyers accept suppliers as long as the supplier’s unit cost is equal to or less than the in-house production of intermediate goods. Value of a Match and Steady-State Match Probability. Given the solution to the Bellman equations, I now define several key moments that are useful to characterize the aggregate steady-state equilibrium. I first derive the ratio of the expected firm sales r j,m of firms in location j and industry m conditional on having a supplier d j,k = 1 in sector k E [r | d =1] relative to being without a supplier d j,k = 0, i.e., φj,km ≡ E[r j,m |d j,k =0] . (The expectation is j,m j,k taken with respect to productivity ϕ and the unit cost of intermediate goods other than in sector k.) In Appendix ??, I show that φj,km is explicitly solved as: φj,km =
−γkm χθ a j,km . 1 − γkm (21) 24 Intuitively, when a j,km is high, firms are more selective about the supplier’s productivity. Therefore, each firm faces a greater increase in sales when they decide to form a relationship with a supplier. Similarly, a higher in-house production iceberg cost (χ) leads to higher sales by firms when they are matched with a supplier. Given a j,km , the steady-state probability that a firm in location j and sector m is forming a relationship with a supplier in sector k is given by: Λ j,km = δ v j,km a j,km , v j,km a j,km + ρkm (22) where δ is again the fraction of firms that can match with an external supplier, v j,k is the matching rate with a new supplier, and ρkm is the exogenous separation rate. 4.5 Gravity Equations of Intermediate Goods Flows Now I derive the spatial structure of the intermediate goods trade. Since buyers’ match and accept probabilities are independent of the suppliers’ production location conditional on the unit cost (net of trade cost), the probability that firms in location j to source from a supplier in location i is given by: πij,m
−θ Γi,m τij,m =  −θ , ∑i0 ∈ N Γi0 ,m τi0 j,m (23) which is same gravity equation for final goods trade flows (9). Furthermore, from the same logic as in Melitz (2003) and Chaney (2008), πij,m also coincides with the expenditure share of intermediate goods consumed in location j that are produced in location i. The characterization so far allows me to derive an analytical expression for the supply capacity Γi,m . In Appendix E.2, I show that: θ Γi,m = $Ai,m wi−θ ∏ (1 + Λi,km (φi,km − 1)) , (24) k∈K where $ ≡ ∏k∈K (ζ k )−θγkm , Ai,m is again the measure of the productivity of the location and sector given by equation (6), and wi are the local wages. Intuitively, the supplier capacity (Γi,m ) is higher if productivity (Ai,m ) is higher, labor cost (wi ) is lower, the probability of matching with a supplier (Λi,km ) is higher, and the benefit of a supplier match (φi,km ) is higher. 25 4.6 Total Expenditure and Trade Balance I ) Aggregate intermediate input sales by firms producing in location i and sector k (Xi,k obeys the following accounting relationship: I Xi,k = ∑ ∑ j∈ N m∈K I Yj,km πij,k , (25) I where Yj,km is the aggregate input goods expenditure by firms in sector m and location j for input sector k. Aggregate final goods sales by firms producing in location i and sector F ) is given by: k (Xi,k F F Xi,k = ∑ Yj,k πij,k (26) j∈ N F is the final goods demand. where Yi,k From the Cobb-Douglas utility function (equation 8), the final goods demand in sector k is given by: F Yi,k = α k wi L i . (27) Furthermore, from the Cobb-Douglas production function assumption, the intermediate input demand by firms in location i and sector m toward input sector k is given by I Yi,km = γkm Ψi,km n I Xi,m + F Xi,m o . (28) where Ψi,km = φi,km Λi,km / (1 − Λi,km + φi,km Λi,km ) is the sales-weighted fraction of firms in location i and sector m that has an ongoing supplier relationship in sector k. Lastly, denoting the trade deficit of location i by Di , the aggregate intermediate goods sales from location i has to balance the aggregate intermediate input purchase up to the trade deficit, i.e., ∑ k∈K   I F Xi,k + Xi,k = ∑ k,m∈K I Yi,km + ∑ Yi,kF − Di . (29) k∈K Following the approach of Caliendo and Parro (2014), I take Di as an exogenous parameter, rather than specifying the sources of trade deficit. 26 4.7 Free Population Mobility I assume that the utility of workers who reside in location i is given by: wj Pj Uj = K j ! 1/υ L− , j where K j is the exogenous residential amenity, w j is the nominal wage, Pj is the consumer price index, and L j is the population size j. Parameter υ governs the dispersion force, which includes housing costs, negative residential spillovers, and idiosyncratic preference heterogeneity. Workers are freely mobile across locations. This implies that the utility is equalized across locations, i.e., Uj = U for all locations j. Therefore, the population size of location j is given by: Lj = K υj  w υ j Pj ∑` K`υ  w` P` υ , (30) and the utility of workers in the economy is given by: U= ∑ K`υ ` 4.8  w` P` υ !1/υ . (31) Steady-State Equilibrium The aggregate steady-state equilibrium is characterized by the following three sets of equilibrium conditions. (i) trade linkages. Given population { Li } and the steady-state matching variables {Λi,km , φi,km }, I , X F }, aggregate intermediate and fiaggregate intermediate and final goods sales { Xi,k i,k I , Y F }, trade flow shares { π }, location and sector productivity nal goods demand {Yi,km ij,k i,k { Ai,m }, supply capacity {Γi,m }, wages {wi }, and consumer price index { Pj } are given by equations (6), (9), (10), (24), (E.11), (E.12), (E.5), (E.6), (E.8). (ii) population mobility. Given {wi } and { Pi }, population size { Li } and worker utility U are given by (30) and (31). (iii) matching in intermediate goods market (for each location i and sector pairs k, m). Given {Γi,m } and {wi }, steady-state probability of having an ongoing relationship with a 27 supplier {Λi,km }, Poisson matching and acceptance rates { ai,km , vi,km }, acceptance threshold of unit cost to form a supplier relationship {ci,km }, and sales increase form a supplier match {φi,km } are determined by equations (D.2), (D.9), (D.10), (D.11), and (D.12).25 In the above equilibrium conditions, the trade and population mobility conditions are standard components of a canonical quantitative spatial model (Allen and Arkolakis 2014, Redding and Rossi-Hansberg 2017). Therefore, this model extends these classes of models to incorporate the endogenous matching outcomes given by the third set of the equilibrium conditions. Despite this extension, this model remains tractable because the matching market outcomes are solved independently for each sector pairs k, m and location i. This property is useful for the sequential estimation procedure in Section 5 and for the counterfactual simulations in Section 6. There are two agglomeration forces in the model. The first channel is the increasing returns to scale in matching technology (λ + ν > 1). In particular, in a special case of the model where λ = ν − 1 = 0 (which implies λ + ν = 1), matching outcomes {Λi,km , ai,km , vi,km } become independent of location (i) given sectors (k, m). The second channel of agglomeration force is the production spillover from local population density (ι > 0). As discussed above, this channel summarizes all other types of agglomeration production spillovers (e.g., labor market pooling, knowledge spillovers). In a typical quantitative spatial model, the second term is the only channel of production agglomeration spillover, assuming that all production spillovers are summarized by this term. Below, I argue that both of these agglomeration spillovers significantly contribute to the observed spatial economic activity and that misattributing these two agglomeration forces biases for counterfactual allocations for the reduction of trade cost. 5 Structural Estimation In this section, I estimate the parameters of the model developed in Section 4 using firmto-firm trade data. The estimation proceeds in four steps, each of which uses different components of equilibrium conditions. In the first step, I start by calibrating a subset of parameters directly from the data or from central estimates in the literature. Second, I esti25 Appendix F shows that an equilibrium always exists. The same appendix further shows that the equilibrium is unique when agglomeration spillovers are sufficiently small, succeeding the property of Allen and Arkolakis (2014). 28 mate the gravity equations of cross-regional trade flows. Third, I estimate matching technology using the impulse responses of unanticipated supplier bankruptcy documented in Section 3 as moment conditions. Fourth, I estimate the productivity of each location and the elasticity of productivity spillovers from local population density. In order to execute the estimation procedure, I need to take a stand on how I map the model’s location and sectors to data. Locations in the model correspond to the 47 prefectures in Japan. Sectors in the model correspond to 33 two-digit sectors in manufacturing, commerce (wholesale and retail), and construction/equipment services. These 33 two-digit sectors together represent over 80% of firms in Japan (Appendix Table ??). 5.1 Calibrating Subset of Parameters (Step 1) First, I calibrate a subset of parameters directly from the data or using central values from the existing empirical literature. I calibrate the intermediate input share {γL,m }, labor share of production {γkm }, and the final goods consumption share {αm }, directly from the input-output table in 2011 created by Japan’s Ministry of International Affairs and Communications in Japan. I set the productivity dispersion parameter θ as 4.3 from Gaubert and Itskhoki (2021), the elasticity of substitution parameter σ as 5 from Broda and Weinstein (2006), and the elasticity of migration with respect to real wage υ as 2 from Kondo and Okubo (2015). I provide a sensitivity analysis of these values in the estimation and counterfactual simulations below. 5.2 Estimating Gravity Equations of Trade Flows (Step 2) Next, I estimate the gravity equations of trade flows (equation 9). I parameterize the
κ̃ iceberg trade cost τij,m as a power function of travel time τij,k = Tij m , where Tij is the travel time from location i to location j and κ̃m is a parameter governing the elasticity of trade cost with travel time. The gravity equation is rewritten as πij,m Γi,m Tij = Ωi,m
κ m , (32) where κm = θ κ̃m . I estimate this equation separately for each sector using the Poisson Pseudo-Maximum Likelihood (PPML) method (Santos Silva and Tenreyro 2006). I use PPML instead of OLS on the log-linear relationship to deal with the presence of zero bilatral cross-region trade 29 flows in the data. I construct the fraction of supplier linkages (πij,m ) from TSR data in 2008, and I obtain the bilateral travel time Tij from the Google Maps API. I use the estimated Γi,m , Ωi,m , and κm in the subsequent estimation procedure. 5.3 Estimating Matching Technology Parameters (Step 3) Third, I estimate the matching technology parameters. I first directly obtain the separation rates ρkm from the TSR firm-to-firm trade data, using the separation rate of a firm and sector m with a supplier in sector k in my sample period (between 2008 and 2016). I next calibrate a subset of parameters. I set the discount rate (ξ) such that ξ = 0.05, and the bargaining weight (β) is set such that β = 0.5. In Appendix Table ??, I show that the subsequent estimation results are not sensitive to the choice of these values.26 The remaining parameters, Θ ={λ, ν, η, χ, δ, {ζ k }}, are estimated in the indirect estimation procedure. The basic idea is to find parameter Θ that closely replicates the reduced-form estimates of the impacts of unanticipated supplier bankruptcies as documented in Section 3.27 More specifically, given parameter values Θ and already estimated Γi,k and Ωi,m from the gravity equations above, I compute the endogenous matching outcomes {vi,km (Θ), ai,km (Θ), ci,km (Θ), φi,km (Θ), Λi,km (Θ), Ψi,km (Θ)} by solving equations (D.2), (D.9), (D.10), (D.11), and (D.12) for each location i and sector pair k, m. Using these objects, I construct the model’s prediction of the impulse responses of the unanticipated supplier bankruptcy (interpreted as exogenous separation in the model) on the new supplier matching and sales growth and obtain the average responses and the heterogeneous responses with respect to supplier and buyer density. (See Appendix ?? for further details of this estimation procedure.) Denoting the collection of these model-predicted average and heterogeneous impulse responses by β(Θ), the indirect inference estimator Θ̂ is defined by the value of Θ that minimizes the Euclidean distance between β(Θ) and 26 Intuitively, ξ and β affect the equilibrium matching outcomes only through the acceptance rates (a j,km in equation D.9). At the same time, a j,km tends to be close to one in most sectors and locations within a reasonable range of these parameters. Hence, ξ and β have limited effects on other parameter estimates. Recall also that firms are owned by foreigners outside of the economy, hence the bargaining share β does not affect the spatial income distribution. 27 I normalize ϑ and ζ for the first sector to one. Because these parameters enter multiplicatively with k η in equation (D.2), these normalizations do not affect any of our counterfactual simulation results. 30 the difference-in-difference effects β using the actual data reported in Section 3. Formally, Θ̂ = argmin k β(Θ) − βk2 + Θ 1 (Ψi,km (Θ) − Ψi,km )2 , ∑ | N ||K | i∈ N; k,m∈K (33) where kk indicates the Euclidean norm. On top of the Euclidean distance of the impulse responses as discussed above, I also include the Euclidean distance in sales-weighted steady-state matching probability (Ψi,km ) between the model-prediction and the data in the second term. Intuitively, this indirect inference identifies each of the structural parameters Θ ={λ, ν, η, χ, δ, {ζ k }} from the following moment conditions. The efficiency of the matching technology (η) is identified from the average impacts of supplier bankruptcy on the number of new suppliers. The elasticities of matching function with supplier density (λ) and buyer density (ν) are identified by this impact heterogeneity. The iceberg cost of in-house intermediate goods production (χ) is identified from the average impacts on sales growth. The fraction of firms that can match with a supplier (δ) and the efficiency of in-house intermediate goods production ({ζ k }) are identified by the steady-state match probability for each sector (Ψi,km ). Table 4 presents the estimation results and the model fit. Panel (A) presents the point estimates and their bootstrapped confidence intervals. I start by discussing the elasticity of matching functions with respect to supplier density (λ) and buyer density (ν). As defined in equation (D.2), λ > 0 implies that the matching rates with suppliers increase in supplier density, ν < 1 implies that the matching rates with suppliers decrease in buyer density, and λ + ν > 1 implies that the matching function exhibits increasing returns to scale (IRS). My estimate of λ = 0.622 (with a bootstrap confidence interval of [0.497, 0.738], significantly above 0) indicates that supplier matching rates are significantly increasing in supplier density. My estimate of ν = 0.974 (with a confidence interval of [0.852, 1.076], not significantly different from 1) indicates the lack of evidence for the crowding-out effects by other buyers. Lastly, I find the point estimate of λ + ν = 1.596, which is significantly greater than one, indicating that the matching technology exhibits increasing returns to scale (IRS). My estimate of matching efficiency parameter η = 0.045 (with a confidence interval of [0.044, 0.047]) supports the evidence that firms indeed match with an alternative supplier after separation of the supplier relationships. My estimate of the iceberg cost of in-house intermediate goods production χ = 1.070 (with a confidence interval of [1.067, 1.071]) 31 indicates that matching with external suppliers indeed benefits firms by significantly reducing the marginal cost. Finally, my estimate of the fraction of firms that can match with a supplier δ = 0.147 indicates that a substantial fraction of firms do not ever match with external suppliers. In Panel (B) of Table 4, I present how closely the estimated model replicates the impulse responses of supplier bankruptcy on new supplier matching and sales decline. Across the board, the model successfully captures the pattern of the actual impacts of unanticipated supplier bankruptcies. In particular, the point estimates of the regression on modelpredicted outcomes (Column 2, 4, 6) are within the confidence intervals of the regression coefficients on actual outcomes in the data (Column 1, 3, 5). Importantly, in Columns 3 and 4, the model successfully replicates the positive heterogeneous impacts with the number of new suppliers by supplier density (row 3 and 4) and the lack of heterogeneous impacts with buyer density (row 5 and 6). Appendix Table ?? shows that these estimation results are not sensitive to calibrated parameters (ξ, β, σ, θ) and by using treatment firms that have no establishments outside the headquarter locations. 5.4 Estimating Productivity Spillovers (Step 4) In the fourth step, I estimate the productivity of each location Ai,m and the productivity spillover elasticity from the local population density (ι). I first estimate Ai,m using the expression for the supply capacity (equation 24). By inverting this equation, I have 1 θ Ai,m = Γi,m " $wi−θ ∏ (1 + Λi,km (φi,km − 1)) #− 1 θ . (34) k∈K I estimate Ai,m using this equation with already estimated Γi,m from the gravity equation of trade flows (Step 2), directly observed wages wi from official statistics, and modelpredicted Λi,km and φi,km using the estimated matching parameters (Step 3). I next estimate the elasticity of productivity spillovers from local population density (ι) using the estimates of Ai,m . By taking the logarithms of the definition of Ai,m in equation (5), I have:   Li log Ai,m = ι log + ϕm + ei,m , (35) Zi 32 Table 4: Estimation Results and Model Fit of Matching Technology Parameters (A) Key Parameter Estimates Parameters θ {γL,m , γkm } {αm } ε υ Description productivity dispersion input share for production final consumption share agglomeration spillover for production migration elasticity λB λS η χ δ ρkm elasticity of matching function with supplier density elasticity of matching function with buyer density efficiency of matching function iceberg cost for in-house intermediate goods production fraction of firms that can match with external suppliers separation rates Point Estimate 4.3 (Gaubert and Itskokhi 2021) IO Table IO Table XX 2 (Kondo and Okubo 2015) Confidence Interval 0.622 0.974 0.045 1.070 0.147 TSR data [0.497, 0.738] [0.852, 1.076] [0.044, 0.047] [1.067, 1.071] [0.130, 0.153] (B) Model Fit 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) 0.111 0.227 (3) 0.133 (0.041) 0.167 (0.056) −0.008 (0.042) −0.071 (0.059) 85,951 0.604 85,939 0.617 Model (4) Sales Growth (Arc-Elasticity) Data Model (5) (6) −0.034 (0.015) −0.042 (0.023) −0.081 −0.076 0.078 0.130 −0.007 −0.007 84,113 0.530 Note: Estimation results and model fit of the matching parameters through indirect inference procedure. Panel (A) reports the point estimates of the indirect inference estimator defined by equation (33). To construct the bootstrapped confidence interval, I resample treatment firms at equal probability (with replacement) to construct 100 sets of bootstrapped samples, and obtain the 5th and 95th percentiles of the estimates of the structural parameters across these bootstrapped samples. Panel (B) reports the estimated effects of unanticipated supplier bankruptcies using actual data (Column 1, 3, 5) and using the model’s prediction (Column 2, 4, 6). Column 1 corresponds to Column 1 of Table 1, Column 3 corresponds to Column 3 of Table 3, and Column 5 corresponds to Column 2 of Table 1, respectively. ∗ ] is the industry fixed effects and e ∗ ∗ where ϕm ≡ E[log Ai,m i,m ≡ log Ai,m − E[log Ai,m ] is the residual exogenous productivity. Estimating this regression by OLS yields a biased estimate of spillover elasticity (ι) because the population density (Li /Zi ) is plausibly correlated with the residual exogenous productivity (ei,m ) through endogenous population mobility. An ideal theory-consistent instrument for the population density in this equation is the component of residential amenities (Ki ) that is unrelated to productiv∗ ). Here, following the approach of Glaeser and Gottlieb (2008) and Allen and ity (Ai,m Donaldson (2020), I use mean temperature and average rainfall as instruments for the population density. For the sectors considered in this section (manufacturing, commerce, and construction/equipment services), these climate characteristics are unlikely to affect ∗ ), and hence they arguably satisfy the exclusion restriction. the productivity (Ai,m Table 5 presents the estimation results. Column (1) presents the first stage regression. 33 Higher temperatures and fewer rainy days are significantly associated with population density, consistent with the interpretation that a warmer and more sunny climate is associated with higher residential amenity. Column (2) reports the IV regression results of my estimation equation (35) using the average temperature and the number of rainy   Li days as an instrument for the population density (log Z ). I obtain the point estimate i of ι = 0.11, which is within the range of parameter values of the productivity spillover estimates in the literature (Melo, Graham, and Noland 2009). To benchmark this result, in Column (3), I estimate regression equation (35) by (falsely) assuming that there were no increasing returns to scale (IRS) in matching function, such that λ = ν − 1 = 0 (unlike the estimated value of 0.622 and 0.974 above). Under these assumptions, the matching probability Λi,km and the match benefit φi,km are constant across locations (i), and hence the term ∏k∈K (1 + Λi,km (φi,km − 1)) in the estimation equation of Ai,m (equation 34) drops out as constant (up to sector m). In Column (3), I show that the elasticity of productivity spillover (ι) is substantially overestimated using this falsely-estimated Ai,m as a dependent variable (ι = 0.18 instead of 0.11). Intuitively, the contribution of IRS in matching technology to location productivity (∏k∈K (1 + Λi,km (φi,km − 1))) in equation (34) is positively correlated with observed population density, and hence the omission of this term creates an upward omitted variable bias (OVB). In the next section, I show that misattributing these two channels substantially biases the welfare gains from trade cost reduction. 6 Counterfactual Simulations In this section, I use the estimated model to undertake a sequence of counterfactual simulations to highlight how the increasing returns to scale (IRS) in firm-to-firm matching affects the spatial distribution of economic activity. In Section 6.1, I hypothetically shut down the model’s two agglomeration forces (IRS in matching technology and productivity spillovers from local population density) and study how the equilibrium changes under these counterfactuals. In Section 6.2, I simulate the hypothetical reduction of crossregional trade cost and study how the predicted welfare gains change by omitting each of these two agglomeration forces. To conduct these counterfactual simulations, I follow the exact-hat algebra approach of Dekle, Eaton, and Kortum (2008) and rewrite the counterfactual equilibrium conditions in terms of the unobserved changes in the endogenous variables between the counterfac- 34 Table 5: Estimation Results of Production Spillover Elasticity Average Annual Temperature Number of Rainy Days log( Li /Zi ) First Stage log Ai,m IV log Ai,m IV (1) (2) (3) 0.11∗∗∗ (0.02) 0.18∗∗∗ (0.02) 1,449 0.87 No IRS in Matching Function (λ = ν − 1 = 0) 1,449 0.88 0.16∗∗∗ (0.01) −0.01∗∗∗ (0.0002) log( Li /Zi ) Specification Observations Adjusted R2 1,449 0.21 Note: Estimation results of the instrumental variable (IV) regression of equation (35). Column (1) presents the results of the first stage regression. Column (2) reports the IV estimates of productivity spillover (ι). Column (3) reports the same IV estimates by (falsely) assuming that there were no increasing returns to scale (IRS) in matching function (such that λ = ν − 1 = 0) when constructing the outcome variable Ai,m in equation (34). *p<0.1; **p<0.05; ***p<0.01. tual and initial equilibria (see Appendix G.1 for the formal system of equilibrium conditions). Compared to a canonical trade and spatial equilibrium model, there are additional equilibrium conditions due to endogenous matching (equilibrium condition (iii) in Section 4.8). Since these additional components of equilibrium conditions can be independently solved for each sector and location, this exact-hat algebra approach remains computationally tractable. 6.1 Shut Down Agglomeration Forces In my first set of counterfactual simulations, I assess the contribution of the two agglomeration forces to the observed spatial differences in wages through three sets of counterfactual simulations. In my first counterfactual, I hypothetically shut down increasing returns to scale (IRS) in matching technology such that λ = ν − 1 = 0. In my second counterfactual, I shut down productivity spillovers from local population density, such that ι = 0. In my third counterfactual, I shut down both types of agglomeration forces, such that λ = ν − 1 = 0 and ι = 0. Table 6 presents the results of these counterfactual simulations. For the observed equi- 35 librium (in Column 1) and for the counterfactual equilibrium (in Columns 2-4), the table shows the regression coefficients of the logarithm of wages on the local population density. Column (1) reports the population-density wage premium observed in the data. The point estimate of 0.076 indicates that a 100 percentage point increase in population density is associated with a 7.6 percentage point increase in observed wages, which is within the range of previous estimates of this relationship (Melo, Graham, and Noland 2009). In Column (2), I report the counterfactual population-density wage premium under my first counterfactual to shut down the IRS in matching technology (λ = ν − 1 = 0). The predicted premium decreases to 0.045, which is a 41 percent reduction from the observed premium (= (0.076 − 0.045)/0.076). In other words, 41 percent of observed wage inequality is explained by IRS in matching technology. To benchmark this result, in Column (3), I report the predicted population-density wage premium under my second counterfactual simulation to shut down the local production spillovers (summarizes all other agglomeration forces) such that ι = 0. The predicted premium decreases to 0.011, or a 86 percent reduction from the observed premium (= (0.076 − 0.011)/0.076). In Column (4), I further shut down both types of agglomeration spillovers (such that λ = ν − 1 = 0 and ι = 0). The predicted premium becomes negative and decreases to −0.013, or a 117 percent reduction from the observed premium (= (0.076 − (−0.013))/0.076). Therefore, the contribution of the IRS in matching technology on observed population-wage premium is nonnegligible to that of the overall agglomeration spillovers. In Appendix Table G.1, I find a similar patterns for the counterfactual changes in population density (in terms of the relative contribution of IRS in matching technology out of overall agglomeration spillovers). Appendix Table G.2 reports that these conclusions are robust to the value of calibrated parameters in Section 5. 6.2 Cross-Regional Trade Cost In my second counterfactual simulation, I study how the welfare gains of the reduction of cross-regional trade costs are affected by these two types of agglomeration spillovers. To consider a realistic spatial pattern of trade cost reduction, I analyze the trade cost reduction that is proportional to travel time reduction from existing highway networks in Japan. In particular, I calibrate the model using the observed travel time with exist- 36 Table 6: Agglomeration Forces and Population-Density Wage Premium Baseline log Population Density C.I. from Bootstrap Parameter Estimates Percentage Difference from Baseline (%) Observations Adjusted R2 Shut Down IRS in Matching Function (λ = ν − 1 = 0) Dependent Variable: log Wages Shut Down Population Productivity Spillovers (ι = 0) Shut Down Both Agglomeration Spillovers (λ = ν − 1 = 0 and ι = 0) (1) (2) (3) (4) 0.076∗∗∗ (0.010) 0.045∗∗∗ (0.009) 0.011 (0.012) −0.013 (0.010) 0 47 0.547 [0.039, 0.055] -41 47 0.323 [0.0096, 0.013] -86 47 −0.002 [-0.018, -0.0043] -117 47 0.011 Note: Results of the counterfactual simulations to shut down agglomeration forces. Each column reports the regression coefficients of the logarithm of wages on the local population density. Column (1) reports this relationship in the observed data, and columns (2)-(4) report these premiums under the three counterfactuals described in Section 6.1. In the fourth to bottom row, I report the confidence intervals using bootstrapped estimates of matching parameters (Table 4). *p<0.1; **p<0.05; ***p<0.01. ing highway networks, and I simulate the counterfactual equilibrium in the absence of κ̃m  nohighway /Tij , highway networks. The increase of trade cost is specified as τ̂ij,k = Tij nohighway where Tij is the travel cost without using highway networks obtained from Google Maps API, and κ̃m = θκm is the estimated elasticity of trade volume with respect to travel time (estimated in Section 5.2). I undertake this counterfactual simulation under four different scenarios. In my first scenario, I incorporate both types of agglomeration spillovers (IRS in matching technology and the local population spillovers). In my second scenario, I (falsely) omit the IRS in matching technology, such that λ = ν − 1 = 0. In my third counterfactual scenario, I (falsely) omit the local population spillover, such that ι = 0. In my fourth counterfactual scenario, I (falsely) omit both types of agglomeration spillovers, such that λ = ν − 1 = 0 and ι = 0. The goal of this counterfactual simulation is not to provide an accurate welfare evaluation of the highway networks in Japan. The model abstracts important realistic features relevant to this question, such as the reduction of migration frictions by highway networks. Instead, the goal of this counterfactual is to highlight how different agglomeration mechanisms lead to different predictions about the welfare gains of cross-regional trade cost reduction, and how the incorrect specifications of agglomeration mechanisms biases the predicted welfare gains. Table 7 presents estimates of the welfare gains under these four counterfactual simulations. The second column reports the percentage point increases in the expected utility. The third column reports their confidence intervals, based on the bootstrapped confidence 37 sets of parameter estimates of matching technology parameters (Table 4). The fourth column reports the estimates of the welfare gains as a percentage of the same estimates under the first scenario of including both types of agglomeration forces. When both types of agglomeration spillovers is present (first row), this trade cost reduction leads to a 6.19 percentage points increase in expected utility. When I (falsely) abstract the IRS in matching technology (λ = ν − 1 = 0, second row), the welfare gains is underestimated as 4.53 percentage points, which is 27 percent smaller (= (6.19 − 4.53)/6.19) than baseline. In other words, about 27 percent of the welfare gains of trade cost reduction is attributed to the IRS in matching technology. Intuitively, these trade cost reductions lead to an increase of the pool of potential suppliers in the affected regions. In the presence of IRS in matching, increasing the supplier pool increases the productivity of firms in these regions. On the other hand, when I (falsely) abstract the production spillovers from population density (ι = 0, third row), the estimated welfare gains are 6.17 percentage points, which is virtually identical to the estimates when I include this type of agglomeration effect (first row). Similarly, the predicted welfare gain when abstracting both types of agglomeration forces (4.53 percentage point, fourth row) is virtually identical to the scenario where I only abstract the IRS in matching technology (second row). Therefore, while the IRS in matching technology significantly contributes to the welfare gains of trade cost reduction, production spillovers from local population density have negligible contributions to the welfare gains in this context. Intuitively, population spillovers only affect productivity through population mobility. Because aggregate population size is fixed, productivity gains in locations with a population increase are largely offset by the productivity losses in locations with a population decrease. Table 7: Welfare Gains of Trade Cost Reduction Specification Welfare Gains (p.p.) Baseline 6.19 Shut Down IRS in Matching Function (λ = ν − 1 = 0) 4.53 Shut Down Population Productivity Spillovers (ι = 0) 6.17 Shut Down Both Agglomeration Spillovers (λ = ν − 1 = 0 and ι = 0) 4.53 Confidence Interval [5.87, 6.33] [4.48, 4.63] [5.87, 6.31] [4.48, 4.63] Relative to Baseline (%) 100 73.2 99.7 73.2 Note: Results of the counterfactual simulations to reduce cross-region trade cost described in Section 6.2.. For each assumption of the agglomeration forces, the table reports the percentage point increase in the expected utility (in the second column), its confidence interval using bootstrapped matching parameter estimates from Table 4 (in the third column), and its ratio to the baseline (assuming both types of agglomeration spillovers). These results indicate that misattributing the agglomeration mechanisms substantially 38 biases the estimates of welfare gains of the trade cost reduction.28 7 Conclusion This paper provides new theory and evidence of how firm-to-firm matching in input trade shapes the agglomeration of economic activity. Using a yearly panel of firm-to-firm trade in Japan, I document that firms gradually match with an alternative supplier after an unanticipated supplier bankruptcy; these rematching rates increase in the geographic density of alternative suppliers, and they do not decrease in the geographic density of other buyers. Motivated by these findings, I develop a quantitative spatial equilibrium model of search and matching frictions in firm-to-firm trade. I structurally estimate the key model parameters and show that the increasing returns to scale (IRS) in matching technology is equally important as other channels of agglomeration spillovers in explaining the spatial productivity spillovers. Undertaking counterfactuals for changes in trade cost, I show that misattributing these agglomeration mechanisms substantially biases the estimates of welfare gains. This paper highlights a particular agglomeration mechanism: IRS of firm-to-firm matching technology in input trade. This is, of course, not the only relevant agglomeration mechanism. Other agglomeration mechanisms, such as labor market pooling or knowledge spillover, are equally 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 its equilibrium implications. 28 Appendix Table G.3 reports that these conclusions are robust to the value of calibrated parameters in Section 5. Appendix Section G.3 discusses the transition dynamics and shows that it takes time for the welfare gains to fully materialize due to a gradual adjustment of supplier matching.