Theory and Evidence from Japanese Firm-to-Firm Trade Data Appendix

1 A Data Appendix In this section of the online appendix, I provide additional information about the firmto-firm trade data from Tokyo Shoko Research (TSR). Table A.1 and Figure A.1 show the coverage rates of TSR data and show that the TSR data is broadly representative across geography. Table A.2 presents the list of reported reasons for bankruptcies from which I identify unanticipated bankruptcies. Figure A.2 presents the frequency of unanticipated bankruptcies across space and time. Table A.1: Sample Size and Coverage of TSR Datasets TSR Economic Census TSR / Economic Census 2009 All Employment <= 4 Employment >= 5 1,241,490 587,611 653,879 1,805,545 1,067,825 737,720 0.68 0.55 0.88 2016 All Employment <= 4 Employment >= 5 1,473,991 793,967 680,024 1,877,438 1,047,189 830,249 0.78 0.75 0.81 Note: This table reports, for year (2009 or 2016) and firm size (over or under 5 employees), the sample size of the TSR dataset (third column), the number of firms in Japan based on economic censuses (fourth column), and the ratio of the third and fourth columns (fifth column). Figure A.1: Coverage of TSR Datasets relative to Economic Census Note: This figure plots the density of firms using two data sources: The economic census on the horizontal axis and TSR data on the vertical axis. Each dot represents a municipality in Japan. All data is from 2009. The straight line in the graph is the linear regression fit between the two variables. The slope of the regression line is 1.04 (with an intercept of 0.27) and the R-squared is 0.98. 2 Table A.2: List of Reasons for Bankruptcies Reason of Bankurptcy Unanticipated Reasons Freq. Freq. (At Least One Buyer) 1588 269 Sales Decline 72437 Accumulation of Debt 10700 Spillovers from Other Bankruptcy 6222 Shortage of Capital 5577 Management Failure 4843 Unknown 3585 Over-Investment in Capital 802 Deterioration of Credit Conditions 547 Difficulty in Collecting Account Receivables 454 Over-Accumulation of Inventory 73 Total 106828 9460 2267 938 1021 601 458 211 161 126 29 15541 Note: This table reports the number of bankruptcies in each category of reported reasons. The second column (“Freq”) reports the number of firms experiencing bankruptcies from 2008 to 2016 for each reason, and the third column (“Freq. (At Least One Buyer)”) reports the number of bankrupt firms with at least one buyer (reported as a supplier by at least one firm). In an internal document by TSR, “unanticipated reasons” is described as “unanticipated accidental problems such as the death of representatives, flood disaster, fire, earthquake, traffic accident, fraud, theft, embezzlement, etc.” Figure A.2: Spatial and Temporal Patterns of Unanticipated Bankruptcies (A) Probability by Firm Density (B) Bankruptcy over Time Note: Panel (A) plots the probability of unanticipated bankruptcies (colored in black; on the left vertical axis) and that of all bankruptcies (colored in gray; on the right vertical axis) against firm density (on the horizontal axis) during the sample period. Each dot represents a prefecture, and the area of the dot represents the log number of firms in the prefecture. Panel (B) plots the frequency of unanticipated bankruptcies (colored in black; on the left vertical axis) and that of all bankruptcies (colored in gray; on the right vertical axis) against year. 3 B Reduced-Form Results Appendix In this section of the online appendix, I present additional results and robustness of the impacts of unanticipated supplier bankrupcies reported in Section 3 of the main paper. Table B.1: Impacts of Supplier Bankruptcy on Additional Firm Outcomes Employment Growth (Arc-Elasticity) Profit / Sales Prob. Continuing Relationship with Existing Suppliers in Baseline Period Sales Growth (Arc-Elasticity) of Existing Suppliers in Baseline Period Number of New Suppliers of Firms in Same Industry and Prefecture Sales Growth (Arc-Elasticity) of Firms in Same Industry and Prefecture (1) (2) (3) (4) (5) (6) −0.007 (0.010) −0.006 (0.008) −0.031 (0.048) −0.007 (0.005) 0.003 (0.016) −0.002 (0.005) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Trt x 1[t - BankruptYear = 0 or 1] −0.017 (0.014) −0.012 (0.008) 0.069∗∗ (0.034) −0.004 (0.008) 0.003 (0.017) 0.002 (0.007) Trt x 1[t - BankruptYear = 2 or 3] −0.025 (0.022) 0.006 (0.013) 0.080∗ (0.044) 0.002 (0.011) 0.001 (0.027) 0.002 (0.009) Observations 85,439 67,756 85,951 76,737 80,838 79,141 Trt x 1[t - BankruptYear = -2 or -3] Trt x 1[t - BankruptYear = -1] Note: Results of the stacked-by-event difference-in-difference regression specification (3) by omitting the terms corresponding to treatment heterogeneity (log S j,k , log Bj,m , X f jkmg ). In Columns (3) and (4), “existing suppliers in baseline period” indicates the set of suppliers (excluding bankrupt ones) that firms are connected to one year prior to the supplier bankruptcy. In Columns (5) and (6), I take the number of new suppliers and sales growth of firms in the same two-digit industry and prefecture as treatment and control firms as outcome variables. The lack of statistically significant results in Columns (5) and (6) provide additional robustness of Column (4) of Table 2 to address the concern that treatment firms’ supplier bankruptcy may have a direct effect on other firms in the same prefecture and industry. See footnote of Table 2 for further details about the specification. Table B.2: Heterogeneous Impacts of Supplier Bankruptcy by Reported Ranking and Firm Size Number of New Suppliers Sales Growth (Arc-Elasticity) (1) (2) (3) (4) (5) (6) (7) (8) Trt x 1[t - BankruptYear = 0 or 1] 0.14∗∗∗ (0.04) 0.11∗∗ (0.05) 0.13∗∗∗ (0.04) 0.12∗ (0.06) −0.03∗∗ (0.02) −0.01 (0.02) −0.03∗∗ (0.01) −0.05∗∗ (0.02) Trt x 1[t - BankruptYear = 2 or 3] 0.19∗∗∗ (0.06) 0.17∗∗ (0.07) 0.19∗∗∗ (0.06) 0.29∗∗∗ (0.09) −0.04∗ (0.02) −0.02 (0.03) −0.04∗ (0.02) −0.05 (0.03) Trt x 1[t - BankruptYear = 0 or 1] x X 0.11 (0.10) −0.11∗∗ (0.04) −0.05 (0.04) −0.11∗∗ (0.05) 0.01 (0.01) −0.01 (0.01) Trt x 1[t - BankruptYear = 2 or 3] x X 0.07 (0.15) −0.10∗ (0.05) 0.02 (0.05) −0.12∗ (0.06) −0.01 (0.02) −0.02 (0.02) Top 1 Supplier Firm Size (Buyer) Firm Size (Supplier) Top 1 Supplier Firm Size (Buyer) Firm Size (Supplier) 85,951 85,596 85,762 84,113 83,897 83,939 X Observations 85,951 84,113 Note: Results of the stacked-by-event difference-in-difference regression specification (3) without supplier and buyer density (log S j,k , log Bj,m ) but with treatment heterogeneity (X f jkmg ) as specified in the bottom row. Columns (2) and (6) include the dummy variable that takes one if the bankrupt supplier is top 1 supplier. Columns (3) and (7) include the log of employment size of firm f (normalized to mean zero), and Columns (4) and (8) include that of bankrupt suppliers. See footnote of Table 2 for further details about the specification. 4 Table B.3: Heterogeneous Impacts of Unanticipated Supplier Bankruptcy: Robustness Number of New Suppliers (1) (2) (3) (4) (5) Trt x 1[t - BankruptYear = 0 or 1] 0.15∗∗∗ (0.04) 0.15∗∗∗ (0.04) 0.16∗∗∗ (0.04) 0.14∗∗∗ (0.04) 0.17∗∗∗ (0.05) Trt x 1[t - BankruptYear = 2 or 3] 0.21∗∗∗ (0.06) 0.22∗∗∗ (0.06) 0.22∗∗∗ (0.06) 0.19∗∗∗ (0.06) 0.25∗∗∗ (0.06) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.15∗∗∗ (0.04) 0.14∗∗∗ (0.05) 0.12∗∗∗ (0.04) 0.15∗∗∗ (0.04) 0.15∗∗∗ (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.17∗∗∗ (0.06) 0.14∗∗ (0.06) 0.10 (0.06) 0.15∗∗ (0.06) 0.15∗∗ (0.06) Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density −0.04 (0.04) −0.03 (0.05) 0.02 (0.04) −0.01 (0.05) 0.01 (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density −0.09 (0.07) −0.11∗ (0.07) −0.05 (0.07) −0.08 (0.06) −0.04 (0.07) Exclude Exit Firms Exclude Bankruptcy in 2009 Exclude Tokyo Exclude Outdated Accounting Year Exclude Firms with Outside-Prefecture Establishments X X X X X X X X X X X X X X X Number of Treatment Firms Number of Bankrupting Suppliers Number of Control Firms 433 181 11,889 380 159 10,094 374 155 7,358 433 181 11,889 334 144 8,311 Observations 83,548 75,758 54,442 81,582 60,554 Panel A: Average Effects Panel B: Heterogeneous Effects 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 Note: Results of the stacked-by-event difference-in-difference regression specification (3). Panel A corresponds to the specification without treatment heterogeneity as in Table 2, and Panel B corresponds to the specification (2) of Table 4. Column (1) excludes firms that drop out during the sample period. Column (2) excludes all supplier bankruptcies in 2009 (the year subsequent to the Great Financial Crisis). Column (3) excludes firms with headquarters in Tokyo prefecture. Column (4) excludes firms whose accounting information is not available after supplier bankruptcy event. Column (5) excludes firms that have establishments outside the headquarter prefecture. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. 5 Table B.4: Heterogeneous Impacts: Alternative Supplier Density Dependent Variable: Number of New Suppliers (1) (2) (3) (4) (5) (6) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.11∗∗∗ (0.03) 0.12∗∗∗ (0.04) 0.13∗∗∗ (0.04) 0.14∗∗∗ (0.04) 0.07∗∗ (0.03) 0.14∗∗∗ (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.15∗∗∗ (0.05) 0.13∗∗ (0.05) 0.14∗∗ (0.06) 0.15∗∗∗ (0.06) 0.08∗∗ (0.03) 0.16∗∗∗ (0.06) Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density −0.04 (0.04) −0.02 (0.04) −0.03 (0.04) −0.03 (0.04) −0.03 (0.04) −0.03 (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density −0.10 (0.06) −0.08 (0.06) −0.08 (0.06) −0.08 (0.06) −0.08 (0.06) −0.08 (0.06) X X X X X X X X X X X X X X X X X X Count Local Suppliers Sum of Suppliers × Travel Time −0.5 Sum of Suppliers × Travel Time −1 Sum of Suppliers × Travel Time −2 Evaluated Right Before Bankrutpcy Two-Digit Supplier Industry 85,395 85,395 85,395 85,395 85,395 85,395 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 Definition of Seller Density Observations Note: Results of the stacked-by-event difference-in-difference regression specification (3) with alternative definitions for supplier density. Column (1) defines the supplier density by the number of suppliers whose headquarters are established in the treatment firms’ prefecture divided by the geographic area. Column (2) to (4) defines it as the sum of the number of suppliers in each prefecture times the power function of travel time between the prefecture and firm f ’s prefecture divided by the geographic area of firm f ’s prefecture. Column (5) evaluates the supplier density one year before each supplier bankruptcy, instead of the value in 2008 in my baseline specification. Column (6) defines the industry of suppliers at the two-digit level, instead of four-digit level in baseline. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. Table B.5: Heterogeneous Impacts: Alternative Buyer Density Dependent Variable: Number of New Suppliers (1) (2) (3) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.13∗∗∗ (0.04) 0.14∗∗∗ (0.04) 0.13∗∗∗ (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.17∗∗∗ (0.06) 0.15∗∗ (0.06) 0.11∗∗ (0.06) Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density −0.01 (0.04) −0.01 (0.03) 0.01 (0.02) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density −0.07 (0.06) 0.02 (0.05) 0.07∗∗ (0.03) Buyers in Same Two-Digit Industry Buyers facing Supplier Separation Buyers facing Unanticipated Supplier Bankruptcies X X X X X X X X X 85,939 85,951 85,951 Definition of Buyers 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 Observations Note: Results of the stacked-by-event difference-in-difference regression specification (3) with alternative definitions for buyer density. Column (1) defines buyer density as the density of firms in the same 2-digit industry and prefecture. Column (2) defines it using the number of firms in the treatment firm’s prefecture that faced an unanticipated supplier bankruptcy in the same two-digit industry up to 3 years prior to the event. Column (3) defines buyer density using the number of firms in the treatment firm’s prefecture that faced separation of supplier linkages in the same two-digit industry up to 3 years prior to the event. Standard errors are clustered at the supplier level. *p<0.1; **p<0.05; ***p<0.01. 6 Table B.6: Heterogeneous Impacts: Inverse Hyperbolic Sine (IHS) Transformation and Reverse Reporting Dependent variable: Number of New Suppliers (IHS) Baseline Include Reverse Reporting (1) (2) (3) Trt x 1[t - BankruptYear = 0 or 1] 0.12∗∗∗ (0.02) 0.06∗∗ (0.03) Trt x 1[t - BankruptYear = 2 or 3] 0.15∗∗∗ (0.03) 0.09∗∗∗ (0.03) (4) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.07∗∗∗ (0.02) 0.06∗∗∗ (0.02) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.05∗ (0.03) 0.02 (0.03) Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density −0.02 (0.02) 0.01 (0.03) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density −0.05∗ (0.03) 0.002 (0.03) X X X X X X 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 Observations 85,951 85,395 85,965 85,409 Note: This table reports the robustness of the results in Table 4 using different transformations of outcome variables and by including the supplier linkages reported by the supplier-side firms to define the number of new suppliers. Columns (1) and (2) apply the inverse hyperbolic sine (IHS) transformation to the number of new suppliers. Columns (3) and (4) define the number of new suppliers by including the supplier linkages reported by the supplier-side firms, in addition to the buyer-reported suppliers, as in the baseline specification. For the latter, I apply the inverse hyperbolic sine (IHS) transformation because of the fat-tailed distribution of the outcome variable, unlike the buyer-reported suppliers which is bounded at 24 (Section 2). Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. Table B.7: Average Impacts of Supplier Bankruptcy due to Management Failure Dependent Variable: Number of New Suppliers (1) (2) (3) (4) (5) (6) (7) (8) (9) 0.05∗∗ (0.02) 0.06∗∗∗ (0.02) 0.01 (0.03) 0.05∗∗ (0.02) 0.13∗∗∗ (0.03) 0.05 (0.03) 0.05 (0.03) 0.08∗∗∗ (0.02) 0.001 (0.01) (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] 0.24∗∗∗ (0.03) 0.25∗∗∗ (0.03) 0.24∗∗∗ (0.03) 0.23∗∗∗ (0.03) 0.35∗∗∗ (0.03) 0.21∗∗∗ (0.03) 0.23∗∗∗ (0.04) 0.27∗∗∗ (0.03) 0.18∗∗∗ (0.02) Trt x 1[t - BankruptYear = 2 or 3] 0.36∗∗∗ (0.04) 0.38∗∗∗ (0.04) 0.34∗∗∗ (0.04) 0.34∗∗∗ (0.04) 0.61∗∗∗ (0.04) 0.32∗∗∗ (0.05) 0.38∗∗∗ (0.05) 0.41∗∗∗ (0.04) 0.30∗∗∗ (0.04) 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 Number of Treatment Firms Number of Bankrupt Suppliers Number of Control Firms Observations 1688 337 32,840 239,292 1688 337 32,840 231,385 1221 258 25,428 182,213 1688 337 31,060 226,239 1688 337 18,797 131,127 1316 301 25,062 182,517 1035 275 18,558 135,297 1688 337 29,933 218,433 1688 337 32,840 231,805 Trt x 1[t - BankruptYear = -2 or -3] Trt x 1[t - BankruptYear = -1] Note: A version of Table 2 where I instead use supplier bankruptcy due to “Management Failure” in Table A.2. 7 Table B.8: Average Impacts of Supplier Bankruptcy due to Management Failure Dependent Variable: Sales Growth (Arc-Elasticity) Exit Sales Growth (Arc-Elasticity) (1) Trt x 1[t - BankruptYear = -2 or -3] (2) (3) −0.001 (0.008) Trt x 1[t - BankruptYear = -1] 0.009 (0.007) (0.000) (0.000) (0.000) Trt x 1[t - BankruptYear = 0 or 1] −0.020∗∗ (0.009) 0.005 (0.004) −0.003 (0.006) Trt x 1[t - BankruptYear = 2 or 3] −0.057∗∗∗ (0.014) 0.017∗∗∗ (0.006) −0.019∗∗ (0.009) All All Cond. on Survival -0.185 1688 337 32,840 234,302 0.052 1688 337 32,840 239,292 -0.086 1688 337 32,840 227,326 Samples Control Mean Number of Treatment Firms Number of Bankrupt Suppliers Number of Control Firms Observations Note: A version of Table 3 where I instead use supplier bankruptcy due to “Management Failure” in Table A.2. Table B.9: Heterogeneous Impacts of Supplier Bankruptcy due to Management Failure 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 0.03 (0.02) 0.005 (0.02) −0.02 (0.04) −0.03 (0.03) 0.01 (0.04) −0.01 (0.03) 0.01 (0.02) 0.02 (0.03) Trt x 1[t - BankruptYear = -3 or -2] x log Buyer Density 0.05∗∗ (0.02) 0.03 (0.02) 0.04 (0.04) 0.02 (0.03) 0.03 (0.03) 0.01 (0.03) 0.03 (0.02) 0.03 (0.03) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.06∗∗∗ (0.02) 0.05∗∗ (0.02) 0.07∗∗ (0.03) 0.05∗∗ (0.02) 0.08∗∗∗ (0.03) 0.08∗∗∗ (0.02) 0.05∗∗ (0.02) 0.07∗∗∗ (0.02) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.10∗∗∗ (0.03) 0.09∗∗∗ (0.03) 0.15∗∗∗ (0.05) 0.11∗∗∗ (0.04) 0.12∗∗∗ (0.04) 0.13∗∗∗ (0.04) 0.08∗∗ (0.03) 0.11∗∗∗ (0.03) Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density 0.03 (0.03) −0.01 (0.03) −0.02 (0.04) −0.02 (0.03) −0.02 (0.04) −0.01 (0.03) −0.02 (0.03) 0.01 (0.03) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density 0.07∗ (0.04) 0.01 (0.04) 0.07 (0.06) −0.02 (0.04) 0.03 (0.05) 0.01 (0.04) −0.01 (0.04) 0.05 (0.04) 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 1688 337 32,840 239,282 1688 337 32,840 237,106 1565 325 30,006 167,314 1316 301 25,062 180,800 1035 275 18,558 133,956 1221 258 25,428 180,347 1688 337 31,060 224,159 1688 337 18,797 134,727 Note: A version of Table 4 where I instead use supplier bankruptcy arising due to “Management Failure” in Table A.2. 8 C Model Derivations This appendix discusses additional details of the model developed in Section 4. C.1 Cutoff for Entry for Sales Let denote the final goods sales of firms in location j with unit cost c (net of trade cost) F ( c ) = $ c−σ+1 , where $ when they enter location j as r j,k j,k j,k is a demand shifter that depends on aggregate equilibrium conditions. Denoting the unit cost threshold of entry as c j,k , the goods market clearing condition is given by: F Yj,k = Z c j,k 0 θ $ j,k c−σ+1 Ω j,k θcθ −1 c− j,k dc =
− σ +1 θ . $ j,k Ω j,k c j,k θ−σ+1 Now, combining with the zero-profit condition for a marginal seller f j,k w j
− σ +1 1 , the entry cutoff c j,k is solved as: σ $ j,k c j,k " c j,k = F θ − σ + 1 Yj,k θσ w j f j,k Ω j,k = #1/θ (C.1) , and I also have S j,k = Ω j,k cθj,k . Furthermore, the aggregate sales fixed cost payment by firms that sell in location j, F j,k , is given by θ−σ+1 F Yj,k . (C.2) F j,k = f j,k w j S j,k = σθ σ +1 Therefore, θ −σθ fraction of aggregate final goods sales are required as sales fixed cost payment. Therefore, the share of profit (subtracting sales fixed cost) to aggregate final σ +1 −1 = σσθ . sales is given by σ1 − θ −σθ C.2 Inverse Cost Shifter Γi,m From the assumption of the measure of firms such that µi,m ( ϕ) = Ni,m ϕ−θ , the measure of firms below unit cost c is given by ! γ L,m Z γkm w p ∏ k∈K k I i Hi,m (c) = Γi,m cθ = µi,m dGi,k ( pk ) ∏ cAi,m p1,..., pK k∈K ! Z = −θγL,m θ Ni,m Ai,m wi ∏ k∈K pk I pk −θγkm dGi,k ( pk ) cθ , (C.3) 9 I (·) is the steady-state distribution of potential suppliers in location i and sector where Gi,k k, which follows the inverse of the Pareto distribution with an upper bound ci,k . Given the steady-state match probability Λi,km , I have: Z I pk −θγkm dGi,k ( pk ) = Λi,km Z c i,k 0 c−γkm θ dGi,k (c) + (1 − Λi,km ) Z c i,k 0 (χc)−γkm θ dGi,k (c) 1 χ−θγkm −θγkm = Λi,km + (1 − Λi,km ) (ci,k ) (ci,k )−θγkm 1 − γkm 1 − γkm  o n − θγ km χ (C.4) = (ci,k )−θγkm 1 + Λi,km χθγkm − 1 , 1 − γkm where Gi,k (·) the CDF of inverse Pareto distribution with upper bound ci,k , i.e., Gi,k (c) = θ cθ /ci,k . (Note that {Γn,m } affect the supplier cost distribution only through ci,k because production costs follow a power law distribution.) Combining equations (C.3) and (C.4) leads to equation (8) of the main paper with $m ≡ ∏k C.3 Derivation for Equation (19)   d log 1 + Λ j,km χθγkm − 1 where and and χ−θγkm 1−γkm .  d log 1 + Λ j,km χθγkm − 1 = d log Λ j,km d log 1 + Λ j,km χθγkm − 1 d log Λ j,km d log Λ j,km d log v j,km , d log v j,km Λ j,km χθγkm − 1

=


1 + Λ j,km (χθγkm − 1) , ρ j,km ρ j,km d log Λ j,km v j,km = = = 1 − Λ j,km ,
2 d log v j,km v j,km + ρ j,km v j,km + ρ j,km Λ j,km S  B  d log v j,km = λ d log S j,k + λ − 1 d log Bj,m . F = d log L . Furthermore, from equation (6) and (12), d log S j,k = d log Yj,k j D Additional Theoretical Results This appendix provides additional theoretical results. Section D.1 derives the equilibrium in a special case of my model with a single sector. Section D.2 provides sufficient conditions for equilibrium existence and uniqueness. Section D.3 analyzes the planner’s problem and highlights the sources of misallocation in the equilibrium. 10 D.1 Single Sector Model In this section, I derive the system of equations for a special case of my model with a single sector (i.e., |K | = 1). Note that in the single sector model final consumption is proportional to intermediate goods absorption (YiF ∝ YiI ) and entry is proportional to population size (Ni ∝ Li ).1 Moreover, from equation (6), ci ∝ ( Li /Ωi )1/θ . The equilibrium is then summarized by two sets of equilibrium conditions, “buyer access” and “supplier access” equations, analogous to Arkolakis et al. (2023). First, it can be shown that the labor market clearing condition (equation 14) becomes wi Li ψi = ∑ w j L j ψj πij . j∈N This equation corresponds to “buyer-access” equation. Together with the gravity equation of πij = Γi τij /Ω j (equation 9) and the expression for Γi (equation 8), I have: 1+θγ −θe+(1−γ) −(1−γ) Li Ωi 1 + Λ i ( χ θ (1− γ ) − 1 ) wi = ∑ j∈N 1 KijB w j L j Ω− j , (D.1) where KijB is a constant that only depends on the exogenous variables and parameters. Using the expression of the steady-state match probability (equation 7) h i S B −1 Λi = 1 + KiΛ L1i −λ −λ , (D.2) where KiΛ is a constant, and I used the fact that Si ∝ Li and Bi ∝ Ni ∝ Li for single-sector model. Second, from the definition of Ω j = ∑i Γi τijθ (equation 6), I have Ωj = −θγ 1+θε−(1−γ) 1−γ Li Ωi ∑ KijS wi i ∈N   1 + Λ i ( χ θ (1− γ ) − 1 ) , (D.3) where KijS is a constant. This equation corresponds to “supplier-access” equation. Lastly, population mobility equation (16) is given by −1+υ θθ (−σσ−+11) wiυ Li υ Ωiθ = ∑ j∈N υ θθ (−σσ−+11) K jL wυj L j υ Ω jθ , (D.4) where K jL is a constant. Together, the single-sector model with exogenous population is characterized by {wi , Ωi } that satisfy equations (D.1), (D.2), (D.3). The equilibrium with endogenous pop1 See Arkolakis, Huneeus, and Miyauchi (2023) for a related derivation. 11 ulation mobility is characterized by {wi , Ωi , Li } that satisfy equations (D.1), (D.2), (D.3), and (D.4). D.2 Equilibrium Existence and Uniqueness In this appendix, I discuss conditions for equilibrium existence and uniqueness. Equilibrium Existence. The equilibrium existence of my multi-sector and location model is immediate from Brower’s fixed point theorem. To see this, equilibrium variables {wi , Li , Λi,km , πij,k } are bounded under normalization ∑i wi = 1, and all the mappings are continuous and differentiable. Equilibrium Uniqueness without Population Mobility. Deriving clear analytical results for the equilibrium uniqueness is challenging using my model with multiple sectors. However, one can show that equilibrium is unique (up to scale) without population mobility with a single sector (Appendix D.1). To see this, from Allen, Arkolakis, and Li (2022), equilibrium is unique if the matrix | BΓ−1 | has a spectral radius equal to or less than one, where     1 + θγ −(1 − γ) 1 −1 Γ= , B= . 0 1 −θγ 1 − γ By invoking the Collatz–Wielandt Formula (see Remark 5 in Allen et al. (2022)), one can show that the largest eigenvalue of | BΓ−1 | is one regardless of the parameter values. Sufficient Conditions for Equilibrium Uniqueness with Population Mobility. I next provide sufficient conditions for uniqueness with population mobility in a single-sector model. Notice that the equilibrium system is not constant elasticity so I apply the results of Allen et al. (2022) that allow for the variable elasticity system. It can be show that equilibrium is unique if the spectral radius of maxΛi ∈[0,1] | BΓ−1 | is equal to or less than one, where maxΛi ∈[0,1] is the element-by-element maximum and   0 0 1 υ −1 + σ̃θ − 1−υ σ − σ̃θ  , Γ= 1 + θγ −θe + (1 − γ) − Λi (λS + λ B − 1) −(1 − γ)   −θγ 1 + θe − (1 − γ) + Λ j (λS + λ B − 1) 1−γ B= υ σ̃θ − 1−υ σ − σ̃θ  , 1 1 −1 where σ̃ = θ − σ +1 θ ( σ −1) and Λ j = Λ j (χθ (1−γ) −1)(1−Λ j ) 1 + Λ j ( χ θ (1− γ ) − 1 ) . Note that while this condition is sufficient, it is not necessary. As discussed in Allen et al. (2022), in the context of a variable elasticity system (as in this case), the sufficient con- 12 dition may be significantly more conservative compared to the necessary and sufficient condition. In fact, although my baseline parametrization in Section 5 does not satisfy the aforementioned condition, I have confirmed that the choice of initial values does not affect my counterfactual simulation results. This indicates that the existence of multiple equilibria is unlikely to pose an issue under my baseline calibration. D.3 Planning Problem and Sources of Misallocation In this appendix, I discuss how thick market and congestion externalities lead to inefficiency in equilibrium entry in a special case of a single location and sector by analyzing an optimal planning problem. I focus on the case with single location and sector to provide a clear and straightforward explanation. Laissez-Faire Equilibrium with a Single Sector and Location. Given that there is single sector, I normalize the wage w = 1. The inverse cost shifter, Γ, is simplified from equation (18) as    (D.5) Γ = Nc−θγ 1 + Λ χθγ − 1 , where I normalized $Aθ = 1.2 The cut-off of marginal cost below which firms enter a location as a seller, c, and the measure of sellers, S, is simplified from equation (6) as c = Γ−1/θ , S = Γcθ , (D.6) σ +1 L 3 where I normalized θ −θσ f = 1. The measure of entrepreneurs, N, is simplified from equation (10) as N = 1, (D.7) −1 L where I normalized σσθ F = 1. Together, the laissez-faire equilibrium is characterized by {Γ, c, N } that satisfy equations (D.5), (D.6), (D.7) above. Planning Problem. I consider an optimal taxation problem where the planner has access to taxes for entrepreneurs’ entry, sellers’ entry, and income tax. First, I assume that the planner imposes labor income tax, τ W , as a fraction of labor income. Keeping the 2 This normalization, and subsequent normalization in this section, has no effect on any of the results presented in this section. Note also that L, and hence A, is exogenous in the single-location model. 3 To derive this equation, note that Y F = wL = L and Ω = Γ in a single location case. 13 same normalization that wage w = 1, individual post-tax income is given by w∗ = 1 − τ W . (D.8) Second, I assume that the planner imposes taxes for sales entry, τ S , as a fraction of fixed cost payment f . Noting that the aggregate sales are also affected by the income tax above, the seller entry cutoff is modified from equation (D.6) as    1/θ c = Γ −1 1 − τ W 1 − τ S , S = Γcθ . (D.9) Third, I assume that the planner imposes taxes for entrepreneurs’ entry, τ N , as a fraction of fixed cost payment F. The entrepreneur entry condition is modified from equation (D.7) as    W N N = 1−τ 1−τ . (D.10) The planner chooses τ W , τ S , τ N subject to the government’s budget constraint τS τN τ L+ Sf + NF = 0, 1 − τN 1 − τS W where 1−1τ S S f and 1−1τ N NF are the aggregate fixed cost payment for entrepreneurs’ entry inclusive of taxes and that for sellers’ entry inclusive of taxes, respectively. From the −1 σ +1 and σσθ fraction of aggregate discussions in Appendix C.1, these two objects are θ −θσ
W final goods sales 1 − τ L, respectively. Together, the government budget constraint is rewritten as τW σ−1 θ−σ+1 = τN + τS . (D.11) W θσ θσ 1−τ The welfare in this economy is given by log U = log   w∗ 1 θ−σ+1 = log 1 − τ W − log Γ − log c, P 1−σ 1−σ (D.12) where the expression for the price index P followed from equation (15). Together, the planner’s problem is given by maximizing equation (D.12) with respect to τ W , τ S , τ N subject to equations (D.8), (D.9), (D.10) and (D.11). Reformulating the first-order conditions, I can show that the optimal tax system satisfies the following two sets of conditions:
   θγ − 1 (1 − Λ ) Λ χ σ − 1 γ 1 ( ) 1 − τ S 1 − τW − 1 = − + λS . (θ − σ + 1) (1 − γ) 1 − γ 1 + Λ (χθγ − 1) | {z } | {z } pooling externality,≤ 0 thick-market externality,≥0 (D.13) 14  1−τ N  1−τ W  γ 1−γ | {z } −1 = input-output externality,≥ 0
 1 Λ χθγ − 1 (1 − Λ)  B + λ −1 . 1 − γ 1 + Λ (χθγ − 1) | {z } congestion externality,≤0 (D.14) These two equations succinctly summarize the sources of misallocation present in the laissez-faire equilibrium, which the planner aims to correct. Equation (D.13) summarizes the inefficiency in sellers’ entry. The first term, labeled “pooling externality,” arises from the fact that increased seller entry raises the average cost of suppliers in the matching market. The second term, labeled “thick market externality,” is positive if γ > 0 and λS > 0. Both of these effects disappear when γ = 0 (i.e., no intermediate inputs), which is consistent with the assumption that there are no matching frictions in the final goods market. Equation (D.14) summarizes the inefficiency in entrepreneur’s entry. The first term, labeled “input-output externality,” arises from firms not internalizing the impact of their entry on reducing intermediate goods costs and generating social surplus. This effect is commonly observed in models with firm entry and input-output linkages (e.g., Krugman and Venables 1995 and Antràs, Fort, Gutiérrez, and Tintelnot 2022). The second term, labeled “congestion externality,” is positive if γ > 0 and λ B < 1. E Model Extensions This appendix provides several extensions of my theoretical framework. I explore these alternative specifications as a part of sensitivity analysis in Appendix H. E.1 Markups in Intermediate Input Sales In this appendix, I consider an alternative setting of the model where firms charge markups for intermediate input sales instead of marginal cost pricing in the main paper. More specifically, I assume that firms apply the same markup ratio σ/(σ − 1) as final goods consumers. This modification changes the condition for the threshold of marginal cost to enter location j and sector k, c j,k , and the measure of suppliers selling in location j, S j,k , as #1/θ " F I θ − σ + 1 Yj,k + ∑m Yj,km , S j,k = Ω j,k cθj,k , c j,k = (E.1) θσ w j f j,k Ω j,k I . Similarly, where the difference from equation (6) is the addition of input demand ∑m Yj,km the condition for the free entry of entrepreneurs is replaced from equation (10) as Nj,m F I σ − 1 X j,m + X j,m , = σθ w j Fj,m (E.2) 15 I I ≡ ∑n π jn,m ∑m0 Yn,mm where X j,m 0 . Market clearing conditions are also replaced from equations (13) and (14) as ! σ−1 I σ−1 F X̃i,m = ∑ (E.3) ∑ σ Yj,mm0 + σ Yj,m0 πij,m0 , j∈N m0 ∈K ! θ − σ + 1 I F I wi Li ψi = ∑ )πij,k + (Yi,k + ∑ Yj,km ) . ∑ (Yj,kF + ∑ Yj,km σθ m m j∈N k (E.4) All other equilibrium conditions remain the same. σ−1 γL,k X̃i,k + σθ E.2 Forward-Looking Acceptance Decision In this appendix, I extend the model by incorporating a forward-looking acceptance decision regarding matching with a supplier. In particular, I assume that buyer-side firms now have the choice to accept or reject a match, leading to the formation of a long-term relationship. This is in contrast to the baseline model discussed in the main text, where firms always form a relationship conditional on a match. I maintain the assumption that suppliers set prices at their marginal cost, meaning that buyers have all the bargaining power. Consequently, suppliers make no profit from the relationships, and the suppliers’ forward-looking decision becomes irrelevant in this context. B ( c ) denote the continuation value of a buyer ω engaged in an ongoing relaLet Vωt,k tionship with a supplier in sector k with a unit cost of c (net of iceberg cost). Additionally, B let Uωt,k represent the value of a firm without an ongoing supplier relationship in sector k. The Bellman equation for the matched buyer can be expressed as follows:   B F B B B ξVωt,k (c) = Πωt,k (c) − ρkm Vωt,k (c) − Uωt,k + V̇ωt,k ( c ), (E.5) 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; ρkm is the Poisson rate at which the relationship B ( c ) indicates the time derivative of the value function V B ( c ). is destroyed; and V̇ωt,k ωt,k The Bellman equation for the unmatched buyer is given by B ξUωt,k = F,U Πωt,k + vi,km aω,k Z c∗  ω,k 0  B B B , Vωt,k (c) − Uωt,k dGω,k (c) + U̇ωt,k (E.6) F,U where Πωt,k is the profit from final goods when the firm does not have a directly matched supplier in sector k; vi,km is the Poisson rate of matching with a supplier; aω,k is the unconditional probability that the buyer accepts a match; c∗ω,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 16 B B . that Gω,k (0) = 0 and Gω,k (c∗ω,k ) = 1); and U̇ωt,k indicates time derivatives of Uωt,k To solve these Bellman equations analytically, I take a limit of sales fixed cost to zero F ( c ) and Π F,U are ( f j,k → 0). This assumption implies that the instantaneous profit Πωt,k ωt,k isoelastic in costs and allows me to derive an analytical solution. Under this assumption, I have F Πωt,k (c) = Ki,m ( pωt,−k cγkm )1−σ , (E.7) F,U Πωt,k = Z c i,k 0 I (c) Ki,m ( pωt,−k (χc)γkm )1−σ dGi,k = Ki,m ( pωt,−k χγkm )1−σ θ (c )γkm (1−σ) , θ + γkm (1 − σ) i,k (E.8) I (·) where ci,k is the entry cutoff of suppliers in market i as defined by equation (6), Gi,k denotes the cost distribution of suppliers in market i (which follows the inverse of Pareto distribution with upperbound ci,k , and pωt,−k indicates the component of marginal cost of firm ω other than the component from input sector k. I assume that buyers set c∗ω,k to maximize the expected value of the unmatched state.4 This implies that c∗ω,k is determined so that firms are in expectation indifferent between accepting or rejecting a match: B B E[Vωt,k (c∗ω,k )] = E[Uωt,k ], (E.9) where the expectation is taken with respect to intermediate input cost other than input sector k. Evaluating equation (E.5) at c = c∗ω,k and taking expectation in the steady state B ] = 0) yields: (i.e. E[V̇ωt,k B F ξE[Vωt,k (c∗ω,k )] = E[Πωt,k (c∗ω,k )] = Ki,m c∗ω,k (1−σ)γkm E[( pωt,−k )1−σ ], (E.10) B ( c ) − U B , equations (E.5) and (E.6) yield: Furthermore, by defining Jωt,k (c) = Vωt,k ωt,k Z   F,U F (ξ + ρkm ) Jωt,k (c) = Πωt,k (c) − Πωt,k − vi,km aω,k c∗ω,k 0 Jωt,k (c)dGω,k (c) + J̇ωt,k (c). Taking the derivative of this equation with respect to c yields: (ξ + ρkm ) ∂ ∂ Jωt,k (c) = γkm (σ − 1)Ki,m ( pωt,−k )1−σ c(1−σ)γkm −1 + J̇ωt,k (c) ∂c ∂c 4 I assume that the cut-off value c∗ ω,k is determined ex-ante. If the cut-off were dependent on input prices in other sectors at each time point, obtaining a closed-form solution would be infeasible. 17 By integrating this expression from c to c∗ω,k , I have: Jωt,k (c) − Jωt,k (c∗ω,k ) = i   Ki,m ( pωt,−k )1−σ h ∗
(1−σ)γkm cω,k − c(1−σ)γkm + J̇ωt,k (c) − J̇ωt,k (c∗ω,k ) ξ + ρkm Using this equation, equation (E.6) is rewritten as: B ξUωt,k = F,U Πωt,k + vi,km aω,k Z c∗ ω,k 0 Jωt,k (c)dGω,k (c) + U̇ωt,k Ki,m ( pωt,−k )1−σ ∗
(1−σ)γkm (1 − σ) γkm = − vi,km aω,k cω,k ξ + ρkm (1 − σ) γkm + θ Z c∗   ω,k B + vi,km aω,k Jωt,k (c∗ω,k ) + J̇ωt,k (c) − J̇ωt,k (c∗ω,k ) dGω,k (c) + U̇ωt,k F,U Πωt,k 0 (E.11) where the last transformation uses the fact that Gω,k (·) is the inverse of the Pareto distribution with dispersion parameter θ and upper bound c∗ω,k . By taking the expectation of this equation, B ξE[Uωt,k ] = K E[( pωt,−k ) F,U E[Πωt,k ] − vi,km aω,k i,m ξ + ρkm 1− σ ] c∗ω,k
(1−σ)γkm (1 − σ) γkm . (1 − σ) γkm + θ (E.12) F,U Together with equations (E.9) and (E.10), and by solving for E[Πωt,k ] using the Pareto distribution of input cost, θ (ci,k )γkm (1−σ) θ + γkm (1 − σ) 1− σ K E[( pωt,−k ) ] ∗
(1−σ)γkm (1 − σ) γkm − vi,km aω,k i,m cω,k ξ + ρkm (1 − σ) γkm + θ Ki,m c∗ω,k (1−σ)γkm E[( pωt,−k )1−σ ] =Ki,m E[( pωt,−k )1−σ ](χγkm )1−σ  vi,km aω,k (σ − 1) γkm θ ⇐⇒ 1− = χγkm (1−σ) (ci,k )(1−σ)γkm ξ + ρkm θ − (σ − 1) γkm θ + γkm (1 − σ)  ∗ θ    θ  θ cω,k γkm (σ−1) (σ−1)γkm vi,km aω,k (σ − 1) γkm θ θ + γkm (1 − σ ) ⇐⇒ =χ 1− ci,k θ ξ + ρkm θ − (σ − 1) γkm (E.13) c∗ω,k (1−σ)γkm  18 Now, noting that aω,k = (  aω,k = min 1, χθ  c∗ θ ω,k ci,k and aω,k ≤ 1, θ + γkm (1 − σ) θ  θ γkm (σ −1)  vi,km aω,k (σ − 1) γkm 1− ξ + ρkm θ − (σ − 1) γkm  θ (σ−1)γkm (E.14) and 1 θ . c∗ω,k = ci,k aω,k (E.15) Furthermore, these expressions imply that c∗ω,k and aω,k depend only on firms’ location i ∗ and the supplier sector k such that c∗ω,k = ci,km and aω,k = ai,km . E.3 Firm Heterogeneity for Supplier Demand In this appendix, I extend the model to incorporate the feature that not all firms possess demand to match with external suppliers. In particular, I assume that only a fraction δj,km of firms in location j and sector m have the demand to match with a supplier in sector k. These exogenous parameters, δj,km , can vary based on j, k, and m. By introducing these additional parameters, I can rationalize the differential patterns of supplier matching between conditional and unconditional on supplier bankruptcy as observed and documented in Appendix G and Table G.1. The model remains largely unchanged, with the only modification being the replacement of the steady-state probability that a firm in location j and sector m has a direct relationship with a supplier in sector k, denoted as Λ j,km in equation (7), with the following expression: v j,km . (E.16) Λ j,km = δj,km v j,km + ρ j,km Similarly, the counterfactual equilibrium remains the same as in Appendix F except that equation (F.7) is replaced with: Λ̂ j,km
λS
λ B −1 (Λ j,km /δj,km ) Ŝi,k B̂i,m =
λ B −1
.
λS (Λ j,km /δj,km ) Ŝi,k B̂i,m + 1 − Λ j,km /δj,km (E.17) To undertake counterfactual simulation using this extended model in my sensitivity analysis (Appendix H), I additionally need to know the values of δj,km . I calibrate δj,km using equation (E.16) with the observed steady-state match probability (Λ j,km ), modelpredicted matching rates (v j,km ), and the observed link separation rates (ρ j,km ). ) , 19 F System of Equations for Counterfactual Equilibrium To conduct these counterfactual simulations, I follow the exact-hat algebra approach of Dekle et al. (2008) and rewrite the counterfactual equilibrium conditions in terms of the unobserved changes in the endogenous variables between the counterfactual and initial equilibria. I denote the value of a variable in the initial equilibrium by x, the value of this variable in the counterfactual equilibrium by x 0 (with a prime), and the relative change in this variable by x̂ = x 0 /x (with a hat). Consider a counterfactual to change exogenous productivity  j,m and iceberg trade costs τ̂ij,m .5 Given the values of the production and preference parameters {αk ,γL,m ,γkm , θ, σ, ε}, baseline population size and trade flows {Li , F }, parameters and baseline variables for firm-to-firm matching {λS , λ B , χ} and πij,k , Yi,k {Λi,km }, counterfactual equilibrium is computed in terms of the changes in the endogenous variables {Λ̂i,km , Γ̂i,k , ĉi,k , ŵ j , N̂j,m , L̂ j ,  j,m }: (i) Production and Trade Linkages: π̂ij,m = Γ̂i,m = Γ̂i,m τ̂ij,m
θ ∑`∈N Γ̂`,m τ̂` j,m −θγ θ ŵi L,m N̂i,m Âi,m Ω̂ j,m =
θ (F.1) π` j,m 0 θγkm −θγkm 1 + Λi,km χ ĉi,k 1 + Λi,km (χθγkm k∈K ∑ ∏ Γ̂`,m τ̂` j,m
θ
−1 − 1) (F.2) (F.3) π` j,m `∈N " ĉ j,k = L̂ j #1/θ (F.4) Ω̂ j,k (ii) Matching: Ŝ j,k = Ω̂ j,k ĉ j,k
θ (F.5) (F.6) B̂j,m = N̂j,m Λ̂ j,km = Λ j,km Ŝ j,k (ii) General Equilibrium: B̂j,m
λ B −1 0 N̂j,m 5 The
λS + 1 − Λ j,km
(F.7) 0 F 1 ∑i π ji,m Yi,m = F ŵ j ∑i π ji,m Yi,m (F.8) counterfactual to change population size L̂ j follows the same procedure, except that equation (F.14) is replaced by the assumed exogenous values. 20 Âi,m = È i,m L̂iε (F.9) F Ŷi,k = ŵi L̂i (F.10) 0 0 I Yi,km = γkm X̃i,m 0 X̃i,m = ∑ j∈N  ŵi = I Yj,mm 0 + m ∈K 0 1 ∑k γL,k X̃i,k +  L̂i ∑k γL,k X̃i,k + L̂ j = G 0 ∑ 0 σ − 1 F0 Yj,m0 σ 0 (F.12) πij,m0 + θ − σ +1 F 0 σθ Yi,k  Fπ ∑ j∈N Yj,k ij,k +
υ ŵ j / P̂j
υ ŵ` / P̂` L` θ − σ +1 F σθ Yi,k  σ −1 σθ F0 π 0 ∑ j∈N Yj,k ij,k σ −1 σθ ∑` (F.11) ! (F.13) (F.14) Model Fit to Untargeted Moments In this appendix, I discuss additional evidence of model fit. New Supplier Link Creation Rates Unconditional on Supplier Bankruptcy. In Section 5, I calibrate the matching function elasticities targeting the spatial heterogeneity of new supplier matching rates conditional on supplier bankruptcy. In this calibration process, I do not specifically target the spatial heterogeneity of supplier matching rates unconditional on supplier bankruptcy. To assess how these untargeted statistics align between the data and model predictions, in Table G.1, I report the regression coefficients of the log of new supplier link creation rates unconditional on supplier loss on the log of supplier density (S∗j,k , as defined in Section 3) for each sector pairs and (buyer) location. For the model predictions (Columns 1 and 3), I calculate the independent variable using the expression S B (1 − exp(ηS∗j,kλ Bλj,m )) × (1 − Λ j,km ), where the multiplication by 1 − Λ j,km reflects the fact that firms with ongoing supplier relationships do not match with new suppliers. For the data (Columns 2 and 4), I calculate the independent variable as the average number of new linkages generated per year and buyer. Both the model prediction and the data reveal a significant positive relationship between new supplier link creation rates and supplier density. However, it is worth noting that the coefficient on the log supplier density is larger in the model prediction (0.52, Column 1) compared to the data (0.20, Column 2). Additionally, the intercepts are larger in the model (-3.07) than in the data (-4.35). This evidence indicates that the model tends to overpredict the new supplier creation rates, particularly for locations and sectors with high supplier density. One potential explanation for this discrepancy lies in the existence of unmodeled het- 21 Table G.1: Supplier Link Creation Rates: Model vs Data log New Supplier Link Creation Rates (Unconditional on Supplier Loss) Model Data Model Data log Supplier Density Constant Prefecture FE Supplier Sector FE Observations Adjusted R2 (1) (2) (3) (4) 0.52∗∗∗ (0.06) 0.20∗∗∗ (0.03) 0.66∗∗∗ (0.06) 0.34∗∗∗ (0.03) −3.07∗∗∗ (0.13) −4.35∗∗∗ (0.14) X X 1,547,590 0.86 X X 1,547,590 0.42 1,547,590 0.81 1,547,590 0.15 Note: Standard errors are clustered at the supplier sector and prefecture level. *p<0.1; **p<0.05; ***p<0.01. erogeneity in the demand for matching with suppliers. To grasp the intuition, consider a scenario where a subset of firms in the economy has no demand for external suppliers. As a result, these firms never encounter supplier bankruptcies, and thus they are excluded from the samples in Section 3 and do not influence the calibrated matching function elasticities (λS and λ B ). However, their presence does impact the heterogeneity of supplier matching rates unconditional on supplier bankruptcy. To address this potential model misspecification, I introduce an extension to the model in Appendix E.3, which incorporates firm heterogeneity regarding the demand for external suppliers across different sectors and locations. In my sensitivity analysis for counterfactual simulations (Appendix H), I demonstrate the robustness of my findings when considering this alternative specification. Aggregate Sales by Location and Sector. Another untargeted moment in my calibration procedure is the aggregate firm sales by sector and location. Unlike the approach taken by Caliendo and Parro (2014), where they precisely match the world input-output tables at a detailed sector and location level, my calibration only targets the aggregate input-output table (aggregated across locations) and the cross-regional trade patterns. Table G.2 presents the regression coefficients for the log aggregate sales, both predicted by the model and observed in the data, against a measure of supplier access

(∏k∈K 1 + Λi,km χθγkm − 1 that appears in equation (8). Both the model prediction and the data exhibit a positive relationship between aggregate sales and the supplier access measure. The regression coefficients are similar, with the model prediction at 0.89 (Column 1) and the data at 0.99 (Column 2). These findings hold even when controlling for prefecture and sector fixed effects (Columns 3 and 4). These results provide further support for the adequacy of my model, particularly the market clearing assumptions used I F. and Yj,k in calibrating the input and final goods demand, Yj,km 22 Table G.2: Aggregate Sales: Model vs Data log Sales (Normalized) Model Data Model Data log Supplier Access Prefecture FE Sector FE Observations Adjusted R2 (1) (2) (3) (4) 0.89∗∗ (0.40) 0.99∗∗ (0.44) 0.52∗∗ (0.24) 0.51∗ (0.26) 3,700 0.64 X X 3,700 0.78 X X 3,700 0.89 3,700 0.42

Note: Standard errors are clustered at the prefecture level. log supplier access is defined by ∏k∈K 1 + Λi,km χθγkm − 1 . *p<0.1; **p<0.05; ***p<0.01. H Additional Results and Sensitivity Analysis for Counterfactual Simulations In this appendix, I provide additional results and sensitivity analysis of the counterfactual simulation in Section 5.2. Increasing Population Size in Tokyo. Figure H.1 presents the results of a counterfactual simulation of increasing Tokyo’s population size on average real wages in Japan and in various prefectures within Japan. Table H.1 presents a sensitivity analysis of the same counterfactual simulation regarding the increase in Tokyo’s population size. For each model specification, Column (a) presents the elasticity of Tokyo’s real wage with population size in baseline specification, Column (b) presents the same elasticity by shutting down thick market externality, and Column (c) presents the same elasticity by shutting down productivity spillovers. Row (1) represents the baseline specification from Table H.1, which serves as the reference point for further analysis. In the subsequent analysis, I specifically focus on the contribution of thick market externality to this elasticity (Column b; “Diff. from Baseline”). Rows (2) to (7) provide a sensitivity analysis of the matching function elasticities (λS and λ B ) and the iceberg cost of indirect sourcing (χ). These alternative parameter values are calibrated based on the same methodology as outlined in Section 5.1.2, targeting plus and minus 1.64 times the standard errors of the point estimates from the reduced-form regression coefficients in Columns (3) and (5) of Table 6. As anticipated, the contribution of thick market externality varies significantly across different parameter values, although it consistently has a negative sign, indicating a positive influence of thick market externality on agglomeration benefits. Rows (8) and (9) consider alternative values for σ and θ, which impact the overall 23 Figure H.1: Effects of Increasing Population Size in Tokyo (A) Average Real Wage in Overall Japan (B) Real Wage in Other Prefectures Note: Results of the counterfactual simulation to increase population size in Tokyo prefecture. Panel (A) plots the log changes in average real wages across Japan against the log change in Tokyo prefecture’s population size. Panel (B) shows the log changes in real wages in each prefecture for a 50 percent increase in Tokyo’s population size against the prefecture’s exposure to Tokyo prefecture, defined as the share of supplier linkages from Tokyo prefecture. agglomeration benefit (Column a) through the love-of-variety externality from firm entry and pooling externality discussed in Section 4.3. Notably, with σ = 5 and θ = 5, the agglomeration benefit becomes negative due to the dominance of pooling externality over other positive agglomeration externalities. However, the contribution of thick market externality remains stable in this scenario. Rows (10) and (11) consider different values of agglomeration productivity spillovers ε. The contribution of thick market externality remains stable. In Row (12), I consider an alternative model specification to incorporate profits from intermediate input sales as discussed in Appendix E.1. In Row (13), I consider an alternative model specification to accommodate forward-looking match acceptance decisions by input buyers as discussed in Appendix E.2.6 In both specifications, the contribution of thick market externality remains similar. In Row (14), I consider an alternative model specification discussed in Appendix E.3, where only a fraction δj,km of firms in location j and sector m have demand to match with a supplier in sector k. Interestingly, I observe a somewhat larger contribution of thick market externality in this specification. This can be attributed to the fact that the changes in matching rates (v j,km ) triggered by an increase in population size have a more pronounced impact on the steady-state match probability (Λ j,km ) in this model specification (as seen in equation E.17). Consequently, the contribution of thick market externality becomes more significant in this alternative model specification. 6 For this simulation, I set the discount rate ζ to 0.03. 24 Table H.1: Sensitivity Analysis: Effects of Increasing Population Size in Tokyo Specification (1) baseline (2) λS = 1.39 (3) λS = 0.46 (4) λ B = 1.29 (5) λ B = 0.60 (6) χ = 2.48 (7) χ = 1.15 (8) σ = 5 and θ = 5 (9) σ = 5 and θ = 10 (10) ε = 0.1 (11) ε = 0.03 (12) Incorporate Profit for Intermediate Goods Sales (13) Introduce Firms without Demand for Suppliers (14) Introduce Forward-Looking Match Acceptance (a) Baseline 0.137 0.157 0.125 0.146 0.128 0.178 0.126 -0.092 0.045 0.167 0.125 0.136 0.136 0.149 (b) λS = 0 0.118 0.118 0.118 0.121 0.116 0.118 0.118 -0.111 0.021 0.148 0.106 0.121 0.118 0.118 Diff. from Baseline (c) ε = 0 -0.019 0.107 -0.039 0.127 -0.006 0.095 -0.025 0.117 -0.012 0.098 -0.060 0.149 -0.008 0.096 -0.019 -0.122 -0.024 0.010 -0.019 0.107 -0.019 0.107 -0.016 0.107 -0.017 0.106 -0.031 0.119 Diff. from Baseline -0.030 -0.030 -0.030 -0.030 -0.030 -0.030 -0.030 -0.030 -0.036 -0.059 -0.018 -0.030 -0.030 -0.030 Note: Results of the counterfactual simulation to increase population size in Tokyo prefecture under alternative specifications indicated by the first column. Column (a) shows the elasticity of Tokyo’s real wages with Tokyo’s population (following the same definition as Table 7) under baseline specification; Column (b) shows the results under no thick market externality (λS = 0); Column (c) shows the results under no agglomeration productivity spillover (ε = 0). See the main text for further details about each specification. Increasing Productivity in Tokyo. Figure H.2 presents the results of a counterfactual simulation to increase Tokyo’s productivity on aggregate welfare in Japan and real wages in different prefectures in Japan. In Table H.2, I provide a sensitivity analysis for the same counterfactual simulation. The patterns observed in the sensitivity analysis closely mirror those observed in Table H.1. Rows (15) and (16) introduce an alternative calibration for the migration elasticity υ. As anticipated, a larger value of υ leads to a greater contribution of thick market externality. This occurs because a higher migration elasticity induces more significant population responses to the productivity shock, thereby amplifying the effects of thick market externality. 25 Figure H.2: Effects of Increasing Productivity in Tokyo (A) Real Wage in Overall Japan (B) Real Wage in Other Prefectures Note: Results of the counterfactual simulation to increase productivity in Tokyo prefecture. Panel (A) plots the log changes in aggregate welfare in Japan (U ) against the log change in Tokyo prefecture’s population size. Panel (B) shows the log changes in real wages in each prefecture for a 20 percent increase in Tokyo’s productivity against the prefecture’s exposure to Tokyo prefecture, defined as the share of supplier linkages from Tokyo prefecture. Table H.2: Sensitivity Analysis: Effects of Increasing Productivity in Tokyo Specification (1) baseline (2) λS = 1.39 (3) λS = 0.46 (4) λ B = 1.29 (5) λ B = 0.60 (6) χ = 2.48 (7) χ = 1.15 (8) σ = 5 and θ = 5 (9) σ = 5 and θ = 10 (10) ε = 0.1 (11) ε = 0.03 (12) Incorporate Profit for Intermediate Goods Sales (13) Introduce Firms without Demand for Suppliers (14) Introduce Forward-Looking Match Acceptance (15) υ = 4 (16) υ = 1 (a) Baseline 0.252 0.251 0.248 0.252 0.249 0.279 0.244 0.181 0.269 0.268 0.246 0.251 0.253 0.246 0.255 0.223 (b) λS = 0 0.238 0.238 0.238 0.242 0.233 0.238 0.238 0.175 0.251 0.252 0.232 0.240 0.238 0.238 0.223 0.220 Diff. from Baseline (c) ε = 0 -0.014 0.238 -0.014 0.238 -0.010 0.234 -0.010 0.238 -0.017 0.236 -0.042 0.262 -0.006 0.231 -0.007 0.175 -0.018 0.251 -0.016 0.238 -0.014 0.238 -0.011 0.238 -0.015 0.239 -0.009 0.233 -0.031 0.232 -0.003 0.218 Diff. from Baseline -0.014 -0.014 -0.014 -0.014 -0.014 -0.017 -0.013 -0.006 -0.018 -0.030 -0.008 -0.013 -0.014 -0.013 -0.022 -0.004 Note: Results of the counterfactual simulation to increase productivity in Tokyo prefecture under alternative specifications indicated by the first column. Column (a) shows the elasticity of Tokyo’s real wages with Tokyo’s productivity (following the same definition as Table 8) under baseline specification; Column (b) shows the results under no thick market externality (λS = 0); Column (c) shows the results under no agglomeration productivity spillover (ε = 0). See the main text for further details about each specification.