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

1 A Additional Information of TSR (Tokyo Shoko Research) Firm-to-Firm Trade Data 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 the reported reasons for bankruptcies from which I identify unanticipated bankruptcies. Figure A.2 shows that the frequency of unanticipated bankruptcies is unrelated to geographic firm density. Table A.1: Sample Size and Coverage of TSR Data Sets TSR Economic Census TSR / Economic Census 2009 All Employment ≤ 4 Employment ≥ 5 1,245,726 589,081 656,645 1,805,545 1,067,825 737,720 0.68 0.55 0.89 2016 All Employment ≤ 4 Employment ≥ 5 1,505,497 808,014 697,483 1,877,438 1,047,189 830,249 0.8 0.77 0.84 Note: By year (2009 or 2016) and Sample size (first column) and firm size (over or under 5 employees), this table reports the sample size of the TSR data set (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 data sets relative to Economic Census Note: This figure plots the density of firms using two data sources: The economic census on the horizontal access and TSR data on the vertical access. 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 0.25) and the R-squared is 0.98. The tight log-linear relationship with a coefficient close to one suggests that the TSR data set is representative across different municipalities in Japan. 2 Table A.2: List of Reasons of Bankruptcies Reason of Bankurptcy Unanticipated Reasons Freq. Freq. (At Least One Buyer) 1589 661 Sales Decline 72483 Accumulation of Debt 10718 Spillovers from Other Bankruptcy 6223 Shortage of Capital 5582 Management Failure 4845 Unknown 3597 Over-Investment in Capital 802 Deterioration of Credit Conditions 547 Difficulty in Collecting Account Receivables 454 Over-Accumulation of Inventory 73 Total 106913 22775 5456 1996 2340 1281 929 368 282 237 38 36363 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 size of the dot represents the 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 Additional Empirical Results for Reduced-Form Evidence In this section of the online appendix, I present additional results for the reduced-form evidence of matching frictions and increasing returns to scale (IRS) in firm-to-firm matching presented in Section 3 of the paper. Table B.1 shows that the characteristics of treatment firms and the control firms are broadly similar. Table B.2 shows that there are no differential pretrends between treatment firms and control firms in the new supplier matching rates and sales growth. Table B.1: Characteristics of Treatment Firms Treatment Firms Control Firms 4 [0, 9] 10 [3, 60.4] 0.28 [0.05, 2.2] 4 [1, 9] 11 [3, 100] 0.32 [0.05, 3.95] (ii) Industry Composition Proportion (Manufacturing) Proportion (Commerce) Proportion (Construction / Equipment Services) Proportion (Others) 0.29 0.28 0.27 0.14 0.26 0.31 0.25 0.17 Sample Size 421 10,842 (i) Firm Size Number of Suppliers Number of Employees Annual Sales (Billion Yen) Note: This table shows the characteristics of the treatment firms (firms that face an unanticipated supplier bankruptcy) and the control firms. Each row of panel (i) shows the median of each statistic, and the bracket shows the 10th and 90th percentile of the statistic. Panel (ii) reports the fraction of firms that fall in each category of industry. Table B.2: Pretrends on New Supplier Matching Rates and Sales Number of New Suppliers Sales (IHS) (1) (2) Trt x 1[t - BankruptYear = -3 or -2] 0.04 (0.03) −0.01 (0.01) Observations 31,861 31,221 Note: This table assesses the pretrends for the difference-in-difference regression (equation 1) by running this regression using only pre-period data and replacing the post dummy with the dummy for 2 or 3 years before the bankruptcy (omitted category is 1 year before the bankruptcy). 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. Tables B.3-B.6 reports additional robustness results for the average effects of the supplier bankruptcy presented in Section 3.1. Table B.3 shows that these supplier bankruptcies do not significantly affect the sales of the treatment firms’ other existing suppliers, 4 indicating that treatment firms primarily respond by rematching with new suppliers instead of simply substituting from existing suppliers. In Table B.4, 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. Table B.5 shows that there are no heterogeneous effects by firm size, suggesting that these matching frictions are relevant for both small and large firms. Table B.6 presents the effects on other firm-level outcome variables. Table B.3: Average Impacts of Supplier Bankruptcy on Other Existing Suppliers Continued Relationships with Other Suppliers log Sales of Other Surviving Suppliers (1) (2) Trt x 1[t - BankruptYear = -2 or -3] 0.02 (0.03) 0.01 (0.01) Trt x 1[t - BankruptYear = -1] −0.06 (0.04) 0.03 (0.02) Observations 76,054 66,784 Note: This table reports the result of the difference-in-difference regression (equation 1). Column (1) reports the effects on the number of continuing relationships with other existing suppliers (suppliers that are connected one year before the bankruptcy). Column (2) reports the effects on the mean of the log sales of the firm’s other existing suppliers. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. Table B.4: Decomposition of Impacts of Supplier Bankruptcy on Newly Matched Suppliers Dependent Variable: Number of New Suppliers within Specified Subset All Within 4-digit Industry Within 2-digit Industry Headquarter in Same Prefecture Has Buyer in Same Prefecture (1) (2) (3) (4) (5) Trt x 1[t - BankruptYear = 0 or 1] 0.17∗∗∗ (0.04) 0.04∗∗∗ (0.01) 0.05∗∗∗ (0.02) 0.08∗∗∗ (0.03) 0.08∗∗∗ (0.03) Trt x 1[t - BankruptYear = 2 or 3] 0.27∗∗∗ (0.06) 0.07∗∗∗ (0.02) 0.10∗∗∗ (0.03) 0.16∗∗∗ (0.04) 0.16∗∗∗ (0.05) 0.27 1 0.001 56 0.007 15 0.012 14 0.004 40 Random Matching Benchmark (Impacts after 2-3 Years) Actual Impacts / Random Matching Benchmark Note: This table reports the result of the difference-in-difference regression (equation 1). The outcome variables are the number of newly matched suppliers (i.e., number of suppliers which are not connected in the baseline period) within a specified subset of potential suppliers in each column. Column (1) reproduces the impacts on the number of all new suppliers (Column 1 of Table 1). Columns (2) and (3) report the impacts on the number of newly matched suppliers within the same four-digit and two-digit industry as the bankrupt suppliers, respectively. Column (4) reports the impacts on the number of newly matched suppliers whose headquarters are located in the same prefecture as the treatment firms. Column (5) reports the impacts on the number of newly matched suppliers that already have existing buyers in the treatment firm’s prefecture (at the point of 2008). The second to bottom row (labelled "Random Matching Benchmark") indicates the hypothetical impacts if supplier matching happens randomly independent of the supplier’s industry or location. The bottom row (labelled "Actual Impacts/Random Matching Benchmark") indicates the ratio of the estimated coefficients and the hypothetical impacts under this random matching benchmark. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. 5 Table B.5: Heterogeneous Impacts of Supplier Bankruptcy by Firm Size Number of New Suppliers Trt x 1[t - BankruptYear = 0 or 1] Trt x 1[t - BankruptYear = 2 or 3] Sales (IHS) (1) (2) (3) (4) 0.17∗∗∗ 0.24∗∗∗ −0.21∗∗ (0.04) (0.09) (0.10) −0.22 (0.18) 0.27∗∗∗ (0.06) 0.21∗∗ (0.09) −0.22 (0.18) −0.25 (0.40) Trt x 1[t - BankruptYear = 0 or 1] x Employment (Medium Tercile) −0.07 (0.14) −0.06 (0.24) Trt x 1[t - BankruptYear = 2 or 3] x Employment (Medium Tercile) 0.07 (0.16) 0.004 (0.54) Trt x 1[t - BankruptYear = 0 or 1] x Employment (Top Tercile) −0.19 (0.13) 0.04 (0.25) Trt x 1[t - BankruptYear = 2 or 3] x Employment (Top Tercile) 0.04 (0.17) −0.04 (0.53) Number of Treatment Firms Number of Bankrupt Suppliers Number of Control Firms Observations 421 161 10,842 76,054 421 161 10,842 75,749 421 161 10,814 74,462 421 161 10,814 74,300 Note: The results of the difference-in-difference regression (equation 1 in the main paper). 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 2, and the inverse-hyperbolic sine (IHS) transformation of sales for Column 3 and 4. “Employment (Medium Tercile)” and “Employment (Top Tercile)” indicate the dummy variables that takes one if the firm’s employment size in baseline period is in the medium tercile and top tercile of the treatment firms. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. Table B.6: Impacts of Supplier Bankruptcy on Additional Firm Outcomes Number of New Suppliers Exit Profit / Sales (1) (2) (3) (4) Employment (IHS) (5) Trt x 1[t - BankruptYear = 0 or 1] 0.170∗∗∗ (0.041) 0.184∗∗∗ (0.043) 0.009 (0.007) 0.005 (0.007) 0.006 (0.013) Trt x 1[t - BankruptYear = 2 or 3] 0.267∗∗∗ (0.063) 0.288∗∗∗ (0.066) 0.010 (0.012) 0.016 (0.014) 0.021 (0.015) Specification Observations All Firms 76,054 Excluding Exit Firms 73,422 76,021 59,167 75,749 Note: This table reports the result of the difference-in-difference regression (equation 1). The outcome variables are the number of newly matched suppliers in Column (1) and (2); the dummy variable that takes one if the firm exits in Column (3); accounting profit divided by sales in Column (4); and the inverse hyperbolic sine (IHS) transformation of employment in Column (5). Column (2) excludes observations if firms drops out from the sample. *p<0.1; **p<0.05; ***p<0.01. Tables B.7-B.10 report the robustness to the heterogeneous effects with supplier density reported in Section 3.2. Table B.7 shows that these empirical results are further robust to various restrictions on samples and adjustment. The results are further robust to alternative definitions of supplier density (Table B.8), splitting samples to manufacturing and non-manufacturing supplier bankruptcies (Table B.9), and including supplier-reported supplier-to-buyer-linkages when constructing the outcome variables (Table B.10). Table B.7: Average Impact Heterogeneity by Supplier Density: Robustness Dependent Variable: Number of New Suppliers Exclude Exclude Exclude No Update Bankruptcy in 2009 Tokyo in Accounting Year Adjust Sampling Rate IV: Seller Density in Birth Prefecture (5) (6) (7) 0.15∗∗∗ (0.04) 0.15∗∗∗ (0.05) 0.27∗∗∗ (0.07) 0.15∗∗∗ (0.05) 0.30∗∗∗ (0.07) 0.25∗∗∗ (0.07) 0.26∗∗∗ (0.07) 0.39∗∗∗ (0.10) 0.25∗∗∗ (0.07) 0.10∗∗ (0.05) 0.10∗∗ (0.05) 0.10∗∗ (0.05) 0.10∗∗ (0.05) 0.16∗∗ (0.07) 0.10∗∗ (0.05) 0.16∗∗ (0.07) 0.12∗ (0.07) 0.11 (0.07) 0.11 (0.07) 0.18∗ (0.10) 0.13∗ (0.07) Exclude Exiting Firms Exclude Tohoku After 2011 (1) (2) (3) (4) Trt x 1[t - BankruptYear = 0 or 1] 0.15∗∗∗ (0.04) 0.15∗∗∗ (0.05) 0.16∗∗∗ (0.05) Trt x 1[t - BankruptYear = 2 or 3] 0.26∗∗∗ (0.07) 0.22∗∗∗ (0.06) Trt x 1[t - BankruptYear = 0 or 1] x log Seller Density (Std.) 0.10∗∗ (0.05) Trt x 1[t - BankruptYear = 2 or 3] x log Seller Density (Std.) 0.16∗∗ (0.07) Panel A: Baseline Panel B: Control for Prefecture Fixed Effects and Supplier Industry Fixed Effects Trt x 1[t - BankruptYear = 0 or 1] x log Seller Density (Std.) 0.12∗∗ (0.06) 0.14∗∗ (0.06) 0.12∗∗ (0.06) 0.16∗∗∗ (0.05) 0.16∗∗ (0.07) 0.22∗∗ (0.09) 0.15∗ (0.08) Trt x 1[t - BankruptYear = 2 or 3] x log Seller Density (Std.) 0.23∗∗∗ (0.09) 0.27∗∗∗ (0.08) 0.18∗∗ (0.09) 0.30∗∗∗ (0.08) 0.30∗∗∗ (0.08) 0.43∗∗∗ (0.12) 0.30∗∗ (0.13) First Stage F-Statistics Number of Treatment Firms Number of Bankrupting Suppliers Number of Control Firms 421 161 10,842 400 154 10,613 368 138 9,137 374 145 6,625 421 161 10,842 421 161 10,842 16301.4 386 154 9,820 Observations 73,422 74,245 66,578 47,208 70,831 76,054 68,497 Note: This table reports the robustness of the regression (2) of the main paper. Panel A controls for the log of the geographic area interacted with treatment and post dummy (corresponding to Column 1 of Table 1), and Panel B controls for prefecture fixed effects and the supplier two-digit industry fixed effects interacted with treatment and post dummy (corresponding to Column 2 of Table 1). In Column (1), I exclude firms that drop out during the sample period (instead of including them as zero in baseline). Column (2) excludes the bankruptcies of suppliers in the Tohoku area after 2011 (the year of the Great Tohoku Earthquake). Column (3) eliminates all supplier bankruptcies in 2009 (the year subsequent to the Financial Crisis). Column (4) excludes firms with headquarters in Tokyo prefecture. Column (5) excludes samples whose accounting information is not available after supplier bankruptcy. Column (6) adjusts the number of reported suppliers by the coverage rates of TSR data set at the suppliers’ headquarter location. Column (7) instruments the supplier density proxy by that evaluated at the birth prefecture of the firm CEO. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. 6 7 Table B.8: Impact Heterogeneity by Supplier Density: Alternative Definitions of Supplier Density Number of New Suppliers (1) (2) (3) Trt x 1[t - BankruptYear = 0 or 1] x log Seller Density (Std.) 0.14∗∗ (0.06) 0.13∗∗ (0.06) 0.07 (0.04) Trt x 1[t - BankruptYear = 2 or 3] x log Seller Density (Std.) 0.28∗∗∗ (0.08) 0.23∗∗∗ (0.08) 0.22∗∗∗ (0.06) X X X X X X Evaluated Right Before Bankrutpcy Count LocallyHeadquarterd Suppliers Defined by Two-Digit Supplier Industry 76,054 76,054 76,054 Trt x Post x Prefecture FE Trt x Post x Supplier 2-digit Industry FE Definition of Seller Density Observations Note: The regression specification follows equation (2) of the main paper with alternative definitions of supplier density. Recall that the baseline definition of the supplier density (in Table 2) is defined as the geographic density of suppliers in the bankrupting suppliers’ four-digit industry that have at least one buyer in firm i’s prefecture in 2008. In Column (1), instead of evaluating the supplier density in 2008, I evaluate the supplier density one year before each supplier bankruptcy. In Column (2), I count the number of suppliers whose headquarters are established in the treatment firms’ prefecture, instead of counting the number of suppliers selling to the treatment firms’ prefecture. Column (3) 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.9: Impact Heterogeneity by Supplier Density: By Bankrupt Supplier Industry Dependent Variable: Number of New Suppliers Manufacturing Bankruptcy Non-manufacturing Bankruptcy (1) (2) (3) Trt x 1[t - BankruptYear = 0 or 1] 0.11 (0.08) 0.23∗∗∗ (0.07) Trt x 1[t - BankruptYear = 2 or 3] 0.20 (0.12) 0.33∗∗∗ (0.12) (4) Trt x 1[t - BankruptYear = 0 or 1] x log Seller Density (Std.) 0.15∗∗∗ (0.04) 0.16∗ (0.08) Trt x 1[t - BankruptYear = 2 or 3] x log Seller Density (Std.) 0.17∗∗ (0.08) 0.27∗∗∗ (0.10) X X X X Trt x Post x Prefecture FE Trt x Post x Supplier 2-digit Industry FE Number of Treatment Firms Number of Bankrupting Suppliers Number of Control Firms 159 50 1,043 159 50 1,043 185 83 1,451 185 83 1,451 Observations 7,552 7,552 65,870 65,870 Note: The regression specification follows equation (2) of the main paper by dividing samples by bankrupt suppliers’ industries. Column (1) and (2) use subsamples where bankrupt suppliers belong to manufacturing sector, and Column (3) and (4) use subsamples where bankrupts suppliers belong to non-manufacturing sector. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. 8 Table B.10: Impact Heterogeneity by Supplier Density: Include Reverse Reporting Number of New Suppliers (IHS) Baseline Include Reverse Reporting (1) (2) (3) Trt x 1[t - BankruptYear = 0 or 1] 0.15∗∗∗ (0.03) 0.10∗∗∗ (0.03) Trt x 1[t - BankruptYear = 2 or 3] 0.18∗∗∗ (0.04) 0.14∗∗∗ (0.04) (4) Trt x 1[t - BankruptYear = 0 or 1] x log Seller Density (Std.) 0.05∗∗ (0.03) 0.03 (0.03) Trt x 1[t - BankruptYear = 2 or 3] x log Seller Density (Std.) 0.13∗∗∗ (0.04) 0.09∗∗∗ (0.03) X X X X Trt x Post x Prefecture FE Trt x Post x Supplier Industry FE Observations 76,054 76,054 76,064 76,064 Note: This table reports the robustness of the results in Table 2 by including the supplier linkages reported by the supplier-side firms, in addition to the buyer-reported suppliers as in baseline, to construct the outcome variable of the number of newly matched suppliers. I apply the inverse hyperbolic transformation (IHT) of this outcome variables to deal with the fat tailed distribution of this outcome variable. Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. Table B.11 shows the heterogeneous effects of supplier matching on sales by the following regression: Yf t = βNumberSuppliers f t + γNumberSuppliers f t × log SupplierDensity g + ηgt + ξ f g + e f t , (B.1) where NumberSuppliers f t is the number of suppliers of firm i in period t, and Yf t is the inverse hyperbolic sine (IHS) transformation of sales, and NumberSuppliers f t (and its interaction term) is instrumented by the unanticipated supplier bankruptcies. I find a significant IV effects of the number of matched suppliers on sales (Column 1), yet these effects are not significantly different by supplier density (Column 2), consistent with the interpretation that firms in a location with denser suppliers may benefit primarily by faster supplier matching but not by better supplier matching. Table B.11: IV Impacts of the Number of Matched Suppliers on Firm Sales Sales (IHS) Number of Suppliers (1) (2) 0.29∗ (0.18) 0.28∗ (0.17) −0.01 (0.16) Number of Suppliers x log Supplier Density (Std.) First Stage F-Statistics Observations 48.8 38.1 44,870 44,870 Note: The results of the IV regression specified in equation (B.1). Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. 9 C Model Appendix In this section of the Online Appendix, I discuss additional details of the model developed in Section 4. Section C.1-C.4 discuss the mathematical derivations. Section C.5 discusses the existence and uniqueness property of the steady-state equilibrium. C.1 Consumer Price Index To derive the consumer price index, I first derive the cutoff of entry as a final good seller in location i. The final goods sales of firms in location i with unit cost c (net of trade cost) when they enter in location i is given by F ri,k (c) = $i,k c−σ+1 where $i,k is a demand shifter that depends on aggregate equilibrium conditions. Denoting the unit cost threshold of entry as c Fj,k , the goods market clearing condition is given by: Z cF j,k  θ θσ F αk w j L j = $ j,k c Ω j,k θc dc = f j,k w j Ω j,k ci,k θ−σ+1 0   − σ +1 F . where I used the zero-profit condition for a marginal seller f j,k w j = σ1 $ j,k ci,k From this equation, the entry cut-off is solved as: − σ +1  c Fj,k θ θ −1 = θ − σ + 1 αk L j θσ f j,k Ω j,k Using this expression, consumer price index of location i for goods in sector k is given by: 1− σ Pj,k = Z cF j,k c1−σ Ω j,k θcθ −1 dc 0   θ − σ +1 θ = Ω cF θ − σ + 1 j,k j,k ! θ − σ +1   − σ +1 θ θ
σ −1
θ − σ +1 1 θ−σ+1 αk = Ω j,k θ L j θ σ θσ f j,k and the consumer price index aggregated across all sector is given by Pj = ∏ k Pj,k
αk = ς Lj
− θ − σ +1 θ ( σ −1) ∏ k f j,k
θ − σ +1 α k θ ( σ −1) Ω j,k
− αk θ (C.1) 10 where ς = C.2  θ − σ +1 θσ 1 θ 1 σ σ −1 ∏ k ( α k ) αk θθ (−σσ−+11) . Solving for Bellman Equations In this section, I solve the Bellman equations of suppliers and buyers to form a relationship. Reproducing the Bellman equations in Section 4.4,   B B B B F ( c ), (C.2) ξVωt,k (c) = Πωt,k (c) − δωt,k (c) − ρkm Vωt,k (c) − Uωt,k + V̇ωt,k Z c  ω,k  B B B dGi (c) + U̇ωt,k , (C.3) Vωt,k (c) − Uωt,k 0   S S S S ξVωt,k (c) = δωt,k (c) − ρkm Vωt,k (c) − Uωt,k + V̇ωt,k ( c ), (C.4) B F ξUωt,k = χ1−σ Πωt,k (ζ k wi ) + vi,km aω,k S Uωt,k = 0, (C.5) S S Vωt,k (c) − Uωt,k = (1 − β) Jωt,k (c), (C.6) B B Vωt,k (c) − Uωt,k = βJωt,k (c). F (c) = K ( p γkm )1−σ is the instantaneous final goods profit by buyer where Πωt,k i,m ωt,−k c ω when the supplier’s unit cost is c when fixed cost for final goods sales is negligible ( f j,k → 0); and pωt,−k indicates the component of marginal cost of firm ω other than the component from input sector k; and Ki,m is the aggregate final goods demand shifter exogenous to the firm. I now solve these set of equations. Using equation (C.6),     S S B B ξ Jωt,k (c) =ξ Vωt,k (c) − Uωt,k + ξ Vωt,k (c) − Uωt,k     F 1− σ F B B S S = Πωt,k (c) − χ Πωt,k (ζ k wi ) − ρkm Vωt,k (c) − Uωt,k + Vωt,k (c) − Uωt,k Z c   ω,k B B Vωt,k (c) − Uωt,k dGω,k (c) + J̇ωt,k (c) ⇐⇒ − vi,km aω,k 0   F F (ξ + ρkm ) Jωt,k (c) = Πωt,k (c) − χ1−σ Πωt,k ( ζ k wi ) Z c   ω,k B B − vi,km aω,k dGω,k (c) + J̇ωt,k (c). Vωt,k (c) − Uωt,k 0 11 By taking the derivative of this with respect to c, I have: (ξ + ρkm ) ∂ F ∂ Jωt,k (c) = Πωt,k (c) + J̇ωt,k (c) ∂c ∂c = γkm (σ − 1)Ki,m ( pωt,−k )1−σ c(1−σ)γkm −1 + ∂ J̇ (c) ∂c ωt,k By integrating this expression from c to cω,k , I have: i   Ki,m ( pωt,−k )1−σ h (1−σ)γkm (1−σ)γkm + J̇ωt,k (c) − J̇ωt,k (cω,k ) −c Jωt,k (c) − Jωt,k (cω,k ) = (cω,k ) ξ + ρkm Using this equation, equation (C.3) is rewritten as: B ξUωt,k = χ1−σ Πωt,k (ζ k wi ) + vi,km aω,k Z c ω,k 0 βJωt,k (c)dGω,k (c) + U̇ωt,k Ki,m ( pωt,−k )1−σ = χ Πωt,k (ζ k wi ) + vi,km aω,k β ξ + ρkm Z c   ω,k B J̇ωt,k (c) − J̇ωt,k (cω,k ) dGω,k (c) + U̇ωt,k + 1− σ Z c  ω,k (1−σ)γkm cω,k 0 −c (1−σ)γkm  dGω,k (c) 0 =χ + 1− σ Ki,m (ζ k wi ) Z c  ω,k 0 γkm pωt,−k
1− σ Ki,m ( pωt,−k )1−σ (1−σ)γkm (1 − σ) γkm − vi,km aω,k β c ξ + ρkm (1 − σ) γkm + θ  B J̇ωt,k (c) − J̇ωt,k (cω,k ) dGω,k (c) + U̇ωt,k (C.7) where the last transformation uses the fact that Gω,k (·) is the inverse of the Pareto distribution with upper bound cω,k (Gω,k (c) = cθ /(cω,k )θ for c < cω,k )). 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: B B E[Vωt,k (cω,k )] = E[Uωt,k ], (C.8) where the expectation is taken with respect to intermediate goods cost other than input sector k. S (c Now, equations (C.6) and (C.8) together imply that E[ Jωt,k (cω,k )] = 0 and E[Vωt,k ω,k )] = B B S E[U ] = 0. Furthermore, E[U˙ B ] = E[V̇ (c)] = E[ J̇ (c)] = 0 in the steadyωt,k ωt,k ωt,k ωt,k state so that firms’ values do not diverge to infinity. Given these observations, evaluating equation (C.2) at c = cω,k and taking expectation yields: (1−σ)γkm B B S F (cω,k )] = ξE[Vωt,k (cω,k ) + Vωt,k (cω,k )] = Ki,m cω,k ξE[Vωt,k (cω,k )] = E[Πωt,k E[( pωt,−k )1−σ ], (C.9) 12 B ] from equation (C.8). Plugging this expression in the left which coincides with E[Uωt,k hand side of equation (C.7), (1−σ)γkm B B ξE[Uωt,k ] = ξE[Vωt,k (cω,k )] = Ki,m cω,k E[( pωt,−k )1−σ ] = Ki,m χ1−σ (ζ k wi )(1−σ)γkm E[( pωt,−k )1−σ ] Ki,m E[( pωt,−k )1−σ ] (1−σ)γkm (1 − σ) γkm cω,k ξ + ρkm (1 − σ) γkm + θ 1 (1 − σ) γkm (1− σ ) γ (1− σ ) γ cω,k km ⇐⇒ cω,k km = χ1−σ (ζ k wi )(1−σ)γkm − vi,km aω,k β ξ + ρkm (1 − σ) γkm + θ   v a (σ − 1) γkm (1− σ ) γ = χ1−σ (ζ k wi )(1−σ)γkm ⇐⇒ cω,k km 1 − β i,km ω,k ξ + ρkm θ − (σ − 1) γkm  θ   θ θ (σ−1)γkm cω,k vi,km aω,k (σ − 1) γkm γkm (C.10) ⇐⇒ =χ 1−β ζ k wi ξ + ρkm θ − (σ − 1) γkm − vi,km aω,k β Now, noting that aω,k =  cω,k ζ k wi θ θ γkm  ( aω,k = min 1, χ and aω,k ≤ 1, vi,km aω,k (σ − 1) γkm 1−β ξ + ρkm θ − (σ − 1) γkm  θ (σ−1)γkm ) , (C.11) and 1 θ cω,k = ζwi aω,k . (C.12) These expressions imply 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 , and they correspond to equations (C.11) and (C.12) of the main paper. C.3 Match Benefit φj,km and Fraction of Unmatched Buyers Φ j,km Given the solution to the Bellman equations, I now define several key moments that are useful to characterize the 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 relative to being without a supplier d j,k = 0, i.e., φj,km ≡ E[r j,m |d j,k =1] . E[r j,m |d j,k =0] (The expectation is taken with respect to productivity ϕ and the unit cost of intermediate goods other than in sector k.) Under the assumption of power law distribution of the unit cost, these sales ratio is entirely driven by the differences in extensive margin (number of matched buyers for intermediate goods sales, entry to sales location for final goods sales), similarly as in Melitz (2003), Chaney (2008). Since this extensive margin responds to the marginal cost shifter with a factor of θ (for both intermediate goods and final goods sales), 13 φj,km is given by R c j,km φj,km = 0 c−γkm θ dGj,km (c)
−γkm θ , χ−θ ζ k w j where Gj,km (c) is the cumulative distribution function for the inverse of the Pareto distribution with upper-bound c j,km . The denominator of this equation is given by: Z c j,km 0 c−γkm θ dGj,km (c) =
−θγkm 1 c j,km 1 − γkm Together with expression (C.12), φj,km
−γkm χθ , = a j,km 1 − γkm (C.13) which corresponds to equation (21) of the main paper. I next derive the steady-state probability that a firm in location j and sector m does not have a supplier in sector k and it is among the δ fraction of firms that can match with a supplier, Φ j,km (which enters in supplier matching rates through the density of buyers (12)). From its definition, Φ j,km = Pr [d j,k = 0, Tω,k |c < ϑ ] = Pr [d j,k = 0|c < ϑ, Tω,k ] Pr [Tω,k |c < ϑ ] where Tω,k indicates the event that firm ω is the δ fraction of firms that can match with a supplier, and d j,k is the probability that the firm has a supplier in sector k. The first term of Φ j,km is given by: Pr [d j,k = 0|c < ϑ, Tω,k ] = = Pr [c < ϑ |d j,k = 0, Tω,k ] Pr [d j,k = 0|Tω,k ] = 0, Tω,k ] Pr [d j,k = 0|Tω,k ] + Pr [c < ϑ|d j,k = 1, Tω,k ] Pr [d j,k = 1|Tω,k ] Pr [c < ϑ |d j,k δ − Λ j,km φj,km Λ j,km + δ − Λ j,km
where the last transformation used the following fact: Z c Pr [c < ϑ |d j,k = 1, Tω,k ] j,km = Pr [c < ϑ |d j,k = 0, Tω,k ] 0 and Pr [d j,k = 1|Tω,k ] = Λ j,km /δ. c ζ k wj !−γkm θ dGj,k (c) = φj,km , 14 The second term of Φ j,km is given by: Pr [Tω,k |c < ϑ ] = 1 + Λ j,km /δ(φj,km − 1) Pr [c < ϑ |Tω,k ] Pr [Tω,k ] = δ, Pr [c < ϑ ] 1 + Λ j,km (φj,km − 1) by noting that Pr [c < ϑ ] = Pr [c < ϑ |d j,k = 1] Pr [d j,k = 1] + Pr [c < ϑ |d j,k = 0] Pr [d j,k = 0] and similarly for Pr [c < ϑ |Tω,k ]. Together, Φ j,km = δ − Λ j,km 1 + Λ j,km (φj,km − 1) (C.14) which corresponds to the expression in footnote 21 in the main paper. C.4 Solving for Supply Capacity Γi,m I now derive the expression for the supply capacity Γi,m . From equation (7), ! Z c I µi,m Hi,m (c) = Γi,m c−θ = dGi,km ( pk ) ∏ γ L,m γ p1,..., pK wi ∏k∈K pk km k∈K ! Z = −θγL,m θ Ai,m wi ∏ k∈K pk I pk −θγkm dGi,km ( pk ) c−θ . (C.15) I (·) is the steady-state distribution of input goods prices in sector k that firms where Gi,km in location i and sector m face. Given the above matching process, I have Z pk pk −θγkm I dGi,km ( pk ) = Λi,km = Λi,km Z c i,km 0 c−γkm θ dGi,km (c) + (1 − Λi,km ) (ζ k wi )−θγkm 1 (ζ w )−θγkm ( ai,km )−γkm + (1 − Λi,km ) (ζ k wi )−θγkm 1 − γkm k i = Λi,km (ζ k wi )−θγkm φi,km + (1 − Λi,km ) (ζ k wi )−θγkm = (ζ k wi )−θγkm {1 + Λi,km (φi,km − 1)} Together, θ Γi,m = $Ai,m wi−θ ∏ {1 + Λi,km (φi,km − 1)} k∈K where $ ≡ ∏k∈K (ζ k )−θγkm . This equation coresponds to (24) of the main paper. (C.16) 15 C.5 Existence and Uniqueness of Equilibrium In this online appendix, I discuss the existence and uniqueness properties of the steadystate equilibrium of the model developed in the main paper. Existence. I first prove the existence of the steady-state equilibrium by applying Brouwer’s fixed point theorem. To apply the Brouwer’s fixed point theorem. I define a mapping in the space of ({wi }, {Li }, {Λi,km }, {1/φi,km }) as follows: 1. Obtain { Ai,m } using equation (6) 2. Obtain {Γi,m } using equation (24) 3. Obtain {πij,m } and {Ωi,m } using equation (9) 4. Obtain {vi,km } using equation (12) 5. Obtain { ai,km } using equation (19) 6. Obtain {Λi,km } using equation (22) 7. Obtain {φi,km } using equation (21) 8. Obtain { Pi } using equation (10) 9. Obtain { Li } using equation (30) I , X F , Y I , Y F , w } using equation (25), (26), (27), (28), (29) under 10. Solve for { Xi,m i i,m i,km i,m ∑i Di = 0. Note that one equation is redundant, so these variables can be only solved up to scale. Therefore, I normalize {wi } such that ∑i wi = 1. From the definition of the steady-state equilibrium, it is clear that the fixed point of this mapping is an equilibrium. Furthermore, this mapping is continuous (because each step of this mapping is a continuous function) in a compact space. Therefore, an equilibrium exists by Brouwer’s fixed point theorem. Uniqueness. Providing a sufficient condition for the uniqueness of the equilibrium in a general model is difficult because the system of equations are highly nonlinear and does not follow the type of equilibrium systems with known sufficient conditions for uniqueness. However, in a special case with a single sector (|K | = 1), no trade deficits (Di = 0 for all i), and if the matching rates are constant within sector pairs (λ = 0 and ν = 1), the system of equations falls into the class of models analyzed by Allen, Arkolakis, and Li (2020). First, combining gravity equations (9), trade balancing conditions (29), and the fact that utility is equalized across locations at U , I have wi−θ L1i −υσ̃ = U −θ ∑ K1,ij w−j θ Lιθj , j∈ N (C.17) 16 θ −(σ−1) where σ̃ = θ (σ−1) , and K2,ij is some constant. Second, by using the free mobility condition (30), I have wi1+θ L1i −θι = U −θ 1−(1−υσ̃) υθ ∑ K2,ij w1j +θ L j . (C.18) j∈ N These two conditions constitute the system of equations of w j and L j . By applying the theorem of Allen, Arkolakis, and Li (2020), one can show that the equilibrium is guaranteed to be unique (up to scale) when υσ̃ < 1 and υι < 1 − υσ̃. Intuitively, these conditions are likely to be satisfied when the dispersion force (commensurate with 1/υ) is sufficiently strong compared to the agglomeration productivity spillovers (ι) and the love-of-variety effects (σ̃). When these conditions are not satisfied, in particular when matching rates are endogenous and exhibits increasing returns to scale (λ + ν > 1), it is possible that there are multiple equilibria. However, the multiple equilibria is not an issue for the estimation in Section 5 because the equilibrium conditions used for the estimation hold for any equilibria. Furthermore, I find that the counterfactual simulation results in Section 6 are not sensitive to the initial values of the iteratition algorithm, suggesting that the presence of multiple equilibria is unlikely to be driving my counterfactual results. D Structural Estimation Appendix In this section of the online appendix, I discuss additional details of the estimation procedure and the estimation results. Section D.1 discusses further details about the estimation of matching parameters in Section 5.3. Section D.2 provides additional results and sensitivity analysis. D.1 Additional Details of the Estimation of Matching Parameters As discussed in Section 5.3, I estimate Θ ={λ, ν, η, χ, δ, {ζ k }} in the following indirect estimation procedure. 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 (12), (19), (20), (21), and (22) 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. I then estimate model parameters that minimize the Euclidean distance between these model-predicted impulse responses and the reducedform estimates with actual data in Section 3, as defined in estimating equation (33). Below, I discuss further details of the construction of the impulse responses of unanticipated supplier bankruptcy on the number of newly matched suppliers and on sales 17 growth. I first construct the impulse responses on the number of new suppliers. Following the standard property of the Poisson process, the expected probability that a firm is matched with a new supplier is approximated by1 NewSuppliersi,km,∆ (Θ) ≡ 1 − exp (−vi,km (Θ) ai,km (Θ)∆) , (D.1) where vi,km (Θ) is the matching rate and ai,km (Θ) is the acceptance probability, and ∆ is the year since the shock. Evaluating this expression for each treatment firm f , I obtain the model-predicted number of new suppliers matched in ∆ years since supplier bankruptcy (NewSuppliers f ,∆ (Θ)). I then take the average value for each time interval (∆) to obtain the average responses of new supplier matching after ∆ years. I next obtain the heterogeneous impulse responses by supplier and buyer density by estimating the following regression equation: NewSuppliers f ,∆ (Θ) = β ∆ (Θ) + γ∆ (Θ) log SupplierDensity f + δ∆ (Θ) log BuyerDensity f + e f ,∆ , (D.2) where log SupplierDensity f and log BuyerDensity f are the empirical proxies for supplier and buyer density as defined in Section 3. This regression corresponds to the differencein-difference regression with actual data (3). Lastly, I construct the impulse response of supplier separation on firm’s sales in model. In the model, when a firm forms a new supplier relationship, the firms’ sales increases by a factor φi,km (Θ) (equation 21). Since the probability that treatment firms have a supplier in sector k after ∆ periods is approximately 1 − NewSuppliersi,km,∆ (Θ) less than that of the control firms, the change of sales after supplier bankruptcy is given by ∆ log Salesi,km,∆ = φi,km (Θ) × ( NewSuppliersi,km,∆ (Θ) − 1) , (D.3) where ( NewSuppliersi,km,∆ (Θ) − 1) is the model-predicted differences in probability that treatment firms have a supplier in sector k compared to the control group. Evaluating this expression for each treatment firm f , I obtain the model-predicted sales reduction in ∆ years since supplier bankruptcy. D.2 Sensitivity Analysis Table D.1 presents the matching technology parameter estimates in Section 5.3 under different values for calibrated parameters. Column (1) reports the baseline point estimates as presented in Table 4. Column (2) increases discount rate ξ to 0.5 instead of 0.05 in the baseline. Column (3) and (4) set 1 This is an approximation because it does not take into account of the possibility that a firm forms a new supplier relationship, becomes separated again, and then rematches one more time. However, for a sufficiently short interval of ∆, the probability of such an event is small. 18 bargaining share β as 0 and 1 instead of 0.5. The estimated parameters are not sensitive to the parameter values of ξ and β. This is because these parameters affect the equilibrium matching outcomes only through the acceptance rates (equation 19), yet in my setting acceptance rates (a j,km ) are close to one in most sectors and locations.2 (Under my estimated parameters, less than one percent of sector-location pairs have acceptance rates ai,km strictly less than one, due to relatively small input coefficient γkm and relatively large iceberg cost of in-house production for intermediate goods χ > 1.) Column (5) sets elasticity of substitution σ as 3 instead of 5, and estimation results are stable. Column (6) sets shape parameter for productivity distribution θ as 6 instead of 4.3. The estimation results are stable except for χ, which is estimated slightly smaller. This is because the sales decrease of supplier separation depends on χθ as evident from equation (21), hence a higher value of θ implies a smaller value of χ. Lastly, Column (7) estimates matching parameters using restricted set of treatment firms whose establishments are located within the same prefecture. The estimates are robust, consistent with the fact that the reduced-form difference-in-difference regressions are robust to this treatment in Column 3 of Table 2. Table D.1: Sensitivity of Matching Technology Estimation to Calibrated Parameters Parameter (1) Baseline λ 0.622 ν 0.974 η 0.045 χ 1.07 δ 0.147 (2) ξ = 0.5 (3) β = 0 0.624 0.623 0.973 0.972 0.045 0.045 1.07 1.07 0.145 0.145 (4) β = 1 0.624 0.986 0.045 1.071 0.143 (5) σ = 3 0.623 0.974 0.045 1.07 0.144 (6) θ = 6 (7) Only firms with local establishments 0.605 0.624 0.994 0.962 0.045 0.045 1.05 1.07 0.143 0.147 Note: Sensitivity of the matching technology parameter estimates in Section 5.3 under different values for calibrated parameters. Columns (1) reports the baseline estimates reported in Table 4. Column (2) sets discount rate ξ as 0.5 instead of 0.05. Column (3) and (4) set bargaining share β as 0 and 1 instead of 0.5. Column (5) sets elasticity of substitution σ as 3 instead of 5. Column (6) sets shape parameter for productivity distribution θ as 6 instead of 4.3. Column (7) estimates matching parameters using restricted set of treatment treatment firms whose establishments are located within the same prefecture. E Counterfactual Simulation Appendix In this section of the online appendix, I provide additional details of the counterfactual simulation in Section 6 of the main paper. Section E.1 details the counterfactual simulation procedure. Section E.2 provides the sensitivity analysis of the estimation results. Section E.3 discusses the transition dynamics of trade cost reduction. E.1 Additional Details of Counterfactual Simulation Procedure 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 2 Recall also that firms are owned by foreigners outside of the economy, hence the bargaining share β does not affect the spatial income distribution. 19 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). Given the values of the endogenous variables in the initial equilibrium ({πij,k , Γi,k , Ωi,k , Li , Di }), the counterfactual equilibrium is determined by the following system of equations in terms of the changes in the endogenous variables ({π̂ij,k , Γ̂i,k , Ω̂i,k , ŵ j , L̂ j , P̂j }) and the values in the new equilibrium 0 0 I 0 , Y F0 , X I 0 , X F0 , v0 ({Yi,km i,k i,k i,k i,km , ai,km , φ j,km }): (i) trade linkages. Γ̂i,m
θ Γ̂i,m τ̂ij,m π̂ij,m =  θ 0 0 Γ̂ τ̂ 0 ∑i ∈ N i ,m i j,m πi0 j,m   0  0 1 + Λi,km φi,km − 1
ιθ   = L̂i ŵi−θ ∏  I k∈K 1 + Λi,km φi,km − 1 Ω̂ j,m =  ∑ Γ̂i0 ,m τ̂i0 j,m (E.1) (E.2) θ πi0 j,m (E.3) i0 ∈ N I0 ( 0 Yi,km = γkm Ψi,km 0I 0 ∑ Yj,km πij,k + 0 0 m∈K 0 0 0 ∑ wi Li πij,k 0 ∑ ∑ j∈ N m∈K F0 0 (E.5) 0 0 I Yj,km πij,k 0 Xi,k = (E.4) j∈ N F Yi,k = α k wi L i I Xi,k = ) 0 ∑ Yj,kF πij,k (E.7) j∈ N ∑   0 0 Xi,kI + Xi,kF = k∈K ∑ 0 0 0 I Yi,km + w i L i − Di (E.8) k,m∈K (ii) population mobility. ŵ j / P̂j L̂ j =
υ ∑` ŵ` / P̂` θ − σ +1 θ ( σ −1) P̂j = L̂ j ∏ k∈K
υ Ω̂ j,k (E.9) L`
− αθk (E.10) 20 (iii) matching in intermediate goods market (for each location i and sector pairs k, m).  λ  0 0 θ Ω ζ w  i,k k i  = η  Zi  0 vi,km 0 ai,km = max   1, χ 0 " θ γkm (σ−1) 1−β 0 0 (σ − 1) γkm θ − (σ − 1) γkm ξ + ρkm  0 −γkm χθ a 1 − γkm j,km 0 φj,km = 0 0 0 1 − Λ j,km + φj,km Λ j,km 0 Λ j,km = δ 0 # (σ−1θ)γ  km  (E.12)  (E.13) (E.14) 0 v j,km a j,km 0 (E.11) 0 0 φj,km Λ j,km 0 ! ν −1 Zi vi,km ai,km  Ψ j,km = 0 Φi,km Γi,m ϑθ 0 (E.15) v j,km a j,km + ρkm 0 δ − Λ j,km 0 Φ j,km = 0 0 1 + Λ j,km (φj,km − 1) (E.16) Calibration of initial equilibrium. In addition to the estimated parameters, I need to construct the proxies for the initial equilibrium for {πij,k , Γi,k , Ωi,k , Li , Di }. To construct {πij,k , Γi,k , Ωi,k }, I use the estimated gravity equations of trade flows in Section 5.2 of the main paper. Using the estimated Γi,m and Ωi,m and κm , I construct πij,k using the same gravity equations. I use this predicted πij,k as the initial equilibrium, instead of the observed πij,k , to deal with the presence of zero trade flows due to the sparsity of firm-tofirm trade data (Dingel and Tintelnot 2020). I obtain Li from the population distribution in each prefecture using the official statistics in Japan. Lastly, I construct Di using the trade (in)balance condition (equation 29). Rewriting this equation, I have   I F I F (E.17) Di = ∑ Yi,km + ∑ Yi,k − ∑ Xi,k + Xi,k k,m∈K k∈K k∈K I , Y F , X I , X F } by solving the set of equations (25), (26), (27), and where I obtain {Yi,km i,k i,k i,k (28), given the observed data of wi and Li (from official statistics) and the predicted trade shares πij,m . 21 E.2 Additional Results and Sensitivity Analysis Table E.1 presents the changes of population density in the counterfactual simulations to shut down agglomeration spillovers (Section 6.1). The table shows the regression coefficients of the counterfactual log population density on the log of observed population density from three counterfactual simulations discussed in Section 6.1 in Columns (2)-(4). Shutting down IRS in matching reduces the concentration of population by 7 percent (Column 2). Shutting down the population-productivity spillovers reduces the concentration of population by 14 percent (Column 3). Lastly, omitting both types of agglomeration spillovers decrease the concentration of population by 19 percent (Column 4). These changes are smaller in absolute terms compared to the counterfactual changes in wages (Table 6). However, the relative magnitudes of Column (2) to Column (3) and (4) remain similar to Table 6. Therefore, IRS in matching is a quantitatively relevant component of overall agglomeration spillovers in terms of its contribution to the observed population density, similarly to the contribution to the observed wages as discussed in Table 6. Table E.1: Agglomeration Forces and Population Density Baseline log Observed Population Density C.I. from Bootstrap Parameter Estimates Percentage Difference from Baseline (%) Observations Adjusted R2 Dependent Variable: log Counterfactual Population Density Shut Down Shut Down Shut Down IRS in Matching Function Population Productivity Spillovers Both Agglomeration Spillovers (ι = 0) (λ = ν − 1 = 0 and ι = 0) (λ = ν − 1 = 0) (1) (2) (3) (4) 1.00∗∗∗ (0.00) 0.93∗∗∗ (0.01) 0.87∗∗∗ (0.01) 0.81∗∗∗ (0.01) 0 47 1.00 [0.92, 0.95] -7 47 0.99 [0.87, 0.87] -14 47 0.99 [0.8, 0.83] -19 47 0.99 Note: Results of the counterfactual simulations to shut down agglomeration forces. Each column reports the regression coefficient of the logarithm of the counterfactual population density on those of the observed data. Columns (1) takes the observed population density as the dependent variable, hence mechanically the regression coefficient is one and R2 is one. and Columns (2)-(4) correspond to the counterfactual simulations of shutting down the increasing returns to scale (IRC) in matching technology such that λ = ν − 1 = 0 (Column 2), shutting down productivity spillovers from local population density such that ι = 0 (Column 3), and shutting down both types of agglomeration forces, such that λ = ν − 1 = 0 and ι = 0 (Column 4). *p<0.1; **p<0.05; ***p<0.01. Table E.2 shows the sensitivity of the results of the counterfactual simulations to shut down the agglomeration spillovers to the value of calibrated parameters. Panel (A) shows the regression coefficients of the log of counterfactual wages on the log of counterfactual population density (corresponding to Table 6), and Panel (B) shows the regression coefficients of the log of counterfactual population density on the log of observed population density (Table E.1). Column (1) is my baseline results. The empirical results are virtually identical for the values of discount rate (ξ, Column 2) and bargaining share (β, Columns 3 and 4), consistent with the findings that these parameter values do not affect the estimation results of matching parameters (Table D.1). 22 When elasticity of substitution (σ) is smaller (in Column 5) or the elasticity of migration (υ) is higher (in Column 7), wages respond less (in Panel A) and population responds more (in Panel B). This is because population movement is more elastic to real wage change under these alternative parameterization. Lastly, the counterfactual responses of wages and population are both greater if the shape parameter for productivity (θ) is greater (Column 6). This is because the supplier capacity term (Γi,m ) is more elastic to the changes in productivity (equation 24). Despite these differences in Column (5)-(7) in absolute values, the relative magnitudes of the three counterfactuals of shutting down agglomeration forces (across different rows) remain stable in Column (5)-(7). Table E.2: Sensitivity Analysis of Counterfactual Simulations to Shut Down Agglomeration Spillovers (A) Wages Specification (1) Baseline (2) ξ = 0.5 (3) β = 0 Baseline 0.076 0.076 0.076 (λ = ν − 1 = 0) 0.045 0.045 0.045 (ι = 0) 0.011 0.011 0.011 (λ = ν − 1 = 0 and ι = 0) -0.013 -0.013 -0.013 (4) β = 1 (5) σ = 3 0.076 0.076 0.045 0.05 0.011 0.025 -0.013 0.008 (6) θ = 6 0.076 0.036 0.001 -0.025 (7) υ = 4 0.076 0.05 0.025 0.008 (6) θ = 6 1 0.899 0.826 0.756 (7) υ = 4 1 0.878 0.786 0.699 (B) log Population Density Specification (1) Baseline (2) ξ = 0.5 (3) β = 0 Baseline 1 1 1 (λ = ν − 1 = 0) 0.932 0.933 0.933 (ι = 0) 0.869 0.868 0.868 (λ = ν − 1 = 0 and ι = 0) 0.815 0.815 0.815 (4) β = 1 (5) σ = 3 1 1 0.932 0.879 0.869 0.787 0.815 0.7 Note: Results of the counterfactual simulations for trade cost reduction presented in Section 6.2. For each type of sensitivity analysis in each column, each entry reports the regression coefficients of the counterfactual wages on the counterfactual population density (Panel A) and the log of counterfactual population density on the observed population density (Panel B), for each type of counterfactual indicated in the first column. Columns (1) reports the baseline estimates reported in Table 4 for Panel (A) and Table E.1 for Panel (B). Column (2) sets discount rate ξ as 0.5 instead of 0.05. Column (3) and (4) set bargaining share β as 0 and 1 instead of 0.5. Column (5) sets elasticity of substitution σ as 3 instead of 5. Column (6) sets shape parameter for productivity distribution θ as 6 instead of 4.3. Column (7) sets the elasticity of migration with respect to real wages υ as 4 instead of 2. Table E.3 shows the same set of sensitivity analyses for the counterfactuals of reducing trade cost. Column (1) is my baseline results from Table 7, and Column (2)-(7) is the same set of sensitivity analysis as above. The predicted welfare gains are stable for the value of discount rate (ξ, Column 2), bargaining share (β, Column 3 and 4), elasticity of substitution (σ, Column 5), and the elasticity of migration (υ, Column 7). The estimates of welfare gains are somewhat smaller with a greater productivity dispersion parameter (θ, Column 6) because of a greater responses in supplier capacity Γi,k (equation 24). Despite these differences, the relative magnitudes of the three counterfactuals (Rows 2-4) remain stable across the board. 23 Table E.3: Sensitivity Analysis of Welfare Gains from Trade Cost Reduction Specification (1) Baseline (2) ξ = 0.5 (3) β = 0 Baseline 6.19 6.19 6.18 (λ = ν − 1 = 0) 4.53 4.55 4.55 (ι = 0) 6.17 6.17 6.17 (λ = ν − 1 = 0 and ι = 0) 4.53 4.55 4.55 (4) β = 1 (5) σ = 3 6.2 6.36 4.56 4.68 6.18 6.33 4.56 4.67 (6) θ = 6 4.72 2.8 4.72 2.81 (7) υ = 4 6.34 4.66 6.31 4.64 Note: Results of the counterfactual simulations for trade cost reduction presented in Section 6.2. For each type of sensitivity analysis in each column, each entry reports the percentage point increase in expected utility from the trade cost reduction under four different assumptions about the agglomeration spillovers indicated in the first column. Columns (1) reports the baseline estimates reported in Table 4. Column (2) sets discount rate ξ as 0.5 instead of 0.05. Column (3) and (4) set bargaining share β as 0 and 1 instead of 0.5. Column (5) sets elasticity of substitution σ as 3 instead of 5. Column (6) sets shape parameter for productivity distribution θ as 6 instead of 4.3. Column (7) sets the elasticity of migration with respect to real wages υ as 4 instead of 2. E.3 Transition Dynamics of Trade Cost Change In this section of the appendix, I discuss the transition dynamics of the model discussed in Section 4. To do so, I focus on a special case of the model where the discount rate ξ → ∞. This corresponds to a special case of the model such that firms always accept a match (ai,km = 1). This special case closely approximates my estimates of baseline model, because my baseline estimates imply that ai,km are close to one for most sector and locations.3 Under this assumption, one can simply simulate the transition dynamics by forward simulation without having to solve for the dynamic changes in option values. More specifically, the law of motion of Λ j,km for a small time interval ∆ is given by Λi,km (t + ∆) = (1 − exp (−∆vi,km )) (δ − Λi,km (t)) + exp (−∆ρkm ) Λi,km (t). (E.18) I assume that all other equilibrium variables flexibly adjust in each period given Λi,km (t). Therefore, the equilibrium allocation is given by the set of equations (E.1)-(E.16) for each period, except that equation (E.15) is replaced by equation (E.18). As an application, I undertake the same counterfactual simulation of cross-regional trade cost change as in Section 6.2 of the main paper. More specifically, I calibrate the baseline model in period t = 0 using the observed equilibrium assuming that it is in the steady state. I then assume that there is a sudden increase of the trade cost that is  κ̃m nohighway nohighway proportional to τ̂ij,k = Tij /Tij , where Tij is the travel cost without using highway networks. Figure E.1 plots the transition dynamics of the expected per-period utility under the same four scenarios discussed in Section 6.2 (incorporate both types of agglomeration spillovers, omit IRS in matching function, omit population-productivity spillovers, omit both types of agglomeration spillovers). noted in Section D.1, less than one percent of sector-location pairs acceptance rates ai,km strictly less than one. Relatedly, the value of ξ has virtually no effects on my counterfactual simulations in Table E.2 and E.3. 3 As 24 Consistent with the findings in Table 7, whether to incorporate population-productivity spillovers or not has a negligible effect. (“Baseline” is on top of “No population-productivity spillovers”, and “No IRS in matching function” is on top of “Shut down both agglomeration spillovers”.) By comparing “Baseline” and “No IRS in matching function,” several things stand out. Right after the shock (t = 1), per-period utilities are relatively similar in both scenarios. When IRS in matching function is incorporated (“Baseline”), per-period utility gradually decreases to the level of steady state. This is because supplier matching (Λ j,km (t)) slowly adjust to the new steady state level. While most of the welfare gains materialize within 10 years, there is still an adjustment in longer time interval. On the other hand, when IRS in matching function is omitted (λ = ν − 1 = 0), per-period utility remains unchanged over time. In this case, matching rates Λ j,km (t) are exogenous and hence does not depend on time t. The analysis of this section suggests that it takes time for the welfare gains from trade cost changes to fully materialize due to a gradual adjustment of supplier matching. Figure E.1: Transition Dynamics of Per-period Utility after Trade Cost Increase Note: This figure plots the percentage point change of per-period utility (relative to the baseline period) due to a sudden increase in trade cost in period. See Appendix Section E.3 for the counterfactual simulation procedure.