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. B Deriving Equation (??) for Empirical Specification In this Appendix, I derive the equation (??) by first-order approximation of the probability that a buyer f in sector m and location j who are looking for supplier in sector k to 3 match with a supplier, N f jkm∆ . From a standard property of the Poisson rates, N f jkm∆ is approximated by:
N f jkm∆ ≈ 1 − exp(− M S j,k , Bj,m /Bj,m × ∆), (B.1)

λS
λ B −1 where M S j,k , Bj,m /Bj,m = η S j,k Bj,m is the Poisson rate of matching with a supplier. The first-order approximation of this equation around ∆ ≈ 0 yields: N f jkm∆ ≈ η S j,k
λS Bj,m
λ B −1 ∆. (B.2) Further taking the first-order approximation of this equation around log S j,k ≈ log Bj,m ≈ 0 yields:     N f jkm∆ ≈ η∆ × 1 + λS log S j,k + λ B − 1 log Bj,m , (B.3) which coincides with equation (??). 4 C 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. Tables C.1-C.3 reports additional robustness results for the average effects of the supplier bankruptcy presented in Section 3.1. Table C.1 shows that these supplier bankruptcies do not significantly affect the sales of the treatment firms’ other existing suppliers, indicating that treatment firms primarily respond by rematching with new suppliers instead of simply substituting from existing suppliers. In Table ??, I show that the newly matched suppliers are significantly more likely to belong to the same industry as bankrupt suppliers, consistent with the interpretation that treatment firms are in search of an alternative supplier. Table C.2 shows that there are no heterogeneous effects by firm size, suggesting that these matching frictions are relevant for both small and large firms. Table C.3 presents the effects on other firm-level outcome variables. Table C.1: Average Impacts of Supplier Bankruptcy on Other Connected Suppliers and on Competitors Fraction of Continued Supplier Relationships Dependent Variable: Number of New Suppliers by Other Firms in Same Industry log Sales of Connected Suppliers Average Sales by Other Firms in Same Industry (1) (2) (3) (4) −0.05 (0.05) −0.01 (0.01) 0.003 (0.02) 0.01 (0.02) (0.00) (0.00) (0.00) (0.00) Trt x 1[t - BankruptYear = 0 or 1] 0.02 (0.03) 0.01 (0.01) 0.004 (0.03) 0.02 (0.04) Trt x 1[t - BankruptYear = 2 or 3] −0.07 (0.04) 0.03 (0.02) 0.03 (0.04) 0.06 (0.08) Observations 76,069 66,802 66,376 65,453 Trt x 1[t - BankruptYear = -2 or -3] Trt x 1[t - BankruptYear = -1] 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. 5 Table C.2: Heterogeneous Impacts of Supplier Bankruptcy by 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: 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 C.3: Impacts of Supplier Bankruptcy on Additional Firm Outcomes Employment Growth (Arc-Elasticity) Trt x 1[t - BankruptYear = -2 or -3] Profit / Sales Prob. of Continued Relationship with Connected Suppliers Sales Growth (Arc-Elasticity) of Connected Suppliers Number of New Suppliers of Competitors Sales Growth (Arc-Elasticity) of Competitors (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) Trt x 1[t - BankruptYear = -1] (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 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 C.4-C.7 report the robustness to the heterogeneous effects with supplier density reported in Section 3.2. Table C.4 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 C.5), splitting samples to manufacturing and non-manufacturing supplier bankruptcies (Table C.6), and including supplier-reported supplier-to-buyer-linkages when constructing the outcome variables (Table C.7). Table C.4: Average Impact Heterogeneity by Supplier Density: Robustness Number of New Suppliers (1) (2) (3) (4) (5) (6) 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) 0.14∗∗∗ (0.04) 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) 0.19∗∗∗ (0.06) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.14∗∗∗ (0.04) 0.14∗∗∗ (0.05) 0.12∗∗∗ (0.04) 0.15∗∗∗ (0.04) 0.15∗∗∗ (0.04) 0.17∗∗∗ (0.05) 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) 0.17∗∗ (0.07) 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) 0.03 (0.06) 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) 0.01 (0.09) Exclude Exit Firms Exclude Bankruptcy in 2009 Exclude Tokyo Exclude Outdated Accounting Year No Outside-Prefecture Establishments IV: Seller Density in Birth Prefecture X X X 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 8,311 405 174 10,933 Observations 83,548 75,758 54,442 81,582 60,554 79,323 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: 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) 7 Table C.5: Impact Heterogeneity by Supplier Density: Alternative Definitions of 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) 0.16∗∗∗ (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) −0.03 (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: 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. 8 Table C.6: 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] 0.09 (0.06) 0.17∗∗∗ (0.05) Trt x 1[t - BankruptYear = 2 or 3] 0.18∗∗ (0.09) 0.20∗∗∗ (0.07) (4) Trt x 1[t - BankruptYear = 0 or 1] x log Supplier Density 0.05 (0.07) 0.28∗∗∗ (0.08) Trt x 1[t - BankruptYear = 2 or 3] x log Supplier Density 0.07 (0.12) 0.40∗∗∗ (0.09) Trt x 1[t - BankruptYear = 0 or 1] x log Buyer Density −0.11∗ (0.06) −0.05 (0.05) Trt x 1[t - BankruptYear = 2 or 3] x log Buyer Density −0.31∗∗∗ (0.11) −0.11 (0.07) 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 Number of Treatment Firms Number of Bankrupting Suppliers Number of Control Firms 166 54 1,227 166 54 1,227 267 127 10,701 267 127 10,701 Observations 9,698 9,621 76,253 75,774 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. Table C.7: Impact Heterogeneity by Supplier Density: Include 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 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. 9 Table C.8: Impact Heterogeneity by Supplier Density: Using Supplier Bankruptcy from Management Failures 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) 0.03 (0.02) 0.004 (0.02) −0.02 (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) 0.02 (0.02) 0.03 (0.02) 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) 0.07∗∗∗ (0.02) 0.05∗∗ (0.02) 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) 0.11∗∗∗ (0.03) 0.09∗∗∗ (0.03) 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) 0.003 (0.03) −0.01 (0.03) −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 0.08∗ (0.04) 0.02 (0.04) 0.01 (0.04) −0.02 (0.04) 0.04 (0.05) 0.01 (0.04) −0.01 (0.04) 0.05 (0.04) 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 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 1688 337 32,840 239,282 1688 337 32,840 237,106 1688 337 32,840 237,106 1221 258 25,428 180,800 1688 337 31,060 133,956 1688 337 18,797 180,347 1316 301 25,062 224,159 1035 275 18,558 134,727 Note: A version of Table 2 except for using supplier bankruptcies due to management failures. Table C.9 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 , (C.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. 10 Table C.9: IV Impacts of the Number of Matched Suppliers on Firm Sales Dependent variable: Sales Growth (Arc Elasticity) Number of Suppliers (1) (2) 0.07∗∗ (0.03) 0.10∗∗ (0.05) Number of Suppliers x log Supplier Density First Stage F-Statistics Observations 0.02 (0.02) 95.7 54.8 62,488 62,488 Note: The results of the IV regression specified in equation (C.1). Standard errors are clustered at the firm level. *p<0.1; **p<0.05; ***p<0.01. 11 D Incorporate Acceptance Decision As mentioned above, there are two possible ways to source input goods: producing them in-house using local labor or forming a direct relationship with a supplier. For the first option, I assume that all firms can produce one unit of sector-k intermediate goods using ζ k unit of local labor. Furthermore, in order to produce in-house intermediate goods, firms have to pay χ − 1(≥ 0) additional unit of transaction cost per unit of output. Below, I first discuss the matching process and its technology. I then derive and solve for Bellman equations for the decision of forming a relationship conditional on a match. Lastly, I derive the steady-state match probabilities. Matching Technology. I first assume that 0 ≤ δ ≤ 1 fraction of firms ever matches with a supplier. When I fit the model to data, this parameter rationalizes the relative magnitudes of the matching rates, separation rates, and the probability of matching in a steady-state. If a potential input buyer (i.e., δ fraction of firms producing in location j) does not currently have a supplier, it stochastically draws a potential supplier. Following the approach of labor search and matching literature (Diamond 1982, Pissarides 1985, Mortensen 1986), I assume that the matching between suppliers and buyers occurs through matching technology summarized by a matching function. The total matches created in one unit of geographic area by buyers producing in location j with suppliers (that can be producing anywhere) is given by

λS
λ B M S j,k , Bj,km = η S j,k Bj,km (D.1) where S j,k is the density of suppliers in location j whose unit cost (net of iceberg trade cost) is below ζ k w j , and Bj,km is the density of buyers in location i whose unit cost is below some sufficiently large threshold ϑ and who is not currently matched with a supplier in sector k. This matching function implies that the Poisson rate for buyers to match with a
supplier is M S j,k , Bj,km /Bj,km . Similarly, the Poisson rate for suppliers to match with
a buyer is M S j,k , Bj,km /S j,k . I allow the matching technology to have a scale effect; i.e., λS + λ B can be different from one. If λS + λ B = 1, the matching function exhibits constant returns to scale. If λS + λ B > 1, the matching function exhibits increasing returns to scale. The measure of suppliers with unit cost below ζ k w j (net of trade cost) in location j  −θ
θ
θ ζ k w j . The measure of buyers whose is given by Ω j,k ζ k w j = ∑i0 ∈ N Γi0 ,k τi0 j,k unit cost for production is below ϑ and who is looking for a supplier is Φ j,km Γ j,m ϑθ , where Φ j,km is the steady-state probability that a firm in location j and sector m has a supplier 12 in sector k.1 Together, the Poisson rate of matching with a supplier v j,km is given by: v j,km = 1 Bj,km
M S j,k , Bj,km = η Ω j,k ζ k w j Zj
θ !λS Φ j,km Γ j,m ϑθ Zj ! λ B −1 , (D.2) where Zj is again the geographic area of location j. Conditional on a match, the two firms jointly determine if they form a long-term relationship. The buyer cannot match with another supplier until the relationship is exogenously destroyed at the Poisson rate ρkm . This creates an option value for buyers to forego the supplier that they draw and wait for a better supplier. On the other hand, suppliers can match with other buyers simultaneously. Therefore, suppliers do not have an option value for waiting, and they always form a relationship whenever the profit is positive. Nash Bargaining and Bellman Equations for Firm-to-Firm Matching. Following the standard search and matching framework, the joint surplus of a match is split between buyers and suppliers according to Nash bargaining. Since Nash bargaining is efficient, intermediate goods are traded at supplier’s unit cost to avoid double marginalization. In addition, there is a lump-sum transfer from the buyer to the supplier in each period based on Nash bargaining. To avoid that this bargaining involves infinitely many third firms indirectly connected through the dynamic firm-to-firm trade network, I assume that the two firms bargain only over the profit from the buyer’s final goods sales. Furthermore, I assume that the relationship-specific unit cost of the supplier is fixed at the beginning of the relationship (by assuming that the supplier can keep receiving the intermediate goods for this specific relationship from its own suppliers until the relationship ends). Under these two assumptions, the bargaining problem can be solved for each pair of suppliers and buyers, independent of the actions of other firms in the economy. I now define the Bellman equations. I first consider the value functions of the buyer. Denote the value of the buyer ω that has an ongoing relationship with a supplier in sector B ( c ), and the value of the firm that does not k at unit cost c (net of iceberg cost) by Vωt,k B . The Bellman equation for the have an ongoing supplier relationship in sector k by Uωt,k buyers when the firm is matched with a supplier is given by   B B B F B (c) − δωt,k (c) − ρkm Vωt,k (c) − Uωt,k + V̇ωt,k ( c ), (D.3) ξVωt,k (c) = Πωt,k F ( c ) is ω’s final goods profit when the unit where ξ is the discount rate of the firm; Πωt,k cost of intermediate goods in sector k is c; δωt,k (c) is the lump-sum transfer to the supplier in sector k; ρkm is again the Poisson rate at which the relationship is destroyed; and     Appendix ??, I analytically derive that Φ j,km = δ − Λ j,km / 1 + Λ j,km (φj,km − 1) , where and φj,km are given by equations (D.11) and (D.12) below. 1 In Λ j,km 13 B ( c ) indicates the time derivative of the value function V B ( c ). As discussed above, V̇ωt,k ωt,k F ( c )), and hence I omit Nash bargaining involves only the buyer’s final goods profit (Πωt,k the profit from intermediate goods sales from the value function.2 The Bellman equation for the buyers in location i when they are not matched with a supplier in sector k is given by B ξUωt,k =χ 1− σ F Πωt,k (ζ k wi ) + vi,km aω,k Z c  ω,k 0  B B B Vωt,k (c) − Uωt,k dGω,k (c) + U̇ωt,k , (D.4) F ( ζ w ) is the where χ is the iceberg cost of producing intermediate goods in-house; Πωt,k k i profit from final goods sales when the unit cost of intermediate goods in sector k is ζ k wi ; vi,km is the Poisson rate of matching with a supplier; aω,k is the probability that the match is accepted (derived below); cω,k is the threshold of the supplier’s unit cost below which the buyer decides to form a relationship; Gω,k (·) is the cumulative distribution function of the suppliers’ unit cost conditional on match acceptance (such that Gω,k (0) = 0 and B B .3 indicates time derivatives of Uωt,k Gω,k (cω,k ) = 1); and U̇ωt,k Turning to the supplier’s side, the value of the supplier with unit cost c from the relationship with buyer ω is given by   S S S S ξVωt,k (c) = δωt,k (c) − ρkm Vωt,k (c) − Uωt,k + V̇ωt,k ( c ), (D.5) where δωt,k (c) is again the lump-sum transfer from the buyer to the supplier, and ρkm is the Poisson rate at which the relationship is destroyed. Since the supplier does not have an option value of waiting (because it can simultaneously form a relationship with other buyers), I normalize the value of the supplier when it is not matched with ω by zero, i.e.,4 S Uωt,k = 0. (D.6)  S (c) − U S Vωt,k ωt,k  Because of the Nash bargaining, the joint surplus Jωt,k (c) = +  B (c) − U B Vωt,k ωt,k is split between the supplier and buyer with a weight of 1 − β and β, where β denotes the bargaining power of the buyer. Therefore:  S S Vωt,k (c) − Uωt,k = (1 − β) Jωt,k (c), (D.7) B B Vωt,k (c) − Uωt,k = βJωt,k (c). B ( c ) and U B normalize Vωt,k ωt,k by omitting the payment to suppliers in other sectors (δωt,k̃ ( c ) for B ( c ) and U B . k̃ 6= k), which equally affects both Vωt,k ωt,k 3 It is straightforward to show that there exists such a threshold. The per-period profit of the buyer is strictly decreasing in c. Therefore, the joint surplus is also strictly decreasing in c, which guarantees the existence of cω,k . 4 Similarly to the normalization of the buyer’s value functions, I normalize V S ( c ) and U S ωt,k ωt,k by omitS ( c ) and U S . ting the profits from other buyers and final consumers, which equally affects Vωt,k ωt,k 2I 14 I now solve these Bellman equations to derive the decision of whether to form a relationship conditional on a match. Because suppliers have no option value, this problem comes down to the buyer’s decision of accepting the match. Buyers decide to form a relationship if the supplier’s unit cost is sufficiently low. I assume that buyers set cω,k to maximize the expected value of unmatched state. This implies that cω,k is determined so that firms are in expectation indifferent between accepting or rejecting a match when the supplier’s unit cost is cω,k : B B E[Vωt,k (cω,k )] = E[Uωt,k ], (D.8) where the expectation is taken with respect to intermediate goods cost other than input sector k. In Appendix D.1, I show that cω,k and aω,k depend only on firms’ location i, sector m, and the supplier sector k, such that cω,k = ci,km and aω,k = ai,km when fixed cost for final goods sales is negligible ( f j,k → 0), and they are analytically solved to give the following expressions: (   θ ) θ (σ−1)γkm v a σ − 1 γ ( ) i,km i,km km ai,km = min 1, χ γkm 1 − β , (D.9) ξ + ρkm θ − (σ − 1) γkm and 1 θ ci,km = ζ k wi ai,km . (D.10) Note that the expression of ai,km in equation (D.9) is implicit because ai,km also appears in both sides of the equations. Because the right-hand side is decreasing in ai,km , equation (D.9) uniquely determines ai,km . These equations are intuitive. In equation (D.9), ai,km is increasing in vi,km because faster matching rates imply that the option value of waiting is higher. ai,km is decreasing in ρkm and ξ because an anticipation of faster separation ρkm and a higher discount rate ξ discourage firms to wait longer. ai,km is decreasing in β because a higher value of β implies a greater option value of buyers to wait for a better supplier. Lastly, ai,km is decreasing in χ because a higher transaction cost for in-house production increases the opportunity cost of waiting. In equation (D.10), ci,km is increasing in ai,km because accepting at higher probability (higher ai,km ) means that firms accept more costly suppliers (higher ci,km ). When ai,km = 1 (always accept the supplier), ci,km = ζ k wi , i.e., buyers accept suppliers as long as the supplier’s unit cost is equal to or less than the in-house production of intermediate goods. Value of a Match and Steady-State Match Probability. Given the solution to the Bellman equations, I now define several key moments that are useful to characterize the aggregate steady-state equilibrium. I first derive the ratio of the expected firm sales r j,m of firms in location j and industry m conditional on having a supplier d j,k = 1 in sector k 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 15 taken with respect to productivity ϕ and the unit cost of intermediate goods other than in sector k.) In Appendix ??, I show that φj,km is explicitly solved as: φj,km =
−γkm χθ a j,km . 1 − γkm (D.11) Intuitively, when a j,km is high, firms are more selective about the supplier’s productivity. Therefore, each firm faces a greater increase in sales when they decide to form a relationship with a supplier. Similarly, a higher in-house production iceberg cost (χ) leads to higher sales by firms when they are matched with a supplier. Given a j,km , the steady-state probability that a firm in location j and sector m is forming a relationship with a supplier in sector k is given by: Λ j,km = δ v j,km a j,km , v j,km a j,km + ρkm (D.12) where δ is again the fraction of firms that can match with an external supplier, v j,k is the matching rate with a new supplier, and ρkm is the exogenous separation rate. D.1 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 D,   B F B B B ξVωt,k (c) = Πωt,k (c) − δωt,k (c) − ρkm Vωt,k (c) − Uωt,k + V̇ωt,k ( c ), (D.13) B ξUωt,k =χ 1− σ F Πωt,k (ζ k wi ) + vi,km aω,k S ξVωt,k (c) = δωt,k (c) − ρkm Z c  ω,k  0 B B Vωt,k (c) − Uωt,k S S Vωt,k (c) − Uωt,k   B dGi (c) + U̇ωt,k , S + V̇ωt,k ( c ), (D.14) (D.15) S Uωt,k = 0, (D.16) S S Vωt,k (c) − Uωt,k = (1 − β) Jωt,k (c), (D.17) B B (c) − Uωt,k = βJωt,k (c). Vωt,k 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. 16 I now solve these set of equations. Using equation (D.17),     S S B B ξ Jωt,k (c) =ξ Vωt,k (c) − Uωt,k + ξ Vωt,k (c) − Uωt,k     F F B B S S = Πωt,k (c) − χ1−σ 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 1− σ F (ξ + ρkm ) Jωt,k (c) = Πωt,k (c) − χ Πωt,k (ζ k wi ) Z c   ω,k B B − vi,km aω,k Vωt,k (c) − Uωt,k dGω,k (c) + J̇ωt,k (c). 0 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 Jωt,k (c) − Jωt,k (cω,k ) = −c + J̇ωt,k (c) − J̇ωt,k (cω,k ) (cω,k ) ξ + ρkm Using this equation, equation (D.14) is rewritten as: B ξUωt,k =χ 1− 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 0 cω,k −c (1−σ)γkm  dGω,k (c) 0 = χ1−σ Ki,m (ζ k wi )γkm pωt,−k + Z c  ω,k 0
1− σ − vi,km aω,k β Ki,m ( pωt,−k )1−σ (1−σ)γkm (1 − σ) γkm c ξ + ρkm (1 − σ) γkm + θ  B J̇ωt,k (c) − J̇ωt,k (cω,k ) dGω,k (c) + U̇ωt,k (D.18) 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 17 accepting or rejecting a match: B B E[Vωt,k (cω,k )] = E[Uωt,k ], (D.19) where the expectation is taken with respect to intermediate goods cost other than input sector k. S (c Now, equations (D.17) and (D.19) together imply that E[ Jωt,k (cω,k )] = 0 and E[Vωt,k ω,k )] = E[U S ] = 0. Furthermore, E[U˙ B ] = E[V̇ B (c)] = E[ J̇ B (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 (D.13) at c = cω,k and taking expectation yields: (1−σ)γkm B B S F (cω,k ) + Vωt,k (cω,k )] = E[Πωt,k (cω,k )] = Ki,m cω,k ξE[Vωt,k (cω,k )] = ξE[Vωt,k E[( pωt,−k )1−σ ], (D.20) B ] from equation (D.19). Plugging this expression in the left which coincides with E[Uωt,k hand side of equation (D.18), (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 ⇐⇒ cω,k km 1 − β i,km ω,k = χ1−σ (ζ k wi )(1−σ)γkm ξ + ρkm θ − (σ − 1) γkm θ    θ θ (σ−1)γkm cω,k vi,km aω,k (σ − 1) γkm γkm =χ ⇐⇒ 1−β (D.21) ζ k wi ξ + ρkm θ − (σ − 1) γkm − vi,km aω,k β Now, noting that aω,k =  ( aω,k = min 1, χ cω,k ζ k wi θ θ γkm  and aω,k ≤ 1, vi,km aω,k (σ − 1) γkm 1−β ξ + ρkm θ − (σ − 1) γkm  θ (σ−1)γkm ) , (D.22) and 1 θ cω,k = ζwi aω,k . (D.23) 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 18 equations (D.22) and (D.23) of the main paper. 19 E Model Appendix In this section of the Online Appendix, I discuss additional details of the model developed in Section 4. Section E.3-E.2 discuss the mathematical derivations. Section F discusses the existence and uniqueness property of the steady-state equilibrium. E.1 Additional Results XXXTEMP XXX Entry into Market as a Seller. The cost threshold of firms to sell in market j, c j,k , is determined through free entry condition. In particular, in order to sell in location j, each firm has to pay a fixed cost f j,k in each period in the unit of output. From the market clearing condition and the zero-profit condition for the marginal entrants, the cost cut-off for entry into market j is given by: " c j,k = F θ − σ + 1 Yj,k θσ f j,k Ω j,k #1/(θ +1) , (E.1) F is the aggregate final goods demand for sector k, which is determined from the where Yj,k market clearing condition below. Final Good Market and Consumer Price Index. I assume that workers do not have access to saving technology and therefore use all of their labor income for consumption in each period. Similarly to the intermediate goods sales, I assume that firms whose unit cost net of iceberg trade cost τnj,k is less than c j,k serve as a potential supplier in location j, where we derive c j,k through firms’ entry decision below. Unlike intermediate goods market, workers in j have access to all firms that sell in j, each of which provide a differentiated final good product with elasticity of substitution σ. Furthermore, workers have Cobb-Douglas utility function over sectors k with expenditure share αk . Together with the Pareto distribution of sellers as discussed in equation (??), the consumer price index Pj is given by: Pj = ∏ k α Pj,kk , 1− σ Pj,k = Z c j,k 0 c1−σ Ω j,k θcθ −1 dc =
θ − σ +1 θ Ω j,k c j,k . θ−σ+1 (E.2) 20 Solving for Supply Capacity Γi,m E.2 I now derive the expression for the supply capacity Γi,m . From equation (7), ! Z c I Hi,m (c) = Γi,m c−θ = µi,m dGi,k ( pk ) ∏ γ L,m γ p1,..., pK wi ∏k∈K pk km k∈K ! Z −θγL,m θ wi Ai,m = ∏ k∈K pk I pk −θγkm dGi,k ( pk ) c−θ . (E.3) 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 a lower bound ci,k . Given the steady-state match probability Λi,km , I have: Z pk pk −θγkm I dGi,k ( pk ) = Λi,km Z c i,k 0 c −γkm θ dGi,k (c) + (1 − Λi,km ) Z c i,k (χc)−γkm θ dGi,k (c) 0 χ−θγkm 1 (ci,k )−θγkm + (1 − Λi,km ) (ci,k )−θγkm 1 − γkm 1 − γkm n  o − θγ km χ = (E.4) (ci,k )−θγkm 1 + Λi,km χθγkm − 1 1 − γkm = Λi,km Combining equations (E.3) and (E.4) leads to equation (24) of the main paper with $m ≡ ∏k χ−θγkm 1−γkm . Market Clearing. Final goods market, intermediate goods market, and labor markets clear in the equilibrium. Final goods market clearing in location i and sector k is given by: F Yi,k = αk wi Li ψi , (E.5) where ψi is the efficiency unit of labor input per population Li , and αk is the expenditure share for final goods in sector k defined in equation (E.2). Intermediate good market clearing condition in location i for sales from sector k to sector m is given by: I = γkm X̃i,m . Yi,km (E.6) where X̃i,m is the total input expenditure required by the production in location i and sector m (all input sectors and labor demand), which is given by: X̃i,m = ∑ ∑ j∈ N m∈K where ς = σ σ +1 + θ − σ +1 σ ( θ +1) I Yj,km πij,k + ς ∑ Yj,kF πij,k , (E.7) j∈ N is the share of total final goods demand to total final goods 21 production cost including the fixed cost payment for entry into market.5 Finally, labor market clearing condition is given by: wi Li ψi = ∑ ∑ γL,m X̃i,m , (E.8) m k∈K where the left hand side is the total labor compensation and the right hand side is the total labor demand. E.3 Entry-Cutoff 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 j,k , the goods market clearing condition is given by: F Yj,k = Z c j,k 0 $ j,k c−σ+1 Ω j,k θcθ −1 dc =
θ − σ +1 θ $ j,k Ω j,k c j,k . θ−σ+1   − σ +1 F Now, combining with the zero-profit condition for a marginal seller f j,k c j,k = σ1 $ j,k ci,k , the entry cutoff c j,k is solved as: " c j,k = F θ − σ + 1 Yj,k θσ f j,k Ω j,k #1/(θ +1) . Lastly, I derive the aggregate fixed cost payment by firms that sells in location j, F̃ j,k 5 See Appendix E.4 for the derivation of X̃i,m . 22 as follows: F̃ j,k = Z c j,k 0 f j,k cΩ j,k θcθ −1 dc θ θ +1 c = f j,k Ω j,k θ + 1 j,k   − σ +1 1 θ θ F = $ j,k ci,k Ω j,k c σ θ + 1 j,k 1 θ θ−σ+1 F = Yj,k σθ+1 θ θ−σ+1 F = Y σ(θ + 1) j,k Therefore, ment. E.4 θ − σ +1 σ ( θ +1) (E.9) fraction of aggregate final goods sales are required as fixed cost pay- Aggregate Input Expenditure X̃i,m Aggregate input expenditure is expressed as: I X̃i,m = Xi,m + σ X F + Fi,m , σ + 1 i,m (E.10) I is the aggregate intermediate input sales by firms producing in location i and where Xi,m F is the aggregate final goods sales by firms producing in location i and sector sector k, Xi,k k, and Fi,m is the aggregate fixed cost payment by firms in location i and sector m. Note F by σ/ ( σ + 1), because 1/ ( σ + 1) fraction of sales go to profit. that we multiply Xi,k I ) Aggregate intermediate input sales by firms producing in location i and sector k (Xi,k obeys the following accounting relationship: I Xi,k = ∑ ∑ j∈ N m∈K I Yj,km πij,k , (E.11) I where Yj,km is the aggregate input goods expenditure by firms in sector m and location j for input sector k. F ) is given Aggregate final goods sales by firms producing in location i and sector k (Xi,k by: F F πij,k (E.12) Xi,k = ∑ Yj,k j∈ N F is the final goods demand. where Yi,k σ +1 fraction of aggregate final goods sales are required as fixed Lastly, recall that σθ − ( θ +1) 23 cost payment. Therefore, Fi,k = F θ−σ+1 F X σ (θ + 1) i,k (E.13) 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 (D.2) 5. Obtain { ai,km } using equation (D.9) 6. Obtain {Λi,km } using equation (D.12) 7. Obtain {φi,km } using equation (D.11) 8. Obtain { Pi } using equation (10) 9. Obtain { Li } using equation (30) I , X F , Y I , Y F , w } using equation (E.11), (E.12), (E.5), (E.6), (E.8) 10. Solve for { Xi,m i i,m i,km i,m under ∑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. 24 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 (λS = 0 and λ B = 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 (E.8), 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 , (F.1) j∈ N 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 ∑ j∈ N 1−(1−υσ̃) υθ K2,ij w1j +θ L j . (F.2) 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 (λS + λ B > 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. G 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 G.1 details the counterfactual simulation procedure. Section G.2 provides the sensitivity analysis of the estimation results. Section G.3 discusses the transition dynamics of trade cost reduction. 25 G.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 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 production and preference parameters (αk ,γL,m ,γL,m , θ, σ, ε, ψi ), parameters related to matching process (η, λS , λ B , δj,km , ρ j,km ), and the endogenous variables in the initial equilibrium ({πij,k , S j,k , Bj,k , wi , Li }), 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 , 0 0 I 0 , Y F0 , X I 0 , X F0 , v0 P̂j }) and the values in the new equilibrium ({Yi,km i,k i,k i,k i,km , ai,km , φ j,km }): (i) Trade linkages: π̂ij,m
θ Γ̂i,m τ̂ij,m =  θ ∑i0 ∈ N Γ̂i0 ,m τ̂i0 j,m πi0 j,m 0 −θγL,m Γ̂i,m = L̂iεθ ŵi Ω̂ j,m θγkm − 1 −θγkm 1 + Λi,km χ ∏ ĉi,k (G.1)
1 + Λi,km (χθγkm − 1) k∈K  θ = ∑ Γ̂i0 ,m τ̂i0 j,m πi0 j,m (G.2) (G.3) i0 ∈ N " ĉ j,k = (ii) Market clearing: F Ŷj,k #1/(θ +1) , Ω̂ j,k 0 0 (G.4) 0 F Yi,k = α k wi L i (G.5) 0 I Yi,km = γkm X̃i,m (G.6) 0 0 X̃i,k 1 ∑m∈K γL,m X̃i,m ŵi = L̂i ∑m∈K γL,m X̃i,m   σ θ−σ+1 I0 F0 = Xi,k + + Xi,k σ+1 σ ( θ + 1) 0 I Xi,k = ∑ ∑ j∈ N m∈K 0 (G.7) (G.8) 0 I Yj,km πij,k (G.9) 26 0 F Xi,k = 0 0 ∑ Yj,kF πij,k (G.10) j∈ N (iii) population mobility. L̂ j = P̂j = ∏  ŵ j / P̂j
υ ∑` ŵ` / P̂` Ω j,k c j,k
υ
θ − σ +1 (G.11) L`  α k / (1− σ ) (G.12) k (iv) matching v̂i,km = Ŝi,k
λS B̂i,k Ŝi,k = Ω̂i,k ĉi,k
λ B −1
θ B̂i,k = Γ̂i,k (G.13) (G.14) (G.15) 0 v j,km 0 Λ j,km = δj,km G.2 0 (G.16) v j,km + ρkm Additional Results and Sensitivity Analysis Table G.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. 27 Table G.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 (λ = ν − 1 = 0) (ι = 0) (λ = ν − 1 = 0 and ι = 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 λS = λ B − 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 λS = λ B − 1 = 0 and ι = 0 (Column 4). *p<0.1; **p<0.05; ***p<0.01. Table G.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 G.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 ??). 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). 28 Table G.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 G.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 G.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. Table G.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. 29 G.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.6 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). (G.17) 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 (G.1)-(??) for each period, except that equation (G.16) is replaced by equation (G.17). 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 G.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). 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 (λS = λ B − 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. ai,km strictly less than one. Relatedly, the value of ξ has virtually no effects on my counterfactual simulations in Table G.2 and G.3. 6 As noted in Section ??, less than one percent of sector-location pairs acceptance rates 30 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 G.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 G.3 for the counterfactual simulation procedure.